diff --git a/fetch_books_and_formal.py b/fetch_books_and_formal.py index c3f76237dd13fc80cdf0ae418c866dc279443421..1d5878db4df43d4c3c225c46c0c3ac6fac37002a 100644 --- a/fetch_books_and_formal.py +++ b/fetch_books_and_formal.py @@ -104,7 +104,7 @@ def lean(creds): { "author": "leanprover-community", "repo": "mathlib", - "sha": "e2ba6de0eb20eed7f16a19c3abf6904fad937fa4", + "sha": "63138639ca195344ae96aa77f3a02b90a3ac5c68", "repo_dir": "src", "save_path": os.path.join(save_dir, "mathlib"), }, @@ -118,7 +118,7 @@ def lean(creds): { "author": "leanprover-community", "repo": "sphere-eversion", - "sha": "3a2e501bbb2512870c98c3e6831a6067b3c826b8", + "sha": "cb378966c3c02d9e4ee83040d20c51782fa351ae", "repo_dir": "src", "save_path": os.path.join(save_dir, "sphere-eversion"), }, @@ -506,20 +506,19 @@ def napkin(creds): def main(): creds = ("zhangir-azerbayev", os.environ["GITHUB_TOKEN"]) - #napkin(creds) - #cring(creds) - # naturalproofs_proofwiki(testing=False) - #stacks(creds) - #mizar(creds) - #afp(testing=False) + napkin(creds) + cring(creds) + stacks(creds) + mizar(creds) + afp(testing=False) setmm(creds) - #trench() - #hott(creds) - #stein(creds) - #coq(creds) - #lean(creds) - #hol() - #cam() + trench() + hott(creds) + stein(creds) + coq(creds) + lean(creds) + hol() + cam() if __name__ == "__main__": diff --git a/formal/lean/mathlib/algebra/algebra/basic.lean b/formal/lean/mathlib/algebra/algebra/basic.lean index 510a7b85e31b301d61d41468405a55ebe983fb2a..bc036801713d99cef29c457bd038c07a2d22e96b 100644 --- a/formal/lean/mathlib/algebra/algebra/basic.lean +++ b/formal/lean/mathlib/algebra/algebra/basic.lean @@ -112,7 +112,7 @@ An associative unital `R`-algebra is a semiring `A` equipped with a map into its See the implementation notes in this file for discussion of the details of this definition. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] class algebra (R : Type u) (A : Type v) [comm_semiring R] [semiring A] extends has_smul R A, R →+* A := (commutes' : ∀ r x, to_fun r * x = x * to_fun r) @@ -261,6 +261,11 @@ search (and was here first). -/ (r • x) * y = r • (x * y) := smul_mul_assoc r x y +@[simp] +lemma _root_.smul_algebra_map {α : Type*} [monoid α] [mul_distrib_mul_action α A] + [smul_comm_class α R A] (a : α) (r : R) : a • algebra_map R A r = algebra_map R A r := +by rw [algebra_map_eq_smul_one, smul_comm a r (1 : A), smul_one] + section variables {r : R} {a : A} @@ -474,7 +479,7 @@ end module set_option old_structure_cmd true /-- Defining the homomorphism in the category R-Alg. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure alg_hom (R : Type u) (A : Type v) (B : Type w) [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] extends ring_hom A B := (commutes' : ∀ r : R, to_fun (algebra_map R A r) = algebra_map R B r) @@ -695,6 +700,17 @@ lemma map_list_prod (s : list A) : φ s.prod = (s.map φ).prod := φ.to_ring_hom.map_list_prod s +@[simps mul one {attrs := []}] instance End : monoid (A →ₐ[R] A) := +{ mul := comp, + mul_assoc := λ ϕ ψ χ, rfl, + one := alg_hom.id R A, + one_mul := λ ϕ, ext $ λ x, rfl, + mul_one := λ ϕ, ext $ λ x, rfl } + +@[simp] lemma one_apply (x : A) : (1 : A →ₐ[R] A) x = x := rfl + +@[simp] lemma mul_apply (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x) := rfl + section prod /-- First projection as `alg_hom`. -/ @@ -797,6 +813,13 @@ instance to_alg_hom_class (F R A B : Type*) map_one := map_one, .. h } +@[priority 100] +instance to_linear_equiv_class (F R A B : Type*) + [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] + [h : alg_equiv_class F R A B] : linear_equiv_class F R A B := +{ map_smulₛₗ := λ f, map_smulₛₗ f, + ..h } + end alg_equiv_class namespace alg_equiv @@ -1225,7 +1248,9 @@ This is a stronger version of `mul_semiring_action.to_ring_hom` and `distrib_mul_action.to_linear_map`. -/ @[simps] def to_alg_hom (m : M) : A →ₐ[R] A := -alg_hom.mk' (mul_semiring_action.to_ring_hom _ _ m) (smul_comm _) +{ to_fun := λ a, m • a, + commutes' := smul_algebra_map _, + ..mul_semiring_action.to_ring_hom _ _ m } theorem to_alg_hom_injective [has_faithful_smul M A] : function.injective (mul_semiring_action.to_alg_hom R A : M → A →ₐ[R] A) := diff --git a/formal/lean/mathlib/algebra/algebra/bilinear.lean b/formal/lean/mathlib/algebra/algebra/bilinear.lean index ab8021367987bb9c99292a757e47aaa62d48805d..b51a110222f82290b59091f0a9cde39ac0524eae 100644 --- a/formal/lean/mathlib/algebra/algebra/bilinear.lean +++ b/formal/lean/mathlib/algebra/algebra/bilinear.lean @@ -5,6 +5,7 @@ Authors: Kenny Lau, Yury Kudryashov -/ import algebra.algebra.basic import algebra.hom.iterate +import algebra.hom.non_unital_alg import linear_algebra.tensor_product /-! @@ -15,143 +16,168 @@ in order to avoid importing `linear_algebra.bilinear_map` and `linear_algebra.tensor_product` unnecessarily. -/ -universes u v w - -namespace algebra - open_locale tensor_product open module -section +namespace linear_map -variables (R A : Type*) [comm_semiring R] [semiring A] [algebra R A] +section non_unital_non_assoc -/-- The multiplication in an algebra is a bilinear map. +variables (R A : Type*) [comm_semiring R] [non_unital_non_assoc_semiring A] + [module R A] [smul_comm_class R A A] [is_scalar_tower R A A] + +/-- The multiplication in a non-unital non-associative algebra is a bilinear map. A weaker version of this for semirings exists as `add_monoid_hom.mul`. -/ -def lmul : A →ₐ[R] (End R A) := -{ map_one' := by { ext a, exact one_mul a }, - map_mul' := by { intros a b, ext c, exact mul_assoc a b c }, - map_zero' := by { ext a, exact zero_mul a }, - commutes' := by { intro r, ext a, dsimp, rw [smul_def] }, - .. (show A →ₗ[R] A →ₗ[R] A, from linear_map.mk₂ R (*) - (λ x y z, add_mul x y z) - (λ c x y, by rw [smul_def, smul_def, mul_assoc _ x y]) - (λ x y z, mul_add x y z) - (λ c x y, by rw [smul_def, smul_def, left_comm])) } +def mul : A →ₗ[R] A →ₗ[R] A := linear_map.mk₂ R (*) add_mul smul_mul_assoc mul_add mul_smul_comm -variables {R A} +/-- The multiplication map on a non-unital algebra, as an `R`-linear map from `A ⊗[R] A` to `A`. -/ +def mul' : A ⊗[R] A →ₗ[R] A := +tensor_product.lift (mul R A) -@[simp] lemma lmul_apply (p q : A) : lmul R A p q = p * q := rfl +variables {A} +/-- The multiplication on the left in a non-unital algebra is a linear map. -/ +def mul_left (a : A) : A →ₗ[R] A := mul R A a -variables (R) +/-- The multiplication on the right in an algebra is a linear map. -/ +def mul_right (a : A) : A →ₗ[R] A := (mul R A).flip a -/-- The multiplication on the left in an algebra is a linear map. -/ -def lmul_left (r : A) : A →ₗ[R] A := -lmul R A r +/-- Simultaneous multiplication on the left and right is a linear map. -/ +def mul_left_right (ab : A × A) : A →ₗ[R] A := (mul_right R ab.snd).comp (mul_left R ab.fst) -@[simp] lemma lmul_left_to_add_monoid_hom (r : A) : - (lmul_left R r : A →+ A) = add_monoid_hom.mul_left r := -fun_like.coe_injective rfl +@[simp] lemma mul_left_to_add_monoid_hom (a : A) : + (mul_left R a : A →+ A) = add_monoid_hom.mul_left a := rfl -/-- The multiplication on the right in an algebra is a linear map. -/ -def lmul_right (r : A) : A →ₗ[R] A := -(lmul R A).to_linear_map.flip r +@[simp] lemma mul_right_to_add_monoid_hom (a : A) : + (mul_right R a : A →+ A) = add_monoid_hom.mul_right a := rfl -@[simp] lemma lmul_right_to_add_monoid_hom (r : A) : - (lmul_right R r : A →+ A) = add_monoid_hom.mul_right r := -fun_like.coe_injective rfl +variables {R} -/-- Simultaneous multiplication on the left and right is a linear map. -/ -def lmul_left_right (vw: A × A) : A →ₗ[R] A := -(lmul_right R vw.2).comp (lmul_left R vw.1) +@[simp] lemma mul_apply' (a b : A) : mul R A a b = a * b := rfl +@[simp] lemma mul_left_apply (a b : A) : mul_left R a b = a * b := rfl +@[simp] lemma mul_right_apply (a b : A) : mul_right R a b = b * a := rfl +@[simp] lemma mul_left_right_apply (a b x : A) : mul_left_right R (a, b) x = a * x * b := rfl -lemma commute_lmul_left_right (a b : A) : - commute (lmul_left R a) (lmul_right R b) := -by { ext c, exact (mul_assoc a c b).symm, } +@[simp] lemma mul'_apply {a b : A} : mul' R A (a ⊗ₜ b) = a * b := +by simp only [linear_map.mul', tensor_product.lift.tmul, mul_apply'] + +@[simp] lemma mul_left_zero_eq_zero : + mul_left R (0 : A) = 0 := +(mul R A).map_zero + +@[simp] lemma mul_right_zero_eq_zero : + mul_right R (0 : A) = 0 := +(mul R A).flip.map_zero + +end non_unital_non_assoc + +section non_unital -/-- The multiplication map on an algebra, as an `R`-linear map from `A ⊗[R] A` to `A`. -/ -def lmul' : A ⊗[R] A →ₗ[R] A := -tensor_product.lift (lmul R A).to_linear_map +variables (R A : Type*) [comm_semiring R] [non_unital_semiring A] + [module R A] [smul_comm_class R A A] [is_scalar_tower R A A] + +/-- The multiplication in a non-unital algebra is a bilinear map. + +A weaker version of this for non-unital non-associative algebras exists as `linear_map.mul`. -/ +def _root_.non_unital_alg_hom.lmul : A →ₙₐ[R] (End R A) := +{ map_mul' := by { intros a b, ext c, exact mul_assoc a b c }, + map_zero' := by { ext a, exact zero_mul a }, + .. (mul R A) } variables {R A} -@[simp] lemma lmul'_apply {x y : A} : lmul' R (x ⊗ₜ y) = x * y := -by simp only [algebra.lmul', tensor_product.lift.tmul, alg_hom.to_linear_map_apply, lmul_apply] +@[simp] +lemma _root_.non_unital_alg_hom.coe_lmul_eq_mul : ⇑(non_unital_alg_hom.lmul R A) = mul R A := rfl + +lemma commute_mul_left_right (a b : A) : + commute (mul_left R a) (mul_right R b) := +by { ext c, exact (mul_assoc a c b).symm, } + +@[simp] lemma mul_left_mul (a b : A) : + mul_left R (a * b) = (mul_left R a).comp (mul_left R b) := +by { ext, simp only [mul_left_apply, comp_apply, mul_assoc] } -@[simp] lemma lmul_left_apply (p q : A) : lmul_left R p q = p * q := rfl -@[simp] lemma lmul_right_apply (p q : A) : lmul_right R p q = q * p := rfl -@[simp] lemma lmul_left_right_apply (vw : A × A) (p : A) : - lmul_left_right R vw p = vw.1 * p * vw.2 := rfl +@[simp] lemma mul_right_mul (a b : A) : + mul_right R (a * b) = (mul_right R b).comp (mul_right R a) := +by { ext, simp only [mul_right_apply, comp_apply, mul_assoc] } -@[simp] lemma lmul_left_one : lmul_left R (1:A) = linear_map.id := -by { ext, simp only [linear_map.id_coe, one_mul, id.def, lmul_left_apply] } +end non_unital -@[simp] lemma lmul_left_mul (a b : A) : - lmul_left R (a * b) = (lmul_left R a).comp (lmul_left R b) := -by { ext, simp only [lmul_left_apply, linear_map.comp_apply, mul_assoc] } +section semiring + +variables (R A : Type*) [comm_semiring R] [semiring A] [algebra R A] -@[simp] lemma lmul_right_one : lmul_right R (1:A) = linear_map.id := -by { ext, simp only [linear_map.id_coe, mul_one, id.def, lmul_right_apply] } +/-- The multiplication in an algebra is an algebra homomorphism into the endomorphisms on +the algebra. -@[simp] lemma lmul_right_mul (a b : A) : - lmul_right R (a * b) = (lmul_right R b).comp (lmul_right R a) := -by { ext, simp only [lmul_right_apply, linear_map.comp_apply, mul_assoc] } +A weaker version of this for non-unital algebras exists as `non_unital_alg_hom.mul`. -/ +def _root_.algebra.lmul : A →ₐ[R] (End R A) := +{ map_one' := by { ext a, exact one_mul a }, + map_mul' := by { intros a b, ext c, exact mul_assoc a b c }, + map_zero' := by { ext a, exact zero_mul a }, + commutes' := by { intro r, ext a, exact (algebra.smul_def r a).symm, }, + .. (linear_map.mul R A) } -@[simp] lemma lmul_left_zero_eq_zero : - lmul_left R (0 : A) = 0 := -(lmul R A).map_zero +variables {R A} -@[simp] lemma lmul_right_zero_eq_zero : - lmul_right R (0 : A) = 0 := -(lmul R A).to_linear_map.flip.map_zero +@[simp] lemma _root_.algebra.coe_lmul_eq_mul : ⇑(algebra.lmul R A) = mul R A := rfl -@[simp] lemma lmul_left_eq_zero_iff (a : A) : - lmul_left R a = 0 ↔ a = 0 := +@[simp] lemma mul_left_eq_zero_iff (a : A) : + mul_left R a = 0 ↔ a = 0 := begin split; intros h, - { rw [← mul_one a, ← lmul_left_apply a 1, h, linear_map.zero_apply], }, - { rw h, exact lmul_left_zero_eq_zero, }, + { rw [← mul_one a, ← mul_left_apply a 1, h, linear_map.zero_apply], }, + { rw h, exact mul_left_zero_eq_zero, }, end -@[simp] lemma lmul_right_eq_zero_iff (a : A) : - lmul_right R a = 0 ↔ a = 0 := +@[simp] lemma mul_right_eq_zero_iff (a : A) : + mul_right R a = 0 ↔ a = 0 := begin split; intros h, - { rw [← one_mul a, ← lmul_right_apply a 1, h, linear_map.zero_apply], }, - { rw h, exact lmul_right_zero_eq_zero, }, + { rw [← one_mul a, ← mul_right_apply a 1, h, linear_map.zero_apply], }, + { rw h, exact mul_right_zero_eq_zero, }, end -@[simp] lemma pow_lmul_left (a : A) (n : ℕ) : - (lmul_left R a) ^ n = lmul_left R (a ^ n) := -((lmul R A).map_pow a n).symm +@[simp] lemma mul_left_one : mul_left R (1:A) = linear_map.id := +by { ext, simp only [linear_map.id_coe, one_mul, id.def, mul_left_apply] } -@[simp] lemma pow_lmul_right (a : A) (n : ℕ) : - (lmul_right R a) ^ n = lmul_right R (a ^ n) := -linear_map.coe_injective $ ((lmul_right R a).coe_pow n).symm ▸ (mul_right_iterate a n) +@[simp] lemma mul_right_one : mul_right R (1:A) = linear_map.id := +by { ext, simp only [linear_map.id_coe, mul_one, id.def, mul_right_apply] } +@[simp] lemma pow_mul_left (a : A) (n : ℕ) : + (mul_left R a) ^ n = mul_left R (a ^ n) := +by simpa only [mul_left, ←algebra.coe_lmul_eq_mul] using ((algebra.lmul R A).map_pow a n).symm + +@[simp] lemma pow_mul_right (a : A) (n : ℕ) : + (mul_right R a) ^ n = mul_right R (a ^ n) := +begin + simp only [mul_right, ←algebra.coe_lmul_eq_mul], + exact linear_map.coe_injective + (((mul_right R a).coe_pow n).symm ▸ (mul_right_iterate a n)), end -section +end semiring + +section ring variables {R A : Type*} [comm_semiring R] [ring A] [algebra R A] -lemma lmul_left_injective [no_zero_divisors A] {x : A} (hx : x ≠ 0) : - function.injective (lmul_left R x) := +lemma mul_left_injective [no_zero_divisors A] {x : A} (hx : x ≠ 0) : + function.injective (mul_left R x) := by { letI : is_domain A := { exists_pair_ne := ⟨x, 0, hx⟩, ..‹ring A›, ..‹no_zero_divisors A› }, exact mul_right_injective₀ hx } -lemma lmul_right_injective [no_zero_divisors A] {x : A} (hx : x ≠ 0) : - function.injective (lmul_right R x) := +lemma mul_right_injective [no_zero_divisors A] {x : A} (hx : x ≠ 0) : + function.injective (mul_right R x) := by { letI : is_domain A := { exists_pair_ne := ⟨x, 0, hx⟩, ..‹ring A›, ..‹no_zero_divisors A› }, exact mul_left_injective₀ hx } -lemma lmul_injective [no_zero_divisors A] {x : A} (hx : x ≠ 0) : - function.injective (lmul R A x) := +lemma mul_injective [no_zero_divisors A] {x : A} (hx : x ≠ 0) : + function.injective (mul R A x) := by { letI : is_domain A := { exists_pair_ne := ⟨x, 0, hx⟩, ..‹ring A›, ..‹no_zero_divisors A› }, exact mul_right_injective₀ hx } -end +end ring -end algebra +end linear_map diff --git a/formal/lean/mathlib/algebra/algebra/operations.lean b/formal/lean/mathlib/algebra/algebra/operations.lean index cffe7f3f3010e5dbc423dcc0f7fe94d7dc2714a5..02ed4b842e6e2e6175b1a099c34ca9054cf74363 100644 --- a/formal/lean/mathlib/algebra/algebra/operations.lean +++ b/formal/lean/mathlib/algebra/algebra/operations.lean @@ -18,7 +18,7 @@ An interface for multiplication and division of sub-R-modules of an R-algebra A ## Main definitions -Let `R` be a commutative ring (or semiring) and aet `A` be an `R`-algebra. +Let `R` be a commutative ring (or semiring) and let `A` be an `R`-algebra. * `1 : submodule R A` : the R-submodule R of the R-algebra A * `has_mul (submodule R A)` : multiplication of two sub-R-modules M and N of A is defined to be @@ -28,6 +28,9 @@ Let `R` be a commutative ring (or semiring) and aet `A` be an `R`-algebra. It is proved that `submodule R A` is a semiring, and also an algebra over `set A`. +Additionally, in the `pointwise` locale we promote `submodule.pointwise_distrib_mul_action` to a +`mul_semiring_action` as `submodule.pointwise_mul_semiring_action`. + ## Tags multiplication of submodules, division of submodules, submodule semiring @@ -118,7 +121,7 @@ by rw [←map_equiv_eq_comap_symm, map_op_one] /-- Multiplication of sub-R-modules of an R-algebra A. The submodule `M * N` is the smallest R-submodule of `A` containing the elements `m * n` for `m ∈ M` and `n ∈ N`. -/ -instance : has_mul (submodule R A) := ⟨submodule.map₂ (algebra.lmul R A).to_linear_map⟩ +instance : has_mul (submodule R A) := ⟨submodule.map₂ $ linear_map.mul R A⟩ theorem mul_mem_mul (hm : m ∈ M) (hn : n ∈ N) : m * n ∈ M * N := apply_mem_map₂ _ hm hn @@ -128,7 +131,7 @@ lemma mul_to_add_submonoid (M N : submodule R A) : (M * N).to_add_submonoid = M.to_add_submonoid * N.to_add_submonoid := begin dsimp [has_mul.mul], - simp_rw [←algebra.lmul_left_to_add_monoid_hom R, algebra.lmul_left, ←map_to_add_submonoid _ N, + simp_rw [←linear_map.mul_left_to_add_monoid_hom R, linear_map.mul_left, ←map_to_add_submonoid _ N, map₂], rw supr_to_add_submonoid, refl, @@ -189,7 +192,7 @@ image2_subset_map₂ (algebra.lmul R A).to_linear_map M N protected lemma map_mul {A'} [semiring A'] [algebra R A'] (f : A →ₐ[R] A') : map f.to_linear_map (M * N) = map f.to_linear_map M * map f.to_linear_map N := calc map f.to_linear_map (M * N) - = ⨆ (i : M), (N.map (lmul R A i)).map f.to_linear_map : map_supr _ _ + = ⨆ (i : M), (N.map (linear_map.mul R A i)).map f.to_linear_map : map_supr _ _ ... = map f.to_linear_map M * map f.to_linear_map N : begin apply congr_arg Sup, @@ -454,6 +457,23 @@ def span.ring_hom : set_semiring A →+* submodule R A := map_add' := span_union, map_mul' := λ s t, by erw [span_mul_span, ← image_mul_prod] } +section +variables {α : Type*} [monoid α] [mul_semiring_action α A] [smul_comm_class α R A] + +/-- The action on a submodule corresponding to applying the action to every element. + +This is available as an instance in the `pointwise` locale. + +This is a stronger version of `submodule.pointwise_distrib_mul_action`. -/ +protected def pointwise_mul_semiring_action : mul_semiring_action α (submodule R A) := +{ smul_mul := λ r x y, submodule.map_mul x y $ mul_semiring_action.to_alg_hom R A r, + smul_one := λ r, submodule.map_one $ mul_semiring_action.to_alg_hom R A r, + ..submodule.pointwise_distrib_mul_action } + +localized "attribute [instance] submodule.pointwise_mul_semiring_action" in pointwise + +end + end ring section comm_ring @@ -512,7 +532,7 @@ lemma smul_le_smul {s t : set_semiring A} {M N : submodule R A} (h₁ : s.down mul_le_mul (span_mono h₁) h₂ lemma smul_singleton (a : A) (M : submodule R A) : - ({a} : set A).up • M = M.map (lmul_left _ a) := + ({a} : set A).up • M = M.map (linear_map.mul_left _ a) := begin conv_lhs {rw ← span_eq M}, change span _ _ * span _ _ = _, diff --git a/formal/lean/mathlib/algebra/algebra/subalgebra/basic.lean b/formal/lean/mathlib/algebra/algebra/subalgebra/basic.lean index e37628792f246cf520156515fdc5e314fcc6c018..7d8d4bb89d90a8b247095138312440a2877a7233 100644 --- a/formal/lean/mathlib/algebra/algebra/subalgebra/basic.lean +++ b/formal/lean/mathlib/algebra/algebra/subalgebra/basic.lean @@ -305,7 +305,7 @@ def to_submodule_equiv (S : subalgebra R A) : S.to_submodule ≃ₗ[R] S := linear_equiv.of_eq _ _ rfl /-- Transport a subalgebra via an algebra homomorphism. -/ -def map (S : subalgebra R A) (f : A →ₐ[R] B) : subalgebra R B := +def map (f : A →ₐ[R] B) (S : subalgebra R A) : subalgebra R B := { algebra_map_mem' := λ r, f.commutes r ▸ set.mem_image_of_mem _ (S.algebra_map_mem r), .. S.to_subsemiring.map (f : A →+* B) } @@ -313,9 +313,9 @@ lemma map_mono {S₁ S₂ : subalgebra R A} {f : A →ₐ[R] B} : S₁ ≤ S₂ → S₁.map f ≤ S₂.map f := set.image_subset f -lemma map_injective {S₁ S₂ : subalgebra R A} (f : A →ₐ[R] B) - (hf : function.injective f) (ih : S₁.map f = S₂.map f) : S₁ = S₂ := -ext $ set.ext_iff.1 $ set.image_injective.2 hf $ set.ext $ set_like.ext_iff.mp ih +lemma map_injective {f : A →ₐ[R] B} (hf : function.injective f) : + function.injective (map f) := +λ S₁ S₂ ih, ext $ set.ext_iff.1 $ set.image_injective.2 hf $ set.ext $ set_like.ext_iff.mp ih @[simp] lemma map_id (S : subalgebra R A) : S.map (alg_hom.id R A) = S := set_like.coe_injective $ set.image_id _ @@ -325,7 +325,7 @@ lemma map_map (S : subalgebra R A) (g : B →ₐ[R] C) (f : A →ₐ[R] B) : set_like.coe_injective $ set.image_image _ _ _ lemma mem_map {S : subalgebra R A} {f : A →ₐ[R] B} {y : B} : - y ∈ map S f ↔ ∃ x ∈ S, f x = y := + y ∈ map f S ↔ ∃ x ∈ S, f x = y := subsemiring.mem_map lemma map_to_submodule {S : subalgebra R A} {f : A →ₐ[R] B} : @@ -341,16 +341,16 @@ set_like.coe_injective rfl rfl /-- Preimage of a subalgebra under an algebra homomorphism. -/ -def comap (S : subalgebra R B) (f : A →ₐ[R] B) : subalgebra R A := +def comap (f : A →ₐ[R] B) (S : subalgebra R B) : subalgebra R A := { algebra_map_mem' := λ r, show f (algebra_map R A r) ∈ S, from (f.commutes r).symm ▸ S.algebra_map_mem r, .. S.to_subsemiring.comap (f : A →+* B) } theorem map_le {S : subalgebra R A} {f : A →ₐ[R] B} {U : subalgebra R B} : - map S f ≤ U ↔ S ≤ comap U f := + map f S ≤ U ↔ S ≤ comap f U := set.image_subset_iff -lemma gc_map_comap (f : A →ₐ[R] B) : galois_connection (λ S, map S f) (λ S, comap S f) := +lemma gc_map_comap (f : A →ₐ[R] B) : galois_connection (map f) (comap f) := λ S U, map_le @[simp] lemma mem_comap (S : subalgebra R B) (f : A →ₐ[R] B) (x : A) : @@ -648,14 +648,14 @@ algebra.eq_top_iff @[simp] theorem range_id : (alg_hom.id R A).range = ⊤ := set_like.coe_injective set.range_id -@[simp] theorem map_top (f : A →ₐ[R] B) : subalgebra.map (⊤ : subalgebra R A) f = f.range := +@[simp] theorem map_top (f : A →ₐ[R] B) : (⊤ : subalgebra R A).map f = f.range := set_like.coe_injective set.image_univ -@[simp] theorem map_bot (f : A →ₐ[R] B) : subalgebra.map (⊥ : subalgebra R A) f = ⊥ := +@[simp] theorem map_bot (f : A →ₐ[R] B) : (⊥ : subalgebra R A).map f = ⊥ := set_like.coe_injective $ by simp only [← set.range_comp, (∘), algebra.coe_bot, subalgebra.coe_map, f.commutes] -@[simp] theorem comap_top (f : A →ₐ[R] B) : subalgebra.comap (⊤ : subalgebra R B) f = ⊤ := +@[simp] theorem comap_top (f : A →ₐ[R] B) : (⊤ : subalgebra R B).comap f = ⊤ := eq_top_iff.2 $ λ x, mem_top /-- `alg_hom` to `⊤ : subalgebra R A`. -/ @@ -810,12 +810,11 @@ variables (S₁ : subalgebra R B) /-- The product of two subalgebras is a subalgebra. -/ def prod : subalgebra R (A × B) := -{ carrier := (S : set A) ×ˢ (S₁ : set B), +{ carrier := S ×ˢ S₁, algebra_map_mem' := λ r, ⟨algebra_map_mem _ _, algebra_map_mem _ _⟩, .. S.to_subsemiring.prod S₁.to_subsemiring } -@[simp] lemma coe_prod : - (prod S S₁ : set (A × B)) = (S : set A) ×ˢ (S₁ : set B):= rfl +@[simp] lemma coe_prod : (prod S S₁ : set (A × B)) = S ×ˢ S₁ := rfl lemma prod_to_submodule : (S.prod S₁).to_submodule = S.to_submodule.prod S₁.to_submodule := rfl diff --git a/formal/lean/mathlib/algebra/algebra/tower.lean b/formal/lean/mathlib/algebra/algebra/tower.lean index accdeb7c1d330857ebca6a16a086485a949c3cb2..7416808106c253283dd70a459d7f3f7dfd5d00ae 100644 --- a/formal/lean/mathlib/algebra/algebra/tower.lean +++ b/formal/lean/mathlib/algebra/algebra/tower.lean @@ -48,7 +48,7 @@ def lsmul : A →ₐ[R] module.End R M := @[simp] lemma lsmul_coe (a : A) : (lsmul R M a : M → M) = (•) a := rfl lemma lmul_algebra_map (x : R) : - lmul R A (algebra_map R A x) = algebra.lsmul R A x := + algebra.lmul R A (algebra_map R A x) = algebra.lsmul R A x := eq.symm $ linear_map.ext $ smul_def x end algebra diff --git a/formal/lean/mathlib/algebra/algebra/unitization.lean b/formal/lean/mathlib/algebra/algebra/unitization.lean index fb95c376eb44f87750c545d7d5d3a201d8f1895b..81dc0b2d05d0276fb541f630613709e8811bb0e9 100644 --- a/formal/lean/mathlib/algebra/algebra/unitization.lean +++ b/formal/lean/mathlib/algebra/algebra/unitization.lean @@ -345,9 +345,8 @@ instance [comm_monoid R] [non_unital_semiring A] [distrib_mul_action R A] [is_sc abel }, ..unitization.mul_one_class } --- This should work for `non_unital_comm_semiring`s, but we don't seem to have those -instance [comm_monoid R] [comm_semiring A] [distrib_mul_action R A] [is_scalar_tower R A A] - [smul_comm_class R A A] : comm_monoid (unitization R A) := +instance [comm_monoid R] [non_unital_comm_semiring A] [distrib_mul_action R A] + [is_scalar_tower R A A] [smul_comm_class R A A] : comm_monoid (unitization R A) := { mul_comm := λ x₁ x₂, ext (mul_comm x₁.1 x₂.1) $ show x₁.1 • x₂.2 + x₂.1 • x₁.2 + x₁.2 * x₂.2 = x₂.1 • x₁.2 + x₁.1 • x₂.2 + x₂.2 * x₁.2, by rw [add_comm (x₁.1 • x₂.2), mul_comm], @@ -358,8 +357,7 @@ instance [comm_semiring R] [non_unital_semiring A] [module R A] [is_scalar_tower { ..unitization.monoid, ..unitization.non_assoc_semiring } --- This should work for `non_unital_comm_semiring`s, but we don't seem to have those -instance [comm_semiring R] [comm_semiring A] [module R A] [is_scalar_tower R A A] +instance [comm_semiring R] [non_unital_comm_semiring A] [module R A] [is_scalar_tower R A A] [smul_comm_class R A A] : comm_semiring (unitization R A) := { ..unitization.comm_monoid, ..unitization.non_assoc_semiring } diff --git a/formal/lean/mathlib/algebra/big_operators/basic.lean b/formal/lean/mathlib/algebra/big_operators/basic.lean index 9636c53e79a5f04124ae17ba1a92ce6a3bb1f90c..84b21f722d20d5d4bd524e32b47cd8c465add2a5 100644 --- a/formal/lean/mathlib/algebra/big_operators/basic.lean +++ b/formal/lean/mathlib/algebra/big_operators/basic.lean @@ -39,7 +39,7 @@ See the documentation of `to_additive.attr` for more information. -/ universes u v w -variables {β : Type u} {α : Type v} {γ : Type w} +variables {ι : Type*} {β : Type u} {α : Type v} {γ : Type w} open fin @@ -512,24 +512,24 @@ eq.trans (by rw one_mul; refl) fold_op_distrib @[to_additive] lemma prod_product {s : finset γ} {t : finset α} {f : γ×α → β} : - (∏ x in s.product t, f x) = ∏ x in s, ∏ y in t, f (x, y) := -prod_finset_product (s.product t) s (λ a, t) (λ p, mem_product) + (∏ x in s ×ˢ t, f x) = ∏ x in s, ∏ y in t, f (x, y) := +prod_finset_product (s ×ˢ t) s (λ a, t) (λ p, mem_product) /-- An uncurried version of `finset.prod_product`. -/ @[to_additive "An uncurried version of `finset.sum_product`"] lemma prod_product' {s : finset γ} {t : finset α} {f : γ → α → β} : - (∏ x in s.product t, f x.1 x.2) = ∏ x in s, ∏ y in t, f x y := + (∏ x in s ×ˢ t, f x.1 x.2) = ∏ x in s, ∏ y in t, f x y := prod_product @[to_additive] lemma prod_product_right {s : finset γ} {t : finset α} {f : γ×α → β} : - (∏ x in s.product t, f x) = ∏ y in t, ∏ x in s, f (x, y) := -prod_finset_product_right (s.product t) t (λ a, s) (λ p, mem_product.trans and.comm) + (∏ x in s ×ˢ t, f x) = ∏ y in t, ∏ x in s, f (x, y) := +prod_finset_product_right (s ×ˢ t) t (λ a, s) (λ p, mem_product.trans and.comm) /-- An uncurried version of `finset.prod_product_right`. -/ @[to_additive "An uncurried version of `finset.prod_product_right`"] lemma prod_product_right' {s : finset γ} {t : finset α} {f : γ → α → β} : - (∏ x in s.product t, f x.1 x.2) = ∏ y in t, ∏ x in s, f x y := + (∏ x in s ×ˢ t, f x.1 x.2) = ∏ y in t, ∏ x in s, f x y := prod_product_right /-- Generalization of `finset.prod_comm` to the case when the inner `finset`s depend on the outer @@ -1015,6 +1015,10 @@ begin rw [count_eq_zero_of_not_mem hx, pow_zero], end +lemma sum_filter_count_eq_countp [decidable_eq α] (p : α → Prop) [decidable_pred p] (l : list α) : + ∑ x in l.to_finset.filter p, l.count x = l.countp p := +by simp [finset.sum, sum_map_count_dedup_filter_eq_countp p l] + open multiset @[to_additive] lemma prod_multiset_map_count [decidable_eq α] (s : multiset α) @@ -1099,6 +1103,16 @@ lemma prod_range_div' {M : Type*} [comm_group M] (f : ℕ → M) (n : ℕ) : ∏ i in range n, (f i / f (i + 1)) = f 0 / f n := by apply prod_range_induction; simp +@[to_additive] +lemma eq_prod_range_div {M : Type*} [comm_group M] (f : ℕ → M) (n : ℕ) : + f n = f 0 * ∏ i in range n, (f (i + 1) / f i) := +by rw [prod_range_div, mul_div_cancel'_right] + +@[to_additive] +lemma eq_prod_range_div' {M : Type*} [comm_group M] (f : ℕ → M) (n : ℕ) : + f n = ∏ i in range (n + 1), if i = 0 then f 0 else f i / f (i - 1) := +by { conv_lhs { rw [finset.eq_prod_range_div f] }, simp [finset.prod_range_succ', mul_comm] } + /-- A telescoping sum along `{0, ..., n-1}` of an `ℕ`-valued function reduces to the difference of the last and first terms @@ -1291,9 +1305,13 @@ begin end /-- Taking a product over `s : finset α` is the same as multiplying the value on a single element -`f a` by the product of `s.erase a`. -/ +`f a` by the product of `s.erase a`. + +See `multiset.prod_map_erase` for the `multiset` version. -/ @[to_additive "Taking a sum over `s : finset α` is the same as adding the value on a single element -`f a` to the sum over `s.erase a`."] +`f a` to the sum over `s.erase a`. + +See `multiset.sum_map_erase` for the `multiset` version."] lemma mul_prod_erase [decidable_eq α] (s : finset α) (f : α → β) {a : α} (h : a ∈ s) : f a * (∏ x in s.erase a, f x) = ∏ x in s, f x := by rw [← prod_insert (not_mem_erase a s), insert_erase h] @@ -1388,17 +1406,6 @@ lemma sum_boole {s : finset α} {p : α → Prop} [non_assoc_semiring β] {hp : (∑ x in s, if p x then (1 : β) else (0 : β)) = (s.filter p).card := by simp [sum_ite] -lemma eq_sum_range_sub [add_comm_group β] (f : ℕ → β) (n : ℕ) : - f n = f 0 + ∑ i in range n, (f (i+1) - f i) := -by rw [finset.sum_range_sub, add_sub_cancel'_right] - -lemma eq_sum_range_sub' [add_comm_group β] (f : ℕ → β) (n : ℕ) : - f n = ∑ i in range (n + 1), if i = 0 then f 0 else f i - f (i - 1) := -begin - conv_lhs { rw [finset.eq_sum_range_sub f] }, - simp [finset.sum_range_succ', add_comm] -end - lemma _root_.commute.sum_right [non_unital_non_assoc_semiring β] (s : finset α) (f : α → β) (b : β) (h : ∀ i ∈ s, commute b (f i)) : commute b (∑ i in s, f i) := @@ -1539,13 +1546,7 @@ end prod_eq_zero lemma prod_unique_nonempty {α β : Type*} [comm_monoid β] [unique α] (s : finset α) (f : α → β) (h : s.nonempty) : (∏ x in s, f x) = f default := -begin - obtain ⟨a, ha⟩ := h, - have : s = {a}, - { ext b, - simpa [subsingleton.elim a b] using ha }, - rw [this, finset.prod_singleton, subsingleton.elim a default] -end +by rw [h.eq_singleton_default, finset.prod_singleton] end finset @@ -1596,8 +1597,8 @@ by rw [univ_unique, prod_singleton] (∏ x : α, f x) = 1 := by rw [eq_empty_of_is_empty (univ : finset α), finset.prod_empty] -@[to_additive] -lemma prod_subsingleton {α β : Type*} [comm_monoid β] [subsingleton α] (f : α → β) (a : α) : +@[to_additive] lemma prod_subsingleton {α β : Type*} [comm_monoid β] [subsingleton α] [fintype α] + (f : α → β) (a : α) : (∏ x : α, f x) = f a := begin haveI : unique α := unique_of_subsingleton a, @@ -1631,8 +1632,85 @@ end list namespace multiset +lemma disjoint_list_sum_left {a : multiset α} {l : list (multiset α)} : + multiset.disjoint l.sum a ↔ ∀ b ∈ l, multiset.disjoint b a := +begin + induction l with b bs ih, + { simp only [zero_disjoint, list.not_mem_nil, is_empty.forall_iff, forall_const, list.sum_nil], }, + { simp_rw [list.sum_cons, disjoint_add_left, list.mem_cons_iff, forall_eq_or_imp], + simp [and.congr_left_iff, iff_self, ih], }, +end + +lemma disjoint_list_sum_right {a : multiset α} {l : list (multiset α)} : + multiset.disjoint a l.sum ↔ ∀ b ∈ l, multiset.disjoint a b := +by simpa only [disjoint_comm] using disjoint_list_sum_left + +lemma disjoint_sum_left {a : multiset α} {i : multiset (multiset α)} : + multiset.disjoint i.sum a ↔ ∀ b ∈ i, multiset.disjoint b a := +quotient.induction_on i $ λ l, begin + rw [quot_mk_to_coe, multiset.coe_sum], + exact disjoint_list_sum_left, +end + +lemma disjoint_sum_right {a : multiset α} {i : multiset (multiset α)} : + multiset.disjoint a i.sum ↔ ∀ b ∈ i, multiset.disjoint a b := +by simpa only [disjoint_comm] using disjoint_sum_left + +lemma disjoint_finset_sum_left {β : Type*} {i : finset β} {f : β → multiset α} {a : multiset α} : + multiset.disjoint (i.sum f) a ↔ ∀ b ∈ i, multiset.disjoint (f b) a := +begin + convert (@disjoint_sum_left _ a) (map f i.val), + simp [finset.mem_def, and.congr_left_iff, iff_self], +end + +lemma disjoint_finset_sum_right {β : Type*} {i : finset β} {f : β → multiset α} {a : multiset α} : + multiset.disjoint a (i.sum f) ↔ ∀ b ∈ i, multiset.disjoint a (f b) := +by simpa only [disjoint_comm] using disjoint_finset_sum_left + variables [decidable_eq α] +lemma add_eq_union_left_of_le {x y z : multiset α} (h : y ≤ x) : + z + x = z ∪ y ↔ z.disjoint x ∧ x = y := +begin + rw ←add_eq_union_iff_disjoint, + split, + { intro h0, + rw and_iff_right_of_imp, + { exact (le_of_add_le_add_left $ h0.trans_le $ union_le_add z y).antisymm h, }, + { rintro rfl, + exact h0, } }, + { rintro ⟨h0, rfl⟩, + exact h0, } +end + +lemma add_eq_union_right_of_le {x y z : multiset α} (h : z ≤ y) : + x + y = x ∪ z ↔ y = z ∧ x.disjoint y := +by simpa only [and_comm] using add_eq_union_left_of_le h + +lemma finset_sum_eq_sup_iff_disjoint {β : Type*} {i : finset β} {f : β → multiset α} : + i.sum f = i.sup f ↔ ∀ x y ∈ i, x ≠ y → multiset.disjoint (f x) (f y) := +begin + induction i using finset.cons_induction_on with z i hz hr, + { simp only [finset.not_mem_empty, is_empty.forall_iff, implies_true_iff, + finset.sum_empty, finset.sup_empty, bot_eq_zero, eq_self_iff_true], }, + { simp_rw [finset.sum_cons hz, finset.sup_cons, finset.mem_cons, multiset.sup_eq_union, + forall_eq_or_imp, ne.def, eq_self_iff_true, not_true, is_empty.forall_iff, true_and, + imp_and_distrib, forall_and_distrib, ←hr, @eq_comm _ z], + have := λ x ∈ i, ne_of_mem_of_not_mem H hz, + simp only [this, not_false_iff, true_implies_iff] {contextual := tt}, + simp_rw [←disjoint_finset_sum_left, ←disjoint_finset_sum_right, disjoint_comm, ←and_assoc, + and_self], + exact add_eq_union_left_of_le (finset.sup_le (λ x hx, le_sum_of_mem (mem_map_of_mem f hx))), }, +end + +lemma sup_powerset_len {α : Type*} [decidable_eq α] (x : multiset α) : + finset.sup (finset.range (x.card + 1)) (λ k, x.powerset_len k) = x.powerset := +begin + convert bind_powerset_len x, + rw [multiset.bind, multiset.join, ←finset.range_coe, ←finset.sum_eq_multiset_sum], + exact eq.symm (finset_sum_eq_sup_iff_disjoint.mpr (λ _ _ _ _ h, disjoint_powerset_len x h)), +end + @[simp] lemma to_finset_sum_count_eq (s : multiset α) : (∑ a in s.to_finset, s.count a) = s.card := multiset.induction_on s rfl @@ -1780,3 +1858,63 @@ begin simp only [his, finset.sum_insert, not_false_iff], exact (int.nat_abs_add_le _ _).trans (add_le_add le_rfl IH) } end + +/-! ### `additive`, `multiplicative` -/ + +open additive multiplicative + +section monoid +variables [monoid α] + +@[simp] lemma of_mul_list_prod (s : list α) : of_mul s.prod = (s.map of_mul).sum := +by simpa [of_mul] + +@[simp] lemma to_mul_list_sum (s : list (additive α)) : + to_mul s.sum = (s.map to_mul).prod := by simpa [to_mul, of_mul] + +end monoid + +section add_monoid +variables [add_monoid α] + +@[simp] lemma of_add_list_prod (s : list α) : of_add s.sum = (s.map of_add).prod := +by simpa [of_add] + +@[simp] lemma to_add_list_sum (s : list (multiplicative α)) : + to_add s.prod = (s.map to_add).sum := by simpa [to_add, of_add] + +end add_monoid + +section comm_monoid +variables [comm_monoid α] + +@[simp] lemma of_mul_multiset_prod (s : multiset α) : + of_mul s.prod = (s.map of_mul).sum := by simpa [of_mul] + +@[simp] lemma to_mul_multiset_sum (s : multiset (additive α)) : + to_mul s.sum = (s.map to_mul).prod := by simpa [to_mul, of_mul] + +@[simp] lemma of_mul_prod (s : finset ι) (f : ι → α) : + of_mul (∏ i in s, f i) = ∑ i in s, of_mul (f i) := rfl + +@[simp] lemma to_mul_sum (s : finset ι) (f : ι → additive α) : + to_mul (∑ i in s, f i) = ∏ i in s, to_mul (f i) := rfl + +end comm_monoid + +section add_comm_monoid +variables [add_comm_monoid α] + +@[simp] lemma of_add_multiset_prod (s : multiset α) : + of_add s.sum = (s.map of_add).prod := by simpa [of_add] + +@[simp] lemma to_add_multiset_sum (s : multiset (multiplicative α)) : + to_add s.prod = (s.map to_add).sum := by simpa [to_add, of_add] + +@[simp] lemma of_add_sum (s : finset ι) (f : ι → α) : + of_add (∑ i in s, f i) = ∏ i in s, of_add (f i) := rfl + +@[simp] lemma to_add_prod (s : finset ι) (f : ι → multiplicative α) : + to_add (∏ i in s, f i) = ∑ i in s, to_add (f i) := rfl + +end add_comm_monoid diff --git a/formal/lean/mathlib/algebra/big_operators/fin.lean b/formal/lean/mathlib/algebra/big_operators/fin.lean index 01763403f776961f086aef10ba571ef5ffddcd49..5967969ed1c49aa1908d241792f11cf8c12cc401 100644 --- a/formal/lean/mathlib/algebra/big_operators/fin.lean +++ b/formal/lean/mathlib/algebra/big_operators/fin.lean @@ -95,6 +95,30 @@ by simp ∏ i, f i = f 0 * f 1 := by simp [prod_univ_succ] +@[to_additive] theorem prod_univ_three [comm_monoid β] (f : fin 3 → β) : + ∏ i, f i = f 0 * f 1 * f 2 := +by { rw [prod_univ_cast_succ, prod_univ_two], refl } + +@[to_additive] theorem prod_univ_four [comm_monoid β] (f : fin 4 → β) : + ∏ i, f i = f 0 * f 1 * f 2 * f 3 := +by { rw [prod_univ_cast_succ, prod_univ_three], refl } + +@[to_additive] theorem prod_univ_five [comm_monoid β] (f : fin 5 → β) : + ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := +by { rw [prod_univ_cast_succ, prod_univ_four], refl } + +@[to_additive] theorem prod_univ_six [comm_monoid β] (f : fin 6 → β) : + ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := +by { rw [prod_univ_cast_succ, prod_univ_five], refl } + +@[to_additive] theorem prod_univ_seven [comm_monoid β] (f : fin 7 → β) : + ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 := +by { rw [prod_univ_cast_succ, prod_univ_six], refl } + +@[to_additive] theorem prod_univ_eight [comm_monoid β] (f : fin 8 → β) : + ∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := +by { rw [prod_univ_cast_succ, prod_univ_seven], refl } + lemma sum_pow_mul_eq_add_pow {n : ℕ} {R : Type*} [comm_semiring R] (a b : R) : ∑ s : finset (fin n), a ^ s.card * b ^ (n - s.card) = (a + b) ^ n := by simpa using fintype.sum_pow_mul_eq_add_pow (fin n) a b @@ -157,6 +181,29 @@ by simp [partial_prod, list.take_succ, list.of_fn_nth_val, dif_pos j.is_lt, ←o partial_prod f j.succ = f 0 * partial_prod (fin.tail f) j := by simpa [partial_prod] +@[to_additive] lemma partial_prod_left_inv {G : Type*} [group G] (f : fin (n + 1) → G) : + f 0 • partial_prod (λ i : fin n, (f i)⁻¹ * f i.succ) = f := +funext $ λ x, fin.induction_on x (by simp) (λ x hx, +begin + simp only [coe_eq_cast_succ, pi.smul_apply, smul_eq_mul] at hx ⊢, + rw [partial_prod_succ, ←mul_assoc, hx, mul_inv_cancel_left], +end) + +@[to_additive] lemma partial_prod_right_inv {G : Type*} [group G] + (g : G) (f : fin n → G) (i : fin n) : + ((g • partial_prod f) i)⁻¹ * (g • partial_prod f) i.succ = f i := +begin + cases i with i hn, + induction i with i hi generalizing hn, + { simp [←fin.succ_mk, partial_prod_succ] }, + { specialize hi (lt_trans (nat.lt_succ_self i) hn), + simp only [mul_inv_rev, fin.coe_eq_cast_succ, fin.succ_mk, fin.cast_succ_mk, + smul_eq_mul, pi.smul_apply] at hi ⊢, + rw [←fin.succ_mk _ _ (lt_trans (nat.lt_succ_self _) hn), ←fin.succ_mk], + simp only [partial_prod_succ, mul_inv_rev, fin.cast_succ_mk], + assoc_rw [hi, inv_mul_cancel_left] } +end + end partial_prod end fin diff --git a/formal/lean/mathlib/algebra/big_operators/multiset.lean b/formal/lean/mathlib/algebra/big_operators/multiset.lean index 15e7e6b55d0c838f5f164b2ad80b19044533992e..385adff005b3d7806393d0dafc4117997b3d1c70 100644 --- a/formal/lean/mathlib/algebra/big_operators/multiset.lean +++ b/formal/lean/mathlib/algebra/big_operators/multiset.lean @@ -58,6 +58,12 @@ lemma prod_cons (a : α) (s) : prod (a ::ₘ s) = a * prod s := foldr_cons _ _ _ lemma prod_erase [decidable_eq α] (h : a ∈ s) : a * (s.erase a).prod = s.prod := by rw [← s.coe_to_list, coe_erase, coe_prod, coe_prod, list.prod_erase ((s.mem_to_list a).2 h)] +@[simp, to_additive] +lemma prod_map_erase [decidable_eq ι] {a : ι} (h : a ∈ m) : + f a * ((m.erase a).map f).prod = (m.map f).prod := +by rw [← m.coe_to_list, coe_erase, coe_map, coe_map, coe_prod, coe_prod, + list.prod_map_erase f ((m.mem_to_list a).2 h)] + @[simp, to_additive] lemma prod_singleton (a : α) : prod {a} = a := by simp only [mul_one, prod_cons, singleton_eq_cons, eq_self_iff_true, prod_zero] @@ -78,6 +84,22 @@ lemma prod_nsmul (m : multiset α) : ∀ (n : ℕ), (n • m).prod = m.prod ^ n @[simp, to_additive] lemma prod_repeat (a : α) (n : ℕ) : (repeat a n).prod = a ^ n := by simp [repeat, list.prod_repeat] +@[to_additive] +lemma prod_map_eq_pow_single [decidable_eq ι] (i : ι) (hf : ∀ i' ≠ i, i' ∈ m → f i' = 1) : + (m.map f).prod = f i ^ m.count i := +begin + induction m using quotient.induction_on with l, + simp [list.prod_map_eq_pow_single i f hf], +end + +@[to_additive] +lemma prod_eq_pow_single [decidable_eq α] (a : α) (h : ∀ a' ≠ a, a' ∈ s → a' = 1) : + s.prod = a ^ (s.count a) := +begin + induction s using quotient.induction_on with l, + simp [list.prod_eq_pow_single a h], +end + @[to_additive] lemma pow_count [decidable_eq α] (a : α) : a ^ s.count a = (s.filter (eq a)).prod := by rw [filter_eq, prod_repeat] @@ -319,6 +341,15 @@ lemma prod_eq_one_iff [canonically_ordered_monoid α] {m : multiset α} : m.prod = 1 ↔ ∀ x ∈ m, x = (1 : α) := quotient.induction_on m $ λ l, by simpa using list.prod_eq_one_iff l +/-- Slightly more general version of `multiset.prod_eq_one_iff` for a non-ordered `monoid` -/ +@[to_additive "Slightly more general version of `multiset.sum_eq_zero_iff` + for a non-ordered `add_monoid`"] +lemma prod_eq_one [comm_monoid α] {m : multiset α} (h : ∀ x ∈ m, x = (1 : α)) : m.prod = 1 := +begin + induction m using quotient.induction_on with l, + simp [list.prod_eq_one h], +end + @[to_additive] lemma le_prod_of_mem [canonically_ordered_monoid α] {m : multiset α} {a : α} (h : a ∈ m) : a ≤ m.prod := diff --git a/formal/lean/mathlib/algebra/big_operators/norm_num.lean b/formal/lean/mathlib/algebra/big_operators/norm_num.lean index f410d99c47343c1da8db7a72bba7fffdc1a9a0e7..07336bf217187b0c68d2c01a27d85af7affa13bb 100644 --- a/formal/lean/mathlib/algebra/big_operators/norm_num.lean +++ b/formal/lean/mathlib/algebra/big_operators/norm_num.lean @@ -83,72 +83,6 @@ meta def list.decide_mem (decide_eq : expr → expr → tactic (bool × expr)) : pf ← i_to_expr ``(list.not_mem_cons %%head_pf %%tail_pf), pure (ff, pf) -lemma finset.insert_eq_coe_list_of_mem {α : Type*} [decidable_eq α] (x : α) (xs : finset α) - {xs' : list α} (h : x ∈ xs') (nd_xs : xs'.nodup) - (hxs' : xs = finset.mk ↑xs' (multiset.coe_nodup.mpr nd_xs)) : - insert x xs = finset.mk ↑xs' (multiset.coe_nodup.mpr nd_xs) := -have h : x ∈ xs, by simpa [hxs'] using h, -by rw [finset.insert_eq_of_mem h, hxs'] - -lemma finset.insert_eq_coe_list_cons {α : Type*} [decidable_eq α] (x : α) (xs : finset α) - {xs' : list α} (h : x ∉ xs') (nd_xs : xs'.nodup) (nd_xxs : (x :: xs').nodup) - (hxs' : xs = finset.mk ↑xs' (multiset.coe_nodup.mpr nd_xs)) : - insert x xs = finset.mk ↑(x :: xs') (multiset.coe_nodup.mpr nd_xxs) := -have h : x ∉ xs, by simpa [hxs'] using h, -by { rw [← finset.val_inj, finset.insert_val_of_not_mem h, hxs'], simp only [multiset.cons_coe] } - -/-- Convert an expression denoting a finset to a list of elements, -a proof that this list is equal to the original finset, -and a proof that the list contains no duplicates. - -We return a list rather than a finset, so we can more easily iterate over it -(without having to prove that our tactics are independent of the order of iteration, -which is in general not true). - -`decide_eq` is a (partial) decision procedure for determining whether two -elements of the finset are equal, for example to parse `{2, 1, 2}` into `[2, 1]`. --/ -meta def eval_finset (decide_eq : expr → expr → tactic (bool × expr)) : - expr → tactic (list expr × expr × expr) -| e@`(has_emptyc.emptyc) := do - eq ← mk_eq_refl e, - nd ← i_to_expr ``(list.nodup_nil), - pure ([], eq, nd) -| e@`(has_singleton.singleton %%x) := do - eq ← mk_eq_refl e, - nd ← i_to_expr ``(list.nodup_singleton %%x), - pure ([x], eq, nd) -| `(@@has_insert.insert (@@finset.has_insert %%dec) %%x %%xs) := do - (exs, xs_eq, xs_nd) ← eval_finset xs, - (is_mem, mem_pf) ← list.decide_mem decide_eq x exs, - if is_mem then do - pf ← i_to_expr ``(finset.insert_eq_coe_list_of_mem %%x %%xs %%mem_pf %%xs_nd %%xs_eq), - pure (exs, pf, xs_nd) - else do - nd ← i_to_expr ``(list.nodup_cons.mpr ⟨%%mem_pf, %%xs_nd⟩), - pf ← i_to_expr ``(finset.insert_eq_coe_list_cons %%x %%xs %%mem_pf %%xs_nd %%nd %%xs_eq), - pure (x :: exs, pf, nd) -| `(@@finset.univ %%ft) := do - -- Convert the fintype instance expression `ft` to a list of its elements. - -- Unfold it to the `fintype.mk` constructor and a list of arguments. - `fintype.mk ← get_app_fn_const_whnf ft - | fail (to_fmt "Unknown fintype expression" ++ format.line ++ to_fmt ft), - [_, args, _] ← get_app_args_whnf ft | fail (to_fmt "Expected 3 arguments to `fintype.mk`"), - eval_finset args -| e@`(finset.range %%en) := do - n ← expr.to_nat en, - eis ← (list.range n).mmap (λ i, expr.of_nat `(ℕ) i), - eq ← mk_eq_refl e, - nd ← i_to_expr ``(list.nodup_range %%en), - pure (eis, eq, nd) -| e@`(finset.fin_range %%en) := do - n ← expr.to_nat en, - eis ← (list.fin_range n).mmap (λ i, expr.of_nat `(fin %%en) i), - eq ← mk_eq_refl e, - nd ← i_to_expr ``(list.nodup_fin_range %%en), - pure (eis, eq, nd) -| e := fail (to_fmt "Unknown finset expression" ++ format.line ++ to_fmt e) - lemma list.map_cons_congr {α β : Type*} (f : α → β) {x : α} {xs : list α} {fx : β} {fxs : list β} (h₁ : f x = fx) (h₂ : xs.map f = fxs) : (x :: xs).map f = fx :: fxs := by rw [list.map_cons, h₁, h₂] @@ -164,6 +98,36 @@ meta def eval_list_map (ef : expr) : list expr → tactic (list expr × expr) eq ← i_to_expr ``(list.map_cons_congr %%ef %%fx_eq %%fxs_eq), pure (fx :: fxs, eq) +lemma list.cons_congr {α : Type*} (x : α) {xs : list α} {xs' : list α} (xs_eq : xs' = xs) : + x :: xs' = x :: xs := +by rw xs_eq + +lemma list.map_congr {α β : Type*} (f : α → β) {xs xs' : list α} + {ys : list β} (xs_eq : xs = xs') (ys_eq : xs'.map f = ys) : + xs.map f = ys := +by rw [← ys_eq, xs_eq] + +/-- Convert an expression denoting a list to a list of elements. -/ +meta def eval_list : expr → tactic (list expr × expr) +| e@`(list.nil) := do + eq ← mk_eq_refl e, + pure ([], eq) +| e@`(list.cons %%x %%xs) := do + (xs, xs_eq) ← eval_list xs, + eq ← i_to_expr ``(list.cons_congr %%x %%xs_eq), + pure (x :: xs, eq) +| e@`(list.range %%en) := do + n ← expr.to_nat en, + eis ← (list.range n).mmap (λ i, expr.of_nat `(ℕ) i), + eq ← mk_eq_refl e, + pure (eis, eq) +| `(@list.map %%α %%β %%ef %%exs) := do + (xs, xs_eq) ← eval_list exs, + (ys, ys_eq) ← eval_list_map ef xs, + eq ← i_to_expr ``(list.map_congr %%ef %%xs_eq %%ys_eq), + pure (ys, eq) +| e := fail (to_fmt "Unknown list expression" ++ format.line ++ to_fmt e) + lemma multiset.cons_congr {α : Type*} (x : α) {xs : multiset α} {xs' : list α} (xs_eq : (xs' : multiset α) = xs) : (list.cons x xs' : multiset α) = x ::ₘ xs := by rw [← xs_eq]; refl @@ -209,35 +173,71 @@ meta def eval_multiset : expr → tactic (list expr × expr) pure (ys, eq) | e := fail (to_fmt "Unknown multiset expression" ++ format.line ++ to_fmt e) -lemma list.cons_congr {α : Type*} (x : α) {xs : list α} {xs' : list α} (xs_eq : xs' = xs) : - x :: xs' = x :: xs := -by rw xs_eq +lemma finset.insert_eq_coe_list_of_mem {α : Type*} [decidable_eq α] (x : α) (xs : finset α) + {xs' : list α} (h : x ∈ xs') (nd_xs : xs'.nodup) + (hxs' : xs = finset.mk ↑xs' (multiset.coe_nodup.mpr nd_xs)) : + insert x xs = finset.mk ↑xs' (multiset.coe_nodup.mpr nd_xs) := +have h : x ∈ xs, by simpa [hxs'] using h, +by rw [finset.insert_eq_of_mem h, hxs'] -lemma list.map_congr {α β : Type*} (f : α → β) {xs xs' : list α} - {ys : list β} (xs_eq : xs = xs') (ys_eq : xs'.map f = ys) : - xs.map f = ys := -by rw [← ys_eq, xs_eq] +lemma finset.insert_eq_coe_list_cons {α : Type*} [decidable_eq α] (x : α) (xs : finset α) + {xs' : list α} (h : x ∉ xs') (nd_xs : xs'.nodup) (nd_xxs : (x :: xs').nodup) + (hxs' : xs = finset.mk ↑xs' (multiset.coe_nodup.mpr nd_xs)) : + insert x xs = finset.mk ↑(x :: xs') (multiset.coe_nodup.mpr nd_xxs) := +have h : x ∉ xs, by simpa [hxs'] using h, +by { rw [← finset.val_inj, finset.insert_val_of_not_mem h, hxs'], simp only [multiset.cons_coe] } -/-- Convert an expression denoting a list to a list of elements. -/ -meta def eval_list : expr → tactic (list expr × expr) -| e@`(list.nil) := do +/-- Convert an expression denoting a finset to a list of elements, +a proof that this list is equal to the original finset, +and a proof that the list contains no duplicates. + +We return a list rather than a finset, so we can more easily iterate over it +(without having to prove that our tactics are independent of the order of iteration, +which is in general not true). + +`decide_eq` is a (partial) decision procedure for determining whether two +elements of the finset are equal, for example to parse `{2, 1, 2}` into `[2, 1]`. +-/ +meta def eval_finset (decide_eq : expr → expr → tactic (bool × expr)) : + expr → tactic (list expr × expr × expr) +| e@`(has_emptyc.emptyc) := do eq ← mk_eq_refl e, - pure ([], eq) -| e@`(list.cons %%x %%xs) := do - (xs, xs_eq) ← eval_list xs, - eq ← i_to_expr ``(list.cons_congr %%x %%xs_eq), - pure (x :: xs, eq) -| e@`(list.range %%en) := do + nd ← i_to_expr ``(list.nodup_nil), + pure ([], eq, nd) +| e@`(has_singleton.singleton %%x) := do + eq ← mk_eq_refl e, + nd ← i_to_expr ``(list.nodup_singleton %%x), + pure ([x], eq, nd) +| `(@@has_insert.insert (@@finset.has_insert %%dec) %%x %%xs) := do + (exs, xs_eq, xs_nd) ← eval_finset xs, + (is_mem, mem_pf) ← list.decide_mem decide_eq x exs, + if is_mem then do + pf ← i_to_expr ``(finset.insert_eq_coe_list_of_mem %%x %%xs %%mem_pf %%xs_nd %%xs_eq), + pure (exs, pf, xs_nd) + else do + nd ← i_to_expr ``(list.nodup_cons.mpr ⟨%%mem_pf, %%xs_nd⟩), + pf ← i_to_expr ``(finset.insert_eq_coe_list_cons %%x %%xs %%mem_pf %%xs_nd %%nd %%xs_eq), + pure (x :: exs, pf, nd) +| `(@@finset.univ %%ft) := do + -- Convert the fintype instance expression `ft` to a list of its elements. + -- Unfold it to the `fintype.mk` constructor and a list of arguments. + `fintype.mk ← get_app_fn_const_whnf ft + | fail (to_fmt "Unknown fintype expression" ++ format.line ++ to_fmt ft), + [_, args, _] ← get_app_args_whnf ft | fail (to_fmt "Expected 3 arguments to `fintype.mk`"), + eval_finset args +| e@`(finset.range %%en) := do n ← expr.to_nat en, eis ← (list.range n).mmap (λ i, expr.of_nat `(ℕ) i), eq ← mk_eq_refl e, - pure (eis, eq) -| `(@list.map %%α %%β %%ef %%exs) := do - (xs, xs_eq) ← eval_list exs, - (ys, ys_eq) ← eval_list_map ef xs, - eq ← i_to_expr ``(list.map_congr %%ef %%xs_eq %%ys_eq), - pure (ys, eq) -| e := fail (to_fmt "Unknown list expression" ++ format.line ++ to_fmt e) + nd ← i_to_expr ``(list.nodup_range %%en), + pure (eis, eq, nd) +| e@`(finset.fin_range %%en) := do + n ← expr.to_nat en, + eis ← (list.fin_range n).mmap (λ i, expr.of_nat `(fin %%en) i), + eq ← mk_eq_refl e, + nd ← i_to_expr ``(list.nodup_fin_range %%en), + pure (eis, eq, nd) +| e := fail (to_fmt "Unknown finset expression" ++ format.line ++ to_fmt e) @[to_additive] lemma list.prod_cons_congr {α : Type*} [monoid α] (xs : list α) (x y z : α) diff --git a/formal/lean/mathlib/algebra/big_operators/ring.lean b/formal/lean/mathlib/algebra/big_operators/ring.lean index fb1e6504e6a795896551f636f55a4da996c6f369..7d5e7727e114e2f89e89155550db5cb82f7336a2 100644 --- a/formal/lean/mathlib/algebra/big_operators/ring.lean +++ b/formal/lean/mathlib/algebra/big_operators/ring.lean @@ -52,7 +52,7 @@ add_monoid_hom.map_sum (add_monoid_hom.mul_left b) _ s lemma sum_mul_sum {ι₁ : Type*} {ι₂ : Type*} (s₁ : finset ι₁) (s₂ : finset ι₂) (f₁ : ι₁ → β) (f₂ : ι₂ → β) : - (∑ x₁ in s₁, f₁ x₁) * (∑ x₂ in s₂, f₂ x₂) = ∑ p in s₁.product s₂, f₁ p.1 * f₂ p.2 := + (∑ x₁ in s₁, f₁ x₁) * (∑ x₂ in s₂, f₂ x₂) = ∑ p in s₁ ×ˢ s₂, f₁ p.1 * f₂ p.2 := by { rw [sum_product, sum_mul, sum_congr rfl], intros, rw mul_sum } end semiring diff --git a/formal/lean/mathlib/algebra/category/FinVect.lean b/formal/lean/mathlib/algebra/category/FinVect.lean index f78a110c0bf5d86ce41ce1f439ec4986396dd768..f3b2583d36e51432f068cb400b58fd17be5523c4 100644 --- a/formal/lean/mathlib/algebra/category/FinVect.lean +++ b/formal/lean/mathlib/algebra/category/FinVect.lean @@ -31,22 +31,21 @@ instance monoidal_predicate_finite_dimensional : { prop_id' := finite_dimensional.finite_dimensional_self K, prop_tensor' := λ X Y hX hY, by exactI module.finite.tensor_product K X Y } +instance closed_predicate_finite_dimensional : + monoidal_category.closed_predicate (λ V : Module.{u} K, finite_dimensional K V) := +{ prop_ihom' := λ X Y hX hY, by exactI @linear_map.finite_dimensional K _ X _ _ hX Y _ _ hY } + /-- Define `FinVect` as the subtype of `Module.{u} K` of finite dimensional vector spaces. -/ -@[derive [large_category, λ α, has_coe_to_sort α (Sort*), concrete_category, monoidal_category, - symmetric_category]] -def FinVect := { V : Module.{u} K // finite_dimensional K V } +@[derive [large_category, concrete_category, monoidal_category, symmetric_category, +monoidal_closed]] +def FinVect := full_subcategory (λ (V : Module.{u} K), finite_dimensional K V) namespace FinVect -instance finite_dimensional (V : FinVect K) : finite_dimensional K V := V.prop +instance finite_dimensional (V : FinVect K) : finite_dimensional K V.obj := V.property instance : inhabited (FinVect K) := ⟨⟨Module.of K K, finite_dimensional.finite_dimensional_self K⟩⟩ -instance : has_coe (FinVect.{u} K) (Module.{u} K) := { coe := λ V, V.1, } - -protected lemma coe_comp {U V W : FinVect K} (f : U ⟶ V) (g : V ⟶ W) : - ((f ≫ g) : U → W) = (g : V → W) ∘ (f : U → V) := rfl - /-- Lift an unbundled vector space to `FinVect K`. -/ def of (V : Type u) [add_comm_group V] [module K V] [finite_dimensional K V] : FinVect K := ⟨Module.of K V, by { change finite_dimensional K V, apply_instance }⟩ @@ -57,46 +56,45 @@ by { dsimp [FinVect], apply_instance, } instance : full (forget₂ (FinVect K) (Module.{u} K)) := { preimage := λ X Y f, f, } -variables (V : FinVect K) +variables (V W : FinVect K) + +@[simp] lemma ihom_obj : (ihom V).obj W = FinVect.of K (V.obj →ₗ[K] W.obj) := rfl /-- The dual module is the dual in the rigid monoidal category `FinVect K`. -/ def FinVect_dual : FinVect K := -⟨Module.of K (module.dual K V), subspace.module.dual.finite_dimensional⟩ - -instance : has_coe_to_fun (FinVect_dual K V) (λ _, V → K) := -{ coe := λ v, by { change V →ₗ[K] K at v, exact v, } } +⟨Module.of K (module.dual K V.obj), subspace.module.dual.finite_dimensional⟩ open category_theory.monoidal_category /-- The coevaluation map is defined in `linear_algebra.coevaluation`. -/ def FinVect_coevaluation : 𝟙_ (FinVect K) ⟶ V ⊗ (FinVect_dual K V) := -by apply coevaluation K V +by apply coevaluation K V.obj lemma FinVect_coevaluation_apply_one : FinVect_coevaluation K V (1 : K) = - ∑ (i : basis.of_vector_space_index K V), - (basis.of_vector_space K V) i ⊗ₜ[K] (basis.of_vector_space K V).coord i := -by apply coevaluation_apply_one K V + ∑ (i : basis.of_vector_space_index K V.obj), + (basis.of_vector_space K V.obj) i ⊗ₜ[K] (basis.of_vector_space K V.obj).coord i := +by apply coevaluation_apply_one K V.obj /-- The evaluation morphism is given by the contraction map. -/ def FinVect_evaluation : (FinVect_dual K V) ⊗ V ⟶ 𝟙_ (FinVect K) := -by apply contract_left K V +by apply contract_left K V.obj @[simp] -lemma FinVect_evaluation_apply (f : (FinVect_dual K V)) (x : V) : - (FinVect_evaluation K V) (f ⊗ₜ x) = f x := +lemma FinVect_evaluation_apply (f : (FinVect_dual K V).obj) (x : V.obj) : + (FinVect_evaluation K V) (f ⊗ₜ x) = f.to_fun x := by apply contract_left_apply f x private theorem coevaluation_evaluation : let V' : FinVect K := FinVect_dual K V in (𝟙 V' ⊗ (FinVect_coevaluation K V)) ≫ (α_ V' V V').inv ≫ (FinVect_evaluation K V ⊗ 𝟙 V') = (ρ_ V').hom ≫ (λ_ V').inv := -by apply contract_left_assoc_coevaluation K V +by apply contract_left_assoc_coevaluation K V.obj private theorem evaluation_coevaluation : (FinVect_coevaluation K V ⊗ 𝟙 V) ≫ (α_ V (FinVect_dual K V) V).hom ≫ (𝟙 V ⊗ FinVect_evaluation K V) = (λ_ V).hom ≫ (ρ_ V).inv := -by apply contract_left_assoc_coevaluation' K V +by apply contract_left_assoc_coevaluation' K V.obj instance exact_pairing : exact_pairing V (FinVect_dual K V) := { coevaluation := FinVect_coevaluation K V, @@ -108,11 +106,11 @@ instance right_dual : has_right_dual V := ⟨FinVect_dual K V⟩ instance right_rigid_category : right_rigid_category (FinVect K) := { } -variables {K V} (W : FinVect K) +variables {K V} /-- Converts and isomorphism in the category `FinVect` to a `linear_equiv` between the underlying vector spaces. -/ -def iso_to_linear_equiv {V W : FinVect K} (i : V ≅ W) : V ≃ₗ[K] W := +def iso_to_linear_equiv {V W : FinVect K} (i : V ≅ W) : V.obj ≃ₗ[K] W.obj := ((forget₂ (FinVect.{u} K) (Module.{u} K)).map_iso i).to_linear_equiv lemma iso.conj_eq_conj {V W : FinVect K} (i : V ≅ W) (f : End V) : diff --git a/formal/lean/mathlib/algebra/category/FinVect/limits.lean b/formal/lean/mathlib/algebra/category/FinVect/limits.lean index e27d7c05aa3c239001bbd87355cb3b7a1e969ced..70188668e71dbc1ad40fa8d6b0c988290f32fe78 100644 --- a/formal/lean/mathlib/algebra/category/FinVect/limits.lean +++ b/formal/lean/mathlib/algebra/category/FinVect/limits.lean @@ -48,7 +48,7 @@ instance (F : J ⥤ FinVect k) : finite_dimensional k (limit (F ⋙ forget₂ (FinVect k) (Module.{v} k)) : Module.{v} k) := begin haveI : ∀ j, finite_dimensional k ((F ⋙ forget₂ (FinVect k) (Module.{v} k)).obj j), - { intro j, change finite_dimensional k (F.obj j), apply_instance, }, + { intro j, change finite_dimensional k (F.obj j).obj, apply_instance, }, exact finite_dimensional.of_injective (limit_subobject_product (F ⋙ forget₂ (FinVect k) (Module.{v} k))) ((Module.mono_iff_injective _).1 (by apply_instance)), diff --git a/formal/lean/mathlib/algebra/category/Group/epi_mono.lean b/formal/lean/mathlib/algebra/category/Group/epi_mono.lean index acc4cac5ed5dd3182b658ffa07ad50f5f87d3253..ca6fc836c22f519050f57143f3fbdd52b57ce35e 100644 --- a/formal/lean/mathlib/algebra/category/Group/epi_mono.lean +++ b/formal/lean/mathlib/algebra/category/Group/epi_mono.lean @@ -3,7 +3,7 @@ Copyright (c) 2022 Jujian Zhang. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jujian Zhang -/ -import algebra.category.Group.basic +import algebra.category.Group.equivalence_Group_AddGroup import category_theory.epi_mono import group_theory.quotient_group @@ -23,6 +23,7 @@ open quotient_group variables {A : Type u} {B : Type v} +section variables [group A] [group B] @[to_additive add_monoid_hom.ker_eq_bot_of_cancel] @@ -30,6 +31,28 @@ lemma ker_eq_bot_of_cancel {f : A →* B} (h : ∀ (u v : f.ker →* A), f.comp f.ker = ⊥ := by simpa using _root_.congr_arg range (h f.ker.subtype 1 (by tidy)) +end + +section +variables [comm_group A] [comm_group B] + +@[to_additive add_monoid_hom.range_eq_top_of_cancel] +lemma range_eq_top_of_cancel {f : A →* B} + (h : ∀ (u v : B →* B ⧸ f.range), u.comp f = v.comp f → u = v) : + f.range = ⊤ := +begin + specialize h 1 (quotient_group.mk' _) _, + { ext1, + simp only [one_apply, coe_comp, coe_mk', function.comp_app], + rw [show (1 : B ⧸ f.range) = (1 : B), from quotient_group.coe_one _, quotient_group.eq, + inv_one, one_mul], + exact ⟨x, rfl⟩, }, + replace h : (quotient_group.mk' _).ker = (1 : B →* B ⧸ f.range).ker := by rw h, + rwa [ker_one, quotient_group.ker_mk] at h, +end + +end + end monoid_hom section @@ -61,7 +84,7 @@ local notation `X` := set.range (function.swap left_coset f.range.carrier) /-- Define `X'` to be the set of all left cosets with an extra point at "infinity". -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] inductive X_with_infinity | from_coset : set.range (function.swap left_coset f.range.carrier) → X_with_infinity | infinity : X_with_infinity @@ -281,4 +304,58 @@ iff.trans (epi_iff_surjective _) (subgroup.eq_top_iff' f.range).symm end Group +namespace AddGroup +variables {A B : AddGroup.{u}} (f : A ⟶ B) + +lemma epi_iff_surjective : epi f ↔ function.surjective f := +begin + have i1 : epi f ↔ epi (Group_AddGroup_equivalence.inverse.map f), + { refine ⟨_, Group_AddGroup_equivalence.inverse.epi_of_epi_map⟩, + introsI e', + apply Group_AddGroup_equivalence.inverse.map_epi }, + rwa Group.epi_iff_surjective at i1, +end + +lemma epi_iff_range_eq_top : epi f ↔ f.range = ⊤ := +iff.trans (epi_iff_surjective _) (add_subgroup.eq_top_iff' f.range).symm + +end AddGroup + +namespace CommGroup +variables {A B : CommGroup.{u}} (f : A ⟶ B) + +@[to_additive AddCommGroup.ker_eq_bot_of_mono] +lemma ker_eq_bot_of_mono [mono f] : f.ker = ⊥ := +monoid_hom.ker_eq_bot_of_cancel $ λ u v, + (@cancel_mono _ _ _ _ _ f _ (show CommGroup.of f.ker ⟶ A, from u) _).1 + +@[to_additive AddCommGroup.mono_iff_ker_eq_bot] +lemma mono_iff_ker_eq_bot : mono f ↔ f.ker = ⊥ := +⟨λ h, @@ker_eq_bot_of_mono f h, + λ h, concrete_category.mono_of_injective _ $ (monoid_hom.ker_eq_bot_iff f).1 h⟩ + +@[to_additive AddCommGroup.mono_iff_injective] +lemma mono_iff_injective : mono f ↔ function.injective f := +iff.trans (mono_iff_ker_eq_bot f) $ monoid_hom.ker_eq_bot_iff f + +@[to_additive] +lemma range_eq_top_of_epi [epi f] : f.range = ⊤ := +monoid_hom.range_eq_top_of_cancel $ λ u v h, + (@cancel_epi _ _ _ _ _ f _ (show B ⟶ ⟨B ⧸ monoid_hom.range f⟩, from u) v).1 h + +@[to_additive] +lemma epi_iff_range_eq_top : epi f ↔ f.range = ⊤ := +⟨λ hf, by exactI range_eq_top_of_epi _, + λ hf, concrete_category.epi_of_surjective _ $ monoid_hom.range_top_iff_surjective.mp hf⟩ + +@[to_additive] +lemma epi_iff_surjective : epi f ↔ function.surjective f := +by rw [epi_iff_range_eq_top, monoid_hom.range_top_iff_surjective] + +@[to_additive] +instance : functor.preserves_epimorphisms (forget₂ CommGroup Group) := +{ preserves := λ X Y f e, by rwa [epi_iff_surjective, ←@Group.epi_iff_surjective ⟨X⟩ ⟨Y⟩ f] at e } + +end CommGroup + end diff --git a/formal/lean/mathlib/algebra/category/Group/equivalence_Group_AddGroup.lean b/formal/lean/mathlib/algebra/category/Group/equivalence_Group_AddGroup.lean new file mode 100644 index 0000000000000000000000000000000000000000..fe392d5a87099718698bf78e6daabec6a0e7e105 --- /dev/null +++ b/formal/lean/mathlib/algebra/category/Group/equivalence_Group_AddGroup.lean @@ -0,0 +1,86 @@ +/- +Copyright (c) 2022 Jujian Zhang. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Jujian Zhang +-/ +import algebra.category.Group.basic + +/-! +# Equivalence between `Group` and `AddGroup` + +This file contains two equivalences: +* `Group_AddGroup_equivalence` : the equivalence between `Group` and `AddGroup` by sending + `X : Group` to `additive X` and `Y : AddGroup` to `multiplicative Y`. +* `CommGroup_AddCommGroup_equivlance` : the equivalence between `CommGroup` and `AddCommGroup` by + sending `X : CommGroup` to `additive X` and `Y : AddCommGroup` to `multiplicative Y`. +-/ + +open category_theory + +namespace Group + +/-- +The functor `Group ⥤ AddGroup` by sending `X ↦ additive X` and `f ↦ f`. +-/ +@[simps] def to_AddGroup : Group ⥤ AddGroup := +{ obj := λ X, AddGroup.of (additive X), + map := λ X Y, monoid_hom.to_additive } + +end Group + +namespace CommGroup + +/-- +The functor `CommGroup ⥤ AddCommGroup` by sending `X ↦ additive X` and `f ↦ f`. +-/ +@[simps] def to_AddCommGroup : CommGroup ⥤ AddCommGroup := +{ obj := λ X, AddCommGroup.of (additive X), + map := λ X Y, monoid_hom.to_additive } + +end CommGroup + +namespace AddGroup + +/-- +The functor `AddGroup ⥤ Group` by sending `X ↦ multiplicative Y` and `f ↦ f`. +-/ +@[simps] def to_Group : AddGroup ⥤ Group := +{ obj := λ X, Group.of (multiplicative X), + map := λ X Y, add_monoid_hom.to_multiplicative } + +end AddGroup + +namespace AddCommGroup + +/-- +The functor `AddCommGroup ⥤ CommGroup` by sending `X ↦ multiplicative Y` and `f ↦ f`. +-/ +@[simps] def to_CommGroup : AddCommGroup ⥤ CommGroup := +{ obj := λ X, CommGroup.of (multiplicative X), + map := λ X Y, add_monoid_hom.to_multiplicative } + +end AddCommGroup + +/-- +The equivalence of categories between `Group` and `AddGroup` +-/ +@[simps] def Group_AddGroup_equivalence : Group ≌ AddGroup := +equivalence.mk Group.to_AddGroup AddGroup.to_Group + (nat_iso.of_components + (λ X, mul_equiv.to_Group_iso (mul_equiv.multiplicative_additive X)) + (λ X Y f, rfl)) + (nat_iso.of_components + (λ X, add_equiv.to_AddGroup_iso (add_equiv.additive_multiplicative X)) + (λ X Y f, rfl)) + +/-- +The equivalence of categories between `CommGroup` and `AddCommGroup`. +-/ +@[simps] def CommGroup_AddCommGroup_equivalence : CommGroup ≌ AddCommGroup := +equivalence.mk CommGroup.to_AddCommGroup AddCommGroup.to_CommGroup + (nat_iso.of_components + (λ X, mul_equiv.to_CommGroup_iso (mul_equiv.multiplicative_additive X)) + (λ X Y f, rfl)) + (nat_iso.of_components + (λ X, add_equiv.to_AddCommGroup_iso (add_equiv.additive_multiplicative X)) + (λ X Y f, rfl)) diff --git a/formal/lean/mathlib/algebra/category/Module/adjunctions.lean b/formal/lean/mathlib/algebra/category/Module/adjunctions.lean index d810bdd2231afa8eaaf5f88d7206ac81cac20a28..e3d6d7980eb0ebbf505f11fc77e42b53896bf2f0 100644 --- a/formal/lean/mathlib/algebra/category/Module/adjunctions.lean +++ b/formal/lean/mathlib/algebra/category/Module/adjunctions.lean @@ -156,7 +156,7 @@ universes v u we will equip with a category structure where the morphisms are formal `R`-linear combinations of the morphisms in `C`. -/ -@[nolint unused_arguments has_inhabited_instance] +@[nolint unused_arguments has_nonempty_instance] def Free (R : Type*) (C : Type u) := C /-- diff --git a/formal/lean/mathlib/algebra/category/Module/change_of_rings.lean b/formal/lean/mathlib/algebra/category/Module/change_of_rings.lean new file mode 100644 index 0000000000000000000000000000000000000000..fc8d97972b2a55538a9ecb857b28c7414ceef941 --- /dev/null +++ b/formal/lean/mathlib/algebra/category/Module/change_of_rings.lean @@ -0,0 +1,65 @@ +/- +Copyright (c) 2022 Jujian Zhang. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Jujian Zhang +-/ +import algebra.category.Module.basic + +/-! +# Change Of Rings + +## Main definitions + +* `category_theory.Module.restrict_scalars`: given rings `R, S` and a ring homomorphism `R ⟶ S`, + then `restrict_scalars : Module S ⥤ Module R` is defined by `M ↦ M` where `M : S-module` is seen + as `R-module` by `r • m := f r • m` and `S`-linear map `l : M ⟶ M'` is `R`-linear as well. +-/ + + +namespace category_theory.Module + +universes v u₁ u₂ + +namespace restrict_scalars + +variables {R : Type u₁} {S : Type u₂} [ring R] [ring S] (f : R →+* S) +variable (M : Module.{v} S) + +/-- Any `S`-module M is also an `R`-module via a ring homomorphism `f : R ⟶ S` by defining + `r • m := f r • m` (`module.comp_hom`). This is called restriction of scalars. -/ +def obj' : Module R := +{ carrier := M, + is_module := module.comp_hom M f } + +/-- +Given an `S`-linear map `g : M → M'` between `S`-modules, `g` is also `R`-linear between `M` and +`M'` by means of restriction of scalars. +-/ +def map' {M M' : Module.{v} S} (g : M ⟶ M') : + obj' f M ⟶ obj' f M' := +{ map_smul' := λ r, g.map_smul (f r), ..g } + +end restrict_scalars + +/-- +The restriction of scalars operation is functorial. For any `f : R →+* S` a ring homomorphism, +* an `S`-module `M` can be considered as `R`-module by `r • m = f r • m` +* an `S`-linear map is also `R`-linear +-/ +def restrict_scalars {R : Type u₁} {S : Type u₂} [ring R] [ring S] (f : R →+* S) : + Module.{v} S ⥤ Module.{v} R := +{ obj := restrict_scalars.obj' f, + map := λ _ _, restrict_scalars.map' f, + map_id' := λ _, linear_map.ext $ λ m, rfl, + map_comp' := λ _ _ _ g h, linear_map.ext $ λ m, rfl } + +@[simp] lemma restrict_scalars.map_apply {R : Type u₁} {S : Type u₂} [ring R] [ring S] (f : R →+* S) + {M M' : Module.{v} S} (g : M ⟶ M') (x) : (restrict_scalars f).map g x = g x := rfl + +@[simp] lemma restrict_scalars.smul_def {R : Type u₁} {S : Type u₂} [ring R] [ring S] (f : R →+* S) + {M : Module.{v} S} (r : R) (m : (restrict_scalars f).obj M) : r • m = (f r • m : M) := rfl + +@[simp] lemma restrict_scalars.smul_def' {R : Type u₁} {S : Type u₂} [ring R] [ring S] (f : R →+* S) + {M : Module.{v} S} (r : R) (m : M) : (r • m : (restrict_scalars f).obj M) = (f r • m : M) := rfl + +end category_theory.Module diff --git a/formal/lean/mathlib/algebra/char_p/algebra.lean b/formal/lean/mathlib/algebra/char_p/algebra.lean index 3b4493df13e96bd7c567648e63d4f9c625b89f06..feebf7234a3854359d3e35d4425e454e8c3d2297 100644 --- a/formal/lean/mathlib/algebra/char_p/algebra.lean +++ b/formal/lean/mathlib/algebra/char_p/algebra.lean @@ -65,6 +65,9 @@ variables (K L : Type*) [field K] [comm_semiring L] [nontrivial L] [algebra K L] lemma algebra.char_p_iff (p : ℕ) : char_p K p ↔ char_p L p := (algebra_map K L).char_p_iff_char_p p +lemma algebra.ring_char_eq : ring_char K = ring_char L := +by { rw [ring_char.eq_iff, algebra.char_p_iff K L], apply ring_char.char_p } + end namespace free_algebra diff --git a/formal/lean/mathlib/algebra/char_p/basic.lean b/formal/lean/mathlib/algebra/char_p/basic.lean index d1a83a49b7a60e9bca049310f88df1a4ac88311a..f2659a1c1744179a082704ec90139a5a9c699c0a 100644 --- a/formal/lean/mathlib/algebra/char_p/basic.lean +++ b/formal/lean/mathlib/algebra/char_p/basic.lean @@ -349,13 +349,16 @@ calc (k : R) = ↑(k % p + p * (k / p)) : by rw [nat.mod_add_div] ... = ↑(k % p) : by simp [cast_eq_zero] /-- The characteristic of a finite ring cannot be zero. -/ -theorem char_ne_zero_of_fintype (p : ℕ) [hc : char_p R p] [fintype R] : p ≠ 0 := -assume h : p = 0, -have char_zero R := @char_p_to_char_zero R _ (h ▸ hc), -absurd (@nat.cast_injective R _ this) (not_injective_infinite_fintype coe) +theorem char_ne_zero_of_finite (p : ℕ) [char_p R p] [finite R] : p ≠ 0 := +begin + unfreezingI { rintro rfl }, + haveI : char_zero R := char_p_to_char_zero R, + casesI nonempty_fintype R, + exact absurd nat.cast_injective (not_injective_infinite_fintype (coe : ℕ → R)) +end -lemma ring_char_ne_zero_of_fintype [fintype R] : ring_char R ≠ 0 := -char_ne_zero_of_fintype R (ring_char R) +lemma ring_char_ne_zero_of_finite [finite R] : ring_char R ≠ 0 := +char_ne_zero_of_finite R (ring_char R) end @@ -432,11 +435,11 @@ end semiring section ring -variables (R) [ring R] [no_zero_divisors R] [nontrivial R] [fintype R] +variables (R) [ring R] [no_zero_divisors R] [nontrivial R] [finite R] theorem char_is_prime (p : ℕ) [char_p R p] : p.prime := -or.resolve_right (char_is_prime_or_zero R p) (char_ne_zero_of_fintype R p) +or.resolve_right (char_is_prime_or_zero R p) (char_ne_zero_of_finite R p) end ring diff --git a/formal/lean/mathlib/algebra/char_p/char_and_card.lean b/formal/lean/mathlib/algebra/char_p/char_and_card.lean index 9581ba6c4602f09359304da464c60fb67de1d351..10d61ec6c37081009365042ffdd1d367056b6ca6 100644 --- a/formal/lean/mathlib/algebra/char_p/char_and_card.lean +++ b/formal/lean/mathlib/algebra/char_p/char_and_card.lean @@ -15,12 +15,14 @@ We prove some results relating characteristic and cardinality of finite rings characterstic, cardinality, ring -/ -/-- A prime `p` is a unit in a finite commutative ring `R` -iff it does not divide the characteristic. -/ -lemma is_unit_iff_not_dvd_char (R : Type*) [comm_ring R] [fintype R] (p : ℕ) [fact p.prime] : +/-- A prime `p` is a unit in a commutative ring `R` of nonzero characterstic iff it does not divide +the characteristic. -/ +lemma is_unit_iff_not_dvd_char_of_ring_char_ne_zero (R : Type*) [comm_ring R] (p : ℕ) [fact p.prime] + (hR : ring_char R ≠ 0) : is_unit (p : R) ↔ ¬ p ∣ ring_char R := begin have hch := char_p.cast_eq_zero R (ring_char R), + have hp : p.prime := fact.out p.prime, split, { rintros h₁ ⟨q, hq⟩, rcases is_unit.exists_left_inv h₁ with ⟨a, ha⟩, @@ -29,8 +31,7 @@ begin rintro ⟨r, hr⟩, rw [hr, ← mul_assoc, mul_comm p, mul_assoc] at hq, nth_rewrite 0 ← mul_one (ring_char R) at hq, - exact nat.prime.not_dvd_one (fact.out p.prime) - ⟨r, mul_left_cancel₀ (char_p.char_ne_zero_of_fintype R (ring_char R)) hq⟩, + exact nat.prime.not_dvd_one hp ⟨r, mul_left_cancel₀ hR hq⟩, end, have h₄ := mt (char_p.int_cast_eq_zero_iff R (ring_char R) q).mp, apply_fun (coe : ℕ → R) at hq, @@ -39,14 +40,19 @@ begin norm_cast at h₄, exact h₄ h₃ hq.symm, }, { intro h, - rcases nat.is_coprime_iff_coprime.mpr ((nat.prime.coprime_iff_not_dvd (fact.out _)).mpr h) - with ⟨a, b, hab⟩, + rcases (hp.coprime_iff_not_dvd.mpr h).is_coprime with ⟨a, b, hab⟩, apply_fun (coe : ℤ → R) at hab, push_cast at hab, rw [hch, mul_zero, add_zero, mul_comm] at hab, exact is_unit_of_mul_eq_one (p : R) a hab, }, end +/-- A prime `p` is a unit in a finite commutative ring `R` +iff it does not divide the characteristic. -/ +lemma is_unit_iff_not_dvd_char (R : Type*) [comm_ring R] (p : ℕ) [fact p.prime] [finite R] : + is_unit (p : R) ↔ ¬ p ∣ ring_char R := +is_unit_iff_not_dvd_char_of_ring_char_ne_zero R p $ char_p.char_ne_zero_of_finite R (ring_char R) + /-- The prime divisors of the characteristic of a finite commutative ring are exactly the prime divisors of its cardinality. -/ lemma prime_dvd_char_iff_dvd_card {R : Type*} [comm_ring R] [fintype R] (p : ℕ) [fact p.prime] : diff --git a/formal/lean/mathlib/algebra/direct_sum/internal.lean b/formal/lean/mathlib/algebra/direct_sum/internal.lean index 5aa5c56af42ef02e662be29f7fbf2149c1a269be..87dbb2e9fb0160c6b087b49250dddccb1e92a4c6 100644 --- a/formal/lean/mathlib/algebra/direct_sum/internal.lean +++ b/formal/lean/mathlib/algebra/direct_sum/internal.lean @@ -152,8 +152,7 @@ lemma direct_sum.coe_mul_apply [add_monoid ι] [semiring R] [set_like σ R] [add_submonoid_class σ R] (A : ι → σ) [set_like.graded_monoid A] [Π (i : ι) (x : A i), decidable (x ≠ 0)] (r r' : ⨁ i, A i) (i : ι) : ((r * r') i : R) = - ∑ ij in finset.filter (λ ij : ι × ι, ij.1 + ij.2 = i) (r.support.product r'.support), - r ij.1 * r' ij.2 := + ∑ ij in (r.support ×ˢ r'.support).filter (λ ij : ι × ι, ij.1 + ij.2 = i), r ij.1 * r' ij.2 := begin rw [direct_sum.mul_eq_sum_support_ghas_mul, dfinsupp.finset_sum_apply, add_submonoid_class.coe_finset_sum], diff --git a/formal/lean/mathlib/algebra/direct_sum/ring.lean b/formal/lean/mathlib/algebra/direct_sum/ring.lean index aae4829ac4f87e4bbecc0a2923421ab694a93096..99e73c97e609fb466c2092a17a9d6322e345256c 100644 --- a/formal/lean/mathlib/algebra/direct_sum/ring.lean +++ b/formal/lean/mathlib/algebra/direct_sum/ring.lean @@ -269,7 +269,7 @@ open_locale big_operators lemma mul_eq_sum_support_ghas_mul [Π (i : ι) (x : A i), decidable (x ≠ 0)] (a a' : ⨁ i, A i) : a * a' = - ∑ (ij : ι × ι) in (dfinsupp.support a).product (dfinsupp.support a'), + ∑ ij in dfinsupp.support a ×ˢ dfinsupp.support a', direct_sum.of _ _ (graded_monoid.ghas_mul.mul (a ij.fst) (a' ij.snd)) := begin change direct_sum.mul_hom _ a a' = _, diff --git a/formal/lean/mathlib/algebra/field/basic.lean b/formal/lean/mathlib/algebra/field/basic.lean index a0cff821c9452aed5fb4155129e59e01031c257c..22b2ecb77fbd74ec6f9667b1e2fcea30ecfb4f8f 100644 --- a/formal/lean/mathlib/algebra/field/basic.lean +++ b/formal/lean/mathlib/algebra/field/basic.lean @@ -144,36 +144,8 @@ by rwa [add_comm, add_div', add_comm] end division_semiring -section division_ring -variables [division_ring K] {a b : K} - -namespace rat - -/-- Construct the canonical injection from `ℚ` into an arbitrary - division ring. If the field has positive characteristic `p`, - we define `1 / p = 1 / 0 = 0` for consistency with our - division by zero convention. -/ --- see Note [coercion into rings] -@[priority 900] instance cast_coe {K : Type*} [has_rat_cast K] : has_coe_t ℚ K := -⟨has_rat_cast.rat_cast⟩ - -theorem cast_mk' (a b h1 h2) : ((⟨a, b, h1, h2⟩ : ℚ) : K) = a * b⁻¹ := -division_ring.rat_cast_mk _ _ _ _ - -theorem cast_def : ∀ (r : ℚ), (r : K) = r.num / r.denom -| ⟨a, b, h1, h2⟩ := (cast_mk' _ _ _ _).trans (div_eq_mul_inv _ _).symm - -@[priority 100] -instance smul_division_ring : has_smul ℚ K := -⟨division_ring.qsmul⟩ - -lemma smul_def (a : ℚ) (x : K) : a • x = ↑a * x := division_ring.qsmul_eq_mul' a x - -end rat - -local attribute [simp] - division_def mul_comm mul_assoc - mul_left_comm mul_inv_cancel inv_mul_cancel +section division_monoid +variables [division_monoid K] [has_distrib_neg K] {a b : K} lemma one_div_neg_one_eq_neg_one : (1:K) / (-1) = -1 := have (-1) * (-1) = (1:K), by rw [neg_mul_neg, one_mul], @@ -202,6 +174,44 @@ by simp [neg_div] lemma neg_div_neg_eq (a b : K) : (-a) / (-b) = a / b := by rw [div_neg_eq_neg_div, neg_div, neg_neg] +lemma neg_inv : - a⁻¹ = (- a)⁻¹ := +by rw [inv_eq_one_div, inv_eq_one_div, div_neg_eq_neg_div] + +lemma div_neg (a : K) : a / -b = -(a / b) := +by rw [← div_neg_eq_neg_div] + +lemma inv_neg : (-a)⁻¹ = -(a⁻¹) := +by rw neg_inv + +end division_monoid + +section division_ring +variables [division_ring K] {a b : K} + +namespace rat + +/-- Construct the canonical injection from `ℚ` into an arbitrary + division ring. If the field has positive characteristic `p`, + we define `1 / p = 1 / 0 = 0` for consistency with our + division by zero convention. -/ +-- see Note [coercion into rings] +@[priority 900] instance cast_coe {K : Type*} [has_rat_cast K] : has_coe_t ℚ K := +⟨has_rat_cast.rat_cast⟩ + +theorem cast_mk' (a b h1 h2) : ((⟨a, b, h1, h2⟩ : ℚ) : K) = a * b⁻¹ := +division_ring.rat_cast_mk _ _ _ _ + +theorem cast_def : ∀ (r : ℚ), (r : K) = r.num / r.denom +| ⟨a, b, h1, h2⟩ := (cast_mk' _ _ _ _).trans (div_eq_mul_inv _ _).symm + +@[priority 100] +instance smul_division_ring : has_smul ℚ K := +⟨division_ring.qsmul⟩ + +lemma smul_def (a : ℚ) (x : K) : a • x = ↑a * x := division_ring.qsmul_eq_mul' a x + +end rat + @[simp] lemma div_neg_self {a : K} (h : a ≠ 0) : a / -a = -1 := by rw [div_neg_eq_neg_div, div_self h] @@ -221,18 +231,9 @@ by simpa only [← @div_self _ _ b h] using (div_sub_div_same a b b).symm lemma div_sub_one {a b : K} (h : b ≠ 0) : a / b - 1 = (a - b) / b := (div_sub_same h).symm -lemma neg_inv : - a⁻¹ = (- a)⁻¹ := -by rw [inv_eq_one_div, inv_eq_one_div, div_neg_eq_neg_div] - lemma sub_div (a b c : K) : (a - b) / c = a / c - b / c := (div_sub_div_same _ _ _).symm -lemma div_neg (a : K) : a / -b = -(a / b) := -by rw [← div_neg_eq_neg_div] - -lemma inv_neg : (-a)⁻¹ = -(a⁻¹) := -by rw neg_inv - lemma one_div_mul_sub_mul_one_div_eq_one_div_add_one_div (ha : a ≠ 0) (hb : b ≠ 0) : (1 / a) * (b - a) * (1 / b) = 1 / a - 1 / b := by rw [(mul_sub_left_distrib (1 / a)), (one_div_mul_cancel ha), mul_sub_right_distrib, diff --git a/formal/lean/mathlib/algebra/field_power.lean b/formal/lean/mathlib/algebra/field_power.lean index 8ebb4bbe7a611dc083c179f37d16f157c43859a9..5f09c5e6fdb54382f6a06ade9007b852ce8a5644 100644 --- a/formal/lean/mathlib/algebra/field_power.lean +++ b/formal/lean/mathlib/algebra/field_power.lean @@ -48,19 +48,11 @@ lemma zpow_pos_of_pos (ha : 0 < a) : ∀ (z : ℤ), 0 < a ^ z lemma zpow_le_of_le (ha : 1 ≤ a) (h : m ≤ n) : a ^ m ≤ a ^ n := begin - induction m with m m; induction n with n n, - { simp only [of_nat_eq_coe, zpow_coe_nat], - exact pow_le_pow ha (le_of_coe_nat_le_coe_nat h) }, - { cases h.not_lt ((neg_succ_lt_zero _).trans_le $ of_nat_nonneg _) }, - { simp only [zpow_neg_succ_of_nat, one_div, of_nat_eq_coe, zpow_coe_nat], - apply le_trans (inv_le_one _); apply one_le_pow_of_one_le ha }, - { simp only [zpow_neg_succ_of_nat], - apply (inv_le_inv _ _).2, - { apply pow_le_pow ha, - have : -(↑(m+1) : ℤ) ≤ -(↑(n+1) : ℤ), from h, - have h' := le_of_neg_le_neg this, - apply le_of_coe_nat_le_coe_nat h' }, - repeat { apply pow_pos (zero_lt_one.trans_le ha) } } + have ha₀ : 0 < a, from one_pos.trans_le ha, + lift n - m to ℕ using sub_nonneg.2 h with k hk, + calc a ^ m = a ^ m * 1 : (mul_one _).symm + ... ≤ a ^ m * a ^ k : mul_le_mul_of_nonneg_left (one_le_pow_of_one_le ha _) (zpow_nonneg ha₀.le _) + ... = a ^ n : by rw [← zpow_coe_nat, ← zpow_add₀ ha₀.ne', hk, add_sub_cancel'_right] end lemma pow_le_max_of_min_le (hx : 1 ≤ x) {a b c : ℤ} (h : min a b ≤ c) : @@ -159,17 +151,7 @@ lemma zpow_two_pos_of_ne_zero (h : a ≠ 0) : 0 < a ^ (2 : ℤ) := zpow_bit0_pos le_iff_le_iff_lt_iff_lt.2 zpow_bit1_neg_iff @[simp] theorem zpow_bit1_nonpos_iff : a ^ bit1 n ≤ 0 ↔ a ≤ 0 := -begin - rw [le_iff_lt_or_eq, zpow_bit1_neg_iff], - split, - { rintro (h | h), - { exact h.le }, - { exact (zpow_eq_zero h).le } }, - { intro h, - rcases eq_or_lt_of_le h with rfl|h, - { exact or.inr (zero_zpow _ (bit1_ne_zero n)) }, - { exact or.inl h } } -end +by rw [le_iff_lt_or_eq, le_iff_lt_or_eq, zpow_bit1_neg_iff, zpow_eq_zero_iff (bit1_ne_zero n)] @[simp] theorem zpow_bit1_pos_iff : 0 < a ^ bit1 n ↔ 0 < a := lt_iff_lt_of_le_iff_le zpow_bit1_nonpos_iff diff --git a/formal/lean/mathlib/algebra/group/prod.lean b/formal/lean/mathlib/algebra/group/prod.lean index 4a56d2187d91845851bc9c879ae3eaf4b9441936..a9056a026776ad5f5d05726f5e66c94baa45379e 100644 --- a/formal/lean/mathlib/algebra/group/prod.lean +++ b/formal/lean/mathlib/algebra/group/prod.lean @@ -495,7 +495,7 @@ open mul_opposite /-- Canonical homomorphism of monoids from `αˣ` into `α × αᵐᵒᵖ`. Used mainly to define the natural topology of `αˣ`. -/ @[to_additive "Canonical homomorphism of additive monoids from `add_units α` into `α × αᵃᵒᵖ`. -Used mainly to define the natural topology of `add_units α`."] +Used mainly to define the natural topology of `add_units α`.", simps] def embed_product (α : Type*) [monoid α] : αˣ →* α × αᵐᵒᵖ := { to_fun := λ x, ⟨x, op ↑x⁻¹⟩, map_one' := by simp only [inv_one, eq_self_iff_true, units.coe_one, op_one, prod.mk_eq_one, diff --git a/formal/lean/mathlib/algebra/group/to_additive.lean b/formal/lean/mathlib/algebra/group/to_additive.lean index cf17f07b07425010d3c5b3d94bda634400496155..a289534609175776ed3d66b37a01b9fc3d23047a 100644 --- a/formal/lean/mathlib/algebra/group/to_additive.lean +++ b/formal/lean/mathlib/algebra/group/to_additive.lean @@ -219,6 +219,8 @@ meta def tr : bool → list string → list string | is_comm ("npow" :: s) := add_comm_prefix is_comm "nsmul" :: tr ff s | is_comm ("zpow" :: s) := add_comm_prefix is_comm "zsmul" :: tr ff s | is_comm ("is" :: "square" :: s) := add_comm_prefix is_comm "even" :: tr ff s +| is_comm ("is" :: "scalar" :: "tower" :: s) := + add_comm_prefix is_comm "vadd_assoc_class" :: tr ff s | is_comm ("is" :: "regular" :: s) := add_comm_prefix is_comm "is_add_regular" :: tr ff s | is_comm ("is" :: "left" :: "regular" :: s) := add_comm_prefix is_comm "is_add_left_regular" :: tr ff s diff --git a/formal/lean/mathlib/algebra/group/units.lean b/formal/lean/mathlib/algebra/group/units.lean index a99a474bcd5928a20ff7ed310a5a44afb2aa2012..3bbc12ec811022f915190f29b4a5c89ed279a270 100644 --- a/formal/lean/mathlib/algebra/group/units.lean +++ b/formal/lean/mathlib/algebra/group/units.lean @@ -1,7 +1,7 @@ /- Copyright (c) 2017 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. -Authors: Kenny Lau, Mario Carneiro, Johannes Hölzl, Chris Hughes, Jens Wagemaker +Authors: Kenny Lau, Mario Carneiro, Johannes Hölzl, Chris Hughes, Jens Wagemaker, Jon Eugster -/ import algebra.group.basic import logic.nontrivial @@ -250,6 +250,10 @@ infix ` /ₚ `:70 := divp theorem divp_assoc (a b : α) (u : αˣ) : a * b /ₚ u = a * (b /ₚ u) := mul_assoc _ _ _ +/-- `field_simp` needs the reverse direction of `divp_assoc` to move all `/ₚ` to the right. -/ +@[field_simps] lemma divp_assoc' (x y : α) (u : αˣ) : x * (y /ₚ u) = (x * y) /ₚ u := +(divp_assoc _ _ _).symm + @[simp] theorem divp_inv (u : αˣ) : a /ₚ u⁻¹ = a * u := rfl @[simp] theorem divp_mul_cancel (a : α) (u : αˣ) : a /ₚ u * u = a := @@ -261,31 +265,60 @@ mul_assoc _ _ _ @[simp] theorem divp_left_inj (u : αˣ) {a b : α} : a /ₚ u = b /ₚ u ↔ a = b := units.mul_left_inj _ -theorem divp_divp_eq_divp_mul (x : α) (u₁ u₂ : αˣ) : (x /ₚ u₁) /ₚ u₂ = x /ₚ (u₂ * u₁) := +@[field_simps] theorem divp_divp_eq_divp_mul (x : α) (u₁ u₂ : αˣ) : + (x /ₚ u₁) /ₚ u₂ = x /ₚ (u₂ * u₁) := by simp only [divp, mul_inv_rev, units.coe_mul, mul_assoc] -theorem divp_eq_iff_mul_eq {x : α} {u : αˣ} {y : α} : x /ₚ u = y ↔ y * u = x := +@[field_simps] theorem divp_eq_iff_mul_eq {x : α} {u : αˣ} {y : α} : x /ₚ u = y ↔ y * u = x := u.mul_left_inj.symm.trans $ by rw [divp_mul_cancel]; exact ⟨eq.symm, eq.symm⟩ +@[field_simps] theorem eq_divp_iff_mul_eq {x : α} {u : αˣ} {y : α} : x = y /ₚ u ↔ x * u = y := +by rw [eq_comm, divp_eq_iff_mul_eq] + theorem divp_eq_one_iff_eq {a : α} {u : αˣ} : a /ₚ u = 1 ↔ a = u := (units.mul_left_inj u).symm.trans $ by rw [divp_mul_cancel, one_mul] @[simp] theorem one_divp (u : αˣ) : 1 /ₚ u = ↑u⁻¹ := one_mul _ +/-- Used for `field_simp` to deal with inverses of units. -/ +@[field_simps] lemma inv_eq_one_divp (u : αˣ) : ↑u⁻¹ = 1 /ₚ u := +by rw one_divp + +/-- +Used for `field_simp` to deal with inverses of units. This form of the lemma +is essential since `field_simp` likes to use `inv_eq_one_div` to rewrite +`↑u⁻¹ = ↑(1 / u)`. +-/ +@[field_simps] lemma inv_eq_one_divp' (u : αˣ) : + ((1 / u : αˣ) : α) = 1 /ₚ u := +by rw [one_div, one_divp] + +/-- +`field_simp` moves division inside `αˣ` to the right, and this lemma +lifts the calculation to `α`. +-/ +@[field_simps] lemma coe_div_eq_divp (u₁ u₂ : αˣ) : ↑(u₁ / u₂) = ↑u₁ /ₚ u₂ := +by rw [divp, division_def, units.coe_mul] + end monoid section comm_monoid variables [comm_monoid α] -theorem divp_eq_divp_iff {x y : α} {ux uy : αˣ} : +@[field_simps] theorem divp_mul_eq_mul_divp (x y : α) (u : αˣ) : x /ₚ u * y = x * y /ₚ u := +by simp_rw [divp, mul_assoc, mul_comm] + +-- Theoretically redundant as `field_simp` lemma. +@[field_simps] lemma divp_eq_divp_iff {x y : α} {ux uy : αˣ} : x /ₚ ux = y /ₚ uy ↔ x * uy = y * ux := -by rw [divp_eq_iff_mul_eq, mul_comm, ← divp_assoc, divp_eq_iff_mul_eq, mul_comm y ux] +by rw [divp_eq_iff_mul_eq, divp_mul_eq_mul_divp, divp_eq_iff_mul_eq] -theorem divp_mul_divp (x y : α) (ux uy : αˣ) : +-- Theoretically redundant as `field_simp` lemma. +@[field_simps] lemma divp_mul_divp (x y : α) (ux uy : αˣ) : (x /ₚ ux) * (y /ₚ uy) = (x * y) /ₚ (ux * uy) := -by rw [← divp_divp_eq_divp_mul, divp_assoc, mul_comm x, divp_assoc, mul_comm] +by rw [divp_mul_eq_mul_divp, divp_assoc', divp_divp_eq_divp_mul] end comm_monoid diff --git a/formal/lean/mathlib/algebra/group_with_zero/power.lean b/formal/lean/mathlib/algebra/group_with_zero/power.lean index a80f4d1b5b06c990848dc3b0ede85ab77858e968..a374d3225621d698ead675dcfbb4826631c07bb8 100644 --- a/formal/lean/mathlib/algebra/group_with_zero/power.lean +++ b/formal/lean/mathlib/algebra/group_with_zero/power.lean @@ -134,13 +134,9 @@ by rw [zpow_bit1₀, (commute.refl a).mul_zpow] lemma zpow_eq_zero {x : G₀} {n : ℤ} (h : x ^ n = 0) : x = 0 := classical.by_contradiction $ λ hx, zpow_ne_zero_of_ne_zero hx n h -lemma zpow_eq_zero_iff {a : G₀} {n : ℤ} (hn : 0 < n) : +lemma zpow_eq_zero_iff {a : G₀} {n : ℤ} (hn : n ≠ 0) : a ^ n = 0 ↔ a = 0 := -begin - refine ⟨zpow_eq_zero, _⟩, - rintros rfl, - exact zero_zpow _ hn.ne' -end +⟨zpow_eq_zero, λ ha, ha.symm ▸ zero_zpow _ hn⟩ lemma zpow_ne_zero {x : G₀} (n : ℤ) : x ≠ 0 → x ^ n ≠ 0 := mt zpow_eq_zero diff --git a/formal/lean/mathlib/algebra/hom/equiv.lean b/formal/lean/mathlib/algebra/hom/equiv.lean index afaa61705f72850d884fd4fea190f14c5c12e2e6..4fcb7c96a05d41c571b11974efa5dfaa23c98d1b 100644 --- a/formal/lean/mathlib/algebra/hom/equiv.lean +++ b/formal/lean/mathlib/algebra/hom/equiv.lean @@ -729,3 +729,16 @@ def mul_equiv.to_additive'' [add_zero_class G] [mul_one_class H] : add_equiv.to_multiplicative''.symm end type_tags + +section +variables (G) (H) + +/-- `additive (multiplicative G)` is just `G`. -/ +def add_equiv.additive_multiplicative [add_zero_class G] : additive (multiplicative G) ≃+ G := +mul_equiv.to_additive'' (mul_equiv.refl (multiplicative G)) + +/-- `multiplicative (additive H)` is just `H`. -/ +def mul_equiv.multiplicative_additive [mul_one_class H] : multiplicative (additive H) ≃* H := +add_equiv.to_multiplicative'' (add_equiv.refl (additive H)) + +end diff --git a/formal/lean/mathlib/algebra/hom/group_action.lean b/formal/lean/mathlib/algebra/hom/group_action.lean index 787da40609da9fc12fd14776fadfe31b396f7aad..cd9109774eeb2ae691ffc448fd62cd96cb56675f 100644 --- a/formal/lean/mathlib/algebra/hom/group_action.lean +++ b/formal/lean/mathlib/algebra/hom/group_action.lean @@ -54,7 +54,7 @@ variables (G : Type*) [group G] (H : subgroup G) set_option old_structure_cmd true /-- Equivariant functions. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure mul_action_hom := (to_fun : X → Y) (map_smul' : ∀ (m : M') (x : X), to_fun (m • x) = m • to_fun x) @@ -258,7 +258,7 @@ end semiring end distrib_mul_action_hom /-- Equivariant ring homomorphisms. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure mul_semiring_action_hom extends R →+[M] S, R →+* S. /-- Reinterpret an equivariant ring homomorphism as a ring homomorphism. -/ diff --git a/formal/lean/mathlib/algebra/hom/group_instances.lean b/formal/lean/mathlib/algebra/hom/group_instances.lean index 17ec75f28ff609b2d94623f560b277c39de73f14..e1b70677df7757f49f49a4a2e6c7a635f1044c8f 100644 --- a/formal/lean/mathlib/algebra/hom/group_instances.lean +++ b/formal/lean/mathlib/algebra/hom/group_instances.lean @@ -69,10 +69,24 @@ instance [add_comm_monoid M] : semiring (add_monoid.End M) := .. add_monoid.End.monoid M, .. add_monoid_hom.add_comm_monoid } +/-- See also `add_monoid.End.nat_cast_def`. -/ +@[simp] lemma add_monoid.End.nat_cast_apply [add_comm_monoid M] (n : ℕ) (m : M) : + (↑n : add_monoid.End M) m = n • m := rfl + +instance [add_comm_group M] : add_comm_group (add_monoid.End M) := +add_monoid_hom.add_comm_group + instance [add_comm_group M] : ring (add_monoid.End M) := -{ .. add_monoid.End.semiring, +{ int_cast := λ z, z • 1, + int_cast_of_nat := of_nat_zsmul _, + int_cast_neg_succ_of_nat := zsmul_neg_succ_of_nat _, + .. add_monoid.End.semiring, .. add_monoid_hom.add_comm_group } +/-- See also `add_monoid.End.int_cast_def`. -/ +@[simp] lemma add_monoid.End.int_cast_apply [add_comm_group M] (z : ℤ) (m : M) : + (↑z : add_monoid.End M) m = z • m := rfl + /-! ### Morphisms of morphisms @@ -208,7 +222,8 @@ variables {R S : Type*} [non_unital_non_assoc_semiring R] [non_unital_non_assoc_ This is a more-strongly bundled version of `add_monoid_hom.mul_left` and `add_monoid_hom.mul_right`. -A stronger version of this exists for algebras as `algebra.lmul`. +Stronger versions of this exists for algebras as `linear_map.mul`, `non_unital_alg_hom.mul` +and `algebra.lmul`. -/ def add_monoid_hom.mul : R →+ R →+ R := { to_fun := add_monoid_hom.mul_left, diff --git a/formal/lean/mathlib/algebra/hom/ring.lean b/formal/lean/mathlib/algebra/hom/ring.lean index 5836a68f5f82fe4ed0a9c0ce583ce2110eceec40..b6dcd88cabb5bd0638d75ce35d11e2f3e7b9e31c 100644 --- a/formal/lean/mathlib/algebra/hom/ring.lean +++ b/formal/lean/mathlib/algebra/hom/ring.lean @@ -413,6 +413,10 @@ mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one lemma domain_nontrivial [nontrivial β] : nontrivial α := ⟨⟨1, 0, mt (λ h, show f 1 = 0, by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ +lemma codomain_trivial (f : α →+* β) [h : subsingleton α] : subsingleton β := +(subsingleton_or_nontrivial β).resolve_right + (λ _, by exactI not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial) + end /-- Ring homomorphisms preserve additive inverse. -/ diff --git a/formal/lean/mathlib/algebra/hom/units.lean b/formal/lean/mathlib/algebra/hom/units.lean index 27925f3728b14c99e0307033434dd9649d8b01df..b1208076e802c2eabdf1eb2b36be63de7a2860f9 100644 --- a/formal/lean/mathlib/algebra/hom/units.lean +++ b/formal/lean/mathlib/algebra/hom/units.lean @@ -76,7 +76,7 @@ variables [division_monoid α] @[simp, norm_cast, to_additive] lemma coe_zpow : ∀ (u : αˣ) (n : ℤ), ((u ^ n : αˣ) : α) = u ^ n := (units.coe_hom α).map_zpow -lemma _root_.divp_eq_div (a : α) (u : αˣ) : a /ₚ u = a / u := +@[field_simps] lemma _root_.divp_eq_div (a : α) (u : αˣ) : a /ₚ u = a / u := by rw [div_eq_mul_inv, divp, u.coe_inv] end division_monoid diff --git a/formal/lean/mathlib/algebra/homology/complex_shape.lean b/formal/lean/mathlib/algebra/homology/complex_shape.lean index a2287f28e970dd0dac4bdbb53fee50f6d8e1c6d2..640e7ce12ade7db1e560ca0346afe349a556ba8a 100644 --- a/formal/lean/mathlib/algebra/homology/complex_shape.lean +++ b/formal/lean/mathlib/algebra/homology/complex_shape.lean @@ -55,7 +55,7 @@ This means that the shape consists of some union of lines, rays, intervals, and Below we define `c.next` and `c.prev` which provide these related elements. -/ -@[ext, nolint has_inhabited_instance] +@[ext, nolint has_nonempty_instance] structure complex_shape (ι : Type*) := (rel : ι → ι → Prop) (next_eq : ∀ {i j j'}, rel i j → rel i j' → j = j') diff --git a/formal/lean/mathlib/algebra/homology/homological_complex.lean b/formal/lean/mathlib/algebra/homology/homological_complex.lean index 8e81c26ccc202d4af104eb9b2df0a4ff05a8fcc1..4d183ab8b5dae49eba11bdfb3c2730039d2f5174 100644 --- a/formal/lean/mathlib/algebra/homology/homological_complex.lean +++ b/formal/lean/mathlib/algebra/homology/homological_complex.lean @@ -27,7 +27,7 @@ and similarly `cochain_complex V α`, with `i = j + 1`. There is a category structure, where morphisms are chain maps. For `C : homological_complex V c`, we define `C.X_next i`, which is either `C.X j` for some -arbitrarily chosen `j` such that `c.r i j`, or the zero object if there is no such `j`. +arbitrarily chosen `j` such that `c.r i j`, or `C.X i` if there is no such `j`. Similarly we have `C.X_prev j`. Defined in terms of these we have `C.d_from i : C.X i ⟶ C.X_next i` and `C.d_to j : C.X_prev j ⟶ C.X j`, which are either defined as `C.d i j`, or zero, as needed. @@ -299,7 +299,7 @@ def X_prev_iso {i j : ι} (r : c.rel i j) : eq_to_iso $ by rw ← c.prev_eq' r /-- If there is no `i` so `c.rel i j`, then `C.X_prev j` is isomorphic to `C.X j`. -/ -def X_prev_iso_zero {j : ι} (h : ¬c.rel (c.prev j) j) : +def X_prev_iso_self {j : ι} (h : ¬c.rel (c.prev j) j) : C.X_prev j ≅ C.X j := eq_to_iso $ congr_arg C.X begin dsimp [complex_shape.prev], @@ -308,7 +308,7 @@ eq_to_iso $ congr_arg C.X begin rw this at h, contradiction, end -/-- Either `C.X j`, if there is some `j` with `c.rel i j`, or `C.X j`. -/ +/-- Either `C.X j`, if there is some `j` with `c.rel i j`, or `C.X i`. -/ abbreviation X_next (i : ι) : V := C.X (c.next i) /-- If `c.rel i j`, then `C.X_next i` is isomorphic to `C.X j`. -/ @@ -316,8 +316,8 @@ def X_next_iso {i j : ι} (r : c.rel i j) : C.X_next i ≅ C.X j := eq_to_iso $ by rw ← c.next_eq' r -/-- If there is no `j` so `c.rel i j`, then `C.X_next i` is isomorphic to `0`. -/ -def X_next_iso_zero {i : ι} (h : ¬c.rel i (c.next i)) : +/-- If there is no `j` so `c.rel i j`, then `C.X_next i` is isomorphic to `C.X i`. -/ +def X_next_iso_self {i : ι} (h : ¬c.rel i (c.next i)) : C.X_next i ≅ C.X i := eq_to_iso $ congr_arg C.X begin dsimp [complex_shape.next], @@ -364,16 +364,16 @@ C.shape _ _ h (C.X_prev_iso r).inv ≫ C.d_to j = C.d i j := by simp [C.d_to_eq r] -@[simp, reassoc] lemma X_prev_iso_zero_comp_d_to {j : ι} (h : ¬c.rel (c.prev j) j) : - (C.X_prev_iso_zero h).inv ≫ C.d_to j = 0 := +@[simp, reassoc] lemma X_prev_iso_self_comp_d_to {j : ι} (h : ¬c.rel (c.prev j) j) : + (C.X_prev_iso_self h).inv ≫ C.d_to j = 0 := by simp [h] @[simp, reassoc] lemma d_from_comp_X_next_iso {i j : ι} (r : c.rel i j) : C.d_from i ≫ (C.X_next_iso r).hom = C.d i j := by simp [C.d_from_eq r] -@[simp, reassoc] lemma d_from_comp_X_next_iso_zero {i : ι} (h : ¬c.rel i (c.next i)) : - C.d_from i ≫ (C.X_next_iso_zero h).hom = 0 := +@[simp, reassoc] lemma d_from_comp_X_next_iso_self {i : ι} (h : ¬c.rel i (c.next i)) : + C.d_from i ≫ (C.X_next_iso_self h).hom = 0 := by simp [h] @[simp] @@ -430,7 +430,7 @@ by { ext, simp, } /-! Lemmas relating chain maps and `d_to`/`d_from`. -/ -/-- `f.prev j` is `f.f i` if there is some `r i j`, and zero otherwise. -/ +/-- `f.prev j` is `f.f i` if there is some `r i j`, and `f.f j` otherwise. -/ abbreviation prev (f : hom C₁ C₂) (j : ι) : C₁.X_prev j ⟶ C₂.X_prev j := f.f _ lemma prev_eq (f : hom C₁ C₂) {i j : ι} (w : c.rel i j) : @@ -440,7 +440,7 @@ begin simp only [X_prev_iso, eq_to_iso_refl, iso.refl_hom, iso.refl_inv, id_comp, comp_id], end -/-- `f.next i` is `f.f j` if there is some `r i j`, and zero otherwise. -/ +/-- `f.next i` is `f.f j` if there is some `r i j`, and `f.f j` otherwise. -/ abbreviation next (f : hom C₁ C₂) (i : ι) : C₁.X_next i ⟶ C₂.X_next i := f.f _ lemma next_eq (f : hom C₁ C₂) {i j : ι} (w : c.rel i j) : @@ -552,7 +552,7 @@ Auxiliary structure for setting up the recursion in `mk`. This is purely an implementation detail: for some reason just using the dependent 6-tuple directly results in `mk_aux` taking much longer (well over the `-T100000` limit) to elaborate. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure mk_struct := (X₀ X₁ X₂ : V) (d₀ : X₁ ⟶ X₀) @@ -745,7 +745,7 @@ Auxiliary structure for setting up the recursion in `mk`. This is purely an implementation detail: for some reason just using the dependent 6-tuple directly results in `mk_aux` taking much longer (well over the `-T100000` limit) to elaborate. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure mk_struct := (X₀ X₁ X₂ : V) (d₀ : X₀ ⟶ X₁) diff --git a/formal/lean/mathlib/algebra/homology/homotopy.lean b/formal/lean/mathlib/algebra/homology/homotopy.lean index 3ad0100890e27bd7aaff50d45eedeb69320c1e38..b733ad12c7c10daf23598b3c28292df5ae4f31f6 100644 --- a/formal/lean/mathlib/algebra/homology/homotopy.lean +++ b/formal/lean/mathlib/algebra/homology/homotopy.lean @@ -113,7 +113,7 @@ end A homotopy `h` between chain maps `f` and `g` consists of components `h i j : C.X i ⟶ D.X j` which are zero unless `c.rel j i`, satisfying the homotopy condition. -/ -@[ext, nolint has_inhabited_instance] +@[ext, nolint has_nonempty_instance] structure homotopy (f g : C ⟶ D) := (hom : Π i j, C.X i ⟶ D.X j) (zero' : ∀ i j, ¬ c.rel j i → hom i j = 0 . obviously) diff --git a/formal/lean/mathlib/algebra/homology/short_exact/preadditive.lean b/formal/lean/mathlib/algebra/homology/short_exact/preadditive.lean index 5fc1314c04831716447c875f2c5b35244798cd33..aea22167cbedc24b204c1dc676edaa975fdaacdf 100644 --- a/formal/lean/mathlib/algebra/homology/short_exact/preadditive.lean +++ b/formal/lean/mathlib/algebra/homology/short_exact/preadditive.lean @@ -167,7 +167,7 @@ end preadditive /-- A *splitting* of a sequence `A -f⟶ B -g⟶ C` is an isomorphism to the short exact sequence `0 ⟶ A ⟶ A ⊞ C ⟶ C ⟶ 0` such that the vertical maps on the left and the right are the identity. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure splitting [has_zero_morphisms 𝒜] [has_binary_biproducts 𝒜] := (iso : B ≅ A ⊞ C) (comp_iso_eq_inl : f ≫ iso.hom = biprod.inl) diff --git a/formal/lean/mathlib/algebra/indicator_function.lean b/formal/lean/mathlib/algebra/indicator_function.lean index 8d697ad85304f0d8bdc0020122f11d30d477a9d4..08f80f48aabd828eb082613b1ae70ed021fae31f 100644 --- a/formal/lean/mathlib/algebra/indicator_function.lean +++ b/formal/lean/mathlib/algebra/indicator_function.lean @@ -457,12 +457,8 @@ lemma inter_indicator_one {s t : set α} : funext (λ _, by simpa only [← inter_indicator_mul, pi.mul_apply, pi.one_apply, one_mul]) lemma indicator_prod_one {s : set α} {t : set β} {x : α} {y : β} : - (s ×ˢ t : set _).indicator (1 : _ → M) (x, y) = s.indicator 1 x * t.indicator 1 y := -begin - letI := classical.dec_pred (∈ s), - letI := classical.dec_pred (∈ t), - simp [indicator_apply, ← ite_and], -end + (s ×ˢ t).indicator (1 : _ → M) (x, y) = s.indicator 1 x * t.indicator 1 y := +by { classical, simp [indicator_apply, ←ite_and] } variables (M) [nontrivial M] diff --git a/formal/lean/mathlib/algebra/lie/base_change.lean b/formal/lean/mathlib/algebra/lie/base_change.lean index 2403b345978e8f85abadcd157dcf4f5603ec8afa..11c0631716dbd191818ff1be7d91f851df409d4d 100644 --- a/formal/lean/mathlib/algebra/lie/base_change.lean +++ b/formal/lean/mathlib/algebra/lie/base_change.lean @@ -40,7 +40,7 @@ support in the tensor product library, it is far easier to bootstrap like this, definition below. -/ private def bracket' : (A ⊗[R] L) →ₗ[R] (A ⊗[R] L) →ₗ[R] A ⊗[R] L := tensor_product.curry $ - (tensor_product.map (algebra.lmul' R) (lie_module.to_module_hom R L L : L ⊗[R] L →ₗ[R] L)) + (tensor_product.map (linear_map.mul' R _) (lie_module.to_module_hom R L L : L ⊗[R] L →ₗ[R] L)) ∘ₗ ↑(tensor_product.tensor_tensor_tensor_comm R A L A L) @[simp] private lemma bracket'_tmul (s t : A) (x y : L) : @@ -115,8 +115,8 @@ begin { simp only [lie_zero, smul_zero], }, { intros a₂ l₂, simp only [bracket_def, bracket', tensor_product.smul_tmul', mul_left_comm a₁ a a₂, - tensor_product.curry_apply, algebra.lmul'_apply, algebra.id.smul_eq_mul, function.comp_app, - linear_equiv.coe_coe, linear_map.coe_comp, tensor_product.map_tmul, + tensor_product.curry_apply, linear_map.mul'_apply, algebra.id.smul_eq_mul, + function.comp_app, linear_equiv.coe_coe, linear_map.coe_comp, tensor_product.map_tmul, tensor_product.tensor_tensor_tensor_comm_tmul], }, { intros z₁ z₂ h₁ h₂, simp only [h₁, h₂, smul_add, lie_add], }, }, diff --git a/formal/lean/mathlib/algebra/lie/nilpotent.lean b/formal/lean/mathlib/algebra/lie/nilpotent.lean index 744ac398569fa8d3dceb0c4b6c4f5450e57505d5..acf0c6fba770bc50797290602966a81b0755068d 100644 --- a/formal/lean/mathlib/algebra/lie/nilpotent.lean +++ b/formal/lean/mathlib/algebra/lie/nilpotent.lean @@ -691,9 +691,12 @@ lemma lie_algebra.ad_nilpotent_of_nilpotent {a : A} (h : is_nilpotent a) : is_nilpotent (lie_algebra.ad R A a) := begin rw lie_algebra.ad_eq_lmul_left_sub_lmul_right, - have hl : is_nilpotent (algebra.lmul_left R a), { rwa algebra.is_nilpotent_lmul_left_iff, }, - have hr : is_nilpotent (algebra.lmul_right R a), { rwa algebra.is_nilpotent_lmul_right_iff, }, - exact (algebra.commute_lmul_left_right R a a).is_nilpotent_sub hl hr, + have hl : is_nilpotent (linear_map.mul_left R a), + { rwa linear_map.is_nilpotent_mul_left_iff, }, + have hr : is_nilpotent (linear_map.mul_right R a), + { rwa linear_map.is_nilpotent_mul_right_iff, }, + have := @linear_map.commute_mul_left_right R A _ _ _ _ _ a a, + exact this.is_nilpotent_sub hl hr, end variables {R} diff --git a/formal/lean/mathlib/algebra/lie/of_associative.lean b/formal/lean/mathlib/algebra/lie/of_associative.lean index 030bde6d006eac0d973dd901bcf410fa7c328af8..f3417892099f7dc75d66662febed3c516817f92c 100644 --- a/formal/lean/mathlib/algebra/lie/of_associative.lean +++ b/formal/lean/mathlib/algebra/lie/of_associative.lean @@ -234,7 +234,7 @@ end lie_submodule open lie_algebra lemma lie_algebra.ad_eq_lmul_left_sub_lmul_right (A : Type v) [ring A] [algebra R A] : - (ad R A : A → module.End R A) = algebra.lmul_left R - algebra.lmul_right R := + (ad R A : A → module.End R A) = linear_map.mul_left R - linear_map.mul_right R := by { ext a b, simp [lie_ring.of_associative_ring_bracket], } lemma lie_subalgebra.ad_comp_incl_eq (K : lie_subalgebra R L) (x : K) : diff --git a/formal/lean/mathlib/algebra/module/basic.lean b/formal/lean/mathlib/algebra/module/basic.lean index 73c64a199c293ee9d801f5abea59c34581eb92d6..64ef7765defbf57515dac420f8696b229918a345 100644 --- a/formal/lean/mathlib/algebra/module/basic.lean +++ b/formal/lean/mathlib/algebra/module/basic.lean @@ -71,6 +71,9 @@ instance add_comm_monoid.nat_module : module ℕ M := zero_smul := zero_nsmul, add_smul := λ r s x, add_nsmul x r s } +lemma add_monoid.End.nat_cast_def (n : ℕ) : + (↑n : add_monoid.End M) = distrib_mul_action.to_add_monoid_End ℕ M n := rfl + theorem add_smul : (r + s) • x = r • x + s • x := module.add_smul r s x lemma convex.combo_self {a b : R} (h : a + b = 1) (x : M) : a • x + b • x = x := @@ -197,11 +200,14 @@ instance add_comm_group.int_module : module ℤ M := zero_smul := zero_zsmul, add_smul := λ r s x, add_zsmul x r s } +lemma add_monoid.End.int_cast_def (z : ℤ) : + (↑z : add_monoid.End M) = distrib_mul_action.to_add_monoid_End ℤ M z := rfl + /-- A structure containing most informations as in a module, except the fields `zero_smul` and `smul_zero`. As these fields can be deduced from the other ones when `M` is an `add_comm_group`, this provides a way to construct a module structure by checking less properties, in `module.of_core`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure module.core extends has_smul R M := (smul_add : ∀(r : R) (x y : M), r • (x + y) = r • x + r • y) (add_smul : ∀(r s : R) (x : M), (r + s) • x = r • x + s • x) diff --git a/formal/lean/mathlib/algebra/module/bimodule.lean b/formal/lean/mathlib/algebra/module/bimodule.lean new file mode 100644 index 0000000000000000000000000000000000000000..7c5acc489e8c978726a0997901caf16a387a80be --- /dev/null +++ b/formal/lean/mathlib/algebra/module/bimodule.lean @@ -0,0 +1,136 @@ +/- +Copyright (c) 2022 Oliver Nash. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Oliver Nash +-/ +import ring_theory.tensor_product + +/-! +# Bimodules + +One frequently encounters situations in which several sets of scalars act on a single space, subject +to compatibility condition(s). A distinguished instance of this is the theory of bimodules: one has +two rings `R`, `S` acting on an additive group `M`, with `R` acting covariantly ("on the left") +and `S` acting contravariantly ("on the right"). The compatibility condition is just: +`(r • m) • s = r • (m • s)` for all `r : R`, `s : S`, `m : M`. + +This situation can be set up in Mathlib as: +```lean +variables (R S M : Type*) [ring R] [ring S] +variables [add_comm_group M] [module R M] [module Sᵐᵒᵖ M] [smul_comm_class R Sᵐᵒᵖ M] +``` +The key fact is: +```lean +example : module (R ⊗[ℕ] Sᵐᵒᵖ) M := tensor_product.algebra.module +``` +Note that the corresponding result holds for the canonically isomorphic ring `R ⊗[ℤ] Sᵐᵒᵖ` but it is +preferable to use the `R ⊗[ℕ] Sᵐᵒᵖ` instance since it works without additive inverses. + +Bimodules are thus just a special case of `module`s and most of their properties follow from the +theory of `module`s`. In particular a two-sided submodule of a bimodule is simply a term of type +`submodule (R ⊗[ℕ] Sᵐᵒᵖ) M`. + +This file is a place to collect results which are specific to bimodules. + +## Main definitions + + * `subbimodule.mk` + * `subbimodule.smul_mem` + * `subbimodule.smul_mem'` + * `subbimodule.to_submodule` + * `subbimodule.to_submodule'` + +## Implementation details + +For many definitions and lemmas it is preferable to set things up without opposites, i.e., as: +`[module S M] [smul_comm_class R S M]` rather than `[module Sᵐᵒᵖ M] [smul_comm_class R Sᵐᵒᵖ M]`. +The corresponding results for opposites then follow automatically and do not require taking +advantage of the fact that `(Sᵐᵒᵖ)ᵐᵒᵖ` is defeq to `S`. + +## TODO + +Develop the theory of two-sided ideals, which have type `submodule (R ⊗[ℕ] Rᵐᵒᵖ) R`. + +-/ + +open_locale tensor_product + +local attribute [instance] tensor_product.algebra.module + +namespace subbimodule + +section algebra + +variables {R A B M : Type*} +variables [comm_semiring R] [add_comm_monoid M] [module R M] +variables [semiring A] [semiring B] [module A M] [module B M] +variables [algebra R A] [algebra R B] +variables [is_scalar_tower R A M] [is_scalar_tower R B M] +variables [smul_comm_class A B M] + +/-- A constructor for a subbimodule which demands closure under the two sets of scalars +individually, rather than jointly via their tensor product. + +Note that `R` plays no role but it is convenient to make this generalisation to support the cases +`R = ℕ` and `R = ℤ` which both show up naturally. See also `base_change`. -/ +@[simps] def mk (p : add_submonoid M) + (hA : ∀ (a : A) {m : M}, m ∈ p → a • m ∈ p) + (hB : ∀ (b : B) {m : M}, m ∈ p → b • m ∈ p) : submodule (A ⊗[R] B) M := +{ carrier := p, + smul_mem' := λ ab m, tensor_product.induction_on ab + (λ hm, by simpa only [zero_smul] using p.zero_mem) + (λ a b hm, by simpa only [tensor_product.algebra.smul_def] using hA a (hB b hm)) + (λ z w hz hw hm, by simpa only [add_smul] using p.add_mem (hz hm) (hw hm)), + .. p } + +lemma smul_mem (p : submodule (A ⊗[R] B) M) (a : A) {m : M} (hm : m ∈ p) : a • m ∈ p := +begin + suffices : a • m = a ⊗ₜ[R] (1 : B) • m, { exact this.symm ▸ p.smul_mem _ hm, }, + simp [tensor_product.algebra.smul_def], +end + +lemma smul_mem' (p : submodule (A ⊗[R] B) M) (b : B) {m : M} (hm : m ∈ p) : b • m ∈ p := +begin + suffices : b • m = (1 : A) ⊗ₜ[R] b • m, { exact this.symm ▸ p.smul_mem _ hm, }, + simp [tensor_product.algebra.smul_def], +end + +/-- If `A` and `B` are also `algebra`s over yet another set of scalars `S` then we may "base change" +from `R` to `S`. -/ +@[simps] def base_change (S : Type*) [comm_semiring S] [module S M] [algebra S A] [algebra S B] + [is_scalar_tower S A M] [is_scalar_tower S B M] (p : submodule (A ⊗[R] B) M) : + submodule (A ⊗[S] B) M := +mk p.to_add_submonoid (smul_mem p) (smul_mem' p) + +/-- Forgetting the `B` action, a `submodule` over `A ⊗[R] B` is just a `submodule` over `A`. -/ +@[simps] def to_submodule (p : submodule (A ⊗[R] B) M) : submodule A M := +{ carrier := p, + smul_mem' := smul_mem p, + .. p } + +/-- Forgetting the `A` action, a `submodule` over `A ⊗[R] B` is just a `submodule` over `B`. -/ +@[simps] def to_submodule' (p : submodule (A ⊗[R] B) M) : submodule B M := +{ carrier := p, + smul_mem' := smul_mem' p, + .. p } + +end algebra + +section ring + +variables (R S M : Type*) [ring R] [ring S] +variables [add_comm_group M] [module R M] [module S M] [smul_comm_class R S M] + +/-- A `submodule` over `R ⊗[ℕ] S` is naturally also a `submodule` over the canonically-isomorphic +ring `R ⊗[ℤ] S`. -/ +@[simps] def to_subbimodule_int (p : submodule (R ⊗[ℕ] S) M) : submodule (R ⊗[ℤ] S) M := +base_change ℤ p + +/-- A `submodule` over `R ⊗[ℤ] S` is naturally also a `submodule` over the canonically-isomorphic +ring `R ⊗[ℕ] S`. -/ +@[simps] def to_subbimodule_nat (p : submodule (R ⊗[ℤ] S) M) : submodule (R ⊗[ℕ] S) M := +base_change ℕ p + +end ring + +end subbimodule diff --git a/formal/lean/mathlib/algebra/module/equiv.lean b/formal/lean/mathlib/algebra/module/equiv.lean index 70322a8407a86ee8f332c1665b0e8e5912a4d56e..7d35aad669c66b583b3d173e5d9fe9644581f7e9 100644 --- a/formal/lean/mathlib/algebra/module/equiv.lean +++ b/formal/lean/mathlib/algebra/module/equiv.lean @@ -45,13 +45,12 @@ section set_option old_structure_cmd true /-- A linear equivalence is an invertible linear map. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure linear_equiv {R : Type*} {S : Type*} [semiring R] [semiring S] (σ : R →+* S) {σ' : S →+* R} [ring_hom_inv_pair σ σ'] [ring_hom_inv_pair σ' σ] (M : Type*) (M₂ : Type*) [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module S M₂] extends linear_map σ M M₂, M ≃+ M₂ -end attribute [nolint doc_blame] linear_equiv.to_linear_map attribute [nolint doc_blame] linear_equiv.to_add_equiv @@ -60,6 +59,49 @@ notation M ` ≃ₛₗ[`:50 σ `] ` M₂ := linear_equiv σ M M₂ notation M ` ≃ₗ[`:50 R `] ` M₂ := linear_equiv (ring_hom.id R) M M₂ notation M ` ≃ₗ⋆[`:50 R `] ` M₂ := linear_equiv (star_ring_end R) M M₂ +/-- `semilinear_equiv_class F σ M M₂` asserts `F` is a type of bundled `σ`-semilinear equivs +`M → M₂`. + +See also `linear_equiv_class F R M M₂` for the case where `σ` is the identity map on `R`. + +A map `f` between an `R`-module and an `S`-module over a ring homomorphism `σ : R →+* S` +is semilinear if it satisfies the two properties `f (x + y) = f x + f y` and +`f (c • x) = (σ c) • f x`. -/ +class semilinear_equiv_class (F : Type*) {R S : out_param Type*} [semiring R] [semiring S] + (σ : out_param $ R →+* S) {σ' : out_param $ S →+* R} + [ring_hom_inv_pair σ σ'] [ring_hom_inv_pair σ' σ] (M M₂ : out_param Type*) + [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module S M₂] + extends add_equiv_class F M M₂ := +(map_smulₛₗ : ∀ (f : F) (r : R) (x : M), f (r • x) = (σ r) • f x) + +-- `R, S, σ, σ'` become metavars, but it's OK since they are outparams. +attribute [nolint dangerous_instance] semilinear_equiv_class.to_add_equiv_class + +/-- `linear_equiv_class F R M M₂` asserts `F` is a type of bundled `R`-linear equivs `M → M₂`. +This is an abbreviation for `semilinear_equiv_class F (ring_hom.id R) M M₂`. +-/ +abbreviation linear_equiv_class (F : Type*) (R M M₂ : out_param Type*) + [semiring R] [add_comm_monoid M] [add_comm_monoid M₂] [module R M] [module R M₂] := +semilinear_equiv_class F (ring_hom.id R) M M₂ + +end + +namespace semilinear_equiv_class + +variables (F : Type*) [semiring R] [semiring S] +variables [add_comm_monoid M] [add_comm_monoid M₁] [add_comm_monoid M₂] +variables [module R M] [module S M₂] {σ : R →+* S} {σ' : S →+* R} + +-- `σ'` becomes a metavariable, but it's OK since it's an outparam +@[priority 100, nolint dangerous_instance] +instance [ring_hom_inv_pair σ σ'] [ring_hom_inv_pair σ' σ] [s : semilinear_equiv_class F σ M M₂] : + semilinear_map_class F σ M M₂ := +{ coe := (coe : F → M → M₂), + coe_injective' := @fun_like.coe_injective F _ _ _, + ..s } + +end semilinear_equiv_class + namespace linear_equiv section add_comm_monoid @@ -101,9 +143,12 @@ lemma to_linear_map_injective : (e₁ : M →ₛₗ[σ] M₂) = e₂ ↔ e₁ = e₂ := to_linear_map_injective.eq_iff -instance : semilinear_map_class (M ≃ₛₗ[σ] M₂) σ M M₂ := +instance : semilinear_equiv_class (M ≃ₛₗ[σ] M₂) σ M M₂ := { coe := linear_equiv.to_fun, - coe_injective' := λ f g h, to_linear_map_injective (fun_like.coe_injective h), + inv := linear_equiv.inv_fun, + coe_injective' := λ f g h₁ h₂, by { cases f, cases g, congr' }, + left_inv := linear_equiv.left_inv, + right_inv := linear_equiv.right_inv, map_add := map_add', map_smulₛₗ := map_smul' } diff --git a/formal/lean/mathlib/algebra/module/linear_map.lean b/formal/lean/mathlib/algebra/module/linear_map.lean index b5df38c3555d4c962eb7b15742b371989f382390..a38d3cf1fd780daf8888aba29ac6f3f5bebc76d2 100644 --- a/formal/lean/mathlib/algebra/module/linear_map.lean +++ b/formal/lean/mathlib/algebra/module/linear_map.lean @@ -811,12 +811,26 @@ instance _root_.module.End.semiring : semiring (module.End R M) := zero_mul := zero_comp, left_distrib := λ f g h, comp_add _ _ _, right_distrib := λ f g h, add_comp _ _ _, + nat_cast := λ n, n • 1, + nat_cast_zero := add_monoid.nsmul_zero' _, + nat_cast_succ := λ n, (add_monoid.nsmul_succ' n 1).trans (add_comm _ _), .. add_monoid_with_one.unary, .. _root_.module.End.monoid, .. linear_map.add_comm_monoid } +/-- See also `module.End.nat_cast_def`. -/ +@[simp] lemma _root_.module.End.nat_cast_apply (n : ℕ) (m : M) : + (↑n : module.End R M) m = n • m := rfl + instance _root_.module.End.ring : ring (module.End R N₁) := -{ ..module.End.semiring, ..linear_map.add_comm_group } +{ int_cast := λ z, z • 1, + int_cast_of_nat := of_nat_zsmul _, + int_cast_neg_succ_of_nat := zsmul_neg_succ_of_nat _, + ..module.End.semiring, ..linear_map.add_comm_group } + +/-- See also `module.End.int_cast_def`. -/ +@[simp] lemma _root_.module.End.int_cast_apply (z : ℤ) (m : N₁) : + (↑z : module.End R N₁) m = z • m := rfl section variables [monoid S] [distrib_mul_action S M] [smul_comm_class R S M] @@ -930,4 +944,10 @@ def module_End_self_op : R ≃+* module.End Rᵐᵒᵖ R := right_inv := λ f, linear_map.ext_ring_op $ mul_one _, ..module.to_module_End _ _ } +lemma End.nat_cast_def (n : ℕ) [add_comm_monoid N₁] [module R N₁] : + (↑n : module.End R N₁) = module.to_module_End R N₁ n := rfl + +lemma End.int_cast_def (z : ℤ) [add_comm_group N₁] [module R N₁] : + (↑z : module.End R N₁) = module.to_module_End R N₁ z := rfl + end module diff --git a/formal/lean/mathlib/algebra/module/localized_module.lean b/formal/lean/mathlib/algebra/module/localized_module.lean index 7cfcf5cd01fe61b44d57e68efb9e8cd99d77aa21..870e117a2ef2ab470fa5e0db347037cd7472cab2 100644 --- a/formal/lean/mathlib/algebra/module/localized_module.lean +++ b/formal/lean/mathlib/algebra/module/localized_module.lean @@ -70,7 +70,7 @@ instance r.setoid : setoid (M × S) := If `S` is a multiplicative subset of a ring `R` and `M` an `R`-module, then we can localize `M` by `S`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def _root_.localized_module : Type (max u v) := quotient (r.setoid S M) section diff --git a/formal/lean/mathlib/algebra/monoid_algebra/basic.lean b/formal/lean/mathlib/algebra/monoid_algebra/basic.lean index 0ad604262a1a7f5df17a6791201872a18bd5bd67..be59f29d52cd23ddc2630194da08548b3cc05206 100644 --- a/formal/lean/mathlib/algebra/monoid_algebra/basic.lean +++ b/formal/lean/mathlib/algebra/monoid_algebra/basic.lean @@ -183,6 +183,8 @@ instance : non_assoc_semiring (monoid_algebra k G) := single_zero, sum_zero, add_zero, mul_one, sum_single], ..monoid_algebra.non_unital_non_assoc_semiring } +lemma nat_cast_def (n : ℕ) : (n : monoid_algebra k G) = single 1 n := rfl + end mul_one_class /-! #### Semiring structure -/ @@ -244,11 +246,16 @@ instance [ring k] [semigroup G] : non_unital_ring (monoid_algebra k G) := .. monoid_algebra.non_unital_semiring } instance [ring k] [mul_one_class G] : non_assoc_ring (monoid_algebra k G) := -{ .. monoid_algebra.add_comm_group, +{ int_cast := λ z, single 1 (z : k), + int_cast_of_nat := λ n, by simpa, + int_cast_neg_succ_of_nat := λ n, by simpa, + .. monoid_algebra.add_comm_group, .. monoid_algebra.non_assoc_semiring } +lemma int_cast_def [ring k] [mul_one_class G] (z : ℤ) : (z : monoid_algebra k G) = single 1 z := rfl + instance [ring k] [monoid G] : ring (monoid_algebra k G) := -{ .. monoid_algebra.non_unital_non_assoc_ring, +{ .. monoid_algebra.non_assoc_ring, .. monoid_algebra.semiring } instance [comm_ring k] [comm_semigroup G] : non_unital_comm_ring (monoid_algebra k G) := @@ -318,8 +325,8 @@ lemma mul_apply_antidiagonal [has_mul G] (f g : monoid_algebra k G) (x : G) (s : let F : G × G → k := λ p, by classical; exact if p.1 * p.2 = x then f p.1 * g p.2 else 0 in calc (f * g) x = (∑ a₁ in f.support, ∑ a₂ in g.support, F (a₁, a₂)) : mul_apply f g x -... = ∑ p in f.support.product g.support, F p : finset.sum_product.symm -... = ∑ p in (f.support.product g.support).filter (λ p : G × G, p.1 * p.2 = x), f p.1 * g p.2 : +... = ∑ p in f.support ×ˢ g.support, F p : finset.sum_product.symm +... = ∑ p in (f.support ×ˢ g.support).filter (λ p : G × G, p.1 * p.2 = x), f p.1 * g p.2 : (finset.sum_filter _ _).symm ... = ∑ p in s.filter (λ p : G × G, p.1 ∈ f.support ∧ p.2 ∈ g.support), f p.1 * g p.2 : sum_congr (by { ext, simp only [mem_filter, mem_product, hs, and_comm] }) (λ _ _, rfl) @@ -1029,6 +1036,8 @@ instance : non_assoc_semiring (add_monoid_algebra k G) := single_zero, sum_zero, add_zero, mul_one, sum_single], .. add_monoid_algebra.non_unital_non_assoc_semiring } +lemma nat_cast_def (n : ℕ) : (n : add_monoid_algebra k G) = single 0 n := rfl + end mul_one_class /-! #### Semiring structure -/ @@ -1091,11 +1100,17 @@ instance [ring k] [add_semigroup G] : non_unital_ring (add_monoid_algebra k G) : .. add_monoid_algebra.non_unital_semiring } instance [ring k] [add_zero_class G] : non_assoc_ring (add_monoid_algebra k G) := -{ .. add_monoid_algebra.add_comm_group, +{ int_cast := λ z, single 0 (z : k), + int_cast_of_nat := λ n, by simpa, + int_cast_neg_succ_of_nat := λ n, by simpa, + .. add_monoid_algebra.add_comm_group, .. add_monoid_algebra.non_assoc_semiring } +lemma int_cast_def [ring k] [add_zero_class G] (z : ℤ) : + (z : add_monoid_algebra k G) = single 0 z := rfl + instance [ring k] [add_monoid G] : ring (add_monoid_algebra k G) := -{ .. add_monoid_algebra.non_unital_non_assoc_ring, +{ .. add_monoid_algebra.non_assoc_ring, .. add_monoid_algebra.semiring } instance [comm_ring k] [add_comm_semigroup G] : non_unital_comm_ring (add_monoid_algebra k G) := diff --git a/formal/lean/mathlib/algebra/monoid_algebra/degree.lean b/formal/lean/mathlib/algebra/monoid_algebra/degree.lean new file mode 100644 index 0000000000000000000000000000000000000000..a15a552b8fba11224d050d1de713583fe74b0b16 --- /dev/null +++ b/formal/lean/mathlib/algebra/monoid_algebra/degree.lean @@ -0,0 +1,154 @@ +/- +Copyright (c) 2022 Damiano Testa. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Damiano Testa +-/ +import algebra.monoid_algebra.basic + +/-! +# Lemmas about the `sup` and `inf` of the support of `add_monoid_algebra` + +## TODO +The current plan is to state and prove lemmas about `finset.sup (finsupp.support f) D` with a +"generic" degree/weight function `D` from the grading Type `A` to a somewhat ordered Type `B`. + +Next, the general lemmas get specialized for some yet-to-be-defined `degree`s. +-/ + +variables {R A T B ι : Type*} + +namespace add_monoid_algebra +open_locale classical big_operators + +/-! ### Results about the `finset.sup` and `finset.inf` of `finsupp.support` -/ + +section general_results_assuming_semilattice_sup +variables [semilattice_sup B] [order_bot B] [semilattice_inf T] [order_top T] + +section semiring +variables [semiring R] + +section explicit_degrees +/-! + +In this section, we use `degb` and `degt` to denote "degree functions" on `A` with values in +a type with *b*ot or *t*op respectively. +-/ +variables (degb : A → B) (degt : A → T) (f g : add_monoid_algebra R A) + +lemma sup_support_add_le : (f + g).support.sup degb ≤ (f.support.sup degb) ⊔ (g.support.sup degb) := +(finset.sup_mono finsupp.support_add).trans_eq finset.sup_union + +lemma le_inf_support_add : f.support.inf degt ⊓ g.support.inf degt ≤ (f + g).support.inf degt := +sup_support_add_le (λ a : A, order_dual.to_dual (degt a)) f g + +end explicit_degrees + +section add_only +variables [has_add A] [has_add B] [has_add T] + [covariant_class B B (+) (≤)] [covariant_class B B (function.swap (+)) (≤)] + [covariant_class T T (+) (≤)] [covariant_class T T (function.swap (+)) (≤)] + +lemma sup_support_mul_le {degb : A → B} (degbm : ∀ {a b}, degb (a + b) ≤ degb a + degb b) + (f g : add_monoid_algebra R A) : + (f * g).support.sup degb ≤ f.support.sup degb + g.support.sup degb := +begin + refine (finset.sup_mono $ support_mul _ _).trans _, + simp_rw [finset.sup_bUnion, finset.sup_singleton], + refine (finset.sup_le $ λ fd fds, finset.sup_le $ λ gd gds, degbm.trans $ add_le_add _ _); + exact finset.le_sup ‹_›, +end + +lemma le_inf_support_mul {degt : A → T} (degtm : ∀ {a b}, degt a + degt b ≤ degt (a + b)) + (f g : add_monoid_algebra R A) : + f.support.inf degt + g.support.inf degt ≤ (f * g).support.inf degt := +order_dual.of_dual_le_of_dual.mpr $ + sup_support_mul_le (λ a b, order_dual.of_dual_le_of_dual.mp degtm) f g + +end add_only + +section add_monoids +variables [add_monoid A] + [add_monoid B] [covariant_class B B (+) (≤)] [covariant_class B B (function.swap (+)) (≤)] + [add_monoid T] [covariant_class T T (+) (≤)] [covariant_class T T (function.swap (+)) (≤)] + {degb : A → B} {degt : A → T} + +lemma sup_support_list_prod_le (degb0 : degb 0 ≤ 0) + (degbm : ∀ a b, degb (a + b) ≤ degb a + degb b) : + ∀ l : list (add_monoid_algebra R A), + l.prod.support.sup degb ≤ (l.map (λ f : add_monoid_algebra R A, f.support.sup degb)).sum +| [] := begin + rw [list.map_nil, finset.sup_le_iff, list.prod_nil, list.sum_nil], + exact λ a ha, by rwa [finset.mem_singleton.mp (finsupp.support_single_subset ha)] + end +| (f::fs) := begin + rw [list.prod_cons, list.map_cons, list.sum_cons], + exact (sup_support_mul_le degbm _ _).trans (add_le_add_left (sup_support_list_prod_le _) _) + end + +lemma le_inf_support_list_prod (degt0 : 0 ≤ degt 0) (degtm : ∀ a b, degt a + degt b ≤ degt (a + b)) + (l : list (add_monoid_algebra R A)) : + (l.map (λ f : add_monoid_algebra R A, f.support.inf degt)).sum ≤ l.prod.support.inf degt := +order_dual.of_dual_le_of_dual.mpr $ sup_support_list_prod_le + (order_dual.of_dual_le_of_dual.mp degt0) (λ a b, order_dual.of_dual_le_of_dual.mp (degtm _ _)) l + +lemma sup_support_pow_le (degb0 : degb 0 ≤ 0) (degbm : ∀ a b, degb (a + b) ≤ degb a + degb b) + (n : ℕ) (f : add_monoid_algebra R A) : + (f ^ n).support.sup degb ≤ n • (f.support.sup degb) := +begin + rw [← list.prod_repeat, ←list.sum_repeat], + refine (sup_support_list_prod_le degb0 degbm _).trans_eq _, + rw list.map_repeat, +end + +lemma le_inf_support_pow (degt0 : 0 ≤ degt 0) (degtm : ∀ a b, degt a + degt b ≤ degt (a + b)) + (n : ℕ) (f : add_monoid_algebra R A) : + n • (f.support.inf degt) ≤ (f ^ n).support.inf degt := +order_dual.of_dual_le_of_dual.mpr $ sup_support_pow_le (order_dual.of_dual_le_of_dual.mp degt0) + (λ a b, order_dual.of_dual_le_of_dual.mp (degtm _ _)) n f + +end add_monoids + +end semiring + +section commutative_lemmas +variables [comm_semiring R] [add_comm_monoid A] + [add_comm_monoid B] [covariant_class B B (+) (≤)] [covariant_class B B (function.swap (+)) (≤)] + [add_comm_monoid T] [covariant_class T T (+) (≤)] [covariant_class T T (function.swap (+)) (≤)] + {degb : A → B} {degt : A → T} + +lemma sup_support_multiset_prod_le + (degb0 : degb 0 ≤ 0) (degbm : ∀ a b, degb (a + b) ≤ degb a + degb b) + (m : multiset (add_monoid_algebra R A)) : + m.prod.support.sup degb ≤ (m.map (λ f : add_monoid_algebra R A, f.support.sup degb)).sum := +begin + induction m using quot.induction_on, + rw [multiset.quot_mk_to_coe'', multiset.coe_map, multiset.coe_sum, multiset.coe_prod], + exact sup_support_list_prod_le degb0 degbm m, +end + +lemma le_inf_support_multiset_prod + (degt0 : 0 ≤ degt 0) (degtm : ∀ a b, degt a + degt b ≤ degt (a + b)) + (m : multiset (add_monoid_algebra R A)) : + (m.map (λ f : add_monoid_algebra R A, f.support.inf degt)).sum ≤ m.prod.support.inf degt := +order_dual.of_dual_le_of_dual.mpr $ + sup_support_multiset_prod_le (order_dual.of_dual_le_of_dual.mp degt0) + (λ a b, order_dual.of_dual_le_of_dual.mp (degtm _ _)) m + +lemma sup_support_finset_prod_le + (degb0 : degb 0 ≤ 0) (degbm : ∀ a b, degb (a + b) ≤ degb a + degb b) + (s : finset ι) (f : ι → add_monoid_algebra R A) : + (∏ i in s, f i).support.sup degb ≤ ∑ i in s, (f i).support.sup degb := +(sup_support_multiset_prod_le degb0 degbm _).trans_eq $ congr_arg _ $ multiset.map_map _ _ _ + +lemma le_inf_support_finset_prod + (degt0 : 0 ≤ degt 0) (degtm : ∀ a b, degt a + degt b ≤ degt (a + b)) + (s : finset ι) (f : ι → add_monoid_algebra R A) : + ∑ i in s, (f i).support.inf degt ≤ (∏ i in s, f i).support.inf degt := +le_of_eq_of_le (by rw [multiset.map_map]; refl) (le_inf_support_multiset_prod degt0 degtm _) + +end commutative_lemmas + +end general_results_assuming_semilattice_sup + +end add_monoid_algebra diff --git a/formal/lean/mathlib/algebra/order/archimedean.lean b/formal/lean/mathlib/algebra/order/archimedean.lean index 3f4b5d4df0fadfe832f750ad0fe07cd858a6344d..39a16c706e9e38406ac3d843ba60a7c3d0191631 100644 --- a/formal/lean/mathlib/algebra/order/archimedean.lean +++ b/formal/lean/mathlib/algebra/order/archimedean.lean @@ -109,26 +109,13 @@ let ⟨n, h⟩ := archimedean.arch x hy in (add_nonneg zero_le_two hy.le) _ ... = (y + 1) ^ n : by rw [add_comm]⟩ -section linear_ordered_ring -variables [linear_ordered_ring α] [archimedean α] +section ordered_ring +variables [ordered_ring α] [nontrivial α] [archimedean α] lemma pow_unbounded_of_one_lt (x : α) {y : α} (hy1 : 1 < y) : ∃ n : ℕ, x < y ^ n := sub_add_cancel y 1 ▸ add_one_pow_unbounded_of_pos _ (sub_pos.2 hy1) -/-- Every x greater than or equal to 1 is between two successive -natural-number powers of every y greater than one. -/ -lemma exists_nat_pow_near {x : α} {y : α} (hx : 1 ≤ x) (hy : 1 < y) : - ∃ n : ℕ, y ^ n ≤ x ∧ x < y ^ (n + 1) := -have h : ∃ n : ℕ, x < y ^ n, from pow_unbounded_of_one_lt _ hy, -by classical; exact let n := nat.find h in - have hn : x < y ^ n, from nat.find_spec h, - have hnp : 0 < n, from pos_iff_ne_zero.2 (λ hn0, - by rw [hn0, pow_zero] at hn; exact (not_le_of_gt hn hx)), - have hnsp : nat.pred n + 1 = n, from nat.succ_pred_eq_of_pos hnp, - have hltn : nat.pred n < n, from nat.pred_lt (ne_of_gt hnp), - ⟨nat.pred n, le_of_not_lt (nat.find_min h hltn), by rwa hnsp⟩ - theorem exists_int_gt (x : α) : ∃ n : ℤ, x < n := let ⟨n, h⟩ := exists_nat_gt x in ⟨n, by rwa int.cast_coe_nat⟩ @@ -149,6 +136,24 @@ begin exact ⟨λ h, le_trans (int.cast_le.2 h) h₁, h₂ z⟩, end +end ordered_ring + +section linear_ordered_ring +variables [linear_ordered_ring α] [archimedean α] + +/-- Every x greater than or equal to 1 is between two successive +natural-number powers of every y greater than one. -/ +lemma exists_nat_pow_near {x : α} {y : α} (hx : 1 ≤ x) (hy : 1 < y) : + ∃ n : ℕ, y ^ n ≤ x ∧ x < y ^ (n + 1) := +have h : ∃ n : ℕ, x < y ^ n, from pow_unbounded_of_one_lt _ hy, +by classical; exact let n := nat.find h in + have hn : x < y ^ n, from nat.find_spec h, + have hnp : 0 < n, from pos_iff_ne_zero.2 (λ hn0, + by rw [hn0, pow_zero] at hn; exact (not_le_of_gt hn hx)), + have hnsp : nat.pred n + 1 = n, from nat.succ_pred_eq_of_pos hnp, + have hltn : nat.pred n < n, from nat.pred_lt (ne_of_gt hnp), + ⟨nat.pred n, le_of_not_lt (nat.find_min h hltn), by rwa hnsp⟩ + end linear_ordered_ring section linear_ordered_field @@ -276,6 +281,22 @@ archimedean_iff_nat_lt.trans λ H x, let ⟨n, h⟩ := H x in ⟨n+1, lt_of_le_of_lt h (nat.cast_lt.2 (lt_add_one _))⟩⟩ +lemma archimedean_iff_int_lt : archimedean α ↔ ∀ x : α, ∃ n : ℤ, x < n := +⟨@exists_int_gt α _ _, +begin + rw archimedean_iff_nat_lt, + intros h x, + obtain ⟨n, h⟩ := h x, + refine ⟨n.to_nat, h.trans_le _⟩, + exact_mod_cast int.le_to_nat _, +end⟩ + +lemma archimedean_iff_int_le : archimedean α ↔ ∀ x : α, ∃ n : ℤ, x ≤ n := +archimedean_iff_int_lt.trans +⟨λ H x, (H x).imp $ λ _, le_of_lt, + λ H x, let ⟨n, h⟩ := H x in ⟨n+1, + lt_of_le_of_lt h (int.cast_lt.2 (lt_add_one _))⟩⟩ + lemma archimedean_iff_rat_lt : archimedean α ↔ ∀ x : α, ∃ q : ℚ, x < q := ⟨@exists_rat_gt α _, λ H, archimedean_iff_nat_lt.2 $ λ x, @@ -308,3 +329,10 @@ noncomputable def archimedean.floor_ring (α) [linear_ordered_ring α] [archimed floor_ring α := floor_ring.of_floor α (λ a, classical.some (exists_floor a)) (λ z a, (classical.some_spec (exists_floor a) z).symm) + +/-- A linear ordered field that is a floor ring is archimedean. -/ +lemma floor_ring.archimedean (α) [linear_ordered_field α] [floor_ring α] : archimedean α := +begin + rw archimedean_iff_int_le, + exact λ x, ⟨⌈x⌉, int.le_ceil x⟩ +end diff --git a/formal/lean/mathlib/algebra/order/field.lean b/formal/lean/mathlib/algebra/order/field.lean index b08aa87da169cacbd9e8481ef709777ee23c93c9..52f79b68efbec684629297c538858af4aaf514bd 100644 --- a/formal/lean/mathlib/algebra/order/field.lean +++ b/formal/lean/mathlib/algebra/order/field.lean @@ -3,12 +3,7 @@ Copyright (c) 2014 Robert Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn -/ -import algebra.field.basic -import algebra.group_power.lemmas -import algebra.group_power.order -import algebra.order.ring -import order.bounds -import tactic.monotonicity.basic +import algebra.order.field_defs /-! # Linear ordered (semi)fields @@ -28,18 +23,6 @@ set_option old_structure_cmd true variables {α β : Type*} -/-- A linear ordered semifield is a field with a linear order respecting the operations. -/ -@[protect_proj] class linear_ordered_semifield (α : Type*) - extends linear_ordered_semiring α, semifield α - -/-- A linear ordered field is a field with a linear order respecting the operations. -/ -@[protect_proj] class linear_ordered_field (α : Type*) extends linear_ordered_comm_ring α, field α - -@[priority 100] -- See note [lower instance priority] -instance linear_ordered_field.to_linear_ordered_semifield [linear_ordered_field α] : - linear_ordered_semifield α := -{ ..linear_ordered_ring.to_linear_ordered_semiring, ..‹linear_ordered_field α› } - namespace function /-- Pullback a `linear_ordered_semifield` under an injective map. -/ diff --git a/formal/lean/mathlib/algebra/order/field_defs.lean b/formal/lean/mathlib/algebra/order/field_defs.lean new file mode 100644 index 0000000000000000000000000000000000000000..a345e440e108c8f2a944e91e913164c30ec3bfe1 --- /dev/null +++ b/formal/lean/mathlib/algebra/order/field_defs.lean @@ -0,0 +1,45 @@ +/- +Copyright (c) 2014 Robert Lewis. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn +-/ +import algebra.field.basic +import algebra.group_power.lemmas +import algebra.group_power.order +import algebra.order.ring +import order.bounds +import tactic.monotonicity.basic + +/-! +# Linear ordered (semi)fields + +A linear ordered (semi)field is a (semi)field equipped with a linear order such that +* addition respects the order: `a ≤ b → c + a ≤ c + b`; +* multiplication of positives is positive: `0 < a → 0 < b → 0 < a * b`; +* `0 < 1`. + +## Main Definitions + +* `linear_ordered_semifield`: Typeclass for linear order semifields. +* `linear_ordered_field`: Typeclass for linear ordered fields. + +## Implementation details + +For olean caching reasons, this file is separate to the main file, algebra.order.field. +The lemmata are instead located there. + +-/ + +set_option old_structure_cmd true + +/-- A linear ordered semifield is a field with a linear order respecting the operations. -/ +@[protect_proj] class linear_ordered_semifield (α : Type*) + extends linear_ordered_semiring α, semifield α + +/-- A linear ordered field is a field with a linear order respecting the operations. -/ +@[protect_proj] class linear_ordered_field (α : Type*) extends linear_ordered_comm_ring α, field α + +@[priority 100] -- See note [lower instance priority] +instance linear_ordered_field.to_linear_ordered_semifield {α} [linear_ordered_field α] : + linear_ordered_semifield α := +{ ..linear_ordered_ring.to_linear_ordered_semiring, ..‹linear_ordered_field α› } diff --git a/formal/lean/mathlib/algebra/order/floor.lean b/formal/lean/mathlib/algebra/order/floor.lean index 25af371180a3d8e9f5e1e48b53b5b989ed0645d9..87ec96e477564b59f83749d609e527f077085edf 100644 --- a/formal/lean/mathlib/algebra/order/floor.lean +++ b/formal/lean/mathlib/algebra/order/floor.lean @@ -260,44 +260,44 @@ by { ext, simp [lt_ceil] } @[simp] lemma preimage_Iic {a : α} (ha : 0 ≤ a) : ((coe : ℕ → α) ⁻¹' (set.Iic a)) = set.Iic ⌊a⌋₊ := by { ext, simp [le_floor_iff, ha] } -end linear_ordered_semiring - -section linear_ordered_ring -variables [linear_ordered_ring α] [floor_semiring α] {a : α} {n : ℕ} - lemma floor_add_nat (ha : 0 ≤ a) (n : ℕ) : ⌊a + n⌋₊ = ⌊a⌋₊ + n := eq_of_forall_le_iff $ λ b, begin - rw [le_floor_iff (add_nonneg ha n.cast_nonneg), ←sub_le_iff_le_add], + rw [le_floor_iff (add_nonneg ha n.cast_nonneg)], obtain hb | hb := le_total n b, - { rw [←cast_sub hb, ←tsub_le_iff_right], - exact (le_floor_iff ha).symm }, - { exact iff_of_true ((sub_nonpos_of_le $ cast_le.2 hb).trans ha) (le_add_left hb) } + { obtain ⟨d, rfl⟩ := exists_add_of_le hb, + rw [nat.cast_add, add_comm n, add_comm (n : α), add_le_add_iff_right, add_le_add_iff_right, + le_floor_iff ha] }, + { obtain ⟨d, rfl⟩ := exists_add_of_le hb, + rw [nat.cast_add, add_left_comm _ b, add_left_comm _ (b : α)], + refine iff_of_true _ le_self_add, + exact (le_add_of_nonneg_right $ ha.trans $ le_add_of_nonneg_right d.cast_nonneg) } end lemma floor_add_one (ha : 0 ≤ a) : ⌊a + 1⌋₊ = ⌊a⌋₊ + 1 := by { convert floor_add_nat ha 1, exact cast_one.symm } -lemma floor_sub_nat (a : α) (n : ℕ) : ⌊a - n⌋₊ = ⌊a⌋₊ - n := +lemma floor_sub_nat [has_sub α] [has_ordered_sub α] [has_exists_add_of_le α] (a : α) (n : ℕ) : + ⌊a - n⌋₊ = ⌊a⌋₊ - n := begin obtain ha | ha := le_total a 0, - { rw [floor_of_nonpos ha, floor_of_nonpos (sub_nonpos_of_le (ha.trans n.cast_nonneg)), + { rw [floor_of_nonpos ha, floor_of_nonpos (tsub_nonpos_of_le (ha.trans n.cast_nonneg)), zero_tsub] }, cases le_total a n, { rw [floor_of_nonpos (tsub_nonpos_of_le h), eq_comm, tsub_eq_zero_iff_le], exact nat.cast_le.1 ((nat.floor_le ha).trans h) }, - { rw [eq_tsub_iff_add_eq_of_le (le_floor h), ←floor_add_nat (sub_nonneg_of_le h), - sub_add_cancel] } + { rw [eq_tsub_iff_add_eq_of_le (le_floor h), ←floor_add_nat _, + tsub_add_cancel_of_le h], + exact le_tsub_of_add_le_left ((add_zero _).trans_le h), } end -lemma sub_one_lt_floor (a : α) : a - 1 < ⌊a⌋₊ := sub_lt_iff_lt_add.2 $ lt_floor_add_one a - lemma ceil_add_nat (ha : 0 ≤ a) (n : ℕ) : ⌈a + n⌉₊ = ⌈a⌉₊ + n := eq_of_forall_ge_iff $ λ b, begin rw [←not_lt, ←not_lt, not_iff_not], rw [lt_ceil], obtain hb | hb := le_or_lt n b, - { rw [←tsub_lt_iff_right hb, ←sub_lt_iff_lt_add, ←cast_sub hb], - exact lt_ceil.symm }, + { obtain ⟨d, rfl⟩ := exists_add_of_le hb, + rw [nat.cast_add, add_comm n, add_comm (n : α), add_lt_add_iff_right, add_lt_add_iff_right, + lt_ceil] }, { exact iff_of_true (lt_add_of_nonneg_of_lt ha $ cast_lt.2 hb) (lt_add_left _ _ _ hb) } end @@ -307,6 +307,13 @@ by { convert ceil_add_nat ha 1, exact cast_one.symm } lemma ceil_lt_add_one (ha : 0 ≤ a) : (⌈a⌉₊ : α) < a + 1 := lt_ceil.1 $ (nat.lt_succ_self _).trans_le (ceil_add_one ha).ge +end linear_ordered_semiring + +section linear_ordered_ring +variables [linear_ordered_ring α] [floor_semiring α] + +lemma sub_one_lt_floor (a : α) : a - 1 < ⌊a⌋₊ := sub_lt_iff_lt_add.2 $ lt_floor_add_one a + end linear_ordered_ring section linear_ordered_semifield @@ -521,6 +528,9 @@ lemma fract_lt_one (a : α) : fract a < 1 := sub_lt.1 $ sub_one_lt_floor _ @[simp] lemma fract_zero : fract (0 : α) = 0 := by rw [fract, floor_zero, cast_zero, sub_self] +@[simp] lemma fract_one : fract (1 : α) = 0 := +by simp [fract] + @[simp] lemma fract_coe (z : ℤ) : fract (z : α) = 0 := by { unfold fract, rw floor_coe, exact sub_self _ } diff --git a/formal/lean/mathlib/algebra/order/group.lean b/formal/lean/mathlib/algebra/order/group.lean index 8106e0424248c72ce357952443745a0f17c6f09a..55a25bb9208bfa957438be06d6643e340b2de87a 100644 --- a/formal/lean/mathlib/algebra/order/group.lean +++ b/formal/lean/mathlib/algebra/order/group.lean @@ -1258,7 +1258,7 @@ namespace add_comm_group /-- A collection of elements in an `add_comm_group` designated as "non-negative". This is useful for constructing an `ordered_add_commm_group` by choosing a positive cone in an exisiting `add_comm_group`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure positive_cone (α : Type*) [add_comm_group α] := (nonneg : α → Prop) (pos : α → Prop := λ a, nonneg a ∧ ¬ nonneg (-a)) @@ -1269,7 +1269,7 @@ structure positive_cone (α : Type*) [add_comm_group α] := /-- A positive cone in an `add_comm_group` induces a linear order if for every `a`, either `a` or `-a` is non-negative. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure total_positive_cone (α : Type*) [add_comm_group α] extends positive_cone α := (nonneg_decidable : decidable_pred nonneg) (nonneg_total : ∀ a : α, nonneg a ∨ nonneg (-a)) diff --git a/formal/lean/mathlib/algebra/order/lattice_group.lean b/formal/lean/mathlib/algebra/order/lattice_group.lean index f53136142fdc7b8eed1ac7539f1005a5fc890981..e90cb24f31213e85f5caef8f5046d7888f4e3a49 100644 --- a/formal/lean/mathlib/algebra/order/lattice_group.lean +++ b/formal/lean/mathlib/algebra/order/lattice_group.lean @@ -395,6 +395,15 @@ end lemma pos_of_one_le (a : α) (h : 1 ≤ a) : a⁺ = a := by { rw m_pos_part_def, exact sup_of_le_left h, } +@[to_additive] -- pos_eq_self_of_pos_pos +lemma pos_eq_self_of_one_lt_pos {α} [linear_order α] [comm_group α] + {x : α} (hx : 1 < x⁺) : x⁺ = x := +begin + rw [m_pos_part_def, right_lt_sup, not_le] at hx, + rw [m_pos_part_def, sup_eq_left], + exact hx.le +end + -- 0 ≤ a implies a⁺ = a @[to_additive] -- pos_of_nonpos lemma pos_of_le_one (a : α) (h : a ≤ 1) : a⁺ = 1 := diff --git a/formal/lean/mathlib/algebra/order/ring.lean b/formal/lean/mathlib/algebra/order/ring.lean index cc7b63bda7912c8454e8d96fa2a89047bca2ee0a..05d628d3dbe31a26b86141fd6a16a22353c472e4 100644 --- a/formal/lean/mathlib/algebra/order/ring.lean +++ b/formal/lean/mathlib/algebra/order/ring.lean @@ -366,6 +366,15 @@ by classical; exact decidable.lt_mul_of_one_lt_left lemma lt_two_mul_self [nontrivial α] (ha : 0 < a) : a < 2 * a := lt_mul_of_one_lt_left ha one_lt_two +lemma lt_mul_left (hn : 0 < a) (hm : 1 < b) : a < b * a := +by { convert mul_lt_mul_of_pos_right hm hn, rw one_mul } + +lemma lt_mul_right (hn : 0 < a) (hm : 1 < b) : a < a * b := +by { convert mul_lt_mul_of_pos_left hm hn, rw mul_one } + +lemma lt_mul_self (hn : 1 < a) : a < a * a := +lt_mul_left (hn.trans_le' zero_le_one) hn + -- See Note [decidable namespace] protected lemma decidable.add_le_mul_two_add [@decidable_rel α (≤)] {a b : α} (a2 : 2 ≤ a) (b0 : 0 ≤ b) : a + (2 + b) ≤ a * (2 + b) := @@ -1435,7 +1444,7 @@ namespace ring /-- A positive cone in a ring consists of a positive cone in underlying `add_comm_group`, which contains `1` and such that the positive elements are closed under multiplication. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure positive_cone (α : Type*) [ring α] extends add_comm_group.positive_cone α := (one_nonneg : nonneg 1) (mul_pos : ∀ (a b), pos a → pos b → pos (a * b)) @@ -1444,7 +1453,7 @@ structure positive_cone (α : Type*) [ring α] extends add_comm_group.positive_c add_decl_doc positive_cone.to_positive_cone /-- A positive cone in a ring induces a linear order if `1` is a positive element. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure total_positive_cone (α : Type*) [ring α] extends positive_cone α, add_comm_group.total_positive_cone α := (one_pos : pos 1) diff --git a/formal/lean/mathlib/algebra/order/sub.lean b/formal/lean/mathlib/algebra/order/sub.lean index b0e25dd9004036bd7008d91dd8195b9dee14e37c..d426fac5f8b78f48de695c40110c0a2e39660b64 100644 --- a/formal/lean/mathlib/algebra/order/sub.lean +++ b/formal/lean/mathlib/algebra/order/sub.lean @@ -670,6 +670,17 @@ lemma tsub_right_inj (hba : b ≤ a) (hca : c ≤ a) : a - b = a - c ↔ b = c : contravariant.add_le_cancellable.tsub_right_inj contravariant.add_le_cancellable contravariant.add_le_cancellable hba hca +variables (α) + +/-- A `canonically_ordered_add_monoid` with ordered subtraction and order-reflecting addition is +cancellative. This is not an instance at it would form a typeclass loop. + +See note [reducible non-instances]. -/ +@[reducible] +def canonically_ordered_add_monoid.to_add_cancel_comm_monoid : add_cancel_comm_monoid α := +{ add_left_cancel := λ a b c h, by simpa only [add_tsub_cancel_left] using congr_arg (λ x, x - a) h, + ..(by apply_instance : add_comm_monoid α) } + end contra end canonically_ordered_add_monoid diff --git a/formal/lean/mathlib/algebra/order/to_interval_mod.lean b/formal/lean/mathlib/algebra/order/to_interval_mod.lean new file mode 100644 index 0000000000000000000000000000000000000000..064c523c4745d66593c9cbc4e63157aa99505cee --- /dev/null +++ b/formal/lean/mathlib/algebra/order/to_interval_mod.lean @@ -0,0 +1,475 @@ +/- +Copyright (c) 2022 Joseph Myers. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Joseph Myers +-/ +import algebra.periodic + +/-! +# Reducing to an interval modulo its length + +This file defines operations that reduce a number (in an `archimedean` +`linear_ordered_add_comm_group`) to a number in a given interval, modulo the length of that +interval. + +## Main definitions + +* `to_Ico_div a hb x` (where `hb : 0 < b`): The unique integer such that this multiple of `b`, + added to `x`, is in `Ico a (a + b)`. +* `to_Ico_mod a hb x` (where `hb : 0 < b`): Reduce `x` to the interval `Ico a (a + b)`. +* `to_Ioc_div a hb x` (where `hb : 0 < b`): The unique integer such that this multiple of `b`, + added to `x`, is in `Ioc a (a + b)`. +* `to_Ioc_mod a hb x` (where `hb : 0 < b`): Reduce `x` to the interval `Ioc a (a + b)`. + +-/ + +noncomputable theory + +section linear_ordered_add_comm_group + +variables {α : Type*} [linear_ordered_add_comm_group α] [archimedean α] + +/-- The unique integer such that this multiple of `b`, added to `x`, is in `Ico a (a + b)`. -/ +def to_Ico_div (a : α) {b : α} (hb : 0 < b) (x : α) : ℤ := +(exists_unique_add_zsmul_mem_Ico hb x a).some + +lemma add_to_Ico_div_zsmul_mem_Ico (a : α) {b : α} (hb : 0 < b) (x : α) : + x + to_Ico_div a hb x • b ∈ set.Ico a (a + b) := +(exists_unique_add_zsmul_mem_Ico hb x a).some_spec.1 + +lemma eq_to_Ico_div_of_add_zsmul_mem_Ico {a b x : α} (hb : 0 < b) {y : ℤ} + (hy : x + y • b ∈ set.Ico a (a + b)) : y = to_Ico_div a hb x := +(exists_unique_add_zsmul_mem_Ico hb x a).some_spec.2 y hy + +/-- The unique integer such that this multiple of `b`, added to `x`, is in `Ioc a (a + b)`. -/ +def to_Ioc_div (a : α) {b : α} (hb : 0 < b) (x : α) : ℤ := +(exists_unique_add_zsmul_mem_Ioc hb x a).some + +lemma add_to_Ioc_div_zsmul_mem_Ioc (a : α) {b : α} (hb : 0 < b) (x : α) : + x + to_Ioc_div a hb x • b ∈ set.Ioc a (a + b) := +(exists_unique_add_zsmul_mem_Ioc hb x a).some_spec.1 + +lemma eq_to_Ioc_div_of_add_zsmul_mem_Ioc {a b x : α} (hb : 0 < b) {y : ℤ} + (hy : x + y • b ∈ set.Ioc a (a + b)) : y = to_Ioc_div a hb x := +(exists_unique_add_zsmul_mem_Ioc hb x a).some_spec.2 y hy + +/-- Reduce `x` to the interval `Ico a (a + b)`. -/ +def to_Ico_mod (a : α) {b : α} (hb : 0 < b) (x : α) : α := x + to_Ico_div a hb x • b + +/-- Reduce `x` to the interval `Ioc a (a + b)`. -/ +def to_Ioc_mod (a : α) {b : α} (hb : 0 < b) (x : α) : α := x + to_Ioc_div a hb x • b + +lemma to_Ico_mod_mem_Ico (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_mod a hb x ∈ set.Ico a (a + b) := +add_to_Ico_div_zsmul_mem_Ico a hb x + +lemma to_Ioc_mod_mem_Ioc (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_mod a hb x ∈ set.Ioc a (a + b) := +add_to_Ioc_div_zsmul_mem_Ioc a hb x + +lemma left_le_to_Ico_mod (a : α) {b : α} (hb : 0 < b) (x : α) : a ≤ to_Ico_mod a hb x := +(set.mem_Ico.1 (to_Ico_mod_mem_Ico a hb x)).1 + +lemma left_lt_to_Ioc_mod (a : α) {b : α} (hb : 0 < b) (x : α) : a < to_Ioc_mod a hb x := +(set.mem_Ioc.1 (to_Ioc_mod_mem_Ioc a hb x)).1 + +lemma to_Ico_mod_lt_right (a : α) {b : α} (hb : 0 < b) (x : α) : to_Ico_mod a hb x < a + b := +(set.mem_Ico.1 (to_Ico_mod_mem_Ico a hb x)).2 + +lemma to_Ioc_mod_le_right (a : α) {b : α} (hb : 0 < b) (x : α) : to_Ioc_mod a hb x ≤ a + b := +(set.mem_Ioc.1 (to_Ioc_mod_mem_Ioc a hb x)).2 + +@[simp] lemma self_add_to_Ico_div_zsmul (a : α) {b : α} (hb : 0 < b) (x : α) : + x + to_Ico_div a hb x • b = to_Ico_mod a hb x := +rfl + +@[simp] lemma self_add_to_Ioc_div_zsmul (a : α) {b : α} (hb : 0 < b) (x : α) : + x + to_Ioc_div a hb x • b = to_Ioc_mod a hb x := +rfl + +@[simp] lemma to_Ico_div_zsmul_add_self (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_div a hb x • b + x = to_Ico_mod a hb x := +by rw [add_comm, to_Ico_mod] + +@[simp] lemma to_Ioc_div_zsmul_add_self (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_div a hb x • b + x = to_Ioc_mod a hb x := +by rw [add_comm, to_Ioc_mod] + +@[simp] lemma to_Ico_mod_sub_self (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_mod a hb x - x = to_Ico_div a hb x • b := +by rw [to_Ico_mod, add_sub_cancel'] + +@[simp] lemma to_Ioc_mod_sub_self (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_mod a hb x - x = to_Ioc_div a hb x • b := +by rw [to_Ioc_mod, add_sub_cancel'] + +@[simp] lemma self_sub_to_Ico_mod (a : α) {b : α} (hb : 0 < b) (x : α) : + x - to_Ico_mod a hb x = -to_Ico_div a hb x • b := +by rw [to_Ico_mod, sub_add_cancel', neg_smul] + +@[simp] lemma self_sub_to_Ioc_mod (a : α) {b : α} (hb : 0 < b) (x : α) : + x - to_Ioc_mod a hb x = -to_Ioc_div a hb x • b := +by rw [to_Ioc_mod, sub_add_cancel', neg_smul] + +@[simp] lemma to_Ico_mod_sub_to_Ico_div_zsmul (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_mod a hb x - to_Ico_div a hb x • b = x := +by rw [to_Ico_mod, add_sub_cancel] + +@[simp] lemma to_Ioc_mod_sub_to_Ioc_div_zsmul (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_mod a hb x - to_Ioc_div a hb x • b = x := +by rw [to_Ioc_mod, add_sub_cancel] + +@[simp] lemma to_Ico_div_zsmul_sub_to_Ico_mod (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_div a hb x • b - to_Ico_mod a hb x = -x := +by rw [←neg_sub, to_Ico_mod_sub_to_Ico_div_zsmul] + +@[simp] lemma to_Ioc_div_zsmul_sub_to_Ioc_mod (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_div a hb x • b - to_Ioc_mod a hb x = -x := +by rw [←neg_sub, to_Ioc_mod_sub_to_Ioc_div_zsmul] + +lemma to_Ico_mod_eq_iff {a b x y : α} (hb : 0 < b) : + to_Ico_mod a hb x = y ↔ a ≤ y ∧ y < a + b ∧ ∃ z : ℤ, y - x = z • b := +begin + refine ⟨λ h, ⟨h ▸ left_le_to_Ico_mod a hb x, + h ▸ to_Ico_mod_lt_right a hb x, + to_Ico_div a hb x, + h ▸ to_Ico_mod_sub_self a hb x⟩, + λ h, _⟩, + rcases h with ⟨ha, hab, z, hz⟩, + rw sub_eq_iff_eq_add' at hz, + subst hz, + rw eq_to_Ico_div_of_add_zsmul_mem_Ico hb (set.mem_Ico.2 ⟨ha, hab⟩), + refl +end + +lemma to_Ioc_mod_eq_iff {a b x y : α} (hb : 0 < b) : + to_Ioc_mod a hb x = y ↔ a < y ∧ y ≤ a + b ∧ ∃ z : ℤ, y - x = z • b := +begin + refine ⟨λ h, ⟨h ▸ left_lt_to_Ioc_mod a hb x, + h ▸ to_Ioc_mod_le_right a hb x, + to_Ioc_div a hb x, + h ▸ to_Ioc_mod_sub_self a hb x⟩, + λ h, _⟩, + rcases h with ⟨ha, hab, z, hz⟩, + rw sub_eq_iff_eq_add' at hz, + subst hz, + rw eq_to_Ioc_div_of_add_zsmul_mem_Ioc hb (set.mem_Ioc.2 ⟨ha, hab⟩), + refl +end + +@[simp] lemma to_Ico_div_apply_left (a : α) {b : α} (hb : 0 < b) : to_Ico_div a hb a = 0 := +begin + refine (eq_to_Ico_div_of_add_zsmul_mem_Ico hb _).symm, + simp [hb] +end + +@[simp] lemma to_Ioc_div_apply_left (a : α) {b : α} (hb : 0 < b) : to_Ioc_div a hb a = 1 := +begin + refine (eq_to_Ioc_div_of_add_zsmul_mem_Ioc hb _).symm, + simp [hb] +end + +@[simp] lemma to_Ico_mod_apply_left (a : α) {b : α} (hb : 0 < b) : to_Ico_mod a hb a = a := +begin + rw to_Ico_mod_eq_iff hb, + refine ⟨le_refl _, lt_add_of_pos_right _ hb, 0, _⟩, + simp +end + +@[simp] lemma to_Ioc_mod_apply_left (a : α) {b : α} (hb : 0 < b) : to_Ioc_mod a hb a = a + b := +begin + rw to_Ioc_mod_eq_iff hb, + refine ⟨lt_add_of_pos_right _ hb, le_refl _, 1, _⟩, + simp +end + +lemma to_Ico_div_apply_right (a : α) {b : α} (hb : 0 < b) : + to_Ico_div a hb (a + b) = -1 := +begin + refine (eq_to_Ico_div_of_add_zsmul_mem_Ico hb _).symm, + simp [hb] +end + +lemma to_Ioc_div_apply_right (a : α) {b : α} (hb : 0 < b) : + to_Ioc_div a hb (a + b) = 0 := +begin + refine (eq_to_Ioc_div_of_add_zsmul_mem_Ioc hb _).symm, + simp [hb] +end + +lemma to_Ico_mod_apply_right (a : α) {b : α} (hb : 0 < b) : to_Ico_mod a hb (a + b) = a := +begin + rw to_Ico_mod_eq_iff hb, + refine ⟨le_refl _, lt_add_of_pos_right _ hb, -1, _⟩, + simp +end + +lemma to_Ioc_mod_apply_right (a : α) {b : α} (hb : 0 < b) : + to_Ioc_mod a hb (a + b) = a + b := +begin + rw to_Ioc_mod_eq_iff hb, + refine ⟨lt_add_of_pos_right _ hb, le_refl _, 0, _⟩, + simp +end + +@[simp] lemma to_Ico_div_add_zsmul (a : α) {b : α} (hb : 0 < b) (x : α) (m : ℤ) : + to_Ico_div a hb (x + m • b) = to_Ico_div a hb x - m := +begin + refine (eq_to_Ico_div_of_add_zsmul_mem_Ico hb _).symm, + convert add_to_Ico_div_zsmul_mem_Ico a hb x using 1, + simp [sub_smul] +end + +@[simp] lemma to_Ioc_div_add_zsmul (a : α) {b : α} (hb : 0 < b) (x : α) (m : ℤ) : + to_Ioc_div a hb (x + m • b) = to_Ioc_div a hb x - m := +begin + refine (eq_to_Ioc_div_of_add_zsmul_mem_Ioc hb _).symm, + convert add_to_Ioc_div_zsmul_mem_Ioc a hb x using 1, + simp [sub_smul] +end + +@[simp] lemma to_Ico_div_zsmul_add (a : α) {b : α} (hb : 0 < b) (x : α) (m : ℤ) : + to_Ico_div a hb (m • b + x) = to_Ico_div a hb x - m := +by rw [add_comm, to_Ico_div_add_zsmul] + +@[simp] lemma to_Ioc_div_zsmul_add (a : α) {b : α} (hb : 0 < b) (x : α) (m : ℤ) : + to_Ioc_div a hb (m • b + x) = to_Ioc_div a hb x - m := +by rw [add_comm, to_Ioc_div_add_zsmul] + +@[simp] lemma to_Ico_div_sub_zsmul (a : α) {b : α} (hb : 0 < b) (x : α) (m : ℤ) : + to_Ico_div a hb (x - m • b) = to_Ico_div a hb x + m := +by rw [sub_eq_add_neg, ←neg_smul, to_Ico_div_add_zsmul, sub_neg_eq_add] + +@[simp] lemma to_Ioc_div_sub_zsmul (a : α) {b : α} (hb : 0 < b) (x : α) (m : ℤ) : + to_Ioc_div a hb (x - m • b) = to_Ioc_div a hb x + m := +by rw [sub_eq_add_neg, ←neg_smul, to_Ioc_div_add_zsmul, sub_neg_eq_add] + +@[simp] lemma to_Ico_div_add_right (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_div a hb (x + b) = to_Ico_div a hb x - 1 := +begin + convert to_Ico_div_add_zsmul a hb x 1, + exact (one_zsmul _).symm +end + +@[simp] lemma to_Ioc_div_add_right (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_div a hb (x + b) = to_Ioc_div a hb x - 1 := +begin + convert to_Ioc_div_add_zsmul a hb x 1, + exact (one_zsmul _).symm +end + +@[simp] lemma to_Ico_div_add_left (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_div a hb (b + x) = to_Ico_div a hb x - 1 := +by rw [add_comm, to_Ico_div_add_right] + +@[simp] lemma to_Ioc_div_add_left (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_div a hb (b + x) = to_Ioc_div a hb x - 1 := +by rw [add_comm, to_Ioc_div_add_right] + +@[simp] lemma to_Ico_div_sub (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_div a hb (x - b) = to_Ico_div a hb x + 1 := +begin + convert to_Ico_div_sub_zsmul a hb x 1, + exact (one_zsmul _).symm +end + +@[simp] lemma to_Ioc_div_sub (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_div a hb (x - b) = to_Ioc_div a hb x + 1 := +begin + convert to_Ioc_div_sub_zsmul a hb x 1, + exact (one_zsmul _).symm +end + +@[simp] lemma to_Ico_mod_add_zsmul (a : α) {b : α} (hb : 0 < b) (x : α) (m : ℤ) : + to_Ico_mod a hb (x + m • b) = to_Ico_mod a hb x := +begin + rw [to_Ico_mod, to_Ico_div_add_zsmul, to_Ico_mod, sub_smul], + abel +end + +@[simp] lemma to_Ioc_mod_add_zsmul (a : α) {b : α} (hb : 0 < b) (x : α) (m : ℤ) : + to_Ioc_mod a hb (x + m • b) = to_Ioc_mod a hb x := +begin + rw [to_Ioc_mod, to_Ioc_div_add_zsmul, to_Ioc_mod, sub_smul], + abel +end + +@[simp] lemma to_Ico_mod_zsmul_add (a : α) {b : α} (hb : 0 < b) (x : α) (m : ℤ) : + to_Ico_mod a hb (m • b + x) = to_Ico_mod a hb x := +by rw [add_comm, to_Ico_mod_add_zsmul] + +@[simp] lemma to_Ioc_mod_zsmul_add (a : α) {b : α} (hb : 0 < b) (x : α) (m : ℤ) : + to_Ioc_mod a hb (m • b + x) = to_Ioc_mod a hb x := +by rw [add_comm, to_Ioc_mod_add_zsmul] + +@[simp] lemma to_Ico_mod_sub_zsmul (a : α) {b : α} (hb : 0 < b) (x : α) (m : ℤ) : + to_Ico_mod a hb (x - m • b) = to_Ico_mod a hb x := +by rw [sub_eq_add_neg, ←neg_smul, to_Ico_mod_add_zsmul] + +@[simp] lemma to_Ioc_mod_sub_zsmul (a : α) {b : α} (hb : 0 < b) (x : α) (m : ℤ) : + to_Ioc_mod a hb (x - m • b) = to_Ioc_mod a hb x := +by rw [sub_eq_add_neg, ←neg_smul, to_Ioc_mod_add_zsmul] + +@[simp] lemma to_Ico_mod_add_right (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_mod a hb (x + b) = to_Ico_mod a hb x := +begin + convert to_Ico_mod_add_zsmul a hb x 1, + exact (one_zsmul _).symm +end + +@[simp] lemma to_Ioc_mod_add_right (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_mod a hb (x + b) = to_Ioc_mod a hb x := +begin + convert to_Ioc_mod_add_zsmul a hb x 1, + exact (one_zsmul _).symm +end + +@[simp] lemma to_Ico_mod_add_left (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_mod a hb (b + x) = to_Ico_mod a hb x := +by rw [add_comm, to_Ico_mod_add_right] + +@[simp] lemma to_Ioc_mod_add_left (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_mod a hb (b + x) = to_Ioc_mod a hb x := +by rw [add_comm, to_Ioc_mod_add_right] + +@[simp] lemma to_Ico_mod_sub (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_mod a hb (x - b) = to_Ico_mod a hb x := +begin + convert to_Ico_mod_sub_zsmul a hb x 1, + exact (one_zsmul _).symm +end + +@[simp] lemma to_Ioc_mod_sub (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_mod a hb (x - b) = to_Ioc_mod a hb x := +begin + convert to_Ioc_mod_sub_zsmul a hb x 1, + exact (one_zsmul _).symm +end + +lemma to_Ico_mod_eq_to_Ico_mod (a : α) {b x y : α} (hb : 0 < b) : + to_Ico_mod a hb x = to_Ico_mod a hb y ↔ ∃ z : ℤ, y - x = z • b := +begin + refine ⟨λ h, ⟨to_Ico_div a hb x - to_Ico_div a hb y, _⟩, λ h, _⟩, + { conv_lhs { rw [←to_Ico_mod_sub_to_Ico_div_zsmul a hb x, + ←to_Ico_mod_sub_to_Ico_div_zsmul a hb y] }, + rw [h, sub_smul], + abel }, + { rcases h with ⟨z, hz⟩, + rw sub_eq_iff_eq_add at hz, + rw [hz, to_Ico_mod_zsmul_add] } +end + +lemma to_Ioc_mod_eq_to_Ioc_mod (a : α) {b x y : α} (hb : 0 < b) : + to_Ioc_mod a hb x = to_Ioc_mod a hb y ↔ ∃ z : ℤ, y - x = z • b := +begin + refine ⟨λ h, ⟨to_Ioc_div a hb x - to_Ioc_div a hb y, _⟩, λ h, _⟩, + { conv_lhs { rw [←to_Ioc_mod_sub_to_Ioc_div_zsmul a hb x, + ←to_Ioc_mod_sub_to_Ioc_div_zsmul a hb y] }, + rw [h, sub_smul], + abel }, + { rcases h with ⟨z, hz⟩, + rw sub_eq_iff_eq_add at hz, + rw [hz, to_Ioc_mod_zsmul_add] } +end + +lemma to_Ico_mod_eq_self {a b x : α} (hb : 0 < b) : to_Ico_mod a hb x = x ↔ a ≤ x ∧ x < a + b := +begin + rw to_Ico_mod_eq_iff, + refine ⟨λ h, ⟨h.1, h.2.1⟩, λ h, ⟨h.1, h.2, 0, _⟩⟩, + simp +end + +lemma to_Ioc_mod_eq_self {a b x : α} (hb : 0 < b) : to_Ioc_mod a hb x = x ↔ a < x ∧ x ≤ a + b := +begin + rw to_Ioc_mod_eq_iff, + refine ⟨λ h, ⟨h.1, h.2.1⟩, λ h, ⟨h.1, h.2, 0, _⟩⟩, + simp +end + +@[simp] lemma to_Ico_mod_to_Ico_mod (a₁ a₂ : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_mod a₁ hb (to_Ico_mod a₂ hb x) = to_Ico_mod a₁ hb x := +begin + rw to_Ico_mod_eq_to_Ico_mod, + exact ⟨-to_Ico_div a₂ hb x, self_sub_to_Ico_mod a₂ hb x⟩ +end + +@[simp] lemma to_Ico_mod_to_Ioc_mod (a₁ a₂ : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_mod a₁ hb (to_Ioc_mod a₂ hb x) = to_Ico_mod a₁ hb x := +begin + rw to_Ico_mod_eq_to_Ico_mod, + exact ⟨-to_Ioc_div a₂ hb x, self_sub_to_Ioc_mod a₂ hb x⟩ +end + +@[simp] lemma to_Ioc_mod_to_Ioc_mod (a₁ a₂ : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_mod a₁ hb (to_Ioc_mod a₂ hb x) = to_Ioc_mod a₁ hb x := +begin + rw to_Ioc_mod_eq_to_Ioc_mod, + exact ⟨-to_Ioc_div a₂ hb x, self_sub_to_Ioc_mod a₂ hb x⟩ +end + +@[simp] lemma to_Ioc_mod_to_Ico_mod (a₁ a₂ : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_mod a₁ hb (to_Ico_mod a₂ hb x) = to_Ioc_mod a₁ hb x := +begin + rw to_Ioc_mod_eq_to_Ioc_mod, + exact ⟨-to_Ico_div a₂ hb x, self_sub_to_Ico_mod a₂ hb x⟩ +end + +lemma to_Ico_mod_periodic (a : α) {b : α} (hb : 0 < b) : function.periodic (to_Ico_mod a hb) b := +to_Ico_mod_add_right a hb + +lemma to_Ioc_mod_periodic (a : α) {b : α} (hb : 0 < b) : function.periodic (to_Ioc_mod a hb) b := +to_Ioc_mod_add_right a hb + +end linear_ordered_add_comm_group + +section linear_ordered_field + +variables {α : Type*} [linear_ordered_field α] [floor_ring α] + +local attribute [instance] floor_ring.archimedean + +lemma to_Ico_div_eq_neg_floor (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_div a hb x = -⌊(x - a) / b⌋ := +begin + refine (eq_to_Ico_div_of_add_zsmul_mem_Ico hb _).symm, + rw [set.mem_Ico, zsmul_eq_mul, int.cast_neg, neg_mul, ←sub_nonneg, add_comm, add_sub_assoc, + add_comm, ←sub_eq_add_neg], + refine ⟨int.sub_floor_div_mul_nonneg _ hb, _⟩, + rw [add_comm a, ←sub_lt_iff_lt_add, add_sub_assoc, add_comm, ←sub_eq_add_neg], + exact int.sub_floor_div_mul_lt _ hb +end + +lemma to_Ioc_div_eq_floor (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_div a hb x = ⌊(a + b - x) / b⌋ := +begin + refine (eq_to_Ioc_div_of_add_zsmul_mem_Ioc hb _).symm, + rw [set.mem_Ioc, zsmul_eq_mul, ←sub_nonneg, sub_add_eq_sub_sub], + refine ⟨_, int.sub_floor_div_mul_nonneg _ hb⟩, + rw [←add_lt_add_iff_right b, add_assoc, add_comm x, ←sub_lt_iff_lt_add, add_comm (_ * _), + ←sub_lt_iff_lt_add], + exact int.sub_floor_div_mul_lt _ hb +end + +lemma to_Ico_div_zero_one (x : α) : to_Ico_div (0 : α) zero_lt_one x = -⌊x⌋ := +by simp [to_Ico_div_eq_neg_floor] + +lemma to_Ico_mod_eq_add_fract_mul (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ico_mod a hb x = a + int.fract ((x - a) / b) * b := +begin + rw [to_Ico_mod, to_Ico_div_eq_neg_floor, int.fract], + field_simp [hb.ne.symm], + ring +end + +lemma to_Ioc_mod_eq_sub_fract_mul (a : α) {b : α} (hb : 0 < b) (x : α) : + to_Ioc_mod a hb x = a + b - int.fract ((a + b - x) / b) * b := +begin + rw [to_Ioc_mod, to_Ioc_div_eq_floor, int.fract], + field_simp [hb.ne.symm], + ring +end + +lemma to_Ico_mod_zero_one (x : α) : to_Ico_mod (0 : α) zero_lt_one x = int.fract x := +by simp [to_Ico_mod_eq_add_fract_mul] + +end linear_ordered_field diff --git a/formal/lean/mathlib/algebra/periodic.lean b/formal/lean/mathlib/algebra/periodic.lean index a3b282c17f46bd6dc82b5e49c896ec4fbf90e610..f7146dd723aaa8e5720fd9dafc1845005cae0390 100644 --- a/formal/lean/mathlib/algebra/periodic.lean +++ b/formal/lean/mathlib/algebra/periodic.lean @@ -177,6 +177,27 @@ lemma periodic.sub_period [add_comm_group α] periodic f (c₁ - c₂) := let h := h2.neg in by simp [*, sub_eq_add_neg, add_comm c₁, ← add_assoc] at * +lemma periodic.const_add [add_semigroup α] (h : periodic f c) (a : α) : + periodic (λ x, f (a + x)) c := +λ x, by simpa [add_assoc] using h (a + x) + +lemma periodic.add_const [add_comm_semigroup α] (h : periodic f c) (a : α) : + periodic (λ x, f (x + a)) c := +λ x, by simpa [add_assoc x c a, add_comm c, ←add_assoc x a c] using h (x + a) + +lemma periodic.const_sub [add_comm_group α] (h : periodic f c) (a : α) : + periodic (λ x, f (a - x)) c := +begin + rw [←neg_neg c], + refine periodic.neg _, + intro x, + simpa [sub_add_eq_sub_sub] using h (a - x) +end + +lemma periodic.sub_const [add_comm_group α] (h : periodic f c) (a : α) : + periodic (λ x, f (x - a)) c := +λ x, by simpa [add_comm x c, add_sub_assoc, add_comm c (x - a)] using h (x - a) + lemma periodic.nsmul [add_monoid α] (h : periodic f c) (n : ℕ) : periodic f (n • c) := @@ -351,14 +372,14 @@ lemma antiperiodic.funext [has_add α] [has_neg β] (λ x, f (x + c)) = -f := funext h -lemma antiperiodic.funext' [has_add α] [add_group β] +lemma antiperiodic.funext' [has_add α] [has_involutive_neg β] (h : antiperiodic f c) : (λ x, -f (x + c)) = f := (eq_neg_iff_eq_neg.mp h.funext).symm /-- If a function is `antiperiodic` with antiperiod `c`, then it is also `periodic` with period `2 * c`. -/ -lemma antiperiodic.periodic [semiring α] [add_group β] +lemma antiperiodic.periodic [semiring α] [has_involutive_neg β] (h : antiperiodic f c) : periodic f (2 * c) := by simp [two_mul, ← add_assoc, h _] @@ -367,27 +388,27 @@ lemma antiperiodic.eq [add_zero_class α] [has_neg β] (h : antiperiodic f c) : f c = -f 0 := by simpa only [zero_add] using h 0 -lemma antiperiodic.nat_even_mul_periodic [semiring α] [add_group β] +lemma antiperiodic.nat_even_mul_periodic [semiring α] [has_involutive_neg β] (h : antiperiodic f c) (n : ℕ) : periodic f (n * (2 * c)) := h.periodic.nat_mul n -lemma antiperiodic.nat_odd_mul_antiperiodic [semiring α] [add_group β] +lemma antiperiodic.nat_odd_mul_antiperiodic [semiring α] [has_involutive_neg β] (h : antiperiodic f c) (n : ℕ) : antiperiodic f (n * (2 * c) + c) := λ x, by rw [← add_assoc, h, h.periodic.nat_mul] -lemma antiperiodic.int_even_mul_periodic [ring α] [add_group β] +lemma antiperiodic.int_even_mul_periodic [ring α] [has_involutive_neg β] (h : antiperiodic f c) (n : ℤ) : periodic f (n * (2 * c)) := h.periodic.int_mul n -lemma antiperiodic.int_odd_mul_antiperiodic [ring α] [add_group β] +lemma antiperiodic.int_odd_mul_antiperiodic [ring α] [has_involutive_neg β] (h : antiperiodic f c) (n : ℤ) : antiperiodic f (n * (2 * c) + c) := λ x, by rw [← add_assoc, h, h.periodic.int_mul] -lemma antiperiodic.nat_mul_eq_of_eq_zero [comm_semiring α] [add_group β] +lemma antiperiodic.nat_mul_eq_of_eq_zero [comm_semiring α] [subtraction_monoid β] (h : antiperiodic f c) (hi : f 0 = 0) (n : ℕ) : f (n * c) = 0 := begin @@ -397,7 +418,7 @@ begin { simpa [add_mul, hk, hi] using (h.nat_odd_mul_antiperiodic k).eq }, end -lemma antiperiodic.int_mul_eq_of_eq_zero [comm_ring α] [add_group β] +lemma antiperiodic.int_mul_eq_of_eq_zero [comm_ring α] [subtraction_monoid β] (h : antiperiodic f c) (hi : f 0 = 0) (n : ℤ) : f (n * c) = 0 := begin @@ -407,26 +428,48 @@ begin { simpa [add_mul, hk, hi] using (h.int_odd_mul_antiperiodic k).eq }, end -lemma antiperiodic.sub_eq [add_group α] [add_group β] +lemma antiperiodic.sub_eq [add_group α] [has_involutive_neg β] (h : antiperiodic f c) (x : α) : f (x - c) = -f x := by simp only [eq_neg_iff_eq_neg.mp (h (x - c)), sub_add_cancel] -lemma antiperiodic.sub_eq' [add_comm_group α] [add_group β] +lemma antiperiodic.sub_eq' [add_comm_group α] [has_neg β] (h : antiperiodic f c) : f (c - x) = -f (-x) := by simpa only [sub_eq_neg_add] using h (-x) -lemma antiperiodic.neg [add_group α] [add_group β] +lemma antiperiodic.neg [add_group α] [has_involutive_neg β] (h : antiperiodic f c) : antiperiodic f (-c) := by simpa only [sub_eq_add_neg, antiperiodic] using h.sub_eq -lemma antiperiodic.neg_eq [add_group α] [add_group β] +lemma antiperiodic.neg_eq [add_group α] [has_involutive_neg β] (h : antiperiodic f c) : f (-c) = -f 0 := by simpa only [zero_add] using h.neg 0 +lemma antiperiodic.const_add [add_semigroup α] [has_neg β] (h : antiperiodic f c) (a : α) : + antiperiodic (λ x, f (a + x)) c := +λ x, by simpa [add_assoc] using h (a + x) + +lemma antiperiodic.add_const [add_comm_semigroup α] [has_neg β] (h : antiperiodic f c) (a : α) : + antiperiodic (λ x, f (x + a)) c := +λ x, by simpa [add_assoc x c a, add_comm c, ←add_assoc x a c] using h (x + a) + +lemma antiperiodic.const_sub [add_comm_group α] [has_involutive_neg β] (h : antiperiodic f c) + (a : α) : + antiperiodic (λ x, f (a - x)) c := +begin + rw [←neg_neg c], + refine antiperiodic.neg _, + intro x, + simpa [sub_add_eq_sub_sub] using h (a - x) +end + +lemma antiperiodic.sub_const [add_comm_group α] [has_neg β] (h : antiperiodic f c) (a : α) : + antiperiodic (λ x, f (x - a)) c := +λ x, by simpa [add_comm x c, add_sub_assoc, add_comm c (x - a)] using h (x - a) + lemma antiperiodic.smul [has_add α] [monoid γ] [add_group β] [distrib_mul_action γ β] (h : antiperiodic f c) (a : γ) : antiperiodic (a • f) c := @@ -482,42 +525,42 @@ lemma antiperiodic.div_inv [division_ring α] [has_neg β] antiperiodic (λ x, f (x / a)) (c * a) := by simpa only [div_eq_mul_inv] using h.mul_const_inv ha -lemma antiperiodic.add [add_group α] [add_group β] +lemma antiperiodic.add [add_group α] [has_involutive_neg β] (h1 : antiperiodic f c₁) (h2 : antiperiodic f c₂) : periodic f (c₁ + c₂) := by simp [*, ← add_assoc] at * -lemma antiperiodic.sub [add_comm_group α] [add_group β] +lemma antiperiodic.sub [add_comm_group α] [has_involutive_neg β] (h1 : antiperiodic f c₁) (h2 : antiperiodic f c₂) : periodic f (c₁ - c₂) := let h := h2.neg in by simp [*, sub_eq_add_neg, add_comm c₁, ← add_assoc] at * -lemma periodic.add_antiperiod [add_group α] [add_group β] +lemma periodic.add_antiperiod [add_group α] [has_neg β] (h1 : periodic f c₁) (h2 : antiperiodic f c₂) : antiperiodic f (c₁ + c₂) := by simp [*, ← add_assoc] at * -lemma periodic.sub_antiperiod [add_comm_group α] [add_group β] +lemma periodic.sub_antiperiod [add_comm_group α] [has_involutive_neg β] (h1 : periodic f c₁) (h2 : antiperiodic f c₂) : antiperiodic f (c₁ - c₂) := let h := h2.neg in by simp [*, sub_eq_add_neg, add_comm c₁, ← add_assoc] at * -lemma periodic.add_antiperiod_eq [add_group α] [add_group β] +lemma periodic.add_antiperiod_eq [add_group α] [has_neg β] (h1 : periodic f c₁) (h2 : antiperiodic f c₂) : f (c₁ + c₂) = -f 0 := (h1.add_antiperiod h2).eq -lemma periodic.sub_antiperiod_eq [add_comm_group α] [add_group β] +lemma periodic.sub_antiperiod_eq [add_comm_group α] [has_involutive_neg β] (h1 : periodic f c₁) (h2 : antiperiodic f c₂) : f (c₁ - c₂) = -f 0 := (h1.sub_antiperiod h2).eq -lemma antiperiodic.mul [has_add α] [ring β] +lemma antiperiodic.mul [has_add α] [has_mul β] [has_distrib_neg β] (hf : antiperiodic f c) (hg : antiperiodic g c) : periodic (f * g) c := by simp * at * -lemma antiperiodic.div [has_add α] [division_ring β] +lemma antiperiodic.div [has_add α] [division_monoid β] [has_distrib_neg β] (hf : antiperiodic f c) (hg : antiperiodic g c) : periodic (f / g) c := by simp [*, neg_div_neg_eq] at * diff --git a/formal/lean/mathlib/algebra/quandle.lean b/formal/lean/mathlib/algebra/quandle.lean index 866d9c0752fe89d7d449250e0af588251cfabc0d..75534effef233c6cdc2915372c167ade1578fe3f 100644 --- a/formal/lean/mathlib/algebra/quandle.lean +++ b/formal/lean/mathlib/algebra/quandle.lean @@ -298,7 +298,7 @@ instance opposite_quandle : quandle Qᵐᵒᵖ := The conjugation quandle of a group. Each element of the group acts by the corresponding inner automorphism. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def conj (G : Type*) := G instance conj.quandle (G : Type*) [group G] : quandle (conj G) := @@ -338,7 +338,7 @@ The dihedral quandle. This is the conjugation quandle of the dihedral group rest Used for Fox n-colorings of knots. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def dihedral (n : ℕ) := zmod n /-- diff --git a/formal/lean/mathlib/algebra/quotient.lean b/formal/lean/mathlib/algebra/quotient.lean index 82c1ccb8eae7d8bee150d5d46cb8d4a819ee682b..0f31cb354613ccf87d1a336f6efeabc54285c1b3 100644 --- a/formal/lean/mathlib/algebra/quotient.lean +++ b/formal/lean/mathlib/algebra/quotient.lean @@ -21,7 +21,7 @@ The actual quotient structures are defined in the following files: The following notation is introduced: -* `G ⧸ H` stands for the quotient of the type `G` by some term `H` +* `G ⧸ H` stands for the quotient of the type `G` by some term `H` (for example, `H` can be a normal subgroup of `G`). To implement this notation for other quotients, you should provide a `has_quotient` instance. Note that since `G` can usually be inferred from `H`, `_ ⧸ H` can also be used, @@ -49,7 +49,7 @@ class has_quotient (A : out_param $ Type u) (B : Type v) := This differs from `has_quotient.quotient'` in that the `A` argument is explicit, which is necessary to make Lean show the notation in the goal state. -/ -@[reducible, nolint has_inhabited_instance] -- Will be provided by e.g. `ideal.quotient.inhabited` +@[reducible, nolint has_nonempty_instance] -- Will be provided by e.g. `ideal.quotient.inhabited` def has_quotient.quotient (A : out_param $ Type u) {B : Type v} [has_quotient A B] (b : B) : Type (max u v) := has_quotient.quotient' b diff --git a/formal/lean/mathlib/algebra/ring/basic.lean b/formal/lean/mathlib/algebra/ring/basic.lean index 11627f0f70a0bd5280c7c061876fd5c5a53851e4..abce6dffdc9bbbc85e0a99700f03b7d4ca31801f 100644 --- a/formal/lean/mathlib/algebra/ring/basic.lean +++ b/formal/lean/mathlib/algebra/ring/basic.lean @@ -838,6 +838,32 @@ instance : has_neg αˣ := ⟨λu, ⟨-↑u, -↑u⁻¹, by simp, by simp⟩ ⟩ instance : has_distrib_neg αˣ := units.ext.has_distrib_neg _ units.coe_neg units.coe_mul +@[field_simps] lemma neg_divp (a : α) (u : αˣ) : -(a /ₚ u) = (-a) /ₚ u := +by simp only [divp, neg_mul] + +@[field_simps] lemma divp_add_divp_same (a b : α) (u : αˣ) : + a /ₚ u + b /ₚ u = (a + b) /ₚ u := +by simp only [divp, add_mul] + +@[field_simps] lemma divp_sub_divp_same (a b : α) (u : αˣ) : + a /ₚ u - b /ₚ u = (a - b) /ₚ u := +by rw [sub_eq_add_neg, sub_eq_add_neg, neg_divp, divp_add_divp_same] + +@[field_simps] lemma add_divp (a b : α) (u : αˣ) : a + b /ₚ u = (a * u + b) /ₚ u := +by simp only [divp, add_mul, units.mul_inv_cancel_right] + +@[field_simps] lemma sub_divp (a b : α) (u : αˣ) : a - b /ₚ u = (a * u - b) /ₚ u := +by simp only [divp, sub_mul, units.mul_inv_cancel_right] + +@[field_simps] lemma divp_add (a b : α) (u : αˣ) : a /ₚ u + b = (a + b * u) /ₚ u := +by simp only [divp, add_mul, units.mul_inv_cancel_right] + +@[field_simps] lemma divp_sub (a b : α) (u : αˣ) : a /ₚ u - b = (a - b * u) /ₚ u := +begin + simp only [divp, sub_mul, sub_right_inj], + assoc_rw [units.mul_inv, mul_one], +end + end units lemma is_unit.neg [ring α] {a : α} : is_unit a → is_unit (-a) @@ -1228,12 +1254,28 @@ lemma mul_self_eq_one_iff [non_assoc_ring R] [no_zero_divisors R] {a : R} : a * a = 1 ↔ a = 1 ∨ a = -1 := by rw [←(commute.one_right a).mul_self_eq_mul_self_iff, mul_one] +namespace units + +@[field_simps] lemma divp_add_divp [comm_ring α] (a b : α) (u₁ u₂ : αˣ) : +a /ₚ u₁ + b /ₚ u₂ = (a * u₂ + u₁ * b) /ₚ (u₁ * u₂) := +begin + simp only [divp, add_mul, mul_inv_rev, coe_mul], + rw [mul_comm (↑u₁ * b), mul_comm b], + assoc_rw [mul_inv, mul_inv, mul_one, mul_one], +end + +@[field_simps] lemma divp_sub_divp [comm_ring α] (a b : α) (u₁ u₂ : αˣ) : + (a /ₚ u₁) - (b /ₚ u₂) = ((a * u₂) - (u₁ * b)) /ₚ (u₁ * u₂) := +by simp_rw [sub_eq_add_neg, neg_divp, divp_add_divp, mul_neg] + /-- In the unit group of an integral domain, a unit is its own inverse iff the unit is one or one's additive inverse. -/ -lemma units.inv_eq_self_iff [ring R] [no_zero_divisors R] (u : Rˣ) : u⁻¹ = u ↔ u = 1 ∨ u = -1 := +lemma inv_eq_self_iff [ring R] [no_zero_divisors R] (u : Rˣ) : u⁻¹ = u ↔ u = 1 ∨ u = -1 := begin rw inv_eq_iff_mul_eq_one, - simp only [units.ext_iff], + simp only [ext_iff], push_cast, exact mul_self_eq_one_iff end + +end units diff --git a/formal/lean/mathlib/algebra/ring/equiv.lean b/formal/lean/mathlib/algebra/ring/equiv.lean index 968535cee52faf25d341e93aee91a3a9ba9268ba..146dcbed6c57e21d79767c819ff44bbfe057b6f2 100644 --- a/formal/lean/mathlib/algebra/ring/equiv.lean +++ b/formal/lean/mathlib/algebra/ring/equiv.lean @@ -210,6 +210,10 @@ symm_bijective.injective $ ext $ λ x, rfl @[simp] lemma trans_apply (e₁ : R ≃+* S) (e₂ : S ≃+* S') (a : R) : e₁.trans e₂ a = e₂ (e₁ a) := rfl +@[simp] +lemma symm_trans_apply (e₁ : R ≃+* S) (e₂ : S ≃+* S') (a : S') : + (e₁.trans e₂).symm a = e₁.symm (e₂.symm a) := rfl + protected lemma bijective (e : R ≃+* S) : function.bijective e := equiv_like.bijective e protected lemma injective (e : R ≃+* S) : function.injective e := equiv_like.injective e protected lemma surjective (e : R ≃+* S) : function.surjective e := equiv_like.surjective e diff --git a/formal/lean/mathlib/algebra/star/chsh.lean b/formal/lean/mathlib/algebra/star/chsh.lean index dc060bed3444081888a4a957236353388b7e77df..e6a99191a1fd3026e0044470c9d776dcbe76d4fb 100644 --- a/formal/lean/mathlib/algebra/star/chsh.lean +++ b/formal/lean/mathlib/algebra/star/chsh.lean @@ -79,7 +79,7 @@ the `Aᵢ` commute with the `Bⱼ`. The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which are spacelike separated from another pair `B₀` and `B₁` of boolean observables. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure is_CHSH_tuple {R} [monoid R] [star_semigroup R] (A₀ A₁ B₀ B₁ : R) := (A₀_inv : A₀^2 = 1) (A₁_inv : A₁^2 = 1) (B₀_inv : B₀^2 = 1) (B₁_inv : B₁^2 = 1) (A₀_sa : star A₀ = A₀) (A₁_sa : star A₁ = A₁) (B₀_sa : star B₀ = B₀) (B₁_sa : star B₁ = B₁) diff --git a/formal/lean/mathlib/algebraic_geometry/AffineScheme.lean b/formal/lean/mathlib/algebraic_geometry/AffineScheme.lean index 4444625b845fe4f0f8c443e931f6160bd9bd3df9..679de7d1bb032fea85e07fe4f7f900cc2804ba47 100644 --- a/formal/lean/mathlib/algebraic_geometry/AffineScheme.lean +++ b/formal/lean/mathlib/algebraic_geometry/AffineScheme.lean @@ -41,7 +41,8 @@ namespace algebraic_geometry open Spec (structure_sheaf) /-- The category of affine schemes -/ -def AffineScheme := Scheme.Spec.ess_image +@[derive category, nolint has_nonempty_instance] +def AffineScheme := Scheme.Spec.ess_image_subcategory /-- A Scheme is affine if the canonical map `X ⟶ Spec Γ(X)` is an isomorphism. -/ class is_affine (X : Scheme) : Prop := @@ -54,18 +55,23 @@ def Scheme.iso_Spec (X : Scheme) [is_affine X] : X ≅ Scheme.Spec.obj (op $ Scheme.Γ.obj $ op X) := as_iso (Γ_Spec.adjunction.unit.app X) -lemma mem_AffineScheme (X : Scheme) : X ∈ AffineScheme ↔ is_affine X := +/-- Construct an affine scheme from a scheme and the information that it is affine. -/ +@[simps] +def AffineScheme.mk (X : Scheme) (h : is_affine X) : AffineScheme := +⟨X, @@mem_ess_image_of_unit_is_iso _ _ _ _ h.1⟩ + +lemma mem_Spec_ess_image (X : Scheme) : X ∈ Scheme.Spec.ess_image ↔ is_affine X := ⟨λ h, ⟨functor.ess_image.unit_is_iso h⟩, λ h, @@mem_ess_image_of_unit_is_iso _ _ _ X h.1⟩ -instance is_affine_AffineScheme (X : AffineScheme.{u}) : is_affine (X : Scheme.{u}) := -(mem_AffineScheme _).mp X.prop +instance is_affine_AffineScheme (X : AffineScheme.{u}) : is_affine X.obj := +⟨functor.ess_image.unit_is_iso X.property⟩ instance Spec_is_affine (R : CommRingᵒᵖ) : is_affine (Scheme.Spec.obj R) := -(mem_AffineScheme _).mp (Scheme.Spec.obj_mem_ess_image R) +algebraic_geometry.is_affine_AffineScheme ⟨_, Scheme.Spec.obj_mem_ess_image R⟩ lemma is_affine_of_iso {X Y : Scheme} (f : X ⟶ Y) [is_iso f] [h : is_affine Y] : is_affine X := -by { rw [← mem_AffineScheme] at h ⊢, exact functor.ess_image.of_iso (as_iso f).symm h } +by { rw [← mem_Spec_ess_image] at h ⊢, exact functor.ess_image.of_iso (as_iso f).symm h } namespace AffineScheme @@ -129,6 +135,10 @@ begin exact set.range_id.symm end +instance Scheme.affine_cover_is_affine (X : Scheme) (i : X.affine_cover.J) : + is_affine (X.affine_cover.obj i) := +algebraic_geometry.Spec_is_affine _ + instance Scheme.affine_basis_cover_is_affine (X : Scheme) (i : X.affine_basis_cover.J) : is_affine (X.affine_basis_cover.obj i) := algebraic_geometry.Spec_is_affine _ @@ -185,6 +195,17 @@ begin exact set.image_univ end +lemma is_affine_open.image_is_open_immersion {X Y : Scheme} {U : opens X.carrier} + (hU : is_affine_open U) + (f : X ⟶ Y) [H : is_open_immersion f] : is_affine_open (H.open_functor.obj U) := +begin + haveI : is_affine _ := hU, + convert range_is_affine_open_of_open_immersion (X.of_restrict U.open_embedding ≫ f), + ext1, + change f.1.base '' U.1 = set.range (f.1.base ∘ coe), + rw [set.range_comp, subtype.range_coe], +end + instance Scheme.quasi_compact_of_affine (X : Scheme) [is_affine X] : compact_space X.carrier := ⟨(top_is_affine_open X).is_compact⟩ @@ -514,7 +535,8 @@ begin end /-- The basic open set of a section `f` on an an affine open as an `X.affine_opens`. -/ -abbreviation Scheme.affine_basic_open (X : Scheme) {U : X.affine_opens} +@[simps] +def Scheme.affine_basic_open (X : Scheme) {U : X.affine_opens} (f : X.presheaf.obj $ op U) : X.affine_opens := ⟨X.basic_open f, U.prop.basic_open_is_affine f⟩ diff --git a/formal/lean/mathlib/algebraic_geometry/EllipticCurve.lean b/formal/lean/mathlib/algebraic_geometry/EllipticCurve.lean index 6bf946c43f7874e7d6a40072a7e34b02b4f1cd62..0b58320030ced89aea6ea753df43c2882bf54d65 100644 --- a/formal/lean/mathlib/algebraic_geometry/EllipticCurve.lean +++ b/formal/lean/mathlib/algebraic_geometry/EllipticCurve.lean @@ -1,11 +1,11 @@ /- Copyright (c) 2021 Kevin Buzzard. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. -Authors: Kevin Buzzard +Authors: Kevin Buzzard, David Kurniadi Angdinata -/ -import data.rat.defs -import tactic.ring +import algebra.cubic_discriminant +import tactic.linear_combination /-! # The category of elliptic curves (over a field or a PID) @@ -18,12 +18,12 @@ in many cases, and also good for computation. Let `S` be a scheme. The actual category of elliptic curves over `S` is a large category, whose objects are schemes `E` equipped with a map `E → S`, a section `S → E`, and some axioms (the map is smooth and proper and the fibres are geometrically connected group varieties -of dimension 1). In the special case where `S` is `Spec R` for some commutative ring `R` +of dimension one). In the special case where `S` is `Spec R` for some commutative ring `R` whose Picard group is trivial (this includes all fields, all principal ideal domains, and many other commutative rings) then it can be shown (using rather a lot of algebro-geometric machinery) that every elliptic curve is, up to isomorphism, a projective plane cubic defined by -the equation `y^2+a₁xy+a₃y=x^3+a₂x^2+a₄x+a₆`, with `aᵢ : R`, and such that the discriminant -of the aᵢ is a unit in `R`. +the equation `y² + a₁xy + a₃y = x³ + a₂x² + a₄x + a₆`, with `aᵢ : R`, and such that +the discriminant of the aᵢ is a unit in `R`. Some more details of the construction can be found on pages 66-69 of [N. Katz and B. Mazur, *Arithmetic moduli of elliptic curves*][katz_mazur] or pages @@ -33,80 +33,180 @@ Some more details of the construction can be found on pages 66-69 of The definition in this file makes sense for all commutative rings `R`, but it only gives a type which can be beefed up to a category which is equivalent to the category of elliptic -curves over `Spec R` in the case that `R` has trivial Picard group or, slightly more generally, -when the 12-torsion of Pic(R) is trivial. The issue is that for a general ring R, there -might be elliptic curves over Spec(R) in the sense of algebraic geometry which are not +curves over `Spec R` in the case that `R` has trivial Picard group `Pic R` or, slightly more +generally, when its `12`-torsion is trivial. The issue is that for a general ring `R`, there +might be elliptic curves over `Spec R` in the sense of algebraic geometry which are not globally defined by a cubic equation valid over the entire base. ## TODO -Define the R-points (or even A-points if A is an R-algebra). Care will be needed -at infinity if R is not a field. Define the group law on the R-points. (hard) prove associativity. - +Define the `R`-points (or even `A`-points if `A` is an `R`-algebra). Care will be needed at infinity +if `R` is not a field. Define the group law on the `R`-points. Prove associativity (hard). -/ -/-- The discriminant of the plane cubic `Y^2+a1*X*Y+a3*Y=X^3+a2*X^2+a4*X+a6`. If `R` is a field -then this polynomial vanishes iff the cubic curve cut out by this equation is singular. -/ -def EllipticCurve.disc_aux {R : Type*} [comm_ring R] (a1 a2 a3 a4 a6 : R) : R := -let b2 := a1^2 + 4*a2, - b4 := 2*a4 + a1*a3, - b6 := a3^2 + 4*a6, - b8 := b2*a6 - a1*a3*a4 + a2*a3^2 - a4^2 in --b2^2*b8 - 8*b4^3 - 27*b6^2 + 9*b2*b4*b6 - -theorem EllipticCurve.disc_aux_def {R : Type*} [comm_ring R] (a1 a2 a3 a4 a6 : R) : - EllipticCurve.disc_aux a1 a2 a3 a4 a6 = - -27*a3^4 + a1^5*a3*a4 + 16*a2^2*a4^2 - 64*a4^3 - a1^6*a6 - 216*a3^2*a6 - 432*a6^2 - - 16*a2^3*(a3^2 + 4*a6) + 72*a2*a4*(a3^2 + 4*a6) + a1^3*a3*(a3^2 + 8*a2*a4 + 36*a6) + - 4*a1*a3*(4*a2^2*a4 - 24*a4^2 + 9*a2*(a3^2 + 4*a6)) + - a1^2*(-30*a3^2*a4 + 8*a2*a4^2 + 72*a4*a6 - 8*a2^2*(a3^2 + 6*a6)) + - a1^4*(a4^2 - a2*(a3^2 + 12*a6)) := -by dsimp only [EllipticCurve.disc_aux]; ring - --- If Pic(R)[12]=0 then this definition is mathematically correct +universes u v + +/-- The discriminant of an elliptic curve given by the long Weierstrass equation + `y² + a₁xy + a₃y = x³ + a₂x² + a₄x + a₆`. If `R` is a field, then this polynomial vanishes iff the + cubic curve cut out by this equation is singular. Sometimes only defined up to sign in the + literature; we choose the sign used by the LMFDB. For more discussion, see + [the LMFDB page on discriminants](https://www.lmfdb.org/knowledge/show/ec.discriminant). -/ +@[simp] def EllipticCurve.Δ_aux {R : Type u} [comm_ring R] (a₁ a₂ a₃ a₄ a₆ : R) : R := +let b₂ : R := a₁ ^ 2 + 4 * a₂, + b₄ : R := 2 * a₄ + a₁ * a₃, + b₆ : R := a₃ ^ 2 + 4 * a₆, + b₈ : R := a₁ ^ 2 * a₆ + 4 * a₂ * a₆ - a₁ * a₃ * a₄ + a₂ * a₃ ^ 2 - a₄ ^ 2 +in -b₂ ^ 2 * b₈ - 8 * b₄ ^ 3 - 27 * b₆ ^ 2 + 9 * b₂ * b₄ * b₆ + /-- The category of elliptic curves over `R` (note that this definition is only mathematically -correct for certain rings, for example if `R` is a field or a PID). -/ -structure EllipticCurve (R : Type*) [comm_ring R] := -(a1 a2 a3 a4 a6 : R) -(disc_unit : Rˣ) -(disc_unit_eq : (disc_unit : R) = EllipticCurve.disc_aux a1 a2 a3 a4 a6) + correct for certain rings `R` with `Pic(R)[12] = 0`, for example if `R` is a field or a PID). -/ +structure EllipticCurve (R : Type u) [comm_ring R] := +(a₁ a₂ a₃ a₄ a₆ : R) (Δ : Rˣ) (Δ_eq : ↑Δ = EllipticCurve.Δ_aux a₁ a₂ a₃ a₄ a₆) namespace EllipticCurve -instance : inhabited (EllipticCurve ℚ) := ⟨⟨0,0,1,-1,0, units.mk0 37 (by norm_num), - by simp only [units.coe_mk, disc_aux_def]; norm_num⟩⟩ +instance : inhabited (EllipticCurve ℚ) := +⟨⟨0, 0, 1, -1, 0, ⟨37, 37⁻¹, by norm_num1, by norm_num1⟩, show (37 : ℚ) = _ + _, by norm_num1 ⟩⟩ + +variables {R : Type u} [comm_ring R] (E : EllipticCurve R) + +/-! ### Standard quantities -/ + +section quantity + +/-- The `b₂` coefficient of an elliptic curve. -/ +@[simp] def b₂ : R := E.a₁ ^ 2 + 4 * E.a₂ + +/-- The `b₄` coefficient of an elliptic curve. -/ +@[simp] def b₄ : R := 2 * E.a₄ + E.a₁ * E.a₃ + +/-- The `b₆` coefficient of an elliptic curve. -/ +@[simp] def b₆ : R := E.a₃ ^ 2 + 4 * E.a₆ + +/-- The `b₈` coefficient of an elliptic curve. -/ +@[simp] def b₈ : R := +E.a₁ ^ 2 * E.a₆ + 4 * E.a₂ * E.a₆ - E.a₁ * E.a₃ * E.a₄ + E.a₂ * E.a₃ ^ 2 - E.a₄ ^ 2 + +lemma b_relation : 4 * E.b₈ = E.b₂ * E.b₆ - E.b₄ ^ 2 := by { simp, ring1 } + +/-- The `c₄` coefficient of an elliptic curve. -/ +@[simp] def c₄ : R := E.b₂ ^ 2 - 24 * E.b₄ + +/-- The `c₆` coefficient of an elliptic curve. -/ +@[simp] def c₆ : R := -E.b₂ ^ 3 + 36 * E.b₂ * E.b₄ - 216 * E.b₆ + +@[simp] lemma coe_Δ : + ↑E.Δ = -E.b₂ ^ 2 * E.b₈ - 8 * E.b₄ ^ 3 - 27 * E.b₆ ^ 2 + 9 * E.b₂ * E.b₄ * E.b₆ := +E.Δ_eq + +lemma c_relation : 1728 * ↑E.Δ = E.c₄ ^ 3 - E.c₆ ^ 2 := by { simp, ring1 } + +/-- The j-invariant of an elliptic curve, which is invariant under isomorphisms over `R`. -/ +@[simp] def j : R := ↑E.Δ⁻¹ * E.c₄ ^ 3 + +end quantity + +/-! ### `2`-torsion polynomials -/ + +section torsion_polynomial + +variables (A : Type v) [comm_ring A] [algebra R A] + +/-- The polynomial whose roots over a splitting field of `R` are the `2`-torsion points of the + elliptic curve when `R` is a field of characteristic different from `2`, and whose discriminant + happens to be a multiple of the discriminant of the elliptic curve. -/ +def two_torsion_polynomial : cubic A := +⟨4, algebra_map R A E.b₂, 2 * algebra_map R A E.b₄, algebra_map R A E.b₆⟩ + +lemma two_torsion_polynomial.disc_eq : + (two_torsion_polynomial E A).disc = 16 * algebra_map R A E.Δ := +begin + simp only [two_torsion_polynomial, cubic.disc, coe_Δ, b₂, b₄, b₆, b₈, map_neg, map_add, map_sub, + map_mul, map_pow, map_one, map_bit0, map_bit1], + ring1 +end + +lemma two_torsion_polynomial.disc_ne_zero {K : Type u} [field K] [invertible (2 : K)] + (E : EllipticCurve K) (A : Type v) [comm_ring A] [nontrivial A] [algebra K A] : + (two_torsion_polynomial E A).disc ≠ 0 := +λ hdisc, E.Δ.ne_zero $ mul_left_cancel₀ (pow_ne_zero 4 $ nonzero_of_invertible (2 : K)) $ + (algebra_map K A).injective +begin + simp only [map_mul, map_pow, map_bit0, map_one, map_zero], + linear_combination hdisc - two_torsion_polynomial.disc_eq E A +end + +end torsion_polynomial + +/-! ### Changes of variables -/ + +variables (u : Rˣ) (r s t : R) + +/-- The elliptic curve over `R` induced by an admissible linear change of variables + `(x, y) ↦ (u²x + r, u³y + u²sx + t)` for some `u ∈ Rˣ` and some `r, s, t ∈ R`. + When `R` is a field, any two isomorphic long Weierstrass equations are related by this. -/ +def change_of_variable : EllipticCurve R := +{ a₁ := ↑u⁻¹ * (E.a₁ + 2 * s), + a₂ := ↑u⁻¹ ^ 2 * (E.a₂ - s * E.a₁ + 3 * r - s ^ 2), + a₃ := ↑u⁻¹ ^ 3 * (E.a₃ + r * E.a₁ + 2 * t), + a₄ := ↑u⁻¹ ^ 4 * (E.a₄ - s * E.a₃ + 2 * r * E.a₂ - (t + r * s) * E.a₁ + 3 * r ^ 2 - 2 * s * t), + a₆ := ↑u⁻¹ ^ 6 * (E.a₆ + r * E.a₄ + r ^ 2 * E.a₂ + r ^ 3 - t * E.a₃ - t ^ 2 - r * t * E.a₁), + Δ := u⁻¹ ^ 12 * E.Δ, + Δ_eq := by { simp [-inv_pow], ring1 } } + +namespace change_of_variable + +@[simp] lemma a₁_eq : (E.change_of_variable u r s t).a₁ = ↑u⁻¹ * (E.a₁ + 2 * s) := rfl + +@[simp] lemma a₂_eq : + (E.change_of_variable u r s t).a₂ = ↑u⁻¹ ^ 2 * (E.a₂ - s * E.a₁ + 3 * r - s ^ 2) := +rfl + +@[simp] lemma a₃_eq : (E.change_of_variable u r s t).a₃ = ↑u⁻¹ ^ 3 * (E.a₃ + r * E.a₁ + 2 * t) := +rfl + +@[simp] lemma a₄_eq : + (E.change_of_variable u r s t).a₄ + = ↑u⁻¹ ^ 4 * (E.a₄ - s * E.a₃ + 2 * r * E.a₂ - (t + r * s) * E.a₁ + 3 * r ^ 2 - 2 * s * t) := +rfl + +@[simp] lemma a₆_eq : + (E.change_of_variable u r s t).a₆ + = ↑u⁻¹ ^ 6 * (E.a₆ + r * E.a₄ + r ^ 2 * E.a₂ + r ^ 3 - t * E.a₃ - t ^ 2 - r * t * E.a₁) := +rfl + +@[simp] lemma b₂_eq : (E.change_of_variable u r s t).b₂ = ↑u⁻¹ ^ 2 * (E.b₂ + 12 * r) := +by { simp [change_of_variable], ring1 } + +@[simp] lemma b₄_eq : + (E.change_of_variable u r s t).b₄ = ↑u⁻¹ ^ 4 * (E.b₄ + r * E.b₂ + 6 * r ^ 2) := +by { simp [change_of_variable], ring1 } -variables {R : Type*} [comm_ring R] (E : EllipticCurve R) +@[simp] lemma b₆_eq : + (E.change_of_variable u r s t).b₆ = ↑u⁻¹ ^ 6 * (E.b₆ + 2 * r * E.b₄ + r ^ 2 * E.b₂ + 4 * r ^ 3) := +by { simp [change_of_variable], ring1 } -/-- The `b_2` coefficient of an elliptic curve. -/ -def b2 : R := E.a1^2 + 4*E.a2 -/-- The `b_4` coefficient of an elliptic curve. -/ -def b4 : R := 2*E.a4 + E.a1*E.a3 -/-- The `b_6` coefficient of an elliptic curve. -/ -def b6 : R := E.a3^2 + 4*E.a6 -/-- The `b_8` coefficient of an elliptic curve. -/ -def b8 : R := E.b2*E.a6 - E.a1*E.a3*E.a4 + E.a2*E.a3^2 - E.a4^2 -/-- The `c_4` coefficient of an elliptic curve. -/ -def c4 : R := E.b2^2 - 24*E.b4 +@[simp] lemma b₈_eq : + (E.change_of_variable u r s t).b₈ + = ↑u⁻¹ ^ 8 * (E.b₈ + 3 * r * E.b₆ + 3 * r ^ 2 * E.b₄ + r ^ 3 * E.b₂ + 3 * r ^ 4) := +by { simp [change_of_variable], ring1 } -theorem c4_def : E.c4 = E.a1^4 + 8*E.a1^2*E.a2 + 16*E.a2^2 - 24*E.a1*E.a3 - 48*E.a4 := -by { unfold c4 b2 b4, ring } +@[simp] lemma c₄_eq : (E.change_of_variable u r s t).c₄ = ↑u⁻¹ ^ 4 * E.c₄ := +by { simp [change_of_variable], ring1 } -/-- The discriminant of an elliptic curve. Sometimes only defined up to sign in the literature; - we choose the sign used by the LMFDB. See - [the LMFDB page on discriminants](https://www.lmfdb.org/knowledge/show/ec.discriminant) - for more discussion. -/ -def disc : R := disc_aux E.a1 E.a2 E.a3 E.a4 E.a6 +@[simp] lemma c₆_eq : (E.change_of_variable u r s t).c₆ = ↑u⁻¹ ^ 6 * E.c₆ := +by { simp [change_of_variable], ring1 } -lemma disc_def : E.disc = -E.b2^2*E.b8 - 8*E.b4^3 - 27*E.b6^2 + 9*E.b2*E.b4*E.b6 := rfl +@[simp] lemma Δ_eq : (E.change_of_variable u r s t).Δ = u⁻¹ ^ 12 * E.Δ := rfl -lemma disc_is_unit : is_unit E.disc := +@[simp] lemma j_eq : (E.change_of_variable u r s t).j = E.j := begin - convert units.is_unit E.disc_unit, - exact E.disc_unit_eq.symm + simp only [j, c₄, Δ_eq, inv_pow, mul_inv_rev, inv_inv, units.coe_mul, units.coe_pow, + c₄_eq, b₂, b₄], + have hu : (u * ↑u⁻¹ : R) ^ 12 = 1 := by rw [u.mul_inv, one_pow], + linear_combination ↑E.Δ⁻¹ * ((E.a₁ ^ 2 + 4 * E.a₂) ^ 2 - 24 * (2 * E.a₄ + E.a₁ * E.a₃)) ^ 3 * hu end -/-- The j-invariant of an elliptic curve. -/ -def j := E.c4^3 /ₚ E.disc_unit +end change_of_variable end EllipticCurve diff --git a/formal/lean/mathlib/algebraic_geometry/gluing.lean b/formal/lean/mathlib/algebraic_geometry/gluing.lean index 8e7a89c701c99ad91fcab4883c8d13453b800ec5..9fc37f88a00ca312ca7bf5a4fded05743a34c127 100644 --- a/formal/lean/mathlib/algebraic_geometry/gluing.lean +++ b/formal/lean/mathlib/algebraic_geometry/gluing.lean @@ -76,7 +76,7 @@ such that We can then glue the schemes `U i` together by identifying `V i j` with `V j i`, such that the `U i`'s are open subschemes of the glued space. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure glue_data extends category_theory.glue_data Scheme := (f_open : ∀ i j, is_open_immersion (f i j)) diff --git a/formal/lean/mathlib/algebraic_geometry/locally_ringed_space.lean b/formal/lean/mathlib/algebraic_geometry/locally_ringed_space.lean index 10096613620f6b305153be7d31658bde1e8b7acf..ab03f1d82675c7b633d05f05db28b3722ff30120 100644 --- a/formal/lean/mathlib/algebraic_geometry/locally_ringed_space.lean +++ b/formal/lean/mathlib/algebraic_geometry/locally_ringed_space.lean @@ -31,7 +31,7 @@ such that all the stalks are local rings. A morphism of locally ringed spaces is a morphism of ringed spaces such that the morphisms induced on stalks are local ring homomorphisms. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure LocallyRingedSpace extends SheafedSpace CommRing := (local_ring : ∀ x, local_ring (presheaf.stalk x)) diff --git a/formal/lean/mathlib/algebraic_geometry/locally_ringed_space/has_colimits.lean b/formal/lean/mathlib/algebraic_geometry/locally_ringed_space/has_colimits.lean index 77a30cfb666e914f7bb45788f51f90b7fd10a521..cdb16170c412f49a5bc82332a9cd129667a7f667 100644 --- a/formal/lean/mathlib/algebraic_geometry/locally_ringed_space/has_colimits.lean +++ b/formal/lean/mathlib/algebraic_geometry/locally_ringed_space/has_colimits.lean @@ -291,7 +291,7 @@ end⟩ end has_coequalizer -instance : has_colimits LocallyRingedSpace := colimits_from_coequalizers_and_coproducts +instance : has_colimits LocallyRingedSpace := has_colimits_of_has_coequalizers_and_coproducts noncomputable instance : preserves_colimits LocallyRingedSpace.forget_to_SheafedSpace := diff --git a/formal/lean/mathlib/algebraic_geometry/morphisms/basic.lean b/formal/lean/mathlib/algebraic_geometry/morphisms/basic.lean index b1774464f194ee4bf69fa9c8392a555d3a86f445..27709c63a54c37c1681f2f1cfa50bb6478871c8d 100644 --- a/formal/lean/mathlib/algebraic_geometry/morphisms/basic.lean +++ b/formal/lean/mathlib/algebraic_geometry/morphisms/basic.lean @@ -12,6 +12,47 @@ import category_theory.morphism_property We provide the basic framework for talking about properties of morphisms between Schemes. +A `morphism_property Scheme` is a predicate on morphisms between schemes, and an +`affine_target_morphism_property` is a predicate on morphisms into affine schemes. Given a +`P : affine_target_morphism_property`, we may construct a `morphism_property` called +`target_affine_locally P` that holds for `f : X ⟶ Y` whenever `P` holds for the +restriction of `f` on every affine open subset of `Y`. + +## Main definitions + +- `algebraic_geometry.affine_target_morphism_property.is_local`: We say that `P.is_local` if `P` +satisfies the assumptions of the affine communication lemma +(`algebraic_geometry.of_affine_open_cover`). That is, +1. `P` respects isomorphisms. +2. If `P` holds for `f : X ⟶ Y`, then `P` holds for `f ∣_ Y.basic_open r` for any + global section `r`. +3. If `P` holds for `f ∣_ Y.basic_open r` for all `r` in a spanning set of the global sections, + then `P` holds for `f`. + +- `algebraic_geometry.property_is_local_at_target`: We say that `property_is_local_at_target P` for +`P : morphism_property Scheme` if +1. `P` respects isomorphisms. +2. If `P` holds for `f : X ⟶ Y`, then `P` holds for `f ∣_ U` for any `U`. +3. If `P` holds for `f ∣_ U` for an open cover `U` of `Y`, then `P` holds for `f`. + +## Main results + +- `algebraic_geometry.affine_target_morphism_property.is_local.affine_open_cover_tfae`: + If `P.is_local`, then `target_affine_locally P f` iff there exists an affine cover `{ Uᵢ }` of `Y` + such that `P` holds for `f ∣_ Uᵢ`. +- `algebraic_geometry.affine_target_morphism_property.is_local_of_open_cover_imply`: + If the existance of an affine cover `{ Uᵢ }` of `Y` such that `P` holds for `f ∣_ Uᵢ` implies + `target_affine_locally P f`, then `P.is_local`. +- `algebraic_geometry.affine_target_morphism_property.is_local.affine_target_iff`: + If `Y` is affine and `f : X ⟶ Y`, then `target_affine_locally P f ↔ P f` provided `P.is_local`. +- `algebraic_geometry.affine_target_morphism_property.is_local.target_affine_locally_is_local` : + If `P.is_local`, then `property_is_local_at_target (target_affine_locally P)`. +- `algebraic_geometry.property_is_local_at_target.open_cover_tfae`: + If `property_is_local_at_target P`, then `P f` iff there exists an open cover `{ Uᵢ }` of `Y` + such that `P` holds for `f ∣_ Uᵢ`. + +These results should not be used directly, and should be ported to each property that is local. + -/ universe u @@ -45,4 +86,296 @@ lemma affine_target_morphism_property.to_property_apply (P : affine_target_morph {X Y : Scheme} (f : X ⟶ Y) [is_affine Y] : P.to_property f ↔ P f := by { delta affine_target_morphism_property.to_property, simp [*] } +lemma affine_cancel_left_is_iso {P : affine_target_morphism_property} + (hP : P.to_property.respects_iso) {X Y Z : Scheme} (f : X ⟶ Y) + (g : Y ⟶ Z) [is_iso f] [is_affine Z] : P (f ≫ g) ↔ P g := +by rw [← P.to_property_apply, ← P.to_property_apply, hP.cancel_left_is_iso] + +lemma affine_cancel_right_is_iso + {P : affine_target_morphism_property} (hP : P.to_property.respects_iso) {X Y Z : Scheme} + (f : X ⟶ Y) (g : Y ⟶ Z) [is_iso g] [is_affine Z] [is_affine Y] : P (f ≫ g) ↔ P f := +by rw [← P.to_property_apply, ← P.to_property_apply, hP.cancel_right_is_iso] + +/-- For a `P : affine_target_morphism_property`, `target_affine_locally P` holds for +`f : X ⟶ Y` whenever `P` holds for the restriction of `f` on every affine open subset of `Y`. -/ +def target_affine_locally (P : affine_target_morphism_property) : morphism_property Scheme := + λ {X Y : Scheme} (f : X ⟶ Y), ∀ (U : Y.affine_opens), @@P (f ∣_ U) U.prop + +lemma is_affine_open.map_is_iso {X Y : Scheme} {U : opens Y.carrier} (hU : is_affine_open U) + (f : X ⟶ Y) [is_iso f] : is_affine_open ((opens.map f.1.base).obj U) := +begin + haveI : is_affine _ := hU, + exact is_affine_of_iso (f ∣_ U), +end + +lemma target_affine_locally_respects_iso {P : affine_target_morphism_property} + (hP : P.to_property.respects_iso) : (target_affine_locally P).respects_iso := +begin + split, + { introv H U, + rw [morphism_restrict_comp, affine_cancel_left_is_iso hP], + exact H U }, + { introv H, + rintro ⟨U, hU : is_affine_open U⟩, dsimp, + haveI : is_affine _ := hU, + haveI : is_affine _ := hU.map_is_iso e.hom, + rw [morphism_restrict_comp, affine_cancel_right_is_iso hP], + exact H ⟨(opens.map e.hom.val.base).obj U, hU.map_is_iso e.hom⟩ } +end + +/-- +We say that `P : affine_target_morphism_property` is a local property if +1. `P` respects isomorphisms. +2. If `P` holds for `f : X ⟶ Y`, then `P` holds for `f ∣_ Y.basic_open r` for any + global section `r`. +3. If `P` holds for `f ∣_ Y.basic_open r` for all `r` in a spanning set of the global sections, + then `P` holds for `f`. +-/ +structure affine_target_morphism_property.is_local (P : affine_target_morphism_property) : Prop := +(respects_iso : P.to_property.respects_iso) +(to_basic_open : ∀ {X Y : Scheme} [is_affine Y] (f : X ⟶ Y) (r : Y.presheaf.obj $ op ⊤), + by exactI P f → + @@P (f ∣_ (Y.basic_open r)) ((top_is_affine_open Y).basic_open_is_affine _)) +(of_basic_open_cover : ∀ {X Y : Scheme} [is_affine Y] (f : X ⟶ Y) + (s : finset (Y.presheaf.obj $ op ⊤)) (hs : ideal.span (s : set (Y.presheaf.obj $ op ⊤)) = ⊤), + by exactI (∀ (r : s), @@P (f ∣_ (Y.basic_open r.1)) + ((top_is_affine_open Y).basic_open_is_affine _)) → P f) + +lemma target_affine_locally_of_open_cover {P : affine_target_morphism_property} + (hP : P.is_local) + {X Y : Scheme} (f : X ⟶ Y) (𝒰 : Y.open_cover) [∀ i, is_affine (𝒰.obj i)] + (h𝒰 : ∀ i, P (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i)) : + target_affine_locally P f := +begin + classical, + let S := λ i, (⟨⟨set.range (𝒰.map i).1.base, (𝒰.is_open i).base_open.open_range⟩, + range_is_affine_open_of_open_immersion (𝒰.map i)⟩ : Y.affine_opens), + intro U, + apply of_affine_open_cover U (set.range S), + { intros U r h, + haveI : is_affine _ := U.2, + have := hP.2 (f ∣_ U.1), + replace this := this (Y.presheaf.map (eq_to_hom U.1.open_embedding_obj_top).op r) h, + rw ← P.to_property_apply at this ⊢, + exact (hP.1.arrow_mk_iso_iff (morphism_restrict_restrict_basic_open f _ r)).mp this }, + { intros U s hs H, + haveI : is_affine _ := U.2, + apply hP.3 (f ∣_ U.1) (s.image (Y.presheaf.map (eq_to_hom U.1.open_embedding_obj_top).op)), + { apply_fun ideal.comap (Y.presheaf.map (eq_to_hom U.1.open_embedding_obj_top.symm).op) at hs, + rw ideal.comap_top at hs, + rw ← hs, + simp only [eq_to_hom_op, eq_to_hom_map, finset.coe_image], + have : ∀ {R S : CommRing} (e : S = R) (s : set S), + (by exactI ideal.span (eq_to_hom e '' s) = ideal.comap (eq_to_hom e.symm) (ideal.span s)), + { intros, subst e, simpa }, + apply this }, + { rintro ⟨r, hr⟩, + obtain ⟨r, hr', rfl⟩ := finset.mem_image.mp hr, + simp_rw ← P.to_property_apply at ⊢ H, + exact + (hP.1.arrow_mk_iso_iff (morphism_restrict_restrict_basic_open f _ r)).mpr (H ⟨r, hr'⟩) } }, + { rw set.eq_univ_iff_forall, + simp only [set.mem_Union], + intro x, + exact ⟨⟨_, ⟨𝒰.f x, rfl⟩⟩, 𝒰.covers x⟩ }, + { rintro ⟨_, i, rfl⟩, + simp_rw ← P.to_property_apply at ⊢ h𝒰, + exact (hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _)).mpr (h𝒰 i) }, +end + +lemma affine_target_morphism_property.is_local.affine_open_cover_tfae + {P : affine_target_morphism_property} + (hP : P.is_local) {X Y : Scheme.{u}} (f : X ⟶ Y) : + tfae [target_affine_locally P f, + ∃ (𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)], ∀ (i : 𝒰.J), + by exactI P (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i), + ∀ (𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)] (i : 𝒰.J), + by exactI P (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i), + ∀ {U : Scheme} (g : U ⟶ Y) [is_affine U] [is_open_immersion g], + by exactI P (pullback.snd : pullback f g ⟶ U), + ∃ {ι : Type u} (U : ι → opens Y.carrier) (hU : supr U = ⊤) (hU' : ∀ i, is_affine_open (U i)), + ∀ i, @@P (f ∣_ (U i)) (hU' i)] := +begin + tfae_have : 1 → 4, + { intros H U g h₁ h₂, + resetI, + replace H := H ⟨⟨_, h₂.base_open.open_range⟩, + range_is_affine_open_of_open_immersion g⟩, + rw ← P.to_property_apply at H ⊢, + rwa ← hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _) }, + tfae_have : 4 → 3, + { intros H 𝒰 h𝒰 i, + resetI, + apply H }, + tfae_have : 3 → 2, + { exact λ H, ⟨Y.affine_cover, infer_instance, H Y.affine_cover⟩ }, + tfae_have : 2 → 1, + { rintro ⟨𝒰, h𝒰, H⟩, exactI target_affine_locally_of_open_cover hP f 𝒰 H }, + tfae_have : 5 → 2, + { rintro ⟨ι, U, hU, hU', H⟩, + refine ⟨Y.open_cover_of_supr_eq_top U hU, hU', _⟩, + intro i, + specialize H i, + rw [← P.to_property_apply, ← hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _)], + rw ← P.to_property_apply at H, + convert H, + all_goals { ext1, exact subtype.range_coe } }, + tfae_have : 1 → 5, + { intro H, + refine ⟨Y.carrier, λ x, is_open_immersion.opens_range (Y.affine_cover.map x), _, + λ i, range_is_affine_open_of_open_immersion _, _⟩, + { rw eq_top_iff, intros x _, erw opens.mem_supr, exact⟨x, Y.affine_cover.covers x⟩ }, + { intro i, exact H ⟨_, range_is_affine_open_of_open_immersion _⟩ } }, + tfae_finish +end + +lemma affine_target_morphism_property.is_local_of_open_cover_imply + (P : affine_target_morphism_property) (hP : P.to_property.respects_iso) + (H : ∀ {X Y : Scheme.{u}} (f : X ⟶ Y), + (∃ (𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)], ∀ (i : 𝒰.J), + by exactI P (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i)) → + (∀ {U : Scheme} (g : U ⟶ Y) [is_affine U] [is_open_immersion g], + by exactI P (pullback.snd : pullback f g ⟶ U))) : P.is_local := +begin + refine ⟨hP, _, _⟩, + { introv h, + resetI, + haveI : is_affine _ := (top_is_affine_open Y).basic_open_is_affine r, + delta morphism_restrict, + rw affine_cancel_left_is_iso hP, + refine @@H f ⟨Scheme.open_cover_of_is_iso (𝟙 Y), _, _⟩ (Y.of_restrict _) _inst _, + { intro i, dsimp, apply_instance }, + { intro i, dsimp, + rwa [← category.comp_id pullback.snd, ← pullback.condition, affine_cancel_left_is_iso hP] } }, + { introv hs hs', + resetI, + replace hs := ((top_is_affine_open Y).basic_open_union_eq_self_iff _).mpr hs, + have := H f ⟨Y.open_cover_of_supr_eq_top _ hs, _, _⟩ (𝟙 _), + rwa [← category.comp_id pullback.snd, ← pullback.condition, + affine_cancel_left_is_iso hP] at this, + { intro i, exact (top_is_affine_open Y).basic_open_is_affine _ }, + { rintro (i : s), + specialize hs' i, + haveI : is_affine _ := (top_is_affine_open Y).basic_open_is_affine i.1, + delta morphism_restrict at hs', + rwa affine_cancel_left_is_iso hP at hs' } } +end + +lemma affine_target_morphism_property.is_local.affine_open_cover_iff + {P : affine_target_morphism_property} (hP : P.is_local) + {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.open_cover.{u} Y) [h𝒰 : ∀ i, is_affine (𝒰.obj i)] : + target_affine_locally P f ↔ ∀ i, @@P (pullback.snd : pullback f (𝒰.map i) ⟶ _) (h𝒰 i) := +⟨λ H, let h := ((hP.affine_open_cover_tfae f).out 0 2).mp H in h 𝒰, + λ H, let h := ((hP.affine_open_cover_tfae f).out 1 0).mp in h ⟨𝒰, infer_instance, H⟩⟩ + +lemma affine_target_morphism_property.is_local.affine_target_iff + {P : affine_target_morphism_property} (hP : P.is_local) + {X Y : Scheme.{u}} (f : X ⟶ Y) [is_affine Y] : + target_affine_locally P f ↔ P f := +begin + rw hP.affine_open_cover_iff f _, + swap, { exact Scheme.open_cover_of_is_iso (𝟙 Y) }, + swap, { intro _, dsimp, apply_instance }, + transitivity (P (pullback.snd : pullback f (𝟙 _) ⟶ _)), + { exact ⟨λ H, H punit.star, λ H _, H⟩ }, + rw [← category.comp_id pullback.snd, ← pullback.condition, affine_cancel_left_is_iso hP.1], +end + +/-- +We say that `P : morphism_property Scheme` is local at the target if +1. `P` respects isomorphisms. +2. If `P` holds for `f : X ⟶ Y`, then `P` holds for `f ∣_ U` for any `U`. +3. If `P` holds for `f ∣_ U` for an open cover `U` of `Y`, then `P` holds for `f`. +-/ +structure property_is_local_at_target (P : morphism_property Scheme) : Prop := +(respects_iso : P.respects_iso) +(restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier), P f → P (f ∣_ U)) +(of_open_cover : ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.open_cover.{u} Y), + (∀ (i : 𝒰.J), P (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i)) → P f) + +lemma affine_target_morphism_property.is_local.target_affine_locally_is_local + {P : affine_target_morphism_property} (hP : P.is_local) : + property_is_local_at_target (target_affine_locally P) := +begin + constructor, + { exact target_affine_locally_respects_iso hP.1 }, + { intros X Y f U H V, + rw [← P.to_property_apply, hP.1.arrow_mk_iso_iff (morphism_restrict_restrict f _ _)], + convert H ⟨_, is_affine_open.image_is_open_immersion V.2 (Y.of_restrict _)⟩, + rw ← P.to_property_apply, + refl }, + { rintros X Y f 𝒰 h𝒰, + rw (hP.affine_open_cover_tfae f).out 0 1, + refine ⟨𝒰.bind (λ _, Scheme.affine_cover _), _, _⟩, + { intro i, dsimp [Scheme.open_cover.bind], apply_instance }, + { intro i, + specialize h𝒰 i.1, + rw (hP.affine_open_cover_tfae (pullback.snd : pullback f (𝒰.map i.fst) ⟶ _)).out 0 2 + at h𝒰, + specialize h𝒰 (Scheme.affine_cover _) i.2, + let e : pullback f ((𝒰.obj i.fst).affine_cover.map i.snd ≫ 𝒰.map i.fst) ⟶ + pullback (pullback.snd : pullback f (𝒰.map i.fst) ⟶ _) + ((𝒰.obj i.fst).affine_cover.map i.snd), + { refine (pullback_symmetry _ _).hom ≫ _, + refine (pullback_right_pullback_fst_iso _ _ _).inv ≫ _, + refine (pullback_symmetry _ _).hom ≫ _, + refine pullback.map _ _ _ _ (pullback_symmetry _ _).hom (𝟙 _) (𝟙 _) _ _; + simp only [category.comp_id, category.id_comp, pullback_symmetry_hom_comp_snd] }, + rw ← affine_cancel_left_is_iso hP.1 e at h𝒰, + convert h𝒰, + simp } }, +end + +lemma property_is_local_at_target.open_cover_tfae + {P : morphism_property Scheme} + (hP : property_is_local_at_target P) + {X Y : Scheme.{u}} (f : X ⟶ Y) : + tfae [P f, + ∃ (𝒰 : Scheme.open_cover.{u} Y), ∀ (i : 𝒰.J), + P (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i), + ∀ (𝒰 : Scheme.open_cover.{u} Y) (i : 𝒰.J), + P (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i), + ∀ (U : opens Y.carrier), P (f ∣_ U), + ∀ {U : Scheme} (g : U ⟶ Y) [is_open_immersion g], + P (pullback.snd : pullback f g ⟶ U), + ∃ {ι : Type u} (U : ι → opens Y.carrier) (hU : supr U = ⊤), (∀ i, P (f ∣_ (U i)))] := +begin + tfae_have : 2 → 1, + { rintro ⟨𝒰, H⟩, exact hP.3 f 𝒰 H }, + tfae_have : 1 → 4, + { intros H U, exact hP.2 f U H }, + tfae_have : 4 → 3, + { intros H 𝒰 i, + rw ← hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _), + exact H (is_open_immersion.opens_range $ 𝒰.map i) }, + tfae_have : 3 → 2, + { exact λ H, ⟨Y.affine_cover, H Y.affine_cover⟩ }, + tfae_have : 4 → 5, + { intros H U g hg, + resetI, + rw ← hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _), + apply H }, + tfae_have : 5 → 4, + { intros H U, + erw hP.1.cancel_left_is_iso, + apply H }, + tfae_have : 4 → 6, + { intro H, exact ⟨punit, λ _, ⊤, csupr_const, λ _, H _⟩ }, + tfae_have : 6 → 2, + { rintro ⟨ι, U, hU, H⟩, + refine ⟨Y.open_cover_of_supr_eq_top U hU, _⟩, + intro i, + rw ← hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _), + convert H i, + all_goals { ext1, exact subtype.range_coe } }, + tfae_finish +end + +lemma affine_target_morphism_property.is_local.open_cover_iff + {P : morphism_property Scheme} (hP : property_is_local_at_target P) + {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.open_cover.{u} Y) : + P f ↔ ∀ i, P (pullback.snd : pullback f (𝒰.map i) ⟶ _) := +⟨λ H, let h := ((hP.open_cover_tfae f).out 0 2).mp H in h 𝒰, + λ H, let h := ((hP.open_cover_tfae f).out 1 0).mp in h ⟨𝒰, H⟩⟩ + end algebraic_geometry diff --git a/formal/lean/mathlib/algebraic_geometry/open_immersion.lean b/formal/lean/mathlib/algebraic_geometry/open_immersion.lean index ef8b042a2c6ef84a53e1b1ad3806f39c00201f8a..f146dc431a34f08b21d550af7b1dfe012b04dea3 100644 --- a/formal/lean/mathlib/algebraic_geometry/open_immersion.lean +++ b/formal/lean/mathlib/algebraic_geometry/open_immersion.lean @@ -1616,6 +1616,11 @@ begin exact (set.preimage_image_eq _ H.base_open.inj).symm end +/-- The image of an open immersion as an open set. -/ +@[simps] +def opens_range (f : X ⟶ Y) [H : is_open_immersion f] : opens Y.carrier := + ⟨_, H.base_open.open_range⟩ + end is_open_immersion /-- The functor taking open subsets of `X` to open subschemes of `X`. -/ @@ -1706,6 +1711,23 @@ def Scheme.open_cover.inter {X : Scheme.{u}} (𝒰₁ : Scheme.open_cover.{v₁} covers := λ x, by { rw is_open_immersion.range_pullback_to_base_of_left, exact ⟨𝒰₁.covers x, 𝒰₂.covers x⟩ } } +/-- If `U` is a family of open sets that covers `X`, then `X.restrict U` forms an `X.open_cover`. -/ +@[simps J obj map] +def Scheme.open_cover_of_supr_eq_top {s : Type*} (X : Scheme) (U : s → opens X.carrier) + (hU : (⨆ i, U i) = ⊤) : X.open_cover := +{ J := s, + obj := λ i, X.restrict (U i).open_embedding, + map := λ i, X.of_restrict (U i).open_embedding, + f := λ x, begin + have : x ∈ ⨆ i, U i := hU.symm ▸ (show x ∈ (⊤ : opens X.carrier), by triv), + exact (opens.mem_supr.mp this).some, + end, + covers := λ x, begin + erw subtype.range_coe, + have : x ∈ ⨆ i, U i := hU.symm ▸ (show x ∈ (⊤ : opens X.carrier), by triv), + exact (opens.mem_supr.mp this).some_spec, + end } + section morphism_restrict /-- Given a morphism `f : X ⟶ Y` and an open set `U ⊆ Y`, we have `X ×[Y] U ≅ X |_{f ⁻¹ U}` -/ @@ -1822,6 +1844,72 @@ begin congr, end +/-- Restricting a morphism onto the the image of an open immersion is isomorphic to the base change +along the immersion. -/ +def morphism_restrict_opens_range + {X Y U : Scheme} (f : X ⟶ Y) (g : U ⟶ Y) [hg : is_open_immersion g] : + arrow.mk (f ∣_ is_open_immersion.opens_range g) ≅ arrow.mk (pullback.snd : pullback f g ⟶ _) := +begin + let V : opens Y.carrier := is_open_immersion.opens_range g, + let e := is_open_immersion.iso_of_range_eq g (Y.of_restrict V.open_embedding) + (by exact subtype.range_coe.symm), + let t : pullback f g ⟶ pullback f (Y.of_restrict V.open_embedding) := + pullback.map _ _ _ _ (𝟙 _) e.hom (𝟙 _) (by rw [category.comp_id, category.id_comp]) + (by rw [category.comp_id, is_open_immersion.iso_of_range_eq_hom, is_open_immersion.lift_fac]), + symmetry, + refine arrow.iso_mk (as_iso t ≪≫ pullback_restrict_iso_restrict f V) e _, + rw [iso.trans_hom, as_iso_hom, ← iso.comp_inv_eq, ← cancel_mono g, arrow.mk_hom, arrow.mk_hom, + is_open_immersion.iso_of_range_eq_inv, category.assoc, category.assoc, category.assoc, + is_open_immersion.lift_fac, ← pullback.condition, morphism_restrict_ι, + pullback_restrict_iso_restrict_hom_restrict_assoc, pullback.lift_fst_assoc, category.comp_id], +end + +/-- The restrictions onto two equal open sets are isomorphic. This currently has bad defeqs when +unfolded, but it should not matter for now. Replace this definition if better defeqs are needed. -/ +def morphism_restrict_eq {X Y : Scheme} (f : X ⟶ Y) {U V : opens Y.carrier} (e : U = V) : + arrow.mk (f ∣_ U) ≅ arrow.mk (f ∣_ V) := eq_to_iso (by subst e) + +/-- Restricting a morphism twice is isomorpic to one restriction. -/ +def morphism_restrict_restrict {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) (V : opens U) : + arrow.mk (f ∣_ U ∣_ V) ≅ arrow.mk (f ∣_ (U.open_embedding.is_open_map.functor.obj V)) := +begin + have : (f ∣_ U ∣_ V) ≫ (iso.refl _).hom = + (as_iso $ (pullback_restrict_iso_restrict (f ∣_ U) V).inv ≫ (pullback_symmetry _ _).hom ≫ + pullback.map _ _ _ _ (𝟙 _) + ((pullback_restrict_iso_restrict f U).inv ≫ (pullback_symmetry _ _).hom) (𝟙 _) + ((category.comp_id _).trans (category.id_comp _).symm) (by simpa) ≫ + (pullback_right_pullback_fst_iso _ _ _).hom ≫ (pullback_symmetry _ _).hom).hom ≫ pullback.snd, + { simpa only [category.comp_id, pullback_right_pullback_fst_iso_hom_fst, iso.refl_hom, + category.assoc, pullback_symmetry_hom_comp_snd, as_iso_hom, pullback.lift_fst, + pullback_symmetry_hom_comp_fst] }, + refine arrow.iso_mk' _ _ _ _ this.symm ≪≫ (morphism_restrict_opens_range _ _).symm ≪≫ + morphism_restrict_eq _ _, + ext1, + dsimp, + rw [coe_comp, set.range_comp], + congr, + exact subtype.range_coe, +end + +/-- Restricting a morphism twice onto a basic open set is isomorphic to one restriction. -/ +def morphism_restrict_restrict_basic_open {X Y : Scheme} (f : X ⟶ Y) (U : opens Y.carrier) + (r : Y.presheaf.obj (op U)) : + arrow.mk (f ∣_ U ∣_ (Y.restrict _).basic_open + (Y.presheaf.map (eq_to_hom U.open_embedding_obj_top).op r)) ≅ arrow.mk (f ∣_ Y.basic_open r) := +begin + refine morphism_restrict_restrict _ _ _ ≪≫ morphism_restrict_eq _ _, + have e := Scheme.preimage_basic_open (Y.of_restrict U.open_embedding) r, + erw [Scheme.of_restrict_coe_c_app, opens.adjunction_counit_app_self, eq_to_hom_op] at e, + rw [← (Y.restrict U.open_embedding).basic_open_res_eq _ + (eq_to_hom U.inclusion_map_eq_top).op, ← comp_apply], + erw ← Y.presheaf.map_comp, + rw [eq_to_hom_op, eq_to_hom_op, eq_to_hom_map, eq_to_hom_trans], + erw ← e, + ext1, dsimp [opens.map, opens.inclusion], + rw [set.image_preimage_eq_inter_range, set.inter_eq_left_iff_subset, subtype.range_coe], + exact Y.basic_open_subset r +end + end morphism_restrict end algebraic_geometry diff --git a/formal/lean/mathlib/algebraic_geometry/presheafed_space/gluing.lean b/formal/lean/mathlib/algebraic_geometry/presheafed_space/gluing.lean index 762d2b64a944087a3e0e0e6c4110ad4a55975512..94ef63d65c7281454dab1ffe224a854a0517b0c7 100644 --- a/formal/lean/mathlib/algebraic_geometry/presheafed_space/gluing.lean +++ b/formal/lean/mathlib/algebraic_geometry/presheafed_space/gluing.lean @@ -87,7 +87,7 @@ such that We can then glue the spaces `U i` together by identifying `V i j` with `V j i`, such that the `U i`'s are open subspaces of the glued space. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure glue_data extends glue_data (PresheafedSpace.{v} C) := (f_open : ∀ i j, is_open_immersion (f i j)) @@ -488,7 +488,7 @@ such that We can then glue the spaces `U i` together by identifying `V i j` with `V j i`, such that the `U i`'s are open subspaces of the glued space. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure glue_data extends glue_data (SheafedSpace.{v} C) := (f_open : ∀ i j, SheafedSpace.is_open_immersion (f i j)) @@ -560,7 +560,7 @@ such that We can then glue the spaces `U i` together by identifying `V i j` with `V j i`, such that the `U i`'s are open subspaces of the glued space. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure glue_data extends glue_data LocallyRingedSpace := (f_open : ∀ i j, LocallyRingedSpace.is_open_immersion (f i j)) diff --git a/formal/lean/mathlib/algebraic_geometry/prime_spectrum/basic.lean b/formal/lean/mathlib/algebraic_geometry/prime_spectrum/basic.lean index f93eb904aa8c6ae37f3f35a05450744d616f5b4c..f3ec6ef8b5dce1cb37b40843c7735ebd55c238af 100644 --- a/formal/lean/mathlib/algebraic_geometry/prime_spectrum/basic.lean +++ b/formal/lean/mathlib/algebraic_geometry/prime_spectrum/basic.lean @@ -57,7 +57,7 @@ is the type of all prime ideals of `R`. It is naturally endowed with a topology (the Zariski topology), and a sheaf of commutative rings (see `algebraic_geometry.structure_sheaf`). It is a fundamental building block in algebraic geometry. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def prime_spectrum := {I : ideal R // I.is_prime} variable {R} diff --git a/formal/lean/mathlib/algebraic_geometry/prime_spectrum/is_open_comap_C.lean b/formal/lean/mathlib/algebraic_geometry/prime_spectrum/is_open_comap_C.lean index 94a5d44e69992dcb1e08118bf69fb41596048706..436f841b76d8f1d6120db1e26a4ccfc0c022a693 100644 --- a/formal/lean/mathlib/algebraic_geometry/prime_spectrum/is_open_comap_C.lean +++ b/formal/lean/mathlib/algebraic_geometry/prime_spectrum/is_open_comap_C.lean @@ -37,11 +37,11 @@ end /-- If a point of `Spec R[x]` is not contained in the vanishing set of `f`, then its image in `Spec R` is contained in the open set where at least one of the coefficients of `f` is non-zero. -This lemma is a reformulation of `exists_coeff_not_mem_C_inverse`. -/ +This lemma is a reformulation of `exists_C_coeff_not_mem`. -/ lemma comap_C_mem_image_of_Df {I : prime_spectrum R[X]} (H : I ∈ (zero_locus {f} : set (prime_spectrum R[X]))ᶜ ) : prime_spectrum.comap (polynomial.C : R →+* R[X]) I ∈ image_of_Df f := -exists_coeff_not_mem_C_inverse (mem_compl_zero_locus_iff_not_mem.mp H) +exists_C_coeff_not_mem (mem_compl_zero_locus_iff_not_mem.mp H) /-- The open set `image_of_Df f` coincides with the image of `basic_open f` under the morphism `C⁺ : Spec R[x] → Spec R`. -/ diff --git a/formal/lean/mathlib/algebraic_geometry/projective_spectrum/topology.lean b/formal/lean/mathlib/algebraic_geometry/projective_spectrum/topology.lean index 4425efc12d9dece2df2acbd004e5f234e79b6f87..a0e04d1a1027790281eb09c396cb54f54545cd8a 100644 --- a/formal/lean/mathlib/algebraic_geometry/projective_spectrum/topology.lean +++ b/formal/lean/mathlib/algebraic_geometry/projective_spectrum/topology.lean @@ -47,7 +47,7 @@ variables (𝒜 : ℕ → submodule R A) [graded_algebra 𝒜] The projective spectrum of a graded commutative ring is the subtype of all homogenous ideals that are prime and do not contain the irrelevant ideal. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def projective_spectrum := {I : homogeneous_ideal 𝒜 // I.to_ideal.is_prime ∧ ¬(homogeneous_ideal.irrelevant 𝒜 ≤ I)} diff --git a/formal/lean/mathlib/algebraic_geometry/properties.lean b/formal/lean/mathlib/algebraic_geometry/properties.lean index 4043ecffb4a343ef4e6e10f2d44d35b6f260c241..0ff5991de7e79fe0650492cb9d68bd79bf5a60c8 100644 --- a/formal/lean/mathlib/algebraic_geometry/properties.lean +++ b/formal/lean/mathlib/algebraic_geometry/properties.lean @@ -31,16 +31,11 @@ variable (X : Scheme) instance : t0_space X.carrier := begin - rw t0_space_iff_inseparable, - intros x y h, + refine t0_space.of_open_cover (λ x, _), obtain ⟨U, R, ⟨e⟩⟩ := X.local_affine x, - have hy : y ∈ U.val := (h.mem_open_iff U.1.2).1 U.2, - erw ← subtype_inseparable_iff (⟨x, U.2⟩ : U.1.1) (⟨y, hy⟩ : U.1.1) at h, let e' : U.1 ≃ₜ prime_spectrum R := homeo_of_iso ((LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget _).map_iso e), - have := t0_space_of_injective_of_continuous e'.injective e'.continuous, - rw t0_space_iff_inseparable at this, - by simpa only [subtype.mk_eq_mk] using this ⟨x, U.2⟩ ⟨y, hy⟩ h + exact ⟨U.1.1, U.2, U.1.2, e'.embedding.t0_space⟩ end instance : quasi_sober X.carrier := diff --git a/formal/lean/mathlib/algebraic_geometry/structure_sheaf.lean b/formal/lean/mathlib/algebraic_geometry/structure_sheaf.lean index dd27bc65946b5f20487623ab8a4be2a40c521882..edae040d083327793efa1191cc2c6290d7173e1b 100644 --- a/formal/lean/mathlib/algebraic_geometry/structure_sheaf.lean +++ b/formal/lean/mathlib/algebraic_geometry/structure_sheaf.lean @@ -688,7 +688,7 @@ begin have n_spec := λ (p : ι × ι), (exists_power p.fst p.snd).some_spec, -- We need one power `(h i * h j) ^ N` that works for *all* pairs `(i,j)` -- Since there are only finitely many indices involved, we can pick the supremum. - let N := (t.product t).sup n, + let N := (t ×ˢ t).sup n, have basic_opens_eq : ∀ i : ι, basic_open ((h i) ^ (N+1)) = basic_open (h i) := λ i, basic_open_pow _ _ (by linarith), -- Expanding the fraction `a i / h i` by the power `(h i) ^ N` gives the desired normalization diff --git a/formal/lean/mathlib/algebraic_topology/dold_kan/p_infty.lean b/formal/lean/mathlib/algebraic_topology/dold_kan/p_infty.lean new file mode 100644 index 0000000000000000000000000000000000000000..d63259f68d3e95a00a0c6c2d47ad56c64d1eeae4 --- /dev/null +++ b/formal/lean/mathlib/algebraic_topology/dold_kan/p_infty.lean @@ -0,0 +1,147 @@ +/- +Copyright (c) 2022 Joël Riou. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Joël Riou +-/ + +import algebraic_topology.dold_kan.projections +import category_theory.idempotents.functor_categories +import category_theory.idempotents.functor_extension + +/-! + +# Construction of the projection `P_infty` for the Dold-Kan correspondence + +TODO (@joelriou) continue adding the various files referenced below + +In this file, we construct the projection `P_infty : K[X] ⟶ K[X]` by passing +to the limit the projections `P q` defined in `projections.lean`. This +projection is a critical tool in this formalisation of the Dold-Kan correspondence, +because in the case of abelian categories, `P_infty` corresponds to the +projection on the normalized Moore subcomplex, with kernel the degenerate subcomplex. +(See `equivalence.lean` for the general strategy of proof.) + +-/ + +open category_theory +open category_theory.category +open category_theory.preadditive +open category_theory.simplicial_object +open category_theory.idempotents +open opposite +open_locale simplicial dold_kan + +noncomputable theory + +namespace algebraic_topology + +namespace dold_kan + +variables {C : Type*} [category C] [preadditive C] {X : simplicial_object C} + +lemma P_is_eventually_constant {q n : ℕ} (hqn : n ≤ q) : + ((P (q+1)).f n : X _[n] ⟶ _ ) = (P q).f n := +begin + cases n, + { simp only [P_f_0_eq], }, + { unfold P, + simp only [add_right_eq_self, comp_add, homological_complex.comp_f, + homological_complex.add_f_apply, comp_id], + exact (higher_faces_vanish.of_P q n).comp_Hσ_eq_zero + (nat.succ_le_iff.mp hqn), }, +end + +lemma Q_is_eventually_constant {q n : ℕ} (hqn : n ≤ q) : + ((Q (q+1)).f n : X _[n] ⟶ _ ) = (Q q).f n := +by simp only [Q, homological_complex.sub_f_apply, P_is_eventually_constant hqn] + +/-- The endomorphism `P_infty : K[X] ⟶ K[X]` obtained from the `P q` by passing to the limit. -/ +def P_infty : K[X] ⟶ K[X] := chain_complex.of_hom _ _ _ _ _ _ + (λ n, ((P n).f n : X _[n] ⟶ _ )) + (λ n, by simpa only [← P_is_eventually_constant (show n ≤ n, by refl), + alternating_face_map_complex.obj_d_eq] using (P (n+1)).comm (n+1) n) + +lemma P_infty_f (n : ℕ) : (P_infty.f n : X _[n] ⟶ X _[n] ) = (P n).f n := rfl + +@[simp, reassoc] +lemma P_infty_f_naturality (n : ℕ) {X Y : simplicial_object C} (f : X ⟶ Y) : + f.app (op [n]) ≫ P_infty.f n = P_infty.f n ≫ f.app (op [n]) := +P_f_naturality n n f + +@[simp, reassoc] +lemma P_infty_f_idem (n : ℕ) : + (P_infty.f n : X _[n] ⟶ _) ≫ (P_infty.f n) = P_infty.f n := +by simp only [P_infty_f, P_f_idem] + +@[simp, reassoc] +lemma P_infty_idem : (P_infty : K[X] ⟶ _) ≫ P_infty = P_infty := +by { ext n, exact P_infty_f_idem n, } + +variable (C) + +/-- `P_infty` induces a natural transformation, i.e. an endomorphism of +the functor `alternating_face_map_complex C`. -/ +@[simps] +def nat_trans_P_infty : + alternating_face_map_complex C ⟶ alternating_face_map_complex C := +{ app := λ _, P_infty, + naturality' := λ X Y f, by { ext n, exact P_infty_f_naturality n f, }, } + +/-- The natural transformation in each degree that is induced by `nat_trans_P_infty`. -/ +@[simps] +def nat_trans_P_infty_f (n : ℕ) := +nat_trans_P_infty C ◫ 𝟙 (homological_complex.eval _ _ n) + +variable {C} + +@[simp] +lemma map_P_infty_f {D : Type*} [category D] [preadditive D] + (G : C ⥤ D) [G.additive] (X : simplicial_object C) (n : ℕ) : + (P_infty : K[((whiskering C D).obj G).obj X] ⟶ _).f n = + G.map ((P_infty : alternating_face_map_complex.obj X ⟶ _).f n) := +by simp only [P_infty_f, map_P] + +/-- Given an object `Y : karoubi (simplicial_object C)`, this lemma +computes `P_infty` for the associated object in `simplicial_object (karoubi C)` +in terms of `P_infty` for `Y.X : simplicial_object C` and `Y.p`. -/ +lemma karoubi_P_infty_f {Y : karoubi (simplicial_object C)} (n : ℕ) : + ((P_infty : K[(karoubi_functor_category_embedding _ _).obj Y] ⟶ _).f n).f = + Y.p.app (op [n]) ≫ (P_infty : K[Y.X] ⟶ _).f n := +begin + -- We introduce P_infty endomorphisms P₁, P₂, P₃, P₄ on various objects Y₁, Y₂, Y₃, Y₄. + let Y₁ := (karoubi_functor_category_embedding _ _).obj Y, + let Y₂ := Y.X, + let Y₃ := (((whiskering _ _).obj (to_karoubi C)).obj Y.X), + let Y₄ := (karoubi_functor_category_embedding _ _).obj ((to_karoubi _).obj Y.X), + let P₁ : K[Y₁] ⟶ _ := P_infty, + let P₂ : K[Y₂] ⟶ _ := P_infty, + let P₃ : K[Y₃] ⟶ _ := P_infty, + let P₄ : K[Y₄] ⟶ _ := P_infty, + -- The statement of lemma relates P₁ and P₂. + change (P₁.f n).f = Y.p.app (op [n]) ≫ P₂.f n, + -- The proof proceeds by obtaining relations h₃₂, h₄₃, h₁₄. + have h₃₂ : (P₃.f n).f = P₂.f n := karoubi.hom_ext.mp (map_P_infty_f (to_karoubi C) Y₂ n), + have h₄₃ : P₄.f n = P₃.f n, + { have h := functor.congr_obj (to_karoubi_comp_karoubi_functor_category_embedding _ _) Y₂, + simp only [← nat_trans_P_infty_f_app], + congr', }, + let τ₁ := 𝟙 (karoubi_functor_category_embedding (simplex_categoryᵒᵖ) C), + let τ₂ := nat_trans_P_infty_f (karoubi C) n, + let τ := τ₁ ◫ τ₂, + have h₁₄ := idempotents.nat_trans_eq τ Y, + dsimp [τ, τ₁, τ₂, nat_trans_P_infty_f] at h₁₄, + rw [id_comp, id_comp, comp_id, comp_id] at h₁₄, + /- We use the three equalities h₃₂, h₄₃, h₁₄. -/ + rw [← h₃₂, ← h₄₃, h₁₄], + simp only [karoubi_functor_category_embedding.map_app_f, karoubi.decomp_id_p_f, + karoubi.decomp_id_i_f, karoubi.comp], + let π : Y₄ ⟶ Y₄ := (to_karoubi _ ⋙ karoubi_functor_category_embedding _ _).map Y.p, + have eq := karoubi.hom_ext.mp (P_infty_f_naturality n π), + simp only [karoubi.comp] at eq, + dsimp [π] at eq, + rw [← eq, reassoc_of (app_idem Y (op [n]))], +end + +end dold_kan + +end algebraic_topology diff --git a/formal/lean/mathlib/algebraic_topology/simplex_category.lean b/formal/lean/mathlib/algebraic_topology/simplex_category.lean index 36815db57f061b34ad9a9d4229fea2aa844a4bec..335a502fd6a9dd11eb7717855b2afb1f8918f24c 100644 --- a/formal/lean/mathlib/algebraic_topology/simplex_category.lean +++ b/formal/lean/mathlib/algebraic_topology/simplex_category.lean @@ -62,7 +62,7 @@ def len (n : simplex_category) : ℕ := n @[simp] lemma mk_len (n : simplex_category) : [n.len] = n := rfl /-- Morphisms in the simplex_category. -/ -@[irreducible, nolint has_inhabited_instance] +@[irreducible, nolint has_nonempty_instance] protected def hom (a b : simplex_category) := fin (a.len + 1) →o fin (b.len + 1) namespace hom @@ -382,7 +382,7 @@ def is_skeleton_of : is_skeleton_of NonemptyFinLinOrd simplex_category skeletal_ /-- The truncated simplex category. -/ @[derive small_category] -def truncated (n : ℕ) := {a : simplex_category // a.len ≤ n} +def truncated (n : ℕ) := full_subcategory (λ a : simplex_category, a.len ≤ n) namespace truncated diff --git a/formal/lean/mathlib/algebraic_topology/simplicial_object.lean b/formal/lean/mathlib/algebraic_topology/simplicial_object.lean index 2c68362906215db1785fb49e8ad6454f115fb195..be384bc4a9b86f57880714d4ca567d7c57c62bfe 100644 --- a/formal/lean/mathlib/algebraic_topology/simplicial_object.lean +++ b/formal/lean/mathlib/algebraic_topology/simplicial_object.lean @@ -31,7 +31,7 @@ variables (C : Type u) [category.