ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- RingHom.map_geom_sum
theorem RingHom.map_geom_sum [Semiring α] [Semiring β] (x : α) (n : ℕ) (f : α →+* β) :
    f (∑ i ∈ range n, x ^ i) = ∑ i ∈ range n, f x ^ i 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [Semiring β] (x : α) (n : ℕ) (f : α →+* β)
Goal: f (∑ i ∈ Finset.range n, x ^ i) = ∑ i ∈ Finset.range n, f x ^ i
-- RingHom.map_geom_sum₂
theorem RingHom.map_geom_sum₂ [Semiring α] [Semiring β] (x y : α) (n : ℕ) (f : α →+* β) :
    f (∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) = ∑ i ∈ range n, f x ^ i * f y ^ (n - 1 - i) 
Arguments: {α : Type u} {β : Type u_1} [Semiring α] [Semiring β] (x y : α) (n : ℕ) (f : α →+* β)
Goal: f (∑ i ∈ Finset.range n, x ^ i * y ^ (n - 1 - i)) = ∑ i ∈ Finset.range n, f x ^ i * f y ^ (n - 1 - i)
-- Nat.pred_mul_geom_sum_le
theorem Nat.pred_mul_geom_sum_le (a b n : ℕ) :
    ((b - 1) * ∑ i ∈ range n.succ, a / b ^ i) ≤ a * b - a / b ^ n 
Arguments: (a b n : ℕ)
Goal: (b - 1) * ∑ i ∈ Finset.range n.succ, a / b ^ i ≤ a * b - a / b ^ n
-- Nat.geom_sum_le
theorem Nat.geom_sum_le {b : ℕ} (hb : 2 ≤ b) (a n : ℕ) :
    ∑ i ∈ range n, a / b ^ i ≤ a * b / (b - 1) 
Arguments: {b : ℕ} (hb : 2 ≤ b) (a n : ℕ)
Goal: ∑ i ∈ Finset.range n, a / b ^ i ≤ a * b / (b - 1)
-- Nat.geom_sum_Ico_le
theorem Nat.geom_sum_Ico_le {b : ℕ} (hb : 2 ≤ b) (a n : ℕ) :
    ∑ i ∈ Ico 1 n, a / b ^ i ≤ a / (b - 1) 
Arguments: {b : ℕ} (hb : 2 ≤ b) (a n : ℕ)
Goal: ∑ i ∈ Finset.Ico 1 n, a / b ^ i ≤ a / (b - 1)
-- geom_sum_pos
theorem geom_sum_pos [StrictOrderedSemiring α] (hx : 0 ≤ x) (hn : n ≠ 0) :
    0 < ∑ i ∈ range n, x ^ i 
Arguments: {α : Type u} {n : ℕ} {x : α} [StrictOrderedSemiring α] (hx : 0 ≤ x) (hn : n ≠ 0)
Goal: 0 < ∑ i ∈ Finset.range n, x ^ i
-- geom_sum_pos_and_lt_one
theorem geom_sum_pos_and_lt_one [StrictOrderedRing α] (hx : x < 0) (hx' : 0 < x + 1) (hn : 1 < n) :
    (0 < ∑ i ∈ range n, x ^ i) ∧ ∑ i ∈ range n, x ^ i < 1 
Arguments: {α : Type u} {n : ℕ} {x : α} [StrictOrderedRing α] (hx : x < 0) (hx' : 0 < x + 1) (hn : 1 < n)
Goal: 0 < ∑ i ∈ Finset.range n, x ^ i ∧ ∑ i ∈ Finset.range n, x ^ i < 1
-- geom_sum_alternating_of_le_neg_one
theorem geom_sum_alternating_of_le_neg_one [StrictOrderedRing α] (hx : x + 1 ≤ 0) (n : ℕ) :
    if Even n then (∑ i ∈ range n, x ^ i) ≤ 0 else 1 ≤ ∑ i ∈ range n, x ^ i 
Arguments: {α : Type u} {x : α} [StrictOrderedRing α] (hx : x + 1 ≤ 0) (n : ℕ)
Goal: if Even n then ∑ i ∈ Finset.range n, x ^ i ≤ 0 else 1 ≤ ∑ i ∈ Finset.range n, x ^ i
-- geom_sum_alternating_of_lt_neg_one
theorem geom_sum_alternating_of_lt_neg_one [StrictOrderedRing α] (hx : x + 1 < 0) (hn : 1 < n) :
    if Even n then (∑ i ∈ range n, x ^ i) < 0 else 1 < ∑ i ∈ range n, x ^ i 
Arguments: {α : Type u} {n : ℕ} {x : α} [StrictOrderedRing α] (hx : x + 1 < 0) (hn : 1 < n)
Goal: if Even n then ∑ i ∈ Finset.range n, x ^ i < 0 else 1 < ∑ i ∈ Finset.range n, x ^ i
