ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsSepClosed.splits_of_separable
class IsSepClosed : Prop where
  splits_of_separable : ∀ p : k[X], p.Separable → (p.Splits <| RingHom.id k)
Arguments: {k : Type u} [Field k] [self : IsSepClosed k] (p : k[X])
Goal: p.Separable → Polynomial.Splits (RingHom.id k) p
-- IsSepClosed.splits_codomain
theorem IsSepClosed.splits_codomain [IsSepClosed K] {f : k →+* K}
    (p : k[X]) (h : p.Separable) : p.Splits f 
Arguments: {k : Type u} [Field k] {K : Type v} [Field K] [IsSepClosed K] {f : k →+* K} (p : k[X]) (h : p.Separable)
Goal: Polynomial.Splits f p
-- IsSepClosed.splits_domain
theorem IsSepClosed.splits_domain [IsSepClosed k] {f : k →+* K}
    (p : k[X]) (h : p.Separable) : p.Splits f 
Arguments: {k : Type u} [Field k] {K : Type v} [Field K] [IsSepClosed k] {f : k →+* K} (p : k[X]) (h : p.Separable)
Goal: Polynomial.Splits f p
-- IsSepClosed.exists_root
theorem exists_root [IsSepClosed k] (p : k[X]) (hp : p.degree ≠ 0) (hsep : p.Separable) :
    ∃ x, IsRoot p x 
Arguments: {k : Type u} [Field k] [IsSepClosed k] (p : k[X]) (hp : p.degree ≠ 0) (hsep : p.Separable)
Goal: ∃ x, p.IsRoot x
-- IsSepClosed.exists_pow_nat_eq
theorem exists_pow_nat_eq [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero (n : k)] :
    ∃ z, z ^ n = x 
Arguments: {k : Type u} [Field k] [IsSepClosed k] (x : k) (n : ℕ) [hn : NeZero ↑n]
Goal: ∃ z, z ^ n = x
-- IsSepClosed.exists_eq_mul_self
theorem exists_eq_mul_self [IsSepClosed k] (x : k) [h2 : NeZero (2 : k)] : ∃ z, x = z * z 
Arguments: {k : Type u} [Field k] [IsSepClosed k] (x : k) [h2 : NeZero 2]
Goal: ∃ z, x = z * z
-- IsSepClosed.roots_eq_zero_iff
theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
    p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) 
Arguments: {k : Type u} [Field k] [IsSepClosed k] {p : k[X]} (hsep : p.Separable)
Goal: p.roots = 0 ↔ p = Polynomial.C (p.coeff 0)
-- IsSepClosed.exists_eval₂_eq_zero
theorem exists_eval₂_eq_zero [IsSepClosed K] (f : k →+* K)
    (p : k[X]) (hp : p.degree ≠ 0) (hsep : p.Separable) :
    ∃ x, p.eval₂ f x = 0 
Arguments: {k : Type u} [Field k] {K : Type v} [Field K] [IsSepClosed K] (f : k →+* K) (p : k[X]) (hp : p.degree ≠ 0) (hsep : p.Separable)
Goal: ∃ x, Polynomial.eval₂ f x p = 0
-- IsSepClosed.exists_aeval_eq_zero
theorem exists_aeval_eq_zero [IsSepClosed K] [Algebra k K] (p : k[X])
    (hp : p.degree ≠ 0) (hsep : p.Separable) : ∃ x : K, aeval x p = 0 
Arguments: {k : Type u} [Field k] (K : Type v) [Field K] [IsSepClosed K] [Algebra k K] (p : k[X]) (hp : p.degree ≠ 0) (hsep : p.Separable)
Goal: ∃ x, (Polynomial.aeval x) p = 0
-- IsSepClosed.of_exists_root
theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → Separable p → ∃ x, p.eval x = 0) :
    IsSepClosed k 
Arguments: (k : Type u) [Field k] (H : ∀ (p : k[X]), p.Monic → Irreducible p → p.Separable → ∃ x, Polynomial.eval x p = 0)
Goal: IsSepClosed k
-- IsSepClosed.degree_eq_one_of_irreducible
theorem degree_eq_one_of_irreducible [IsSepClosed k] {p : k[X]}
    (hp : Irreducible p) (hsep : p.Separable) : p.degree = 1 
Arguments: (k : Type u) [Field k] [IsSepClosed k] {p : k[X]} (hp : Irreducible p) (hsep : p.Separable)
