ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- BialgHom.congr_arg
protected theorem congr_arg (φ : A →ₐc[R] B) {x y : A} (h : x = y) : φ x = φ y 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₐc[R] B) {x y : A} (h : x = y)
Goal: φ x = φ y
-- BialgHom.ext
@[ext]
theorem ext {φ₁ φ₂ : A →ₐc[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ φ₂ : A →ₐc[R] B} (H : ∀ (x : A), φ₁ x = φ₂ x)
Goal: φ₁ = φ₂
-- BialgHom.ext_iff
theorem ext_iff {φ₁ φ₂ : A →ₐc[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ φ₂ : A →ₐc[R] B}
Goal: φ₁ = φ₂ ↔ ∀ (x : A), φ₁ x = φ₂ x
-- BialgHom.ext_of_ring
@[ext high]
theorem ext_of_ring {f g : R →ₐc[R] A} (h : f 1 = g 1) : f = g 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] {f g : R →ₐc[R] A} (h : f 1 = g 1)
Goal: f = g
-- BialgHom.mk_coe
@[simp]
theorem mk_coe {f : A →ₐc[R] B} (h₀ h₁ h₂ h₃ h₄ h₅) :
    (⟨⟨⟨⟨f, h₀⟩, h₁⟩, h₂, h₃⟩, h₄, h₅⟩ : A →ₐc[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A →ₐc[R] B} (h₀ : ∀ (x y : A), f (x + y) = f x + f y) (h₁ :
  ∀ (m : R) (x : A),
    { toFun := ⇑f, map_add' := h₀ }.toFun (m • x) = (RingHom.id R) m • { toFun := ⇑f, map_add' := h₀ }.toFun x) (h₂ : Coalgebra.counit ∘ₗ { toFun := ⇑f, map_add' := h₀, map_smul' := h₁ } = Coalgebra.counit) (h₃ :
  TensorProduct.map { toFun := ⇑f, map_add' := h₀, map_smul' := h₁ } { toFun := ⇑f, map_add' := h₀, map_smul' := h₁ } ∘ₗ
      Coalgebra.comul =
    Coalgebra.comul ∘ₗ { toFun := ⇑f, map_add' := h₀, map_smul' := h₁ }) (h₄ : { toFun := ⇑f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun 1 = 1) (h₅ :
  ∀ (x y : A),
    { toFun := ⇑f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun (x * y) =
      { toFun := ⇑f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun x *
        { toFun := ⇑f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun y)
Goal: { toFun := ⇑f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃, map_one' := h₄,
    map_mul' := h₅ } =
  f
-- BialgHom.coe_copy
@[simp]
theorem coe_copy (f : A →ₗc[R] B) (f' : A → B) (h : f' = ⇑f) : ⇑(f.copy f' h) = f' 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B) (f' : A → B) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- BialgHom.copy_eq
theorem copy_eq (f : A →ₗc[R] B) (f' : A → B) (h : f' = ⇑f) : f.copy f' h = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₗc[R] B) (f' : A → B) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- BialgHom.id_apply
@[simps!] protected def id : A →ₐc[R] A 
Arguments: (R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] (a : A)
Goal: (BialgHom.id R A) a = a
-- BialgHom.coe_id
@[simp]
theorem coe_id : ⇑(BialgHom.id R A) = id 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A]
Goal: ⇑(BialgHom.id R A) = id
-- BialgHom.id_toCoalgHom
@[simp]
theorem id_toCoalgHom : BialgHom.id R A = CoalgHom.id R A 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A]
Goal: ↑(BialgHom.id R A) = CoalgHom.id R A
-- BialgHom.id_toAlgHom
@[simp]
theorem id_toAlgHom : BialgHom.id R A = AlgHom.id R A 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A]
Goal: ↑(BialgHom.id R A) = AlgHom.id R A
-- BialgHom.comp_apply
@[simps!] def comp (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B) : A →ₐc[R] C 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B)
Goal: ∀ (a : A), (φ₁.comp φ₂) a = φ₁ (φ₂ a)
-- BialgHom.coe_comp
@[simp]
theorem coe_comp (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B) : ⇑(φ₁.comp φ₂) = φ₁ ∘ φ₂ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B)
Goal: ⇑(φ₁.comp φ₂) = ⇑φ₁ ∘ ⇑φ₂
-- BialgHom.comp_toCoalgHom