{v} C] /-- The category of simplicial objects valued in a category `C`. This is the category of contravariant functors from `simplex_category` to `C`. -/ -@[derive category, nolint has_inhabited_instance] +@[derive category, nolint has_nonempty_instance] def simplicial_object := simplex_categoryᵒᵖ ⥤ C namespace simplicial_object @@ -119,7 +119,7 @@ def whiskering (D : Type*) [category D] : whiskering_right _ _ _ /-- Truncated simplicial objects. -/ -@[derive category, nolint has_inhabited_instance] +@[derive category, nolint has_nonempty_instance] def truncated (n : ℕ) := (simplex_category.truncated n)ᵒᵖ ⥤ C variable {C} @@ -163,7 +163,7 @@ variable (C) abbreviation const : C ⥤ simplicial_object C := category_theory.functor.const _ /-- The category of augmented simplicial objects, defined as a comma category. -/ -@[derive category, nolint has_inhabited_instance] +@[derive category, nolint has_nonempty_instance] def augmented := comma (𝟭 (simplicial_object C)) (const C) variable {C} @@ -261,7 +261,7 @@ by { dsimp, rw [simplex_category.hom_zero_zero ([0].const 0), op_id, X.map_id, c end simplicial_object /-- Cosimplicial objects. -/ -@[derive category, nolint has_inhabited_instance] +@[derive category, nolint has_nonempty_instance] def cosimplicial_object := simplex_category ⥤ C namespace cosimplicial_object @@ -349,7 +349,7 @@ def whiskering (D : Type*) [category D] : whiskering_right _ _ _ /-- Truncated cosimplicial objects. -/ -@[derive category, nolint has_inhabited_instance] +@[derive category, nolint has_nonempty_instance] def truncated (n : ℕ) := simplex_category.truncated n ⥤ C variable {C} @@ -394,7 +394,7 @@ variable (C) abbreviation const : C ⥤ cosimplicial_object C := category_theory.functor.const _ /-- Augmented cosimplicial objects. -/ -@[derive category, nolint has_inhabited_instance] +@[derive category, nolint has_nonempty_instance] def augmented := comma (const C) (𝟭 (cosimplicial_object C)) variable {C} diff --git a/formal/lean/mathlib/analysis/calculus/cont_diff.lean b/formal/lean/mathlib/analysis/calculus/cont_diff.lean index aa6a6ca3041d04393a2cfb9f2d98e27e25add1df..9a83247cbaeb8e2f723337920db96fd300f10e98 100644 --- a/formal/lean/mathlib/analysis/calculus/cont_diff.lean +++ b/formal/lean/mathlib/analysis/calculus/cont_diff.lean @@ -1160,10 +1160,10 @@ hf.fderiv_within' (by { rw [insert_eq_of_mem hxs], exact eventually_of_mem self_ continuous. -/ lemma cont_diff_on.continuous_on_fderiv_within_apply (h : cont_diff_on 𝕜 n f s) (hs : unique_diff_on 𝕜 s) (hn : 1 ≤ n) : - continuous_on (λp : E × E, (fderiv_within 𝕜 f s p.1 : E → F) p.2) (s ×ˢ (univ : set E)) := + continuous_on (λp : E × E, (fderiv_within 𝕜 f s p.1 : E → F) p.2) (s ×ˢ univ) := begin have A : continuous (λq : (E →L[𝕜] F) × E, q.1 q.2) := is_bounded_bilinear_map_apply.continuous, - have B : continuous_on (λp : E × E, (fderiv_within 𝕜 f s p.1, p.2)) (s ×ˢ (univ : set E)), + have B : continuous_on (λp : E × E, (fderiv_within 𝕜 f s p.1, p.2)) (s ×ˢ univ), { apply continuous_on.prod _ continuous_snd.continuous_on, exact continuous_on.comp (h.continuous_on_fderiv_within hs hn) continuous_fst.continuous_on (prod_subset_preimage_fst _ _) }, @@ -1519,6 +1519,16 @@ by simp [cont_diff_on_univ.symm, continuous_iff_continuous_on_univ, differentiable_on_univ.symm, iterated_fderiv_within_univ, cont_diff_on_iff_continuous_on_differentiable_on unique_diff_on_univ] +/-- If `f` is `C^n` then its `m`-times iterated derivative is continuous for `m ≤ n`. -/ +lemma cont_diff.continuous_iterated_fderiv {m : ℕ} (hm : (m : with_top ℕ) ≤ n) + (hf : cont_diff 𝕜 n f) : continuous (λ x, iterated_fderiv 𝕜 m f x) := +(cont_diff_iff_continuous_differentiable.mp hf).1 m hm + +/-- If `f` is `C^n` then its `m`-times iterated derivative is differentiable for `m < n`. -/ +lemma cont_diff.differentiable_iterated_fderiv {m : ℕ} (hm : (m : with_top ℕ) < n) + (hf : cont_diff 𝕜 n f) : differentiable 𝕜 (λ x, iterated_fderiv 𝕜 m f x) := +(cont_diff_iff_continuous_differentiable.mp hf).2 m hm + lemma cont_diff_of_differentiable_iterated_fderiv (h : ∀(m : ℕ), (m : with_top ℕ) ≤ n → differentiable 𝕜 (iterated_fderiv 𝕜 m f)) : cont_diff 𝕜 n f := @@ -1568,7 +1578,7 @@ end /-! ### Constants -/ -lemma iterated_fderiv_within_zero_fun {n : ℕ} : +@[simp] lemma iterated_fderiv_zero_fun {n : ℕ} : iterated_fderiv 𝕜 n (λ x : E, (0 : F)) = 0 := begin induction n with n IH, @@ -1584,7 +1594,7 @@ lemma cont_diff_zero_fun : cont_diff 𝕜 n (λ x : E, (0 : F)) := begin apply cont_diff_of_differentiable_iterated_fderiv (λm hm, _), - rw iterated_fderiv_within_zero_fun, + rw iterated_fderiv_zero_fun, apply differentiable_const (0 : (E [×m]→L[𝕜] F)) end @@ -2253,8 +2263,7 @@ lemma cont_diff_prod_assoc_symm : cont_diff 𝕜 ⊤ $ (equiv.prod_assoc E F G). /-- The bundled derivative of a `C^{n+1}` function is `C^n`. -/ lemma cont_diff_on_fderiv_within_apply {m n : with_top ℕ} {s : set E} {f : E → F} (hf : cont_diff_on 𝕜 n f s) (hs : unique_diff_on 𝕜 s) (hmn : m + 1 ≤ n) : - cont_diff_on 𝕜 m (λp : E × E, (fderiv_within 𝕜 f s p.1 : E →L[𝕜] F) p.2) - (s ×ˢ (univ : set E)) := + cont_diff_on 𝕜 m (λp : E × E, (fderiv_within 𝕜 f s p.1 : E →L[𝕜] F) p.2) (s ×ˢ univ) := begin have A : cont_diff 𝕜 m (λp : (E →L[𝕜] F) × E, p.1 p.2), { apply is_bounded_bilinear_map.cont_diff, @@ -2604,6 +2613,41 @@ lemma cont_diff_on.smul {s : set E} {f : E → 𝕜} {g : E → F} cont_diff_on 𝕜 n (λ x, f x • g x) s := λ x hx, (hf x hx).smul (hg x hx) +/-! ### Constant scalar multiplication -/ + +section const_smul + +variables {R : Type*} [semiring R] [module R F] [smul_comm_class 𝕜 R F] +variables [has_continuous_const_smul R F] + +/- The scalar multiplication with a constant is smooth. -/ +lemma cont_diff_const_smul (c : R) : cont_diff 𝕜 n (λ p : F, c • p) := +(c • continuous_linear_map.id 𝕜 F).cont_diff + +/-- The scalar multiplication of a constant and a `C^n` function within a set at a point is `C^n` +within this set at this point. -/ +lemma cont_diff_within_at.const_smul {s : set E} {f : E → F} {x : E} (c : R) + (hf : cont_diff_within_at 𝕜 n f s x) : cont_diff_within_at 𝕜 n (λ y, c • f y) s x := +(cont_diff_const_smul c).cont_diff_at.comp_cont_diff_within_at x hf + +/-- The scalar multiplication of a constant and a `C^n` function at a point is `C^n` at this +point. -/ +lemma cont_diff_at.const_smul {f : E → F} {x : E} (c : R) + (hf : cont_diff_at 𝕜 n f x) : cont_diff_at 𝕜 n (λ y, c • f y) x := +by rw [←cont_diff_within_at_univ] at *; exact hf.const_smul c + +/-- The scalar multiplication of a constant and a `C^n` function is `C^n`. -/ +lemma cont_diff.const_smul {f : E → F} (c : R) + (hf : cont_diff 𝕜 n f) : cont_diff 𝕜 n (λ y, c • f y) := +(cont_diff_const_smul c).comp hf + +/-- The scalar multiplication of a constant and a `C^n` on a domain is `C^n`. -/ +lemma cont_diff_on.const_smul {s : set E} {f : E → F} (c : R) + (hf : cont_diff_on 𝕜 n f s) : cont_diff_on 𝕜 n (λ y, c • f y) s := +λ x hx, (hf x hx).const_smul c + +end const_smul + /-! ### Cartesian product of two functions -/ section prod_map diff --git a/formal/lean/mathlib/analysis/calculus/fderiv.lean b/formal/lean/mathlib/analysis/calculus/fderiv.lean index bee2e8b8abe3a5edb7f849bd56dd4a4091c12eda..2a18bd143f570afa19e09f1165eee2c20947c408 100644 --- a/formal/lean/mathlib/analysis/calculus/fderiv.lean +++ b/formal/lean/mathlib/analysis/calculus/fderiv.lean @@ -496,6 +496,11 @@ begin exact ⟨f', hf'.mono st⟩ end +lemma differentiable_within_at.mono_of_mem (h : differentiable_within_at 𝕜 f s x) {t : set E} + (hst : s ∈ nhds_within x t) : + differentiable_within_at 𝕜 f t x := +(h.has_fderiv_within_at.mono_of_mem hst).differentiable_within_at + lemma differentiable_within_at_univ : differentiable_within_at 𝕜 f univ x ↔ differentiable_at 𝕜 f x := by simp only [differentiable_within_at, has_fderiv_within_at_univ, differentiable_at] diff --git a/formal/lean/mathlib/analysis/calculus/implicit.lean b/formal/lean/mathlib/analysis/calculus/implicit.lean index 47aad7899c0a5c3522347ac76094eca8bc066968..622726d32387ca0a5c62000ec597907cebdc56ef 100644 --- a/formal/lean/mathlib/analysis/calculus/implicit.lean +++ b/formal/lean/mathlib/analysis/calculus/implicit.lean @@ -87,7 +87,7 @@ such that * both functions are strictly differentiable at `a`; * the derivatives are surjective; * the kernels of the derivatives are complementary subspaces of `E`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure implicit_function_data (𝕜 : Type*) [nontrivially_normed_field 𝕜] (E : Type*) [normed_add_comm_group E] [normed_space 𝕜 E] [complete_space E] (F : Type*) [normed_add_comm_group F] [normed_space 𝕜 F] [complete_space F] diff --git a/formal/lean/mathlib/analysis/complex/basic.lean b/formal/lean/mathlib/analysis/complex/basic.lean index fd4b701a61d4fd8dc48ae1b9ded615168a2d7521..b7f9f9f6324fe1724825668eff450d03098b295a 100644 --- a/formal/lean/mathlib/analysis/complex/basic.lean +++ b/formal/lean/mathlib/analysis/complex/basic.lean @@ -50,8 +50,10 @@ instance : normed_field ℂ := norm_mul' := abs_mul, .. complex.field, .. complex.normed_add_comm_group } -instance : nontrivially_normed_field ℂ := -{ non_trivial := ⟨2, by simp; norm_num⟩ } +instance : densely_normed_field ℂ := +{ lt_norm_lt := λ r₁ r₂ h₀ hr, let ⟨x, h⟩ := normed_field.exists_lt_norm_lt ℝ h₀ hr in + have this : ∥(∥x∥ : ℂ)∥ = ∥(∥x∥)∥, by simp only [norm_eq_abs, abs_of_real, real.norm_eq_abs], + ⟨∥x∥, by rwa [this, norm_norm]⟩ } instance {R : Type*} [normed_field R] [normed_algebra R ℝ] : normed_algebra R ℂ := { norm_smul_le := λ r x, begin diff --git a/formal/lean/mathlib/analysis/convex/basic.lean b/formal/lean/mathlib/analysis/convex/basic.lean index 5a4cf4f7b91f8001e20eb9ec541e980b1a64e728..afa93ca3e9711a03c00db8ad95b07332e1f6a9a1 100644 --- a/formal/lean/mathlib/analysis/convex/basic.lean +++ b/formal/lean/mathlib/analysis/convex/basic.lean @@ -8,6 +8,7 @@ import algebra.order.module import linear_algebra.affine_space.midpoint import linear_algebra.affine_space.affine_subspace import linear_algebra.ray +import tactic.positivity /-! # Convex sets and functions in vector spaces @@ -331,8 +332,8 @@ begin use [a, b, ha, hb], rw [hab, div_one, div_one] }, { rintro ⟨a, b, ha, hb, rfl⟩, - have hab : 0 < a + b, from add_pos ha hb, - refine ⟨a / (a + b), b / (a + b), div_pos ha hab, div_pos hb hab, _, rfl⟩, + have hab : 0 < a + b := by positivity, + refine ⟨a / (a + b), b / (a + b), by positivity, by positivity, _, rfl⟩, rw [← add_div, div_self hab.ne'] } end @@ -1005,9 +1006,9 @@ begin exact ⟨p • v, q • v, smul_mem_smul_set hv, smul_mem_smul_set hv, (add_smul _ _ _).symm⟩ }, { rintro ⟨v₁, v₂, ⟨v₁₁, h₁₂, rfl⟩, ⟨v₂₁, h₂₂, rfl⟩, rfl⟩, have hpq := add_pos hp' hq', - exact mem_smul_set.2 ⟨_, h_conv h₁₂ h₂₂ (div_pos hp' hpq).le (div_pos hq' hpq).le + refine mem_smul_set.2 ⟨_, h_conv h₁₂ h₂₂ _ _ (by rw [←div_self hpq.ne', add_div] : p / (p + q) + q / (p + q) = 1), - by simp only [← mul_smul, smul_add, mul_div_cancel' _ hpq.ne']⟩ } + by simp only [← mul_smul, smul_add, mul_div_cancel' _ hpq.ne']⟩; positivity } end end add_comm_group diff --git a/formal/lean/mathlib/analysis/convex/combination.lean b/formal/lean/mathlib/analysis/convex/combination.lean index 6787f8a41fd3ee399d8f2cac21b36e7f1d1272fa..9955898d66203c369f83f85cc43aa58f8420bbda 100644 --- a/formal/lean/mathlib/analysis/convex/combination.lean +++ b/formal/lean/mathlib/analysis/convex/combination.lean @@ -352,7 +352,7 @@ begin rw convex_hull_eq at ⊢ hx hy, obtain ⟨ι, a, w, S, hw, hw', hS, hSp⟩ := hx, obtain ⟨κ, b, v, T, hv, hv', hT, hTp⟩ := hy, - have h_sum : ∑ (i : ι × κ) in a.product b, w i.fst * v i.snd = 1, + have h_sum : ∑ (i : ι × κ) in a ×ˢ b, w i.fst * v i.snd = 1, { rw [finset.sum_product, ← hw'], congr, ext i, @@ -360,7 +360,7 @@ begin { congr, ext, simp [mul_comm] }, rw [this, ← finset.sum_mul, hv'], simp }, - refine ⟨ι × κ, a.product b, λ p, (w p.1) * (v p.2), λ p, (S p.1, T p.2), + refine ⟨ι × κ, a ×ˢ b, λ p, (w p.1) * (v p.2), λ p, (S p.1, T p.2), λ p hp, _, h_sum, λ p hp, _, _⟩, { rw mem_product at hp, exact mul_nonneg (hw p.1 hp.1) (hv p.2 hp.2) }, diff --git a/formal/lean/mathlib/analysis/convex/cone.lean b/formal/lean/mathlib/analysis/convex/cone.lean index 66c018ce84c98392987f8601220d549bb297d96e..9ce0f7558baf31d238d0be6f281375bad09a5813 100644 --- a/formal/lean/mathlib/analysis/convex/cone.lean +++ b/formal/lean/mathlib/analysis/convex/cone.lean @@ -74,35 +74,24 @@ variables [ordered_semiring 𝕜] [add_comm_monoid E] section has_smul variables [has_smul 𝕜 E] (S T : convex_cone 𝕜 E) -instance : has_coe (convex_cone 𝕜 E) (set E) := ⟨convex_cone.carrier⟩ - -instance : has_mem E (convex_cone 𝕜 E) := ⟨λ m S, m ∈ S.carrier⟩ - -instance : has_le (convex_cone 𝕜 E) := ⟨λ S T, (S : set E) ⊆ T⟩ - -instance : has_lt (convex_cone 𝕜 E) := ⟨λ S T, (S : set E) ⊂ T⟩ - -@[simp, norm_cast] lemma mem_coe {x : E} : x ∈ (S : set E) ↔ x ∈ S := iff.rfl +instance : set_like (convex_cone 𝕜 E) E := +{ coe := carrier, + coe_injective' := λ S T h, by cases S; cases T; congr' } @[simp] lemma coe_mk {s : set E} {h₁ h₂} : ↑(@mk 𝕜 _ _ _ _ s h₁ h₂) = s := rfl @[simp] lemma mem_mk {s : set E} {h₁ h₂ x} : x ∈ @mk 𝕜 _ _ _ _ s h₁ h₂ ↔ x ∈ s := iff.rfl -/-- Two `convex_cone`s are equal if the underlying sets are equal. -/ -theorem ext' {S T : convex_cone 𝕜 E} (h : (S : set E) = T) : S = T := -by cases S; cases T; congr' - -/-- Two `convex_cone`s are equal if and only if the underlying sets are equal. -/ -protected theorem ext'_iff {S T : convex_cone 𝕜 E} : (S : set E) = T ↔ S = T := -⟨ext', λ h, h ▸ rfl⟩ - /-- Two `convex_cone`s are equal if they have the same elements. -/ -@[ext] theorem ext {S T : convex_cone 𝕜 E} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T := ext' $ set.ext h +@[ext] theorem ext {S T : convex_cone 𝕜 E} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T := set_like.ext h lemma smul_mem {c : 𝕜} {x : E} (hc : 0 < c) (hx : x ∈ S) : c • x ∈ S := S.smul_mem' hc hx lemma add_mem ⦃x⦄ (hx : x ∈ S) ⦃y⦄ (hy : y ∈ S) : x + y ∈ S := S.add_mem' hx hy +instance : add_mem_class (convex_cone 𝕜 E) E := +{ add_mem := λ c a b ha hb, add_mem c ha hb } + instance : has_inf (convex_cone 𝕜 E) := ⟨λ S T, ⟨S ∩ T, λ c hc x hx, ⟨S.smul_mem hc hx.1, T.smul_mem hc hx.2⟩, λ x hx y hy, ⟨S.add_mem hx.1 hy.1, T.add_mem hx.2 hy.2⟩⟩⟩ @@ -161,7 +150,7 @@ instance : complete_lattice (convex_cone 𝕜 E) := Sup_le := λ s p hs x hx, mem_Inf.1 hx p hs, le_Inf := λ s a ha x hx, mem_Inf.2 $ λ t ht, ha t ht hx, Inf_le := λ s a ha x hx, mem_Inf.1 hx _ ha, - .. partial_order.lift (coe : convex_cone 𝕜 E → set E) (λ a b, ext') } + .. set_like.partial_order } instance : inhabited (convex_cone 𝕜 E) := ⟨⊥⟩ @@ -202,11 +191,16 @@ def map (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 E) : convex_cone 𝕜 F := add_mem' := λ y₁ ⟨x₁, hx₁, hy₁⟩ y₂ ⟨x₂, hx₂, hy₂⟩, hy₁ ▸ hy₂ ▸ f.map_add x₁ x₂ ▸ mem_image_of_mem f (S.add_mem hx₁ hx₂) } +@[simp] lemma mem_map {f : E →ₗ[𝕜] F} {S : convex_cone 𝕜 E} {y : F} : + y ∈ S.map f ↔ ∃ x ∈ S, f x = y := +mem_image_iff_bex + lemma map_map (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 E) : (S.map f).map g = S.map (g.comp f) := -ext' $ image_image g f S +set_like.coe_injective $ image_image g f S -@[simp] lemma map_id (S : convex_cone 𝕜 E) : S.map linear_map.id = S := ext' $ image_id _ +@[simp] lemma map_id (S : convex_cone 𝕜 E) : S.map linear_map.id = S := +set_like.coe_injective $ image_id _ /-- The preimage of a convex cone under a `𝕜`-linear map is a convex cone. -/ def comap (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 F) : convex_cone 𝕜 E := @@ -214,11 +208,14 @@ def comap (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 F) : convex_cone 𝕜 E : smul_mem' := λ c hc x hx, by { rw [mem_preimage, f.map_smul c], exact S.smul_mem hc hx }, add_mem' := λ x hx y hy, by { rw [mem_preimage, f.map_add], exact S.add_mem hx hy } } -@[simp] lemma comap_id (S : convex_cone 𝕜 E) : S.comap linear_map.id = S := ext' preimage_id +@[simp] lemma coe_comap (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 F) : (S.comap f : set E) = f ⁻¹' S := rfl + +@[simp] lemma comap_id (S : convex_cone 𝕜 E) : S.comap linear_map.id = S := +set_like.coe_injective preimage_id lemma comap_comap (g : F →ₗ[𝕜] G) (f : E →ₗ[𝕜] F) (S : convex_cone 𝕜 G) : (S.comap g).comap f = S.comap (g.comp f) := -ext' $ preimage_comp.symm +set_like.coe_injective $ preimage_comp.symm @[simp] lemma mem_comap {f : E →ₗ[𝕜] F} {S : convex_cone 𝕜 F} {x : E} : x ∈ S.comap f ↔ f x ∈ S := iff.rfl @@ -265,6 +262,10 @@ lemma pointed_iff_not_blunt (S : convex_cone 𝕜 E) : S.pointed ↔ ¬S.blunt : lemma blunt_iff_not_pointed (S : convex_cone 𝕜 E) : S.blunt ↔ ¬S.pointed := by rw [pointed_iff_not_blunt, not_not] +lemma pointed.mono {S T : convex_cone 𝕜 E} (h : S ≤ T) : S.pointed → T.pointed := @h _ + +lemma blunt.anti {S T : convex_cone 𝕜 E} (h : T ≤ S) : S.blunt → T.blunt := (∘ @@h) + end add_comm_monoid section add_comm_group @@ -287,6 +288,12 @@ begin exact h } end +lemma flat.mono {S T : convex_cone 𝕜 E} (h : S ≤ T) : S.flat → T.flat +| ⟨x, hxS, hx, hnxS⟩ := ⟨x, h hxS, hx, h hnxS⟩ + +lemma salient.anti {S T : convex_cone 𝕜 E} (h : T ≤ S) : S.salient → T.salient := +λ hS x hxT hx hnT, hS x (h hxT) hx (h hnT) + /-- A flat cone is always pointed (contains `0`). -/ lemma flat.pointed {S : convex_cone 𝕜 E} (hS : S.flat) : S.pointed := begin @@ -346,18 +353,16 @@ variables (𝕜 E) [ordered_semiring 𝕜] [ordered_add_comm_group E] [module The positive cone is the convex cone formed by the set of nonnegative elements in an ordered module. -/ -def positive_cone : convex_cone 𝕜 E := -{ carrier := {x | 0 ≤ x}, - smul_mem' := - begin - rintro c hc x (hx : _ ≤ _), - rw ←smul_zero c, - exact smul_le_smul_of_nonneg hx hc.le, - end, +def positive : convex_cone 𝕜 E := +{ carrier := set.Ici 0, + smul_mem' := λ c hc x (hx : _ ≤ _), smul_nonneg hc.le hx, add_mem' := λ x (hx : _ ≤ _) y (hy : _ ≤ _), add_nonneg hx hy } +@[simp] lemma mem_positive {x : E} : x ∈ positive 𝕜 E ↔ 0 ≤ x := iff.rfl +@[simp] lemma coe_positive : ↑(positive 𝕜 E) = set.Ici (0 : E) := rfl + /-- The positive cone of an ordered module is always salient. -/ -lemma salient_positive_cone : salient (positive_cone 𝕜 E) := +lemma salient_positive : salient (positive 𝕜 E) := λ x xs hx hx', lt_irrefl (0 : E) (calc 0 < x : lt_of_le_of_ne xs hx.symm @@ -365,7 +370,28 @@ lemma salient_positive_cone : salient (positive_cone 𝕜 E) := ... = 0 : add_neg_self x) /-- The positive cone of an ordered module is always pointed. -/ -lemma pointed_positive_cone : pointed (positive_cone 𝕜 E) := le_refl 0 +lemma pointed_positive : pointed (positive 𝕜 E) := le_refl 0 + +/-- The cone of strictly positive elements. + +Note that this naming diverges from the mathlib convention of `pos` and `nonneg` due to "positive +cone" (`convex_cone.positive`) being established terminology for the non-negative elements. -/ +def strictly_positive : convex_cone 𝕜 E := +{ carrier := set.Ioi 0, + smul_mem' := λ c hc x (hx : _ < _), smul_pos hc hx, + add_mem' := λ x hx y hy, add_pos hx hy } + +@[simp] lemma mem_strictly_positive {x : E} : x ∈ strictly_positive 𝕜 E ↔ 0 < x := iff.rfl +@[simp] lemma coe_strictly_positive : ↑(strictly_positive 𝕜 E) = set.Ioi (0 : E) := rfl + +lemma positive_le_strictly_positive : strictly_positive 𝕜 E ≤ positive 𝕜 E := λ x, le_of_lt + +/-- The strictly positive cone of an ordered module is always salient. -/ +lemma salient_strictly_positive : salient (strictly_positive 𝕜 E) := +(salient_positive 𝕜 E).anti $ positive_le_strictly_positive 𝕜 E + +/-- The strictly positive cone of an ordered module is always blunt. -/ +lemma blunt_strictly_positive : blunt (strictly_positive 𝕜 E) := lt_irrefl 0 end positive_cone end convex_cone @@ -373,7 +399,7 @@ end convex_cone /-! ### Cone over a convex set -/ section cone_from_convex -variables [linear_ordered_field 𝕜] [ordered_add_comm_group E] [module 𝕜 E] +variables [linear_ordered_field 𝕜] [add_comm_group E] [module 𝕜 E] namespace convex @@ -430,7 +456,7 @@ end lemma convex_hull_to_cone_eq_Inf (s : set E) : (convex_convex_hull 𝕜 s).to_cone _ = Inf {t : convex_cone 𝕜 E | s ⊆ t} := -(convex_hull_to_cone_is_least s).is_glb.Inf_eq.symm +eq.symm $ is_glb.Inf_eq $ is_least.is_glb $ convex_hull_to_cone_is_least s end cone_from_convex @@ -616,12 +642,11 @@ def set.inner_dual_cone (s : set H) : convex_cone ℝ H := exact add_nonneg (hu x hx) (hv x hx) end } -lemma mem_inner_dual_cone (y : H) (s : set H) : - y ∈ s.inner_dual_cone ↔ ∀ x ∈ s, 0 ≤ ⟪ x, y ⟫ := by refl +@[simp] lemma mem_inner_dual_cone (y : H) (s : set H) : + y ∈ s.inner_dual_cone ↔ ∀ x ∈ s, 0 ≤ ⟪ x, y ⟫ := iff.rfl @[simp] lemma inner_dual_cone_empty : (∅ : set H).inner_dual_cone = ⊤ := -convex_cone.ext' (eq_univ_of_forall - (λ x y hy, false.elim (set.not_mem_empty _ hy))) +eq_top_iff.mpr $ λ x hy y, false.elim lemma inner_dual_cone_le_inner_dual_cone (h : t ⊆ s) : s.inner_dual_cone ≤ t.inner_dual_cone := @@ -630,17 +655,56 @@ lemma inner_dual_cone_le_inner_dual_cone (h : t ⊆ s) : lemma pointed_inner_dual_cone : s.inner_dual_cone.pointed := λ x hx, by rw inner_zero_right +/-- The inner dual cone of a singleton is given by the preimage of the positive cone under the +linear map `λ y, ⟪x, y⟫`. -/ +lemma inner_dual_cone_singleton (x : H) : + ({x} : set H).inner_dual_cone = (convex_cone.positive ℝ ℝ).comap (innerₛₗ x) := +convex_cone.ext $ λ i, forall_eq + +lemma inner_dual_cone_union (s t : set H) : + (s ∪ t).inner_dual_cone = s.inner_dual_cone ⊓ t.inner_dual_cone := +le_antisymm + (le_inf (λ x hx y hy, hx _ $ or.inl hy) (λ x hx y hy, hx _ $ or.inr hy)) + (λ x hx y, or.rec (hx.1 _) (hx.2 _)) + +lemma inner_dual_cone_insert (x : H) (s : set H) : + (insert x s).inner_dual_cone = set.inner_dual_cone {x} ⊓ s.inner_dual_cone := +by rw [insert_eq, inner_dual_cone_union] + +lemma inner_dual_cone_Union {ι : Sort*} (f : ι → set H) : + (⋃ i, f i).inner_dual_cone = ⨅ i, (f i).inner_dual_cone := +begin + refine le_antisymm (le_infi $ λ i x hx y hy, hx _ $ mem_Union_of_mem _ hy) _, + intros x hx y hy, + rw [convex_cone.mem_infi] at hx, + obtain ⟨j, hj⟩ := mem_Union.mp hy, + exact hx _ _ hj, +end + +lemma inner_dual_cone_sUnion (S : set (set H)) : + (⋃₀ S).inner_dual_cone = Inf (set.inner_dual_cone '' S) := +by simp_rw [Inf_image, sUnion_eq_bUnion, inner_dual_cone_Union] + /-- The dual cone of `s` equals the intersection of dual cones of the points in `s`. -/ lemma inner_dual_cone_eq_Inter_inner_dual_cone_singleton : (s.inner_dual_cone : set H) = ⋂ i : s, (({i} : set H).inner_dual_cone : set H) := +by rw [←convex_cone.coe_infi, ←inner_dual_cone_Union, Union_of_singleton_coe] + +lemma is_closed_inner_dual_cone : is_closed (s.inner_dual_cone : set H) := begin - simp_rw [set.Inter_coe_set, subtype.coe_mk], - refine set.ext (λ x, iff.intro (λ hx, _) _), - { refine set.mem_Inter.2 (λ i, set.mem_Inter.2 (λ hi _, _)), - rintro ⟨ ⟩, - exact hx i hi }, - { simp only [set.mem_Inter, convex_cone.mem_coe, mem_inner_dual_cone, - set.mem_singleton_iff, forall_eq, imp_self] } + -- reduce the problem to showing that dual cone of a singleton `{x}` is closed + rw inner_dual_cone_eq_Inter_inner_dual_cone_singleton, + apply is_closed_Inter, + intros x, + + -- the dual cone of a singleton `{x}` is the preimage of `[0, ∞)` under `inner x` + have h : ↑({x} : set H).inner_dual_cone = (inner x : H → ℝ) ⁻¹' set.Ici 0, + { rw [inner_dual_cone_singleton, convex_cone.coe_comap, convex_cone.coe_positive, + innerₛₗ_apply_coe] }, + + -- the preimage is closed as `inner x` is continuous and `[0, ∞)` is closed + rw h, + exact is_closed_Ici.preimage (by continuity), end end dual diff --git a/formal/lean/mathlib/analysis/convex/gauge.lean b/formal/lean/mathlib/analysis/convex/gauge.lean index 33907add5532f812640f6f46ea57c2473983d75b..46f967daad3d7c1ac157ffa89b3ace29451efcf4 100644 --- a/formal/lean/mathlib/analysis/convex/gauge.lean +++ b/formal/lean/mathlib/analysis/convex/gauge.lean @@ -6,6 +6,7 @@ Authors: Yaël Dillies, Bhavik Mehta import analysis.convex.star import analysis.normed_space.pointwise import analysis.seminorm +import data.complex.is_R_or_C import tactic.congrm /-! @@ -40,7 +41,7 @@ open_locale pointwise noncomputable theory -variables {E : Type*} +variables {𝕜 E F : Type*} section add_comm_group variables [add_comm_group E] [module ℝ E] @@ -239,18 +240,6 @@ begin exact smul_mem_smul_set hx } end -/-- In textbooks, this is the homogeneity of the Minkowksi functional. -/ -lemma gauge_smul [module α E] [is_scalar_tower α ℝ (set E)] {s : set E} - (symmetric : ∀ x ∈ s, -x ∈ s) (r : α) (x : E) : - gauge s (r • x) = abs r • gauge s x := -begin - rw ←gauge_smul_of_nonneg (abs_nonneg r), - obtain h | h := abs_choice r, - { rw h }, - { rw [h, neg_smul, gauge_neg symmetric] }, - { apply_instance } -end - lemma gauge_smul_left_of_nonneg [mul_action_with_zero α E] [smul_comm_class α ℝ ℝ] [is_scalar_tower α ℝ ℝ] [is_scalar_tower α ℝ E] {s : set E} {a : α} (ha : 0 ≤ a) : gauge (a • s) = a⁻¹ • gauge s := @@ -292,6 +281,26 @@ end end linear_ordered_field +section is_R_or_C +variables [is_R_or_C 𝕜] [module 𝕜 E] [is_scalar_tower ℝ 𝕜 E] + +lemma gauge_norm_smul (hs : balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (∥r∥ • x) = gauge s (r • x) := +begin + rw @is_R_or_C.real_smul_eq_coe_smul 𝕜, + obtain rfl | hr := eq_or_ne r 0, + { simp only [norm_zero, is_R_or_C.of_real_zero] }, + unfold gauge, + congr' with θ, + refine and_congr_right (λ hθ, (hs.smul _).mem_smul_iff _), + rw [is_R_or_C.norm_of_real, norm_norm], +end + +/-- If `s` is balanced, then the Minkowski functional is ℂ-homogeneous. -/ +lemma gauge_smul (hs : balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (r • x) = ∥r∥ * gauge s x := +by { rw [←smul_eq_mul, ←gauge_smul_of_nonneg (norm_nonneg r), gauge_norm_smul hs], apply_instance } + +end is_R_or_C + section topological_space variables [topological_space E] [has_continuous_smul ℝ E] @@ -365,24 +374,22 @@ begin mul_inv_cancel hb.ne', ←smul_add, one_div, ←mem_smul_set_iff_inv_smul_mem₀ hab.ne'] at this, end -/-- `gauge s` as a seminorm when `s` is symmetric, convex and absorbent. -/ -@[simps] def gauge_seminorm (hs₀ : ∀ x ∈ s, -x ∈ s) (hs₁ : convex ℝ s) (hs₂ : absorbent ℝ s) : - seminorm ℝ E := -seminorm.of (gauge s) (gauge_add_le hs₁ hs₂) - (λ r x, by rw [gauge_smul hs₀, real.norm_eq_abs, smul_eq_mul]; apply_instance) +section is_R_or_C +variables [is_R_or_C 𝕜] [module 𝕜 E] [is_scalar_tower ℝ 𝕜 E] -section gauge_seminorm -variables {hs₀ : ∀ x ∈ s, -x ∈ s} {hs₁ : convex ℝ s} {hs₂ : absorbent ℝ s} +/-- `gauge s` as a seminorm when `s` is balanced, convex and absorbent. -/ +@[simps] def gauge_seminorm (hs₀ : balanced 𝕜 s) (hs₁ : convex ℝ s) (hs₂ : absorbent ℝ s) : + seminorm 𝕜 E := +seminorm.of (gauge s) (gauge_add_le hs₁ hs₂) (gauge_smul hs₀) -section topological_space -variables [topological_space E] [has_continuous_smul ℝ E] +variables {hs₀ : balanced 𝕜 s} {hs₁ : convex ℝ s} {hs₂ : absorbent ℝ s} [topological_space E] + [has_continuous_smul ℝ E] lemma gauge_seminorm_lt_one_of_open (hs : is_open s) {x : E} (hx : x ∈ s) : gauge_seminorm hs₀ hs₁ hs₂ x < 1 := gauge_lt_one_of_mem_of_open hs₁ hs₂.zero_mem hs hx -end topological_space -end gauge_seminorm +end is_R_or_C /-- Any seminorm arises as the gauge of its unit ball. -/ @[simp] protected lemma seminorm.gauge_ball (p : seminorm ℝ E) : gauge (p.ball 0 1) = p := @@ -410,7 +417,7 @@ begin end lemma seminorm.gauge_seminorm_ball (p : seminorm ℝ E) : - gauge_seminorm (λ x, p.symmetric_ball_zero 1) (p.convex_ball 0 1) + gauge_seminorm (p.balanced_ball_zero 1) (p.convex_ball 0 1) (p.absorbent_ball_zero zero_lt_one) = p := fun_like.coe_injective p.gauge_ball end add_comm_group diff --git a/formal/lean/mathlib/analysis/inner_product_space/adjoint.lean b/formal/lean/mathlib/analysis/inner_product_space/adjoint.lean index d9f6ea3d84a1a560d3c34b78dd30cb2a95ab21ff..3d98aedfa5af2527fbc6462b8f4a210fee333ea4 100644 --- a/formal/lean/mathlib/analysis/inner_product_space/adjoint.lean +++ b/formal/lean/mathlib/analysis/inner_product_space/adjoint.lean @@ -378,8 +378,9 @@ local attribute [instance, priority 20] finite_dimensional.complete /-- The adjoint of an operator from the finite-dimensional inner product space E to the finite- dimensional inner product space F. -/ def adjoint : (E →ₗ[𝕜] F) ≃ₗ⋆[𝕜] (F →ₗ[𝕜] E) := - (linear_map.to_continuous_linear_map.trans continuous_linear_map.adjoint.to_linear_equiv).trans - linear_map.to_continuous_linear_map.symm +((linear_map.to_continuous_linear_map : (E →ₗ[𝕜] F) ≃ₗ[𝕜] (E →L[𝕜] F)).trans + continuous_linear_map.adjoint.to_linear_equiv).trans + linear_map.to_continuous_linear_map.symm lemma adjoint_to_continuous_linear_map (A : E →ₗ[𝕜] F) : A.adjoint.to_continuous_linear_map = A.to_continuous_linear_map.adjoint := rfl diff --git a/formal/lean/mathlib/analysis/inner_product_space/basic.lean b/formal/lean/mathlib/analysis/inner_product_space/basic.lean index 66177b2cc0ef1f899fef05f976111f141c2690ce..94fa118b3e1c4769c7ab633d6d56abe87a58d294 100644 --- a/formal/lean/mathlib/analysis/inner_product_space/basic.lean +++ b/formal/lean/mathlib/analysis/inner_product_space/basic.lean @@ -128,7 +128,7 @@ instance defined on it, otherwise this will create a second non-defeq norm insta /-- A structure requiring that a scalar product is positive definite and symmetric, from which one can construct an `inner_product_space` instance in `inner_product_space.of_core`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure inner_product_space.core (𝕜 : Type*) (F : Type*) [is_R_or_C 𝕜] [add_comm_group F] [module 𝕜 F] := @@ -1684,7 +1684,8 @@ linear_map.mk_continuous₂ innerₛₗ 1 `inner_product_space.dual` as `to_dual_map`. -/ @[simp] lemma innerSL_apply_norm {x : E} : ∥(innerSL x : E →L[𝕜] 𝕜)∥ = ∥x∥ := begin - refine le_antisymm ((innerSL x).op_norm_le_bound (norm_nonneg _) (λ y, norm_inner_le_norm _ _)) _, + refine le_antisymm ((innerSL x : E →L[𝕜] 𝕜).op_norm_le_bound (norm_nonneg _) + (λ y, norm_inner_le_norm _ _)) _, cases eq_or_lt_of_le (norm_nonneg x) with h h, { have : x = 0 := norm_eq_zero.mp (eq.symm h), simp [this] }, @@ -1693,7 +1694,7 @@ begin ... = re ⟪x, x⟫ : norm_sq_eq_inner _ ... ≤ abs ⟪x, x⟫ : re_le_abs _ ... = ∥innerSL x x∥ : by { rw [←is_R_or_C.norm_eq_abs], refl } - ... ≤ ∥innerSL x∥ * ∥x∥ : (innerSL x).le_op_norm _ } + ... ≤ ∥innerSL x∥ * ∥x∥ : (innerSL x : E →L[𝕜] 𝕜).le_op_norm _ } end /-- The inner product as a continuous sesquilinear map, with the two arguments flipped. -/ @@ -2132,6 +2133,7 @@ lemma continuous_on.inner (hf : continuous_on f s) (hg : continuous_on g s) : continuous_on (λ t, ⟪f t, g t⟫) s := λ x hx, (hf x hx).inner (hg x hx) +@[continuity] lemma continuous.inner (hf : continuous f) (hg : continuous g) : continuous (λ t, ⟪f t, g t⟫) := continuous_iff_continuous_at.2 $ λ x, hf.continuous_at.inner hg.continuous_at @@ -2311,4 +2313,12 @@ begin rwa [h, inf_comm, top_inf_eq] at this end +lemma submodule.orthogonal_family_self : + @orthogonal_family 𝕜 E _ _ _ (λ b, ((cond b K Kᗮ : submodule 𝕜 E) : Type*)) _ + (λ b, (cond b K Kᗮ).subtypeₗᵢ) +| tt tt := absurd rfl +| tt ff := λ _ x y, submodule.inner_right_of_mem_orthogonal x.prop y.prop +| ff tt := λ _ x y, submodule.inner_left_of_mem_orthogonal y.prop x.prop +| ff ff := absurd rfl + end orthogonal diff --git a/formal/lean/mathlib/analysis/inner_product_space/calculus.lean b/formal/lean/mathlib/analysis/inner_product_space/calculus.lean index 35910f4e3d76bfd1e72b892ac97df8339123f442..845bde105be51c20e53310d467a720313c07485b 100644 --- a/formal/lean/mathlib/analysis/inner_product_space/calculus.lean +++ b/formal/lean/mathlib/analysis/inner_product_space/calculus.lean @@ -191,7 +191,7 @@ cont_diff_iff_cont_diff_at.2 $ omit 𝕜 lemma has_strict_fderiv_at_norm_sq (x : F) : - has_strict_fderiv_at (λ x, ∥x∥ ^ 2) (bit0 (innerSL x)) x := + has_strict_fderiv_at (λ x, ∥x∥ ^ 2) (bit0 (innerSL x : F →L[ℝ] ℝ)) x := begin simp only [sq, ← inner_self_eq_norm_mul_norm], convert (has_strict_fderiv_at_id x).inner (has_strict_fderiv_at_id x), diff --git a/formal/lean/mathlib/analysis/inner_product_space/l2_space.lean b/formal/lean/mathlib/analysis/inner_product_space/l2_space.lean index 0b26b94f4c823e0315efd1f012dde740e5d6d4aa..1a07d1c490c3aab0a03c749f77952b3df8534a8c 100644 --- a/formal/lean/mathlib/analysis/inner_product_space/l2_space.lean +++ b/formal/lean/mathlib/analysis/inner_product_space/l2_space.lean @@ -16,6 +16,9 @@ dependent functions `f : Π i, G i` for which `∑' i, ∥f i∥ ^ 2`, the sum o summable. This construction is sometimes called the *Hilbert sum* of the family `G`. By choosing `G` to be `ι → 𝕜`, the Hilbert space `ℓ²(ι, 𝕜)` may be seen as a special case of this construction. +We also define a *predicate* `is_hilbert_sum 𝕜 E V`, where `V : Π i, G i →ₗᵢ[𝕜] E`, expressing that +`V` is an `orthogonal_family` and that the associated map `lp G 2 →ₗᵢ[𝕜] E` is surjective. + ## Main definitions * `orthogonal_family.linear_isometry`: Given a Hilbert space `E`, a family `G` of inner product @@ -23,10 +26,14 @@ summable. This construction is sometimes called the *Hilbert sum* of the family mutually-orthogonal images, there is an induced isometric embedding of the Hilbert sum of `G` into `E`. -* `orthogonal_family.linear_isometry_equiv`: Given a Hilbert space `E`, a family `G` of inner - product spaces and a family `V : Π i, G i →ₗᵢ[𝕜] E` of isometric embeddings of the `G i` into `E` - with mutually-orthogonal images whose span is dense in `E`, there is an induced isometric - isomorphism of the Hilbert sum of `G` with `E`. +* `is_hilbert_sum`: Given a Hilbert space `E`, a family `G` of inner product + spaces and a family `V : Π i, G i →ₗᵢ[𝕜] E` of isometric embeddings of the `G i` into `E`, + `is_hilbert_sum 𝕜 E V` means that `V` is an `orthogonal_family` and that the above + linear isometry is surjective. + +* `is_hilbert_sum.linear_isometry_equiv`: If a Hilbert space `E` is a Hilbert sum of the + inner product spaces `G i` with respect to the family `V : Π i, G i →ₗᵢ[𝕜] E`, then the + corresponding `orthogonal_family.linear_isometry` can be upgraded to a `linear_isometry_equiv`. * `hilbert_basis`: We define a *Hilbert basis* of a Hilbert space `E` to be a structure whose single field `hilbert_basis.repr` is an isometric isomorphism of `E` with `ℓ²(ι, 𝕜)` (i.e., the Hilbert @@ -242,69 +249,128 @@ begin exact hV.linear_isometry.isometry.uniform_inducing.is_complete_range.is_closed } end -/-- A mutually orthogonal family of complete subspaces of `E`, whose range is dense in `E`, induces -a isometric isomorphism from E to `lp 2` of the subspaces. +end orthogonal_family + +section is_hilbert_sum + +variables (𝕜 E) (V : Π i, G i →ₗᵢ[𝕜] E) (F : ι → submodule 𝕜 E) +include cplt + +/-- Given a family of Hilbert spaces `G : ι → Type*`, a Hilbert sum of `G` consists of a Hilbert +space `E` and an orthogonal family `V : Π i, G i →ₗᵢ[𝕜] E` such that the induced isometry +`Φ : lp G 2 → E` is surjective. + +Keeping in mind that `lp G 2` is "the" external Hilbert sum of `G : ι → Type*`, this is analogous +to `direct_sum.is_internal`, except that we don't express it in terms of actual submodules. -/ +@[protect_proj] structure is_hilbert_sum : Prop := of_surjective :: +(orthogonal_family : orthogonal_family 𝕜 V) +(surjective_isometry : function.surjective (orthogonal_family.linear_isometry)) + +variables {𝕜 E V} + +/-- If `V : Π i, G i →ₗᵢ[𝕜] E` is an orthogonal family such that the supremum of the ranges of +`V i` is dense, then `(E, V)` is a Hilbert sum of `G`. -/ +lemma is_hilbert_sum.mk [Π i, complete_space $ G i] + (hVortho : orthogonal_family 𝕜 V) + (hVtotal : ⊤ ≤ (⨆ i, (V i).to_linear_map.range).topological_closure) : + is_hilbert_sum 𝕜 E V := +{ orthogonal_family := hVortho, + surjective_isometry := + begin + rw [←linear_isometry.coe_to_linear_map], + exact linear_map.range_eq_top.mp (eq_top_iff.mpr $ + hVtotal.trans_eq hVortho.range_linear_isometry.symm) + end } + +/-- This is `orthogonal_family.is_hilbert_sum` in the case of actual inclusions from subspaces. -/ +lemma is_hilbert_sum.mk_internal [Π i, complete_space $ F i] + (hFortho : @orthogonal_family 𝕜 E _ _ _ (λ i, F i) _ (λ i, (F i).subtypeₗᵢ)) + (hFtotal : ⊤ ≤ (⨆ i, (F i)).topological_closure) : + @is_hilbert_sum _ 𝕜 _ E _ _ (λ i, F i) _ (λ i, (F i).subtypeₗᵢ) := +is_hilbert_sum.mk hFortho (by simpa [subtypeₗᵢ_to_linear_map, range_subtype] using hFtotal) + +/-- *A* Hilbert sum `(E, V)` of `G` is canonically isomorphic to *the* Hilbert sum of `G`, +i.e `lp G 2`. Note that this goes in the opposite direction from `orthogonal_family.linear_isometry`. -/ -noncomputable def linear_isometry_equiv [Π i, complete_space (G i)] - (hV' : ⊤ ≤ (⨆ i, (V i).to_linear_map.range).topological_closure) : +noncomputable def is_hilbert_sum.linear_isometry_equiv (hV : is_hilbert_sum 𝕜 E V) : E ≃ₗᵢ[𝕜] lp G 2 := linear_isometry_equiv.symm $ linear_isometry_equiv.of_surjective -hV.linear_isometry -begin - rw [←linear_isometry.coe_to_linear_map], - exact linear_map.range_eq_top.mp (eq_top_iff.mpr $ hV'.trans_eq hV.range_linear_isometry.symm) -end +hV.orthogonal_family.linear_isometry hV.surjective_isometry -/-- In the canonical isometric isomorphism `E ≃ₗᵢ[𝕜] lp G 2` induced by an orthogonal family `G`, +/-- In the canonical isometric isomorphism between a Hilbert sum `E` of `G` and `lp G 2`, a vector `w : lp G 2` is the image of the infinite sum of the associated elements in `E`. -/ -protected lemma linear_isometry_equiv_symm_apply [Π i, complete_space (G i)] - (hV' : ⊤ ≤ (⨆ i, (V i).to_linear_map.range).topological_closure) (w : lp G 2) : - (hV.linear_isometry_equiv hV').symm w = ∑' i, V i (w i) := -by simp [orthogonal_family.linear_isometry_equiv, orthogonal_family.linear_isometry_apply] +protected lemma is_hilbert_sum.linear_isometry_equiv_symm_apply + (hV : is_hilbert_sum 𝕜 E V) (w : lp G 2) : + hV.linear_isometry_equiv.symm w = ∑' i, V i (w i) := +by simp [is_hilbert_sum.linear_isometry_equiv, orthogonal_family.linear_isometry_apply] -/-- In the canonical isometric isomorphism `E ≃ₗᵢ[𝕜] lp G 2` induced by an orthogonal family `G`, +/-- In the canonical isometric isomorphism between a Hilbert sum `E` of `G` and `lp G 2`, a vector `w : lp G 2` is the image of the infinite sum of the associated elements in `E`, and this sum indeed converges. -/ -protected lemma has_sum_linear_isometry_equiv_symm [Π i, complete_space (G i)] - (hV' : ⊤ ≤ (⨆ i, (V i).to_linear_map.range).topological_closure) (w : lp G 2) : - has_sum (λ i, V i (w i)) ((hV.linear_isometry_equiv hV').symm w) := -by simp [orthogonal_family.linear_isometry_equiv, orthogonal_family.has_sum_linear_isometry] +protected lemma is_hilbert_sum.has_sum_linear_isometry_equiv_symm + (hV : is_hilbert_sum 𝕜 E V) (w : lp G 2) : + has_sum (λ i, V i (w i)) (hV.linear_isometry_equiv.symm w) := +by simp [is_hilbert_sum.linear_isometry_equiv, orthogonal_family.has_sum_linear_isometry] -/-- In the canonical isometric isomorphism `E ≃ₗᵢ[𝕜] lp G 2` induced by an `ι`-indexed orthogonal -family `G`, an "elementary basis vector" in `lp G 2` supported at `i : ι` is the image of the +/-- In the canonical isometric isomorphism between a Hilbert sum `E` of `G : ι → Type*` and +`lp G 2`, an "elementary basis vector" in `lp G 2` supported at `i : ι` is the image of the associated element in `E`. -/ -@[simp] protected lemma linear_isometry_equiv_symm_apply_single [Π i, complete_space (G i)] - (hV' : ⊤ ≤ (⨆ i, (V i).to_linear_map.range).topological_closure) {i : ι} (x : G i) : - (hV.linear_isometry_equiv hV').symm (lp.single 2 i x) = V i x := -by simp [orthogonal_family.linear_isometry_equiv, orthogonal_family.linear_isometry_apply_single] +@[simp] protected lemma is_hilbert_sum.linear_isometry_equiv_symm_apply_single + (hV : is_hilbert_sum 𝕜 E V) {i : ι} (x : G i) : + hV.linear_isometry_equiv.symm (lp.single 2 i x) = V i x := +by simp [is_hilbert_sum.linear_isometry_equiv, orthogonal_family.linear_isometry_apply_single] -/-- In the canonical isometric isomorphism `E ≃ₗᵢ[𝕜] lp G 2` induced by an `ι`-indexed orthogonal -family `G`, a finitely-supported vector in `lp G 2` is the image of the associated finite sum of +/-- In the canonical isometric isomorphism between a Hilbert sum `E` of `G : ι → Type*` and +`lp G 2`, a finitely-supported vector in `lp G 2` is the image of the associated finite sum of elements of `E`. -/ -@[simp] protected lemma linear_isometry_equiv_symm_apply_dfinsupp_sum_single - [Π i, complete_space (G i)] - (hV' : ⊤ ≤ (⨆ i, (V i).to_linear_map.range).topological_closure) (W₀ : Π₀ (i : ι), G i) : - (hV.linear_isometry_equiv hV').symm (W₀.sum (lp.single 2)) = (W₀.sum (λ i, V i)) := -by simp [orthogonal_family.linear_isometry_equiv, +@[simp] protected lemma is_hilbert_sum.linear_isometry_equiv_symm_apply_dfinsupp_sum_single + (hV : is_hilbert_sum 𝕜 E V) (W₀ : Π₀ (i : ι), G i) : + hV.linear_isometry_equiv.symm (W₀.sum (lp.single 2)) = (W₀.sum (λ i, V i)) := +by simp [is_hilbert_sum.linear_isometry_equiv, orthogonal_family.linear_isometry_apply_dfinsupp_sum_single] -/-- In the canonical isometric isomorphism `E ≃ₗᵢ[𝕜] lp G 2` induced by an `ι`-indexed orthogonal -family `G`, a finitely-supported vector in `lp G 2` is the image of the associated finite sum of +/-- In the canonical isometric isomorphism between a Hilbert sum `E` of `G : ι → Type*` and +`lp G 2`, a finitely-supported vector in `lp G 2` is the image of the associated finite sum of elements of `E`. -/ -@[simp] protected lemma linear_isometry_equiv_apply_dfinsupp_sum_single - [Π i, complete_space (G i)] - (hV' : ⊤ ≤ (⨆ i, (V i).to_linear_map.range).topological_closure) (W₀ : Π₀ (i : ι), G i) : - (hV.linear_isometry_equiv hV' (W₀.sum (λ i, V i)) : Π i, G i) = W₀ := +@[simp] protected lemma is_hilbert_sum.linear_isometry_equiv_apply_dfinsupp_sum_single + (hV : is_hilbert_sum 𝕜 E V) (W₀ : Π₀ (i : ι), G i) : + (hV.linear_isometry_equiv (W₀.sum (λ i, V i)) : Π i, G i) = W₀ := begin - rw ← hV.linear_isometry_equiv_symm_apply_dfinsupp_sum_single hV', + rw ← hV.linear_isometry_equiv_symm_apply_dfinsupp_sum_single, rw linear_isometry_equiv.apply_symm_apply, ext i, simp [dfinsupp.sum, lp.single_apply] {contextual := tt}, end -end orthogonal_family +/-- Given a total orthonormal family `v : ι → E`, `E` is a Hilbert sum of `λ i : ι, 𝕜` relative to +the family of linear isometries `λ i, λ k, k • v i`. -/ +lemma orthonormal.is_hilbert_sum {v : ι → E} (hv : orthonormal 𝕜 v) + (hsp : ⊤ ≤ (span 𝕜 (set.range v)).topological_closure) : + @is_hilbert_sum _ 𝕜 _ _ _ _ (λ i : ι, 𝕜) _ + (λ i, linear_isometry.to_span_singleton 𝕜 E (hv.1 i)) := +is_hilbert_sum.mk hv.orthogonal_family +begin + convert hsp, + simp [← linear_map.span_singleton_eq_range, ← submodule.span_Union], +end + +lemma submodule.is_hilbert_sum_orthogonal (K : submodule 𝕜 E) [hK : complete_space K] : + @is_hilbert_sum _ 𝕜 _ E _ _ (λ b, ((cond b K Kᗮ : submodule 𝕜 E) : Type*)) _ + (λ b, (cond b K Kᗮ).subtypeₗᵢ) := +begin + haveI : Π b, complete_space ((cond b K Kᗮ : submodule 𝕜 E) : Type*), + { intro b, + cases b; + exact orthogonal.complete_space K <|> assumption }, + refine is_hilbert_sum.mk_internal _ K.orthogonal_family_self _, + refine le_trans _ (submodule.submodule_topological_closure _), + rw supr_bool_eq, + exact submodule.is_compl_orthogonal_of_complete_space.2 +end + +end is_hilbert_sum /-! ### Hilbert bases -/ @@ -426,15 +492,11 @@ include hv cplt protected def mk (hsp : ⊤ ≤ (span 𝕜 (set.range v)).topological_closure) : hilbert_basis ι 𝕜 E := hilbert_basis.of_repr $ -hv.orthogonal_family.linear_isometry_equiv -begin - convert hsp, - simp [← linear_map.span_singleton_eq_range, ← submodule.span_Union], -end +(hv.is_hilbert_sum hsp).linear_isometry_equiv lemma _root_.orthonormal.linear_isometry_equiv_symm_apply_single_one (h i) : - (hv.orthogonal_family.linear_isometry_equiv h).symm (lp.single 2 i 1) = v i := -by rw [orthogonal_family.linear_isometry_equiv_symm_apply_single, + (hv.is_hilbert_sum h).linear_isometry_equiv.symm (lp.single 2 i 1) = v i := +by rw [is_hilbert_sum.linear_isometry_equiv_symm_apply_single, linear_isometry.to_span_singleton_apply, one_smul] @[simp] protected lemma coe_mk (hsp : ⊤ ≤ (span 𝕜 (set.range v)).topological_closure) : diff --git a/formal/lean/mathlib/analysis/inner_product_space/projection.lean b/formal/lean/mathlib/analysis/inner_product_space/projection.lean index c74cf11b4e0d1d2e6e5efa5c7b2a87ee71ca8514..ff2da3c426d081be101024e16b83aa2ae96812ff 100644 --- a/formal/lean/mathlib/analysis/inner_product_space/projection.lean +++ b/formal/lean/mathlib/analysis/inner_product_space/projection.lean @@ -318,7 +318,7 @@ Then point `v` minimizes the distance `∥u - v∥` over points in `K` if and on for all `w ∈ K`, `⟪u - v, w⟫ = 0` (i.e., `u - v` is orthogonal to the subspace `K`) -/ theorem norm_eq_infi_iff_inner_eq_zero {u : E} {v : E} - (hv : v ∈ K) : ∥u - v∥ = (⨅ w : (↑K : set E), ∥u - w∥) ↔ ∀ w ∈ K, ⟪u - v, w⟫ = 0 := + (hv : v ∈ K) : ∥u - v∥ = (⨅ w : K, ∥u - w∥) ↔ ∀ w ∈ K, ⟪u - v, w⟫ = 0 := begin letI : inner_product_space ℝ E := inner_product_space.is_R_or_C_to_real 𝕜 E, letI : module ℝ E := restrict_scalars.module ℝ 𝕜 E, @@ -465,6 +465,14 @@ lemma eq_orthogonal_projection_of_mem_of_inner_eq_zero (orthogonal_projection K u : E) = v := eq_orthogonal_projection_fn_of_mem_of_inner_eq_zero hvm hvo +/-- The orthogonal projection of `y` on `U` minimizes the distance `∥y - x∥` for `x ∈ U`. -/ +lemma orthogonal_projection_minimal {U : submodule 𝕜 E} [complete_space U] (y : E) : + ∥y - orthogonal_projection U y∥ = ⨅ x : U, ∥y - x∥ := +begin + rw norm_eq_infi_iff_inner_eq_zero _ (submodule.coe_mem _), + exact orthogonal_projection_inner_eq_zero _ +end + /-- The orthogonal projections onto equal subspaces are coerced back to the same point in `E`. -/ lemma eq_orthogonal_projection_of_eq_submodule {K' : submodule 𝕜 E} [complete_space K'] (h : K = K') (u : E) : @@ -763,6 +771,61 @@ lemma orthogonal_projection_mem_subspace_orthogonal_precomplement_eq_zero orthogonal_projection Kᗮ v = 0 := orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero (K.le_orthogonal_orthogonal hv) +/-- If `U ≤ V`, then projecting on `V` and then on `U` is the same as projecting on `U`. -/ +lemma orthogonal_projection_orthogonal_projection_of_le {U V : submodule 𝕜 E} [complete_space U] + [complete_space V] (h : U ≤ V) (x : E) : + orthogonal_projection U (orthogonal_projection V x) = orthogonal_projection U x := +eq.symm $ by simpa only [sub_eq_zero, map_sub] using + orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero + (submodule.orthogonal_le h (sub_orthogonal_projection_mem_orthogonal x)) + +/-- Given a monotone family `U` of complete submodules of `E` and a fixed `x : E`, +the orthogonal projection of `x` on `U i` tends to the orthogonal projection of `x` on +`(⨆ i, U i).topological_closure` along `at_top`. -/ +lemma orthogonal_projection_tendsto_closure_supr [complete_space E] {ι : Type*} + [semilattice_sup ι] (U : ι → submodule 𝕜 E) [∀ i, complete_space (U i)] + (hU : monotone U) (x : E) : + filter.tendsto (λ i, (orthogonal_projection (U i) x : E)) at_top + (𝓝 (orthogonal_projection (⨆ i, U i).topological_closure x : E)) := +begin + casesI is_empty_or_nonempty ι, + { rw filter_eq_bot_of_is_empty (at_top : filter ι), + exact tendsto_bot }, + let y := (orthogonal_projection (⨆ i, U i).topological_closure x : E), + have proj_x : ∀ i, orthogonal_projection (U i) x = orthogonal_projection (U i) y := + λ i, (orthogonal_projection_orthogonal_projection_of_le + ((le_supr U i).trans (supr U).submodule_topological_closure) _).symm, + suffices : ∀ ε > 0, ∃ I, ∀ i ≥ I, ∥(orthogonal_projection (U i) y : E) - y∥ < ε, + { simpa only [proj_x, normed_add_comm_group.tendsto_at_top] using this }, + intros ε hε, + obtain ⟨a, ha, hay⟩ : ∃ a ∈ ⨆ i, U i, dist y a < ε, + { have y_mem : y ∈ (⨆ i, U i).topological_closure := submodule.coe_mem _, + rw [← set_like.mem_coe, submodule.topological_closure_coe, metric.mem_closure_iff] at y_mem, + exact y_mem ε hε }, + rw dist_eq_norm at hay, + obtain ⟨I, hI⟩ : ∃ I, a ∈ U I, + { rwa [submodule.mem_supr_of_directed _ (hU.directed_le)] at ha }, + refine ⟨I, λ i (hi : I ≤ i), _⟩, + rw [norm_sub_rev, orthogonal_projection_minimal], + refine lt_of_le_of_lt _ hay, + change _ ≤ ∥y - (⟨a, hU hi hI⟩ : U i)∥, + exact cinfi_le ⟨0, set.forall_range_iff.mpr $ λ _, norm_nonneg _⟩ _, +end + +/-- Given a monotone family `U` of complete submodules of `E` with dense span supremum, +and a fixed `x : E`, the orthogonal projection of `x` on `U i` tends to `x` along `at_top`. -/ +lemma orthogonal_projection_tendsto_self [complete_space E] {ι : Type*} [semilattice_sup ι] + (U : ι → submodule 𝕜 E) [∀ t, complete_space (U t)] (hU : monotone U) + (x : E) (hU' : ⊤ ≤ (⨆ t, U t).topological_closure) : + filter.tendsto (λ t, (orthogonal_projection (U t) x : E)) at_top (𝓝 x) := +begin + rw ← eq_top_iff at hU', + convert orthogonal_projection_tendsto_closure_supr U hU x, + rw orthogonal_projection_eq_self_iff.mpr _, + rw hU', + trivial +end + /-- The orthogonal complement satisfies `Kᗮᗮᗮ = Kᗮ`. -/ lemma submodule.triorthogonal_eq_orthogonal [complete_space E] : Kᗮᗮᗮ = Kᗮ := begin diff --git a/formal/lean/mathlib/analysis/inner_product_space/rayleigh.lean b/formal/lean/mathlib/analysis/inner_product_space/rayleigh.lean index 4ef2624b746dae6b9577583ac0a661eba432d222..d307153745f235c569e1729946562177e850fc22 100644 --- a/formal/lean/mathlib/analysis/inner_product_space/rayleigh.lean +++ b/formal/lean/mathlib/analysis/inner_product_space/rayleigh.lean @@ -94,7 +94,7 @@ variables {F : Type*} [inner_product_space ℝ F] lemma has_strict_fderiv_at_re_apply_inner_self {T : F →L[ℝ] F} (hT : is_self_adjoint (T : F →ₗ[ℝ] F)) (x₀ : F) : - has_strict_fderiv_at T.re_apply_inner_self (bit0 (innerSL (T x₀))) x₀ := + has_strict_fderiv_at T.re_apply_inner_self (bit0 (innerSL (T x₀) : F →L[ℝ] ℝ)) x₀ := begin convert T.has_strict_fderiv_at.inner (has_strict_fderiv_at_id x₀), ext y, diff --git a/formal/lean/mathlib/analysis/locally_convex/basic.lean b/formal/lean/mathlib/analysis/locally_convex/basic.lean index e892e70092d59e0641324349d8f8ee9428ad026a..04a2c5f9c0648181ce9720d0afd6891d97f94aa1 100644 --- a/formal/lean/mathlib/analysis/locally_convex/basic.lean +++ b/formal/lean/mathlib/analysis/locally_convex/basic.lean @@ -3,7 +3,9 @@ Copyright (c) 2019 Jean Lo. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jean Lo, Bhavik Mehta, Yaël Dillies -/ -import analysis.normed_space.basic +import analysis.convex.basic +import analysis.normed_space.lattice_ordered_group +import analysis.normed_space.ordered /-! # Local convexity @@ -341,3 +343,17 @@ lemma absorbent.zero_mem (hs : absorbent 𝕜 s) : (0 : E) ∈ s := absorbs_zero_iff.1 $ absorbent_iff_forall_absorbs_singleton.1 hs _ end nontrivially_normed_field + +section real +variables [add_comm_group E] [module ℝ E] {s : set E} + +lemma balanced_iff_neg_mem (hs : convex ℝ s) : balanced ℝ s ↔ ∀ ⦃x⦄, x ∈ s → -x ∈ s := +begin + refine ⟨λ h x, h.neg_mem_iff.2, λ h a ha, smul_set_subset_iff.2 $ λ x hx, _⟩, + rw [real.norm_eq_abs, abs_le] at ha, + rw [show a = -((1 - a) / 2) + (a - -1)/2, by ring, add_smul, neg_smul, ←smul_neg], + exact hs (h hx) hx (div_nonneg (sub_nonneg_of_le ha.2) zero_le_two) + (div_nonneg (sub_nonneg_of_le ha.1) zero_le_two) (by ring), +end + +end real diff --git a/formal/lean/mathlib/analysis/mean_inequalities_pow.lean b/formal/lean/mathlib/analysis/mean_inequalities_pow.lean index ea49f3cc8f0c267ede09070b85c21ad77fc97d18..0865f42f60bc18dee84d3192c37719d6bc4ddbb7 100644 --- a/formal/lean/mathlib/analysis/mean_inequalities_pow.lean +++ b/formal/lean/mathlib/analysis/mean_inequalities_pow.lean @@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne -/ import analysis.convex.specific_functions +import tactic.positivity /-! # Mean value inequalities @@ -71,7 +72,7 @@ theorem arith_mean_le_rpow_mean (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) : ∑ i in s, w i * z i ≤ (∑ i in s, (w i * z i ^ p)) ^ (1 / p) := begin - have : 0 < p := lt_of_lt_of_le zero_lt_one hp, + have : 0 < p := by positivity, rw [← rpow_le_rpow_iff _ _ this, ← rpow_mul, one_div_mul_cancel (ne_of_gt this), rpow_one], exact rpow_arith_mean_le_arith_mean_rpow s w z hw hw' hz hp, all_goals { apply_rules [sum_nonneg, rpow_nonneg_of_nonneg], @@ -133,7 +134,7 @@ end lemma add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) : a ^ p + b ^ p ≤ (a + b) ^ p := begin - have hp_pos : 0 < p := lt_of_lt_of_le zero_lt_one hp1, + have hp_pos : 0 < p := by positivity, by_cases h_zero : a + b = 0, { simp [add_eq_zero_iff.mp h_zero, hp_pos.ne'] }, have h_nonzero : ¬(a = 0 ∧ b = 0), by rwa add_eq_zero_iff at h_zero, @@ -171,9 +172,11 @@ begin rwa one_div_div at h_rpow_add_rpow_le_add, end -lemma rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp_pos : 0 < p) (hp1 : p ≤ 1) : +lemma rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) : (a + b) ^ p ≤ a ^ p + b ^ p := begin + rcases hp.eq_or_lt with rfl|hp_pos, + { simp }, have h := rpow_add_rpow_le a b hp_pos hp1, rw one_div_one at h, repeat { rw nnreal.rpow_one at h }, @@ -190,8 +193,8 @@ theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ (hp : 1 ≤ p) : (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, (w i * z i ^ p) := begin - have hp_pos : 0 < p, from lt_of_lt_of_le zero_lt_one hp, - have hp_nonneg : 0 ≤ p, from le_of_lt hp_pos, + have hp_pos : 0 < p, positivity, + have hp_nonneg : 0 ≤ p, positivity, have hp_not_nonpos : ¬ p ≤ 0, by simp [hp_pos], have hp_not_neg : ¬ p < 0, by simp [hp_nonneg], have h_top_iff_rpow_top : ∀ (i : ι) (hi : i ∈ s), w i * z i = ⊤ ↔ w i * (z i) ^ p = ⊤, @@ -252,7 +255,7 @@ namespace ennreal lemma add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p) : a ^ p + b ^ p ≤ (a + b) ^ p := begin - have hp_pos : 0 < p := lt_of_lt_of_le zero_lt_one hp1, + have hp_pos : 0 < p := by positivity, by_cases h_top : a + b = ⊤, { rw ←@ennreal.rpow_eq_top_iff_of_pos (a + b) p hp_pos at h_top, rw h_top, @@ -285,9 +288,14 @@ begin rwa one_div_div at h_rpow_add_rpow_le_add, end -lemma rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0∞) (hp_pos : 0 < p) (hp1 : p ≤ 1) : +lemma rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0∞) (hp : 0 ≤ p) (hp1 : p ≤ 1) : (a + b) ^ p ≤ a ^ p + b ^ p := begin + rcases hp.eq_or_lt with rfl|hp_pos, + { suffices : (1 : ℝ≥0∞) ≤ 1 + 1, + { simpa using this }, + norm_cast, + norm_num }, have h := rpow_add_rpow_le a b hp_pos hp1, rw one_div_one at h, repeat { rw ennreal.rpow_one at h }, diff --git a/formal/lean/mathlib/analysis/normed/field/basic.lean b/formal/lean/mathlib/analysis/normed/field/basic.lean index 6dc141b00ac35c100721ebd181dd4b3a5118be42..9df7c465f7af835fedd9f5415b05f4538337db0e 100644 --- a/formal/lean/mathlib/analysis/normed/field/basic.lean +++ b/formal/lean/mathlib/analysis/normed/field/basic.lean @@ -494,8 +494,22 @@ multiplication by the powers of any element, and thus to relate algebra and topo class nontrivially_normed_field (α : Type*) extends normed_field α := (non_trivial : ∃ x : α, 1 < ∥x∥) +/-- A densely normed field is a normed field for which the image of the norm is dense in `ℝ≥0`, +which means it is also nontrivially normed. However, not all nontrivally normed fields are densely +normed; in particular, the `padic`s exhibit this fact. -/ +class densely_normed_field (α : Type*) extends normed_field α := +(lt_norm_lt : ∀ x y : ℝ, 0 ≤ x → x < y → ∃ a : α, x < ∥a∥ ∧ ∥a∥ < y) + section normed_field +/-- A densely normed field is always a nontrivially normed field. +See note [lower instance priority]. -/ +@[priority 100] +instance densely_normed_field.to_nontrivially_normed_field [densely_normed_field α] : + nontrivially_normed_field α := +{ non_trivial := let ⟨a, h, _⟩ := densely_normed_field.lt_norm_lt 1 2 zero_le_one one_lt_two in + ⟨a, h⟩ } + variables [normed_field α] @[priority 100] -- see Note [lower instance priority] @@ -518,6 +532,8 @@ end normed_field namespace normed_field +section nontrivially + variables (α) [nontrivially_normed_field α] lemma exists_one_lt_norm : ∃x : α, 1 < ∥x∥ := ‹nontrivially_normed_field α›.non_trivial @@ -550,14 +566,46 @@ end lemma nhds_within_is_unit_ne_bot : ne_bot (𝓝[{x : α | is_unit x}] 0) := by simpa only [is_unit_iff_ne_zero] using punctured_nhds_ne_bot (0:α) +end nontrivially + +section densely + +variables (α) [densely_normed_field α] + +lemma exists_lt_norm_lt {r₁ r₂ : ℝ} (h₀ : 0 ≤ r₁) (h : r₁ < r₂) : ∃ x : α, r₁ < ∥x∥ ∧ ∥x∥ < r₂ := +densely_normed_field.lt_norm_lt r₁ r₂ h₀ h + +lemma exists_lt_nnnorm_lt {r₁ r₂ : ℝ≥0} (h : r₁ < r₂) : ∃ x : α, r₁ < ∥x∥₊ ∧ ∥x∥₊ < r₂ := +by exact_mod_cast exists_lt_norm_lt α r₁.prop h + +instance densely_ordered_range_norm : densely_ordered (set.range (norm : α → ℝ)) := +{ dense := + begin + rintro ⟨-, x, rfl⟩ ⟨-, y, rfl⟩ hxy, + exact let ⟨z, h⟩ := exists_lt_norm_lt α (norm_nonneg _) hxy in ⟨⟨∥z∥, z, rfl⟩, h⟩, + end } + +instance densely_ordered_range_nnnorm : densely_ordered (set.range (nnnorm : α → ℝ≥0)) := +{ dense := + begin + rintro ⟨-, x, rfl⟩ ⟨-, y, rfl⟩ hxy, + exact let ⟨z, h⟩ := exists_lt_nnnorm_lt α hxy in ⟨⟨∥z∥₊, z, rfl⟩, h⟩, + end } + +lemma dense_range_nnnorm : dense_range (nnnorm : α → ℝ≥0) := +dense_of_exists_between $ λ _ _ hr, let ⟨x, h⟩ := exists_lt_nnnorm_lt α hr in ⟨∥x∥₊, ⟨x, rfl⟩, h⟩ + +end densely + end normed_field instance : normed_field ℝ := { norm_mul' := abs_mul, .. real.normed_add_comm_group } -instance : nontrivially_normed_field ℝ := -{ non_trivial := ⟨2, by { unfold norm, rw abs_of_nonneg; norm_num }⟩ } +instance : densely_normed_field ℝ := +{ lt_norm_lt := λ _ _ h₀ hr, let ⟨x, h⟩ := exists_between hr in + ⟨x, by rwa [real.norm_eq_abs, abs_of_nonneg (h₀.trans h.1.le)]⟩ } namespace real @@ -664,8 +712,9 @@ instance : normed_field ℚ := norm_mul' := λ r₁ r₂, by simp only [norm, rat.cast_mul, abs_mul], dist_eq := λ r₁ r₂, by simp only [rat.dist_eq, norm, rat.cast_sub] } -instance : nontrivially_normed_field ℚ := -{ non_trivial := ⟨2, by { unfold norm, rw abs_of_nonneg; norm_num }⟩ } +instance : densely_normed_field ℚ := +{ lt_norm_lt := λ r₁ r₂ h₀ hr, let ⟨q, h⟩ := exists_rat_btwn hr in + ⟨q, by { unfold norm, rwa abs_of_pos (h₀.trans_lt h.1) } ⟩ } @[norm_cast, simp] lemma rat.norm_cast_real (r : ℚ) : ∥(r : ℝ)∥ = ∥r∥ := rfl @@ -735,7 +784,7 @@ suffices this : ∀ u : finset (ι × ι'), ∑ x in u, f x.1 * g x.2 ≤ s*t, from summable_of_sum_le (λ x, mul_nonneg (hf' _) (hg' _)) this, assume u, calc ∑ x in u, f x.1 * g x.2 - ≤ ∑ x in (u.image prod.fst).product (u.image prod.snd), f x.1 * g x.2 : + ≤ ∑ x in u.image prod.fst ×ˢ u.image prod.snd, f x.1 * g x.2 : sum_mono_set_of_nonneg (λ x, mul_nonneg (hf' _) (hg' _)) subset_product ... = ∑ x in u.image prod.fst, ∑ y in u.image prod.snd, f x * g y : sum_product ... = ∑ x in u.image prod.fst, f x * ∑ y in u.image prod.snd, g y : diff --git a/formal/lean/mathlib/analysis/normed_space/add_torsor_bases.lean b/formal/lean/mathlib/analysis/normed_space/add_torsor_bases.lean index f219e92a065d036bbe736b9113cef651203b29c2..86479d528ff06f28e758722f853773c79444e2c6 100644 --- a/formal/lean/mathlib/analysis/normed_space/add_torsor_bases.lean +++ b/formal/lean/mathlib/analysis/normed_space/add_torsor_bases.lean @@ -72,7 +72,7 @@ begin have : convex_hull ℝ (range b.points) = ⋂ i, (b.coord i)⁻¹' Ici 0, { rw convex_hull_affine_basis_eq_nonneg_barycentric b, ext, simp, }, ext, - simp only [this, interior_Inter_of_fintype, ← is_open_map.preimage_interior_eq_interior_preimage + simp only [this, interior_Inter, ← is_open_map.preimage_interior_eq_interior_preimage (is_open_map_barycentric_coord b _) (continuous_barycentric_coord b _), interior_Ici, mem_Inter, mem_set_of_eq, mem_Ioi, mem_preimage], }, end diff --git a/formal/lean/mathlib/analysis/normed_space/linear_isometry.lean b/formal/lean/mathlib/analysis/normed_space/linear_isometry.lean index adeaf29e23ac714b1c7a10d5521972ec01fa3363..85a246a181f5d0f3f9dab2fef276998d50149675 100644 --- a/formal/lean/mathlib/analysis/normed_space/linear_isometry.lean +++ b/formal/lean/mathlib/analysis/normed_space/linear_isometry.lean @@ -50,6 +50,77 @@ notation E ` →ₛₗᵢ[`:25 σ₁₂:25 `] `:0 E₂:0 := linear_isometry σ notation E ` →ₗᵢ[`:25 R:25 `] `:0 E₂:0 := linear_isometry (ring_hom.id R) E E₂ notation E ` →ₗᵢ⋆[`:25 R:25 `] `:0 E₂:0 := linear_isometry (star_ring_end R) E E₂ +set_option old_structure_cmd true +/-- `semilinear_isometry_class F σ E E₂` asserts `F` is a type of bundled `σ`-semilinear isometries +`E → E₂`. + +See also `linear_isometry_class F R E E₂` for the case where `σ` is the identity map on `R`. + +A map `f` between an `R`-module and an `S`-module over a ring homomorphism `σ : R →+* S` +is semilinear if it satisfies the two properties `f (x + y) = f x + f y` and +`f (c • x) = (σ c) • f x`. -/ +class semilinear_isometry_class (𝓕 : Type*) {R R₂ : out_param Type*} [semiring R] [semiring R₂] + (σ₁₂ : out_param $ R →+* R₂) (E E₂ : out_param Type*) [seminormed_add_comm_group E] + [seminormed_add_comm_group E₂] [module R E] [module R₂ E₂] + extends semilinear_map_class 𝓕 σ₁₂ E E₂ := +(norm_map : ∀ (f : 𝓕) (x : E), ∥f x∥ = ∥x∥) + +/-- `linear_isometry_class F R E E₂` asserts `F` is a type of bundled `R`-linear isometries +`M → M₂`. + +This is an abbreviation for `semilinear_isometry_class F (ring_hom.id R) E E₂`. +-/ +abbreviation linear_isometry_class (𝓕 : Type*) (R E E₂ : out_param Type*) [semiring R] + [seminormed_add_comm_group E] [seminormed_add_comm_group E₂] [module R E] [module R E₂] := +semilinear_isometry_class 𝓕 (ring_hom.id R) E E₂ + +set_option old_structure_cmd false + +namespace semilinear_isometry_class +variables {𝓕 : Type*} + +protected lemma isometry [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) : isometry f := +add_monoid_hom_class.isometry_of_norm _ (norm_map _) + +@[continuity] protected lemma continuous [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) : + continuous f := +(semilinear_isometry_class.isometry f).continuous + +@[simp] lemma nnnorm_map [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) (x : E) : + ∥f x∥₊ = ∥x∥₊ := +nnreal.eq $ norm_map f x + +protected lemma lipschitz [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) : + lipschitz_with 1 f := +(semilinear_isometry_class.isometry f).lipschitz + +protected lemma antilipschitz [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) : + antilipschitz_with 1 f := +(semilinear_isometry_class.isometry f).antilipschitz + +lemma ediam_image [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) (s : set E) : + emetric.diam (f '' s) = emetric.diam s := +(semilinear_isometry_class.isometry f).ediam_image s + +lemma ediam_range [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) : + emetric.diam (range f) = emetric.diam (univ : set E) := +(semilinear_isometry_class.isometry f).ediam_range + +lemma diam_image [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) (s : set E) : + metric.diam (f '' s) = metric.diam s := +(semilinear_isometry_class.isometry f).diam_image s + +lemma diam_range [semilinear_isometry_class 𝓕 σ₁₂ E E₂] (f : 𝓕) : + metric.diam (range f) = metric.diam (univ : set E) := +(semilinear_isometry_class.isometry f).diam_range + +@[priority 100] +instance [s : semilinear_isometry_class 𝓕 σ₁₂ E E₂] : continuous_semilinear_map_class 𝓕 σ₁₂ E E₂ := +{ map_continuous := semilinear_isometry_class.continuous, + ..s } + +end semilinear_isometry_class + namespace linear_isometry /-- We use `f₁` when we need the domain to be a `normed_space`. -/ @@ -61,11 +132,12 @@ lemma to_linear_map_injective : injective (to_linear_map : (E →ₛₗᵢ[σ₁ @[simp] lemma to_linear_map_inj {f g : E →ₛₗᵢ[σ₁₂] E₂} : f.to_linear_map = g.to_linear_map ↔ f = g := to_linear_map_injective.eq_iff -instance : add_monoid_hom_class (E →ₛₗᵢ[σ₁₂] E₂) E E₂ := -{ coe := λ e, e.to_fun, +instance : semilinear_isometry_class (E →ₛₗᵢ[σ₁₂] E₂) σ₁₂ E E₂ := +{ coe := λ f, f.to_fun, coe_injective' := λ f g h, to_linear_map_injective (fun_like.coe_injective h), map_add := λ f, map_add f.to_linear_map, - map_zero := λ f, map_zero f.to_linear_map } + map_smulₛₗ := λ f, map_smulₛₗ f.to_linear_map, + norm_map := λ f, f.norm_map' } /-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun` directly. @@ -94,17 +166,19 @@ protected lemma congr_arg {f : E →ₛₗᵢ[σ₁₂] E₂} : Π {x x' : E}, x protected lemma congr_fun {f g : E →ₛₗᵢ[σ₁₂] E₂} (h : f = g) (x : E) : f x = g x := h ▸ rfl -@[simp] lemma map_zero : f 0 = 0 := f.to_linear_map.map_zero +@[simp] protected lemma map_zero : f 0 = 0 := f.to_linear_map.map_zero -@[simp] lemma map_add (x y : E) : f (x + y) = f x + f y := f.to_linear_map.map_add x y +@[simp] protected lemma map_add (x y : E) : f (x + y) = f x + f y := f.to_linear_map.map_add x y -@[simp] lemma map_neg (x : E) : f (- x) = - f x := f.to_linear_map.map_neg x +@[simp] protected lemma map_neg (x : E) : f (- x) = - f x := f.to_linear_map.map_neg x -@[simp] lemma map_sub (x y : E) : f (x - y) = f x - f y := f.to_linear_map.map_sub x y +@[simp] protected lemma map_sub (x y : E) : f (x - y) = f x - f y := f.to_linear_map.map_sub x y -@[simp] lemma map_smulₛₗ (c : R) (x : E) : f (c • x) = σ₁₂ c • f x := f.to_linear_map.map_smulₛₗ c x +@[simp] protected lemma map_smulₛₗ (c : R) (x : E) : f (c • x) = σ₁₂ c • f x := +f.to_linear_map.map_smulₛₗ c x -@[simp] lemma map_smul [module R E₂] (f : E →ₗᵢ[R] E₂) (c : R) (x : E) : f (c • x) = c • f x := +@[simp] protected lemma map_smul [module R E₂] (f : E →ₗᵢ[R] E₂) (c : R) (x : E) : + f (c • x) = c • f x := f.to_linear_map.map_smul c x @[simp] lemma norm_map (x : E) : ∥f x∥ = ∥x∥ := f.norm_map' x @@ -140,11 +214,6 @@ protected lemma antilipschitz : antilipschitz_with 1 f := f.isometry.antilipschi @[continuity] protected lemma continuous : continuous f := f.isometry.continuous -instance : continuous_semilinear_map_class (E →ₛₗᵢ[σ₁₂] E₂) σ₁₂ E E₂ := -{ map_smulₛₗ := λ f, f.map_smulₛₗ, - map_continuous := λ f, f.continuous, - ..linear_isometry.add_monoid_hom_class } - @[simp] lemma preimage_ball (x : E) (r : ℝ) : f ⁻¹' (metric.ball (f x) r) = metric.ball x r := f.isometry.preimage_ball x r @@ -267,6 +336,48 @@ notation E ` ≃ₗᵢ[`:25 R:25 `] `:0 E₂:0 := linear_isometry_equiv (ring_ho notation E ` ≃ₗᵢ⋆[`:25 R:25 `] `:0 E₂:0 := linear_isometry_equiv (star_ring_end R) E E₂ +set_option old_structure_cmd true +/-- `semilinear_isometry_equiv_class F σ E E₂` asserts `F` is a type of bundled `σ`-semilinear +isometric equivs `E → E₂`. + +See also `linear_isometry_equiv_class F R E E₂` for the case where `σ` is the identity map on `R`. + +A map `f` between an `R`-module and an `S`-module over a ring homomorphism `σ : R →+* S` +is semilinear if it satisfies the two properties `f (x + y) = f x + f y` and +`f (c • x) = (σ c) • f x`. -/ +class semilinear_isometry_equiv_class (𝓕 : Type*) {R R₂ : out_param Type*} + [semiring R] [semiring R₂] (σ₁₂ : out_param $ R →+* R₂) {σ₂₁ : out_param $ R₂ →+* R} + [ring_hom_inv_pair σ₁₂ σ₂₁] [ring_hom_inv_pair σ₂₁ σ₁₂] (E E₂ : out_param Type*) + [seminormed_add_comm_group E] [seminormed_add_comm_group E₂] [module R E] [module R₂ E₂] + extends semilinear_equiv_class 𝓕 σ₁₂ E E₂ := +(norm_map : ∀ (f : 𝓕) (x : E), ∥f x∥ = ∥x∥) + +/-- `linear_isometry_equiv_class F R E E₂` asserts `F` is a type of bundled `R`-linear isometries +`M → M₂`. + +This is an abbreviation for `semilinear_isometry_equiv_class F (ring_hom.id R) E E₂`. +-/ +abbreviation linear_isometry_equiv_class (𝓕 : Type*) (R E E₂ : out_param Type*) [semiring R] + [seminormed_add_comm_group E] [seminormed_add_comm_group E₂] [module R E] [module R E₂] := +semilinear_isometry_equiv_class 𝓕 (ring_hom.id R) E E₂ + +set_option old_structure_cmd false + +namespace semilinear_isometry_equiv_class +variables (𝓕 : Type*) + +include σ₂₁ +-- `σ₂₁` becomes a metavariable, but it's OK since it's an outparam +@[priority 100, nolint dangerous_instance] +instance [s : semilinear_isometry_equiv_class 𝓕 σ₁₂ E E₂] : semilinear_isometry_class 𝓕 σ₁₂ E E₂ := +{ coe := (coe : 𝓕 → E → E₂), + coe_injective' := @fun_like.coe_injective 𝓕 _ _ _, + ..s } +omit σ₂₁ + +end semilinear_isometry_equiv_class + + namespace linear_isometry_equiv variables (e : E ≃ₛₗᵢ[σ₁₂] E₂) diff --git a/formal/lean/mathlib/analysis/normed_space/lp_space.lean b/formal/lean/mathlib/analysis/normed_space/lp_space.lean index 6a644b03008e632f63b4717776dad0fb0399d1b0..f20c1d8efe1bd3bec065af313b04b4ef01fb5deb 100644 --- a/formal/lean/mathlib/analysis/normed_space/lp_space.lean +++ b/formal/lean/mathlib/analysis/normed_space/lp_space.lean @@ -229,7 +229,7 @@ begin { refine (real.rpow_le_rpow (norm_nonneg _) (norm_add_le _ _) hp.le).trans _, dsimp [C], split_ifs with h h, - { simpa using nnreal.coe_le_coe.2 (nnreal.rpow_add_le_add_rpow (∥f i∥₊) (∥g i∥₊) hp h.le) }, + { simpa using nnreal.coe_le_coe.2 (nnreal.rpow_add_le_add_rpow (∥f i∥₊) (∥g i∥₊) hp.le h.le) }, { let F : fin 2 → ℝ≥0 := ![∥f i∥₊, ∥g i∥₊], have : ∀ i, (0:ℝ) ≤ F i := λ i, (F i).coe_nonneg, simp only [not_lt] at h, diff --git a/formal/lean/mathlib/analysis/normed_space/multilinear.lean b/formal/lean/mathlib/analysis/normed_space/multilinear.lean index 725c85f90e5ee5adb91b8cea33fa7be3105ef0fb..6536964a175d8dd4ad40a14f925149034a80be28 100644 --- a/formal/lean/mathlib/analysis/normed_space/multilinear.lean +++ b/formal/lean/mathlib/analysis/normed_space/multilinear.lean @@ -956,6 +956,17 @@ linear_map.mk_continuous_norm_le _ (prod_nonneg $ λ i _, norm_nonneg _) _ end continuous_multilinear_map +section smul + +variables {R : Type*} [semiring R] [module R G] [smul_comm_class 𝕜 R G] + [has_continuous_const_smul R G] + +instance : has_continuous_const_smul R (continuous_multilinear_map 𝕜 E G) := +⟨λ c, (continuous_linear_map.comp_continuous_multilinear_mapL 𝕜 _ G G + (c • continuous_linear_map.id 𝕜 G)).2⟩ + +end smul + section currying /-! ### Currying diff --git a/formal/lean/mathlib/analysis/normed_space/operator_norm.lean b/formal/lean/mathlib/analysis/normed_space/operator_norm.lean index 06feeeb22c30cf88f2392cd5782b76905f0a2e02..016744c5072427948e5783638ffae850b8d050ef 100644 --- a/formal/lean/mathlib/analysis/normed_space/operator_norm.lean +++ b/formal/lean/mathlib/analysis/normed_space/operator_norm.lean @@ -819,11 +819,14 @@ end prod variables {𝕜 E Fₗ Gₗ} section multiplication_linear -variables (𝕜) (𝕜' : Type*) [normed_ring 𝕜'] [normed_algebra 𝕜 𝕜'] + +section non_unital +variables (𝕜) (𝕜' : Type*) [non_unital_semi_normed_ring 𝕜'] [normed_space 𝕜 𝕜'] + [is_scalar_tower 𝕜 𝕜' 𝕜'] [smul_comm_class 𝕜 𝕜' 𝕜'] /-- Left multiplication in a normed algebra as a continuous bilinear map. -/ def lmul : 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' := -(algebra.lmul 𝕜 𝕜').to_linear_map.mk_continuous₂ 1 $ +(linear_map.mul 𝕜 𝕜').mk_continuous₂ 1 $ λ x y, by simpa using norm_mul_le x y @[simp] lemma lmul_apply (x y : 𝕜') : lmul 𝕜 𝕜' x y = x * y := rfl @@ -831,19 +834,6 @@ def lmul : 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' := @[simp] lemma op_norm_lmul_apply_le (x : 𝕜') : ∥lmul 𝕜 𝕜' x∥ ≤ ∥x∥ := (op_norm_le_bound _ (norm_nonneg x) (norm_mul_le x)) -/-- Left multiplication in a normed algebra as a linear isometry to the space of -continuous linear maps. -/ -def lmulₗᵢ [norm_one_class 𝕜'] : 𝕜' →ₗᵢ[𝕜] 𝕜' →L[𝕜] 𝕜' := -{ to_linear_map := lmul 𝕜 𝕜', - norm_map' := λ x, le_antisymm (op_norm_lmul_apply_le _ _ _) - (by { convert ratio_le_op_norm _ (1 : 𝕜'), simp [norm_one], - apply_instance }) } - -@[simp] lemma coe_lmulₗᵢ [norm_one_class 𝕜'] : ⇑(lmulₗᵢ 𝕜 𝕜') = lmul 𝕜 𝕜' := rfl - -@[simp] lemma op_norm_lmul_apply [norm_one_class 𝕜'] (x : 𝕜') : ∥lmul 𝕜 𝕜' x∥ = ∥x∥ := -(lmulₗᵢ 𝕜 𝕜').norm_map x - /-- Right-multiplication in a normed algebra, considered as a continuous linear map. -/ def lmul_right : 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' := (lmul 𝕜 𝕜').flip @@ -852,20 +842,6 @@ def lmul_right : 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' := (lmul 𝕜 𝕜').fl @[simp] lemma op_norm_lmul_right_apply_le (x : 𝕜') : ∥lmul_right 𝕜 𝕜' x∥ ≤ ∥x∥ := op_norm_le_bound _ (norm_nonneg x) (λ y, (norm_mul_le y x).trans_eq (mul_comm _ _)) -@[simp] lemma op_norm_lmul_right_apply [norm_one_class 𝕜'] (x : 𝕜') : ∥lmul_right 𝕜 𝕜' x∥ = ∥x∥ := -le_antisymm - (op_norm_lmul_right_apply_le _ _ _) - (by { convert ratio_le_op_norm _ (1 : 𝕜'), simp [norm_one], - apply_instance }) - -/-- Right-multiplication in a normed algebra, considered as a linear isometry to the space of -continuous linear maps. -/ -def lmul_rightₗᵢ [norm_one_class 𝕜'] : 𝕜' →ₗᵢ[𝕜] 𝕜' →L[𝕜] 𝕜' := -{ to_linear_map := lmul_right 𝕜 𝕜', - norm_map' := op_norm_lmul_right_apply 𝕜 𝕜' } - -@[simp] lemma coe_lmul_rightₗᵢ [norm_one_class 𝕜'] : ⇑(lmul_rightₗᵢ 𝕜 𝕜') = lmul_right 𝕜 𝕜' := rfl - /-- Simultaneous left- and right-multiplication in a normed algebra, considered as a continuous trilinear map. -/ def lmul_left_right : 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' := @@ -888,6 +864,40 @@ lemma op_norm_lmul_left_right_le : ∥lmul_left_right 𝕜 𝕜'∥ ≤ 1 := op_norm_le_bound _ zero_le_one (λ x, (one_mul ∥x∥).symm ▸ op_norm_lmul_left_right_apply_le 𝕜 𝕜' x) +end non_unital + +section unital +variables (𝕜) (𝕜' : Type*) [semi_normed_ring 𝕜'] [normed_algebra 𝕜 𝕜'] [norm_one_class 𝕜'] + +/-- Left multiplication in a normed algebra as a linear isometry to the space of +continuous linear maps. -/ +def lmulₗᵢ : 𝕜' →ₗᵢ[𝕜] 𝕜' →L[𝕜] 𝕜' := +{ to_linear_map := lmul 𝕜 𝕜', + norm_map' := λ x, le_antisymm (op_norm_lmul_apply_le _ _ _) + (by { convert ratio_le_op_norm _ (1 : 𝕜'), simp [norm_one], + apply_instance }) } + +@[simp] lemma coe_lmulₗᵢ : ⇑(lmulₗᵢ 𝕜 𝕜') = lmul 𝕜 𝕜' := rfl + +@[simp] lemma op_norm_lmul_apply (x : 𝕜') : ∥lmul 𝕜 𝕜' x∥ = ∥x∥ := +(lmulₗᵢ 𝕜 𝕜').norm_map x + +@[simp] lemma op_norm_lmul_right_apply (x : 𝕜') : ∥lmul_right 𝕜 𝕜' x∥ = ∥x∥ := +le_antisymm + (op_norm_lmul_right_apply_le _ _ _) + (by { convert ratio_le_op_norm _ (1 : 𝕜'), simp [norm_one], + apply_instance }) + +/-- Right-multiplication in a normed algebra, considered as a linear isometry to the space of +continuous linear maps. -/ +def lmul_rightₗᵢ : 𝕜' →ₗᵢ[𝕜] 𝕜' →L[𝕜] 𝕜' := +{ to_linear_map := lmul_right 𝕜 𝕜', + norm_map' := op_norm_lmul_right_apply 𝕜 𝕜' } + +@[simp] lemma coe_lmul_rightₗᵢ : ⇑(lmul_rightₗᵢ 𝕜 𝕜') = lmul_right 𝕜 𝕜' := rfl + +end unital + end multiplication_linear section smul_linear @@ -1649,7 +1659,7 @@ variables [normed_ring 𝕜'] [normed_algebra 𝕜 𝕜'] by haveI := norm_one_class.nontrivial 𝕜'; exact (lmulₗᵢ 𝕜 𝕜').norm_to_continuous_linear_map @[simp] lemma op_norm_lmul_right [norm_one_class 𝕜'] : ∥lmul_right 𝕜 𝕜'∥ = 1 := -(op_norm_flip (@lmul 𝕜 _ 𝕜' _ _)).trans (op_norm_lmul _ _) +(op_norm_flip (lmul 𝕜 𝕜')).trans (op_norm_lmul _ _) end /-- The norm of `lsmul` equals 1 in any nontrivial normed group. diff --git a/formal/lean/mathlib/analysis/special_functions/complex/arg.lean b/formal/lean/mathlib/analysis/special_functions/complex/arg.lean index 787c96a7bd4cf0ca65b8fb66d080ad8ec8c8f6c8..ce2eb9bd195aaa59785b80a3a40aff6bea34faad 100644 --- a/formal/lean/mathlib/analysis/special_functions/complex/arg.lean +++ b/formal/lean/mathlib/analysis/special_functions/complex/arg.lean @@ -3,6 +3,7 @@ Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ +import algebra.order.to_interval_mod import analysis.special_functions.trigonometric.angle import analysis.special_functions.trigonometric.inverse @@ -389,34 +390,26 @@ begin real.angle.sub_coe_pi_eq_add_coe_pi] } end -lemma arg_mul_cos_add_sin_mul_I_eq_mul_fract {r : ℝ} (hr : 0 < r) (θ : ℝ) : - arg (r * (cos θ + sin θ * I)) = π - 2 * π * int.fract ((π - θ) / (2 * π)) := +lemma arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod {r : ℝ} (hr : 0 < r) (θ : ℝ) : + arg (r * (cos θ + sin θ * I)) = to_Ioc_mod (-π) real.two_pi_pos θ := begin - have hi : π - 2 * π * int.fract ((π - θ) / (2 * π)) ∈ Ioc (-π) π, - { rw [←mem_preimage, preimage_const_sub_Ioc, ←mem_preimage, - preimage_const_mul_Ico _ _ real.two_pi_pos, sub_self, zero_div, sub_neg_eq_add, - ←two_mul, div_self real.two_pi_pos.ne.symm], - refine set.mem_of_mem_of_subset (set.mem_range_self _) _, - rw [←image_univ, int.image_fract], - simp }, - have hs : π - 2 * π * int.fract ((π - θ) / (2 * π)) = 2 * π * ⌊(π - θ) / (2 * π)⌋ + θ, - { rw [int.fract, mul_sub, mul_div_cancel' _ real.two_pi_pos.ne.symm], - abel }, + have hi : to_Ioc_mod (-π) real.two_pi_pos θ ∈ Ioc (-π) π, + { convert to_Ioc_mod_mem_Ioc _ real.two_pi_pos _, + ring }, convert arg_mul_cos_add_sin_mul_I hr hi using 3, - simp_rw [hs, mul_comm (2 * π), add_comm _ θ, ←of_real_cos, ←of_real_sin, - real.cos_add_int_mul_two_pi, real.sin_add_int_mul_two_pi] + simp [to_Ioc_mod, cos_add_int_mul_two_pi, sin_add_int_mul_two_pi] end -lemma arg_cos_add_sin_mul_I_eq_mul_fract (θ : ℝ) : - arg (cos θ + sin θ * I) = π - 2 * π * int.fract ((π - θ) / (2 * π)) := -by rw [←one_mul (_ + _), ←of_real_one, arg_mul_cos_add_sin_mul_I_eq_mul_fract zero_lt_one] +lemma arg_cos_add_sin_mul_I_eq_to_Ioc_mod (θ : ℝ) : + arg (cos θ + sin θ * I) = to_Ioc_mod (-π) real.two_pi_pos θ := +by rw [←one_mul (_ + _), ←of_real_one, arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod zero_lt_one] lemma arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := begin - rw [arg_mul_cos_add_sin_mul_I_eq_mul_fract hr, int.fract, mul_sub, - mul_div_cancel' _ real.two_pi_pos.ne.symm], - abel + rw [arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod hr, to_Ioc_mod_sub_self, to_Ioc_div_eq_floor, + zsmul_eq_mul], + ring_nf end lemma arg_cos_add_sin_mul_I_sub (θ : ℝ) : diff --git a/formal/lean/mathlib/analysis/special_functions/trigonometric/angle.lean b/formal/lean/mathlib/analysis/special_functions/trigonometric/angle.lean index 1830a5ad901cce542948cf5dc1952df5feb0a95c..45751f3bf13f20d60f88078bd2670492f46b671a 100644 --- a/formal/lean/mathlib/analysis/special_functions/trigonometric/angle.lean +++ b/formal/lean/mathlib/analysis/special_functions/trigonometric/angle.lean @@ -5,6 +5,7 @@ Authors: Calle Sönne -/ import analysis.special_functions.trigonometric.basic import algebra.char_zero.quotient +import data.sign /-! # The type of angles @@ -177,7 +178,7 @@ def sin (θ : angle) : ℝ := sin_periodic.lift θ rfl @[continuity] lemma continuous_sin : continuous sin := -continuous_quotient_lift_on' _ real.continuous_sin +real.continuous_sin.quotient_lift_on' _ /-- The cosine of a `real.angle`. -/ def cos (θ : angle) : ℝ := cos_periodic.lift θ @@ -186,7 +187,7 @@ def cos (θ : angle) : ℝ := cos_periodic.lift θ rfl @[continuity] lemma continuous_cos : continuous cos := -continuous_quotient_lift_on' _ real.continuous_cos +real.continuous_cos.quotient_lift_on' _ lemma cos_eq_real_cos_iff_eq_or_eq_neg {θ : angle} {ψ : ℝ} : cos θ = real.cos ψ ↔ θ = ψ ∨ θ = -ψ := begin @@ -213,6 +214,95 @@ begin exact sin_eq_real_sin_iff_eq_or_add_eq_pi end +@[simp] lemma sin_zero : sin (0 : angle) = 0 := +by rw [←coe_zero, sin_coe, real.sin_zero] + +@[simp] lemma sin_coe_pi : sin (π : angle) = 0 := +by rw [sin_coe, real.sin_pi] + +lemma sin_eq_zero_iff {θ : angle} : sin θ = 0 ↔ θ = 0 ∨ θ = π := +begin + nth_rewrite 0 ←sin_zero, + rw sin_eq_iff_eq_or_add_eq_pi, + simp +end + +@[simp] lemma sin_neg (θ : angle) : sin (-θ) = -sin θ := +begin + induction θ using real.angle.induction_on, + exact real.sin_neg _ +end + +lemma sin_antiperiodic : function.antiperiodic sin (π : angle) := +begin + intro θ, + induction θ using real.angle.induction_on, + exact real.sin_antiperiodic θ +end + +@[simp] lemma sin_add_pi (θ : angle) : sin (θ + π) = -sin θ := +sin_antiperiodic θ + +@[simp] lemma sin_sub_pi (θ : angle) : sin (θ - π) = -sin θ := +sin_antiperiodic.sub_eq θ + +@[simp] lemma cos_zero : cos (0 : angle) = 1 := +by rw [←coe_zero, cos_coe, real.cos_zero] + +@[simp] lemma cos_coe_pi : cos (π : angle) = -1 := +by rw [cos_coe, real.cos_pi] + +@[simp] lemma cos_neg (θ : angle) : cos (-θ) = cos θ := +begin + induction θ using real.angle.induction_on, + exact real.cos_neg _ +end + +lemma cos_antiperiodic : function.antiperiodic cos (π : angle) := +begin + intro θ, + induction θ using real.angle.induction_on, + exact real.cos_antiperiodic θ +end + +@[simp] lemma cos_add_pi (θ : angle) : cos (θ + π) = -cos θ := +cos_antiperiodic θ + +@[simp] lemma cos_sub_pi (θ : angle) : cos (θ - π) = -cos θ := +cos_antiperiodic.sub_eq θ + +/-- The sign of a `real.angle` is `0` if the angle is `0` or `π`, `1` if the angle is strictly +between `0` and `π` and `-1` is the angle is strictly between `-π` and `0`. It is defined as the +sign of the sine of the angle. -/ +def sign (θ : angle) : sign_type := sign (sin θ) + +@[simp] lemma sign_zero : (0 : angle).sign = 0 := +by rw [sign, sin_zero, sign_zero] + +@[simp] lemma sign_coe_pi : (π : angle).sign = 0 := +by rw [sign, sin_coe_pi, _root_.sign_zero] + +@[simp] lemma sign_neg (θ : angle) : (-θ).sign = - θ.sign := +by simp_rw [sign, sin_neg, left.sign_neg] + +lemma sign_antiperiodic : function.antiperiodic sign (π : angle) := +λ θ, by rw [sign, sign, sin_add_pi, left.sign_neg] + +@[simp] lemma sign_add_pi (θ : angle) : (θ + π).sign = -θ.sign := +sign_antiperiodic θ + +@[simp] lemma sign_pi_add (θ : angle) : ((π : angle) + θ).sign = -θ.sign := +by rw [add_comm, sign_add_pi] + +@[simp] lemma sign_sub_pi (θ : angle) : (θ - π).sign = -θ.sign := +sign_antiperiodic.sub_eq θ + +@[simp] lemma sign_pi_sub (θ : angle) : ((π : angle) - θ).sign = θ.sign := +by simp [sign_antiperiodic.sub_eq'] + +lemma sign_eq_zero_iff {θ : angle} : θ.sign = 0 ↔ θ = 0 ∨ θ = π := +by rw [sign, sign_eq_zero_iff, sin_eq_zero_iff] + end angle end real diff --git a/formal/lean/mathlib/analysis/special_functions/trigonometric/chebyshev.lean b/formal/lean/mathlib/analysis/special_functions/trigonometric/chebyshev.lean index 69dd2162d0021cf838dfd9d85bcb5f3934f0ea0a..78ddae6ca6b6f5046b29bc049dc27ff8f3a74b3e 100644 --- a/formal/lean/mathlib/analysis/special_functions/trigonometric/chebyshev.lean +++ b/formal/lean/mathlib/analysis/special_functions/trigonometric/chebyshev.lean @@ -4,24 +4,52 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin -/ import analysis.complex.basic -import ring_theory.polynomial.chebyshev import data.complex.exponential +import data.polynomial.algebra_map +import ring_theory.polynomial.chebyshev /-! # Multiple angle formulas in terms of Chebyshev polynomials -* `polynomial.chebyshev.T_complex_cos`: the `n`-th Chebyshev polynomial evaluates on `complex.cos θ` - to the value `complex.cos (n * θ)`. +This file gives the trigonometric characterizations of Chebyshev polynomials, for both the real +(`real.cos`) and complex (`complex.cos`) cosine. -/ namespace polynomial.chebyshev +open polynomial + +variables {R A : Type*} [comm_ring R] [comm_ring A] [algebra R A] + +@[simp] lemma aeval_T (x : A) (n : ℕ) : aeval x (T R n) = (T A n).eval x := +by rw [aeval_def, eval₂_eq_eval_map, map_T] + +@[simp] lemma aeval_U (x : A) (n : ℕ) : aeval x (U R n) = (U A n).eval x := +by rw [aeval_def, eval₂_eq_eval_map, map_U] + +@[simp] lemma algebra_map_eval_T (x : R) (n : ℕ) : + algebra_map R A ((T R n).eval x) = (T A n).eval (algebra_map R A x) := +by rw [←aeval_algebra_map_apply, aeval_T] + +@[simp] lemma algebra_map_eval_U (x : R) (n : ℕ) : + algebra_map R A ((U R n).eval x) = (U A n).eval (algebra_map R A x) := +by rw [←aeval_algebra_map_apply, aeval_U] + +@[simp, norm_cast] lemma complex_of_real_eval_T : ∀ x n, ((T ℝ n).eval x : ℂ) = (T ℂ n).eval x := +@algebra_map_eval_T ℝ ℂ _ _ _ + +@[simp, norm_cast] lemma complex_of_real_eval_U : ∀ x n, ((U ℝ n).eval x : ℂ) = (U ℂ n).eval x := +@algebra_map_eval_U ℝ ℂ _ _ _ + +/-! ### Complex versions -/ + +section complex +open complex -open polynomial complex +variable (θ : ℂ) /-- The `n`-th Chebyshev polynomial of the first kind evaluates on `cos θ` to the value `cos (n * θ)`. -/ -lemma T_complex_cos (θ : ℂ) : - ∀ n, (T ℂ n).eval (cos θ) = cos (n * θ) +@[simp] lemma T_complex_cos : ∀ n, (T ℂ n).eval (cos θ) = cos (n * θ) | 0 := by simp only [T_zero, eval_one, nat.cast_zero, zero_mul, cos_zero] | 1 := by simp only [eval_X, one_mul, T_one, nat.cast_one] | (n + 2) := @@ -34,16 +62,9 @@ begin ring, end -/-- `cos (n * θ)` is equal to the `n`-th Chebyshev polynomial of the first kind evaluated -on `cos θ`. -/ -lemma cos_nat_mul (n : ℕ) (θ : ℂ) : - cos (n * θ) = (T ℂ n).eval (cos θ) := -(T_complex_cos θ n).symm - /-- The `n`-th Chebyshev polynomial of the second kind evaluates on `cos θ` to the -value `sin ((n+1) * θ) / sin θ`. -/ -lemma U_complex_cos (θ : ℂ) (n : ℕ) : - (U ℂ n).eval (cos θ) * sin θ = sin ((n+1) * θ) := +value `sin ((n + 1) * θ) / sin θ`. -/ +@[simp] lemma U_complex_cos (n : ℕ) : (U ℂ n).eval (cos θ) * sin θ = sin ((n + 1) * θ) := begin induction n with d hd, { simp only [U_zero, nat.cast_zero, eval_one, mul_one, zero_add, one_mul] }, @@ -54,10 +75,25 @@ begin simp only [add_mul, one_mul] } end -/-- `sin ((n + 1) * θ)` is equal to `sin θ` multiplied with the `n`-th Chebyshev polynomial of the -second kind evaluated on `cos θ`. -/ -lemma sin_nat_succ_mul (n : ℕ) (θ : ℂ) : - sin ((n + 1) * θ) = (U ℂ n).eval (cos θ) * sin θ := -(U_complex_cos θ n).symm +end complex + +/- ### Real versions -/ + +section real +open real + +variables (θ : ℝ) (n : ℕ) + +/-- The `n`-th Chebyshev polynomial of the first kind evaluates on `cos θ` to the +value `cos (n * θ)`. -/ +@[simp] lemma T_real_cos : (T ℝ n).eval (cos θ) = cos (n * θ) := +by exact_mod_cast T_complex_cos θ n + +/-- The `n`-th Chebyshev polynomial of the second kind evaluates on `cos θ` to the +value `sin ((n + 1) * θ) / sin θ`. -/ +@[simp] lemma U_real_cos : (U ℝ n).eval (cos θ) * sin θ = sin ((n + 1) * θ) := +by exact_mod_cast U_complex_cos θ n + +end real end polynomial.chebyshev diff --git a/formal/lean/mathlib/analysis/specific_limits/normed.lean b/formal/lean/mathlib/analysis/specific_limits/normed.lean index 30eebbd5ef473638a9b67275a31eefd72950de68..6ae1ad1fc27b0718522551b78380aae043ef6083 100644 --- a/formal/lean/mathlib/analysis/specific_limits/normed.lean +++ b/formal/lean/mathlib/analysis/specific_limits/normed.lean @@ -561,7 +561,7 @@ begin apply (cauchy_seq_range_of_norm_bounded _ _ (_ : ∀ n, _ ≤ b * |f(n+1) - f(n)|)).neg, { exact normed_uniform_group }, { simp_rw [abs_of_nonneg (sub_nonneg_of_le (hfa (nat.le_succ _))), ← mul_sum], - apply real.uniform_continuous_mul_const.comp_cauchy_seq, + apply real.uniform_continuous_const_mul.comp_cauchy_seq, simp_rw [sum_range_sub, sub_eq_add_neg], exact (tendsto.cauchy_seq hf0).add_const }, { intro n, diff --git a/formal/lean/mathlib/analysis/von_neumann_algebra/basic.lean b/formal/lean/mathlib/analysis/von_neumann_algebra/basic.lean index 720b1dd2b040193b3cbf8e9a5a9bdfec757d9332..e6c427a9f13017bfcc74953eef6f17afa06b102d 100644 --- a/formal/lean/mathlib/analysis/von_neumann_algebra/basic.lean +++ b/formal/lean/mathlib/analysis/von_neumann_algebra/basic.lean @@ -64,7 +64,7 @@ Thus we can't say that the bounded operators `H →L[ℂ] H` form a `von_neumann (although we will later construct the instance `wstar_algebra (H →L[ℂ] H)`), and instead will use `⊤ : von_neumann_algebra H`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure von_neumann_algebra (H : Type u) [inner_product_space ℂ H] [complete_space H] extends star_subalgebra ℂ (H →L[ℂ] H) := (double_commutant : set.centralizer (set.centralizer carrier) = carrier) diff --git a/formal/lean/mathlib/category_theory/abelian/basic.lean b/formal/lean/mathlib/category_theory/abelian/basic.lean index f720095bfa1e22b7508ee6f0f273add14ebd1183..1d0d3e8df9ed45badaa8a48119ee47e9c92f1d21 100644 --- a/formal/lean/mathlib/category_theory/abelian/basic.lean +++ b/formal/lean/mathlib/category_theory/abelian/basic.lean @@ -460,11 +460,11 @@ has_pushouts_of_has_binary_coproducts_of_has_coequalizers C @[priority 100] instance has_finite_limits : has_finite_limits C := -limits.finite_limits_from_equalizers_and_finite_products +limits.has_finite_limits_of_has_equalizers_and_finite_products @[priority 100] instance has_finite_colimits : has_finite_colimits C := -limits.finite_colimits_from_coequalizers_and_finite_coproducts +limits.has_finite_colimits_of_has_coequalizers_and_finite_coproducts end diff --git a/formal/lean/mathlib/category_theory/abelian/exact.lean b/formal/lean/mathlib/category_theory/abelian/exact.lean index 7fcfc6ebf5cf057f1ff49ad6e952fbc08fd77f01..2d3e9b1c1d974d47685a02f059740a7f19db5487 100644 --- a/formal/lean/mathlib/category_theory/abelian/exact.lean +++ b/formal/lean/mathlib/category_theory/abelian/exact.lean @@ -7,6 +7,7 @@ import category_theory.abelian.opposite import category_theory.limits.constructions.finite_products_of_binary_products import category_theory.limits.preserves.shapes.zero import category_theory.limits.preserves.shapes.kernels +import category_theory.preadditive.left_exact import category_theory.adjunction.limits import algebra.homology.exact import tactic.tfae @@ -28,8 +29,8 @@ true in more general settings. sequences. * `X ⟶ Y ⟶ Z ⟶ 0` is exact if and only if the second map is a cokernel of the first, and `0 ⟶ X ⟶ Y ⟶ Z` is exact if and only if the first map is a kernel of the second. -* An exact functor preserves exactness, more specifically, if `F` preserves finite colimits and - limits, then `exact f g` implies `exact (F.map f) (F.map g)` +* An exact functor preserves exactness, more specifically, `F` preserves finite colimits and + finite limits, if and only if `exact f g` implies `exact (F.map f) (F.map g)`. -/ universes v₁ v₂ u₁ u₂ @@ -330,10 +331,16 @@ namespace functor open limits abelian variables {A : Type u₁} {B : Type u₂} [category.{v₁} A] [category.{v₂} B] -variables [has_zero_object A] [has_zero_morphisms A] [has_images A] [has_equalizers A] -variables [has_cokernels A] [abelian B] -variables (L : A ⥤ B) [preserves_finite_limits L] [preserves_finite_colimits L] +variables [abelian A] [abelian B] +variables (L : A ⥤ B) +section + +variables [preserves_finite_limits L] [preserves_finite_colimits L] + +/-- A functor preserving finite limits and finite colimits preserves exactness. The converse +result is also true, see `functor.preserves_finite_limits_of_map_exact` and +`functor.preserves_finite_colimits_of_map_exact`. -/ lemma map_exact {X Y Z : A} (f : X ⟶ Y) (g : Y ⟶ Z) (e1 : exact f g) : exact (L.map f) (L.map g) := begin @@ -343,6 +350,93 @@ begin rw [fork.ι_of_ι, cofork.π_of_π, ← L.map_comp, kernel_comp_cokernel _ _ e1, L.map_zero] end +end + +section + +variables (h : ∀ ⦃X Y Z : A⦄ {f : X ⟶ Y} {g : Y ⟶ Z}, exact f g → exact (L.map f) (L.map g)) +include h + +open_locale zero_object + +/-- A functor which preserves exactness preserves zero morphisms. -/ +lemma preserves_zero_morphisms_of_map_exact : L.preserves_zero_morphisms := +begin + replace h := (h (exact_of_zero (𝟙 0) (𝟙 0))).w, + rw [L.map_id, category.comp_id] at h, + exact preserves_zero_morphisms_of_map_zero_object (id_zero_equiv_iso_zero _ h), +end + +/-- A functor which preserves exactness preserves monomorphisms. -/ +lemma preserves_monomorphisms_of_map_exact : L.preserves_monomorphisms := +{ preserves := λ X Y f hf, + begin + letI := preserves_zero_morphisms_of_map_exact L h, + apply ((tfae_mono (L.obj 0) (L.map f)).out 2 0).mp, + rw ←L.map_zero, + exact h (((tfae_mono 0 f).out 0 2).mp hf) + end } + +/-- A functor which preserves exactness preserves epimorphisms. -/ +lemma preserves_epimorphisms_of_map_exact : L.preserves_epimorphisms := +{ preserves := λ X Y f hf, + begin + letI := preserves_zero_morphisms_of_map_exact L h, + apply ((tfae_epi (L.obj 0) (L.map f)).out 2 0).mp, + rw ←L.map_zero, + exact h (((tfae_epi 0 f).out 0 2).mp hf) + end } + +/-- A functor which preserves exactness preserves kernels. -/ +def preserves_kernels_of_map_exact (X Y : A) (f : X ⟶ Y) : + preserves_limit (parallel_pair f 0) L := +{ preserves := λ c ic, + begin + letI := preserves_zero_morphisms_of_map_exact L h, + letI := preserves_monomorphisms_of_map_exact L h, + letI := mono_of_is_limit_fork ic, + have hf := (is_limit_map_cone_fork_equiv' L (kernel_fork.condition c)).symm + (is_limit_of_exact_of_mono (L.map (fork.ι c)) (L.map f) + (h (exact_of_is_kernel (fork.ι c) f (kernel_fork.condition c) + (ic.of_iso_limit (iso_of_ι _))))), + exact hf.of_iso_limit ((cones.functoriality _ L).map_iso (iso_of_ι _).symm), + end } + +/-- A functor which preserves exactness preserves zero cokernels. -/ +def preserves_cokernels_of_map_exact (X Y : A) (f : X ⟶ Y) : + preserves_colimit (parallel_pair f 0) L := +{ preserves := λ c ic, + begin + letI := preserves_zero_morphisms_of_map_exact L h, + letI := preserves_epimorphisms_of_map_exact L h, + letI := epi_of_is_colimit_cofork ic, + have hf := (is_colimit_map_cocone_cofork_equiv' L (cokernel_cofork.condition c)).symm + (is_colimit_of_exact_of_epi (L.map f) (L.map (cofork.π c)) + (h (exact_of_is_cokernel f (cofork.π c) (cokernel_cofork.condition c) + (ic.of_iso_colimit (iso_of_π _))))), + exact hf.of_iso_colimit ((cocones.functoriality _ L).map_iso (iso_of_π _).symm), + end } + +/-- A functor which preserves exactness is left exact, i.e. preserves finite limits. +This is part of the inverse implication to `functor.map_exact`. -/ +def preserves_finite_limits_of_map_exact : limits.preserves_finite_limits L := +begin + letI := preserves_zero_morphisms_of_map_exact L h, + letI := preserves_kernels_of_map_exact L h, + apply preserves_finite_limits_of_preserves_kernels, +end + +/-- A functor which preserves exactness is right exact, i.e. preserves finite colimits. +This is part of the inverse implication to `functor.map_exact`. -/ +def preserves_finite_colimits_of_map_exact : limits.preserves_finite_colimits L := +begin + letI := preserves_zero_morphisms_of_map_exact L h, + letI := preserves_cokernels_of_map_exact L h, + apply preserves_finite_colimits_of_preserves_cokernels, +end + +end + end functor end category_theory diff --git a/formal/lean/mathlib/category_theory/abelian/generator.lean b/formal/lean/mathlib/category_theory/abelian/generator.lean new file mode 100644 index 0000000000000000000000000000000000000000..e5ff33f877a57fb11dd87cc01234634acff8277f --- /dev/null +++ b/formal/lean/mathlib/category_theory/abelian/generator.lean @@ -0,0 +1,58 @@ +/- +Copyright (c) 2022 Markus Himmel. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Markus Himmel +-/ +import category_theory.abelian.subobject +import category_theory.limits.essentially_small +import category_theory.preadditive.injective +import category_theory.preadditive.generator + +/-! +# A complete abelian category with enough injectives and a separator has an injective coseparator + +## Future work +* Once we know that Grothendieck categories have enough injectives, we can use this to conclude + that Grothendieck categories have an injective coseparator. + +## References +* [Peter J Freyd, *Abelian Categories* (Theorem 3.37)][freyd1964abelian] + +-/ + +open category_theory category_theory.limits opposite + +universes v u + +namespace category_theory.abelian +variables {C : Type u} [category.{v} C] [abelian C] + +theorem has_injective_coseparator [has_limits C] [enough_injectives C] (G : C) + (hG : is_separator G) : ∃ G : C, injective G ∧ is_coseparator G := +begin + haveI : well_powered C := well_powered_of_is_detector G hG.is_detector, + haveI : has_products_of_shape (subobject (op G)) C := has_products_of_shape_of_small _ _, + let T : C := injective.under (pi_obj (λ P : subobject (op G), unop P)), + refine ⟨T, infer_instance, (preadditive.is_coseparator_iff _).2 (λ X Y f hf, _)⟩, + refine (preadditive.is_separator_iff _).1 hG _ (λ h, _), + suffices hh : factor_thru_image (h ≫ f) = 0, + { rw [← limits.image.fac (h ≫ f), hh, zero_comp] }, + let R := subobject.mk (factor_thru_image (h ≫ f)).op, + let q₁ : image (h ≫ f) ⟶ unop R := + (subobject.underlying_iso (factor_thru_image (h ≫ f)).op).unop.hom, + let q₂ : unop (R : Cᵒᵖ) ⟶ pi_obj (λ P : subobject (op G), unop P) := + section_ (pi.π (λ P : subobject (op G), unop P) R), + let q : image (h ≫ f) ⟶ T := q₁ ≫ q₂ ≫ injective.ι _, + exact zero_of_comp_mono q (by rw [← injective.comp_factor_thru q (limits.image.ι (h ≫ f)), + limits.image.fac_assoc, category.assoc, hf, comp_zero]) +end + +theorem has_projective_separator [has_colimits C] [enough_projectives C] (G : C) + (hG : is_coseparator G) : ∃ G : C, projective G ∧ is_separator G := +begin + haveI : has_limits Cᵒᵖ := has_limits_op_of_has_colimits, + obtain ⟨T, hT₁, hT₂⟩ := has_injective_coseparator (op G) ((is_separator_op_iff _).2 hG), + exactI ⟨unop T, infer_instance, (is_separator_unop_iff _).2 hT₂⟩ +end + +end category_theory.abelian diff --git a/formal/lean/mathlib/category_theory/abelian/opposite.lean b/formal/lean/mathlib/category_theory/abelian/opposite.lean index 3dd2cf2967c26c2d0bd0131ccfa09d40bb7d368a..5f7725253b888cfa84a71723e62c7aaefa94a116 100644 --- a/formal/lean/mathlib/category_theory/abelian/opposite.lean +++ b/formal/lean/mathlib/category_theory/abelian/opposite.lean @@ -21,8 +21,8 @@ open category_theory.limits variables (C : Type*) [category C] [abelian C] local attribute [instance] - finite_limits_from_equalizers_and_finite_products - finite_colimits_from_coequalizers_and_finite_coproducts + has_finite_limits_of_has_equalizers_and_finite_products + has_finite_colimits_of_has_coequalizers_and_finite_coproducts has_finite_limits_opposite has_finite_colimits_opposite has_finite_products_opposite instance : abelian Cᵒᵖ := diff --git a/formal/lean/mathlib/category_theory/abelian/subobject.lean b/formal/lean/mathlib/category_theory/abelian/subobject.lean new file mode 100644 index 0000000000000000000000000000000000000000..da3e05db28aab9843d83a368f9e7e4be1aef8475 --- /dev/null +++ b/formal/lean/mathlib/category_theory/abelian/subobject.lean @@ -0,0 +1,64 @@ +/- +Copyright (c) 2022 Markus Himmel. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Markus Himmel +-/ +import category_theory.abelian.opposite +import category_theory.subobject.limits + +/-! +# Equivalence between subobjects and quotients in an abelian category + +-/ + +open category_theory category_theory.limits opposite + +universes v u + +noncomputable theory + +namespace category_theory.abelian +variables {C : Type u} [category.{v} C] + +/-- In an abelian category, the subobjects and quotient objects of an object `X` are + order-isomorphic via taking kernels and cokernels. + Implemented here using subobjects in the opposite category, + since mathlib does not have a notion of quotient objects at the time of writing. -/ +@[simps] +def subobject_iso_subobject_op [abelian C] (X : C) : subobject X ≃o (subobject (op X))ᵒᵈ := +begin + refine order_iso.of_hom_inv (cokernel_order_hom X) (kernel_order_hom X) _ _, + { change (cokernel_order_hom X).comp (kernel_order_hom X) = _, + refine order_hom.ext _ _ (funext (subobject.ind _ _)), + introsI A f hf, + dsimp only [order_hom.comp_coe, function.comp_app, kernel_order_hom_coe, subobject.lift_mk, + cokernel_order_hom_coe, order_hom.id_coe, id.def], + refine subobject.mk_eq_mk_of_comm _ _ ⟨_, _, quiver.hom.unop_inj _, quiver.hom.unop_inj _⟩ _, + { exact (abelian.epi_desc f.unop _ (cokernel.condition (kernel.ι f.unop))).op }, + { exact (cokernel.desc _ _ (kernel.condition f.unop)).op }, + { simp only [← cancel_epi (cokernel.π (kernel.ι f.unop)), unop_comp, quiver.hom.unop_op, + unop_id_op, cokernel.π_desc_assoc, comp_epi_desc, category.comp_id] }, + { simp only [← cancel_epi f.unop, unop_comp, quiver.hom.unop_op, unop_id, comp_epi_desc_assoc, + cokernel.π_desc, category.comp_id] }, + { exact quiver.hom.unop_inj (by simp only [unop_comp, quiver.hom.unop_op, comp_epi_desc]) } }, + { change (kernel_order_hom X).comp (cokernel_order_hom X) = _, + refine order_hom.ext _ _ (funext (subobject.ind _ _)), + introsI A f hf, + dsimp only [order_hom.comp_coe, function.comp_app, cokernel_order_hom_coe, subobject.lift_mk, + kernel_order_hom_coe, order_hom.id_coe, id.def, unop_op, quiver.hom.unop_op], + refine subobject.mk_eq_mk_of_comm _ _ ⟨_, _, _, _⟩ _, + { exact abelian.mono_lift f _ (kernel.condition (cokernel.π f)) }, + { exact kernel.lift _ _ (cokernel.condition f) }, + { simp only [← cancel_mono (kernel.ι (cokernel.π f)), category.assoc, image.fac, mono_lift_comp, + category.id_comp, auto_param_eq] }, + { simp only [← cancel_mono f, category.assoc, mono_lift_comp, image.fac, category.id_comp, + auto_param_eq] }, + { simp only [mono_lift_comp] } } +end + +/-- A well-powered abelian category is also well-copowered. -/ +instance well_powered_opposite [abelian C] [well_powered C] : well_powered Cᵒᵖ := +{ subobject_small := λ X, + (small_congr (subobject_iso_subobject_op (unop X)).to_equiv).1 infer_instance } + +end category_theory.abelian diff --git a/formal/lean/mathlib/category_theory/adjunction/basic.lean b/formal/lean/mathlib/category_theory/adjunction/basic.lean index 5bdfc3a91b088ad32d4365fbed9e669abe012f8b..0df5be9436e539a0e528de2813ede0c634abbfe9 100644 --- a/formal/lean/mathlib/category_theory/adjunction/basic.lean +++ b/formal/lean/mathlib/category_theory/adjunction/basic.lean @@ -170,7 +170,7 @@ This is an auxiliary data structure useful for constructing adjunctions. See `adjunction.mk_of_hom_equiv`. This structure won't typically be used anywhere else. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure core_hom_equiv (F : C ⥤ D) (G : D ⥤ C) := (hom_equiv : Π (X Y), (F.obj X ⟶ Y) ≃ (X ⟶ G.obj Y)) (hom_equiv_naturality_left_symm' : Π {X' X Y} (f : X' ⟶ X) (g : X ⟶ G.obj Y), @@ -201,7 +201,7 @@ This is an auxiliary data structure useful for constructing adjunctions. See `adjunction.mk_of_unit_counit`. This structure won't typically be used anywhere else. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure core_unit_counit (F : C ⥤ D) (G : D ⥤ C) := (unit : 𝟭 C ⟶ F.comp G) (counit : G.comp F ⟶ 𝟭 D) diff --git a/formal/lean/mathlib/category_theory/adjunction/reflective.lean b/formal/lean/mathlib/category_theory/adjunction/reflective.lean index 67ce6368ea73d684b22a7b4d95ff23d0c244585d..3febb0a8d88f7e10e4af8f975b4b0c60eff63522 100644 --- a/formal/lean/mathlib/category_theory/adjunction/reflective.lean +++ b/formal/lean/mathlib/category_theory/adjunction/reflective.lean @@ -89,8 +89,8 @@ lemma mem_ess_image_of_unit_is_iso [is_right_adjoint i] (A : C) ⟨(left_adjoint i).obj A, ⟨(as_iso ((of_right_adjoint i).unit.app A)).symm⟩⟩ /-- If `η_A` is a split monomorphism, then `A` is in the reflective subcategory. -/ -lemma mem_ess_image_of_unit_split_mono [reflective i] {A : C} - [split_mono ((of_right_adjoint i).unit.app A)] : A ∈ i.ess_image := +lemma mem_ess_image_of_unit_is_split_mono [reflective i] {A : C} + [is_split_mono ((of_right_adjoint i).unit.app A)] : A ∈ i.ess_image := begin let η : 𝟭 C ⟶ left_adjoint i ⋙ i := (of_right_adjoint i).unit, haveI : is_iso (η.app (i.obj ((left_adjoint i).obj A))) := (i.obj_mem_ess_image _).unit_is_iso, @@ -99,7 +99,7 @@ begin rw (show retraction _ ≫ η.app A = _, from η.naturality (retraction (η.app A))), apply epi_comp (η.app (i.obj ((left_adjoint i).obj A))) }, resetI, - haveI := is_iso_of_epi_of_split_mono (η.app A), + haveI := is_iso_of_epi_of_is_split_mono (η.app A), exact mem_ess_image_of_unit_is_iso A, end @@ -156,17 +156,19 @@ by rw [←equiv.eq_symm_apply, unit_comp_partial_bijective_symm_natural A h, equ /-- If `i : D ⥤ C` is reflective, the inverse functor of `i ≌ F.ess_image` can be explicitly defined by the reflector. -/ @[simps] -def equiv_ess_image_of_reflective [reflective i] : D ≌ i.ess_image := +def equiv_ess_image_of_reflective [reflective i] : D ≌ i.ess_image_subcategory := { functor := i.to_ess_image, inverse := i.ess_image_inclusion ⋙ (left_adjoint i : _), unit_iso := nat_iso.of_components (λ X, (as_iso $ (of_right_adjoint i).counit.app X).symm) (by { intros X Y f, dsimp, simp only [is_iso.eq_inv_comp, is_iso.comp_inv_eq, category.assoc], exact ((of_right_adjoint i).counit.naturality _).symm }), - counit_iso := nat_iso.of_components - (λ X, by { refine (iso.symm $ as_iso _), exact (of_right_adjoint i).unit.app X, + counit_iso := + nat_iso.of_components + (λ X, by { refine (iso.symm $ as_iso _), exact (of_right_adjoint i).unit.app X.obj, apply_with (is_iso_of_reflects_iso _ i.ess_image_inclusion) { instances := ff }, - exact functor.ess_image.unit_is_iso X.prop }) - (by { intros X Y f, dsimp, simp only [is_iso.eq_inv_comp, is_iso.comp_inv_eq, category.assoc], - exact ((of_right_adjoint i).unit.naturality f).symm }) } + exact functor.ess_image.unit_is_iso X.property }) + (by { intros X Y f, dsimp, rw [is_iso.comp_inv_eq, assoc], + have h := ((of_right_adjoint i).unit.naturality f).symm, + rw [functor.id_map] at h, erw [← h, is_iso.inv_hom_id_assoc, functor.comp_map] }) } end category_theory diff --git a/formal/lean/mathlib/category_theory/arrow.lean b/formal/lean/mathlib/category_theory/arrow.lean index d20e8c476c75cdb882867b6d7e57990b2f566238..e95092f6dfb3fde6cabe9c1637db6027221c3144 100644 --- a/formal/lean/mathlib/category_theory/arrow.lean +++ b/formal/lean/mathlib/category_theory/arrow.lean @@ -100,6 +100,12 @@ and a proof that the square commutes. -/ f ≅ g := comma.iso_mk l r h +/-- A variant of `arrow.iso_mk` that creates an iso between two `arrow.mk`s with a better type +signature. -/ +abbreviation iso_mk' {W X Y Z : T} (f : W ⟶ X) (g : Y ⟶ Z) + (e₁ : W ≅ Y) (e₂ : X ≅ Z) (h : e₁.hom ≫ g = f ≫ e₂.hom) : arrow.mk f ≅ arrow.mk g := +arrow.iso_mk e₁ e₂ h + section variables {f g : arrow T} (sq : f ⟶ g) @@ -162,74 +168,6 @@ lemma square_from_iso_invert {X Y : T} (i : X ≅ Y) (p : arrow T) (sq : arrow.m i.inv ≫ sq.left ≫ p.hom = sq.right := by simp only [iso.inv_hom_id_assoc, arrow.w, arrow.mk_hom] -/-- A lift of a commutative square is a diagonal morphism making the two triangles commute. -/ -@[ext] structure lift_struct {f g : arrow T} (sq : f ⟶ g) := -(lift : f.right ⟶ g.left) -(fac_left' : f.hom ≫ lift = sq.left . obviously) -(fac_right' : lift ≫ g.hom = sq.right . obviously) - -restate_axiom lift_struct.fac_left' -restate_axiom lift_struct.fac_right' - -instance lift_struct_inhabited {X : T} : inhabited (lift_struct (𝟙 (arrow.mk (𝟙 X)))) := -⟨⟨𝟙 _, category.id_comp _, category.comp_id _⟩⟩ - -/-- `has_lift sq` says that there is some `lift_struct sq`, i.e., that it is possible to find a - diagonal morphism making the two triangles commute. -/ -class has_lift {f g : arrow T} (sq : f ⟶ g) : Prop := -mk' :: (exists_lift : nonempty (lift_struct sq)) - -lemma has_lift.mk {f g : arrow T} {sq : f ⟶ g} (s : lift_struct sq) : has_lift sq := -⟨nonempty.intro s⟩ - -attribute [simp, reassoc] lift_struct.fac_left lift_struct.fac_right - -/-- Given `has_lift sq`, obtain a lift. -/ -noncomputable def has_lift.struct {f g : arrow T} (sq : f ⟶ g) [has_lift sq] : lift_struct sq := -classical.choice has_lift.exists_lift - -/-- If there is a lift of a commutative square `sq`, we can access it by saying `lift sq`. -/ -noncomputable abbreviation lift {f g : arrow T} (sq : f ⟶ g) [has_lift sq] : f.right ⟶ g.left := -(has_lift.struct sq).lift - -lemma lift.fac_left {f g : arrow T} (sq : f ⟶ g) [has_lift sq] : f.hom ≫ lift sq = sq.left := -by simp - -lemma lift.fac_right {f g : arrow T} (sq : f ⟶ g) [has_lift sq] : lift sq ≫ g.hom = sq.right := -by simp - -@[simp, reassoc] -lemma lift.fac_right_of_to_mk {X Y : T} {f : arrow T} {g : X ⟶ Y} (sq : f ⟶ mk g) [has_lift sq] : - lift sq ≫ g = sq.right := -by simp only [←mk_hom g, lift.fac_right] - -@[simp, reassoc] -lemma lift.fac_left_of_from_mk {X Y : T} {f : X ⟶ Y} {g : arrow T} (sq : mk f ⟶ g) [has_lift sq] : - f ≫ lift sq = sq.left := -by simp only [←mk_hom f, lift.fac_left] - -@[simp, reassoc] -lemma lift_mk'_left {X Y P Q : T} {f : X ⟶ Y} {g : P ⟶ Q} {u : X ⟶ P} {v : Y ⟶ Q} - (h : u ≫ g = f ≫ v) [has_lift $ arrow.hom_mk' h] : f ≫ lift (arrow.hom_mk' h) = u := -by simp only [←arrow.mk_hom f, lift.fac_left, arrow.hom_mk'_left] - -@[simp, reassoc] -lemma lift_mk'_right {X Y P Q : T} {f : X ⟶ Y} {g : P ⟶ Q} {u : X ⟶ P} {v : Y ⟶ Q} - (h : u ≫ g = f ≫ v) [has_lift $ arrow.hom_mk' h] : lift (arrow.hom_mk' h) ≫ g = v := -by simp only [←arrow.mk_hom g, lift.fac_right, arrow.hom_mk'_right] - -section - -instance subsingleton_lift_struct_of_epi {f g : arrow T} (sq : f ⟶ g) [epi f.hom] : - subsingleton (lift_struct sq) := -subsingleton.intro $ λ a b, lift_struct.ext a b $ (cancel_epi f.hom).1 $ by simp - -instance subsingleton_lift_struct_of_mono {f g : arrow T} (sq : f ⟶ g) [mono g.hom] : - subsingleton (lift_struct sq) := -subsingleton.intro $ λ a b, lift_struct.ext a b $ (cancel_mono g.hom).1 $ by simp - -end - variables {C : Type u} [category.{v} C] /-- A helper construction: given a square between `i` and `f ≫ g`, produce a square between `i` and `g`, whose top leg uses `f`: diff --git a/formal/lean/mathlib/category_theory/bicategory/free.lean b/formal/lean/mathlib/category_theory/bicategory/free.lean index 6b78babfdbf1cfbeb1616c1fb499a6ada23c0307..6528cd11f2bf2ca75b8d2ad6e17dac818cf5aaa8 100644 --- a/formal/lean/mathlib/category_theory/bicategory/free.lean +++ b/formal/lean/mathlib/category_theory/bicategory/free.lean @@ -45,7 +45,7 @@ inductive hom : B → B → Type (max u v) instance (a b : B) [inhabited (a ⟶ b)] : inhabited (hom a b) := ⟨hom.of default⟩ /-- Representatives of 2-morphisms in the free bicategory. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] inductive hom₂ : Π {a b : B}, hom a b → hom a b → Type (max u v) | id {a b} (f : hom a b) : hom₂ f f | vcomp {a b} {f g h : hom a b} (η : hom₂ f g) (θ : hom₂ g h) : hom₂ f h diff --git a/formal/lean/mathlib/category_theory/bicategory/functor.lean b/formal/lean/mathlib/category_theory/bicategory/functor.lean index e2697618d3fecc2380f060a32fa550e0f7bb00d9..ad5757a9bb35b2ae44909972c5c33b486b916f73 100644 --- a/formal/lean/mathlib/category_theory/bicategory/functor.lean +++ b/formal/lean/mathlib/category_theory/bicategory/functor.lean @@ -240,7 +240,7 @@ def comp (F : oplax_functor B C) (G : oplax_functor C D) : oplax_functor B D := A structure on an oplax functor that promotes an oplax functor to a pseudofunctor. See `pseudofunctor.mk_of_oplax`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure pseudo_core (F : oplax_functor B C) := (map_id_iso (a : B) : F.map (𝟙 a) ≅ 𝟙 (F.obj a)) (map_comp_iso {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : F.map (f ≫ g) ≅ F.map f ≫ F.map g) diff --git a/formal/lean/mathlib/category_theory/category/PartialFun.lean b/formal/lean/mathlib/category_theory/category/PartialFun.lean index 2f1349a549b6caa12069890bcc9231fb4ca989dc..bef9af2a87470ef3a9a3a562df0b0d13d96a5e43 100644 --- a/formal/lean/mathlib/category_theory/category/PartialFun.lean +++ b/formal/lean/mathlib/category_theory/category/PartialFun.lean @@ -39,7 +39,7 @@ namespace PartialFun instance : has_coe_to_sort PartialFun Type* := ⟨id⟩ /-- Turns a type into a `PartialFun`. -/ -@[nolint has_inhabited_instance] def of : Type* → PartialFun := id +@[nolint has_nonempty_instance] def of : Type* → PartialFun := id @[simp] lemma coe_of (X : Type*) : ↥(of X) = X := rfl diff --git a/formal/lean/mathlib/category_theory/closed/cartesian.lean b/formal/lean/mathlib/category_theory/closed/cartesian.lean index b3ea9b725fd7cd3612c0a55037ae1ab73c9c368e..848aeeb9213a2e762614494163d8de467d426139 100644 --- a/formal/lean/mathlib/category_theory/closed/cartesian.lean +++ b/formal/lean/mathlib/category_theory/closed/cartesian.lean @@ -306,8 +306,8 @@ lemma strict_initial {I : C} (t : is_initial I) (f : A ⟶ I) : is_iso f := begin haveI : mono (limits.prod.lift (𝟙 A) f ≫ (zero_mul t).hom) := mono_comp _ _, rw [zero_mul_hom, prod.lift_snd] at _inst, - haveI: split_epi f := ⟨t.to _, t.hom_ext _ _⟩, - apply is_iso_of_mono_of_split_epi + haveI: is_split_epi f := is_split_epi.mk' ⟨t.to _, t.hom_ext _ _⟩, + apply is_iso_of_mono_of_is_split_epi end instance to_initial_is_iso [has_initial C] (f : A ⟶ ⊥_ C) : is_iso f := diff --git a/formal/lean/mathlib/category_theory/closed/ideal.lean b/formal/lean/mathlib/category_theory/closed/ideal.lean index 50ba92dc0dbc453164bbe7ed04c6e426b9d605e8..7d8c309392d38ec454c3b2d06a60db0b5e01172e 100644 --- a/formal/lean/mathlib/category_theory/closed/ideal.lean +++ b/formal/lean/mathlib/category_theory/closed/ideal.lean @@ -143,8 +143,8 @@ begin ir.hom_equiv_apply_eq, assoc, assoc, prod_comparison_natural_assoc, L.map_id, ← prod.map_id_comp_assoc, ir.left_triangle_components, prod.map_id_id, id_comp], apply is_iso.hom_inv_id_assoc }, - haveI : split_mono (η.app (A ⟹ i.obj B)) := ⟨_, this⟩, - apply mem_ess_image_of_unit_split_mono, + haveI : is_split_mono (η.app (A ⟹ i.obj B)) := is_split_mono.mk' ⟨_, this⟩, + apply mem_ess_image_of_unit_is_split_mono, end variables [exponential_ideal i] diff --git a/formal/lean/mathlib/category_theory/comm_sq.lean b/formal/lean/mathlib/category_theory/comm_sq.lean index 95b3d528e095e3a47ca28d22776dd59053c042bf..19e70cbc27cbefd7bfb1b8909d601b4630a21fb8 100644 --- a/formal/lean/mathlib/category_theory/comm_sq.lean +++ b/formal/lean/mathlib/category_theory/comm_sq.lean @@ -1,11 +1,10 @@ /- Copyright (c) 2022 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. -Authors: Scott Morrison +Authors: Scott Morrison, Joël Riou -/ import category_theory.arrow - /-! # Commutative squares @@ -75,4 +74,111 @@ end functor alias functor.map_comm_sq ← comm_sq.map +namespace comm_sq + +variables {A B X Y : C} {f : A ⟶ X} {i : A ⟶ B} {p : X ⟶ Y} {g : B ⟶ Y} + +/-- The datum of a lift in a commutative square, i.e. a up-right-diagonal +morphism which makes both triangles commute. -/ +@[ext, nolint has_nonempty_instance] +structure lift_struct (sq : comm_sq f i p g) := +(l : B ⟶ X) (fac_left' : i ≫ l = f) (fac_right' : l ≫ p = g) + +namespace lift_struct + +restate_axiom fac_left' +restate_axiom fac_right' + +/-- A `lift_struct` for a commutative square gives a `lift_struct` for the +corresponding square in the opposite category. -/ +@[simps] +def op {sq : comm_sq f i p g} (l : lift_struct sq) : lift_struct sq.op := +{ l := l.l.op, + fac_left' := by rw [← op_comp, l.fac_right], + fac_right' := by rw [← op_comp, l.fac_left], } + +/-- A `lift_struct` for a commutative square in the opposite category +gives a `lift_struct` for the corresponding square in the original category. -/ +@[simps] +def unop {A B X Y : Cᵒᵖ} {f : A ⟶ X} {i : A ⟶ B} {p : X ⟶ Y} {g : B ⟶ Y} {sq : comm_sq f i p g} + (l : lift_struct sq) : lift_struct sq.unop := +{ l := l.l.unop, + fac_left' := by rw [← unop_comp, l.fac_right], + fac_right' := by rw [← unop_comp, l.fac_left], } + +/-- Equivalences of `lift_struct` for a square and the corresponding square +in the opposite category. -/ +@[simps] +def op_equiv (sq : comm_sq f i p g) : lift_struct sq ≃ lift_struct sq.op := +{ to_fun := op, + inv_fun := unop, + left_inv := by tidy, + right_inv := by tidy, } + +/-- Equivalences of `lift_struct` for a square in the oppositive category and +the corresponding square in the original category. -/ +def unop_equiv {A B X Y : Cᵒᵖ} {f : A ⟶ X} {i : A ⟶ B} {p : X ⟶ Y} {g : B ⟶ Y} + (sq : comm_sq f i p g) : lift_struct sq ≃ lift_struct sq.unop := +{ to_fun := unop, + inv_fun := op, + left_inv := by tidy, + right_inv := by tidy, } + +end lift_struct + +instance subsingleton_lift_struct_of_epi (sq : comm_sq f i p g) [epi i] : + subsingleton (lift_struct sq) := +⟨λ l₁ l₂, by { ext, simp only [← cancel_epi i, lift_struct.fac_left], }⟩ + +instance subsingleton_lift_struct_of_mono (sq : comm_sq f i p g) [mono p] : + subsingleton (lift_struct sq) := +⟨λ l₁ l₂, by { ext, simp only [← cancel_mono p, lift_struct.fac_right], }⟩ + +variable (sq : comm_sq f i p g) + +/-- The assertion that a square has a `lift_struct`. -/ +class has_lift : Prop := (exists_lift : nonempty sq.lift_struct) + +namespace has_lift + +variable {sq} + +lemma mk' (l : sq.lift_struct) : has_lift sq := ⟨nonempty.intro l⟩ + +variable (sq) + +lemma iff : has_lift sq ↔ nonempty sq.lift_struct := +by { split, exacts [λ h, h.exists_lift, λ h, mk h], } + +lemma iff_op : has_lift sq ↔ has_lift sq.op := +begin + rw [iff, iff], + exact nonempty.congr (lift_struct.op_equiv sq).to_fun (lift_struct.op_equiv sq).inv_fun, +end + +lemma iff_unop {A B X Y : Cᵒᵖ} {f : A ⟶ X} {i : A ⟶ B} {p : X ⟶ Y} {g : B ⟶ Y} + (sq : comm_sq f i p g) : has_lift sq ↔ has_lift sq.unop := +begin + rw [iff, iff], + exact nonempty.congr (lift_struct.unop_equiv sq).to_fun (lift_struct.unop_equiv sq).inv_fun, +end + +end has_lift + +/-- A choice of a diagonal morphism that is part of a `lift_struct` when +the square has a lift. -/ +noncomputable +def lift [hsq : has_lift sq] : B ⟶ X := +hsq.exists_lift.some.l + +@[simp, reassoc] +lemma fac_left [hsq : has_lift sq] : i ≫ sq.lift = f := +hsq.exists_lift.some.fac_left + +@[simp, reassoc] +lemma fac_right [hsq : has_lift sq] : sq.lift ≫ p = g := +hsq.exists_lift.some.fac_right + +end comm_sq + end category_theory diff --git a/formal/lean/mathlib/category_theory/concrete_category/basic.lean b/formal/lean/mathlib/category_theory/concrete_category/basic.lean index bf12a8c81f6648dd726c78aa57e51e0559ecb80d..29e9eb2079c0bc89c3f21cfcaf33f6ddc0148725 100644 --- a/formal/lean/mathlib/category_theory/concrete_category/basic.lean +++ b/formal/lean/mathlib/category_theory/concrete_category/basic.lean @@ -171,11 +171,11 @@ instance induced_category.has_forget₂ {C : Type v} {D : Type v'} [category D] forget_comp := rfl } instance full_subcategory.concrete_category {C : Type v} [category C] [concrete_category C] - (Z : C → Prop) : concrete_category {X : C // Z X} := + (Z : C → Prop) : concrete_category (full_subcategory Z) := { forget := full_subcategory_inclusion Z ⋙ forget C } instance full_subcategory.has_forget₂ {C : Type v} [category C] [concrete_category C] - (Z : C → Prop) : has_forget₂ {X : C // Z X} C := + (Z : C → Prop) : has_forget₂ (full_subcategory Z) C := { forget₂ := full_subcategory_inclusion Z, forget_comp := rfl } diff --git a/formal/lean/mathlib/category_theory/concrete_category/bundled.lean b/formal/lean/mathlib/category_theory/concrete_category/bundled.lean index 419621b09a5bf99852bf177e6a7b9838301ac114..de675d9604c4005dfda3e786ba30dac92e8b64f5 100644 --- a/formal/lean/mathlib/category_theory/concrete_category/bundled.lean +++ b/formal/lean/mathlib/category_theory/concrete_category/bundled.lean @@ -22,7 +22,7 @@ variables {c d : Type u → Type v} {α : Type u} /-- `bundled` is a type bundled with a type class instance for that type. Only the type class is exposed as a parameter. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure bundled (c : Type u → Type v) : Type (max (u+1) v) := (α : Type u) (str : c α . tactic.apply_instance) diff --git a/formal/lean/mathlib/category_theory/connected_components.lean b/formal/lean/mathlib/category_theory/connected_components.lean index e4f1e2bc3d2f1ec83cf53fed5cdd73816e86e13a..ba3c0b93c162f27b9cca985668e0fa017ff15809 100644 --- a/formal/lean/mathlib/category_theory/connected_components.lean +++ b/formal/lean/mathlib/category_theory/connected_components.lean @@ -41,7 +41,7 @@ instance [inhabited J] : inhabited (connected_components J) := ⟨quotient.mk' d /-- Given an index for a connected component, produce the actual component as a full subcategory. -/ @[derive category] -def component (j : connected_components J) : Type u₁ := {k : J // quotient.mk' k = j} +def component (j : connected_components J) : Type u₁ := full_subcategory (λ k, quotient.mk' k = j) /-- The inclusion functor from a connected component to the whole category. -/ @[derive [full, faithful], simps {rhs_md := semireducible}] @@ -82,7 +82,7 @@ begin { refine @@list.chain_pmap_of_chain _ _ _ f (λ x y _ _ h, _) hl₁ h₁₂ _, exact zag_of_zag_obj (component.ι _) h }, { erw list.last_pmap _ f (j₁ :: l) (by simpa [h₁₂] using hf) (list.cons_ne_nil _ _), - exact subtype.ext hl₂ }, + exact full_subcategory.ext _ _ hl₂ }, end /-- diff --git a/formal/lean/mathlib/category_theory/core.lean b/formal/lean/mathlib/category_theory/core.lean index 5f583ecb9e1f6faf5ad8e4d8aabb4c8c20aba823..8c55b963516401551495965273e58a2419a35fbb 100644 --- a/formal/lean/mathlib/category_theory/core.lean +++ b/formal/lean/mathlib/category_theory/core.lean @@ -26,7 +26,7 @@ universes v₁ v₂ u₁ u₂ -- morphism levels before object levels. See note /-- The core of a category C is the groupoid whose morphisms are all the isomorphisms of C. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def core (C : Type u₁) := C variables {C : Type u₁} [category.{v₁} C] diff --git a/formal/lean/mathlib/category_theory/differential_object.lean b/formal/lean/mathlib/category_theory/differential_object.lean index 268eb4e9ff21ac3a2e45ee3054108b4281d3d1ea..e6a656e86a25cdabf070b26e5f286b7cf19b895a 100644 --- a/formal/lean/mathlib/category_theory/differential_object.lean +++ b/formal/lean/mathlib/category_theory/differential_object.lean @@ -35,7 +35,7 @@ A differential object in a category with zero morphisms and a shift is an object `X` equipped with a morphism `d : X ⟶ X⟦1⟧`, such that `d^2 = 0`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure differential_object := (X : C) (d : X ⟶ X⟦1⟧) @@ -51,7 +51,7 @@ namespace differential_object /-- A morphism of differential objects is a morphism commuting with the differentials. -/ -@[ext, nolint has_inhabited_instance] +@[ext, nolint has_nonempty_instance] structure hom (X Y : differential_object C) := (f : X.X ⟶ Y.X) (comm' : X.d ≫ f⟦1⟧' = f ≫ Y.d . obviously) diff --git a/formal/lean/mathlib/category_theory/elements.lean b/formal/lean/mathlib/category_theory/elements.lean index 9a8cb9b3f8d14050041119a8b8e6a9da50598f94..2229113de827ea7c004146937f00980bb3ddc904 100644 --- a/formal/lean/mathlib/category_theory/elements.lean +++ b/formal/lean/mathlib/category_theory/elements.lean @@ -40,7 +40,7 @@ variables {C : Type u} [category.{v} C] The type of objects for the category of elements of a functor `F : C ⥤ Type` is a pair `(X : C, x : F.obj X)`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def functor.elements (F : C ⥤ Type w) := (Σ c : C, F.obj c) /-- The category structure on `F.elements`, for `F : C ⥤ Type`. diff --git a/formal/lean/mathlib/category_theory/enriched/basic.lean b/formal/lean/mathlib/category_theory/enriched/basic.lean index 1627a7ae4dd6a7e31934058415c35b1820e6f2f9..dcb15e7119b03adbc6f0f6bfbe85fb5353b5e766 100644 --- a/formal/lean/mathlib/category_theory/enriched/basic.lean +++ b/formal/lean/mathlib/category_theory/enriched/basic.lean @@ -93,7 +93,7 @@ A type synonym for `C`, which should come equipped with a `V`-enriched category In a moment we will equip this with the `W`-enriched category structure obtained by applying the functor `F : lax_monoidal_functor V W` to each hom object. -/ -@[nolint has_inhabited_instance unused_arguments] +@[nolint has_nonempty_instance unused_arguments] def transport_enrichment (F : lax_monoidal_functor V W) (C : Type u₁) := C instance (F : lax_monoidal_functor V W) : @@ -180,7 +180,7 @@ When `V` is any of `Type`, `Top`, `AddCommGroup`, or `Module R`, For `V = Algebra R`, the usual forgetful functor is coyoneda of `polynomial R`, not of `R`. (Perhaps we should have a typeclass for this situation: `concrete_monoidal`?) -/ -@[nolint has_inhabited_instance unused_arguments] +@[nolint has_nonempty_instance unused_arguments] def forget_enrichment (W : Type (v+1)) [category.{v} W] [monoidal_category W] (C : Type u₁) [enriched_category W C] := C @@ -354,7 +354,7 @@ coming from the ambient braiding on `V`.) The type of `A`-graded natural transformations between `V`-functors `F` and `G`. This is the type of morphisms in `V` from `A` to the `V`-object of natural transformations. -/ -@[ext, nolint has_inhabited_instance] +@[ext, nolint has_nonempty_instance] structure graded_nat_trans (A : center V) (F G : enriched_functor V C D) := (app : Π (X : C), A.1 ⟶ (F.obj X ⟶[V] G.obj X)) (naturality : diff --git a/formal/lean/mathlib/category_theory/epi_mono.lean b/formal/lean/mathlib/category_theory/epi_mono.lean index 002a0a4857b37081ca1d4cc56057d48a946a9683..ff2d5f37c00a90ee2069e93c970781139f789d8f 100644 --- a/formal/lean/mathlib/category_theory/epi_mono.lean +++ b/formal/lean/mathlib/category_theory/epi_mono.lean @@ -33,92 +33,146 @@ instance op_epi_of_mono {A B : C} (f : A ⟶ B) [mono f] : epi f.op := ⟨λ Z g h eq, quiver.hom.unop_inj ((cancel_mono f).1 (quiver.hom.op_inj eq))⟩ /-- -A split monomorphism is a morphism `f : X ⟶ Y` admitting a retraction `retraction f : Y ⟶ X` +A split monomorphism is a morphism `f : X ⟶ Y` with a given retraction `retraction f : Y ⟶ X` such that `f ≫ retraction f = 𝟙 X`. Every split monomorphism is a monomorphism. -/ -class split_mono {X Y : C} (f : X ⟶ Y) := +@[ext, nolint has_nonempty_instance] +structure split_mono {X Y : C} (f : X ⟶ Y) := (retraction : Y ⟶ X) (id' : f ≫ retraction = 𝟙 X . obviously) +restate_axiom split_mono.id' +attribute [simp, reassoc] split_mono.id + +/-- `is_split_mono f` is the assertion that `f` admits a retraction -/ +class is_split_mono {X Y : C} (f : X ⟶ Y) : Prop := +(exists_split_mono : nonempty (split_mono f)) + +/-- A constructor for `is_split_mono f` taking a `split_mono f` as an argument -/ +lemma is_split_mono.mk' {X Y : C} {f : X ⟶ Y} (sm : split_mono f) : + is_split_mono f := ⟨nonempty.intro sm⟩ + /-- -A split epimorphism is a morphism `f : X ⟶ Y` admitting a section `section_ f : Y ⟶ X` +A split epimorphism is a morphism `f : X ⟶ Y` with a given section `section_ f : Y ⟶ X` such that `section_ f ≫ f = 𝟙 Y`. (Note that `section` is a reserved keyword, so we append an underscore.) Every split epimorphism is an epimorphism. -/ -class split_epi {X Y : C} (f : X ⟶ Y) := +@[ext, nolint has_nonempty_instance] +structure split_epi {X Y : C} (f : X ⟶ Y) := (section_ : Y ⟶ X) (id' : section_ ≫ f = 𝟙 Y . obviously) +restate_axiom split_epi.id' +attribute [simp, reassoc] split_epi.id + +/-- `is_split_epi f` is the assertion that `f` admits a section -/ +class is_split_epi {X Y : C} (f : X ⟶ Y) : Prop := +(exists_split_epi : nonempty (split_epi f)) + +/-- A constructor for `is_split_epi f` taking a `split_epi f` as an argument -/ +lemma is_split_epi.mk' {X Y : C} {f : X ⟶ Y} (se : split_epi f) : + is_split_epi f := ⟨nonempty.intro se⟩ + /-- The chosen retraction of a split monomorphism. -/ -def retraction {X Y : C} (f : X ⟶ Y) [split_mono f] : Y ⟶ X := split_mono.retraction f +noncomputable def retraction {X Y : C} (f : X ⟶ Y) [hf : is_split_mono f] : Y ⟶ X := +hf.exists_split_mono.some.retraction + @[simp, reassoc] -lemma split_mono.id {X Y : C} (f : X ⟶ Y) [split_mono f] : f ≫ retraction f = 𝟙 X := -split_mono.id' +lemma is_split_mono.id {X Y : C} (f : X ⟶ Y) [hf : is_split_mono f] : f ≫ retraction f = 𝟙 X := +hf.exists_split_mono.some.id + +/-- The retraction of a split monomorphism has an obvious section. -/ +def split_mono.split_epi {X Y : C} {f : X ⟶ Y} (sm : split_mono f) : split_epi (sm.retraction) := +{ section_ := f, } + /-- The retraction of a split monomorphism is itself a split epimorphism. -/ -instance retraction_split_epi {X Y : C} (f : X ⟶ Y) [split_mono f] : split_epi (retraction f) := -{ section_ := f } +instance retraction_is_split_epi {X Y : C} (f : X ⟶ Y) [hf : is_split_mono f] : +is_split_epi (retraction f) := +is_split_epi.mk' (split_mono.split_epi _) /-- A split mono which is epi is an iso. -/ -lemma is_iso_of_epi_of_split_mono {X Y : C} (f : X ⟶ Y) [split_mono f] [epi f] : is_iso f := +lemma is_iso_of_epi_of_is_split_mono {X Y : C} (f : X ⟶ Y) [is_split_mono f] [epi f] : is_iso f := ⟨⟨retraction f, ⟨by simp, by simp [← cancel_epi f]⟩⟩⟩ /-- The chosen section of a split epimorphism. (Note that `section` is a reserved keyword, so we append an underscore.) -/ -def section_ {X Y : C} (f : X ⟶ Y) [split_epi f] : Y ⟶ X := split_epi.section_ f +noncomputable def section_ {X Y : C} (f : X ⟶ Y) [hf : is_split_epi f] : Y ⟶ X := +hf.exists_split_epi.some.section_ + @[simp, reassoc] -lemma split_epi.id {X Y : C} (f : X ⟶ Y) [split_epi f] : section_ f ≫ f = 𝟙 Y := -split_epi.id' +lemma is_split_epi.id {X Y : C} (f : X ⟶ Y) [hf : is_split_epi f] : section_ f ≫ f = 𝟙 Y := +hf.exists_split_epi.some.id + +/-- The section of a split epimorphism has an obvious retraction. -/ +def split_epi.split_mono {X Y : C} {f : X ⟶ Y} (se : split_epi f) : split_mono (se.section_) := +{ retraction := f, } + /-- The section of a split epimorphism is itself a split monomorphism. -/ -instance section_split_mono {X Y : C} (f : X ⟶ Y) [split_epi f] : split_mono (section_ f) := -{ retraction := f } +instance section_is_split_mono {X Y : C} (f : X ⟶ Y) [hf : is_split_epi f] : + is_split_mono (section_ f) := +is_split_mono.mk' (split_epi.split_mono _) /-- A split epi which is mono is an iso. -/ -lemma is_iso_of_mono_of_split_epi {X Y : C} (f : X ⟶ Y) [mono f] [split_epi f] : is_iso f := +lemma is_iso_of_mono_of_is_split_epi {X Y : C} (f : X ⟶ Y) [mono f] [is_split_epi f] : is_iso f := ⟨⟨section_ f, ⟨by simp [← cancel_mono f], by simp⟩⟩⟩ /-- Every iso is a split mono. -/ @[priority 100] -noncomputable -instance split_mono.of_iso {X Y : C} (f : X ⟶ Y) [is_iso f] : split_mono f := -{ retraction := inv f } +instance is_split_mono.of_iso {X Y : C} (f : X ⟶ Y) [is_iso f] : is_split_mono f := +is_split_mono.mk' { retraction := inv f } /-- Every iso is a split epi. -/ @[priority 100] -noncomputable -instance split_epi.of_iso {X Y : C} (f : X ⟶ Y) [is_iso f] : split_epi f := -{ section_ := inv f } +instance is_split_epi.of_iso {X Y : C} (f : X ⟶ Y) [is_iso f] : is_split_epi f := +is_split_epi.mk' { section_ := inv f } + +lemma split_mono.mono {X Y : C} {f : X ⟶ Y} (sm : split_mono f) : mono f := +{ right_cancellation := λ Z g h w, begin replace w := w =≫ sm.retraction, simpa using w, end } /-- Every split mono is a mono. -/ @[priority 100] -instance split_mono.mono {X Y : C} (f : X ⟶ Y) [split_mono f] : mono f := -{ right_cancellation := λ Z g h w, begin replace w := w =≫ retraction f, simpa using w, end } +instance is_split_mono.mono {X Y : C} (f : X ⟶ Y) [hf : is_split_mono f] : mono f := +hf.exists_split_mono.some.mono + +lemma split_epi.epi {X Y : C} {f : X ⟶ Y} (se : split_epi f) : epi f := +{ left_cancellation := λ Z g h w, begin replace w := se.section_ ≫= w, simpa using w, end } /-- Every split epi is an epi. -/ @[priority 100] -instance split_epi.epi {X Y : C} (f : X ⟶ Y) [split_epi f] : epi f := -{ left_cancellation := λ Z g h w, begin replace w := section_ f ≫= w, simpa using w, end } +instance is_split_epi.epi {X Y : C} (f : X ⟶ Y) [hf : is_split_epi f] : epi f := +hf.exists_split_epi.some.epi + +/-- Every split mono whose retraction is mono is an iso. -/ +lemma is_iso.of_mono_retraction' {X Y : C} {f : X ⟶ Y} (hf : split_mono f) + [mono $ hf.retraction] : is_iso f := +⟨⟨hf.retraction, ⟨by simp, (cancel_mono_id $ hf.retraction).mp (by simp)⟩⟩⟩ /-- Every split mono whose retraction is mono is an iso. -/ -lemma is_iso.of_mono_retraction {X Y : C} {f : X ⟶ Y} [split_mono f] [mono $ retraction f] - : is_iso f := -⟨⟨retraction f, ⟨by simp, (cancel_mono_id $ retraction f).mp (by simp)⟩⟩⟩ +lemma is_iso.of_mono_retraction {X Y : C} (f : X ⟶ Y) [hf : is_split_mono f] + [hf' : mono $ retraction f] : is_iso f := +@is_iso.of_mono_retraction' _ _ _ _ _ hf.exists_split_mono.some hf' /-- Every split epi whose section is epi is an iso. -/ -lemma is_iso.of_epi_section {X Y : C} {f : X ⟶ Y} [split_epi f] [epi $ section_ f] - : is_iso f := -⟨⟨section_ f, ⟨(cancel_epi_id $ section_ f).mp (by simp), by simp⟩⟩⟩ +lemma is_iso.of_epi_section' {X Y : C} {f : X ⟶ Y} (hf : split_epi f) + [epi $ hf.section_] : is_iso f := +⟨⟨hf.section_, ⟨(cancel_epi_id $ hf.section_).mp (by simp), by simp⟩⟩⟩ + +/-- Every split epi whose section is epi is an iso. -/ +lemma is_iso.of_epi_section {X Y : C} (f : X ⟶ Y) [hf : is_split_epi f] + [hf' : epi $ section_ f] : is_iso f := +@is_iso.of_epi_section' _ _ _ _ _ hf.exists_split_epi.some hf' /-- A category where every morphism has a `trunc` retraction is computably a groupoid. -/ -- FIXME this has unnecessarily become noncomputable! noncomputable def groupoid.of_trunc_split_mono - (all_split_mono : ∀ {X Y : C} (f : X ⟶ Y), trunc (split_mono f)) : + (all_split_mono : ∀ {X Y : C} (f : X ⟶ Y), trunc (is_split_mono f)) : groupoid.{v₁} C := begin apply groupoid.of_is_iso, @@ -133,36 +187,48 @@ variables (C) /-- A split mono category is a category in which every monomorphism is split. -/ class split_mono_category := -(split_mono_of_mono : ∀ {X Y : C} (f : X ⟶ Y) [mono f], split_mono f) +(is_split_mono_of_mono : ∀ {X Y : C} (f : X ⟶ Y) [mono f], is_split_mono f) /-- A split epi category is a category in which every epimorphism is split. -/ class split_epi_category := -(split_epi_of_epi : ∀ {X Y : C} (f : X ⟶ Y) [epi f], split_epi f) +(is_split_epi_of_epi : ∀ {X Y : C} (f : X ⟶ Y) [epi f], is_split_epi f) end /-- In a category in which every monomorphism is split, every monomorphism splits. This is not an instance because it would create an instance loop. -/ -def split_mono_of_mono [split_mono_category C] {X Y : C} (f : X ⟶ Y) [mono f] : split_mono f := -split_mono_category.split_mono_of_mono _ +lemma is_split_mono_of_mono [split_mono_category C] {X Y : C} (f : X ⟶ Y) [mono f] : + is_split_mono f := +split_mono_category.is_split_mono_of_mono _ /-- In a category in which every epimorphism is split, every epimorphism splits. This is not an instance because it would create an instance loop. -/ -def split_epi_of_epi [split_epi_category C] {X Y : C} (f : X ⟶ Y) [epi f] : split_epi f := -split_epi_category.split_epi_of_epi _ +lemma is_split_epi_of_epi [split_epi_category C] {X Y : C} (f : X ⟶ Y) [epi f] : + is_split_epi f := split_epi_category.is_split_epi_of_epi _ section variables {D : Type u₂} [category.{v₂} D] /-- Split monomorphisms are also absolute monomorphisms. -/ -instance {X Y : C} (f : X ⟶ Y) [split_mono f] (F : C ⥤ D) : split_mono (F.map f) := -{ retraction := F.map (retraction f), +@[simps] +def split_mono.map {X Y : C} {f : X ⟶ Y} (sm : split_mono f) (F : C ⥤ D ) : + split_mono (F.map f) := +{ retraction := F.map (sm.retraction), id' := by { rw [←functor.map_comp, split_mono.id, functor.map_id], } } /-- Split epimorphisms are also absolute epimorphisms. -/ -instance {X Y : C} (f : X ⟶ Y) [split_epi f] (F : C ⥤ D) : split_epi (F.map f) := -{ section_ := F.map (section_ f), +@[simps] +def split_epi.map {X Y : C} {f : X ⟶ Y} (se : split_epi f) (F : C ⥤ D ) : + split_epi (F.map f) := +{ section_ := F.map (se.section_), id' := by { rw [←functor.map_comp, split_epi.id, functor.map_id], } } + +instance {X Y : C} (f : X ⟶ Y) [hf : is_split_mono f] (F : C ⥤ D) : is_split_mono (F.map f) := +is_split_mono.mk' (hf.exists_split_mono.some.map F) + +instance {X Y : C} (f : X ⟶ Y) [hf : is_split_epi f] (F : C ⥤ D) : is_split_epi (F.map f) := +is_split_epi.mk' (hf.exists_split_epi.some.map F) + end end category_theory diff --git a/formal/lean/mathlib/category_theory/eq_to_hom.lean b/formal/lean/mathlib/category_theory/eq_to_hom.lean index 19e66aee7315dfb42e9acaf6d8f55f7a5f723883..c3c30dea4260aceeb88d799d0fec5989bcabb3b1 100644 --- a/formal/lean/mathlib/category_theory/eq_to_hom.lean +++ b/formal/lean/mathlib/category_theory/eq_to_hom.lean @@ -147,6 +147,9 @@ F.map e.inv = eq_to_hom (by rw hY) ≫ G.map e.inv ≫ eq_to_hom (by rw hX) := by simp only [← is_iso.iso.inv_hom e, functor.map_inv, h₂, is_iso.inv_comp, inv_eq_to_hom, category.assoc] +lemma congr_map (F : C ⥤ D) {X Y : C} {f g : X ⟶ Y} (h : f = g) : + F.map f = F.map g := by rw h + section heq /- Composition of functors and maps w.r.t. heq -/ diff --git a/formal/lean/mathlib/category_theory/essential_image.lean b/formal/lean/mathlib/category_theory/essential_image.lean index 7a832885783c3ecd97380b7e63b7b8aa672f309c..e5cbab18144e608dd0a5cad26434c780695c4844 100644 --- a/formal/lean/mathlib/category_theory/essential_image.lean +++ b/formal/lean/mathlib/category_theory/essential_image.lean @@ -66,11 +66,13 @@ set.ext $ λ A, ⟨ess_image.of_nat_iso h, ess_image.of_nat_iso h.symm⟩ /-- An object in the image is in the essential image. -/ lemma obj_mem_ess_image (F : D ⥤ C) (Y : D) : F.obj Y ∈ ess_image F := ⟨Y, ⟨iso.refl _⟩⟩ -instance : category F.ess_image := category_theory.full_subcategory _ +/-- The essential image of a functor, interpreted of a full subcategory of the target category. -/ +@[derive category, nolint has_nonempty_instance] +def ess_image_subcategory (F : C ⥤ D) := full_subcategory F.ess_image /-- The essential image as a subcategory has a fully faithful inclusion into the target category. -/ @[derive [full, faithful], simps] -def ess_image_inclusion (F : C ⥤ D) : F.ess_image ⥤ D := +def ess_image_inclusion (F : C ⥤ D) : F.ess_image_subcategory ⥤ D := full_subcategory_inclusion _ /-- @@ -78,9 +80,8 @@ Given a functor `F : C ⥤ D`, we have an (essentially surjective) functor from image of `F`. -/ @[simps] -def to_ess_image (F : C ⥤ D) : C ⥤ F.ess_image := -{ obj := λ X, ⟨_, obj_mem_ess_image _ X⟩, - map := λ X Y f, (ess_image_inclusion F).preimage (F.map f) } +def to_ess_image (F : C ⥤ D) : C ⥤ F.ess_image_subcategory := +full_subcategory.lift _ F (obj_mem_ess_image _) /-- The functor `F` factorises through its essential image, where the first functor is essentially @@ -89,7 +90,7 @@ surjective and the second is fully faithful. @[simps] def to_ess_image_comp_essential_image_inclusion (F : C ⥤ D) : F.to_ess_image ⋙ F.ess_image_inclusion ≅ F := -nat_iso.of_components (λ X, iso.refl _) (by tidy) +full_subcategory.lift_comp_inclusion _ _ _ end functor diff --git a/formal/lean/mathlib/category_theory/essentially_small.lean b/formal/lean/mathlib/category_theory/essentially_small.lean index 82779c2c980b960f62fb9219792a17014f0a8a73..9e43ff04afaeac421e4c25e0a9776a5f6f730d22 100644 --- a/formal/lean/mathlib/category_theory/essentially_small.lean +++ b/formal/lean/mathlib/category_theory/essentially_small.lean @@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison -/ import logic.small +import category_theory.category.ulift import category_theory.skeletal /-! @@ -39,7 +40,7 @@ lemma essentially_small.mk' {C : Type u} [category.{v} C] {S : Type w} [small_ca /-- An arbitrarily chosen small model for an essentially small category. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def small_model (C : Type u) [category.{v} C] [essentially_small.{w} C] : Type w := classical.some (@essentially_small.equiv_small_category C _ _) @@ -73,6 +74,9 @@ lemma discrete.essentially_small_of_small {α : Type u} [small.{w} α] : essentially_small.{w} (discrete α) := ⟨⟨discrete (shrink α), ⟨infer_instance, ⟨discrete.equivalence (equiv_shrink _)⟩⟩⟩⟩ +lemma essentially_small_self : essentially_small.{max w v u} C := +essentially_small.mk' (as_small.equiv : C ≌ as_small.{w} C) + /-- A category is `w`-locally small if every hom set is `w`-small. @@ -115,7 +119,7 @@ instance locally_small_of_essentially_small We define a type alias `shrink_homs C` for `C`. When we have `locally_small.{w} C`, we'll put a `category.{w}` instance on `shrink_homs C`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def shrink_homs (C : Type u) := C namespace shrink_homs diff --git a/formal/lean/mathlib/category_theory/full_subcategory.lean b/formal/lean/mathlib/category_theory/full_subcategory.lean index 8bbbfda80519aebd373dd731e850bf1716e2527d..52397e39da4f60797ba01ea321a743397504e2b5 100644 --- a/formal/lean/mathlib/category_theory/full_subcategory.lean +++ b/formal/lean/mathlib/category_theory/full_subcategory.lean @@ -47,7 +47,7 @@ include F which provides a category structure so that the morphisms `X ⟶ Y` are the morphisms in `D` from `F X` to `F Y`. -/ -@[nolint has_inhabited_instance unused_arguments] +@[nolint has_nonempty_instance unused_arguments] def induced_category : Type u₁ := C variables {D} @@ -81,35 +81,41 @@ variables {C : Type u₁} [category.{v} C] variables (Z : C → Prop) /-- -The category structure on a subtype; morphisms just ignore the property. +A subtype-like structure for full subcategories. Morphisms just ignore the property. We don't use +actual subtypes since the simp-normal form `↑X` of `X.val` does not work well for full +subcategories. See . We do not define 'strictly full' subcategories. -/ -instance full_subcategory : category.{v} {X : C // Z X} := -induced_category.category subtype.val +@[ext, nolint has_nonempty_instance] structure full_subcategory := +(obj : C) +(property : Z obj) + +instance full_subcategory.category : category.{v} (full_subcategory Z) := +induced_category.category full_subcategory.obj /-- The forgetful functor from a full subcategory into the original category ("forgetting" the condition). -/ -def full_subcategory_inclusion : {X : C // Z X} ⥤ C := -induced_functor subtype.val +def full_subcategory_inclusion : full_subcategory Z ⥤ C := +induced_functor full_subcategory.obj @[simp] lemma full_subcategory_inclusion.obj {X} : - (full_subcategory_inclusion Z).obj X = X.val := rfl + (full_subcategory_inclusion Z).obj X = X.obj := rfl @[simp] lemma full_subcategory_inclusion.map {X Y} {f : X ⟶ Y} : (full_subcategory_inclusion Z).map f = f := rfl instance full_subcategory.full : full (full_subcategory_inclusion Z) := -induced_category.full subtype.val +induced_category.full _ instance full_subcategory.faithful : faithful (full_subcategory_inclusion Z) := -induced_category.faithful subtype.val +induced_category.faithful _ variables {Z} {Z' : C → Prop} /-- An implication of predicates `Z → Z'` induces a functor between full subcategories. -/ @[simps] -def full_subcategory.map (h : ∀ ⦃X⦄, Z X → Z' X) : {X // Z X} ⥤ {X // Z' X} := +def full_subcategory.map (h : ∀ ⦃X⦄, Z X → Z' X) : full_subcategory Z ⥤ full_subcategory Z' := { obj := λ X, ⟨X.1, h X.2⟩, map := λ X Y f, f } @@ -128,7 +134,7 @@ variables {D : Type u₂} [category.{v₂} D] (P Q : D → Prop) /-- A functor which maps objects to objects satisfying a certain property induces a lift through the full subcategory of objects satisfying that property. -/ @[simps] -def full_subcategory.lift (F : C ⥤ D) (hF : ∀ X, P (F.obj X)) : C ⥤ {X // P X} := +def full_subcategory.lift (F : C ⥤ D) (hF : ∀ X, P (F.obj X)) : C ⥤ full_subcategory P := { obj := λ X, ⟨F.obj X, hF X⟩, map := λ X Y f, F.map f } diff --git a/formal/lean/mathlib/category_theory/functor/epi_mono.lean b/formal/lean/mathlib/category_theory/functor/epi_mono.lean index 2ed2ad6f30291514adee3e5c205352d1c148320e..cb3bff689fb2a6f8dc52d256876754c5bc4f3fc7 100644 --- a/formal/lean/mathlib/category_theory/functor/epi_mono.lean +++ b/formal/lean/mathlib/category_theory/functor/epi_mono.lean @@ -167,4 +167,54 @@ instance reflects_epimorphisms_of_faithful (F : C ⥤ D) [faithful F] : reflects { reflects := λ X Y f hf, ⟨λ Z g h hgh, by exactI F.map_injective ((cancel_epi (F.map f)).1 (by rw [← F.map_comp, hgh, F.map_comp]))⟩ } +section + +variables (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) + +/-- If `F` is a fully faithful functor, split epimorphisms are preserved and reflected by `F`. -/ +def split_epi_equiv [full F] [faithful F] : split_epi f ≃ split_epi (F.map f) := +{ to_fun := λ f, f.map F, + inv_fun := λ s, begin + refine ⟨F.preimage s.section_, _⟩, + apply F.map_injective, + simp only [map_comp, image_preimage, map_id], + apply split_epi.id, + end, + left_inv := by tidy, + right_inv := by tidy, } + +/-- If `F` is a fully faithful functor, split monomorphisms are preserved and reflected by `F`. -/ +def split_mono_equiv [full F] [faithful F] : split_mono f ≃ split_mono (F.map f) := +{ to_fun := λ f, f.map F, + inv_fun := λ s, begin + refine ⟨F.preimage s.retraction, _⟩, + apply F.map_injective, + simp only [map_comp, image_preimage, map_id], + apply split_mono.id, + end, + left_inv := by tidy, + right_inv := by tidy, } + +@[simp] +lemma epi_map_iff_epi [hF₁ : preserves_epimorphisms F] [hF₂ : reflects_epimorphisms F] : + epi (F.map f) ↔ epi f := +begin + split, + { exact F.epi_of_epi_map, }, + { introI h, + exact F.map_epi f, }, +end + +@[simp] +lemma mono_map_iff_mono [hF₁ : preserves_monomorphisms F] [hF₂ : reflects_monomorphisms F] : + mono (F.map f) ↔ mono f := +begin + split, + { exact F.mono_of_mono_map, }, + { introI h, + exact F.map_mono f, }, +end + +end + end category_theory.functor diff --git a/formal/lean/mathlib/category_theory/generator.lean b/formal/lean/mathlib/category_theory/generator.lean index 7861e931a2a7c4eff32a576853bccf5fdbd925cd..e398a80921da10eeeff500f8a488924e6c24a3ff 100644 --- a/formal/lean/mathlib/category_theory/generator.lean +++ b/formal/lean/mathlib/category_theory/generator.lean @@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Markus Himmel -/ import category_theory.balanced +import category_theory.limits.essentially_small import category_theory.limits.opposites import category_theory.limits.shapes.zero_morphisms import category_theory.subobject.lattice @@ -47,8 +48,6 @@ We * We currently don't have any examples yet. * We will want typeclasses `has_separator C` and similar. -* To state the Special Adjoint Functor Theorem, we will need to be able to talk about *small* - separating sets. -/ @@ -261,6 +260,43 @@ begin simpa using hh j.as.1.1 j.as.1.2 j.as.2 } end +/-- An ingredient of the proof of the Special Adjoint Functor Theorem: a complete well-powered + category with a small coseparating set has an initial object. + + In fact, it follows from the Special Adjoint Functor Theorem that `C` is already cocomplete. -/ +lemma has_initial_of_is_cosepatating [well_powered C] [has_limits C] {𝒢 : set C} [small.{v} 𝒢] + (h𝒢 : is_coseparating 𝒢) : has_initial C := +begin + haveI := has_products_of_shape_of_small C 𝒢, + haveI := λ A, has_products_of_shape_of_small.{v} C (Σ G : 𝒢, A ⟶ (G : C)), + letI := complete_lattice_of_complete_semilattice_Inf (subobject (pi_obj (coe : 𝒢 → C))), + suffices : ∀ A : C, unique (((⊥ : subobject (pi_obj (coe : 𝒢 → C))) : C) ⟶ A), + { exactI has_initial_of_unique ((⊥ : subobject (pi_obj (coe : 𝒢 → C))) : C) }, + refine λ A, ⟨⟨_⟩, λ f, _⟩, + { let s := pi.lift (λ f : Σ G : 𝒢, A ⟶ (G : C), id (pi.π (coe : 𝒢 → C)) f.1), + let t := pi.lift (@sigma.snd 𝒢 (λ G, A ⟶ (G : C))), + haveI : mono t := (is_coseparating_iff_mono 𝒢).1 h𝒢 A, + exact subobject.of_le_mk _ (pullback.fst : pullback s t ⟶ _) bot_le ≫ pullback.snd }, + { generalize : default = g, + suffices : is_split_epi (equalizer.ι f g), + { exactI eq_of_epi_equalizer }, + exact is_split_epi.mk' ⟨subobject.of_le_mk _ (equalizer.ι f g ≫ subobject.arrow _) + bot_le, by { ext, simp }⟩ } +end + +/-- An ingredient of the proof of the Special Adjoint Functor Theorem: a cocomplete well-copowered + category with a small separating set has a terminal object. + + In fact, it follows from the Special Adjoint Functor Theorem that `C` is already complete. -/ +lemma has_terminal_of_is_separating [well_powered Cᵒᵖ] [has_colimits C] {𝒢 : set C} [small.{v} 𝒢] + (h𝒢 : is_separating 𝒢) : has_terminal C := +begin + haveI : has_limits Cᵒᵖ := has_limits_op_of_has_colimits, + haveI : small.{v} 𝒢.op := small_of_injective (set.op_equiv_self 𝒢).injective, + haveI : has_initial Cᵒᵖ := has_initial_of_is_cosepatating ((is_coseparating_op_iff _).2 h𝒢), + exact has_terminal_of_has_initial_op +end + section well_powered namespace subobject diff --git a/formal/lean/mathlib/category_theory/glue_data.lean b/formal/lean/mathlib/category_theory/glue_data.lean index bd2220c9a6b922620d39181cc6370861422dba79..15b95f35be6b71d3ae5f23b19ed37f0df3c164de 100644 --- a/formal/lean/mathlib/category_theory/glue_data.lean +++ b/formal/lean/mathlib/category_theory/glue_data.lean @@ -42,7 +42,7 @@ such that `t' : V i j ×[U i] V i k ⟶ V j k ×[U j] V j i`. 10. `t' i j k ≫ t' j k i ≫ t' k i j = 𝟙 _`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure glue_data := (J : Type v) (U : J → C) diff --git a/formal/lean/mathlib/category_theory/grothendieck.lean b/formal/lean/mathlib/category_theory/grothendieck.lean index 12d8105a9653372df7e05c3b00c56c74d0071dc1..c304cd2f89a7f19ed2125a4be5c1b8662d2ca664 100644 --- a/formal/lean/mathlib/category_theory/grothendieck.lean +++ b/formal/lean/mathlib/category_theory/grothendieck.lean @@ -48,7 +48,7 @@ gives a category whose `base : X.base ⟶ Y.base` and `f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber` -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure grothendieck := (base : C) (fiber : F.obj base) diff --git a/formal/lean/mathlib/category_theory/idempotents/functor_categories.lean b/formal/lean/mathlib/category_theory/idempotents/functor_categories.lean index 910bdec18751e7d6ad18b6f6e3c7288d9cf5b33f..baf4b966076a876ac75e8bf2cc584779e0312f33 100644 --- a/formal/lean/mathlib/category_theory/idempotents/functor_categories.lean +++ b/formal/lean/mathlib/category_theory/idempotents/functor_categories.lean @@ -161,6 +161,24 @@ begin refl, }, } end +variables {J C} (P Q : karoubi (J ⥤ C)) (f : P ⟶ Q) (X : J) + + +@[simp, reassoc] +lemma app_idem (X : J) : + P.p.app X ≫ P.p.app X = P.p.app X := congr_app P.idem X + +variables {P Q} + +@[simp, reassoc] +lemma app_p_comp : P.p.app X ≫ f.f.app X = f.f.app X := congr_app (p_comp f) X + +@[simp, reassoc] +lemma app_comp_p : f.f.app X ≫ Q.p.app X = f.f.app X := congr_app (comp_p f) X + +@[reassoc] +lemma app_p_comm : P.p.app X ≫ f.f.app X = f.f.app X ≫ Q.p.app X := congr_app (p_comm f) X + end idempotents end category_theory diff --git a/formal/lean/mathlib/category_theory/idempotents/functor_extension.lean b/formal/lean/mathlib/category_theory/idempotents/functor_extension.lean new file mode 100644 index 0000000000000000000000000000000000000000..b1fb977c16832d4e025a39fd6fb72b6a94a4e890 --- /dev/null +++ b/formal/lean/mathlib/category_theory/idempotents/functor_extension.lean @@ -0,0 +1,130 @@ +/- +Copyright (c) 2022 Joël Riou. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Joël Riou +-/ + +import category_theory.idempotents.karoubi +import category_theory.natural_isomorphism + +/-! +# Extension of functors to the idempotent completion + +In this file, we construct an extension `functor_extension₁` +of functors `C ⥤ karoubi D` to functors `karoubi C ⥤ karoubi D`. + +TODO : Obtain the equivalences +`karoubi_universal₁ C D : C ⥤ karoubi D ≌ karoubi C ⥤ karoubi D` +for all categories, and +`karoubi_universal C D : C ⥤ D ≌ karoubi C ⥤ D`. +when `D` is idempotent complete +-/ + +open category_theory.category +open category_theory.idempotents.karoubi + +namespace category_theory + +namespace idempotents + +variables {C D E : Type*} [category C] [category D] [category E] + +/-- A natural transformation between functors `karoubi C ⥤ D` is determined +by its value on objects coming from `C`. -/ +lemma nat_trans_eq {F G : karoubi C ⥤ D} (φ : F ⟶ G) (P : karoubi C) : + φ.app P = F.map (decomp_id_i P) ≫ φ.app P.X ≫ G.map (decomp_id_p P) := +begin + rw [← φ.naturality, ← assoc, ← F.map_comp], + conv { to_lhs, rw [← id_comp (φ.app P), ← F.map_id], }, + congr, + apply decomp_id, +end + +namespace functor_extension₁ + +/-- The canonical extension of a functor `C ⥤ karoubi D` to a functor +`karoubi C ⥤ karoubi D` -/ +@[simps] +def obj (F : C ⥤ karoubi D) : karoubi C ⥤ karoubi D := +{ obj := λ P, ⟨(F.obj P.X).X, (F.map P.p).f, + by simpa only [F.map_comp, hom_ext] using F.congr_map P.idem⟩, + map := λ P Q f, ⟨(F.map f.f).f, + by simpa only [F.map_comp, hom_ext] using F.congr_map f.comm⟩, } + +/-- Extension of a natural transformation `φ` between functors +`C ⥤ karoubi D` to a natural transformation between the +extension of these functors to `karoubi C ⥤ karoubi D` -/ +@[simps] +def map {F G : C ⥤ karoubi D} (φ : F ⟶ G) : obj F ⟶ obj G := +{ app := λ P, + { f := (F.map P.p).f ≫ (φ.app P.X).f, + comm := begin + have h := φ.naturality P.p, + have h' := F.congr_map P.idem, + simp only [hom_ext, karoubi.comp, F.map_comp] at h h', + simp only [obj_obj_p, assoc, ← h], + slice_rhs 1 3 { rw [h', h'], }, + end, }, + naturality' := λ P Q f, begin + ext, + dsimp [obj], + have h := φ.naturality f.f, + have h' := F.congr_map (comp_p f), + have h'' := F.congr_map (p_comp f), + simp only [hom_ext, functor.map_comp, comp] at ⊢ h h' h'', + slice_rhs 2 3 { rw ← h, }, + slice_lhs 1 2 { rw h', }, + slice_rhs 1 2 { rw h'', }, + end } + +end functor_extension₁ + +variables (C D E) + +/-- The canonical functor `(C ⥤ karoubi D) ⥤ (karoubi C ⥤ karoubi D)` -/ +@[simps] +def functor_extension₁ : (C ⥤ karoubi D) ⥤ (karoubi C ⥤ karoubi D) := +{ obj := functor_extension₁.obj, + map := λ F G, functor_extension₁.map, + map_id' := λ F, by { ext P, exact comp_p (F.map P.p), }, + map_comp' := λ F G H φ φ', begin + ext P, + simp only [comp, functor_extension₁.map_app_f, nat_trans.comp_app, assoc], + have h := φ.naturality P.p, + have h' := F.congr_map P.idem, + simp only [hom_ext, comp, F.map_comp] at h h', + slice_rhs 2 3 { rw ← h, }, + slice_rhs 1 2 { rw h', }, + simp only [assoc], + end, } + +lemma functor_extension₁_comp_whiskering_left_to_karoubi : + functor_extension₁ C D ⋙ + (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) = 𝟭 _ := +begin + refine functor.ext _ _, + { intro F, + refine functor.ext _ _, + { intro X, + ext, + { dsimp, + rw [id_comp, comp_id, F.map_id, id_eq], }, + { refl, }, }, + { intros X Y f, + ext, + dsimp, + simp only [comp_id, eq_to_hom_f, eq_to_hom_refl, comp_p, functor_extension₁.obj_obj_p, + to_karoubi_obj_p, comp], + dsimp, + simp only [functor.map_id, id_eq, p_comp], }, }, + { intros F G φ, + ext X, + dsimp, + simp only [eq_to_hom_app, F.map_id, karoubi.comp, eq_to_hom_f, id_eq, p_comp, + eq_to_hom_refl, comp_id, comp_p, functor_extension₁.obj_obj_p, + to_karoubi_obj_p, F.map_id X], }, +end + +end idempotents + +end category_theory diff --git a/formal/lean/mathlib/category_theory/idempotents/karoubi.lean b/formal/lean/mathlib/category_theory/idempotents/karoubi.lean index 40f9de580a7cc247feff7673db7435b8721ed2c4..45d790af537daeb0fa4e56989e1061282a74973f 100644 --- a/formal/lean/mathlib/category_theory/idempotents/karoubi.lean +++ b/formal/lean/mathlib/category_theory/idempotents/karoubi.lean @@ -41,7 +41,7 @@ obvious idempotent `X ⟶ P ⟶ X` which is the projection onto `P` with kernel one may define a formal direct factor of an object `X : C` : it consists of an idempotent `p : X ⟶ X` which is thought as the "formal image" of `p`. The type `karoubi C` shall be the type of the objects of the karoubi enveloppe of `C`. It makes sense for any category `C`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure karoubi := (X : C) (p : X ⟶ X) (idem : p ≫ p = p) namespace karoubi diff --git a/formal/lean/mathlib/category_theory/lifting_properties.lean b/formal/lean/mathlib/category_theory/lifting_properties.lean deleted file mode 100644 index 1fc25dd152767bb385c022f67c8e88ea10efd057..0000000000000000000000000000000000000000 --- a/formal/lean/mathlib/category_theory/lifting_properties.lean +++ /dev/null @@ -1,126 +0,0 @@ -/- -Copyright (c) 2021 Jakob Scholbach. All rights reserved. -Released under Apache 2.0 license as described in the file LICENSE. -Authors: Jakob Scholbach --/ -import category_theory.limits.shapes.terminal -import category_theory.arrow - -/-! -# Lifting properties - -This file defines the lifting property of two arrows in a category and shows basic properties of -this notion. -We also construct the subcategory consisting of those morphisms which have the right lifting -property with respect to arrows in a given diagram. - -## Main results -- `has_lifting_property`: the definition of the lifting property -- `iso_has_right_lifting_property`: any isomorphism satisfies the right lifting property (rlp) -- `id_has_right_lifting_property`: any identity has the rlp -- `right_lifting_property_initial_iff`: spells out the rlp with respect to a map whose source is an - initial object -- `right_lifting_subcat`: given a set of arrows `F : D → arrow C`, we construct the subcategory - of those morphisms `p` in `C` that satisfy the rlp w.r.t. `F i`, for any element `i` of `D`. - -## Tags -lifting property --/ - -open category_theory.limits - -namespace category_theory - -universes v u v₁ -variables {C : Type u} [category.{v} C] -variables {D : Type v₁} - -variables {X Y Z : C} - -/-- The lifting property of a morphism `i` with respect to a morphism `p`. -This can be interpreted as the right lifting property of `i` with respect to `p`, -or the left lifting property of `p` with respect to `i`. -/ -class has_lifting_property (i p : arrow C) : Prop := -(sq_has_lift : ∀ (sq : i ⟶ p), arrow.has_lift sq) - -@[priority 100] -- see Note [lower instance priority] -instance has_lifting_property' {i p : arrow C} [has_lifting_property i p] (sq : i ⟶ p) : - arrow.has_lift sq := -has_lifting_property.sq_has_lift sq - -/-- Any isomorphism has the right lifting property with respect to any map. -A → X -↓i ↓p≅ -B → Y --/ -lemma iso_has_right_lifting_property (i : arrow C) (p : X ≅ Y) : - has_lifting_property i (arrow.mk p.hom) := -⟨λ sq, ⟨⟨{ lift := sq.right ≫ p.inv, }⟩⟩⟩ -- the lift is obtained by p⁻¹ ∘ (B → Y) - -/-- Any identity has the right lifting property with respect to any map. -/ -lemma id_has_right_lifting_property (i : arrow C) : has_lifting_property i (arrow.mk (𝟙 X)) := -iso_has_right_lifting_property i (iso.refl _) - -/-- An equivalent characterization for right lifting with respect to a map `i` whose source is -initial. -∅ → X -↓ ↓ -B → Y has a lifting iff there is a map B → X making the right part commute. --/ -lemma right_lifting_property_initial_iff (i p : arrow C) (h : is_initial i.left) : - has_lifting_property i p ↔ ∀ {e : i.right ⟶ p.right}, ∃ l : i.right ⟶ p.left, l ≫ p.hom = e := -begin - fsplit, - { introsI hlift e, - have comm : (is_initial.to h p.left) ≫ p.hom = i.hom ≫ e := - is_initial.hom_ext h _ _, - use arrow.lift (arrow.hom_mk comm), - simp }, - { refine λ hlift, ⟨λ sq, _⟩, - obtain ⟨l, hl⟩ : ∃ (l : i.right ⟶ p.left), l ≫ p.hom = sq.right := hlift, - exact arrow.has_lift.mk ⟨l, is_initial.hom_ext h _ _⟩, } -end - -/-- The condition of having the rlp with respect to a morphism `i` is stable under composition. -/ -lemma has_right_lifting_property_comp {i : arrow C} {f : X ⟶ Y} {g : Y ⟶ Z} - (hf : has_lifting_property i (arrow.mk f)) - (hg : has_lifting_property i (arrow.mk g)) : - has_lifting_property i (arrow.mk (f ≫ g)) := -{ sq_has_lift := λ sq1, - -- construct a square i ⟶ f - let sq2 : i ⟶ (arrow.mk f) := ⟨sq1.left, arrow.lift (arrow.square_to_snd sq1)⟩ in - -- show that the lift of this square is a lift of i with respect to g ∘ f - ⟨⟨⟨(arrow.lift sq2 : _ ⟶ _), by simp⟩⟩⟩ } - -/-- The objects of the subcategory `right_lifting_subcategory` are the ones in the -underlying category. -/ -def right_lifting_subcat (R : Type u) := R - -instance right_lifting_subcat.inhabited (R : Type u) [inhabited R] : - inhabited (right_lifting_subcat R) := -{ default := (default : R) } - -/-- The objects of the subcategory `right_lifting_subcategory` are the ones in the -underlying category. -/ -def right_lifting_subcat.X {R : Type u} (x : right_lifting_subcat R) : R := x - -lemma id_has_right_lifting_property' {F : D → arrow C} (X : C) : - ∀ i : D, has_lifting_property (F i) (arrow.mk (𝟙 X)) := -λ i, id_has_right_lifting_property (F i) - -lemma has_right_lifting_property_comp' - {F : D → arrow C} {f : X ⟶ Y} (hf : ∀ i : D, has_lifting_property (F i) (arrow.mk f)) - {g : Y ⟶ Z} (hg : ∀ i : D, has_lifting_property (F i) (arrow.mk g)) : - ∀ i : D, has_lifting_property (F i) (arrow.mk (f ≫ g)) := -λ i, has_right_lifting_property_comp (hf i) (hg i) - -/-- Given a set of arrows in C, indexed by `F : D → arrow C`, -we construct the (non-full) subcategory of `C` -spanned by those morphisms that have the right lifting property relative to all maps -of the form `F i`, where `i` is any element in `D`. -/ -def right_lifting_subcategory (F : D → arrow C) : category (right_lifting_subcat C) := -{ hom := λ X Y, { p : X ⟶ Y // ∀ {i : D}, has_lifting_property (F i) (arrow.mk p) }, - id := λ X, ⟨𝟙 X, id_has_right_lifting_property' X⟩, - comp := λ X Y Z f g, ⟨f.val ≫ g.val, has_right_lifting_property_comp' f.property g.property⟩ } - -end category_theory diff --git a/formal/lean/mathlib/category_theory/lifting_properties/basic.lean b/formal/lean/mathlib/category_theory/lifting_properties/basic.lean new file mode 100644 index 0000000000000000000000000000000000000000..9171c760b84d69954ead34e573886c244f8c017d --- /dev/null +++ b/formal/lean/mathlib/category_theory/lifting_properties/basic.lean @@ -0,0 +1,109 @@ +/- +Copyright (c) 2021 Jakob Scholbach. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Jakob Scholbach, Joël Riou +-/ +import category_theory.comm_sq + +/-! +# Lifting properties + +This file defines the lifting property of two morphisms in a category and +shows basic properties of this notion. + +## Main results +- `has_lifting_property`: the definition of the lifting property + +## Tags +lifting property + +@TODO : +1) define llp/rlp with respect to a `morphism_property` +2) retracts, direct/inverse images, (co)products, adjunctions + +-/ + +universe v + +namespace category_theory + +open category + +variables {C : Type*} [category C] {A B B' X Y Y' : C} + (i : A ⟶ B) (i' : B ⟶ B') (p : X ⟶ Y) (p' : Y ⟶ Y') + +/-- `has_lifting_property i p` means that `i` has the left lifting +property with respect to `p`, or equivalently that `p` has +the right lifting property with respect to `i`. -/ +class has_lifting_property : Prop := +(sq_has_lift : ∀ {f : A ⟶ X} {g : B ⟶ Y} (sq : comm_sq f i p g), sq.has_lift) + +@[priority 100] +instance sq_has_lift_of_has_lifting_property {f : A ⟶ X} {g : B ⟶ Y} (sq : comm_sq f i p g) + [hip : has_lifting_property i p] : sq.has_lift := by apply hip.sq_has_lift + +namespace has_lifting_property + +variables {i p} + +lemma op (h : has_lifting_property i p) : has_lifting_property p.op i.op := +⟨λ f g sq, begin + simp only [comm_sq.has_lift.iff_unop, quiver.hom.unop_op], + apply_instance, +end⟩ + +lemma unop {A B X Y : Cᵒᵖ} {i : A ⟶ B} {p : X ⟶ Y} + (h : has_lifting_property i p) : has_lifting_property p.unop i.unop := +⟨λ f g sq, begin + rw comm_sq.has_lift.iff_op, + simp only [quiver.hom.op_unop], + apply_instance, +end⟩ + +lemma iff_op : has_lifting_property i p ↔ has_lifting_property p.op i.op := ⟨op, unop⟩ + +lemma iff_unop {A B X Y : Cᵒᵖ} (i : A ⟶ B) (p : X ⟶ Y) : + has_lifting_property i p ↔ has_lifting_property p.unop i.unop := ⟨unop, op⟩ + +variables (i p) + +@[priority 100] +instance of_left_iso [is_iso i] : has_lifting_property i p := +⟨λ f g sq, comm_sq.has_lift.mk' + { l := inv i ≫ f, + fac_left' := by simp only [is_iso.hom_inv_id_assoc], + fac_right' := by simp only [sq.w, assoc, is_iso.inv_hom_id_assoc], }⟩ + +@[priority 100] +instance of_right_iso [is_iso p] : has_lifting_property i p := +⟨λ f g sq, comm_sq.has_lift.mk' + { l := g ≫ inv p, + fac_left' := by simp only [← sq.w_assoc, is_iso.hom_inv_id, comp_id], + fac_right' := by simp only [assoc, is_iso.inv_hom_id, comp_id], }⟩ + +instance of_comp_left [has_lifting_property i p] [has_lifting_property i' p] : + has_lifting_property (i ≫ i') p := +⟨λ f g sq, begin + have fac := sq.w, + rw assoc at fac, + exact comm_sq.has_lift.mk' + { l := (comm_sq.mk (comm_sq.mk fac).fac_right).lift, + fac_left' := by simp only [assoc, comm_sq.fac_left], + fac_right' := by simp only [comm_sq.fac_right], }, +end⟩ + +instance of_comp_right [has_lifting_property i p] [has_lifting_property i p'] : + has_lifting_property i (p ≫ p') := +⟨λ f g sq, begin + have fac := sq.w, + rw ← assoc at fac, + let sq₂ := (comm_sq.mk ((comm_sq.mk fac).fac_left.symm)).lift, + exact comm_sq.has_lift.mk' + { l := (comm_sq.mk ((comm_sq.mk fac).fac_left.symm)).lift, + fac_left' := by simp only [comm_sq.fac_left], + fac_right' := by simp only [comm_sq.fac_right_assoc, comm_sq.fac_right], }, +end⟩ + +end has_lifting_property + +end category_theory diff --git a/formal/lean/mathlib/category_theory/limits/cone_category.lean b/formal/lean/mathlib/category_theory/limits/cone_category.lean index a511297fa5ecf24d2390b1d46031d3ac9366a989..e67014e07dc183a10d96bd7d20e421fb0292d721 100644 --- a/formal/lean/mathlib/category_theory/limits/cone_category.lean +++ b/formal/lean/mathlib/category_theory/limits/cone_category.lean @@ -3,30 +3,56 @@ Copyright (c) 2021 Andrew Yang. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Andrew Yang -/ - import category_theory.limits.preserves.shapes.terminal +import category_theory.structured_arrow /-! - # Limits and the category of (co)cones This files contains results that stem from the limit API. For the definition and the category instance of `cone`, please refer to `category_theory/limits/cones.lean`. -A cone is limiting iff it is terminal in the category of cones. As a corollary, an equivalence of -categories of cones preserves limiting properties. We also provide the dual. +## Main results +* The category of cones on `F : J ⥤ C` is equivalent to the category + `costructured_arrow (const J) F`. +* A cone is limiting iff it is terminal in the category of cones. As a corollary, an equivalence of + categories of cones preserves limiting properties. -/ namespace category_theory.limits -open category_theory +open category_theory category_theory.functor universes v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄ variables {J : Type u₁} [category.{v₁} J] {K : Type u₂} [category.{v₂} K] variables {C : Type u₃} [category.{v₃} C] {D : Type u₄} [category.{v₄} D] +/-- Construct an object of the category `(Δ ↓ F)` from a cone on `F`. This is part of an + equivalence, see `cone.equiv_costructured_arrow`. -/ +@[simps] +def cone.to_costructured_arrow (F : J ⥤ C) : cone F ⥤ costructured_arrow (const J) F := +{ obj := λ c, costructured_arrow.mk c.π, + map := λ c d f, costructured_arrow.hom_mk f.hom $ by { ext, simp } } + +/-- Construct a cone on `F` from an object of the category `(Δ ↓ F)`. This is part of an + equivalence, see `cone.equiv_costructured_arrow`. -/ +@[simps] +def cone.from_costructured_arrow (F : J ⥤ C) : costructured_arrow (const J) F ⥤ cone F := +{ obj := λ c, ⟨c.left, c.hom⟩, + map := λ c d f, + { hom := f.left, + w' := λ j, by { convert (congr_fun (congr_arg nat_trans.app f.w) j), dsimp, simp } } } + +/-- The category of cones on `F` is just the comma category `(Δ ↓ F)`, where `Δ` is the constant + functor. -/ +@[simps] +def cone.equiv_costructured_arrow (F : J ⥤ C) : cone F ≌ costructured_arrow (const J) F := +equivalence.mk (cone.to_costructured_arrow F) (cone.from_costructured_arrow F) + (nat_iso.of_components cones.eta (by tidy)) + (nat_iso.of_components (λ c, (costructured_arrow.eta _).symm) (by tidy)) + /-- A cone is a limit cone iff it is terminal. -/ def cone.is_limit_equiv_is_terminal {F : J ⥤ C} (c : cone F) : is_limit c ≃ is_terminal c := is_limit.iso_unique_cone_morphism.to_equiv.trans @@ -35,6 +61,10 @@ is_limit.iso_unique_cone_morphism.to_equiv.trans left_inv := by tidy, right_inv := by tidy } +lemma has_limit_iff_has_terminal_cone (F : J ⥤ C) : has_limit F ↔ has_terminal (cone F) := +⟨λ h, by exactI (cone.is_limit_equiv_is_terminal _ (limit.is_limit F)).has_terminal, + λ h, ⟨⟨by exactI ⟨⊤_ _, (cone.is_limit_equiv_is_terminal _).symm terminal_is_terminal⟩⟩⟩⟩ + lemma is_limit.lift_cone_morphism_eq_is_terminal_from {F : J ⥤ C} {c : cone F} (hc : is_limit c) (s : cone F) : hc.lift_cone_morphism s = is_terminal.from (cone.is_limit_equiv_is_terminal _ hc) _ := rfl @@ -58,6 +88,30 @@ def is_limit.of_reflects_cone_terminal {F : J ⥤ C} {F' : K ⥤ D} (G : cone F (cone.is_limit_equiv_is_terminal _).symm $ (cone.is_limit_equiv_is_terminal _ hc).is_terminal_of_obj _ _ +/-- Construct an object of the category `(F ↓ Δ)` from a cocone on `F`. This is part of an + equivalence, see `cocone.equiv_structured_arrow`. -/ +@[simps] +def cocone.to_structured_arrow (F : J ⥤ C) : cocone F ⥤ structured_arrow F (const J) := +{ obj := λ c, structured_arrow.mk c.ι, + map := λ c d f, structured_arrow.hom_mk f.hom $ by { ext, simp } } + +/-- Construct a cocone on `F` from an object of the category `(F ↓ Δ)`. This is part of an + equivalence, see `cocone.equiv_structured_arrow`. -/ +@[simps] +def cocone.from_structured_arrow (F : J ⥤ C) : structured_arrow F (const J) ⥤ cocone F := +{ obj := λ c, ⟨c.right, c.hom⟩, + map := λ c d f, + { hom := f.right, + w' := λ j, by { convert (congr_fun (congr_arg nat_trans.app f.w) j).symm, dsimp, simp } } } + +/-- The category of cocones on `F` is just the comma category `(F ↓ Δ)`, where `Δ` is the constant + functor. -/ +@[simps] +def cocone.equiv_structured_arrow (F : J ⥤ C) : cocone F ≌ structured_arrow F (const J) := +equivalence.mk (cocone.to_structured_arrow F) (cocone.from_structured_arrow F) + (nat_iso.of_components cocones.eta (by tidy)) + (nat_iso.of_components (λ c, (structured_arrow.eta _).symm) (by tidy)) + /-- A cocone is a colimit cocone iff it is initial. -/ def cocone.is_colimit_equiv_is_initial {F : J ⥤ C} (c : cocone F) : is_colimit c ≃ is_initial c := is_colimit.iso_unique_cocone_morphism.to_equiv.trans @@ -66,6 +120,10 @@ is_colimit.iso_unique_cocone_morphism.to_equiv.trans left_inv := by tidy, right_inv := by tidy } +lemma has_colimit_iff_has_initial_cocone (F : J ⥤ C) : has_colimit F ↔ has_initial (cocone F) := +⟨λ h, by exactI (cocone.is_colimit_equiv_is_initial _ (colimit.is_colimit F)).has_initial, + λ h, ⟨⟨by exactI ⟨⊥_ _, (cocone.is_colimit_equiv_is_initial _).symm initial_is_initial⟩⟩⟩⟩ + lemma is_colimit.desc_cocone_morphism_eq_is_initial_to {F : J ⥤ C} {c : cocone F} (hc : is_colimit c) (s : cocone F) : hc.desc_cocone_morphism s = diff --git a/formal/lean/mathlib/category_theory/limits/cones.lean b/formal/lean/mathlib/category_theory/limits/cones.lean index d08eed316f623ca817b6123b6d7629dbd5769be3..b34b822e37d54be38d3a8e43d7a4fe25d303f7fe 100644 --- a/formal/lean/mathlib/category_theory/limits/cones.lean +++ b/formal/lean/mathlib/category_theory/limits/cones.lean @@ -29,6 +29,8 @@ We define morphisms of cones, and the category of cones. We define `cone.postcompose α : cone F ⥤ cone G` for `α` a natural transformation `F ⟶ G`. And, of course, we dualise all this to cocones as well. + +For more results about the category of cones, see `cone_category.lean`. -/ -- morphism levels before object levels. See note [category_theory universes]. @@ -220,6 +222,10 @@ namespace cones { hom := { hom := φ.hom }, inv := { hom := φ.inv, w' := λ j, φ.inv_comp_eq.mpr (w j) } } +/-- Eta rule for cones. -/ +@[simps] def eta (c : cone F) : c ≅ ⟨c.X, c.π⟩ := +cones.ext (iso.refl _) (by tidy) + /-- Given a cone morphism whose object part is an isomorphism, produce an isomorphism of cones. @@ -379,6 +385,10 @@ namespace cocones { hom := { hom := φ.hom }, inv := { hom := φ.inv, w' := λ j, φ.comp_inv_eq.mpr (w j).symm } } +/-- Eta rule for cocones. -/ +@[simps] def eta (c : cocone F) : c ≅ ⟨c.X, c.ι⟩ := +cocones.ext (iso.refl _) (by tidy) + /-- Given a cocone morphism whose object part is an isomorphism, produce an isomorphism of cocones. diff --git a/formal/lean/mathlib/category_theory/limits/constructions/binary_products.lean b/formal/lean/mathlib/category_theory/limits/constructions/binary_products.lean index bd282853b05af4551b6e3f66cd88decdc087b76f..87d68c8022ab4837c347e34227755009fb460a96 100644 --- a/formal/lean/mathlib/category_theory/limits/constructions/binary_products.lean +++ b/formal/lean/mathlib/category_theory/limits/constructions/binary_products.lean @@ -90,7 +90,7 @@ variable (C) /-- Any category with pullbacks and terminal object has binary products. -/ -- This is not an instance, as it is not always how one wants to construct binary products! -lemma has_binary_products_of_terminal_and_pullbacks +lemma has_binary_products_of_has_terminal_and_pullbacks [has_terminal C] [has_pullbacks C] : has_binary_products C := { has_limit := λ F, has_limit.mk (limit_cone_of_terminal_and_pullbacks F) } @@ -187,7 +187,7 @@ variable (C) /-- Any category with pushouts and initial object has binary coproducts. -/ -- This is not an instance, as it is not always how one wants to construct binary coproducts! -lemma has_binary_coproducts_of_initial_and_pushouts +lemma has_binary_coproducts_of_has_initial_and_pushouts [has_initial C] [has_pushouts C] : has_binary_coproducts C := { has_colimit := λ F, has_colimit.mk (colimit_cocone_of_initial_and_pushouts F) } diff --git a/formal/lean/mathlib/category_theory/limits/constructions/equalizers.lean b/formal/lean/mathlib/category_theory/limits/constructions/equalizers.lean index eb32cf33463d37b80b90c2f28357bb995b3935d1..fb51d341ed8106d8628610f89f14888ff7d5f848 100644 --- a/formal/lean/mathlib/category_theory/limits/constructions/equalizers.lean +++ b/formal/lean/mathlib/category_theory/limits/constructions/equalizers.lean @@ -29,7 +29,7 @@ variables {C : Type u} [category.{v} C] variables {D : Type u'} [category.{v'} D] (G : C ⥤ D) -- We hide the "implementation details" inside a namespace -namespace has_equalizers_of_pullbacks_and_binary_products +namespace has_equalizers_of_has_pullbacks_and_binary_products variables [has_binary_products C] [has_pullbacks C] @@ -76,12 +76,13 @@ def equalizer_cone_is_limit (F : walking_parallel_pair ⥤ C) : is_limit (equali { erw [limit.lift_π, ← J0, pullback_fst_eq_pullback_snd] } end } -end has_equalizers_of_pullbacks_and_binary_products +end has_equalizers_of_has_pullbacks_and_binary_products -open has_equalizers_of_pullbacks_and_binary_products +open has_equalizers_of_has_pullbacks_and_binary_products /-- Any category with pullbacks and binary products, has equalizers. -/ -- This is not an instance, as it is not always how one wants to construct equalizers! -lemma has_equalizers_of_pullbacks_and_binary_products [has_binary_products C] [has_pullbacks C] : +lemma has_equalizers_of_has_pullbacks_and_binary_products + [has_binary_products C] [has_pullbacks C] : has_equalizers C := { has_limit := λ F, has_limit.mk { cone := equalizer_cone F, @@ -90,7 +91,7 @@ lemma has_equalizers_of_pullbacks_and_binary_products [has_binary_products C] [h local attribute[instance] has_pullback_of_preserves_pullback /-- A functor that preserves pullbacks and binary products also presrves equalizers. -/ -def preserves_equalizers_of_pullbacks_and_binary_products +def preserves_equalizers_of_preserves_pullbacks_and_binary_products [has_binary_products C] [has_pullbacks C] [preserves_limits_of_shape (discrete walking_pair) G] [preserves_limits_of_shape walking_cospan G] : @@ -128,7 +129,7 @@ def preserves_equalizers_of_pullbacks_and_binary_products -- We hide the "implementation details" inside a namespace -namespace has_coequalizers_of_pushouts_and_binary_coproducts +namespace has_coequalizers_of_has_pushouts_and_binary_coproducts variables [has_binary_coproducts C] [has_pushouts C] @@ -177,12 +178,12 @@ def coequalizer_cocone_is_colimit (F : walking_parallel_pair ⥤ C) : { rw [colimit.ι_desc, ← pushout_inl_eq_pushout_inr], exact J1 } end } -end has_coequalizers_of_pushouts_and_binary_coproducts +end has_coequalizers_of_has_pushouts_and_binary_coproducts -open has_coequalizers_of_pushouts_and_binary_coproducts +open has_coequalizers_of_has_pushouts_and_binary_coproducts /-- Any category with pullbacks and binary products, has equalizers. -/ -- This is not an instance, as it is not always how one wants to construct equalizers! -lemma has_coequalizers_of_pushouts_and_binary_coproducts +lemma has_coequalizers_of_has_pushouts_and_binary_coproducts [has_binary_coproducts C] [has_pushouts C] : has_coequalizers C := { has_colimit := λ F, has_colimit.mk { cocone := coequalizer_cocone F, @@ -191,7 +192,7 @@ lemma has_coequalizers_of_pushouts_and_binary_coproducts local attribute[instance] has_pushout_of_preserves_pushout /-- A functor that preserves pushouts and binary coproducts also presrves coequalizers. -/ -def preserves_coequalizers_of_pushouts_and_binary_coproducts +def preserves_coequalizers_of_preserves_pushouts_and_binary_coproducts [has_binary_coproducts C] [has_pushouts C] [preserves_colimits_of_shape (discrete walking_pair) G] [preserves_colimits_of_shape walking_span G] : diff --git a/formal/lean/mathlib/category_theory/limits/constructions/finite_products_of_binary_products.lean b/formal/lean/mathlib/category_theory/limits/constructions/finite_products_of_binary_products.lean index 3506c733d254869d6bba59c8fdd38339cfd96869..6ca3bb7ede315bf5ab8dd0b77cad520ccec1d9e0 100644 --- a/formal/lean/mathlib/category_theory/limits/constructions/finite_products_of_binary_products.lean +++ b/formal/lean/mathlib/category_theory/limits/constructions/finite_products_of_binary_products.lean @@ -303,7 +303,7 @@ begin end } /-- If `C` has an initial object and binary coproducts, then it has finite coproducts. -/ -lemma has_finite_coproducts_of_has_binary_and_terminal : has_finite_coproducts C := +lemma has_finite_coproducts_of_has_binary_and_initial : has_finite_coproducts C := ⟨λ J 𝒥, begin resetI, apply has_colimits_of_shape_of_equivalence (discrete.equivalence (fintype.equiv_fin J)).symm, diff --git a/formal/lean/mathlib/category_theory/limits/constructions/limits_of_products_and_equalizers.lean b/formal/lean/mathlib/category_theory/limits/constructions/limits_of_products_and_equalizers.lean index c97bd217a843772455d6ce87f6a765a0a48c97c5..a5acd213bbb0816b530af7603fe99f57e110ab2b 100644 --- a/formal/lean/mathlib/category_theory/limits/constructions/limits_of_products_and_equalizers.lean +++ b/formal/lean/mathlib/category_theory/limits/constructions/limits_of_products_and_equalizers.lean @@ -133,7 +133,7 @@ Any category with products and equalizers has all limits. See . -/ -lemma limits_from_equalizers_and_products +lemma has_limits_of_has_equalizers_and_products [has_products.{w} C] [has_equalizers C] : has_limits_of_size.{w w} C := { has_limits_of_shape := λ J 𝒥, { has_limit := λ F, by exactI has_limit_of_equalizer_and_product F } } @@ -143,7 +143,7 @@ Any category with finite products and equalizers has all finite limits. See . -/ -lemma finite_limits_from_equalizers_and_finite_products +lemma has_finite_limits_of_has_equalizers_and_finite_products [has_finite_products C] [has_equalizers C] : has_finite_limits C := ⟨λ J _ _, { has_limit := λ F, by exactI has_limit_of_equalizer_and_product F }⟩ @@ -219,11 +219,11 @@ preserves_limits_of_size.{w w} G := lemma has_finite_limits_of_has_terminal_and_pullbacks [has_terminal C] [has_pullbacks C] : has_finite_limits C := -@@finite_limits_from_equalizers_and_finite_products _ +@@has_finite_limits_of_has_equalizers_and_finite_products _ (@@has_finite_products_of_has_binary_and_terminal _ - (has_binary_products_of_terminal_and_pullbacks C) infer_instance) - (@@has_equalizers_of_pullbacks_and_binary_products _ - (has_binary_products_of_terminal_and_pullbacks C) infer_instance) + (has_binary_products_of_has_terminal_and_pullbacks C) infer_instance) + (@@has_equalizers_of_has_pullbacks_and_binary_products _ + (has_binary_products_of_has_terminal_and_pullbacks C) infer_instance) /-- If G preserves terminal objects and pullbacks, it preserves all finite limits. -/ def preserves_finite_limits_of_preserves_terminal_and_pullbacks @@ -236,7 +236,7 @@ begin haveI : preserves_limits_of_shape (discrete walking_pair) G := preserves_binary_products_of_preserves_terminal_and_pullbacks G, exact @@preserves_finite_limits_of_preserves_equalizers_and_finite_products _ _ _ _ G - (preserves_equalizers_of_pullbacks_and_binary_products G) + (preserves_equalizers_of_preserves_pullbacks_and_binary_products G) (preserves_finite_products_of_preserves_binary_and_terminal G), end @@ -341,7 +341,7 @@ Any category with coproducts and coequalizers has all colimits. See . -/ -lemma colimits_from_coequalizers_and_coproducts +lemma has_colimits_of_has_coequalizers_and_coproducts [has_coproducts.{w} C] [has_coequalizers C] : has_colimits_of_size.{w w} C := { has_colimits_of_shape := λ J 𝒥, { has_colimit := λ F, by exactI has_colimit_of_coequalizer_and_coproduct F } } @@ -351,7 +351,7 @@ Any category with finite coproducts and coequalizers has all finite colimits. See . -/ -lemma finite_colimits_from_coequalizers_and_finite_coproducts +lemma has_finite_colimits_of_has_coequalizers_and_finite_coproducts [has_finite_coproducts C] [has_coequalizers C] : has_finite_colimits C := ⟨λ J _ _, { has_colimit := λ F, by exactI has_colimit_of_coequalizer_and_coproduct F }⟩ @@ -426,11 +426,11 @@ preserves_colimits_of_size.{w} G := lemma has_finite_colimits_of_has_initial_and_pushouts [has_initial C] [has_pushouts C] : has_finite_colimits C := -@@finite_colimits_from_coequalizers_and_finite_coproducts _ - (@@has_finite_coproducts_of_has_binary_and_terminal _ - (has_binary_coproducts_of_initial_and_pushouts C) infer_instance) - (@@has_coequalizers_of_pushouts_and_binary_coproducts _ - (has_binary_coproducts_of_initial_and_pushouts C) infer_instance) +@@has_finite_colimits_of_has_coequalizers_and_finite_coproducts _ + (@@has_finite_coproducts_of_has_binary_and_initial _ + (has_binary_coproducts_of_has_initial_and_pushouts C) infer_instance) + (@@has_coequalizers_of_has_pushouts_and_binary_coproducts _ + (has_binary_coproducts_of_has_initial_and_pushouts C) infer_instance) /-- If G preserves initial objects and pushouts, it preserves all finite colimits. -/ def preserves_finite_colimits_of_preserves_initial_and_pushouts @@ -443,7 +443,7 @@ begin haveI : preserves_colimits_of_shape (discrete walking_pair) G := preserves_binary_coproducts_of_preserves_initial_and_pushouts G, exact @@preserves_finite_colimits_of_preserves_coequalizers_and_finite_coproducts _ _ _ _ G - (preserves_coequalizers_of_pushouts_and_binary_coproducts G) + (preserves_coequalizers_of_preserves_pushouts_and_binary_coproducts G) (preserves_finite_coproducts_of_preserves_binary_and_initial G), end diff --git a/formal/lean/mathlib/category_theory/limits/constructions/over/default.lean b/formal/lean/mathlib/category_theory/limits/constructions/over/default.lean index 09025a23cc049bc4eb3b804e37d32ed50ab27221..e4085355f2824c00c686844d9d468db9e7ccc221 100644 --- a/formal/lean/mathlib/category_theory/limits/constructions/over/default.lean +++ b/formal/lean/mathlib/category_theory/limits/constructions/over/default.lean @@ -44,9 +44,9 @@ end instance has_finite_limits {B : C} [has_finite_wide_pullbacks C] : has_finite_limits (over B) := begin - apply @finite_limits_from_equalizers_and_finite_products _ _ _ _, + apply @has_finite_limits_of_has_equalizers_and_finite_products _ _ _ _, { exact construct_products.over_finite_products_of_finite_wide_pullbacks, }, - { apply @has_equalizers_of_pullbacks_and_binary_products _ _ _ _, + { apply @has_equalizers_of_has_pullbacks_and_binary_products _ _ _ _, { haveI : has_pullbacks C := ⟨by apply_instance⟩, exact construct_products.over_binary_product_of_pullback }, { apply_instance, } } @@ -54,9 +54,9 @@ end instance has_limits {B : C} [has_wide_pullbacks.{w} C] : has_limits_of_size.{w} (over B) := begin - apply @limits_from_equalizers_and_products _ _ _ _, + apply @has_limits_of_has_equalizers_and_products _ _ _ _, { exact construct_products.over_products_of_wide_pullbacks }, - { apply @has_equalizers_of_pullbacks_and_binary_products _ _ _ _, + { apply @has_equalizers_of_has_pullbacks_and_binary_products _ _ _ _, { haveI : has_pullbacks C := ⟨infer_instance⟩, exact construct_products.over_binary_product_of_pullback }, { apply_instance, } } diff --git a/formal/lean/mathlib/category_theory/limits/constructions/weakly_initial.lean b/formal/lean/mathlib/category_theory/limits/constructions/weakly_initial.lean index 8dc5249777940286a5cdd8b9a4e1d6b7377aa4e0..2995d575ff221d6ff66d0ed4a410c3dadacd0da3 100644 --- a/formal/lean/mathlib/category_theory/limits/constructions/weakly_initial.lean +++ b/formal/lean/mathlib/category_theory/limits/constructions/weakly_initial.lean @@ -56,7 +56,7 @@ begin { rw [category.assoc, category.assoc], apply wide_equalizer.condition (id : endos → endos) (h ≫ e ≫ i) }, rw [category.comp_id, cancel_mono_id i] at this, - haveI : split_epi e := ⟨i ≫ h, this⟩, + haveI : is_split_epi e := is_split_epi.mk' ⟨i ≫ h, this⟩, rw ←cancel_epi e, apply equalizer.condition }, exactI has_initial_of_unique (wide_equalizer (id : endos → endos)), diff --git a/formal/lean/mathlib/category_theory/limits/essentially_small.lean b/formal/lean/mathlib/category_theory/limits/essentially_small.lean index 28d3a6488773f1f75a22eb0b890d1ebf7fe33c77..2c04a775c663d802eca96c22078c327dce0c2a94 100644 --- a/formal/lean/mathlib/category_theory/limits/essentially_small.lean +++ b/formal/lean/mathlib/category_theory/limits/essentially_small.lean @@ -30,11 +30,11 @@ lemma has_colimits_of_shape_of_essentially_small [essentially_small.{w₁} J] [has_colimits_of_size.{w₁ w₁} C] : has_colimits_of_shape J C := has_colimits_of_shape_of_equivalence $ equivalence.symm $ equiv_small_model.{w₁} J -lemma has_products_of_shape_of_small (β : Type w₁) [small.{w₂} β] [has_products.{w₂} C] : +lemma has_products_of_shape_of_small (β : Type w₂) [small.{w₁} β] [has_products.{w₁} C] : has_products_of_shape β C := has_limits_of_shape_of_equivalence $ discrete.equivalence $ equiv.symm $ equiv_shrink β -lemma has_coproducts_of_shape_of_small (β : Type w₁) [small.{w₂} β] [has_coproducts.{w₂} C] : +lemma has_coproducts_of_shape_of_small (β : Type w₂) [small.{w₁} β] [has_coproducts.{w₁} C] : has_coproducts_of_shape β C := has_colimits_of_shape_of_equivalence $ discrete.equivalence $ equiv.symm $ equiv_shrink β diff --git a/formal/lean/mathlib/category_theory/limits/exact_functor.lean b/formal/lean/mathlib/category_theory/limits/exact_functor.lean new file mode 100644 index 0000000000000000000000000000000000000000..d7974bcafefeb109dd0f2dce4849888f4021b879 --- /dev/null +++ b/formal/lean/mathlib/category_theory/limits/exact_functor.lean @@ -0,0 +1,130 @@ +/- +Copyright (c) 2022 Markus Himmel. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Markus Himmel +-/ +import category_theory.limits.preserves.finite + +/-! +# Bundled exact functors + +We say that a functor `F` is left exact if it preserves finite limits, it is right exact if it +preserves finite colimits, and it is exact if it is both left exact and right exact. + +In this file, we define the categories of bundled left exact, right exact and exact functors. + +-/ + +universes v₁ v₂ u₁ u₂ + +open category_theory.limits + +namespace category_theory +variables {C : Type u₁} [category.{v₁} C] {D : Type u₂} [category.{v₂} D] + +section +variables (C) (D) + +/-- Bundled left-exact functors. -/ +@[derive category, nolint has_nonempty_instance] +def LeftExactFunctor := +full_subcategory (λ F : C ⥤ D, nonempty (preserves_finite_limits F)) + +infixr ` ⥤ₗ `:26 := LeftExactFunctor + +/-- A left exact functor is in particular a functor. -/ +@[derive full, derive faithful] +def LeftExactFunctor.forget : (C ⥤ₗ D) ⥤ (C ⥤ D) := +full_subcategory_inclusion _ + +/-- Bundled right-exact functors. -/ +@[derive category, nolint has_nonempty_instance] +def RightExactFunctor := +full_subcategory (λ F : C ⥤ D, nonempty (preserves_finite_colimits F)) + +infixr ` ⥤ᵣ `:26 := RightExactFunctor + +/-- A right exact functor is in particular a functor. -/ +@[derive full, derive faithful] +def RightExactFunctor.forget : (C ⥤ᵣ D) ⥤ (C ⥤ D) := +full_subcategory_inclusion _ + +/-- Bundled exact functors. -/ +@[derive category, nolint has_nonempty_instance] +def ExactFunctor := full_subcategory + (λ F : C ⥤ D, nonempty (preserves_finite_limits F) ∧ nonempty (preserves_finite_colimits F)) + +infixr ` ⥤ₑ `:26 := ExactFunctor + +/-- An exact functor is in particular a functor. -/ +@[derive full, derive faithful] +def ExactFunctor.forget : (C ⥤ₑ D) ⥤ (C ⥤ D) := +full_subcategory_inclusion _ + +/-- Turn an exact functor into a left exact functor. -/ +@[derive full, derive faithful] +def LeftExactFunctor.of_exact : (C ⥤ₑ D) ⥤ (C ⥤ₗ D) := +full_subcategory.map (λ X, and.left) + +/-- Turn an exact functor into a left exact functor. -/ +@[derive full, derive faithful] +def RightExactFunctor.of_exact : (C ⥤ₑ D) ⥤ (C ⥤ᵣ D) := +full_subcategory.map (λ X, and.right) + +variables {C D} + +@[simp] lemma LeftExactFunctor.of_exact_obj (F : C ⥤ₑ D) : + (LeftExactFunctor.of_exact C D).obj F = ⟨F.1, F.2.1⟩ := rfl +@[simp] lemma RightExactFunctor.of_exact_obj (F : C ⥤ₑ D) : + (RightExactFunctor.of_exact C D).obj F = ⟨F.1, F.2.2⟩ := rfl + +@[simp] lemma LeftExactFunctor.of_exact_map {F G : C ⥤ₑ D} (α : F ⟶ G) : + (LeftExactFunctor.of_exact C D).map α = α := rfl +@[simp] lemma RightExactFunctor.of_exact_map {F G : C ⥤ₑ D} (α : F ⟶ G) : + (RightExactFunctor.of_exact C D).map α = α := rfl + +@[simp] lemma LeftExactFunctor.forget_obj (F : C ⥤ₗ D) : + (LeftExactFunctor.forget C D).obj F = F.1 := rfl +@[simp] lemma RightExactFunctor.forget_obj (F : C ⥤ᵣ D) : + (RightExactFunctor.forget C D).obj F = F.1 := rfl +@[simp] lemma ExactFunctor.forget_obj (F : C ⥤ₑ D) : + (ExactFunctor.forget C D).obj F = F.1 := rfl + +@[simp] lemma LeftExactFunctor.forget_map {F G : C ⥤ₗ D} (α : F ⟶ G) : + (LeftExactFunctor.forget C D).map α = α := rfl +@[simp] lemma RightExactFunctor.forget_map {F G : C ⥤ᵣ D} (α : F ⟶ G) : + (RightExactFunctor.forget C D).map α = α := rfl +@[simp] lemma ExactFunctor.forget_map {F G : C ⥤ₑ D} (α : F ⟶ G) : + (ExactFunctor.forget C D).map α = α := rfl + +/-- Turn a left exact functor into an object of the category `LeftExactFunctor C D`. -/ +def LeftExactFunctor.of (F : C ⥤ D) [preserves_finite_limits F] : C ⥤ₗ D := ⟨F, ⟨infer_instance⟩⟩ +/-- Turn a right exact functor into an object of the category `RightExactFunctor C D`. -/ +def RightExactFunctor.of (F : C ⥤ D) [preserves_finite_colimits F] : C ⥤ᵣ D := +⟨F, ⟨infer_instance⟩⟩ +/-- Turn an exact functor into an object of the category `ExactFunctor C D`. -/ +def ExactFunctor.of (F : C ⥤ D) [preserves_finite_limits F] [preserves_finite_colimits F] : + C ⥤ₑ D := ⟨F, ⟨⟨infer_instance⟩, ⟨infer_instance⟩⟩⟩ + +@[simp] lemma LeftExactFunctor.of_fst (F : C ⥤ D) [preserves_finite_limits F] : + (LeftExactFunctor.of F).obj = F := rfl +@[simp] lemma RightExactFunctor.of_fst (F : C ⥤ D) [preserves_finite_colimits F] : + (RightExactFunctor.of F).obj = F := rfl +@[simp] lemma ExactFunctor.of_fst (F : C ⥤ D) [preserves_finite_limits F] + [preserves_finite_colimits F] : (ExactFunctor.of F).obj = F := rfl + +lemma LeftExactFunctor.forget_obj_of (F : C ⥤ D) [preserves_finite_limits F] : + (LeftExactFunctor.forget C D).obj (LeftExactFunctor.of F) = F := rfl +lemma RightExactFunctor.forget_obj_of (F : C ⥤ D) [preserves_finite_colimits F] : + (RightExactFunctor.forget C D).obj (RightExactFunctor.of F) = F := rfl +lemma ExactFunctor.forget_obj_of (F : C ⥤ D) [preserves_finite_limits F] + [preserves_finite_colimits F] : (ExactFunctor.forget C D).obj (ExactFunctor.of F) = F := rfl + +noncomputable instance (F : C ⥤ₗ D) : preserves_finite_limits F.obj := F.property.some +noncomputable instance (F : C ⥤ᵣ D) : preserves_finite_colimits F.obj := F.property.some +noncomputable instance (F : C ⥤ₑ D) : preserves_finite_limits F.obj := F.property.1.some +noncomputable instance (F : C ⥤ₑ D) : preserves_finite_colimits F.obj := F.property.2.some + +end + +end category_theory diff --git a/formal/lean/mathlib/category_theory/limits/functor_category.lean b/formal/lean/mathlib/category_theory/limits/functor_category.lean index b0d8ece00f57bab8c2c374075a186bfeab848557..ec1cb9b3364e825a66b5a2e95cbbb3621636b359 100644 --- a/formal/lean/mathlib/category_theory/limits/functor_category.lean +++ b/formal/lean/mathlib/category_theory/limits/functor_category.lean @@ -19,10 +19,10 @@ We also show that `F : D ⥤ K ⥤ C` preserves (co)limits if it does so for eac `category_theory.limits.preserves_colimits_of_evaluation`). -/ -open category_theory category_theory.category +open category_theory category_theory.category category_theory.functor -- morphism levels before object levels. See note [category_theory universes]. -universes v₁ v₂ u₁ u₂ v v' u u' +universes w' w v₁ v₂ u₁ u₂ v v' u u' namespace category_theory.limits @@ -308,12 +308,17 @@ def preserves_limits_of_shape_of_evaluation (F : D ⥤ K ⥤ C) (J : Type*) [cat ⟨λ G, preserves_limit_of_evaluation F G (λ k, preserves_limits_of_shape.preserves_limit)⟩ /-- `F : D ⥤ K ⥤ C` preserves all limits if it does for each `k : K`. -/ -def {w' w} preserves_limits_of_evaluation (F : D ⥤ K ⥤ C) +def preserves_limits_of_evaluation (F : D ⥤ K ⥤ C) (H : Π (k : K), preserves_limits_of_size.{w' w} (F ⋙ (evaluation K C).obj k)) : preserves_limits_of_size.{w' w} F := ⟨λ L hL, by exactI preserves_limits_of_shape_of_evaluation F L (λ k, preserves_limits_of_size.preserves_limits_of_shape)⟩ +/-- The constant functor `C ⥤ (D ⥤ C)` preserves limits. -/ +instance preserves_limits_const : preserves_limits_of_size.{w' w} (const D : C ⥤ _) := +preserves_limits_of_evaluation _ $ λ X, preserves_limits_of_nat_iso $ iso.symm $ + const_comp_evaluation_obj _ _ + instance evaluation_preserves_colimits [has_colimits C] (k : K) : preserves_colimits ((evaluation K C).obj k) := { preserves_colimits_of_shape := λ J 𝒥, by resetI; apply_instance } @@ -336,11 +341,17 @@ def preserves_colimits_of_shape_of_evaluation (F : D ⥤ K ⥤ C) (J : Type*) [c ⟨λ G, preserves_colimit_of_evaluation F G (λ k, preserves_colimits_of_shape.preserves_colimit)⟩ /-- `F : D ⥤ K ⥤ C` preserves all colimits if it does for each `k : K`. -/ -def {w' w} preserves_colimits_of_evaluation (F : D ⥤ K ⥤ C) +def preserves_colimits_of_evaluation (F : D ⥤ K ⥤ C) (H : Π (k : K), preserves_colimits_of_size.{w' w} (F ⋙ (evaluation K C).obj k)) : preserves_colimits_of_size.{w' w} F := ⟨λ L hL, by exactI preserves_colimits_of_shape_of_evaluation F L (λ k, preserves_colimits_of_size.preserves_colimits_of_shape)⟩ + +/-- The constant functor `C ⥤ (D ⥤ C)` preserves colimits. -/ +instance preserves_colimits_const : preserves_colimits_of_size.{w' w} (const D : C ⥤ _) := +preserves_colimits_of_evaluation _ $ λ X, preserves_colimits_of_nat_iso $ iso.symm $ + const_comp_evaluation_obj _ _ + open category_theory.prod /-- The limit of a diagram `F : J ⥤ K ⥤ C` is isomorphic to the functor given by diff --git a/formal/lean/mathlib/category_theory/limits/has_limits.lean b/formal/lean/mathlib/category_theory/limits/has_limits.lean index 37f6c18894d68d3a70c4aac5b1b3272e8568b26c..d2d87212c65e4a94b4ade41c2c9a408bc383c23f 100644 --- a/formal/lean/mathlib/category_theory/limits/has_limits.lean +++ b/formal/lean/mathlib/category_theory/limits/has_limits.lean @@ -69,7 +69,7 @@ variables {F : J ⥤ C} section limit /-- `limit_cone F` contains a cone over `F` together with the information that it is a limit. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure limit_cone (F : J ⥤ C) := (cone : cone F) (is_limit : is_limit cone) @@ -519,7 +519,7 @@ section colimit /-- `colimit_cocone F` contains a cocone over `F` together with the information that it is a colimit. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure colimit_cocone (F : J ⥤ C) := (cocone : cocone F) (is_colimit : is_colimit cocone) diff --git a/formal/lean/mathlib/category_theory/limits/is_limit.lean b/formal/lean/mathlib/category_theory/limits/is_limit.lean index 9b220c75b965ffca12fe7c7ed9998141054ce782..56ce006946f2a9b7e8b953e92fa9609437daf8eb 100644 --- a/formal/lean/mathlib/category_theory/limits/is_limit.lean +++ b/formal/lean/mathlib/category_theory/limits/is_limit.lean @@ -50,7 +50,7 @@ cone morphism to `t`. See . -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure is_limit (t : cone F) := (lift : Π (s : cone F), s.X ⟶ t.X) (fac' : ∀ (s : cone F) (j : J), lift s ≫ t.π.app j = s.π.app j . obviously) @@ -504,7 +504,7 @@ cocone morphism from `t`. See . -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure is_colimit (t : cocone F) := (desc : Π (s : cocone F), t.X ⟶ s.X) (fac' : ∀ (s : cocone F) (j : J), t.ι.app j ≫ desc s = s.ι.app j . obviously) diff --git a/formal/lean/mathlib/category_theory/limits/preserves/shapes/biproducts.lean b/formal/lean/mathlib/category_theory/limits/preserves/shapes/biproducts.lean index 9d6601853f40923d7a3a1d349075609d5b06adbd..5727f969e1b36fa080fccdb59d6fb8668a319aab 100644 --- a/formal/lean/mathlib/category_theory/limits/preserves/shapes/biproducts.lean +++ b/formal/lean/mathlib/category_theory/limits/preserves/shapes/biproducts.lean @@ -216,17 +216,21 @@ variables [preserves_zero_morphisms F] biproduct_comparison' F f ≫ biproduct_comparison F f = 𝟙 (⨁ (F.obj ∘ f)) := by { classical, ext, simp [biproduct.ι_π, ← functor.map_comp, eq_to_hom_map] } -instance : split_epi (biproduct_comparison F f) := +/-- `biproduct_comparison F f` is a split epimorphism. -/ +@[simps] +def split_epi_biproduct_comparison : split_epi (biproduct_comparison F f) := ⟨biproduct_comparison' F f⟩ -@[simp] lemma section_biproduct_comparison : - section_ (biproduct_comparison F f) = biproduct_comparison' F f := rfl +instance : is_split_epi (biproduct_comparison F f) := +is_split_epi.mk' (split_epi_biproduct_comparison F f) -instance : split_mono (biproduct_comparison' F f) := +/-- `biproduct_comparison' F f` is a split monomorphism. -/ +@[simps] +def split_mono_biproduct_comparison' : split_mono (biproduct_comparison' F f) := ⟨biproduct_comparison F f⟩ -@[simp] lemma retraction_biproduct_comparison' : - retraction (biproduct_comparison' F f) = biproduct_comparison F f := rfl +instance : is_split_mono (biproduct_comparison' F f) := +is_split_mono.mk' (split_mono_biproduct_comparison' F f) end @@ -290,17 +294,21 @@ variables [preserves_zero_morphisms F] biprod_comparison' F X Y ≫ biprod_comparison F X Y = 𝟙 (F.obj X ⊞ F.obj Y) := by { ext; simp [← functor.map_comp] } -instance : split_epi (biprod_comparison F X Y) := +/-- `biprod_comparison F X Y` is a split epi. -/ +@[simps] +def split_epi_biprod_comparison : split_epi (biprod_comparison F X Y) := ⟨biprod_comparison' F X Y⟩ -@[simp] lemma section_biprod_comparison : - section_ (biprod_comparison F X Y) = biprod_comparison' F X Y := rfl +instance : is_split_epi (biprod_comparison F X Y) := +is_split_epi.mk' (split_epi_biprod_comparison F X Y) -instance : split_mono (biprod_comparison' F X Y) := +/-- `biprod_comparison' F X Y` is a split mono. -/ +@[simps] +def split_mono_biprod_comparison' : split_mono (biprod_comparison' F X Y) := ⟨biprod_comparison F X Y⟩ -@[simp] lemma retraction_biprod_comparison' : - retraction (biprod_comparison' F X Y) = biprod_comparison F X Y := rfl +instance : is_split_mono (biprod_comparison' F X Y) := +is_split_mono.mk' (split_mono_biprod_comparison' F X Y) end @@ -421,7 +429,7 @@ begin have : pi_comparison F f = (F.map_iso (biproduct.iso_product f)).inv ≫ biproduct_comparison F f ≫ (biproduct.iso_product _).hom, { ext, convert pi_comparison_comp_π F f j.as; simp [← functor.map_comp] }, - haveI : is_iso (biproduct_comparison F f) := is_iso_of_mono_of_split_epi _, + haveI : is_iso (biproduct_comparison F f) := is_iso_of_mono_of_is_split_epi _, haveI : is_iso (pi_comparison F f) := by { rw this, apply_instance }, haveI := preserves_product.of_iso_comparison F f, apply preserves_biproduct_of_preserves_product @@ -432,8 +440,9 @@ end def preserves_biproduct_of_epi_biproduct_comparison' {f : J → C} [has_biproduct f] [has_biproduct (F.obj ∘ f)] [epi (biproduct_comparison' F f)] : preserves_biproduct f F := begin - haveI : epi (section_ (biproduct_comparison F f)) := by simpa, - haveI : is_iso (biproduct_comparison F f) := is_iso.of_epi_section, + haveI : epi ((split_epi_biproduct_comparison F f).section_) := by simpa, + haveI : is_iso (biproduct_comparison F f) := is_iso.of_epi_section' + (split_epi_biproduct_comparison F f), apply preserves_biproduct_of_mono_biproduct_comparison end @@ -519,7 +528,7 @@ def preserves_binary_biproduct_of_mono_biprod_comparison {X Y : C} [has_binary_b begin have : prod_comparison F X Y = (F.map_iso (biprod.iso_prod X Y)).inv ≫ biprod_comparison F X Y ≫ (biprod.iso_prod _ _).hom := by { ext; simp [← functor.map_comp] }, - haveI : is_iso (biprod_comparison F X Y) := is_iso_of_mono_of_split_epi _, + haveI : is_iso (biprod_comparison F X Y) := is_iso_of_mono_of_is_split_epi _, haveI : is_iso (prod_comparison F X Y) := by { rw this, apply_instance }, haveI := preserves_limit_pair.of_iso_prod_comparison F X Y, apply preserves_binary_biproduct_of_preserves_binary_product @@ -531,8 +540,9 @@ def preserves_binary_biproduct_of_epi_biprod_comparison' {X Y : C} [has_binary_b [has_binary_biproduct (F.obj X) (F.obj Y)] [epi (biprod_comparison' F X Y)] : preserves_binary_biproduct X Y F := begin - haveI : epi (section_ (biprod_comparison F X Y)) := by simpa, - haveI : is_iso (biprod_comparison F X Y) := is_iso.of_epi_section, + haveI : epi ((split_epi_biprod_comparison F X Y).section_) := by simpa, + haveI : is_iso (biprod_comparison F X Y) := is_iso.of_epi_section' + (split_epi_biprod_comparison F X Y), apply preserves_binary_biproduct_of_mono_biprod_comparison end diff --git a/formal/lean/mathlib/category_theory/limits/shapes/binary_products.lean b/formal/lean/mathlib/category_theory/limits/shapes/binary_products.lean index 663d82320607f92c5d22e7b7dcdb04a03a44c703..76698c5a4a4829c44fdb315f4002b676a00a27e8 100644 --- a/formal/lean/mathlib/category_theory/limits/shapes/binary_products.lean +++ b/formal/lean/mathlib/category_theory/limits/shapes/binary_products.lean @@ -471,8 +471,8 @@ lemma prod.diag_map_fst_snd_comp [has_limits_of_shape (discrete walking_pair) C diag (X ⨯ X') ≫ prod.map (prod.fst ≫ g) (prod.snd ≫ g') = prod.map g g' := by simp -instance {X : C} [has_binary_product X X] : split_mono (diag X) := -{ retraction := prod.fst } +instance {X : C} [has_binary_product X X] : is_split_mono (diag X) := +is_split_mono.mk' { retraction := prod.fst } end prod_lemmas diff --git a/formal/lean/mathlib/category_theory/limits/shapes/biproducts.lean b/formal/lean/mathlib/category_theory/limits/shapes/biproducts.lean index 6cec24a59572bf019bfe37cd0c14a8a6febaa6ef..6e3f996250607b172400b5d37ac5018ba03b6677 100644 --- a/formal/lean/mathlib/category_theory/limits/shapes/biproducts.lean +++ b/formal/lean/mathlib/category_theory/limits/shapes/biproducts.lean @@ -74,7 +74,7 @@ A `c : bicone F` is: * morphisms `π j : X ⟶ F j` and `ι j : F j ⟶ X` for each `j`, * such that `ι j ≫ π j'` is the identity when `j = j'` and zero otherwise. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure bicone (F : J → C) := (X : C) (π : Π j, X ⟶ F j) @@ -145,7 +145,7 @@ lemma π_of_is_colimit {f : J → C} {t : bicone f} (ht : is_colimit t.to_cocone ht.hom_ext (λ j', by { rw ht.fac, discrete_cases, simp [t.ι_π] }) /-- Structure witnessing that a bicone is both a limit cone and a colimit cocone. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure is_bilimit {F : J → C} (B : bicone F) := (is_limit : is_limit B.to_cone) (is_colimit : is_colimit B.to_cocone) @@ -212,7 +212,7 @@ end bicone /-- A bicone over `F : J → C`, which is both a limit cone and a colimit cocone. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure limit_bicone (F : J → C) := (bicone : bicone F) (is_bilimit : bicone.is_bilimit) @@ -665,10 +665,12 @@ end finite_biproducts variables {J : Type w} {C : Type u} [category.{v} C] [has_zero_morphisms C] -instance biproduct.ι_mono (f : J → C) [has_biproduct f] (b : J) : split_mono (biproduct.ι f b) := +instance biproduct.ι_mono (f : J → C) [has_biproduct f] (b : J) : + is_split_mono (biproduct.ι f b) := is_split_mono.mk' { retraction := biproduct.desc $ pi.single b _ } -instance biproduct.π_epi (f : J → C) [has_biproduct f] (b : J) : split_epi (biproduct.π f b) := +instance biproduct.π_epi (f : J → C) [has_biproduct f] (b : J) : + is_split_epi (biproduct.π f b) := is_split_epi.mk' { section_ := biproduct.lift $ pi.single b _ } /-- Auxiliary lemma for `biproduct.unique_up_to_iso`. -/ @@ -740,7 +742,7 @@ A binary bicone for a pair of objects `P Q : C` consists of the cone point `X`, maps from `X` to both `P` and `Q`, and maps from both `P` and `Q` to `X`, so that `inl ≫ fst = 𝟙 P`, `inl ≫ snd = 0`, `inr ≫ fst = 0`, and `inr ≫ snd = 𝟙 Q` -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure binary_bicone (P Q : C) := (X : C) (fst : X ⟶ P) @@ -802,13 +804,17 @@ rfl lemma binary_cofan_inr_to_cocone (c : binary_bicone P Q) : binary_cofan.inr c.to_cocone = c.inr := rfl -instance (c : binary_bicone P Q) : split_mono c.inl := { retraction := c.fst, id' := c.inl_fst } +instance (c : binary_bicone P Q) : is_split_mono c.inl := +is_split_mono.mk' { retraction := c.fst, id' := c.inl_fst } -instance (c : binary_bicone P Q) : split_mono c.inr := { retraction := c.snd, id' := c.inr_snd } +instance (c : binary_bicone P Q) : is_split_mono c.inr := +is_split_mono.mk' { retraction := c.snd, id' := c.inr_snd } -instance (c : binary_bicone P Q) : split_epi c.fst := { section_ := c.inl, id' := c.inl_fst } +instance (c : binary_bicone P Q) : is_split_epi c.fst := +is_split_epi.mk' { section_ := c.inl, id' := c.inl_fst } -instance (c : binary_bicone P Q) : split_epi c.snd := { section_ := c.inr, id' := c.inr_snd } +instance (c : binary_bicone P Q) : is_split_epi c.snd := +is_split_epi.mk' { section_ := c.inr, id' := c.inr_snd } /-- Convert a `binary_bicone` into a `bicone` over a pair. -/ @[simps] @@ -861,7 +867,7 @@ is_colimit.equiv_iso_colimit $ cocones.ext (iso.refl _) (λ j, by { rcases j wit end bicone /-- Structure witnessing that a binary bicone is a limit cone and a limit cocone. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure binary_bicone.is_bilimit {P Q : C} (b : binary_bicone P Q) := (is_limit : is_limit b.to_cone) (is_colimit : is_colimit b.to_cocone) @@ -888,7 +894,7 @@ def bicone.to_binary_bicone_is_bilimit {X Y : C} (b : bicone (pair_function X Y) /-- A bicone over `P Q : C`, which is both a limit cone and a colimit cocone. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure binary_biproduct_data (P Q : C) := (bicone : binary_bicone P Q) (is_bilimit : bicone.is_bilimit) @@ -1146,20 +1152,20 @@ begin end instance biprod.inl_mono {X Y : C} [has_binary_biproduct X Y] : - split_mono (biprod.inl : X ⟶ X ⊞ Y) := -{ retraction := biprod.fst } + is_split_mono (biprod.inl : X ⟶ X ⊞ Y) := +is_split_mono.mk' { retraction := biprod.fst } instance biprod.inr_mono {X Y : C} [has_binary_biproduct X Y] : - split_mono (biprod.inr : Y ⟶ X ⊞ Y) := -{ retraction := biprod.snd } + is_split_mono (biprod.inr : Y ⟶ X ⊞ Y) := +is_split_mono.mk' { retraction := biprod.snd } instance biprod.fst_epi {X Y : C} [has_binary_biproduct X Y] : - split_epi (biprod.fst : X ⊞ Y ⟶ X) := -{ section_ := biprod.inl } + is_split_epi (biprod.fst : X ⊞ Y ⟶ X) := +is_split_epi.mk' { section_ := biprod.inl } instance biprod.snd_epi {X Y : C} [has_binary_biproduct X Y] : - split_epi (biprod.snd : X ⊞ Y ⟶ Y) := -{ section_ := biprod.inr } + is_split_epi (biprod.snd : X ⊞ Y ⟶ Y) := +is_split_epi.mk' { section_ := biprod.inr } @[simp,reassoc] lemma biprod.map_fst {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z] @@ -1819,7 +1825,7 @@ the cokernel map as its `snd`. We will show in `is_bilimit_binary_bicone_of_split_mono_of_cokernel` that this binary bicone is in fact already a biproduct. -/ @[simps] -def binary_bicone_of_split_mono_of_cokernel {X Y : C} {f : X ⟶ Y} [split_mono f] +def binary_bicone_of_is_split_mono_of_cokernel {X Y : C} {f : X ⟶ Y} [is_split_mono f] {c : cokernel_cofork f} (i : is_colimit c) : binary_bicone X c.X := { X := Y, fst := retraction f, @@ -1841,8 +1847,8 @@ def binary_bicone_of_split_mono_of_cokernel {X Y : C} {f : X ⟶ Y} [split_mono dsimp only [cokernel_cofork_of_cofork_of_π], letI := epi_of_is_colimit_cofork i, apply zero_of_epi_comp c.π, - simp only [sub_comp, comp_sub, category.comp_id, category.assoc, split_mono.id, sub_self, - cofork.is_colimit.π_desc_assoc, cokernel_cofork.π_of_π, split_mono.id_assoc], + simp only [sub_comp, comp_sub, category.comp_id, category.assoc, is_split_mono.id, sub_self, + cofork.is_colimit.π_desc_assoc, cokernel_cofork.π_of_π, is_split_mono.id_assoc], apply sub_eq_zero_of_eq, apply category.id_comp end, @@ -1850,17 +1856,17 @@ def binary_bicone_of_split_mono_of_cokernel {X Y : C} {f : X ⟶ Y} [split_mono /-- The bicone constructed in `binary_bicone_of_split_mono_of_cokernel` is a bilimit. This is a version of the splitting lemma that holds in all preadditive categories. -/ -def is_bilimit_binary_bicone_of_split_mono_of_cokernel {X Y : C} {f : X ⟶ Y} [split_mono f] +def is_bilimit_binary_bicone_of_is_split_mono_of_cokernel {X Y : C} {f : X ⟶ Y} [is_split_mono f] {c : cokernel_cofork f} (i : is_colimit c) : - (binary_bicone_of_split_mono_of_cokernel i).is_bilimit := + (binary_bicone_of_is_split_mono_of_cokernel i).is_bilimit := is_binary_bilimit_of_total _ begin - simp only [binary_bicone_of_split_mono_of_cokernel_fst, - binary_bicone_of_split_mono_of_cokernel_inr, binary_bicone_of_split_mono_of_cokernel_snd, + simp only [binary_bicone_of_is_split_mono_of_cokernel_fst, + binary_bicone_of_is_split_mono_of_cokernel_inr, binary_bicone_of_is_split_mono_of_cokernel_snd, split_epi_of_idempotent_of_is_colimit_cofork_section_], - dsimp only [binary_bicone_of_split_mono_of_cokernel_X], + dsimp only [binary_bicone_of_is_split_mono_of_cokernel_X], rw [is_colimit_cofork_of_cokernel_cofork_desc, is_cokernel_epi_comp_desc], - simp only [binary_bicone_of_split_mono_of_cokernel_inl, cofork.is_colimit.π_desc, + simp only [binary_bicone_of_is_split_mono_of_cokernel_inl, cofork.is_colimit.π_desc, cokernel_cofork_of_cofork_π, cofork.π_of_π, add_sub_cancel'_right] end @@ -1923,10 +1929,10 @@ is_binary_bilimit_of_is_colimit _ $ binary_cofan.is_colimit.mk _ /-- Every split epi `f` with a kernel induces a binary bicone with `f` as its `snd` and the kernel map as its `inl`. -We will show in `binary_bicone_of_split_mono_of_cokernel` that this binary bicone is in fact +We will show in `binary_bicone_of_is_split_mono_of_cokernel` that this binary bicone is in fact already a biproduct. -/ @[simps] -def binary_bicone_of_split_epi_of_kernel {X Y : C} {f : X ⟶ Y} [split_epi f] +def binary_bicone_of_is_split_epi_of_kernel {X Y : C} {f : X ⟶ Y} [is_split_epi f] {c : kernel_fork f} (i : is_limit c) : binary_bicone c.X Y := { X := X, fst := @@ -1949,15 +1955,15 @@ def binary_bicone_of_split_epi_of_kernel {X Y : C} {f : X ⟶ Y} [split_epi f] letI := mono_of_is_limit_fork i, apply zero_of_comp_mono c.ι, simp only [comp_sub, category.comp_id, category.assoc, sub_self, fork.is_limit.lift_ι, - fork.ι_of_ι, split_epi.id_assoc] + fork.ι_of_ι, is_split_epi.id_assoc] end, inr_snd' := by simp } -/-- The bicone constructed in `binary_bicone_of_split_epi_of_kernel` is a bilimit. +/-- The bicone constructed in `binary_bicone_of_is_split_epi_of_kernel` is a bilimit. This is a version of the splitting lemma that holds in all preadditive categories. -/ -def is_bilimit_binary_bicone_of_split_epi_of_kernel {X Y : C} {f : X ⟶ Y} [split_epi f] +def is_bilimit_binary_bicone_of_is_split_epi_of_kernel {X Y : C} {f : X ⟶ Y} [is_split_epi f] {c : kernel_fork f} (i : is_limit c) : - (binary_bicone_of_split_epi_of_kernel i).is_bilimit := + (binary_bicone_of_is_split_epi_of_kernel i).is_bilimit := binary_bicone.is_bilimit_of_kernel_inl _ $ i.of_iso_limit $ fork.ext (iso.refl _) (by simp) end diff --git a/formal/lean/mathlib/category_theory/limits/shapes/equalizers.lean b/formal/lean/mathlib/category_theory/limits/shapes/equalizers.lean index 1126c479f388793e5ecdbba8c9a4640ef722542d..0e534c02717730d50af3e80868a7615c2af6c73e 100644 --- a/formal/lean/mathlib/category_theory/limits/shapes/equalizers.lean +++ b/formal/lean/mathlib/category_theory/limits/shapes/equalizers.lean @@ -911,22 +911,23 @@ lemma has_coequalizers_of_has_colimit_parallel_pair section -- In this section we show that a split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`. -variables {C} [split_mono f] +variables {C} [is_split_mono f] /-- A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`. -Here we build the cone, and show in `split_mono_equalizes` that it is a limit cone. +Here we build the cone, and show in `is_split_mono_equalizes` that it is a limit cone. -/ @[simps {rhs_md := semireducible}] -def cone_of_split_mono : fork (𝟙 Y) (retraction f ≫ f) := +def cone_of_is_split_mono : fork (𝟙 Y) (retraction f ≫ f) := fork.of_ι f (by simp) -@[simp] lemma cone_of_split_mono_ι : (cone_of_split_mono f).ι = f := rfl +@[simp] lemma cone_of_is_split_mono_ι : (cone_of_is_split_mono f).ι = f := rfl /-- A split mono `f` equalizes `(retraction f ≫ f)` and `(𝟙 Y)`. -/ -def split_mono_equalizes {X Y : C} (f : X ⟶ Y) [split_mono f] : is_limit (cone_of_split_mono f) := +def is_split_mono_equalizes {X Y : C} (f : X ⟶ Y) [is_split_mono f] : + is_limit (cone_of_is_split_mono f) := fork.is_limit.mk' _ $ λ s, ⟨s.ι ≫ retraction f, by { dsimp, rw [category.assoc, ←s.condition], apply category.comp_id }, @@ -934,7 +935,7 @@ fork.is_limit.mk' _ $ λ s, end -/-- We show that the converse to `split_mono_equalizes` is true: +/-- We show that the converse to `is_split_mono_equalizes` is true: Whenever `f` equalizes `(r ≫ f)` and `(𝟙 Y)`, then `r` is a retraction of `f`. -/ def split_mono_of_equalizer {X Y : C} {f : X ⟶ Y} {r : Y ⟶ X} (hr : f ≫ r ≫ f = f) (h : is_limit (fork.of_ι f (hr.trans (category.comp_id _).symm : f ≫ r ≫ f = f ≫ 𝟙 Y))) : @@ -979,23 +980,23 @@ split_mono_of_idempotent_of_is_limit_fork _ hf (limit.is_limit _) section -- In this section we show that a split epi `f` coequalizes `(f ≫ section_ f)` and `(𝟙 X)`. -variables {C} [split_epi f] +variables {C} [is_split_epi f] /-- A split epi `f` coequalizes `(f ≫ section_ f)` and `(𝟙 X)`. -Here we build the cocone, and show in `split_epi_coequalizes` that it is a colimit cocone. +Here we build the cocone, and show in `is_split_epi_coequalizes` that it is a colimit cocone. -/ @[simps {rhs_md := semireducible}] -def cocone_of_split_epi : cofork (𝟙 X) (f ≫ section_ f) := +def cocone_of_is_split_epi : cofork (𝟙 X) (f ≫ section_ f) := cofork.of_π f (by simp) -@[simp] lemma cocone_of_split_epi_π : (cocone_of_split_epi f).π = f := rfl +@[simp] lemma cocone_of_is_split_epi_π : (cocone_of_is_split_epi f).π = f := rfl /-- A split epi `f` coequalizes `(f ≫ section_ f)` and `(𝟙 X)`. -/ -def split_epi_coequalizes {X Y : C} (f : X ⟶ Y) [split_epi f] : - is_colimit (cocone_of_split_epi f) := +def is_split_epi_coequalizes {X Y : C} (f : X ⟶ Y) [is_split_epi f] : + is_colimit (cocone_of_is_split_epi f) := cofork.is_colimit.mk' _ $ λ s, ⟨section_ f ≫ s.π, by { dsimp, rw [← category.assoc, ← s.condition, category.id_comp] }, @@ -1003,7 +1004,7 @@ cofork.is_colimit.mk' _ $ λ s, end -/-- We show that the converse to `split_epi_equalizes` is true: +/-- We show that the converse to `is_split_epi_equalizes` is true: Whenever `f` coequalizes `(f ≫ s)` and `(𝟙 X)`, then `s` is a section of `f`. -/ def split_epi_of_coequalizer {X Y : C} {f : X ⟶ Y} {s : Y ⟶ X} (hs : f ≫ s ≫ f = f) (h : is_colimit (cofork.of_π f ((category.assoc _ _ _).trans $ diff --git a/formal/lean/mathlib/category_theory/limits/shapes/equivalence.lean b/formal/lean/mathlib/category_theory/limits/shapes/equivalence.lean new file mode 100644 index 0000000000000000000000000000000000000000..8adbd7bec47738268060f99cd677bda332016b99 --- /dev/null +++ b/formal/lean/mathlib/category_theory/limits/shapes/equivalence.lean @@ -0,0 +1,37 @@ +/- +Copyright (c) Markus Himmel. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Markus Himmel +-/ +import category_theory.adjunction.limits +import category_theory.limits.shapes.terminal + +/-! +# Transporting existence of specific limits across equivalences + +For now, we only treat the case of initial and terminal objects, but other special shapes can be +added in the future. +-/ + +open category_theory category_theory.limits + +namespace category_theory +universes v₁ v₂ u₁ u₂ +variables {C : Type u₁} [category.{v₁} C] {D : Type u₂} [category.{v₂} D] + +lemma has_initial_of_equivalence (e : D ⥤ C) [is_equivalence e] [has_initial C] : has_initial D := +adjunction.has_colimits_of_shape_of_equivalence e + +lemma equivalence.has_initial_iff (e : C ≌ D) : has_initial C ↔ has_initial D := +⟨λ h, by exactI has_initial_of_equivalence e.inverse, + λ h, by exactI has_initial_of_equivalence e.functor⟩ + +lemma has_terminal_of_equivalence (e : D ⥤ C) [is_equivalence e] [has_terminal C] : + has_terminal D := +adjunction.has_limits_of_shape_of_equivalence e + +lemma equivalence.has_terminal_iff (e : C ≌ D) : has_terminal C ↔ has_terminal D := +⟨λ h, by exactI has_terminal_of_equivalence e.inverse, + λ h, by exactI has_terminal_of_equivalence e.functor⟩ + +end category_theory diff --git a/formal/lean/mathlib/category_theory/limits/shapes/images.lean b/formal/lean/mathlib/category_theory/limits/shapes/images.lean index df25feb3385a46536c95ac524c9ec290a552a44b..1457e0257f1e31eba8dce8f18e0c52601579d64b 100644 --- a/formal/lean/mathlib/category_theory/limits/shapes/images.lean +++ b/formal/lean/mathlib/category_theory/limits/shapes/images.lean @@ -706,8 +706,8 @@ instance strong_epi_mono_factorisation_inhabited {X Y : C} (f : X ⟶ Y) [strong property of the image. -/ def strong_epi_mono_factorisation.to_mono_is_image {X Y : C} {f : X ⟶ Y} (F : strong_epi_mono_factorisation f) : is_image F.to_mono_factorisation := -{ lift := λ G, arrow.lift $ arrow.hom_mk' $ - show G.e ≫ G.m = F.e ≫ F.m, by rw [F.to_mono_factorisation.fac, G.fac] } +{ lift := λ G, (comm_sq.mk (show G.e ≫ G.m = F.e ≫ F.m, + by rw [F.to_mono_factorisation.fac, G.fac])).lift, } variable (C) @@ -776,22 +776,24 @@ variables [has_images C] instance has_image_maps_of_has_strong_epi_images [has_strong_epi_images C] : has_image_maps C := { has_image_map := λ f g st, has_image_map.mk - { map := arrow.lift $ arrow.hom_mk' $ show (st.left ≫ factor_thru_image g.hom) ≫ image.ι g.hom = - factor_thru_image f.hom ≫ (image.ι f.hom ≫ st.right), by simp } } + { map := (comm_sq.mk (show (st.left ≫ factor_thru_image g.hom) ≫ image.ι g.hom = + factor_thru_image f.hom ≫ (image.ι f.hom ≫ st.right), by simp)).lift, }, } /-- If a category has images, equalizers and pullbacks, then images are automatically strong epi images. -/ @[priority 100] instance has_strong_epi_images_of_has_pullbacks_of_has_equalizers [has_pullbacks C] [has_equalizers C] : has_strong_epi_images C := -{ strong_factor_thru_image := λ X Y f, - { epi := by apply_instance, - has_lift := λ A B x y h h_mono w, arrow.has_lift.mk - { lift := image.lift - { I := pullback h y, - m := pullback.snd ≫ image.ι f, - m_mono := by exactI mono_comp _ _, - e := pullback.lift _ _ w } ≫ pullback.fst } } } +{ strong_factor_thru_image := λ X Y f, strong_epi.mk' + (λ A B h h_mono x y sq, comm_sq.has_lift.mk' + { l := image.lift + { I := pullback h y, + m := pullback.snd ≫ image.ι f, + m_mono := by exactI mono_comp _ _, + e := pullback.lift _ _ sq.w } ≫ pullback.fst, + fac_left' := by simp only [image.fac_lift_assoc, pullback.lift_fst], + fac_right' := by { ext, simp only [sq.w, category.assoc, + image.fac_lift_assoc, pullback.lift_fst_assoc], }, }) } end has_strong_epi_images diff --git a/formal/lean/mathlib/category_theory/limits/shapes/multiequalizer.lean b/formal/lean/mathlib/category_theory/limits/shapes/multiequalizer.lean index 7292893ebb255d366bb2bb02b86adecdec33b600..5cc5edaf285dbfbc1b7d38249f8085ed6d5fd3bf 100644 --- a/formal/lean/mathlib/category_theory/limits/shapes/multiequalizer.lean +++ b/formal/lean/mathlib/category_theory/limits/shapes/multiequalizer.lean @@ -112,7 +112,7 @@ instance : small_category (walking_multispan fst snd) := end walking_multispan /-- This is a structure encapsulating the data necessary to define a `multicospan`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure multicospan_index (C : Type u) [category.{v} C] := (L R : Type w) (fst_to snd_to : R → L) @@ -122,7 +122,7 @@ structure multicospan_index (C : Type u) [category.{v} C] := (snd : Π b, left (snd_to b) ⟶ right b) /-- This is a structure encapsulating the data necessary to define a `multispan`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure multispan_index (C : Type u) [category.{v} C] := (L R : Type w) (fst_from snd_from : L → R) @@ -252,11 +252,11 @@ end multispan_index variables {C : Type u} [category.{v} C] /-- A multifork is a cone over a multicospan. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] abbreviation multifork (I : multicospan_index C) := cone I.multicospan /-- A multicofork is a cocone over a multispan. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] abbreviation multicofork (I : multispan_index C) := cocone I.multispan namespace multifork diff --git a/formal/lean/mathlib/category_theory/limits/shapes/regular_mono.lean b/formal/lean/mathlib/category_theory/limits/shapes/regular_mono.lean index 5713672ddc2bcb0852f54fc24f3487edf56b2ea3..1efd3c2609519e5d59ca3e8913d731eb98b829f8 100644 --- a/formal/lean/mathlib/category_theory/limits/shapes/regular_mono.lean +++ b/formal/lean/mathlib/category_theory/limits/shapes/regular_mono.lean @@ -13,7 +13,7 @@ import category_theory.limits.shapes.equalizers A regular monomorphism is a morphism that is the equalizer of some parallel pair. We give the constructions -* `split_mono → regular_mono` and +* `is_split_mono → regular_mono` and * `regular_mono → mono` as well as the dual constructions for regular epimorphisms. Additionally, we give the construction * `regular_epi ⟶ strong_epi`. @@ -59,12 +59,12 @@ instance equalizer_regular (g h : X ⟶ Y) [has_limit (parallel_pair g h)] : /-- Every split monomorphism is a regular monomorphism. -/ @[priority 100] -instance regular_mono.of_split_mono (f : X ⟶ Y) [split_mono f] : regular_mono f := +instance regular_mono.of_is_split_mono (f : X ⟶ Y) [is_split_mono f] : regular_mono f := { Z := Y, left := 𝟙 Y, right := retraction f ≫ f, w := by tidy, - is_limit := split_mono_equalizes f } + is_limit := is_split_mono_equalizes f } /-- If `f` is a regular mono, then any map `k : W ⟶ Y` equalizing `regular_mono.left` and `regular_mono.right` induces a morphism `l : W ⟶ X` such that `l ≫ f = k`. -/ @@ -118,17 +118,15 @@ regular_of_is_pullback_snd_of_regular comm.symm (pullback_cone.flip_is_limit t) @[priority 100] instance strong_mono_of_regular_mono (f : X ⟶ Y) [regular_mono f] : strong_mono f := -{ mono := by apply_instance, - has_lift := - begin - introsI, - have : v ≫ (regular_mono.left : Y ⟶ regular_mono.Z f) = v ≫ regular_mono.right, - { apply (cancel_epi z).1, - simp only [regular_mono.w, ← reassoc_of h] }, - obtain ⟨t, ht⟩ := regular_mono.lift' _ _ this, - refine arrow.has_lift.mk ⟨t, (cancel_mono f).1 _, ht⟩, - simp only [arrow.mk_hom, arrow.hom_mk'_left, category.assoc, ht, h] - end } +strong_mono.mk' begin + introsI A B z hz u v sq, + have : v ≫ (regular_mono.left : Y ⟶ regular_mono.Z f) = v ≫ regular_mono.right, + { apply (cancel_epi z).1, + simp only [regular_mono.w, ← reassoc_of sq.w] }, + obtain ⟨t, ht⟩ := regular_mono.lift' _ _ this, + refine comm_sq.has_lift.mk' ⟨t, (cancel_mono f).1 _, ht⟩, + simp only [arrow.mk_hom, arrow.hom_mk'_left, category.assoc, ht, sq.w], +end /-- A regular monomorphism is an isomorphism if it is an epimorphism. -/ lemma is_iso_of_regular_mono_of_epi (f : X ⟶ Y) [regular_mono f] [e : epi f] : is_iso f := @@ -152,7 +150,7 @@ regular_mono_category.regular_mono_of_mono _ instance regular_mono_category_of_split_mono_category [split_mono_category C] : regular_mono_category C := { regular_mono_of_mono := λ _ _ f _, - by { haveI := by exactI split_mono_of_mono f, apply_instance } } + by { haveI := by exactI is_split_mono_of_mono f, apply_instance } } @[priority 100] instance strong_mono_category_of_regular_mono_category [regular_mono_category C] : @@ -185,12 +183,12 @@ instance coequalizer_regular (g h : X ⟶ Y) [has_colimit (parallel_pair g h)] : /-- Every split epimorphism is a regular epimorphism. -/ @[priority 100] -instance regular_epi.of_split_epi (f : X ⟶ Y) [split_epi f] : regular_epi f := +instance regular_epi.of_split_epi (f : X ⟶ Y) [is_split_epi f] : regular_epi f := { W := X, left := 𝟙 X, right := f ≫ section_ f, w := by tidy, - is_colimit := split_epi_coequalizes f } + is_colimit := is_split_epi_coequalizes f } /-- If `f` is a regular epi, then every morphism `k : X ⟶ W` coequalizing `regular_epi.left` and `regular_epi.right` induces `l : Y ⟶ W` such that `f ≫ l = k`. -/ @@ -244,17 +242,15 @@ regular_of_is_pushout_snd_of_regular comm.symm (pushout_cocone.flip_is_colimit t @[priority 100] instance strong_epi_of_regular_epi (f : X ⟶ Y) [regular_epi f] : strong_epi f := -{ epi := by apply_instance, - has_lift := - begin - introsI, - have : (regular_epi.left : regular_epi.W f ⟶ X) ≫ u = regular_epi.right ≫ u, - { apply (cancel_mono z).1, - simp only [category.assoc, h, regular_epi.w_assoc] }, - obtain ⟨t, ht⟩ := regular_epi.desc' f u this, - exact arrow.has_lift.mk ⟨t, ht, (cancel_epi f).1 - (by simp only [←category.assoc, ht, ←h, arrow.mk_hom, arrow.hom_mk'_right])⟩, - end } +strong_epi.mk' begin + introsI A B z hz u v sq, + have : (regular_epi.left : regular_epi.W f ⟶ X) ≫ u = regular_epi.right ≫ u, + { apply (cancel_mono z).1, + simp only [category.assoc, sq.w, regular_epi.w_assoc] }, + obtain ⟨t, ht⟩ := regular_epi.desc' f u this, + exact comm_sq.has_lift.mk' ⟨t, ht, (cancel_epi f).1 + (by simp only [←category.assoc, ht, ←sq.w, arrow.mk_hom, arrow.hom_mk'_right])⟩, +end /-- A regular epimorphism is an isomorphism if it is a monomorphism. -/ lemma is_iso_of_regular_epi_of_mono (f : X ⟶ Y) [regular_epi f] [m : mono f] : is_iso f := @@ -277,7 +273,7 @@ regular_epi_category.regular_epi_of_epi _ @[priority 100] instance regular_epi_category_of_split_epi_category [split_epi_category C] : regular_epi_category C := -{ regular_epi_of_epi := λ _ _ f _, by { haveI := by exactI split_epi_of_epi f, apply_instance } } +{ regular_epi_of_epi := λ _ _ f _, by { haveI := by exactI is_split_epi_of_epi f, apply_instance } } @[priority 100] instance strong_epi_category_of_regular_epi_category [regular_epi_category C] : diff --git a/formal/lean/mathlib/category_theory/limits/shapes/strong_epi.lean b/formal/lean/mathlib/category_theory/limits/shapes/strong_epi.lean index 81e03fdfddf7a50e9e54bffa76d43cc91fc65424..90efac5332580a04b43e56c89ca4e5d036ec4002 100644 --- a/formal/lean/mathlib/category_theory/limits/shapes/strong_epi.lean +++ b/formal/lean/mathlib/category_theory/limits/shapes/strong_epi.lean @@ -3,16 +3,16 @@ Copyright (c) 2020 Markus Himmel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Markus Himmel -/ -import category_theory.arrow import category_theory.balanced +import category_theory.lifting_properties.basic /-! # Strong epimorphisms -In this file, we define strong epimorphisms. A strong epimorphism is an epimorphism `f`, such -that for every commutative square with `f` at the top and a monomorphism at the bottom, there is -a diagonal morphism making the two triangles commute. This lift is necessarily unique (as shown in -`comma.lean`). +In this file, we define strong epimorphisms. A strong epimorphism is an epimorphism `f` +which has the (unique) left lifting property with respect to monomorphisms. Similarly, +a strong monomorphisms in a monomorphism which has the (unique) right lifting property +with respect to epimorphisms. ## Main results @@ -40,22 +40,32 @@ variables {C : Type u} [category.