ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Unitization.inl_smul
@[simp]
theorem inl_smul [Monoid S] [AddMonoid A] [SMul S R] [DistribMulAction S A] (s : S) (r : R) :
    (inl (s • r) : Unitization R A) = s • inl r 
Arguments: {S : Type u_2} {R : Type u_3} (A : Type u_4) [Monoid S] [AddMonoid A] [SMul S R] [DistribMulAction S A] (s : S) (r : R)
Goal: Unitization.inl (s • r) = s • Unitization.inl r
-- Unitization.inr_zero
@[simp]
theorem inr_zero [Zero R] [Zero A] : ↑(0 : A) = (0 : Unitization R A) 
Arguments: (R : Type u_3) {A : Type u_4} [Zero R] [Zero A]
Goal: ↑0 = 0
-- Unitization.inr_add
@[simp]
theorem inr_add [AddZeroClass R] [Add A] (m₁ m₂ : A) : (↑(m₁ + m₂) : Unitization R A) = m₁ + m₂ 
Arguments: (R : Type u_3) {A : Type u_4} [AddZeroClass R] [Add A] (m₁ m₂ : A)
Goal: ↑(m₁ + m₂) = ↑m₁ + ↑m₂
-- Unitization.inr_neg
@[simp]
theorem inr_neg [AddGroup R] [Neg A] (m : A) : (↑(-m) : Unitization R A) = -m 
Arguments: (R : Type u_3) {A : Type u_4} [AddGroup R] [Neg A] (m : A)
Goal: ↑(-m) = -↑m
-- Unitization.inr_sub
@[simp]
theorem inr_sub [AddGroup R] [AddGroup A] (m₁ m₂ : A) : (↑(m₁ - m₂) : Unitization R A) = m₁ - m₂ 
Arguments: (R : Type u_3) {A : Type u_4} [AddGroup R] [AddGroup A] (m₁ m₂ : A)
Goal: ↑(m₁ - m₂) = ↑m₁ - ↑m₂
-- Unitization.inr_smul
@[simp]
theorem inr_smul [Zero R] [Zero S] [SMulWithZero S R] [SMul S A] (r : S) (m : A) :
    (↑(r • m) : Unitization R A) = r • (m : Unitization R A) 
Arguments: {S : Type u_2} (R : Type u_3) {A : Type u_4} [Zero R] [Zero S] [SMulWithZero S R] [SMul S A] (r : S) (m : A)
Goal: ↑(r • m) = r • ↑m
-- Unitization.inl_fst_add_inr_snd_eq
theorem inl_fst_add_inr_snd_eq [AddZeroClass R] [AddZeroClass A] (x : Unitization R A) :
    inl x.fst + (x.snd : Unitization R A) = x 
Arguments: {R : Type u_3} {A : Type u_4} [AddZeroClass R] [AddZeroClass A] (x : Unitization R A)
Goal: Unitization.inl x.fst + ↑x.snd = x
-- Unitization.ind
@[elab_as_elim, induction_eliminator, cases_eliminator]
theorem ind {R A} [AddZeroClass R] [AddZeroClass A] {P : Unitization R A → Prop}
    (inl_add_inr : ∀ (r : R) (a : A), P (inl r + (a : Unitization R A))) (x) : P x 
Arguments: {R : Type u_5} {A : Type u_6} [AddZeroClass R] [AddZeroClass A] {P : Unitization R A → Prop} (inl_add_inr : ∀ (r : R) (a : A), P (Unitization.inl r + ↑a)) (x : Unitization R A)
Goal: P x
-- Unitization.linearMap_ext
theorem linearMap_ext {N} [Semiring S] [AddCommMonoid R] [AddCommMonoid A] [AddCommMonoid N]
    [Module S R] [Module S A] [Module S N] ⦃f g : Unitization R A →ₗ[S] N⦄
    (hl : ∀ r, f (inl r) = g (inl r)) (hr : ∀ a : A, f a = g a) : f = g 