@[simp]
theorem comp_toCoalgHom (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B) :
    φ₁.comp φ₂ = (φ₁ : B →ₗc[R] C).comp (φ₂ : A →ₗc[R] B) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B)
Goal: ↑(φ₁.comp φ₂) = (↑φ₁).comp ↑φ₂
-- BialgHom.comp_toAlgHom
@[simp]
theorem comp_toAlgHom (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B) :
    φ₁.comp φ₂ = (φ₁ : B →ₐ[R] C).comp (φ₂ : A →ₐ[R] B) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (φ₁ : B →ₐc[R] C) (φ₂ : A →ₐc[R] B)
Goal: ↑(φ₁.comp φ₂) = (↑φ₁).comp ↑φ₂
-- BialgHom.comp_id
@[simp]
theorem comp_id : φ.comp (BialgHom.id R A) = φ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₐc[R] B)
Goal: φ.comp (BialgHom.id R A) = φ
-- BialgHom.id_comp
@[simp]
theorem id_comp : (BialgHom.id R B).comp φ = φ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₐc[R] B)
Goal: (BialgHom.id R B).comp φ = φ
-- BialgHom.comp_assoc
theorem comp_assoc (φ₁ : C →ₐc[R] D) (φ₂ : B →ₐc[R] C) (φ₃ : A →ₐc[R] B) :
    (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {C : Type u_4} {D : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Semiring D] [Algebra R D] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] [CoalgebraStruct R D] (φ₁ : C →ₐc[R] D) (φ₂ : B →ₐc[R] C) (φ₃ : A →ₐc[R] B)
Goal: (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃)
-- BialgHom.map_smul_of_tower
theorem map_smul_of_tower {R'} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R')
    (x : A) : φ (r • x) = r • φ x 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (φ : A →ₐc[R] B) {R' : Type u_6} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R') (x : A)
Goal: φ (r • x) = r • φ x
-- BialgHom.End_toOne_one
@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]
instance End : Monoid (A →ₐc[R] A) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A]
Goal: 1 = BialgHom.id R A
-- BialgHom.End_toSemigroup_toMul_mul
@[simps (config := .lemmasOnly) toSemigroup_toMul_mul toOne_one]
instance End : Monoid (A →ₐc[R] A) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] (φ₁ φ₂ : A →ₐc[R] A)
Goal: φ₁ * φ₂ = φ₁.comp φ₂
-- BialgHom.one_apply
@[simp]
theorem one_apply (x : A) : (1 : A →ₐc[R] A) x = x 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] (x : A)
Goal: 1 x = x
-- BialgHom.mul_apply
@[simp]
theorem mul_apply (φ ψ : A →ₐc[R] A) (x : A) : (φ * ψ) x = φ (ψ x) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A] (φ ψ : A →ₐc[R] A) (x : A)
Goal: (φ * ψ) x = φ (ψ x)
-- Bialgebra.counitBialgHom_apply
@[simp]
theorem counitBialgHom_apply (x : A) :
    counitBialgHom R A x = Coalgebra.counit x 
Arguments: (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Bialgebra R A] (x : A)
Goal: (Bialgebra.counitBialgHom R A) x = Coalgebra.counit x
-- Bialgebra.counitBialgHom_toCoalgHom
@[simp]
theorem counitBialgHom_toCoalgHom :
    counitBialgHom R A = Coalgebra.counitCoalgHom R A 
Arguments: (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Bialgebra R A]
Goal: ↑(Bialgebra.counitBialgHom R A) = Coalgebra.counitCoalgHom R A
-- Bialgebra.ext_to_ring
@[ext high]
theorem ext_to_ring (f g : A →ₐc[R] R) : f = g 
Arguments: {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Bialgebra R A] (f g : A →ₐc[R] R)
Goal: f = g
-- Ideal.eq_span_singleton_of_mem_of_not_mem_sq_of_not_mem_prime_ne
theorem Ideal.eq_span_singleton_of_mem_of_not_mem_sq_of_not_mem_prime_ne {P : Ideal R}
    (hP : P.IsPrime) [IsDedekindDomain R] {x : R} (x_mem : x ∈ P) (hxP2 : x ∉ P ^ 2)
    (hxQ : ∀ Q : Ideal R, IsPrime Q → Q ≠ P → x ∉ Q) : P = Ideal.span {x} 
