ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LinearMap.BilinForm.IsRefl.smul
protected theorem smul {α} [CommSemiring α] [Module α R] [SMulCommClass R α R]
    [NoZeroSMulDivisors α R] (a : α) {B : BilinForm R M} (hB : B.IsRefl) :
    (a • B).IsRefl 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {α : Type u_9} [CommSemiring α] [Module α R] [SMulCommClass R α R] [NoZeroSMulDivisors α R] (a : α) {B : LinearMap.BilinForm R M} (hB : B.IsRefl)
Goal: (a • B).IsRefl
-- LinearMap.BilinForm.IsRefl.groupSMul
protected theorem groupSMul {α} [Group α] [DistribMulAction α R] [SMulCommClass R α R] (a : α)
    {B : BilinForm R M} (hB : B.IsRefl) : (a • B).IsRefl 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {α : Type u_9} [Group α] [DistribMulAction α R] [SMulCommClass R α R] (a : α) {B : LinearMap.BilinForm R M} (hB : B.IsRefl)
Goal: (a • B).IsRefl
-- LinearMap.BilinForm.isRefl_zero
@[simp]
theorem isRefl_zero : (0 : BilinForm R M).IsRefl 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M]
Goal: LinearMap.BilinForm.IsRefl 0
-- LinearMap.BilinForm.isRefl_neg
@[simp]
theorem isRefl_neg {B : BilinForm R₁ M₁} : (-B).IsRefl ↔ B.IsRefl 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] {B : LinearMap.BilinForm R₁ M₁}
Goal: (-B).IsRefl ↔ B.IsRefl
-- LinearMap.BilinForm.IsSymm.eq
protected theorem eq (x y : M) : B x y = B y x 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} (H : B.IsSymm) (x y : M)
Goal: (B x) y = (B y) x
-- LinearMap.BilinForm.IsSymm.isRefl
theorem isRefl : B.IsRefl 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} (H : B.IsSymm)
Goal: B.IsRefl
-- LinearMap.BilinForm.IsSymm.add
protected theorem add {B₁ B₂ : BilinForm R M} (hB₁ : B₁.IsSymm) (hB₂ : B₂.IsSymm) :
    (B₁ + B₂).IsSymm 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B₁ B₂ : LinearMap.BilinForm R M} (hB₁ : B₁.IsSymm) (hB₂ : B₂.IsSymm)
Goal: (B₁ + B₂).IsSymm
-- LinearMap.BilinForm.IsSymm.sub
protected theorem sub {B₁ B₂ : BilinForm R₁ M₁} (hB₁ : B₁.IsSymm) (hB₂ : B₂.IsSymm) :
    (B₁ - B₂).IsSymm 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] {B₁ B₂ : LinearMap.BilinForm R₁ M₁} (hB₁ : B₁.IsSymm) (hB₂ : B₂.IsSymm)
Goal: (B₁ - B₂).IsSymm
-- LinearMap.BilinForm.IsSymm.neg
protected theorem neg {B : BilinForm R₁ M₁} (hB : B.IsSymm) : (-B).IsSymm 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] {B : LinearMap.BilinForm R₁ M₁} (hB : B.IsSymm)
Goal: (-B).IsSymm
-- LinearMap.BilinForm.IsSymm.smul
protected theorem smul {α} [Monoid α] [DistribMulAction α R] [SMulCommClass R α R] (a : α)
    {B : BilinForm R M} (hB : B.IsSymm) : (a • B).IsSymm 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {α : Type u_9} [Monoid α] [DistribMulAction α R] [SMulCommClass R α R] (a : α) {B : LinearMap.BilinForm R M} (hB : B.IsSymm)
Goal: (a • B).IsSymm
