ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LocalRing.ker_eq_maximalIdeal
theorem ker_eq_maximalIdeal [Field K] (φ : R →+* K) (hφ : Function.Surjective φ) :
    RingHom.ker φ = maximalIdeal R 
Arguments: {R : Type u_1} {K : Type u_3} [CommRing R] [LocalRing R] [Field K] (φ : R →+* K) (hφ : Function.Surjective ⇑φ)
Goal: RingHom.ker φ = LocalRing.maximalIdeal R
-- LocalRing.maximalIdeal_eq_bot
theorem LocalRing.maximalIdeal_eq_bot {R : Type*} [Field R] : LocalRing.maximalIdeal R = ⊥ 
Arguments: {R : Type u_4} [Field R]
Goal: LocalRing.maximalIdeal R = ⊥
-- IsLocalization.ker_map
lemma IsLocalization.ker_map (hT : Submonoid.map g M = T) :
    RingHom.ker (IsLocalization.map Q g (hT.symm ▸ M.le_comap_map) : S →+* Q) =
      (RingHom.ker g).map (algebraMap R S) 
Arguments: {R : Type u_1} {S : Type u_2} {P : Type u_3} (Q : Type u_4) [CommSemiring R] [CommSemiring S] [CommSemiring P] [CommSemiring Q] {M : Submonoid R} {T : Submonoid P} [Algebra R S] [Algebra P Q] [IsLocalization M S] [IsLocalization T Q] (g : R →+* P) (hT : Submonoid.map g M = T)
Goal: RingHom.ker (IsLocalization.map Q g ⋯) = Ideal.map (algebraMap R S) (RingHom.ker g)
-- RingHom.toKerIsLocalization_apply
@[simp]
lemma RingHom.toKerIsLocalization_apply (hy : M ≤ Submonoid.comap g T) (r : RingHom.ker g) :
    (RingHom.toKerIsLocalization S Q g hy r).val = algebraMap R S r 
Arguments: {R : Type u_1} {S : Type u_2} {P : Type u_3} (Q : Type u_4) [CommSemiring R] [CommSemiring S] [CommSemiring P] [CommSemiring Q] {M : Submonoid R} {T : Submonoid P} [Algebra R S] [Algebra P Q] [IsLocalization M S] [IsLocalization T Q] (g : R →+* P) (hy : M ≤ Submonoid.comap g T) (r : ↥(RingHom.ker g))
Goal: ↑((RingHom.toKerIsLocalization S Q g hy) r) = (algebraMap R S) ↑r
-- RingHom.toKerIsLocalization_isLocalizedModule
lemma RingHom.toKerIsLocalization_isLocalizedModule (hT : Submonoid.map g M = T) :
    IsLocalizedModule M (toKerIsLocalization S Q g (hT.symm ▸ Submonoid.le_comap_map M)) 
Arguments: {R : Type u_1} {S : Type u_2} {P : Type u_3} (Q : Type u_4) [CommSemiring R] [CommSemiring S] [CommSemiring P] [CommSemiring Q] {M : Submonoid R} {T : Submonoid P} [Algebra R S] [Algebra P Q] [IsLocalization M S] [IsLocalization T Q] (g : R →+* P) (hT : Submonoid.map g M = T)
Goal: IsLocalizedModule M (RingHom.toKerIsLocalization S Q g ⋯)
-- IsLocalization.mapₐ_coe
@[simp]
lemma mapₐ_coe (f : A →ₐ[R] B) :
    (mapₐ M Rₚ Aₚ Bₚ f : Aₚ → Bₚ) = map Bₚ f.toRingHom (algebraMapSubmonoid_le_comap M f)  
Arguments: {R : Type u} [CommRing R] (M : Submonoid R) {A : Type v} [CommRing A] [Algebra R A] {B : Type w} [CommRing B] [Algebra R B] (Rₚ : Type u') [CommRing Rₚ] [Algebra R Rₚ] [IsLocalization M Rₚ] (Aₚ : Type v') [CommRing Aₚ] [Algebra R Aₚ] [Algebra A Aₚ] [IsScalarTower R A Aₚ] [IsLocalization (Algebra.algebraMapSubmonoid A M) Aₚ] (Bₚ : Type v') [CommRing Bₚ] [Algebra R Bₚ] [Algebra B Bₚ] [IsScalarTower R B Bₚ] [IsLocalization (Algebra.algebraMapSubmonoid B M) Bₚ] [Algebra Rₚ Aₚ] [Algebra Rₚ Bₚ] [IsScalarTower R Rₚ Aₚ] [IsScalarTower R Rₚ Bₚ] (f : A →ₐ[R] B)
