ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LinearMap.trace_conj
@[simp]
theorem trace_conj (g : M →ₗ[R] M) (f : (M →ₗ[R] M)ˣ) :
    trace R M (↑f * g * ↑f⁻¹) = trace R M g 
Arguments: (R : Type u) [CommSemiring R] {M : Type v} [AddCommMonoid M] [Module R M] (g : M →ₗ[R] M) (f : (M →ₗ[R] M)ˣ)
Goal: (LinearMap.trace R M) (↑f * g * ↑f⁻¹) = (LinearMap.trace R M) g
-- LinearMap.trace_lie
@[simp]
lemma trace_lie {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M] (f g : Module.End R M) :
    trace R M ⁅f, g⁆ = 0 
Arguments: {R : Type u_2} {M : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] (f g : Module.End R M)
Goal: (LinearMap.trace R M) ⁅f, g⁆ = 0
-- LinearMap.trace_eq_contract_of_basis
theorem trace_eq_contract_of_basis [Finite ι] (b : Basis ι R M) :
    LinearMap.trace R M ∘ₗ dualTensorHom R M M = contractLeft R M 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_5} [Finite ι] (b : Basis ι R M)
Goal: LinearMap.trace R M ∘ₗ dualTensorHom R M M = contractLeft R M
-- LinearMap.trace_eq_contract_of_basis'
theorem trace_eq_contract_of_basis' [Fintype ι] [DecidableEq ι] (b : Basis ι R M) :
    LinearMap.trace R M = contractLeft R M ∘ₗ (dualTensorHomEquivOfBasis b).symm.toLinearMap 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_5} [Fintype ι] [DecidableEq ι] (b : Basis ι R M)
Goal: LinearMap.trace R M = contractLeft R M ∘ₗ ↑(dualTensorHomEquivOfBasis b).symm
-- LinearMap.trace_eq_contract
@[simp]
theorem trace_eq_contract : LinearMap.trace R M ∘ₗ dualTensorHom R M M = contractLeft R M 
Arguments: (R : Type u_1) [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M]
Goal: LinearMap.trace R M ∘ₗ dualTensorHom R M M = contractLeft R M
-- LinearMap.trace_eq_contract_apply
@[simp]
theorem trace_eq_contract_apply (x : Module.Dual R M ⊗[R] M) :
    (LinearMap.trace R M) ((dualTensorHom R M M) x) = contractLeft R M x 
Arguments: (R : Type u_1) [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (x : TensorProduct R (Module.Dual R M) M)
Goal: (LinearMap.trace R M) ((dualTensorHom R M M) x) = (contractLeft R M) x
-- LinearMap.trace_eq_contract'
theorem trace_eq_contract' :
    LinearMap.trace R M = contractLeft R M ∘ₗ (dualTensorHomEquiv R M M).symm.toLinearMap 
Arguments: (R : Type u_1) [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M]
Goal: LinearMap.trace R M = contractLeft R M ∘ₗ ↑(dualTensorHomEquiv R M M).symm
-- LinearMap.trace_one
@[simp]
theorem trace_one : trace R M 1 = (finrank R M : R) 
Arguments: (R : Type u_1) [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M]
Goal: (LinearMap.trace R M) 1 = ↑(FiniteDimensional.finrank R M)
-- LinearMap.trace_id
@[simp]
theorem trace_id : trace R M id = (finrank R M : R) 
Arguments: (R : Type u_1) [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M]
Goal: (LinearMap.trace R M) LinearMap.id = ↑(FiniteDimensional.finrank R M)
-- LinearMap.trace_transpose
@[simp]
theorem trace_transpose : trace R (Module.Dual R M) ∘ₗ Module.Dual.transpose = trace R M 
Arguments: (R : Type u_1) [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M]
Goal: LinearMap.trace R (Module.Dual R M) ∘ₗ Module.Dual.transpose = LinearMap.trace R M