{v} C] variables {P Q : C} -/-- A strong epimorphism `f` is an epimorphism such that every commutative square with `f` at the - top and a monomorphism at the bottom has a lift. -/ +/-- A strong epimorphism `f` is an epimorphism which has the left lifting property +with respect to monomorphisms. -/ class strong_epi (f : P ⟶ Q) : Prop := (epi : epi f) -(has_lift : Π {X Y : C} {u : P ⟶ X} {v : Q ⟶ Y} {z : X ⟶ Y} [mono z] (h : u ≫ z = f ≫ v), - arrow.has_lift $ arrow.hom_mk' h) +(llp : ∀ ⦃X Y : C⦄ (z : X ⟶ Y) [mono z], has_lifting_property f z) -/-- A strong monomorphism `f` is a monomorphism such that every commutative square with `f` at the - bottom and an epimorphism at the top has a lift. -/ +lemma strong_epi.mk' {f : P ⟶ Q} [epi f] + (hf : ∀ (X Y : C) (z : X ⟶ Y) (hz : mono z) (u : P ⟶ X) (v : Q ⟶ Y) + (sq : comm_sq u f z v), sq.has_lift) : strong_epi f := +{ epi := infer_instance, + llp := λ X Y z hz, ⟨λ u v sq, hf X Y z hz u v sq⟩, } + +/-- A strong monomorphism `f` is a monomorphism which has the right lifting property +with respect to epimorphisms. -/ class strong_mono (f : P ⟶ Q) : Prop := (mono : mono f) -(has_lift : Π {X Y : C} {u : X ⟶ P} {v : Y ⟶ Q} {z : X ⟶ Y} [epi z] (h : u ≫ f = z ≫ v), - arrow.has_lift $ arrow.hom_mk' h) +(rlp : ∀ ⦃X Y : C⦄ (z : X ⟶ Y) [epi z], has_lifting_property z f) + +lemma strong_mono.mk' {f : P ⟶ Q} [mono f] + (hf : ∀ (X Y : C) (z : X ⟶ Y) (hz : epi z) (u : X ⟶ P) (v : Y ⟶ Q) + (sq : comm_sq u z f v), sq.has_lift) : strong_mono f := +{ mono := infer_instance, + rlp := λ X Y z hz, ⟨λ u v sq, hf X Y z hz u v sq⟩, } -attribute [instance] strong_epi.has_lift -attribute [instance] strong_mono.has_lift +attribute [instance, priority 100] strong_epi.llp +attribute [instance, priority 100] strong_mono.rlp @[priority 100] instance epi_of_strong_epi (f : P ⟶ Q) [strong_epi f] : epi f := strong_epi.epi @@ -69,69 +79,55 @@ variables {R : C} (f : P ⟶ Q) (g : Q ⟶ R) /-- The composition of two strong epimorphisms is a strong epimorphism. -/ lemma strong_epi_comp [strong_epi f] [strong_epi g] : strong_epi (f ≫ g) := { epi := epi_comp _ _, - has_lift := - begin - introsI, - have h₀ : u ≫ z = f ≫ g ≫ v, by simpa [category.assoc] using h, - let w : Q ⟶ X := arrow.lift (arrow.hom_mk' h₀), - have h₁ : w ≫ z = g ≫ v, by rw arrow.lift_mk'_right, - exact arrow.has_lift.mk ⟨(arrow.lift (arrow.hom_mk' h₁) : R ⟶ X), by simp, by simp⟩ - end } + llp := by { introsI, apply_instance, }, } /-- The composition of two strong monomorphisms is a strong monomorphism. -/ lemma strong_mono_comp [strong_mono f] [strong_mono g] : strong_mono (f ≫ g) := { mono := mono_comp _ _, - has_lift := - begin - introsI, - have h₀ : (u ≫ f) ≫ g = z ≫ v, by simpa [category.assoc] using h, - let w : Y ⟶ Q := arrow.lift (arrow.hom_mk' h₀), - have h₁ : u ≫ f = z ≫ w, by rw arrow.lift_mk'_left, - exact arrow.has_lift.mk ⟨(arrow.lift (arrow.hom_mk' h₁) : Y ⟶ P), by simp, by simp⟩ - end } + rlp := by { introsI, apply_instance, }, } /-- If `f ≫ g` is a strong epimorphism, then so is `g`. -/ lemma strong_epi_of_strong_epi [strong_epi (f ≫ g)] : strong_epi g := { epi := epi_of_epi f g, - has_lift := - begin + llp := begin introsI, - have h₀ : (f ≫ u) ≫ z = (f ≫ g) ≫ v, by simp only [category.assoc, h], - exact arrow.has_lift.mk - ⟨(arrow.lift (arrow.hom_mk' h₀) : R ⟶ X), (cancel_mono z).1 (by simp [h]), by simp⟩, - end } + constructor, + intros u v sq, + have h₀ : (f ≫ u) ≫ z = (f ≫ g) ≫ v, by simp only [category.assoc, sq.w], + exact comm_sq.has_lift.mk' ⟨(comm_sq.mk h₀).lift, + by simp only [← cancel_mono z, category.assoc, comm_sq.fac_right, sq.w], by simp⟩, + end, } /-- If `f ≫ g` is a strong monomorphism, then so is `f`. -/ lemma strong_mono_of_strong_mono [strong_mono (f ≫ g)] : strong_mono f := { mono := mono_of_mono f g, - has_lift := - begin + rlp := begin introsI, - have h₀ : u ≫ f ≫ g = z ≫ v ≫ g, by rw reassoc_of h, - exact arrow.has_lift.mk - ⟨(arrow.lift (arrow.hom_mk' h₀) : Y ⟶ P), by simp, (cancel_epi z).1 (by simp [h])⟩ - end } + constructor, + intros u v sq, + have h₀ : u ≫ f ≫ g = z ≫ v ≫ g, by rw reassoc_of sq.w, + exact comm_sq.has_lift.mk' ⟨(comm_sq.mk h₀).lift, by simp, by simp [← cancel_epi z, sq.w]⟩, + end, } /-- An isomorphism is in particular a strong epimorphism. -/ @[priority 100] instance strong_epi_of_is_iso [is_iso f] : strong_epi f := { epi := by apply_instance, - has_lift := λ X Y u v z _ h, arrow.has_lift.mk ⟨inv f ≫ u, by simp, by simp [h]⟩ } + llp := λ X Y z hz, has_lifting_property.of_left_iso _ _, } /-- An isomorphism is in particular a strong monomorphism. -/ @[priority 100] instance strong_mono_of_is_iso [is_iso f] : strong_mono f := { mono := by apply_instance, - has_lift := λ X Y u v z _ h, arrow.has_lift.mk - ⟨v ≫ inv f, by simp [← category.assoc, ← h], by simp⟩ } + rlp := λ X Y z hz, has_lifting_property.of_right_iso _ _, } end /-- A strong epimorphism that is a monomorphism is an isomorphism. -/ lemma is_iso_of_mono_of_strong_epi (f : P ⟶ Q) [mono f] [strong_epi f] : is_iso f := -⟨⟨arrow.lift $ arrow.hom_mk' $ show 𝟙 P ≫ f = f ≫ 𝟙 Q, by simp, by tidy⟩⟩ +⟨⟨(comm_sq.mk (show 𝟙 P ≫ f = f ≫ 𝟙 Q, by simp)).lift, by tidy⟩⟩ /-- A strong monomorphism that is an epimorphism is an isomorphism. -/ lemma is_iso_of_epi_of_strong_mono (f : P ⟶ Q) [epi f] [strong_mono f] : is_iso f := -⟨⟨arrow.lift $ arrow.hom_mk' $ show 𝟙 P ≫ f = f ≫ 𝟙 Q, by simp, by tidy⟩⟩ +⟨⟨(comm_sq.mk (show 𝟙 P ≫ f = f ≫ 𝟙 Q, by simp)).lift, by tidy⟩⟩ section variables (C) diff --git a/formal/lean/mathlib/category_theory/limits/shapes/terminal.lean b/formal/lean/mathlib/category_theory/limits/shapes/terminal.lean index 2498ca63c71921bfe0e7b821dd9f542f61f94c35..3c8cd3fa89c1dcbd09e682a3391e2b185dacddfd 100644 --- a/formal/lean/mathlib/category_theory/limits/shapes/terminal.lean +++ b/formal/lean/mathlib/category_theory/limits/shapes/terminal.lean @@ -116,20 +116,20 @@ t.hom_ext _ _ t.hom_ext _ _ /-- Any morphism from a terminal object is split mono. -/ -def is_terminal.split_mono_from {X Y : C} (t : is_terminal X) (f : X ⟶ Y) : split_mono f := -⟨t.from _, t.hom_ext _ _⟩ +lemma is_terminal.is_split_mono_from {X Y : C} (t : is_terminal X) (f : X ⟶ Y) : + is_split_mono f := is_split_mono.mk' ⟨t.from _, t.hom_ext _ _⟩ /-- Any morphism to an initial object is split epi. -/ -def is_initial.split_epi_to {X Y : C} (t : is_initial X) (f : Y ⟶ X) : split_epi f := -⟨t.to _, t.hom_ext _ _⟩ +lemma is_initial.is_split_epi_to {X Y : C} (t : is_initial X) (f : Y ⟶ X) : + is_split_epi f := is_split_epi.mk' ⟨t.to _, t.hom_ext _ _⟩ /-- Any morphism from a terminal object is mono. -/ lemma is_terminal.mono_from {X Y : C} (t : is_terminal X) (f : X ⟶ Y) : mono f := -by haveI := t.split_mono_from f; apply_instance +by haveI := t.is_split_mono_from f; apply_instance /-- Any morphism to an initial object is epi. -/ lemma is_initial.epi_to {X Y : C} (t : is_initial X) (f : Y ⟶ X) : epi f := -by haveI := t.split_epi_to f; apply_instance +by haveI := t.is_split_epi_to f; apply_instance /-- If `T` and `T'` are terminal, they are isomorphic. -/ @[simps] @@ -235,11 +235,18 @@ and showing there is a unique morphism to it from any other object. -/ lemma has_terminal_of_unique (X : C) [h : Π Y : C, unique (Y ⟶ X)] : has_terminal C := { has_limit := λ F, has_limit.mk ⟨_, (is_terminal_equiv_unique F X).inv_fun h⟩ } +lemma is_terminal.has_terminal {X : C} (h : is_terminal X) : has_terminal C := +{ has_limit := λ F, has_limit.mk ⟨⟨X, by tidy⟩, is_limit_change_empty_cone _ h _ (iso.refl _)⟩ } + /-- We can more explicitly show that a category has an initial object by specifying the object, and showing there is a unique morphism from it to any other object. -/ lemma has_initial_of_unique (X : C) [h : Π Y : C, unique (X ⟶ Y)] : has_initial C := { has_colimit := λ F, has_colimit.mk ⟨_, (is_initial_equiv_unique F X).inv_fun h⟩ } +lemma is_initial.has_initial {X : C} (h : is_initial X) : has_initial C := +{ has_colimit := λ F, has_colimit.mk + ⟨⟨X, by tidy⟩, is_colimit_change_empty_cocone _ h _ (iso.refl _)⟩ } + /-- The map from an object to the terminal object. -/ abbreviation terminal.from [has_terminal C] (P : C) : P ⟶ ⊤_ C := limit.lift (functor.empty C) (as_empty_cone P) @@ -277,12 +284,12 @@ initial_is_initial.unique_up_to_iso t terminal_is_terminal.unique_up_to_iso t /-- Any morphism from a terminal object is split mono. -/ -instance terminal.split_mono_from {Y : C} [has_terminal C] (f : ⊤_ C ⟶ Y) : split_mono f := -is_terminal.split_mono_from terminal_is_terminal _ +instance terminal.is_split_mono_from {Y : C} [has_terminal C] (f : ⊤_ C ⟶ Y) : is_split_mono f := +is_terminal.is_split_mono_from terminal_is_terminal _ /-- Any morphism to an initial object is split epi. -/ -instance initial.split_epi_to {Y : C} [has_initial C] (f : Y ⟶ ⊥_ C) : split_epi f := -is_initial.split_epi_to initial_is_initial _ +instance initial.is_split_epi_to {Y : C} [has_initial C] (f : Y ⟶ ⊥_ C) : is_split_epi f := +is_initial.is_split_epi_to initial_is_initial _ /-- An initial object is terminal in the opposite category. -/ def terminal_op_of_initial {X : C} (t : is_initial X) : is_terminal (opposite.op X) := @@ -304,6 +311,18 @@ def initial_unop_of_terminal {X : Cᵒᵖ} (t : is_terminal X) : is_initial X.un { desc := λ s, (t.from (opposite.op s.X)).unop, uniq' := λ s m w, quiver.hom.op_inj (t.hom_ext _ _) } +instance has_initial_op_of_has_terminal [has_terminal C] : has_initial Cᵒᵖ := +(initial_op_of_terminal terminal_is_terminal).has_initial + +instance has_terminal_op_of_has_initial [has_initial C] : has_terminal Cᵒᵖ := +(terminal_op_of_initial initial_is_initial).has_terminal + +lemma has_terminal_of_has_initial_op [has_initial Cᵒᵖ] : has_terminal C := +(terminal_unop_of_initial initial_is_initial).has_terminal + +lemma has_initial_of_has_terminal_op [has_terminal Cᵒᵖ] : has_initial C := +(initial_unop_of_terminal terminal_is_terminal).has_initial + instance {J : Type*} [category J] {C : Type*} [category C] [has_terminal C] : has_limit ((category_theory.functor.const J).obj (⊤_ C)) := has_limit.mk diff --git a/formal/lean/mathlib/category_theory/limits/shapes/types.lean b/formal/lean/mathlib/category_theory/limits/shapes/types.lean index 480590a1516415fc9b9bf9bbd52ca6739c855525..1fb3086376ac7c03654368aacf810655790d53b5 100644 --- a/formal/lean/mathlib/category_theory/limits/shapes/types.lean +++ b/formal/lean/mathlib/category_theory/limits/shapes/types.lean @@ -388,7 +388,7 @@ variables (f : X ⟶ Z) (g : Y ⟶ Z) The usual explicit pullback in the category of types, as a subtype of the product. The full `limit_cone` data is bundled as `pullback_limit_cone f g`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] abbreviation pullback_obj : Type u := { p : X × Y // f p.1 = g p.2 } -- `pullback_obj f g` comes with a coercion to the product type `X × Y`. diff --git a/formal/lean/mathlib/category_theory/limits/shapes/zero_morphisms.lean b/formal/lean/mathlib/category_theory/limits/shapes/zero_morphisms.lean index e74ed7ee776edea6ba372d914975d4e3e3197f9a..00d1e6267f0edeff59679cf94551f4c7f0f39cb3 100644 --- a/formal/lean/mathlib/category_theory/limits/shapes/zero_morphisms.lean +++ b/formal/lean/mathlib/category_theory/limits/shapes/zero_morphisms.lean @@ -160,20 +160,20 @@ by { unfreezingI { subst h, }, apply of_mono_zero X Y, } lemma of_epi_eq_zero {X Y : C} (f : X ⟶ Y) [epi f] (h : f = 0) : is_zero Y := by { unfreezingI { subst h, }, apply of_epi_zero X Y, } -lemma iff_split_mono_eq_zero {X Y : C} (f : X ⟶ Y) [split_mono f] : is_zero X ↔ f = 0 := +lemma iff_is_split_mono_eq_zero {X Y : C} (f : X ⟶ Y) [is_split_mono f] : is_zero X ↔ f = 0 := begin rw iff_id_eq_zero, split, { intro h, rw [←category.id_comp f, h, zero_comp], }, - { intro h, rw [←split_mono.id f], simp [h], }, + { intro h, rw [←is_split_mono.id f], simp [h], }, end -lemma iff_split_epi_eq_zero {X Y : C} (f : X ⟶ Y) [split_epi f] : is_zero Y ↔ f = 0 := +lemma iff_is_split_epi_eq_zero {X Y : C} (f : X ⟶ Y) [is_split_epi f] : is_zero Y ↔ f = 0 := begin rw iff_id_eq_zero, split, { intro h, rw [←category.comp_id f, h, comp_zero], }, - { intro h, rw [←split_epi.id f], simp [h], }, + { intro h, rw [←is_split_epi.id f], simp [h], }, end lemma of_mono {X Y : C} (f : X ⟶ Y) [mono f] (i : is_zero Y) : is_zero X := @@ -519,31 +519,31 @@ by { rw image.eq_fac h, simp } end image /-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/ -instance split_mono_sigma_ι {β : Type u'} [has_zero_morphisms C] (f : β → C) - [has_colimit (discrete.functor f)] (b : β) : split_mono (sigma.ι f b) := +instance is_split_mono_sigma_ι {β : Type u'} [has_zero_morphisms C] (f : β → C) + [has_colimit (discrete.functor f)] (b : β) : is_split_mono (sigma.ι f b) := is_split_mono.mk' { retraction := sigma.desc $ pi.single b (𝟙 _) } /-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/ -instance split_epi_pi_π {β : Type u'} [has_zero_morphisms C] (f : β → C) - [has_limit (discrete.functor f)] (b : β) : split_epi (pi.π f b) := +instance is_split_epi_pi_π {β : Type u'} [has_zero_morphisms C] (f : β → C) + [has_limit (discrete.functor f)] (b : β) : is_split_epi (pi.π f b) := is_split_epi.mk' { section_ := pi.lift $ pi.single b (𝟙 _) } /-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/ -instance split_mono_coprod_inl [has_zero_morphisms C] {X Y : C} [has_colimit (pair X Y)] : - split_mono (coprod.inl : X ⟶ X ⨿ Y) := +instance is_split_mono_coprod_inl [has_zero_morphisms C] {X Y : C} [has_colimit (pair X Y)] : + is_split_mono (coprod.inl : X ⟶ X ⨿ Y) := is_split_mono.mk' { retraction := coprod.desc (𝟙 X) 0, } /-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/ -instance split_mono_coprod_inr [has_zero_morphisms C] {X Y : C} [has_colimit (pair X Y)] : - split_mono (coprod.inr : Y ⟶ X ⨿ Y) := +instance is_split_mono_coprod_inr [has_zero_morphisms C] {X Y : C} [has_colimit (pair X Y)] : + is_split_mono (coprod.inr : Y ⟶ X ⨿ Y) := is_split_mono.mk' { retraction := coprod.desc 0 (𝟙 Y), } /-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/ -instance split_epi_prod_fst [has_zero_morphisms C] {X Y : C} [has_limit (pair X Y)] : - split_epi (prod.fst : X ⨯ Y ⟶ X) := +instance is_split_epi_prod_fst [has_zero_morphisms C] {X Y : C} [has_limit (pair X Y)] : + is_split_epi (prod.fst : X ⨯ Y ⟶ X) := is_split_epi.mk' { section_ := prod.lift (𝟙 X) 0, } /-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/ -instance split_epi_prod_snd [has_zero_morphisms C] {X Y : C} [has_limit (pair X Y)] : - split_epi (prod.snd : X ⨯ Y ⟶ Y) := +instance is_split_epi_prod_snd [has_zero_morphisms C] {X Y : C} [has_limit (pair X Y)] : + is_split_epi (prod.snd : X ⨯ Y ⟶ Y) := is_split_epi.mk' { section_ := prod.lift 0 (𝟙 Y), } end category_theory.limits diff --git a/formal/lean/mathlib/category_theory/limits/types.lean b/formal/lean/mathlib/category_theory/limits/types.lean index adf4696e1c695c09e790ed458f4d3179ca442825..f02267b171972abcb31ec0581d97b6ae03359e7d 100644 --- a/formal/lean/mathlib/category_theory/limits/types.lean +++ b/formal/lean/mathlib/category_theory/limits/types.lean @@ -194,7 +194,7 @@ A quotient type implementing the colimit of a functor `F : J ⥤ Type u`, as pairs `⟨j, x⟩` where `x : F.obj j`, modulo the equivalence relation generated by `⟨j, x⟩ ~ ⟨j', x'⟩` whenever there is a morphism `f : j ⟶ j'` so `F.map f x = x'`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def quot (F : J ⥤ Type (max v u)) : Type (max v u) := @quot (Σ j, F.obj j) (quot.rel F) diff --git a/formal/lean/mathlib/category_theory/localization/construction.lean b/formal/lean/mathlib/category_theory/localization/construction.lean index 112084a4bf505cc71a1c34ddcc35449f38fa0a45..7e86810b73ce34a3f2114ed67ad1ad0a1f513a41 100644 --- a/formal/lean/mathlib/category_theory/localization/construction.lean +++ b/formal/lean/mathlib/category_theory/localization/construction.lean @@ -75,7 +75,7 @@ namespace construction /-- If `W : morphism_property C`, `loc_quiver W` is a quiver with the same objects as `C`, and whose morphisms are those in `C` and placeholders for formal inverses of the morphisms in `W`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure loc_quiver (W : morphism_property C) := (obj : C) instance : quiver (loc_quiver W) := @@ -115,7 +115,7 @@ open localization.construction /-- The localized category obtained by formally inverting the morphisms in `W : morphism_property C` -/ -@[derive category, nolint has_inhabited_instance] +@[derive category, nolint has_nonempty_instance] def localization := category_theory.quotient (localization.construction.relations W) /-- The obvious functor `C ⥤ W.localization` -/ @@ -131,6 +131,8 @@ namespace localization namespace construction +variable {W} + /-- The isomorphism in `W.localization` associated to a morphism `w` in W -/ def Wiso {X Y : C} (w : X ⟶ Y) (hw : W w) : iso (W.Q.obj X) (W.Q.obj Y) := { hom := W.Q.map w, @@ -138,6 +140,9 @@ def Wiso {X Y : C} (w : X ⟶ Y) (hw : W w) : iso (W.Q.obj X) (W.Q.obj Y) := hom_inv_id' := quotient.sound _ (relations.Winv₁ w hw), inv_hom_id' := quotient.sound _ (relations.Winv₂ w hw), } +/-- The formal inverse in `W.localization` of a morphism `w` in `W`. -/ +abbreviation Winv {X Y : C} (w : X ⟶ Y) (hw : W w) := (Wiso w hw).inv + end construction end localization @@ -199,7 +204,7 @@ begin apply functor.congr_obj h, }, { rintro ⟨X⟩ ⟨Y⟩ (f|⟨w, hw⟩), { simpa only using functor.congr_hom h f, }, - { have hw : W.Q.map w = (Wiso W w hw).hom := rfl, + { have hw : W.Q.map w = (Wiso w hw).hom := rfl, have hw' := functor.congr_hom h w, simp only [functor.comp_map, hw] at hw', refine functor.congr_inv_of_congr_hom _ _ _ _ _ hw', @@ -207,6 +212,109 @@ begin { apply functor.congr_obj h, }, }, }, }, end +variable (W) + +/-- The canonical bijection between objects in a category and its +localization with respect to a morphism_property `W` -/ +@[simps] +def obj_equiv : C ≃ W.localization := +{ to_fun := W.Q.obj, + inv_fun := λ X, X.as.obj, + left_inv := λ X, rfl, + right_inv := by { rintro ⟨⟨X⟩⟩, refl, }, } + +variable {W} + +/-- A `morphism_property` in `W.localization` is satisfied by all +morphisms in the localized category if it contains the image of the +morphisms in the original category, the inverses of the morphisms +in `W` and if it is stable under composition -/ +lemma morphism_property_is_top + (P : morphism_property W.localization) + (hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)) + (hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (Winv w hw)) + (hP₃ : P.stable_under_composition) : P = ⊤ := +begin + ext X Y f, + split, + { intro hf, + simp only [pi.top_apply], }, + { intro hf, clear hf, + let G : _ ⥤ W.localization := quotient.functor _, + suffices : ∀ (X₁ X₂ : C) (p : localization.construction.ι_paths W X₁ ⟶ + localization.construction.ι_paths W X₂), P (G.map p), + { rcases X with ⟨⟨X⟩⟩, + rcases Y with ⟨⟨Y⟩⟩, + simpa only [functor.image_preimage] using this _ _ (G.preimage f), }, + intros X₁ X₂ p, + induction p with X₂ X₃ p g hp, + { simpa only [functor.map_id] using hP₁ (𝟙 X₁), }, + { cases X₂, + cases X₃, + let p' : ι_paths W X₁ ⟶ ι_paths W X₂ := p, + rw [show p.cons g = p' ≫ quiver.hom.to_path g, by refl, G.map_comp], + refine hP₃ _ _ hp _, + rcases g with (g | ⟨g, hg⟩), + { apply hP₁, }, + { apply hP₂, }, }, }, +end + +/-- A `morphism_property` in `W.localization` is satisfied by all +morphisms in the localized category if it contains the image of the +morphisms in the original category, if is stable under composition +and if the property is stable by passing to inverses. -/ +lemma morphism_property_is_top' + (P : morphism_property W.localization) + (hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)) + (hP₂ : ∀ ⦃X Y : W.localization⦄ (e : X ≅ Y) (he : P e.hom), P e.inv) + (hP₃ : P.stable_under_composition) : P = ⊤ := +morphism_property_is_top P hP₁ (λ X Y w hw, hP₂ _ (by exact hP₁ w)) hP₃ + +namespace nat_trans_extension + +variables {F₁ F₂ : W.localization ⥤ D} (τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂) +include τ + +/-- If `F₁` and `F₂` are functors `W.localization ⥤ D` and if we have +`τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂`, we shall define a natural transformation `F₁ ⟶ F₂`. +This is the `app` field of this natural transformation. -/ +def app (X : W.localization) : F₁.obj X ⟶ F₂.obj X := +eq_to_hom (congr_arg F₁.obj ((obj_equiv W).right_inv X).symm) ≫ + τ.app ((obj_equiv W).inv_fun X) ≫ eq_to_hom (congr_arg F₂.obj ((obj_equiv W).right_inv X)) + +@[simp] +lemma app_eq (X : C) : (app τ) (W.Q.obj X) = τ.app X := +by simpa only [app, eq_to_hom_refl, comp_id, id_comp] + +end nat_trans_extension + +/-- If `F₁` and `F₂` are functors `W.localization ⥤ D`, a natural transformation `F₁ ⟶ F₂` +can be obtained from a natural transformation `W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂`. -/ +@[simps] +def nat_trans_extension {F₁ F₂ : W.localization ⥤ D} (τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂) : + F₁ ⟶ F₂ := +{ app := nat_trans_extension.app τ, + naturality' := λ X Y f, begin + have h := morphism_property_is_top' + (morphism_property.naturality_property (nat_trans_extension.app τ)) _ + (morphism_property.naturality_property.is_stable_under_inverse _) + (morphism_property.naturality_property.is_stable_under_composition _), swap, + { intros X Y f, + simpa only [morphism_property.naturality_property, nat_trans_extension.app_eq] + using τ.naturality f, }, + have hf : (⊤ : morphism_property _) f := by simp only [pi.top_apply], + simpa only [← h] using hf, + end, } + +@[simp] +lemma nat_trans_extension_hcomp {F G : W.localization ⥤ D} (τ : W.Q ⋙ F ⟶ W.Q ⋙ G) : + (𝟙 W.Q) ◫ nat_trans_extension τ = τ := +begin + ext X, + simp only [nat_trans.hcomp_app, nat_trans.id_app, G.map_id, comp_id, + nat_trans_extension_app, nat_trans_extension.app_eq], +end + end construction end localization diff --git a/formal/lean/mathlib/category_theory/monad/algebra.lean b/formal/lean/mathlib/category_theory/monad/algebra.lean index 0c62df9578e4e56cc3a7776d57f3566b358e1287..198f5f83f93e2422db0bde1c4217bfa1610ee61e 100644 --- a/formal/lean/mathlib/category_theory/monad/algebra.lean +++ b/formal/lean/mathlib/category_theory/monad/algebra.lean @@ -241,7 +241,7 @@ end monad namespace comonad /-- An Eilenberg-Moore coalgebra for a comonad `T`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure coalgebra (G : comonad C) : Type (max u₁ v₁) := (A : C) (a : A ⟶ (G : C ⥤ C).obj A) @@ -256,7 +256,7 @@ namespace coalgebra variables {G : comonad C} /-- A morphism of Eilenberg-Moore coalgebras for the comonad `G`. -/ -@[ext, nolint has_inhabited_instance] structure hom (A B : coalgebra G) := +@[ext, nolint has_nonempty_instance] structure hom (A B : coalgebra G) := (f : A.A ⟶ B.A) (h' : A.a ≫ (G : C ⥤ C).map f = f ≫ B.a . obviously) diff --git a/formal/lean/mathlib/category_theory/monoidal/center.lean b/formal/lean/mathlib/category_theory/monoidal/center.lean index 4ffea5d7850a467a2d69e10183b1c87b72eb433e..4c6eb069a90e36581f7262c0b68c053179f9072d 100644 --- a/formal/lean/mathlib/category_theory/monoidal/center.lean +++ b/formal/lean/mathlib/category_theory/monoidal/center.lean @@ -47,7 +47,7 @@ Thinking of `C` as a 2-category with a single `0`-morphism, these are the same a transformations (in the pseudo- sense) of the identity 2-functor on `C`, which send the unique `0`-morphism to `X`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure half_braiding (X : C) := (β : Π U, X ⊗ U ≅ U ⊗ X) (monoidal' : ∀ U U', (β (U ⊗ U')).hom = @@ -65,7 +65,7 @@ variables (C) The Drinfeld center of a monoidal category `C` has as objects pairs `⟨X, b⟩`, where `X : C` and `b` is a half-braiding on `X`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def center := Σ X : C, half_braiding X namespace center @@ -73,7 +73,7 @@ namespace center variables {C} /-- A morphism in the Drinfeld center of `C`. -/ -@[ext, nolint has_inhabited_instance] +@[ext, nolint has_nonempty_instance] structure hom (X Y : center C) := (f : X.1 ⟶ Y.1) (comm' : ∀ U, (f ⊗ 𝟙 U) ≫ (Y.2.β U).hom = (X.2.β U).hom ≫ (𝟙 U ⊗ f) . obviously) diff --git a/formal/lean/mathlib/category_theory/monoidal/free/basic.lean b/formal/lean/mathlib/category_theory/monoidal/free/basic.lean index 4b94068049a12fc74f99f37366bded2bc189e463..dd133c7f02b1baca83f6dea5a473a56f6a67038a 100644 --- a/formal/lean/mathlib/category_theory/monoidal/free/basic.lean +++ b/formal/lean/mathlib/category_theory/monoidal/free/basic.lean @@ -52,7 +52,7 @@ namespace free_monoidal_category /-- Formal compositions and tensor products of identities, unitors and associators. The morphisms of the free monoidal category are obtained as a quotient of these formal morphisms by the relations defining a monoidal category. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] inductive hom : F C → F C → Type u | id (X) : hom X X | α_hom (X Y Z : F C) : hom ((X.tensor Y).tensor Z) (X.tensor (Y.tensor Z)) diff --git a/formal/lean/mathlib/category_theory/monoidal/free/coherence.lean b/formal/lean/mathlib/category_theory/monoidal/free/coherence.lean index 2d28aae09bc78bcb084c2b5d4581a05f2c67a301..a1314d9cb820a5727a50dc03f6275060a0a6ec97 100644 --- a/formal/lean/mathlib/category_theory/monoidal/free/coherence.lean +++ b/formal/lean/mathlib/category_theory/monoidal/free/coherence.lean @@ -49,7 +49,7 @@ variables (C) /-- We say an object in the free monoidal category is in normal form if it is of the form `(((𝟙_ C) ⊗ X₁) ⊗ X₂) ⊗ ⋯`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] inductive normal_monoidal_object : Type u | unit : normal_monoidal_object | tensor : normal_monoidal_object → C → normal_monoidal_object diff --git a/formal/lean/mathlib/category_theory/monoidal/internal/Module.lean b/formal/lean/mathlib/category_theory/monoidal/internal/Module.lean index 0b5fc52a0a1e33a143eeca5ae1fbb6125d81bfde..e71d858040d5ebddf137194aa331389eb2846e24 100644 --- a/formal/lean/mathlib/category_theory/monoidal/internal/Module.lean +++ b/formal/lean/mathlib/category_theory/monoidal/internal/Module.lean @@ -91,7 +91,7 @@ Converting a bundled algebra to a monoid object in `Module R`. def inverse_obj (A : Algebra.{u} R) : Mon_ (Module.{u} R) := { X := Module.of R A, one := algebra.linear_map R A, - mul := @algebra.lmul' R A _ _ _, + mul := linear_map.mul' R A, one_mul' := begin ext x, @@ -99,7 +99,7 @@ def inverse_obj (A : Algebra.{u} R) : Mon_ (Module.{u} R) := algebra.linear_map_apply, linear_map.compr₂_apply, function.comp_app, ring_hom.map_one, Module.monoidal_category.hom_apply, Algebra.coe_comp, Module.monoidal_category.left_unitor_hom_apply], - rw [algebra.lmul'_apply, monoidal_category.left_unitor_hom_apply, ← algebra.smul_def] + rw [linear_map.mul'_apply, monoidal_category.left_unitor_hom_apply, ← algebra.smul_def] end, mul_one' := begin @@ -107,7 +107,7 @@ def inverse_obj (A : Algebra.{u} R) : Mon_ (Module.{u} R) := dsimp only [Algebra.id_apply, tensor_product.mk_apply, algebra.linear_map_apply, linear_map.compr₂_apply, function.comp_app, Module.monoidal_category.hom_apply, Algebra.coe_comp], - rw [algebra.lmul'_apply, Module.monoidal_category.right_unitor_hom_apply, + rw [linear_map.mul'_apply, Module.monoidal_category.right_unitor_hom_apply, ← algebra.commutes, ← algebra.smul_def] end, mul_assoc' := @@ -116,7 +116,7 @@ def inverse_obj (A : Algebra.{u} R) : Mon_ (Module.{u} R) := dsimp only [Algebra.id_apply, tensor_product.mk_apply, linear_map.compr₂_apply, function.comp_app, Module.monoidal_category.hom_apply, Algebra.coe_comp, monoidal_category.associator_hom_apply], - simp only [algebra.lmul'_apply, mul_assoc] + simp only [linear_map.mul'_apply, mul_assoc] end } /-- @@ -130,7 +130,7 @@ def inverse : Algebra.{u} R ⥤ Mon_ (Module.{u} R) := one_hom' := by { ext, dsimp, simp only [ring_hom.map_one, alg_hom.map_one] }, mul_hom' := - by { ext, dsimp, simp only [algebra.lmul'_apply, ring_hom.map_mul, alg_hom.map_mul] } } }. + by { ext, dsimp, simp only [linear_map.mul'_apply, ring_hom.map_mul, alg_hom.map_mul] } } }. end Mon_Module_equivalence_Algebra @@ -147,10 +147,10 @@ def Mon_Module_equivalence_Algebra : Mon_ (Module.{u} R) ≌ Algebra R := (λ A, { hom := { hom := { to_fun := id, map_add' := λ x y, rfl, map_smul' := λ r a, rfl, }, mul_hom' := by { ext, dsimp at *, - simp only [algebra.lmul'_apply, Mon_.X.ring_mul] } }, + simp only [linear_map.mul'_apply, Mon_.X.ring_mul] } }, inv := { hom := { to_fun := id, map_add' := λ x y, rfl, map_smul' := λ r a, rfl, }, mul_hom' := by { ext, dsimp at *, - simp only [algebra.lmul'_apply, Mon_.X.ring_mul]} } }) + simp only [linear_map.mul'_apply, Mon_.X.ring_mul]} } }) (by tidy), counit_iso := nat_iso.of_components (λ A, { hom := @@ -158,14 +158,14 @@ def Mon_Module_equivalence_Algebra : Mon_ (Module.{u} R) ≌ Algebra R := map_zero' := rfl, map_add' := λ x y, rfl, map_one' := (algebra_map R A).map_one, - map_mul' := λ x y, algebra.lmul'_apply, + map_mul' := λ x y, linear_map.mul'_apply, commutes' := λ r, rfl, }, inv := { to_fun := id, map_zero' := rfl, map_add' := λ x y, rfl, map_one' := (algebra_map R A).map_one.symm, - map_mul' := λ x y, algebra.lmul'_apply.symm, + map_mul' := λ x y, (@linear_map.mul'_apply R _ _ _ _ _ _ x y).symm, commutes' := λ r, rfl } }) (by { intros, refl }), }. /-- diff --git a/formal/lean/mathlib/category_theory/monoidal/of_chosen_finite_products.lean b/formal/lean/mathlib/category_theory/monoidal/of_chosen_finite_products.lean index 9e822085b515c5ed07a6cb7792461d5fb8a235e0..9191730e2f0575e738335aeeeebc9dc37bc765ec 100644 --- a/formal/lean/mathlib/category_theory/monoidal/of_chosen_finite_products.lean +++ b/formal/lean/mathlib/category_theory/monoidal/of_chosen_finite_products.lean @@ -321,7 +321,7 @@ a fixed choice of limit data for the empty functor, and for `pair X Y` for every This is an implementation detail for `symmetric_of_chosen_finite_products`. -/ -@[derive category, nolint unused_arguments has_inhabited_instance] +@[derive category, nolint unused_arguments has_nonempty_instance] def monoidal_of_chosen_finite_products_synonym (𝒯 : limit_cone (functor.empty.{v} C)) (ℬ : Π (X Y : C), limit_cone (pair X Y)):= C diff --git a/formal/lean/mathlib/category_theory/monoidal/opposite.lean b/formal/lean/mathlib/category_theory/monoidal/opposite.lean index 28f98b1404a0dcf9c0987067ad4fec41949ab0a0..fcb952d416984d0f37b1d3099b6d56a2144614a2 100644 --- a/formal/lean/mathlib/category_theory/monoidal/opposite.lean +++ b/formal/lean/mathlib/category_theory/monoidal/opposite.lean @@ -21,7 +21,7 @@ namespace category_theory open category_theory.monoidal_category /-- A type synonym for the monoidal opposite. Use the notation `Cᴹᵒᵖ`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def monoidal_opposite (C : Type u₁) := C namespace monoidal_opposite diff --git a/formal/lean/mathlib/category_theory/monoidal/rigid/basic.lean b/formal/lean/mathlib/category_theory/monoidal/rigid/basic.lean index bb43e26be3124907ba6b60058fa2ece84256667f..be984afca94973f730014c5f348cf77b0196237e 100644 --- a/formal/lean/mathlib/category_theory/monoidal/rigid/basic.lean +++ b/formal/lean/mathlib/category_theory/monoidal/rigid/basic.lean @@ -372,8 +372,19 @@ adjunction.mk_of_hom_equiv hom_equiv_naturality_right' := λ X Z Z' f g, tensor_right_hom_equiv_naturality f g, } +/-- +If `Y` has a left dual `ᘁY`, then it is a closed object, with the internal hom functor `Y ⟶[C] -` +given by left tensoring by `ᘁY`. +This has to be a definition rather than an instance to avoid diamonds, for example between +`category_theory.monoidal_closed.functor_closed` and +`category_theory.monoidal.functor_has_left_dual`. Moreover, in concrete applications there is often +a more useful definition of the internal hom object than `ᘁY ⊗ X`, in which case the closed +structure shouldn't come from `has_left_dual` (e.g. in the category `FinVect k`, it is more +convenient to define the internal hom as `Y →ₗ[k] X` rather than `ᘁY ⊗ X` even though these are +naturally isomorphic). +-/ @[priority 100] -instance closed_of_has_left_dual (Y : C) [has_left_dual Y] : closed Y := +def closed_of_has_left_dual (Y : C) [has_left_dual Y] : closed Y := { is_adj := ⟨_, tensor_left_adjunction (ᘁY) Y⟩, } /-- `tensor_left_hom_equiv` commutes with tensoring on the right -/ @@ -618,11 +629,19 @@ class left_rigid_category (C : Type u) [category.{v} C] [monoidal_category.{v} C attribute [instance, priority 100] right_rigid_category.right_dual attribute [instance, priority 100] left_rigid_category.left_dual +/-- Any left rigid category is monoidal closed, with the internal hom `X ⟶[C] Y = ᘁX ⊗ Y`. +This has to be a definition rather than an instance to avoid diamonds, for example between +`category_theory.monoidal_closed.functor_category` and +`category_theory.monoidal.left_rigid_functor_category`. Moreover, in concrete applications there is +often a more useful definition of the internal hom object than `ᘁY ⊗ X`, in which case the monoidal +closed structure shouldn't come the rigid structure (e.g. in the category `FinVect k`, it is more +convenient to define the internal hom as `Y →ₗ[k] X` rather than `ᘁY ⊗ X` even though these are +naturally isomorphic). -/ @[priority 100] -instance monoidal_closed_of_left_rigid_category +def monoidal_closed_of_left_rigid_category (C : Type u) [category.{v} C] [monoidal_category.{v} C] [left_rigid_category C] : monoidal_closed C := -{ closed' := λ X, by apply_instance, } +{ closed' := λ X, closed_of_has_left_dual X, } /-- A rigid monoidal category is a monoidal category which is left rigid and right rigid. -/ class rigid_category (C : Type u) [category.{v} C] [monoidal_category.{v} C] diff --git a/formal/lean/mathlib/category_theory/monoidal/subcategory.lean b/formal/lean/mathlib/category_theory/monoidal/subcategory.lean index 8fef005cad002f024b4dfb79a9a64ba9b840a9b7..5bf28be48ca431b2ebe64391033ede18d2743649 100644 --- a/formal/lean/mathlib/category_theory/monoidal/subcategory.lean +++ b/formal/lean/mathlib/category_theory/monoidal/subcategory.lean @@ -5,6 +5,7 @@ Authors: Antoine Labelle -/ import category_theory.monoidal.braided import category_theory.concrete_category.basic +import category_theory.closed.monoidal /-! # Full monoidal subcategories @@ -33,7 +34,7 @@ variables {C : Type u} [category.{v} C] [monoidal_category C] (P : C → Prop) /-- A property `C → Prop` is a monoidal predicate if it is closed under `𝟙_` and `⊗`. -/ -class monoidal_predicate := +class monoidal_predicate : Prop := (prop_id' : P (𝟙_ C) . obviously) (prop_tensor' : ∀ {X Y}, P X → P Y → P (X ⊗ Y) . obviously) @@ -45,11 +46,11 @@ open monoidal_predicate variables [monoidal_predicate P] /-- -When `P` is a monoidal predicate, the full subcategory `{X : C // P X}` inherits the monoidal -structure of `C` +When `P` is a monoidal predicate, the full subcategory for `P` inherits the monoidal structure of + `C`. -/ -instance full_monoidal_subcategory : monoidal_category {X : C // P X} := -{ tensor_obj := λ X Y, ⟨X ⊗ Y, prop_tensor X.2 Y.2⟩, +instance full_monoidal_subcategory : monoidal_category (full_subcategory P) := +{ tensor_obj := λ X Y, ⟨X.1 ⊗ Y.1, prop_tensor X.2 Y.2⟩, tensor_hom := λ X₁ Y₁ X₂ Y₂ f g, by { change X₁.1 ⊗ X₂.1 ⟶ Y₁.1 ⊗ Y₂.1, change X₁.1 ⟶ Y₁.1 at f, change X₂.1 ⟶ Y₂.1 at g, exact f ⊗ g }, tensor_unit := ⟨𝟙_ C, prop_id⟩, @@ -71,7 +72,7 @@ The forgetful monoidal functor from a full monoidal subcategory into the origina ("forgetting" the condition). -/ @[simps] -def full_monoidal_subcategory_inclusion : monoidal_functor {X : C // P X} C := +def full_monoidal_subcategory_inclusion : monoidal_functor (full_subcategory P) C := { to_functor := full_subcategory_inclusion P, ε := 𝟙 _, μ := λ X Y, 𝟙 _ } @@ -87,7 +88,7 @@ variables {P} {P' : C → Prop} [monoidal_predicate P'] subcategories. -/ @[simps] def full_monoidal_subcategory.map (h : ∀ ⦃X⦄, P X → P' X) : - monoidal_functor {X : C // P X} {X : C // P' X} := + monoidal_functor (full_subcategory P) (full_subcategory P') := { to_functor := full_subcategory.map h, ε := 𝟙 _, μ := λ X Y, 𝟙 _ } @@ -102,9 +103,9 @@ section braided variables (P) [braided_category C] /-- -The braided structure on `{X : C // P X}` inherited by the braided structure on `C`. +The braided structure on a full subcategory inherited by the braided structure on `C`. -/ -instance full_braided_subcategory : braided_category {X : C // P X} := +instance full_braided_subcategory : braided_category (full_subcategory P) := braided_category_of_faithful (full_monoidal_subcategory_inclusion P) (λ X Y, ⟨(β_ X.1 Y.1).hom, (β_ X.1 Y.1).inv, (β_ X.1 Y.1).hom_inv_id, (β_ X.1 Y.1).inv_hom_id⟩) (λ X Y, by tidy) @@ -114,7 +115,7 @@ The forgetful braided functor from a full braided subcategory into the original ("forgetting" the condition). -/ @[simps] -def full_braided_subcategory_inclusion : braided_functor {X : C // P X} C := +def full_braided_subcategory_inclusion : braided_functor (full_subcategory P) C := { to_monoidal_functor := full_monoidal_subcategory_inclusion P, braided' := λ X Y, by { rw [is_iso.eq_inv_comp], tidy } } @@ -129,7 +130,7 @@ variables {P} subcategories. -/ @[simps] def full_braided_subcategory.map (h : ∀ ⦃X⦄, P X → P' X) : - braided_functor {X : C // P X} {X : C // P' X} := + braided_functor (full_subcategory P) (full_subcategory P') := { to_monoidal_functor := full_monoidal_subcategory.map h, braided' := λ X Y, by { rw [is_iso.eq_inv_comp], tidy } } @@ -144,11 +145,49 @@ section symmetric variables (P) [symmetric_category C] -instance full_symmetric_subcategory : symmetric_category {X : C // P X} := +instance full_symmetric_subcategory : symmetric_category (full_subcategory P) := symmetric_category_of_faithful (full_braided_subcategory_inclusion P) end symmetric +section closed + +variables (P) [monoidal_closed C] + +/-- +A property `C → Prop` is a closed predicate if it is closed under taking internal homs +-/ +class closed_predicate : Prop := +(prop_ihom' : ∀ {X Y}, P X → P Y → P ((ihom X).obj Y) . obviously) + +restate_axiom closed_predicate.prop_ihom' + +open closed_predicate + +variable [closed_predicate P] + +instance full_monoidal_closed_subcategory : monoidal_closed (full_subcategory P) := +{ closed' := λ X, + { is_adj := + { right := full_subcategory.lift P (full_subcategory_inclusion P ⋙ (ihom X.1)) + (λ Y, prop_ihom X.2 Y.2), + adj := adjunction.mk_of_unit_counit + { unit := { app := λ Y, (ihom.coev X.1).app Y.1, + naturality' := λ Y Z f, ihom.coev_naturality X.1 f }, + counit := { app := λ Y, (ihom.ev X.1).app Y.1, + naturality' := λ Y Z f, ihom.ev_naturality X.1 f }, + left_triangle' := by { ext Y, simp, exact ihom.ev_coev X.1 Y.1 }, + right_triangle' := by { ext Y, simp, exact ihom.coev_ev X.1 Y.1 } } } } } + +@[simp] lemma full_monoidal_closed_subcategory_ihom_obj (X Y : full_subcategory P) : + ((ihom X).obj Y).obj = (ihom (X.obj)).obj Y.obj := rfl + +@[simp] lemma full_monoidal_closed_subcategory_ihom_map (X : full_subcategory P) + {Y Z : full_subcategory P} + (f : Y ⟶ Z) : (ihom X).map f = (ihom (X.obj)).map f := rfl + +end closed + end monoidal_category end category_theory diff --git a/formal/lean/mathlib/category_theory/morphism_property.lean b/formal/lean/mathlib/category_theory/morphism_property.lean index 102616acd5aeb592b4b3488854448730b258c804..e05ab6edd756bea1bf933af0f2530809b7da542b 100644 --- a/formal/lean/mathlib/category_theory/morphism_property.lean +++ b/formal/lean/mathlib/category_theory/morphism_property.lean @@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Andrew Yang -/ import category_theory.limits.shapes.pullbacks +import category_theory.arrow /-! # Properties of morphisms @@ -48,7 +49,12 @@ still falls in the class. -/ def stable_under_composition (P : morphism_property C) : Prop := ∀ ⦃X Y Z⦄ (f : X ⟶ Y) (g : Y ⟶ Z), P f → P g → P (f ≫ g) -/-- A morphism property is `stable_under_composition` if the base change of such a morphism +/-- A morphism property is `stable_under_inverse` if the inverse of a morphism satisfying +the property still falls in the class. -/ +def stable_under_inverse (P : morphism_property C) : Prop := +∀ ⦃X Y⦄ (e : X ≅ Y), P e.hom → P e.inv + +/-- A morphism property is `stable_under_base_change` if the base change of such a morphism still falls in the class. -/ def stable_under_base_change [has_pullbacks C] (P : morphism_property C) : Prop := ∀ ⦃X Y S : C⦄ (f : X ⟶ S) (g : Y ⟶ S), P g → P (pullback.fst : pullback f g ⟶ X) @@ -67,6 +73,15 @@ lemma respects_iso.cancel_right_is_iso {P : morphism_property C} P (f ≫ g) ↔ P f := ⟨λ h, by simpa using hP.2 (as_iso g).symm (f ≫ g) h, hP.2 (as_iso g) f⟩ +lemma respects_iso.arrow_iso_iff {P : morphism_property C} + (hP : respects_iso P) {f g : arrow C} (e : f ≅ g) : P f.hom ↔ P g.hom := +by { rw [← arrow.inv_left_hom_right e.hom, hP.cancel_left_is_iso, hP.cancel_right_is_iso], refl } + +lemma respects_iso.arrow_mk_iso_iff {P : morphism_property C} + (hP : respects_iso P) {W X Y Z : C} {f : W ⟶ X} {g : Y ⟶ Z} (e : arrow.mk f ≅ arrow.mk g) : + P f ↔ P g := +hP.arrow_iso_iff e + -- This is here to mirror `stable_under_base_change.snd`. @[nolint unused_arguments] lemma stable_under_base_change.fst [has_pullbacks C] {P : morphism_property C} @@ -127,6 +142,36 @@ to isomorphisms in `D`. -/ def is_inverted_by (P : morphism_property C) (F : C ⥤ D) : Prop := ∀ ⦃X Y : C⦄ (f : X ⟶ Y) (hf : P f), is_iso (F.map f) +/-- Given `app : Π X, F₁.obj X ⟶ F₂.obj X` where `F₁` and `F₂` are two functors, +this is the `morphism_property C` satisfied by the morphisms in `C` with respect +to whom `app` is natural. -/ +@[simp] +def naturality_property {F₁ F₂ : C ⥤ D} (app : Π X, F₁.obj X ⟶ F₂.obj X) : + morphism_property C := λ X Y f, F₁.map f ≫ app Y = app X ≫ F₂.map f + +namespace naturality_property + +lemma is_stable_under_composition {F₁ F₂ : C ⥤ D} (app : Π X, F₁.obj X ⟶ F₂.obj X) : + (naturality_property app).stable_under_composition := λ X Y Z f g hf hg, +begin + simp only [naturality_property] at ⊢ hf hg, + simp only [functor.map_comp, category.assoc, hg], + slice_lhs 1 2 { rw hf }, + rw category.assoc, +end + +lemma is_stable_under_inverse {F₁ F₂ : C ⥤ D} (app : Π X, F₁.obj X ⟶ F₂.obj X) : + (naturality_property app).stable_under_inverse := λ X Y e he, +begin + simp only [naturality_property] at ⊢ he, + rw ← cancel_epi (F₁.map e.hom), + slice_rhs 1 2 { rw he }, + simp only [category.assoc, ← F₁.map_comp_assoc, ← F₂.map_comp, + e.hom_inv_id, functor.map_id, category.id_comp, category.comp_id], +end + +end naturality_property + end morphism_property end category_theory diff --git a/formal/lean/mathlib/category_theory/preadditive/Mat.lean b/formal/lean/mathlib/category_theory/preadditive/Mat.lean index 6e0685e0c24c758fd8b26d55e3104c037ae258a8..e3a1f83a302988080890b495629b34cb95c4ad07 100644 --- a/formal/lean/mathlib/category_theory/preadditive/Mat.lean +++ b/formal/lean/mathlib/category_theory/preadditive/Mat.lean @@ -71,7 +71,7 @@ namespace Mat_ variables {C} /-- A morphism in `Mat_ C` is a dependently typed matrix of morphisms. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def hom (M N : Mat_ C) : Type v₁ := dmatrix M.ι N.ι (λ i j, M.X i ⟶ N.X j) namespace hom diff --git a/formal/lean/mathlib/category_theory/preadditive/additive_functor.lean b/formal/lean/mathlib/category_theory/preadditive/additive_functor.lean index 78590a36a8d41455fa01648936f649e7338dd812..d5119cad4094ff057cc0301956b96a452879ef2b 100644 --- a/formal/lean/mathlib/category_theory/preadditive/additive_functor.lean +++ b/formal/lean/mathlib/category_theory/preadditive/additive_functor.lean @@ -145,9 +145,9 @@ section variables (C D : Type*) [category C] [category D] [preadditive C] [preadditive D] /-- Bundled additive functors. -/ -@[derive category, nolint has_inhabited_instance] +@[derive category, nolint has_nonempty_instance] def AdditiveFunctor := -{ F : C ⥤ D // functor.additive F } +full_subcategory (λ (F : C ⥤ D), F.additive) infixr ` ⥤+ `:26 := AdditiveFunctor diff --git a/formal/lean/mathlib/category_theory/preadditive/injective.lean b/formal/lean/mathlib/category_theory/preadditive/injective.lean index 69c7841196cbc67afa56cca91dd5119f31844c16..93388285bfa9faebeb82e7fb04cd1f37dcd377ca 100644 --- a/formal/lean/mathlib/category_theory/preadditive/injective.lean +++ b/formal/lean/mathlib/category_theory/preadditive/injective.lean @@ -21,10 +21,10 @@ open category_theory open category_theory.limits open opposite -universes v u +universes v v₁ v₂ u₁ u₂ namespace category_theory -variables {C : Type u} [category.{v} C] +variables {C : Type u₁} [category.{v₁} C] /-- An object `J` is injective iff every morphism into `J` can be obtained by extending a monomorphism. @@ -37,7 +37,7 @@ section An injective presentation of an object `X` consists of a monomorphism `f : X ⟶ J` to some injective object `J`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure injective_presentation (X : C) := (J : C) (injective : injective J . tactic.apply_instance) @@ -84,7 +84,7 @@ lemma iso_iff {P Q : C} (i : P ≅ Q) : injective P ↔ injective Q := ⟨of_iso i, of_iso i.symm⟩ /-- The axiom of choice says that every nonempty type is an injective object in `Type`. -/ -instance (X : Type u) [nonempty X] : injective X := +instance (X : Type u₁) [nonempty X] : injective X := { factors := λ Y Z g f mono, ⟨λ z, by classical; exact if h : z ∈ set.range f @@ -98,7 +98,7 @@ instance (X : Type u) [nonempty X] : injective X := { exact false.elim (h ⟨y, rfl⟩) }, end⟩ } -instance Type.enough_injectives : enough_injectives (Type u) := +instance Type.enough_injectives : enough_injectives (Type u₁) := { presentation := λ X, nonempty.intro { J := with_bot X, injective := infer_instance, @@ -174,6 +174,18 @@ begin exact functor.preserves_epimorphisms.iso_iff (coyoneda.obj_op_op _) end +section adjunction +open category_theory.functor + +variables {D : Type u₂} [category.{v₂} D] +variables {L : C ⥤ D} {R : D ⥤ C} [preserves_monomorphisms L] + +lemma injective_of_adjoint (adj : L ⊣ R) (J : D) [injective J] : injective $ R.obj J := +⟨λ A A' g f im, by exactI ⟨adj.hom_equiv _ _ (factor_thru ((adj.hom_equiv A J).symm g) (L.map f)), + (adj.hom_equiv _ _).symm.injective (by simp)⟩⟩ + +end adjunction + section enough_injectives variable [enough_injectives C] diff --git a/formal/lean/mathlib/category_theory/preadditive/injective_resolution.lean b/formal/lean/mathlib/category_theory/preadditive/injective_resolution.lean index 1ef24b230c5a945f801afe8eaa56ec8f48b1149c..d002bb53a9b1e3333c83d63b158c8f7ac086aef6 100644 --- a/formal/lean/mathlib/category_theory/preadditive/injective_resolution.lean +++ b/formal/lean/mathlib/category_theory/preadditive/injective_resolution.lean @@ -49,7 +49,7 @@ you will not typically need to use this bundled object, and will instead use `injective_resolution Z` (all the components are equipped with `mono` instances, and when the category is `abelian` we will show `ι` is a quasi-iso). -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure InjectiveResolution (Z : C) := (cocomplex : cochain_complex C ℕ) (ι: ((cochain_complex.single₀ C).obj Z) ⟶ cocomplex) diff --git a/formal/lean/mathlib/category_theory/preadditive/projective.lean b/formal/lean/mathlib/category_theory/preadditive/projective.lean index aa8ecaf35dcc07446e16c0ad01465203925cad89..5fb7122a8b5224d761217f9b3c6ce2f510a777ff 100644 --- a/formal/lean/mathlib/category_theory/preadditive/projective.lean +++ b/formal/lean/mathlib/category_theory/preadditive/projective.lean @@ -45,7 +45,7 @@ section A projective presentation of an object `X` consists of an epimorphism `f : P ⟶ X` from some projective object `P`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure projective_presentation (X : C) := (P : C) (projective : projective P . tactic.apply_instance) diff --git a/formal/lean/mathlib/category_theory/preadditive/projective_resolution.lean b/formal/lean/mathlib/category_theory/preadditive/projective_resolution.lean index c61f01a3c8343b7b3638881571ba98e9bcf37d9d..d9f1d8c7636c5c587a07a57692f63a61f11838ae 100644 --- a/formal/lean/mathlib/category_theory/preadditive/projective_resolution.lean +++ b/formal/lean/mathlib/category_theory/preadditive/projective_resolution.lean @@ -66,7 +66,7 @@ you will not typically need to use this bundled object, and will instead use `(single C _ 0).obj Z` (all the components are equipped with `epi` instances, and when the category is `abelian` we will show `π` is a quasi-iso). -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure ProjectiveResolution (Z : C) := (complex : chain_complex C ℕ) (π : homological_complex.hom complex ((chain_complex.single₀ C).obj Z)) diff --git a/formal/lean/mathlib/category_theory/products/basic.lean b/formal/lean/mathlib/category_theory/products/basic.lean index 5bd7e8228e7e87344490c6f9a7531c6df94328c9..f73e539ed68aeadaef0db947a297057c42191d23 100644 --- a/formal/lean/mathlib/category_theory/products/basic.lean +++ b/formal/lean/mathlib/category_theory/products/basic.lean @@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Stephen Morgan, Scott Morrison -/ import category_theory.eq_to_hom +import category_theory.functor.const /-! # Cartesian products of categories @@ -168,6 +169,13 @@ as a functor `C × (C ⥤ D) ⥤ D`. category.assoc, nat_trans.naturality], end } +variables {C} + +/-- The constant functor followed by the evalutation functor is just the identity. -/ +@[simps] def functor.const_comp_evaluation_obj (X : C) : + functor.const C ⋙ (evaluation C D).obj X ≅ 𝟭 D := +nat_iso.of_components (λ Y, iso.refl _) (λ Y Z f, by simp) + end variables {A : Type u₁} [category.{v₁} A] diff --git a/formal/lean/mathlib/category_theory/shift.lean b/formal/lean/mathlib/category_theory/shift.lean index c61713108faab18d2d52aac4b9f96d65e030a4af..51d671b5b7f8c342c6899b3deba2130e6d856d82 100644 --- a/formal/lean/mathlib/category_theory/shift.lean +++ b/formal/lean/mathlib/category_theory/shift.lean @@ -77,7 +77,7 @@ class has_shift (C : Type u) (A : Type*) [category.{v} C] [add_monoid A] := (shift : monoidal_functor (discrete A) (C ⥤ C)) /-- A helper structure to construct the shift functor `(discrete A) ⥤ (C ⥤ C)`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure shift_mk_core := (F : A → (C ⥤ C)) (ε : 𝟭 C ≅ F 0) diff --git a/formal/lean/mathlib/category_theory/simple.lean b/formal/lean/mathlib/category_theory/simple.lean index 544a488f7055517fb0ac342b4e5a8fb0033896f1..c7f2433aeaccdc6f6357ada20d31d94cbe2a315f 100644 --- a/formal/lean/mathlib/category_theory/simple.lean +++ b/formal/lean/mathlib/category_theory/simple.lean @@ -190,8 +190,9 @@ begin rw [biprod.is_iso_inl_iff_id_eq_fst_comp_inl, ←biprod.total, add_right_eq_self], split, { intro h, replace h := h =≫ biprod.snd, - simpa [←is_zero.iff_split_epi_eq_zero (biprod.snd : X ⊞ Y ⟶ Y)] using h, }, - { intro h, rw is_zero.iff_split_epi_eq_zero (biprod.snd : X ⊞ Y ⟶ Y) at h, rw [h, zero_comp], }, + simpa [←is_zero.iff_is_split_epi_eq_zero (biprod.snd : X ⊞ Y ⟶ Y)] using h, }, + { intro h, rw is_zero.iff_is_split_epi_eq_zero (biprod.snd : X ⊞ Y ⟶ Y) at h, + rw [h, zero_comp], }, end /-- Any simple object in a preadditive category is indecomposable. -/ @@ -199,7 +200,7 @@ lemma indecomposable_of_simple (X : C) [simple X] : indecomposable X := ⟨simple.not_is_zero X, λ Y Z i, begin refine or_iff_not_imp_left.mpr (λ h, _), - rw is_zero.iff_split_mono_eq_zero (biprod.inl : Y ⟶ Y ⊞ Z) at h, + rw is_zero.iff_is_split_mono_eq_zero (biprod.inl : Y ⟶ Y ⊞ Z) at h, change biprod.inl ≠ 0 at h, rw ←(simple.mono_is_iso_iff_nonzero biprod.inl) at h, { rwa biprod.is_iso_inl_iff_is_zero at h, }, diff --git a/formal/lean/mathlib/category_theory/single_obj.lean b/formal/lean/mathlib/category_theory/single_obj.lean index 033642514465a991f0711a52a9b9821e6d2cc261..4e7c97bd837d88048e04de0c8215f5863c467b2b 100644 --- a/formal/lean/mathlib/category_theory/single_obj.lean +++ b/formal/lean/mathlib/category_theory/single_obj.lean @@ -38,7 +38,7 @@ universes u v w namespace category_theory /-- Type tag on `unit` used to define single-object categories and groupoids. -/ -@[nolint unused_arguments has_inhabited_instance] +@[nolint unused_arguments has_nonempty_instance] def single_obj (α : Type u) : Type := unit namespace single_obj diff --git a/formal/lean/mathlib/category_theory/sites/cover_lifting.lean b/formal/lean/mathlib/category_theory/sites/cover_lifting.lean index 6e3eb8f8408672d1f5a6febfb75f426f8b066922..deca91fbd087b9b556f6c586f04be7d0fe12eadf 100644 --- a/formal/lean/mathlib/category_theory/sites/cover_lifting.lean +++ b/formal/lean/mathlib/category_theory/sites/cover_lifting.lean @@ -60,7 +60,7 @@ variables {L : grothendieck_topology E} A functor `G : (C, J) ⥤ (D, K)` between sites is called to have the cover-lifting property if for all covering sieves `R` in `D`, `R.pullback G` is a covering sieve in `C`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure cover_lifting (G : C ⥤ D) : Prop := (cover_lift : ∀ {U : C} {S : sieve (G.obj U)} (hS : S ∈ K (G.obj U)), S.functor_pullback G ∈ J U) diff --git a/formal/lean/mathlib/category_theory/sites/cover_preserving.lean b/formal/lean/mathlib/category_theory/sites/cover_preserving.lean index 753a7a0ad6e21247a1eef2f63445e330173d3d72..c0d61ab09942777925274d0948a174fd218a609d 100644 --- a/formal/lean/mathlib/category_theory/sites/cover_preserving.lean +++ b/formal/lean/mathlib/category_theory/sites/cover_preserving.lean @@ -62,7 +62,7 @@ variables {L : grothendieck_topology A} A functor `G : (C, J) ⥤ (D, K)` between sites is *cover-preserving* if for all covering sieves `R` in `C`, `R.pushforward_functor G` is a covering sieve in `D`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure cover_preserving (G : C ⥤ D) : Prop := (cover_preserve : ∀ {U : C} {S : sieve U} (hS : S ∈ J U), S.functor_pushforward G ∈ K (G.obj U)) @@ -86,7 +86,7 @@ compatible family of elements at `C` and valued in `G.op ⋙ ℱ`, and each comm This is actually stronger than merely preserving compatible families because of the definition of `functor_pushforward` used. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure compatible_preserving (K : grothendieck_topology D) (G : C ⥤ D) : Prop := (compatible : ∀ (ℱ : SheafOfTypes.{w} K) {Z} {T : presieve Z} diff --git a/formal/lean/mathlib/category_theory/sites/dense_subsite.lean b/formal/lean/mathlib/category_theory/sites/dense_subsite.lean index d7498c315fffa972cab2b7e5e04b78a5246c147b..ceac0e5660ff710d6e69f85d97575bab9c4c0377 100644 --- a/formal/lean/mathlib/category_theory/sites/dense_subsite.lean +++ b/formal/lean/mathlib/category_theory/sites/dense_subsite.lean @@ -51,7 +51,7 @@ variables {L : grothendieck_topology E} /-- An auxiliary structure that witnesses the fact that `f` factors through an image object of `G`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure presieve.cover_by_image_structure (G : C ⥤ D) {V U : D} (f : V ⟶ U) := (obj : C) (lift : V ⟶ G.obj obj) diff --git a/formal/lean/mathlib/category_theory/sites/grothendieck.lean b/formal/lean/mathlib/category_theory/sites/grothendieck.lean index ebba468fb34db36f80a6ecfd233e78ab505f2c64..658de4af3c30f417ebfb18a0b08266b1ba362b18 100644 --- a/formal/lean/mathlib/category_theory/sites/grothendieck.lean +++ b/formal/lean/mathlib/category_theory/sites/grothendieck.lean @@ -406,14 +406,14 @@ instance : semilattice_inf (J.cover X) := instance : inhabited (J.cover X) := ⟨⊤⟩ /-- An auxiliary structure, used to define `S.index` in `plus.lean`. -/ -@[nolint has_inhabited_instance, ext] +@[nolint has_nonempty_instance, ext] structure arrow (S : J.cover X) := (Y : C) (f : Y ⟶ X) (hf : S f) /-- An auxiliary structure, used to define `S.index` in `plus.lean`. -/ -@[nolint has_inhabited_instance, ext] +@[nolint has_nonempty_instance, ext] structure relation (S : J.cover X) := (Y₁ Y₂ Z : C) (g₁ : Z ⟶ Y₁) diff --git a/formal/lean/mathlib/category_theory/sites/sheaf.lean b/formal/lean/mathlib/category_theory/sites/sheaf.lean index 060e5ad495dd1c672db002ca98181ae8ee88634f..6e8c10b4a65a22e9c2591b9ad026860ac49a2d1f 100644 --- a/formal/lean/mathlib/category_theory/sites/sheaf.lean +++ b/formal/lean/mathlib/category_theory/sites/sheaf.lean @@ -257,6 +257,14 @@ def Sheaf_to_presheaf : Sheaf J A ⥤ (Cᵒᵖ ⥤ A) := instance : full (Sheaf_to_presheaf J A) := { preimage := λ X Y f, ⟨f⟩ } instance : faithful (Sheaf_to_presheaf J A) := {} +/--This is stated as a lemma to prevent class search from forming a loop since a sheaf morphism is +monic if and only if it is monic as a presheaf morphism (under suitable assumption).-/ +lemma Sheaf.hom.mono_of_presheaf_mono {F G : Sheaf J A} (f : F ⟶ G) [h : mono f.1] : mono f := +(Sheaf_to_presheaf J A).mono_of_mono_map h + +instance Sheaf.hom.epi_of_presheaf_epi {F G : Sheaf J A} (f : F ⟶ G) [h : epi f.1] : epi f := +(Sheaf_to_presheaf J A).epi_of_epi_map h + /-- The sheaf of sections guaranteed by the sheaf condition. -/ @[simps] def sheaf_over {A : Type u₂} [category.{v₂} A] {J : grothendieck_topology C} (ℱ : Sheaf J A) (E : A) : SheafOfTypes J := ⟨ℱ.val ⋙ coyoneda.obj (op E), ℱ.cond E⟩ diff --git a/formal/lean/mathlib/category_theory/sites/sheaf_of_types.lean b/formal/lean/mathlib/category_theory/sites/sheaf_of_types.lean index 5a9368fd1a50253dbdec7f7965f11993c1e484c5..fd0b2296cd3ac4f54440fed3a29807e14f795b65 100644 --- a/formal/lean/mathlib/category_theory/sites/sheaf_of_types.lean +++ b/formal/lean/mathlib/category_theory/sites/sheaf_of_types.lean @@ -663,7 +663,7 @@ Every presheaf is a sheaf for the maximal sieve. lemma is_sheaf_for_top_sieve (P : Cᵒᵖ ⥤ Type w) : is_sheaf_for P ((⊤ : sieve X) : presieve X) := begin - rw ← generate_of_singleton_split_epi (𝟙 X), + rw ← generate_of_singleton_is_split_epi (𝟙 X), rw ← is_sheaf_for_iff_generate, apply is_sheaf_for_singleton_iso, end diff --git a/formal/lean/mathlib/category_theory/sites/sheafification.lean b/formal/lean/mathlib/category_theory/sites/sheafification.lean index fc5fb5a349270d0c4ffdf2754c70fb225d22b1a4..792eb964d8788f8839778a2c4de390dd6fe17538 100644 --- a/formal/lean/mathlib/category_theory/sites/sheafification.lean +++ b/formal/lean/mathlib/category_theory/sites/sheafification.lean @@ -34,7 +34,7 @@ local attribute [instance] concrete_category.has_coe_to_fun /-- A concrete version of the multiequalizer, to be used below. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def meq {X : C} (P : Cᵒᵖ ⥤ D) (S : J.cover X) := { x : Π (I : S.arrow), P.obj (op I.Y) // ∀ (I : S.relation), P.map I.g₁.op (x I.fst) = P.map I.g₂.op (x I.snd) } @@ -621,6 +621,16 @@ adjunction.mk_of_hom_equiv end, hom_equiv_naturality_right' := λ P Q R η γ, by { dsimp, rw category.assoc } } +instance Sheaf_to_presheaf_is_right_adjoint : is_right_adjoint (Sheaf_to_presheaf J D) := +⟨_, sheafification_adjunction J D⟩ + +instance presheaf_mono_of_mono {F G : Sheaf J D} (f : F ⟶ G) [mono f] : mono f.1 := +(Sheaf_to_presheaf J D).map_mono _ + +lemma Sheaf.hom.mono_iff_presheaf_mono {F G : Sheaf J D} (f : F ⟶ G) : mono f ↔ mono f.1 := +⟨λ m, by { resetI, apply_instance }, + λ m, by { resetI, exact Sheaf.hom.mono_of_presheaf_mono J D f }⟩ + variables {J D} /-- A sheaf `P` is isomorphic to its own sheafification. -/ @[simps] diff --git a/formal/lean/mathlib/category_theory/sites/sieves.lean b/formal/lean/mathlib/category_theory/sites/sieves.lean index 41522c7061f28dac001cd2ab3c139b7a5a44d307..fcb14e3bb6273b8948a16ef711132ab503f2a3e5 100644 --- a/formal/lean/mathlib/category_theory/sites/sieves.lean +++ b/formal/lean/mathlib/category_theory/sites/sieves.lean @@ -44,7 +44,7 @@ instance : inhabited (presieve X) := ⟨⊤⟩ /-- Given a sieve `S` on `X : C`, its associated diagram `S.diagram` is defined to be the natural functor from the full subcategory of the over category `C/X` consisting of arrows in `S` to `C`. -/ -abbreviation diagram (S : presieve X) : {f : over X // S f.hom} ⥤ C := +abbreviation diagram (S : presieve X) : full_subcategory (λ (f : over X), S f.hom) ⥤ C := full_subcategory_inclusion _ ⋙ over.forget X /-- Given a sieve `S` on `X : C`, its associated cocone `S.cocone` is defined to be @@ -170,7 +170,7 @@ def functor_pushforward (S : presieve X) : presieve (F.obj X) := /-- An auxillary definition in order to fix the choice of the preimages between various definitions. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure functor_pushforward_structure (S : presieve X) {Y} (f : Y ⟶ F.obj X) := (preobj : C) (premap : preobj ⟶ X) (lift : Y ⟶ F.obj preobj) (cover : S premap) (fac : f = lift ≫ F.map premap) @@ -357,7 +357,7 @@ lemma id_mem_iff_eq_top : S (𝟙 X) ↔ S = ⊤ := λ h, h.symm ▸ trivial⟩ /-- If an arrow set contains a split epi, it generates the maximal sieve. -/ -lemma generate_of_contains_split_epi {R : presieve X} (f : Y ⟶ X) [split_epi f] +lemma generate_of_contains_is_split_epi {R : presieve X} (f : Y ⟶ X) [is_split_epi f] (hf : R f) : generate R = ⊤ := begin rw ← id_mem_iff_eq_top, @@ -365,13 +365,13 @@ begin end @[simp] -lemma generate_of_singleton_split_epi (f : Y ⟶ X) [split_epi f] : +lemma generate_of_singleton_is_split_epi (f : Y ⟶ X) [is_split_epi f] : generate (presieve.singleton f) = ⊤ := -generate_of_contains_split_epi f (presieve.singleton_self _) +generate_of_contains_is_split_epi f (presieve.singleton_self _) @[simp] lemma generate_top : generate (⊤ : presieve X) = ⊤ := -generate_of_contains_split_epi (𝟙 _) ⟨⟩ +generate_of_contains_is_split_epi (𝟙 _) ⟨⟩ /-- Given a morphism `h : Y ⟶ X`, send a sieve S on X to a sieve on Y as the inverse image of S with `_ ≫ h`. @@ -470,7 +470,7 @@ begin end /-- If `f` is a split epi, the pushforward-pullback adjunction on sieves is reflective. -/ -def galois_insertion_of_split_epi (f : Y ⟶ X) [split_epi f] : +def galois_insertion_of_is_split_epi (f : Y ⟶ X) [is_split_epi f] : galois_insertion (sieve.pushforward f) (sieve.pullback f) := begin apply (galois_connection f).to_galois_insertion, @@ -599,7 +599,7 @@ lemma functor_pullback_inter (S R : sieve (F.obj X)) : (⊤ : sieve X).functor_pushforward F = ⊤ := begin refine (generate_sieve _).symm.trans _, - apply generate_of_contains_split_epi (𝟙 (F.obj X)), + apply generate_of_contains_is_split_epi (𝟙 (F.obj X)), refine ⟨X, 𝟙 _, 𝟙 _, trivial, by simp⟩ end diff --git a/formal/lean/mathlib/category_theory/sites/subsheaf.lean b/formal/lean/mathlib/category_theory/sites/subsheaf.lean new file mode 100644 index 0000000000000000000000000000000000000000..74582be734a374c154c378695c4d0ddaf6e69121 --- /dev/null +++ b/formal/lean/mathlib/category_theory/sites/subsheaf.lean @@ -0,0 +1,237 @@ +/- +Copyright (c) 2022 Andrew Yang. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Andrew Yang +-/ +import category_theory.elementwise +import category_theory.sites.compatible_sheafification +import category_theory.limits.constructions.epi_mono + +/-! + +# Subsheaf of types + +We define the sub(pre)sheaf of a type valued presheaf. + +## Main results + +- `category_theory.grothendieck_topology.subpresheaf` : + A subpresheaf of a presheaf of types. +- `category_theory.grothendieck_topology.subpresheaf.sheafify` : + The sheafification of a subpresheaf as a subpresheaf. Note that this is a sheaf only when the + whole sheaf is. +- `category_theory.grothendieck_topology.subpresheaf.sheafify_is_sheaf` : + The sheafification is a sheaf +- `category_theory.grothendieck_topology.subpresheaf.sheafify_lift` : + The descent of a map into a sheaf to the sheafification. +-/ + +universes w v u + +open opposite category_theory + +namespace category_theory.grothendieck_topology + +variables {C : Type u} [category.{v} C] (J : grothendieck_topology C) + +/-- A subpresheaf of a presheaf consists of a subset of `F.obj U` for every `U`, +compatible with the restriction maps `F.map i`. -/ +@[ext] +structure subpresheaf (F : Cᵒᵖ ⥤ Type w) := +(obj : Π U, set (F.obj U)) +(map : Π {U V : Cᵒᵖ} (i : U ⟶ V), (obj U) ⊆ (F.map i) ⁻¹' (obj V)) + +variables {F F' : Cᵒᵖ ⥤ Type w} (G G' : subpresheaf F) + +instance : partial_order (subpresheaf F) := +partial_order.lift subpresheaf.obj subpresheaf.ext + +instance : has_top (subpresheaf F) := +⟨⟨λ U, ⊤, λ U V i x h, _root_.trivial⟩⟩ + +instance : nonempty (subpresheaf F) := infer_instance + +/-- The subpresheaf as a presheaf. -/ +@[simps] +def subpresheaf.to_presheaf : Cᵒᵖ ⥤ Type w := +{ obj := λ U, G.obj U, + map := λ U V i x, ⟨F.map i x, G.map i x.prop⟩, + map_id' := λ X, by { ext ⟨x, _⟩, dsimp, rw F.map_id, refl }, + map_comp' := λ X Y Z i j, by { ext ⟨x, _⟩, dsimp, rw F.map_comp, refl } } + +instance {U} : has_coe (G.to_presheaf.obj U) (F.obj U) := +coe_subtype + +/-- The inclusion of a subpresheaf to the original presheaf. -/ +@[simps] +def subpresheaf.ι : G.to_presheaf ⟶ F := +{ app := λ U x, x } + +instance : mono G.ι := +⟨λ H f₁ f₂ e, nat_trans.ext f₁ f₂ $ funext $ λ U, + funext $ λ x, subtype.ext $ congr_fun (congr_app e U) x⟩ + +/-- The inclusion of a subpresheaf to a larger subpresheaf -/ +@[simps] +def subpresheaf.hom_of_le {G G' : subpresheaf F} (h : G ≤ G') : G.to_presheaf ⟶ G'.to_presheaf := +{ app := λ U x, ⟨x, h U x.prop⟩ } + +instance {G G' : subpresheaf F} (h : G ≤ G') : mono (subpresheaf.hom_of_le h) := +⟨λ H f₁ f₂ e, nat_trans.ext f₁ f₂ $ funext $ λ U, + funext $ λ x, subtype.ext $ (congr_arg subtype.val $ (congr_fun (congr_app e U) x : _) : _)⟩ + +/-- Given a subpresheaf `G` of `F`, an `F`-section `s` on `U`, we may define a sieve of `U` +consisting of all `f : V ⟶ U` such that the restriction of `s` along `f` is in `G`. -/ +@[simps] +def subpresheaf.sieve_of_section {U : Cᵒᵖ} (s : F.obj U) : sieve (unop U) := +{ arrows := λ V f, F.map f.op s ∈ G.obj (op V), + downward_closed' := λ V W i hi j, + by { rw [op_comp, functor_to_types.map_comp_apply], exact G.map _ hi } } + +/-- Given a `F`-section `s` on `U` and a subpresheaf `G`, we may define a family of elements in +`G` consisting of the restrictions of `s` -/ +def subpresheaf.family_of_elements_of_section {U : Cᵒᵖ} (s : F.obj U) : + (G.sieve_of_section s).1.family_of_elements G.to_presheaf := +λ V i hi, ⟨F.map i.op s, hi⟩ + +lemma subpresheaf.family_of_elements_compatible {U : Cᵒᵖ} (s : F.obj U) : + (G.family_of_elements_of_section s).compatible := +begin + intros Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ e, + ext1, + change F.map g₁.op (F.map f₁.op s) = F.map g₂.op (F.map f₂.op s), + rw [← functor_to_types.map_comp_apply, ← functor_to_types.map_comp_apply, + ← op_comp, ← op_comp, e], +end + +include J + +/-- The sheafification of a subpresheaf as a subpresheaf. +Note that this is a sheaf only when the whole presheaf is a sheaf. -/ +def subpresheaf.sheafify : subpresheaf F := +{ obj := λ U, { s | G.sieve_of_section s ∈ J (unop U) }, + map := begin + rintros U V i s hs, + refine J.superset_covering _ (J.pullback_stable i.unop hs), + intros _ _ h, + dsimp at h ⊢, + rwa ← functor_to_types.map_comp_apply, + end } + +lemma subpresheaf.le_sheafify : G ≤ G.sheafify J := +begin + intros U s hs, + change _ ∈ J _, + convert J.top_mem _, + rw eq_top_iff, + rintros V i -, + exact G.map i.op hs, +end + +variable {J} + +lemma subpresheaf.eq_sheafify (h : presieve.is_sheaf J F) + (hG : presieve.is_sheaf J G.to_presheaf) : G = G.sheafify J := +begin + apply (G.le_sheafify J).antisymm, + intros U s hs, + suffices : ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).1 = s, + { rw ← this, exact ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).2 }, + apply (h _ hs).is_separated_for.ext, + intros V i hi, + exact (congr_arg subtype.val ((hG _ hs).valid_glue (G.family_of_elements_compatible s) _ hi) : _) +end + +lemma subpresheaf.sheafify_is_sheaf (hF : presieve.is_sheaf J F) : + presieve.is_sheaf J (G.sheafify J).to_presheaf := +begin + intros U S hS x hx, + let S' := sieve.bind S (λ Y f hf, G.sieve_of_section (x f hf).1), + have := λ {V} {i : V ⟶ U} (hi : S' i), hi, + choose W i₁ i₂ hi₂ h₁ h₂, + dsimp [-sieve.bind_apply] at *, + let x'' : presieve.family_of_elements F S' := + λ V i hi, F.map (i₁ hi).op (x _ (hi₂ hi)), + have H : ∀ s, x.is_amalgamation s ↔ x''.is_amalgamation s.1, + { intro s, + split, + { intros H V i hi, + dsimp only [x''], + conv_lhs { rw ← h₂ hi }, + rw ← H _ (hi₂ hi), + exact functor_to_types.map_comp_apply F (i₂ hi).op (i₁ hi).op _ }, + { intros H V i hi, + ext1, + apply (hF _ (x i hi).2).is_separated_for.ext, + intros V' i' hi', + have hi'' : S' (i' ≫ i) := ⟨_, _, _, hi, hi', rfl⟩, + have := H _ hi'', + rw [op_comp, F.map_comp] at this, + refine this.trans (congr_arg subtype.val (hx _ _ (hi₂ hi'') hi (h₂ hi''))) } }, + have : x''.compatible, + { intros V₁ V₂ V₃ g₁ g₂ g₃ g₄ S₁ S₂ e, + rw [← functor_to_types.map_comp_apply, ← functor_to_types.map_comp_apply], + exact congr_arg subtype.val + (hx (g₁ ≫ i₁ S₁) (g₂ ≫ i₁ S₂) (hi₂ S₁) (hi₂ S₂) (by simp only [category.assoc, h₂, e])) }, + obtain ⟨t, ht, ht'⟩ := hF _ (J.bind_covering hS (λ V i hi, (x i hi).2)) _ this, + refine ⟨⟨t, _⟩, (H ⟨t, _⟩).mpr ht, λ y hy, subtype.ext (ht' _ ((H _).mp hy))⟩, + show G.sieve_of_section t ∈ J _, + refine J.superset_covering _ (J.bind_covering hS (λ V i hi, (x i hi).2)), + intros V i hi, + dsimp, + rw ht _ hi, + exact h₁ hi +end + +lemma subpresheaf.eq_sheafify_iff (h : presieve.is_sheaf J F) : + G = G.sheafify J ↔ presieve.is_sheaf J G.to_presheaf := +⟨λ e, e.symm ▸ G.sheafify_is_sheaf h, G.eq_sheafify h⟩ + +lemma subpresheaf.is_sheaf_iff (h : presieve.is_sheaf J F) : + presieve.is_sheaf J G.to_presheaf ↔ + ∀ U (s : F.obj U), G.sieve_of_section s ∈ J (unop U) → s ∈ G.obj U := +begin + rw ← G.eq_sheafify_iff h, + change _ ↔ G.sheafify J ≤ G, + exact ⟨eq.ge, (G.le_sheafify J).antisymm⟩ +end + +lemma subpresheaf.sheafify_sheafify (h : presieve.is_sheaf J F) : + (G.sheafify J).sheafify J = G.sheafify J := +((subpresheaf.eq_sheafify_iff _ h).mpr $ G.sheafify_is_sheaf h).symm + +/-- The lift of a presheaf morphism onto the sheafification subpresheaf. -/ +noncomputable +def subpresheaf.sheafify_lift (f : G.to_presheaf ⟶ F') (h : presieve.is_sheaf J F') : + (G.sheafify J).to_presheaf ⟶ F' := +{ app := λ U s, + (h _ s.prop).amalgamate _ ((G.family_of_elements_compatible ↑s).comp_presheaf_map f), + naturality' := + begin + intros U V i, + ext s, + apply (h _ ((subpresheaf.sheafify J G).to_presheaf.map i s).prop).is_separated_for.ext, + intros W j hj, + refine (presieve.is_sheaf_for.valid_glue _ _ _ hj).trans _, + dsimp, + conv_rhs { rw ← functor_to_types.map_comp_apply }, + change _ = F'.map (j ≫ i.unop).op _, + refine eq.trans _ (presieve.is_sheaf_for.valid_glue _ _ _ _).symm, + { dsimp at ⊢ hj, rwa functor_to_types.map_comp_apply }, + { dsimp [presieve.family_of_elements.comp_presheaf_map], + congr' 1, + ext1, + exact (functor_to_types.map_comp_apply _ _ _ _).symm } + end } + +lemma subpresheaf.to_sheafify_lift (f : G.to_presheaf ⟶ F') (h : presieve.is_sheaf J F') : + subpresheaf.hom_of_le (G.le_sheafify J) ≫ G.sheafify_lift f h = f := +begin + ext U s, + apply (h _ ((subpresheaf.hom_of_le (G.le_sheafify J)).app U s).prop).is_separated_for.ext, + intros V i hi, + have := elementwise_of f.naturality, + exact (presieve.is_sheaf_for.valid_glue _ _ _ hi).trans (this _ _) +end + +end category_theory.grothendieck_topology diff --git a/formal/lean/mathlib/category_theory/structured_arrow.lean b/formal/lean/mathlib/category_theory/structured_arrow.lean index 5b765d7c15d3b46fde85dd8701a7c91925c95e0a..b6a5803f7abe0cf16ee0a20d338f954beabec075 100644 --- a/formal/lean/mathlib/category_theory/structured_arrow.lean +++ b/formal/lean/mathlib/category_theory/structured_arrow.lean @@ -29,7 +29,7 @@ The category of `T`-structured arrows with domain `S : D` (here `T : C ⥤ D`), has as its objects `D`-morphisms of the form `S ⟶ T Y`, for some `Y : C`, and morphisms `C`-morphisms `Y ⟶ Y'` making the obvious triangle commute. -/ -@[derive category, nolint has_inhabited_instance] +@[derive category, nolint has_nonempty_instance] def structured_arrow (S : D) (T : C ⥤ D) := comma (functor.from_punit S) T namespace structured_arrow @@ -50,9 +50,6 @@ def mk (f : S ⟶ T.obj Y) : structured_arrow S T := ⟨⟨⟨⟩⟩, Y, f⟩ @[simp, reassoc] lemma w {A B : structured_arrow S T} (f : A ⟶ B) : A.hom ≫ T.map f.right = B.hom := by { have := f.w; tidy } -lemma eq_mk (f : structured_arrow S T) : f = mk f.hom := -by { cases f, congr, ext, } - /-- To construct a morphism of structured arrows, we need a morphism of the objects underlying the target, @@ -83,6 +80,16 @@ def iso_mk {f f' : structured_arrow S T} (g : f.right ≅ f'.right) (w : f.hom ≫ T.map g.hom = f'.hom) : f ≅ f' := comma.iso_mk (eq_to_iso (by ext)) g (by simpa [eq_to_hom_map] using w.symm) +/-- Eta rule for structured arrows. Prefer `structured_arrow.eta`, since equality of objects tends + to cause problems. -/ +lemma eq_mk (f : structured_arrow S T) : f = mk f.hom := +by { cases f, congr, ext, } + +/-- Eta rule for structured arrows. -/ +@[simps] +def eta (f : structured_arrow S T) : f ≅ mk f.hom := +iso_mk (iso.refl _) (by tidy) + /-- A morphism between source objects `S ⟶ S'` contravariantly induces a functor between structured arrows, @@ -145,7 +152,7 @@ The category of `S`-costructured arrows with target `T : D` (here `S : C ⥤ D`) has as its objects `D`-morphisms of the form `S Y ⟶ T`, for some `Y : C`, and morphisms `C`-morphisms `Y ⟶ Y'` making the obvious triangle commute. -/ -@[derive category, nolint has_inhabited_instance] +@[derive category, nolint has_nonempty_instance] def costructured_arrow (S : C ⥤ D) (T : D) := comma S (functor.from_punit T) namespace costructured_arrow @@ -167,9 +174,6 @@ def mk (f : S.obj Y ⟶ T) : costructured_arrow S T := ⟨Y, ⟨⟨⟩⟩, f⟩ S.map f.left ≫ B.hom = A.hom := by tidy -lemma eq_mk (f : costructured_arrow S T) : f = mk f.hom := -by { cases f, congr, ext, } - /-- To construct a morphism of costructured arrows, we need a morphism of the objects underlying the source, @@ -192,6 +196,16 @@ def iso_mk {f f' : costructured_arrow S T} (g : f.left ≅ f'.left) (w : S.map g.hom ≫ f'.hom = f.hom) : f ≅ f' := comma.iso_mk g (eq_to_iso (by ext)) (by simpa [eq_to_hom_map] using w) +/-- Eta rule for costructured arrows. Prefer `costructured_arrow.eta`, as equality of objects tends + to cause problems. -/ +lemma eq_mk (f : costructured_arrow S T) : f = mk f.hom := +by { cases f, congr, ext, } + +/-- Eta rule for costructured arrows. -/ +@[simps] +def eta (f : costructured_arrow S T) : f ≅ mk f.hom := +iso_mk (iso.refl _) (by tidy) + /-- A morphism between target objects `T ⟶ T'` covariantly induces a functor between costructured arrows, diff --git a/formal/lean/mathlib/category_theory/subobject/basic.lean b/formal/lean/mathlib/category_theory/subobject/basic.lean index 5a8913a0d881107277a743dc09d3271893cb2fba..10c301af39121dcd92ed50a103274ba70d7e7c9c 100644 --- a/formal/lean/mathlib/category_theory/subobject/basic.lean +++ b/formal/lean/mathlib/category_theory/subobject/basic.lean @@ -98,6 +98,43 @@ namespace subobject abbreviation mk {X A : C} (f : A ⟶ X) [mono f] : subobject X := (to_thin_skeleton _).obj (mono_over.mk' f) +section +local attribute [ext] category_theory.comma + +protected lemma ind {X : C} (p : subobject X → Prop) + (h : ∀ ⦃A : C⦄ (f : A ⟶ X) [mono f], by exactI p (subobject.mk f)) (P : subobject X) : p P := +begin + apply quotient.induction_on', + intro a, + convert h a.arrow, + ext; refl +end + +protected lemma ind₂ {X : C} (p : subobject X → subobject X → Prop) + (h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [mono f] [mono g], + by exactI p (subobject.mk f) (subobject.mk g)) (P Q : subobject X) : p P Q := +begin + apply quotient.induction_on₂', + intros a b, + convert h a.arrow b.arrow; + ext; refl +end + +end + +/-- Declare a function on subobjects of `X` by specifying a function on monomorphisms with + codomain `X`. -/ +protected def lift {α : Sort*} {X : C} (F : Π ⦃A : C⦄ (f : A ⟶ X) [mono f], α) + (h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [mono f] [mono g] (i : A ≅ B), + i.hom ≫ g = f → by exactI F f = F g) : subobject X → α := +λ P, quotient.lift_on' P (λ m, by exactI F m.arrow) $ λ m n ⟨i⟩, + h m.arrow n.arrow ((mono_over.forget X ⋙ over.forget X).map_iso i) (over.w i.hom) + +@[simp] +protected lemma lift_mk {α : Sort*} {X : C} (F : Π ⦃A : C⦄ (f : A ⟶ X) [mono f], α) {h A} + (f : A ⟶ X) [mono f] : subobject.lift F h (subobject.mk f) = F f := +rfl + /-- The category of subobjects is equivalent to the `mono_over` category. It is more convenient to use the former due to the partial order instance, but oftentimes it is easier to define structures on the latter. -/ @@ -149,7 +186,7 @@ The morphism in `C` from the arbitrarily chosen underlying object to the ambient -/ noncomputable def arrow {X : C} (Y : subobject X) : (Y : C) ⟶ X := -(representative.obj Y).val.hom +(representative.obj Y).obj.hom instance arrow_mono {X : C} (Y : subobject X) : mono (Y.arrow) := (representative.obj Y).property diff --git a/formal/lean/mathlib/category_theory/subobject/limits.lean b/formal/lean/mathlib/category_theory/subobject/limits.lean index e880716772f180326c74f6a1cbc231617a2ee561..3bc6056120a1c2a30673d447df6b3b9174412080 100644 --- a/formal/lean/mathlib/category_theory/subobject/limits.lean +++ b/formal/lean/mathlib/category_theory/subobject/limits.lean @@ -22,6 +22,7 @@ universes v u noncomputable theory open category_theory category_theory.category category_theory.limits category_theory.subobject + opposite variables {C : Type u} [category.{v} C] {X Y Z : C} @@ -203,6 +204,50 @@ begin { simp, }, end +/-- Taking cokernels is an order-reversing map from the subobjects of `X` to the quotient objects + of `X`. -/ +@[simps] +def cokernel_order_hom [has_cokernels C] (X : C) : subobject X →o (subobject (op X))ᵒᵈ := +{ to_fun := subobject.lift (λ A f hf, subobject.mk (cokernel.π f).op) + begin + rintros A B f g hf hg i rfl, + refine subobject.mk_eq_mk_of_comm _ _ (iso.op _) (quiver.hom.unop_inj _), + { exact (is_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit _) + (is_cokernel_epi_comp (colimit.is_colimit _) i.hom rfl)).symm }, + { simp only [iso.comp_inv_eq, iso.op_hom, iso.symm_hom, unop_comp, quiver.hom.unop_op, + colimit.comp_cocone_point_unique_up_to_iso_hom, cofork.of_π_ι_app, coequalizer.cofork_π] } + end, + monotone' := subobject.ind₂ _ $ + begin + introsI A B f g hf hg h, + dsimp only [subobject.lift_mk], + refine subobject.mk_le_mk_of_comm (cokernel.desc f (cokernel.π g) _).op _, + { rw [← subobject.of_mk_le_mk_comp h, category.assoc, cokernel.condition, comp_zero] }, + { exact quiver.hom.unop_inj (cokernel.π_desc _ _ _) } + end } + +/-- Taking kernels is an order-reversing map from the quotient objects of `X` to the subobjects of + `X`. -/ +@[simps] +def kernel_order_hom [has_kernels C] (X : C) : (subobject (op X))ᵒᵈ →o subobject X := +{ to_fun := subobject.lift (λ A f hf, subobject.mk (kernel.ι f.unop)) + begin + rintros A B f g hf hg i rfl, + refine subobject.mk_eq_mk_of_comm _ _ _ _, + { exact is_limit.cone_point_unique_up_to_iso (limit.is_limit _) + (is_kernel_comp_mono (limit.is_limit (parallel_pair g.unop 0)) i.unop.hom rfl) }, + { dsimp, + simp only [←iso.eq_inv_comp, limit.cone_point_unique_up_to_iso_inv_comp, fork.of_ι_π_app] } + end, + monotone' := subobject.ind₂ _ $ + begin + introsI A B f g hf hg h, + dsimp only [subobject.lift_mk], + refine subobject.mk_le_mk_of_comm (kernel.lift g.unop (kernel.ι f.unop) _) _, + { rw [← subobject.of_mk_le_mk_comp h, unop_comp, kernel.condition_assoc, zero_comp] }, + { exact quiver.hom.op_inj (by simp) } + end } + end kernel section image diff --git a/formal/lean/mathlib/category_theory/subobject/mono_over.lean b/formal/lean/mathlib/category_theory/subobject/mono_over.lean index 0325ade408c0438f8fc44c6e0e2c375875c2d5ee..98ccbe1a4b4e159c706ae3bb3e7216c52a269976 100644 --- a/formal/lean/mathlib/category_theory/subobject/mono_over.lean +++ b/formal/lean/mathlib/category_theory/subobject/mono_over.lean @@ -49,19 +49,19 @@ This isn't skeletal, so it's not a partial order. Later we define `subobject X` as the quotient of this by isomorphisms. -/ @[derive [category]] -def mono_over (X : C) := {f : over X // mono f.hom} +def mono_over (X : C) := full_subcategory (λ (f : over X), mono f.hom) namespace mono_over /-- Construct a `mono_over X`. -/ @[simps] -def mk' {X A : C} (f : A ⟶ X) [hf : mono f] : mono_over X := { val := over.mk f, property := hf } +def mk' {X A : C} (f : A ⟶ X) [hf : mono f] : mono_over X := { obj := over.mk f, property := hf } /-- The inclusion from monomorphisms over X to morphisms over X. -/ def forget (X : C) : mono_over X ⥤ over X := full_subcategory_inclusion _ instance : has_coe (mono_over X) C := -{ coe := λ Y, Y.val.left, } +{ coe := λ Y, Y.obj.left, } @[simp] lemma forget_obj_left {f} : ((forget X).obj f).left = (f : C) := rfl @@ -93,13 +93,13 @@ end⟩ @[reassoc] lemma w {f g : mono_over X} (k : f ⟶ g) : k.left ≫ g.arrow = f.arrow := over.w _ /-- Convenience constructor for a morphism in monomorphisms over `X`. -/ -abbreviation hom_mk {f g : mono_over X} (h : f.val.left ⟶ g.val.left) (w : h ≫ g.arrow = f.arrow) : +abbreviation hom_mk {f g : mono_over X} (h : f.obj.left ⟶ g.obj.left) (w : h ≫ g.arrow = f.arrow) : f ⟶ g := over.hom_mk h w /-- Convenience constructor for an isomorphism in monomorphisms over `X`. -/ @[simps] -def iso_mk {f g : mono_over X} (h : f.val.left ≅ g.val.left) (w : h.hom ≫ g.arrow = f.arrow) : +def iso_mk {f g : mono_over X} (h : f.obj.left ≅ g.obj.left) (w : h.hom ≫ g.arrow = f.arrow) : f ≅ g := { hom := hom_mk h.hom w, inv := hom_mk h.inv (by rw [h.inv_comp_eq, w]) } @@ -217,7 +217,7 @@ def map_id : map (𝟙 X) ≅ 𝟭 _ := lift_iso _ _ over.map_id ≪≫ lift_id @[simp] lemma map_obj_left (f : X ⟶ Y) [mono f] (g : mono_over X) : - (((map f).obj g) : C) = g.val.left := + (((map f).obj g) : C) = g.obj.left := rfl @[simp] @@ -324,7 +324,7 @@ adjunction.mk_of_hom_equiv inv_fun := λ k, begin refine over.hom_mk _ _, - refine image.lift {I := g.val.left, m := g.arrow, e := k.left, fac' := over.w k}, + refine image.lift {I := g.obj.left, m := g.arrow, e := k.left, fac' := over.w k}, apply image.lift_fac, end, left_inv := λ k, subsingleton.elim _ _, diff --git a/formal/lean/mathlib/category_theory/subterminal.lean b/formal/lean/mathlib/category_theory/subterminal.lean index ddf8015a321e3661a46a53a263767d72e3605c31..5adb0eedb5aaef2ceaf544be62135bd5b772180c 100644 --- a/formal/lean/mathlib/category_theory/subterminal.lean +++ b/formal/lean/mathlib/category_theory/subterminal.lean @@ -117,7 +117,7 @@ to the lattice of open subsets of `X`. More generally, if `C` is a topos, this i -/ @[derive category] def subterminals (C : Type u₁) [category.{v₁} C] := -{A : C // is_subterminal A} +full_subcategory (λ (A : C), is_subterminal A) instance [has_terminal C] : inhabited (subterminals C) := ⟨⟨⊤_ C, is_subterminal_of_terminal⟩⟩ @@ -140,7 +140,7 @@ def subterminals_equiv_mono_over_terminal [has_terminal C] : { obj := λ X, ⟨over.mk (terminal.from X.1), X.2.mono_terminal_from⟩, map := λ X Y f, mono_over.hom_mk f (by ext1 ⟨⟨⟩⟩) }, inverse := - { obj := λ X, ⟨X.val.left, λ Z f g, by { rw ← cancel_mono X.arrow, apply subsingleton.elim }⟩, + { obj := λ X, ⟨X.obj.left, λ Z f g, by { rw ← cancel_mono X.arrow, apply subsingleton.elim }⟩, map := λ X Y f, f.1 }, unit_iso := { hom := { app := λ X, 𝟙 _ }, diff --git a/formal/lean/mathlib/category_theory/types.lean b/formal/lean/mathlib/category_theory/types.lean index e1074994e2618519fdc280a0de49a57f52c6dba6..41c59d98a3a8b492037ad6c173a051d45c05f18e 100644 --- a/formal/lean/mathlib/category_theory/types.lean +++ b/formal/lean/mathlib/category_theory/types.lean @@ -278,8 +278,8 @@ iff.intro (λ i, (by exactI as_iso f : X ≅ Y).to_equiv.bijective) (λ b, is_iso.of_iso (equiv.of_bijective f b).to_iso) -noncomputable instance : split_epi_category (Type u) := -{ split_epi_of_epi := λ X Y f hf, +instance : split_epi_category (Type u) := +{ is_split_epi_of_epi := λ X Y f hf, is_split_epi.mk' { section_ := function.surj_inv $ (epi_iff_surjective f).1 hf, id' := funext $ function.right_inverse_surj_inv $ (epi_iff_surjective f).1 hf } } diff --git a/formal/lean/mathlib/category_theory/with_terminal.lean b/formal/lean/mathlib/category_theory/with_terminal.lean index b3e3142e36c5cf2197cde89d18a8b70eb026334f..28b0f21f327636abafb57c05d4e2055bf771f475 100644 --- a/formal/lean/mathlib/category_theory/with_terminal.lean +++ b/formal/lean/mathlib/category_theory/with_terminal.lean @@ -56,7 +56,7 @@ local attribute [tidy] tactic.case_bash variable {C} /-- Morphisms for `with_terminal C`. -/ -@[simp, nolint has_inhabited_instance] +@[simp, nolint has_nonempty_instance] def hom : with_terminal C → with_terminal C → Type v | (of X) (of Y) := X ⟶ Y | star (of X) := pempty @@ -213,7 +213,7 @@ local attribute [tidy] tactic.case_bash variable {C} /-- Morphisms for `with_initial C`. -/ -@[simp, nolint has_inhabited_instance] +@[simp, nolint has_nonempty_instance] def hom : with_initial C → with_initial C → Type v | (of X) (of Y) := X ⟶ Y | (of X) _ := pempty diff --git a/formal/lean/mathlib/combinatorics/additive/salem_spencer.lean b/formal/lean/mathlib/combinatorics/additive/salem_spencer.lean index fd646013d86b3b06fc28bd409aec16a279a78e23..9050f8ee6a0aa057160b02b320a69a8817092ca9 100644 --- a/formal/lean/mathlib/combinatorics/additive/salem_spencer.lean +++ b/formal/lean/mathlib/combinatorics/additive/salem_spencer.lean @@ -345,7 +345,7 @@ calc @[to_additive] lemma le_mul_roth_number_product (s : finset α) (t : finset β) : - mul_roth_number s * mul_roth_number t ≤ mul_roth_number (s.product t) := + mul_roth_number s * mul_roth_number t ≤ mul_roth_number (s ×ˢ t) := begin obtain ⟨u, hus, hucard, hu⟩ := mul_roth_number_spec s, obtain ⟨v, hvt, hvcard, hv⟩ := mul_roth_number_spec t, diff --git a/formal/lean/mathlib/combinatorics/quiver/basic.lean b/formal/lean/mathlib/combinatorics/quiver/basic.lean index c0f0e947c10b3b6cd8e672c367708a80f6ecd53d..37c9b5a9eb9457d9bac9eaeb361ce84a4b8dbcd3 100644 --- a/formal/lean/mathlib/combinatorics/quiver/basic.lean +++ b/formal/lean/mathlib/combinatorics/quiver/basic.lean @@ -92,7 +92,7 @@ def hom.unop {V} [quiver V] {X Y : Vᵒᵖ} (f : X ⟶ Y) : unop Y ⟶ unop X := attribute [irreducible] quiver.opposite /-- A type synonym for a quiver with no arrows. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def empty (V) : Type u := V instance empty_quiver (V : Type u) : quiver.{u} (empty V) := ⟨λ a b, pempty⟩ diff --git a/formal/lean/mathlib/combinatorics/quiver/connected_component.lean b/formal/lean/mathlib/combinatorics/quiver/connected_component.lean index b43d6b4b177bced0bf2ececc5e532d0d764578dc..6ce1276737e98ff8b6a4cd6e39abfca544c7312f 100644 --- a/formal/lean/mathlib/combinatorics/quiver/connected_component.lean +++ b/formal/lean/mathlib/combinatorics/quiver/connected_component.lean @@ -23,7 +23,7 @@ namespace quiver /-- A type synonym for the symmetrized quiver (with an arrow both ways for each original arrow). NB: this does not work for `Prop`-valued quivers. It requires `[quiver.{v+1} V]`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def symmetrify (V) : Type u := V instance symmetrify_quiver (V : Type u) [quiver V] : quiver (symmetrify V) := diff --git a/formal/lean/mathlib/combinatorics/quiver/subquiver.lean b/formal/lean/mathlib/combinatorics/quiver/subquiver.lean index 812fff2a73b8eb1f12d17fa6823d0d37eb309f2c..551a2a2eaad7e5eee7b96f8bd995265243fc4321 100644 --- a/formal/lean/mathlib/combinatorics/quiver/subquiver.lean +++ b/formal/lean/mathlib/combinatorics/quiver/subquiver.lean @@ -25,7 +25,7 @@ def wide_subquiver (V) [quiver.{v+1} V] := /-- A type synonym for `V`, when thought of as a quiver having only the arrows from some `wide_subquiver`. -/ -@[nolint unused_arguments has_inhabited_instance] +@[nolint unused_arguments has_nonempty_instance] def wide_subquiver.to_Type (V) [quiver V] (H : wide_subquiver V) : Type u := V instance wide_subquiver_has_coe_to_sort {V} [quiver V] : @@ -44,7 +44,7 @@ instance {V} [quiver V] : inhabited (wide_subquiver V) := ⟨⊤⟩ /-- `total V` is the type of _all_ arrows of `V`. -/ -- TODO Unify with `category_theory.arrow`? (The fields have been named to match.) -@[ext, nolint has_inhabited_instance] +@[ext, nolint has_nonempty_instance] structure total (V : Type u) [quiver.{v} V] : Sort (max (u+1) v) := (left : V) (right : V) diff --git a/formal/lean/mathlib/combinatorics/set_family/compression/down.lean b/formal/lean/mathlib/combinatorics/set_family/compression/down.lean new file mode 100644 index 0000000000000000000000000000000000000000..b9470340b6671b53ac6ac67f34e4093258094868 --- /dev/null +++ b/formal/lean/mathlib/combinatorics/set_family/compression/down.lean @@ -0,0 +1,189 @@ +/- +Copyright (c) 2022 Yaël Dillies. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Yaël Dillies +-/ +import data.finset.card +import data.fintype.basic + +/-! +# Down-compressions + +This file defines down-compression. + +Down-compressing `𝒜 : finset (finset α)` along `a : α` means removing `a` from the elements of `𝒜`, +when the resulting set is not already in `𝒜`. + +## Main declarations + +* `finset.non_member_subfamily`: `𝒜.non_member_subfamily a` is the subfamily of sets not containing + `a`. +* `finset.member_subfamily`: `𝒜.member_subfamily a` is the image of the subfamily of sets containing + `a` under removing `a`. +* `down.compression`: Down-compression. + +## Notation + +`𝓓 a 𝒜` is notation for `down.compress a 𝒜` in locale `set_family`. + +## References + +* https://github.com/b-mehta/maths-notes/blob/master/iii/mich/combinatorics.pdf + +## Tags + +compression, down-compression +-/ + +variables {α : Type*} [decidable_eq α] {𝒜 ℬ : finset (finset α)} {s : finset α} {a : α} + +namespace finset + +/-- Elements of `𝒜` that do not contain `a`. -/ +def non_member_subfamily (a : α) (𝒜 : finset (finset α)) : finset (finset α) := +𝒜.filter $ λ s, a ∉ s + +/-- Image of the elements of `𝒜` which contain `a` under removing `a`. Finsets that do not contain +`a` such that `insert a s ∈ 𝒜`. -/ +def member_subfamily (a : α) (𝒜 : finset (finset α)) : finset (finset α) := +(𝒜.filter $ λ s, a ∈ s).image $ λ s, erase s a + +@[simp] lemma mem_non_member_subfamily : s ∈ 𝒜.non_member_subfamily a ↔ s ∈ 𝒜 ∧ a ∉ s := mem_filter +@[simp] lemma mem_member_subfamily : s ∈ 𝒜.member_subfamily a ↔ insert a s ∈ 𝒜 ∧ a ∉ s := +begin + simp_rw [member_subfamily, mem_image, mem_filter], + refine ⟨_, λ h, ⟨insert a s, ⟨h.1, mem_insert_self _ _⟩, erase_insert h.2⟩⟩, + rintro ⟨s, hs, rfl⟩, + rw insert_erase hs.2, + exact ⟨hs.1, not_mem_erase _ _⟩, +end + +lemma non_member_subfamily_inter (a : α) (𝒜 ℬ : finset (finset α)) : + (𝒜 ∩ ℬ).non_member_subfamily a = 𝒜.non_member_subfamily a ∩ ℬ.non_member_subfamily a := +filter_inter_distrib _ _ _ + +lemma member_subfamily_inter (a : α) (𝒜 ℬ : finset (finset α)) : + (𝒜 ∩ ℬ).member_subfamily a = 𝒜.member_subfamily a ∩ ℬ.member_subfamily a := +begin + unfold member_subfamily, + rw [filter_inter_distrib, image_inter_of_inj_on _ _ ((erase_inj_on' _).mono _)], + rw [←coe_union, ←filter_union, coe_filter], + exact set.inter_subset_right _ _, +end + +lemma non_member_subfamily_union (a : α) (𝒜 ℬ : finset (finset α)) : + (𝒜 ∪ ℬ).non_member_subfamily a = 𝒜.non_member_subfamily a ∪ ℬ.non_member_subfamily a := +filter_union _ _ _ + +lemma member_subfamily_union (a : α) (𝒜 ℬ : finset (finset α)) : + (𝒜 ∪ ℬ).member_subfamily a = 𝒜.member_subfamily a ∪ ℬ.member_subfamily a := +by simp_rw [member_subfamily, filter_union, image_union] + +lemma card_member_subfamily_add_card_non_member_subfamily (a : α) (𝒜 : finset (finset α)) : + (𝒜.member_subfamily a).card + (𝒜.non_member_subfamily a).card = 𝒜.card := +begin + rw [member_subfamily, non_member_subfamily, card_image_of_inj_on, + filter_card_add_filter_neg_card_eq_card], + exact (erase_inj_on' _).mono (λ s hs, (mem_filter.1 hs).2), +end + +lemma member_subfamily_union_non_member_subfamily (a : α) (𝒜 : finset (finset α)) : + 𝒜.member_subfamily a ∪ 𝒜.non_member_subfamily a = 𝒜.image (λ s, s.erase a) := +begin + ext s, + simp only [mem_union, mem_member_subfamily, mem_non_member_subfamily, mem_image, exists_prop], + split, + { rintro (h | h), + { exact ⟨_, h.1, erase_insert h.2⟩ }, + { exact ⟨_, h.1, erase_eq_of_not_mem h.2⟩ } }, + { rintro ⟨s, hs, rfl⟩, + by_cases ha : a ∈ s, + { exact or.inl ⟨by rwa insert_erase ha, not_mem_erase _ _⟩ }, + { exact or.inr ⟨by rwa erase_eq_of_not_mem ha, not_mem_erase _ _⟩ } } +end + +@[simp] lemma member_subfamily_member_subfamily : (𝒜.member_subfamily a).member_subfamily a = ∅ := +by { ext, simp } + +@[simp] lemma member_subfamily_non_member_subfamily : + (𝒜.non_member_subfamily a).member_subfamily a = ∅ := +by { ext, simp } + +@[simp] lemma non_member_subfamily_member_subfamily : + (𝒜.member_subfamily a).non_member_subfamily a = 𝒜.member_subfamily a := +by { ext, simp } + +@[simp] lemma non_member_subfamily_non_member_subfamily : + (𝒜.non_member_subfamily a).non_member_subfamily a = 𝒜.non_member_subfamily a := +by { ext, simp } + +end finset + +open finset + +-- The namespace is here to distinguish from other compressions. +namespace down + +/-- `a`-down-compressing `𝒜` means removing `a` from the elements of `𝒜` that contain it, when the +resulting finset is not already in `𝒜`. -/ +def compression (a : α) (𝒜 : finset (finset α)) : finset (finset α) := +(𝒜.filter $ λ s, erase s a ∈ 𝒜).disj_union ((𝒜.image $ λ s, erase s a).filter $ λ s, s ∉ 𝒜) $ + λ s h₁ h₂, (mem_filter.1 h₂).2 (mem_filter.1 h₁).1 + +localized "notation `𝓓 ` := down.compression" in finset_family + +/-- `a` is in the down-compressed family iff it's in the original and its compression is in the +original, or it's not in the original but it's the compression of something in the original. -/ +lemma mem_compression : s ∈ 𝓓 a 𝒜 ↔ s ∈ 𝒜 ∧ s.erase a ∈ 𝒜 ∨ s ∉ 𝒜 ∧ insert a s ∈ 𝒜 := +begin + simp_rw [compression, mem_disj_union, mem_filter, mem_image, and_comm (s ∉ 𝒜)], + refine or_congr_right' (and_congr_left $ λ hs, + ⟨_, λ h, ⟨_, h, erase_insert $ insert_ne_self.1 $ ne_of_mem_of_not_mem h hs⟩⟩), + rintro ⟨t, ht, rfl⟩, + rwa insert_erase (erase_ne_self.1 (ne_of_mem_of_not_mem ht hs).symm), +end + +lemma erase_mem_compression (hs : s ∈ 𝒜) : s.erase a ∈ 𝓓 a 𝒜 := +begin + simp_rw [mem_compression, erase_idem, and_self], + refine (em _).imp_right (λ h, ⟨h, _⟩), + rwa insert_erase (erase_ne_self.1 (ne_of_mem_of_not_mem hs h).symm), +end + +-- This is a special case of `erase_mem_compression` once we have `compression_idem`. +lemma erase_mem_compression_of_mem_compression : s ∈ 𝓓 a 𝒜 → s.erase a ∈ 𝓓 a 𝒜 := +begin + simp_rw [mem_compression, erase_idem], + refine or.imp (λ h, ⟨h.2, h.2⟩) (λ h, _), + rwa [erase_eq_of_not_mem (insert_ne_self.1 $ ne_of_mem_of_not_mem h.2 h.1)], +end + +lemma mem_compression_of_insert_mem_compression (h : insert a s ∈ 𝓓 a 𝒜) : s ∈ 𝓓 a 𝒜 := +begin + by_cases ha : a ∈ s, + { rwa insert_eq_of_mem ha at h }, + { rw ←erase_insert ha, + exact erase_mem_compression_of_mem_compression h } +end + +/-- Down-compressing a family is idempotent. -/ +@[simp] lemma compression_idem (a : α) (𝒜 : finset (finset α)) : 𝓓 a (𝓓 a 𝒜) = 𝓓 a 𝒜 := +begin + ext s, + refine mem_compression.trans ⟨_, λ h, or.inl ⟨h, erase_mem_compression_of_mem_compression h⟩⟩, + rintro (h | h), + { exact h.1 }, + { cases h.1 (mem_compression_of_insert_mem_compression h.2) } +end + +/-- Down-compressing a family doesn't change its size. -/ +@[simp] lemma card_compression (a : α) (𝒜 : finset (finset α)) : (𝓓 a 𝒜).card = 𝒜.card := +begin + rw [compression, card_disj_union, image_filter, card_image_of_inj_on ((erase_inj_on' _).mono $ + λ s hs, _), ←card_disjoint_union, filter_union_filter_neg_eq], + { exact disjoint_filter_filter_neg _ _ }, + rw [mem_coe, mem_filter] at hs, + exact not_imp_comm.1 erase_eq_of_not_mem (ne_of_mem_of_not_mem hs.1 hs.2).symm, +end + +end down diff --git a/formal/lean/mathlib/combinatorics/set_family/harris_kleitman.lean b/formal/lean/mathlib/combinatorics/set_family/harris_kleitman.lean index 153ea58531502ff4573d98f71bf6888d40976c1b..4130df8b0cbc7b44a6faa405f7efdff3cacc5f06 100644 --- a/formal/lean/mathlib/combinatorics/set_family/harris_kleitman.lean +++ b/formal/lean/mathlib/combinatorics/set_family/harris_kleitman.lean @@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Yaël Dillies -/ import algebra.big_operators.basic +import combinatorics.set_family.compression.down import order.upper_lower /-! @@ -16,10 +17,6 @@ correlate in the uniform measure. ## Main declarations -* `finset.non_member_subfamily`: `𝒜.non_member_subfamily a` is the subfamily of sets not containing - `a`. -* `finset.member_subfamily`: `𝒜.member_subfamily a` is the image of the subfamily of sets - containing `a` under removing `a`. * `is_lower_set.le_card_inter_finset`: One form of the Harris-Kleitman inequality. ## References @@ -27,56 +24,11 @@ correlate in the uniform measure. * [D. J. Kleitman, *Families of non-disjoint subsets*][kleitman1966] -/ +open finset open_locale big_operators variables {α : Type*} [decidable_eq α] {𝒜 ℬ : finset (finset α)} {s : finset α} {a : α} -namespace finset - -/-- ELements of `𝒜` that do not contain `a`. -/ -def non_member_subfamily (𝒜 : finset (finset α)) (a : α) : finset (finset α) := -𝒜.filter $ λ s, a ∉ s - -/-- Image of the elements of `𝒜` which contain `a` under removing `a`. Finsets that do not contain -`a` such that `insert a s ∈ 𝒜`. -/ -def member_subfamily (𝒜 : finset (finset α)) (a : α) : finset (finset α) := -(𝒜.filter $ λ s, a ∈ s).image $ λ s, erase s a - -@[simp] lemma mem_non_member_subfamily : s ∈ 𝒜.non_member_subfamily a ↔ s ∈ 𝒜 ∧ a ∉ s := mem_filter -@[simp] lemma mem_member_subfamily : s ∈ 𝒜.member_subfamily a ↔ insert a s ∈ 𝒜 ∧ a ∉ s := -begin - simp_rw [member_subfamily, mem_image, mem_filter], - refine ⟨_, λ h, ⟨insert a s, ⟨h.1, mem_insert_self _ _⟩, erase_insert h.2⟩⟩, - rintro ⟨s, hs, rfl⟩, - rw insert_erase hs.2, - exact ⟨hs.1, not_mem_erase _ _⟩, -end - -lemma non_member_subfamily_inter (𝒜 ℬ : finset (finset α)) (a : α) : - (𝒜 ∩ ℬ).non_member_subfamily a = 𝒜.non_member_subfamily a ∩ ℬ.non_member_subfamily a := -filter_inter_distrib _ _ _ - -lemma member_subfamily_inter (𝒜 ℬ : finset (finset α)) (a : α) : - (𝒜 ∩ ℬ).member_subfamily a = 𝒜.member_subfamily a ∩ ℬ.member_subfamily a := -begin - unfold member_subfamily, - rw [filter_inter_distrib, image_inter_of_inj_on _ _ ((erase_inj_on' _).mono _)], - rw [←coe_union, ←filter_union, coe_filter], - exact set.inter_subset_right _ _, -end - -lemma card_member_subfamily_add_card_non_member_subfamily (𝒜 : finset (finset α)) (a : α) : - (𝒜.member_subfamily a).card + (𝒜.non_member_subfamily a).card = 𝒜.card := -begin - rw [member_subfamily, non_member_subfamily, card_image_of_inj_on, - filter_card_add_filter_neg_card_eq_card], - exact (erase_inj_on' _).mono (λ s hs, (mem_filter.1 hs).2), -end - -end finset - -open finset - lemma is_lower_set.non_member_subfamily (h : is_lower_set (𝒜 : set (finset α))) : is_lower_set (𝒜.non_member_subfamily a : set (finset α)) := λ s t hts, by { simp_rw [mem_coe, mem_non_member_subfamily], exact and.imp (h hts) (mt $ @hts _) } @@ -108,8 +60,8 @@ begin obtain rfl | rfl := hℬs, { simp only [card_empty, inter_empty, mul_zero, zero_mul] }, { simp only [card_empty, pow_zero, inter_singleton_of_mem, mem_singleton, card_singleton] } }, - rw [card_insert_of_not_mem hs, ←card_member_subfamily_add_card_non_member_subfamily 𝒜 a, - ←card_member_subfamily_add_card_non_member_subfamily ℬ a, add_mul, mul_add, mul_add, + rw [card_insert_of_not_mem hs, ←card_member_subfamily_add_card_non_member_subfamily a 𝒜, + ←card_member_subfamily_add_card_non_member_subfamily a ℬ, add_mul, mul_add, mul_add, add_comm (_ * _), add_add_add_comm], refine (add_le_add_right (mul_add_mul_le_mul_add_mul (card_le_of_subset h𝒜.member_subfamily_subset_non_member_subfamily) $ diff --git a/formal/lean/mathlib/combinatorics/simple_graph/density.lean b/formal/lean/mathlib/combinatorics/simple_graph/density.lean index 5931847b14f168900c623746452deef124f4420f..e46e30d36ad9edd00ca08bc0c8dbf9532b5f12cc 100644 --- a/formal/lean/mathlib/combinatorics/simple_graph/density.lean +++ b/formal/lean/mathlib/combinatorics/simple_graph/density.lean @@ -33,8 +33,7 @@ variables (r : α → β → Prop) [Π a, decidable_pred (r a)] {s s₁ s₂ : f {a : α} {b : β} {δ : ℚ} /-- Finset of edges of a relation between two finsets of vertices. -/ -def interedges (s : finset α) (t : finset β) : finset (α × β) := -(s.product t).filter $ λ e, r e.1 e.2 +def interedges (s : finset α) (t : finset β) : finset (α × β) := (s ×ˢ t).filter $ λ e, r e.1 e.2 /-- Edge density of a relation between two finsets of vertices. -/ def edge_density (s : finset α) (t : finset β) : ℚ := (interedges r s t).card / (s.card * t.card) @@ -93,7 +92,7 @@ ext $ λ a, by simp only [mem_interedges_iff, mem_bUnion, ←exists_and_distrib_ lemma interedges_bUnion (s : finset ι) (t : finset κ) (f : ι → finset α) (g : κ → finset β) : interedges r (s.bUnion f) (t.bUnion g) = - (s.product t).bUnion (λ ab, interedges r (f ab.1) (g ab.2)) := + (s ×ˢ t).bUnion (λ ab, interedges r (f ab.1) (g ab.2)) := by simp_rw [product_bUnion, interedges_bUnion_left, interedges_bUnion_right] end decidable_eq @@ -143,7 +142,7 @@ end lemma card_interedges_finpartition [decidable_eq α] [decidable_eq β] (P : finpartition s) (Q : finpartition t) : - (interedges r s t).card = ∑ ab in P.parts.product Q.parts, (interedges r ab.1 ab.2).card := + (interedges r s t).card = ∑ ab in P.parts ×ˢ Q.parts, (interedges r ab.1 ab.2).card := by simp_rw [card_interedges_finpartition_left _ P, card_interedges_finpartition_right _ _ Q, sum_product] @@ -262,7 +261,7 @@ def interedges (s t : finset α) : finset (α × α) := interedges G.adj s t def edge_density : finset α → finset α → ℚ := edge_density G.adj lemma interedges_def (s t : finset α) : - G.interedges s t = (s.product t).filter (λ e, G.adj e.1 e.2) := rfl + G.interedges s t = (s ×ˢ t).filter (λ e, G.adj e.1 e.2) := rfl lemma edge_density_def (s t : finset α) : G.edge_density s t = (G.interedges s t).card / (s.card * t.card) := rfl @@ -302,14 +301,14 @@ interedges_bUnion_right _ _ _ _ lemma interedges_bUnion (s : finset ι) (t : finset κ) (f : ι → finset α) (g : κ → finset α) : G.interedges (s.bUnion f) (t.bUnion g) = - (s.product t).bUnion (λ ab, G.interedges (f ab.1) (g ab.2)) := + (s ×ˢ t).bUnion (λ ab, G.interedges (f ab.1) (g ab.2)) := interedges_bUnion _ _ _ _ _ lemma card_interedges_add_card_interedges_compl (h : disjoint s t) : (G.interedges s t).card + (Gᶜ.interedges s t).card = s.card * t.card := begin rw [←card_product, interedges_def, interedges_def], - have : (s.product t).filter (λ e , Gᶜ.adj e.1 e.2) = (s.product t).filter (λ e , ¬ G.adj e.1 e.2), + have : (s ×ˢ t).filter (λ e , Gᶜ.adj e.1 e.2) = (s ×ˢ t).filter (λ e , ¬ G.adj e.1 e.2), { refine filter_congr (λ x hx, _), rw mem_product at hx, rw [compl_adj, and_iff_right (h.forall_ne_finset hx.1 hx.2)] }, diff --git a/formal/lean/mathlib/computability/turing_machine.lean b/formal/lean/mathlib/computability/turing_machine.lean index fbdc015692744cc46c0a317f5d83b5ca7654ca28..87f740cd64b8a204fb00bde7927252accf3c74c8 100644 --- a/formal/lean/mathlib/computability/turing_machine.lean +++ b/formal/lean/mathlib/computability/turing_machine.lean @@ -1284,8 +1284,7 @@ end variables [fintype σ] /-- Given a finite set of accessible `Λ` machine states, there is a finite set of accessible machine states in the target (even though the type `Λ'` is infinite). -/ -noncomputable def tr_stmts (S : finset Λ) : finset Λ' := -(TM1.stmts M S).product finset.univ +noncomputable def tr_stmts (S : finset Λ) : finset Λ' := TM1.stmts M S ×ˢ finset.univ open_locale classical local attribute [simp] TM1.stmts₁_self diff --git a/formal/lean/mathlib/data/bundle.lean b/formal/lean/mathlib/data/bundle.lean index 5f9810be8eca69aefde0ec190120dae4d1aa38b7..946d03f62ac00213f9527f43f6d282846d373b4a 100644 --- a/formal/lean/mathlib/data/bundle.lean +++ b/formal/lean/mathlib/data/bundle.lean @@ -89,7 +89,7 @@ variable {B' : Type*} /-- The pullback of a bundle `E` over a base `B` under a map `f : B' → B`, denoted by `pullback f E` or `f *ᵖ E`, is the bundle over `B'` whose fiber over `b'` is `E (f b')`. -/ -@[nolint has_inhabited_instance] def pullback (f : B' → B) (E : B → Type*) := λ x, E (f x) +@[nolint has_nonempty_instance] def pullback (f : B' → B) (E : B → Type*) := λ x, E (f x) notation f ` *ᵖ ` E := pullback f E diff --git a/formal/lean/mathlib/data/complex/is_R_or_C.lean b/formal/lean/mathlib/data/complex/is_R_or_C.lean index 1395d096a85bcfdf9cf4014f60ec952dd8f15de6..ad1c371b32cc8f7da5bbbdcb049d1812f564fe5c 100644 --- a/formal/lean/mathlib/data/complex/is_R_or_C.lean +++ b/formal/lean/mathlib/data/complex/is_R_or_C.lean @@ -47,7 +47,7 @@ open_locale complex_conjugate This typeclass captures properties shared by ℝ and ℂ, with an API that closely matches that of ℂ. -/ class is_R_or_C (K : Type*) - extends nontrivially_normed_field K, star_ring K, normed_algebra ℝ K, complete_space K := + extends densely_normed_field K, star_ring K, normed_algebra ℝ K, complete_space K := (re : K →+ ℝ) (im : K →+ ℝ) (I : K) -- Meant to be set to 0 for K=ℝ @@ -70,7 +70,7 @@ end mk_simp_attribute is_R_or_C_simps "Simp attribute for lemmas about `is_R_or_C`" -variables {K : Type*} [is_R_or_C K] +variables {K E : Type*} [is_R_or_C K] namespace is_R_or_C @@ -83,6 +83,13 @@ See Note [coercion into rings], or `data/nat/cast.lean` for more details. -/ lemma of_real_alg (x : ℝ) : (x : K) = x • (1 : K) := algebra.algebra_map_eq_smul_one x +lemma real_smul_eq_coe_mul (r : ℝ) (z : K) : r • z = (r : K) * z := +by rw [is_R_or_C.of_real_alg, ←smul_eq_mul, smul_assoc, smul_eq_mul, one_mul] + +lemma real_smul_eq_coe_smul [add_comm_group E] [module K E] [module ℝ E] [is_scalar_tower ℝ K E] + (r : ℝ) (x : E) : r • x = (r : K) • x := +by rw [is_R_or_C.of_real_alg, smul_one_smul] + lemma algebra_map_eq_of_real : ⇑(algebra_map ℝ K) = coe := rfl @[simp, is_R_or_C_simps] lemma re_add_im (z : K) : ((re z) : K) + (im z) * I = z := @@ -699,9 +706,9 @@ library_note "is_R_or_C instance" simp [re_add_im a, algebra.smul_def, algebra_map_eq_of_real] end⟩⟩ -variables (K) (E : Type*) [normed_add_comm_group E] [normed_space K E] +variables (K E) [normed_add_comm_group E] [normed_space K E] -/-- A finite dimensional vector space Over an `is_R_or_C` is a proper metric space. +/-- A finite dimensional vector space over an `is_R_or_C` is a proper metric space. This is not an instance because it would cause a search for `finite_dimensional ?x E` before `is_R_or_C ?x`. -/ @@ -745,7 +752,7 @@ noncomputable instance real.is_R_or_C : is_R_or_C ℝ := abs_mul_abs_self, ←div_eq_mul_inv, algebra.id.map_eq_id, id.def, ring_hom.id_apply, div_self_mul_self'], div_I_ax := λ z, by simp only [div_zero, mul_zero, neg_zero], - .. real.nontrivially_normed_field, .. real.metric_space } + .. real.densely_normed_field, .. real.metric_space } end instances diff --git a/formal/lean/mathlib/data/countable/small.lean b/formal/lean/mathlib/data/countable/small.lean index bea5bd047170d6fcc8ffcd3bd648364388166b0c..7629e4784ead6a24a00ea7a0a056821c16d8226f 100644 --- a/formal/lean/mathlib/data/countable/small.lean +++ b/formal/lean/mathlib/data/countable/small.lean @@ -7,9 +7,9 @@ import data.countable.basic import logic.small /-! -# All encodable types are small. +# All countable types are small. -That is, any encodable type is equivalent to a type in any universe. +That is, any countable type is equivalent to a type in any universe. -/ universes w v diff --git a/formal/lean/mathlib/data/fin/basic.lean b/formal/lean/mathlib/data/fin/basic.lean index f00f9a20db11ee35640ad745beb794e23dac6aef..69ba83256b4af0e94dc8fede1a6f0de98a869b1a 100644 --- a/formal/lean/mathlib/data/fin/basic.lean +++ b/formal/lean/mathlib/data/fin/basic.lean @@ -1234,26 +1234,44 @@ begin rwa subtype.ext_iff at h end +lemma coe_sub_iff_le {n : ℕ} {a b : fin n} : + (↑(a - b) : ℕ) = a - b ↔ b ≤ a := +begin + cases n, {exact fin_zero_elim a}, + rw [le_iff_coe_le_coe, fin.coe_sub, ←add_tsub_assoc_of_le b.is_lt.le], + cases le_or_lt (b : ℕ) a with h h, + { simp [←tsub_add_eq_add_tsub h, h, nat.mod_eq_of_lt ((nat.sub_le _ _).trans_lt a.is_lt)] }, + { rw [nat.mod_eq_of_lt, tsub_eq_zero_of_le h.le, tsub_eq_zero_iff_le, ←not_iff_not], + { simpa [b.is_lt.trans_le (le_add_self)] using h }, + { rwa [tsub_lt_iff_left (b.is_lt.le.trans (le_add_self)), add_lt_add_iff_right] } } +end + +lemma coe_sub_iff_lt {n : ℕ} {a b : fin n} : + (↑(a - b) : ℕ) = n + a - b ↔ a < b := +begin + cases n, {exact fin_zero_elim a}, + rw [lt_iff_coe_lt_coe, fin.coe_sub, add_comm], + cases le_or_lt (b : ℕ) a with h h, + { simpa [add_tsub_assoc_of_le h, ←not_le, h] + using ((nat.mod_lt _ (nat.succ_pos _)).trans_le le_self_add).ne }, + { simp [←tsub_tsub_assoc b.is_lt.le h.le, ←tsub_add_eq_add_tsub b.is_lt.le, + nat.mod_eq_of_lt (tsub_lt_self (nat.succ_pos _) (tsub_pos_of_lt h)), h] } +end + /-- By sending `x` to `last n - x`, `fin n` is order-equivalent to its `order_dual`. -/ def _root_.order_iso.fin_equiv : ∀ {n}, (fin n)ᵒᵈ ≃o fin n | 0 := ⟨⟨elim0, elim0, elim0, elim0⟩, elim0⟩ | (n+1) := order_iso.symm $ -{ to_fun := λ x, last n - x, - inv_fun := λ x, last n - x, +{ to_fun := λ x, order_dual.to_dual (last n - x), + inv_fun := λ x, last n - x.of_dual, left_inv := sub_sub_cancel _, right_inv := sub_sub_cancel _, map_rel_iff' := λ a b, begin - rw [order_dual.has_le], - simp only [equiv.coe_fn_mk], - rw [le_iff_coe_le_coe, fin.coe_sub, fin.coe_sub, coe_last], - have : (n - ↑b) % (n + 1) ≤ (n - ↑a) % (n + 1) ↔ a ≤ b, - { rw [nat.mod_eq_of_lt, nat.mod_eq_of_lt, tsub_le_tsub_iff_left a.is_le, - le_iff_coe_le_coe]; exact tsub_le_self.trans_lt n.lt_succ_self }, - suffices key : ∀ {x : fin (n + 1)}, (n + (n + 1 - x)) % (n + 1) = (n - x) % (n + 1), - { convert this using 2; exact key }, - intro x, - rw [add_comm, tsub_add_eq_add_tsub x.is_lt.le, add_tsub_assoc_of_le x.is_le, nat.add_mod_left] + simp only [equiv.coe_fn_mk, order_dual.to_dual_le_to_dual], + rw [le_iff_coe_le_coe, coe_sub_iff_le.mpr (le_last b), coe_sub_iff_le.mpr (le_last _), + tsub_le_tsub_iff_left, le_iff_coe_le_coe], + exact le_last _, end } lemma _root_.order_iso.fin_equiv_apply (a) : order_iso.fin_equiv a = last n - a.of_dual := rfl diff --git a/formal/lean/mathlib/data/fin_enum.lean b/formal/lean/mathlib/data/fin_enum.lean index 338b1b1bb5db00c2819bf267e3fd946c7f325f53..5bb070e8df17f520f83ff41eba388122b7d9c961 100644 --- a/formal/lean/mathlib/data/fin_enum.lean +++ b/formal/lean/mathlib/data/fin_enum.lean @@ -86,7 +86,7 @@ instance punit : fin_enum punit := of_list [punit.star] (λ x, by cases x; simp) instance prod {β} [fin_enum α] [fin_enum β] : fin_enum (α × β) := -of_list ( (to_list α).product (to_list β) ) (λ x, by cases x; simp) +of_list (to_list α ×ˢ to_list β) (λ x, by cases x; simp) instance sum {β} [fin_enum α] [fin_enum β] : fin_enum (α ⊕ β) := of_list ( (to_list α).map sum.inl ++ (to_list β).map sum.inr ) (λ x, by cases x; simp) diff --git a/formal/lean/mathlib/data/finite/basic.lean b/formal/lean/mathlib/data/finite/basic.lean index 34dc04a93081898fa0e6ff028c1e837bd0fafdf7..f3700752eea3933f295dbf3c2509de7329a8e047 100644 --- a/formal/lean/mathlib/data/finite/basic.lean +++ b/formal/lean/mathlib/data/finite/basic.lean @@ -64,11 +64,6 @@ not_finite_iff_infinite.mp h lemma not_infinite_iff_finite {α : Type*} : ¬ infinite α ↔ finite α := not_finite_iff_infinite.not_right.symm -lemma of_subsingleton {α : Sort*} [subsingleton α] : finite α := finite.of_equiv _ equiv.plift - -@[nolint instance_priority] -instance finite.prop (p : Prop) : finite p := of_subsingleton - namespace finite lemma exists_max [finite α] [nonempty α] [linear_order β] (f : α → β) : @@ -80,7 +75,11 @@ lemma exists_min [finite α] [nonempty α] [linear_order β] (f : α → β) : by { haveI := fintype.of_finite α, exact fintype.exists_min f } @[priority 100] -- see Note [lower instance priority] -instance of_is_empty {α : Sort*} [is_empty α] : finite α := finite.of_equiv _ equiv.plift +instance of_subsingleton {α : Sort*} [subsingleton α] : finite α := +of_injective (function.const α ()) $ function.injective_of_subsingleton _ + +@[nolint instance_priority] -- Higher priority for `Prop`s +instance prop (p : Prop) : finite p := finite.of_subsingleton instance [finite α] [finite β] : finite (α × β) := by { haveI := fintype.of_finite α, haveI := fintype.of_finite β, apply_instance } diff --git a/formal/lean/mathlib/data/finset/basic.lean b/formal/lean/mathlib/data/finset/basic.lean index 3c9773f8ccc69cfe2b8864ca52fb27edd8a1b32a..c45861de58ec9238dfb2b3673957f43bca921d73 100644 --- a/formal/lean/mathlib/data/finset/basic.lean +++ b/formal/lean/mathlib/data/finset/basic.lean @@ -146,8 +146,9 @@ namespace finset theorem eq_of_veq : ∀ {s t : finset α}, s.1 = t.1 → s = t | ⟨s, _⟩ ⟨t, _⟩ rfl := rfl -@[simp] theorem val_inj {s t : finset α} : s.1 = t.1 ↔ s = t := -⟨eq_of_veq, congr_arg _⟩ +theorem val_injective : injective (val : finset α → multiset α) := λ _ _, eq_of_veq + +@[simp] theorem val_inj {s t : finset α} : s.1 = t.1 ↔ s = t := val_injective.eq_iff @[simp] theorem dedup_eq_self [decidable_eq α] (s : finset α) : dedup s.1 = s.1 := s.2.dedup @@ -468,6 +469,12 @@ begin rw ← h_uniq hne.some hne.some_spec, exact hne.some_spec } end +lemma nonempty_iff_eq_singleton_default [unique α] {s : finset α} : + s.nonempty ↔ s = {default} := +by simp [eq_singleton_iff_nonempty_unique_mem] + +alias nonempty_iff_eq_singleton_default ↔ nonempty.eq_singleton_default _ + lemma singleton_iff_unique_mem (s : finset α) : (∃ a, s = {a}) ↔ ∃! a, a ∈ s := by simp only [eq_singleton_iff_unique_mem, exists_unique] @@ -1296,6 +1303,18 @@ by { ext, rw [mem_erase, mem_sdiff, mem_singleton], tauto } @[simp] lemma sdiff_singleton_not_mem_eq_self (s : finset α) {a : α} (ha : a ∉ s) : s \ {a} = s := by simp only [sdiff_singleton_eq_erase, ha, erase_eq_of_not_mem, not_false_iff] +lemma sdiff_sdiff_left' (s t u : finset α) : + (s \ t) \ u = (s \ t) ∩ (s \ u) := sdiff_sdiff_left' + +lemma sdiff_insert (s t : finset α) (x : α) : + s \ insert x t = (s \ t).erase x := +by simp_rw [← sdiff_singleton_eq_erase, insert_eq, + sdiff_sdiff_left', sdiff_union_distrib, inter_comm] + +lemma sdiff_insert_insert_of_mem_of_not_mem {s t : finset α} {x : α} (hxs : x ∈ s) (hxt : x ∉ t) : + insert x (s \ insert x t) = s \ t := +by rw [sdiff_insert, insert_erase (mem_sdiff.mpr ⟨hxs, hxt⟩)] + lemma sdiff_erase {x : α} (hx : x ∈ s) : s \ s.erase x = {x} := begin rw [← sdiff_singleton_eq_erase, sdiff_sdiff_right_self], diff --git a/formal/lean/mathlib/data/finset/functor.lean b/formal/lean/mathlib/data/finset/functor.lean index d3073272295ded2497f13c9b440201cd2435521f..5cdbde7163ccbb0296fe2c584a4b19fdacb90621 100644 --- a/formal/lean/mathlib/data/finset/functor.lean +++ b/formal/lean/mathlib/data/finset/functor.lean @@ -110,7 +110,7 @@ instance : is_comm_applicative finset := { commutative_prod := λ α β s t, begin simp_rw [seq_def, fmap_def, sup_image, sup_eq_bUnion], change s.bUnion (λ a, t.image $ λ b, (a, b)) = t.bUnion (λ b, s.image $ λ a, (a, b)), - transitivity s.product t; + transitivity s ×ˢ t; [rw product_eq_bUnion, rw product_eq_bUnion_right]; congr; ext; simp_rw mem_image, end, .. finset.is_lawful_applicative } diff --git a/formal/lean/mathlib/data/finset/lattice.lean b/formal/lean/mathlib/data/finset/lattice.lean index a5acdb750bd9ed75711a10efa7518fdd6fc4fd1b..ad60c8a44f2a0c4349df6b1a9e61987442b51d4f 100644 --- a/formal/lean/mathlib/data/finset/lattice.lean +++ b/formal/lean/mathlib/data/finset/lattice.lean @@ -116,7 +116,7 @@ end /-- See also `finset.product_bUnion`. -/ lemma sup_product_left (s : finset β) (t : finset γ) (f : β × γ → α) : - (s.product t).sup f = s.sup (λ i, t.sup $ λ i', f ⟨i, i'⟩) := + (s ×ˢ t).sup f = s.sup (λ i, t.sup $ λ i', f ⟨i, i'⟩) := begin refine le_antisymm _ (sup_le (λ i hi, sup_le $ λ i' hi', le_sup $ mem_product.2 ⟨hi, hi'⟩)), refine sup_le _, @@ -127,7 +127,7 @@ begin end lemma sup_product_right (s : finset β) (t : finset γ) (f : β × γ → α) : - (s.product t).sup f = t.sup (λ i', s.sup $ λ i, f ⟨i, i'⟩) := + (s ×ˢ t).sup f = t.sup (λ i', s.sup $ λ i, f ⟨i, i'⟩) := by rw [sup_product_left, sup_comm] @[simp] lemma sup_erase_bot [decidable_eq α] (s : finset α) : (s.erase ⊥).sup id = s.sup id := @@ -322,11 +322,11 @@ lemma inf_comm (s : finset β) (t : finset γ) (f : β → γ → α) : @sup_comm αᵒᵈ _ _ _ _ _ _ _ lemma inf_product_left (s : finset β) (t : finset γ) (f : β × γ → α) : - (s.product t).inf f = s.inf (λ i, t.inf $ λ i', f ⟨i, i'⟩) := + (s ×ˢ t).inf f = s.inf (λ i, t.inf $ λ i', f ⟨i, i'⟩) := @sup_product_left αᵒᵈ _ _ _ _ _ _ _ lemma inf_product_right (s : finset β) (t : finset γ) (f : β × γ → α) : - (s.product t).inf f = t.inf (λ i', s.inf $ λ i, f ⟨i, i'⟩) := + (s ×ˢ t).inf f = t.inf (λ i', s.inf $ λ i, f ⟨i, i'⟩) := @sup_product_right αᵒᵈ _ _ _ _ _ _ _ @[simp] lemma inf_erase_top [decidable_eq α] (s : finset α) : (s.erase ⊤).inf id = s.inf id := diff --git a/formal/lean/mathlib/data/finset/n_ary.lean b/formal/lean/mathlib/data/finset/n_ary.lean index 174012d5c2e3537cb949f481241ddb27ca1de893..b28c3a645e4be7a84e7f223ed48ba7b870d8600d 100644 --- a/formal/lean/mathlib/data/finset/n_ary.lean +++ b/formal/lean/mathlib/data/finset/n_ary.lean @@ -33,7 +33,7 @@ variables {α α' β β' γ γ' δ δ' ε ε' : Type*} /-- The image of a binary function `f : α → β → γ` as a function `finset α → finset β → finset γ`. Mathematically this should be thought of as the image of the corresponding function `α × β → γ`. -/ def image₂ (f : α → β → γ) (s : finset α) (t : finset β) : finset γ := -(s.product t).image $ uncurry f +(s ×ˢ t).image $ uncurry f @[simp] lemma mem_image₂ : c ∈ image₂ f s t ↔ ∃ a b, a ∈ s ∧ b ∈ t ∧ f a b = c := by simp [image₂, and_assoc] @@ -47,8 +47,7 @@ lemma card_image₂_le (f : α → β → γ) (s : finset α) (t : finset β) : card_image_le.trans_eq $ card_product _ _ lemma card_image₂_iff : - (image₂ f s t).card = s.card * t.card ↔ - ((s : set α) ×ˢ (t : set β) : set (α × β)).inj_on (λ x, f x.1 x.2) := + (image₂ f s t).card = s.card * t.card ↔ (s ×ˢ t : set (α × β)).inj_on (λ x, f x.1 x.2) := by { rw [←card_product, ←coe_product], exact card_image_iff } lemma card_image₂ (hf : injective2 f) (s : finset α) (t : finset β) : diff --git a/formal/lean/mathlib/data/finset/pointwise.lean b/formal/lean/mathlib/data/finset/pointwise.lean index cd6fed1ef11455bfb23e784df137385bb37d45af..5457384c38c7fd79ccd8cc4bb9814fd2c20bde3d 100644 --- a/formal/lean/mathlib/data/finset/pointwise.lean +++ b/formal/lean/mathlib/data/finset/pointwise.lean @@ -151,10 +151,10 @@ protected def has_mul : has_mul (finset α) := ⟨image₂ (*)⟩ localized "attribute [instance] finset.has_mul finset.has_add" in pointwise @[to_additive] -lemma mul_def : s * t = (s.product t).image (λ p : α × α, p.1 * p.2) := rfl +lemma mul_def : s * t = (s ×ˢ t).image (λ p : α × α, p.1 * p.2) := rfl @[to_additive] -lemma image_mul_product : (s.product t).image (λ x : α × α, x.fst * x.snd) = s * t := rfl +lemma image_mul_product : (s ×ˢ t).image (λ x : α × α, x.fst * x.snd) = s * t := rfl @[to_additive] lemma mem_mul {x : α} : x ∈ s * t ↔ ∃ y z, y ∈ s ∧ z ∈ t ∧ y * z = x := mem_image₂ @@ -167,7 +167,7 @@ lemma coe_mul (s t : finset α) : (↑(s * t) : set α) = ↑s * ↑t := coe_ima @[to_additive] lemma card_mul_iff : (s * t).card = s.card * t.card ↔ - ((s : set α) ×ˢ (t : set α) : set (α × α)).inj_on (λ p, p.1 * p.2) := card_image₂_iff + (s ×ˢ t : set (α × α)).inj_on (λ p, p.1 * p.2) := card_image₂_iff @[simp, to_additive] lemma empty_mul (s : finset α) : ∅ * s = ∅ := image₂_empty_left @[simp, to_additive] lemma mul_empty (s : finset α) : s * ∅ = ∅ := image₂_empty_right @@ -231,10 +231,10 @@ protected def has_div : has_div (finset α) := ⟨image₂ (/)⟩ localized "attribute [instance] finset.has_div finset.has_sub" in pointwise @[to_additive] -lemma div_def : s / t = (s.product t).image (λ p : α × α, p.1 / p.2) := rfl +lemma div_def : s / t = (s ×ˢ t).image (λ p : α × α, p.1 / p.2) := rfl @[to_additive add_image_prod] -lemma image_div_prod : (s.product t).image (λ x : α × α, x.fst / x.snd) = s / t := rfl +lemma image_div_prod : (s ×ˢ t).image (λ x : α × α, x.fst / x.snd) = s / t := rfl @[to_additive] lemma mem_div : a ∈ s / t ↔ ∃ b c, b ∈ s ∧ c ∈ t ∧ b / c = a := mem_image₂ @@ -392,6 +392,18 @@ begin exact ih.trans (subset_mul_right _ hs) } end +@[simp, norm_cast, to_additive] +lemma coe_list_prod (s : list (finset α)) : (↑s.prod : set α) = (s.map coe).prod := +map_list_prod (coe_monoid_hom : finset α →* set α) _ + +@[to_additive] lemma mem_prod_list_of_fn {a : α} {s : fin n → finset α} : + a ∈ (list.of_fn s).prod ↔ ∃ f : (Π i : fin n, s i), (list.of_fn (λ i, (f i : α))).prod = a := +by { rw [←mem_coe, coe_list_prod, list.map_of_fn, set.mem_prod_list_of_fn], refl } + +@[to_additive] lemma mem_pow {a : α} {n : ℕ} : + a ∈ s ^ n ↔ ∃ f : fin n → s, (list.of_fn (λ i, ↑(f i))).prod = a := +by { simp_rw [←mem_coe, coe_pow, set.mem_pow], refl } + @[simp, to_additive] lemma empty_pow (hn : n ≠ 0) : (∅ : finset α) ^ n = ∅ := by rw [←tsub_add_cancel_of_le (nat.succ_le_of_lt $ nat.pos_of_ne_zero hn), pow_succ, empty_mul] @@ -618,10 +630,10 @@ protected def has_smul : has_smul (finset α) (finset β) := ⟨image₂ (•) localized "attribute [instance] finset.has_smul finset.has_vadd" in pointwise -@[to_additive] lemma smul_def : s • t = (s.product t).image (λ p : α × β, p.1 • p.2) := rfl +@[to_additive] lemma smul_def : s • t = (s ×ˢ t).image (λ p : α × β, p.1 • p.2) := rfl @[to_additive] -lemma image_smul_product : (s.product t).image (λ x : α × β, x.fst • x.snd) = s • t := rfl +lemma image_smul_product : (s ×ˢ t).image (λ x : α × β, x.fst • x.snd) = s • t := rfl @[to_additive] lemma mem_smul {x : β} : x ∈ s • t ↔ ∃ y z, y ∈ s ∧ z ∈ t ∧ y • z = x := mem_image₂ @@ -810,20 +822,20 @@ instance smul_comm_class [has_smul α γ] [has_smul β γ] [smul_comm_class α smul_comm_class (finset α) (finset β) (finset γ) := ⟨λ s t u, coe_injective $ by simp_rw [coe_smul, smul_comm]⟩ -instance is_scalar_tower [has_smul α β] [has_smul α γ] [has_smul β γ] - [is_scalar_tower α β γ] : +@[to_additive] +instance is_scalar_tower [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] : is_scalar_tower α β (finset γ) := ⟨λ a b s, by simp only [←image_smul, image_image, smul_assoc]⟩ variables [decidable_eq β] -instance is_scalar_tower' [has_smul α β] [has_smul α γ] [has_smul β γ] - [is_scalar_tower α β γ] : +@[to_additive] +instance is_scalar_tower' [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] : is_scalar_tower α (finset β) (finset γ) := ⟨λ a s t, coe_injective $ by simp only [coe_smul_finset, coe_smul, smul_assoc]⟩ -instance is_scalar_tower'' [has_smul α β] [has_smul α γ] [has_smul β γ] - [is_scalar_tower α β γ] : +@[to_additive] +instance is_scalar_tower'' [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] : is_scalar_tower (finset α) (finset β) (finset γ) := ⟨λ a s t, coe_injective $ by simp only [coe_smul_finset, coe_smul, smul_assoc]⟩ @@ -890,7 +902,7 @@ section left_cancel_semigroup variables [left_cancel_semigroup α] [decidable_eq α] (s t : finset α) (a : α) @[to_additive] lemma pairwise_disjoint_smul_iff {s : set α} {t : finset α} : - s.pairwise_disjoint (• t) ↔ ((s : set α) ×ˢ (t : set α) : set (α × α)).inj_on (λ p, p.1 * p.2) := + s.pairwise_disjoint (• t) ↔ (s ×ˢ t : set (α × α)).inj_on (λ p, p.1 * p.2) := by simp_rw [←pairwise_disjoint_coe, coe_smul_finset, set.pairwise_disjoint_smul_iff] @[simp, to_additive] lemma card_singleton_mul : ({a} * t).card = t.card := diff --git a/formal/lean/mathlib/data/finset/prod.lean b/formal/lean/mathlib/data/finset/prod.lean index 9b4c0756134960f798108d6d4d5a1c501537fda4..885a9af98772b701e68fc665998cf4d6412fa7fa 100644 --- a/formal/lean/mathlib/data/finset/prod.lean +++ b/formal/lean/mathlib/data/finset/prod.lean @@ -33,55 +33,58 @@ variables {s s' : finset α} {t t' : finset β} {a : α} {b : β} /-- `product s t` is the set of pairs `(a, b)` such that `a ∈ s` and `b ∈ t`. -/ protected def product (s : finset α) (t : finset β) : finset (α × β) := ⟨_, s.nodup.product t.nodup⟩ -@[simp] lemma product_val : (s.product t).1 = s.1.product t.1 := rfl +/- This notation binds more strongly than (pre)images, unions and intersections. -/ +infixr ` ×ˢ `:82 := finset.product -@[simp] lemma mem_product {p : α × β} : p ∈ s.product t ↔ p.1 ∈ s ∧ p.2 ∈ t := mem_product +@[simp] lemma product_val : (s ×ˢ t).1 = s.1 ×ˢ t.1 := rfl -lemma mk_mem_product (ha : a ∈ s) (hb : b ∈ t) : (a, b) ∈ s.product t := mem_product.2 ⟨ha, hb⟩ +@[simp] lemma mem_product {p : α × β} : p ∈ s ×ˢ t ↔ p.1 ∈ s ∧ p.2 ∈ t := mem_product + +lemma mk_mem_product (ha : a ∈ s) (hb : b ∈ t) : (a, b) ∈ s ×ˢ t := mem_product.2 ⟨ha, hb⟩ @[simp, norm_cast] lemma coe_product (s : finset α) (t : finset β) : - (s.product t : set (α × β)) = (s : set α) ×ˢ (t : set β) := + (↑(s ×ˢ t) : set (α × β)) = s ×ˢ t := set.ext $ λ x, finset.mem_product lemma subset_product [decidable_eq α] [decidable_eq β] {s : finset (α × β)} : - s ⊆ (s.image prod.fst).product (s.image prod.snd) := + s ⊆ s.image prod.fst ×ˢ s.image prod.snd := λ p hp, mem_product.2 ⟨mem_image_of_mem _ hp, mem_image_of_mem _ hp⟩ -lemma product_subset_product (hs : s ⊆ s') (ht : t ⊆ t') : s.product t ⊆ s'.product t' := +lemma product_subset_product (hs : s ⊆ s') (ht : t ⊆ t') : s ×ˢ t ⊆ s' ×ˢ t' := λ ⟨x,y⟩ h, mem_product.2 ⟨hs (mem_product.1 h).1, ht (mem_product.1 h).2⟩ -lemma product_subset_product_left (hs : s ⊆ s') : s.product t ⊆ s'.product t := +lemma product_subset_product_left (hs : s ⊆ s') : s ×ˢ t ⊆ s' ×ˢ t := product_subset_product hs (subset.refl _) -lemma product_subset_product_right (ht : t ⊆ t') : s.product t ⊆ s.product t' := +lemma product_subset_product_right (ht : t ⊆ t') : s ×ˢ t ⊆ s ×ˢ t' := product_subset_product (subset.refl _) ht lemma product_eq_bUnion [decidable_eq α] [decidable_eq β] (s : finset α) (t : finset β) : - s.product t = s.bUnion (λa, t.image $ λb, (a, b)) := + s ×ˢ t = s.bUnion (λa, t.image $ λb, (a, b)) := ext $ λ ⟨x, y⟩, by simp only [mem_product, mem_bUnion, mem_image, exists_prop, prod.mk.inj_iff, and.left_comm, exists_and_distrib_left, exists_eq_right, exists_eq_left] lemma product_eq_bUnion_right [decidable_eq α] [decidable_eq β] (s : finset α) (t : finset β) : - s.product t = t.bUnion (λ b, s.image $ λ a, (a, b)) := + s ×ˢ t = t.bUnion (λ b, s.image $ λ a, (a, b)) := ext $ λ ⟨x, y⟩, by simp only [mem_product, mem_bUnion, mem_image, exists_prop, prod.mk.inj_iff, and.left_comm, exists_and_distrib_left, exists_eq_right, exists_eq_left] /-- See also `finset.sup_product_left`. -/ @[simp] lemma product_bUnion [decidable_eq γ] (s : finset α) (t : finset β) (f : α × β → finset γ) : - (s.product t).bUnion f = s.bUnion (λ a, t.bUnion (λ b, f (a, b))) := + (s ×ˢ t).bUnion f = s.bUnion (λ a, t.bUnion (λ b, f (a, b))) := by { classical, simp_rw [product_eq_bUnion, bUnion_bUnion, image_bUnion] } -@[simp] lemma card_product (s : finset α) (t : finset β) : card (s.product t) = card s * card t := +@[simp] lemma card_product (s : finset α) (t : finset β) : card (s ×ˢ t) = card s * card t := multiset.card_product _ _ lemma filter_product (p : α → Prop) (q : β → Prop) [decidable_pred p] [decidable_pred q] : - (s.product t).filter (λ (x : α × β), p x.1 ∧ q x.2) = (s.filter p).product (t.filter q) := + (s ×ˢ t).filter (λ (x : α × β), p x.1 ∧ q x.2) = s.filter p ×ˢ t.filter q := by { ext ⟨a, b⟩, simp only [mem_filter, mem_product], exact and_and_and_comm (a ∈ s) (b ∈ t) (p a) (q b) } lemma filter_product_card (s : finset α) (t : finset β) (p : α → Prop) (q : β → Prop) [decidable_pred p] [decidable_pred q] : - ((s.product t).filter (λ (x : α × β), p x.1 ↔ q x.2)).card = + ((s ×ˢ t).filter (λ (x : α × β), p x.1 ↔ q x.2)).card = (s.filter p).card * (t.filter q).card + (s.filter (not ∘ p)).card * (t.filter (not ∘ q)).card := begin classical, @@ -95,45 +98,45 @@ begin not_mem_empty], intros, assumption } end -lemma empty_product (t : finset β) : (∅ : finset α).product t = ∅ := rfl +lemma empty_product (t : finset β) : (∅ : finset α) ×ˢ t = ∅ := rfl -lemma product_empty (s : finset α) : s.product (∅ : finset β) = ∅ := +lemma product_empty (s : finset α) : s ×ˢ (∅ : finset β) = ∅ := eq_empty_of_forall_not_mem (λ x h, (finset.mem_product.1 h).2) -lemma nonempty.product (hs : s.nonempty) (ht : t.nonempty) : (s.product t).nonempty := +lemma nonempty.product (hs : s.nonempty) (ht : t.nonempty) : (s ×ˢ t).nonempty := let ⟨x, hx⟩ := hs, ⟨y, hy⟩ := ht in ⟨(x, y), mem_product.2 ⟨hx, hy⟩⟩ -lemma nonempty.fst (h : (s.product t).nonempty) : s.nonempty := +lemma nonempty.fst (h : (s ×ˢ t).nonempty) : s.nonempty := let ⟨xy, hxy⟩ := h in ⟨xy.1, (mem_product.1 hxy).1⟩ -lemma nonempty.snd (h : (s.product t).nonempty) : t.nonempty := +lemma nonempty.snd (h : (s ×ˢ t).nonempty) : t.nonempty := let ⟨xy, hxy⟩ := h in ⟨xy.2, (mem_product.1 hxy).2⟩ -@[simp] lemma nonempty_product : (s.product t).nonempty ↔ s.nonempty ∧ t.nonempty := +@[simp] lemma nonempty_product : (s ×ˢ t).nonempty ↔ s.nonempty ∧ t.nonempty := ⟨λ h, ⟨h.fst, h.snd⟩, λ h, h.1.product h.2⟩ -@[simp] lemma product_eq_empty {s : finset α} {t : finset β} : s.product t = ∅ ↔ s = ∅ ∨ t = ∅ := +@[simp] lemma product_eq_empty {s : finset α} {t : finset β} : s ×ˢ t = ∅ ↔ s = ∅ ∨ t = ∅ := by rw [←not_nonempty_iff_eq_empty, nonempty_product, not_and_distrib, not_nonempty_iff_eq_empty, not_nonempty_iff_eq_empty] @[simp] lemma singleton_product {a : α} : - ({a} : finset α).product t = t.map ⟨prod.mk a, prod.mk.inj_left _⟩ := + ({a} : finset α) ×ˢ t = t.map ⟨prod.mk a, prod.mk.inj_left _⟩ := by { ext ⟨x, y⟩, simp [and.left_comm, eq_comm] } @[simp] lemma product_singleton {b : β} : - s.product {b} = s.map ⟨λ i, (i, b), prod.mk.inj_right _⟩ := + s ×ˢ {b} = s.map ⟨λ i, (i, b), prod.mk.inj_right _⟩ := by { ext ⟨x, y⟩, simp [and.left_comm, eq_comm] } lemma singleton_product_singleton {a : α} {b : β} : - ({a} : finset α).product ({b} : finset β) = {(a, b)} := + ({a} : finset α) ×ˢ ({b} : finset β) = {(a, b)} := by simp only [product_singleton, function.embedding.coe_fn_mk, map_singleton] @[simp] lemma union_product [decidable_eq α] [decidable_eq β] : - (s ∪ s').product t = s.product t ∪ s'.product t := + (s ∪ s') ×ˢ t = s ×ˢ t ∪ s' ×ˢ t := by { ext ⟨x, y⟩, simp only [or_and_distrib_right, mem_union, mem_product] } @[simp] lemma product_union [decidable_eq α] [decidable_eq β] : - s.product (t ∪ t') = s.product t ∪ s.product t' := + s ×ˢ (t ∪ t') = s ×ˢ t ∪ s ×ˢ t' := by { ext ⟨x, y⟩, simp only [and_or_distrib_left, mem_union, mem_product] } end prod @@ -143,11 +146,11 @@ variables (s t : finset α) [decidable_eq α] /-- Given a finite set `s`, the diagonal, `s.diag` is the set of pairs of the form `(a, a)` for `a ∈ s`. -/ -def diag := (s.product s).filter (λ (a : α × α), a.fst = a.snd) +def diag := (s ×ˢ s).filter (λ a : α × α, a.fst = a.snd) /-- Given a finite set `s`, the off-diagonal, `s.off_diag` is the set of pairs `(a, b)` with `a ≠ b` for `a, b ∈ s`. -/ -def off_diag := (s.product s).filter (λ (a : α × α), a.fst ≠ a.snd) +def off_diag := (s ×ˢ s).filter (λ (a : α × α), a.fst ≠ a.snd) @[simp] lemma mem_diag (x : α × α) : x ∈ s.diag ↔ x.1 ∈ s ∧ x.1 = x.2 := by { simp only [diag, mem_filter, mem_product], split; intros h; @@ -178,16 +181,16 @@ end @[simp] lemma off_diag_empty : (∅ : finset α).off_diag = ∅ := rfl -@[simp] lemma diag_union_off_diag : s.diag ∪ s.off_diag = s.product s := +@[simp] lemma diag_union_off_diag : s.diag ∪ s.off_diag = s ×ˢ s := filter_union_filter_neg_eq _ _ @[simp] lemma disjoint_diag_off_diag : disjoint s.diag s.off_diag := disjoint_filter_filter_neg _ _ -lemma product_sdiff_diag : s.product s \ s.diag = s.off_diag := +lemma product_sdiff_diag : s ×ˢ s \ s.diag = s.off_diag := by rw [←diag_union_off_diag, union_comm, union_sdiff_self, sdiff_eq_self_of_disjoint (disjoint_diag_off_diag _).symm] -lemma product_sdiff_off_diag : s.product s \ s.off_diag = s.diag := +lemma product_sdiff_off_diag : s ×ˢ s \ s.off_diag = s.diag := by rw [←diag_union_off_diag, union_sdiff_self, sdiff_eq_self_of_disjoint (disjoint_diag_off_diag _)] lemma diag_union : (s ∪ t).diag = s.diag ∪ t.diag := @@ -196,10 +199,10 @@ by { ext ⟨i, j⟩, simp only [mem_diag, mem_union, or_and_distrib_right] } variables {s t} lemma off_diag_union (h : disjoint s t) : - (s ∪ t).off_diag = s.off_diag ∪ t.off_diag ∪ s.product t ∪ t.product s := + (s ∪ t).off_diag = s.off_diag ∪ t.off_diag ∪ s ×ˢ t ∪ t ×ˢ s := begin - rw [off_diag, union_product, product_union, product_union, union_comm _ (t.product t), - union_assoc, union_left_comm (s.product t), ←union_assoc, filter_union, filter_union, ←off_diag, + rw [off_diag, union_product, product_union, product_union, union_comm _ (t ×ˢ t), + union_assoc, union_left_comm (s ×ˢ t), ←union_assoc, filter_union, filter_union, ←off_diag, ←off_diag, filter_true_of_mem, ←union_assoc], simp only [mem_union, mem_product, ne.def, prod.forall], rintro i j (⟨hi, hj⟩ | ⟨hi, hj⟩), @@ -219,7 +222,7 @@ lemma diag_insert : (insert a s).diag = insert (a, a) s.diag := by rw [insert_eq, insert_eq, diag_union, diag_singleton] lemma off_diag_insert (has : a ∉ s) : - (insert a s).off_diag = s.off_diag ∪ ({a} : finset α).product s ∪ s.product {a} := + (insert a s).off_diag = s.off_diag ∪ {a} ×ˢ s ∪ s ×ˢ {a} := by rw [insert_eq, union_comm, off_diag_union (disjoint_singleton_right.2 has), off_diag_singleton, union_empty, union_right_comm] diff --git a/formal/lean/mathlib/data/finset/sym.lean b/formal/lean/mathlib/data/finset/sym.lean index 73d53a96cc41ad824779a7928790f91fad7cb49a..83c74fe19d923bf40fe8857be4a8122c7dfe9e30 100644 --- a/formal/lean/mathlib/data/finset/sym.lean +++ b/formal/lean/mathlib/data/finset/sym.lean @@ -37,7 +37,7 @@ section sym2 variables {m : sym2 α} /-- Lifts a finset to `sym2 α`. `s.sym2` is the finset of all pairs with elements in `s`. -/ -protected def sym2 (s : finset α) : finset (sym2 α) := (s.product s).image quotient.mk +protected def sym2 (s : finset α) : finset (sym2 α) := (s ×ˢ s).image quotient.mk @[simp] lemma mem_sym2_iff : m ∈ s.sym2 ↔ ∀ a ∈ m, a ∈ s := begin diff --git a/formal/lean/mathlib/data/finsupp/basic.lean b/formal/lean/mathlib/data/finsupp/basic.lean index e4b7bfbff3da9b3b95112840c48992b3da8de1ab..67b0753c1ff728b6b6d86687cefe886971085b9a 100644 --- a/formal/lean/mathlib/data/finsupp/basic.lean +++ b/formal/lean/mathlib/data/finsupp/basic.lean @@ -6,6 +6,7 @@ Authors: Johannes Hölzl, Scott Morrison import algebra.hom.group_action import algebra.indicator_function import data.finset.preimage +import data.list.alist /-! # Type of functions with finite support @@ -774,8 +775,71 @@ end @[simp] lemma graph_eq_empty {f : α →₀ M} : f.graph = ∅ ↔ f = 0 := (graph_injective α M).eq_iff' graph_zero +/-- Produce an association list for the finsupp over its support using choice. -/ +@[simps] def to_alist (f : α →₀ M) : alist (λ x : α, M) := +⟨f.graph.to_list.map prod.to_sigma, begin + rw [list.nodupkeys, list.keys, list.map_map, prod.fst_comp_to_sigma, list.nodup_map_iff_inj_on], + { rintros ⟨b, m⟩ hb ⟨c, n⟩ hc (rfl : b = c), + rw [mem_to_list, finsupp.mem_graph_iff] at hb hc, + dsimp at hb hc, + rw [←hc.1, hb.1] }, + { apply nodup_to_list } +end⟩ + +@[simp] lemma to_alist_keys_to_finset (f : α →₀ M) : f.to_alist.keys.to_finset = f.support := +by { ext, simp [to_alist, alist.mem_keys, alist.keys, list.keys] } + +@[simp] lemma mem_to_alist {f : α →₀ M} {x : α} : x ∈ f.to_alist ↔ f x ≠ 0 := +by rw [alist.mem_keys, ←list.mem_to_finset, to_alist_keys_to_finset, mem_support_iff] + end graph +end finsupp + +/-! ### Declarations about `alist.lookup_finsupp` -/ + +section lookup_finsupp + +variable [has_zero M] + +namespace alist +open list + +/-- Converts an association list into a finitely supported function via `alist.lookup`, sending +absent keys to zero. -/ +@[simps] def lookup_finsupp (l : alist (λ x : α, M)) : α →₀ M := +{ support := (l.1.filter $ λ x, sigma.snd x ≠ 0).keys.to_finset, + to_fun := λ a, (l.lookup a).get_or_else 0, + mem_support_to_fun := λ a, begin + simp_rw [mem_to_finset, list.mem_keys, list.mem_filter, ←mem_lookup_iff], + cases lookup a l; + simp + end } + +alias lookup_finsupp_to_fun ← lookup_finsupp_apply + +@[simp] lemma to_alist_lookup_finsupp (f : α →₀ M) : f.to_alist.lookup_finsupp = f := +begin + ext, + by_cases h : f a = 0, + { suffices : f.to_alist.lookup a = none, + { simp [h, this] }, + { simp [lookup_eq_none, h] } }, + { suffices : f.to_alist.lookup a = some (f a), + { simp [h, this] }, + { apply mem_lookup_iff.2, + simpa using h } } +end + +lemma lookup_finsupp_surjective : surjective (@lookup_finsupp α M _) := +λ f, ⟨_, to_alist_lookup_finsupp f⟩ + +end alist + +end lookup_finsupp + +namespace finsupp + /-! ### Declarations about `sum` and `prod` @@ -1471,6 +1535,18 @@ have ∀ {f1 f2 : α →₀ M}, disjoint f1.support f2.support → by simp_rw [← this hd, ← this hd.symm, add_comm (f2 _), finsupp.prod, support_add_eq hd, prod_union hd, add_apply] +lemma prod_dvd_prod_of_subset_of_dvd [add_comm_monoid M] [comm_monoid N] + {f1 f2 : α →₀ M} {g1 g2 : α → M → N} (h1 : f1.support ⊆ f2.support) + (h2 : ∀ (a : α), a ∈ f1.support → g1 a (f1 a) ∣ g2 a (f2 a)) : + f1.prod g1 ∣ f2.prod g2 := +begin + simp only [finsupp.prod, finsupp.prod_mul], + rw [←sdiff_union_of_subset h1, prod_union sdiff_disjoint], + apply dvd_mul_of_dvd_right, + apply prod_dvd_prod_of_dvd, + exact h2, +end + section map_range section equiv diff --git a/formal/lean/mathlib/data/fintype/basic.lean b/formal/lean/mathlib/data/fintype/basic.lean index 526416a162bec707b283f0be7417e48d0cd0a08f..564a7e6accf020145534d4ce5331c87285671e96 100644 --- a/formal/lean/mathlib/data/fintype/basic.lean +++ b/formal/lean/mathlib/data/fintype/basic.lean @@ -619,17 +619,6 @@ arbitrary `fintype` instances, use `finset.univ_eq_empty`. -/ arbitrary `fintype` instances, use `fintype.card_eq_zero_iff`. -/ @[simp] theorem card_of_is_empty [is_empty α] : fintype.card α = 0 := rfl -open_locale classical -variables (α) - -/-- Any subsingleton type is (noncomputably) a fintype (with zero or one term). -/ -@[priority 5] -- see Note [lower instance priority] -noncomputable instance of_subsingleton' [subsingleton α] : fintype α := -if h : nonempty α then - of_subsingleton (nonempty.some h) -else - @fintype.of_is_empty _ $ not_nonempty_iff.mp h - end fintype namespace set @@ -918,10 +907,10 @@ instance {α : Type*} (β : α → Type*) (univ : finset α).sigma (λ a, (univ : finset (β a))) = univ := rfl instance (α β : Type*) [fintype α] [fintype β] : fintype (α × β) := -⟨univ.product univ, λ ⟨a, b⟩, by simp⟩ +⟨univ ×ˢ univ, λ ⟨a, b⟩, by simp⟩ @[simp] lemma finset.univ_product_univ {α β : Type*} [fintype α] [fintype β] : - (univ : finset α).product (univ : finset β) = univ := + (univ : finset α) ×ˢ (univ : finset β) = univ := rfl @[simp] theorem fintype.card_prod (α β : Type*) [fintype α] [fintype β] : @@ -1488,11 +1477,8 @@ fintype.of_surjective quotient.mk (λ x, quotient.induction_on x (λ x, ⟨x, rf instance finset.fintype [fintype α] : fintype (finset α) := ⟨univ.powerset, λ x, finset.mem_powerset.2 (finset.subset_univ _)⟩ --- irreducible due to this conversation on Zulip: --- https://leanprover.zulipchat.com/#narrow/stream/113488-general/ --- topic/.60simp.60.20ignoring.20lemmas.3F/near/241824115 -@[irreducible] instance function.embedding.fintype {α β} [fintype α] [fintype β] - [decidable_eq α] [decidable_eq β] : fintype (α ↪ β) := +instance function.embedding.fintype {α β} [fintype α] [fintype β] [decidable_eq α] + [decidable_eq β] : fintype (α ↪ β) := fintype.of_equiv _ (equiv.subtype_injective_equiv_embedding α β) instance [decidable_eq α] [fintype α] {n : ℕ} : fintype (sym.sym' α n) := @@ -2067,10 +2053,27 @@ end noncomputable def nat_embedding (α : Type*) [infinite α] : ℕ ↪ α := ⟨_, nat_embedding_aux_injective α⟩ +/-- See `infinite.exists_superset_card_eq` for a version that, for a `s : finset α`, +provides a superset `t : finset α`, `s ⊆ t` such that `t.card` is fixed. -/ lemma exists_subset_card_eq (α : Type*) [infinite α] (n : ℕ) : ∃ s : finset α, s.card = n := ⟨(range n).map (nat_embedding α), by rw [card_map, card_range]⟩ +/-- See `infinite.exists_subset_card_eq` for a version that provides an arbitrary +`s : finset α` for any cardinality. -/ +lemma exists_superset_card_eq [infinite α] (s : finset α) (n : ℕ) (hn : s.card ≤ n) : + ∃ t : finset α, s ⊆ t ∧ t.card = n := +begin + induction n with n IH generalizing s, + { exact ⟨s, subset_refl _, nat.eq_zero_of_le_zero hn⟩ }, + { cases hn.eq_or_lt with hn' hn', + { exact ⟨s, subset_refl _, hn'⟩ }, + obtain ⟨t, hs, ht⟩ := IH _ (nat.le_of_lt_succ hn'), + obtain ⟨x, hx⟩ := exists_not_mem_finset t, + refine ⟨finset.cons x t hx, hs.trans (finset.subset_cons _), _⟩, + simp [hx, ht] } +end + end infinite /-- If every finset in a type has bounded cardinality, that type is finite. -/ @@ -2104,17 +2107,9 @@ begin intros x y, contrapose, apply hf, end --- irreducible due to this conversation on Zulip: --- https://leanprover.zulipchat.com/#narrow/stream/113488-general/ --- topic/.60simp.60.20ignoring.20lemmas.3F/near/241824115 - -@[irreducible] instance function.embedding.is_empty {α β} [infinite α] [fintype β] : is_empty (α ↪ β) := - ⟨λ f, let ⟨x, y, ne, feq⟩ := fintype.exists_ne_map_eq_of_infinite f in ne $ f.injective feq⟩ +⟨λ f, let ⟨x, y, ne, feq⟩ := fintype.exists_ne_map_eq_of_infinite f in ne $ f.injective feq⟩ -@[priority 100] -noncomputable instance function.embedding.fintype' {α β : Type*} [fintype β] : fintype (α ↪ β) := -by casesI fintype_or_infinite α; apply_instance /-- The strong pigeonhole principle for infinitely many pigeons in finitely many pigeonholes. If there are infinitely many pigeons in diff --git a/formal/lean/mathlib/data/int/modeq.lean b/formal/lean/mathlib/data/int/modeq.lean index 3fd8d7e07c7b157cb3b6ec8a33a341f25099dcb2..f9f9160004dd2098ccca78226d302eb411d0ec11 100644 --- a/formal/lean/mathlib/data/int/modeq.lean +++ b/formal/lean/mathlib/data/int/modeq.lean @@ -36,6 +36,8 @@ namespace modeq protected theorem rfl : a ≡ a [ZMOD n] := modeq.refl _ +instance : is_refl _ (modeq n) := ⟨modeq.refl⟩ + @[symm] protected theorem symm : a ≡ b [ZMOD n] → b ≡ a [ZMOD n] := eq.symm @[trans] protected theorem trans : a ≡ b [ZMOD n] → b ≡ c [ZMOD n] → a ≡ c [ZMOD n] := eq.trans diff --git a/formal/lean/mathlib/data/list/alist.lean b/formal/lean/mathlib/data/list/alist.lean index 37ca52e6a9c6c0558dc431c2c5b346cdd0f36ae8..35045ccea9b8a95e36a876949937d7e319404fea 100644 --- a/formal/lean/mathlib/data/list/alist.lean +++ b/formal/lean/mathlib/data/list/alist.lean @@ -122,6 +122,10 @@ theorem lookup_eq_none {a : α} {s : alist β} : lookup a s = none ↔ a ∉ s := lookup_eq_none +theorem mem_lookup_iff {a : α} {b : β a} {s : alist β} : + b ∈ lookup a s ↔ sigma.mk a b ∈ s.entries := +mem_lookup_iff s.nodupkeys + theorem perm_lookup {a : α} {s₁ s₂ : alist β} (p : s₁.entries ~ s₂.entries) : s₁.lookup a = s₂.lookup a := perm_lookup _ s₁.nodupkeys s₂.nodupkeys p diff --git a/formal/lean/mathlib/data/list/basic.lean b/formal/lean/mathlib/data/list/basic.lean index c1f62824fa76bbc6a670fa656080c75b21e9c921..74a9bc59ad26ea3b2294281a795f7c365fd0be42 100644 --- a/formal/lean/mathlib/data/list/basic.lean +++ b/formal/lean/mathlib/data/list/basic.lean @@ -1157,6 +1157,8 @@ theorem nth_len_le : ∀ {l : list α} {n}, length l ≤ n → nth l n = none | [] n h := rfl | (a :: l) (n+1) h := nth_len_le (le_of_succ_le_succ h) +@[simp] theorem nth_length (l : list α) : l.nth l.length = none := nth_len_le le_rfl + theorem nth_eq_some {l : list α} {n a} : nth l n = some a ↔ ∃ h, nth_le l n h = a := ⟨λ e, have h : n < length l, from lt_of_not_ge $ λ hn, @@ -4084,6 +4086,9 @@ lemma inth_eq_iget_nth (n : ℕ) : l.inth n = (l.nth n).iget := by rw [←nthd_default_eq_inth, nthd_eq_get_or_else_nth, option.get_or_else_default_eq_iget] +lemma inth_zero_eq_head : l.inth 0 = l.head := +by { cases l; refl, } + end inth end list diff --git a/formal/lean/mathlib/data/list/big_operators.lean b/formal/lean/mathlib/data/list/big_operators.lean index 8bbb57426bf0ba3441a5eb3820c6a7dd6533c4e4..492101040a6f9d5366e180ad1a6b7e07deb8eae6 100644 --- a/formal/lean/mathlib/data/list/big_operators.lean +++ b/formal/lean/mathlib/data/list/big_operators.lean @@ -434,6 +434,12 @@ le_antisymm (hl₂ ▸ single_le_prod hl₁ _ hx) (hl₁ x hx) ⟨all_one_of_le_one_le_of_prod_eq_one (λ _ _, one_le _), λ h, by rw [eq_repeat.2 ⟨rfl, h⟩, prod_repeat, one_pow]⟩ +/-- Slightly more general version of `list.prod_eq_one_iff` for a non-ordered `monoid` -/ +@[to_additive "Slightly more general version of `list.sum_eq_zero_iff` + for a non-ordered `add_monoid`"] +lemma prod_eq_one [monoid M] {l : list M} (hl : ∀ (x ∈ l), x = (1 : M)) : l.prod = 1 := +trans (prod_eq_pow_card l 1 hl) (one_pow l.length) + /-- If all elements in a list are bounded below by `1`, then the length of the list is bounded by the sum of the elements. -/ lemma length_le_sum_of_one_le (L : list ℕ) (h : ∀ i ∈ L, 1 ≤ i) : L.length ≤ L.sum := @@ -464,6 +470,17 @@ lemma prod_erase [decidable_eq M] [comm_monoid M] {a} : { simp only [list.erase, if_neg (mt eq.symm ne), prod_cons, prod_erase h, mul_left_comm a b] } end +@[simp, to_additive] +lemma prod_map_erase [decidable_eq ι] [comm_monoid M] (f : ι → M) {a} : + ∀ {l : list ι}, a ∈ l → f a * ((l.erase a).map f).prod = (l.map f).prod +| (b :: l) h := + begin + obtain rfl | ⟨ne, h⟩ := decidable.list.eq_or_ne_mem_of_mem h, + { simp only [map, erase_cons_head, prod_cons] }, + { simp only [map, erase_cons_tail _ ne.symm, prod_cons, prod_map_erase h, + mul_left_comm (f a) (f b)], } + end + lemma dvd_prod [comm_monoid M] {a} {l : list M} (ha : a ∈ l) : a ∣ l.prod := let ⟨s, t, h⟩ := mem_split ha in by { rw [h, prod_append, prod_cons, mul_left_comm], exact dvd_mul_right _ _ } diff --git a/formal/lean/mathlib/data/list/count.lean b/formal/lean/mathlib/data/list/count.lean index 4bbbbe6b36dce0c073090d7998f8c8b67337c6ed..b4479638e6b4737d1f9215ecc35fb5a74bcfbd6d 100644 --- a/formal/lean/mathlib/data/list/count.lean +++ b/formal/lean/mathlib/data/list/count.lean @@ -207,6 +207,24 @@ begin { rw [count_cons', count_cons', count_erase_of_ne] } end +@[to_additive] +lemma prod_map_eq_pow_single [monoid β] {l : list α} (a : α) (f : α → β) + (hf : ∀ a' ≠ a, a' ∈ l → f a' = 1) : (l.map f).prod = (f a) ^ (l.count a) := +begin + induction l with a' as h generalizing a, + { rw [map_nil, prod_nil, count_nil, pow_zero] }, + { specialize h a (λ a' ha' hfa', hf a' ha' (mem_cons_of_mem _ hfa')), + rw [list.map_cons, list.prod_cons, count_cons, h], + split_ifs with ha', + { rw [ha', pow_succ] }, + { rw [hf a' (ne.symm ha') (list.mem_cons_self a' as), one_mul] } } +end + +@[to_additive] +lemma prod_eq_pow_single [monoid α] {l : list α} (a : α) + (h : ∀ a' ≠ a, a' ∈ l → a' = 1) : l.prod = a ^ (l.count a) := +trans (by rw [map_id'']) (prod_map_eq_pow_single a id h) + end count end list diff --git a/formal/lean/mathlib/data/list/dedup.lean b/formal/lean/mathlib/data/list/dedup.lean index cbd44cd603d3917fffe66dde907d41fc1108859b..20dfcd565439e774c98c01729aca574db3b4dc24 100644 --- a/formal/lean/mathlib/data/list/dedup.lean +++ b/formal/lean/mathlib/data/list/dedup.lean @@ -77,4 +77,35 @@ lemma repeat_dedup {x : α} : ∀ {k}, k ≠ 0 → (repeat x k).dedup = [x] | (n+2) _ := by rw [repeat_succ, dedup_cons_of_mem (mem_repeat.2 ⟨n.succ_ne_zero, rfl⟩), repeat_dedup n.succ_ne_zero] +lemma count_dedup (l : list α) (a : α) : + l.dedup.count a = if a ∈ l then 1 else 0 := +by simp_rw [count_eq_of_nodup $ nodup_dedup l, mem_dedup] + +/-- Summing the count of `x` over a list filtered by some `p` is just `countp` applied to `p` -/ +lemma sum_map_count_dedup_filter_eq_countp (p : α → Prop) [decidable_pred p] + (l : list α) : ((l.dedup.filter p).map $ λ x, l.count x).sum = l.countp p := +begin + induction l with a as h, + { simp }, + { simp_rw [list.countp_cons, list.count_cons', list.sum_map_add], + congr' 1, + { refine trans _ h, + by_cases ha : a ∈ as, + { simp [dedup_cons_of_mem ha] }, + { simp only [dedup_cons_of_not_mem ha, list.filter], + split_ifs with hp; simp [list.map_cons, list.sum_cons, + list.count_eq_zero.2 ha, zero_add] } }, + { by_cases hp : p a, + { refine trans (sum_map_eq_nsmul_single a _ (λ _ h _, by simp [h])) _, + simp [hp, count_dedup] }, + { refine trans (list.sum_eq_zero $ λ n hn, _) (by simp [hp]), + obtain ⟨a', ha'⟩ := list.mem_map.1 hn, + simp only [(λ h, hp (h ▸ (list.mem_filter.1 ha'.1).2) : a' ≠ a), if_false] at ha', + exact ha'.2.symm } } }, +end + +lemma sum_map_count_dedup_eq_length (l : list α) : + (l.dedup.map $ λ x, l.count x).sum = l.length := +by simpa using sum_map_count_dedup_filter_eq_countp (λ _, true) l + end list diff --git a/formal/lean/mathlib/data/list/defs.lean b/formal/lean/mathlib/data/list/defs.lean index 0ace872e6c219166c83e21eeff3eafaa4e0da0b0..6c8c0a3cf96a044433396c8b9c52cd8a3a09decb 100644 --- a/formal/lean/mathlib/data/list/defs.lean +++ b/formal/lean/mathlib/data/list/defs.lean @@ -540,6 +540,9 @@ def revzip (l : list α) : list (α × α) := zip l l.reverse def product (l₁ : list α) (l₂ : list β) : list (α × β) := l₁.bind $ λ a, l₂.map $ prod.mk a +/- This notation binds more strongly than (pre)images, unions and intersections. -/ +infixr ` ×ˢ `:82 := list.product + /-- `sigma l₁ l₂` is the list of dependent pairs `(a, b)` where `a ∈ l₁` and `b ∈ l₂ a`. sigma [1, 2] (λ_, [(5 : ℕ), 6]) = [(1, 5), (1, 6), (2, 5), (2, 6)] -/ diff --git a/formal/lean/mathlib/data/list/nodup.lean b/formal/lean/mathlib/data/list/nodup.lean index 266f073427c39bd57730147659cd5a502ebce4d8..1a77d853637ca5aa08e27a0e87ddba533e92932f 100644 --- a/formal/lean/mathlib/data/list/nodup.lean +++ b/formal/lean/mathlib/data/list/nodup.lean @@ -135,6 +135,14 @@ theorem nodup_repeat (a : α) : ∀ {n : ℕ}, nodup (repeat a n) ↔ n ≤ 1 (d : nodup l) (h : a ∈ l) : count a l = 1 := le_antisymm (nodup_iff_count_le_one.1 d a) (count_pos.2 h) +lemma count_eq_of_nodup [decidable_eq α] {a : α} {l : list α} + (d : nodup l) : count a l = if a ∈ l then 1 else 0 := +begin + split_ifs with h, + { exact count_eq_one_of_mem d h }, + { exact count_eq_zero_of_not_mem h }, +end + lemma nodup.of_append_left : nodup (l₁ ++ l₂) → nodup l₁ := nodup.sublist (sublist_append_left l₁ l₂) diff --git a/formal/lean/mathlib/data/list/of_fn.lean b/formal/lean/mathlib/data/list/of_fn.lean index 4c695e90a6889e8feeed997c50fb1d40113d3789..f131ed93579ed08f167801adbba7d2e496305d83 100644 --- a/formal/lean/mathlib/data/list/of_fn.lean +++ b/formal/lean/mathlib/data/list/of_fn.lean @@ -109,6 +109,20 @@ begin simp_rw [fin.cast_succ_fin_succ], } end +@[simp] lemma of_fn_eq_nil_iff {n : ℕ} {f : fin n → α} : + of_fn f = [] ↔ n = 0 := +by cases n; simp only [of_fn_zero, of_fn_succ, eq_self_iff_true, nat.succ_ne_zero] + +lemma last_of_fn {n : ℕ} (f : fin n → α) (h : of_fn f ≠ []) + (hn : n - 1 < n := nat.pred_lt $ of_fn_eq_nil_iff.not.mp h) : + last (of_fn f) h = f ⟨n - 1, hn⟩ := +by simp [last_eq_nth_le] + +lemma last_of_fn_succ {n : ℕ} (f : fin n.succ → α) + (h : of_fn f ≠ [] := mt of_fn_eq_nil_iff.mp (nat.succ_ne_zero _)) : + last (of_fn f) h = f (fin.last _) := +last_of_fn f h + /-- Note this matches the convention of `list.of_fn_succ'`, putting the `fin m` elements first. -/ theorem of_fn_add {m n} (f : fin (m + n) → α) : list.of_fn f = list.of_fn (λ i, f (fin.cast_add n i)) ++ list.of_fn (λ j, f (fin.nat_add m j)) := diff --git a/formal/lean/mathlib/data/list/perm.lean b/formal/lean/mathlib/data/list/perm.lean index c3421132bfcfd1e997810b652c813743ca1ecd6b..88df2a51d0e59ddc35d594abe917b01d193129db 100644 --- a/formal/lean/mathlib/data/list/perm.lean +++ b/formal/lean/mathlib/data/list/perm.lean @@ -7,6 +7,7 @@ import data.list.dedup import data.list.lattice import data.list.permutation import data.list.zip +import data.list.range import logic.relation /-! @@ -924,6 +925,10 @@ begin exact perm_append_comm.append_right _ end +theorem map_append_bind_perm (l : list α) (f : α → β) (g : α → list β) : + l.map f ++ l.bind g ~ l.bind (λ x, f x :: g x) := +by simpa [←map_eq_bind] using bind_append_perm l (λ x, [f x]) g + theorem perm.product_right {l₁ l₂ : list α} (t₁ : list β) (p : l₁ ~ l₂) : product l₁ t₁ ~ product l₂ t₁ := p.bind_right _ @@ -983,6 +988,24 @@ begin { exact (IH _ _ h).cons _ } } end +lemma range_bind_sublists_len_perm {α : Type*} (l : list α) : + (list.range (l.length + 1)).bind (λ n, sublists_len n l) ~ sublists' l := +begin + induction l with h tl, + { simp [range_succ] }, + { simp_rw [range_succ_eq_map, length, cons_bind, map_bind, sublists_len_succ_cons, + sublists'_cons, list.sublists_len_zero, list.singleton_append], + refine ((bind_append_perm (range (tl.length + 1)) _ _).symm.cons _).trans _, + simp_rw [←list.bind_map, ←cons_append], + rw [←list.singleton_append, ←list.sublists_len_zero tl], + refine perm.append _ (l_ih.map _), + rw [list.range_succ, append_bind, bind_singleton, + sublists_len_of_length_lt (nat.lt_succ_self _), append_nil, + ←list.map_bind (λ n, sublists_len n tl) nat.succ, ←cons_bind 0 _ (λ n, sublists_len n tl), + ←range_succ_eq_map], + exact l_ih } +end + theorem perm_lookmap (f : α → option α) {l₁ l₂ : list α} (H : pairwise (λ a b, ∀ (c ∈ f a) (d ∈ f b), a = b ∧ c = d) l₁) (p : l₁ ~ l₂) : lookmap f l₁ ~ lookmap f l₂ := diff --git a/formal/lean/mathlib/data/list/sublists.lean b/formal/lean/mathlib/data/list/sublists.lean index b8cae5900962779a88a9d9c57246334893adddae..a23405897b56aec0544a4140bca5288cbca7c8fa 100644 --- a/formal/lean/mathlib/data/list/sublists.lean +++ b/formal/lean/mathlib/data/list/sublists.lean @@ -282,4 +282,13 @@ end length_of_sublists_len h⟩, λ ⟨h₁, h₂⟩, h₂ ▸ mem_sublists_len_self h₁⟩ +lemma sublists_len_of_length_lt {n} {l : list α} (h : l.length < n) : sublists_len n l = [] := +eq_nil_iff_forall_not_mem.mpr $ λ x, mem_sublists_len.not.mpr $ λ ⟨hs, hl⟩, + (h.trans_eq hl.symm).not_le (length_le_of_sublist hs) + +@[simp] lemma sublists_len_length : ∀ (l : list α), sublists_len l.length l = [l] +| [] := rfl +| (a::l) := by rw [length, sublists_len_succ_cons, sublists_len_length, map_singleton, + sublists_len_of_length_lt (lt_succ_self _), nil_append] + end list diff --git a/formal/lean/mathlib/data/matrix/kronecker.lean b/formal/lean/mathlib/data/matrix/kronecker.lean index a810b104a51e3eb7aa65488e3536b7f11f4f8d18..4b3c9b9eab590976026b2307743a33fa1efe4992 100644 --- a/formal/lean/mathlib/data/matrix/kronecker.lean +++ b/formal/lean/mathlib/data/matrix/kronecker.lean @@ -206,7 +206,7 @@ lemma kronecker_apply [has_mul α] (A : matrix l m α) (B : matrix n p α) (i₁ /-- `matrix.kronecker` as a bilinear map. -/ def kronecker_bilinear [comm_semiring R] [semiring α] [algebra R α] : matrix l m α →ₗ[R] matrix n p α →ₗ[R] matrix (l × n) (m × p) α := -kronecker_map_bilinear (algebra.lmul R α).to_linear_map +kronecker_map_bilinear (algebra.lmul R α) /-! What follows is a copy, in order, of every `matrix.kronecker_map` lemma above that has hypotheses which can be filled by properties of `*`. -/ diff --git a/formal/lean/mathlib/data/multiset/antidiagonal.lean b/formal/lean/mathlib/data/multiset/antidiagonal.lean index ad9f28add138d6c04928ddd9bfd5ea80597a7f10..11de3bec9c3edeb01a2d7f4a68d76358b73d0b1c 100644 --- a/formal/lean/mathlib/data/multiset/antidiagonal.lean +++ b/formal/lean/mathlib/data/multiset/antidiagonal.lean @@ -67,10 +67,23 @@ quotient.induction_on s $ λ l, begin {congr; simp}, {simp} end +theorem antidiagonal_eq_map_powerset [decidable_eq α] (s : multiset α) : + s.antidiagonal = s.powerset.map (λ t, (s - t, t)) := +begin + induction s using multiset.induction_on with a s hs, + { simp only [antidiagonal_zero, powerset_zero, zero_tsub, map_singleton] }, + { simp_rw [antidiagonal_cons, powerset_cons, map_add, hs, map_map, function.comp, prod.map_mk, + id.def, sub_cons, erase_cons_head], + rw add_comm, + congr' 1, + refine multiset.map_congr rfl (λ x hx, _), + rw cons_sub_of_le _ (mem_powerset.mp hx) } +end + @[simp] theorem card_antidiagonal (s : multiset α) : card (antidiagonal s) = 2 ^ card s := by have := card_powerset s; - rwa [← antidiagonal_map_fst, card_map] at this + rwa [← antidiagonal_map_fst, card_map] at this lemma prod_map_add [comm_semiring β] {s : multiset α} {f g : α → β} : prod (s.map (λa, f a + g a)) = diff --git a/formal/lean/mathlib/data/multiset/basic.lean b/formal/lean/mathlib/data/multiset/basic.lean index 38fc3f9dda8180f029ae9219050c149327a091a7..8be28fdcbfbed825681ec2ee95b16fda567886b7 100644 --- a/formal/lean/mathlib/data/multiset/basic.lean +++ b/formal/lean/mathlib/data/multiset/basic.lean @@ -218,7 +218,27 @@ end end mem +/-! ### Singleton -/ + +instance : has_singleton α (multiset α) := ⟨λ a, a ::ₘ 0⟩ + +instance : is_lawful_singleton α (multiset α) := ⟨λ a, rfl⟩ + +theorem singleton_eq_cons (a : α) : singleton a = a ::ₘ 0 := rfl + +@[simp] theorem mem_singleton {a b : α} : b ∈ ({a} : multiset α) ↔ b = a := +by simp only [singleton_eq_cons, mem_cons, iff_self, or_false, not_mem_zero] + +theorem mem_singleton_self (a : α) : a ∈ ({a} : multiset α) := +by { rw singleton_eq_cons, exact mem_cons_self _ _ } + +theorem singleton_inj {a b : α} : ({a} : multiset α) = {b} ↔ a = b := +by { simp_rw [singleton_eq_cons], exact cons_inj_left _ } + +theorem pair_comm (x y : α) : ({x, y} : multiset α) = {y, x} := cons_swap x y 0 + /-! ### `multiset.subset` -/ + section subset /-- `s ⊆ t` is the lift of the list subset relation. It means that any @@ -268,6 +288,8 @@ lemma induction_on' {p : multiset α → Prop} (S : multiset α) end subset +/-! ### `multiset.to_list` -/ + section to_list /-- Produces a list of the elements in the multiset using choice. -/ @@ -323,7 +345,12 @@ quotient.induction_on₂ s t (λ l₁ l₂ ⟨l, p, s⟩, theorem zero_le (s : multiset α) : 0 ≤ s := quot.induction_on s $ λ l, (nil_sublist l).subperm -lemma le_zero : s ≤ 0 ↔ s = 0 := ⟨λ h, le_antisymm h (zero_le _), le_of_eq⟩ +instance : order_bot (multiset α) := ⟨0, zero_le⟩ + +/-- This is a `rfl` and `simp` version of `bot_eq_zero`. -/ +@[simp] theorem bot_eq_zero : (⊥ : multiset α) = 0 := rfl + +lemma le_zero : s ≤ 0 ↔ s = 0 := le_bot_iff theorem lt_cons_self (s : multiset α) (a : α) : s < a ::ₘ s := quot.induction_on s $ λ l, @@ -352,24 +379,6 @@ begin ((sublist_or_mem_of_sublist s).resolve_right m₁).subperm) end -end - -/-! ### Singleton -/ -instance : has_singleton α (multiset α) := ⟨λ a, a ::ₘ 0⟩ - -instance : is_lawful_singleton α (multiset α) := ⟨λ a, rfl⟩ - -theorem singleton_eq_cons (a : α) : singleton a = a ::ₘ 0 := rfl - -@[simp] theorem mem_singleton {a b : α} : b ∈ ({a} : multiset α) ↔ b = a := -by simp only [singleton_eq_cons, mem_cons, iff_self, or_false, not_mem_zero] - -theorem mem_singleton_self (a : α) : a ∈ ({a} : multiset α) := -by { rw singleton_eq_cons, exact mem_cons_self _ _ } - -theorem singleton_inj {a b : α} : ({a} : multiset α) = {b} ↔ a = b := -by { simp_rw [singleton_eq_cons], exact cons_inj_left _ } - @[simp] theorem singleton_ne_zero (a : α) : ({a} : multiset α) ≠ 0 := ne_of_gt (lt_cons_self _ _) @@ -377,7 +386,7 @@ ne_of_gt (lt_cons_self _ _) ⟨λ h, mem_of_le h (mem_singleton_self _), λ h, let ⟨t, e⟩ := exists_cons_of_mem h in e.symm ▸ cons_le_cons _ (zero_le _)⟩ -theorem pair_comm (x y : α) : ({x, y} : multiset α) = {y, x} := cons_swap x y 0 +end /-! ### Additive monoid -/ @@ -426,10 +435,6 @@ theorem le_iff_exists_add {s t : multiset α} : s ≤ t ↔ ∃ u, t = s + u := let ⟨l, p⟩ := s.exists_perm_append in ⟨l, quot.sound p⟩, λ ⟨u, e⟩, e.symm ▸ le_add_right _ _⟩ -instance : order_bot (multiset α) := -{ bot := 0, - bot_le := multiset.zero_le } - instance : canonically_ordered_add_monoid (multiset α) := { le_self_add := le_add_right, exists_add_of_le := λ a b h, le_induction_on h $ λ l₁ l₂ s, @@ -437,9 +442,6 @@ instance : canonically_ordered_add_monoid (multiset α) := ..multiset.order_bot, ..multiset.ordered_cancel_add_comm_monoid } -/-- This is a `rfl` and `simp` version of `bot_eq_zero`. -/ -@[simp] theorem bot_eq_zero : (⊥ : multiset α) = 0 := rfl - @[simp] theorem cons_add (a : α) (s t : multiset α) : a ::ₘ s + t = a ::ₘ (s + t) := by rw [← singleton_add, ← singleton_add, add_assoc] @@ -1159,6 +1161,10 @@ multiset.induction_on t (by simp [multiset.sub_zero]) instance : has_ordered_sub (multiset α) := ⟨λ n m k, multiset.sub_le_iff_le_add⟩ +lemma cons_sub_of_le (a : α) {s t : multiset α} (h : t ≤ s) : + a ::ₘ s - t = a ::ₘ (s - t) := +by rw [←singleton_add, ←singleton_add, add_tsub_assoc_of_le h] + theorem sub_eq_fold_erase (s t : multiset α) : s - t = foldl erase erase_comm s t := quotient.induction_on₂ s t $ λ l₁ l₂, show ↑(l₁.diff l₂) = foldl erase erase_comm ↑l₁ ↑l₂, diff --git a/formal/lean/mathlib/data/multiset/bind.lean b/formal/lean/mathlib/data/multiset/bind.lean index 7e04601843c552c23abe2564ffd4fefa4b3ccd19..9191a8111b7434b2dadc26168338e193a200ebaf 100644 --- a/formal/lean/mathlib/data/multiset/bind.lean +++ b/formal/lean/mathlib/data/multiset/bind.lean @@ -146,35 +146,36 @@ end bind section product variables (a : α) (b : β) (s : multiset α) (t : multiset β) -/-- The multiplicity of `(a, b)` in `s.product t` is +/-- The multiplicity of `(a, b)` in `s ×ˢ t` is the product of the multiplicity of `a` in `s` and `b` in `t`. -/ def product (s : multiset α) (t : multiset β) : multiset (α × β) := s.bind $ λ a, t.map $ prod.mk a +/- This notation binds more strongly than (pre)images, unions and intersections. -/ +infixr ` ×ˢ `:82 := multiset.product + @[simp] lemma coe_product (l₁ : list α) (l₂ : list β) : @product α β l₁ l₂ = l₁.product l₂ := by { rw [product, list.product, ←coe_bind], simp } @[simp] lemma zero_product : @product α β 0 t = 0 := rfl --TODO: Add `product_zero` -@[simp] lemma cons_product : (a ::ₘ s).product t = map (prod.mk a) t + s.product t := +@[simp] lemma cons_product : (a ::ₘ s) ×ˢ t = map (prod.mk a) t + s ×ˢ t := by simp [product] -@[simp] lemma product_singleton : ({a} : multiset α).product ({b} : multiset β) = {(a, b)} := +@[simp] lemma product_singleton : ({a} : multiset α) ×ˢ ({b} : multiset β) = {(a, b)} := by simp only [product, bind_singleton, map_singleton] -@[simp] lemma add_product (s t : multiset α) (u : multiset β) : - (s + t).product u = s.product u + product t u := +@[simp] lemma add_product (s t : multiset α) (u : multiset β) : (s + t) ×ˢ u = s ×ˢ u + t ×ˢ u := by simp [product] -@[simp] lemma product_add (s : multiset α) : ∀ t u : multiset β, - s.product (t + u) = s.product t + s.product u := +@[simp] lemma product_add (s : multiset α) : ∀ t u : multiset β, s ×ˢ (t + u) = s ×ˢ t + s ×ˢ u := multiset.induction_on s (λ t u, rfl) $ λ a s IH t u, by rw [cons_product, IH]; simp; cc @[simp] lemma mem_product {s t} : ∀ {p : α × β}, p ∈ @product α β s t ↔ p.1 ∈ s ∧ p.2 ∈ t | (a, b) := by simp [product, and.left_comm] -@[simp] lemma card_product : (s.product t).card = s.card * t.card := +@[simp] lemma card_product : (s ×ˢ t).card = s.card * t.card := by simp [product, repeat, (∘), mul_comm] end product diff --git a/formal/lean/mathlib/data/multiset/fintype.lean b/formal/lean/mathlib/data/multiset/fintype.lean index f9e8388c856ba09db80ed74b5f271b068c9cdfb5..70830824d1bb2a86ea67c676e538696c7a74f888 100644 --- a/formal/lean/mathlib/data/multiset/fintype.lean +++ b/formal/lean/mathlib/data/multiset/fintype.lean @@ -37,7 +37,7 @@ variables {α : Type*} [decidable_eq α] {m : multiset α} /-- Auxiliary definition for the `has_coe_to_sort` instance. This prevents the `has_coe m α` instance from inadverently applying to other sigma types. One should not use this definition directly. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def multiset.to_type (m : multiset α) : Type* := Σ (x : α), fin (m.count x) /-- Create a type that has the same number of elements as the multiset. diff --git a/formal/lean/mathlib/data/multiset/nodup.lean b/formal/lean/mathlib/data/multiset/nodup.lean index 09b5860fe59e1665df34f194adf4d22e16bc3335..9044e8d5c91238f3590cfb557e814fdf338ac092 100644 --- a/formal/lean/mathlib/data/multiset/nodup.lean +++ b/formal/lean/mathlib/data/multiset/nodup.lean @@ -60,6 +60,14 @@ quot.induction_on s $ λ l, nodup_iff_count_le_one (d : nodup s) (h : a ∈ s) : count a s = 1 := le_antisymm (nodup_iff_count_le_one.1 d a) (count_pos.2 h) +lemma count_eq_of_nodup [decidable_eq α] {a : α} {s : multiset α} + (d : nodup s) : count a s = if a ∈ s then 1 else 0 := +begin + split_ifs with h, + { exact count_eq_one_of_mem d h }, + { exact count_eq_zero_of_not_mem h }, +end + lemma nodup_iff_pairwise {α} {s : multiset α} : nodup s ↔ pairwise (≠) s := quotient.induction_on s $ λ l, (pairwise_coe_iff_pairwise (by exact λ a b, ne.symm)).symm diff --git a/formal/lean/mathlib/data/multiset/powerset.lean b/formal/lean/mathlib/data/multiset/powerset.lean index f712941507367c030f54098c9d85faec6f01631a..675cee9ba82be9a44387ece1ff08095bb93efeff 100644 --- a/formal/lean/mathlib/data/multiset/powerset.lean +++ b/formal/lean/mathlib/data/multiset/powerset.lean @@ -4,7 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro -/ import data.multiset.basic - +import data.multiset.range +import data.multiset.bind /-! # The powerset of a multiset -/ @@ -253,4 +254,18 @@ begin { cases n; simp [ih, map_comp_cons], }, end +lemma disjoint_powerset_len (s : multiset α) {i j : ℕ} (h : i ≠ j) : + multiset.disjoint (s.powerset_len i) (s.powerset_len j) := +λ x hi hj, h (eq.trans (multiset.mem_powerset_len.mp hi).right.symm + (multiset.mem_powerset_len.mp hj).right) + +lemma bind_powerset_len {α : Type*} (S : multiset α) : + bind (multiset.range (S.card + 1)) (λ k, S.powerset_len k) = S.powerset := +begin + induction S using quotient.induction_on, + simp_rw [quot_mk_to_coe, powerset_coe', powerset_len_coe, ←coe_range, coe_bind, ←list.bind_map, + coe_card], + exact coe_eq_coe.mpr ((list.range_bind_sublists_len_perm S).map _), +end + end multiset diff --git a/formal/lean/mathlib/data/mv_polynomial/basic.lean b/formal/lean/mathlib/data/mv_polynomial/basic.lean index 379204a439a1cab8de7645792b9e780021ffd586..2c2b9d802150ab1c6362b13f89491f5b5df11365 100644 --- a/formal/lean/mathlib/data/mv_polynomial/basic.lean +++ b/formal/lean/mathlib/data/mv_polynomial/basic.lean @@ -28,39 +28,27 @@ corresponding to the terms in `σ`, and coefficients in `R`. In the definitions below, we use the following notation: + `σ : Type*` (indexing the variables) - + `R : Type*` `[comm_semiring R]` (the coefficients) - + `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set. -This will give rise to a monomial in `mv_polynomial σ R` which mathematicians might call `X^s` - + This will give rise to a monomial in `mv_polynomial σ R` which mathematicians might call `X^s` + `a : R` - + `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians - + `p : mv_polynomial σ R` ### Definitions * `mv_polynomial σ R` : the type of polynomials with variables of type `σ` and coefficients in the commutative semiring `R` - * `monomial s a` : the monomial which mathematically would be denoted `a * X^s` - * `C a` : the constant polynomial with value `a` - * `X i` : the degree one monomial corresponding to i; mathematically this might be denoted `Xᵢ`. - * `coeff s p` : the coefficient of `s` in `p`. - * `eval₂ (f : R → S₁) (g : σ → S₁) p` : given a semiring homomorphism from `R` to another semiring `S₁`, and a map `σ → S₁`, evaluates `p` at this valuation, returning a term of type `S₁`. Note that `eval₂` can be made using `eval` and `map` (see below), and it has been suggested that sticking to `eval` and `map` might make the code less brittle. - * `eval (g : σ → R) p` : given a map `σ → R`, evaluates `p` at this valuation, returning a term of type `R` - * `map (f : R → S₁) p` : returns the multivariate polynomial obtained from `p` by the change of coefficient semiring corresponding to `f` @@ -101,34 +89,45 @@ section instances instance decidable_eq_mv_polynomial [comm_semiring R] [decidable_eq σ] [decidable_eq R] : decidable_eq (mv_polynomial σ R) := finsupp.decidable_eq + instance [comm_semiring R] : comm_semiring (mv_polynomial σ R) := add_monoid_algebra.comm_semiring + instance [comm_semiring R] : inhabited (mv_polynomial σ R) := ⟨0⟩ + instance [monoid R] [comm_semiring S₁] [distrib_mul_action R S₁] : distrib_mul_action R (mv_polynomial σ S₁) := add_monoid_algebra.distrib_mul_action + instance [monoid R] [comm_semiring S₁] [distrib_mul_action R S₁] [has_faithful_smul R S₁] : has_faithful_smul R (mv_polynomial σ S₁) := add_monoid_algebra.has_faithful_smul + instance [semiring R] [comm_semiring S₁] [module R S₁] : module R (mv_polynomial σ S₁) := add_monoid_algebra.module + instance [monoid R] [monoid S₁] [comm_semiring S₂] [has_smul R S₁] [distrib_mul_action R S₂] [distrib_mul_action S₁ S₂] [is_scalar_tower R S₁ S₂] : is_scalar_tower R S₁ (mv_polynomial σ S₂) := add_monoid_algebra.is_scalar_tower + instance [monoid R] [monoid S₁][comm_semiring S₂] [distrib_mul_action R S₂] [distrib_mul_action S₁ S₂] [smul_comm_class R S₁ S₂] : smul_comm_class R S₁ (mv_polynomial σ S₂) := add_monoid_algebra.smul_comm_class + instance [monoid R] [comm_semiring S₁] [distrib_mul_action R S₁] [distrib_mul_action Rᵐᵒᵖ S₁] [is_central_scalar R S₁] : is_central_scalar R (mv_polynomial σ S₁) := add_monoid_algebra.is_central_scalar + instance [comm_semiring R] [comm_semiring S₁] [algebra R S₁] : algebra R (mv_polynomial σ S₁) := add_monoid_algebra.algebra + -- Register with high priority to avoid timeout in `data.mv_polynomial.pderiv` instance is_scalar_tower' [comm_semiring R] [comm_semiring S₁] [algebra R S₁] : is_scalar_tower R (mv_polynomial σ S₁) (mv_polynomial σ S₁) := is_scalar_tower.right + -- TODO[gh-6025]: make this an instance once safe to do so /-- If `R` is a subsingleton, then `mv_polynomial σ R` has a unique element -/ protected def unique [comm_semiring R] [subsingleton R] : unique (mv_polynomial σ R) := @@ -156,6 +155,14 @@ variables {R σ} /-- `X n` is the degree `1` monomial $X_n$. -/ def X (n : σ) : mv_polynomial σ R := monomial (single n 1) 1 +lemma monomial_left_injective {r : R} (hr : r ≠ 0) : + function.injective (λ s : σ →₀ ℕ, monomial s r) := +finsupp.single_left_injective hr + +@[simp] lemma monomial_left_inj {s t : σ →₀ ℕ} {r : R} (hr : r ≠ 0) : + monomial s r = monomial t r ↔ s = t := +finsupp.single_left_inj hr + lemma C_apply : (C a : mv_polynomial σ R) = monomial 0 a := rfl @[simp] lemma C_0 : C 0 = (0 : mv_polynomial σ R) := by simp [C_apply, monomial] @@ -195,8 +202,7 @@ instance infinite_of_nonempty (σ : Type*) (R : Type*) [nonempty σ] [comm_semir [nontrivial R] : infinite (mv_polynomial σ R) := infinite.of_injective ((λ s : σ →₀ ℕ, monomial s 1) ∘ single (classical.arbitrary σ)) $ - function.injective.comp - (λ m n, (finsupp.single_left_inj one_ne_zero).mp) (finsupp.single_injective _) + (monomial_left_injective $ @one_ne_zero R _ _).comp (finsupp.single_injective _) lemma C_eq_coe_nat (n : ℕ) : (C ↑n : mv_polynomial σ R) = n := by induction n; simp [nat.succ_eq_add_one, *] @@ -209,6 +215,12 @@ lemma smul_eq_C_mul (p : mv_polynomial σ R) (a : R) : a • p = C a * p := C_mu lemma C_eq_smul_one : (C a : mv_polynomial σ R) = a • 1 := by rw [← C_mul', mul_one] +lemma X_injective [nontrivial R] : function.injective (X : σ → mv_polynomial σ R) := +(monomial_left_injective $ @one_ne_zero R _ _).comp (finsupp.single_left_injective one_ne_zero) + +@[simp] lemma X_inj [nontrivial R] (m n : σ) : X m = (X n : mv_polynomial σ R) ↔ m = n := +X_injective.eq_iff + lemma monomial_pow : monomial s a ^ e = monomial (e • s) (a ^ e) := add_monoid_algebra.single_pow e @@ -1109,26 +1121,44 @@ by { ext i, simp } φ (aeval g p) = (eval₂_hom (φ.comp (algebra_map R S₁)) (λ i, φ (g i)) p) := by { rw ← comp_eval₂_hom, refl } -@[simp] lemma eval₂_hom_zero (f : R →+* S₂) (p : mv_polynomial σ R) : +@[simp] lemma eval₂_hom_zero (f : R →+* S₂) : + eval₂_hom f (0 : σ → S₂) = f.comp constant_coeff := +by { ext; simp } + +@[simp] lemma eval₂_hom_zero' (f : R →+* S₂) : + eval₂_hom f (λ _, 0 : σ → S₂) = f.comp constant_coeff := +eval₂_hom_zero f + +lemma eval₂_hom_zero_apply (f : R →+* S₂) (p : mv_polynomial σ R) : eval₂_hom f (0 : σ → S₂) p = f (constant_coeff p) := -begin - suffices : eval₂_hom f (0 : σ → S₂) = f.comp constant_coeff, - from ring_hom.congr_fun this p, - ext; simp -end +ring_hom.congr_fun (eval₂_hom_zero f) p -@[simp] lemma eval₂_hom_zero' (f : R →+* S₂) (p : mv_polynomial σ R) : +lemma eval₂_hom_zero'_apply (f : R →+* S₂) (p : mv_polynomial σ R) : eval₂_hom f (λ _, 0 : σ → S₂) p = f (constant_coeff p) := -eval₂_hom_zero f p +eval₂_hom_zero_apply f p + +@[simp] lemma eval₂_zero_apply (f : R →+* S₂) (p : mv_polynomial σ R) : + eval₂ f (0 : σ → S₂) p = f (constant_coeff p) := +eval₂_hom_zero_apply _ _ + +@[simp] lemma eval₂_zero'_apply (f : R →+* S₂) (p : mv_polynomial σ R) : + eval₂ f (λ _, 0 : σ → S₂) p = f (constant_coeff p) := +eval₂_zero_apply f p @[simp] lemma aeval_zero (p : mv_polynomial σ R) : aeval (0 : σ → S₁) p = algebra_map _ _ (constant_coeff p) := -eval₂_hom_zero (algebra_map R S₁) p +eval₂_hom_zero_apply (algebra_map R S₁) p @[simp] lemma aeval_zero' (p : mv_polynomial σ R) : aeval (λ _, 0 : σ → S₁) p = algebra_map _ _ (constant_coeff p) := aeval_zero p +@[simp] lemma eval_zero : eval (0 : σ → R) = constant_coeff := +eval₂_hom_zero _ + +@[simp] lemma eval_zero' : eval (λ _, 0 : σ → R) = constant_coeff := +eval₂_hom_zero _ + lemma aeval_monomial (g : σ → S₁) (d : σ →₀ ℕ) (r : R) : aeval g (monomial d r) = algebra_map _ _ r * d.prod (λ i k, g i ^ k) := eval₂_hom_monomial _ _ _ _ diff --git a/formal/lean/mathlib/data/nat/basic.lean b/formal/lean/mathlib/data/nat/basic.lean index 8348fe0d6571ceba331e4de21c0709c1b057ad55..adc1c3fe923adf1b840df5ada261ece20e4f17cd 100644 --- a/formal/lean/mathlib/data/nat/basic.lean +++ b/formal/lean/mathlib/data/nat/basic.lean @@ -1020,6 +1020,15 @@ end /-! ### `mod`, `dvd` -/ +lemma mod_eq_iff_lt {a b : ℕ} (h : b ≠ 0) : a % b = a ↔ a < b := +begin + cases b, contradiction, + exact ⟨λ h, h.ge.trans_lt (mod_lt _ (succ_pos _)), mod_eq_of_lt⟩, +end + +@[simp] lemma mod_succ_eq_iff_lt {a b : ℕ} : a % b.succ = a ↔ a < b.succ := +mod_eq_iff_lt (succ_ne_zero _) + lemma div_add_mod (m k : ℕ) : k * (m / k) + m % k = m := (nat.add_comm _ _).trans (mod_add_div _ _) diff --git a/formal/lean/mathlib/data/nat/bits.lean b/formal/lean/mathlib/data/nat/bits.lean new file mode 100644 index 0000000000000000000000000000000000000000..91df832d1e5a601b8f13f0f3ec98a33f4f8a5809 --- /dev/null +++ b/formal/lean/mathlib/data/nat/bits.lean @@ -0,0 +1,94 @@ +/- +Copyright (c) 2022 Praneeth Kolichala. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Praneeth Kolichala +-/ +import tactic.generalize_proofs +import tactic.norm_num + +/-! +# Additional properties of binary recursion on `nat` + +This file documents additional properties of binary recursion, +which allows us to more easily work with operations which do depend +on the number of leading zeros in the binary representation of `n`. +For example, we can more easily work with `nat.bits` and `nat.size`. + +See also: `nat.bitwise`, `nat.pow` (for various lemmas about `size` and `shiftl`/`shiftr`), +and `nat.digits`. +-/ + +namespace nat + +lemma bit_eq_zero_iff {n : ℕ} {b : bool} : bit b n = 0 ↔ n = 0 ∧ b = ff := +by { split, { cases b; simp [nat.bit], }, rintro ⟨rfl, rfl⟩, refl, } + +/-- The same as binary_rec_eq, but that one unfortunately requires `f` to be the identity when + appending `ff` to `0`. Here, we allow you to explicitly say that that case is not happening, i.e. + supplying `n = 0 → b = tt`. -/ +lemma binary_rec_eq' {C : ℕ → Sort*} {z : C 0} {f : ∀ b n, C n → C (bit b n)} + (b n) (h : f ff 0 z = z ∨ (n = 0 → b = tt)) : + binary_rec z f (bit b n) = f b n (binary_rec z f n) := +begin + rw [binary_rec], + split_ifs with h', + { rcases bit_eq_zero_iff.mp h' with ⟨rfl, rfl⟩, + rw binary_rec_zero, + simp only [imp_false, or_false, eq_self_iff_true, not_true] at h, + exact h.symm }, + { generalize_proofs e, revert e, + rw [bodd_bit, div2_bit], + intros, refl, } +end + +/-- The same as `binary_rec`, but the induction step can assume that if `n=0`, + the bit being appended is `tt`-/ +@[elab_as_eliminator] +def binary_rec' {C : ℕ → Sort*} (z : C 0) (f : ∀ b n, (n = 0 → b = tt) → C n → C (bit b n)) : + ∀ n, C n := +binary_rec z (λ b n ih, if h : n = 0 → b = tt then f b n h ih else + by { convert z, rw bit_eq_zero_iff, simpa using h, }) + +/-- The same as `binary_rec`, but special casing both 0 and 1 as base cases -/ +@[elab_as_eliminator] +def binary_rec_from_one {C : ℕ → Sort*} (z₀ : C 0) (z₁ : C 1) + (f : ∀ b n, n ≠ 0 → C n → C (bit b n)) : ∀ n, C n := +binary_rec' z₀ (λ b n h ih, if h' : n = 0 then by { rw [h', h h'], exact z₁ } else f b n h' ih) + +@[simp] lemma zero_bits : bits 0 = [] := by simp [nat.bits] + +@[simp] lemma bits_append_bit (n : ℕ) (b : bool) (hn : n = 0 → b = tt) : + (bit b n).bits = b :: n.bits := +by { rw [nat.bits, binary_rec_eq'], simpa, } + +@[simp] lemma bit0_bits (n : ℕ) (hn : n ≠ 0) : (bit0 n).bits = ff :: n.bits := +bits_append_bit n ff (λ hn', absurd hn' hn) + +@[simp] lemma bit1_bits (n : ℕ) : (bit1 n).bits = tt :: n.bits := +bits_append_bit n tt (λ _, rfl) + +@[simp] lemma one_bits : nat.bits 1 = [tt] := by { convert bit1_bits 0, simp, } + +example : bits 3423 = [tt, tt, tt, tt, tt, ff, tt, ff, tt, ff, tt, tt] := by norm_num + +lemma bodd_eq_bits_head (n : ℕ) : n.bodd = n.bits.head := +begin + induction n using nat.binary_rec' with b n h ih, { simp, }, + simp [bodd_bit, bits_append_bit _ _ h], +end + +lemma div2_bits_eq_tail (n : ℕ) : n.div2.bits = n.bits.tail := +begin + induction n using nat.binary_rec' with b n h ih, { simp, }, + simp [div2_bit, bits_append_bit _ _ h], +end + +lemma size_eq_bits_len (n : ℕ) : n.bits.length = n.size := +begin + induction n using nat.binary_rec' with b n h ih, { simp, }, + rw [size_bit, bits_append_bit _ _ h], + { simp [ih], }, + { simpa [bit_eq_zero_iff], } +end + +end nat diff --git a/formal/lean/mathlib/data/nat/bitwise.lean b/formal/lean/mathlib/data/nat/bitwise.lean index 569082b3ec093aa4bba070c0b3cde68ac9438465..159359c85cfa2bb75ffceee14f938b6dc6d824e3 100644 --- a/formal/lean/mathlib/data/nat/bitwise.lean +++ b/formal/lean/mathlib/data/nat/bitwise.lean @@ -3,6 +3,7 @@ Copyright (c) 2020 Markus Himmel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Markus Himmel -/ +import data.nat.bits import tactic.linarith /-! @@ -51,6 +52,16 @@ end @[simp] lemma zero_test_bit (i : ℕ) : test_bit 0 i = ff := by simp [test_bit] +/-- The ith bit is the ith element of `n.bits`. -/ +lemma test_bit_eq_inth (n i : ℕ) : n.test_bit i = n.bits.inth i := +begin + induction i with i ih generalizing n, + { simp [test_bit, shiftr, bodd_eq_bits_head, list.inth_zero_eq_head], }, + conv_lhs { rw ← bit_decomp n, }, + rw [test_bit_succ, ih n.div2, div2_bits_eq_tail], + cases n.bits; simp, +end + /-- Bitwise extensionality: Two numbers agree if they agree at every bit position. -/ lemma eq_of_test_bit_eq {n m : ℕ} (h : ∀ i, test_bit n i = test_bit m i) : n = m := begin diff --git a/formal/lean/mathlib/data/nat/choose/central.lean b/formal/lean/mathlib/data/nat/choose/central.lean index 220d0d421fab84ee51602f3ce78f3fd4da4b5436..d5e9bebe5f13c67b71501fb5431bf5d55a65b6fa 100644 --- a/formal/lean/mathlib/data/nat/choose/central.lean +++ b/formal/lean/mathlib/data/nat/choose/central.lean @@ -86,7 +86,7 @@ end /-- An exponential lower bound on the central binomial coefficient. -This bound is weaker than `four_pow_n_lt_n_mul_central_binom`, but it is of historical interest +This bound is weaker than `nat.four_pow_lt_mul_central_binom`, but it is of historical interest because it appears in Erdős's proof of Bertrand's postulate. -/ lemma four_pow_le_two_mul_self_mul_central_binom : ∀ (n : ℕ) (n_pos : 0 < n), diff --git a/formal/lean/mathlib/data/nat/choose/factorization.lean b/formal/lean/mathlib/data/nat/choose/factorization.lean index 8a13ee9a1891086d16594ee2a5223a5aeee9391e..0dc2dfedbecdec13eba70c0f0889dee5c29c458f 100644 --- a/formal/lean/mathlib/data/nat/choose/factorization.lean +++ b/formal/lean/mathlib/data/nat/choose/factorization.lean @@ -26,6 +26,8 @@ do not appear in the factorization of the `n`th central binomial coefficient. These results appear in the [Erdős proof of Bertrand's postulate](aigner1999proofs). -/ +open_locale big_operators + namespace nat variables {p n k : ℕ} @@ -137,4 +139,31 @@ lemma le_two_mul_of_factorization_central_binom_pos (h_pos : 0 < (central_binom n).factorization p) : p ≤ 2 * n := le_of_not_lt (pos_iff_ne_zero.mp h_pos ∘ factorization_central_binom_eq_zero_of_two_mul_lt) +/-- A binomial coefficient is the product of its prime factors, which are at most `n`. -/ +lemma prod_pow_factorization_choose (n k : ℕ) (hkn : k ≤ n) : + ∏ p in (finset.range (n + 1)), + p ^ ((nat.choose n k).factorization p) + = choose n k := +begin + nth_rewrite_rhs 0 ←factorization_prod_pow_eq_self (choose_pos hkn).ne', + rw eq_comm, + apply finset.prod_subset, + { intros p hp, + rw finset.mem_range, + contrapose! hp, + rw [finsupp.mem_support_iff, not_not, factorization_choose_eq_zero_of_lt hp] }, + { intros p _ h2, simp [not_not.1 (mt finsupp.mem_support_iff.2 h2)] }, +end + +/-- The `n`th central binomial coefficient is the product of its prime factors, which are +at most `2n`. -/ +lemma prod_pow_factorization_central_binom (n : ℕ) : + ∏ p in (finset.range (2 * n + 1)), + p ^ ((central_binom n).factorization p) + = central_binom n := +begin + apply prod_pow_factorization_choose, + linarith, +end + end nat diff --git a/formal/lean/mathlib/data/nat/digits.lean b/formal/lean/mathlib/data/nat/digits.lean index 651d8479406d38a2fda5015601e1a6fd4d1d0016..a3021a07aa3ee3fa23f2a3f6c721c2aaa0b567d7 100644 --- a/formal/lean/mathlib/data/nat/digits.lean +++ b/formal/lean/mathlib/data/nat/digits.lean @@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro -/ import data.int.modeq +import data.nat.bits import data.nat.log import data.nat.parity import data.list.indexes @@ -471,6 +472,22 @@ begin exact base_pow_length_digits_le' b m, end +/-! ### Binary -/ +lemma digits_two_eq_bits (n : ℕ) : digits 2 n = n.bits.map (λ b, cond b 1 0) := +begin + induction n using nat.binary_rec_from_one with b n h ih, + { simp, }, + { simp, }, + rw bits_append_bit _ _ (λ hn, absurd hn h), + cases b, + { rw digits_def' (le_refl 2), + { simpa [nat.bit, nat.bit0_val n], }, + { simpa [pos_iff_ne_zero, bit_eq_zero_iff], }, }, + { simpa [nat.bit, nat.bit1_val n, add_comm, digits_add 2 le_rfl 1 n (by norm_num) + (by norm_num)] }, +end + + /-! ### Modular Arithmetic -/ -- This is really a theorem about polynomials. diff --git a/formal/lean/mathlib/data/nat/factorization/basic.lean b/formal/lean/mathlib/data/nat/factorization/basic.lean index 8846a22839aefaf032e485a36675aad85a6025f8..fa807937a1fecf060841fa1f9157c8748eaa4e74 100644 --- a/formal/lean/mathlib/data/nat/factorization/basic.lean +++ b/formal/lean/mathlib/data/nat/factorization/basic.lean @@ -187,6 +187,10 @@ begin refl, end +/-- The only prime factor of prime `p` is `p` itself, with multiplicity `1` -/ +@[simp] lemma prime.factorization_self {p : ℕ} (hp : prime p) : p.factorization p = 1 := +by simp [hp] + /-- For prime `p` the only prime factor of `p^k` is `p` with multiplicity `k` -/ lemma prime.factorization_pow {p k : ℕ} (hp : prime p) : factorization (p ^ k) = single p k := diff --git a/formal/lean/mathlib/data/nat/modeq.lean b/formal/lean/mathlib/data/nat/modeq.lean index 7bc8f6ead6859d77ad672790e52ca5d1c14338ad..53ee3cded32c970d9eb12dae9367d1f6c152ce4d 100644 --- a/formal/lean/mathlib/data/nat/modeq.lean +++ b/formal/lean/mathlib/data/nat/modeq.lean @@ -39,6 +39,8 @@ namespace modeq protected theorem rfl : a ≡ a [MOD n] := modeq.refl _ +instance : is_refl _ (modeq n) := ⟨modeq.refl⟩ + @[symm] protected theorem symm : a ≡ b [MOD n] → b ≡ a [MOD n] := eq.symm @[trans] protected theorem trans : a ≡ b [MOD n] → b ≡ c [MOD n] → a ≡ c [MOD n] := eq.trans diff --git a/formal/lean/mathlib/data/nat/totient.lean b/formal/lean/mathlib/data/nat/totient.lean index 8526eaab5b8be307cb8cee95f65383f17f089668..ded01f637e46c7a8eaba5c059690382a85ebfc6d 100644 --- a/formal/lean/mathlib/data/nat/totient.lean +++ b/formal/lean/mathlib/data/nat/totient.lean @@ -336,6 +336,19 @@ begin apply mul_le_mul_left' (nat.totient_le d), end +lemma totient_dvd_of_dvd {a b : ℕ} (h : a ∣ b) : φ a ∣ φ b := +begin + rcases eq_or_ne a 0 with rfl | ha0, { simp [zero_dvd_iff.1 h] }, + rcases eq_or_ne b 0 with rfl | hb0, { simp }, + have hab' : a.factorization.support ⊆ b.factorization.support, + { intro p, + simp only [support_factorization, list.mem_to_finset], + apply factors_subset_of_dvd h hb0 }, + rw [totient_eq_prod_factorization ha0, totient_eq_prod_factorization hb0], + refine finsupp.prod_dvd_prod_of_subset_of_dvd hab' (λ p hp, mul_dvd_mul _ dvd_rfl), + exact pow_dvd_pow p (tsub_le_tsub_right ((factorization_le_iff_dvd ha0 hb0).2 h p) 1), +end + lemma totient_mul_of_prime_of_dvd {p n : ℕ} (hp : p.prime) (h : p ∣ n) : (p * n).totient = p * n.totient := begin diff --git a/formal/lean/mathlib/data/pfunctor/multivariate/W.lean b/formal/lean/mathlib/data/pfunctor/multivariate/W.lean index dbb2b2165b607e5974f3e101fc77e7f96e54e8a6..3923df2b0c67f6d4852bda6b8f56e8206b6298a1 100644 --- a/formal/lean/mathlib/data/pfunctor/multivariate/W.lean +++ b/formal/lean/mathlib/data/pfunctor/multivariate/W.lean @@ -114,7 +114,7 @@ def Wp : mvpfunctor n := { A := P.last.W, B := P.W_path } /-- W-type of `P` -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def W (α : typevec n) : Type* := P.Wp.obj α instance mvfunctor_W : mvfunctor P.W := by delta mvpfunctor.W; apply_instance diff --git a/formal/lean/mathlib/data/pfunctor/univariate/basic.lean b/formal/lean/mathlib/data/pfunctor/univariate/basic.lean index e4aeca9a721caca886b8663976b450460413921e..ba63040d31e2b59c630e2caf42d1178bd0981c0a 100644 --- a/formal/lean/mathlib/data/pfunctor/univariate/basic.lean +++ b/formal/lean/mathlib/data/pfunctor/univariate/basic.lean @@ -65,7 +65,7 @@ def W := _root_.W_type P.B /- inhabitants of W types is awkward to encode as an instance assumption because there needs to be a value `a : P.A` such that `P.B a` is empty to yield a finite tree -/ -attribute [nolint has_inhabited_instance] W +attribute [nolint has_nonempty_instance] W variables {P} /-- root element of a W tree -/ diff --git a/formal/lean/mathlib/data/polynomial/basic.lean b/formal/lean/mathlib/data/polynomial/basic.lean index 3841580de0b4824bdca2b6eed75dfc22ac624968..848965f374573d0ed15e76260e6545022626295a 100644 --- a/formal/lean/mathlib/data/polynomial/basic.lean +++ b/formal/lean/mathlib/data/polynomial/basic.lean @@ -32,7 +32,7 @@ the polynomials. For instance, Polynomials are defined using `add_monoid_algebra R ℕ`, where `R` is a semiring. The variable `X` commutes with every polynomial `p`: lemma `X_mul` proves the identity -`X * p = `p * X`. The relationship to `add_monoid_algebra R ℕ` is through a structure +`X * p = p * X`. The relationship to `add_monoid_algebra R ℕ` is through a structure to make polynomials irreducible from the point of view of the kernel. Most operations are irreducible since Lean can not compute anyway with `add_monoid_algebra`. There are two exceptions that we make semireducible: @@ -83,8 +83,6 @@ they unfold around `polynomial.of_finsupp` and `polynomial.to_finsupp`. -/ section add_monoid_algebra -/-- The function version of `monomial`. Use `monomial` instead of this one. -/ -@[irreducible] def monomial_fun (n : ℕ) (a : R) : R[X] := ⟨finsupp.single n a⟩ @[irreducible] private def add : R[X] → R[X] → R[X] | ⟨a⟩ ⟨b⟩ := ⟨a + b⟩ @[irreducible] private def neg {R : Type u} [ring R] : R[X] → R[X] @@ -93,7 +91,7 @@ section add_monoid_algebra | ⟨a⟩ ⟨b⟩ := ⟨a * b⟩ instance : has_zero R[X] := ⟨⟨0⟩⟩ -instance : has_one R[X] := ⟨monomial_fun 0 (1 : R)⟩ +instance : has_one R[X] := ⟨⟨1⟩⟩ instance : has_add R[X] := ⟨add⟩ instance {R : Type u} [ring R] : has_neg R[X] := ⟨neg⟩ instance {R : Type u} [ring R] : has_sub R[X] := ⟨λ a b, a + -b⟩ @@ -103,15 +101,8 @@ instance {S : Type*} [monoid S] [distrib_mul_action S R] : has_smul S R[X] := @[priority 1] -- to avoid a bug in the `ring` tactic instance has_pow : has_pow R[X] ℕ := { pow := λ p n, npow_rec n p } -@[simp] lemma of_finsupp_zero : (⟨0⟩ : R[X]) = 0 := -rfl - -@[simp] lemma of_finsupp_one : (⟨1⟩ : R[X]) = 1 := -begin - change (⟨1⟩ : R[X]) = monomial_fun 0 (1 : R), - rw [monomial_fun], - refl -end +@[simp] lemma of_finsupp_zero : (⟨0⟩ : R[X]) = 0 := rfl +@[simp] lemma of_finsupp_one : (⟨1⟩ : R[X]) = 1 := rfl @[simp] lemma of_finsupp_add {a b} : (⟨a + b⟩ : R[X]) = ⟨a⟩ + ⟨b⟩ := show _ = add _ _, by rw add @[simp] lemma of_finsupp_neg {R : Type u} [ring R] {a} : (⟨-a⟩ : R[X]) = -⟨a⟩ := @@ -133,12 +124,7 @@ end @[simp] lemma to_finsupp_zero : (0 : R[X]).to_finsupp = 0 := rfl -@[simp] lemma to_finsupp_one : (1 : R[X]).to_finsupp = 1 := -begin - change to_finsupp (monomial_fun _ _) = _, - rw monomial_fun, - refl, -end +@[simp] lemma to_finsupp_one : (1 : R[X]).to_finsupp = 1 := rfl @[simp] lemma to_finsupp_add (a b : R[X]) : (a + b).to_finsupp = a.to_finsupp + b.to_finsupp := by { cases a, cases b, rw ←of_finsupp_add } @@ -263,17 +249,17 @@ by simp /-- `monomial s a` is the monomial `a * X^s` -/ def monomial (n : ℕ) : R →ₗ[R] R[X] := -{ to_fun := monomial_fun n, - map_add' := by simp [monomial_fun], - map_smul' := by simp [monomial_fun, ←of_finsupp_smul] } +{ to_fun := λ t, ⟨finsupp.single n t⟩, + map_add' := by simp, + map_smul' := by simp [←of_finsupp_smul] } @[simp] lemma to_finsupp_monomial (n : ℕ) (r : R) : (monomial n r).to_finsupp = finsupp.single n r := -by simp [monomial, monomial_fun] +by simp [monomial] @[simp] lemma of_finsupp_single (n : ℕ) (r : R) : (⟨finsupp.single n r⟩ : R[X]) = monomial n r := -by simp [monomial, monomial_fun] +by simp [monomial] @[simp] lemma monomial_zero_right (n : ℕ) : @@ -308,11 +294,7 @@ to_finsupp_injective $ by simp lemma monomial_injective (n : ℕ) : function.injective (monomial n : R → R[X]) := -begin - convert (to_finsupp_iso R).symm.injective.comp (single_injective n), - ext, - simp -end +(to_finsupp_iso R).symm.injective.comp (single_injective n) @[simp] lemma monomial_eq_zero_iff (t : R) (n : ℕ) : monomial n t = 0 ↔ t = 0 := @@ -498,6 +480,15 @@ lemma monomial_eq_C_mul_X : ∀{n}, monomial n a = C a * X^n calc C a = 0 ↔ C a = C 0 : by rw C_0 ... ↔ a = 0 : C_inj +lemma subsingleton_iff_subsingleton : + subsingleton R[X] ↔ subsingleton R := +⟨λ h, subsingleton_iff.mpr (λ a b, C_inj.mp (subsingleton_iff.mp h _ _)), + by { introI, apply_instance } ⟩ + +lemma forall_eq_iff_forall_eq : + (∀ f g : R[X], f = g) ↔ (∀ a b : R, a = b) := +by simpa only [← subsingleton_iff] using subsingleton_iff_subsingleton + theorem ext_iff {p q : R[X]} : p = q ↔ ∀ n, coeff p n = coeff q n := by { rcases p, rcases q, simp [coeff, finsupp.ext_iff] } @@ -684,6 +675,12 @@ begin simpa [sum, support, coeff] using finsupp.sum_smul_index hf, end +lemma sum_monomial_eq : ∀ p : R[X], p.sum (λ n a, monomial n a) = p +| ⟨p⟩ := (of_finsupp_sum _ _).symm.trans (congr_arg _ $ finsupp.sum_single _) + +lemma sum_C_mul_X_eq (p : R[X]) : p.sum (λn a, C a * X^n) = p := +by simp_rw [←monomial_eq_C_mul_X, sum_monomial_eq] + /-- `erase p n` is the polynomial `p` in which the `X^n` term has been erased. -/ @[irreducible] definition erase (n : ℕ) : R[X] → R[X] | ⟨p⟩ := ⟨p.erase n⟩ diff --git a/formal/lean/mathlib/data/polynomial/coeff.lean b/formal/lean/mathlib/data/polynomial/coeff.lean index f787cc51eb32e58fbd626a0f1238d2b92dcbded1..8a11187bef5e05edf1fdafb4beb6f01a2d73dd83 100644 --- a/formal/lean/mathlib/data/polynomial/coeff.lean +++ b/formal/lean/mathlib/data/polynomial/coeff.lean @@ -205,6 +205,24 @@ by simpa only [pow_one] using coeff_mul_X_pow p 1 n @[simp] theorem coeff_X_mul (p : R[X]) (n : ℕ) : coeff (X * p) (n + 1) = coeff p n := by rw [(commute_X p).eq, coeff_mul_X] +theorem coeff_mul_monomial (p : R[X]) (n d : ℕ) (r : R) : + coeff (p * monomial n r) (d + n) = coeff p d * r := +by rw [monomial_eq_C_mul_X, ←X_pow_mul, ←mul_assoc, coeff_mul_C, coeff_mul_X_pow] + +theorem coeff_monomial_mul (p : R[X]) (n d : ℕ) (r : R) : + coeff (monomial n r * p) (d + n) = r * coeff p d := +by rw [monomial_eq_C_mul_X, mul_assoc, coeff_C_mul, X_pow_mul, coeff_mul_X_pow] + +-- This can already be proved by `simp`. +theorem coeff_mul_monomial_zero (p : R[X]) (d : ℕ) (r : R) : + coeff (p * monomial 0 r) d = coeff p d * r := +coeff_mul_monomial p 0 d r + +-- This can already be proved by `simp`. +theorem coeff_monomial_zero_mul (p : R[X]) (d : ℕ) (r : R) : + coeff (monomial 0 r * p) d = r * coeff p d := +coeff_monomial_mul p 0 d r + theorem mul_X_pow_eq_zero {p : R[X]} {n : ℕ} (H : p * X ^ n = 0) : p = 0 := ext $ λ k, (coeff_mul_X_pow p n k).symm.trans $ ext_iff.1 H (k+n) diff --git a/formal/lean/mathlib/data/polynomial/degree/definitions.lean b/formal/lean/mathlib/data/polynomial/degree/definitions.lean index f29b94445b6595c8742e05b2f7cfb45e87a5dbb4..f8d1c695c5469002ddde847dea1535286b5f37eb 100644 --- a/formal/lean/mathlib/data/polynomial/degree/definitions.lean +++ b/formal/lean/mathlib/data/polynomial/degree/definitions.lean @@ -127,6 +127,14 @@ lemma le_nat_degree_of_mem_supp (a : ℕ) : a ∈ p.support → a ≤ nat_degree p:= le_nat_degree_of_ne_zero ∘ mem_support_iff.mp +lemma degree_eq_of_le_of_coeff_ne_zero (pn : p.degree ≤ n) (p1 : p.coeff n ≠ 0) : + p.degree = n := +pn.antisymm (le_degree_of_ne_zero p1) + +lemma nat_degree_eq_of_le_of_coeff_ne_zero (pn : p.nat_degree ≤ n) (p1 : p.coeff n ≠ 0) : + p.nat_degree = n := +pn.antisymm (le_nat_degree_of_ne_zero p1) + lemma degree_mono [semiring S] {f : R[X]} {g : S[X]} (h : f.support ⊆ g.support) : f.degree ≤ g.degree := finset.sup_mono h @@ -640,6 +648,18 @@ lemma monic.ne_zero_of_ne (h : (0:R) ≠ 1) {p : R[X]} (hp : p.monic) : p ≠ 0 := by { nontriviality R, exact hp.ne_zero } +lemma monic_of_nat_degree_le_of_coeff_eq_one (n : ℕ) (pn : p.nat_degree ≤ n) (p1 : p.coeff n = 1) : + monic p := +begin + nontriviality, + refine (congr_arg _ $ nat_degree_eq_of_le_of_coeff_ne_zero pn _).trans p1, + exact ne_of_eq_of_ne p1 one_ne_zero, +end + +lemma monic_of_degree_le_of_coeff_eq_one (n : ℕ) (pn : p.degree ≤ n) (p1 : p.coeff n = 1) : + monic p := +monic_of_nat_degree_le_of_coeff_eq_one n (nat_degree_le_of_degree_le pn) p1 + lemma monic.ne_zero_of_polynomial_ne {r} (hp : monic p) (hne : q ≠ r) : p ≠ 0 := by { haveI := nontrivial.of_polynomial_ne hne, exact hp.ne_zero } @@ -819,12 +839,6 @@ begin exact coeff_mul_degree_add_degree _ _ } } end -lemma subsingleton_of_monic_zero (h : monic (0 : R[X])) : - (∀ p q : R[X], p = q) ∧ (∀ a b : R, a = b) := -by rw [monic.def, leading_coeff_zero] at h; - exact ⟨λ p q, by rw [← mul_one p, ← mul_one q, ← C_1, ← h, C_0, mul_zero, mul_zero], - λ a b, by rw [← mul_one a, ← mul_one b, ← h, mul_zero, mul_zero]⟩ - lemma zero_le_degree_iff {p : R[X]} : 0 ≤ degree p ↔ p ≠ 0 := by rw [ne.def, ← degree_eq_bot]; cases degree p; exact dec_trivial diff --git a/formal/lean/mathlib/data/polynomial/degree/lemmas.lean b/formal/lean/mathlib/data/polynomial/degree/lemmas.lean index 6d2370c227de16b52fc036ccbbedddf36374ef91..5bf2503324f5d42a1db3809277ed85731c397ab6 100644 --- a/formal/lean/mathlib/data/polynomial/degree/lemmas.lean +++ b/formal/lean/mathlib/data/polynomial/degree/lemmas.lean @@ -139,6 +139,38 @@ lemma nat_degree_lt_coeff_mul (h : p.nat_degree + q.nat_degree < m + n) : (p * q).coeff (m + n) = 0 := coeff_eq_zero_of_nat_degree_lt (nat_degree_mul_le.trans_lt h) +lemma coeff_mul_of_nat_degree_le (pm : p.nat_degree ≤ m) (qn : q.nat_degree ≤ n) : + (p * q).coeff (m + n) = p.coeff m * q.coeff n := +begin + rcases eq_or_lt_of_le pm with rfl | hm; + rcases eq_or_lt_of_le qn with rfl | hn, + { exact nat_degree_add_coeff_mul _ _ }, + { rw [coeff_eq_zero_of_nat_degree_lt hn, mul_zero], + exact nat_degree_lt_coeff_mul (add_lt_add_left hn _) }, + { rw [coeff_eq_zero_of_nat_degree_lt hm, zero_mul], + exact nat_degree_lt_coeff_mul (add_lt_add_right hm _) }, + { rw [coeff_eq_zero_of_nat_degree_lt hn, mul_zero], + exact nat_degree_lt_coeff_mul (add_lt_add hm hn) }, +end + +lemma coeff_pow_of_nat_degree_le (pn : p.nat_degree ≤ n) : + (p ^ m).coeff (n * m) = (p.coeff n) ^ m := +begin + induction m with m hm, + { simp }, + { rw [pow_succ', pow_succ', ← hm, nat.mul_succ, coeff_mul_of_nat_degree_le _ pn], + refine nat_degree_pow_le.trans (le_trans _ (mul_comm _ _).le), + exact mul_le_mul_of_nonneg_left pn m.zero_le } +end + +lemma coeff_add_eq_left_of_lt (qn : q.nat_degree < n) : + (p + q).coeff n = p.coeff n := +(coeff_add _ _ _).trans $ (congr_arg _ $ coeff_eq_zero_of_nat_degree_lt $ qn).trans $ add_zero _ + +lemma coeff_add_eq_right_of_lt (pn : p.nat_degree < n) : + (p + q).coeff n = q.coeff n := +by { rw add_comm, exact coeff_add_eq_left_of_lt pn } + lemma degree_sum_eq_of_disjoint (f : S → R[X]) (s : finset S) (h : set.pairwise { i | i ∈ s ∧ f i ≠ 0 } (ne on (degree ∘ f))) : degree (s.sum f) = s.sup (λ i, degree (f i)) := @@ -204,13 +236,6 @@ lemma nat_degree_bit0 (a : R[X]) : (bit0 a).nat_degree ≤ a.nat_degree := lemma nat_degree_bit1 (a : R[X]) : (bit1 a).nat_degree ≤ a.nat_degree := (nat_degree_add_le _ _).trans (by simp [nat_degree_bit0]) -lemma nat_degree_sub_le_iff_left {R} [ring R] {n : ℕ} (p q : polynomial R) (qn : q.nat_degree ≤ n) : - (p - q).nat_degree ≤ n ↔ p.nat_degree ≤ n := -begin - rw [sub_eq_add_neg, nat_degree_add_le_iff_left], - rwa nat_degree_neg, -end - variables [semiring S] lemma nat_degree_pos_of_eval₂_root {p : R[X]} (hp : p ≠ 0) (f : R →+* S) @@ -239,6 +264,37 @@ end end degree end semiring +section ring + +variables [ring R] {p q : R[X]} + +lemma nat_degree_sub : (p - q).nat_degree = (q - p).nat_degree := +by rw [← nat_degree_neg, neg_sub] + +lemma nat_degree_sub_le_iff_left (qn : q.nat_degree ≤ n) : + (p - q).nat_degree ≤ n ↔ p.nat_degree ≤ n := +begin + rw ← nat_degree_neg at qn, + rw [sub_eq_add_neg, nat_degree_add_le_iff_left _ _ qn], +end + +lemma nat_degree_sub_le_iff_right (pn : p.nat_degree ≤ n) : + (p - q).nat_degree ≤ n ↔ q.nat_degree ≤ n := +by rwa [nat_degree_sub, nat_degree_sub_le_iff_left] + +lemma coeff_sub_eq_left_of_lt (dg : q.nat_degree < n) : + (p - q).coeff n = p.coeff n := +begin + rw ← nat_degree_neg at dg, + rw [sub_eq_add_neg, coeff_add_eq_left_of_lt dg], +end + +lemma coeff_sub_eq_neg_right_of_lt (df : p.nat_degree < n) : + (p - q).coeff n = - q.coeff n := +by rwa [sub_eq_add_neg, coeff_add_eq_right_of_lt, coeff_neg] + +end ring + section no_zero_divisors variables [semiring R] [no_zero_divisors R] {p q : R[X]} diff --git a/formal/lean/mathlib/data/polynomial/derivative.lean b/formal/lean/mathlib/data/polynomial/derivative.lean index 440e2d4b2d542e55545b51fd664258f3f7d82798..fbc1819285c936d61a50a738e54599e4bdbe9cc5 100644 --- a/formal/lean/mathlib/data/polynomial/derivative.lean +++ b/formal/lean/mathlib/data/polynomial/derivative.lean @@ -158,7 +158,7 @@ begin { exact nat.le_pred_of_lt (nat_degree_derivative_lt p0) } end -@[simp] lemma derivative_cast_nat {n : ℕ} : derivative (n : R[X]) = 0 := +@[simp] lemma derivative_nat_cast {n : ℕ} : derivative (n : R[X]) = 0 := begin rw ← map_nat_cast C n, exact derivative_C, @@ -248,7 +248,7 @@ begin { simp only [ih, function.iterate_succ, polynomial.derivative_map, function.comp_app], }, end -@[simp] lemma iterate_derivative_cast_nat_mul {n k : ℕ} {f : R[X]} : +@[simp] lemma iterate_derivative_nat_cast_mul {n k : ℕ} {f : R[X]} : derivative^[k] (n * f) = n * (derivative^[k] f) := by induction k with k ih generalizing f; simp* diff --git a/formal/lean/mathlib/data/polynomial/div.lean b/formal/lean/mathlib/data/polynomial/div.lean index 50aad0659c3975b8469f61975d5c8a49806e12dc..0ee1356baca2ed37efb0d5cd55490ebfcd5a1a48 100644 --- a/formal/lean/mathlib/data/polynomial/div.lean +++ b/formal/lean/mathlib/data/polynomial/div.lean @@ -150,12 +150,12 @@ begin end @[simp] lemma mod_by_monic_zero (p : R[X]) : p %ₘ 0 = p := -if h : monic (0 : R[X]) then (subsingleton_of_monic_zero h).1 _ _ else -by unfold mod_by_monic div_mod_by_monic_aux; rw dif_neg h +if h : monic (0 : R[X]) then by { haveI := monic_zero_iff_subsingleton.mp h, simp } +else by unfold mod_by_monic div_mod_by_monic_aux; rw dif_neg h @[simp] lemma div_by_monic_zero (p : R[X]) : p /ₘ 0 = 0 := -if h : monic (0 : R[X]) then (subsingleton_of_monic_zero h).1 _ _ else -by unfold div_by_monic div_mod_by_monic_aux; rw dif_neg h +if h : monic (0 : R[X]) then by { haveI := monic_zero_iff_subsingleton.mp h, simp } +else by unfold div_by_monic div_mod_by_monic_aux; rw dif_neg h lemma div_by_monic_eq_of_not_monic (p : R[X]) (hq : ¬monic q) : p /ₘ q = 0 := dif_neg hq diff --git a/formal/lean/mathlib/data/polynomial/expand.lean b/formal/lean/mathlib/data/polynomial/expand.lean index c5d4f12bd692db16fe882d9241b0622b7ba6b0a0..22d0a2234866d12429f352db143ff464c940235d 100644 --- a/formal/lean/mathlib/data/polynomial/expand.lean +++ b/formal/lean/mathlib/data/polynomial/expand.lean @@ -240,7 +240,7 @@ variable {R} theorem of_irreducible_expand {p : ℕ} (hp : p ≠ 0) {f : R[X]} (hf : irreducible (expand R p f)) : irreducible f := -@@of_irreducible_map _ _ _ (is_local_ring_hom_expand R hp.bot_lt) hf +let _ := is_local_ring_hom_expand R hp.bot_lt in by exactI of_irreducible_map ↑(expand R p) hf theorem of_irreducible_expand_pow {p : ℕ} (hp : p ≠ 0) {f : R[X]} {n : ℕ} : irreducible (expand R (p ^ n) f) → irreducible f := diff --git a/formal/lean/mathlib/data/polynomial/induction.lean b/formal/lean/mathlib/data/polynomial/induction.lean index b1c9de63a0e8b4cb233cf6171919a404dc7b6370..10f7f2a91162fe554df8fa1939154667fe4e270d 100644 --- a/formal/lean/mathlib/data/polynomial/induction.lean +++ b/formal/lean/mathlib/data/polynomial/induction.lean @@ -4,11 +4,15 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker -/ import data.polynomial.coeff +import ring_theory.ideal.basic /-! -# Theory of univariate polynomials +# Induction on polynomials -The main results are `induction_on` and `as_sum`. +This file contains lemmas dealing with different flavours of induction on polynomials. +See also `data/polynomial/inductions.lean` (with an `s`!). + +The main result is `polynomial.induction_on`. -/ noncomputable theory @@ -24,17 +28,6 @@ variables {R : Type u} {S : Type v} {T : Type w} {ι : Type x} {k : Type y} {A : section semiring variables [semiring R] {p q r : R[X]} -lemma sum_C_mul_X_eq (p : R[X]) : p.sum (λn a, C a * X^n) = p := -begin - ext n, - simp only [polynomial.sum, X_pow_eq_monomial, coeff_monomial, mul_one, finset_sum_coeff, - C_mul_monomial, not_not, mem_support_iff, finset.sum_ite_eq', ite_eq_left_iff], - exact λ h, h.symm -end - -lemma sum_monomial_eq (p : R[X]) : p.sum (λn a, monomial n a) = p := -by simp only [monomial_eq_C_mul_X, sum_C_mul_X_eq] - @[elab_as_eliminator] protected lemma induction_on {M : R[X] → Prop} (p : R[X]) (h_C : ∀a, M (C a)) (h_add : ∀p q, M p → M q → M (p + q)) @@ -66,45 +59,22 @@ and it holds for monomials. polynomial.induction_on p (h_monomial 0) h_add (λ n a h, by { rw ← monomial_eq_C_mul_X at ⊢, exact h_monomial _ _ }) -section coeff - -theorem coeff_mul_monomial (p : R[X]) (n d : ℕ) (r : R) : - coeff (p * monomial n r) (d + n) = coeff p d * r := -by rw [monomial_eq_C_mul_X, ←X_pow_mul, ←mul_assoc, coeff_mul_C, coeff_mul_X_pow] - -theorem coeff_monomial_mul (p : R[X]) (n d : ℕ) (r : R) : - coeff (monomial n r * p) (d + n) = r * coeff p d := -by rw [monomial_eq_C_mul_X, mul_assoc, coeff_C_mul, X_pow_mul, coeff_mul_X_pow] - --- This can already be proved by `simp`. -theorem coeff_mul_monomial_zero (p : R[X]) (d : ℕ) (r : R) : - coeff (p * monomial 0 r) d = coeff p d * r := -coeff_mul_monomial p 0 d r - --- This can already be proved by `simp`. -theorem coeff_monomial_zero_mul (p : R[X]) (d : ℕ) (r : R) : - coeff (monomial 0 r * p) d = r * coeff p d := -coeff_monomial_mul p 0 d r - -end coeff open submodule polynomial set -variables {f : R[X]} {I : submodule R[X] R[X]} +variables {f : R[X]} {I : ideal R[X]} -/-- If the coefficients of a polynomial belong to n ideal contains the submodule span of the -coefficients of a polynomial. -/ -lemma span_le_of_coeff_mem_C_inverse (cf : ∀ (i : ℕ), f.coeff i ∈ (C ⁻¹' I.carrier)) : - (span R[X] {g | ∃ i, g = C (f.coeff i)}) ≤ I := +/-- If the coefficients of a polynomial belong to an ideal, then that ideal contains +the ideal spanned by the coefficients of the polynomial. -/ +lemma span_le_of_C_coeff_mem (cf : ∀ (i : ℕ), C (f.coeff i) ∈ I) : + ideal.span {g | ∃ i, g = C (f.coeff i)} ≤ I := begin - refine bInter_subset_of_mem _, - rintros _ ⟨i, rfl⟩, - exact set_like.mem_coe.mpr (cf i), + simp only [@eq_comm _ _ (C _)] {single_pass := tt}, + exact (ideal.span_le.trans range_subset_iff).mpr cf, end -lemma mem_span_C_coeff : - f ∈ span R[X] {g : R[X] | ∃ i : ℕ, g = (C (coeff f i))} := +lemma mem_span_C_coeff : f ∈ ideal.span {g : R[X] | ∃ i : ℕ, g = C (coeff f i)} := begin - let p := span R[X] {g : R[X] | ∃ i : ℕ, g = (C (coeff f i))}, + let p := ideal.span {g : R[X] | ∃ i : ℕ, g = C (coeff f i)}, nth_rewrite 0 (sum_C_mul_X_eq f).symm, refine submodule.sum_mem _ (λ n hn, _), dsimp, @@ -115,10 +85,8 @@ begin rw monomial_eq_C_mul_X, end -lemma exists_coeff_not_mem_C_inverse : - f ∉ I → ∃ i : ℕ , coeff f i ∉ (C ⁻¹' I.carrier) := -imp_of_not_imp_not _ _ - (λ cf, not_not.mpr ((span_le_of_coeff_mem_C_inverse (not_exists_not.mp cf)) mem_span_C_coeff)) +lemma exists_C_coeff_not_mem : f ∉ I → ∃ i : ℕ, C (coeff f i) ∉ I := +not.imp_symm $ λ cf, span_le_of_C_coeff_mem (not_exists_not.mp cf) mem_span_C_coeff end semiring diff --git a/formal/lean/mathlib/data/polynomial/lifts.lean b/formal/lean/mathlib/data/polynomial/lifts.lean index 52a3a3b89f0798b03403ed09e8b1722f31b8bf16..6487f49a26b134631d843050ad36809a97328e6a 100644 --- a/formal/lean/mathlib/data/polynomial/lifts.lean +++ b/formal/lean/mathlib/data/polynomial/lifts.lean @@ -209,6 +209,16 @@ begin { rw [degree_add_eq_right_of_degree_lt hdeg, degree_X_pow, degree_eq_nat_degree hp.ne_zero] } end +lemma lifts_and_nat_degree_eq_and_monic {p : S[X]} (hlifts : p ∈ lifts f) + (hp : p.monic) : ∃ (q : R[X]), map f q = p ∧ q.nat_degree = p.nat_degree ∧ q.monic := +begin + casesI subsingleton_or_nontrivial S with hR hR, + { obtain (rfl : p = 1) := subsingleton.elim _ _, + refine ⟨1, subsingleton.elim _ _, by simp, by simp⟩ }, + obtain ⟨p', h₁, h₂, h₃⟩ := lifts_and_degree_eq_and_monic hlifts hp, + exact ⟨p', h₁, nat_degree_eq_of_degree_eq h₂, h₃⟩ +end + end monic end semiring diff --git a/formal/lean/mathlib/data/polynomial/monic.lean b/formal/lean/mathlib/data/polynomial/monic.lean index 360d21a747eba47bc7da097b7fca85491cb90a65..86e959970fdc9604b73ff6a1f42c7635beaa950a 100644 --- a/formal/lean/mathlib/data/polynomial/monic.lean +++ b/formal/lean/mathlib/data/polynomial/monic.lean @@ -26,6 +26,17 @@ variables {R : Type u} {S : Type v} {a b : R} {m n : ℕ} {ι : Type y} section semiring variables [semiring R] {p q r : R[X]} +lemma monic_zero_iff_subsingleton : monic (0 : R[X]) ↔ subsingleton R := +subsingleton_iff_zero_eq_one + +lemma not_monic_zero_iff : ¬ monic (0 : R[X]) ↔ (0 : R) ≠ 1 := +(monic_zero_iff_subsingleton.trans subsingleton_iff_zero_eq_one.symm).not + +lemma monic_zero_iff_subsingleton' : + monic (0 : R[X]) ↔ (∀ f g : R[X], f = g) ∧ (∀ a b : R, a = b) := +polynomial.monic_zero_iff_subsingleton.trans ⟨by { introI, simp }, + λ h, subsingleton_iff.mpr h.2⟩ + lemma monic.as_sum (hp : p.monic) : p = X^(p.nat_degree) + (∑ i in range p.nat_degree, C (p.coeff i) * X^i) := begin @@ -392,7 +403,7 @@ section nonzero_semiring variables [semiring R] [nontrivial R] {p q : R[X]} @[simp] lemma not_monic_zero : ¬monic (0 : R[X]) := -by simpa only [monic, leading_coeff_zero] using (zero_ne_one : (0 : R) ≠ 1) +not_monic_zero_iff.mp zero_ne_one end nonzero_semiring diff --git a/formal/lean/mathlib/data/polynomial/ring_division.lean b/formal/lean/mathlib/data/polynomial/ring_division.lean index 3d4d974f07d0adf9091b06ae2f161f26c427c2ff..5635f594bf255bffa3f35d3e2ae54c8500be3c36 100644 --- a/formal/lean/mathlib/data/polynomial/ring_division.lean +++ b/formal/lean/mathlib/data/polynomial/ring_division.lean @@ -378,6 +378,12 @@ end @[simp] lemma mem_roots (hp : p ≠ 0) : a ∈ p.roots ↔ is_root p a := by rw [← count_pos, count_roots p, root_multiplicity_pos hp] +lemma ne_zero_of_mem_roots (h : a ∈ p.roots) : p ≠ 0 := +λ hp, by rwa [hp, roots_zero] at h + +lemma is_root_of_mem_roots (h : a ∈ p.roots) : is_root p a := +(mem_roots $ ne_zero_of_mem_roots h).mp h + theorem card_le_degree_of_subset_roots {p : R[X]} {Z : finset R} (h : Z.val ⊆ p.roots) : Z.card ≤ p.nat_degree := (multiset.card_le_of_le (finset.val_le_iff_val_subset.2 h)).trans (polynomial.card_roots' p) @@ -678,6 +684,23 @@ begin exact hp (map_injective _ (no_zero_smul_divisors.algebra_map_injective T S) h) end +lemma root_set_maps_to {p : T[X]} {S S'} [comm_ring S] [is_domain S] [algebra T S] + [comm_ring S'] [is_domain S'] [algebra T S'] (hp : p.map (algebra_map T S') ≠ 0) + (f : S →ₐ[T] S') : (p.root_set S).maps_to f (p.root_set S') := +λ x hx, begin + rw [mem_root_set_iff' hp, ← f.comp_algebra_map, ← map_map, eval_map], + erw [eval₂_hom, (mem_root_set_iff' (mt (λ h, _) hp) x).1 hx, _root_.map_zero], + rw [← f.comp_algebra_map, ← map_map, h, polynomial.map_zero], +end + +lemma ne_zero_of_mem_root_set {p : T[X]} [comm_ring S] [is_domain S] [algebra T S] {a : S} + (h : a ∈ p.root_set S) : p ≠ 0 := +λ hf, by rwa [hf, root_set_zero] at h + +lemma aeval_eq_zero_of_mem_root_set {p : T[X]} [comm_ring S] [is_domain S] [algebra T S] + [no_zero_smul_divisors T S] {a : S} (hx : a ∈ p.root_set S) : aeval a p = 0 := +(mem_root_set_iff (ne_zero_of_mem_root_set hx) a).mp hx + end roots theorem is_unit_iff {f : R[X]} : is_unit f ↔ ∃ r : R, is_unit r ∧ C r = f := diff --git a/formal/lean/mathlib/data/qpf/multivariate/constructions/fix.lean b/formal/lean/mathlib/data/qpf/multivariate/constructions/fix.lean index 2d6733aa52772788205e0795de9806d19730ac5c..ef1470eae2ebb88d53f3d71b6ae2fc061fd745a6 100644 --- a/formal/lean/mathlib/data/qpf/multivariate/constructions/fix.lean +++ b/formal/lean/mathlib/data/qpf/multivariate/constructions/fix.lean @@ -184,7 +184,7 @@ fix F a b = F a b (fix F a b) def fix {n : ℕ} (F : typevec (n+1) → Type*) [mvfunctor F] [q : mvqpf F] (α : typevec n) := quotient (W_setoid α : setoid (q.P.W α)) -attribute [nolint has_inhabited_instance] fix +attribute [nolint has_nonempty_instance] fix /-- `fix F` is a functor -/ def fix.map {α β : typevec n} (g : α ⟹ β) : fix F α → fix F β := diff --git a/formal/lean/mathlib/data/qpf/univariate/basic.lean b/formal/lean/mathlib/data/qpf/univariate/basic.lean index 271577b624e48576477c90dd9ce7f06e0c561f97..94c474a2a20557892e429e4fe25fe31402119fba 100644 --- a/formal/lean/mathlib/data/qpf/univariate/basic.lean +++ b/formal/lean/mathlib/data/qpf/univariate/basic.lean @@ -213,7 +213,7 @@ def W_setoid : setoid q.P.W := local attribute [instance] W_setoid /-- inductive type defined as initial algebra of a Quotient of Polynomial Functor -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def fix (F : Type u → Type u) [functor F] [q : qpf F] := quotient (W_setoid : setoid q.P.W) /-- recursor of a type defined by a qpf -/ diff --git a/formal/lean/mathlib/data/real/ennreal.lean b/formal/lean/mathlib/data/real/ennreal.lean index 759fa233032afbc2091379b1964017888a4a0886..ecda571346a9f79a3273456f1f5ee785839bdad3 100644 --- a/formal/lean/mathlib/data/real/ennreal.lean +++ b/formal/lean/mathlib/data/real/ennreal.lean @@ -94,6 +94,12 @@ canonically_ordered_comm_semiring.to_covariant_mul_le instance covariant_class_add_le : covariant_class ℝ≥0∞ ℝ≥0∞ (+) (≤) := ordered_add_comm_monoid.to_covariant_class_left ℝ≥0∞ +noncomputable instance : linear_ordered_comm_monoid_with_zero ℝ≥0∞ := +{ mul_le_mul_left := λ a b, mul_le_mul_left', + zero_le_one := zero_le 1, + .. ennreal.linear_ordered_add_comm_monoid_with_top, + .. (show comm_semiring ℝ≥0∞, from infer_instance) } + instance : inhabited ℝ≥0∞ := ⟨0⟩ instance : has_coe ℝ≥0 ℝ≥0∞ := ⟨ option.some ⟩ @@ -221,7 +227,6 @@ protected lemma zero_lt_one : 0 < (1 : ℝ≥0∞) := @[simp] lemma one_lt_two : (1 : ℝ≥0∞) < 2 := coe_one ▸ coe_two ▸ by exact_mod_cast (@one_lt_two ℕ _ _) -lemma one_le_two : (1 : ℝ≥0∞) ≤ 2 := one_lt_two.le @[simp] lemma zero_lt_two : (0:ℝ≥0∞) < 2 := lt_trans ennreal.zero_lt_one one_lt_two lemma two_ne_zero : (2:ℝ≥0∞) ≠ 0 := (ne_of_lt zero_lt_two).symm lemma two_ne_top : (2:ℝ≥0∞) ≠ ∞ := coe_two ▸ coe_ne_top @@ -230,8 +235,7 @@ lemma two_ne_top : (2:ℝ≥0∞) ≠ ∞ := coe_two ▸ coe_ne_top instance _root_.fact_one_le_one_ennreal : fact ((1 : ℝ≥0∞) ≤ 1) := ⟨le_rfl⟩ /-- `(1 : ℝ≥0∞) ≤ 2`, recorded as a `fact` for use with `Lp` spaces. -/ -instance _root_.fact_one_le_two_ennreal : fact ((1 : ℝ≥0∞) ≤ 2) := -⟨ennreal.coe_le_coe.2 (show (1 : ℝ≥0) ≤ 2, by norm_num)⟩ +instance _root_.fact_one_le_two_ennreal : fact ((1 : ℝ≥0∞) ≤ 2) := ⟨one_le_two⟩ /-- `(1 : ℝ≥0∞) ≤ ∞`, recorded as a `fact` for use with `Lp` spaces. -/ instance _root_.fact_one_le_top_ennreal : fact ((1 : ℝ≥0∞) ≤ ∞) := ⟨le_top⟩ @@ -560,7 +564,6 @@ lemma coe_nat_lt_coe {n : ℕ} : (n : ℝ≥0∞) < r ↔ ↑n < r := ennreal.co lemma coe_lt_coe_nat {n : ℕ} : (r : ℝ≥0∞) < n ↔ r < n := ennreal.coe_nat n ▸ coe_lt_coe @[simp, norm_cast] lemma coe_nat_lt_coe_nat {m n : ℕ} : (m : ℝ≥0∞) < n ↔ m < n := ennreal.coe_nat n ▸ coe_nat_lt_coe.trans nat.cast_lt -lemma coe_nat_ne_top {n : ℕ} : (n : ℝ≥0∞) ≠ ∞ := ennreal.coe_nat n ▸ coe_ne_top lemma coe_nat_mono : strict_mono (coe : ℕ → ℝ≥0∞) := λ _ _, coe_nat_lt_coe_nat.2 @[simp, norm_cast] lemma coe_nat_le_coe_nat {m n : ℕ} : (m : ℝ≥0∞) ≤ n ↔ m ≤ n := coe_nat_mono.le_iff_le @@ -582,7 +585,7 @@ begin end @[simp] lemma Union_Iic_coe_nat : (⋃ n : ℕ, Iic (n : ℝ≥0∞)) = {∞}ᶜ := -subset.antisymm (Union_subset $ λ n x hx, ne_top_of_le_ne_top coe_nat_ne_top hx) $ +subset.antisymm (Union_subset $ λ n x hx, ne_top_of_le_ne_top (nat_ne_top n) hx) $ Union_Iio_coe_nat ▸ Union_mono (λ n, Iio_subset_Iic_self) @[simp] lemma Union_Ioc_coe_nat : (⋃ n : ℕ, Ioc a n) = Ioi a \ {∞} := @@ -1347,8 +1350,8 @@ begin rcases exists_nat_pos_mul_gt hb ha with ⟨n, npos, hn⟩, have : (n : ℝ≥0∞) ≠ 0 := nat.cast_ne_zero.2 npos.lt.ne', use [n, npos], - rwa [← one_mul b, ← inv_mul_cancel this coe_nat_ne_top, - mul_assoc, mul_lt_mul_left (inv_ne_zero.2 coe_nat_ne_top) (inv_ne_top.2 this)] + rwa [← one_mul b, ← inv_mul_cancel this (nat_ne_top n), + mul_assoc, mul_lt_mul_left (inv_ne_zero.2 $ nat_ne_top _) (inv_ne_top.2 this)] end lemma exists_nnreal_pos_mul_lt (ha : a ≠ ∞) (hb : b ≠ 0) : diff --git a/formal/lean/mathlib/data/seq/computation.lean b/formal/lean/mathlib/data/seq/computation.lean index 06bd8c3c010087dc796b1b97766a8d5bb601b48a..819eb5174e804b6f410d6133a65bba466f7cd885 100644 --- a/formal/lean/mathlib/data/seq/computation.lean +++ b/formal/lean/mathlib/data/seq/computation.lean @@ -22,21 +22,21 @@ coinductive computation (α : Type u) : Type u An element of `computation α` is an infinite sequence of `option α` such that if `f n = some a` for some `n` then it is constantly `some a` after that. -/ def computation (α : Type u) : Type u := -{ f : stream (option α) // ∀ {n a}, f n = some a → f (n+1) = some a } +{f : stream (option α) // ∀ {n a}, f n = some a → f (n + 1) = some a} namespace computation variables {α : Type u} {β : Type v} {γ : Type w} -- constructors /-- `return a` is the computation that immediately terminates with result `a`. -/ -def return (a : α) : computation α := ⟨stream.const (some a), λn a', id⟩ +def return (a : α) : computation α := ⟨stream.const (some a), λ n a', id⟩ instance : has_coe_t α (computation α) := ⟨return⟩ -- note [use has_coe_t] /-- `think c` is the computation that delays for one "tick" and then performs computation `c`. -/ def think (c : computation α) : computation α := -⟨none :: c.1, λn a h, by {cases n with n, contradiction, exact c.2 h}⟩ +⟨none :: c.1, λ n a h, by {cases n with n, contradiction, exact c.2 h}⟩ /-- `thinkN c n` is the computation that delays for `n` ticks and then performs computation `c`. -/ @@ -57,7 +57,7 @@ def tail (c : computation α) : computation α := /-- `empty α` is the computation that never returns, an infinite sequence of `think`s. -/ -def empty (α) : computation α := ⟨stream.const none, λn a', id⟩ +def empty (α) : computation α := ⟨stream.const none, λ n a', id⟩ instance : inhabited (computation α) := ⟨empty _⟩ @@ -147,7 +147,7 @@ def corec.F (f : β → α ⊕ β) : α ⊕ β → option α × (α ⊕ β) `corec f b = think (corec f b')`. -/ def corec (f : β → α ⊕ β) (b : β) : computation α := begin - refine ⟨stream.corec' (corec.F f) (sum.inr b), λn a' h, _⟩, + refine ⟨stream.corec' (corec.F f) (sum.inr b), λ n a' h, _⟩, rw stream.corec'_eq, change stream.corec' (corec.F f) (corec.F f (sum.inr b)).2 n = some a', revert h, generalize : sum.inr b = o, revert o, @@ -203,12 +203,12 @@ section bisim theorem eq_of_bisim (bisim : is_bisimulation R) {s₁ s₂} (r : s₁ ~ s₂) : s₁ = s₂ := begin apply subtype.eq, - apply stream.eq_of_bisim (λx y, ∃ s s' : computation α, s.1 = x ∧ s'.1 = y ∧ R s s'), + apply stream.eq_of_bisim (λ x y, ∃ s s' : computation α, s.1 = x ∧ s'.1 = y ∧ R s s'), dsimp [stream.is_bisimulation], intros t₁ t₂ e, exact match t₁, t₂, e with ._, ._, ⟨s, s', rfl, rfl, r⟩ := suffices head s = head s' ∧ R (tail s) (tail s'), from - and.imp id (λr, ⟨tail s, tail s', + and.imp id (λ r, ⟨tail s, tail s', by cases s; refl, by cases s'; refl, r⟩) this, begin have := bisim r, revert r this, @@ -467,8 +467,8 @@ mem_rec_on (get_mem s) (h1 _) h2 /-- Map a function on the result of a computation. -/ def map (f : α → β) : computation α → computation β -| ⟨s, al⟩ := ⟨s.map (λo, option.cases_on o none (some ∘ f)), -λn b, begin +| ⟨s, al⟩ := ⟨s.map (λ o, option.cases_on o none (some ∘ f)), +λ n b, begin dsimp [stream.map, stream.nth], induction e : s n with a; intro h, { contradiction }, { rw [al e, ←h] } @@ -511,7 +511,7 @@ by apply s.cases_on; intro; simp @[simp] theorem map_id : ∀ (s : computation α), map id s = s | ⟨f, al⟩ := begin apply subtype.eq; simp [map, function.comp], - have e : (@option.rec α (λ_, option α) none some) = id, + have e : (@option.rec α (λ _, option α) none some) = id, { ext ⟨⟩; refl }, simp [e, stream.map_id] end @@ -528,7 +528,7 @@ end @[simp] theorem ret_bind (a) (f : α → computation β) : bind (return a) f = f a := begin - apply eq_of_bisim (λc₁ c₂, + apply eq_of_bisim (λ c₁ c₂, c₁ = bind (return a) f ∧ c₂ = f a ∨ c₁ = corec (bind.F f) (sum.inr c₂)), { intros c₁ c₂ h, @@ -550,7 +550,7 @@ destruct_eq_think $ by simp [bind, bind.F] @[simp] theorem bind_ret (f : α → β) (s) : bind s (return ∘ f) = map f s := begin - apply eq_of_bisim (λc₁ c₂, c₁ = c₂ ∨ + apply eq_of_bisim (λ c₁ c₂, c₁ = c₂ ∨ ∃ s, c₁ = bind s (return ∘ f) ∧ c₂ = map f s), { intros c₁ c₂ h, exact match c₁, c₂, h with @@ -568,7 +568,7 @@ by rw bind_ret; change (λ x : α, x) with @id α; rw map_id @[simp] theorem bind_assoc (s : computation α) (f : α → computation β) (g : β → computation γ) : bind (bind s f) g = bind s (λ (x : α), bind (f x) g) := begin - apply eq_of_bisim (λc₁ c₂, c₁ = c₂ ∨ + apply eq_of_bisim (λ c₁ c₂, c₁ = c₂ ∨ ∃ s, c₁ = bind (bind s f) g ∧ c₂ = bind s (λ (x : α), bind (f x) g)), { intros c₁ c₂ h, exact match c₁, c₂, h with @@ -678,7 +678,7 @@ theorem terminates_map_iff (f : α → β) (s : computation α) : the first one that gives a result. -/ def orelse (c₁ c₂ : computation α) : computation α := @computation.corec α (computation α × computation α) - (λ⟨c₁, c₂⟩, match destruct c₁ with + (λ ⟨c₁, c₂⟩, match destruct c₁ with | sum.inl a := sum.inl a | sum.inr c₁' := match destruct c₂ with | sum.inl a := sum.inl a @@ -703,7 +703,7 @@ destruct_eq_think $ by unfold has_orelse.orelse; simp [orelse] @[simp] theorem empty_orelse (c) : (empty α <|> c) = c := begin - apply eq_of_bisim (λc₁ c₂, (empty α <|> c₂) = c₁) _ rfl, + apply eq_of_bisim (λ c₁ c₂, (empty α <|> c₂) = c₁) _ rfl, intros s' s h, rw ←h, apply cases_on s; intros s; rw think_empty; simp, rw ←think_empty, @@ -711,7 +711,7 @@ end @[simp] theorem orelse_empty (c : computation α) : (c <|> empty α) = c := begin - apply eq_of_bisim (λc₁ c₂, (c₂ <|> empty α) = c₁) _ rfl, + apply eq_of_bisim (λ c₁ c₂, (c₂ <|> empty α) = c₁) _ rfl, intros s' s h, rw ←h, apply cases_on s; intros s; rw think_empty; simp, rw←think_empty, @@ -723,20 +723,20 @@ def equiv (c₁ c₂ : computation α) : Prop := ∀ a, a ∈ c₁ ↔ a ∈ c infix ` ~ `:50 := equiv -@[refl] theorem equiv.refl (s : computation α) : s ~ s := λ_, iff.rfl +@[refl] theorem equiv.refl (s : computation α) : s ~ s := λ _, iff.rfl @[symm] theorem equiv.symm {s t : computation α} : s ~ t → t ~ s := -λh a, (h a).symm +λ h a, (h a).symm @[trans] theorem equiv.trans {s t u : computation α} : s ~ t → t ~ u → s ~ u := -λh1 h2 a, (h1 a).trans (h2 a) +λ h1 h2 a, (h1 a).trans (h2 a) theorem equiv.equivalence : equivalence (@equiv α) := ⟨@equiv.refl _, @equiv.symm _, @equiv.trans _⟩ theorem equiv_of_mem {s t : computation α} {a} (h1 : a ∈ s) (h2 : a ∈ t) : s ~ t := -λa', ⟨λma, by rw mem_unique ma h1; exact h2, - λma, by rw mem_unique ma h2; exact h1⟩ +λ a', ⟨λ ma, by rw mem_unique ma h1; exact h2, + λ ma, by rw mem_unique ma h2; exact h1⟩ theorem terminates_congr {c₁ c₂ : computation α} (h : c₁ ~ c₂) : terminates c₁ ↔ terminates c₂ := @@ -744,7 +744,7 @@ by simp only [terminates_iff, exists_congr h] theorem promises_congr {c₁ c₂ : computation α} (h : c₁ ~ c₂) (a) : c₁ ~> a ↔ c₂ ~> a := -forall_congr (λa', imp_congr (h a') iff.rfl) +forall_congr (λ a', imp_congr (h a') iff.rfl) theorem get_equiv {c₁ c₂ : computation α} (h : c₁ ~ c₂) [terminates c₁] [terminates c₂] : get c₁ = get c₂ := @@ -758,9 +758,9 @@ theorem thinkN_equiv (s : computation α) (n) : thinkN s n ~ s := theorem bind_congr {s1 s2 : computation α} {f1 f2 : α → computation β} (h1 : s1 ~ s2) (h2 : ∀ a, f1 a ~ f2 a) : bind s1 f1 ~ bind s2 f2 := -λ b, ⟨λh, let ⟨a, ha, hb⟩ := exists_of_mem_bind h in +λ b, ⟨λ h, let ⟨a, ha, hb⟩ := exists_of_mem_bind h in mem_bind ((h1 a).1 ha) ((h2 a b).1 hb), - λh, let ⟨a, ha, hb⟩ := exists_of_mem_bind h in + λ h, let ⟨a, ha, hb⟩ := exists_of_mem_bind h in mem_bind ((h1 a).2 ha) ((h2 a b).2 hb)⟩ theorem equiv_ret_of_mem {s : computation α} {a} (h : a ∈ s) : s ~ return a := @@ -779,10 +779,10 @@ theorem lift_rel.swap (R : α → β → Prop) (ca : computation α) (cb : compu and_comm _ _ theorem lift_eq_iff_equiv (c₁ c₂ : computation α) : lift_rel (=) c₁ c₂ ↔ c₁ ~ c₂ := -⟨λ⟨h1, h2⟩ a, +⟨λ ⟨h1, h2⟩ a, ⟨λ a1, let ⟨b, b2, ab⟩ := h1 a1 in by rwa ab, λ a2, let ⟨b, b1, ab⟩ := h2 a2 in by rwa ←ab⟩, -λe, ⟨λ a a1, ⟨a, (e _).1 a1, rfl⟩, λ a a2, ⟨a, (e _).2 a2, rfl⟩⟩⟩ +λ e, ⟨λ a a1, ⟨a, (e _).1 a1, rfl⟩, λ a a2, ⟨a, (e _).2 a2, rfl⟩⟩⟩ theorem lift_rel.refl (R : α → α → Prop) (H : reflexive R) : reflexive (lift_rel R) := λ s, ⟨λ a as, ⟨a, as, H a⟩, λ b bs, ⟨b, bs, H b⟩⟩ @@ -831,9 +831,9 @@ H.right h theorem lift_rel_def {R : α → β → Prop} {ca cb} : lift_rel R ca cb ↔ (terminates ca ↔ terminates cb) ∧ ∀ {a b}, a ∈ ca → b ∈ cb → R a b := -⟨λh, ⟨terminates_of_lift_rel h, λ a b ma mb, +⟨λ h, ⟨terminates_of_lift_rel h, λ a b ma mb, let ⟨b', mb', ab⟩ := h.left ma in by rwa mem_unique mb mb'⟩, -λ⟨l, r⟩, +λ ⟨l, r⟩, ⟨λ a ma, let ⟨⟨b, mb⟩⟩ := l.1 ⟨⟨_, ma⟩⟩ in ⟨b, mb, r ma mb⟩, λ b mb, let ⟨⟨a, ma⟩⟩ := l.2 ⟨⟨_, mb⟩⟩ in ⟨a, ma, r ma mb⟩⟩⟩ @@ -858,8 +858,8 @@ let ⟨l1, r1⟩ := h1 in @[simp] theorem lift_rel_return_left (R : α → β → Prop) (a : α) (cb : computation β) : lift_rel R (return a) cb ↔ ∃ {b}, b ∈ cb ∧ R a b := -⟨λ⟨l, r⟩, l (ret_mem _), - λ⟨b, mb, ab⟩, +⟨λ ⟨l, r⟩, l (ret_mem _), + λ ⟨b, mb, ab⟩, ⟨λ a' ma', by rw eq_of_ret_mem ma'; exact ⟨b, mb, ab⟩, λ b' mb', ⟨_, ret_mem _, by rw mem_unique mb' mb; exact ab⟩⟩⟩ @@ -870,13 +870,13 @@ by rw [lift_rel.swap, lift_rel_return_left] @[simp] theorem lift_rel_return (R : α → β → Prop) (a : α) (b : β) : lift_rel R (return a) (return b) ↔ R a b := by rw [lift_rel_return_left]; exact -⟨λ⟨b', mb', ab'⟩, by rwa eq_of_ret_mem mb' at ab', - λab, ⟨_, ret_mem _, ab⟩⟩ +⟨λ ⟨b', mb', ab'⟩, by rwa eq_of_ret_mem mb' at ab', + λ ab, ⟨_, ret_mem _, ab⟩⟩ @[simp] theorem lift_rel_think_left (R : α → β → Prop) (ca : computation α) (cb : computation β) : lift_rel R (think ca) cb ↔ lift_rel R ca cb := -and_congr (forall_congr $ λb, imp_congr ⟨of_think_mem, think_mem⟩ iff.rfl) - (forall_congr $ λb, imp_congr iff.rfl $ +and_congr (forall_congr $ λ b, imp_congr ⟨of_think_mem, think_mem⟩ iff.rfl) + (forall_congr $ λ b, imp_congr iff.rfl $ exists_congr $ λ b, and_congr ⟨of_think_mem, think_mem⟩ iff.rfl) @[simp] theorem lift_rel_think_right (R : α → β → Prop) (ca : computation α) (cb : computation β) : diff --git a/formal/lean/mathlib/data/seq/parallel.lean b/formal/lean/mathlib/data/seq/parallel.lean index cb1a1ea565b589dd8392b268b8d3b87e27ecc797..32ecf2e561c6f5c3b6b4e39fc34e4fbc8e14addd 100644 --- a/formal/lean/mathlib/data/seq/parallel.lean +++ b/formal/lean/mathlib/data/seq/parallel.lean @@ -18,7 +18,7 @@ open wseq variables {α : Type u} {β : Type v} def parallel.aux2 : list (computation α) → α ⊕ list (computation α) := -list.foldr (λc o, match o with +list.foldr (λ c o, match o with | sum.inl a := sum.inl a | sum.inr ls := rmap (λ c', c' :: ls) (destruct c) end) (sum.inr []) @@ -204,7 +204,7 @@ def parallel_rec {S : wseq (computation α)} (C : α → Sort v) (H : ∀ s ∈ S, ∀ a ∈ s, C a) {a} (h : a ∈ parallel S) : C a := begin let T : wseq (computation (α × computation α)) := - S.map (λc, c.map (λ a, (a, c))), + S.map (λ c, c.map (λ a, (a, c))), have : S = T.map (map (λ c, c.1)), { rw [←wseq.map_comp], refine (wseq.map_id _).symm.trans (congr_arg (λ f, wseq.map f S) _), funext c, dsimp [id, function.comp], rw [←map_comp], exact (map_id _).symm }, @@ -243,11 +243,11 @@ theorem parallel_congr_lem {S T : wseq (computation α)} {a} theorem parallel_congr_left {S T : wseq (computation α)} {a} (h1 : ∀ s ∈ S, s ~> a) (H : S.lift_rel equiv T) : parallel S ~ parallel T := let h2 := (parallel_congr_lem H).1 h1 in -λ a', ⟨λh, by have aa := parallel_promises h1 h; rw ←aa; rw ←aa at h; exact +λ a', ⟨λ h, by have aa := parallel_promises h1 h; rw ←aa; rw ←aa at h; exact let ⟨s, sS, as⟩ := exists_of_mem_parallel h, ⟨t, tT, st⟩ := wseq.exists_of_lift_rel_left H sS, aT := (st _).1 as in mem_parallel h2 tT aT, -λh, by have aa := parallel_promises h2 h; rw ←aa; rw ←aa at h; exact +λ h, by have aa := parallel_promises h2 h; rw ←aa; rw ←aa at h; exact let ⟨s, sS, as⟩ := exists_of_mem_parallel h, ⟨t, tT, st⟩ := wseq.exists_of_lift_rel_right H sS, aT := (st _).2 as in mem_parallel h1 tT aT⟩ diff --git a/formal/lean/mathlib/data/seq/seq.lean b/formal/lean/mathlib/data/seq/seq.lean index 05f1631891f2ac21aaa8233ee1ab5b39e6a8a1fa..17c086e5bcc02bc0177e0f71c77cb9c94d7b06b2 100644 --- a/formal/lean/mathlib/data/seq/seq.lean +++ b/formal/lean/mathlib/data/seq/seq.lean @@ -25,7 +25,7 @@ def stream.is_seq {α : Type u} (s : stream (option α)) : Prop := /-- `seq α` is the type of possibly infinite lists (referred here as sequences). It is encoded as an infinite stream of options such that if `f n = none`, then `f m = none` for all `m ≥ n`. -/ -def seq (α : Type u) : Type u := { f : stream (option α) // f.is_seq } +def seq (α : Type u) : Type u := {f : stream (option α) // f.is_seq} /-- `seq1 α` is the type of nonempty sequences. -/ def seq1 (α) := α × seq α @@ -34,13 +34,17 @@ namespace seq variables {α : Type u} {β : Type v} {γ : Type w} /-- The empty sequence -/ -def nil : seq α := ⟨stream.const none, λn h, rfl⟩ +def nil : seq α := ⟨stream.const none, λ n h, rfl⟩ instance : inhabited (seq α) := ⟨nil⟩ /-- Prepend an element to a sequence -/ -def cons (a : α) : seq α → seq α -| ⟨f, al⟩ := ⟨some a :: f, λn h, by {cases n with n, contradiction, exact al h}⟩ +def cons (a : α) (s : seq α) : seq α := +⟨some a :: s.1, begin + rintros (n | _) h, + { contradiction }, + { exact s.2 h } +end⟩ /-- Get the nth element of a sequence (if it exists) -/ def nth : seq α → ℕ → option α := subtype.val @@ -55,6 +59,9 @@ decidable_of_iff' (s.nth n).is_none $ by unfold terminated_at; cases s.nth n; si /-- A sequence terminates if there is some position `n` at which it has terminated. -/ def terminates (s : seq α) : Prop := ∃ (n : ℕ), s.terminated_at n +theorem not_terminates_iff {s : seq α} : ¬ s.terminates ↔ ∀ n, (s.nth n).is_some := +by simp [terminates, terminated_at, ←ne.def, option.ne_none_iff_is_some] + /-- Functorial action of the functor `option (α × _)` -/ @[simp] def omap (f : β → γ) : option (α × β) → option (α × γ) | none := none @@ -64,23 +71,20 @@ def terminates (s : seq α) : Prop := ∃ (n : ℕ), s.terminated_at n def head (s : seq α) : option α := nth s 0 /-- Get the tail of a sequence (or `nil` if the sequence is `nil`) -/ -def tail : seq α → seq α -| ⟨f, al⟩ := ⟨f.tail, λ n, al⟩ +def tail (s : seq α) : seq α := ⟨s.1.tail, λ n, by { cases s with f al, exact al }⟩ protected def mem (a : α) (s : seq α) := some a ∈ s.1 instance : has_mem α (seq α) := ⟨seq.mem⟩ -theorem le_stable (s : seq α) {m n} (h : m ≤ n) : - s.nth m = none → s.nth n = none := -by {cases s with f al, induction h with n h IH, exacts [id, λ h2, al (IH h2)]} +theorem le_stable (s : seq α) {m n} (h : m ≤ n) : s.nth m = none → s.nth n = none := +by { cases s with f al, induction h with n h IH, exacts [id, λ h2, al (IH h2)] } /-- If a sequence terminated at position `n`, it also terminated at `m ≥ n `. -/ -lemma terminated_stable (s : seq α) {m n : ℕ} (m_le_n : m ≤ n) -(terminated_at_m : s.terminated_at m) : +lemma terminated_stable : ∀ (s : seq α) {m n : ℕ}, m ≤ n → s.terminated_at m → s.terminated_at n := -le_stable s m_le_n terminated_at_m +le_stable /-- If `s.nth n = some aₙ` for some value `aₙ`, then there is also some value `aₘ` such @@ -103,16 +107,15 @@ theorem mem_cons_of_mem (y : α) {a : α} : ∀ {s : seq α}, a ∈ s → a ∈ | ⟨f, al⟩ := stream.mem_cons_of_mem (some y) theorem eq_or_mem_of_mem_cons {a b : α} : ∀ {s : seq α}, a ∈ cons b s → a = b ∨ a ∈ s -| ⟨f, al⟩ h := (stream.eq_or_mem_of_mem_cons h).imp_left (λh, by injection h) +| ⟨f, al⟩ h := (stream.eq_or_mem_of_mem_cons h).imp_left (λ h, by injection h) @[simp] theorem mem_cons_iff {a b : α} {s : seq α} : a ∈ cons b s ↔ a = b ∨ a ∈ s := -⟨eq_or_mem_of_mem_cons, λo, by cases o with e m; - [{rw e, apply mem_cons}, exact mem_cons_of_mem _ m]⟩ +⟨eq_or_mem_of_mem_cons, by rintro (rfl|m); [apply mem_cons, exact mem_cons_of_mem _ m]⟩ /-- Destructor for a sequence, resulting in either `none` (for `nil`) or `some (a, s)` (for `cons a s`). -/ def destruct (s : seq α) : option (seq1 α) := -(λa', (a', s.tail)) <$> nth s 0 +(λ a', (a', s.tail)) <$> nth s 0 theorem destruct_eq_nil {s : seq α} : destruct s = none → s = nil := begin @@ -189,7 +192,7 @@ def corec.F (f : β → option (α × β)) : option β → option α × option of the sequence until `none` is obtained. -/ def corec (f : β → option (α × β)) (b : β) : seq α := begin - refine ⟨stream.corec' (corec.F f) (some b), λn h, _⟩, + refine ⟨stream.corec' (corec.F f) (some b), λ n h, _⟩, rw stream.corec'_eq, change stream.corec' (corec.F f) (corec.F f (some b)).2 n = none, revert h, generalize : some b = o, revert o, @@ -221,7 +224,7 @@ end /-- Embed a list as a sequence -/ def of_list (l : list α) : seq α := -⟨list.nth l, λn h, begin +⟨list.nth l, λ n h, begin induction l with a l IH generalizing n, refl, dsimp [list.nth], cases n with n; dsimp [list.nth] at h, { contradiction }, @@ -247,12 +250,12 @@ section bisim theorem eq_of_bisim (bisim : is_bisimulation R) {s₁ s₂} (r : s₁ ~ s₂) : s₁ = s₂ := begin apply subtype.eq, - apply stream.eq_of_bisim (λx y, ∃ s s' : seq α, s.1 = x ∧ s'.1 = y ∧ R s s'), + apply stream.eq_of_bisim (λ x y, ∃ s s' : seq α, s.1 = x ∧ s'.1 = y ∧ R s s'), dsimp [stream.is_bisimulation], intros t₁ t₂ e, exact match t₁, t₂, e with ._, ._, ⟨s, s', rfl, rfl, r⟩ := suffices head s = head s' ∧ R (tail s) (tail s'), from - and.imp id (λr, ⟨tail s, tail s', + and.imp id (λ r, ⟨tail s, tail s', by cases s; refl, by cases s'; refl, r⟩) this, begin have := bisim r, revert r this, @@ -276,7 +279,7 @@ theorem coinduction : ∀ {s₁ s₂ : seq α}, head s₁ = head s₂ → (∀ (β : Type u) (fr : seq α → β), fr s₁ = fr s₂ → fr (tail s₁) = fr (tail s₂)) → s₁ = s₂ | ⟨f₁, a₁⟩ ⟨f₂, a₂⟩ hh ht := - subtype.eq (stream.coinduction hh (λ β fr, ht β (λs, fr s.1))) + subtype.eq (stream.coinduction hh (λ β fr, ht β (λ s, fr s.1))) theorem coinduction2 (s) (f g : seq α → seq β) (H : ∀ s, bisim_o (λ (s1 s2 : seq β), ∃ (s : seq α), s1 = f s ∧ s2 = g s) @@ -290,7 +293,7 @@ end /-- Embed an infinite stream as a sequence -/ def of_stream (s : stream α) : seq α := -⟨s.map some, λn h, by contradiction⟩ +⟨s.map some, λ n h, by contradiction⟩ instance coe_stream : has_coe (stream α) (seq α) := ⟨of_stream⟩ @@ -298,7 +301,7 @@ instance coe_stream : has_coe (stream α) (seq α) := ⟨of_stream⟩ is non-meta, it will produce infinite sequences if used with cyclic `lazy_list`s created by meta constructions. -/ def of_lazy_list : lazy_list α → seq α := -corec (λl, match l with +corec (λ l, match l with | lazy_list.nil := none | lazy_list.cons a l' := some (a, l' ()) end) @@ -326,16 +329,16 @@ lemma nats_nth (n : ℕ) : nats.nth n = some n := rfl /-- Append two sequences. If `s₁` is infinite, then `s₁ ++ s₂ = s₁`, otherwise it puts `s₂` at the location of the `nil` in `s₁`. -/ def append (s₁ s₂ : seq α) : seq α := -@corec α (seq α × seq α) (λ⟨s₁, s₂⟩, +@corec α (seq α × seq α) (λ ⟨s₁, s₂⟩, match destruct s₁ with - | none := omap (λs₂, (nil, s₂)) (destruct s₂) + | none := omap (λ s₂, (nil, s₂)) (destruct s₂) | some (a, s₁') := some (a, s₁', s₂) end) (s₁, s₂) /-- Map a function over a sequence. -/ def map (f : α → β) : seq α → seq β | ⟨s, al⟩ := ⟨s.map (option.map f), -λn, begin +λ n, begin dsimp [stream.map, stream.nth], induction e : s n; intro, { rw al e, assumption }, { contradiction } @@ -346,7 +349,7 @@ end⟩ of an infinite sequence of `nil`, the first element is never generated.) -/ def join : seq (seq1 α) → seq α := -corec (λS, match destruct S with +corec (λ S, match destruct S with | none := none | some ((a, s), S') := some (a, match destruct s with | none := S' @@ -381,12 +384,12 @@ section zip_with /-- Combine two