Goal: ⇑(IsLocalization.mapₐ M Rₚ Aₚ Bₚ f) = ⇑(IsLocalization.map Bₚ f.toRingHom ⋯)
-- IsLocalization.mapₐ_injective_of_injective
lemma mapₐ_injective_of_injective (f : A →ₐ[R] B) (hf : Function.Injective f) :
    Function.Injective (mapₐ M Rₚ Aₚ Bₚ f) 
Arguments: {R : Type u} [CommRing R] (M : Submonoid R) {A : Type v} [CommRing A] [Algebra R A] {B : Type w} [CommRing B] [Algebra R B] (Rₚ : Type u') [CommRing Rₚ] [Algebra R Rₚ] [IsLocalization M Rₚ] (Aₚ : Type v') [CommRing Aₚ] [Algebra R Aₚ] [Algebra A Aₚ] [IsScalarTower R A Aₚ] [IsLocalization (Algebra.algebraMapSubmonoid A M) Aₚ] (Bₚ : Type v') [CommRing Bₚ] [Algebra R Bₚ] [Algebra B Bₚ] [IsScalarTower R B Bₚ] [IsLocalization (Algebra.algebraMapSubmonoid B M) Bₚ] [Algebra Rₚ Aₚ] [Algebra Rₚ Bₚ] [IsScalarTower R Rₚ Aₚ] [IsScalarTower R Rₚ Bₚ] (f : A →ₐ[R] B) (hf : Function.Injective ⇑f)
Goal: Function.Injective ⇑(IsLocalization.mapₐ M Rₚ Aₚ Bₚ f)
-- IsLocalization.mapₐ_surjective_of_surjective
lemma mapₐ_surjective_of_surjective (f : A →ₐ[R] B) (hf : Function.Surjective f) :
    Function.Surjective (mapₐ M Rₚ Aₚ Bₚ f) 
Arguments: {R : Type u} [CommRing R] (M : Submonoid R) {A : Type v} [CommRing A] [Algebra R A] {B : Type w} [CommRing B] [Algebra R B] (Rₚ : Type u') [CommRing Rₚ] [Algebra R Rₚ] [IsLocalization M Rₚ] (Aₚ : Type v') [CommRing Aₚ] [Algebra R Aₚ] [Algebra A Aₚ] [IsScalarTower R A Aₚ] [IsLocalization (Algebra.algebraMapSubmonoid A M) Aₚ] (Bₚ : Type v') [CommRing Bₚ] [Algebra R Bₚ] [Algebra B Bₚ] [IsScalarTower R B Bₚ] [IsLocalization (Algebra.algebraMapSubmonoid B M) Bₚ] [Algebra Rₚ Aₚ] [Algebra Rₚ Bₚ] [IsScalarTower R Rₚ Aₚ] [IsScalarTower R Rₚ Bₚ] (f : A →ₐ[R] B) (hf : Function.Surjective ⇑f)
Goal: Function.Surjective ⇑(IsLocalization.mapₐ M Rₚ Aₚ Bₚ f)
-- AlgHom.toKerIsLocalization_apply
@[simp]
lemma AlgHom.toKerIsLocalization_apply (f : A →ₐ[R] B) (x : RingHom.ker f) :
    AlgHom.toKerIsLocalization M Rₚ Aₚ Bₚ f x =
      RingHom.toKerIsLocalization Aₚ Bₚ f.toRingHom (algebraMapSubmonoid_le_comap M f) x 
Arguments: {R : Type u} [CommRing R] (M : Submonoid R) {A : Type v} [CommRing A] [Algebra R A] {B : Type w} [CommRing B] [Algebra R B] (Rₚ : Type u') [CommRing Rₚ] [Algebra R Rₚ] [IsLocalization M Rₚ] (Aₚ : Type v') [CommRing Aₚ] [Algebra R Aₚ] [Algebra A Aₚ] [IsScalarTower R A Aₚ] [IsLocalization (Algebra.algebraMapSubmonoid A M) Aₚ] (Bₚ : Type v') [CommRing Bₚ] [Algebra R Bₚ] [Algebra B Bₚ] [IsScalarTower R B Bₚ] [IsLocalization (Algebra.algebraMapSubmonoid B M) Bₚ] [Algebra Rₚ Aₚ] [Algebra Rₚ Bₚ] [IsScalarTower R Rₚ Aₚ] [IsScalarTower R Rₚ Bₚ] (f : A →ₐ[R] B) (x : ↥(RingHom.ker f))
Goal: (AlgHom.toKerIsLocalization M Rₚ Aₚ Bₚ f) x = (RingHom.toKerIsLocalization Aₚ Bₚ f.toRingHom ⋯) x
-- AlgHom.toKerIsLocalization_isLocalizedModule
lemma AlgHom.toKerIsLocalization_isLocalizedModule (f : A →ₐ[R] B) :
    IsLocalizedModule (Algebra.algebraMapSubmonoid A M)
      (AlgHom.toKerIsLocalization M Rₚ Aₚ Bₚ f) 