-- LinearMap.trace_prodMap
theorem trace_prodMap :
    trace R (M × N) ∘ₗ prodMapLinear R M N M N R =
      (coprod id id : R × R →ₗ[R] R) ∘ₗ prodMap (trace R M) (trace R N) 
Arguments: (R : Type u_1) [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] (N : Type u_3) [AddCommGroup N] [Module R N] [Module.Free R M] [Module.Finite R M] [Module.Free R N] [Module.Finite R N]
Goal: LinearMap.trace R (M × N) ∘ₗ LinearMap.prodMapLinear R M N M N R =
  LinearMap.id.coprod LinearMap.id ∘ₗ (LinearMap.trace R M).prodMap (LinearMap.trace R N)
-- LinearMap.trace_prodMap'
theorem trace_prodMap' (f : M →ₗ[R] M) (g : N →ₗ[R] N) :
    trace R (M × N) (prodMap f g) = trace R M f + trace R N g 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {N : Type u_3} [AddCommGroup N] [Module R N] [Module.Free R M] [Module.Finite R M] [Module.Free R N] [Module.Finite R N] (f : M →ₗ[R] M) (g : N →ₗ[R] N)
Goal: (LinearMap.trace R (M × N)) (f.prodMap g) = (LinearMap.trace R M) f + (LinearMap.trace R N) g
-- LinearMap.trace_tensorProduct
theorem trace_tensorProduct : compr₂ (mapBilinear R M N M N) (trace R (M ⊗ N)) =
    compl₁₂ (lsmul R R : R →ₗ[R] R →ₗ[R] R) (trace R M) (trace R N) 
Arguments: (R : Type u_1) [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] (N : Type u_3) [AddCommGroup N] [Module R N] [Module.Free R M] [Module.Finite R M] [Module.Free R N] [Module.Finite R N]
Goal: (TensorProduct.mapBilinear R M N M N).compr₂ (LinearMap.trace R (TensorProduct R M N)) =
  (LinearMap.lsmul R R).compl₁₂ (LinearMap.trace R M) (LinearMap.trace R N)
-- LinearMap.trace_comp_comm
theorem trace_comp_comm :
    compr₂ (llcomp R M N M) (trace R M) = compr₂ (llcomp R N M N).flip (trace R N) 
Arguments: (R : Type u_1) [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] (N : Type u_3) [AddCommGroup N] [Module R N] [Module.Free R M] [Module.Finite R M] [Module.Free R N] [Module.Finite R N]
Goal: (LinearMap.llcomp R M N M).compr₂ (LinearMap.trace R M) = (LinearMap.llcomp R N M N).flip.compr₂ (LinearMap.trace R N)
-- LinearMap.trace_transpose'
@[simp]
theorem trace_transpose' (f : M →ₗ[R] M) :
    trace R _ (Module.Dual.transpose (R := R) f) = trace R M f 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
Goal: (LinearMap.trace R (Module.Dual R M)) (Module.Dual.transpose f) = (LinearMap.trace R M) f
-- LinearMap.trace_tensorProduct'
theorem trace_tensorProduct' (f : M →ₗ[R] M) (g : N →ₗ[R] N) :
    trace R (M ⊗ N) (map f g) = trace R M f * trace R N g 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {N : Type u_3} [AddCommGroup N] [Module R N] [Module.Free R M] [Module.Finite R M] [Module.Free R N] [Module.Finite R N] (f : M →ₗ[R] M) (g : N →ₗ[R] N)
Goal: (LinearMap.trace R (TensorProduct R M N)) (TensorProduct.map f g) = (LinearMap.trace R M) f * (LinearMap.trace R N) g
-- LinearMap.trace_comp_comm'
theorem trace_comp_comm' (f : M →ₗ[R] N) (g : N →ₗ[R] M) :
    trace R M (g ∘ₗ f) = trace R N (f ∘ₗ g) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {N : Type u_3} [AddCommGroup N] [Module R N] [Module.Free R M] [Module.Finite R M] [Module.Free R N] [Module.Finite R N] (f : M →ₗ[R] N) (g : N →ₗ[R] M)