sequences with a function -/ def zip_with (f : α → β → γ) : seq α → seq β → seq γ -| ⟨f₁, a₁⟩ ⟨f₂, a₂⟩ := ⟨λn, +| ⟨f₁, a₁⟩ ⟨f₂, a₂⟩ := ⟨λ n, match f₁ n, f₂ n with | some a, some b := some (f a b) | _, _ := none end, - λn, begin + λ n, begin induction h1 : f₁ n, { intro H, simp only [(a₁ h1)], refl }, induction h2 : f₂ n; dsimp [seq.zip_with._match_1]; intro H, @@ -436,21 +439,20 @@ def zip : seq α → seq β → seq (α × β) := zip_with prod.mk def unzip (s : seq (α × β)) : seq α × seq β := (map prod.fst s, map prod.snd s) /-- Convert a sequence which is known to terminate into a list -/ -def to_list (s : seq α) (h : ∃ n, ¬ (nth s n).is_some) : list α := +def to_list (s : seq α) (h : s.terminates) : list α := take (nat.find h) s /-- Convert a sequence which is known not to terminate into a stream -/ -def to_stream (s : seq α) (h : ∀ n, (nth s n).is_some) : stream α := -λn, option.get (h n) +def to_stream (s : seq α) (h : ¬ s.terminates) : stream α := +λ n, option.get $ not_terminates_iff.1 h n /-- Convert a sequence into either a list or a stream depending on whether it is finite or infinite. (Without decidability of the infiniteness predicate, this is not constructively possible.) -/ -def to_list_or_stream (s : seq α) [decidable (∃ n, ¬ (nth s n).is_some)] : - list α ⊕ stream α := -if h : ∃ n, ¬ (nth s n).is_some +def to_list_or_stream (s : seq α) [decidable s.terminates] : list α ⊕ stream α := +if h : s.terminates then sum.inl (to_list s h) -else sum.inr (to_stream s (λn, decidable.by_contradiction (λ hn, h ⟨n, hn⟩))) +else sum.inr (to_stream s h) @[simp] theorem nil_append (s : seq α) : append nil s = s := begin @@ -483,7 +485,7 @@ end @[simp] theorem append_assoc (s t u : seq α) : append (append s t) u = append s (append t u) := begin - apply eq_of_bisim (λs1 s2, ∃ s t u, + apply eq_of_bisim (λ s1 s2, ∃ s t u, s1 = append (append s t) u ∧ s2 = append s (append t u)), { intros s1 s2 h, exact match s1, s2, h with ._, ._, ⟨s, t, u, rfl, rfl⟩ := begin apply cases_on s; simp, @@ -520,7 +522,7 @@ end @[simp] theorem map_append (f : α → β) (s t) : map f (append s t) = append (map f s) (map f t) := begin - apply eq_of_bisim (λs1 s2, ∃ s t, + apply eq_of_bisim (λ s1 s2, ∃ s t, s1 = map f (append s t) ∧ s2 = append (map f s) (map f t)) _ ⟨s, t, rfl, rfl⟩, intros s1 s2 h, exact match s1, s2, h with ._, ._, ⟨s, t, rfl, rfl⟩ := begin apply cases_on s; simp, @@ -551,7 +553,7 @@ destruct_eq_cons $ by simp [join] @[simp, priority 990] theorem join_cons (a : α) (s S) : join (cons (a, s) S) = cons a (append s (join S)) := begin - apply eq_of_bisim (λs1 s2, s1 = s2 ∨ + apply eq_of_bisim (λ s1 s2, s1 = s2 ∨ ∃ a s S, s1 = join (cons (a, s) S) ∧ s2 = cons a (append s (join S))) _ (or.inr ⟨a, s, S, rfl, rfl⟩), intros s1 s2 h, @@ -571,7 +573,7 @@ end @[simp] theorem join_append (S T : seq (seq1 α)) : join (append S T) = append (join S) (join T) := begin - apply eq_of_bisim (λs1 s2, ∃ s S T, + apply eq_of_bisim (λ s1 s2, ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T))), { intros s1 s2 h, exact match s1, s2, h with ._, ._, ⟨s, S, T, rfl, rfl⟩ := begin @@ -609,7 +611,7 @@ by induction l; simp [*, stream.nil_append_stream, stream.cons_append_stream] the possibility of infinite sequences (in which case the computation never returns anything). -/ def to_list' {α} (s : seq α) : computation (list α) := -@computation.corec (list α) (list α × seq α) (λ⟨l, s⟩, +@computation.corec (list α) (list α × seq α) (λ ⟨l, s⟩, match destruct s with | none := sum.inl l.reverse | some (a, s') := sum.inr (a::l, s') @@ -725,7 +727,7 @@ by apply coinduction2 s; intro s; apply cases_on s; simp [ret] @[simp] theorem bind_ret (f : α → β) : ∀ s, bind s (ret ∘ f) = map f s | ⟨a, s⟩ := begin - dsimp [bind, map], change (λx, ret (f x)) with (ret ∘ f), + dsimp [bind, map], change (λ x, ret (f x)) with (ret ∘ f), rw [map_comp], simp [function.comp, ret] end @@ -739,7 +741,7 @@ end @[simp] theorem map_join' (f : α → β) (S) : seq.map f (seq.join S) = seq.join (seq.map (map f) S) := begin - apply eq_of_bisim (λs1 s2, + apply eq_of_bisim (λ s1 s2, ∃ s S, s1 = append s (seq.map f (seq.join S)) ∧ s2 = append s (seq.join (seq.map (map f) S))), { intros s1 s2 h, exact match s1, s2, h with ._, ._, ⟨s, S, rfl, rfl⟩ := begin @@ -758,7 +760,7 @@ end @[simp] theorem join_join (SS : seq (seq1 (seq1 α))) : seq.join (seq.join SS) = seq.join (seq.map join SS) := begin - apply eq_of_bisim (λs1 s2, + apply eq_of_bisim (λ s1 s2, ∃ s SS, s1 = seq.append s (seq.join (seq.join SS)) ∧ s2 = seq.append s (seq.join (seq.map join SS))), { intros s1 s2 h, exact match s1, s2, h with ._, ._, ⟨s, SS, rfl, rfl⟩ := begin diff --git a/formal/lean/mathlib/data/seq/wseq.lean b/formal/lean/mathlib/data/seq/wseq.lean index b3a090a28752ca67e277ab823b2ce5a8869033c6..842bfc06f6f783357ebd8dde02c6a073904b3740 100644 --- a/formal/lean/mathlib/data/seq/wseq.lean +++ b/formal/lean/mathlib/data/seq/wseq.lean @@ -60,7 +60,7 @@ def think : wseq α → wseq α := seq.cons none /-- Destruct a weak sequence, to (eventually possibly) produce either `none` for `nil` or `some (a, s)` if an element is produced. -/ def destruct : wseq α → computation (option (α × wseq α)) := -computation.corec (λs, match seq.destruct s with +computation.corec (λ s, match seq.destruct s with | none := sum.inl none | some (none, s') := sum.inr s' | some (some a, s') := sum.inl (some (a, s')) @@ -85,7 +85,7 @@ computation.map ((<$>) prod.fst) (destruct s) /-- Encode a computation yielding a weak sequence into additional `think` constructors in a weak sequence -/ def flatten : computation (wseq α) → wseq α := -seq.corec (λc, match computation.destruct c with +seq.corec (λ c, match computation.destruct c with | sum.inl s := seq.omap return (seq.destruct s) | sum.inr c' := some (none, c') end) @@ -94,7 +94,7 @@ seq.corec (λc, match computation.destruct c with wrapper, unlike `head`, because `flatten` allows us to hide this in the construction of the weak sequence itself. -/ def tail (s : wseq α) : wseq α := -flatten $ (λo, option.rec_on o nil prod.snd) <$> destruct s +flatten $ (λ o, option.rec_on o nil prod.snd) <$> destruct s /-- drop the first `n` elements from `s`. -/ def drop (s : wseq α) : ℕ → wseq α @@ -107,7 +107,7 @@ def nth (s : wseq α) (n : ℕ) : computation (option α) := head (drop s n) /-- Convert `s` to a list (if it is finite and completes in finite time). -/ def to_list (s : wseq α) : computation (list α) := -@computation.corec (list α) (list α × wseq α) (λ⟨l, s⟩, +@computation.corec (list α) (list α × wseq α) (λ ⟨l, s⟩, match seq.destruct s with | none := sum.inl l.reverse | some (none, s') := sum.inr (l, s') @@ -116,7 +116,7 @@ def to_list (s : wseq α) : computation (list α) := /-- Get the length of `s` (if it is finite and completes in finite time). -/ def length (s : wseq α) : computation ℕ := -@computation.corec ℕ (ℕ × wseq α) (λ⟨n, s⟩, +@computation.corec ℕ (ℕ × wseq α) (λ ⟨n, s⟩, match seq.destruct s with | none := sum.inl n | some (none, s') := sum.inr (n, s') @@ -148,7 +148,7 @@ instance head_terminates (s : wseq α) [productive s] : /-- Replace the `n`th element of `s` with `a`. -/ def update_nth (s : wseq α) (n : ℕ) (a : α) : wseq α := -@seq.corec (option α) (ℕ × wseq α) (λ⟨n, s⟩, +@seq.corec (option α) (ℕ × wseq α) (λ ⟨n, s⟩, match seq.destruct s, n with | none, n := none | some (none, s'), n := some (none, n, s') @@ -159,7 +159,7 @@ def update_nth (s : wseq α) (n : ℕ) (a : α) : wseq α := /-- Remove the `n`th element of `s`. -/ def remove_nth (s : wseq α) (n : ℕ) : wseq α := -@seq.corec (option α) (ℕ × wseq α) (λ⟨n, s⟩, +@seq.corec (option α) (ℕ × wseq α) (λ ⟨n, s⟩, match seq.destruct s, n with | none, n := none | some (none, s'), n := some (none, n, s') @@ -170,7 +170,7 @@ def remove_nth (s : wseq α) (n : ℕ) : wseq α := /-- Map the elements of `s` over `f`, removing any values that yield `none`. -/ def filter_map (f : α → option β) : wseq α → wseq β := -seq.corec (λs, match seq.destruct s with +seq.corec (λ s, match seq.destruct s with | none := none | some (none, s') := some (none, s') | some (some a, s') := some (f a, s') @@ -178,7 +178,7 @@ seq.corec (λs, match seq.destruct s with /-- Select the elements of `s` that satisfy `p`. -/ def filter (p : α → Prop) [decidable_pred p] : wseq α → wseq α := -filter_map (λa, if p a then some a else none) +filter_map (λ a, if p a then some a else none) -- example of infinite list manipulations /-- Get the first element of `s` satisfying `p`. -/ @@ -187,7 +187,7 @@ head $ filter p s /-- Zip a function over two weak sequences -/ def zip_with (f : α → β → γ) (s1 : wseq α) (s2 : wseq β) : wseq γ := -@seq.corec (option γ) (wseq α × wseq β) (λ⟨s1, s2⟩, +@seq.corec (option γ) (wseq α × wseq β) (λ ⟨s1, s2⟩, match seq.destruct s1, seq.destruct s2 with | some (none, s1'), some (none, s2') := some (none, s1', s2') | some (some a1, s1'), some (none, s2') := some (none, s1, s2') @@ -217,7 +217,7 @@ def indexes_of [decidable_eq α] (a : α) : wseq α → wseq ℕ := find_indexes /-- `union s1 s2` is a weak sequence which interleaves `s1` and `s2` in some order (nondeterministically). -/ def union (s1 s2 : wseq α) : wseq α := -@seq.corec (option α) (wseq α × wseq α) (λ⟨s1, s2⟩, +@seq.corec (option α) (wseq α × wseq α) (λ ⟨s1, s2⟩, match seq.destruct s1, seq.destruct s2 with | none, none := none | some (a1, s1'), none := some (a1, s1', nil) @@ -241,7 +241,7 @@ end /-- Get the first `n` elements of a weak sequence -/ def take (s : wseq α) (n : ℕ) : wseq α := -@seq.corec (option α) (ℕ × wseq α) (λ⟨n, s⟩, +@seq.corec (option α) (ℕ × wseq α) (λ ⟨n, s⟩, match n, seq.destruct s with | 0, _ := none | m+1, none := none @@ -252,7 +252,7 @@ def take (s : wseq α) (n : ℕ) : wseq α := /-- Split the sequence at position `n` into a finite initial segment and the weak sequence tail -/ def split_at (s : wseq α) (n : ℕ) : computation (list α × wseq α) := -@computation.corec (list α × wseq α) (ℕ × list α × wseq α) (λ⟨n, l, s⟩, +@computation.corec (list α × wseq α) (ℕ × list α × wseq α) (λ ⟨n, l, s⟩, match n, seq.destruct s with | 0, _ := sum.inl (l.reverse, s) | m+1, none := sum.inl (l.reverse, s) @@ -262,7 +262,7 @@ def split_at (s : wseq α) (n : ℕ) : computation (list α × wseq α) := /-- Returns `tt` if any element of `s` satisfies `p` -/ def any (s : wseq α) (p : α → bool) : computation bool := -computation.corec (λs : wseq α, +computation.corec (λ s : wseq α, match seq.destruct s with | none := sum.inl ff | some (none, s') := sum.inr s' @@ -271,7 +271,7 @@ computation.corec (λs : wseq α, /-- Returns `tt` if every element of `s` satisfies `p` -/ def all (s : wseq α) (p : α → bool) : computation bool := -computation.corec (λs : wseq α, +computation.corec (λ s : wseq α, match seq.destruct s with | none := sum.inl tt | some (none, s') := sum.inr s' @@ -282,7 +282,7 @@ computation.corec (λs : wseq α, of partial results. (There is no `scanr` because this would require working from the end of the sequence, which may not exist.) -/ def scanl (f : α → β → α) (a : α) (s : wseq β) : wseq α := -cons a $ @seq.corec (option α) (α × wseq β) (λ⟨a, s⟩, +cons a $ @seq.corec (option α) (α × wseq β) (λ ⟨a, s⟩, match seq.destruct s with | none := none | some (none, s') := some (none, a, s') @@ -314,7 +314,7 @@ def map (f : α → β) : wseq α → wseq β := seq.map (option.map f) /-- Flatten a sequence of weak sequences. (Note that this allows empty sequences, unlike `seq.join`.) -/ def join (S : wseq (wseq α)) : wseq α := -seq.join ((λo : option (wseq α), match o with +seq.join ((λ o : option (wseq α), match o with | none := seq1.ret none | some s := (none, s) end) <$> S) @@ -483,7 +483,7 @@ seq.destruct_cons _ _ @[simp] theorem flatten_ret (s : wseq α) : flatten (return s) = s := begin - refine seq.eq_of_bisim (λs1 s2, flatten (return s2) = s1) _ rfl, + refine seq.eq_of_bisim (λ s1 s2, flatten (return s2) = s1) _ rfl, intros s' s h, rw ←h, simp [flatten], cases seq.destruct s, { simp }, { cases val with o s', simp } @@ -495,12 +495,12 @@ seq.destruct_eq_cons $ by simp [flatten, think] @[simp] theorem destruct_flatten (c : computation (wseq α)) : destruct (flatten c) = c >>= destruct := begin - refine computation.eq_of_bisim (λc1 c2, c1 = c2 ∨ + refine computation.eq_of_bisim (λ c1 c2, c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = computation.bind c destruct) _ (or.inr ⟨c, rfl, rfl⟩), intros c1 c2 h, exact match c1, c2, h with | _, _, (or.inl $ eq.refl c) := by cases c.destruct; simp | _, _, (or.inr ⟨c, rfl, rfl⟩) := begin - apply c.cases_on (λa, _) (λc', _); repeat {simp}, + apply c.cases_on (λ a, _) (λ c', _); repeat {simp}, { cases (destruct a).destruct; simp }, { exact or.inr ⟨c', rfl, rfl⟩ } end end @@ -630,7 +630,7 @@ instance productive_dropn (s : wseq α) [productive s] (n) : productive (drop s /-- Given a productive weak sequence, we can collapse all the `think`s to produce a sequence. -/ def to_seq (s : wseq α) [productive s] : seq α := -⟨λ n, (nth s n).get, λn h, +⟨λ n, (nth s n).get, λ n h, begin cases e : computation.get (nth s (n + 1)), {assumption}, have := mem_of_get_eq _ e, @@ -685,7 +685,7 @@ begin unfold cons has_mem.mem wseq.mem seq.mem seq.cons, simp, have h_a_eq_a' : a = a' ↔ some (some a) = some (some a'), {simp}, rw [h_a_eq_a'], - refine ⟨stream.eq_or_mem_of_mem_cons, λo, _⟩, + refine ⟨stream.eq_or_mem_of_mem_cons, λ o, _⟩, { cases o with e m, { rw e, apply stream.mem_cons }, { exact stream.mem_cons_of_mem _ m } } }, @@ -704,7 +704,7 @@ theorem mem_cons (s : wseq α) (a) : a ∈ cons a s := theorem mem_of_mem_tail {s : wseq α} {a} : a ∈ tail s → a ∈ s := begin intro h, have := h, cases h with n e, revert s, simp [stream.nth], - induction n with n IH; intro s; apply s.cases_on _ (λx s, _) (λ s, _); + induction n with n IH; intro s; apply s.cases_on _ (λ x s, _) (λ s, _); repeat{simp}; intros m e; injections, { exact or.inr m }, { exact or.inr m }, @@ -877,7 +877,7 @@ by simp only [productive_iff]; exact forall_congr (λ n, terminates_congr $ nth_congr h _) theorem equiv.ext {s t : wseq α} (h : ∀ n, nth s n ~ nth t n) : s ~ t := -⟨λ s t, ∀ n, nth s n ~ nth t n, h, λs t h, begin +⟨λ s t, ∀ n, nth s n ~ nth t n, h, λ s t h, begin refine lift_rel_def.2 ⟨_, _⟩, { rw [←head_terminates_iff, ←head_terminates_iff], exact terminates_congr (h 0) }, @@ -1194,7 +1194,7 @@ theorem lift_rel_join (R : α → β → Prop) {S : wseq (wseq α)} {T : wseq (w s1 = append s (join S) ∧ s2 = append t (join T) ∧ lift_rel R s t ∧ lift_rel (lift_rel R) S T, ⟨nil, nil, S, T, by simp, by simp, by simp, h⟩, -λs1 s2 ⟨s, t, S, T, h1, h2, st, ST⟩, begin +λ s1 s2 ⟨s, t, S, T, h1, h2, st, ST⟩, begin clear _fun_match _x, rw [h1, h2], rw [destruct_append, destruct_append], apply computation.lift_rel_bind _ _ (lift_rel_destruct st), @@ -1277,7 +1277,7 @@ end @[simp] theorem bind_ret (f : α → β) (s) : bind s (ret ∘ f) ~ map f s := begin - dsimp [bind], change (λx, ret (f x)) with (ret ∘ f), + dsimp [bind], change (λ x, ret (f x)) with (ret ∘ f), rw [map_comp], apply join_map_ret end @@ -1287,7 +1287,7 @@ end @[simp] theorem map_join (f : α → β) (S) : map f (join S) = join (map (map f) S) := begin - apply seq.eq_of_bisim (λs1 s2, + apply seq.eq_of_bisim (λ s1 s2, ∃ s S, s1 = append s (map f (join S)) ∧ s2 = append s (join (map (map f) S))), { intros s1 s2 h, diff --git a/formal/lean/mathlib/data/set/basic.lean b/formal/lean/mathlib/data/set/basic.lean index 8e361d17937bbc9f3e486d42b97e9478dedcac86..1b70b62dfaf37eecaf6a01e928f2c1a454f8351a 100644 --- a/formal/lean/mathlib/data/set/basic.lean +++ b/formal/lean/mathlib/data/set/basic.lean @@ -2025,6 +2025,14 @@ range_subset_iff.2 $ λ x, rfl | ⟨x⟩ c := subset.antisymm range_const_subset $ assume y hy, (mem_singleton_iff.1 hy).symm ▸ mem_range_self x +lemma range_subtype_map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ x, p x → q (f x)) : + range (subtype.map f h) = coe ⁻¹' (f '' {x | p x}) := +begin + ext ⟨x, hx⟩, + simp_rw [mem_preimage, mem_range, mem_image, subtype.exists, subtype.map, subtype.coe_mk, + mem_set_of, exists_prop] +end + lemma image_swap_eq_preimage_swap : image (@prod.swap α β) = preimage prod.swap := image_eq_preimage_of_inverse prod.swap_left_inverse prod.swap_right_inverse diff --git a/formal/lean/mathlib/data/set/finite.lean b/formal/lean/mathlib/data/set/finite.lean index 90f5bf779bd572af58ff5a90ae14b83aa442d01e..b59a266730b6460258a1b0be30a135999b8cbb33 100644 --- a/formal/lean/mathlib/data/set/finite.lean +++ b/formal/lean/mathlib/data/set/finite.lean @@ -307,7 +307,7 @@ set.fintype_lt_nat n instance fintype_prod (s : set α) (t : set β) [fintype s] [fintype t] : fintype (s ×ˢ t : set (α × β)) := -fintype.of_finset (s.to_finset.product t.to_finset) $ by simp +fintype.of_finset (s.to_finset ×ˢ t.to_finset) $ by simp /-- `image2 f s t` is `fintype` if `s` and `t` are. -/ instance fintype_image2 [decidable_eq γ] (f : α → β → γ) (s : set α) (t : set β) @@ -606,10 +606,11 @@ lemma finite.finite_subsets {α : Type u} {a : set α} (h : a.finite) : {b | b ← and.assoc] using h.subset⟩ /-- Finite product of finite sets is finite -/ -lemma finite.pi {δ : Type*} [fintype δ] {κ : δ → Type*} {t : Π d, set (κ d)} +lemma finite.pi {δ : Type*} [finite δ] {κ : δ → Type*} {t : Π d, set (κ d)} (ht : ∀ d, (t d).finite) : (pi univ t).finite := begin + casesI nonempty_fintype δ, lift t to Π d, finset (κ d) using ht, classical, rw ← fintype.coe_pi_finset, @@ -676,7 +677,7 @@ lemma finite_image_fst_and_snd_iff {s : set (α × β)} : ⟨λ h, (h.1.prod h.2).subset $ λ x h, ⟨mem_image_of_mem _ h, mem_image_of_mem _ h⟩, λ h, ⟨h.image _, h.image _⟩⟩ -lemma forall_finite_image_eval_iff {δ : Type*} [fintype δ] {κ : δ → Type*} {s : set (Π d, κ d)} : +lemma forall_finite_image_eval_iff {δ : Type*} [finite δ] {κ : δ → Type*} {s : set (Π d, κ d)} : (∀ d, (eval d '' s).finite) ↔ s.finite := ⟨λ h, (finite.pi h).subset $ subset_pi_eval_image _ _, λ h d, h.image _⟩ @@ -1000,46 +1001,46 @@ lemma finite.infi_bsupr_of_antitone {ι ι' α : Type*} [preorder ι'] [nonempty (⨅ j, ⨆ i ∈ s, f i j) = ⨆ i ∈ s, ⨅ j, f i j := hs.supr_binfi_of_monotone (λ i hi, (hf i hi).dual_right) -lemma _root_.supr_infi_of_monotone {ι ι' α : Type*} [fintype ι] [preorder ι'] [nonempty ι'] +lemma _root_.supr_infi_of_monotone {ι ι' α : Type*} [finite ι] [preorder ι'] [nonempty ι'] [is_directed ι' (≤)] [order.frame α] {f : ι → ι' → α} (hf : ∀ i, monotone (f i)) : (⨆ j, ⨅ i, f i j) = ⨅ i, ⨆ j, f i j := by simpa only [infi_univ] using finite_univ.supr_binfi_of_monotone (λ i hi, hf i) -lemma _root_.supr_infi_of_antitone {ι ι' α : Type*} [fintype ι] [preorder ι'] [nonempty ι'] +lemma _root_.supr_infi_of_antitone {ι ι' α : Type*} [finite ι] [preorder ι'] [nonempty ι'] [is_directed ι' (swap (≤))] [order.frame α] {f : ι → ι' → α} (hf : ∀ i, antitone (f i)) : (⨆ j, ⨅ i, f i j) = ⨅ i, ⨆ j, f i j := @supr_infi_of_monotone ι ι'ᵒᵈ α _ _ _ _ _ _ (λ i, (hf i).dual_left) -lemma _root_.infi_supr_of_monotone {ι ι' α : Type*} [fintype ι] [preorder ι'] [nonempty ι'] +lemma _root_.infi_supr_of_monotone {ι ι' α : Type*} [finite ι] [preorder ι'] [nonempty ι'] [is_directed ι' (swap (≤))] [order.coframe α] {f : ι → ι' → α} (hf : ∀ i, monotone (f i)) : (⨅ j, ⨆ i, f i j) = ⨆ i, ⨅ j, f i j := supr_infi_of_antitone (λ i, (hf i).dual_right) -lemma _root_.infi_supr_of_antitone {ι ι' α : Type*} [fintype ι] [preorder ι'] [nonempty ι'] +lemma _root_.infi_supr_of_antitone {ι ι' α : Type*} [finite ι] [preorder ι'] [nonempty ι'] [is_directed ι' (≤)] [order.coframe α] {f : ι → ι' → α} (hf : ∀ i, antitone (f i)) : (⨅ j, ⨆ i, f i j) = ⨆ i, ⨅ j, f i j := supr_infi_of_monotone (λ i, (hf i).dual_right) /-- An increasing union distributes over finite intersection. -/ -lemma Union_Inter_of_monotone {ι ι' α : Type*} [fintype ι] [preorder ι'] [is_directed ι' (≤)] +lemma Union_Inter_of_monotone {ι ι' α : Type*} [finite ι] [preorder ι'] [is_directed ι' (≤)] [nonempty ι'] {s : ι → ι' → set α} (hs : ∀ i, monotone (s i)) : (⋃ j : ι', ⋂ i : ι, s i j) = ⋂ i : ι, ⋃ j : ι', s i j := supr_infi_of_monotone hs /-- A decreasing union distributes over finite intersection. -/ -lemma Union_Inter_of_antitone {ι ι' α : Type*} [fintype ι] [preorder ι'] [is_directed ι' (swap (≤))] +lemma Union_Inter_of_antitone {ι ι' α : Type*} [finite ι] [preorder ι'] [is_directed ι' (swap (≤))] [nonempty ι'] {s : ι → ι' → set α} (hs : ∀ i, antitone (s i)) : (⋃ j : ι', ⋂ i : ι, s i j) = ⋂ i : ι, ⋃ j : ι', s i j := supr_infi_of_antitone hs /-- An increasing intersection distributes over finite union. -/ -lemma Inter_Union_of_monotone {ι ι' α : Type*} [fintype ι] [preorder ι'] [is_directed ι' (swap (≤))] +lemma Inter_Union_of_monotone {ι ι' α : Type*} [finite ι] [preorder ι'] [is_directed ι' (swap (≤))] [nonempty ι'] {s : ι → ι' → set α} (hs : ∀ i, monotone (s i)) : (⋂ j : ι', ⋃ i : ι, s i j) = ⋃ i : ι, ⋂ j : ι', s i j := infi_supr_of_monotone hs /-- A decreasing intersection distributes over finite union. -/ -lemma Inter_Union_of_antitone {ι ι' α : Type*} [fintype ι] [preorder ι'] [is_directed ι' (≤)] +lemma Inter_Union_of_antitone {ι ι' α : Type*} [finite ι] [preorder ι'] [is_directed ι' (≤)] [nonempty ι'] {s : ι → ι' → set α} (hs : ∀ i, antitone (s i)) : (⋂ j : ι', ⋃ i : ι, s i j) = ⋃ i : ι, ⋂ j : ι', s i j := infi_supr_of_antitone hs @@ -1053,7 +1054,7 @@ begin exact Union_Inter_of_monotone (λ i j₁ j₂ h, preimage_mono $ hs i i.2 h) end -lemma Union_univ_pi_of_monotone {ι ι' : Type*} [linear_order ι'] [nonempty ι'] [fintype ι] +lemma Union_univ_pi_of_monotone {ι ι' : Type*} [linear_order ι'] [nonempty ι'] [finite ι] {α : ι → Type*} {s : Π i, ι' → set (α i)} (hs : ∀ i, monotone (s i)) : (⋃ j : ι', pi univ (λ i, s i j)) = pi univ (λ i, ⋃ j, s i j) := Union_pi_of_monotone finite_univ (λ i _, hs i) diff --git a/formal/lean/mathlib/data/set/function.lean b/formal/lean/mathlib/data/set/function.lean index 43fad68aed2e549cb5f1d1cca4e4ba149311e356..3751674f66ac4e08b6fb156692c999f9ae13c0f3 100644 --- a/formal/lean/mathlib/data/set/function.lean +++ b/formal/lean/mathlib/data/set/function.lean @@ -261,6 +261,13 @@ subtype.map f h @[simp] lemma maps_to.coe_restrict_apply (h : maps_to f s t) (x : s) : (h.restrict f s t x : β) = f x := rfl +lemma maps_to.coe_restrict (h : set.maps_to f s t) : + coe ∘ h.restrict f s t = s.restrict f := rfl + +lemma maps_to.range_restrict (f : α → β) (s : set α) (t : set β) (h : maps_to f s t) : + range (h.restrict f s t) = coe ⁻¹' (f '' s) := +set.range_subtype_map f h + lemma maps_to_iff_exists_map_subtype : maps_to f s t ↔ ∃ g : s → t, ∀ x : s, f x = g x := ⟨λ h, ⟨h.restrict f s t, λ _, rfl⟩, λ ⟨g, hg⟩ x hx, by { erw [hg ⟨x, hx⟩], apply subtype.coe_prop }⟩ @@ -367,6 +374,26 @@ theorem surjective_maps_to_image_restrict (f : α → β) (s : set α) : theorem maps_to.mem_iff (h : maps_to f s t) (hc : maps_to f sᶜ tᶜ) {x} : f x ∈ t ↔ x ∈ s := ⟨λ ht, by_contra $ λ hs, hc hs ht, λ hx, h hx⟩ +/-! ### Restriction onto preimage -/ + +section + +variables (t f) + +/-- The restriction of a function onto the preimage of a set. -/ +@[simps] def restrict_preimage : f ⁻¹' t → t := +(set.maps_to_preimage f t).restrict _ _ _ + +lemma range_restrict_preimage : + range (t.restrict_preimage f) = coe ⁻¹' (range f) := +begin + delta set.restrict_preimage, + rw [maps_to.range_restrict, set.image_preimage_eq_inter_range, + set.preimage_inter, subtype.coe_preimage_self, set.univ_inter], +end + +end + /-! ### Injectivity on a set -/ /-- `f` is injective on `a` if the restriction of `f` to `a` is injective. -/ diff --git a/formal/lean/mathlib/data/set/lattice.lean b/formal/lean/mathlib/data/set/lattice.lean index e4b4042b491a315ac3f237b696aa3bde102b16ce..a29d0eb26cd68881c3b719f88da9e2c273ef1544 100644 --- a/formal/lean/mathlib/data/set/lattice.lean +++ b/formal/lean/mathlib/data/set/lattice.lean @@ -1090,6 +1090,50 @@ lemma image_sInter_subset (S : set (set α)) (f : α → β) : f '' (⋂₀ S) ⊆ ⋂ s ∈ S, f '' s := by { rw sInter_eq_bInter, apply image_Inter₂_subset } +/-! ### `restrict_preimage` -/ +section + +open function + +variables (s : set β) {f : α → β} {U : ι → set β} (hU : Union U = univ) + +lemma restrict_preimage_injective (hf : injective f) : injective (s.restrict_preimage f) := +λ x y e, subtype.mk.inj_arrow e (λ e, subtype.coe_injective (hf e)) + +lemma restrict_preimage_surjective (hf : surjective f) : surjective (s.restrict_preimage f) := +λ x, ⟨⟨_, (show f (hf x).some ∈ s, from (hf x).some_spec.symm ▸ x.2)⟩, subtype.ext (hf x).some_spec⟩ + +lemma restrict_preimage_bijective (hf : bijective f) : bijective (s.restrict_preimage f) := +⟨s.restrict_preimage_injective hf.1, s.restrict_preimage_surjective hf.2⟩ + +alias set.restrict_preimage_injective ← _root_.function.injective.restrict_preimage +alias set.restrict_preimage_surjective ← _root_.function.surjective.restrict_preimage +alias set.restrict_preimage_bijective ← _root_.function.bijective.restrict_preimage + +include hU + +lemma injective_iff_injective_of_Union_eq_univ : + injective f ↔ ∀ i, injective ((U i).restrict_preimage f) := +begin + refine ⟨λ H i, (U i).restrict_preimage_injective H, λ H x y e, _⟩, + obtain ⟨i, hi⟩ := set.mem_Union.mp (show f x ∈ set.Union U, by { rw hU, triv }), + injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i, from e ▸ hi⟩ (subtype.ext e) +end + +lemma surjective_iff_surjective_of_Union_eq_univ : + surjective f ↔ ∀ i, surjective ((U i).restrict_preimage f) := +begin + refine ⟨λ H i, (U i).restrict_preimage_surjective H, λ H x, _⟩, + obtain ⟨i, hi⟩ := set.mem_Union.mp (show x ∈ set.Union U, by { rw hU, triv }), + exact ⟨_, congr_arg subtype.val (H i ⟨x, hi⟩).some_spec⟩ +end + +lemma bijective_iff_bijective_of_Union_eq_univ : + bijective f ↔ ∀ i, bijective ((U i).restrict_preimage f) := +by simp_rw [bijective, forall_and_distrib, injective_iff_injective_of_Union_eq_univ hU, + surjective_iff_surjective_of_Union_eq_univ hU] +end + /-! ### `inj_on` -/ lemma inj_on.image_inter {f : α → β} {s t u : set α} (hf : inj_on f u) (hs : s ⊆ u) (ht : t ⊆ u) : diff --git a/formal/lean/mathlib/data/set/opposite.lean b/formal/lean/mathlib/data/set/opposite.lean index f2866a025d0995f8655a32538dc593ac551ca88d..5db33b36caeff10434a6d839ae2c4f3301b286bd 100644 --- a/formal/lean/mathlib/data/set/opposite.lean +++ b/formal/lean/mathlib/data/set/opposite.lean @@ -44,6 +44,10 @@ ext (by simp only [mem_unop, op_mem_op, iff_self, implies_true_iff]) @[simp] lemma unop_op (s : set αᵒᵖ) : s.unop.op = s := ext (by simp only [mem_op, unop_mem_unop, iff_self, implies_true_iff]) +/-- The members of the opposite of a set are in bijection with the members of the set itself. -/ +@[simps] def op_equiv_self (s : set α) : s.op ≃ s := +⟨λ x, ⟨unop x, x.2⟩, λ x, ⟨op x, x.2⟩, λ x, by simp, λ x, by simp⟩ + /-- Taking opposites as an equivalence of powersets. -/ @[simps] def op_equiv : set α ≃ set αᵒᵖ := ⟨set.op, set.unop, op_unop, unop_op⟩ diff --git a/formal/lean/mathlib/data/set/pairwise.lean b/formal/lean/mathlib/data/set/pairwise.lean index db4d17d4d8ba2fa3192d6d19f068383556227a29..e5fee97a3f25bbea1f596c81187095f3625f8a15 100644 --- a/formal/lean/mathlib/data/set/pairwise.lean +++ b/formal/lean/mathlib/data/set/pairwise.lean @@ -389,7 +389,7 @@ noncomputable def bUnion_eq_sigma_of_disjoint {s : set ι} {f : ι → set α} disjoint iff `f` is injective . -/ lemma pairwise_disjoint_image_right_iff {f : α → β → γ} {s : set α} {t : set β} (hf : ∀ a ∈ s, injective (f a)) : - s.pairwise_disjoint (λ a, f a '' t) ↔ (s ×ˢ t : set (α × β)).inj_on (λ p, f p.1 p.2) := + s.pairwise_disjoint (λ a, f a '' t) ↔ (s ×ˢ t).inj_on (λ p, f p.1 p.2) := begin refine ⟨λ hs x hx y hy (h : f _ _ = _), _, λ hs x hx y hy h, _⟩, { suffices : x.1 = y.1, @@ -405,7 +405,7 @@ end disjoint iff `f` is injective . -/ lemma pairwise_disjoint_image_left_iff {f : α → β → γ} {s : set α} {t : set β} (hf : ∀ b ∈ t, injective (λ a, f a b)) : - t.pairwise_disjoint (λ b, (λ a, f a b) '' s) ↔ (s ×ˢ t : set (α × β)).inj_on (λ p, f p.1 p.2) := + t.pairwise_disjoint (λ b, (λ a, f a b) '' s) ↔ (s ×ˢ t).inj_on (λ p, f p.1 p.2) := begin refine ⟨λ ht x hx y hy (h : f _ _ = _), _, λ ht x hx y hy h, _⟩, { suffices : x.2 = y.2, diff --git a/formal/lean/mathlib/data/set/pointwise.lean b/formal/lean/mathlib/data/set/pointwise.lean index 96eff9b880847563014f6eab407912d3b41c4c61..3bea849b349ed80a84d9c97b6fab4252e6e0e856 100644 --- a/formal/lean/mathlib/data/set/pointwise.lean +++ b/formal/lean/mathlib/data/set/pointwise.lean @@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Floris van Doorn -/ import algebra.module.basic +import data.fin.tuple.basic import data.set.finite import group_theory.submonoid.basic @@ -471,6 +472,32 @@ begin exact ih.trans (subset_mul_right _ hs) } end +@[to_additive] lemma mem_prod_list_of_fn {a : α} {s : fin n → set α} : + a ∈ (list.of_fn s).prod ↔ ∃ f : (Π i : fin n, s i), (list.of_fn (λ i, (f i : α))).prod = a := +begin + induction n with n ih generalizing a, + { simp_rw [list.of_fn_zero, list.prod_nil, fin.exists_fin_zero_pi, eq_comm, set.mem_one] }, + { simp_rw [list.of_fn_succ, list.prod_cons, fin.exists_fin_succ_pi, fin.cons_zero, fin.cons_succ, + mem_mul, @ih, exists_and_distrib_left, exists_exists_eq_and, set_coe.exists, subtype.coe_mk, + exists_prop] } +end + +@[to_additive] lemma mem_list_prod {l : list (set α)} {a : α} : + a ∈ l.prod ↔ ∃ l' : list (Σ s : set α, ↥s), + list.prod (l'.map (λ x, (sigma.snd x : α))) = a ∧ l'.map sigma.fst = l := +begin + induction l using list.of_fn_rec with n f, + simp_rw [list.exists_iff_exists_tuple, list.map_of_fn, list.of_fn_inj', and.left_comm, + exists_and_distrib_left, exists_eq_left, heq_iff_eq, function.comp, mem_prod_list_of_fn], + split, + { rintros ⟨fi, rfl⟩, exact ⟨λ i, ⟨_, fi i⟩, rfl, rfl⟩, }, + { rintros ⟨fi, rfl, rfl⟩, exact ⟨λ i, _, rfl⟩, }, +end + +@[to_additive] lemma mem_pow {a : α} {n : ℕ} : + a ∈ s ^ n ↔ ∃ f : fin n → s, (list.of_fn (λ i, (f i : α))).prod = a := +by rw [←mem_prod_list_of_fn, list.of_fn_const, list.prod_repeat] + @[simp, to_additive] lemma empty_pow {n : ℕ} (hn : n ≠ 0) : (∅ : set α) ^ n = ∅ := by rw [← tsub_add_cancel_of_le (nat.succ_le_of_lt $ nat.pos_of_ne_zero hn), pow_succ, empty_mul] @@ -1016,18 +1043,18 @@ instance smul_comm_class [has_smul α γ] [has_smul β γ] [smul_comm_class α smul_comm_class (set α) (set β) (set γ) := ⟨λ _ _ _, image2_left_comm smul_comm⟩ -instance is_scalar_tower [has_smul α β] [has_smul α γ] [has_smul β γ] - [is_scalar_tower α β γ] : +@[to_additive] +instance is_scalar_tower [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] : is_scalar_tower α β (set γ) := { smul_assoc := λ a b T, by simp only [←image_smul, image_image, smul_assoc] } -instance is_scalar_tower' [has_smul α β] [has_smul α γ] [has_smul β γ] - [is_scalar_tower α β γ] : +@[to_additive] +instance is_scalar_tower' [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] : is_scalar_tower α (set β) (set γ) := ⟨λ _ _ _, image2_image_left_comm $ smul_assoc _⟩ -instance is_scalar_tower'' [has_smul α β] [has_smul α γ] [has_smul β γ] - [is_scalar_tower α β γ] : +@[to_additive] +instance is_scalar_tower'' [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] : is_scalar_tower (set α) (set β) (set γ) := { smul_assoc := λ T T' T'', image2_assoc smul_assoc } @@ -1241,7 +1268,7 @@ section left_cancel_semigroup variables [left_cancel_semigroup α] {s t : set α} @[to_additive] lemma pairwise_disjoint_smul_iff : - s.pairwise_disjoint (• t) ↔ (s ×ˢ t : set (α × α)).inj_on (λ p, p.1 * p.2) := + s.pairwise_disjoint (• t) ↔ (s ×ˢ t).inj_on (λ p, p.1 * p.2) := pairwise_disjoint_image_right_iff $ λ _ _, mul_right_injective _ end left_cancel_semigroup @@ -1368,6 +1395,7 @@ le_antisymm (λ k hk, subset_closure ⟨1, k, H.one_mem, hk, one_mul k⟩)) (by conv_rhs { rw [← closure_eq H, ← closure_eq K] }; apply closure_mul_le) +@[to_additive] lemma pow_smul_mem_closure_smul {N : Type*} [comm_monoid N] [mul_action M N] [is_scalar_tower M N N] (r : M) (s : set N) {x : N} (hx : x ∈ closure s) : ∃ n : ℕ, r ^ n • x ∈ closure (r • s) := diff --git a/formal/lean/mathlib/data/set/prod.lean b/formal/lean/mathlib/data/set/prod.lean index aceb213dab742e037e637c77ca0ee5cc83521387..9ca9fe0628b4fa48cc0ce8f3a34d6376b8ca63bf 100644 --- a/formal/lean/mathlib/data/set/prod.lean +++ b/formal/lean/mathlib/data/set/prod.lean @@ -21,13 +21,6 @@ type. open function -/-- Notation class for product of subobjects (sets, submonoids, subgroups, etc). -/ -class has_set_prod (α β : Type*) (γ : out_param Type*) := -(prod : α → β → γ) - -/- This notation binds more strongly than (pre)images, unions and intersections. -/ -infixr ` ×ˢ `:82 := has_set_prod.prod - namespace set /-! ### Cartesian binary product of sets -/ @@ -35,10 +28,11 @@ namespace set section prod variables {α β γ δ : Type*} {s s₁ s₂ : set α} {t t₁ t₂ : set β} {a : α} {b : β} -/-- The cartesian product `prod s t` is the set of `(a, b)` - such that `a ∈ s` and `b ∈ t`. -/ -instance : has_set_prod (set α) (set β) (set (α × β)) := -⟨λ s t, {p | p.1 ∈ s ∧ p.2 ∈ t}⟩ +/-- The cartesian product `prod s t` is the set of `(a, b)` such that `a ∈ s` and `b ∈ t`. -/ +def prod (s : set α) (t : set β) : set (α × β) := {p | p.1 ∈ s ∧ p.2 ∈ t} + +/- This notation binds more strongly than (pre)images, unions and intersections. -/ +infixr ` ×ˢ `:82 := set.prod lemma prod_eq (s : set α) (t : set β) : s ×ˢ t = prod.fst ⁻¹' s ∩ prod.snd ⁻¹' t := rfl @@ -177,13 +171,13 @@ lemma range_pair_subset (f : α → β) (g : α → γ) : have (λ x, (f x, g x)) = prod.map f g ∘ (λ x, (x, x)), from funext (λ x, rfl), by { rw [this, ← range_prod_map], apply range_comp_subset_range } -lemma nonempty.prod : s.nonempty → t.nonempty → (s ×ˢ t : set _).nonempty := +lemma nonempty.prod : s.nonempty → t.nonempty → (s ×ˢ t).nonempty := λ ⟨x, hx⟩ ⟨y, hy⟩, ⟨(x, y), ⟨hx, hy⟩⟩ -lemma nonempty.fst : (s ×ˢ t : set _).nonempty → s.nonempty := λ ⟨x, hx⟩, ⟨x.1, hx.1⟩ -lemma nonempty.snd : (s ×ˢ t : set _).nonempty → t.nonempty := λ ⟨x, hx⟩, ⟨x.2, hx.2⟩ +lemma nonempty.fst : (s ×ˢ t).nonempty → s.nonempty := λ ⟨x, hx⟩, ⟨x.1, hx.1⟩ +lemma nonempty.snd : (s ×ˢ t).nonempty → t.nonempty := λ ⟨x, hx⟩, ⟨x.2, hx.2⟩ -lemma prod_nonempty_iff : (s ×ˢ t : set _).nonempty ↔ s.nonempty ∧ t.nonempty := +lemma prod_nonempty_iff : (s ×ˢ t).nonempty ↔ s.nonempty ∧ t.nonempty := ⟨λ h, ⟨h.fst, h.snd⟩, λ h, h.1.prod h.2⟩ lemma prod_eq_empty_iff : s ×ˢ t = ∅ ↔ s = ∅ ∨ t = ∅ := @@ -224,7 +218,7 @@ by { ext x, by_cases h₁ : x.1 ∈ s₁; by_cases h₂ : x.2 ∈ t₁; simp * } first set is empty. -/ lemma prod_subset_prod_iff : s ×ˢ t ⊆ s₁ ×ˢ t₁ ↔ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅ := begin - cases (s ×ˢ t : set _).eq_empty_or_nonempty with h h, + cases (s ×ˢ t).eq_empty_or_nonempty with h h, { simp [h, prod_eq_empty_iff.1 h] }, have st : s.nonempty ∧ t.nonempty, by rwa [prod_nonempty_iff] at h, refine ⟨λ H, or.inl ⟨_, _⟩, _⟩, @@ -237,7 +231,7 @@ begin exact prod_mono H.1 H.2 } end -lemma prod_eq_prod_iff_of_nonempty (h : (s ×ˢ t : set _).nonempty) : +lemma prod_eq_prod_iff_of_nonempty (h : (s ×ˢ t).nonempty) : s ×ˢ t = s₁ ×ˢ t₁ ↔ s = s₁ ∧ t = t₁ := begin split, diff --git a/formal/lean/mathlib/data/sigma/basic.lean b/formal/lean/mathlib/data/sigma/basic.lean index 954537a403716bc524b28ed9dcb0bfbfa063a870..45d544b02a1cadd8ede5d7a55e66aef81b5352d2 100644 --- a/formal/lean/mathlib/data/sigma/basic.lean +++ b/formal/lean/mathlib/data/sigma/basic.lean @@ -143,6 +143,7 @@ rfl /-- Convert a product type to a Σ-type. -/ def prod.to_sigma {α β} (p : α × β) : Σ _ : α, β := ⟨p.1, p.2⟩ +@[simp] lemma prod.fst_comp_to_sigma {α β} : sigma.fst ∘ @prod.to_sigma α β = prod.fst := rfl @[simp] lemma prod.fst_to_sigma {α β} (x : α × β) : (prod.to_sigma x).fst = x.fst := rfl @[simp] lemma prod.snd_to_sigma {α β} (x : α × β) : (prod.to_sigma x).snd = x.snd := rfl @[simp] lemma prod.to_sigma_mk {α β} (x : α) (y : β) : (x, y).to_sigma = ⟨x, y⟩ := rfl diff --git a/formal/lean/mathlib/data/sign.lean b/formal/lean/mathlib/data/sign.lean index 214ca9403e4583205b9e2c30d9826b5eb9bff3be..54653c2946217384e2cfd856a9cec7778fa782ef 100644 --- a/formal/lean/mathlib/data/sign.lean +++ b/formal/lean/mathlib/data/sign.lean @@ -35,15 +35,12 @@ end⟩ @[simp] lemma neg_eq_neg_one : neg = -1 := rfl @[simp] lemma pos_eq_one : pos = 1 := rfl -/-- The multiplication on `sign_type`. -/ -def mul : sign_type → sign_type → sign_type -| neg neg := pos -| neg zero := zero -| neg pos := neg -| zero _ := zero -| pos h := h - -instance : has_mul sign_type := ⟨mul⟩ +instance : has_mul sign_type := +⟨λ x y, match x with +| neg := -y +| zero := zero +| pos := y +end⟩ /-- The less-than relation on signs. -/ inductive le : sign_type → sign_type → Prop @@ -81,6 +78,12 @@ instance : linear_order sign_type := le_trans := λ a b c hab hbc, by casesm* _; constructor, decidable_le := le.decidable_rel } +instance : bounded_order sign_type := +{ top := 1, + le_top := le.of_pos, + bot := -1, + bot_le := le.of_neg } + instance : has_distrib_neg sign_type := { neg_neg := λ x, by cases x; refl, neg_mul := λ x y, by casesm* _; refl, @@ -105,6 +108,44 @@ def fin3_equiv : sign_type ≃* fin 3 := end, map_mul' := λ x y, by casesm* _; refl } +section case_bashing + +lemma nonneg_iff {a : sign_type} : 0 ≤ a ↔ a = 0 ∨ a = 1 := by dec_trivial! + +lemma nonneg_iff_ne_neg_one {a : sign_type} : 0 ≤ a ↔ a ≠ -1 := by dec_trivial! + +lemma neg_one_lt_iff {a : sign_type} : -1 < a ↔ 0 ≤ a := by dec_trivial! + +lemma nonpos_iff {a : sign_type} : a ≤ 0 ↔ a = -1 ∨ a = 0 := by dec_trivial! + +lemma nonpos_iff_ne_one {a : sign_type} : a ≤ 0 ↔ a ≠ 1 := by dec_trivial! + +lemma lt_one_iff {a : sign_type} : a < 1 ↔ a ≤ 0 := by dec_trivial! + +@[simp] lemma neg_iff {a : sign_type} : a < 0 ↔ a = -1 := by dec_trivial! + +@[simp] lemma le_neg_one_iff {a : sign_type} : a ≤ -1 ↔ a = -1 := le_bot_iff + +@[simp] lemma pos_iff {a : sign_type} : 0 < a ↔ a = 1 := by dec_trivial! + +@[simp] lemma one_le_iff {a : sign_type} : 1 ≤ a ↔ a = 1 := top_le_iff + +@[simp] lemma neg_one_le (a : sign_type) : -1 ≤ a := bot_le + +@[simp] lemma le_one (a : sign_type) : a ≤ 1 := le_top + +@[simp] lemma not_lt_neg_one (a : sign_type) : ¬ a < -1 := not_lt_bot + +@[simp] lemma not_one_lt (a : sign_type) : ¬ 1 < a := not_top_lt + +@[simp] lemma self_eq_neg_iff (a : sign_type) : a = -a ↔ a = 0 := by dec_trivial! + +@[simp] lemma neg_eq_self_iff (a : sign_type) : -a = a ↔ a = 0 := by dec_trivial! + +@[simp] lemma neg_one_lt_one : (-1 : sign_type) < 1 := bot_lt_top + +end case_bashing + section cast variables {α : Type*} [has_zero α] [has_one α] [has_neg α] @@ -160,6 +201,25 @@ lemma sign_apply : sign a = ite (0 < a) 1 (ite (a < 0) (-1) 0) := rfl @[simp] lemma sign_pos (ha : 0 < a) : sign a = 1 := by rwa [sign_apply, if_pos] @[simp] lemma sign_neg (ha : a < 0) : sign a = -1 := by rwa [sign_apply, if_neg $ asymm ha, if_pos] +lemma sign_eq_one_iff : sign a = 1 ↔ 0 < a := +begin + refine ⟨λ h, _, λ h, sign_pos h⟩, + by_contra hn, + rw [sign_apply, if_neg hn] at h, + split_ifs at h; + simpa using h +end + +lemma sign_eq_neg_one_iff : sign a = -1 ↔ a < 0 := +begin + refine ⟨λ h, _, λ h, sign_neg h⟩, + rw sign_apply at h, + split_ifs at h, + { simpa using h }, + { exact h_2 }, + { simpa using h } +end + end preorder section linear_order @@ -177,6 +237,22 @@ end lemma sign_ne_zero : sign a ≠ 0 ↔ a ≠ 0 := sign_eq_zero_iff.not +@[simp] lemma sign_nonneg_iff : 0 ≤ sign a ↔ 0 ≤ a := +begin + rcases lt_trichotomy 0 a with (h|rfl|h), + { simp [h, h.le] }, + { simp }, + { simpa [h, h.not_le] } +end + +@[simp] lemma sign_nonpos_iff : sign a ≤ 0 ↔ a ≤ 0 := +begin + rcases lt_trichotomy 0 a with (h|rfl|h), + { simp [h, h.not_le] }, + { simp }, + { simp [h, h.le] } +end + end linear_order section linear_ordered_ring diff --git a/formal/lean/mathlib/data/stream/defs.lean b/formal/lean/mathlib/data/stream/defs.lean index 52a07a844f43bb689fe793da7512c0893c6e397f..67ae6bcd57d83de5868953f9d26770ff45fb1473 100644 --- a/formal/lean/mathlib/data/stream/defs.lean +++ b/formal/lean/mathlib/data/stream/defs.lean @@ -22,12 +22,9 @@ namespace stream variables {α : Type u} {β : Type v} {δ : Type w} /-- Prepend an element to a stream. -/ -def cons (a : α) (s : stream α) : stream α := -λ i, - match i with - | 0 := a - | succ n := s n - end +def cons (a : α) (s : stream α) : stream α +| 0 := a +| (n + 1) := s n notation h :: t := cons h t diff --git a/formal/lean/mathlib/data/sym/sym2.lean b/formal/lean/mathlib/data/sym/sym2.lean index f6c68e55179a3393673bd3855eff24e6ab49c544..d31ec392e4dc0c58883c230204400326fdcd6430 100644 --- a/formal/lean/mathlib/data/sym/sym2.lean +++ b/formal/lean/mathlib/data/sym/sym2.lean @@ -94,9 +94,14 @@ protected lemma ind {f : sym2 α → Prop} (h : ∀ x y, f ⟦(x, y)⟧) : ∀ i quotient.ind $ prod.rec $ by exact h @[elab_as_eliminator] -protected lemma induction_on {f : sym2 α → Prop} (i : sym2 α) (hf : ∀ x y, f ⟦(x,y)⟧) : f i := +protected lemma induction_on {f : sym2 α → Prop} (i : sym2 α) (hf : ∀ x y, f ⟦(x, y)⟧) : f i := i.ind hf +@[elab_as_eliminator] +protected lemma induction_on₂ {f : sym2 α → sym2 β → Prop} (i : sym2 α) (j : sym2 β) + (hf : ∀ a₁ a₂ b₁ b₂, f ⟦(a₁, a₂)⟧ ⟦(b₁, b₂)⟧) : f i j := +quotient.induction_on₂ i j $ by { rintros ⟨a₁, a₂⟩ ⟨b₁, b₂⟩, exact hf _ _ _ _ } + protected lemma «exists» {α : Sort*} {f : sym2 α → Prop} : (∃ (x : sym2 α), f x) ↔ ∃ x y, f ⟦(x, y)⟧ := (surjective_quotient_mk _).exists.trans prod.exists @@ -142,6 +147,27 @@ lemma lift_mk (f : {f : α → α → β // ∀ a₁ a₂, f a₁ a₂ = f a₂ lemma coe_lift_symm_apply (F : sym2 α → β) (a₁ a₂ : α) : (lift.symm F : α → α → β) a₁ a₂ = F ⟦(a₁, a₂)⟧ := rfl +/-- A two-argument version of `sym2.lift`. -/ +def lift₂ : {f : α → α → β → β → γ // ∀ a₁ a₂ b₁ b₂, + f a₁ a₂ b₁ b₂ = f a₂ a₁ b₁ b₂ ∧ f a₁ a₂ b₁ b₂ = f a₁ a₂ b₂ b₁} ≃ (sym2 α → sym2 β → γ) := +{ to_fun := λ f, quotient.lift₂ (λ (a : α × α) (b : β × β), f.1 a.1 a.2 b.1 b.2) begin + rintro _ _ _ _ ⟨⟩ ⟨⟩, + exacts [rfl, (f.2 _ _ _ _).2, (f.2 _ _ _ _).1, (f.2 _ _ _ _).1.trans (f.2 _ _ _ _).2] + end, + inv_fun := λ F, ⟨λ a₁ a₂ b₁ b₂, F ⟦(a₁, a₂)⟧ ⟦(b₁, b₂)⟧, λ a₁ a₂ b₁ b₂, + by { split, exacts [congr_arg2 F eq_swap rfl, congr_arg2 F rfl eq_swap] }⟩, + left_inv := λ f, subtype.ext rfl, + right_inv := λ F, funext₂ $ λ a b, sym2.induction_on₂ a b $ λ _ _ _ _, rfl } + +@[simp] +lemma lift₂_mk (f : {f : α → α → β → β → γ // ∀ a₁ a₂ b₁ b₂, + f a₁ a₂ b₁ b₂ = f a₂ a₁ b₁ b₂ ∧ f a₁ a₂ b₁ b₂ = f a₁ a₂ b₂ b₁}) (a₁ a₂ : α) (b₁ b₂ : β) : + lift₂ f ⟦(a₁, a₂)⟧ ⟦(b₁, b₂)⟧ = (f : α → α → β → β → γ) a₁ a₂ b₁ b₂ := rfl + +@[simp] +lemma coe_lift₂_symm_apply (F : sym2 α → sym2 β → γ) (a₁ a₂ : α) (b₁ b₂ : β) : + (lift₂.symm F : α → α → β → β → γ) a₁ a₂ b₁ b₂ = F ⟦(a₁, a₂)⟧ ⟦(b₁, b₂)⟧ := rfl + /-- The functor `sym2` is functorial, and this function constructs the induced maps. -/ @@ -521,7 +547,7 @@ begin end lemma filter_image_quotient_mk_is_diag [decidable_eq α] (s : finset α) : - ((s.product s).image quotient.mk).filter is_diag = s.diag.image quotient.mk := + ((s ×ˢ s).image quotient.mk).filter is_diag = s.diag.image quotient.mk := begin ext z, induction z using quotient.induction_on, @@ -537,7 +563,7 @@ begin end lemma filter_image_quotient_mk_not_is_diag [decidable_eq α] (s : finset α) : - ((s.product s).image quotient.mk).filter (λ a : sym2 α, ¬a.is_diag) = + ((s ×ˢ s).image quotient.mk).filter (λ a : sym2 α, ¬a.is_diag) = s.off_diag.image quotient.mk := begin ext z, diff --git a/formal/lean/mathlib/data/zmod/basic.lean b/formal/lean/mathlib/data/zmod/basic.lean index 9a49c33fa81a595c56243996ec9d5ce73a919cd0..52752b7e99160bd3fc2c8830358b3795d491290c 100644 --- a/formal/lean/mathlib/data/zmod/basic.lean +++ b/formal/lean/mathlib/data/zmod/basic.lean @@ -138,7 +138,7 @@ nat_cast_right_inverse.surjective /-- So-named because the outer coercion is `int.cast` into `zmod`. For `int.cast` into an arbitrary ring, see `zmod.int_cast_cast`. -/ -lemma int_cast_zmod_cast (a : zmod n) : ((a : ℤ) : zmod n) = a := +@[norm_cast] lemma int_cast_zmod_cast (a : zmod n) : ((a : ℤ) : zmod n) = a := begin cases n, { rw [int.cast_id a, int.cast_id a], }, diff --git a/formal/lean/mathlib/field_theory/adjoin.lean b/formal/lean/mathlib/field_theory/adjoin.lean index d890a38fb2cdede145422464edf438f2ca15b860..79c5c08d7d0d88b667a0b90bfae3cd44f6a5cd56 100644 --- a/formal/lean/mathlib/field_theory/adjoin.lean +++ b/formal/lean/mathlib/field_theory/adjoin.lean @@ -365,32 +365,27 @@ adjoin_adjoin_left _ _ _ lemma adjoin_simple_comm (β : E) : F⟮α⟯⟮β⟯.restrict_scalars F = F⟮β⟯⟮α⟯.restrict_scalars F := adjoin_adjoin_comm _ _ _ --- TODO: develop the API for `subalgebra.is_field_of_algebraic` so it can be used here +variables {F} {α} + +lemma adjoin_algebraic_to_subalgebra + {S : set E} (hS : ∀ x ∈ S, is_algebraic F x) : + (intermediate_field.adjoin F S).to_subalgebra = algebra.adjoin F S := +begin + simp only [is_algebraic_iff_is_integral] at hS, + have : algebra.is_integral F (algebra.adjoin F S) := + by rwa [←le_integral_closure_iff_is_integral, algebra.adjoin_le_iff], + have := is_field_of_is_integral_of_is_field' this (field.to_is_field F), + rw ← ((algebra.adjoin F S).to_intermediate_field' this).eq_adjoin_of_eq_algebra_adjoin F S; refl, +end + lemma adjoin_simple_to_subalgebra_of_integral (hα : is_integral F α) : (F⟮α⟯).to_subalgebra = algebra.adjoin F {α} := begin - apply adjoin_eq_algebra_adjoin, - intros x hx, - by_cases x = 0, - { rw [h, inv_zero], exact subalgebra.zero_mem (algebra.adjoin F {α}) }, - - let ϕ := alg_equiv.adjoin_singleton_equiv_adjoin_root_minpoly F α, - haveI := fact.mk (minpoly.irreducible hα), - suffices : ϕ ⟨x, hx⟩ * (ϕ ⟨x, hx⟩)⁻¹ = 1, - { convert subtype.mem (ϕ.symm (ϕ ⟨x, hx⟩)⁻¹), - refine inv_eq_of_mul_eq_one_right _, - apply_fun ϕ.symm at this, - rw [alg_equiv.map_one, alg_equiv.map_mul, alg_equiv.symm_apply_apply] at this, - rw [←subsemiring.coe_one, ←this, subsemiring.coe_mul, subtype.coe_mk] }, - - rw mul_inv_cancel (mt (λ key, _) h), - rw ← ϕ.map_zero at key, - change ↑(⟨x, hx⟩ : algebra.adjoin F {α}) = _, - rw [ϕ.injective key, subalgebra.coe_zero] + apply adjoin_algebraic_to_subalgebra, + rintro x (rfl : x = α), + rwa is_algebraic_iff_is_integral, end -variables {F} {α} - open set complete_lattice @[simp] lemma adjoin_simple_le_iff {K : intermediate_field F E} : F⟮α⟯ ≤ K ↔ α ∈ K := @@ -956,9 +951,9 @@ lemma is_algebraic_supr {ι : Type*} {f : ι → intermediate_field K L} begin rintros ⟨x, hx⟩, obtain ⟨s, hx⟩ := exists_finset_of_mem_supr' hx, - rw [algebraic_iff, subtype.coe_mk, ←subtype.coe_mk x hx, ←algebraic_iff], + rw [is_algebraic_iff, subtype.coe_mk, ←subtype.coe_mk x hx, ←is_algebraic_iff], haveI : ∀ i : (Σ i, f i), finite_dimensional K K⟮(i.2 : L)⟯ := - λ ⟨i, x⟩, adjoin.finite_dimensional (is_algebraic_iff_is_integral.mp (algebraic_iff.mp (h i x))), + λ ⟨i, x⟩, adjoin.finite_dimensional (is_integral_iff.1 (is_algebraic_iff_is_integral.1 (h i x))), apply algebra.is_algebraic_of_finite, end diff --git a/formal/lean/mathlib/field_theory/finite/basic.lean b/formal/lean/mathlib/field_theory/finite/basic.lean index 4411ccf3054e05b45c15b664841a2f4f1f1946e3..8a18cebbeec1141044c2eafae8989ec4f14b5d3c 100644 --- a/formal/lean/mathlib/field_theory/finite/basic.lean +++ b/formal/lean/mathlib/field_theory/finite/basic.lean @@ -353,18 +353,21 @@ open zmod /-- The **Fermat-Euler totient theorem**. `nat.modeq.pow_totient` is an alternative statement of the same theorem. -/ -@[simp] lemma zmod.pow_totient {n : ℕ} [fact (0 < n)] (x : (zmod n)ˣ) : x ^ φ n = 1 := -by rw [← card_units_eq_totient, pow_card_eq_one] +@[simp] lemma zmod.pow_totient {n : ℕ} (x : (zmod n)ˣ) : x ^ φ n = 1 := +begin + cases n, + { rw [nat.totient_zero, pow_zero] }, + { rw [← card_units_eq_totient, pow_card_eq_one] } +end /-- The **Fermat-Euler totient theorem**. `zmod.pow_totient` is an alternative statement of the same theorem. -/ lemma nat.modeq.pow_totient {x n : ℕ} (h : nat.coprime x n) : x ^ φ n ≡ 1 [MOD n] := begin - cases n, {simp}, rw ← zmod.eq_iff_modeq_nat, - let x' : units (zmod (n+1)) := zmod.unit_of_coprime _ h, + let x' : units (zmod n) := zmod.unit_of_coprime _ h, have := zmod.pow_totient x', - apply_fun (coe : units (zmod (n+1)) → zmod (n+1)) at this, + apply_fun (coe : units (zmod n) → zmod n) at this, simpa only [-zmod.pow_totient, nat.succ_eq_add_one, nat.cast_pow, units.coe_one, nat.cast_one, coe_unit_of_coprime, units.coe_pow], end diff --git a/formal/lean/mathlib/field_theory/galois.lean b/formal/lean/mathlib/field_theory/galois.lean index a43d724c1f7ce9539feb1bc1203c394eb4202d01..bda2410f5a67cc4165467b159608b27c79b7af31 100644 --- a/formal/lean/mathlib/field_theory/galois.lean +++ b/formal/lean/mathlib/field_theory/galois.lean @@ -208,7 +208,7 @@ begin (set.inclusion H_le)).mpr ⟨set.inclusion_injective H_le, this⟩).2).symm }, apply fintype.card_congr, refine (fixed_points.to_alg_hom_equiv H E).trans _, - refine (alg_equiv_equiv_alg_hom (fixed_field H) E).symm.trans _, + refine (alg_equiv_equiv_alg_hom (fixed_field H) E).to_equiv.symm.trans _, exact (fixing_subgroup_equiv (fixed_field H)).to_equiv.symm end @@ -297,7 +297,7 @@ begin cases field.exists_primitive_element F E with α h1, use [minpoly F α, separable F α, is_galois.splits F α], rw [eq_top_iff, ←intermediate_field.top_to_subalgebra, ←h1], - rw intermediate_field.adjoin_simple_to_subalgebra_of_integral F α (integral F α), + rw intermediate_field.adjoin_simple_to_subalgebra_of_integral (integral F α), apply algebra.adjoin_mono, rw [set.singleton_subset_iff, finset.mem_coe, multiset.mem_to_finset, polynomial.mem_roots], { dsimp only [polynomial.is_root], @@ -380,8 +380,8 @@ begin { rw adjoin_root at this, apply of_card_aut_eq_finrank, rw ← eq.trans this (linear_equiv.finrank_eq intermediate_field.top_equiv.to_linear_equiv), - exact fintype.card_congr (equiv.trans (alg_equiv_equiv_alg_hom F E) - (alg_equiv.arrow_congr intermediate_field.top_equiv.symm alg_equiv.refl)) }, + exact fintype.card_congr ((alg_equiv_equiv_alg_hom F E).to_equiv.trans + (intermediate_field.top_equiv.symm.arrow_congr alg_equiv.refl)) }, apply intermediate_field.induction_on_adjoin_finset s P, { have key := intermediate_field.card_alg_hom_adjoin_integral F (show is_integral F (0 : E), by exact is_integral_zero), diff --git a/formal/lean/mathlib/field_theory/intermediate_field.lean b/formal/lean/mathlib/field_theory/intermediate_field.lean index f2933d2f4e68fc40824d2050e0e2385b386268b8..321ad4289349a7571bd1ec1306a905fe551fb68f 100644 --- a/formal/lean/mathlib/field_theory/intermediate_field.lean +++ b/formal/lean/mathlib/field_theory/intermediate_field.lean @@ -4,9 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Anne Baanen -/ +import field_theory.minpoly import field_theory.subfield import field_theory.tower -import ring_theory.algebraic /-! # Intermediate fields @@ -287,7 +287,7 @@ variables {L' : Type*} [field L'] [algebra K L'] /-- If `f : L →+* L'` fixes `K`, `S.map f` is the intermediate field between `L'` and `K` such that `x ∈ S ↔ f x ∈ S.map f`. -/ -def map (f : L →ₐ[K] L') : intermediate_field K L' := +def map (f : L →ₐ[K] L') (S : intermediate_field K L) : intermediate_field K L' := { inv_mem' := by { rintros _ ⟨x, hx, rfl⟩, exact ⟨x⁻¹, S.inv_mem hx, f.map_inv x⟩ }, neg_mem' := λ x hx, (S.to_subalgebra.map f).neg_mem hx, .. S.to_subalgebra.map f} @@ -397,7 +397,7 @@ section tower /-- Lift an intermediate_field of an intermediate_field -/ def lift {F : intermediate_field K L} (E : intermediate_field K F) : intermediate_field K L := -map E (val F) +E.map (val F) instance has_lift {F : intermediate_field K L} : has_lift_t (intermediate_field K F) (intermediate_field K L) := ⟨lift⟩ @@ -484,9 +484,19 @@ eq_of_le_of_finrank_le' h_le h_finrank.le end finite_dimensional -lemma algebraic_iff {x : S} : is_algebraic K x ↔ is_algebraic K (x : L) := +lemma is_algebraic_iff {x : S} : is_algebraic K x ↔ is_algebraic K (x : L) := (is_algebraic_algebra_map_iff (algebra_map S L).injective).symm +lemma is_integral_iff {x : S} : is_integral K x ↔ is_integral K (x : L) := +by rw [←is_algebraic_iff_is_integral, is_algebraic_iff, is_algebraic_iff_is_integral] + +lemma minpoly_eq (x : S) : minpoly K x = minpoly K (x : L) := +begin + by_cases hx : is_integral K x, + { exact minpoly.eq_of_algebra_map_eq (algebra_map S L).injective hx rfl }, + { exact (minpoly.eq_zero hx).trans (minpoly.eq_zero (mt is_integral_iff.mpr hx)).symm }, +end + end intermediate_field /-- If `L/K` is algebraic, the `K`-subalgebras of `L` are all fields. -/ diff --git a/formal/lean/mathlib/field_theory/polynomial_galois_group.lean b/formal/lean/mathlib/field_theory/polynomial_galois_group.lean index 9d7a487de5b6cc0bd11eecef5427ef801d434988..cb29cbc08bc77b5dd067d12071a5431d292f9e0b 100644 --- a/formal/lean/mathlib/field_theory/polynomial_galois_group.lean +++ b/formal/lean/mathlib/field_theory/polynomial_galois_group.lean @@ -56,6 +56,9 @@ namespace gal instance : has_coe_to_fun p.gal (λ _, p.splitting_field → p.splitting_field) := alg_equiv.has_coe_to_fun +instance apply_mul_semiring_action : mul_semiring_action p.gal p.splitting_field := +alg_equiv.apply_mul_semiring_action + @[ext] lemma ext {σ τ : p.gal} (h : ∀ x ∈ p.root_set p.splitting_field, σ x = τ x) : σ = τ := begin refine alg_equiv.ext (λ x, (alg_hom.mem_equalizer σ.to_alg_hom τ.to_alg_hom x).mp @@ -185,10 +188,7 @@ variables (p E) /-- `polynomial.gal.gal_action` as a permutation representation -/ def gal_action_hom [fact (p.splits (algebra_map F E))] : p.gal →* equiv.perm (root_set p E) := -{ to_fun := λ ϕ, equiv.mk (λ x, ϕ • x) (λ x, ϕ⁻¹ • x) - (λ x, inv_smul_smul ϕ x) (λ x, smul_inv_smul ϕ x), - map_one' := by { ext1 x, exact mul_action.one_smul x }, - map_mul' := λ x y, by { ext1 z, exact mul_action.mul_smul x y z } } +mul_action.to_perm_hom _ _ lemma gal_action_hom_restrict [fact (p.splits (algebra_map F E))] (ϕ : E ≃ₐ[F] E) (x : root_set p E) : ↑(gal_action_hom p E (restrict p E ϕ) x) = ϕ x := diff --git a/formal/lean/mathlib/field_theory/splitting_field.lean b/formal/lean/mathlib/field_theory/splitting_field.lean index a38f86aded4b2aa0ed2a582cf9d71c8a85c050d4..fe7d679dbdcaa9f0dcf6ec91ae9886549f04828c 100644 --- a/formal/lean/mathlib/field_theory/splitting_field.lean +++ b/formal/lean/mathlib/field_theory/splitting_field.lean @@ -3,6 +3,7 @@ Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes -/ +import field_theory.intermediate_field import ring_theory.adjoin_root /-! @@ -229,6 +230,22 @@ theorem roots_map {f : K[X]} (hf : f.splits $ ring_hom.id K) : (roots_map_of_injective_card_eq_total_degree i.injective $ by { convert (nat_degree_eq_card_roots hf).symm, rw map_id }).symm +lemma image_root_set [algebra F K] [algebra F L] {p : F[X]} (h : p.splits (algebra_map F K)) + (f : K →ₐ[F] L) : f '' p.root_set K = p.root_set L := +begin + classical, + rw [root_set, ←finset.coe_image, ←multiset.to_finset_map, ←f.coe_to_ring_hom, ←roots_map ↑f + ((splits_id_iff_splits (algebra_map F K)).mpr h), map_map, f.comp_algebra_map, ←root_set], +end + +lemma adjoin_root_set_eq_range [algebra F K] [algebra F L] {p : F[X]} + (h : p.splits (algebra_map F K)) (f : K →ₐ[F] L) : + algebra.adjoin F (p.root_set L) = f.range ↔ algebra.adjoin F (p.root_set K) = ⊤ := +begin + rw [←image_root_set h f, algebra.adjoin_image, ←algebra.map_top], + exact (subalgebra.map_injective f.to_ring_hom.injective).eq_iff, +end + lemma eq_prod_roots_of_splits {p : K[X]} {i : K →+* L} (hsplit : splits i p) : p.map i = C (i p.leading_coeff) * ((p.map i).roots.map (λ a, X - C a)).prod := begin @@ -357,7 +374,6 @@ end splits end polynomial - section embeddings variables (F) [field F] @@ -784,8 +800,41 @@ begin exact ring_hom.injective (lift L f $ splits (splitting_field f) f : L →+* f.splitting_field) end +lemma of_alg_equiv [algebra K F] (p : K[X]) (f : F ≃ₐ[K] L) [is_splitting_field K F p] : + is_splitting_field K L p := +begin + split, + { rw ← f.to_alg_hom.comp_algebra_map, + exact splits_comp_of_splits _ _ (splits F p) }, + { rw [←(algebra.range_top_iff_surjective f.to_alg_hom).mpr f.surjective, + ←root_set, adjoin_root_set_eq_range (splits F p), root_set, adjoin_roots F p] }, +end + end is_splitting_field end splitting_field end polynomial + +namespace intermediate_field + +open polynomial + +variables [field K] [field L] [algebra K L] {p : polynomial K} + +lemma splits_of_splits {F : intermediate_field K L} (h : p.splits (algebra_map K L)) + (hF : ∀ x ∈ p.root_set L, x ∈ F) : p.splits (algebra_map K F) := +begin + simp_rw [root_set, finset.mem_coe, multiset.mem_to_finset] at hF, + rw splits_iff_exists_multiset, + refine ⟨multiset.pmap subtype.mk _ hF, map_injective _ (algebra_map F L).injective _⟩, + conv_lhs { rw [polynomial.map_map, ←is_scalar_tower.algebra_map_eq, + eq_prod_roots_of_splits h, ←multiset.pmap_eq_map _ _ _ hF] }, + simp_rw [polynomial.map_mul, polynomial.map_multiset_prod, + multiset.map_pmap, polynomial.map_sub, map_C, map_X], + refl, +end + +-- TODO (Thomas): If `p` splits in `L/K`, then `p.is_splitting_field K (adjoin K (p.root_set L))` + +end intermediate_field diff --git a/formal/lean/mathlib/geometry/euclidean/oriented_angle.lean b/formal/lean/mathlib/geometry/euclidean/oriented_angle.lean index ca13248f0fdf69cfbb8cae3c3d16a4760df8a411..472c924018e6e10af3e2af51ef2ddeba31c98c00 100644 --- a/formal/lean/mathlib/geometry/euclidean/oriented_angle.lean +++ b/formal/lean/mathlib/geometry/euclidean/oriented_angle.lean @@ -6,6 +6,7 @@ Authors: Joseph Myers import analysis.inner_product_space.orientation import analysis.inner_product_space.pi_L2 import analysis.special_functions.complex.circle +import geometry.euclidean.basic /-! # Oriented angles. @@ -39,6 +40,7 @@ generally use the definitions and results in the `orientation` namespace instead noncomputable theory open_locale real +open_locale real_inner_product_space namespace orthonormal @@ -788,6 +790,50 @@ begin rw [←submonoid.coe_mul, mul_left_inv, submonoid.coe_one, mul_one] end +/-- The inner product of two vectors is the product of the norms and the cosine of the oriented +angle between the vectors. -/ +lemma inner_eq_norm_mul_norm_mul_cos_oangle (x y : V) : + ⟪x, y⟫ = ∥x∥ * ∥y∥ * real.angle.cos (hb.oangle x y) := +begin + by_cases hx : x = 0, { simp [hx] }, + by_cases hy : y = 0, { simp [hy] }, + rw [oangle, real.angle.cos_coe, complex.cos_arg], swap, { simp [hx, hy] }, + simp_rw [complex.abs_div, ←complex.norm_eq_abs, linear_isometry_equiv.norm_map, complex.div_re, + ←complex.sq_abs, ←complex.norm_eq_abs, linear_isometry_equiv.norm_map, + complex.isometry_of_orthonormal_symm_apply, complex.add_re, complex.add_im, + is_R_or_C.I, complex.mul_I_re, complex.mul_I_im, complex.of_real_re, + complex.of_real_im, basis.coord_apply, neg_zero, zero_add, add_zero], + conv_lhs { rw [←b.sum_repr x, ←b.sum_repr y] }, + simp_rw [hb.inner_sum, (dec_trivial : (finset.univ : finset (fin 2)) = {0, 1}), + star_ring_end_apply, star_trivial], + rw [finset.sum_insert (dec_trivial : (0 : fin 2) ∉ ({1} : finset (fin 2))), + finset.sum_singleton], + field_simp [norm_ne_zero_iff.2 hx, norm_ne_zero_iff.2 hy], + ring +end + +/-- The cosine of the oriented angle between two nonzero vectors is the inner product divided by +the product of the norms. -/ +lemma cos_oangle_eq_inner_div_norm_mul_norm {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) : + real.angle.cos (hb.oangle x y) = ⟪x, y⟫ / (∥x∥ * ∥y∥) := +begin + rw hb.inner_eq_norm_mul_norm_mul_cos_oangle, + field_simp [norm_ne_zero_iff.2 hx, norm_ne_zero_iff.2 hy], + ring +end + +/-- The cosine of the oriented angle between two nonzero vectors equals that of the unoriented +angle. -/ +lemma cos_oangle_eq_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) : + real.angle.cos (hb.oangle x y) = real.cos (inner_product_geometry.angle x y) := +by rw [hb.cos_oangle_eq_inner_div_norm_mul_norm hx hy, inner_product_geometry.cos_angle] + +/-- The oriented angle between two nonzero vectors is plus or minus the unoriented angle. -/ +lemma oangle_eq_angle_or_eq_neg_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) : + hb.oangle x y = inner_product_geometry.angle x y ∨ + hb.oangle x y = -inner_product_geometry.angle x y := +real.angle.cos_eq_real_cos_iff_eq_or_eq_neg.1 $ hb.cos_oangle_eq_cos_angle hx hy + end orthonormal namespace orientation @@ -1215,4 +1261,28 @@ begin simp_rw orientation.fin_orthonormal_basis_orientation end +/-- The inner product of two vectors is the product of the norms and the cosine of the oriented +angle between the vectors. -/ +lemma inner_eq_norm_mul_norm_mul_cos_oangle (x y : V) : + ⟪x, y⟫ = ∥x∥ * ∥y∥ * real.angle.cos (o.oangle x y) := +(ob).inner_eq_norm_mul_norm_mul_cos_oangle x y + +/-- The cosine of the oriented angle between two nonzero vectors is the inner product divided by +the product of the norms. -/ +lemma cos_oangle_eq_inner_div_norm_mul_norm {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) : + real.angle.cos (o.oangle x y) = ⟪x, y⟫ / (∥x∥ * ∥y∥) := +(ob).cos_oangle_eq_inner_div_norm_mul_norm hx hy + +/-- The cosine of the oriented angle between two nonzero vectors equals that of the unoriented +angle. -/ +lemma cos_oangle_eq_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) : + real.angle.cos (o.oangle x y) = real.cos (inner_product_geometry.angle x y) := +(ob).cos_oangle_eq_cos_angle hx hy + +/-- The oriented angle between two nonzero vectors is plus or minus the unoriented angle. -/ +lemma oangle_eq_angle_or_eq_neg_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) : + o.oangle x y = inner_product_geometry.angle x y ∨ + o.oangle x y = -inner_product_geometry.angle x y := +(ob).oangle_eq_angle_or_eq_neg_angle hx hy + end orientation diff --git a/formal/lean/mathlib/geometry/manifold/charted_space.lean b/formal/lean/mathlib/geometry/manifold/charted_space.lean index e57ae70149fcba5e60d57ab521558c3ddddbd4d0..1787d7f5117b1bc3fcb0d150b8593ffdf7e46458 100644 --- a/formal/lean/mathlib/geometry/manifold/charted_space.lean +++ b/formal/lean/mathlib/geometry/manifold/charted_space.lean @@ -503,6 +503,16 @@ lemma chart_source_mem_nhds (x : M) : (chart_at H x).source ∈ 𝓝 x := lemma chart_target_mem_nhds (x : M) : (chart_at H x).target ∈ 𝓝 (chart_at H x x) := (chart_at H x).open_target.mem_nhds $ mem_chart_target H x +/-- `achart H x` is the chart at `x`, considered as an element of the atlas. +Especially useful for working with `basic_smooth_vector_bundle_core` -/ +def achart (x : M) : atlas H M := ⟨chart_at H x, chart_mem_atlas H x⟩ + +lemma achart_def (x : M) : achart H x = ⟨chart_at H x, chart_mem_atlas H x⟩ := rfl +@[simp, mfld_simps] +lemma coe_achart (x : M) : (achart H x : local_homeomorph M H) = chart_at H x := rfl +@[simp, mfld_simps] +lemma achart_val (x : M) : (achart H x).1 = chart_at H x := rfl + open topological_space lemma charted_space.second_countable_of_countable_cover [second_countable_topology H] @@ -544,6 +554,23 @@ begin exact h₂.image_of_continuous_on ((chart_at H x).continuous_on_symm.mono h₃) end +/-- If a topological space admits an atlas with locally connected charts, then the space itself is +locally connected. -/ +lemma charted_space.locally_connected_space [locally_connected_space H] : + locally_connected_space M := +begin + let E : M → local_homeomorph M H := chart_at H, + refine locally_connected_space_of_connected_bases + (λ x s, (E x).symm '' s) + (λ x s, (is_open s ∧ E x x ∈ s ∧ is_connected s) ∧ s ⊆ (E x).target) _ _, + { intros x, + simpa only [local_homeomorph.symm_map_nhds_eq, mem_chart_source] using + ((locally_connected_space.open_connected_basis (E x x)).restrict_subset + ((E x).open_target.mem_nhds (mem_chart_target H x))).map (E x).symm }, + { rintros x s ⟨⟨-, -, hsconn⟩, hssubset⟩, + exact hsconn.is_preconnected.image _ ((E x).continuous_on_symm.mono hssubset) }, +end + end /-- For technical reasons we introduce two type tags: @@ -648,7 +675,7 @@ end charted_space have a topological structure, where the topology would come from the charts. For this, one needs charts that are only local equivs, and continuity properties for their composition. This is formalised in `charted_space_core`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure charted_space_core (H : Type*) [topological_space H] (M : Type*) := (atlas : set (local_equiv M H)) (chart_at : M → local_equiv M H) @@ -792,13 +819,13 @@ def structure_groupoid.maximal_atlas : set (local_homeomorph M H) := variable {M} /-- The elements of the atlas belong to the maximal atlas for any structure groupoid -/ -lemma structure_groupoid.mem_maximal_atlas_of_mem_atlas [has_groupoid M G] - {e : local_homeomorph M H} (he : e ∈ atlas H M) : e ∈ G.maximal_atlas M := -λ e' he', ⟨G.compatible he he', G.compatible he' he⟩ +lemma structure_groupoid.subset_maximal_atlas [has_groupoid M G] : + atlas H M ⊆ G.maximal_atlas M := +λ e he e' he', ⟨G.compatible he he', G.compatible he' he⟩ lemma structure_groupoid.chart_mem_maximal_atlas [has_groupoid M G] (x : M) : chart_at H x ∈ G.maximal_atlas M := -G.mem_maximal_atlas_of_mem_atlas (chart_mem_atlas H x) +G.subset_maximal_atlas (chart_mem_atlas H x) variable {G} @@ -835,7 +862,7 @@ variable (G) /-- In the model space, the identity is in any maximal atlas. -/ lemma structure_groupoid.id_mem_maximal_atlas : local_homeomorph.refl H ∈ G.maximal_atlas H := -G.mem_maximal_atlas_of_mem_atlas (by simp) +G.subset_maximal_atlas $ by simp end maximal_atlas @@ -939,7 +966,7 @@ end topological_space.opens /-- A `G`-diffeomorphism between two charted spaces is a homeomorphism which, when read in the charts, belongs to `G`. We avoid the word diffeomorph as it is too related to the smooth category, and use structomorph instead. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure structomorph (G : structure_groupoid H) (M : Type*) (M' : Type*) [topological_space M] [topological_space M'] [charted_space H M] [charted_space H M'] extends homeomorph M M' := diff --git a/formal/lean/mathlib/geometry/manifold/complex.lean b/formal/lean/mathlib/geometry/manifold/complex.lean new file mode 100644 index 0000000000000000000000000000000000000000..1061e5d0fdd175e18e47abcaa7038540c384b840 --- /dev/null +++ b/formal/lean/mathlib/geometry/manifold/complex.lean @@ -0,0 +1,130 @@ +/- +Copyright (c) 2022 Heather Macbeth. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Heather Macbeth +-/ +import analysis.complex.abs_max +import analysis.locally_convex.with_seminorms +import geometry.manifold.mfderiv +import topology.locally_constant.basic + +/-! # Holomorphic functions on complex manifolds + +Thanks to the rigidity of complex-differentiability compared to real-differentiability, there are +many results about complex manifolds with no analogue for manifolds over a general normed field. For +now, this file contains just two (closely related) such results: + +## Main results + +* `mdifferentiable.is_locally_constant`: A complex-differentiable function on a compact complex + manifold is locally constant. +* `mdifferentiable.exists_eq_const_of_compact_space`: A complex-differentiable function on a compact + preconnected complex manifold is constant. + +## TODO + +There is a whole theory to develop here. Maybe a next step would be to develop a theory of +holomorphic vector/line bundles, including: +* the finite-dimensionality of the space of sections of a holomorphic vector bundle +* Siegel's theorem: for any `n + 1` formal ratios `g 0 / h 0`, `g 1 / h 1`, .... `g n / h n` of + sections of a fixed line bundle `L` over a complex `n`-manifold, there exists a polynomial + relationship `P (g 0 / h 0, g 1 / h 1, .... g n / h n) = 0` + +Another direction would be to develop the relationship with sheaf theory, building the sheaves of +holomorphic and meromorphic functions on a complex manifold and proving algebraic results about the +stalks, such as the Weierstrass preparation theorem. + +-/ + +open_locale manifold topological_space +open complex + +namespace mdifferentiable + +variables {E : Type*} [normed_add_comm_group E] [normed_space ℂ E] +variables {F : Type*} [normed_add_comm_group F] [normed_space ℂ F] [strict_convex_space ℝ F] + +variables {M : Type*} [topological_space M] [charted_space E M] + [smooth_manifold_with_corners 𝓘(ℂ, E) M] + +/-- A holomorphic function on a complex manifold is constant on every compact, preconnected, clopen +subset. -/ +lemma apply_eq_of_is_compact {s : set M} (hs₁ : is_compact s) (hs₂ : is_preconnected s) + (hs₃ : is_clopen s) + {f : M → F} (hf : mdifferentiable 𝓘(ℂ, E) 𝓘(ℂ, F) f) {a b : M} (ha : a ∈ s) (hb : b ∈ s) : + f a = f b := +begin + -- for an empty set this fact is trivial + rcases s.eq_empty_or_nonempty with rfl | hs', + { exact false.rec _ ha }, + -- otherwise, let `p₀` be a point where the value of `f` has maximal norm + obtain ⟨p₀, hp₀s, hp₀⟩ := hs₁.exists_forall_ge hs' hf.continuous.norm.continuous_on, + -- we will show `f` agrees everywhere with `f p₀` + suffices : s ⊆ {r : M | f r = f p₀} ∩ s, + { exact (this ha).1.trans (this hb).1.symm }, clear ha hb a b, + refine hs₂.subset_clopen _ ⟨p₀, hp₀s, ⟨rfl, hp₀s⟩⟩, + -- closedness of the set of points sent to `f p₀` + refine ⟨_, (is_closed_singleton.preimage hf.continuous).inter hs₃.2⟩, + -- we will show this set is open by showing it is a neighbourhood of each of its members + rw is_open_iff_mem_nhds, + rintros p ⟨hp : f p = _, hps⟩, -- let `p` be in this set + have hps' : s ∈ 𝓝 p := hs₃.1.mem_nhds hps, + have key₁ : (chart_at E p).symm ⁻¹' s ∈ 𝓝 (chart_at E p p), + { rw [← filter.mem_map, (chart_at E p).symm_map_nhds_eq (mem_chart_source E p)], + exact hps' }, + have key₂ : (chart_at E p).target ∈ 𝓝 (chart_at E p p) := + (local_homeomorph.open_target _).mem_nhds (mem_chart_target E p), + -- `f` pulled back by the chart at `p` is differentiable around `chart_at E p p` + have hf' : ∀ᶠ (z : E) in 𝓝 (chart_at E p p), differentiable_at ℂ (f ∘ (chart_at E p).symm) z, + { refine filter.eventually_of_mem key₂ (λ z hz, _), + have H₁ : (chart_at E p).symm z ∈ (chart_at E p).source := (chart_at E p).map_target hz, + have H₂ : f ((chart_at E p).symm z) ∈ (chart_at F (0:F)).source := trivial, + have H := (mdifferentiable_at_iff_of_mem_source H₁ H₂).mp (hf ((chart_at E p).symm z)), + simp only [differentiable_within_at_univ] with mfld_simps at H, + simpa [local_homeomorph.right_inv _ hz] using H.2, }, + -- `f` pulled back by the chart at `p` has a local max at `chart_at E p p` + have hf'' : is_local_max (norm ∘ f ∘ (chart_at E p).symm) (chart_at E p p), + { refine filter.eventually_of_mem key₁ (λ z hz, _), + refine (hp₀ ((chart_at E p).symm z) hz).trans (_ : ∥f p₀∥ ≤ ∥f _∥), + rw [← hp, local_homeomorph.left_inv _ (mem_chart_source E p)] }, + -- so by the maximum principle `f` is equal to `f p` near `p` + obtain ⟨U, hU, hUf⟩ := (complex.eventually_eq_of_is_local_max_norm hf' hf'').exists_mem, + have H₁ : (chart_at E p) ⁻¹' U ∈ 𝓝 p := (chart_at E p).continuous_at (mem_chart_source E p) hU, + have H₂ : (chart_at E p).source ∈ 𝓝 p := + (local_homeomorph.open_source _).mem_nhds (mem_chart_source E p), + apply filter.mem_of_superset (filter.inter_mem hps' (filter.inter_mem H₁ H₂)), + rintros q ⟨hqs, hq : chart_at E p q ∈ _, hq'⟩, + refine ⟨_, hqs⟩, + simpa [local_homeomorph.left_inv _ hq', hp, -norm_eq_abs] using hUf (chart_at E p q) hq, +end + +/-- A holomorphic function on a compact complex manifold is locally constant. -/ +protected lemma is_locally_constant [compact_space M] + {f : M → F} (hf : mdifferentiable 𝓘(ℂ, E) 𝓘(ℂ, F) f) : + is_locally_constant f := +begin + haveI : locally_connected_space M := charted_space.locally_connected_space E M, + apply is_locally_constant.of_constant_on_preconnected_clopens, + intros s hs hs' x hx y hy, + exact hf.apply_eq_of_is_compact hs'.2.is_compact hs hs' hy hx, +end + +/-- A holomorphic function on a compact connected complex manifold is constant. -/ +lemma apply_eq_of_compact_space [compact_space M] [preconnected_space M] + {f : M → F} (hf : mdifferentiable 𝓘(ℂ, E) 𝓘(ℂ, F) f) (a b : M) : + f a = f b := +hf.apply_eq_of_is_compact compact_univ is_preconnected_univ is_clopen_univ (set.mem_univ _) + (set.mem_univ _) + +/-- A holomorphic function on a compact connected complex manifold is the constant function `f ≡ v`, +for some value `v`. -/ +lemma exists_eq_const_of_compact_space [compact_space M] [preconnected_space M] + {f : M → F} (hf : mdifferentiable 𝓘(ℂ, E) 𝓘(ℂ, F) f) : + ∃ v : F, f = function.const M v := +begin + casesI is_empty_or_nonempty M, + { exact ⟨0, funext $ λ a, h.elim a⟩ }, + { inhabit M, exact ⟨f default, funext $ λ a, hf.apply_eq_of_compact_space a default⟩ }, +end + +end mdifferentiable diff --git a/formal/lean/mathlib/geometry/manifold/cont_mdiff.lean b/formal/lean/mathlib/geometry/manifold/cont_mdiff.lean index 56be1c95a03ff0481a4f20de1e06b2f8b11d8777..e1d310f87da4d599874dd442cca19826e6672038 100644 --- a/formal/lean/mathlib/geometry/manifold/cont_mdiff.lean +++ b/formal/lean/mathlib/geometry/manifold/cont_mdiff.lean @@ -78,6 +78,22 @@ functions between manifolds. -/ def cont_diff_within_at_prop (n : with_top ℕ) (f : H → H') (s : set H) (x : H) : Prop := cont_diff_within_at 𝕜 n (I' ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) (I x) +lemma cont_diff_within_at_prop_self_source {f : E → H'} {s : set E} {x : E} : + cont_diff_within_at_prop 𝓘(𝕜, E) I' n f s x ↔ cont_diff_within_at 𝕜 n (I' ∘ f) s x := +begin + simp_rw [cont_diff_within_at_prop, model_with_corners_self_coe, range_id, inter_univ], + refl +end + +lemma cont_diff_within_at_prop_self {f : E → E'} {s : set E} {x : E} : + cont_diff_within_at_prop 𝓘(𝕜, E) 𝓘(𝕜, E') n f s x ↔ cont_diff_within_at 𝕜 n f s x := +cont_diff_within_at_prop_self_source 𝓘(𝕜, E') + +lemma cont_diff_within_at_prop_self_target {f : H → E'} {s : set H} {x : H} : + cont_diff_within_at_prop I 𝓘(𝕜, E') n f s x ↔ + cont_diff_within_at 𝕜 n (f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) (I x) := +iff.rfl + /-- Being `Cⁿ` in the model space is a local property, invariant under smooth maps. Therefore, it will lift nicely to manifolds. -/ lemma cont_diff_within_at_local_invariant_prop (n : with_top ℕ) : @@ -95,7 +111,7 @@ lemma cont_diff_within_at_local_invariant_prop (n : with_top ℕ) : by { rw [model_with_corners.left_inv], exact is_open.mem_nhds u_open xu }, apply continuous_at.preimage_mem_nhds I.continuous_symm.continuous_at this, end, - right_invariance := + right_invariance' := begin assume s x f e he hx h, rw cont_diff_within_at_prop at h ⊢, @@ -306,6 +322,15 @@ lemma smooth_within_at_iff_target : smooth_within_at I 𝓘(𝕜, E') (ext_chart_at I' (f x) ∘ f) s x := cont_mdiff_within_at_iff_target +lemma cont_mdiff_at_iff_target {x : M} : + cont_mdiff_at I I' n f x ↔ + continuous_at f x ∧ cont_mdiff_at I 𝓘(𝕜, E') n (ext_chart_at I' (f x) ∘ f) x := +by rw [cont_mdiff_at, cont_mdiff_at, cont_mdiff_within_at_iff_target, continuous_within_at_univ] + +lemma smooth_at_iff_target {x : M} : + smooth_at I I' f x ↔ continuous_at f x ∧ smooth_at I 𝓘(𝕜, E') (ext_chart_at I' (f x) ∘ f) x := +cont_mdiff_at_iff_target + include Is I's /-- One can reformulate smoothness within a set at a point as continuity within this set at this @@ -347,7 +372,80 @@ lemma cont_mdiff_at_iff_of_mem_source {x' : M} {y : M'} (hx : x' ∈ (chart_at H (cont_mdiff_within_at_iff_of_mem_source hx hy).trans $ by rw [continuous_within_at_univ, preimage_univ, univ_inter] +omit Is + +lemma cont_mdiff_within_at_iff_target_of_mem_source + {x : M} {y : M'} (hy : f x ∈ (chart_at H' y).source) : + cont_mdiff_within_at I I' n f s x ↔ continuous_within_at f s x ∧ + cont_mdiff_within_at I 𝓘(𝕜, E') n (ext_chart_at I' y ∘ f) s x := +begin + simp_rw [cont_mdiff_within_at], + rw [(cont_diff_within_at_local_invariant_prop I I' n).lift_prop_within_at_indep_chart_target + (chart_mem_maximal_atlas I' y) hy, and_congr_right], + intro hf, + simp_rw [structure_groupoid.lift_prop_within_at_self_target], + simp_rw [((chart_at H' y).continuous_at hy).comp_continuous_within_at hf], + rw [← ext_chart_at_source I'] at hy, + simp_rw [(ext_chart_at_continuous_at' I' _ hy).comp_continuous_within_at hf], + refl, +end + +lemma cont_mdiff_at_iff_target_of_mem_source + {x : M} {y : M'} (hy : f x ∈ (chart_at H' y).source) : + cont_mdiff_at I I' n f x ↔ continuous_at f x ∧ + cont_mdiff_at I 𝓘(𝕜, E') n (ext_chart_at I' y ∘ f) x := +begin + rw [cont_mdiff_at, cont_mdiff_within_at_iff_target_of_mem_source hy, + continuous_within_at_univ, cont_mdiff_at], + apply_instance +end + omit I's +variable (I) +lemma model_with_corners.symm_continuous_within_at_comp_right_iff {X} [topological_space X] + {f : H → X} {s : set H} {x : H} : + continuous_within_at (f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) (I x) ↔ continuous_within_at f s x := +begin + refine ⟨λ h, _, λ h, _⟩, + { have := h.comp I.continuous_within_at (maps_to_preimage _ _), + simp_rw [preimage_inter, preimage_preimage, I.left_inv, preimage_id', preimage_range, + inter_univ] at this, + rwa [function.comp.assoc, I.symm_comp_self] at this }, + { rw [← I.left_inv x] at h, exact h.comp I.continuous_within_at_symm (inter_subset_left _ _) } +end +variable {I} + +lemma ext_chart_at_symm_continuous_within_at_comp_right_iff {X} [topological_space X] {f : M → X} + {s : set M} {x x' : M} : + continuous_within_at (f ∘ (ext_chart_at I x).symm) ((ext_chart_at I x).symm ⁻¹' s ∩ range I) + (ext_chart_at I x x') ↔ + continuous_within_at (f ∘ (chart_at H x).symm) ((chart_at H x).symm ⁻¹' s) (chart_at H x x') := +by convert I.symm_continuous_within_at_comp_right_iff; refl + +include Is + +lemma cont_mdiff_within_at_iff_source_of_mem_source + {x' : M} (hx' : x' ∈ (chart_at H x).source) : + cont_mdiff_within_at I I' n f s x' ↔ + cont_mdiff_within_at 𝓘(𝕜, E) I' n (f ∘ (ext_chart_at I x).symm) + ((ext_chart_at I x).symm ⁻¹' s ∩ range I) (ext_chart_at I x x') := +begin + have h2x' := hx', rw [← ext_chart_at_source I] at h2x', + simp_rw [cont_mdiff_within_at, + (cont_diff_within_at_local_invariant_prop I I' n).lift_prop_within_at_indep_chart_source + (chart_mem_maximal_atlas I x) hx', structure_groupoid.lift_prop_within_at_self_source, + ext_chart_at_symm_continuous_within_at_comp_right_iff, cont_diff_within_at_prop_self_source, + cont_diff_within_at_prop, function.comp, (chart_at H x).left_inv hx', + (ext_chart_at I x).left_inv h2x'], + refl, +end + +lemma cont_mdiff_at_iff_source_of_mem_source + {x' : M} (hx' : x' ∈ (chart_at H x).source) : + cont_mdiff_at I I' n f x' ↔ cont_mdiff_within_at 𝓘(𝕜, E) I' n (f ∘ (ext_chart_at I x).symm) + (range I) (ext_chart_at I x x') := +by simp_rw [cont_mdiff_at, cont_mdiff_within_at_iff_source_of_mem_source hx', preimage_univ, + univ_inter] lemma cont_mdiff_at_ext_chart_at' {x' : M} (h : x' ∈ (chart_at H x).source) : cont_mdiff_at I 𝓘(𝕜, E) n (ext_chart_at I x) x' := @@ -366,6 +464,36 @@ cont_mdiff_at_ext_chart_at' $ mem_chart_source H x include I's +/-- If the set where you want `f` to be smooth lies entirely in a single chart, and `f` maps it + into a single chart, the smoothness of `f` on that set can be expressed by purely looking in + these charts. + Note: this lemma uses `ext_chart_at I x '' s` instead of `(ext_chart_at I x).symm ⁻¹' s` to ensure + that this set lies in `(ext_chart_at I x).target`. -/ +lemma cont_mdiff_on_iff_of_subset_source {x : M} {y : M'} + (hs : s ⊆ (chart_at H x).source) + (h2s : maps_to f s (chart_at H' y).source) : + cont_mdiff_on I I' n f s ↔ continuous_on f s ∧ + cont_diff_on 𝕜 n (ext_chart_at I' y ∘ f ∘ (ext_chart_at I x).symm) + (ext_chart_at I x '' s) := +begin + split, + { refine λ H, ⟨λ x hx, (H x hx).1, _⟩, + rintro _ ⟨x', hx', rfl⟩, + exact ((cont_mdiff_within_at_iff_of_mem_source (hs hx') + (h2s.image_subset $ mem_image_of_mem f hx')).mp (H _ hx')).2.mono + (maps_to_ext_chart_at I x hs).image_subset }, + { rintro ⟨h1, h2⟩ x' hx', + refine (cont_mdiff_within_at_iff_of_mem_source (hs hx') + (h2s.image_subset $ mem_image_of_mem f hx')).mpr + ⟨h1.continuous_within_at hx', _⟩, + refine (h2 _ $ mem_image_of_mem _ hx').mono_of_mem _, + rw [← ext_chart_at_source I] at hs, + rw [(ext_chart_at I x).image_eq_target_inter_inv_preimage hs], + refine inter_mem _ (ext_chart_preimage_mem_nhds_within' I x (hs hx') self_mem_nhds_within), + have := ext_chart_at_target_mem_nhds_within' I x (hs hx'), + refine nhds_within_mono _ (inter_subset_right _ _) this } +end + /-- One can reformulate smoothness on a set as continuity on this set, and smoothness in any extended chart. -/ lemma cont_mdiff_on_iff : @@ -495,7 +623,7 @@ lemma cont_mdiff.of_succ {n : ℕ} (h : cont_mdiff I I' n.succ f) : cont_mdiff I I' n f := λ x, (h x).of_succ -/-! ### Deducing continuity from smoothness-/ +/-! ### Deducing continuity from smoothness -/ lemma cont_mdiff_within_at.continuous_within_at (hf : cont_mdiff_within_at I I' n f s x) : continuous_within_at f s x := @@ -719,6 +847,18 @@ lemma cont_mdiff_at_iff_cont_mdiff_on_nhds {n : ℕ} : by simp [← cont_mdiff_within_at_univ, cont_mdiff_within_at_iff_cont_mdiff_on_nhds, nhds_within_univ] +/-- Note: This does not hold for `n = ∞`. `f` being `C^∞` at `x` means that for every `n`, `f` is +`C^n` on some neighborhood of `x`, but this neighborhood can depend on `n`. -/ +lemma cont_mdiff_at_iff_cont_mdiff_at_nhds {n : ℕ} : + cont_mdiff_at I I' n f x ↔ ∀ᶠ x' in 𝓝 x, cont_mdiff_at I I' n f x' := +begin + refine ⟨_, λ h, h.self_of_nhds⟩, + rw [cont_mdiff_at_iff_cont_mdiff_on_nhds], + rintro ⟨u, hu, h⟩, + refine (eventually_mem_nhds.mpr hu).mono (λ x' hx', _), + exact (h x' $ mem_of_mem_nhds hx').cont_mdiff_at hx' +end + omit Is I's /-! ### Congruence lemmas -/ @@ -1146,33 +1286,31 @@ begin { assume y hy, simpa only [unique_mdiff_on, unique_mdiff_within_at, hy.1, inter_comm] with mfld_simps using hs (I.symm y) hy.2 }, - have U : unique_diff_on 𝕜 ((range I ∩ I.symm ⁻¹' s) ×ˢ (univ : set E)) := - U'.prod unique_diff_on_univ, rw cont_mdiff_on_iff, refine ⟨hf.continuous_on_tangent_map_within_aux one_le_n hs, λp q, _⟩, - have A : range I ×ˢ (univ : set E) ∩ + have A : range I ×ˢ univ ∩ ((equiv.sigma_equiv_prod H E).symm ∘ λ (p : E × E), ((I.symm) p.fst, p.snd)) ⁻¹' (tangent_bundle.proj I H ⁻¹' s) - = (range I ∩ I.symm ⁻¹' s) ×ˢ (univ : set E), + = (range I ∩ I.symm ⁻¹' s) ×ˢ univ, by { ext ⟨x, v⟩, simp only with mfld_simps }, suffices h : cont_diff_on 𝕜 m (((λ (p : H' × E'), (I' p.fst, p.snd)) ∘ (equiv.sigma_equiv_prod H' E')) ∘ tangent_map_within I I' f s ∘ ((equiv.sigma_equiv_prod H E).symm) ∘ λ (p : E × E), (I.symm p.fst, p.snd)) - ((range ⇑I ∩ ⇑(I.symm) ⁻¹' s) ×ˢ (univ : set E)), + ((range ⇑I ∩ ⇑(I.symm) ⁻¹' s) ×ˢ univ), by simpa [A] using h, change cont_diff_on 𝕜 m (λ (p : E × E), ((I' (f (I.symm p.fst)), ((mfderiv_within I I' f s (I.symm p.fst)) : E → E') p.snd) : E' × E')) - ((range I ∩ I.symm ⁻¹' s) ×ˢ (univ : set E)), + ((range I ∩ I.symm ⁻¹' s) ×ˢ univ), -- check that all bits in this formula are `C^n` have hf' := cont_mdiff_on_iff.1 hf, have A : cont_diff_on 𝕜 m (I' ∘ f ∘ I.symm) (range I ∩ I.symm ⁻¹' s) := by simpa only with mfld_simps using (hf'.2 (I.symm 0) (I'.symm 0)).of_le m_le_n, have B : cont_diff_on 𝕜 m ((I' ∘ f ∘ I.symm) ∘ prod.fst) - ((range I ∩ I.symm ⁻¹' s) ×ˢ (univ : set E)) := + ((range I ∩ I.symm ⁻¹' s) ×ˢ univ) := A.comp (cont_diff_fst.cont_diff_on) (prod_subset_preimage_fst _ _), suffices C : cont_diff_on 𝕜 m (λ (p : E × E), ((fderiv_within 𝕜 (I' ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) p.1 : _) p.2)) - ((range I ∩ I.symm ⁻¹' s) ×ˢ (univ : set E)), + ((range I ∩ I.symm ⁻¹' s) ×ˢ univ), { apply cont_diff_on.prod B _, apply C.congr (λp hp, _), simp only with mfld_simps at hp, @@ -1411,6 +1549,20 @@ namespace basic_smooth_vector_bundle_core variables (Z : basic_smooth_vector_bundle_core I M E') +/-- A version of `cont_mdiff_at_iff_target` when the codomain is the total space of + a `basic_smooth_vector_bundle_core`. The continuity condition in the RHS is weaker. -/ +lemma cont_mdiff_at_iff_target {f : N → Z.to_topological_vector_bundle_core.total_space} + {x : N} {n : with_top ℕ} : + cont_mdiff_at J (I.prod 𝓘(𝕜, E')) n f x ↔ continuous_at (bundle.total_space.proj ∘ f) x ∧ + cont_mdiff_at J 𝓘(𝕜, E × E') n (ext_chart_at (I.prod 𝓘(𝕜, E')) (f x) ∘ f) x := +begin + let Z' := Z.to_topological_vector_bundle_core, + rw [cont_mdiff_at_iff_target, and.congr_left_iff], + refine λ hf, ⟨λ h, Z'.continuous_proj.continuous_at.comp h, λ h, _⟩, + exact (Z'.local_triv ⟨chart_at _ (f x).1, chart_mem_atlas _ _⟩).to_fiber_bundle_trivialization + .continuous_at_of_comp_left h (mem_chart_source _ _) (h.prod hf.continuous_at.snd) +end + lemma cont_mdiff_proj : cont_mdiff (I.prod 𝓘(𝕜, E')) I n Z.to_topological_vector_bundle_core.proj := begin diff --git a/formal/lean/mathlib/geometry/manifold/diffeomorph.lean b/formal/lean/mathlib/geometry/manifold/diffeomorph.lean index 72a80945f79fb0f8ea600f4e91bd5260976f804d..a1b654d4ce246ade685ac42223733ebcd9cf8f8a 100644 --- a/formal/lean/mathlib/geometry/manifold/diffeomorph.lean +++ b/formal/lean/mathlib/geometry/manifold/diffeomorph.lean @@ -70,7 +70,7 @@ variables (I I' M M' n) /-- `n`-times continuously differentiable diffeomorphism between `M` and `M'` with respect to I and I' -/ -@[protect_proj, nolint has_inhabited_instance] +@[protect_proj, nolint has_nonempty_instance] structure diffeomorph extends M ≃ M' := (cont_mdiff_to_fun : cont_mdiff I I' n to_equiv) (cont_mdiff_inv_fun : cont_mdiff I' I n to_equiv.symm) diff --git a/formal/lean/mathlib/geometry/manifold/instances/sphere.lean b/formal/lean/mathlib/geometry/manifold/instances/sphere.lean index 0c89673dd021e0a1f8fb789ed5d077c6ee6d4c41..f63a5beffe4c52451fce10f33cb5c360804cebe5 100644 --- a/formal/lean/mathlib/geometry/manifold/instances/sphere.lean +++ b/formal/lean/mathlib/geometry/manifold/instances/sphere.lean @@ -88,7 +88,7 @@ begin refine cont_diff_on.smul _ (orthogonal_projection ((ℝ ∙ v)ᗮ)).cont_diff.cont_diff_on, refine cont_diff_const.cont_diff_on.div _ _, - { exact (cont_diff_const.sub (innerSL v).cont_diff).cont_diff_on }, + { exact (cont_diff_const.sub (innerSL v : E →L[ℝ] ℝ).cont_diff).cont_diff_on }, { intros x h h', exact h (sub_eq_zero.mp h').symm } end diff --git a/formal/lean/mathlib/geometry/manifold/local_invariant_properties.lean b/formal/lean/mathlib/geometry/manifold/local_invariant_properties.lean index d93795dcc5cbb11d76f0b3580c9d08b66c1157eb..4e3a8114291bf76909a5c5ebf37f4f4f9727721c 100644 --- a/formal/lean/mathlib/geometry/manifold/local_invariant_properties.lean +++ b/formal/lean/mathlib/geometry/manifold/local_invariant_properties.lean @@ -47,8 +47,10 @@ open_locale classical manifold topological_space open set filter -variables {H : Type*} {M : Type*} [topological_space H] [topological_space M] [charted_space H M] -{H' : Type*} {M' : Type*} [topological_space H'] [topological_space M'] [charted_space H' M'] +variables {H M H' M' X : Type*} +variables [topological_space H] [topological_space M] [charted_space H M] +variables [topological_space H'] [topological_space M'] [charted_space H' M'] +variables [topological_space X] namespace structure_groupoid @@ -60,7 +62,7 @@ variables (G : structure_groupoid H) (G' : structure_groupoid H') to charted spaces admitting these groupoids will inherit the good behavior. -/ structure local_invariant_prop (P : (H → H') → (set H) → H → Prop) : Prop := (is_local : ∀ {s x u} {f : H → H'}, is_open u → x ∈ u → (P f s x ↔ P f (s ∩ u) x)) -(right_invariance : ∀ {s x f} {e : local_homeomorph H H}, e ∈ G → x ∈ e.source → P f s x → +(right_invariance' : ∀ {s x f} {e : local_homeomorph H H}, e ∈ G → x ∈ e.source → P f s x → P (f ∘ e.symm) (e.symm ⁻¹' s) (e x)) (congr_of_forall : ∀ {s x} {f g : H → H'}, (∀ y ∈ s, f y = g y) → f x = g x → P f s x → P g s x) (left_invariance' : ∀ {s x f} {e' : local_homeomorph H' H'}, e' ∈ G' → s ⊆ f ⁻¹' e'.source → @@ -85,14 +87,6 @@ lemma is_local_nhds {s u : set H} {x : H} {f : H → H'} (hu : u ∈ 𝓝[s] x) P f s x ↔ P f (s ∩ u) x := hG.congr_set $ mem_nhds_within_iff_eventually_eq.mp hu -lemma left_invariance {s : set H} {x : H} {f : H → H'} {e' : local_homeomorph H' H'} - (he' : e' ∈ G') (hfs : continuous_within_at f s x) (hxe' : f x ∈ e'.source) (hP : P f s x) : - P (e' ∘ f) s x := -begin - rw [hG.is_local_nhds (hfs.preimage_mem_nhds_within $ e'.open_source.mem_nhds hxe')] at hP ⊢, - exact hG.left_invariance' he' (inter_subset_right _ _) hxe' hP -end - lemma congr_iff_nhds_within {s : set H} {x : H} {f g : H → H'} (h1 : f =ᶠ[𝓝[s] x] g) (h2 : f x = g x) : P f s x ↔ P g s x := by { simp_rw [hG.is_local_nhds h1], @@ -115,6 +109,39 @@ lemma congr {s : set H} {x : H} {f g : H → H'} (h : f =ᶠ[𝓝 x] g) (hP : P lemma congr' {s : set H} {x : H} {f g : H → H'} (h : f =ᶠ[𝓝 x] g) (hP : P g s x) : P f s x := hG.congr h.symm hP +lemma left_invariance {s : set H} {x : H} {f : H → H'} {e' : local_homeomorph H' H'} + (he' : e' ∈ G') (hfs : continuous_within_at f s x) (hxe' : f x ∈ e'.source) : + P (e' ∘ f) s x ↔ P f s x := +begin + have h2f := hfs.preimage_mem_nhds_within (e'.open_source.mem_nhds hxe'), + have h3f := (((e'.continuous_at hxe').comp_continuous_within_at hfs).preimage_mem_nhds_within $ + e'.symm.open_source.mem_nhds $ e'.maps_to hxe'), + split, + { intro h, + rw [hG.is_local_nhds h3f] at h, + have h2 := hG.left_invariance' (G'.symm he') (inter_subset_right _ _) + (by exact e'.maps_to hxe') h, + rw [← hG.is_local_nhds h3f] at h2, + refine hG.congr_nhds_within _ (e'.left_inv hxe') h2, + exact eventually_of_mem h2f (λ x', e'.left_inv) }, + { simp_rw [hG.is_local_nhds h2f], + exact hG.left_invariance' he' (inter_subset_right _ _) hxe' } +end + +lemma right_invariance {s : set H} {x : H} {f : H → H'} {e : local_homeomorph H H} + (he : e ∈ G) (hxe : x ∈ e.source) : P (f ∘ e.symm) (e.symm ⁻¹' s) (e x) ↔ P f s x := +begin + refine ⟨λ h, _, hG.right_invariance' he hxe⟩, + have := hG.right_invariance' (G.symm he) (e.maps_to hxe) h, + rw [e.symm_symm, e.left_inv hxe] at this, + refine hG.congr _ ((hG.congr_set _).mp this), + { refine eventually_of_mem (e.open_source.mem_nhds hxe) (λ x' hx', _), + simp_rw [function.comp_apply, e.left_inv hx'] }, + { rw [eventually_eq_set], + refine eventually_of_mem (e.open_source.mem_nhds hxe) (λ x' hx', _), + simp_rw [mem_preimage, e.left_inv hx'] }, +end + end local_invariant_prop end structure_groupoid @@ -175,6 +202,20 @@ iff.rfl lemma lift_prop_on_univ : lift_prop_on P g univ ↔ lift_prop P g := by simp [lift_prop_on, lift_prop, lift_prop_at] +lemma lift_prop_within_at_self {f : H → H'} {s : set H} {x : H} : + lift_prop_within_at P f s x ↔ continuous_within_at f s x ∧ P f s x := +iff.rfl + +lemma lift_prop_within_at_self_source {f : H → M'} {s : set H} {x : H} : + lift_prop_within_at P f s x ↔ continuous_within_at f s x ∧ P (chart_at H' (f x) ∘ f) s x := +iff.rfl + +lemma lift_prop_within_at_self_target {f : M → H'} : + lift_prop_within_at P f s x ↔ + continuous_within_at f s x ∧ + P (f ∘ (chart_at H x).symm) ((chart_at H x).symm ⁻¹' s) (chart_at H x x) := +iff.rfl + namespace local_invariant_prop variable (hG : G.local_invariant_prop G' P) @@ -193,6 +234,52 @@ begin (mem_chart_source H x) (chart_source_mem_nhds H' (f x)) end +lemma lift_prop_within_at_indep_chart_source_aux (g : M → H') + (he : e ∈ G.maximal_atlas M) (xe : x ∈ e.source) + (he' : e' ∈ G.maximal_atlas M) (xe' : x ∈ e'.source) : + P (g ∘ e.symm) (e.symm ⁻¹' s) (e x) ↔ P (g ∘ e'.symm) (e'.symm ⁻¹' s) (e' x) := +begin + rw [← hG.right_invariance (compatible_of_mem_maximal_atlas he he')], + swap, { simp only [xe, xe'] with mfld_simps }, + simp_rw [local_homeomorph.trans_apply, e.left_inv xe], + rw [hG.congr_iff], + { refine hG.congr_set _, + refine (eventually_of_mem _ $ λ y (hy : y ∈ e'.symm ⁻¹' e.source), _).set_eq, + { refine (e'.symm.continuous_at $ e'.maps_to xe').preimage_mem_nhds (e.open_source.mem_nhds _), + simp_rw [e'.left_inv xe', xe] }, + simp_rw [mem_preimage, local_homeomorph.coe_trans_symm, local_homeomorph.symm_symm, + function.comp_apply, e.left_inv hy] }, + { refine ((e'.eventually_nhds' _ xe').mpr $ e.eventually_left_inverse xe).mono (λ y hy, _), + simp only with mfld_simps, + rw [hy] }, +end + +lemma lift_prop_within_at_indep_chart_target_aux2 (g : H → M') {x : H} {s : set H} + (hf : f ∈ G'.maximal_atlas M') (xf : g x ∈ f.source) + (hf' : f' ∈ G'.maximal_atlas M') (xf' : g x ∈ f'.source) + (hgs : continuous_within_at g s x) : + P (f ∘ g) s x ↔ P (f' ∘ g) s x := +begin + have hcont : continuous_within_at (f ∘ g) s x := + (f.continuous_at xf).comp_continuous_within_at hgs, + rw [← hG.left_invariance (compatible_of_mem_maximal_atlas hf hf') hcont + (by simp only [xf, xf'] with mfld_simps)], + refine hG.congr_iff_nhds_within _ (by simp only [xf] with mfld_simps), + exact (hgs.eventually $ f.eventually_left_inverse xf).mono (λ y, congr_arg f') +end + +lemma lift_prop_within_at_indep_chart_target_aux {g : X → M'} {e : local_homeomorph X H} {x : X} + {s : set X} (xe : x ∈ e.source) + (hf : f ∈ G'.maximal_atlas M') (xf : g x ∈ f.source) + (hf' : f' ∈ G'.maximal_atlas M') (xf' : g x ∈ f'.source) + (hgs : continuous_within_at g s x) : + P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x) ↔ P (f' ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x) := +begin + rw [← e.left_inv xe] at xf xf' hgs, + refine hG.lift_prop_within_at_indep_chart_target_aux2 (g ∘ e.symm) hf xf hf' xf' _, + exact hgs.comp (e.symm.continuous_at $ e.maps_to xe).continuous_within_at subset.rfl +end + /-- If a property of a germ of function `g` on a pointed set `(s, x)` is invariant under the structure groupoid (by composition in the source space and in the target space), then expressing it in charted spaces does not depend on the element of the maximal atlas one uses @@ -204,53 +291,61 @@ lemma lift_prop_within_at_indep_chart_aux (he' : e' ∈ G.maximal_atlas M) (xe' : x ∈ e'.source) (hf : f ∈ G'.maximal_atlas M') (xf : g x ∈ f.source) (hf' : f' ∈ G'.maximal_atlas M') (xf' : g x ∈ f'.source) - (hgs : continuous_within_at g s x) - (h : P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x)) : - P (f' ∘ g ∘ e'.symm) (e'.symm ⁻¹' s) (e' x) := -begin - have hcont : continuous_within_at (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x), - { rw [← e.left_inv xe] at hgs xf, - refine (f.continuous_at $ by exact xf).comp_continuous_within_at _, - exact hgs.comp (e.symm.continuous_at $ e.maps_to xe).continuous_within_at subset.rfl }, - have A : P ((f.symm ≫ₕ f') ∘ (f ∘ g ∘ e.symm)) (e.symm ⁻¹' s) (e x), - { refine hG.left_invariance (compatible_of_mem_maximal_atlas hf hf') hcont - (by simp only [xe, xf, xf'] with mfld_simps) h }, - have B : P (f' ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x), - { refine hG.congr_nhds_within _ (by simp only [xe, xf] with mfld_simps) A, - simp_rw [local_homeomorph.coe_trans, eventually_eq], - have := (e.eventually_nhds_within' _ xe).mpr (hgs.eventually $ f.eventually_left_inverse xf), - exact this.mono (λ y, congr_arg f') }, - let w := e.symm ≫ₕ e', - let ow := w.symm ⁻¹' (e.symm ⁻¹' s), - have wG : w ∈ G := compatible_of_mem_maximal_atlas he he', - have C : P ((f' ∘ g ∘ e.symm) ∘ w.symm) ow (w (e x)) := - hG.right_invariance wG (by simp only [w, xe, xe'] with mfld_simps) B, - have : ∀ y ∈ e.source, w (e y) = e' y := λ y hy, by simp only [w, hy] with mfld_simps, - rw [this x xe] at C, - have D : P (f' ∘ g ∘ e'.symm) ow (e' x), - { refine hG.congr _ C, - refine ((e'.eventually_nhds' _ xe').mpr $ e.eventually_left_inverse xe).mono (λ y hy, _), - simp only [w] with mfld_simps, - rw [hy] }, - refine (hG.congr_set _).2 D, - refine (eventually_of_mem _ $ λ y (hy : y ∈ e'.symm ⁻¹' e.source), _).set_eq, - { refine (e'.symm.continuous_at $ e'.maps_to xe').preimage_mem_nhds (e.open_source.mem_nhds _), - simp_rw [e'.left_inv xe', xe] }, - simp_rw [ow, mem_preimage, w, local_homeomorph.coe_trans_symm, local_homeomorph.symm_symm, - function.comp_apply, e.left_inv hy] -end + (hgs : continuous_within_at g s x) : + P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x) ↔ P (f' ∘ g ∘ e'.symm) (e'.symm ⁻¹' s) (e' x) := +by rw [hG.lift_prop_within_at_indep_chart_source_aux (f ∘ g) he xe he' xe', + hG.lift_prop_within_at_indep_chart_target_aux xe' hf xf hf' xf' hgs] lemma lift_prop_within_at_indep_chart [has_groupoid M G] [has_groupoid M' G'] (he : e ∈ G.maximal_atlas M) (xe : x ∈ e.source) (hf : f ∈ G'.maximal_atlas M') (xf : g x ∈ f.source) : lift_prop_within_at P g s x ↔ continuous_within_at g s x ∧ P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x) := -⟨λ H, ⟨H.1, - hG.lift_prop_within_at_indep_chart_aux (chart_mem_maximal_atlas _ _) (mem_chart_source _ _) he xe - (chart_mem_maximal_atlas _ _) (mem_chart_source _ _) hf xf H.1 H.2⟩, -λ H, ⟨H.1, - hG.lift_prop_within_at_indep_chart_aux he xe (chart_mem_maximal_atlas _ _) (mem_chart_source _ _) - hf xf (chart_mem_maximal_atlas _ _) (mem_chart_source _ _) H.1 H.2⟩⟩ +and_congr_right $ hG.lift_prop_within_at_indep_chart_aux (chart_mem_maximal_atlas _ _) + (mem_chart_source _ _) he xe (chart_mem_maximal_atlas _ _) (mem_chart_source _ _) hf xf + +/-- A version of `lift_prop_within_at_indep_chart`, only for the source. -/ +lemma lift_prop_within_at_indep_chart_source [has_groupoid M G] + (he : e ∈ G.maximal_atlas M) (xe : x ∈ e.source) : + lift_prop_within_at P g s x ↔ lift_prop_within_at P (g ∘ e.symm) (e.symm ⁻¹' s) (e x) := +begin + have := e.symm.continuous_within_at_iff_continuous_within_at_comp_right xe, + rw [e.symm_symm] at this, + rw [lift_prop_within_at_self_source, lift_prop_within_at, ← this], + simp_rw [function.comp_app, e.left_inv xe], + refine and_congr iff.rfl _, + rw hG.lift_prop_within_at_indep_chart_source_aux (chart_at H' (g x) ∘ g) + (chart_mem_maximal_atlas G x) (mem_chart_source H x) he xe, +end + +/-- A version of `lift_prop_within_at_indep_chart`, only for the target. -/ +lemma lift_prop_within_at_indep_chart_target [has_groupoid M' G'] + (hf : f ∈ G'.maximal_atlas M') (xf : g x ∈ f.source) : + lift_prop_within_at P g s x ↔ continuous_within_at g s x ∧ lift_prop_within_at P (f ∘ g) s x := +begin + rw [lift_prop_within_at_self_target, lift_prop_within_at, and.congr_right_iff], + intro hg, + simp_rw [(f.continuous_at xf).comp_continuous_within_at hg, true_and], + exact hG.lift_prop_within_at_indep_chart_target_aux (mem_chart_source _ _) + (chart_mem_maximal_atlas _ _) (mem_chart_source _ _) hf xf hg +end + +/-- A version of `lift_prop_within_at_indep_chart`, that uses `lift_prop_within_at` on both sides. +-/ +lemma lift_prop_within_at_indep_chart' [has_groupoid M G] [has_groupoid M' G'] + (he : e ∈ G.maximal_atlas M) (xe : x ∈ e.source) + (hf : f ∈ G'.maximal_atlas M') (xf : g x ∈ f.source) : + lift_prop_within_at P g s x ↔ + continuous_within_at g s x ∧ lift_prop_within_at P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x) := +begin + rw [hG.lift_prop_within_at_indep_chart he xe hf xf, lift_prop_within_at_self, and.left_comm, + iff.comm, and_iff_right_iff_imp], + intro h, + have h1 := (e.symm.continuous_within_at_iff_continuous_within_at_comp_right xe).mp h.1, + have : continuous_at f ((g ∘ e.symm) (e x)), + { simp_rw [function.comp, e.left_inv xe, f.continuous_at xf] }, + exact this.comp_continuous_within_at h1, +end lemma lift_prop_on_indep_chart [has_groupoid M G] [has_groupoid M' G'] (he : e ∈ G.maximal_atlas M) (hf : f ∈ G'.maximal_atlas M') (h : lift_prop_on P g s) @@ -277,17 +372,11 @@ hG.lift_prop_within_at_inter' (mem_nhds_within_of_mem_nhds ht) lemma lift_prop_at_of_lift_prop_within_at (h : lift_prop_within_at P g s x) (hs : s ∈ 𝓝 x) : lift_prop_at P g x := -begin - have : s = univ ∩ s, by rw univ_inter, - rwa [this, hG.lift_prop_within_at_inter hs] at h, -end +by rwa [← univ_inter s, hG.lift_prop_within_at_inter hs] at h lemma lift_prop_within_at_of_lift_prop_at_of_mem_nhds (h : lift_prop_at P g x) (hs : s ∈ 𝓝 x) : lift_prop_within_at P g s x := -begin - have : s = univ ∩ s, by rw univ_inter, - rwa [this, hG.lift_prop_within_at_inter hs], -end +by rwa [← univ_inter s, hG.lift_prop_within_at_inter hs] lemma lift_prop_on_of_locally_lift_prop_on (h : ∀ x ∈ s, ∃ u, is_open u ∧ x ∈ u ∧ lift_prop_on P g (s ∩ u)) : @@ -489,7 +578,7 @@ lemma is_local_structomorph_within_at_local_invariant_prop [closed_under_restric simpa only [this, interior_interior, hu.interior_eq] with mfld_simps using hef }, { simp only [*, interior_interior, hu.interior_eq] with mfld_simps } } end, - right_invariance := begin + right_invariance' := begin intros s x f e' he'G he'x h hx, have hxs : x ∈ s := by simpa only [e'.left_inv he'x] with mfld_simps using hx, rcases h hxs with ⟨e, heG, hef, hex⟩, diff --git a/formal/lean/mathlib/geometry/manifold/mfderiv.lean b/formal/lean/mathlib/geometry/manifold/mfderiv.lean index c43b8c20ea839cdc07b5b0e7d5332ae3c3d45a2e..00f5ce29857f6683f11e57641a4257d563f00b6c 100644 --- a/formal/lean/mathlib/geometry/manifold/mfderiv.lean +++ b/formal/lean/mathlib/geometry/manifold/mfderiv.lean @@ -3,6 +3,7 @@ Copyright (c) 2020 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ +import geometry.manifold.local_invariant_properties import geometry.manifold.tangent_bundle /-! @@ -118,6 +119,64 @@ variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] {H' : Type*} [topological_space H'] (I' : model_with_corners 𝕜 E' H') {M' : Type*} [topological_space M'] [charted_space H' M'] +/-- Property in the model space of a model with corners of being differentiable within at set at a +point, when read in the model vector space. This property will be lifted to manifolds to define +differentiable functions between manifolds. -/ +def differentiable_within_at_prop (f : H → H') (s : set H) (x : H) : Prop := +differentiable_within_at 𝕜 (I' ∘ f ∘ (I.symm)) (⇑(I.symm) ⁻¹' s ∩ set.range I) (I x) + +/-- Being differentiable in the model space is a local property, invariant under smooth maps. +Therefore, it will lift nicely to manifolds. -/ +lemma differentiable_within_at_local_invariant_prop : + (cont_diff_groupoid ⊤ I).local_invariant_prop (cont_diff_groupoid ⊤ I') + (differentiable_within_at_prop I I') := +{ is_local := + begin + assume s x u f u_open xu, + have : I.symm ⁻¹' (s ∩ u) ∩ set.range I = (I.symm ⁻¹' s ∩ set.range I) ∩ I.symm ⁻¹' u, + by simp only [set.inter_right_comm, set.preimage_inter], + rw [differentiable_within_at_prop, differentiable_within_at_prop, this], + symmetry, + apply differentiable_within_at_inter, + have : u ∈ 𝓝 (I.symm (I x)), + by { rw [model_with_corners.left_inv], exact is_open.mem_nhds u_open xu }, + apply continuous_at.preimage_mem_nhds I.continuous_symm.continuous_at this, + end, + right_invariance' := + begin + assume s x f e he hx h, + rw differentiable_within_at_prop at h ⊢, + have : I x = (I ∘ e.symm ∘ I.symm) (I (e x)), by simp only [hx] with mfld_simps, + rw this at h, + have : I (e x) ∈ (I.symm) ⁻¹' e.target ∩ set.range I, by simp only [hx] with mfld_simps, + have := ((mem_groupoid_of_pregroupoid.2 he).2.cont_diff_within_at this), + convert (h.comp' _ (this.differentiable_within_at le_top)).mono_of_mem _ using 1, + { ext y, simp only with mfld_simps }, + refine mem_nhds_within.mpr ⟨I.symm ⁻¹' e.target, e.open_target.preimage I.continuous_symm, + by simp_rw [set.mem_preimage, I.left_inv, e.maps_to hx], _⟩, + mfld_set_tac + end, + congr_of_forall := + begin + assume s x f g h hx hf, + apply hf.congr, + { assume y hy, + simp only with mfld_simps at hy, + simp only [h, hy] with mfld_simps }, + { simp only [hx] with mfld_simps } + end, + left_invariance' := + begin + assume s x f e' he' hs hx h, + rw differentiable_within_at_prop at h ⊢, + have A : (I' ∘ f ∘ I.symm) (I x) ∈ (I'.symm ⁻¹' e'.source ∩ set.range I'), + by simp only [hx] with mfld_simps, + have := ((mem_groupoid_of_pregroupoid.2 he').1.cont_diff_within_at A), + convert (this.differentiable_within_at le_top).comp _ h _, + { ext y, simp only with mfld_simps }, + { assume y hy, simp only with mfld_simps at hy, simpa only [hy] with mfld_simps using hs hy.1 } + end } + /-- Predicate ensuring that, at a point and within a set, a function can have at most one derivative. This is expressed using the preferred chart at the considered point. -/ def unique_mdiff_within_at (s : set M) (x : M) := @@ -145,6 +204,11 @@ continuous_within_at f s x ∧ differentiable_within_at 𝕜 (written_in_ext_chart_at I I' x f) ((ext_chart_at I x).symm ⁻¹' s ∩ range I) ((ext_chart_at I x) x) +lemma mdifferentiable_within_at_iff_lift_prop_within_at (f : M → M') (s : set M) (x : M) : + mdifferentiable_within_at I I' f s x + ↔ lift_prop_within_at (differentiable_within_at_prop I I') f s x := +by refl + /-- `mdifferentiable_at I I' f x` indicates that the function `f` between manifolds has a derivative at the point `x`. This is a generalization of `differentiable_at` to manifolds. @@ -158,6 +222,15 @@ continuous_at f x ∧ differentiable_within_at 𝕜 (written_in_ext_chart_at I I' x f) (range I) ((ext_chart_at I x) x) +lemma mdifferentiable_at_iff_lift_prop_at (f : M → M') (x : M) : + mdifferentiable_at I I' f x + ↔ lift_prop_at (differentiable_within_at_prop I I') f x := +begin + congrm _ ∧ _, + { rw continuous_within_at_univ }, + { simp [differentiable_within_at_prop, set.univ_inter] } +end + /-- `mdifferentiable_on I I' f s` indicates that the function `f` between manifolds has a derivative within `s` at all points of `s`. This is a generalization of `differentiable_on` to manifolds. -/ @@ -345,6 +418,22 @@ end include Is I's +/-- One can reformulate differentiability within a set at a point as continuity within this set at +this point, and differentiability in any chart containing that point. -/ +lemma mdifferentiable_within_at_iff_of_mem_source + {x' : M} {y : M'} + (hx : x' ∈ (charted_space.chart_at H x).source) + (hy : f x' ∈ (charted_space.chart_at H' y).source) : + mdifferentiable_within_at I I' f s x' + ↔ continuous_within_at f s x' + ∧ differentiable_within_at 𝕜 + ((ext_chart_at I' y) ∘ f ∘ ((ext_chart_at I x).symm)) + (((ext_chart_at I x).symm) ⁻¹' s ∩ set.range I) + ((ext_chart_at I x) x') := +(differentiable_within_at_local_invariant_prop I I').lift_prop_within_at_indep_chart + (structure_groupoid.chart_mem_maximal_atlas _ x) hx + (structure_groupoid.chart_mem_maximal_atlas _ y) hy + lemma mfderiv_within_zero_of_not_mdifferentiable_within_at (h : ¬ mdifferentiable_within_at I I' f s x) : mfderiv_within I I' f s x = 0 := by simp only [mfderiv_within, h, dif_neg, not_false_iff] @@ -538,6 +627,19 @@ lemma mfderiv_within_inter (ht : t ∈ 𝓝 x) (hs : unique_mdiff_within_at I s by rw [mfderiv_within, mfderiv_within, ext_chart_preimage_inter_eq, mdifferentiable_within_at_inter ht, fderiv_within_inter (ext_chart_preimage_mem_nhds I x ht) hs] +lemma mdifferentiable_at_iff_of_mem_source {x' : M} {y : M'} + (hx : x' ∈ (charted_space.chart_at H x).source) + (hy : f x' ∈ (charted_space.chart_at H' y).source) : + mdifferentiable_at I I' f x' + ↔ continuous_at f x' + ∧ differentiable_within_at 𝕜 + ((ext_chart_at I' y) ∘ f ∘ ((ext_chart_at I x).symm)) + (set.range I) + ((ext_chart_at I x) x') := +mdifferentiable_within_at_univ.symm.trans $ + (mdifferentiable_within_at_iff_of_mem_source hx hy).trans $ + by rw [continuous_within_at_univ, set.preimage_univ, set.univ_inter] + omit Is I's /-! ### Deriving continuity from differentiability on manifolds -/ @@ -1549,8 +1651,8 @@ begin -- rewrite the relevant set in the chart as a direct product have : (λ (p : E × F), (I.symm p.1, p.snd)) ⁻¹' e.target ∩ (λ (p : E × F), (I.symm p.1, p.snd)) ⁻¹' (e.symm ⁻¹' (sigma.fst ⁻¹' s)) ∩ - (range I ×ˢ (univ : set F)) - = (I.symm ⁻¹' (e₀.target ∩ e₀.symm⁻¹' s) ∩ range I) ×ˢ (univ : set F), + (range I ×ˢ univ) + = (I.symm ⁻¹' (e₀.target ∩ e₀.symm⁻¹' s) ∩ range I) ×ˢ univ, by mfld_set_tac, assume q hq, replace hq : q.1 ∈ (chart_at H p.1).target ∧ ((chart_at H p.1).symm : H → M) q.1 ∈ s, diff --git a/formal/lean/mathlib/geometry/manifold/partition_of_unity.lean b/formal/lean/mathlib/geometry/manifold/partition_of_unity.lean index 778490f7c89005bc45aeba1009c3ef46a10d3062..d5c795f33cc344ea80692409fd48adb1d3c58dad 100644 --- a/formal/lean/mathlib/geometry/manifold/partition_of_unity.lean +++ b/formal/lean/mathlib/geometry/manifold/partition_of_unity.lean @@ -97,7 +97,7 @@ subordinate to `U`, see `smooth_bump_covering.exists_is_subordinate`. This covering can be used, e.g., to construct a partition of unity and to prove the weak Whitney embedding theorem. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure smooth_bump_covering (s : set M := univ) := (c : ι → M) (to_fun : Π i, smooth_bump_function I (c i)) diff --git a/formal/lean/mathlib/geometry/manifold/smooth_manifold_with_corners.lean b/formal/lean/mathlib/geometry/manifold/smooth_manifold_with_corners.lean index abc1771348f2c782a6bfd78d56731e7363856910..7125c5da404aabcb3e42160c7620805074e8c0eb 100644 --- a/formal/lean/mathlib/geometry/manifold/smooth_manifold_with_corners.lean +++ b/formal/lean/mathlib/geometry/manifold/smooth_manifold_with_corners.lean @@ -113,19 +113,18 @@ noncomputable theory universes u v w u' v' w' -open set filter +open set filter function open_locale manifold filter topological_space localized "notation `∞` := (⊤ : with_top ℕ)" in manifold -section model_with_corners /-! ### Models with corners. -/ /-- A structure containing informations on the way a space `H` embeds in a model vector space `E` over the field `𝕜`. This is all what is needed to define a smooth manifold with model space `H`, and model vector space `E`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure model_with_corners (𝕜 : Type*) [nontrivially_normed_field 𝕜] (E : Type*) [normed_add_comm_group E] [normed_space 𝕜 E] (H : Type*) [topological_space H] extends local_equiv H E := @@ -213,7 +212,10 @@ protected lemma unique_diff : unique_diff_on 𝕜 (range I) := I.target_eq ▸ I @[simp, mfld_simps] protected lemma left_inv (x : H) : I.symm (I x) = x := by { refine I.left_inv' _, simp } -protected lemma left_inverse : function.left_inverse I.symm I := I.left_inv +protected lemma left_inverse : left_inverse I.symm I := I.left_inv + +lemma injective : injective I := +I.left_inverse.injective @[simp, mfld_simps] lemma symm_comp_self : I.symm ∘ I = id := I.left_inverse.comp_eq_id @@ -542,8 +544,6 @@ end end cont_diff_groupoid -end model_with_corners - section smooth_manifold_with_corners /-! ### Smooth manifolds with corners -/ @@ -603,9 +603,9 @@ def maximal_atlas := (cont_diff_groupoid ∞ I).maximal_atlas M variable {M} -lemma mem_maximal_atlas_of_mem_atlas [smooth_manifold_with_corners I M] - {e : local_homeomorph M H} (he : e ∈ atlas H M) : e ∈ maximal_atlas I M := -structure_groupoid.mem_maximal_atlas_of_mem_atlas _ he +lemma subset_maximal_atlas [smooth_manifold_with_corners I M] : + atlas H M ⊆ maximal_atlas I M := +structure_groupoid.subset_maximal_atlas _ lemma chart_mem_maximal_atlas [smooth_manifold_with_corners I M] (x : M) : chart_at H x ∈ maximal_atlas I M := @@ -708,10 +708,23 @@ by { rw ext_chart_at_source, exact (chart_at H x).open_source } lemma mem_ext_chart_source : x ∈ (ext_chart_at I x).source := by simp only [ext_chart_at_source, mem_chart_source] +lemma ext_chart_at_target (x : M) : (ext_chart_at I x).target = + I.symm ⁻¹' (chart_at H x).target ∩ range I := +by simp_rw [ext_chart_at, local_equiv.trans_target, I.target_eq, I.to_local_equiv_coe_symm, + inter_comm] + lemma ext_chart_at_to_inv : (ext_chart_at I x).symm ((ext_chart_at I x) x) = x := (ext_chart_at I x).left_inv (mem_ext_chart_source I x) +lemma maps_to_ext_chart_at (hs : s ⊆ (chart_at H x).source) : + maps_to (ext_chart_at I x) s ((ext_chart_at I x).symm ⁻¹' s ∩ range I) := +begin + rw [maps_to', ext_chart_at_coe, ext_chart_at_coe_symm, preimage_comp, ← I.image_eq, image_comp, + (chart_at H x).image_eq_target_inter_inv_preimage hs], + exact image_subset _ (inter_subset_right _ _) +end + lemma ext_chart_at_source_mem_nhds' {x' : M} (h : x' ∈ (ext_chart_at I x).source) : (ext_chart_at I x).source ∈ 𝓝 x' := is_open.mem_nhds (ext_chart_at_open_source I x) h @@ -873,6 +886,13 @@ lemma ext_chart_preimage_mem_nhds_within (ht : t ∈ 𝓝[s] x) : 𝓝[(ext_chart_at I x).symm ⁻¹' s ∩ range I] ((ext_chart_at I x) x) := ext_chart_preimage_mem_nhds_within' I x (mem_ext_chart_source I x) ht +lemma ext_chart_preimage_mem_nhds' {x' : M} (h : x' ∈ (ext_chart_at I x).source) (ht : t ∈ 𝓝 x') : + (ext_chart_at I x).symm ⁻¹' t ∈ 𝓝 (ext_chart_at I x x') := +begin + apply (ext_chart_continuous_at_symm' I x h).preimage_mem_nhds, + rwa (ext_chart_at I x).left_inv h +end + /-- Technical lemma ensuring that the preimage under an extended chart of a neighborhood of a point is a neighborhood of the preimage. -/ lemma ext_chart_preimage_mem_nhds (ht : t ∈ 𝓝 x) : @@ -889,11 +909,41 @@ lemma ext_chart_preimage_inter_eq : = ((ext_chart_at I x).symm ⁻¹' s ∩ range I) ∩ ((ext_chart_at I x).symm ⁻¹' t) := by mfld_set_tac -end extended_charts +/-! We use the name `ext_coord_change` for `(ext_chart_at I x').symm ≫ ext_chart_at I x`. -/ + +lemma ext_coord_change_source (x x' : M) : + ((ext_chart_at I x').symm ≫ ext_chart_at I x).source = + I '' ((chart_at H x').symm ≫ₕ (chart_at H x)).source := +by { simp_rw [local_equiv.trans_source, I.image_eq, ext_chart_at_source, local_equiv.symm_source, + ext_chart_at_target, inter_right_comm _ (range I)], refl } + +lemma cont_diff_on_ext_coord_change [smooth_manifold_with_corners I M] (x x' : M) : + cont_diff_on 𝕜 ⊤ (ext_chart_at I x ∘ (ext_chart_at I x').symm) + ((ext_chart_at I x').symm ≫ ext_chart_at I x).source := +by { rw [ext_coord_change_source, I.image_eq], exact (has_groupoid.compatible + (cont_diff_groupoid ⊤ I) (chart_mem_atlas H x') (chart_mem_atlas H x)).1 } + +lemma cont_diff_within_at_ext_coord_change [smooth_manifold_with_corners I M] (x x' : M) {y : E} + (hy : y ∈ ((ext_chart_at I x').symm ≫ ext_chart_at I x).source) : + cont_diff_within_at 𝕜 ⊤ (ext_chart_at I x ∘ (ext_chart_at I x').symm) (range I) y := +begin + apply (cont_diff_on_ext_coord_change I x x' y hy).mono_of_mem, + rw [ext_coord_change_source] at hy ⊢, + obtain ⟨z, hz, rfl⟩ := hy, + exact I.image_mem_nhds_within ((local_homeomorph.open_source _).mem_nhds hz) +end + +variable (𝕜) + +lemma ext_chart_self_eq {x : H} : ⇑(ext_chart_at I x) = I := rfl +lemma ext_chart_self_apply {x y : H} : ext_chart_at I x y = I y := rfl /-- In the case of the manifold structure on a vector space, the extended charts are just the identity.-/ -lemma ext_chart_model_space_eq_id (𝕜 : Type*) [nontrivially_normed_field 𝕜] - {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] (x : E) : - ext_chart_at (model_with_corners_self 𝕜 E) x = local_equiv.refl E := +lemma ext_chart_model_space_eq_id (x : E) : ext_chart_at 𝓘(𝕜, E) x = local_equiv.refl E := by simp only with mfld_simps + +lemma ext_chart_model_space_apply {x y : E} : ext_chart_at 𝓘(𝕜, E) x y = y := rfl + + +end extended_charts diff --git a/formal/lean/mathlib/geometry/manifold/tangent_bundle.lean b/formal/lean/mathlib/geometry/manifold/tangent_bundle.lean index a0771fdfc9d5d3627c52165fe77026599affe6e3..88b4ba20ef87a5c8fc65057c9e13134e047df879 100644 --- a/formal/lean/mathlib/geometry/manifold/tangent_bundle.lean +++ b/formal/lean/mathlib/geometry/manifold/tangent_bundle.lean @@ -134,13 +134,13 @@ end lemma coord_change_smooth (i j : atlas H M) : cont_diff_on 𝕜 ∞ (λ p : E × F, Z.coord_change i j (I.symm p.1) p.2) - ((I '' (i.1.symm.trans j.1).source) ×ˢ (univ : set F)) := + ((I '' (i.1.symm.trans j.1).source) ×ˢ univ) := begin have A : cont_diff 𝕜 ∞ (λ p : (F →L[𝕜] F) × F, p.1 p.2), { apply is_bounded_bilinear_map.cont_diff, exact is_bounded_bilinear_map_apply }, have B : cont_diff_on 𝕜 ∞ (λ (p : E × F), (Z.coord_change i j (I.symm p.1), p.snd)) - ((I '' (i.1.symm.trans j.1).source) ×ˢ (univ : set F)), + ((I '' (i.1.symm.trans j.1).source) ×ˢ univ), { apply cont_diff_on.prod _ _, { exact (Z.coord_change_smooth_clm i j).comp cont_diff_fst.cont_diff_on (prod_subset_preimage_fst _ _) }, @@ -153,7 +153,7 @@ end def to_topological_vector_bundle_core : topological_vector_bundle_core 𝕜 M F (atlas H M) := { base_set := λ i, i.1.source, is_open_base_set := λ i, i.1.open_source, - index_at := λ x, ⟨chart_at H x, chart_mem_atlas H x⟩, + index_at := achart H, mem_base_set_at := λ x, mem_chart_source H x, coord_change := λ i j x, Z.coord_change i j (i.1 x), coord_change_self := λ i x hx v, Z.coord_change_self i (i.1 x) (i.1.map_source hx) v, @@ -174,7 +174,7 @@ def to_topological_vector_bundle_core : topological_vector_bundle_core 𝕜 M F (Z.to_topological_vector_bundle_core.local_triv i).base_set = i.1.source := rfl @[simp, mfld_simps] lemma target (i : atlas H M) : - (Z.to_topological_vector_bundle_core.local_triv i).target = i.1.source ×ˢ (univ : set F) := rfl + (Z.to_topological_vector_bundle_core.local_triv i).target = i.1.source ×ˢ univ := rfl /-- Local chart for the total space of a basic smooth bundle -/ def chart {e : local_homeomorph M H} (he : e ∈ atlas H M) : @@ -187,7 +187,7 @@ def chart {e : local_homeomorph M H} (he : e ∈ atlas H M) : by { simp only [chart, mem_prod], mfld_set_tac } @[simp, mfld_simps] lemma chart_target (e : local_homeomorph M H) (he : e ∈ atlas H M) : - (Z.chart he).target = e.target ×ˢ (univ : set F) := + (Z.chart he).target = e.target ×ˢ univ := by { simp only [chart], mfld_set_tac } /-- The total space of a basic smooth bundle is endowed with a charted space structure, where the @@ -240,20 +240,20 @@ begin (J.symm ⁻¹' ((Z.chart he).symm.trans (Z.chart he')).source ∩ range J), { assume e e' he he', have : J.symm ⁻¹' ((chart Z he).symm.trans (chart Z he')).source ∩ range J = - (I.symm ⁻¹' (e.symm.trans e').source ∩ range I) ×ˢ (univ : set F), + (I.symm ⁻¹' (e.symm.trans e').source ∩ range I) ×ˢ univ, by { simp only [J, chart, model_with_corners.prod], mfld_set_tac }, rw this, -- check separately that the two components of the coordinate change are smooth apply cont_diff_on.prod, show cont_diff_on 𝕜 ∞ (λ (p : E × F), (I ∘ e' ∘ e.symm ∘ I.symm) p.1) - ((I.symm ⁻¹' (e.symm.trans e').source ∩ range I) ×ˢ (univ : set F)), + ((I.symm ⁻¹' (e.symm.trans e').source ∩ range I) ×ˢ univ), { -- the coordinate change on the base is just a coordinate change for `M`, smooth since -- `M` is smooth have A : cont_diff_on 𝕜 ∞ (I ∘ (e.symm.trans e') ∘ I.symm) (I.symm ⁻¹' (e.symm.trans e').source ∩ range I) := (has_groupoid.compatible (cont_diff_groupoid ∞ I) he he').1, have B : cont_diff_on 𝕜 ∞ (λ p : E × F, p.1) - ((I.symm ⁻¹' (e.symm.trans e').source ∩ range I) ×ˢ (univ : set F)) := + ((I.symm ⁻¹' (e.symm.trans e').source ∩ range I) ×ˢ univ) := cont_diff_fst.cont_diff_on, exact cont_diff_on.comp A B (prod_subset_preimage_fst _ _) }, show cont_diff_on 𝕜 ∞ (λ (p : E × F), @@ -261,7 +261,7 @@ begin ((chart_at H (e.symm (I.symm p.1)) : M → H) (e.symm (I.symm p.1))) (Z.coord_change ⟨e, he⟩ ⟨chart_at H (e.symm (I.symm p.1)), _⟩ (e (e.symm (I.symm p.1))) p.2)) - ((I.symm ⁻¹' (e.symm.trans e').source ∩ range I) ×ˢ (univ : set F)), + ((I.symm ⁻¹' (e.symm.trans e').source ∩ range I) ×ˢ univ), { /- The coordinate change in the fiber is more complicated as its definition involves the reference chart chosen at each point. However, it appears with its inverse, so using the cocycle property one can get rid of it, and then conclude using the smoothness of the @@ -311,7 +311,7 @@ fiber corresponds to the derivative of the coordinate change in `M`. -/ have C : cont_diff_on 𝕜 ∞ (λ (p : E × E), (fderiv_within 𝕜 (I ∘ j.1 ∘ i.1.symm ∘ I.symm) (I.symm ⁻¹' (i.1.symm.trans j.1).source ∩ range I) p.1 : E → E) p.2) - ((I.symm ⁻¹' (i.1.symm.trans j.1).source ∩ range I) ×ˢ (univ : set E)) := + ((I.symm ⁻¹' (i.1.symm.trans j.1).source ∩ range I) ×ˢ univ) := cont_diff_on_fderiv_within_apply A B le_top, have D : ∀ x ∈ (I.symm ⁻¹' (i.1.symm.trans j.1).source ∩ range I), fderiv_within 𝕜 (I ∘ j.1 ∘ i.1.symm ∘ I.symm) @@ -473,7 +473,7 @@ variable (M) /-- The tangent bundle to a smooth manifold, as a Sigma type. Defined in terms of `bundle.total_space` to be able to put a suitable topology on it. -/ -@[nolint has_inhabited_instance, reducible] -- is empty if the base manifold is empty +@[nolint has_nonempty_instance, reducible] -- is empty if the base manifold is empty def tangent_bundle := bundle.total_space (tangent_space I : M → Type*) local notation `TM` := tangent_bundle I M diff --git a/formal/lean/mathlib/group_theory/free_product.lean b/formal/lean/mathlib/group_theory/free_product.lean index beb73b8706d7e536732d26419de2bef174a68328..a20b6ee73241f4d2a915bf5cd2eee86b95afe91a 100644 --- a/formal/lean/mathlib/group_theory/free_product.lean +++ b/formal/lean/mathlib/group_theory/free_product.lean @@ -380,7 +380,7 @@ variable (M) /-- A `neword M i j` is a representation of a non-empty reduced words where the first letter comes from `M i` and the last letter comes from `M j`. It can be constructed from singletons and via concatentation, and thus provides a useful induction principle. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] inductive neword : ι → ι → Type (max u_1 u_2) | singleton : ∀ {i} (x : M i) (hne1 : x ≠ 1), neword i i | append : ∀ {i j k l} (w₁ : neword i j) (hne : j ≠ k) (w₂ : neword k l), neword i l diff --git a/formal/lean/mathlib/group_theory/group_action/conj_act.lean b/formal/lean/mathlib/group_theory/group_action/conj_act.lean index b371a48913d10c53916285e84d12ef3ff04110e2..bd48b96ae352736117ac82124c794e8d7bcc5c17 100644 --- a/formal/lean/mathlib/group_theory/group_action/conj_act.lean +++ b/formal/lean/mathlib/group_theory/group_action/conj_act.lean @@ -30,7 +30,7 @@ is that some theorems about the group actions will not apply when since this -/ -variables (M G G₀ R K : Type*) +variables (α M G G₀ R K : Type*) /-- A type alias for a group `G`. `conj_act G` acts on `G` by conjugation -/ def conj_act : Type* := G @@ -62,6 +62,9 @@ def to_conj_act : G ≃* conj_act G := of_conj_act.symm /-- A recursor for `conj_act`, for use as `induction x using conj_act.rec` when `x : conj_act G`. -/ protected def rec {C : conj_act G → Sort*} (h : Π g, C (to_conj_act g)) : Π g, C g := h +@[simp] lemma «forall» (p : conj_act G → Prop) : + (∀ (x : conj_act G), p x) ↔ ∀ x : G, p (to_conj_act x) := iff.rfl + @[simp] lemma of_mul_symm_eq : (@of_conj_act G _).symm = to_conj_act := rfl @[simp] lemma to_mul_symm_eq : (@to_conj_act G _).symm = of_conj_act := rfl @[simp] lemma to_conj_act_of_conj_act (x : conj_act G) : to_conj_act (of_conj_act x) = x := rfl @@ -80,9 +83,6 @@ instance : has_smul (conj_act G) G := lemma smul_def (g : conj_act G) (h : G) : g • h = of_conj_act g * h * (of_conj_act g)⁻¹ := rfl -@[simp] lemma «forall» (p : conj_act G → Prop) : - (∀ (x : conj_act G), p x) ↔ ∀ x : G, p (to_conj_act x) := iff.rfl - end div_inv_monoid section units @@ -103,6 +103,14 @@ instance units_mul_distrib_mul_action : mul_distrib_mul_action (conj_act Mˣ) M smul_mul := by simp [units_smul_def, mul_assoc], smul_one := by simp [units_smul_def], } +instance units_smul_comm_class [has_smul α M] [smul_comm_class α M M] [is_scalar_tower α M M] : + smul_comm_class α (conj_act Mˣ) M := +{ smul_comm := λ a um m, by rw [units_smul_def, units_smul_def, mul_smul_comm, smul_mul_assoc] } + +instance units_smul_comm_class' [has_smul α M] [smul_comm_class M α M] [is_scalar_tower α M M] : + smul_comm_class (conj_act Mˣ) α M := +by { haveI : smul_comm_class α M M := smul_comm_class.symm _ _ _, exact smul_comm_class.symm _ _ _ } + end monoid section semiring @@ -129,6 +137,14 @@ instance mul_action₀ : mul_action (conj_act G₀) G₀ := one_smul := by simp [smul_def], mul_smul := by simp [smul_def, mul_assoc, mul_inv_rev] } +instance smul_comm_class₀ [has_smul α G₀] [smul_comm_class α G₀ G₀] [is_scalar_tower α G₀ G₀] : + smul_comm_class α (conj_act G₀) G₀ := +{ smul_comm := λ a ug g, by rw [smul_def, smul_def, mul_smul_comm, smul_mul_assoc] } + +instance smul_comm_class₀' [has_smul α G₀] [smul_comm_class G₀ α G₀] [is_scalar_tower α G₀ G₀] : + smul_comm_class (conj_act G₀) α G₀ := +by { haveI := smul_comm_class.symm G₀ α G₀, exact smul_comm_class.symm _ _ _ } + end group_with_zero section division_ring @@ -151,6 +167,14 @@ instance : mul_distrib_mul_action (conj_act G) G := one_smul := by simp [smul_def], mul_smul := by simp [smul_def, mul_assoc] } +instance smul_comm_class [has_smul α G] [smul_comm_class α G G] [is_scalar_tower α G G] : + smul_comm_class α (conj_act G) G := +{ smul_comm := λ a ug g, by rw [smul_def, smul_def, mul_smul_comm, smul_mul_assoc] } + +instance smul_comm_class' [has_smul α G] [smul_comm_class G α G] [is_scalar_tower α G G] : + smul_comm_class (conj_act G) α G := +by { haveI := smul_comm_class.symm G α G, exact smul_comm_class.symm _ _ _ } + lemma smul_eq_mul_aut_conj (g : conj_act G) (h : G) : g • h = mul_aut.conj (of_conj_act g) h := rfl /-- The set of fixed points of the conjugation action of `G` on itself is the center of `G`. -/ diff --git a/formal/lean/mathlib/group_theory/group_action/defs.lean b/formal/lean/mathlib/group_theory/group_action/defs.lean index ed6da272ce2a1e831dfcc1749d43c55f39f082cb..c7d062d5ffbb70ba83693c9f8f131f1960d50225 100644 --- a/formal/lean/mathlib/group_theory/group_action/defs.lean +++ b/formal/lean/mathlib/group_theory/group_action/defs.lean @@ -69,7 +69,8 @@ lemma smul_left_injective' [has_smul M α] [has_faithful_smul M α] : λ m₁ m₂ h, has_faithful_smul.eq_of_smul_eq_smul (congr_fun h) /-- See also `monoid.to_mul_action` and `mul_zero_class.to_smul_with_zero`. -/ -@[priority 910, to_additive] -- see Note [lower instance priority] +@[priority 910, -- see Note [lower instance priority] +to_additive "See also `add_monoid.to_add_action`"] instance has_mul.to_has_smul (α : Type*) [has_mul α] : has_smul α α := ⟨(*)⟩ @[simp, to_additive] lemma smul_eq_mul (α : Type*) [has_mul α] {a a' : α} : a • a' = a * a' := rfl @@ -136,7 +137,8 @@ is_pretransitive.exists_smul_eq x y @[to_additive] lemma surjective_smul (x : α) : surjective (λ c : M, c • x) := exists_smul_eq M x /-- The regular action of a group on itself is transitive. -/ -@[to_additive] instance regular.is_pretransitive [group G] : is_pretransitive G G := +@[to_additive "The regular action of a group on itself is transitive."] +instance regular.is_pretransitive [group G] : is_pretransitive G G := ⟨λ x y, ⟨y * x⁻¹, inv_mul_cancel_right _ _⟩⟩ end mul_action @@ -188,17 +190,24 @@ add_decl_doc vadd_comm_class.symm smul_comm_class M M α := ⟨λ a a' b, by rw [← mul_smul, mul_comm, mul_smul]⟩ +/-- An instance of `vadd_assoc_class M N α` states that the additive action of `M` on `α` is +determined by the additive actions of `M` on `N` and `N` on `α`. -/ +class vadd_assoc_class (M N α : Type*) [has_vadd M N] [has_vadd N α] [has_vadd M α] : Prop := +(vadd_assoc : ∀ (x : M) (y : N) (z : α), (x +ᵥ y) +ᵥ z = x +ᵥ (y +ᵥ z)) + /-- An instance of `is_scalar_tower M N α` states that the multiplicative action of `M` on `α` is determined by the multiplicative actions of `M` on `N` and `N` on `α`. -/ +@[to_additive] class is_scalar_tower (M N α : Type*) [has_smul M N] [has_smul N α] [has_smul M α] : Prop := (smul_assoc : ∀ (x : M) (y : N) (z : α), (x • y) • z = x • (y • z)) -@[simp] lemma smul_assoc {M N} [has_smul M N] [has_smul N α] [has_smul M α] +@[simp, to_additive] lemma smul_assoc {M N} [has_smul M N] [has_smul N α] [has_smul M α] [is_scalar_tower M N α] (x : M) (y : N) (z : α) : (x • y) • z = x • y • z := is_scalar_tower.smul_assoc x y z +@[to_additive] instance semigroup.is_scalar_tower [semigroup α] : is_scalar_tower α α α := ⟨mul_assoc⟩ /-- A typeclass indicating that the right (aka `mul_opposite`) and left actions by `M` on `α` are @@ -268,9 +277,13 @@ tower of scalar actions `N → α → β`. This cannot be an instance because it can cause infinite loops whenever the `has_smul` arguments are still metavariables. -/ -@[priority 100] -lemma comp.is_scalar_tower [has_smul M β] [has_smul α β] [is_scalar_tower M α β] - (g : N → M) : +@[priority 100, to_additive "Given a tower of additive actions `M → α → β`, if we use +`has_smul.comp` to pull back both of `M`'s actions by a map `g : N → M`, then we obtain a new tower +of scalar actions `N → α → β`. + +This cannot be an instance because it can cause infinite loops whenever the `has_smul` arguments +are still metavariables."] +lemma comp.is_scalar_tower [has_smul M β] [has_smul α β] [is_scalar_tower M α β] (g : N → M) : (by haveI := comp α g; haveI := comp β g; exact is_scalar_tower N α β) := by exact {smul_assoc := λ n, @smul_assoc _ _ _ _ _ _ _ (g n) } @@ -298,17 +311,19 @@ section /-- Note that the `smul_comm_class α β β` typeclass argument is usually satisfied by `algebra α β`. -/ -@[to_additive] +@[to_additive, nolint to_additive_doc] lemma mul_smul_comm [has_mul β] [has_smul α β] [smul_comm_class α β β] (s : α) (x y : β) : x * (s • y) = s • (x * y) := (smul_comm s x y).symm /-- Note that the `is_scalar_tower α β β` typeclass argument is usually satisfied by `algebra α β`. -/ +@[to_additive, nolint to_additive_doc] lemma smul_mul_assoc [has_mul β] [has_smul α β] [is_scalar_tower α β β] (r : α) (x y : β) : (r • x) * y = r • (x * y) := smul_assoc r x y +@[to_additive] lemma smul_smul_smul_comm [has_smul α β] [has_smul α γ] [has_smul β δ] [has_smul α δ] [has_smul γ δ] [is_scalar_tower α β δ] [is_scalar_tower α γ δ] [smul_comm_class β γ δ] (a : α) (b : β) (c : γ) (d : δ) : (a • b) • (c • d) = (a • c) • b • d := @@ -316,11 +331,13 @@ by { rw [smul_assoc, smul_assoc, smul_comm b], apply_instance } variables [has_smul M α] +@[to_additive] lemma commute.smul_right [has_mul α] [smul_comm_class M α α] [is_scalar_tower M α α] {a b : α} (h : commute a b) (r : M) : commute a (r • b) := (mul_smul_comm _ _ _).trans ((congr_arg _ h).trans $ (smul_mul_assoc _ _ _).symm) +@[to_additive] lemma commute.smul_left [has_mul α] [smul_comm_class M α α] [is_scalar_tower M α α] {a b : α} (h : commute a b) (r : M) : commute (r • a) b := @@ -349,11 +366,11 @@ variable (M) @[simp, to_additive] theorem one_smul (b : α) : (1 : M) • b = b := mul_action.one_smul _ /-- `has_smul` version of `one_mul_eq_id` -/ -@[to_additive] +@[to_additive "`has_vadd` version of `zero_add_eq_id`"] lemma one_smul_eq_id : ((•) (1 : M) : α → α) = id := funext $ one_smul _ /-- `has_smul` version of `comp_mul_left` -/ -@[to_additive] +@[to_additive "`has_vadd` version of `comp_add_left`"] lemma comp_smul_left (a₁ a₂ : M) : (•) a₁ ∘ (•) a₂ = ((•) (a₁ * a₂) : α → α) := funext $ λ _, (mul_smul _ _ _).symm @@ -411,13 +428,14 @@ instance monoid.to_mul_action : mul_action M M := This is promoted to an `add_torsor` by `add_group_is_add_torsor`. -/ add_decl_doc add_monoid.to_add_action -instance is_scalar_tower.left : is_scalar_tower M M α := +@[to_additive] instance is_scalar_tower.left : is_scalar_tower M M α := ⟨λ x y z, mul_smul x y z⟩ variables {M} /-- Note that the `is_scalar_tower M α α` and `smul_comm_class M α α` typeclass arguments are usually satisfied by `algebra M α`. -/ +@[to_additive, nolint to_additive_doc] lemma smul_mul_smul [has_mul α] (r s : M) (x y : α) [is_scalar_tower M α α] [smul_comm_class M α α] : (r • x) * (s • y) = (r * s) • (x * y) := @@ -465,13 +483,13 @@ end section compatible_scalar -@[simp] lemma smul_one_smul {M} (N) [monoid N] [has_smul M N] [mul_action N α] [has_smul M α] - [is_scalar_tower M N α] (x : M) (y : α) : +@[simp, to_additive] lemma smul_one_smul {M} (N) [monoid N] [has_smul M N] [mul_action N α] + [has_smul M α] [is_scalar_tower M N α] (x : M) (y : α) : (x • (1 : N)) • y = x • y := by rw [smul_assoc, one_smul] -@[simp] lemma smul_one_mul {M N} [mul_one_class N] [has_smul M N] [is_scalar_tower M N N] (x : M) - (y : N) : (x • 1) * y = x • y := +@[simp, to_additive] lemma smul_one_mul {M N} [mul_one_class N] [has_smul M N] + [is_scalar_tower M N N] (x : M) (y : N) : (x • 1) * y = x • y := by rw [smul_mul_assoc, one_mul] @[simp, to_additive] lemma mul_smul_one @@ -479,6 +497,7 @@ by rw [smul_mul_assoc, one_mul] y * (x • 1) = x • y := by rw [← smul_eq_mul, ← smul_comm, smul_eq_mul, mul_one] +@[to_additive] lemma is_scalar_tower.of_smul_one_mul {M N} [monoid N] [has_smul M N] (h : ∀ (x : M) (y : N), (x • (1 : N)) * y = x • y) : is_scalar_tower M N N := diff --git a/formal/lean/mathlib/group_theory/monoid_localization.lean b/formal/lean/mathlib/group_theory/monoid_localization.lean index 3b6394173eb0e339d22f11dd2da7cfa9313bd345..b797ad903101ecf70610b95fff17954b1974c393 100644 --- a/formal/lean/mathlib/group_theory/monoid_localization.lean +++ b/formal/lean/mathlib/group_theory/monoid_localization.lean @@ -68,7 +68,7 @@ variables {M : Type*} [add_comm_monoid M] (S : add_submonoid M) (N : Type*) [add /-- The type of add_monoid homomorphisms satisfying the characteristic predicate: if `f : M →+ N` satisfies this predicate, then `N` is isomorphic to the localization of `M` at `S`. -/ -@[nolint has_inhabited_instance] structure localization_map +@[nolint has_nonempty_instance] structure localization_map extends add_monoid_hom M N := (map_add_units' : ∀ y : S, is_add_unit (to_fun y)) (surj' : ∀ z : N, ∃ x : M × S, z + to_fun x.2 = to_fun x.1) @@ -88,7 +88,7 @@ namespace submonoid /-- The type of monoid homomorphisms satisfying the characteristic predicate: if `f : M →* N` satisfies this predicate, then `N` is isomorphic to the localization of `M` at `S`. -/ -@[nolint has_inhabited_instance] structure localization_map +@[nolint has_nonempty_instance] structure localization_map extends monoid_hom M N := (map_units' : ∀ y : S, is_unit (to_fun y)) (surj' : ∀ z : N, ∃ x : M × S, z * to_fun x.2 = to_fun x.1) @@ -1334,7 +1334,7 @@ namespace submonoid /-- The type of homomorphisms between monoids with zero satisfying the characteristic predicate: if `f : M →*₀ N` satisfies this predicate, then `N` is isomorphic to the localization of `M` at `S`. -/ -@[nolint has_inhabited_instance] structure localization_with_zero_map +@[nolint has_nonempty_instance] structure localization_with_zero_map extends localization_map S N := (map_zero' : to_fun 0 = 0) diff --git a/formal/lean/mathlib/group_theory/nielsen_schreier.lean b/formal/lean/mathlib/group_theory/nielsen_schreier.lean index 6d863349658e4bbb7a57bff86e3143951856a4d3..c36283d3646de7bb2509c0c55f6777a4a9e9ad28 100644 --- a/formal/lean/mathlib/group_theory/nielsen_schreier.lean +++ b/formal/lean/mathlib/group_theory/nielsen_schreier.lean @@ -56,7 +56,7 @@ open category_theory category_theory.action_category category_theory.single_obj /-- `is_free_groupoid.generators G` is a type synonym for `G`. We think of this as the vertices of the generating quiver of `G` when `G` is free. We can't use `G` directly, since `G` already has a quiver instance from being a groupoid. -/ -@[nolint unused_arguments has_inhabited_instance] +@[nolint unused_arguments has_nonempty_instance] def is_free_groupoid.generators (G) [groupoid G] := G /-- A groupoid `G` is free when we have the following data: diff --git a/formal/lean/mathlib/group_theory/schreier.lean b/formal/lean/mathlib/group_theory/schreier.lean index f55c3633a087bc41deca061746e1de43e22d75b3..c787ff596e72fd6b0c0b56299435e71977b817eb 100644 --- a/formal/lean/mathlib/group_theory/schreier.lean +++ b/formal/lean/mathlib/group_theory/schreier.lean @@ -115,7 +115,7 @@ begin replace hR1 : (1 : G) ∈ R := by rwa set.mem_to_finset, refine ⟨_, _, closure_mul_image_eq_top' hR hR1 hS⟩, calc _ ≤ (R * S).card : finset.card_image_le - ... ≤ (R.product S).card : finset.card_image_le + ... ≤ (R ×ˢ S).card : finset.card_image_le ... = R.card * S.card : R.card_product S ... = H.index * S.card : congr_arg (* S.card) _, calc R.card = fintype.card R : (fintype.card_coe R).symm diff --git a/formal/lean/mathlib/group_theory/subgroup/basic.lean b/formal/lean/mathlib/group_theory/subgroup/basic.lean index fa30d218469e8d8be59695bc5eafbd18a05a4cdc..97174d54c780b3efe68eee80413860d6c565a877 100644 --- a/formal/lean/mathlib/group_theory/subgroup/basic.lean +++ b/formal/lean/mathlib/group_theory/subgroup/basic.lean @@ -1312,8 +1312,7 @@ def prod (H : subgroup G) (K : subgroup N) : subgroup (G × N) := .. submonoid.prod H.to_submonoid K.to_submonoid} @[to_additive coe_prod] -lemma coe_prod (H : subgroup G) (K : subgroup N) : - (H.prod K : set (G × N)) = (H : set G) ×ˢ (K : set N) := rfl +lemma coe_prod (H : subgroup G) (K : subgroup N) : (H.prod K : set (G × N)) = H ×ˢ K := rfl @[to_additive mem_prod] lemma mem_prod {H : subgroup G} {K : subgroup N} {p : G × N} : diff --git a/formal/lean/mathlib/group_theory/submonoid/operations.lean b/formal/lean/mathlib/group_theory/submonoid/operations.lean index 6516c992520255cd354c20483b3242ec1ae613ae..e0d307a9f5b8934cd287ce6f189a67d37bf9b3bd 100644 --- a/formal/lean/mathlib/group_theory/submonoid/operations.lean +++ b/formal/lean/mathlib/group_theory/submonoid/operations.lean @@ -621,14 +621,12 @@ of `M × N`. -/ @[to_additive prod "Given `add_submonoid`s `s`, `t` of `add_monoid`s `A`, `B` respectively, `s × t` as an `add_submonoid` of `A × B`."] def prod (s : submonoid M) (t : submonoid N) : submonoid (M × N) := -{ carrier := (s : set M) ×ˢ (t : set N), +{ carrier := s ×ˢ t, one_mem' := ⟨s.one_mem, t.one_mem⟩, mul_mem' := λ p q hp hq, ⟨s.mul_mem hp.1 hq.1, t.mul_mem hp.2 hq.2⟩ } @[to_additive coe_prod] -lemma coe_prod (s : submonoid M) (t : submonoid N) : - (s.prod t : set (M × N)) = (s : set M) ×ˢ (t : set N) := -rfl +lemma coe_prod (s : submonoid M) (t : submonoid N) : (s.prod t : set (M × N)) = s ×ˢ t := rfl @[to_additive mem_prod] lemma mem_prod {s : submonoid M} {t : submonoid N} {p : M × N} : diff --git a/formal/lean/mathlib/group_theory/subsemigroup/operations.lean b/formal/lean/mathlib/group_theory/subsemigroup/operations.lean index c975e1cb6002fb7033efbb1d3558221a5fbc9bcb..f7d869856605f1731ed082727c73abcb4506ee0c 100644 --- a/formal/lean/mathlib/group_theory/subsemigroup/operations.lean +++ b/formal/lean/mathlib/group_theory/subsemigroup/operations.lean @@ -470,13 +470,11 @@ of `M × N`. -/ @[to_additive prod "Given `add_subsemigroup`s `s`, `t` of `add_semigroup`s `A`, `B` respectively, `s × t` as an `add_subsemigroup` of `A × B`."] def prod (s : subsemigroup M) (t : subsemigroup N) : subsemigroup (M × N) := -{ carrier := (s : set M) ×ˢ (t : set N), +{ carrier := s ×ˢ t, mul_mem' := λ p q hp hq, ⟨s.mul_mem hp.1 hq.1, t.mul_mem hp.2 hq.2⟩ } @[to_additive coe_prod] -lemma coe_prod (s : subsemigroup M) (t : subsemigroup N) : - (s.prod t : set (M × N)) = (s : set M) ×ˢ (t : set N) := -rfl +lemma coe_prod (s : subsemigroup M) (t : subsemigroup N) : (s.prod t : set (M × N)) = s ×ˢ t := rfl @[to_additive mem_prod] lemma mem_prod {s : subsemigroup M} {t : subsemigroup N} {p : M × N} : diff --git a/formal/lean/mathlib/linear_algebra/affine_space/affine_equiv.lean b/formal/lean/mathlib/linear_algebra/affine_space/affine_equiv.lean index 7177d57104ed53a133e11f72824738e50aa251a4..646f3b099e4444dc43fa9a9aa509e2cf1f060a38 100644 --- a/formal/lean/mathlib/linear_algebra/affine_space/affine_equiv.lean +++ b/formal/lean/mathlib/linear_algebra/affine_space/affine_equiv.lean @@ -39,7 +39,7 @@ and inverse maps are affine. We define it using an `equiv` for the map and a `linear_equiv` for the linear part in order to allow affine equivalences with good definitional equalities. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure affine_equiv (k P₁ P₂ : Type*) {V₁ V₂ : Type*} [ring k] [add_comm_group V₁] [module k V₁] [add_torsor V₁ P₁] [add_comm_group V₂] [module k V₂] [add_torsor V₂ P₂] extends P₁ ≃ P₂ := diff --git a/formal/lean/mathlib/linear_algebra/alternating.lean b/formal/lean/mathlib/linear_algebra/alternating.lean index 14d0b96d0024b48bc6d57bac38fc7002dc2a84d4..62ac65c6d40276870da325727f543a83c268a346 100644 --- a/formal/lean/mathlib/linear_algebra/alternating.lean +++ b/formal/lean/mathlib/linear_algebra/alternating.lean @@ -808,7 +808,7 @@ Here, we generalize this by replacing: * the additions in the subscripts of $\sigma$ with an index of type `sum` The specialized version can be obtained by combining this definition with `fin_sum_fin_equiv` and -`algebra.lmul'`. +`linear_map.mul'`. -/ @[simps] def dom_coprod diff --git a/formal/lean/mathlib/linear_algebra/basic.lean b/formal/lean/mathlib/linear_algebra/basic.lean index 66a4104bbc7f5145a2e4abf3dbeab1d3f8953207..94c9ca5c8599084ad07b0a347f87eb63a2e830e4 100644 --- a/formal/lean/mathlib/linear_algebra/basic.lean +++ b/formal/lean/mathlib/linear_algebra/basic.lean @@ -117,6 +117,22 @@ end (linear_equiv_fun_on_fintype R M α).symm f = f := by { ext, simp [linear_equiv_fun_on_fintype], } +/-- If `α` has a unique term, then the type of finitely supported functions `α →₀ M` is +`R`-linearly equivalent to `M`. -/ +noncomputable def linear_equiv.finsupp_unique (α : Type*) [unique α] : (α →₀ M) ≃ₗ[R] M := +{ map_add' := λ x y, rfl, + map_smul' := λ r x, rfl, + ..finsupp.equiv_fun_on_fintype.trans (equiv.fun_unique α M) } + +variables {R M α} + +@[simp] lemma linear_equiv.finsupp_unique_apply (α : Type*) [unique α] (f : α →₀ M) : + linear_equiv.finsupp_unique R M α f = f default := rfl + +@[simp] lemma linear_equiv.finsupp_unique_symm_apply {α : Type*} [unique α] (m : M) : + (linear_equiv.finsupp_unique R M α).symm m = finsupp.single default m := +by ext; simp [linear_equiv.finsupp_unique] + end finsupp /-- decomposing `x : ι → R` as a sum along the canonical basis -/ diff --git a/formal/lean/mathlib/linear_algebra/clifford_algebra/basic.lean b/formal/lean/mathlib/linear_algebra/clifford_algebra/basic.lean index 11cba46a02f1fa14ea322f37ccd40acec5e0c8dd..25c90d1484e0686aea0bd990c0fd273add50343d 100644 --- a/formal/lean/mathlib/linear_algebra/clifford_algebra/basic.lean +++ b/formal/lean/mathlib/linear_algebra/clifford_algebra/basic.lean @@ -16,12 +16,12 @@ a quadratic_form `Q`. ## Notation -The Clifford algebra of the `R`-module `M` equipped with a quadratic_form `Q` is denoted as -`clifford_algebra Q`. +The Clifford algebra of the `R`-module `M` equipped with a quadratic_form `Q` is +an `R`-algebra denoted `clifford_algebra Q`. Given a linear morphism `f : M → A` from a module `M` to another `R`-algebra `A`, such that `cond : ∀ m, f m * f m = algebra_map _ _ (Q m)`, there is a (unique) lift of `f` to an `R`-algebra -morphism, which is denoted `clifford_algebra.lift Q f cond`. +morphism from `clifford_algebra Q` to `A`, which is denoted `clifford_algebra.lift Q f cond`. The canonical linear map `M → clifford_algebra Q` is denoted `clifford_algebra.ι Q`. diff --git a/formal/lean/mathlib/linear_algebra/clifford_algebra/equivs.lean b/formal/lean/mathlib/linear_algebra/clifford_algebra/equivs.lean index 97143343bc02f58cebd598788fae6b38ab41df42..a7a3bbbcb4dc314f7bde4f7fbeb36d6df237d484 100644 --- a/formal/lean/mathlib/linear_algebra/clifford_algebra/equivs.lean +++ b/formal/lean/mathlib/linear_algebra/clifford_algebra/equivs.lean @@ -3,10 +3,11 @@ Copyright (c) 2021 Eric Wieser. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Eric Wieser -/ +import algebra.dual_number import algebra.quaternion_basis import data.complex.module import linear_algebra.clifford_algebra.conjugation -import algebra.dual_number +import linear_algebra.clifford_algebra.star import linear_algebra.quadratic_form.prod /-! @@ -46,7 +47,7 @@ is the same as `clifford_algebra.involute`. We show additionally that this equivalence sends `quaternion_algebra.conj` to the clifford conjugate and vice-versa: -* `clifford_algebra_quaternion.to_quaternion_involute_reverse` +* `clifford_algebra_quaternion.to_quaternion_star` * `clifford_algebra_quaternion.of_quaternion_conj` ## Dual numbers @@ -270,11 +271,11 @@ lemma to_quaternion_ι (v : R × R) : to_quaternion (ι (Q c₁ c₂) v) = (⟨0, v.1, v.2, 0⟩ : ℍ[R,c₁,c₂]) := clifford_algebra.lift_ι_apply _ _ v -/-- The "clifford conjugate" (aka `involute ∘ reverse = reverse ∘ involute`) maps to the quaternion -conjugate. -/ -lemma to_quaternion_involute_reverse (c : clifford_algebra (Q c₁ c₂)) : - to_quaternion (involute (reverse c)) = quaternion_algebra.conj (to_quaternion c) := +/-- The "clifford conjugate" maps to the quaternion conjugate. -/ +lemma to_quaternion_star (c : clifford_algebra (Q c₁ c₂)) : + to_quaternion (star c) = quaternion_algebra.conj (to_quaternion c) := begin + simp only [clifford_algebra.star_def'], induction c using clifford_algebra.induction, case h_grade0 : r { simp only [reverse.commutes, alg_hom.commutes, quaternion_algebra.coe_algebra_map, @@ -337,12 +338,11 @@ alg_equiv.of_alg_hom to_quaternion of_quaternion to_quaternion_comp_of_quaternion of_quaternion_comp_to_quaternion -/-- The quaternion conjugate maps to the "clifford conjugate" (aka -`involute ∘ reverse = reverse ∘ involute`). -/ +/-- The quaternion conjugate maps to the "clifford conjugate" (aka `star`). -/ @[simp] lemma of_quaternion_conj (q : ℍ[R,c₁,c₂]) : - of_quaternion (q.conj) = (of_quaternion q).reverse.involute := + of_quaternion (q.conj) = star (of_quaternion q) := clifford_algebra_quaternion.equiv.injective $ - by rw [equiv_apply, equiv_apply, to_quaternion_involute_reverse, to_quaternion_of_quaternion, + by rw [equiv_apply, equiv_apply, to_quaternion_star, to_quaternion_of_quaternion, to_quaternion_of_quaternion] -- this name is too short for us to want it visible after `open clifford_algebra_quaternion` diff --git a/formal/lean/mathlib/linear_algebra/clifford_algebra/even.lean b/formal/lean/mathlib/linear_algebra/clifford_algebra/even.lean index 665fe2a12f5477c812a85aa3139aa7110d14bf85..e977571460879a33570a4a97087c06e615d7a1e0 100644 --- a/formal/lean/mathlib/linear_algebra/clifford_algebra/even.lean +++ b/formal/lean/mathlib/linear_algebra/clifford_algebra/even.lean @@ -120,7 +120,8 @@ namespace even.lift /-- An auxiliary submodule used to store the half-applied values of `f`. This is the span of elements `f'` such that `∃ x m₂, ∀ m₁, f' m₁ = f m₁ m₂ * x`. -/ private def S : submodule R (M →ₗ[R] A) := -submodule.span R {f' | ∃ x m₂, f' = linear_map.lcomp R _ (f.bilin.flip m₂) (algebra.lmul_right R x)} +submodule.span R + {f' | ∃ x m₂, f' = linear_map.lcomp R _ (f.bilin.flip m₂) (linear_map.mul_right R x)} /-- An auxiliary bilinear map that is later passed into `clifford_algebra.fold`. Our desired result is stored in the `A` part of the accumulator, while auxiliary recursion state is stored in the `S f` @@ -134,7 +135,7 @@ linear_map.mk₂ R (λ m acc, (linear_map.llcomp R M A A).flip.comp f.flip : M →ₗ[R] A →ₗ[R] M →ₗ[R] A) ``` -/ - (acc.2 m, ⟨(algebra.lmul_right R acc.1).comp (f.bilin.flip m), + (acc.2 m, ⟨(linear_map.mul_right R acc.1).comp (f.bilin.flip m), submodule.subset_span $ ⟨_, _, rfl⟩⟩)) (λ m₁ m₂ a, prod.ext (linear_map.map_add _ m₁ m₂) diff --git a/formal/lean/mathlib/linear_algebra/clifford_algebra/even_equiv.lean b/formal/lean/mathlib/linear_algebra/clifford_algebra/even_equiv.lean index b9ec5de35dad80895826d835f26780cec58a83eb..c16e72cfe02b9dc339538eef332ac46665ddc85e 100644 --- a/formal/lean/mathlib/linear_algebra/clifford_algebra/even_equiv.lean +++ b/formal/lean/mathlib/linear_algebra/clifford_algebra/even_equiv.lean @@ -93,12 +93,12 @@ def to_even : clifford_algebra Q →ₐ[R] clifford_algebra.even (Q' Q) := begin refine clifford_algebra.lift Q ⟨_, λ m, _⟩, { refine linear_map.cod_restrict _ _ (λ m, submodule.mem_supr_of_mem ⟨2, rfl⟩ _), - exact (algebra.lmul_left R $ e0 Q).comp (v Q), + exact (linear_map.mul_left R $ e0 Q).comp (v Q), rw [subtype.coe_mk, pow_two], exact submodule.mul_mem_mul (linear_map.mem_range_self _ _) (linear_map.mem_range_self _ _), }, { ext1, dsimp only [subalgebra.coe_mul, linear_map.cod_restrict_apply, linear_map.comp_apply, - algebra.lmul_left_apply, linear_map.inl_apply, subalgebra.coe_algebra_map], + linear_map.mul_left_apply, linear_map.inl_apply, subalgebra.coe_algebra_map], rw [←mul_assoc, e0_mul_v_mul_e0, v_sq_scalar] } end diff --git a/formal/lean/mathlib/linear_algebra/clifford_algebra/star.lean b/formal/lean/mathlib/linear_algebra/clifford_algebra/star.lean new file mode 100644 index 0000000000000000000000000000000000000000..9c340fcbcb04f27cba593af0e99d686c7f616833 --- /dev/null +++ b/formal/lean/mathlib/linear_algebra/clifford_algebra/star.lean @@ -0,0 +1,55 @@ +/- +Copyright (c) 2022 Eric Wieser. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Eric Wieser +-/ +import linear_algebra.clifford_algebra.conjugation + +/-! +# Star structure on `clifford_algebra` + +This file defines the "clifford conjugation", equal to `reverse (involute x)`, and assigns it the +`star` notation. + +This choice is somewhat non-canonical; a star structure is also possible under `reverse` alone. +However, defining it gives us access to constructions like `unitary`. + +Most results about `star` can be obtained by unfolding it via `clifford_algebra.star_def`. + +## Main definitions + +* `clifford_algebra.star_ring` + +-/ + +variables {R : Type*} [comm_ring R] +variables {M : Type*} [add_comm_group M] [module R M] +variables {Q : quadratic_form R M} + +namespace clifford_algebra + +instance : star_ring (clifford_algebra Q) := +{ star := λ x, reverse (involute x), + star_involutive := λ x, + by simp only [reverse_involute_commute.eq, reverse_reverse, involute_involute], + star_mul := λ x y, by simp only [map_mul, reverse.map_mul], + star_add := λ x y, by simp only [map_add] } + +lemma star_def (x : clifford_algebra Q) : star x = reverse (involute x) := rfl +lemma star_def' (x : clifford_algebra Q) : star x = involute (reverse x) := reverse_involute _ + +@[simp] lemma star_ι (m : M) : star (ι Q m) = -ι Q m := +by rw [star_def, involute_ι, map_neg, reverse_ι] + +/-- Note that this not match the `star_smul` implied by `star_module`; it certainly could if we +also conjugated all the scalars, but there appears to be nothing in the literature that advocates +doing this. -/ +@[simp] lemma star_smul (r : R) (x : clifford_algebra Q) : + star (r • x) = r • star x := +by rw [star_def, star_def, map_smul, map_smul] + +@[simp] lemma star_algebra_map (r : R) : + star (algebra_map R (clifford_algebra Q) r) = algebra_map R (clifford_algebra Q) r := +by rw [star_def, involute.commutes, reverse.commutes] + +end clifford_algebra diff --git a/formal/lean/mathlib/linear_algebra/dual.lean b/formal/lean/mathlib/linear_algebra/dual.lean index 96ef0d5e4fdbd125ffc3c10425c110c5aaaf1bc0..854ecd539f6827f4c83b2d908c4e1c1b3f2b6404 100644 --- a/formal/lean/mathlib/linear_algebra/dual.lean +++ b/formal/lean/mathlib/linear_algebra/dual.lean @@ -354,7 +354,7 @@ variables {R M ι : Type*} variables [comm_semiring R] [add_comm_monoid M] [module R M] [decidable_eq ι] /-- `e` and `ε` have characteristic properties of a basis and its dual -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure dual_pair (e : ι → M) (ε : ι → (dual R M)) := (eval : ∀ i j : ι, ε i (e j) = if i = j then 1 else 0) (total : ∀ {m : M}, (∀ i, ε i m = 0) → m = 0) diff --git a/formal/lean/mathlib/linear_algebra/exterior_algebra/basic.lean b/formal/lean/mathlib/linear_algebra/exterior_algebra/basic.lean index faff4b56c2d68e1df2f461f52f42578dce518002..38a0b71a4e7ae5548ab66a1e3583d8e60f9d9be3 100644 --- a/formal/lean/mathlib/linear_algebra/exterior_algebra/basic.lean +++ b/formal/lean/mathlib/linear_algebra/exterior_algebra/basic.lean @@ -268,7 +268,7 @@ lemma ι_multi_apply {n : ℕ} (v : fin n → M) : lemma ι_multi_succ_curry_left {n : ℕ} (m : M) : (ι_multi R n.succ).curry_left m = - (algebra.lmul_left R (ι R m)).comp_alternating_map (ι_multi R n) := + (linear_map.mul_left R (ι R m)).comp_alternating_map (ι_multi R n) := alternating_map.ext $ λ v, (ι_multi_succ_apply _).trans $ begin simp_rw matrix.tail_cons, refl, diff --git a/formal/lean/mathlib/linear_algebra/exterior_algebra/of_alternating.lean b/formal/lean/mathlib/linear_algebra/exterior_algebra/of_alternating.lean index c3ea223fc6c831afb2649ce6e1f7ee0bbf29b445..2ad764e901130c0ba9490b93b79cdb238e76a6d6 100644 --- a/formal/lean/mathlib/linear_algebra/exterior_algebra/of_alternating.lean +++ b/formal/lean/mathlib/linear_algebra/exterior_algebra/of_alternating.lean @@ -137,7 +137,7 @@ begin { rw [lift_alternating_algebra_map, ι_multi_zero_apply, algebra.algebra_map_eq_smul_one] }, { rw [map_add, hx, hy] }, { simp_rw [lift_alternating_ι_mul, ι_multi_succ_curry_left, lift_alternating_comp, - linear_map.comp_apply, algebra.lmul_left_apply, hx] }, + linear_map.comp_apply, linear_map.mul_left_apply, hx] }, end /-- `exterior_algebra.lift_alternating` is an equivalence. -/ diff --git a/formal/lean/mathlib/linear_algebra/finite_dimensional.lean b/formal/lean/mathlib/linear_algebra/finite_dimensional.lean index 78a5476bd9dfb4d159250d59a534221c0167584b..c62ab32f3d842c8992a835f2b25f2b6c4ab01a55 100644 --- a/formal/lean/mathlib/linear_algebra/finite_dimensional.lean +++ b/formal/lean/mathlib/linear_algebra/finite_dimensional.lean @@ -1191,23 +1191,6 @@ linear_equiv.of_bijective f hf $ end linear_map -namespace alg_hom - -lemma bijective {F : Type*} [field F] {E : Type*} [field E] [algebra F E] - [finite_dimensional F E] (ϕ : E →ₐ[F] E) : function.bijective ϕ := -have inj : function.injective ϕ.to_linear_map := ϕ.to_ring_hom.injective, -⟨inj, (linear_map.injective_iff_surjective_of_finrank_eq_finrank rfl).mp inj⟩ - -end alg_hom - -/-- Bijection between algebra equivalences and algebra homomorphisms -/ -noncomputable def alg_equiv_equiv_alg_hom (F : Type u) [field F] (E : Type v) [field E] - [algebra F E] [finite_dimensional F E] : (E ≃ₐ[F] E) ≃ (E →ₐ[F] E) := -{ to_fun := λ ϕ, ϕ.to_alg_hom, - inv_fun := λ ϕ, alg_equiv.of_bijective ϕ ϕ.bijective, - left_inv := λ _, by {ext, refl}, - right_inv := λ _, by {ext, refl} } - section /-- A domain that is module-finite as an algebra over a field is a division ring. -/ @@ -1215,10 +1198,10 @@ noncomputable def division_ring_of_finite_dimensional (F K : Type*) [field F] [ring K] [is_domain K] [algebra F K] [finite_dimensional F K] : division_ring K := { inv := λ x, if H : x = 0 then 0 else classical.some $ - (show function.surjective (algebra.lmul_left F x), from + (show function.surjective (linear_map.mul_left F x), from linear_map.injective_iff_surjective.1 $ λ _ _, (mul_right_inj' H).1) 1, mul_inv_cancel := λ x hx, show x * dite _ _ _ = _, by { rw dif_neg hx, - exact classical.some_spec ((show function.surjective (algebra.lmul_left F x), from + exact classical.some_spec ((show function.surjective (linear_map.mul_left F x), from linear_map.injective_iff_surjective.1 $ λ _ _, (mul_right_inj' hx).1) 1) }, inv_zero := dif_pos rfl, .. ‹is_domain K›, @@ -1314,6 +1297,10 @@ lemma finrank_span_finset_le_card (s : finset V) : calc (s : set V).finrank K ≤ (s : set V).to_finset.card : finrank_span_le_card s ... = s.card : by simp +lemma finrank_range_le_card {ι : Type*} [fintype ι] {b : ι → V} : + (set.range b).finrank K ≤ fintype.card ι := +(finrank_span_le_card _).trans $ by { rw set.to_finset_range, exact finset.card_image_le } + lemma finrank_span_eq_card {ι : Type*} [fintype ι] {b : ι → V} (hb : linear_independent K b) : finrank K (span K (set.range b)) = fintype.card ι := @@ -1450,6 +1437,10 @@ begin convert (linear_independent_of_top_le_span_of_card_eq_finrank hs hc).map' _ hi } end +lemma linear_independent_iff_card_le_finrank_span {ι : Type*} [fintype ι] {b : ι → V} : + linear_independent K b ↔ fintype.card ι ≤ (set.range b).finrank K := +by rw [linear_independent_iff_card_eq_finrank_span, finrank_range_le_card.le_iff_eq] + /-- A family of `finrank K V` vectors forms a basis if they span the whole space. -/ noncomputable def basis_of_top_le_span_of_card_eq_finrank {ι : Type*} [fintype ι] (b : ι → V) (le_span : ⊤ ≤ span K (set.range b)) (card_eq : fintype.card ι = finrank K V) : diff --git a/formal/lean/mathlib/linear_algebra/finsupp_vector_space.lean b/formal/lean/mathlib/linear_algebra/finsupp_vector_space.lean index 7c5200df64ca3e8edd6014f268f3b4f6c12c2a9c..efc5670b9683f69230cf0c838d207403ca394bc4 100644 --- a/formal/lean/mathlib/linear_algebra/finsupp_vector_space.lean +++ b/formal/lean/mathlib/linear_algebra/finsupp_vector_space.lean @@ -57,6 +57,12 @@ begin apply range_comp_subset_range } } end +end ring + +section semiring +variables {R : Type*} {M : Type*} {ι : Type*} +variables [semiring R] [add_comm_monoid M] [module R M] + open linear_map submodule /-- The basis on `ι →₀ M` with basis vectors `λ ⟨i, x⟩, single i (b i x)`. -/ @@ -116,7 +122,7 @@ basis.of_repr (linear_equiv.refl _ _) (finsupp.basis_single_one : ι → (ι →₀ R)) = λ i, finsupp.single i 1 := funext $ λ i, basis.apply_eq_iff.mpr rfl -end ring +end semiring section dim variables {K : Type u} {V : Type v} {ι : Type v} diff --git a/formal/lean/mathlib/linear_algebra/free_module/basic.lean b/formal/lean/mathlib/linear_algebra/free_module/basic.lean index f90e7bc658b6a6aee1fd9eca658a6eb652da39ef..15a86b4839cb24d9ed9582f8e3b92a2f155fdb55 100644 --- a/formal/lean/mathlib/linear_algebra/free_module/basic.lean +++ b/formal/lean/mathlib/linear_algebra/free_module/basic.lean @@ -7,6 +7,7 @@ Authors: Riccardo Brasca import linear_algebra.direct_sum.finsupp import logic.small import linear_algebra.std_basis +import linear_algebra.finsupp_vector_space /-! @@ -25,7 +26,7 @@ Use `finsupp.total_id_surjective` to prove that any module is the quotient of a universes u v w z -variables (R : Type u) (M : Type v) (N : Type z) +variables {ι : Type*} (R : Type u) (M : Type v) (N : Type z) open_locale tensor_product direct_sum big_operators @@ -67,7 +68,7 @@ variables [add_comm_monoid N] [module R N] /-- If `module.free R M` then `choose_basis_index R M` is the `ι` which indexes the basis `ι → M`. -/ -@[nolint has_inhabited_instance] def choose_basis_index := (exists_basis R M).some.1 +@[nolint has_nonempty_instance] def choose_basis_index := (exists_basis R M).some.1 /-- If `module.free R M` then `choose_basis : ι → M` is the basis. Here `ι = choose_basis_index R M`. -/ @@ -91,15 +92,6 @@ noncomputable def constr {S : Type z} [semiring S] [module S N] [smul_comm_class instance no_zero_smul_divisors [no_zero_divisors R] : no_zero_smul_divisors R M := let ⟨⟨_, b⟩⟩ := exists_basis R M in b.no_zero_smul_divisors -/-- The product of finitely many free modules is free. -/ -instance pi {ι : Type*} [fintype ι] {M : ι → Type*} [Π (i : ι), add_comm_group (M i)] - [Π (i : ι), module R (M i)] [Π (i : ι), module.free R (M i)] : module.free R (Π i, M i) := -of_basis $ pi.basis $ λ i, choose_basis R (M i) - -/-- The module of finite matrices is free. -/ -instance matrix {m n : Type*} [fintype m] [fintype n] : module.free R (matrix m n R) := -of_basis $ matrix.std_basis R m n - variables {R M N} lemma of_equiv (e : M ≃ₗ[R] N) : module.free R N := @@ -113,16 +105,31 @@ of_equiv e variables (R M N) -instance {ι : Type v} : module.free R (ι →₀ R) := -of_basis (basis.of_repr (linear_equiv.refl _ _)) - -instance {ι : Type v} [fintype ι] : module.free R (ι → R) := -of_equiv (basis.of_repr $ linear_equiv.refl _ _).equiv_fun +/-- The module structure provided by `semiring.to_module` is free. -/ +instance self : module.free R R := of_basis (basis.singleton unit R) instance prod [module.free R N] : module.free R (M × N) := of_basis $ (choose_basis R M).prod (choose_basis R N) -instance self : module.free R R := of_basis $ basis.singleton unit R +/-- The product of finitely many free modules is free. -/ +instance pi (M : ι → Type*) [finite ι] [Π (i : ι), add_comm_monoid (M i)] + [Π (i : ι), module R (M i)] [Π (i : ι), module.free R (M i)] : module.free R (Π i, M i) := +let ⟨_⟩ := nonempty_fintype ι in by exactI (of_basis $ pi.basis $ λ i, choose_basis R (M i)) + +/-- The module of finite matrices is free. -/ +instance matrix {m n : Type*} [finite m] [finite n] : module.free R (matrix m n M) := +module.free.pi R _ + +variables (ι) + +/-- The product of finitely many free modules is free (non-dependent version to help with typeclass +search). -/ +instance function [finite ι] : module.free R (ι → M) := free.pi _ _ + +instance finsupp : module.free R (ι →₀ M) := +of_basis (finsupp.basis $ λ i, choose_basis R M) + +variables {ι} @[priority 100] instance of_subsingleton [subsingleton N] : module.free R N := @@ -148,7 +155,7 @@ variables [comm_ring R] [add_comm_group M] [module R M] [module.free R M] variables [add_comm_group N] [module R N] [module.free R N] instance tensor : module.free R (M ⊗[R] N) := -of_equiv' (of_equiv' (finsupp.free R) (finsupp_tensor_finsupp' R _ _).symm) +of_equiv' (of_equiv' (free.finsupp _ R _) (finsupp_tensor_finsupp' R _ _).symm) (tensor_product.congr (choose_basis R M).repr (choose_basis R N).repr).symm end comm_ring diff --git a/formal/lean/mathlib/linear_algebra/free_module/pid.lean b/formal/lean/mathlib/linear_algebra/free_module/pid.lean index 5f1726359916bcf4ba632d83a06a2aafb7cb0f21..73df194265e5949cf6a262f02271f0d78806a23f 100644 --- a/formal/lean/mathlib/linear_algebra/free_module/pid.lean +++ b/formal/lean/mathlib/linear_algebra/free_module/pid.lean @@ -405,7 +405,7 @@ section smith_normal /-- A Smith normal form basis for a submodule `N` of a module `M` consists of bases for `M` and `N` such that the inclusion map `N → M` can be written as a (rectangular) matrix with `a` along the diagonal: in Smith normal form. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure basis.smith_normal_form (N : submodule R M) (ι : Type*) (n : ℕ) := (bM : basis ι R M) (bN : basis (fin n) R N) diff --git a/formal/lean/mathlib/linear_algebra/free_module/strong_rank_condition.lean b/formal/lean/mathlib/linear_algebra/free_module/strong_rank_condition.lean index be1fc8420365359a16bfc0a0aa734d7ccff40b96..9b5a50c17bdff8acebc29a7adb61cd22489e8148 100644 --- a/formal/lean/mathlib/linear_algebra/free_module/strong_rank_condition.lean +++ b/formal/lean/mathlib/linear_algebra/free_module/strong_rank_condition.lean @@ -43,9 +43,10 @@ begin { rwa strong_rank_condition_iff_succ R }, intros n f, by_contradiction hf, - -- Lean is unable to find this instance without help, either via this `letI`, or via a duplicate - -- instance with unecessarily strong typeclasses on `R` and `M`. + -- Lean is unable to find these instances without help, either via this `letI`, or via duplicate + -- instances with unecessarily strong typeclasses on `R` and `M`. letI : module.finite R (fin n.succ → R) := module.finite.pi, + letI : module.free R (fin n.succ → R) := module.free.pi _ _, let g : (fin (n + 1) → R) →ₗ[R] fin (n + 1) → R := (extend_by_zero.linear_map R cast_succ).comp f, diff --git a/formal/lean/mathlib/linear_algebra/lagrange.lean b/formal/lean/mathlib/linear_algebra/lagrange.lean index fae4f3c345c356e7795551def79e85df7ca5b724..72cbe39d45cb514135fc7123a70be8cb9d3c01c8 100644 --- a/formal/lean/mathlib/linear_algebra/lagrange.lean +++ b/formal/lean/mathlib/linear_algebra/lagrange.lean @@ -1,224 +1,566 @@ /- Copyright (c) 2020 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. -Authors: Kenny Lau +Authors: Kenny Lau, Wrenna Robson -/ import algebra.big_operators.basic +import linear_algebra.vandermonde +import logic.lemmas import ring_theory.polynomial.basic /-! # Lagrange interpolation ## Main definitions - -* `lagrange.basis s x` where `s : finset F` and `x : F`: the Lagrange basis polynomial - that evaluates to `1` at `x` and `0` at other elements of `s`. -* `lagrange.interpolate s f` where `s : finset F` and `f : F → F`: the Lagrange interpolant - that evaluates to `f x` at `x` for `x ∈ s`. +* In everything that follows, `s : finset ι` is a finite set of indexes, with `v : ι → F` an +indexing of the field over some type. We call the image of v on s the interpolation nodes, +though strictly unique nodes are only defined when v is injective on s. +* `lagrange.basis_divisor x y`, with `x y : F`. These are the normalised irreducible factors of +the Lagrange basis polynomials. They evaluate to `1` at `x` and `0` at `y` when `x` and `y` +are distinct. +* `lagrange.basis v i` with `i : ι`: the Lagrange basis polynomial that evaluates to `1` at `v i` +and `0` at `v j` for `i ≠ j`. +* `lagrange.interpolate v r` where `r : ι → F` is a function from the fintype to the field: the +Lagrange interpolant that evaluates to `r i` at `x i` for all `i : ι`. The `r i` are the _values_ +associated with the _nodes_`x i`. +* `lagrange.interpolate_at v f`, where `v : ι ↪ F` and `ι` is a fintype, and `f : F → F` is a +function from the field to itself: this is the Lagrange interpolant that evaluates to `f (x i)` +at `x i`, and so approximates the function `f`. This is just a special case of the general +interpolation, where the values are given by a known function `f`. -/ -noncomputable theory -open_locale big_operators classical polynomial +open_locale polynomial big_operators -universe u +section polynomial_determination -namespace lagrange +namespace polynomial +variables {R : Type*} [comm_ring R] [is_domain R] {f g : R[X]} -variables {F : Type u} [decidable_eq F] [field F] (s : finset F) -variables {F' : Type u} [field F'] (s' : finset F') +section finset +open function fintype +variables (s : finset R) -open polynomial +theorem eq_zero_of_degree_lt_of_eval_finset_eq_zero (degree_f_lt : f.degree < s.card) + (eval_f : ∀ x ∈ s, f.eval x = 0) : f = 0 := +begin + rw ← mem_degree_lt at degree_f_lt, + simp_rw eval_eq_sum_degree_lt_equiv degree_f_lt at eval_f, + rw ← degree_lt_equiv_eq_zero_iff_eq_zero degree_f_lt, + exact matrix.eq_zero_of_forall_index_sum_mul_pow_eq_zero + (injective.comp (embedding.subtype _).inj' (equiv_fin_of_card_eq (card_coe _)).symm.injective) + (λ _, eval_f _ (finset.coe_mem _)) +end + +theorem eq_of_degree_sub_lt_of_eval_finset_eq (degree_fg_lt : (f - g).degree < s.card) + (eval_fg : ∀ x ∈ s, f.eval x = g.eval x) : f = g := +begin + rw ← sub_eq_zero, + refine eq_zero_of_degree_lt_of_eval_finset_eq_zero _ degree_fg_lt _, + simp_rw [eval_sub, sub_eq_zero], + exact eval_fg +end + +theorem eq_of_degrees_lt_of_eval_finset_eq (degree_f_lt : f.degree < s.card) + (degree_g_lt : g.degree < s.card) (eval_fg : ∀ x ∈ s, f.eval x = g.eval x) : f = g := +begin + rw ← mem_degree_lt at degree_f_lt degree_g_lt, + refine eq_of_degree_sub_lt_of_eval_finset_eq _ _ eval_fg, + rw ← mem_degree_lt, exact submodule.sub_mem _ degree_f_lt degree_g_lt +end + +end finset + +section indexed +open finset +variables {ι : Type*} {v : ι → R} (s : finset ι) + +theorem eq_zero_of_degree_lt_of_eval_index_eq_zero (hvs : set.inj_on v s) + (degree_f_lt : f.degree < s.card) (eval_f : ∀ i ∈ s, f.eval (v i) = 0) : f = 0 := +begin + classical, + rw ← card_image_of_inj_on hvs at degree_f_lt, + refine eq_zero_of_degree_lt_of_eval_finset_eq_zero _ degree_f_lt _, + intros x hx, + rcases mem_image.mp hx with ⟨_, hj, rfl⟩, + exact eval_f _ hj +end + +theorem eq_of_degree_sub_lt_of_eval_index_eq (hvs : set.inj_on v s) + (degree_fg_lt : (f - g).degree < s.card) (eval_fg : ∀ i ∈ s, f.eval (v i) = g.eval (v i)) : + f = g := +begin + rw ← sub_eq_zero, + refine eq_zero_of_degree_lt_of_eval_index_eq_zero _ hvs degree_fg_lt _, + simp_rw [eval_sub, sub_eq_zero], + exact eval_fg +end + +theorem eq_of_degrees_lt_of_eval_index_eq (hvs : set.inj_on v s) (degree_f_lt : f.degree < s.card) + (degree_g_lt : g.degree < s.card) (eval_fg : ∀ i ∈ s, f.eval (v i) = g.eval (v i)) : f = g := +begin + refine eq_of_degree_sub_lt_of_eval_index_eq _ hvs _ eval_fg, + rw ← mem_degree_lt at degree_f_lt degree_g_lt ⊢, + exact submodule.sub_mem _ degree_f_lt degree_g_lt +end + +end indexed + +end polynomial + +end polynomial_determination + +noncomputable theory -/-- Lagrange basis polynomials that evaluate to 1 at `x` and 0 at other elements of `s`. -/ -def basis (x : F) : F[X] := -∏ y in s.erase x, C (x - y)⁻¹ * (X - C y) +namespace lagrange +open polynomial +variables {F : Type*} [field F] -@[simp] theorem basis_empty (x : F) : basis ∅ x = 1 := -rfl +section basis_divisor +variables {x y : F} +/-- `basis_divisor x y` is the unique linear or constant polynomial such that +when evaluated at `x` it gives `1` and `y` it gives `0` (where when `x = y` it is identically `0`). +Such polynomials are the building blocks for the Lagrange interpolants. -/ +def basis_divisor (x y : F) : F[X] := C ((x - y)⁻¹) * (X - C (y)) -@[simp] theorem basis_singleton_self (x : F) : basis {x} x = 1 := -by rw [basis, finset.erase_singleton, finset.prod_empty] +lemma basis_divisor_self : basis_divisor x x = 0 := +by simp only [basis_divisor, sub_self, inv_zero, map_zero, zero_mul] -@[simp] theorem eval_basis_self (x : F) : (basis s x).eval x = 1 := +lemma basis_divisor_inj (hxy : basis_divisor x y = 0) : x = y := begin - rw [basis, ← coe_eval_ring_hom, (eval_ring_hom x).map_prod, coe_eval_ring_hom, - finset.prod_eq_one], - intros y hy, simp_rw [eval_mul, eval_sub, eval_C, eval_X], - exact inv_mul_cancel (sub_ne_zero_of_ne (finset.ne_of_mem_erase hy).symm) + simp_rw [basis_divisor, mul_eq_zero, X_sub_C_ne_zero, or_false, + C_eq_zero, inv_eq_zero, sub_eq_zero] at hxy, + exact hxy end -@[simp] theorem eval_basis_ne (x y : F) (h1 : y ∈ s) (h2 : y ≠ x) : (basis s x).eval y = 0 := +@[simp] lemma basis_divisor_eq_zero_iff : basis_divisor x y = 0 ↔ x = y := +⟨basis_divisor_inj, λ H, H ▸ basis_divisor_self⟩ + +lemma basis_divisor_ne_zero_iff : basis_divisor x y ≠ 0 ↔ x ≠ y := +by rw [ne.def, basis_divisor_eq_zero_iff] + +lemma degree_basis_divisor_of_ne (hxy : x ≠ y) : (basis_divisor x y).degree = 1 := begin - rw [basis, - ← coe_eval_ring_hom, (eval_ring_hom y).map_prod, coe_eval_ring_hom, - finset.prod_eq_zero (finset.mem_erase.2 ⟨h2, h1⟩)], - simp_rw [eval_mul, eval_sub, eval_C, eval_X, sub_self, mul_zero] + rw [basis_divisor, degree_mul, degree_X_sub_C, degree_C, zero_add], + exact inv_ne_zero (sub_ne_zero_of_ne hxy) end -theorem eval_basis (x y : F) (h : y ∈ s) : (basis s x).eval y = if y = x then 1 else 0 := -by { split_ifs with H, { subst H, apply eval_basis_self }, { exact eval_basis_ne s x y h H } } +@[simp] lemma degree_basis_divisor_self : (basis_divisor x x).degree = ⊥ := +by rw [basis_divisor_self, degree_zero] + +lemma nat_degree_basis_divisor_self : (basis_divisor x x).nat_degree = 0 := +by rw [basis_divisor_self, nat_degree_zero] + +lemma nat_degree_basis_divisor_of_ne (hxy : x ≠ y) : (basis_divisor x y).nat_degree = 1 := +nat_degree_eq_of_degree_eq_some (degree_basis_divisor_of_ne hxy) + +@[simp] lemma eval_basis_divisor_right : eval y (basis_divisor x y) = 0 := +by simp only [basis_divisor, eval_mul, eval_C, eval_sub, eval_X, sub_self, mul_zero] -@[simp] theorem nat_degree_basis (x : F) (hx : x ∈ s) : (basis s x).nat_degree = s.card - 1 := +lemma eval_basis_divisor_left_of_ne (hxy : x ≠ y) : eval x (basis_divisor x y) = 1 := begin - unfold basis, generalize hsx : s.erase x = sx, - have : x ∉ sx := hsx ▸ finset.not_mem_erase x s, - rw [← finset.insert_erase hx, hsx, finset.card_insert_of_not_mem this, add_tsub_cancel_right], - clear hx hsx s, revert this, apply sx.induction_on, - { intros hx, rw [finset.prod_empty, nat_degree_one], refl }, - { intros y s hys ih hx, rw [finset.mem_insert, not_or_distrib] at hx, - have h1 : C (x - y)⁻¹ ≠ C 0 := λ h, hx.1 (eq_of_sub_eq_zero $ inv_eq_zero.1 $ C_inj.1 h), - have h2 : X ^ 1 - C y ≠ 0 := by convert X_pow_sub_C_ne_zero zero_lt_one y, - rw C_0 at h1, rw pow_one at h2, - rw [finset.prod_insert hys, nat_degree_mul (mul_ne_zero h1 h2), ih hx.2, - finset.card_insert_of_not_mem hys, nat_degree_mul h1 h2, - nat_degree_C, zero_add, nat_degree, degree_X_sub_C, add_comm], refl, - rw [ne, finset.prod_eq_zero_iff], rintro ⟨z, hzs, hz⟩, - rw mul_eq_zero at hz, cases hz with hz hz, - { rw [← C_0, C_inj, inv_eq_zero, sub_eq_zero] at hz, exact hx.2 (hz.symm ▸ hzs) }, - { rw ← pow_one (X : F[X]) at hz, exact X_pow_sub_C_ne_zero zero_lt_one _ hz } } + simp only [basis_divisor, eval_mul, eval_C, eval_sub, eval_X], + exact inv_mul_cancel (sub_ne_zero_of_ne hxy) end -variables (f : F → F) +end basis_divisor -/-- Lagrange interpolation: given a finset `s` and a function `f : F → F`, -`interpolate s f` is the unique polynomial of degree `< s.card` -that takes value `f x` on all `x` in `s`. -/ -def interpolate : F[X] := -∑ x in s, C (f x) * basis s x +section basis +open finset +variables {ι : Type*} [decidable_eq ι] {s : finset ι} {v : ι → F} {i j : ι} + +/-- Lagrange basis polynomials indexed by `s : finset ι`, defined at nodes `v i` for a +map `v : ι → F`. For `i, j ∈ s`, `basis s v i` evaluates to 0 at `v j` for `i ≠ j`. When +`v` is injective on `s`, `basis s v i` evaluates to 1 at `v i`. -/ +protected def basis (s : finset ι) (v : ι → F) (i : ι) : F[X] := +∏ j in s.erase i, basis_divisor (v i) (v j) + +@[simp] theorem basis_empty : lagrange.basis ∅ v i = 1 := rfl + +@[simp] theorem basis_singleton (i : ι) : lagrange.basis {i} v i = 1 := +by rw [lagrange.basis, erase_singleton, prod_empty] + +@[simp] theorem basis_pair_left (hij : i ≠ j) : + lagrange.basis {i, j} v i = basis_divisor (v i) (v j) := +by simp only [lagrange.basis, hij, erase_insert_eq_erase, erase_eq_of_not_mem, + mem_singleton, not_false_iff, prod_singleton] + +@[simp] theorem basis_pair_right (hij : i ≠ j) : + lagrange.basis {i, j} v j = basis_divisor (v j) (v i) := +by { rw pair_comm, exact basis_pair_left hij.symm } + +lemma basis_ne_zero (hvs : set.inj_on v s) (hi : i ∈ s) : lagrange.basis s v i ≠ 0 := +begin + simp_rw [lagrange.basis, prod_ne_zero_iff, ne.def, mem_erase], + rintros j ⟨hij, hj⟩, + rw [basis_divisor_eq_zero_iff, hvs.eq_iff hi hj], + exact hij.symm +end -@[simp] theorem interpolate_empty (f) : interpolate (∅ : finset F) f = 0 := -rfl +@[simp] theorem eval_basis_self (hvs : set.inj_on v s) (hi : i ∈ s) : + (lagrange.basis s v i).eval (v i) = 1 := +begin + rw [lagrange.basis, eval_prod], + refine prod_eq_one (λ j H, _), + rw eval_basis_divisor_left_of_ne, + rcases mem_erase.mp H with ⟨hij, hj⟩, + exact mt (hvs hi hj) hij.symm +end -@[simp] theorem interpolate_singleton (f) (x : F) : interpolate {x} f = C (f x) := -by rw [interpolate, finset.sum_singleton, basis_singleton_self, mul_one] +@[simp] theorem eval_basis_of_ne (hij : i ≠ j) (hj : j ∈ s) : + (lagrange.basis s v i).eval (v j) = 0 := +begin + simp_rw [lagrange.basis, eval_prod, prod_eq_zero_iff], + exact ⟨j, ⟨mem_erase.mpr ⟨hij.symm, hj⟩, eval_basis_divisor_right⟩⟩ +end -@[simp] theorem eval_interpolate (x) (H : x ∈ s) : eval x (interpolate s f) = f x := +@[simp] theorem nat_degree_basis (hvs : set.inj_on v s) (hi : i ∈ s) : + (lagrange.basis s v i).nat_degree = s.card - 1 := begin - rw [interpolate, ←coe_eval_ring_hom, ring_hom.map_sum, coe_eval_ring_hom, finset.sum_eq_single x], - { simp }, - { intros y hy hxy, simp [eval_basis_ne s y x H hxy.symm] }, - { intro h, exact (h H).elim } + have H : ∀ j, j ∈ s.erase i → basis_divisor (v i) (v j) ≠ 0, + { simp_rw [ne.def, mem_erase, basis_divisor_eq_zero_iff], + exact λ j ⟨hij₁, hj⟩ hij₂, hij₁ (hvs hj hi hij₂.symm) }, + rw [← card_erase_of_mem hi, card_eq_sum_ones], + convert nat_degree_prod _ _ H using 1, + refine sum_congr rfl (λ j hj, (nat_degree_basis_divisor_of_ne _).symm), + rw [ne.def, ← basis_divisor_eq_zero_iff], + exact H _ hj end -theorem degree_interpolate_lt : (interpolate s f).degree < s.card := -if H : s = ∅ then by { subst H, rw [interpolate_empty, degree_zero], exact with_bot.bot_lt_coe _ } -else (degree_sum_le _ _).trans_lt $ (finset.sup_lt_iff $ with_bot.bot_lt_coe s.card).2 $ λ b _, -calc (C (f b) * basis s b).degree - ≤ (C (f b)).degree + (basis s b).degree : degree_mul_le _ _ -... ≤ 0 + (basis s b).degree : add_le_add_right degree_C_le _ -... = (basis s b).degree : zero_add _ -... ≤ (basis s b).nat_degree : degree_le_nat_degree -... = (s.card - 1 : ℕ) : by { rwa nat_degree_basis } -... < s.card : with_bot.coe_lt_coe.2 (nat.pred_lt $ mt finset.card_eq_zero.1 H) +theorem degree_basis (hvs : set.inj_on v s) (hi : i ∈ s) : + (lagrange.basis s v i).degree = ↑(s.card - 1) := +by rw [degree_eq_nat_degree (basis_ne_zero hvs hi), nat_degree_basis hvs hi] -theorem degree_interpolate_erase {x} (hx : x ∈ s) : - (interpolate (s.erase x) f).degree < (s.card - 1 : ℕ) := +lemma sum_basis (hvs : set.inj_on v s) (hs : s.nonempty) : ∑ j in s, (lagrange.basis s v j) = 1 := begin - convert degree_interpolate_lt (s.erase x) f, - rw finset.card_erase_of_mem hx, + refine eq_of_degrees_lt_of_eval_index_eq s hvs (lt_of_le_of_lt (degree_sum_le _ _) _) _ _, + { rw finset.sup_lt_iff (with_bot.bot_lt_coe s.card), + intros i hi, + rw [degree_basis hvs hi, with_bot.coe_lt_coe], + exact nat.pred_lt (card_ne_zero_of_mem hi) }, + { rw [degree_one, ← with_bot.coe_zero, with_bot.coe_lt_coe], + exact nonempty.card_pos hs }, + { intros i hi, + rw [eval_finset_sum, eval_one, ← add_sum_erase _ _ hi, + eval_basis_self hvs hi, add_right_eq_self], + refine sum_eq_zero (λ j hj, _), + rcases mem_erase.mp hj with ⟨hij, hj⟩, + rw eval_basis_of_ne hij hi } end -theorem interpolate_eq_of_eval_eq (f g : F → F) {s : finset F} (hs : ∀ x ∈ s, f x = g x) : - interpolate s f = interpolate s g := +lemma basis_divisor_add_symm {x y : F} (hxy : x ≠ y) : basis_divisor x y + basis_divisor y x = 1 := begin - rw [interpolate, interpolate], - refine finset.sum_congr rfl (λ x hx, _), - rw hs x hx, + classical, + rw [←sum_basis (set.inj_on_of_injective function.injective_id _) ⟨x, mem_insert_self _ {y}⟩, + sum_insert (not_mem_singleton.mpr hxy), sum_singleton, basis_pair_left hxy, + basis_pair_right hxy, id, id] end -/-- Linear version of `interpolate`. -/ -def linterpolate : (F → F) →ₗ[F] polynomial F := -{ to_fun := interpolate s, - map_add' := λ f g, by { simp_rw [interpolate, ← finset.sum_add_distrib, ← add_mul, ← C_add], - refl }, - map_smul' := λ c f, by { simp_rw [interpolate, finset.smul_sum, C_mul', smul_smul], refl } } +end basis + +section interpolate +open finset +variables {ι : Type*} [decidable_eq ι] {s t : finset ι} {i j : ι} {v : ι → F} (r r' : ι → F) + +/-- Lagrange interpolation: given a finset `s : finset ι`, a nodal map `v : ι → F` injective on +`s` and a value function `r : ι → F`, `interpolate s v r` is the unique +polynomial of degree `< s.card` that takes value `r i` on `v i` for all `i` in `s`. -/ +@[simps] +def interpolate (s : finset ι) (v : ι → F) : (ι → F) →ₗ[F] F[X] := +{ to_fun := λ r, ∑ i in s, C (r i) * (lagrange.basis s v i), + map_add' := λ f g, by simp_rw [← finset.sum_add_distrib, ← add_mul, + ← C_add, pi.add_apply], + map_smul' := λ c f, by simp_rw [finset.smul_sum, C_mul', smul_smul, + pi.smul_apply, ring_hom.id_apply, smul_eq_mul] } -@[simp] lemma interpolate_add (f g) : interpolate s (f + g) = interpolate s f + interpolate s g := -(linterpolate s).map_add f g +@[simp] theorem interpolate_empty : interpolate ∅ v r = 0 := +by rw [interpolate_apply, sum_empty] -@[simp] lemma interpolate_zero : interpolate s 0 = 0 := -(linterpolate s).map_zero +@[simp] theorem interpolate_singleton : interpolate {i} v r = C (r i) := +by rw [interpolate_apply, sum_singleton, basis_singleton, mul_one] -@[simp] lemma interpolate_neg (f) : interpolate s (-f) = -interpolate s f := -(linterpolate s).map_neg f +theorem interpolate_one (hvs : set.inj_on v s) (hs : s.nonempty) : interpolate s v 1 = 1 := +by { simp_rw [interpolate_apply, pi.one_apply, map_one, one_mul], exact sum_basis hvs hs } -@[simp] lemma interpolate_sub (f g) : interpolate s (f - g) = interpolate s f - interpolate s g := -(linterpolate s).map_sub f g +theorem eval_interpolate_at_node (hvs : set.inj_on v s) (hi : i ∈ s) : + eval (v i) (interpolate s v r) = r i := +begin + rw [interpolate_apply, eval_finset_sum, ← add_sum_erase _ _ hi], + simp_rw [eval_mul, eval_C, eval_basis_self hvs hi, mul_one, add_right_eq_self], + refine sum_eq_zero (λ j H, _), + rw [eval_basis_of_ne (mem_erase.mp H).1 hi, mul_zero] +end -@[simp] lemma interpolate_smul (c : F) (f) : interpolate s (c • f) = c • interpolate s f := -(linterpolate s).map_smul c f - -theorem eq_zero_of_eval_eq_zero {f : F'[X]} (hf1 : f.degree < s'.card) - (hf2 : ∀ x ∈ s', f.eval x = 0) : f = 0 := -by_contradiction $ λ hf3, not_le_of_lt hf1 $ -calc (s'.card : with_bot ℕ) - ≤ f.roots.to_finset.card : with_bot.coe_le_coe.2 $ finset.card_le_of_subset $ λ x hx, - (multiset.mem_to_finset).mpr $ (mem_roots hf3).2 $ hf2 x hx -... ≤ f.roots.card : with_bot.coe_le_coe.2 $ f.roots.to_finset_card_le -... ≤ f.degree : card_roots hf3 - -theorem eq_of_eval_eq {f g : F'[X]} (hf : f.degree < s'.card) (hg : g.degree < s'.card) - (hfg : ∀ x ∈ s', f.eval x = g.eval x) : f = g := -eq_of_sub_eq_zero $ eq_zero_of_eval_eq_zero s' - (lt_of_le_of_lt (degree_sub_le f g) $ max_lt hf hg) - (λ x hx, by rw [eval_sub, hfg x hx, sub_self]) - -theorem eq_interpolate_of_eval_eq {g : F[X]} (hg : g.degree < s.card) - (hgf : ∀ x ∈ s, g.eval x = f x) : interpolate s f = g := -eq_of_eval_eq s (degree_interpolate_lt _ _) hg $ λ x hx, begin - rw hgf x hx, - exact eval_interpolate _ _ _ hx, +theorem degree_interpolate_le (hvs : set.inj_on v s) : (interpolate s v r).degree ≤ ↑(s.card - 1) := +begin + refine (degree_sum_le _ _).trans _, + rw finset.sup_le_iff, + intros i hi, + rw [degree_mul, degree_basis hvs hi], + by_cases hr : r i = 0, + { simpa only [hr, map_zero, degree_zero, with_bot.bot_add] using bot_le }, + { rw [degree_C hr, zero_add, with_bot.coe_le_coe] } end -theorem eq_interpolate (f : F[X]) (hf : f.degree < s.card) : - interpolate s (λ x, f.eval x) = f := -eq_of_eval_eq s (degree_interpolate_lt s _) hf $ λ x hx, eval_interpolate s _ x hx +theorem degree_interpolate_lt (hvs : set.inj_on v s) : (interpolate s v r).degree < s.card := +begin + rcases eq_empty_or_nonempty s with rfl | h, + { rw [interpolate_empty, degree_zero, card_empty], + exact with_bot.bot_lt_coe _ }, + { refine lt_of_le_of_lt (degree_interpolate_le _ hvs) _, + rw with_bot.coe_lt_coe, + exact nat.sub_lt (nonempty.card_pos h) zero_lt_one } +end -/-- Lagrange interpolation induces isomorphism between functions from `s` and polynomials -of degree less than `s.card`. -/ -def fun_equiv_degree_lt : degree_lt F s.card ≃ₗ[F] (s → F) := -{ to_fun := λ f x, f.1.eval x, - map_add' := λ f g, funext $ λ x, eval_add, +theorem degree_interpolate_erase_lt (hvs : set.inj_on v s) (hi : i ∈ s) : + (interpolate (s.erase i) v r).degree < ↑(s.card - 1) := +begin + rw ← finset.card_erase_of_mem hi, + exact degree_interpolate_lt _ (set.inj_on.mono (coe_subset.mpr (erase_subset _ _)) hvs), +end + +theorem values_eq_on_of_interpolate_eq (hvs : set.inj_on v s) + (hrr' : interpolate s v r = interpolate s v r') : ∀ i ∈ s, r i = r' i := +λ _ hi, by rw [← eval_interpolate_at_node r hvs hi, hrr', eval_interpolate_at_node r' hvs hi] + +theorem interpolate_eq_of_values_eq_on (hrr' : ∀ i ∈ s, r i = r' i) : + interpolate s v r = interpolate s v r' := +sum_congr rfl (λ i hi, (by rw hrr' _ hi)) + +theorem interpolate_eq_iff_values_eq_on (hvs : set.inj_on v s) : + interpolate s v r = interpolate s v r' ↔ ∀ i ∈ s, r i = r' i := +⟨values_eq_on_of_interpolate_eq _ _ hvs, interpolate_eq_of_values_eq_on _ _⟩ + +theorem eq_interpolate {f : F[X]} (hvs : set.inj_on v s) (degree_f_lt : f.degree < s.card) : + f = interpolate s v (λ i, f.eval (v i)) := +eq_of_degrees_lt_of_eval_index_eq _ hvs degree_f_lt (degree_interpolate_lt _ hvs) $ +λ i hi, (eval_interpolate_at_node _ hvs hi).symm + +theorem eq_interpolate_of_eval_eq {f : F[X]} (hvs : set.inj_on v s) + (degree_f_lt : f.degree < s.card) (eval_f : ∀ i ∈ s, f.eval (v i) = r i) : + f = interpolate s v r := +by { rw eq_interpolate hvs degree_f_lt, exact interpolate_eq_of_values_eq_on _ _ eval_f } + +/-- +This is the characteristic property of the interpolation: the interpolation is the +unique polynomial of `degree < fintype.card ι` which takes the value of the `r i` on the `v i`. +-/ +theorem eq_interpolate_iff {f : F[X]} (hvs : set.inj_on v s) : + (f.degree < s.card ∧ ∀ i ∈ s, eval (v i) f = r i) ↔ f = interpolate s v r := +begin + split; intro h, + { exact eq_interpolate_of_eval_eq _ hvs h.1 h.2 }, + { rw h, exact ⟨degree_interpolate_lt _ hvs, λ _ hi, eval_interpolate_at_node _ hvs hi⟩ } +end + +/-- Lagrange interpolation induces isomorphism between functions from `s` +and polynomials of degree less than `fintype.card ι`.-/ +def fun_equiv_degree_lt (hvs : set.inj_on v s) : degree_lt F s.card ≃ₗ[F] (s → F) := +{ to_fun := λ f i, f.1.eval (v i), + map_add' := λ f g, funext $ λ v, eval_add, map_smul' := λ c f, funext $ by simp, - inv_fun := λ f, ⟨interpolate s (λ x, if hx : x ∈ s then f ⟨x, hx⟩ else 0), - mem_degree_lt.2 $ degree_interpolate_lt _ _⟩, - left_inv := λ f, begin apply subtype.eq, - simp only [subtype.coe_mk, subtype.val_eq_coe, dite_eq_ite], - convert eq_interpolate s f (mem_degree_lt.1 f.2) using 1, - rw interpolate_eq_of_eval_eq, - intros x hx, - rw if_pos hx end, - right_inv := λ f, funext $ λ ⟨x, hx⟩, begin - convert eval_interpolate s _ x hx, - simp_rw dif_pos hx end } - -theorem interpolate_eq_interpolate_erase_add {x y : F} (hx : x ∈ s) (hy : y ∈ s) (hxy : x ≠ y) : - interpolate s f = - C (y - x)⁻¹ * ((X - C x) * interpolate (s.erase x) f + (C y - X) * interpolate (s.erase y) f) := -begin - refine eq_interpolate_of_eval_eq _ _ _ (λ z hz, _), - { rw [degree_mul, degree_C (inv_ne_zero (sub_ne_zero.2 hxy.symm)), zero_add], - refine lt_of_le_of_lt (degree_add_le _ _) (max_lt _ _), - { rw [degree_mul, degree_X_sub_C], - convert (with_bot.add_lt_add_iff_left with_bot.coe_ne_bot).2 - (degree_interpolate_erase s f hx), - simp [nat.one_add, nat.sub_one, nat.succ_pred_eq_of_pos (finset.card_pos.2 ⟨x, hx⟩)] }, - { rw [degree_mul, ←neg_sub, degree_neg, degree_X_sub_C], - convert (with_bot.add_lt_add_iff_left with_bot.coe_ne_bot).2 - (degree_interpolate_erase s f hy), - simp [nat.one_add, nat.sub_one, nat.succ_pred_eq_of_pos (finset.card_pos.2 ⟨y, hy⟩)] } }, - { by_cases hzx : z = x, - { simp [hzx, eval_interpolate (s.erase y) f x (finset.mem_erase_of_ne_of_mem hxy hx), - inv_mul_eq_iff_eq_mul₀ (sub_ne_zero_of_ne hxy.symm)] }, - { by_cases hzy : z = y, - { simp [hzy, eval_interpolate (s.erase x) f y (finset.mem_erase_of_ne_of_mem hxy.symm hy), - inv_mul_eq_iff_eq_mul₀ (sub_ne_zero_of_ne hxy.symm)] }, - { simp only [eval_interpolate (s.erase x) f z (finset.mem_erase_of_ne_of_mem hzx hz), - eval_interpolate (s.erase y) f z (finset.mem_erase_of_ne_of_mem hzy hz), - inv_mul_eq_iff_eq_mul₀ (sub_ne_zero_of_ne hxy.symm), eval_mul, eval_C, eval_add, - eval_sub, eval_X], - ring } } } + inv_fun := λ r, ⟨interpolate s v (λ x, if hx : x ∈ s then r ⟨x, hx⟩ else 0), + mem_degree_lt.2 $ degree_interpolate_lt _ hvs⟩, + left_inv := + begin + rintros ⟨f, hf⟩, + simp only [subtype.mk_eq_mk, subtype.coe_mk, dite_eq_ite], + rw mem_degree_lt at hf, + nth_rewrite_rhs 0 eq_interpolate hvs hf, + exact interpolate_eq_of_values_eq_on _ _ (λ _ hi, if_pos hi) + end, + right_inv := + begin + intro f, + ext ⟨i, hi⟩, + simp only [subtype.coe_mk, eval_interpolate_at_node _ hvs hi], + exact dif_pos hi, + end } + +theorem interpolate_eq_sum_interpolate_insert_sdiff (hvt : set.inj_on v t) (hs : s.nonempty) + (hst : s ⊆ t) : interpolate t v r = + ∑ i in s, (interpolate (insert i (t \ s)) v r) * lagrange.basis s v i := +begin + symmetry, + refine eq_interpolate_of_eval_eq _ hvt (lt_of_le_of_lt (degree_sum_le _ _) _) (λ i hi, _), + { simp_rw [(finset.sup_lt_iff (with_bot.bot_lt_coe t.card)), degree_mul], + intros i hi, + have hs : 1 ≤ s.card := nonempty.card_pos ⟨_, hi⟩, + have hst' : s.card ≤ t.card := card_le_of_subset hst, + have H : t.card = (1 + (t.card - s.card)) + (s.card - 1), + { rw [add_assoc, tsub_add_tsub_cancel hst' hs, ← add_tsub_assoc_of_le (hs.trans hst'), + nat.succ_add_sub_one, zero_add] }, + rw [degree_basis (set.inj_on.mono hst hvt) hi, H, with_bot.coe_add, + with_bot.add_lt_add_iff_right (@with_bot.coe_ne_bot _ (s.card - 1))], + convert degree_interpolate_lt _ (hvt.mono (coe_subset.mpr (insert_subset.mpr + ⟨hst hi, sdiff_subset _ _⟩))), + rw [card_insert_of_not_mem (not_mem_sdiff_of_mem_right hi), card_sdiff hst, add_comm] }, + + { simp_rw [eval_finset_sum, eval_mul], + by_cases hi' : i ∈ s, + { rw [← add_sum_erase _ _ hi', eval_basis_self (hvt.mono hst) hi', + eval_interpolate_at_node _ (hvt.mono (coe_subset.mpr + (insert_subset.mpr ⟨hi, sdiff_subset _ _⟩))) (mem_insert_self _ _), + mul_one, add_right_eq_self], + refine sum_eq_zero (λ j hj, _), + rcases mem_erase.mp hj with ⟨hij, hj⟩, + rw [eval_basis_of_ne hij hi', mul_zero] }, + { have H : ∑ j in s, eval (v i) (lagrange.basis s v j) = 1, + { rw [← eval_finset_sum, sum_basis (hvt.mono hst) hs, eval_one] }, + rw [← mul_one (r i), ← H, mul_sum], + refine sum_congr rfl (λ j hj, _), + congr, + exact eval_interpolate_at_node _ (hvt.mono (insert_subset.mpr ⟨hst hj, sdiff_subset _ _⟩)) + (mem_insert.mpr (or.inr (mem_sdiff.mpr ⟨hi, hi'⟩))) } } +end + +theorem interpolate_eq_add_interpolate_erase (hvs : set.inj_on v s) (hi : i ∈ s) (hj : j ∈ s) + (hij : i ≠ j) : interpolate s v r = interpolate (s.erase j) v r * basis_divisor (v i) (v j) + + interpolate (s.erase i) v r * basis_divisor (v j) (v i) := +begin + rw [interpolate_eq_sum_interpolate_insert_sdiff _ hvs ⟨i, (mem_insert_self i {j})⟩ _, + sum_insert (not_mem_singleton.mpr hij), sum_singleton, basis_pair_left hij, + basis_pair_right hij, + sdiff_insert_insert_of_mem_of_not_mem hi (not_mem_singleton.mpr hij), + sdiff_singleton_eq_erase, pair_comm, + sdiff_insert_insert_of_mem_of_not_mem hj (not_mem_singleton.mpr hij.symm), + sdiff_singleton_eq_erase], + { exact insert_subset.mpr ⟨hi, singleton_subset_iff.mpr hj⟩ }, end +end interpolate + +section nodal +open finset polynomial +variables {ι : Type*} {s : finset ι} {v : ι → F} {i : ι} (r : ι → F) {x : F} + +/-- +`nodal s v` is the unique monic polynomial whose roots are the nodes defined by `v` and `s`. + +That is, the roots of `nodal s v` are exactly the image of `v` on `s`, +with appropriate multiplicity. + +We can use `nodal` to define the barycentric forms of the evaluated interpolant. +-/ +def nodal (s : finset ι) (v : ι → F) : F[X] := ∏ i in s, (X - C (v i)) + +lemma nodal_eq (s : finset ι) (v : ι → F) : nodal s v = ∏ i in s, (X - C (v i)) := rfl + +@[simp] lemma nodal_empty : nodal ∅ v = 1 := rfl + +lemma degree_nodal : (nodal s v).degree = s.card := +by simp_rw [nodal, degree_prod, degree_X_sub_C, sum_const, nat.smul_one_eq_coe] + +lemma eval_nodal {x : F} : (nodal s v).eval x = ∏ i in s, (x - v i) := +by simp_rw [nodal, eval_prod, eval_sub, eval_X, eval_C] + +lemma eval_nodal_at_node (hi : i ∈ s) : eval (v i) (nodal s v) = 0 := +by { rw [eval_nodal, prod_eq_zero_iff], exact ⟨i, hi, sub_eq_zero_of_eq rfl⟩ } + +lemma eval_nodal_not_at_node (hx : ∀ i ∈ s, x ≠ v i) : eval x (nodal s v) ≠ 0 := +by { simp_rw [nodal, eval_prod, prod_ne_zero_iff, eval_sub, eval_X, eval_C, sub_ne_zero], exact hx } + +lemma nodal_eq_mul_nodal_erase [decidable_eq ι] (hi : i ∈ s) : + nodal s v = (X - C (v i)) * nodal (s.erase i) v := by simp_rw [nodal, mul_prod_erase _ _ hi] + +lemma X_sub_C_dvd_nodal (v : ι → F) (hi : i ∈ s) : (X - C (v i)) ∣ nodal s v := +⟨_, by { classical, exact nodal_eq_mul_nodal_erase hi }⟩ + +variable [decidable_eq ι] + +lemma nodal_erase_eq_nodal_div (hi : i ∈ s) : + nodal (s.erase i) v = nodal s v / (X - C (v i)) := +begin + rw [nodal_eq_mul_nodal_erase hi, euclidean_domain.mul_div_cancel_left], + exact X_sub_C_ne_zero _ +end + +lemma nodal_insert_eq_nodal (hi : i ∉ s) : + nodal (insert i s) v = (X - C (v i)) * (nodal s v) := by simp_rw [nodal, prod_insert hi] + +lemma derivative_nodal : (nodal s v).derivative = ∑ i in s, nodal (s.erase i) v := +begin + refine finset.induction_on s _ (λ _ _ hit IH, _), + { rw [nodal_empty, derivative_one, sum_empty] }, + { rw [nodal_insert_eq_nodal hit, derivative_mul, IH, derivative_sub, + derivative_X, derivative_C, sub_zero, one_mul, sum_insert hit, + mul_sum, erase_insert hit, add_right_inj], + refine sum_congr rfl (λ j hjt, _), + rw [nodal_erase_eq_nodal_div (mem_insert_of_mem hjt), nodal_insert_eq_nodal hit, + euclidean_domain.mul_div_assoc _ (X_sub_C_dvd_nodal v hjt), + nodal_erase_eq_nodal_div hjt] } +end + +lemma eval_nodal_derivative_eval_node_eq (hi : i ∈ s) : + eval (v i) (nodal s v).derivative = eval (v i) (nodal (s.erase i) v) := +begin + rw [derivative_nodal, eval_finset_sum, ← add_sum_erase _ _ hi, add_right_eq_self], + refine sum_eq_zero (λ j hj, _), + simp_rw [nodal, eval_prod, eval_sub, eval_X, eval_C, prod_eq_zero_iff, mem_erase], + exact ⟨i, ⟨(mem_erase.mp hj).1.symm, hi⟩, sub_eq_zero_of_eq rfl⟩ +end + +/-- This defines the nodal weight for a given set of node indexes and node mapping function `v`. -/ +def nodal_weight (s : finset ι) (v : ι → F) (i : ι) := ∏ j in s.erase i, (v i - v j)⁻¹ + +lemma nodal_weight_eq_eval_nodal_erase_inv : nodal_weight s v i = + (eval (v i) (nodal (s.erase i) v))⁻¹ := +by rw [eval_nodal, nodal_weight, prod_inv_distrib] + +lemma nodal_weight_eq_eval_nodal_derative (hi : i ∈ s) : nodal_weight s v i = + (eval (v i) (nodal s v).derivative)⁻¹ := +by rw [eval_nodal_derivative_eval_node_eq hi, nodal_weight_eq_eval_nodal_erase_inv] + +lemma nodal_weight_ne_zero (hvs : set.inj_on v s) (hi : i ∈ s) : nodal_weight s v i ≠ 0 := +begin + rw [nodal_weight, prod_ne_zero_iff], + intros j hj, + rcases mem_erase.mp hj with ⟨hij, hj⟩, + refine inv_ne_zero (sub_ne_zero_of_ne (mt (hvs.eq_iff hi hj).mp hij.symm)), +end + +lemma basis_eq_prod_sub_inv_mul_nodal_div (hi : i ∈ s) : + lagrange.basis s v i = C (nodal_weight s v i) * ( nodal s v / (X - C (v i)) ) := +by simp_rw [lagrange.basis, basis_divisor, nodal_weight, prod_mul_distrib, + map_prod, ← nodal_erase_eq_nodal_div hi, nodal] + +lemma eval_basis_not_at_node (hi : i ∈ s) (hxi : x ≠ v i) : + eval x (lagrange.basis s v i) = (eval x (nodal s v)) * (nodal_weight s v i * (x - v i)⁻¹) := +by rw [mul_comm, basis_eq_prod_sub_inv_mul_nodal_div hi, eval_mul, eval_C, + ← nodal_erase_eq_nodal_div hi, eval_nodal, eval_nodal, mul_assoc, ← mul_prod_erase _ _ hi, + ← mul_assoc (x - v i)⁻¹, inv_mul_cancel (sub_ne_zero_of_ne hxi), one_mul] + +lemma interpolate_eq_nodal_weight_mul_nodal_div_X_sub_C : + interpolate s v r = ∑ i in s, C (nodal_weight s v i) * (nodal s v / (X - C (v i))) * C (r i) := +sum_congr rfl (λ j hj, by rw [mul_comm, basis_eq_prod_sub_inv_mul_nodal_div hj]) + +/-- This is the first barycentric form of the Lagrange interpolant. -/ +lemma eval_interpolate_not_at_node (hx : ∀ i ∈ s, x ≠ v i) : eval x (interpolate s v r) = + eval x (nodal s v) * ∑ i in s, nodal_weight s v i * (x - v i)⁻¹ * r i := +begin + simp_rw [interpolate_apply, mul_sum, eval_finset_sum, eval_mul, eval_C], + refine sum_congr rfl (λ i hi, _), + rw [← mul_assoc, mul_comm, eval_basis_not_at_node hi (hx _ hi)] +end + +lemma sum_nodal_weight_mul_inv_sub_ne_zero (hvs : set.inj_on v s) + (hx : ∀ i ∈ s, x ≠ v i) (hs : s.nonempty) : + ∑ i in s, nodal_weight s v i * (x - v i)⁻¹ ≠ 0 := +@right_ne_zero_of_mul_eq_one _ _ _ (eval x (nodal s v)) _ $ + by simpa only [pi.one_apply, interpolate_one hvs hs, eval_one, mul_one] + using (eval_interpolate_not_at_node 1 hx).symm + +/-- This is the second barycentric form of the Lagrange interpolant. -/ +lemma eval_interpolate_not_at_node' (hvs : set.inj_on v s) (hs : s.nonempty) + (hx : ∀ i ∈ s, x ≠ v i) : eval x (interpolate s v r) = + (∑ i in s, nodal_weight s v i * (x - v i)⁻¹ * r i) / + ∑ i in s, nodal_weight s v i * (x - v i)⁻¹ := +begin + rw [← div_one (eval x (interpolate s v r)), ← @eval_one _ _ x, ← interpolate_one hvs hs, + eval_interpolate_not_at_node r hx, eval_interpolate_not_at_node 1 hx], + simp only [mul_div_mul_left _ _ (eval_nodal_not_at_node hx), pi.one_apply, mul_one] +end + +end nodal + end lagrange diff --git a/formal/lean/mathlib/linear_algebra/matrix/nonsingular_inverse.lean b/formal/lean/mathlib/linear_algebra/matrix/nonsingular_inverse.lean index 246435a355e1b80d06a3c04a6015895c9910f9ec..2db66ef059bb472945f344f44a49144a5d050f5c 100644 --- a/formal/lean/mathlib/linear_algebra/matrix/nonsingular_inverse.lean +++ b/formal/lean/mathlib/linear_algebra/matrix/nonsingular_inverse.lean @@ -413,7 +413,8 @@ invertible.map (diagonal_ring_hom n α) v lemma inv_of_diagonal_eq {α} [semiring α] (v : n → α) [invertible v] [invertible (diagonal v)] : ⅟(diagonal v) = diagonal (⅟v) := -by { letI := diagonal_invertible v, convert (rfl : ⅟(diagonal v) = _) } +by { letI := diagonal_invertible v, convert (rfl : ⅟(diagonal v) = _), + convert subsingleton.elim _ _, apply invertible.subsingleton } /-- `v` is invertible if `diagonal v` is -/ def invertible_of_diagonal_invertible (v : n → α) [invertible (diagonal v)] : invertible v := diff --git a/formal/lean/mathlib/linear_algebra/matrix/to_lin.lean b/formal/lean/mathlib/linear_algebra/matrix/to_lin.lean index c1ac511e877a5159ad3f0aafc43d4104a2d1fcc2..6389869a8837140ac4d15f69e922126b54b6176a 100644 --- a/formal/lean/mathlib/linear_algebra/matrix/to_lin.lean +++ b/formal/lean/mathlib/linear_algebra/matrix/to_lin.lean @@ -268,6 +268,11 @@ lemma matrix.ker_to_lin'_eq_bot_iff {M : matrix n n R} : M.to_lin'.ker = ⊥ ↔ ∀ v, M.mul_vec v = 0 → v = 0 := by simp only [submodule.eq_bot_iff, linear_map.mem_ker, matrix.to_lin'_apply] +lemma matrix.range_to_lin' (M : matrix m n R) : M.to_lin'.range = span R (range Mᵀ) := +by simp_rw [range_eq_map, ←supr_range_std_basis, map_supr, range_eq_map, ←ideal.span_singleton_one, + ideal.span, submodule.map_span, image_image, image_singleton, matrix.to_lin'_apply, + M.mul_vec_std_basis_apply, supr_span, range_eq_Union] + /-- If `M` and `M'` are each other's inverse matrices, they provide an equivalence between `m → A` and `n → A` corresponding to `M.mul_vec` and `M'.mul_vec`. -/ @[simps] @@ -623,7 +628,7 @@ open algebra lemma to_matrix_lmul' (x : S) (i j) : linear_map.to_matrix b b (lmul R S x) i j = b.repr (x * b j) i := -by rw [linear_map.to_matrix_apply', lmul_apply] +by simp only [linear_map.to_matrix_apply', coe_lmul_eq_mul, linear_map.mul_apply'] @[simp] lemma to_matrix_lsmul (x : R) (i j) : linear_map.to_matrix b b (algebra.lsmul R S x) i j = if i = j then x else 0 := @@ -635,7 +640,7 @@ by { rw [linear_map.to_matrix_apply', algebra.lsmul_coe, linear_equiv.map_smul, `left_mul_matrix_eq_repr_mul` gives a formula for the entries of `left_mul_matrix`. -This definition is useful for doing (more) explicit computations with `algebra.lmul`, +This definition is useful for doing (more) explicit computations with `linear_map.mul_left`, such as the trace form or norm map for algebras. -/ noncomputable def left_mul_matrix : S →ₐ[R] matrix m m R := @@ -658,10 +663,10 @@ by rw [left_mul_matrix_apply, to_matrix_lmul' b x i j] lemma left_mul_matrix_mul_vec_repr (x y : S) : (left_mul_matrix b x).mul_vec (b.repr y) = b.repr (x * y) := -linear_map.to_matrix_mul_vec_repr b b (algebra.lmul R S x) y +(linear_map.mul_left R x).to_matrix_mul_vec_repr b b y @[simp] lemma to_matrix_lmul_eq (x : S) : - linear_map.to_matrix b b (lmul R S x) = left_mul_matrix b x := + linear_map.to_matrix b b (linear_map.mul_left R x) = left_mul_matrix b x := rfl lemma left_mul_matrix_injective : function.injective (left_mul_matrix b) := @@ -675,8 +680,8 @@ lemma smul_left_mul_matrix (x) (ik jk) : left_mul_matrix (b.smul c) x ik jk = left_mul_matrix b (left_mul_matrix c x ik.2 jk.2) ik.1 jk.1 := by simp only [left_mul_matrix_apply, linear_map.to_matrix_apply, mul_comm, basis.smul_apply, - basis.smul_repr, finsupp.smul_apply, algebra.lmul_apply, id.smul_eq_mul, - linear_equiv.map_smul, mul_smul_comm] + basis.smul_repr, finsupp.smul_apply, id.smul_eq_mul, linear_equiv.map_smul, mul_smul_comm, + coe_lmul_eq_mul, linear_map.mul_apply'] lemma smul_left_mul_matrix_algebra_map (x : S) : left_mul_matrix (b.smul c) (algebra_map _ _ x) = block_diagonal (λ k, left_mul_matrix b x) := diff --git a/formal/lean/mathlib/linear_algebra/matrix/transvection.lean b/formal/lean/mathlib/linear_algebra/matrix/transvection.lean index 2e130f81624ba71333de3459c29e6d293cc049ce..1265eb6799ad442027a146370d8be024138bc1b6 100644 --- a/formal/lean/mathlib/linear_algebra/matrix/transvection.lean +++ b/formal/lean/mathlib/linear_algebra/matrix/transvection.lean @@ -132,7 +132,7 @@ variables (R n) /-- A structure containing all the information from which one can build a nontrivial transvection. This structure is easier to manipulate than transvections as one has a direct access to all the relevant fields. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure transvection_struct := (i j : n) (hij : i ≠ j) diff --git a/formal/lean/mathlib/linear_algebra/projective_space/basic.lean b/formal/lean/mathlib/linear_algebra/projective_space/basic.lean index 4be3ed2e0e4a049a9a67161a6b72a93c8ee73530..cde97cc906b6b144cc28ffd532a27591bc8ef6f8 100644 --- a/formal/lean/mathlib/linear_algebra/projective_space/basic.lean +++ b/formal/lean/mathlib/linear_algebra/projective_space/basic.lean @@ -43,7 +43,7 @@ def projectivization_setoid : setoid { v : V // v ≠ 0 } := /-- The projectivization of the `K`-vector space `V`. The notation `ℙ K V` is preferred. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def projectivization := quotient (projectivization_setoid K V) notation `ℙ` := projectivization diff --git a/formal/lean/mathlib/linear_algebra/quadratic_form/isometry.lean b/formal/lean/mathlib/linear_algebra/quadratic_form/isometry.lean index f51eef6dfcb0e339881e471efe1eb31332202d3f..36af1c374f979e5851d7b1c4e51bbf19e318a8b5 100644 --- a/formal/lean/mathlib/linear_algebra/quadratic_form/isometry.lean +++ b/formal/lean/mathlib/linear_algebra/quadratic_form/isometry.lean @@ -30,7 +30,7 @@ variables [module R M] [module R M₁] [module R M₂] [module R M₃] /-- An isometry between two quadratic spaces `M₁, Q₁` and `M₂, Q₂` over a ring `R`, is a linear equivalence between `M₁` and `M₂` that commutes with the quadratic forms. -/ -@[nolint has_inhabited_instance] structure isometry +@[nolint has_nonempty_instance] structure isometry (Q₁ : quadratic_form R M₁) (Q₂ : quadratic_form R M₂) extends M₁ ≃ₗ[R] M₂ := (map_app' : ∀ m, Q₂ (to_fun m) = Q₁ m) diff --git a/formal/lean/mathlib/linear_algebra/quadratic_form/real.lean b/formal/lean/mathlib/linear_algebra/quadratic_form/real.lean index 506ee8226d9c6cb466f1258604be15f8f1ae5ec4..8e172a1e9c1c58fbe933e5c4bcd8ac4d1456fb38 100644 --- a/formal/lean/mathlib/linear_algebra/quadratic_form/real.lean +++ b/formal/lean/mathlib/linear_algebra/quadratic_form/real.lean @@ -55,7 +55,7 @@ begin erw [hsum], simp only [u, function.comp, smul_eq_mul], split_ifs, - { simp only [h, zero_smul, zero_mul, sign_zero] }, + { simp only [h, zero_smul, zero_mul, real.sign_zero] }, have hwu : w j = u j, { simp only [u, dif_neg h, units.coe_mk0] }, simp only [hwu, units.coe_mk0], diff --git a/formal/lean/mathlib/linear_algebra/ray.lean b/formal/lean/mathlib/linear_algebra/ray.lean index e6637cf101f870284934309dc10b44a9d814beac..5692a3e5fc97f81472305fff7fe6f5a808e21369 100644 --- a/formal/lean/mathlib/linear_algebra/ray.lean +++ b/formal/lean/mathlib/linear_algebra/ray.lean @@ -159,7 +159,7 @@ end same_ray /-- Nonzero vectors, as used to define rays. This type depends on an unused argument `R` so that `ray_vector.setoid` can be an instance. -/ -@[nolint unused_arguments has_inhabited_instance] +@[nolint unused_arguments has_nonempty_instance] def ray_vector (R M : Type*) [has_zero M] := {v : M // v ≠ 0} instance ray_vector.has_coe {R M : Type*} [has_zero M] : @@ -177,7 +177,7 @@ instance : setoid (ray_vector R M) := λ x y z hxy hyz, hxy.trans hyz $ λ hy, (y.2 hy).elim⟩ } /-- A ray (equivalence class of nonzero vectors with common positive multiples) in a module. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def module.ray := quotient (ray_vector.setoid R M) variables {R M} diff --git a/formal/lean/mathlib/linear_algebra/smodeq.lean b/formal/lean/mathlib/linear_algebra/smodeq.lean index 7e74a8766c50452510dedd7c47511a38fac6da73..edba4d40fa9ee0cc0aea8804b722b2cec285d83b 100644 --- a/formal/lean/mathlib/linear_algebra/smodeq.lean +++ b/formal/lean/mathlib/linear_algebra/smodeq.lean @@ -44,7 +44,11 @@ by rw [smodeq.def, submodule.quotient.eq, mem_bot, sub_eq_zero] @[mono] theorem mono (HU : U₁ ≤ U₂) (hxy : x ≡ y [SMOD U₁]) : x ≡ y [SMOD U₂] := (submodule.quotient.eq U₂).2 $ HU $ (submodule.quotient.eq U₁).1 hxy -@[refl] theorem refl : x ≡ x [SMOD U] := eq.refl _ +@[refl] protected theorem refl (x : M) : x ≡ x [SMOD U] := @rfl _ _ + +protected theorem rfl : x ≡ x [SMOD U] := smodeq.refl _ + +instance : is_refl _ (smodeq U) := ⟨smodeq.refl⟩ @[symm] theorem symm (hxy : x ≡ y [SMOD U]) : y ≡ x [SMOD U] := hxy.symm diff --git a/formal/lean/mathlib/linear_algebra/span.lean b/formal/lean/mathlib/linear_algebra/span.lean index fd7d3f49fe80bc2a6b8f6a6bba48102a62f5892d..740ade5b903ca2eb415a7fbce64cbcb92510e3d1 100644 --- a/formal/lean/mathlib/linear_algebra/span.lean +++ b/formal/lean/mathlib/linear_algebra/span.lean @@ -505,11 +505,12 @@ lemma not_mem_span_of_apply_not_mem_span_image x ∉ submodule.span R s := h.imp (apply_mem_span_image_of_mem_span f) -lemma supr_eq_span {ι : Sort*} (p : ι → submodule R M) : - (⨆ (i : ι), p i) = submodule.span R (⋃ (i : ι), ↑(p i)) := -le_antisymm - (supr_le $ assume i, subset.trans (assume m hm, set.mem_Union.mpr ⟨i, hm⟩) subset_span) - (span_le.mpr $ Union_subset_iff.mpr $ assume i m hm, mem_supr_of_mem i hm) +lemma supr_span {ι : Sort*} (p : ι → set M) : (⨆ i, span R (p i)) = span R (⋃ i, p i) := +le_antisymm (supr_le $ λ i, span_mono $ subset_Union _ i) $ + span_le.mpr $ Union_subset $ λ i m hm, mem_supr_of_mem i $ subset_span hm + +lemma supr_eq_span {ι : Sort*} (p : ι → submodule R M) : (⨆ i, p i) = span R (⋃ i, ↑(p i)) := +by simp_rw [← supr_span, span_eq] lemma supr_to_add_submonoid {ι : Sort*} (p : ι → submodule R M) : (⨆ i, p i).to_add_submonoid = ⨆ i, (p i).to_add_submonoid := @@ -667,12 +668,11 @@ variables {M' : Type*} [add_comm_monoid M'] [module R M'] (q₁ q₁' : submodul /-- The product of two submodules is a submodule. -/ def prod : submodule R (M × M') := -{ carrier := (p : set M) ×ˢ (q₁ : set M'), +{ carrier := p ×ˢ q₁, smul_mem' := by rintro a ⟨x, y⟩ ⟨hx, hy⟩; exact ⟨smul_mem _ a hx, smul_mem _ a hy⟩, .. p.to_add_submonoid.prod q₁.to_add_submonoid } -@[simp] lemma prod_coe : - (prod p q₁ : set (M × M')) = (p : set M) ×ˢ (q₁ : set M') := rfl +@[simp] lemma prod_coe : (prod p q₁ : set (M × M')) = p ×ˢ q₁ := rfl @[simp] lemma mem_prod {p : submodule R M} {q : submodule R M'} {x : M × M'} : x ∈ prod p q ↔ x.1 ∈ p ∧ x.2 ∈ q := set.mem_prod diff --git a/formal/lean/mathlib/linear_algebra/vandermonde.lean b/formal/lean/mathlib/linear_algebra/vandermonde.lean index b548a2cb00e323ade9ff69840477ccc09dc96f73..17df0625a00a589d36fed6b2c4856bcae333f97b 100644 --- a/formal/lean/mathlib/linear_algebra/vandermonde.lean +++ b/formal/lean/mathlib/linear_algebra/vandermonde.lean @@ -3,11 +3,11 @@ Copyright (c) 2020 Anne Baanen. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Anne Baanen -/ - import algebra.big_operators.fin import algebra.geom_sum import group_theory.perm.fin import linear_algebra.matrix.determinant +import linear_algebra.matrix.nondegenerate /-! # Vandermonde matrix @@ -136,4 +136,19 @@ lemma det_vandermonde_ne_zero_iff [is_domain R] {n : ℕ} {v : fin n → R} : det (vandermonde v) ≠ 0 ↔ function.injective v := by simpa only [det_vandermonde_eq_zero_iff, ne.def, not_exists, not_and, not_not] +theorem eq_zero_of_forall_index_sum_pow_mul_eq_zero {R : Type*} [comm_ring R] + [is_domain R] {n : ℕ} {f v : fin n → R} (hf : function.injective f) + (hfv : ∀ j, ∑ i : fin n, (f j ^ (i : ℕ)) * v i = 0) : v = 0 := +eq_zero_of_mul_vec_eq_zero (det_vandermonde_ne_zero_iff.mpr hf) (funext hfv) + +theorem eq_zero_of_forall_index_sum_mul_pow_eq_zero {R : Type*} [comm_ring R] + [is_domain R] {n : ℕ} {f v : fin n → R} (hf : function.injective f) + (hfv : ∀ j, ∑ i, v i * (f j ^ (i : ℕ)) = 0) : v = 0 := +by { apply eq_zero_of_forall_index_sum_pow_mul_eq_zero hf, simp_rw mul_comm, exact hfv } + +theorem eq_zero_of_forall_pow_sum_mul_pow_eq_zero {R : Type*} [comm_ring R] + [is_domain R] {n : ℕ} {f v : fin n → R} (hf : function.injective f) + (hfv : ∀ i : fin n, ∑ j : fin n, v j * (f j ^ (i : ℕ)) = 0) : v = 0 := +eq_zero_of_vec_mul_eq_zero (det_vandermonde_ne_zero_iff.mpr hf) (funext hfv) + end matrix diff --git a/formal/lean/mathlib/logic/basic.lean b/formal/lean/mathlib/logic/basic.lean index ec97e2eee05f841c8656efc3724084cb3aea2342..8482772a34807c22673e79fa94f133f5eba49671 100644 --- a/formal/lean/mathlib/logic/basic.lean +++ b/formal/lean/mathlib/logic/basic.lean @@ -914,10 +914,10 @@ lemma congr_fun₃ {f g : Π a b c, δ a b c} (h : f = g) (a : α) (b : β a) (c f a b c = g a b c := congr_fun₂ (congr_fun h _) _ _ -lemma funext₂ {f g : Π a, β a → Prop} (h : ∀ a b, f a b = g a b) : f = g := +lemma funext₂ {f g : Π a b, γ a b} (h : ∀ a b, f a b = g a b) : f = g := funext $ λ _, funext $ h _ -lemma funext₃ {f g : Π a b, γ a b → Prop} (h : ∀ a b c, f a b c = g a b c) : f = g := +lemma funext₃ {f g : Π a b c, δ a b c} (h : ∀ a b c, f a b c = g a b c) : f = g := funext $ λ _, funext₂ $ h _ end equality diff --git a/formal/lean/mathlib/logic/embedding.lean b/formal/lean/mathlib/logic/embedding.lean index 2e6de95a72255cadf19cf47b35b060d23791353f..c7ee1e4a6625cfa8b55696c16afc6798087394c3 100644 --- a/formal/lean/mathlib/logic/embedding.lean +++ b/formal/lean/mathlib/logic/embedding.lean @@ -18,7 +18,7 @@ universes u v w x namespace function /-- `α ↪ β` is a bundled injective function. -/ -@[nolint has_inhabited_instance] -- depending on cardinalities, an injective function may not exist +@[nolint has_nonempty_instance] -- depending on cardinalities, an injective function may not exist structure embedding (α : Sort*) (β : Sort*) := (to_fun : α → β) (inj' : injective to_fun) @@ -194,6 +194,12 @@ def subtype {α} (p : α → Prop) : subtype p ↪ α := @[simp] lemma coe_subtype {α} (p : α → Prop) : ⇑(subtype p) = coe := rfl +/-- `quotient.out` as an embedding. -/ +noncomputable def quotient_out (α) [s : setoid α] : quotient s ↪ α := +⟨_, quotient.out_injective⟩ + +@[simp] theorem coe_quotient_out (α) [s : setoid α] : ⇑(quotient_out α) = quotient.out := rfl + /-- Choosing an element `b : β` gives an embedding of `punit` into `β`. -/ def punit {β : Sort*} (b : β) : punit ↪ β := ⟨λ _, b, by { rintros ⟨⟩ ⟨⟩ _, refl, }⟩ diff --git a/formal/lean/mathlib/logic/equiv/basic.lean b/formal/lean/mathlib/logic/equiv/basic.lean index c8b51114f46ecbb22141769ff9fbc15495b86273..af1fdde196ec5cbe660cbde99f62d364adc7f171 100644 --- a/formal/lean/mathlib/logic/equiv/basic.lean +++ b/formal/lean/mathlib/logic/equiv/basic.lean @@ -629,10 +629,28 @@ calc punit × α ≃ α × punit : prod_comm _ _ def prod_unique (α β : Type*) [unique β] : α × β ≃ α := ((equiv.refl α).prod_congr $ equiv_punit β).trans $ prod_punit α +@[simp] lemma coe_prod_unique {α β : Type*} [unique β] : + ⇑(prod_unique α β) = prod.fst := rfl + +lemma prod_unique_apply {α β : Type*} [unique β] (x : α × β) : + prod_unique α β x = x.1 := rfl + +@[simp] lemma prod_unique_symm_apply {α β : Type*} [unique β] (x : α) : + (prod_unique α β).symm x = (x, default) := rfl + /-- Any `unique` type is a left identity for type product up to equivalence. -/ def unique_prod (α β : Type*) [unique β] : β × α ≃ α := ((equiv_punit β).prod_congr $ equiv.refl α).trans $ punit_prod α +@[simp] lemma coe_unique_prod {α β : Type*} [unique β] : + ⇑(unique_prod α β) = prod.snd := rfl + +lemma unique_prod_apply {α β : Type*} [unique β] (x : β × α) : + unique_prod α β x = x.2 := rfl + +@[simp] lemma unique_prod_symm_apply {α β : Type*} [unique β] (x : α) : + (unique_prod α β).symm x = (default, x) := rfl + /-- `empty` type is a right absorbing element for type product up to an equivalence. -/ def prod_empty (α : Type*) : α × empty ≃ empty := equiv_empty _ diff --git a/formal/lean/mathlib/logic/equiv/list.lean b/formal/lean/mathlib/logic/equiv/list.lean index ccfe8e61d4b7fb5e3343121a7c6e5799933803e8..6c73ce175a80efceb0efe8a0282144b2c477c8d5 100644 --- a/formal/lean/mathlib/logic/equiv/list.lean +++ b/formal/lean/mathlib/logic/equiv/list.lean @@ -41,6 +41,9 @@ instance _root_.list.encodable : encodable (list α) := ⟨encode_list, decode_list, λ l, by induction l with a l IH; simp [encode_list, decode_list, unpair_mkpair, encodek, *]⟩ +instance _root_.list.countable {α : Type*} [countable α] : countable (list α) := +by { haveI := encodable.of_countable α, apply_instance } + @[simp] theorem encode_list_nil : encode (@nil α) = 0 := rfl @[simp] theorem encode_list_cons (a : α) (l : list α) : encode (a :: l) = succ (mkpair (encode a) (encode l)) := rfl @@ -88,6 +91,10 @@ instance _root_.multiset.encodable : encodable (multiset α) := ⟨encode_multiset, decode_multiset, λ s, by simp [encode_multiset, decode_multiset, encodek]⟩ +/-- If `α` is countable, then so is `multiset α`. -/ +instance _root_.multiset.countable {α : Type*} [countable α] : countable (multiset α) := +quotient.countable + end finset /-- A listable type with decidable equality is encodable. -/ @@ -112,6 +119,9 @@ by { classical, exact (fintype.trunc_encodable α).out } /-- If `α` is encodable, then so is `vector α n`. -/ instance _root_.vector.encodable [encodable α] {n} : encodable (vector α n) := subtype.encodable +/-- If `α` is countable, then so is `vector α n`. -/ +instance _root_.vector.countable [countable α] {n} : countable (vector α n) := subtype.countable + /-- If `α` is encodable, then so is `fin n → α`. -/ instance fin_arrow [encodable α] {n} : encodable (fin n → α) := of_equiv _ (equiv.vector_equiv_fin _ _).symm @@ -123,12 +133,20 @@ of_equiv _ (equiv.pi_equiv_subtype_sigma (fin n) π) instance _root_.array.encodable [encodable α] {n} : encodable (array n α) := of_equiv _ (equiv.array_equiv_fin _ _) +/-- If `α` is countable, then so is `array n α`. -/ +instance _root_.array.countable [countable α] {n} : countable (array n α) := +countable.of_equiv _ (equiv.vector_equiv_array _ _) + /-- If `α` is encodable, then so is `finset α`. -/ instance _root_.finset.encodable [encodable α] : encodable (finset α) := by haveI := decidable_eq_of_encodable α; exact of_equiv {s : multiset α // s.nodup} ⟨λ ⟨a, b⟩, ⟨a, b⟩, λ ⟨a, b⟩, ⟨a, b⟩, λ ⟨a, b⟩, rfl, λ ⟨a, b⟩, rfl⟩ +/-- If `α` is countable, then so is `finset α`. -/ +instance _root_.finset.countable [countable α] : countable (finset α) := +finset.val_injective.countable + -- TODO: Unify with `fintype_pi` and find a better name /-- When `α` is finite and `β` is encodable, `α → β` is encodable too. Because the encoding is not unique, we wrap it in `trunc` to preserve computability. -/ diff --git a/formal/lean/mathlib/logic/equiv/set.lean b/formal/lean/mathlib/logic/equiv/set.lean index 2ae28c23b6f059b9e00a120970e9ad604dae8c2d..4d6c03580206ae04fcc0cb86cf1cce4ae2619d3d 100644 --- a/formal/lean/mathlib/logic/equiv/set.lean +++ b/formal/lean/mathlib/logic/equiv/set.lean @@ -503,7 +503,7 @@ e.injective.preimage_surjective.forall lemma preimage_pi_equiv_pi_subtype_prod_symm_pi {α : Type*} {β : α → Type*} (p : α → Prop) [decidable_pred p] (s : Π i, set (β i)) : (pi_equiv_pi_subtype_prod p β).symm ⁻¹' pi univ s = - (pi univ (λ i : {i // p i}, s i)) ×ˢ (pi univ (λ i : {i // ¬p i}, s i)) := + (pi univ (λ i : {i // p i}, s i)) ×ˢ pi univ (λ i : {i // ¬p i}, s i) := begin ext ⟨f, g⟩, simp only [mem_preimage, mem_univ_pi, prod_mk_mem_set_prod_eq, subtype.forall, diff --git a/formal/lean/mathlib/logic/function/basic.lean b/formal/lean/mathlib/logic/function/basic.lean index 7af69f462df5073ea4ac68f3f0989f74a2e65814..c80b3d209d686ae61222484556c1b91fabb7bbba 100644 --- a/formal/lean/mathlib/logic/function/basic.lean +++ b/formal/lean/mathlib/logic/function/basic.lean @@ -86,7 +86,7 @@ h ▸ hf.ne_iff /-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then the domain `α` also has decidable equality. -/ -def injective.decidable_eq [decidable_eq β] (I : injective f) : decidable_eq α := +protected def injective.decidable_eq [decidable_eq β] (I : injective f) : decidable_eq α := λ a b, decidable_of_iff _ I.eq_iff lemma injective.of_comp {g : γ → α} (I : injective (f ∘ g)) : injective g := diff --git a/formal/lean/mathlib/logic/small.lean b/formal/lean/mathlib/logic/small.lean index e2dc1c96593d108d9887823356e119bfab1aad20..ae110d8e29457b2c5d9db58b53e4e9de1e6f24a0 100644 --- a/formal/lean/mathlib/logic/small.lean +++ b/formal/lean/mathlib/logic/small.lean @@ -34,7 +34,7 @@ lemma small.mk' {α : Type v} {S : Type w} (e : α ≃ S) : small.{w} α := /-- An arbitrarily chosen model in `Type w` for a `w`-small type. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def shrink (α : Type v) [small.{w} α] : Type w := classical.some (@small.equiv_small α _) diff --git a/formal/lean/mathlib/measure_theory/covering/besicovitch.lean b/formal/lean/mathlib/measure_theory/covering/besicovitch.lean index 2231aa2ef485448cece924bb9a7ad83e71fa5bc9..838ed37d0037124c69a37247ec6c5346a040dc0f 100644 --- a/formal/lean/mathlib/measure_theory/covering/besicovitch.lean +++ b/formal/lean/mathlib/measure_theory/covering/besicovitch.lean @@ -557,7 +557,7 @@ begin simp only [finset.card_fin, finset.sum_const, nsmul_eq_mul], rw ennreal.mul_div_cancel', { simp only [Npos, ne.def, nat.cast_eq_zero, not_false_iff] }, - { exact ennreal.coe_nat_ne_top } + { exact (ennreal.nat_ne_top _) } end ... ≤ ∑ i, μ (s ∩ v i) : by { conv_lhs { rw A }, apply measure_Union_fintype_le }, -- choose an index `i` of a subfamily covering at least a proportion `1/N` of `s`. diff --git a/formal/lean/mathlib/measure_theory/covering/vitali.lean b/formal/lean/mathlib/measure_theory/covering/vitali.lean index a618a36cee83bce2a02e2345a80887751834e6f9..7836842778ca8ad9b8ebb1176a3ea51ee2218ff0 100644 --- a/formal/lean/mathlib/measure_theory/covering/vitali.lean +++ b/formal/lean/mathlib/measure_theory/covering/vitali.lean @@ -303,6 +303,7 @@ begin let v := {a ∈ u | (a ∩ ball x (r x)).nonempty }, have vu : v ⊆ u := λ a ha, ha.1, -- they are all contained in a fixed ball of finite measure, thanks to our choice of `t'` + obtain ⟨R, μR, hR⟩ : ∃ R, μ (closed_ball x R) < ∞ ∧ ∀ a ∈ u, (a ∩ ball x (r x)).nonempty → a ⊆ closed_ball x R, { have : ∀ a ∈ u, ∃ y, a ⊆ closed_ball y (r y) := λ a hau, (ut' hau).2, @@ -348,8 +349,8 @@ begin nonpos_iff_eq_zero.mp (le_of_forall_le_of_dense (λ ε εpos, _))⟩, -- the elements of `v` are disjoint and all contained in a finite volume ball, hence the sum -- of their measures is finite. - have I : ∑' (a : v), μ a < ∞, - { calc ∑' (a : v), μ a = μ (⋃ (a ∈ v), a) : begin + have I : ∑' (a : v), μ (↑a) < ∞, + { calc ∑' (a : v), μ (↑a) = μ (⋃ (a ∈ v), a) : begin rw measure_bUnion (u_count.mono vu) _ (λ a ha, (h't _ (vu.trans ut ha)).measurable_set), exact u_disj.subset vu end @@ -357,7 +358,7 @@ begin ... < ∞ : μR }, -- we can obtain a finite subfamily of `v`, such that the measures of the remaining elements -- add up to an arbitrarily small number, say `ε / C`. - obtain ⟨w, hw⟩ : ∃ (w : finset ↥v), ∑' (a : {a // a ∉ w}), μ a < ε / C, + obtain ⟨w, hw⟩ : ∃ (w : finset ↥v), ∑' (a : {a // a ∉ w}), μ (↑a) < ε / C, { haveI : ne_bot (at_top : filter (finset v)) := at_top_ne_bot, have : 0 < ε / C, by simp only [ennreal.div_pos_iff, εpos.ne', ennreal.coe_ne_top, ne.def, not_false_iff, and_self], @@ -368,7 +369,7 @@ begin have M : (s \ ⋃ (a : set α) (H : a ∈ u), a) ∩ ball x (r x) ⊆ ⋃ (a : {a // a ∉ w}), closed_ball (y a) (3 * diam (a : set α)), { assume z hz, - set k := ⋃ (a : v) (ha : a ∈ w), (a : set α) with hk, + set k := ⋃ (a : v) (ha : a ∈ w), (↑a : set α) with hk, have k_closed : is_closed k := is_closed_bUnion w.finite_to_set (λ i hi, h't _ (ut (vu i.2))), have z_notmem_k : z ∉ k, @@ -401,7 +402,7 @@ begin -- contrary to `b` have b'_notmem_w : b' ∉ w, { assume b'w, - have b'k : (b' : set α) ⊆ k := finset.subset_set_bUnion_of_mem b'w, + have b'k : (↑b' : set α) ⊆ k := finset.subset_set_bUnion_of_mem b'w, have : ((ball x (r x) \ k) ∩ k).nonempty := ab.mono (inter_subset_inter (ad.trans hd) b'k), simpa only [diff_inter_self, not_nonempty_empty] }, let b'' : {a // a ∉ w} := ⟨b', b'_notmem_w⟩, @@ -415,19 +416,20 @@ begin apply dist_le_diam_of_mem (bounded_closed_ball.mono hc) eb (hy b (ut bu)).1 }, simp only [mem_closed_ball], linarith [dist_triangle z e (y b)] }, - suffices H : closed_ball (y (b'' : set α)) (3 * diam (b'' : set α)) - ⊆ ⋃ (a : {a // a ∉ w}), closed_ball (y (a : set α)) (3 * diam (a : set α)), from H zb, - exact subset_Union (λ (a : {a // a ∉ w}), closed_ball (y a) (3 * diam (a : set α))) b'' }, + suffices H : closed_ball (y (↑b'')) (3 * diam (↑b'' : set α)) + ⊆ ⋃ (a : {a // a ∉ w}), closed_ball (y (↑a)) (3 * diam (↑a : set α)), from H zb, + exact subset_Union (λ (a : {a // a ∉ w}), closed_ball (y (↑a)) (3 * diam (↑a : set α))) b'' }, -- now that we have proved our main inclusion, we can use it to estimate the measure of the points -- in `ball x (r x)` not covered by `u`. haveI : encodable v := (u_count.mono vu).to_encodable, calc μ ((s \ ⋃ (a : set α) (H : a ∈ u), a) ∩ ball x (r x)) - ≤ μ (⋃ (a : {a // a ∉ w}), closed_ball (y a) (3 * diam (a : set α))) : measure_mono M - ... ≤ ∑' (a : {a // a ∉ w}), μ (closed_ball (y a) (3 * diam (a : set α))) : measure_Union_le _ - ... ≤ ∑' (a : {a // a ∉ w}), C * μ a : ennreal.tsum_le_tsum (λ a, (hy a (ut (vu a.1.2))).2) - ... = C * ∑' (a : {a // a ∉ w}), μ a : ennreal.tsum_mul_left + ≤ μ (⋃ (a : {a // a ∉ w}), closed_ball (y (↑a)) (3 * diam (↑a : set α))) : measure_mono M + ... ≤ ∑' (a : {a // a ∉ w}), μ (closed_ball (y (↑a)) (3 * diam (↑a : set α))) : + measure_Union_le _ + ... ≤ ∑' (a : {a // a ∉ w}), C * μ (↑a) : ennreal.tsum_le_tsum (λ a, (hy a (ut (vu a.1.2))).2) + ... = C * ∑' (a : {a // a ∉ w}), μ (↑a) : ennreal.tsum_mul_left ... ≤ C * (ε / C) : ennreal.mul_le_mul le_rfl hw.le - ... ≤ ε : ennreal.mul_div_le, + ... ≤ ε : ennreal.mul_div_le end /-- Assume that around every point there are arbitrarily small scales at which the measure is diff --git a/formal/lean/mathlib/measure_theory/covering/vitali_family.lean b/formal/lean/mathlib/measure_theory/covering/vitali_family.lean index 192072a166498ec25a05eb65e098a3b5deefe8ac..bd0d9afdbf04ad37c74f4f3698f048f8e8018a95 100644 --- a/formal/lean/mathlib/measure_theory/covering/vitali_family.lean +++ b/formal/lean/mathlib/measure_theory/covering/vitali_family.lean @@ -60,7 +60,7 @@ Vitali families are provided by covering theorems such as the Besicovitch coveri Vitali covering theorem. They make it possible to formulate general versions of theorems on differentiations of measure that apply in both contexts. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure vitali_family {m : measurable_space α} (μ : measure α) := (sets_at : Π (x : α), set (set α)) (measurable_set' : ∀ (x : α), ∀ (a : set α), a ∈ sets_at x → measurable_set a) diff --git a/formal/lean/mathlib/measure_theory/function/ae_eq_of_integral.lean b/formal/lean/mathlib/measure_theory/function/ae_eq_of_integral.lean index 137691bae68987a429998f31584b2185c700c1f0..3b634734019f781b4534a5fdb1741ef70ff4081d 100644 --- a/formal/lean/mathlib/measure_theory/function/ae_eq_of_integral.lean +++ b/formal/lean/mathlib/measure_theory/function/ae_eq_of_integral.lean @@ -236,14 +236,12 @@ end ennreal section real -section real_finite_measure +variables {f : α → ℝ} -variables [is_finite_measure μ] {f : α → ℝ} - -/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg_of_finite_measure`. -/ -lemma ae_nonneg_of_forall_set_integral_nonneg_of_finite_measure_of_strongly_measurable +/-- Don't use this lemma. Use `ae_nonneg_of_forall_set_integral_nonneg`. -/ +lemma ae_nonneg_of_forall_set_integral_nonneg_of_strongly_measurable (hfm : strongly_measurable f) - (hf : integrable f μ) (hf_zero : ∀ s, measurable_set s → 0 ≤ ∫ x in s, f x ∂μ) : + (hf : integrable f μ) (hf_zero : ∀ s, measurable_set s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f := begin simp_rw [eventually_le, pi.zero_apply], @@ -251,57 +249,70 @@ begin intros b hb_neg, let s := {x | f x ≤ b}, have hs : measurable_set s, from hfm.measurable_set_le strongly_measurable_const, + have mus : μ s < ∞, + { let c : ℝ≥0 := ⟨|b|, abs_nonneg _⟩, + have c_pos : (c : ℝ≥0∞) ≠ 0, by simpa using hb_neg.ne, + calc μ s ≤ μ {x | (c : ℝ≥0∞) ≤ ∥f x∥₊} : + begin + apply measure_mono, + assume x hx, + simp only [set.mem_set_of_eq] at hx, + simpa only [nnnorm, abs_of_neg hb_neg, abs_of_neg (hx.trans_lt hb_neg), real.norm_eq_abs, + subtype.mk_le_mk, neg_le_neg_iff, set.mem_set_of_eq, ennreal.coe_le_coe] using hx, + end + ... ≤ (∫⁻ x, ∥f x∥₊ ∂μ) / c : + meas_ge_le_lintegral_div hfm.ae_measurable.ennnorm c_pos ennreal.coe_ne_top + ... < ∞ : ennreal.div_lt_top (ne_of_lt hf.2) c_pos }, have h_int_gt : ∫ x in s, f x ∂μ ≤ b * (μ s).to_real, { have h_const_le : ∫ x in s, f x ∂μ ≤ ∫ x in s, b ∂μ, { refine set_integral_mono_ae_restrict hf.integrable_on - (integrable_on_const.mpr (or.inr (measure_lt_top μ s))) _, + (integrable_on_const.mpr (or.inr mus)) _, rw [eventually_le, ae_restrict_iff hs], exact eventually_of_forall (λ x hxs, hxs), }, rwa [set_integral_const, smul_eq_mul, mul_comm] at h_const_le, }, by_contra, refine (lt_self_iff_false (∫ x in s, f x ∂μ)).mp (h_int_gt.trans_lt _), refine (mul_neg_iff.mpr (or.inr ⟨hb_neg, _⟩)).trans_le _, - swap, { simp_rw measure.restrict_restrict hs, exact hf_zero s hs, }, + swap, { simp_rw measure.restrict_restrict hs, exact hf_zero s hs mus, }, refine (ennreal.to_real_nonneg).lt_of_ne (λ h_eq, h _), cases (ennreal.to_real_eq_zero_iff _).mp h_eq.symm with hμs_eq_zero hμs_eq_top, { exact hμs_eq_zero, }, - { exact absurd hμs_eq_top (measure_lt_top μ s).ne, }, + { exact absurd hμs_eq_top mus.ne, }, end -lemma ae_nonneg_of_forall_set_integral_nonneg_of_finite_measure (hf : integrable f μ) - (hf_zero : ∀ s, measurable_set s → 0 ≤ ∫ x in s, f x ∂μ) : +lemma ae_nonneg_of_forall_set_integral_nonneg (hf : integrable f μ) + (hf_zero : ∀ s, measurable_set s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f := begin rcases hf.1 with ⟨f', hf'_meas, hf_ae⟩, have hf'_integrable : integrable f' μ, from integrable.congr hf hf_ae, - have hf'_zero : ∀ s, measurable_set s → 0 ≤ ∫ x in s, f' x ∂μ, - { intros s hs, + have hf'_zero : ∀ s, measurable_set s → μ s < ∞ → 0 ≤ ∫ x in s, f' x ∂μ, + { intros s hs h's, rw set_integral_congr_ae hs (hf_ae.mono (λ x hx hxs, hx.symm)), - exact hf_zero s hs, }, - exact (ae_nonneg_of_forall_set_integral_nonneg_of_finite_measure_of_strongly_measurable hf'_meas + exact hf_zero s hs h's, }, + exact (ae_nonneg_of_forall_set_integral_nonneg_of_strongly_measurable hf'_meas hf'_integrable hf'_zero).trans hf_ae.symm.le, end lemma ae_le_of_forall_set_integral_le {f g : α → ℝ} (hf : integrable f μ) (hg : integrable g μ) - (hf_le : ∀ s, measurable_set s → ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ) : + (hf_le : ∀ s, measurable_set s → μ s < ∞ → ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ) : f ≤ᵐ[μ] g := begin rw ← eventually_sub_nonneg, - refine ae_nonneg_of_forall_set_integral_nonneg_of_finite_measure (hg.sub hf) (λ s hs, _), + refine ae_nonneg_of_forall_set_integral_nonneg (hg.sub hf) (λ s hs, _), rw [integral_sub' hg.integrable_on hf.integrable_on, sub_nonneg], exact hf_le s hs end -end real_finite_measure - -lemma ae_nonneg_restrict_of_forall_set_integral_nonneg_inter {f : α → ℝ} {t : set α} (hμt : μ t ≠ ∞) - (hf : integrable_on f t μ) (hf_zero : ∀ s, measurable_set s → 0 ≤ ∫ x in (s ∩ t), f x ∂μ) : +lemma ae_nonneg_restrict_of_forall_set_integral_nonneg_inter {f : α → ℝ} {t : set α} + (hf : integrable_on f t μ) + (hf_zero : ∀ s, measurable_set s → μ (s ∩ t) < ∞ → 0 ≤ ∫ x in (s ∩ t), f x ∂μ) : 0 ≤ᵐ[μ.restrict t] f := begin - haveI : fact (μ t < ∞) := ⟨lt_top_iff_ne_top.mpr hμt⟩, - refine ae_nonneg_of_forall_set_integral_nonneg_of_finite_measure hf (λ s hs, _), + refine ae_nonneg_of_forall_set_integral_nonneg hf (λ s hs h's, _), simp_rw measure.restrict_restrict hs, - exact hf_zero s hs, + apply hf_zero s hs, + rwa measure.restrict_apply hs at h's, end lemma ae_nonneg_of_forall_set_integral_nonneg_of_sigma_finite [sigma_finite μ] {f : α → ℝ} @@ -311,9 +322,9 @@ lemma ae_nonneg_of_forall_set_integral_nonneg_of_sigma_finite [sigma_finite μ] begin apply ae_of_forall_measure_lt_top_ae_restrict, assume t t_meas t_lt_top, - apply ae_nonneg_restrict_of_forall_set_integral_nonneg_inter t_lt_top.ne + apply ae_nonneg_restrict_of_forall_set_integral_nonneg_inter (hf_int_finite t t_meas t_lt_top), - assume s s_meas, + assume s s_meas hs, exact hf_zero _ (s_meas.inter t_meas) (lt_of_le_of_lt (measure_mono (set.inter_subset_right _ _)) t_lt_top) end @@ -338,20 +349,14 @@ begin exact hf_zero (s ∩ t) (hs.inter hf.measurable_set) hμts, }, end -lemma integrable.ae_nonneg_of_forall_set_integral_nonneg {f : α → ℝ} (hf : integrable f μ) - (hf_zero : ∀ s, measurable_set s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : - 0 ≤ᵐ[μ] f := -ae_fin_strongly_measurable.ae_nonneg_of_forall_set_integral_nonneg hf.ae_fin_strongly_measurable - (λ s hs hμs, hf.integrable_on) hf_zero - lemma ae_nonneg_restrict_of_forall_set_integral_nonneg {f : α → ℝ} (hf_int_finite : ∀ s, measurable_set s → μ s < ∞ → integrable_on f s μ) (hf_zero : ∀ s, measurable_set s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) {t : set α} (ht : measurable_set t) (hμt : μ t ≠ ∞) : 0 ≤ᵐ[μ.restrict t] f := begin - refine ae_nonneg_restrict_of_forall_set_integral_nonneg_inter hμt - (hf_int_finite t ht (lt_top_iff_ne_top.mpr hμt)) (λ s hs, _), + refine ae_nonneg_restrict_of_forall_set_integral_nonneg_inter + (hf_int_finite t ht (lt_top_iff_ne_top.mpr hμt)) (λ s hs h's, _), refine (hf_zero (s ∩ t) (hs.inter ht) _), exact (measure_mono (set.inter_subset_right s t)).trans_lt (lt_top_iff_ne_top.mpr hμt), end diff --git a/formal/lean/mathlib/measure_theory/function/conditional_expectation.lean b/formal/lean/mathlib/measure_theory/function/conditional_expectation/basic.lean similarity index 82% rename from formal/lean/mathlib/measure_theory/function/conditional_expectation.lean rename to formal/lean/mathlib/measure_theory/function/conditional_expectation/basic.lean index 9c14580e7f849b59fb09048b55ac1d542683c497..2005c88e1d8c9f490a7a3a8f32f461c3c50b2a46 100644 --- a/formal/lean/mathlib/measure_theory/function/conditional_expectation.lean +++ b/formal/lean/mathlib/measure_theory/function/conditional_expectation/basic.lean @@ -69,9 +69,6 @@ However, some lemmas also use `𝕜 : is_R_or_C`: * results about scalar multiplication are stated not only for `ℝ` but also for `𝕜` if we happen to have `normed_space 𝕜 F`. -TODO: split this file in two with one containing constructions and the other with basic -properties. - ## Tags conditional expectation, conditional expected value @@ -2189,446 +2186,6 @@ begin exact tendsto_nhds_unique_of_eventually_eq hcond_gs hcond_fs (eventually_of_forall hn_eq), end -section indicator - -lemma condexp_ae_eq_restrict_zero (hs : measurable_set[m] s) (hf : f =ᵐ[μ.restrict s] 0) : - μ[f | m] =ᵐ[μ.restrict s] 0 := -begin - by_cases hm : m ≤ m0, - swap, { simp_rw condexp_of_not_le hm, }, - by_cases hμm : sigma_finite (μ.trim hm), - swap, { simp_rw condexp_of_not_sigma_finite hm hμm, }, - haveI : sigma_finite (μ.trim hm) := hμm, - haveI : sigma_finite ((μ.restrict s).trim hm), - { rw ← restrict_trim hm _ hs, - exact restrict.sigma_finite _ s, }, - by_cases hf_int : integrable f μ, - swap, { exact ae_restrict_of_ae (condexp_undef hf_int), }, - refine ae_eq_of_forall_set_integral_eq_of_sigma_finite' hm _ _ _ _ _, - { exact λ t ht hμt, integrable_condexp.integrable_on.integrable_on, }, - { exact λ t ht hμt, (integrable_zero _ _ _).integrable_on, }, - { intros t ht hμt, - rw [measure.restrict_restrict (hm _ ht), set_integral_condexp hm hf_int (ht.inter hs), - ← measure.restrict_restrict (hm _ ht)], - refine set_integral_congr_ae (hm _ ht) _, - filter_upwards [hf] with x hx h using hx, }, - { exact strongly_measurable_condexp.ae_strongly_measurable', }, - { exact strongly_measurable_zero.ae_strongly_measurable', }, -end - -/-- Auxiliary lemma for `condexp_indicator`. -/ -lemma condexp_indicator_aux (hs : measurable_set[m] s) (hf : f =ᵐ[μ.restrict sᶜ] 0) : - μ[s.indicator f | m] =ᵐ[μ] s.indicator (μ[f | m]) := -begin - by_cases hm : m ≤ m0, - swap, { simp_rw [condexp_of_not_le hm, set.indicator_zero'], }, - have hsf_zero : ∀ g : α → F', g =ᵐ[μ.restrict sᶜ] 0 → s.indicator g =ᵐ[μ] g, - from λ g, indicator_ae_eq_of_restrict_compl_ae_eq_zero (hm _ hs), - refine ((hsf_zero (μ[f | m]) (condexp_ae_eq_restrict_zero hs.compl hf)).trans _).symm, - exact condexp_congr_ae (hsf_zero f hf).symm, -end - -/-- The conditional expectation of the indicator of a function over an `m`-measurable set with -respect to the σ-algebra `m` is a.e. equal to the indicator of the conditional expectation. -/ -lemma condexp_indicator (hf_int : integrable f μ) (hs : measurable_set[m] s) : - μ[s.indicator f | m] =ᵐ[μ] s.indicator (μ[f | m]) := -begin - by_cases hm : m ≤ m0, - swap, { simp_rw [condexp_of_not_le hm, set.indicator_zero'], }, - by_cases hμm : sigma_finite (μ.trim hm), - swap, { simp_rw [condexp_of_not_sigma_finite hm hμm, set.indicator_zero'], }, - haveI : sigma_finite (μ.trim hm) := hμm, - -- use `have` to perform what should be the first calc step because of an error I don't - -- understand - have : s.indicator (μ[f|m]) =ᵐ[μ] s.indicator (μ[s.indicator f + sᶜ.indicator f|m]), - by rw set.indicator_self_add_compl s f, - refine (this.trans _).symm, - calc s.indicator (μ[s.indicator f + sᶜ.indicator f|m]) - =ᵐ[μ] s.indicator (μ[s.indicator f|m] + μ[sᶜ.indicator f|m]) : - begin - have : μ[s.indicator f + sᶜ.indicator f|m] =ᵐ[μ] μ[s.indicator f|m] + μ[sᶜ.indicator f|m], - from condexp_add (hf_int.indicator (hm _ hs)) (hf_int.indicator (hm _ hs.compl)), - filter_upwards [this] with x hx, - classical, - rw [set.indicator_apply, set.indicator_apply, hx], - end - ... = s.indicator (μ[s.indicator f|m]) + s.indicator (μ[sᶜ.indicator f|m]) : - s.indicator_add' _ _ - ... =ᵐ[μ] s.indicator (μ[s.indicator f|m]) + s.indicator (sᶜ.indicator (μ[sᶜ.indicator f|m])) : - begin - refine filter.eventually_eq.rfl.add _, - have : sᶜ.indicator (μ[sᶜ.indicator f|m]) =ᵐ[μ] μ[sᶜ.indicator f|m], - { refine (condexp_indicator_aux hs.compl _).symm.trans _, - { exact indicator_ae_eq_restrict_compl (hm _ hs.compl), }, - { rw [set.indicator_indicator, set.inter_self], }, }, - filter_upwards [this] with x hx, - by_cases hxs : x ∈ s, - { simp only [hx, hxs, set.indicator_of_mem], }, - { simp only [hxs, set.indicator_of_not_mem, not_false_iff], }, - end - ... =ᵐ[μ] s.indicator (μ[s.indicator f|m]) : - by rw [set.indicator_indicator, set.inter_compl_self, set.indicator_empty', add_zero] - ... =ᵐ[μ] μ[s.indicator f|m] : - begin - refine (condexp_indicator_aux hs _).symm.trans _, - { exact indicator_ae_eq_restrict_compl (hm _ hs), }, - { rw [set.indicator_indicator, set.inter_self], }, - end -end - -lemma condexp_restrict_ae_eq_restrict (hm : m ≤ m0) [sigma_finite (μ.trim hm)] - (hs_m : measurable_set[m] s) (hf_int : integrable f μ) : - (μ.restrict s)[f | m] =ᵐ[μ.restrict s] μ[f | m] := -begin - haveI : sigma_finite ((μ.restrict s).trim hm), - { rw ← restrict_trim hm _ hs_m, apply_instance, }, - rw ae_eq_restrict_iff_indicator_ae_eq (hm _ hs_m), - swap, { apply_instance, }, - refine eventually_eq.trans _ (condexp_indicator hf_int hs_m), - refine ae_eq_condexp_of_forall_set_integral_eq hm (hf_int.indicator (hm _ hs_m)) _ _ _, - { intros t ht hμt, - rw [← integrable_indicator_iff (hm _ ht), set.indicator_indicator, set.inter_comm, - ← set.indicator_indicator], - suffices h_int_restrict : integrable (t.indicator ((μ.restrict s)[f|m])) (μ.restrict s), - { rw [integrable_indicator_iff (hm _ hs_m), integrable_on], - rw [integrable_indicator_iff (hm _ ht), integrable_on] at h_int_restrict ⊢, - exact h_int_restrict, }, - exact integrable_condexp.indicator (hm _ ht), }, - { intros t ht hμt, - calc ∫ x in t, s.indicator ((μ.restrict s)[f|m]) x ∂μ - = ∫ x in t, ((μ.restrict s)[f|m]) x ∂(μ.restrict s) : - by rw [integral_indicator (hm _ hs_m), measure.restrict_restrict (hm _ hs_m), - measure.restrict_restrict (hm _ ht), set.inter_comm] - ... = ∫ x in t, f x ∂(μ.restrict s) : set_integral_condexp hm hf_int.integrable_on ht - ... = ∫ x in t, s.indicator f x ∂μ : - by rw [integral_indicator (hm _ hs_m), measure.restrict_restrict (hm _ hs_m), - measure.restrict_restrict (hm _ ht), set.inter_comm], }, - { exact (strongly_measurable_condexp.indicator hs_m).ae_strongly_measurable', }, -end - -/-- If the restriction to a `m`-measurable set `s` of a σ-algebra `m` is equal to the restriction -to `s` of another σ-algebra `m₂` (hypothesis `hs`), then `μ[f | m] =ᵐ[μ.restrict s] μ[f | m₂]`. -/ -lemma condexp_ae_eq_restrict_of_measurable_space_eq_on {m m₂ m0 : measurable_space α} - {μ : measure α} (hm : m ≤ m0) (hm₂ : m₂ ≤ m0) - [sigma_finite (μ.trim hm)] [sigma_finite (μ.trim hm₂)] - (hs_m : measurable_set[m] s) (hs : ∀ t, measurable_set[m] (s ∩ t) ↔ measurable_set[m₂] (s ∩ t)) : - μ[f | m] =ᵐ[μ.restrict s] μ[f | m₂] := -begin - rw ae_eq_restrict_iff_indicator_ae_eq (hm _ hs_m), - have hs_m₂ : measurable_set[m₂] s, - { rwa [← set.inter_univ s, ← hs set.univ, set.inter_univ], }, - by_cases hf_int : integrable f μ, - swap, - { filter_upwards [@condexp_undef _ _ _ _ _ m _ μ _ hf_int, - @condexp_undef _ _ _ _ _ m₂ _ μ _ hf_int] with x hxm hxm₂, - simp only [set.indicator_apply, hxm, hxm₂], }, - refine ((condexp_indicator hf_int hs_m).symm.trans _).trans (condexp_indicator hf_int hs_m₂), - refine ae_eq_of_forall_set_integral_eq_of_sigma_finite' hm₂ - (λ s hs hμs, integrable_condexp.integrable_on) - (λ s hs hμs, integrable_condexp.integrable_on) _ _ - strongly_measurable_condexp.ae_strongly_measurable', - swap, - { have : strongly_measurable[m] (μ[s.indicator f | m]) := strongly_measurable_condexp, - refine this.ae_strongly_measurable'.ae_strongly_measurable'_of_measurable_space_le_on - hm hs_m (λ t, (hs t).mp) _, - exact condexp_ae_eq_restrict_zero hs_m.compl (indicator_ae_eq_restrict_compl (hm _ hs_m)), }, - intros t ht hμt, - have : ∫ x in t, μ[s.indicator f|m] x ∂μ = ∫ x in s ∩ t, μ[s.indicator f|m] x ∂μ, - { rw ← integral_add_compl (hm _ hs_m) integrable_condexp.integrable_on, - suffices : ∫ x in sᶜ, μ[s.indicator f|m] x ∂μ.restrict t = 0, - by rw [this, add_zero, measure.restrict_restrict (hm _ hs_m)], - rw measure.restrict_restrict (measurable_set.compl (hm _ hs_m)), - suffices : μ[s.indicator f|m] =ᵐ[μ.restrict sᶜ] 0, - { rw [set.inter_comm, ← measure.restrict_restrict (hm₂ _ ht)], - calc ∫ (x : α) in t, μ[s.indicator f|m] x ∂μ.restrict sᶜ - = ∫ (x : α) in t, 0 ∂μ.restrict sᶜ : begin - refine set_integral_congr_ae (hm₂ _ ht) _, - filter_upwards [this] with x hx h using hx, - end - ... = 0 : integral_zero _ _, }, - refine condexp_ae_eq_restrict_zero hs_m.compl _, - exact indicator_ae_eq_restrict_compl (hm _ hs_m), }, - have hst_m : measurable_set[m] (s ∩ t) := (hs _).mpr (hs_m₂.inter ht), - simp_rw [this, set_integral_condexp hm₂ (hf_int.indicator (hm _ hs_m)) ht, - set_integral_condexp hm (hf_int.indicator (hm _ hs_m)) hst_m, - integral_indicator (hm _ hs_m), measure.restrict_restrict (hm _ hs_m), - ← set.inter_assoc, set.inter_self], -end - -end indicator - -section pull_out --- TODO: this section could be generalized beyond multiplication, to any bounded bilinear map. - -/-- Auxiliary lemma for `condexp_measurable_mul`. -/ -lemma condexp_strongly_measurable_simple_func_mul (hm : m ≤ m0) - (f : @simple_func α m ℝ) {g : α → ℝ} (hg : integrable g μ) : - μ[f * g | m] =ᵐ[μ] f * μ[g | m] := -begin - have : ∀ s c (f : α → ℝ), set.indicator s (function.const α c) * f = s.indicator (c • f), - { intros s c f, - ext1 x, - by_cases hx : x ∈ s, - { simp only [hx, pi.mul_apply, set.indicator_of_mem, pi.smul_apply, algebra.id.smul_eq_mul] }, - { simp only [hx, pi.mul_apply, set.indicator_of_not_mem, not_false_iff, zero_mul], }, }, - refine @simple_func.induction _ _ m _ _ (λ c s hs, _) (λ g₁ g₂ h_disj h_eq₁ h_eq₂, _) f, - { simp only [simple_func.const_zero, simple_func.coe_piecewise, simple_func.coe_const, - simple_func.coe_zero, set.piecewise_eq_indicator], - rw [this, this], - refine (condexp_indicator (hg.smul c) hs).trans _, - filter_upwards [@condexp_smul α ℝ ℝ _ _ _ _ _ m m0 μ c g] with x hx, - classical, - simp_rw [set.indicator_apply, hx], }, - { have h_add := @simple_func.coe_add _ _ m _ g₁ g₂, - calc μ[⇑(g₁ + g₂) * g|m] =ᵐ[μ] μ[(⇑g₁ + ⇑g₂) * g|m] : - by { refine condexp_congr_ae (eventually_eq.mul _ eventually_eq.rfl), rw h_add, } - ... =ᵐ[μ] μ[⇑g₁ * g|m] + μ[⇑g₂ * g|m] : - by { rw add_mul, exact condexp_add (hg.simple_func_mul' hm _) (hg.simple_func_mul' hm _), } - ... =ᵐ[μ] ⇑g₁ * μ[g|m] + ⇑g₂ * μ[g|m] : eventually_eq.add h_eq₁ h_eq₂ - ... =ᵐ[μ] ⇑(g₁ + g₂) * μ[g|m] : by rw [h_add, add_mul], }, -end - -lemma condexp_strongly_measurable_mul_of_bound (hm : m ≤ m0) [is_finite_measure μ] - {f g : α → ℝ} (hf : strongly_measurable[m] f) (hg : integrable g μ) (c : ℝ) - (hf_bound : ∀ᵐ x ∂μ, ∥f x∥ ≤ c) : - μ[f * g | m] =ᵐ[μ] f * μ[g | m] := -begin - let fs := hf.approx_bounded c, - have hfs_tendsto : ∀ᵐ x ∂μ, tendsto (λ n, fs n x) at_top (𝓝 (f x)), - from hf.tendsto_approx_bounded_ae hf_bound, - by_cases hμ : μ = 0, - { simp only [hμ, ae_zero], }, - haveI : μ.ae.ne_bot, by simp only [hμ, ae_ne_bot, ne.def, not_false_iff], - have hc : 0 ≤ c, - { have h_exists : ∃ x, ∥f x∥ ≤ c := eventually.exists hf_bound, - exact (norm_nonneg _).trans h_exists.some_spec, }, - have hfs_bound : ∀ n x, ∥fs n x∥ ≤ c := hf.norm_approx_bounded_le hc, - have hn_eq : ∀ n, μ[fs n * g | m] =ᵐ[μ] fs n * μ[g | m], - from λ n, condexp_strongly_measurable_simple_func_mul hm _ hg, - have : μ[f * μ[g|m]|m] = f * μ[g|m], - { refine condexp_of_strongly_measurable hm (hf.mul strongly_measurable_condexp) _, - exact integrable_condexp.bdd_mul' ((hf.mono hm).ae_strongly_measurable) hf_bound, }, - rw ← this, - refine tendsto_condexp_unique (λ n x, fs n x * g x) (λ n x, fs n x * μ[g|m] x) (f * g) - (f * μ[g|m]) _ _ _ _ (λ x, c * ∥g x∥) _ (λ x, c * ∥μ[g|m] x∥) _ _ _ _, - { exact λ n, hg.bdd_mul' - ((simple_func.strongly_measurable (fs n)).mono hm).ae_strongly_measurable - (eventually_of_forall (hfs_bound n)), }, - { exact λ n, integrable_condexp.bdd_mul' - ((simple_func.strongly_measurable (fs n)).mono hm).ae_strongly_measurable - (eventually_of_forall (hfs_bound n)), }, - { filter_upwards [hfs_tendsto] with x hx, - rw pi.mul_apply, - exact tendsto.mul hx tendsto_const_nhds, }, - { filter_upwards [hfs_tendsto] with x hx, - rw pi.mul_apply, - exact tendsto.mul hx tendsto_const_nhds, }, - { exact hg.norm.const_mul c, }, - { exact integrable_condexp.norm.const_mul c, }, - { refine λ n, eventually_of_forall (λ x, _), - exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (hfs_bound n x) (norm_nonneg _)), }, - { refine λ n, eventually_of_forall (λ x, _), - exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (hfs_bound n x) (norm_nonneg _)), }, - { intros n, - simp_rw ← pi.mul_apply, - refine (condexp_strongly_measurable_simple_func_mul hm _ hg).trans _, - rw condexp_of_strongly_measurable hm - ((simple_func.strongly_measurable _).mul strongly_measurable_condexp) _, - { apply_instance, }, - { apply_instance, }, - exact integrable_condexp.bdd_mul' - ((simple_func.strongly_measurable (fs n)).mono hm).ae_strongly_measurable - (eventually_of_forall (hfs_bound n)), }, -end - -lemma condexp_strongly_measurable_mul_of_bound₀ (hm : m ≤ m0) [is_finite_measure μ] - {f g : α → ℝ} (hf : ae_strongly_measurable' m f μ) (hg : integrable g μ) (c : ℝ) - (hf_bound : ∀ᵐ x ∂μ, ∥f x∥ ≤ c) : - μ[f * g | m] =ᵐ[μ] f * μ[g | m] := -begin - have : μ[f * g | m] =ᵐ[μ] μ[hf.mk f * g | m], - from condexp_congr_ae (eventually_eq.mul hf.ae_eq_mk eventually_eq.rfl), - refine this.trans _, - have : f * μ[g | m] =ᵐ[μ] hf.mk f * μ[g | m] := eventually_eq.mul hf.ae_eq_mk eventually_eq.rfl, - refine eventually_eq.trans _ this.symm, - refine condexp_strongly_measurable_mul_of_bound hm hf.strongly_measurable_mk hg c _, - filter_upwards [hf_bound, hf.ae_eq_mk] with x hxc hx_eq, - rw ← hx_eq, - exact hxc, -end - -/-- Pull-out property of the conditional expectation. -/ -lemma condexp_strongly_measurable_mul {f g : α → ℝ} (hf : strongly_measurable[m] f) - (hfg : integrable (f * g) μ) (hg : integrable g μ) : - μ[f * g | m] =ᵐ[μ] f * μ[g | m] := -begin - by_cases hm : m ≤ m0, swap, { simp_rw condexp_of_not_le hm, rw mul_zero, }, - by_cases hμm : sigma_finite (μ.trim hm), - swap, { simp_rw condexp_of_not_sigma_finite hm hμm, rw mul_zero, }, - haveI : sigma_finite (μ.trim hm) := hμm, - obtain ⟨sets, sets_prop, h_univ⟩ := hf.exists_spanning_measurable_set_norm_le hm μ, - simp_rw forall_and_distrib at sets_prop, - obtain ⟨h_meas, h_finite, h_norm⟩ := sets_prop, - - suffices : ∀ n, ∀ᵐ x ∂μ, x ∈ sets n → μ[f * g|m] x = f x * μ[g|m] x, - { rw ← ae_all_iff at this, - filter_upwards [this] with x hx, - rw pi.mul_apply, - obtain ⟨i, hi⟩ : ∃ i, x ∈ sets i, - { have h_mem : x ∈ ⋃ i, sets i, - { rw h_univ, exact set.mem_univ _, }, - simpa using h_mem, }, - exact hx i hi, }, - refine λ n, ae_imp_of_ae_restrict _, - suffices : (μ.restrict (sets n))[f * g | m] - =ᵐ[μ.restrict (sets n)] f * (μ.restrict (sets n))[g | m], - { simp_rw ← pi.mul_apply, - refine (condexp_restrict_ae_eq_restrict hm (h_meas n) hfg).symm.trans _, - exact this.trans (eventually_eq.rfl.mul (condexp_restrict_ae_eq_restrict hm (h_meas n) hg)), }, - suffices : (μ.restrict (sets n))[((sets n).indicator f) * g | m] - =ᵐ[μ.restrict (sets n)] ((sets n).indicator f) * (μ.restrict (sets n))[g | m], - { refine eventually_eq.trans _ (this.trans _), - { exact condexp_congr_ae - ((indicator_ae_eq_restrict (hm _ (h_meas n))).symm.mul eventually_eq.rfl), }, - { exact (indicator_ae_eq_restrict (hm _ (h_meas n))).mul eventually_eq.rfl, }, }, - haveI : is_finite_measure (μ.restrict (sets n)), - { constructor, - rw measure.restrict_apply_univ, - exact h_finite n, }, - refine condexp_strongly_measurable_mul_of_bound hm (hf.indicator (h_meas n)) hg.integrable_on n _, - refine eventually_of_forall (λ x, _), - by_cases hxs : x ∈ sets n, - { simp only [hxs, set.indicator_of_mem], - exact h_norm n x hxs, }, - { simp only [hxs, set.indicator_of_not_mem, not_false_iff, _root_.norm_zero, nat.cast_nonneg], }, -end - -/-- Pull-out property of the conditional expectation. -/ -lemma condexp_strongly_measurable_mul₀ {f g : α → ℝ} (hf : ae_strongly_measurable' m f μ) - (hfg : integrable (f * g) μ) (hg : integrable g μ) : - μ[f * g | m] =ᵐ[μ] f * μ[g | m] := -begin - have : μ[f * g | m] =ᵐ[μ] μ[hf.mk f * g | m], - from condexp_congr_ae (eventually_eq.mul hf.ae_eq_mk eventually_eq.rfl), - refine this.trans _, - have : f * μ[g | m] =ᵐ[μ] hf.mk f * μ[g | m] := eventually_eq.mul hf.ae_eq_mk eventually_eq.rfl, - refine eventually_eq.trans _ this.symm, - refine condexp_strongly_measurable_mul hf.strongly_measurable_mk _ hg, - refine (integrable_congr _).mp hfg, - exact eventually_eq.mul hf.ae_eq_mk eventually_eq.rfl, -end - -end pull_out - -section real - -lemma rn_deriv_ae_eq_condexp {hm : m ≤ m0} [hμm : sigma_finite (μ.trim hm)] {f : α → ℝ} - (hf : integrable f μ) : - signed_measure.rn_deriv ((μ.with_densityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f | m] := -begin - refine ae_eq_condexp_of_forall_set_integral_eq hm hf _ _ _, - { exact λ _ _ _, (integrable_of_integrable_trim hm (signed_measure.integrable_rn_deriv - ((μ.with_densityᵥ f).trim hm) (μ.trim hm))).integrable_on }, - { intros s hs hlt, - conv_rhs { rw [← hf.with_densityᵥ_trim_eq_integral hm hs, - ← signed_measure.with_densityᵥ_rn_deriv_eq ((μ.with_densityᵥ f).trim hm) (μ.trim hm) - (hf.with_densityᵥ_trim_absolutely_continuous hm)], }, - rw [with_densityᵥ_apply - (signed_measure.integrable_rn_deriv ((μ.with_densityᵥ f).trim hm) (μ.trim hm)) hs, - ← set_integral_trim hm _ hs], - exact (signed_measure.measurable_rn_deriv _ _).strongly_measurable }, - { exact strongly_measurable.ae_strongly_measurable' - (signed_measure.measurable_rn_deriv _ _).strongly_measurable }, -end - -/-- TODO: this should be generalized and proved using Jensen's inequality -for the conditional expectation (not in mathlib yet) .-/ -lemma snorm_one_condexp_le_snorm (f : α → ℝ) : - snorm (μ[f | m]) 1 μ ≤ snorm f 1 μ := -begin - by_cases hf : integrable f μ, - swap, { rw [snorm_congr_ae (condexp_undef hf), snorm_zero], exact zero_le _ }, - by_cases hm : m ≤ m0, - swap, { rw [condexp_of_not_le hm, snorm_zero], exact zero_le _ }, - by_cases hsig : sigma_finite (μ.trim hm), - swap, { rw [condexp_of_not_sigma_finite hm hsig, snorm_zero], exact zero_le _ }, - calc snorm (μ[f | m]) 1 μ ≤ snorm (μ[|f| | m]) 1 μ : - begin - refine snorm_mono_ae _, - filter_upwards [@condexp_mono _ m m0 _ _ _ _ _ _ _ _ hf hf.abs - (@ae_of_all _ m0 _ μ (λ x, le_abs_self (f x) : ∀ x, f x ≤ |f x|)), - eventually_le.trans (condexp_neg f).symm.le - (@condexp_mono _ m m0 _ _ _ _ _ _ _ _ hf.neg hf.abs - (@ae_of_all _ m0 _ μ (λ x, neg_le_abs_self (f x) : ∀ x, -f x ≤ |f x|)))] with x hx₁ hx₂, - exact abs_le_abs hx₁ hx₂, - end - ... = snorm f 1 μ : - begin - rw [snorm_one_eq_lintegral_nnnorm, snorm_one_eq_lintegral_nnnorm, - ← ennreal.to_real_eq_to_real (ne_of_lt integrable_condexp.2) (ne_of_lt hf.2), - ← integral_norm_eq_lintegral_nnnorm - (strongly_measurable_condexp.mono hm).ae_strongly_measurable, - ← integral_norm_eq_lintegral_nnnorm hf.1], - simp_rw [real.norm_eq_abs], - rw ← @integral_condexp _ _ _ _ _ m m0 μ _ hm hsig hf.abs, - refine integral_congr_ae _, - have : 0 ≤ᵐ[μ] μ[|f| | m], - { rw ← @condexp_zero α ℝ _ _ _ m m0 μ, - exact condexp_mono (integrable_zero _ _ _) hf.abs - (@ae_of_all _ m0 _ μ (λ x, abs_nonneg (f x) : ∀ x, 0 ≤ |f x|)) }, - filter_upwards [this] with x hx, - exact abs_eq_self.2 hx - end -end - -/-- Given a integrable function `g`, the conditional expectations of `g` with respect to -a sequence of sub-σ-algebras is uniformly integrable. -/ -lemma integrable.uniform_integrable_condexp {ι : Type*} [is_finite_measure μ] - {g : α → ℝ} (hint : integrable g μ) {ℱ : ι → measurable_space α} (hℱ : ∀ i, ℱ i ≤ m0) : - uniform_integrable (λ i, μ[g | ℱ i]) 1 μ := -begin - have hmeas : ∀ n, ∀ C, measurable_set {x | C ≤ ∥μ[g | ℱ n] x∥₊} := - λ n C, measurable_set_le measurable_const - (strongly_measurable_condexp.mono (hℱ n)).measurable.nnnorm, - have hg : mem_ℒp g 1 μ := mem_ℒp_one_iff_integrable.2 hint, - refine uniform_integrable_of le_rfl ennreal.one_ne_top - (λ n, (strongly_measurable_condexp.mono (hℱ n)).ae_strongly_measurable) (λ ε hε, _), - by_cases hne : snorm g 1 μ = 0, - { rw snorm_eq_zero_iff hg.1 one_ne_zero at hne, - refine ⟨0, λ n, (le_of_eq $ (snorm_eq_zero_iff - ((strongly_measurable_condexp.mono (hℱ n)).ae_strongly_measurable.indicator (hmeas n 0)) - one_ne_zero).2 _).trans (zero_le _)⟩, - filter_upwards [@condexp_congr_ae _ _ _ _ _ (ℱ n) m0 μ _ _ hne] with x hx, - simp only [zero_le', set.set_of_true, set.indicator_univ, pi.zero_apply, hx, condexp_zero] }, - obtain ⟨δ, hδ, h⟩ := hg.snorm_indicator_le μ le_rfl ennreal.one_ne_top hε, - set C : ℝ≥0 := ⟨δ, hδ.le⟩⁻¹ * (snorm g 1 μ).to_nnreal with hC, - have hCpos : 0 < C := - mul_pos (nnreal.inv_pos.2 hδ) (ennreal.to_nnreal_pos hne hg.snorm_lt_top.ne), - have : ∀ n, μ {x : α | C ≤ ∥μ[g | ℱ n] x∥₊} ≤ ennreal.of_real δ, - { intro n, - have := mul_meas_ge_le_pow_snorm' μ one_ne_zero ennreal.one_ne_top - ((@strongly_measurable_condexp _ _ _ _ _ (ℱ n) _ μ g).mono - (hℱ n)).ae_strongly_measurable C, - rw [ennreal.one_to_real, ennreal.rpow_one, ennreal.rpow_one, mul_comm, - ← ennreal.le_div_iff_mul_le (or.inl (ennreal.coe_ne_zero.2 hCpos.ne.symm)) - (or.inl ennreal.coe_lt_top.ne)] at this, - simp_rw [ennreal.coe_le_coe] at this, - refine this.trans _, - rw [ennreal.div_le_iff_le_mul (or.inl (ennreal.coe_ne_zero.2 hCpos.ne.symm)) - (or.inl ennreal.coe_lt_top.ne), hC, nonneg.inv_mk, ennreal.coe_mul, - ennreal.coe_to_nnreal hg.snorm_lt_top.ne, ← mul_assoc, ← ennreal.of_real_eq_coe_nnreal, - ← ennreal.of_real_mul hδ.le, mul_inv_cancel hδ.ne.symm, ennreal.of_real_one, one_mul], - exact snorm_one_condexp_le_snorm _ }, - refine ⟨C, λ n, le_trans _ (h {x : α | C ≤ ∥μ[g | ℱ n] x∥₊} (hmeas n C) (this n))⟩, - have hmeasℱ : measurable_set[ℱ n] {x : α | C ≤ ∥μ[g | ℱ n] x∥₊} := - @measurable_set_le _ _ _ _ _ (ℱ n) _ _ _ _ _ measurable_const - (@measurable.nnnorm _ _ _ _ _ (ℱ n) _ strongly_measurable_condexp.measurable), - rw ← snorm_congr_ae (condexp_indicator hint hmeasℱ), - exact snorm_one_condexp_le_snorm _, -end - -end real - end condexp end measure_theory diff --git a/formal/lean/mathlib/measure_theory/function/conditional_expectation/indicator.lean b/formal/lean/mathlib/measure_theory/function/conditional_expectation/indicator.lean new file mode 100644 index 0000000000000000000000000000000000000000..fdba24c63553a88c3c6dc5b31cd32976b7d21d2c --- /dev/null +++ b/formal/lean/mathlib/measure_theory/function/conditional_expectation/indicator.lean @@ -0,0 +1,200 @@ +/- +Copyright (c) 2022 Rémy Degenne. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Rémy Degenne +-/ + +import measure_theory.function.conditional_expectation.basic + +/-! + +# Conditional expectation of indicator functions + +This file proves some results about the conditional expectation of an indicator function and +as a corollary, also proves several results about the behaviour of the conditional expectation on +a restricted measure. + +## Main result + +* `measure_theory.condexp_indicator`: If `s` is a `m`-measurable set, then the conditional + expectation of the indicator function of `s` is almost everywhere equal to the indicator + of `s` of the conditional expectation. Namely, `𝔼[s.indicator f | m] = s.indicator 𝔼[f | m]` a.e. + +-/ + +noncomputable theory +open topological_space measure_theory.Lp filter continuous_linear_map +open_locale nnreal ennreal topological_space big_operators measure_theory + +namespace measure_theory + +variables {α 𝕜 E : Type*} {m m0 : measurable_space α} + [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] + {μ : measure α} {f : α → E} {s : set α} + +lemma condexp_ae_eq_restrict_zero (hs : measurable_set[m] s) (hf : f =ᵐ[μ.restrict s] 0) : + μ[f | m] =ᵐ[μ.restrict s] 0 := +begin + by_cases hm : m ≤ m0, + swap, { simp_rw condexp_of_not_le hm, }, + by_cases hμm : sigma_finite (μ.trim hm), + swap, { simp_rw condexp_of_not_sigma_finite hm hμm, }, + haveI : sigma_finite (μ.trim hm) := hμm, + haveI : sigma_finite ((μ.restrict s).trim hm), + { rw ← restrict_trim hm _ hs, + exact restrict.sigma_finite _ s, }, + by_cases hf_int : integrable f μ, + swap, { exact ae_restrict_of_ae (condexp_undef hf_int), }, + refine ae_eq_of_forall_set_integral_eq_of_sigma_finite' hm _ _ _ _ _, + { exact λ t ht hμt, integrable_condexp.integrable_on.integrable_on, }, + { exact λ t ht hμt, (integrable_zero _ _ _).integrable_on, }, + { intros t ht hμt, + rw [measure.restrict_restrict (hm _ ht), set_integral_condexp hm hf_int (ht.inter hs), + ← measure.restrict_restrict (hm _ ht)], + refine set_integral_congr_ae (hm _ ht) _, + filter_upwards [hf] with x hx h using hx, }, + { exact strongly_measurable_condexp.ae_strongly_measurable', }, + { exact strongly_measurable_zero.ae_strongly_measurable', }, +end + +/-- Auxiliary lemma for `condexp_indicator`. -/ +lemma condexp_indicator_aux (hs : measurable_set[m] s) (hf : f =ᵐ[μ.restrict sᶜ] 0) : + μ[s.indicator f | m] =ᵐ[μ] s.indicator (μ[f | m]) := +begin + by_cases hm : m ≤ m0, + swap, { simp_rw [condexp_of_not_le hm, set.indicator_zero'], }, + have hsf_zero : ∀ g : α → E, g =ᵐ[μ.restrict sᶜ] 0 → s.indicator g =ᵐ[μ] g, + from λ g, indicator_ae_eq_of_restrict_compl_ae_eq_zero (hm _ hs), + refine ((hsf_zero (μ[f | m]) (condexp_ae_eq_restrict_zero hs.compl hf)).trans _).symm, + exact condexp_congr_ae (hsf_zero f hf).symm, +end + +/-- The conditional expectation of the indicator of a function over an `m`-measurable set with +respect to the σ-algebra `m` is a.e. equal to the indicator of the conditional expectation. -/ +lemma condexp_indicator (hf_int : integrable f μ) (hs : measurable_set[m] s) : + μ[s.indicator f | m] =ᵐ[μ] s.indicator (μ[f | m]) := +begin + by_cases hm : m ≤ m0, + swap, { simp_rw [condexp_of_not_le hm, set.indicator_zero'], }, + by_cases hμm : sigma_finite (μ.trim hm), + swap, { simp_rw [condexp_of_not_sigma_finite hm hμm, set.indicator_zero'], }, + haveI : sigma_finite (μ.trim hm) := hμm, + -- use `have` to perform what should be the first calc step because of an error I don't + -- understand + have : s.indicator (μ[f|m]) =ᵐ[μ] s.indicator (μ[s.indicator f + sᶜ.indicator f|m]), + by rw set.indicator_self_add_compl s f, + refine (this.trans _).symm, + calc s.indicator (μ[s.indicator f + sᶜ.indicator f|m]) + =ᵐ[μ] s.indicator (μ[s.indicator f|m] + μ[sᶜ.indicator f|m]) : + begin + have : μ[s.indicator f + sᶜ.indicator f|m] =ᵐ[μ] μ[s.indicator f|m] + μ[sᶜ.indicator f|m], + from condexp_add (hf_int.indicator (hm _ hs)) (hf_int.indicator (hm _ hs.compl)), + filter_upwards [this] with x hx, + classical, + rw [set.indicator_apply, set.indicator_apply, hx], + end + ... = s.indicator (μ[s.indicator f|m]) + s.indicator (μ[sᶜ.indicator f|m]) : + s.indicator_add' _ _ + ... =ᵐ[μ] s.indicator (μ[s.indicator f|m]) + s.indicator (sᶜ.indicator (μ[sᶜ.indicator f|m])) : + begin + refine filter.eventually_eq.rfl.add _, + have : sᶜ.indicator (μ[sᶜ.indicator f|m]) =ᵐ[μ] μ[sᶜ.indicator f|m], + { refine (condexp_indicator_aux hs.compl _).symm.trans _, + { exact indicator_ae_eq_restrict_compl (hm _ hs.compl), }, + { rw [set.indicator_indicator, set.inter_self], }, }, + filter_upwards [this] with x hx, + by_cases hxs : x ∈ s, + { simp only [hx, hxs, set.indicator_of_mem], }, + { simp only [hxs, set.indicator_of_not_mem, not_false_iff], }, + end + ... =ᵐ[μ] s.indicator (μ[s.indicator f|m]) : + by rw [set.indicator_indicator, set.inter_compl_self, set.indicator_empty', add_zero] + ... =ᵐ[μ] μ[s.indicator f|m] : + begin + refine (condexp_indicator_aux hs _).symm.trans _, + { exact indicator_ae_eq_restrict_compl (hm _ hs), }, + { rw [set.indicator_indicator, set.inter_self], }, + end +end + +lemma condexp_restrict_ae_eq_restrict (hm : m ≤ m0) [sigma_finite (μ.trim hm)] + (hs_m : measurable_set[m] s) (hf_int : integrable f μ) : + (μ.restrict s)[f | m] =ᵐ[μ.restrict s] μ[f | m] := +begin + haveI : sigma_finite ((μ.restrict s).trim hm), + { rw ← restrict_trim hm _ hs_m, apply_instance, }, + rw ae_eq_restrict_iff_indicator_ae_eq (hm _ hs_m), + swap, { apply_instance, }, + refine eventually_eq.trans _ (condexp_indicator hf_int hs_m), + refine ae_eq_condexp_of_forall_set_integral_eq hm (hf_int.indicator (hm _ hs_m)) _ _ _, + { intros t ht hμt, + rw [← integrable_indicator_iff (hm _ ht), set.indicator_indicator, set.inter_comm, + ← set.indicator_indicator], + suffices h_int_restrict : integrable (t.indicator ((μ.restrict s)[f|m])) (μ.restrict s), + { rw [integrable_indicator_iff (hm _ hs_m), integrable_on], + rw [integrable_indicator_iff (hm _ ht), integrable_on] at h_int_restrict ⊢, + exact h_int_restrict, }, + exact integrable_condexp.indicator (hm _ ht), }, + { intros t ht hμt, + calc ∫ x in t, s.indicator ((μ.restrict s)[f|m]) x ∂μ + = ∫ x in t, ((μ.restrict s)[f|m]) x ∂(μ.restrict s) : + by rw [integral_indicator (hm _ hs_m), measure.restrict_restrict (hm _ hs_m), + measure.restrict_restrict (hm _ ht), set.inter_comm] + ... = ∫ x in t, f x ∂(μ.restrict s) : set_integral_condexp hm hf_int.integrable_on ht + ... = ∫ x in t, s.indicator f x ∂μ : + by rw [integral_indicator (hm _ hs_m), measure.restrict_restrict (hm _ hs_m), + measure.restrict_restrict (hm _ ht), set.inter_comm], }, + { exact (strongly_measurable_condexp.indicator hs_m).ae_strongly_measurable', }, +end + +/-- If the restriction to a `m`-measurable set `s` of a σ-algebra `m` is equal to the restriction +to `s` of another σ-algebra `m₂` (hypothesis `hs`), then `μ[f | m] =ᵐ[μ.restrict s] μ[f | m₂]`. -/ +lemma condexp_ae_eq_restrict_of_measurable_space_eq_on {m m₂ m0 : measurable_space α} + {μ : measure α} (hm : m ≤ m0) (hm₂ : m₂ ≤ m0) + [sigma_finite (μ.trim hm)] [sigma_finite (μ.trim hm₂)] + (hs_m : measurable_set[m] s) (hs : ∀ t, measurable_set[m] (s ∩ t) ↔ measurable_set[m₂] (s ∩ t)) : + μ[f | m] =ᵐ[μ.restrict s] μ[f | m₂] := +begin + rw ae_eq_restrict_iff_indicator_ae_eq (hm _ hs_m), + have hs_m₂ : measurable_set[m₂] s, + { rwa [← set.inter_univ s, ← hs set.univ, set.inter_univ], }, + by_cases hf_int : integrable f μ, + swap, + { filter_upwards [@condexp_undef _ _ _ _ _ m _ μ _ hf_int, + @condexp_undef _ _ _ _ _ m₂ _ μ _ hf_int] with x hxm hxm₂, + rw pi.zero_apply at hxm hxm₂, + rw [set.indicator_apply_eq_zero.2 (λ _, hxm), set.indicator_apply_eq_zero.2 (λ _, hxm₂)] }, + refine ((condexp_indicator hf_int hs_m).symm.trans _).trans (condexp_indicator hf_int hs_m₂), + refine ae_eq_of_forall_set_integral_eq_of_sigma_finite' hm₂ + (λ s hs hμs, integrable_condexp.integrable_on) + (λ s hs hμs, integrable_condexp.integrable_on) _ _ + strongly_measurable_condexp.ae_strongly_measurable', + swap, + { have : strongly_measurable[m] (μ[s.indicator f | m]) := strongly_measurable_condexp, + refine this.ae_strongly_measurable'.ae_strongly_measurable'_of_measurable_space_le_on + hm hs_m (λ t, (hs t).mp) _, + exact condexp_ae_eq_restrict_zero hs_m.compl (indicator_ae_eq_restrict_compl (hm _ hs_m)), }, + intros t ht hμt, + have : ∫ x in t, μ[s.indicator f|m] x ∂μ = ∫ x in s ∩ t, μ[s.indicator f|m] x ∂μ, + { rw ← integral_add_compl (hm _ hs_m) integrable_condexp.integrable_on, + suffices : ∫ x in sᶜ, μ[s.indicator f|m] x ∂μ.restrict t = 0, + by rw [this, add_zero, measure.restrict_restrict (hm _ hs_m)], + rw measure.restrict_restrict (measurable_set.compl (hm _ hs_m)), + suffices : μ[s.indicator f|m] =ᵐ[μ.restrict sᶜ] 0, + { rw [set.inter_comm, ← measure.restrict_restrict (hm₂ _ ht)], + calc ∫ (x : α) in t, μ[s.indicator f|m] x ∂μ.restrict sᶜ + = ∫ (x : α) in t, 0 ∂μ.restrict sᶜ : begin + refine set_integral_congr_ae (hm₂ _ ht) _, + filter_upwards [this] with x hx h using hx, + end + ... = 0 : integral_zero _ _, }, + refine condexp_ae_eq_restrict_zero hs_m.compl _, + exact indicator_ae_eq_restrict_compl (hm _ hs_m), }, + have hst_m : measurable_set[m] (s ∩ t) := (hs _).mpr (hs_m₂.inter ht), + simp_rw [this, set_integral_condexp hm₂ (hf_int.indicator (hm _ hs_m)) ht, + set_integral_condexp hm (hf_int.indicator (hm _ hs_m)) hst_m, + integral_indicator (hm _ hs_m), measure.restrict_restrict (hm _ hs_m), + ← set.inter_assoc, set.inter_self], +end + +end measure_theory diff --git a/formal/lean/mathlib/measure_theory/function/conditional_expectation/real.lean b/formal/lean/mathlib/measure_theory/function/conditional_expectation/real.lean new file mode 100644 index 0000000000000000000000000000000000000000..6360daf577c2358de9a5f0e5b71b80e9f14d0293 --- /dev/null +++ b/formal/lean/mathlib/measure_theory/function/conditional_expectation/real.lean @@ -0,0 +1,302 @@ +/- +Copyright (c) 2022 Rémy Degenne. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Rémy Degenne, Kexing Ying +-/ + +import measure_theory.function.conditional_expectation.indicator + +/-! + +# Conditional expectation of real-valued functions + +This file proves some results regarding the conditional expectation of real-valued functions. + +## Main results + +* `measure_theory.rn_deriv_ae_eq_condexp`: the conditional expectation `μ[f | m]` is equal to the + Radon-Nikodym derivative of `fμ` restricted on `m` with respect to `μ` restricted on `m`. +* `measure_theory.integrable.uniform_integrable_condexp`: the conditional expectation of a function + form a uniformly integrable class. +* `measure_theory.condexp_strongly_measurable_mul`: the pull-out property of the conditional + expectation. + +-/ + +noncomputable theory +open topological_space measure_theory.Lp filter continuous_linear_map +open_locale nnreal ennreal topological_space big_operators measure_theory + +namespace measure_theory + +variables {α : Type*} {m m0 : measurable_space α} {μ : measure α} + +lemma rn_deriv_ae_eq_condexp {hm : m ≤ m0} [hμm : sigma_finite (μ.trim hm)] {f : α → ℝ} + (hf : integrable f μ) : + signed_measure.rn_deriv ((μ.with_densityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f | m] := +begin + refine ae_eq_condexp_of_forall_set_integral_eq hm hf _ _ _, + { exact λ _ _ _, (integrable_of_integrable_trim hm (signed_measure.integrable_rn_deriv + ((μ.with_densityᵥ f).trim hm) (μ.trim hm))).integrable_on }, + { intros s hs hlt, + conv_rhs { rw [← hf.with_densityᵥ_trim_eq_integral hm hs, + ← signed_measure.with_densityᵥ_rn_deriv_eq ((μ.with_densityᵥ f).trim hm) (μ.trim hm) + (hf.with_densityᵥ_trim_absolutely_continuous hm)], }, + rw [with_densityᵥ_apply + (signed_measure.integrable_rn_deriv ((μ.with_densityᵥ f).trim hm) (μ.trim hm)) hs, + ← set_integral_trim hm _ hs], + exact (signed_measure.measurable_rn_deriv _ _).strongly_measurable }, + { exact strongly_measurable.ae_strongly_measurable' + (signed_measure.measurable_rn_deriv _ _).strongly_measurable }, +end + +/-- TODO: this should be generalized and proved using Jensen's inequality +for the conditional expectation (not in mathlib yet) .-/ +lemma snorm_one_condexp_le_snorm (f : α → ℝ) : + snorm (μ[f | m]) 1 μ ≤ snorm f 1 μ := +begin + by_cases hf : integrable f μ, + swap, { rw [snorm_congr_ae (condexp_undef hf), snorm_zero], exact zero_le _ }, + by_cases hm : m ≤ m0, + swap, { rw [condexp_of_not_le hm, snorm_zero], exact zero_le _ }, + by_cases hsig : sigma_finite (μ.trim hm), + swap, { rw [condexp_of_not_sigma_finite hm hsig, snorm_zero], exact zero_le _ }, + calc snorm (μ[f | m]) 1 μ ≤ snorm (μ[|f| | m]) 1 μ : + begin + refine snorm_mono_ae _, + filter_upwards [@condexp_mono _ m m0 _ _ _ _ _ _ _ _ hf hf.abs + (@ae_of_all _ m0 _ μ (λ x, le_abs_self (f x) : ∀ x, f x ≤ |f x|)), + eventually_le.trans (condexp_neg f).symm.le + (@condexp_mono _ m m0 _ _ _ _ _ _ _ _ hf.neg hf.abs + (@ae_of_all _ m0 _ μ (λ x, neg_le_abs_self (f x) : ∀ x, -f x ≤ |f x|)))] with x hx₁ hx₂, + exact abs_le_abs hx₁ hx₂, + end + ... = snorm f 1 μ : + begin + rw [snorm_one_eq_lintegral_nnnorm, snorm_one_eq_lintegral_nnnorm, + ← ennreal.to_real_eq_to_real (ne_of_lt integrable_condexp.2) (ne_of_lt hf.2), + ← integral_norm_eq_lintegral_nnnorm + (strongly_measurable_condexp.mono hm).ae_strongly_measurable, + ← integral_norm_eq_lintegral_nnnorm hf.1], + simp_rw [real.norm_eq_abs], + rw ← @integral_condexp _ _ _ _ _ m m0 μ _ hm hsig hf.abs, + refine integral_congr_ae _, + have : 0 ≤ᵐ[μ] μ[|f| | m], + { rw ← @condexp_zero α ℝ _ _ _ m m0 μ, + exact condexp_mono (integrable_zero _ _ _) hf.abs + (@ae_of_all _ m0 _ μ (λ x, abs_nonneg (f x) : ∀ x, 0 ≤ |f x|)) }, + filter_upwards [this] with x hx, + exact abs_eq_self.2 hx + end +end + +/-- Given a integrable function `g`, the conditional expectations of `g` with respect to +a sequence of sub-σ-algebras is uniformly integrable. -/ +lemma integrable.uniform_integrable_condexp {ι : Type*} [is_finite_measure μ] + {g : α → ℝ} (hint : integrable g μ) {ℱ : ι → measurable_space α} (hℱ : ∀ i, ℱ i ≤ m0) : + uniform_integrable (λ i, μ[g | ℱ i]) 1 μ := +begin + have hmeas : ∀ n, ∀ C, measurable_set {x | C ≤ ∥μ[g | ℱ n] x∥₊} := + λ n C, measurable_set_le measurable_const + (strongly_measurable_condexp.mono (hℱ n)).measurable.nnnorm, + have hg : mem_ℒp g 1 μ := mem_ℒp_one_iff_integrable.2 hint, + refine uniform_integrable_of le_rfl ennreal.one_ne_top + (λ n, (strongly_measurable_condexp.mono (hℱ n)).ae_strongly_measurable) (λ ε hε, _), + by_cases hne : snorm g 1 μ = 0, + { rw snorm_eq_zero_iff hg.1 one_ne_zero at hne, + refine ⟨0, λ n, (le_of_eq $ (snorm_eq_zero_iff + ((strongly_measurable_condexp.mono (hℱ n)).ae_strongly_measurable.indicator (hmeas n 0)) + one_ne_zero).2 _).trans (zero_le _)⟩, + filter_upwards [@condexp_congr_ae _ _ _ _ _ (ℱ n) m0 μ _ _ hne] with x hx, + simp only [zero_le', set.set_of_true, set.indicator_univ, pi.zero_apply, hx, condexp_zero] }, + obtain ⟨δ, hδ, h⟩ := hg.snorm_indicator_le μ le_rfl ennreal.one_ne_top hε, + set C : ℝ≥0 := ⟨δ, hδ.le⟩⁻¹ * (snorm g 1 μ).to_nnreal with hC, + have hCpos : 0 < C := + mul_pos (nnreal.inv_pos.2 hδ) (ennreal.to_nnreal_pos hne hg.snorm_lt_top.ne), + have : ∀ n, μ {x : α | C ≤ ∥μ[g | ℱ n] x∥₊} ≤ ennreal.of_real δ, + { intro n, + have := mul_meas_ge_le_pow_snorm' μ one_ne_zero ennreal.one_ne_top + ((@strongly_measurable_condexp _ _ _ _ _ (ℱ n) _ μ g).mono + (hℱ n)).ae_strongly_measurable C, + rw [ennreal.one_to_real, ennreal.rpow_one, ennreal.rpow_one, mul_comm, + ← ennreal.le_div_iff_mul_le (or.inl (ennreal.coe_ne_zero.2 hCpos.ne.symm)) + (or.inl ennreal.coe_lt_top.ne)] at this, + simp_rw [ennreal.coe_le_coe] at this, + refine this.trans _, + rw [ennreal.div_le_iff_le_mul (or.inl (ennreal.coe_ne_zero.2 hCpos.ne.symm)) + (or.inl ennreal.coe_lt_top.ne), hC, nonneg.inv_mk, ennreal.coe_mul, + ennreal.coe_to_nnreal hg.snorm_lt_top.ne, ← mul_assoc, ← ennreal.of_real_eq_coe_nnreal, + ← ennreal.of_real_mul hδ.le, mul_inv_cancel hδ.ne.symm, ennreal.of_real_one, one_mul], + exact snorm_one_condexp_le_snorm _ }, + refine ⟨C, λ n, le_trans _ (h {x : α | C ≤ ∥μ[g | ℱ n] x∥₊} (hmeas n C) (this n))⟩, + have hmeasℱ : measurable_set[ℱ n] {x : α | C ≤ ∥μ[g | ℱ n] x∥₊} := + @measurable_set_le _ _ _ _ _ (ℱ n) _ _ _ _ _ measurable_const + (@measurable.nnnorm _ _ _ _ _ (ℱ n) _ strongly_measurable_condexp.measurable), + rw ← snorm_congr_ae (condexp_indicator hint hmeasℱ), + exact snorm_one_condexp_le_snorm _, +end + +section pull_out +-- TODO: this section could be generalized beyond multiplication, to any bounded bilinear map. + +/-- Auxiliary lemma for `condexp_measurable_mul`. -/ +lemma condexp_strongly_measurable_simple_func_mul (hm : m ≤ m0) + (f : @simple_func α m ℝ) {g : α → ℝ} (hg : integrable g μ) : + μ[f * g | m] =ᵐ[μ] f * μ[g | m] := +begin + have : ∀ s c (f : α → ℝ), set.indicator s (function.const α c) * f = s.indicator (c • f), + { intros s c f, + ext1 x, + by_cases hx : x ∈ s, + { simp only [hx, pi.mul_apply, set.indicator_of_mem, pi.smul_apply, algebra.id.smul_eq_mul] }, + { simp only [hx, pi.mul_apply, set.indicator_of_not_mem, not_false_iff, zero_mul], }, }, + refine @simple_func.induction _ _ m _ _ (λ c s hs, _) (λ g₁ g₂ h_disj h_eq₁ h_eq₂, _) f, + { simp only [simple_func.const_zero, simple_func.coe_piecewise, simple_func.coe_const, + simple_func.coe_zero, set.piecewise_eq_indicator], + rw [this, this], + refine (condexp_indicator (hg.smul c) hs).trans _, + filter_upwards [@condexp_smul α ℝ ℝ _ _ _ _ _ m m0 μ c g] with x hx, + classical, + simp_rw [set.indicator_apply, hx], }, + { have h_add := @simple_func.coe_add _ _ m _ g₁ g₂, + calc μ[⇑(g₁ + g₂) * g|m] =ᵐ[μ] μ[(⇑g₁ + ⇑g₂) * g|m] : + by { refine condexp_congr_ae (eventually_eq.mul _ eventually_eq.rfl), rw h_add, } + ... =ᵐ[μ] μ[⇑g₁ * g|m] + μ[⇑g₂ * g|m] : + by { rw add_mul, exact condexp_add (hg.simple_func_mul' hm _) (hg.simple_func_mul' hm _), } + ... =ᵐ[μ] ⇑g₁ * μ[g|m] + ⇑g₂ * μ[g|m] : eventually_eq.add h_eq₁ h_eq₂ + ... =ᵐ[μ] ⇑(g₁ + g₂) * μ[g|m] : by rw [h_add, add_mul], }, +end + +lemma condexp_strongly_measurable_mul_of_bound (hm : m ≤ m0) [is_finite_measure μ] + {f g : α → ℝ} (hf : strongly_measurable[m] f) (hg : integrable g μ) (c : ℝ) + (hf_bound : ∀ᵐ x ∂μ, ∥f x∥ ≤ c) : + μ[f * g | m] =ᵐ[μ] f * μ[g | m] := +begin + let fs := hf.approx_bounded c, + have hfs_tendsto : ∀ᵐ x ∂μ, tendsto (λ n, fs n x) at_top (𝓝 (f x)), + from hf.tendsto_approx_bounded_ae hf_bound, + by_cases hμ : μ = 0, + { simp only [hμ, ae_zero], }, + haveI : μ.ae.ne_bot, by simp only [hμ, ae_ne_bot, ne.def, not_false_iff], + have hc : 0 ≤ c, + { have h_exists : ∃ x, ∥f x∥ ≤ c := eventually.exists hf_bound, + exact (norm_nonneg _).trans h_exists.some_spec, }, + have hfs_bound : ∀ n x, ∥fs n x∥ ≤ c := hf.norm_approx_bounded_le hc, + have hn_eq : ∀ n, μ[fs n * g | m] =ᵐ[μ] fs n * μ[g | m], + from λ n, condexp_strongly_measurable_simple_func_mul hm _ hg, + have : μ[f * μ[g|m]|m] = f * μ[g|m], + { refine condexp_of_strongly_measurable hm (hf.mul strongly_measurable_condexp) _, + exact integrable_condexp.bdd_mul' ((hf.mono hm).ae_strongly_measurable) hf_bound, }, + rw ← this, + refine tendsto_condexp_unique (λ n x, fs n x * g x) (λ n x, fs n x * μ[g|m] x) (f * g) + (f * μ[g|m]) _ _ _ _ (λ x, c * ∥g x∥) _ (λ x, c * ∥μ[g|m] x∥) _ _ _ _, + { exact λ n, hg.bdd_mul' + ((simple_func.strongly_measurable (fs n)).mono hm).ae_strongly_measurable + (eventually_of_forall (hfs_bound n)), }, + { exact λ n, integrable_condexp.bdd_mul' + ((simple_func.strongly_measurable (fs n)).mono hm).ae_strongly_measurable + (eventually_of_forall (hfs_bound n)), }, + { filter_upwards [hfs_tendsto] with x hx, + rw pi.mul_apply, + exact tendsto.mul hx tendsto_const_nhds, }, + { filter_upwards [hfs_tendsto] with x hx, + rw pi.mul_apply, + exact tendsto.mul hx tendsto_const_nhds, }, + { exact hg.norm.const_mul c, }, + { exact integrable_condexp.norm.const_mul c, }, + { refine λ n, eventually_of_forall (λ x, _), + exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (hfs_bound n x) (norm_nonneg _)), }, + { refine λ n, eventually_of_forall (λ x, _), + exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (hfs_bound n x) (norm_nonneg _)), }, + { intros n, + simp_rw ← pi.mul_apply, + refine (condexp_strongly_measurable_simple_func_mul hm _ hg).trans _, + rw condexp_of_strongly_measurable hm + ((simple_func.strongly_measurable _).mul strongly_measurable_condexp) _, + { apply_instance, }, + { apply_instance, }, + exact integrable_condexp.bdd_mul' + ((simple_func.strongly_measurable (fs n)).mono hm).ae_strongly_measurable + (eventually_of_forall (hfs_bound n)), }, +end + +lemma condexp_strongly_measurable_mul_of_bound₀ (hm : m ≤ m0) [is_finite_measure μ] + {f g : α → ℝ} (hf : ae_strongly_measurable' m f μ) (hg : integrable g μ) (c : ℝ) + (hf_bound : ∀ᵐ x ∂μ, ∥f x∥ ≤ c) : + μ[f * g | m] =ᵐ[μ] f * μ[g | m] := +begin + have : μ[f * g | m] =ᵐ[μ] μ[hf.mk f * g | m], + from condexp_congr_ae (eventually_eq.mul hf.ae_eq_mk eventually_eq.rfl), + refine this.trans _, + have : f * μ[g | m] =ᵐ[μ] hf.mk f * μ[g | m] := eventually_eq.mul hf.ae_eq_mk eventually_eq.rfl, + refine eventually_eq.trans _ this.symm, + refine condexp_strongly_measurable_mul_of_bound hm hf.strongly_measurable_mk hg c _, + filter_upwards [hf_bound, hf.ae_eq_mk] with x hxc hx_eq, + rw ← hx_eq, + exact hxc, +end + +/-- Pull-out property of the conditional expectation. -/ +lemma condexp_strongly_measurable_mul {f g : α → ℝ} (hf : strongly_measurable[m] f) + (hfg : integrable (f * g) μ) (hg : integrable g μ) : + μ[f * g | m] =ᵐ[μ] f * μ[g | m] := +begin + by_cases hm : m ≤ m0, swap, { simp_rw condexp_of_not_le hm, rw mul_zero, }, + by_cases hμm : sigma_finite (μ.trim hm), + swap, { simp_rw condexp_of_not_sigma_finite hm hμm, rw mul_zero, }, + haveI : sigma_finite (μ.trim hm) := hμm, + obtain ⟨sets, sets_prop, h_univ⟩ := hf.exists_spanning_measurable_set_norm_le hm μ, + simp_rw forall_and_distrib at sets_prop, + obtain ⟨h_meas, h_finite, h_norm⟩ := sets_prop, + + suffices : ∀ n, ∀ᵐ x ∂μ, x ∈ sets n → μ[f * g|m] x = f x * μ[g|m] x, + { rw ← ae_all_iff at this, + filter_upwards [this] with x hx, + rw pi.mul_apply, + obtain ⟨i, hi⟩ : ∃ i, x ∈ sets i, + { have h_mem : x ∈ ⋃ i, sets i, + { rw h_univ, exact set.mem_univ _, }, + simpa using h_mem, }, + exact hx i hi, }, + refine λ n, ae_imp_of_ae_restrict _, + suffices : (μ.restrict (sets n))[f * g | m] + =ᵐ[μ.restrict (sets n)] f * (μ.restrict (sets n))[g | m], + { simp_rw ← pi.mul_apply, + refine (condexp_restrict_ae_eq_restrict hm (h_meas n) hfg).symm.trans _, + exact this.trans (eventually_eq.rfl.mul (condexp_restrict_ae_eq_restrict hm (h_meas n) hg)), }, + suffices : (μ.restrict (sets n))[((sets n).indicator f) * g | m] + =ᵐ[μ.restrict (sets n)] ((sets n).indicator f) * (μ.restrict (sets n))[g | m], + { refine eventually_eq.trans _ (this.trans _), + { exact condexp_congr_ae + ((indicator_ae_eq_restrict (hm _ (h_meas n))).symm.mul eventually_eq.rfl), }, + { exact (indicator_ae_eq_restrict (hm _ (h_meas n))).mul eventually_eq.rfl, }, }, + haveI : is_finite_measure (μ.restrict (sets n)), + { constructor, + rw measure.restrict_apply_univ, + exact h_finite n, }, + refine condexp_strongly_measurable_mul_of_bound hm (hf.indicator (h_meas n)) hg.integrable_on n _, + refine eventually_of_forall (λ x, _), + by_cases hxs : x ∈ sets n, + { simp only [hxs, set.indicator_of_mem], + exact h_norm n x hxs, }, + { simp only [hxs, set.indicator_of_not_mem, not_false_iff, _root_.norm_zero, nat.cast_nonneg], }, +end + +/-- Pull-out property of the conditional expectation. -/ +lemma condexp_strongly_measurable_mul₀ {f g : α → ℝ} (hf : ae_strongly_measurable' m f μ) + (hfg : integrable (f * g) μ) (hg : integrable g μ) : + μ[f * g | m] =ᵐ[μ] f * μ[g | m] := +begin + have : μ[f * g | m] =ᵐ[μ] μ[hf.mk f * g | m], + from condexp_congr_ae (eventually_eq.mul hf.ae_eq_mk eventually_eq.rfl), + refine this.trans _, + have : f * μ[g | m] =ᵐ[μ] hf.mk f * μ[g | m] := eventually_eq.mul hf.ae_eq_mk eventually_eq.rfl, + refine eventually_eq.trans _ this.symm, + refine condexp_strongly_measurable_mul hf.strongly_measurable_mk _ hg, + refine (integrable_congr _).mp hfg, + exact eventually_eq.mul hf.ae_eq_mk eventually_eq.rfl, +end + +end pull_out + +end measure_theory diff --git a/formal/lean/mathlib/measure_theory/function/lp_space.lean b/formal/lean/mathlib/measure_theory/function/lp_space.lean index adf23f5ebe61e266e8d04793fec8a8503dd3316f..c4ecc2d2109e2c30b608fff95d648a05b5fd8980 100644 --- a/formal/lean/mathlib/measure_theory/function/lp_space.lean +++ b/formal/lean/mathlib/measure_theory/function/lp_space.lean @@ -760,7 +760,7 @@ end ... ≤ (∫⁻ a, (∥f a∥₊ : ℝ≥0∞) ^ q + (∥g a∥₊ : ℝ≥0∞) ^ q ∂μ) ^ (1 / q) : begin refine ennreal.rpow_le_rpow (lintegral_mono (λ a, _)) (by simp [hq_pos.le] : 0 ≤ 1 / q), - exact ennreal.rpow_add_le_add_rpow _ _ hq_pos hq1, + exact ennreal.rpow_add_le_add_rpow _ _ hq_pos.le hq1, end ... < ∞ : begin @@ -1320,6 +1320,63 @@ hf.of_le_mul (ae_strongly_measurable.inner hf.1 ae_strongly_measurable_const) end inner_product +section liminf + +variables [measurable_space E] [opens_measurable_space E] {R : ℝ≥0} + +lemma ae_bdd_liminf_at_top_rpow_of_snorm_bdd {p : ℝ≥0∞} + {f : ℕ → α → E} (hfmeas : ∀ n, measurable (f n)) (hbdd : ∀ n, snorm (f n) p μ ≤ R) : + ∀ᵐ x ∂μ, liminf at_top (λ n, (∥f n x∥₊ ^ p.to_real : ℝ≥0∞)) < ∞ := +begin + by_cases hp0 : p.to_real = 0, + { simp only [hp0, ennreal.rpow_zero], + refine eventually_of_forall (λ x, _), + rw liminf_const (1 : ℝ≥0∞), + exacts [ennreal.one_lt_top, at_top_ne_bot] }, + have hp : p ≠ 0 := λ h, by simpa [h] using hp0, + have hp' : p ≠ ∞ := λ h, by simpa [h] using hp0, + refine ae_lt_top + (measurable_liminf (λ n, (hfmeas n).nnnorm.coe_nnreal_ennreal.pow_const p.to_real)) + (lt_of_le_of_lt (lintegral_liminf_le + (λ n, (hfmeas n).nnnorm.coe_nnreal_ennreal.pow_const p.to_real)) + (lt_of_le_of_lt _ (ennreal.rpow_lt_top_of_nonneg + ennreal.to_real_nonneg ennreal.coe_ne_top : ↑R ^ p.to_real < ∞))).ne, + simp_rw snorm_eq_lintegral_rpow_nnnorm hp hp' at hbdd, + simp_rw [liminf_eq, eventually_at_top], + exact Sup_le (λ b ⟨a, ha⟩, (ha a le_rfl).trans + ((ennreal.rpow_one_div_le_iff (ennreal.to_real_pos hp hp')).1 (hbdd _))), +end + +lemma ae_bdd_liminf_at_top_of_snorm_bdd {p : ℝ≥0∞} (hp : p ≠ 0) + {f : ℕ → α → E} (hfmeas : ∀ n, measurable (f n)) (hbdd : ∀ n, snorm (f n) p μ ≤ R) : + ∀ᵐ x ∂μ, liminf at_top (λ n, (∥f n x∥₊ : ℝ≥0∞)) < ∞ := +begin + by_cases hp' : p = ∞, + { subst hp', + simp_rw snorm_exponent_top at hbdd, + have : ∀ n, ∀ᵐ x ∂μ, (∥f n x∥₊ : ℝ≥0∞) < R + 1 := + λ n, ae_lt_of_ess_sup_lt (lt_of_le_of_lt (hbdd n) $ + ennreal.lt_add_right ennreal.coe_ne_top one_ne_zero), + rw ← ae_all_iff at this, + filter_upwards [this] with x hx using lt_of_le_of_lt + (liminf_le_of_frequently_le' $ frequently_of_forall $ λ n, (hx n).le) + (ennreal.add_lt_top.2 ⟨ennreal.coe_lt_top, ennreal.one_lt_top⟩) }, + filter_upwards [ae_bdd_liminf_at_top_rpow_of_snorm_bdd hfmeas hbdd] with x hx, + have hppos : 0 < p.to_real := ennreal.to_real_pos hp hp', + have : liminf at_top (λ n, (∥f n x∥₊ ^ p.to_real : ℝ≥0∞)) = + liminf at_top (λ n, (∥f n x∥₊ : ℝ≥0∞)) ^ p.to_real, + { change liminf at_top (λ n, ennreal.order_iso_rpow p.to_real hppos (∥f n x∥₊ : ℝ≥0∞)) = + ennreal.order_iso_rpow p.to_real hppos (liminf at_top (λ n, (∥f n x∥₊ : ℝ≥0∞))), + refine (order_iso.liminf_apply (ennreal.order_iso_rpow p.to_real _) _ _ _ _).symm; + is_bounded_default }, + rw this at hx, + rw [← ennreal.rpow_one (liminf at_top (λ n, ∥f n x∥₊)), ← mul_inv_cancel hppos.ne.symm, + ennreal.rpow_mul], + exact ennreal.rpow_lt_top_of_nonneg (inv_nonneg.2 hppos.le) hx.ne, +end + +end liminf + end ℒp /-! diff --git a/formal/lean/mathlib/measure_theory/function/uniform_integrable.lean b/formal/lean/mathlib/measure_theory/function/uniform_integrable.lean index 7288f0efe7dff9275ec1afd71fe57bc6fab3011e..12a46e9d26327c9b3e1fb3b8ebbafc786c960b90 100644 --- a/formal/lean/mathlib/measure_theory/function/uniform_integrable.lean +++ b/formal/lean/mathlib/measure_theory/function/uniform_integrable.lean @@ -973,7 +973,7 @@ begin exact finset.sum_le_card_nsmul _ _ _ (λ i hi, hδ₂ _ _ hs hle) }, { simp only [ennreal.coe_eq_zero, inv_eq_zero, nat.cast_eq_zero] at hn, rw [nsmul_eq_mul, ← mul_assoc, ennreal.coe_inv, ennreal.coe_nat, - ennreal.inv_mul_cancel _ ennreal.coe_nat_ne_top, one_mul], + ennreal.inv_mul_cancel _ (ennreal.nat_ne_top _), one_mul], { exact le_rfl }, all_goals { simpa only [ne.def, nat.cast_eq_zero] } } }, { obtain ⟨C, hC⟩ := hf₃, @@ -992,7 +992,7 @@ begin exact finset.sum_le_card_nsmul _ _ _ (λ i hi, hC i) }, { simp only [ennreal.coe_eq_zero, inv_eq_zero, nat.cast_eq_zero] at hn, rw [nsmul_eq_mul, ← mul_assoc, ennreal.coe_inv, ennreal.coe_nat, - ennreal.inv_mul_cancel _ ennreal.coe_nat_ne_top, one_mul], + ennreal.inv_mul_cancel _ (ennreal.nat_ne_top _), one_mul], { exact le_rfl }, all_goals { simpa only [ne.def, nat.cast_eq_zero] } } } end diff --git a/formal/lean/mathlib/measure_theory/group/measure.lean b/formal/lean/mathlib/measure_theory/group/measure.lean index 16e6d6bfd37c1a24f4791669d4b060da8a6df82e..2ade894b17f6192601da7bddfc9c13b5fe432b14 100644 --- a/formal/lean/mathlib/measure_theory/group/measure.lean +++ b/formal/lean/mathlib/measure_theory/group/measure.lean @@ -384,7 +384,7 @@ begin calc μ K ≤ μ (⋃ (g : G) (H : g ∈ t), (λ (h : G), g * h) ⁻¹' U) : measure_mono hKt ... ≤ ∑ g in t, μ ((λ (h : G), g * h) ⁻¹' U) : measure_bUnion_finset_le _ _ ... = finset.card t * μ U : by simp only [measure_preimage_mul, finset.sum_const, nsmul_eq_mul] - ... < ∞ : ennreal.mul_lt_top ennreal.coe_nat_ne_top h + ... < ∞ : ennreal.mul_lt_top (ennreal.nat_ne_top _) h end /-- If a left-invariant measure gives finite mass to a set with nonempty interior, then diff --git a/formal/lean/mathlib/measure_theory/group/prod.lean b/formal/lean/mathlib/measure_theory/group/prod.lean index 9e84e468f01ffbe8302c870fdca123f6e6e54b1e..296ab73148c1a34ead79e82cb8a7e5b388f17ad9 100644 --- a/formal/lean/mathlib/measure_theory/group/prod.lean +++ b/formal/lean/mathlib/measure_theory/group/prod.lean @@ -79,7 +79,7 @@ lemma measurable_measure_mul_right (hE : measurable_set E) : measurable (λ x, μ ((λ y, y * x) ⁻¹' E)) := begin suffices : measurable (λ y, - μ ((λ x, (x, y)) ⁻¹' ((λ z : G × G, ((1 : G), z.1 * z.2)) ⁻¹' ((univ : set G) ×ˢ E)))), + μ ((λ x, (x, y)) ⁻¹' ((λ z : G × G, ((1 : G), z.1 * z.2)) ⁻¹' (univ ×ˢ E)))), { convert this, ext1 x, congr' 1 with y : 1, simp }, apply measurable_measure_prod_mk_right, exact measurable_const.prod_mk (measurable_fst.mul measurable_snd) (measurable_set.univ.prod hE) diff --git a/formal/lean/mathlib/measure_theory/integral/circle_integral_transform.lean b/formal/lean/mathlib/measure_theory/integral/circle_integral_transform.lean index 81ce0c737c03bab84bac7b1b6db032fde7d04009..36b42bcf906b16da337764a72f8c43abb7744693 100644 --- a/formal/lean/mathlib/measure_theory/integral/circle_integral_transform.lean +++ b/formal/lean/mathlib/measure_theory/integral/circle_integral_transform.lean @@ -97,8 +97,7 @@ def circle_transform_bounding_function (R : ℝ) (z : ℂ) (w : ℂ × ℝ) : circle_transform_deriv R z w.1 (λ x, 1) w.2 lemma continuous_on_prod_circle_transform_function {R r : ℝ} (hr : r < R) {z : ℂ} : - continuous_on (λ (w : ℂ × ℝ), ((circle_map z R w.snd - w.fst)⁻¹) ^ 2) - ((closed_ball z r) ×ˢ (⊤ : set ℝ)) := + continuous_on (λ w : ℂ × ℝ, ((circle_map z R w.snd - w.fst)⁻¹) ^ 2) (closed_ball z r ×ˢ univ) := begin simp_rw ←one_div, apply_rules [continuous_on.pow, continuous_on.div, continuous_on_const], @@ -111,8 +110,7 @@ begin end lemma continuous_on_abs_circle_transform_bounding_function {R r : ℝ} (hr : r < R) (z : ℂ) : - continuous_on (abs ∘ (λ t, circle_transform_bounding_function R z t)) - ((closed_ball z r) ×ˢ (⊤ : set ℝ) : set $ ℂ × ℝ) := + continuous_on (abs ∘ (λ t, circle_transform_bounding_function R z t)) (closed_ball z r ×ˢ univ) := begin have : continuous_on (circle_transform_bounding_function R z) (closed_ball z r ×ˢ (⊤ : set ℝ)), { apply_rules [continuous_on.smul, continuous_on_const], @@ -127,12 +125,12 @@ begin end lemma abs_circle_transform_bounding_function_le {R r : ℝ} (hr : r < R) (hr' : 0 ≤ r) (z : ℂ) : - ∃ (x : ((closed_ball z r) ×ˢ [0, 2 * π] : set $ ℂ × ℝ)), - ∀ (y : ((closed_ball z r) ×ˢ [0, 2 * π] : set $ ℂ × ℝ)), + ∃ x : closed_ball z r ×ˢ [0, 2 * π], + ∀ y : closed_ball z r ×ˢ [0, 2 * π], abs (circle_transform_bounding_function R z y) ≤ abs (circle_transform_bounding_function R z x) := begin have cts := continuous_on_abs_circle_transform_bounding_function hr z, - have comp : is_compact (((closed_ball z r) ×ˢ [0, 2 * π]) : set (ℂ × ℝ)), + have comp : is_compact (closed_ball z r ×ˢ [0, 2 * π]), { apply_rules [is_compact.prod, proper_space.is_compact_closed_ball z r, is_compact_interval], }, have none := (nonempty_closed_ball.2 hr').prod nonempty_interval, simpa using is_compact.exists_forall_ge comp none (cts.mono (by { intro z, simp, tauto })), diff --git a/formal/lean/mathlib/measure_theory/integral/lebesgue.lean b/formal/lean/mathlib/measure_theory/integral/lebesgue.lean index c44276bbd9a16cb09efa44877fdb5ccd50f92002..c6221acf215c966fcd3eb41d1844dde0aab2c1eb 100644 --- a/formal/lean/mathlib/measure_theory/integral/lebesgue.lean +++ b/formal/lean/mathlib/measure_theory/integral/lebesgue.lean @@ -1680,7 +1680,7 @@ begin have : ∫⁻ x, f x ∂μ + (↑n)⁻¹ * μ {x : α | f x + n⁻¹ ≤ g x} ≤ ∫⁻ x, f x ∂μ, from (lintegral_add_mul_meas_add_le_le_lintegral hfg hg n⁻¹).trans hgf, rw [(ennreal.cancel_of_ne hf).add_le_iff_nonpos_right, nonpos_iff_eq_zero, mul_eq_zero] at this, - exact this.resolve_left (ennreal.inv_ne_zero.2 ennreal.coe_nat_ne_top) }, + exact this.resolve_left (ennreal.inv_ne_zero.2 (ennreal.nat_ne_top _)) }, refine hfg.mp ((ae_all_iff.2 this).mono (λ x hlt hle, hle.antisymm _)), suffices : tendsto (λ n : ℕ, f x + n⁻¹) at_top (𝓝 (f x)), from ge_of_tendsto' this (λ i, (hlt i).le), diff --git a/formal/lean/mathlib/measure_theory/measurable_space.lean b/formal/lean/mathlib/measure_theory/measurable_space.lean index 214a70853781fa47826d46ce8a682d7f096d8ce6..386c3000b628f5922407ce405623b9db84aa68ff 100644 --- a/formal/lean/mathlib/measure_theory/measurable_space.lean +++ b/formal/lean/mathlib/measure_theory/measurable_space.lean @@ -552,7 +552,7 @@ lemma measurable_set.prod {s : set α} {t : set β} (hs : measurable_set s) (ht measurable_set (s ×ˢ t) := measurable_set.inter (measurable_fst hs) (measurable_snd ht) -lemma measurable_set_prod_of_nonempty {s : set α} {t : set β} (h : (s ×ˢ t : set _).nonempty) : +lemma measurable_set_prod_of_nonempty {s : set α} {t : set β} (h : (s ×ˢ t).nonempty) : measurable_set (s ×ˢ t) ↔ measurable_set s ∧ measurable_set t := begin rcases h with ⟨⟨x, y⟩, hx, hy⟩, @@ -565,7 +565,7 @@ end lemma measurable_set_prod {s : set α} {t : set β} : measurable_set (s ×ˢ t) ↔ (measurable_set s ∧ measurable_set t) ∨ s = ∅ ∨ t = ∅ := begin - cases (s ×ˢ t : set _).eq_empty_or_nonempty with h h, + cases (s ×ˢ t).eq_empty_or_nonempty with h h, { simp [h, prod_eq_empty_iff.mp h] }, { simp [←not_nonempty_iff_eq_empty, prod_nonempty_iff.mp h, measurable_set_prod_of_nonempty h] } end diff --git a/formal/lean/mathlib/measure_theory/measure/hausdorff.lean b/formal/lean/mathlib/measure_theory/measure/hausdorff.lean index 10076b104abbb2d38ccb906a4669780162185c64..0fadb3eee711c1d2bcc8e3342cb4d5156c63dd85 100644 --- a/formal/lean/mathlib/measure_theory/measure/hausdorff.lean +++ b/formal/lean/mathlib/measure_theory/measure/hausdorff.lean @@ -163,7 +163,7 @@ begin rw [borel_eq_generate_from_is_closed], refine measurable_space.generate_from_le (λ t ht, μ.is_caratheodory_iff_le.2 $ λ s, _), set S : ℕ → set X := λ n, {x ∈ s | (↑n)⁻¹ ≤ inf_edist x t}, - have n0 : ∀ {n : ℕ}, (n⁻¹ : ℝ≥0∞) ≠ 0, from λ n, ennreal.inv_ne_zero.2 ennreal.coe_nat_ne_top, + have n0 : ∀ {n : ℕ}, (n⁻¹ : ℝ≥0∞) ≠ 0, from λ n, ennreal.inv_ne_zero.2 (ennreal.nat_ne_top _), have Ssep : ∀ n, is_metric_separated (S n) t, from λ n, ⟨n⁻¹, n0, λ x hx y hy, hx.2.trans $ inf_edist_le_edist_of_mem hy⟩, have Ssep' : ∀ n, is_metric_separated (S n) (s ∩ t), diff --git a/formal/lean/mathlib/measure_theory/measure/lebesgue.lean b/formal/lean/mathlib/measure_theory/measure/lebesgue.lean index bb2065e5b5fdd2f37ede96f7533a6c3c0bf9f148..0249bbb9a3c6c0f096ad6fa92c5bc6babdd55bb2 100644 --- a/formal/lean/mathlib/measure_theory/measure/lebesgue.lean +++ b/formal/lean/mathlib/measure_theory/measure/lebesgue.lean @@ -396,8 +396,7 @@ variable {α : Type*} def region_between (f g : α → ℝ) (s : set α) : set (α × ℝ) := {p : α × ℝ | p.1 ∈ s ∧ p.2 ∈ Ioo (f p.1) (g p.1)} -lemma region_between_subset (f g : α → ℝ) (s : set α) : - region_between f g s ⊆ s ×ˢ (univ : set ℝ) := +lemma region_between_subset (f g : α → ℝ) (s : set α) : region_between f g s ⊆ s ×ˢ univ := by simpa only [prod_univ, region_between, set.preimage, set_of_subset_set_of] using λ a, and.left variables [measurable_space α] {μ : measure α} {f g : α → ℝ} {s : set α} diff --git a/formal/lean/mathlib/measure_theory/measure/measure_space.lean b/formal/lean/mathlib/measure_theory/measure/measure_space.lean index 431bbcde4d5b0335abe38572b2e28bb2f8c8a5d6..529820cdfca5f3083562c2b6023bfab09595c90e 100644 --- a/formal/lean/mathlib/measure_theory/measure/measure_space.lean +++ b/formal/lean/mathlib/measure_theory/measure/measure_space.lean @@ -1414,6 +1414,14 @@ begin outer_measure.restrict_apply] end +lemma exists_mem_of_measure_ne_zero_of_ae (hs : μ s ≠ 0) + {p : α → Prop} (hp : ∀ᵐ x ∂μ.restrict s, p x) : + ∃ x, x ∈ s ∧ p x := +begin + rw [← μ.restrict_apply_self, ← frequently_ae_mem_iff] at hs, + exact (hs.and_eventually hp).exists, +end + /-! ### Extensionality results -/ /-- Two measures are equal if they have equal restrictions on a spanning collection of sets @@ -2483,7 +2491,7 @@ lemma finite_at_bot {m0 : measurable_space α} (μ : measure α) : μ.finite_at_ about the sets, such as that they are monotone. `sigma_finite` is defined in terms of this: `μ` is σ-finite if there exists a sequence of finite spanning sets in the collection of all measurable sets. -/ -@[protect_proj, nolint has_inhabited_instance] +@[protect_proj, nolint has_nonempty_instance] structure finite_spanning_sets_in {m0 : measurable_space α} (μ : measure α) (C : set (set α)) := (set : ℕ → set α) (set_mem : ∀ i, set i ∈ C) @@ -2863,6 +2871,13 @@ protected lemma is_finite_measure_on_compacts.smul [topological_space α] (μ : is_finite_measure_on_compacts (c • μ) := ⟨λ K hK, ennreal.mul_lt_top hc (hK.measure_lt_top).ne⟩ +/-- Note this cannot be an instance because it would form a typeclass loop with +`is_finite_measure_on_compacts_of_is_locally_finite_measure`. -/ +lemma compact_space.is_finite_measure + [topological_space α] [compact_space α] [is_finite_measure_on_compacts μ] : + is_finite_measure μ := +⟨is_finite_measure_on_compacts.lt_top_of_is_compact compact_univ⟩ + omit m0 @[priority 100] -- see Note [lower instance priority] @@ -3323,6 +3338,15 @@ instance is_finite_measure_on_compacts_of_is_locally_finite_measure [is_locally_finite_measure μ] : is_finite_measure_on_compacts μ := ⟨λ s hs, hs.measure_lt_top_of_nhds_within $ λ x hx, μ.finite_at_nhds_within _ _⟩ +lemma is_finite_measure_iff_is_finite_measure_on_compacts_of_compact_space + [topological_space α] [measurable_space α] {μ : measure α} [compact_space α] : + is_finite_measure μ ↔ is_finite_measure_on_compacts μ := +begin + split; introsI, + { apply_instance, }, + { exact compact_space.is_finite_measure, }, +end + /-- Compact covering of a `σ`-compact topological space as `measure_theory.measure.finite_spanning_sets_in`. -/ def measure_theory.measure.finite_spanning_sets_in_compact [topological_space α] diff --git a/formal/lean/mathlib/model_theory/basic.lean b/formal/lean/mathlib/model_theory/basic.lean index 4b4bcf682c69b094f18ea0ad55cbff5dd5a5294d..2c3df278de426b3ae386813d459ee837baf4e6c9 100644 --- a/formal/lean/mathlib/model_theory/basic.lean +++ b/formal/lean/mathlib/model_theory/basic.lean @@ -112,14 +112,14 @@ protected def sum (L : language.{u v}) (L' : language.{u' v'}) : language := variable (L : language.{u v}) /-- The type of constants in a given language. -/ -@[nolint has_inhabited_instance] protected def «constants» := L.functions 0 +@[nolint has_nonempty_instance] protected def «constants» := L.functions 0 @[simp] lemma constants_mk₂ (c f₁ f₂ : Type u) (r₁ r₂ : Type v) : (language.mk₂ c f₁ f₂ r₁ r₂).constants = c := rfl /-- The type of symbols in a given language. -/ -@[nolint has_inhabited_instance] def symbols := (Σl, L.functions l) ⊕ (Σl, L.relations l) +@[nolint has_nonempty_instance] def symbols := (Σl, L.functions l) ⊕ (Σl, L.relations l) /-- The cardinality of a language is the cardinality of its type of symbols. -/ def card : cardinal := # L.symbols diff --git a/formal/lean/mathlib/model_theory/language_map.lean b/formal/lean/mathlib/model_theory/language_map.lean index 9c759c5a40d4d3cb95918818f4f279f200f43bf4..8bbbe065e63ab3721d353e401b03c69dd9eda682 100644 --- a/formal/lean/mathlib/model_theory/language_map.lean +++ b/formal/lean/mathlib/model_theory/language_map.lean @@ -27,7 +27,7 @@ the continuum hypothesis*][flypitch_itp] -/ -universes u v u' v' w +universes u v u' v' w w' namespace first_order namespace language @@ -179,7 +179,17 @@ class is_expansion_on (M : Type*) [L.Structure M] [L'.Structure M] : Prop := (map_on_relation : ∀ {n} (R : L.relations n) (x : fin n → M), rel_map (ϕ.on_relation R) x = rel_map R x) -attribute [simp] is_expansion_on.map_on_function is_expansion_on.map_on_relation +@[simp] lemma map_on_function {M : Type*} + [L.Structure M] [L'.Structure M] [ϕ.is_expansion_on M] + {n} (f : L.functions n) (x : fin n → M) : + fun_map (ϕ.on_function f) x = fun_map f x := +is_expansion_on.map_on_function f x + +@[simp] lemma map_on_relation {M : Type*} + [L.Structure M] [L'.Structure M] [ϕ.is_expansion_on M] + {n} (R : L.relations n) (x : fin n → M) : + rel_map (ϕ.on_relation R) x = rel_map R x := +is_expansion_on.map_on_relation R x instance id_is_expansion_on (M : Type*) [L.Structure M] : is_expansion_on (Lhom.id L) M := ⟨λ _ _ _, rfl, λ _ _ _, rfl⟩ @@ -211,6 +221,18 @@ instance sum_inr_is_expansion_on (M : Type*) (Lhom.sum_inr : L' →ᴸ L.sum L').is_expansion_on M := ⟨λ _ f _, rfl, λ _ R _, rfl⟩ +@[simp] lemma fun_map_sum_inl [(L.sum L').Structure M] + [(Lhom.sum_inl : L →ᴸ L.sum L').is_expansion_on M] + {n} {f : L.functions n} {x : fin n → M} : + @fun_map (L.sum L') M _ n (sum.inl f) x = fun_map f x := +(Lhom.sum_inl : L →ᴸ L.sum L').map_on_function f x + +@[simp] lemma fun_map_sum_inr [(L'.sum L).Structure M] + [(Lhom.sum_inr : L →ᴸ L'.sum L).is_expansion_on M] + {n} {f : L.functions n} {x : fin n → M} : + @fun_map (L'.sum L) M _ n (sum.inr f) x = fun_map f x := +(Lhom.sum_inr : L →ᴸ L'.sum L).map_on_function f x + @[priority 100] instance is_expansion_on_reduct (ϕ : L →ᴸ L') (M : Type*) [L'.Structure M] : @is_expansion_on L L' ϕ M (ϕ.reduct M) _ := begin @@ -307,15 +329,15 @@ section with_constants variable (L) section -variables (α : Type w) +variables (α : Type w') /-- Extends a language with a constant for each element of a parameter set in `M`. -/ -def with_constants : language.{(max u w) v} := L.sum (constants_on α) +def with_constants : language.{(max u w') v} := L.sum (constants_on α) localized "notation L`[[`:95 α`]]`:90 := L.with_constants α" in first_order @[simp] lemma card_with_constants : - (L[[α]]).card = cardinal.lift.{w} L.card + cardinal.lift.{max u v} (# α) := + (L[[α]]).card = cardinal.lift.{w'} L.card + cardinal.lift.{max u v} (# α) := by rw [with_constants, card_sum, card_constants_on] /-- The language map adding constants. -/ @@ -346,6 +368,18 @@ variables (L) (α) variables {α} {β : Type*} +@[simp] lemma with_constants_fun_map_sum_inl [L[[α]].Structure M] + [(Lhom_with_constants L α).is_expansion_on M] + {n} {f : L.functions n} {x : fin n → M} : + @fun_map (L[[α]]) M _ n (sum.inl f) x = fun_map f x := +(Lhom_with_constants L α).map_on_function f x + +@[simp] lemma with_constants_rel_map_sum_inl [L[[α]].Structure M] + [(Lhom_with_constants L α).is_expansion_on M] + {n} {R : L.relations n} {x : fin n → M} : + @rel_map (L[[α]]) M _ n (sum.inl R) x = rel_map R x := +(Lhom_with_constants L α).map_on_relation R x + /-- The language map extending the constant set. -/ def Lhom_with_constants_map (f : α → β) : L[[α]] →ᴸ L[[β]] := Lhom.sum_map (Lhom.id L) (Lhom.constants_on_map f) @@ -388,6 +422,13 @@ instance add_constants_expansion {L' : language} [L'.Structure M] (φ : L →ᴸ (φ.add_constants α).is_expansion_on M := Lhom.sum_map_is_expansion_on _ _ M +@[simp] lemma with_constants_fun_map_sum_inr {a : α} {x : fin 0 → M} : + @fun_map (L[[α]]) M _ 0 (sum.inr a : L[[α]].functions 0) x = L.con a := +begin + rw unique.eq_default x, + exact (Lhom.sum_inr : (constants_on α) →ᴸ L.sum _).map_on_function _ _, +end + variables {α} (A : set M) @[simp] lemma coe_con {a : A} : ((L.con a) : M) = a := rfl diff --git a/formal/lean/mathlib/model_theory/order.lean b/formal/lean/mathlib/model_theory/order.lean index f483e7782e1962b61f7ad7e97226645c7a6bb62e..640ce8e9546ed9fadf58fbfdcf9004bd9b1038fd 100644 --- a/formal/lean/mathlib/model_theory/order.lean +++ b/formal/lean/mathlib/model_theory/order.lean @@ -169,7 +169,7 @@ variables [is_ordered L] [L.Structure M] @[simp] lemma rel_map_le_symb [has_le M] [L.is_ordered_structure M] {a b : M} : rel_map (le_symb : L.relations 2) ![a, b] ↔ a ≤ b := begin - rw [← order_Lhom_le_symb, Lhom.is_expansion_on.map_on_relation], + rw [← order_Lhom_le_symb, Lhom.map_on_relation], refl, end diff --git a/formal/lean/mathlib/model_theory/semantics.lean b/formal/lean/mathlib/model_theory/semantics.lean index cab8779e403c5b8103b8ca8859392c3af9edcac3..c4ed8a4946d183140ed8d257642b2d1a22fe35b9 100644 --- a/formal/lean/mathlib/model_theory/semantics.lean +++ b/formal/lean/mathlib/model_theory/semantics.lean @@ -134,6 +134,48 @@ begin exact congr rfl (funext (λ i, ih i (h i (finset.mem_univ i)))) }, end +@[simp] lemma realize_constants_to_vars [L[[α]].Structure M] + [(Lhom_with_constants L α).is_expansion_on M] + {t : L[[α]].term β} {v : β → M} : + t.constants_to_vars.realize (sum.elim (λ a, ↑(L.con a)) v) = t.realize v := +begin + induction t with _ n f _ ih, + { simp }, + { cases n, + { cases f, + { simp [ih], }, + { simp only [realize, constants_to_vars, sum.elim_inl, fun_map_eq_coe_constants], + refl } }, + { cases f, + { simp [ih] }, + { exact is_empty_elim f } } } +end + +@[simp] lemma realize_vars_to_constants [L[[α]].Structure M] + [(Lhom_with_constants L α).is_expansion_on M] + {t : L.term (α ⊕ β)} {v : β → M} : + t.vars_to_constants.realize v = t.realize (sum.elim (λ a, ↑(L.con a)) v) := +begin + induction t with ab n f ts ih, + { cases ab; + simp [language.con], }, + { simp [ih], } +end + +lemma realize_constants_vars_equiv_left [L[[α]].Structure M] + [(Lhom_with_constants L α).is_expansion_on M] + {n} {t : L[[α]].term (β ⊕ fin n)} {v : β → M} {xs : fin n → M} : + (constants_vars_equiv_left t).realize (sum.elim (sum.elim (λ a, ↑(L.con a)) v) xs) = + t.realize (sum.elim v xs) := +begin + simp only [constants_vars_equiv_left, realize_relabel, equiv.coe_trans, function.comp_app, + constants_vars_equiv_apply, relabel_equiv_symm_apply], + refine trans _ (realize_constants_to_vars), + rcongr, + rcases x with (a | (b | i)); + simp, +end + end term namespace Lhom @@ -144,7 +186,7 @@ namespace Lhom begin induction t with _ n f ts ih, { refl }, - { simp only [term.realize, Lhom.on_term, Lhom.is_expansion_on.map_on_function, ih] } + { simp only [term.realize, Lhom.on_term, Lhom.map_on_function, ih] } end end Lhom @@ -393,6 +435,20 @@ begin { simp [restrict_free_var, realize, ih3] }, end +lemma realize_constants_vars_equiv [L[[α]].Structure M] + [(Lhom_with_constants L α).is_expansion_on M] + {n} {φ : L[[α]].bounded_formula β n} {v : β → M} {xs : fin n → M} : + (constants_vars_equiv φ).realize (sum.elim (λ a, ↑(L.con a)) v) xs ↔ φ.realize v xs := +begin + refine realize_map_term_rel_id (λ n t xs, realize_constants_vars_equiv_left) (λ n R xs, _), + rw ← (Lhom_with_constants L α).map_on_relation (equiv.sum_empty (L.relations n) + ((constants_on α).relations n) R) xs, + rcongr, + cases R, + { simp, }, + { exact is_empty_elim R } +end + variables [nonempty M] lemma realize_all_lift_at_one_self {n : ℕ} {φ : L.bounded_formula α n} @@ -486,7 +542,7 @@ begin { refl }, { simp only [on_bounded_formula, realize_bd_equal, realize_on_term], refl, }, - { simp only [on_bounded_formula, realize_rel, realize_on_term, is_expansion_on.map_on_relation], + { simp only [on_bounded_formula, realize_rel, realize_on_term, Lhom.map_on_relation], refl, }, { simp only [on_bounded_formula, ih1, ih2, realize_imp], }, { simp only [on_bounded_formula, ih3, realize_all], }, diff --git a/formal/lean/mathlib/model_theory/substructures.lean b/formal/lean/mathlib/model_theory/substructures.lean index 68187258f94971ebe1c1b74d5b714b48c2e2e1f3..7104f63c491a32fefa62c18d275555f5cb9fdc19 100644 --- a/formal/lean/mathlib/model_theory/substructures.lean +++ b/formal/lean/mathlib/model_theory/substructures.lean @@ -577,7 +577,7 @@ def substructure_reduct : L'.substructure M ↪o L.substructure M := { to_fun := λ S, { carrier := S, fun_mem := λ n f x hx, begin have h := S.fun_mem (φ.on_function f) x hx, - simp only [is_expansion_on.map_on_function, substructure.mem_carrier] at h, + simp only [Lhom.map_on_function, substructure.mem_carrier] at h, exact h, end }, inj' := λ S T h, begin diff --git a/formal/lean/mathlib/model_theory/syntax.lean b/formal/lean/mathlib/model_theory/syntax.lean index 7d7c4c35869aaf3d606a366c944b3d885ad69400..afa5ab848bd720285c5f0213687c9c06bedbe86b 100644 --- a/formal/lean/mathlib/model_theory/syntax.lean +++ b/formal/lean/mathlib/model_theory/syntax.lean @@ -31,6 +31,9 @@ above a particular index. variables with given terms. * Language maps can act on syntactic objects with functions such as `first_order.language.Lhom.on_formula`. +* `first_order.language.term.constants_vars_equiv` and +`first_order.language.bounded_formula.constants_vars_equiv` switch terms and formulas between having +constants in the language and having extra variables indexed by the same type. ## Implementation Notes * Formulas use a modified version of de Bruijn variables. Specifically, a `L.bounded_formula α n` @@ -112,6 +115,10 @@ end (term.relabel g ∘ term.relabel f : L.term α → L.term γ) = term.relabel (g ∘ f) := funext (relabel_relabel f g) +/-- Relabels a term's variables along a bijection. -/ +@[simps] def relabel_equiv (g : α ≃ β) : L.term α ≃ L.term β := +⟨relabel g, relabel g.symm, λ t, by simp, λ t, by simp⟩ + /-- Restricts a term to use only a set of the given variables. -/ def restrict_var [decidable_eq α] : Π (t : L.term α) (f : t.var_finset → β), L.term β | (var a) f := var (f ⟨a, mem_singleton_self a⟩) @@ -140,6 +147,54 @@ def functions.apply₂ (f : L.functions 2) (t₁ t₂ : L.term α) : L.term α : namespace term +/-- Sends a term with constants to a term with extra variables. -/ +@[simp] def constants_to_vars : L[[γ]].term α → L.term (γ ⊕ α) +| (var a) := var (sum.inr a) +| (@func _ _ 0 f ts) := sum.cases_on f (λ f, func f (λ i, (ts i).constants_to_vars)) + (λ c, var (sum.inl c)) +| (@func _ _ (n + 1) f ts) := sum.cases_on f (λ f, func f (λ i, (ts i).constants_to_vars)) + (λ c, is_empty_elim c) + +/-- Sends a term with extra variables to a term with constants. -/ +@[simp] def vars_to_constants : L.term (γ ⊕ α) → L[[γ]].term α +| (var (sum.inr a)) := var a +| (var (sum.inl c)) := constants.term (sum.inr c) +| (func f ts) := func (sum.inl f) (λ i, (ts i).vars_to_constants) + +/-- A bijection between terms with constants and terms with extra variables. -/ +@[simps] def constants_vars_equiv : L[[γ]].term α ≃ L.term (γ ⊕ α) := +⟨constants_to_vars, vars_to_constants, begin + intro t, + induction t with _ n f _ ih, + { refl }, + { cases n, + { cases f, + { simp [constants_to_vars, vars_to_constants, ih] }, + { simp [constants_to_vars, vars_to_constants, constants.term] } }, + { cases f, + { simp [constants_to_vars, vars_to_constants, ih] }, + { exact is_empty_elim f } } } +end, begin + intro t, + induction t with x n f _ ih, + { cases x; + refl }, + { cases n; + { simp [vars_to_constants, constants_to_vars, ih] } } +end⟩ + +/-- A bijection between terms with constants and terms with extra variables. -/ +def constants_vars_equiv_left : L[[γ]].term (α ⊕ β) ≃ L.term ((γ ⊕ α) ⊕ β) := +constants_vars_equiv.trans (relabel_equiv (equiv.sum_assoc _ _ _)).symm + +@[simp] lemma constants_vars_equiv_left_apply (t : L[[γ]].term (α ⊕ β)) : + constants_vars_equiv_left t = (constants_to_vars t).relabel (equiv.sum_assoc _ _ _).symm := +rfl + +@[simp] lemma constants_vars_equiv_left_symm_apply (t : L.term ((γ ⊕ α) ⊕ β)) : + constants_vars_equiv_left.symm t = vars_to_constants (t.relabel (equiv.sum_assoc _ _ _)) := +rfl + instance inhabited_of_var [inhabited α] : inhabited (L.term α) := ⟨var default⟩ @@ -485,6 +540,10 @@ end φ.map_term_rel (λ _ t, t.subst (sum.elim (term.relabel sum.inl ∘ f) (var ∘ sum.inr))) (λ _, id) (λ _, id) +/-- A bijection sending formulas with constants to formulas with extra variables. -/ +def constants_vars_equiv : L[[γ]].bounded_formula α n ≃ L.bounded_formula (γ ⊕ α) n := +map_term_rel_equiv (λ _, term.constants_vars_equiv_left) (λ _, equiv.sum_empty _ _) + /-- Turns the extra variables of a bounded formula into free variables. -/ @[simp] def to_formula : ∀ {n : ℕ}, L.bounded_formula α n → L.formula (α ⊕ fin n) | n falsum := falsum @@ -864,7 +923,7 @@ def nonempty_theory : L.Theory := {sentence.card_ge L 1} /-- A theory indicating that each of a set of constants is distinct. -/ def distinct_constants_theory (s : set α) : L[[α]].Theory := -(λ ab : α × α, (((L.con ab.1).term.equal (L.con ab.2).term).not)) '' ((s ×ˢ s) ∩ (set.diagonal α)ᶜ) +(λ ab : α × α, (((L.con ab.1).term.equal (L.con ab.2).term).not)) '' (s ×ˢ s ∩ (set.diagonal α)ᶜ) variables {L} {α} diff --git a/formal/lean/mathlib/number_theory/bertrand.lean b/formal/lean/mathlib/number_theory/bertrand.lean new file mode 100644 index 0000000000000000000000000000000000000000..27480499ca9b31167d3bc4c1f46653343101bc1f --- /dev/null +++ b/formal/lean/mathlib/number_theory/bertrand.lean @@ -0,0 +1,234 @@ +/- +Copyright (c) 2020 Patrick Stevens. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Patrick Stevens, Bolton Bailey +-/ +import data.nat.choose.factorization +import number_theory.primorial +import analysis.convex.specific_functions + +/-! +# Bertrand's Postulate + +This file contains a proof of Bertrand's postulate: That between any positive number and its +double there is a prime. + +The proof follows the outline of the Erdős proof presented in "Proofs from THE BOOK": One considers +the prime factorization of `(2 * n).choose n`, and splits the constituent primes up into various +groups, then upper bounds the contribution of each group. This upper bounds the central binomial +coefficient, and if the postulate does not hold, this upper bound conflicts with a simple lower +bound for large enough `n`. This proves the result holds for large enough `n`, and for smaller `n` +an explicit list of primes is provided which covers the remaining cases. + +As in the [Metamath implementation](carneiro2015arithmetic), we rely on some optimizations from +[Shigenori Tochiori](tochiori_bertrand). In particular we use the cleaner bound on the central +binomial coefficient given in `nat.four_pow_lt_mul_central_binom`. + +## References + +* [M. Aigner and G. M. Ziegler _Proofs from THE BOOK_][aigner1999proofs] +* [S. Tochiori, _Considering the Proof of “There is a Prime between n and 2n”_][tochiori_bertrand] +* [M. Carneiro, _Arithmetic in Metamath, Case Study: Bertrand's Postulate_][carneiro2015arithmetic] + +## Tags + +Bertrand, prime, binomial coefficients +-/ + +open_locale big_operators + +section real + +open real + +namespace bertrand + +/-- +A reified version of the `bertrand.main_inequality` below. +This is not best possible: it actually holds for 464 ≤ x. +-/ +lemma real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) : + x * (2 * x) ^ (sqrt (2 * x)) * 4 ^ (2 * x / 3) ≤ 4 ^ x := +begin + let f : ℝ → ℝ := λ x, log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x, + have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := + λ x h, div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _), + have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)), + { intros x h5, + have h6 := mul_pos two_pos h5, + have h7 := rpow_pos_of_pos h6 (sqrt (2 * x)), + rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne', + log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x] }, + have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large, + rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, + ← mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3), + mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5], + have h : concave_on ℝ (set.Ioi 0.5) f, + { refine ((strict_concave_on_log_Ioi.concave_on.subset (set.Ioi_subset_Ioi _) + (convex_Ioi 0.5)).add ((strict_concave_on_sqrt_mul_log_Ioi.concave_on.comp_linear_map + ((2 : ℝ) • linear_map.id)).subset + (λ a ha, lt_of_eq_of_lt _ ((mul_lt_mul_left two_pos).mpr ha)) (convex_Ioi 0.5))).sub + ((convex_on_id (convex_Ioi 0.5)).smul (div_nonneg (log_nonneg _) _)); norm_num1 }, + suffices : ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0, + { obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this, + exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 }, + refine ⟨18, 512, by norm_num1, by norm_num1, le_trans (by norm_num1) n_large, _, _⟩, + { have : sqrt (2 * 18) = 6 := + (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1), + rw [hf, log_nonneg_iff (hf' 18 _), this]; norm_num1 }, + { have : sqrt (2 * 512) = 32, + { exact (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1) }, + rw [hf, log_nonpos_iff (hf' _ _), this, div_le_one (rpow_pos_of_pos four_pos _), + ← rpow_le_rpow_iff _ (rpow_pos_of_pos four_pos _).le three_pos, ← rpow_mul]; norm_num1 }, +end + +end bertrand + +end real + +section nat + +open nat + +/-- +The inequality which contradicts Bertrand's postulate, for large enough `n`. +-/ +lemma bertrand_main_inequality {n : ℕ} (n_large : 512 ≤ n) : + n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := +begin + rw ← @cast_le ℝ, + simp only [cast_bit0, cast_add, cast_one, cast_mul, cast_pow, ← real.rpow_nat_cast], + have n_pos : 0 < n := (dec_trivial : 0 < 512).trans_le n_large, + have n2_pos : 1 ≤ 2 * n := mul_pos dec_trivial n_pos, + refine trans (mul_le_mul _ _ _ _) (bertrand.real_main_inequality (by exact_mod_cast n_large)), + { refine mul_le_mul_of_nonneg_left _ (nat.cast_nonneg _), + refine real.rpow_le_rpow_of_exponent_le (by exact_mod_cast n2_pos) _, + exact_mod_cast real.nat_sqrt_le_real_sqrt }, + { exact real.rpow_le_rpow_of_exponent_le (by norm_num1) (cast_div_le.trans (by norm_cast)) }, + { exact real.rpow_nonneg_of_nonneg (by norm_num1) _ }, + { refine mul_nonneg (nat.cast_nonneg _) _, + exact real.rpow_nonneg_of_nonneg (mul_nonneg zero_le_two (nat.cast_nonneg _)) _, }, +end + +/-- +A lemma that tells us that, in the case where Bertrand's postulate does not hold, the prime +factorization of the central binomial coefficent only has factors at most `2 * n / 3 + 1`. +-/ +lemma central_binom_factorization_small (n : ℕ) (n_large : 2 < n) + (no_prime: ¬∃ (p : ℕ), p.prime ∧ n < p ∧ p ≤ 2 * n) : + central_binom n = ∏ p in finset.range (2 * n / 3 + 1), p ^ ((central_binom n).factorization p) := +begin + refine (eq.trans _ n.prod_pow_factorization_central_binom).symm, + apply finset.prod_subset, + { exact finset.range_subset.2 (add_le_add_right (nat.div_le_self _ _) _) }, + intros x hx h2x, + rw [finset.mem_range, lt_succ_iff] at hx h2x, + rw [not_le, div_lt_iff_lt_mul' three_pos, mul_comm x] at h2x, + replace no_prime := not_exists.mp no_prime x, + rw [←and_assoc, not_and', not_and_distrib, not_lt] at no_prime, + cases no_prime hx with h h, + { rw [factorization_eq_zero_of_non_prime n.central_binom h, pow_zero] }, + { rw [factorization_central_binom_of_two_mul_self_lt_three_mul n_large h h2x, pow_zero] }, +end + +/-- +An upper bound on the central binomial coefficient used in the proof of Bertrand's postulate. +The bound splits the prime factors of `central_binom n` into those +1. At most `sqrt (2 * n)`, which contribute at most `2 * n` for each such prime. +2. Between `sqrt (2 * n)` and `2 * n / 3`, which contribute at most `4^(2 * n / 3)` in total. +3. Between `2 * n / 3` and `n`, which do not exist. +4. Between `n` and `2 * n`, which would not exist in the case where Bertrand's postulate is false. +5. Above `2 * n`, which do not exist. +-/ +lemma central_binom_le_of_no_bertrand_prime (n : ℕ) (n_big : 2 < n) + (no_prime : ¬∃ (p : ℕ), nat.prime p ∧ n < p ∧ p ≤ 2 * n) : + central_binom n ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := +begin + have n_pos : 0 < n := (nat.zero_le _).trans_lt n_big, + have n2_pos : 1 ≤ 2 * n := mul_pos two_pos n_pos, + let S := (finset.range (2 * n / 3 + 1)).filter nat.prime, + let f := λ x, x ^ n.central_binom.factorization x, + have : ∏ (x : ℕ) in S, f x = ∏ (x : ℕ) in finset.range (2 * n / 3 + 1), f x, + { refine finset.prod_filter_of_ne (λ p hp h, _), + contrapose! h, dsimp only [f], + rw [factorization_eq_zero_of_non_prime n.central_binom h, pow_zero] }, + rw [central_binom_factorization_small n n_big no_prime, ← this, + ← finset.prod_filter_mul_prod_filter_not S (≤ sqrt (2 * n))], + apply mul_le_mul', + { refine (finset.prod_le_prod'' (λ p hp, (_ : f p ≤ 2 * n))).trans _, + { exact pow_factorization_choose_le (mul_pos two_pos n_pos) }, + have : (finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n), + { rw [card_Icc, nat.add_sub_cancel] }, + rw finset.prod_const, + refine pow_le_pow n2_pos ((finset.card_le_of_subset (λ x hx, _)).trans this.le), + obtain ⟨h1, h2⟩ := finset.mem_filter.1 hx, + exact finset.mem_Icc.mpr ⟨(finset.mem_filter.1 h1).2.one_lt.le, h2⟩ }, + { refine le_trans _ (primorial_le_4_pow (2 * n / 3)), + refine (finset.prod_le_prod' (λ p hp, (_ : f p ≤ p))).trans _, + { obtain ⟨h1, h2⟩ := finset.mem_filter.1 hp, + refine (pow_le_pow (finset.mem_filter.1 h1).2.one_lt.le _).trans (pow_one p).le, + exact nat.factorization_choose_le_one (sqrt_lt'.mp $ not_le.1 h2) }, + refine finset.prod_le_prod_of_subset_of_one_le' (finset.filter_subset _ _) _, + exact λ p hp _, (finset.mem_filter.1 hp).2.one_lt.le } +end + +namespace nat + +/-- +Proves that Bertrand's postulate holds for all sufficiently large `n`. +-/ +lemma exists_prime_lt_and_le_two_mul_eventually (n : ℕ) (n_big : 512 ≤ n) : + ∃ (p : ℕ), p.prime ∧ n < p ∧ p ≤ 2 * n := +begin + -- Assume there is no prime in the range. + by_contradiction no_prime, + -- Then we have the above sub-exponential bound on the size of this central binomial coefficient. + -- We now couple this bound with an exponential lower bound on the central binomial coefficient, + -- yielding an inequality which we have seen is false for large enough n. + have H1 : n * (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) ≤ 4 ^ n := bertrand_main_inequality n_big, + have H2 : 4 ^ n < n * n.central_binom := + nat.four_pow_lt_mul_central_binom n (le_trans (by norm_num1) n_big), + have H3 : n.central_binom ≤ (2 * n) ^ sqrt (2 * n) * 4 ^ (2 * n / 3) := + central_binom_le_of_no_bertrand_prime n (lt_of_lt_of_le (by norm_num1) n_big) no_prime, + rw mul_assoc at H1, exact not_le.2 H2 ((mul_le_mul_left' H3 n).trans H1), +end + +/-- +Proves that Bertrand's postulate holds over all positive naturals less than n by identifying a +descending list of primes, each no more than twice the next, such that the list contains a witness +for each number ≤ n. +-/ +lemma exists_prime_lt_and_le_two_mul_succ {n} (q) + {p : ℕ} (prime_p : nat.prime p) (covering : p ≤ 2 * q) + (H : n < q → ∃ (p : ℕ), p.prime ∧ n < p ∧ p ≤ 2 * n) + (hn : n < p) : ∃ (p : ℕ), p.prime ∧ n < p ∧ p ≤ 2 * n := +begin + by_cases p ≤ 2 * n, { exact ⟨p, prime_p, hn, h⟩ }, + exact H (lt_of_mul_lt_mul_left' (lt_of_lt_of_le (not_le.1 h) covering)) +end + +/-- +**Bertrand's Postulate**: For any positive natural number, there is a prime which is greater than +it, but no more than twice as large. +-/ +theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) : + ∃ p, nat.prime p ∧ n < p ∧ p ≤ 2 * n := +begin + -- Split into cases whether `n` is large or small + cases lt_or_le 511 n, + -- If `n` is large, apply the lemma derived from the inequalities on the central binomial + -- coefficient. + { exact exists_prime_lt_and_le_two_mul_eventually n h, }, + replace h : n < 521 := h.trans_lt (by norm_num1), + revert h, + -- For small `n`, supply a list of primes to cover the initial cases. + ([317, 163, 83, 43, 23, 13, 7, 5, 3, 2].mmap' $ λ n, + `[refine exists_prime_lt_and_le_two_mul_succ %%(reflect n) (by norm_num1) (by norm_num1) _]), + exact λ h2, ⟨2, prime_two, h2, nat.mul_le_mul_left 2 (nat.pos_of_ne_zero hn0)⟩, +end + +alias nat.exists_prime_lt_and_le_two_mul ← bertrand + +end nat + +end nat diff --git a/formal/lean/mathlib/number_theory/class_number/finite.lean b/formal/lean/mathlib/number_theory/class_number/finite.lean index c201beff98c05ec22ff40f985d69a9fae638b133..c1014c786715da08a482be94c249b5fb5085407d 100644 --- a/formal/lean/mathlib/number_theory/class_number/finite.lean +++ b/formal/lean/mathlib/number_theory/class_number/finite.lean @@ -165,9 +165,7 @@ variables [decidable_eq R] /-- `finset_approx` is a finite set such that each fractional ideal in the integral closure contains an element close to `finset_approx`. -/ noncomputable def finset_approx : finset R := -((finset.univ.product finset.univ) - .image (λ (xy : _ × _), distinct_elems bS adm xy.1 - distinct_elems bS adm xy.2)) - .erase 0 +(finset.univ.image $ λ xy : _ × _, distinct_elems bS adm xy.1 - distinct_elems bS adm xy.2).erase 0 lemma finset_approx.zero_not_mem : (0 : R) ∉ finset_approx bS adm := finset.not_mem_erase _ _ @@ -183,7 +181,7 @@ begin rintro rfl, simpa using hx }, { rintros ⟨i, j, hij, rfl⟩, - refine ⟨_, ⟨i, j⟩, finset.mem_product.mpr ⟨finset.mem_univ _, finset.mem_univ _⟩, rfl⟩, + refine ⟨_, ⟨i, j⟩, finset.mem_univ _, rfl⟩, rw [ne.def, sub_eq_zero], exact λ h, hij ((distinct_elems bS adm).injective h) } end diff --git a/formal/lean/mathlib/number_theory/cyclotomic/gal.lean b/formal/lean/mathlib/number_theory/cyclotomic/gal.lean index d4f9dbd21fcfc5e9d13764419702bb64fd9dffde..42b0de42698bf1c2c7668f7f16f59d2a447f83ad 100644 --- a/formal/lean/mathlib/number_theory/cyclotomic/gal.lean +++ b/formal/lean/mathlib/number_theory/cyclotomic/gal.lean @@ -39,8 +39,7 @@ it is always a subgroup, and if the `n`th cyclotomic polynomial is irreducible, local attribute [instance] pnat.fact_pos -variables {n : ℕ+} (K : Type*) [field K] {L : Type*} [field L] {μ : L} (hμ : is_primitive_root μ n) - [algebra K L] [is_cyclotomic_extension {n} K L] +variables {n : ℕ+} (K : Type*) [field K] {L : Type*} {μ : L} open polynomial is_cyclotomic_extension @@ -48,6 +47,9 @@ open_locale cyclotomic namespace is_primitive_root +variables [comm_ring L] [is_domain L] (hμ : is_primitive_root μ n) [algebra K L] + [is_cyclotomic_extension {n} K L] + /-- `is_primitive_root.aut_to_pow` is injective in the case that it's considered over a cyclotomic field extension. -/ lemma aut_to_pow_injective : function.injective $ hμ.aut_to_pow K := @@ -78,6 +80,9 @@ end is_primitive_root namespace is_cyclotomic_extension +variables [comm_ring L] [is_domain L] (hμ : is_primitive_root μ n) [algebra K L] + [is_cyclotomic_extension {n} K L] + /-- Cyclotomic extensions are abelian. -/ noncomputable def aut.comm_group : comm_group (L ≃ₐ[K] L) := ((zeta_spec n K L).aut_to_pow_injective K).comm_group _ @@ -110,7 +115,7 @@ let hζ := zeta_spec n K L, end, right_inv := λ x, begin simp only [monoid_hom.to_fun_eq_coe], - generalize_proofs _ _ _ h, + generalize_proofs _ _ h, have key := hζ.aut_to_pow_spec K ((hζ.power_basis K).equiv_of_minpoly ((hμ x).power_basis K) h), have := (hζ.power_basis K).equiv_of_minpoly_gen ((hμ x).power_basis K) h, @@ -139,7 +144,7 @@ let hζ := (zeta_spec n K L).eq_pow_of_pow_eq_one hμ.pow_eq_one n.pos in lemma from_zeta_aut_spec : from_zeta_aut hμ h (zeta n K L) = μ := begin simp_rw [from_zeta_aut, aut_equiv_pow_symm_apply], - generalize_proofs _ _ hζ h _ hμ _, + generalize_proofs _ hζ h _ hμ _, rw [←hζ.power_basis_gen K] {occs := occurrences.pos [4]}, rw [power_basis.equiv_of_minpoly_gen, hμ.power_basis_gen K], convert h.some_spec.some_spec, @@ -150,7 +155,8 @@ end is_cyclotomic_extension section gal -variables (h : irreducible (cyclotomic n K)) {K} +variables [field L] (hμ : is_primitive_root μ n) [algebra K L] + [is_cyclotomic_extension {n} K L] (h : irreducible (cyclotomic n K)) {K} /-- `is_cyclotomic_extension.aut_equiv_pow` repackaged in terms of `gal`. Asserts that the Galois group of `cyclotomic n K` is equivalent to `(zmod n)ˣ` if `cyclotomic n K` is irreducible in diff --git a/formal/lean/mathlib/number_theory/cyclotomic/primitive_roots.lean b/formal/lean/mathlib/number_theory/cyclotomic/primitive_roots.lean index 3be3b8787f591f8ed7bce18b2675e0c85e0645ca..750eafe822ea9e2d5db9ec7599ce3c202cf181f2 100644 --- a/formal/lean/mathlib/number_theory/cyclotomic/primitive_roots.lean +++ b/formal/lean/mathlib/number_theory/cyclotomic/primitive_roots.lean @@ -130,8 +130,10 @@ end variables {K} (C) +-- We are not using @[simps] to avoid a timeout. + /-- The equivalence between `L →ₐ[K] C` and `primitive_roots n C` given by a primitive root `ζ`. -/ -@[simps] noncomputable def embeddings_equiv_primitive_roots (C : Type*) [comm_ring C] [is_domain C] +noncomputable def embeddings_equiv_primitive_roots (C : Type*) [comm_ring C] [is_domain C] [algebra K C] (hirr : irreducible (cyclotomic n K)) : (L →ₐ[K] C) ≃ primitive_roots n C := ((hζ.power_basis K).lift_equiv).trans { to_fun := λ x, @@ -157,6 +159,11 @@ variables {K} (C) left_inv := λ x, subtype.ext rfl, right_inv := λ x, subtype.ext rfl } +@[simp] +lemma embeddings_equiv_primitive_roots_apply_coe (C : Type*) [comm_ring C] [is_domain C] + [algebra K C] (hirr : irreducible (cyclotomic n K)) (φ : L →ₐ[K] C) : + (hζ.embeddings_equiv_primitive_roots C hirr φ : C) = φ ζ := rfl + end is_primitive_root namespace is_cyclotomic_extension @@ -312,7 +319,7 @@ begin haveI : is_cyclotomic_extension {p ^ (k - s + 1)} K K⟮η⟯, { suffices : is_cyclotomic_extension {p ^ (k - s + 1)} K K⟮η + 1⟯.to_subalgebra, { have H : K⟮η + 1⟯.to_subalgebra = K⟮η⟯.to_subalgebra, - { simp only [intermediate_field.adjoin_simple_to_subalgebra_of_integral _ _ + { simp only [intermediate_field.adjoin_simple_to_subalgebra_of_integral (is_cyclotomic_extension.integral {p ^ (k + 1)} K L _)], refine subalgebra.ext (λ x, ⟨λ hx, adjoin_le _ hx, λ hx, adjoin_le _ hx⟩), { simp only [set.singleton_subset_iff, set_like.mem_coe], @@ -322,7 +329,7 @@ begin refine subalgebra.sub_mem _ (subset_adjoin (mem_singleton _)) (subalgebra.one_mem _) } }, rw [H] at this, exact this }, - rw [intermediate_field.adjoin_simple_to_subalgebra_of_integral _ _ + rw [intermediate_field.adjoin_simple_to_subalgebra_of_integral (is_cyclotomic_extension.integral {p ^ (k + 1)} K L _)], have hη' : is_primitive_root (η + 1) ↑(p ^ (k + 1 - s)) := by simpa using hη, convert hη'.adjoin_is_cyclotomic_extension K, diff --git a/formal/lean/mathlib/number_theory/cyclotomic/rat.lean b/formal/lean/mathlib/number_theory/cyclotomic/rat.lean index e053cb23bc0da8f9c1201bad6564ed05f86214d9..e71acd34ee2c974095378bc414109465f4a36ec3 100644 --- a/formal/lean/mathlib/number_theory/cyclotomic/rat.lean +++ b/formal/lean/mathlib/number_theory/cyclotomic/rat.lean @@ -9,7 +9,8 @@ import ring_theory.polynomial.eisenstein /-! # Ring of integers of `p ^ n`-th cyclotomic fields -We compute the ring of integers of a `p ^ n`-th cyclotomic extension of `ℚ`. +We gather results about cyclotomic extensions of `ℚ`. In particular, we compute the ring of +integers of a `p ^ n`-th cyclotomic extension of `ℚ`. ## Main results * `is_cyclotomic_extension.rat.is_integral_closure_adjoing_singleton_of_prime_pow`: if `K` is a @@ -21,16 +22,16 @@ We compute the ring of integers of a `p ^ n`-th cyclotomic extension of `ℚ`. universes u -open algebra is_cyclotomic_extension polynomial +open algebra is_cyclotomic_extension polynomial number_field -open_locale cyclotomic - -namespace is_cyclotomic_extension.rat +open_locale cyclotomic number_field nat variables {p : ℕ+} {k : ℕ} {K : Type u} [field K] [char_zero K] {ζ : K} [hp : fact (p : ℕ).prime] include hp +namespace is_cyclotomic_extension.rat + /-- The discriminant of the power basis given by `ζ - 1`. -/ lemma discr_prime_pow_ne_two' [is_cyclotomic_extension {p ^ (k + 1)} ℚ K] (hζ : is_primitive_root ζ ↑(p ^ (k + 1))) (hk : p ^ (k + 1) ≠ 2) : @@ -166,3 +167,90 @@ begin end end is_cyclotomic_extension.rat + +section power_basis + +open is_cyclotomic_extension.rat + +namespace is_primitive_root + +/-- The algebra isomorphism `adjoin ℤ {ζ} ≃ₐ[ℤ] (𝓞 K)`, where `ζ` is a primitive `p ^ k`-th root of +unity and `K` is a `p ^ k`-th cyclotomic extension of `ℚ`. -/ +@[simps] noncomputable def _root_.is_primitive_root.adjoin_equiv_ring_of_integers + [hcycl : is_cyclotomic_extension {p ^ k} ℚ K] (hζ : is_primitive_root ζ ↑(p ^ k)) : + adjoin ℤ ({ζ} : set K) ≃ₐ[ℤ] (𝓞 K) := +let _ := is_integral_closure_adjoing_singleton_of_prime_pow hζ in + by exactI (is_integral_closure.equiv ℤ (adjoin ℤ ({ζ} : set K)) K (𝓞 K)) + +/-- The integral `power_basis` of `𝓞 K` given by a primitive root of unity, where `K` is a `p ^ k` +cyclotomic extension of `ℚ`. -/ +noncomputable def integral_power_basis [hcycl : is_cyclotomic_extension {p ^ k} ℚ K] + (hζ : is_primitive_root ζ ↑(p ^ k)) : power_basis ℤ (𝓞 K) := +(adjoin.power_basis' (hζ.is_integral (p ^ k).pos)).map hζ.adjoin_equiv_ring_of_integers + +@[simp] lemma integral_power_basis_gen [hcycl : is_cyclotomic_extension {p ^ k} ℚ K] + (hζ : is_primitive_root ζ ↑(p ^ k)) : + hζ.integral_power_basis.gen = ⟨ζ, hζ.is_integral (p ^ k).pos⟩ := +subtype.ext $ show algebra_map _ K hζ.integral_power_basis.gen = _, by simpa [integral_power_basis] + +@[simp] lemma integral_power_basis_dim [hcycl : is_cyclotomic_extension {p ^ k} ℚ K] + (hζ : is_primitive_root ζ ↑(p ^ k)) : hζ.integral_power_basis.dim = φ (p ^ k) := +by simp [integral_power_basis, ←cyclotomic_eq_minpoly hζ, nat_degree_cyclotomic] + +/-- The algebra isomorphism `adjoin ℤ {ζ} ≃ₐ[ℤ] (𝓞 K)`, where `ζ` is a primitive `p`-th root of +unity and `K` is a `p`-th cyclotomic extension of `ℚ`. -/ +@[simps] noncomputable def _root_.is_primitive_root.adjoin_equiv_ring_of_integers' + [hcycl : is_cyclotomic_extension {p} ℚ K] (hζ : is_primitive_root ζ p) : + adjoin ℤ ({ζ} : set K) ≃ₐ[ℤ] (𝓞 K) := +@adjoin_equiv_ring_of_integers p 1 K _ _ _ _ (by { convert hcycl, rw pow_one }) (by rwa pow_one) + +/-- The integral `power_basis` of `𝓞 K` given by a primitive root of unity, where `K` is a `p`-th +cyclotomic extension of `ℚ`. -/ +noncomputable def integral_power_basis' [hcycl : is_cyclotomic_extension {p} ℚ K] + (hζ : is_primitive_root ζ p) : power_basis ℤ (𝓞 K) := +@integral_power_basis p 1 K _ _ _ _ (by { convert hcycl, rw pow_one }) (by rwa pow_one) + +@[simp] lemma integral_power_basis'_gen [hcycl : is_cyclotomic_extension {p} ℚ K] + (hζ : is_primitive_root ζ p) : hζ.integral_power_basis'.gen = ⟨ζ, hζ.is_integral p.pos⟩ := +@integral_power_basis_gen p 1 K _ _ _ _ (by { convert hcycl, rw pow_one }) (by rwa pow_one) + +@[simp] lemma power_basis_int'_dim [hcycl : is_cyclotomic_extension {p} ℚ K] + (hζ : is_primitive_root ζ p) : hζ.integral_power_basis'.dim = φ p := +by erw [@integral_power_basis_dim p 1 K _ _ _ _ (by { convert hcycl, rw pow_one }) + (by rwa pow_one), pow_one] + +/-- The integral `power_basis` of `𝓞 K` given by `ζ - 1`, where `K` is a `p ^ k` cyclotomic +extension of `ℚ`. -/ +noncomputable def sub_one_integral_power_basis [is_cyclotomic_extension {p ^ k} ℚ K] + (hζ : is_primitive_root ζ ↑(p ^ k)) : power_basis ℤ (𝓞 K) := +power_basis.of_gen_mem_adjoin' hζ.integral_power_basis (is_integral_of_mem_ring_of_integers $ + subalgebra.sub_mem _ (hζ.is_integral (p ^ k).pos) (subalgebra.one_mem _)) +begin + simp only [integral_power_basis_gen], + convert subalgebra.add_mem _ + (self_mem_adjoin_singleton ℤ (⟨ζ - 1, _⟩ : 𝓞 K)) + (subalgebra.one_mem _), + simp +end + +@[simp] lemma sub_one_integral_power_basis_gen [is_cyclotomic_extension {p ^ k} ℚ K] + (hζ : is_primitive_root ζ ↑(p ^ k)) : + hζ.sub_one_integral_power_basis.gen = + ⟨ζ - 1, subalgebra.sub_mem _ (hζ.is_integral (p ^ k).pos) (subalgebra.one_mem _)⟩ := +by simp [sub_one_integral_power_basis] + +/-- The integral `power_basis` of `𝓞 K` given by `ζ - 1`, where `K` is a `p`-th cyclotomic +extension of `ℚ`. -/ +noncomputable def sub_one_integral_power_basis' [hcycl : is_cyclotomic_extension {p} ℚ K] + (hζ : is_primitive_root ζ p) : power_basis ℤ (𝓞 K) := +@sub_one_integral_power_basis p 1 K _ _ _ _ (by { convert hcycl, rw pow_one }) (by rwa pow_one) + +@[simp] lemma sub_one_integral_power_basis'_gen [hcycl : is_cyclotomic_extension {p} ℚ K] + (hζ : is_primitive_root ζ p) : + hζ.sub_one_integral_power_basis'.gen = + ⟨ζ - 1, subalgebra.sub_mem _ (hζ.is_integral p.pos) (subalgebra.one_mem _)⟩ := +@sub_one_integral_power_basis_gen p 1 K _ _ _ _ (by { convert hcycl, rw pow_one }) (by rwa pow_one) + +end is_primitive_root + +end power_basis diff --git a/formal/lean/mathlib/number_theory/divisors.lean b/formal/lean/mathlib/number_theory/divisors.lean index edfb0e5ced1eb3d42728e11f715acea1e30159cd..22fc2c46c48fd4c2637359d68c9db0533af890d2 100644 --- a/formal/lean/mathlib/number_theory/divisors.lean +++ b/formal/lean/mathlib/number_theory/divisors.lean @@ -45,7 +45,7 @@ def proper_divisors : finset ℕ := finset.filter (λ x : ℕ, x ∣ n) (finset. /-- `divisors_antidiagonal n` is the `finset` of pairs `(x,y)` such that `x * y = n`. As a special case, `divisors_antidiagonal 0 = ∅`. -/ def divisors_antidiagonal : finset (ℕ × ℕ) := -((finset.Ico 1 (n + 1)).product (finset.Ico 1 (n + 1))).filter (λ x, x.fst * x.snd = n) +(Ico 1 (n + 1) ×ˢ Ico 1 (n + 1)).filter (λ x, x.fst * x.snd = n) variable {n} diff --git a/formal/lean/mathlib/number_theory/legendre_symbol/add_character.lean b/formal/lean/mathlib/number_theory/legendre_symbol/add_character.lean index e94d14798370895d41ff92d1adfface2d0dbe7d9..f9b1793a19702114a13d1318100878cb94140674 100644 --- a/formal/lean/mathlib/number_theory/legendre_symbol/add_character.lean +++ b/formal/lean/mathlib/number_theory/legendre_symbol/add_character.lean @@ -15,6 +15,8 @@ Let `R` be a finite commutative ring. An *additive character* of `R` with values in another commutative ring `R'` is simply a morphism from the additive group of `R` into the multiplicative monoid of `R'`. +The additive characters on `R` with values in `R'` form a commutative group. + We use the namespace `add_char`. ## Main definitions and results @@ -25,7 +27,7 @@ if `mul_shift ψ a` is trivial only when `a = 0`. We show that when `ψ` is primitive, then the map `a ↦ mul_shift ψ a` is injective (`add_char.to_mul_shift_inj_of_is_primitive`) and that `ψ` is primitive when `R` is a field -and `ψ` is nontrivial (`add_char.is_primitive_of_is_nontrivial`). +and `ψ` is nontrivial (`add_char.is_nontrivial.is_primitive`). We also show that there are primitive additive characters on `R` (with suitable target `R'`) when `R` is a field or `R = zmod n` (`add_char.primitive_char_finite_field` @@ -50,16 +52,57 @@ universes u v -- The domain of our additive characters variables (R : Type u) [add_monoid R] -- The target -variables (R' : Type v) [monoid R'] +variables (R' : Type v) [comm_monoid R'] /-- Define `add_char R R'` as `(multiplicative R) →* R'`. The definition works for an additive monoid `R` and a monoid `R'`, but we will restrict to the case that both are commutative rings below. +We assume right away that `R'` is commutative, so that `add_char R R'` carries +a structure of commutative monoid. The trivial additive character (sending everything to `1`) is `(1 : add_char R R').` -/ -abbreviation add_char : Type (max u v) := (multiplicative R) →* R' +@[derive [comm_monoid, inhabited]] +def add_char : Type (max u v) := (multiplicative R) →* R' + +instance add_char.has_coe_to_fun : has_coe_to_fun (add_char R R') (λ x, multiplicative R → R') := +monoid_hom.has_coe_to_fun + +instance add_char.monoid_hom_class : monoid_hom_class (add_char R R') (multiplicative R) R' := +monoid_hom.monoid_hom_class end add_char_def +section group_structure + +universes u v + +namespace add_char + +open multiplicative + +variables {R : Type u} [add_comm_group R] {R' : Type v} [comm_monoid R'] + +/-- An additive character on a commutative additive group has an inverse. + +Note that this is a different inverse to the one provided by `monoid_hom.has_inv`, +as it acts on the domain instead of the codomain. -/ +instance has_inv : has_inv (add_char R R') := ⟨λ ψ, ψ.comp inv_monoid_hom⟩ + +lemma inv_apply (ψ : add_char R R') (x : multiplicative R) : ψ⁻¹ x = ψ (of_add (-(to_add x))) := +rfl + +lemma inv_apply' (ψ : add_char R R') (x : R) : ψ⁻¹ (of_add x) = ψ (of_add (-x)) := rfl + +/-- The additive characters on a commutative additive group form a commutative group. -/ +instance comm_group : comm_group (add_char R R') := +{ inv := has_inv.inv, + mul_left_inv := + λ ψ, by { ext, rw [monoid_hom.mul_apply, monoid_hom.one_apply, inv_apply, ← map_mul, of_add_neg, + of_add_to_add, mul_left_inv, map_one], }, + ..monoid_hom.comm_monoid } + +end add_char + +end group_structure section additive @@ -93,11 +136,25 @@ def mul_shift (ψ : add_char R R') (a : R) : add_char R R' := @[simp] lemma mul_shift_apply {ψ : add_char R R'} {a : R} {x : multiplicative R} : mul_shift ψ a x = ψ (of_add (a * to_add x)) := rfl +/-- `ψ⁻¹ = mul_shift ψ (-1))`. -/ +lemma inv_mul_shift (ψ : add_char R R') : ψ⁻¹ = mul_shift ψ (-1) := +begin + ext, + rw [inv_apply, mul_shift_apply, neg_mul, one_mul], +end + /-- If `n` is a natural number, then `mul_shift ψ n x = (ψ x) ^ n`. -/ lemma mul_shift_spec' (ψ : add_char R R') (n : ℕ) (x : multiplicative R) : mul_shift ψ n x = (ψ x) ^ n := by rw [mul_shift_apply, ← nsmul_eq_mul, of_add_nsmul, map_pow, of_add_to_add] +/-- If `n` is a natural number, then `ψ ^ n = mul_shift ψ n`. -/ +lemma pow_mul_shift (ψ : add_char R R') (n : ℕ) : ψ ^ n = mul_shift ψ n := +begin + ext x, + rw [show (ψ ^ n) x = (ψ x) ^ n, from rfl, ← mul_shift_spec'], +end + /-- The product of `mul_shift ψ a` and `mul_shift ψ b` is `mul_shift ψ (a + b)`. -/ lemma mul_shift_mul (ψ : add_char R R') (a b : R) : mul_shift ψ a * mul_shift ψ b = mul_shift ψ (a + b) := @@ -127,7 +184,7 @@ begin intros a b h, apply_fun (λ x, x * mul_shift ψ (-b)) at h, simp only [mul_shift_mul, mul_shift_zero, add_right_neg] at h, - have h₂ := hψ (a + (- b)), + have h₂ := hψ (a + (-b)), rw [h, is_nontrivial_iff_ne_trivial, ← sub_eq_add_neg, sub_ne_zero] at h₂, exact not_not.mp (λ h, h₂ h rfl), end @@ -151,7 +208,7 @@ end /-- Structure for a primitive additive character on a finite ring `R` into a cyclotomic extension of a field `R'`. It records which cyclotomic extension it is, the character, and the fact that the character is primitive. -/ -@[nolint has_inhabited_instance] -- can't prove that they always exist +@[nolint has_nonempty_instance] -- can't prove that they always exist structure primitive_add_char (R : Type u) [comm_ring R] [fintype R] (R' : Type v) [field R'] := (n : pnat) (char : add_char R (cyclotomic_field n R')) @@ -315,6 +372,21 @@ lemma sum_eq_card_of_is_trivial' {ψ : add_char R R'} (hψ : ¬ is_nontrivial ψ ∑ a, ψ a = fintype.card R := sum_eq_card_of_is_trivial hψ +/-- The sum over the values of `mul_shift ψ b` for `ψ` primitive is zero when `b ≠ 0` +and `#R` otherwise. -/ +lemma sum_mul_shift [decidable_eq R] [is_domain R'] (ψ : add_char R R') (b : R) + (hψ : is_primitive ψ) : + ∑ (x : R), ψ (of_add (x * b)) = if b = 0 then fintype.card R else 0 := +begin + split_ifs with h, + { -- case `b = 0` + simp only [h, sub_self, mul_zero, of_add_zero, map_one, finset.sum_const, nat.smul_one_eq_coe], + refl, }, + { -- case `b ≠ 0` + simp_rw mul_comm, + exact sum_eq_zero_of_is_nontrivial (hψ b h), }, +end + end add_char end additive diff --git a/formal/lean/mathlib/number_theory/legendre_symbol/gauss_eisenstein_lemmas.lean b/formal/lean/mathlib/number_theory/legendre_symbol/gauss_eisenstein_lemmas.lean index bd60ea818edcfb50e3ff0351d4283c5f73b7fd71..7bb9f0735124f707a960ec0b4800803c34963a6d 100644 --- a/formal/lean/mathlib/number_theory/legendre_symbol/gauss_eisenstein_lemmas.lean +++ b/formal/lean/mathlib/number_theory/legendre_symbol/gauss_eisenstein_lemmas.lean @@ -206,9 +206,8 @@ calc a / b = (Ico 1 (a / b).succ).card : by simp /-- The given sum is the number of integer points in the triangle formed by the diagonal of the rectangle `(0, p/2) × (0, q/2)` -/ private lemma sum_Ico_eq_card_lt {p q : ℕ} : - ∑ a in Ico 1 (p / 2).succ, (a * q) / p = - (((Ico 1 (p / 2).succ).product (Ico 1 (q / 2).succ)).filter - (λ x : ℕ × ℕ, x.2 * p ≤ x.1 * q)).card := + ∑ a in Ico 1 (p / 2).succ, (a * q) / p = ((Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ).filter $ + λ x : ℕ × ℕ, x.2 * p ≤ x.1 * q).card := if hp0 : p = 0 then by simp [hp0, finset.ext_iff] else calc ∑ a in Ico 1 (p / 2).succ, (a * q) / p = @@ -240,9 +239,9 @@ lemma sum_mul_div_add_sum_mul_div_eq_mul (p q : ℕ) [hp : fact p.prime] ∑ a in Ico 1 (q / 2).succ, (a * p) / q = (p / 2) * (q / 2) := begin - have hswap : (((Ico 1 (q / 2).succ).product (Ico 1 (p / 2).succ)).filter + have hswap : ((Ico 1 (q / 2).succ ×ˢ Ico 1 (p / 2).succ).filter (λ x : ℕ × ℕ, x.2 * q ≤ x.1 * p)).card = - (((Ico 1 (p / 2).succ).product (Ico 1 (q / 2).succ)).filter + ((Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ).filter (λ x : ℕ × ℕ, x.1 * q ≤ x.2 * p)).card := card_congr (λ x _, prod.swap x) (λ ⟨_, _⟩, by simp only [mem_filter, and_self, prod.swap_prod_mk, forall_true_iff, mem_product] @@ -252,9 +251,9 @@ begin (λ ⟨x₁, x₂⟩ h, ⟨⟨x₂, x₁⟩, by revert h; simp only [mem_filter, eq_self_iff_true, and_self, exists_prop_of_true, prod.swap_prod_mk, forall_true_iff, mem_product] {contextual := tt}⟩), have hdisj : disjoint - (((Ico 1 (p / 2).succ).product (Ico 1 (q / 2).succ)).filter + ((Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ).filter (λ x : ℕ × ℕ, x.2 * p ≤ x.1 * q)) - (((Ico 1 (p / 2).succ).product (Ico 1 (q / 2).succ)).filter + ((Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ).filter (λ x : ℕ × ℕ, x.1 * q ≤ x.2 * p)), { apply disjoint_filter.2 (λ x hx hpq hqp, _), have hxp : x.1 < p, from lt_of_le_of_lt @@ -265,11 +264,11 @@ begin apply_fun zmod.val at this, rw [val_cast_of_lt hxp, val_zero] at this, simpa only [this, nonpos_iff_eq_zero, mem_Ico, one_ne_zero, false_and, mem_product] using hx }, - have hunion : ((Ico 1 (p / 2).succ).product (Ico 1 (q / 2).succ)).filter + have hunion : (Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ).filter (λ x : ℕ × ℕ, x.2 * p ≤ x.1 * q) ∪ - ((Ico 1 (p / 2).succ).product (Ico 1 (q / 2).succ)).filter + (Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ).filter (λ x : ℕ × ℕ, x.1 * q ≤ x.2 * p) = - ((Ico 1 (p / 2).succ).product (Ico 1 (q / 2).succ)), + (Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ), from finset.ext (λ x, by have := le_total (x.2 * p) (x.1 * q); simp only [mem_union, mem_filter, mem_Ico, mem_product]; tauto), rw [sum_Ico_eq_card_lt, sum_Ico_eq_card_lt, hswap, ← card_disjoint_union hdisj, hunion, diff --git a/formal/lean/mathlib/number_theory/legendre_symbol/gauss_sum.lean b/formal/lean/mathlib/number_theory/legendre_symbol/gauss_sum.lean new file mode 100644 index 0000000000000000000000000000000000000000..d8b88ea14b5d476e42a2538e24998f153f333f29 --- /dev/null +++ b/formal/lean/mathlib/number_theory/legendre_symbol/gauss_sum.lean @@ -0,0 +1,317 @@ +/- +Copyright (c) 2022 Michael Stoll. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Michael Stoll +-/ +import number_theory.legendre_symbol.add_character +import number_theory.legendre_symbol.zmod_char +import algebra.char_p.char_and_card + +/-! +# Gauss sums + +We define the Gauss sum associated to a multiplicative and an additive +character of a finite field and prove some results about them. + +## Main definition + +Let `R` be a finite commutative ring and let `R'` be another commutative ring. +If `χ` is a multiplicative character `R → R'` (type `mul_char R R'`) and `ψ` +is an additive character `R → R'` (type `add_char R R'`, which abbreviates +`(multiplicative R) →* R'`), then the *Gauss sum* of `χ` and `ψ` is `∑ a, χ a * ψ a`. + +## Main results + +Some important results are as follows. + +* `gauss_sum_mul_gauss_sum_eq_card`: The product of the Gauss + sums of `χ` and `ψ` and that of `χ⁻¹` and `ψ⁻¹` is the cardinality + of the source ring `R` (if `χ` is nontrivial, `ψ` is primitive and `R` is a field). +* `gauss_sum_sq`: The square of the Gauss sum is `χ(-1)` times + the cardinality of `R` if in addition `χ` is a quadratic character. +* `quad_gauss_sum_frob`: For a quadratic character `χ`, raising + the Gauss sum to the `p`th power (where `p` is the characteristic of + the target ring `R'`) multiplies it by `χ p`. +* `char.card_pow_card`: When `F` and `F'` are finite fields and `χ : F → F'` + is a nontrivial quadratic character, then `(χ (-1) * #F)^(#F'/2) = χ (#F')`. +* `finite_field.two_pow_card`: For every finite field `F` of odd characteristic, + we have `2^(#F/2) = χ₈(#F)` in `F`. + +This machinery can be used to derive (a generalization of) the Law of +Quadratic Reciprocity. + +## Tags + +additive character, multiplicative character, Gauss sum +-/ + +universes u v + +open_locale big_operators + +open add_char mul_char multiplicative + +section gauss_sum_def + +-- `R` is the domain of the characters +variables {R : Type u} [comm_ring R] [fintype R] +-- `R'` is the target of the characters +variables {R' : Type v} [comm_ring R'] + +/-! +### Definition and first properties +-/ + +/-- Definition of the Gauss sum associated to a multiplicative and an additive character. -/ +def gauss_sum (χ : mul_char R R') (ψ : add_char R R') : R' := ∑ a, χ a * ψ (of_add a) + +/-- Replacing `ψ` by `mul_shift ψ a` and multiplying the Gauss sum by `χ a` does not change it. -/ +lemma gauss_sum_mul_shift (χ : mul_char R R') (ψ : add_char R R') (a : Rˣ) : + χ a * gauss_sum χ (mul_shift ψ a) = gauss_sum χ ψ := +begin + simp only [gauss_sum, mul_shift, monoid_hom.coe_comp, function.comp_app, to_add_of_add, + add_monoid_hom.to_multiplicative_apply_apply, add_monoid_hom.coe_mul_left], + rw [finset.mul_sum], + simp_rw [← mul_assoc, ← map_mul], + exact fintype.sum_bijective _ a.mul_left_bijective _ _ (λ x, rfl), +end + +end gauss_sum_def + +/-! +### The product of two Gauss sums +-/ + +section gauss_sum_prod + +-- In the following, we need `R` to be a finite field and `R'` to be a domain. +variables {R : Type u} [field R] [fintype R] {R' : Type v} [comm_ring R'] [is_domain R'] + +-- A helper lemma for `gauss_sum_mul_gauss_sum_eq_card` below +-- Is this useful enough in other contexts to be public? +private +lemma gauss_sum_mul_aux {χ : mul_char R R'} (hχ : is_nontrivial χ) (ψ : add_char R R') (b : R) : + ∑ a, χ (a * b⁻¹) * ψ (of_add (a - b)) = ∑ c, χ c * ψ (of_add $ b * (c - 1)) := +begin + cases eq_or_ne b 0 with hb hb, + { -- case `b = 0` + simp only [hb, inv_zero, mul_zero, mul_char.map_zero, zero_mul, finset.sum_const_zero, + of_add_zero, monoid_hom.map_one, mul_one], + exact hχ.sum_eq_zero.symm, }, + { -- case `b ≠ 0` + refine (fintype.sum_bijective _ (mul_left_bijective₀ b hb) _ _ $ λ x, _).symm, + rw [mul_assoc, mul_comm x, ← mul_assoc, mul_inv_cancel hb, one_mul, mul_sub, mul_one] }, +end + +/-- We have `gauss_sum χ ψ * gauss_sum χ⁻¹ ψ⁻¹ = fintype.card R` +when `χ` is nontrivial and `ψ` is primitive (and `R` is a field). -/ +lemma gauss_sum_mul_gauss_sum_eq_card {χ : mul_char R R'} (hχ : is_nontrivial χ) + {ψ : add_char R R'} (hψ : is_primitive ψ) : + gauss_sum χ ψ * gauss_sum χ⁻¹ ψ⁻¹ = fintype.card R := +begin + simp only [gauss_sum, inv_mul_shift, mul_shift, monoid_hom.coe_comp, + function.comp_app, add_monoid_hom.to_multiplicative_apply_apply, finset.mul_sum, + add_monoid_hom.coe_mul_left, neg_mul, one_mul, of_add_neg, of_add_to_add], + simp_rw [finset.sum_mul, mul_mul_mul_comm, inv_apply' χ, ← map_mul χ, ← map_mul ψ], + convert_to ∑ y x, χ (x * y⁻¹) * ψ (of_add (x - y)) = _, + { simp_rw sub_eq_add_neg, refl }, + simp_rw gauss_sum_mul_aux hχ ψ, + rw [finset.sum_comm], + classical, -- to get `[decidable_eq R]` for `gauss_sum_mul_aux₂` + simp_rw [← finset.mul_sum, sum_mul_shift _ _ hψ, sub_eq_zero, mul_ite, mul_zero], + rw [finset.sum_ite_eq' finset.univ (1 : R)], + simp only [finset.mem_univ, map_one, one_mul, if_true], +end + +/-- When `χ` is a nontrivial quadratic character, then the square of `gauss_sum χ ψ` +is `χ(-1)` times the cardinality of `R`. -/ +lemma gauss_sum_sq {χ : mul_char R R'} (hχ₁ : is_nontrivial χ) (hχ₂ : is_quadratic χ) + {ψ : add_char R R'} (hψ : is_primitive ψ) : + (gauss_sum χ ψ) ^ 2 = χ (-1) * fintype.card R := +begin + rw [pow_two, ← gauss_sum_mul_gauss_sum_eq_card hχ₁ hψ, hχ₂.inv, mul_rotate'], + congr, + rw [mul_comm, ← gauss_sum_mul_shift _ _ (-1 : Rˣ), inv_mul_shift], + refl, +end + +end gauss_sum_prod + +/-! +### Gauss sums and Frobenius +-/ + +section gauss_sum_frob + +variables {R : Type u} [comm_ring R] [fintype R] {R' : Type v} [comm_ring R'] +-- We assume that the target ring `R'` has prime characteristic `p`. +variables (p : ℕ) [fp : fact p.prime] [hch : char_p R' p] +include fp hch + +/-- When `R'` has prime characteristic `p`, then the `p`th power of the Gauss sum +of `χ` and `ψ` is the Gauss sum of `χ^p` and `mul_shift ψ p`. -/ +lemma gauss_sum_frob (χ : mul_char R R') (ψ : add_char R R') : + gauss_sum χ ψ ^ p = gauss_sum (χ ^ p) (ψ ^ p) := +begin + rw [← frobenius_def, gauss_sum, gauss_sum, map_sum], + simp_rw [pow_apply' χ fp.1.pos, map_mul, frobenius_def], + refl, +end + +/-- For a quadratic character `χ` and when the characteristic `p` of the target ring +is a unit in the source ring, the `p`th power of the Gauss sum of`χ` and `ψ` is +`χ p` times the original Gauss sum. -/ +lemma mul_char.is_quadratic.gauss_sum_frob (hp : is_unit (p : R)) {χ : mul_char R R'} + (hχ : is_quadratic χ) (ψ : add_char R R') : + gauss_sum χ ψ ^ p = χ p * gauss_sum χ ψ := +by rw [gauss_sum_frob, pow_mul_shift, hχ.pow_char p, ← gauss_sum_mul_shift χ ψ hp.unit, + ← mul_assoc, hp.unit_spec, ← pow_two, ← pow_apply' _ (by norm_num : 0 < 2), + hχ.sq_eq_one, ← hp.unit_spec, one_apply_coe, one_mul] + +/-- For a quadratic character `χ` and when the characteristic `p` of the target ring +is a unit in the source ring and `n` is a natural number, the `p^n`th power of the Gauss +sum of`χ` and `ψ` is `χ (p^n)` times the original Gauss sum. -/ +lemma mul_char.is_quadratic.gauss_sum_frob_iter (n : ℕ) (hp : is_unit (p : R)) + {χ : mul_char R R'} (hχ : is_quadratic χ) (ψ : add_char R R') : + gauss_sum χ ψ ^ (p ^ n) = χ (p ^ n) * gauss_sum χ ψ := +begin + induction n with n ih, + { rw [pow_zero, pow_one, pow_zero, mul_char.map_one, one_mul], }, + { rw [pow_succ, mul_comm p, pow_mul, ih, mul_pow, hχ.gauss_sum_frob _ hp, + ← mul_assoc, pow_succ, mul_comm (p : R), map_mul, + ← pow_apply' χ fp.1.pos (p ^ n), hχ.pow_char p], }, +end + +end gauss_sum_frob + +/-! +### Values of quadratic characters +-/ + +section gauss_sum_values + +variables {R : Type u} [comm_ring R] [fintype R] {R' : Type v} [comm_ring R'] [is_domain R'] + +/-- If the square of the Gauss sum of a quadratic character is `χ(-1) * #R`, +then we get, for all `n : ℕ`, the relation `(χ(-1) * #R) ^ (p^n/2) = χ(p^n)`, +where `p` is the (odd) characteristic of the target ring `R'`. +This version can be used when `R` is not a field, e.g., `ℤ/8ℤ`. -/ +lemma char.card_pow_char_pow {χ : mul_char R R'} (hχ : is_quadratic χ) (ψ : add_char R R') (p n : ℕ) + [fp : fact p.prime] [hch : char_p R' p] (hp : is_unit (p : R)) (hp' : p ≠ 2) + (hg : (gauss_sum χ ψ) ^ 2 = χ (-1) * fintype.card R) : + (χ (-1) * fintype.card R) ^ (p ^ n / 2) = χ (p ^ n) := +begin + have : gauss_sum χ ψ ≠ 0, + { intro hf, rw [hf, zero_pow (by norm_num : 0 < 2), eq_comm, mul_eq_zero] at hg, + exact not_is_unit_prime_of_dvd_card p + ((char_p.cast_eq_zero_iff R' p _).mp $ hg.resolve_left (is_unit_one.neg.map χ).ne_zero) hp }, + rw ← hg, apply mul_right_cancel₀ this, + rw [← hχ.gauss_sum_frob_iter p n hp ψ, ← pow_mul, mul_comm, ← pow_succ, + nat.two_mul_div_two_add_one_of_odd ((fp.1.eq_two_or_odd').resolve_left hp').pow], +end + +/-- When `F` and `F'` are finite fields and `χ : F → F'` is a nontrivial quadratic character, +then `(χ(-1) * #F)^(#F'/2) = χ(#F')`. -/ +lemma char.card_pow_card {F : Type} [field F] [fintype F] {F' : Type} [field F'] [fintype F'] + {χ : mul_char F F'} (hχ₁ : is_nontrivial χ) (hχ₂ : is_quadratic χ) + (hch₁ : ring_char F' ≠ ring_char F) (hch₂ : ring_char F' ≠ 2) : + (χ (-1) * fintype.card F) ^ (fintype.card F' / 2) = χ (fintype.card F') := +begin + obtain ⟨n, hp, hc⟩ := finite_field.card F (ring_char F), + obtain ⟨n', hp', hc'⟩ := finite_field.card F' (ring_char F'), + let ψ := primitive_char_finite_field F F' hch₁, + let FF' := cyclotomic_field ψ.n F', + have hchar := algebra.ring_char_eq F' FF', + apply (algebra_map F' FF').injective, + rw [map_pow, map_mul, map_nat_cast, hc', hchar, nat.cast_pow], + simp only [← mul_char.ring_hom_comp_apply], + haveI := fact.mk hp', + haveI := fact.mk (hchar.subst hp'), + rw [ne, ← nat.prime_dvd_prime_iff_eq hp' hp, ← is_unit_iff_not_dvd_char, hchar] at hch₁, + exact char.card_pow_char_pow (hχ₂.comp _) ψ.char (ring_char FF') n' hch₁ (hchar ▸ hch₂) + (gauss_sum_sq (hχ₁.comp $ ring_hom.injective _) (hχ₂.comp _) ψ.prim), +end + +end gauss_sum_values + +section gauss_sum_two + +/-! +### The quadratic character of 2 + +This section proves the following result. + +For every finite field `F` of odd characteristic, we have `2^(#F/2) = χ₈(#F)` in `F`. +This can be used to show that the quadratic character of `F` takes the value +`χ₈(#F)` at `2`. + +The proof uses the Gauss sum of `χ₈` and a primitive additive character on `ℤ/8ℤ`; +in this way, the result is reduced to `card_pow_char_pow`. +-/ + +open zmod + +/-- For every finite field `F` of odd characteristic, we have `2^(#F/2) = χ₈(#F)` in `F`. -/ +lemma finite_field.two_pow_card {F : Type*} [fintype F] [field F] (hF : ring_char F ≠ 2) : + (2 : F) ^ (fintype.card F / 2) = χ₈ (fintype.card F) := +begin + have hp2 : ∀ (n : ℕ), (2 ^ n : F) ≠ 0 := λ n, pow_ne_zero n (ring.two_ne_zero hF), + obtain ⟨n, hp, hc⟩ := finite_field.card F (ring_char F), + + -- we work in `FF`, the eighth cyclotomic field extension of `F` + let FF := (polynomial.cyclotomic 8 F).splitting_field, + haveI : finite_dimensional F FF := + polynomial.is_splitting_field.finite_dimensional FF (polynomial.cyclotomic 8 F), + haveI : fintype FF := finite_dimensional.fintype_of_fintype F FF, + have hchar := algebra.ring_char_eq F FF, + have FFp := hchar.subst hp, + haveI := fact.mk FFp, + have hFF := ne_of_eq_of_ne hchar.symm hF, -- `ring_char FF ≠ 2` + have hu : is_unit (ring_char FF : zmod 8), + { rw [is_unit_iff_not_dvd_char, ring_char_zmod_n], + rw [ne, ← nat.prime_dvd_prime_iff_eq FFp nat.prime_two] at hFF, + change ¬ _ ∣ 2 ^ 3, + exact mt FFp.dvd_of_dvd_pow hFF }, + + -- there is a primitive additive character `ℤ/8ℤ → FF`, sending `a + 8ℤ ↦ τ^a` + -- with a primitive eighth root of unity `τ` + let ψ₈ := primitive_zmod_char 8 F (by convert hp2 3; norm_num), + let τ : FF := ψ₈.char (of_add 1), + have τ_spec : τ ^ 4 = -1, + { refine (sq_eq_one_iff.1 _).resolve_left _; + { simp only [τ, ← map_pow], + erw add_char.is_primitive.zmod_char_eq_one_iff 8 ψ₈.prim, + dec_trivial } }, + + -- we consider `χ₈` as a multiplicative character `ℤ/8ℤ → FF` + let χ := χ₈.ring_hom_comp (int.cast_ring_hom FF), + have hχ : χ (-1) = 1 := norm_num.int_cast_one, + have hq : is_quadratic χ := is_quadratic_χ₈.comp _, + + -- we now show that the Gauss sum of `χ` and `ψ₈` has the relevant property + have hg : gauss_sum χ ψ₈.char ^ 2 = χ (-1) * fintype.card (zmod 8), + { rw [hχ, one_mul, card, gauss_sum], + convert ← congr_arg (^ 2) (fin.sum_univ_eight $ λ x, (χ₈ x : FF) * τ ^ x.val), + { ext, congr, apply pow_one }, + convert_to (0 + 1 * τ ^ 1 + 0 + (-1) * τ ^ 3 + 0 + (-1) * τ ^ 5 + 0 + 1 * τ ^ 7) ^ 2 = _, + { simp only [χ₈_apply, matrix.cons_val_zero, matrix.cons_val_one, matrix.head_cons, + matrix.cons_vec_bit0_eq_alt0, matrix.cons_vec_bit1_eq_alt1, matrix.cons_append, + matrix.cons_vec_alt0, matrix.cons_vec_alt1, int.cast_zero, int.cast_one, int.cast_neg, + zero_mul], refl }, + convert_to 8 + (τ ^ 4 + 1) * (τ ^ 10 - 2 * τ ^ 8 - 2 * τ ^ 6 + 6 * τ ^ 4 + τ ^ 2 - 8) = _, + { ring }, { rw τ_spec, norm_num } }, + + -- this allows us to apply `card_pow_char_pow` to our situation + have h := char.card_pow_char_pow hq ψ₈.char (ring_char FF) n hu hFF hg, + rw [card, ← hchar, hχ, one_mul, ← hc, ← nat.cast_pow (ring_char F), ← hc] at h, + + -- finally, we change `2` to `8` on the left hand side + convert_to (8 : F) ^ (fintype.card F / 2) = _, + { rw [(by norm_num : (8 : F) = 2 ^ 2 * 2), mul_pow, + (finite_field.is_square_iff hF $ hp2 2).mp ⟨2, pow_two 2⟩, one_mul] }, + apply (algebra_map F FF).injective, + simp only [map_pow, map_bit0, map_one, ring_hom.map_int_cast], + convert h, norm_num, +end + +end gauss_sum_two diff --git a/formal/lean/mathlib/number_theory/legendre_symbol/mul_character.lean b/formal/lean/mathlib/number_theory/legendre_symbol/mul_character.lean index 820fc0682d8136fa9c1809b789c47e416f64aa67..117f6c465394c541b669cde174df16cd047c2426 100644 --- a/formal/lean/mathlib/number_theory/legendre_symbol/mul_character.lean +++ b/formal/lean/mathlib/number_theory/legendre_symbol/mul_character.lean @@ -15,6 +15,18 @@ that sends non-units to zero. We use the namespace `mul_char` for the definitions and results. +## Main results + +We show that the multiplicative characters form a group (if `R'` is commutative); +see `mul_char.comm_group`. We also provide an equivalence with the +homomorphisms `Rˣ →* R'ˣ`; see `mul_char.equiv_to_unit_hom`. + +We define a multiplicative character to be *quadratic* if its values +are among `0`, `1` and `-1`, and we prove some properties of quadratic characters. + +Finally, we show that the sum of all values of a nontrivial multiplicative +character vanishes; see `mul_char.is_nontrivial.sum_eq_zero`. + ## Tags multiplicative character @@ -221,6 +233,11 @@ lemma map_zero {R : Type u} [comm_monoid_with_zero R] [nontrivial R] (χ : mul_c χ (0 : R) = 0 := by rw [map_nonunit χ not_is_unit_zero] +/-- If the domain is a ring `R`, then `χ (ring_char R) = 0`. -/ +lemma map_ring_char {R : Type u} [comm_ring R] [nontrivial R] (χ : mul_char R R') : + χ (ring_char R) = 0 := +by rw [ring_char.nat.cast_ring_char, χ.map_zero] + noncomputable instance has_one : has_one (mul_char R R') := ⟨trivial R R'⟩ @@ -266,12 +283,14 @@ lemma inv_apply_eq_inv (χ : mul_char R R') (a : R) : χ⁻¹ a = ring.inverse (χ a) := eq.refl $ inv χ a -/-- Variant when the target is a field -/ +/-- The inverse of a multiplicative character `χ`, applied to `a`, is the inverse of `χ a`. +Variant when the target is a field -/ lemma inv_apply_eq_inv' {R' : Type v} [field R'] (χ : mul_char R R') (a : R) : χ⁻¹ a = (χ a)⁻¹ := (inv_apply_eq_inv χ a).trans $ ring.inverse_eq_inv (χ a) -/-- When the domain has a zero, we can as well take the inverse first. -/ +/-- When the domain has a zero, then the inverse of a multiplicative character `χ`, +applied to `a`, is `χ` applied to the inverse of `a`. -/ lemma inv_apply {R : Type u} [comm_monoid_with_zero R] (χ : mul_char R R') (a : R) : χ⁻¹ a = χ (ring.inverse a) := begin @@ -284,7 +303,8 @@ begin rw [map_nonunit _ ha, ring.inverse_non_unit a ha, mul_char.map_zero χ], }, end -/-- When the domain is a field, we can use the field inverse instead. -/ +/-- When the domain has a zero, then the inverse of a multiplicative character `χ`, +applied to `a`, is `χ` applied to the inverse of `a`. -/ lemma inv_apply' {R : Type u} [field R] (χ : mul_char R R') (a : R) : χ⁻¹ a = χ a⁻¹ := (inv_apply χ a).trans $ congr_arg _ (ring.inverse_eq_inv a) @@ -297,7 +317,7 @@ begin ring.inverse_mul_cancel _ (is_unit.map _ x.is_unit), one_apply_coe], end -/-- Finally, the commutative group structure on `mul_char R R'`. -/ +/-- The commutative group structure on `mul_char R R'`. -/ noncomputable instance comm_group : comm_group (mul_char R R') := { one := 1, @@ -360,6 +380,14 @@ by simp only [is_nontrivial, ne.def, ext_iff, not_forall, one_apply_coe] /-- A multiplicative character is *quadratic* if it takes only the values `0`, `1`, `-1`. -/ def is_quadratic (χ : mul_char R R') : Prop := ∀ a, χ a = 0 ∨ χ a = 1 ∨ χ a = -1 +/-- If two values of quadratic characters with target `ℤ` agree after coercion into a ring +of characteristic not `2`, then they agree in `ℤ`. -/ +lemma is_quadratic.eq_of_eq_coe {χ : mul_char R ℤ} (hχ : is_quadratic χ) + {χ' : mul_char R' ℤ} (hχ' : is_quadratic χ') [nontrivial R''] (hR'' : ring_char R'' ≠ 2) + {a : R} {a' : R'} (h : (χ a : R'') = χ' a') : + χ a = χ' a' := +int.cast_inj_on_of_ring_char_ne_two hR'' (hχ a) (hχ' a') h + /-- We can post-compose a multiplicative character with a ring homomorphism. -/ @[simps] def ring_hom_comp (χ : mul_char R R') (f : R' →+* R'') : mul_char R R'' := diff --git a/formal/lean/mathlib/number_theory/legendre_symbol/quadratic_char.lean b/formal/lean/mathlib/number_theory/legendre_symbol/quadratic_char.lean index 798696ff0eb4204d3d47f2418cb903f75611a3a7..97b3738473c1fc80e74c1cdc3d9ca6cf2e4e4b91 100644 --- a/formal/lean/mathlib/number_theory/legendre_symbol/quadratic_char.lean +++ b/formal/lean/mathlib/number_theory/legendre_symbol/quadratic_char.lean @@ -5,6 +5,7 @@ Authors: Michael Stoll -/ import number_theory.legendre_symbol.zmod_char import field_theory.finite.basic +import number_theory.legendre_symbol.gauss_sum /-! # Quadratic characters of finite fields @@ -17,8 +18,6 @@ some basic statements about it. quadratic character -/ -namespace char - /-! ### Definition of the quadratic character @@ -80,9 +79,7 @@ lemma quadratic_char_eq_one_of_char_two' (hF : ring_char F = 2) {a : F} (ha : a quadratic_char_fun F a = 1 := begin simp only [quadratic_char_fun, ha, if_false, ite_eq_left_iff], - intro h, - exfalso, - exact h (finite_field.is_square_of_char_two hF a), + exact λ h, false.rec _ (h (finite_field.is_square_of_char_two hF a)) end /-- If `ring_char F` is odd, then `quadratic_char_fun F a` can be computed in @@ -159,7 +156,7 @@ lemma quadratic_char_dichotomy {a : F} (ha : a ≠ 0) : quadratic_char F a = 1 ∨ quadratic_char F a = -1 := sq_eq_one_iff.1 $ quadratic_char_sq_one ha -/-- A variant -/ +/-- The quadratic character is `1` or `-1` on nonzero arguments. -/ lemma quadratic_char_eq_neg_one_iff_not_one {a : F} (ha : a ≠ 0) : quadratic_char F a = -1 ↔ ¬ quadratic_char F a = 1 := begin @@ -192,6 +189,20 @@ lemma quadratic_char_eq_pow_of_char_ne_two (hF : ring_char F ≠ 2) {a : F} (ha quadratic_char F a = if a ^ (fintype.card F / 2) = 1 then 1 else -1 := quadratic_char_eq_pow_of_char_ne_two' hF ha +lemma quadratic_char_eq_pow_of_char_ne_two'' (hF : ring_char F ≠ 2) (a : F) : + (quadratic_char F a : F) = a ^ (fintype.card F / 2) := +begin + by_cases ha : a = 0, + { have : 0 < fintype.card F / 2 := nat.div_pos fintype.one_lt_card two_pos, + simp only [ha, zero_pow this, quadratic_char_apply, quadratic_char_zero, int.cast_zero], }, + { rw [quadratic_char_eq_pow_of_char_ne_two hF ha], + by_cases ha' : a ^ (fintype.card F / 2) = 1, + { simp only [ha', eq_self_iff_true, if_true, int.cast_one], }, + { have ha'' := or.resolve_left (finite_field.pow_dichotomy hF ha) ha', + simp only [ha'', int.cast_ite, int.cast_one, int.cast_neg, ite_eq_right_iff], + exact eq.symm, } } +end + variables (F) /-- The quadratic character is quadratic as a multiplicative character. -/ @@ -261,8 +272,6 @@ is_nontrivial.sum_eq_zero (quadratic_char_is_nontrivial hF) end quadratic_char -end char - /-! ### Special values of the quadratic character @@ -271,11 +280,9 @@ We express `quadratic_char F (-1)` in terms of `χ₄`. section special_values -namespace char +open zmod mul_char -open zmod - -variables {F : Type*} [field F] [fintype F] +variables {F : Type} [field F] [fintype F] /-- The value of the quadratic character at `-1` -/ lemma quadratic_char_neg_one [decidable_eq F] (hF : ring_char F ≠ 2) : @@ -290,27 +297,136 @@ begin exact λ hf, false.rec (1 = -1) (ring.neg_one_ne_one_of_char_ne_two hF hf), }, end -/-- The interpretation in terms of whether `-1` is a square in `F` -/ -lemma is_square_neg_one_iff : is_square (-1 : F) ↔ fintype.card F % 4 ≠ 3 := +/-- `-1` is a square in `F` iff `#F` is not congruent to `3` mod `4`. -/ +lemma finite_field.is_square_neg_one_iff : is_square (-1 : F) ↔ fintype.card F % 4 ≠ 3 := begin classical, -- suggested by the linter (instead of `[decidable_eq F]`) by_cases hF : ring_char F = 2, { simp only [finite_field.is_square_of_char_two hF, ne.def, true_iff], - exact (λ hf, one_ne_zero ((nat.odd_of_mod_four_eq_three hf).symm.trans - (finite_field.even_card_of_char_two hF)))}, - { have h₁ : (-1 : F) ≠ 0 := neg_ne_zero.mpr one_ne_zero, - have h₂ := finite_field.odd_card_of_char_ne_two hF, - rw [← quadratic_char_one_iff_is_square h₁, quadratic_char_neg_one hF, - χ₄_nat_eq_if_mod_four, h₂], - simp only [nat.one_ne_zero, if_false, ite_eq_left_iff, ne.def], - norm_num, - split, - { intros h h', - have t := (of_not_not h).symm.trans h', - norm_num at t, }, - exact λ h h', h' ((or_iff_left h).mp (nat.odd_mod_four_iff.mp h₂)), }, + exact (λ hf, one_ne_zero $ (nat.odd_of_mod_four_eq_three hf).symm.trans + $ finite_field.even_card_of_char_two hF) }, + { have h₁ := finite_field.odd_card_of_char_ne_two hF, + rw [← quadratic_char_one_iff_is_square (neg_ne_zero.mpr (@one_ne_zero F _ _)), + quadratic_char_neg_one hF, χ₄_nat_eq_if_mod_four, h₁], + simp only [nat.one_ne_zero, if_false, ite_eq_left_iff, ne.def, (dec_trivial : (-1 : ℤ) ≠ 1), + imp_false, not_not], + exact ⟨λ h, ne_of_eq_of_ne h (dec_trivial : 1 ≠ 3), + or.resolve_right (nat.odd_mod_four_iff.mp h₁)⟩, }, +end + +/-- The value of the quadratic character at `2` -/ +lemma quadratic_char_two [decidable_eq F] (hF : ring_char F ≠ 2) : + quadratic_char F 2 = χ₈ (fintype.card F) := +is_quadratic.eq_of_eq_coe (quadratic_char_is_quadratic F) is_quadratic_χ₈ hF + ((quadratic_char_eq_pow_of_char_ne_two'' hF 2).trans (finite_field.two_pow_card hF)) + +/-- `2` is a square in `F` iff `#F` is not congruent to `3` or `5` mod `8`. -/ +lemma finite_field.is_square_two_iff : + is_square (2 : F) ↔ fintype.card F % 8 ≠ 3 ∧ fintype.card F % 8 ≠ 5 := +begin + classical, + by_cases hF : ring_char F = 2, + focus + { have h := finite_field.even_card_of_char_two hF, + simp only [finite_field.is_square_of_char_two hF, true_iff], }, + rotate, focus + { have h := finite_field.odd_card_of_char_ne_two hF, + rw [← quadratic_char_one_iff_is_square (ring.two_ne_zero hF), quadratic_char_two hF, + χ₈_nat_eq_if_mod_eight], + simp only [h, nat.one_ne_zero, if_false, ite_eq_left_iff, ne.def, (dec_trivial : (-1 : ℤ) ≠ 1), + imp_false, not_not], }, + all_goals + { rw [← nat.mod_mod_of_dvd _ (by norm_num : 2 ∣ 8)] at h, + have h₁ := nat.mod_lt (fintype.card F) (dec_trivial : 0 < 8), + revert h₁ h, + generalize : fintype.card F % 8 = n, + dec_trivial!, } +end + +/-- The value of the quadratic character at `-2` -/ +lemma quadratic_char_neg_two [decidable_eq F] (hF : ring_char F ≠ 2) : + quadratic_char F (-2) = χ₈' (fintype.card F) := +begin + rw [(by norm_num : (-2 : F) = (-1) * 2), map_mul, χ₈'_eq_χ₄_mul_χ₈, quadratic_char_neg_one hF, + quadratic_char_two hF, @cast_nat_cast _ (zmod 4) _ _ _ (by norm_num : 4 ∣ 8)], +end + +/-- `-2` is a square in `F` iff `#F` is not congruent to `5` or `7` mod `8`. -/ +lemma finite_field.is_square_neg_two_iff : + is_square (-2 : F) ↔ fintype.card F % 8 ≠ 5 ∧ fintype.card F % 8 ≠ 7 := +begin + classical, + by_cases hF : ring_char F = 2, + focus + { have h := finite_field.even_card_of_char_two hF, + simp only [finite_field.is_square_of_char_two hF, true_iff], }, + rotate, focus + { have h := finite_field.odd_card_of_char_ne_two hF, + rw [← quadratic_char_one_iff_is_square (neg_ne_zero.mpr (ring.two_ne_zero hF)), + quadratic_char_neg_two hF, χ₈'_nat_eq_if_mod_eight], + simp only [h, nat.one_ne_zero, if_false, ite_eq_left_iff, ne.def, (dec_trivial : (-1 : ℤ) ≠ 1), + imp_false, not_not], }, + all_goals + { rw [← nat.mod_mod_of_dvd _ (by norm_num : 2 ∣ 8)] at h, + have h₁ := nat.mod_lt (fintype.card F) (dec_trivial : 0 < 8), + revert h₁ h, + generalize : fintype.card F % 8 = n, + dec_trivial! } +end + +/-- The relation between the values of the quadratic character of one field `F` at the +cardinality of another field `F'` and of the quadratic character of `F'` at the cardinality +of `F`. -/ +lemma quadratic_char_card_card [decidable_eq F] (hF : ring_char F ≠ 2) {F' : Type} [field F'] + [fintype F'] [decidable_eq F'] (hF' : ring_char F' ≠ 2) (h : ring_char F' ≠ ring_char F) : + quadratic_char F (fintype.card F') = quadratic_char F' (quadratic_char F (-1) * fintype.card F) := +begin + let χ := (quadratic_char F).ring_hom_comp (algebra_map ℤ F'), + have hχ₁ : χ.is_nontrivial, + { obtain ⟨a, ha⟩ := quadratic_char_exists_neg_one hF, + have hu : is_unit a, + { contrapose ha, + exact ne_of_eq_of_ne (map_nonunit (quadratic_char F) ha) + (mt zero_eq_neg.mp one_ne_zero), }, + use hu.unit, + simp only [is_unit.unit_spec, ring_hom_comp_apply, ring_hom.eq_int_cast, ne.def, ha], + rw [int.cast_neg, int.cast_one], + exact ring.neg_one_ne_one_of_char_ne_two hF', }, + have hχ₂ : χ.is_quadratic := is_quadratic.comp (quadratic_char_is_quadratic F) _, + have h := char.card_pow_card hχ₁ hχ₂ h hF', + rw [← quadratic_char_eq_pow_of_char_ne_two'' hF'] at h, + exact (is_quadratic.eq_of_eq_coe (quadratic_char_is_quadratic F') + (quadratic_char_is_quadratic F) hF' h).symm, end -end char +/-- The value of the quadratic character at an odd prime `p` different from `ring_char F`. -/ +lemma quadratic_char_odd_prime [decidable_eq F] (hF : ring_char F ≠ 2) {p : ℕ} [fact p.prime] + (hp₁ : p ≠ 2) (hp₂ : ring_char F ≠ p) : + quadratic_char F p = quadratic_char (zmod p) (χ₄ (fintype.card F) * fintype.card F) := +begin + rw [← quadratic_char_neg_one hF], + have h := quadratic_char_card_card hF (ne_of_eq_of_ne (ring_char_zmod_n p) hp₁) + (ne_of_eq_of_ne (ring_char_zmod_n p) hp₂.symm), + rwa [card p] at h, +end + +/-- An odd prime `p` is a square in `F` iff the quadratic character of `zmod p` does not +take the value `-1` on `χ₄(#F) * #F`. -/ +lemma finite_field.is_square_odd_prime_iff (hF : ring_char F ≠ 2) {p : ℕ} [fact p.prime] + (hp : p ≠ 2) : + is_square (p : F) ↔ quadratic_char (zmod p) (χ₄ (fintype.card F) * fintype.card F) ≠ -1 := +begin + classical, + by_cases hFp : ring_char F = p, + { rw [show (p : F) = 0, by { rw ← hFp, exact ring_char.nat.cast_ring_char }], + simp only [is_square_zero, ne.def, true_iff, map_mul], + obtain ⟨n, _, hc⟩ := finite_field.card F (ring_char F), + have hchar : ring_char F = ring_char (zmod p) := by {rw hFp, exact (ring_char_zmod_n p).symm}, + conv {congr, to_lhs, congr, skip, rw [hc, nat.cast_pow, map_pow, hchar, map_ring_char], }, + simp only [zero_pow n.pos, mul_zero, zero_eq_neg, one_ne_zero, not_false_iff], }, + { rw [← iff.not_left (@quadratic_char_neg_one_iff_not_is_square F _ _ _ _), + quadratic_char_odd_prime hF hp], + exact hFp, }, +end end special_values diff --git a/formal/lean/mathlib/number_theory/legendre_symbol/quadratic_reciprocity.lean b/formal/lean/mathlib/number_theory/legendre_symbol/quadratic_reciprocity.lean index 126151850c4387eca2c9e4e1c1111e6d5943f1a2..e3c70794f1c4b13c5bb1b93588d291fecf78a463 100644 --- a/formal/lean/mathlib/number_theory/legendre_symbol/quadratic_reciprocity.lean +++ b/formal/lean/mathlib/number_theory/legendre_symbol/quadratic_reciprocity.lean @@ -64,7 +64,7 @@ end lemma exists_sq_eq_neg_one_iff : is_square (-1 : zmod p) ↔ p % 4 ≠ 3 := begin - have h := @is_square_neg_one_iff (zmod p) _ _, + have h := @finite_field.is_square_neg_one_iff (zmod p) _ _, rw card p at h, exact h, end diff --git a/formal/lean/mathlib/number_theory/legendre_symbol/zmod_char.lean b/formal/lean/mathlib/number_theory/legendre_symbol/zmod_char.lean index 4d0f109f8742b80911e0332aa12354b2d992a6d2..899d2f9403f4d53bafd6b229f28ffe740a2b5089 100644 --- a/formal/lean/mathlib/number_theory/legendre_symbol/zmod_char.lean +++ b/formal/lean/mathlib/number_theory/legendre_symbol/zmod_char.lean @@ -120,13 +120,7 @@ lemma χ₈'_eq_χ₄_mul_χ₈ (a : zmod 8) : χ₈' a = χ₄ a * χ₈ a := b lemma χ₈'_int_eq_χ₄_mul_χ₈ (a : ℤ) : χ₈' a = χ₄ a * χ₈ a := begin - have h : (a : zmod 4) = (a : zmod 8), - { have help : ∀ m : ℤ, 0 ≤ m → m < 8 → ((m % 4 : ℤ) : zmod 4) = (m : zmod 8) := dec_trivial, - rw [← zmod.int_cast_mod a 8, ← zmod.int_cast_mod a 4, - (by norm_cast : ((8 : ℕ) : ℤ) = 8), (by norm_cast : ((4 : ℕ) : ℤ) = 4), - ← int.mod_mod_of_dvd a (by norm_num : (4 : ℤ) ∣ 8)], - exact help (a % 8) (int.mod_nonneg a (by norm_num)) (int.mod_lt a (by norm_num)), }, - rw h, + rw ← @cast_int_cast 8 (zmod 4) _ 4 _ (by norm_num) a, exact χ₈'_eq_χ₄_mul_χ₈ a, end diff --git a/formal/lean/mathlib/number_theory/number_field.lean b/formal/lean/mathlib/number_theory/number_field.lean index 4289b651064a65c63c123b1106aa9a06b0b304dd..ca2642b27dc1426427200d7dfa70fd2fcb8eaa32 100644 --- a/formal/lean/mathlib/number_theory/number_field.lean +++ b/formal/lean/mathlib/number_theory/number_field.lean @@ -70,6 +70,15 @@ localized "notation `𝓞` := number_field.ring_of_integers" in number_field lemma mem_ring_of_integers (x : K) : x ∈ 𝓞 K ↔ is_integral ℤ x := iff.rfl +lemma is_integral_of_mem_ring_of_integers {K : Type*} [field K] {x : K} (hx : x ∈ 𝓞 K) : + is_integral ℤ (⟨x, hx⟩ : 𝓞 K) := +begin + obtain ⟨P, hPm, hP⟩ := hx, + refine ⟨P, hPm, _⟩, + rw [← polynomial.aeval_def, ← subalgebra.coe_eq_zero, polynomial.aeval_subalgebra_coe, + polynomial.aeval_def, subtype.coe_mk, hP] +end + /-- Given an algebra between two fields, create an algebra between their two rings of integers. For now, this is not an instance by default as it creates an equal-but-not-defeq diamond with @@ -159,21 +168,18 @@ is a number field. -/ instance {f : ℚ[X]} [hf : fact (irreducible f)] : number_field (adjoin_root f) := { to_char_zero := char_zero_of_injective_algebra_map (algebra_map ℚ _).injective, to_finite_dimensional := by convert (adjoin_root.power_basis hf.out.ne_zero).finite_dimensional } - end end adjoin_root namespace number_field.embeddings -section number_field +section fintype -open set finite_dimensional polynomial +open finite_dimensional -variables {K L : Type*} [field K] [field L] -variables [number_field K] [number_field L] (x : K) - -variables {A : Type*} [field A] [char_zero A] +variables (K : Type*) [field K] [number_field K] +variables (A : Type*) [field A] [char_zero A] /-- There are finitely many embeddings of a number field. -/ noncomputable instance : fintype (K →+* A) := fintype.of_equiv (K →ₐ[ℚ] A) @@ -181,47 +187,66 @@ ring_hom.equiv_rat_alg_hom.symm variables [is_alg_closed A] -/-- The number of embeddings of a number field is its finrank. -/ +/-- The number of embeddings of a number field is equal to its finrank. -/ lemma card : fintype.card (K →+* A) = finrank ℚ K := by rw [fintype.of_equiv_card ring_hom.equiv_rat_alg_hom.symm, alg_hom.card] -/-- For `x ∈ K`, with `K` a number field, the images of `x` by the embeddings of `K` are exactly -the roots of the minimal polynomial of `x` over `ℚ` -/ -lemma eq_roots : range (λ φ : K →+* A, φ x) = (minpoly ℚ x).root_set A := +end fintype + +section roots + +open set polynomial + +/-- Let `A` an algebraically closed field and let `x ∈ K`, with `K` a number field. For `F`, +subfield of `K`, the images of `x` by the `F`-algebra morphisms from `K` to `A` are exactly +the roots in `A` of the minimal polynomial of `x` over `F` -/ +lemma range_eq_roots (F K A : Type*) [field F] [number_field F] [field K] [number_field K] + [field A] [is_alg_closed A] [algebra F K] [algebra F A] (x : K) : + range (λ ψ : K →ₐ[F] A, ψ x) = (minpoly F x).root_set A := begin - have hx : is_integral ℚ x := is_separable.is_integral ℚ x, + haveI : finite_dimensional F K := finite_dimensional.right ℚ _ _ , + have hx : is_integral F x := is_separable.is_integral F x, ext a, split, - { rintro ⟨φ, hφ⟩, - rw [mem_root_set_iff, ←hφ], - { let ψ := ring_hom.equiv_rat_alg_hom φ, - show (aeval (ψ x)) (minpoly ℚ x) = 0, - rw aeval_alg_hom_apply ψ x (minpoly ℚ x), + { rintro ⟨ψ, hψ⟩, + rw [mem_root_set_iff, ←hψ], + { rw aeval_alg_hom_apply ψ x (minpoly F x), simp only [minpoly.aeval, map_zero], }, exact minpoly.ne_zero hx, }, { intro ha, - let Qx := adjoin_root (minpoly ℚ x), - haveI : fact (irreducible $ minpoly ℚ x) := ⟨minpoly.irreducible hx⟩, - have hK : (aeval x) (minpoly ℚ x) = 0 := minpoly.aeval _ _, - have hA : (aeval a) (minpoly ℚ x) = 0, - { rw [aeval_def, ←eval_map, ←mem_root_set_iff'], - exact ha, - refine polynomial.monic.ne_zero _, - exact polynomial.monic.map (algebra_map ℚ A) (minpoly.monic hx), }, - let ψ : Qx →+* A := by convert adjoin_root.lift (algebra_map ℚ A) a hA, - letI : algebra Qx A := ring_hom.to_algebra ψ, - letI : algebra Qx K := ring_hom.to_algebra (by convert adjoin_root.lift (algebra_map ℚ K) x hK), - let φ₀ : K →ₐ[Qx] A, - { refine is_alg_closed.lift (algebra.is_algebraic_of_larger_base ℚ Qx _), - exact number_field.is_algebraic _, }, - let φ := φ₀.to_ring_hom, - use φ, - rw (_ : x = (algebra_map Qx K) (adjoin_root.root (minpoly ℚ x))), - { rw (_ : a = ψ (adjoin_root.root (minpoly ℚ x))), - refine alg_hom.commutes _ _, - exact (adjoin_root.lift_root hA).symm, }, + let Fx := adjoin_root (minpoly F x), + haveI : fact (irreducible $ minpoly F x) := ⟨minpoly.irreducible hx⟩, + have hK : (aeval x) (minpoly F x) = 0 := minpoly.aeval _ _, + have hA : (aeval a) (minpoly F x) = 0, + { rwa [aeval_def, ←eval_map, ←mem_root_set_iff'], + exact monic.ne_zero (monic.map (algebra_map F A) (minpoly.monic hx)), }, + letI : algebra Fx A := ring_hom.to_algebra (by convert adjoin_root.lift (algebra_map F A) a hA), + letI : algebra Fx K := ring_hom.to_algebra (by convert adjoin_root.lift (algebra_map F K) x hK), + haveI : finite_dimensional Fx K := finite_dimensional.right ℚ _ _ , + let ψ₀ : K →ₐ[Fx] A := is_alg_closed.lift (algebra.is_algebraic_of_finite _ _), + haveI : is_scalar_tower F Fx K := is_scalar_tower.of_ring_hom (adjoin_root.lift_hom _ _ hK), + haveI : is_scalar_tower F Fx A := is_scalar_tower.of_ring_hom (adjoin_root.lift_hom _ _ hA), + let ψ : K →ₐ[F] A := alg_hom.restrict_scalars F ψ₀, + refine ⟨ψ, _⟩, + rw (_ : x = (algebra_map Fx K) (adjoin_root.root (minpoly F x))), + rw (_ : a = (algebra_map Fx A) (adjoin_root.root (minpoly F x))), + exact alg_hom.commutes _ _, + exact (adjoin_root.lift_root hA).symm, exact (adjoin_root.lift_root hK).symm, }, end -end number_field +variables (K A : Type*) [field K] [number_field K] [field A] [char_zero A] [is_alg_closed A] (x : K) + +/-- Let `A` be an algebraically closed field and let `x ∈ K`, with `K` a number field. +The images of `x` by the embeddings of `K` in `A` are exactly the roots in `A` of +the minimal polynomial of `x` over `ℚ` -/ +lemma rat_range_eq_roots : +range (λ φ : K →+* A, φ x) = (minpoly ℚ x).root_set A := +begin + convert range_eq_roots ℚ K A x using 1, + ext a, + exact ⟨λ ⟨φ, hφ⟩, ⟨φ.to_rat_alg_hom, hφ⟩, λ ⟨φ, hφ⟩, ⟨φ.to_ring_hom, hφ⟩⟩, +end + +end roots end number_field.embeddings diff --git a/formal/lean/mathlib/number_theory/padics/hensel.lean b/formal/lean/mathlib/number_theory/padics/hensel.lean index f0d7e079859284c9a54f068f8ffffd12168e8e76..499605aa266135cca3d4d20dd00eed7f8a7f3ab0 100644 --- a/formal/lean/mathlib/number_theory/padics/hensel.lean +++ b/formal/lean/mathlib/number_theory/padics/hensel.lean @@ -164,12 +164,12 @@ have ∥(↑(F.derivative.eval z) * (↑(F.eval z) / ↑(F.derivative.eval z)) : have F.derivative.eval z * (-z1) = -F.eval z, from calc F.derivative.eval z * (-z1) = (F.derivative.eval z) * -⟨↑(F.eval z) / ↑(F.derivative.eval z), h1⟩ : by rw [hzeq] -... = -((F.derivative.eval z) * ⟨↑(F.eval z) / ↑(F.derivative.eval z), h1⟩) : - by simp [subtype.ext_iff_val] +... = -((F.derivative.eval z) * ⟨↑(F.eval z) / ↑(F.derivative.eval z), h1⟩) : mul_neg _ _ ... = -(⟨↑(F.derivative.eval z) * (↑(F.eval z) / ↑(F.derivative.eval z)), this⟩) : - subtype.ext $ by simp -... = -(F.eval z) : by simp [mul_div_cancel' _ hdzne'], -have heq : F.eval z' = q * z1^2, by simpa [sub_eq_add_neg, this, hz'] using hq, + subtype.ext $ by simp only [padic_int.coe_neg, padic_int.coe_mul, subtype.coe_mk] +... = -(F.eval z) : by simp only [mul_div_cancel' _ hdzne', subtype.coe_eta], +have heq : F.eval z' = q * z1^2, by simpa only [sub_eq_add_neg, this, hz', add_right_neg, neg_sq, + zero_add] using hq, ⟨q, heq⟩ private def calc_eval_z'_norm {z z' z1 : ℤ_[p]} {n} (hz : ih n z) {q} diff --git a/formal/lean/mathlib/number_theory/ramification_inertia.lean b/formal/lean/mathlib/number_theory/ramification_inertia.lean index 8b4c99f64392d9a572a9dbc71498b9fffc8b4773..9eb6d1dee8984b4f06779e690209d6f1bb4c7678 100644 --- a/formal/lean/mathlib/number_theory/ramification_inertia.lean +++ b/formal/lean/mathlib/number_theory/ramification_inertia.lean @@ -37,6 +37,11 @@ Often the above theory is set up in the case where: * `P` is an ideal lying over `p` We will try to relax the above hypotheses as much as possible. +## Notation + +In this file, `e` stands for the ramification index and `f` for the inertia degree of `P` over `p`, +leaving `p` and `P` implicit. + -/ namespace ideal @@ -123,6 +128,9 @@ lemma le_pow_ramification_idx : map f p ≤ P ^ ramification_idx f p P := le_pow_of_le_ramification_idx (le_refl _) +lemma le_comap_of_ramification_idx_ne_zero (h : ramification_idx f p P ≠ 0) : p ≤ comap f P := +ideal.map_le_iff_le_comap.mp $ le_pow_ramification_idx.trans $ ideal.pow_le_self $ h + namespace is_dedekind_domain variables [is_domain S] [is_dedekind_domain S] @@ -206,6 +214,8 @@ end end dec_eq +section finrank_quotient_map + open_locale big_operators open_locale non_zero_divisors @@ -301,13 +311,12 @@ begin -- we can write the elements of `a` as `p`-linear combinations of other elements of `a`. have exists_sum : ∀ x : (S ⧸ M), ∃ a' : fin n → R, (∀ i, a' i ∈ p) ∧ ∑ i, a' i • a i = x, { intro x, - obtain ⟨a'', ha'', hx⟩ := submodule.exists_sum_of_mem_ideal_smul_span p set.univ a x _, - refine ⟨λ i, a'' ⟨i, set.mem_univ _⟩, λ i, ha'' _, _⟩, - rwa [finsupp.sum_fintype, fintype.sum_equiv (equiv.set.univ (fin n))] at hx, - { intros, simp only [eq_self_iff_true, subtype.coe_eta, equiv.set.univ_apply] }, - { intros, simp only [zero_smul] }, - rw [set.image_univ, ha, smul_top_eq], - exact submodule.mem_top }, + obtain ⟨a'', ha'', hx⟩ := (submodule.mem_ideal_smul_span_iff_exists_sum p a x).1 _, + { refine ⟨λ i, a'' i, λ i, ha'' _, _⟩, + rw [← hx, finsupp.sum_fintype], + exact λ _, zero_smul _ _ }, + { rw [ha, smul_top_eq], + exact submodule.mem_top } }, choose A' hA'p hA' using λ i, exists_sum (a i), -- This gives us a(n invertible) matrix `A` such that `det A ∈ (M = span R b)`, let A : matrix (fin n) (fin n) R := A' - 1, @@ -428,4 +437,237 @@ begin { exact is_fraction_ring.injective S L } }, end +end finrank_quotient_map + +section fact_le_comap + +local notation `e` := ramification_idx f p P + +/-- `R / p` has a canonical map to `S / (P ^ e)`, where `e` is the ramification index +of `P` over `p`. -/ +noncomputable instance quotient.algebra_quotient_pow_ramification_idx : + algebra (R ⧸ p) (S ⧸ (P ^ e)) := +quotient.algebra_quotient_of_le_comap (ideal.map_le_iff_le_comap.mp le_pow_ramification_idx) + +@[simp] lemma quotient.algebra_map_quotient_pow_ramification_idx (x : R) : + algebra_map (R ⧸ p) (S ⧸ P ^ e) (ideal.quotient.mk p x) = ideal.quotient.mk _ (f x) := +rfl + +variables [hfp : fact (ramification_idx f p P ≠ 0)] +include hfp + +/-- If `P` lies over `p`, then `R / p` has a canonical map to `S / P`. + +This can't be an instance since the map `f : R → S` is generally not inferrable. +-/ +def quotient.algebra_quotient_of_ramification_idx_ne_zero : + algebra (R ⧸ p) (S ⧸ P) := +quotient.algebra_quotient_of_le_comap (le_comap_of_ramification_idx_ne_zero hfp.out) + +-- In this file, the value for `f` can be inferred. +local attribute [instance] ideal.quotient.algebra_quotient_of_ramification_idx_ne_zero + +@[simp] lemma quotient.algebra_map_quotient_of_ramification_idx_ne_zero (x : R) : + algebra_map (R ⧸ p) (S ⧸ P) (ideal.quotient.mk p x) = ideal.quotient.mk _ (f x) := +rfl + +omit hfp + +/-- The inclusion `(P^(i + 1) / P^e) ⊂ (P^i / P^e)`. -/ +@[simps] +def pow_quot_succ_inclusion (i : ℕ) : + ideal.map (P^e)^.quotient.mk (P ^ (i + 1)) →ₗ[R ⧸ p] ideal.map (P^e)^.quotient.mk (P ^ i) := +{ to_fun := λ x, ⟨x, ideal.map_mono (ideal.pow_le_pow i.le_succ) x.2⟩, + map_add' := λ x y, rfl, + map_smul' := λ c x, rfl } + +lemma pow_quot_succ_inclusion_injective (i : ℕ) : + function.injective (pow_quot_succ_inclusion f p P i) := +begin + rw [← linear_map.ker_eq_bot, linear_map.ker_eq_bot'], + rintro ⟨x, hx⟩ hx0, + rw subtype.ext_iff at hx0 ⊢, + rwa pow_quot_succ_inclusion_apply_coe at hx0 +end + +/-- `S ⧸ P` embeds into the quotient by `P^(i+1) ⧸ P^e` as a subspace of `P^i ⧸ P^e`. +See `quotient_to_quotient_range_pow_quot_succ` for this as a linear map, +and `quotient_range_pow_quot_succ_inclusion_equiv` for this as a linear equivalence. +-/ +noncomputable def quotient_to_quotient_range_pow_quot_succ_aux {i : ℕ} {a : S} (a_mem : a ∈ P^i) : + S ⧸ P → ((P ^ i).map (P ^ e)^.quotient.mk ⧸ (pow_quot_succ_inclusion f p P i).range) := +quotient.map' (λ (x : S), ⟨_, ideal.mem_map_of_mem _ (ideal.mul_mem_left _ x a_mem)⟩) + (λ x y h, begin + rw submodule.quotient_rel_r_def at ⊢ h, + simp only [_root_.map_mul, linear_map.mem_range], + refine ⟨⟨_, ideal.mem_map_of_mem _ (ideal.mul_mem_mul h a_mem)⟩, _⟩, + ext, + rw [pow_quot_succ_inclusion_apply_coe, subtype.coe_mk, submodule.coe_sub, subtype.coe_mk, + subtype.coe_mk, _root_.map_mul, map_sub, sub_mul] + end) + +lemma quotient_to_quotient_range_pow_quot_succ_aux_mk {i : ℕ} {a : S} (a_mem : a ∈ P^i) (x : S) : + quotient_to_quotient_range_pow_quot_succ_aux f p P a_mem (submodule.quotient.mk x) = + submodule.quotient.mk ⟨_, ideal.mem_map_of_mem _ (ideal.mul_mem_left _ x a_mem)⟩ := +by apply quotient.map'_mk' + +include hfp + +/-- `S ⧸ P` embeds into the quotient by `P^(i+1) ⧸ P^e` as a subspace of `P^i ⧸ P^e`. -/ +noncomputable def quotient_to_quotient_range_pow_quot_succ {i : ℕ} {a : S} (a_mem : a ∈ P^i) : + S ⧸ P →ₗ[R ⧸ p] ((P ^ i).map (P ^ e)^.quotient.mk ⧸ (pow_quot_succ_inclusion f p P i).range) := +{ to_fun := quotient_to_quotient_range_pow_quot_succ_aux f p P a_mem, + map_add' := begin + intros x y, refine quotient.induction_on' x (λ x, quotient.induction_on' y (λ y, _)), + simp only [submodule.quotient.mk'_eq_mk, ← submodule.quotient.mk_add, + quotient_to_quotient_range_pow_quot_succ_aux_mk, add_mul], + refine congr_arg submodule.quotient.mk _, + ext, + refl + end, + map_smul' := begin + intros x y, refine quotient.induction_on' x (λ x, quotient.induction_on' y (λ y, _)), + simp only [submodule.quotient.mk'_eq_mk, ← submodule.quotient.mk_add, + quotient_to_quotient_range_pow_quot_succ_aux_mk, ring_hom.id_apply], + refine congr_arg submodule.quotient.mk _, + ext, + simp only [subtype.coe_mk, _root_.map_mul, algebra.smul_def, submodule.coe_mk, mul_assoc, + ideal.quotient.mk_eq_mk, submodule.coe_smul_of_tower, + ideal.quotient.algebra_map_quotient_pow_ramification_idx] + end } + +lemma quotient_to_quotient_range_pow_quot_succ_mk {i : ℕ} {a : S} (a_mem : a ∈ P^i) (x : S) : + quotient_to_quotient_range_pow_quot_succ f p P a_mem (submodule.quotient.mk x) = + submodule.quotient.mk ⟨_, ideal.mem_map_of_mem _ (ideal.mul_mem_left _ x a_mem)⟩ := +quotient_to_quotient_range_pow_quot_succ_aux_mk f p P a_mem x + +lemma quotient_to_quotient_range_pow_quot_succ_injective [is_domain S] [is_dedekind_domain S] + [P.is_prime] {i : ℕ} (hi : i < e) {a : S} (a_mem : a ∈ P^i) (a_not_mem : a ∉ P^(i + 1)) : + function.injective (quotient_to_quotient_range_pow_quot_succ f p P a_mem) := +λ x, quotient.induction_on' x $ λ x y, quotient.induction_on' y $ λ y h, +begin + have Pe_le_Pi1 : P^e ≤ P^(i + 1) := ideal.pow_le_pow hi, + simp only [submodule.quotient.mk'_eq_mk, quotient_to_quotient_range_pow_quot_succ_mk, + submodule.quotient.eq, linear_map.mem_range, subtype.ext_iff, subtype.coe_mk, + submodule.coe_sub] at ⊢ h, + rcases h with ⟨⟨⟨z⟩, hz⟩, h⟩, + rw [submodule.quotient.quot_mk_eq_mk, ideal.quotient.mk_eq_mk, ideal.mem_quotient_iff_mem_sup, + sup_eq_left.mpr Pe_le_Pi1] at hz, + rw [pow_quot_succ_inclusion_apply_coe, subtype.coe_mk, submodule.quotient.quot_mk_eq_mk, + ideal.quotient.mk_eq_mk, ← map_sub, ideal.quotient.eq, ← sub_mul] at h, + exact (ideal.is_prime.mul_mem_pow _ + ((submodule.sub_mem_iff_right _ hz).mp (Pe_le_Pi1 h))).resolve_right a_not_mem, +end + +lemma quotient_to_quotient_range_pow_quot_succ_surjective [is_domain S] [is_dedekind_domain S] + (hP0 : P ≠ ⊥) [hP : P.is_prime] {i : ℕ} (hi : i < e) + {a : S} (a_mem : a ∈ P^i) (a_not_mem : a ∉ P^(i + 1)) : + function.surjective (quotient_to_quotient_range_pow_quot_succ f p P a_mem) := +begin + rintro ⟨⟨⟨x⟩, hx⟩⟩, + have Pe_le_Pi : P^e ≤ P^i := ideal.pow_le_pow hi.le, + have Pe_le_Pi1 : P^e ≤ P^(i + 1) := ideal.pow_le_pow hi, + rw [submodule.quotient.quot_mk_eq_mk, ideal.quotient.mk_eq_mk, ideal.mem_quotient_iff_mem_sup, + sup_eq_left.mpr Pe_le_Pi] at hx, + suffices hx' : x ∈ ideal.span {a} ⊔ P^(i+1), + { obtain ⟨y', hy', z, hz, rfl⟩ := submodule.mem_sup.mp hx', + obtain ⟨y, rfl⟩ := ideal.mem_span_singleton.mp hy', + refine ⟨submodule.quotient.mk y, _⟩, + simp only [submodule.quotient.quot_mk_eq_mk, quotient_to_quotient_range_pow_quot_succ_mk, + submodule.quotient.eq, linear_map.mem_range, subtype.ext_iff, subtype.coe_mk, + submodule.coe_sub], + refine ⟨⟨_, ideal.mem_map_of_mem _ (submodule.neg_mem _ hz)⟩, _⟩, + rw [pow_quot_succ_inclusion_apply_coe, subtype.coe_mk, ideal.quotient.mk_eq_mk, map_add, + mul_comm y a, sub_add_cancel', map_neg] }, + letI := classical.dec_eq (ideal S), + rw [sup_eq_prod_inf_factors _ (pow_ne_zero _ hP0), normalized_factors_pow, + normalized_factors_irreducible ((ideal.prime_iff_is_prime hP0).mpr hP).irreducible, + normalize_eq, multiset.nsmul_singleton, multiset.inter_repeat, multiset.prod_repeat], + rw [← submodule.span_singleton_le_iff_mem, ideal.submodule_span_eq] at a_mem a_not_mem, + rwa [ideal.count_normalized_factors_eq a_mem a_not_mem, min_eq_left i.le_succ], + { intro ha, + rw ideal.span_singleton_eq_bot.mp ha at a_not_mem, + have := (P^(i+1)).zero_mem, + contradiction }, +end + +/-- Quotienting `P^i / P^e` by its subspace `P^(i+1) ⧸ P^e` is +`R ⧸ p`-linearly isomorphic to `S ⧸ P`. -/ +noncomputable def quotient_range_pow_quot_succ_inclusion_equiv [is_domain S] [is_dedekind_domain S] + [P.is_prime] (hP : P ≠ ⊥) {i : ℕ} (hi : i < e) : + ((P ^ i).map (P ^ e)^.quotient.mk ⧸ (pow_quot_succ_inclusion f p P i).range) ≃ₗ[R ⧸ p] S ⧸ P := +begin + choose a a_mem a_not_mem using set_like.exists_of_lt + (ideal.strict_anti_pow P hP (ideal.is_prime.ne_top infer_instance) (le_refl i.succ)), + refine (linear_equiv.of_bijective _ _ _).symm, + { exact quotient_to_quotient_range_pow_quot_succ f p P a_mem }, + { exact quotient_to_quotient_range_pow_quot_succ_injective f p P hi a_mem a_not_mem }, + { exact quotient_to_quotient_range_pow_quot_succ_surjective f p P hP hi a_mem a_not_mem } +end + +/-- Since the inclusion `(P^(i + 1) / P^e) ⊂ (P^i / P^e)` has a kernel isomorphic to `P / S`, +`[P^i / P^e : R / p] = [P^(i+1) / P^e : R / p] + [P / S : R / p]` -/ +lemma dim_pow_quot_aux [is_domain S] [is_dedekind_domain S] [p.is_maximal] [P.is_prime] + (hP0 : P ≠ ⊥) {i : ℕ} (hi : i < e) : + module.rank (R ⧸ p) (ideal.map (P^e)^.quotient.mk (P ^ i)) = + module.rank (R ⧸ p) (S ⧸ P) + module.rank (R ⧸ p) (ideal.map (P^e)^.quotient.mk (P ^ (i + 1))) := +begin + letI : field (R ⧸ p) := ideal.quotient.field _, + rw [dim_eq_of_injective _ (pow_quot_succ_inclusion_injective f p P i), + (quotient_range_pow_quot_succ_inclusion_equiv f p P hP0 hi).symm.dim_eq], + exact (dim_quotient_add_dim (linear_map.range (pow_quot_succ_inclusion f p P i))).symm, +end + +lemma dim_pow_quot [is_domain S] [is_dedekind_domain S] [p.is_maximal] [P.is_prime] + (hP0 : P ≠ ⊥) (i : ℕ) (hi : i ≤ e) : + module.rank (R ⧸ p) (ideal.map (P^e)^.quotient.mk (P ^ i)) = + (e - i) • module.rank (R ⧸ p) (S ⧸ P) := +begin + refine @nat.decreasing_induction' _ i e (λ j lt_e le_j ih, _) hi _, + { rw [dim_pow_quot_aux f p P _ lt_e, ih, ← succ_nsmul, nat.sub_succ, ← nat.succ_eq_add_one, + nat.succ_pred_eq_of_pos (nat.sub_pos_of_lt lt_e)], + assumption }, + { rw [nat.sub_self, zero_nsmul, map_quotient_self], + exact dim_bot (R ⧸ p) (S ⧸ (P^e)) } +end + +omit hfp + +/-- If `p` is a maximal ideal of `R`, `S` extends `R` and `P^e` lies over `p`, +then the dimension `[S/(P^e) : R/p]` is equal to `e * [S/P : R/p]`. -/ +lemma dim_prime_pow_ramification_idx [is_domain S] [is_dedekind_domain S] [p.is_maximal] + [P.is_prime] (hP0 : P ≠ ⊥) (he : e ≠ 0) : + module.rank (R ⧸ p) (S ⧸ P^e) = + e • @module.rank (R ⧸ p) (S ⧸ P) _ _ (@algebra.to_module _ _ _ _ $ + @@quotient.algebra_quotient_of_ramification_idx_ne_zero _ _ _ _ _ ⟨he⟩) := +begin + letI : fact (e ≠ 0) := ⟨he⟩, + have := dim_pow_quot f p P hP0 0 (nat.zero_le e), + rw [pow_zero, nat.sub_zero, ideal.one_eq_top, ideal.map_top] at this, + exact (dim_top (R ⧸ p) _).symm.trans this +end + +/-- If `p` is a maximal ideal of `R`, `S` extends `R` and `P^e` lies over `p`, +then the dimension `[S/(P^e) : R/p]`, as a natural number, is equal to `e * [S/P : R/p]`. -/ +lemma finrank_prime_pow_ramification_idx [is_domain S] [is_dedekind_domain S] + (hP0 : P ≠ ⊥) [p.is_maximal] [P.is_prime] (he : e ≠ 0) : + finrank (R ⧸ p) (S ⧸ P^e) = + e * @finrank (R ⧸ p) (S ⧸ P) _ _ (@algebra.to_module _ _ _ _ $ + @@quotient.algebra_quotient_of_ramification_idx_ne_zero _ _ _ _ _ ⟨he⟩) := +begin + letI : fact (e ≠ 0) := ⟨he⟩, + letI : algebra (R ⧸ p) (S ⧸ P) := quotient.algebra_quotient_of_ramification_idx_ne_zero f p P, + letI := ideal.quotient.field p, + have hdim := dim_prime_pow_ramification_idx _ _ _ hP0 he, + by_cases hP : finite_dimensional (R ⧸ p) (S ⧸ P), + { haveI := hP, + haveI := (finite_dimensional_iff_of_rank_eq_nsmul he hdim).mpr hP, + refine cardinal.nat_cast_injective _, + rw [finrank_eq_dim, nat.cast_mul, finrank_eq_dim, hdim, nsmul_eq_mul] }, + have hPe := mt (finite_dimensional_iff_of_rank_eq_nsmul he hdim).mp hP, + simp only [finrank_of_infinite_dimensional hP, finrank_of_infinite_dimensional hPe, mul_zero], +end + +end fact_le_comap + end ideal diff --git a/formal/lean/mathlib/number_theory/wilson.lean b/formal/lean/mathlib/number_theory/wilson.lean index 4d4e9de4c8c5c0e03d01ff5c0aca4aa2891f3e1a..3a6463cd1b53d9e62449b53a7bd50ec2de8627d9 100644 --- a/formal/lean/mathlib/number_theory/wilson.lean +++ b/formal/lean/mathlib/number_theory/wilson.lean @@ -29,10 +29,13 @@ open_locale nat namespace nat variable {n : ℕ} -/-- For `n > 1`, `(n-1)!` is congruent to `-1` modulo `n` only if n is prime. --/ +/-- For `n ≠ 1`, `(n-1)!` is congruent to `-1` modulo `n` only if n is prime. --/ lemma prime_of_fac_equiv_neg_one - (h : ((n - 1)! : zmod n) = -1) (h1 : 1 < n) : prime n := + (h : ((n - 1)! : zmod n) = -1) (h1 : n ≠ 1) : prime n := begin + rcases eq_or_ne n 0 with rfl | h0, + { norm_num at h }, + replace h1 : 1 < n := n.two_le_iff.mpr ⟨h0, h1⟩, by_contradiction h2, obtain ⟨m, hm1, hm2 : 1 < m, hm3⟩ := exists_dvd_of_not_prime2 h1 h2, have hm : m ∣ (n - 1)! := nat.dvd_factorial (pos_of_gt hm2) (le_pred_of_lt hm3), @@ -40,8 +43,8 @@ begin rw [←zmod.nat_coe_zmod_eq_zero_iff_dvd, cast_add, cast_one, h, add_left_neg], end -/-- **Wilson's Theorem**: For `n > 1`, `(n-1)!` is congruent to `-1` modulo `n` iff n is prime. --/ -theorem prime_iff_fac_equiv_neg_one (h : 1 < n) : +/-- **Wilson's Theorem**: For `n ≠ 1`, `(n-1)!` is congruent to `-1` modulo `n` iff n is prime. --/ +theorem prime_iff_fac_equiv_neg_one (h : n ≠ 1) : prime n ↔ ((n - 1)! : zmod n) = -1 := begin refine ⟨λ h1, _, λ h2, prime_of_fac_equiv_neg_one h2 h⟩, diff --git a/formal/lean/mathlib/number_theory/zsqrtd/gaussian_int.lean b/formal/lean/mathlib/number_theory/zsqrtd/gaussian_int.lean index e6243ec06cec5f84e223bf313a14ec5bb67236e9..b2560a8959462d2274029585dd0944ba744506a5 100644 --- a/formal/lean/mathlib/number_theory/zsqrtd/gaussian_int.lean +++ b/formal/lean/mathlib/number_theory/zsqrtd/gaussian_int.lean @@ -83,10 +83,10 @@ by cases x; cases y; simp [to_complex_def₂] by rw [← to_complex_zero, to_complex_inj] @[simp] lemma nat_cast_real_norm (x : ℤ[i]) : (x.norm : ℝ) = (x : ℂ).norm_sq := -by rw [norm, norm_sq]; simp +by rw [zsqrtd.norm, norm_sq]; simp @[simp] lemma nat_cast_complex_norm (x : ℤ[i]) : (x.norm : ℂ) = (x : ℂ).norm_sq := -by cases x; rw [norm, norm_sq]; simp +by cases x; rw [zsqrtd.norm, norm_sq]; simp lemma norm_nonneg (x : ℤ[i]) : 0 ≤ norm x := norm_nonneg (by norm_num) _ @@ -105,7 +105,7 @@ by rw [← int.cast_coe_nat, coe_nat_abs_norm] lemma nat_abs_norm_eq (x : ℤ[i]) : x.norm.nat_abs = x.re.nat_abs * x.re.nat_abs + x.im.nat_abs * x.im.nat_abs := -int.coe_nat_inj $ begin simp, simp [norm] end +int.coe_nat_inj $ begin simp, simp [zsqrtd.norm] end instance : has_div ℤ[i] := ⟨λ x y, let n := (rat.of_int (norm y))⁻¹, c := y.conj in @@ -154,12 +154,12 @@ lemma mod_def (x y : ℤ[i]) : x % y = x - y * (x / y) := rfl lemma norm_mod_lt (x : ℤ[i]) {y : ℤ[i]} (hy : y ≠ 0) : (x % y).norm < y.norm := have (y : ℂ) ≠ 0, by rwa [ne.def, ← to_complex_zero, to_complex_inj], (@int.cast_lt ℝ _ _ _ _).1 $ - calc ↑(norm (x % y)) = (x - y * (x / y : ℤ[i]) : ℂ).norm_sq : by simp [mod_def] + calc ↑(zsqrtd.norm (x % y)) = (x - y * (x / y : ℤ[i]) : ℂ).norm_sq : by simp [mod_def] ... = (y : ℂ).norm_sq * (((x / y) - (x / y : ℤ[i])) : ℂ).norm_sq : by rw [← norm_sq_mul, mul_sub, mul_div_cancel' _ this] ... < (y : ℂ).norm_sq * 1 : mul_lt_mul_of_pos_left (norm_sq_div_sub_div_lt_one _ _) (norm_sq_pos.2 this) - ... = norm y : by simp + ... = zsqrtd.norm y : by simp lemma nat_abs_norm_mod_lt (x : ℤ[i]) {y : ℤ[i]} (hy : y ≠ 0) : (x % y).norm.nat_abs < y.norm.nat_abs := @@ -167,7 +167,7 @@ int.coe_nat_lt.1 (by simp [-int.coe_nat_lt, norm_mod_lt x hy]) lemma norm_le_norm_mul_left (x : ℤ[i]) {y : ℤ[i]} (hy : y ≠ 0) : (norm x).nat_abs ≤ (norm (x * y)).nat_abs := -by rw [norm_mul, int.nat_abs_mul]; +by rw [zsqrtd.norm_mul, int.nat_abs_mul]; exact le_mul_of_one_le_right (nat.zero_le _) (int.coe_nat_le.1 (by rw [coe_nat_abs_norm]; exact int.add_one_le_of_lt (norm_pos.2 hy))) @@ -224,15 +224,15 @@ hp.1.eq_two_or_odd.elim ... < p * p : mul_lt_mul k_lt_p k_lt_p (nat.succ_pos _) (nat.zero_le _), have hpk₁ : ¬ (p : ℤ[i]) ∣ ⟨k, -1⟩ := λ ⟨x, hx⟩, lt_irrefl (p * x : ℤ[i]).norm.nat_abs $ - calc (norm (p * x : ℤ[i])).nat_abs = (norm ⟨k, -1⟩).nat_abs : by rw hx - ... < (norm (p : ℤ[i])).nat_abs : by simpa [add_comm, norm] using hkltp + calc (norm (p * x : ℤ[i])).nat_abs = (zsqrtd.norm ⟨k, -1⟩).nat_abs : by rw hx + ... < (norm (p : ℤ[i])).nat_abs : by simpa [add_comm, zsqrtd.norm] using hkltp ... ≤ (norm (p * x : ℤ[i])).nat_abs : norm_le_norm_mul_left _ (λ hx0, (show (-1 : ℤ) ≠ 0, from dec_trivial) $ by simpa [hx0] using congr_arg zsqrtd.im hx), have hpk₂ : ¬ (p : ℤ[i]) ∣ ⟨k, 1⟩ := λ ⟨x, hx⟩, lt_irrefl (p * x : ℤ[i]).norm.nat_abs $ - calc (norm (p * x : ℤ[i])).nat_abs = (norm ⟨k, 1⟩).nat_abs : by rw hx - ... < (norm (p : ℤ[i])).nat_abs : by simpa [add_comm, norm] using hkltp + calc (norm (p * x : ℤ[i])).nat_abs = (zsqrtd.norm ⟨k, 1⟩).nat_abs : by rw hx + ... < (norm (p : ℤ[i])).nat_abs : by simpa [add_comm, zsqrtd.norm] using hkltp ... ≤ (norm (p * x : ℤ[i])).nat_abs : norm_le_norm_mul_left _ (λ hx0, (show (1 : ℤ) ≠ 0, from dec_trivial) $ by simpa [hx0] using congr_arg zsqrtd.im hx), diff --git a/formal/lean/mathlib/order/basic.lean b/formal/lean/mathlib/order/basic.lean index 99ca82e76f1fef3185e8a076e3c5175556007fd4..f8fa2c8e0e14273224084297bf009532fd633c27 100644 --- a/formal/lean/mathlib/order/basic.lean +++ b/formal/lean/mathlib/order/basic.lean @@ -770,6 +770,23 @@ instance : densely_ordered punit := ⟨λ _ _, false.elim⟩ end punit +section prop + +/-- Propositions form a complete boolean algebra, where the `≤` relation is given by implication. -/ +instance Prop.has_le : has_le Prop := ⟨(→)⟩ + +@[simp] lemma le_Prop_eq : ((≤) : Prop → Prop → Prop) = (→) := rfl + +lemma subrelation_iff_le {r s : α → α → Prop} : subrelation r s ↔ r ≤ s := iff.rfl + +instance Prop.partial_order : partial_order Prop := +{ le_refl := λ _, id, + le_trans := λ a b c f g, g ∘ f, + le_antisymm := λ a b Hab Hba, propext ⟨Hab, Hba⟩, + ..Prop.has_le } + +end prop + variables {s : β → β → Prop} {t : γ → γ → Prop} /-! ### Linear order from a total partial order -/ diff --git a/formal/lean/mathlib/order/bounded_order.lean b/formal/lean/mathlib/order/bounded_order.lean index 7a4e984d79b3f4739f2403755c0b5a7a91b3c006..78c422d7937fabceaf40d15f6eb2c1dbfcd128ec 100644 --- a/formal/lean/mathlib/order/bounded_order.lean +++ b/formal/lean/mathlib/order/bounded_order.lean @@ -346,12 +346,7 @@ end /-- Propositions form a distributive lattice. -/ instance Prop.distrib_lattice : distrib_lattice Prop := -{ le := λ a b, a → b, - le_refl := λ _, id, - le_trans := λ a b c f g, g ∘ f, - le_antisymm := λ a b Hab Hba, propext ⟨Hab, Hba⟩, - - sup := or, +{ sup := or, le_sup_left := @or.inl, le_sup_right := @or.inr, sup_le := λ a b c, or.rec, @@ -361,7 +356,8 @@ instance Prop.distrib_lattice : distrib_lattice Prop := inf_le_right := @and.right, le_inf := λ a b c Hab Hac Ha, and.intro (Hab Ha) (Hac Ha), le_sup_inf := λ a b c H, or_iff_not_imp_left.2 $ - λ Ha, ⟨H.1.resolve_left Ha, H.2.resolve_left Ha⟩ } + λ Ha, ⟨H.1.resolve_left Ha, H.2.resolve_left Ha⟩, + ..Prop.partial_order } /-- Propositions form a bounded order. -/ instance Prop.bounded_order : bounded_order Prop := @@ -376,9 +372,6 @@ instance Prop.le_is_total : is_total Prop (≤) := noncomputable instance Prop.linear_order : linear_order Prop := by classical; exact lattice.to_linear_order Prop -lemma subrelation_iff_le {r s : α → α → Prop} : subrelation r s ↔ r ≤ s := iff.rfl - -@[simp] lemma le_Prop_eq : ((≤) : Prop → Prop → Prop) = (→) := rfl @[simp] lemma sup_Prop_eq : (⊔) = (∨) := rfl @[simp] lemma inf_Prop_eq : (⊓) = (∧) := rfl @@ -1268,6 +1261,10 @@ lemma disjoint.comm : disjoint a b ↔ disjoint b a := by rw [disjoint, disjoint lemma symmetric_disjoint : symmetric (disjoint : α → α → Prop) := disjoint.symm lemma disjoint_assoc : disjoint (a ⊓ b) c ↔ disjoint a (b ⊓ c) := by rw [disjoint, disjoint, inf_assoc] +lemma disjoint_left_comm : disjoint a (b ⊓ c) ↔ disjoint b (a ⊓ c) := +by simp_rw [disjoint, inf_left_comm] +lemma disjoint_right_comm : disjoint (a ⊓ b) c ↔ disjoint (a ⊓ c) b := +by simp_rw [disjoint, inf_right_comm] @[simp] lemma disjoint_bot_left : disjoint ⊥ a := inf_le_left @[simp] lemma disjoint_bot_right : disjoint a ⊥ := inf_le_right @@ -1355,6 +1352,10 @@ lemma codisjoint.comm : codisjoint a b ↔ codisjoint b a := by rw [codisjoint, lemma symmetric_codisjoint : symmetric (codisjoint : α → α → Prop) := codisjoint.symm lemma codisjoint_assoc : codisjoint (a ⊔ b) c ↔ codisjoint a (b ⊔ c) := by rw [codisjoint, codisjoint, sup_assoc] +lemma codisjoint_left_comm : codisjoint a (b ⊔ c) ↔ codisjoint b (a ⊔ c) := +by simp_rw [codisjoint, sup_left_comm] +lemma codisjoint_right_comm : codisjoint (a ⊔ b) c ↔ codisjoint (a ⊔ c) b := +by simp_rw [codisjoint, sup_right_comm] @[simp] lemma codisjoint_top_left : codisjoint ⊤ a := le_sup_left @[simp] lemma codisjoint_top_right : codisjoint a ⊤ := le_sup_right @@ -1445,6 +1446,17 @@ lemma codisjoint.dual [semilattice_sup α] [order_top α] {a b : α} : @[simp] lemma codisjoint_of_dual_iff [semilattice_sup α] [order_top α] {a b : αᵒᵈ} : codisjoint (of_dual a) (of_dual b) ↔ disjoint a b := iff.rfl +section distrib_lattice +variables [distrib_lattice α] [bounded_order α] {a b c : α} + +lemma disjoint.le_of_codisjoint (hab : disjoint a b) (hbc : codisjoint b c) : a ≤ c := +begin + rw [←@inf_top_eq _ _ _ a, ←@bot_sup_eq _ _ _ c, ←hab.eq_bot, ←hbc.eq_top, sup_inf_right], + exact inf_le_inf_right _ le_sup_left, +end + +end distrib_lattice + section is_compl /-- Two elements `x` and `y` are complements of each other if `x ⊔ y = ⊤` and `x ⊓ y = ⊥`. -/ diff --git a/formal/lean/mathlib/order/category/omega_complete_partial_order.lean b/formal/lean/mathlib/order/category/omega_complete_partial_order.lean index dffcf5b9076fa70fd525c8380420c143f82c1f28..6f1c756e8de05a211da6ea85b9345dab81df74e9 100644 --- a/formal/lean/mathlib/order/category/omega_complete_partial_order.lean +++ b/formal/lean/mathlib/order/category/omega_complete_partial_order.lean @@ -131,7 +131,7 @@ has_limit.mk ⟨_, has_equalizers.is_equalizer f g⟩ instance : has_equalizers ωCPO.{v} := has_equalizers_of_has_limit_parallel_pair _ -instance : has_limits ωCPO.{v} := limits_from_equalizers_and_products +instance : has_limits ωCPO.{v} := has_limits_of_has_equalizers_and_products end diff --git a/formal/lean/mathlib/order/complete_lattice.lean b/formal/lean/mathlib/order/complete_lattice.lean index 0dfaaca61b75701cd15e64d079e33969e1d17226..b82d56608ee027814159d8b38243f43af09f9518 100644 --- a/formal/lean/mathlib/order/complete_lattice.lean +++ b/formal/lean/mathlib/order/complete_lattice.lean @@ -412,10 +412,18 @@ lemma function.surjective.supr_comp {f : ι → ι'} (hf : surjective f) (g : ι (⨆ x, g (f x)) = ⨆ y, g y := by simp only [supr, hf.range_comp] +lemma equiv.supr_comp {g : ι' → α} (e : ι ≃ ι') : + (⨆ x, g (e x)) = ⨆ y, g y := +e.surjective.supr_comp _ + protected lemma function.surjective.supr_congr {g : ι' → α} (h : ι → ι') (h1 : surjective h) (h2 : ∀ x, g (h x) = f x) : (⨆ x, f x) = ⨆ y, g y := by { convert h1.supr_comp g, exact (funext h2).symm } +protected lemma equiv.supr_congr {g : ι' → α} (e : ι ≃ ι') (h : ∀ x, g (e x) = f x) : + (⨆ x, f x) = ⨆ y, g y := +e.surjective.supr_congr _ h + @[congr] lemma supr_congr_Prop {p q : Prop} {f₁ : p → α} {f₂ : q → α} (pq : p ↔ q) (f : ∀ x, f₁ (pq.mpr x) = f₂ x) : supr f₁ = supr f₂ := by { obtain rfl := propext pq, congr' with x, apply f } @@ -440,10 +448,18 @@ lemma function.surjective.infi_comp {f : ι → ι'} (hf : surjective f) (g : ι (⨅ x, g (f x)) = ⨅ y, g y := @function.surjective.supr_comp αᵒᵈ _ _ _ f hf g -lemma function.surjective.infi_congr {g : ι' → α} (h : ι → ι') (h1 : surjective h) +lemma equiv.infi_comp {g : ι' → α} (e : ι ≃ ι') : + (⨅ x, g (e x)) = ⨅ y, g y := +@equiv.supr_comp αᵒᵈ _ _ _ _ e + +protected lemma function.surjective.infi_congr {g : ι' → α} (h : ι → ι') (h1 : surjective h) (h2 : ∀ x, g (h x) = f x) : (⨅ x, f x) = ⨅ y, g y := @function.surjective.supr_congr αᵒᵈ _ _ _ _ _ h h1 h2 +protected lemma equiv.infi_congr {g : ι' → α} (e : ι ≃ ι') (h : ∀ x, g (e x) = f x) : + (⨅ x, f x) = ⨅ y, g y := +@equiv.supr_congr αᵒᵈ _ _ _ _ _ e h + @[congr]lemma infi_congr_Prop {p q : Prop} {f₁ : p → α} {f₂ : q → α} (pq : p ↔ q) (f : ∀ x, f₁ (pq.mpr x) = f₂ x) : infi f₁ = infi f₂ := @supr_congr_Prop αᵒᵈ _ p q f₁ f₂ pq f diff --git a/formal/lean/mathlib/order/conditionally_complete_lattice.lean b/formal/lean/mathlib/order/conditionally_complete_lattice.lean index b823acb29e85f15428c0229f607facb994972148..dfa2ac6a0a5a0f2043f2b57647aadbe33e78bf4b 100644 --- a/formal/lean/mathlib/order/conditionally_complete_lattice.lean +++ b/formal/lean/mathlib/order/conditionally_complete_lattice.lean @@ -59,10 +59,66 @@ noncomputable instance {α : Type*} [preorder α] [has_Inf α] : has_Inf (with_b theorem with_top.cInf_empty {α : Type*} [has_Inf α] : Inf (∅ : set (with_top α)) = ⊤ := if_pos $ set.empty_subset _ +@[simp] +theorem with_top.cinfi_empty {α : Type*} [is_empty ι] [has_Inf α] (f : ι → with_top α) : + (⨅ i, f i) = ⊤ := +by rw [infi, range_eq_empty, with_top.cInf_empty] + +lemma with_top.coe_Inf' [has_Inf α] {s : set α} (hs : s.nonempty) : + ↑(Inf s) = (Inf (coe '' s) : with_top α) := +begin + obtain ⟨x, hx⟩ := hs, + change _ = ite _ _ _, + split_ifs, + { cases h (mem_image_of_mem _ hx) }, + { rw preimage_image_eq, + exact option.some_injective _ }, +end + +@[norm_cast] lemma with_top.coe_infi [nonempty ι] [has_Inf α] (f : ι → α) : + ↑(⨅ i, f i) = (⨅ i, f i : with_top α) := +by rw [infi, infi, with_top.coe_Inf' (range_nonempty f), range_comp] + +theorem with_top.coe_Sup' [preorder α] [has_Sup α] {s : set α} (hs : bdd_above s) : + ↑(Sup s) = (Sup (coe '' s) : with_top α) := +begin + change _ = ite _ _ _, + rw [if_neg, preimage_image_eq, if_pos hs], + { exact option.some_injective _ }, + { rintro ⟨x, h, ⟨⟩⟩ }, +end + +@[norm_cast] lemma with_top.coe_supr [preorder α] [has_Sup α] (f : ι → α) + (h : bdd_above (set.range f)) : + ↑(⨆ i, f i) = (⨆ i, f i : with_top α) := +by rw [supr, supr, with_top.coe_Sup' h, range_comp] + @[simp] theorem with_bot.cSup_empty {α : Type*} [has_Sup α] : Sup (∅ : set (with_bot α)) = ⊥ := if_pos $ set.empty_subset _ +@[simp] +theorem with_bot.csupr_empty {α : Type*} [is_empty ι] [has_Sup α] (f : ι → with_bot α) : + (⨆ i, f i) = ⊥ := +@with_top.cinfi_empty _ αᵒᵈ _ _ _ + +@[norm_cast] lemma with_bot.coe_Sup' [has_Sup α] {s : set α} (hs : s.nonempty) : + ↑(Sup s) = (Sup (coe '' s) : with_bot α) := +@with_top.coe_Inf' αᵒᵈ _ _ hs + +@[norm_cast] lemma with_bot.coe_supr [nonempty ι] [has_Sup α] (f : ι → α) : + ↑(⨆ i, f i) = (⨆ i, f i : with_bot α) := +@with_top.coe_infi αᵒᵈ _ _ _ _ + +@[norm_cast] theorem with_bot.coe_Inf' [preorder α] [has_Inf α] {s : set α} (hs : bdd_below s) : + ↑(Inf s) = (Inf (coe '' s) : with_bot α) := +@with_top.coe_Sup' αᵒᵈ _ _ _ hs + +@[norm_cast] lemma with_bot.coe_infi [preorder α] [has_Inf α] (f : ι → α) + (h : bdd_below (set.range f)) : + ↑(⨅ i, f i) = (⨅ i, f i : with_bot α) := +@with_top.coe_supr αᵒᵈ _ _ _ _ h + end -- section /-- A conditionally complete lattice is a lattice in which @@ -827,28 +883,15 @@ noncomputable instance : complete_linear_order (with_top α) := Inf := Inf, le_Inf := λ s, (is_glb_Inf s).2, Inf_le := λ s, (is_glb_Inf s).1, .. with_top.linear_order, ..with_top.lattice, ..with_top.order_top, ..with_top.order_bot } -lemma coe_Sup {s : set α} (hb : bdd_above s) : (↑(Sup s) : with_top α) = ⨆ a ∈ s, ↑a := -begin - cases s.eq_empty_or_nonempty with hs hs, - { rw [hs, cSup_empty], simp only [set.mem_empty_eq, supr_bot, supr_false], refl }, - apply le_antisymm, - { refine (coe_le_iff.2 $ λ b hb, cSup_le hs $ λ a has, coe_le_coe.1 $ hb ▸ _), - exact le_supr₂_of_le a has le_rfl }, - { exact supr₂_le (λ a ha, coe_le_coe.2 $ le_cSup hb ha) } -end +/-- A version of `with_top.coe_Sup'` with a more convenient but less general statement. -/ +@[norm_cast] lemma coe_Sup {s : set α} (hb : bdd_above s) : + ↑(Sup s) = (⨆ a ∈ s, ↑a : with_top α) := +by rw [coe_Sup' hb, Sup_image] -lemma coe_Inf {s : set α} (hs : s.nonempty) : (↑(Inf s) : with_top α) = ⨅ a ∈ s, ↑a := -begin - obtain ⟨x, hx⟩ := hs, - have : (⨅ a ∈ s, ↑a : with_top α) ≤ x := infi₂_le_of_le x hx le_rfl, - rcases le_coe_iff.1 this with ⟨r, r_eq, hr⟩, - refine le_antisymm - (le_infi₂ $ λ a ha, coe_le_coe.2 $ cInf_le (order_bot.bdd_below s) ha) _, - { rw r_eq, - apply coe_le_coe.2 (le_cInf ⟨x, hx⟩ (λ a has, coe_le_coe.1 _)), - rw ←r_eq, - exact infi₂_le_of_le a has le_rfl } -end +/-- A version of `with_top.coe_Inf'` with a more convenient but less general statement. -/ +@[norm_cast] lemma coe_Inf {s : set α} (hs : s.nonempty) : + ↑(Inf s) = (⨅ a ∈ s, ↑a : with_top α) := +by rw [coe_Inf' hs, Inf_image] end with_top diff --git a/formal/lean/mathlib/order/filter/bases.lean b/formal/lean/mathlib/order/filter/bases.lean index 994ab6953cd52fc6df0a71288035e930704d6cc5..9dad20ab3b65683e10a903665135059c59cb9605 100644 --- a/formal/lean/mathlib/order/filter/bases.lean +++ b/formal/lean/mathlib/order/filter/bases.lean @@ -5,7 +5,7 @@ Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot -/ import data.prod.pprod import data.set.countable -import order.filter.basic +import order.filter.prod /-! # Filter bases diff --git a/formal/lean/mathlib/order/filter/basic.lean b/formal/lean/mathlib/order/filter/basic.lean index f04b69bb48b9071453184f95cd5ce92e0d2359d4..0cb5388330dc7606557a7d723f315697b92c3e12 100644 --- a/formal/lean/mathlib/order/filter/basic.lean +++ b/formal/lean/mathlib/order/filter/basic.lean @@ -15,7 +15,7 @@ import tactic.monotonicity * `filter` : filters on a set; * `at_top`, `at_bot`, `cofinite`, `principal` : specific filters; -* `map`, `comap`, `prod` : operations on filters; +* `map`, `comap` : operations on filters; * `tendsto` : limit with respect to filters; * `eventually` : `f.eventually p` means `{x | p x} ∈ f`; * `frequently` : `f.frequently p` means `{x | ¬p x} ∉ f`; @@ -39,7 +39,6 @@ the smallest filter containing it in the other direction. We also prove `filter` is a monadic functor, with a push-forward operation `filter.map` and a pull-back operation `filter.comap` that form a Galois connections for the order on filters. -Finally we describe a product operation `filter X → filter Y → filter (X × Y)`. The examples of filters appearing in the description of the two motivating ideas are: * `(at_top : filter ℕ)` : made of sets of `ℕ` containing `{n | n ≥ N}` for some `N` @@ -66,7 +65,6 @@ which is a special case of `mem_closure_of_tendsto` from topology.basic. * `∃ᶠ x in f, p x` : `f.frequently p`; * `f =ᶠ[l] g` : `∀ᶠ x in l, f x = g x`; * `f ≤ᶠ[l] g` : `∀ᶠ x in l, f x ≤ g x`; -* `f ×ᶠ g` : `filter.prod f g`, localized in `filter`; * `𝓟 s` : `principal s`, localized in `filter`. ## References @@ -165,7 +163,7 @@ attribute [protected] finset.Inter_mem_sets ⋂₀ s ∈ f ↔ ∀ U ∈ s, U ∈ f := by rw [sInter_eq_bInter, bInter_mem hfin] -@[simp] lemma Inter_mem {β : Type v} {s : β → set α} [fintype β] : +@[simp] lemma Inter_mem {β : Type v} {s : β → set α} [finite β] : (⋂ i, s i) ∈ f ↔ ∀ i, s i ∈ f := by simpa using bInter_mem finite_univ @@ -557,7 +555,7 @@ lemma exists_Inter_of_mem_infi {ι : Type*} {α : Type*} {f : ι → filter α} (hs : s ∈ ⨅ i, f i) : ∃ t : ι → set α, (∀ i, t i ∈ f i) ∧ s = ⋂ i, t i := let ⟨I, If, V, hVs, hV', hVU, hVU'⟩ := mem_infi'.1 hs in ⟨V, hVs, hVU'⟩ -lemma mem_infi_of_fintype {ι : Type*} [fintype ι] {α : Type*} {f : ι → filter α} (s) : +lemma mem_infi_of_finite {ι : Type*} [finite ι] {α : Type*} {f : ι → filter α} (s) : s ∈ (⨅ i, f i) ↔ ∃ t : ι → set α, (∀ i, t i ∈ f i) ∧ s = ⋂ i, t i := begin refine ⟨exists_Inter_of_mem_infi, _⟩, @@ -774,7 +772,7 @@ lemma mem_infi_finset {s : finset α} {f : α → filter β} {t : set β} : begin simp only [← finset.set_bInter_coe, bInter_eq_Inter, infi_subtype'], refine ⟨λ h, _, _⟩, - { rcases (mem_infi_of_fintype _).1 h with ⟨p, hp, rfl⟩, + { rcases (mem_infi_of_finite _).1 h with ⟨p, hp, rfl⟩, refine ⟨λ a, if h : a ∈ s then p ⟨a, h⟩ else univ, λ a ha, by simpa [ha] using hp ⟨a, ha⟩, _⟩, refine Inter_congr_of_surjective id surjective_id _, rintro ⟨a, ha⟩, simp [ha] }, @@ -893,9 +891,9 @@ begin { rw [finset.infi_insert, finset.set_bInter_insert, hs, inf_principal] }, end -@[simp] lemma infi_principal_fintype {ι : Type w} [fintype ι] (f : ι → set α) : +@[simp] lemma infi_principal {ι : Type w} [finite ι] (f : ι → set α) : (⨅ i, 𝓟 (f i)) = 𝓟 (⋂ i, f i) := -by simpa using infi_principal_finset finset.univ f +by { casesI nonempty_fintype ι, simpa using infi_principal_finset finset.univ f } lemma infi_principal_finite {ι : Type w} {s : set ι} (hs : s.finite) (f : ι → set α) : (⨅ i ∈ s, 𝓟 (f i)) = 𝓟 (⋂ i ∈ s, f i) := @@ -986,9 +984,9 @@ lemma eventually_congr {f : filter α} {p q : α → Prop} (h : ∀ᶠ x in f, p (∀ᶠ x in f, p x) ↔ (∀ᶠ x in f, q x) := ⟨λ hp, hp.congr h, λ hq, hq.congr $ by simpa only [iff.comm] using h⟩ -@[simp] lemma eventually_all {ι} [fintype ι] {l} {p : ι → α → Prop} : +@[simp] lemma eventually_all {ι : Type*} [finite ι] {l} {p : ι → α → Prop} : (∀ᶠ x in l, ∀ i, p i x) ↔ ∀ i, ∀ᶠ x in l, p i x := -by simpa only [filter.eventually, set_of_forall] using Inter_mem +by { casesI nonempty_fintype ι, simpa only [filter.eventually, set_of_forall] using Inter_mem } @[simp] lemma eventually_all_finite {ι} {I : set ι} (hI : I.finite) {l} {p : ι → α → Prop} : (∀ᶠ x in l, ∀ i ∈ I, p i x) ↔ (∀ i ∈ I, ∀ᶠ x in l, p i x) := @@ -2514,398 +2512,6 @@ lemma tendsto.piecewise {l₁ : filter α} {l₂ : filter β} {f g : α → β} tendsto (piecewise s f g) l₁ l₂ := h₀.if h₁ -/-! ### Products of filters -/ - -section prod -variables {s : set α} {t : set β} {f : filter α} {g : filter β} -/- The product filter cannot be defined using the monad structure on filters. For example: - - F := do {x ← seq, y ← top, return (x, y)} - hence: - s ∈ F ↔ ∃ n, [n..∞] × univ ⊆ s - - G := do {y ← top, x ← seq, return (x, y)} - hence: - s ∈ G ↔ ∀ i:ℕ, ∃ n, [n..∞] × {i} ⊆ s - - Now ⋃ i, [i..∞] × {i} is in G but not in F. - - As product filter we want to have F as result. --/ - -/-- Product of filters. This is the filter generated by cartesian products - of elements of the component filters. -/ -protected def prod (f : filter α) (g : filter β) : filter (α × β) := -f.comap prod.fst ⊓ g.comap prod.snd - -localized "infix ` ×ᶠ `:60 := filter.prod" in filter - -lemma prod_mem_prod {s : set α} {t : set β} {f : filter α} {g : filter β} - (hs : s ∈ f) (ht : t ∈ g) : s ×ˢ t ∈ f ×ᶠ g := -inter_mem_inf (preimage_mem_comap hs) (preimage_mem_comap ht) - -lemma mem_prod_iff {s : set (α×β)} {f : filter α} {g : filter β} : - s ∈ f ×ᶠ g ↔ (∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ×ˢ t₂ ⊆ s) := -begin - simp only [filter.prod], - split, - { rintro ⟨t₁, ⟨s₁, hs₁, hts₁⟩, t₂, ⟨s₂, hs₂, hts₂⟩, rfl⟩, - exact ⟨s₁, hs₁, s₂, hs₂, λ p ⟨h, h'⟩, ⟨hts₁ h, hts₂ h'⟩⟩ }, - { rintro ⟨t₁, ht₁, t₂, ht₂, h⟩, - exact mem_inf_of_inter (preimage_mem_comap ht₁) (preimage_mem_comap ht₂) h } -end - -@[simp] lemma prod_mem_prod_iff {s : set α} {t : set β} {f : filter α} {g : filter β} - [f.ne_bot] [g.ne_bot] : - s ×ˢ t ∈ f ×ᶠ g ↔ s ∈ f ∧ t ∈ g := -⟨λ h, let ⟨s', hs', t', ht', H⟩ := mem_prod_iff.1 h in (prod_subset_prod_iff.1 H).elim - (λ ⟨hs's, ht't⟩, ⟨mem_of_superset hs' hs's, mem_of_superset ht' ht't⟩) - (λ h, h.elim - (λ hs'e, absurd hs'e (nonempty_of_mem hs').ne_empty) - (λ ht'e, absurd ht'e (nonempty_of_mem ht').ne_empty)), - λ h, prod_mem_prod h.1 h.2⟩ - -lemma mem_prod_principal {f : filter α} {s : set (α × β)} {t : set β}: - s ∈ f ×ᶠ 𝓟 t ↔ {a | ∀ b ∈ t, (a, b) ∈ s} ∈ f := -begin - rw [← @exists_mem_subset_iff _ f, mem_prod_iff], - refine exists₂_congr (λ u u_in, ⟨_, λ h, ⟨t, mem_principal_self t, _⟩⟩), - { rintros ⟨v, v_in, hv⟩ a a_in b b_in, - exact hv (mk_mem_prod a_in $ v_in b_in) }, - { rintro ⟨x, y⟩ ⟨hx, hy⟩, - exact h hx y hy } -end - -lemma mem_prod_top {f : filter α} {s : set (α × β)} : - s ∈ f ×ᶠ (⊤ : filter β) ↔ {a | ∀ b, (a, b) ∈ s} ∈ f := -begin - rw [← principal_univ, mem_prod_principal], - simp only [mem_univ, forall_true_left] -end - -lemma comap_prod (f : α → β × γ) (b : filter β) (c : filter γ) : - comap f (b ×ᶠ c) = (comap (prod.fst ∘ f) b) ⊓ (comap (prod.snd ∘ f) c) := -by erw [comap_inf, filter.comap_comap, filter.comap_comap] - -lemma prod_top {f : filter α} : f ×ᶠ (⊤ : filter β) = f.comap prod.fst := -by rw [filter.prod, comap_top, inf_top_eq] - -lemma sup_prod (f₁ f₂ : filter α) (g : filter β) : (f₁ ⊔ f₂) ×ᶠ g = (f₁ ×ᶠ g) ⊔ (f₂ ×ᶠ g) := -by rw [filter.prod, comap_sup, inf_sup_right, ← filter.prod, ← filter.prod] - -lemma prod_sup (f : filter α) (g₁ g₂ : filter β) : f ×ᶠ (g₁ ⊔ g₂) = (f ×ᶠ g₁) ⊔ (f ×ᶠ g₂) := -by rw [filter.prod, comap_sup, inf_sup_left, ← filter.prod, ← filter.prod] - -lemma eventually_prod_iff {p : α × β → Prop} {f : filter α} {g : filter β} : - (∀ᶠ x in f ×ᶠ g, p x) ↔ ∃ (pa : α → Prop) (ha : ∀ᶠ x in f, pa x) - (pb : β → Prop) (hb : ∀ᶠ y in g, pb y), ∀ {x}, pa x → ∀ {y}, pb y → p (x, y) := -by simpa only [set.prod_subset_iff] using @mem_prod_iff α β p f g - -lemma tendsto_fst {f : filter α} {g : filter β} : tendsto prod.fst (f ×ᶠ g) f := -tendsto_inf_left tendsto_comap - -lemma tendsto_snd {f : filter α} {g : filter β} : tendsto prod.snd (f ×ᶠ g) g := -tendsto_inf_right tendsto_comap - -lemma tendsto.prod_mk {f : filter α} {g : filter β} {h : filter γ} {m₁ : α → β} {m₂ : α → γ} - (h₁ : tendsto m₁ f g) (h₂ : tendsto m₂ f h) : tendsto (λ x, (m₁ x, m₂ x)) f (g ×ᶠ h) := -tendsto_inf.2 ⟨tendsto_comap_iff.2 h₁, tendsto_comap_iff.2 h₂⟩ - -lemma tendsto_prod_swap {α1 α2 : Type*} {a1 : filter α1} {a2 : filter α2} : - tendsto (prod.swap : α1 × α2 → α2 × α1) (a1 ×ᶠ a2) (a2 ×ᶠ a1) := -tendsto_snd.prod_mk tendsto_fst - -lemma eventually.prod_inl {la : filter α} {p : α → Prop} (h : ∀ᶠ x in la, p x) (lb : filter β) : - ∀ᶠ x in la ×ᶠ lb, p (x : α × β).1 := -tendsto_fst.eventually h - -lemma eventually.prod_inr {lb : filter β} {p : β → Prop} (h : ∀ᶠ x in lb, p x) (la : filter α) : - ∀ᶠ x in la ×ᶠ lb, p (x : α × β).2 := -tendsto_snd.eventually h - -lemma eventually.prod_mk {la : filter α} {pa : α → Prop} (ha : ∀ᶠ x in la, pa x) - {lb : filter β} {pb : β → Prop} (hb : ∀ᶠ y in lb, pb y) : - ∀ᶠ p in la ×ᶠ lb, pa (p : α × β).1 ∧ pb p.2 := -(ha.prod_inl lb).and (hb.prod_inr la) - -lemma eventually_eq.prod_map {δ} {la : filter α} {fa ga : α → γ} (ha : fa =ᶠ[la] ga) - {lb : filter β} {fb gb : β → δ} (hb : fb =ᶠ[lb] gb) : - prod.map fa fb =ᶠ[la ×ᶠ lb] prod.map ga gb := -(eventually.prod_mk ha hb).mono $ λ x h, prod.ext h.1 h.2 - -lemma eventually_le.prod_map {δ} [has_le γ] [has_le δ] {la : filter α} {fa ga : α → γ} - (ha : fa ≤ᶠ[la] ga) {lb : filter β} {fb gb : β → δ} (hb : fb ≤ᶠ[lb] gb) : - prod.map fa fb ≤ᶠ[la ×ᶠ lb] prod.map ga gb := -eventually.prod_mk ha hb - -lemma eventually.curry {la : filter α} {lb : filter β} {p : α × β → Prop} - (h : ∀ᶠ x in la ×ᶠ lb, p x) : - ∀ᶠ x in la, ∀ᶠ y in lb, p (x, y) := -begin - rcases eventually_prod_iff.1 h with ⟨pa, ha, pb, hb, h⟩, - exact ha.mono (λ a ha, hb.mono $ λ b hb, h ha hb) -end - -/-- A fact that is eventually true about all pairs `l ×ᶠ l` is eventually true about -all diagonal pairs `(i, i)` -/ -lemma eventually.diag_of_prod {f : filter α} {p : α × α → Prop} - (h : ∀ᶠ i in f ×ᶠ f, p i) : (∀ᶠ i in f, p (i, i)) := -begin - obtain ⟨t, ht, s, hs, hst⟩ := eventually_prod_iff.1 h, - apply (ht.and hs).mono (λ x hx, hst hx.1 hx.2), -end - -lemma tendsto_diag : tendsto (λ i, (i, i)) f (f ×ᶠ f) := -tendsto_iff_eventually.mpr (λ _ hpr, hpr.diag_of_prod) - -lemma prod_infi_left [nonempty ι] {f : ι → filter α} {g : filter β}: - (⨅ i, f i) ×ᶠ g = (⨅ i, (f i) ×ᶠ g) := -by { rw [filter.prod, comap_infi, infi_inf], simp only [filter.prod, eq_self_iff_true] } - -lemma prod_infi_right [nonempty ι] {f : filter α} {g : ι → filter β} : - f ×ᶠ (⨅ i, g i) = (⨅ i, f ×ᶠ (g i)) := -by { rw [filter.prod, comap_infi, inf_infi], simp only [filter.prod, eq_self_iff_true] } - -@[mono] lemma prod_mono {f₁ f₂ : filter α} {g₁ g₂ : filter β} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) : - f₁ ×ᶠ g₁ ≤ f₂ ×ᶠ g₂ := -inf_le_inf (comap_mono hf) (comap_mono hg) - -lemma prod_comap_comap_eq {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x} - {f₁ : filter α₁} {f₂ : filter α₂} {m₁ : β₁ → α₁} {m₂ : β₂ → α₂} : - (comap m₁ f₁) ×ᶠ (comap m₂ f₂) = comap (λ p : β₁×β₂, (m₁ p.1, m₂ p.2)) (f₁ ×ᶠ f₂) := -by simp only [filter.prod, comap_comap, eq_self_iff_true, comap_inf] - -lemma prod_comm' : f ×ᶠ g = comap (prod.swap) (g ×ᶠ f) := -by simp only [filter.prod, comap_comap, (∘), inf_comm, prod.fst_swap, - eq_self_iff_true, prod.snd_swap, comap_inf] - -lemma prod_comm : f ×ᶠ g = map (λ p : β×α, (p.2, p.1)) (g ×ᶠ f) := -by { rw [prod_comm', ← map_swap_eq_comap_swap], refl } - -lemma prod_assoc (f : filter α) (g : filter β) (h : filter γ) : - map (equiv.prod_assoc α β γ) ((f ×ᶠ g) ×ᶠ h) = f ×ᶠ (g ×ᶠ h) := -by simp_rw [← comap_equiv_symm, filter.prod, comap_inf, comap_comap, inf_assoc, function.comp, - equiv.prod_assoc_symm_apply] - -theorem prod_assoc_symm (f : filter α) (g : filter β) (h : filter γ) : -map (equiv.prod_assoc α β γ).symm (f ×ᶠ (g ×ᶠ h)) = (f ×ᶠ g) ×ᶠ h := -by simp_rw [map_equiv_symm, filter.prod, comap_inf, comap_comap, inf_assoc, function.comp, - equiv.prod_assoc_apply] - -lemma tendsto_prod_assoc {f : filter α} {g : filter β} {h : filter γ} : - tendsto (equiv.prod_assoc α β γ) (f ×ᶠ g ×ᶠ h) (f ×ᶠ (g ×ᶠ h)) := -(prod_assoc f g h).le - -lemma tendsto_prod_assoc_symm {f : filter α} {g : filter β} {h : filter γ} : - tendsto (equiv.prod_assoc α β γ).symm (f ×ᶠ (g ×ᶠ h)) (f ×ᶠ g ×ᶠ h) := -(prod_assoc_symm f g h).le - -/-- A useful lemma when dealing with uniformities. -/ -lemma map_swap4_prod {f : filter α} {g : filter β} {h : filter γ} {k : filter δ} : - map (λ p : (α × β) × (γ × δ), ((p.1.1, p.2.1), (p.1.2, p.2.2))) ((f ×ᶠ g) ×ᶠ (h ×ᶠ k)) = - (f ×ᶠ h) ×ᶠ (g ×ᶠ k) := -by simp_rw [map_swap4_eq_comap, filter.prod, comap_inf, comap_comap, inf_assoc, inf_left_comm] - -lemma tendsto_swap4_prod {f : filter α} {g : filter β} {h : filter γ} {k : filter δ} : - tendsto (λ p : (α × β) × (γ × δ), ((p.1.1, p.2.1), (p.1.2, p.2.2))) - ((f ×ᶠ g) ×ᶠ (h ×ᶠ k)) ((f ×ᶠ h) ×ᶠ (g ×ᶠ k)) := -map_swap4_prod.le - -lemma prod_map_map_eq {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x} - {f₁ : filter α₁} {f₂ : filter α₂} {m₁ : α₁ → β₁} {m₂ : α₂ → β₂} : - (map m₁ f₁) ×ᶠ (map m₂ f₂) = map (λ p : α₁×α₂, (m₁ p.1, m₂ p.2)) (f₁ ×ᶠ f₂) := -le_antisymm - (λ s hs, - let ⟨s₁, hs₁, s₂, hs₂, h⟩ := mem_prod_iff.mp hs in - filter.sets_of_superset _ (prod_mem_prod (image_mem_map hs₁) (image_mem_map hs₂)) $ - calc (m₁ '' s₁) ×ˢ (m₂ '' s₂) = (λ p : α₁×α₂, (m₁ p.1, m₂ p.2)) '' s₁ ×ˢ s₂ : - set.prod_image_image_eq - ... ⊆ _ : by rwa [image_subset_iff]) - ((tendsto.comp le_rfl tendsto_fst).prod_mk (tendsto.comp le_rfl tendsto_snd)) - -lemma prod_map_map_eq' {α₁ : Type*} {α₂ : Type*} {β₁ : Type*} {β₂ : Type*} - (f : α₁ → α₂) (g : β₁ → β₂) (F : filter α₁) (G : filter β₁) : - (map f F) ×ᶠ (map g G) = map (prod.map f g) (F ×ᶠ G) := -prod_map_map_eq - -lemma le_prod_map_fst_snd {f : filter (α × β)} : f ≤ map prod.fst f ×ᶠ map prod.snd f := -le_inf le_comap_map le_comap_map - -lemma tendsto.prod_map {δ : Type*} {f : α → γ} {g : β → δ} {a : filter α} {b : filter β} - {c : filter γ} {d : filter δ} (hf : tendsto f a c) (hg : tendsto g b d) : - tendsto (prod.map f g) (a ×ᶠ b) (c ×ᶠ d) := -begin - erw [tendsto, ← prod_map_map_eq], - exact filter.prod_mono hf hg, -end - -protected lemma map_prod (m : α × β → γ) (f : filter α) (g : filter β) : - map m (f ×ᶠ g) = (f.map (λ a b, m (a, b))).seq g := -begin - simp [filter.ext_iff, mem_prod_iff, mem_map_seq_iff], - intro s, - split, - exact λ ⟨t, ht, s, hs, h⟩, ⟨s, hs, t, ht, λ x hx y hy, @h ⟨x, y⟩ ⟨hx, hy⟩⟩, - exact λ ⟨s, hs, t, ht, h⟩, ⟨t, ht, s, hs, λ ⟨x, y⟩ ⟨hx, hy⟩, h x hx y hy⟩ -end - -lemma prod_eq {f : filter α} {g : filter β} : f ×ᶠ g = (f.map prod.mk).seq g := -have h : _ := f.map_prod id g, by rwa [map_id] at h - -lemma prod_inf_prod {f₁ f₂ : filter α} {g₁ g₂ : filter β} : - (f₁ ×ᶠ g₁) ⊓ (f₂ ×ᶠ g₂) = (f₁ ⊓ f₂) ×ᶠ (g₁ ⊓ g₂) := -by simp only [filter.prod, comap_inf, inf_comm, inf_assoc, inf_left_comm] - -@[simp] lemma prod_bot {f : filter α} : f ×ᶠ (⊥ : filter β) = ⊥ := by simp [filter.prod] -@[simp] lemma bot_prod {g : filter β} : (⊥ : filter α) ×ᶠ g = ⊥ := by simp [filter.prod] - -@[simp] lemma prod_principal_principal {s : set α} {t : set β} : - (𝓟 s) ×ᶠ (𝓟 t) = 𝓟 (s ×ˢ t) := -by simp only [filter.prod, comap_principal, principal_eq_iff_eq, comap_principal, inf_principal]; - refl - -@[simp] lemma pure_prod {a : α} {f : filter β} : pure a ×ᶠ f = map (prod.mk a) f := -by rw [prod_eq, map_pure, pure_seq_eq_map] - -lemma map_pure_prod (f : α → β → γ) (a : α) (B : filter β) : - filter.map (function.uncurry f) (pure a ×ᶠ B) = filter.map (f a) B := -by { rw filter.pure_prod, refl } - -@[simp] lemma prod_pure {f : filter α} {b : β} : f ×ᶠ pure b = map (λ a, (a, b)) f := -by rw [prod_eq, seq_pure, map_map] - -lemma prod_pure_pure {a : α} {b : β} : (pure a) ×ᶠ (pure b) = pure (a, b) := -by simp - -lemma prod_eq_bot {f : filter α} {g : filter β} : f ×ᶠ g = ⊥ ↔ (f = ⊥ ∨ g = ⊥) := -begin - split, - { intro h, - rcases mem_prod_iff.1 (empty_mem_iff_bot.2 h) with ⟨s, hs, t, ht, hst⟩, - rw [subset_empty_iff, set.prod_eq_empty_iff] at hst, - cases hst with s_eq t_eq, - { left, exact empty_mem_iff_bot.1 (s_eq ▸ hs) }, - { right, exact empty_mem_iff_bot.1 (t_eq ▸ ht) } }, - { rintro (rfl | rfl), - exact bot_prod, - exact prod_bot } -end - -lemma prod_ne_bot {f : filter α} {g : filter β} : ne_bot (f ×ᶠ g) ↔ (ne_bot f ∧ ne_bot g) := -by simp only [ne_bot_iff, ne, prod_eq_bot, not_or_distrib] - -lemma ne_bot.prod {f : filter α} {g : filter β} (hf : ne_bot f) (hg : ne_bot g) : - ne_bot (f ×ᶠ g) := -prod_ne_bot.2 ⟨hf, hg⟩ - -instance prod_ne_bot' {f : filter α} {g : filter β} [hf : ne_bot f] [hg : ne_bot g] : - ne_bot (f ×ᶠ g) := -hf.prod hg - -lemma tendsto_prod_iff {f : α × β → γ} {x : filter α} {y : filter β} {z : filter γ} : - filter.tendsto f (x ×ᶠ y) z ↔ - ∀ W ∈ z, ∃ U ∈ x, ∃ V ∈ y, ∀ x y, x ∈ U → y ∈ V → f (x, y) ∈ W := -by simp only [tendsto_def, mem_prod_iff, prod_sub_preimage_iff, exists_prop, iff_self] - -lemma tendsto_prod_iff' {f : filter α} {g : filter β} {g' : filter γ} - {s : α → β × γ} : - tendsto s f (g ×ᶠ g') ↔ tendsto (λ n, (s n).1) f g ∧ tendsto (λ n, (s n).2) f g' := -by { unfold filter.prod, simp only [tendsto_inf, tendsto_comap_iff, iff_self] } - -end prod - -/-! ### Coproducts of filters -/ - -section coprod -variables {f : filter α} {g : filter β} - -/-- Coproduct of filters. -/ -protected def coprod (f : filter α) (g : filter β) : filter (α × β) := -f.comap prod.fst ⊔ g.comap prod.snd - -lemma mem_coprod_iff {s : set (α×β)} {f : filter α} {g : filter β} : - s ∈ f.coprod g ↔ ((∃ t₁ ∈ f, prod.fst ⁻¹' t₁ ⊆ s) ∧ (∃ t₂ ∈ g, prod.snd ⁻¹' t₂ ⊆ s)) := -by simp [filter.coprod] - -@[simp] lemma bot_coprod (l : filter β) : (⊥ : filter α).coprod l = comap prod.snd l := -by simp [filter.coprod] - -@[simp] lemma coprod_bot (l : filter α) : l.coprod (⊥ : filter β) = comap prod.fst l := -by simp [filter.coprod] - -lemma bot_coprod_bot : (⊥ : filter α).coprod (⊥ : filter β) = ⊥ := by simp - -lemma compl_mem_coprod {s : set (α × β)} {la : filter α} {lb : filter β} : - sᶜ ∈ la.coprod lb ↔ (prod.fst '' s)ᶜ ∈ la ∧ (prod.snd '' s)ᶜ ∈ lb := -by simp only [filter.coprod, mem_sup, compl_mem_comap] - -@[mono] lemma coprod_mono {f₁ f₂ : filter α} {g₁ g₂ : filter β} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) : - f₁.coprod g₁ ≤ f₂.coprod g₂ := -sup_le_sup (comap_mono hf) (comap_mono hg) - -lemma coprod_ne_bot_iff : (f.coprod g).ne_bot ↔ f.ne_bot ∧ nonempty β ∨ nonempty α ∧ g.ne_bot := -by simp [filter.coprod] - -@[instance] lemma coprod_ne_bot_left [ne_bot f] [nonempty β] : (f.coprod g).ne_bot := -coprod_ne_bot_iff.2 (or.inl ⟨‹_›, ‹_›⟩) - -@[instance] lemma coprod_ne_bot_right [ne_bot g] [nonempty α] : (f.coprod g).ne_bot := -coprod_ne_bot_iff.2 (or.inr ⟨‹_›, ‹_›⟩) - -lemma principal_coprod_principal (s : set α) (t : set β) : - (𝓟 s).coprod (𝓟 t) = 𝓟 (sᶜ ×ˢ tᶜ)ᶜ := -by rw [filter.coprod, comap_principal, comap_principal, sup_principal, set.prod_eq, compl_inter, - preimage_compl, preimage_compl, compl_compl, compl_compl] - --- this inequality can be strict; see `map_const_principal_coprod_map_id_principal` and --- `map_prod_map_const_id_principal_coprod_principal` below. -lemma map_prod_map_coprod_le {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x} - {f₁ : filter α₁} {f₂ : filter α₂} {m₁ : α₁ → β₁} {m₂ : α₂ → β₂} : - map (prod.map m₁ m₂) (f₁.coprod f₂) ≤ (map m₁ f₁).coprod (map m₂ f₂) := -begin - intros s, - simp only [mem_map, mem_coprod_iff], - rintro ⟨⟨u₁, hu₁, h₁⟩, u₂, hu₂, h₂⟩, - refine ⟨⟨m₁ ⁻¹' u₁, hu₁, λ _ hx, h₁ _⟩, ⟨m₂ ⁻¹' u₂, hu₂, λ _ hx, h₂ _⟩⟩; convert hx -end - -/-- Characterization of the coproduct of the `filter.map`s of two principal filters `𝓟 {a}` and -`𝓟 {i}`, the first under the constant function `λ a, b` and the second under the identity function. -Together with the next lemma, `map_prod_map_const_id_principal_coprod_principal`, this provides an -example showing that the inequality in the lemma `map_prod_map_coprod_le` can be strict. -/ -lemma map_const_principal_coprod_map_id_principal {α β ι : Type*} (a : α) (b : β) (i : ι) : - (map (λ _ : α, b) (𝓟 {a})).coprod (map id (𝓟 {i})) - = 𝓟 (({b} : set β) ×ˢ (univ : set ι) ∪ (univ : set β) ×ˢ ({i} : set ι)) := -by simp only [map_principal, filter.coprod, comap_principal, sup_principal, image_singleton, - image_id, prod_univ, univ_prod] - -/-- Characterization of the `filter.map` of the coproduct of two principal filters `𝓟 {a}` and -`𝓟 {i}`, under the `prod.map` of two functions, respectively the constant function `λ a, b` and the -identity function. Together with the previous lemma, -`map_const_principal_coprod_map_id_principal`, this provides an example showing that the inequality -in the lemma `map_prod_map_coprod_le` can be strict. -/ -lemma map_prod_map_const_id_principal_coprod_principal {α β ι : Type*} (a : α) (b : β) (i : ι) : - map (prod.map (λ _ : α, b) id) ((𝓟 {a}).coprod (𝓟 {i})) - = 𝓟 (({b} : set β) ×ˢ (univ : set ι)) := -begin - rw [principal_coprod_principal, map_principal], - congr, - ext ⟨b', i'⟩, - split, - { rintro ⟨⟨a'', i''⟩, h₁, h₂, h₃⟩, - simp }, - { rintro ⟨h₁, h₂⟩, - use (a, i'), - simpa using h₁.symm } -end - -lemma tendsto.prod_map_coprod {δ : Type*} {f : α → γ} {g : β → δ} {a : filter α} {b : filter β} - {c : filter γ} {d : filter δ} (hf : tendsto f a c) (hg : tendsto g b d) : - tendsto (prod.map f g) (a.coprod b) (c.coprod d) := -map_prod_map_coprod_le.trans (coprod_mono hf hg) - -end coprod - end filter open_locale filter diff --git a/formal/lean/mathlib/order/filter/cofinite.lean b/formal/lean/mathlib/order/filter/cofinite.lean index 02f6be134ff89be18304127b3c016b9ebbf90bad..6706656e87e2185284be8ddc84c98264c0ea4663 100644 --- a/formal/lean/mathlib/order/filter/cofinite.lean +++ b/formal/lean/mathlib/order/filter/cofinite.lean @@ -95,7 +95,7 @@ filter.coext $ λ s, by simp only [compl_mem_coprod, mem_cofinite, compl_compl, finite_image_fst_and_snd_iff] /-- Finite product of finite sets is finite -/ -lemma Coprod_cofinite {α : ι → Type*} [fintype ι] : +lemma Coprod_cofinite {α : ι → Type*} [finite ι] : filter.Coprod (λ i, (cofinite : filter (α i))) = cofinite := filter.coext $ λ s, by simp only [compl_mem_Coprod, mem_cofinite, compl_compl, forall_finite_image_eval_iff] diff --git a/formal/lean/mathlib/order/filter/n_ary.lean b/formal/lean/mathlib/order/filter/n_ary.lean index 398dfa3bab70c3706e8a95b0e0bec12c55406dca..6a9515f4282badbe927a9fff4ea898f7d8a6743c 100644 --- a/formal/lean/mathlib/order/filter/n_ary.lean +++ b/formal/lean/mathlib/order/filter/n_ary.lean @@ -3,7 +3,7 @@ Copyright (c) 2022 Yaël Dillies. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yaël Dillies -/ -import order.filter.basic +import order.filter.prod /-! # N-ary maps of filter diff --git a/formal/lean/mathlib/order/filter/pointwise.lean b/formal/lean/mathlib/order/filter/pointwise.lean index 886f3e75e2ec2a155eff6df55fd50dbd64bfb370..0b3e90b529f1c9a1854b21933c7d9d3b49d41d9c 100644 --- a/formal/lean/mathlib/order/filter/pointwise.lean +++ b/formal/lean/mathlib/order/filter/pointwise.lean @@ -636,18 +636,18 @@ instance smul_comm_class [has_smul α γ] [has_smul β γ] [smul_comm_class α smul_comm_class (filter α) (filter β) (filter γ) := ⟨λ f g h, map₂_left_comm smul_comm⟩ -instance is_scalar_tower [has_smul α β] [has_smul α γ] [has_smul β γ] - [is_scalar_tower α β γ] : +@[to_additive] +instance is_scalar_tower [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] : is_scalar_tower α β (filter γ) := ⟨λ a b f, by simp only [←map_smul, map_map, smul_assoc]⟩ -instance is_scalar_tower' [has_smul α β] [has_smul α γ] [has_smul β γ] - [is_scalar_tower α β γ] : +@[to_additive] +instance is_scalar_tower' [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] : is_scalar_tower α (filter β) (filter γ) := ⟨λ a f g, by { refine (map_map₂_distrib_left $ λ _ _, _).symm, exact (smul_assoc a _ _).symm }⟩ -instance is_scalar_tower'' [has_smul α β] [has_smul α γ] [has_smul β γ] - [is_scalar_tower α β γ] : +@[to_additive] +instance is_scalar_tower'' [has_smul α β] [has_smul α γ] [has_smul β γ] [is_scalar_tower α β γ] : is_scalar_tower (filter α) (filter β) (filter γ) := ⟨λ f g h, map₂_assoc smul_assoc⟩ diff --git a/formal/lean/mathlib/order/filter/prod.lean b/formal/lean/mathlib/order/filter/prod.lean new file mode 100644 index 0000000000000000000000000000000000000000..7c93895fa46c37e72b957d3bb8721723298d72af --- /dev/null +++ b/formal/lean/mathlib/order/filter/prod.lean @@ -0,0 +1,420 @@ +/- +Copyright (c) 2022 Johannes Hölzl. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Johanes Hölzl, Patrick Massot, Yury Kudryashov, Kevin Wilson, Heather Macbeth +-/ +import order.filter.basic + +/-! +# Product and coproduct filters + +In this file we define `filter.prod f g` (notation: `f ×ᶠ g`) and `filter.coprod f g`. The product +of two filters is the largest filter `l` such that `filter.tendsto prod.fst l f` and +`filter.tendsto prod.snd l g`. + +## Implementation details + +The product filter cannot be defined using the monad structure on filters. For example: + +```lean +F := do {x ← seq, y ← top, return (x, y)} +G := do {y ← top, x ← seq, return (x, y)} +``` +hence: +```lean +s ∈ F ↔ ∃ n, [n..∞] × univ ⊆ s +s ∈ G ↔ ∀ i:ℕ, ∃ n, [n..∞] × {i} ⊆ s +``` +Now `⋃ i, [i..