ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- FractionalIdeal.mem_spanSingleton
@[simp]
theorem mem_spanSingleton {x y : P} : x ∈ spanSingleton S y ↔ ∃ z : R, z • y = x 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] {x y : P}
Goal: x ∈ FractionalIdeal.spanSingleton S y ↔ ∃ z, z • y = x
-- FractionalIdeal.mem_spanSingleton_self
theorem mem_spanSingleton_self (x : P) : x ∈ spanSingleton S x 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] (x : P)
Goal: x ∈ FractionalIdeal.spanSingleton S x
-- FractionalIdeal.den_mul_self_eq_num'
theorem den_mul_self_eq_num' (I : FractionalIdeal S P) :
    spanSingleton S (algebraMap R P I.den) * I = I.num 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) (P : Type u_2) [CommRing P] [Algebra R P] [loc : IsLocalization S P] (I : FractionalIdeal S P)
Goal: FractionalIdeal.spanSingleton S ((algebraMap R P) ↑I.den) * I = ↑I.num
-- FractionalIdeal.spanSingleton_le_iff_mem
@[simp]
theorem spanSingleton_le_iff_mem {x : P} {I : FractionalIdeal S P} :
    spanSingleton S x ≤ I ↔ x ∈ I 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] {x : P} {I : FractionalIdeal S P}
Goal: FractionalIdeal.spanSingleton S x ≤ I ↔ x ∈ I
-- FractionalIdeal.spanSingleton_eq_spanSingleton
theorem spanSingleton_eq_spanSingleton [NoZeroSMulDivisors R P] {x y : P} :
    spanSingleton S x = spanSingleton S y ↔ ∃ z : Rˣ, z • x = y 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] [NoZeroSMulDivisors R P] {x y : P}
Goal: FractionalIdeal.spanSingleton S x = FractionalIdeal.spanSingleton S y ↔ ∃ z, z • x = y
-- FractionalIdeal.eq_spanSingleton_of_principal
theorem eq_spanSingleton_of_principal (I : FractionalIdeal S P) [IsPrincipal (I : Submodule R P)] :
    I = spanSingleton S (generator (I : Submodule R P)) 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] (I : FractionalIdeal S P) [(↑I).IsPrincipal]
Goal: I = FractionalIdeal.spanSingleton S (Submodule.IsPrincipal.generator ↑I)
-- FractionalIdeal.isPrincipal_iff
theorem isPrincipal_iff (I : FractionalIdeal S P) :
    IsPrincipal (I : Submodule R P) ↔ ∃ x, I = spanSingleton S x 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] (I : FractionalIdeal S P)
Goal: (↑I).IsPrincipal ↔ ∃ x, I = FractionalIdeal.spanSingleton S x
-- FractionalIdeal.spanSingleton_zero
@[simp]
theorem spanSingleton_zero : spanSingleton S (0 : P) = 0 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P]
Goal: FractionalIdeal.spanSingleton S 0 = 0
-- FractionalIdeal.spanSingleton_eq_zero_iff
theorem spanSingleton_eq_zero_iff {y : P} : spanSingleton S y = 0 ↔ y = 0 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] {y : P}
Goal: FractionalIdeal.spanSingleton S y = 0 ↔ y = 0
-- FractionalIdeal.spanSingleton_ne_zero_iff
theorem spanSingleton_ne_zero_iff {y : P} : spanSingleton S y ≠ 0 ↔ y ≠ 0 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] {y : P}
Goal: FractionalIdeal.spanSingleton S y ≠ 0 ↔ y ≠ 0
-- FractionalIdeal.spanSingleton_one
@[simp]
theorem spanSingleton_one : spanSingleton S (1 : P) = 1 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P]
Goal: FractionalIdeal.spanSingleton S 1 = 1
-- FractionalIdeal.spanSingleton_mul_spanSingleton
@[simp]