Goal: (LinearMap.trace R M) (g ∘ₗ f) = (LinearMap.trace R N) (f ∘ₗ g)
-- LinearMap.trace_comp_cycle
lemma trace_comp_cycle (f : M →ₗ[R] N) (g : N →ₗ[R] P) (h : P →ₗ[R] M) :
    trace R P (g ∘ₗ f ∘ₗ h) = trace R N (f ∘ₗ h ∘ₗ g) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {N : Type u_3} {P : Type u_4} [AddCommGroup N] [Module R N] [AddCommGroup P] [Module R P] [Module.Free R N] [Module.Finite R N] [Module.Free R P] [Module.Finite R P] (f : M →ₗ[R] N) (g : N →ₗ[R] P) (h : P →ₗ[R] M)
Goal: (LinearMap.trace R P) (g ∘ₗ f ∘ₗ h) = (LinearMap.trace R N) (f ∘ₗ h ∘ₗ g)
-- LinearMap.trace_comp_cycle'
lemma trace_comp_cycle' (f : M →ₗ[R] N) (g : N →ₗ[R] P) (h : P →ₗ[R] M) :
    trace R P ((g ∘ₗ f) ∘ₗ h) = trace R M ((h ∘ₗ g) ∘ₗ f) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {N : Type u_3} {P : Type u_4} [AddCommGroup N] [Module R N] [AddCommGroup P] [Module R P] [Module.Free R M] [Module.Finite R M] [Module.Free R P] [Module.Finite R P] (f : M →ₗ[R] N) (g : N →ₗ[R] P) (h : P →ₗ[R] M)
Goal: (LinearMap.trace R P) ((g ∘ₗ f) ∘ₗ h) = (LinearMap.trace R M) ((h ∘ₗ g) ∘ₗ f)
-- LinearMap.trace_conj'
@[simp]
theorem trace_conj' (f : M →ₗ[R] M) (e : M ≃ₗ[R] N) : trace R N (e.conj f) = trace R M f 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {N : Type u_3} [AddCommGroup N] [Module R N] (f : M →ₗ[R] M) (e : M ≃ₗ[R] N)
Goal: (LinearMap.trace R N) (e.conj f) = (LinearMap.trace R M) f
-- LinearMap.IsProj.trace
theorem IsProj.trace {p : Submodule R M} {f : M →ₗ[R] M} (h : IsProj p f) [Module.Free R p]
    [Module.Finite R p] [Module.Free R (ker f)] [Module.Finite R (ker f)] :
    trace R M f = (finrank R p : R) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {p : Submodule R M} {f : M →ₗ[R] M} (h : LinearMap.IsProj p f) [Module.Free R ↥p] [Module.Finite R ↥p] [Module.Free R ↥(LinearMap.ker f)] [Module.Finite R ↥(LinearMap.ker f)]
Goal: (LinearMap.trace R M) f = ↑(FiniteDimensional.finrank R ↥p)
-- LinearMap.isNilpotent_trace_of_isNilpotent
lemma isNilpotent_trace_of_isNilpotent {f : M →ₗ[R] M} (hf : IsNilpotent f) :
    IsNilpotent (trace R M f) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] {f : M →ₗ[R] M} (hf : IsNilpotent f)
Goal: IsNilpotent ((LinearMap.trace R M) f)
-- LinearMap.trace_comp_eq_mul_of_commute_of_isNilpotent
lemma trace_comp_eq_mul_of_commute_of_isNilpotent [IsReduced R] {f g : Module.End R M}
    (μ : R) (h_comm : Commute f g) (hg : IsNilpotent (g - algebraMap R _ μ)) :
    trace R M (f ∘ₗ g) = μ * trace R M f 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] [IsReduced R] {f g : Module.End R M} (μ : R) (h_comm : Commute f g) (hg : IsNilpotent (g - (algebraMap R (Module.End R M)) μ))
Goal: (LinearMap.trace R M) (f ∘ₗ g) = μ * (LinearMap.trace R M) f
-- LinearMap.trace_baseChange
@[simp]
lemma trace_baseChange [Module.Free R M] [Module.Finite R M]
    (f : M →ₗ[R] M) (A : Type*) [CommRing A] [Algebra R A] :
    trace A _ (f.baseChange A) = algebraMap R A (trace R _ f) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M) (A : Type u_6) [CommRing A] [Algebra R A]