∞] × {i}` is in `G` but not in `F`. +As product filter we want to have `F` as result. + +## Notations + +* `f ×ᶠ g` : `filter.prod f g`, localized in `filter`. + +-/ + +open set +open_locale filter + +namespace filter + +variables {α β γ δ : Type*} {ι : Sort*} + +section prod +variables {s : set α} {t : set β} {f : filter α} {g : filter β} + +/-- Product of filters. This is the filter generated by cartesian products +of elements of the component filters. -/ +protected def prod (f : filter α) (g : filter β) : filter (α × β) := +f.comap prod.fst ⊓ g.comap prod.snd + +localized "infix ` ×ᶠ `:60 := filter.prod" in filter + +lemma prod_mem_prod {s : set α} {t : set β} {f : filter α} {g : filter β} + (hs : s ∈ f) (ht : t ∈ g) : s ×ˢ t ∈ f ×ᶠ g := +inter_mem_inf (preimage_mem_comap hs) (preimage_mem_comap ht) + +lemma mem_prod_iff {s : set (α×β)} {f : filter α} {g : filter β} : + s ∈ f ×ᶠ g ↔ (∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ×ˢ t₂ ⊆ s) := +begin + simp only [filter.prod], + split, + { rintro ⟨t₁, ⟨s₁, hs₁, hts₁⟩, t₂, ⟨s₂, hs₂, hts₂⟩, rfl⟩, + exact ⟨s₁, hs₁, s₂, hs₂, λ p ⟨h, h'⟩, ⟨hts₁ h, hts₂ h'⟩⟩ }, + { rintro ⟨t₁, ht₁, t₂, ht₂, h⟩, + exact mem_inf_of_inter (preimage_mem_comap ht₁) (preimage_mem_comap ht₂) h } +end + +@[simp] lemma prod_mem_prod_iff {s : set α} {t : set β} {f : filter α} {g : filter β} + [f.ne_bot] [g.ne_bot] : + s ×ˢ t ∈ f ×ᶠ g ↔ s ∈ f ∧ t ∈ g := +⟨λ h, let ⟨s', hs', t', ht', H⟩ := mem_prod_iff.1 h in (prod_subset_prod_iff.1 H).elim + (λ ⟨hs's, ht't⟩, ⟨mem_of_superset hs' hs's, mem_of_superset ht' ht't⟩) + (λ h, h.elim + (λ hs'e, absurd hs'e (nonempty_of_mem hs').ne_empty) + (λ ht'e, absurd ht'e (nonempty_of_mem ht').ne_empty)), + λ h, prod_mem_prod h.1 h.2⟩ + +lemma mem_prod_principal {f : filter α} {s : set (α × β)} {t : set β}: + s ∈ f ×ᶠ 𝓟 t ↔ {a | ∀ b ∈ t, (a, b) ∈ s} ∈ f := +begin + rw [← @exists_mem_subset_iff _ f, mem_prod_iff], + refine exists₂_congr (λ u u_in, ⟨_, λ h, ⟨t, mem_principal_self t, _⟩⟩), + { rintros ⟨v, v_in, hv⟩ a a_in b b_in, + exact hv (mk_mem_prod a_in $ v_in b_in) }, + { rintro ⟨x, y⟩ ⟨hx, hy⟩, + exact h hx y hy } +end + +lemma mem_prod_top {f : filter α} {s : set (α × β)} : + s ∈ f ×ᶠ (⊤ : filter β) ↔ {a | ∀ b, (a, b) ∈ s} ∈ f := +begin + rw [← principal_univ, mem_prod_principal], + simp only [mem_univ, forall_true_left] +end + +lemma comap_prod (f : α → β × γ) (b : filter β) (c : filter γ) : + comap f (b ×ᶠ c) = (comap (prod.fst ∘ f) b) ⊓ (comap (prod.snd ∘ f) c) := +by erw [comap_inf, filter.comap_comap, filter.comap_comap] + +lemma prod_top {f : filter α} : f ×ᶠ (⊤ : filter β) = f.comap prod.fst := +by rw [filter.prod, comap_top, inf_top_eq] + +lemma sup_prod (f₁ f₂ : filter α) (g : filter β) : (f₁ ⊔ f₂) ×ᶠ g = (f₁ ×ᶠ g) ⊔ (f₂ ×ᶠ g) := +by rw [filter.prod, comap_sup, inf_sup_right, ← filter.prod, ← filter.prod] + +lemma prod_sup (f : filter α) (g₁ g₂ : filter β) : f ×ᶠ (g₁ ⊔ g₂) = (f ×ᶠ g₁) ⊔ (f ×ᶠ g₂) := +by rw [filter.prod, comap_sup, inf_sup_left, ← filter.prod, ← filter.prod] + +lemma eventually_prod_iff {p : α × β → Prop} {f : filter α} {g : filter β} : + (∀ᶠ x in f ×ᶠ g, p x) ↔ ∃ (pa : α → Prop) (ha : ∀ᶠ x in f, pa x) + (pb : β → Prop) (hb : ∀ᶠ y in g, pb y), ∀ {x}, pa x → ∀ {y}, pb y → p (x, y) := +by simpa only [set.prod_subset_iff] using @mem_prod_iff α β p f g + +lemma tendsto_fst {f : filter α} {g : filter β} : tendsto prod.fst (f ×ᶠ g) f := +tendsto_inf_left tendsto_comap + +lemma tendsto_snd {f : filter α} {g : filter β} : tendsto prod.snd (f ×ᶠ g) g := +tendsto_inf_right tendsto_comap + +lemma tendsto.prod_mk {f : filter α} {g : filter β} {h : filter γ} {m₁ : α → β} {m₂ : α → γ} + (h₁ : tendsto m₁ f g) (h₂ : tendsto m₂ f h) : tendsto (λ x, (m₁ x, m₂ x)) f (g ×ᶠ h) := +tendsto_inf.2 ⟨tendsto_comap_iff.2 h₁, tendsto_comap_iff.2 h₂⟩ + +lemma tendsto_prod_swap {α1 α2 : Type*} {a1 : filter α1} {a2 : filter α2} : + tendsto (prod.swap : α1 × α2 → α2 × α1) (a1 ×ᶠ a2) (a2 ×ᶠ a1) := +tendsto_snd.prod_mk tendsto_fst + +lemma eventually.prod_inl {la : filter α} {p : α → Prop} (h : ∀ᶠ x in la, p x) (lb : filter β) : + ∀ᶠ x in la ×ᶠ lb, p (x : α × β).1 := +tendsto_fst.eventually h + +lemma eventually.prod_inr {lb : filter β} {p : β → Prop} (h : ∀ᶠ x in lb, p x) (la : filter α) : + ∀ᶠ x in la ×ᶠ lb, p (x : α × β).2 := +tendsto_snd.eventually h + +lemma eventually.prod_mk {la : filter α} {pa : α → Prop} (ha : ∀ᶠ x in la, pa x) + {lb : filter β} {pb : β → Prop} (hb : ∀ᶠ y in lb, pb y) : + ∀ᶠ p in la ×ᶠ lb, pa (p : α × β).1 ∧ pb p.2 := +(ha.prod_inl lb).and (hb.prod_inr la) + +lemma eventually_eq.prod_map {δ} {la : filter α} {fa ga : α → γ} (ha : fa =ᶠ[la] ga) + {lb : filter β} {fb gb : β → δ} (hb : fb =ᶠ[lb] gb) : + prod.map fa fb =ᶠ[la ×ᶠ lb] prod.map ga gb := +(eventually.prod_mk ha hb).mono $ λ x h, prod.ext h.1 h.2 + +lemma eventually_le.prod_map {δ} [has_le γ] [has_le δ] {la : filter α} {fa ga : α → γ} + (ha : fa ≤ᶠ[la] ga) {lb : filter β} {fb gb : β → δ} (hb : fb ≤ᶠ[lb] gb) : + prod.map fa fb ≤ᶠ[la ×ᶠ lb] prod.map ga gb := +eventually.prod_mk ha hb + +lemma eventually.curry {la : filter α} {lb : filter β} {p : α × β → Prop} + (h : ∀ᶠ x in la ×ᶠ lb, p x) : + ∀ᶠ x in la, ∀ᶠ y in lb, p (x, y) := +begin + rcases eventually_prod_iff.1 h with ⟨pa, ha, pb, hb, h⟩, + exact ha.mono (λ a ha, hb.mono $ λ b hb, h ha hb) +end + +/-- A fact that is eventually true about all pairs `l ×ᶠ l` is eventually true about +all diagonal pairs `(i, i)` -/ +lemma eventually.diag_of_prod {f : filter α} {p : α × α → Prop} + (h : ∀ᶠ i in f ×ᶠ f, p i) : (∀ᶠ i in f, p (i, i)) := +begin + obtain ⟨t, ht, s, hs, hst⟩ := eventually_prod_iff.1 h, + apply (ht.and hs).mono (λ x hx, hst hx.1 hx.2), +end + +lemma tendsto_diag : tendsto (λ i, (i, i)) f (f ×ᶠ f) := +tendsto_iff_eventually.mpr (λ _ hpr, hpr.diag_of_prod) + +lemma prod_infi_left [nonempty ι] {f : ι → filter α} {g : filter β}: + (⨅ i, f i) ×ᶠ g = (⨅ i, (f i) ×ᶠ g) := +by { rw [filter.prod, comap_infi, infi_inf], simp only [filter.prod, eq_self_iff_true] } + +lemma prod_infi_right [nonempty ι] {f : filter α} {g : ι → filter β} : + f ×ᶠ (⨅ i, g i) = (⨅ i, f ×ᶠ (g i)) := +by { rw [filter.prod, comap_infi, inf_infi], simp only [filter.prod, eq_self_iff_true] } + +@[mono] lemma prod_mono {f₁ f₂ : filter α} {g₁ g₂ : filter β} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) : + f₁ ×ᶠ g₁ ≤ f₂ ×ᶠ g₂ := +inf_le_inf (comap_mono hf) (comap_mono hg) + +lemma {u v w x} prod_comap_comap_eq {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x} + {f₁ : filter α₁} {f₂ : filter α₂} {m₁ : β₁ → α₁} {m₂ : β₂ → α₂} : + (comap m₁ f₁) ×ᶠ (comap m₂ f₂) = comap (λ p : β₁×β₂, (m₁ p.1, m₂ p.2)) (f₁ ×ᶠ f₂) := +by simp only [filter.prod, comap_comap, eq_self_iff_true, comap_inf] + +lemma prod_comm' : f ×ᶠ g = comap (prod.swap) (g ×ᶠ f) := +by simp only [filter.prod, comap_comap, (∘), inf_comm, prod.fst_swap, + eq_self_iff_true, prod.snd_swap, comap_inf] + +lemma prod_comm : f ×ᶠ g = map (λ p : β×α, (p.2, p.1)) (g ×ᶠ f) := +by { rw [prod_comm', ← map_swap_eq_comap_swap], refl } + +lemma prod_assoc (f : filter α) (g : filter β) (h : filter γ) : + map (equiv.prod_assoc α β γ) ((f ×ᶠ g) ×ᶠ h) = f ×ᶠ (g ×ᶠ h) := +by simp_rw [← comap_equiv_symm, filter.prod, comap_inf, comap_comap, inf_assoc, function.comp, + equiv.prod_assoc_symm_apply] + +theorem prod_assoc_symm (f : filter α) (g : filter β) (h : filter γ) : +map (equiv.prod_assoc α β γ).symm (f ×ᶠ (g ×ᶠ h)) = (f ×ᶠ g) ×ᶠ h := +by simp_rw [map_equiv_symm, filter.prod, comap_inf, comap_comap, inf_assoc, function.comp, + equiv.prod_assoc_apply] + +lemma tendsto_prod_assoc {f : filter α} {g : filter β} {h : filter γ} : + tendsto (equiv.prod_assoc α β γ) (f ×ᶠ g ×ᶠ h) (f ×ᶠ (g ×ᶠ h)) := +(prod_assoc f g h).le + +lemma tendsto_prod_assoc_symm {f : filter α} {g : filter β} {h : filter γ} : + tendsto (equiv.prod_assoc α β γ).symm (f ×ᶠ (g ×ᶠ h)) (f ×ᶠ g ×ᶠ h) := +(prod_assoc_symm f g h).le + +/-- A useful lemma when dealing with uniformities. -/ +lemma map_swap4_prod {f : filter α} {g : filter β} {h : filter γ} {k : filter δ} : + map (λ p : (α × β) × (γ × δ), ((p.1.1, p.2.1), (p.1.2, p.2.2))) ((f ×ᶠ g) ×ᶠ (h ×ᶠ k)) = + (f ×ᶠ h) ×ᶠ (g ×ᶠ k) := +by simp_rw [map_swap4_eq_comap, filter.prod, comap_inf, comap_comap, inf_assoc, inf_left_comm] + +lemma tendsto_swap4_prod {f : filter α} {g : filter β} {h : filter γ} {k : filter δ} : + tendsto (λ p : (α × β) × (γ × δ), ((p.1.1, p.2.1), (p.1.2, p.2.2))) + ((f ×ᶠ g) ×ᶠ (h ×ᶠ k)) ((f ×ᶠ h) ×ᶠ (g ×ᶠ k)) := +map_swap4_prod.le + +lemma {u v w x} prod_map_map_eq {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x} + {f₁ : filter α₁} {f₂ : filter α₂} {m₁ : α₁ → β₁} {m₂ : α₂ → β₂} : + (map m₁ f₁) ×ᶠ (map m₂ f₂) = map (λ p : α₁×α₂, (m₁ p.1, m₂ p.2)) (f₁ ×ᶠ f₂) := +le_antisymm + (λ s hs, + let ⟨s₁, hs₁, s₂, hs₂, h⟩ := mem_prod_iff.mp hs in + filter.sets_of_superset _ (prod_mem_prod (image_mem_map hs₁) (image_mem_map hs₂)) $ + calc (m₁ '' s₁) ×ˢ (m₂ '' s₂) = (λ p : α₁×α₂, (m₁ p.1, m₂ p.2)) '' s₁ ×ˢ s₂ : + set.prod_image_image_eq + ... ⊆ _ : by rwa [image_subset_iff]) + ((tendsto.comp le_rfl tendsto_fst).prod_mk (tendsto.comp le_rfl tendsto_snd)) + +lemma prod_map_map_eq' {α₁ : Type*} {α₂ : Type*} {β₁ : Type*} {β₂ : Type*} + (f : α₁ → α₂) (g : β₁ → β₂) (F : filter α₁) (G : filter β₁) : + (map f F) ×ᶠ (map g G) = map (prod.map f g) (F ×ᶠ G) := +prod_map_map_eq + +lemma le_prod_map_fst_snd {f : filter (α × β)} : f ≤ map prod.fst f ×ᶠ map prod.snd f := +le_inf le_comap_map le_comap_map + +lemma tendsto.prod_map {δ : Type*} {f : α → γ} {g : β → δ} {a : filter α} {b : filter β} + {c : filter γ} {d : filter δ} (hf : tendsto f a c) (hg : tendsto g b d) : + tendsto (prod.map f g) (a ×ᶠ b) (c ×ᶠ d) := +begin + erw [tendsto, ← prod_map_map_eq], + exact filter.prod_mono hf hg, +end + +protected lemma map_prod (m : α × β → γ) (f : filter α) (g : filter β) : + map m (f ×ᶠ g) = (f.map (λ a b, m (a, b))).seq g := +begin + simp [filter.ext_iff, mem_prod_iff, mem_map_seq_iff], + intro s, + split, + exact λ ⟨t, ht, s, hs, h⟩, ⟨s, hs, t, ht, λ x hx y hy, @h ⟨x, y⟩ ⟨hx, hy⟩⟩, + exact λ ⟨s, hs, t, ht, h⟩, ⟨t, ht, s, hs, λ ⟨x, y⟩ ⟨hx, hy⟩, h x hx y hy⟩ +end + +lemma prod_eq {f : filter α} {g : filter β} : f ×ᶠ g = (f.map prod.mk).seq g := +have h : _ := f.map_prod id g, by rwa [map_id] at h + +lemma prod_inf_prod {f₁ f₂ : filter α} {g₁ g₂ : filter β} : + (f₁ ×ᶠ g₁) ⊓ (f₂ ×ᶠ g₂) = (f₁ ⊓ f₂) ×ᶠ (g₁ ⊓ g₂) := +by simp only [filter.prod, comap_inf, inf_comm, inf_assoc, inf_left_comm] + +@[simp] lemma prod_bot {f : filter α} : f ×ᶠ (⊥ : filter β) = ⊥ := by simp [filter.prod] +@[simp] lemma bot_prod {g : filter β} : (⊥ : filter α) ×ᶠ g = ⊥ := by simp [filter.prod] + +@[simp] lemma prod_principal_principal {s : set α} {t : set β} : + (𝓟 s) ×ᶠ (𝓟 t) = 𝓟 (s ×ˢ t) := +by simp only [filter.prod, comap_principal, principal_eq_iff_eq, comap_principal, inf_principal]; + refl + +@[simp] lemma pure_prod {a : α} {f : filter β} : pure a ×ᶠ f = map (prod.mk a) f := +by rw [prod_eq, map_pure, pure_seq_eq_map] + +lemma map_pure_prod (f : α → β → γ) (a : α) (B : filter β) : + filter.map (function.uncurry f) (pure a ×ᶠ B) = filter.map (f a) B := +by { rw filter.pure_prod, refl } + +@[simp] lemma prod_pure {f : filter α} {b : β} : f ×ᶠ pure b = map (λ a, (a, b)) f := +by rw [prod_eq, seq_pure, map_map] + +lemma prod_pure_pure {a : α} {b : β} : (pure a) ×ᶠ (pure b) = pure (a, b) := +by simp + +lemma prod_eq_bot {f : filter α} {g : filter β} : f ×ᶠ g = ⊥ ↔ (f = ⊥ ∨ g = ⊥) := +begin + split, + { intro h, + rcases mem_prod_iff.1 (empty_mem_iff_bot.2 h) with ⟨s, hs, t, ht, hst⟩, + rw [subset_empty_iff, set.prod_eq_empty_iff] at hst, + cases hst with s_eq t_eq, + { left, exact empty_mem_iff_bot.1 (s_eq ▸ hs) }, + { right, exact empty_mem_iff_bot.1 (t_eq ▸ ht) } }, + { rintro (rfl | rfl), + exact bot_prod, + exact prod_bot } +end + +lemma prod_ne_bot {f : filter α} {g : filter β} : ne_bot (f ×ᶠ g) ↔ (ne_bot f ∧ ne_bot g) := +by simp only [ne_bot_iff, ne, prod_eq_bot, not_or_distrib] + +lemma ne_bot.prod {f : filter α} {g : filter β} (hf : ne_bot f) (hg : ne_bot g) : + ne_bot (f ×ᶠ g) := +prod_ne_bot.2 ⟨hf, hg⟩ + +instance prod_ne_bot' {f : filter α} {g : filter β} [hf : ne_bot f] [hg : ne_bot g] : + ne_bot (f ×ᶠ g) := +hf.prod hg + +lemma tendsto_prod_iff {f : α × β → γ} {x : filter α} {y : filter β} {z : filter γ} : + filter.tendsto f (x ×ᶠ y) z ↔ + ∀ W ∈ z, ∃ U ∈ x, ∃ V ∈ y, ∀ x y, x ∈ U → y ∈ V → f (x, y) ∈ W := +by simp only [tendsto_def, mem_prod_iff, prod_sub_preimage_iff, exists_prop, iff_self] + +lemma tendsto_prod_iff' {f : filter α} {g : filter β} {g' : filter γ} + {s : α → β × γ} : + tendsto s f (g ×ᶠ g') ↔ tendsto (λ n, (s n).1) f g ∧ tendsto (λ n, (s n).2) f g' := +by { unfold filter.prod, simp only [tendsto_inf, tendsto_comap_iff, iff_self] } + +end prod + +/-! ### Coproducts of filters -/ + +section coprod +variables {f : filter α} {g : filter β} + +/-- Coproduct of filters. -/ +protected def coprod (f : filter α) (g : filter β) : filter (α × β) := +f.comap prod.fst ⊔ g.comap prod.snd + +lemma mem_coprod_iff {s : set (α×β)} {f : filter α} {g : filter β} : + s ∈ f.coprod g ↔ ((∃ t₁ ∈ f, prod.fst ⁻¹' t₁ ⊆ s) ∧ (∃ t₂ ∈ g, prod.snd ⁻¹' t₂ ⊆ s)) := +by simp [filter.coprod] + +@[simp] lemma bot_coprod (l : filter β) : (⊥ : filter α).coprod l = comap prod.snd l := +by simp [filter.coprod] + +@[simp] lemma coprod_bot (l : filter α) : l.coprod (⊥ : filter β) = comap prod.fst l := +by simp [filter.coprod] + +lemma bot_coprod_bot : (⊥ : filter α).coprod (⊥ : filter β) = ⊥ := by simp + +lemma compl_mem_coprod {s : set (α × β)} {la : filter α} {lb : filter β} : + sᶜ ∈ la.coprod lb ↔ (prod.fst '' s)ᶜ ∈ la ∧ (prod.snd '' s)ᶜ ∈ lb := +by simp only [filter.coprod, mem_sup, compl_mem_comap] + +@[mono] lemma coprod_mono {f₁ f₂ : filter α} {g₁ g₂ : filter β} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) : + f₁.coprod g₁ ≤ f₂.coprod g₂ := +sup_le_sup (comap_mono hf) (comap_mono hg) + +lemma coprod_ne_bot_iff : (f.coprod g).ne_bot ↔ f.ne_bot ∧ nonempty β ∨ nonempty α ∧ g.ne_bot := +by simp [filter.coprod] + +@[instance] lemma coprod_ne_bot_left [ne_bot f] [nonempty β] : (f.coprod g).ne_bot := +coprod_ne_bot_iff.2 (or.inl ⟨‹_›, ‹_›⟩) + +@[instance] lemma coprod_ne_bot_right [ne_bot g] [nonempty α] : (f.coprod g).ne_bot := +coprod_ne_bot_iff.2 (or.inr ⟨‹_›, ‹_›⟩) + +lemma principal_coprod_principal (s : set α) (t : set β) : + (𝓟 s).coprod (𝓟 t) = 𝓟 (sᶜ ×ˢ tᶜ)ᶜ := +by rw [filter.coprod, comap_principal, comap_principal, sup_principal, set.prod_eq, compl_inter, + preimage_compl, preimage_compl, compl_compl, compl_compl] + +-- this inequality can be strict; see `map_const_principal_coprod_map_id_principal` and +-- `map_prod_map_const_id_principal_coprod_principal` below. +lemma {u v w x} map_prod_map_coprod_le {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x} + {f₁ : filter α₁} {f₂ : filter α₂} {m₁ : α₁ → β₁} {m₂ : α₂ → β₂} : + map (prod.map m₁ m₂) (f₁.coprod f₂) ≤ (map m₁ f₁).coprod (map m₂ f₂) := +begin + intros s, + simp only [mem_map, mem_coprod_iff], + rintro ⟨⟨u₁, hu₁, h₁⟩, u₂, hu₂, h₂⟩, + refine ⟨⟨m₁ ⁻¹' u₁, hu₁, λ _ hx, h₁ _⟩, ⟨m₂ ⁻¹' u₂, hu₂, λ _ hx, h₂ _⟩⟩; convert hx +end + +/-- Characterization of the coproduct of the `filter.map`s of two principal filters `𝓟 {a}` and +`𝓟 {i}`, the first under the constant function `λ a, b` and the second under the identity function. +Together with the next lemma, `map_prod_map_const_id_principal_coprod_principal`, this provides an +example showing that the inequality in the lemma `map_prod_map_coprod_le` can be strict. -/ +lemma map_const_principal_coprod_map_id_principal {α β ι : Type*} (a : α) (b : β) (i : ι) : + (map (λ _ : α, b) (𝓟 {a})).coprod (map id (𝓟 {i})) + = 𝓟 (({b} : set β) ×ˢ univ ∪ univ ×ˢ ({i} : set ι)) := +by simp only [map_principal, filter.coprod, comap_principal, sup_principal, image_singleton, + image_id, prod_univ, univ_prod] + +/-- Characterization of the `filter.map` of the coproduct of two principal filters `𝓟 {a}` and +`𝓟 {i}`, under the `prod.map` of two functions, respectively the constant function `λ a, b` and the +identity function. Together with the previous lemma, +`map_const_principal_coprod_map_id_principal`, this provides an example showing that the inequality +in the lemma `map_prod_map_coprod_le` can be strict. -/ +lemma map_prod_map_const_id_principal_coprod_principal {α β ι : Type*} (a : α) (b : β) (i : ι) : + map (prod.map (λ _ : α, b) id) ((𝓟 {a}).coprod (𝓟 {i})) + = 𝓟 (({b} : set β) ×ˢ (univ : set ι)) := +begin + rw [principal_coprod_principal, map_principal], + congr, + ext ⟨b', i'⟩, + split, + { rintro ⟨⟨a'', i''⟩, h₁, h₂, h₃⟩, + simp }, + { rintro ⟨h₁, h₂⟩, + use (a, i'), + simpa using h₁.symm } +end + +lemma tendsto.prod_map_coprod {δ : Type*} {f : α → γ} {g : β → δ} {a : filter α} {b : filter β} + {c : filter γ} {d : filter δ} (hf : tendsto f a c) (hg : tendsto g b d) : + tendsto (prod.map f g) (a.coprod b) (c.coprod d) := +map_prod_map_coprod_le.trans (coprod_mono hf hg) + +end coprod + +end filter diff --git a/formal/lean/mathlib/order/galois_connection.lean b/formal/lean/mathlib/order/galois_connection.lean index c52a6a1d7ae7353a131fabadc733a8abecef6ffd..e38dd6af59b1f1c8b4929f81da0172cd7e79c576 100644 --- a/formal/lean/mathlib/order/galois_connection.lean +++ b/formal/lean/mathlib/order/galois_connection.lean @@ -389,7 +389,7 @@ end nat /-- A Galois insertion is a Galois connection where `l ∘ u = id`. It also contains a constructive choice function, to give better definitional equalities when lifting order structures. Dual to `galois_coinsertion` -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure galois_insertion {α β : Type*} [preorder α] [preorder β] (l : α → β) (u : β → α) := (choice : Πx : α, u (l x) ≤ x → β) (gc : galois_connection l u) @@ -582,7 +582,7 @@ end galois_insertion /-- A Galois coinsertion is a Galois connection where `u ∘ l = id`. It also contains a constructive choice function, to give better definitional equalities when lifting order structures. Dual to `galois_insertion` -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure galois_coinsertion [preorder α] [preorder β] (l : α → β) (u : β → α) := (choice : Πx : β, x ≤ l (u x) → α) (gc : galois_connection l u) diff --git a/formal/lean/mathlib/order/initial_seg.lean b/formal/lean/mathlib/order/initial_seg.lean index 405b90051df034ba9382d2b18ce4e7d3ce3c9e2d..20c1eca7531c71b25389f134a59cc7f18d3f6b85 100644 --- a/formal/lean/mathlib/order/initial_seg.lean +++ b/formal/lean/mathlib/order/initial_seg.lean @@ -169,10 +169,10 @@ segments. /-- If `r` is a relation on `α` and `s` in a relation on `β`, then `f : r ≺i s` is an order embedding whose range is an open interval `(-∞, top)` for some element `top` of `β`. Such order embeddings are called principal segments -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure principal_seg {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends r ↪r s := (top : β) -(down : ∀ b, s b top ↔ ∃ a, to_rel_embedding a = b) +(down' : ∀ b, s b top ↔ ∃ a, to_rel_embedding a = b) localized "infix ` ≺i `:25 := principal_seg" in initial_seg @@ -188,14 +188,12 @@ instance : has_coe_to_fun (r ≺i s) (λ _, α → β) := ⟨λ f, f⟩ @[simp] theorem coe_coe_fn (f : r ≺i s) : ((f : r ↪r s) : α → β) = f := rfl -theorem down' (f : r ≺i s) {b : β} : s b f.top ↔ ∃ a, f a = b := -f.down _ +theorem down (f : r ≺i s) : ∀ {b : β}, s b f.top ↔ ∃ a, f a = b := f.down' -theorem lt_top (f : r ≺i s) (a : α) : s (f a) f.top := -f.down'.2 ⟨_, rfl⟩ +theorem lt_top (f : r ≺i s) (a : α) : s (f a) f.top := f.down.2 ⟨_, rfl⟩ theorem init [is_trans β s] (f : r ≺i s) {a : α} {b : β} (h : s b (f a)) : ∃ a', f a' = b := -f.down'.1 $ trans h $ f.lt_top _ +f.down.1 $ trans h $ f.lt_top _ /-- A principal segment is in particular an initial segment. -/ instance has_coe_initial_seg [is_trans β s] : has_coe (r ≺i s) (r ≼i s) := @@ -207,7 +205,7 @@ theorem init_iff [is_trans β s] (f : r ≺i s) {a : α} {b : β} : s b (f a) ↔ ∃ a', f a' = b ∧ r a' a := @initial_seg.init_iff α β r s f a b -theorem irrefl (r : α → α → Prop) [is_well_order α r] (f : r ≺i r) : false := +theorem irrefl {r : α → α → Prop} [is_well_order α r] (f : r ≺i r) : false := begin have := f.lt_top f.top, rw [show f f.top = f.top, from @@ -215,6 +213,8 @@ begin exact irrefl _ this end +instance (r : α → α → Prop) [is_well_order α r] : is_empty (r ≺i r) := ⟨λ f, f.irrefl⟩ + /-- Composition of a principal segment with an initial segment, as a principal segment -/ def lt_le (f : r ≺i s) (g : s ≼i t) : r ≺i t := ⟨@rel_embedding.trans _ _ _ r s t f g, g f.top, λ a, @@ -296,7 +296,7 @@ def of_element {α : Type*} (r : α → α → Prop) (a : α) : subrel r {b | r def cod_restrict (p : set β) (f : r ≺i s) (H : ∀ a, f a ∈ p) (H₂ : f.top ∈ p) : r ≺i subrel s p := ⟨rel_embedding.cod_restrict p f H, ⟨f.top, H₂⟩, λ ⟨b, h⟩, - f.down'.trans $ exists_congr $ λ a, + f.down.trans $ exists_congr $ λ a, show (⟨f a, H a⟩ : p).1 = _ ↔ _, from ⟨subtype.eq, congr_arg _⟩⟩ @[simp] @@ -308,7 +308,7 @@ theorem cod_restrict_top (p) (f : r ≺i s) (H H₂) : (cod_restrict p f H H₂) /-- Principal segment from an empty type into a type with a minimal element. -/ def of_is_empty (r : α → α → Prop) [is_empty α] {b : β} (H : ∀ b', ¬ s b' b) : r ≺i s := { top := b, - down := by simp [H], + down' := by simp [H], ..rel_embedding.of_is_empty r s } @[simp] theorem of_is_empty_top (r : α → α → Prop) [is_empty α] {b : β} (H : ∀ b', ¬ s b' b) : diff --git a/formal/lean/mathlib/order/locally_finite.lean b/formal/lean/mathlib/order/locally_finite.lean index 2cdd96d39832b3e413e29eeda932a4e979aed018..a7d3bcd28f630a03a77fd858191e38d81170c01b 100644 --- a/formal/lean/mathlib/order/locally_finite.lean +++ b/formal/lean/mathlib/order/locally_finite.lean @@ -660,20 +660,20 @@ instance [locally_finite_order α] [locally_finite_order β] [decidable_rel ((≤) : α × β → α × β → Prop)] : locally_finite_order (α × β) := locally_finite_order.of_Icc' (α × β) - (λ a b, (Icc a.fst b.fst).product (Icc a.snd b.snd)) + (λ a b, Icc a.fst b.fst ×ˢ Icc a.snd b.snd) (λ a b x, by { rw [mem_product, mem_Icc, mem_Icc, and_and_and_comm], refl }) instance [locally_finite_order_top α] [locally_finite_order_top β] [decidable_rel ((≤) : α × β → α × β → Prop)] : locally_finite_order_top (α × β) := locally_finite_order_top.of_Ici' (α × β) - (λ a, (Ici a.fst).product (Ici a.snd)) (λ a x, by { rw [mem_product, mem_Ici, mem_Ici], refl }) + (λ a, Ici a.fst ×ˢ Ici a.snd) (λ a x, by { rw [mem_product, mem_Ici, mem_Ici], refl }) instance [locally_finite_order_bot α] [locally_finite_order_bot β] [decidable_rel ((≤) : α × β → α × β → Prop)] : locally_finite_order_bot (α × β) := locally_finite_order_bot.of_Iic' (α × β) - (λ a, (Iic a.fst).product (Iic a.snd)) (λ a x, by { rw [mem_product, mem_Iic, mem_Iic], refl }) + (λ a, Iic a.fst ×ˢ Iic a.snd) (λ a x, by { rw [mem_product, mem_Iic, mem_Iic], refl }) end preorder diff --git a/formal/lean/mathlib/order/max.lean b/formal/lean/mathlib/order/max.lean index 277283b83c401891823ae5ca269d8da6d759167d..465f1fd7e27811c57b342b6f864fec65e661b8ff 100644 --- a/formal/lean/mathlib/order/max.lean +++ b/formal/lean/mathlib/order/max.lean @@ -79,6 +79,12 @@ instance no_min_order.to_no_bot_order (α : Type*) [preorder α] [no_min_order instance no_max_order.to_no_top_order (α : Type*) [preorder α] [no_max_order α] : no_top_order α := ⟨λ a, (exists_gt a).imp $ λ _, not_le_of_lt⟩ +theorem no_min_order.not_acc [has_lt α] [no_min_order α] (a : α) : ¬ acc (<) a := +λ h, acc.rec_on h $ λ x _, (exists_lt x).rec_on + +theorem no_max_order.not_acc [has_lt α] [no_max_order α] (a : α) : ¬ acc (>) a := +λ h, acc.rec_on h $ λ x _, (exists_gt x).rec_on + section has_le variables [has_le α] {a b : α} diff --git a/formal/lean/mathlib/order/partition/finpartition.lean b/formal/lean/mathlib/order/partition/finpartition.lean index af8bf00615867b99d254dfeccb244f65c52a2d27..3d351d98f15ba76ca62e7842940ee87d33489ada 100644 --- a/formal/lean/mathlib/order/partition/finpartition.lean +++ b/formal/lean/mathlib/order/partition/finpartition.lean @@ -229,7 +229,7 @@ section inf variables [decidable_eq α] {a b c : α} instance : has_inf (finpartition a) := -⟨λ P Q, of_erase ((P.parts.product Q.parts).image $ λ bc, bc.1 ⊓ bc.2) +⟨λ P Q, of_erase ((P.parts ×ˢ Q.parts).image $ λ bc, bc.1 ⊓ bc.2) begin rw sup_indep_iff_disjoint_erase, simp only [mem_image, and_imp, exists_prop, forall_exists_index, id.def, prod.exists, @@ -250,7 +250,7 @@ instance : has_inf (finpartition a) := end⟩ @[simp] lemma parts_inf (P Q : finpartition a) : - (P ⊓ Q).parts = ((P.parts.product Q.parts).image $ λ bc : α × α, bc.1 ⊓ bc.2).erase ⊥ := rfl + (P ⊓ Q).parts = ((P.parts ×ˢ Q.parts).image $ λ bc : α × α, bc.1 ⊓ bc.2).erase ⊥ := rfl instance : semilattice_inf (finpartition a) := { inf_le_left := λ P Q b hb, begin diff --git a/formal/lean/mathlib/order/prime_ideal.lean b/formal/lean/mathlib/order/prime_ideal.lean index 09e5457b2ea8054d71bb992b2b50e30f0f807119..d113dabce9f8efc3e88dd9b92c432f2b729c30b1 100644 --- a/formal/lean/mathlib/order/prime_ideal.lean +++ b/formal/lean/mathlib/order/prime_ideal.lean @@ -40,7 +40,7 @@ namespace ideal /-- A pair of an `ideal` and a `pfilter` which form a partition of `P`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure prime_pair (P : Type*) [preorder P] := (I : ideal P) (F : pfilter P) diff --git a/formal/lean/mathlib/order/rel_iso.lean b/formal/lean/mathlib/order/rel_iso.lean index 60c811dfbeef126ad3ce10d025adde97802f03b3..17eb6b62f5a5d40e9b2e8c88e1f341a7fd2e1ce0 100644 --- a/formal/lean/mathlib/order/rel_iso.lean +++ b/formal/lean/mathlib/order/rel_iso.lean @@ -36,11 +36,12 @@ isomorphisms. open function universes u v w -variables {α β γ : Type*} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} +variables {α β γ δ : Type*} + {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} {u : δ → δ → Prop} /-- A relation homomorphism with respect to a given pair of relations `r` and `s` is a function `f : α → β` such that `r a b → s (f a) (f b)`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure rel_hom {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) := (to_fun : α → β) (map_rel' : ∀ {a b}, r a b → s (to_fun a) (to_fun b)) @@ -212,6 +213,8 @@ initialize_simps_projections rel_embedding (to_embedding_to_fun → apply, -to_e theorem injective (f : r ↪r s) : injective f := f.inj' +@[simp] theorem inj (f : r ↪r s) {a b} : f a = f b ↔ a = b := f.injective.eq_iff + theorem map_rel_iff (f : r ↪r s) : ∀ {a b}, s (f a) (f b) ↔ r a b := f.map_rel_iff' @[simp] theorem coe_fn_mk (f : α ↪ β) (o) : @@ -304,6 +307,29 @@ protected theorem well_founded : ∀ (f : r ↪r s) (h : well_founded s), well_f protected theorem is_well_order : ∀ (f : r ↪r s) [is_well_order β s], is_well_order α r | f H := by exactI {wf := f.well_founded H.wf, ..f.is_strict_total_order'} +/-- `quotient.out` as a relation embedding between the lift of a relation and the relation. -/ +@[simps] noncomputable def _root_.quotient.out_rel_embedding [s : setoid α] {r : α → α → Prop} (H) : + quotient.lift₂ r H ↪r r := +⟨embedding.quotient_out α, begin + refine λ x y, quotient.induction_on₂ x y (λ a b, _), + apply iff_iff_eq.2 (H _ _ _ _ _ _); + apply quotient.mk_out +end⟩ + +/-- A relation is well founded iff its lift to a quotient is. -/ +@[simp] theorem _root_.well_founded_lift₂_iff [s : setoid α] {r : α → α → Prop} {H} : + well_founded (quotient.lift₂ r H) ↔ well_founded r := +⟨λ hr, begin + suffices : ∀ {x : quotient s} {a : α}, ⟦a⟧ = x → acc r a, + { exact ⟨λ a, this rfl⟩ }, + { refine λ x, hr.induction x _, + rintros x IH a rfl, + exact ⟨_, λ b hb, IH ⟦b⟧ hb rfl⟩ } +end, (quotient.out_rel_embedding H).well_founded⟩ + +alias _root_.well_founded_lift₂_iff ↔ + _root_.well_founded.of_quotient_lift₂ _root_.well_founded.quotient_lift₂ + /-- To define an relation embedding from an antisymmetric relation `r` to a reflexive relation `s` it suffices to give a function together with a proof that it satisfies `s (f a) (f b) ↔ r a b`. @@ -341,6 +367,60 @@ end def of_is_empty (r : α → α → Prop) (s : β → β → Prop) [is_empty α] : r ↪r s := ⟨embedding.of_is_empty, is_empty_elim⟩ +/-- `sum.inl` as a relation embedding into `sum.lift_rel r s`. -/ +@[simps] def sum_lift_rel_inl (r : α → α → Prop) (s : β → β → Prop) : r ↪r sum.lift_rel r s := +{ to_fun := sum.inl, + inj' := sum.inl_injective, + map_rel_iff' := λ a b, sum.lift_rel_inl_inl } + +/-- `sum.inr` as a relation embedding into `sum.lift_rel r s`. -/ +@[simps] def sum_lift_rel_inr (r : α → α → Prop) (s : β → β → Prop) : s ↪r sum.lift_rel r s := +{ to_fun := sum.inr, + inj' := sum.inr_injective, + map_rel_iff' := λ a b, sum.lift_rel_inr_inr } + +/-- `sum.map` as a relation embedding between `sum.lift_rel` relations. -/ +@[simps] def sum_lift_rel_map (f : r ↪r s) (g : t ↪r u) : sum.lift_rel r t ↪r sum.lift_rel s u := +{ to_fun := sum.map f g, + inj' := f.injective.sum_map g.injective, + map_rel_iff' := by { rintro (a | b) (c | d); simp [f.map_rel_iff, g.map_rel_iff] } } + +/-- `sum.inl` as a relation embedding into `sum.lex r s`. -/ +@[simps] def sum_lex_inl (r : α → α → Prop) (s : β → β → Prop) : r ↪r sum.lex r s := +{ to_fun := sum.inl, + inj' := sum.inl_injective, + map_rel_iff' := λ a b, sum.lex_inl_inl } + +/-- `sum.inr` as a relation embedding into `sum.lex r s`. -/ +@[simps] def sum_lex_inr (r : α → α → Prop) (s : β → β → Prop) : s ↪r sum.lex r s := +{ to_fun := sum.inr, + inj' := sum.inr_injective, + map_rel_iff' := λ a b, sum.lex_inr_inr } + +/-- `sum.map` as a relation embedding between `sum.lex` relations. -/ +@[simps] def sum_lex_map (f : r ↪r s) (g : t ↪r u) : sum.lex r t ↪r sum.lex s u := +{ to_fun := sum.map f g, + inj' := f.injective.sum_map g.injective, + map_rel_iff' := by { rintro (a | b) (c | d); simp [f.map_rel_iff, g.map_rel_iff] } } + +/-- `λ b, prod.mk a b` as a relation embedding. -/ +@[simps] def prod_lex_mk_left (s : β → β → Prop) {a : α} (h : ¬ r a a) : s ↪r prod.lex r s := +{ to_fun := prod.mk a, + inj' := prod.mk.inj_left a, + map_rel_iff' := λ b₁ b₂, by simp [prod.lex_def, h] } + +/-- `λ a, prod.mk a b` as a relation embedding. -/ +@[simps] def prod_lex_mk_right (r : α → α → Prop) {b : β} (h : ¬ s b b) : r ↪r prod.lex r s := +{ to_fun := λ a, (a, b), + inj' := prod.mk.inj_right b, + map_rel_iff' := λ a₁ a₂, by simp [prod.lex_def, h] } + +/-- `prod.map` as a relation embedding. -/ +@[simps] def prod_lex_map (f : r ↪r s) (g : t ↪r u) : prod.lex r t ↪r prod.lex s u := +{ to_fun := prod.map f g, + inj' := f.injective.prod_map g.injective, + map_rel_iff' := λ a b, by simp [prod.lex_def, f.map_rel_iff, g.map_rel_iff] } + end rel_embedding /-- A relation isomorphism is an equivalence that is also a relation embedding. -/ diff --git a/formal/lean/mathlib/order/succ_pred/basic.lean b/formal/lean/mathlib/order/succ_pred/basic.lean index c3a1904915278b5ac13e9915695284c619c50891..5f731b900659682928c895ca1826389f44dbeeaa 100644 --- a/formal/lean/mathlib/order/succ_pred/basic.lean +++ b/formal/lean/mathlib/order/succ_pred/basic.lean @@ -362,7 +362,15 @@ lt_succ_iff_not_is_max.trans not_is_max_iff_ne_top end order_top section order_bot -variables [order_bot α] [nontrivial α] +variable [order_bot α] + +@[simp] lemma lt_succ_bot_iff [no_max_order α] : a < succ ⊥ ↔ a = ⊥ := +by rw [lt_succ_iff, le_bot_iff] + +lemma le_succ_bot_iff : a ≤ succ ⊥ ↔ a = ⊥ ∨ a = succ ⊥ := +by rw [le_succ_iff_eq_or_le, le_bot_iff, or_comm] + +variable [nontrivial α] lemma bot_lt_succ (a : α) : ⊥ < succ a := (lt_succ_of_not_is_max not_is_max_bot).trans_le $ succ_mono bot_le @@ -575,7 +583,15 @@ variables [order_bot α] end order_bot section order_top -variables [order_top α] [nontrivial α] + +variable [order_top α] + +@[simp] lemma pred_top_lt_iff [no_min_order α] : pred ⊤ < a ↔ a = ⊤ := +@lt_succ_bot_iff αᵒᵈ _ _ _ _ _ + +lemma pred_top_le_iff : pred ⊤ ≤ a ↔ a = ⊤ ∨ a = pred ⊤ := @le_succ_bot_iff αᵒᵈ _ _ _ _ + +variable [nontrivial α] lemma pred_lt_top (a : α) : pred a < ⊤ := (pred_mono le_top).trans_lt $ pred_lt_of_not_is_min not_is_min_top diff --git a/formal/lean/mathlib/order/well_founded.lean b/formal/lean/mathlib/order/well_founded.lean index 6ebd7fa9314f70e4c0fd6148377cade5a4af7180..2ad3a0b402a0d3566cf4578cd19cebd6116289b3 100644 --- a/formal/lean/mathlib/order/well_founded.lean +++ b/formal/lean/mathlib/order/well_founded.lean @@ -46,32 +46,26 @@ If you're working with a nonempty linear order, consider defining a `conditionally_complete_linear_order_bot` instance via `well_founded.conditionally_complete_linear_order_with_bot` and using `Inf` instead. -/ noncomputable def min {r : α → α → Prop} (H : well_founded r) - (p : set α) (h : p.nonempty) : α := -classical.some (H.has_min p h) + (s : set α) (h : s.nonempty) : α := +classical.some (H.has_min s h) theorem min_mem {r : α → α → Prop} (H : well_founded r) - (p : set α) (h : p.nonempty) : H.min p h ∈ p := -let ⟨h, _⟩ := classical.some_spec (H.has_min p h) in h + (s : set α) (h : s.nonempty) : H.min s h ∈ s := +let ⟨h, _⟩ := classical.some_spec (H.has_min s h) in h theorem not_lt_min {r : α → α → Prop} (H : well_founded r) - (p : set α) (h : p.nonempty) {x} (xp : x ∈ p) : ¬ r x (H.min p h) := -let ⟨_, h'⟩ := classical.some_spec (H.has_min p h) in h' _ xp + (s : set α) (h : s.nonempty) {x} (hx : x ∈ s) : ¬ r x (H.min s h) := +let ⟨_, h'⟩ := classical.some_spec (H.has_min s h) in h' _ hx theorem well_founded_iff_has_min {r : α → α → Prop} : (well_founded r) ↔ - ∀ (p : set α), p.nonempty → ∃ m ∈ p, ∀ x ∈ p, ¬ r x m := + ∀ (s : set α), s.nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬ r x m := begin - classical, - split, - { exact has_min, }, - { set counterexamples := { x : α | ¬ acc r x}, - intro exists_max, - fconstructor, - intro x, - by_contra hx, - obtain ⟨m, m_mem, hm⟩ := exists_max counterexamples ⟨x, hx⟩, - refine m_mem (acc.intro _ ( λ y y_gt_m, _)), - by_contra hy, - exact hm y hy y_gt_m, }, + refine ⟨λ h, h.has_min, λ h, ⟨λ x, _⟩⟩, + by_contra hx, + obtain ⟨m, hm, hm'⟩ := h _ ⟨x, hx⟩, + refine hm ⟨_, λ y hy, _⟩, + by_contra hy', + exact hm' y hy' hy end lemma eq_iff_not_lt_of_le {α} [partial_order α] {x y : α} : x ≤ y → y = x ↔ ¬ x < y := @@ -163,8 +157,8 @@ theorem eq_strict_mono_iff_eq_range {f g : β → γ} (hf : strict_mono f) (eq_strict_mono_iff_eq_range_aux hg hf hfg.symm (λ a hab, (H a hab).symm)) end, congr_arg _⟩ -theorem self_le_of_strict_mono {φ : β → β} (hφ : strict_mono φ) : ∀ n, n ≤ φ n := -by { by_contra' h₁, have h₂ := h.min_mem _ h₁, exact h.not_lt_min _ h₁ (hφ h₂) h₂ } +theorem self_le_of_strict_mono {f : β → β} (hf : strict_mono f) : ∀ n, n ≤ f n := +by { by_contra' h₁, have h₂ := h.min_mem _ h₁, exact h.not_lt_min _ h₁ (hf h₂) h₂ } end linear_order diff --git a/formal/lean/mathlib/order/well_founded_set.lean b/formal/lean/mathlib/order/well_founded_set.lean index eb1ef542aca4d6e7594ac9f7ff2a1785ad9a4d65..ccf5bec5dbd20acd6763d98debf6ee272b516dd3 100644 --- a/formal/lean/mathlib/order/well_founded_set.lean +++ b/formal/lean/mathlib/order/well_founded_set.lean @@ -337,7 +337,7 @@ partially_well_ordered_on_iff_exists_monotone_subseq lemma is_pwo.prod (hs : s.is_pwo) (ht : t.is_pwo) : - (s ×ˢ t : set _).is_pwo := + (s ×ˢ t).is_pwo := begin classical, rw is_pwo_iff_exists_monotone_subseq at *, diff --git a/formal/lean/mathlib/probability/cond_count.lean b/formal/lean/mathlib/probability/cond_count.lean index 021fb06fdc57273da6d1190da0fd8fe9fa4f9795..97ec5c4acea2ba728820ed4cbf54a4c09b52d08c 100644 --- a/formal/lean/mathlib/probability/cond_count.lean +++ b/formal/lean/mathlib/probability/cond_count.lean @@ -40,31 +40,31 @@ open measure_theory measurable_space namespace probability_theory -variables {α : Type*} [measurable_space α] +variables {Ω : Type*} [measurable_space Ω] /-- Given a set `s`, `cond_count s` is the counting measure conditioned on `s`. In particular, `cond_count s t` is the proportion of `s` that is contained in `t`. This is a probability measure when `s` is finite and nonempty and is given by `probability_theory.cond_count_is_probability_measure`. -/ -def cond_count (s : set α) : measure α := measure.count[|s] +def cond_count (s : set Ω) : measure Ω := measure.count[|s] -@[simp] lemma cond_count_empty_meas : (cond_count ∅ : measure α) = 0 := +@[simp] lemma cond_count_empty_meas : (cond_count ∅ : measure Ω) = 0 := by simp [cond_count] -lemma cond_count_empty {s : set α} : cond_count s ∅ = 0 := +lemma cond_count_empty {s : set Ω} : cond_count s ∅ = 0 := by simp -lemma finite_of_cond_count_ne_zero {s t : set α} (h : cond_count s t ≠ 0) : +lemma finite_of_cond_count_ne_zero {s t : set Ω} (h : cond_count s t ≠ 0) : s.finite := begin by_contra hs', simpa [cond_count, cond, measure.count_apply_infinite hs'] using h, end -variables [measurable_singleton_class α] +variables [measurable_singleton_class Ω] -lemma cond_count_is_probability_measure {s : set α} (hs : s.finite) (hs' : s.nonempty) : +lemma cond_count_is_probability_measure {s : set Ω} (hs : s.finite) (hs' : s.nonempty) : is_probability_measure (cond_count s) := { measure_univ := begin @@ -73,17 +73,17 @@ lemma cond_count_is_probability_measure {s : set α} (hs : s.finite) (hs' : s.no { exact (measure.count_apply_lt_top.2 hs).ne } end } -lemma cond_count_singleton (a : α) (t : set α) [decidable (a ∈ t)] : - cond_count {a} t = if a ∈ t then 1 else 0 := +lemma cond_count_singleton (ω : Ω) (t : set Ω) [decidable (ω ∈ t)] : + cond_count {ω} t = if ω ∈ t then 1 else 0 := begin - rw [cond_count, cond_apply _ (measurable_set_singleton a), measure.count_singleton, + rw [cond_count, cond_apply _ (measurable_set_singleton ω), measure.count_singleton, ennreal.inv_one, one_mul], split_ifs, - { rw [(by simpa : ({a} : set α) ∩ t = {a}), measure.count_singleton] }, - { rw [(by simpa : ({a} : set α) ∩ t = ∅), measure.count_empty] }, + { rw [(by simpa : ({ω} : set Ω) ∩ t = {ω}), measure.count_singleton] }, + { rw [(by simpa : ({ω} : set Ω) ∩ t = ∅), measure.count_empty] }, end -variables {s t u : set α} +variables {s t u : set Ω} lemma cond_count_inter_self (hs : s.finite): cond_count s (s ∩ t) = cond_count s t := @@ -159,7 +159,7 @@ begin exacts [htu.mono inf_le_right inf_le_right, (hs.inter_of_left _).measurable_set], end -lemma cond_count_compl (t : set α) (hs : s.finite) (hs' : s.nonempty) : +lemma cond_count_compl (t : set Ω) (hs : s.finite) (hs' : s.nonempty) : cond_count s t + cond_count s tᶜ = 1 := begin rw [← cond_count_union hs disjoint_compl_right, set.union_compl_self, @@ -190,7 +190,7 @@ begin end /-- A version of the law of total probability for counting probabilites. -/ -lemma cond_count_add_compl_eq (u t : set α) (hs : s.finite) : +lemma cond_count_add_compl_eq (u t : set Ω) (hs : s.finite) : cond_count (s ∩ u) t * cond_count s u + cond_count (s ∩ uᶜ) t * cond_count s uᶜ = cond_count s t := begin diff --git a/formal/lean/mathlib/probability/conditional_expectation.lean b/formal/lean/mathlib/probability/conditional_expectation.lean index 77acd08812dae9c24b487c16eea6b766aa557156..ec903d56ad21d8bf1d7789a71ddf2c4cadb4abd5 100644 --- a/formal/lean/mathlib/probability/conditional_expectation.lean +++ b/formal/lean/mathlib/probability/conditional_expectation.lean @@ -27,8 +27,8 @@ namespace measure_theory open probability_theory -variables {α E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] - {m₁ m₂ m : measurable_space α} {μ : measure α} {f : α → E} +variables {Ω E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] + {m₁ m₂ m : measurable_space Ω} {μ : measure Ω} {f : Ω → E} /-- If `m₁, m₂` are independent σ-algebras and `f` is `m₁`-measurable, then `𝔼[f | m₂] = 𝔼[f]` almost everywhere. -/ diff --git a/formal/lean/mathlib/probability/conditional_probability.lean b/formal/lean/mathlib/probability/conditional_probability.lean index 278b0bfdd21bdc6bf3c20154544a14cee63bcf98..dfa312ffd476d68b509971026892aa017ed0788f 100644 --- a/formal/lean/mathlib/probability/conditional_probability.lean +++ b/formal/lean/mathlib/probability/conditional_probability.lean @@ -10,13 +10,13 @@ import probability.independence This file defines conditional probability and includes basic results relating to it. -Given some measure `μ` defined on a measure space on some type `α` and some `s : set α`, +Given some measure `μ` defined on a measure space on some type `Ω` and some `s : set Ω`, we define the measure of `μ` conditioned on `s` as the restricted measure scaled by the inverse of the measure of `s`: `cond μ s = (μ s)⁻¹ • μ.restrict s`. The scaling ensures that this is a probability measure (when `μ` is a finite measure). From this definition, we derive the "axiomatic" definition of conditional probability -based on application: for any `s t : set α`, we have `μ[t|s] = (μ s)⁻¹ * μ (s ∩ t)`. +based on application: for any `s t : set Ω`, we have `μ[t|s] = (μ s)⁻¹ * μ (s ∩ t)`. ## Main Statements @@ -59,7 +59,7 @@ open_locale ennreal open measure_theory measurable_space -variables {α : Type*} {m : measurable_space α} (μ : measure α) {s t : set α} +variables {Ω : Type*} {m : measurable_space Ω} (μ : measure Ω) {s t : set Ω} namespace probability_theory @@ -68,7 +68,7 @@ section definitions /-- The conditional probability measure of measure `μ` on set `s` is `μ` restricted to `s` and scaled by the inverse of `μ s` (to make it a probability measure): `(μ s)⁻¹ • μ.restrict s`. -/ -def cond (s : set α) : measure α := +def cond (s : set Ω) : measure Ω := (μ s)⁻¹ • μ.restrict s end definitions @@ -93,11 +93,11 @@ by simp [cond] by simp [cond, measure_univ, measure.restrict_univ] /-- The axiomatic definition of conditional probability derived from a measure-theoretic one. -/ -lemma cond_apply (hms : measurable_set s) (t : set α) : +lemma cond_apply (hms : measurable_set s) (t : set Ω) : μ[t|s] = (μ s)⁻¹ * μ (s ∩ t) := by { rw [cond, measure.smul_apply, measure.restrict_apply' hms, set.inter_comm], refl } -lemma cond_inter_self (hms : measurable_set s) (t : set α) : +lemma cond_inter_self (hms : measurable_set s) (t : set Ω) : μ[s ∩ t|s] = μ[t|s] := by rw [cond_apply _ hms, ← set.inter_assoc, set.inter_self, ← cond_apply _ hms] @@ -138,13 +138,13 @@ lemma cond_cond_eq_cond_inter [is_finite_measure μ] cond_cond_eq_cond_inter' μ hms hmt (measure_ne_top μ s) hci lemma cond_mul_eq_inter' - (hms : measurable_set s) (hcs : μ s ≠ 0) (hcs' : μ s ≠ ∞) (t : set α) : + (hms : measurable_set s) (hcs : μ s ≠ 0) (hcs' : μ s ≠ ∞) (t : set Ω) : μ[t|s] * μ s = μ (s ∩ t) := by rw [cond_apply μ hms t, mul_comm, ←mul_assoc, ennreal.mul_inv_cancel hcs hcs', one_mul] lemma cond_mul_eq_inter [is_finite_measure μ] - (hms : measurable_set s) (hcs : μ s ≠ 0) (t : set α) : + (hms : measurable_set s) (hcs : μ s ≠ 0) (t : set Ω) : μ[t|s] * μ s = μ (s ∩ t) := cond_mul_eq_inter' μ hms hcs (measure_ne_top _ s) t diff --git a/formal/lean/mathlib/probability/density.lean b/formal/lean/mathlib/probability/density.lean index 1ffefadfb1bad5d1806e978863c354e544d0c4a1..7dcbe2022356095f3c3136a661e4fd5ca122d1df 100644 --- a/formal/lean/mathlib/probability/density.lean +++ b/formal/lean/mathlib/probability/density.lean @@ -21,8 +21,8 @@ random variables with this distribution. ## Main definitions -* `measure_theory.has_pdf` : A random variable `X : α → E` is said to `has_pdf` with - respect to the measure `ℙ` on `α` and `μ` on `E` if there exists a measurable function `f` +* `measure_theory.has_pdf` : A random variable `X : Ω → E` is said to `has_pdf` with + respect to the measure `ℙ` on `Ω` and `μ` on `E` if there exists a measurable function `f` such that the push-forward measure of `ℙ` along `X` equals `μ.with_density f`. * `measure_theory.pdf` : If `X` is a random variable that `has_pdf X ℙ μ`, then `pdf X` is the measurable function `f` such that the push-forward measure of `ℙ` along `X` equals @@ -33,7 +33,7 @@ random variables with this distribution. ## Main results * `measure_theory.pdf.integral_fun_mul_eq_integral` : Law of the unconscious statistician, - i.e. if a random variable `X : α → E` has pdf `f`, then `𝔼(g(X)) = ∫ x, g x * f x dx` for + i.e. if a random variable `X : Ω → E` has pdf `f`, then `𝔼(g(X)) = ∫ x, g x * f x dx` for all measurable `g : E → ℝ`. * `measure_theory.pdf.integral_mul_eq_integral` : A real-valued random variable `X` with pdf `f` has expectation `∫ x, x * f x dx`. @@ -55,32 +55,32 @@ namespace measure_theory open topological_space measure_theory.measure -variables {α E : Type*} [measurable_space E] +variables {Ω E : Type*} [measurable_space E] -/-- A random variable `X : α → E` is said to `has_pdf` with respect to the measure `ℙ` on `α` and +/-- A random variable `X : Ω → E` is said to `has_pdf` with respect to the measure `ℙ` on `Ω` and `μ` on `E` if there exists a measurable function `f` such that the push-forward measure of `ℙ` along `X` equals `μ.with_density f`. -/ -class has_pdf {m : measurable_space α} (X : α → E) - (ℙ : measure α) (μ : measure E . volume_tac) : Prop := +class has_pdf {m : measurable_space Ω} (X : Ω → E) + (ℙ : measure Ω) (μ : measure E . volume_tac) : Prop := (pdf' : measurable X ∧ ∃ (f : E → ℝ≥0∞), measurable f ∧ map X ℙ = μ.with_density f) @[measurability] -lemma has_pdf.measurable {m : measurable_space α} - (X : α → E) (ℙ : measure α) (μ : measure E . volume_tac) [hX : has_pdf X ℙ μ] : +lemma has_pdf.measurable {m : measurable_space Ω} + (X : Ω → E) (ℙ : measure Ω) (μ : measure E . volume_tac) [hX : has_pdf X ℙ μ] : measurable X := hX.pdf'.1 /-- If `X` is a random variable that `has_pdf X ℙ μ`, then `pdf X` is the measurable function `f` such that the push-forward measure of `ℙ` along `X` equals `μ.with_density f`. -/ -def pdf {m : measurable_space α} (X : α → E) (ℙ : measure α) (μ : measure E . volume_tac) := +def pdf {m : measurable_space Ω} (X : Ω → E) (ℙ : measure Ω) (μ : measure E . volume_tac) := if hX : has_pdf X ℙ μ then classical.some hX.pdf'.2 else 0 -lemma pdf_undef {m : measurable_space α} {ℙ : measure α} {μ : measure E} {X : α → E} +lemma pdf_undef {m : measurable_space Ω} {ℙ : measure Ω} {μ : measure E} {X : Ω → E} (h : ¬ has_pdf X ℙ μ) : pdf X ℙ μ = 0 := by simp only [pdf, dif_neg h] -lemma has_pdf_of_pdf_ne_zero {m : measurable_space α} {ℙ : measure α} {μ : measure E} {X : α → E} +lemma has_pdf_of_pdf_ne_zero {m : measurable_space Ω} {ℙ : measure Ω} {μ : measure E} {X : Ω → E} (h : pdf X ℙ μ ≠ 0) : has_pdf X ℙ μ := begin by_contra hpdf, @@ -88,19 +88,19 @@ begin exact hpdf (false.rec (has_pdf X ℙ μ) (h rfl)) end -lemma pdf_eq_zero_of_not_measurable {m : measurable_space α} - {ℙ : measure α} {μ : measure E} {X : α → E} (hX : ¬ measurable X) : +lemma pdf_eq_zero_of_not_measurable {m : measurable_space Ω} + {ℙ : measure Ω} {μ : measure E} {X : Ω → E} (hX : ¬ measurable X) : pdf X ℙ μ = 0 := pdf_undef (λ hpdf, hX hpdf.pdf'.1) -lemma measurable_of_pdf_ne_zero {m : measurable_space α} - {ℙ : measure α} {μ : measure E} (X : α → E) (h : pdf X ℙ μ ≠ 0) : +lemma measurable_of_pdf_ne_zero {m : measurable_space Ω} + {ℙ : measure Ω} {μ : measure E} (X : Ω → E) (h : pdf X ℙ μ ≠ 0) : measurable X := by { by_contra hX, exact h (pdf_eq_zero_of_not_measurable hX) } @[measurability] -lemma measurable_pdf {m : measurable_space α} - (X : α → E) (ℙ : measure α) (μ : measure E . volume_tac) : +lemma measurable_pdf {m : measurable_space Ω} + (X : Ω → E) (ℙ : measure Ω) (μ : measure E . volume_tac) : measurable (pdf X ℙ μ) := begin by_cases hX : has_pdf X ℙ μ, @@ -110,32 +110,32 @@ begin exact measurable_zero } end -lemma map_eq_with_density_pdf {m : measurable_space α} - (X : α → E) (ℙ : measure α) (μ : measure E . volume_tac) [hX : has_pdf X ℙ μ] : +lemma map_eq_with_density_pdf {m : measurable_space Ω} + (X : Ω → E) (ℙ : measure Ω) (μ : measure E . volume_tac) [hX : has_pdf X ℙ μ] : measure.map X ℙ = μ.with_density (pdf X ℙ μ) := begin rw [pdf, dif_pos hX], exact (classical.some_spec hX.pdf'.2).2 end -lemma map_eq_set_lintegral_pdf {m : measurable_space α} - (X : α → E) (ℙ : measure α) (μ : measure E . volume_tac) [hX : has_pdf X ℙ μ] +lemma map_eq_set_lintegral_pdf {m : measurable_space Ω} + (X : Ω → E) (ℙ : measure Ω) (μ : measure E . volume_tac) [hX : has_pdf X ℙ μ] {s : set E} (hs : measurable_set s) : measure.map X ℙ s = ∫⁻ x in s, pdf X ℙ μ x ∂μ := by rw [← with_density_apply _ hs, map_eq_with_density_pdf X ℙ μ] namespace pdf -variables {m : measurable_space α} {ℙ : measure α} {μ : measure E} +variables {m : measurable_space Ω} {ℙ : measure Ω} {μ : measure E} -lemma lintegral_eq_measure_univ {X : α → E} [has_pdf X ℙ μ] : +lemma lintegral_eq_measure_univ {X : Ω → E} [has_pdf X ℙ μ] : ∫⁻ x, pdf X ℙ μ x ∂μ = ℙ set.univ := begin rw [← set_lintegral_univ, ← map_eq_set_lintegral_pdf X ℙ μ measurable_set.univ, measure.map_apply (has_pdf.measurable X ℙ μ) measurable_set.univ, set.preimage_univ], end -lemma ae_lt_top [is_finite_measure ℙ] {μ : measure E} {X : α → E} : +lemma ae_lt_top [is_finite_measure ℙ] {μ : measure E} {X : Ω → E} : ∀ᵐ x ∂μ, pdf X ℙ μ x < ∞ := begin by_cases hpdf : has_pdf X ℙ μ, @@ -147,11 +147,11 @@ begin exact filter.eventually_of_forall (λ x, with_top.zero_lt_top) } end -lemma of_real_to_real_ae_eq [is_finite_measure ℙ] {X : α → E} : +lemma of_real_to_real_ae_eq [is_finite_measure ℙ] {X : Ω → E} : (λ x, ennreal.of_real (pdf X ℙ μ x).to_real) =ᵐ[μ] pdf X ℙ μ := of_real_to_real_ae_eq ae_lt_top -lemma integrable_iff_integrable_mul_pdf [is_finite_measure ℙ] {X : α → E} [has_pdf X ℙ μ] +lemma integrable_iff_integrable_mul_pdf [is_finite_measure ℙ] {X : Ω → E} [has_pdf X ℙ μ] {f : E → ℝ} (hf : measurable f) : integrable (λ x, f (X x)) ℙ ↔ integrable (λ x, f x * (pdf X ℙ μ x).to_real) μ := begin @@ -165,7 +165,7 @@ end function `f`, `f ∘ X` is a random variable with expectation `∫ x, f x * pdf X ∂μ` where `μ` is a measure on the codomain of `X`. -/ lemma integral_fun_mul_eq_integral [is_finite_measure ℙ] - {X : α → E} [has_pdf X ℙ μ] {f : E → ℝ} (hf : measurable f) : + {X : Ω → E} [has_pdf X ℙ μ] {f : E → ℝ} (hf : measurable f) : ∫ x, f x * (pdf X ℙ μ x).to_real ∂μ = ∫ x, f (X x) ∂ℙ := begin by_cases hpdf : integrable (λ x, f x * (pdf X ℙ μ x).to_real) μ, @@ -206,21 +206,21 @@ begin all_goals { apply_instance } } end -lemma map_absolutely_continuous {X : α → E} [has_pdf X ℙ μ] : map X ℙ ≪ μ := +lemma map_absolutely_continuous {X : Ω → E} [has_pdf X ℙ μ] : map X ℙ ≪ μ := by { rw map_eq_with_density_pdf X ℙ μ, exact with_density_absolutely_continuous _ _, } /-- A random variable that `has_pdf` is quasi-measure preserving. -/ -lemma to_quasi_measure_preserving {X : α → E} [has_pdf X ℙ μ] : quasi_measure_preserving X ℙ μ := +lemma to_quasi_measure_preserving {X : Ω → E} [has_pdf X ℙ μ] : quasi_measure_preserving X ℙ μ := { measurable := has_pdf.measurable X ℙ μ, absolutely_continuous := map_absolutely_continuous, } -lemma have_lebesgue_decomposition_of_has_pdf {X : α → E} [hX' : has_pdf X ℙ μ] : +lemma have_lebesgue_decomposition_of_has_pdf {X : Ω → E} [hX' : has_pdf X ℙ μ] : (map X ℙ).have_lebesgue_decomposition μ := ⟨⟨⟨0, pdf X ℙ μ⟩, by simp only [zero_add, measurable_pdf X ℙ μ, true_and, mutually_singular.zero_left, map_eq_with_density_pdf X ℙ μ] ⟩⟩ -lemma has_pdf_iff {X : α → E} : +lemma has_pdf_iff {X : Ω → E} : has_pdf X ℙ μ ↔ measurable X ∧ (map X ℙ).have_lebesgue_decomposition μ ∧ map X ℙ ≪ μ := begin split, @@ -232,7 +232,7 @@ begin rwa with_density_rn_deriv_eq } end -lemma has_pdf_iff_of_measurable {X : α → E} (hX : measurable X) : +lemma has_pdf_iff_of_measurable {X : Ω → E} (hX : measurable X) : has_pdf X ℙ μ ↔ (map X ℙ).have_lebesgue_decomposition μ ∧ map X ℙ ≪ μ := by { rw has_pdf_iff, simp only [hX, true_and], } @@ -245,7 +245,7 @@ map also `has_pdf` if `(map g (map X ℙ)).have_lebesgue_decomposition μ`. `quasi_measure_preserving_has_pdf'` is more useful in the case we are working with a probability measure and a real-valued random variable. -/ -lemma quasi_measure_preserving_has_pdf {X : α → E} [has_pdf X ℙ μ] +lemma quasi_measure_preserving_has_pdf {X : Ω → E} [has_pdf X ℙ μ] {g : E → F} (hg : quasi_measure_preserving g μ ν) (hmap : (map g (map X ℙ)).have_lebesgue_decomposition ν) : has_pdf (g ∘ X) ℙ ν := @@ -261,7 +261,7 @@ begin end lemma quasi_measure_preserving_has_pdf' [is_finite_measure ℙ] [sigma_finite ν] - {X : α → E} [has_pdf X ℙ μ] {g : E → F} (hg : quasi_measure_preserving g μ ν) : + {X : Ω → E} [has_pdf X ℙ μ] {g : E → F} (hg : quasi_measure_preserving g μ ν) : has_pdf (g ∘ X) ℙ ν := quasi_measure_preserving_has_pdf hg infer_instance @@ -269,7 +269,7 @@ end section real -variables [is_finite_measure ℙ] {X : α → ℝ} +variables [is_finite_measure ℙ] {X : Ω → ℝ} /-- A real-valued random variable `X` `has_pdf X ℙ λ` (where `λ` is the Lebesgue measure) if and only if the push-forward measure of `ℙ` along `X` is absolutely continuous with respect to `λ`. -/ @@ -318,21 +318,21 @@ section /-- A random variable `X` has uniform distribution if it has a probability density function `f` with support `s` such that `f = (μ s)⁻¹ 1ₛ` a.e. where `1ₛ` is the indicator function for `s`. -/ -def is_uniform {m : measurable_space α} (X : α → E) (support : set E) - (ℙ : measure α) (μ : measure E . volume_tac) := +def is_uniform {m : measurable_space Ω} (X : Ω → E) (support : set E) + (ℙ : measure Ω) (μ : measure E . volume_tac) := pdf X ℙ μ =ᵐ[μ] support.indicator ((μ support)⁻¹ • 1) namespace is_uniform -lemma has_pdf {m : measurable_space α} {X : α → E} {ℙ : measure α} {μ : measure E} - {support : set E} (hns : μ support ≠ 0) (hnt : μ support ≠ ⊤) (hu : is_uniform X support ℙ μ) : +lemma has_pdf {m : measurable_space Ω} {X : Ω → E} {ℙ : measure Ω} {μ : measure E} + {s : set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hu : is_uniform X s ℙ μ) : has_pdf X ℙ μ := has_pdf_of_pdf_ne_zero begin intro hpdf, rw [is_uniform, hpdf] at hu, - suffices : μ (support ∩ function.support ((μ support)⁻¹ • 1)) = 0, - { have heq : function.support ((μ support)⁻¹ • (1 : E → ℝ≥0∞)) = set.univ, + suffices : μ (s ∩ function.support ((μ s)⁻¹ • 1)) = 0, + { have heq : function.support ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) = set.univ, { ext x, rw [function.mem_support], simp [hnt] }, @@ -341,18 +341,41 @@ begin exact set.indicator_ae_eq_zero hu.symm, end -lemma pdf_to_real_ae_eq {m : measurable_space α} - {X : α → E} {ℙ : measure α} {μ : measure E} {s : set E} (hX : is_uniform X s ℙ μ) : +lemma pdf_to_real_ae_eq {m : measurable_space Ω} + {X : Ω → E} {ℙ : measure Ω} {μ : measure E} {s : set E} (hX : is_uniform X s ℙ μ) : (λ x, (pdf X ℙ μ x).to_real) =ᵐ[μ] (λ x, (s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) x).to_real) := filter.eventually_eq.fun_comp hX ennreal.to_real -variables [is_finite_measure ℙ] {X : α → ℝ} -variables {s : set ℝ} (hms : measurable_set s) (hns : volume s ≠ 0) +lemma measure_preimage {m : measurable_space Ω} {X : Ω → E} {ℙ : measure Ω} {μ : measure E} + {s : set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hms : measurable_set s) + (hu : is_uniform X s ℙ μ) + {A : set E} (hA : measurable_set A) : + ℙ (X ⁻¹' A) = μ (s ∩ A) / μ s := +begin + haveI := hu.has_pdf hns hnt, + rw [←measure.map_apply (has_pdf.measurable X ℙ μ) hA, map_eq_set_lintegral_pdf X ℙ μ hA, + lintegral_congr_ae hu.restrict], + simp only [hms, hA, lintegral_indicator, pi.smul_apply, pi.one_apply, algebra.id.smul_eq_mul, + mul_one, lintegral_const, restrict_apply', set.univ_inter], + rw ennreal.div_eq_inv_mul, +end + +lemma is_probability_measure {m : measurable_space Ω} {X : Ω → E} {ℙ : measure Ω} {μ : measure E} + {s : set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hms : measurable_set s) + (hu : is_uniform X s ℙ μ) : + is_probability_measure ℙ := +⟨begin + have : X ⁻¹' set.univ = set.univ, { simp only [set.preimage_univ] }, + rw [←this, hu.measure_preimage hns hnt hms measurable_set.univ, set.inter_univ, + ennreal.div_self hns hnt], +end⟩ + +variables {X : Ω → ℝ} {s : set ℝ} (hms : measurable_set s) (hns : volume s ≠ 0) include hms hns -lemma mul_pdf_integrable (hcs : is_compact s) (huX : is_uniform X s ℙ) : +lemma mul_pdf_integrable [is_finite_measure ℙ] (hcs : is_compact s) (huX : is_uniform X s ℙ) : integrable (λ x : ℝ, x * (pdf X ℙ volume x).to_real) := begin by_cases hsupp : volume s = ∞, @@ -379,12 +402,12 @@ end /-- A real uniform random variable `X` with support `s` has expectation `(λ s)⁻¹ * ∫ x in s, x ∂λ` where `λ` is the Lebesgue measure. -/ -lemma integral_eq (hnt : volume s ≠ ⊤) (huX : is_uniform X s ℙ) : +lemma integral_eq (hnt : volume s ≠ ∞) (huX : is_uniform X s ℙ) : ∫ x, X x ∂ℙ = (volume s)⁻¹.to_real * ∫ x in s, x := begin haveI := has_pdf hns hnt huX, + haveI := huX.is_probability_measure hns hnt hms, rw ← integral_mul_eq_integral, - all_goals { try { apply_instance } }, rw integral_congr_ae (filter.eventually_eq.mul (ae_eq_refl _) (pdf_to_real_ae_eq huX)), have : ∀ x, x * (s.indicator ((volume s)⁻¹ • (1 : ℝ → ℝ≥0∞)) x).to_real = x * (s.indicator ((volume s)⁻¹.to_real • (1 : ℝ → ℝ)) x), diff --git a/formal/lean/mathlib/probability/hitting_time.lean b/formal/lean/mathlib/probability/hitting_time.lean index 70924d70c016fce6715a076281d0904abe92ebeb..6e148b94e46b7dcc661dc326495f16720ff5d1e6 100644 --- a/formal/lean/mathlib/probability/hitting_time.lean +++ b/formal/lean/mathlib/probability/hitting_time.lean @@ -37,24 +37,24 @@ open_locale classical measure_theory nnreal ennreal topological_space big_operat namespace measure_theory -variables {α β ι : Type*} {m : measurable_space α} +variables {Ω β ι : Type*} {m : measurable_space Ω} /-- Hitting time: given a stochastic process `u` and a set `s`, `hitting u s n m` is the first time `u` is in `s` after time `n` and before time `m` (if `u` does not hit `s` after time `n` and before `m` then the hitting time is simply `m`). The hitting time is a stopping time if the process is adapted and discrete. -/ -noncomputable def hitting [preorder ι] [has_Inf ι] (u : ι → α → β) (s : set β) (n m : ι) : α → ι := +noncomputable def hitting [preorder ι] [has_Inf ι] (u : ι → Ω → β) (s : set β) (n m : ι) : Ω → ι := λ x, if ∃ j ∈ set.Icc n m, u j x ∈ s then Inf (set.Icc n m ∩ {i : ι | u i x ∈ s}) else m section inequalities -variables [conditionally_complete_linear_order ι] {u : ι → α → β} {s : set β} {n i : ι} {x : α} +variables [conditionally_complete_linear_order ι] {u : ι → Ω → β} {s : set β} {n i : ι} {ω : Ω} -lemma hitting_of_lt {m : ι} (h : m < n) : hitting u s n m x = m := +lemma hitting_of_lt {m : ι} (h : m < n) : hitting u s n m ω = m := begin simp_rw [hitting], - have h_not : ¬∃ (j : ι) (H : j ∈ set.Icc n m), u j x ∈ s, + have h_not : ¬ ∃ (j : ι) (H : j ∈ set.Icc n m), u j ω ∈ s, { push_neg, intro j, rw set.Icc_eq_empty_of_lt h, @@ -62,7 +62,7 @@ begin simp only [h_not, if_false], end -lemma hitting_le {m : ι} (x : α) : hitting u s n m x ≤ m := +lemma hitting_le {m : ι} (ω : Ω) : hitting u s n m ω ≤ m := begin cases le_or_lt n m with h_le h_lt, { simp only [hitting], @@ -73,7 +73,7 @@ begin { rw hitting_of_lt h_lt, }, end -lemma le_hitting {m : ι} (hnm : n ≤ m) (x : α) : n ≤ hitting u s n m x := +lemma le_hitting {m : ι} (hnm : n ≤ m) (ω : Ω) : n ≤ hitting u s n m ω := begin simp only [hitting], split_ifs, @@ -85,23 +85,23 @@ begin { exact hnm }, end -lemma le_hitting_of_exists {m : ι} (h_exists : ∃ j ∈ set.Icc n m, u j x ∈ s) : - n ≤ hitting u s n m x := +lemma le_hitting_of_exists {m : ι} (h_exists : ∃ j ∈ set.Icc n m, u j ω ∈ s) : + n ≤ hitting u s n m ω := begin - refine le_hitting _ x, + refine le_hitting _ ω, by_contra, rw set.Icc_eq_empty_of_lt (not_le.mp h) at h_exists, simpa using h_exists, end -lemma hitting_mem_Icc {m : ι} (hnm : n ≤ m) (x : α) : hitting u s n m x ∈ set.Icc n m := -⟨le_hitting hnm x, hitting_le x⟩ +lemma hitting_mem_Icc {m : ι} (hnm : n ≤ m) (ω : Ω) : hitting u s n m ω ∈ set.Icc n m := +⟨le_hitting hnm ω, hitting_le ω⟩ -lemma hitting_mem_set [is_well_order ι (<)] {m : ι} (h_exists : ∃ j ∈ set.Icc n m, u j x ∈ s) : - u (hitting u s n m x) x ∈ s := +lemma hitting_mem_set [is_well_order ι (<)] {m : ι} (h_exists : ∃ j ∈ set.Icc n m, u j ω ∈ s) : + u (hitting u s n m ω) ω ∈ s := begin simp_rw [hitting, if_pos h_exists], - have h_nonempty : (set.Icc n m ∩ {i : ι | u i x ∈ s}).nonempty, + have h_nonempty : (set.Icc n m ∩ {i : ι | u i ω ∈ s}).nonempty, { obtain ⟨k, hk₁, hk₂⟩ := h_exists, exact ⟨k, set.mem_inter hk₁ hk₂⟩, }, have h_mem := Inf_mem h_nonempty, @@ -109,34 +109,34 @@ begin exact h_mem.2, end -lemma hitting_le_of_mem {m : ι} (hin : n ≤ i) (him : i ≤ m) (his : u i x ∈ s) : - hitting u s n m x ≤ i := +lemma hitting_le_of_mem {m : ι} (hin : n ≤ i) (him : i ≤ m) (his : u i ω ∈ s) : + hitting u s n m ω ≤ i := begin - have h_exists : ∃ k ∈ set.Icc n m, u k x ∈ s := ⟨i, ⟨hin, him⟩, his⟩, + have h_exists : ∃ k ∈ set.Icc n m, u k ω ∈ s := ⟨i, ⟨hin, him⟩, his⟩, simp_rw [hitting, if_pos h_exists], exact cInf_le (bdd_below.inter_of_left bdd_below_Icc) (set.mem_inter ⟨hin, him⟩ his), end lemma hitting_le_iff_of_exists [is_well_order ι (<)] {m : ι} - (h_exists : ∃ j ∈ set.Icc n m, u j x ∈ s) : - hitting u s n m x ≤ i ↔ ∃ j ∈ set.Icc n i, u j x ∈ s := + (h_exists : ∃ j ∈ set.Icc n m, u j ω ∈ s) : + hitting u s n m ω ≤ i ↔ ∃ j ∈ set.Icc n i, u j ω ∈ s := begin split; intro h', - { exact ⟨hitting u s n m x, ⟨le_hitting_of_exists h_exists, h'⟩, hitting_mem_set h_exists⟩, }, - { have h'' : ∃ k ∈ set.Icc n (min m i), u k x ∈ s, + { exact ⟨hitting u s n m ω, ⟨le_hitting_of_exists h_exists, h'⟩, hitting_mem_set h_exists⟩, }, + { have h'' : ∃ k ∈ set.Icc n (min m i), u k ω ∈ s, { obtain ⟨k₁, hk₁_mem, hk₁_s⟩ := h_exists, obtain ⟨k₂, hk₂_mem, hk₂_s⟩ := h', refine ⟨min k₁ k₂, ⟨le_min hk₁_mem.1 hk₂_mem.1, min_le_min hk₁_mem.2 hk₂_mem.2⟩, _⟩, - exact min_rec' (λ j, u j x ∈ s) hk₁_s hk₂_s, }, + exact min_rec' (λ j, u j ω ∈ s) hk₁_s hk₂_s, }, obtain ⟨k, hk₁, hk₂⟩ := h'', refine le_trans _ (hk₁.2.trans (min_le_right _ _)), exact hitting_le_of_mem hk₁.1 (hk₁.2.trans (min_le_left _ _)) hk₂, }, end lemma hitting_le_iff_of_lt [is_well_order ι (<)] {m : ι} (i : ι) (hi : i < m) : - hitting u s n m x ≤ i ↔ ∃ j ∈ set.Icc n i, u j x ∈ s := + hitting u s n m ω ≤ i ↔ ∃ j ∈ set.Icc n i, u j ω ∈ s := begin - by_cases h_exists : ∃ j ∈ set.Icc n m, u j x ∈ s, + by_cases h_exists : ∃ j ∈ set.Icc n m, u j ω ∈ s, { rw hitting_le_iff_of_exists h_exists, }, { simp_rw [hitting, if_neg h_exists], push_neg at h_exists, @@ -145,34 +145,51 @@ begin end lemma hitting_lt_iff [is_well_order ι (<)] {m : ι} (i : ι) (hi : i ≤ m) : - hitting u s n m x < i ↔ ∃ j ∈ set.Ico n i, u j x ∈ s := + hitting u s n m ω < i ↔ ∃ j ∈ set.Ico n i, u j ω ∈ s := begin split; intro h', - { have h : ∃ j ∈ set.Icc n m, u j x ∈ s, + { have h : ∃ j ∈ set.Icc n m, u j ω ∈ s, { by_contra, simp_rw [hitting, if_neg h, ← not_le] at h', exact h' hi, }, - exact ⟨hitting u s n m x, ⟨le_hitting_of_exists h, h'⟩, hitting_mem_set h⟩, }, + exact ⟨hitting u s n m ω, ⟨le_hitting_of_exists h, h'⟩, hitting_mem_set h⟩, }, { obtain ⟨k, hk₁, hk₂⟩ := h', refine lt_of_le_of_lt _ hk₁.2, exact hitting_le_of_mem hk₁.1 (hk₁.2.le.trans hi) hk₂, }, end +lemma hitting_eq_hitting_of_exists + {m₁ m₂ : ι} (h : m₁ ≤ m₂) (h' : ∃ j ∈ set.Icc n m₁, u j ω ∈ s) : + hitting u s n m₁ ω = hitting u s n m₂ ω := +begin + simp only [hitting, if_pos h'], + obtain ⟨j, hj₁, hj₂⟩ := h', + rw if_pos, + { refine le_antisymm _ (cInf_le_cInf bdd_below_Icc.inter_of_left ⟨j, hj₁, hj₂⟩ + (set.inter_subset_inter_left _ (set.Icc_subset_Icc_right h))), + refine le_cInf ⟨j, set.Icc_subset_Icc_right h hj₁, hj₂⟩ (λ i hi, _), + by_cases hi' : i ≤ m₁, + { exact cInf_le bdd_below_Icc.inter_of_left ⟨⟨hi.1.1, hi'⟩, hi.2⟩ }, + { exact ((cInf_le bdd_below_Icc.inter_of_left ⟨hj₁, hj₂⟩).trans (hj₁.2.trans le_rfl)).trans + (le_of_lt (not_le.1 hi')) } }, + exact ⟨j, ⟨hj₁.1, hj₁.2.trans h⟩, hj₂⟩, +end + end inequalities /-- A discrete hitting time is a stopping time. -/ lemma hitting_is_stopping_time [conditionally_complete_linear_order ι] [is_well_order ι (<)] [encodable ι] [topological_space β] [pseudo_metrizable_space β] [measurable_space β] [borel_space β] - {f : filtration ι m} {u : ι → α → β} {s : set β} {n n' : ι} + {f : filtration ι m} {u : ι → Ω → β} {s : set β} {n n' : ι} (hu : adapted f u) (hs : measurable_set s) : is_stopping_time f (hitting u s n n') := begin intro i, cases le_or_lt n' i with hi hi, - { have h_le : ∀ x, hitting u s n n' x ≤ i := λ x, (hitting_le x).trans hi, + { have h_le : ∀ ω, hitting u s n n' ω ≤ i := λ x, (hitting_le x).trans hi, simp [h_le], }, - { have h_set_eq_Union : {x | hitting u s n n' x ≤ i} = ⋃ j ∈ set.Icc n i, u j ⁻¹' s, + { have h_set_eq_Union : {ω | hitting u s n n' ω ≤ i} = ⋃ j ∈ set.Icc n i, u j ⁻¹' s, { ext x, rw [set.mem_set_of_eq, hitting_le_iff_of_lt _ hi], simp only [set.mem_Icc, exists_prop, set.mem_Union, set.mem_preimage], }, @@ -182,21 +199,48 @@ begin end lemma stopped_value_hitting_mem [conditionally_complete_linear_order ι] [is_well_order ι (<)] - {u : ι → α → β} {s : set β} {n m : ι} {x : α} (h : ∃ j ∈ set.Icc n m, u j x ∈ s) : - stopped_value u (hitting u s n m) x ∈ s := + {u : ι → Ω → β} {s : set β} {n m : ι} {ω : Ω} (h : ∃ j ∈ set.Icc n m, u j ω ∈ s) : + stopped_value u (hitting u s n m) ω ∈ s := begin simp only [stopped_value, hitting, if_pos h], obtain ⟨j, hj₁, hj₂⟩ := h, - have : Inf (set.Icc n m ∩ {i | u i x ∈ s}) ∈ set.Icc n m ∩ {i | u i x ∈ s} := + have : Inf (set.Icc n m ∩ {i | u i ω ∈ s}) ∈ set.Icc n m ∩ {i | u i ω ∈ s} := Inf_mem (set.nonempty_of_mem ⟨hj₁, hj₂⟩), exact this.2, end +/-- The hitting time of a discrete process with the starting time indexed by a stopping time +is a stopping time. -/ +lemma is_stopping_time_hitting_is_stopping_time + [conditionally_complete_linear_order ι] [is_well_order ι (<)] [encodable ι] + [topological_space ι] [order_topology ι] [first_countable_topology ι] + [topological_space β] [pseudo_metrizable_space β] [measurable_space β] [borel_space β] + {f : filtration ι m} {u : ι → Ω → β} {τ : Ω → ι} (hτ : is_stopping_time f τ) + {N : ι} (hτbdd : ∀ x, τ x ≤ N) {s : set β} (hs : measurable_set s) (hf : adapted f u) : + is_stopping_time f (λ x, hitting u s (τ x) N x) := +begin + intro n, + have h₁ : {x | hitting u s (τ x) N x ≤ n} = + (⋃ i ≤ n, {x | τ x = i} ∩ {x | hitting u s i N x ≤ n}) ∪ + (⋃ i > n, {x | τ x = i} ∩ {x | hitting u s i N x ≤ n}), + { ext x, + simp [← exists_or_distrib, ← or_and_distrib_right, le_or_lt] }, + have h₂ : (⋃ i > n, {x | τ x = i} ∩ {x | hitting u s i N x ≤ n}) = ∅, + { ext x, + simp only [gt_iff_lt, set.mem_Union, set.mem_inter_eq, set.mem_set_of_eq, + exists_prop, set.mem_empty_eq, iff_false, not_exists, not_and, not_le], + rintro m hm rfl, + exact lt_of_lt_of_le hm (le_hitting (hτbdd _) _) }, + rw [h₁, h₂, set.union_empty], + exact measurable_set.Union (λ i, measurable_set.Union_Prop + (λ hi, (f.mono hi _ (hτ.measurable_set_eq i)).inter (hitting_is_stopping_time hf hs n))), +end + section complete_lattice -variables [complete_lattice ι] {u : ι → α → β} {s : set β} {f : filtration ι m} +variables [complete_lattice ι] {u : ι → Ω → β} {s : set β} {f : filtration ι m} -lemma hitting_eq_Inf (x : α) : hitting u s ⊥ ⊤ x = Inf {i : ι | u i x ∈ s} := +lemma hitting_eq_Inf (ω : Ω) : hitting u s ⊥ ⊤ ω = Inf {i : ι | u i ω ∈ s} := begin simp only [hitting, set.mem_Icc, bot_le, le_top, and_self, exists_true_left, set.Icc_bot, set.Iic_top, set.univ_inter, ite_eq_left_iff, not_exists], @@ -211,15 +255,15 @@ end complete_lattice section conditionally_complete_linear_order_bot variables [conditionally_complete_linear_order_bot ι] [is_well_order ι (<)] -variables {u : ι → α → β} {s : set β} {f : filtration ℕ m} +variables {u : ι → Ω → β} {s : set β} {f : filtration ℕ m} -lemma hitting_bot_le_iff {i n : ι} {x : α} (hx : ∃ j, j ≤ n ∧ u j x ∈ s) : - hitting u s ⊥ n x ≤ i ↔ ∃ j ≤ i, u j x ∈ s := +lemma hitting_bot_le_iff {i n : ι} {ω : Ω} (hx : ∃ j, j ≤ n ∧ u j ω ∈ s) : + hitting u s ⊥ n ω ≤ i ↔ ∃ j ≤ i, u j ω ∈ s := begin cases lt_or_le i n with hi hi, { rw hitting_le_iff_of_lt _ hi, simp, }, - { simp only [(hitting_le x).trans hi, true_iff], + { simp only [(hitting_le ω).trans hi, true_iff], obtain ⟨j, hj₁, hj₂⟩ := hx, exact ⟨j, hj₁.trans hi, hj₂⟩, }, end diff --git a/formal/lean/mathlib/probability/independence.lean b/formal/lean/mathlib/probability/independence.lean index 205b2eee2a0f153d97d843a72ae5e89e072dc75b..e43fcb7cb62ee89d1bc6ee48ed9ad5fb2ff4a7dc 100644 --- a/formal/lean/mathlib/probability/independence.lean +++ b/formal/lean/mathlib/probability/independence.lean @@ -9,12 +9,12 @@ import measure_theory.constructions.pi /-! # Independence of sets of sets and measure spaces (σ-algebras) -* A family of sets of sets `π : ι → set (set α)` is independent with respect to a measure `μ` if for +* A family of sets of sets `π : ι → set (set Ω)` is independent with respect to a measure `μ` if for any finite set of indices `s = {i_1, ..., i_n}`, for any sets `f i_1 ∈ π i_1, ..., f i_n ∈ π i_n`, `μ (⋂ i in s, f i) = ∏ i in s, μ (f i) `. It will be used for families of π-systems. * A family of measurable space structures (i.e. of σ-algebras) is independent with respect to a measure `μ` (typically defined on a finer σ-algebra) if the family of sets of measurable sets they - define is independent. I.e., `m : ι → measurable_space α` is independent with respect to a + define is independent. I.e., `m : ι → measurable_space Ω` is independent with respect to a measure `μ` if for any finite set of indices `s = {i_1, ..., i_n}`, for any sets `f i_1 ∈ m i_1, ..., f i_n ∈ m i_n`, then `μ (⋂ i in s, f i) = ∏ i in s, μ (f i)`. * Independence of sets (or events in probabilistic parlance) is defined as independence of the @@ -33,12 +33,12 @@ measurable space structures they generate are independent. ## Implementation notes We provide one main definition of independence: -* `Indep_sets`: independence of a family of sets of sets `pi : ι → set (set α)`. +* `Indep_sets`: independence of a family of sets of sets `pi : ι → set (set Ω)`. Three other independence notions are defined using `Indep_sets`: -* `Indep`: independence of a family of measurable space structures `m : ι → measurable_space α`, -* `Indep_set`: independence of a family of sets `s : ι → set α`, +* `Indep`: independence of a family of measurable space structures `m : ι → measurable_space Ω`, +* `Indep_set`: independence of a family of sets `s : ι → set Ω`, * `Indep_fun`: independence of a family of functions. For measurable spaces - `m : Π (i : ι), measurable_space (β i)`, we consider functions `f : Π (i : ι), α → β i`. + `m : Π (i : ι), measurable_space (β i)`, we consider functions `f : Π (i : ι), Ω → β i`. Additionally, we provide four corresponding statements for two measurable space structures (resp. sets of sets, sets, functions) instead of a family. These properties are denoted by the same names @@ -51,10 +51,10 @@ TODO: prove that equivalence. Most of the definitions and lemma in this file list all variables instead of using the `variables` keyword at the beginning of a section, for example -`lemma indep.symm {α} {m₁ m₂ : measurable_space α} [measurable_space α] {μ : measure α} ...` . +`lemma indep.symm {Ω} {m₁ m₂ : measurable_space Ω} [measurable_space Ω] {μ : measure Ω} ...` . This is intentional, to be able to control the order of the `measurable_space` variables. Indeed when defining `μ` in the example above, the measurable space used is the last one defined, here -`[measurable_space α]`, and not `m₁` or `m₂`. +`[measurable_space Ω]`, and not `m₁` or `m₂`. ## References @@ -69,95 +69,95 @@ namespace probability_theory section definitions -/-- A family of sets of sets `π : ι → set (set α)` is independent with respect to a measure `μ` if +/-- A family of sets of sets `π : ι → set (set Ω)` is independent with respect to a measure `μ` if for any finite set of indices `s = {i_1, ..., i_n}`, for any sets `f i_1 ∈ π i_1, ..., f i_n ∈ π i_n`, then `μ (⋂ i in s, f i) = ∏ i in s, μ (f i) `. It will be used for families of pi_systems. -/ -def Indep_sets {α ι} [measurable_space α] (π : ι → set (set α)) (μ : measure α . volume_tac) : +def Indep_sets {Ω ι} [measurable_space Ω] (π : ι → set (set Ω)) (μ : measure Ω . volume_tac) : Prop := -∀ (s : finset ι) {f : ι → set α} (H : ∀ i, i ∈ s → f i ∈ π i), μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i) +∀ (s : finset ι) {f : ι → set Ω} (H : ∀ i, i ∈ s → f i ∈ π i), μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i) /-- Two sets of sets `s₁, s₂` are independent with respect to a measure `μ` if for any sets `t₁ ∈ p₁, t₂ ∈ s₂`, then `μ (t₁ ∩ t₂) = μ (t₁) * μ (t₂)` -/ -def indep_sets {α} [measurable_space α] (s1 s2 : set (set α)) (μ : measure α . volume_tac) : Prop := -∀ t1 t2 : set α, t1 ∈ s1 → t2 ∈ s2 → μ (t1 ∩ t2) = μ t1 * μ t2 +def indep_sets {Ω} [measurable_space Ω] (s1 s2 : set (set Ω)) (μ : measure Ω . volume_tac) : Prop := +∀ t1 t2 : set Ω, t1 ∈ s1 → t2 ∈ s2 → μ (t1 ∩ t2) = μ t1 * μ t2 /-- A family of measurable space structures (i.e. of σ-algebras) is independent with respect to a measure `μ` (typically defined on a finer σ-algebra) if the family of sets of measurable sets they -define is independent. `m : ι → measurable_space α` is independent with respect to measure `μ` if +define is independent. `m : ι → measurable_space Ω` is independent with respect to measure `μ` if for any finite set of indices `s = {i_1, ..., i_n}`, for any sets `f i_1 ∈ m i_1, ..., f i_n ∈ m i_n`, then `μ (⋂ i in s, f i) = ∏ i in s, μ (f i) `. -/ -def Indep {α ι} (m : ι → measurable_space α) [measurable_space α] (μ : measure α . volume_tac) : +def Indep {Ω ι} (m : ι → measurable_space Ω) [measurable_space Ω] (μ : measure Ω . volume_tac) : Prop := Indep_sets (λ x, {s | measurable_set[m x] s}) μ /-- Two measurable space structures (or σ-algebras) `m₁, m₂` are independent with respect to a measure `μ` (defined on a third σ-algebra) if for any sets `t₁ ∈ m₁, t₂ ∈ m₂`, `μ (t₁ ∩ t₂) = μ (t₁) * μ (t₂)` -/ -def indep {α} (m₁ m₂ : measurable_space α) [measurable_space α] (μ : measure α . volume_tac) : +def indep {Ω} (m₁ m₂ : measurable_space Ω) [measurable_space Ω] (μ : measure Ω . volume_tac) : Prop := indep_sets {s | measurable_set[m₁] s} {s | measurable_set[m₂] s} μ /-- A family of sets is independent if the family of measurable space structures they generate is independent. For a set `s`, the generated measurable space has measurable sets `∅, s, sᶜ, univ`. -/ -def Indep_set {α ι} [measurable_space α] (s : ι → set α) (μ : measure α . volume_tac) : Prop := +def Indep_set {Ω ι} [measurable_space Ω] (s : ι → set Ω) (μ : measure Ω . volume_tac) : Prop := Indep (λ i, generate_from {s i}) μ /-- Two sets are independent if the two measurable space structures they generate are independent. For a set `s`, the generated measurable space structure has measurable sets `∅, s, sᶜ, univ`. -/ -def indep_set {α} [measurable_space α] (s t : set α) (μ : measure α . volume_tac) : Prop := +def indep_set {Ω} [measurable_space Ω] (s t : set Ω) (μ : measure Ω . volume_tac) : Prop := indep (generate_from {s}) (generate_from {t}) μ -/-- A family of functions defined on the same space `α` and taking values in possibly different +/-- A family of functions defined on the same space `Ω` and taking values in possibly different spaces, each with a measurable space structure, is independent if the family of measurable space -structures they generate on `α` is independent. For a function `g` with codomain having measurable +structures they generate on `Ω` is independent. For a function `g` with codomain having measurable space structure `m`, the generated measurable space structure is `measurable_space.comap g m`. -/ -def Indep_fun {α ι} [measurable_space α] {β : ι → Type*} (m : Π (x : ι), measurable_space (β x)) - (f : Π (x : ι), α → β x) (μ : measure α . volume_tac) : Prop := +def Indep_fun {Ω ι} [measurable_space Ω] {β : ι → Type*} (m : Π (x : ι), measurable_space (β x)) + (f : Π (x : ι), Ω → β x) (μ : measure Ω . volume_tac) : Prop := Indep (λ x, measurable_space.comap (f x) (m x)) μ /-- Two functions are independent if the two measurable space structures they generate are independent. For a function `f` with codomain having measurable space structure `m`, the generated measurable space structure is `measurable_space.comap f m`. -/ -def indep_fun {α β γ} [measurable_space α] [mβ : measurable_space β] [mγ : measurable_space γ] - (f : α → β) (g : α → γ) (μ : measure α . volume_tac) : Prop := +def indep_fun {Ω β γ} [measurable_space Ω] [mβ : measurable_space β] [mγ : measurable_space γ] + (f : Ω → β) (g : Ω → γ) (μ : measure Ω . volume_tac) : Prop := indep (measurable_space.comap f mβ) (measurable_space.comap g mγ) μ end definitions section indep -lemma indep_sets.symm {α} {s₁ s₂ : set (set α)} [measurable_space α] {μ : measure α} +lemma indep_sets.symm {Ω} {s₁ s₂ : set (set Ω)} [measurable_space Ω] {μ : measure Ω} (h : indep_sets s₁ s₂ μ) : indep_sets s₂ s₁ μ := by { intros t1 t2 ht1 ht2, rw [set.inter_comm, mul_comm], exact h t2 t1 ht2 ht1, } -lemma indep.symm {α} {m₁ m₂ : measurable_space α} [measurable_space α] {μ : measure α} +lemma indep.symm {Ω} {m₁ m₂ : measurable_space Ω} [measurable_space Ω] {μ : measure Ω} (h : indep m₁ m₂ μ) : indep m₂ m₁ μ := indep_sets.symm h -lemma indep_sets_of_indep_sets_of_le_left {α} {s₁ s₂ s₃: set (set α)} [measurable_space α] - {μ : measure α} (h_indep : indep_sets s₁ s₂ μ) (h31 : s₃ ⊆ s₁) : +lemma indep_sets_of_indep_sets_of_le_left {Ω} {s₁ s₂ s₃: set (set Ω)} [measurable_space Ω] + {μ : measure Ω} (h_indep : indep_sets s₁ s₂ μ) (h31 : s₃ ⊆ s₁) : indep_sets s₃ s₂ μ := λ t1 t2 ht1 ht2, h_indep t1 t2 (set.mem_of_subset_of_mem h31 ht1) ht2 -lemma indep_sets_of_indep_sets_of_le_right {α} {s₁ s₂ s₃: set (set α)} [measurable_space α] - {μ : measure α} (h_indep : indep_sets s₁ s₂ μ) (h32 : s₃ ⊆ s₂) : +lemma indep_sets_of_indep_sets_of_le_right {Ω} {s₁ s₂ s₃: set (set Ω)} [measurable_space Ω] + {μ : measure Ω} (h_indep : indep_sets s₁ s₂ μ) (h32 : s₃ ⊆ s₂) : indep_sets s₁ s₃ μ := λ t1 t2 ht1 ht2, h_indep t1 t2 ht1 (set.mem_of_subset_of_mem h32 ht2) -lemma indep_of_indep_of_le_left {α} {m₁ m₂ m₃: measurable_space α} [measurable_space α] - {μ : measure α} (h_indep : indep m₁ m₂ μ) (h31 : m₃ ≤ m₁) : +lemma indep_of_indep_of_le_left {Ω} {m₁ m₂ m₃: measurable_space Ω} [measurable_space Ω] + {μ : measure Ω} (h_indep : indep m₁ m₂ μ) (h31 : m₃ ≤ m₁) : indep m₃ m₂ μ := λ t1 t2 ht1 ht2, h_indep t1 t2 (h31 _ ht1) ht2 -lemma indep_of_indep_of_le_right {α} {m₁ m₂ m₃: measurable_space α} [measurable_space α] - {μ : measure α} (h_indep : indep m₁ m₂ μ) (h32 : m₃ ≤ m₂) : +lemma indep_of_indep_of_le_right {Ω} {m₁ m₂ m₃: measurable_space Ω} [measurable_space Ω] + {μ : measure Ω} (h_indep : indep m₁ m₂ μ) (h32 : m₃ ≤ m₂) : indep m₁ m₃ μ := λ t1 t2 ht1 ht2, h_indep t1 t2 ht1 (h32 _ ht2) -lemma indep_sets.union {α} [measurable_space α] {s₁ s₂ s' : set (set α)} {μ : measure α} +lemma indep_sets.union {Ω} [measurable_space Ω] {s₁ s₂ s' : set (set Ω)} {μ : measure Ω} (h₁ : indep_sets s₁ s' μ) (h₂ : indep_sets s₂ s' μ) : indep_sets (s₁ ∪ s₂) s' μ := begin @@ -167,15 +167,15 @@ begin { exact h₂ t1 t2 ht1₂ ht2, }, end -@[simp] lemma indep_sets.union_iff {α} [measurable_space α] {s₁ s₂ s' : set (set α)} - {μ : measure α} : +@[simp] lemma indep_sets.union_iff {Ω} [measurable_space Ω] {s₁ s₂ s' : set (set Ω)} + {μ : measure Ω} : indep_sets (s₁ ∪ s₂) s' μ ↔ indep_sets s₁ s' μ ∧ indep_sets s₂ s' μ := ⟨λ h, ⟨indep_sets_of_indep_sets_of_le_left h (set.subset_union_left s₁ s₂), indep_sets_of_indep_sets_of_le_left h (set.subset_union_right s₁ s₂)⟩, λ h, indep_sets.union h.left h.right⟩ -lemma indep_sets.Union {α ι} [measurable_space α] {s : ι → set (set α)} {s' : set (set α)} - {μ : measure α} (hyp : ∀ n, indep_sets (s n) s' μ) : +lemma indep_sets.Union {Ω ι} [measurable_space Ω] {s : ι → set (set Ω)} {s' : set (set Ω)} + {μ : measure Ω} (hyp : ∀ n, indep_sets (s n) s' μ) : indep_sets (⋃ n, s n) s' μ := begin intros t1 t2 ht1 ht2, @@ -184,17 +184,17 @@ begin exact hyp n t1 t2 ht1 ht2, end -lemma indep_sets.inter {α} [measurable_space α] {s₁ s' : set (set α)} (s₂ : set (set α)) - {μ : measure α} (h₁ : indep_sets s₁ s' μ) : +lemma indep_sets.inter {Ω} [measurable_space Ω] {s₁ s' : set (set Ω)} (s₂ : set (set Ω)) + {μ : measure Ω} (h₁ : indep_sets s₁ s' μ) : indep_sets (s₁ ∩ s₂) s' μ := λ t1 t2 ht1 ht2, h₁ t1 t2 ((set.mem_inter_iff _ _ _).mp ht1).left ht2 -lemma indep_sets.Inter {α ι} [measurable_space α] {s : ι → set (set α)} {s' : set (set α)} - {μ : measure α} (h : ∃ n, indep_sets (s n) s' μ) : +lemma indep_sets.Inter {Ω ι} [measurable_space Ω] {s : ι → set (set Ω)} {s' : set (set Ω)} + {μ : measure Ω} (h : ∃ n, indep_sets (s n) s' μ) : indep_sets (⋂ n, s n) s' μ := by {intros t1 t2 ht1 ht2, cases h with n h, exact h t1 t2 (set.mem_Inter.mp ht1 n) ht2 } -lemma indep_sets_singleton_iff {α} [measurable_space α] {s t : set α} {μ : measure α} : +lemma indep_sets_singleton_iff {Ω} [measurable_space Ω] {s t : set Ω} {μ : measure Ω} : indep_sets {s} {t} μ ↔ μ (s ∩ t) = μ s * μ t := ⟨λ h, h s t rfl rfl, λ h s1 t1 hs1 ht1, by rwa [set.mem_singleton_iff.mp hs1, set.mem_singleton_iff.mp ht1]⟩ @@ -204,7 +204,7 @@ end indep /-! ### Deducing `indep` from `Indep` -/ section from_Indep_to_indep -lemma Indep_sets.indep_sets {α ι} {s : ι → set (set α)} [measurable_space α] {μ : measure α} +lemma Indep_sets.indep_sets {Ω ι} {s : ι → set (set Ω)} [measurable_space Ω] {μ : measure Ω} (h_indep : Indep_sets s μ) {i j : ι} (hij : i ≠ j) : indep_sets (s i) (s j) μ := begin @@ -229,7 +229,7 @@ begin rw [←h_inter, ←h_prod, h_indep {i, j} hf_m], end -lemma Indep.indep {α ι} {m : ι → measurable_space α} [measurable_space α] {μ : measure α} +lemma Indep.indep {Ω ι} {m : ι → measurable_space Ω} [measurable_space Ω] {μ : measure Ω} (h_indep : Indep m μ) {i j : ι} (hij : i ≠ j) : indep (m i) (m j) μ := begin @@ -237,8 +237,8 @@ begin exact Indep_sets.indep_sets h_indep hij, end -lemma Indep_fun.indep_fun {α ι : Type*} {m₀ : measurable_space α} {μ : measure α} {β : ι → Type*} - {m : Π x, measurable_space (β x)} {f : Π i, α → β i} (hf_Indep : Indep_fun m f μ) +lemma Indep_fun.indep_fun {Ω ι : Type*} {m₀ : measurable_space Ω} {μ : measure Ω} {β : ι → Type*} + {m : Π x, measurable_space (β x)} {f : Π i, Ω → β i} (hf_Indep : Indep_fun m f μ) {i j : ι} (hij : i ≠ j) : indep_fun (f i) (f j) μ := hf_Indep.indep hij @@ -254,14 +254,14 @@ Independence of measurable spaces is equivalent to independence of generating π section from_measurable_spaces_to_sets_of_sets /-! ### Independence of measurable space structures implies independence of generating π-systems -/ -lemma Indep.Indep_sets {α ι} [measurable_space α] {μ : measure α} {m : ι → measurable_space α} - {s : ι → set (set α)} (hms : ∀ n, m n = generate_from (s n)) +lemma Indep.Indep_sets {Ω ι} [measurable_space Ω] {μ : measure Ω} {m : ι → measurable_space Ω} + {s : ι → set (set Ω)} (hms : ∀ n, m n = generate_from (s n)) (h_indep : Indep m μ) : Indep_sets s μ := λ S f hfs, h_indep S $ λ x hxS, ((hms x).symm ▸ measurable_set_generate_from (hfs x hxS) : measurable_set[m x] (f x)) -lemma indep.indep_sets {α} [measurable_space α] {μ : measure α} {s1 s2 : set (set α)} +lemma indep.indep_sets {Ω} [measurable_space Ω] {μ : measure Ω} {s1 s2 : set (set Ω)} (h_indep : indep (generate_from s1) (generate_from s2) μ) : indep_sets s1 s2 μ := λ t1 t2 ht1 ht2, h_indep t1 t2 (measurable_set_generate_from ht1) (measurable_set_generate_from ht2) @@ -271,18 +271,18 @@ end from_measurable_spaces_to_sets_of_sets section from_pi_systems_to_measurable_spaces /-! ### Independence of generating π-systems implies independence of measurable space structures -/ -private lemma indep_sets.indep_aux {α} {m2 : measurable_space α} - {m : measurable_space α} {μ : measure α} [is_probability_measure μ] {p1 p2 : set (set α)} +private lemma indep_sets.indep_aux {Ω} {m2 : measurable_space Ω} + {m : measurable_space Ω} {μ : measure Ω} [is_probability_measure μ] {p1 p2 : set (set Ω)} (h2 : m2 ≤ m) (hp2 : is_pi_system p2) (hpm2 : m2 = generate_from p2) - (hyp : indep_sets p1 p2 μ) {t1 t2 : set α} (ht1 : t1 ∈ p1) (ht2m : measurable_set[m2] t2) : + (hyp : indep_sets p1 p2 μ) {t1 t2 : set Ω} (ht1 : t1 ∈ p1) (ht2m : measurable_set[m2] t2) : μ (t1 ∩ t2) = μ t1 * μ t2 := begin let μ_inter := μ.restrict t1, let ν := (μ t1) • μ, have h_univ : μ_inter set.univ = ν set.univ, by rw [measure.restrict_apply_univ, measure.smul_apply, smul_eq_mul, measure_univ, mul_one], - haveI : is_finite_measure μ_inter := @restrict.is_finite_measure α _ t1 μ ⟨measure_lt_top μ t1⟩, - rw [set.inter_comm, ←@measure.restrict_apply α _ μ t1 t2 (h2 t2 ht2m)], + haveI : is_finite_measure μ_inter := @restrict.is_finite_measure Ω _ t1 μ ⟨measure_lt_top μ t1⟩, + rw [set.inter_comm, ←@measure.restrict_apply Ω _ μ t1 t2 (h2 t2 ht2m)], refine ext_on_measurable_space_of_generate_finite m p2 (λ t ht, _) h2 hpm2 hp2 h_univ ht2m, have ht2 : measurable_set[m] t, { refine h2 _ _, @@ -292,8 +292,8 @@ begin exact hyp t1 t ht1 ht, end -lemma indep_sets.indep {α} {m1 m2 : measurable_space α} {m : measurable_space α} - {μ : measure α} [is_probability_measure μ] {p1 p2 : set (set α)} (h1 : m1 ≤ m) (h2 : m2 ≤ m) +lemma indep_sets.indep {Ω} {m1 m2 : measurable_space Ω} {m : measurable_space Ω} + {μ : measure Ω} [is_probability_measure μ] {p1 p2 : set (set Ω)} (h1 : m1 ≤ m) (h2 : m2 ≤ m) (hp1 : is_pi_system p1) (hp2 : is_pi_system p2) (hpm1 : m1 = generate_from p1) (hpm2 : m2 = generate_from p2) (hyp : indep_sets p1 p2 μ) : indep m1 m2 μ := @@ -303,8 +303,8 @@ begin let ν := (μ t2) • μ, have h_univ : μ_inter set.univ = ν set.univ, by rw [measure.restrict_apply_univ, measure.smul_apply, smul_eq_mul, measure_univ, mul_one], - haveI : is_finite_measure μ_inter := @restrict.is_finite_measure α _ t2 μ ⟨measure_lt_top μ t2⟩, - rw [mul_comm, ←@measure.restrict_apply α _ μ t2 t1 (h1 t1 ht1)], + haveI : is_finite_measure μ_inter := @restrict.is_finite_measure Ω _ t2 μ ⟨measure_lt_top μ t2⟩, + rw [mul_comm, ←@measure.restrict_apply Ω _ μ t2 t1 (h1 t1 ht1)], refine ext_on_measurable_space_of_generate_finite m p1 (λ t ht, _) h1 hpm1 hp1 h_univ ht1, have ht1 : measurable_set[m] t, { refine h1 _ _, @@ -314,9 +314,9 @@ begin exact indep_sets.indep_aux h2 hp2 hpm2 hyp ht ht2, end -variables {α ι : Type*} {m0 : measurable_space α} {μ : measure α} +variables {Ω ι : Type*} {m0 : measurable_space Ω} {μ : measure Ω} -lemma Indep_sets.pi_Union_Inter_singleton {π : ι → set (set α)} {a : ι} {S : finset ι} +lemma Indep_sets.pi_Union_Inter_singleton {π : ι → set (set Ω)} {a : ι} {S : finset ι} (hp_ind : Indep_sets π μ) (haS : a ∉ S) : indep_sets (pi_Union_Inter π {S}) (π a) μ := begin @@ -351,8 +351,8 @@ begin end /-- Auxiliary lemma for `Indep_sets.Indep`. -/ -theorem Indep_sets.Indep_aux [is_probability_measure μ] (m : ι → measurable_space α) - (h_le : ∀ i, m i ≤ m0) (π : ι → set (set α)) (h_pi : ∀ n, is_pi_system (π n)) +theorem Indep_sets.Indep_aux [is_probability_measure μ] (m : ι → measurable_space Ω) + (h_le : ∀ i, m i ≤ m0) (π : ι → set (set Ω)) (h_pi : ∀ n, is_pi_system (π n)) (hp_univ : ∀ i, set.univ ∈ π i) (h_generate : ∀ i, m i = generate_from (π i)) (h_ind : Indep_sets π μ) : Indep m μ := @@ -379,8 +379,8 @@ begin end /-- The measurable space structures generated by independent pi-systems are independent. -/ -theorem Indep_sets.Indep [is_probability_measure μ] (m : ι → measurable_space α) - (h_le : ∀ i, m i ≤ m0) (π : ι → set (set α)) (h_pi : ∀ n, is_pi_system (π n)) +theorem Indep_sets.Indep [is_probability_measure μ] (m : ι → measurable_space Ω) + (h_le : ∀ i, m i ≤ m0) (π : ι → set (set Ω)) (h_pi : ∀ n, is_pi_system (π n)) (h_generate : ∀ i, m i = generate_from (π i)) (h_ind : Indep_sets π μ) : Indep m μ := begin @@ -437,10 +437,10 @@ We prove the following equivalences on `indep_set`, for measurable sets `s, t`. * `indep_set s t μ ↔ indep_sets {s} {t} μ`. -/ -variables {α : Type*} [measurable_space α] {s t : set α} (S T : set (set α)) +variables {Ω : Type*} [measurable_space Ω] {s t : set Ω} (S T : set (set Ω)) lemma indep_set_iff_indep_sets_singleton (hs_meas : measurable_set s) (ht_meas : measurable_set t) - (μ : measure α . volume_tac) [is_probability_measure μ] : + (μ : measure Ω . volume_tac) [is_probability_measure μ] : indep_set s t μ ↔ indep_sets {s} {t} μ := ⟨indep.indep_sets, λ h, indep_sets.indep (generate_from_le (λ u hu, by rwa set.mem_singleton_iff.mp hu)) @@ -448,12 +448,12 @@ lemma indep_set_iff_indep_sets_singleton (hs_meas : measurable_set s) (ht_meas : (is_pi_system.singleton t) rfl rfl h⟩ lemma indep_set_iff_measure_inter_eq_mul (hs_meas : measurable_set s) (ht_meas : measurable_set t) - (μ : measure α . volume_tac) [is_probability_measure μ] : + (μ : measure Ω . volume_tac) [is_probability_measure μ] : indep_set s t μ ↔ μ (s ∩ t) = μ s * μ t := (indep_set_iff_indep_sets_singleton hs_meas ht_meas μ).trans indep_sets_singleton_iff lemma indep_sets.indep_set_of_mem (hs : s ∈ S) (ht : t ∈ T) (hs_meas : measurable_set s) - (ht_meas : measurable_set t) (μ : measure α . volume_tac) [is_probability_measure μ] + (ht_meas : measurable_set t) (μ : measure Ω . volume_tac) [is_probability_measure μ] (h_indep : indep_sets S T μ) : indep_set s t μ := (indep_set_iff_measure_inter_eq_mul hs_meas ht_meas μ).mpr (h_indep s t hs ht) @@ -466,7 +466,7 @@ section indep_fun -/ -variables {α β β' γ γ' : Type*} {mα : measurable_space α} {μ : measure α} {f : α → β} {g : α → β'} +variables {Ω β β' γ γ' : Type*} {mΩ : measurable_space Ω} {μ : measure Ω} {f : Ω → β} {g : Ω → β'} lemma indep_fun_iff_measure_inter_preimage_eq_mul {mβ : measurable_space β} {mβ' : measurable_space β'} : @@ -480,30 +480,30 @@ begin end lemma Indep_fun_iff_measure_inter_preimage_eq_mul {ι : Type*} {β : ι → Type*} - (m : Π x, measurable_space (β x)) (f : Π i, α → β i) : + (m : Π x, measurable_space (β x)) (f : Π i, Ω → β i) : Indep_fun m f μ ↔ ∀ (S : finset ι) {sets : Π i : ι, set (β i)} (H : ∀ i, i ∈ S → measurable_set[m i] (sets i)), μ (⋂ i ∈ S, (f i) ⁻¹' (sets i)) = ∏ i in S, μ ((f i) ⁻¹' (sets i)) := begin refine ⟨λ h S sets h_meas, h _ (λ i hi_mem, ⟨sets i, h_meas i hi_mem, rfl⟩), _⟩, - intros h S setsα h_meas, + intros h S setsΩ h_meas, let setsβ : (Π i : ι, set (β i)) := λ i, dite (i ∈ S) (λ hi_mem, (h_meas i hi_mem).some) (λ _, set.univ), have h_measβ : ∀ i ∈ S, measurable_set[m i] (setsβ i), { intros i hi_mem, simp_rw [setsβ, dif_pos hi_mem], exact (h_meas i hi_mem).some_spec.1, }, - have h_preim : ∀ i ∈ S, setsα i = (f i) ⁻¹' (setsβ i), + have h_preim : ∀ i ∈ S, setsΩ i = (f i) ⁻¹' (setsβ i), { intros i hi_mem, simp_rw [setsβ, dif_pos hi_mem], exact (h_meas i hi_mem).some_spec.2.symm, }, - have h_left_eq : μ (⋂ i ∈ S, setsα i) = μ (⋂ i ∈ S, (f i) ⁻¹' (setsβ i)), + have h_left_eq : μ (⋂ i ∈ S, setsΩ i) = μ (⋂ i ∈ S, (f i) ⁻¹' (setsβ i)), { congr' with i x, simp only [set.mem_Inter], split; intros h hi_mem; specialize h hi_mem, { rwa h_preim i hi_mem at h, }, { rwa h_preim i hi_mem, }, }, - have h_right_eq : (∏ i in S, μ (setsα i)) = ∏ i in S, μ ((f i) ⁻¹' (setsβ i)), + have h_right_eq : (∏ i in S, μ (setsΩ i)) = ∏ i in S, μ ((f i) ⁻¹' (setsβ i)), { refine finset.prod_congr rfl (λ i hi_mem, _), rw h_preim i hi_mem, }, rw [h_left_eq, h_right_eq], @@ -520,7 +520,7 @@ begin { rwa ← indep_set_iff_measure_inter_eq_mul (hf hs) (hg ht) μ, }, end -lemma indep_fun.ae_eq {mβ : measurable_space β} {f g f' g' : α → β} +lemma indep_fun.ae_eq {mβ : measurable_space β} {f g f' g' : Ω → β} (hfg : indep_fun f g μ) (hf : f =ᵐ[μ] f') (hg : g =ᵐ[μ] g') : indep_fun f' g' μ := begin @@ -547,14 +547,14 @@ two disjoint finite index sets, then the tuple formed by `f i` for `i ∈ S` is tuple `(f i)_i` for `i ∈ T`. -/ lemma Indep_fun.indep_fun_finset [is_probability_measure μ] {ι : Type*} {β : ι → Type*} {m : Π i, measurable_space (β i)} - {f : Π i, α → β i} (S T : finset ι) (hST : disjoint S T) (hf_Indep : Indep_fun m f μ) + {f : Π i, Ω → β i} (S T : finset ι) (hST : disjoint S T) (hf_Indep : Indep_fun m f μ) (hf_meas : ∀ i, measurable (f i)) : indep_fun (λ a (i : S), f i a) (λ a (i : T), f i a) μ := begin -- We introduce π-systems, build from the π-system of boxes which generates `measurable_space.pi`. let πSβ := (set.pi (set.univ : set S) '' (set.pi (set.univ : set S) (λ i, {s : set (β i) | measurable_set[m i] s}))), - let πS := {s : set α | ∃ t ∈ πSβ, (λ a (i : S), f i a) ⁻¹' t = s}, + let πS := {s : set Ω | ∃ t ∈ πSβ, (λ a (i : S), f i a) ⁻¹' t = s}, have hπS_pi : is_pi_system πS := is_pi_system_pi.comap (λ a i, f i a), have hπS_gen : measurable_space.pi.comap (λ a (i : S), f i a) = generate_from πS, { rw [generate_from_pi.symm, comap_generate_from], @@ -563,7 +563,7 @@ begin { exact finset.fintype_coe_sort S, }, }, let πTβ := (set.pi (set.univ : set T) '' (set.pi (set.univ : set T) (λ i, {s : set (β i) | measurable_set[m i] s}))), - let πT := {s : set α | ∃ t ∈ πTβ, (λ a (i : T), f i a) ⁻¹' t = s}, + let πT := {s : set Ω | ∃ t ∈ πTβ, (λ a (i : T), f i a) ⁻¹' t = s}, have hπT_pi : is_pi_system πT := is_pi_system_pi.comap (λ a i, f i a), have hπT_gen : measurable_space.pi.comap (λ a (i : T), f i a) = generate_from πT, { rw [generate_from_pi.symm, comap_generate_from], @@ -590,14 +590,14 @@ begin { intros i hi, rw h_sets_s'_eq hi, exact hs1 _, }, have h_meas_t' : ∀ i ∈ T, measurable_set (sets_t' i), { intros i hi, simp_rw [sets_t', dif_pos hi], exact ht1 _, }, - have h_eq_inter_S : (λ (a : α) (i : ↥S), f ↑i a) ⁻¹' set.pi set.univ sets_s + have h_eq_inter_S : (λ (ω : Ω) (i : ↥S), f ↑i ω) ⁻¹' set.pi set.univ sets_s = ⋂ i ∈ S, (f i) ⁻¹' (sets_s' i), { ext1 x, simp only [set.mem_preimage, set.mem_univ_pi, set.mem_Inter], split; intro h, { intros i hi, rw [h_sets_s'_eq hi], exact h ⟨i, hi⟩, }, { rintros ⟨i, hi⟩, specialize h i hi, rw [h_sets_s'_eq hi] at h, exact h, }, }, - have h_eq_inter_T : (λ (a : α) (i : ↥T), f ↑i a) ⁻¹' set.pi set.univ sets_t + have h_eq_inter_T : (λ (ω : Ω) (i : ↥T), f ↑i ω) ⁻¹' set.pi set.univ sets_t = ⋂ i ∈ T, (f i) ⁻¹' (sets_t' i), { ext1 x, simp only [set.mem_preimage, set.mem_univ_pi, set.mem_Inter], @@ -632,7 +632,7 @@ end lemma Indep_fun.indep_fun_prod [is_probability_measure μ] {ι : Type*} {β : ι → Type*} {m : Π i, measurable_space (β i)} - {f : Π i, α → β i} (hf_Indep : Indep_fun m f μ) (hf_meas : ∀ i, measurable (f i)) + {f : Π i, Ω → β i} (hf_Indep : Indep_fun m f μ) (hf_meas : ∀ i, measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) : indep_fun (λ a, (f i a, f j a)) (f k) μ := begin @@ -667,7 +667,7 @@ end @[to_additive] lemma Indep_fun.mul [is_probability_measure μ] {ι : Type*} {β : Type*} {m : measurable_space β} [has_mul β] [has_measurable_mul₂ β] - {f : ι → α → β} (hf_Indep : Indep_fun (λ _, m) f μ) (hf_meas : ∀ i, measurable (f i)) + {f : ι → Ω → β} (hf_Indep : Indep_fun (λ _, m) f μ) (hf_meas : ∀ i, measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) : indep_fun (f i * f j) (f k) μ := begin @@ -679,7 +679,7 @@ end @[to_additive] lemma Indep_fun.indep_fun_finset_prod_of_not_mem [is_probability_measure μ] {ι : Type*} {β : Type*} {m : measurable_space β} [comm_monoid β] [has_measurable_mul₂ β] - {f : ι → α → β} (hf_Indep : Indep_fun (λ _, m) f μ) (hf_meas : ∀ i, measurable (f i)) + {f : ι → Ω → β} (hf_Indep : Indep_fun (λ _, m) f μ) (hf_meas : ∀ i, measurable (f i)) {s : finset ι} {i : ι} (hi : i ∉ s) : indep_fun (∏ j in s, f j) (f i) μ := begin @@ -704,7 +704,7 @@ end @[to_additive] lemma Indep_fun.indep_fun_prod_range_succ [is_probability_measure μ] {β : Type*} {m : measurable_space β} [comm_monoid β] [has_measurable_mul₂ β] - {f : ℕ → α → β} (hf_Indep : Indep_fun (λ _, m) f μ) (hf_meas : ∀ i, measurable (f i)) + {f : ℕ → Ω → β} (hf_Indep : Indep_fun (λ _, m) f μ) (hf_meas : ∀ i, measurable (f i)) (n : ℕ) : indep_fun (∏ j in finset.range n, f j) (f n) μ := hf_Indep.indep_fun_finset_prod_of_not_mem hf_meas finset.not_mem_range_self diff --git a/formal/lean/mathlib/probability/integration.lean b/formal/lean/mathlib/probability/integration.lean index d91f922d314fee5da36e5773a226a972c0009781..7a28eaab17b53e2e4480b7d15f440def121ad9f8 100644 --- a/formal/lean/mathlib/probability/integration.lean +++ b/formal/lean/mathlib/probability/integration.lean @@ -20,8 +20,8 @@ will always pick the later typeclass in this situation, and does not care whethe `[]`, `{}`, or `()`. All of these use the `measurable_space` `M2` to define `μ`: ```lean -example {M1 : measurable_space α} [M2 : measurable_space α] {μ : measure α} : sorry := sorry -example [M1 : measurable_space α] {M2 : measurable_space α} {μ : measure α} : sorry := sorry +example {M1 : measurable_space Ω} [M2 : measurable_space Ω] {μ : measure Ω} : sorry := sorry +example [M1 : measurable_space Ω] {M2 : measurable_space Ω} {μ : measure Ω} : sorry := sorry ``` -/ @@ -30,17 +30,17 @@ noncomputable theory open set measure_theory open_locale ennreal measure_theory -variables {α : Type*} {mα : measurable_space α} {μ : measure α} {f g : α → ℝ≥0∞} {X Y : α → ℝ} +variables {Ω : Type*} {mΩ : measurable_space Ω} {μ : measure Ω} {f g : Ω → ℝ≥0∞} {X Y : Ω → ℝ} namespace probability_theory /-- If a random variable `f` in `ℝ≥0∞` is independent of an event `T`, then if you restrict the random variable to `T`, then `E[f * indicator T c 0]=E[f] * E[indicator T c 0]`. It is useful for `lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurable_space`. -/ -lemma lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator {Mf mα : measurable_space α} - {μ : measure α} (hMf : Mf ≤ mα) (c : ℝ≥0∞) {T : set α} (h_meas_T : measurable_set T) +lemma lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator {Mf mΩ : measurable_space Ω} + {μ : measure Ω} (hMf : Mf ≤ mΩ) (c : ℝ≥0∞) {T : set Ω} (h_meas_T : measurable_set T) (h_ind : indep_sets {s | measurable_set[Mf] s} {T} μ) (h_meas_f : measurable[Mf] f) : - ∫⁻ a, f a * T.indicator (λ _, c) a ∂μ = ∫⁻ a, f a ∂μ * ∫⁻ a, T.indicator (λ _, c) a ∂μ := + ∫⁻ ω, f ω * T.indicator (λ _, c) ω ∂μ = ∫⁻ ω, f ω ∂μ * ∫⁻ ω, T.indicator (λ _, c) ω ∂μ := begin revert f, have h_mul_indicator : ∀ g, measurable g → measurable (λ a, g a * T.indicator (λ x, c) a), @@ -77,10 +77,10 @@ end independence. See `lintegral_mul_eq_lintegral_mul_lintegral_of_independent_fn` for a more common variant of the product of independent variables. -/ lemma lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurable_space - {Mf Mg mα : measurable_space α} {μ : measure α} - (hMf : Mf ≤ mα) (hMg : Mg ≤ mα) (h_ind : indep Mf Mg μ) + {Mf Mg mΩ : measurable_space Ω} {μ : measure Ω} + (hMf : Mf ≤ mΩ) (hMg : Mg ≤ mΩ) (h_ind : indep Mf Mg μ) (h_meas_f : measurable[Mf] f) (h_meas_g : measurable[Mg] g) : - ∫⁻ a, f a * g a ∂μ = ∫⁻ a, f a ∂μ * ∫⁻ a, g a ∂μ := + ∫⁻ ω, f ω * g ω ∂μ = ∫⁻ ω, f ω ∂μ * ∫⁻ ω, g ω ∂μ := begin revert g, have h_measM_f : measurable f, from h_meas_f.mono hMf le_rfl, @@ -108,7 +108,7 @@ end then `E[f * g] = E[f] * E[g]`. -/ lemma lintegral_mul_eq_lintegral_mul_lintegral_of_indep_fun (h_meas_f : measurable f) (h_meas_g : measurable g) (h_indep_fun : indep_fun f g μ) : - ∫⁻ a, (f * g) a ∂μ = ∫⁻ a, f a ∂μ * ∫⁻ a, g a ∂μ := + ∫⁻ ω, (f * g) ω ∂μ = ∫⁻ ω, f ω ∂μ * ∫⁻ ω, g ω ∂μ := lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurable_space (measurable_iff_comap_le.1 h_meas_f) (measurable_iff_comap_le.1 h_meas_g) h_indep_fun (measurable.of_comap_le le_rfl) (measurable.of_comap_le le_rfl) @@ -118,7 +118,7 @@ lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurable_space `lintegral_mul_eq_lintegral_mul_lintegral_of_indep_fun`). -/ lemma lintegral_mul_eq_lintegral_mul_lintegral_of_indep_fun' (h_meas_f : ae_measurable f μ) (h_meas_g : ae_measurable g μ) (h_indep_fun : indep_fun f g μ) : - ∫⁻ a, (f * g) a ∂μ = ∫⁻ a, f a ∂μ * ∫⁻ a, g a ∂μ := + ∫⁻ ω, (f * g) ω ∂μ = ∫⁻ ω, f ω ∂μ * ∫⁻ ω, g ω ∂μ := begin have fg_ae : f * g =ᵐ[μ] (h_meas_f.mk _) * (h_meas_g.mk _), from h_meas_f.ae_eq_mk.mul h_meas_g.ae_eq_mk, @@ -130,13 +130,13 @@ begin end /-- The product of two independent, integrable, real_valued random variables is integrable. -/ -lemma indep_fun.integrable_mul {β : Type*} [measurable_space β] {X Y : α → β} +lemma indep_fun.integrable_mul {β : Type*} [measurable_space β] {X Y : Ω → β} [normed_division_ring β] [borel_space β] (hXY : indep_fun X Y μ) (hX : integrable X μ) (hY : integrable Y μ) : integrable (X * Y) μ := begin - let nX : α → ennreal := λ a, ∥X a∥₊, - let nY : α → ennreal := λ a, ∥Y a∥₊, + let nX : Ω → ennreal := λ a, ∥X a∥₊, + let nY : Ω → ennreal := λ a, ∥Y a∥₊, have hXY' : indep_fun (λ a, ∥X a∥₊) (λ a, ∥Y a∥₊) μ := hXY.comp measurable_nnnorm measurable_nnnorm, @@ -244,7 +244,7 @@ hXY.integral_mul_of_integrable hX hY satisfying appropriate integrability conditions. -/ theorem indep_fun_iff_integral_comp_mul [is_finite_measure μ] {β β' : Type*} {mβ : measurable_space β} {mβ' : measurable_space β'} - {f : α → β} {g : α → β'} {hfm : measurable f} {hgm : measurable g} : + {f : Ω → β} {g : Ω → β'} {hfm : measurable f} {hgm : measurable g} : indep_fun f g μ ↔ ∀ {φ : β → ℝ} {ψ : β' → ℝ}, measurable φ → measurable ψ → integrable (φ ∘ f) μ → integrable (ψ ∘ g) μ → diff --git a/formal/lean/mathlib/probability/martingale.lean b/formal/lean/mathlib/probability/martingale/basic.lean similarity index 76% rename from formal/lean/mathlib/probability/martingale.lean rename to formal/lean/mathlib/probability/martingale/basic.lean index a793f225030b4c55dc631a9d8a45c413c6ecb637..e5a83bcbccf4931fa11c2d1be5459c614fe21499 100644 --- a/formal/lean/mathlib/probability/martingale.lean +++ b/formal/lean/mathlib/probability/martingale/basic.lean @@ -9,11 +9,11 @@ import probability.hitting_time /-! # Martingales -A family of functions `f : ι → α → E` is a martingale with respect to a filtration `ℱ` if every +A family of functions `f : ι → Ω → E` is a martingale with respect to a filtration `ℱ` if every `f i` is integrable, `f` is adapted with respect to `ℱ` and for all `i ≤ j`, -`μ[f j | ℱ i] =ᵐ[μ] f i`. On the other hand, `f : ι → α → E` is said to be a supermartingale +`μ[f j | ℱ i] =ᵐ[μ] f i`. On the other hand, `f : ι → Ω → E` is said to be a supermartingale with respect to the filtration `ℱ` if `f i` is integrable, `f` is adapted with resepct to `ℱ` -and for all `i ≤ j`, `μ[f j | ℱ i] ≤ᵐ[μ] f i`. Finally, `f : ι → α → E` is said to be a +and for all `i ≤ j`, `μ[f j | ℱ i] ≤ᵐ[μ] f i`. Finally, `f : ι → Ω → E` is said to be a submartingale with respect to the filtration `ℱ` if `f i` is integrable, `f` is adapted with resepct to `ℱ` and for all `i ≤ j`, `f i ≤ᵐ[μ] μ[f j | ℱ i]`. @@ -40,31 +40,35 @@ open_locale nnreal ennreal measure_theory probability_theory big_operators namespace measure_theory -variables {α E ι : Type*} [preorder ι] - {m0 : measurable_space α} {μ : measure α} +variables {Ω E ι : Type*} [preorder ι] + {m0 : measurable_space Ω} {μ : measure Ω} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] - {f g : ι → α → E} {ℱ : filtration ι m0} + {f g : ι → Ω → E} {ℱ : filtration ι m0} -/-- A family of functions `f : ι → α → E` is a martingale with respect to a filtration `ℱ` if `f` +/-- A family of functions `f : ι → Ω → E` is a martingale with respect to a filtration `ℱ` if `f` is adapted with respect to `ℱ` and for all `i ≤ j`, `μ[f j | ℱ i] =ᵐ[μ] f i`. -/ -def martingale (f : ι → α → E) (ℱ : filtration ι m0) (μ : measure α) : Prop := +def martingale (f : ι → Ω → E) (ℱ : filtration ι m0) (μ : measure Ω) : Prop := adapted ℱ f ∧ ∀ i j, i ≤ j → μ[f j | ℱ i] =ᵐ[μ] f i -/-- A family of integrable functions `f : ι → α → E` is a supermartingale with respect to a +/-- A family of integrable functions `f : ι → Ω → E` is a supermartingale with respect to a filtration `ℱ` if `f` is adapted with respect to `ℱ` and for all `i ≤ j`, `μ[f j | ℱ.le i] ≤ᵐ[μ] f i`. -/ -def supermartingale [has_le E] (f : ι → α → E) (ℱ : filtration ι m0) (μ : measure α) : Prop := +def supermartingale [has_le E] (f : ι → Ω → E) (ℱ : filtration ι m0) (μ : measure Ω) : Prop := adapted ℱ f ∧ (∀ i j, i ≤ j → μ[f j | ℱ i] ≤ᵐ[μ] f i) ∧ ∀ i, integrable (f i) μ -/-- A family of integrable functions `f : ι → α → E` is a submartingale with respect to a +/-- A family of integrable functions `f : ι → Ω → E` is a submartingale with respect to a filtration `ℱ` if `f` is adapted with respect to `ℱ` and for all `i ≤ j`, `f i ≤ᵐ[μ] μ[f j | ℱ.le i]`. -/ -def submartingale [has_le E] (f : ι → α → E) (ℱ : filtration ι m0) (μ : measure α) : Prop := +def submartingale [has_le E] (f : ι → Ω → E) (ℱ : filtration ι m0) (μ : measure Ω) : Prop := adapted ℱ f ∧ (∀ i j, i ≤ j → f i ≤ᵐ[μ] μ[f j | ℱ i]) ∧ ∀ i, integrable (f i) μ +lemma martingale_const (ℱ : filtration ι m0) (μ : measure Ω) [is_finite_measure μ] (x : E) : + martingale (λ _ _, x) ℱ μ := +⟨adapted_const ℱ _, λ i j hij, by rw condexp_const (ℱ.le _)⟩ + variables (E) -lemma martingale_zero (ℱ : filtration ι m0) (μ : measure α) : - martingale (0 : ι → α → E) ℱ μ := +lemma martingale_zero (ℱ : filtration ι m0) (μ : measure Ω) : + martingale (0 : ι → Ω → E) ℱ μ := ⟨adapted_zero E ℱ, λ i j hij, by { rw [pi.zero_apply, condexp_zero], simp, }⟩ variables {E} @@ -86,7 +90,7 @@ lemma integrable (hf : martingale f ℱ μ) (i : ι) : integrable (f i) μ := integrable_condexp.congr (hf.condexp_ae_eq (le_refl i)) lemma set_integral_eq [sigma_finite_filtration μ ℱ] (hf : martingale f ℱ μ) {i j : ι} (hij : i ≤ j) - {s : set α} (hs : measurable_set[ℱ i] s) : + {s : set Ω} (hs : measurable_set[ℱ i] s) : ∫ x in s, f i x ∂μ = ∫ x in s, f j x ∂μ := begin rw ← @set_integral_condexp _ _ _ _ _ (ℱ i) m0 _ _ _ (ℱ.le i) _ (hf.integrable j) hs, @@ -127,7 +131,7 @@ lemma martingale_iff [partial_order E] : martingale f ℱ μ ↔ ⟨λ hf, ⟨hf.supermartingale, hf.submartingale⟩, λ ⟨hf₁, hf₂⟩, ⟨hf₁.1, λ i j hij, (hf₁.2.1 i j hij).antisymm (hf₂.2.1 i j hij)⟩⟩ -lemma martingale_condexp (f : α → E) (ℱ : filtration ι m0) (μ : measure α) +lemma martingale_condexp (f : Ω → E) (ℱ : filtration ι m0) (μ : measure Ω) [sigma_finite_filtration μ ℱ] : martingale (λ i, μ[f | ℱ i]) ℱ μ := ⟨λ i, strongly_measurable_condexp, λ i j hij, condexp_condexp_of_le (ℱ.mono hij) (ℱ.le j)⟩ @@ -149,8 +153,8 @@ lemma condexp_ae_le [has_le E] (hf : supermartingale f ℱ μ) {i j : ι} (hij : μ[f j | ℱ i] ≤ᵐ[μ] f i := hf.2.1 i j hij -lemma set_integral_le [sigma_finite_filtration μ ℱ] {f : ι → α → ℝ} (hf : supermartingale f ℱ μ) - {i j : ι} (hij : i ≤ j) {s : set α} (hs : measurable_set[ℱ i] s) : +lemma set_integral_le [sigma_finite_filtration μ ℱ] {f : ι → Ω → ℝ} (hf : supermartingale f ℱ μ) + {i j : ι} (hij : i ≤ j) {s : set Ω} (hs : measurable_set[ℱ i] s) : ∫ x in s, f j x ∂μ ≤ ∫ x in s, f i x ∂μ := begin rw ← set_integral_condexp (ℱ.le i) (hf.integrable j) hs, @@ -223,8 +227,8 @@ begin end /-- The converse of this lemma is `measure_theory.submartingale_of_set_integral_le`. -/ -lemma set_integral_le [sigma_finite_filtration μ ℱ] {f : ι → α → ℝ} (hf : submartingale f ℱ μ) - {i j : ι} (hij : i ≤ j) {s : set α} (hs : measurable_set[ℱ i] s) : +lemma set_integral_le [sigma_finite_filtration μ ℱ] {f : ι → Ω → ℝ} (hf : submartingale f ℱ μ) + {i j : ι} (hij : i ≤ j) {s : set Ω} (hs : measurable_set[ℱ i] s) : ∫ x in s, f i x ∂μ ≤ ∫ x in s, f j x ∂μ := begin rw [← neg_le_neg_iff, ← integral_neg, ← integral_neg], @@ -239,7 +243,7 @@ lemma sub_martingale [preorder E] [covariant_class E E (+) (≤)] (hf : submartingale f ℱ μ) (hg : martingale g ℱ μ) : submartingale (f - g) ℱ μ := hf.sub_supermartingale hg.supermartingale -protected lemma sup {f g : ι → α → ℝ} (hf : submartingale f ℱ μ) (hg : submartingale g ℱ μ) : +protected lemma sup {f g : ι → Ω → ℝ} (hf : submartingale f ℱ μ) (hg : submartingale g ℱ μ) : submartingale (f ⊔ g) ℱ μ := begin refine ⟨λ i, @strongly_measurable.sup _ _ _ _ (ℱ i) _ _ _ (hf.adapted i) (hg.adapted i), @@ -253,7 +257,7 @@ begin (eventually_of_forall (λ x, le_max_right _ _))) } end -protected lemma pos {f : ι → α → ℝ} (hf : submartingale f ℱ μ) : +protected lemma pos {f : ι → Ω → ℝ} (hf : submartingale f ℱ μ) : submartingale (f⁺) ℱ μ := hf.sup (martingale_zero _ _ _).submartingale @@ -262,8 +266,8 @@ end submartingale section submartingale lemma submartingale_of_set_integral_le [is_finite_measure μ] - {f : ι → α → ℝ} (hadp : adapted ℱ f) (hint : ∀ i, integrable (f i) μ) - (hf : ∀ i j : ι, i ≤ j → ∀ s : set α, measurable_set[ℱ i] s → + {f : ι → Ω → ℝ} (hadp : adapted ℱ f) (hint : ∀ i, integrable (f i) μ) + (hf : ∀ i j : ι, i ≤ j → ∀ s : set Ω, measurable_set[ℱ i] s → ∫ x in s, f i x ∂μ ≤ ∫ x in s, f j x ∂μ) : submartingale f ℱ μ := begin @@ -273,9 +277,9 @@ begin suffices : 0 ≤ᵐ[μ.trim (ℱ.le i)] μ[f j| ℱ i] - f i, { filter_upwards [this] with x hx, rwa ← sub_nonneg }, - refine ae_nonneg_of_forall_set_integral_nonneg_of_finite_measure + refine ae_nonneg_of_forall_set_integral_nonneg ((integrable_condexp.sub (hint i)).trim _ (strongly_measurable_condexp.sub $ hadp i)) - (λ s hs, _), + (λ s hs h's, _), specialize hf i j hij s hs, rwa [← set_integral_trim _ (strongly_measurable_condexp.sub $ hadp i) hs, integral_sub' integrable_condexp.integrable_on (hint i).integrable_on, sub_nonneg, @@ -283,7 +287,7 @@ begin end lemma submartingale_of_condexp_sub_nonneg [is_finite_measure μ] - {f : ι → α → ℝ} (hadp : adapted ℱ f) (hint : ∀ i, integrable (f i) μ) + {f : ι → Ω → ℝ} (hadp : adapted ℱ f) (hint : ∀ i, integrable (f i) μ) (hf : ∀ i j, i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i | ℱ i]) : submartingale f ℱ μ := begin @@ -293,18 +297,20 @@ begin apply_instance end -lemma submartingale.condexp_sub_nonneg [is_finite_measure μ] - {f : ι → α → ℝ} (hf : submartingale f ℱ μ) {i j : ι} (hij : i ≤ j) : +lemma submartingale.condexp_sub_nonneg + {f : ι → Ω → ℝ} (hf : submartingale f ℱ μ) {i j : ι} (hij : i ≤ j) : 0 ≤ᵐ[μ] μ[f j - f i | ℱ i] := begin + by_cases h : sigma_finite (μ.trim (ℱ.le i)), + swap, { rw condexp_of_not_sigma_finite (ℱ.le i) h }, refine eventually_le.trans _ (condexp_sub (hf.integrable _) (hf.integrable _)).symm.le, rw [eventually_sub_nonneg, condexp_of_strongly_measurable (ℱ.le _) (hf.adapted _) (hf.integrable _)], - exact hf.2.1 i j hij, - apply_instance + { exact hf.2.1 i j hij }, + { exact h } end -lemma submartingale_iff_condexp_sub_nonneg [is_finite_measure μ] {f : ι → α → ℝ} : +lemma submartingale_iff_condexp_sub_nonneg [is_finite_measure μ] {f : ι → Ω → ℝ} : submartingale f ℱ μ ↔ adapted ℱ f ∧ (∀ i, integrable (f i) μ) ∧ ∀ i j, i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i | ℱ i] := ⟨λ h, ⟨h.adapted, h.integrable, λ i j, h.condexp_sub_nonneg⟩, @@ -327,7 +333,7 @@ section variables {F : Type*} [normed_lattice_add_comm_group F] [normed_space ℝ F] [complete_space F] [ordered_smul ℝ F] -lemma smul_nonneg {f : ι → α → F} +lemma smul_nonneg {f : ι → Ω → F} {c : ℝ} (hc : 0 ≤ c) (hf : supermartingale f ℱ μ) : supermartingale (c • f) ℱ μ := begin @@ -338,7 +344,7 @@ begin exact smul_le_smul_of_nonneg hle hc, end -lemma smul_nonpos {f : ι → α → F} +lemma smul_nonpos {f : ι → Ω → F} {c : ℝ} (hc : c ≤ 0) (hf : supermartingale f ℱ μ) : submartingale (c • f) ℱ μ := begin @@ -357,7 +363,7 @@ section variables {F : Type*} [normed_lattice_add_comm_group F] [normed_space ℝ F] [complete_space F] [ordered_smul ℝ F] -lemma smul_nonneg {f : ι → α → F} +lemma smul_nonneg {f : ι → Ω → F} {c : ℝ} (hc : 0 ≤ c) (hf : submartingale f ℱ μ) : submartingale (c • f) ℱ μ := begin @@ -365,7 +371,7 @@ begin exact supermartingale.neg (hf.neg.smul_nonneg hc), end -lemma smul_nonpos {f : ι → α → F} +lemma smul_nonpos {f : ι → Ω → F} {c : ℝ} (hc : c ≤ 0) (hf : submartingale f ℱ μ) : supermartingale (c • f) ℱ μ := begin @@ -382,8 +388,8 @@ section nat variables {𝒢 : filtration ℕ m0} lemma submartingale_of_set_integral_le_succ [is_finite_measure μ] - {f : ℕ → α → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) - (hf : ∀ i, ∀ s : set α, measurable_set[𝒢 i] s → ∫ x in s, f i x ∂μ ≤ ∫ x in s, f (i + 1) x ∂μ) : + {f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) + (hf : ∀ i, ∀ s : set Ω, measurable_set[𝒢 i] s → ∫ x in s, f i x ∂μ ≤ ∫ x in s, f (i + 1) x ∂μ) : submartingale f 𝒢 μ := begin refine submartingale_of_set_integral_le hadp hint (λ i j hij s hs, _), @@ -392,8 +398,26 @@ begin { exact le_trans hk₂ (hf k s (𝒢.mono hk₁ _ hs)) } end +lemma supermartingale_of_set_integral_succ_le [is_finite_measure μ] + {f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) + (hf : ∀ i, ∀ s : set Ω, measurable_set[𝒢 i] s → ∫ x in s, f (i + 1) x ∂μ ≤ ∫ x in s, f i x ∂μ) : + supermartingale f 𝒢 μ := +begin + rw ← neg_neg f, + refine (submartingale_of_set_integral_le_succ hadp.neg (λ i, (hint i).neg) _).neg, + simpa only [integral_neg, pi.neg_apply, neg_le_neg_iff], +end + +lemma martingale_of_set_integral_eq_succ [is_finite_measure μ] + {f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) + (hf : ∀ i, ∀ s : set Ω, measurable_set[𝒢 i] s → ∫ x in s, f i x ∂μ = ∫ x in s, f (i + 1) x ∂μ) : + martingale f 𝒢 μ := +martingale_iff.2 + ⟨supermartingale_of_set_integral_succ_le hadp hint $ λ i s hs, (hf i s hs).ge, + submartingale_of_set_integral_le_succ hadp hint $ λ i s hs, (hf i s hs).le⟩ + lemma submartingale_nat [is_finite_measure μ] - {f : ℕ → α → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) + {f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) (hf : ∀ i, f i ≤ᵐ[μ] μ[f (i + 1) | 𝒢 i]) : submartingale f 𝒢 μ := begin @@ -404,8 +428,26 @@ begin exact set_integral_mono_ae (hint i).integrable_on integrable_condexp.integrable_on (hf i), end +lemma supermartingale_nat [is_finite_measure μ] + {f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) + (hf : ∀ i, μ[f (i + 1) | 𝒢 i] ≤ᵐ[μ] f i) : + supermartingale f 𝒢 μ := +begin + rw ← neg_neg f, + refine (submartingale_nat hadp.neg (λ i, (hint i).neg) $ λ i, + eventually_le.trans _ (condexp_neg _).symm.le).neg, + filter_upwards [hf i] with x hx using neg_le_neg hx, +end + +lemma martingale_nat [is_finite_measure μ] + {f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) + (hf : ∀ i, f i =ᵐ[μ] μ[f (i + 1) | 𝒢 i]) : + martingale f 𝒢 μ := +martingale_iff.2 ⟨supermartingale_nat hadp hint $ λ i, (hf i).symm.le, + submartingale_nat hadp hint $ λ i, (hf i).le⟩ + lemma submartingale_of_condexp_sub_nonneg_nat [is_finite_measure μ] - {f : ℕ → α → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) + {f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) (hf : ∀ i, 0 ≤ᵐ[μ] μ[f (i + 1) - f i | 𝒢 i]) : submartingale f 𝒢 μ := begin @@ -415,9 +457,32 @@ begin apply_instance end +lemma supermartingale_of_condexp_sub_nonneg_nat [is_finite_measure μ] + {f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) + (hf : ∀ i, 0 ≤ᵐ[μ] μ[f i - f (i + 1) | 𝒢 i]) : + supermartingale f 𝒢 μ := +begin + rw ← neg_neg f, + refine (submartingale_of_condexp_sub_nonneg_nat hadp.neg (λ i, (hint i).neg) _).neg, + simpa only [pi.zero_apply, pi.neg_apply, neg_sub_neg] +end + +lemma martingale_of_condexp_sub_eq_zero_nat [is_finite_measure μ] + {f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) + (hf : ∀ i, μ[f (i + 1) - f i | 𝒢 i] =ᵐ[μ] 0) : + martingale f 𝒢 μ := +begin + refine martingale_iff.2 ⟨supermartingale_of_condexp_sub_nonneg_nat hadp hint $ λ i, _, + submartingale_of_condexp_sub_nonneg_nat hadp hint $ λ i, (hf i).symm.le⟩, + rw ← neg_sub, + refine (eventually_eq.trans _ (condexp_neg _).symm).le, + filter_upwards [hf i] with x hx, + simpa only [pi.zero_apply, pi.neg_apply, zero_eq_neg], +end + namespace submartingale -lemma integrable_stopped_value [has_le E] {f : ℕ → α → E} (hf : submartingale f 𝒢 μ) {τ : α → ℕ} +lemma integrable_stopped_value [has_le E] {f : ℕ → Ω → E} (hf : submartingale f 𝒢 μ) {τ : Ω → ℕ} (hτ : is_stopping_time 𝒢 τ) {N : ℕ} (hbdd : ∀ x, τ x ≤ N) : integrable (stopped_value f τ) μ := integrable_stopped_value hτ hf.integrable hbdd @@ -429,14 +494,14 @@ integrable_stopped_value hτ hf.integrable hbdd expectation of `stopped_value f τ` is less than or equal to the expectation of `stopped_value f π`. This is the forward direction of the optional stopping theorem. -/ lemma expected_stopped_value_mono [sigma_finite_filtration μ 𝒢] - {f : ℕ → α → ℝ} (hf : submartingale f 𝒢 μ) {τ π : α → ℕ} + {f : ℕ → Ω → ℝ} (hf : submartingale f 𝒢 μ) {τ π : Ω → ℕ} (hτ : is_stopping_time 𝒢 τ) (hπ : is_stopping_time 𝒢 π) (hle : τ ≤ π) {N : ℕ} (hbdd : ∀ x, π x ≤ N) : μ[stopped_value f τ] ≤ μ[stopped_value f π] := begin rw [← sub_nonneg, ← integral_sub', stopped_value_sub_eq_sum' hle hbdd], { simp only [finset.sum_apply], - have : ∀ i, measurable_set[𝒢 i] {x : α | τ x ≤ i ∧ i < π x}, + have : ∀ i, measurable_set[𝒢 i] {ω : Ω | τ ω ≤ i ∧ i < π ω}, { intro i, refine (hτ i).inter _, convert (hπ i).compl, @@ -461,8 +526,8 @@ end submartingale is a submartingale if for all bounded stopping times `τ` and `π` such that `τ ≤ π`, the stopped value of `f` at `τ` has expectation smaller than its stopped value at `π`. -/ lemma submartingale_of_expected_stopped_value_mono [is_finite_measure μ] - {f : ℕ → α → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) - (hf : ∀ τ π : α → ℕ, is_stopping_time 𝒢 τ → is_stopping_time 𝒢 π → τ ≤ π → (∃ N, ∀ x, π x ≤ N) → + {f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) + (hf : ∀ τ π : Ω → ℕ, is_stopping_time 𝒢 τ → is_stopping_time 𝒢 π → τ ≤ π → (∃ N, ∀ x, π x ≤ N) → μ[stopped_value f τ] ≤ μ[stopped_value f π]) : submartingale f 𝒢 μ := begin @@ -481,9 +546,9 @@ end is a submartingale if and only if for all bounded stopping times `τ` and `π` such that `τ ≤ π`, the stopped value of `f` at `τ` has expectation smaller than its stopped value at `π`. -/ lemma submartingale_iff_expected_stopped_value_mono [is_finite_measure μ] - {f : ℕ → α → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) : + {f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) : submartingale f 𝒢 μ ↔ - ∀ τ π : α → ℕ, is_stopping_time 𝒢 τ → is_stopping_time 𝒢 π → τ ≤ π → (∃ N, ∀ x, π x ≤ N) → + ∀ τ π : Ω → ℕ, is_stopping_time 𝒢 τ → is_stopping_time 𝒢 π → τ ≤ π → (∃ N, ∀ x, π x ≤ N) → μ[stopped_value f τ] ≤ μ[stopped_value f π] := ⟨λ hf _ _ hτ hπ hle ⟨N, hN⟩, hf.expected_stopped_value_mono hτ hπ hle hN, submartingale_of_expected_stopped_value_mono hadp hint⟩ @@ -493,7 +558,7 @@ section maximal open finset lemma smul_le_stopped_value_hitting [is_finite_measure μ] - {f : ℕ → α → ℝ} (hsub : submartingale f 𝒢 μ) {ε : ℝ≥0} (n : ℕ) : + {f : ℕ → Ω → ℝ} (hsub : submartingale f 𝒢 μ) {ε : ℝ≥0} (n : ℕ) : ε • μ {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)} ≤ ennreal.of_real (∫ x in {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)}, stopped_value f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) x ∂μ) := @@ -524,7 +589,7 @@ we have `ε • μ {ε ≤ f* n} ≤ ∫ x in {ε ≤ f* n}, f n` where `f* n x In some literature, the Doob's maximal inequality refers to what we call Doob's Lp inequality (which is a corollary of this lemma and will be proved in an upcomming PR). -/ lemma maximal_ineq [is_finite_measure μ] - {f : ℕ → α → ℝ} (hsub : submartingale f 𝒢 μ) (hnonneg : 0 ≤ f) {ε : ℝ≥0} (n : ℕ) : + {f : ℕ → Ω → ℝ} (hsub : submartingale f 𝒢 μ) (hnonneg : 0 ≤ f) {ε : ℝ≥0} (n : ℕ) : ε • μ {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)} ≤ ennreal.of_real (∫ x in {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)}, f n x ∂μ) := @@ -532,31 +597,31 @@ begin suffices : ε • μ {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)} + ennreal.of_real (∫ x in {x | ((range (n + 1)).sup' nonempty_range_succ (λ k, f k x)) < ε}, f n x ∂μ) ≤ ennreal.of_real (μ[f n]), - { have hadd : ennreal.of_real (∫ (x : α), f n x ∂μ) = - ennreal.of_real (∫ (x : α) in - {x : α | ↑ε ≤ ((range (n + 1)).sup' nonempty_range_succ (λ k, f k x))}, f n x ∂μ) + - ennreal.of_real (∫ (x : α) in - {x : α | ((range (n + 1)).sup' nonempty_range_succ (λ k, f k x)) < ↑ε}, f n x ∂μ), + { have hadd : ennreal.of_real (∫ ω, f n ω ∂μ) = + ennreal.of_real (∫ ω in + {ω | ↑ε ≤ ((range (n + 1)).sup' nonempty_range_succ (λ k, f k ω))}, f n ω ∂μ) + + ennreal.of_real (∫ ω in + {ω | ((range (n + 1)).sup' nonempty_range_succ (λ k, f k ω)) < ↑ε}, f n ω ∂μ), { rw [← ennreal.of_real_add, ← integral_union], { conv_lhs { rw ← integral_univ }, convert rfl, - ext x, + ext ω, change (ε : ℝ) ≤ _ ∨ _ < (ε : ℝ) ↔ _, simp only [le_or_lt, true_iff] }, - { rintro x ⟨hx₁ : _ ≤ _, hx₂ : _ < _⟩, - exact (not_le.2 hx₂) hx₁ }, + { rintro ω ⟨hω₁ : _ ≤ _, hω₂ : _ < _⟩, + exact (not_le.2 hω₂) hω₁ }, { exact (measurable_set_lt (finset.measurable_range_sup'' (λ n _, (hsub.strongly_measurable n).measurable.le (𝒢.le n))) measurable_const) }, exacts [(hsub.integrable _).integrable_on, (hsub.integrable _).integrable_on, integral_nonneg (hnonneg _), integral_nonneg (hnonneg _)] }, rwa [hadd, ennreal.add_le_add_iff_right ennreal.of_real_ne_top] at this }, - calc ε • μ {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)} - + ennreal.of_real (∫ x in {x | ((range (n + 1)).sup' nonempty_range_succ (λ k, f k x)) < ε}, - f n x ∂μ) - ≤ ennreal.of_real (∫ x in {x | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k x)}, - stopped_value f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) x ∂μ) - + ennreal.of_real (∫ x in {x | ((range (n + 1)).sup' nonempty_range_succ (λ k, f k x)) < ε}, - stopped_value f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) x ∂μ) : + calc ε • μ {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k ω)} + + ennreal.of_real (∫ ω in {ω | ((range (n + 1)).sup' nonempty_range_succ (λ k, f k ω)) < ε}, + f n ω ∂μ) + ≤ ennreal.of_real (∫ ω in {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k ω)}, + stopped_value f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) ω ∂μ) + + ennreal.of_real (∫ ω in {ω | ((range (n + 1)).sup' nonempty_range_succ (λ k, f k ω)) < ε}, + stopped_value f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) ω ∂μ) : begin refine add_le_add (smul_le_stopped_value_hitting hsub _) (ennreal.of_real_le_of_real (set_integral_mono_on (hsub.integrable n).integrable_on @@ -564,26 +629,26 @@ begin (hitting_is_stopping_time hsub.adapted measurable_set_Ici) hsub.integrable hitting_le)) (measurable_set_lt (finset.measurable_range_sup'' (λ n _, (hsub.strongly_measurable n).measurable.le (𝒢.le n))) measurable_const) _)), - intros x hx, - rw set.mem_set_of_eq at hx, - have : hitting f {y : ℝ | ↑ε ≤ y} 0 n x = n, + intros ω hω, + rw set.mem_set_of_eq at hω, + have : hitting f {y : ℝ | ↑ε ≤ y} 0 n ω = n, { simp only [hitting, set.mem_set_of_eq, exists_prop, pi.coe_nat, nat.cast_id, ite_eq_right_iff, forall_exists_index, and_imp], intros m hm hεm, - exact false.elim ((not_le.2 hx) + exact false.elim ((not_le.2 hω) ((le_sup'_iff _).2 ⟨m, mem_range.2 (nat.lt_succ_of_le hm.2), hεm⟩)) }, simp_rw [stopped_value, this], end - ... = ennreal.of_real (∫ x, stopped_value f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) x ∂μ) : + ... = ennreal.of_real (∫ ω, stopped_value f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) ω ∂μ) : begin rw [← ennreal.of_real_add, ← integral_union], { conv_rhs { rw ← integral_univ }, convert rfl, - ext x, + ext ω, change _ ↔ (ε : ℝ) ≤ _ ∨ _ < (ε : ℝ), simp only [le_or_lt, iff_true] }, - { rintro x ⟨hx₁ : _ ≤ _, hx₂ : _ < _⟩, - exact (not_le.2 hx₂) hx₁ }, + { rintro ω ⟨hω₁ : _ ≤ _, hω₂ : _ < _⟩, + exact (not_le.2 hω₂) hω₁ }, { exact (measurable_set_lt (finset.measurable_range_sup'' (λ n _, (hsub.strongly_measurable n).measurable.le (𝒢.le n))) measurable_const) }, { exact (integrable.integrable_on (integrable_stopped_value @@ -598,24 +663,24 @@ begin rw ← stopped_value_const f n, exact hsub.expected_stopped_value_mono (hitting_is_stopping_time hsub.adapted measurable_set_Ici) - (is_stopping_time_const _ _) (λ x, hitting_le x) (λ x, le_rfl : ∀ x, n ≤ n), + (is_stopping_time_const _ _) (λ ω, hitting_le ω) (λ ω, le_rfl : ∀ ω, n ≤ n), end end end maximal -lemma submartingale.sum_mul_sub [is_finite_measure μ] {R : ℝ} {ξ f : ℕ → α → ℝ} +lemma submartingale.sum_mul_sub [is_finite_measure μ] {R : ℝ} {ξ f : ℕ → Ω → ℝ} (hf : submartingale f 𝒢 μ) (hξ : adapted 𝒢 ξ) - (hbdd : ∀ n x, ξ n x ≤ R) (hnonneg : ∀ n x, 0 ≤ ξ n x) : - submartingale (λ n : ℕ, ∑ k in finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ := + (hbdd : ∀ n ω, ξ n ω ≤ R) (hnonneg : ∀ n ω, 0 ≤ ξ n ω) : + submartingale (λ n, ∑ k in finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ := begin - have hξbdd : ∀ i, ∃ (C : ℝ), ∀ (x : α), |ξ i x| ≤ C := - λ i, ⟨R, λ x, (abs_of_nonneg (hnonneg i x)).trans_le (hbdd i x)⟩, + have hξbdd : ∀ i, ∃ C, ∀ ω, |ξ i ω| ≤ C := + λ i, ⟨R, λ ω, (abs_of_nonneg (hnonneg i ω)).trans_le (hbdd i ω)⟩, have hint : ∀ m, integrable (∑ k in finset.range m, ξ k * (f (k + 1) - f k)) μ := λ m, integrable_finset_sum' _ (λ i hi, integrable.bdd_mul ((hf.integrable _).sub (hf.integrable _)) hξ.strongly_measurable.ae_strongly_measurable (hξbdd _)), - have hadp : adapted 𝒢 (λ (n : ℕ), ∑ (k : ℕ) in finset.range n, ξ k * (f (k + 1) - f k)), + have hadp : adapted 𝒢 (λ n, ∑ k in finset.range n, ξ k * (f (k + 1) - f k)), { intro m, refine finset.strongly_measurable_sum' _ (λ i hi, _), rw finset.mem_range at hi, @@ -635,10 +700,10 @@ end /-- Given a discrete submartingale `f` and a predictable process `ξ` (i.e. `ξ (n + 1)` is adapted) the process defined by `λ n, ∑ k in finset.range n, ξ (k + 1) * (f (k + 1) - f k)` is also a submartingale. -/ -lemma submartingale.sum_mul_sub' [is_finite_measure μ] {R : ℝ} {ξ f : ℕ → α → ℝ} +lemma submartingale.sum_mul_sub' [is_finite_measure μ] {R : ℝ} {ξ f : ℕ → Ω → ℝ} (hf : submartingale f 𝒢 μ) (hξ : adapted 𝒢 (λ n, ξ (n + 1))) - (hbdd : ∀ n x, ξ n x ≤ R) (hnonneg : ∀ n x, 0 ≤ ξ n x) : - submartingale (λ n : ℕ, ∑ k in finset.range n, ξ (k + 1) * (f (k + 1) - f k)) 𝒢 μ := + (hbdd : ∀ n ω, ξ n ω ≤ R) (hnonneg : ∀ n ω, 0 ≤ ξ n ω) : + submartingale (λ n, ∑ k in finset.range n, ξ (k + 1) * (f (k + 1) - f k)) 𝒢 μ := hf.sum_mul_sub hξ (λ n, hbdd _) (λ n, hnonneg _) end nat diff --git a/formal/lean/mathlib/probability/martingale/convergence.lean b/formal/lean/mathlib/probability/martingale/convergence.lean new file mode 100644 index 0000000000000000000000000000000000000000..e7f36feeb71de3187eb90371605ab0a9bdf023f8 --- /dev/null +++ b/formal/lean/mathlib/probability/martingale/convergence.lean @@ -0,0 +1,234 @@ +/- +Copyright (c) 2022 Kexing Ying. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Kexing Ying +-/ +import probability.martingale.upcrossing +import measure_theory.function.uniform_integrable +import measure_theory.constructions.polish + +/-! + +# Martingale convergence theorems + +The martingale convergence theorems are a collection of theorems characterizing the convergence +of a martingale provided it satisfies some boundedness conditions. This file contains the +almost everywhere martingale convergence theorem which provides an almost everywhere limit to +an L¹ bounded submartingale. + +## Main results + +* `measure_theory.submartingale.ae_tendsto_limit_process`: the almost everywhere martingale + convergence theorem: an L¹-bounded submartingale adapted to the filtration `ℱ` converges almost + everywhere to its limit process. +* `measure_theory.submartingale.mem_ℒ1_limit_process`: the limit process of an L¹-bounded + submartingale is integrable. + +-/ + +open topological_space filter measure_theory.filtration +open_locale nnreal ennreal measure_theory probability_theory big_operators topological_space + +namespace measure_theory + +variables {Ω ι : Type*} {m0 : measurable_space Ω} {μ : measure Ω} {ℱ : filtration ℕ m0} +variables {a b : ℝ} {f : ℕ → Ω → ℝ} {ω : Ω} {R : ℝ≥0} + +section ae_convergence + +/-! + +### Almost everywhere martingale convergence theorem + +We will now prove the almost everywhere martingale convergence theorem. + +The a.e. martingale convergence theorem states: if `f` is an L¹-bounded `ℱ`-submartingale, then +it converges almost everywhere to an integrable function which is measurable with respect to +the σ-algebra `ℱ∞ := ⨆ n, ℱ n`. + +Mathematically, we proceed by first noting that a real sequence $(x_n)$ converges if +(a) $\limsup_{n \to \infty} |x_n| < \infty$, (b) for all $a < b \in \mathbb{Q}$ we have the +number of upcrossings of $(x_n)$ from below $a$ to above $b$ is finite. +Thus, for all $\omega$ satisfying $\limsup_{n \to \infty} |f_n(\omega)| < \infty$ and the number of +upcrossings of $(f_n(\omega))$ from below $a$ to above $b$ is finite for all $a < b \in \mathbb{Q}$, +we have $(f_n(\omega))$ is convergent. + +Hence, assuming $(f_n)$ is L¹-bounded, using Fatou's lemma, we have +$$ + \mathbb{E] \limsup_{n \to \infty} |f_n| \le \limsup_{n \to \infty} \mathbb{E}|f_n| < \infty +$$ +implying $\limsup_{n \to \infty} |f_n| < \infty$ a.e. Furthermore, by the upcrossing estimate, +the number of upcrossings is finite almost everywhere implying $f$ converges pointwise almost +everywhere. + +Thus, denoting $g$ the a.e. limit of $(f_n)$, $g$ is $\mathcal{F}_\infty$-measurable as for all +$n$, $f_n$ is $\mathcal{F}_n$-measurable and $\mathcal{F}_n \le \mathcal{F}_\infty$. Finally, $g$ +is integrable as $|g| \le \liminf_{n \to \infty} |f_n|$ so +$$ + \mathbb{E}|g| \le \mathbb{E} \limsup_{n \to \infty} |f_n| \le + \limsup_{n \to \infty} \mathbb{E}|f_n| < \infty +$$ +as required. + +Implementation wise, we have `tendsto_of_no_upcrossings` which showed that +a bounded sequence converges if it does not visit below $a$ and above $b$ infinitely often +for all $a, b ∈ s$ for some dense set $s$. So, we may skip the first step provided we can prove +that the realizations are bounded almost everywhere. Indeed, suppose $(|f_n(\omega)|)$ is not +bounded, then either $f_n(\omega) \to \pm \infty$ or one of $\limsup f_n(\omega)$ or +$\liminf f_n(\omega)$ equals $\pm \infty$ while the other is finite. But the first case +contradicts $\liminf |f_n(\omega)| < \infty$ while the second case contradicts finite upcrossings. + +-/ + +/-- If a stochastic process has bounded upcrossing from below `a` to above `b`, +then it does not frequently visit both below `a` and above `b`. -/ +lemma not_frequently_of_upcrossings_lt_top (hab : a < b) (hω : upcrossings a b f ω ≠ ∞) : + ¬((∃ᶠ n in at_top, f n ω < a) ∧ (∃ᶠ n in at_top, b < f n ω)) := +begin + rw [← lt_top_iff_ne_top, upcrossings_lt_top_iff] at hω, + replace hω : ∃ k, ∀ N, upcrossings_before a b f N ω < k, + { obtain ⟨k, hk⟩ := hω, + exact ⟨k + 1, λ N, lt_of_le_of_lt (hk N) k.lt_succ_self⟩ }, + rintro ⟨h₁, h₂⟩, + rw frequently_at_top at h₁ h₂, + refine not_not.2 hω _, + push_neg, + intro k, + induction k with k ih, + { simp only [zero_le', exists_const] }, + { obtain ⟨N, hN⟩ := ih, + obtain ⟨N₁, hN₁, hN₁'⟩ := h₁ N, + obtain ⟨N₂, hN₂, hN₂'⟩ := h₂ N₁, + exact ⟨(N₂ + 1), nat.succ_le_of_lt $ lt_of_le_of_lt hN + (upcrossings_before_lt_of_exists_upcrossing hab hN₁ hN₁' hN₂ hN₂')⟩ } +end + +/-- A stochastic process that frequently visits below `a` and above `b` have infinite +upcrossings. -/ +lemma upcrossings_eq_top_of_frequently_lt (hab : a < b) + (h₁ : ∃ᶠ n in at_top, f n ω < a) (h₂ : ∃ᶠ n in at_top, b < f n ω) : + upcrossings a b f ω = ∞ := +classical.by_contradiction (λ h, not_frequently_of_upcrossings_lt_top hab h ⟨h₁, h₂⟩) + +/-- A realization of a stochastic process with bounded upcrossings and bounded liminfs is +convergent. + +We use the spelling `< ∞` instead of the standard `≠ ∞` in the assumptions since it is not as easy +to change `<` to `≠` under binders. -/ +lemma tendsto_of_uncrossing_lt_top + (hf₁ : liminf at_top (λ n, (∥f n ω∥₊ : ℝ≥0∞)) < ∞) + (hf₂ : ∀ a b : ℚ, a < b → upcrossings a b f ω < ∞) : + ∃ c, tendsto (λ n, f n ω) at_top (𝓝 c) := +begin + by_cases h : is_bounded_under (≤) at_top (λ n, |f n ω|), + { rw is_bounded_under_le_abs at h, + refine tendsto_of_no_upcrossings rat.dense_range_cast _ h.1 h.2, + { intros a ha b hb hab, + obtain ⟨⟨a, rfl⟩, ⟨b, rfl⟩⟩ := ⟨ha, hb⟩, + exact not_frequently_of_upcrossings_lt_top hab (hf₂ a b (rat.cast_lt.1 hab)).ne } }, + { obtain ⟨a, b, hab, h₁, h₂⟩ := ennreal.exists_upcrossings_of_not_bounded_under hf₁.ne h, + exact false.elim ((hf₂ a b hab).ne + (upcrossings_eq_top_of_frequently_lt (rat.cast_lt.2 hab) h₁ h₂)) } +end + +/-- An L¹-bounded submartingale has bounded upcrossings almost everywhere. -/ +lemma submartingale.upcrossings_ae_lt_top' [is_finite_measure μ] + (hf : submartingale f ℱ μ) (hbdd : ∀ n, snorm (f n) 1 μ ≤ R) (hab : a < b) : + ∀ᵐ ω ∂μ, upcrossings a b f ω < ∞ := +begin + refine ae_lt_top (hf.adapted.measurable_upcrossings hab) _, + have := hf.mul_lintegral_upcrossings_le_lintegral_pos_part a b, + rw [mul_comm, ← ennreal.le_div_iff_mul_le] at this, + { refine (lt_of_le_of_lt this (ennreal.div_lt_top _ _)).ne, + { have hR' : ∀ n, ∫⁻ ω, ∥f n ω - a∥₊ ∂μ ≤ R + ∥a∥₊ * μ set.univ, + { simp_rw snorm_one_eq_lintegral_nnnorm at hbdd, + intro n, + refine (lintegral_mono _ : ∫⁻ ω, ∥f n ω - a∥₊ ∂μ ≤ ∫⁻ ω, ∥f n ω∥₊ + ∥a∥₊ ∂μ).trans _, + { intro ω, + simp_rw [sub_eq_add_neg, ← nnnorm_neg a, ← ennreal.coe_add, ennreal.coe_le_coe], + exact nnnorm_add_le _ _ }, + { simp_rw [ lintegral_add_right _ measurable_const, lintegral_const], + exact add_le_add (hbdd _) le_rfl } }, + refine ne_of_lt (supr_lt_iff.2 ⟨R + ∥a∥₊ * μ set.univ, ennreal.add_lt_top.2 + ⟨ennreal.coe_lt_top, ennreal.mul_lt_top ennreal.coe_lt_top.ne (measure_ne_top _ _)⟩, + λ n, le_trans _ (hR' n)⟩), + refine lintegral_mono (λ ω, _), + rw [ennreal.of_real_le_iff_le_to_real, ennreal.coe_to_real, coe_nnnorm], + by_cases hnonneg : 0 ≤ f n ω - a, + { rw [lattice_ordered_comm_group.pos_of_nonneg _ hnonneg, + real.norm_eq_abs, abs_of_nonneg hnonneg] }, + { rw lattice_ordered_comm_group.pos_of_nonpos _ (not_le.1 hnonneg).le, + exact norm_nonneg _ }, + { simp only [ne.def, ennreal.coe_ne_top, not_false_iff] } }, + { simp only [hab, ne.def, ennreal.of_real_eq_zero, sub_nonpos, not_le] } }, + { simp only [hab, ne.def, ennreal.of_real_eq_zero, sub_nonpos, not_le, true_or]}, + { simp only [ne.def, ennreal.of_real_ne_top, not_false_iff, true_or] } +end + +lemma submartingale.upcrossings_ae_lt_top [is_finite_measure μ] + (hf : submartingale f ℱ μ) (hbdd : ∀ n, snorm (f n) 1 μ ≤ R) : + ∀ᵐ ω ∂μ, ∀ a b : ℚ, a < b → upcrossings a b f ω < ∞ := +begin + simp only [ae_all_iff, eventually_imp_distrib_left], + rintro a b hab, + exact hf.upcrossings_ae_lt_top' hbdd (rat.cast_lt.2 hab), +end + +/-- An L¹-bounded submartingale converges almost everywhere. -/ +lemma submartingale.exists_ae_tendsto_of_bdd [is_finite_measure μ] + (hf : submartingale f ℱ μ) (hbdd : ∀ n, snorm (f n) 1 μ ≤ R) : + ∀ᵐ ω ∂μ, ∃ c, tendsto (λ n, f n ω) at_top (𝓝 c) := +begin + filter_upwards [hf.upcrossings_ae_lt_top hbdd, ae_bdd_liminf_at_top_of_snorm_bdd one_ne_zero + (λ n, (hf.strongly_measurable n).measurable.mono (ℱ.le n) le_rfl) hbdd] with ω h₁ h₂, + exact tendsto_of_uncrossing_lt_top h₂ h₁, +end + +lemma submartingale.exists_ae_trim_tendsto_of_bdd [is_finite_measure μ] + (hf : submartingale f ℱ μ) (hbdd : ∀ n, snorm (f n) 1 μ ≤ R) : + ∀ᵐ ω ∂(μ.trim (Sup_le (λ m ⟨n, hn⟩, hn ▸ ℱ.le _) : (⨆ n, ℱ n) ≤ m0)), + ∃ c, tendsto (λ n, f n ω) at_top (𝓝 c) := +begin + rw [ae_iff, trim_measurable_set_eq], + { exact hf.exists_ae_tendsto_of_bdd hbdd }, + { exact measurable_set.compl (@measurable_set_exists_tendsto _ _ _ _ _ _ (⨆ n, ℱ n) _ _ _ _ _ + (λ n, ((hf.strongly_measurable n).measurable.mono (le_Sup ⟨n, rfl⟩) le_rfl))) } +end + +/-- **Almost everywhere martingale convergence theorem**: An L¹-bounded submartingale converges +almost everywhere to a `⨆ n, ℱ n`-measurable function. -/ +lemma submartingale.ae_tendsto_limit_process [is_finite_measure μ] + (hf : submartingale f ℱ μ) (hbdd : ∀ n, snorm (f n) 1 μ ≤ R) : + ∀ᵐ ω ∂μ, tendsto (λ n, f n ω) at_top (𝓝 (ℱ.limit_process f μ ω)) := +begin + classical, + suffices : ∃ g, strongly_measurable[⨆ n, ℱ n] g ∧ ∀ᵐ ω ∂μ, tendsto (λ n, f n ω) at_top (𝓝 (g ω)), + { rw [limit_process, dif_pos this], + exact (classical.some_spec this).2 }, + set g' : Ω → ℝ := λ ω, if h : ∃ c, tendsto (λ n, f n ω) at_top (𝓝 c) then h.some else 0, + have hle : (⨆ n, ℱ n) ≤ m0 := Sup_le (λ m ⟨n, hn⟩, hn ▸ ℱ.le _), + have hg' : ∀ᵐ ω ∂(μ.trim hle), tendsto (λ n, f n ω) at_top (𝓝 (g' ω)), + { filter_upwards [hf.exists_ae_trim_tendsto_of_bdd hbdd] with ω hω, + simp_rw [g', dif_pos hω], + exact hω.some_spec }, + have hg'm : @ae_strongly_measurable _ _ _ (⨆ n, ℱ n) g' (μ.trim hle) := + (@ae_measurable_of_tendsto_metrizable_ae' _ _ (⨆ n, ℱ n) _ _ _ _ _ _ _ + (λ n, ((hf.strongly_measurable n).measurable.mono + (le_Sup ⟨n, rfl⟩ : ℱ n ≤ ⨆ n, ℱ n) le_rfl).ae_measurable) hg').ae_strongly_measurable, + obtain ⟨g, hgm, hae⟩ := hg'm, + have hg : ∀ᵐ ω ∂μ.trim hle, tendsto (λ n, f n ω) at_top (𝓝 (g ω)), + { filter_upwards [hae, hg'] with ω hω hg'ω, + exact hω ▸ hg'ω }, + exact ⟨g, hgm, measure_eq_zero_of_trim_eq_zero hle hg⟩, +end + +/-- The limiting process of an Lᵖ-bounded submartingale is Lᵖ. -/ +lemma submartingale.mem_ℒp_limit_process {p : ℝ≥0∞} + (hf : submartingale f ℱ μ) (hbdd : ∀ n, snorm (f n) p μ ≤ R) : + mem_ℒp (ℱ.limit_process f μ) p μ := +mem_ℒp_limit_process_of_snorm_bdd + (λ n, ((hf.strongly_measurable n).mono (ℱ.le n)).ae_strongly_measurable) hbdd + +end ae_convergence + +end measure_theory diff --git a/formal/lean/mathlib/probability/martingale/upcrossing.lean b/formal/lean/mathlib/probability/martingale/upcrossing.lean new file mode 100644 index 0000000000000000000000000000000000000000..40c21806a1d45543b35eb24a14a22a8c5833bedd --- /dev/null +++ b/formal/lean/mathlib/probability/martingale/upcrossing.lean @@ -0,0 +1,921 @@ +/- +Copyright (c) 2022 Kexing Ying. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Kexing Ying +-/ +import probability.hitting_time +import probability.martingale.basic + +/-! + +# Doob's upcrossing estimate + +Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting $U_N(a, b)$ the +number of times $f_n$ crossed from below $a$ to above $b$ before time $N$, Doob's upcrossing +estimate (also known as Doob's inequality) states that +$$(b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(f_N - a)^+].$$ +Doob's upcrossing estimate is an important inequality and is central in proving the martingale +convergence theorems. + +## Main definitions + +* `measure_theory.upper_crossing_time a b f N n`: is the stopping time corresponding to `f` + crossing above `b` the `n`-th time before time `N` (if this does not occur then the value is + taken to be `N`). +* `measure_theory.lower_crossing_time a b f N n`: is the stopping time corresponding to `f` + crossing below `a` the `n`-th time before time `N` (if this does not occur then the value is + taken to be `N`). +* `measure_theory.upcrossing_strat a b f N`: is the predictable process which is 1 if `n` is + between a consecutive pair of lower and upper crossing and is 0 otherwise. Intuitively + one might think of the `upcrossing_strat` as the strategy of buying 1 share whenever the process + crosses below `a` for the first time after selling and selling 1 share whenever the process + crosses above `b` for the first time after buying. +* `measure_theory.upcrossings_before a b f N`: is the number of times `f` crosses from below `a` to + above `b` before time `N`. +* `measure_theory.upcrossings a b f`: is the number of times `f` crosses from below `a` to above + `b`. This takes value in `ℝ≥0∞` and so is allowed to be `∞`. + +## Main results + +* `measure_theory.adapted.is_stopping_time_upper_crossing_time`: `upper_crossing_time` is a + stopping time whenever the process it is associated to is adapted. +* `measure_theory.adapted.is_stopping_time_lower_crossing_time`: `lower_crossing_time` is a + stopping time whenever the process it is associated to is adapted. +* `measure_theory.submartingale.mul_integral_upcrossings_before_le_integral_pos_part`: Doob's + upcrossing estimate. +* `measure_theory.submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part`: the inequality + obtained by taking the supremum on both sides of Doob's upcrossing estimate. + +### References + +We mostly follow the proof from [Kallenberg, *Foundations of modern probability*][kallenberg2021] + +-/ + +open topological_space filter +open_locale nnreal ennreal measure_theory probability_theory big_operators topological_space + +namespace measure_theory + +variables {Ω ι : Type*} {m0 : measurable_space Ω} {μ : measure Ω} + +/-! + +## Proof outline + +In this section, we will denote $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$ to +above $b$ before time $N$. + +To define $U_N(a, b)$, we will construct two stopping times corresponding to when $(f_n)$ crosses +below $a$ and above $b$. Namely, we define +$$ + \sigma_n := \inf \{n \ge \tau_n \mid f_n \le a\} \wedge N; +$$ +$$ + \tau_{n + 1} := \inf \{n \ge \sigma_n \mid f_n \ge b\} \wedge N. +$$ +These are `lower_crossing_time` and `upper_crossing_time` in our formalization which are defined +using `measure_theory.hitting` allowing us to specify a starting and ending time. +Then, we may simply define $U_N(a, b) := \sup \{n \mid \tau_n < N\}$. + +Fixing $a < b \in \mathbb{R}$, we will first prove the theorem in the special case that +$0 \le f_0$ and $a \le f_N$. In particular, we will show +$$ + (b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[f_N]. +$$ +This is `measure_theory.integral_mul_upcrossings_before_le_integral` in our formalization. + +To prove this, we use the fact that given a non-negative, bounded, predictable process $(C_n)$ +(i.e. $(C_{n + 1})$ is adapted), $(C \bullet f)_n := \sum_{k \le n} C_{k + 1}(f_{k + 1} - f_k)$ is +a submartingale if $(f_n)$ is. + +Define $C_n := \sum_{k \le n} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)$. It is easy to see that +$(1 - C_n)$ is non-negative, bounded and predictable, and hence, given a submartingale $(f_n)$, +$(1 - C) \bullet f$ is also a submartingale. Thus, by the submartingale property, +$0 \le \mathbb{E}[((1 - C) \bullet f)_0] \le \mathbb{E}[((1 - C) \bullet f)_N]$ implying +$$ + \mathbb{E}[(C \bullet f)_N] \le \mathbb{E}[(1 \bullet f)_N] = \mathbb{E}[f_N] - \mathbb{E}[f_0]. +$$ + +Furthermore, +\begin{align} + (C \bullet f)_N & = + \sum_{n \le N} \sum_{k \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\ + & = \sum_{k \le N} \sum_{n \le N} \mathbf{1}_{[\sigma_k, \tau_{k + 1})}(n)(f_{n + 1} - f_n)\\ + & = \sum_{k \le N} (f_{\sigma_k + 1} - f_{\sigma_k} + f_{\sigma_k + 2} - f_{\sigma_k + 1} + + \cdots + f_{\tau_{k + 1}} - f_{\tau_{k + 1} - 1})\\ + & = \sum_{k \le N} (f_{\tau_{k + 1}} - f_{\sigma_k}) + \ge \sum_{k < U_N(a, b)} (b - a) = (b - a) U_N(a, b) +\end{align} +where the inequality follows since for all $k < U_N(a, b)$, +$f_{\tau_{k + 1}} - f_{\sigma_k} \ge b - a$ while for all $k > U_N(a, b)$, +$f_{\tau_{k + 1}} = f_{\sigma_k} = f_N$ and +$f_{\tau_{U_N(a, b) + 1}} - f_{\sigma_{U_N(a, b)}} = f_N - a \ge 0$. Hence, we have +$$ + (b - a) \mathbb{E}[U_N(a, b)] \le \mathbb{E}[(C \bullet f)_N] + \le \mathbb{E}[f_N] - \mathbb{E}[f_0] \le \mathbb{E}[f_N], +$$ +as required. + +To obtain the general case, we simply apply the above to $((f_n - a)^+)_n$. + +-/ + +/-- `lower_crossing_time_aux a f c N` is the first time `f` reached below `a` after time `c` before +time `N`. -/ +noncomputable +def lower_crossing_time_aux [preorder ι] [has_Inf ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) : Ω → ι := +hitting f (set.Iic a) c N + +/-- `upper_crossing_time a b f N n` is the first time before time `N`, `f` reaches +above `b` after `f` reached below `a` for the `n - 1`-th time. -/ +noncomputable +def upper_crossing_time [preorder ι] [order_bot ι] [has_Inf ι] + (a b : ℝ) (f : ι → Ω → ℝ) (N : ι) : ℕ → Ω → ι +| 0 := ⊥ +| (n + 1) := λ ω, hitting f (set.Ici b) + (lower_crossing_time_aux a f (upper_crossing_time n ω) N ω) N ω + +/-- `lower_crossing_time a b f N n` is the first time before time `N`, `f` reaches +below `a` after `f` reached above `b` for the `n`-th time. -/ +noncomputable +def lower_crossing_time [preorder ι] [order_bot ι] [has_Inf ι] + (a b : ℝ) (f : ι → Ω → ℝ) (N : ι) (n : ℕ) : Ω → ι := +λ ω, hitting f (set.Iic a) (upper_crossing_time a b f N n ω) N ω + +section + +variables [preorder ι] [order_bot ι] [has_Inf ι] +variables {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω} + +@[simp] +lemma upper_crossing_time_zero : upper_crossing_time a b f N 0 = ⊥ := rfl + +@[simp] +lemma lower_crossing_time_zero : lower_crossing_time a b f N 0 = hitting f (set.Iic a) ⊥ N := rfl + +lemma upper_crossing_time_succ : + upper_crossing_time a b f N (n + 1) ω = + hitting f (set.Ici b) (lower_crossing_time_aux a f (upper_crossing_time a b f N n ω) N ω) N ω := +by rw upper_crossing_time + +lemma upper_crossing_time_succ_eq (ω : Ω) : + upper_crossing_time a b f N (n + 1) ω = + hitting f (set.Ici b) (lower_crossing_time a b f N n ω) N ω := +begin + simp only [upper_crossing_time_succ], + refl, +end + +end + +section conditionally_complete_linear_order_bot + +variables [conditionally_complete_linear_order_bot ι] +variables {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω} + +lemma upper_crossing_time_le : upper_crossing_time a b f N n ω ≤ N := +begin + cases n, + { simp only [upper_crossing_time_zero, pi.bot_apply, bot_le] }, + { simp only [upper_crossing_time_succ, hitting_le] }, +end + +@[simp] +lemma upper_crossing_time_zero' : upper_crossing_time a b f ⊥ n ω = ⊥ := +eq_bot_iff.2 upper_crossing_time_le + +lemma lower_crossing_time_le : lower_crossing_time a b f N n ω ≤ N := +by simp only [lower_crossing_time, hitting_le ω] + +lemma upper_crossing_time_le_lower_crossing_time : + upper_crossing_time a b f N n ω ≤ lower_crossing_time a b f N n ω := +by simp only [lower_crossing_time, le_hitting upper_crossing_time_le ω] + +lemma lower_crossing_time_le_upper_crossing_time_succ : + lower_crossing_time a b f N n ω ≤ upper_crossing_time a b f N (n + 1) ω := +begin + rw upper_crossing_time_succ, + exact le_hitting lower_crossing_time_le ω, +end + +lemma lower_crossing_time_mono (hnm : n ≤ m) : + lower_crossing_time a b f N n ω ≤ lower_crossing_time a b f N m ω := +begin + suffices : monotone (λ n, lower_crossing_time a b f N n ω), + { exact this hnm }, + exact monotone_nat_of_le_succ + (λ n, le_trans lower_crossing_time_le_upper_crossing_time_succ + upper_crossing_time_le_lower_crossing_time) +end + +lemma upper_crossing_time_mono (hnm : n ≤ m) : + upper_crossing_time a b f N n ω ≤ upper_crossing_time a b f N m ω := +begin + suffices : monotone (λ n, upper_crossing_time a b f N n ω), + { exact this hnm }, + exact monotone_nat_of_le_succ + (λ n, le_trans upper_crossing_time_le_lower_crossing_time + lower_crossing_time_le_upper_crossing_time_succ), +end + +end conditionally_complete_linear_order_bot + +variables {a b : ℝ} {f : ℕ → Ω → ℝ} {N : ℕ} {n m : ℕ} {ω : Ω} + +lemma stopped_value_lower_crossing_time (h : lower_crossing_time a b f N n ω ≠ N) : + stopped_value f (lower_crossing_time a b f N n) ω ≤ a := +begin + obtain ⟨j, hj₁, hj₂⟩ := + (hitting_le_iff_of_lt _ (lt_of_le_of_ne lower_crossing_time_le h)).1 le_rfl, + exact stopped_value_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lower_crossing_time_le⟩, hj₂⟩, +end + +lemma stopped_value_upper_crossing_time (h : upper_crossing_time a b f N (n + 1) ω ≠ N) : + b ≤ stopped_value f (upper_crossing_time a b f N (n + 1)) ω := +begin + obtain ⟨j, hj₁, hj₂⟩ := + (hitting_le_iff_of_lt _ (lt_of_le_of_ne upper_crossing_time_le h)).1 le_rfl, + exact stopped_value_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩, +end + +lemma upper_crossing_time_lt_lower_crossing_time + (hab : a < b) (hn : lower_crossing_time a b f N (n + 1) ω ≠ N) : + upper_crossing_time a b f N (n + 1) ω < lower_crossing_time a b f N (n + 1) ω := +begin + refine lt_of_le_of_ne upper_crossing_time_le_lower_crossing_time + (λ h, not_le.2 hab $ le_trans _ (stopped_value_lower_crossing_time hn)), + simp only [stopped_value], + rw ← h, + exact stopped_value_upper_crossing_time (h.symm ▸ hn), +end + +lemma lower_crossing_time_lt_upper_crossing_time + (hab : a < b) (hn : upper_crossing_time a b f N (n + 1) ω ≠ N) : + lower_crossing_time a b f N n ω < upper_crossing_time a b f N (n + 1) ω := +begin + refine lt_of_le_of_ne lower_crossing_time_le_upper_crossing_time_succ + (λ h, not_le.2 hab $ le_trans (stopped_value_upper_crossing_time hn) _), + simp only [stopped_value], + rw ← h, + exact stopped_value_lower_crossing_time (h.symm ▸ hn), +end + +lemma upper_crossing_time_lt_succ (hab : a < b) (hn : upper_crossing_time a b f N (n + 1) ω ≠ N) : + upper_crossing_time a b f N n ω < upper_crossing_time a b f N (n + 1) ω := +lt_of_le_of_lt upper_crossing_time_le_lower_crossing_time + (lower_crossing_time_lt_upper_crossing_time hab hn) + +lemma lower_crossing_time_stabilize (hnm : n ≤ m) (hn : lower_crossing_time a b f N n ω = N) : + lower_crossing_time a b f N m ω = N := +le_antisymm lower_crossing_time_le (le_trans (le_of_eq hn.symm) (lower_crossing_time_mono hnm)) + +lemma upper_crossing_time_stabilize (hnm : n ≤ m) (hn : upper_crossing_time a b f N n ω = N) : + upper_crossing_time a b f N m ω = N := +le_antisymm upper_crossing_time_le (le_trans (le_of_eq hn.symm) (upper_crossing_time_mono hnm)) + +lemma lower_crossing_time_stabilize' (hnm : n ≤ m) (hn : N ≤ lower_crossing_time a b f N n ω) : + lower_crossing_time a b f N m ω = N := +lower_crossing_time_stabilize hnm (le_antisymm lower_crossing_time_le hn) + +lemma upper_crossing_time_stabilize' (hnm : n ≤ m) (hn : N ≤ upper_crossing_time a b f N n ω) : + upper_crossing_time a b f N m ω = N := +upper_crossing_time_stabilize hnm (le_antisymm upper_crossing_time_le hn) + +-- `upper_crossing_time_bound_eq` provides an explicit bound +lemma exists_upper_crossing_time_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) : + ∃ n, upper_crossing_time a b f N n ω = N := +begin + by_contra h, push_neg at h, + have : strict_mono (λ n, upper_crossing_time a b f N n ω) := + strict_mono_nat_of_lt_succ (λ n, upper_crossing_time_lt_succ hab (h _)), + obtain ⟨_, ⟨k, rfl⟩, hk⟩ : + ∃ m (hm : m ∈ set.range (λ n, upper_crossing_time a b f N n ω)), N < m := + ⟨upper_crossing_time a b f N (N + 1) ω, ⟨N + 1, rfl⟩, + lt_of_lt_of_le (N.lt_succ_self) (strict_mono.id_le this (N + 1))⟩, + exact not_le.2 hk upper_crossing_time_le +end + +lemma upper_crossing_time_lt_bdd_above (hab : a < b) : + bdd_above {n | upper_crossing_time a b f N n ω < N} := +begin + obtain ⟨k, hk⟩ := exists_upper_crossing_time_eq f N ω hab, + refine ⟨k, λ n (hn : upper_crossing_time a b f N n ω < N), _⟩, + by_contra hn', + exact hn.ne (upper_crossing_time_stabilize (not_le.1 hn').le hk) +end + +lemma upper_crossing_time_lt_nonempty (hN : 0 < N) : + {n | upper_crossing_time a b f N n ω < N}.nonempty := +⟨0, hN⟩ + +lemma upper_crossing_time_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) : + upper_crossing_time a b f N N ω = N := +begin + by_cases hN' : N < nat.find (exists_upper_crossing_time_eq f N ω hab), + { refine le_antisymm upper_crossing_time_le _, + have hmono : strict_mono_on (λ n, upper_crossing_time a b f N n ω) + (set.Iic (nat.find (exists_upper_crossing_time_eq f N ω hab)).pred), + { refine strict_mono_on_Iic_of_lt_succ (λ m hm, upper_crossing_time_lt_succ hab _), + rw nat.lt_pred_iff at hm, + convert nat.find_min _ hm }, + convert strict_mono_on.Iic_id_le hmono N (nat.le_pred_of_lt hN') }, + { rw not_lt at hN', + exact upper_crossing_time_stabilize hN' + (nat.find_spec (exists_upper_crossing_time_eq f N ω hab)) } +end + +lemma upper_crossing_time_eq_of_bound_le (hab : a < b) (hn : N ≤ n) : + upper_crossing_time a b f N n ω = N := +le_antisymm upper_crossing_time_le + ((le_trans (upper_crossing_time_bound_eq f N ω hab).symm.le (upper_crossing_time_mono hn))) + +variables {ℱ : filtration ℕ m0} + +lemma adapted.is_stopping_time_crossing (hf : adapted ℱ f) : + is_stopping_time ℱ (upper_crossing_time a b f N n) ∧ + is_stopping_time ℱ (lower_crossing_time a b f N n) := +begin + induction n with k ih, + { refine ⟨is_stopping_time_const _ 0, _⟩, + simp [hitting_is_stopping_time hf measurable_set_Iic] }, + { obtain ⟨ih₁, ih₂⟩ := ih, + have : is_stopping_time ℱ (upper_crossing_time a b f N (k + 1)), + { intro n, + simp_rw upper_crossing_time_succ_eq, + exact is_stopping_time_hitting_is_stopping_time ih₂ (λ _, lower_crossing_time_le) + measurable_set_Ici hf _ }, + refine ⟨this, _⟩, + { intro n, + exact is_stopping_time_hitting_is_stopping_time this (λ _, upper_crossing_time_le) + measurable_set_Iic hf _ } } +end + +lemma adapted.is_stopping_time_upper_crossing_time (hf : adapted ℱ f) : + is_stopping_time ℱ (upper_crossing_time a b f N n) := +hf.is_stopping_time_crossing.1 + +lemma adapted.is_stopping_time_lower_crossing_time (hf : adapted ℱ f) : + is_stopping_time ℱ (lower_crossing_time a b f N n) := +hf.is_stopping_time_crossing.2 + +/-- `upcrossing_strat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper +crossings and is 0 otherwise. `upcrossing_strat` is shifted by one index so that it is adapted +rather than predictable. -/ +noncomputable +def upcrossing_strat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ := +∑ k in finset.range N, + (set.Ico (lower_crossing_time a b f N k ω) (upper_crossing_time a b f N (k + 1) ω)).indicator 1 n + +lemma upcrossing_strat_nonneg : 0 ≤ upcrossing_strat a b f N n ω := +finset.sum_nonneg (λ i hi, set.indicator_nonneg (λ ω hω, zero_le_one) _) + +lemma upcrossing_strat_le_one : upcrossing_strat a b f N n ω ≤ 1 := +begin + rw [upcrossing_strat, ← set.indicator_finset_bUnion_apply], + { exact set.indicator_le_self' (λ _ _, zero_le_one) _ }, + { intros i hi j hj hij, + rw set.Ico_disjoint_Ico, + obtain (hij' | hij') := lt_or_gt_of_ne hij, + { rw [min_eq_left ((upper_crossing_time_mono (nat.succ_le_succ hij'.le)) : + upper_crossing_time a b f N _ ω ≤ upper_crossing_time a b f N _ ω), + max_eq_right (lower_crossing_time_mono hij'.le : + lower_crossing_time a b f N _ _ ≤ lower_crossing_time _ _ _ _ _ _)], + refine le_trans upper_crossing_time_le_lower_crossing_time (lower_crossing_time_mono + (nat.succ_le_of_lt hij')) }, + { rw gt_iff_lt at hij', + rw [min_eq_right ((upper_crossing_time_mono (nat.succ_le_succ hij'.le)) : + upper_crossing_time a b f N _ ω ≤ upper_crossing_time a b f N _ ω), + max_eq_left (lower_crossing_time_mono hij'.le : + lower_crossing_time a b f N _ _ ≤ lower_crossing_time _ _ _ _ _ _)], + refine le_trans upper_crossing_time_le_lower_crossing_time + (lower_crossing_time_mono (nat.succ_le_of_lt hij')) } } +end + +lemma adapted.upcrossing_strat_adapted (hf : adapted ℱ f) : + adapted ℱ (upcrossing_strat a b f N) := +begin + intro n, + change strongly_measurable[ℱ n] (λ ω, ∑ k in finset.range N, + ({n | lower_crossing_time a b f N k ω ≤ n} ∩ + {n | n < upper_crossing_time a b f N (k + 1) ω}).indicator 1 n), + refine finset.strongly_measurable_sum _ (λ i hi, + strongly_measurable_const.indicator ((hf.is_stopping_time_lower_crossing_time n).inter _)), + simp_rw ← not_le, + exact (hf.is_stopping_time_upper_crossing_time n).compl, +end + +lemma submartingale.sum_upcrossing_strat_mul [is_finite_measure μ] (hf : submartingale f ℱ μ) + (a b : ℝ) (N : ℕ) : + submartingale + (λ n : ℕ, ∑ k in finset.range n, upcrossing_strat a b f N k * (f (k + 1) - f k)) ℱ μ := +hf.sum_mul_sub hf.adapted.upcrossing_strat_adapted + (λ _ _, upcrossing_strat_le_one) (λ _ _, upcrossing_strat_nonneg) + +lemma submartingale.sum_sub_upcrossing_strat_mul [is_finite_measure μ] (hf : submartingale f ℱ μ) + (a b : ℝ) (N : ℕ) : + submartingale + (λ n : ℕ, ∑ k in finset.range n, (1 - upcrossing_strat a b f N k) * (f (k + 1) - f k)) ℱ μ := +begin + refine hf.sum_mul_sub (λ n, (adapted_const ℱ 1 n).sub (hf.adapted.upcrossing_strat_adapted n)) + (_ : ∀ n ω, (1 - upcrossing_strat a b f N n) ω ≤ 1) _, + { refine λ n ω, sub_le.1 _, + simp [upcrossing_strat_nonneg] }, + { intros n ω, + simp [upcrossing_strat_le_one] } +end + +lemma submartingale.sum_mul_upcrossing_strat_le [is_finite_measure μ] (hf : submartingale f ℱ μ) : + μ[∑ k in finset.range n, upcrossing_strat a b f N k * (f (k + 1) - f k)] ≤ + μ[f n] - μ[f 0] := +begin + have h₁ : (0 : ℝ) ≤ + μ[∑ k in finset.range n, (1 - upcrossing_strat a b f N k) * (f (k + 1) - f k)], + { have := (hf.sum_sub_upcrossing_strat_mul a b N).set_integral_le (zero_le n) measurable_set.univ, + rw [integral_univ, integral_univ] at this, + refine le_trans _ this, + simp only [finset.range_zero, finset.sum_empty, integral_zero'] }, + have h₂ : μ[∑ k in finset.range n, (1 - upcrossing_strat a b f N k) * (f (k + 1) - f k)] = + μ[∑ k in finset.range n, (f (k + 1) - f k)] - + μ[∑ k in finset.range n, upcrossing_strat a b f N k * (f (k + 1) - f k)], + { simp only [sub_mul, one_mul, finset.sum_sub_distrib, pi.sub_apply, + finset.sum_apply, pi.mul_apply], + refine integral_sub (integrable.sub (integrable_finset_sum _ (λ i hi, hf.integrable _)) + (integrable_finset_sum _ (λ i hi, hf.integrable _))) _, + convert (hf.sum_upcrossing_strat_mul a b N).integrable n, + ext, simp }, + rw [h₂, sub_nonneg] at h₁, + refine le_trans h₁ _, + simp_rw [finset.sum_range_sub, integral_sub' (hf.integrable _) (hf.integrable _)], +end + +/-- The number of upcrossings (strictly) before time `N`. -/ +noncomputable +def upcrossings_before [preorder ι] [order_bot ι] [has_Inf ι] + (a b : ℝ) (f : ι → Ω → ℝ) (N : ι) (ω : Ω) : ℕ := +Sup {n | upper_crossing_time a b f N n ω < N} + +@[simp] +lemma upcrossings_before_bot [preorder ι] [order_bot ι] [has_Inf ι] + {a b : ℝ} {f : ι → Ω → ℝ} {ω : Ω} : + upcrossings_before a b f ⊥ ω = ⊥ := +by simp [upcrossings_before] + +lemma upcrossings_before_zero : + upcrossings_before a b f 0 ω = 0 := +by simp [upcrossings_before] + +@[simp] lemma upcrossings_before_zero' : + upcrossings_before a b f 0 = 0 := +by { ext ω, exact upcrossings_before_zero } + +lemma upper_crossing_time_lt_of_le_upcrossings_before + (hN : 0 < N) (hab : a < b) (hn : n ≤ upcrossings_before a b f N ω) : + upper_crossing_time a b f N n ω < N := +begin + have : upper_crossing_time a b f N (upcrossings_before a b f N ω) ω < N := + (upper_crossing_time_lt_nonempty hN).cSup_mem + ((order_bot.bdd_below _).finite_of_bdd_above (upper_crossing_time_lt_bdd_above hab)), + exact lt_of_le_of_lt (upper_crossing_time_mono hn) this, +end + +lemma upper_crossing_time_eq_of_upcrossings_before_lt + (hab : a < b) (hn : upcrossings_before a b f N ω < n) : + upper_crossing_time a b f N n ω = N := +begin + refine le_antisymm upper_crossing_time_le (not_lt.1 _), + convert not_mem_of_cSup_lt hn (upper_crossing_time_lt_bdd_above hab), +end + +lemma upcrossings_before_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) : + upcrossings_before a b f N ω ≤ N := +begin + by_cases hN : N = 0, + { subst hN, + rw upcrossings_before_zero }, + { refine cSup_le ⟨0, zero_lt_iff.2 hN⟩ (λ n (hn : _ < _), _), + by_contra hnN, + exact hn.ne (upper_crossing_time_eq_of_bound_le hab (not_le.1 hnN).le) }, +end + +lemma crossing_eq_crossing_of_lower_crossing_time_lt {M : ℕ} (hNM : N ≤ M) + (h : lower_crossing_time a b f N n ω < N) : + upper_crossing_time a b f M n ω = upper_crossing_time a b f N n ω ∧ + lower_crossing_time a b f M n ω = lower_crossing_time a b f N n ω := +begin + have h' : upper_crossing_time a b f N n ω < N := + lt_of_le_of_lt upper_crossing_time_le_lower_crossing_time h, + induction n with k ih, + { simp only [nat.nat_zero_eq_zero, upper_crossing_time_zero, bot_eq_zero', eq_self_iff_true, + lower_crossing_time_zero, true_and, eq_comm], + refine hitting_eq_hitting_of_exists hNM _, + simp only [lower_crossing_time, hitting_lt_iff] at h, + obtain ⟨j, hj₁, hj₂⟩ := h, + exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩ }, + { specialize ih (lt_of_le_of_lt (lower_crossing_time_mono (nat.le_succ _)) h) + (lt_of_le_of_lt (upper_crossing_time_mono (nat.le_succ _)) h'), + have : upper_crossing_time a b f M k.succ ω = upper_crossing_time a b f N k.succ ω, + { simp only [upper_crossing_time_succ_eq, hitting_lt_iff] at h' ⊢, + obtain ⟨j, hj₁, hj₂⟩ := h', + rw [eq_comm, ih.2], + exact hitting_eq_hitting_of_exists hNM ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩ }, + refine ⟨this, _⟩, + simp only [lower_crossing_time, eq_comm, this], + refine hitting_eq_hitting_of_exists hNM _, + rw [lower_crossing_time, hitting_lt_iff _ le_rfl] at h, + swap, { apply_instance }, + obtain ⟨j, hj₁, hj₂⟩ := h, + exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩ } +end + +lemma crossing_eq_crossing_of_upper_crossing_time_lt {M : ℕ} (hNM : N ≤ M) + (h : upper_crossing_time a b f N (n + 1) ω < N) : + upper_crossing_time a b f M (n + 1) ω = upper_crossing_time a b f N (n + 1) ω ∧ + lower_crossing_time a b f M n ω = lower_crossing_time a b f N n ω := +begin + have := (crossing_eq_crossing_of_lower_crossing_time_lt hNM + (lt_of_le_of_lt lower_crossing_time_le_upper_crossing_time_succ h)).2, + refine ⟨_, this⟩, + rw [upper_crossing_time_succ_eq, upper_crossing_time_succ_eq, eq_comm, this], + refine hitting_eq_hitting_of_exists hNM _, + simp only [upper_crossing_time_succ_eq, hitting_lt_iff] at h, + obtain ⟨j, hj₁, hj₂⟩ := h, + exact ⟨j, ⟨hj₁.1, hj₁.2.le⟩, hj₂⟩ +end + +lemma upper_crossing_time_eq_upper_crossing_time_of_lt {M : ℕ} (hNM : N ≤ M) + (h : upper_crossing_time a b f N n ω < N) : + upper_crossing_time a b f M n ω = upper_crossing_time a b f N n ω := +begin + cases n, + { simp }, + { exact (crossing_eq_crossing_of_upper_crossing_time_lt hNM h).1 } +end + +lemma upcrossings_before_mono (hab : a < b) : + monotone (λ N ω, upcrossings_before a b f N ω) := +begin + intros N M hNM ω, + simp only [upcrossings_before], + by_cases hemp : {n : ℕ | upper_crossing_time a b f N n ω < N}.nonempty, + { refine cSup_le_cSup (upper_crossing_time_lt_bdd_above hab) hemp (λ n hn, _), + rw [set.mem_set_of_eq, upper_crossing_time_eq_upper_crossing_time_of_lt hNM hn], + exact lt_of_lt_of_le hn hNM }, + { rw set.not_nonempty_iff_eq_empty at hemp, + simp [hemp, cSup_empty, bot_eq_zero', zero_le'] } +end + +lemma upcrossings_before_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} + (hN₁: N ≤ N₁) (hN₁': f N₁ ω < a) (hN₂: N₁ ≤ N₂) (hN₂': b < f N₂ ω) : + upcrossings_before a b f N ω < upcrossings_before a b f (N₂ + 1) ω := +begin + refine lt_of_lt_of_le (nat.lt_succ_self _) (le_cSup (upper_crossing_time_lt_bdd_above hab) _), + rw [set.mem_set_of_eq, upper_crossing_time_succ_eq, hitting_lt_iff _ le_rfl], + swap, + { apply_instance }, + { refine ⟨N₂, ⟨_, nat.lt_succ_self _⟩, hN₂'.le⟩, + rw [lower_crossing_time, hitting_le_iff_of_lt _ (nat.lt_succ_self _)], + refine ⟨N₁, ⟨le_trans _ hN₁, hN₂⟩, hN₁'.le⟩, + by_cases hN : 0 < N, + { have : upper_crossing_time a b f N (upcrossings_before a b f N ω) ω < N := + nat.Sup_mem (upper_crossing_time_lt_nonempty hN) (upper_crossing_time_lt_bdd_above hab), + rw upper_crossing_time_eq_upper_crossing_time_of_lt + (hN₁.trans (hN₂.trans $ nat.le_succ _)) this, + exact this.le }, + { rw [not_lt, le_zero_iff] at hN, + rw [hN, upcrossings_before_zero, upper_crossing_time_zero], + refl } }, +end + +lemma lower_crossing_time_lt_of_lt_upcrossings_before + (hN : 0 < N) (hab : a < b) (hn : n < upcrossings_before a b f N ω) : + lower_crossing_time a b f N n ω < N := +lt_of_le_of_lt lower_crossing_time_le_upper_crossing_time_succ + (upper_crossing_time_lt_of_le_upcrossings_before hN hab hn) + +lemma le_sub_of_le_upcrossings_before + (hN : 0 < N) (hab : a < b) (hn : n < upcrossings_before a b f N ω) : + b - a ≤ + stopped_value f (upper_crossing_time a b f N (n + 1)) ω - + stopped_value f (lower_crossing_time a b f N n) ω := +sub_le_sub (stopped_value_upper_crossing_time + (upper_crossing_time_lt_of_le_upcrossings_before hN hab hn).ne) + (stopped_value_lower_crossing_time (lower_crossing_time_lt_of_lt_upcrossings_before hN hab hn).ne) + +lemma sub_eq_zero_of_upcrossings_before_lt (hab : a < b) (hn : upcrossings_before a b f N ω < n) : + stopped_value f (upper_crossing_time a b f N (n + 1)) ω - + stopped_value f (lower_crossing_time a b f N n) ω = 0 := +begin + have : N ≤ upper_crossing_time a b f N n ω, + { rw upcrossings_before at hn, + rw ← not_lt, + exact λ h, not_le.2 hn (le_cSup (upper_crossing_time_lt_bdd_above hab) h) }, + simp [stopped_value, upper_crossing_time_stabilize' (nat.le_succ n) this, + lower_crossing_time_stabilize' le_rfl + (le_trans this upper_crossing_time_le_lower_crossing_time)] +end + +lemma mul_upcrossings_before_le (hf : a ≤ f N ω) (hab : a < b) : + (b - a) * upcrossings_before a b f N ω ≤ + ∑ k in finset.range N, upcrossing_strat a b f N k ω * (f (k + 1) - f k) ω := +begin + classical, + by_cases hN : N = 0, + { simp [hN] }, + simp_rw [upcrossing_strat, finset.sum_mul, ← set.indicator_mul_left, pi.one_apply, + pi.sub_apply, one_mul], + rw finset.sum_comm, + have h₁ : ∀ k, ∑ n in finset.range N, + (set.Ico (lower_crossing_time a b f N k ω) (upper_crossing_time a b f N (k + 1) ω)).indicator + (λ m, f (m + 1) ω - f m ω) n = + stopped_value f (upper_crossing_time a b f N (k + 1)) ω - + stopped_value f (lower_crossing_time a b f N k) ω, + { intro k, + rw [finset.sum_indicator_eq_sum_filter, (_ : (finset.filter + (λ i, i ∈ set.Ico (lower_crossing_time a b f N k ω) (upper_crossing_time a b f N (k + 1) ω)) + (finset.range N)) = + finset.Ico (lower_crossing_time a b f N k ω) (upper_crossing_time a b f N (k + 1) ω)), + finset.sum_Ico_eq_add_neg _ lower_crossing_time_le_upper_crossing_time_succ, + finset.sum_range_sub (λ n, f n ω), finset.sum_range_sub (λ n, f n ω), neg_sub, + sub_add_sub_cancel], + { refl }, + { ext i, + simp only [set.mem_Ico, finset.mem_filter, finset.mem_range, finset.mem_Ico, + and_iff_right_iff_imp, and_imp], + exact λ _ h, lt_of_lt_of_le h upper_crossing_time_le } }, + simp_rw [h₁], + have h₂ : ∑ k in finset.range (upcrossings_before a b f N ω), (b - a) ≤ + ∑ k in finset.range N, + (stopped_value f (upper_crossing_time a b f N (k + 1)) ω - + stopped_value f (lower_crossing_time a b f N k) ω), + { calc ∑ k in finset.range (upcrossings_before a b f N ω), (b - a) + ≤ ∑ k in finset.range (upcrossings_before a b f N ω), + (stopped_value f (upper_crossing_time a b f N (k + 1)) ω - + stopped_value f (lower_crossing_time a b f N k) ω) : + begin + refine finset.sum_le_sum (λ i hi, le_sub_of_le_upcrossings_before (zero_lt_iff.2 hN) hab _), + rwa finset.mem_range at hi, + end + ...≤ ∑ k in finset.range N, + (stopped_value f (upper_crossing_time a b f N (k + 1)) ω - + stopped_value f (lower_crossing_time a b f N k) ω) : + begin + refine finset.sum_le_sum_of_subset_of_nonneg + (finset.range_subset.2 (upcrossings_before_le f ω hab)) (λ i _ hi, _), + by_cases hi' : i = upcrossings_before a b f N ω, + { subst hi', + simp only [stopped_value], + rw upper_crossing_time_eq_of_upcrossings_before_lt hab (nat.lt_succ_self _), + by_cases heq : lower_crossing_time a b f N (upcrossings_before a b f N ω) ω = N, + { rw [heq, sub_self] }, + { rw sub_nonneg, + exact le_trans (stopped_value_lower_crossing_time heq) hf } }, + { rw sub_eq_zero_of_upcrossings_before_lt hab, + rw [finset.mem_range, not_lt] at hi, + exact lt_of_le_of_ne hi (ne.symm hi') }, + end }, + refine le_trans _ h₂, + rw [finset.sum_const, finset.card_range, nsmul_eq_mul, mul_comm], +end + +lemma integral_mul_upcrossings_before_le_integral [is_finite_measure μ] + (hf : submartingale f ℱ μ) (hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) : + (b - a) * μ[upcrossings_before a b f N] ≤ μ[f N] := +calc (b - a) * μ[upcrossings_before a b f N] + ≤ μ[∑ k in finset.range N, upcrossing_strat a b f N k * (f (k + 1) - f k)] : +begin + rw ← integral_mul_left, + refine integral_mono_of_nonneg _ ((hf.sum_upcrossing_strat_mul a b N).integrable N) _, + { exact eventually_of_forall (λ ω, mul_nonneg (sub_nonneg.2 hab.le) (nat.cast_nonneg _)) }, + { refine eventually_of_forall (λ ω, _), + simpa using mul_upcrossings_before_le (hfN ω) hab }, +end + ...≤ μ[f N] - μ[f 0] : hf.sum_mul_upcrossing_strat_le + ...≤ μ[f N] : (sub_le_self_iff _).2 (integral_nonneg hfzero) + +lemma crossing_pos_eq (hab : a < b) : + upper_crossing_time 0 (b - a) (λ n ω, (f n ω - a)⁺) N n = upper_crossing_time a b f N n ∧ + lower_crossing_time 0 (b - a) (λ n ω, (f n ω - a)⁺) N n = lower_crossing_time a b f N n := +begin + have hab' : 0 < b - a := sub_pos.2 hab, + have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω, + { intros i ω, + refine ⟨λ h, _, λ h, _⟩, + { rwa [← sub_le_sub_iff_right a, + ← lattice_ordered_comm_group.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)] }, + { rw ← sub_le_sub_iff_right a at h, + rwa lattice_ordered_comm_group.pos_of_nonneg _ (le_trans hab'.le h) } }, + have hf' : ∀ ω i, (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a, + { intros ω i, + rw [lattice_ordered_comm_group.pos_nonpos_iff, sub_nonpos] }, + induction n with k ih, + { refine ⟨rfl, _⟩, + simp only [lower_crossing_time_zero, hitting, set.mem_Icc, set.mem_Iic], + ext ω, + split_ifs with h₁ h₂ h₂, + { simp_rw [hf'] }, + { simp_rw [set.mem_Iic, ← hf' _ _] at h₂, + exact false.elim (h₂ h₁) }, + { simp_rw [set.mem_Iic, hf' _ _] at h₁, + exact false.elim (h₁ h₂) }, + { refl } }, + { have : upper_crossing_time 0 (b - a) (λ n ω, (f n ω - a)⁺) N (k + 1) = + upper_crossing_time a b f N (k + 1), + { ext ω, + simp only [upper_crossing_time_succ_eq, ← ih.2, hitting, set.mem_Ici, tsub_le_iff_right], + split_ifs with h₁ h₂ h₂, + { simp_rw [← sub_le_iff_le_add, hf ω] }, + { simp_rw [set.mem_Ici, ← hf _ _] at h₂, + exact false.elim (h₂ h₁) }, + { simp_rw [set.mem_Ici, hf _ _] at h₁, + exact false.elim (h₁ h₂) }, + { refl } }, + refine ⟨this, _⟩, + ext ω, + simp only [lower_crossing_time, this, hitting, set.mem_Iic], + split_ifs with h₁ h₂ h₂, + { simp_rw [hf' ω] }, + { simp_rw [set.mem_Iic, ← hf' _ _] at h₂, + exact false.elim (h₂ h₁) }, + { simp_rw [set.mem_Iic, hf' _ _] at h₁, + exact false.elim (h₁ h₂) }, + { refl } } +end + +lemma upcrossings_before_pos_eq (hab : a < b) : + upcrossings_before 0 (b - a) (λ n ω, (f n ω - a)⁺) N ω = upcrossings_before a b f N ω := +by simp_rw [upcrossings_before, (crossing_pos_eq hab).1] + +lemma mul_integral_upcrossings_before_le_integral_pos_part_aux1 [is_finite_measure μ] + (hf : submartingale f ℱ μ) (hab : a < b) : + (b - a) * μ[upcrossings_before a b f N] ≤ μ[λ ω, (f N ω - a)⁺] := +begin + refine le_trans (le_of_eq _) (integral_mul_upcrossings_before_le_integral + (hf.sub_martingale (martingale_const _ _ _)).pos + (λ ω, lattice_ordered_comm_group.pos_nonneg _) + (λ ω, lattice_ordered_comm_group.pos_nonneg _) (sub_pos.2 hab)), + simp_rw [sub_zero, ← upcrossings_before_pos_eq hab], + refl, +end + +lemma mul_integral_upcrossings_before_le_integral_pos_part_aux2 [is_finite_measure μ] + (hf : submartingale f ℱ μ) (hab : a < b) : + (b - a) * μ[upcrossings_before a b f N] ≤ μ[λ ω, (f N ω - a)⁺] := +mul_integral_upcrossings_before_le_integral_pos_part_aux1 hf hab + +/-- **Doob's upcrossing estimate**: given a real valued discrete submartingale `f` and real +values `a` and `b`, we have `(b - a) * 𝔼[upcrossings_before a b f N] ≤ 𝔼[(f N - a)⁺]` where +`upcrossings_before a b f N` is the number of times the process `f` crossed from below `a` to above +`b` before the time `N`. -/ +theorem submartingale.mul_integral_upcrossings_before_le_integral_pos_part [is_finite_measure μ] + (a b : ℝ) (hf : submartingale f ℱ μ) (N : ℕ) : + (b - a) * μ[upcrossings_before a b f N] ≤ μ[λ ω, (f N ω - a)⁺] := +begin + by_cases hab : a < b, + { exact mul_integral_upcrossings_before_le_integral_pos_part_aux2 hf hab }, + { rw [not_lt, ← sub_nonpos] at hab, + exact le_trans (mul_nonpos_of_nonpos_of_nonneg hab (integral_nonneg (λ ω, nat.cast_nonneg _))) + (integral_nonneg (λ ω, lattice_ordered_comm_group.pos_nonneg _)) } +end + +/-! + +### Variant of the upcrossing estimate + +Now, we would like to prove a variant of the upcrossing estimate obtained by taking the supremum +over $N$ of the original upcrossing estimate. Namely, we want the inequality +$$ + (b - a) \sup_N \mathbb{E}[U_N(a, b)] \le \sup_N \mathbb{E}[f_N]. +$$ +This inequality is central for the martingale convergence theorem as it provides a uniform bound +for the upcrossings. + +We note that on top of taking the supremum on both sides of the inequality, we had also used +the monotone convergence theorem on the left hand side to take the supremum outside of the +integral. To do this, we need to make sure $U_N(a, b)$ is measurable and integrable. Integrability +is easy to check as $U_N(a, b) ≤ N$ and so it suffices to show measurability. Indeed, by +noting that +$$ + U_N(a, b) = \sum_{i = 1}^N \mathbf{1}_{\{U_N(a, b) < N\}} +$$ +$U_N(a, b)$ is measurable as $\{U_N(a, b) < N\}$ is a measurable set since $U_N(a, b)$ is a +stopping time. + +-/ + +lemma upcrossings_before_eq_sum (hab : a < b) : + upcrossings_before a b f N ω = + ∑ i in finset.Ico 1 (N + 1), {n | upper_crossing_time a b f N n ω < N}.indicator 1 i := +begin + by_cases hN : N = 0, + { simp [hN] }, + rw ← finset.sum_Ico_consecutive _ (nat.succ_le_succ zero_le') + (nat.succ_le_succ (upcrossings_before_le f ω hab)), + have h₁ : ∀ k ∈ finset.Ico 1 (upcrossings_before a b f N ω + 1), + {n : ℕ | upper_crossing_time a b f N n ω < N}.indicator 1 k = 1, + { rintro k hk, + rw finset.mem_Ico at hk, + rw set.indicator_of_mem, + { refl }, + { exact upper_crossing_time_lt_of_le_upcrossings_before (zero_lt_iff.2 hN) hab + (nat.lt_succ_iff.1 hk.2) } }, + have h₂ : ∀ k ∈ finset.Ico (upcrossings_before a b f N ω + 1) (N + 1), + {n : ℕ | upper_crossing_time a b f N n ω < N}.indicator 1 k = 0, + { rintro k hk, + rw [finset.mem_Ico, nat.succ_le_iff] at hk, + rw set.indicator_of_not_mem, + simp only [set.mem_set_of_eq, not_lt], + exact (upper_crossing_time_eq_of_upcrossings_before_lt hab hk.1).symm.le }, + rw [finset.sum_congr rfl h₁, finset.sum_congr rfl h₂, finset.sum_const, finset.sum_const, + smul_eq_mul, mul_one, smul_eq_mul, mul_zero, nat.card_Ico, nat.add_succ_sub_one, + add_zero, add_zero], +end + +lemma adapted.measurable_upcrossings_before (hf : adapted ℱ f) (hab : a < b) : + measurable (upcrossings_before a b f N) := +begin + have : upcrossings_before a b f N = + λ ω, ∑ i in finset.Ico 1 (N + 1), {n | upper_crossing_time a b f N n ω < N}.indicator 1 i, + { ext ω, + exact upcrossings_before_eq_sum hab }, + rw this, + exact finset.measurable_sum _ (λ i hi, measurable.indicator measurable_const $ + ℱ.le N _ (hf.is_stopping_time_upper_crossing_time.measurable_set_lt_of_pred N)) +end + +lemma adapted.integrable_upcrossings_before [is_finite_measure μ] + (hf : adapted ℱ f) (hab : a < b) : + integrable (λ ω, (upcrossings_before a b f N ω : ℝ)) μ := +begin + have : ∀ᵐ ω ∂μ, ∥(upcrossings_before a b f N ω : ℝ)∥ ≤ N, + { refine eventually_of_forall (λ ω, _), + rw [real.norm_eq_abs, nat.abs_cast, nat.cast_le], + refine upcrossings_before_le _ _ hab }, + exact ⟨measurable.ae_strongly_measurable + (measurable_from_top.comp (hf.measurable_upcrossings_before hab)), + has_finite_integral_of_bounded this⟩ +end + +/-- The number of upcrossings of a realization of a stochastic process (`upcrossing` takes value +in `ℝ≥0∞` and so is allowed to be `∞`). -/ +noncomputable def upcrossings [preorder ι] [order_bot ι] [has_Inf ι] + (a b : ℝ) (f : ι → Ω → ℝ) (ω : Ω) : ℝ≥0∞ := +⨆ N, (upcrossings_before a b f N ω : ℝ≥0∞) + +lemma adapted.measurable_upcrossings (hf : adapted ℱ f) (hab : a < b) : + measurable (upcrossings a b f) := +measurable_supr (λ N, measurable_from_top.comp (hf.measurable_upcrossings_before hab)) + +lemma upcrossings_lt_top_iff : + upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossings_before a b f N ω ≤ k := +begin + have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k, + { split, + { intro h, + lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr, + exact ⟨r, le_rfl⟩ }, + { rintro ⟨k, hk⟩, + exact lt_of_le_of_lt hk ennreal.coe_lt_top } }, + simp_rw [this, upcrossings, supr_le_iff], + split; rintro ⟨k, hk⟩, + { obtain ⟨m, hm⟩ := exists_nat_ge k, + refine ⟨m, λ N, ennreal.coe_nat_le_coe_nat.1 ((hk N).trans _)⟩, + rwa [← ennreal.coe_nat, ennreal.coe_le_coe] }, + { refine ⟨k, λ N, _⟩, + simp only [ennreal.coe_nat, ennreal.coe_nat_le_coe_nat, hk N] } +end + +/-- A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. -/ +lemma submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [is_finite_measure μ] + (a b : ℝ) (hf : submartingale f ℱ μ) : + ennreal.of_real (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤ + ⨆ N, ∫⁻ ω, ennreal.of_real ((f N ω - a)⁺) ∂μ := +begin + by_cases hab : a < b, + { simp_rw [upcrossings], + have : ∀ N, ∫⁻ ω, ennreal.of_real ((f N ω - a)⁺) ∂μ = ennreal.of_real (∫ ω, (f N ω - a)⁺ ∂μ), + { intro N, + rw of_real_integral_eq_lintegral_of_real, + { exact (hf.sub_martingale (martingale_const _ _ _)).pos.integrable _ }, + { exact eventually_of_forall (λ ω, lattice_ordered_comm_group.pos_nonneg _) } }, + rw lintegral_supr', + { simp_rw [this, ennreal.mul_supr, supr_le_iff], + intro N, + rw [(by simp : ∫⁻ ω, upcrossings_before a b f N ω ∂μ = + ∫⁻ ω, ↑(upcrossings_before a b f N ω : ℝ≥0) ∂μ), lintegral_coe_eq_integral, + ← ennreal.of_real_mul (sub_pos.2 hab).le], + { simp_rw [nnreal.coe_nat_cast], + exact (ennreal.of_real_le_of_real + (hf.mul_integral_upcrossings_before_le_integral_pos_part a b N)).trans (le_supr _ N) }, + { simp only [nnreal.coe_nat_cast, hf.adapted.integrable_upcrossings_before hab] } }, + { exact λ n, measurable_from_top.comp_ae_measurable + (hf.adapted.measurable_upcrossings_before hab).ae_measurable }, + { refine eventually_of_forall (λ ω N M hNM, _), + rw ennreal.coe_nat_le_coe_nat, + exact upcrossings_before_mono hab hNM ω } }, + { rw [not_lt, ← sub_nonpos] at hab, + rw [ennreal.of_real_of_nonpos hab, zero_mul], + exact zero_le _ } +end + +end measure_theory diff --git a/formal/lean/mathlib/probability/notation.lean b/formal/lean/mathlib/probability/notation.lean index 9f167533e0d9d7bce8ac49da960d2aba33927f2d..882efd388369cdbef292ac7e328fe9693469db73 100644 --- a/formal/lean/mathlib/probability/notation.lean +++ b/formal/lean/mathlib/probability/notation.lean @@ -3,7 +3,7 @@ Copyright (c) 2021 Rémy Degenne. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Rémy Degenne -/ -import measure_theory.function.conditional_expectation +import measure_theory.function.conditional_expectation.real /-! # Notations for probability theory @@ -34,10 +34,10 @@ localized "notation P `[` X `]` := ∫ x, X x ∂P" in probability_theory localized "notation `𝔼[` X `]` := ∫ a, X a" in probability_theory -localized "notation X `=ₐₛ`:50 Y:50 := X =ᵐ[measure_theory.measure_space.volume] Y" +localized "notation X ` =ₐₛ `:50 Y:50 := X =ᵐ[measure_theory.measure_space.volume] Y" in probability_theory -localized "notation X `≤ₐₛ`:50 Y:50 := X ≤ᵐ[measure_theory.measure_space.volume] Y" +localized "notation X ` ≤ₐₛ `:50 Y:50 := X ≤ᵐ[measure_theory.measure_space.volume] Y" in probability_theory localized "notation `∂` P `/∂`:50 Q:50 := P.rn_deriv Q" in probability_theory diff --git a/formal/lean/mathlib/probability/stopping.lean b/formal/lean/mathlib/probability/stopping.lean index cba03e7f8f1e94ae4e71fd0cff4ea451ea5f6249..b45d42c26a784b5e1aa6bfabc2eb3dd42655ca56 100644 --- a/formal/lean/mathlib/probability/stopping.lean +++ b/formal/lean/mathlib/probability/stopping.lean @@ -3,7 +3,7 @@ Copyright (c) 2021 Kexing Ying. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kexing Ying -/ -import measure_theory.function.conditional_expectation +import measure_theory.function.conditional_expectation.real import topology.instances.discrete /-! @@ -20,8 +20,8 @@ at a specific time and is the first step in formalizing stochastic processes. filtration `f` if at each point in time `i`, `u i` is `f i`-strongly measurable * `measure_theory.prog_measurable`: a sequence of functions `u` is said to be progressively measurable with respect to a filtration `f` if at each point in time `i`, `u` restricted to - `set.Iic i × α` is strongly measurable with respect to the product `measurable_space` structure - where the σ-algebra used for `α` is `f i`. + `set.Iic i × Ω` is strongly measurable with respect to the product `measurable_space` structure + where the σ-algebra used for `Ω` is `f i`. * `measure_theory.filtration.natural`: the natural filtration with respect to a sequence of measurable functions is the smallest filtration to which it is adapted to * `measure_theory.is_stopping_time`: a stopping time with respect to some filtration `f` is a @@ -50,16 +50,16 @@ namespace measure_theory /-! ### Filtrations -/ -/-- A `filtration` on measurable space `α` with σ-algebra `m` is a monotone +/-- A `filtration` on measurable space `Ω` with σ-algebra `m` is a monotone sequence of sub-σ-algebras of `m`. -/ -structure filtration {α : Type*} (ι : Type*) [preorder ι] (m : measurable_space α) := -(seq : ι → measurable_space α) +structure filtration {Ω : Type*} (ι : Type*) [preorder ι] (m : measurable_space Ω) := +(seq : ι → measurable_space Ω) (mono' : monotone seq) (le' : ∀ i : ι, seq i ≤ m) -variables {α β ι : Type*} {m : measurable_space α} +variables {Ω β ι : Type*} {m : measurable_space Ω} -instance [preorder ι] : has_coe_to_fun (filtration ι m) (λ _, ι → measurable_space α) := +instance [preorder ι] : has_coe_to_fun (filtration ι m) (λ _, ι → measurable_space Ω) := ⟨λ f, f.seq⟩ namespace filtration @@ -69,17 +69,17 @@ protected lemma mono {i j : ι} (f : filtration ι m) (hij : i ≤ j) : f i ≤ protected lemma le (f : filtration ι m) (i : ι) : f i ≤ m := f.le' i -@[ext] protected lemma ext {f g : filtration ι m} (h : (f : ι → measurable_space α) = g) : f = g := +@[ext] protected lemma ext {f g : filtration ι m} (h : (f : ι → measurable_space Ω) = g) : f = g := by { cases f, cases g, simp only, exact h, } variable (ι) /-- The constant filtration which is equal to `m` for all `i : ι`. -/ -def const (m' : measurable_space α) (hm' : m' ≤ m) : filtration ι m := +def const (m' : measurable_space Ω) (hm' : m' ≤ m) : filtration ι m := ⟨λ _, m', monotone_const, λ _, hm'⟩ variable {ι} @[simp] -lemma const_apply {m' : measurable_space α} {hm' : m' ≤ m} (i : ι) : const ι m' hm' i = m' := rfl +lemma const_apply {m' : measurable_space Ω} {hm' : m' ≤ m} (i : ι) : const ι m' hm' i = m' := rfl instance : inhabited (filtration ι m) := ⟨const ι m le_rfl⟩ @@ -198,22 +198,22 @@ instance : complete_lattice (filtration ι m) := end filtration -lemma measurable_set_of_filtration [preorder ι] {f : filtration ι m} {s : set α} {i : ι} +lemma measurable_set_of_filtration [preorder ι] {f : filtration ι m} {s : set Ω} {i : ι} (hs : measurable_set[f i] s) : measurable_set[m] s := f.le i s hs /-- A measure is σ-finite with respect to filtration if it is σ-finite with respect to all the sub-σ-algebra of the filtration. -/ -class sigma_finite_filtration [preorder ι] (μ : measure α) (f : filtration ι m) : Prop := +class sigma_finite_filtration [preorder ι] (μ : measure Ω) (f : filtration ι m) : Prop := (sigma_finite : ∀ i : ι, sigma_finite (μ.trim (f.le i))) -instance sigma_finite_of_sigma_finite_filtration [preorder ι] (μ : measure α) (f : filtration ι m) +instance sigma_finite_of_sigma_finite_filtration [preorder ι] (μ : measure Ω) (f : filtration ι m) [hf : sigma_finite_filtration μ f] (i : ι) : sigma_finite (μ.trim (f.le i)) := by apply hf.sigma_finite -- can't exact here @[priority 100] -instance is_finite_measure.sigma_finite_filtration [preorder ι] (μ : measure α) (f : filtration ι m) +instance is_finite_measure.sigma_finite_filtration [preorder ι] (μ : measure Ω) (f : filtration ι m) [is_finite_measure μ] : sigma_finite_filtration μ f := ⟨λ n, by apply_instance⟩ @@ -221,19 +221,19 @@ instance is_finite_measure.sigma_finite_filtration [preorder ι] (μ : measure /-- Given a integrable function `g`, the conditional expectations of `g` with respect to a filtration is uniformly integrable. -/ lemma integrable.uniform_integrable_condexp_filtration - [preorder ι] {μ : measure α} [is_finite_measure μ] {f : filtration ι m} - {g : α → ℝ} (hg : integrable g μ) : + [preorder ι] {μ : measure Ω} [is_finite_measure μ] {f : filtration ι m} + {g : Ω → ℝ} (hg : integrable g μ) : uniform_integrable (λ i, μ[g | f i]) 1 μ := hg.uniform_integrable_condexp f.le section adapted_process variables [topological_space β] [preorder ι] - {u v : ι → α → β} {f : filtration ι m} + {u v : ι → Ω → β} {f : filtration ι m} /-- A sequence of functions `u` is adapted to a filtration `f` if for all `i`, `u i` is `f i`-measurable. -/ -def adapted (f : filtration ι m) (u : ι → α → β) : Prop := +def adapted (f : filtration ι m) (u : ι → Ω → β) : Prop := ∀ i : ι, strongly_measurable[f i] (u i) namespace adapted @@ -261,22 +261,25 @@ lemma strongly_measurable_le {i j : ι} (hf : adapted f u) (hij : i ≤ j) : end adapted +lemma adapted_const (f : filtration ι m) (x : β) : adapted f (λ _ _, x) := +λ i, strongly_measurable_const + variable (β) -lemma adapted_zero [has_zero β] (f : filtration ι m) : adapted f (0 : ι → α → β) := -λ i, @strongly_measurable_zero α β (f i) _ _ +lemma adapted_zero [has_zero β] (f : filtration ι m) : adapted f (0 : ι → Ω → β) := +λ i, @strongly_measurable_zero Ω β (f i) _ _ variable {β} /-- Progressively measurable process. A sequence of functions `u` is said to be progressively measurable with respect to a filtration `f` if at each point in time `i`, `u` restricted to -`set.Iic i × α` is measurable with respect to the product `measurable_space` structure where the -σ-algebra used for `α` is `f i`. +`set.Iic i × Ω` is measurable with respect to the product `measurable_space` structure where the +σ-algebra used for `Ω` is `f i`. The usual definition uses the interval `[0,i]`, which we replace by `set.Iic i`. We recover the usual definition for index types `ℝ≥0` or `ℕ`. -/ -def prog_measurable [measurable_space ι] (f : filtration ι m) (u : ι → α → β) : Prop := -∀ i, strongly_measurable[subtype.measurable_space.prod (f i)] (λ p : set.Iic i × α, u p.1 p.2) +def prog_measurable [measurable_space ι] (f : filtration ι m) (u : ι → Ω → β) : Prop := +∀ i, strongly_measurable[subtype.measurable_space.prod (f i)] (λ p : set.Iic i × Ω, u p.1 p.2) lemma prog_measurable_const [measurable_space ι] (f : filtration ι m) (b : β) : - prog_measurable f ((λ _ _, b) : ι → α → β) := + prog_measurable f ((λ _ _, b) : ι → Ω → β) := λ i, @strongly_measurable_const _ _ (subtype.measurable_space.prod (f i)) _ _ namespace prog_measurable @@ -286,19 +289,19 @@ variables [measurable_space ι] protected lemma adapted (h : prog_measurable f u) : adapted f u := begin intro i, - have : u i = (λ p : set.Iic i × α, u p.1 p.2) ∘ (λ x, (⟨i, set.mem_Iic.mpr le_rfl⟩, x)) := rfl, + have : u i = (λ p : set.Iic i × Ω, u p.1 p.2) ∘ (λ x, (⟨i, set.mem_Iic.mpr le_rfl⟩, x)) := rfl, rw this, exact (h i).comp_measurable measurable_prod_mk_left, end -protected lemma comp {t : ι → α → ι} [topological_space ι] [borel_space ι] [metrizable_space ι] +protected lemma comp {t : ι → Ω → ι} [topological_space ι] [borel_space ι] [metrizable_space ι] (h : prog_measurable f u) (ht : prog_measurable f t) - (ht_le : ∀ i x, t i x ≤ i) : - prog_measurable f (λ i x, u (t i x) x) := + (ht_le : ∀ i ω, t i ω ≤ i) : + prog_measurable f (λ i ω, u (t i ω) ω) := begin intro i, - have : (λ p : ↥(set.Iic i) × α, u (t (p.fst : ι) p.snd) p.snd) - = (λ p : ↥(set.Iic i) × α, u (p.fst : ι) p.snd) ∘ (λ p : ↥(set.Iic i) × α, + have : (λ p : ↥(set.Iic i) × Ω, u (t (p.fst : ι) p.snd) p.snd) + = (λ p : ↥(set.Iic i) × Ω, u (p.fst : ι) p.snd) ∘ (λ p : ↥(set.Iic i) × Ω, (⟨t (p.fst : ι) p.snd, set.mem_Iic.mpr ((ht_le _ _).trans p.fst.prop)⟩, p.snd)) := rfl, rw this, exact (h i).comp_measurable ((ht i).measurable.subtype_mk.prod_mk measurable_snd), @@ -308,27 +311,27 @@ section arithmetic @[to_additive] protected lemma mul [has_mul β] [has_continuous_mul β] (hu : prog_measurable f u) (hv : prog_measurable f v) : - prog_measurable f (λ i x, u i x * v i x) := + prog_measurable f (λ i ω, u i ω * v i ω) := λ i, (hu i).mul (hv i) @[to_additive] protected lemma finset_prod' {γ} [comm_monoid β] [has_continuous_mul β] - {U : γ → ι → α → β} {s : finset γ} (h : ∀ c ∈ s, prog_measurable f (U c)) : + {U : γ → ι → Ω → β} {s : finset γ} (h : ∀ c ∈ s, prog_measurable f (U c)) : prog_measurable f (∏ c in s, U c) := finset.prod_induction U (prog_measurable f) (λ _ _, prog_measurable.mul) (prog_measurable_const _ 1) h @[to_additive] protected lemma finset_prod {γ} [comm_monoid β] [has_continuous_mul β] - {U : γ → ι → α → β} {s : finset γ} (h : ∀ c ∈ s, prog_measurable f (U c)) : + {U : γ → ι → Ω → β} {s : finset γ} (h : ∀ c ∈ s, prog_measurable f (U c)) : prog_measurable f (λ i a, ∏ c in s, U c i a) := by { convert prog_measurable.finset_prod' h, ext i a, simp only [finset.prod_apply], } @[to_additive] protected lemma inv [group β] [topological_group β] (hu : prog_measurable f u) : - prog_measurable f (λ i x, (u i x)⁻¹) := + prog_measurable f (λ i ω, (u i ω)⁻¹) := λ i, (hu i).inv @[to_additive] protected lemma div [group β] [topological_group β] (hu : prog_measurable f u) (hv : prog_measurable f v) : - prog_measurable f (λ i x, u i x / v i x) := + prog_measurable f (λ i ω, u i ω / v i ω) := λ i, (hu i).div (hv i) end arithmetic @@ -336,12 +339,12 @@ end arithmetic end prog_measurable lemma prog_measurable_of_tendsto' {γ} [measurable_space ι] [metrizable_space β] - (fltr : filter γ) [fltr.ne_bot] [fltr.is_countably_generated] {U : γ → ι → α → β} + (fltr : filter γ) [fltr.ne_bot] [fltr.is_countably_generated] {U : γ → ι → Ω → β} (h : ∀ l, prog_measurable f (U l)) (h_tendsto : tendsto U fltr (𝓝 u)) : prog_measurable f u := begin assume i, - apply @strongly_measurable_of_tendsto (set.Iic i × α) β γ (measurable_space.prod _ (f i)) + apply @strongly_measurable_of_tendsto (set.Iic i × Ω) β γ (measurable_space.prod _ (f i)) _ _ fltr _ _ _ _ (λ l, h l i), rw tendsto_pi_nhds at h_tendsto ⊢, intro x, @@ -351,7 +354,7 @@ begin end lemma prog_measurable_of_tendsto [measurable_space ι] [metrizable_space β] - {U : ℕ → ι → α → β} + {U : ℕ → ι → Ω → β} (h : ∀ l, prog_measurable f (U l)) (h_tendsto : tendsto U at_top (𝓝 u)) : prog_measurable f u := prog_measurable_of_tendsto' at_top h h_tendsto @@ -361,10 +364,10 @@ prog_measurable_of_tendsto' at_top h h_tendsto theorem adapted.prog_measurable_of_continuous [topological_space ι] [metrizable_space ι] [measurable_space ι] [second_countable_topology ι] [opens_measurable_space ι] [metrizable_space β] - (h : adapted f u) (hu_cont : ∀ x, continuous (λ i, u i x)) : + (h : adapted f u) (hu_cont : ∀ ω, continuous (λ i, u i ω)) : prog_measurable f u := λ i, @strongly_measurable_uncurry_of_continuous_of_strongly_measurable _ _ (set.Iic i) _ _ _ _ _ _ _ - (f i) _ (λ x, (hu_cont x).comp continuous_induced_dom) (λ j, (h j).mono (f.mono j.prop)) + (f i) _ (λ ω, (hu_cont ω).comp continuous_induced_dom) (λ j, (h j).mono (f.mono j.prop)) end adapted_process @@ -377,7 +380,7 @@ include mβ /-- Given a sequence of functions, the natural filtration is the smallest sequence of σ-algebras such that that sequence of functions is measurable with respect to the filtration. -/ -def natural (u : ι → α → β) (hum : ∀ i, strongly_measurable (u i)) : filtration ι m := +def natural (u : ι → Ω → β) (hum : ∀ i, strongly_measurable (u i)) : filtration ι m := { seq := λ i, ⨆ j ≤ i, measurable_space.comap (u j) mβ, mono' := λ i j hij, bsupr_mono $ λ k, ge_trans hij, le' := λ i, @@ -387,7 +390,7 @@ def natural (u : ι → α → β) (hum : ∀ i, strongly_measurable (u i)) : fi exact (hum j).measurable ht, end } -lemma adapted_natural {u : ι → α → β} (hum : ∀ i, strongly_measurable[m] (u i)) : +lemma adapted_natural {u : ι → Ω → β} (hum : ∀ i, strongly_measurable[m] (u i)) : adapted (natural u hum) u := begin assume i, @@ -396,6 +399,54 @@ begin exact ⟨measurable_iff_comap_le.2 le_rfl, (hum i).is_separable_range⟩ end +section limit + +omit mβ + +variables {E : Type*} [has_zero E] [topological_space E] + {ℱ : filtration ι m} {f : ι → Ω → E} {μ : measure Ω} + +/-- Given a process `f` and a filtration `ℱ`, if `f` converges to some `g` almost everywhere and +`g` is `⨆ n, ℱ n`-measurable, then `limit_process f ℱ μ` chooses said `g`, else it returns 0. + +This definition is used to phrase the a.e. martingale convergence theorem +`submartingale.ae_tendsto_limit_process` where an L¹-bounded submartingale `f` adapted to `ℱ` +converges to `limit_process f ℱ μ` `μ`-almost everywhere. -/ +noncomputable +def limit_process (f : ι → Ω → E) (ℱ : filtration ι m) (μ : measure Ω . volume_tac) := +if h : ∃ g : Ω → E, strongly_measurable[⨆ n, ℱ n] g ∧ + ∀ᵐ ω ∂μ, tendsto (λ n, f n ω) at_top (𝓝 (g ω)) then classical.some h else 0 + +lemma strongly_measurable_limit_process : + strongly_measurable[⨆ n, ℱ n] (limit_process f ℱ μ) := +begin + rw limit_process, + split_ifs with h h, + exacts [(classical.some_spec h).1, strongly_measurable_zero] +end + +lemma strongly_measurable_limit_process' : + strongly_measurable[m] (limit_process f ℱ μ) := +strongly_measurable_limit_process.mono (Sup_le (λ m ⟨n, hn⟩, hn ▸ ℱ.le _)) + +lemma mem_ℒp_limit_process_of_snorm_bdd {R : ℝ≥0} {p : ℝ≥0∞} + {F : Type*} [normed_add_comm_group F] {ℱ : filtration ℕ m} {f : ℕ → Ω → F} + (hfm : ∀ n, ae_strongly_measurable (f n) μ) (hbdd : ∀ n, snorm (f n) p μ ≤ R) : + mem_ℒp (limit_process f ℱ μ) p μ := +begin + rw limit_process, + split_ifs with h, + { refine ⟨strongly_measurable.ae_strongly_measurable + ((classical.some_spec h).1.mono (Sup_le (λ m ⟨n, hn⟩, hn ▸ ℱ.le _))), + lt_of_le_of_lt (Lp.snorm_lim_le_liminf_snorm hfm _ (classical.some_spec h).2) + (lt_of_le_of_lt _ (ennreal.coe_lt_top : ↑R < ∞))⟩, + simp_rw [liminf_eq, eventually_at_top], + exact Sup_le (λ b ⟨a, ha⟩, (ha a le_rfl).trans (hbdd _)) }, + { exact zero_mem_ℒp } +end + +end limit + end filtration /-! ### Stopping times -/ @@ -406,8 +457,8 @@ with respect to `f i`. Intuitively, the stopping time `τ` describes some stopping rule such that at time `i`, we may determine it with the information we have at time `i`. -/ -def is_stopping_time [preorder ι] (f : filtration ι m) (τ : α → ι) := -∀ i : ι, measurable_set[f i] $ {x | τ x ≤ i} +def is_stopping_time [preorder ι] (f : filtration ι m) (τ : Ω → ι) := +∀ i : ι, measurable_set[f i] $ {ω | τ ω ≤ i} lemma is_stopping_time_const [preorder ι] (f : filtration ι m) (i : ι) : is_stopping_time f (λ x, i) := @@ -416,23 +467,23 @@ lemma is_stopping_time_const [preorder ι] (f : filtration ι m) (i : ι) : section measurable_set section preorder -variables [preorder ι] {f : filtration ι m} {τ : α → ι} +variables [preorder ι] {f : filtration ι m} {τ : Ω → ι} protected lemma is_stopping_time.measurable_set_le (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[f i] {x | τ x ≤ i} := + measurable_set[f i] {ω | τ ω ≤ i} := hτ i lemma is_stopping_time.measurable_set_lt_of_pred [pred_order ι] (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[f i] {x | τ x < i} := + measurable_set[f i] {ω | τ ω < i} := begin by_cases hi_min : is_min i, - { suffices : {x : α | τ x < i} = ∅, by { rw this, exact @measurable_set.empty _ (f i), }, - ext1 x, + { suffices : {ω : Ω | τ ω < i} = ∅, by { rw this, exact @measurable_set.empty _ (f i), }, + ext1 ω, simp only [set.mem_set_of_eq, set.mem_empty_eq, iff_false], rw is_min_iff_forall_not_lt at hi_min, - exact hi_min (τ x), }, - have : {x : α | τ x < i} = τ ⁻¹' (set.Iio i) := rfl, + exact hi_min (τ ω), }, + have : {ω : Ω | τ ω < i} = τ ⁻¹' (set.Iio i) := rfl, rw [this, ←Iic_pred_of_not_is_min hi_min], exact f.mono (pred_le i) _ (hτ.measurable_set_le $ pred i), end @@ -443,13 +494,13 @@ section countable_stopping_time namespace is_stopping_time -variables [partial_order ι] {τ : α → ι} {f : filtration ι m} +variables [partial_order ι] {τ : Ω → ι} {f : filtration ι m} protected lemma measurable_set_eq_of_countable (hτ : is_stopping_time f τ) (h_countable : (set.range τ).countable) (i : ι) : - measurable_set[f i] {a | τ a = i} := + measurable_set[f i] {ω | τ ω = i} := begin - have : {a | τ a = i} = {a | τ a ≤ i} \ (⋃ (j ∈ set.range τ) (hj : j < i), {a | τ a ≤ j}), + have : {ω | τ ω = i} = {ω | τ ω ≤ i} \ (⋃ (j ∈ set.range τ) (hj : j < i), {ω | τ ω ≤ j}), { ext1 a, simp only [set.mem_set_of_eq, set.mem_range, set.Union_exists, set.Union_Union_eq', set.mem_diff, set.mem_Union, exists_prop, not_exists, not_and, not_le], @@ -471,36 +522,36 @@ begin end protected lemma measurable_set_eq_of_encodable [encodable ι] (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[f i] {a | τ a = i} := + measurable_set[f i] {ω | τ ω = i} := hτ.measurable_set_eq_of_countable (set.to_countable _) i protected lemma measurable_set_lt_of_countable (hτ : is_stopping_time f τ) (h_countable : (set.range τ).countable) (i : ι) : - measurable_set[f i] {a | τ a < i} := + measurable_set[f i] {ω | τ ω < i} := begin - have : {a | τ a < i} = {a | τ a ≤ i} \ {a | τ a = i}, - { ext1 x, simp [lt_iff_le_and_ne], }, + have : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i}, + { ext1 ω, simp [lt_iff_le_and_ne], }, rw this, exact (hτ.measurable_set_le i).diff (hτ.measurable_set_eq_of_countable h_countable i), end protected lemma measurable_set_lt_of_encodable [encodable ι] (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[f i] {a | τ a < i} := + measurable_set[f i] {ω | τ ω < i} := hτ.measurable_set_lt_of_countable (set.to_countable _) i -protected lemma measurable_set_ge_of_countable {ι} [linear_order ι] {τ : α → ι} {f : filtration ι m} +protected lemma measurable_set_ge_of_countable {ι} [linear_order ι] {τ : Ω → ι} {f : filtration ι m} (hτ : is_stopping_time f τ) (h_countable : (set.range τ).countable) (i : ι) : - measurable_set[f i] {a | i ≤ τ a} := + measurable_set[f i] {ω | i ≤ τ ω} := begin - have : {x | i ≤ τ x} = {x | τ x < i}ᶜ, - { ext1 x, simp only [set.mem_set_of_eq, set.mem_compl_eq, not_lt], }, + have : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ, + { ext1 ω, simp only [set.mem_set_of_eq, set.mem_compl_eq, not_lt], }, rw this, exact (hτ.measurable_set_lt_of_countable h_countable i).compl, end -protected lemma measurable_set_ge_of_encodable {ι} [linear_order ι] {τ : α → ι} {f : filtration ι m} +protected lemma measurable_set_ge_of_encodable {ι} [linear_order ι] {τ : Ω → ι} {f : filtration ι m} [encodable ι] (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[f i] {a | i ≤ τ a} := + measurable_set[f i] {ω | i ≤ τ ω} := hτ.measurable_set_ge_of_countable (set.to_countable _) i end is_stopping_time @@ -508,13 +559,13 @@ end is_stopping_time end countable_stopping_time section linear_order -variables [linear_order ι] {f : filtration ι m} {τ : α → ι} +variables [linear_order ι] {f : filtration ι m} {τ : Ω → ι} lemma is_stopping_time.measurable_set_gt (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[f i] {x | i < τ x} := + measurable_set[f i] {ω | i < τ ω} := begin - have : {x | i < τ x} = {x | τ x ≤ i}ᶜ, - { ext1 x, simp only [set.mem_set_of_eq, set.mem_compl_eq, not_le], }, + have : {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ, + { ext1 ω, simp only [set.mem_set_of_eq, set.mem_compl_eq, not_le], }, rw this, exact (hτ.measurable_set_le i).compl, end @@ -526,17 +577,17 @@ variables [topological_space ι] [order_topology ι] [first_countable_topology /-- Auxiliary lemma for `is_stopping_time.measurable_set_lt`. -/ lemma is_stopping_time.measurable_set_lt_of_is_lub (hτ : is_stopping_time f τ) (i : ι) (h_lub : is_lub (set.Iio i) i) : - measurable_set[f i] {x | τ x < i} := + measurable_set[f i] {ω | τ ω < i} := begin by_cases hi_min : is_min i, - { suffices : {x : α | τ x < i} = ∅, by { rw this, exact @measurable_set.empty _ (f i), }, - ext1 x, + { suffices : {ω | τ ω < i} = ∅, by { rw this, exact @measurable_set.empty _ (f i), }, + ext1 ω, simp only [set.mem_set_of_eq, set.mem_empty_eq, iff_false], - exact is_min_iff_forall_not_lt.mp hi_min (τ x), }, + exact is_min_iff_forall_not_lt.mp hi_min (τ ω), }, obtain ⟨seq, -, -, h_tendsto, h_bound⟩ : ∃ seq : ℕ → ι, monotone seq ∧ (∀ j, seq j ≤ i) ∧ tendsto seq at_top (𝓝 i) ∧ (∀ j, seq j < i), from h_lub.exists_seq_monotone_tendsto (not_is_min_iff.mp hi_min), - have h_Ioi_eq_Union : set.Iio i = ⋃ j, { k | k ≤ seq j}, + have h_Ioi_eq_Union : set.Iio i = ⋃ j, {k | k ≤ seq j}, { ext1 k, simp only [set.mem_Iio, set.mem_Union, set.mem_set_of_eq], refine ⟨λ hk_lt_i, _, λ h_exists_k_le_seq, _⟩, @@ -547,8 +598,8 @@ begin exact ⟨a, ha a le_rfl⟩, }, { obtain ⟨j, hk_seq_j⟩ := h_exists_k_le_seq, exact hk_seq_j.trans_lt (h_bound j), }, }, - have h_lt_eq_preimage : {x : α | τ x < i} = τ ⁻¹' (set.Iio i), - { ext1 x, simp only [set.mem_set_of_eq, set.mem_preimage, set.mem_Iio], }, + have h_lt_eq_preimage : {ω | τ ω < i} = τ ⁻¹' (set.Iio i), + { ext1 ω, simp only [set.mem_set_of_eq, set.mem_preimage, set.mem_Iio], }, rw [h_lt_eq_preimage, h_Ioi_eq_Union], simp only [set.preimage_Union, set.preimage_set_of_eq], exact measurable_set.Union @@ -556,41 +607,41 @@ begin end lemma is_stopping_time.measurable_set_lt (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[f i] {x | τ x < i} := + measurable_set[f i] {ω | τ ω < i} := begin obtain ⟨i', hi'_lub⟩ : ∃ i', is_lub (set.Iio i) i', from exists_lub_Iio i, cases lub_Iio_eq_self_or_Iio_eq_Iic i hi'_lub with hi'_eq_i h_Iio_eq_Iic, { rw ← hi'_eq_i at hi'_lub ⊢, exact hτ.measurable_set_lt_of_is_lub i' hi'_lub, }, - { have h_lt_eq_preimage : {x : α | τ x < i} = τ ⁻¹' (set.Iio i) := rfl, + { have h_lt_eq_preimage : {ω : Ω | τ ω < i} = τ ⁻¹' (set.Iio i) := rfl, rw [h_lt_eq_preimage, h_Iio_eq_Iic], exact f.mono (lub_Iio_le i hi'_lub) _ (hτ.measurable_set_le i'), }, end lemma is_stopping_time.measurable_set_ge (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[f i] {x | i ≤ τ x} := + measurable_set[f i] {ω | i ≤ τ ω} := begin - have : {x | i ≤ τ x} = {x | τ x < i}ᶜ, - { ext1 x, simp only [set.mem_set_of_eq, set.mem_compl_eq, not_lt], }, + have : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ, + { ext1 ω, simp only [set.mem_set_of_eq, set.mem_compl_eq, not_lt], }, rw this, exact (hτ.measurable_set_lt i).compl, end lemma is_stopping_time.measurable_set_eq (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[f i] {x | τ x = i} := + measurable_set[f i] {ω | τ ω = i} := begin - have : {x | τ x = i} = {x | τ x ≤ i} ∩ {x | τ x ≥ i}, - { ext1 x, simp only [set.mem_set_of_eq, ge_iff_le, set.mem_inter_eq, le_antisymm_iff], }, + have : {ω | τ ω = i} = {ω | τ ω ≤ i} ∩ {ω | τ ω ≥ i}, + { ext1 ω, simp only [set.mem_set_of_eq, ge_iff_le, set.mem_inter_eq, le_antisymm_iff], }, rw this, exact (hτ.measurable_set_le i).inter (hτ.measurable_set_ge i), end lemma is_stopping_time.measurable_set_eq_le (hτ : is_stopping_time f τ) {i j : ι} (hle : i ≤ j) : - measurable_set[f j] {x | τ x = i} := + measurable_set[f j] {ω | τ ω = i} := f.mono hle _ $ hτ.measurable_set_eq i lemma is_stopping_time.measurable_set_lt_le (hτ : is_stopping_time f τ) {i j : ι} (hle : i ≤ j) : - measurable_set[f j] {x | τ x < i} := + measurable_set[f j] {ω | τ ω < i} := f.mono hle _ $ hτ.measurable_set_lt i end topological_space @@ -600,11 +651,11 @@ end linear_order section encodable lemma is_stopping_time_of_measurable_set_eq [preorder ι] [encodable ι] - {f : filtration ι m} {τ : α → ι} (hτ : ∀ i, measurable_set[f i] {x | τ x = i}) : + {f : filtration ι m} {τ : Ω → ι} (hτ : ∀ i, measurable_set[f i] {ω | τ ω = i}) : is_stopping_time f τ := begin intro i, - rw show {x | τ x ≤ i} = ⋃ k ≤ i, {x | τ x = k}, by { ext, simp }, + rw show {ω | τ ω ≤ i} = ⋃ k ≤ i, {ω | τ ω = k}, by { ext, simp }, refine measurable_set.bUnion (set.to_countable _) (λ k hk, _), exact f.mono hk _ (hτ k), end @@ -615,38 +666,38 @@ end measurable_set namespace is_stopping_time -protected lemma max [linear_order ι] {f : filtration ι m} {τ π : α → ι} +protected lemma max [linear_order ι] {f : filtration ι m} {τ π : Ω → ι} (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) : - is_stopping_time f (λ x, max (τ x) (π x)) := + is_stopping_time f (λ ω, max (τ ω) (π ω)) := begin intro i, simp_rw [max_le_iff, set.set_of_and], exact (hτ i).inter (hπ i), end -protected lemma max_const [linear_order ι] {f : filtration ι m} {τ : α → ι} +protected lemma max_const [linear_order ι] {f : filtration ι m} {τ : Ω → ι} (hτ : is_stopping_time f τ) (i : ι) : - is_stopping_time f (λ x, max (τ x) i) := + is_stopping_time f (λ ω, max (τ ω) i) := hτ.max (is_stopping_time_const f i) -protected lemma min [linear_order ι] {f : filtration ι m} {τ π : α → ι} +protected lemma min [linear_order ι] {f : filtration ι m} {τ π : Ω → ι} (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) : - is_stopping_time f (λ x, min (τ x) (π x)) := + is_stopping_time f (λ ω, min (τ ω) (π ω)) := begin intro i, simp_rw [min_le_iff, set.set_of_or], exact (hτ i).union (hπ i), end -protected lemma min_const [linear_order ι] {f : filtration ι m} {τ : α → ι} +protected lemma min_const [linear_order ι] {f : filtration ι m} {τ : Ω → ι} (hτ : is_stopping_time f τ) (i : ι) : - is_stopping_time f (λ x, min (τ x) i) := + is_stopping_time f (λ ω, min (τ ω) i) := hτ.min (is_stopping_time_const f i) lemma add_const [add_group ι] [preorder ι] [covariant_class ι ι (function.swap (+)) (≤)] [covariant_class ι ι (+) (≤)] - {f : filtration ι m} {τ : α → ι} (hτ : is_stopping_time f τ) {i : ι} (hi : 0 ≤ i) : - is_stopping_time f (λ x, τ x + i) := + {f : filtration ι m} {τ : Ω → ι} (hτ : is_stopping_time f τ) {i : ι} (hi : 0 ≤ i) : + is_stopping_time f (λ ω, τ ω + i) := begin intro j, simp_rw [← le_sub_iff_add_le], @@ -654,8 +705,8 @@ begin end lemma add_const_nat - {f : filtration ℕ m} {τ : α → ℕ} (hτ : is_stopping_time f τ) {i : ℕ} : - is_stopping_time f (λ x, τ x + i) := + {f : filtration ℕ m} {τ : Ω → ℕ} (hτ : is_stopping_time f τ) {i : ℕ} : + is_stopping_time f (λ ω, τ ω + i) := begin refine is_stopping_time_of_measurable_set_eq (λ j, _), by_cases hij : i ≤ j, @@ -663,40 +714,40 @@ begin exact f.mono (j.sub_le i) _ (hτ.measurable_set_eq (j - i)) }, { rw not_le at hij, convert measurable_set.empty, - ext x, + ext ω, simp only [set.mem_empty_eq, iff_false], - rintro (hx : τ x + i = j), + rintro (hx : τ ω + i = j), linarith }, end -- generalize to certain encodable type? lemma add - {f : filtration ℕ m} {τ π : α → ℕ} (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) : + {f : filtration ℕ m} {τ π : Ω → ℕ} (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) : is_stopping_time f (τ + π) := begin intro i, - rw (_ : {x | (τ + π) x ≤ i} = ⋃ k ≤ i, {x | π x = k} ∩ {x | τ x + k ≤ i}), + rw (_ : {ω | (τ + π) ω ≤ i} = ⋃ k ≤ i, {ω | π ω = k} ∩ {ω | τ ω + k ≤ i}), { exact measurable_set.Union (λ k, measurable_set.Union_Prop (λ hk, (hπ.measurable_set_eq_le hk).inter (hτ.add_const_nat i))) }, - ext, + ext ω, simp only [pi.add_apply, set.mem_set_of_eq, set.mem_Union, set.mem_inter_eq, exists_prop], - refine ⟨λ h, ⟨π x, by linarith, rfl, h⟩, _⟩, + refine ⟨λ h, ⟨π ω, by linarith, rfl, h⟩, _⟩, rintro ⟨j, hj, rfl, h⟩, assumption end section preorder -variables [preorder ι] {f : filtration ι m} {τ π : α → ι} +variables [preorder ι] {f : filtration ι m} {τ π : Ω → ι} /-- The associated σ-algebra with a stopping time. -/ -protected def measurable_space (hτ : is_stopping_time f τ) : measurable_space α := -{ measurable_set' := λ s, ∀ i : ι, measurable_set[f i] (s ∩ {x | τ x ≤ i}), +protected def measurable_space (hτ : is_stopping_time f τ) : measurable_space Ω := +{ measurable_set' := λ s, ∀ i : ι, measurable_set[f i] (s ∩ {ω | τ ω ≤ i}), measurable_set_empty := - λ i, (set.empty_inter {x | τ x ≤ i}).symm ▸ @measurable_set.empty _ (f i), + λ i, (set.empty_inter {ω | τ ω ≤ i}).symm ▸ @measurable_set.empty _ (f i), measurable_set_compl := λ s hs i, begin - rw (_ : sᶜ ∩ {x | τ x ≤ i} = (sᶜ ∪ {x | τ x ≤ i}ᶜ) ∩ {x | τ x ≤ i}), + rw (_ : sᶜ ∩ {ω | τ ω ≤ i} = (sᶜ ∪ {ω | τ ω ≤ i}ᶜ) ∩ {ω | τ ω ≤ i}), { refine measurable_set.inter _ _, { rw ← set.compl_inter, exact (hs i).compl }, @@ -711,9 +762,9 @@ protected def measurable_space (hτ : is_stopping_time f τ) : measurable_space exact measurable_set.Union (hs i), end } -protected lemma measurable_set (hτ : is_stopping_time f τ) (s : set α) : +protected lemma measurable_set (hτ : is_stopping_time f τ) (s : set Ω) : measurable_set[hτ.measurable_space] s ↔ - ∀ i : ι, measurable_set[f i] (s ∩ {x | τ x ≤ i}) := + ∀ i : ι, measurable_set[f i] (s ∩ {ω | τ ω ≤ i}) := iff.rfl lemma measurable_space_mono @@ -721,7 +772,7 @@ lemma measurable_space_mono hτ.measurable_space ≤ hπ.measurable_space := begin intros s hs i, - rw (_ : s ∩ {x | π x ≤ i} = s ∩ {x | τ x ≤ i} ∩ {x | π x ≤ i}), + rw (_ : s ∩ {ω | π ω ≤ i} = s ∩ {ω | τ ω ≤ i} ∩ {ω | π ω ≤ i}), { exact (hs i).inter (hπ i) }, { ext, simp only [set.mem_inter_eq, iff_self_and, and.congr_left_iff, set.mem_set_of_eq], @@ -733,11 +784,11 @@ lemma measurable_space_le_of_encodable [encodable ι] (hτ : is_stopping_time f hτ.measurable_space ≤ m := begin intros s hs, - change ∀ i, measurable_set[f i] (s ∩ {x | τ x ≤ i}) at hs, - rw (_ : s = ⋃ i, s ∩ {x | τ x ≤ i}), + change ∀ i, measurable_set[f i] (s ∩ {ω | τ ω ≤ i}) at hs, + rw (_ : s = ⋃ i, s ∩ {ω | τ ω ≤ i}), { exact measurable_set.Union (λ i, f.le i _ (hs i)) }, - { ext x, split; rw set.mem_Union, - { exact λ hx, ⟨τ x, hx, le_rfl⟩ }, + { ext ω, split; rw set.mem_Union, + { exact λ hx, ⟨τ ω, hx, le_rfl⟩ }, { rintro ⟨_, hx, _⟩, exact hx } } end @@ -747,37 +798,37 @@ lemma measurable_space_le' [is_countably_generated (at_top : filter ι)] [(at_to hτ.measurable_space ≤ m := begin intros s hs, - change ∀ i, measurable_set[f i] (s ∩ {x | τ x ≤ i}) at hs, + change ∀ i, measurable_set[f i] (s ∩ {ω | τ ω ≤ i}) at hs, obtain ⟨seq : ℕ → ι, h_seq_tendsto⟩ := at_top.exists_seq_tendsto, - rw (_ : s = ⋃ n, s ∩ {x | τ x ≤ seq n}), + rw (_ : s = ⋃ n, s ∩ {ω | τ ω ≤ seq n}), { exact measurable_set.Union (λ i, f.le (seq i) _ (hs (seq i))), }, - { ext x, split; rw set.mem_Union, + { ext ω, split; rw set.mem_Union, { intros hx, - suffices : ∃ i, τ x ≤ seq i, from ⟨this.some, hx, this.some_spec⟩, + suffices : ∃ i, τ ω ≤ seq i, from ⟨this.some, hx, this.some_spec⟩, rw tendsto_at_top at h_seq_tendsto, - exact (h_seq_tendsto (τ x)).exists, }, + exact (h_seq_tendsto (τ ω)).exists, }, { rintro ⟨_, hx, _⟩, exact hx }, }, all_goals { apply_instance, }, end -lemma measurable_space_le {ι} [semilattice_sup ι] {f : filtration ι m} {τ : α → ι} +lemma measurable_space_le {ι} [semilattice_sup ι] {f : filtration ι m} {τ : Ω → ι} [is_countably_generated (at_top : filter ι)] (hτ : is_stopping_time f τ) : hτ.measurable_space ≤ m := begin casesI is_empty_or_nonempty ι, - { haveI : is_empty α := ⟨λ x, is_empty.false (τ x)⟩, + { haveI : is_empty Ω := ⟨λ ω, is_empty.false (τ ω)⟩, intros s hsτ, suffices hs : s = ∅, by { rw hs, exact measurable_set.empty, }, - haveI : unique (set α) := set.unique_empty, + haveI : unique (set Ω) := set.unique_empty, rw [unique.eq_default s, unique.eq_default ∅], }, exact measurable_space_le' hτ, end -example {f : filtration ℕ m} {τ : α → ℕ} (hτ : is_stopping_time f τ) : hτ.measurable_space ≤ m := +example {f : filtration ℕ m} {τ : Ω → ℕ} (hτ : is_stopping_time f τ) : hτ.measurable_space ≤ m := hτ.measurable_space_le -example {f : filtration ℝ m} {τ : α → ℝ} (hτ : is_stopping_time f τ) : hτ.measurable_space ≤ m := +example {f : filtration ℝ m} {τ : Ω → ℝ} (hτ : is_stopping_time f τ) : hτ.measurable_space ≤ m := hτ.measurable_space_le @[simp] lemma measurable_space_const (f : filtration ι m) (i : ι) : @@ -796,13 +847,13 @@ begin { simp only [hij, set.set_of_false, set.inter_empty, measurable_set.empty], }, }, end -lemma measurable_set_inter_eq_iff (hτ : is_stopping_time f τ) (s : set α) (i : ι) : - measurable_set[hτ.measurable_space] (s ∩ {x | τ x = i}) - ↔ measurable_set[f i] (s ∩ {x | τ x = i}) := +lemma measurable_set_inter_eq_iff (hτ : is_stopping_time f τ) (s : set Ω) (i : ι) : + measurable_set[hτ.measurable_space] (s ∩ {ω | τ ω = i}) + ↔ measurable_set[f i] (s ∩ {ω | τ ω = i}) := begin - have : ∀ j, ({x : α | τ x = i} ∩ {x : α | τ x ≤ j}) = {x : α | τ x = i} ∩ {x | i ≤ j}, + have : ∀ j, ({ω : Ω | τ ω = i} ∩ {ω : Ω | τ ω ≤ j}) = {ω : Ω | τ ω = i} ∩ {ω | i ≤ j}, { intro j, - ext1 x, + ext1 ω, simp only [set.mem_inter_eq, set.mem_set_of_eq, and.congr_right_iff], intro hxi, rw hxi, }, @@ -817,11 +868,11 @@ begin { simp [hij], }, }, end -lemma measurable_space_le_of_le_const (hτ : is_stopping_time f τ) {i : ι} (hτ_le : ∀ x, τ x ≤ i) : +lemma measurable_space_le_of_le_const (hτ : is_stopping_time f τ) {i : ι} (hτ_le : ∀ ω, τ ω ≤ i) : hτ.measurable_space ≤ f i := (measurable_space_mono hτ _ hτ_le).trans (measurable_space_const _ _).le -lemma le_measurable_space_of_const_le (hτ : is_stopping_time f τ) {i : ι} (hτ_le : ∀ x, i ≤ τ x) : +lemma le_measurable_space_of_const_le (hτ : is_stopping_time f τ) {i : ι} (hτ_le : ∀ ω, i ≤ τ ω) : f i ≤ hτ.measurable_space := (measurable_space_const _ _).symm.le.trans (measurable_space_mono _ hτ hτ_le) @@ -829,7 +880,7 @@ end preorder instance sigma_finite_stopping_time {ι} [semilattice_sup ι] [order_bot ι] [(filter.at_top : filter ι).is_countably_generated] - {μ : measure α} {f : filtration ι m} {τ : α → ι} + {μ : measure Ω} {f : filtration ι m} {τ : Ω → ι} [sigma_finite_filtration μ f] (hτ : is_stopping_time f τ) : sigma_finite (μ.trim hτ.measurable_space_le) := begin @@ -841,22 +892,22 @@ end section linear_order -variables [linear_order ι] {f : filtration ι m} {τ π : α → ι} +variables [linear_order ι] {f : filtration ι m} {τ π : Ω → ι} protected lemma measurable_set_le' (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[hτ.measurable_space] {x | τ x ≤ i} := + measurable_set[hτ.measurable_space] {ω | τ ω ≤ i} := begin intro j, - have : {x : α | τ x ≤ i} ∩ {x : α | τ x ≤ j} = {x : α | τ x ≤ min i j}, - { ext1 x, simp only [set.mem_inter_eq, set.mem_set_of_eq, le_min_iff], }, + have : {ω : Ω | τ ω ≤ i} ∩ {ω : Ω | τ ω ≤ j} = {ω : Ω | τ ω ≤ min i j}, + { ext1 ω, simp only [set.mem_inter_eq, set.mem_set_of_eq, le_min_iff], }, rw this, exact f.mono (min_le_right i j) _ (hτ _), end protected lemma measurable_set_gt' (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[hτ.measurable_space] {x | i < τ x} := + measurable_set[hτ.measurable_space] {ω | i < τ ω} := begin - have : {x : α | i < τ x} = {x : α | τ x ≤ i}ᶜ, by { ext1 x, simp, }, + have : {ω : Ω | i < τ ω} = {ω : Ω | τ ω ≤ i}ᶜ, by { ext1 ω, simp, }, rw this, exact (hτ.measurable_set_le' i).compl, end @@ -864,19 +915,19 @@ end protected lemma measurable_set_eq' [topological_space ι] [order_topology ι] [first_countable_topology ι] (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[hτ.measurable_space] {x | τ x = i} := + measurable_set[hτ.measurable_space] {ω | τ ω = i} := begin - rw [← set.univ_inter {x | τ x = i}, measurable_set_inter_eq_iff, set.univ_inter], + rw [← set.univ_inter {ω | τ ω = i}, measurable_set_inter_eq_iff, set.univ_inter], exact hτ.measurable_set_eq i, end protected lemma measurable_set_ge' [topological_space ι] [order_topology ι] [first_countable_topology ι] (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[hτ.measurable_space] {x | i ≤ τ x} := + measurable_set[hτ.measurable_space] {ω | i ≤ τ ω} := begin - have : {x | i ≤ τ x} = {x | τ x = i} ∪ {x | i < τ x}, - { ext1 x, + have : {ω | i ≤ τ ω} = {ω | τ ω = i} ∪ {ω | i < τ ω}, + { ext1 ω, simp only [le_iff_lt_or_eq, set.mem_set_of_eq, set.mem_union_eq], rw [@eq_comm _ i, or_comm], }, rw this, @@ -886,10 +937,10 @@ end protected lemma measurable_set_lt' [topological_space ι] [order_topology ι] [first_countable_topology ι] (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[hτ.measurable_space] {x | τ x < i} := + measurable_set[hτ.measurable_space] {ω | τ ω < i} := begin - have : {x | τ x < i} = {x | τ x ≤ i} \ {x | τ x = i}, - { ext1 x, + have : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i}, + { ext1 ω, simp only [lt_iff_le_and_ne, set.mem_set_of_eq, set.mem_diff], }, rw this, exact (hτ.measurable_set_le' i).diff (hτ.measurable_set_eq' i), @@ -899,22 +950,22 @@ section countable protected lemma measurable_set_eq_of_countable' (hτ : is_stopping_time f τ) (h_countable : (set.range τ).countable) (i : ι) : - measurable_set[hτ.measurable_space] {x | τ x = i} := + measurable_set[hτ.measurable_space] {ω | τ ω = i} := begin - rw [← set.univ_inter {x | τ x = i}, measurable_set_inter_eq_iff, set.univ_inter], + rw [← set.univ_inter {ω | τ ω = i}, measurable_set_inter_eq_iff, set.univ_inter], exact hτ.measurable_set_eq_of_countable h_countable i, end protected lemma measurable_set_eq_of_encodable' [encodable ι] (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[hτ.measurable_space] {a | τ a = i} := + measurable_set[hτ.measurable_space] {ω | τ ω = i} := hτ.measurable_set_eq_of_countable' (set.to_countable _) i protected lemma measurable_set_ge_of_countable' (hτ : is_stopping_time f τ) (h_countable : (set.range τ).countable) (i : ι) : - measurable_set[hτ.measurable_space] {x | i ≤ τ x} := + measurable_set[hτ.measurable_space] {ω | i ≤ τ ω} := begin - have : {x | i ≤ τ x} = {x | τ x = i} ∪ {x | i < τ x}, - { ext1 x, + have : {ω | i ≤ τ ω} = {ω | τ ω = i} ∪ {ω | i < τ ω}, + { ext1 ω, simp only [le_iff_lt_or_eq, set.mem_set_of_eq, set.mem_union_eq], rw [@eq_comm _ i, or_comm], }, rw this, @@ -922,22 +973,22 @@ begin end protected lemma measurable_set_ge_of_encodable' [encodable ι] (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[hτ.measurable_space] {a | i ≤ τ a} := + measurable_set[hτ.measurable_space] {ω | i ≤ τ ω} := hτ.measurable_set_ge_of_countable' (set.to_countable _) i protected lemma measurable_set_lt_of_countable' (hτ : is_stopping_time f τ) (h_countable : (set.range τ).countable) (i : ι) : - measurable_set[hτ.measurable_space] {x | τ x < i} := + measurable_set[hτ.measurable_space] {ω | τ ω < i} := begin - have : {x | τ x < i} = {x | τ x ≤ i} \ {x | τ x = i}, - { ext1 x, + have : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i}, + { ext1 ω, simp only [lt_iff_le_and_ne, set.mem_set_of_eq, set.mem_diff], }, rw this, exact (hτ.measurable_set_le' i).diff (hτ.measurable_set_eq_of_countable' h_countable i), end protected lemma measurable_set_lt_of_encodable' [encodable ι] (hτ : is_stopping_time f τ) (i : ι) : - measurable_set[hτ.measurable_space] {a | τ a < i} := + measurable_set[hτ.measurable_space] {ω | τ ω < i} := hτ.measurable_set_lt_of_countable' (set.to_countable _) i protected lemma measurable_space_le_of_countable (hτ : is_stopping_time f τ) @@ -945,12 +996,12 @@ protected lemma measurable_space_le_of_countable (hτ : is_stopping_time f τ) hτ.measurable_space ≤ m := begin intros s hs, - change ∀ i, measurable_set[f i] (s ∩ {x | τ x ≤ i}) at hs, - rw (_ : s = ⋃ (i ∈ set.range τ), s ∩ {x | τ x ≤ i}), + change ∀ i, measurable_set[f i] (s ∩ {ω | τ ω ≤ i}) at hs, + rw (_ : s = ⋃ (i ∈ set.range τ), s ∩ {ω | τ ω ≤ i}), { exact measurable_set.bUnion h_countable (λ i _, f.le i _ (hs i)), }, - { ext x, + { ext ω, split; rw set.mem_Union, - { exact λ hx, ⟨τ x, by simpa using hx⟩,}, + { exact λ hx, ⟨τ ω, by simpa using hx⟩,}, { rintro ⟨i, hx⟩, simp only [set.mem_range, set.Union_exists, set.mem_Union, set.mem_inter_eq, set.mem_set_of_eq, exists_prop, exists_and_distrib_right] at hx, @@ -963,11 +1014,11 @@ protected lemma measurable [topological_space ι] [measurable_space ι] [borel_space ι] [order_topology ι] [second_countable_topology ι] (hτ : is_stopping_time f τ) : measurable[hτ.measurable_space] τ := -@measurable_of_Iic ι α _ _ _ hτ.measurable_space _ _ _ _ (λ i, hτ.measurable_set_le' i) +@measurable_of_Iic ι Ω _ _ _ hτ.measurable_space _ _ _ _ (λ i, hτ.measurable_set_le' i) protected lemma measurable_of_le [topological_space ι] [measurable_space ι] [borel_space ι] [order_topology ι] [second_countable_topology ι] - (hτ : is_stopping_time f τ) {i : ι} (hτ_le : ∀ x, τ x ≤ i) : + (hτ : is_stopping_time f τ) {i : ι} (hτ_le : ∀ ω, τ ω ≤ i) : measurable[f i] τ := hτ.measurable.mono (measurable_space_le_of_le_const _ hτ_le) le_rfl @@ -981,13 +1032,13 @@ begin change measurable_set[hτ.measurable_space] s ∧ measurable_set[hπ.measurable_space] s → measurable_set[(hτ.min hπ).measurable_space] s, simp_rw is_stopping_time.measurable_set, - have : ∀ i, {x | min (τ x) (π x) ≤ i} = {x | τ x ≤ i} ∪ {x | π x ≤ i}, - { intro i, ext1 x, simp, }, + have : ∀ i, {ω | min (τ ω) (π ω) ≤ i} = {ω | τ ω ≤ i} ∪ {ω | π ω ≤ i}, + { intro i, ext1 ω, simp, }, simp_rw [this, set.inter_union_distrib_left], exact λ h i, (h.left i).union (h.right i), }, end -lemma measurable_set_min_iff (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) (s : set α) : +lemma measurable_set_min_iff (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) (s : set Ω) : measurable_set[(hτ.min hπ).measurable_space] s ↔ measurable_set[hτ.measurable_space] s ∧ measurable_set[hπ.measurable_space] s := by { rw measurable_space_min, refl, } @@ -996,7 +1047,7 @@ lemma measurable_space_min_const (hτ : is_stopping_time f τ) {i : ι} : (hτ.min_const i).measurable_space = hτ.measurable_space ⊓ f i := by rw [hτ.measurable_space_min (is_stopping_time_const _ i), measurable_space_const] -lemma measurable_set_min_const_iff (hτ : is_stopping_time f τ) (s : set α) +lemma measurable_set_min_const_iff (hτ : is_stopping_time f τ) (s : set Ω) {i : ι} : measurable_set[(hτ.min_const i).measurable_space] s ↔ measurable_set[hτ.measurable_space] s ∧ measurable_set[f i] s := @@ -1004,18 +1055,18 @@ by rw [measurable_space_min_const, measurable_space.measurable_set_inf] lemma measurable_set_inter_le [topological_space ι] [second_countable_topology ι] [order_topology ι] [measurable_space ι] [borel_space ι] - (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) (s : set α) + (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) (s : set Ω) (hs : measurable_set[hτ.measurable_space] s) : - measurable_set[(hτ.min hπ).measurable_space] (s ∩ {x | τ x ≤ π x}) := + measurable_set[(hτ.min hπ).measurable_space] (s ∩ {ω | τ ω ≤ π ω}) := begin simp_rw is_stopping_time.measurable_set at ⊢ hs, intro i, - have : (s ∩ {x | τ x ≤ π x} ∩ {x | min (τ x) (π x) ≤ i}) - = (s ∩ {x | τ x ≤ i}) ∩ {x | min (τ x) (π x) ≤ i} ∩ {x | min (τ x) i ≤ min (min (τ x) (π x)) i}, - { ext1 x, + have : (s ∩ {ω | τ ω ≤ π ω} ∩ {ω | min (τ ω) (π ω) ≤ i}) + = (s ∩ {ω | τ ω ≤ i}) ∩ {ω | min (τ ω) (π ω) ≤ i} ∩ {ω | min (τ ω) i ≤ min (min (τ ω) (π ω)) i}, + { ext1 ω, simp only [min_le_iff, set.mem_inter_eq, set.mem_set_of_eq, le_min_iff, le_refl, true_and, and_true, true_or, or_true], - by_cases hτi : τ x ≤ i, + by_cases hτi : τ ω ≤ i, { simp only [hτi, true_or, and_true, and.congr_right_iff], intro hx, split; intro h, @@ -1037,12 +1088,12 @@ end lemma measurable_set_inter_le_iff [topological_space ι] [second_countable_topology ι] [order_topology ι] [measurable_space ι] [borel_space ι] (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) - (s : set α) : - measurable_set[hτ.measurable_space] (s ∩ {x | τ x ≤ π x}) - ↔ measurable_set[(hτ.min hπ).measurable_space] (s ∩ {x | τ x ≤ π x}) := + (s : set Ω) : + measurable_set[hτ.measurable_space] (s ∩ {ω | τ ω ≤ π ω}) + ↔ measurable_set[(hτ.min hπ).measurable_space] (s ∩ {ω | τ ω ≤ π ω}) := begin split; intro h, - { have : s ∩ {x | τ x ≤ π x} = s ∩ {x | τ x ≤ π x} ∩ {x | τ x ≤ π x}, + { have : s ∩ {ω | τ ω ≤ π ω} = s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ π ω}, by rw [set.inter_assoc, set.inter_self], rw this, exact measurable_set_inter_le _ _ _ h, }, @@ -1053,17 +1104,17 @@ end lemma measurable_set_le_stopping_time [topological_space ι] [second_countable_topology ι] [order_topology ι] [measurable_space ι] [borel_space ι] (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) : - measurable_set[hτ.measurable_space] {x | τ x ≤ π x} := + measurable_set[hτ.measurable_space] {ω | τ ω ≤ π ω} := begin rw hτ.measurable_set, intro j, - have : {x | τ x ≤ π x} ∩ {x | τ x ≤ j} = {x | min (τ x) j ≤ min (π x) j} ∩ {x | τ x ≤ j}, - { ext1 x, + have : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j}, + { ext1 ω, simp only [set.mem_inter_eq, set.mem_set_of_eq, min_le_iff, le_min_iff, le_refl, and_true, and.congr_left_iff], intro h, simp only [h, or_self, and_true], - by_cases hj : j ≤ π x, + by_cases hj : j ≤ π ω, { simp only [hj, h.trans hj, or_self], }, { simp only [hj, or_false], }, }, rw this, @@ -1076,11 +1127,11 @@ end lemma measurable_set_stopping_time_le [topological_space ι] [second_countable_topology ι] [order_topology ι] [measurable_space ι] [borel_space ι] (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) : - measurable_set[hπ.measurable_space] {x | τ x ≤ π x} := + measurable_set[hπ.measurable_space] {ω | τ ω ≤ π ω} := begin - suffices : measurable_set[(hτ.min hπ).measurable_space] {x : α | τ x ≤ π x}, + suffices : measurable_set[(hτ.min hπ).measurable_space] {ω : Ω | τ ω ≤ π ω}, by { rw measurable_set_min_iff hτ hπ at this, exact this.2, }, - rw [← set.univ_inter {x : α | τ x ≤ π x}, ← hτ.measurable_set_inter_le_iff hπ, set.univ_inter], + rw [← set.univ_inter {ω : Ω | τ ω ≤ π ω}, ← hτ.measurable_set_inter_le_iff hπ, set.univ_inter], exact measurable_set_le_stopping_time hτ hπ, end @@ -1088,13 +1139,13 @@ lemma measurable_set_eq_stopping_time [add_group ι] [topological_space ι] [measurable_space ι] [borel_space ι] [order_topology ι] [measurable_singleton_class ι] [second_countable_topology ι] [has_measurable_sub₂ ι] (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) : - measurable_set[hτ.measurable_space] {x | τ x = π x} := + measurable_set[hτ.measurable_space] {ω | τ ω = π ω} := begin rw hτ.measurable_set, intro j, - have : {x | τ x = π x} ∩ {x | τ x ≤ j} - = {x | min (τ x) j = min (π x) j} ∩ {x | τ x ≤ j} ∩ {x | π x ≤ j}, - { ext1 x, + have : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} + = {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j}, + { ext1 ω, simp only [set.mem_inter_eq, set.mem_set_of_eq], refine ⟨λ h, ⟨⟨_, h.2⟩, _⟩, λ h, ⟨_, h.1.2⟩⟩, { rw h.1, }, @@ -1114,13 +1165,13 @@ lemma measurable_set_eq_stopping_time_of_encodable [encodable ι] [topological_space ι] [measurable_space ι] [borel_space ι] [order_topology ι] [measurable_singleton_class ι] [second_countable_topology ι] (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) : - measurable_set[hτ.measurable_space] {x | τ x = π x} := + measurable_set[hτ.measurable_space] {ω | τ ω = π ω} := begin rw hτ.measurable_set, intro j, - have : {x | τ x = π x} ∩ {x | τ x ≤ j} - = {x | min (τ x) j = min (π x) j} ∩ {x | τ x ≤ j} ∩ {x | π x ≤ j}, - { ext1 x, + have : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} + = {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j}, + { ext1 ω, simp only [set.mem_inter_eq, set.mem_set_of_eq], refine ⟨λ h, ⟨⟨_, h.2⟩, _⟩, λ h, ⟨_, h.1.2⟩⟩, { rw h.1, }, @@ -1144,29 +1195,29 @@ section linear_order /-! ## Stopped value and stopped process -/ -/-- Given a map `u : ι → α → E`, its stopped value with respect to the stopping -time `τ` is the map `x ↦ u (τ x) x`. -/ -def stopped_value (u : ι → α → β) (τ : α → ι) : α → β := -λ x, u (τ x) x +/-- Given a map `u : ι → Ω → E`, its stopped value with respect to the stopping +time `τ` is the map `x ↦ u (τ ω) x`. -/ +def stopped_value (u : ι → Ω → β) (τ : Ω → ι) : Ω → β := +λ ω, u (τ ω) ω -lemma stopped_value_const (u : ι → α → β) (i : ι) : stopped_value u (λ x, i) = u i := +lemma stopped_value_const (u : ι → Ω → β) (i : ι) : stopped_value u (λ ω, i) = u i := rfl variable [linear_order ι] -/-- Given a map `u : ι → α → E`, the stopped process with respect to `τ` is `u i x` if -`i ≤ τ x`, and `u (τ x) x` otherwise. +/-- Given a map `u : ι → Ω → E`, the stopped process with respect to `τ` is `u i x` if +`i ≤ τ ω`, and `u (τ ω) x` otherwise. Intuitively, the stopped process stops evolving once the stopping time has occured. -/ -def stopped_process (u : ι → α → β) (τ : α → ι) : ι → α → β := -λ i x, u (min i (τ x)) x +def stopped_process (u : ι → Ω → β) (τ : Ω → ι) : ι → Ω → β := +λ i ω, u (min i (τ ω)) ω -lemma stopped_process_eq_of_le {u : ι → α → β} {τ : α → ι} - {i : ι} {x : α} (h : i ≤ τ x) : stopped_process u τ i x = u i x := +lemma stopped_process_eq_of_le {u : ι → Ω → β} {τ : Ω → ι} + {i : ι} {ω : Ω} (h : i ≤ τ ω) : stopped_process u τ i ω = u i ω := by simp [stopped_process, min_eq_left h] -lemma stopped_process_eq_of_ge {u : ι → α → β} {τ : α → ι} - {i : ι} {x : α} (h : τ x ≤ i) : stopped_process u τ i x = u (τ x) x := +lemma stopped_process_eq_of_ge {u : ι → Ω → β} {τ : Ω → ι} + {i : ι} {ω : Ω} (h : τ ω ≤ i) : stopped_process u τ i ω = u (τ ω) ω := by simp [stopped_process, min_eq_right h] section prog_measurable @@ -1174,42 +1225,42 @@ section prog_measurable variables [measurable_space ι] [topological_space ι] [order_topology ι] [second_countable_topology ι] [borel_space ι] [topological_space β] - {u : ι → α → β} {τ : α → ι} {f : filtration ι m} + {u : ι → Ω → β} {τ : Ω → ι} {f : filtration ι m} lemma prog_measurable_min_stopping_time [metrizable_space ι] (hτ : is_stopping_time f τ) : - prog_measurable f (λ i x, min i (τ x)) := + prog_measurable f (λ i ω, min i (τ ω)) := begin intro i, - let m_prod : measurable_space (set.Iic i × α) := measurable_space.prod _ (f i), - let m_set : ∀ t : set (set.Iic i × α), measurable_space t := - λ _, @subtype.measurable_space (set.Iic i × α) _ m_prod, - let s := {p : set.Iic i × α | τ p.2 ≤ i}, - have hs : measurable_set[m_prod] s, from @measurable_snd (set.Iic i) α _ (f i) _ (hτ i), - have h_meas_fst : ∀ t : set (set.Iic i × α), - measurable[m_set t] (λ x : t, ((x : set.Iic i × α).fst : ι)), - from λ t, (@measurable_subtype_coe (set.Iic i × α) m_prod _).fst.subtype_coe, + let m_prod : measurable_space (set.Iic i × Ω) := measurable_space.prod _ (f i), + let m_set : ∀ t : set (set.Iic i × Ω), measurable_space t := + λ _, @subtype.measurable_space (set.Iic i × Ω) _ m_prod, + let s := {p : set.Iic i × Ω | τ p.2 ≤ i}, + have hs : measurable_set[m_prod] s, from @measurable_snd (set.Iic i) Ω _ (f i) _ (hτ i), + have h_meas_fst : ∀ t : set (set.Iic i × Ω), + measurable[m_set t] (λ x : t, ((x : set.Iic i × Ω).fst : ι)), + from λ t, (@measurable_subtype_coe (set.Iic i × Ω) m_prod _).fst.subtype_coe, apply measurable.strongly_measurable, refine measurable_of_restrict_of_restrict_compl hs _ _, { refine @measurable.min _ _ _ _ _ (m_set s) _ _ _ _ _ (h_meas_fst s) _, refine @measurable_of_Iic ι s _ _ _ (m_set s) _ _ _ _ (λ j, _), - have h_set_eq : (λ x : s, τ (x : set.Iic i × α).snd) ⁻¹' set.Iic j - = (λ x : s, (x : set.Iic i × α).snd) ⁻¹' {x | τ x ≤ min i j}, - { ext1 x, + have h_set_eq : (λ x : s, τ (x : set.Iic i × Ω).snd) ⁻¹' set.Iic j + = (λ x : s, (x : set.Iic i × Ω).snd) ⁻¹' {ω | τ ω ≤ min i j}, + { ext1 ω, simp only [set.mem_preimage, set.mem_Iic, iff_and_self, le_min_iff, set.mem_set_of_eq], - exact λ _, x.prop, }, + exact λ _, ω.prop, }, rw h_set_eq, - suffices h_meas : @measurable _ _ (m_set s) (f i) (λ x : s, (x : set.Iic i × α).snd), + suffices h_meas : @measurable _ _ (m_set s) (f i) (λ x : s, (x : set.Iic i × Ω).snd), from h_meas (f.mono (min_le_left _ _) _ (hτ.measurable_set_le (min i j))), exact measurable_snd.comp (@measurable_subtype_coe _ m_prod _), }, - { suffices h_min_eq_left : (λ x : sᶜ, min ↑((x : set.Iic i × α).fst) (τ (x : set.Iic i × α).snd)) - = λ x : sᶜ, ↑((x : set.Iic i × α).fst), + { suffices h_min_eq_left : (λ x : sᶜ, min ↑((x : set.Iic i × Ω).fst) (τ (x : set.Iic i × Ω).snd)) + = λ x : sᶜ, ↑((x : set.Iic i × Ω).fst), { rw [set.restrict, h_min_eq_left], exact h_meas_fst _, }, - ext1 x, + ext1 ω, rw min_eq_left, - have hx_fst_le : ↑(x : set.Iic i × α).fst ≤ i, from (x : set.Iic i × α).fst.prop, + have hx_fst_le : ↑(ω : set.Iic i × Ω).fst ≤ i, from (ω : set.Iic i × Ω).fst.prop, refine hx_fst_le.trans (le_of_lt _), - convert x.prop, + convert ω.prop, simp only [not_le, set.mem_compl_eq, set.mem_set_of_eq], }, end @@ -1229,11 +1280,11 @@ lemma prog_measurable.strongly_measurable_stopped_process [metrizable_space ι] (hu.adapted_stopped_process hτ i).mono (f.le _) lemma strongly_measurable_stopped_value_of_le - (h : prog_measurable f u) (hτ : is_stopping_time f τ) {n : ι} (hτ_le : ∀ x, τ x ≤ n) : + (h : prog_measurable f u) (hτ : is_stopping_time f τ) {n : ι} (hτ_le : ∀ ω, τ ω ≤ n) : strongly_measurable[f n] (stopped_value u τ) := begin - have : stopped_value u τ = (λ (p : set.Iic n × α), u ↑(p.fst) p.snd) ∘ (λ x, (⟨τ x, hτ_le x⟩, x)), - { ext1 x, simp only [stopped_value, function.comp_app, subtype.coe_mk], }, + have : stopped_value u τ = (λ (p : set.Iic n × Ω), u ↑(p.fst) p.snd) ∘ (λ ω, (⟨τ ω, hτ_le ω⟩, ω)), + { ext1 ω, simp only [stopped_value, function.comp_app, subtype.coe_mk], }, rw this, refine strongly_measurable.comp_measurable (h n) _, exact (hτ.measurable_of_le hτ_le).subtype_mk.prod_mk measurable_id, @@ -1243,14 +1294,14 @@ lemma measurable_stopped_value [metrizable_space β] [measurable_space β] [bore (hf_prog : prog_measurable f u) (hτ : is_stopping_time f τ) : measurable[hτ.measurable_space] (stopped_value u τ) := begin - have h_str_meas : ∀ i, strongly_measurable[f i] (stopped_value u (λ x, min (τ x) i)), + have h_str_meas : ∀ i, strongly_measurable[f i] (stopped_value u (λ ω, min (τ ω) i)), from λ i, strongly_measurable_stopped_value_of_le hf_prog (hτ.min_const i) (λ _, min_le_right _ _), intros t ht i, - suffices : stopped_value u τ ⁻¹' t ∩ {x : α | τ x ≤ i} - = stopped_value u (λ x, min (τ x) i) ⁻¹' t ∩ {x : α | τ x ≤ i}, + suffices : stopped_value u τ ⁻¹' t ∩ {ω : Ω | τ ω ≤ i} + = stopped_value u (λ ω, min (τ ω) i) ⁻¹' t ∩ {ω : Ω | τ ω ≤ i}, by { rw this, exact ((h_str_meas i).measurable ht).inter (hτ.measurable_set_le i), }, - ext1 x, + ext1 ω, simp only [stopped_value, set.mem_inter_eq, set.mem_preimage, set.mem_set_of_eq, and.congr_left_iff], intro h, @@ -1266,24 +1317,24 @@ section nat open filtration -variables {f : filtration ℕ m} {u : ℕ → α → β} {τ π : α → ℕ} +variables {f : filtration ℕ m} {u : ℕ → Ω → β} {τ π : Ω → ℕ} lemma stopped_value_sub_eq_sum [add_comm_group β] (hle : τ ≤ π) : stopped_value u π - stopped_value u τ = - λ x, (∑ i in finset.Ico (τ x) (π x), (u (i + 1) - u i)) x := + λ ω, (∑ i in finset.Ico (τ ω) (π ω), (u (i + 1) - u i)) ω := begin - ext x, - rw [finset.sum_Ico_eq_sub _ (hle x), finset.sum_range_sub, finset.sum_range_sub], + ext ω, + rw [finset.sum_Ico_eq_sub _ (hle ω), finset.sum_range_sub, finset.sum_range_sub], simp [stopped_value], end -lemma stopped_value_sub_eq_sum' [add_comm_group β] (hle : τ ≤ π) {N : ℕ} (hbdd : ∀ x, π x ≤ N) : +lemma stopped_value_sub_eq_sum' [add_comm_group β] (hle : τ ≤ π) {N : ℕ} (hbdd : ∀ ω, π ω ≤ N) : stopped_value u π - stopped_value u τ = - λ x, (∑ i in finset.range (N + 1), - set.indicator {x | τ x ≤ i ∧ i < π x} (u (i + 1) - u i)) x := + λ ω, (∑ i in finset.range (N + 1), + set.indicator {ω | τ ω ≤ i ∧ i < π ω} (u (i + 1) - u i)) ω := begin rw stopped_value_sub_eq_sum hle, - ext x, + ext ω, simp only [finset.sum_apply, finset.sum_indicator_eq_sum_filter], refine finset.sum_congr _ (λ _ _, rfl), ext i, @@ -1302,8 +1353,8 @@ lemma adapted.prog_measurable_of_nat [topological_space β] [has_continuous_add (h : adapted f u) : prog_measurable f u := begin intro i, - have : (λ p : ↥(set.Iic i) × α, u ↑(p.fst) p.snd) - = λ p : ↥(set.Iic i) × α, ∑ j in finset.range (i + 1), if ↑p.fst = j then u j p.snd else 0, + have : (λ p : ↥(set.Iic i) × Ω, u ↑(p.fst) p.snd) + = λ p : ↥(set.Iic i) × Ω, ∑ j in finset.range (i + 1), if ↑p.fst = j then u j p.snd else 0, { ext1 p, rw finset.sum_ite_eq, have hp_mem : (p.fst : ℕ) ∈ finset.range (i + 1) := finset.mem_range_succ_iff.mpr p.fst.prop, @@ -1311,7 +1362,7 @@ begin rw this, refine finset.strongly_measurable_sum _ (λ j hj, strongly_measurable.ite _ _ _), { suffices h_meas : measurable[measurable_space.prod _ (f i)] - (λ a : ↥(set.Iic i) × α, (a.fst : ℕ)), + (λ a : ↥(set.Iic i) × Ω, (a.fst : ℕ)), from h_meas (measurable_set_singleton j), exact measurable_fst.subtype_coe, }, { have h_le : j ≤ i, from finset.mem_range_succ_iff.mp hj, @@ -1332,9 +1383,9 @@ lemma adapted.strongly_measurable_stopped_process_of_nat [topological_space β] strongly_measurable (stopped_process u τ n) := hu.prog_measurable_of_nat.strongly_measurable_stopped_process hτ n -lemma stopped_value_eq {N : ℕ} (hbdd : ∀ x, τ x ≤ N) : +lemma stopped_value_eq {N : ℕ} (hbdd : ∀ ω, τ ω ≤ N) : stopped_value u τ = - λ x, (∑ i in finset.range (N + 1), set.indicator {x | τ x = i} (u i)) x := + λ x, (∑ i in finset.range (N + 1), set.indicator {ω | τ ω = i} (u i)) x := begin ext y, rw [stopped_value, finset.sum_apply, finset.sum_eq_single (τ y)], @@ -1349,17 +1400,17 @@ end lemma stopped_process_eq (n : ℕ) : stopped_process u τ n = set.indicator {a | n ≤ τ a} (u n) + - ∑ i in finset.range n, set.indicator {a | τ a = i} (u i) := + ∑ i in finset.range n, set.indicator {ω | τ ω = i} (u i) := begin - ext x, + ext ω, rw [pi.add_apply, finset.sum_apply], - cases le_or_lt n (τ x), + cases le_or_lt n (τ ω), { rw [stopped_process_eq_of_le h, set.indicator_of_mem, finset.sum_eq_zero, add_zero], { intros m hm, rw finset.mem_range at hm, exact set.indicator_of_not_mem ((lt_of_lt_of_le hm h).ne.symm) _ }, { exact h } }, - { rw [stopped_process_eq_of_ge (le_of_lt h), finset.sum_eq_single_of_mem (τ x)], + { rw [stopped_process_eq_of_ge (le_of_lt h), finset.sum_eq_single_of_mem (τ ω)], { rw [set.indicator_of_not_mem, zero_add, set.indicator_of_mem], { exact rfl }, -- refl does not work { exact not_le.2 h } }, @@ -1373,18 +1424,18 @@ end add_comm_monoid section normed_add_comm_group -variables [normed_add_comm_group β] {p : ℝ≥0∞} {μ : measure α} +variables [normed_add_comm_group β] {p : ℝ≥0∞} {μ : measure Ω} lemma mem_ℒp_stopped_process (hτ : is_stopping_time f τ) (hu : ∀ n, mem_ℒp (u n) p μ) (n : ℕ) : mem_ℒp (stopped_process u τ n) p μ := begin rw stopped_process_eq, refine mem_ℒp.add _ _, - { exact mem_ℒp.indicator (f.le n {a : α | n ≤ τ a} (hτ.measurable_set_ge n)) (hu n) }, - { suffices : mem_ℒp (λ x, ∑ (i : ℕ) in finset.range n, {a : α | τ a = i}.indicator (u i) x) p μ, - { convert this, ext1 x, simp only [finset.sum_apply] }, + { exact mem_ℒp.indicator (f.le n {a : Ω | n ≤ τ a} (hτ.measurable_set_ge n)) (hu n) }, + { suffices : mem_ℒp (λ ω, ∑ (i : ℕ) in finset.range n, {a : Ω | τ a = i}.indicator (u i) ω) p μ, + { convert this, ext1 ω, simp only [finset.sum_apply] }, refine mem_ℒp_finset_sum _ (λ i hi, mem_ℒp.indicator _ (hu i)), - exact f.le i {a : α | τ a = i} (hτ.measurable_set_eq i) }, + exact f.le i {a : Ω | τ a = i} (hτ.measurable_set_eq i) }, end lemma integrable_stopped_process (hτ : is_stopping_time f τ) @@ -1393,19 +1444,19 @@ lemma integrable_stopped_process (hτ : is_stopping_time f τ) by { simp_rw ← mem_ℒp_one_iff_integrable at hu ⊢, exact mem_ℒp_stopped_process hτ hu n, } lemma mem_ℒp_stopped_value (hτ : is_stopping_time f τ) - (hu : ∀ n, mem_ℒp (u n) p μ) {N : ℕ} (hbdd : ∀ x, τ x ≤ N) : + (hu : ∀ n, mem_ℒp (u n) p μ) {N : ℕ} (hbdd : ∀ ω, τ ω ≤ N) : mem_ℒp (stopped_value u τ) p μ := begin rw stopped_value_eq hbdd, suffices : mem_ℒp (λ x, ∑ (i : ℕ) in finset.range (N + 1), - {a : α | τ a = i}.indicator (u i) x) p μ, - { convert this, ext1 x, simp only [finset.sum_apply] }, + {a : Ω | τ a = i}.indicator (u i) x) p μ, + { convert this, ext1 ω, simp only [finset.sum_apply] }, refine mem_ℒp_finset_sum _ (λ i hi, mem_ℒp.indicator _ (hu i)), - exact f.le i {a : α | τ a = i} (hτ.measurable_set_eq i) + exact f.le i {a : Ω | τ a = i} (hτ.measurable_set_eq i) end lemma integrable_stopped_value (hτ : is_stopping_time f τ) - (hu : ∀ n, integrable (u n) μ) {N : ℕ} (hbdd : ∀ x, τ x ≤ N) : + (hu : ∀ n, integrable (u n) μ) {N : ℕ} (hbdd : ∀ ω, τ ω ≤ N) : integrable (stopped_value u τ) μ := by { simp_rw ← mem_ℒp_one_iff_integrable at hu ⊢, exact mem_ℒp_stopped_value hτ hu hbdd, } @@ -1415,28 +1466,28 @@ end nat section piecewise_const -variables [preorder ι] {𝒢 : filtration ι m} {τ η : α → ι} {i j : ι} {s : set α} +variables [preorder ι] {𝒢 : filtration ι m} {τ η : Ω → ι} {i j : ι} {s : set Ω} [decidable_pred (∈ s)] /-- Given stopping times `τ` and `η` which are bounded below, `set.piecewise s τ η` is also a stopping time with respect to the same filtration. -/ lemma is_stopping_time.piecewise_of_le (hτ_st : is_stopping_time 𝒢 τ) - (hη_st : is_stopping_time 𝒢 η) (hτ : ∀ x, i ≤ τ x) (hη : ∀ x, i ≤ η x) + (hη_st : is_stopping_time 𝒢 η) (hτ : ∀ ω, i ≤ τ ω) (hη : ∀ x, i ≤ η x) (hs : measurable_set[𝒢 i] s) : is_stopping_time 𝒢 (s.piecewise τ η) := begin intro n, have : {x | s.piecewise τ η x ≤ n} - = (s ∩ {x | τ x ≤ n}) ∪ (sᶜ ∩ {x | η x ≤ n}), - { ext1 x, + = (s ∩ {ω | τ ω ≤ n}) ∪ (sᶜ ∩ {x | η x ≤ n}), + { ext1 ω, simp only [set.piecewise, set.mem_inter_eq, set.mem_set_of_eq, and.congr_right_iff], - by_cases hx : x ∈ s; simp [hx], }, + by_cases hx : ω ∈ s; simp [hx], }, rw this, by_cases hin : i ≤ n, { have hs_n : measurable_set[𝒢 n] s, from 𝒢.mono hin _ hs, exact (hs_n.inter (hτ_st n)).union (hs_n.compl.inter (hη_st n)), }, - { have hτn : ∀ x, ¬ τ x ≤ n := λ x hτn, hin ((hτ x).trans hτn), - have hηn : ∀ x, ¬ η x ≤ n := λ x hηn, hin ((hη x).trans hηn), + { have hτn : ∀ ω, ¬ τ ω ≤ n := λ ω hτn, hin ((hτ ω).trans hτn), + have hηn : ∀ ω, ¬ η ω ≤ n := λ ω hηn, hin ((hη ω).trans hηn), simp [hτn, hηn], }, end @@ -1445,13 +1496,13 @@ lemma is_stopping_time_piecewise_const (hij : i ≤ j) (hs : measurable_set[𝒢 (is_stopping_time_const 𝒢 i).piecewise_of_le (is_stopping_time_const 𝒢 j) (λ x, le_rfl) (λ _, hij) hs -lemma stopped_value_piecewise_const {ι' : Type*} {i j : ι'} {f : ι' → α → ℝ} : +lemma stopped_value_piecewise_const {ι' : Type*} {i j : ι'} {f : ι' → Ω → ℝ} : stopped_value f (s.piecewise (λ _, i) (λ _, j)) = s.piecewise (f i) (f j) := -by { ext x, rw stopped_value, by_cases hx : x ∈ s; simp [hx] } +by { ext ω, rw stopped_value, by_cases hx : ω ∈ s; simp [hx] } -lemma stopped_value_piecewise_const' {ι' : Type*} {i j : ι'} {f : ι' → α → ℝ} : +lemma stopped_value_piecewise_const' {ι' : Type*} {i j : ι'} {f : ι' → Ω → ℝ} : stopped_value f (s.piecewise (λ _, i) (λ _, j)) = s.indicator (f i) + sᶜ.indicator (f j) := -by { ext x, rw stopped_value, by_cases hx : x ∈ s; simp [hx] } +by { ext ω, rw stopped_value, by_cases hx : ω ∈ s; simp [hx] } end piecewise_const diff --git a/formal/lean/mathlib/probability/variance.lean b/formal/lean/mathlib/probability/variance.lean index 4b43f159ed94907d7fa3a41736215af0921de565..55c010f4ed306153bcbac7d260e375e4c0eb7691 100644 --- a/formal/lean/mathlib/probability/variance.lean +++ b/formal/lean/mathlib/probability/variance.lean @@ -33,7 +33,7 @@ namespace probability_theory /-- The variance of a random variable is `𝔼[X^2] - 𝔼[X]^2` or, equivalently, `𝔼[(X - 𝔼[X])^2]`. We use the latter as the definition, to ensure better behavior even in garbage situations. -/ def variance {Ω : Type*} {m : measurable_space Ω} (f : Ω → ℝ) (μ : measure Ω) : ℝ := -μ[(f - (λ x, μ[f])) ^ 2] +μ[(f - (λ ω, μ[f])) ^ 2] @[simp] lemma variance_zero {Ω : Type*} {m : measurable_space Ω} (μ : measure Ω) : variance 0 μ = 0 := @@ -41,12 +41,12 @@ by simp [variance] lemma variance_nonneg {Ω : Type*} {m : measurable_space Ω} (f : Ω → ℝ) (μ : measure Ω) : 0 ≤ variance f μ := -integral_nonneg (λ x, sq_nonneg _) +integral_nonneg (λ ω, sq_nonneg _) lemma variance_mul {Ω : Type*} {m : measurable_space Ω} (c : ℝ) (f : Ω → ℝ) (μ : measure Ω) : - variance (λ x, c * f x) μ = c^2 * variance f μ := + variance (λ ω, c * f ω) μ = c^2 * variance f μ := calc -variance (λ x, c * f x) μ +variance (λ ω, c * f ω) μ = ∫ x, (c * f x - ∫ y, c * f y ∂μ) ^ 2 ∂μ : rfl ... = ∫ x, (c * (f x - ∫ y, f y ∂μ)) ^ 2 ∂μ : by { congr' 1 with x, simp_rw [integral_mul_left, mul_sub] } @@ -82,7 +82,7 @@ begin { apply hX.integrable_sq.add, convert integrable_const (𝔼[X] ^ 2), apply_instance }, - { exact ((hX.integrable ennreal.one_le_two).const_mul 2).mul_const' _ }, + { exact ((hX.integrable one_le_two).const_mul 2).mul_const' _ }, simp only [integral_mul_right, pi.pow_apply, pi.mul_apply, pi.bit0_apply, pi.one_apply, integral_const (integral ℙ X ^ 2), integral_mul_left (2 : ℝ), one_mul, variance, pi.pow_apply, measure_univ, ennreal.one_to_real, algebra.id.smul_eq_mul], @@ -100,9 +100,9 @@ begin { rw [variance, integral_undef], { exact integral_nonneg (λ a, sq_nonneg _) }, { assume h, - have A : mem_ℒp (X - λ (x : Ω), 𝔼[X]) 2 ℙ := (mem_ℒp_two_iff_integrable_sq + have A : mem_ℒp (X - λ (ω : Ω), 𝔼[X]) 2 ℙ := (mem_ℒp_two_iff_integrable_sq (h_int.ae_strongly_measurable.sub ae_strongly_measurable_const)).2 h, - have B : mem_ℒp (λ (x : Ω), 𝔼[X]) 2 ℙ := mem_ℒp_const _, + have B : mem_ℒp (λ (ω : Ω), 𝔼[X]) 2 ℙ := mem_ℒp_const _, apply hX, convert A.add B, simp } } @@ -149,19 +149,19 @@ Var[X + Y] = 𝔼[λ a, (X a)^2 + (Y a)^2 + 2 * X a * Y a] - 𝔼[X+Y]^2 : begin simp only [pi.add_apply, pi.pow_apply, pi.mul_apply, mul_assoc], rw [integral_add, integral_add, integral_add, integral_mul_left], - { exact hX.integrable ennreal.one_le_two }, - { exact hY.integrable ennreal.one_le_two }, + { exact hX.integrable one_le_two }, + { exact hY.integrable one_le_two }, { exact hX.integrable_sq }, { exact hY.integrable_sq }, { exact hX.integrable_sq.add hY.integrable_sq }, { apply integrable.const_mul, - exact h.integrable_mul (hX.integrable ennreal.one_le_two) (hY.integrable ennreal.one_le_two) } + exact h.integrable_mul (hX.integrable one_le_two) (hY.integrable one_le_two) } end ... = (𝔼[X^2] + 𝔼[Y^2] + 2 * (𝔼[X] * 𝔼[Y])) - (𝔼[X] + 𝔼[Y])^2 : begin congr, exact h.integral_mul_of_integrable - (hX.integrable ennreal.one_le_two) (hY.integrable ennreal.one_le_two), + (hX.integrable one_le_two) (hY.integrable one_le_two), end ... = Var[X] + Var[Y] : by { simp only [variance_def', hX, hY, pi.pow_apply], ring } @@ -182,9 +182,9 @@ begin - (𝔼[X k] + 𝔼[∑ i in s, X i]) ^ 2 : begin rw [integral_add', integral_add', integral_add'], - { exact mem_ℒp.integrable ennreal.one_le_two (hs _ (mem_insert_self _ _)) }, + { exact mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _)) }, { apply integrable_finset_sum' _ (λ i hi, _), - exact mem_ℒp.integrable ennreal.one_le_two (hs _ (mem_insert_of_mem hi)) }, + exact mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)) }, { exact mem_ℒp.integrable_sq (hs _ (mem_insert_self _ _)) }, { apply mem_ℒp.integrable_sq, exact mem_ℒp_finset_sum' _ (λ i hi, (hs _ (mem_insert_of_mem hi))) }, @@ -197,8 +197,8 @@ begin simp only [mul_sum, sum_apply, pi.mul_apply], apply integrable_finset_sum _ (λ i hi, _), apply indep_fun.integrable_mul _ - (mem_ℒp.integrable ennreal.one_le_two (hs _ (mem_insert_self _ _))) - (mem_ℒp.integrable ennreal.one_le_two (hs _ (mem_insert_of_mem hi))), + (mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _))) + (mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi))), apply h (mem_insert_self _ _) (mem_insert_of_mem hi), exact (λ hki, ks (hki.symm ▸ hi)) } end @@ -216,19 +216,19 @@ begin rw integral_finset_sum s (λ i hi, _), swap, { apply integrable.const_mul _ 2, apply indep_fun.integrable_mul _ - (mem_ℒp.integrable ennreal.one_le_two (hs _ (mem_insert_self _ _))) - (mem_ℒp.integrable ennreal.one_le_two (hs _ (mem_insert_of_mem hi))), + (mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _))) + (mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi))), apply h (mem_insert_self _ _) (mem_insert_of_mem hi), exact (λ hki, ks (hki.symm ▸ hi)) }, rw [integral_finset_sum s - (λ i hi, (mem_ℒp.integrable ennreal.one_le_two (hs _ (mem_insert_of_mem hi)))), + (λ i hi, (mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)))), mul_sum, mul_sum, ← sum_sub_distrib], apply finset.sum_eq_zero (λ i hi, _), rw [integral_mul_left, indep_fun.integral_mul_of_integrable', sub_self], { apply h (mem_insert_self _ _) (mem_insert_of_mem hi), exact (λ hki, ks (hki.symm ▸ hi)) }, - { exact mem_ℒp.integrable ennreal.one_le_two (hs _ (mem_insert_self _ _)) }, - { exact mem_ℒp.integrable ennreal.one_le_two (hs _ (mem_insert_of_mem hi)) } + { exact mem_ℒp.integrable one_le_two (hs _ (mem_insert_self _ _)) }, + { exact mem_ℒp.integrable one_le_two (hs _ (mem_insert_of_mem hi)) } end ... = Var[X k] + ∑ i in s, Var[X i] : by rw IH (λ i hi, hs i (mem_insert_of_mem hi)) diff --git a/formal/lean/mathlib/representation_theory/basic.lean b/formal/lean/mathlib/representation_theory/basic.lean index 80d5f2988061bb204b073ac2dad59de99f63e92b..cc21f6da13798611dcb40af74c32be9f0f967883 100644 --- a/formal/lean/mathlib/representation_theory/basic.lean +++ b/formal/lean/mathlib/representation_theory/basic.lean @@ -93,7 +93,7 @@ which we equip with an instance `module (monoid_algebra k G) ρ.as_module`. You should use `as_module_equiv : ρ.as_module ≃+ V` to translate terms. -/ -@[nolint unused_arguments, derive add_comm_monoid] +@[nolint unused_arguments, derive [add_comm_monoid, module (module.End k V)]] def as_module (ρ : representation k G V) := V instance : inhabited ρ.as_module := ⟨0⟩ @@ -103,10 +103,7 @@ A `k`-linear representation of `G` on `V` can be thought of as a module over `monoid_algebra k G`. -/ noncomputable instance as_module_module : module (monoid_algebra k G) ρ.as_module := -begin - change module (monoid_algebra k G) V, - exact module.comp_hom V (as_algebra_hom ρ).to_ring_hom, -end +module.comp_hom V (as_algebra_hom ρ).to_ring_hom /-- The additive equivalence from the `module (monoid_algebra k G)` to the original vector space @@ -264,6 +261,20 @@ begin simp only [of_mul_action_def, finsupp.lmap_domain_apply, finsupp.map_domain_apply, hg], end +lemma of_mul_action_self_smul_eq_mul + (x : monoid_algebra k G) (y : (of_mul_action k G G).as_module) : + x • y = (x * y : monoid_algebra k G) := +x.induction_on (λ g, by show as_algebra_hom _ _ _ = _; ext; simp) + (λ x y hx hy, by simp only [hx, hy, add_mul, add_smul]) + (λ r x hx, by show as_algebra_hom _ _ _ = _; simpa [←hx]) + +/-- If we equip `k[G]` with the `k`-linear `G`-representation induced by the left regular action of +`G` on itself, the resulting object is isomorphic as a `k[G]`-module to `k[G]` with its natural +`k[G]`-module structure. -/ +@[simps] noncomputable def of_mul_action_self_as_module_equiv : + (of_mul_action k G G).as_module ≃ₗ[monoid_algebra k G] monoid_algebra k G := +{ map_smul' := of_mul_action_self_smul_eq_mul, ..as_module_equiv _ } + /-- When `G` is a group, a `k`-linear representation of `G` on `V` can be thought of as a group homomorphism from `G` into the invertible `k`-linear endomorphisms of `V`. @@ -298,6 +309,27 @@ local notation ρV ` ⊗ ` ρW := tprod ρV ρW @[simp] lemma tprod_apply (g : G) : (ρV ⊗ ρW) g = tensor_product.map (ρV g) (ρW g) := rfl +lemma smul_tprod_one_as_module (r : monoid_algebra k G) (x : V) (y : W) : + (r • (x ⊗ₜ y) : (ρV.tprod 1).as_module) = (r • x : ρV.as_module) ⊗ₜ y := +begin + show as_algebra_hom _ _ _ = as_algebra_hom _ _ _ ⊗ₜ _, + simp only [as_algebra_hom_def, monoid_algebra.lift_apply, + tprod_apply, monoid_hom.one_apply, linear_map.finsupp_sum_apply, + linear_map.smul_apply, tensor_product.map_tmul, linear_map.one_apply], + simp only [finsupp.sum, tensor_product.sum_tmul], + refl, +end + +lemma smul_one_tprod_as_module (r : monoid_algebra k G) (x : V) (y : W) : + (r • (x ⊗ₜ y) : ((1 : representation k G V).tprod ρW).as_module) = x ⊗ₜ (r • y : ρW.as_module) := +begin + show as_algebra_hom _ _ _ = _ ⊗ₜ as_algebra_hom _ _ _, + simp only [as_algebra_hom_def, monoid_algebra.lift_apply, + tprod_apply, monoid_hom.one_apply, linear_map.finsupp_sum_apply, + linear_map.smul_apply, tensor_product.map_tmul, linear_map.one_apply], + simp only [finsupp.sum, tensor_product.tmul_sum, tensor_product.tmul_smul], +end + end tensor_product section linear_hom diff --git a/formal/lean/mathlib/representation_theory/fdRep.lean b/formal/lean/mathlib/representation_theory/fdRep.lean index bdc8703f7707cb2e3dbf9d2252dcac16c8a11031..054983741e6a12891774139376c4b57d31c7dba1 100644 --- a/formal/lean/mathlib/representation_theory/fdRep.lean +++ b/formal/lean/mathlib/representation_theory/fdRep.lean @@ -43,13 +43,13 @@ variables {k G : Type u} [field k] [monoid G] instance : has_coe_to_sort (fdRep k G) (Type u) := concrete_category.has_coe_to_sort _ instance (V : fdRep k G) : add_comm_group V := -by { change add_comm_group ((forget₂ (fdRep k G) (FinVect k)).obj V), apply_instance, } +by { change add_comm_group ((forget₂ (fdRep k G) (FinVect k)).obj V).obj, apply_instance, } instance (V : fdRep k G) : module k V := -by { change module k ((forget₂ (fdRep k G) (FinVect k)).obj V), apply_instance, } +by { change module k ((forget₂ (fdRep k G) (FinVect k)).obj V).obj, apply_instance, } instance (V : fdRep k G) : finite_dimensional k V := -by { change finite_dimensional k ((forget₂ (fdRep k G) (FinVect k)).obj V), apply_instance, } +by { change finite_dimensional k ((forget₂ (fdRep k G) (FinVect k)).obj V).obj, apply_instance, } /-- The monoid homomorphism corresponding to the action of `G` onto `V : fdRep k G`. -/ def ρ (V : fdRep k G) : G →* (V →ₗ[k] V) := V.ρ diff --git a/formal/lean/mathlib/representation_theory/group_cohomology_resolution.lean b/formal/lean/mathlib/representation_theory/group_cohomology_resolution.lean index 3536193182c0a2540dd863817399b4ccd875ac59..0c953185cae1c4e5df693bf8c8acd9a3efa34bec 100644 --- a/formal/lean/mathlib/representation_theory/group_cohomology_resolution.lean +++ b/formal/lean/mathlib/representation_theory/group_cohomology_resolution.lean @@ -13,20 +13,22 @@ This file contains facts about an important `k[G]`-module structure on `k[Gⁿ]` commutative ring and `G` is a group. The module structure arises from the representation `G →* End(k[Gⁿ])` induced by the diagonal action of `G` on `Gⁿ.