Arguments: {S : Type u_2} {R : Type u_3} {A : Type u_4} {N : Type u_5} [Semiring S] [AddCommMonoid R] [AddCommMonoid A] [AddCommMonoid N] [Module S R] [Module S A] [Module S N] ⦃f g : Unitization R A →ₗ[S] N⦄ (hl : ∀ (r : R), f (Unitization.inl r) = g (Unitization.inl r)) (hr : ∀ (a : A), f ↑a = g ↑a)
Goal: f = g
-- Unitization.inrHom_apply
@[simps apply]
def inrHom [Semiring R] [AddCommMonoid A] [Module R A] : A →ₗ[R] Unitization R A 
Arguments: (R : Type u_3) (A : Type u_4) [Semiring R] [AddCommMonoid A] [Module R A] (a : A)
Goal: (Unitization.inrHom R A) a = ↑a
-- Unitization.sndHom_apply
@[simps apply]
def sndHom [Semiring R] [AddCommMonoid A] [Module R A] : Unitization R A →ₗ[R] A 
Arguments: (R : Type u_3) (A : Type u_4) [Semiring R] [AddCommMonoid A] [Module R A] (x : Unitization R A)
Goal: (Unitization.sndHom R A) x = x.snd
-- Unitization.fst_one
@[simp]
theorem fst_one [One R] [Zero A] : (1 : Unitization R A).fst = 1 
Arguments: {R : Type u_1} {A : Type u_2} [One R] [Zero A]
Goal: Unitization.fst 1 = 1
-- Unitization.snd_one
@[simp]
theorem snd_one [One R] [Zero A] : (1 : Unitization R A).snd = 0 
Arguments: {R : Type u_1} {A : Type u_2} [One R] [Zero A]
Goal: Unitization.snd 1 = 0
-- Unitization.fst_mul
@[simp]
theorem fst_mul [Mul R] [Add A] [Mul A] [SMul R A] (x₁ x₂ : Unitization R A) :
    (x₁ * x₂).fst = x₁.fst * x₂.fst 
Arguments: {R : Type u_1} {A : Type u_2} [Mul R] [Add A] [Mul A] [SMul R A] (x₁ x₂ : Unitization R A)
Goal: (x₁ * x₂).fst = x₁.fst * x₂.fst
-- Unitization.snd_mul
@[simp]
theorem snd_mul [Mul R] [Add A] [Mul A] [SMul R A] (x₁ x₂ : Unitization R A) :
    (x₁ * x₂).snd = x₁.fst • x₂.snd + x₂.fst • x₁.snd + x₁.snd * x₂.snd 
Arguments: {R : Type u_1} {A : Type u_2} [Mul R] [Add A] [Mul A] [SMul R A] (x₁ x₂ : Unitization R A)
Goal: (x₁ * x₂).snd = x₁.fst • x₂.snd + x₂.fst • x₁.snd + x₁.snd * x₂.snd
-- Unitization.inl_one
@[simp]
theorem inl_one [One R] [Zero A] : (inl 1 : Unitization R A) = 1 
Arguments: {R : Type u_1} (A : Type u_2) [One R] [Zero A]
Goal: Unitization.inl 1 = 1
-- Unitization.inl_mul
@[simp]
theorem inl_mul [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] (r₁ r₂ : R) :
    (inl (r₁ * r₂) : Unitization R A) = inl r₁ * inl r₂ 
Arguments: {R : Type u_1} (A : Type u_2) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] (r₁ r₂ : R)
Goal: Unitization.inl (r₁ * r₂) = Unitization.inl r₁ * Unitization.inl r₂
-- Unitization.inl_mul_inl
theorem inl_mul_inl [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] (r₁ r₂ : R) :
    (inl r₁ * inl r₂ : Unitization R A) = inl (r₁ * r₂) 
Arguments: {R : Type u_1} (A : Type u_2) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] (r₁ r₂ : R)
Goal: Unitization.inl r₁ * Unitization.inl r₂ = Unitization.inl (r₁ * r₂)
-- Unitization.inr_mul
@[simp]
theorem inr_mul [Semiring R] [AddCommMonoid A] [Mul A] [SMulWithZero R A] (a₁ a₂ : A) :
