ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MonoidAlgebra.fg_of_finiteType
theorem fg_of_finiteType [CommRing R] [Nontrivial R] [h : FiniteType R (MonoidAlgebra R M)] :
    Monoid.FG M 
Arguments: {R : Type u_1} {M : Type u_2} [Monoid M] [CommRing R] [Nontrivial R] [h : Algebra.FiniteType R (MonoidAlgebra R M)]
Goal: Monoid.FG M
-- MonoidAlgebra.finiteType_iff_group_fg
theorem finiteType_iff_group_fg {G : Type*} [Group G] [CommRing R] [Nontrivial R] :
    FiniteType R (MonoidAlgebra R G) ↔ Group.FG G 
Arguments: {R : Type u_1} {G : Type u_3} [Group G] [CommRing R] [Nontrivial R]
Goal: Algebra.FiniteType R (MonoidAlgebra R G) ↔ Group.FG G
-- Module.Finite.injective_of_surjective_endomorphism
@[deprecated OrzechProperty.injective_of_surjective_endomorphism (since := "2024-05-30")]
theorem Module.Finite.injective_of_surjective_endomorphism {R : Type*} [CommRing R] {M : Type*}
    [AddCommGroup M] [Module R M] [Finite R M] (f : M →ₗ[R] M)
    (f_surj : Function.Surjective f) : Function.Injective f 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] [Module.Finite R M] (f : M →ₗ[R] M) (f_surj : Function.Surjective ⇑f)
Goal: Function.Injective ⇑f
-- PowerSeries.coeff_derivativeFun
theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
    coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) 
Arguments: {R : Type u_1} [CommSemiring R] (f : PowerSeries R) (n : ℕ)
Goal: (PowerSeries.coeff R n) f.derivativeFun = (PowerSeries.coeff R (n + 1)) f * (↑n + 1)
-- PowerSeries.derivativeFun_coe
theorem derivativeFun_coe (f : R[X]) : (f : R⟦X⟧).derivativeFun = derivative f 
Arguments: {R : Type u_1} [CommSemiring R] (f : Polynomial R)
Goal: (↑f).derivativeFun = ↑(Polynomial.derivative f)
-- PowerSeries.derivativeFun_add
theorem derivativeFun_add (f g : R⟦X⟧) :
    derivativeFun (f + g) = derivativeFun f + derivativeFun g 
Arguments: {R : Type u_1} [CommSemiring R] (f g : PowerSeries R)
Goal: (f + g).derivativeFun = f.derivativeFun + g.derivativeFun
-- PowerSeries.derivativeFun_C
theorem derivativeFun_C (r : R) : derivativeFun (C R r) = 0 
Arguments: {R : Type u_1} [CommSemiring R] (r : R)
Goal: ((PowerSeries.C R) r).derivativeFun = 0
-- PowerSeries.trunc_derivativeFun
theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
    trunc n f.derivativeFun = derivative (trunc (n + 1) f) 
Arguments: {R : Type u_1} [CommSemiring R] (f : PowerSeries R) (n : ℕ)
Goal: PowerSeries.trunc n f.derivativeFun = Polynomial.derivative (PowerSeries.trunc (n + 1) f)
-- _private.Mathlib.RingTheory.PowerSeries.Derivative.0.PowerSeries.derivativeFun_coe_mul_coe
private theorem derivativeFun_coe_mul_coe (f g : R[X]) : derivativeFun (f * g : R⟦X⟧) =
    f * derivative g + g * derivative f  
Arguments: {R : Type u_1} [CommSemiring R] (f g : Polynomial R)
Goal: (↑f * ↑g).derivativeFun = ↑f * ↑(Polynomial.derivative g) + ↑g * ↑(Polynomial.derivative f)
-- PowerSeries.derivativeFun_mul
theorem derivativeFun_mul (f g : R⟦X⟧) :
    derivativeFun (f * g) = f • g.derivativeFun + g • f.derivativeFun 
Arguments: {R : Type u_1} [CommSemiring R] (f g : PowerSeries R)
Goal: (f * g).derivativeFun = f • g.derivativeFun + g • f.derivativeFun
-- PowerSeries.derivativeFun_one