theorem spanSingleton_mul_spanSingleton (x y : P) :
    spanSingleton S x * spanSingleton S y = spanSingleton S (x * y) 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] (x y : P)
Goal: FractionalIdeal.spanSingleton S x * FractionalIdeal.spanSingleton S y = FractionalIdeal.spanSingleton S (x * y)
-- FractionalIdeal.spanSingleton_pow
@[simp]
theorem spanSingleton_pow (x : P) (n : ℕ) : spanSingleton S x ^ n = spanSingleton S (x ^ n) 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] (x : P) (n : ℕ)
Goal: FractionalIdeal.spanSingleton S x ^ n = FractionalIdeal.spanSingleton S (x ^ n)
-- FractionalIdeal.coeIdeal_span_singleton
@[simp]
theorem coeIdeal_span_singleton (x : R) :
    (↑(Ideal.span {x} : Ideal R) : FractionalIdeal S P) = spanSingleton S (algebraMap R P x) 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] (x : R)
Goal: ↑(Ideal.span {x}) = FractionalIdeal.spanSingleton S ((algebraMap R P) x)
-- FractionalIdeal.canonicalEquiv_spanSingleton
@[simp]
theorem canonicalEquiv_spanSingleton {P'} [CommRing P'] [Algebra R P'] [IsLocalization S P']
    (x : P) :
    canonicalEquiv S P P' (spanSingleton S x) =
      spanSingleton S
        (IsLocalization.map P' (RingHom.id R)
          (fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy) x) 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] {P' : Type u_5} [CommRing P'] [Algebra R P'] [IsLocalization S P'] (x : P)
Goal: (FractionalIdeal.canonicalEquiv S P P') (FractionalIdeal.spanSingleton S x) =
  FractionalIdeal.spanSingleton S ((IsLocalization.map P' (RingHom.id R) ⋯) x)
-- FractionalIdeal.mem_singleton_mul
theorem mem_singleton_mul {x y : P} {I : FractionalIdeal S P} :
    y ∈ spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y' 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] {x y : P} {I : FractionalIdeal S P}
Goal: y ∈ FractionalIdeal.spanSingleton S x * I ↔ ∃ y' ∈ I, y = x * y'
-- FractionalIdeal.mk'_mul_coeIdeal_eq_coeIdeal
theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
    spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * I = (J : FractionalIdeal R₁⁰ K) ↔
      Ideal.span {x} * I = Ideal.span {y} * J 
Arguments: {R₁ : Type u_3} [CommRing R₁] (K : Type u_4) [Field K] [Algebra R₁ K] [IsFractionRing R₁ K] {I J : Ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰)
Goal: FractionalIdeal.spanSingleton R₁⁰ (IsLocalization.mk' K x ⟨y, hy⟩) * ↑I = ↑J ↔ Ideal.span {x} * I = Ideal.span {y} * J
-- FractionalIdeal.spanSingleton_mul_coeIdeal_eq_coeIdeal
theorem spanSingleton_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {z : K} :
    spanSingleton R₁⁰ z * (I : FractionalIdeal R₁⁰ K) = J ↔
      Ideal.span {((IsLocalization.sec R₁⁰ z).1 : R₁)} * I =
        Ideal.span {((IsLocalization.sec R₁⁰ z).2 : R₁)} * J 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [IsFractionRing R₁ K] {I J : Ideal R₁} {z : K}
Goal: FractionalIdeal.spanSingleton R₁⁰ z * ↑I = ↑J ↔
  Ideal.span {(IsLocalization.sec R₁⁰ z).1} * I = Ideal.span {↑(IsLocalization.sec R₁⁰ z).2} * J
-- FractionalIdeal.one_div_spanSingleton
theorem one_div_spanSingleton (x : K) : 1 / spanSingleton R₁⁰ x = spanSingleton R₁⁰ x⁻¹ 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [IsFractionRing R₁ K] [IsDomain R₁] (x : K)