    (↑(a₁ * a₂) : Unitization R A) = a₁ * a₂ 
Arguments: (R : Type u_1) {A : Type u_2} [Semiring R] [AddCommMonoid A] [Mul A] [SMulWithZero R A] (a₁ a₂ : A)
Goal: ↑(a₁ * a₂) = ↑a₁ * ↑a₂
-- Unitization.inl_mul_inr
theorem inl_mul_inr [Semiring R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] (r : R)
    (a : A) : ((inl r : Unitization R A) * a) = ↑(r • a) 
Arguments: {R : Type u_1} {A : Type u_2} [Semiring R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] (r : R) (a : A)
Goal: Unitization.inl r * ↑a = ↑(r • a)
-- Unitization.inr_mul_inl
theorem inr_mul_inl [Semiring R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] (r : R)
    (a : A) : a * (inl r : Unitization R A) = ↑(r • a) 
Arguments: {R : Type u_1} {A : Type u_2} [Semiring R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] (r : R) (a : A)
Goal: ↑a * Unitization.inl r = ↑(r • a)
-- Unitization.inlRingHom_apply
@[simps apply]
def inlRingHom [Semiring R] [NonUnitalSemiring A] [Module R A] : R →+* Unitization R A 
Arguments: (R : Type u_1) (A : Type u_2) [Semiring R] [NonUnitalSemiring A] [Module R A] (r : R)
Goal: (Unitization.inlRingHom R A) r = Unitization.inl r
-- Unitization.fst_star
@[simp]
theorem fst_star [Star R] [Star A] (x : Unitization R A) : (star x).fst = star x.fst 
Arguments: {R : Type u_1} {A : Type u_2} [Star R] [Star A] (x : Unitization R A)
Goal: (star x).fst = star x.fst
-- Unitization.snd_star
@[simp]
theorem snd_star [Star R] [Star A] (x : Unitization R A) : (star x).snd = star x.snd 
Arguments: {R : Type u_1} {A : Type u_2} [Star R] [Star A] (x : Unitization R A)
Goal: (star x).snd = star x.snd
-- Unitization.inl_star
@[simp]
theorem inl_star [Star R] [AddMonoid A] [StarAddMonoid A] (r : R) :
    inl (star r) = star (inl r : Unitization R A) 
Arguments: {R : Type u_1} {A : Type u_2} [Star R] [AddMonoid A] [StarAddMonoid A] (r : R)
Goal: Unitization.inl (star r) = star (Unitization.inl r)
-- Unitization.inr_star
@[simp]
theorem inr_star [AddMonoid R] [StarAddMonoid R] [Star A] (a : A) :
    ↑(star a) = star (a : Unitization R A) 
Arguments: {R : Type u_1} {A : Type u_2} [AddMonoid R] [StarAddMonoid R] [Star A] (a : A)
Goal: ↑(star a) = star ↑a
-- Unitization.algebraMap_eq_inl_comp
theorem algebraMap_eq_inl_comp : ⇑(algebraMap S (Unitization R A)) = inl ∘ algebraMap S R 
Arguments: (S : Type u_1) (R : Type u_2) (A : Type u_3) [CommSemiring S] [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [Algebra S R] [DistribMulAction S A] [IsScalarTower S R A]
Goal: ⇑(algebraMap S (Unitization R A)) = Unitization.inl ∘ ⇑(algebraMap S R)
-- Unitization.algebraMap_eq_inlRingHom_comp
theorem algebraMap_eq_inlRingHom_comp :
