ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- StarModule.decomposeProdAdjoint_symm_apply
-- Porting note: This attribute causes a `timeout at 'whnf'`.
@[simps!]
def StarModule.decomposeProdAdjoint : A ≃ₗ[R] selfAdjoint A × skewAdjoint A 
Arguments: (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] (a : ↥(selfAdjoint A) × ↥(skewAdjoint A))
Goal: (StarModule.decomposeProdAdjoint R A).symm a =
  (selfAdjoint.submodule R A).subtype a.1 + (skewAdjoint.submodule R A).subtype a.2
-- algebraMap_star_comm
@[simp]
theorem algebraMap_star_comm (r : R) : algebraMap R A (star r) = star (algebraMap R A r) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [Semiring A] [StarMul A] [Algebra R A] [StarModule R A] (r : R)
Goal: (algebraMap R A) (star r) = star ((algebraMap R A) r)
-- IsSelfAdjoint.algebraMap
protected lemma IsSelfAdjoint.algebraMap {r : R} (hr : IsSelfAdjoint r) :
    IsSelfAdjoint (algebraMap R A r) 
Arguments: {R : Type u_1} (A : Type u_2) [CommSemiring R] [StarRing R] [Semiring A] [StarMul A] [Algebra R A] [StarModule R A] {r : R} (hr : IsSelfAdjoint r)
Goal: IsSelfAdjoint ((algebraMap R A) r)
-- isSelfAdjoint_algebraMap_iff
lemma isSelfAdjoint_algebraMap_iff {r : R} (h : Function.Injective (algebraMap R A)) :
    IsSelfAdjoint (algebraMap R A r) ↔ IsSelfAdjoint r 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [StarRing R] [Semiring A] [StarMul A] [Algebra R A] [StarModule R A] {r : R} (h : Function.Injective ⇑(algebraMap R A))
Goal: IsSelfAdjoint ((algebraMap R A) r) ↔ IsSelfAdjoint r
-- MonCat.Colimits.quot_one
@[simp]
theorem quot_one : Quot.mk Setoid.r one = (1 : ColimitType F) 
Arguments: {J : Type v} [CategoryTheory.SmallCategory J] (F : J ⥤ MonCat)
Goal: Quot.mk Setoid.r MonCat.Colimits.Prequotient.one = 1
-- MonCat.Colimits.quot_mul
@[simp]
theorem quot_mul (x y : Prequotient F) : Quot.mk Setoid.r (mul x y) =
    @HMul.hMul (ColimitType F) (ColimitType F) (ColimitType F) _
      (Quot.mk Setoid.r x) (Quot.mk Setoid.r y) 
Arguments: {J : Type v} [CategoryTheory.SmallCategory J] (F : J ⥤ MonCat) (x y : MonCat.Colimits.Prequotient F)
Goal: Quot.mk Setoid.r (x.mul y) = Quot.mk Setoid.r x * Quot.mk Setoid.r y
-- MonCat.Colimits.cocone_naturality
@[simp]
theorem cocone_naturality {j j' : J} (f : j ⟶ j') :
    F.map f ≫ coconeMorphism F j' = coconeMorphism F j 
Arguments: {J : Type v} [CategoryTheory.SmallCategory J] (F : J ⥤ MonCat) {j j' : J} (f : j ⟶ j')
Goal: F.map f ≫ MonCat.Colimits.coconeMorphism F j' = MonCat.Colimits.coconeMorphism F j
-- MonCat.Colimits.cocone_naturality_components
@[simp]
theorem cocone_naturality_components (j j' : J) (f : j ⟶ j') (x : F.obj j) :
    (coconeMorphism F j') (F.map f x) = (coconeMorphism F j) x 
Arguments: {J : Type v} [CategoryTheory.SmallCategory J] (F : J ⥤ MonCat) (j j' : J) (f : j ⟶ j') (x : ↑(F.obj j))
Goal: (MonCat.Colimits.coconeMorphism F j') ((F.map f) x) = (MonCat.Colimits.coconeMorphism F j) x
-- IsStarNormal.star_comm_self
@[mk_iff]
class IsStarNormal [Mul R] [Star R] (x : R) : Prop where
