ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- BialgEquiv.refl_toCoalgEquiv
@[simp]
theorem refl_toCoalgEquiv : refl R A = CoalgEquiv.refl R A 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A]
Goal: ↑(BialgEquiv.refl R A) = CoalgEquiv.refl R A
-- BialgEquiv.refl_toBialgHom
@[simp]
theorem refl_toBialgHom : refl R A = BialgHom.id R A 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] [CoalgebraStruct R A]
Goal: ↑(BialgEquiv.refl R A) = BialgHom.id R A
-- BialgEquiv.symm_toCoalgEquiv
@[simp]
theorem symm_toCoalgEquiv (e : A ≃ₐc[R] B) :
    e.symm = (e : A ≃ₗc[R] B).symm 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (e : A ≃ₐc[R] B)
Goal: ↑e.symm = (↑e).symm
-- BialgEquiv.invFun_eq_symm
theorem invFun_eq_symm : e.invFun = e.symm 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (e : A ≃ₐc[R] B)
Goal: e.invFun = ⇑e.symm
-- BialgEquiv.coe_toEquiv_symm
@[simp]
theorem coe_toEquiv_symm : e.toEquiv.symm = e.symm 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (e : A ≃ₐc[R] B)
Goal: e.toEquiv.symm = ↑e.symm
-- BialgEquiv.trans_invFun
@[trans, simps!]
def trans (e₁₂ : A ≃ₐc[R] B) (e₂₃ : B ≃ₐc[R] C) : A ≃ₐc[R] C 
Arguments: {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (e₁₂ : A ≃ₐc[R] B) (e₂₃ : B ≃ₐc[R] C)
Goal: ∀ (a : C), (e₁₂.trans e₂₃).invFun a = (↑↑e₁₂).symm ((↑↑e₂₃).symm a)
-- BialgEquiv.trans_apply
@[trans, simps!]
def trans (e₁₂ : A ≃ₐc[R] B) (e₂₃ : B ≃ₐc[R] C) : A ≃ₐc[R] C 
Arguments: {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] (e₁₂ : A ≃ₐc[R] B) (e₂₃ : B ≃ₐc[R] C)
Goal: ∀ (a : A), (e₁₂.trans e₂₃) a = ↑e₂₃ (↑e₁₂ a)
-- BialgEquiv.trans_toCoalgEquiv
@[simp]
theorem trans_toCoalgEquiv :
    (e₁₂.trans e₂₃ : A ≃ₗc[R] C) = (e₁₂ : A ≃ₗc[R] B).trans (e₂₃ : B ≃ₗc[R] C) 
Arguments: {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] {e₁₂ : A ≃ₐc[R] B} {e₂₃ : B ≃ₐc[R] C}
Goal: ↑(e₁₂.trans e₂₃) = (↑e₁₂).trans ↑e₂₃
-- BialgEquiv.trans_toBialgHom
@[simp]
theorem trans_toBialgHom :
    (e₁₂.trans e₂₃ : A →ₐc[R] C) = (e₂₃ : B →ₐc[R] C).comp e₁₂ 
Arguments: {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] {e₁₂ : A ≃ₐc[R] B} {e₂₃ : B ≃ₐc[R] C}
Goal: ↑(e₁₂.trans e₂₃) = (↑e₂₃).comp ↑e₁₂
-- BialgEquiv.coe_toEquiv_trans
@[simp]
theorem coe_toEquiv_trans : (e₁₂ : A ≃ B).trans e₂₃ = (e₁₂.trans e₂₃ : A ≃ C) 
Arguments: {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] {e₁₂ : A ≃ₐc[R] B} {e₂₃ : B ≃ₐc[R] C}
Goal: (↑e₁₂).trans ↑e₂₃ = ↑(e₁₂.trans e₂₃)
-- DiscreteValuationRing.not_a_field'
class DiscreteValuationRing (R : Type u) [CommRing R] [IsDomain R]
    extends IsPrincipalIdealRing R, LocalRing R : Prop where
  not_a_field' : maximalIdeal R ≠ ⊥
Arguments: {R : Type u} [CommRing R] [IsDomain R] [self : DiscreteValuationRing R]
Goal: LocalRing.maximalIdeal R ≠ ⊥
-- DiscreteValuationRing.not_a_field
theorem not_a_field : maximalIdeal R ≠ ⊥ 
Arguments: (R : Type u) [CommRing R] [IsDomain R] [DiscreteValuationRing R]
Goal: LocalRing.maximalIdeal R ≠ ⊥
-- DiscreteValuationRing.not_isField
theorem not_isField : ¬IsField R 
Arguments: (R : Type u) [CommRing R] [IsDomain R] [DiscreteValuationRing R]
Goal: ¬IsField R
-- DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal
theorem irreducible_of_span_eq_maximalIdeal {R : Type*} [CommRing R] [LocalRing R] [IsDomain R]
    (ϖ : R) (hϖ : ϖ ≠ 0) (h : maximalIdeal R = Ideal.span {ϖ}) : Irreducible ϖ 
Arguments: {R : Type u_1} [CommRing R] [LocalRing R] [IsDomain R] (ϖ : R) (hϖ : ϖ ≠ 0) (h : LocalRing.maximalIdeal R = Ideal.span {ϖ})
Goal: Irreducible ϖ
-- DiscreteValuationRing.irreducible_iff_uniformizer
theorem irreducible_iff_uniformizer (ϖ : R) : Irreducible ϖ ↔ maximalIdeal R = Ideal.span {ϖ} 