-- LinearMap.BilinForm.IsSymm.restrict
theorem restrict {B : BilinForm R M} (b : B.IsSymm) (W : Submodule R M) :
    (B.restrict W).IsSymm 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} (b : B.IsSymm) (W : Submodule R M)
Goal: (B.restrict W).IsSymm
-- LinearMap.BilinForm.isSymm_zero
@[simp]
theorem isSymm_zero : (0 : BilinForm R M).IsSymm 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M]
Goal: LinearMap.BilinForm.IsSymm 0
-- LinearMap.BilinForm.isSymm_neg
@[simp]
theorem isSymm_neg {B : BilinForm R₁ M₁} : (-B).IsSymm ↔ B.IsSymm 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] {B : LinearMap.BilinForm R₁ M₁}
Goal: (-B).IsSymm ↔ B.IsSymm
-- LinearMap.BilinForm.isSymm_iff_flip
theorem isSymm_iff_flip : B.IsSymm ↔ flipHom B = B 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M}
Goal: B.IsSymm ↔ LinearMap.BilinForm.flipHom B = B
-- LinearMap.BilinForm.IsAlt.self_eq_zero
theorem self_eq_zero (H : B.IsAlt) (x : M) : B x x = 0 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} (H : B.IsAlt) (x : M)
Goal: (B x) x = 0
-- LinearMap.BilinForm.IsAlt.neg_eq
theorem neg_eq (H : B₁.IsAlt) (x y : M₁) : -B₁ x y = B₁ y x 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] {B₁ : LinearMap.BilinForm R₁ M₁} (H : B₁.IsAlt) (x y : M₁)
Goal: -(B₁ x) y = (B₁ y) x
-- LinearMap.BilinForm.IsAlt.isRefl
theorem isRefl (H : B₁.IsAlt) : B₁.IsRefl 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] {B₁ : LinearMap.BilinForm R₁ M₁} (H : B₁.IsAlt)
Goal: B₁.IsRefl
-- LinearMap.BilinForm.IsAlt.eq_of_add_add_eq_zero
theorem eq_of_add_add_eq_zero [IsCancelAdd R] {a b c : M} (H : B.IsAlt) (hAdd : a + b + c = 0) :
    B a b = B b c 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} [IsCancelAdd R] {a b c : M} (H : B.IsAlt) (hAdd : a + b + c = 0)
Goal: (B a) b = (B b) c
-- LinearMap.BilinForm.IsAlt.add
protected theorem add {B₁ B₂ : BilinForm R M} (hB₁ : B₁.IsAlt) (hB₂ : B₂.IsAlt) : (B₁ + B₂).IsAlt 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B₁ B₂ : LinearMap.BilinForm R M} (hB₁ : B₁.IsAlt) (hB₂ : B₂.IsAlt)
Goal: (B₁ + B₂).IsAlt
-- LinearMap.BilinForm.IsAlt.sub
protected theorem sub {B₁ B₂ : BilinForm R₁ M₁} (hB₁ : B₁.IsAlt) (hB₂ : B₂.IsAlt) :
    (B₁ - B₂).IsAlt 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] {B₁ B₂ : LinearMap.BilinForm R₁ M₁} (hB₁ : B₁.IsAlt) (hB₂ : B₂.IsAlt)
Goal: (B₁ - B₂).IsAlt
-- LinearMap.BilinForm.IsAlt.neg
protected theorem neg {B : BilinForm R₁ M₁} (hB : B.IsAlt) : (-B).IsAlt 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] {B : LinearMap.BilinForm R₁ M₁} (hB : B.IsAlt)
Goal: (-B).IsAlt
-- LinearMap.BilinForm.IsAlt.smul
protected theorem smul {α} [Monoid α] [DistribMulAction α R] [SMulCommClass R α R] (a : α)