Goal: 1 / FractionalIdeal.spanSingleton R₁⁰ x = FractionalIdeal.spanSingleton R₁⁰ x⁻¹
-- FractionalIdeal.div_spanSingleton
@[simp]
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
    J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [IsFractionRing R₁ K] [IsDomain R₁] (J : FractionalIdeal R₁⁰ K) (d : K)
Goal: J / FractionalIdeal.spanSingleton R₁⁰ d = FractionalIdeal.spanSingleton R₁⁰ d⁻¹ * J
-- FractionalIdeal.exists_eq_spanSingleton_mul
theorem exists_eq_spanSingleton_mul (I : FractionalIdeal R₁⁰ K) :
    ∃ (a : R₁) (aI : Ideal R₁), a ≠ 0 ∧ I = spanSingleton R₁⁰ (algebraMap R₁ K a)⁻¹ * aI 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [IsFractionRing R₁ K] [IsDomain R₁] (I : FractionalIdeal R₁⁰ K)
Goal: ∃ a aI, a ≠ 0 ∧ I = FractionalIdeal.spanSingleton R₁⁰ ((algebraMap R₁ K) a)⁻¹ * ↑aI
-- FractionalIdeal.ideal_factor_ne_zero
theorem ideal_factor_ne_zero {R} [CommRing R] {K : Type*} [Field K] [Algebra R K]
    [IsFractionRing R K] {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) {a : R} {J : Ideal R}
    (haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J) : J ≠ 0 
Arguments: {R : Type u_6} [CommRing R] {K : Type u_5} [Field K] [Algebra R K] [IsFractionRing R K] {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) {a : R} {J : Ideal R} (haJ : I = FractionalIdeal.spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J)
Goal: J ≠ 0
-- FractionalIdeal.constant_factor_ne_zero
theorem constant_factor_ne_zero {R} [CommRing R] {K : Type*} [Field K] [Algebra R K]
    [IsFractionRing R K] {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) {a : R} {J : Ideal R}
    (haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J) :
    (Ideal.span {a} : Ideal R) ≠ 0 
Arguments: {R : Type u_6} [CommRing R] {K : Type u_5} [Field K] [Algebra R K] [IsFractionRing R K] {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) {a : R} {J : Ideal R} (haJ : I = FractionalIdeal.spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J)
Goal: Ideal.span {a} ≠ 0
-- FractionalIdeal.le_spanSingleton_mul_iff
theorem le_spanSingleton_mul_iff {x : P} {I J : FractionalIdeal S P} :
    I ≤ spanSingleton S x * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] {x : P} {I J : FractionalIdeal S P}
Goal: I ≤ FractionalIdeal.spanSingleton S x * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI
-- FractionalIdeal.spanSingleton_mul_le_iff
theorem spanSingleton_mul_le_iff {x : P} {I J : FractionalIdeal S P} :
    spanSingleton _ x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] {x : P} {I J : FractionalIdeal S P}
Goal: FractionalIdeal.spanSingleton S x * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J
-- FractionalIdeal.eq_spanSingleton_mul
theorem eq_spanSingleton_mul {x : P} {I J : FractionalIdeal S P} :
    I = spanSingleton _ x * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] {x : P} {I J : FractionalIdeal S P}
Goal: I = FractionalIdeal.spanSingleton S x * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I
-- FractionalIdeal.num_le
theorem num_le (I : FractionalIdeal S P) :
    (I.num : FractionalIdeal S P) ≤ I 