-- geom_sum_pos'
theorem geom_sum_pos' [LinearOrderedRing α] (hx : 0 < x + 1) (hn : n ≠ 0) :
    0 < ∑ i ∈ range n, x ^ i 
Arguments: {α : Type u} {n : ℕ} {x : α} [LinearOrderedRing α] (hx : 0 < x + 1) (hn : n ≠ 0)
Goal: 0 < ∑ i ∈ Finset.range n, x ^ i
-- Odd.geom_sum_pos
theorem Odd.geom_sum_pos [LinearOrderedRing α] (h : Odd n) : 0 < ∑ i ∈ range n, x ^ i 
Arguments: {α : Type u} {n : ℕ} {x : α} [LinearOrderedRing α] (h : Odd n)
Goal: 0 < ∑ i ∈ Finset.range n, x ^ i
-- geom_sum_pos_iff
theorem geom_sum_pos_iff [LinearOrderedRing α] (hn : n ≠ 0) :
    (0 < ∑ i ∈ range n, x ^ i) ↔ Odd n ∨ 0 < x + 1 
Arguments: {α : Type u} {n : ℕ} {x : α} [LinearOrderedRing α] (hn : n ≠ 0)
Goal: 0 < ∑ i ∈ Finset.range n, x ^ i ↔ Odd n ∨ 0 < x + 1
-- geom_sum_ne_zero
theorem geom_sum_ne_zero [LinearOrderedRing α] (hx : x ≠ -1) (hn : n ≠ 0) :
    ∑ i ∈ range n, x ^ i ≠ 0 
Arguments: {α : Type u} {n : ℕ} {x : α} [LinearOrderedRing α] (hx : x ≠ -1) (hn : n ≠ 0)
Goal: ∑ i ∈ Finset.range n, x ^ i ≠ 0
-- geom_sum_eq_zero_iff_neg_one
theorem geom_sum_eq_zero_iff_neg_one [LinearOrderedRing α] (hn : n ≠ 0) :
    ∑ i ∈ range n, x ^ i = 0 ↔ x = -1 ∧ Even n 
Arguments: {α : Type u} {n : ℕ} {x : α} [LinearOrderedRing α] (hn : n ≠ 0)
Goal: ∑ i ∈ Finset.range n, x ^ i = 0 ↔ x = -1 ∧ Even n
-- geom_sum_neg_iff
theorem geom_sum_neg_iff [LinearOrderedRing α] (hn : n ≠ 0) :
    ∑ i ∈ range n, x ^ i < 0 ↔ Even n ∧ x + 1 < 0 
Arguments: {α : Type u} {n : ℕ} {x : α} [LinearOrderedRing α] (hn : n ≠ 0)
Goal: ∑ i ∈ Finset.range n, x ^ i < 0 ↔ Even n ∧ x + 1 < 0
-- Nat.geomSum_eq
lemma Nat.geomSum_eq (hm : 2 ≤ m) (n : ℕ) :
    ∑ k ∈ range n, m ^ k = (m ^ n - 1) / (m - 1) 
Arguments: {m : ℕ} (hm : 2 ≤ m) (n : ℕ)
Goal: ∑ k ∈ Finset.range n, m ^ k = (m ^ n - 1) / (m - 1)
-- Nat.geomSum_lt
lemma Nat.geomSum_lt (hm : 2 ≤ m) (hs : ∀ k ∈ s, k < n) : ∑ k ∈ s, m ^ k < m ^ n 
Arguments: {m n : ℕ} {s : Finset ℕ} (hm : 2 ≤ m) (hs : ∀ k ∈ s, k < n)
Goal: ∑ k ∈ s, m ^ k < m ^ n
-- ModuleCat.hasKernels_moduleCat
theorem hasKernels_moduleCat : HasKernels (ModuleCat R) 
Arguments: {R : Type u} [Ring R]
Goal: CategoryTheory.Limits.HasKernels (ModuleCat R)
-- ModuleCat.hasCokernels_moduleCat
theorem hasCokernels_moduleCat : HasCokernels (ModuleCat R) 
Arguments: {R : Type u} [Ring R]
Goal: CategoryTheory.Limits.HasCokernels (ModuleCat R)
-- ModuleCat.kernelIsoKer_inv_kernel_ι
@[simp, elementwise]
    -- Porting note (#11036): broken dot notation
theorem kernelIsoKer_inv_kernel_ι : (kernelIsoKer f).inv ≫ kernel.ι f =
    (LinearMap.ker f).subtype 
Arguments: {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H)
Goal: (ModuleCat.kernelIsoKer f).inv ≫ CategoryTheory.Limits.kernel.ι f = (LinearMap.ker f).subtype
-- ModuleCat.kernelIsoKer_hom_ker_subtype
@[simp, elementwise]
theorem kernelIsoKer_hom_ker_subtype :
    -- Porting note (#11036): broken dot notation