    {B : BilinForm R M} (hB : B.IsAlt) : (a • B).IsAlt 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {α : Type u_9} [Monoid α] [DistribMulAction α R] [SMulCommClass R α R] (a : α) {B : LinearMap.BilinForm R M} (hB : B.IsAlt)
Goal: (a • B).IsAlt
-- LinearMap.BilinForm.isAlt_zero
@[simp]
theorem isAlt_zero : (0 : BilinForm R M).IsAlt 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M]
Goal: LinearMap.BilinForm.IsAlt 0
-- LinearMap.BilinForm.isAlt_neg
@[simp]
theorem isAlt_neg {B : BilinForm R₁ M₁} : (-B).IsAlt ↔ B.IsAlt 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] {B : LinearMap.BilinForm R₁ M₁}
Goal: (-B).IsAlt ↔ B.IsAlt
-- LinearMap.BilinForm.IsAdjointPair.eq
theorem IsAdjointPair.eq (h : IsAdjointPair B B' f g) : ∀ {x y}, B' (f x) y = B x (g y) 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} {M' : Type u_9} [AddCommMonoid M'] [Module R M'] {B' : LinearMap.BilinForm R M'} {f : M →ₗ[R] M'} {g : M' →ₗ[R] M} (h : B.IsAdjointPair B' f g) {x : M} {y : M'}
Goal: (B' (f x)) y = (B x) (g y)
-- LinearMap.BilinForm.isAdjointPair_iff_compLeft_eq_compRight
theorem isAdjointPair_iff_compLeft_eq_compRight (f g : Module.End R M) :
    IsAdjointPair B F f g ↔ F.compLeft f = B.compRight g 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} (F : LinearMap.BilinForm R M) (f g : Module.End R M)
Goal: B.IsAdjointPair F f g ↔ F.compLeft f = B.compRight g
-- LinearMap.BilinForm.isAdjointPair_zero
theorem isAdjointPair_zero : IsAdjointPair B B' 0 0 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} {M' : Type u_9} [AddCommMonoid M'] [Module R M'] {B' : LinearMap.BilinForm R M'}
Goal: B.IsAdjointPair B' 0 0
-- LinearMap.BilinForm.isAdjointPair_id
theorem isAdjointPair_id : IsAdjointPair B B 1 1 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M}
Goal: B.IsAdjointPair B 1 1
-- LinearMap.BilinForm.IsAdjointPair.add
theorem IsAdjointPair.add (h : IsAdjointPair B B' f g) (h' : IsAdjointPair B B' f' g') :
    IsAdjointPair B B' (f + f') (g + g') 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} {M' : Type u_9} [AddCommMonoid M'] [Module R M'] {B' : LinearMap.BilinForm R M'} {f f' : M →ₗ[R] M'} {g g' : M' →ₗ[R] M} (h : B.IsAdjointPair B' f g) (h' : B.IsAdjointPair B' f' g')
Goal: B.IsAdjointPair B' (f + f') (g + g')
-- LinearMap.BilinForm.IsAdjointPair.sub
theorem IsAdjointPair.sub (h : IsAdjointPair B₁ B₁' f₁ g₁) (h' : IsAdjointPair B₁ B₁' f₁' g₁') :
    IsAdjointPair B₁ B₁' (f₁ - f₁') (g₁ - g₁') 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] {B₁ : LinearMap.BilinForm R₁ M₁} {M₁' : Type u_10} [AddCommGroup M₁'] [Module R₁ M₁'] {B₁' : LinearMap.BilinForm R₁ M₁'} {f₁ f₁' : M₁ →ₗ[R₁] M₁'} {g₁ g₁' : M₁' →ₗ[R₁] M₁} (h : B₁.IsAdjointPair B₁' f₁ g₁) (h' : B₁.IsAdjointPair B₁' f₁' g₁')
Goal: B₁.IsAdjointPair B₁' (f₁ - f₁') (g₁ - g₁')
-- LinearMap.BilinForm.IsAdjointPair.smul