    star_comm_self : Commute (star x) x
Arguments: {R : Type u_1} [Mul R] [Star R] {x : R} [self : IsStarNormal x]
Goal: Commute (star x) x
-- isStarNormal_iff
@[mk_iff]
class IsStarNormal [Mul R] [Star R] (x : R) : Prop where
    star_comm_self : Commute (star x) x
Arguments: {R : Type u_1} [Mul R] [Star R] (x : R)
Goal: IsStarNormal x ↔ Commute (star x) x
-- star_comm_self'
theorem star_comm_self' [Mul R] [Star R] (x : R) [IsStarNormal x] : star x * x = x * star x 
Arguments: {R : Type u_1} [Mul R] [Star R] (x : R) [IsStarNormal x]
Goal: star x * x = x * star x
-- IsSelfAdjoint.all
theorem all [Star R] [TrivialStar R] (r : R) : IsSelfAdjoint r 
Arguments: {R : Type u_1} [Star R] [TrivialStar R] (r : R)
Goal: IsSelfAdjoint r
-- IsSelfAdjoint.star_eq
theorem star_eq [Star R] {x : R} (hx : IsSelfAdjoint x) : star x = x 
Arguments: {R : Type u_1} [Star R] {x : R} (hx : IsSelfAdjoint x)
Goal: star x = x
-- isSelfAdjoint_iff
theorem _root_.isSelfAdjoint_iff [Star R] {x : R} : IsSelfAdjoint x ↔ star x = x 
Arguments: {R : Type u_1} [Star R] {x : R}
Goal: IsSelfAdjoint x ↔ star x = x
-- IsSelfAdjoint.star_iff
@[simp]
theorem star_iff [InvolutiveStar R] {x : R} : IsSelfAdjoint (star x) ↔ IsSelfAdjoint x 
Arguments: {R : Type u_1} [InvolutiveStar R] {x : R}
Goal: IsSelfAdjoint (star x) ↔ IsSelfAdjoint x
-- IsSelfAdjoint.star_mul_self
@[simp]
theorem star_mul_self [Mul R] [StarMul R] (x : R) : IsSelfAdjoint (star x * x) 
Arguments: {R : Type u_1} [Mul R] [StarMul R] (x : R)
Goal: IsSelfAdjoint (star x * x)
-- IsSelfAdjoint.mul_star_self
@[simp]
theorem mul_star_self [Mul R] [StarMul R] (x : R) : IsSelfAdjoint (x * star x) 
Arguments: {R : Type u_1} [Mul R] [StarMul R] (x : R)
Goal: IsSelfAdjoint (x * star x)
-- IsSelfAdjoint.commute_iff
lemma commute_iff {R : Type*} [Mul R] [StarMul R] {x y : R}
    (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y) : Commute x y ↔ IsSelfAdjoint (x * y) 
Arguments: {R : Type u_3} [Mul R] [StarMul R] {x y : R} (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y)
Goal: Commute x y ↔ IsSelfAdjoint (x * y)
-- IsSelfAdjoint.starHom_apply
theorem starHom_apply {F R S : Type*} [Star R] [Star S] [FunLike F R S] [StarHomClass F R S]
    {x : R} (hx : IsSelfAdjoint x) (f : F) : IsSelfAdjoint (f x) 
Arguments: {F : Type u_3} {R : Type u_4} {S : Type u_5} [Star R] [Star S] [FunLike F R S] [StarHomClass F R S] {x : R} (hx : IsSelfAdjoint x) (f : F)
Goal: IsSelfAdjoint (f x)
-- isSelfAdjoint_starHom_apply
theorem _root_.isSelfAdjoint_starHom_apply {F R S : Type*} [Star R] [Star S] [FunLike F R S]
    [StarHomClass F R S] [TrivialStar R] (f : F) (x : R) : IsSelfAdjoint (f x) 
Arguments: {F : Type u_3} {R : Type u_4} {S : Type u_5} [Star R] [Star S] [FunLike F R S] [StarHomClass F R S] [TrivialStar R] (f : F) (x : R)
Goal: IsSelfAdjoint (f x)
-- IsSelfAdjoint.zero
@[simp] protected theorem zero : IsSelfAdjoint (0 : R) 
Arguments: (R : Type u_1) [AddMonoid R] [StarAddMonoid R]
Goal: IsSelfAdjoint 0
-- IsSelfAdjoint.add
@[aesop 90% apply]