Arguments: {R : Type u} [CommRing R] [IsDomain R] [DiscreteValuationRing R] (ϖ : R)
Goal: Irreducible ϖ ↔ LocalRing.maximalIdeal R = Ideal.span {ϖ}
-- Irreducible.maximalIdeal_eq
theorem _root_.Irreducible.maximalIdeal_eq {ϖ : R} (h : Irreducible ϖ) :
    maximalIdeal R = Ideal.span {ϖ} 
Arguments: {R : Type u} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {ϖ : R} (h : Irreducible ϖ)
Goal: LocalRing.maximalIdeal R = Ideal.span {ϖ}
-- DiscreteValuationRing.exists_irreducible
theorem exists_irreducible : ∃ ϖ : R, Irreducible ϖ 
Arguments: (R : Type u) [CommRing R] [IsDomain R] [DiscreteValuationRing R]
Goal: ∃ ϖ, Irreducible ϖ
-- DiscreteValuationRing.exists_prime
theorem exists_prime : ∃ ϖ : R, Prime ϖ 
Arguments: (R : Type u) [CommRing R] [IsDomain R] [DiscreteValuationRing R]
Goal: ∃ ϖ, Prime ϖ
-- DiscreteValuationRing.iff_pid_with_one_nonzero_prime
theorem iff_pid_with_one_nonzero_prime (R : Type u) [CommRing R] [IsDomain R] :
    DiscreteValuationRing R ↔ IsPrincipalIdealRing R ∧ ∃! P : Ideal R, P ≠ ⊥ ∧ IsPrime P 
Arguments: (R : Type u) [CommRing R] [IsDomain R]
Goal: DiscreteValuationRing R ↔ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime
-- DiscreteValuationRing.associated_of_irreducible
theorem associated_of_irreducible {a b : R} (ha : Irreducible a) (hb : Irreducible b) :
    Associated a b 
Arguments: (R : Type u) [CommRing R] [IsDomain R] [DiscreteValuationRing R] {a b : R} (ha : Irreducible a) (hb : Irreducible b)
Goal: Associated a b
-- DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.unique_irreducible
theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
    Associated p q 
Arguments: {R : Type u_1} [CommRing R] (hR : DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization R) ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q)
Goal: Associated p q
-- DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid
theorem toUniqueFactorizationMonoid : UniqueFactorizationMonoid R 
Arguments: {R : Type u_1} [CommRing R] (hR : DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization R) [IsDomain R]
Goal: UniqueFactorizationMonoid R
-- DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.of_ufd_of_unique_irreducible
theorem of_ufd_of_unique_irreducible [UniqueFactorizationMonoid R] (h₁ : ∃ p : R, Irreducible p)
    (h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q) :
    HasUnitMulPowIrreducibleFactorization R 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [UniqueFactorizationMonoid R] (h₁ : ∃ p, Irreducible p) (h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q)
Goal: DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization R
-- DiscreteValuationRing.aux_pid_of_ufd_of_unique_irreducible
theorem aux_pid_of_ufd_of_unique_irreducible (R : Type u) [CommRing R] [IsDomain R]
    [UniqueFactorizationMonoid R] (h₁ : ∃ p : R, Irreducible p)
    (h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q) :
    IsPrincipalIdealRing R 
Arguments: (R : Type u) [CommRing R] [IsDomain R] [UniqueFactorizationMonoid R] (h₁ : ∃ p, Irreducible p) (h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q)
Goal: IsPrincipalIdealRing R
-- DiscreteValuationRing.of_ufd_of_unique_irreducible
theorem of_ufd_of_unique_irreducible {R : Type u} [CommRing R] [IsDomain R]
    [UniqueFactorizationMonoid R] (h₁ : ∃ p : R, Irreducible p)
    (h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q) :
    DiscreteValuationRing R 
Arguments: {R : Type u} [CommRing R] [IsDomain R] [UniqueFactorizationMonoid R] (h₁ : ∃ p, Irreducible p) (h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q)
Goal: DiscreteValuationRing R
-- DiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization
theorem ofHasUnitMulPowIrreducibleFactorization {R : Type u} [CommRing R] [IsDomain R]
    (hR : HasUnitMulPowIrreducibleFactorization R) : DiscreteValuationRing R 
Arguments: {R : Type u} [CommRing R] [IsDomain R] (hR : DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization R)
Goal: DiscreteValuationRing R
-- DiscreteValuationRing.associated_pow_irreducible
theorem associated_pow_irreducible {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : Irreducible ϖ) :
    ∃ n : ℕ, Associated x (ϖ ^ n) 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : Irreducible ϖ)
Goal: ∃ n, Associated x (ϖ ^ n)
-- DiscreteValuationRing.eq_unit_mul_pow_irreducible
theorem eq_unit_mul_pow_irreducible {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : Irreducible ϖ) :
    ∃ (n : ℕ) (u : Rˣ), x = u * ϖ ^ n 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : Irreducible ϖ)
Goal: ∃ n u, x = ↑u * ϖ ^ n
-- DiscreteValuationRing.ideal_eq_span_pow_irreducible
theorem ideal_eq_span_pow_irreducible {s : Ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : Irreducible ϖ) :
    ∃ n : ℕ, s = Ideal.span {ϖ ^ n} 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {s : Ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : Irreducible ϖ)
Goal: ∃ n, s = Ideal.span {ϖ ^ n}
-- DiscreteValuationRing.unit_mul_pow_congr_pow
theorem unit_mul_pow_congr_pow {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ)
    (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) : m = n 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {p q : R} (hp : Irreducible p) (hq : Irreducible q) (u v : Rˣ) (m n : ℕ) (h : ↑u * p ^ m = ↑v * q ^ n)
Goal: m = n
-- DiscreteValuationRing.unit_mul_pow_congr_unit
theorem unit_mul_pow_congr_unit {ϖ : R} (hirr : Irreducible ϖ) (u v : Rˣ) (m n : ℕ)
    (h : ↑u * ϖ ^ m = v * ϖ ^ n) : u = v 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {ϖ : R} (hirr : Irreducible ϖ) (u v : Rˣ) (m n : ℕ) (h : ↑u * ϖ ^ m = ↑v * ϖ ^ n)
Goal: u = v
-- DiscreteValuationRing.addVal_def
theorem addVal_def (r : R) (u : Rˣ) {ϖ : R} (hϖ : Irreducible ϖ) (n : ℕ) (hr : r = u * ϖ ^ n) :
    addVal R r = n 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] (r : R) (u : Rˣ) {ϖ : R} (hϖ : Irreducible ϖ) (n : ℕ) (hr : r = ↑u * ϖ ^ n)
Goal: (DiscreteValuationRing.addVal R) r = ↑n
-- DiscreteValuationRing.addVal_def'
theorem addVal_def' (u : Rˣ) {ϖ : R} (hϖ : Irreducible ϖ) (n : ℕ) :
    addVal R ((u : R) * ϖ ^ n) = n 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] (u : Rˣ) {ϖ : R} (hϖ : Irreducible ϖ) (n : ℕ)
Goal: (DiscreteValuationRing.addVal R) (↑u * ϖ ^ n) = ↑n
-- DiscreteValuationRing.addVal_zero
theorem addVal_zero : addVal R 0 = ⊤ 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R]
Goal: (DiscreteValuationRing.addVal R) 0 = ⊤
-- DiscreteValuationRing.addVal_one
theorem addVal_one : addVal R 1 = 0 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R]
Goal: (DiscreteValuationRing.addVal R) 1 = 0
-- DiscreteValuationRing.addVal_uniformizer
@[simp]
theorem addVal_uniformizer {ϖ : R} (hϖ : Irreducible ϖ) : addVal R ϖ = 1 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {ϖ : R} (hϖ : Irreducible ϖ)
Goal: (DiscreteValuationRing.addVal R) ϖ = 1
-- DiscreteValuationRing.addVal_mul
theorem addVal_mul {a b : R} :
    addVal R (a * b) = addVal R a + addVal R b 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {a b : R}
Goal: (DiscreteValuationRing.addVal R) (a * b) = (DiscreteValuationRing.addVal R) a + (DiscreteValuationRing.addVal R) b
-- DiscreteValuationRing.addVal_pow
theorem addVal_pow (a : R) (n : ℕ) : addVal R (a ^ n) = n • addVal R a 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] (a : R) (n : ℕ)