theorem IsAdjointPair.smul (c : R) (h : IsAdjointPair B B₂' f₂ g₂) :
    IsAdjointPair B B₂' (c • f₂) (c • g₂) 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} {M' : Type u_9} [AddCommMonoid M'] [Module R M'] {B₂' : LinearMap.BilinForm R M'} {f₂ : M →ₗ[R] M'} {g₂ : M' →ₗ[R] M} (c : R) (h : B.IsAdjointPair B₂' f₂ g₂)
Goal: B.IsAdjointPair B₂' (c • f₂) (c • g₂)
-- LinearMap.BilinForm.IsAdjointPair.comp
theorem IsAdjointPair.comp {f' : M' →ₗ[R] M''} {g' : M'' →ₗ[R] M'} (h : IsAdjointPair B B' f g)
    (h' : IsAdjointPair B' B'' f' g') : IsAdjointPair B B'' (f'.comp f) (g.comp g') 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} {M' : Type u_9} [AddCommMonoid M'] [Module R M'] {B' : LinearMap.BilinForm R M'} {f : M →ₗ[R] M'} {g : M' →ₗ[R] M} {M'' : Type u_11} [AddCommMonoid M''] [Module R M''] (B'' : LinearMap.BilinForm R M'') {f' : M' →ₗ[R] M''} {g' : M'' →ₗ[R] M'} (h : B.IsAdjointPair B' f g) (h' : B'.IsAdjointPair B'' f' g')
Goal: B.IsAdjointPair B'' (f' ∘ₗ f) (g ∘ₗ g')
-- LinearMap.BilinForm.IsAdjointPair.mul
theorem IsAdjointPair.mul {f g f' g' : Module.End R M} (h : IsAdjointPair B B f g)
    (h' : IsAdjointPair B B f' g') : IsAdjointPair B B (f * f') (g' * g) 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} {f g f' g' : Module.End R M} (h : B.IsAdjointPair B f g) (h' : B.IsAdjointPair B f' g')
Goal: B.IsAdjointPair B (f * f') (g' * g)
-- LinearMap.BilinForm.mem_isPairSelfAdjointSubmodule
@[simp]
theorem mem_isPairSelfAdjointSubmodule (f : Module.End R M) :
    f ∈ isPairSelfAdjointSubmodule B₂ F₂ ↔ IsPairSelfAdjoint B₂ F₂ f 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (B₂ F₂ : LinearMap.BilinForm R M) (f : Module.End R M)
Goal: f ∈ B₂.isPairSelfAdjointSubmodule F₂ ↔ B₂.IsPairSelfAdjoint F₂ f
-- LinearMap.BilinForm.isPairSelfAdjoint_equiv
theorem isPairSelfAdjoint_equiv (e : M' ≃ₗ[R] M) (f : Module.End R M) :
    IsPairSelfAdjoint B₂ F₂ f ↔
      IsPairSelfAdjoint (B₂.comp ↑e ↑e) (F₂.comp ↑e ↑e) (e.symm.conj f) 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {M' : Type u_9} [AddCommMonoid M'] [Module R M'] (B₂ F₂ : LinearMap.BilinForm R M) (e : M' ≃ₗ[R] M) (f : Module.End R M)
Goal: B₂.IsPairSelfAdjoint F₂ f ↔ (B₂.comp ↑e ↑e).IsPairSelfAdjoint (F₂.comp ↑e ↑e) (e.symm.conj f)
-- LinearMap.BilinForm.isSkewAdjoint_iff_neg_self_adjoint
theorem isSkewAdjoint_iff_neg_self_adjoint (f : Module.End R₁ M₁) :
    B₁.IsSkewAdjoint f ↔ IsAdjointPair (-B₁) B₁ f f 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] (B₁ : LinearMap.BilinForm R₁ M₁) (f : Module.End R₁ M₁)
Goal: B₁.IsSkewAdjoint f ↔ (-B₁).IsAdjointPair B₁ f f
-- LinearMap.BilinForm.mem_selfAdjointSubmodule
@[simp]
theorem mem_selfAdjointSubmodule (f : Module.End R M) :
    f ∈ B.selfAdjointSubmodule ↔ B.IsSelfAdjoint f 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (B : LinearMap.BilinForm R M) (f : Module.End R M)