theorem derivativeFun_one : derivativeFun (1 : R⟦X⟧) = 0 
Arguments: {R : Type u_1} [CommSemiring R]
Goal: PowerSeries.derivativeFun 1 = 0
-- PowerSeries.derivativeFun_smul
theorem derivativeFun_smul (r : R) (f : R⟦X⟧) : derivativeFun (r • f) = r • derivativeFun f 
Arguments: {R : Type u_1} [CommSemiring R] (r : R) (f : PowerSeries R)
Goal: (r • f).derivativeFun = r • f.derivativeFun
-- PowerSeries.derivative_C
@[simp] theorem derivative_C (r : R) : d⁄dX R (C R r) = 0 
Arguments: {R : Type u_1} [CommSemiring R] (r : R)
Goal: (PowerSeries.derivative R) ((PowerSeries.C R) r) = 0
-- PowerSeries.coeff_derivative
theorem coeff_derivative (f : R⟦X⟧) (n : ℕ) :
    coeff R n (d⁄dX R f) = coeff R (n + 1) f * (n + 1) 
Arguments: {R : Type u_1} [CommSemiring R] (f : PowerSeries R) (n : ℕ)
Goal: (PowerSeries.coeff R n) ((PowerSeries.derivative R) f) = (PowerSeries.coeff R (n + 1)) f * (↑n + 1)
-- PowerSeries.derivative_coe
theorem derivative_coe (f : R[X]) : d⁄dX R f = Polynomial.derivative f 
Arguments: {R : Type u_1} [CommSemiring R] (f : Polynomial R)
Goal: (PowerSeries.derivative R) ↑f = ↑(Polynomial.derivative f)
-- PowerSeries.derivative_X
@[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 
Arguments: {R : Type u_1} [CommSemiring R]
Goal: (PowerSeries.derivative R) PowerSeries.X = 1
-- PowerSeries.trunc_derivative
theorem trunc_derivative (f : R⟦X⟧) (n : ℕ) :
    trunc n (d⁄dX R f) = Polynomial.derivative (trunc (n + 1) f) 
Arguments: {R : Type u_1} [CommSemiring R] (f : PowerSeries R) (n : ℕ)
Goal: PowerSeries.trunc n ((PowerSeries.derivative R) f) = Polynomial.derivative (PowerSeries.trunc (n + 1) f)
-- PowerSeries.trunc_derivative'
theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
    trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) 
Arguments: {R : Type u_1} [CommSemiring R] (f : PowerSeries R) (n : ℕ)
Goal: PowerSeries.trunc (n - 1) ((PowerSeries.derivative R) f) = Polynomial.derivative (PowerSeries.trunc n f)
-- PowerSeries.derivative.ext
theorem derivative.ext {R} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g} (hD : d⁄dX R f = d⁄dX R g)
    (hc : constantCoeff R f = constantCoeff R g) : f = g 
Arguments: {R : Type u_1} [CommRing R] [NoZeroSMulDivisors ℕ R] {f g : PowerSeries R} (hD : (PowerSeries.derivative R) f = (PowerSeries.derivative R) g) (hc : (PowerSeries.constantCoeff R) f = (PowerSeries.constantCoeff R) g)
Goal: f = g
-- PowerSeries.derivative_inv
@[simp] theorem derivative_inv {R} [CommRing R] (f : R⟦X⟧ˣ) :
    d⁄dX R ↑f⁻¹ = -(↑f⁻¹ : R⟦X⟧) ^ 2 * d⁄dX R f 
Arguments: {R : Type u_1} [CommRing R] (f : (PowerSeries R)ˣ)
Goal: (PowerSeries.derivative R) ↑f⁻¹ = -↑f⁻¹ ^ 2 * (PowerSeries.derivative R) ↑f
-- PowerSeries.derivative_invOf
@[simp] theorem derivative_invOf {R} [CommRing R] (f : R⟦X⟧) [Invertible f] :
    d⁄dX R ⅟f = - ⅟f ^ 2 * d⁄dX R f 
Arguments: {R : Type u_1} [CommRing R] (f : PowerSeries R) [Invertible f]
Goal: (PowerSeries.derivative R) ⅟f = -⅟f ^ 2 * (PowerSeries.derivative R) f
-- PowerSeries.derivative_inv'
@[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f 
Arguments: {R : Type u_1} [Field R] (f : PowerSeries R)
Goal: (PowerSeries.derivative R) f⁻¹ = -f⁻¹ ^ 2 * (PowerSeries.derivative R) f