    (kernelIsoKer f).hom ≫ (LinearMap.ker f).subtype = kernel.ι f 
Arguments: {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H)
Goal: (ModuleCat.kernelIsoKer f).hom ≫ (LinearMap.ker f).subtype = CategoryTheory.Limits.kernel.ι f
-- ModuleCat.cokernel_π_cokernelIsoRangeQuotient_hom
@[simp, elementwise]
theorem cokernel_π_cokernelIsoRangeQuotient_hom :
    cokernel.π f ≫ (cokernelIsoRangeQuotient f).hom = f.range.mkQ 
Arguments: {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H)
Goal: CategoryTheory.Limits.cokernel.π f ≫ (ModuleCat.cokernelIsoRangeQuotient f).hom = (LinearMap.range f).mkQ
-- ModuleCat.range_mkQ_cokernelIsoRangeQuotient_inv
@[simp, elementwise]
theorem range_mkQ_cokernelIsoRangeQuotient_inv :
    ↿f.range.mkQ ≫ (cokernelIsoRangeQuotient f).inv = cokernel.π f 
Arguments: {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H)
Goal: ModuleCat.asHomLeft (LinearMap.range f).mkQ ≫ (ModuleCat.cokernelIsoRangeQuotient f).inv =
  CategoryTheory.Limits.cokernel.π f
-- ModuleCat.cokernel_π_ext
theorem cokernel_π_ext {M N : ModuleCat.{u} R} (f : M ⟶ N) {x y : N} (m : M) (w : x = y + f m) :
    cokernel.π f x = cokernel.π f y 
Arguments: {R : Type u} [Ring R] {M N : ModuleCat R} (f : M ⟶ N) {x y : ↑N} (m : ↑M) (w : x = y + f m)
Goal: (CategoryTheory.Limits.cokernel.π f) x = (CategoryTheory.Limits.cokernel.π f) y
-- ModuleCat.Derivation.d_add
@[simp]
lemma d_add (b b' : B) : D.d (b + b') = D.d b + D.d b' 
Arguments: {A B : CommRingCat} {M : ModuleCat ↑B} {f : A ⟶ B} (D : M.Derivation f) (b b' : ↑B)
Goal: D.d (b + b') = D.d b + D.d b'
-- ModuleCat.Derivation.d_mul
@[simp]
lemma d_mul (b b' : B) : D.d (b * b') = b • D.d b' + b' • D.d b 
Arguments: {A B : CommRingCat} {M : ModuleCat ↑B} {f : A ⟶ B} (D : M.Derivation f) (b b' : ↑B)
Goal: D.d (b * b') = b • D.d b' + b' • D.d b
-- ModuleCat.Derivation.d_map
@[simp]
lemma d_map (a : A) : D.d (f a) = 0 
Arguments: {A B : CommRingCat} {M : ModuleCat ↑B} {f : A ⟶ B} (D : M.Derivation f) (a : ↑A)
Goal: D.d (f a) = 0
-- CommRingCat.KaehlerDifferential.ext
@[ext]
lemma ext {M : ModuleCat B} {α β : KaehlerDifferential f ⟶ M}
    (h : ∀ (b : B), α (d b) = β (d b)) : α = β 
Arguments: {A B : CommRingCat} {f : A ⟶ B} {M : ModuleCat ↑B} {α β : CommRingCat.KaehlerDifferential f ⟶ M} (h : ∀ (b : ↑B), α (CommRingCat.KaehlerDifferential.d b) = β (CommRingCat.KaehlerDifferential.d b))
Goal: α = β
-- CommRingCat.KaehlerDifferential.map_d
@[simp]
lemma map_d (b : B) : map fac (d b) = d (g' b) 
Arguments: {A B A' B' : CommRingCat} {f : A ⟶ B} {f' : A' ⟶ B'} {g : A ⟶ A'} {g' : B ⟶ B'} (fac : g ≫ f' = f ≫ g') (b : ↑B)
Goal: (CommRingCat.KaehlerDifferential.map fac) (CommRingCat.KaehlerDifferential.d b) =
  CommRingCat.KaehlerDifferential.d (g' b)
-- ModuleCat.Derivation.desc_d
@[simp]
lemma desc_d (b : B) : D.desc (CommRingCat.KaehlerDifferential.d b) = D.d b 