Goal: p.degree = 1
-- IsSepClosed.algebraMap_surjective
theorem algebraMap_surjective
    [IsSepClosed k] [Algebra k K] [Algebra.IsSeparable k K] :
    Function.Surjective (algebraMap k K) 
Arguments: (k : Type u) [Field k] (K : Type v) [Field K] [IsSepClosed k] [Algebra k K] [Algebra.IsSeparable k K]
Goal: Function.Surjective ⇑(algebraMap k K)
-- IntermediateField.eq_bot_of_isSepClosed_of_isSeparable
theorem IntermediateField.eq_bot_of_isSepClosed_of_isSeparable [IsSepClosed k] [Algebra k K]
    (L : IntermediateField k K) [Algebra.IsSeparable k L] : L = ⊥ 
Arguments: {k : Type u} [Field k] {K : Type v} [Field K] [IsSepClosed k] [Algebra k K] (L : IntermediateField k K) [Algebra.IsSeparable k ↥L]
Goal: L = ⊥
-- IsSepClosure.sep_closed
class IsSepClosure [Algebra k K] : Prop where
  sep_closed : IsSepClosed K
  separable : Algebra.IsSeparable k K
Arguments: (k : Type u) [Field k] {K : Type v} [Field K] [Algebra k K] [self : IsSepClosure k K]
Goal: IsSepClosed K
-- IsSepClosure.separable
class IsSepClosure [Algebra k K] : Prop where
  sep_closed : IsSepClosed K
  separable : Algebra.IsSeparable k K
Arguments: {k : Type u} [Field k] {K : Type v} [Field K] [Algebra k K] [self : IsSepClosure k K]
Goal: Algebra.IsSeparable k K
-- isSepClosure_iff
theorem isSepClosure_iff [Algebra k K] :
    IsSepClosure k K ↔ IsSepClosed K ∧ Algebra.IsSeparable k K 
Arguments: {k : Type u} [Field k] {K : Type v} [Field K] [Algebra k K]
Goal: IsSepClosure k K ↔ IsSepClosed K ∧ Algebra.IsSeparable k K
-- IsSepClosed.surjective_comp_algebraMap_of_isSeparable
theorem surjective_comp_algebraMap_of_isSeparable {E : Type*}
    [Field E] [Algebra K E] [Algebra L E] [IsScalarTower K L E] [Algebra.IsSeparable L E] :
    Function.Surjective fun φ : E →ₐ[K] M ↦ φ.comp (IsScalarTower.toAlgHom K L E) 
Arguments: {K : Type u} (L : Type v) {M : Type w} [Field K] [Field L] [Algebra K L] [Field M] [Algebra K M] [IsSepClosed M] {E : Type u_1} [Field E] [Algebra K E] [Algebra L E] [IsScalarTower K L E] [Algebra.IsSeparable L E]
Goal: Function.Surjective fun φ => φ.comp (IsScalarTower.toAlgHom K L E)
-- IsSepClosed.lift_def
noncomputable irreducible_def lift : L →ₐ[K] M 
Arguments: {K : Type u_1} {L : Type u_2} {M : Type u_3} [Field K] [Field L] [Algebra K L] [Field M] [Algebra K M] [IsSepClosed M] [Algebra.IsSeparable K L]
Goal: IsSepClosed.lift = Classical.choice ⋯
-- minpoly.monic
theorem monic (hx : IsIntegral A x) : Monic (minpoly A x) 
Arguments: {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] {x : B} (hx : IsIntegral A x)
Goal: (minpoly A x).Monic
-- minpoly.ne_zero
theorem ne_zero [Nontrivial A] (hx : IsIntegral A x) : minpoly A x ≠ 0 
Arguments: {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] {x : B} [Nontrivial A] (hx : IsIntegral A x)
Goal: minpoly A x ≠ 0
-- minpoly.eq_zero
theorem eq_zero (hx : ¬IsIntegral A x) : minpoly A x = 0 
Arguments: {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] {x : B} (hx : ¬IsIntegral A x)
Goal: minpoly A x = 0
-- minpoly.algHom_eq
theorem algHom_eq (f : B →ₐ[A] B') (hf : Function.Injective f) (x : B) :