-- Bialgebra.comul_one
class Bialgebra (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] extends
    Algebra R A, Coalgebra R A where
  -- The counit is an algebra morphism
    counit_one : counit 1 = 1
    mul_compr₂_counit : (LinearMap.mul R A).compr₂ counit = (LinearMap.mul R R).compl₁₂ counit counit
  -- The comultiplication is an algebra morphism
    comul_one : comul 1 = 1
    mul_compr₂_comul :
    (LinearMap.mul R A).compr₂ comul = (LinearMap.mul R (A ⊗[R] A)).compl₁₂ comul comul
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [self : Bialgebra R A]
Goal: Coalgebra.comul 1 = 1
-- Bialgebra.mul_compr₂_comul
class Bialgebra (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] extends
    Algebra R A, Coalgebra R A where
  -- The counit is an algebra morphism
    counit_one : counit 1 = 1
    mul_compr₂_counit : (LinearMap.mul R A).compr₂ counit = (LinearMap.mul R R).compl₁₂ counit counit
  -- The comultiplication is an algebra morphism
    comul_one : comul 1 = 1
    mul_compr₂_comul :
    (LinearMap.mul R A).compr₂ comul = (LinearMap.mul R (A ⊗[R] A)).compl₁₂ comul comul
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [self : Bialgebra R A]
Goal: (LinearMap.mul R A).compr₂ Coalgebra.comul = (LinearMap.mul R (A ⊗[R] A)).compl₁₂ Coalgebra.comul Coalgebra.comul
-- Bialgebra.mul_compr₂_counit
class Bialgebra (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] extends
    Algebra R A, Coalgebra R A where
  -- The counit is an algebra morphism
    counit_one : counit 1 = 1
    mul_compr₂_counit : (LinearMap.mul R A).compr₂ counit = (LinearMap.mul R R).compl₁₂ counit counit
  -- The comultiplication is an algebra morphism
    comul_one : comul 1 = 1
    mul_compr₂_comul :
    (LinearMap.mul R A).compr₂ comul = (LinearMap.mul R (A ⊗[R] A)).compl₁₂ comul comul
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [self : Bialgebra R A]
Goal: (LinearMap.mul R A).compr₂ Coalgebra.counit = (LinearMap.mul R R).compl₁₂ Coalgebra.counit Coalgebra.counit
-- Bialgebra.counit_one
class Bialgebra (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] extends
    Algebra R A, Coalgebra R A where
  -- The counit is an algebra morphism
    counit_one : counit 1 = 1
    mul_compr₂_counit : (LinearMap.mul R A).compr₂ counit = (LinearMap.mul R R).compl₁₂ counit counit
  -- The comultiplication is an algebra morphism
    comul_one : comul 1 = 1
    mul_compr₂_comul :
    (LinearMap.mul R A).compr₂ comul = (LinearMap.mul R (A ⊗[R] A)).compl₁₂ comul comul
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [self : Bialgebra R A]
Goal: Coalgebra.counit 1 = 1
-- Bialgebra.counit_mul
lemma counit_mul (a b : A) : counit (R := R) (a * b) = counit a * counit b 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Bialgebra R A] (a b : A)
Goal: Coalgebra.counit (a * b) = Coalgebra.counit a * Coalgebra.counit b
-- Bialgebra.comul_mul
lemma comul_mul (a b : A) : comul (R := R) (a * b) = comul a * comul b 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Bialgebra R A] (a b : A)
Goal: Coalgebra.comul (a * b) = Coalgebra.comul a * Coalgebra.comul b
-- Bialgebra.counitAlgHom_apply
@[simps!]