Arguments: {A B : CommRingCat} {f : A ⟶ B} {M : ModuleCat ↑B} (D : M.Derivation f) (b : ↑B)
Goal: D.desc (CommRingCat.KaehlerDifferential.d b) = D.d b
-- StarSubalgebra.star_mem'
structure StarSubalgebra (R : Type u) (A : Type v) [CommSemiring R] [StarRing R] [Semiring A]
  [StarRing A] [Algebra R A] [StarModule R A] extends Subalgebra R A : Type v where
    star_mem' {a} : a ∈ carrier → star a ∈ carrier
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (self : StarSubalgebra R A) {a : A}
Goal: a ∈ self.carrier → star a ∈ self.carrier
-- StarSubalgebra.mem_carrier
@[simp, nolint simpNF] -- porting note (#10618): `simpNF` says `simp` can prove this, but it can't
theorem mem_carrier {s : StarSubalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] {s : StarSubalgebra R A} {x : A}
Goal: x ∈ s.carrier ↔ x ∈ s
-- StarSubalgebra.ext
@[ext]
theorem ext {S T : StarSubalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] {S T : StarSubalgebra R A} (h : ∀ (x : A), x ∈ S ↔ x ∈ T)
Goal: S = T
-- StarSubalgebra.coe_mk
@[simp]
lemma coe_mk (S : Subalgebra R A) (h) : ((⟨S, h⟩ : StarSubalgebra R A) : Set A) = S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : Subalgebra R A) (h : ∀ {a : A}, a ∈ S.carrier → star a ∈ S.carrier)
Goal: ↑{ toSubalgebra := S, star_mem' := h } = ↑S
-- StarSubalgebra.mem_toSubalgebra
@[simp]
theorem mem_toSubalgebra {S : StarSubalgebra R A} {x} : x ∈ S.toSubalgebra ↔ x ∈ S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] {S : StarSubalgebra R A} {x : A}
Goal: x ∈ S.toSubalgebra ↔ x ∈ S
-- StarSubalgebra.coe_toSubalgebra
@[simp]
theorem coe_toSubalgebra (S : StarSubalgebra R A) : (S.toSubalgebra : Set A) = S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A)
Goal: ↑S.toSubalgebra = ↑S
-- StarSubalgebra.toSubalgebra_injective
theorem toSubalgebra_injective :
    Function.Injective (toSubalgebra : StarSubalgebra R A → Subalgebra R A) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A]
Goal: Function.Injective StarSubalgebra.toSubalgebra
-- StarSubalgebra.toSubalgebra_inj
theorem toSubalgebra_inj {S U : StarSubalgebra R A} : S.toSubalgebra = U.toSubalgebra ↔ S = U 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] {S U : StarSubalgebra R A}
Goal: S.toSubalgebra = U.toSubalgebra ↔ S = U
-- StarSubalgebra.toSubalgebra_le_iff
theorem toSubalgebra_le_iff {S₁ S₂ : StarSubalgebra R A} :
    S₁.toSubalgebra ≤ S₂.toSubalgebra ↔ S₁ ≤ S₂ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] {S₁ S₂ : StarSubalgebra R A}
Goal: S₁.toSubalgebra ≤ S₂.toSubalgebra ↔ S₁ ≤ S₂
-- StarSubalgebra.coe_copy
@[simp]
theorem coe_copy (S : StarSubalgebra R A) (s : Set A) (hs : s = ↑S) : (S.copy s hs : Set A) = s 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A) (s : Set A) (hs : s = ↑S)
Goal: ↑(S.copy s hs) = s
-- StarSubalgebra.copy_eq