    minpoly A (f x) = minpoly A x 
Arguments: {A : Type u_1} {B : Type u_2} {B' : Type u_3} [CommRing A] [Ring B] [Ring B'] [Algebra A B] [Algebra A B'] (f : B →ₐ[A] B') (hf : Function.Injective ⇑f) (x : B)
Goal: minpoly A (f x) = minpoly A x
-- minpoly.algebraMap_eq
theorem algebraMap_eq {B} [CommRing B] [Algebra A B] [Algebra B B'] [IsScalarTower A B B']
    (h : Function.Injective (algebraMap B B')) (x : B) :
    minpoly A (algebraMap B B' x) = minpoly A x 
Arguments: {A : Type u_1} {B' : Type u_3} [CommRing A] [Ring B'] [Algebra A B'] {B : Type u_4} [CommRing B] [Algebra A B] [Algebra B B'] [IsScalarTower A B B'] (h : Function.Injective ⇑(algebraMap B B')) (x : B)
Goal: minpoly A ((algebraMap B B') x) = minpoly A x
-- minpoly.algEquiv_eq
@[simp]
theorem algEquiv_eq (f : B ≃ₐ[A] B') (x : B) : minpoly A (f x) = minpoly A x 
Arguments: {A : Type u_1} {B : Type u_2} {B' : Type u_3} [CommRing A] [Ring B] [Ring B'] [Algebra A B] [Algebra A B'] (f : B ≃ₐ[A] B') (x : B)
Goal: minpoly A (f x) = minpoly A x
-- minpoly.aeval
@[simp]
theorem aeval : aeval x (minpoly A x) = 0 
Arguments: (A : Type u_1) {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] (x : B)
Goal: (Polynomial.aeval x) (minpoly A x) = 0
-- minpoly.aeval_algHom
@[simp]
theorem aeval_algHom (f : B →ₐ[A] B') (x : B) : (Polynomial.aeval (f x)) (minpoly A x) = 0 
Arguments: (A : Type u_1) {B : Type u_2} {B' : Type u_3} [CommRing A] [Ring B] [Ring B'] [Algebra A B] [Algebra A B'] (f : B →ₐ[A] B') (x : B)
Goal: (Polynomial.aeval (f x)) (minpoly A x) = 0
-- minpoly.ne_one
theorem ne_one [Nontrivial B] : minpoly A x ≠ 1 
Arguments: (A : Type u_1) {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] (x : B) [Nontrivial B]
Goal: minpoly A x ≠ 1
-- minpoly.map_ne_one
theorem map_ne_one [Nontrivial B] {R : Type*} [Semiring R] [Nontrivial R] (f : A →+* R) :
    (minpoly A x).map f ≠ 1 
Arguments: (A : Type u_1) {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] (x : B) [Nontrivial B] {R : Type u_4} [Semiring R] [Nontrivial R] (f : A →+* R)
Goal: Polynomial.map f (minpoly A x) ≠ 1
-- minpoly.not_isUnit
theorem not_isUnit [Nontrivial B] : ¬IsUnit (minpoly A x) 
Arguments: (A : Type u_1) {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] (x : B) [Nontrivial B]
Goal: ¬IsUnit (minpoly A x)
-- minpoly.mem_range_of_degree_eq_one
theorem mem_range_of_degree_eq_one (hx : (minpoly A x).degree = 1) :
    x ∈ (algebraMap A B).range 
Arguments: (A : Type u_1) {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] (x : B) (hx : (minpoly A x).degree = 1)
Goal: x ∈ (algebraMap A B).range
-- minpoly.min
theorem min {p : A[X]} (pmonic : p.Monic) (hp : Polynomial.aeval x p = 0) :
    degree (minpoly A x) ≤ degree p 
Arguments: (A : Type u_1) {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] (x : B) {p : A[X]} (pmonic : p.Monic) (hp : (Polynomial.aeval x) p = 0)
Goal: (minpoly A x).degree ≤ p.degree
-- minpoly.unique'
theorem unique' {p : A[X]} (hm : p.Monic) (hp : Polynomial.aeval x p = 0)
    (hl : ∀ q : A[X], degree q < degree p → q = 0 ∨ Polynomial.aeval x q ≠ 0) :
    p = minpoly A x 