def counitAlgHom : A →ₐ[R] R 
Arguments: (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Bialgebra R A] (a : A)
Goal: (Bialgebra.counitAlgHom R A) a = Coalgebra.counit a
-- Bialgebra.comulAlgHom_apply
@[simps!]
def comulAlgHom : A →ₐ[R] A ⊗[R] A 
Arguments: (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Bialgebra R A] (a : A)
Goal: (Bialgebra.comulAlgHom R A) a = Coalgebra.comul a
-- Bialgebra.counit_algebraMap
@[simp] lemma counit_algebraMap (r : R) : counit (R := R) (algebraMap R A r) = r 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Bialgebra R A] (r : R)
Goal: Coalgebra.counit ((algebraMap R A) r) = r
-- Bialgebra.comul_algebraMap
@[simp] lemma comul_algebraMap (r : R) :
    comul (R := R) (algebraMap R A r) = algebraMap R (A ⊗[R] A) r 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Bialgebra R A] (r : R)
Goal: Coalgebra.comul ((algebraMap R A) r) = (algebraMap R (A ⊗[R] A)) r
-- Bialgebra.counit_natCast
@[simp] lemma counit_natCast (n : ℕ) : counit (R := R) (n : A) = n 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Bialgebra R A] (n : ℕ)
Goal: Coalgebra.counit ↑n = ↑n
-- Bialgebra.comul_natCast
@[simp] lemma comul_natCast (n : ℕ) : comul (R := R) (n : A) = n 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Bialgebra R A] (n : ℕ)
Goal: Coalgebra.comul ↑n = ↑n
-- Bialgebra.counit_pow
@[simp] lemma counit_pow (a : A) (n : ℕ) : counit (R := R) (a ^ n) = counit a ^ n 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Bialgebra R A] (a : A) (n : ℕ)
Goal: Coalgebra.counit (a ^ n) = Coalgebra.counit a ^ n
-- Bialgebra.comul_pow
@[simp] lemma comul_pow (a : A) (n : ℕ) : comul (R := R) (a ^ n) = comul a ^ n 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Bialgebra R A] (a : A) (n : ℕ)
Goal: Coalgebra.comul (a ^ n) = Coalgebra.comul a ^ n
-- IsIntegralClosure.isLocalization
theorem IsIntegralClosure.isLocalization [Algebra.IsAlgebraic K L] :
    IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L 
Arguments: (A : Type u_2) (K : Type u_3) [CommRing A] [Field K] [IsDomain A] [Algebra A K] [IsFractionRing A K] (L : Type u_4) [Field L] (C : Type u_5) [CommRing C] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L] [Algebra.IsAlgebraic K L]
Goal: IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L
-- IsIntegralClosure.isLocalization_of_isSeparable
theorem IsIntegralClosure.isLocalization_of_isSeparable [Algebra.IsSeparable K L] :
    IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L 
Arguments: (A : Type u_2) (K : Type u_3) [CommRing A] [Field K] [IsDomain A] [Algebra A K] [IsFractionRing A K] (L : Type u_4) [Field L] (C : Type u_5) [CommRing C] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L] [Algebra.IsSeparable K L]