Arguments: {R : Type u_1} [CommRing R] {S : Submonoid R} {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] (I : FractionalIdeal S P)
Goal: ↑I.num ≤ I
-- FractionalIdeal.isNoetherian_zero
theorem isNoetherian_zero : IsNoetherian R₁ (0 : FractionalIdeal R₁⁰ K) 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K]
Goal: IsNoetherian R₁ ↥↑0
-- FractionalIdeal.isNoetherian_iff
theorem isNoetherian_iff {I : FractionalIdeal R₁⁰ K} :
    IsNoetherian R₁ I ↔ ∀ J ≤ I, (J : Submodule R₁ K).FG 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] {I : FractionalIdeal R₁⁰ K}
Goal: IsNoetherian R₁ ↥↑I ↔ ∀ J ≤ I, (↑J).FG
-- FractionalIdeal.isNoetherian_coeIdeal
theorem isNoetherian_coeIdeal [IsNoetherianRing R₁] (I : Ideal R₁) :
    IsNoetherian R₁ (I : FractionalIdeal R₁⁰ K) 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [IsNoetherianRing R₁] (I : Ideal R₁)
Goal: IsNoetherian R₁ ↥↑↑I
-- FractionalIdeal.isNoetherian_spanSingleton_inv_to_map_mul
theorem isNoetherian_spanSingleton_inv_to_map_mul (x : R₁) {I : FractionalIdeal R₁⁰ K}
    (hI : IsNoetherian R₁ I) :
    IsNoetherian R₁ (spanSingleton R₁⁰ (algebraMap R₁ K x)⁻¹ * I : FractionalIdeal R₁⁰ K) 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] (x : R₁) {I : FractionalIdeal R₁⁰ K} (hI : IsNoetherian R₁ ↥↑I)
Goal: IsNoetherian R₁ ↥↑(FractionalIdeal.spanSingleton R₁⁰ ((algebraMap R₁ K) x)⁻¹ * I)
-- FractionalIdeal.isNoetherian
theorem isNoetherian [IsNoetherianRing R₁] (I : FractionalIdeal R₁⁰ K) : IsNoetherian R₁ I 
Arguments: {R₁ : Type u_3} [CommRing R₁] {K : Type u_4} [Field K] [Algebra R₁ K] [frac : IsFractionRing R₁ K] [IsDomain R₁] [IsNoetherianRing R₁] (I : FractionalIdeal R₁⁰ K)
Goal: IsNoetherian R₁ ↥↑I
-- FractionalIdeal.isFractional_adjoin_integral
theorem isFractional_adjoin_integral :
    IsFractional S (Subalgebra.toSubmodule (Algebra.adjoin R ({x} : Set P))) 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] (x : P) (hx : IsIntegral R x)
Goal: IsFractional S (Subalgebra.toSubmodule (Algebra.adjoin R {x}))
-- FractionalIdeal.adjoinIntegral_coe
@[simp]
theorem adjoinIntegral_coe :
    (adjoinIntegral S x hx : Submodule R P) =
      (Subalgebra.toSubmodule (Algebra.adjoin R ({x} : Set P))) 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] (x : P) (hx : IsIntegral R x)
Goal: ↑(FractionalIdeal.adjoinIntegral S x hx) = Subalgebra.toSubmodule (Algebra.adjoin R {x})
-- FractionalIdeal.mem_adjoinIntegral_self
theorem mem_adjoinIntegral_self : x ∈ adjoinIntegral S x hx 
Arguments: {R : Type u_1} [CommRing R] (S : Submonoid R) {P : Type u_2} [CommRing P] [Algebra R P] [loc : IsLocalization S P] (x : P) (hx : IsIntegral R x)
Goal: x ∈ FractionalIdeal.adjoinIntegral S x hx
-- Algebra.map_leftMulMatrix_localization
theorem Algebra.map_leftMulMatrix_localization {ι : Type*} [Fintype ι] [DecidableEq ι]
    (b : Basis ι R S) (a : S) :
    (algebraMap R Rₘ).mapMatrix (leftMulMatrix b a) =
    leftMulMatrix (b.localizationLocalization Rₘ M Sₘ) (algebraMap S Sₘ a) 
Arguments: (R : Type u_1) {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {Rₘ : Type u_3} {Sₘ : Type u_4} [CommRing Rₘ] [Algebra R Rₘ] [CommRing Sₘ] [Algebra S Sₘ] (M : Submonoid R) [IsLocalization M Rₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] [Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ] {ι : Type u_5} [Fintype ι] [DecidableEq ι] (b : Basis ι R S) (a : S)