Arguments: {R : Type u_1} [CommRing R] {P : Ideal R} (hP : P.IsPrime) [IsDedekindDomain R] {x : R} (x_mem : x ∈ P) (hxP2 : x ∉ P ^ 2) (hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q)
Goal: P = Ideal.span {x}
-- FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_top
theorem FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_top {R A : Type*}
    [CommRing R] [CommRing A] [Algebra R A] {S : Submonoid R} [IsLocalization S A]
    (I : (FractionalIdeal S A)ˣ) {v : A} (hv : v ∈ (↑I⁻¹ : FractionalIdeal S A))
    (h : Submodule.comap (Algebra.linearMap R A) ((I : Submodule R A) * Submodule.span R {v}) = ⊤) :
    Submodule.IsPrincipal (I : Submodule R A) 
Arguments: {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {S : Submonoid R} [IsLocalization S A] (I : (FractionalIdeal S A)ˣ) {v : A} (hv : v ∈ ↑I⁻¹) (h : Submodule.comap (Algebra.linearMap R A) (↑↑I * Submodule.span R {v}) = ⊤)
Goal: (↑↑I).IsPrincipal
-- FractionalIdeal.isPrincipal.of_finite_maximals_of_inv
theorem FractionalIdeal.isPrincipal.of_finite_maximals_of_inv {A : Type*} [CommRing A]
    [Algebra R A] {S : Submonoid R} [IsLocalization S A] (hS : S ≤ R⁰)
    (hf : {I : Ideal R | I.IsMaximal}.Finite) (I I' : FractionalIdeal S A) (hinv : I * I' = 1) :
    Submodule.IsPrincipal (I : Submodule R A) 
Arguments: {R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {S : Submonoid R} [IsLocalization S A] (hS : S ≤ R⁰) (hf : {I | I.IsMaximal}.Finite) (I I' : FractionalIdeal S A) (hinv : I * I' = 1)
Goal: (↑I).IsPrincipal
-- Ideal.IsPrincipal.of_finite_maximals_of_isUnit
theorem Ideal.IsPrincipal.of_finite_maximals_of_isUnit (hf : {I : Ideal R | I.IsMaximal}.Finite)
    {I : Ideal R} (hI : IsUnit (I : FractionalIdeal R⁰ (FractionRing R))) : I.IsPrincipal 
Arguments: {R : Type u_1} [CommRing R] (hf : {I | I.IsMaximal}.Finite) {I : Ideal R} (hI : IsUnit ↑I)
Goal: Submodule.IsPrincipal I
-- IsPrincipalIdealRing.of_finite_primes
theorem IsPrincipalIdealRing.of_finite_primes [IsDedekindDomain R]
    (h : {I : Ideal R | I.IsPrime}.Finite) : IsPrincipalIdealRing R 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (h : {I | I.IsPrime}.Finite)
Goal: IsPrincipalIdealRing R
-- IsLocalization.OverPrime.mem_normalizedFactors_of_isPrime
theorem IsLocalization.OverPrime.mem_normalizedFactors_of_isPrime [IsDomain S]
    {P : Ideal Sₚ} (hP : IsPrime P) (hP0 : P ≠ ⊥) :
    P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p) 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (S : Type u_2) [CommRing S] [Algebra R S] [Module.Free R S] [Module.Finite R S] (p : Ideal R) (hp0 : p ≠ ⊥) [p.IsPrime] {Sₚ : Type u_3} [CommRing Sₚ] [Algebra S Sₚ] [IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] [Algebra R Sₚ] [IsScalarTower R S Sₚ] [IsDedekindDomain Sₚ] [IsDomain S] {P : Ideal Sₚ} (hP : P.IsPrime) (hP0 : P ≠ ⊥)
Goal: P ∈ UniqueFactorizationMonoid.normalizedFactors (Ideal.map (algebraMap R Sₚ) p)
-- IsDedekindDomain.isPrincipalIdealRing_localization_over_prime
theorem IsDedekindDomain.isPrincipalIdealRing_localization_over_prime [IsDomain S] :
    IsPrincipalIdealRing Sₚ 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (S : Type u_2) [CommRing S] [Algebra R S] [Module.Free R S] [Module.Finite R S] (p : Ideal R) (hp0 : p ≠ ⊥) [p.IsPrime] {Sₚ : Type u_3} [CommRing Sₚ] [Algebra S Sₚ] [IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] [Algebra R Sₚ] [IsScalarTower R S Sₚ] [IsDedekindDomain Sₚ] [IsDomain S]
Goal: IsPrincipalIdealRing Sₚ
-- RingHom.PropertyIsLocal.HoldsForLocalizationAway
structure RingHom.PropertyIsLocal : Prop where