Goal: IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L
-- IsIntegralClosure.range_le_span_dualBasis
theorem IsIntegralClosure.range_le_span_dualBasis [Algebra.IsSeparable K L] {ι : Type*} [Fintype ι]
    [DecidableEq ι] (b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
    LinearMap.range ((Algebra.linearMap C L).restrictScalars A) ≤
    Submodule.span A (Set.range <| (traceForm K L).dualBasis (traceForm_nondegenerate K L) b) 
Arguments: {A : Type u_2} {K : Type u_3} [CommRing A] [Field K] [Algebra A K] [IsFractionRing A K] {L : Type u_4} [Field L] (C : Type u_5) [CommRing C] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L] [FiniteDimensional K L] [Algebra.IsSeparable K L] {ι : Type u_6} [Fintype ι] [DecidableEq ι] (b : Basis ι K L) (hb_int : ∀ (i : ι), IsIntegral A (b i)) [IsIntegrallyClosed A]
Goal: LinearMap.range (↑A (Algebra.linearMap C L)) ≤ Submodule.span A (Set.range ⇑((Algebra.traceForm K L).dualBasis ⋯ b))
-- integralClosure_le_span_dualBasis
theorem integralClosure_le_span_dualBasis [Algebra.IsSeparable K L] {ι : Type*} [Fintype ι]
    [DecidableEq ι] (b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
    Subalgebra.toSubmodule (integralClosure A L) ≤
    Submodule.span A (Set.range <| (traceForm K L).dualBasis (traceForm_nondegenerate K L) b) 
Arguments: {A : Type u_2} {K : Type u_3} [CommRing A] [Field K] [Algebra A K] [IsFractionRing A K] {L : Type u_4} [Field L] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [FiniteDimensional K L] [Algebra.IsSeparable K L] {ι : Type u_6} [Fintype ι] [DecidableEq ι] (b : Basis ι K L) (hb_int : ∀ (i : ι), IsIntegral A (b i)) [IsIntegrallyClosed A]
Goal: Subalgebra.toSubmodule (integralClosure A L) ≤ Submodule.span A (Set.range ⇑((Algebra.traceForm K L).dualBasis ⋯ b))
-- exists_integral_multiples
theorem exists_integral_multiples (s : Finset L) :
    ∃ y ≠ (0 : A), ∀ x ∈ s, IsIntegral A (y • x) 
Arguments: (A : Type u_2) (K : Type u_3) [CommRing A] [Field K] [IsDomain A] [Algebra A K] [IsFractionRing A K] {L : Type u_4} [Field L] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [FiniteDimensional K L] (s : Finset L)
Goal: ∃ y, y ≠ 0 ∧ ∀ x ∈ s, IsIntegral A (y • x)
-- FiniteDimensional.exists_is_basis_integral
theorem FiniteDimensional.exists_is_basis_integral :
    ∃ (s : Finset L) (b : Basis s K L), ∀ x, IsIntegral A (b x) 
Arguments: (A : Type u_2) (K : Type u_3) [CommRing A] [Field K] [IsDomain A] [Algebra A K] [IsFractionRing A K] (L : Type u_4) [Field L] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [FiniteDimensional K L]
Goal: ∃ s b, ∀ (x : { x // x ∈ s }), IsIntegral A (b x)
-- IsIntegralClosure.isNoetherian
theorem IsIntegralClosure.isNoetherian [IsIntegrallyClosed A] [IsNoetherianRing A] :
    IsNoetherian A C 