theorem copy_eq (S : StarSubalgebra R A) (s : Set A) (hs : s = ↑S) : S.copy s hs = S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A) (s : Set A) (hs : s = ↑S)
Goal: S.copy s hs = S
-- StarSubalgebra.algebraMap_mem
theorem algebraMap_mem (r : R) : algebraMap R A r ∈ S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A) (r : R)
Goal: (algebraMap R A) r ∈ S
-- StarSubalgebra.rangeS_le
theorem rangeS_le : (algebraMap R A).rangeS ≤ S.toSubalgebra.toSubsemiring 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A)
Goal: (algebraMap R A).rangeS ≤ S.toSubsemiring
-- StarSubalgebra.range_subset
theorem range_subset : Set.range (algebraMap R A) ⊆ S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A)
Goal: Set.range ⇑(algebraMap R A) ⊆ ↑S
-- StarSubalgebra.range_le
theorem range_le : Set.range (algebraMap R A) ≤ S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A)
Goal: Set.range ⇑(algebraMap R A) ≤ ↑S
-- StarSubalgebra.smul_mem
protected theorem smul_mem {x : A} (hx : x ∈ S) (r : R) : r • x ∈ S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A) {x : A} (hx : x ∈ S) (r : R)
Goal: r • x ∈ S
-- StarSubalgebra.coe_subtype
@[simp]
theorem coe_subtype : (S.subtype : S → A) = Subtype.val 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A)
Goal: ⇑S.subtype = Subtype.val
-- StarSubalgebra.subtype_apply
theorem subtype_apply (x : S) : S.subtype x = (x : A) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A) (x : ↥S)
Goal: S.subtype x = ↑x
-- StarSubalgebra.toSubalgebra_subtype
@[simp]
theorem toSubalgebra_subtype : S.toSubalgebra.val = S.subtype.toAlgHom 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A)
Goal: S.val = S.subtype.toAlgHom
-- StarSubalgebra.inclusion_apply
@[simps]
def inclusion {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂) : S₁ →⋆ₐ[R] S₂ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂)
Goal: ∀ (a : ↥S₁), (StarSubalgebra.inclusion h) a = Subtype.map id h a
-- StarSubalgebra.inclusion_injective
theorem inclusion_injective {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂) :
    Function.Injective <| inclusion h 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂)
Goal: Function.Injective ⇑(StarSubalgebra.inclusion h)
-- StarSubalgebra.subtype_comp_inclusion
@[simp]
theorem subtype_comp_inclusion {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂) :
    S₂.subtype.comp (inclusion h) = S₁.subtype 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂)
Goal: S₂.subtype.comp (StarSubalgebra.inclusion h) = S₁.subtype
-- StarSubalgebra.map_mono
theorem map_mono {S₁ S₂ : StarSubalgebra R A} {f : A →⋆ₐ[R] B} : S₁ ≤ S₂ → S₁.map f ≤ S₂.map f 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] {S₁ S₂ : StarSubalgebra R A} {f : A →⋆ₐ[R] B}
Goal: S₁ ≤ S₂ → StarSubalgebra.map f S₁ ≤ StarSubalgebra.map f S₂
-- StarSubalgebra.map_injective