theorem add {x y : R} (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y) : IsSelfAdjoint (x + y) 
Arguments: {R : Type u_1} [AddMonoid R] [StarAddMonoid R] {x y : R} (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y)
Goal: IsSelfAdjoint (x + y)
-- IsSelfAdjoint.neg
@[aesop safe apply]
theorem neg {x : R} (hx : IsSelfAdjoint x) : IsSelfAdjoint (-x) 
Arguments: {R : Type u_1} [AddGroup R] [StarAddMonoid R] {x : R} (hx : IsSelfAdjoint x)
Goal: IsSelfAdjoint (-x)
-- IsSelfAdjoint.sub
@[aesop 90% apply]
theorem sub {x y : R} (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y) : IsSelfAdjoint (x - y) 
Arguments: {R : Type u_1} [AddGroup R] [StarAddMonoid R] {x y : R} (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y)
Goal: IsSelfAdjoint (x - y)
-- IsSelfAdjoint.add_star_self
@[simp]
theorem add_star_self (x : R) : IsSelfAdjoint (x + star x) 
Arguments: {R : Type u_1} [AddCommMonoid R] [StarAddMonoid R] (x : R)
Goal: IsSelfAdjoint (x + star x)
-- IsSelfAdjoint.star_add_self
@[simp]
theorem star_add_self (x : R) : IsSelfAdjoint (star x + x) 
Arguments: {R : Type u_1} [AddCommMonoid R] [StarAddMonoid R] (x : R)
Goal: IsSelfAdjoint (star x + x)
-- IsSelfAdjoint.conjugate
@[aesop safe apply]
theorem conjugate {x : R} (hx : IsSelfAdjoint x) (z : R) : IsSelfAdjoint (z * x * star z) 
Arguments: {R : Type u_1} [Semigroup R] [StarMul R] {x : R} (hx : IsSelfAdjoint x) (z : R)
Goal: IsSelfAdjoint (z * x * star z)
-- IsSelfAdjoint.conjugate'
@[aesop safe apply]
theorem conjugate' {x : R} (hx : IsSelfAdjoint x) (z : R) : IsSelfAdjoint (star z * x * z) 
Arguments: {R : Type u_1} [Semigroup R] [StarMul R] {x : R} (hx : IsSelfAdjoint x) (z : R)
Goal: IsSelfAdjoint (star z * x * z)
-- IsSelfAdjoint.isStarNormal
@[aesop 10% apply]
theorem isStarNormal {x : R} (hx : IsSelfAdjoint x) : IsStarNormal x 
Arguments: {R : Type u_1} [Semigroup R] [StarMul R] {x : R} (hx : IsSelfAdjoint x)
Goal: IsStarNormal x
-- IsSelfAdjoint.one
@[simp] protected theorem one : IsSelfAdjoint (1 : R) 
Arguments: (R : Type u_1) [MulOneClass R] [StarMul R]
Goal: IsSelfAdjoint 1
-- IsSelfAdjoint.pow
@[aesop safe apply]
theorem pow {x : R} (hx : IsSelfAdjoint x) (n : ℕ) : IsSelfAdjoint (x ^ n) 
Arguments: {R : Type u_1} [Monoid R] [StarMul R] {x : R} (hx : IsSelfAdjoint x) (n : ℕ)
Goal: IsSelfAdjoint (x ^ n)
-- IsSelfAdjoint.natCast
@[simp]
protected theorem natCast (n : ℕ) : IsSelfAdjoint (n : R) 
Arguments: {R : Type u_1} [Semiring R] [StarRing R] (n : ℕ)
Goal: IsSelfAdjoint ↑n
-- IsSelfAdjoint.ofNat
@[simp]
protected theorem ofNat (n : ℕ) [n.AtLeastTwo] :
    IsSelfAdjoint (no_index (OfNat.ofNat n : R)) 
Arguments: {R : Type u_1} [Semiring R] [StarRing R] (n : ℕ) [n.AtLeastTwo]
Goal: IsSelfAdjoint (OfNat.ofNat n)
-- IsSelfAdjoint.mul
theorem mul {x y : R} (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y) : IsSelfAdjoint (x * y) 
Arguments: {R : Type u_1} [CommSemigroup R] [StarMul R] {x y : R} (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y)
Goal: IsSelfAdjoint (x * y)
-- IsSelfAdjoint.conj_eq
lemma conj_eq (ha : IsSelfAdjoint a) : conj a = a 