` -In particular, we define morphisms of `k`-linear `G`-representations between `k[Gⁿ⁺¹]` and +In particular, we define an isomorphism of `k`-linear `G`-representations between `k[Gⁿ⁺¹]` and `k[G] ⊗ₖ k[Gⁿ]` (on which `G` acts by `ρ(g₁)(g₂ ⊗ x) = (g₁ * g₂) ⊗ x`). +This allows us to define a `k[G]`-basis on `k[Gⁿ⁺¹]`, by mapping the natural `k[G]`-basis of +`k[G] ⊗ₖ k[Gⁿ]` along the isomorphism. + ## Main definitions * `group_cohomology.resolution.to_tensor` * `group_cohomology.resolution.of_tensor` * `Rep.of_mul_action` + * `group_cohomology.resolution.equiv_tensor` + * `group_cohomology.resolution.of_mul_action_basis` ## TODO - * Show that `group_cohomology.resolution.to_tensor` and `group_cohomology.resolution.of_tensor` are - mutually inverse. - * Use the above to deduce that `k[Gⁿ⁺¹]` is free over `k[G]`. * Use the freeness of `k[Gⁿ⁺¹]` to build a projective resolution of the (trivial) `k[G]`-module `k`, and so develop group cohomology. @@ -79,8 +81,7 @@ variables {k G n} lemma to_tensor_aux_single (f : Gⁿ⁺¹) (m : k) : to_tensor_aux k G n (single f m) = single (f 0) m ⊗ₜ single (λ i, (f i)⁻¹ * f i.succ) 1 := begin - erw [lift_apply, sum_single_index, tensor_product.smul_tmul'], - { simp }, + simp only [to_tensor_aux, lift_apply, sum_single_index, tensor_product.smul_tmul'], { simp }, end @@ -99,10 +100,36 @@ lemma of_tensor_aux_comm_of_mul_action (g h : G) (x : Gⁿ) : (1 : module.End k (Gⁿ →₀ k)) (single h (1 : k) ⊗ₜ single x (1 : k))) = of_mul_action k G Gⁿ⁺¹ g (of_tensor_aux k G n (single h 1 ⊗ₜ single x 1)) := begin - dsimp, simp [of_mul_action_def, of_tensor_aux_single, mul_smul], end +lemma to_tensor_aux_left_inv (x : Gⁿ⁺¹ →₀ k) : + of_tensor_aux _ _ _ (to_tensor_aux _ _ _ x) = x := +begin + refine linear_map.ext_iff.1 (@finsupp.lhom_ext _ _ _ k _ _ _ _ _ + (linear_map.comp (of_tensor_aux _ _ _) (to_tensor_aux _ _ _)) linear_map.id (λ x y, _)) x, + dsimp, + rw [to_tensor_aux_single x y, of_tensor_aux_single, finsupp.lift_apply, finsupp.sum_single_index, + one_smul, fin.partial_prod_left_inv], + { rw zero_smul } +end + +lemma to_tensor_aux_right_inv (x : (G →₀ k) ⊗[k] (Gⁿ →₀ k)) : + to_tensor_aux _ _ _ (of_tensor_aux _ _ _ x) = x := +begin + refine tensor_product.induction_on x (by simp) (λ y z, _) (λ z w hz hw, by simp [hz, hw]), + rw [←finsupp.sum_single y, finsupp.sum, tensor_product.sum_tmul], + simp only [finset.smul_sum, linear_map.map_sum], + refine finset.sum_congr rfl (λ f hf, _), + simp only [of_tensor_aux_single, finsupp.lift_apply, finsupp.smul_single', + linear_map.map_finsupp_sum, to_tensor_aux_single, fin.partial_prod_right_inv], + dsimp, + simp only [fin.partial_prod_zero, mul_one], + conv_rhs {rw [←finsupp.sum_single z, finsupp.sum, tensor_product.tmul_sum]}, + exact finset.sum_congr rfl (λ g hg, show _ ⊗ₜ _ = _, by + rw [←finsupp.smul_single', tensor_product.smul_tmul, finsupp.smul_single_one]) +end + variables (k G n) /-- Given a `G`-action on `H`, this is `k[H]` bundled with the natural representation @@ -144,4 +171,55 @@ lemma of_tensor_single' (g : G →₀ k) (x : Gⁿ) (m : k) : finsupp.lift _ k G (λ a, single (a • partial_prod x) m) g := by simp [of_tensor, of_tensor_aux] +variables (k G n) + +/-- An isomorphism of `k`-linear representations of `G` from `k[Gⁿ⁺¹]` to `k[G] ⊗ₖ k[Gⁿ]` (on +which `G` acts by `ρ(g₁)(g₂ ⊗ x) = (g₁ * g₂) ⊗ x`) sending `(g₀, ..., gₙ)` to +`g₀ ⊗ (g₀⁻¹g₁, g₁⁻¹g₂, ..., gₙ₋₁⁻¹gₙ)`. -/ +def equiv_tensor : (Rep.of_mul_action k G (fin (n + 1) → G)) ≅ Rep.of + ((representation.of_mul_action k G G).tprod (1 : representation k G ((fin n → G) →₀ k))) := +Action.mk_iso (linear_equiv.to_Module_iso +{ inv_fun := of_tensor_aux k G n, + left_inv := to_tensor_aux_left_inv, + right_inv := λ x, by convert to_tensor_aux_right_inv x, + ..to_tensor_aux k G n }) (to_tensor k G n).comm + +-- not quite sure which simp lemmas to make here +@[simp] lemma equiv_tensor_def : + (equiv_tensor k G n).hom = to_tensor k G n := rfl + +@[simp] lemma equiv_tensor_inv_def : + (equiv_tensor k G n).inv = of_tensor k G n := rfl + +/-- The `k[G]`-linear isomorphism `k[G] ⊗ₖ k[Gⁿ] ≃ k[Gⁿ⁺¹]`, where the `k[G]`-module structure on +the lefthand side is `tensor_product.left_module`, whilst that of the righthand side comes from +`representation.as_module`. Allows us to use `basis.algebra_tensor_product` to get a `k[G]`-basis +of the righthand side. -/ +def of_mul_action_basis_aux : (monoid_algebra k G ⊗[k] ((fin n → G) →₀ k)) ≃ₗ[monoid_algebra k G] + (of_mul_action k G (fin (n + 1) → G)).as_module := +{ map_smul' := λ r x, + begin + rw [ring_hom.id_apply, linear_equiv.to_fun_eq_coe, ←linear_equiv.map_smul], + congr' 1, + refine x.induction_on _ (λ x y, _) (λ y z hy hz, _), + { simp only [smul_zero] }, + { simp only [tensor_product.smul_tmul'], + show (r * x) ⊗ₜ y = _, + rw [←of_mul_action_self_smul_eq_mul, smul_tprod_one_as_module] }, + { rw [smul_add, hz, hy, smul_add], } + end, .. ((Rep.equivalence_Module_monoid_algebra.1).map_iso + (equiv_tensor k G n).symm).to_linear_equiv } + +/-- A `k[G]`-basis of `k[Gⁿ⁺¹]`, coming from the `k[G]`-linear isomorphism +`k[G] ⊗ₖ k[Gⁿ] ≃ k[Gⁿ⁺¹].` -/ +def of_mul_action_basis : + basis (fin n → G) (monoid_algebra k G) (of_mul_action k G (fin (n + 1) → G)).as_module := +@basis.map _ (monoid_algebra k G) (monoid_algebra k G ⊗[k] ((fin n → G) →₀ k)) + _ _ _ _ _ _ (@algebra.tensor_product.basis k _ (monoid_algebra k G) _ _ ((fin n → G) →₀ k) _ _ + (fin n → G) (⟨linear_equiv.refl k _⟩)) (of_mul_action_basis_aux k G n) + +lemma of_mul_action_free : + module.free (monoid_algebra k G) (of_mul_action k G (fin (n + 1) → G)).as_module := +module.free.of_basis (of_mul_action_basis k G n) + end group_cohomology.resolution diff --git a/formal/lean/mathlib/ring_theory/adjoin/fg.lean b/formal/lean/mathlib/ring_theory/adjoin/fg.lean index 29c97f8ff97dd5efd3e97abf18b16d3b59fe7985..a997fc1c51b345a97a3508e0daa767762af31a61 100644 --- a/formal/lean/mathlib/ring_theory/adjoin/fg.lean +++ b/formal/lean/mathlib/ring_theory/adjoin/fg.lean @@ -126,7 +126,7 @@ end lemma fg_of_fg_map (S : subalgebra R A) (f : A →ₐ[R] B) (hf : function.injective f) (hs : (S.map f).fg) : S.fg := -let ⟨s, hs⟩ := hs in ⟨s.preimage f $ λ _ _ _ _ h, hf h, map_injective f hf $ +let ⟨s, hs⟩ := hs in ⟨s.preimage f $ λ _ _ _ _ h, hf h, map_injective hf $ by { rw [← algebra.adjoin_image, finset.coe_preimage, set.image_preimage_eq_of_subset, hs], rw [← alg_hom.coe_range, ← algebra.adjoin_le_iff, hs, ← algebra.map_top], exact map_mono le_top }⟩ diff --git a/formal/lean/mathlib/ring_theory/adjoin_root.lean b/formal/lean/mathlib/ring_theory/adjoin_root.lean index ef8f030aa82c591da4d4a75d3509025fefb238c1..4732be2449d0f72727560deb8c83e38cc99d6c8f 100644 --- a/formal/lean/mathlib/ring_theory/adjoin_root.lean +++ b/formal/lean/mathlib/ring_theory/adjoin_root.lean @@ -107,7 +107,7 @@ def root : adjoin_root f := mk f X variables {f} -instance adjoin_root.has_coe_t : has_coe_t R (adjoin_root f) := ⟨of f⟩ +instance has_coe_t : has_coe_t R (adjoin_root f) := ⟨of f⟩ @[simp] lemma mk_eq_mk {g h : R[X]} : mk f g = mk f h ↔ f ∣ g - h := ideal.quotient.eq.trans ideal.mem_span_singleton @@ -227,6 +227,13 @@ mul_div_eq_iff_is_root.2 $ is_root_root _ end irreducible +section is_noetherian_ring + +instance [comm_ring R] [is_noetherian_ring R] {f : R[X]} : is_noetherian_ring (adjoin_root f) := +ideal.quotient.is_noetherian_ring _ + +end is_noetherian_ring + section power_basis variables [comm_ring R] {g : R[X]} @@ -522,6 +529,14 @@ lemma quot_map_of_equiv_quot_map_C_map_span_mk_mk (x : adjoin_root f) : ideal.quotient.mk _ x := rfl +--this lemma should have the simp tag but this causes a lint issue +lemma quot_map_of_equiv_quot_map_C_map_span_mk_symm_mk (x : adjoin_root f) : + (quot_map_of_equiv_quot_map_C_map_span_mk I f).symm + (ideal.quotient.mk ((I.map (C : R →+* R[X])).map (span {f})^.quotient.mk) x) = + ideal.quotient.mk (I.map (of f)) x := +by rw [quot_map_of_equiv_quot_map_C_map_span_mk, ideal.quot_equiv_of_eq_symm, + ideal.quot_equiv_of_eq_mk ] + /-- The natural isomorphism `R[α]/((I[x] ⊔ (f)) / (f)) ≅ (R[x]/I[x])/((f) ⊔ I[x] / I[x])` for `α` a root of `f : polynomial R` and `I : ideal R`-/ def quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk : @@ -536,6 +551,12 @@ lemma quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk_mk (p : R[X]) : quot_quot_mk (I.map C) (span {f}) p := rfl +@[simp] +lemma quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk_symm_quot_quot_mk (p : R[X]) : + (quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk I f).symm + (quot_quot_mk (I.map C) (span {f}) p) = (ideal.quotient.mk _ (mk f p)) := +rfl + /-- The natural isomorphism `(R/I)[x]/(f mod I) ≅ (R[x]/I*R[x])/(f mod I[x])` where `f : polynomial R` and `I : ideal R`-/ def polynomial.quot_quot_equiv_comm : @@ -548,7 +569,14 @@ quotient_equiv (span ({f.map (I^.quotient.mk)} : set (polynomial (R ⧸ I)))) polynomial_quotient_equiv_quotient_polynomial_map_mk I f]) @[simp] -lemma polynomial.quot_quot_equiv_comm_mk_mk (p : R[X]) : +lemma polynomial.quot_quot_equiv_comm_mk (p : R[X]) : + (polynomial.quot_quot_equiv_comm I f) (ideal.quotient.mk _ (p.map I^.quotient.mk)) = + (ideal.quotient.mk _ (ideal.quotient.mk _ p)) := +by simp only [polynomial.quot_quot_equiv_comm, quotient_equiv_mk, + polynomial_quotient_equiv_quotient_polynomial_map_mk ] + +@[simp] +lemma polynomial.quot_quot_equiv_comm_symm_mk_mk (p : R[X]) : (polynomial.quot_quot_equiv_comm I f).symm (ideal.quotient.mk _ (ideal.quotient.mk _ p)) = (ideal.quotient.mk _ (p.map I^.quotient.mk)) := by simp only [polynomial.quot_quot_equiv_comm, quotient_equiv_symm_mk, @@ -556,7 +584,7 @@ by simp only [polynomial.quot_quot_equiv_comm, quotient_equiv_symm_mk, /-- The natural isomorphism `R[α]/I[α] ≅ (R/I)[X]/(f mod I)` for `α` a root of `f : polynomial R` and `I : ideal R`-/ -def quot_map_of_equiv : (adjoin_root f) ⧸ (I.map (of f)) ≃+* +def quot_adjoin_root_equiv_quot_polynomial_quot : (adjoin_root f) ⧸ (I.map (of f)) ≃+* polynomial (R ⧸ I) ⧸ (span ({f.map (I^.quotient.mk)} : set (polynomial (R ⧸ I)))) := (quot_map_of_equiv_quot_map_C_map_span_mk I f).trans ((quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk I f).trans @@ -568,13 +596,25 @@ def quot_map_of_equiv : (adjoin_root f) ⧸ (I.map (of f)) ≃+* @[simp] lemma quot_adjoin_root_equiv_quot_polynomial_quot_mk_of (p : R[X]) : - quot_map_of_equiv I f (ideal.quotient.mk (I.map (of f)) (mk f p)) = + quot_adjoin_root_equiv_quot_polynomial_quot I f (ideal.quotient.mk (I.map (of f)) (mk f p)) = ideal.quotient.mk (span ({f.map (I^.quotient.mk)} : set (polynomial (R ⧸ I)))) (p.map I^.quotient.mk) := -by rw [quot_map_of_equiv, ring_equiv.trans_apply, ring_equiv.trans_apply, ring_equiv.trans_apply, - quot_map_of_equiv_quot_map_C_map_span_mk_mk, - quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk_mk,quot_quot_mk, ring_hom.comp_apply, - quot_equiv_of_eq_mk, polynomial.quot_quot_equiv_comm_mk_mk] +by rw [quot_adjoin_root_equiv_quot_polynomial_quot, ring_equiv.trans_apply, ring_equiv.trans_apply, + ring_equiv.trans_apply, quot_map_of_equiv_quot_map_C_map_span_mk_mk, + quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk_mk, quot_quot_mk, ring_hom.comp_apply, + quot_equiv_of_eq_mk, polynomial.quot_quot_equiv_comm_symm_mk_mk] + +@[simp] +lemma quot_adjoin_root_equiv_quot_polynomial_quot_symm_mk_mk (p : R[X]) : + (quot_adjoin_root_equiv_quot_polynomial_quot I f).symm + (ideal.quotient.mk (span ({f.map (I^.quotient.mk)} : set (polynomial (R ⧸ I)))) + (p.map I^.quotient.mk)) = (ideal.quotient.mk (I.map (of f)) (mk f p)) := +by rw [quot_adjoin_root_equiv_quot_polynomial_quot, ring_equiv.symm_trans_apply, + ring_equiv.symm_trans_apply, ring_equiv.symm_trans_apply, ring_equiv.symm_symm, + polynomial.quot_quot_equiv_comm_mk, ideal.quot_equiv_of_eq_symm, + ideal.quot_equiv_of_eq_mk, ← ring_hom.comp_apply, ← double_quot.quot_quot_mk, + quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk_symm_quot_quot_mk, + quot_map_of_equiv_quot_map_C_map_span_mk_symm_mk] end diff --git a/formal/lean/mathlib/ring_theory/algebra_tower.lean b/formal/lean/mathlib/ring_theory/algebra_tower.lean index 8d019859161ba669221e53eea67bf5205ef49970..99878f3a6a48ca3984c4c4742bce1063a244d3b4 100644 --- a/formal/lean/mathlib/ring_theory/algebra_tower.lean +++ b/formal/lean/mathlib/ring_theory/algebra_tower.lean @@ -68,8 +68,7 @@ namespace algebra theorem adjoin_algebra_map (R : Type u) (S : Type v) (A : Type w) [comm_semiring R] [comm_semiring S] [semiring A] [algebra R S] [algebra S A] [algebra R A] [is_scalar_tower R S A] (s : set S) : - adjoin R (algebra_map S A '' s) = - subalgebra.map (adjoin R s) (is_scalar_tower.to_alg_hom R S A) := + adjoin R (algebra_map S A '' s) = (adjoin R s).map (is_scalar_tower.to_alg_hom R S A) := le_antisymm (adjoin_le $ set.image_subset_iff.2 $ λ y hy, ⟨y, subset_adjoin hy, rfl⟩) (subalgebra.map_le.2 $ adjoin_le $ λ y hy, subset_adjoin ⟨y, hy, rfl⟩) @@ -151,7 +150,7 @@ theorem linear_independent_smul {ι : Type v₁} {b : ι → S} {ι' : Type w₁ begin rw linear_independent_iff' at hb hc, rw linear_independent_iff'', rintros s g hg hsg ⟨i, k⟩, by_cases hik : (i, k) ∈ s, - { have h1 : ∑ i in (s.image prod.fst).product (s.image prod.snd), g i • b i.1 • c i.2 = 0, + { have h1 : ∑ i in s.image prod.fst ×ˢ s.image prod.snd, g i • b i.1 • c i.2 = 0, { rw ← hsg, exact (finset.sum_subset finset.subset_product $ λ p _ hp, show g p • b p.1 • c p.2 = 0, by rw [hg p hp, zero_smul]).symm }, rw finset.sum_product_right at h1, @@ -227,18 +226,12 @@ begin cases hAC with x hx, cases hBC with y hy, have := hy, simp_rw [eq_top_iff', mem_span_finset] at this, choose f hf, - let s : finset B := (finset.product (x ∪ (y * y)) y).image (function.uncurry f), - have hsx : ∀ (xi ∈ x) (yj ∈ y), f xi yj ∈ s := λ xi hxi yj hyj, - show function.uncurry f (xi, yj) ∈ s, - from mem_image_of_mem _ $ mem_product.2 ⟨mem_union_left _ hxi, hyj⟩, - have hsy : ∀ (yi yj yk ∈ y), f (yi * yj) yk ∈ s := λ yi hyi yj hyj yk hyk, - show function.uncurry f (yi * yj, yk) ∈ s, - from mem_image_of_mem _ $ mem_product.2 ⟨mem_union_right _ $ finset.mul_mem_mul hyi hyj, hyk⟩, + let s : finset B := finset.image₂ f (x ∪ (y * y)) y, have hxy : ∀ xi ∈ x, xi ∈ span (algebra.adjoin A (↑s : set B)) (↑(insert 1 y : finset C) : set C) := λ xi hxi, hf xi ▸ sum_mem (λ yj hyj, smul_mem (span (algebra.adjoin A (↑s : set B)) (↑(insert 1 y : finset C) : set C)) - ⟨f xi yj, algebra.subset_adjoin $ hsx xi hxi yj hyj⟩ + ⟨f xi yj, algebra.subset_adjoin $ mem_image₂_of_mem (mem_union_left _ hxi) hyj⟩ (subset_span $ mem_insert_of_mem hyj)), have hyy : span (algebra.adjoin A (↑s : set B)) (↑(insert 1 y : finset C) : set C) * span (algebra.adjoin A (↑s : set B)) (↑(insert 1 y : finset C) : set C) ≤ @@ -249,7 +242,8 @@ begin { rw mul_one, exact subset_span (set.mem_insert_of_mem _ hyi) }, { rw ← hf (yi * yj), exact set_like.mem_coe.2 (sum_mem $ λ yk hyk, smul_mem (span (algebra.adjoin A (↑s : set B)) (insert 1 ↑y : set C)) - ⟨f (yi * yj) yk, algebra.subset_adjoin $ hsy yi hyi yj hyj yk hyk⟩ + ⟨f (yi * yj) yk, algebra.subset_adjoin $ mem_image₂_of_mem (mem_union_right _ $ + mul_mem_mul hyi hyj) hyk⟩ (subset_span $ set.mem_insert_of_mem _ hyk : yk ∈ _)) } }, refine ⟨algebra.adjoin A (↑s : set B), subalgebra.fg_adjoin_finset _, insert 1 y, _⟩, refine restrict_scalars_injective A _ _ _, diff --git a/formal/lean/mathlib/ring_theory/algebraic.lean b/formal/lean/mathlib/ring_theory/algebraic.lean index dc992c142e6d4edf9e341fd9337e8aacd4130b04..f47818adcc25273c92f57d11ddff9a6bffb9e490 100644 --- a/formal/lean/mathlib/ring_theory/algebraic.lean +++ b/formal/lean/mathlib/ring_theory/algebraic.lean @@ -104,12 +104,30 @@ lemma is_algebraic_rat (R : Type u) {A : Type v} [division_ring A] [field R] [ch [algebra R A] (n : ℚ) : is_algebraic R (n : A) := by { rw ←map_rat_cast (algebra_map R A), exact is_algebraic_algebra_map n } +lemma is_algebraic_of_mem_root_set {R : Type u} {A : Type v} [field R] [field A] [algebra R A] + {p : R[X]} {x : A} (hx : x ∈ p.root_set A) : is_algebraic R x := +⟨p, ne_zero_of_mem_root_set hx, aeval_eq_zero_of_mem_root_set hx⟩ + open is_scalar_tower lemma is_algebraic_algebra_map_of_is_algebraic {a : S} : is_algebraic R a → is_algebraic R (algebra_map S A a) := λ ⟨f, hf₁, hf₂⟩, ⟨f, hf₁, by rw [←algebra_map_aeval, hf₂, map_zero]⟩ +/-- This is slightly more general than `is_algebraic_algebra_map_of_is_algebraic` in that it + allows noncommutative intermediate rings `A`. -/ +lemma is_algebraic_alg_hom_of_is_algebraic {B} [ring B] [algebra R B] + (f : A →ₐ[R] B) {a : A} (h : is_algebraic R a) : is_algebraic R (f a) := +let ⟨p, hp, ha⟩ := h in ⟨p, hp, by rw [aeval_alg_hom, f.comp_apply, ha, map_zero]⟩ + +/-- Transfer `algebra.is_algebraic` across an `alg_equiv`. -/ +lemma _root_.alg_equiv.is_algebraic {B} [ring B] [algebra R B] (e : A ≃ₐ[R] B) + (h : algebra.is_algebraic R A) : algebra.is_algebraic R B := +λ b, by convert ← is_algebraic_alg_hom_of_is_algebraic e.to_alg_hom (h _); apply e.apply_symm_apply + +lemma _root_.alg_equiv.is_algebraic_iff {B} [ring B] [algebra R B] (e : A ≃ₐ[R] B) : + algebra.is_algebraic R A ↔ algebra.is_algebraic R B := ⟨e.is_algebraic, e.symm.is_algebraic⟩ + lemma is_algebraic_algebra_map_iff {a : S} (h : function.injective (algebra_map S A)) : is_algebraic R (algebra_map S A a) ↔ is_algebraic R a := ⟨λ ⟨p, hp0, hp⟩, ⟨p, hp0, h (by rwa [map_zero, algebra_map_aeval])⟩, @@ -178,7 +196,7 @@ _root_.is_algebraic_of_larger_base_of_injective (algebra_map K L).injective A_al lemma is_algebraic_of_larger_base (A_alg : is_algebraic K A) : is_algebraic L A := is_algebraic_of_larger_base_of_injective (algebra_map K L).injective A_alg -variables {R S} (K L) +variables (K L) /-- A field extension is integral if it is finite. -/ lemma is_integral_of_finite [finite_dimensional K L] : algebra.is_integral K L := @@ -189,6 +207,40 @@ lemma is_integral_of_finite [finite_dimensional K L] : algebra.is_integral K L : lemma is_algebraic_of_finite [finite : finite_dimensional K L] : is_algebraic K L := algebra.is_algebraic_iff_is_integral.mpr (is_integral_of_finite K L) +variables {K L} + +theorem is_algebraic.alg_hom_bijective + (ha : algebra.is_algebraic K L) (f : L →ₐ[K] L) : function.bijective f := +begin + refine ⟨f.to_ring_hom.injective, λ b, _⟩, + obtain ⟨p, hp, he⟩ := ha b, + let f' : p.root_set L → p.root_set L := + set.maps_to.restrict f _ _ (root_set_maps_to (map_ne_zero hp) f), + have : function.surjective f' := fintype.injective_iff_surjective.1 + (λ _ _ h, subtype.eq $ f.to_ring_hom.injective $ subtype.ext_iff.1 h), + obtain ⟨a, ha⟩ := this ⟨b, (mem_root_set_iff hp b).2 he⟩, + exact ⟨a, subtype.ext_iff.1 ha⟩, +end + +theorem _root_.alg_hom.bijective [finite_dimensional K L] (ϕ : L →ₐ[K] L) : function.bijective ϕ := +(algebra.is_algebraic_of_finite K L).alg_hom_bijective ϕ + +variables (K L) + +/-- Bijection between algebra equivalences and algebra homomorphisms -/ +@[simps] noncomputable +def is_algebraic.alg_equiv_equiv_alg_hom + (ha : algebra.is_algebraic K L) : (L ≃ₐ[K] L) ≃* (L →ₐ[K] L) := +{ to_fun := λ ϕ, ϕ.to_alg_hom, + inv_fun := λ ϕ, alg_equiv.of_bijective ϕ (ha.alg_hom_bijective ϕ), + left_inv := λ _, by {ext, refl}, + right_inv := λ _, by {ext, refl}, + map_mul' := λ _ _, rfl } + +/-- Bijection between algebra equivalences and algebra homomorphisms -/ +@[reducible] noncomputable def _root_.alg_equiv_equiv_alg_hom [finite_dimensional K L] : + (L ≃ₐ[K] L) ≃* (L →ₐ[K] L) := (algebra.is_algebraic_of_finite K L).alg_equiv_equiv_alg_hom K L + end algebra variables {R S : Type*} [comm_ring R] [is_domain R] [comm_ring S] diff --git a/formal/lean/mathlib/ring_theory/artinian.lean b/formal/lean/mathlib/ring_theory/artinian.lean index 1681526123d6a0625a5c23a371ee4aa0ef59992e..c748bec23fd58488498042a9c22c47834c92049c 100644 --- a/formal/lean/mathlib/ring_theory/artinian.lean +++ b/formal/lean/mathlib/ring_theory/artinian.lean @@ -3,17 +3,8 @@ Copyright (c) 2021 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes -/ -import linear_algebra.basic -import linear_algebra.prod -import linear_algebra.pi -import data.set_like.fintype -import linear_algebra.linear_independent -import tactic.linarith -import algebra.algebra.basic -import ring_theory.noetherian -import ring_theory.jacobson_ideal -import ring_theory.nilpotent import ring_theory.nakayama +import data.set_like.fintype /-! # Artinian rings and modules @@ -55,7 +46,8 @@ class is_artinian (R M) [semiring R] [add_comm_monoid M] [module R M] : Prop := (well_founded_submodule_lt [] : well_founded ((<) : submodule R M → submodule R M → Prop)) section -variables {R : Type*} {M : Type*} {P : Type*} {N : Type*} +variables {R M P N : Type*} + variables [ring R] [add_comm_group M] [add_comm_group P] [add_comm_group N] variables [module R M] [module R P] [module R N] open is_artinian @@ -117,8 +109,9 @@ is_artinian_of_range_eq_ker linear_map.snd_surjective (linear_map.range_inl R M P) -@[instance, priority 100] -lemma is_artinian_of_fintype [fintype M] : is_artinian R M := +@[priority 100] +instance is_artinian_of_finite [finite M] : is_artinian R M := +let ⟨_⟩ := nonempty_fintype M in by exactI ⟨fintype.well_founded_of_trans_of_irrefl _⟩ local attribute [elab_as_eliminator] fintype.induction_empty_option @@ -151,7 +144,8 @@ end open is_artinian submodule function -section +section ring + variables {R M : Type*} [ring R] [add_comm_group M] [module R M] theorem is_artinian_iff_well_founded : @@ -201,24 +195,26 @@ theorem monotone_stabilizes_iff_artinian : (∀ (f : ℕ →o (submodule R M)ᵒᵈ), ∃ n, ∀ m, n ≤ m → f n = f m) ↔ is_artinian R M := by { rw is_artinian_iff_well_founded, exact well_founded.monotone_chain_condition.symm } -theorem is_artinian.monotone_stabilizes [is_artinian R M] (f : ℕ →o (submodule R M)ᵒᵈ) : - ∃ n, ∀ m, n ≤ m → f n = f m := +namespace is_artinian + +variables [is_artinian R M] + +theorem monotone_stabilizes (f : ℕ →o (submodule R M)ᵒᵈ) : ∃ n, ∀ m, n ≤ m → f n = f m := monotone_stabilizes_iff_artinian.mpr ‹_› f /-- If `∀ I > J, P I` implies `P J`, then `P` holds for all submodules. -/ -lemma is_artinian.induction [is_artinian R M] {P : submodule R M → Prop} - (hgt : ∀ I, (∀ J < I, P J) → P I) (I : submodule R M) : P I := -well_founded.recursion (well_founded_submodule_lt R M) I hgt +lemma induction {P : submodule R M → Prop} (hgt : ∀ I, (∀ J < I, P J) → P I) (I : submodule R M) : + P I := +(well_founded_submodule_lt R M).recursion I hgt /-- For any endomorphism of a Artinian module, there is some nontrivial iterate with disjoint kernel and range. -/ -theorem is_artinian.exists_endomorphism_iterate_ker_sup_range_eq_top - [I : is_artinian R M] (f : M →ₗ[R] M) : ∃ n : ℕ, n ≠ 0 ∧ (f ^ n).ker ⊔ (f ^ n).range = ⊤ := +theorem exists_endomorphism_iterate_ker_sup_range_eq_top (f : M →ₗ[R] M) : + ∃ n : ℕ, n ≠ 0 ∧ (f ^ n).ker ⊔ (f ^ n).range = ⊤ := begin - obtain ⟨n, w⟩ := monotone_stabilizes_iff_artinian.mpr I - (f.iterate_range.comp ⟨λ n, n+1, λ n m w, by linarith⟩), + obtain ⟨n, w⟩ := monotone_stabilizes (f.iterate_range.comp ⟨λ n, n+1, λ n m w, by linarith⟩), specialize w ((n + 1) + n) (by linarith), dsimp at w, refine ⟨n + 1, nat.succ_ne_zero _, _⟩, @@ -236,28 +232,25 @@ begin end /-- Any injective endomorphism of an Artinian module is surjective. -/ -theorem is_artinian.surjective_of_injective_endomorphism [is_artinian R M] - (f : M →ₗ[R] M) (s : injective f) : surjective f := +theorem surjective_of_injective_endomorphism (f : M →ₗ[R] M) (s : injective f) : surjective f := begin - obtain ⟨n, ne, w⟩ := is_artinian.exists_endomorphism_iterate_ker_sup_range_eq_top f, + obtain ⟨n, ne, w⟩ := exists_endomorphism_iterate_ker_sup_range_eq_top f, rw [linear_map.ker_eq_bot.mpr (linear_map.iterate_injective s n), bot_sup_eq, linear_map.range_eq_top] at w, exact linear_map.surjective_of_iterate_surjective ne w, end /-- Any injective endomorphism of an Artinian module is bijective. -/ -theorem is_artinian.bijective_of_injective_endomorphism [is_artinian R M] - (f : M →ₗ[R] M) (s : injective f) : bijective f := -⟨s, is_artinian.surjective_of_injective_endomorphism f s⟩ +theorem bijective_of_injective_endomorphism (f : M →ₗ[R] M) (s : injective f) : bijective f := +⟨s, surjective_of_injective_endomorphism f s⟩ /-- A sequence `f` of submodules of a artinian module, with the supremum `f (n+1)` and the infinum of `f 0`, ..., `f n` being ⊤, is eventually ⊤. -/ -lemma is_artinian.disjoint_partial_infs_eventually_top [I : is_artinian R M] - (f : ℕ → submodule R M) (h : ∀ n, disjoint - (partial_sups (order_dual.to_dual ∘ f) n) (order_dual.to_dual (f (n+1)))) : +lemma disjoint_partial_infs_eventually_top (f : ℕ → submodule R M) + (h : ∀ n, disjoint (partial_sups (order_dual.to_dual ∘ f) n) (order_dual.to_dual (f (n+1)))) : ∃ n : ℕ, ∀ m, n ≤ m → f m = ⊤ := begin -- A little off-by-one cleanup first: @@ -269,18 +262,48 @@ begin { apply w, exact nat.succ_le_succ_iff.mp p }, }, - obtain ⟨n, w⟩ := monotone_stabilizes_iff_artinian.mpr I (partial_sups (order_dual.to_dual ∘ f)), - exact ⟨n, λ m p, (h m).eq_bot_of_ge $ sup_eq_left.1 $ (w (m + 1) $ le_add_right p).symm.trans $ - w m p⟩ + obtain ⟨n, w⟩ := monotone_stabilizes (partial_sups (order_dual.to_dual ∘ f)), + refine ⟨n, λ m p, _⟩, + exact (h m).eq_bot_of_ge (sup_eq_left.1 $ (w (m + 1) $ le_add_right p).symm.trans $ w m p) end -universe w -variables {N : Type w} [add_comm_group N] [module R N] +end is_artinian + +end ring + +section comm_ring + +variables {R : Type*} (M : Type*) [comm_ring R] [add_comm_group M] [module R M] [is_artinian R M] + +namespace is_artinian + +lemma range_smul_pow_stabilizes (r : R) : ∃ n : ℕ, ∀ m, n ≤ m → + (r^n • linear_map.id : M →ₗ[R] M).range = (r^m • linear_map.id).range := +monotone_stabilizes +⟨λ n, (r^n • linear_map.id : M →ₗ[R] M).range, + λ n m h x ⟨y, hy⟩, ⟨r ^ (m - n) • y, + by { dsimp at ⊢ hy, rw [←smul_assoc, smul_eq_mul, ←pow_add, ←hy, add_tsub_cancel_of_le h] }⟩⟩ + +variables {M} + +lemma exists_pow_succ_smul_dvd (r : R) (x : M) : + ∃ (n : ℕ) (y : M), r ^ n.succ • y = r ^ n • x := +begin + obtain ⟨n, hn⟩ := is_artinian.range_smul_pow_stabilizes M r, + simp_rw [set_like.ext_iff] at hn, + exact ⟨n, by simpa using hn n.succ n.le_succ (r ^ n • x)⟩, +end + +end is_artinian + +end comm_ring -- TODO: Prove this for artinian modules -- /-- -- If `M ⊕ N` embeds into `M`, for `M` noetherian over `R`, then `N` is trivial. -- -/ +-- universe w +-- variables {N : Type w} [add_comm_group N] [module R N] -- noncomputable def is_noetherian.equiv_punit_of_prod_injective [is_noetherian R M] -- (f : M × N →ₗ[R] M) (i : injective f) : N ≃ₗ[R] punit.{w+1} := -- begin @@ -294,14 +317,16 @@ variables {N : Type w} [add_comm_group N] [module R N] -- exact submodule.bot_equiv_punit, -- end -end - /-- A ring is Artinian if it is Artinian as a module over itself. Strictly speaking, this should be called `is_left_artinian_ring` but we omit the `left_` for convenience in the commutative case. For a right Artinian ring, use `is_artinian Rᵐᵒᵖ R`. -/ class is_artinian_ring (R) [ring R] extends is_artinian R R : Prop +-- TODO: Can we define `is_artinian_ring` in a different way so this isn't needed? +@[priority 100] +instance is_artinian_ring_of_finite (R) [ring R] [finite R] : is_artinian_ring R := ⟨⟩ + theorem is_artinian_ring_iff {R} [ring R] : is_artinian_ring R ↔ is_artinian R R := ⟨λ h, h.1, @is_artinian_ring.mk _ _⟩ @@ -369,22 +394,16 @@ theorem is_artinian_span_of_finite (R) {M} [ring R] [add_comm_group M] [module R [is_artinian_ring R] {A : set M} (hA : A.finite) : is_artinian R (submodule.span R A) := is_artinian_of_fg_of_artinian _ (submodule.fg_def.mpr ⟨A, hA, rfl⟩) -theorem is_artinian_ring_of_surjective (R) [ring R] (S) [ring S] - (f : R →+* S) (hf : function.surjective f) - [H : is_artinian_ring R] : is_artinian_ring S := +theorem function.surjective.is_artinian_ring {R} [ring R] {S} [ring S] {F} [ring_hom_class F R S] + {f : F} (hf : function.surjective f) [H : is_artinian_ring R] : is_artinian_ring S := begin rw [is_artinian_ring_iff, is_artinian_iff_well_founded] at H ⊢, - exact order_embedding.well_founded (ideal.order_embedding_of_surjective f hf) H, + exact (ideal.order_embedding_of_surjective f hf).well_founded H, end -instance is_artinian_ring_range {R} [ring R] {S} [ring S] (f : R →+* S) - [is_artinian_ring R] : is_artinian_ring f.range := -is_artinian_ring_of_surjective R f.range f.range_restrict - f.range_restrict_surjective - -theorem is_artinian_ring_of_ring_equiv (R) [ring R] {S} [ring S] - (f : R ≃+* S) [is_artinian_ring R] : is_artinian_ring S := -is_artinian_ring_of_surjective R S f.to_ring_hom f.to_equiv.surjective +instance is_artinian_ring_range {R} [ring R] {S} [ring S] (f : R →+* S) [is_artinian_ring R] : + is_artinian_ring f.range := +f.range_restrict_surjective.is_artinian_ring namespace is_artinian_ring @@ -428,4 +447,35 @@ begin rwa [← hn (n + 1) (nat.le_succ _)] end +section localization + +variables (S : submonoid R) (L : Type*) [comm_ring L] [algebra R L] [is_localization S L] + +include S + +/-- Localizing an artinian ring can only reduce the amount of elements. -/ +theorem localization_surjective : function.surjective (algebra_map R L) := +begin + intro r', + obtain ⟨r₁, s, rfl⟩ := is_localization.mk'_surjective S r', + obtain ⟨r₂, h⟩ : ∃ r : R, is_localization.mk' L 1 s = algebra_map R L r, + swap, { exact ⟨r₁ * r₂, by rw [is_localization.mk'_eq_mul_mk'_one, map_mul, h]⟩ }, + obtain ⟨n, r, hr⟩ := is_artinian.exists_pow_succ_smul_dvd (s : R) (1 : R), + use r, + rw [smul_eq_mul, smul_eq_mul, pow_succ', mul_assoc] at hr, + apply_fun algebra_map R L at hr, + simp only [map_mul, ←submonoid.coe_pow] at hr, + rw [←is_localization.mk'_one L, is_localization.mk'_eq_iff_eq, one_mul, submonoid.coe_one, + ←(is_localization.map_units L (s ^ n)).mul_left_cancel hr, map_mul, mul_comm], +end + +lemma localization_artinian : is_artinian_ring L := +(localization_surjective S L).is_artinian_ring + +/-- `is_artinian_ring.localization_artinian` can't be made an instance, as it would make `S` + `R` +into metavariables. However, this is safe. -/ +instance : is_artinian_ring (localization S) := localization_artinian S _ + +end localization + end is_artinian_ring diff --git a/formal/lean/mathlib/ring_theory/coprime/lemmas.lean b/formal/lean/mathlib/ring_theory/coprime/lemmas.lean index 34901b4106736e0d76898c9cfede2674e4aad786..c6334f80f3d0c68cd051f547eb359bdaa86fb623 100644 --- a/formal/lean/mathlib/ring_theory/coprime/lemmas.lean +++ b/formal/lean/mathlib/ring_theory/coprime/lemmas.lean @@ -33,6 +33,8 @@ theorem nat.is_coprime_iff_coprime {m n : ℕ} : is_coprime (m : ℤ) n ↔ nat. λ H, ⟨nat.gcd_a m n, nat.gcd_b m n, by rw [mul_comm _ (m : ℤ), mul_comm _ (n : ℤ), ← nat.gcd_eq_gcd_ab, show _ = _, from H, int.coe_nat_one]⟩⟩ +alias nat.is_coprime_iff_coprime ↔ is_coprime.nat_coprime nat.coprime.is_coprime + theorem is_coprime.prod_left : (∀ i ∈ t, is_coprime (s i) x) → is_coprime (∏ i in t, s i) x := finset.induction_on t (λ _, is_coprime_one_left) $ λ b t hbt ih H, by { rw finset.prod_insert hbt, rw finset.forall_mem_insert at H, exact H.1.mul_left (ih H.2) } diff --git a/formal/lean/mathlib/ring_theory/dedekind_domain/ideal.lean b/formal/lean/mathlib/ring_theory/dedekind_domain/ideal.lean index f75b732dd923a65b2c65e438a106e72b6d5f5470..75728323de8be94ad14d5a35e4b204dd02f0fd30 100644 --- a/formal/lean/mathlib/ring_theory/dedekind_domain/ideal.lean +++ b/formal/lean/mathlib/ring_theory/dedekind_domain/ideal.lean @@ -575,8 +575,8 @@ variables {R A} [is_dedekind_domain A] [algebra A K] [is_fraction_ring A K] open fractional_ideal open ideal -noncomputable instance fractional_ideal.comm_group_with_zero : - comm_group_with_zero (fractional_ideal A⁰ K) := +noncomputable instance fractional_ideal.semifield : + semifield (fractional_ideal A⁰ K) := { inv := λ I, I⁻¹, inv_zero := inv_zero' _, div := (/), @@ -586,11 +586,23 @@ noncomputable instance fractional_ideal.comm_group_with_zero : mul_inv_cancel := λ I, fractional_ideal.mul_inv_cancel, .. fractional_ideal.comm_semiring } -noncomputable instance ideal.cancel_comm_monoid_with_zero : +/-- Fractional ideals have cancellative multiplication in a Dedekind domain. + +Although this instance is a direct consequence of the instance +`fractional_ideal.comm_group_with_zero`, we define this instance to provide +a computable alternative. +-/ +instance fractional_ideal.cancel_comm_monoid_with_zero : + cancel_comm_monoid_with_zero (fractional_ideal A⁰ K) := +{ .. fractional_ideal.comm_semiring, -- Project out the computable fields first. + .. (by apply_instance : cancel_comm_monoid_with_zero (fractional_ideal A⁰ K)) } + +instance ideal.cancel_comm_monoid_with_zero : cancel_comm_monoid_with_zero (ideal A) := -function.injective.cancel_comm_monoid_with_zero (coe_ideal_hom A⁰ (fraction_ring A)) - coe_ideal_injective (ring_hom.map_zero _) (ring_hom.map_one _) (ring_hom.map_mul _) - (ring_hom.map_pow _) +{ .. ideal.comm_semiring, + .. function.injective.cancel_comm_monoid_with_zero (coe_ideal_hom A⁰ (fraction_ring A)) + coe_ideal_injective (ring_hom.map_zero _) (ring_hom.map_one _) (ring_hom.map_mul _) + (ring_hom.map_pow _) } /-- For ideals in a Dedekind domain, to divide is to contain. -/ lemma ideal.dvd_iff_le {I J : ideal A} : (I ∣ J) ↔ J ≤ I := @@ -645,7 +657,7 @@ instance ideal.unique_factorization_monoid : prime.irreducible⟩, .. ideal.wf_dvd_monoid } -noncomputable instance ideal.normalization_monoid : normalization_monoid (ideal A) := +instance ideal.normalization_monoid : normalization_monoid (ideal A) := normalization_monoid_of_unique_units @[simp] lemma ideal.dvd_span_singleton {I : ideal A} {x : A} : @@ -831,7 +843,7 @@ variables [is_domain R] [is_dedekind_domain R] /-- The height one prime spectrum of a Dedekind domain `R` is the type of nonzero prime ideals of `R`. Note that this equals the maximal spectrum if `R` has Krull dimension 1. -/ -@[ext, nolint has_inhabited_instance unused_arguments] +@[ext, nolint has_nonempty_instance unused_arguments] structure height_one_spectrum := (as_ideal : ideal R) (is_prime : as_ideal.is_prime) diff --git a/formal/lean/mathlib/ring_theory/derivation.lean b/formal/lean/mathlib/ring_theory/derivation.lean index 806f3376bb97c5c021d2518122dc0cc8642dc26f..2d0bb2a9ec069b607fce9d3cfa4d5640b62182eb 100644 --- a/formal/lean/mathlib/ring_theory/derivation.lean +++ b/formal/lean/mathlib/ring_theory/derivation.lean @@ -1,11 +1,12 @@ /- Copyright © 2020 Nicolò Cavalleri. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. -Authors: Nicolò Cavalleri +Authors: Nicolò Cavalleri, Andrew Yang -/ import ring_theory.adjoin.basic import algebra.lie.of_associative +import ring_theory.tensor_product /-! # Derivations @@ -220,6 +221,12 @@ rfl (f.comp_der D : A → N) = (f : M →ₗ[R] N).comp (D : A →ₗ[R] M) := rfl +/-- The composition of a derivation with a linear map as a bilinear map -/ +def llcomp : (M →ₗ[A] N) →ₗ[A] derivation R A M →ₗ[R] derivation R A N := +{ to_fun := λ f, f.comp_der, + map_add' := λ f₁ f₂, by { ext, refl }, + map_smul' := λ r D, by { ext, refl } } + end push_forward end @@ -440,3 +447,97 @@ begin end end to_square_zero + +section derivation_module + +open_locale tensor_product + +variables (R S : Type*) [comm_ring R] [comm_ring S] [algebra R S] + +/-- The kernel of the multiplication map `S ⊗[R] S →ₐ[R] S`. -/ +abbreviation derivation_module.ideal : ideal (S ⊗[R] S) := +ring_hom.ker (tensor_product.lmul' R : S ⊗[R] S →ₐ[R] S) + +variable {S} + +lemma derivation_module.one_smul_sub_smul_one_mem_ideal (a : S) : + (1 : S) ⊗ₜ[R] a - a ⊗ₜ[R] (1 : S) ∈ derivation_module.ideal R S := +by simp [ring_hom.mem_ker] + +variables {R} + +variables {M : Type*} [add_comm_group M] [module R M] [module S M] [is_scalar_tower R S M] + +/-- For a `R`-derivation `S → M`, this is the map `S ⊗[R] S →ₗ[S] M` sending `s ⊗ₜ t ↦ s • D t`. -/ +def derivation.tensor_product_to (D : derivation R S M) : S ⊗[R] S →ₗ[S] M := +tensor_product.algebra_tensor_module.lift ((linear_map.lsmul S (S →ₗ[R] M)).flip D.to_linear_map) + +lemma derivation.tensor_product_to_tmul (D : derivation R S M) (s t : S) : + D.tensor_product_to (s ⊗ₜ t) = s • D t := +tensor_product.lift.tmul s t + +lemma derivation.tensor_product_to_mul (D : derivation R S M) (x y : S ⊗[R] S) : + D.tensor_product_to (x * y) = tensor_product.lmul' R x • D.tensor_product_to y + + tensor_product.lmul' R y • D.tensor_product_to x := +begin + apply tensor_product.induction_on x, + { rw [zero_mul, map_zero, map_zero, zero_smul, smul_zero, add_zero] }, + swap, { rintros, simp only [add_mul, map_add, add_smul, *, smul_add], rw add_add_add_comm }, + intros x₁ x₂, + apply tensor_product.induction_on y, + { rw [mul_zero, map_zero, map_zero, zero_smul, smul_zero, add_zero] }, + swap, { rintros, simp only [mul_add, map_add, add_smul, *, smul_add], rw add_add_add_comm }, + intros x y, + simp only [tensor_product.tmul_mul_tmul, derivation.tensor_product_to, + tensor_product.algebra_tensor_module.lift_apply, tensor_product.lift.tmul', + tensor_product.lmul'_apply_tmul], + dsimp, + rw D.leibniz, + simp only [smul_smul, smul_add, mul_comm (x * y) x₁, mul_right_comm x₁ x₂, ← mul_assoc], +end + +variables (R S) + +/-- The kernel of `S ⊗[R] S →ₐ[R] S` is generated by `1 ⊗ s - s ⊗ 1` as a `S`-module. -/ +lemma derivation_module.submodule_span_range_eq_ideal : + submodule.span S (set.range $ λ s : S, (1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) = + (derivation_module.ideal R S).restrict_scalars S := +begin + apply le_antisymm, + { rw submodule.span_le, + rintros _ ⟨s, rfl⟩, + exact derivation_module.one_smul_sub_smul_one_mem_ideal _ _ }, + { rintros x (hx : _ = _), + have : x - (tensor_product.lmul' R x) ⊗ₜ[R] (1 : S) = x, + { rw [hx, tensor_product.zero_tmul, sub_zero] }, + rw ← this, + clear this hx, + apply tensor_product.induction_on x; clear x, + { rw [map_zero, tensor_product.zero_tmul, sub_zero], exact zero_mem _ }, + { intros x y, + convert_to x • (1 ⊗ₜ y - y ⊗ₜ 1) ∈ _, + { rw [tensor_product.lmul'_apply_tmul, smul_sub, tensor_product.smul_tmul', + tensor_product.smul_tmul', smul_eq_mul, smul_eq_mul, mul_one] }, + { refine submodule.smul_mem _ x _, + apply submodule.subset_span, + exact set.mem_range_self y } }, + { intros x y hx hy, + rw [map_add, tensor_product.add_tmul, ← sub_add_sub_comm], + exact add_mem hx hy } } +end + +lemma derivation_module.span_range_eq_ideal : + ideal.span (set.range $ λ s : S, (1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) = + derivation_module.ideal R S := +begin + apply le_antisymm, + { rw ideal.span_le, + rintros _ ⟨s, rfl⟩, + exact derivation_module.one_smul_sub_smul_one_mem_ideal _ _ }, + { change (derivation_module.ideal R S).restrict_scalars S ≤ (ideal.span _).restrict_scalars S, + rw [← derivation_module.submodule_span_range_eq_ideal, ideal.span], + conv_rhs { rw ← submodule.span_span_of_tower S }, + exact submodule.subset_span } +end + +end derivation_module diff --git a/formal/lean/mathlib/ring_theory/finiteness.lean b/formal/lean/mathlib/ring_theory/finiteness.lean index 3e70cf5d43427b3e597b273eded08a50eaad50ab..54b7af03d3f13e74fc01a4aaed9efece19137ba6 100644 --- a/formal/lean/mathlib/ring_theory/finiteness.lean +++ b/formal/lean/mathlib/ring_theory/finiteness.lean @@ -517,6 +517,11 @@ begin end hf hg +lemma of_finite {f : A →+* B} (hf : f.finite) : f.finite_type := +@module.finite.finite_type _ _ _ _ f.to_algebra hf + +alias of_finite ← _root_.ring_hom.finite.to_finite_type + lemma of_finite_presentation {f : A →+* B} (hf : f.finite_presentation) : f.finite_type := @algebra.finite_type.of_finite_presentation A B _ _ f.to_algebra hf diff --git a/formal/lean/mathlib/ring_theory/graded_algebra/homogeneous_localization.lean b/formal/lean/mathlib/ring_theory/graded_algebra/homogeneous_localization.lean index 4b8d5d8e9f1e606153ca5a9e29f851ded6e2134a..ee1f4e61e9b0ef8a84bb2f63851d131599c3f5fb 100644 --- a/formal/lean/mathlib/ring_theory/graded_algebra/homogeneous_localization.lean +++ b/formal/lean/mathlib/ring_theory/graded_algebra/homogeneous_localization.lean @@ -79,7 +79,7 @@ section Let `x` be a prime ideal, then `num_denom_same_deg 𝒜 x` is a structure with a numerator and a denominator with same grading such that the denominator is not contained in `x`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure num_denom_same_deg := (deg : ι) (num denom : 𝒜 deg) @@ -214,7 +214,7 @@ For `x : prime ideal of A`, `homogeneous_localization 𝒜 x` is `num_denom_same kernel of `embedding 𝒜 x`. This is essentially the subring of `Aₓ` where the numerator and denominator share the same grading. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def homogeneous_localization : Type* := quotient (setoid.ker $ homogeneous_localization.num_denom_same_deg.embedding 𝒜 x) diff --git a/formal/lean/mathlib/ring_theory/graded_algebra/radical.lean b/formal/lean/mathlib/ring_theory/graded_algebra/radical.lean index f8935fb7c04cca3b791c945c23e41180ca12f945..a4953692a153bf58ce1b86b16283610c6103304a 100644 --- a/formal/lean/mathlib/ring_theory/graded_algebra/radical.lean +++ b/formal/lean/mathlib/ring_theory/graded_algebra/radical.lean @@ -85,7 +85,7 @@ lemma ideal.is_homogeneous.is_prime_of_homogeneous_mem_or_mem have mem_I : proj 𝒜 max₁ x * proj 𝒜 max₂ y ∈ I, { set antidiag := - ((decompose 𝒜 x).support.product (decompose 𝒜 y).support) + ((decompose 𝒜 x).support ×ˢ (decompose 𝒜 y).support) .filter (λ z : ι × ι, z.1 + z.2 = max₁ + max₂) with ha, have mem_antidiag : (max₁, max₂) ∈ antidiag, { simp only [add_sum_erase, mem_filter, mem_product], diff --git a/formal/lean/mathlib/ring_theory/henselian.lean b/formal/lean/mathlib/ring_theory/henselian.lean index 453fb3c8464a1a9070024f8d749f47d0614206fe..38d366b1512370bb22e7fd1f0c3ebd9b2c381055 100644 --- a/formal/lean/mathlib/ring_theory/henselian.lean +++ b/formal/lean/mathlib/ring_theory/henselian.lean @@ -250,7 +250,7 @@ instance is_adic_complete.henselian_ring specialize ha 1, rw [hc, pow_one, ← ideal.one_eq_top, ideal.smul_eq_mul, mul_one, sub_eq_add_neg] at ha, rw [← smodeq.sub_mem, ← add_zero a₀], - refine ha.symm.trans (smodeq.refl.add _), + refine ha.symm.trans (smodeq.rfl.add _), rw [smodeq.zero, ideal.neg_mem_iff], exact ideal.mul_mem_right _ _ h₁, } end } diff --git a/formal/lean/mathlib/ring_theory/ideal/cotangent.lean b/formal/lean/mathlib/ring_theory/ideal/cotangent.lean new file mode 100644 index 0000000000000000000000000000000000000000..e3f499ab4111957bf8c844283a858b346c4d58b9 --- /dev/null +++ b/formal/lean/mathlib/ring_theory/ideal/cotangent.lean @@ -0,0 +1,161 @@ +/- +Copyright (c) 2022 Andrew Yang. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Andrew Yang +-/ +import ring_theory.ideal.operations +import algebra.module.torsion +import algebra.ring.idempotents +import linear_algebra.finite_dimensional +import ring_theory.ideal.local_ring +import ring_theory.nakayama + +/-! +# The module `I ⧸ I ^ 2` + +In this file, we provide special API support for the module `I ⧸ I ^ 2`. The official +definition is a quotient module of `I`, but the alternative definition as an ideal of `R ⧸ I ^ 2` is +also given, and the two are `R`-equivalent as in `ideal.cotangent_equiv_ideal`. + +Additional support is also given to the cotangent space `m ⧸ m ^ 2` of a local ring. + +-/ + +namespace ideal + +variables {R S : Type*} [comm_ring R] [comm_semiring S] [algebra S R] (I : ideal R) + +/-- `I ⧸ I ^ 2` as a quotient of `I`. -/ +@[derive [add_comm_group, module (R ⧸ I)]] +def cotangent : Type* := I ⧸ (I • ⊤ : submodule R I) + +instance : inhabited I.cotangent := ⟨0⟩ + +instance cotangent.module_of_tower : module S I.cotangent := +submodule.quotient.module' _ + +instance : is_scalar_tower S R I.cotangent := by { delta cotangent, apply_instance } + +instance [is_noetherian R I] : is_noetherian R I.cotangent := by { delta cotangent, apply_instance } + +/-- The quotient map from `I` to `I ⧸ I ^ 2`. -/ +@[simps apply (lemmas_only)] +def to_cotangent : I →ₗ[R] I.cotangent := submodule.mkq _ + +lemma map_to_cotangent_ker : I.to_cotangent.ker.map I.subtype = I ^ 2 := +by simp [ideal.to_cotangent, submodule.map_smul'', pow_two] + +lemma mem_to_cotangent_ker {x : I} : x ∈ I.to_cotangent.ker ↔ (x : R) ∈ I ^ 2 := +begin + rw ← I.map_to_cotangent_ker, + simp, +end + +lemma to_cotangent_eq {x y : I} : I.to_cotangent x = I.to_cotangent y ↔ (x - y : R) ∈ I ^ 2 := +begin + rw [← sub_eq_zero, ← map_sub], + exact I.mem_to_cotangent_ker +end + +lemma to_cotangent_eq_zero (x : I) : I.to_cotangent x = 0 ↔ (x : R) ∈ I ^ 2 := +I.mem_to_cotangent_ker + +lemma to_cotangent_surjective : function.surjective I.to_cotangent := +submodule.mkq_surjective _ + +lemma to_cotangent_range : I.to_cotangent.range = ⊤ := +submodule.range_mkq _ + +lemma cotangent_subsingleton_iff : + subsingleton I.cotangent ↔ is_idempotent_elem I := +begin + split, + { introI H, + refine (pow_two I).symm.trans (le_antisymm (ideal.pow_le_self two_ne_zero) _), + exact λ x hx, (I.to_cotangent_eq_zero ⟨x, hx⟩).mp (subsingleton.elim _ _) }, + { exact λ e, ⟨λ x y, quotient.induction_on₂' x y $ λ x y, + I.to_cotangent_eq.mpr $ ((pow_two I).trans e).symm ▸ I.sub_mem x.prop y.prop⟩ } +end + +/-- The inclusion map `I ⧸ I ^ 2` to `R ⧸ I ^ 2`. -/ +def cotangent_to_quotient_square : I.cotangent →ₗ[R] R ⧸ I ^ 2 := +submodule.mapq (I • ⊤) (I ^ 2) I.subtype + (by { rw [← submodule.map_le_iff_le_comap, submodule.map_smul'', submodule.map_top, + submodule.range_subtype, smul_eq_mul, pow_two], exact rfl.le }) + +lemma to_quotient_square_comp_to_cotangent : I.cotangent_to_quotient_square.comp I.to_cotangent = + (I ^ 2).mkq.comp (submodule.subtype I) := +linear_map.ext $ λ _, rfl + +@[simp] +lemma to_cotangent_to_quotient_square (x : I) : I.cotangent_to_quotient_square (I.to_cotangent x) = + (I ^ 2).mkq x := rfl + +/-- `I ⧸ I ^ 2` as an ideal of `R ⧸ I ^ 2`. -/ +def cotangent_ideal (I : ideal R) : ideal (R ⧸ I ^ 2) := +begin + haveI : @ring_hom_surjective R (R ⧸ I ^ 2) _ _ _ := ⟨ideal.quotient.mk_surjective⟩, + exact submodule.map (ring_hom.to_semilinear_map (I ^ 2)^.quotient.mk) I, +end + +lemma to_quotient_square_range : + I.cotangent_to_quotient_square.range = I.cotangent_ideal.restrict_scalars R := +begin + transitivity (I.cotangent_to_quotient_square.comp I.to_cotangent).range, + { rw [linear_map.range_comp, I.to_cotangent_range, submodule.map_top] }, + { rw [to_quotient_square_comp_to_cotangent, linear_map.range_comp, I.range_subtype], ext, refl } +end + +/-- The equivalence of the two definitions of `I / I ^ 2`, either as the quotient of `I` or the +ideal of `R / I ^ 2`. -/ +noncomputable +def cotangent_equiv_ideal : I.cotangent ≃ₗ[R] I.cotangent_ideal := +begin + refine + { ..(I.cotangent_to_quotient_square.cod_restrict (I.cotangent_ideal.restrict_scalars R) + (λ x, by { rw ← to_quotient_square_range, exact linear_map.mem_range_self _ _ })), + ..(equiv.of_bijective _ ⟨_, _⟩) }, + { rintros x y e, + replace e := congr_arg subtype.val e, + obtain ⟨x, rfl⟩ := I.to_cotangent_surjective x, + obtain ⟨y, rfl⟩ := I.to_cotangent_surjective y, + rw I.to_cotangent_eq, + dsimp only [to_cotangent_to_quotient_square, submodule.mkq_apply] at e, + rwa submodule.quotient.eq at e }, + { rintro ⟨_, x, hx, rfl⟩, + refine ⟨I.to_cotangent ⟨x, hx⟩, subtype.ext rfl⟩ } +end + +@[simp, nolint simp_nf] +lemma cotangent_equiv_ideal_apply (x : I.cotangent) : + ↑(I.cotangent_equiv_ideal x) = I.cotangent_to_quotient_square x := rfl + +lemma cotangent_equiv_ideal_symm_apply (x : R) (hx : x ∈ I) : + I.cotangent_equiv_ideal.symm ⟨(I ^ 2).mkq x, submodule.mem_map_of_mem hx⟩ = + I.to_cotangent ⟨x, hx⟩ := +begin + apply I.cotangent_equiv_ideal.injective, + rw I.cotangent_equiv_ideal.apply_symm_apply, + ext, + refl +end + +end ideal + +namespace local_ring + +variables (R : Type*) [comm_ring R] [local_ring R] + +/-- The `A ⧸ I`-vector space `I ⧸ I ^ 2`. -/ +@[reducible] def cotangent_space : Type* := (maximal_ideal R).cotangent + +instance : module (residue_field R) (cotangent_space R) := +ideal.cotangent.module _ + +instance : is_scalar_tower R (residue_field R) (cotangent_space R) := +module.is_torsion_by_set.is_scalar_tower _ + +instance [is_noetherian_ring R] : finite_dimensional (residue_field R) (cotangent_space R) := +module.finite.of_restrict_scalars_finite R _ _ + +end local_ring diff --git a/formal/lean/mathlib/ring_theory/ideal/local_ring.lean b/formal/lean/mathlib/ring_theory/ideal/local_ring.lean index 4d446c7349bff1fa170dfd35149363541d262541..c1dccf7c0f3c7a8e81459596f5c32995a50c52ea 100644 --- a/formal/lean/mathlib/ring_theory/ideal/local_ring.lean +++ b/formal/lean/mathlib/ring_theory/ideal/local_ring.lean @@ -307,6 +307,16 @@ begin f.is_unit_map f.is_unit_map end +/-- If `f : R →+* S` is a surjective local ring hom, then the induced units map is surjective. -/ +lemma surjective_units_map_of_local_ring_hom [comm_ring R] [comm_ring S] + (f : R →+* S) (hf : function.surjective f) (h : is_local_ring_hom f) : + function.surjective (units.map $ f.to_monoid_hom) := +begin + intro a, + obtain ⟨b,hb⟩ := hf (a : S), + use (is_unit_of_map_unit f _ (by { rw hb, exact units.is_unit _})).unit, ext, exact hb, +end + section variables (R) [comm_ring R] [local_ring R] [comm_ring S] [local_ring S] @@ -345,6 +355,16 @@ lemma ker_eq_maximal_ideal [field K] (φ : R →+* K) (hφ : function.surjective φ.ker = maximal_ideal R := local_ring.eq_maximal_ideal $ (ring_hom.ker_is_maximal_of_surjective φ) hφ +lemma is_local_ring_hom_residue : + is_local_ring_hom (local_ring.residue R) := +begin + constructor, + intros a ha, + by_contra, + erw ideal.quotient.eq_zero_iff_mem.mpr ((local_ring.mem_maximal_ideal _).mpr h) at ha, + exact ha.ne_zero rfl, +end + end end local_ring diff --git a/formal/lean/mathlib/ring_theory/ideal/operations.lean b/formal/lean/mathlib/ring_theory/ideal/operations.lean index 9e8d92e6e8df1060487047f8b89260cb03719fc4..cae7baf6d4b55e331e7b1ac75d17e919ac6692fb 100644 --- a/formal/lean/mathlib/ring_theory/ideal/operations.lean +++ b/formal/lean/mathlib/ring_theory/ideal/operations.lean @@ -228,17 +228,20 @@ variables (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ -lemma exists_sum_of_mem_ideal_smul_span {ι : Type*} (s : set ι) (f : ι → M) (x : M) - (hx : x ∈ I • span R (f '' s)) : - ∃ (a : s →₀ R) (ha : ∀ i, a i ∈ I), a.sum (λ i c, c • f i) = x := +lemma mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : + x ∈ I • span R (set.range f) ↔ + ∃ (a : ι →₀ R) (ha : ∀ i, a i ∈ I), a.sum (λ i c, c • f i) = x := begin - refine span_induction (mem_smul_span.mp hx) _ _ _ _, + split, swap, + { rintro ⟨a, ha, rfl⟩, + exact submodule.sum_mem _ (λ c _, smul_mem_smul (ha c) $ subset_span $ set.mem_range_self _) }, + refine λ hx, span_induction (mem_smul_span.mp hx) _ _ _ _, { simp only [set.mem_Union, set.mem_range, set.mem_singleton_iff], - rintros x ⟨y, hy, x, ⟨i, hi, rfl⟩, rfl⟩, - refine ⟨finsupp.single ⟨i, hi⟩ y, λ j, _, _⟩, - { letI := classical.dec_eq s, + rintros x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩, + refine ⟨finsupp.single i y, λ j, _, _⟩, + { letI := classical.dec_eq ι, rw finsupp.single_apply, split_ifs, { assumption }, { exact I.zero_mem } }, - refine @finsupp.sum_single_index s R M _ _ ⟨i, hi⟩ _ (λ i y, y • f i) _, + refine @finsupp.sum_single_index ι R M _ _ i _ (λ i y, y • f i) _, simp }, { exact ⟨0, λ i, I.zero_mem, finsupp.sum_zero_index⟩ }, { rintros x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩, @@ -250,6 +253,11 @@ begin intros; simp only [zero_smul, mul_smul] }, end +theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : set ι) (f : ι → M) (x : M) : + x ∈ I • span R (f '' s) ↔ + ∃ (a : s →₀ R) (ha : ∀ i, a i ∈ I), a.sum (λ i c, c • f i) = x := +by rw [← submodule.mem_ideal_smul_span_iff_exists_sum, ← set.image_eq_range] + @[simp] lemma smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : submodule R M') (I : ideal R) : I • S.comap f ≤ (I • S).comap f := begin @@ -1397,6 +1405,50 @@ end⟩ end is_primary +section total + +variables (ι : Type*) +variables (M : Type*) [add_comm_group M] {R : Type*} [comm_ring R] [module R M] (I : ideal R) +variables (v : ι → M) (hv : submodule.span R (set.range v) = ⊤) + + +open_locale big_operators + +/-- A variant of `finsupp.total` that takes in vectors valued in `I`. -/ +noncomputable +def finsupp_total : (ι →₀ I) →ₗ[R] M := +(finsupp.total ι M R v).comp (finsupp.map_range.linear_map I.subtype) + +variables {ι M v} + +lemma finsupp_total_apply (f : ι →₀ I) : + finsupp_total ι M I v f = f.sum (λ i x, (x : R) • v i) := +begin + dsimp [finsupp_total], + rw [finsupp.total_apply, finsupp.sum_map_range_index], + exact λ _, zero_smul _ _ +end + +lemma finsupp_total_apply_eq_of_fintype [fintype ι] (f : ι →₀ I) : + finsupp_total ι M I v f = ∑ i, (f i : R) • v i := +by { rw [finsupp_total_apply, finsupp.sum_fintype], exact λ _, zero_smul _ _ } + +lemma range_finsupp_total : + (finsupp_total ι M I v).range = I • (submodule.span R (set.range v)) := +begin + ext, + rw submodule.mem_ideal_smul_span_iff_exists_sum, + refine ⟨λ ⟨f, h⟩, ⟨finsupp.map_range.linear_map I.subtype f, λ i, (f i).2, h⟩, _⟩, + rintro ⟨a, ha, rfl⟩, + classical, + refine ⟨a.map_range (λ r, if h : r ∈ I then ⟨r, h⟩ else 0) (by split_ifs; refl), _⟩, + rw [finsupp_total_apply, finsupp.sum_map_range_index], + { apply finsupp.sum_congr, intros i _, rw dif_pos (ha i), refl }, + { exact λ _, zero_smul _ _ }, +end + +end total + end ideal lemma associates.mk_ne_zero' {R : Type*} [comm_semiring R] {r : R} : diff --git a/formal/lean/mathlib/ring_theory/ideal/quotient.lean b/formal/lean/mathlib/ring_theory/ideal/quotient.lean index e810dbd7ae5bbeccfba695c72e68805b11308bb7..1eafbe3513dfb33893f122f87c3e1df964136def 100644 --- a/formal/lean/mathlib/ring_theory/ideal/quotient.lean +++ b/formal/lean/mathlib/ring_theory/ideal/quotient.lean @@ -236,6 +236,11 @@ lemma quot_equiv_of_eq_mk {R : Type*} [comm_ring R] {I J : ideal R} (h : I = J) quot_equiv_of_eq h (ideal.quotient.mk I x) = ideal.quotient.mk J x := rfl +@[simp] +lemma quot_equiv_of_eq_symm {R : Type*} [comm_ring R] {I J : ideal R} (h : I = J) : + (ideal.quot_equiv_of_eq h).symm = ideal.quot_equiv_of_eq h.symm := +by ext; refl + section pi variables (ι : Type v) diff --git a/formal/lean/mathlib/ring_theory/integral_closure.lean b/formal/lean/mathlib/ring_theory/integral_closure.lean index 9bbae86e7bd233bb5a5730607e62b9715afb0cda..013dc970075cfb70308816fee0d04351ea8edbf4 100644 --- a/formal/lean/mathlib/ring_theory/integral_closure.lean +++ b/formal/lean/mathlib/ring_theory/integral_closure.lean @@ -219,7 +219,7 @@ begin have hyS : ∀ {p}, p ∈ y → p ∈ S := λ p hp, show p ∈ S.to_submodule, by { rw ← hy, exact subset_span hp }, -- Now `S` is a subalgebra so the product of two elements of `y` is also in `S`. - have : ∀ (jk : (↑(y.product y) : set (A × A))), jk.1.1 * jk.1.2 ∈ S.to_submodule := + have : ∀ (jk : (↑(y ×ˢ y) : set (A × A))), jk.1.1 * jk.1.2 ∈ S.to_submodule := λ jk, S.mul_mem (hyS (finset.mem_product.1 jk.2).1) (hyS (finset.mem_product.1 jk.2).2), rw [← hy, ← set.image_id ↑y] at this, simp only [finsupp.mem_span_image_iff_total] at this, -- Say `yᵢyⱼ = ∑rᵢⱼₖ yₖ` @@ -286,6 +286,51 @@ begin exact subalgebra.smul_mem _ (algebra.subset_adjoin $ hlx1 hr) _ end +variables {f} + +lemma ring_hom.finite.to_is_integral (h : f.finite) : f.is_integral := +by { letI := f.to_algebra, exact λ x, is_integral_of_mem_of_fg ⊤ h.1 _ trivial } + +alias ring_hom.finite.to_is_integral ← ring_hom.is_integral.of_finite + +lemma ring_hom.is_integral.to_finite (h : f.is_integral) (h' : f.finite_type) : f.finite := +begin + letI := f.to_algebra, + unfreezingI { obtain ⟨s, hs⟩ := h' }, + constructor, + change (⊤ : subalgebra R S).to_submodule.fg, + rw ← hs, + exact fg_adjoin_of_finite (set.to_finite _) (λ x _, h x) +end + +alias ring_hom.is_integral.to_finite ← ring_hom.finite.of_is_integral_of_finite_type + +/-- finite = integral + finite type -/ +lemma ring_hom.finite_iff_is_integral_and_finite_type : + f.finite ↔ f.is_integral ∧ f.finite_type := +⟨λ h, ⟨h.to_is_integral, h.to_finite_type⟩, λ ⟨h, h'⟩, h.to_finite h'⟩ + +lemma algebra.is_integral.finite (h : algebra.is_integral R A) [h' : algebra.finite_type R A] : + module.finite R A := +begin + have := h.to_finite + (by { delta ring_hom.finite_type, convert h', ext, exact (algebra.smul_def _ _).symm }), + delta ring_hom.finite at this, convert this, ext, exact algebra.smul_def _ _, +end + +lemma algebra.is_integral.of_finite [h : module.finite R A] : algebra.is_integral R A := +begin + apply ring_hom.finite.to_is_integral, + delta ring_hom.finite, convert h, ext, exact (algebra.smul_def _ _).symm, +end + +/-- finite = integral + finite type -/ +lemma algebra.finite_iff_is_integral_and_finite_type : + module.finite R A ↔ algebra.is_integral R A ∧ algebra.finite_type R A := +⟨λ h, by exactI ⟨algebra.is_integral.of_finite, infer_instance⟩, λ ⟨h, h'⟩, by exactI h.finite⟩ + +variables (f) + lemma ring_hom.is_integral_of_mem_closure {x y z : S} (hx : f.is_integral_elem x) (hy : f.is_integral_elem y) (hz : z ∈ subring.closure ({x, y} : set S)) : @@ -884,7 +929,7 @@ begin haveI : is_noetherian R A := is_noetherian_of_fg_of_noetherian A.to_submodule (fg_adjoin_singleton_of_integral x (H x)), haveI : module.finite R A := module.is_noetherian.finite R A, - obtain ⟨y, hy⟩ := linear_map.surjective_of_injective (@lmul_left_injective R A _ _ _ _ + obtain ⟨y, hy⟩ := linear_map.surjective_of_injective (@linear_map.mul_left_injective R A _ _ _ _ ⟨x, subset_adjoin (set.mem_singleton x)⟩ (λ h, hx (subtype.ext_iff.mp h))) 1, exact ⟨y, subtype.ext_iff.mp hy⟩, end diff --git a/formal/lean/mathlib/ring_theory/is_tensor_product.lean b/formal/lean/mathlib/ring_theory/is_tensor_product.lean index d79925faff7465df344381ceeab76fdcdd9174a1..e2d58cab89499ba6b69c063ec6fe2ad1dcb08911 100644 --- a/formal/lean/mathlib/ring_theory/is_tensor_product.lean +++ b/formal/lean/mathlib/ring_theory/is_tensor_product.lean @@ -5,6 +5,8 @@ Authors: Andrew Yang -/ import ring_theory.tensor_product +import algebra.module.ulift +import logic.equiv.transfer_instance /-! # The characteristice predicate of tensor product @@ -117,7 +119,8 @@ end is_tensor_product section is_base_change -variables {R M N : Type*} (S : Type*) [add_comm_monoid M] [add_comm_monoid N] [comm_ring R] +variables {R : Type*} {M : Type v₁} {N : Type v₂} (S : Type v₃) +variables [add_comm_monoid M] [add_comm_monoid N] [comm_ring R] variables [comm_ring S] [algebra R S] [module R M] [module R N] [module S N] [is_scalar_tower R S N] variables (f : M →ₗ[R] N) @@ -131,7 +134,11 @@ def is_base_change : Prop := is_tensor_product variables {S f} (h : is_base_change S f) variables {P Q : Type*} [add_comm_monoid P] [module R P] -variables[add_comm_monoid Q] [module R Q] [module S Q] [is_scalar_tower R S Q] +variables [add_comm_monoid Q] [module S Q] + +section + +variables [module R Q] [is_scalar_tower R S Q] /-- Suppose `f : M →ₗ[R] N` is the base change of `M` along `R → S`. Then any `R`-linear map from `M` to an `S`-module factors thorugh `f`. -/ @@ -161,9 +168,36 @@ end lemma is_base_change.lift_comp (g : M →ₗ[R] Q) : ((h.lift g).restrict_scalars R).comp f = g := linear_map.ext (h.lift_eq g) +end +include h + +@[elab_as_eliminator] +lemma is_base_change.induction_on (x : N) (P : N → Prop) + (h₁ : P 0) + (h₂ : ∀ m : M, P (f m)) + (h₃ : ∀ (s : S) n, P n → P (s • n)) + (h₄ : ∀ n₁ n₂, P n₁ → P n₂ → P (n₁ + n₂)) : P x := +h.induction_on x h₁ (λ s y, h₃ _ _ (h₂ _)) h₄ + +lemma is_base_change.alg_hom_ext (g₁ g₂ : N →ₗ[S] Q) (e : ∀ x, g₁ (f x) = g₂ (f x)) : + g₁ = g₂ := +begin + ext x, + apply h.induction_on x, + { rw [map_zero, map_zero] }, + { assumption }, + { intros s n e', rw [g₁.map_smul, g₂.map_smul, e'] }, + { intros x y e₁ e₂, rw [map_add, map_add, e₁, e₂] } +end + +lemma is_base_change.alg_hom_ext' [module R Q] [is_scalar_tower R S Q] (g₁ g₂ : N →ₗ[S] Q) + (e : (g₁.restrict_scalars R).comp f = (g₂.restrict_scalars R).comp f) : + g₁ = g₂ := +h.alg_hom_ext g₁ g₂ (linear_map.congr_fun e) + variables (R M N S) -omit f +omit h f lemma tensor_product.is_base_change : is_base_change S (tensor_product.mk R S M 1) := begin @@ -188,4 +222,144 @@ tensor_product.lift.tmul s m lemma is_base_change.equiv_symm_apply (m : M) : h.equiv.symm (f m) = 1 ⊗ₜ m := by rw [h.equiv.symm_apply_eq, h.equiv_tmul, one_smul] + +variable (f) + +lemma is_base_change.of_lift_unique + (h : ∀ (Q : Type (max v₁ v₂ v₃)) [add_comm_monoid Q], by exactI ∀ [module R Q] [module S Q], + by exactI ∀ [is_scalar_tower R S Q], by exactI ∀ (g : M →ₗ[R] Q), + ∃! (g' : N →ₗ[S] Q), (g'.restrict_scalars R).comp f = g) : is_base_change S f := +begin + delta is_base_change is_tensor_product, + obtain ⟨g, hg, hg'⟩ := h (ulift.{v₂} $ S ⊗[R] M) + (ulift.module_equiv.symm.to_linear_map.comp $ tensor_product.mk R S M 1), + let f' : S ⊗[R] M →ₗ[R] N := _, change function.bijective f', + let f'' : S ⊗[R] M →ₗ[S] N, + { refine { map_smul' := λ r x, _, ..f' }, + apply tensor_product.induction_on x, + { simp only [map_zero, smul_zero, linear_map.to_fun_eq_coe] }, + { intros x y, + simp only [algebra.of_id_apply, algebra.id.smul_eq_mul, + alg_hom.to_linear_map_apply, linear_map.mul_apply, tensor_product.lift.tmul', + linear_map.smul_apply, ring_hom.id_apply, module.algebra_map_End_apply, f', + _root_.map_mul, tensor_product.smul_tmul', linear_map.coe_restrict_scalars_eq_coe, + linear_map.flip_apply] }, + { intros x y hx hy, dsimp at hx hy ⊢, simp only [hx, hy, smul_add, map_add] } }, + change function.bijective f'', + split, + { apply function.has_left_inverse.injective, + refine ⟨ulift.module_equiv.to_linear_map.comp g, λ x, _⟩, + apply tensor_product.induction_on x, + { simp only [map_zero] }, + { intros x y, + have := (congr_arg (λ a, x • a) (linear_map.congr_fun hg y)).trans + (ulift.module_equiv.symm.map_smul x _).symm, + apply (ulift.module_equiv : ulift.{v₂} (S ⊗ M) ≃ₗ[S] S ⊗ M) + .to_equiv.apply_eq_iff_eq_symm_apply.mpr, + any_goals { apply_instance }, + simpa only [algebra.of_id_apply, smul_tmul', algebra.id.smul_eq_mul, lift.tmul', + linear_map.coe_restrict_scalars_eq_coe, linear_map.flip_apply, alg_hom.to_linear_map_apply, + module.algebra_map_End_apply, linear_map.smul_apply, linear_map.coe_mk, + linear_map.map_smulₛₗ, mk_apply, mul_one] using this }, + { intros x y hx hy, simp only [map_add, hx, hy] } }, + { apply function.has_right_inverse.surjective, + refine ⟨ulift.module_equiv.to_linear_map.comp g, λ x, _⟩, + obtain ⟨g', hg₁, hg₂⟩ := h (ulift.{max v₁ v₃} N) (ulift.module_equiv.symm.to_linear_map.comp f), + have : g' = ulift.module_equiv.symm.to_linear_map := by { refine (hg₂ _ _).symm, refl }, + subst this, + apply (ulift.module_equiv : ulift.{max v₁ v₃} N ≃ₗ[S] N).symm.injective, + simp_rw [← linear_equiv.coe_to_linear_map, ← linear_map.comp_apply], + congr' 1, + apply hg₂, + ext y, + have := linear_map.congr_fun hg y, + dsimp [ulift.module_equiv] at this ⊢, + rw this, + simp only [lift.tmul, linear_map.coe_restrict_scalars_eq_coe, linear_map.flip_apply, + alg_hom.to_linear_map_apply, _root_.map_one, linear_map.one_apply] } +end + +variable {f} + +lemma is_base_change.iff_lift_unique : + is_base_change S f ↔ + ∀ (Q : Type (max v₁ v₂ v₃)) [add_comm_monoid Q], by exactI ∀ [module R Q] [module S Q], + by exactI ∀ [is_scalar_tower R S Q], by exactI ∀ (g : M →ₗ[R] Q), + ∃! (g' : N →ₗ[S] Q), (g'.restrict_scalars R).comp f = g := +⟨λ h, by { introsI, + exact ⟨h.lift g, h.lift_comp g, λ g' e, h.alg_hom_ext' _ _ (e.trans (h.lift_comp g).symm)⟩ }, + is_base_change.of_lift_unique f⟩ + +lemma is_base_change.of_equiv (e : M ≃ₗ[R] N) : is_base_change R e.to_linear_map := +begin + apply is_base_change.of_lift_unique, + introsI Q I₁ I₂ I₃ I₄ g, + have : I₂ = I₃, + { ext r q, + rw [← one_smul R q, smul_smul, ← smul_assoc, smul_eq_mul, mul_one] }, + unfreezingI { cases this }, + refine ⟨g.comp e.symm.to_linear_map, by { ext, simp }, _⟩, + rintros y (rfl : _ = _), + ext, + simp, +end + +variables {T O : Type*} [comm_ring T] [algebra R T] [algebra S T] [is_scalar_tower R S T] +variables [add_comm_monoid O] [module R O] [module S O] [module T O] [is_scalar_tower S T O] +variables [is_scalar_tower R S O] [is_scalar_tower R T O] + +lemma is_base_change.comp {f : M →ₗ[R] N} (hf : is_base_change S f) {g : N →ₗ[S] O} + (hg : is_base_change T g) : is_base_change T ((g.restrict_scalars R).comp f) := +begin + apply is_base_change.of_lift_unique, + introsI Q _ _ _ _ i, + letI := module.comp_hom Q (algebra_map S T), + haveI : is_scalar_tower S T Q := ⟨λ x y z, by { rw [algebra.smul_def, mul_smul], refl }⟩, + haveI : is_scalar_tower R S Q, + { refine ⟨λ x y z, _⟩, + change (is_scalar_tower.to_alg_hom R S T) (x • y) • z = x • (algebra_map S T y • z), + rw [alg_hom.map_smul, smul_assoc], + refl }, + refine ⟨hg.lift (hf.lift i), by { ext, simp [is_base_change.lift_eq] }, _⟩, + rintros g' (e : _ = _), + refine hg.alg_hom_ext' _ _ (hf.alg_hom_ext' _ _ _), + rw [is_base_change.lift_comp, is_base_change.lift_comp, ← e], + ext, + refl +end + +variables {R' S' : Type*} [comm_ring R'] [comm_ring S'] +variables [algebra R R'] [algebra S S'] [algebra R' S'] [algebra R S'] +variables [is_scalar_tower R R' S'] [is_scalar_tower R S S'] + +lemma is_base_change.symm + (h : is_base_change S (is_scalar_tower.to_alg_hom R R' S').to_linear_map) : + is_base_change R' (is_scalar_tower.to_alg_hom R S S').to_linear_map := +begin + letI := (algebra.tensor_product.include_right : R' →ₐ[R] S ⊗ R').to_ring_hom.to_algebra, + let e : R' ⊗[R] S ≃ₗ[R'] S', + { refine { map_smul' := _, ..((tensor_product.comm R R' S).trans $ h.equiv.restrict_scalars R) }, + intros r x, + change + h.equiv (tensor_product.comm R R' S (r • x)) = r • h.equiv (tensor_product.comm R R' S x), + apply tensor_product.induction_on x, + { simp only [smul_zero, map_zero] }, + { intros x y, + simp [smul_tmul', algebra.smul_def, ring_hom.algebra_map_to_algebra, h.equiv_tmul], + ring }, + { intros x y hx hy, simp only [map_add, smul_add, hx, hy] } }, + have : (is_scalar_tower.to_alg_hom R S S').to_linear_map + = (e.to_linear_map.restrict_scalars R).comp (tensor_product.mk R R' S 1), + { ext, simp [e, h.equiv_tmul, algebra.smul_def] }, + rw this, + exact (tensor_product.is_base_change R S R').comp (is_base_change.of_equiv e), +end + +variables (R S R' S') + +lemma is_base_change.comm : + is_base_change S (is_scalar_tower.to_alg_hom R R' S').to_linear_map ↔ + is_base_change R' (is_scalar_tower.to_alg_hom R S S').to_linear_map := +⟨is_base_change.symm, is_base_change.symm⟩ + end is_base_change diff --git a/formal/lean/mathlib/ring_theory/localization/cardinality.lean b/formal/lean/mathlib/ring_theory/localization/cardinality.lean index e2e7617bffe01633a30e4492e83884902c165783..84dc2115277249c01dbe2b62595212dafa04edfd 100644 --- a/formal/lean/mathlib/ring_theory/localization/cardinality.lean +++ b/formal/lean/mathlib/ring_theory/localization/cardinality.lean @@ -3,9 +3,8 @@ Copyright (c) 2022 Eric Rodriguez. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Eric Rodriguez -/ -import ring_theory.integral_domain -import ring_theory.localization.basic import set_theory.cardinal.ordinal +import ring_theory.artinian /-! # Cardinality of localizations @@ -23,7 +22,6 @@ submonoid, then your localization is trivial (see `is_localization.unique_of_zer -/ - open_locale cardinal non_zero_divisors universes u v @@ -34,25 +32,12 @@ variables {R : Type u} [comm_ring R] (S : submonoid R) {L : Type u} [comm_ring L [algebra R L] [is_localization S L] include S -/-- Localizing a finite ring can only reduce the amount of elements. -/ -lemma algebra_map_surjective_of_fintype [fintype R] : function.surjective (algebra_map R L) := -begin - classical, - haveI : fintype L := is_localization.fintype' S L, - intro x, - obtain ⟨⟨r, s⟩, h : x * (algebra_map R L) ↑s = (algebra_map R L) r⟩ := is_localization.surj S x, - obtain ⟨n, hn, hp⟩ := - (is_of_fin_order_iff_pow_eq_one _).1 (exists_pow_eq_one (is_localization.map_units L s).unit), - rw [units.ext_iff, units.coe_pow, is_unit.unit_spec, ←nat.succ_pred_eq_of_pos hn, pow_succ] at hp, - exact ⟨r * s ^ (n - 1), by erw [map_mul, map_pow, ←h, mul_assoc, hp, mul_one]⟩ -end - /-- A localization always has cardinality less than or equal to the base ring. -/ lemma card_le : #L ≤ #R := begin classical, casesI fintype_or_infinite R, - { exact cardinal.mk_le_of_surjective (algebra_map_surjective_of_fintype S) }, + { exact cardinal.mk_le_of_surjective (is_artinian_ring.localization_surjective S _) }, erw [←cardinal.mul_eq_self $ cardinal.aleph_0_le_mk R], set f : R × R → L := λ aa, is_localization.mk' _ aa.1 (if h : aa.2 ∈ S then ⟨aa.2, h⟩ else 1), refine @cardinal.mk_le_of_surjective _ _ f (λ a, _), diff --git a/formal/lean/mathlib/ring_theory/localization/integral.lean b/formal/lean/mathlib/ring_theory/localization/integral.lean index a47c23d4295b28f7e94e6762c68fac676f822e04..1c94f56cd59210c15aded1372f3d55a165604c6a 100644 --- a/formal/lean/mathlib/ring_theory/localization/integral.lean +++ b/formal/lean/mathlib/ring_theory/localization/integral.lean @@ -240,6 +240,51 @@ lemma is_integral_localization' {R S : Type*} [comm_ring R] [comm_ring S] (M.le_comap_map : _ ≤ submonoid.comap (f : R →* S) _) : localization M →+* _).is_integral := @is_integral_localization R _ M S _ f.to_algebra _ _ _ _ _ _ _ _ hf +variable (M) + +lemma is_localization.scale_roots_common_denom_mem_lifts (p : Rₘ[X]) + (hp : p.leading_coeff ∈ (algebra_map R Rₘ).range) : + p.scale_roots (algebra_map R Rₘ $ is_localization.common_denom M p.support p.coeff) ∈ + polynomial.lifts (algebra_map R Rₘ) := +begin + rw polynomial.lifts_iff_coeff_lifts, + intro n, + rw [polynomial.coeff_scale_roots], + by_cases h₁ : n ∈ p.support, + by_cases h₂ : n = p.nat_degree, + { rwa [h₂, polynomial.coeff_nat_degree, tsub_self, pow_zero, _root_.mul_one] }, + { have : n + 1 ≤ p.nat_degree := lt_of_le_of_ne (polynomial.le_nat_degree_of_mem_supp _ h₁) h₂, + rw [← tsub_add_cancel_of_le (le_tsub_of_add_le_left this), pow_add, pow_one, mul_comm, + _root_.mul_assoc, ← map_pow], + change _ ∈ (algebra_map R Rₘ).range, + apply mul_mem, + { exact ring_hom.mem_range_self _ _ }, + { rw ← algebra.smul_def, + exact ⟨_, is_localization.map_integer_multiple M p.support p.coeff ⟨n, h₁⟩⟩ } }, + { rw polynomial.not_mem_support_iff at h₁, + rw [h₁, zero_mul], + exact zero_mem (algebra_map R Rₘ).range } +end + +lemma is_integral.exists_multiple_integral_of_is_localization + [algebra Rₘ S] [is_scalar_tower R Rₘ S] (x : S) (hx : is_integral Rₘ x) : + ∃ m : M, is_integral R (m • x) := +begin + cases subsingleton_or_nontrivial Rₘ with _ nontriv; resetI, + { haveI := (algebra_map Rₘ S).codomain_trivial, + exact ⟨1, polynomial.X, polynomial.monic_X, subsingleton.elim _ _⟩ }, + obtain ⟨p, hp₁, hp₂⟩ := hx, + obtain ⟨p', hp'₁, -, hp'₂⟩ := lifts_and_nat_degree_eq_and_monic + (is_localization.scale_roots_common_denom_mem_lifts M p _) _, + { refine ⟨is_localization.common_denom M p.support p.coeff, p', hp'₂, _⟩, + rw [is_scalar_tower.algebra_map_eq R Rₘ S, ← polynomial.eval₂_map, hp'₁, + submonoid.smul_def, algebra.smul_def, is_scalar_tower.algebra_map_apply R Rₘ S], + exact polynomial.scale_roots_eval₂_eq_zero _ hp₂ }, + { rw hp₁.leading_coeff, exact one_mem _ }, + { rwa polynomial.monic_scale_roots_iff }, +end + + end is_integral variables {A K : Type*} [comm_ring A] [is_domain A] diff --git a/formal/lean/mathlib/ring_theory/nilpotent.lean b/formal/lean/mathlib/ring_theory/nilpotent.lean index dc7d78edce0bfd3d796db54b7f195dffb4c07156..345b1912435cc343fe0061f25586a0f975db22c0 100644 --- a/formal/lean/mathlib/ring_theory/nilpotent.lean +++ b/formal/lean/mathlib/ring_theory/nilpotent.lean @@ -159,27 +159,27 @@ ideal.ext $ λ _, is_nilpotent_iff_eq_zero end comm_semiring -namespace algebra +namespace linear_map variables (R) {A : Type v} [comm_semiring R] [semiring A] [algebra R A] -@[simp] lemma is_nilpotent_lmul_left_iff (a : A) : - is_nilpotent (lmul_left R a) ↔ is_nilpotent a := +@[simp] lemma is_nilpotent_mul_left_iff (a : A) : + is_nilpotent (mul_left R a) ↔ is_nilpotent a := begin split; rintros ⟨n, hn⟩; use n; - simp only [lmul_left_eq_zero_iff, pow_lmul_left] at ⊢ hn; + simp only [mul_left_eq_zero_iff, pow_mul_left] at ⊢ hn; exact hn, end -@[simp] lemma is_nilpotent_lmul_right_iff (a : A) : - is_nilpotent (lmul_right R a) ↔ is_nilpotent a := +@[simp] lemma is_nilpotent_mul_right_iff (a : A) : + is_nilpotent (mul_right R a) ↔ is_nilpotent a := begin split; rintros ⟨n, hn⟩; use n; - simp only [lmul_right_eq_zero_iff, pow_lmul_right] at ⊢ hn; + simp only [mul_right_eq_zero_iff, pow_mul_right] at ⊢ hn; exact hn, end -end algebra +end linear_map namespace module.End diff --git a/formal/lean/mathlib/ring_theory/noetherian.lean b/formal/lean/mathlib/ring_theory/noetherian.lean index 25594cff210405d70323d36592646907161571f2..cb08552d3328880f8a7ede4df68e64f24233d1d1 100644 --- a/formal/lean/mathlib/ring_theory/noetherian.lean +++ b/formal/lean/mathlib/ring_theory/noetherian.lean @@ -774,17 +774,15 @@ lemma is_noetherian_ring_iff_ideal_fg (R : Type*) [semiring R] : is_noetherian_ring_iff.trans is_noetherian_def @[priority 80] -- see Note [lower instance priority] -instance is_noetherian_of_fintype (R M) [fintype M] [semiring R] [add_comm_monoid M] [module R M] : +instance is_noetherian_of_finite (R M) [finite M] [semiring R] [add_comm_monoid M] [module R M] : is_noetherian R M := -by letI := classical.dec; exact -⟨assume s, ⟨to_finset s, by rw [set.coe_to_finset, submodule.span_eq]⟩⟩ +⟨λ s, ⟨(s : set M).to_finite.to_finset, by rw [set.finite.coe_to_finset, submodule.span_eq]⟩⟩ /-- Modules over the trivial ring are Noetherian. -/ @[priority 100] -- see Note [lower instance priority] instance is_noetherian_of_subsingleton (R M) [subsingleton R] [semiring R] [add_comm_monoid M] [module R M] : is_noetherian R M := -by haveI := module.subsingleton R M; - exact is_noetherian_of_fintype R M +by { haveI := module.subsingleton R M, exact is_noetherian_of_finite R M } @[priority 100] -- see Note [lower instance priority] instance ring.is_noetherian_of_subsingleton {R} [semiring R] [subsingleton R] : diff --git a/formal/lean/mathlib/ring_theory/norm.lean b/formal/lean/mathlib/ring_theory/norm.lean index 0d94b6619a7e92778baa5b396c3d8dd455455238..7fb06fda724de4e06351d4205fcd7cfe96d77dc4 100644 --- a/formal/lean/mathlib/ring_theory/norm.lean +++ b/formal/lean/mathlib/ring_theory/norm.lean @@ -26,7 +26,7 @@ The current definition is as general as possible and the assumption that we have fields or that the extension is finite is added to the lemmas as needed. We only define the norm for left multiplication (`algebra.left_mul_matrix`, -i.e. `algebra.lmul_left`). +i.e. `linear_map.mul_left`). For now, the definitions assume `S` is commutative, so the choice doesn't matter anyway. @@ -73,7 +73,7 @@ variables {R} -- Can't be a `simp` lemma because it depends on a choice of basis lemma norm_eq_matrix_det [decidable_eq ι] (b : basis ι R S) (s : S) : norm R s = matrix.det (algebra.left_mul_matrix b s) := -by rw [norm_apply, ← linear_map.det_to_matrix b, to_matrix_lmul_eq] +by { rwa [norm_apply, ← linear_map.det_to_matrix b, ← to_matrix_lmul_eq], refl } /-- If `x` is in the base field `K`, then the norm is `x ^ [L : K]`. -/ lemma norm_algebra_map_of_basis (b : basis ι R S) (x : R) : @@ -163,7 +163,7 @@ algebra.norm_eq_zero_iff_of_basis (basis.of_vector_space K L) /-- This is `algebra.norm_eq_zero_iff` composed with `algebra.norm_apply`. -/ @[simp] lemma norm_eq_zero_iff' {K L : Type*} [field K] [comm_ring L] [algebra K L] [is_domain L] - [finite_dimensional K L] {x : L} : linear_map.det (algebra.lmul K L x) = 0 ↔ x = 0 := + [finite_dimensional K L] {x : L} : linear_map.det (linear_map.mul K L x) = 0 ↔ x = 0 := algebra.norm_eq_zero_iff_of_basis (basis.of_vector_space K L) end eq_zero_iff diff --git a/formal/lean/mathlib/ring_theory/perfection.lean b/formal/lean/mathlib/ring_theory/perfection.lean index c4f8b0bb7bb69d571370d19e842512c2598f237a..f2ee9c480787b2da4d1dd10cd11f7f61da697020 100644 --- a/formal/lean/mathlib/ring_theory/perfection.lean +++ b/formal/lean/mathlib/ring_theory/perfection.lean @@ -201,7 +201,7 @@ end perfection /-- A perfection map to a ring of characteristic `p` is a map that is isomorphic to its perfection. -/ -@[nolint has_inhabited_instance] structure perfection_map (p : ℕ) [fact p.prime] +@[nolint has_nonempty_instance] structure perfection_map (p : ℕ) [fact p.prime] {R : Type u₁} [comm_semiring R] [char_p R p] {P : Type u₂} [comm_semiring P] [char_p P p] [perfect_ring P p] (π : P →+* R) : Prop := (injective : ∀ ⦃x y : P⦄, (∀ n, π (pth_root P p ^[n] x) = π (pth_root P p ^[n] y)) → x = y) @@ -321,7 +321,7 @@ variables (p : ℕ) include hv /-- `O/(p)` for `O`, ring of integers of `K`. -/ -@[nolint unused_arguments has_inhabited_instance] def mod_p := +@[nolint unused_arguments has_nonempty_instance] def mod_p := O ⧸ (ideal.span {p} : ideal O) variables [hp : fact p.prime] [hvp : fact (v p ≠ 1)] @@ -431,7 +431,7 @@ end classical end mod_p /-- Perfection of `O/(p)` where `O` is the ring of integers of `K`. -/ -@[nolint has_inhabited_instance] def pre_tilt := +@[nolint has_nonempty_instance] def pre_tilt := ring.perfection (mod_p K v O hv p) p include hp hvp @@ -559,7 +559,7 @@ end pre_tilt /-- The tilt of a field, as defined in Perfectoid Spaces by Peter Scholze, as in [scholze2011perfectoid]. Given a field `K` with valuation `K → ℝ≥0` and ring of integers `O`, this is implemented as the fraction field of the perfection of `O/(p)`. -/ -@[nolint has_inhabited_instance] def tilt := +@[nolint has_nonempty_instance] def tilt := fraction_ring (pre_tilt K v O hv p) namespace tilt diff --git a/formal/lean/mathlib/ring_theory/polynomial/basic.lean b/formal/lean/mathlib/ring_theory/polynomial/basic.lean index affcf102a72bfcdc960b7c7ab906f1f8b6893cb5..dff704c08562ab38c7d9b7a7c5aa861706920fdd 100644 --- a/formal/lean/mathlib/ring_theory/polynomial/basic.lean +++ b/formal/lean/mathlib/ring_theory/polynomial/basic.lean @@ -131,6 +131,17 @@ def degree_lt_equiv (R) [semiring R] (n : ℕ) : degree_lt R n ≃ₗ[R] (fin n { intro h, exact (h (finset.mem_univ _)).elim } end } +@[simp] theorem degree_lt_equiv_eq_zero_iff_eq_zero {n : ℕ} {p : R[X]} (hp : p ∈ degree_lt R n) : + degree_lt_equiv _ _ ⟨p, hp⟩ = 0 ↔ p = 0 := +by rw [linear_equiv.map_eq_zero_iff, submodule.mk_eq_zero] + +theorem eval_eq_sum_degree_lt_equiv {n : ℕ} {p : R[X]} (hp : p ∈ degree_lt R n) (x : R) : + p.eval x = ∑ i, degree_lt_equiv _ _ ⟨p, hp⟩ i * (x ^ (i : ℕ)) := +begin + simp_rw [eval_eq_sum], + exact (sum_fin _ (by simp_rw [zero_mul, forall_const]) (mem_degree_lt.mp hp)).symm +end + /-- The finset of nonzero coefficients of a polynomial. -/ def frange (p : R[X]) : finset R := finset.image (λ n, p.coeff n) p.support diff --git a/formal/lean/mathlib/ring_theory/polynomial/bernstein.lean b/formal/lean/mathlib/ring_theory/polynomial/bernstein.lean index ecac38edff1b240a328f74ce3c45b61ae174581d..3fc5d067c27d408b9eadd7e7bdf5633d643b1507 100644 --- a/formal/lean/mathlib/ring_theory/polynomial/bernstein.lean +++ b/formal/lean/mathlib/ring_theory/polynomial/bernstein.lean @@ -153,7 +153,7 @@ begin { simp [eval_at_0], }, { simp only [derivative_succ, int.coe_nat_eq_zero, mul_eq_zero, function.comp_app, function.iterate_succ, - polynomial.iterate_derivative_sub, polynomial.iterate_derivative_cast_nat_mul, + polynomial.iterate_derivative_sub, polynomial.iterate_derivative_nat_cast_mul, polynomial.eval_mul, polynomial.eval_nat_cast, polynomial.eval_sub], intro h, apply mul_eq_zero_of_right, @@ -180,7 +180,7 @@ begin { have h' : ν ≤ n-1 := le_tsub_of_add_le_right h, simp only [derivative_succ, ih (n-1) h', iterate_derivative_succ_at_0_eq_zero, nat.succ_sub_succ_eq_sub, tsub_zero, sub_zero, - iterate_derivative_sub, iterate_derivative_cast_nat_mul, + iterate_derivative_sub, iterate_derivative_nat_cast_mul, eval_one, eval_mul, eval_add, eval_sub, eval_X, eval_comp, eval_nat_cast, function.comp_app, function.iterate_succ, pochhammer_succ_left], obtain rfl | h'' := ν.eq_zero_or_pos, diff --git a/formal/lean/mathlib/ring_theory/polynomial/chebyshev.lean b/formal/lean/mathlib/ring_theory/polynomial/chebyshev.lean index c9730bd60a07f19577a653f364bd5a38314711db..568cf21e85310cbc913c8ef0d2c36b8164f90bb1 100644 --- a/formal/lean/mathlib/ring_theory/polynomial/chebyshev.lean +++ b/formal/lean/mathlib/ring_theory/polynomial/chebyshev.lean @@ -65,35 +65,17 @@ noncomputable def T : ℕ → R[X] @[simp] lemma T_zero : T R 0 = 1 := rfl @[simp] lemma T_one : T R 1 = X := rfl -lemma T_two : T R 2 = 2 * X ^ 2 - 1 := -by simp only [T, sub_left_inj, sq, mul_assoc] -@[simp] lemma T_add_two (n : ℕ) : - T R (n + 2) = 2 * X * T R (n + 1) - T R n := -by rw T - -lemma T_of_two_le (n : ℕ) (h : 2 ≤ n) : - T R n = 2 * X * T R (n - 1) - T R (n - 2) := +@[simp] lemma T_add_two (n : ℕ) : T R (n + 2) = 2 * X * T R (n + 1) - T R n := by rw T + +lemma T_two : T R 2 = 2 * X ^ 2 - 1 := by simp only [T, sub_left_inj, sq, mul_assoc] + +lemma T_of_two_le (n : ℕ) (h : 2 ≤ n) : T R n = 2 * X * T R (n - 1) - T R (n - 2) := begin obtain ⟨n, rfl⟩ := nat.exists_eq_add_of_le h, rw add_comm, exact T_add_two R n end -variables {R S} - -lemma map_T (f : R →+* S) : - ∀ (n : ℕ), map f (T R n) = T S n -| 0 := by simp only [T_zero, polynomial.map_one] -| 1 := by simp only [T_one, map_X] -| (n + 2) := -begin - simp only [T_add_two, polynomial.map_mul, polynomial.map_sub, map_X, bit0, - polynomial.map_add, polynomial.map_one], - rw [map_T (n + 1), map_T n], -end - -variables (R S) - /-- `U n` is the `n`-th Chebyshev polynomial of the second kind -/ noncomputable def U : ℕ → R[X] | 0 := 1 @@ -102,23 +84,18 @@ noncomputable def U : ℕ → R[X] @[simp] lemma U_zero : U R 0 = 1 := rfl @[simp] lemma U_one : U R 1 = 2 * X := rfl -lemma U_two : U R 2 = 4 * X ^ 2 - 1 := -by { simp only [U], ring, } +@[simp] lemma U_add_two (n : ℕ) : U R (n + 2) = 2 * X * U R (n + 1) - U R n := by rw U -@[simp] lemma U_add_two (n : ℕ) : - U R (n + 2) = 2 * X * U R (n + 1) - U R n := -by rw U +lemma U_two : U R 2 = 4 * X ^ 2 - 1 := by { simp only [U], ring, } -lemma U_of_two_le (n : ℕ) (h : 2 ≤ n) : - U R n = 2 * X * U R (n - 1) - U R (n - 2) := +lemma U_of_two_le (n : ℕ) (h : 2 ≤ n) : U R n = 2 * X * U R (n - 1) - U R (n - 2) := begin obtain ⟨n, rfl⟩ := nat.exists_eq_add_of_le h, rw add_comm, exact U_add_two R n end -lemma U_eq_X_mul_U_add_T : - ∀ (n : ℕ), U R (n+1) = X * U R n + T R (n+1) +lemma U_eq_X_mul_U_add_T : ∀ n : ℕ, U R (n + 1) = X * U R n + T R (n + 1) | 0 := by { simp only [U_zero, U_one, T_one], ring } | 1 := by { simp only [U_one, T_two, U_two], ring } | (n + 2) := @@ -127,13 +104,10 @@ lemma U_eq_X_mul_U_add_T : ... = X * (2 * X * U R (n + 1) - U R n) + (2 * X * T R (n + 2) - T R (n + 1)) : by ring ... = X * U R (n + 2) + T R (n + 2 + 1) : by simp only [U_add_two, T_add_two] -lemma T_eq_U_sub_X_mul_U (n : ℕ) : - T R (n+1) = U R (n+1) - X * U R n := +lemma T_eq_U_sub_X_mul_U (n : ℕ) : T R (n + 1) = U R (n + 1) - X * U R n := by rw [U_eq_X_mul_U_add_T, add_comm (X * U R n), add_sub_cancel] - -lemma T_eq_X_mul_T_sub_pol_U : - ∀ (n : ℕ), T R (n+2) = X * T R (n+1) - (1 - X ^ 2) * U R n +lemma T_eq_X_mul_T_sub_pol_U : ∀ n : ℕ, T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n | 0 := by { simp only [T_one, T_two, U_zero], ring } | 1 := by { simp only [T_add_two, T_zero, T_add_two, U_one, T_one], ring } @@ -152,13 +126,22 @@ by rw [T_eq_X_mul_T_sub_pol_U, ←sub_add, sub_self, zero_add] variables {R S} -@[simp] lemma map_U (f : R →+* S) : - ∀ (n : ℕ), map f (U R n) = U S n +@[simp] lemma map_T (f : R →+* S) : ∀ n : ℕ, map f (T R n) = T S n +| 0 := by simp only [T_zero, polynomial.map_one] +| 1 := by simp only [T_one, map_X] +| (n + 2) := +begin + simp only [T_add_two, polynomial.map_mul, polynomial.map_sub, map_X, bit0, + polynomial.map_add, polynomial.map_one], + rw [map_T (n + 1), map_T n], +end + +@[simp] lemma map_U (f : R →+* S) : ∀ n : ℕ, map f (U R n) = U S n | 0 := by simp only [U_zero, polynomial.map_one] | 1 := begin simp only [U_one, map_X, polynomial.map_mul, polynomial.map_add, polynomial.map_one], - change map f (1+1) * X = 2 * X, + change map f (1 + 1) * X = 2 * X, simpa only [polynomial.map_add, polynomial.map_one] end | (n + 2) := @@ -168,8 +151,7 @@ begin rw [map_U (n + 1), map_U n], end -lemma T_derivative_eq_U : - ∀ (n : ℕ), derivative (T R (n + 1)) = (n + 1) * U R n +lemma T_derivative_eq_U : ∀ n : ℕ, derivative (T R (n + 1)) = (n + 1) * U R n | 0 := by simp only [T_one, U_zero, derivative_X, nat.cast_zero, zero_add, mul_one] | 1 := by { simp only [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, nat.cast_one, nat.cast_two], @@ -188,19 +170,17 @@ lemma T_derivative_eq_U : ... = (↑(n + 2) + 1) * U R (n + 2) : by norm_cast lemma one_sub_X_sq_mul_derivative_T_eq_poly_in_T (n : ℕ) : - (1 - X ^ 2) * (derivative (T R (n+1))) = - (n + 1) * (T R n - X * T R (n+1)) := + (1 - X ^ 2) * (derivative (T R (n + 1))) = (n + 1) * (T R n - X * T R (n+1)) := calc - (1 - X ^ 2) * (derivative (T R (n+1))) = (1 - X ^ 2 ) * ((n + 1) * U R n) : + (1 - X ^ 2) * (derivative (T R (n + 1))) = (1 - X ^ 2) * ((n + 1) * U R n) : by rw T_derivative_eq_U ... = (n + 1) * ((1 - X ^ 2) * U R n) : by ring ... = (n + 1) * (X * T R (n + 1) - (2 * X * T R (n + 1) - T R n)) : by rw [one_sub_X_sq_mul_U_eq_pol_in_T, T_add_two] - ... = (n + 1) * (T R n - X * T R (n+1)) : by ring + ... = (n + 1) * (T R n - X * T R (n + 1)) : by ring lemma add_one_mul_T_eq_poly_in_U (n : ℕ) : - ((n : R[X]) + 1) * T R (n+1) = - X * U R n - (1 - X ^ 2) * derivative ( U R n) := + ((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := begin have h : derivative (T R (n + 2)) = (U R (n + 1) - X * U R n) + X * derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * derivative (U R n), @@ -224,9 +204,7 @@ end variables (R) /-- The product of two Chebyshev polynomials is the sum of two other Chebyshev polynomials. -/ -lemma mul_T : - ∀ m : ℕ, ∀ k, - 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k +lemma mul_T : ∀ m k, 2 * T R m * T R (m + k) = T R (2 * m + k) + T R k | 0 := by simp [two_mul, add_mul] | 1 := by simp [add_comm] | (m + 2) := begin @@ -256,8 +234,7 @@ lemma mul_T : end /-- The `(m * n)`-th Chebyshev polynomial is the composition of the `m`-th and `n`-th -/ -lemma T_mul : - ∀ m : ℕ, ∀ n : ℕ, T R (m * n) = (T R m).comp (T R n) +lemma T_mul : ∀ m n, T R (m * n) = (T R m).comp (T R n) | 0 := by simp | 1 := by simp | (m + 2) := begin diff --git a/formal/lean/mathlib/ring_theory/power_basis.lean b/formal/lean/mathlib/ring_theory/power_basis.lean index 65d72d8a8f8744439e612b5d29fb303f75fbccb6..349cf5064473d411ce011da31f382cdbc1091149 100644 --- a/formal/lean/mathlib/ring_theory/power_basis.lean +++ b/formal/lean/mathlib/ring_theory/power_basis.lean @@ -55,7 +55,7 @@ This is a structure, not a class, since the same algebra can have many power bas For the common case where `S` is defined by adjoining an integral element to `R`, the canonical power basis is given by `{algebra,intermediate_field}.adjoin.power_basis`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure power_basis (R S : Type*) [comm_ring R] [ring S] [algebra R S] := (gen : S) (dim : ℕ) diff --git a/formal/lean/mathlib/ring_theory/roots_of_unity.lean b/formal/lean/mathlib/ring_theory/roots_of_unity.lean index ebf88f4b6a39062aa3e4db9a68b745ac4091ea27..34227ce168ade13463e87461b0f1bd925aefcbdc 100644 --- a/formal/lean/mathlib/ring_theory/roots_of_unity.lean +++ b/formal/lean/mathlib/ring_theory/roots_of_unity.lean @@ -67,8 +67,8 @@ noncomputable theory open polynomial open finset -variables {M N G G₀ R S F : Type*} -variables [comm_monoid M] [comm_monoid N] [comm_group G] [comm_group_with_zero G₀] +variables {M N G R S F : Type*} +variables [comm_monoid M] [comm_monoid N] [division_comm_monoid G] section roots_of_unity @@ -301,7 +301,7 @@ end section comm_monoid -variables {ζ : M} (h : is_primitive_root ζ k) +variables {ζ : M} {f : F} (h : is_primitive_root ζ k) @[nontriviality] lemma of_subsingleton [subsingleton M] (x : M) : is_primitive_root x 1 := ⟨subsingleton.elim _ _, λ _ _, one_dvd _⟩ @@ -439,6 +439,37 @@ begin exact h.dvd_of_pow_eq_one _ hl end +section maps + +open function + +lemma map_of_injective [monoid_hom_class F M N] (h : is_primitive_root ζ k) (hf : injective f) : + is_primitive_root (f ζ) k := +{ pow_eq_one := by rw [←map_pow, h.pow_eq_one, _root_.map_one], + dvd_of_pow_eq_one := begin + rw h.eq_order_of, + intros l hl, + rw [←map_pow, ←map_one f] at hl, + exact order_of_dvd_of_pow_eq_one (hf hl) + end } + +lemma of_map_of_injective [monoid_hom_class F M N] (h : is_primitive_root (f ζ) k) + (hf : injective f) : is_primitive_root ζ k := +{ pow_eq_one := by { apply_fun f, rw [map_pow, _root_.map_one, h.pow_eq_one] }, + dvd_of_pow_eq_one := begin + rw h.eq_order_of, + intros l hl, + apply_fun f at hl, + rw [map_pow, _root_.map_one] at hl, + exact order_of_dvd_of_pow_eq_one hl + end } + +lemma map_iff_of_injective [monoid_hom_class F M N] (hf : injective f) : + is_primitive_root (f ζ) k ↔ is_primitive_root ζ k := +⟨λ h, h.of_map_of_injective hf, λ h, h.map_of_injective hf⟩ + +end maps + end comm_monoid section comm_monoid_with_zero @@ -453,7 +484,7 @@ mt $ λ hn, h.unique (hn.symm ▸ is_primitive_root.zero) end comm_monoid_with_zero -section comm_group +section division_comm_monoid variables {ζ : G} @@ -499,88 +530,7 @@ begin exact hi end -end comm_group - -section comm_group_with_zero - -variables {ζ : G₀} - -lemma zpow_eq_one₀ (h : is_primitive_root ζ k) : ζ ^ (k : ℤ) = 1 := -by { rw zpow_coe_nat, exact h.pow_eq_one } - -lemma zpow_eq_one_iff_dvd₀ (h : is_primitive_root ζ k) (l : ℤ) : - ζ ^ l = 1 ↔ (k : ℤ) ∣ l := -begin - by_cases h0 : 0 ≤ l, - { lift l to ℕ using h0, rw [zpow_coe_nat], norm_cast, exact h.pow_eq_one_iff_dvd l }, - { have : 0 ≤ -l, { simp only [not_le, neg_nonneg] at h0 ⊢, exact le_of_lt h0 }, - lift -l to ℕ using this with l' hl', - rw [← dvd_neg, ← hl'], - norm_cast, - rw [← h.pow_eq_one_iff_dvd, ← inv_inj, ← zpow_neg, ← hl', zpow_coe_nat, inv_one] } -end - -lemma inv' (h : is_primitive_root ζ k) : is_primitive_root ζ⁻¹ k := -{ pow_eq_one := by simp only [h.pow_eq_one, inv_one, eq_self_iff_true, inv_pow], - dvd_of_pow_eq_one := - begin - intros l hl, - apply h.dvd_of_pow_eq_one l, - rw [← inv_inj, ← inv_pow, hl, inv_one] - end } - -@[simp] lemma inv_iff' : is_primitive_root ζ⁻¹ k ↔ is_primitive_root ζ k := -by { refine ⟨_, λ h, inv' h⟩, intro h, rw [← inv_inv ζ], exact inv' h } - -lemma zpow_of_gcd_eq_one₀ (h : is_primitive_root ζ k) (i : ℤ) (hi : i.gcd k = 1) : - is_primitive_root (ζ ^ i) k := -begin - by_cases h0 : 0 ≤ i, - { lift i to ℕ using h0, - rw zpow_coe_nat, - exact h.pow_of_coprime i hi }, - have : 0 ≤ -i, { simp only [not_le, neg_nonneg] at h0 ⊢, exact le_of_lt h0 }, - lift -i to ℕ using this with i' hi', - rw [← inv_iff', ← zpow_neg, ← hi', zpow_coe_nat], - apply h.pow_of_coprime, - rw [int.gcd, ← int.nat_abs_neg, ← hi'] at hi, - exact hi -end - -end comm_group_with_zero - -section comm_semiring - -variables [comm_semiring R] [comm_semiring S] {f : F} {ζ : R} - -open function - -lemma map_of_injective [monoid_hom_class F R S] (h : is_primitive_root ζ k) (hf : injective f) : - is_primitive_root (f ζ) k := -{ pow_eq_one := by rw [←map_pow, h.pow_eq_one, _root_.map_one], - dvd_of_pow_eq_one := begin - rw h.eq_order_of, - intros l hl, - rw [←map_pow, ←map_one f] at hl, - exact order_of_dvd_of_pow_eq_one (hf hl) - end } - -lemma of_map_of_injective [monoid_hom_class F R S] (h : is_primitive_root (f ζ) k) - (hf : injective f) : is_primitive_root ζ k := -{ pow_eq_one := by { apply_fun f, rw [map_pow, _root_.map_one, h.pow_eq_one] }, - dvd_of_pow_eq_one := begin - rw h.eq_order_of, - intros l hl, - apply_fun f at hl, - rw [map_pow, _root_.map_one] at hl, - exact order_of_dvd_of_pow_eq_one hl - end } - -lemma map_iff_of_injective [monoid_hom_class F R S] (hf : injective f) : - is_primitive_root (f ζ) k ↔ is_primitive_root ζ k := -⟨λ h, h.of_map_of_injective hf, λ h, h.map_of_injective hf⟩ - -end comm_semiring +end division_comm_monoid section is_domain @@ -944,21 +894,19 @@ begin { simp only [((is_primitive_root.iff_def μ n).mp h).left, eval₂_one, eval₂_X_pow, eval₂_sub, sub_self] } end -end comm_ring - -variables {n : ℕ} {K : Type*} [field K] {μ : K} (h : is_primitive_root μ n) (hpos : 0 < n) -include n μ h hpos +section is_domain -variables [char_zero K] +variables [is_domain K] [char_zero K] omit hpos + /--The minimal polynomial of a root of unity `μ` divides `X ^ n - 1`. -/ lemma minpoly_dvd_X_pow_sub_one : minpoly ℤ μ ∣ X ^ n - 1 := begin - by_cases hpos : n = 0, { simp [hpos], }, - apply minpoly.gcd_domain_dvd (is_integral h (nat.pos_of_ne_zero hpos)) - (monic_X_pow_sub_C 1 (ne_of_lt (nat.pos_of_ne_zero hpos)).symm).ne_zero, + rcases n.eq_zero_or_pos with rfl | hpos, + { simp }, + apply minpoly.gcd_domain_dvd (is_integral h hpos) (monic_X_pow_sub_C 1 hpos.ne').ne_zero, simp only [((is_primitive_root.iff_def μ n).mp h).left, aeval_X_pow, ring_hom.eq_int_cast, int.cast_one, aeval_one, alg_hom.map_sub, sub_self] end @@ -985,11 +933,10 @@ lemma squarefree_minpoly_mod {p : ℕ} [fact p.prime] (hdiv : ¬ p ∣ n) : /- Let `P` be the minimal polynomial of a root of unity `μ` and `Q` be the minimal polynomial of `μ ^ p`, where `p` is a prime that does not divide `n`. Then `P` divides `expand ℤ p Q`. -/ lemma minpoly_dvd_expand {p : ℕ} (hprime : nat.prime p) (hdiv : ¬ p ∣ n) : - minpoly ℤ μ ∣ - expand ℤ p (minpoly ℤ (μ ^ p)) := + minpoly ℤ μ ∣ expand ℤ p (minpoly ℤ (μ ^ p)) := begin - by_cases hn : n = 0, { simp * at *, }, - have hpos := nat.pos_of_ne_zero hn, + rcases n.eq_zero_or_pos with rfl | hpos, + { simp * at *, }, refine minpoly.gcd_domain_dvd (h.is_integral hpos) _ _, { apply monic.ne_zero, rw [polynomial.monic, leading_coeff, nat_degree_expand, mul_comm, coeff_expand_mul' @@ -1133,6 +1080,10 @@ n.totient = (primitive_roots n K).card : h.card_primitive_roots.symm ... ≤ P_K.nat_degree : card_roots' _ ... ≤ P.nat_degree : nat_degree_map_le _ _ +end is_domain + +end comm_ring + end minpoly section automorphisms diff --git a/formal/lean/mathlib/ring_theory/subring/basic.lean b/formal/lean/mathlib/ring_theory/subring/basic.lean index 5de9780f72b5f8734bc3108df8d315a492bbedd9..865b5371541186179c266b329efb2c930508b348 100644 --- a/formal/lean/mathlib/ring_theory/subring/basic.lean +++ b/formal/lean/mathlib/ring_theory/subring/basic.lean @@ -816,13 +816,11 @@ lemma comap_infi {ι : Sort*} (f : R →+* S) (s : ι → subring S) : /-- Given `subring`s `s`, `t` of rings `R`, `S` respectively, `s.prod t` is `s ×̂ t` as a subring of `R × S`. -/ def prod (s : subring R) (t : subring S) : subring (R × S) := -{ carrier := (s : set R) ×ˢ (t : set S), +{ carrier := s ×ˢ t, .. s.to_submonoid.prod t.to_submonoid, .. s.to_add_subgroup.prod t.to_add_subgroup} @[norm_cast] -lemma coe_prod (s : subring R) (t : subring S) : - (s.prod t : set (R × S)) = (s : set R) ×ˢ (t : set S) := -rfl +lemma coe_prod (s : subring R) (t : subring S) : (s.prod t : set (R × S)) = s ×ˢ t := rfl lemma mem_prod {s : subring R} {t : subring S} {p : R × S} : p ∈ s.prod t ↔ p.1 ∈ s ∧ p.2 ∈ t := iff.rfl diff --git a/formal/lean/mathlib/ring_theory/subsemiring/basic.lean b/formal/lean/mathlib/ring_theory/subsemiring/basic.lean index b39772122725f6f0908fdf148a9c482ed074b6e3..e693a75715df0ab47bd873c2b9dac76d0ab4e6f6 100644 --- a/formal/lean/mathlib/ring_theory/subsemiring/basic.lean +++ b/formal/lean/mathlib/ring_theory/subsemiring/basic.lean @@ -760,12 +760,11 @@ lemma comap_infi {ι : Sort*} (f : R →+* S) (s : ι → subsemiring S) : /-- Given `subsemiring`s `s`, `t` of semirings `R`, `S` respectively, `s.prod t` is `s × t` as a subsemiring of `R × S`. -/ def prod (s : subsemiring R) (t : subsemiring S) : subsemiring (R × S) := -{ carrier := (s : set R) ×ˢ (t : set S), +{ carrier := s ×ˢ t, .. s.to_submonoid.prod t.to_submonoid, .. s.to_add_submonoid.prod t.to_add_submonoid} @[norm_cast] -lemma coe_prod (s : subsemiring R) (t : subsemiring S) : - (s.prod t : set (R × S)) = (s : set R) ×ˢ (t : set S) := +lemma coe_prod (s : subsemiring R) (t : subsemiring S) : (s.prod t : set (R × S)) = s ×ˢ t := rfl lemma mem_prod {s : subsemiring R} {t : subsemiring S} {p : R × S} : diff --git a/formal/lean/mathlib/ring_theory/tensor_product.lean b/formal/lean/mathlib/ring_theory/tensor_product.lean index 23ab3776198249c559dfefc307bff6266157fd0b..f769f1c12deeb9dc77c71ce1006c77f49cbf9ffe 100644 --- a/formal/lean/mathlib/ring_theory/tensor_product.lean +++ b/formal/lean/mathlib/ring_theory/tensor_product.lean @@ -200,8 +200,8 @@ def base_change (f : M →ₗ[R] N) : A ⊗[R] M →ₗ[A] A ⊗[R] N := { to_fun := f.ltensor A, map_add' := (f.ltensor A).map_add, map_smul' := λ a x, - show (f.ltensor A) (rtensor M (algebra.lmul R A a) x) = - (rtensor N ((algebra.lmul R A) a)) ((ltensor A f) x), + show (f.ltensor A) (rtensor M (linear_map.mul R A a) x) = + (rtensor N ((linear_map.mul R A) a)) ((ltensor A f) x), by { rw [← comp_apply, ← comp_apply], simp only [ltensor_comp_rtensor, rtensor_comp_ltensor] } } @@ -270,7 +270,7 @@ for a fixed pure tensor in the first argument, as an `R`-linear map. -/ def mul_aux (a₁ : A) (b₁ : B) : (A ⊗[R] B) →ₗ[R] (A ⊗[R] B) := -tensor_product.map (lmul_left R a₁) (lmul_left R b₁) +tensor_product.map (linear_map.mul_left R a₁) (linear_map.mul_left R b₁) @[simp] lemma mul_aux_apply (a₁ a₂ : A) (b₁ b₂ : B) : @@ -445,6 +445,12 @@ def include_right : B →ₐ[R] A ⊗[R] B := @[simp] lemma include_right_apply (b : B) : (include_right : B →ₐ[R] A ⊗[R] B) b = 1 ⊗ₜ b := rfl +lemma include_left_comp_algebra_map {R S T : Type*} [comm_ring R] [comm_ring S] [comm_ring T] + [algebra R S] [algebra R T] : + (include_left.to_ring_hom.comp (algebra_map R S) : R →+* S ⊗[R] T) = + include_right.to_ring_hom.comp (algebra_map R T) := +by { ext, simp } + end semiring section ring @@ -749,17 +755,17 @@ variables (f : A →ₐ[R] S) (g : B →ₐ[R] S) variables (R) -/-- `algebra.lmul'` is an alg_hom on commutative rings. -/ +/-- `linear_map.mul'` is an alg_hom on commutative rings. -/ def lmul' : S ⊗[R] S →ₐ[R] S := -alg_hom_of_linear_map_tensor_product (algebra.lmul' R) - (λ a₁ a₂ b₁ b₂, by simp only [algebra.lmul'_apply, mul_mul_mul_comm]) - (λ r, by simp only [algebra.lmul'_apply, _root_.mul_one]) +alg_hom_of_linear_map_tensor_product (linear_map.mul' R S) + (λ a₁ a₂ b₁ b₂, by simp only [linear_map.mul'_apply, mul_mul_mul_comm]) + (λ r, by simp only [linear_map.mul'_apply, _root_.mul_one]) variables {R} -lemma lmul'_to_linear_map : (lmul' R : _ →ₐ[R] S).to_linear_map = algebra.lmul' R := rfl +lemma lmul'_to_linear_map : (lmul' R : _ →ₐ[R] S).to_linear_map = linear_map.mul' R S := rfl -@[simp] lemma lmul'_apply_tmul (a b : S) : lmul' R (a ⊗ₜ[R] b) = a * b := lmul'_apply +@[simp] lemma lmul'_apply_tmul (a b : S) : lmul' R (a ⊗ₜ[R] b) = a * b := linear_map.mul'_apply @[simp] lemma lmul'_comp_include_left : (lmul' R : _ →ₐ[R] S).comp include_left = alg_hom.id R S := @@ -793,7 +799,6 @@ by rw [product_map, alg_hom.range_comp, map_range, map_sup, ←alg_hom.range_com lmul'_comp_include_right, alg_hom.id_comp, alg_hom.id_comp] end - section variables {R A A' B S : Type*} @@ -808,7 +813,50 @@ and `g : B →ₐ[R] S` is an `A`-algebra homomorphism. -/ ..(product_map (f.restrict_scalars R) g).to_ring_hom } end +section basis + +variables {k : Type*} [comm_ring k] (R : Type*) [ring R] [algebra k R] {M : Type*} + [add_comm_monoid M] [module k M] {ι : Type*} (b : basis ι k M) + +/-- Given a `k`-algebra `R` and a `k`-basis of `M,` this is a `k`-linear isomorphism +`R ⊗[k] M ≃ (ι →₀ R)` (which is in fact `R`-linear). -/ +noncomputable def basis_aux : R ⊗[k] M ≃ₗ[k] (ι →₀ R) := +(_root_.tensor_product.congr (finsupp.linear_equiv.finsupp_unique k R punit).symm b.repr) ≪≫ₗ + (finsupp_tensor_finsupp k R k punit ι).trans (finsupp.lcongr (equiv.unique_prod ι punit) + (_root_.tensor_product.rid k R)) + +variables {R} + +lemma basis_aux_tmul (r : R) (m : M) : + basis_aux R b (r ⊗ₜ m) = r • (finsupp.map_range (algebra_map k R) + (map_zero _) (b.repr m)) := +begin + ext, + simp [basis_aux, ←algebra.commutes, algebra.smul_def], +end + +lemma basis_aux_map_smul (r : R) (x : R ⊗[k] M) : + basis_aux R b (r • x) = r • basis_aux R b x := +tensor_product.induction_on x (by simp) (λ x y, by simp only [tensor_product.smul_tmul', + basis_aux_tmul, smul_assoc]) (λ x y hx hy, by simp [hx, hy]) +variables (R) + +/-- Given a `k`-algebra `R`, this is the `R`-basis of `R ⊗[k] M` induced by a `k`-basis of `M`. -/ +noncomputable def basis : basis ι R (R ⊗[k] M) := +{ repr := { map_smul' := basis_aux_map_smul b, .. basis_aux R b } } + +variables {R} + +@[simp] lemma basis_repr_tmul (r : R) (m : M) : + (basis R b).repr (r ⊗ₜ m) = r • (finsupp.map_range (algebra_map k R) (map_zero _) (b.repr m)) := +basis_aux_tmul _ _ _ + +@[simp] lemma basis_repr_symm_apply (r : R) (i : ι) : + (basis R b).repr.symm (finsupp.single i r) = r ⊗ₜ b.repr.symm (finsupp.single i 1) := +by simp [basis, equiv.unique_prod_symm_apply, basis_aux] + +end basis end tensor_product end algebra @@ -845,3 +893,75 @@ begin rw [←E1.range_val, ←E2.range_val, ←algebra.tensor_product.product_map_range], exact (algebra.tensor_product.product_map E1.val E2.val).to_linear_map.finite_dimensional_range, end + +namespace tensor_product.algebra + +variables {R A B M : Type*} +variables [comm_semiring R] [add_comm_monoid M] [module R M] +variables [semiring A] [semiring B] [module A M] [module B M] +variables [algebra R A] [algebra R B] +variables [is_scalar_tower R A M] [is_scalar_tower R B M] + +/-- An auxiliary definition, used for constructing the `module (A ⊗[R] B) M` in +`tensor_product.algebra.module` below. -/ +def module_aux : A ⊗[R] B →ₗ[R] M →ₗ[R] M := +tensor_product.lift +{ to_fun := λ a, a • (algebra.lsmul R M : B →ₐ[R] module.End R M).to_linear_map, + map_add' := λ r t, by { ext, simp only [add_smul, linear_map.add_apply] }, + map_smul' := λ n r, by { ext, simp only [ring_hom.id_apply, linear_map.smul_apply, smul_assoc] } } + +lemma module_aux_apply (a : A) (b : B) (m : M) : + module_aux (a ⊗ₜ[R] b) m = a • b • m := +by simp [module_aux] + +variables [smul_comm_class A B M] + +/-- If `M` is a representation of two different `R`-algebras `A` and `B` whose actions commute, +then it is a representation the `R`-algebra `A ⊗[R] B`. + +An important example arises from a semiring `S`; allowing `S` to act on itself via left and right +multiplication, the roles of `R`, `A`, `B`, `M` are played by `ℕ`, `S`, `Sᵐᵒᵖ`, `S`. This example +is important because a submodule of `S` as a `module` over `S ⊗[ℕ] Sᵐᵒᵖ` is a two-sided ideal. + +NB: This is not an instance because in the case `B = A` and `M = A ⊗[R] A` we would have a diamond +of `smul` actions. Furthermore, this would not be a mere definitional diamond but a true +mathematical diamond in which `A ⊗[R] A` had two distinct scalar actions on itself: one from its +multiplication, and one from this would-be instance. Arguably we could live with this but in any +case the real fix is to address the ambiguity in notation, probably along the lines outlined here: +https://leanprover.zulipchat.com/#narrow/stream/144837-PR-reviews/topic/.234773.20base.20change/near/240929258 +-/ +protected def module : module (A ⊗[R] B) M := +{ smul := λ x m, module_aux x m, + zero_smul := λ m, by simp only [map_zero, linear_map.zero_apply], + smul_zero := λ x, by simp only [map_zero], + smul_add := λ x m₁ m₂, by simp only [map_add], + add_smul := λ x y m, by simp only [map_add, linear_map.add_apply], + one_smul := λ m, by simp only [module_aux_apply, algebra.tensor_product.one_def, one_smul], + mul_smul := λ x y m, + begin + apply tensor_product.induction_on x; + apply tensor_product.induction_on y, + { simp only [mul_zero, map_zero, linear_map.zero_apply], }, + { intros a b, simp only [zero_mul, map_zero, linear_map.zero_apply], }, + { intros z w hz hw, simp only [zero_mul, map_zero, linear_map.zero_apply], }, + { intros a b, simp only [mul_zero, map_zero, linear_map.zero_apply], }, + { intros a₁ b₁ a₂ b₂, + simp only [module_aux_apply, mul_smul, smul_comm a₁ b₂, algebra.tensor_product.tmul_mul_tmul, + linear_map.mul_apply], }, + { intros z w hz hw a b, + simp only at hz hw, + simp only [mul_add, hz, hw, map_add, linear_map.add_apply], }, + { intros z w hz hw, simp only [mul_zero, map_zero, linear_map.zero_apply], }, + { intros a b z w hz hw, + simp only at hz hw, + simp only [map_add, add_mul, linear_map.add_apply, hz, hw], }, + { intros u v hu hv z w hz hw, + simp only at hz hw, + simp only [add_mul, hz, hw, map_add, linear_map.add_apply], }, + end } + +local attribute [instance] tensor_product.algebra.module + +lemma smul_def (a : A) (b : B) (m : M) : (a ⊗ₜ[R] b) • m = a • b • m := module_aux_apply a b m + +end tensor_product.algebra diff --git a/formal/lean/mathlib/ring_theory/trace.lean b/formal/lean/mathlib/ring_theory/trace.lean index fc9b832baf97a8dcaf922cc16f96f8ab74bab835..e552de7478814a0801a463caf486f45b3ecfda26 100644 --- a/formal/lean/mathlib/ring_theory/trace.lean +++ b/formal/lean/mathlib/ring_theory/trace.lean @@ -50,7 +50,7 @@ The definition is as general as possible and the assumption that we have fields or that the extension is finite is added to the lemmas as needed. We only define the trace for left multiplication (`algebra.left_mul_matrix`, -i.e. `algebra.lmul_left`). +i.e. `linear_map.mul_left`). For now, the definitions assume `S` is commutative, so the choice doesn't matter anyway. ## References @@ -101,7 +101,7 @@ variables {R} -- Can't be a `simp` lemma because it depends on a choice of basis lemma trace_eq_matrix_trace [decidable_eq ι] (b : basis ι R S) (s : S) : trace R S s = matrix.trace (algebra.left_mul_matrix b s) := -by rw [trace_apply, linear_map.trace_eq_matrix_trace _ b, to_matrix_lmul_eq] +by { rw [trace_apply, linear_map.trace_eq_matrix_trace _ b, ←to_matrix_lmul_eq], refl } /-- If `x` is in the base field `K`, then the trace is `[L : K] * x`. -/ lemma trace_algebra_map_of_basis (x : R) : @@ -111,7 +111,7 @@ begin rw [trace_apply, linear_map.trace_eq_matrix_trace R b, matrix.trace], convert finset.sum_const _, ext i, - simp, + simp [-coe_lmul_eq_mul], end omit b diff --git a/formal/lean/mathlib/ring_theory/valuation/basic.lean b/formal/lean/mathlib/ring_theory/valuation/basic.lean index 0174b62776a02f9c1c4ecb7af3184054148fb6f5..172abda537bf445a869b3fc7b45ae71ab3e660d5 100644 --- a/formal/lean/mathlib/ring_theory/valuation/basic.lean +++ b/formal/lean/mathlib/ring_theory/valuation/basic.lean @@ -75,7 +75,7 @@ variables (F R) (Γ₀ : Type*) [linear_ordered_comm_monoid_with_zero Γ₀] [ri /-- The type of `Γ₀`-valued valuations on `R`. When you extend this structure, make sure to extend `valuation_class`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure valuation extends R →*₀ Γ₀ := (map_add_le_max' : ∀ x y, to_fun (x + y) ≤ max (to_fun x) (to_fun y)) @@ -580,7 +580,7 @@ section add_monoid variables (R) [ring R] (Γ₀ : Type*) [linear_ordered_add_comm_monoid_with_top Γ₀] /-- The type of `Γ₀`-valued additive valuations on `R`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def add_valuation := valuation R (multiplicative Γ₀ᵒᵈ) end add_monoid diff --git a/formal/lean/mathlib/ring_theory/valuation/valuation_subring.lean b/formal/lean/mathlib/ring_theory/valuation/valuation_subring.lean index f26a3abd9eb69afec7f0201d5892f90c9e419657..3a87ae6fba22e307fafbfb274c9e215a7252e8b8 100644 --- a/formal/lean/mathlib/ring_theory/valuation/valuation_subring.lean +++ b/formal/lean/mathlib/ring_theory/valuation/valuation_subring.lean @@ -1,7 +1,7 @@ /- Copyright (c) 2022 Adam Topaz. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. -Authors: Adam Topaz, Junyan Xu +Authors: Adam Topaz, Junyan Xu, Jack McKoen -/ import ring_theory.valuation.valuation_ring import ring_theory.localization.as_subring @@ -23,7 +23,7 @@ noncomputable theory variables (K : Type*) [field K] /-- A valuation subring of a field `K` is a subring `A` such that for every `x : K`, -either `x ∈ A` or `x⁻¹ ∈ K`. -/ +either `x ∈ A` or `x⁻¹ ∈ A`. -/ structure valuation_subring extends subring K := (mem_or_inv_mem' : ∀ x : K, x ∈ carrier ∨ x⁻¹ ∈ carrier) @@ -367,21 +367,7 @@ section unit_group def unit_group : subgroup Kˣ := (A.valuation.to_monoid_with_zero_hom.to_monoid_hom.comp (units.coe_hom K)).ker -lemma mem_unit_group_iff (x : Kˣ) : x ∈ A.unit_group ↔ A.valuation x = 1 := iff.refl _ - -lemma unit_group_injective : function.injective (unit_group : valuation_subring K → subgroup _) := -λ A B h, begin - rw [← A.valuation_subring_valuation, ← B.valuation_subring_valuation, - ← valuation.is_equiv_iff_valuation_subring, valuation.is_equiv_iff_val_eq_one], - rw set_like.ext_iff at h, - intros x, - by_cases hx : x = 0, { simp only [hx, valuation.map_zero, zero_ne_one] }, - exact h (units.mk0 x hx) -end - -lemma eq_iff_unit_group {A B : valuation_subring K} : - A = B ↔ A.unit_group = B.unit_group := -unit_group_injective.eq_iff.symm +lemma mem_unit_group_iff (x : Kˣ) : x ∈ A.unit_group ↔ A.valuation x = 1 := iff.rfl /-- For a valuation subring `A`, `A.unit_group` agrees with the units of `A`. -/ def unit_group_mul_equiv : A.unit_group ≃* Aˣ := @@ -403,6 +389,7 @@ lemma coe_unit_group_mul_equiv_apply (a : A.unit_group) : lemma coe_unit_group_mul_equiv_symm_apply (a : Aˣ) : (A.unit_group_mul_equiv.symm a : K) = a := rfl + lemma unit_group_le_unit_group {A B : valuation_subring K} : A.unit_group ≤ B.unit_group ↔ A ≤ B := begin @@ -424,6 +411,13 @@ begin simpa using hx } end +lemma unit_group_injective : function.injective (unit_group : valuation_subring K → subgroup _) := +λ A B h, by { simpa only [le_antisymm_iff, unit_group_le_unit_group] using h} + +lemma eq_iff_unit_group {A B : valuation_subring K} : + A = B ↔ A.unit_group = B.unit_group := +unit_group_injective.eq_iff.symm + /-- The map on valuation subrings to their unit groups is an order embedding. -/ def unit_group_order_embedding : valuation_subring K ↪o subgroup Kˣ := { to_fun := λ A, A.unit_group, @@ -442,18 +436,7 @@ def nonunits : subsemigroup K := { carrier := { x | A.valuation x < 1 }, mul_mem' := λ a b ha hb, (mul_lt_mul₀ ha hb).trans_eq $ mul_one _ } -lemma mem_nonunits_iff {x : K} : x ∈ A.nonunits ↔ A.valuation x < 1 := iff.refl _ - -lemma nonunits_injective : - function.injective (nonunits : valuation_subring K → subsemigroup _) := -λ A B h, begin - rw [← A.valuation_subring_valuation, ← B.valuation_subring_valuation, - ← valuation.is_equiv_iff_valuation_subring, valuation.is_equiv_iff_val_lt_one], - exact set_like.ext_iff.1 h, -end - -lemma nonunits_inj {A B : valuation_subring K} : A.nonunits = B.nonunits ↔ A = B := -nonunits_injective.eq_iff +lemma mem_nonunits_iff {x : K} : x ∈ A.nonunits ↔ A.valuation x < 1 := iff.rfl lemma nonunits_le_nonunits {A B : valuation_subring K} : B.nonunits ≤ A.nonunits ↔ A ≤ B := @@ -467,6 +450,13 @@ begin by_contra h_1, from not_lt.2 (monotone_map_of_le _ _ h (not_lt.1 h_1)) hx } end +lemma nonunits_injective : + function.injective (nonunits : valuation_subring K → subsemigroup _) := +λ A B h, by { simpa only [le_antisymm_iff, nonunits_le_nonunits] using h.symm } + +lemma nonunits_inj {A B : valuation_subring K} : A.nonunits = B.nonunits ↔ A = B := +nonunits_injective.eq_iff + /-- The map on valuation subrings to their nonunits is a dual order embedding. -/ def nonunits_order_embedding : valuation_subring K ↪o (subsemigroup K)ᵒᵈ := @@ -510,4 +500,136 @@ end end nonunits +section principal_unit_group + +/-- The principal unit group of a valuation subring, as a subgroup of `Kˣ`. -/ +def principal_unit_group : subgroup Kˣ := +{ carrier := { x | A.valuation (x - 1) < 1 }, + mul_mem' := begin + intros a b ha hb, + refine lt_of_le_of_lt _ (max_lt hb ha), + rw [← one_mul (A.valuation (b - 1)), ← A.valuation.map_one_add_of_lt ha, add_sub_cancel'_right, + ← valuation.map_mul, mul_sub_one, ← sub_add_sub_cancel], + exact A.valuation.map_add _ _, + end, + one_mem' := by simpa using zero_lt_one₀, + inv_mem' := begin + dsimp, + intros a ha, + conv {to_lhs, rw [← mul_one (A.valuation _), ← A.valuation.map_one_add_of_lt ha]}, + rwa [add_sub_cancel'_right, ← valuation.map_mul, sub_mul, units.inv_mul, ← neg_sub, one_mul, + valuation.map_neg], + end } + +lemma principal_units_le_units : A.principal_unit_group ≤ A.unit_group := +λ a h, by simpa only [add_sub_cancel'_right] using A.valuation.map_one_add_of_lt h + +lemma mem_principal_unit_group_iff (x : Kˣ) : + x ∈ A.principal_unit_group ↔ A.valuation ((x : K) - 1) < 1 := iff.rfl + +lemma principal_unit_group_le_principal_unit_group {A B : valuation_subring K} : + B.principal_unit_group ≤ A.principal_unit_group ↔ A ≤ B := +begin + split, + { intros h x hx, + by_cases h_1 : x = 0, { simp only [h_1, zero_mem] }, + by_cases h_2 : x⁻¹ + 1 = 0, + { rw [add_eq_zero_iff_eq_neg, inv_eq_iff_inv_eq, inv_neg, inv_one] at h_2, + simpa only [h_2] using B.neg_mem _ B.one_mem }, + { rw [← valuation_le_one_iff, ← not_lt, valuation.one_lt_val_iff _ h_1, ← add_sub_cancel x⁻¹, + ← units.coe_mk0 h_2, ← mem_principal_unit_group_iff] at hx ⊢, + simpa only [hx] using @h (units.mk0 (x⁻¹ + 1) h_2) } }, + { intros h x hx, + by_contra h_1, from not_lt.2 (monotone_map_of_le _ _ h (not_lt.1 h_1)) hx } +end + +lemma principal_unit_group_injective : + function.injective (principal_unit_group : valuation_subring K → subgroup _) := +λ A B h, by { simpa [le_antisymm_iff, principal_unit_group_le_principal_unit_group] using h.symm } + +lemma eq_iff_principal_unit_group {A B : valuation_subring K} : + A = B ↔ A.principal_unit_group = B.principal_unit_group := +principal_unit_group_injective.eq_iff.symm + +/-- The map on valuation subrings to their principal unit groups is an order embedding. -/ +def principal_unit_group_order_embedding : + valuation_subring K ↪o (subgroup Kˣ)ᵒᵈ := +{ to_fun := λ A, A.principal_unit_group, + inj' := principal_unit_group_injective, + map_rel_iff' := λ A B, principal_unit_group_le_principal_unit_group } + +lemma coe_mem_principal_unit_group_iff {x : A.unit_group} : + (x : Kˣ) ∈ A.principal_unit_group ↔ + A.unit_group_mul_equiv x ∈ (units.map (local_ring.residue A).to_monoid_hom).ker := +begin + rw [monoid_hom.mem_ker, units.ext_iff], + dsimp, + let π := ideal.quotient.mk (local_ring.maximal_ideal A), change _ ↔ π _ = _, + rw [← π.map_one, ← sub_eq_zero, ← π.map_sub, ideal.quotient.eq_zero_iff_mem, + valuation_lt_one_iff], + simpa, +end + +/-- The principal unit group agrees with the kernel of the canonical map from +the units of `A` to the units of the residue field of `A`. -/ +def principal_unit_group_equiv : + A.principal_unit_group ≃* (units.map (local_ring.residue A).to_monoid_hom).ker := +{ to_fun := λ x, ⟨A.unit_group_mul_equiv ⟨_, A.principal_units_le_units x.2⟩, + A.coe_mem_principal_unit_group_iff.1 x.2⟩, + inv_fun := λ x, ⟨A.unit_group_mul_equiv.symm x, + by { rw A.coe_mem_principal_unit_group_iff, simpa using set_like.coe_mem x }⟩, + left_inv := λ x, by simp, + right_inv := λ x, by simp, + map_mul' := λ x y, by refl, } + +@[simp] +lemma principal_unit_group_equiv_apply (a : A.principal_unit_group) : + (principal_unit_group_equiv A a : K) = a := rfl + +@[simp] +lemma principal_unit_group_symm_apply + (a : (units.map (local_ring.residue A).to_monoid_hom).ker) : + (A.principal_unit_group_equiv.symm a : K) = a := rfl + +/-- The canonical map from the unit group of `A` to the units of the residue field of `A`. -/ +def unit_group_to_residue_field_units : + A.unit_group →* (local_ring.residue_field A)ˣ := +monoid_hom.comp (units.map $ (ideal.quotient.mk _).to_monoid_hom) + A.unit_group_mul_equiv.to_monoid_hom + +@[simp] +lemma coe_unit_group_to_residue_field_units_apply (x : A.unit_group) : + (A.unit_group_to_residue_field_units x : (local_ring.residue_field A) ) = + (ideal.quotient.mk _ (A.unit_group_mul_equiv x : A)) := rfl + +lemma ker_unit_group_to_residue_field_units : + A.unit_group_to_residue_field_units.ker = A.principal_unit_group.comap A.unit_group.subtype := +by { ext, simpa only [subgroup.mem_comap, subgroup.coe_subtype, coe_mem_principal_unit_group_iff] } + +lemma surjective_unit_group_to_residue_field_units : + function.surjective A.unit_group_to_residue_field_units := +(local_ring.surjective_units_map_of_local_ring_hom _ +ideal.quotient.mk_surjective local_ring.is_local_ring_hom_residue).comp (mul_equiv.surjective _) + +/-- The quotient of the unit group of `A` by the principal unit group of `A` agrees with +the units of the residue field of `A`. -/ +def units_mod_principal_units_equiv_residue_field_units : + (A.unit_group ⧸ (A.principal_unit_group.comap A.unit_group.subtype)) ≃* + (local_ring.residue_field A)ˣ := +mul_equiv.trans (quotient_group.equiv_quotient_of_eq A.ker_unit_group_to_residue_field_units.symm) + (quotient_group.quotient_ker_equiv_of_surjective _ A.surjective_unit_group_to_residue_field_units) + +@[simp] +lemma units_mod_principal_units_equiv_residue_field_units_comp_quotient_group_mk : + A.units_mod_principal_units_equiv_residue_field_units.to_monoid_hom.comp + (quotient_group.mk' _) = A.unit_group_to_residue_field_units := rfl + +@[simp] +lemma units_mod_principal_units_equiv_residue_field_units_comp_quotient_group_mk_apply + (x : A.unit_group) : + A.units_mod_principal_units_equiv_residue_field_units.to_monoid_hom + (quotient_group.mk x) = A.unit_group_to_residue_field_units x := rfl + +end principal_unit_group + end valuation_subring diff --git a/formal/lean/mathlib/ring_theory/witt_vector/defs.lean b/formal/lean/mathlib/ring_theory/witt_vector/defs.lean index 61a2f2b150848014c6e9b7848be3a27f82cd8b04..07648860fc3011ce075e83e21923bf3ba586795e 100644 --- a/formal/lean/mathlib/ring_theory/witt_vector/defs.lean +++ b/formal/lean/mathlib/ring_theory/witt_vector/defs.lean @@ -369,32 +369,28 @@ by simp [mul_coeff, peval] end coeff -lemma witt_add_vars (n : ℕ) : - (witt_add p n).vars ⊆ finset.univ.product (finset.range (n + 1)) := +lemma witt_add_vars (n : ℕ) : (witt_add p n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) := witt_structure_int_vars _ _ _ -lemma witt_sub_vars (n : ℕ) : - (witt_sub p n).vars ⊆ finset.univ.product (finset.range (n + 1)) := +lemma witt_sub_vars (n : ℕ) : (witt_sub p n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) := witt_structure_int_vars _ _ _ -lemma witt_mul_vars (n : ℕ) : - (witt_mul p n).vars ⊆ finset.univ.product (finset.range (n + 1)) := +lemma witt_mul_vars (n : ℕ) : (witt_mul p n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) := witt_structure_int_vars _ _ _ -lemma witt_neg_vars (n : ℕ) : - (witt_neg p n).vars ⊆ finset.univ.product (finset.range (n + 1)) := +lemma witt_neg_vars (n : ℕ) : (witt_neg p n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) := witt_structure_int_vars _ _ _ lemma witt_nsmul_vars (m : ℕ) (n : ℕ) : - (witt_nsmul p m n).vars ⊆ finset.univ.product (finset.range (n + 1)) := + (witt_nsmul p m n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) := witt_structure_int_vars _ _ _ lemma witt_zsmul_vars (m : ℤ) (n : ℕ) : - (witt_zsmul p m n).vars ⊆ finset.univ.product (finset.range (n + 1)) := + (witt_zsmul p m n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) := witt_structure_int_vars _ _ _ lemma witt_pow_vars (m : ℕ) (n : ℕ) : - (witt_pow p m n).vars ⊆ finset.univ.product (finset.range (n + 1)) := + (witt_pow p m n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) := witt_structure_int_vars _ _ _ end witt_vector diff --git a/formal/lean/mathlib/ring_theory/witt_vector/isocrystal.lean b/formal/lean/mathlib/ring_theory/witt_vector/isocrystal.lean index 6ad6d3fb1904efff903a44050b9273d780e7f161..7323a6d8f3312aba2d12c4c2b6454110f57ca753 100644 --- a/formal/lean/mathlib/ring_theory/witt_vector/isocrystal.lean +++ b/formal/lean/mathlib/ring_theory/witt_vector/isocrystal.lean @@ -110,12 +110,12 @@ variables (V) localized "notation `Φ(` p`,` k`)` := witt_vector.isocrystal.frobenius p k" in isocrystal /-- A homomorphism between isocrystals respects the Frobenius map. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure isocrystal_hom extends V →ₗ[K(p, k)] V₂ := ( frob_equivariant : ∀ x : V, Φ(p, k) (to_linear_map x) = to_linear_map (Φ(p, k) x) ) /-- An isomorphism between isocrystals respects the Frobenius map. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure isocrystal_equiv extends V ≃ₗ[K(p, k)] V₂ := ( frob_equivariant : ∀ x : V, Φ(p, k) (to_linear_equiv x) = to_linear_equiv (Φ(p, k) x) ) @@ -139,7 +139,7 @@ def fraction_ring.module : module K(p, k) K(p, k) := semiring.to_module Type synonym for `K(p, k)` to carry the standard 1-dimensional isocrystal structure of slope `m : ℤ`. -/ -@[nolint unused_arguments has_inhabited_instance, derive [add_comm_group, module K(p, k)]] +@[nolint unused_arguments has_nonempty_instance, derive [add_comm_group, module K(p, k)]] def standard_one_dim_isocrystal (m : ℤ) : Type* := K(p, k) diff --git a/formal/lean/mathlib/ring_theory/witt_vector/mul_coeff.lean b/formal/lean/mathlib/ring_theory/witt_vector/mul_coeff.lean index 274b845d5b2ca124aa4ba40407b0b2ae2cdde42a..5a2bed5e2b75c4ce47ca59603e31b1cb4840be98 100644 --- a/formal/lean/mathlib/ring_theory/witt_vector/mul_coeff.lean +++ b/formal/lean/mathlib/ring_theory/witt_vector/mul_coeff.lean @@ -48,8 +48,7 @@ rename (prod.mk (0 : fin 2)) (witt_polynomial p ℤ n) * include hp -lemma witt_poly_prod_vars (n : ℕ) : - (witt_poly_prod p n).vars ⊆ finset.univ.product (finset.range (n + 1)) := +lemma witt_poly_prod_vars (n : ℕ) : (witt_poly_prod p n).vars ⊆ univ ×ˢ range (n + 1) := begin rw [witt_poly_prod], apply subset.trans (vars_mul _ _), @@ -63,7 +62,7 @@ def witt_poly_prod_remainder (n : ℕ) : mv_polynomial (fin 2 × ℕ) ℤ := ∑ i in range n, p^i * (witt_mul p i)^(p^(n-i)) lemma witt_poly_prod_remainder_vars (n : ℕ) : - (witt_poly_prod_remainder p n).vars ⊆ finset.univ.product (finset.range n) := + (witt_poly_prod_remainder p n).vars ⊆ univ ×ˢ range n := begin rw [witt_poly_prod_remainder], apply subset.trans (vars_sum_subset _ _), @@ -100,7 +99,7 @@ def remainder (n : ℕ) : mv_polynomial (fin 2 × ℕ) ℤ := include hp -lemma remainder_vars (n : ℕ) : (remainder p n).vars ⊆ univ.product (range (n+1)) := +lemma remainder_vars (n : ℕ) : (remainder p n).vars ⊆ univ ×ˢ range (n + 1) := begin rw [remainder], apply subset.trans (vars_mul _ _), @@ -205,7 +204,7 @@ end lemma mul_poly_of_interest_vars (n : ℕ) : ((p ^ (n + 1) : mv_polynomial (fin 2 × ℕ) ℤ) * poly_of_interest p n).vars ⊆ - univ.product (range (n+1)) := + univ ×ˢ range (n + 1) := begin rw mul_poly_of_interest_aux5, apply subset.trans (vars_sub_subset _ _), @@ -228,7 +227,7 @@ begin exact_mod_cast hp.out.ne_zero end -lemma poly_of_interest_vars (n : ℕ) : (poly_of_interest p n).vars ⊆ univ.product (range (n+1)) := +lemma poly_of_interest_vars (n : ℕ) : (poly_of_interest p n).vars ⊆ univ ×ˢ (range (n+1)) := by rw poly_of_interest_vars_eq; apply mul_poly_of_interest_vars lemma peval_poly_of_interest (n : ℕ) (x y : 𝕎 k) : diff --git a/formal/lean/mathlib/ring_theory/witt_vector/structure_polynomial.lean b/formal/lean/mathlib/ring_theory/witt_vector/structure_polynomial.lean index ad89d24cb533f30f2cd64d7de3a88dcde3ba6bfb..94f53c9710b236b358278c9b399ada4473069a9d 100644 --- a/formal/lean/mathlib/ring_theory/witt_vector/structure_polynomial.lean +++ b/formal/lean/mathlib/ring_theory/witt_vector/structure_polynomial.lean @@ -345,12 +345,12 @@ lemma constant_coeff_witt_structure_rat_zero (Φ : mv_polynomial idx ℚ) : constant_coeff (witt_structure_rat p Φ 0) = constant_coeff Φ := by simp only [witt_structure_rat, bind₁, map_aeval, X_in_terms_of_W_zero, constant_coeff_rename, constant_coeff_witt_polynomial, aeval_X, constant_coeff_comp_algebra_map, - eval₂_hom_zero', ring_hom.id_apply] + eval₂_hom_zero'_apply, ring_hom.id_apply] lemma constant_coeff_witt_structure_rat (Φ : mv_polynomial idx ℚ) (h : constant_coeff Φ = 0) (n : ℕ) : constant_coeff (witt_structure_rat p Φ n) = 0 := -by simp only [witt_structure_rat, eval₂_hom_zero', h, bind₁, map_aeval, constant_coeff_rename, +by simp only [witt_structure_rat, eval₂_hom_zero'_apply, h, bind₁, map_aeval, constant_coeff_rename, constant_coeff_witt_polynomial, constant_coeff_comp_algebra_map, ring_hom.id_apply, constant_coeff_X_in_terms_of_W] @@ -382,7 +382,7 @@ variable (R) -- we could relax the fintype on `idx`, but then we need to cast from finset to set. -- for our applications `idx` is always finite. lemma witt_structure_rat_vars [fintype idx] (Φ : mv_polynomial idx ℚ) (n : ℕ) : - (witt_structure_rat p Φ n).vars ⊆ finset.univ.product (finset.range (n + 1)) := + (witt_structure_rat p Φ n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) := begin rw witt_structure_rat, intros x hx, @@ -399,7 +399,7 @@ end -- we could relax the fintype on `idx`, but then we need to cast from finset to set. -- for our applications `idx` is always finite. lemma witt_structure_int_vars [fintype idx] (Φ : mv_polynomial idx ℤ) (n : ℕ) : - (witt_structure_int p Φ n).vars ⊆ finset.univ.product (finset.range (n + 1)) := + (witt_structure_int p Φ n).vars ⊆ finset.univ ×ˢ finset.range (n + 1) := begin have : function.injective (int.cast_ring_hom ℚ) := int.cast_injective, rw [← vars_map_of_injective _ this, map_witt_structure_int], diff --git a/formal/lean/mathlib/set_theory/cardinal/finite.lean b/formal/lean/mathlib/set_theory/cardinal/finite.lean index 8014e39334561f10c0a02978737821b6cbe45816..7b0d8ab267cd9dc6204f5268f23181cb8eb6b1b9 100644 --- a/formal/lean/mathlib/set_theory/cardinal/finite.lean +++ b/formal/lean/mathlib/set_theory/cardinal/finite.lean @@ -55,8 +55,8 @@ end lemma card_of_subsingleton (a : α) [subsingleton α] : nat.card α = 1 := begin - rw [card_eq_fintype_card], - convert fintype.card_of_subsingleton a, + letI := fintype.of_subsingleton a, + rw [card_eq_fintype_card, fintype.card_of_subsingleton a] end @[simp] lemma card_unique [unique α] : nat.card α = 1 := diff --git a/formal/lean/mathlib/set_theory/game/nim.lean b/formal/lean/mathlib/set_theory/game/nim.lean index 58935edcf070dfde08bdba72a58445d9708e2a53..ef5a43a5db66f2afd8d76674b4a023314401b712 100644 --- a/formal/lean/mathlib/set_theory/game/nim.lean +++ b/formal/lean/mathlib/set_theory/game/nim.lean @@ -32,6 +32,10 @@ noncomputable theory universe u +open_locale pgame + +namespace pgame + /-- The definition of single-heap nim, which can be viewed as a pile of stones where each player can take a positive number of stones from it on their turn. -/ -- Uses `noncomputable!` to avoid `rec_fn_macro only allowed in meta definitions` VM error @@ -43,15 +47,6 @@ noncomputable! def nim : ordinal.{u} → pgame.{u} in ⟨o₁.out.α, o₁.out.α, f, f⟩ using_well_founded { dec_tac := tactic.assumption } -namespace pgame - -local infix ` ⧏ `:50 := lf -local infix ` ≈ ` := equiv -local infix ` ∥ `:50 := fuzzy -local infix ` ≡r `:50 := relabelling - -namespace nim - open ordinal lemma nim_def (o : ordinal) : nim o = pgame.mk o.out.α o.out.α @@ -101,23 +96,45 @@ lemma move_right_nim {o : ordinal} (i) : (nim o).move_right (to_right_moves_nim i) = nim i := by simp -instance : is_empty (nim 0).left_moves := +instance is_empty_nim_zero_left_moves : is_empty (nim 0).left_moves := by { rw nim_def, exact ordinal.is_empty_out_zero } -instance : is_empty (nim 0).right_moves := +instance is_empty_nim_zero_right_moves : is_empty (nim 0).right_moves := by { rw nim_def, exact ordinal.is_empty_out_zero } -instance : unique (nim 1).left_moves := -by { rw nim_def, exact ordinal.unique_out_one } - -instance : unique (nim 1).right_moves := -by { rw nim_def, exact ordinal.unique_out_one } - /-- `nim 0` has exactly the same moves as `0`. -/ def nim_zero_relabelling : nim 0 ≡r 0 := relabelling.is_empty _ theorem nim_zero_equiv : nim 0 ≈ 0 := equiv.is_empty _ +noncomputable instance unique_nim_one_left_moves : unique (nim 1).left_moves := +(equiv.cast $ left_moves_nim 1).unique + +noncomputable instance unique_nim_one_right_moves : unique (nim 1).right_moves := +(equiv.cast $ right_moves_nim 1).unique + +@[simp] theorem default_nim_one_left_moves_eq : + (default : (nim 1).left_moves) = @to_left_moves_nim 1 ⟨0, ordinal.zero_lt_one⟩ := +rfl + +@[simp] theorem default_nim_one_right_moves_eq : + (default : (nim 1).right_moves) = @to_right_moves_nim 1 ⟨0, ordinal.zero_lt_one⟩ := +rfl + +@[simp] theorem to_left_moves_nim_one_symm (i) : + (@to_left_moves_nim 1).symm i = ⟨0, ordinal.zero_lt_one⟩ := +by simp + +@[simp] theorem to_right_moves_nim_one_symm (i) : + (@to_right_moves_nim 1).symm i = ⟨0, ordinal.zero_lt_one⟩ := +by simp + +theorem nim_one_move_left (x) : (nim 1).move_left x = nim 0 := +by simp + +theorem nim_one_move_right (x) : (nim 1).move_right x = nim 0 := +by simp + /-- `nim 1` has exactly the same moves as `star`. -/ def nim_one_relabelling : nim 1 ≡r star := begin @@ -161,14 +178,14 @@ lemma exists_ordinal_move_left_eq {o : ordinal} (i) : ∃ o' < o, (nim o).move_l lemma exists_move_left_eq {o o' : ordinal} (h : o' < o) : ∃ i, (nim o).move_left i = nim o' := ⟨to_left_moves_nim ⟨o', h⟩, by simp⟩ -lemma non_zero_first_wins {o : ordinal} (ho : o ≠ 0) : nim o ∥ 0 := +lemma nim_fuzzy_zero_of_ne_zero {o : ordinal} (ho : o ≠ 0) : nim o ∥ 0 := begin rw [impartial.fuzzy_zero_iff_lf, nim_def, lf_zero_le], rw ←ordinal.pos_iff_ne_zero at ho, exact ⟨(ordinal.principal_seg_out ho).top, by simp⟩ end -@[simp] lemma add_equiv_zero_iff_eq (o₁ o₂ : ordinal) : nim o₁ + nim o₂ ≈ 0 ↔ o₁ = o₂ := +@[simp] lemma nim_add_equiv_zero_iff (o₁ o₂ : ordinal) : nim o₁ + nim o₂ ≈ 0 ↔ o₁ = o₂ := begin split, { contrapose, @@ -186,13 +203,11 @@ begin exact impartial.add_self (nim o₁) } end -@[simp] lemma add_fuzzy_zero_iff_ne (o₁ o₂ : ordinal) : nim o₁ + nim o₂ ∥ 0 ↔ o₁ ≠ o₂ := -by rw [iff_not_comm, impartial.not_fuzzy_zero_iff, add_equiv_zero_iff_eq] +@[simp] lemma nim_add_fuzzy_zero_iff {o₁ o₂ : ordinal} : nim o₁ + nim o₂ ∥ 0 ↔ o₁ ≠ o₂ := +by rw [iff_not_comm, impartial.not_fuzzy_zero_iff, nim_add_equiv_zero_iff] -@[simp] lemma equiv_iff_eq (o₁ o₂ : ordinal) : nim o₁ ≈ nim o₂ ↔ o₁ = o₂ := -by rw [impartial.equiv_iff_add_equiv_zero, add_equiv_zero_iff_eq] - -end nim +@[simp] lemma nim_equiv_iff_eq {o₁ o₂ : ordinal} : nim o₁ ≈ nim o₂ ↔ o₁ = o₂ := +by rw [impartial.equiv_iff_add_equiv_zero, nim_add_equiv_zero_iff] /-- The Grundy value of an impartial game, the ordinal which corresponds to the game of nim that the game is equivalent to -/ @@ -200,7 +215,7 @@ noncomputable def grundy_value : Π (G : pgame.{u}), ordinal.{u} | G := ordinal.mex.{u u} (λ i, grundy_value (G.move_left i)) using_well_founded { dec_tac := pgame_wf_tac } -lemma grundy_value_def (G : pgame) : +lemma grundy_value_eq_mex_left (G : pgame) : grundy_value G = ordinal.mex.{u u} (λ i, grundy_value (G.move_left i)) := by rw grundy_value @@ -216,22 +231,22 @@ begin { intro i₁, rw add_move_left_inl, apply (fuzzy_congr_left (add_congr_left (equiv_nim_grundy_value (G.move_left i₁)).symm)).1, - rw nim.add_fuzzy_zero_iff_ne, + rw nim_add_fuzzy_zero_iff, intro heq, - rw [eq_comm, grundy_value_def G] at heq, + rw [eq_comm, grundy_value_eq_mex_left G] at heq, have h := ordinal.ne_mex _, rw heq at h, exact (h i₁).irrefl }, { intro i₂, rw [add_move_left_inr, ←impartial.exists_left_move_equiv_iff_fuzzy_zero], revert i₂, - rw nim.nim_def, + rw nim_def, intro i₂, have h' : ∃ i : G.left_moves, (grundy_value (G.move_left i)) = ordinal.typein (quotient.out (grundy_value G)).r i₂, { revert i₂, - rw grundy_value_def, + rw grundy_value_eq_mex_left, intros i₂, have hnotin : _ ∉ _ := λ hin, (le_not_le_of_lt (ordinal.typein_lt_self i₂)).2 (cInf_le' hin), simpa using hnotin}, @@ -247,9 +262,9 @@ using_well_founded { dec_tac := pgame_wf_tac } lemma grundy_value_eq_iff_equiv_nim {G : pgame} [G.impartial] {o : ordinal} : grundy_value G = o ↔ G ≈ nim o := ⟨by { rintro rfl, exact equiv_nim_grundy_value G }, - by { intro h, rw ←nim.equiv_iff_eq, exact (equiv_nim_grundy_value G).symm.trans h }⟩ + by { intro h, rw ←nim_equiv_iff_eq, exact (equiv_nim_grundy_value G).symm.trans h }⟩ -@[simp] lemma nim.grundy_value (o : ordinal.{u}) : grundy_value (nim o) = o := +@[simp] lemma nim_grundy_value (o : ordinal.{u}) : grundy_value (nim o) = o := grundy_value_eq_iff_equiv_nim.2 pgame.equiv_rfl lemma grundy_value_eq_iff_equiv (G H : pgame) [G.impartial] [H.impartial] : @@ -257,20 +272,34 @@ lemma grundy_value_eq_iff_equiv (G H : pgame) [G.impartial] [H.impartial] : grundy_value_eq_iff_equiv_nim.trans (equiv_congr_left.1 (equiv_nim_grundy_value H) _).symm @[simp] lemma grundy_value_zero : grundy_value 0 = 0 := -grundy_value_eq_iff_equiv_nim.2 nim.nim_zero_equiv.symm +grundy_value_eq_iff_equiv_nim.2 nim_zero_equiv.symm lemma grundy_value_iff_equiv_zero (G : pgame) [G.impartial] : grundy_value G = 0 ↔ G ≈ 0 := by rw [←grundy_value_eq_iff_equiv, grundy_value_zero] @[simp] lemma grundy_value_star : grundy_value star = 1 := -grundy_value_eq_iff_equiv_nim.2 nim.nim_one_equiv.symm +grundy_value_eq_iff_equiv_nim.2 nim_one_equiv.symm + +@[simp] lemma grundy_value_neg (G : pgame) [G.impartial] : grundy_value (-G) = grundy_value G := +by rw [grundy_value_eq_iff_equiv_nim, neg_equiv_iff, neg_nim, ←grundy_value_eq_iff_equiv_nim] + +lemma grundy_value_eq_mex_right : ∀ (G : pgame) [G.impartial], + grundy_value G = ordinal.mex.{u u} (λ i, grundy_value (G.move_right i)) +| ⟨l, r, L, R⟩ := begin + introI H, + rw [←grundy_value_neg, grundy_value_eq_mex_left], + congr, + ext i, + haveI : (R i).impartial := @impartial.move_right_impartial ⟨l, r, L, R⟩ _ i, + apply grundy_value_neg +end @[simp] lemma grundy_value_nim_add_nim (n m : ℕ) : grundy_value (nim.{u} n + nim.{u} m) = nat.lxor n m := begin induction n using nat.strong_induction_on with n hn generalizing m, induction m using nat.strong_induction_on with m hm, - rw [grundy_value_def], + rw [grundy_value_eq_mex_left], -- We want to show that `n xor m` is the smallest unreachable Grundy value. We will do this in two -- steps: @@ -288,7 +317,7 @@ begin all_goals { -- One of the piles is reduced to `k` stones, with `k < n` or `k < m`. intro a, - obtain ⟨ok, hk, hk'⟩ := nim.exists_ordinal_move_left_eq a, + obtain ⟨ok, hk, hk'⟩ := exists_ordinal_move_left_eq a, obtain ⟨k, rfl⟩ := ordinal.lt_omega.1 (lt_trans hk (ordinal.nat_lt_omega _)), replace hk := ordinal.nat_cast_lt.1 hk, @@ -319,11 +348,11 @@ begin -- Therefore, we can play the corresponding move, and by the inductive hypothesis the new state -- is `(u xor m) xor m = u` or `n xor (u xor n) = u` as required. - { obtain ⟨i, hi⟩ := nim.exists_move_left_eq (ordinal.nat_cast_lt.2 h), + { obtain ⟨i, hi⟩ := exists_move_left_eq (ordinal.nat_cast_lt.2 h), refine ⟨to_left_moves_add (sum.inl i), _⟩, simp only [hi, add_move_left_inl], rw [hn _ h, nat.lxor_assoc, nat.lxor_self, nat.lxor_zero] }, - { obtain ⟨i, hi⟩ := nim.exists_move_left_eq (ordinal.nat_cast_lt.2 h), + { obtain ⟨i, hi⟩ := exists_move_left_eq (ordinal.nat_cast_lt.2 h), refine ⟨to_left_moves_add (sum.inr i), _⟩, simp only [hi, add_move_left_inr], rw [hm _ h, nat.lxor_comm, nat.lxor_assoc, nat.lxor_self, nat.lxor_zero] } }, @@ -340,7 +369,7 @@ by rw [←grundy_value_eq_iff_equiv_nim, grundy_value_nim_add_nim] lemma grundy_value_add (G H : pgame) [G.impartial] [H.impartial] {n m : ℕ} (hG : grundy_value G = n) (hH : grundy_value H = m) : grundy_value (G + H) = nat.lxor n m := begin - rw [←nim.grundy_value (nat.lxor n m), grundy_value_eq_iff_equiv], + rw [←nim_grundy_value (nat.lxor n m), grundy_value_eq_iff_equiv], refine equiv.trans _ nim_add_nim_equiv, convert add_congr (equiv_nim_grundy_value G) (equiv_nim_grundy_value H); simp only [hG, hH] diff --git a/formal/lean/mathlib/set_theory/ordinal/arithmetic.lean b/formal/lean/mathlib/set_theory/ordinal/arithmetic.lean index 5524d109f77ebd25e1409f40268345e2d4a57464..5bfc553e3f012cf79d0b6c75c376e5fba33683ac 100644 --- a/formal/lean/mathlib/set_theory/ordinal/arithmetic.lean +++ b/formal/lean/mathlib/set_theory/ordinal/arithmetic.lean @@ -120,8 +120,7 @@ theorem nat_cast_succ (n : ℕ) : ↑n.succ = succ (n : ordinal) := rfl theorem add_left_cancel (a) {b c : ordinal} : a + b = a + c ↔ b = c := by simp only [le_antisymm_iff, add_le_add_iff_left] -theorem lt_one_iff_zero {a : ordinal} : a < 1 ↔ a = 0 := -by rw [←succ_zero, lt_succ_iff, ordinal.le_zero] +theorem lt_one_iff_zero {a : ordinal} : a < 1 ↔ a = 0 := by simpa using @lt_succ_bot_iff _ _ _ a _ _ private theorem add_lt_add_iff_left' (a) {b c : ordinal} : a + b < a + c ↔ b < c := by rw [← not_le, ← not_le, add_le_add_iff_left] @@ -185,13 +184,7 @@ unique.eq_default x by rw [one_out_eq x, typein_enum] theorem le_one_iff {a : ordinal} : a ≤ 1 ↔ a = 0 ∨ a = 1 := -begin - refine ⟨λ ha, _, _⟩, - { rcases eq_or_lt_of_le ha with rfl | ha, - exacts [or.inr rfl, or.inl (lt_one_iff_zero.1 ha)], }, - { rintro (rfl | rfl), - exacts [zero_le_one, le_rfl], } -end +by simpa using @le_succ_bot_iff _ _ _ a _ theorem add_eq_zero_iff {a b : ordinal} : a + b = 0 ↔ (a = 0 ∧ b = 0) := induction_on a $ λ α r _, induction_on b $ λ β s _, begin @@ -431,8 +424,8 @@ by simp only [le_antisymm_iff, H.le_iff] theorem is_normal.self_le {f} (H : is_normal f) (a) : a ≤ f a := lt_wf.self_le_of_strict_mono H.strict_mono a -theorem is_normal.le_set {f} (H : is_normal f) (p : set ordinal) (p0 : p.nonempty) (b) - (H₂ : ∀ o, b ≤ o ↔ ∀ a ∈ p, a ≤ o) {o : ordinal} : f b ≤ o ↔ ∀ a ∈ p, f a ≤ o := +theorem is_normal.le_set {f o} (H : is_normal f) (p : set ordinal) (p0 : p.nonempty) (b) + (H₂ : ∀ o, b ≤ o ↔ ∀ a ∈ p, a ≤ o) : f b ≤ o ↔ ∀ a ∈ p, f a ≤ o := ⟨λ h a pa, (H.le_iff.2 ((H₂ _).1 le_rfl _ pa)).trans h, λ h, begin revert H₂, refine limit_rec_on b (λ H₂, _) (λ S _ H₂, _) (λ S L _ H₂, (H.2 _ L _).2 (λ a h', _)), @@ -445,30 +438,21 @@ theorem is_normal.le_set {f} (H : is_normal f) (p : set ordinal) (p0 : p.nonempt exact (H.le_iff.2 $ (not_le.1 h₂).le).trans (h _ h₁) } end⟩ -theorem is_normal.le_set' {f} (H : is_normal f) (p : set α) (g : α → ordinal) (p0 : p.nonempty) (b) - (H₂ : ∀ o, b ≤ o ↔ ∀ a ∈ p, g a ≤ o) {o} : f b ≤ o ↔ ∀ a ∈ p, f (g a) ≤ o := -(H.le_set (λ x, ∃ y, p y ∧ x = g y) - (let ⟨x, px⟩ := p0 in ⟨_, _, px, rfl⟩) _ - (λ o, (H₂ o).trans ⟨λ H a ⟨y, h1, h2⟩, h2.symm ▸ H y h1, - λ H a h1, H (g a) ⟨a, h1, rfl⟩⟩)).trans -⟨λ H a h, H (g a) ⟨a, h, rfl⟩, λ H a ⟨y, h1, h2⟩, h2.symm ▸ H y h1⟩ +theorem is_normal.le_set' {f o} (H : is_normal f) (p : set α) (p0 : p.nonempty) (g : α → ordinal) + (b) (H₂ : ∀ o, b ≤ o ↔ ∀ a ∈ p, g a ≤ o) : f b ≤ o ↔ ∀ a ∈ p, f (g a) ≤ o := +by simpa [H₂] using H.le_set (g '' p) (p0.image g) b -theorem is_normal.refl : is_normal id := -⟨lt_succ, λ o l a, limit_le l⟩ +theorem is_normal.refl : is_normal id := ⟨lt_succ, λ o l a, limit_le l⟩ -theorem is_normal.trans {f g} (H₁ : is_normal f) (H₂ : is_normal g) : - is_normal (λ x, f (g x)) := -⟨λ x, H₁.lt_iff.2 (H₂.1 _), - λ o l a, H₁.le_set' (< o) g ⟨_, l.pos⟩ _ (λ c, H₂.2 _ l _)⟩ +theorem is_normal.trans {f g} (H₁ : is_normal f) (H₂ : is_normal g) : is_normal (f ∘ g) := +⟨λ x, H₁.lt_iff.2 (H₂.1 _), λ o l a, H₁.le_set' (< o) ⟨_, l.pos⟩ g _ (λ c, H₂.2 _ l _)⟩ -theorem is_normal.is_limit {f} (H : is_normal f) {o} (l : is_limit o) : - is_limit (f o) := +theorem is_normal.is_limit {f} (H : is_normal f) {o} (l : is_limit o) : is_limit (f o) := ⟨ne_of_gt $ (ordinal.zero_le _).trans_lt $ H.lt_iff.2 l.pos, λ a h, let ⟨b, h₁, h₂⟩ := (H.limit_lt l).1 h in (succ_le_of_lt h₂).trans_lt (H.lt_iff.2 h₁)⟩ -theorem is_normal.le_iff_eq {f} (H : is_normal f) {a} : f a ≤ a ↔ f a = a := -(H.self_le a).le_iff_eq +theorem is_normal.le_iff_eq {f} (H : is_normal f) {a} : f a ≤ a ↔ f a = a := (H.self_le a).le_iff_eq theorem add_le_of_limit {a b c : ordinal} (h : is_limit b) : a + b ≤ c ↔ ∀ b' < b, a + b' ≤ c := ⟨λ h b' l, (add_le_add_left l.le _).trans h, @@ -1080,11 +1064,10 @@ begin exact le_sup f i end -theorem is_normal.sup {f} (H : is_normal f) {ι} (g : ι → ordinal) (h : nonempty ι) : +theorem is_normal.sup {f} (H : is_normal f) {ι} (g : ι → ordinal) [nonempty ι] : f (sup g) = sup (f ∘ g) := eq_of_forall_ge_iff $ λ a, -by rw [sup_le_iff, comp, H.le_set' (λ_:ι, true) g (let ⟨i⟩ := h in ⟨i, ⟨⟩⟩)]; - intros; simp only [sup_le_iff, true_implies_iff]; tauto +by rw [sup_le_iff, comp, H.le_set' set.univ set.univ_nonempty g]; simp [sup_le_iff] @[simp] theorem sup_empty {ι} [is_empty ι] (f : ι → ordinal) : sup f = 0 := csupr_of_empty f @@ -1207,8 +1190,12 @@ by simpa only [not_forall, not_le] using not_congr (@bsup_le_iff _ f a) theorem is_normal.bsup {f} (H : is_normal f) {o} : ∀ (g : Π a < o, ordinal) (h : o ≠ 0), f (bsup o g) = bsup o (λ a h, f (g a h)) := -induction_on o $ λ α r _ g h, -by { resetI, rw [←sup_eq_bsup' r, H.sup _ (type_ne_zero_iff_nonempty.1 h), ←sup_eq_bsup' r]; refl } +induction_on o $ λ α r _ g h, begin + resetI, + haveI := type_ne_zero_iff_nonempty.1 h, + rw [←sup_eq_bsup' r, H.sup, ←sup_eq_bsup' r]; + refl +end theorem lt_bsup_of_ne_bsup {o : ordinal} {f : Π a < o, ordinal} : (∀ i h, f i h ≠ o.bsup f) ↔ ∀ i h, f i h < o.bsup f := @@ -1956,6 +1943,7 @@ begin { intros c l IH, refine eq_of_forall_ge_iff (λ d, (((opow_is_normal a1).trans (add_is_normal b)).limit_le l).trans _), + dsimp only [function.comp], simp only [IH] {contextual := tt}, exact (((mul_is_normal $ opow_pos b (ordinal.pos_iff_ne_zero.2 a0)).trans (opow_is_normal a1)).limit_le l).symm } @@ -1991,6 +1979,7 @@ begin { intros c l IH, refine eq_of_forall_ge_iff (λ d, (((opow_is_normal a1).trans (mul_is_normal (ordinal.pos_iff_ne_zero.2 b0))).limit_le l).trans _), + dsimp only [function.comp], simp only [IH] {contextual := tt}, exact (opow_le_of_limit (opow_ne_zero _ a0) l).symm } end @@ -2104,8 +2093,12 @@ else by simp only [log_of_not_one_lt_left hb, ordinal.zero_le] @[simp] theorem log_one_right (b : ordinal) : log b 1 = 0 := if hb : 1 < b then log_eq_zero hb else log_of_not_one_lt_left hb 1 -theorem mod_opow_log_lt_self {b o : ordinal} (hb : b ≠ 0) (ho : o ≠ 0) : o % b ^ log b o < o := -(mod_lt _ $ opow_ne_zero _ hb).trans_le (opow_log_le_self _ ho) +theorem mod_opow_log_lt_self (b : ordinal) {o : ordinal} (ho : o ≠ 0) : o % b ^ log b o < o := +begin + rcases eq_or_ne b 0 with rfl | hb, + { simpa using ordinal.pos_iff_ne_zero.2 ho }, + { exact (mod_lt _ $ opow_ne_zero _ hb).trans_le (opow_log_le_self _ ho) } +end theorem log_mod_opow_log_lt_log_self {b o : ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : log b (o % b ^ log b o) < log b o := @@ -2346,7 +2339,7 @@ end theorem is_normal.apply_omega {f : ordinal.{u} → ordinal.{u}} (hf : is_normal f) : sup.{0 u} (f ∘ nat.cast) = f ω := -by rw [←sup_nat_cast, is_normal.sup.{0 u u} hf _ ⟨0⟩] +by rw [←sup_nat_cast, is_normal.sup.{0 u u} hf] @[simp] theorem sup_add_nat (o : ordinal) : sup (λ n : ℕ, o + n) = o + ω := (add_is_normal o).apply_omega diff --git a/formal/lean/mathlib/set_theory/ordinal/basic.lean b/formal/lean/mathlib/set_theory/ordinal/basic.lean index 0a4f52e70e0c9813f47d783c3ca571dd084e49df..0817e24c9a2a5294cf03531cbf5c06c8d8e447cc 100644 --- a/formal/lean/mathlib/set_theory/ordinal/basic.lean +++ b/formal/lean/mathlib/set_theory/ordinal/basic.lean @@ -182,7 +182,7 @@ instance : partial_order ordinal := λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩ ⟨f⟩ ⟨g⟩, ⟨f.trans g⟩, lt_iff_le_not_le := λ a b, quotient.induction_on₂ a b $ λ ⟨α, r, _⟩ ⟨β, s, _⟩, by exactI - ⟨λ ⟨f⟩, ⟨⟨f⟩, λ ⟨g⟩, (f.lt_le g).irrefl _⟩, + ⟨λ ⟨f⟩, ⟨⟨f⟩, λ ⟨g⟩, (f.lt_le g).irrefl⟩, λ ⟨⟨f⟩, h⟩, sum.rec_on f.lt_or_eq (λ g, ⟨g⟩) (λ g, (h ⟨initial_seg.of_iso g.symm⟩).elim)⟩, le_antisymm := λ a b, @@ -251,7 +251,7 @@ by { simp_rw ←type_lt o, apply typein_lt_type } typein s f.top = type r := eq.symm $ quot.sound ⟨rel_iso.of_surjective (rel_embedding.cod_restrict _ f f.lt_top) - (λ ⟨a, h⟩, by rcases f.down'.1 h with ⟨b, rfl⟩; exact ⟨b, rfl⟩)⟩ + (λ ⟨a, h⟩, by rcases f.down.1 h with ⟨b, rfl⟩; exact ⟨b, rfl⟩)⟩ @[simp] theorem typein_apply {α β} {r : α → α → Prop} {s : β → β → Prop} [is_well_order α r] [is_well_order β s] (f : r ≼i s) (a : α) : @@ -408,11 +408,9 @@ induction_on o $ λ α r _, ⟨⟨⟨embedding.of_is_empty, is_empty_elim⟩, is instance : order_bot ordinal := ⟨0, ordinal.zero_le⟩ -@[simp] protected theorem le_zero {o : ordinal} : o ≤ 0 ↔ o = 0 := -by simp only [le_antisymm_iff, ordinal.zero_le, and_true] +@[simp] protected theorem le_zero {o : ordinal} : o ≤ 0 ↔ o = 0 := le_bot_iff -protected theorem pos_iff_ne_zero {o : ordinal} : 0 < o ↔ o ≠ 0 := -by simp only [lt_iff_le_and_ne, ordinal.zero_le, true_and, ne.def, eq_comm] +protected theorem pos_iff_ne_zero {o : ordinal} : 0 < o ↔ o ≠ 0 := bot_lt_iff_ne_bot @[simp] theorem out_empty_iff_eq_zero {o : ordinal} : is_empty o.out.α ↔ o = 0 := by rw [←@type_eq_zero_iff_is_empty o.out.α (<), type_lt] @@ -967,7 +965,7 @@ by simpa only [lift_lift, lift_univ, univ_umax] using @[simp] theorem ord_univ : ord univ.{u v} = ordinal.univ.{u v} := le_antisymm (ord_card_le _) $ le_of_forall_lt $ λ o h, lt_ord.2 begin - rcases lift.principal_seg.{u v}.down'.1 + rcases lift.principal_seg.{u v}.down.1 (by simpa only [lift.principal_seg_coe] using h) with ⟨o', rfl⟩, simp only [lift.principal_seg_coe], rw [← lift_card], apply lift_lt_univ' @@ -977,7 +975,7 @@ theorem lt_univ {c} : c < univ.{u (u+1)} ↔ ∃ c', c = lift.{(u+1) u} c' := ⟨λ h, begin have := ord_lt_ord.2 h, rw ord_univ at this, - cases lift.principal_seg.{u (u+1)}.down'.1 + cases lift.principal_seg.{u (u+1)}.down.1 (by simpa only [lift.principal_seg_top]) with o e, have := card_ord c, rw [← e, lift.principal_seg_coe, ← lift_card] at this, diff --git a/formal/lean/mathlib/set_theory/ordinal/cantor_normal_form.lean b/formal/lean/mathlib/set_theory/ordinal/cantor_normal_form.lean index 05d255eee48f2053934a39890af7f6d0824d6018..148bc847319ef3dd03d752e38a8de96fbab172b5 100644 --- a/formal/lean/mathlib/set_theory/ordinal/cantor_normal_form.lean +++ b/formal/lean/mathlib/set_theory/ordinal/cantor_normal_form.lean @@ -37,116 +37,118 @@ open order namespace ordinal /-- Inducts on the base `b` expansion of an ordinal. -/ -@[elab_as_eliminator] noncomputable def CNF_rec {b : ordinal} (hb : b ≠ 0) +@[elab_as_eliminator] noncomputable def CNF_rec (b : ordinal) {C : ordinal → Sort*} (H0 : C 0) (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o | o := if ho : o = 0 then by rwa ho else - have _, from mod_opow_log_lt_self hb ho, - H o ho (CNF_rec (o % b ^ log b o)) + let hwf := mod_opow_log_lt_self b ho in H o ho (CNF_rec (o % b ^ log b o)) using_well_founded {dec_tac := `[assumption]} -@[simp] theorem CNF_rec_zero {b} (hb) {C H0 H} : @CNF_rec b hb C H0 H 0 = H0 := -by rw [CNF_rec, dif_pos rfl]; refl +@[simp] theorem CNF_rec_zero {C : ordinal → Sort*} (b : ordinal) + (H0 : C 0) (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNF_rec b C H0 H 0 = H0 := +by { rw [CNF_rec, dif_pos rfl], refl } -@[simp] theorem CNF_rec_ne_zero {b} (hb) {C H0 H o} (ho) : - @CNF_rec b hb C H0 H o = H o ho (@CNF_rec b hb C H0 H _) := +theorem CNF_rec_pos (b : ordinal) {o : ordinal} {C : ordinal → Sort*} (ho : o ≠ 0) + (H0 : C 0) (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : + @CNF_rec b C H0 H o = H o ho (@CNF_rec b C H0 H _) := by rw [CNF_rec, dif_neg ho] /-- The Cantor normal form of an ordinal `o` is the list of coefficients and exponents in the base-`b` expansion of `o`. -We special-case `CNF 0 o = []`, `CNF b 0 = []`, and `CNF 1 o = [(0, o)]` for `o ≠ 0`. +We special-case `CNF 0 o = CNF 1 o = [(0, o)]` for `o ≠ 0`. `CNF b (b ^ u₁ * v₁ + b ^ u₂ * v₂) = [(u₁, v₁), (u₂, v₂)]` -/ @[pp_nodot] def CNF (b o : ordinal) : list (ordinal × ordinal) := -if hb : b = 0 then [] else -CNF_rec hb [] (λ o o0 IH, (log b o, o / b ^ log b o) :: IH) o +CNF_rec b [] (λ o ho IH, (log b o, o / b ^ log b o) :: IH) o -@[simp] theorem zero_CNF (o) : CNF 0 o = [] := -dif_pos rfl - -@[simp] theorem CNF_zero (b) : CNF b 0 = [] := -if hb : b = 0 then dif_pos hb else (dif_neg hb).trans $ CNF_rec_zero _ +@[simp] theorem CNF_zero (b : ordinal) : CNF b 0 = [] := CNF_rec_zero b _ _ /-- Recursive definition for the Cantor normal form. -/ -theorem CNF_ne_zero {b o : ordinal} (hb : b ≠ 0) (ho : o ≠ 0) : +theorem CNF_ne_zero {b o : ordinal} (ho : o ≠ 0) : CNF b o = (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) := -by unfold CNF; rw [dif_neg hb, dif_neg hb, CNF_rec_ne_zero hb ho] +CNF_rec_pos b ho _ _ -@[simp] theorem one_CNF {o : ordinal} (ho : o ≠ 0) : CNF 1 o = [(0, o)] := -by rw [CNF_ne_zero ordinal.one_ne_zero ho, log_of_not_one_lt_left (irrefl _), opow_zero, mod_one, - CNF_zero, div_one] +theorem zero_CNF {o : ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩] := by simp [CNF_ne_zero ho] -/-- Evaluating the Cantor normal form of an ordinal returns the ordinal. -/ -theorem CNF_foldr {b : ordinal} (hb : b ≠ 0) (o) : - (CNF b o).foldr (λ p r, b ^ p.1 * p.2 + r) 0 = o := -CNF_rec hb (by rw CNF_zero; refl) - (λ o ho IH, by rw [CNF_ne_zero hb ho, list.foldr_cons, IH, div_add_mod]) o - -/-- This theorem exists to factor out commonalities between the proofs of `ordinal.CNF_pairwise` and -`ordinal.CNF_fst_le_log`. -/ -private theorem CNF_pairwise_aux (b o : ordinal.{u}) : - (∀ p : ordinal × ordinal, p ∈ CNF b o → p.1 ≤ log b o) ∧ (CNF b o).pairwise (λ p q, q.1 < p.1) := +theorem one_CNF {o : ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩] := by simp [CNF_ne_zero ho] + +theorem CNF_of_le_one {b o : ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩] := begin - by_cases hb : b = 0, - { simp only [hb, zero_CNF, list.pairwise.nil, and_true], exact λ _, false.elim }, - cases lt_or_eq_of_le (one_le_iff_ne_zero.2 hb) with hb' hb', - { refine CNF_rec hb _ _ o, - { simp only [CNF_zero, list.pairwise.nil, and_true], exact λ _, false.elim }, - intros o ho IH, cases IH with IH₁ IH₂, - simp only [CNF_ne_zero hb ho, list.forall_mem_cons, list.pairwise_cons, IH₂, and_true], - refine ⟨⟨le_rfl, λ p m, _⟩, λ p m, _⟩, - { exact (IH₁ p m).trans (log_mono_right _ $ le_of_lt $ mod_opow_log_lt_self hb ho) }, - { refine (IH₁ p m).trans_lt ((lt_opow_iff_log_lt hb' _).1 _), - { intro e, - rw e at m, simpa only [CNF_zero] using m }, - { exact mod_lt _ (opow_ne_zero _ hb) } } }, - { by_cases ho : o = 0, - { simp only [ho, CNF_zero, list.pairwise.nil, and_true], exact λ _, false.elim }, - rw [← hb', one_CNF ho], - simp only [list.mem_singleton, log_one_left, forall_eq, le_refl, true_and, - list.pairwise_singleton] } + rcases le_one_iff.1 hb with rfl | rfl, + { exact zero_CNF ho }, + { exact one_CNF ho } end -/-- The exponents of the Cantor normal form are decreasing. -/ -theorem CNF_pairwise (b o : ordinal.{u}) : - (CNF b o).pairwise (λ p q : ordinal × ordinal, q.1 < p.1) := -(CNF_pairwise_aux _ _).2 +theorem CNF_of_lt {b o : ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩] := +by simp [CNF_ne_zero ho, log_eq_zero hb] + +/-- Evaluating the Cantor normal form of an ordinal returns the ordinal. -/ +theorem CNF_foldr (b o : ordinal) : (CNF b o).foldr (λ p r, b ^ p.1 * p.2 + r) 0 = o := +CNF_rec b (by { rw CNF_zero, refl }) + (λ o ho IH, by rw [CNF_ne_zero ho, list.foldr_cons, IH, div_add_mod]) o /-- Every exponent in the Cantor normal form `CNF b o` is less or equal to `log b o`. -/ -theorem CNF_fst_le_log {b o : ordinal.{u}} : - ∀ {p : ordinal × ordinal}, p ∈ CNF b o → p.1 ≤ log b o := -(CNF_pairwise_aux _ _).1 +theorem CNF_fst_le_log {b o : ordinal.{u}} {x : ordinal × ordinal} : + x ∈ CNF b o → x.1 ≤ log b o := +begin + refine CNF_rec b _ (λ o ho H, _) o, + { rw CNF_zero, + exact false.elim }, + { rw [CNF_ne_zero ho, list.mem_cons_iff], + rintro (rfl | h), + { exact le_rfl }, + { exact (H h).trans (log_mono_right _ (mod_opow_log_lt_self b ho).le) } } +end /-- Every exponent in the Cantor normal form `CNF b o` is less or equal to `o`. -/ -theorem CNF_fst_le {b o : ordinal.{u}} {p : ordinal × ordinal} (hp : p ∈ CNF b o) : p.1 ≤ o := -(CNF_fst_le_log hp).trans (log_le_self _ _) +theorem CNF_fst_le {b o : ordinal.{u}} {x : ordinal × ordinal} (h : x ∈ CNF b o) : x.1 ≤ o := +(CNF_fst_le_log h).trans $ log_le_self _ _ -/-- This theorem exists to factor out commonalities between the proofs of `ordinal.CNF_snd_lt` and -`ordinal.CNF_lt_snd`. -/ -private theorem CNF_snd_lt_aux {b o : ordinal.{u}} (hb' : 1 < b) : - ∀ {p : ordinal × ordinal}, p ∈ CNF b o → p.2 < b ∧ 0 < p.2 := +/-- Every coefficient in a Cantor normal form is positive. -/ +theorem CNF_lt_snd {b o : ordinal.{u}} {x : ordinal × ordinal} : x ∈ CNF b o → 0 < x.2 := begin - have hb := (zero_lt_one.trans hb').ne', - refine CNF_rec hb (λ _, by { rw CNF_zero, exact false.elim }) (λ o ho IH, _) o, - simp only [CNF_ne_zero hb ho, list.mem_cons_iff, forall_eq_or_imp, iff_true_intro @IH, and_true], - nth_rewrite 1 ←@succ_le_iff, - rw [div_lt (opow_ne_zero _ hb), ←opow_succ, le_div (opow_ne_zero _ hb), succ_zero, mul_one], - exact ⟨lt_opow_succ_log_self hb' _, opow_log_le_self _ ho⟩ + refine CNF_rec b _ (λ o ho IH, _) o, + { simp }, + { rcases eq_zero_or_pos b with rfl | hb, + { rw [zero_CNF ho, list.mem_singleton], + rintro rfl, + exact ordinal.pos_iff_ne_zero.2 ho }, + { rw CNF_ne_zero ho, + rintro (rfl | h), + { simp, + rw div_pos, + { exact opow_log_le_self _ ho }, + { exact (opow_pos _ hb).ne' } }, + { exact IH h } } } end /-- Every coefficient in the Cantor normal form `CNF b o` is less than `b`. -/ -theorem CNF_snd_lt {b o : ordinal.{u}} (hb : 1 < b) {p : ordinal × ordinal} (hp : p ∈ CNF b o) : - p.2 < b := -(CNF_snd_lt_aux hb hp).1 - -/-- Every coefficient in a Cantor normal form is positive. -/ -theorem CNF_lt_snd {b o : ordinal.{u}} (hb : 1 < b) {p : ordinal × ordinal} (hp : p ∈ CNF b o) : - 0 < p.2 := -(CNF_snd_lt_aux hb hp).2 +theorem CNF_snd_lt {b o : ordinal.{u}} (hb : 1 < b) {x : ordinal × ordinal} : + x ∈ CNF b o → x.2 < b := +begin + refine CNF_rec b _ (λ o ho IH, _) o, + { simp }, + { rw CNF_ne_zero ho, + rintro (rfl | h), + { simpa using div_opow_log_lt o hb }, + { exact IH h } } +end /-- The exponents of the Cantor normal form are decreasing. -/ theorem CNF_sorted (b o : ordinal) : ((CNF b o).map prod.fst).sorted (>) := -by { rw [list.sorted, list.pairwise_map], exact CNF_pairwise b o } +begin + refine CNF_rec b _ (λ o ho IH, _) o, + { simp }, + { cases le_or_lt b 1 with hb hb, + { simp [CNF_of_le_one hb ho] }, + { cases lt_or_le o b with hob hbo, + { simp [CNF_of_lt ho hob] }, + { rw [CNF_ne_zero ho, list.map_cons, list.sorted_cons], + refine ⟨λ a H, _, IH⟩, + rw list.mem_map at H, + rcases H with ⟨⟨a, a'⟩, H, rfl⟩, + exact (CNF_fst_le_log H).trans_lt (log_mod_opow_log_lt_log_self hb ho hbo) } } } +end end ordinal diff --git a/formal/lean/mathlib/set_theory/ordinal/topology.lean b/formal/lean/mathlib/set_theory/ordinal/topology.lean index e7ff8dd8663696c087766004fef86a026219fdbc..00475ad72e637f426721e83f9d3fe674969ddb33 100644 --- a/formal/lean/mathlib/set_theory/ordinal/topology.lean +++ b/formal/lean/mathlib/set_theory/ordinal/topology.lean @@ -234,7 +234,7 @@ begin { let g : ι → ordinal.{u} := λ i, (enum_ord_order_iso hs).symm ⟨_, hf i⟩, suffices : enum_ord s (sup.{u u} g) = sup.{u u} f, { rw ←this, exact enum_ord_mem hs _ }, - rw is_normal.sup.{u u u} h g hι, + rw @is_normal.sup.{u u u} _ h ι g hι, congr, ext, change ((enum_ord_order_iso hs) _).val = f x, rw order_iso.apply_symm_apply }, diff --git a/formal/lean/mathlib/set_theory/surreal/basic.lean b/formal/lean/mathlib/set_theory/surreal/basic.lean index 1781d11f732772046566b232ff79267809763336..a96fb78270fdae226df352238ff29be5812c9cc0 100644 --- a/formal/lean/mathlib/set_theory/surreal/basic.lean +++ b/formal/lean/mathlib/set_theory/surreal/basic.lean @@ -233,16 +233,10 @@ def surreal.equiv (x y : {x // pgame.numeric x}) : Prop := x.1.equiv y.1 open pgame -instance surreal.setoid : setoid {x // pgame.numeric x} := -⟨λ x y, x.1 ≈ y.1, - λ x, equiv_rfl, - λ x y, pgame.equiv.symm, - λ x y z, pgame.equiv.trans⟩ - /-- The type of surreal numbers. These are the numeric pre-games quotiented by the equivalence relation `x ≈ y ↔ x ≤ y ∧ y ≤ x`. In the quotient, the order becomes a total order. -/ -def surreal := quotient surreal.setoid +def surreal := quotient (subtype.setoid numeric) namespace surreal diff --git a/formal/lean/mathlib/set_theory/zfc/basic.lean b/formal/lean/mathlib/set_theory/zfc/basic.lean index 3fb1c9e86102a77b49ee9f25d62a934a71235366..7b7600f04ced7acfac026a4e369214ac12b29f4c 100644 --- a/formal/lean/mathlib/set_theory/zfc/basic.lean +++ b/formal/lean/mathlib/set_theory/zfc/basic.lean @@ -109,13 +109,15 @@ equivalent to some element of the second family and vice-versa. -/ def equiv (x y : pSet) : Prop := pSet.rec (λ α z m ⟨β, B⟩, (∀ a, ∃ b, m a (B b)) ∧ (∀ b, ∃ a, m a (B b))) x y -theorem exists_equiv_left : Π {x y : pSet} (h : equiv x y) (i : x.type), - ∃ j, equiv (x.func i) (y.func j) -| ⟨α, A⟩ ⟨β, B⟩ h := h.1 +theorem equiv_iff : Π {x y : pSet}, equiv x y ↔ + (∀ i, ∃ j, equiv (x.func i) (y.func j)) ∧ (∀ j, ∃ i, equiv (x.func i) (y.func j)) +| ⟨α, A⟩ ⟨β, B⟩ := iff.rfl -theorem exists_equiv_right : Π {x y : pSet} (h : equiv x y) (j : y.type), - ∃ i, equiv (x.func i) (y.func j) -| ⟨α, A⟩ ⟨β, B⟩ h := h.2 +theorem equiv.exists_left {x y : pSet} (h : equiv x y) : ∀ i, ∃ j, equiv (x.func i) (y.func j) := +(equiv_iff.1 h).1 + +theorem equiv.exists_right {x y : pSet} (h : equiv x y) : ∀ j, ∃ i, equiv (x.func i) (y.func j) := +(equiv_iff.1 h).2 @[refl] protected theorem equiv.refl (x) : equiv x x := pSet.rec_on x $ λ α A IH, ⟨λ a, ⟨a, IH a⟩, λ a, ⟨a, IH a⟩⟩ @@ -133,6 +135,9 @@ pSet.rec_on x $ λ α A IH y, pSet.cases_on y $ λ β B ⟨γ, Γ⟩ ⟨αβ, β @[trans] protected theorem equiv.trans {x y z} (h1 : equiv x y) (h2 : equiv y z) : equiv x z := h1.euc h2.symm +protected theorem equiv_of_is_empty (x y : pSet) [is_empty x.type] [is_empty y.type] : equiv x y := +equiv_iff.2 $ by simp + instance setoid : setoid pSet := ⟨pSet.equiv, equiv.refl, λ x y, equiv.symm, λ x y z, equiv.trans⟩ @@ -198,7 +203,7 @@ theorem mem.congr_left : Π {x y : pSet.{u}}, equiv x y → (∀ {w : pSet.{u}}, private theorem mem_wf_aux : Π {x y : pSet.{u}}, equiv x y → acc (∈) y | ⟨α, A⟩ ⟨β, B⟩ H := ⟨_, begin rintros ⟨γ, C⟩ ⟨b, hc⟩, - cases exists_equiv_right H b with a ha, + cases H.exists_right b with a ha, have H := ha.trans hc.symm, rw mk_func at H, exact mem_wf_aux H @@ -215,6 +220,8 @@ theorem mem_irrefl (x : pSet) : x ∉ x := irrefl x /-- Convert a pre-set to a `set` of pre-sets. -/ def to_set (u : pSet.{u}) : set pSet.{u} := {x | x ∈ u} +@[simp] theorem mem_to_set (a u : pSet.{u}) : a ∈ u.to_set ↔ a ∈ u := iff.rfl + /-- Two pre-sets are equivalent iff they have the same members. -/ theorem equiv.eq {x y : pSet} : equiv x y ↔ to_set x = to_set y := equiv_iff_mem.trans set.ext_iff.symm @@ -232,8 +239,13 @@ instance : is_empty (type (∅)) := pempty.is_empty @[simp] theorem mem_empty (x : pSet.{u}) : x ∉ (∅ : pSet.{u}) := is_empty.exists_iff.1 +@[simp] theorem to_set_empty : to_set ∅ = ∅ := by simp [to_set] + @[simp] theorem empty_subset (x : pSet.{u}) : (∅ : pSet) ⊆ x := λ x, x.elim +protected theorem equiv_empty (x : pSet) [is_empty x.type] : equiv x ∅ := +pSet.equiv_of_is_empty x _ + /-- Insert an element into a pre-set -/ protected def insert (x y : pSet) : pSet := ⟨option y.type, λ o, option.rec x y.func o⟩ @@ -280,6 +292,9 @@ prefix `⋃₀ `:110 := pSet.sUnion by { rw ←(eta (A a)) at e, exact let ⟨βt, tβ⟩ := e, ⟨c, bc⟩ := βt b in ⟨⟨a, c⟩, yb.trans bc⟩ }⟩ +@[simp] theorem to_set_sUnion (x : pSet.{u}) : (⋃₀ x).to_set = ⋃₀ (to_set '' x.to_set) := +by { ext, simp } + /-- The image of a function from pre-sets to pre-sets. -/ def image (f : pSet.{u} → pSet.{u}) (x : pSet.{u}) : pSet := ⟨x.type, f ∘ x.func⟩ @@ -432,9 +447,13 @@ quotient.lift₂ pSet.mem instance : has_mem Set Set := ⟨Set.mem⟩ +@[simp] theorem mk_mem_iff {x y : pSet} : mk x ∈ mk y ↔ x ∈ y := iff.rfl + /-- Convert a ZFC set into a `set` of ZFC sets -/ def to_set (u : Set.{u}) : set Set.{u} := {x | x ∈ u} +@[simp] theorem mem_to_set (a u : Set.{u}) : a ∈ u.to_set ↔ a ∈ u := iff.rfl + instance small_to_set (x : Set.{u}) : small.{u} x.to_set := quotient.induction_on x $ λ a, begin let f : a.type → (mk a).to_set := λ i, ⟨mk $ a.func i, func_mem a i⟩, @@ -462,11 +481,19 @@ instance : is_trans Set (⊆) := ⟨λ x y z hxy hyz a ha, hyz (hxy ha)⟩ | ⟨α, A⟩ ⟨β, B⟩ := ⟨λ h a, @h ⟦A a⟧ (mem.mk A a), λ h z, quotient.induction_on z (λ z ⟨a, za⟩, let ⟨b, ab⟩ := h a in ⟨b, za.trans ab⟩)⟩ -theorem ext {x y : Set.{u}} : (∀ z : Set.{u}, z ∈ x ↔ z ∈ y) → x = y := +@[simp] theorem to_set_subset_iff {x y : Set} : x.to_set ⊆ y.to_set ↔ x ⊆ y := +by simp [subset_def, set.subset_def] + +@[ext] theorem ext {x y : Set.{u}} : (∀ z : Set.{u}, z ∈ x ↔ z ∈ y) → x = y := quotient.induction_on₂ x y (λ u v h, quotient.sound (mem.ext (λ w, h ⟦w⟧))) -theorem ext_iff {x y : Set.{u}} : (∀ z : Set.{u}, z ∈ x ↔ z ∈ y) ↔ x = y := -⟨ext, λ h, by simp [h]⟩ +theorem ext_iff {x y : Set.{u}} : x = y ↔ (∀ z : Set.{u}, z ∈ x ↔ z ∈ y) := +⟨λ h, by simp [h], ext⟩ + +theorem to_set_injective : function.injective to_set := λ x y h, ext $ set.ext_iff.1 h + +@[simp] theorem to_set_inj {x y : Set} : x.to_set = y.to_set ↔ x = y := +to_set_injective.eq_iff instance : is_antisymm Set (⊆) := ⟨λ a b hab hba, ext $ λ c, ⟨@hab c, @hba c⟩⟩ @@ -478,6 +505,8 @@ instance : inhabited Set := ⟨∅⟩ @[simp] theorem mem_empty (x) : x ∉ (∅ : Set.{u}) := quotient.induction_on x pSet.mem_empty +@[simp] theorem to_set_empty : to_set ∅ = ∅ := by simp [to_set] + @[simp] theorem empty_subset (x : Set.{u}) : (∅ : Set) ⊆ x := quotient.induction_on x $ λ y, subset_iff.2 $ pSet.empty_subset y @@ -517,9 +546,15 @@ quotient.induction_on₃ x y z theorem mem_insert (x y : Set) : x ∈ insert x y := mem_insert_iff.2 $ or.inl rfl theorem mem_insert_of_mem {y z : Set} (x) (h : z ∈ y): z ∈ insert x y := mem_insert_iff.2 $ or.inr h +@[simp] theorem to_set_insert (x y : Set) : (insert x y).to_set = insert x y.to_set := +by { ext, simp } + @[simp] theorem mem_singleton {x y : Set.{u}} : x ∈ @singleton Set.{u} Set.{u} _ y ↔ x = y := iff.trans mem_insert_iff ⟨λ o, or.rec (λ h, h) (λ n, absurd n (mem_empty _)) o, or.inl⟩ +@[simp] theorem to_set_singleton (x : Set) : ({x} : Set).to_set = {x} := +by { ext, simp } + @[simp] theorem mem_pair {x y z : Set.{u}} : x ∈ ({y, z} : Set) ↔ x = y ∨ x = z := iff.trans mem_insert_iff $ or_congr iff.rfl mem_singleton @@ -546,6 +581,10 @@ quotient.induction_on₂ x y (λ ⟨α, A⟩ y, ⟨λ ⟨⟨a, pa⟩, h⟩, ⟨⟨a, h⟩, by rwa (@quotient.sound pSet _ _ _ h)⟩, λ ⟨⟨a, h⟩, pa⟩, ⟨⟨a, by { rw mk_func at h, rwa [mk_func, ←Set.sound h] }⟩, h⟩⟩) +@[simp] theorem to_set_sep (a : Set) (p : Set → Prop) : + {x ∈ a | p x}.to_set = {x ∈ a.to_set | p x} := +by { ext, simp } + /-- The powerset operation, the collection of subsets of a ZFC set -/ def powerset : Set → Set := resp.eval 1 ⟨powerset, λ ⟨α, A⟩ ⟨β, B⟩ ⟨αβ, βα⟩, @@ -599,6 +638,9 @@ theorem singleton_injective : function.injective (@singleton Set Set _) := @[simp] theorem singleton_inj {x y : Set} : ({x} : Set) = {y} ↔ x = y := singleton_injective.eq_iff +@[simp] theorem to_set_sUnion (x : Set.{u}) : (⋃₀ x).to_set = ⋃₀ (to_set '' x.to_set) := +by { ext, simp } + /-- The binary union operation -/ protected def union (x y : Set.{u}) : Set.{u} := ⋃₀ {x, y} @@ -612,15 +654,17 @@ instance : has_union Set := ⟨Set.union⟩ instance : has_inter Set := ⟨Set.inter⟩ instance : has_sdiff Set := ⟨Set.diff⟩ +@[simp] theorem to_set_union (x y : Set.{u}) : (x ∪ y).to_set = x.to_set ∪ y.to_set := +by { unfold has_union.union, rw Set.union, simp } + +@[simp] theorem to_set_inter (x y : Set.{u}) : (x ∩ y).to_set = x.to_set ∩ y.to_set := +by { unfold has_inter.inter, rw Set.inter, ext, simp } + +@[simp] theorem to_set_sdiff (x y : Set.{u}) : (x \ y).to_set = x.to_set \ y.to_set := +by { change {z ∈ x | z ∉ y}.to_set = _, ext, simp } + @[simp] theorem mem_union {x y z : Set.{u}} : z ∈ x ∪ y ↔ z ∈ x ∨ z ∈ y := -iff.trans mem_sUnion - ⟨λ ⟨w, wxy, zw⟩, match mem_pair.1 wxy with - | or.inl wx := or.inl (by rwa ←wx) - | or.inr wy := or.inr (by rwa ←wy) - end, λ zxy, match zxy with - | or.inl zx := ⟨x, mem_pair.2 (or.inl rfl), zx⟩ - | or.inr zy := ⟨y, mem_pair.2 (or.inr rfl), zy⟩ - end⟩ +by { rw ←mem_to_set, simp } @[simp] theorem mem_inter {x y z : Set.{u}} : z ∈ x ∩ y ↔ z ∈ x ∧ z ∈ y := @@mem_sep (λ z : Set.{u}, z ∈ y) @@ -668,9 +712,15 @@ theorem image.mk : ⟨λ ⟨a, ya⟩, ⟨⟦A a⟧, mem.mk A a, eq.symm $ quotient.sound ya⟩, λ ⟨z, hz, e⟩, e ▸ image.mk _ _ hz⟩ +@[simp] theorem to_set_image (f : Set → Set) [H : definable 1 f] (x : Set) : + (image f x).to_set = f '' x.to_set := +by { ext, simp } + /-- Kuratowski ordered pair -/ def pair (x y : Set.{u}) : Set.{u} := {{x}, {x, y}} +@[simp] theorem to_set_pair (x y : Set.{u}) : (pair x y).to_set = {{x}, {x, y}} := by simp [pair] + /-- A subset of pairs `{(a, b) ∈ x × y | p a b}` -/ def pair_sep (p : Set.{u} → Set.{u} → Prop) (x y : Set.{u}) : Set.{u} := {z ∈ powerset (powerset (x ∪ y)) | ∃ a ∈ x, ∃ b ∈ y, z = pair a b ∧ p a b} @@ -688,7 +738,7 @@ end theorem pair_injective : function.injective2 pair := λ x x' y y' H, begin - have ae := ext_iff.2 H, + have ae := ext_iff.1 H, simp only [pair, mem_pair] at ae, obtain rfl : x = x', { cases (ae {x}).1 (by simp) with h h, @@ -701,11 +751,11 @@ theorem pair_injective : function.injective2 pair := cases (ae {x, y'}).2 (by simp only [eq_self_iff_true, or_true]) with xy'x xy'xx, { rw [eq_comm, ←mem_singleton, ←xy'x, mem_pair], exact or.inr rfl }, - { simpa [eq_comm] using (ext_iff.2 xy'xx y').1 (by simp) } }, + { simpa [eq_comm] using (ext_iff.1 xy'xx y').1 (by simp) } }, obtain xyx | xyy' := (ae {x, y}).1 (by simp), - { obtain rfl := mem_singleton.mp ((ext_iff.2 xyx y).1 $ by simp), + { obtain rfl := mem_singleton.mp ((ext_iff.1 xyx y).1 $ by simp), simp [he rfl] }, - { obtain rfl | yy' := mem_pair.mp ((ext_iff.2 xyy' y).1 $ by simp), + { obtain rfl | yy' := mem_pair.mp ((ext_iff.1 xyy' y).1 $ by simp), { simp [he rfl] }, { simp [yy'] } } end diff --git a/formal/lean/mathlib/topology/algebra/constructions.lean b/formal/lean/mathlib/topology/algebra/constructions.lean index cf61c93cbee106ccaccf58b34748e33fbd162d7d..406f3ebd3a3ee42902b5768dbb573df843bf56d1 100644 --- a/formal/lean/mathlib/topology/algebra/constructions.lean +++ b/formal/lean/mathlib/topology/algebra/constructions.lean @@ -36,16 +36,16 @@ continuous_induced_dom @[continuity, to_additive] lemma continuous_op : continuous (op : M → Mᵐᵒᵖ) := continuous_induced_rng.2 continuous_id -@[to_additive] instance [t2_space M] : t2_space Mᵐᵒᵖ := -⟨λ x y h, separated_by_continuous mul_opposite.continuous_unop $ unop_injective.ne h⟩ - /-- `mul_opposite.op` as a homeomorphism. -/ -@[to_additive "`add_opposite.op` as a homeomorphism."] +@[to_additive "`add_opposite.op` as a homeomorphism.", simps] def op_homeomorph : M ≃ₜ Mᵐᵒᵖ := { to_equiv := op_equiv, continuous_to_fun := continuous_op, continuous_inv_fun := continuous_unop } +@[to_additive] instance [t2_space M] : t2_space Mᵐᵒᵖ := +op_homeomorph.symm.embedding.t2_space + @[simp, to_additive] lemma map_op_nhds (x : M) : map (op : M → Mᵐᵒᵖ) (𝓝 x) = 𝓝 (op x) := op_homeomorph.map_nhds_eq x @@ -64,7 +64,7 @@ namespace units open mul_opposite -variables [topological_space M] [monoid M] +variables [topological_space M] [monoid M] [topological_space X] /-- The units of a monoid are equipped with a topology, via the embedding into `M × M`. -/ @[to_additive] instance : topological_space Mˣ := @@ -81,4 +81,12 @@ continuous_induced_dom @[to_additive] lemma continuous_coe : continuous (coe : Mˣ → M) := (@continuous_embed_product M _ _).fst +@[to_additive] protected lemma continuous_iff {f : X → Mˣ} : + continuous f ↔ continuous (coe ∘ f : X → M) ∧ continuous (λ x, ↑(f x)⁻¹ : X → M) := +by simp only [inducing_embed_product.continuous_iff, embed_product_apply, (∘), continuous_prod_mk, + op_homeomorph.symm.inducing.continuous_iff, op_homeomorph_symm_apply, unop_op] + +@[to_additive] lemma continuous_coe_inv : continuous (λ u, ↑u⁻¹ : Mˣ → M) := +(units.continuous_iff.1 continuous_id).2 + end units diff --git a/formal/lean/mathlib/topology/algebra/group.lean b/formal/lean/mathlib/topology/algebra/group.lean index 3b4404977f83bbceb8ee96457ae173caca0ca1e0..74e98d40487a7d2eee403bae7d2dd83fd6773589 100644 --- a/formal/lean/mathlib/topology/algebra/group.lean +++ b/formal/lean/mathlib/topology/algebra/group.lean @@ -231,12 +231,12 @@ end end pointwise_limits -instance additive.has_continuous_neg [h : topological_space H] [has_inv H] - [has_continuous_inv H] : @has_continuous_neg (additive H) h _ := +instance [topological_space H] [has_inv H] [has_continuous_inv H] : + has_continuous_neg (additive H) := { continuous_neg := @continuous_inv H _ _ _ } -instance multiplicative.has_continuous_inv [h : topological_space H] [has_neg H] - [has_continuous_neg H] : @has_continuous_inv (multiplicative H) h _ := +instance [topological_space H] [has_neg H] [has_continuous_neg H] : + has_continuous_inv (multiplicative H) := { continuous_inv := @continuous_neg H _ _ _ } end continuous_inv @@ -746,11 +746,7 @@ instance topological_group_quotient [N.normal] : topological_group (G ⧸ N) := { exact (surjective_quot_mk _).prod_map (surjective_quot_mk _) } }, exact (quotient_map.continuous_iff quot).2 cont, end, - continuous_inv := begin - have : continuous ((coe : G → G ⧸ N) ∘ (λ (a : G), a⁻¹)) := - continuous_quot_mk.comp continuous_inv, - convert continuous_quotient_lift _ this, - end } + continuous_inv := by convert (@continuous_inv G _ _ _).quotient_map' _ } end quotient_topological_group @@ -1131,12 +1127,12 @@ end end filter_mul -instance additive.topological_add_group {G} [h : topological_space G] - [group G] [topological_group G] : @topological_add_group (additive G) h _ := +instance {G} [topological_space G] [group G] [topological_group G] : + topological_add_group (additive G) := { continuous_neg := @continuous_inv G _ _ _ } -instance multiplicative.topological_group {G} [h : topological_space G] - [add_group G] [topological_add_group G] : @topological_group (multiplicative G) h _ := +instance {G} [topological_space G] [add_group G] [topological_add_group G] : + topological_group (multiplicative G) := { continuous_inv := @continuous_neg G _ _ _ } section quotient @@ -1144,19 +1140,14 @@ variables [group G] [topological_space G] [topological_group G] {Γ : subgroup G @[to_additive] instance quotient_group.has_continuous_const_smul : has_continuous_const_smul G (G ⧸ Γ) := -{ continuous_const_smul := λ g₀, begin - apply continuous_coinduced_dom.2, - change continuous (λ g : G, quotient_group.mk (g₀ * g)), - exact continuous_coinduced_rng.comp (continuous_mul_left g₀), - end } +{ continuous_const_smul := λ g, + by convert ((@continuous_const _ _ _ _ g).mul continuous_id).quotient_map' _ } @[to_additive] lemma quotient_group.continuous_smul₁ (x : G ⧸ Γ) : continuous (λ g : G, g • x) := begin - obtain ⟨g₀, rfl⟩ : ∃ g₀, quotient_group.mk g₀ = x, - { exact @quotient.exists_rep _ (quotient_group.left_rel Γ) x }, - change continuous (λ g, quotient_group.mk (g * g₀)), - exact continuous_coinduced_rng.comp (continuous_mul_right g₀) + induction x using quotient_group.induction_on, + exact continuous_quotient_mk.comp (continuous_mul_right x) end @[to_additive] @@ -1177,69 +1168,45 @@ namespace units open mul_opposite (continuous_op continuous_unop) -variables [monoid α] [topological_space α] [has_continuous_mul α] [monoid β] [topological_space β] - [has_continuous_mul β] +variables [monoid α] [topological_space α] [monoid β] [topological_space β] -@[to_additive] instance : topological_group αˣ := -{ continuous_inv := continuous_induced_rng.2 ((continuous_unop.comp - (@continuous_embed_product α _ _).snd).prod_mk (continuous_op.comp continuous_coe)) } +@[to_additive] instance [has_continuous_mul α] : topological_group αˣ := +{ continuous_inv := units.continuous_iff.2 $ ⟨continuous_coe_inv, continuous_coe⟩ } /-- The topological group isomorphism between the units of a product of two monoids, and the product - of the units of each monoid. -/ -def homeomorph.prod_units : homeomorph (α × β)ˣ (αˣ × βˣ) := -{ continuous_to_fun := - begin - show continuous (λ i : (α × β)ˣ, (map (monoid_hom.fst α β) i, map (monoid_hom.snd α β) i)), - refine continuous.prod_mk _ _, - { refine continuous_induced_rng.2 ((continuous_fst.comp units.continuous_coe).prod_mk _), - refine mul_opposite.continuous_op.comp (continuous_fst.comp _), - simp_rw units.inv_eq_coe_inv, - exact units.continuous_coe.comp continuous_inv, }, - { refine continuous_induced_rng.2 ((continuous_snd.comp units.continuous_coe).prod_mk _), - simp_rw units.coe_map_inv, - exact continuous_op.comp (continuous_snd.comp (units.continuous_coe.comp continuous_inv)), } - end, - continuous_inv_fun := - begin - refine continuous_induced_rng.2 (continuous.prod_mk _ _), - { exact (units.continuous_coe.comp continuous_fst).prod_mk - (units.continuous_coe.comp continuous_snd), }, - { refine continuous_op.comp - (units.continuous_coe.comp $ continuous_induced_rng.2 $ continuous.prod_mk _ _), - { exact (units.continuous_coe.comp (continuous_inv.comp continuous_fst)).prod_mk - (units.continuous_coe.comp (continuous_inv.comp continuous_snd)) }, - { exact continuous_op.comp ((units.continuous_coe.comp continuous_fst).prod_mk - (units.continuous_coe.comp continuous_snd)) }} - end, - ..mul_equiv.prod_units } +of the units of each monoid. -/ +@[to_additive "The topological group isomorphism between the additive units of a product of two +additive monoids, and the product of the additive units of each additive monoid."] +def homeomorph.prod_units : (α × β)ˣ ≃ₜ (αˣ × βˣ) := +{ continuous_to_fun := (continuous_fst.units_map (monoid_hom.fst α β)).prod_mk + (continuous_snd.units_map (monoid_hom.snd α β)), + continuous_inv_fun := units.continuous_iff.2 ⟨continuous_coe.fst'.prod_mk continuous_coe.snd', + continuous_coe_inv.fst'.prod_mk continuous_coe_inv.snd'⟩, + to_equiv := mul_equiv.prod_units.to_equiv } end units section lattice_ops -variables {ι : Sort*} [group G] [group H] - {t : topological_space H} [topological_group H] {F : Type*} - [monoid_hom_class F G H] (f : F) +variables {ι : Sort*} [group G] @[to_additive] lemma topological_group_Inf {ts : set (topological_space G)} (h : ∀ t ∈ ts, @topological_group G t _) : @topological_group G (Inf ts) _ := -{ continuous_inv := @has_continuous_inv.continuous_inv G (Inf ts) _ - (@has_continuous_inv_Inf _ _ _ - (λ t ht, @topological_group.to_has_continuous_inv G t _ (h t ht))), - continuous_mul := @has_continuous_mul.continuous_mul G (Inf ts) _ - (@has_continuous_mul_Inf _ _ _ - (λ t ht, @topological_group.to_has_continuous_mul G t _ (h t ht))) } +{ to_has_continuous_inv := @has_continuous_inv_Inf _ _ _ $ + λ t ht, @topological_group.to_has_continuous_inv G t _ $ h t ht, + to_has_continuous_mul := @has_continuous_mul_Inf _ _ _ $ + λ t ht, @topological_group.to_has_continuous_mul G t _ $ h t ht } @[to_additive] lemma topological_group_infi {ts' : ι → topological_space G} (h' : ∀ i, @topological_group G (ts' i) _) : @topological_group G (⨅ i, ts' i) _ := -by {rw ← Inf_range, exact topological_group_Inf (set.forall_range_iff.mpr h')} +by { rw ← Inf_range, exact topological_group_Inf (set.forall_range_iff.mpr h') } @[to_additive] lemma topological_group_inf {t₁ t₂ : topological_space G} (h₁ : @topological_group G t₁ _) (h₂ : @topological_group G t₂ _) : @topological_group G (t₁ ⊓ t₂) _ := -by {rw inf_eq_infi, refine topological_group_infi (λ b, _), cases b; assumption} +by { rw inf_eq_infi, refine topological_group_infi (λ b, _), cases b; assumption } end lattice_ops diff --git a/formal/lean/mathlib/topology/algebra/module/basic.lean b/formal/lean/mathlib/topology/algebra/module/basic.lean index acf06b2b4f516d066d3f24faa02102998db7ff7f..dd51e5e7a73db139841f887b93b4b77a8416fe77 100644 --- a/formal/lean/mathlib/topology/algebra/module/basic.lean +++ b/formal/lean/mathlib/topology/algebra/module/basic.lean @@ -173,13 +173,12 @@ variables {R : Type u} {M : Type v} [module R M] [has_continuous_smul R M] lemma submodule.closure_smul_self_subset (s : submodule R M) : - (λ p : R × M, p.1 • p.2) '' ((set.univ : set R) ×ˢ closure (s : set M)) - ⊆ closure (s : set M) := + (λ p : R × M, p.1 • p.2) '' (set.univ ×ˢ closure s) ⊆ closure s := calc -(λ p : R × M, p.1 • p.2) '' ((set.univ : set R) ×ˢ closure (s : set M)) - = (λ p : R × M, p.1 • p.2) '' (closure ((set.univ : set R) ×ˢ (s : set M))) : +(λ p : R × M, p.1 • p.2) '' (set.univ ×ˢ closure s) + = (λ p : R × M, p.1 • p.2) '' closure (set.univ ×ˢ s) : by simp [closure_prod_eq] -... ⊆ closure ((λ p : R × M, p.1 • p.2) '' ((set.univ : set R) ×ˢ (s : set M))) : +... ⊆ closure ((λ p : R × M, p.1 • p.2) '' (set.univ ×ˢ s)) : image_closure_subset_closure_image continuous_smul ... = closure s : begin congr, @@ -190,10 +189,8 @@ calc end lemma submodule.closure_smul_self_eq (s : submodule R M) : - (λ p : R × M, p.1 • p.2) '' ((set.univ : set R) ×ˢ closure (s : set M)) - = closure (s : set M) := -set.subset.antisymm s.closure_smul_self_subset - (λ x hx, ⟨⟨1, x⟩, ⟨set.mem_univ _, hx⟩, one_smul R _⟩) + (λ p : R × M, p.1 • p.2) '' (set.univ ×ˢ closure s) = closure s := +s.closure_smul_self_subset.antisymm $ λ x hx, ⟨⟨1, x⟩, ⟨set.mem_univ _, hx⟩, one_smul R _⟩ variables [has_continuous_add M] @@ -236,6 +233,11 @@ lemma submodule.dense_iff_topological_closure_eq_top {s : submodule R M} : dense (s : set M) ↔ s.topological_closure = ⊤ := by { rw [←set_like.coe_set_eq, dense_iff_closure_eq], simp } +instance {M' : Type*} [add_comm_monoid M'] [module R M'] [uniform_space M'] + [has_continuous_add M'] [has_continuous_smul R M'] [complete_space M'] (U : submodule R M') : + complete_space U.topological_closure := +is_closed_closure.complete_space_coe + end closure /-- Continuous linear maps between modules. We only put the type classes that are necessary for the @@ -268,8 +270,6 @@ class continuous_semilinear_map_class (F : Type*) {R S : out_param Type*} [semir -- `σ`, `R` and `S` become metavariables, but they are all outparams so it's OK attribute [nolint dangerous_instance] continuous_semilinear_map_class.to_continuous_map_class -set_option old_structure_cmd false - /-- `continuous_linear_map_class F R M M₂` asserts `F` is a type of bundled continuous `R`-linear maps `M → M₂`. This is an abbreviation for `continuous_semilinear_map_class F (ring_hom.id R) M M₂`. -/ @@ -280,10 +280,12 @@ abbreviation continuous_linear_map_class (F : Type*) [module R M] [module R M₂] := continuous_semilinear_map_class F (ring_hom.id R) M M₂ +set_option old_structure_cmd false + /-- Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications `M` and `M₂` will be topological modules over the topological semiring `R`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure continuous_linear_equiv {R : Type*} {S : Type*} [semiring R] [semiring S] (σ : R →+* S) {σ' : S →+* R} [ring_hom_inv_pair σ σ'] [ring_hom_inv_pair σ' σ] @@ -298,6 +300,54 @@ notation M ` ≃SL[`:50 σ `] ` M₂ := continuous_linear_equiv σ M M₂ notation M ` ≃L[`:50 R `] ` M₂ := continuous_linear_equiv (ring_hom.id R) M M₂ notation M ` ≃L⋆[`:50 R `] ` M₂ := continuous_linear_equiv (star_ring_end R) M M₂ +set_option old_structure_cmd true +/-- `continuous_semilinear_equiv_class F σ M M₂` asserts `F` is a type of bundled continuous +`σ`-semilinear equivs `M → M₂`. See also `continuous_linear_equiv_class F R M M₂` for the case +where `σ` is the identity map on `R`. A map `f` between an `R`-module and an `S`-module over a ring +homomorphism `σ : R →+* S` is semilinear if it satisfies the two properties `f (x + y) = f x + f y` +and `f (c • x) = (σ c) • f x`. -/ +class continuous_semilinear_equiv_class (F : Type*) + {R : out_param Type*} {S : out_param Type*} [semiring R] [semiring S] (σ : out_param $ R →+* S) + {σ' : out_param $ S →+* R} [ring_hom_inv_pair σ σ'] [ring_hom_inv_pair σ' σ] + (M : out_param Type*) [topological_space M] [add_comm_monoid M] + (M₂ : out_param Type*) [topological_space M₂] [add_comm_monoid M₂] + [module R M] [module S M₂] + extends semilinear_equiv_class F σ M M₂ := +(map_continuous : ∀ (f : F), continuous f . tactic.interactive.continuity') +(inv_continuous : ∀ (f : F), continuous (inv f) . tactic.interactive.continuity') + +/-- `continuous_linear_equiv_class F σ M M₂` asserts `F` is a type of bundled continuous +`R`-linear equivs `M → M₂`. This is an abbreviation for +`continuous_semilinear_equiv_class F (ring_hom.id) M M₂`. -/ +abbreviation continuous_linear_equiv_class (F : Type*) + (R : out_param Type*) [semiring R] + (M : out_param Type*) [topological_space M] [add_comm_monoid M] + (M₂ : out_param Type*) [topological_space M₂] [add_comm_monoid M₂] + [module R M] [module R M₂] := +continuous_semilinear_equiv_class F (ring_hom.id R) M M₂ + +set_option old_structure_cmd false + +namespace continuous_semilinear_equiv_class +variables (F : Type*) + {R : Type*} {S : Type*} [semiring R] [semiring S] (σ : R →+* S) + {σ' : S →+* R} [ring_hom_inv_pair σ σ'] [ring_hom_inv_pair σ' σ] + (M : Type*) [topological_space M] [add_comm_monoid M] + (M₂ : Type*) [topological_space M₂] [add_comm_monoid M₂] + [module R M] [module S M₂] + +include σ' +-- `σ'` becomes a metavariable, but it's OK since it's an outparam +@[priority 100, nolint dangerous_instance] +instance [s: continuous_semilinear_equiv_class F σ M M₂] : + continuous_semilinear_map_class F σ M M₂ := +{ coe := (coe : F → M → M₂), + coe_injective' := @fun_like.coe_injective F _ _ _, + ..s } +omit σ' + +end continuous_semilinear_equiv_class + section pointwise_limits variables @@ -987,13 +1037,13 @@ end pi section ring variables -{R : Type*} [ring R] {R₂ : Type*} [ring R₂] +{R : Type*} [ring R] {R₂ : Type*} [ring R₂] {R₃ : Type*} [ring R₃] {M : Type*} [topological_space M] [add_comm_group M] {M₂ : Type*} [topological_space M₂] [add_comm_group M₂] {M₃ : Type*} [topological_space M₃] [add_comm_group M₃] {M₄ : Type*} [topological_space M₄] [add_comm_group M₄] -[module R M] [module R₂ M₂] -{σ₁₂ : R →+* R₂} +[module R M] [module R₂ M₂] [module R₃ M₃] +{σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} section @@ -1053,6 +1103,26 @@ lemma sub_apply (f g : M →SL[σ₁₂] M₂) (x : M) : (f - g) x = f x - g x : end +@[simp] lemma comp_neg [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] [topological_add_group M₂] + [topological_add_group M₃] (g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) : + g.comp (-f) = -g.comp f := +by { ext, simp } + +@[simp] lemma neg_comp [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] [topological_add_group M₃] + (g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) : + (-g).comp f = -g.comp f := +by { ext, simp } + +@[simp] lemma comp_sub [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] [topological_add_group M₂] + [topological_add_group M₃] (g : M₂ →SL[σ₂₃] M₃) (f₁ f₂ : M →SL[σ₁₂] M₂) : + g.comp (f₁ - f₂) = g.comp f₁ - g.comp f₂ := +by { ext, simp } + +@[simp] lemma sub_comp [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] [topological_add_group M₃] + (g₁ g₂ : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) : + (g₁ - g₂).comp f = g₁.comp f - g₂.comp f := +by { ext, simp } + instance [topological_add_group M] : ring (M →L[R] M) := { mul := (*), one := 1, @@ -1341,6 +1411,18 @@ def to_continuous_linear_map (e : M₁ ≃SL[σ₁₂] M₂) : M₁ →SL[σ₁ /-- Coerce continuous linear equivs to continuous linear maps. -/ instance : has_coe (M₁ ≃SL[σ₁₂] M₂) (M₁ →SL[σ₁₂] M₂) := ⟨to_continuous_linear_map⟩ +instance : continuous_semilinear_equiv_class (M₁ ≃SL[σ₁₂] M₂) σ₁₂ M₁ M₂ := +{ coe := λ f, f, + inv := λ f, f.inv_fun, + coe_injective' := λ f g h₁ h₂, by { cases f with f' _, cases g with g' _, cases f', cases g', + congr' }, + left_inv := λ f, f.left_inv, + right_inv := λ f, f.right_inv, + map_add := λ f, f.map_add', + map_smulₛₗ := λ f, f.map_smul', + map_continuous := continuous_to_fun, + inv_continuous := continuous_inv_fun } + /-- Coerce continuous linear equivs to maps. -/ -- see Note [function coercion] instance : has_coe_to_fun (M₁ ≃SL[σ₁₂] M₂) (λ _, M₁ → M₂) := ⟨λ f, f⟩ diff --git a/formal/lean/mathlib/topology/algebra/module/finite_dimension.lean b/formal/lean/mathlib/topology/algebra/module/finite_dimension.lean index d4d7c22e7667bb2e981f0a1382bff30f5bd090c1..92ef28cc3524400f6518912983e11876f04ebd5e 100644 --- a/formal/lean/mathlib/topology/algebra/module/finite_dimension.lean +++ b/formal/lean/mathlib/topology/algebra/module/finite_dimension.lean @@ -47,13 +47,14 @@ open_locale big_operators section semiring -variables {ι 𝕜 F : Type*} [fintype ι] [semiring 𝕜] [topological_space 𝕜] +variables {ι 𝕜 F : Type*} [finite ι] [semiring 𝕜] [topological_space 𝕜] [add_comm_monoid F] [module 𝕜 F] [topological_space F] [has_continuous_add F] [has_continuous_smul 𝕜 F] -/-- A linear map on `ι → 𝕜` (where `ι` is a fintype) is continuous -/ +/-- A linear map on `ι → 𝕜` (where `ι` is finite) is continuous -/ lemma linear_map.continuous_on_pi (f : (ι → 𝕜) →ₗ[𝕜] F) : continuous f := begin + casesI nonempty_fintype ι, classical, -- for the proof, write `f` in the standard basis, and use that each coordinate is a continuous -- function. @@ -69,9 +70,8 @@ end semiring section field -variables {ι 𝕜 E F : Type*} [fintype ι] [field 𝕜] [topological_space 𝕜] - [add_comm_group E] [module 𝕜 E] [topological_space E] - [add_comm_group F] [module 𝕜 F] [topological_space F] +variables {𝕜 E F : Type*} [field 𝕜] [topological_space 𝕜] [add_comm_group E] [module 𝕜 E] + [topological_space E] [add_comm_group F] [module 𝕜 F] [topological_space F] [topological_add_group F] [has_continuous_smul 𝕜 F] /-- The space of continuous linear maps between finite-dimensional spaces is finite-dimensional. -/ diff --git a/formal/lean/mathlib/topology/algebra/module/locally_convex.lean b/formal/lean/mathlib/topology/algebra/module/locally_convex.lean index 3e8443f25d3e6e3482bd86707000a4fd1bc0cc2a..3f2133b60152f778dc0231aae717d4ddac5953ba 100644 --- a/formal/lean/mathlib/topology/algebra/module/locally_convex.lean +++ b/formal/lean/mathlib/topology/algebra/module/locally_convex.lean @@ -91,6 +91,14 @@ lemma locally_convex_space_iff_exists_convex_subset_zero : ∀ U ∈ (𝓝 0 : filter E), ∃ S ∈ (𝓝 0 : filter E), convex 𝕜 S ∧ S ⊆ U := (locally_convex_space_iff_zero 𝕜 E).trans has_basis_self +-- see Note [lower instance priority] +@[priority 100] instance locally_convex_space.to_locally_connected_space [module ℝ E] + [has_continuous_smul ℝ E] [locally_convex_space ℝ E] : + locally_connected_space E := +locally_connected_space_of_connected_bases _ _ + (λ x, @locally_convex_space.convex_basis ℝ _ _ _ _ _ _ x) + (λ x s hs, hs.2.is_preconnected) + end module section lattice_ops diff --git a/formal/lean/mathlib/topology/algebra/module/weak_dual.lean b/formal/lean/mathlib/topology/algebra/module/weak_dual.lean index 81c938b9b92d85f78890a255fda9b7ba9bd72343..122106d3c27f9c7f38d2304be70ead43ac5428b2 100644 --- a/formal/lean/mathlib/topology/algebra/module/weak_dual.lean +++ b/formal/lean/mathlib/topology/algebra/module/weak_dual.lean @@ -68,7 +68,7 @@ section weak_topology /-- The space `E` equipped with the weak topology induced by the bilinear form `B`. -/ @[derive [add_comm_monoid, module 𝕜], -nolint has_inhabited_instance unused_arguments] +nolint has_nonempty_instance unused_arguments] def weak_bilin [comm_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] [add_comm_monoid F] [module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) := E @@ -96,6 +96,7 @@ variables (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) instance : topological_space (weak_bilin B) := topological_space.induced (λ x y, B x y) Pi.topological_space +/-- The coercion `(λ x y, B x y) : E → (F → 𝕜)` is continuous. -/ lemma coe_fn_continuous : continuous (λ (x : weak_bilin B) y, B x y) := continuous_induced_dom @@ -242,7 +243,7 @@ end weak_dual /-- The weak topology is the topology coarsest topology on `E` such that all functionals `λ x, top_dual_pairing 𝕜 E v x` are continuous. -/ @[derive [add_comm_monoid, module 𝕜, topological_space, has_continuous_add], -nolint has_inhabited_instance] +nolint has_nonempty_instance] def weak_space (𝕜 E) [comm_semiring 𝕜] [topological_space 𝕜] [has_continuous_add 𝕜] [has_continuous_const_smul 𝕜 𝕜] [add_comm_monoid E] [module 𝕜 E] [topological_space E] := weak_bilin (top_dual_pairing 𝕜 E).flip diff --git a/formal/lean/mathlib/topology/algebra/monoid.lean b/formal/lean/mathlib/topology/algebra/monoid.lean index 646fdbe5dc95293f61370729d4b5a161b8390b42..8a0f6eb83a0d96f4ed8f371d19527f3cbdb5fb56 100644 --- a/formal/lean/mathlib/topology/algebra/monoid.lean +++ b/formal/lean/mathlib/topology/algebra/monoid.lean @@ -249,17 +249,17 @@ variables [topological_space M] [monoid M] [has_continuous_mul M] @[to_additive] lemma submonoid.top_closure_mul_self_subset (s : submonoid M) : - (closure (s : set M)) * closure (s : set M) ⊆ closure (s : set M) := + closure (s : set M) * closure s ⊆ closure s := calc -(closure (s : set M)) * closure (s : set M) - = (λ p : M × M, p.1 * p.2) '' (closure ((s : set M) ×ˢ (s : set M))) : by simp [closure_prod_eq] -... ⊆ closure ((λ p : M × M, p.1 * p.2) '' ((s : set M) ×ˢ (s : set M))) : +closure (s : set M) * closure s + = (λ p : M × M, p.1 * p.2) '' closure (s ×ˢ s) : by simp [closure_prod_eq] +... ⊆ closure ((λ p : M × M, p.1 * p.2) '' s ×ˢ s) : image_closure_subset_closure_image continuous_mul ... = closure s : by simp [s.coe_mul_self_eq] @[to_additive] lemma submonoid.top_closure_mul_self_eq (s : submonoid M) : - (closure (s : set M)) * closure (s : set M) = closure (s : set M) := + closure (s : set M) * closure s = closure s := subset.antisymm s.top_closure_mul_self_subset (λ x hx, ⟨x, 1, hx, subset_closure s.one_mem, mul_one _⟩) @@ -300,7 +300,7 @@ def submonoid.comm_monoid_topological_closure [t2_space M] (s : submonoid M) have h₁ : (s.topological_closure : set M) = closure s := rfl, let f₁ := λ (x : M × M), x.1 * x.2, let f₂ := λ (x : M × M), x.2 * x.1, - let S : set (M × M) := (s : set M) ×ˢ (s : set M), + let S : set (M × M) := s ×ˢ s, have h₃ : set.eq_on f₁ f₂ (closure S), { refine set.eq_on.closure _ continuous_mul (by continuity), intros x hx, @@ -479,6 +479,10 @@ instance : has_continuous_mul αˣ := inducing_embed_product.has_continuous_mul end units +@[to_additive] lemma continuous.units_map [monoid M] [monoid N] [topological_space M] + [topological_space N] (f : M →* N) (hf : continuous f) : continuous (units.map f) := +units.continuous_iff.2 ⟨hf.comp units.continuous_coe, hf.comp units.continuous_coe_inv⟩ + section variables [topological_space M] [comm_monoid M] @@ -550,12 +554,12 @@ end end -instance additive.has_continuous_add {M} [h : topological_space M] [has_mul M] - [has_continuous_mul M] : @has_continuous_add (additive M) h _ := +instance [topological_space M] [has_mul M] [has_continuous_mul M] : + has_continuous_add (additive M) := { continuous_add := @continuous_mul M _ _ _ } -instance multiplicative.has_continuous_mul {M} [h : topological_space M] [has_add M] - [has_continuous_add M] : @has_continuous_mul (multiplicative M) h _ := +instance [topological_space M] [has_add M] [has_continuous_add M] : + has_continuous_mul (multiplicative M) := { continuous_mul := @continuous_add M _ _ _ } section lattice_ops diff --git a/formal/lean/mathlib/topology/algebra/open_subgroup.lean b/formal/lean/mathlib/topology/algebra/open_subgroup.lean index 145f6a278c9658346d5f1ec3953951a0d6947699..61c5049e76d7a1f8d2404c32a41d20faadd63806 100644 --- a/formal/lean/mathlib/topology/algebra/open_subgroup.lean +++ b/formal/lean/mathlib/topology/algebra/open_subgroup.lean @@ -124,7 +124,7 @@ variables {H : Type*} [group H] [topological_space H] /-- The product of two open subgroups as an open subgroup of the product group. -/ @[to_additive "The product of two open subgroups as an open subgroup of the product group."] def prod (U : open_subgroup G) (V : open_subgroup H) : open_subgroup (G × H) := -{ carrier := (U : set G) ×ˢ (V : set H), +{ carrier := U ×ˢ V, is_open' := U.is_open.prod V.is_open, .. (U : subgroup G).prod (V : subgroup H) } diff --git a/formal/lean/mathlib/topology/algebra/order/basic.lean b/formal/lean/mathlib/topology/algebra/order/basic.lean index 6bd6d1c141859e02c1772d839171ad3337423df9..9ec7b4ba4999babc1dfaef5c9ea9ca4d01171e18 100644 --- a/formal/lean/mathlib/topology/algebra/order/basic.lean +++ b/formal/lean/mathlib/topology/algebra/order/basic.lean @@ -9,6 +9,7 @@ import order.filter.interval import topology.algebra.field import tactic.linarith import tactic.tfae +import tactic.positivity /-! # Theory of topology on ordered spaces @@ -90,8 +91,6 @@ generally, and suffices to derive many interesting properties relating order and class order_closed_topology (α : Type*) [topological_space α] [preorder α] : Prop := (is_closed_le' : is_closed {p:α×α | p.1 ≤ p.2}) -instance : Π [topological_space α], topological_space αᵒᵈ := id - instance [topological_space α] [h : first_countable_topology α] : first_countable_topology αᵒᵈ := h instance [topological_space α] [h : second_countable_topology α] : second_countable_topology αᵒᵈ := @@ -1795,8 +1794,7 @@ section continuous_mul lemma mul_tendsto_nhds_zero_right (x : α) : tendsto (uncurry ((*) : α → α → α)) (𝓝 0 ×ᶠ 𝓝 x) $ 𝓝 0 := begin - have hx : 0 < 2 * (1 + |x|) := (mul_pos (zero_lt_two) $ - lt_of_lt_of_le zero_lt_one $ le_add_of_le_of_nonneg le_rfl (abs_nonneg x)), + have hx : 0 < 2 * (1 + |x|) := by positivity, rw ((nhds_basis_zero_abs_sub_lt α).prod $ nhds_basis_abs_sub_lt x).tendsto_iff (nhds_basis_zero_abs_sub_lt α), refine λ ε ε_pos, ⟨(ε/(2 * (1 + |x|)), 1), ⟨div_pos ε_pos hx, zero_lt_one⟩, _⟩, @@ -1806,7 +1804,6 @@ begin refine lt_of_le_of_lt (mul_le_mul_of_nonneg_left _ (abs_nonneg a)) ((lt_div_iff hx).1 h), calc |b| = |(b - x) + x| : by rw sub_add_cancel b x ... ≤ |b - x| + |x| : abs_add (b - x) x - ... ≤ 1 + |x| : add_le_add_right (le_of_lt h') (|x|) ... ≤ 2 * (1 + |x|) : by linarith, end @@ -1833,7 +1830,7 @@ begin refine ⟨i / (|x₀|), div_pos hi (abs_pos.2 hx₀), λ x hx, hit _⟩, calc |x₀ * x - x₀| = |x₀ * (x - 1)| : congr_arg abs (by ring_nf) ... = |x₀| * |x - 1| : abs_mul x₀ (x - 1) - ... < |x₀| * (i / |x₀|) : mul_lt_mul' le_rfl hx (abs_nonneg (x - 1)) (abs_pos.2 hx₀) + ... < |x₀| * (i / |x₀|) : mul_lt_mul' le_rfl hx (by positivity) (abs_pos.2 hx₀) ... = |x₀| * i / |x₀| : by ring ... = i : mul_div_cancel_left i (λ h, hx₀ (abs_eq_zero.1 h)) }, { obtain ⟨i, hi, hit⟩ := h, @@ -1842,8 +1839,7 @@ begin calc |x / x₀ - 1| = |x / x₀ - x₀ / x₀| : (by rw div_self hx₀) ... = |(x - x₀) / x₀| : congr_arg abs (sub_div x x₀ x₀).symm ... = |x - x₀| / |x₀| : abs_div (x - x₀) x₀ - ... < i * |x₀| / |x₀| : div_lt_div hx le_rfl - (mul_nonneg (le_of_lt hi) (abs_nonneg x₀)) (abs_pos.2 hx₀) + ... < i * |x₀| / |x₀| : div_lt_div_of_lt (abs_pos.2 hx₀) hx ... = i : by rw [← mul_div_assoc', div_self (ne_of_lt $ abs_pos.2 hx₀).symm, mul_one], specialize hit (x / x₀) this, rwa [mul_div_assoc', mul_div_cancel_left x hx₀] at hit } @@ -1869,7 +1865,7 @@ begin refine ⟨lt_of_le_of_lt _ (mul_lt_mul'' ha hb hε' hε'), lt_of_lt_of_le (mul_lt_mul'' ha' hb' ha0 hb0) _⟩, { calc 1 - ε = 1 - ε / 2 - ε/2 : by ring_nf - ... ≤ 1 - ε/2 - ε/2 + (ε/2)*(ε/2) : le_add_of_nonneg_right (le_of_lt (mul_pos ε_pos' ε_pos')) + ... ≤ 1 - ε/2 - ε/2 + (ε/2)*(ε/2) : le_add_of_nonneg_right (by positivity) ... = (1 - ε/2) * (1 - ε/2) : by ring_nf ... ≤ (1 - ε/4) * (1 - ε/4) : mul_le_mul (by linarith) (by linarith) (by linarith) hε' }, { calc (1 + ε/4) * (1 + ε/4) = 1 + ε/2 + (ε/4)*(ε/4) : by ring_nf @@ -1976,7 +1972,7 @@ by simpa only [mul_comm] using hg.at_bot_mul_neg hC hf lemma tendsto_inv_zero_at_top : tendsto (λx:α, x⁻¹) (𝓝[>] (0:α)) at_top := begin refine (at_top_basis' 1).tendsto_right_iff.2 (λ b hb, _), - have hb' : 0 < b := zero_lt_one.trans_le hb, + have hb' : 0 < b := by positivity, filter_upwards [Ioc_mem_nhds_within_Ioi ⟨le_rfl, inv_pos.2 hb'⟩] with x hx using (le_inv hx.1 hb').1 hx.2, end @@ -2073,18 +2069,17 @@ instance linear_ordered_field.to_topological_division_ring : topological_divisio rw [continuous_at, (nhds_basis_Ioo_pos t).tendsto_iff $ nhds_basis_Ioo_pos_of_pos $ inv_pos.2 ht], rintros ε ⟨hε : ε > 0, hεt : ε ≤ t⁻¹⟩, - refine ⟨min (t ^ 2 * ε / 2) (t / 2), - lt_min (half_pos $ mul_pos (by nlinarith) hε) $ by linarith, λ x h, _⟩, + refine ⟨min (t ^ 2 * ε / 2) (t / 2), by positivity, λ x h, _⟩, have hx : t / 2 < x, { rw [set.mem_Ioo, sub_lt, lt_min_iff] at h, nlinarith }, have hx' : 0 < x := (half_pos ht).trans hx, - have aux : 0 < 2 / t ^ 2 := div_pos zero_lt_two (sq_pos_of_pos ht), + have aux : 0 < 2 / t ^ 2 := by positivity, rw [set.mem_Ioo, ←sub_lt_iff_lt_add', sub_lt, ←abs_sub_lt_iff] at h ⊢, rw [inv_sub_inv ht.ne' hx'.ne', abs_div, div_eq_mul_inv], suffices : |t * x|⁻¹ < 2 / t ^ 2, { rw [←abs_neg, neg_sub], - refine (mul_lt_mul'' h this (abs_nonneg _) $ inv_nonneg.mpr $ abs_nonneg _).trans_le _, + refine (mul_lt_mul'' h this (by positivity) (by positivity)).trans_le _, rw [mul_comm, mul_min_of_nonneg _ _ aux.le], apply min_le_of_left_le, rw [←mul_div, ←mul_assoc, div_mul_cancel _ (sq_pos_of_pos ht).ne', diff --git a/formal/lean/mathlib/topology/algebra/order/floor.lean b/formal/lean/mathlib/topology/algebra/order/floor.lean index 837b14a8f142308cca7b601633749054fc6bd614..26adfe819ab00a8d7ae836f56e16274649dcbc38 100644 --- a/formal/lean/mathlib/topology/algebra/order/floor.lean +++ b/formal/lean/mathlib/topology/algebra/order/floor.lean @@ -158,7 +158,7 @@ variables [order_topology α] [topological_add_group α] [topological_space β] /-- Do not use this, use `continuous_on.comp_fract` instead. -/ lemma continuous_on.comp_fract' {f : β → α → γ} - (h : continuous_on (uncurry f) $ (univ : set β) ×ˢ I) (hf : ∀ s, f s 0 = f s 1) : + (h : continuous_on (uncurry f) $ univ ×ˢ I) (hf : ∀ s, f s 0 = f s 1) : continuous (λ st : β × α, f st.1 $ fract st.2) := begin change continuous ((uncurry f) ∘ (prod.map id (fract))), @@ -167,7 +167,7 @@ begin by_cases ht : t = floor t, { rw ht, rw ← continuous_within_at_univ, - have : (univ : set (β × α)) ⊆ ((univ : set β) ×ˢ Iio ↑⌊t⌋) ∪ ((univ : set β) ×ˢ Ici ↑⌊t⌋), + have : (univ : set (β × α)) ⊆ (univ ×ˢ Iio ↑⌊t⌋) ∪ (univ ×ˢ Ici ↑⌊t⌋), { rintros p -, rw ← prod_union, exact ⟨trivial, lt_or_le p.2 _⟩ }, @@ -203,7 +203,7 @@ end lemma continuous_on.comp_fract {s : β → α} {f : β → α → γ} - (h : continuous_on (uncurry f) $ (univ : set β) ×ˢ (Icc 0 1 : set α)) + (h : continuous_on (uncurry f) $ univ ×ˢ Icc 0 1) (hs : continuous s) (hf : ∀ s, f s 0 = f s 1) : continuous (λ x : β, f x $ int.fract (s x)) := diff --git a/formal/lean/mathlib/topology/algebra/uniform_group.lean b/formal/lean/mathlib/topology/algebra/uniform_group.lean index 689a6bd99ef03c6ee1755c964bae170a93dd4b58..ee738e1bfb15d5cdbbbf2496dfb1f093a8c2c3e0 100644 --- a/formal/lean/mathlib/topology/algebra/uniform_group.lean +++ b/formal/lean/mathlib/topology/algebra/uniform_group.lean @@ -343,12 +343,12 @@ variables {ι : Type*} {l : filter ι} {f f' : ι → β → α} {g g' : β → end uniform_convergence end uniform_group -section topological_comm_group +section topological_group open filter -variables (G : Type*) [comm_group G] [topological_space G] [topological_group G] +variables (G : Type*) [group G] [topological_space G] [topological_group G] /-- The right uniformity on a topological group. -/ -@[to_additive "The right uniformity on a topological group"] +@[to_additive "The right uniformity on a topological additive group"] def topological_group.to_uniform_space : uniform_space G := { uniformity := comap (λp:G×G, p.2 / p.1) (𝓝 1), refl := @@ -375,7 +375,7 @@ def topological_group.to_uniform_space : uniform_space G := begin intros p p_comp_rel, rcases p_comp_rel with ⟨z, ⟨Hz1, Hz2⟩⟩, - simpa [sub_eq_add_neg, add_comm, add_left_comm] using V_sum _ Hz1 _ Hz2 + simpa using V_sum _ Hz2 _ Hz1, end, exact set.subset.trans comp_rel_sub U_sub }, { exact monotone_comp_rel monotone_id monotone_id } @@ -395,6 +395,11 @@ def topological_group.to_uniform_space : uniform_space G := { rintros h x hx, exact @h (a, x) hx rfl } end } +local attribute [instance] topological_group.to_uniform_space + +@[to_additive] lemma uniformity_eq_comap_nhds_one' : + 𝓤 G = comap (λp:G×G, p.2 / p.1) (𝓝 (1 : G)) := rfl + variables {G} @[to_additive] lemma topological_group.tendsto_uniformly_iff @@ -425,7 +430,7 @@ variables {G} ⟨λ h u hu, h _ ⟨u, hu, λ _, id⟩, λ h v ⟨u, hu, hv⟩ x, exists_imp_exists (by exact λ a, exists_imp_exists (λ ha hp, mem_of_superset hp (λ i hi a ha, hv (by exact hi a ha)))) ∘ h u hu x⟩ -end topological_comm_group +end topological_group section topological_comm_group universes u v w x @@ -436,9 +441,6 @@ variables (G : Type*) [comm_group G] [topological_space G] [topological_group G] section local attribute [instance] topological_group.to_uniform_space -@[to_additive] lemma uniformity_eq_comap_nhds_one' : - 𝓤 G = comap (λp:G×G, p.2 / p.1) (𝓝 (1 : G)) := rfl - variable {G} @[to_additive] lemma topological_group_is_uniform : uniform_group G := have tendsto @@ -494,7 +496,7 @@ end end @[to_additive] lemma uniform_group.to_uniform_space_eq {G : Type*} [u : uniform_space G] - [comm_group G] [uniform_group G] : topological_group.to_uniform_space G = u := + [group G] [uniform_group G] : topological_group.to_uniform_space G = u := begin ext : 1, show @uniformity G (topological_group.to_uniform_space G) = 𝓤 G, @@ -503,14 +505,14 @@ end end topological_comm_group -open comm_group filter set function +open filter set function section variables {α : Type*} {β : Type*} {hom : Type*} -variables [topological_space α] [comm_group α] [topological_group α] +variables [topological_space α] [group α] [topological_group α] -- β is a dense subgroup of α, inclusion is denoted by e -variables [topological_space β] [comm_group β] +variables [topological_space β] [group β] variables [monoid_hom_class hom β α] {e : hom} (de : dense_inducing e) include de diff --git a/formal/lean/mathlib/topology/algebra/uniform_mul_action.lean b/formal/lean/mathlib/topology/algebra/uniform_mul_action.lean index a541a115c4b66196e027c89e2595a46017b84707..9dd7a19bba2c910cc2d51f553818b25c25435afc 100644 --- a/formal/lean/mathlib/topology/algebra/uniform_mul_action.lean +++ b/formal/lean/mathlib/topology/algebra/uniform_mul_action.lean @@ -47,6 +47,25 @@ instance add_group.has_uniform_continuous_const_smul_int [add_group X] [uniform_ has_uniform_continuous_const_smul ℤ X := ⟨uniform_continuous_const_zsmul⟩ +/-- A `distrib_mul_action` that is continuous on a uniform group is uniformly continuous. +This can't be an instance due to it forming a loop with +`has_uniform_continuous_const_smul.to_has_continuous_const_smul` -/ +lemma has_uniform_continuous_const_smul_of_continuous_const_smul [monoid R] [add_comm_group M] + [distrib_mul_action R M] [uniform_space M] [uniform_add_group M] [has_continuous_const_smul R M] : + has_uniform_continuous_const_smul R M := +⟨λ r, uniform_continuous_of_continuous_at_zero (distrib_mul_action.to_add_monoid_hom M r) + (continuous.continuous_at (continuous_const_smul r))⟩ + +/-- The action of `semiring.to_module` is uniformly continuous. -/ +instance ring.has_uniform_continuous_const_smul [ring R] [uniform_space R] + [uniform_add_group R] [has_continuous_mul R] : has_uniform_continuous_const_smul R R := +has_uniform_continuous_const_smul_of_continuous_const_smul _ _ + +/-- The action of `semiring.to_opposite_module` is uniformly continuous. -/ +instance ring.has_uniform_continuous_const_op_smul [ring R] [uniform_space R] + [uniform_add_group R] [has_continuous_mul R] : has_uniform_continuous_const_smul Rᵐᵒᵖ R := +has_uniform_continuous_const_smul_of_continuous_const_smul _ _ + section has_smul variable [has_smul M X] diff --git a/formal/lean/mathlib/topology/basic.lean b/formal/lean/mathlib/topology/basic.lean index 1ac0b45759c1b10a4775f2f508d05ad963d0d7b2..c711427dc78b0c40bd3f30dc1dad72d4fabf70dd 100644 --- a/formal/lean/mathlib/topology/basic.lean +++ b/formal/lean/mathlib/topology/basic.lean @@ -18,8 +18,6 @@ Then `set α` gets predicates `is_open`, `is_closed` and functions `interior`, ` along `F : filter ι` if `map_cluster_pt x F f : cluster_pt x (map f F)`. In particular the notion of cluster point of a sequence `u` is `map_cluster_pt x at_top u`. -This file also defines locally finite families of subsets of `α`. - For topological spaces `α` and `β`, a function `f : α → β` and a point `a : α`, `continuous_at f a` means `f` is continuous at `a`, and global continuity is `continuous f`. There is also a version of continuity `pcontinuous` for @@ -141,8 +139,7 @@ finite.induction_on hs (λ a s has hs ih h, by rw bInter_insert; exact is_open.inter (h a (mem_insert _ _)) (ih (λ i hi, h i (mem_insert_of_mem _ hi)))) -lemma is_open_Inter [fintype β] {s : β → set α} - (h : ∀ i, is_open (s i)) : is_open (⋂ i, s i) := +lemma is_open_Inter [finite β] {s : β → set α} (h : ∀ i, is_open (s i)) : is_open (⋂ i, s i) := suffices is_open (⋂ (i : β) (hi : i ∈ @univ β), s i), by simpa, is_open_bInter finite_univ (λ i _, h i) @@ -206,8 +203,8 @@ finite.induction_on hs (λ a s has hs ih h, by rw bUnion_insert; exact is_closed.union (h a (mem_insert _ _)) (ih (λ i hi, h i (mem_insert_of_mem _ hi)))) -lemma is_closed_Union [fintype β] {s : β → set α} - (h : ∀ i, is_closed (s i)) : is_closed (Union s) := +lemma is_closed_Union [finite β] {s : β → set α} (h : ∀ i, is_closed (s i)) : + is_closed (⋃ i, s i) := suffices is_closed (⋃ (i : β) (hi : i ∈ @univ β), s i), by convert this; simp [set.ext_iff], is_closed_bUnion finite_univ (λ i _, h i) @@ -291,9 +288,9 @@ begin simp [h₂], end -@[simp] lemma interior_Inter_of_fintype {ι : Type*} [fintype ι] (f : ι → set α) : +@[simp] lemma interior_Inter {ι : Type*} [finite ι] (f : ι → set α) : interior (⋂ i, f i) = ⋂ i, interior (f i) := -by { convert finset.univ.interior_Inter f; simp, } +by { casesI nonempty_fintype ι, convert finset.univ.interior_Inter f; simp } lemma interior_union_is_closed_of_interior_empty {s t : set α} (h₁ : is_closed s) (h₂ : interior t = ∅) : @@ -422,9 +419,9 @@ begin simp [h₂], end -@[simp] lemma closure_Union_of_fintype {ι : Type*} [fintype ι] (f : ι → set α) : +@[simp] lemma closure_Union {ι : Type*} [finite ι] (f : ι → set α) : closure (⋃ i, f i) = ⋃ i, closure (f i) := -by { convert finset.univ.closure_bUnion f; simp, } +by { casesI nonempty_fintype ι, convert finset.univ.closure_bUnion f; simp } lemma interior_subset_closure {s : set α} : interior s ⊆ closure s := subset.trans interior_subset subset_closure diff --git a/formal/lean/mathlib/topology/bornology/basic.lean b/formal/lean/mathlib/topology/bornology/basic.lean index f292dcc2b0317d46827f6f90297e2226a79e854f..80f7ae0d71c7abf9187864b61eef897e736fedab 100644 --- a/formal/lean/mathlib/topology/bornology/basic.lean +++ b/formal/lean/mathlib/topology/bornology/basic.lean @@ -176,7 +176,7 @@ bInter_mem hs is_cobounded (⋂ i ∈ s, f i) ↔ ∀ i ∈ s, is_cobounded (f i) := bInter_finset_mem s -@[simp] lemma is_cobounded_Inter [fintype ι] {f : ι → set α} : +@[simp] lemma is_cobounded_Inter [finite ι] {f : ι → set α} : is_cobounded (⋂ i, f i) ↔ ∀ i, is_cobounded (f i) := Inter_mem @@ -196,7 +196,7 @@ lemma is_bounded_sUnion {S : set (set α)} (hs : S.finite) : is_bounded (⋃₀ S) ↔ (∀ s ∈ S, is_bounded s) := by rw [sUnion_eq_bUnion, is_bounded_bUnion hs] -@[simp] lemma is_bounded_Union [fintype ι] {s : ι → set α} : +@[simp] lemma is_bounded_Union [finite ι] {s : ι → set α} : is_bounded (⋃ i, s i) ↔ ∀ i, is_bounded (s i) := by rw [← sUnion_range, is_bounded_sUnion (finite_range s), forall_range_iff] diff --git a/formal/lean/mathlib/topology/bornology/constructions.lean b/formal/lean/mathlib/topology/bornology/constructions.lean index 1886ce80562f209e6e2f6095c6c377f4fc594e00..1cbbd2438f29fba8623d7986d3f17df28616d64a 100644 --- a/formal/lean/mathlib/topology/bornology/constructions.lean +++ b/formal/lean/mathlib/topology/bornology/constructions.lean @@ -145,3 +145,23 @@ alias bounded_space_coe_set_iff ↔ _ bornology.is_bounded.bounded_space_coe instance [bounded_space α] {p : α → Prop} : bounded_space (subtype p) := (is_bounded.all {x | p x}).bounded_space_subtype + +/-! +### `additive`, `multiplicative` + +The bornology on those type synonyms is inherited without change. +-/ + +instance : bornology (additive α) := ‹bornology α› +instance : bornology (multiplicative α) := ‹bornology α› +instance [bounded_space α] : bounded_space (additive α) := ‹bounded_space α› +instance [bounded_space α] : bounded_space (multiplicative α) := ‹bounded_space α› + +/-! +### Order dual + +The bornology on this type synonym is inherited without change. +-/ + +instance : bornology αᵒᵈ := ‹bornology α› +instance [bounded_space α] : bounded_space αᵒᵈ := ‹bounded_space α› diff --git a/formal/lean/mathlib/topology/category/Top/open_nhds.lean b/formal/lean/mathlib/topology/category/Top/open_nhds.lean index 5fad2e3daddcc3eb9a7e46d53c8513a2a2498170..19d3a759f5b6c435fe229ccdd3c1f43e6b2cf1cc 100644 --- a/formal/lean/mathlib/topology/category/Top/open_nhds.lean +++ b/formal/lean/mathlib/topology/category/Top/open_nhds.lean @@ -35,7 +35,7 @@ variables {X Y : Top.{u}} (f : X ⟶ Y) namespace topological_space /-- The type of open neighbourhoods of a point `x` in a (bundled) topological space. -/ -def open_nhds (x : X) := { U : opens X // x ∈ U } +def open_nhds (x : X) := full_subcategory (λ (U : opens X), x ∈ U) namespace open_nhds @@ -43,7 +43,7 @@ instance (x : X) : partial_order (open_nhds x) := { le := λ U V, U.1 ≤ V.1, le_refl := λ _, le_rfl, le_trans := λ _ _ _, le_trans, - le_antisymm := λ _ _ i j, subtype.eq $ le_antisymm i j } + le_antisymm := λ _ _ i j, full_subcategory.ext _ _ $ le_antisymm i j } instance (x : X) : lattice (open_nhds x) := { inf := λ U V, ⟨U.1 ⊓ V.1, ⟨U.2, V.2⟩⟩, @@ -92,7 +92,7 @@ lemma open_embedding {x : X} (U : open_nhds x) : open_embedding (U.1.inclusion) U.1.open_embedding def map (x : X) : open_nhds (f x) ⥤ open_nhds x := -{ obj := λ U, ⟨(opens.map f).obj U.1, by tidy⟩, +{ obj := λ U, ⟨(opens.map f).obj U.1, U.2⟩, map := λ U V i, (opens.map f).map i } @[simp] lemma map_obj (x : X) (U) (q) : (map f x).obj ⟨U, q⟩ = ⟨(opens.map f).obj U, by tidy⟩ := diff --git a/formal/lean/mathlib/topology/connected.lean b/formal/lean/mathlib/topology/connected.lean index 8e4c4755dfecf5415b6b5e31d07d4f009fe12391..0e161357a59a61b881e6d9baa1f245f27a99b32b 100644 --- a/formal/lean/mathlib/topology/connected.lean +++ b/formal/lean/mathlib/topology/connected.lean @@ -8,6 +8,7 @@ import data.nat.succ_pred import order.partial_sups import order.succ_pred.relation import topology.subset_properties +import tactic.congrm /-! # Connected subsets of topological spaces @@ -571,26 +572,85 @@ that contains this point. -/ def connected_component (x : α) : set α := ⋃₀ { s : set α | is_preconnected s ∧ x ∈ s } -/-- The connected component of a point inside a set. -/ -def connected_component_in (F : set α) (x : F) : set α := coe '' (connected_component x) +/-- Given a set `F` in a topological space `α` and a point `x : α`, the connected +component of `x` in `F` is the connected component of `x` in the subtype `F` seen as +a set in `α`. This definition does not make sense if `x` is not in `F` so we return the +empty set in this case. -/ +def connected_component_in (F : set α) (x : α) : set α := +if h : x ∈ F then coe '' (connected_component (⟨x, h⟩ : F)) else ∅ + +lemma connected_component_in_eq_image {F : set α} {x : α} (h : x ∈ F) : + connected_component_in F x = coe '' (connected_component (⟨x, h⟩ : F)) := +dif_pos h + +lemma connected_component_in_eq_empty {F : set α} {x : α} (h : x ∉ F) : + connected_component_in F x = ∅ := +dif_neg h theorem mem_connected_component {x : α} : x ∈ connected_component x := mem_sUnion_of_mem (mem_singleton x) ⟨is_connected_singleton.is_preconnected, mem_singleton x⟩ +theorem mem_connected_component_in {x : α} {F : set α} (hx : x ∈ F) : + x ∈ connected_component_in F x := +by simp [connected_component_in_eq_image hx, mem_connected_component, hx] + +theorem connected_component_nonempty {x : α} : + (connected_component x).nonempty := +⟨x, mem_connected_component⟩ + +theorem connected_component_in_nonempty_iff {x : α} {F : set α} : + (connected_component_in F x).nonempty ↔ x ∈ F := +by { rw [connected_component_in], split_ifs; simp [connected_component_nonempty, h] } + +theorem connected_component_in_subset (F : set α) (x : α) : + connected_component_in F x ⊆ F := +by { rw [connected_component_in], split_ifs; simp } + theorem is_preconnected_connected_component {x : α} : is_preconnected (connected_component x) := is_preconnected_sUnion x _ (λ _, and.right) (λ _, and.left) +lemma is_preconnected_connected_component_in {x : α} {F : set α} : + is_preconnected (connected_component_in F x) := +begin + rw [connected_component_in], split_ifs, + { exact embedding_subtype_coe.to_inducing.is_preconnected_image.mpr + is_preconnected_connected_component }, + { exact is_preconnected_empty }, +end + theorem is_connected_connected_component {x : α} : is_connected (connected_component x) := ⟨⟨x, mem_connected_component⟩, is_preconnected_connected_component⟩ +lemma is_connected_connected_component_in_iff {x : α} {F : set α} : + is_connected (connected_component_in F x) ↔ x ∈ F := +by simp_rw [← connected_component_in_nonempty_iff, is_connected, + is_preconnected_connected_component_in, and_true] + theorem is_preconnected.subset_connected_component {x : α} {s : set α} (H1 : is_preconnected s) (H2 : x ∈ s) : s ⊆ connected_component x := λ z hz, mem_sUnion_of_mem hz ⟨H1, H2⟩ +lemma is_preconnected.subset_connected_component_in {x : α} {F : set α} (hs : is_preconnected s) + (hxs : x ∈ s) (hsF : s ⊆ F) : s ⊆ connected_component_in F x := +begin + have : is_preconnected ((coe : F → α) ⁻¹' s), + { refine embedding_subtype_coe.to_inducing.is_preconnected_image.mp _, + rwa [subtype.image_preimage_coe, inter_eq_left_iff_subset.mpr hsF] }, + have h2xs : (⟨x, hsF hxs⟩ : F) ∈ coe ⁻¹' s := by { rw [mem_preimage], exact hxs }, + have := this.subset_connected_component h2xs, + rw [connected_component_in_eq_image (hsF hxs)], + refine subset.trans _ (image_subset _ this), + rw [subtype.image_preimage_coe, inter_eq_left_iff_subset.mpr hsF] +end + theorem is_connected.subset_connected_component {x : α} {s : set α} (H1 : is_connected s) (H2 : x ∈ s) : s ⊆ connected_component x := H1.2.subset_connected_component H2 +lemma is_preconnected.connected_component_in {x : α} {F : set α} (h : is_preconnected F) + (hx : x ∈ F) : connected_component_in F x = F := +(connected_component_in_subset F x).antisymm (h.subset_connected_component_in hx subset_rfl) + theorem connected_component_eq {x y : α} (h : y ∈ connected_component x) : connected_component x = connected_component y := eq_of_subset_of_subset @@ -599,6 +659,23 @@ eq_of_subset_of_subset (set.mem_of_mem_of_subset mem_connected_component (is_connected_connected_component.subset_connected_component h))) +lemma connected_component_in_eq {x y : α} {F : set α} (h : y ∈ connected_component_in F x) : + connected_component_in F x = connected_component_in F y := +begin + have hx : x ∈ F := connected_component_in_nonempty_iff.mp ⟨y, h⟩, + simp_rw [connected_component_in_eq_image hx] at h ⊢, + obtain ⟨⟨y, hy⟩, h2y, rfl⟩ := h, + simp_rw [subtype.coe_mk, connected_component_in_eq_image hy, connected_component_eq h2y] +end + +theorem connected_component_in_univ (x : α) : + connected_component_in univ x = connected_component x := +subset_antisymm + (is_preconnected_connected_component_in.subset_connected_component $ + mem_connected_component_in trivial) + (is_preconnected_connected_component.subset_connected_component_in mem_connected_component $ + subset_univ _) + lemma connected_component_disjoint {x y : α} (h : connected_component x ≠ connected_component y) : disjoint (connected_component x) (connected_component y) := set.disjoint_left.2 (λ a h1 h2, h @@ -625,6 +702,18 @@ theorem irreducible_component_subset_connected_component {x : α} : is_irreducible_irreducible_component.is_connected.subset_connected_component mem_irreducible_component +@[mono] +lemma connected_component_in_mono (x : α) {F G : set α} (h : F ⊆ G) : + connected_component_in F x ⊆ connected_component_in G x := +begin + by_cases hx : x ∈ F, + { rw [connected_component_in_eq_image hx, connected_component_in_eq_image (h hx), + ← show (coe : G → α) ∘ inclusion h = coe, by ext ; refl, image_comp], + exact image_subset coe ((continuous_inclusion h).image_connected_component_subset ⟨x, hx⟩) }, + { rw connected_component_in_eq_empty hx, + exact set.empty_subset _ }, +end + /-- A preconnected space is one where there is no non-trivial open partition. -/ class preconnected_space (α : Type u) [topological_space α] : Prop := (is_preconnected_univ : is_preconnected (univ : set α)) @@ -1035,6 +1124,117 @@ by rw [← hf.preimage_connected_component h_fibers, image_preimage_eq _ hf.surj end preconnected +section locally_connected_space + +/-- A topological space is **locally connected** if each neighborhood filter admits a basis +of connected *open* sets. Note that it is equivalent to each point having a basis of connected +(non necessarily open) sets but in a non-trivial way, so we choose this definition and prove the +equivalence later in `locally_connected_space_iff_connected_basis`. -/ +class locally_connected_space (α : Type*) [topological_space α] : Prop := +(open_connected_basis : ∀ x, (𝓝 x).has_basis (λ s : set α, is_open s ∧ x ∈ s ∧ is_connected s) id) + +lemma locally_connected_space_iff_open_connected_basis : locally_connected_space α ↔ + ∀ x, (𝓝 x).has_basis (λ s : set α, is_open s ∧ x ∈ s ∧ is_connected s) id := +⟨@locally_connected_space.open_connected_basis _ _, locally_connected_space.mk⟩ + +lemma locally_connected_space_iff_open_connected_subsets : + locally_connected_space α ↔ ∀ (x : α) (U ∈ 𝓝 x), ∃ V ⊆ U, is_open V ∧ x ∈ V ∧ is_connected V := +begin + rw locally_connected_space_iff_open_connected_basis, + congrm ∀ x, (_ : Prop), + split, + { intros h U hU, + rcases h.mem_iff.mp hU with ⟨V, hV, hVU⟩, + exact ⟨V, hVU, hV⟩ }, + { exact λ h, ⟨λ U, ⟨λ hU, let ⟨V, hVU, hV⟩ := h U hU in ⟨V, hV, hVU⟩, + λ ⟨V, ⟨hV, hxV, _⟩, hVU⟩, mem_nhds_iff.mpr ⟨V, hVU, hV, hxV⟩⟩⟩ } +end + +lemma connected_component_in_mem_nhds [locally_connected_space α] {F : set α} {x : α} + (h : F ∈ 𝓝 x) : + connected_component_in F x ∈ 𝓝 x := +begin + rw (locally_connected_space.open_connected_basis x).mem_iff at h, + rcases h with ⟨s, ⟨h1s, hxs, h2s⟩, hsF⟩, + exact mem_nhds_iff.mpr ⟨s, h2s.is_preconnected.subset_connected_component_in hxs hsF, h1s, hxs⟩ +end + +lemma is_open.connected_component_in [locally_connected_space α] {F : set α} {x : α} + (hF : is_open F) : + is_open (connected_component_in F x) := +begin + rw [is_open_iff_mem_nhds], + intros y hy, + rw [connected_component_in_eq hy], + exact connected_component_in_mem_nhds (is_open_iff_mem_nhds.mp hF y $ + connected_component_in_subset F x hy) +end + +lemma is_open_connected_component [locally_connected_space α] {x : α} : + is_open (connected_component x) := +begin + rw ← connected_component_in_univ, + exact is_open_univ.connected_component_in +end + +lemma is_clopen_connected_component [locally_connected_space α] {x : α} : + is_clopen (connected_component x) := +⟨is_open_connected_component, is_closed_connected_component⟩ + +lemma locally_connected_space_iff_connected_component_in_open : + locally_connected_space α ↔ ∀ F : set α, is_open F → ∀ x ∈ F, + is_open (connected_component_in F x) := +begin + split, + { introI h, + exact λ F hF x _, hF.connected_component_in }, + { intro h, + rw locally_connected_space_iff_open_connected_subsets, + refine (λ x U hU, ⟨connected_component_in (interior U) x, + (connected_component_in_subset _ _).trans interior_subset, h _ is_open_interior x _, + mem_connected_component_in _, is_connected_connected_component_in_iff.mpr _⟩); + exact (mem_interior_iff_mem_nhds.mpr hU) } +end + +lemma locally_connected_space_iff_connected_subsets : + locally_connected_space α ↔ ∀ (x : α) (U ∈ 𝓝 x), ∃ V ∈ 𝓝 x, is_preconnected V ∧ V ⊆ U := +begin + split, + { rw locally_connected_space_iff_open_connected_subsets, + intros h x U hxU, + rcases h x U hxU with ⟨V, hVU, hV₁, hxV, hV₂⟩, + exact ⟨V, hV₁.mem_nhds hxV, hV₂.is_preconnected, hVU⟩ }, + { rw locally_connected_space_iff_connected_component_in_open, + refine λ h U hU x hxU, is_open_iff_mem_nhds.mpr (λ y hy, _), + rw connected_component_in_eq hy, + rcases h y U (hU.mem_nhds $ (connected_component_in_subset _ _) hy) with ⟨V, hVy, hV, hVU⟩, + exact filter.mem_of_superset hVy + (hV.subset_connected_component_in (mem_of_mem_nhds hVy) hVU) } +end + +lemma locally_connected_space_iff_connected_basis : + locally_connected_space α ↔ + ∀ x, (𝓝 x).has_basis (λ s : set α, s ∈ 𝓝 x ∧ is_preconnected s) id := +begin + rw locally_connected_space_iff_connected_subsets, + congrm ∀ x, (_ : Prop), + exact filter.has_basis_self.symm +end + +lemma locally_connected_space_of_connected_bases {ι : Type*} (b : α → ι → set α) (p : α → ι → Prop) + (hbasis : ∀ x, (𝓝 x).has_basis (p x) (b x)) + (hconnected : ∀ x i, p x i → is_preconnected (b x i)) : + locally_connected_space α := +begin + rw locally_connected_space_iff_connected_basis, + exact λ x, (hbasis x).to_has_basis + (λ i hi, ⟨b x i, ⟨(hbasis x).mem_of_mem hi, hconnected x i hi⟩, subset_rfl⟩) + (λ s hs, ⟨(hbasis x).index s hs.1, + ⟨(hbasis x).property_index hs.1, (hbasis x).set_index_subset hs.1⟩⟩) +end + +end locally_connected_space + section totally_disconnected /-- A set `s` is called totally disconnected if every subset `t ⊆ s` which is preconnected is @@ -1280,7 +1480,7 @@ def continuous.connected_components_lift (h : continuous f) : @[continuity] lemma continuous.connected_components_lift_continuous (h : continuous f) : continuous h.connected_components_lift := -continuous_quotient_lift_on' h.image_eq_of_connected_component_eq h +h.quotient_lift_on' h.image_eq_of_connected_component_eq @[simp] lemma continuous.connected_components_lift_apply_coe (h : continuous f) (x : α) : h.connected_components_lift x = f x := rfl diff --git a/formal/lean/mathlib/topology/constructions.lean b/formal/lean/mathlib/topology/constructions.lean index 8292f143f06a4d7ac4cc62880063ac54ebd241db..d83c88863d9a87ccb3c95572097b61321609738e 100644 --- a/formal/lean/mathlib/topology/constructions.lean +++ b/formal/lean/mathlib/topology/constructions.lean @@ -67,6 +67,76 @@ instance Pi.topological_space {β : α → Type v} [t₂ : Πa, topological_spac instance ulift.topological_space [t : topological_space α] : topological_space (ulift.