Goal: f ∈ B.selfAdjointSubmodule ↔ B.IsSelfAdjoint f
-- LinearMap.BilinForm.mem_skewAdjointSubmodule
@[simp]
theorem mem_skewAdjointSubmodule (f : Module.End R₁ M₁) :
    f ∈ B₁.skewAdjointSubmodule ↔ B₁.IsSkewAdjoint f 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] (B₁ : LinearMap.BilinForm R₁ M₁) (f : Module.End R₁ M₁)
Goal: f ∈ B₁.skewAdjointSubmodule ↔ B₁.IsSkewAdjoint f
-- LinearMap.BilinForm.not_nondegenerate_zero
theorem not_nondegenerate_zero [Nontrivial M] : ¬(0 : BilinForm R M).Nondegenerate 
Arguments: (R : Type u_1) (M : Type u_2) [CommSemiring R] [AddCommMonoid M] [Module R M] [Nontrivial M]
Goal: ¬LinearMap.BilinForm.Nondegenerate 0
-- LinearMap.BilinForm.Nondegenerate.ne_zero
theorem Nondegenerate.ne_zero [Nontrivial M] {B : BilinForm R M} (h : B.Nondegenerate) : B ≠ 0 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] [Nontrivial M] {B : LinearMap.BilinForm R M} (h : B.Nondegenerate)
Goal: B ≠ 0
-- LinearMap.BilinForm.Nondegenerate.congr
theorem Nondegenerate.congr {B : BilinForm R M} (e : M ≃ₗ[R] M') (h : B.Nondegenerate) :
    (congr e B).Nondegenerate 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {M' : Type u_9} [AddCommMonoid M'] [Module R M'] {B : LinearMap.BilinForm R M} (e : M ≃ₗ[R] M') (h : B.Nondegenerate)
Goal: ((LinearMap.BilinForm.congr e) B).Nondegenerate
-- LinearMap.BilinForm.nondegenerate_congr_iff
@[simp]
theorem nondegenerate_congr_iff {B : BilinForm R M} (e : M ≃ₗ[R] M') :
    (congr e B).Nondegenerate ↔ B.Nondegenerate 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {M' : Type u_9} [AddCommMonoid M'] [Module R M'] {B : LinearMap.BilinForm R M} (e : M ≃ₗ[R] M')
Goal: ((LinearMap.BilinForm.congr e) B).Nondegenerate ↔ B.Nondegenerate
-- LinearMap.BilinForm.nondegenerate_iff_ker_eq_bot
theorem nondegenerate_iff_ker_eq_bot {B : BilinForm R M} :
    B.Nondegenerate ↔ LinearMap.ker B = ⊥ 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M}
Goal: B.Nondegenerate ↔ LinearMap.ker B = ⊥
-- LinearMap.BilinForm.Nondegenerate.ker_eq_bot
theorem Nondegenerate.ker_eq_bot {B : BilinForm R M} (h : B.Nondegenerate) :
    LinearMap.ker B = ⊥ 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} (h : B.Nondegenerate)
Goal: LinearMap.ker B = ⊥
-- LinearMap.BilinForm.compLeft_injective
theorem compLeft_injective (B : BilinForm R₁ M₁) (b : B.Nondegenerate) :
    Function.Injective B.compLeft 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] (B : LinearMap.BilinForm R₁ M₁) (b : B.Nondegenerate)
Goal: Function.Injective B.compLeft
-- LinearMap.BilinForm.isAdjointPair_unique_of_nondegenerate
theorem isAdjointPair_unique_of_nondegenerate (B : BilinForm R₁ M₁) (b : B.Nondegenerate)
    (φ ψ₁ ψ₂ : M₁ →ₗ[R₁] M₁) (hψ₁ : IsAdjointPair B B ψ₁ φ) (hψ₂ : IsAdjointPair B B ψ₂ φ) :