    algebraMap S (Unitization R A) = (inlRingHom R A).comp (algebraMap S R) 
Arguments: (S : Type u_1) (R : Type u_2) (A : Type u_3) [CommSemiring S] [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [Algebra S R] [DistribMulAction S A] [IsScalarTower S R A]
Goal: algebraMap S (Unitization R A) = (Unitization.inlRingHom R A).comp (algebraMap S R)
-- Unitization.algebraMap_eq_inl
theorem algebraMap_eq_inl : ⇑(algebraMap R (Unitization R A)) = inl 
Arguments: (R : Type u_2) (A : Type u_3) [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: ⇑(algebraMap R (Unitization R A)) = Unitization.inl
-- Unitization.algebraMap_eq_inlRingHom
theorem algebraMap_eq_inlRingHom : algebraMap R (Unitization R A) = inlRingHom R A 
Arguments: (R : Type u_2) (A : Type u_3) [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: algebraMap R (Unitization R A) = Unitization.inlRingHom R A
-- Unitization.fstHom_apply
@[simps]
def fstHom : Unitization R A →ₐ[R] R 
Arguments: (R : Type u_2) (A : Type u_3) [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (x : Unitization R A)
Goal: (Unitization.fstHom R A) x = x.fst
-- Unitization.inrNonUnitalAlgHom_apply
@[simps]
def inrNonUnitalAlgHom (R A : Type*) [CommSemiring R] [NonUnitalSemiring A] [Module R A] :
    A →ₙₐ[R] Unitization R A 
Arguments: (R : Type u_1) (A : Type u_2) [CommSemiring R] [NonUnitalSemiring A] [Module R A] (a : A)
Goal: (Unitization.inrNonUnitalAlgHom R A) a = ↑a
-- Unitization.inrNonUnitalAlgHom_toFun
@[simps]
def inrNonUnitalAlgHom (R A : Type*) [CommSemiring R] [NonUnitalSemiring A] [Module R A] :
    A →ₙₐ[R] Unitization R A 
Arguments: (R : Type u_1) (A : Type u_2) [CommSemiring R] [NonUnitalSemiring A] [Module R A] (a : A)
Goal: (Unitization.inrNonUnitalAlgHom R A) a = ↑a
-- Unitization.inrNonUnitalStarAlgHom_apply
@[simps!]
def inrNonUnitalStarAlgHom (R A : Type*) [CommSemiring R] [StarAddMonoid R]
    [NonUnitalSemiring A] [Star A] [Module R A] :
    A →⋆ₙₐ[R] Unitization R A 
Arguments: (R : Type u_1) (A : Type u_2) [CommSemiring R] [StarAddMonoid R] [NonUnitalSemiring A] [Star A] [Module R A] (a : A)
Goal: (Unitization.inrNonUnitalStarAlgHom R A) a = ↑a
-- Unitization.inrRangeEquiv_apply_coe
@[simps!]
def inrRangeEquiv (R A : Type*) [CommSemiring R] [StarAddMonoid R] [NonUnitalSemiring A]
    [Star A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] :
    A ≃⋆ₐ[R] NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom R A) 
Arguments: (R : Type u_1) (A : Type u_2) [CommSemiring R] [StarAddMonoid R] [NonUnitalSemiring A] [Star A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (a : A)
Goal: ↑((Unitization.inrRangeEquiv R A) a) = ↑(Unitization.inrNonUnitalStarAlgHom R A) a
-- Unitization.inrRangeEquiv_symm_apply
@[simps!]