Goal: (LinearMap.trace A (TensorProduct R A M)) (LinearMap.baseChange A f) = (algebraMap R A) ((LinearMap.trace R M) f)
-- coevaluation_apply_one
theorem coevaluation_apply_one :
    (coevaluation K V) (1 : K) =
      let bV := Basis.ofVectorSpace K V
      ∑ i : Basis.ofVectorSpaceIndex K V, bV i ⊗ₜ[K] bV.coord i 
Arguments: (K : Type u) [Field K] (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
Goal: (coevaluation K V) 1 =
  let bV := Basis.ofVectorSpace K V;
  ∑ i : ↑(Basis.ofVectorSpaceIndex K V), bV i ⊗ₜ[K] bV.coord i
-- contractLeft_assoc_coevaluation
theorem contractLeft_assoc_coevaluation :
    (contractLeft K V).rTensor _ ∘ₗ
        (TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
          (coevaluation K V).lTensor (Module.Dual K V) =
      (TensorProduct.lid K _).symm.toLinearMap ∘ₗ (TensorProduct.rid K _).toLinearMap 
Arguments: (K : Type u) [Field K] (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
Goal: LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ∘ₗ
    ↑(TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)).symm ∘ₗ
      LinearMap.lTensor (Module.Dual K V) (coevaluation K V) =
  ↑(TensorProduct.lid K (Module.Dual K V)).symm ∘ₗ ↑(TensorProduct.rid K (Module.Dual K V))
-- contractLeft_assoc_coevaluation'
theorem contractLeft_assoc_coevaluation' :
    (contractLeft K V).lTensor _ ∘ₗ
        (TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
      (TensorProduct.rid K _).symm.toLinearMap ∘ₗ (TensorProduct.lid K _).toLinearMap 
Arguments: (K : Type u) [Field K] (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V]
Goal: LinearMap.lTensor V (contractLeft K V) ∘ₗ
    ↑(TensorProduct.assoc K V (Module.Dual K V) V) ∘ₗ LinearMap.rTensor V (coevaluation K V) =
  ↑(TensorProduct.rid K V).symm ∘ₗ ↑(TensorProduct.lid K V)
-- IsIntegral.map_of_comp_eq
theorem IsIntegral.map_of_comp_eq {R S T U : Type*} [CommRing R] [Ring S]
    [CommRing T] [Ring U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
    (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a) :
    IsIntegral T (ψ a) 
Arguments: {R : Type u_5} {S : Type u_6} {T : Type u_7} {U : Type u_8} [CommRing R] [Ring S] [CommRing T] [Ring U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U) (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a)
Goal: IsIntegral T (ψ a)
-- Algebra.IsIntegral.of_injective
theorem Algebra.IsIntegral.of_injective [Algebra.IsIntegral R B] : Algebra.IsIntegral R A 
Arguments: {R : Type u_1} [CommRing R] {A : Type u_5} {B : Type u_6} [Ring A] [Ring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : Function.Injective ⇑f) [Algebra.IsIntegral R B]
Goal: Algebra.IsIntegral R A
-- isIntegral_algEquiv
@[simp]
theorem isIntegral_algEquiv {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
    (f : A ≃ₐ[R] B) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x 
Arguments: {R : Type u_1} [CommRing R] {A : Type u_5} {B : Type u_6} [Ring A] [Ring B] [Algebra R A] [Algebra R B] (f : A ≃ₐ[R] B) {x : A}
Goal: IsIntegral R (f x) ↔ IsIntegral R x
-- IsIntegral.tower_top
theorem IsIntegral.tower_top [Algebra A B] [IsScalarTower R A B] {x : B}