Arguments: {α : Type u_3} [CommSemiring α] [StarRing α] {a : α} (ha : IsSelfAdjoint a)
Goal: (starRingEnd α) a = a
-- IsSelfAdjoint.intCast
@[simp]
protected theorem intCast (z : ℤ) : IsSelfAdjoint (z : R) 
Arguments: {R : Type u_1} [Ring R] [StarRing R] (z : ℤ)
Goal: IsSelfAdjoint ↑z
-- IsSelfAdjoint.inv
@[aesop safe apply]
theorem inv {x : R} (hx : IsSelfAdjoint x) : IsSelfAdjoint x⁻¹ 
Arguments: {R : Type u_1} [DivisionSemiring R] [StarRing R] {x : R} (hx : IsSelfAdjoint x)
Goal: IsSelfAdjoint x⁻¹
-- IsSelfAdjoint.zpow
@[aesop safe apply]
theorem zpow {x : R} (hx : IsSelfAdjoint x) (n : ℤ) : IsSelfAdjoint (x ^ n) 
Arguments: {R : Type u_1} [DivisionSemiring R] [StarRing R] {x : R} (hx : IsSelfAdjoint x) (n : ℤ)
Goal: IsSelfAdjoint (x ^ n)
-- IsSelfAdjoint.nnratCast
@[simp]
protected lemma nnratCast (q : ℚ≥0) : IsSelfAdjoint (q : R) 
Arguments: {R : Type u_1} [DivisionSemiring R] [StarRing R] (q : ℚ≥0)
Goal: IsSelfAdjoint ↑q
-- IsSelfAdjoint.ratCast
@[simp]
protected theorem ratCast (x : ℚ) : IsSelfAdjoint (x : R) 
Arguments: {R : Type u_1} [DivisionRing R] [StarRing R] (x : ℚ)
Goal: IsSelfAdjoint ↑x
-- IsSelfAdjoint.div
theorem div {x y : R} (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y) : IsSelfAdjoint (x / y) 
Arguments: {R : Type u_1} [Semifield R] [StarRing R] {x y : R} (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y)
Goal: IsSelfAdjoint (x / y)
-- IsSelfAdjoint.smul
@[aesop safe apply]
theorem smul {r : R} (hr : IsSelfAdjoint r) {x : A} (hx : IsSelfAdjoint x) :
    IsSelfAdjoint (r • x) 
Arguments: {R : Type u_1} {A : Type u_2} [Star R] [AddMonoid A] [StarAddMonoid A] [SMul R A] [StarModule R A] {r : R} (hr : IsSelfAdjoint r) {x : A} (hx : IsSelfAdjoint x)
Goal: IsSelfAdjoint (r • x)
-- selfAdjoint.mem_iff
theorem mem_iff {x : R} : x ∈ selfAdjoint R ↔ star x = x 
Arguments: {R : Type u_1} [AddGroup R] [StarAddMonoid R] {x : R}
Goal: x ∈ selfAdjoint R ↔ star x = x
-- selfAdjoint.star_val_eq
@[simp, norm_cast]
theorem star_val_eq {x : selfAdjoint R} : star (x : R) = x 
Arguments: {R : Type u_1} [AddGroup R] [StarAddMonoid R] {x : ↥(selfAdjoint R)}
Goal: star ↑x = ↑x
-- selfAdjoint.val_one
@[simp, norm_cast]
theorem val_one : ↑(1 : selfAdjoint R) = (1 : R) 
Arguments: {R : Type u_1} [Ring R] [StarRing R]
Goal: ↑1 = 1
-- selfAdjoint.val_pow
@[simp, norm_cast]
theorem val_pow (x : selfAdjoint R) (n : ℕ) : ↑(x ^ n) = (x : R) ^ n 
Arguments: {R : Type u_1} [Ring R] [StarRing R] (x : ↥(selfAdjoint R)) (n : ℕ)
Goal: ↑(x ^ n) = ↑x ^ n
-- selfAdjoint.val_mul
@[simp, norm_cast]
theorem val_mul (x y : selfAdjoint R) : ↑(x * y) = (x : R) * y 
Arguments: {R : Type u_1} [NonUnitalCommRing R] [StarRing R] (x y : ↥(selfAdjoint R))
Goal: ↑(x * y) = ↑x * ↑y
-- selfAdjoint.val_inv
@[simp, norm_cast]
theorem val_inv (x : selfAdjoint R) : ↑x⁻¹ = (x : R)⁻¹ 
Arguments: {R : Type u_1} [Field R] [StarRing R] (x : ↥(selfAdjoint R))
Goal: ↑x⁻¹ = (↑x)⁻¹
-- selfAdjoint.val_div
@[simp, norm_cast]
theorem val_div (x y : selfAdjoint R) : ↑(x / y) = (x / y : R) 