Goal: (DiscreteValuationRing.addVal R) (a ^ n) = n • (DiscreteValuationRing.addVal R) a
-- Irreducible.addVal_pow
nonrec theorem _root_.Irreducible.addVal_pow {ϖ : R} (h : Irreducible ϖ) (n : ℕ) :
    addVal R (ϖ ^ n) = n 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {ϖ : R} (h : Irreducible ϖ) (n : ℕ)
Goal: (DiscreteValuationRing.addVal R) (ϖ ^ n) = ↑n
-- DiscreteValuationRing.addVal_eq_top_iff
theorem addVal_eq_top_iff {a : R} : addVal R a = ⊤ ↔ a = 0 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {a : R}
Goal: (DiscreteValuationRing.addVal R) a = ⊤ ↔ a = 0
-- DiscreteValuationRing.addVal_le_iff_dvd
theorem addVal_le_iff_dvd {a b : R} : addVal R a ≤ addVal R b ↔ a ∣ b 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {a b : R}
Goal: (DiscreteValuationRing.addVal R) a ≤ (DiscreteValuationRing.addVal R) b ↔ a ∣ b
-- DiscreteValuationRing.addVal_add
theorem addVal_add {a b : R} : min (addVal R a) (addVal R b) ≤ addVal R (a + b) 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] [DiscreteValuationRing R] {a b : R}
Goal: min ((DiscreteValuationRing.addVal R) a) ((DiscreteValuationRing.addVal R) b) ≤ (DiscreteValuationRing.addVal R) (a + b)
-- diffToIdealOfQuotientCompEq_apply
@[simp]
theorem diffToIdealOfQuotientCompEq_apply (f₁ f₂ : A →ₐ[R] B)
    (e : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂) (x : A) :
    ((diffToIdealOfQuotientCompEq I f₁ f₂ e) x : B) = f₁ x - f₂ x 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommRing B] [Algebra R A] [Algebra R B] (I : Ideal B) (f₁ f₂ : A →ₐ[R] B) (e : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂) (x : A)
Goal: ↑((diffToIdealOfQuotientCompEq I f₁ f₂ e) x) = f₁ x - f₂ x
-- derivationToSquareZeroOfLift_apply
theorem derivationToSquareZeroOfLift_apply (f : A →ₐ[R] B)
    (e : (Ideal.Quotient.mkₐ R I).comp f = IsScalarTower.toAlgHom R A (B ⧸ I)) (x : A) :
    (derivationToSquareZeroOfLift I hI f e x : B) = f x - algebraMap A B x 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommRing B] [Algebra R A] [Algebra R B] (I : Ideal B) (hI : I ^ 2 = ⊥) [Algebra A B] [IsScalarTower R A B] (f : A →ₐ[R] B) (e : (Ideal.Quotient.mkₐ R I).comp f = IsScalarTower.toAlgHom R A (B ⧸ I)) (x : A)
Goal: ↑((derivationToSquareZeroOfLift I hI f e) x) = f x - (algebraMap A B) x
-- liftOfDerivationToSquareZero_apply
@[simps (config := .lemmasOnly)]
def liftOfDerivationToSquareZero (f : Derivation R A I) : A →ₐ[R] B 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommRing B] [Algebra R A] [Algebra R B] (I : Ideal B) (hI : I ^ 2 = ⊥) [Algebra A B] [IsScalarTower R A B] (f : Derivation R A ↥I) (x : A)
Goal: (liftOfDerivationToSquareZero I hI f) x = ↑(f x) + (algebraMap A B) x
-- liftOfDerivationToSquareZero_mk_apply
theorem liftOfDerivationToSquareZero_mk_apply (d : Derivation R A I) (x : A) :
    Ideal.Quotient.mk I (liftOfDerivationToSquareZero I hI d x) = algebraMap A (B ⧸ I) x 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommRing B] [Algebra R A] [Algebra R B] (I : Ideal B) (hI : I ^ 2 = ⊥) [Algebra A B] [IsScalarTower R A B] (d : Derivation R A ↥I) (x : A)
Goal: (Ideal.Quotient.mk I) ((liftOfDerivationToSquareZero I hI d) x) = (algebraMap A (B ⧸ I)) x
-- liftOfDerivationToSquareZero_mk_apply'
@[simp]
theorem liftOfDerivationToSquareZero_mk_apply' (d : Derivation R A I) (x : A) :
    (Ideal.Quotient.mk I) (d x) + (algebraMap A (B ⧸ I)) x = algebraMap A (B ⧸ I) x 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommRing B] [Algebra R A] [Algebra R B] (I : Ideal B) [Algebra A B] (d : Derivation R A ↥I) (x : A)
Goal: (Ideal.Quotient.mk I) ↑(d x) + (algebraMap A (B ⧸ I)) x = (algebraMap A (B ⧸ I)) x
-- derivationToSquareZeroEquivLift_symm_apply_apply_coe
@[simps!]