theorem map_injective {f : A →⋆ₐ[R] B} (hf : Function.Injective f) : Function.Injective (map f) 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] {f : A →⋆ₐ[R] B} (hf : Function.Injective ⇑f)
Goal: Function.Injective (StarSubalgebra.map f)
-- StarSubalgebra.map_id
@[simp]
theorem map_id (S : StarSubalgebra R A) : S.map (StarAlgHom.id R A) = S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A)
Goal: StarSubalgebra.map (StarAlgHom.id R A) S = S
-- StarSubalgebra.map_map
theorem map_map (S : StarSubalgebra R A) (g : B →⋆ₐ[R] C) (f : A →⋆ₐ[R] B) :
    (S.map f).map g = S.map (g.comp f) 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] [Semiring C] [StarRing C] [Algebra R C] [StarModule R C] (S : StarSubalgebra R A) (g : B →⋆ₐ[R] C) (f : A →⋆ₐ[R] B)
Goal: StarSubalgebra.map g (StarSubalgebra.map f S) = StarSubalgebra.map (g.comp f) S
-- StarSubalgebra.mem_map
@[simp]
theorem mem_map {S : StarSubalgebra R A} {f : A →⋆ₐ[R] B} {y : B} :
    y ∈ map f S ↔ ∃ x ∈ S, f x = y 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] {S : StarSubalgebra R A} {f : A →⋆ₐ[R] B} {y : B}
Goal: y ∈ StarSubalgebra.map f S ↔ ∃ x ∈ S, f x = y
-- StarSubalgebra.map_toSubalgebra
theorem map_toSubalgebra {S : StarSubalgebra R A} {f : A →⋆ₐ[R] B} :
    (S.map f).toSubalgebra = S.toSubalgebra.map f.toAlgHom 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] {S : StarSubalgebra R A} {f : A →⋆ₐ[R] B}
Goal: (StarSubalgebra.map f S).toSubalgebra = Subalgebra.map f.toAlgHom S.toSubalgebra
-- StarSubalgebra.coe_map
@[simp]
theorem coe_map (S : StarSubalgebra R A) (f : A →⋆ₐ[R] B) : (S.map f : Set B) = f '' S 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] (S : StarSubalgebra R A) (f : A →⋆ₐ[R] B)
Goal: ↑(StarSubalgebra.map f S) = ⇑f '' ↑S
-- StarSubalgebra.map_le_iff_le_comap
theorem map_le_iff_le_comap {S : StarSubalgebra R A} {f : A →⋆ₐ[R] B} {U : StarSubalgebra R B} :
    map f S ≤ U ↔ S ≤ comap f U 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] {S : StarSubalgebra R A} {f : A →⋆ₐ[R] B} {U : StarSubalgebra R B}
Goal: StarSubalgebra.map f S ≤ U ↔ S ≤ StarSubalgebra.comap f U
-- StarSubalgebra.gc_map_comap
theorem gc_map_comap (f : A →⋆ₐ[R] B) : GaloisConnection (map f) (comap f) 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] (f : A →⋆ₐ[R] B)
Goal: GaloisConnection (StarSubalgebra.map f) (StarSubalgebra.comap f)
-- StarSubalgebra.comap_mono
theorem comap_mono {S₁ S₂ : StarSubalgebra R B} {f : A →⋆ₐ[R] B} :
    S₁ ≤ S₂ → S₁.comap f ≤ S₂.comap f 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] {S₁ S₂ : StarSubalgebra R B} {f : A →⋆ₐ[R] B}
Goal: S₁ ≤ S₂ → StarSubalgebra.comap f S₁ ≤ StarSubalgebra.comap f S₂
-- StarSubalgebra.comap_injective
theorem comap_injective {f : A →⋆ₐ[R] B} (hf : Function.Surjective f) :
    Function.Injective (comap f) 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] {f : A →⋆ₐ[R] B} (hf : Function.Surjective ⇑f)