Arguments: (A : Type u_2) (K : Type u_3) [CommRing A] [Field K] [IsDomain A] [Algebra A K] [IsFractionRing A K] (L : Type u_4) [Field L] (C : Type u_5) [CommRing C] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsNoetherianRing A]
Goal: IsNoetherian A C
-- IsIntegralClosure.isNoetherianRing
theorem IsIntegralClosure.isNoetherianRing [IsIntegrallyClosed A] [IsNoetherianRing A] :
    IsNoetherianRing C 
Arguments: (A : Type u_2) (K : Type u_3) [CommRing A] [Field K] [IsDomain A] [Algebra A K] [IsFractionRing A K] (L : Type u_4) [Field L] (C : Type u_5) [CommRing C] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsNoetherianRing A]
Goal: IsNoetherianRing C
-- IsIntegralClosure.module_free
theorem IsIntegralClosure.module_free [NoZeroSMulDivisors A L] [IsPrincipalIdealRing A] :
    Module.Free A C 
Arguments: (A : Type u_2) (K : Type u_3) [CommRing A] [Field K] [IsDomain A] [Algebra A K] [IsFractionRing A K] (L : Type u_4) [Field L] (C : Type u_5) [CommRing C] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [NoZeroSMulDivisors A L] [IsPrincipalIdealRing A]
Goal: Module.Free A C
-- IsIntegralClosure.rank
theorem IsIntegralClosure.rank [IsPrincipalIdealRing A] [NoZeroSMulDivisors A L] :
    FiniteDimensional.finrank A C = FiniteDimensional.finrank K L 
Arguments: (A : Type u_2) (K : Type u_3) [CommRing A] [Field K] [IsDomain A] [Algebra A K] [IsFractionRing A K] (L : Type u_4) [Field L] (C : Type u_5) [CommRing C] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsPrincipalIdealRing A] [NoZeroSMulDivisors A L]
Goal: FiniteDimensional.finrank A C = FiniteDimensional.finrank K L
-- integralClosure.isNoetherianRing
theorem integralClosure.isNoetherianRing [IsIntegrallyClosed A] [IsNoetherianRing A] :
    IsNoetherianRing (integralClosure A L) 
Arguments: {A : Type u_2} {K : Type u_3} [CommRing A] [Field K] [IsDomain A] [Algebra A K] [IsFractionRing A K] (L : Type u_4) [Field L] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegrallyClosed A] [IsNoetherianRing A]
Goal: IsNoetherianRing ↥(integralClosure A L)
-- IsIntegralClosure.isDedekindDomain
theorem IsIntegralClosure.isDedekindDomain [IsDedekindDomain A] : IsDedekindDomain C 
Arguments: (A : Type u_2) (K : Type u_3) [CommRing A] [Field K] [IsDomain A] [Algebra A K] [IsFractionRing A K] (L : Type u_4) [Field L] (C : Type u_5) [CommRing C] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsDomain C] [IsDedekindDomain A]
Goal: IsDedekindDomain C
-- integralClosure.isDedekindDomain
theorem integralClosure.isDedekindDomain [IsDedekindDomain A] :
    IsDedekindDomain (integralClosure A L) 
Arguments: (A : Type u_2) (K : Type u_3) [CommRing A] [Field K] [IsDomain A] [Algebra A K] [IsFractionRing A K] (L : Type u_4) [Field L] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsDedekindDomain A]