Arguments: {R : Type u_1} [Field R] [StarRing R] (x y : ↥(selfAdjoint R))
Goal: ↑(x / y) = ↑x / ↑y
-- selfAdjoint.val_zpow
@[simp, norm_cast]
theorem val_zpow (x : selfAdjoint R) (z : ℤ) : ↑(x ^ z) = (x : R) ^ z 
Arguments: {R : Type u_1} [Field R] [StarRing R] (x : ↥(selfAdjoint R)) (z : ℤ)
Goal: ↑(x ^ z) = ↑x ^ z
-- selfAdjoint.val_nnratCast
@[simp, norm_cast] lemma val_nnratCast (q : ℚ≥0) : (q : selfAdjoint R) = (q : R) 
Arguments: {R : Type u_1} [Field R] [StarRing R] (q : ℚ≥0)
Goal: ↑↑q = ↑q
-- selfAdjoint.val_ratCast
@[simp, norm_cast] lemma val_ratCast (q : ℚ) : (q : selfAdjoint R) = (q : R) 
Arguments: {R : Type u_1} [Field R] [StarRing R] (q : ℚ)
Goal: ↑↑q = ↑q
-- selfAdjoint.val_nnqsmul
@[simp, norm_cast] lemma val_nnqsmul (q : ℚ≥0) (x : selfAdjoint R) : ↑(q • x) = q • (x : R) 
Arguments: {R : Type u_1} [Field R] [StarRing R] (q : ℚ≥0) (x : ↥(selfAdjoint R))
Goal: ↑(q • x) = q • ↑x
-- selfAdjoint.val_qsmul
@[simp, norm_cast] lemma val_qsmul (q : ℚ) (x : selfAdjoint R) : ↑(q • x) = q • (x : R) 
Arguments: {R : Type u_1} [Field R] [StarRing R] (q : ℚ) (x : ↥(selfAdjoint R))
Goal: ↑(q • x) = q • ↑x
-- selfAdjoint.val_smul
@[simp, norm_cast]
theorem val_smul [SMul R A] [StarModule R A] (r : R) (x : selfAdjoint A) : ↑(r • x) = r • (x : A) 
Arguments: {R : Type u_1} {A : Type u_2} [Star R] [TrivialStar R] [AddGroup A] [StarAddMonoid A] [SMul R A] [StarModule R A] (r : R) (x : ↥(selfAdjoint A))
Goal: ↑(r • x) = r • ↑x
-- skewAdjoint.mem_iff
theorem mem_iff {x : R} : x ∈ skewAdjoint R ↔ star x = -x 
Arguments: {R : Type u_1} [AddCommGroup R] [StarAddMonoid R] {x : R}
Goal: x ∈ skewAdjoint R ↔ star x = -x
-- skewAdjoint.star_val_eq
@[simp, norm_cast]
theorem star_val_eq {x : skewAdjoint R} : star (x : R) = -x 
Arguments: {R : Type u_1} [AddCommGroup R] [StarAddMonoid R] {x : ↥(skewAdjoint R)}
Goal: star ↑x = -↑x
-- skewAdjoint.conjugate
theorem conjugate {x : R} (hx : x ∈ skewAdjoint R) (z : R) : z * x * star z ∈ skewAdjoint R 
Arguments: {R : Type u_1} [Ring R] [StarRing R] {x : R} (hx : x ∈ skewAdjoint R) (z : R)
Goal: z * x * star z ∈ skewAdjoint R
-- skewAdjoint.conjugate'
theorem conjugate' {x : R} (hx : x ∈ skewAdjoint R) (z : R) : star z * x * z ∈ skewAdjoint R 
Arguments: {R : Type u_1} [Ring R] [StarRing R] {x : R} (hx : x ∈ skewAdjoint R) (z : R)
Goal: star z * x * z ∈ skewAdjoint R
-- skewAdjoint.isStarNormal_of_mem
theorem isStarNormal_of_mem {x : R} (hx : x ∈ skewAdjoint R) : IsStarNormal x 
Arguments: {R : Type u_1} [Ring R] [StarRing R] {x : R} (hx : x ∈ skewAdjoint R)
Goal: IsStarNormal x
-- skewAdjoint.smul_mem
@[aesop safe apply (rule_sets := [SetLike])]
theorem smul_mem [Monoid R] [DistribMulAction R A] [StarModule R A] (r : R) {x : A}
    (h : x ∈ skewAdjoint A) : r • x ∈ skewAdjoint A 
Arguments: {R : Type u_1} {A : Type u_2} [Star R] [TrivialStar R] [AddCommGroup A] [StarAddMonoid A] [Monoid R] [DistribMulAction R A] [StarModule R A] (r : R) {x : A} (h : x ∈ skewAdjoint A)
Goal: r • x ∈ skewAdjoint A
-- skewAdjoint.val_smul
@[simp, norm_cast]