Arguments: {R : Type u} [CommRing R] (M : Submonoid R) {A : Type v} [CommRing A] [Algebra R A] {B : Type w} [CommRing B] [Algebra R B] (Rₚ : Type u') [CommRing Rₚ] [Algebra R Rₚ] [IsLocalization M Rₚ] (Aₚ : Type v') [CommRing Aₚ] [Algebra R Aₚ] [Algebra A Aₚ] [IsScalarTower R A Aₚ] [IsLocalization (Algebra.algebraMapSubmonoid A M) Aₚ] (Bₚ : Type v') [CommRing Bₚ] [Algebra R Bₚ] [Algebra B Bₚ] [IsScalarTower R B Bₚ] [IsLocalization (Algebra.algebraMapSubmonoid B M) Bₚ] [Algebra Rₚ Aₚ] [Algebra Rₚ Bₚ] [IsScalarTower R Rₚ Aₚ] [IsScalarTower R Rₚ Bₚ] (f : A →ₐ[R] B)
Goal: IsLocalizedModule (Algebra.algebraMapSubmonoid A M) (AlgHom.toKerIsLocalization M Rₚ Aₚ Bₚ f)
-- Algebra.traceForm_toMatrix_powerBasis
theorem Algebra.traceForm_toMatrix_powerBasis (h : PowerBasis R S) :
    BilinForm.toMatrix h.basis (traceForm R S) = of fun i j => trace R S (h.gen ^ (i.1 + j.1)) 
Arguments: {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] (h : PowerBasis R S)
Goal: (BilinForm.toMatrix h.basis) (Algebra.traceForm R S) = Matrix.of fun i j => (Algebra.trace R S) (h.gen ^ (↑i + ↑j))
-- PowerBasis.trace_gen_eq_nextCoeff_minpoly
theorem PowerBasis.trace_gen_eq_nextCoeff_minpoly [Nontrivial S] (pb : PowerBasis K S) :
    Algebra.trace K S pb.gen = -(minpoly K pb.gen).nextCoeff 
Arguments: {S : Type u_2} [CommRing S] {K : Type u_4} [Field K] [Algebra K S] [Nontrivial S] (pb : PowerBasis K S)
Goal: (Algebra.trace K S) pb.gen = -(minpoly K pb.gen).nextCoeff
-- PowerBasis.trace_gen_eq_sum_roots
theorem PowerBasis.trace_gen_eq_sum_roots [Nontrivial S] (pb : PowerBasis K S)
    (hf : (minpoly K pb.gen).Splits (algebraMap K F)) :
    algebraMap K F (trace K S pb.gen) = ((minpoly K pb.gen).aroots F).sum 
Arguments: {S : Type u_2} [CommRing S] {K : Type u_4} [Field K] {F : Type u_6} [Field F] [Algebra K S] [Algebra K F] [Nontrivial S] (pb : PowerBasis K S) (hf : Polynomial.Splits (algebraMap K F) (minpoly K pb.gen))
Goal: (algebraMap K F) ((Algebra.trace K S) pb.gen) = ((minpoly K pb.gen).aroots F).sum
-- IntermediateField.AdjoinSimple.trace_gen_eq_zero
theorem trace_gen_eq_zero {x : L} (hx : ¬IsIntegral K x) :
    Algebra.trace K K⟮x⟯ (AdjoinSimple.gen K x) = 0 
Arguments: {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K L] {x : L} (hx : ¬IsIntegral K x)
Goal: (Algebra.trace K ↥K⟮x⟯) (IntermediateField.AdjoinSimple.gen K x) = 0
-- IntermediateField.AdjoinSimple.trace_gen_eq_sum_roots
theorem trace_gen_eq_sum_roots (x : L) (hf : (minpoly K x).Splits (algebraMap K F)) :
    algebraMap K F (trace K K⟮x⟯ (AdjoinSimple.gen K x)) =
      ((minpoly K x).aroots F).sum 
Arguments: {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K L] {F : Type u_6} [Field F] [Algebra K F] (x : L) (hf : Polynomial.Splits (algebraMap K F) (minpoly K x))
Goal: (algebraMap K F) ((Algebra.trace K ↥K⟮x⟯) (IntermediateField.AdjoinSimple.gen K x)) = ((minpoly K x).aroots F).sum
-- trace_eq_trace_adjoin
theorem trace_eq_trace_adjoin [FiniteDimensional K L] (x : L) :
    Algebra.trace K L x = finrank K⟮x⟯ L • trace K K⟮x⟯ (AdjoinSimple.gen K x) 