Arguments: (A : Type u_1) {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] (x : B) {p : A[X]} (hm : p.Monic) (hp : (Polynomial.aeval x) p = 0) (hl : ∀ (q : A[X]), q.degree < p.degree → q = 0 ∨ (Polynomial.aeval x) q ≠ 0)
Goal: p = minpoly A x
-- minpoly.subsingleton
@[nontriviality]
theorem subsingleton [Subsingleton B] : minpoly A x = 1 
Arguments: (A : Type u_1) {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] (x : B) [Subsingleton B]
Goal: minpoly A x = 1
-- minpoly.natDegree_pos
theorem natDegree_pos [Nontrivial B] (hx : IsIntegral A x) : 0 < natDegree (minpoly A x) 
Arguments: {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] {x : B} [Nontrivial B] (hx : IsIntegral A x)
Goal: 0 < (minpoly A x).natDegree
-- minpoly.degree_pos
theorem degree_pos [Nontrivial B] (hx : IsIntegral A x) : 0 < degree (minpoly A x) 
Arguments: {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] {x : B} [Nontrivial B] (hx : IsIntegral A x)
Goal: 0 < (minpoly A x).degree
-- minpoly.degree_eq_one_iff
theorem degree_eq_one_iff : (minpoly A x).degree = 1 ↔ x ∈ (algebraMap A B).range 
Arguments: {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] {x : B} [Nontrivial B]
Goal: (minpoly A x).degree = 1 ↔ x ∈ (algebraMap A B).range
-- minpoly.natDegree_eq_one_iff
theorem natDegree_eq_one_iff :
    (minpoly A x).natDegree = 1 ↔ x ∈ (algebraMap A B).range 
Arguments: {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] {x : B} [Nontrivial B]
Goal: (minpoly A x).natDegree = 1 ↔ x ∈ (algebraMap A B).range
-- minpoly.two_le_natDegree_iff
theorem two_le_natDegree_iff (int : IsIntegral A x) :
    2 ≤ (minpoly A x).natDegree ↔ x ∉ (algebraMap A B).range 
Arguments: {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] {x : B} [Nontrivial B] (int : IsIntegral A x)
Goal: 2 ≤ (minpoly A x).natDegree ↔ x ∉ (algebraMap A B).range
-- minpoly.two_le_natDegree_subalgebra
theorem two_le_natDegree_subalgebra {B} [CommRing B] [Algebra A B] [Nontrivial B]
    {S : Subalgebra A B} {x : B} (int : IsIntegral S x) : 2 ≤ (minpoly S x).natDegree ↔ x ∉ S 
Arguments: {A : Type u_1} [CommRing A] {B : Type u_4} [CommRing B] [Algebra A B] [Nontrivial B] {S : Subalgebra A B} {x : B} (int : IsIntegral (↥S) x)
Goal: 2 ≤ (minpoly (↥S) x).natDegree ↔ x ∉ S
-- minpoly.eq_X_sub_C_of_algebraMap_inj
theorem eq_X_sub_C_of_algebraMap_inj (a : A) (hf : Function.Injective (algebraMap A B)) :
    minpoly A (algebraMap A B a) = X - C a 
Arguments: {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] (a : A) (hf : Function.Injective ⇑(algebraMap A B))
Goal: minpoly A ((algebraMap A B) a) = Polynomial.X - Polynomial.C a
-- minpoly.aeval_ne_zero_of_dvdNotUnit_minpoly
theorem aeval_ne_zero_of_dvdNotUnit_minpoly {a : A[X]} (hx : IsIntegral A x) (hamonic : a.Monic)
    (hdvd : DvdNotUnit a (minpoly A x)) : Polynomial.aeval x a ≠ 0 
Arguments: {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] {x : B} {a : A[X]} (hx : IsIntegral A x) (hamonic : a.Monic) (hdvd : DvdNotUnit a (minpoly A x))
Goal: (Polynomial.aeval x) a ≠ 0
-- minpoly.irreducible
theorem irreducible (hx : IsIntegral A x) : Irreducible (minpoly A x) 