theorem val_smul [Monoid R] [DistribMulAction R A] [StarModule R A] (r : R) (x : skewAdjoint A) :
    ↑(r • x) = r • (x : A) 
Arguments: {R : Type u_1} {A : Type u_2} [Star R] [TrivialStar R] [AddCommGroup A] [StarAddMonoid A] [Monoid R] [DistribMulAction R A] [StarModule R A] (r : R) (x : ↥(skewAdjoint A))
Goal: ↑(r • x) = r • ↑x
-- IsSelfAdjoint.smul_mem_skewAdjoint
theorem IsSelfAdjoint.smul_mem_skewAdjoint [Ring R] [AddCommGroup A] [Module R A] [StarAddMonoid R]
    [StarAddMonoid A] [StarModule R A] {r : R} (hr : r ∈ skewAdjoint R) {a : A}
    (ha : IsSelfAdjoint a) : r • a ∈ skewAdjoint A 
Arguments: {R : Type u_1} {A : Type u_2} [Ring R] [AddCommGroup A] [Module R A] [StarAddMonoid R] [StarAddMonoid A] [StarModule R A] {r : R} (hr : r ∈ skewAdjoint R) {a : A} (ha : IsSelfAdjoint a)
Goal: r • a ∈ skewAdjoint A
-- isSelfAdjoint_smul_of_mem_skewAdjoint
theorem isSelfAdjoint_smul_of_mem_skewAdjoint [Ring R] [AddCommGroup A] [Module R A]
    [StarAddMonoid R] [StarAddMonoid A] [StarModule R A] {r : R} (hr : r ∈ skewAdjoint R) {a : A}
    (ha : a ∈ skewAdjoint A) : IsSelfAdjoint (r • a) 
Arguments: {R : Type u_1} {A : Type u_2} [Ring R] [AddCommGroup A] [Module R A] [StarAddMonoid R] [StarAddMonoid A] [StarModule R A] {r : R} (hr : r ∈ skewAdjoint R) {a : A} (ha : a ∈ skewAdjoint A)
Goal: IsSelfAdjoint (r • a)
-- Pi.isSelfAdjoint
protected lemma isSelfAdjoint : IsSelfAdjoint f ↔ ∀ i, IsSelfAdjoint (f i) 
Arguments: {ι : Type u_3} {α : ι → Type u_4} [(i : ι) → Star (α i)] {f : (i : ι) → α i}
Goal: IsSelfAdjoint f ↔ ∀ (i : ι), IsSelfAdjoint (f i)
-- obj_carrier
@[simp] lemma obj_carrier (M : FGModuleCat R) : M.obj.carrier = M.carrier 
Arguments: {R : Type u} [Ring R] (M : FGModuleCat R)
Goal: ↑M.obj = ↑M
-- LinearEquiv.toFGModuleCatIso_inv
@[simps]
def _root_.LinearEquiv.toFGModuleCatIso
    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]
    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :
    FGModuleCat.of R V ≅ FGModuleCat.of R W 
Arguments: {R : Type u} [Ring R] {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V] [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W)
Goal: e.toFGModuleCatIso.inv = ↑e.symm
-- LinearEquiv.toFGModuleCatIso_hom
@[simps]
def _root_.LinearEquiv.toFGModuleCatIso
    {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V]
    [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W) :
    FGModuleCat.of R V ≅ FGModuleCat.of R W 
Arguments: {R : Type u} [Ring R] {V W : Type u} [AddCommGroup V] [Module R V] [Module.Finite R V] [AddCommGroup W] [Module R W] [Module.Finite R W] (e : V ≃ₗ[R] W)
Goal: e.toFGModuleCatIso.hom = ↑e
-- FGModuleCat.tensorUnit_obj
@[simp] lemma tensorUnit_obj : (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R) 
Arguments: (R : Type u) [CommRing R]
Goal: (𝟙_ (FGModuleCat R)).obj = 𝟙_ (ModuleCat R)
-- FGModuleCat.tensorObj_obj
@[simp] lemma tensorObj_obj (M N : FGModuleCat.{u} R) : (M ⊗ N).obj = (M.obj ⊗ N.obj) 
Arguments: (R : Type u) [CommRing R] (M N : FGModuleCat R)
Goal: (CategoryTheory.MonoidalCategory.tensorObj M N).obj = CategoryTheory.MonoidalCategory.tensorObj M.obj N.obj