Arguments: (K : Type u_4) {L : Type u_5} [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] (x : L)
Goal: (Algebra.trace K L) x =
  FiniteDimensional.finrank (↥K⟮x⟯) L • (Algebra.trace K ↥K⟮x⟯) (IntermediateField.AdjoinSimple.gen K x)
-- trace_eq_sum_roots
theorem trace_eq_sum_roots [FiniteDimensional K L] {x : L}
    (hF : (minpoly K x).Splits (algebraMap K F)) :
    algebraMap K F (Algebra.trace K L x) =
      finrank K⟮x⟯ L • ((minpoly K x).aroots F).sum 
Arguments: {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K L] {F : Type u_6} [Field F] [Algebra K F] [FiniteDimensional K L] {x : L} (hF : Polynomial.Splits (algebraMap K F) (minpoly K x))
Goal: (algebraMap K F) ((Algebra.trace K L) x) = FiniteDimensional.finrank (↥K⟮x⟯) L • ((minpoly K x).aroots F).sum
-- Algebra.isIntegral_trace
theorem Algebra.isIntegral_trace [FiniteDimensional L F] {x : F} (hx : IsIntegral R x) :
    IsIntegral R (Algebra.trace L F x) 
Arguments: {R : Type u_1} [CommRing R] {L : Type u_5} [Field L] {F : Type u_6} [Field F] [Algebra R L] [Algebra L F] [Algebra R F] [IsScalarTower R L F] [FiniteDimensional L F] {x : F} (hx : IsIntegral R x)
Goal: IsIntegral R ((Algebra.trace L F) x)
-- Algebra.trace_eq_of_algEquiv
lemma Algebra.trace_eq_of_algEquiv {A B C : Type*} [CommRing A] [CommRing B] [CommRing C]
    [Algebra A B] [Algebra A C] (e : B ≃ₐ[A] C) (x) :
    Algebra.trace A C (e x) = Algebra.trace A B x 
Arguments: {A : Type u_7} {B : Type u_8} {C : Type u_9} [CommRing A] [CommRing B] [CommRing C] [Algebra A B] [Algebra A C] (e : B ≃ₐ[A] C) (x : B)
Goal: (Algebra.trace A C) (e x) = (Algebra.trace A B) x
-- Algebra.trace_eq_of_ringEquiv
lemma Algebra.trace_eq_of_ringEquiv {A B C : Type*} [CommRing A] [CommRing B] [CommRing C]
    [Algebra A C] [Algebra B C] (e : A ≃+* B) (he : (algebraMap B C).comp e = algebraMap A C) (x) :
    e (Algebra.trace A C x) = Algebra.trace B C x 
Arguments: {A : Type u_7} {B : Type u_8} {C : Type u_9} [CommRing A] [CommRing B] [CommRing C] [Algebra A C] [Algebra B C] (e : A ≃+* B) (he : (algebraMap B C).comp ↑e = algebraMap A C) (x : C)
Goal: e ((Algebra.trace A C) x) = (Algebra.trace B C) x
-- Algebra.trace_eq_of_equiv_equiv
lemma Algebra.trace_eq_of_equiv_equiv {A₁ B₁ A₂ B₂ : Type*} [CommRing A₁] [CommRing B₁]
    [CommRing A₂] [CommRing B₂] [Algebra A₁ B₁] [Algebra A₂ B₂] (e₁ : A₁ ≃+* A₂) (e₂ : B₁ ≃+* B₂)
    (he : RingHom.comp (algebraMap A₂ B₂) ↑e₁ = RingHom.comp ↑e₂ (algebraMap A₁ B₁)) (x) :
    Algebra.trace A₁ B₁ x = e₁.symm (Algebra.trace A₂ B₂ (e₂ x)) 
Arguments: {A₁ : Type u_7} {B₁ : Type u_8} {A₂ : Type u_9} {B₂ : Type u_10} [CommRing A₁] [CommRing B₁] [CommRing A₂] [CommRing B₂] [Algebra A₁ B₁] [Algebra A₂ B₂] (e₁ : A₁ ≃+* A₂) (e₂ : B₁ ≃+* B₂) (he : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)) (x : B₁)
Goal: (Algebra.trace A₁ B₁) x = e₁.symm ((Algebra.trace A₂ B₂) (e₂ x))
-- trace_eq_sum_embeddings_gen
theorem trace_eq_sum_embeddings_gen (pb : PowerBasis K L)
    (hE : (minpoly K pb.gen).Splits (algebraMap K E)) (hfx : IsSeparable K pb.gen) :