def derivationToSquareZeroEquivLift : Derivation R A I ≃
    { f : A →ₐ[R] B // (Ideal.Quotient.mkₐ R I).comp f = IsScalarTower.toAlgHom R A (B ⧸ I) } 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommRing B] [Algebra R A] [Algebra R B] (I : Ideal B) (hI : I ^ 2 = ⊥) [Algebra A B] [IsScalarTower R A B] (f : { f // (Ideal.Quotient.mkₐ R I).comp f = IsScalarTower.toAlgHom R A (B ⧸ I) }) (c : A)
Goal: ↑(((derivationToSquareZeroEquivLift I hI).symm f) c) = ↑f c - (algebraMap A B) c
-- derivationToSquareZeroEquivLift_apply_coe_apply
@[simps!]
def derivationToSquareZeroEquivLift : Derivation R A I ≃
    { f : A →ₐ[R] B // (Ideal.Quotient.mkₐ R I).comp f = IsScalarTower.toAlgHom R A (B ⧸ I) } 
Arguments: {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommRing B] [Algebra R A] [Algebra R B] (I : Ideal B) (hI : I ^ 2 = ⊥) [Algebra A B] [IsScalarTower R A B] (d : Derivation R A ↥I) (x : A)
Goal: ↑((derivationToSquareZeroEquivLift I hI) d) x = ↑(d x) + (algebraMap A B) x
-- BialgHomClass.counitAlgHom_comp
@[simp]
theorem counitAlgHom_comp (f : F) :
    (counitAlgHom R B).comp (f : A →ₐ[R] B) = counitAlgHom R A 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [Semiring A] [Bialgebra R A] [Semiring B] [Bialgebra R B] [FunLike F A B] [BialgHomClass F R A B] (f : F)
Goal: (Bialgebra.counitAlgHom R B).comp ↑f = Bialgebra.counitAlgHom R A
-- BialgHomClass.map_comp_comulAlgHom
@[simp]
theorem map_comp_comulAlgHom (f : F) :
    (Algebra.TensorProduct.map f f).comp (comulAlgHom R A) = (comulAlgHom R B).comp f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [Semiring A] [Bialgebra R A] [Semiring B] [Bialgebra R B] [FunLike F A B] [BialgHomClass F R A B] (f : F)
Goal: (Algebra.TensorProduct.map ↑f ↑f).comp (Bialgebra.comulAlgHom R A) = (Bialgebra.comulAlgHom R B).comp ↑f
-- BialgHom.coe_coe
@[simp]
protected theorem coe_coe {F : Type*} [FunLike F A B] [BialgHomClass F R A B] (f : F) :
    ⇑(f : A →ₐc[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {F : Type u_6} [FunLike F A B] [BialgHomClass F R A B] (f : F)
Goal: ⇑↑f = ⇑f
-- BialgHom.coe_mk
@[simp]
theorem coe_mk {f : A →ₗc[R] B} (h h₁) : ((⟨f, h, h₁⟩ : A →ₐc[R] B) : A → B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A →ₗc[R] B} (h : f.toFun 1 = 1) (h₁ : ∀ (x y : A), f.toFun (x * y) = f.toFun x * f.toFun y)
Goal: ⇑{ toCoalgHom := f, map_one' := h, map_mul' := h₁ } = ⇑f
-- BialgHom.coe_mks
@[norm_cast]
theorem coe_mks {f : A → B} (h₀ h₁ h₂ h₃ h₄ h₅) :
    ⇑(⟨⟨⟨⟨f, h₀⟩, h₁⟩, h₂, h₃⟩, h₄, h₅⟩ : A →ₐc[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A → B} (h₀ : ∀ (x y : A), f (x + y) = f x + f y) (h₁ :
  ∀ (m : R) (x : A),
    { toFun := f, map_add' := h₀ }.toFun (m • x) = (RingHom.id R) m • { toFun := f, map_add' := h₀ }.toFun x) (h₂ : Coalgebra.counit ∘ₗ { toFun := f, map_add' := h₀, map_smul' := h₁ } = Coalgebra.counit) (h₃ :
  TensorProduct.map { toFun := f, map_add' := h₀, map_smul' := h₁ } { toFun := f, map_add' := h₀, map_smul' := h₁ } ∘ₗ