Goal: (algebraMap R Rₘ).mapMatrix ((Algebra.leftMulMatrix b) a) =
  (Algebra.leftMulMatrix (Basis.localizationLocalization Rₘ M Sₘ b)) ((algebraMap S Sₘ) a)
-- Algebra.norm_localization
theorem Algebra.norm_localization [Module.Free R S] [Module.Finite R S] (a : S) :
    Algebra.norm Rₘ (algebraMap S Sₘ a) = algebraMap R Rₘ (Algebra.norm R a) 
Arguments: (R : Type u_1) {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {Rₘ : Type u_3} {Sₘ : Type u_4} [CommRing Rₘ] [Algebra R Rₘ] [CommRing Sₘ] [Algebra S Sₘ] (M : Submonoid R) [IsLocalization M Rₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] [Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ] [Module.Free R S] [Module.Finite R S] (a : S)
Goal: (Algebra.norm Rₘ) ((algebraMap S Sₘ) a) = (algebraMap R Rₘ) ((Algebra.norm R) a)
-- Algebra.norm_eq_iff
lemma Algebra.norm_eq_iff [Module.Free R S] [Module.Finite R S] {a : S} {b : R}
    (hM : M ≤ nonZeroDivisors R) : Algebra.norm R a = b ↔
      (Algebra.norm Rₘ) ((algebraMap S Sₘ) a) = algebraMap R Rₘ b 
Arguments: (R : Type u_1) {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {Rₘ : Type u_3} {Sₘ : Type u_4} [CommRing Rₘ] [Algebra R Rₘ] [CommRing Sₘ] [Algebra S Sₘ] {M : Submonoid R} [IsLocalization M Rₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] [Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ] [Module.Free R S] [Module.Finite R S] {a : S} {b : R} (hM : M ≤ R⁰)
Goal: (Algebra.norm R) a = b ↔ (Algebra.norm Rₘ) ((algebraMap S Sₘ) a) = (algebraMap R Rₘ) b
-- Algebra.trace_localization
theorem Algebra.trace_localization [Module.Free R S] [Module.Finite R S] (a : S) :
    Algebra.trace Rₘ Sₘ (algebraMap S Sₘ a) = algebraMap R Rₘ (Algebra.trace R S a) 
Arguments: (R : Type u_1) {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {Rₘ : Type u_3} {Sₘ : Type u_4} [CommRing Rₘ] [Algebra R Rₘ] [CommRing Sₘ] [Algebra S Sₘ] (M : Submonoid R) [IsLocalization M Rₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] [Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ] [Module.Free R S] [Module.Finite R S] (a : S)
Goal: (Algebra.trace Rₘ Sₘ) ((algebraMap S Sₘ) a) = (algebraMap R Rₘ) ((Algebra.trace R S) a)
-- Algebra.traceMatrix_localizationLocalization
theorem Algebra.traceMatrix_localizationLocalization (b : Basis ι R S) :
    Algebra.traceMatrix Rₘ (b.localizationLocalization Rₘ M Sₘ) =
      (algebraMap R Rₘ).mapMatrix (Algebra.traceMatrix R b) 
Arguments: (R : Type u_1) {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {Rₘ : Type u_3} [CommRing Rₘ] [Algebra R Rₘ] (M : Submonoid R) [IsLocalization M Rₘ] (Sₘ : Type u_5) [CommRing Sₘ] [Algebra S Sₘ] [Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] {ι : Type u_6} [Fintype ι] [DecidableEq ι] (b : Basis ι R S)
Goal: Algebra.traceMatrix Rₘ ⇑(Basis.localizationLocalization Rₘ M Sₘ b) =
  (algebraMap R Rₘ).mapMatrix (Algebra.traceMatrix R ⇑b)
-- Algebra.discr_localizationLocalization
theorem Algebra.discr_localizationLocalization (b : Basis ι R S) :
    Algebra.discr Rₘ (b.localizationLocalization Rₘ M Sₘ) =
    algebraMap R Rₘ (Algebra.discr R b) 