Arguments: {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] {x : B} [IsDomain A] [IsDomain B] (hx : IsIntegral A x)
Goal: Irreducible (minpoly A x)
-- LinearPMap.toFun_eq_coe
@[simp]
theorem toFun_eq_coe (f : E →ₗ.[R] F) (x : f.domain) : f.toFun x = f x 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : ↥f.domain)
Goal: f.toFun x = ↑f x
-- LinearPMap.ext
@[ext]
theorem ext {f g : E →ₗ.[R] F} (h : f.domain = g.domain)
    (h' : ∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄ (_h : (x : E) = y), f x = g y) : f = g 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f g : E →ₗ.[R] F} (h : f.domain = g.domain) (h' : ∀ ⦃x : ↥f.domain⦄ ⦃y : ↥g.domain⦄, ↑x = ↑y → ↑f x = ↑g y)
Goal: f = g
-- LinearPMap.map_zero
@[simp]
theorem map_zero (f : E →ₗ.[R] F) : f 0 = 0 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F)
Goal: ↑f 0 = 0
-- LinearPMap.ext_iff
theorem ext_iff {f g : E →ₗ.[R] F} :
    f = g ↔
      ∃ _domain_eq : f.domain = g.domain,
        ∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄ (_h : (x : E) = y), f x = g y 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f g : E →ₗ.[R] F}
Goal: f = g ↔ ∃ (_ : f.domain = g.domain), ∀ ⦃x : ↥f.domain⦄ ⦃y : ↥g.domain⦄, ↑x = ↑y → ↑f x = ↑g y
-- LinearPMap.ext'
theorem ext' {s : Submodule R E} {f g : s →ₗ[R] F} (h : f = g) : mk s f = mk s g 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {s : Submodule R E} {f g : ↥s →ₗ[R] F} (h : f = g)
Goal: { domain := s, toFun := f } = { domain := s, toFun := g }
-- LinearPMap.map_add
theorem map_add (f : E →ₗ.[R] F) (x y : f.domain) : f (x + y) = f x + f y 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x y : ↥f.domain)
Goal: ↑f (x + y) = ↑f x + ↑f y
-- LinearPMap.map_neg
theorem map_neg (f : E →ₗ.[R] F) (x : f.domain) : f (-x) = -f x 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : ↥f.domain)
Goal: ↑f (-x) = -↑f x
-- LinearPMap.map_sub
theorem map_sub (f : E →ₗ.[R] F) (x y : f.domain) : f (x - y) = f x - f y 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x y : ↥f.domain)
Goal: ↑f (x - y) = ↑f x - ↑f y
-- LinearPMap.map_smul
theorem map_smul (f : E →ₗ.[R] F) (c : R) (x : f.domain) : f (c • x) = c • f x 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (c : R) (x : ↥f.domain)
Goal: ↑f (c • x) = c • ↑f x
-- LinearPMap.mk_apply
@[simp]
theorem mk_apply (p : Submodule R E) (f : p →ₗ[R] F) (x : p) : mk p f x = f x 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (p : Submodule R E) (f : ↥p →ₗ[R] F) (x : ↥p)
Goal: ↑{ domain := p, toFun := f } x = f x
-- LinearPMap.domain_mkSpanSingleton
@[simp]
theorem domain_mkSpanSingleton (x : E) (y : F) (H : ∀ c : R, c • x = 0 → c • y = 0) :
    (mkSpanSingleton' x y H).domain = R ∙ x 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : E) (y : F) (H : ∀ (c : R), c • x = 0 → c • y = 0)
Goal: (LinearPMap.mkSpanSingleton' x y H).domain = Submodule.span R {x}
-- LinearPMap.mkSpanSingleton'_apply
@[simp]
theorem mkSpanSingleton'_apply (x : E) (y : F) (H : ∀ c : R, c • x = 0 → c • y = 0) (c : R) (h) :
    mkSpanSingleton' x y H ⟨c • x, h⟩ = c • y 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : E) (y : F) (H : ∀ (c : R), c • x = 0 → c • y = 0) (c : R) (h : c • x ∈ (LinearPMap.mkSpanSingleton' x y H).domain)