    algebraMap K E (Algebra.trace K L pb.gen) =
      (@Finset.univ _ (PowerBasis.AlgHom.fintype pb)).sum fun σ => σ pb.gen 
Arguments: {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K L] (E : Type u_7) [Field E] [Algebra K E] (pb : PowerBasis K L) (hE : Polynomial.Splits (algebraMap K E) (minpoly K pb.gen)) (hfx : IsSeparable K pb.gen)
Goal: (algebraMap K E) ((Algebra.trace K L) pb.gen) = ∑ σ : L →ₐ[K] E, σ pb.gen
-- sum_embeddings_eq_finrank_mul
theorem sum_embeddings_eq_finrank_mul [FiniteDimensional K F] [Algebra.IsSeparable K F]
    (pb : PowerBasis K L) :
    ∑ σ : F →ₐ[K] E, σ (algebraMap L F pb.gen) =
      finrank L F •
        (@Finset.univ _ (PowerBasis.AlgHom.fintype pb)).sum fun σ : L →ₐ[K] E => σ pb.gen 
Arguments: {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K L] (F : Type u_6) [Field F] [Algebra L F] [Algebra K F] [IsScalarTower K L F] (E : Type u_7) [Field E] [Algebra K E] [IsAlgClosed E] [FiniteDimensional K F] [Algebra.IsSeparable K F] (pb : PowerBasis K L)
Goal: ∑ σ : F →ₐ[K] E, σ ((algebraMap L F) pb.gen) = FiniteDimensional.finrank L F • ∑ σ : L →ₐ[K] E, σ pb.gen
-- trace_eq_sum_embeddings
theorem trace_eq_sum_embeddings [FiniteDimensional K L] [Algebra.IsSeparable K L] {x : L} :
    algebraMap K E (Algebra.trace K L x) = ∑ σ : L →ₐ[K] E, σ x 
Arguments: {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K L] (E : Type u_7) [Field E] [Algebra K E] [IsAlgClosed E] [FiniteDimensional K L] [Algebra.IsSeparable K L] {x : L}
Goal: (algebraMap K E) ((Algebra.trace K L) x) = ∑ σ : L →ₐ[K] E, σ x
-- trace_eq_sum_automorphisms
theorem trace_eq_sum_automorphisms (x : L) [FiniteDimensional K L] [IsGalois K L] :
    algebraMap K L (Algebra.trace K L x) = ∑ σ : L ≃ₐ[K] L, σ x 
Arguments: {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K L] (x : L) [FiniteDimensional K L] [IsGalois K L]
Goal: (algebraMap K L) ((Algebra.trace K L) x) = ∑ σ : L ≃ₐ[K] L, σ x
-- Algebra.traceMatrix_apply
@[simp]
theorem traceMatrix_apply (b : κ → B) (i j) : traceMatrix A b i j = traceForm A B (b i) (b j) 
Arguments: {κ : Type w} (A : Type u) {B : Type v} [CommRing A] [CommRing B] [Algebra A B] (b : κ → B) (i j : κ)
Goal: Algebra.traceMatrix A b i j = ((Algebra.traceForm A B) (b i)) (b j)
-- Algebra.traceMatrix_reindex
theorem traceMatrix_reindex {κ' : Type*} (b : Basis κ A B) (f : κ ≃ κ') :
    traceMatrix A (b.reindex f) = reindex f f (traceMatrix A b) 
Arguments: {κ : Type w} (A : Type u) {B : Type v} [CommRing A] [CommRing B] [Algebra A B] {κ' : Type u_7} (b : Basis κ A B) (f : κ ≃ κ')
Goal: Algebra.traceMatrix A ⇑(b.reindex f) = (Matrix.reindex f f) (Algebra.traceMatrix A ⇑b)
-- Algebra.traceMatrix_of_matrix_vecMul
theorem traceMatrix_of_matrix_vecMul [Fintype κ] (b : κ → B) (P : Matrix κ κ A) :
    traceMatrix A (b ᵥ* P.map (algebraMap A B)) = Pᵀ * traceMatrix A b * P 
Arguments: {κ : Type w} {A : Type u} {B : Type v} [CommRing A] [CommRing B] [Algebra A B] [Fintype κ] (b : κ → B) (P : Matrix κ κ A)
Goal: Algebra.traceMatrix A (b ᵥ* P.map ⇑(algebraMap A B)) = Pᵀ * Algebra.traceMatrix A b * P