Arguments: (R : Type u_1) {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {Rₘ : Type u_3} [CommRing Rₘ] [Algebra R Rₘ] (M : Submonoid R) [IsLocalization M Rₘ] (Sₘ : Type u_5) [CommRing Sₘ] [Algebra S Sₘ] [Algebra Rₘ Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ] [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ] {ι : Type u_6} [Fintype ι] [DecidableEq ι] (b : Basis ι R S)
Goal: Algebra.discr Rₘ ⇑(Basis.localizationLocalization Rₘ M Sₘ b) = (algebraMap R Rₘ) (Algebra.discr R ⇑b)
-- _private.Mathlib.RingTheory.AdicCompletion.AsTensorProduct.0.AdicCompletion.ofTensorProductBil_apply_apply
@[simp]
private lemma ofTensorProductBil_apply_apply (r : AdicCompletion I R) (x : M) :
    ((AdicCompletion.ofTensorProductBil I M) r) x = r • (of I M) x 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R) (M : Type u_2) [AddCommGroup M] [Module R M] (r : AdicCompletion I R) (x : M)
Goal: ((AdicCompletion.ofTensorProductBil I M) r) x = r • (AdicCompletion.of I M) x
-- AdicCompletion.ofTensorProduct_tmul
@[simp]
lemma ofTensorProduct_tmul (r : AdicCompletion I R) (x : M) :
    ofTensorProduct I M (r ⊗ₜ x) = r • of I M x 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R) (M : Type u_2) [AddCommGroup M] [Module R M] (r : AdicCompletion I R) (x : M)
Goal: (AdicCompletion.ofTensorProduct I M) (r ⊗ₜ[R] x) = r • (AdicCompletion.of I M) x
-- AdicCompletion.ofTensorProduct_naturality
lemma ofTensorProduct_naturality (f : M →ₗ[R] N) :
    map I f ∘ₗ ofTensorProduct I M =
      ofTensorProduct I N ∘ₗ AlgebraTensorModule.map LinearMap.id f 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R) {M : Type u_2} [AddCommGroup M] [Module R M] {N : Type u_3} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N)
Goal: AdicCompletion.map I f ∘ₗ AdicCompletion.ofTensorProduct I M =
  AdicCompletion.ofTensorProduct I N ∘ₗ TensorProduct.AlgebraTensorModule.map LinearMap.id f
-- _private.Mathlib.RingTheory.AdicCompletion.AsTensorProduct.0.AdicCompletion.piEquivOfFintype_comp_ofTensorProduct_eq
private lemma piEquivOfFintype_comp_ofTensorProduct_eq :
    piEquivOfFintype I (fun _ : ι ↦ R) ∘ₗ ofTensorProduct I (ι → R) =
      (TensorProduct.piScalarRight R (AdicCompletion I R) (AdicCompletion I R) ι).toLinearMap 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R) (ι : Type u_4) [Fintype ι] [DecidableEq ι]
Goal: ↑(AdicCompletion.piEquivOfFintype I fun x => R) ∘ₗ AdicCompletion.ofTensorProduct I (ι → R) =
  ↑(TensorProduct.piScalarRight R (AdicCompletion I R) (AdicCompletion I R) ι)
-- _private.Mathlib.RingTheory.AdicCompletion.AsTensorProduct.0.AdicCompletion.ofTensorProduct_eq
private lemma ofTensorProduct_eq :
    ofTensorProduct I (ι → R) = (piEquivOfFintype I (ι := ι) (fun _ : ι ↦ R)).symm.toLinearMap ∘ₗ
      (TensorProduct.piScalarRight R (AdicCompletion I R) (AdicCompletion I R) ι).toLinearMap 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R) (ι : Type u_4) [Fintype ι] [DecidableEq ι]
Goal: AdicCompletion.ofTensorProduct I (ι → R) =
  ↑(AdicCompletion.piEquivOfFintype I fun x => R).symm ∘ₗ
    ↑(TensorProduct.piScalarRight R (AdicCompletion I R) (AdicCompletion I R) ι)