    ψ₁ = ψ₂ 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] (B : LinearMap.BilinForm R₁ M₁) (b : B.Nondegenerate) (φ ψ₁ ψ₂ : M₁ →ₗ[R₁] M₁) (hψ₁ : B.IsAdjointPair B ψ₁ φ) (hψ₂ : B.IsAdjointPair B ψ₂ φ)
Goal: ψ₁ = ψ₂
-- LinearMap.BilinForm.toDual_def
theorem toDual_def {B : BilinForm K V} (b : B.SeparatingLeft) {m n : V} : B.toDual b m n = B m n 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {B : LinearMap.BilinForm K V} (b : LinearMap.SeparatingLeft B) {m n : V}
Goal: ((B.toDual b) m) n = (B m) n
-- LinearMap.BilinForm.apply_toDual_symm_apply
@[simp]
lemma apply_toDual_symm_apply {B : BilinForm K V} {hB : B.Nondegenerate}
    (f : Module.Dual K V) (v : V) :
    B ((B.toDual hB).symm f) v = f v 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {B : LinearMap.BilinForm K V} {hB : B.Nondegenerate} (f : Module.Dual K V) (v : V)
Goal: (B ((B.toDual hB).symm f)) v = f v
-- LinearMap.BilinForm.Nondegenerate.flip
lemma Nondegenerate.flip {B : BilinForm K V} (hB : B.Nondegenerate) :
    B.flip.Nondegenerate 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {B : LinearMap.BilinForm K V} (hB : B.Nondegenerate)
Goal: B.flip.Nondegenerate
-- LinearMap.BilinForm.nonDegenerateFlip_iff
lemma nonDegenerateFlip_iff {B : BilinForm K V} :
    B.flip.Nondegenerate ↔ B.Nondegenerate 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {B : LinearMap.BilinForm K V}
Goal: B.flip.Nondegenerate ↔ B.Nondegenerate
-- LinearMap.BilinForm.dualBasis_repr_apply
@[simp]
theorem dualBasis_repr_apply (B : BilinForm K V) (hB : B.Nondegenerate) (b : Basis ι K V) (x i) :
    (B.dualBasis hB b).repr x i = B x (b i) 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] {ι : Type u_10} [DecidableEq ι] [Finite ι] (B : LinearMap.BilinForm K V) (hB : B.Nondegenerate) (b : Basis ι K V) (x : V) (i : ι)
Goal: ((B.dualBasis hB b).repr x) i = (B x) (b i)
-- LinearMap.BilinForm.apply_dualBasis_left
theorem apply_dualBasis_left (B : BilinForm K V) (hB : B.Nondegenerate) (b : Basis ι K V) (i j) :
    B (B.dualBasis hB b i) (b j) = if j = i then 1 else 0 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] {ι : Type u_10} [DecidableEq ι] [Finite ι] (B : LinearMap.BilinForm K V) (hB : B.Nondegenerate) (b : Basis ι K V) (i j : ι)
Goal: (B ((B.dualBasis hB b) i)) (b j) = if j = i then 1 else 0
-- LinearMap.BilinForm.apply_dualBasis_right
theorem apply_dualBasis_right (B : BilinForm K V) (hB : B.Nondegenerate) (sym : B.IsSymm)
    (b : Basis ι K V) (i j) : B (b i) (B.dualBasis hB b j) = if i = j then 1 else 0 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] {ι : Type u_10} [DecidableEq ι] [Finite ι] (B : LinearMap.BilinForm K V) (hB : B.Nondegenerate) (sym : B.IsSymm) (b : Basis ι K V) (i j : ι)
Goal: (B (b i)) ((B.dualBasis hB b) j) = if i = j then 1 else 0