    (hx : IsIntegral R x) : IsIntegral A x 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B] {x : B} (hx : IsIntegral R x)
Goal: IsIntegral A x
-- map_isIntegral_int
theorem map_isIntegral_int {B C F : Type*} [Ring B] [Ring C] {b : B}
    [FunLike F B C] [RingHomClass F B C] (f : F)
    (hb : IsIntegral ℤ b) : IsIntegral ℤ (f b) 
Arguments: {B : Type u_5} {C : Type u_6} {F : Type u_7} [Ring B] [Ring C] {b : B} [FunLike F B C] [RingHomClass F B C] (f : F) (hb : IsIntegral ℤ b)
Goal: IsIntegral ℤ (f b)
-- IsIntegral.of_subring
theorem IsIntegral.of_subring {x : B} (T : Subring R) (hx : IsIntegral T x) : IsIntegral R x 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (T : Subring R) (hx : IsIntegral (↥T) x)
Goal: IsIntegral R x
-- IsIntegral.algebraMap
protected theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A}
    (h : IsIntegral R x) : IsIntegral R (algebraMap A B x) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B] {x : A} (h : IsIntegral R x)
Goal: IsIntegral R ((algebraMap A B) x)
-- isIntegral_algebraMap_iff
theorem isIntegral_algebraMap_iff [Algebra A B] [IsScalarTower R A B] {x : A}
    (hAB : Function.Injective (algebraMap A B)) :
    IsIntegral R (algebraMap A B x) ↔ IsIntegral R x 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B] {x : A} (hAB : Function.Injective ⇑(algebraMap A B))
Goal: IsIntegral R ((algebraMap A B) x) ↔ IsIntegral R x
-- isIntegral_iff_isIntegral_closure_finite
theorem isIntegral_iff_isIntegral_closure_finite {r : B} :
    IsIntegral R r ↔ ∃ s : Set R, s.Finite ∧ IsIntegral (Subring.closure s) r 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {r : B}
Goal: IsIntegral R r ↔ ∃ s, s.Finite ∧ IsIntegral (↥(Subring.closure s)) r
-- fg_adjoin_of_finite
theorem fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
    (Algebra.adjoin R s).toSubmodule.FG 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x)
Goal: (Subalgebra.toSubmodule (Algebra.adjoin R s)).FG
-- isNoetherian_adjoin_finset
theorem isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
    (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (s : Set A)) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] [IsNoetherianRing R] (s : Finset A) (hs : ∀ x ∈ s, IsIntegral R x)
Goal: IsNoetherian R ↥(Algebra.adjoin R ↑s)
-- isIntegral_of_noetherian
theorem isIntegral_of_noetherian (_ : IsNoetherian R B) (x : B) : IsIntegral R x 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B]
Goal: IsNoetherian R B → ∀ (x : B), IsIntegral R x
-- isIntegral_of_submodule_noetherian
theorem isIntegral_of_submodule_noetherian (S : Subalgebra R B)
    (H : IsNoetherian R (Subalgebra.toSubmodule S)) (x : B) (hx : x ∈ S) : IsIntegral R x 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] (S : Subalgebra R B) (H : IsNoetherian R ↥(Subalgebra.toSubmodule S)) (x : B) (hx : x ∈ S)
Goal: IsIntegral R x
-- isIntegral_of_smul_mem_submodule
theorem isIntegral_of_smul_mem_submodule {M : Type*} [AddCommGroup M] [Module R M] [Module A M]
    [IsScalarTower R A M] [NoZeroSMulDivisors A M] (N : Submodule R M) (hN : N ≠ ⊥) (hN' : N.FG)