Goal: IsDedekindDomain ↥(integralClosure A L)
-- PowerSeries.coeff_inv_aux
theorem coeff_inv_aux (n : ℕ) (a : R) (φ : R⟦X⟧) :
    coeff R n (inv.aux a φ) =
      if n = 0 then a
      else
        -a *
          ∑ x ∈ antidiagonal n,
            if x.2 < n then coeff R x.1 φ * coeff R x.2 (inv.aux a φ) else 0 
Arguments: {R : Type u_1} [Ring R] (n : ℕ) (a : R) (φ : PowerSeries R)
Goal: (PowerSeries.coeff R n) (PowerSeries.inv.aux a φ) =
  if n = 0 then a
  else
    -a *
      ∑ x ∈ Finset.antidiagonal n,
        if x.2 < n then (PowerSeries.coeff R x.1) φ * (PowerSeries.coeff R x.2) (PowerSeries.inv.aux a φ) else 0
-- PowerSeries.coeff_invOfUnit
theorem coeff_invOfUnit (n : ℕ) (φ : R⟦X⟧) (u : Rˣ) :
    coeff R n (invOfUnit φ u) =
      if n = 0 then ↑u⁻¹
      else
        -↑u⁻¹ *
          ∑ x ∈ antidiagonal n,
            if x.2 < n then coeff R x.1 φ * coeff R x.2 (invOfUnit φ u) else 0 
Arguments: {R : Type u_1} [Ring R] (n : ℕ) (φ : PowerSeries R) (u : Rˣ)
Goal: (PowerSeries.coeff R n) (φ.invOfUnit u) =
  if n = 0 then ↑u⁻¹
  else
    -↑u⁻¹ *
      ∑ x ∈ Finset.antidiagonal n,
        if x.2 < n then (PowerSeries.coeff R x.1) φ * (PowerSeries.coeff R x.2) (φ.invOfUnit u) else 0
-- PowerSeries.constantCoeff_invOfUnit
@[simp]
theorem constantCoeff_invOfUnit (φ : R⟦X⟧) (u : Rˣ) :
    constantCoeff R (invOfUnit φ u) = ↑u⁻¹ 
Arguments: {R : Type u_1} [Ring R] (φ : PowerSeries R) (u : Rˣ)
Goal: (PowerSeries.constantCoeff R) (φ.invOfUnit u) = ↑u⁻¹
-- PowerSeries.mul_invOfUnit
@[simp]
theorem mul_invOfUnit (φ : R⟦X⟧) (u : Rˣ) (h : constantCoeff R φ = u) :
    φ * invOfUnit φ u = 1 
Arguments: {R : Type u_1} [Ring R] (φ : PowerSeries R) (u : Rˣ) (h : (PowerSeries.constantCoeff R) φ = ↑u)
Goal: φ * φ.invOfUnit u = 1
-- PowerSeries.invOfUnit_mul
@[simp]
theorem invOfUnit_mul (φ : R⟦X⟧) (u : Rˣ) (h : constantCoeff R φ = u) :
    invOfUnit φ u * φ = 1 
Arguments: {R : Type u_1} [Ring R] (φ : PowerSeries R) (u : Rˣ) (h : (PowerSeries.constantCoeff R) φ = ↑u)
Goal: φ.invOfUnit u * φ = 1
-- PowerSeries.isUnit_iff_constantCoeff
theorem isUnit_iff_constantCoeff {φ : R⟦X⟧} :
    IsUnit φ ↔ IsUnit (constantCoeff R φ) 
Arguments: {R : Type u_1} [Ring R] {φ : PowerSeries R}
Goal: IsUnit φ ↔ IsUnit ((PowerSeries.constantCoeff R) φ)
-- PowerSeries.sub_const_eq_shift_mul_X
theorem sub_const_eq_shift_mul_X (φ : R⟦X⟧) :
    φ - C R (constantCoeff R φ) = (mk fun p ↦ coeff R (p + 1) φ) * X 
Arguments: {R : Type u_1} [Ring R] (φ : PowerSeries R)
Goal: φ - (PowerSeries.C R) ((PowerSeries.constantCoeff R) φ) =
  (PowerSeries.mk fun p => (PowerSeries.coeff R (p + 1)) φ) * PowerSeries.X
-- PowerSeries.sub_const_eq_X_mul_shift