  LocalizationPreserves : RingHom.LocalizationPreserves @P
  OfLocalizationSpanTarget : RingHom.OfLocalizationSpanTarget @P
  StableUnderComposition : RingHom.StableUnderComposition @P
  HoldsForLocalizationAway : RingHom.HoldsForLocalizationAway @P
Arguments: {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} (self : RingHom.PropertyIsLocal P)
Goal: RingHom.HoldsForLocalizationAway P
-- RingHom.PropertyIsLocal.OfLocalizationSpanTarget
structure RingHom.PropertyIsLocal : Prop where
  LocalizationPreserves : RingHom.LocalizationPreserves @P
  OfLocalizationSpanTarget : RingHom.OfLocalizationSpanTarget @P
  StableUnderComposition : RingHom.StableUnderComposition @P
  HoldsForLocalizationAway : RingHom.HoldsForLocalizationAway @P
Arguments: {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} (self : RingHom.PropertyIsLocal P)
Goal: RingHom.OfLocalizationSpanTarget P
-- RingHom.PropertyIsLocal.StableUnderComposition
structure RingHom.PropertyIsLocal : Prop where
  LocalizationPreserves : RingHom.LocalizationPreserves @P
  OfLocalizationSpanTarget : RingHom.OfLocalizationSpanTarget @P
  StableUnderComposition : RingHom.StableUnderComposition @P
  HoldsForLocalizationAway : RingHom.HoldsForLocalizationAway @P
Arguments: {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} (self : RingHom.PropertyIsLocal P)
Goal: RingHom.StableUnderComposition P
-- RingHom.PropertyIsLocal.LocalizationPreserves
structure RingHom.PropertyIsLocal : Prop where
  LocalizationPreserves : RingHom.LocalizationPreserves @P
  OfLocalizationSpanTarget : RingHom.OfLocalizationSpanTarget @P
  StableUnderComposition : RingHom.StableUnderComposition @P
  HoldsForLocalizationAway : RingHom.HoldsForLocalizationAway @P
Arguments: {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} (self : RingHom.PropertyIsLocal P)
Goal: RingHom.LocalizationPreserves P
-- RingHom.ofLocalizationSpan_iff_finite
theorem RingHom.ofLocalizationSpan_iff_finite :
    RingHom.OfLocalizationSpan @P ↔ RingHom.OfLocalizationFiniteSpan @P 
Arguments: (P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop)
Goal: RingHom.OfLocalizationSpan P ↔ RingHom.OfLocalizationFiniteSpan P
-- RingHom.ofLocalizationSpanTarget_iff_finite
theorem RingHom.ofLocalizationSpanTarget_iff_finite :
    RingHom.OfLocalizationSpanTarget @P ↔ RingHom.OfLocalizationFiniteSpanTarget @P 
Arguments: (P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop)
Goal: RingHom.OfLocalizationSpanTarget P ↔ RingHom.OfLocalizationFiniteSpanTarget P
-- RingHom.HoldsForLocalizationAway.of_bijective
theorem RingHom.HoldsForLocalizationAway.of_bijective
    (H : RingHom.HoldsForLocalizationAway P) (hf : Function.Bijective f) :
    P f 
Arguments: {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) (P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop) (H : RingHom.HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] => P) (hf : Function.Bijective ⇑f)
Goal: P f
-- RingHom.PropertyIsLocal.respectsIso
theorem RingHom.PropertyIsLocal.respectsIso (hP : RingHom.PropertyIsLocal @P) :
    RingHom.RespectsIso @P 
Arguments: {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} (hP : RingHom.PropertyIsLocal P)
Goal: RingHom.RespectsIso P
-- RingHom.LocalizationPreserves.away
theorem RingHom.LocalizationPreserves.away (H : RingHom.LocalizationPreserves @P) (r : R)
    [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] (hf : P f) :
    P (IsLocalization.Away.map R' S' f r) 