    (x : A) (hx : ∀ n ∈ N, x • n ∈ N) : IsIntegral R x 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {M : Type u_5} [AddCommGroup M] [Module R M] [Module A M] [IsScalarTower R A M] [NoZeroSMulDivisors A M] (N : Submodule R M) (hN : N ≠ ⊥) (hN' : N.FG) (x : A) (hx : ∀ n ∈ N, x • n ∈ N)
Goal: IsIntegral R x
-- RingHom.Finite.to_isIntegral
theorem RingHom.Finite.to_isIntegral (h : f.Finite) : f.IsIntegral 
Arguments: {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] {f : R →+* S} (h : f.Finite)
Goal: f.IsIntegral
-- Algebra.IsIntegral.finite
theorem Algebra.IsIntegral.finite [Algebra.IsIntegral R A] [h' : Algebra.FiniteType R A] :
    Module.Finite R A 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] [h' : Algebra.FiniteType R A]
Goal: Module.Finite R A
-- Algebra.finite_iff_isIntegral_and_finiteType
theorem Algebra.finite_iff_isIntegral_and_finiteType :
    Module.Finite R A ↔ Algebra.IsIntegral R A ∧ Algebra.FiniteType R A 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A]
Goal: Module.Finite R A ↔ Algebra.IsIntegral R A ∧ Algebra.FiniteType R A
-- RingHom.IsIntegral.to_finite
theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.Finite 
Arguments: {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] {f : R →+* S} (h : f.IsIntegral) (h' : f.FiniteType)
Goal: f.Finite
-- RingHom.finite_iff_isIntegral_and_finiteType
theorem RingHom.finite_iff_isIntegral_and_finiteType : f.Finite ↔ f.IsIntegral ∧ f.FiniteType 
Arguments: {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] {f : R →+* S}
Goal: f.Finite ↔ f.IsIntegral ∧ f.FiniteType
-- mem_integralClosure_iff_mem_fg
theorem mem_integralClosure_iff_mem_fg {r : A} :
    r ∈ integralClosure R A ↔ ∃ M : Subalgebra R A, M.toSubmodule.FG ∧ r ∈ M 
Arguments: (R : Type u_1) (A : Type u_2) [CommRing R] [CommRing A] [Algebra R A] {r : A}
Goal: r ∈ integralClosure R A ↔ ∃ M, (Subalgebra.toSubmodule M).FG ∧ r ∈ M
-- adjoin_le_integralClosure
theorem adjoin_le_integralClosure {x : A} (hx : IsIntegral R x) :
    Algebra.adjoin R {x} ≤ integralClosure R A 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {x : A} (hx : IsIntegral R x)
Goal: Algebra.adjoin R {x} ≤ integralClosure R A
-- le_integralClosure_iff_isIntegral
theorem le_integralClosure_iff_isIntegral {S : Subalgebra R A} :
    S ≤ integralClosure R A ↔ Algebra.IsIntegral R S 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {S : Subalgebra R A}
Goal: S ≤ integralClosure R A ↔ Algebra.IsIntegral R ↥S
-- Algebra.isIntegral_sup
theorem Algebra.isIntegral_sup {S T : Subalgebra R A} :
    Algebra.IsIntegral R (S ⊔ T : Subalgebra R A) ↔
      Algebra.IsIntegral R S ∧ Algebra.IsIntegral R T 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {S T : Subalgebra R A}
Goal: Algebra.IsIntegral R ↥(S ⊔ T) ↔ Algebra.IsIntegral R ↥S ∧ Algebra.IsIntegral R ↥T
-- integralClosure_map_algEquiv
theorem integralClosure_map_algEquiv [Algebra R S] (f : A ≃ₐ[R] S) :
    (integralClosure R A).map (f : A →ₐ[R] S) = integralClosure R S 
Arguments: {R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A ≃ₐ[R] S)
Goal: Subalgebra.map (↑f) (integralClosure R A) = integralClosure R S
-- AlgHom.coe_mapIntegralClosure
@[simp]
theorem AlgHom.coe_mapIntegralClosure [Algebra R S] (f : A →ₐ[R] S)