def inrRangeEquiv (R A : Type*) [CommSemiring R] [StarAddMonoid R] [NonUnitalSemiring A]
    [Star A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] :
    A ≃⋆ₐ[R] NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom R A) 
Arguments: (R : Type u_1) (A : Type u_2) [CommSemiring R] [StarAddMonoid R] [NonUnitalSemiring A] [Star A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: ∀ (a : ↥(NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom R A))),
  (Unitization.inrRangeEquiv R A).symm a = (↑a).snd
-- Unitization.algHom_ext
theorem algHom_ext {F : Type*}
    [FunLike F (Unitization R A) B] [AlgHomClass F S (Unitization R A) B] {φ ψ : F}
    (h : ∀ a : A, φ a = ψ a)
    (h' : ∀ r, φ (algebraMap R (Unitization R A) r) = ψ (algebraMap R (Unitization R A) r)) :
    φ = ψ 
Arguments: {S : Type u_1} {R : Type u_2} {A : Type u_3} [CommSemiring S] [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] {B : Type u_4} [Semiring B] [Algebra S B] [Algebra S R] [DistribMulAction S A] [IsScalarTower S R A] {F : Type u_6} [FunLike F (Unitization R A) B] [AlgHomClass F S (Unitization R A) B] {φ ψ : F} (h : ∀ (a : A), φ ↑a = ψ ↑a) (h' : ∀ (r : R), φ ((algebraMap R (Unitization R A)) r) = ψ ((algebraMap R (Unitization R A)) r))
Goal: φ = ψ
-- Unitization.algHom_ext''
lemma algHom_ext'' {F : Type*}
    [FunLike F (Unitization R A) C] [AlgHomClass F R (Unitization R A) C] {φ ψ : F}
    (h : ∀ a : A, φ a = ψ a) : φ = ψ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] {C : Type u_5} [Semiring C] [Algebra R C] {F : Type u_6} [FunLike F (Unitization R A) C] [AlgHomClass F R (Unitization R A) C] {φ ψ : F} (h : ∀ (a : A), φ ↑a = ψ ↑a)
Goal: φ = ψ
-- Unitization.algHom_ext'
@[ext 1100]
theorem algHom_ext' {φ ψ : Unitization R A →ₐ[R] C}
    (h :
      φ.toNonUnitalAlgHom.comp (inrNonUnitalAlgHom R A) =
        ψ.toNonUnitalAlgHom.comp (inrNonUnitalAlgHom R A)) :
    φ = ψ 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] {C : Type u_5} [Semiring C] [Algebra R C] {φ ψ : Unitization R A →ₐ[R] C} (h : (↑φ).comp (Unitization.inrNonUnitalAlgHom R A) = (↑ψ).comp (Unitization.inrNonUnitalAlgHom R A))
Goal: φ = ψ
-- NonUnitalAlgHom.toAlgHom_apply
@[simps]
def _root_.NonUnitalAlgHom.toAlgHom (φ : A →ₙₐ[R] C) : Unitization R A →ₐ[R] C 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] {C : Type u_5} [Semiring C] [Algebra R C] (φ : A →ₙₐ[R] C) (x : Unitization R A)
Goal: φ.toAlgHom x = (algebraMap R C) x.fst + φ x.snd
-- Unitization.lift_apply
@[simps! apply symm_apply apply_apply]
def lift : (A →ₙₐ[R] C) ≃ (Unitization R A →ₐ[R] C) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] {C : Type u_5} [Semiring C] [Algebra R C] (φ : A →ₙₐ[R] C)
Goal: Unitization.lift φ = φ.toAlgHom
-- Unitization.lift_apply_apply
@[simps! apply symm_apply apply_apply]
def lift : (A →ₙₐ[R] C) ≃ (Unitization R A →ₐ[R] C) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] {C : Type u_5} [Semiring C] [Algebra R C] (φ : A →ₙₐ[R] C) (x : Unitization R A)