Arguments: {R S : Type u} [CommRing R] [CommRing S] {R' S' : Type u} [CommRing R'] [CommRing S'] {f : R →+* S} [Algebra R R'] [Algebra S S'] {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} (H : RingHom.LocalizationPreserves P) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] (hf : P f)
Goal: P (IsLocalization.Away.map R' S' f r)
-- RingHom.PropertyIsLocal.ofLocalizationSpan
theorem RingHom.PropertyIsLocal.ofLocalizationSpan (hP : RingHom.PropertyIsLocal @P) :
    RingHom.OfLocalizationSpan @P 
Arguments: {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} (hP : RingHom.PropertyIsLocal P)
Goal: RingHom.OfLocalizationSpan P
-- RingHom.OfLocalizationSpanTarget.ofIsLocalization
lemma RingHom.OfLocalizationSpanTarget.ofIsLocalization
    (hP : RingHom.OfLocalizationSpanTarget P) (hP' : RingHom.RespectsIso P)
    {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) (s : Set S) (hs : Ideal.span s = ⊤)
    (hT : ∀ r : s, ∃ (T : Type u) (_ : CommRing T) (_ : Algebra S T)
      (_ : IsLocalization.Away (r : S) T), P ((algebraMap S T).comp f)) : P f 
Arguments: {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} (hP : RingHom.OfLocalizationSpanTarget fun {R S} [CommRing R] [CommRing S] => P) (hP' : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => P) {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) (s : Set S) (hs : Ideal.span s = ⊤) (hT : ∀ (r : ↑s), ∃ T x x_1, ∃ (_ : IsLocalization.Away (↑r) T), P ((algebraMap S T).comp f))
Goal: P f
-- Ideal.le_of_localization_maximal
theorem Ideal.le_of_localization_maximal {I J : Ideal R}
    (h : ∀ (P : Ideal R) (hP : P.IsMaximal),
      Ideal.map (algebraMap R (Localization.AtPrime P)) I ≤
        Ideal.map (algebraMap R (Localization.AtPrime P)) J) :
    I ≤ J 
Arguments: {R : Type u} [CommRing R] {I J : Ideal R} (h :
  ∀ (P : Ideal R) (hP : P.IsMaximal),
    Ideal.map (algebraMap R (Localization.AtPrime P)) I ≤ Ideal.map (algebraMap R (Localization.AtPrime P)) J)
Goal: I ≤ J
-- Ideal.eq_of_localization_maximal
theorem Ideal.eq_of_localization_maximal {I J : Ideal R}
    (h : ∀ (P : Ideal R) (_ : P.IsMaximal),
      Ideal.map (algebraMap R (Localization.AtPrime P)) I =
        Ideal.map (algebraMap R (Localization.AtPrime P)) J) :
    I = J 
Arguments: {R : Type u} [CommRing R] {I J : Ideal R} (h :
  ∀ (P : Ideal R) (x : P.IsMaximal),
    Ideal.map (algebraMap R (Localization.AtPrime P)) I = Ideal.map (algebraMap R (Localization.AtPrime P)) J)
Goal: I = J
-- ideal_eq_bot_of_localization'
theorem ideal_eq_bot_of_localization' (I : Ideal R)
    (h : ∀ (J : Ideal R) (hJ : J.IsMaximal),
      Ideal.map (algebraMap R (Localization.AtPrime J)) I = ⊥) :
    I = ⊥ 
Arguments: {R : Type u} [CommRing R] (I : Ideal R) (h : ∀ (J : Ideal R) (hJ : J.IsMaximal), Ideal.map (algebraMap R (Localization.AtPrime J)) I = ⊥)
Goal: I = ⊥
-- ideal_eq_bot_of_localization
theorem ideal_eq_bot_of_localization (I : Ideal R)
    (h : ∀ (J : Ideal R) (hJ : J.IsMaximal),
      IsLocalization.coeSubmodule (Localization.AtPrime J) I = ⊥) :
    I = ⊥ 
Arguments: {R : Type u} [CommRing R] (I : Ideal R) (h : ∀ (J : Ideal R) (hJ : J.IsMaximal), IsLocalization.coeSubmodule (Localization.AtPrime J) I = ⊥)
Goal: I = ⊥
-- eq_zero_of_localization
theorem eq_zero_of_localization (r : R)
    (h : ∀ (J : Ideal R) (hJ : J.IsMaximal), algebraMap R (Localization.AtPrime J) r = 0) :
    r = 0 
Arguments: {R : Type u} [CommRing R] (r : R) (h : ∀ (J : Ideal R) (hJ : J.IsMaximal), (algebraMap R (Localization.AtPrime J)) r = 0)
Goal: r = 0
-- localization_isReduced
theorem localization_isReduced : LocalizationPreserves fun R hR => IsReduced R 