    (x : integralClosure R A) : (f.mapIntegralClosure x : S) = f (x : A) 
Arguments: {R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A →ₐ[R] S) (x : ↥(integralClosure R A))
Goal: ↑(f.mapIntegralClosure x) = f ↑x
-- AlgEquiv.coe_mapIntegralClosure
@[simp]
theorem AlgEquiv.coe_mapIntegralClosure [Algebra R S] (f : A ≃ₐ[R] S)
    (x : integralClosure R A) : (f.mapIntegralClosure x : S) = f (x : A) 
Arguments: {R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A ≃ₐ[R] S) (x : ↥(integralClosure R A))
Goal: ↑(f.mapIntegralClosure x) = f ↑x
-- integralClosure.isIntegral
theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (x : ↥(integralClosure R A))
Goal: IsIntegral R x
-- IsIntegral.of_mul_unit
theorem IsIntegral.of_mul_unit {x y : B} {r : R} (hr : algebraMap R B r * y = 1)
    (hx : IsIntegral R (x * y)) : IsIntegral R x 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x y : B} {r : R} (hr : (algebraMap R B) r * y = 1) (hx : IsIntegral R (x * y))
Goal: IsIntegral R x
-- RingHom.IsIntegralElem.of_mul_unit
theorem RingHom.IsIntegralElem.of_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
    (hx : f.IsIntegralElem (x * y)) : f.IsIntegralElem x 
Arguments: {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) (x y : S) (r : R) (hr : f r * y = 1) (hx : f.IsIntegralElem (x * y))
Goal: f.IsIntegralElem x
-- IsIntegral.of_mem_closure'
theorem IsIntegral.of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x) :
    ∀ x ∈ Subring.closure G, IsIntegral R x 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x) (x : A)
Goal: x ∈ Subring.closure G → IsIntegral R x
-- IsIntegral.of_mem_closure''
theorem IsIntegral.of_mem_closure'' {S : Type*} [CommRing S] {f : R →+* S} (G : Set S)
    (hG : ∀ x ∈ G, f.IsIntegralElem x) : ∀ x ∈ Subring.closure G, f.IsIntegralElem x 
Arguments: {R : Type u_1} [CommRing R] {S : Type u_5} [CommRing S] {f : R →+* S} (G : Set S) (hG : ∀ x ∈ G, f.IsIntegralElem x) (x : S)
Goal: x ∈ Subring.closure G → f.IsIntegralElem x
-- IsIntegral.pow
theorem IsIntegral.pow {x : B} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (x ^ n) 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (h : IsIntegral R x) (n : ℕ)
Goal: IsIntegral R (x ^ n)
-- IsIntegral.nsmul
theorem IsIntegral.nsmul {x : B} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (n • x) 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (h : IsIntegral R x) (n : ℕ)
Goal: IsIntegral R (n • x)
-- IsIntegral.zsmul
theorem IsIntegral.zsmul {x : B} (h : IsIntegral R x) (n : ℤ) : IsIntegral R (n • x) 
Arguments: {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (h : IsIntegral R x) (n : ℤ)
Goal: IsIntegral R (n • x)
-- IsIntegral.multiset_prod
theorem IsIntegral.multiset_prod {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x) :
    IsIntegral R s.prod 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x)
Goal: IsIntegral R s.prod
-- IsIntegral.multiset_sum
theorem IsIntegral.multiset_sum {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x) :
    IsIntegral R s.sum 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x)
Goal: IsIntegral R s.sum