      Coalgebra.comul =
    Coalgebra.comul ∘ₗ { toFun := f, map_add' := h₀, map_smul' := h₁ }) (h₄ : { toFun := f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun 1 = 1) (h₅ :
  ∀ (x y : A),
    { toFun := f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun (x * y) =
      { toFun := f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun x *
        { toFun := f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃ }.toFun y)
Goal: ⇑{ toFun := f, map_add' := h₀, map_smul' := h₁, counit_comp := h₂, map_comp_comul := h₃, map_one' := h₄,
      map_mul' := h₅ } =
  f
-- BialgHom.coe_coalgHom_mk
@[simp, norm_cast]
theorem coe_coalgHom_mk {f : A →ₗc[R] B} (h h₁) :
    ((⟨f, h, h₁⟩ : A →ₐc[R] B) : A →ₗc[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {f : A →ₗc[R] B} (h : f.toFun 1 = 1) (h₁ : ∀ (x y : A), f.toFun (x * y) = f.toFun x * f.toFun y)
Goal: ↑{ toCoalgHom := f, map_one' := h, map_mul' := h₁ } = f
-- BialgHom.coe_toCoalgHom
@[norm_cast]
theorem coe_toCoalgHom (f : A →ₐc[R] B) : ⇑(f : A →ₗc[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₐc[R] B)
Goal: ⇑↑f = ⇑f
-- BialgHom.coe_toLinearMap
@[simp, norm_cast]
theorem coe_toLinearMap (f : A →ₐc[R] B) : ⇑(f : A →ₗ[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₐc[R] B)
Goal: ⇑↑f = ⇑f
-- BialgHom.coe_toAlgHom
@[norm_cast]
theorem coe_toAlgHom (f : A →ₐc[R] B) : ⇑(f : A →ₐ[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₐc[R] B)
Goal: ⇑↑f = ⇑f
-- BialgHom.toAlgHom_toLinearMap
theorem toAlgHom_toLinearMap (f : A →ₐc[R] B) :
    ((f : A →ₐ[R] B) : A →ₗ[R] B) = f 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] (f : A →ₐc[R] B)
Goal: (↑↑f).toLinearMap = ↑f
-- BialgHom.coe_fn_injective
theorem coe_fn_injective : @Function.Injective (A →ₐc[R] B) (A → B) (↑) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B]
Goal: Function.Injective DFunLike.coe
-- BialgHom.coe_fn_inj
theorem coe_fn_inj {φ₁ φ₂ : A →ₐc[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ φ₂ : A →ₐc[R] B}
Goal: ⇑φ₁ = ⇑φ₂ ↔ φ₁ = φ₂
-- BialgHom.coe_coalgHom_injective
theorem coe_coalgHom_injective : Function.Injective ((↑) : (A →ₐc[R] B) → A →ₗc[R] B) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B]
Goal: Function.Injective CoalgHomClass.toCoalgHom
-- BialgHom.coe_algHom_injective
theorem coe_algHom_injective : Function.Injective ((↑) : (A →ₐc[R] B) → A →ₐ[R] B) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B]
Goal: Function.Injective AlgHomClass.toAlgHom
-- BialgHom.coe_linearMap_injective
theorem coe_linearMap_injective : Function.Injective ((↑) : (A →ₐc[R] B) → A →ₗ[R] B) 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B]
Goal: Function.Injective fun x => ↑x
-- BialgHom.congr_fun
protected theorem congr_fun {φ₁ φ₂ : A →ₐc[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x 
Arguments: {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CoalgebraStruct R A] [CoalgebraStruct R B] {φ₁ φ₂ : A →ₐc[R] B} (H : φ₁ = φ₂) (x : A)
Goal: φ₁ x = φ₂ x