Goal: (Unitization.lift φ) x = (algebraMap R C) x.fst + φ x.snd
-- Unitization.lift_symm_apply
@[simps! apply symm_apply apply_apply]
def lift : (A →ₙₐ[R] C) ≃ (Unitization R A →ₐ[R] C) 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] {C : Type u_5} [Semiring C] [Algebra R C] (φ : Unitization R A →ₐ[R] C)
Goal: Unitization.lift.symm φ = (NonUnitalAlgHomClass.toNonUnitalAlgHom φ).comp (Unitization.inrNonUnitalAlgHom R A)
-- Unitization.lift_symm_apply_apply
theorem lift_symm_apply_apply (φ : Unitization R A →ₐ[R] C) (a : A) :
    Unitization.lift.symm φ a = φ a 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] {C : Type u_5} [Semiring C] [Algebra R C] (φ : Unitization R A →ₐ[R] C) (a : A)
Goal: (Unitization.lift.symm φ) a = φ ↑a
-- NonUnitalAlgHom.toAlgHom_zero
@[simp]
lemma _root_.NonUnitalAlgHom.toAlgHom_zero :
    ⇑(0 : A →ₙₐ[R] R).toAlgHom = Unitization.fst 
Arguments: {R : Type u_2} {A : Type u_3} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A]
Goal: ⇑(NonUnitalAlgHom.toAlgHom 0) = Unitization.fst
-- Unitization.starAlgHom_ext
@[ext]
theorem starAlgHom_ext {φ ψ : Unitization R A →⋆ₐ[R] C}
    (h : (φ : Unitization R A →⋆ₙₐ[R] C).comp (Unitization.inrNonUnitalStarAlgHom R A) =
      (ψ : Unitization R A →⋆ₙₐ[R] C).comp (Unitization.inrNonUnitalStarAlgHom R A)) :
    φ = ψ 
Arguments: {R : Type u_1} {A : Type u_2} {C : Type u_3} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [Semiring C] [Algebra R C] [StarRing C] {φ ψ : Unitization R A →⋆ₐ[R] C} (h : (↑φ).comp (Unitization.inrNonUnitalStarAlgHom R A) = (↑ψ).comp (Unitization.inrNonUnitalStarAlgHom R A))
Goal: φ = ψ
-- Unitization.starLift_symm_apply
@[simps! apply symm_apply apply_apply]
def starLift : (A →⋆ₙₐ[R] C) ≃ (Unitization R A →⋆ₐ[R] C) 
Arguments: {R : Type u_1} {A : Type u_2} {C : Type u_3} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [Semiring C] [Algebra R C] [StarRing C] [StarModule R C] (φ : Unitization R A →⋆ₐ[R] C)
Goal: Unitization.starLift.symm φ = φ.toNonUnitalStarAlgHom.comp (Unitization.inrNonUnitalStarAlgHom R A)
-- Unitization.starLift_apply_apply
@[simps! apply symm_apply apply_apply]
def starLift : (A →⋆ₙₐ[R] C) ≃ (Unitization R A →⋆ₐ[R] C) 
Arguments: {R : Type u_1} {A : Type u_2} {C : Type u_3} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [Semiring C] [Algebra R C] [StarRing C] [StarModule R C] (φ : A →⋆ₙₐ[R] C) (x : Unitization R A)
Goal: (Unitization.starLift φ) x = (algebraMap R C) x.fst + φ x.snd
-- Unitization.starLift_apply
@[simps! apply symm_apply apply_apply]
def starLift : (A →⋆ₙₐ[R] C) ≃ (Unitization R A →⋆ₐ[R] C) 
Arguments: {R : Type u_1} {A : Type u_2} {C : Type u_3} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [Semiring C] [Algebra R C] [StarRing C] [StarModule R C] (φ : A →⋆ₙₐ[R] C)
Goal: Unitization.starLift φ = { toAlgHom := φ.toAlgHom, map_star' := ⋯ }
-- Unitization.starLift_symm_apply_apply