Goal: ↑(LinearPMap.mkSpanSingleton' x y H) ⟨c • x, h⟩ = c • y
-- LinearPMap.mkSpanSingleton'_apply_self
@[simp]
theorem mkSpanSingleton'_apply_self (x : E) (y : F) (H : ∀ c : R, c • x = 0 → c • y = 0) (h) :
    mkSpanSingleton' x y H ⟨x, h⟩ = y 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : E) (y : F) (H : ∀ (c : R), c • x = 0 → c • y = 0) (h : x ∈ (LinearPMap.mkSpanSingleton' x y H).domain)
Goal: ↑(LinearPMap.mkSpanSingleton' x y H) ⟨x, h⟩ = y
-- LinearPMap.mkSpanSingleton_apply
theorem mkSpanSingleton_apply (K : Type*) {E F : Type*} [DivisionRing K] [AddCommGroup E]
    [Module K E] [AddCommGroup F] [Module K F] {x : E} (hx : x ≠ 0) (y : F) :
    mkSpanSingleton x y hx ⟨x, (Submodule.mem_span_singleton_self x : x ∈ Submodule.span K {x})⟩ =
      y 
Arguments: (K : Type u_5) {E : Type u_6} {F : Type u_7} [DivisionRing K] [AddCommGroup E] [Module K E] [AddCommGroup F] [Module K F] {x : E} (hx : x ≠ 0) (y : F)
Goal: ↑(LinearPMap.mkSpanSingleton x y hx) ⟨x, ⋯⟩ = y
-- LinearPMap.fst_apply
@[simp]
theorem fst_apply (p : Submodule R E) (p' : Submodule R F) (x : p.prod p') :
    LinearPMap.fst p p' x = (x : E × F).1 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (p : Submodule R E) (p' : Submodule R F) (x : ↥(p.prod p'))
Goal: ↑(LinearPMap.fst p p') x = (↑x).1
-- LinearPMap.snd_apply
@[simp]
theorem snd_apply (p : Submodule R E) (p' : Submodule R F) (x : p.prod p') :
    LinearPMap.snd p p' x = (x : E × F).2 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (p : Submodule R E) (p' : Submodule R F) (x : ↥(p.prod p'))
Goal: ↑(LinearPMap.snd p p') x = (↑x).2
-- LinearPMap.apply_comp_inclusion
theorem apply_comp_inclusion {T S : E →ₗ.[R] F} (h : T ≤ S) (x : T.domain) :
    T x = S (Submodule.inclusion h.1 x) 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {T S : E →ₗ.[R] F} (h : T ≤ S) (x : ↥T.domain)
Goal: ↑T x = ↑S ((Submodule.inclusion ⋯) x)
-- LinearPMap.exists_of_le
theorem exists_of_le {T S : E →ₗ.[R] F} (h : T ≤ S) (x : T.domain) :
    ∃ y : S.domain, (x : E) = y ∧ T x = S y 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {T S : E →ₗ.[R] F} (h : T ≤ S) (x : ↥T.domain)
Goal: ∃ y, ↑x = ↑y ∧ ↑T x = ↑S y
-- LinearPMap.eq_of_le_of_domain_eq
theorem eq_of_le_of_domain_eq {f g : E →ₗ.[R] F} (hle : f ≤ g) (heq : f.domain = g.domain) :
    f = g 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f g : E →ₗ.[R] F} (hle : f ≤ g) (heq : f.domain = g.domain)
Goal: f = g
-- LinearPMap.le_of_eqLocus_ge
theorem le_of_eqLocus_ge {f g : E →ₗ.[R] F} (H : f.domain ≤ f.eqLocus g) : f ≤ g 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f g : E →ₗ.[R] F} (H : f.domain ≤ f.eqLocus g)
Goal: f ≤ g
-- LinearPMap.domain_mono
theorem domain_mono : StrictMono (@domain R _ E _ _ F _ _) 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F]
Goal: StrictMono LinearPMap.domain
-- _private.Mathlib.LinearAlgebra.LinearPMap.0.LinearPMap.sup_aux
private theorem sup_aux (f g : E →ₗ.[R] F)
    (h : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y) :