-- Algebra.traceMatrix_of_matrix_mulVec
theorem traceMatrix_of_matrix_mulVec [Fintype κ] (b : κ → B) (P : Matrix κ κ A) :
    traceMatrix A (P.map (algebraMap A B) *ᵥ b) = P * traceMatrix A b * Pᵀ 
Arguments: {κ : Type w} {A : Type u} {B : Type v} [CommRing A] [CommRing B] [Algebra A B] [Fintype κ] (b : κ → B) (P : Matrix κ κ A)
Goal: Algebra.traceMatrix A (P.map ⇑(algebraMap A B) *ᵥ b) = P * Algebra.traceMatrix A b * Pᵀ
-- Algebra.traceMatrix_of_basis
theorem traceMatrix_of_basis [Fintype κ] [DecidableEq κ] (b : Basis κ A B) :
    traceMatrix A b = BilinForm.toMatrix b (traceForm A B) 
Arguments: {κ : Type w} {A : Type u} {B : Type v} [CommRing A] [CommRing B] [Algebra A B] [Fintype κ] [DecidableEq κ] (b : Basis κ A B)
Goal: Algebra.traceMatrix A ⇑b = (BilinForm.toMatrix b) (Algebra.traceForm A B)
-- Algebra.traceMatrix_of_basis_mulVec
theorem traceMatrix_of_basis_mulVec (b : Basis ι A B) (z : B) :
    traceMatrix A b *ᵥ b.equivFun z = fun i => trace A B (z * b i) 
Arguments: {ι : Type w} [Fintype ι] {A : Type u} {B : Type v} [CommRing A] [CommRing B] [Algebra A B] (b : Basis ι A B) (z : B)
Goal: Algebra.traceMatrix A ⇑b *ᵥ b.equivFun z = fun i => (Algebra.trace A B) (z * b i)
-- Algebra.embeddingsMatrix_apply
@[simp]
theorem embeddingsMatrix_apply (b : κ → B) (i) (σ : B →ₐ[A] C) :
    embeddingsMatrix A C b i σ = σ (b i) 
Arguments: {κ : Type w} (A : Type u) {B : Type v} (C : Type z) [CommRing A] [CommRing B] [Algebra A B] [CommRing C] [Algebra A C] (b : κ → B) (i : κ) (σ : B →ₐ[A] C)
Goal: Algebra.embeddingsMatrix A C b i σ = σ (b i)
-- Algebra.embeddingsMatrixReindex_eq_vandermonde
theorem embeddingsMatrixReindex_eq_vandermonde (pb : PowerBasis A B)
    (e : Fin pb.dim ≃ (B →ₐ[A] C)) :
    embeddingsMatrixReindex A C pb.basis e = (vandermonde fun i => e i pb.gen)ᵀ 
Arguments: {A : Type u} {B : Type v} (C : Type z) [CommRing A] [CommRing B] [Algebra A B] [CommRing C] [Algebra A C] (pb : PowerBasis A B) (e : Fin pb.dim ≃ (B →ₐ[A] C))
Goal: Algebra.embeddingsMatrixReindex A C (⇑pb.basis) e = (Matrix.vandermonde fun i => (e i) pb.gen)ᵀ
-- Algebra.traceMatrix_eq_embeddingsMatrix_mul_trans
theorem traceMatrix_eq_embeddingsMatrix_mul_trans : (traceMatrix K b).map (algebraMap K E) =
    embeddingsMatrix K E b * (embeddingsMatrix K E b)ᵀ 
Arguments: (K : Type u_4) {L : Type u_5} [Field K] [Field L] [Algebra K L] {κ : Type w} (E : Type z) [Field E] [Algebra K E] [Module.Finite K L] [Algebra.IsSeparable K L] [IsAlgClosed E] (b : κ → L)
Goal: (Algebra.traceMatrix K b).map ⇑(algebraMap K E) = Algebra.embeddingsMatrix K E b * (Algebra.embeddingsMatrix K E b)ᵀ
-- Algebra.traceMatrix_eq_embeddingsMatrixReindex_mul_trans
theorem traceMatrix_eq_embeddingsMatrixReindex_mul_trans [Fintype κ] (e : κ ≃ (L →ₐ[K] E)) :
    (traceMatrix K b).map (algebraMap K E) =
      embeddingsMatrixReindex K E b e * (embeddingsMatrixReindex K E b e)ᵀ 