-- _private.Mathlib.RingTheory.AdicCompletion.AsTensorProduct.0.AdicCompletion.ofTensorProductInvOfPiFintype_comp_ofTensorProduct
private lemma ofTensorProductInvOfPiFintype_comp_ofTensorProduct :
    ofTensorProductInvOfPiFintype I ι ∘ₗ ofTensorProduct I (ι → R) = LinearMap.id 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R) (ι : Type u_4) [Fintype ι] [DecidableEq ι]
Goal: ↑(AdicCompletion.ofTensorProductInvOfPiFintype I ι) ∘ₗ AdicCompletion.ofTensorProduct I (ι → R) = LinearMap.id
-- _private.Mathlib.RingTheory.AdicCompletion.AsTensorProduct.0.AdicCompletion.ofTensorProduct_comp_ofTensorProductInvOfPiFintype
private lemma ofTensorProduct_comp_ofTensorProductInvOfPiFintype :
    ofTensorProduct I (ι → R) ∘ₗ ofTensorProductInvOfPiFintype I ι = LinearMap.id 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R) (ι : Type u_4) [Fintype ι] [DecidableEq ι]
Goal: AdicCompletion.ofTensorProduct I (ι → R) ∘ₗ ↑(AdicCompletion.ofTensorProductInvOfPiFintype I ι) = LinearMap.id
-- AdicCompletion.ofTensorProduct_bijective_of_pi_of_fintype
lemma ofTensorProduct_bijective_of_pi_of_fintype [Finite ι] :
    Function.Bijective (ofTensorProduct I (ι → R)) 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R) (ι : Type u_4) [Finite ι]
Goal: Function.Bijective ⇑(AdicCompletion.ofTensorProduct I (ι → R))
-- AdicCompletion.ofTensorProduct_surjective_of_finite
lemma ofTensorProduct_surjective_of_finite [Module.Finite R M] :
    Function.Surjective (ofTensorProduct I M) 
Arguments: {R : Type u_1} [CommRing R] (I : Ideal R) (M : Type u_2) [AddCommGroup M] [Module R M] [Module.Finite R M]
Goal: Function.Surjective ⇑(AdicCompletion.ofTensorProduct I M)
-- _private.Mathlib.RingTheory.AdicCompletion.AsTensorProduct.0.AdicCompletion.if_exact
private lemma if_exact : Function.Exact (LinearMap.ker f).subtype f 
Arguments: {R : Type u} [CommRing R] (M : Type u) [AddCommGroup M] [Module R M] {ι : Type} (f : (ι → R) →ₗ[R] M)
Goal: Function.Exact ⇑(LinearMap.ker f).subtype ⇑f
-- _private.Mathlib.RingTheory.AdicCompletion.AsTensorProduct.0.AdicCompletion.tens_exact
private lemma tens_exact : Function.Exact (lTensorKerIncl I M f) (lTensorf I M f) 
Arguments: {R : Type u} [CommRing R] (I : Ideal R) (M : Type u) [AddCommGroup M] [Module R M] {ι : Type} (f : (ι → R) →ₗ[R] M) (hf : Function.Surjective ⇑f)
Goal: Function.Exact ⇑(AdicCompletion.lTensorKerIncl I M f) ⇑(AdicCompletion.lTensorf I M f)
-- _private.Mathlib.RingTheory.AdicCompletion.AsTensorProduct.0.AdicCompletion.tens_surj
private lemma tens_surj : Function.Surjective (lTensorf I M f) 
Arguments: {R : Type u} [CommRing R] (I : Ideal R) (M : Type u) [AddCommGroup M] [Module R M] {ι : Type} (f : (ι → R) →ₗ[R] M) (hf : Function.Surjective ⇑f)
Goal: Function.Surjective ⇑(AdicCompletion.lTensorf I M f)
-- _private.Mathlib.RingTheory.AdicCompletion.AsTensorProduct.0.AdicCompletion.adic_exact
private lemma adic_exact : Function.Exact (map I (LinearMap.ker f).subtype) (map I f) 
Arguments: {R : Type u} [CommRing R] (I : Ideal R) (M : Type u) [AddCommGroup M] [Module R M] [IsNoetherianRing R] {ι : Type} [Fintype ι] (f : (ι → R) →ₗ[R] M) (hf : Function.Surjective ⇑f)
Goal: Function.Exact ⇑(AdicCompletion.map I (LinearMap.ker f).subtype) ⇑(AdicCompletion.map I f)