    ∃ fg : ↥(f.domain ⊔ g.domain) →ₗ[R] F,
      ∀ (x : f.domain) (y : g.domain) (z : ↥(f.domain ⊔ g.domain)),
        (x : E) + y = ↑z → fg z = f x + g y 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f g : E →ₗ.[R] F) (h : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y)
Goal: ∃ fg, ∀ (x : ↥f.domain) (y : ↥g.domain) (z : ↥(f.domain ⊔ g.domain)), ↑x + ↑y = ↑z → fg z = ↑f x + ↑g y
-- LinearPMap.domain_sup
@[simp]
theorem domain_sup (f g : E →ₗ.[R] F)
    (h : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y) :
    (f.sup g h).domain = f.domain ⊔ g.domain 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f g : E →ₗ.[R] F) (h : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y)
Goal: (f.sup g h).domain = f.domain ⊔ g.domain
-- LinearPMap.sup_apply
theorem sup_apply {f g : E →ₗ.[R] F} (H : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y)
    (x : f.domain) (y : g.domain) (z : ↥(f.domain ⊔ g.domain)) (hz : (↑x : E) + ↑y = ↑z) :
    f.sup g H z = f x + g y 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f g : E →ₗ.[R] F} (H : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y) (x : ↥f.domain) (y : ↥g.domain) (z : ↥(f.domain ⊔ g.domain)) (hz : ↑x + ↑y = ↑z)
Goal: ↑(f.sup g H) z = ↑f x + ↑g y
-- LinearPMap.left_le_sup
protected theorem left_le_sup (f g : E →ₗ.[R] F)
    (h : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y) : f ≤ f.sup g h 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f g : E →ₗ.[R] F) (h : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y)
Goal: f ≤ f.sup g h
-- LinearPMap.right_le_sup
protected theorem right_le_sup (f g : E →ₗ.[R] F)
    (h : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y) : g ≤ f.sup g h 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f g : E →ₗ.[R] F) (h : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y)
Goal: g ≤ f.sup g h
-- LinearPMap.sup_le
protected theorem sup_le {f g h : E →ₗ.[R] F}
    (H : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y) (fh : f ≤ h) (gh : g ≤ h) :
    f.sup g H ≤ h 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f g h : E →ₗ.[R] F} (H : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y) (fh : f ≤ h) (gh : g ≤ h)
Goal: f.sup g H ≤ h
-- LinearPMap.sup_h_of_disjoint
theorem sup_h_of_disjoint (f g : E →ₗ.[R] F) (h : Disjoint f.domain g.domain) (x : f.domain)
    (y : g.domain) (hxy : (x : E) = y) : f x = g y 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f g : E →ₗ.[R] F) (h : Disjoint f.domain g.domain) (x : ↥f.domain) (y : ↥g.domain) (hxy : ↑x = ↑y)
Goal: ↑f x = ↑g y
-- LinearPMap.zero_domain
@[simp]
theorem zero_domain : (0 : E →ₗ.[R] F).domain = ⊤ 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F]
Goal: LinearPMap.domain 0 = ⊤
-- LinearPMap.zero_apply
@[simp]
theorem zero_apply (x : (⊤ : Submodule R E)) : (0 : E →ₗ.[R] F) x = 0 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : ↥⊤)
Goal: ↑0 x = 0
-- LinearPMap.smul_domain
@[simp]
theorem smul_domain (a : M) (f : E →ₗ.[R] F) : (a • f).domain = f.domain 
Arguments: {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] (a : M) (f : E →ₗ.[R] F)
Goal: (a • f).domain = f.domain