{v u} α) := t.induced ulift.down +/-! +### `additive`, `multiplicative` + +The topology on those type synonyms is inherited without change. +-/ + +section +variables [topological_space α] + +open additive multiplicative + +instance : topological_space (additive α) := ‹topological_space α› +instance : topological_space (multiplicative α) := ‹topological_space α› +instance [discrete_topology α] : discrete_topology (additive α) := ‹discrete_topology α› +instance [discrete_topology α] : discrete_topology (multiplicative α) := ‹discrete_topology α› + +lemma continuous_of_mul : continuous (of_mul : α → additive α) := continuous_id +lemma continuous_to_mul : continuous (to_mul : additive α → α) := continuous_id +lemma continuous_of_add : continuous (of_add : α → multiplicative α) := continuous_id +lemma continuous_to_add : continuous (to_add : multiplicative α → α) := continuous_id + +lemma is_open_map_of_mul : is_open_map (of_mul : α → additive α) := is_open_map.id +lemma is_open_map_to_mul : is_open_map (to_mul : additive α → α) := is_open_map.id +lemma is_open_map_of_add : is_open_map (of_add : α → multiplicative α) := is_open_map.id +lemma is_open_map_to_add : is_open_map (to_add : multiplicative α → α) := is_open_map.id + +lemma is_closed_map_of_mul : is_closed_map (of_mul : α → additive α) := is_closed_map.id +lemma is_closed_map_to_mul : is_closed_map (to_mul : additive α → α) := is_closed_map.id +lemma is_closed_map_of_add : is_closed_map (of_add : α → multiplicative α) := is_closed_map.id +lemma is_closed_map_to_add : is_closed_map (to_add : multiplicative α → α) := is_closed_map.id + +local attribute [semireducible] nhds + +lemma nhds_of_mul (a : α) : 𝓝 (of_mul a) = map of_mul (𝓝 a) := rfl +lemma nhds_of_add (a : α) : 𝓝 (of_add a) = map of_add (𝓝 a) := rfl +lemma nhds_to_mul (a : additive α) : 𝓝 (to_mul a) = map to_mul (𝓝 a) := rfl +lemma nhds_to_add (a : multiplicative α) : 𝓝 (to_add a) = map to_add (𝓝 a) := rfl + +end + +/-! +### Order dual + +The topology on this type synonym is inherited without change. +-/ + +section +variables [topological_space α] + +open order_dual + +instance : topological_space αᵒᵈ := ‹topological_space α› +instance [discrete_topology α] : discrete_topology (αᵒᵈ) := ‹discrete_topology α› + +lemma continuous_to_dual : continuous (to_dual : α → αᵒᵈ) := continuous_id +lemma continuous_of_dual : continuous (of_dual : αᵒᵈ → α) := continuous_id + +lemma is_open_map_to_dual : is_open_map (to_dual : α → αᵒᵈ) := is_open_map.id +lemma is_open_map_of_dual : is_open_map (of_dual : αᵒᵈ → α) := is_open_map.id + +lemma is_closed_map_to_dual : is_closed_map (to_dual : α → αᵒᵈ) := is_closed_map.id +lemma is_closed_map_of_dual : is_closed_map (of_dual : αᵒᵈ → α) := is_closed_map.id + +local attribute [semireducible] nhds + +lemma nhds_to_dual (a : α) : 𝓝 (to_dual a) = map to_dual (𝓝 a) := rfl +lemma nhds_of_dual (a : α) : 𝓝 (of_dual a) = map of_dual (𝓝 a) := rfl + +end + lemma quotient.preimage_mem_nhds [topological_space α] [s : setoid α] {V : set $ quotient s} {a : α} (hs : V ∈ 𝓝 (quotient.mk a)) : quotient.mk ⁻¹' V ∈ 𝓝 a := preimage_nhds_coinduced hs @@ -233,6 +303,10 @@ hf.comp continuous_at_snd (hf : continuous f) (hg : continuous g) : continuous (λx, (f x, g x)) := continuous_inf_rng.2 ⟨continuous_induced_rng.2 hf, continuous_induced_rng.2 hg⟩ +@[simp] lemma continuous_prod_mk {f : α → β} {g : α → γ} : + continuous (λ x, (f x, g x)) ↔ continuous f ∧ continuous g := +⟨λ h, ⟨h.fst, h.snd⟩, λ h, h.1.prod_mk h.2⟩ + @[continuity] lemma continuous.prod.mk (a : α) : continuous (λ b : β, (a, b)) := continuous_const.prod_mk continuous_id' @@ -524,7 +598,7 @@ empty -/ lemma is_open_prod_iff' {s : set α} {t : set β} : is_open (s ×ˢ t) ↔ (is_open s ∧ is_open t) ∨ (s = ∅) ∨ (t = ∅) := begin - cases (s ×ˢ t : set _).eq_empty_or_nonempty with h h, + cases (s ×ˢ t).eq_empty_or_nonempty with h h, { simp [h, prod_eq_empty_iff.1 h] }, { have st : s.nonempty ∧ t.nonempty, from prod_nonempty_iff.1 h, split, @@ -557,11 +631,11 @@ lemma frontier_prod_eq (s : set α) (t : set β) : by simp only [frontier, closure_prod_eq, interior_prod_eq, prod_diff_prod] @[simp] lemma frontier_prod_univ_eq (s : set α) : - frontier (s ×ˢ (univ : set β)) = frontier s ×ˢ (univ : set β) := + frontier (s ×ˢ (univ : set β)) = frontier s ×ˢ univ := by simp [frontier_prod_eq] @[simp] lemma frontier_univ_prod_eq (s : set β) : - frontier ((univ : set α) ×ˢ s) = (univ : set α) ×ˢ (frontier s) := + frontier ((univ : set α) ×ˢ s) = univ ×ˢ frontier s := by simp [frontier_prod_eq] lemma map_mem_closure2 {s : set α} {t : set β} {u : set γ} {f : α → β → γ} {a : α} {b : β} @@ -823,6 +897,20 @@ lemma closure_subtype {x : {a // p a}} {s : set {a // p a}}: x ∈ closure s ↔ (x : α) ∈ closure ((coe : _ → α) '' s) := closure_induced +lemma continuous_at_cod_restrict_iff {f : α → β} {t : set β} (h1 : ∀ x, f x ∈ t) {x : α} : + continuous_at (cod_restrict f t h1) x ↔ continuous_at f x := +by simp_rw [inducing_coe.continuous_at_iff, function.comp, coe_cod_restrict_apply] + +alias continuous_at_cod_restrict_iff ↔ _ continuous_at.cod_restrict + +lemma continuous_at.restrict {f : α → β} {s : set α} {t : set β} (h1 : maps_to f s t) {x : s} + (h2 : continuous_at f x) : continuous_at (h1.restrict f s t) x := +(h2.comp continuous_at_subtype_coe).cod_restrict _ + +lemma continuous_at.restrict_preimage {f : α → β} {s : set β} {x : f ⁻¹' s} + (h : continuous_at f x) : continuous_at (s.restrict_preimage f) x := +h.restrict _ + @[continuity] lemma continuous.cod_restrict {f : α → β} {s : set β} (hf : continuous f) (hs : ∀ a, f a ∈ s) : continuous (s.cod_restrict f hs) := continuous_subtype_mk hs hf @@ -856,13 +944,18 @@ quotient_map_quot_mk lemma continuous_quotient_mk : continuous (@quotient.mk α s) := continuous_coinduced_rng -lemma continuous_quotient_lift {f : α → β} (hs : ∀ a b, a ≈ b → f a = f b) - (h : continuous f) : continuous (quotient.lift f hs : quotient s → β) := +lemma continuous.quotient_lift {f : α → β} (h : continuous f) (hs : ∀ a b, a ≈ b → f a = f b) : + continuous (quotient.lift f hs : quotient s → β) := continuous_coinduced_dom.2 h -lemma continuous_quotient_lift_on' {f : α → β} (hs : ∀ a b, a ≈ b → f a = f b) - (h : continuous f) : continuous (λ x, quotient.lift_on' x f hs : quotient s → β) := -continuous_coinduced_dom.2 h +lemma continuous.quotient_lift_on' {f : α → β} (h : continuous f) + (hs : ∀ a b, @setoid.r _ s a b → f a = f b) : + continuous (λ x, quotient.lift_on' x f hs : quotient s → β) := +h.quotient_lift hs + +lemma continuous.quotient_map' {t : setoid β} {f : α → β} (hf : continuous f) + (H : (s.r ⇒ t.r) f f) : continuous (quotient.map' f H) := +(continuous_quotient_mk.comp hf).quotient_lift _ end quotient @@ -1051,7 +1144,7 @@ begin erw induced_compose, end -variables [fintype ι] [∀ i, topological_space (π i)] [∀ i, discrete_topology (π i)] +variables [finite ι] [∀ i, topological_space (π i)] [∀ i, discrete_topology (π i)] /-- A finite product of discrete spaces is discrete. -/ instance Pi.discrete_topology : discrete_topology (Π i, π i) := diff --git a/formal/lean/mathlib/topology/continuous_function/basic.lean b/formal/lean/mathlib/topology/continuous_function/basic.lean index 49f46cc646133b7a20a8ee5caf1552879cda3e94..f4f97f381c459dab8c2d5afd86a383ba5f853edf 100644 --- a/formal/lean/mathlib/topology/continuous_function/basic.lean +++ b/formal/lean/mathlib/topology/continuous_function/basic.lean @@ -211,6 +211,11 @@ def restrict (f : C(α, β)) : C(s, β) := ⟨f ∘ coe⟩ @[simp] lemma coe_restrict (f : C(α, β)) : ⇑(f.restrict s) = f ∘ coe := rfl +/-- The restriction of a continuous map onto the preimage of a set. -/ +@[simps] +def restrict_preimage (f : C(α, β)) (s : set β) : C(f ⁻¹' s, s) := +⟨s.restrict_preimage f, continuous_iff_continuous_at.mpr $ λ x, f.2.continuous_at.restrict_preimage⟩ + end restrict section gluing @@ -286,9 +291,31 @@ end gluing end continuous_map -/-- -The forward direction of a homeomorphism, as a bundled continuous map. --/ +namespace homeomorph +variables {α β γ : Type*} [topological_space α] [topological_space β] [topological_space γ] +variables (f : α ≃ₜ β) (g : β ≃ₜ γ) + +/-- The forward direction of a homeomorphism, as a bundled continuous map. -/ @[simps] -def homeomorph.to_continuous_map {α β : Type*} [topological_space α] [topological_space β] - (e : α ≃ₜ β) : C(α, β) := ⟨e⟩ +def to_continuous_map (e : α ≃ₜ β) : C(α, β) := ⟨e⟩ + +/--`homeomorph.to_continuous_map` as a coercion. -/ +instance : has_coe (α ≃ₜ β) C(α, β) := ⟨homeomorph.to_continuous_map⟩ + +lemma to_continuous_map_as_coe : f.to_continuous_map = f := rfl + +@[simp] lemma coe_refl : (homeomorph.refl α : C(α, α)) = continuous_map.id α := rfl + +@[simp] lemma coe_trans : (f.trans g : C(α, γ)) = (g : C(β, γ)).comp f := rfl + +/-- Left inverse to a continuous map from a homeomorphism, mirroring `equiv.symm_comp_self`. -/ +@[simp] lemma symm_comp_to_continuous_map : + (f.symm : C(β, α)).comp (f : C(α, β)) = continuous_map.id α := +by rw [← coe_trans, self_trans_symm, coe_refl] + +/-- Right inverse to a continuous map from a homeomorphism, mirroring `equiv.self_comp_symm`. -/ +@[simp] lemma to_continuous_map_comp_symm : + (f : C(α, β)).comp (f.symm : C(β, α)) = continuous_map.id β := +by rw [← coe_trans, symm_trans_self, coe_refl] + +end homeomorph diff --git a/formal/lean/mathlib/topology/continuous_on.lean b/formal/lean/mathlib/topology/continuous_on.lean index 33faa2bdbcd2ec601fa65675e3c121ee44d6e65d..492859e552e2212640eae5efaa895fcdad6dcc69 100644 --- a/formal/lean/mathlib/topology/continuous_on.lean +++ b/formal/lean/mathlib/topology/continuous_on.lean @@ -283,7 +283,7 @@ begin simp only [infi_inf_eq] end -lemma nhds_within_pi_univ_eq {ι : Type*} {α : ι → Type*} [fintype ι] [Π i, topological_space (α i)] +lemma nhds_within_pi_univ_eq {ι : Type*} {α : ι → Type*} [finite ι] [Π i, topological_space (α i)] (s : Π i, set (α i)) (x : Π i, α i) : 𝓝[pi univ s] x = ⨅ i, comap (λ x, x i) 𝓝[s i] (x i) := by simpa [nhds_within] using nhds_within_pi_eq finite_univ s x diff --git a/formal/lean/mathlib/topology/fiber_bundle.lean b/formal/lean/mathlib/topology/fiber_bundle.lean index baae1192a3ec64d7a490b8a1251a82097f43c2e5..821fc8a1f5b7b61c227318cc3e04d97cdf5a7042 100644 --- a/formal/lean/mathlib/topology/fiber_bundle.lean +++ b/formal/lean/mathlib/topology/fiber_bundle.lean @@ -164,13 +164,13 @@ below as `trivialization F proj`) if the total space has not been given a topolo have a topology on both the fiber and the base space. Through the construction `topological_fiber_prebundle F proj` it will be possible to promote a `pretrivialization F proj` to a `trivialization F proj`. -/ -@[ext, nolint has_inhabited_instance] +@[ext, nolint has_nonempty_instance] structure topological_fiber_bundle.pretrivialization (proj : Z → B) extends local_equiv Z (B × F) := (open_target : is_open target) (base_set : set B) (open_base_set : is_open base_set) (source_eq : source = proj ⁻¹' base_set) -(target_eq : target = base_set ×ˢ (univ : set F)) +(target_eq : target = base_set ×ˢ univ) (proj_to_fun : ∀ p ∈ source, (to_fun p).1 = proj p) open topological_fiber_bundle @@ -233,8 +233,7 @@ begin end @[simp, mfld_simps] lemma preimage_symm_proj_inter (s : set B) : - (e.to_local_equiv.symm ⁻¹' (proj ⁻¹' s)) ∩ e.base_set ×ˢ (univ : set F) = - (s ∩ e.base_set) ×ˢ (univ : set F) := + (e.to_local_equiv.symm ⁻¹' (proj ⁻¹' s)) ∩ e.base_set ×ˢ univ = (s ∩ e.base_set) ×ˢ univ := begin ext ⟨x, y⟩, suffices : x ∈ e.base_set → (proj (e.to_local_equiv.symm (x, y)) ∈ s ↔ x ∈ s), @@ -244,13 +243,11 @@ begin end lemma target_inter_preimage_symm_source_eq (e f : pretrivialization F proj) : - f.target ∩ (f.to_local_equiv.symm) ⁻¹' e.source - = (e.base_set ∩ f.base_set) ×ˢ (univ : set F) := + f.target ∩ (f.to_local_equiv.symm) ⁻¹' e.source = (e.base_set ∩ f.base_set) ×ˢ univ := by rw [inter_comm, f.target_eq, e.source_eq, f.preimage_symm_proj_inter] lemma trans_source (e f : pretrivialization F proj) : - (f.to_local_equiv.symm.trans e.to_local_equiv).source - = (e.base_set ∩ f.base_set) ×ˢ (univ : set F) := + (f.to_local_equiv.symm.trans e.to_local_equiv).source = (e.base_set ∩ f.base_set) ×ˢ univ := by rw [local_equiv.trans_source, local_equiv.symm_source, e.target_inter_preimage_symm_source_eq] lemma symm_trans_symm (e e' : pretrivialization F proj) : @@ -259,14 +256,12 @@ lemma symm_trans_symm (e e' : pretrivialization F proj) : by rw [local_equiv.trans_symm_eq_symm_trans_symm, local_equiv.symm_symm] lemma symm_trans_source_eq (e e' : pretrivialization F proj) : - (e.to_local_equiv.symm.trans e'.to_local_equiv).source = - (e.base_set ∩ e'.base_set) ×ˢ (univ : set F) := + (e.to_local_equiv.symm.trans e'.to_local_equiv).source = (e.base_set ∩ e'.base_set) ×ˢ univ := by rw [local_equiv.trans_source, e'.source_eq, local_equiv.symm_source, e.target_eq, inter_comm, e.preimage_symm_proj_inter, inter_comm] lemma symm_trans_target_eq (e e' : pretrivialization F proj) : - (e.to_local_equiv.symm.trans e'.to_local_equiv).target = - (e.base_set ∩ e'.base_set) ×ˢ (univ : set F) := + (e.to_local_equiv.symm.trans e'.to_local_equiv).target = (e.base_set ∩ e'.base_set) ×ˢ univ := by rw [← local_equiv.symm_source, symm_trans_symm, symm_trans_source_eq, inter_comm] end topological_fiber_bundle.pretrivialization @@ -278,13 +273,13 @@ A structure extending local homeomorphisms, defining a local trivialization of a `proj : Z → B` with fiber `F`, as a local homeomorphism between `Z` and `B × F` defined between two sets of the form `proj ⁻¹' base_set` and `base_set × F`, acting trivially on the first coordinate. -/ -@[ext, nolint has_inhabited_instance] +@[ext, nolint has_nonempty_instance] structure topological_fiber_bundle.trivialization (proj : Z → B) extends local_homeomorph Z (B × F) := (base_set : set B) (open_base_set : is_open base_set) (source_eq : source = proj ⁻¹' base_set) -(target_eq : target = base_set ×ˢ (univ : set F)) +(target_eq : target = base_set ×ˢ univ) (proj_to_fun : ∀ p ∈ source, (to_local_homeomorph p).1 = proj p) open topological_fiber_bundle @@ -349,13 +344,11 @@ e.to_pretrivialization.apply_symm_apply' hx e.to_pretrivialization.symm_apply_mk_proj ex lemma symm_trans_source_eq (e e' : trivialization F proj) : - (e.to_local_equiv.symm.trans e'.to_local_equiv).source - = (e.base_set ∩ e'.base_set) ×ˢ (univ : set F) := + (e.to_local_equiv.symm.trans e'.to_local_equiv).source = (e.base_set ∩ e'.base_set) ×ˢ univ := pretrivialization.symm_trans_source_eq e.to_pretrivialization e' lemma symm_trans_target_eq (e e' : trivialization F proj) : - (e.to_local_equiv.symm.trans e'.to_local_equiv).target - = (e.base_set ∩ e'.base_set) ×ˢ (univ : set F) := + (e.to_local_equiv.symm.trans e'.to_local_equiv).target = (e.base_set ∩ e'.base_set) ×ˢ univ := pretrivialization.symm_trans_target_eq e.to_pretrivialization e' lemma coe_fst_eventually_eq_proj (ex : x ∈ e.source) : prod.fst ∘ e =ᶠ[𝓝 x] proj := @@ -644,7 +637,7 @@ end comap namespace topological_fiber_bundle.trivialization lemma is_image_preimage_prod (e : trivialization F proj) (s : set B) : - e.to_local_homeomorph.is_image (proj ⁻¹' s) (s ×ˢ (univ : set F)) := + e.to_local_homeomorph.is_image (proj ⁻¹' s) (s ×ˢ univ) := λ x hx, by simp [e.coe_fst', hx] /-- Restrict a `trivialization` to an open set in the base. `-/ @@ -675,7 +668,7 @@ noncomputable def piecewise (e e' : trivialization F proj) (s : set B) (Heq : eq_on e e' $ proj ⁻¹' (e.base_set ∩ frontier s)) : trivialization F proj := { to_local_homeomorph := e.to_local_homeomorph.piecewise e'.to_local_homeomorph - (proj ⁻¹' s) (s ×ˢ (univ : set F)) (e.is_image_preimage_prod s) (e'.is_image_preimage_prod s) + (proj ⁻¹' s) (s ×ˢ univ) (e.is_image_preimage_prod s) (e'.is_image_preimage_prod s) (by rw [e.frontier_preimage, e'.frontier_preimage, Hs]) (by rwa e.frontier_preimage), base_set := s.ite e.base_set e'.base_set, @@ -833,7 +826,7 @@ Trivialization changes from `i` to `j` are given by continuous maps `coord_chang `base_set i ∩ base_set j` to the set of homeomorphisms of `F`, but we express them as maps `B → F → F` and require continuity on `(base_set i ∩ base_set j) × F` to avoid the topology on the space of continuous maps on `F`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure topological_fiber_bundle_core (ι : Type*) (B : Type*) [topological_space B] (F : Type*) [topological_space F] := (base_set : ι → set B) @@ -843,7 +836,7 @@ structure topological_fiber_bundle_core (ι : Type*) (B : Type*) [topological_sp (coord_change : ι → ι → B → F → F) (coord_change_self : ∀ i, ∀ x ∈ base_set i, ∀ v, coord_change i i x v = v) (coord_change_continuous : ∀ i j, continuous_on (λp : B × F, coord_change i j p.1 p.2) - (((base_set i) ∩ (base_set j)) ×ˢ (univ : set F))) + (((base_set i) ∩ (base_set j)) ×ˢ univ)) (coord_change_comp : ∀ i j k, ∀ x ∈ (base_set i) ∩ (base_set j) ∩ (base_set k), ∀ v, (coord_change j k x) (coord_change i j x v) = coord_change i k x v) @@ -854,7 +847,7 @@ variables [topological_space B] [topological_space F] (Z : topological_fiber_bun include Z /-- The index set of a topological fiber bundle core, as a convenience function for dot notation -/ -@[nolint unused_arguments has_inhabited_instance] +@[nolint unused_arguments has_nonempty_instance] def index := ι /-- The base space of a topological fiber bundle core, as a convenience function for dot notation -/ @@ -863,7 +856,7 @@ def base := B /-- The fiber of a topological fiber bundle core, as a convenience function for dot notation and typeclass inference -/ -@[nolint unused_arguments has_inhabited_instance] +@[nolint unused_arguments has_nonempty_instance] def fiber (x : B) := F section fiber_instances @@ -884,8 +877,8 @@ def total_space := bundle.total_space Z.fiber /-- Local homeomorphism version of the trivialization change. -/ def triv_change (i j : ι) : local_homeomorph (B × F) (B × F) := -{ source := (Z.base_set i ∩ Z.base_set j) ×ˢ (univ : set F), - target := (Z.base_set i ∩ Z.base_set j) ×ˢ (univ : set F), +{ source := (Z.base_set i ∩ Z.base_set j) ×ˢ univ, + target := (Z.base_set i ∩ Z.base_set j) ×ˢ univ, to_fun := λp, ⟨p.1, Z.coord_change i j p.1 p.2⟩, inv_fun := λp, ⟨p.1, Z.coord_change j i p.1 p.2⟩, map_source' := λp hp, by simpa using hp, @@ -927,7 +920,7 @@ and use `Z.local_triv` instead. -/ def local_triv_as_local_equiv (i : ι) : local_equiv Z.total_space (B × F) := { source := Z.proj ⁻¹' (Z.base_set i), - target := Z.base_set i ×ˢ (univ : set F), + target := Z.base_set i ×ˢ univ, inv_fun := λp, ⟨p.1, Z.coord_change i (Z.index_at p.1) p.1 p.2⟩, to_fun := λp, ⟨p.1, Z.coord_change (Z.index_at p.1) i p.1 p.2⟩, map_source' := λp hp, @@ -992,7 +985,7 @@ lemma open_source' (i : ι) : is_open (Z.local_triv_as_local_equiv i).source := begin apply topological_space.generate_open.basic, simp only [exists_prop, mem_Union, mem_singleton_iff], - refine ⟨i, Z.base_set i ×ˢ (univ : set F), (Z.is_open_base_set i).prod is_open_univ, _⟩, + refine ⟨i, Z.base_set i ×ˢ univ, (Z.is_open_base_set i).prod is_open_univ, _⟩, ext p, simp only [local_triv_as_local_equiv_apply, prod_mk_mem_set_prod_eq, mem_inter_eq, and_self, mem_local_triv_as_local_equiv_source, and_true, mem_univ, mem_preimage], @@ -1166,7 +1159,7 @@ open topological_fiber_bundle equivalences but there is not yet a topology on the total space. The total space is hence given a topology in such a way that there is a fiber bundle structure for which the local equivalences are also local homeomorphism and hence local trivializations. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure topological_fiber_prebundle (proj : Z → B) := (pretrivialization_atlas : set (pretrivialization F proj)) (pretrivialization_at : B → pretrivialization F proj) diff --git a/formal/lean/mathlib/topology/gluing.lean b/formal/lean/mathlib/topology/gluing.lean index 25543d9719b0b06f795ddd11a32d9dfa0675cc2a..934a506d6500bfd5e59a7e45e5e0ec8016186af5 100644 --- a/formal/lean/mathlib/topology/gluing.lean +++ b/formal/lean/mathlib/topology/gluing.lean @@ -77,7 +77,7 @@ that the `U i`'s are open subspaces of the glued space. Most of the times it would be easier to use the constructor `Top.glue_data.mk'` where the conditions are stated in a less categorical way. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure glue_data extends glue_data Top := (f_open : ∀ i j, open_embedding (f i j)) (f_mono := λ i j, (Top.mono_iff_injective _).mpr (f_open i j).to_embedding.inj) @@ -279,7 +279,7 @@ such that We can then glue the topological spaces `U i` together by identifying `V i j` with `V j i`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure mk_core := {J : Type u} (U : J → Top.{u}) diff --git a/formal/lean/mathlib/topology/homeomorph.lean b/formal/lean/mathlib/topology/homeomorph.lean index 8dbee1460d6d002e5e27e5d62a3433fe99ffa402..20f2d0e48b7016e02975342752bf36a483f04feb 100644 --- a/formal/lean/mathlib/topology/homeomorph.lean +++ b/formal/lean/mathlib/topology/homeomorph.lean @@ -32,7 +32,7 @@ open_locale topological_space variables {α : Type*} {β : Type*} {γ : Type*} {δ : Type*} /-- Homeomorphism between `α` and `β`, also called topological isomorphism -/ -@[nolint has_inhabited_instance] -- not all spaces are homeomorphic to each other +@[nolint has_nonempty_instance] -- not all spaces are homeomorphic to each other structure homeomorph (α : Type*) (β : Type*) [topological_space α] [topological_space β] extends α ≃ β := (continuous_to_fun : continuous to_fun . tactic.interactive.continuity') @@ -106,6 +106,12 @@ h.to_equiv.apply_symm_apply x @[simp] lemma symm_apply_apply (h : α ≃ₜ β) (x : α) : h.symm (h x) = x := h.to_equiv.symm_apply_apply x +@[simp] lemma self_trans_symm (h : α ≃ₜ β) : h.trans h.symm = homeomorph.refl α := +by { ext, apply symm_apply_apply } + +@[simp] lemma symm_trans_self (h : α ≃ₜ β) : h.symm.trans h = homeomorph.refl β := +by { ext, apply apply_symm_apply } + protected lemma bijective (h : α ≃ₜ β) : function.bijective h := h.to_equiv.bijective protected lemma injective (h : α ≃ₜ β) : function.injective h := h.to_equiv.injective protected lemma surjective (h : α ≃ₜ β) : function.surjective h := h.to_equiv.surjective @@ -442,6 +448,15 @@ def set.univ (α : Type*) [topological_space α] : (univ : set α) ≃ₜ α := continuous_to_fun := continuous_subtype_coe, continuous_inv_fun := continuous_subtype_mk _ continuous_id } +/-- `s ×ˢ t` is homeomorphic to `s × t`. -/ +@[simps] def set.prod (s : set α) (t : set β) : ↥(s ×ˢ t) ≃ₜ s × t := +{ to_equiv := equiv.set.prod s t, + continuous_to_fun := continuous.prod_mk + (continuous_subtype_mk _ (continuous_fst.comp continuous_induced_dom)) + (continuous_subtype_mk _ (continuous_snd.comp continuous_induced_dom)), + continuous_inv_fun := continuous_subtype_mk _ (continuous.prod_mk + (continuous_induced_dom.comp continuous_fst) (continuous_induced_dom.comp continuous_snd)) } + end homeomorph /-- An inducing equiv between topological spaces is a homeomorphism. -/ diff --git a/formal/lean/mathlib/topology/instances/ennreal.lean b/formal/lean/mathlib/topology/instances/ennreal.lean index ce78ebe87cd997e7e877a794aaabaa422eb5bc85..ce2e438aba1692b16880965d5ddb32dcf778c5a7 100644 --- a/formal/lean/mathlib/topology/instances/ennreal.lean +++ b/formal/lean/mathlib/topology/instances/ennreal.lean @@ -4,9 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl -/ import topology.instances.nnreal -import order.liminf_limsup -import topology.metric_space.lipschitz import topology.algebra.order.monotone_continuity +import analysis.normed.group.basic /-! # Extended non-negative reals -/ @@ -652,6 +651,60 @@ end end topological_space +section liminf + +lemma exists_frequently_lt_of_liminf_ne_top + {ι : Type*} {l : filter ι} {x : ι → ℝ} (hx : liminf l (λ n, (∥x n∥₊ : ℝ≥0∞)) ≠ ∞) : + ∃ R, ∃ᶠ n in l, x n < R := +begin + by_contra h, + simp_rw [not_exists, not_frequently, not_lt] at h, + refine hx (ennreal.eq_top_of_forall_nnreal_le $ λ r, le_Liminf_of_le (by is_bounded_default) _), + simp only [eventually_map, ennreal.coe_le_coe], + filter_upwards [h r] with i hi using hi.trans ((coe_nnnorm (x i)).symm ▸ le_abs_self (x i)), +end + +lemma exists_frequently_lt_of_liminf_ne_top' + {ι : Type*} {l : filter ι} {x : ι → ℝ} (hx : liminf l (λ n, (∥x n∥₊ : ℝ≥0∞)) ≠ ∞) : + ∃ R, ∃ᶠ n in l, R < x n := +begin + by_contra h, + simp_rw [not_exists, not_frequently, not_lt] at h, + refine hx (ennreal.eq_top_of_forall_nnreal_le $ λ r, le_Liminf_of_le (by is_bounded_default) _), + simp only [eventually_map, ennreal.coe_le_coe], + filter_upwards [h (-r)] with i hi using (le_neg.1 hi).trans (neg_le_abs_self _), +end + +lemma exists_upcrossings_of_not_bounded_under + {ι : Type*} {l : filter ι} {x : ι → ℝ} + (hf : liminf l (λ i, (∥x i∥₊ : ℝ≥0∞)) ≠ ∞) + (hbdd : ¬ is_bounded_under (≤) l (λ i, |x i|)) : + ∃ a b : ℚ, a < b ∧ (∃ᶠ i in l, x i < a) ∧ (∃ᶠ i in l, ↑b < x i) := +begin + rw [is_bounded_under_le_abs, not_and_distrib] at hbdd, + obtain hbdd | hbdd := hbdd, + { obtain ⟨R, hR⟩ := exists_frequently_lt_of_liminf_ne_top hf, + obtain ⟨q, hq⟩ := exists_rat_gt R, + refine ⟨q, q + 1, (lt_add_iff_pos_right _).2 zero_lt_one, _, _⟩, + { refine λ hcon, hR _, + filter_upwards [hcon] with x hx using not_lt.2 (lt_of_lt_of_le hq (not_lt.1 hx)).le }, + { simp only [is_bounded_under, is_bounded, eventually_map, eventually_at_top, + ge_iff_le, not_exists, not_forall, not_le, exists_prop] at hbdd, + refine λ hcon, hbdd ↑(q + 1) _, + filter_upwards [hcon] with x hx using not_lt.1 hx } }, + { obtain ⟨R, hR⟩ := exists_frequently_lt_of_liminf_ne_top' hf, + obtain ⟨q, hq⟩ := exists_rat_lt R, + refine ⟨q - 1, q, (sub_lt_self_iff _).2 zero_lt_one, _, _⟩, + { simp only [is_bounded_under, is_bounded, eventually_map, eventually_at_top, + ge_iff_le, not_exists, not_forall, not_le, exists_prop] at hbdd, + refine λ hcon, hbdd ↑(q - 1) _, + filter_upwards [hcon] with x hx using not_lt.1 hx }, + { refine λ hcon, hR _, + filter_upwards [hcon] with x hx using not_lt.2 ((not_lt.1 hx).trans hq.le) } } +end + +end liminf + section tsum variables {f g : α → ℝ≥0∞} diff --git a/formal/lean/mathlib/topology/instances/real.lean b/formal/lean/mathlib/topology/instances/real.lean index 515ecfb0f9c1b7ea9de9f286fdd595c9c5841f99..1bef704d733ae1c25aeef95376fdaff623843a29 100644 --- a/formal/lean/mathlib/topology/instances/real.lean +++ b/formal/lean/mathlib/topology/instances/real.lean @@ -5,6 +5,7 @@ Authors: Johannes Hölzl, Mario Carneiro -/ import topology.metric_space.basic import topology.algebra.uniform_group +import topology.algebra.uniform_mul_action import topology.algebra.ring import topology.algebra.star import ring_theory.subring.basic @@ -101,14 +102,8 @@ lemma real.continuous.inv [topological_space α] {f : α → ℝ} (h : ∀a, f a show continuous ((has_inv.inv ∘ @subtype.val ℝ (λr, r ≠ 0)) ∘ λa, ⟨f a, h a⟩), from real.continuous_inv.comp (continuous_subtype_mk _ hf) -lemma real.uniform_continuous_mul_const {x : ℝ} : uniform_continuous ((*) x) := -metric.uniform_continuous_iff.2 $ λ ε ε0, begin - cases exists_gt (|x|) with y xy, - have y0 := lt_of_le_of_lt (abs_nonneg _) xy, - refine ⟨_, div_pos ε0 y0, λ a b h, _⟩, - rw [real.dist_eq, ← mul_sub, abs_mul, ← mul_div_cancel' ε (ne_of_gt y0)], - exact mul_lt_mul' (le_of_lt xy) h (abs_nonneg _) y0 -end +lemma real.uniform_continuous_const_mul {x : ℝ} : uniform_continuous ((*) x) := +uniform_continuous_const_smul x lemma real.uniform_continuous_mul (s : set (ℝ × ℝ)) {r₁ r₂ : ℝ} (H : ∀ x ∈ s, |(x : ℝ × ℝ).1| < r₁ ∧ |x.2| < r₂) : diff --git a/formal/lean/mathlib/topology/instances/sign.lean b/formal/lean/mathlib/topology/instances/sign.lean new file mode 100644 index 0000000000000000000000000000000000000000..5ed4798f7a5bb295fb5768419d2ccbb89b006504 --- /dev/null +++ b/formal/lean/mathlib/topology/instances/sign.lean @@ -0,0 +1,53 @@ +/- +Copyright (c) 2022 Joseph Myers. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Joseph Myers +-/ +import data.sign +import topology.algebra.order.basic + +/-! +# Topology on `sign_type` + +This file gives `sign_type` the discrete topology, and proves continuity results for `sign` in +an `order_topology`. + +-/ + +instance : topological_space sign_type := ⊥ +instance : discrete_topology sign_type := ⟨rfl⟩ + +variables {α : Type*} [has_zero α] [topological_space α] + +section partial_order + +variables [partial_order α] [decidable_rel ((<) : α → α → Prop)] [order_topology α] + +lemma continuous_at_sign_of_pos {a : α} (h : 0 < a) : continuous_at sign a := +begin + refine (continuous_at_const : continuous_at (λ x, (1 : sign_type)) a).congr _, + rw [filter.eventually_eq, eventually_nhds_iff], + exact ⟨{x | 0 < x}, λ x hx, (sign_pos hx).symm, is_open_lt' 0, h⟩ +end + +lemma continuous_at_sign_of_neg {a : α} (h : a < 0) : continuous_at sign a := +begin + refine (continuous_at_const : continuous_at (λ x, (-1 : sign_type)) a).congr _, + rw [filter.eventually_eq, eventually_nhds_iff], + exact ⟨{x | x < 0}, λ x hx, (sign_neg hx).symm, is_open_gt' 0, h⟩ +end + +end partial_order + +section linear_order + +variables [linear_order α] [order_topology α] + +lemma continuous_at_sign_of_ne_zero {a : α} (h : a ≠ 0) : continuous_at sign a := +begin + rcases h.lt_or_lt with h_neg|h_pos, + { exact continuous_at_sign_of_neg h_neg }, + { exact continuous_at_sign_of_pos h_pos } +end + +end linear_order diff --git a/formal/lean/mathlib/topology/local_at_target.lean b/formal/lean/mathlib/topology/local_at_target.lean new file mode 100644 index 0000000000000000000000000000000000000000..8506d806e0a7b7d0ae22c1509844ee226685b82e --- /dev/null +++ b/formal/lean/mathlib/topology/local_at_target.lean @@ -0,0 +1,125 @@ +/- +Copyright (c) 2022 Andrew Yang. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Andrew Yang +-/ +import topology.sets.opens + +/-! +# Properties of maps that are local at the target. + +We show that the following properties of continuous maps are local at the target : +- `inducing` +- `embedding` +- `open_embedding` +- `closed_embedding` + +-/ + +open topological_space set filter +open_locale topological_space filter + +variables {α β : Type*} [topological_space α] [topological_space β] {f : α → β} +variables {s : set β} {ι : Type*} {U : ι → opens β} (hU : supr U = ⊤) + +lemma set.restrict_preimage_inducing (s : set β) (h : inducing f) : + inducing (s.restrict_preimage f) := +begin + simp_rw [inducing_coe.inducing_iff, inducing_iff_nhds, restrict_preimage, maps_to.coe_restrict, + restrict_eq, ← @filter.comap_comap _ _ _ _ coe f] at h ⊢, + intros a, + rw [← h, ← inducing_coe.nhds_eq_comap], +end + +alias set.restrict_preimage_inducing ← inducing.restrict_preimage + +lemma set.restrict_preimage_embedding (s : set β) (h : embedding f) : + embedding (s.restrict_preimage f) := +⟨h.1.restrict_preimage s, h.2.restrict_preimage s⟩ + +alias set.restrict_preimage_embedding ← embedding.restrict_preimage + +lemma set.restrict_preimage_open_embedding (s : set β) (h : open_embedding f) : + open_embedding (s.restrict_preimage f) := +⟨h.1.restrict_preimage s, + (s.range_restrict_preimage f).symm ▸ continuous_subtype_coe.is_open_preimage _ h.2⟩ + +alias set.restrict_preimage_open_embedding ← open_embedding.restrict_preimage + +lemma set.restrict_preimage_closed_embedding (s : set β) (h : closed_embedding f) : + closed_embedding (s.restrict_preimage f) := +⟨h.1.restrict_preimage s, + (s.range_restrict_preimage f).symm ▸ inducing_coe.is_closed_preimage _ h.2⟩ + +alias set.restrict_preimage_closed_embedding ← closed_embedding.restrict_preimage + +include hU + +lemma open_iff_inter_of_supr_eq_top (s : set β) : + is_open s ↔ ∀ i, is_open (s ∩ U i) := +begin + split, + { exact λ H i, H.inter (U i).2 }, + { intro H, + have : (⋃ i, (U i : set β)) = set.univ := by { convert (congr_arg coe hU), simp }, + rw [← s.inter_univ, ← this, set.inter_Union], + exact is_open_Union H } +end + +lemma open_iff_coe_preimage_of_supr_eq_top (s : set β) : + is_open s ↔ ∀ i, is_open (coe ⁻¹' s : set (U i)) := +begin + simp_rw [(U _).2.open_embedding_subtype_coe.open_iff_image_open, + set.image_preimage_eq_inter_range, subtype.range_coe], + apply open_iff_inter_of_supr_eq_top, + assumption +end + +lemma closed_iff_coe_preimage_of_supr_eq_top (s : set β) : + is_closed s ↔ ∀ i, is_closed (coe ⁻¹' s : set (U i)) := +by simpa using open_iff_coe_preimage_of_supr_eq_top hU sᶜ + +lemma inducing_iff_inducing_of_supr_eq_top (h : continuous f) : + inducing f ↔ ∀ i, inducing ((U i).1.restrict_preimage f) := +begin + simp_rw [inducing_coe.inducing_iff, inducing_iff_nhds, restrict_preimage, maps_to.coe_restrict, + restrict_eq, ← @filter.comap_comap _ _ _ _ coe f], + split, + { intros H i x, rw [← H, ← inducing_coe.nhds_eq_comap] }, + { intros H x, + obtain ⟨i, hi⟩ := opens.mem_supr.mp (show f x ∈ supr U, by { rw hU, triv }), + erw ← open_embedding.map_nhds_eq (h.1 _ (U i).2).open_embedding_subtype_coe ⟨x, hi⟩, + rw [(H i) ⟨x, hi⟩, filter.subtype_coe_map_comap, function.comp_apply, subtype.coe_mk, + inf_eq_left, filter.le_principal_iff], + exact filter.preimage_mem_comap ((U i).2.mem_nhds hi) } +end + +lemma embedding_iff_embedding_of_supr_eq_top (h : continuous f) : + embedding f ↔ ∀ i, embedding ((U i).1.restrict_preimage f) := +begin + simp_rw embedding_iff, + rw forall_and_distrib, + apply and_congr, + { apply inducing_iff_inducing_of_supr_eq_top; assumption }, + { apply set.injective_iff_injective_of_Union_eq_univ, convert (congr_arg coe hU), simp } +end + +lemma open_embedding_iff_open_embedding_of_supr_eq_top (h : continuous f) : + open_embedding f ↔ ∀ i, open_embedding ((U i).1.restrict_preimage f) := +begin + simp_rw open_embedding_iff, + rw forall_and_distrib, + apply and_congr, + { apply embedding_iff_embedding_of_supr_eq_top; assumption }, + { simp_rw set.range_restrict_preimage, apply open_iff_coe_preimage_of_supr_eq_top hU } +end + +lemma closed_embedding_iff_closed_embedding_of_supr_eq_top (h : continuous f) : + closed_embedding f ↔ ∀ i, closed_embedding ((U i).1.restrict_preimage f) := +begin + simp_rw closed_embedding_iff, + rw forall_and_distrib, + apply and_congr, + { apply embedding_iff_embedding_of_supr_eq_top; assumption }, + { simp_rw set.range_restrict_preimage, apply closed_iff_coe_preimage_of_supr_eq_top hU } +end diff --git a/formal/lean/mathlib/topology/local_homeomorph.lean b/formal/lean/mathlib/topology/local_homeomorph.lean index 4c1bd0645075e8c8cab6f50d486513b15013d49d..1ac0bf05a418f75c5867daa9be4f650589bf9f73 100644 --- a/formal/lean/mathlib/topology/local_homeomorph.lean +++ b/formal/lean/mathlib/topology/local_homeomorph.lean @@ -49,7 +49,7 @@ variables {α : Type*} {β : Type*} {γ : Type*} {δ : Type*} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] /-- local homeomorphisms, defined on open subsets of the space -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure local_homeomorph (α : Type*) (β : Type*) [topological_space α] [topological_space β] extends local_equiv α β := (open_source : is_open source) @@ -947,7 +947,7 @@ lemma continuous_within_at_iff_continuous_within_at_comp_left {f : γ → α} {s : set γ} {x : γ} (hx : f x ∈ e.source) (h : f ⁻¹' e.source ∈ 𝓝[s] x) : continuous_within_at f s x ↔ continuous_within_at (e ∘ f) s x := begin - refine ⟨(e.continuous_at hx).tendsto.comp, λ fe_cont, _⟩, + refine ⟨(e.continuous_at hx).comp_continuous_within_at, λ fe_cont, _⟩, rw [← continuous_within_at_inter' h] at fe_cont ⊢, have : continuous_within_at (e.symm ∘ (e ∘ f)) (s ∩ f ⁻¹' e.source) x, { have : continuous_within_at e.symm univ (e (f x)) diff --git a/formal/lean/mathlib/topology/locally_constant/basic.lean b/formal/lean/mathlib/topology/locally_constant/basic.lean index 061d0cad2b9226e24f66ba69a5378033c8658f9d..8829076a3582c70772fb531984ced9a6721f09fb 100644 --- a/formal/lean/mathlib/topology/locally_constant/basic.lean +++ b/formal/lean/mathlib/topology/locally_constant/basic.lean @@ -185,6 +185,20 @@ begin apply h, end +lemma of_constant_on_connected_components [locally_connected_space X] {f : X → Y} + (h : ∀ x, ∀ y ∈ connected_component x, f y = f x) : + is_locally_constant f := +begin + rw iff_exists_open, + exact λ x, ⟨connected_component x, is_open_connected_component, mem_connected_component, h x⟩, +end + +lemma of_constant_on_preconnected_clopens [locally_connected_space X] {f : X → Y} + (h : ∀ U : set X, is_preconnected U → is_clopen U → ∀ x ∈ U, ∀ y ∈ U, f y = f x) : + is_locally_constant f := +of_constant_on_connected_components (λ x, h (connected_component x) + is_preconnected_connected_component is_clopen_connected_component x mem_connected_component) + end is_locally_constant /-- A (bundled) locally constant function from a topological space `X` to a type `Y`. -/ diff --git a/formal/lean/mathlib/topology/maps.lean b/formal/lean/mathlib/topology/maps.lean index 611876f4304272a1c22918f088dffd15f7a190c8..53dac03657330717aa8b667764c25efa77a16d8a 100644 --- a/formal/lean/mathlib/topology/maps.lean +++ b/formal/lean/mathlib/topology/maps.lean @@ -129,6 +129,10 @@ lemma inducing.is_closed_iff' {f : α → β} (hf : inducing f) {s : set α} : is_closed s ↔ ∀ x, f x ∈ closure (f '' s) → x ∈ s := by rw [hf.induced, is_closed_induced_iff'] +lemma inducing.is_closed_preimage {f : α → β} (h : inducing f) (s : set β) (hs : is_closed s) : + is_closed (f ⁻¹' s) := +(inducing.is_closed_iff h).mpr ⟨s, hs, rfl⟩ + lemma inducing.is_open_iff {f : α → β} (hf : inducing f) {s : set α} : is_open s ↔ ∃ t, is_open t ∧ f ⁻¹' t = s := by rw [hf.induced, is_open_induced_iff] @@ -429,6 +433,7 @@ section open_embedding variables [topological_space α] [topological_space β] [topological_space γ] /-- An open embedding is an embedding with open image. -/ +@[mk_iff] structure open_embedding (f : α → β) extends _root_.embedding f : Prop := (open_range : is_open $ range f) @@ -510,6 +515,7 @@ section closed_embedding variables [topological_space α] [topological_space β] [topological_space γ] /-- A closed embedding is an embedding with closed image. -/ +@[mk_iff] structure closed_embedding (f : α → β) extends _root_.embedding f : Prop := (closed_range : is_closed $ range f) diff --git a/formal/lean/mathlib/topology/metric_space/algebra.lean b/formal/lean/mathlib/topology/metric_space/algebra.lean index db5792f8f9fb5f3695de2fe2526a2c2b91ae7d01..d46ec93004b1f40f26267458d906cbeb44e2fe18 100644 --- a/formal/lean/mathlib/topology/metric_space/algebra.lean +++ b/formal/lean/mathlib/topology/metric_space/algebra.lean @@ -176,3 +176,12 @@ instance has_bounded_smul.op [has_smul αᵐᵒᵖ β] [is_central_scalar α β] by simpa only [op_smul_eq_smul] using dist_pair_smul x₁ x₂ y } end has_bounded_smul + +instance [monoid α] [has_lipschitz_mul α] : has_lipschitz_add (additive α) := +⟨@has_lipschitz_mul.lipschitz_mul α _ _ _⟩ + +instance [add_monoid α] [has_lipschitz_add α] : has_lipschitz_mul (multiplicative α) := +⟨@has_lipschitz_add.lipschitz_add α _ _ _⟩ + +@[to_additive] instance [monoid α] [has_lipschitz_mul α] : has_lipschitz_mul αᵒᵈ := +‹has_lipschitz_mul α› diff --git a/formal/lean/mathlib/topology/metric_space/basic.lean b/formal/lean/mathlib/topology/metric_space/basic.lean index abf1f2204afccc4276675d2c675e0aaf1141be95..a08b776e6efbe01edbc95c97417d396d9490bbda 100644 --- a/formal/lean/mathlib/topology/metric_space/basic.lean +++ b/formal/lean/mathlib/topology/metric_space/basic.lean @@ -54,7 +54,7 @@ open set filter topological_space bornology open_locale uniformity topological_space big_operators filter nnreal ennreal universes u v w -variables {α : Type u} {β : Type v} +variables {α : Type u} {β : Type v} {X : Type*} /-- Construct a uniform structure core from a distance function and metric space axioms. This is a technical construction that can be immediately used to construct a uniform structure @@ -2881,3 +2881,75 @@ instance metric_space_quot {α : Type u} [pseudo_metric_space α] : λxc yc zc, quotient.induction_on₃ xc yc zc (λx y z, pseudo_metric_space.dist_triangle _ _ _) } end eq_rel + +/-! +### `additive`, `multiplicative` + +The distance on those type synonyms is inherited without change. +-/ + +open additive multiplicative + +section +variables [has_dist X] + +instance : has_dist (additive X) := ‹has_dist X› +instance : has_dist (multiplicative X) := ‹has_dist X› + +@[simp] lemma dist_of_mul (a b : X) : dist (of_mul a) (of_mul b) = dist a b := rfl +@[simp] lemma dist_of_add (a b : X) : dist (of_add a) (of_add b) = dist a b := rfl +@[simp] lemma dist_to_mul (a b : additive X) : dist (to_mul a) (to_mul b) = dist a b := rfl +@[simp] lemma dist_to_add (a b : multiplicative X) : dist (to_add a) (to_add b) = dist a b := rfl + +end + +section +variables [pseudo_metric_space X] + +instance : pseudo_metric_space (additive X) := ‹pseudo_metric_space X› +instance : pseudo_metric_space (multiplicative X) := ‹pseudo_metric_space X› + +@[simp] lemma nndist_of_mul (a b : X) : nndist (of_mul a) (of_mul b) = nndist a b := rfl +@[simp] lemma nndist_of_add (a b : X) : nndist (of_add a) (of_add b) = nndist a b := rfl +@[simp] lemma nndist_to_mul (a b : additive X) : nndist (to_mul a) (to_mul b) = nndist a b := rfl +@[simp] lemma nndist_to_add (a b : multiplicative X) : nndist (to_add a) (to_add b) = nndist a b := +rfl + +end + +instance [metric_space X] : metric_space (additive X) := ‹metric_space X› +instance [metric_space X] : metric_space (multiplicative X) := ‹metric_space X› +instance [pseudo_metric_space X] [proper_space X] : proper_space (additive X) := ‹proper_space X› +instance [pseudo_metric_space X] [proper_space X] : proper_space (multiplicative X) := +‹proper_space X› + +/-! +### Order dual + +The distance on this type synonym is inherited without change. +-/ + +open order_dual + +section +variables [has_dist X] + +instance : has_dist Xᵒᵈ := ‹has_dist X› + +@[simp] lemma dist_to_dual (a b : X) : dist (to_dual a) (to_dual b) = dist a b := rfl +@[simp] lemma dist_of_dual (a b : Xᵒᵈ) : dist (of_dual a) (of_dual b) = dist a b := rfl + +end + +section +variables [pseudo_metric_space X] + +instance : pseudo_metric_space Xᵒᵈ := ‹pseudo_metric_space X› + +@[simp] lemma nndist_to_dual (a b : X) : nndist (to_dual a) (to_dual b) = nndist a b := rfl +@[simp] lemma nndist_of_dual (a b : Xᵒᵈ) : nndist (of_dual a) (of_dual b) = nndist a b := rfl + +end + +instance [metric_space X] : metric_space Xᵒᵈ := ‹metric_space X› +instance [pseudo_metric_space X] [proper_space X] : proper_space Xᵒᵈ := ‹proper_space X› diff --git a/formal/lean/mathlib/topology/metric_space/emetric_space.lean b/formal/lean/mathlib/topology/metric_space/emetric_space.lean index ced3fdf605dfce7cb7b2ebef0cfcba12cf4a7bce..3a65a7af6808db1ab7716e861e3359103d5931a1 100644 --- a/formal/lean/mathlib/topology/metric_space/emetric_space.lean +++ b/formal/lean/mathlib/topology/metric_space/emetric_space.lean @@ -33,7 +33,7 @@ noncomputable theory open_locale uniformity topological_space big_operators filter nnreal ennreal universes u v w -variables {α : Type u} {β : Type v} +variables {α : Type u} {β : Type v} {X : Type*} /-- Characterizing uniformities associated to a (generalized) distance function `D` in terms of the elements of the uniformity. -/ @@ -1015,3 +1015,51 @@ by simp only [pos_iff_ne_zero, ne.def, diam_eq_zero_iff, set.subsingleton, not_f end diam end emetric + +/-! +### `additive`, `multiplicative` + +The distance on those type synonyms is inherited without change. +-/ + +open additive multiplicative + +section +variables [has_edist X] + +instance : has_edist (additive X) := ‹has_edist X› +instance : has_edist (multiplicative X) := ‹has_edist X› + +@[simp] lemma edist_of_mul (a b : X) : edist (of_mul a) (of_mul b) = edist a b := rfl +@[simp] lemma edist_of_add (a b : X) : edist (of_add a) (of_add b) = edist a b := rfl +@[simp] lemma edist_to_mul (a b : additive X) : edist (to_mul a) (to_mul b) = edist a b := rfl +@[simp] lemma edist_to_add (a b : multiplicative X) : edist (to_add a) (to_add b) = edist a b := rfl + +end + +instance [pseudo_emetric_space X] : pseudo_emetric_space (additive X) := ‹pseudo_emetric_space X› +instance [pseudo_emetric_space X] : pseudo_emetric_space (multiplicative X) := +‹pseudo_emetric_space X› +instance [emetric_space X] : emetric_space (additive X) := ‹emetric_space X› +instance [emetric_space X] : emetric_space (multiplicative X) := ‹emetric_space X› + +/-! +### Order dual + +The distance on this type synonym is inherited without change. +-/ + +open order_dual + +section +variables [has_edist X] + +instance : has_edist Xᵒᵈ := ‹has_edist X› + +@[simp] lemma edist_to_dual (a b : X) : edist (to_dual a) (to_dual b) = edist a b := rfl +@[simp] lemma edist_of_dual (a b : Xᵒᵈ) : edist (of_dual a) (of_dual b) = edist a b := rfl + +end + +instance [pseudo_emetric_space X] : pseudo_emetric_space Xᵒᵈ := ‹pseudo_emetric_space X› +instance [emetric_space X] : emetric_space Xᵒᵈ := ‹emetric_space X› diff --git a/formal/lean/mathlib/topology/metric_space/gromov_hausdorff.lean b/formal/lean/mathlib/topology/metric_space/gromov_hausdorff.lean index fd9c2b517a402d604f37d1e597224ceaeabb6417..61f22c5b4d64ed7f972c85683399034177389dd7 100644 --- a/formal/lean/mathlib/topology/metric_space/gromov_hausdorff.lean +++ b/formal/lean/mathlib/topology/metric_space/gromov_hausdorff.lean @@ -83,7 +83,7 @@ definition to_GH_space (X : Type u) [metric_space X] [compact_space X] [nonempty instance : inhabited GH_space := ⟨quot.mk _ ⟨⟨{0}, is_compact_singleton⟩, singleton_nonempty _⟩⟩ /-- A metric space representative of any abstract point in `GH_space` -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def GH_space.rep (p : GH_space) : Type := (quotient.out p : nonempty_compacts ℓ_infty_ℝ) lemma eq_to_GH_space_iff {X : Type u} [metric_space X] [compact_space X] [nonempty X] diff --git a/formal/lean/mathlib/topology/metric_space/gromov_hausdorff_realized.lean b/formal/lean/mathlib/topology/metric_space/gromov_hausdorff_realized.lean index f0623b8f2058b76a7794aca3621c5ae85847b010..c5f09a4aaed7da21b48d7ff65939ec2c74b74dac 100644 --- a/formal/lean/mathlib/topology/metric_space/gromov_hausdorff_realized.lean +++ b/formal/lean/mathlib/topology/metric_space/gromov_hausdorff_realized.lean @@ -451,7 +451,7 @@ def premetric_optimal_GH_dist : pseudo_metric_space (X ⊕ Y) := local attribute [instance] premetric_optimal_GH_dist pseudo_metric.dist_setoid /-- A metric space which realizes the optimal coupling between `X` and `Y` -/ -@[derive metric_space, nolint has_inhabited_instance] +@[derive metric_space, nolint has_nonempty_instance] definition optimal_GH_coupling : Type* := pseudo_metric_quot (X ⊕ Y) diff --git a/formal/lean/mathlib/topology/metric_space/isometry.lean b/formal/lean/mathlib/topology/metric_space/isometry.lean index d92a4530380cd01eb47af1938f6e31f1bf9954ab..37e013925d5fe0c3fcc5a50f8d436694aac9b29e 100644 --- a/formal/lean/mathlib/topology/metric_space/isometry.lean +++ b/formal/lean/mathlib/topology/metric_space/isometry.lean @@ -232,7 +232,7 @@ begin end /-- `α` and `β` are isometric if there is an isometric bijection between them. -/ -@[nolint has_inhabited_instance] -- such a bijection need not exist +@[nolint has_nonempty_instance] -- such a bijection need not exist structure isometric (α : Type*) (β : Type*) [pseudo_emetric_space α] [pseudo_emetric_space β] extends α ≃ β := (isometry_to_fun : isometry to_fun) diff --git a/formal/lean/mathlib/topology/metric_space/metrizable.lean b/formal/lean/mathlib/topology/metric_space/metrizable.lean index 70dcfdee9c335aee5075e21e5f7eda46df799515..5106db3a45fef6a324c177e7d665249126b99673 100644 --- a/formal/lean/mathlib/topology/metric_space/metrizable.lean +++ b/formal/lean/mathlib/topology/metric_space/metrizable.lean @@ -24,7 +24,7 @@ open_locale bounded_continuous_function filter topological_space namespace topological_space variables {ι X Y : Type*} {π : ι → Type*} [topological_space X] [topological_space Y] - [fintype ι] [Π i, topological_space (π i)] + [finite ι] [Π i, topological_space (π i)] /-- A topological space is *pseudo metrizable* if there exists a pseudo metric space structure compatible with the topology. To endow such a space with a compatible distance, use @@ -68,7 +68,8 @@ inducing_coe.pseudo_metrizable_space instance pseudo_metrizable_space_pi [Π i, pseudo_metrizable_space (π i)] : pseudo_metrizable_space (Π i, π i) := -by { letI := λ i, pseudo_metrizable_space_pseudo_metric (π i), apply_instance } +by { casesI nonempty_fintype ι, letI := λ i, pseudo_metrizable_space_pseudo_metric (π i), + apply_instance } /-- A topological space is metrizable if there exists a metric space structure compatible with the topology. To endow such a space with a compatible distance, use @@ -117,7 +118,7 @@ instance metrizable_space.subtype [metrizable_space X] (s : set X) : metrizable_ embedding_subtype_coe.metrizable_space instance metrizable_space_pi [Π i, metrizable_space (π i)] : metrizable_space (Π i, π i) := -by { letI := λ i, metrizable_space_metric (π i), apply_instance } +by { casesI nonempty_fintype ι, letI := λ i, metrizable_space_metric (π i), apply_instance } variables (X) [t3_space X] [second_countable_topology X] diff --git a/formal/lean/mathlib/topology/metric_space/polish.lean b/formal/lean/mathlib/topology/metric_space/polish.lean index b65a2b0d9ef48aba229fd5bd575552688d488d4a..d30db5f66922496d15001a66f6a87090269fea23 100644 --- a/formal/lean/mathlib/topology/metric_space/polish.lean +++ b/formal/lean/mathlib/topology/metric_space/polish.lean @@ -180,7 +180,7 @@ lemma _root_.is_closed.polish_space {α : Type*} [topological_space α] [polish_ /-- A sequence of type synonyms of a given type `α`, useful in the proof of `exists_polish_space_forall_le` to endow each copy with a different topology. -/ -@[nolint unused_arguments has_inhabited_instance] +@[nolint unused_arguments has_nonempty_instance] def aux_copy (α : Type*) {ι : Type*} (i : ι) : Type* := α /-- Given a Polish space, and countably many finer Polish topologies, there exists another Polish @@ -259,7 +259,7 @@ variables [metric_space α] {s : set α} /-- A type synonym for a subset `s` of a metric space, on which we will construct another metric for which it will be complete. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] def complete_copy {α : Type*} (s : set α) : Type* := s /-- A distance on a subset `s` of a metric space, designed to make it complete if `s` is open. diff --git a/formal/lean/mathlib/topology/nhds_set.lean b/formal/lean/mathlib/topology/nhds_set.lean index 23a632815070e13a02d4836890129b8667f5b739..9b4dded88b23ab3fad39229cab9b2efed7d0e39a 100644 --- a/formal/lean/mathlib/topology/nhds_set.lean +++ b/formal/lean/mathlib/topology/nhds_set.lean @@ -24,7 +24,7 @@ Furthermore, we have the following results: -/ open set filter -open_locale topological_space +open_locale topological_space filter variables {α β : Type*} [topological_space α] [topological_space β] {s t s₁ s₂ t₁ t₂ : set α} {x : α} @@ -50,6 +50,18 @@ lemma has_basis_nhds_set (s : set α) : (𝓝ˢ s).has_basis (λ U, is_open U lemma is_open.mem_nhds_set (hU : is_open s) : s ∈ 𝓝ˢ t ↔ t ⊆ s := by rw [← subset_interior_iff_mem_nhds_set, interior_eq_iff_open.mpr hU] +lemma principal_le_nhds_set : 𝓟 s ≤ 𝓝ˢ s := +λ s hs, (subset_interior_iff_mem_nhds_set.mpr hs).trans interior_subset + +@[simp] lemma nhds_set_eq_principal_iff : 𝓝ˢ s = 𝓟 s ↔ is_open s := +by rw [← principal_le_nhds_set.le_iff_eq, le_principal_iff, mem_nhds_set_iff_forall, + is_open_iff_mem_nhds] + +alias nhds_set_eq_principal_iff ↔ _ is_open.nhds_set_eq + +@[simp] lemma nhds_set_interior : 𝓝ˢ (interior s) = 𝓟 (interior s) := +is_open_interior.nhds_set_eq + @[simp] lemma nhds_set_singleton : 𝓝ˢ {x} = 𝓝 x := by { ext, rw [← subset_interior_iff_mem_nhds_set, ← mem_interior_iff_mem_nhds, singleton_subset_iff] } @@ -57,14 +69,13 @@ by { ext, lemma mem_nhds_set_interior : s ∈ 𝓝ˢ (interior s) := subset_interior_iff_mem_nhds_set.mp subset.rfl -lemma mem_nhds_set_empty : s ∈ 𝓝ˢ (∅ : set α) := -subset_interior_iff_mem_nhds_set.mp $ empty_subset _ - @[simp] lemma nhds_set_empty : 𝓝ˢ (∅ : set α) = ⊥ := -by { ext, simp [mem_nhds_set_empty] } +by rw [is_open_empty.nhds_set_eq, principal_empty] + +lemma mem_nhds_set_empty : s ∈ 𝓝ˢ (∅ : set α) := by simp @[simp] lemma nhds_set_univ : 𝓝ˢ (univ : set α) = ⊤ := -by { ext, rw [← subset_interior_iff_mem_nhds_set, univ_subset_iff, interior_eq_univ, mem_top] } +by rw [is_open_univ.nhds_set_eq, principal_univ] lemma monotone_nhds_set : monotone (𝓝ˢ : set α → filter α) := λ s t hst, Sup_le_Sup $ image_subset _ hst diff --git a/formal/lean/mathlib/topology/path_connected.lean b/formal/lean/mathlib/topology/path_connected.lean index 52d02aceb005d863f76092a999595584ece2b545..4397b7b123f04eb34a6113b0358fc09d80188519 100644 --- a/formal/lean/mathlib/topology/path_connected.lean +++ b/formal/lean/mathlib/topology/path_connected.lean @@ -67,7 +67,7 @@ variables {X Y : Type*} [topological_space X] [topological_space Y] {x y z : X} /-! ### Paths -/ /-- Continuous path connecting two points `x` and `y` in a topological space -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure path (x y : X) extends C(I, X) := (source' : to_fun 0 = x) (target' : to_fun 1 = y) diff --git a/formal/lean/mathlib/topology/separation.lean b/formal/lean/mathlib/topology/separation.lean index 28f14db63faef0f9e254be3dfcd8e9138c4df48e..0e8c86ca5e98d4af37a5e51541fc2d66f311abb9 100644 --- a/formal/lean/mathlib/topology/separation.lean +++ b/formal/lean/mathlib/topology/separation.lean @@ -258,7 +258,7 @@ begin exact ⟨t, ssubset_iff_subset_ne.2 ⟨hts, mt finset.coe_inj.2 hts'⟩, htne, hto⟩ } end -theorem exists_open_singleton_of_fintype [t0_space α] [fintype α] [nonempty α] : +theorem exists_open_singleton_of_fintype [t0_space α] [finite α] [nonempty α] : ∃ x : α, is_open ({x} : set α) := let ⟨x, _, h⟩ := exists_open_singleton_of_open_finite (set.to_finite _) univ_nonempty is_open_univ in ⟨x, h⟩ @@ -285,6 +285,20 @@ instance {ι : Type*} {π : ι → Type*} [Π i, topological_space (π i)] [Π i t0_space (Π i, π i) := ⟨λ x y h, funext $ λ i, (h.map (continuous_apply i)).eq⟩ +lemma t0_space.of_cover (h : ∀ x y, inseparable x y → ∃ s : set α, x ∈ s ∧ y ∈ s ∧ t0_space s) : + t0_space α := +begin + refine ⟨λ x y hxy, _⟩, + rcases h x y hxy with ⟨s, hxs, hys, hs⟩, resetI, + lift x to s using hxs, lift y to s using hys, + rw ← subtype_inseparable_iff at hxy, + exact congr_arg coe hxy.eq +end + +lemma t0_space.of_open_cover (h : ∀ x, ∃ s : set α, x ∈ s ∧ is_open s ∧ t0_space s) : t0_space α := +t0_space.of_cover $ λ x y hxy, + let ⟨s, hxs, hso, hs⟩ := h x in ⟨s, hxs, (hxy.mem_open_iff hso).1 hxs, hs⟩ + /-- A T₁ space, also known as a Fréchet space, is a topological space where every singleton set is closed. Equivalently, for every pair `x ≠ y`, there is an open set containing `x` and not `y`. -/ diff --git a/formal/lean/mathlib/topology/sets/compacts.lean b/formal/lean/mathlib/topology/sets/compacts.lean index cf0a8b8e63cefa733f06b0bb2014cfaad2b4e5a6..606e7844bb914f705da5830a1cc944678e7ede2a 100644 --- a/formal/lean/mathlib/topology/sets/compacts.lean +++ b/formal/lean/mathlib/topology/sets/compacts.lean @@ -108,11 +108,10 @@ congr_fun (image_eq_preimage_of_inverse f.left_inv f.right_inv) K.1 /-- The product of two `compacts`, as a `compacts` in the product space. -/ protected def prod (K : compacts α) (L : compacts β) : compacts (α × β) := -{ carrier := (K : set α) ×ˢ (L : set β), +{ carrier := K ×ˢ L, compact' := is_compact.prod K.2 L.2 } -@[simp] lemma coe_prod (K : compacts α) (L : compacts β) : - (K.prod L : set (α × β)) = (K : set α) ×ˢ (L : set β) := rfl +@[simp] lemma coe_prod (K : compacts α) (L : compacts β) : (K.prod L : set (α × β)) = K ×ˢ L := rfl end compacts @@ -172,7 +171,7 @@ protected def prod (K : nonempty_compacts α) (L : nonempty_compacts β) : .. K.to_compacts.prod L.to_compacts } @[simp] lemma coe_prod (K : nonempty_compacts α) (L : nonempty_compacts β) : - (K.prod L : set (α × β)) = (K : set α) ×ˢ (L : set β) := rfl + (K.prod L : set (α × β)) = K ×ˢ L := rfl end nonempty_compacts @@ -245,7 +244,7 @@ protected def prod (K : positive_compacts α) (L : positive_compacts β) : .. K.to_compacts.prod L.to_compacts } @[simp] lemma coe_prod (K : positive_compacts α) (L : positive_compacts β) : - (K.prod L : set (α × β)) = (K : set α) ×ˢ (L : set β) := rfl + (K.prod L : set (α × β)) = K ×ˢ L := rfl end positive_compacts @@ -326,7 +325,7 @@ protected def prod (K : compact_opens α) (L : compact_opens β) : .. K.to_compacts.prod L.to_compacts } @[simp] lemma coe_prod (K : compact_opens α) (L : compact_opens β) : - (K.prod L : set (α × β)) = (K : set α) ×ˢ (L : set β) := rfl + (K.prod L : set (α × β)) = K ×ˢ L := rfl end compact_opens end topological_space diff --git a/formal/lean/mathlib/topology/sheaves/presheaf.lean b/formal/lean/mathlib/topology/sheaves/presheaf.lean index b27deb2fb58cb02b9ed665ce89b2bb17393e56c9..06147fc2c2434ce0be4988a8d0d7e03203b1ba38 100644 --- a/formal/lean/mathlib/topology/sheaves/presheaf.lean +++ b/formal/lean/mathlib/topology/sheaves/presheaf.lean @@ -37,7 +37,7 @@ variables (C : Type u) [category.{v} C] namespace Top /-- The category of `C`-valued presheaves on a (bundled) topological space `X`. -/ -@[derive category, nolint has_inhabited_instance] +@[derive category, nolint has_nonempty_instance] def presheaf (X : Top.{w}) := (opens X)ᵒᵖ ⥤ C variables {C} diff --git a/formal/lean/mathlib/topology/sheaves/sheaf_condition/opens_le_cover.lean b/formal/lean/mathlib/topology/sheaves/sheaf_condition/opens_le_cover.lean index 7af25b4afdc980eba13195a51f6a2b3395d0357d..081f85a57a0bd29ac6b2c0d1349856933f5acaef 100644 --- a/formal/lean/mathlib/topology/sheaves/sheaf_condition/opens_le_cover.lean +++ b/formal/lean/mathlib/topology/sheaves/sheaf_condition/opens_le_cover.lean @@ -47,13 +47,12 @@ namespace sheaf_condition /-- The category of open sets contained in some element of the cover. -/ -def opens_le_cover : Type v := { V : opens X // ∃ i, V ≤ U i } +@[derive category] +def opens_le_cover : Type v := full_subcategory (λ (V : opens X), ∃ i, V ≤ U i) instance [inhabited ι] : inhabited (opens_le_cover U) := ⟨⟨⊥, default, bot_le⟩⟩ -instance : category (opens_le_cover U) := category_theory.full_subcategory _ - namespace opens_le_cover variables {U} @@ -66,7 +65,7 @@ def index (V : opens_le_cover U) : ι := V.property.some /-- The morphism from `V` to `U i` for some `i`. -/ -def hom_to_index (V : opens_le_cover U) : V.val ⟶ U (index V) := +def hom_to_index (V : opens_le_cover U) : V.obj ⟶ U (index V) := (V.property.some_spec).hom end opens_le_cover @@ -244,7 +243,7 @@ variables {Y : opens X} (hY : Y = supr U) in the sieve. This full subcategory is equivalent to `opens_le_cover U`, the (poset) category of opens contained in some `U i`. -/ @[simps] def generate_equivalence_opens_le : - {f : over Y // (sieve.generate (presieve_of_covering_aux U Y)).arrows f.hom} ≌ + full_subcategory (λ (f : over Y), (sieve.generate (presieve_of_covering_aux U Y)).arrows f.hom) ≌ opens_le_cover U := { functor := { obj := λ f, ⟨f.1.left, let ⟨_,h,_,⟨i,hY⟩,_⟩ := f.2 in ⟨i, hY ▸ h.le⟩⟩, diff --git a/formal/lean/mathlib/topology/sheaves/sheafify.lean b/formal/lean/mathlib/topology/sheaves/sheafify.lean index 46368756c0bce96c23ccad201d8e082875b3494f..3c707fbc5aa5ef650814d889dfdd4efb9dbe551a 100644 --- a/formal/lean/mathlib/topology/sheaves/sheafify.lean +++ b/formal/lean/mathlib/topology/sheaves/sheafify.lean @@ -104,9 +104,9 @@ begin dsimp at e', use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩, refine ⟨_, _, _⟩, - { change W ⊓ (U' ⊓ V') ⟶ U.val, + { change W ⊓ (U' ⊓ V') ⟶ U.obj, exact (opens.inf_le_right _ _) ≫ (opens.inf_le_left _ _) ≫ iU, }, - { change W ⊓ (U' ⊓ V') ⟶ V.val, + { change W ⊓ (U' ⊓ V') ⟶ V.obj, exact (opens.inf_le_right _ _) ≫ (opens.inf_le_right _ _) ≫ iV, }, { intro w, dsimp, diff --git a/formal/lean/mathlib/topology/sheaves/stalks.lean b/formal/lean/mathlib/topology/sheaves/stalks.lean index 5ff3e104a9df6b274a36afb53dc9669998b51480..90f4f1ce8cb0cd558136457e275de7e21d1d0597 100644 --- a/formal/lean/mathlib/topology/sheaves/stalks.lean +++ b/formal/lean/mathlib/topology/sheaves/stalks.lean @@ -155,7 +155,7 @@ begin dsimp [stalk_pushforward, stalk_functor], ext1, tactic.op_induction', - cases j, cases j_val, + rcases j with ⟨⟨_, _⟩, _⟩, rw [colimit.ι_map_assoc, colimit.ι_map, colimit.ι_pre, whisker_left_app, whisker_right_app, pushforward.id_hom_app, eq_to_hom_map, eq_to_hom_refl], dsimp, @@ -172,8 +172,7 @@ begin dsimp [stalk_pushforward, stalk_functor], ext U, induction U using opposite.rec, - cases U, - cases U_val, + rcases U with ⟨⟨_, _⟩, _⟩, simp only [colimit.ι_map_assoc, colimit.ι_pre_assoc, whisker_right_app, category.assoc], dsimp, diff --git a/formal/lean/mathlib/topology/shrinking_lemma.lean b/formal/lean/mathlib/topology/shrinking_lemma.lean index 7c346e2dd844338ec1a07b46f59a8f2ea293b883..cc359edf4b3cedc3326c15434c2c121168529b29 100644 --- a/formal/lean/mathlib/topology/shrinking_lemma.lean +++ b/formal/lean/mathlib/topology/shrinking_lemma.lean @@ -45,7 +45,7 @@ namespace shrinking_lemma This type is equipped with the folowing partial order: `v ≤ v'` if `v.carrier ⊆ v'.carrier` and `v i = v' i` for `i ∈ v.carrier`. We will use Zorn's lemma to prove that this type has a maximal element, then show that the maximal element must have `carrier = univ`. -/ -@[nolint has_inhabited_instance] -- the trivial refinement needs `u` to be a covering +@[nolint has_nonempty_instance] -- the trivial refinement needs `u` to be a covering structure partial_refinement (u : ι → set X) (s : set X) := (to_fun : ι → set X) (carrier : set ι) diff --git a/formal/lean/mathlib/topology/subset_properties.lean b/formal/lean/mathlib/topology/subset_properties.lean index 853e00436baea27b2114cccf48f41d084e46f118..c10575b739b9bb8fbfd2d0a22c6282177fc125fd 100644 --- a/formal/lean/mathlib/topology/subset_properties.lean +++ b/formal/lean/mathlib/topology/subset_properties.lean @@ -413,8 +413,8 @@ lemma compact_accumulate {K : ℕ → set α} (hK : ∀ n, is_compact (K n)) (n is_compact (accumulate K n) := (finite_le_nat n).compact_bUnion $ λ k _, hK k -lemma compact_Union {f : ι → set α} [fintype ι] - (h : ∀ i, is_compact (f i)) : is_compact (⋃ i, f i) := +lemma compact_Union {f : ι → set α} [finite ι] (h : ∀ i, is_compact (f i)) : + is_compact (⋃ i, f i) := by rw ← bUnion_univ; exact finite_univ.compact_bUnion (λ i _, h i) lemma set.finite.is_compact (hs : s.finite) : is_compact s := @@ -751,11 +751,10 @@ lemma not_compact_space_iff : ¬compact_space α ↔ noncompact_space α := instance : noncompact_space ℤ := noncompact_space_of_ne_bot $ by simp only [filter.cocompact_eq_cofinite, filter.cofinite_ne_bot] +-- Note: We can't make this into an instance because it loops with `finite.compact_space`. /-- A compact discrete space is finite. -/ -noncomputable -def fintype_of_compact_of_discrete [compact_space α] [discrete_topology α] : - fintype α := -fintype_of_finite_univ $ compact_univ.finite_of_discrete +lemma finite_of_compact_of_discrete [compact_space α] [discrete_topology α] : finite α := +finite.of_finite_univ $ compact_univ.finite_of_discrete lemma finite_cover_nhds_interior [compact_space α] {U : α → set α} (hU : ∀ x, U x ∈ 𝓝 x) : ∃ t : finset α, (⋃ x ∈ t, interior (U x)) = univ := @@ -915,7 +914,7 @@ begin end /-- Finite topological spaces are compact. -/ -@[priority 100] instance fintype.compact_space [fintype α] : compact_space α := +@[priority 100] instance finite.compact_space [finite α] : compact_space α := { compact_univ := finite_univ.is_compact } /-- The product of two compact spaces is compact. -/ @@ -929,7 +928,7 @@ instance [compact_space α] [compact_space β] : compact_space (α ⊕ β) := exact (is_compact_range continuous_inl).union (is_compact_range continuous_inr) end⟩ -instance [fintype ι] [Π i, topological_space (π i)] [∀ i, compact_space (π i)] : +instance [finite ι] [Π i, topological_space (π i)] [∀ i, compact_space (π i)] : compact_space (Σ i, π i) := begin refine ⟨_⟩, @@ -1418,8 +1417,8 @@ lemma is_clopen.prod {s : set α} {t : set β} (hs : is_clopen s) (ht : is_clope is_clopen (s ×ˢ t) := ⟨hs.1.prod ht.1, hs.2.prod ht.2⟩ -lemma is_clopen_Union {β : Type*} [fintype β] {s : β → set α} - (h : ∀ i, is_clopen (s i)) : is_clopen (⋃ i, s i) := +lemma is_clopen_Union {β : Type*} [finite β] {s : β → set α} (h : ∀ i, is_clopen (s i)) : + is_clopen (⋃ i, s i) := ⟨is_open_Union (forall_and_distrib.1 h).1, is_closed_Union (forall_and_distrib.1 h).2⟩ lemma is_clopen_bUnion {β : Type*} {s : set β} {f : β → set α} (hs : s.finite) @@ -1432,8 +1431,8 @@ lemma is_clopen_bUnion_finset {β : Type*} {s : finset β} {f : β → set α} is_clopen (⋃ i ∈ s, f i) := is_clopen_bUnion s.finite_to_set h -lemma is_clopen_Inter {β : Type*} [fintype β] {s : β → set α} - (h : ∀ i, is_clopen (s i)) : is_clopen (⋂ i, s i) := +lemma is_clopen_Inter {β : Type*} [finite β] {s : β → set α} (h : ∀ i, is_clopen (s i)) : + is_clopen (⋂ i, s i) := ⟨(is_open_Inter (forall_and_distrib.1 h).1), (is_closed_Inter (forall_and_distrib.1 h).2)⟩ lemma is_clopen_bInter {β : Type*} {s : set β} (hs : s.finite) {f : β → set α} diff --git a/formal/lean/mathlib/topology/uniform_space/basic.lean b/formal/lean/mathlib/topology/uniform_space/basic.lean index a9b7e29604558b4216c110f0cd1654c14927ce29..5aad3035aa56cd20ed5c26698703f047de3c3a1b 100644 --- a/formal/lean/mathlib/topology/uniform_space/basic.lean +++ b/formal/lean/mathlib/topology/uniform_space/basic.lean @@ -1241,6 +1241,31 @@ instance : uniform_space bool := ⊥ instance : uniform_space ℕ := ⊥ instance : uniform_space ℤ := ⊥ +section +variables [uniform_space α] + +open additive multiplicative + +instance : uniform_space (additive α) := ‹uniform_space α› +instance : uniform_space (multiplicative α) := ‹uniform_space α› + +lemma uniform_continuous_of_mul : uniform_continuous (of_mul : α → additive α) := +uniform_continuous_id +lemma uniform_continuous_to_mul : uniform_continuous (to_mul : additive α → α) := +uniform_continuous_id +lemma uniform_continuous_of_add : uniform_continuous (of_add : α → multiplicative α) := +uniform_continuous_id +lemma uniform_continuous_to_add : uniform_continuous (to_add : multiplicative α → α) := +uniform_continuous_id + +lemma uniformity_additive : 𝓤 (additive α) = (𝓤 α).map (prod.map of_mul of_mul) := +by { convert map_id.symm, exact prod.map_id } + +lemma uniformity_multiplicative : 𝓤 (multiplicative α) = (𝓤 α).map (prod.map of_add of_add) := +by { convert map_id.symm, exact prod.map_id } + +end + instance {p : α → Prop} [t : uniform_space α] : uniform_space (subtype p) := uniform_space.comap subtype.val t diff --git a/formal/lean/mathlib/topology/uniform_space/compact_separated.lean b/formal/lean/mathlib/topology/uniform_space/compact_separated.lean index e0d5d3d3836a55c182b6b04a1264b9795ebeee15..e226a3b7e569e873739f73d7c8ec72a7ad10e011 100644 --- a/formal/lean/mathlib/topology/uniform_space/compact_separated.lean +++ b/formal/lean/mathlib/topology/uniform_space/compact_separated.lean @@ -221,11 +221,11 @@ hs.uniform_continuous_on_of_continuous' (is_separated_of_separated_space s) hf neighborhood `U` of `x`. -/ lemma continuous_on.tendsto_uniformly [locally_compact_space α] [compact_space β] [separated_space β] [uniform_space γ] {f : α → β → γ} {x : α} {U : set α} - (hxU : U ∈ 𝓝 x) (hU : is_separated U) (h : continuous_on ↿f (U ×ˢ (univ : set β))) : + (hxU : U ∈ 𝓝 x) (hU : is_separated U) (h : continuous_on ↿f (U ×ˢ univ)) : tendsto_uniformly f (f x) (𝓝 x) := begin rcases locally_compact_space.local_compact_nhds _ _ hxU with ⟨K, hxK, hKU, hK⟩, - have : uniform_continuous_on ↿f (K ×ˢ (univ : set β)), + have : uniform_continuous_on ↿f (K ×ˢ univ), { refine is_compact.uniform_continuous_on_of_continuous' (hK.prod compact_univ) _ (h.mono $ prod_mono hKU subset.rfl), exact (hU.mono hKU).prod (is_separated_of_separated_space _) }, diff --git a/formal/lean/mathlib/topology/uniform_space/equiv.lean b/formal/lean/mathlib/topology/uniform_space/equiv.lean index 9ebf5a9481b64e291e5bf20acaf67c2c92399ca8..037462f508c2a553366b0a52cadc32c8cc43b703 100644 --- a/formal/lean/mathlib/topology/uniform_space/equiv.lean +++ b/formal/lean/mathlib/topology/uniform_space/equiv.lean @@ -27,7 +27,7 @@ open_locale variables {α : Type*} {β : Type*} {γ : Type*} {δ : Type*} /-- Uniform isomorphism between `α` and `β` -/ -@[nolint has_inhabited_instance] -- not all spaces are homeomorphic to each other +@[nolint has_nonempty_instance] -- not all spaces are homeomorphic to each other structure uniform_equiv (α : Type*) (β : Type*) [uniform_space α] [uniform_space β] extends α ≃ β := (uniform_continuous_to_fun : uniform_continuous to_fun) diff --git a/formal/lean/mathlib/topology/vector_bundle/basic.lean b/formal/lean/mathlib/topology/vector_bundle/basic.lean index 1f421ccfd5d5ea40ad375d6c512a77e9f51e8d5e..a5152e0f1972f9e6aea00cbcf3c1c97217671335 100644 --- a/formal/lean/mathlib/topology/vector_bundle/basic.lean +++ b/formal/lean/mathlib/topology/vector_bundle/basic.lean @@ -56,7 +56,7 @@ variables [semiring R] [∀ x, add_comm_monoid (E x)] [∀ x, module R (E x)] /-- A pretrivialization for a (yet to be defined) topological vector bundle `total_space E` is a local equiv between sets of the form `proj ⁻¹' base_set` and `base_set × F` which respects the first coordinate, and is linear in each fiber. -/ -@[ext, nolint has_inhabited_instance] +@[ext, nolint has_nonempty_instance] structure topological_vector_bundle.pretrivialization extends to_fiber_bundle_pretrivialization : topological_fiber_bundle.pretrivialization F (@total_space.proj B E) := (linear' : ∀ x ∈ base_set, is_linear_map R (λ y : E x, (to_fun (total_space_mk x y)).2)) @@ -234,7 +234,7 @@ A structure extending local homeomorphisms, defining a local trivialization of t and `B × F` defined between two sets of the form `proj ⁻¹' base_set` and `base_set × F`, acting trivially on the first coordinate and linear in the fibers. -/ -@[ext, nolint has_inhabited_instance] +@[ext, nolint has_nonempty_instance] structure topological_vector_bundle.trivialization extends to_fiber_bundle_trivialization : topological_fiber_bundle.trivialization F (@total_space.proj B E) := (linear' : ∀ x ∈ base_set, is_linear_map R (λ y : E x, (to_fun (total_space_mk x y)).2)) @@ -351,8 +351,7 @@ lemma apply_mk_symm (e : trivialization R F E) {b : B} (hb : b ∈ e.base_set) ( e.to_pretrivialization.apply_mk_symm hb y lemma continuous_on_symm (e : trivialization R F E) : - continuous_on (λ z : B × F, total_space_mk z.1 (e.symm z.1 z.2)) - (e.base_set ×ˢ (univ : set F)) := + continuous_on (λ z : B × F, total_space_mk z.1 (e.symm z.1 z.2)) (e.base_set ×ˢ univ) := begin have : ∀ (z : B × F) (hz : z ∈ e.base_set ×ˢ (univ : set F)), total_space_mk z.1 (e.symm z.1 z.2) = e.to_local_homeomorph.symm z, @@ -490,7 +489,7 @@ target both `s ×ˢ univ`, which on this set is of the form `λ (b, v), (b, ε b map `ε` from `s` to `F ≃L[R] F`. Here continuity is with respect to the operator norm on `F →L[R] F`. -/ def continuous_transitions (e : local_equiv (B × F) (B × F)) : Prop := -∃ s : set B, e.source = s ×ˢ (univ : set F) ∧ e.target = s ×ˢ (univ : set F) +∃ s : set B, e.source = s ×ˢ univ ∧ e.target = s ×ˢ univ ∧ ∃ ε : B → (F ≃L[R] F), continuous_on (λ b, (ε b : F →L[R] F)) s ∧ ∀ b ∈ s, ∀ v : F, e (b, v) = (b, ε b v) @@ -765,7 +764,7 @@ lemma coord_change_linear_comp (i j k : ι): ∀ x ∈ (Z.base_set i) ∩ (Z.bas λ x hx, by { ext v, exact Z.coord_change_comp i j k x hx v } /-- The index set of a topological vector bundle core, as a convenience function for dot notation -/ -@[nolint unused_arguments has_inhabited_instance] +@[nolint unused_arguments has_nonempty_instance] def index := ι /-- The base space of a topological vector bundle core, as a convenience function for dot notation-/ @@ -774,7 +773,7 @@ def base := B /-- The fiber of a topological vector bundle core, as a convenience function for dot notation and typeclass inference -/ -@[nolint unused_arguments has_inhabited_instance] +@[nolint unused_arguments has_nonempty_instance] def fiber (x : B) := F instance topological_space_fiber (x : B) : topological_space (Z.fiber x) := @@ -934,7 +933,7 @@ The field `exists_coord_change` is stated as an existential statement (instead o fields), since it depends on propositional information (namely `e e' ∈ pretrivialization_atlas`). This makes it inconvenient to explicitly define a `coord_change` function when constructing a `topological_vector_prebundle`. -/ -@[nolint has_inhabited_instance] +@[nolint has_nonempty_instance] structure topological_vector_prebundle := (pretrivialization_atlas : set (pretrivialization R F E)) (pretrivialization_at : B → pretrivialization R F E) @@ -992,8 +991,7 @@ def to_topological_fiber_prebundle (a : topological_vector_prebundle R F E) : ((a.continuous_on_coord_change he' he).prod_map continuous_on_id), have H : e'.to_fiber_bundle_pretrivialization.to_local_equiv.target ∩ e'.to_fiber_bundle_pretrivialization.to_local_equiv.symm ⁻¹' - e.to_fiber_bundle_pretrivialization.to_local_equiv.source = - (e'.base_set ∩ e.base_set) ×ˢ (univ : set F), + e.to_fiber_bundle_pretrivialization.to_local_equiv.source =(e'.base_set ∩ e.base_set) ×ˢ univ, { rw [e'.target_eq, e.source_eq], ext ⟨b, f⟩, simp only [-total_space.proj, and.congr_right_iff, e'.proj_symm_apply', iff_self, diff --git a/formal/lean/mathlib/topology/vector_bundle/hom.lean b/formal/lean/mathlib/topology/vector_bundle/hom.lean index d9a29f2ce10ac99e62edb9a181a043cfd57f56ea..a21c54f6ffe811e93a1dc0049434ef23c6f80837 100644 --- a/formal/lean/mathlib/topology/vector_bundle/hom.lean +++ b/formal/lean/mathlib/topology/vector_bundle/hom.lean @@ -134,7 +134,7 @@ def continuous_linear_map : { to_fun := λ p, ⟨p.1, (e₂.continuous_linear_map_at p.1).comp $ p.2.comp $ e₁.symmL p.1⟩, inv_fun := λ p, ⟨p.1, (e₂.symmL p.1).comp $ p.2.comp $ e₁.continuous_linear_map_at p.1⟩, source := (bundle.total_space.proj) ⁻¹' (e₁.base_set ∩ e₂.base_set), - target := (e₁.base_set ∩ e₂.base_set) ×ˢ (set.univ : set (F₁ →SL[σ] F₂)), + target := (e₁.base_set ∩ e₂.base_set) ×ˢ set.univ, map_source' := λ ⟨x, L⟩ h, ⟨h, set.mem_univ _⟩, map_target' := λ ⟨x, f⟩ h, h.1, left_inv' := λ ⟨x, L⟩ ⟨h₁, h₂⟩, diff --git a/formal/lean/mathlib/topology/vector_bundle/prod.lean b/formal/lean/mathlib/topology/vector_bundle/prod.lean index 43ced74eaa23e20741cd71071392db92c18876c9..a7486d3642dc5828dc2bbbc1fdf28ab70eb314c9 100644 --- a/formal/lean/mathlib/topology/vector_bundle/prod.lean +++ b/formal/lean/mathlib/topology/vector_bundle/prod.lean @@ -130,7 +130,7 @@ begin end lemma prod.continuous_inv_fun : - continuous_on (prod.inv_fun' e₁ e₂) ((e₁.base_set ∩ e₂.base_set) ×ˢ (univ : set (F₁ × F₂))) := + continuous_on (prod.inv_fun' e₁ e₂) ((e₁.base_set ∩ e₂.base_set) ×ˢ univ) := begin rw (prod.inducing_diag E₁ E₂).continuous_on_iff, have H₁ : continuous (λ p : B × F₁ × F₂, ((p.1, p.2.1), (p.1, p.2.2))) := @@ -151,7 +151,7 @@ def prod : trivialization R (F₁ × F₂) (E₁ ×ᵇ E₂) := { to_fun := prod.to_fun' e₁ e₂, inv_fun := prod.inv_fun' e₁ e₂, source := (@total_space.proj B (E₁ ×ᵇ E₂)) ⁻¹' (e₁.base_set ∩ e₂.base_set), - target := (e₁.base_set ∩ e₂.base_set) ×ˢ (set.univ : set (F₁ × F₂)), + target := (e₁.base_set ∩ e₂.base_set) ×ˢ set.univ, map_source' := λ x h, ⟨h, set.mem_univ _⟩, map_target' := λ x h, h.1, left_inv' := λ x, prod.left_inv, diff --git a/formal/lean/mathlib/topology/vector_bundle/pullback.lean b/formal/lean/mathlib/topology/vector_bundle/pullback.lean index 8074fd62b983cea75af2e19ba8a9d323e507b973..68abd4579383474abcbd7f36d3f2cc5c2daf36e7 100644 --- a/formal/lean/mathlib/topology/vector_bundle/pullback.lean +++ b/formal/lean/mathlib/topology/vector_bundle/pullback.lean @@ -88,7 +88,7 @@ def topological_vector_bundle.trivialization.pullback (e : trivialization 𝕜 F inv_fun := λ y, @total_space_mk _ (f *ᵖ E) y.1 (e.symm (f y.1) y.2), source := pullback.lift f ⁻¹' e.source, base_set := f ⁻¹' e.base_set, - target := (f ⁻¹' e.base_set) ×ˢ (univ : set F), + target := (f ⁻¹' e.base_set) ×ˢ univ, map_source' := λ x h, by { simp_rw [e.source_eq, mem_preimage, pullback.proj_lift] at h, simp_rw [prod_mk_mem_set_prod_eq, mem_univ, and_true, mem_preimage, h] }, map_target' := λ y h, by { rw [mem_prod, mem_preimage] at h, diff --git a/formal/lean/sphere-eversion/global/gromov.lean b/formal/lean/sphere-eversion/global/gromov.lean new file mode 100644 index 0000000000000000000000000000000000000000..f14c46cdeedf941f7741342bf9bf2950bef90063 --- /dev/null +++ b/formal/lean/sphere-eversion/global/gromov.lean @@ -0,0 +1,37 @@ +import global.relation + +/-! +# Gromov's theorem + +We prove the h-principle for open and ample first order differential relations. +-/ + +noncomputable theory + +open set +open_locale topological_space manifold + +variables +{E : Type*} [normed_add_comm_group E] [normed_space ℝ E] +{H : Type*} [topological_space H] {I : model_with_corners ℝ E H} +{M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] +{E' : Type*} [normed_add_comm_group E'] [normed_space ℝ E'] +{H' : Type*} [topological_space H'] {I' : model_with_corners ℝ E' H'} +{M' : Type*} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] +{F : Type*} [normed_add_comm_group F] [normed_space ℝ F] +{G : Type*} [topological_space G] (J : model_with_corners ℝ F G) +(N : Type*) [topological_space N] [charted_space G N] [smooth_manifold_with_corners J N] +{F' : Type*} [normed_add_comm_group F'] [normed_space ℝ F'] +{G' : Type*} [topological_space G'] (J' : model_with_corners ℝ F' G') +(N' : Type*) [topological_space N'] [charted_space G' N'] [smooth_manifold_with_corners J' N'] +{R : rel_mfld I M I' M'} + +/-- The non-parametric version of Gromov's theorem -/ +lemma rel_mfld.ample.satisfies_h_principle (h1 : R.ample) (h2 : is_open R) : + R.satisfies_h_principle := +sorry + +/-- **Gromov's Theorem** -/ +theorem rel_mfld.ample.satisfies_h_principle_with (h1 : R.ample) (h2 : is_open R) : + R.satisfies_h_principle_with J N := +sorry diff --git a/formal/lean/sphere-eversion/global/immersion.lean b/formal/lean/sphere-eversion/global/immersion.lean index 0b712c889525ed66aea4710f4173fbce0eb57a28..982fcabe73357c129a2a73e6f44e63705136c8a6 100644 --- a/formal/lean/sphere-eversion/global/immersion.lean +++ b/formal/lean/sphere-eversion/global/immersion.lean @@ -1,6 +1,6 @@ import geometry.manifold.instances.sphere -import global.relation +import global.gromov noncomputable theory @@ -10,14 +10,18 @@ open_locale manifold section general variables -{E : Type*} [normed_group E] [normed_space ℝ E] +{E : Type*} [normed_add_comm_group E] [normed_space ℝ E] {H : Type*} [topological_space H] (I : model_with_corners ℝ E H) {M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] -{E' : Type*} [normed_group E'] [normed_space ℝ E'] +{E' : Type*} [normed_add_comm_group E'] [normed_space ℝ E'] {H' : Type*} [topological_space H'] (I' : model_with_corners ℝ E' H') {M' : Type*} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] +{F : Type*} [normed_add_comm_group F] [normed_space ℝ F] +{G : Type*} [topological_space G] (J : model_with_corners ℝ F G) +(N : Type*) [topological_space N] [charted_space G N] [smooth_manifold_with_corners J N] + local notation `TM` := tangent_space I local notation `TM'` := tangent_space I' @@ -38,9 +42,20 @@ lemma immersion_iff_one_jet (f : M → M') : immersion I I' f ↔ ∀ m, one_jet_ext I I' f m ∈ immersion_rel I M I' M' := by simp [immersion, one_jet_ext, immersion_rel] -/- -TODO: state the h-principle for immersions in positive codimension. --/ +variables [finite_dimensional ℝ E] [finite_dimensional ℝ E'] + +lemma immersion_rel_ample (h : finrank ℝ E < finrank ℝ E') : + (immersion_rel I M I' M').ample := +sorry + +/-- parametric h-principle for immersions. -/ +theorem immersion_rel_satisfies_h_principle_with (h : finrank ℝ E < finrank ℝ E') : + (immersion_rel I M I' M').satisfies_h_principle_with J N := +begin + apply (immersion_rel_ample I I' h).satisfies_h_principle_with J N, + have : is_open {L : E →L[ℝ] E' | injective L} := continuous_linear_map.is_open_injective, + sorry +end end general diff --git a/formal/lean/sphere-eversion/global/indexing.lean b/formal/lean/sphere-eversion/global/indexing.lean index ad2eff6c224f5002c569a4930a9e190afd55b48d..7a3977103025d2e7b1748ead1c8301a051243b99 100644 --- a/formal/lean/sphere-eversion/global/indexing.lean +++ b/formal/lean/sphere-eversion/global/indexing.lean @@ -1,6 +1,6 @@ import tactic.linarith import algebra.order.with_zero -import topology.basic +import topology.locally_finite import to_mathlib.set_theory.cardinal.basic diff --git a/formal/lean/sphere-eversion/global/localisation_data.lean b/formal/lean/sphere-eversion/global/localisation_data.lean index d5fbd0c67d20914b206ca5e576b1a1aecda6fea0..b50262d1480ab699a38dba9fbaad4efa09a9301d 100644 --- a/formal/lean/sphere-eversion/global/localisation_data.lean +++ b/formal/lean/sphere-eversion/global/localisation_data.lean @@ -8,12 +8,12 @@ open_locale manifold open metric section -variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] +variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type*} [topological_space H] (I : model_with_corners 𝕜 E H) {M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] - {E' : Type*} [normed_group E'] [normed_space 𝕜 E'] + {E' : Type*} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type*} [topological_space H'] (I' : model_with_corners 𝕜 E' H') {M' : Type*} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] @@ -33,11 +33,11 @@ structure localisation_data (f : continuous_map M M') := end section -variables (𝕜 : Type*) [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] +variables (𝕜 : Type*) [nontrivially_normed_field 𝕜] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] {M : Type*} [topological_space M] [sigma_compact_space M] [locally_compact_space M] [t2_space M] [nonempty M] [charted_space E M] [smooth_manifold_with_corners 𝓘(𝕜, E) M] - (E' : Type*) [normed_group E'] [normed_space 𝕜 E'] + (E' : Type*) [normed_add_comm_group E'] [normed_space 𝕜 E'] {M' : Type*} [topological_space M'] [sigma_compact_space M'] [locally_compact_space M'] [t2_space M'] [nonempty M'] [charted_space E' M'] [smooth_manifold_with_corners 𝓘(𝕜, E') M'] diff --git a/formal/lean/sphere-eversion/global/one_jet_bundle.lean b/formal/lean/sphere-eversion/global/one_jet_bundle.lean index f22bced0bc0478c413aebc9b9a0133493d4b4cfb..ba0de713fcd4600ec49b902632c8e900e1146408 100644 --- a/formal/lean/sphere-eversion/global/one_jet_bundle.lean +++ b/formal/lean/sphere-eversion/global/one_jet_bundle.lean @@ -5,24 +5,38 @@ Authors: Patrick Massot, Floris van Doorn -/ import to_mathlib.geometry.manifold.vector_bundle.basic_core_constructions +/-! +# 1-jet bundles + +This file contains the definition of the 1-jet bundle `J¹(M, M')`, also known as +`one_jet_bundle I M I' M'`. + +We also define +* `one_jet_ext I I' f : M → J¹(M, M')`: the 1-jet extension `j¹f` of a map `f : M → M'` + +We prove +* If `f` is smooth, `j¹f` is smooth. +* If `x ↦ (f₁ x, f₂ x, ϕ₁ x) : N → J¹(M₁, M₂)` and `x ↦ (f₂ x, f₃ x, ϕ₂ x) : N → J¹(M₂, M₃)` are smooth, then so is `x ↦ (f₁ x, f₃ x, ϕ₂ x ∘ ϕ₁ x) : N → J¹(M₁, M₃)`. +-/ + noncomputable theory -open set equiv +open filter set equiv basic_smooth_vector_bundle_core open_locale manifold -variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] +variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type*} [topological_space H] (I : model_with_corners 𝕜 E H) (M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] - {E' : Type*} [normed_group E'] [normed_space 𝕜 E'] + {E' : Type*} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type*} [topological_space H'] (I' : model_with_corners 𝕜 E' H') (M' : Type*) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] - {F : Type*} [normed_group F] [normed_space 𝕜 F] + {F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F] {G : Type*} [topological_space G] (J : model_with_corners 𝕜 F G) {N : Type*} [topological_space N] [charted_space G N] [smooth_manifold_with_corners J N] - {F' : Type*} [normed_group F'] [normed_space 𝕜 F'] + {F' : Type*} [normed_add_comm_group F'] [normed_space 𝕜 F'] {G' : Type*} [topological_space G'] (J' : model_with_corners 𝕜 F' G') {N' : Type*} [topological_space N'] [charted_space G' N'] [smooth_manifold_with_corners J' N'] @@ -35,7 +49,7 @@ variables {M M'} /-- The fibers of the one jet-bundle. -/ @[nolint unused_arguments] -def one_jet_space (p : M × M') : Type* := E →L[𝕜] E' +def one_jet_space (p : M × M') : Type* := tangent_space I p.1 →L[𝕜] tangent_space I' p.2 instance (p : M × M') : has_coe_to_fun (one_jet_space I I' p) (λ σ, tangent_space I p.1 → tangent_space I' p.2) := ⟨λ φ, φ.to_fun⟩ @@ -73,7 +87,7 @@ section variables {M} (p : M × M') -instance : normed_group (one_jet_space I I' p) := by delta_instance one_jet_space +instance : normed_add_comm_group (one_jet_space I I' p) := by delta_instance one_jet_space instance : normed_space 𝕜 (one_jet_space I I' p) := by delta_instance one_jet_space instance : inhabited (one_jet_space I I' p) := ⟨0⟩ @@ -107,6 +121,36 @@ lemma one_jet_bundle_proj_continuous : continuous (one_jet_bundle.proj I M I' M' lemma one_jet_bundle_proj_open : is_open_map (one_jet_bundle.proj I M I' M') := ((one_jet_bundle_core I M I' M').to_topological_vector_bundle_core).is_open_map_proj +/-- Computing the value of a chart around `v` at point `v'` in `J¹(M, M')`. + The last component equals the continuous linear map `v'.2`, composed on both sides by an + appropriate coordinate change function. -/ +lemma one_jet_bundle_chart_at {v v' : one_jet_bundle I M I' M'} : + chart_at (model_prod (model_prod H H') (E →L[𝕜] E')) v v' = + ((chart_at H v.1.1 v'.1.1, chart_at H' v.1.2 v'.1.2), + ((tangent_bundle_core I' M').coord_change (achart H' v'.1.2) (achart H' v.1.2) + (chart_at H' v'.1.2 v'.1.2)).comp $ v'.2.comp $ + (tangent_bundle_core I M).coord_change (achart H v.1.1) (achart H v'.1.1) + (chart_at H v.1.1 v'.1.1)) := +begin + simp_rw [to_charted_space_chart_at], + dsimp only [one_jet_bundle_core], + simp_rw [hom_chart, ← achart_def, pullback_fst_coord_change_at, + pullback_snd_coord_change_at, prod_charted_space_chart_at, local_homeomorph.prod_apply], +end + +/-- Computing the value of an extended chart around `v` at point `v'` in `J¹(M, M')`. + The last component equals the continuous linear map `v'.2`, composed on both sides by an + appropriate coordinate change function. -/ +lemma one_jet_bundle_ext_chart_at {v v' : one_jet_bundle I M I' M'} : + ext_chart_at ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) v v' = + ((I (chart_at H v.1.1 v'.1.1), I' (chart_at H' v.1.2 v'.1.2)), + ((tangent_bundle_core I' M').coord_change (achart H' v'.1.2) (achart H' v.1.2) + (chart_at H' v'.1.2 v'.1.2)).comp $ v'.2.comp $ + (tangent_bundle_core I M).coord_change (achart H v.1.1) (achart H v'.1.1) + (chart_at H v.1.1 v'.1.1)) := +by simp_rw [ext_chart_at_coe, function.comp_apply, one_jet_bundle_chart_at, + model_with_corners.prod_apply, model_with_corners_self_coe, id] + section maps variables {M M'} @@ -115,13 +159,18 @@ variables {M M'} def one_jet_ext (f : M → M') : M → one_jet_bundle I M I' M' := λ x, ⟨(x, f x), (mfderiv I I' f x : tangent_space I x →L[𝕜] tangent_space I' (f x))⟩ -variables {I I'} +variables {I I' J J'} /-- The constructor of one_jet_bundle, in case `sigma.mk` will not give the right type. -/ @[simp] def one_jet_bundle.mk (x : M) (y : M') (f : one_jet_space I I' (x, y)) : one_jet_bundle I M I' M' := ⟨(x, y), f⟩ +@[simp, mfld_simps] lemma one_jet_bundle_mk_fst {x : M} {y : M'} {f : one_jet_space I I' (x, y)} : + (one_jet_bundle.mk x y f).1 = (x, y) := rfl + +@[simp, mfld_simps] lemma one_jet_bundle_mk_snd {x : M} {y : M'} {f : one_jet_space I I' (x, y)} : + (one_jet_bundle.mk x y f).2 = f := rfl @[simp, mfld_simps] lemma one_jet_ext_one_jet_bundle_proj {f : M → M'} {x : M} : one_jet_bundle.proj I M I' M' (one_jet_ext I I' f x) = (x, f x) := rfl @@ -129,41 +178,91 @@ variables {I I'} @[simp, mfld_simps] lemma one_jet_ext_proj {f : M → M'} {x : M} : (one_jet_ext I I' f x).1 = (x, f x) := rfl -open basic_smooth_vector_bundle_core - -lemma smooth_at.one_jet_ext {f : M → M'} {x : M} (hf : smooth_at I I' f x) : - smooth_at I ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) (one_jet_ext I I' f) x := +/-- A family of one-jet extensions indexed by a parameter is smooth. -/ +lemma smooth_at.one_jet_ext' {f : N → M → M'} {g : N → M} {n : N} + (hf : smooth_at (J.prod I) I' (function.uncurry f) (n, g n)) (hg : smooth_at J I g n) : + smooth_at J ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) (λ x, one_jet_ext I I' (f x) (g x)) n := begin + -- it is so horrible to work with `cont_mdiff_at.comp` + have : smooth_at J I' (λ x, f x (g x)) n, + { exact cont_mdiff_at.comp n hf (smooth_at_id.prod_mk hg) }, rw [smooth_at, (one_jet_bundle_core I M I' M').cont_mdiff_at_iff_target], - refine ⟨continuous_at_id.prod hf.continuous_at, _⟩, - simp_rw [ext_chart_at, local_equiv.coe_trans, function.comp, to_charted_space_chart_at], - dsimp only [one_jet_bundle_core], - simp_rw [local_homeomorph.coe_coe, hom_chart, ← achart_def, pullback_fst_coord_change_at, - pullback_snd_coord_change_at, model_with_corners.to_local_equiv_coe, - model_with_corners.prod_apply, model_with_corners_self_coe, id, prod_charted_space_chart_at, - local_homeomorph.prod_apply], - refine (cont_mdiff_at_ext_chart_at.prod_mk_space $ cont_mdiff_at_ext_chart_at.comp _ hf) - .prod_mk_space _, - exact hf.mfderiv' le_rfl + refine ⟨hg.continuous_at.prod this.continuous_at, _⟩, + simp_rw [function.comp, one_jet_bundle_ext_chart_at], + dsimp only [one_jet_ext_proj], + refine ((cont_mdiff_at_ext_chart_at.comp _ hg).prod_mk_space $ + cont_mdiff_at_ext_chart_at.comp _ this).prod_mk_space _, + sorry + -- exact hf.mfderiv' le_rfl end +lemma smooth_at.one_jet_ext {f : M → M'} {x : M} (hf : smooth_at I I' f x) : + smooth_at I ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) (one_jet_ext I I' f) x := +smooth_at.one_jet_ext' (cont_mdiff_at.comp (x, x) (by exact hf) smooth_at_snd) smooth_at_id + +lemma smooth.one_jet_ext' {f : N → M → M'} {g : N → M} + (hf : smooth (J.prod I) I' (function.uncurry f)) (hg : smooth J I g) : + smooth J ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) (λ x, one_jet_ext I I' (f x) (g x)) := +λ x, hf.smooth_at.one_jet_ext' hg.smooth_at + lemma smooth.one_jet_ext {f : M → M'} (hf : smooth I I' f) : smooth I ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) (one_jet_ext I I' f) := λ x, (hf x).smooth_at.one_jet_ext -def one_jet_comp {x : M} {y : M'} {z : N} (h : one_jet_space I' J (y, z)) - (g : one_jet_space I I' (x, y)) : one_jet_bundle I M J N := -one_jet_bundle.mk x z $ h.comp g - -variables (I I' J J') -lemma smooth.one_jet_comp - {f1 : N' → M} (f2 : N' → M') {f3 : N' → N} +variables (I') +lemma smooth.one_jet_comp {f1 : N' → M} (f2 : N' → M') {f3 : N' → N} {h : ∀ x : N', one_jet_space I' J (f2 x, f3 x)} {g : ∀ x : N', one_jet_space I I' (f1 x, f2 x)} (hh : smooth J' ((I'.prod J).prod 𝓘(𝕜, E' →L[𝕜] F)) (λ x, one_jet_bundle.mk _ _ (h x))) (hg : smooth J' ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) (λ x, one_jet_bundle.mk _ _ (g x))) : smooth J' ((I.prod J).prod 𝓘(𝕜, E →L[𝕜] F)) (λ x, one_jet_bundle.mk (f1 x) (f3 x) (h x ∘L g x) : N' → one_jet_bundle I M J N) := -sorry +begin + rw [basic_smooth_vector_bundle_core.smooth_iff_target] at hh hg ⊢, + refine ⟨hg.1.fst.prod_mk hh.1.snd, λ x, _⟩, + have hf2 : continuous_at f2 x := hg.1.snd.continuous_at, + simp_rw [function.comp, one_jet_bundle_ext_chart_at], + refine ((cont_diff_at_fst.fst.cont_mdiff_at.comp _ (hg.2 x)).prod_mk_space $ + cont_diff_at_fst.snd.cont_mdiff_at.comp _ (hh.2 x)).prod_mk_space _, + have h1 := (cont_diff_at_snd.cont_mdiff_at.comp _ (hg.2 x)), + have h2 := (cont_diff_at_snd.cont_mdiff_at.comp _ (hh.2 x)), + refine (h2.clm_comp h1).congr_of_eventually_eq _, + refine eventually_of_mem (hf2.preimage_mem_nhds $ (achart H' (f2 x)).1.open_source.mem_nhds $ + mem_achart_source H' (f2 x)) (λ x' hx', _), + ext v, + simp_rw [function.comp_apply, one_jet_bundle_ext_chart_at, + one_jet_bundle_mk_fst, one_jet_bundle_mk_snd, continuous_linear_map.comp_apply], + congr' 2, + symmetry, + exact (tangent_bundle_core I' M').coord_change_comp_eq_self' (mem_achart_source H' (f2 x')) hx' _ +end + +variables {I'} +lemma smooth.one_jet_add {f : N → M} {g : N → M'} + {ϕ ϕ' : ∀ x : N, one_jet_space I I' (f x, g x)} + (hϕ : smooth J ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) (λ x, one_jet_bundle.mk _ _ (ϕ x))) + (hϕ' : smooth J ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) (λ x, one_jet_bundle.mk _ _ (ϕ' x))) : + smooth J ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) + (λ x, one_jet_bundle.mk (f x) (g x) (ϕ x + ϕ' x)) := +begin + rw [basic_smooth_vector_bundle_core.smooth_iff_target] at hϕ hϕ' ⊢, + refine ⟨hϕ.1.fst.prod_mk hϕ.1.snd, λ x, _⟩, + simp_rw [function.comp, one_jet_bundle_ext_chart_at], + refine ((cont_diff_at_fst.fst.cont_mdiff_at.comp _ (hϕ.2 x)).prod_mk_space $ + cont_diff_at_fst.snd.cont_mdiff_at.comp _ (hϕ.2 x)).prod_mk_space _, + have h1 := (cont_diff_at_snd.cont_mdiff_at.comp _ (hϕ.2 x)), + have h2 := (cont_diff_at_snd.cont_mdiff_at.comp _ (hϕ'.2 x)), + refine (h1.add h2).congr_of_eventually_eq (eventually_of_forall $ λ x', _), + ext v, + simp_rw [pi.add_apply, function.comp_apply, one_jet_bundle_ext_chart_at, + one_jet_bundle_mk_fst, one_jet_bundle_mk_snd, continuous_linear_map.add_apply, + continuous_linear_map.comp_apply, continuous_linear_map.add_apply, + continuous_linear_map.map_add] +end + +/-- Used for `Ψ` in the notes. -/ +def map_left (f : M → N) (Dfinv : ∀ x : M, tangent_space J (f x) →L[𝕜] tangent_space I x) : + one_jet_bundle I M I' M' → one_jet_bundle J N I' M' := +λ p, one_jet_bundle.mk (f p.1.1) p.1.2 (p.2 ∘L Dfinv p.1.1) end maps diff --git a/formal/lean/sphere-eversion/global/one_jet_sec.lean b/formal/lean/sphere-eversion/global/one_jet_sec.lean index d2946dae9dc651b6526b2d7b5dcc64b50d96f203..7b7c0236bf47961098baa69e21d29e98134234be 100644 --- a/formal/lean/sphere-eversion/global/one_jet_sec.lean +++ b/formal/lean/sphere-eversion/global/one_jet_sec.lean @@ -26,13 +26,13 @@ open_locale topological_space manifold section general -variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] +variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] -- declare a smooth manifold `M` over the pair `(E, H)`. -{E : Type*} [normed_group E] [normed_space 𝕜 E] +{E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type*} [topological_space H] (I : model_with_corners 𝕜 E H) {M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] -- declare a smooth manifold `M'` over the pair `(E', H')`. -{E' : Type*} [normed_group E'] [normed_space 𝕜 E'] +{E' : Type*} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type*} [topological_space H'] (I' : model_with_corners 𝕜 E' H') {M' : Type*} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] @@ -40,26 +40,22 @@ variables (I I' M M') /-- A section of a 1-jet bundle seen as a bundle over the source manifold. -/ structure one_jet_sec := -(to_fun : M → one_jet_bundle I M I' M') -(is_sec' : ∀ x : M, (to_fun x).1.1 = x) -(smooth' : smooth I ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) to_fun) +(bs : M → M') +(ϕ : ∀ x : M, tangent_space I x →L[𝕜] tangent_space I' (bs x)) +(smooth' : smooth I ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) (λ x, one_jet_bundle.mk x (bs x) (ϕ x))) -instance : has_coe_to_fun (one_jet_sec I M I' M') (λ S, M → one_jet_bundle I M I' M'):= -⟨one_jet_sec.to_fun⟩ +instance : has_coe_to_fun (one_jet_sec I M I' M') (λ S, M → one_jet_bundle I M I' M') := +⟨λ S x, one_jet_bundle.mk x (S.bs x) (S.ϕ x)⟩ variables {I I' M M'} namespace one_jet_sec -lemma is_sec (F : one_jet_sec I M I' M') (x : M) : (F x).1.1 = x := -F.is_sec' x - -/-- The base map corresponding to a section of J¹(M, M') → M. -/ -def bs (F : one_jet_sec I M I' M') : M → M' := -λ x, (F x).1.2 - -lemma fst_eq (F : one_jet_sec I M I' M') (x : M) : (F x).1 = (x, F.bs x) := -prod.ext (F.is_sec x) rfl +lemma coe_apply (F : one_jet_sec I M I' M') (x : M) : F x = ⟨(x, F.bs x), (F.ϕ x)⟩ := rfl +lemma fst_eq (F : one_jet_sec I M I' M') (x : M) : (F x).1 = (x, F.bs x) := rfl +lemma snd_eq (F : one_jet_sec I M I' M') (x : M) : (F x).2 = F.ϕ x := rfl +lemma is_sec (F : one_jet_sec I M I' M') (x : M) : (F x).1.1 = x := rfl +lemma bs_eq (F : one_jet_sec I M I' M') (x : M) : F.bs x = (F x).1.2 := rfl protected lemma smooth (F : one_jet_sec I M I' M') : smooth I ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) F := @@ -67,7 +63,7 @@ F.smooth' lemma smooth_eta (F : one_jet_sec I M I' M') : smooth I ((I.prod I').prod 𝓘(𝕜, E →L[𝕜] E')) (λ x, one_jet_bundle.mk x (F.bs x) (F x).2 : M → one_jet_bundle I M I' M') := -by { convert F.smooth, ext1, rw [one_jet_bundle.mk, ← F.fst_eq x, sigma.eta] } +F.smooth lemma smooth_bs (F : one_jet_sec I M I' M') : smooth I I' F.bs := (smooth_snd.comp $ basic_smooth_vector_bundle_core.smooth_proj _).comp F.smooth @@ -93,53 +89,90 @@ end one_jet_sec /-- The one-jet extension of a function, seen as a section of the 1-jet bundle. -/ def one_jet_ext_sec (f : C^∞⟮I, M; I', M'⟯) : one_jet_sec I M I' M' := -⟨one_jet_ext I I' f, λ x, rfl, f.smooth.one_jet_ext⟩ +⟨f, mfderiv I I' f, f.smooth.one_jet_ext⟩ end general section real variables -{E : Type*} [normed_group E] [normed_space ℝ E] +{E : Type*} [normed_add_comm_group E] [normed_space ℝ E] {H : Type*} [topological_space H] (I : model_with_corners ℝ E H) (M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] -{E' : Type*} [normed_group E'] [normed_space ℝ E'] +{E' : Type*} [normed_add_comm_group E'] [normed_space ℝ E'] {H' : Type*} [topological_space H'] (I' : model_with_corners ℝ E' H') (M' : Type*) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] -{F : Type*} [normed_group F] [normed_space ℝ F] +{F : Type*} [normed_add_comm_group F] [normed_space ℝ F] {G : Type*} [topological_space G] (J : model_with_corners ℝ F G) (N : Type*) [topological_space N] [charted_space G N] [smooth_manifold_with_corners J N] -{F' : Type*} [normed_group F'] [normed_space ℝ F'] +{F' : Type*} [normed_add_comm_group F'] [normed_space ℝ F'] {G' : Type*} [topological_space G'] (J' : model_with_corners ℝ F' G') (N' : Type*) [topological_space N'] [charted_space G' N'] [smooth_manifold_with_corners J' N'] /-- A family of jet sections indexed by manifold `N` is a function from `N` into jet sections in such a way that the function is smooth as a function of all arguments. -/ structure family_one_jet_sec := -(to_fun : N → M → one_jet_bundle I M I' M') -(is_sec' : ∀ (t : N) (x : M), (to_fun t x).1.1 = x) -(smooth' : smooth (J.prod I) ((I.prod I').prod 𝓘(ℝ, E →L[ℝ] E')) (uncurry to_fun)) +(bs : N → M → M') +(ϕ : ∀ (n : N) (m : M), tangent_space I m →L[ℝ] tangent_space I' (bs n m)) +(smooth' : smooth (J.prod I) ((I.prod I').prod 𝓘(ℝ, E →L[ℝ] E')) + (λ p : N × M, one_jet_bundle.mk p.2 (bs p.1 p.2) (ϕ p.1 p.2))) instance : has_coe_to_fun (family_one_jet_sec I M I' M' J N) (λ S, N → one_jet_sec I M I' M') := ⟨λ S t, - { to_fun := S.to_fun t, - is_sec' := S.is_sec' t, + { bs := S.bs t, + ϕ := S.ϕ t, smooth' := λ x, (S.smooth' (t, x)).comp x $ smooth_at_const.prod_mk smooth_at_id }⟩ namespace family_one_jet_sec variables {I M I' M' J N J' N'} +protected lemma smooth (S : family_one_jet_sec I M I' M' J N) : + smooth (J.prod I) ((I.prod I').prod 𝓘(ℝ, E →L[ℝ] E')) (λ p : N × M, S p.1 p.2) := S.smooth' + +lemma smooth_bs (S : family_one_jet_sec I M I' M' J N) : + smooth (J.prod I) I' (λ p : N × M, S.bs p.1 p.2) := +(smooth_snd.comp $ basic_smooth_vector_bundle_core.smooth_proj _).comp S.smooth + /-- Reindex a family along a smooth function `f`. -/ def reindex (S : family_one_jet_sec I M I' M' J' N') (f : C^∞⟮J, N; J', N'⟯) : family_one_jet_sec I M I' M' J N := -{ to_fun := λ t, S (f t), - is_sec' := λ t, S.is_sec' (f t), +{ bs := λ t, S.bs (f t), + ϕ := λ t, S.ϕ (f t), smooth' := λ x, (S.smooth' (f x.1, x.2)).comp x $ (f.smooth.smooth_at).prod_map' smooth_at_id } +def prod (S : family_one_jet_sec I M I' M' J N) : one_jet_sec (I.prod J) (M × N) I' M' := +{ bs := λ p, S.bs p.2 p.1, + ϕ := λ p, mfderiv J I' (λ z, S.bs z p.1) p.2 ∘L mfderiv (I.prod J) J prod.snd p + + S.ϕ p.2 p.1 ∘L mfderiv (I.prod J) I prod.fst p, + smooth' := begin + refine smooth.one_jet_add _ _, + { refine smooth.one_jet_comp J (λ p, p.2) _ smooth_snd.one_jet_ext, + -- have := S.smooth_bs.comp (smooth_id.prod_mk smooth_const), dsimp [function.comp] at this, + -- have := smooth.one_jet_ext this, + sorry + }, + { refine smooth.one_jet_comp I (λ p, p.1) _ smooth_fst.one_jet_ext, + exact S.smooth.comp (smooth_snd.prod_mk smooth_fst) } + end } + +/- -- attempted version with one one `mfderiv` left of addition +def prod (S : family_one_jet_sec I M I' M' J N) : one_jet_sec (I.prod J) (M × N) I' M' := +{ bs := λ p, S.bs p.2 p.1, + ϕ := λ p, (mfderiv (I.prod J) I' (λ z : M × N, S.bs z.2 p.1) p : _) + + S.ϕ p.2 p.1 ∘L mfderiv (I.prod J) I prod.fst p, + smooth' := begin + refine smooth.one_jet_add _ _, + { refine smooth.one_jet_ext _, -- nope + }, + { refine smooth.one_jet_comp I (λ p, p.1) _ smooth_fst.one_jet_ext, + exact S.smooth.comp (smooth_snd.prod_mk smooth_fst) } + end } + +-/ end family_one_jet_sec /-- A homotopy of formal solutions is a family indexed by `ℝ` -/ -abbreviation htpy_one_jet_sec := family_one_jet_sec I M I' M' 𝓘(ℝ, ℝ) ℝ +@[reducible] def htpy_one_jet_sec := family_one_jet_sec I M I' M' 𝓘(ℝ, ℝ) ℝ example : has_coe_to_fun (htpy_one_jet_sec I M I' M') (λ S, ℝ → one_jet_sec I M I' M') := by apply_instance diff --git a/formal/lean/sphere-eversion/global/relation.lean b/formal/lean/sphere-eversion/global/relation.lean index 793d7b5cd566b6509a9b7ac9bcb956cb4260a7b6..04740c7a81fbf9221d90df0b74b3a57fd35718e1 100644 --- a/formal/lean/sphere-eversion/global/relation.lean +++ b/formal/lean/sphere-eversion/global/relation.lean @@ -1,11 +1,7 @@ import local.relation import global.one_jet_sec import global.smooth_embedding - -noncomputable theory - -open set function filter charted_space smooth_manifold_with_corners -open_locale topological_space manifold +import to_mathlib.topology.algebra.module /-! # First order partial differential relations for maps between manifolds @@ -17,27 +13,27 @@ relations for maps between vector spaces. Given manifolds `M` and `M'` modelled on `I` and `I'`, a first order partial differential relation for maps from `M` to `M'` is a set in the 1-jet bundle J¹(M, M'), also known as `one_jet_bundle I M I' M'`. +-/ -We also define -* `one_jet_ext I I' f`: the 1-jet extension of a map `f : M → M'` - +noncomputable theory --/ +open set function filter charted_space smooth_manifold_with_corners +open_locale topological_space manifold section defs /-! ## Fundamental definitions -/ variables -{E : Type*} [normed_group E] [normed_space ℝ E] +{E : Type*} [normed_add_comm_group E] [normed_space ℝ E] {H : Type*} [topological_space H] (I : model_with_corners ℝ E H) (M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] -{E' : Type*} [normed_group E'] [normed_space ℝ E'] +{E' : Type*} [normed_add_comm_group E'] [normed_space ℝ E'] {H' : Type*} [topological_space H'] (I' : model_with_corners ℝ E' H') (M' : Type*) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] -{F : Type*} [normed_group F] [normed_space ℝ F] +{F : Type*} [normed_add_comm_group F] [normed_space ℝ F] {G : Type*} [topological_space G] (J : model_with_corners ℝ F G) (N : Type*) [topological_space N] [charted_space G N] [smooth_manifold_with_corners J N] -{F' : Type*} [normed_group F'] [normed_space ℝ F'] +{F' : Type*} [normed_add_comm_group F'] [normed_space ℝ F'] {G' : Type*} [topological_space G'] (J' : model_with_corners ℝ F' G') (N' : Type*) [topological_space N'] [charted_space G' N'] [smooth_manifold_with_corners J' N'] @@ -45,9 +41,7 @@ local notation `TM` := tangent_space I local notation `TM'` := tangent_space I' /-- A first-order differential relation for maps from `M` to `N` is a subset of the 1-jet bundle. -/ -def rel_mfld := set (one_jet_bundle I M I' M') - -instance : has_mem (one_jet_bundle I M I' M') (rel_mfld I M I' M') := set.has_mem +@[reducible] def rel_mfld := set (one_jet_bundle I M I' M') variables {I M I' M'} {R : rel_mfld I M I' M'} @@ -58,8 +52,9 @@ variables {I M I' M'} {R : rel_mfld I M I' M'} (is_sol : ∀ x, one_jet_ext I I' f x ∈ R) /-- A formal solution to a local relation `R` over a set `U`. -/ -@[ext] structure formal_sol (R : rel_mfld I M I' M') extends one_jet_sec I M I' M' := -(is_sol' : ∀ x : M, to_fun x ∈ R) +@[ext] structure formal_sol (R : rel_mfld I M I' M') extends + to_one_jet_sec : one_jet_sec I M I' M' := +(is_sol' : ∀ x : M, to_one_jet_sec x ∈ R) instance (R : rel_mfld I M I' M') : has_coe_to_fun (formal_sol R) (λ S, M → one_jet_bundle I M I' M') := @@ -68,17 +63,31 @@ instance (R : rel_mfld I M I' M') : lemma formal_sol.is_sol (F : formal_sol R) : ∀ x, F x ∈ R := F.is_sol' +/-- part of the construction of the slice `R(σ,p)`. -/ +def rel_mfld.preslice (R : rel_mfld I M I' M') (σ : one_jet_bundle I M I' M') + (p : dual_pair' $ TM σ.1.1) : set (TM' σ.1.2) := +{w : TM' σ.1.2 | one_jet_bundle.mk σ.1.1 σ.1.2 (p.update σ.2 w) ∈ R} + +/-- For some reason `rw [mem_set_of_eq]` fails after unfolding `preslice`, +but rewriting with this lemma works. -/ +lemma mem_preslice {R : rel_mfld I M I' M'} {σ : one_jet_bundle I M I' M'} + {p : dual_pair' $ TM σ.1.1} {w : TM' σ.1.2} : + w ∈ R.preslice σ p ↔ one_jet_bundle.mk σ.1.1 σ.1.2 (p.update σ.2 w) ∈ R := +iff.rfl + +/-- the slice `R(σ,p)`. -/ def rel_mfld.slice (R : rel_mfld I M I' M') (σ : one_jet_bundle I M I' M') (p : dual_pair' $ TM σ.1.1) : set (TM' σ.1.2) := -{w | (⟨⟨σ.1.1, σ.1.2⟩, p.update σ.2 w⟩ : one_jet_bundle I M I' M') ∈ R} +connected_component_in (R.preslice σ p) (σ.2 p.v) def rel_mfld.ample (R : rel_mfld I M I' M') : Prop := -∀ (σ : one_jet_bundle I M I' M') (p : dual_pair' $ TM σ.1.1), ample_set (R.slice σ p) +∀ ⦃σ : one_jet_bundle I M I' M'⦄ (p : dual_pair' $ TM σ.1.1), σ ∈ R → ample_set (R.slice σ p) /-- A family of formal solutions indexed by manifold `N` is a function from `N` into formal solutions in such a way that the function is smooth as a function of all arguments. -/ -structure family_formal_sol (R : rel_mfld I M I' M') extends family_one_jet_sec I M I' M' J N := -(is_sol' : ∀ (t : N) (x : M), to_fun t x ∈ R) +structure family_formal_sol (R : rel_mfld I M I' M') extends + to_family_one_jet_sec : family_one_jet_sec I M I' M' J N := +(is_sol' : ∀ (t : N) (x : M), to_family_one_jet_sec t x ∈ R) instance : has_coe_to_fun (family_formal_sol J N R) (λ S, N → formal_sol R) := ⟨λ S t, ⟨S.to_family_one_jet_sec t, S.is_sol' t⟩⟩ @@ -95,7 +104,7 @@ def reindex (S : family_formal_sol J' N' R) (f : C^∞⟮J, N; J', N'⟯) : end family_formal_sol /-- A homotopy of formal solutions is a family indexed by `ℝ` -/ -abbreviation htpy_formal_sol (R : rel_mfld I M I' M') := family_formal_sol 𝓘(ℝ, ℝ) ℝ R +@[reducible] def htpy_formal_sol (R : rel_mfld I M I' M') := family_formal_sol 𝓘(ℝ, ℝ) ℝ R /-- A relation `R` satisfies the (non-parametric) h-principle if all its formal solutions are homotopic to a holonomic one. -/ @@ -133,19 +142,19 @@ Note: Patrick doesn't know whether we really need to allow different `E`, `H` an manifolds `X` and `M` (and for `Y` and `N`). We use maximal generality just in case. -/ variables - {EX : Type*} [normed_group EX] [normed_space ℝ EX] + {EX : Type*} [normed_add_comm_group EX] [normed_space ℝ EX] {HX : Type*} [topological_space HX] {IX : model_with_corners ℝ EX HX} {X : Type*} [topological_space X] [charted_space HX X] [smooth_manifold_with_corners IX X] - {EM : Type*} [normed_group EM] [normed_space ℝ EM] + {EM : Type*} [normed_add_comm_group EM] [normed_space ℝ EM] {HM : Type*} [topological_space HM] {IM : model_with_corners ℝ EM HM} {M : Type*} [topological_space M] [charted_space HM M] [smooth_manifold_with_corners IM M] - {EY : Type*} [normed_group EY] [normed_space ℝ EY] + {EY : Type*} [normed_add_comm_group EY] [normed_space ℝ EY] {HY : Type*} [topological_space HY] {IY : model_with_corners ℝ EY HY} {Y : Type*} [topological_space Y] [charted_space HY Y] [smooth_manifold_with_corners IY Y] - {EN : Type*} [normed_group EN] [normed_space ℝ EN] + {EN : Type*} [normed_add_comm_group EN] [normed_space ℝ EN] {HN : Type*} [topological_space HN] {IN : model_with_corners ℝ EN HN} {N : Type*} [topological_space N] [charted_space HN N] [smooth_manifold_with_corners IN N] @@ -160,6 +169,7 @@ local notation `TY` := tangent_space IY /-- Transfer map between one jet bundles induced by open smooth embedding into the source and targets. -/ +@[simps fst_fst fst_snd] def one_jet_bundle.transfer : one_jet_bundle IX X IY Y → one_jet_bundle IM M IN N := λ σ, ⟨⟨h σ.1.1, g σ.1.2⟩, ((g.fderiv σ.1.2 : TY σ.1.2 →L[ℝ] TN (g σ.1.2)).comp σ.2).comp @@ -177,82 +187,96 @@ sorry def rel_mfld.localize (R : rel_mfld IM M IN N) : rel_mfld IX X IY Y := one_jet_bundle.transfer g h ⁻¹' R -/-- Underlying map of `one_jet_sec.localize`. It maps `x` to `(x, y, (D_y(g))⁻¹ ∘ F_φ(h x) ∘ D_x(h))` - where `y : M := g⁻¹(F_{bs}(h x))`. -/ -@[simps fst_fst fst_snd] -def one_jet_sec.localize_fun : X → one_jet_bundle IX X IY Y := -λ x, let y := g.inv_fun (F.bs $ h x) in -⟨(x, y), ((g.fderiv y).symm : TN (g y) →L[ℝ] TY y) ∘L - ((F $ h x).2 ∘L (h.fderiv x : TX x →L[ℝ] TM (h x)))⟩ - open basic_smooth_vector_bundle_core -/-- Localize a one-jet section in two open embeddings. -/ + +/-- Localize a one-jet section in two open embeddings. + It maps `x` to `(x, y, (D_y(g))⁻¹ ∘ F_φ(h x) ∘ D_x(h))` where `y : M := g⁻¹(F_{bs}(h x))`. -/ @[simps] def one_jet_sec.localize (hF : range (F.bs ∘ h) ⊆ range g) : one_jet_sec IX X IY Y := -{ to_fun := F.localize_fun g h, - is_sec' := λ x, rfl, +{ bs := λ x, g.inv_fun (F.bs $ h x), + ϕ := λ x, let y := g.inv_fun (F.bs $ h x) in + (↑(g.fderiv y).symm : TN (g y) →L[ℝ] TY y) ∘L ((F $ h x).2 ∘L (h.fderiv x : TX x →L[ℝ] TM (h x))), smooth' := begin - simp_rw [one_jet_sec.localize_fun, h.fderiv_coe, g.fderiv_symm_coe], - convert smooth.one_jet_comp IX IN IY IX (λ x', F.bs (h x')) _ _, - { have := λ x, g.right_inv (hF $ mem_range_self x), - simp_rw [function.comp] at this, - conv { congr, skip, skip, funext, rw [this x] }, -- simp_rw doesn't do this - intro x, - have : smooth IN IY g.inv_fun, sorry, -- false. to do: deal with the partial smoothness of g⁻¹ - - have := this.one_jet_ext.comp (F.smooth_bs.comp h.smooth_to), - -- simp_rw [function.comp] at this, - convert this, - simp_rw [one_jet_bundle.mk, function.comp, one_jet_ext], - sorry -- why is this not refl? - - -- refine g.smooth_inv.one_jet_ext.comp_smooth _ _, - -- refine ((g.smooth_inv _ _).smooth_within_at.smooth_at _).one_jet_ext.comp _ _, - }, - refine smooth.one_jet_comp IX IM IN IX h _ _, - { exact F.smooth_eta.comp h.smooth_to }, - { exact h.smooth_to.one_jet_ext }, - -- have h1 : smooth_at IX IY (λ x', g.inv_fun (F.bs $ h x')) x, - -- sorry, - -- rw [(one_jet_bundle_core IX X IY Y).cont_mdiff_at_iff_target], - -- refine ⟨continuous_at_id.prod h1.continuous_at, _⟩, - -- simp_rw [ext_chart_at, local_equiv.coe_trans, function.comp, to_charted_space_chart_at], - -- dsimp only [one_jet_bundle_core], - -- simp_rw [local_homeomorph.coe_coe, hom_chart, ← achart_def, pullback_fst_coord_change_at, - -- pullback_snd_coord_change_at, model_with_corners.to_local_equiv_coe, - -- model_with_corners.prod_apply, model_with_corners_self_coe, id, prod_charted_space_chart_at, - -- local_homeomorph.prod_apply], - -- refine (cont_mdiff_at_ext_chart_at.prod_mk_space $ cont_mdiff_at_ext_chart_at.comp _ h1) - -- .prod_mk_space _, - -- simp_rw [F.localize_fun_fst_fst, F.localize_fun_fst_snd], - -- sorry - -- exact h1.mfderiv' le_rfl + simp_rw [h.fderiv_coe, g.fderiv_symm_coe, + mfderiv_congr_point (g.right_inv (hF $ mem_range_self _))], + refine smooth.one_jet_comp IN (λ x', F.bs (h x')) _ _, + { exact λ x, (g.smooth_at_inv $ hF $ mem_range_self x).one_jet_ext.comp _ + (F.smooth_bs.comp h.smooth_to).cont_mdiff_at }, + apply smooth.one_jet_comp IM h (F.smooth_eta.comp h.smooth_to) h.smooth_to.one_jet_ext end } lemma transfer_localize (hF : range (F.bs ∘ h) ⊆ range g) (x : X) : (F.localize g h hF x).transfer g h = F (h x) := begin - rw [one_jet_sec.localize_to_fun, one_jet_sec.localize_fun, one_jet_bundle.transfer], + rw [one_jet_sec.coe_apply, one_jet_sec.localize_bs, one_jet_sec.localize_ϕ, + one_jet_bundle.transfer], + dsimp only, ext, - { simp_rw [F.is_sec] }, - { simp_rw [g.right_inv (hF $ mem_range_self x), one_jet_sec.bs] }, + { refl }, + { simp_rw [g.right_inv (hF $ mem_range_self x), function.comp_apply, F.bs_eq] }, { apply heq_of_eq, dsimp only, ext v, simp_rw [continuous_linear_map.comp_apply, continuous_linear_equiv.coe_coe, continuous_linear_equiv.apply_symm_apply] }, end -/-- Underlying map of `htpy_one_jet_sec.unlocalize`. -/ -def htpy_one_jet_sec.unlocalize_fun (F : htpy_one_jet_sec IX X IY Y) (t : ℝ) (m : M) : - one_jet_bundle IM M IN N := -⟨⟨m, g $ (F t).bs (h.inv_fun m)⟩, - (g.fderiv $ (F t).bs (h.inv_fun m)).to_continuous_linear_map ∘L - ((F t $ h.inv_fun m).2 ∘L (h.fderiv $ h.inv_fun m).symm.to_continuous_linear_map)⟩ +lemma one_jet_sec.localize_bs_fun (hF : range (F.bs ∘ h) ⊆ range g) : + (F.localize g h hF).bs = g.inv_fun ∘ F.bs ∘ h := +rfl + +lemma one_jet_sec.localize_mem_iff (hF : range (F.bs ∘ h) ⊆ range g) {x : X} : + F.localize g h hF x ∈ R.localize g h ↔ F (h x) ∈ R := +by rw [rel_mfld.localize, mem_preimage, transfer_localize F g h hF] + +lemma rel_mfld.ample.localize (hR : R.ample) : (R.localize g h).ample := +begin + intros x p hx, + have : (rel_mfld.localize g h R).slice x p = + (g.fderiv x.1.2).symm '' R.slice (x.transfer g h) (p.map (h.fderiv x.1.1)), + { simp_rw [rel_mfld.slice, rel_mfld.localize], + symmetry, + refine ((g.fderiv x.1.2).symm.to_homeomorph.image_connected_component_in _).trans _, + { rw [mem_preslice, dual_pair'.update_self], exact hx }, + simp_rw [continuous_linear_equiv.coe_to_homeomorph, + continuous_linear_equiv.image_symm_eq_preimage], + congr' 1, + { ext v, simp_rw [mem_preimage, mem_preslice, mem_preimage], + dsimp only [one_jet_bundle.transfer, one_jet_bundle_mk_fst, one_jet_bundle_mk_snd], + simp_rw [p.map_update_comp_right, ← p.update_comp_left, continuous_linear_equiv.coe_coe, + one_jet_bundle.mk] }, + { dsimp only [one_jet_bundle.transfer], + simp_rw [continuous_linear_map.comp_apply, continuous_linear_equiv.coe_coe, p.map_v, + continuous_linear_equiv.symm_apply_apply] } }, + rw [this], + exact (hR _ hx).image (g.fderiv x.1.2).symm +end + +lemma is_holonomic_at_localize_iff (hF : range (F.bs ∘ h) ⊆ range g) (x : X) : + (F.localize g h hF).is_holonomic_at x ↔ F.is_holonomic_at (h x) := +begin + have : mfderiv IX IY (g.inv_fun ∘ F.bs ∘ h) x = + (g.fderiv (g.inv_fun (F.bs (h x)))).symm.to_continuous_linear_map.comp + ((mfderiv IM IN F.bs (h x)).comp (h.fderiv x).to_continuous_linear_map), + { have h1 : mdifferentiable_at IN IY g.inv_fun (F.bs (h x)) := + (g.smooth_at_inv $ hF $ mem_range_self _).mdifferentiable_at le_top, + have h2 : mdifferentiable_at IM IN F.bs (h x) := + F.smooth_bs.smooth_at.mdifferentiable_at le_top, + have h3 : mdifferentiable_at IX IM h x := + h.smooth_to.smooth_at.mdifferentiable_at le_top, + rw [mfderiv_comp x h1 (h2.comp x h3), mfderiv_comp x h2 h3, + ← g.fderiv_symm_coe' (hF $ mem_range_self _)], + refl, }, + simp_rw [one_jet_sec.is_holonomic_at], + rw [mfderiv_congr (F.localize_bs_fun g h hF), one_jet_sec.snd_eq, F.localize_ϕ, this], + simp_rw [← continuous_linear_equiv.coe_def_rev, + continuous_linear_equiv.cancel_left, continuous_linear_equiv.cancel_right] +end /-- Un-localize a homotopy of one-jet sections from two open embeddings. -/ -- Note(F): this is only well-defined on `univ × range h`, right? def htpy_one_jet_sec.unlocalize (F : htpy_one_jet_sec IX X IY Y) : htpy_one_jet_sec IM M IN N := -{ to_fun := F.unlocalize_fun g h, - is_sec' := λ x m, rfl, +{ bs := λ t m , g $ (F t).bs (h.inv_fun m), + ϕ := λ t m, (g.fderiv $ (F t).bs (h.inv_fun m)).to_continuous_linear_map ∘L + ((F t $ h.inv_fun m).2 ∘L (h.fderiv $ h.inv_fun m).symm.to_continuous_linear_map), smooth' := sorry } lemma one_jet_sec.unlocalize_localize (G : htpy_one_jet_sec IX X IY Y) @@ -260,17 +284,6 @@ lemma one_jet_sec.unlocalize_localize (G : htpy_one_jet_sec IX X IY Y) (hFG : G 0 = F.localize g h hF) : G.unlocalize g h 0 = F := sorry -lemma one_jet_sec.localize_mem_iff (hF : range (F.bs ∘ h) ⊆ range g) {x : X} : - F.localize g h hF x ∈ R.localize g h ↔ F (h x) ∈ R := -by rw [rel_mfld.localize, mem_preimage, transfer_localize F g h hF] - -lemma rel_mfld.ample.localize (hR : R.ample) : (R.localize g h).ample := -sorry - -lemma is_holonomic_at_localize_iff (hF : range (F.bs ∘ h) ⊆ range g) (x : X) : - (F.localize g h hF).is_holonomic_at x ↔ F.is_holonomic_at (h x) := -by { simp_rw [one_jet_sec.is_holonomic_at], sorry } - /-- Localize a formal solution. -/ def transfer (hF : range (F.bs ∘ h) ⊆ range g) (h2F : ∀ x, F (h x) ∈ R) : formal_sol (R.localize g h) := @@ -284,8 +297,8 @@ section loc Now we really bridge the gap all the way to vector spaces. -/ -variables {E : Type*} [normed_group E] [normed_space ℝ E] -variables {E' : Type*} [normed_group E'] [normed_space ℝ E'] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] +variables {E' : Type*} [normed_add_comm_group E'] [normed_space ℝ E'] /-- For maps between vector spaces, `one_jet_ext` is the obvious thing. -/ @[simp] theorem one_jet_ext_eq_fderiv {f : E → E'} {x : E} : diff --git a/formal/lean/sphere-eversion/global/smooth_embedding.lean b/formal/lean/sphere-eversion/global/smooth_embedding.lean index 4678eea9872bafcfc6e8c03ab429ac055da05ce2..6a0458b799be2a26bd70314accb983d440fa11c8 100644 --- a/formal/lean/sphere-eversion/global/smooth_embedding.lean +++ b/formal/lean/sphere-eversion/global/smooth_embedding.lean @@ -1,6 +1,7 @@ import geometry.manifold.cont_mdiff import global.indexing import to_mathlib.topology.paracompact +import to_mathlib.topology.local_homeomorph import to_mathlib.geometry.manifold.charted_space noncomputable theory @@ -9,12 +10,12 @@ open set equiv open_locale manifold topological_space section general -variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] +variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type*} [topological_space H] (I : model_with_corners 𝕜 E H) (M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] - {E' : Type*} [normed_group E'] [normed_space 𝕜 E'] + {E' : Type*} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type*} [topological_space H'] (I' : model_with_corners 𝕜 E' H') (M' : Type*) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] @@ -44,6 +45,12 @@ variables {I I' M M'} (f : open_smooth_embedding I M I' M') lemma coe_comp_inv_fun_eventually_eq (x : M) : f ∘ f.inv_fun =ᶠ[𝓝 (f x)] id := filter.eventually_of_mem (f.open_map.range_mem_nhds x) $ λ y hy, f.right_inv' hy +lemma is_open_range : is_open (range f) := +f.open_map.is_open_range + +lemma smooth_at_inv {y : M'} (hy : y ∈ range f) : smooth_at I' I f.inv_fun y := +(f.smooth_inv y hy).cont_mdiff_at $ f.is_open_range.mem_nhds hy + /- Note that we are slightly abusing the fact that `tangent_space I x` and `tangent_space I (f.inv_fun (f x))` are both definitionally `E` below. -/ def fderiv (x : M) : tangent_space I x ≃L[𝕜] tangent_space I' (f x) := @@ -93,8 +100,8 @@ open metric (hiding mem_nhds_iff) function universe u -variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] +variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] (M : Type u) [topological_space M] [charted_space E M] [smooth_manifold_with_corners 𝓘(𝕜, E) M] [t2_space M] [locally_compact_space M] [sigma_compact_space M] @@ -133,18 +140,20 @@ rfl variables (𝕜) -/-- A diffeomorphism from `E` onto the open ball of radius `r` in `E` centred at a point `c`, -sending the open ball of radius 1 centered at 0 to the open ball of radius `r/2` centred at `c`. -/ -def open_smooth_embedding_to_ball (c : E) {r : ℝ} (h : 0 < r) : +/-- Provided `0 < r`, this is a diffeomorphism from `E` onto the open ball of radius `r` in `E` +centred at a point `c` and sending `0` to `c`. + +The values for `r ≤ 0` are junk. -/ +def open_smooth_embedding_to_ball (c : E) (r : ℝ) : open_smooth_embedding 𝓘(𝕜, E) E 𝓘(𝕜, E) E := sorry -@[simp] lemma range_open_smooth_embedding_to_ball (c : E) {r : ℝ} (h : 0 < r) : - range (open_smooth_embedding_to_ball 𝕜 c h) = (ball c r : set E) := +@[simp] lemma open_smooth_embedding_to_ball_apply_zero (c : E) {r : ℝ} (h : 0 < r) : + open_smooth_embedding_to_ball 𝕜 c r 0 = c := sorry -@[simp] lemma open_smooth_embedding_to_ball_image_unit_ball (c : E) {r : ℝ} (h : 0 < r) : - open_smooth_embedding_to_ball 𝕜 c h '' ball 0 1 = (ball c (r/2) : set E) := +@[simp] lemma range_open_smooth_embedding_to_ball (c : E) {r : ℝ} (h : 0 < r) : + range (open_smooth_embedding_to_ball 𝕜 c r) = ball c r := sorry variables (E) {M} @@ -157,41 +166,41 @@ lemma nice_atlas' locally_finite (λ i, range (φ i)) ∧ (⋃ i, φ i '' ball 0 1) = univ := begin + let W : M → ℝ → set M := λ x r, + (chart_at E x).symm ∘ open_smooth_embedding_to_ball 𝕜 (chart_at E x x) r '' (ball 0 1), let B : M → ℝ → set M := charted_space.ball E, let p : M → ℝ → Prop := λ x r, 0 < r ∧ ball (chart_at E x x) r ⊆ (chart_at E x).target ∧ ∃ j, B x r ⊆ s j, - have hB₀ : ∀ x r, p x r → is_open (B x r), - { rintros x r ⟨hr, hx, -⟩, - change ball (chart_at E x x) r ⊆ (chart_at E x).symm.source at hx, - replace hr : is_open (ball (chart_at E x x) r) := is_open_ball, - exact (chart_at E x).symm.image_open_of_open hr hx, }, - have hB₁ : ∀ x r, p x r → x ∈ B x r, - { rintros x r ⟨hr, hx, -⟩, - exact ⟨chart_at E x x, by simp [hr], by simp⟩, }, - have hB₂ : ∀ x, (𝓝 x).has_basis (p x) (B x) := + have hW₀ : ∀ x r, p x r → x ∈ W x r := λ x r h, ⟨0, by simp, by simp [h.1]⟩, + have hW₁ : ∀ x r, p x r → is_open (W x r), + { rintros x r ⟨h₁, h₂, -, -⟩, + simp only [W], + have aux : + open_smooth_embedding_to_ball 𝕜 (chart_at E x x) r '' ball 0 1 ⊆ (chart_at E x).target := + subset.trans ((image_subset_range _ _).trans (by simp [h₁])) h₂, + rw [image_comp, local_homeomorph.is_open_symm_image_iff_of_subset_target _ aux], + exact open_smooth_embedding.open_map _ _ is_open_ball, }, + have hB : ∀ x, (𝓝 x).has_basis (p x) (B x) := λ x, charted_space.nhds_has_basis_balls_of_open_cov E x s_op cov, have hp : ∀ i r, p i r → 0 < r := λ i r h, h.1, - have hp' : ∀ i r r', 0 < r → r < r' → p i r' → p i r, - { rintros x r r' hr hr' ⟨h₁, h₂, j, hj⟩, - exact ⟨hr, (ball_subset_ball hr'.le).trans h₂, j, - (monotone_image (ball_subset_ball hr'.le)).trans hj⟩, }, obtain ⟨t, ht₁, ht₂, ht₃, ht₄⟩ := - exists_countable_locally_finite_cover surjective_id hp hp' hB₀ hB₁ hB₂, + exists_countable_locally_finite_cover surjective_id hp hW₀ hW₁ hB, refine ⟨M × ℝ, t, λ z, _, ht₁, λ z, _, _, _⟩, - { have h : range (open_smooth_embedding_to_ball 𝕜 (chart_at E z.1.1 z.1.1) $ hp _ _ $ ht₂ _ z.2) ⊆ + { have h : range (open_smooth_embedding_to_ball 𝕜 (chart_at E z.1.1 z.1.1) z.1.2) ⊆ (chart_at E z.1.1).target, - { simpa only [range_open_smooth_embedding_to_ball] using (ht₂ _ z.2).2.1, }, + { have aux : 0 < z.val.snd := hp _ _ (ht₂ _ z.2), + simpa only [range_open_smooth_embedding_to_ball, aux] using (ht₂ _ z.2).2.1, }, exact open_smooth_embedding_of_subset_chart_target M h, }, - { simp only [subtype.val_eq_coe, coe_open_smooth_embedding_of_subset_chart_target], - simp only [range_comp, range_open_smooth_embedding_to_ball], + { have aux : 0 < (z : M × ℝ).snd := hp _ _ (ht₂ _ z.2), + simp only [subtype.val_eq_coe, coe_open_smooth_embedding_of_subset_chart_target], + simp only [range_comp, range_open_smooth_embedding_to_ball, aux], exact (ht₂ z.1 z.2).2.2, }, - { convert ht₃, - ext1, - simp only [subtype.val_eq_coe, coe_open_smooth_embedding_of_subset_chart_target, comp_app], - simpa only [range_comp, range_open_smooth_embedding_to_ball], }, - { simpa only [subtype.val_eq_coe, subtype.coe_mk, coe_open_smooth_embedding_of_subset_chart_target, - Union_coe_set, image_comp (chart_at E _).symm (open_smooth_embedding_to_ball 𝕜 _ _), - open_smooth_embedding_to_ball_image_unit_ball] using ht₄, }, + { convert ht₄, + ext1 z, + have aux : 0 < (z : M × ℝ).snd := hp _ _ (ht₂ _ z.2), + simp only [subtype.val_eq_coe, coe_open_smooth_embedding_of_subset_chart_target], + simpa only [range_comp, range_open_smooth_embedding_to_ball, aux], }, + { simpa only [Union_coe_set] using ht₃, }, end variables [nonempty M] diff --git a/formal/lean/sphere-eversion/local/ample.lean b/formal/lean/sphere-eversion/local/ample.lean index 7e21215f3c8454775fdbc5da4a2e293e0e14a6d3..0c31c0ad43f7880f1178df5cd4a72e9c450052ed 100644 --- a/formal/lean/sphere-eversion/local/ample.lean +++ b/formal/lean/sphere-eversion/local/ample.lean @@ -24,12 +24,31 @@ open set affine_map open_locale convex matrix -variables {F : Type*} [add_comm_group F] [module ℝ F] [topological_space F] +variables {E F : Type*} [add_comm_group F] [module ℝ F] [topological_space F] +variables [add_comm_group E] [module ℝ E] [topological_space E] /-- A subset of a topological real vector space is ample if the convex hull of each of its connected components is the full space. -/ -def ample_set (s : set F) := -∀ x ∈ s, convex_hull ℝ (connected_comp_in s x) = univ +def ample_set (s : set F) : Prop := +∀ x ∈ s, convex_hull ℝ (connected_component_in s x) = univ + +/-- images of ample sets under continuous linear equivalences are ample. -/ +lemma ample_set.image {s : set E} (h : ample_set s) (L : E ≃L[ℝ] F) : ample_set (L '' s) := +begin + intros x hx, + rw [L.image_eq_preimage] at hx, + have : L '' connected_component_in s (L.symm x) = connected_component_in (L '' s) x, + { conv_rhs { rw [← L.apply_symm_apply x] }, + exact L.to_homeomorph.image_connected_component_in hx }, + rw [← this], + refine (L.to_linear_equiv.to_linear_map.convex_hull_image _).trans _, + rw [h (L.symm x) hx, image_univ], + exact L.to_linear_equiv.to_equiv.range_eq_univ, +end + +/-- preimages of ample sets under continuous linear equivalences are ample. -/ +lemma ample_set.preimage {s : set F} (h : ample_set s) (L : E ≃L[ℝ] F) : ample_set (L ⁻¹' s) := +by { rw [← L.image_symm_eq_preimage], exact h.image L.symm } section lemma_2_13 @@ -149,8 +168,8 @@ lemma ample_of_two_le_codim [topological_add_group F] [has_continuous_smul ℝ F begin haveI : connected_space (Eᶜ : set F) := connected_space_compl_of_two_le_codim hcodim, intros x hx, - have : connected_comp_in (↑E)ᶜ x = (↑E)ᶜ, - from is_preconnected.connected_comp_in (is_connected_compl_of_two_le_codim hcodim).2 hx, + have : connected_component_in (↑E)ᶜ x = (↑E)ᶜ, + from is_preconnected.connected_component_in (is_connected_compl_of_two_le_codim hcodim).2 hx, rw [this, eq_univ_iff_forall], intro y, by_cases h : y ∈ E, diff --git a/formal/lean/sphere-eversion/local/corrugation.lean b/formal/lean/sphere-eversion/local/corrugation.lean index 567a95ade80c46976b3cf14bc860a88d46342c8f..7a9cdc7a09b6d5abfef4f9a0494ffcfeb598b0cd 100644 --- a/formal/lean/sphere-eversion/local/corrugation.lean +++ b/formal/lean/sphere-eversion/local/corrugation.lean @@ -4,7 +4,6 @@ import measure_theory.integral.interval_integral import analysis.calculus.parametric_integral import to_mathlib.topology.periodic -import to_mathlib.topology.bases import to_mathlib.analysis.calculus import to_mathlib.measure_theory.parametric_interval_integral @@ -22,11 +21,11 @@ open set function finite_dimensional asymptotics filter topological_space int me open_locale topological_space unit_interval -variables {E : Type*} [normed_group E] [normed_space ℝ E] - {F : Type*} [normed_group F] [normed_space ℝ F] [measurable_space F] [borel_space F] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] + {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] [measurable_space F] [borel_space F] [finite_dimensional ℝ F] - {G : Type*} [normed_group G] [normed_space ℝ G] [finite_dimensional ℝ G] - {H : Type*} [normed_group H] [normed_space ℝ H] [finite_dimensional ℝ H] + {G : Type*} [normed_add_comm_group G] [normed_space ℝ G] [finite_dimensional ℝ G] + {H : Type*} [normed_add_comm_group H] [normed_space ℝ H] [finite_dimensional ℝ H] {π : E →L[ℝ] ℝ} (N : ℝ) (γ : E → loop F) @@ -129,7 +128,7 @@ begin { exact hγ_diff.comp₃ hg.fst' hx.fst' cont_diff_snd }, { apply cont_diff_average, exact hγ_diff.comp₃ hg.fst'.fst' hx.fst'.fst' cont_diff_snd } }, - { apply (π.cont_diff.comp hx).const_smul }, + { apply cont_diff_const.mul (π.cont_diff.comp hx) }, end /-- @@ -206,7 +205,7 @@ begin apply hγ_diff.comp₃ hg.fst'.snd' cont_diff_fst cont_diff_snd.snd, apply cont_diff_average, exact hγ_diff.comp₃ hg.fst'.snd'.fst' cont_diff_fst.fst' cont_diff_snd }, - { apply (π.cont_diff.comp hx).const_smul }, + { exact cont_diff_const.mul (π.cont_diff.comp hx) }, end lemma remainder_c0_small_on {K : set E} (hK : is_compact K) diff --git a/formal/lean/sphere-eversion/local/dual_pair.lean b/formal/lean/sphere-eversion/local/dual_pair.lean index f63edc6ffa716326eedf4e3748bd9f4fa701b4c4..48e0d4ce3a95fb55f40da58d9d47da928303d090 100644 --- a/formal/lean/sphere-eversion/local/dual_pair.lean +++ b/formal/lean/sphere-eversion/local/dual_pair.lean @@ -10,11 +10,13 @@ import to_mathlib.linear_algebra.basic noncomputable theory -open function +open function continuous_linear_map section no_norm -variables (E : Type*) [add_comm_group E] [module ℝ E] [topological_space E] (F : Type*) - [normed_group F] [normed_space ℝ F] +variables (E : Type*) {E' F G : Type*} +variables [add_comm_group E] [module ℝ E] [topological_space E] +variables [add_comm_group E'] [module ℝ E'] [topological_space E'] +variables [normed_add_comm_group F] [normed_space ℝ F] [normed_add_comm_group G] [normed_space ℝ G] -- TODO: move mathlib's dual_pair out of the root namespace! @@ -33,7 +35,7 @@ local attribute [simp] continuous_linear_map.to_span_singleton_apply lemma ker_pi_ne_top (p : dual_pair' E) : p.π.ker ≠ ⊤ := begin intro H, - have : (p.π : E →ₗ[ℝ] ℝ) p.v = 1 := p.pairing, + have : (p.π : E →ₗ[ℝ] ℝ) p.v = 1 := p.pairing, simpa [linear_map.ker_eq_top.mp H] end @@ -99,18 +101,48 @@ begin use [u, hu, t, rfl] end +/-- Map a dual pair under a linear equivalence. -/ +@[simps] def map (p : dual_pair' E) (L : E ≃L[ℝ] E') : dual_pair' E' := +⟨p.π ∘L ↑L.symm, L p.v, (congr_arg p.π $ L.symm_apply_apply p.v).trans p.pairing⟩ + +lemma update_comp_left (p : dual_pair' E) (ψ : F →L[ℝ] G) (φ : E →L[ℝ] F) (w : F) : + p.update (ψ ∘L φ) (ψ w) = ψ ∘L p.update φ w := +begin + ext1 u, + simp only [update, add_apply, continuous_linear_map.comp_apply, to_span_singleton_apply, + ψ.map_add, ψ.map_smul, ψ.map_sub], +end + +lemma update_comp_right (p : dual_pair' E) (ψ : E' →L[ℝ] F) (φ : E ≃L[ℝ] E') (w : F) : + p.update (ψ ∘L ↑φ) w = (p.map φ).update ψ w ∘L ↑φ := +begin + ext1 u, + simp only [update, add_apply, continuous_linear_map.comp_apply, to_span_singleton_apply, map, + continuous_linear_equiv.coe_coe, φ.symm_apply_apply], +end + +lemma map_update_comp_right (p : dual_pair' E) (ψ : E →L[ℝ] F) (φ : E ≃L[ℝ] E') (w : F) : + (p.map φ).update (ψ ∘L ↑φ.symm) w = p.update ψ w ∘L ↑φ.symm := +begin + -- todo: use `update_comp_right` + ext1 u, + simp only [update, add_apply, continuous_linear_map.comp_apply, to_span_singleton_apply, map, + continuous_linear_equiv.coe_coe, φ.symm_apply_apply], +end + end dual_pair' end no_norm namespace dual_pair' -variables {E : Type*} [normed_group E] [normed_space ℝ E] - {F : Type*} [normed_group F] [normed_space ℝ F] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] + {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] /- In the next two lemmas, finite dimensionality of `E` is clearly uneeded, but allows to use `cont_diff_clm_apply` and `continuous_clm_apply`. -/ -lemma smooth_update [finite_dimensional ℝ E] (p : dual_pair' E) {G : Type*} [normed_group G] [normed_space ℝ G] +lemma smooth_update [finite_dimensional ℝ E] (p : dual_pair' E) + {G : Type*} [normed_add_comm_group G] [normed_space ℝ G] {φ : G → (E →L[ℝ] F)} (hφ : 𝒞 ∞ φ) {w : G → F} (hw : 𝒞 ∞ w) : 𝒞 ∞ (λ g, p.update (φ g) (w g)) := begin @@ -120,7 +152,8 @@ begin exact (hw.sub (cont_diff_clm_apply.mp hφ p.v)).const_smul _, end -lemma continuous_update [finite_dimensional ℝ E] (p : dual_pair' E) {X : Type*} [topological_space X] +lemma continuous_update [finite_dimensional ℝ E] (p : dual_pair' E) + {X : Type*} [topological_space X] {φ : X → (E →L[ℝ] F)} (hφ : continuous φ) {w : X → F} (hw : continuous w) : continuous (λ g, p.update (φ g) (w g)) := begin diff --git a/formal/lean/sphere-eversion/local/h_principle.lean b/formal/lean/sphere-eversion/local/h_principle.lean index 0d3741ca84bee658e38299afc371bc89f4208cbb..bf703899dcaf76346132872a67b66a7406aecabb 100644 --- a/formal/lean/sphere-eversion/local/h_principle.lean +++ b/formal/lean/sphere-eversion/local/h_principle.lean @@ -19,10 +19,10 @@ noncomputable theory open_locale unit_interval classical filter topological_space open filter set rel_loc -variables (E : Type*) [normed_group E] [normed_space ℝ E] [finite_dimensional ℝ E] - {F : Type*} [normed_group F] [normed_space ℝ F] [measurable_space F] [borel_space F] +variables (E : Type*) [normed_add_comm_group E] [normed_space ℝ E] [finite_dimensional ℝ E] + {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] [measurable_space F] [borel_space F] [finite_dimensional ℝ F] - {G : Type*} [normed_group G] [normed_space ℝ G] + {G : Type*} [normed_add_comm_group G] [normed_space ℝ G] open_locale unit_interval /-- @@ -71,7 +71,7 @@ structure accepts (L : step_landscape E) (𝓕 : jet_sec E F) : Prop := /-- The union of all slices of `R` corresponding to `𝓕`. -/ def Ω (L : step_landscape E) (𝓕 : jet_sec E F) : set (E × F) := {p | p.2 ∈ 𝓕.slice_at R L.p p.1} ---⋃ x, ({x} : set E) ×ˢ (connected_comp_in (𝓕.slice_at R L.p x) $ 𝓕.φ x L.p.v) +--⋃ x, ({x} : set E) ×ˢ (connected_component_in (𝓕.slice_at R L.p x) $ 𝓕.φ x L.p.v) def π (L : step_landscape E) : E →L[ℝ] ℝ := L.p.π diff --git a/formal/lean/sphere-eversion/local/relation.lean b/formal/lean/sphere-eversion/local/relation.lean index 4802f37fe8e670b0b474159866652887953d57c3..e97dc7372b05107940bac4fb4c74c23f35561d1e 100644 --- a/formal/lean/sphere-eversion/local/relation.lean +++ b/formal/lean/sphere-eversion/local/relation.lean @@ -35,8 +35,8 @@ noncomputable theory open set function module (dual) real filter open_locale unit_interval topological_space -variables (E : Type*) [normed_group E] [normed_space ℝ E] (F : Type*) - [normed_group F] [normed_space ℝ F] +variables (E : Type*) [normed_add_comm_group E] [normed_space ℝ E] (F : Type*) + [normed_add_comm_group F] [normed_space ℝ F] @[derive metric_space] @@ -63,7 +63,7 @@ ample_set (R.slice p θ) /- FIXME: the proof below is awful. -/ lemma is_ample.mem_hull {R : rel_loc E F} (h : is_ample R) {θ : E × F × (E →L[ℝ] F)} - (hθ : θ ∈ R) (v : F) (p) : v ∈ hull (connected_comp_in (R.slice p θ) (θ.2.2 p.v)) := + (hθ : θ ∈ R) (v : F) (p) : v ∈ hull (connected_component_in (R.slice p θ) (θ.2.2 p.v)) := begin rw h p θ (θ.2.2 p.v) _, exact mem_univ _, @@ -82,7 +82,7 @@ end variables (E) -@[ext] structure jet_sec (F : Type*) [normed_group F] [normed_space ℝ F] := +@[ext] structure jet_sec (F : Type*) [normed_add_comm_group F] [normed_space ℝ F] := (f : E → F) (f_diff : 𝒞 ∞ f) (φ : E → E →L[ℝ] F) @@ -254,10 +254,10 @@ by simpa [rel_loc.formal_sol.slice_at, rel_loc.slice] using 𝓕.is_sol x the function `𝓕.f` at `x` is in the convex hull of the relevant connected component of the corresponding slice. -/ def is_short_at (𝓕 : jet_sec E F) (R : rel_loc E F) (p : dual_pair' E) (x : E) : Prop := -D 𝓕.f x p.v ∈ hull (connected_comp_in (𝓕.slice_at R p x) $ 𝓕.φ x p.v) +D 𝓕.f x p.v ∈ hull (connected_component_in (𝓕.slice_at R p x) $ 𝓕.φ x p.v) def _root_.rel_loc.formal_sol.is_short_at (𝓕 : formal_sol R)(p : dual_pair' E) (x : E) : Prop := -D 𝓕.f x p.v ∈ hull (connected_comp_in (𝓕.slice_at p x) $ 𝓕.φ x p.v) +D 𝓕.f x p.v ∈ hull (connected_component_in (𝓕.slice_at p x) $ 𝓕.φ x p.v) lemma _root_.rel_loc.is_ample.is_short_at {R : rel_loc E F} (hR : is_ample R) (𝓕 : formal_sol R) (p : dual_pair' E) (x : E) : 𝓕.is_short_at p x := @@ -314,7 +314,7 @@ def rel_loc.jet_sec.const_htpy (𝓕 : jet_sec E F) : htpy_jet_sec E F := def smooth_step : ℝ → ℝ := λ t, smooth_transition (2 * t - 1/2) lemma smooth_step.smooth : 𝒞 ∞ smooth_step := -smooth_transition.cont_diff.comp $ (cont_diff_id.const_smul 2).sub cont_diff_const +smooth_transition.cont_diff.comp $ (cont_diff_id.const_smul (2 : ℝ)).sub cont_diff_const @[simp] lemma smooth_step.zero : smooth_step 0 = 0 := @@ -357,7 +357,7 @@ begin { change 𝒞 ∞ ((prod.map smooth_step id) ∘ (λ p : ℝ × E, (2*p.1, p.2))), apply (smooth_step.smooth.prod_map cont_diff_id).comp, apply cont_diff.prod, - apply cont_diff_fst.const_smul, + apply cont_diff_const.mul cont_diff_fst, apply cont_diff_snd }, replace hf := hf.comp s₁, have s₂ : 𝒞 ∞ (λ p : ℝ × E, (smooth_step $ 2*p.1 - 1, p.2)), @@ -365,7 +365,7 @@ begin apply (smooth_step.smooth.prod_map cont_diff_id).comp, apply cont_diff.prod, apply cont_diff.sub, - apply cont_diff_fst.const_smul, + apply cont_diff_const.mul cont_diff_fst, apply cont_diff_const, apply cont_diff_snd }, replace hg := hg.comp s₂, diff --git a/formal/lean/sphere-eversion/loops/basic.lean b/formal/lean/sphere-eversion/loops/basic.lean index b6a7a37d493f1e1091c84fc9e70b5ab0d6047ff8..5d82fefdb6d44c53633c7e2b9cceeb5a0c7eff23 100644 --- a/formal/lean/sphere-eversion/loops/basic.lean +++ b/formal/lean/sphere-eversion/loops/basic.lean @@ -22,9 +22,9 @@ noncomputable theory variables {K X X' Y Z : Type*} -- variables [topological_space X'] [topological_space Y] [topological_space Z] -variables {E : Type*} [normed_group E] [normed_space ℝ E] - {F : Type*} [normed_group F] [normed_space ℝ F] - {F' : Type*} [normed_group F'] [normed_space ℝ F'] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] + {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] + {F' : Type*} [normed_add_comm_group F'] [normed_space ℝ F'] set_option old_structure_cmd true diff --git a/formal/lean/sphere-eversion/loops/delta_mollifier.lean b/formal/lean/sphere-eversion/loops/delta_mollifier.lean index 1d80993de8c928e396d3b0c00641117105802fe9..30b10deddf061b3aa62fbfa9fde288ac090e64e8 100644 --- a/formal/lean/sphere-eversion/loops/delta_mollifier.lean +++ b/formal/lean/sphere-eversion/loops/delta_mollifier.lean @@ -17,7 +17,7 @@ open_locale topological_space big_operators filter convolution -variables {F : Type*} [normed_group F] [normed_space ℝ F] [finite_dimensional ℝ F] +variables {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] [finite_dimensional ℝ F] variables [measurable_space F] [borel_space F] section @@ -100,7 +100,7 @@ begin { rwa finsum_of_infinite_support }, end -variables {E : Type*} [normed_group E] [normed_space ℝ E] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] lemma cont_diff.periodize {f : ℝ → E} {n : with_top ℕ} (h : cont_diff ℝ n f) (h' : has_compact_support f) : cont_diff ℝ n (periodize f) := diff --git a/formal/lean/sphere-eversion/loops/exists.lean b/formal/lean/sphere-eversion/loops/exists.lean index ffe0be4574af59d8f79a881e6172ec0e6e25158f..8098f236edb73a4ce71f6af94f2e9052d4b5973c 100644 --- a/formal/lean/sphere-eversion/loops/exists.lean +++ b/formal/lean/sphere-eversion/loops/exists.lean @@ -9,8 +9,8 @@ open set function finite_dimensional prod int topological_space metric filter open measure_theory measure_theory.measure real open_locale topological_space unit_interval -variables {E : Type*} [normed_group E] [normed_space ℝ E] - {F : Type*} [normed_group F] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] + {F : Type*} [normed_add_comm_group F] {g b : E → F} {Ω : set (E × F)} {U K C : set E} variables [normed_space ℝ F] [measurable_space F] [borel_space F] [finite_dimensional ℝ F] @@ -33,13 +33,13 @@ lemma exists_loops_aux1 [finite_dimensional ℝ E] (hΩ_op : is_open Ω) (hg : 𝒞 ∞ g) (hb : 𝒞 ∞ b) (hgK : ∀ᶠ x near K, g x = b x) - (hconv : ∀ x, g x ∈ hull (connected_comp_in (prod.mk x ⁻¹' Ω) $ b x)) : + (hconv : ∀ x, g x ∈ hull (connected_component_in (prod.mk x ⁻¹' Ω) $ b x)) : ∃ (γ : E → ℝ → loop F) (V ∈ 𝓝ˢ K) (ε > 0), surrounding_family_in g b γ V Ω ∧ (∀ (x ∈ V), ball (x, b x) (ε + ε) ⊆ Ω) ∧ ∀ (x ∈ V) t s, dist (γ x t s) (b x) < ε := begin have b_in : ∀ x, (x, b x) ∈ Ω := - λ x, (connected_comp_in_nonempty_iff.mp (convex_hull_nonempty_iff.mp ⟨g x, hconv x⟩) : _), + λ x, (connected_component_in_nonempty_iff.mp (convex_hull_nonempty_iff.mp ⟨g x, hconv x⟩) : _), have h2Ω : is_open (Ω ∩ fst ⁻¹' univ), { rwa [preimage_univ, inter_univ] }, -- we could probably get away with something simpler to get γ₀. @@ -129,7 +129,7 @@ lemma exists_loops_aux2 [finite_dimensional ℝ E] (hΩ_op : is_open Ω) (hg : 𝒞 ∞ g) (hb : 𝒞 ∞ b) (hgK : ∀ᶠ x near K, g x = b x) - (hconv : ∀ x, g x ∈ hull (connected_comp_in (prod.mk x ⁻¹' Ω) $ b x)) : + (hconv : ∀ x, g x ∈ hull (connected_component_in (prod.mk x ⁻¹' Ω) $ b x)) : ∃ (γ : E → ℝ → loop F), surrounding_family_in g b γ univ Ω ∧ 𝒞 ∞ ↿γ ∧ ∀ᶠ x near K, ∀ t s, closed_ball (x, b x) (dist (γ x t s) (b x)) ⊆ Ω := begin @@ -250,7 +250,7 @@ theorem exists_loops [finite_dimensional ℝ E] (hΩ_op : is_open Ω) (hg : 𝒞 ∞ g) (hb : 𝒞 ∞ b) (hgK : ∀ᶠ x near K, g x = b x) - (hconv : ∀ x, g x ∈ hull (connected_comp_in (prod.mk x ⁻¹' Ω) $ b x)) : + (hconv : ∀ x, g x ∈ hull (connected_component_in (prod.mk x ⁻¹' Ω) $ b x)) : ∃ γ : ℝ → E → loop F, nice_loop g b Ω K γ := begin obtain ⟨γ₁, hγ₁, hsγ₁, h2γ₁⟩ := exists_loops_aux2 hK hΩ_op hg hb hgK hconv, diff --git a/formal/lean/sphere-eversion/loops/reparametrization.lean b/formal/lean/sphere-eversion/loops/reparametrization.lean index ad9af6d7ba65360132b5d0062fa887db1df41efa..8fea9243cdac32c4c96cdc9c8d501ad424b8dba5 100644 --- a/formal/lean/sphere-eversion/loops/reparametrization.lean +++ b/formal/lean/sphere-eversion/loops/reparametrization.lean @@ -17,8 +17,8 @@ open set function measure_theory interval_integral filter open_locale topological_space unit_interval manifold big_operators variables {E F : Type*} -variables [normed_group E] [normed_space ℝ E] [finite_dimensional ℝ E] -variables [normed_group F] [normed_space ℝ F] [finite_dimensional ℝ F] +variables [normed_add_comm_group E] [normed_space ℝ E] [finite_dimensional ℝ E] +variables [normed_add_comm_group F] [normed_space ℝ F] [finite_dimensional ℝ F] variables [measurable_space F] [borel_space F] local notation `ι` := fin (finite_dimensional.finrank ℝ F + 1) @@ -272,7 +272,7 @@ begin have h₁ := smooth_barycentric ι ℝ F (fintype.card_fin _), have h₂ : 𝒞 ∞ (eval i : (ι → ℝ) → ℝ) := cont_diff_apply _ _ i, refine (h₂.comp_cont_diff_on h₁).comp _ _, - { have h₃ := (diag_preimage_prod_sellf (γ.local_centering_density_nhd x)).symm.subset, + { have h₃ := (diag_preimage_prod_self (γ.local_centering_density_nhd x)).symm.subset, refine cont_diff_on.comp _ (cont_diff_id.prod cont_diff_id).cont_diff_on h₃, refine (γ.smooth_surrounded).cont_diff_on.prod_map (cont_diff.cont_diff_on _), exact γ.approx_surrounding_points_at_smooth x _, }, diff --git a/formal/lean/sphere-eversion/loops/surrounding.lean b/formal/lean/sphere-eversion/loops/surrounding.lean index f82bcb2530b9c00aeb9b90083f79a309c689de5b..f7ac98912c1a919899a901a3c60427e626293c07 100644 --- a/formal/lean/sphere-eversion/loops/surrounding.lean +++ b/formal/lean/sphere-eversion/loops/surrounding.lean @@ -1,5 +1,6 @@ import loops.basic import tactic.fin_cases +import analysis.locally_convex.with_seminorms -- to obtain that normed spaces are locally connected import topology.metric_space.emetric_paracompact import topology.shrinking_lemma import to_mathlib.partition @@ -59,8 +60,8 @@ end is_path_connected noncomputable theory -variables {E : Type*} [normed_group E] [normed_space ℝ E] - {F : Type*} [normed_group F] [normed_space ℝ F] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] + {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] local notation `d` := finrank ℝ F local notation `smooth_on` := cont_diff_on ℝ ⊤ @@ -731,21 +732,21 @@ lemma local_loops [finite_dimensional ℝ F] {x₀ : E} (hΩ_op : ∃ U ∈ 𝓝 x₀, is_open (Ω ∩ fst ⁻¹' U)) (hg : continuous_at g x₀) (hb : continuous b) - (hconv : g x₀ ∈ convex_hull ℝ (connected_comp_in (prod.mk x₀ ⁻¹' Ω) $ b x₀)) : + (hconv : g x₀ ∈ convex_hull ℝ (connected_component_in (prod.mk x₀ ⁻¹' Ω) $ b x₀)) : ∃ (γ : E → ℝ → loop F) (U ∈ 𝓝 x₀), surrounding_family_in g b γ U Ω := begin have hbx₀ : continuous_at b x₀ := hb.continuous_at, - have hΩ_op_x₀ : is_open (connected_comp_in (prod.mk x₀ ⁻¹' Ω) $ b x₀) := - (is_open_slice_of_is_open_over hΩ_op).connected_comp_in, + have hΩ_op_x₀ : is_open (connected_component_in (prod.mk x₀ ⁻¹' Ω) $ b x₀) := + (is_open_slice_of_is_open_over hΩ_op).connected_component_in, have b_in : b x₀ ∈ prod.mk x₀ ⁻¹' Ω := - connected_comp_in_nonempty_iff.mp (convex_hull_nonempty_iff.mp ⟨g x₀, hconv⟩), - have hΩ_conn : is_connected (connected_comp_in (prod.mk x₀ ⁻¹' Ω) $ b x₀) := - is_connected_connected_comp_in.mpr b_in, - have hb_in : b x₀ ∈ (connected_comp_in (prod.mk x₀ ⁻¹' Ω) $ b x₀) := - mem_connected_comp_in_self b_in, + connected_component_in_nonempty_iff.mp (convex_hull_nonempty_iff.mp ⟨g x₀, hconv⟩), + have hΩ_conn : is_connected (connected_component_in (prod.mk x₀ ⁻¹' Ω) $ b x₀) := + is_connected_connected_component_in_iff.mpr b_in, + have hb_in : b x₀ ∈ (connected_component_in (prod.mk x₀ ⁻¹' Ω) $ b x₀) := + mem_connected_component_in b_in, rcases surrounding_loop_of_convex_hull hΩ_op_x₀ hΩ_conn hconv hb_in with ⟨γ, h1γ, h2γ, h3γ, h4γ, h5γ, h6γ⟩, - have h5γ : ∀ (t s : ℝ), γ t s ∈ mk x₀ ⁻¹' Ω := λ t s, connected_comp_in_subset _ _ (h5γ t s), + have h5γ : ∀ (t s : ℝ), γ t s ∈ mk x₀ ⁻¹' Ω := λ t s, connected_component_in_subset _ _ (h5γ t s), let δ : E → ℝ → loop F := λ x t, b x - b x₀ +ᵥ γ t, have hδ : continuous ↿δ, { dsimp only [δ, has_uncurry.uncurry, loop.vadd_apply], @@ -790,7 +791,7 @@ lemma local_loops_open [finite_dimensional ℝ F] {x₀ : E} (hΩ_op : ∃ U ∈ 𝓝 x₀, is_open (Ω ∩ fst ⁻¹' U)) (hg : continuous_at g x₀) (hb : continuous b) - (hconv : g x₀ ∈ convex_hull ℝ (connected_comp_in (prod.mk x₀ ⁻¹' Ω) $ b x₀)) : + (hconv : g x₀ ∈ convex_hull ℝ (connected_component_in (prod.mk x₀ ⁻¹' Ω) $ b x₀)) : ∃ (γ : E → ℝ → loop F) (U : set E), is_open U ∧ x₀ ∈ U ∧ surrounding_family_in g b γ U Ω := begin obtain ⟨γ, U, hU, hγ⟩ := local_loops hΩ_op hg hb hconv, @@ -1239,7 +1240,7 @@ lemma exists_surrounding_loops [finite_dimensional ℝ F] (hK : is_compact K) (hC : is_closed C) (hU : is_open U) (hCU : C ⊆ U) (hΩ_op : is_open (Ω ∩ fst ⁻¹' U)) (hg : ∀ x ∈ C, continuous_at g x) (hb : continuous b) - (hconv : ∀ x ∈ C, g x ∈ convex_hull ℝ (connected_comp_in (prod.mk x ⁻¹' Ω) $ b x)) + (hconv : ∀ x ∈ C, g x ∈ convex_hull ℝ (connected_component_in (prod.mk x ⁻¹' Ω) $ b x)) {γ₀ : E → ℝ → loop F} (hγ₀_surr : ∃ V ∈ 𝓝ˢ K, surrounding_family_in g b γ₀ V Ω) : ∃ γ : E → ℝ → loop F, surrounding_family_in g b γ C Ω ∧ ∀ᶠ x in 𝓝ˢ K, γ x = γ₀ x := diff --git a/formal/lean/sphere-eversion/to_mathlib/analysis/calculus.lean b/formal/lean/sphere-eversion/to_mathlib/analysis/calculus.lean index a7871d184407e89123fbcf997bc8c30606fa2f7d..bb1c946b0dc240f5869b3b3ed6c50f5f65a27b50 100644 --- a/formal/lean/sphere-eversion/to_mathlib/analysis/calculus.lean +++ b/formal/lean/sphere-eversion/to_mathlib/analysis/calculus.lean @@ -6,7 +6,7 @@ noncomputable theory open set function filter open_locale topological_space -lemma is_compact.bdd_above_norm {X : Type*} [topological_space X] {E : Type*} [normed_group E] +lemma is_compact.bdd_above_norm {X : Type*} [topological_space X] {E : Type*} [normed_add_comm_group E] {s : set X} (hs : is_compact s) {f : X → E} (hf : continuous f) : ∃ M > 0, ∀ x ∈ s, ∥f x∥ ≤ M := begin cases (hs.image (continuous_norm.comp hf)).bdd_above with M hM, @@ -35,10 +35,10 @@ end real section calculus open continuous_linear_map -variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] - {F : Type*} [normed_group F] [normed_space 𝕜 F] - {G : Type*} [normed_group G] [normed_space 𝕜 G] +variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] + {F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F] + {G : Type*} [normed_add_comm_group G] [normed_space 𝕜 G] {n : with_top ℕ} lemma has_fderiv_at.partial_fst {φ : E → F → G} {φ' : E × F →L[𝕜] G} {e₀ : E} {f₀ : F} @@ -208,8 +208,8 @@ end calculus section real_calculus open continuous_linear_map -variables {E : Type*} [normed_group E] [normed_space ℝ E] - {F : Type*} [normed_group F] [normed_space ℝ F] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] + {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] lemma cont_diff.lipschitz_on_with {s : set E} {f : E → F} (hf : cont_diff ℝ 1 f) (hs : convex ℝ s) (hs' : is_compact s) : ∃ K, lipschitz_on_with K f s := @@ -229,23 +229,13 @@ lemma of_eventually_nhds {X : Type*} [topological_space X] {P : X → Prop} {x (h : ∀ᶠ x in 𝓝 x₀, P x) : P x₀ := mem_of_mem_nhds h - - -/- Move this next to cont_diff_smul -/ -lemma cont_diff.const_smul {𝕜 : Type*} [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] - {F : Type*} [normed_group F] [normed_space 𝕜 F] - {f : E → F} {n : with_top ℕ} (hf : cont_diff 𝕜 n f) (a : 𝕜) : - cont_diff 𝕜 n (λ x, a • f x) := -cont_diff_const.smul hf - section open asymptotics continuous_linear_map filter open_locale filter -variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] - {E : Type*} {F : Type*} [normed_group F] +variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] + {E : Type*} {F : Type*} [normed_add_comm_group F] lemma filter.eventually_le.is_O {f g h : E → F} {l : filter E} (hfg : (λ x, ∥f x∥) ≤ᶠ[l] λ x, ∥g x∥) (hh : g =O[l] h) : f =O[l] h := @@ -265,8 +255,8 @@ begin rwa [real.norm_eq_abs, abs_of_nonneg ((norm_nonneg $ f x).trans h), one_mul] end -variables [normed_group E] [normed_space 𝕜 E] [normed_space 𝕜 F] - {G : Type*} [normed_group G] [normed_space 𝕜 G] +variables [normed_add_comm_group E] [normed_space 𝕜 E] [normed_space 𝕜 F] + {G : Type*} [normed_add_comm_group G] [normed_space 𝕜 G] lemma asymptotics.is_O.eq_zero {f : E → F} {x₀ : E} {n : ℕ} (h : f =O[𝓝 x₀] λ x, ∥x - x₀∥^n) (hn : 0 < n) : f x₀ = 0 := @@ -374,7 +364,7 @@ begin simp only [sub_zero] at this, have key := this.const_mul C, rw mul_zero at key, - apply (normed_group.tendsto_nhds_zero.mp key ε ε_pos).mono, + apply (normed_add_comm_group.tendsto_nhds_zero.mp key ε ε_pos).mono, intros N hN, cases le_or_lt (C * ∥1 / N∥) 0 with h h, { exact h.trans_lt ε_pos }, diff --git a/formal/lean/sphere-eversion/to_mathlib/analysis/cont_diff.lean b/formal/lean/sphere-eversion/to_mathlib/analysis/cont_diff.lean index 7f7989c528835f1af8f8c8aac9257bd5a7de0f94..045526d97e649ac80729b8aa595622efeb257141 100644 --- a/formal/lean/sphere-eversion/to_mathlib/analysis/cont_diff.lean +++ b/formal/lean/sphere-eversion/to_mathlib/analysis/cont_diff.lean @@ -16,11 +16,11 @@ universes u₁ u₂ u₃ u₄ u₅ open continuous_linear_map -variables {𝕜 : Type u₁} [nondiscrete_normed_field 𝕜] - {M₁ : Type u₂} [normed_group M₁] [normed_space 𝕜 M₁] - {M₂ : Type u₃} [normed_group M₂] [normed_space 𝕜 M₂] - {M₃ : Type u₄} [normed_group M₃] [normed_space 𝕜 M₃] - {M₄ : Type u₅} [normed_group M₄] [normed_space 𝕜 M₄] +variables {𝕜 : Type u₁} [nontrivially_normed_field 𝕜] + {M₁ : Type u₂} [normed_add_comm_group M₁] [normed_space 𝕜 M₁] + {M₂ : Type u₃} [normed_add_comm_group M₂] [normed_space 𝕜 M₂] + {M₃ : Type u₄} [normed_add_comm_group M₃] [normed_space 𝕜 M₃] + {M₄ : Type u₅} [normed_add_comm_group M₄] [normed_space 𝕜 M₄] -- The next definition won't be used here, it's practice before the next one. @@ -60,10 +60,10 @@ lemma continuous_linear_equiv.continuous_lower_triangular {X : Type*} [topologic end section -variables (𝕜 : Type*) [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] - {F : Type*} [normed_group F] [normed_space 𝕜 F] - {G : Type*} [normed_group G] [normed_space 𝕜 G] +variables (𝕜 : Type*) [nontrivially_normed_field 𝕜] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] + {F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F] + {G : Type*} [normed_add_comm_group G] [normed_space 𝕜 G] {n : with_top ℕ} -- The next two definitions aren't used in the end, but they may still go to mathlib @@ -148,9 +148,9 @@ end section variables {𝕜 : Type*} [is_R_or_C 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] - {F : Type*} [normed_group F] [normed_space 𝕜 F] - {G : Type*} [normed_group G] [normed_space 𝕜 G] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] + {F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F] + {G : Type*} [normed_add_comm_group G] [normed_space 𝕜 G] {n : with_top ℕ} @@ -205,7 +205,7 @@ end end section -variables {E : Type*} [normed_group E] [normed_space ℝ E] [complete_space E] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] lemma cont_diff_parametric_symm_of_deriv_pos {f : E → ℝ → ℝ} (hf : cont_diff ℝ ⊤ ↿f) (hderiv : ∀ x t, partial_deriv_snd f x t > 0) (hsurj : ∀ x, surjective $ f x) : @@ -228,9 +228,9 @@ end end section -variables (𝕜 : Type*) [nondiscrete_normed_field 𝕜] +variables (𝕜 : Type*) [nontrivially_normed_field 𝕜] -lemma cont_diff_to_span_singleton (E : Type*) [normed_group E] [normed_space 𝕜 E] : +lemma cont_diff_to_span_singleton (E : Type*) [normed_add_comm_group E] [normed_space 𝕜 E] : cont_diff 𝕜 ⊤ (continuous_linear_map.to_span_singleton 𝕜 : E → 𝕜 →L[𝕜] E) := (continuous_linear_map.lsmul 𝕜 𝕜 : 𝕜 →L[𝕜] E →L[𝕜] E).flip.cont_diff diff --git a/formal/lean/sphere-eversion/to_mathlib/analysis/cut_off.lean b/formal/lean/sphere-eversion/to_mathlib/analysis/cut_off.lean index 00f0d6843b8e759e7f09a682356128e20a1acd23..82ccab857a9ebed4a5d2d4faab5e352f50316d46 100644 --- a/formal/lean/sphere-eversion/to_mathlib/analysis/cut_off.lean +++ b/formal/lean/sphere-eversion/to_mathlib/analysis/cut_off.lean @@ -6,7 +6,7 @@ import to_mathlib.topology.nhds_set open set filter open_locale manifold topological_space -lemma exists_cont_diff_zero_one {E : Type*} [normed_group E] +lemma exists_cont_diff_zero_one {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [finite_dimensional ℝ E] {s t : set E} (hs : is_closed s) (ht : is_closed t) (hd : disjoint s t) : ∃ f : E → ℝ, cont_diff ℝ ⊤ f ∧ eq_on f 0 s ∧ eq_on f 1 t ∧ @@ -14,7 +14,7 @@ lemma exists_cont_diff_zero_one {E : Type*} [normed_group E] let ⟨f, hfs, hft, hf01⟩ := exists_smooth_zero_one_of_closed 𝓘(ℝ, E) hs ht hd in ⟨f, f.smooth.cont_diff, hfs, hft, hf01⟩ -lemma exists_cont_diff_zero_one_nhds {E : Type*} [normed_group E] +lemma exists_cont_diff_zero_one_nhds {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [finite_dimensional ℝ E] {s t : set E} (hs : is_closed s) (ht : is_closed t) (hd : disjoint s t) : ∃ f : E → ℝ, cont_diff ℝ ⊤ f ∧ (∀ᶠ x in 𝓝ˢ s, f x = 0) ∧ (∀ᶠ x in 𝓝ˢ t, f x = 1) ∧ @@ -33,7 +33,7 @@ begin apply eventually_of_mem (mem_of_superset (v_op.mem_nhds_set.mpr htv) subset_closure) hfv end -lemma exists_cont_diff_one_nhds_of_interior {E : Type*} [normed_group E] +lemma exists_cont_diff_one_nhds_of_interior {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [finite_dimensional ℝ E] {s t : set E} (hs : is_closed s) (hd : s ⊆ interior t) : ∃ f : E → ℝ, cont_diff ℝ ⊤ f ∧ (∀ᶠ x in 𝓝ˢ s, f x = 1) ∧ (∀ x ∉ t, f x = 0) ∧ diff --git a/formal/lean/sphere-eversion/to_mathlib/analysis/normed_group.lean b/formal/lean/sphere-eversion/to_mathlib/analysis/normed_group.lean index 577b974aa501cbd045c7635cf4a57dc9ac1ec8d1..1cfea521793edcf6406fef179673eee8ca4108a7 100644 --- a/formal/lean/sphere-eversion/to_mathlib/analysis/normed_group.lean +++ b/formal/lean/sphere-eversion/to_mathlib/analysis/normed_group.lean @@ -1,8 +1,8 @@ import analysis.normed.group.basic -lemma norm_sub_le_add {G : Type*} [normed_group G] (a b c : G) : ∥a - b∥ ≤ ∥a - c∥ + ∥c - b∥ := +lemma norm_sub_le_add {G : Type*} [normed_add_comm_group G] (a b c : G) : ∥a - b∥ ≤ ∥a - c∥ + ∥c - b∥ := by simp [← dist_eq_norm, ← dist_eq_norm, ← dist_eq_norm, dist_triangle] -lemma norm_sub_le_add_of_le {G : Type*} [normed_group G] {a b c : G} {d d' : ℝ} +lemma norm_sub_le_add_of_le {G : Type*} [normed_add_comm_group G] {a b c : G} {d d' : ℝ} (h : ∥a - c∥ ≤ d) (h' : ∥c - b∥ ≤ d') : ∥a - b∥ ≤ d + d' := (norm_sub_le_add a b c).trans $ add_le_add h h' diff --git a/formal/lean/sphere-eversion/to_mathlib/analysis/normed_space/operator_norm.lean b/formal/lean/sphere-eversion/to_mathlib/analysis/normed_space/operator_norm.lean index 11168876cf40dae5eac6889b34aefff88d1954ff..a8c4891a8ae0e21205a524000287e70b9315f2af 100644 --- a/formal/lean/sphere-eversion/to_mathlib/analysis/normed_space/operator_norm.lean +++ b/formal/lean/sphere-eversion/to_mathlib/analysis/normed_space/operator_norm.lean @@ -5,8 +5,15 @@ noncomputable theory local notation u ` ⬝ `:70 φ:65 := continuous_linear_map.comp (continuous_linear_map.to_span_singleton ℝ u) φ + +variables {𝕜 E F G Fₗ Gₗ X : Type*} [nontrivially_normed_field 𝕜] [normed_add_comm_group E] + [normed_add_comm_group Fₗ] [normed_add_comm_group Gₗ] [normed_add_comm_group F] + [normed_add_comm_group G] + [normed_space 𝕜 E] [normed_space 𝕜 Fₗ] [normed_space 𝕜 Gₗ] [normed_space 𝕜 F] [normed_space 𝕜 G] + [topological_space X] + @[simp] -lemma continuous_linear_map.to_span_singleton_zero (𝕜 : Type*) {E : Type*} [semi_normed_group E] [nondiscrete_normed_field 𝕜] +lemma continuous_linear_map.to_span_singleton_zero (𝕜 : Type*) {E : Type*} [seminormed_add_comm_group E] [nontrivially_normed_field 𝕜] [normed_space 𝕜 E] : continuous_linear_map.to_span_singleton 𝕜 (0 : E) = 0 := begin ext u, @@ -15,8 +22,8 @@ begin end @[simp] -lemma continuous_linear_map.comp_to_span_singleton_apply {E : Type*} [normed_group E] [normed_space ℝ E] - {F : Type*} [normed_group F] [normed_space ℝ F] +lemma continuous_linear_map.comp_to_span_singleton_apply {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] + {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] (φ : E →L[ℝ] ℝ) (v : E) (u : F) : (u ⬝ φ) v = (φ v) • u:= rfl @@ -45,9 +52,9 @@ lemma add_le_twice_max (a b : ℝ) : a + b ≤ 2*max a b := calc a + b ≤ max a b + max a b : add_le_add (le_max_left a b) (le_max_right a b) ... = _ : by ring -lemma is_bounded_linear_map_coprod (𝕜 : Type*) [nondiscrete_normed_field 𝕜] (E : Type*) [normed_group E] - [normed_space 𝕜 E] (F : Type*) [normed_group F] [normed_space 𝕜 F] - (G : Type*) [normed_group G] [normed_space 𝕜 G] : is_bounded_linear_map 𝕜 +lemma is_bounded_linear_map_coprod (𝕜 : Type*) [nontrivially_normed_field 𝕜] (E : Type*) [normed_add_comm_group E] + [normed_space 𝕜 E] (F : Type*) [normed_add_comm_group F] [normed_space 𝕜 F] + (G : Type*) [normed_add_comm_group G] [normed_space 𝕜 G] : is_bounded_linear_map 𝕜 (λ p : (E →L[𝕜] G) × (F →L[𝕜] G), p.1.coprod p.2) := { map_add := begin intros, @@ -82,23 +89,18 @@ lemma is_bounded_linear_map_coprod (𝕜 : Type*) [nondiscrete_normed_field 𝕜 end } -def continuous_linear_map.coprodL {𝕜 : Type*} [nondiscrete_normed_field 𝕜] {E : Type*} [normed_group E] - [normed_space 𝕜 E] {F : Type*} [normed_group F] [normed_space 𝕜 F] - {G : Type*} [normed_group G] [normed_space 𝕜 G] : +def continuous_linear_map.coprodL : ((E →L[𝕜] G) × (F →L[𝕜] G)) →L[𝕜] (E × F →L[𝕜] G) := (is_bounded_linear_map_coprod 𝕜 E F G).to_continuous_linear_map @[continuity] -lemma continuous.coprodL {𝕜 : Type*} [nondiscrete_normed_field 𝕜] {E : Type*} [normed_group E] - [normed_space 𝕜 E] {F : Type*} [normed_group F] [normed_space 𝕜 F] - {G : Type*} [normed_group G] [normed_space 𝕜 G] - {X : Type*} [topological_space X] +lemma continuous.coprodL {f : X → E →L[𝕜] G} {g : X → F →L[𝕜] G} (hf : continuous f) (hg : continuous g) : continuous (λ x, (f x).coprod (g x)) := continuous_linear_map.coprodL.continuous.comp₂ hf hg -lemma continuous.prodL' {𝕜 : Type*} {E : Type*} {Fₗ : Type*} {Gₗ : Type*} [semi_normed_group E] - [semi_normed_group Fₗ] [semi_normed_group Gₗ] [nondiscrete_normed_field 𝕜] +lemma continuous.prodL' {𝕜 : Type*} {E : Type*} {Fₗ : Type*} {Gₗ : Type*} [seminormed_add_comm_group E] + [seminormed_add_comm_group Fₗ] [seminormed_add_comm_group Gₗ] [nontrivially_normed_field 𝕜] [normed_space 𝕜 E] [normed_space 𝕜 Fₗ] [normed_space 𝕜 Gₗ] (R : Type*) [semiring R] [module R Fₗ] [module R Gₗ] [has_continuous_const_smul R Fₗ] [has_continuous_const_smul R Gₗ] @@ -109,8 +111,8 @@ lemma continuous.prodL' {𝕜 : Type*} {E : Type*} {Fₗ : Type*} {Gₗ : Type*} (continuous_linear_map.prodₗᵢ 𝕜).continuous.comp₂ hf hg @[continuity] -lemma continuous.prodL {𝕜 : Type*} {E : Type*} {Fₗ : Type*} {Gₗ : Type*} [semi_normed_group E] - [semi_normed_group Fₗ] [semi_normed_group Gₗ] [nondiscrete_normed_field 𝕜] +lemma continuous.prodL {𝕜 : Type*} {E : Type*} {Fₗ : Type*} {Gₗ : Type*} [seminormed_add_comm_group E] + [seminormed_add_comm_group Fₗ] [seminormed_add_comm_group Gₗ] [nontrivially_normed_field 𝕜] [normed_space 𝕜 E] [normed_space 𝕜 Fₗ] [normed_space 𝕜 Gₗ] {X : Type*} [topological_space X] {f : X → E →L[𝕜] Fₗ} {g : X → E →L[𝕜] Gₗ} @@ -118,11 +120,12 @@ lemma continuous.prodL {𝕜 : Type*} {E : Type*} {Fₗ : Type*} {Gₗ : Type*} hf.prodL' 𝕜 hg @[continuity] -lemma continuous.compL {𝕜 : Type*} {E : Type*} {Fₗ : Type*} {Gₗ : Type*} [normed_group E] - [normed_group Fₗ] [normed_group Gₗ] [nondiscrete_normed_field 𝕜] - [normed_space 𝕜 E] [normed_space 𝕜 Fₗ] [normed_space 𝕜 Gₗ] - {X : Type*} [topological_space X] {f : X → Fₗ →L[𝕜] Gₗ} {g : X → E →L[𝕜] Fₗ} +lemma continuous.compL {f : X → Fₗ →L[𝕜] Gₗ} {g : X → E →L[𝕜] Fₗ} (hf : continuous f) (hg : continuous g) : continuous (λ x, (f x).comp (g x)) := (continuous_linear_map.apply 𝕜 (E →L[𝕜] Gₗ) : (E →L[𝕜] Fₗ) →L[𝕜] ((E →L[𝕜] Fₗ) →L[𝕜] E →L[𝕜] Gₗ) →L[𝕜] E →L[𝕜] Gₗ).is_bounded_bilinear_map.continuous.comp₂ hg $ (continuous_linear_map.compL 𝕜 E Fₗ Gₗ).continuous.comp hf + +-- this might not be sufficient to prove that `immersion_rel` is open. +lemma continuous_linear_map.is_open_injective : is_open {L : E →L[𝕜] F | function.injective L} := +sorry diff --git a/formal/lean/sphere-eversion/to_mathlib/convolution.lean b/formal/lean/sphere-eversion/to_mathlib/convolution.lean index 7e88519781d2ae86de8d8827c48bfcc61926f8cc..81e17cf6ee1c1d6a0b1adf514991f717cb850e37 100644 --- a/formal/lean/sphere-eversion/to_mathlib/convolution.lean +++ b/formal/lean/sphere-eversion/to_mathlib/convolution.lean @@ -76,12 +76,12 @@ open continuous_linear_map metric open_locale pointwise topological_space variables {𝕜 G E E' E'' F F' F'' : Type*} -variables [normed_group E] [normed_group E'] [normed_group E''] [normed_group F] +variables [normed_add_comm_group E] [normed_add_comm_group E'] [normed_add_comm_group E''] [normed_add_comm_group F] variables {f f' : G → E} {g g' : G → E'} {x x' : G} {y y' : E} -section nondiscrete_normed_field +section nontrivially_normed_field -variables [nondiscrete_normed_field 𝕜] +variables [nontrivially_normed_field 𝕜] variables [normed_space 𝕜 E] [normed_space 𝕜 E'] [normed_space 𝕜 E''] [normed_space 𝕜 F] variables (L : E →L[𝕜] E' →L[𝕜] F) @@ -537,9 +537,9 @@ lemma has_compact_support.continuous_convolution_left_of_integrable end comm_group -section normed_group +section normed_add_comm_group -variables [semi_normed_group G] +variables [seminormed_add_comm_group G] /-- We can simplify the RHS further if we assume `f` is integrable, but also if `L = (•)`. TODO: add a version that assumes `antilipschitz_with` on `L`. -/ @@ -637,7 +637,7 @@ begin .trans_lt (half_lt_self hε) end -end normed_group +end normed_add_comm_group namespace cont_diff_bump_of_inner @@ -676,7 +676,7 @@ lemma convolution_tendsto_right' {ι} {φ : ι → cont_diff_bump_of_inner (0 : begin refine convolution_tendsto_right (λ i, (φ i).nonneg_normed) (λ i, (φ i).integral_normed) _ hmg hcg, - rw [normed_group.tendsto_nhds_zero] at hφ, + rw [normed_add_comm_group.tendsto_nhds_zero] at hφ, rw [tendsto_small_sets_iff], intros t ht, rcases metric.mem_nhds_iff.mp ht with ⟨ε, hε, ht⟩, @@ -696,7 +696,7 @@ end cont_diff_bump_of_inner end measurability -end nondiscrete_normed_field +end nontrivially_normed_field open_locale convolution @@ -714,8 +714,8 @@ variables [measurable_space G] {μ : measure G} variables (L : E →L[𝕜] E' →L[𝕜] F) section assoc -variables [normed_group F'] [normed_space ℝ F'] [normed_space 𝕜 F'] [complete_space F'] -variables [normed_group F''] [normed_space ℝ F''] [normed_space 𝕜 F''] [complete_space F''] +variables [normed_add_comm_group F'] [normed_space ℝ F'] [normed_space 𝕜 F'] [complete_space F'] +variables [normed_add_comm_group F''] [normed_space ℝ F''] [normed_space 𝕜 F''] [complete_space F''] variables {k : G → E''} variables (L₂ : F →L[𝕜] E'' →L[𝕜] F') variables (L₃ : E →L[𝕜] F'' →L[𝕜] F') @@ -745,7 +745,7 @@ end end assoc -variables [normed_group G] [borel_space G] +variables [normed_add_comm_group G] [borel_space G] variables [second_countable_topology G] [sigma_compact_space G] lemma convolution_precompR_apply {g : G → E'' →L[𝕜] E'} diff --git a/formal/lean/sphere-eversion/to_mathlib/equivariant.lean b/formal/lean/sphere-eversion/to_mathlib/equivariant.lean index 84a4142ae3fc47833d25ff2fb7deddaebc25d46e..522a24c2e00c383b39cd20412dc2191cfb5b46f5 100644 --- a/formal/lean/sphere-eversion/to_mathlib/equivariant.lean +++ b/formal/lean/sphere-eversion/to_mathlib/equivariant.lean @@ -192,7 +192,7 @@ end begin rw [← linear_reparam.fract_fract], rcases h with h|h, { rw [linear_reparam_eq_zero' (fract_nonneg _) h, fract_zero] }, - { rw [linear_reparam_eq_one' h (fract_lt_one _).le, fract_one] }, + { rw [linear_reparam_eq_one' h (fract_lt_one _).le, _root_.fract_one] }, end lemma continuous_linear_reparam : continuous linear_reparam := diff --git a/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/charted_space.lean b/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/charted_space.lean index b8fbe46a437ba347314446507cc3bf97bda3624e..49ddc4a82a77aefeba39fa0de92b0e92dacdd749 100644 --- a/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/charted_space.lean +++ b/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/charted_space.lean @@ -6,7 +6,7 @@ namespace charted_space open_locale topological_space open metric (hiding mem_nhds_iff ball) set -variables (E : Type*) {M : Type*} [normed_group E] [topological_space M] [charted_space E M] +variables (E : Type*) {M : Type*} [normed_add_comm_group E] [topological_space M] [charted_space E M] def ball (x : M) (r : ℝ) := (chart_at E x).symm '' metric.ball (chart_at E x x) r diff --git a/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/misc.lean b/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/misc.lean index 1b31c3b8fb94c0f47de4130c33fd1562864c45b2..71f0797cf6d42b2e3c54f1b78a37f3407a7e251f 100644 --- a/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/misc.lean +++ b/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/misc.lean @@ -3,9 +3,9 @@ import geometry.manifold.cont_mdiff_map open_locale manifold variables {𝕜 E E' H H' M M' N : Type*} -variables [nondiscrete_normed_field 𝕜] -variables [normed_group E] [normed_space 𝕜 E] [topological_space H] -variables [normed_group E'] [normed_space 𝕜 E'] [topological_space H'] +variables [nontrivially_normed_field 𝕜] +variables [normed_add_comm_group E] [normed_space 𝕜 E] [topological_space H] +variables [normed_add_comm_group E'] [normed_space 𝕜 E'] [topological_space H'] variables (I : model_with_corners 𝕜 E H) variables [topological_space M] [charted_space H M] variables [topological_space M'] [charted_space H M'] diff --git a/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/misc_manifold.lean b/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/misc_manifold.lean index 346e6e707b09da579164dc51354f87a4fe328d68..6fe73169594babcb5c4eb32ae9bae63cbf8e778a 100644 --- a/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/misc_manifold.lean +++ b/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/misc_manifold.lean @@ -1,4 +1,5 @@ import geometry.manifold.diffeomorph +import geometry.manifold.algebra.monoid open bundle set function filter open_locale manifold topological_space @@ -33,29 +34,30 @@ section charted_space variables {M H : Type*} [topological_space M] [topological_space H] [charted_space H M] (G : structure_groupoid H) -variable (H) -/-- `achart H x` is the chart at `x`, considered as an element of the atlas. -/ -def achart (x : M) : atlas H M := ⟨chart_at H x, chart_mem_atlas H x⟩ - -lemma achart_def (x : M) : achart H x = ⟨chart_at H x, chart_mem_atlas H x⟩ := rfl -@[simp] lemma coe_achart (x : M) : (achart H x : local_homeomorph M H) = chart_at H x := rfl -@[simp] lemma achart_val (x : M) : (achart H x).1 = chart_at H x := rfl - -variable {H} +variables (H) +lemma mem_achart_source (x : M) : x ∈ (achart H x).1.source := +mem_chart_source H x +variables {H} end charted_space +@[simp] +lemma local_equiv.refl_prod_refl {α β : Type*} : + (local_equiv.refl α).prod (local_equiv.refl β) = local_equiv.refl (α × β) := +by { ext1 ⟨x, y⟩, { refl }, { rintro ⟨x, y⟩, refl }, exact univ_prod_univ } + +@[simp] +lemma local_homeomorph.refl_prod_refl {α β : Type*} [topological_space α] [topological_space β] : + (local_homeomorph.refl α).prod (local_homeomorph.refl β) = local_homeomorph.refl (α × β) := +by { ext1 ⟨x, y⟩, { refl }, { rintro ⟨x, y⟩, refl }, exact univ_prod_univ } namespace model_with_corners -variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] +variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type*} [topological_space H] {M : Type*} [topological_space M] (f : local_homeomorph M H) (I : model_with_corners 𝕜 E H) -lemma injective : injective I := -I.left_inverse.injective - lemma preimage_image (s : set H) : I ⁻¹' (I '' s) = s := I.injective.preimage_image s @@ -64,30 +66,26 @@ end model_with_corners section smooth_manifold_with_corners open smooth_manifold_with_corners -variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] - {E' : Type*} [normed_group E'] [normed_space 𝕜 E'] - {F : Type*} [normed_group F] [normed_space 𝕜 F] +variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] + {E' : Type*} [normed_add_comm_group E'] [normed_space 𝕜 E'] + {F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F] {H : Type*} [topological_space H] {I : model_with_corners 𝕜 E H} {H' : Type*} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {G : Type*} [topological_space G] {J : model_with_corners 𝕜 F G} {M : Type*} [topological_space M] [charted_space H M] {M' : Type*} [topological_space M'] [charted_space H' M'] {N : Type*} [topological_space N] [charted_space G N] + {F' : Type*} [normed_add_comm_group F'] [normed_space 𝕜 F'] + {F'' : Type*} [normed_add_comm_group F''] [normed_space 𝕜 F''] variables {f : M → M'} {m n : with_top ℕ} {s : set M} {x : M} -lemma ext_chart_preimage_mem_nhds' {x' : M} {t : set M} - (h : x' ∈ (ext_chart_at I x).source) (ht : t ∈ 𝓝 x') : - (ext_chart_at I x).symm ⁻¹' t ∈ 𝓝 (ext_chart_at I x x') := -begin - apply (ext_chart_continuous_at_symm' I x h).preimage_mem_nhds, - rwa (ext_chart_at I x).left_inv h -end +attribute [ext] model_with_corners charted_space +lemma model_with_corners_self_prod : 𝓘(𝕜, E × F) = 𝓘(𝕜, E).prod 𝓘(𝕜, F) := +by { ext1, simp } -lemma cont_mdiff_at_iff_target {x : M} : - cont_mdiff_at I I' n f x ↔ - continuous_at f x ∧ cont_mdiff_at I 𝓘(𝕜, E') n (ext_chart_at I' (f x) ∘ f) x := -by rw [cont_mdiff_at, cont_mdiff_at, cont_mdiff_within_at_iff_target, continuous_within_at_univ] +lemma charted_space_self_prod : prod_charted_space E E F F = charted_space_self (E × F) := +by { ext1, simp [prod_charted_space, atlas], ext1, simp, } section boundary @@ -110,86 +108,121 @@ end boundary namespace basic_smooth_vector_bundle_core variables [smooth_manifold_with_corners I M] (Z : basic_smooth_vector_bundle_core I M E') +lemma coord_change_comp' {i j k : atlas H M} {x : M} + (hi : x ∈ i.1.source) (hj : x ∈ j.1.source) (hk : x ∈ k.1.source) (v : E') : + Z.coord_change j k (j x) (Z.coord_change i j (i x) v) = Z.coord_change i k (i x) v := +begin + rw [show j x = _, by rw [← i.1.left_inv hi]], + apply Z.coord_change_comp, + simp only [hi, hj, hk] with mfld_simps +end + +lemma coord_change_self' {i : atlas H M} {x : M} (hi : x ∈ i.1.source) (v : E') : + Z.coord_change i i (i x) v = v := +Z.coord_change_self i (i x) (i.1.maps_to hi) v + +lemma coord_change_comp_eq_self (i j : atlas H M) {x : H} + (hx : x ∈ (i.1.symm.trans j.1).source) (v : E') : + Z.coord_change j i (i.1.symm.trans j.1 x) (Z.coord_change i j x v) = v := +begin + rw [Z.coord_change_comp i j i x _ v, Z.coord_change_self _ _ hx.1], + simp only with mfld_simps at hx, + simp only [hx.1, hx.2] with mfld_simps +end + +lemma coord_change_comp_eq_self' {i j : atlas H M} {x : M} + (hi : x ∈ i.1.source) (hj : x ∈ j.1.source) (v : E') : + Z.coord_change j i (j x) (Z.coord_change i j (i x) v) = v := +by rw [Z.coord_change_comp' hi hj hi v, Z.coord_change_self' hi] + lemma chart_apply (z : Z.to_topological_vector_bundle_core.total_space) : Z.chart (chart_mem_atlas H x) z = (chart_at H x z.proj, Z.coord_change (achart H z.proj) (achart H x) (achart H z.proj z.proj) z.2) := rfl -/-- A version of `cont_mdiff_at_iff_target` when the codomain is the total space of - a `basic_smooth_vector_bundle_core`. The continuity condition in the RHS is weaker. -/ -lemma cont_mdiff_at_iff_target {f : N → Z.to_topological_vector_bundle_core.total_space} - {x : N} {n : with_top ℕ} : - cont_mdiff_at J (I.prod 𝓘(𝕜, E')) n f x ↔ continuous_at (bundle.total_space.proj ∘ f) x ∧ - cont_mdiff_at J 𝓘(𝕜, E × E') n (ext_chart_at (I.prod 𝓘(𝕜, E')) (f x) ∘ f) x := +lemma smooth_iff_target {f : N → Z.to_topological_vector_bundle_core.total_space} : + smooth J (I.prod 𝓘(𝕜, E')) f ↔ continuous (bundle.total_space.proj ∘ f) ∧ + ∀ x, smooth_at J 𝓘(𝕜, E × E') (ext_chart_at (I.prod 𝓘(𝕜, E')) (f x) ∘ f) x := +by simp_rw [smooth, smooth_at, cont_mdiff, Z.cont_mdiff_at_iff_target, forall_and_distrib, + continuous_iff_continuous_at] + +end basic_smooth_vector_bundle_core + +lemma cont_diff_within_at.comp_cont_mdiff_within_at + {g : F → F''} {f : M → F} {s : set M} {t : set F} {x : M} + (hg : cont_diff_within_at 𝕜 n g t (f x)) + (hf : cont_mdiff_within_at I 𝓘(𝕜, F) n f s x) (h : s ⊆ f ⁻¹' t) : + cont_mdiff_within_at I 𝓘(𝕜, F'') n (g ∘ f) s x := begin - let Z' := Z.to_topological_vector_bundle_core, - rw [cont_mdiff_at_iff_target, and.congr_left_iff], - refine λ hf, ⟨λ h, Z'.continuous_proj.continuous_at.comp h, λ h, _⟩, - exact (Z'.local_triv ⟨chart_at _ (f x).1, chart_mem_atlas _ _⟩).to_fiber_bundle_trivialization - .continuous_at_of_comp_left h (mem_chart_source _ _) (h.prod hf.continuous_at.snd) + rw cont_mdiff_within_at_iff at *, + refine ⟨hg.continuous_within_at.comp hf.1 h, _⟩, + rw [← (ext_chart_at I x).left_inv (mem_ext_chart_source I x)] at hg, + apply cont_diff_within_at.comp _ (by exact hg) hf.2 _, + exact (inter_subset_left _ _).trans (preimage_mono h) end -end basic_smooth_vector_bundle_core +lemma cont_diff_at.comp_cont_mdiff_at {g : F → F''} {f : M → F} {x : M} + (hg : cont_diff_at 𝕜 n g (f x)) (hf : cont_mdiff_at I 𝓘(𝕜, F) n f x) : + cont_mdiff_at I 𝓘(𝕜, F'') n (g ∘ f) x := +hg.comp_cont_mdiff_within_at hf subset.rfl + +lemma cont_diff.comp_cont_mdiff {g : F → F''} {f : M → F} + (hg : cont_diff 𝕜 n g) (hf : cont_mdiff I 𝓘(𝕜, F) n f) : + cont_mdiff I 𝓘(𝕜, F'') n (g ∘ f) := +λ x, hg.cont_diff_at.comp_cont_mdiff_at (hf x) + +variables [smooth_manifold_with_corners I M] [smooth_manifold_with_corners I' M'] -lemma tangent_bundle_core_coord_change_achart [smooth_manifold_with_corners I M] - (x x' : M) (z : H) : +-- this can be useful to see where we (ab)use definitional equalities +-- local attribute [irreducible] tangent_space + +/-! +I don't know if these instances were intentionally not declared for `tangent_space` +(maybe to not endow it with a particular norm), but if we don't want them we need to redesign some +other things. +Note that `dual_pair.update` wants `F` to be a `normed_add_comm_group` (which seems to be pretty +necessary for the definition -- although maybe we can get away with `has_continuous_smul` by +redesigning some things?). +In `rel_mfld.slice` we use `dual_pair.update` applied to `tangent_space`. If we don't add these +instances, it is a miracle that Lean still accepts the definition, but what is going on is that Lean +is unfolding the definition of `tangent_space`, realizing that `tangent_space I x = E` and +`tangent_space I' y = E'` and using the `normed_add_comm_group` instances of these types. +Note that this still uses these instances in a very sneaky way for the tangent space, but with +additional detriment that up to reducible transparancy, the term is not type-correct +(in other words: you have to unfold `tangent_space` to realize that the term is type-correct). +This means that many tactics, like `simp`, `rw` and `dsimp` fail to rewrite within this term, +because the result is not type correct up to reducible transparancy. +This is a nightmare, so we declare these instances. +(at least, this is what I think was going on, but unfortunately some issues still persisted after +this.) -/ +instance {x : M} : normed_add_comm_group (tangent_space I x) := by delta_instance tangent_space +instance {x : M} : normed_space 𝕜 (tangent_space I x) := by delta_instance tangent_space + +lemma tangent_bundle_core_coord_change_achart (x x' : M) (z : H) : (tangent_bundle_core I M).coord_change (achart H x) (achart H x') z = fderiv_within 𝕜 (ext_chart_at I x' ∘ (ext_chart_at I x).symm) (range I) (I z) := rfl variables (I) -lemma cont_diff_on_coord_change' [smooth_manifold_with_corners I M] - {e e' : local_homeomorph M H} (h : e ∈ atlas H M) (h' : e' ∈ atlas H M) : +lemma cont_diff_on_coord_change' {e e' : local_homeomorph M H} + (h : e ∈ atlas H M) (h' : e' ∈ atlas H M) : cont_diff_on 𝕜 ⊤ (I ∘ (e.symm ≫ₕ e') ∘ I.symm) (I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ range I) := (has_groupoid.compatible (cont_diff_groupoid ⊤ I) h h').1 -lemma ext_chart_at_target (x : M) : (ext_chart_at I x).target = - I.symm ⁻¹' (chart_at H x).target ∩ range I := -by simp_rw [ext_chart_at, local_equiv.trans_target, I.target_eq, I.to_local_equiv_coe_symm, - inter_comm] - -lemma ext_coord_change_source (x x' : M) : - ((ext_chart_at I x').symm ≫ ext_chart_at I x).source = - I '' ((chart_at H x').symm ≫ₕ (chart_at H x)).source := -by { simp_rw [local_equiv.trans_source, I.image_eq, ext_chart_at_source, local_equiv.symm_source, - ext_chart_at_target, inter_right_comm _ (range I)], refl } +variables {I} +/-- A congruence lemma for `mfderiv`, (ab)using the fact that `tangent_space I' (f x)` is +definitionally equal to `E'`. -/ +lemma mfderiv_congr_point {x' : M} (h : x = x') : + @eq (E →L[𝕜] E') (mfderiv I I' f x) (mfderiv I I' f x') := +by subst h -variables [smooth_manifold_with_corners I M] [smooth_manifold_with_corners I' M'] +/-- A congruence lemma for `mfderiv`, (ab)using the fact that `tangent_space I' (f x)` is +definitionally equal to `E'`. -/ +lemma mfderiv_congr {f' : M → M'} (h : f = f') : + @eq (E →L[𝕜] E') (mfderiv I I' f x) (mfderiv I I' f' x) := +by subst h -lemma cont_diff_on_ext_coord_change (x x' : M) : - cont_diff_on 𝕜 ⊤ (ext_chart_at I x ∘ (ext_chart_at I x').symm) - ((ext_chart_at I x').symm ≫ ext_chart_at I x).source := -by { rw [ext_coord_change_source, I.image_eq], exact (has_groupoid.compatible - (cont_diff_groupoid ⊤ I) (chart_mem_atlas H x') (chart_mem_atlas H x)).1 } - -lemma cont_diff_within_at_ext_coord_change (x x' : M) {y : E} - (hy : y ∈ ((ext_chart_at I x').symm ≫ ext_chart_at I x).source) : - cont_diff_within_at 𝕜 ⊤ (ext_chart_at I x ∘ (ext_chart_at I x').symm) (range I) y := -begin - apply (cont_diff_on_ext_coord_change I x x' y hy).mono_of_mem, - rw [ext_coord_change_source] at hy ⊢, - obtain ⟨z, hz, rfl⟩ := hy, - exact I.image_mem_nhds_within ((local_homeomorph.open_source _).mem_nhds hz) -end - -variables (𝕜) -lemma ext_chart_self_eq {x : H} : ⇑(ext_chart_at I x) = I := rfl -lemma ext_chart_self_apply {x y : H} : ext_chart_at I x y = I y := rfl -lemma ext_chart_model_space_apply {x y : E} : ext_chart_at 𝓘(𝕜, E) x y = y := rfl -variables {𝕜 I} - -/-- Note: does not hold for `n = ∞`. -/ -lemma cont_mdiff_at_iff_cont_mdiff_at_nhds {n : ℕ} : - cont_mdiff_at I I' n f x ↔ ∀ᶠ x' in 𝓝 x, cont_mdiff_at I I' n f x' := -begin - refine ⟨_, λ h, h.self_of_nhds⟩, - rw [cont_mdiff_at_iff_cont_mdiff_on_nhds], - rintro ⟨u, hu, h⟩, - obtain ⟨v, hvu, hv, hxv⟩ := mem_nhds_iff.mp hu, - refine eventually_of_mem (hv.mem_nhds hxv) (λ x' hx', _), - exact (h x' (hvu hx')).cont_mdiff_at (mem_of_superset (hv.mem_nhds hx') hvu) -end lemma cont_mdiff_at.mfderiv' {f : M → M'} (hf : cont_mdiff_at I I' n f x) (hmn : m + 1 ≤ n) : @@ -228,8 +261,8 @@ begin rw [(ext_chart_at I x₂).left_inv hx', (ext_chart_at I' (f x₂)).left_inv h2x'] }, refine filter.eventually_eq.fderiv_within_eq_nhds (I.unique_diff _ $ mem_range_self _) _, refine eventually_of_mem (inter_mem _ _) this, - { exact ext_chart_preimage_mem_nhds' hx₂ (ext_chart_at_source_mem_nhds I x₂) }, - refine ext_chart_preimage_mem_nhds' hx₂ _, + { exact ext_chart_preimage_mem_nhds' _ _ hx₂ (ext_chart_at_source_mem_nhds I x₂) }, + refine ext_chart_preimage_mem_nhds' _ _ hx₂ _, exact (h2x₂.continuous_at).preimage_mem_nhds (ext_chart_at_source_mem_nhds _ _) }, /- The conclusion is the same as the following, when unfolding coord_change of `tangent_bundle_core` -/ @@ -258,14 +291,34 @@ begin { simp_rw [function.comp_apply, (ext_chart_at I x).left_inv hx₂] } end +-- the following proof takes very long in pure term mode +lemma cont_mdiff_at.clm_comp {g : M → F →L[𝕜] F''} {f : M → F' →L[𝕜] F} {x : M} + (hg : cont_mdiff_at I 𝓘(𝕜, F →L[𝕜] F'') n g x) (hf : cont_mdiff_at I 𝓘(𝕜, F' →L[𝕜] F) n f x) : + cont_mdiff_at I 𝓘(𝕜, F' →L[𝕜] F'') n (λ x, (g x).comp (f x)) x := +@cont_diff_at.comp_cont_mdiff_at 𝕜 _ E _ _ ((F →L[𝕜] F'') × (F' →L[𝕜] F)) _ _ _ _ _ _ _ _ + _ _ _ _ + (λ x, x.1.comp x.2) (λ x, (g x, f x)) x + (by { apply cont_diff.cont_diff_at, apply is_bounded_bilinear_map.cont_diff, + exact is_bounded_bilinear_map_comp }) + (hg.prod_mk_space hf) + +lemma cont_mdiff.clm_comp {g : M → F →L[𝕜] F''} {f : M → F' →L[𝕜] F} + (hg : cont_mdiff I 𝓘(𝕜, F →L[𝕜] F'') n g) (hf : cont_mdiff I 𝓘(𝕜, F' →L[𝕜] F) n f) : + cont_mdiff I 𝓘(𝕜, F' →L[𝕜] F'') n (λ x, (g x).comp (f x)) := +λ x, (hg x).clm_comp (hf x) + +instance has_smooth_add_self : has_smooth_add 𝓘(𝕜, F) F := +⟨by { convert cont_diff_add.cont_mdiff, exact model_with_corners_self_prod.symm, + exact charted_space_self_prod }⟩ + end smooth_manifold_with_corners section maps -variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] -{E : Type*} [normed_group E] [normed_space 𝕜 E] -{E' : Type*} [normed_group E'] [normed_space 𝕜 E'] -{F : Type*} [normed_group F] [normed_space 𝕜 F] +variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] +{E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] +{E' : Type*} [normed_add_comm_group E'] [normed_space 𝕜 E'] +{F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F] {H : Type*} [topological_space H] {H' : Type*} [topological_space H'] {G : Type*} [topological_space G] diff --git a/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/vector_bundle/basic_core_constructions.lean b/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/vector_bundle/basic_core_constructions.lean index 9464ef35db387d9d3c0591f7ae8d0696cf9d7072..a9916ca15025032d3daa84c4dcd2704918196cc8 100644 --- a/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/vector_bundle/basic_core_constructions.lean +++ b/formal/lean/sphere-eversion/to_mathlib/geometry/manifold/vector_bundle/basic_core_constructions.lean @@ -17,14 +17,14 @@ open bundle set topological_space topological_vector_bundle local_homeomorph open_locale classical manifold variables {𝕜 B B' M VB VB' VM HB HB' HM : Type*} -variables [nondiscrete_normed_field 𝕜] -variables [normed_group VB] [normed_space 𝕜 VB] [normed_group VB'] [normed_space 𝕜 VB'] -variables [normed_group VM] [normed_space 𝕜 VM] +variables [nontrivially_normed_field 𝕜] +variables [normed_add_comm_group VB] [normed_space 𝕜 VB] [normed_add_comm_group VB'] [normed_space 𝕜 VB'] +variables [normed_add_comm_group VM] [normed_space 𝕜 VM] variables [topological_space HB] [topological_space HB'] [topological_space HM] variables {IB : model_with_corners 𝕜 VB HB} {IB' : model_with_corners 𝕜 VB' HB'} variables {IM : model_with_corners 𝕜 VM HM} variables {F F' : Type*} -variables [normed_group F] [normed_space 𝕜 F] [normed_group F'] [normed_space 𝕜 F'] +variables [normed_add_comm_group F] [normed_space 𝕜 F] [normed_add_comm_group F'] [normed_space 𝕜 F'] variables [topological_space B] [charted_space HB B] [smooth_manifold_with_corners IB B] variables [topological_space B'] [charted_space HB' B'] variables [topological_space M] [charted_space HM M] [smooth_manifold_with_corners IM M] @@ -102,7 +102,7 @@ lemma pullback_chart {v : VB' → VB} {hv : cont_diff 𝕜 ∞ v} {h : HB' → H by simp_rw [chart, trans_apply, prod_apply, trivialization.coe_coe, local_homeomorph.refl_apply, function.id_def, topological_vector_bundle_core.local_triv_apply, to_topological_vector_bundle_core_coord_change, to_topological_vector_bundle_core_index_at, - pullback_coord_change, ← achart_def, g_at, h_at, subtype.eta] + pullback_coord_change, g_at, achart_def, h_at, subtype.eta] variables (IB' B') @@ -309,7 +309,7 @@ lemma hom_chart by simp_rw [chart, trans_apply, local_homeomorph.prod_apply, trivialization.coe_coe, local_homeomorph.refl_apply, function.id_def, topological_vector_bundle_core.local_triv_apply, to_topological_vector_bundle_core_coord_change, to_topological_vector_bundle_core_index_at, - hom_coord_change, comp_apply, flip_apply, compL_apply, ← achart_def, + hom_coord_change, comp_apply, flip_apply, compL_apply, achart_def, (chart_at HB x.1).left_inv (mem_chart_source HB x.1)] end basic_smooth_vector_bundle_core diff --git a/formal/lean/sphere-eversion/to_mathlib/measure_theory/basic.lean b/formal/lean/sphere-eversion/to_mathlib/measure_theory/basic.lean index 9710846981a922096c4ab27ddd1470b1a2c4c155..867328fd513f32dca6a1d981be6d471bd1cd9a35 100644 --- a/formal/lean/sphere-eversion/to_mathlib/measure_theory/basic.lean +++ b/formal/lean/sphere-eversion/to_mathlib/measure_theory/basic.lean @@ -7,7 +7,7 @@ open measure_theory set classical filter open_locale classical topological_space filter interval -variables {α β E F : Type*} [measurable_space α] {μ : measure α} [normed_group E] +variables {α β E F : Type*} [measurable_space α] {μ : measure α} [normed_add_comm_group E] [complete_space E] [normed_space ℝ E] namespace measure_theory @@ -40,7 +40,7 @@ begin end lemma is_compact.integrable_const {α : Type*} [measurable_space α] [topological_space α] - {E : Type*} [normed_group E] [measurable_space E] + {E : Type*} [normed_add_comm_group E] [measurable_space E] {s : set α} (hs : is_compact s) (c : E) (μ : measure α) [is_locally_finite_measure μ] : integrable (λ (x : α), c) (μ.restrict s) := by simp_rw [integrable_const_iff, measure.restrict_apply_univ, hs.measure_lt_top, or_true] diff --git a/formal/lean/sphere-eversion/to_mathlib/measure_theory/interval_integral.lean b/formal/lean/sphere-eversion/to_mathlib/measure_theory/interval_integral.lean index a984efee555c4169ddf82887c8e55faeb233e726..1a00920fe6bb9ab2dfa4c7cbb0da7204a12dbf72 100644 --- a/formal/lean/sphere-eversion/to_mathlib/measure_theory/interval_integral.lean +++ b/formal/lean/sphere-eversion/to_mathlib/measure_theory/interval_integral.lean @@ -14,11 +14,11 @@ namespace continuous_linear_map open interval_integral variables {𝕜 E H F : Type*} variables [is_R_or_C 𝕜] {μ : measure ℝ} -variables [normed_group E] [normed_space 𝕜 E] [complete_space E] +variables [normed_add_comm_group E] [normed_space 𝕜 E] [complete_space E] variables [normed_space ℝ E] [is_scalar_tower ℝ 𝕜 E] -variables [normed_group F] [normed_space 𝕜 F] [complete_space F] +variables [normed_add_comm_group F] [normed_space 𝕜 F] [complete_space F] variables [normed_space ℝ F] [is_scalar_tower ℝ 𝕜 F] -variables [normed_group H] [normed_space 𝕜 H] +variables [normed_add_comm_group H] [normed_space 𝕜 H] lemma interval_integral_apply {a b : ℝ} {φ : ℝ → H →L[𝕜] E} (φ_int : interval_integrable φ μ a b) (v : H) : (∫ x in a..b, φ x ∂μ) v = ∫ x in a..b, φ x v ∂μ := @@ -29,7 +29,7 @@ end continuous_linear_map section -variables {E : Type*} [normed_group E] +variables {E : Type*} [normed_add_comm_group E] lemma interval_integrable_of_integral_ne_zero [complete_space E] [normed_space ℝ E] {a b : ℝ} @@ -143,7 +143,7 @@ end interval_integral /- This should replace interval_integrable.mono_set in mathlib -/ lemma interval_integrable.mono_set' {E : Type*} - [normed_group E] {f : ℝ → E} {a b c d : ℝ} {μ : measure ℝ} + [normed_add_comm_group E] {f : ℝ → E} {a b c d : ℝ} {μ : measure ℝ} (hf : interval_integrable f μ a b) (hsub : Ι c d ⊆ Ι a b) : interval_integrable f μ c d := interval_integrable_iff.mpr (hf.def.mono hsub le_rfl) @@ -197,7 +197,7 @@ begin simp } } end -lemma interval_integrable_of_norm_sub_le {β : Type*} [normed_group β] +lemma interval_integrable_of_norm_sub_le {β : Type*} [normed_add_comm_group β] {f₀ f₁ : ℝ → β} {g : ℝ → ℝ} {a b : ℝ} (hf₁_m : ae_strongly_measurable f₁ (μ.restrict $ Ι a b)) @@ -218,7 +218,7 @@ end section -variables {E : Type*} [normed_group E] [normed_space ℝ E] [complete_space E] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] open interval_integral diff --git a/formal/lean/sphere-eversion/to_mathlib/measure_theory/parametric_interval_integral.lean b/formal/lean/sphere-eversion/to_mathlib/measure_theory/parametric_interval_integral.lean index 4a5dda62630c543d72d16d71e5b3ffb60036406d..8be12f421786ceffa7abc9cd7ea89eb11e6c409d 100644 --- a/formal/lean/sphere-eversion/to_mathlib/measure_theory/parametric_interval_integral.lean +++ b/formal/lean/sphere-eversion/to_mathlib/measure_theory/parametric_interval_integral.lean @@ -19,8 +19,8 @@ lemma ae_strongly_measurable_interval_oc_iff {α β : Type*} [measurable_space by rw [interval_oc_eq_union, ae_strongly_measurable_union_iff] section -variables {E : Type*} [normed_group E] [normed_space ℝ E] [complete_space E] - {H : Type*} [normed_group H] [normed_space ℝ H] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] + {H : Type*} [normed_add_comm_group H] [normed_space ℝ H] (ν : measure ℝ) /-- Interval version of `has_fderiv_at_of_dominated_of_fderiv_le` -/ @@ -76,7 +76,7 @@ section open function theorem continuous_parametric_integral_of_continuous - {E : Type*} [normed_group E] [normed_space ℝ E] + {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] {α : Type*} [topological_space α] [measurable_space α] [opens_measurable_space α] [second_countable_topology_either α E] @@ -111,7 +111,7 @@ open measure_theory variables {μ : measure ℝ} {X : Type*} [topological_space X] [first_countable_topology X] - {E : Type*} [normed_group E] [normed_space ℝ E] [complete_space E] + {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] lemma continuous_at_parametric_primitive_of_dominated {F : X → ℝ → E} (bound : ℝ → ℝ) (a b : ℝ) {a₀ b₀ : ℝ} {x₀ : X} @@ -191,7 +191,7 @@ section variables {μ : measure ℝ} [is_locally_finite_measure μ] [has_no_atoms μ] {X : Type*} [topological_space X] [first_countable_topology X] - {E : Type*} [normed_group E] [normed_space ℝ E] [complete_space E] + {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] lemma continuous_parametric_primitive_of_continuous [locally_compact_space X] @@ -245,9 +245,9 @@ end section -variables {E : Type*} [normed_group E] [normed_space ℝ E] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] - {H : Type*} [normed_group H] [normed_space ℝ H] + {H : Type*} [normed_add_comm_group H] [normed_space ℝ H] /-! We could weaken `finite_dimensional ℝ H` with `second_countable (H →L[ℝ] E)` if needed, @@ -435,6 +435,7 @@ begin intros t t_in, rw [show (λ (x : H), F x t) = (uncurry F) ∘ (λ x : H, (x, t)), by { ext, simp }, ← mul_one K], apply hK.comp ((lipschitz_with_prod_mk_right t).lipschitz_on_with $ ball x₀ 1), + rw maps_to', rintros ⟨x, s⟩ ⟨x', hx, h⟩, cases h, refine ⟨ball_subset_closed_ball hx, mem_Icc_of_Ioo t_in⟩ }, have cont_x : ∀ x, continuous (F x), @@ -474,9 +475,9 @@ end end section -variables {E : Type*} [normed_group E] [normed_space ℝ E] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] - {H : Type*} [normed_group H] [normed_space ℝ H] + {H : Type*} [normed_add_comm_group H] [normed_space ℝ H] [finite_dimensional ℝ H] open real continuous_linear_map asymptotics @@ -528,9 +529,9 @@ section universe variables v u variables {E : Type u} -variables [normed_group E] [normed_space ℝ E] +variables [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] - {H : Type v} [normed_group H] [normed_space ℝ H] + {H : Type v} [normed_add_comm_group H] [normed_space ℝ H] [finite_dimensional ℝ H] /- Should we directly prove the version below?-/ diff --git a/formal/lean/sphere-eversion/to_mathlib/misc.lean b/formal/lean/sphere-eversion/to_mathlib/misc.lean index f63f41c4ab9bf4df63c9976ffcc0012f63fe695a..f470de009c049ec08cd65b47544d1cbc5207d088 100644 --- a/formal/lean/sphere-eversion/to_mathlib/misc.lean +++ b/formal/lean/sphere-eversion/to_mathlib/misc.lean @@ -31,8 +31,6 @@ lemma finite_of_finite_preimage {s : set β} {f : α → β} (h : (f ⁻¹' s).f (hs : s ⊆ range f) : s.finite := by { rw [← image_preimage_eq_of_subset hs], exact finite.image f h } -alias ord_interval.interval_oc_subset ← ord_connected.interval_oc_subset -- waiting for https://github.com/leanprover-community/mathlib/pull/15627 - end set diff --git a/formal/lean/sphere-eversion/to_mathlib/partition.lean b/formal/lean/sphere-eversion/to_mathlib/partition.lean index 4d74cdb6e0daf19dc3eb41c6333d0687bc22e78d..cf07bb7a1a50ab8c03afb4368ac8b995c514fc92 100644 --- a/formal/lean/sphere-eversion/to_mathlib/partition.lean +++ b/formal/lean/sphere-eversion/to_mathlib/partition.lean @@ -129,9 +129,9 @@ end section variables - {𝕜 : Type*} [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] - {F : Type*} [normed_group F] [normed_space 𝕜 F] + {𝕜 : Type*} [nontrivially_normed_field 𝕜] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] + {F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F] lemma cont_diff_within_at_finsum {ι : Type*} {f : ι → E → F} (lf : locally_finite (λ i, support $ f i)) {n : with_top ℕ} {s : set E} {x₀ : E} @@ -151,11 +151,10 @@ end section variables - {ι : Type*} {E : Type*} [normed_group E] [normed_space ℝ E] + {ι : Type*} {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] {H : Type*} [topological_space H] {I : model_with_corners ℝ E H} {M : Type*} [topological_space M] [charted_space H M] - {s : set M} {F : Type*} [normed_group F] [normed_space ℝ F] - + {s : set M} {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] lemma cont_mdiff_within_at_of_not_mem {f : M → F} {x : M} (hx : x ∉ tsupport f) (n : with_top ℕ) (s : set M) : @@ -165,28 +164,10 @@ lemma cont_mdiff_within_at_of_not_mem {f : M → F} {x : M} (hx : x ∉ tsupport (eventually_nhds_within_of_eventually_nhds $ not_mem_tsupport_iff_eventually_eq.mp hx) (image_eq_zero_of_nmem_tsupport hx) - lemma cont_mdiff_at_of_not_mem {f : M → F} {x : M} (hx : x ∉ tsupport f) (n : with_top ℕ) : cont_mdiff_at I 𝓘(ℝ, F) n f x := cont_mdiff_within_at_of_not_mem hx n univ -@[simp] -lemma local_equiv.refl_prod_refl {α β : Type*} : - (local_equiv.refl α).prod (local_equiv.refl β) = local_equiv.refl (α × β) := -by { ext1 ⟨x, y⟩, { refl }, { rintro ⟨x, y⟩, refl }, exact univ_prod_univ } - -@[simp] -lemma local_homeomorph.refl_prod_refl {α β : Type*} [topological_space α] [topological_space β] : - (local_homeomorph.refl α).prod (local_homeomorph.refl β) = local_homeomorph.refl (α × β) := -by { ext1 ⟨x, y⟩, { refl }, { rintro ⟨x, y⟩, refl }, exact univ_prod_univ } - -attribute [ext] model_with_corners charted_space -lemma model_with_corners_self_prod : 𝓘(ℝ, E × F) = 𝓘(ℝ, E).prod 𝓘(ℝ, F) := -by { ext1, simp } - -lemma charted_space_self_prod : prod_charted_space E E F F = charted_space_self (E × F) := -by { ext1, simp [prod_charted_space, atlas], ext1, simp, } - lemma cont_mdiff_within_at.sum {ι : Type*} {f : ι → M → F} {J : finset ι} {n : with_top ℕ} {s : set M} {x₀ : M} (h : ∀ i ∈ J, cont_mdiff_within_at I 𝓘(ℝ, F) n (f i) s x₀) : @@ -240,19 +221,19 @@ end end variables - {E : Type*} [normed_group E] [normed_space ℝ E] [finite_dimensional ℝ E] - {F : Type*} [normed_group F] [normed_space ℝ F] + {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [finite_dimensional ℝ E] + {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] -- Not used here, but should be in mathlib -lemma has_fderiv_at_of_not_mem (𝕜 : Type*) {E : Type*} {F : Type*} [nondiscrete_normed_field 𝕜] - [normed_group E] [normed_space 𝕜 E] [normed_group F] [normed_space 𝕜 F] +lemma has_fderiv_at_of_not_mem (𝕜 : Type*) {E : Type*} {F : Type*} [nontrivially_normed_field 𝕜] + [normed_add_comm_group E] [normed_space 𝕜 E] [normed_add_comm_group F] [normed_space 𝕜 F] {f : E → F} {x} (hx : x ∉ tsupport f) : has_fderiv_at f (0 : E →L[𝕜] F) x := (has_fderiv_at_const (0 : F) x).congr_of_eventually_eq (not_mem_tsupport_iff_eventually_eq.mp hx) -- Not used here, but should be in mathlib -lemma cont_diff_at_of_not_mem (𝕜 : Type*) {E : Type*} {F : Type*} [nondiscrete_normed_field 𝕜] - [normed_group E] [normed_space 𝕜 E] [normed_group F] [normed_space 𝕜 F] +lemma cont_diff_at_of_not_mem (𝕜 : Type*) {E : Type*} {F : Type*} [nontrivially_normed_field 𝕜] + [normed_add_comm_group E] [normed_space 𝕜 E] [normed_add_comm_group F] [normed_space 𝕜 F] {f : E → F} {x} (hx : x ∉ tsupport f) (n : with_top ℕ) : cont_diff_at 𝕜 n f x := (cont_diff_at_const : cont_diff_at 𝕜 n (λ x, (0 : F)) x).congr_of_eventually_eq (not_mem_tsupport_iff_eventually_eq.mp hx) diff --git a/formal/lean/sphere-eversion/to_mathlib/partition2.lean b/formal/lean/sphere-eversion/to_mathlib/partition2.lean index 765e216be0d7b1768eecae8da24d1ce20c83f4f0..a50ba7b0ac7cbcd57bfd998f53ae70afd979d200 100644 --- a/formal/lean/sphere-eversion/to_mathlib/partition2.lean +++ b/formal/lean/sphere-eversion/to_mathlib/partition2.lean @@ -1,4 +1,5 @@ import to_mathlib.partition +import to_mathlib.geometry.manifold.misc_manifold noncomputable theory @@ -21,13 +22,13 @@ begin end -- See: https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Typeclass.20resolution.20under.20binders/near/281296989 -instance real.normed_space.to_module (F : Type*) [normed_group F] [normed_space ℝ F] : module ℝ F := +instance real.normed_space.to_module (F : Type*) [normed_add_comm_group F] [normed_space ℝ F] : module ℝ F := by apply_instance variables {E₁ E₂ F : Type*} -variables [normed_group E₁] [normed_space ℝ E₁] [finite_dimensional ℝ E₁] -variables [normed_group E₂] [normed_space ℝ E₂] [finite_dimensional ℝ E₂] -variables [normed_group F] [normed_space ℝ F] +variables [normed_add_comm_group E₁] [normed_space ℝ E₁] [finite_dimensional ℝ E₁] +variables [normed_add_comm_group E₂] [normed_space ℝ E₂] [finite_dimensional ℝ E₂] +variables [normed_add_comm_group F] [normed_space ℝ F] variables {H₁ M₁ H₂ M₂ : Type*} variables [topological_space H₁] (I₁ : model_with_corners ℝ E₁ H₁) diff --git a/formal/lean/sphere-eversion/to_mathlib/smooth_barycentric.lean b/formal/lean/sphere-eversion/to_mathlib/smooth_barycentric.lean index e763160eb6387d7a1b9ad77a378b7cdf789c9c2d..916af429a4b54006d65edf21ab7bd749e0b8dc8a 100644 --- a/formal/lean/sphere-eversion/to_mathlib/smooth_barycentric.lean +++ b/formal/lean/sphere-eversion/to_mathlib/smooth_barycentric.lean @@ -80,9 +80,9 @@ end barycentric_det namespace matrix -variables (ι k : Type*) [fintype ι] [decidable_eq ι] [nondiscrete_normed_field k] +variables (ι k : Type*) [fintype ι] [decidable_eq ι] [nontrivially_normed_field k] -attribute [instance] matrix.normed_group matrix.normed_space +attribute [instance] matrix.normed_add_comm_group matrix.normed_space -- This should really be deduced from general results about continuous multilinear maps. lemma smooth_det (m : with_top ℕ) : @@ -115,8 +115,8 @@ end matrix section smooth_barycentric variables (ι 𝕜 F : Type*) -variables [fintype ι] [decidable_eq ι] [nondiscrete_normed_field 𝕜] [complete_space 𝕜] -variables [normed_group F] [normed_space 𝕜 F] +variables [fintype ι] [decidable_eq ι] [nontrivially_normed_field 𝕜] [complete_space 𝕜] +variables [normed_add_comm_group F] [normed_space 𝕜 F] -- An alternative approach would be to prove the affine version of `cont_diff_at_map_inverse` -- and prove that barycentric coordinates give a continuous affine equivalence to diff --git a/formal/lean/sphere-eversion/to_mathlib/topology/algebra/module.lean b/formal/lean/sphere-eversion/to_mathlib/topology/algebra/module.lean new file mode 100644 index 0000000000000000000000000000000000000000..c7953542e65a327c9a96c8118d8300ba8ca50b98 --- /dev/null +++ b/formal/lean/sphere-eversion/to_mathlib/topology/algebra/module.lean @@ -0,0 +1,44 @@ +import topology.algebra.module.basic + +open filter continuous_linear_map +open_locale topological_space big_operators filter + +variables {R₁ : Type*} {R₂ : Type*} {R₃ : Type*} [semiring R₁] [semiring R₂] [semiring R₃] +{σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [ring_hom_inv_pair σ₁₂ σ₂₁] [ring_hom_inv_pair σ₂₁ σ₁₂] +{σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [ring_hom_inv_pair σ₂₃ σ₃₂] [ring_hom_inv_pair σ₃₂ σ₂₃] +{σ₁₃ : R₁ →+* R₃} {σ₃₁ : R₃ →+* R₁} [ring_hom_inv_pair σ₁₃ σ₃₁] [ring_hom_inv_pair σ₃₁ σ₁₃] +[ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] [ring_hom_comp_triple σ₃₂ σ₂₁ σ₃₁] +{M₁ : Type*} [topological_space M₁] [add_comm_monoid M₁] +{M'₁ : Type*} [topological_space M'₁] [add_comm_monoid M'₁] +{M₂ : Type*} [topological_space M₂] [add_comm_monoid M₂] +{M₃ : Type*} [topological_space M₃] [add_comm_monoid M₃] +{M₄ : Type*} [topological_space M₄] [add_comm_monoid M₄] +[module R₁ M₁] [module R₁ M'₁] [module R₂ M₂] [module R₃ M₃] + +namespace continuous_linear_equiv + +include σ₂₁ σ₁₃ +@[simp] theorem cancel_right {f f' : M₂ →SL[σ₂₃] M₃} {e : M₁ ≃SL[σ₁₂] M₂} : + f.comp e.to_continuous_linear_map = f'.comp e.to_continuous_linear_map ↔ f = f' := +begin + split, + { simp_rw [continuous_linear_map.ext_iff, continuous_linear_map.comp_apply, coe_def_rev, coe_coe], + intros h v, rw [← e.apply_symm_apply v, h] }, + { rintro rfl, refl } +end + +omit σ₂₁ +include σ₃₂ + +@[simp] theorem cancel_left {e : M₂ ≃SL[σ₂₃] M₃} {f f' : M₁ →SL[σ₁₂] M₂} : + e.to_continuous_linear_map.comp f = e.to_continuous_linear_map.comp f' ↔ f = f' := +begin + split, + { simp_rw [continuous_linear_map.ext_iff, continuous_linear_map.comp_apply, coe_def_rev, coe_coe], + intros h v, rw [← e.symm_apply_apply (f v), h, e.symm_apply_apply] }, + { rintro rfl, refl } +end + +omit σ₃₂ + +end continuous_linear_equiv diff --git a/formal/lean/sphere-eversion/to_mathlib/topology/bases.lean b/formal/lean/sphere-eversion/to_mathlib/topology/bases.lean deleted file mode 100644 index d9903c308fe8a9db9bdfad6bb286066014ec6783..0000000000000000000000000000000000000000 --- a/formal/lean/sphere-eversion/to_mathlib/topology/bases.lean +++ /dev/null @@ -1,13 +0,0 @@ -import topology.bases - -open_locale filter topological_space - -instance topological_space.prod.first_countable_topology (α : Type*) [t : topological_space α] - {β : Type*} [topological_space β] - [topological_space.first_countable_topology α] [topological_space.first_countable_topology β] : - topological_space.first_countable_topology (α × β) := -⟨begin - rintros ⟨a, b⟩, - rw nhds_prod_eq, - apply_instance -end⟩ diff --git a/formal/lean/sphere-eversion/to_mathlib/topology/local_homeomorph.lean b/formal/lean/sphere-eversion/to_mathlib/topology/local_homeomorph.lean new file mode 100644 index 0000000000000000000000000000000000000000..48700ae8253b1e397afcca36800cbfd737635001 --- /dev/null +++ b/formal/lean/sphere-eversion/to_mathlib/topology/local_homeomorph.lean @@ -0,0 +1,18 @@ +import topology.local_homeomorph + +variables {α β : Type*} [topological_space α] [topological_space β] (e : local_homeomorph α β) + +namespace local_homeomorph + +-- TODO When migrating this, strengthen Mathlib's `local_homeomorph.image_open_of_open` +lemma is_open_symm_image_iff_of_subset_target {s : set β} (hs : s ⊆ e.target) : + is_open (e.symm '' s) ↔ is_open s := +begin + refine ⟨λ h, _, λ h, e.symm.image_open_of_open h hs⟩, + have hs' : e.symm '' s ⊆ e.source, + { rw e.symm_image_eq_source_inter_preimage hs, apply set.inter_subset_left, }, + rw ← e.to_local_equiv.image_symm_image_of_subset_target hs, + exact e.image_open_of_open h hs', +end + +end local_homeomorph diff --git a/formal/lean/sphere-eversion/to_mathlib/topology/metric_space.lean b/formal/lean/sphere-eversion/to_mathlib/topology/metric_space.lean index 2f6d23f25f5951f60e2bf16a0f91b38c64f94aef..44d32b29bf5fde74b81c4fba45bc76be9526b8c6 100644 --- a/formal/lean/sphere-eversion/to_mathlib/topology/metric_space.lean +++ b/formal/lean/sphere-eversion/to_mathlib/topology/metric_space.lean @@ -17,15 +17,6 @@ section lipschitz variables {α β γ : Type*} [pseudo_emetric_space α] [pseudo_emetric_space β] [pseudo_emetric_space γ] -protected lemma lipschitz_on_with.comp {Kf Kg : ℝ≥0} {f : β → γ} {g : α → β} {s : set α} {t : set β} - (hf : lipschitz_on_with Kf f t) (hg : lipschitz_on_with Kg g s) (hst : g '' s ⊆ t) : - lipschitz_on_with (Kf * Kg) (f ∘ g) s := -assume x x_in y y_in, -calc edist (f (g x)) (f (g y)) - ≤ Kf * edist (g x) (g y) : hf (hst $ mem_image_of_mem g x_in) (hst $ mem_image_of_mem g y_in) -... ≤ Kf * (Kg * edist x y) : ennreal.mul_left_mono (hg x_in y_in) -... = (Kf * Kg : ℝ≥0) * edist x y : by rw [← mul_assoc, ennreal.coe_mul] - lemma lipschitz_with_prod_mk_left (a : α) : lipschitz_with 1 (prod.mk a : β → α × β) := λ x y, show max _ _ ≤ _, by simp diff --git a/formal/lean/sphere-eversion/to_mathlib/topology/misc.lean b/formal/lean/sphere-eversion/to_mathlib/topology/misc.lean index f906b9226cb855762a6d4d437a29fe382460af87..302d78d0657717618ace2bdaf8e3375f6a4d2928 100644 --- a/formal/lean/sphere-eversion/to_mathlib/topology/misc.lean +++ b/formal/lean/sphere-eversion/to_mathlib/topology/misc.lean @@ -30,7 +30,7 @@ end to_specific_limits section -lemma support_norm {α E : Type*} [normed_group E] (f : α → E) : support (λ a, ∥f a∥) = support f := +lemma support_norm {α E : Type*} [normed_add_comm_group E] (f : α → E) : support (λ a, ∥f a∥) = support f := function.support_comp_eq norm (λ x, norm_eq_zero) f @[to_additive] @@ -208,7 +208,7 @@ end fract section -- to normed_space -variables {E F : Type*} [normed_group E] [normed_group F] +variables {E F : Type*} [normed_add_comm_group E] [normed_add_comm_group F] variables [normed_space ℝ E] [normed_space ℝ F] lemma dist_smul_add_one_sub_smul_le {r : ℝ} {x y : E} (h : r ∈ unit_interval) : @@ -420,8 +420,8 @@ needs import linear_algebra.affine_space.independent import analysis.normed_space.finite_dimension -/ -lemma is_open_affine_independent (𝕜 E : Type*) {ι : Type*} [nondiscrete_normed_field 𝕜] - [normed_group E] [normed_space 𝕜 E] [complete_space 𝕜] [fintype ι] : +lemma is_open_affine_independent (𝕜 E : Type*) {ι : Type*} [nontrivially_normed_field 𝕜] + [normed_add_comm_group E] [normed_space 𝕜 E] [complete_space 𝕜] [fintype ι] : is_open {p : ι → E | affine_independent 𝕜 p} := begin classical, @@ -438,33 +438,6 @@ end end -section -/-- A locally connected space is a space where every neighborhood filter has a basis of open - connected sets. -/ -class locally_connected_space (α : Type*) [topological_space α] : Prop := -(has_basis : ∀ x, (𝓝 x).has_basis (λ s : set α, is_open s ∧ x ∈ s ∧ is_connected s) id) - --- class locally_connected_space (α : Type*) [topological_space α] : Prop := --- (exists_connected_neihborhoods : ∃ (b : set (set α)), is_topological_basis b ∧ --- ∀ s ∈ b, is_connected s) - -variables {α : Type*} [topological_space α] - -lemma locally_connected_space_of_connected_subsets - (h : ∀ (x : α) (U ∈ 𝓝 x), ∃ V ⊆ U, is_open V ∧ x ∈ V ∧ is_connected V) : - locally_connected_space α := -begin - constructor, - intro x, - constructor, - intro t, - split, - { intro ht, obtain ⟨V, hVU, hV⟩ := h x t ht, exact ⟨V, hV, hVU⟩ }, - { rintro ⟨V, ⟨hV, hxV, -⟩, hVU⟩, refine mem_nhds_iff.mpr ⟨V, hVU, hV, hxV⟩ } -end - -end - section convex variables {E : Type*} [add_comm_group E] [module ℝ E] [topological_space E] @@ -489,7 +462,7 @@ end section normed_space open metric -variables {E : Type*} [normed_group E] [normed_space ℝ E] +variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] lemma is_preconnected_ball (x : E) (r : ℝ) : is_preconnected (ball x r) := (convex_ball x r).is_preconnected' @@ -500,96 +473,32 @@ begin refine ⟨λ h, h.nonempty, λ h, ((convex_ball x r).is_path_connected $ h).is_connected⟩ end --- make metric.mem_nhds_iff protected -instance normed_space.locally_connected_space : locally_connected_space E := -begin - apply locally_connected_space_of_connected_subsets, - intros x U hU, - obtain ⟨ε, hε, hU⟩ := metric.mem_nhds_iff.mp hU, - refine ⟨_, hU, is_open_ball, mem_ball_self hε, is_connected_ball.mpr hε⟩ -end +-- todo: make metric.mem_nhds_iff protected end normed_space --- TODO: replace mathlib's `connected_component_in`, which is never used, by the following. - -section connected_comp_in - -variables {α : Type*} [topological_space α] {F s : set α} {x y : α} +section connected_component_in -/-- Given a set `F` in a topological space `α` and a point `x : α`, the connected -component of `x` in `F` is the connected component of `x` in the subtype `F` seen as -a set in `α`. This definition does not make sense if `x` is not in `F` so we return the -empty set in this case. -/ -def connected_comp_in (F : set α) (x : α) : set α := -if h : x ∈ F then coe '' (connected_component (⟨x, h⟩ : F)) else ∅ +variables {α β : Type*} [topological_space α] [topological_space β] -lemma connected_comp_in_subset (F : set α) (x : α) : - connected_comp_in F x ⊆ F := -by { rw [connected_comp_in], split_ifs; simp } +lemma continuous.image_connected_component_in_subset {f : α → β} {s : set α} {x : α} + (hf : continuous f) (hx : x ∈ s) : + f '' connected_component_in s x ⊆ connected_component_in (f '' s) (f x) := +(is_preconnected_connected_component_in.image _ hf.continuous_on).subset_connected_component_in + (mem_image_of_mem _ $ mem_connected_component_in hx) + (image_subset _ $ connected_component_in_subset _ _) -lemma mem_connected_comp_in_self (h : x ∈ F) : x ∈ connected_comp_in F x := -by simp [connected_comp_in, mem_connected_component, h] - -lemma connected_comp_in_nonempty_iff : (connected_comp_in F x).nonempty ↔ x ∈ F := -by { rw [connected_comp_in], split_ifs; simp [is_connected_connected_component.nonempty, h] } - -lemma is_preconnected.subset_connected_comp_in (hs : is_preconnected s) (hxs : x ∈ s) - (hsF : s ⊆ F) : s ⊆ connected_comp_in F x := -begin - have : is_preconnected ((coe : F → α) ⁻¹' s), - { refine embedding_subtype_coe.to_inducing.is_preconnected_image.mp _, - rwa [subtype.image_preimage_coe, inter_eq_left_iff_subset.mpr hsF] }, - have h2xs : (⟨x, hsF hxs⟩ : F) ∈ coe ⁻¹' s := by { rw [mem_preimage], exact hxs }, - have := this.subset_connected_component h2xs, - rw [connected_comp_in, dif_pos (hsF hxs)], - refine subset.trans _ (image_subset _ this), - rw [subtype.image_preimage_coe, inter_eq_left_iff_subset.mpr hsF] -end - -lemma is_preconnected_connected_comp_in : is_preconnected (connected_comp_in F x) := -begin - rw [connected_comp_in], split_ifs, - { refine embedding_subtype_coe.to_inducing.is_preconnected_image.mpr - is_preconnected_connected_component }, - { exact is_preconnected_empty }, -end - -lemma is_connected_connected_comp_in : is_connected (connected_comp_in F x) ↔ x ∈ F := -by simp_rw [← connected_comp_in_nonempty_iff, is_connected, is_preconnected_connected_comp_in, - and_true] - -lemma is_preconnected.connected_comp_in (h : is_preconnected F) (hx : x ∈ F) : - connected_comp_in F x = F := -(connected_comp_in_subset F x).antisymm (h.subset_connected_comp_in hx subset_rfl) - -lemma connected_comp_in_eq (h : y ∈ connected_comp_in F x) : - connected_comp_in F x = connected_comp_in F y := -begin - have hx : x ∈ F := connected_comp_in_nonempty_iff.mp ⟨y, h⟩, - simp_rw [connected_comp_in, dif_pos hx] at h ⊢, - obtain ⟨⟨y, hy⟩, h2y, rfl⟩ := h, - simp_rw [subtype.coe_mk, dif_pos hy, connected_component_eq h2y] -end - -lemma connected_comp_in_mem_nhds [locally_connected_space α] (h : F ∈ 𝓝 x) : - connected_comp_in F x ∈ 𝓝 x := +lemma homeomorph.image_connected_component_in (f : α ≃ₜ β) {s : set α} {x : α} (hx : x ∈ s) : + f '' connected_component_in s x = connected_component_in (f '' s) (f x) := begin - rw (locally_connected_space.has_basis x).mem_iff at h, - obtain ⟨s, ⟨h1s, hxs, h2s⟩, hsF⟩ := h, - exact mem_nhds_iff.mpr ⟨s, h2s.is_preconnected.subset_connected_comp_in hxs hsF, h1s, hxs⟩ + refine (f.continuous.image_connected_component_in_subset hx).antisymm _, + have := f.symm.continuous.image_connected_component_in_subset (mem_image_of_mem _ hx), + rwa [image_subset_iff, f.preimage_symm, f.image_symm, f.preimage_image, f.symm_apply_apply] + at this, end -lemma is_open.connected_comp_in [locally_connected_space α] {F : set α} {x : α} (hF : is_open F) : - is_open (connected_comp_in F x) := -begin - rw [is_open_iff_mem_nhds], - intros y hy, - rw [connected_comp_in_eq hy], - exact connected_comp_in_mem_nhds (is_open_iff_mem_nhds.mp hF y $ connected_comp_in_subset F x hy), -end -end connected_comp_in +end connected_component_in namespace topological_space -- to topology.bases lemma cover_nat_nhds_within {α} [topological_space α] [second_countable_topology α] {f : α → set α} diff --git a/formal/lean/sphere-eversion/to_mathlib/topology/nhds_set.lean b/formal/lean/sphere-eversion/to_mathlib/topology/nhds_set.lean index e498a6fa74be89e7369dfa46b75e93ce0f47d0b5..d858ca49d2577b214e7f4530e0f0a3a3f98e4745 100644 --- a/formal/lean/sphere-eversion/to_mathlib/topology/nhds_set.lean +++ b/formal/lean/sphere-eversion/to_mathlib/topology/nhds_set.lean @@ -5,13 +5,9 @@ variables {α : Type*} [topological_space α] {s t s₁ s₂ t₁ t₂ : set α} open filter open_locale filter topological_space -lemma principal_le_nhds_set (s : set α) : 𝓟 s ≤ 𝓝ˢ s := -λ u u_in x x_in, mem_of_mem_nhds (mem_nhds_set_iff_forall.mp u_in x x_in) - - lemma is_open.nhds_set_eq_principal {s : set α} (h : is_open s) : 𝓝ˢ s = 𝓟 s := begin - apply le_antisymm _ (principal_le_nhds_set s), + apply le_antisymm _ principal_le_nhds_set, rw [filter.le_principal_iff, h.mem_nhds_set] end @@ -48,4 +44,4 @@ lemma filter.eventually.eventually_nhds_set {p : α → Prop} (h : ∀ᶠ y in eventually_nhds_set_iff.mpr (λ x x_in, (eventually_nhds_set_iff.mp h x x_in).eventually_nhds) lemma filter.eventually.on_set {p : α → Prop} (h : ∀ᶠ y in 𝓝ˢ s, p y) : ∀ x ∈ s, p x := -eventually_principal.mp $ eventually.filter_mono (principal_le_nhds_set s) h +eventually_principal.mp $ eventually.filter_mono principal_le_nhds_set h diff --git a/formal/lean/sphere-eversion/to_mathlib/topology/paracompact.lean b/formal/lean/sphere-eversion/to_mathlib/topology/paracompact.lean index 34ee41807c8b834962ec28e323ebee62a200cf1e..c6ab2e29a66ba91de889d3b439564d5e85f957d1 100644 --- a/formal/lean/sphere-eversion/to_mathlib/topology/paracompact.lean +++ b/formal/lean/sphere-eversion/to_mathlib/topology/paracompact.lean @@ -12,18 +12,17 @@ open set function worth it. Proof partly based on `refinement_of_locally_compact_sigma_compact_of_nhds_basis_set`. -/ lemma exists_countable_locally_finite_cover {ι X : Type*} [topological_space X] [t2_space X] [locally_compact_space X] [sigma_compact_space X] - {c : ι → X} {B : ι → ℝ → set X} {p : ι → ℝ → Prop} + {c : ι → X} {W : ι → ℝ → set X} {B : ι → ℝ → set X} {p : ι → ℝ → Prop} (hc : surjective c) - (hp : ∀ i r, p i r → 0 < r) - (hp' : ∀ i r r', 0 < r → r < r' → p i r' → p i r) - (hB₀ : ∀ i r, p i r → is_open (B i r)) - (hB₁ : ∀ i r, p i r → c i ∈ B i r) - (hB₂ : ∀ i, (𝓝 (c i)).has_basis (p i) (B i)) : + (hp : ∀ i r, p i r → 0 < r) + (hW₀ : ∀ i r, p i r → c i ∈ W i r) + (hW₁ : ∀ i r, p i r → is_open (W i r)) + (hB : ∀ i, (𝓝 (c i)).has_basis (p i) (B i)) : ∃ (s : set (ι × ℝ)), s.countable ∧ (∀ z ∈ s, ↿p z) ∧ - locally_finite (↿B ∘ (coe : s → ι × ℝ)) ∧ - (⋃ z ∈ s, B (z : ι × ℝ).fst ((z : ι × ℝ).snd / 2)) = univ := + (⋃ z ∈ s, ↿W z) = univ ∧ + locally_finite (↿B ∘ (coe : s → ι × ℝ)) := begin let K' := compact_exhaustion.choice X, let K := K'.shiftr.shiftr, @@ -51,28 +50,32 @@ begin (set.diff_eq_empty.mpr (K.subset hab)), }, have hU'' : ∀ n x, x ∈ C n → U n ∈ 𝓝 x := λ n x hx, mem_nhds_iff.mpr ⟨U n, subset.rfl, hU n, hCU n hx⟩, - have : ∀ n (x : C n), ∃ i r, ↑x ∈ B i (r/2) ∧ B i r ⊆ U n ∧ p i r, + have : ∀ n (x : C n), ∃ i r, ↑x ∈ W i r ∧ B i r ⊆ U n ∧ p i r, { rintros n ⟨x, hx⟩, obtain ⟨i, rfl⟩ := hc x, - obtain ⟨r, hr₁, hr₂⟩ := (hB₂ i).mem_iff.mp (hU'' n _ hx), - have hr₃ : 0 < r := hp i r hr₁, - exact ⟨i, r, hB₁ i _ (hp' i (r/2) r (half_pos hr₃) (half_lt_self hr₃) hr₁), hr₂, hr₁⟩, }, + obtain ⟨r, hr₁, hr₂⟩ := (hB i).mem_iff.mp (hU'' n _ hx), + exact ⟨i, r, hW₀ i r hr₁, hr₂, hr₁⟩, }, choose i r h₁ h₂ h₃ using λ n, this n, have hr : ∀ n (x : C n), 0 < r n x := λ n x, hp _ _ (h₃ n x), - have hpr : ∀ n (x : C n), p (i n x) (r n x / 2) := - λ n x, hp' (i n x) _ _ (half_pos (hr n x)) (half_lt_self (hr n x)) (h₃ n x), - let V : Π n, C n → set X := λ n x, B (i n x) (r n x / 2), - have hV₁ : ∀ n x, is_open (V n x) := λ n x, hB₀ _ _ (hpr n x), + let V : Π n, C n → set X := λ n x, W (i n x) (r n x), + have hV₁ : ∀ n x, is_open (V n x) := λ n x, hW₁ _ _ (h₃ n x), have hV₂ : ∀ n, C n ⊆ ⋃ (x : C n), V n x := λ n x hx, mem_Union.mpr ⟨⟨x, hx⟩, h₁ _ _⟩, choose f hf using λ n, (hC n).elim_finite_subcover (V n) (hV₁ n) (hV₂ n), classical, let s : set (ι × ℝ) := ⋃ n, (f n).image (pi.prod (i n) (r n)), - refine ⟨s, countable_Union (λ n, finset.countable_to_set _), λ z hz, _, λ x, _, - set.univ_subset_iff.mp (λ x hx, _)⟩, + refine ⟨s, countable_Union (λ n, finset.countable_to_set _), λ z hz, _, + set.univ_subset_iff.mp (λ x hx, _), λ x, _⟩, { simp only [pi.prod, mem_Union, finset.coe_image, mem_image, finset.mem_coe, set_coe.exists] at hz, obtain ⟨n, x, hx, -, rfl⟩ := hz, apply h₃, }, + { obtain ⟨n, hn⟩ := Union_eq_univ_iff.mp hC' x, + specialize hf n hn, + simp only [Union_coe_set, mem_Union, exists_prop] at hf, + obtain ⟨y, hy₁, hy₂, hy₃⟩ := hf, + simp only [pi.prod, mem_Union, finset.mem_coe, finset.mem_image, exists_prop, set_coe.exists, + Union_exists, exists_and_distrib_right, prod.exists, prod.mk.inj_iff], + exact ⟨i n ⟨y, hy₁⟩, r n ⟨y, hy₁⟩, ⟨n, y, hy₁, hy₂, rfl, rfl⟩, hy₃⟩, }, { obtain ⟨n, hn⟩ := Union_eq_univ_iff.mp hC' x, refine ⟨U n, hU'' n x hn, _⟩, let P : ι × ℝ → Prop := λ z, (↿B (z : ι × ℝ) ∩ U n).nonempty, @@ -87,11 +90,4 @@ begin rw set.not_nonempty_iff_eq_empty, have := set.inter_subset_inter_left (U n) (h₂ m ⟨x, hx₁⟩), rwa [set.not_nonempty_iff_eq_empty.mp hm, set.subset_empty_iff] at this, }, - { obtain ⟨n, hn⟩ := Union_eq_univ_iff.mp hC' x, - specialize hf n hn, - simp only [Union_coe_set, mem_Union, exists_prop] at hf, - obtain ⟨y, hy₁, hy₂, hy₃⟩ := hf, - simp only [pi.prod, mem_Union, finset.mem_coe, finset.mem_image, exists_prop, set_coe.exists, - Union_exists, exists_and_distrib_right, prod.exists, prod.mk.inj_iff], - exact ⟨i n ⟨y, hy₁⟩, r n ⟨y, hy₁⟩, ⟨n, y, hy₁, hy₂, rfl, rfl⟩, hy₃⟩, }, end diff --git a/formal/lean/sphere-eversion/to_mathlib/topology/periodic.lean b/formal/lean/sphere-eversion/to_mathlib/topology/periodic.lean index 271d6077f9fd189bf41de596755f93c015666b8a..6fe765266dd5b7351ca80e19298c81a289b255a0 100644 --- a/formal/lean/sphere-eversion/to_mathlib/topology/periodic.lean +++ b/formal/lean/sphere-eversion/to_mathlib/topology/periodic.lean @@ -150,7 +150,7 @@ begin exact is_closed.preimage (continuous_snd.sub continuous_fst) is_closed_int end -variables {X E : Type*} [topological_space X] [normed_group E] [t2_space X] +variables {X E : Type*} [topological_space X] [normed_add_comm_group E] [t2_space X] lemma continuous.bounded_on_compact_of_one_periodic {f : X → ℝ → E} (cont : continuous ↿f) (hper : ∀ x, one_periodic (f x)) {K : set X} (hK : is_compact K) : diff --git a/formal/lean/sphere-eversion/to_mathlib/unused/cont_mdiff.lean b/formal/lean/sphere-eversion/to_mathlib/unused/cont_mdiff.lean index c51733c49beb01539955afeb3a7c314cfb1af602..58e4be91e1c76d071b11006dcf3036e28b1be350 100644 --- a/formal/lean/sphere-eversion/to_mathlib/unused/cont_mdiff.lean +++ b/formal/lean/sphere-eversion/to_mathlib/unused/cont_mdiff.lean @@ -33,8 +33,8 @@ open_locale topological_space filter manifold open filter topological_space function set variables - {E : Type*} [normed_group E] [normed_space ℝ E] - {F : Type*} [normed_group F] [normed_space ℝ F] + {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] + {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] lemma cont_mdiff_iff_cont_diff_on {s : opens E} {f : E → F} {n : with_top ℕ} : cont_mdiff 𝓘(ℝ, E) 𝓘(ℝ, F) n (f ∘ (coe : s → E)) ↔ cont_diff_on ℝ n f s := diff --git a/formal/lean/sphere-eversion/to_mathlib/unused/misc.lean b/formal/lean/sphere-eversion/to_mathlib/unused/misc.lean index a64ae0b62cd6cf4f9156856e207c937f9bd0eef5..b61a5064ee5968f4b4452079211de703aefaf59d 100644 --- a/formal/lean/sphere-eversion/to_mathlib/unused/misc.lean +++ b/formal/lean/sphere-eversion/to_mathlib/unused/misc.lean @@ -59,7 +59,7 @@ section -- logic.function (c : γ) : ↿f (x, c) = ↿(f x) c := rfl --- @[simp] lemma uncurry_loop_apply {F : Type*} [normed_group F] [normed_space ℝ F] +-- @[simp] lemma uncurry_loop_apply {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] -- [finite_dimensional ℝ F] {α : Type*} (f : α → loop F) (x : α) (t : ℝ) : -- ↿f (x, t) = f x t := -- rfl @@ -174,16 +174,16 @@ def continuous_linear_equiv.ulift (R M : Type*) [semiring R] [topological_space [add_comm_monoid M] [module R M] : ulift.{v} M ≃L[R] M := { ..linear_equiv.ulift R M, ..homeomorph.ulift } -lemma cont_diff_up {F X : Type*} [nondiscrete_normed_field F] [normed_group X] +lemma cont_diff_up {F X : Type*} [nontrivially_normed_field F] [normed_add_comm_group X] [normed_space F X] {n : with_top ℕ} : cont_diff F n (@up X) := (continuous_linear_equiv.ulift F X).symm.cont_diff -lemma cont_diff_down {F X : Type*} [nondiscrete_normed_field F] [normed_group X] +lemma cont_diff_down {F X : Type*} [nontrivially_normed_field F] [normed_add_comm_group X] [normed_space F X] {n : with_top ℕ} : cont_diff F n (@down X) := (continuous_linear_equiv.ulift F X).cont_diff -lemma cont_diff_up_iff {F X Y : Type*} [nondiscrete_normed_field F] [normed_group X] - [normed_space F X] [normed_group Y] [normed_space F Y] {n : with_top ℕ} (f : X → Y) : +lemma cont_diff_up_iff {F X Y : Type*} [nontrivially_normed_field F] [normed_add_comm_group X] + [normed_space F X] [normed_add_comm_group Y] [normed_space F Y] {n : with_top ℕ} (f : X → Y) : cont_diff F n (λ x, up (f x)) ↔ cont_diff F n f := (continuous_linear_equiv.ulift F Y).symm.comp_cont_diff_iff @@ -249,18 +249,18 @@ section continuous_linear -- two lemmas about continuous bilinear maps, not that useful variables {X 𝕜 E E' F : Type*} -variables [nondiscrete_normed_field 𝕜] - [normed_group E] [normed_group E'] [normed_group F] +variables [nontrivially_normed_field 𝕜] + [normed_add_comm_group E] [normed_add_comm_group E'] [normed_add_comm_group F] [normed_space 𝕜 E] [normed_space 𝕜 E'] [normed_space 𝕜 F] namespace continuous_linear_map -lemma has_fderiv_at_const_left [normed_group X] [normed_space 𝕜 X] +lemma has_fderiv_at_const_left [normed_add_comm_group X] [normed_space 𝕜 X] (L : E →L[𝕜] E' →L[𝕜] F) {f : X → E'} {f' : X →L[𝕜] E'} (x : X) {c : E} (hf : has_fderiv_at f f' x) : has_fderiv_at (λ x, L c (f x)) ((L c).comp f') x := (L c).has_fderiv_at.comp x hf -lemma has_fderiv_at_const_right [normed_group X] [normed_space 𝕜 X] +lemma has_fderiv_at_const_right [normed_add_comm_group X] [normed_space 𝕜 X] (L : E →L[𝕜] E' →L[𝕜] F) {f : X → E} {f' : X →L[𝕜] E} (x : X) {c : E'} (hf : has_fderiv_at f f' x) : has_fderiv_at (λ x, L (f x) c) ((flip L c).comp f') x := diff --git a/formal/lean/sphere-eversion/to_mathlib/unused/misc_manifold.lean b/formal/lean/sphere-eversion/to_mathlib/unused/misc_manifold.lean index 005bfdb18c5d0df241faffbdd54784bd3d748497..865d222dc1ac06d573a88368bcac010f30e41a1c 100644 --- a/formal/lean/sphere-eversion/to_mathlib/unused/misc_manifold.lean +++ b/formal/lean/sphere-eversion/to_mathlib/unused/misc_manifold.lean @@ -17,8 +17,8 @@ end local_equiv namespace topological_vector_bundle variables {R : Type*} {B : Type*} {F : Type*} {E : B → Type*} -variables [nondiscrete_normed_field R] [∀ x, add_comm_monoid (E x)] [∀ x, module R (E x)] - [normed_group F] [normed_space R F] [topological_space B] +variables [nontrivially_normed_field R] [∀ x, add_comm_monoid (E x)] [∀ x, module R (E x)] + [normed_add_comm_group F] [normed_space R F] [topological_space B] [topological_space (total_space E)] variables {HB : Type*} [topological_space HB] @@ -45,21 +45,10 @@ def total_space.to_charted_space [topological_vector_bundle R F E] [charted_spac end topological_vector_bundle -section charted_space - -variables {M H : Type*} [topological_space M] [topological_space H] [charted_space H M] - (G : structure_groupoid H) - -lemma structure_groupoid.subset_maximal_atlas [has_groupoid M G] : - atlas H M ⊆ G.maximal_atlas M := -λ e he e' he', ⟨G.compatible he he', G.compatible he' he⟩ - -end charted_space - namespace model_with_corners -variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] +variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type*} [topological_space H] {M : Type*} [topological_space M] (f : local_homeomorph M H) (I : model_with_corners 𝕜 E H) @@ -90,10 +79,10 @@ end model_with_corners namespace basic_smooth_vector_bundle_core -variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] - {E' : Type*} [normed_group E'] [normed_space 𝕜 E'] - {F : Type*} [normed_group F] [normed_space 𝕜 F] +variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] + {E' : Type*} [normed_add_comm_group E'] [normed_space 𝕜 E'] + {F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F] {H : Type*} [topological_space H] {I : model_with_corners 𝕜 E H} {H' : Type*} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {G : Type*} [topological_space G] {J : model_with_corners 𝕜 F G} @@ -124,22 +113,16 @@ end basic_smooth_vector_bundle_core section smooth_manifold_with_corners open smooth_manifold_with_corners -variables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] - {E : Type*} [normed_group E] [normed_space 𝕜 E] - {E' : Type*} [normed_group E'] [normed_space 𝕜 E'] +variables {𝕜 : Type*} [nontrivially_normed_field 𝕜] + {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E] + {E' : Type*} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H : Type*} [topological_space H] {I : model_with_corners 𝕜 E H} {H' : Type*} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M : Type*} [topological_space M] [charted_space H M] {M' : Type*} [topological_space M'] [charted_space H' M'] variables {f : M → M'} {m n : with_top ℕ} {s : set M} {x : M} -lemma smooth_manifold_with_corners.subset_maximal_atlas [smooth_manifold_with_corners I M] : - atlas H M ⊆ maximal_atlas I M := -structure_groupoid.subset_maximal_atlas _ - -variables (I) - -variables {I} [smooth_manifold_with_corners I M] [smooth_manifold_with_corners I' M'] +variables [smooth_manifold_with_corners I M] [smooth_manifold_with_corners I' M'] /-- One can reformulate smoothness within a set at a point as continuity within this set at this point, and smoothness in any chart containing that point. -/ @@ -167,55 +150,18 @@ everywhere, which gives some nice properties. variables {F G F' : Type*} -variables [normed_group F] [normed_group G] [normed_group F'] +variables [normed_add_comm_group F] [normed_add_comm_group G] [normed_add_comm_group F'] variables [normed_space 𝕜 F] [normed_space 𝕜 G] [normed_space 𝕜 F'] -lemma cont_diff_within_at.comp_cont_mdiff_within_at {g : F → G} {f : M → F} {s : set M} {t : set F} - {x : M} - (hg : cont_diff_within_at 𝕜 n g t (f x)) - (hf : cont_mdiff_within_at I 𝓘(𝕜, F) n f s x) (h : s ⊆ f ⁻¹' t) : - cont_mdiff_within_at I 𝓘(𝕜, G) n (g ∘ f) s x := -begin - rw cont_mdiff_within_at_iff at *, - refine ⟨hg.continuous_within_at.comp hf.1 h, _⟩, - rw [← (ext_chart_at I x).left_inv (mem_ext_chart_source I x)] at hg, - apply cont_diff_within_at.comp _ (by exact hg) hf.2 _, - exact (inter_subset_left _ _).trans (preimage_mono h) -end - -lemma cont_diff_at.comp_cont_mdiff_at {g : F → G} {f : M → F} {x : M} - (hg : cont_diff_at 𝕜 n g (f x)) (hf : cont_mdiff_at I 𝓘(𝕜, F) n f x) : - cont_mdiff_at I 𝓘(𝕜, G) n (g ∘ f) x := -hg.comp_cont_mdiff_within_at hf subset.rfl - -lemma cont_diff.comp_cont_mdiff {g : F → G} {f : M → F} - (hg : cont_diff 𝕜 n g) (hf : cont_mdiff I 𝓘(𝕜, F) n f) : - cont_mdiff I 𝓘(𝕜, G) n (g ∘ f) := -λ x, hg.cont_diff_at.comp_cont_mdiff_at (hf x) - -- lemma cont_mdiff_within_at.clm_comp {g : M → F →L[𝕜] G} {f : M → E →L[𝕜] F} {s : set M} {x : M} -- (hg : cont_mdiff_within_at I 𝓘(𝕜, F →L[𝕜] G) n g s x) -- (hf : cont_mdiff_within_at I 𝓘(𝕜, E →L[𝕜] F) n f s x) : -- cont_mdiff_within_at I 𝓘(𝕜, E →L[𝕜] G) n (λ x, (g x).comp (f x)) s x := -- sorry --- the following proof takes very long in pure term mode -lemma cont_mdiff_at.clm_comp {g : M → F →L[𝕜] G} {f : M → F' →L[𝕜] F} {x : M} - (hg : cont_mdiff_at I 𝓘(𝕜, F →L[𝕜] G) n g x) (hf : cont_mdiff_at I 𝓘(𝕜, F' →L[𝕜] F) n f x) : - cont_mdiff_at I 𝓘(𝕜, F' →L[𝕜] G) n (λ x, (g x).comp (f x)) x := -@cont_diff_at.comp_cont_mdiff_at 𝕜 _ E _ _ H _ I M _ _ n _ ((F →L[𝕜] G) × (F' →L[𝕜] F)) - _ _ _ _ _ - (λ x, continuous_linear_map.comp x.1 x.2) (λ x, (g x, f x)) x - (by { apply cont_diff.cont_diff_at, apply is_bounded_bilinear_map.cont_diff, exact is_bounded_bilinear_map_comp, }) (hg.prod_mk_space hf) - -lemma cont_mdiff.clm_comp {g : M → F →L[𝕜] G} {f : M → F' →L[𝕜] F} - (hg : cont_mdiff I 𝓘(𝕜, F →L[𝕜] G) n g) (hf : cont_mdiff I 𝓘(𝕜, F' →L[𝕜] F) n f) : - cont_mdiff I 𝓘(𝕜, F' →L[𝕜] G) n (λ x, (g x).comp (f x)) := -λ x, (hg x).clm_comp (hf x) - lemma ext_chart_preimage_mem_nhds_within_range {x' : M} {t : set M} (h : x' ∈ (ext_chart_at I x).source) (ht : t ∈ 𝓝 x') : (ext_chart_at I x).symm ⁻¹' t ∈ 𝓝[range I] ((ext_chart_at I x) x') := -nhds_within_le_nhds $ ext_chart_preimage_mem_nhds' h ht +nhds_within_le_nhds $ ext_chart_preimage_mem_nhds' _ _ h ht end smooth_manifold_with_corners diff --git a/formal/lean/sphere-eversion/to_mathlib/unused/parition_of_unity.lean b/formal/lean/sphere-eversion/to_mathlib/unused/parition_of_unity.lean index 78a690debd85682b1af532aa934aa6693725a794..4c687bdb5d655984998df2368dd97f868c146cb9 100644 --- a/formal/lean/sphere-eversion/to_mathlib/unused/parition_of_unity.lean +++ b/formal/lean/sphere-eversion/to_mathlib/unused/parition_of_unity.lean @@ -5,8 +5,8 @@ open_locale topological_space filter manifold open topological_space filter variables - {E : Type*} [normed_group E] [normed_space ℝ E] [finite_dimensional ℝ E] - {F : Type*} [normed_group F] [normed_space ℝ F] + {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [finite_dimensional ℝ E] + {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] universes uH uM diff --git a/formal/lean/sphere-eversion/to_mathlib/unused/smoothness.lean b/formal/lean/sphere-eversion/to_mathlib/unused/smoothness.lean index a3fedaf3b8bc686f7e02a9485ee88dd893df37f3..712003124d73cc21a1030ae9592ebc1f134302e9 100644 --- a/formal/lean/sphere-eversion/to_mathlib/unused/smoothness.lean +++ b/formal/lean/sphere-eversion/to_mathlib/unused/smoothness.lean @@ -47,9 +47,9 @@ sorry_ax section C1_real variables {E E' F : Type*} -variables [normed_group E] [normed_space ℝ E] -variables [normed_group E'] [normed_space ℝ E'] -variables [normed_group F] [normed_space ℝ F] +variables [normed_add_comm_group E] [normed_space ℝ E] +variables [normed_add_comm_group E'] [normed_space ℝ E'] +variables [normed_add_comm_group F] [normed_space ℝ F] open filter asymptotics metric open_locale topological_space filter @@ -60,7 +60,7 @@ to prove them in the middle of some serious proof. Maybe there is a more general that would still be useful enough to combine is_o.comp_tendsto and is_o.trans_is_O. -/ -lemma asymptotics.is_o.comp_fst' {E E' F : Type*} [normed_group E] [normed_group E'] [normed_group F] +lemma asymptotics.is_o.comp_fst' {E E' F : Type*} [normed_add_comm_group E] [normed_add_comm_group E'] [normed_add_comm_group F] {f : E → F} (h : is_o (𝓝 0) f id) : is_o (𝓝 0) (λ p : E × E', f p.1) id := begin @@ -70,7 +70,7 @@ begin exact is_O_fst_prod' end -lemma asymptotics.is_o.comp_fst {E E' F : Type*} [normed_group E] [normed_group E'] [normed_group F] +lemma asymptotics.is_o.comp_fst {E E' F : Type*} [normed_add_comm_group E] [normed_add_comm_group E'] [normed_add_comm_group F] {f : E → F} {e : E} (h : is_o (𝓝 e) f (λ x, x - e)) (e' : E') : is_o (𝓝 (e, e')) (λ p : E × E', f p.1) (λ p, p - (e, e')) := begin @@ -80,7 +80,7 @@ begin exact is_O_fst_prod end -lemma asymptotics.is_o.comp_snd' {E E' F : Type*} [normed_group E] [normed_group E'] [normed_group F] +lemma asymptotics.is_o.comp_snd' {E E' F : Type*} [normed_add_comm_group E] [normed_add_comm_group E'] [normed_add_comm_group F] {f : E' → F} (h : is_o (𝓝 0) f id) : is_o (𝓝 0) (λ p : E × E', f p.2) id := begin @@ -90,7 +90,7 @@ begin exact is_O_snd_prod' end -lemma asymptotics.is_o.comp_snd {E E' F : Type*} [normed_group E] [normed_group E'] [normed_group F] +lemma asymptotics.is_o.comp_snd {E E' F : Type*} [normed_add_comm_group E] [normed_add_comm_group E'] [normed_add_comm_group F] {f : E' → F} {e' : E'} (h : is_o (𝓝 e') f (λ x, x - e')) (e : E) : is_o (𝓝 (e, e')) (λ p : E × E', f p.2) (λ p, p - (e, e')) := begin @@ -123,16 +123,16 @@ end -lemma has_fderiv_at.comp' {𝕜 : Type*} [nondiscrete_normed_field 𝕜] {E : Type*} [normed_group E] - [normed_space 𝕜 E] {F : Type*} [normed_group F] [normed_space 𝕜 F] - {G : Type*} [normed_group G] [normed_space 𝕜 G] {f : E → F} {f' : E →L[𝕜] F} {x : E} +lemma has_fderiv_at.comp' {𝕜 : Type*} [nontrivially_normed_field 𝕜] {E : Type*} [normed_add_comm_group E] + [normed_space 𝕜 E] {F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F] + {G : Type*} [normed_add_comm_group G] [normed_space 𝕜 G] {f : E → F} {f' : E →L[𝕜] F} {x : E} {g : F → G} {g' : F →L[𝕜] G} (hg : has_fderiv_at g g' (f x)) (hf : has_fderiv_at f f' x) {gf' : E →L[𝕜] G} (h : gf' = g'.comp f') : has_fderiv_at (g ∘ f) gf' x := h.symm ▸ hg.comp x hf -lemma has_fderiv_at.sub' {𝕜 : Type*} [ nondiscrete_normed_field 𝕜] {E : Type*} [normed_group E] - [normed_space 𝕜 E] {F : Type*} [normed_group F] [normed_space 𝕜 F] +lemma has_fderiv_at.sub' {𝕜 : Type*} [ nontrivially_normed_field 𝕜] {E : Type*} [normed_add_comm_group E] + [normed_space 𝕜 E] {F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F] {f g : E → F} {f' g' fg' : E →L[𝕜] F} {x : E} (hf : has_fderiv_at f f' x) (hg : has_fderiv_at g g' x) (h : fg' = f' - g') : has_fderiv_at (λ (x : E), f x - g x) fg' x := @@ -218,13 +218,13 @@ end C1_real /- The lemmas below are maybe-true lemmas about iterated derivatives, that are useful to have (though we probably don't need them in this project) -/ section smooth variables {𝕜 E E' F F' G H K : Type*} -variables [nondiscrete_normed_field 𝕜] -variables [normed_group E] [normed_space 𝕜 E] -variables [normed_group E'] [normed_space 𝕜 E'] -variables [normed_group F] [normed_space 𝕜 F] -variables [normed_group G] [normed_space 𝕜 G] -variables [normed_group H] [normed_space 𝕜 H] -variables [normed_group K] [normed_space 𝕜 K] +variables [nontrivially_normed_field 𝕜] +variables [normed_add_comm_group E] [normed_space 𝕜 E] +variables [normed_add_comm_group E'] [normed_space 𝕜 E'] +variables [normed_add_comm_group F] [normed_space 𝕜 F] +variables [normed_add_comm_group G] [normed_space 𝕜 G] +variables [normed_add_comm_group H] [normed_space 𝕜 H] +variables [normed_add_comm_group K] [normed_space 𝕜 K] variables [normed_linear_ordered_field F'] [normed_space 𝕜 F'] variables {n : with_top ℕ} diff --git a/splits.json b/splits.json index 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"formal/afp/Network_Security_Policy_Verification/Security_Invariants/SINVAR_DomainHierarchyNG_impl.thy", - "formal/afp/JinjaThreads/Examples/BufferExample.thy" + "formal/afp/Abstract-Hoare-Logics/Proc/PLang.thy", + "formal/afp/Pi_Calculus/Strong_Late_Bisim_SC.thy", + "formal/afp/Higher_Order_Terms/Fresh_Monad.thy", + "formal/afp/Network_Security_Policy_Verification/Examples/Tainting/MeasrDroid.thy", + "formal/afp/JinjaThreads/DFA/Opt.thy" ] } \ No newline at end of file diff --git a/test.py b/test.py index 70f2c369aa0a4142e255e8636403427c443592da..9d7b670ed3e378a6d4ef6a6daf8e64e029a77206 100644 --- a/test.py +++ b/test.py @@ -3,30 +3,43 @@ from itertools import islice import sys dataset = load_dataset("./proof-pile.py", "arxiv") +for x in dataset["train"]: + print(x) + break print('ARXIV') print(dataset) -""" + dataset = load_dataset("./proof-pile.py", "books") +for x in dataset["train"]: + print(x) + break print("BOOKS") print(dataset) -for x in dataset["train"]: - print(x["text"]) - sys.exit() dataset = load_dataset("./proof-pile.py", "formal") +for x in dataset["train"]: + print(x) + break print("FORMAL") print(dataset) dataset = load_dataset("./proof-pile.py", "stack-exchange") -print("STACK EXCHANGE") for x in dataset["train"]: print(x) break +print("STACK EXCHANGE") print(dataset) + dataset = load_dataset("./proof-pile.py", "wiki") +for x in dataset["train"]: + print(x) + break print("WIKI") print(dataset) + dataset = load_dataset("./proof-pile.py", "math-dataset", download_mode='force_redownload') +for x in dataset["train"]: + print(x) + break print("MATH DATASET") print(dataset) -"""