Arguments: (K : Type u_4) {L : Type u_5} [Field K] [Field L] [Algebra K L] {κ : Type w} (E : Type z) [Field E] [Algebra K E] [Module.Finite K L] [Algebra.IsSeparable K L] [IsAlgClosed E] (b : κ → L) [Fintype κ] (e : κ ≃ (L →ₐ[K] E))
Goal: (Algebra.traceMatrix K b).map ⇑(algebraMap K E) =
  Algebra.embeddingsMatrixReindex K E b e * (Algebra.embeddingsMatrixReindex K E b e)ᵀ
-- det_traceMatrix_ne_zero'
theorem det_traceMatrix_ne_zero' [Algebra.IsSeparable K L] : det (traceMatrix K pb.basis) ≠ 0 
Arguments: {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K L] (pb : PowerBasis K L) [Algebra.IsSeparable K L]
Goal: (Algebra.traceMatrix K ⇑pb.basis).det ≠ 0
-- det_traceForm_ne_zero
theorem det_traceForm_ne_zero [Algebra.IsSeparable K L] [DecidableEq ι] (b : Basis ι K L) :
    det (BilinForm.toMatrix b (traceForm K L)) ≠ 0 
Arguments: {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K L] {ι : Type w} [Fintype ι] [Algebra.IsSeparable K L] [DecidableEq ι] (b : Basis ι K L)
Goal: ((BilinForm.toMatrix b) (Algebra.traceForm K L)).det ≠ 0
-- traceForm_nondegenerate
theorem traceForm_nondegenerate [FiniteDimensional K L] [Algebra.IsSeparable K L] :
    (traceForm K L).Nondegenerate 
Arguments: (K : Type u_4) (L : Type u_5) [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] [Algebra.IsSeparable K L]
Goal: (Algebra.traceForm K L).Nondegenerate
-- Algebra.trace_ne_zero
theorem Algebra.trace_ne_zero [FiniteDimensional K L] [Algebra.IsSeparable K L] :
    Algebra.trace K L ≠ 0 
Arguments: (K : Type u_4) (L : Type u_5) [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] [Algebra.IsSeparable K L]
Goal: Algebra.trace K L ≠ 0
-- Algebra.trace_surjective
theorem Algebra.trace_surjective [FiniteDimensional K L] [Algebra.IsSeparable K L] :
    Function.Surjective (Algebra.trace K L) 
Arguments: (K : Type u_4) (L : Type u_5) [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] [Algebra.IsSeparable K L]
Goal: Function.Surjective ⇑(Algebra.trace K L)
-- traceForm_dualBasis_powerBasis_eq
lemma traceForm_dualBasis_powerBasis_eq [FiniteDimensional K L] [Algebra.IsSeparable K L]
    (pb : PowerBasis K L) (i) :
    (Algebra.traceForm K L).dualBasis (traceForm_nondegenerate K L) pb.basis i =
      (minpolyDiv K pb.gen).coeff i / aeval pb.gen (derivative <| minpoly K pb.gen) 
Arguments: {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] [Algebra.IsSeparable K L] (pb : PowerBasis K L) (i : Fin pb.dim)
Goal: ((Algebra.traceForm K L).dualBasis ⋯ pb.basis) i =
  (minpolyDiv K pb.gen).coeff ↑i / (Polynomial.aeval pb.gen) (Polynomial.derivative (minpoly K pb.gen))
-- KaehlerDifferential.mvPolynomialBasis_repr_comp_D
lemma KaehlerDifferential.mvPolynomialBasis_repr_comp_D (σ) :
    (mvPolynomialBasis R σ).repr.toLinearMap.compDer (D _ _) =
      MvPolynomial.mkDerivation _ (Finsupp.single · 1) 
Arguments: (R : Type u) [CommRing R] (σ : Type u_1)
Goal: (↑(KaehlerDifferential.mvPolynomialBasis R σ).repr).compDer (KaehlerDifferential.D R (MvPolynomial σ R)) =
  MvPolynomial.mkDerivation R fun x => Finsupp.single x 1
-- KaehlerDifferential.mvPolynomialBasis_repr_D