Goal: LocalizationPreserves fun R hR => IsReduced R
-- isReduced_ofLocalizationMaximal
theorem isReduced_ofLocalizationMaximal : OfLocalizationMaximal fun R hR => IsReduced R 
Goal: OfLocalizationMaximal fun R hR => IsReduced R
-- localizationPreserves_surjective
theorem localizationPreserves_surjective :
    RingHom.LocalizationPreserves fun {R S} _ _ f => Function.Surjective f 
Goal: RingHom.LocalizationPreserves fun {R S} x x_1 f => Function.Surjective ⇑f
-- surjective_ofLocalizationSpan
theorem surjective_ofLocalizationSpan :
    RingHom.OfLocalizationSpan fun {R S} _ _ f => Function.Surjective f 
Goal: RingHom.OfLocalizationSpan fun {R S} x x_1 f => Function.Surjective ⇑f
-- surjective_localRingHom_of_surjective
theorem surjective_localRingHom_of_surjective (h : Function.Surjective f) (P : Ideal S)
    [P.IsPrime] : Function.Surjective (Localization.localRingHom (P.comap f) P f rfl) 
Arguments: {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) (h : Function.Surjective ⇑f) (P : Ideal S) [P.IsPrime]
Goal: Function.Surjective ⇑(Localization.localRingHom (Ideal.comap f P) P f ⋯)
-- surjective_respectsIso
lemma surjective_respectsIso : RingHom.RespectsIso (fun f ↦ Function.Surjective f) 
Goal: RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] f => Function.Surjective ⇑f
-- Module.Finite_of_isLocalization
lemma Module.Finite_of_isLocalization (R S Rₚ Sₚ) [CommSemiring R] [CommRing S] [CommRing Rₚ]
    [CommRing Sₚ] [Algebra R S] [Algebra R Rₚ] [Algebra R Sₚ] [Algebra S Sₚ] [Algebra Rₚ Sₚ]
    [IsScalarTower R S Sₚ] [IsScalarTower R Rₚ Sₚ] (M : Submonoid R) [IsLocalization M Rₚ]
    [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₚ] [hRS : Module.Finite R S] :
    Module.Finite Rₚ Sₚ 
Arguments: (R : Type u_1) (S : Type u_2) (Rₚ : Type u_3) (Sₚ : Type u_4) [CommSemiring R] [CommRing S] [CommRing Rₚ] [CommRing Sₚ] [Algebra R S] [Algebra R Rₚ] [Algebra R Sₚ] [Algebra S Sₚ] [Algebra Rₚ Sₚ] [IsScalarTower R S Sₚ] [IsScalarTower R Rₚ Sₚ] (M : Submonoid R) [IsLocalization M Rₚ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₚ] [hRS : Module.Finite R S]
Goal: Module.Finite Rₚ Sₚ
-- localization_finite
theorem localization_finite : RingHom.LocalizationPreserves @RingHom.Finite 
Goal: RingHom.LocalizationPreserves @RingHom.Finite
-- localization_away_map_finite
theorem localization_away_map_finite (r : R) [IsLocalization.Away r R']
    [IsLocalization.Away (f r) S'] (hf : f.Finite) : (IsLocalization.Away.map R' S' f r).Finite 
Arguments: {R S : Type u} [CommRing R] [CommRing S] (R' S' : Type u) [CommRing R'] [CommRing S'] (f : R →+* S) [Algebra R R'] [Algebra S S'] (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] (hf : f.Finite)
Goal: (IsLocalization.Away.map R' S' f r).Finite
-- IsLocalization.smul_mem_finsetIntegerMultiple_span
theorem IsLocalization.smul_mem_finsetIntegerMultiple_span [Algebra R S] [Algebra R S']
    [IsScalarTower R S S'] [IsLocalization (M.map (algebraMap R S)) S'] (x : S) (s : Finset S')
    (hx : algebraMap S S' x ∈ Submodule.span R (s : Set S')) :
    ∃ m : M, m • x ∈
      Submodule.span R
        (IsLocalization.finsetIntegerMultiple (M.map (algebraMap R S)) s : Set S) 
Arguments: {R S : Type u} [CommRing R] [CommRing S] (M : Submonoid R) (S' : Type u) [CommRing S'] [Algebra S S'] [Algebra R S] [Algebra R S'] [IsScalarTower R S S'] [IsLocalization (Submonoid.map (algebraMap R S) M) S'] (x : S) (s : Finset S') (hx : (algebraMap S S') x ∈ Submodule.span R ↑s)
Goal: ∃ m, m • x ∈ Submodule.span R ↑(IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)