theorem sub_const_eq_X_mul_shift (φ : R⟦X⟧) :
    φ - C R (constantCoeff R φ) = X * mk fun p ↦ coeff R (p + 1) φ 
Arguments: {R : Type u_1} [Ring R] (φ : PowerSeries R)
Goal: φ - (PowerSeries.C R) ((PowerSeries.constantCoeff R) φ) =
  PowerSeries.X * PowerSeries.mk fun p => (PowerSeries.coeff R (p + 1)) φ
-- PowerSeries.inv_eq_inv_aux
theorem inv_eq_inv_aux (φ : k⟦X⟧) : φ⁻¹ = inv.aux (constantCoeff k φ)⁻¹ φ 
Arguments: {k : Type u_2} [Field k] (φ : PowerSeries k)
Goal: φ⁻¹ = PowerSeries.inv.aux ((PowerSeries.constantCoeff k) φ)⁻¹ φ
-- PowerSeries.coeff_inv
theorem coeff_inv (n) (φ : k⟦X⟧) :
    coeff k n φ⁻¹ =
      if n = 0 then (constantCoeff k φ)⁻¹
      else
        -(constantCoeff k φ)⁻¹ *
          ∑ x ∈ antidiagonal n,
            if x.2 < n then coeff k x.1 φ * coeff k x.2 φ⁻¹ else 0 
Arguments: {k : Type u_2} [Field k] (n : ℕ) (φ : PowerSeries k)
Goal: (PowerSeries.coeff k n) φ⁻¹ =
  if n = 0 then ((PowerSeries.constantCoeff k) φ)⁻¹
  else
    -((PowerSeries.constantCoeff k) φ)⁻¹ *
      ∑ x ∈ Finset.antidiagonal n, if x.2 < n then (PowerSeries.coeff k x.1) φ * (PowerSeries.coeff k x.2) φ⁻¹ else 0
-- PowerSeries.constantCoeff_inv
@[simp]
theorem constantCoeff_inv (φ : k⟦X⟧) : constantCoeff k φ⁻¹ = (constantCoeff k φ)⁻¹ 
Arguments: {k : Type u_2} [Field k] (φ : PowerSeries k)
Goal: (PowerSeries.constantCoeff k) φ⁻¹ = ((PowerSeries.constantCoeff k) φ)⁻¹
-- PowerSeries.inv_eq_zero
theorem inv_eq_zero {φ : k⟦X⟧} : φ⁻¹ = 0 ↔ constantCoeff k φ = 0 
Arguments: {k : Type u_2} [Field k] {φ : PowerSeries k}
Goal: φ⁻¹ = 0 ↔ (PowerSeries.constantCoeff k) φ = 0
-- PowerSeries.zero_inv
theorem zero_inv : (0 : k⟦X⟧)⁻¹ = 0 
Arguments: {k : Type u_2} [Field k]
Goal: 0⁻¹ = 0
-- PowerSeries.invOfUnit_eq
theorem invOfUnit_eq (φ : k⟦X⟧) (h : constantCoeff k φ ≠ 0) :
    invOfUnit φ (Units.mk0 _ h) = φ⁻¹ 
Arguments: {k : Type u_2} [Field k] (φ : PowerSeries k) (h : (PowerSeries.constantCoeff k) φ ≠ 0)
Goal: φ.invOfUnit (Units.mk0 ((PowerSeries.constantCoeff k) φ) h) = φ⁻¹
-- PowerSeries.invOfUnit_eq'
@[simp]
theorem invOfUnit_eq' (φ : k⟦X⟧) (u : Units k) (h : constantCoeff k φ = u) :
    invOfUnit φ u = φ⁻¹ 
Arguments: {k : Type u_2} [Field k] (φ : PowerSeries k) (u : kˣ) (h : (PowerSeries.constantCoeff k) φ = ↑u)
Goal: φ.invOfUnit u = φ⁻¹
-- PowerSeries.mul_inv_cancel
@[simp]
protected theorem mul_inv_cancel (φ : k⟦X⟧) (h : constantCoeff k φ ≠ 0) : φ * φ⁻¹ = 1 
Arguments: {k : Type u_2} [Field k] (φ : PowerSeries k) (h : (PowerSeries.constantCoeff k) φ ≠ 0)
Goal: φ * φ⁻¹ = 1
-- PowerSeries.inv_mul_cancel
@[simp]
protected theorem inv_mul_cancel (φ : k⟦X⟧) (h : constantCoeff k φ ≠ 0) : φ⁻¹ * φ = 1 
Arguments: {k : Type u_2} [Field k] (φ : PowerSeries k) (h : (PowerSeries.constantCoeff k) φ ≠ 0)
Goal: φ⁻¹ * φ = 1