lemma KaehlerDifferential.mvPolynomialBasis_repr_D (σ) (x) :
    (mvPolynomialBasis R σ).repr (D _ _ x) =
      MvPolynomial.mkDerivation R (Finsupp.single · (1 : MvPolynomial σ R)) x 
Arguments: (R : Type u) [CommRing R] (σ : Type u_1) (x : MvPolynomial σ R)
Goal: (KaehlerDifferential.mvPolynomialBasis R σ).repr ((KaehlerDifferential.D R (MvPolynomial σ R)) x) =
  (MvPolynomial.mkDerivation R fun x => Finsupp.single x 1) x
-- KaehlerDifferential.mvPolynomialBasis_repr_D_X
@[simp]
lemma KaehlerDifferential.mvPolynomialBasis_repr_D_X (σ) (i) :
    (mvPolynomialBasis R σ).repr (D _ _ (.X i)) = Finsupp.single i 1 
Arguments: (R : Type u) [CommRing R] (σ : Type u_1) (i : σ)
Goal: (KaehlerDifferential.mvPolynomialBasis R σ).repr ((KaehlerDifferential.D R (MvPolynomial σ R)) (MvPolynomial.X i)) =
  Finsupp.single i 1
-- KaehlerDifferential.mvPolynomialBasis_repr_apply
@[simp]
lemma KaehlerDifferential.mvPolynomialBasis_repr_apply (σ) (x) (i) :
    (mvPolynomialBasis R σ).repr (D _ _ x) i = MvPolynomial.pderiv i x 
Arguments: (R : Type u) [CommRing R] (σ : Type u_1) (x : MvPolynomial σ R) (i : σ)
Goal: ((KaehlerDifferential.mvPolynomialBasis R σ).repr ((KaehlerDifferential.D R (MvPolynomial σ R)) x)) i =
  (MvPolynomial.pderiv i) x
-- KaehlerDifferential.mvPolynomialBasis_repr_symm_single
lemma KaehlerDifferential.mvPolynomialBasis_repr_symm_single (σ) (i) (x) :
    (mvPolynomialBasis R σ).repr.symm (Finsupp.single i x) = x • D R (MvPolynomial σ R) (.X i) 
Arguments: (R : Type u) [CommRing R] (σ : Type u_1) (i : σ) (x : MvPolynomial σ R)
Goal: (KaehlerDifferential.mvPolynomialBasis R σ).repr.symm (Finsupp.single i x) =
  x • (KaehlerDifferential.D R (MvPolynomial σ R)) (MvPolynomial.X i)
-- KaehlerDifferential.mvPolynomialBasis_apply
@[simp]
lemma KaehlerDifferential.mvPolynomialBasis_apply (σ) (i) :
    mvPolynomialBasis R σ i = D R (MvPolynomial σ R) (.X i) 
Arguments: (R : Type u) [CommRing R] (σ : Type u_1) (i : σ)
Goal: (KaehlerDifferential.mvPolynomialBasis R σ) i = (KaehlerDifferential.D R (MvPolynomial σ R)) (MvPolynomial.X i)
-- KaehlerDifferential.polynomial_D_apply
lemma KaehlerDifferential.polynomial_D_apply (P : R[X]) :
    D R R[X] P = derivative P • D R R[X] X 
Arguments: (R : Type u) [CommRing R] (P : Polynomial R)
Goal: (KaehlerDifferential.D R (Polynomial R)) P =
  Polynomial.derivative P • (KaehlerDifferential.D R (Polynomial R)) Polynomial.X
-- KaehlerDifferential.polynomialEquiv_comp_D
lemma KaehlerDifferential.polynomialEquiv_comp_D :
    (polynomialEquiv R).compDer (D R R[X]) = derivative' 
Arguments: (R : Type u) [CommRing R]
Goal: (KaehlerDifferential.polynomialEquiv R).compDer (KaehlerDifferential.D R (Polynomial R)) = Polynomial.derivative'
-- KaehlerDifferential.polynomialEquiv_D
@[simp]
lemma KaehlerDifferential.polynomialEquiv_D (P) :
    polynomialEquiv R (D R R[X] P) = derivative P 
Arguments: (R : Type u) [CommRing R] (P : Polynomial R)
Goal: (KaehlerDifferential.polynomialEquiv R) ((KaehlerDifferential.D R (Polynomial R)) P) = Polynomial.derivative P