-- FGModuleCat.Iso.conj_eq_conj
theorem Iso.conj_eq_conj {V W : FGModuleCat R} (i : V ≅ W) (f : End V) :
    Iso.conj i f = LinearEquiv.conj (isoToLinearEquiv i) f 
Arguments: (R : Type u) [CommRing R] {V W : FGModuleCat R} (i : V ≅ W) (f : CategoryTheory.End V)
Goal: i.conj f = (FGModuleCat.isoToLinearEquiv i).conj f
-- FGModuleCat.ihom_obj
@[simp]
theorem ihom_obj : (ihom V).obj W = FGModuleCat.of K (V →ₗ[K] W) 
Arguments: (K : Type u) [Field K] (V W : FGModuleCat K)
Goal: (CategoryTheory.ihom V).obj W = FGModuleCat.of K (↑V →ₗ[K] ↑W)
-- FGModuleCat.FGModuleCatDual_obj
@[simp] lemma FGModuleCatDual_obj : (FGModuleCatDual K V).obj = ModuleCat.of K (Module.Dual K V) 
Arguments: (K : Type u) [Field K] (V : FGModuleCat K)
Goal: (FGModuleCat.FGModuleCatDual K V).obj = ModuleCat.of K (Module.Dual K ↑V)
-- FGModuleCat.FGModuleCatDual_coe
@[simp] lemma FGModuleCatDual_coe : (FGModuleCatDual K V : Type u) = Module.Dual K V 
Arguments: (K : Type u) [Field K] (V : FGModuleCat K)
Goal: ↑(FGModuleCat.FGModuleCatDual K V) = Module.Dual K ↑V
-- FGModuleCat.FGModuleCatCoevaluation_apply_one
theorem FGModuleCatCoevaluation_apply_one :
    FGModuleCatCoevaluation K V (1 : K) =
      ∑ i : Basis.ofVectorSpaceIndex K V,
        (Basis.ofVectorSpace K V) i ⊗ₜ[K] (Basis.ofVectorSpace K V).coord i 
Arguments: (K : Type u) [Field K] (V : FGModuleCat K)
Goal: (FGModuleCat.FGModuleCatCoevaluation K V) 1 =
  ∑ i : ↑(Basis.ofVectorSpaceIndex K ↑V), (Basis.ofVectorSpace K ↑V) i ⊗ₜ[K] (Basis.ofVectorSpace K ↑V).coord i
-- FGModuleCat.FGModuleCatEvaluation_apply
@[simp]
theorem FGModuleCatEvaluation_apply (f : FGModuleCatDual K V) (x : V) :
    (FGModuleCatEvaluation K V) (f ⊗ₜ x) = f.toFun x 
Arguments: (K : Type u) [Field K] (V : FGModuleCat K) (f : ↑(FGModuleCat.FGModuleCatDual K V)) (x : ↑V)
Goal: (FGModuleCat.FGModuleCatEvaluation K V) (f ⊗ₜ[K] x) = f.toFun x
-- _private.Mathlib.Algebra.Category.FGModuleCat.Basic.0.FGModuleCat.coevaluation_evaluation
private theorem coevaluation_evaluation :
    letI V' : FGModuleCat K := FGModuleCatDual K V
    V' ◁ FGModuleCatCoevaluation K V ≫ (α_ V' V V').inv ≫ FGModuleCatEvaluation K V ▷ V' =
      (ρ_ V').hom ≫ (λ_ V').inv 
Arguments: (K : Type u) [Field K] (V : FGModuleCat K)
Goal: CategoryTheory.MonoidalCategory.whiskerLeft (FGModuleCat.FGModuleCatDual K V)
      (FGModuleCat.FGModuleCatCoevaluation K V) ≫
    (CategoryTheory.MonoidalCategory.associator (FGModuleCat.FGModuleCatDual K V) V
          (FGModuleCat.FGModuleCatDual K V)).inv ≫
      CategoryTheory.MonoidalCategory.whiskerRight (FGModuleCat.FGModuleCatEvaluation K V)
        (FGModuleCat.FGModuleCatDual K V) =
  (CategoryTheory.MonoidalCategory.rightUnitor (FGModuleCat.FGModuleCatDual K V)).hom ≫
    (CategoryTheory.MonoidalCategory.leftUnitor (FGModuleCat.FGModuleCatDual K V)).inv
-- _private.Mathlib.Algebra.Category.FGModuleCat.Basic.0.FGModuleCat.evaluation_coevaluation
private theorem evaluation_coevaluation :
    FGModuleCatCoevaluation K V ▷ V ≫