@[simp high, nolint simpNF]
theorem starLift_symm_apply_apply (φ : Unitization R A →ₐ[R] C) (a : A) :
    Unitization.lift.symm φ a = φ a 
Arguments: {R : Type u_1} {A : Type u_2} {C : Type u_3} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [Semiring C] [Algebra R C] (φ : Unitization R A →ₐ[R] C) (a : A)
Goal: (Unitization.lift.symm φ) a = φ ↑a
-- Unitization.isStarNormal_inr
@[simp]
lemma isStarNormal_inr : IsStarNormal (a : Unitization R A) ↔ IsStarNormal a 
Arguments: {R : Type u_1} {A : Type u_2} [Semiring R] [AddCommMonoid A] [Mul A] [SMulWithZero R A] [StarAddMonoid R] [Star A] {a : A}
Goal: IsStarNormal ↑a ↔ IsStarNormal a
-- Unitization.isSelfAdjoint_inr
@[simp]
lemma isSelfAdjoint_inr : IsSelfAdjoint (a : Unitization R A) ↔ IsSelfAdjoint a 
Arguments: {R : Type u_1} {A : Type u_2} [Semiring R] [StarAddMonoid R] [Star A] {a : A}
Goal: IsSelfAdjoint ↑a ↔ IsSelfAdjoint a
-- IsSelfAdjoint.inr
lemma _root_.IsSelfAdjoint.inr (ha : IsSelfAdjoint a) : IsSelfAdjoint (a : Unitization R A) 
Arguments: (R : Type u_1) {A : Type u_2} [Semiring R] [StarAddMonoid R] [Star A] {a : A} (ha : IsSelfAdjoint a)
Goal: IsSelfAdjoint ↑a
-- PreQuasiregular.equiv_symm_apply
@[simps]
def equiv : R ≃ PreQuasiregular R 
Arguments: {R : Type u_1} (self : PreQuasiregular R)
Goal: PreQuasiregular.equiv.symm self = self.val
-- PreQuasiregular.equiv_apply_val
@[simps]
def equiv : R ≃ PreQuasiregular R 
Arguments: {R : Type u_1} (val : R)
Goal: (PreQuasiregular.equiv val).val = val
-- PreQuasiregular.val_one
@[simp]
lemma val_one : (1 : PreQuasiregular R).val = 0 
Arguments: {R : Type u_1} [NonUnitalSemiring R]
Goal: PreQuasiregular.val 1 = 0
-- PreQuasiregular.val_mul
@[simp]
lemma val_mul (x y : PreQuasiregular R) : (x * y).val = y.val + x.val + x.val * y.val 
Arguments: {R : Type u_1} [NonUnitalSemiring R] (x y : PreQuasiregular R)
Goal: (x * y).val = y.val + x.val + x.val * y.val
-- PreQuasiregular.inv_add_add_mul_eq_zero
@[simp]
lemma inv_add_add_mul_eq_zero (u : (PreQuasiregular R)ˣ) :
    u⁻¹.val.val + u.val.val + u.val.val * u⁻¹.val.val = 0 
Arguments: {R : Type u_1} [NonUnitalSemiring R] (u : (PreQuasiregular R)ˣ)
Goal: (↑u⁻¹).val + (↑u).val + (↑u).val * (↑u⁻¹).val = 0
-- PreQuasiregular.add_inv_add_mul_eq_zero
@[simp]
lemma add_inv_add_mul_eq_zero (u : (PreQuasiregular R)ˣ) :
    u.val.val + u⁻¹.val.val + u⁻¹.val.val * u.val.val = 0 
Arguments: {R : Type u_1} [NonUnitalSemiring R] (u : (PreQuasiregular R)ˣ)
Goal: (↑u).val + (↑u⁻¹).val + (↑u⁻¹).val * (↑u).val = 0
-- Unitization.mem_unitsFstOne
@[simp]
lemma mem_unitsFstOne {x : (Unitization R A)ˣ} : x ∈ unitsFstOne R A ↔ x.val.fst = 1 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {x : (Unitization R A)ˣ}
Goal: x ∈ Unitization.unitsFstOne R A ↔ (↑x).fst = 1
-- Unitization.unitsFstOne_val_val_fst
@[simp]
lemma unitsFstOne_val_val_fst (x : (unitsFstOne R A)) : x.val.val.fst = 1 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (x : ↥(Unitization.unitsFstOne R A))