Goal: Function.Injective (StarSubalgebra.comap f)
-- StarSubalgebra.comap_id
@[simp]
theorem comap_id (S : StarSubalgebra R A) : S.comap (StarAlgHom.id R A) = S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (S : StarSubalgebra R A)
Goal: StarSubalgebra.comap (StarAlgHom.id R A) S = S
-- StarSubalgebra.comap_comap
theorem comap_comap (S : StarSubalgebra R C) (g : B →⋆ₐ[R] C) (f : A →⋆ₐ[R] B) :
    (S.comap g).comap f = S.comap (g.comp f) 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} {C : Type u_5} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] [Semiring C] [StarRing C] [Algebra R C] [StarModule R C] (S : StarSubalgebra R C) (g : B →⋆ₐ[R] C) (f : A →⋆ₐ[R] B)
Goal: StarSubalgebra.comap f (StarSubalgebra.comap g S) = StarSubalgebra.comap (g.comp f) S
-- StarSubalgebra.mem_comap
@[simp]
theorem mem_comap (S : StarSubalgebra R B) (f : A →⋆ₐ[R] B) (x : A) : x ∈ S.comap f ↔ f x ∈ S 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] (S : StarSubalgebra R B) (f : A →⋆ₐ[R] B) (x : A)
Goal: x ∈ StarSubalgebra.comap f S ↔ f x ∈ S
-- StarSubalgebra.coe_comap
@[simp, norm_cast]
theorem coe_comap (S : StarSubalgebra R B) (f : A →⋆ₐ[R] B) :
    (S.comap f : Set A) = f ⁻¹' (S : Set B) 
Arguments: {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] (S : StarSubalgebra R B) (f : A →⋆ₐ[R] B)
Goal: ↑(StarSubalgebra.comap f S) = ⇑f ⁻¹' ↑S
-- StarSubalgebra.coe_centralizer
@[simp, norm_cast]
theorem coe_centralizer (s : Set A) : (centralizer R s : Set A) = (s ∪ star s).centralizer 
Arguments: (R : Type u_2) {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (s : Set A)
Goal: ↑(StarSubalgebra.centralizer R s) = (s ∪ star s).centralizer
-- StarSubalgebra.mem_centralizer_iff
theorem mem_centralizer_iff {s : Set A} {z : A} :
    z ∈ centralizer R s ↔ ∀ g ∈ s, g * z = z * g ∧ star g * z = z * star g 
Arguments: (R : Type u_2) {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] {s : Set A} {z : A}
Goal: z ∈ StarSubalgebra.centralizer R s ↔ ∀ g ∈ s, g * z = z * g ∧ star g * z = z * star g
-- StarSubalgebra.centralizer_le
theorem centralizer_le (s t : Set A) (h : s ⊆ t) : centralizer R t ≤ centralizer R s 
Arguments: (R : Type u_2) {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] (s t : Set A) (h : s ⊆ t)
Goal: StarSubalgebra.centralizer R t ≤ StarSubalgebra.centralizer R s
-- Subalgebra.mem_star_iff
@[simp]
theorem mem_star_iff (S : Subalgebra R A) (x : A) : x ∈ star S ↔ star x ∈ S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (S : Subalgebra R A) (x : A)
Goal: x ∈ star S ↔ star x ∈ S
-- Subalgebra.star_mem_star_iff
theorem star_mem_star_iff (S : Subalgebra R A) (x : A) : star x ∈ star S ↔ x ∈ S 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [StarRing R] [Semiring A] [Algebra R A] [StarRing A] [StarModule R A] (S : Subalgebra R A) (x : A)
Goal: star x ∈ star S ↔ x ∈ S