        (α_ V (FGModuleCatDual K V) V).hom ≫ V ◁ FGModuleCatEvaluation K V =
      (λ_ V).hom ≫ (ρ_ V).inv 
Arguments: (K : Type u) [Field K] (V : FGModuleCat K)
Goal: CategoryTheory.MonoidalCategory.whiskerRight (FGModuleCat.FGModuleCatCoevaluation K V) V ≫
    (CategoryTheory.MonoidalCategory.associator V (FGModuleCat.FGModuleCatDual K V) V).hom ≫
      CategoryTheory.MonoidalCategory.whiskerLeft V (FGModuleCat.FGModuleCatEvaluation K V) =
  (CategoryTheory.MonoidalCategory.leftUnitor V).hom ≫ (CategoryTheory.MonoidalCategory.rightUnitor V).inv
-- LinearMap.comp_id_fgModuleCat
@[simp] theorem LinearMap.comp_id_fgModuleCat
    {R} [Ring R] {G : FGModuleCat.{u} R} {H : Type u} [AddCommGroup H] [Module R H]
    (f : G →ₗ[R] H) : f.comp (𝟙 G) = f 
Arguments: {R : Type u} [Ring R] {G : FGModuleCat R} {H : Type u} [AddCommGroup H] [Module R H] (f : ↑G →ₗ[R] H)
Goal: f ∘ₗ 𝟙 G = f
-- LinearMap.id_fgModuleCat_comp
@[simp] theorem LinearMap.id_fgModuleCat_comp
    {R} [Ring R] {G : Type u} [AddCommGroup G] [Module R G] {H : FGModuleCat.{u} R}
    (f : G →ₗ[R] H) : LinearMap.comp (𝟙 H) f = f 
Arguments: {R : Type u} [Ring R] {G : Type u} [AddCommGroup G] [Module R G] {H : FGModuleCat R} (f : G →ₗ[R] ↑H)
Goal: 𝟙 H ∘ₗ f = f
-- BoolRing.coe_of
@[simp]
theorem coe_of (α : Type*) [BooleanRing α] : ↥(of α) = α 
Arguments: (α : Type u_1) [BooleanRing α]
Goal: ↑(BoolRing.of α) = α
-- BoolRing.Iso.mk_hom
@[simps]
def Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β 
Arguments: {α β : BoolRing} (e : ↑α ≃+* ↑β)
Goal: (BoolRing.Iso.mk e).hom = ↑e
-- BoolRing.Iso.mk_inv
@[simps]
def Iso.mk {α β : BoolRing.{u}} (e : α ≃+* β) : α ≅ β 
Arguments: {α β : BoolRing} (e : ↑α ≃+* ↑β)
Goal: (BoolRing.Iso.mk e).inv = ↑e.symm
-- BoolRing.hasForgetToBoolAlg_forget₂_map
@[simps]
instance BoolRing.hasForgetToBoolAlg : HasForget₂ BoolRing BoolAlg 
Arguments: {X Y : BoolRing} (f : ↑X →+* ↑Y)
Goal: CategoryTheory.HasForget₂.forget₂.map f = f.asBoolAlg
-- BoolRing.hasForgetToBoolAlg_forget₂_obj
@[simps]
instance BoolRing.hasForgetToBoolAlg : HasForget₂ BoolRing BoolAlg 
Arguments: (X : BoolRing)
Goal: CategoryTheory.HasForget₂.forget₂.obj X = BoolAlg.of (AsBoolAlg ↑X)
-- BoolAlg.hasForgetToBoolRing_forget₂_obj
@[simps]
instance BoolAlg.hasForgetToBoolRing : HasForget₂ BoolAlg BoolRing 
Arguments: (X : BoolAlg)
Goal: CategoryTheory.HasForget₂.forget₂.obj X = BoolRing.of (AsBoolRing ↑X)
-- BoolAlg.hasForgetToBoolRing_forget₂_map
@[simps]
instance BoolAlg.hasForgetToBoolRing : HasForget₂ BoolAlg BoolRing 
Arguments: {X Y : BoolAlg} (f : BoundedLatticeHom ↑X.toBddDistLat.toBddLat.toLat ↑Y.toBddDistLat.toBddLat.toLat)
Goal: CategoryTheory.HasForget₂.forget₂.map f = f.asBoolRing
-- boolRingCatEquivBoolAlg_inverse
@[simps functor inverse]
def boolRingCatEquivBoolAlg : BoolRing ≌ BoolAlg 
Goal: boolRingCatEquivBoolAlg.inverse = CategoryTheory.forget₂ BoolAlg BoolRing
-- boolRingCatEquivBoolAlg_functor
@[simps functor inverse]
def boolRingCatEquivBoolAlg : BoolRing ≌ BoolAlg 
Goal: boolRingCatEquivBoolAlg.functor = CategoryTheory.forget₂ BoolRing BoolAlg