Goal: (↑↑x).fst = 1
-- Unitization.unitsFstOne_val_inv_val_fst
@[simp]
lemma unitsFstOne_val_inv_val_fst (x : (unitsFstOne R A)) : x.val⁻¹.val.fst = 1 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (x : ↥(Unitization.unitsFstOne R A))
Goal: (↑(↑x)⁻¹).fst = 1
-- Unitization.val_unitsFstOne_mulEquiv_quasiregular_apply
@[simps]
def unitsFstOne_mulEquiv_quasiregular : unitsFstOne R A ≃* (PreQuasiregular A)ˣ 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (x : ↥(Unitization.unitsFstOne R A))
Goal: ↑((Unitization.unitsFstOne_mulEquiv_quasiregular R) x) = PreQuasiregular.equiv (↑↑x).snd
-- Unitization.val_inv_unitsFstOne_mulEquiv_quasiregular_symm_apply_coe
@[simps]
def unitsFstOne_mulEquiv_quasiregular : unitsFstOne R A ≃* (PreQuasiregular A)ˣ 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (x : (PreQuasiregular A)ˣ)
Goal: ↑(↑((Unitization.unitsFstOne_mulEquiv_quasiregular R).symm x))⁻¹ = 1 + ↑(PreQuasiregular.equiv.symm ↑x⁻¹)
-- Unitization.val_unitsFstOne_mulEquiv_quasiregular_symm_apply_coe
@[simps]
def unitsFstOne_mulEquiv_quasiregular : unitsFstOne R A ≃* (PreQuasiregular A)ˣ 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (x : (PreQuasiregular A)ˣ)
Goal: ↑↑((Unitization.unitsFstOne_mulEquiv_quasiregular R).symm x) = 1 + ↑(PreQuasiregular.equiv.symm ↑x)
-- Unitization.val_inv_unitsFstOne_mulEquiv_quasiregular_apply
@[simps]
def unitsFstOne_mulEquiv_quasiregular : unitsFstOne R A ≃* (PreQuasiregular A)ˣ 
Arguments: (R : Type u_1) {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (x : ↥(Unitization.unitsFstOne R A))
Goal: ↑((Unitization.unitsFstOne_mulEquiv_quasiregular R) x)⁻¹ = PreQuasiregular.equiv (↑↑x⁻¹).snd
-- isQuasiregular_zero
@[simp]
lemma isQuasiregular_zero : IsQuasiregular 0 
Goal: IsQuasiregular 0
-- isQuasiregular_iff
lemma isQuasiregular_iff {x : R} :
    IsQuasiregular x ↔ ∃ y, y + x + x * y = 0 ∧ x + y + y * x = 0 
Arguments: {R : Type u_1} [NonUnitalSemiring R] {x : R}
Goal: IsQuasiregular x ↔ ∃ y, y + x + x * y = 0 ∧ x + y + y * x = 0
-- IsQuasiregular.map
lemma IsQuasiregular.map {F R S : Type*} [NonUnitalSemiring R] [NonUnitalSemiring S]
    [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) {x : R} (hx : IsQuasiregular x) :
    IsQuasiregular (f x) 
Arguments: {F : Type u_1} {R : Type u_2} {S : Type u_3} [NonUnitalSemiring R] [NonUnitalSemiring S] [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) {x : R} (hx : IsQuasiregular x)
Goal: IsQuasiregular (f x)
-- IsQuasiregular.isUnit_one_add
lemma IsQuasiregular.isUnit_one_add {R : Type*} [Semiring R] {x : R} (hx : IsQuasiregular x) :
    IsUnit (1 + x) 
Arguments: {R : Type u_1} [Semiring R] {x : R} (hx : IsQuasiregular x)
Goal: IsUnit (1 + x)
-- isQuasiregular_iff_isUnit
lemma isQuasiregular_iff_isUnit {R : Type*} [Ring R] {x : R} :
    IsQuasiregular x ↔ IsUnit (1 + x) 
Arguments: {R : Type u_1} [Ring R] {x : R}
Goal: IsQuasiregular x ↔ IsUnit (1 + x)