-- LinearMap.BilinForm.dualBasis_dualBasis_flip
@[simp]
lemma dualBasis_dualBasis_flip (B : BilinForm K V) (hB : B.Nondegenerate) {ι}
    [Finite ι] [DecidableEq ι] (b : Basis ι K V) :
    B.dualBasis hB (B.flip.dualBasis hB.flip b) = b 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (B : LinearMap.BilinForm K V) (hB : B.Nondegenerate) {ι : Type u_11} [Finite ι] [DecidableEq ι] (b : Basis ι K V)
Goal: B.dualBasis hB (B.flip.dualBasis ⋯ b) = b
-- LinearMap.BilinForm.dualBasis_flip_dualBasis
@[simp]
lemma dualBasis_flip_dualBasis (B : BilinForm K V) (hB : B.Nondegenerate) {ι}
    [Finite ι] [DecidableEq ι] [FiniteDimensional K V] (b : Basis ι K V) :
    B.flip.dualBasis hB.flip (B.dualBasis hB b) = b 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] (B : LinearMap.BilinForm K V) (hB : B.Nondegenerate) {ι : Type u_11} [Finite ι] [DecidableEq ι] [FiniteDimensional K V] (b : Basis ι K V)
Goal: B.flip.dualBasis ⋯ (B.dualBasis hB b) = b
-- LinearMap.BilinForm.dualBasis_dualBasis
@[simp]
lemma dualBasis_dualBasis (B : BilinForm K V) (hB : B.Nondegenerate) (hB' : B.IsSymm) {ι}
    [Finite ι] [DecidableEq ι] [FiniteDimensional K V] (b : Basis ι K V) :
    B.dualBasis hB (B.dualBasis hB b) = b 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] (B : LinearMap.BilinForm K V) (hB : B.Nondegenerate) (hB' : B.IsSymm) {ι : Type u_11} [Finite ι] [DecidableEq ι] [FiniteDimensional K V] (b : Basis ι K V)
Goal: B.dualBasis hB (B.dualBasis hB b) = b
-- LinearMap.BilinForm.comp_symmCompOfNondegenerate_apply
theorem comp_symmCompOfNondegenerate_apply (B₁ : BilinForm K V) {B₂ : BilinForm K V}
    (b₂ : B₂.Nondegenerate) (v : V) :
    B₂ (B₁.symmCompOfNondegenerate B₂ b₂ v) = B₁ v 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (B₁ : LinearMap.BilinForm K V) {B₂ : LinearMap.BilinForm K V} (b₂ : B₂.Nondegenerate) (v : V)
Goal: B₂ ((B₁.symmCompOfNondegenerate B₂ b₂) v) = B₁ v
-- LinearMap.BilinForm.symmCompOfNondegenerate_left_apply
@[simp]
theorem symmCompOfNondegenerate_left_apply (B₁ : BilinForm K V) {B₂ : BilinForm K V}
    (b₂ : B₂.Nondegenerate) (v w : V) : B₂ (symmCompOfNondegenerate B₁ B₂ b₂ w) v = B₁ w v 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (B₁ : LinearMap.BilinForm K V) {B₂ : LinearMap.BilinForm K V} (b₂ : B₂.Nondegenerate) (v w : V)
Goal: (B₂ ((B₁.symmCompOfNondegenerate B₂ b₂) w)) v = (B₁ w) v
-- LinearMap.BilinForm.isAdjointPairLeftAdjointOfNondegenerate
theorem isAdjointPairLeftAdjointOfNondegenerate (B : BilinForm K V) (b : B.Nondegenerate)
    (φ : V →ₗ[K] V) : IsAdjointPair B B (B.leftAdjointOfNondegenerate b φ) φ 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (B : LinearMap.BilinForm K V) (b : B.Nondegenerate) (φ : V →ₗ[K] V)
Goal: B.IsAdjointPair B (B.leftAdjointOfNondegenerate b φ) φ
-- LinearMap.BilinForm.isAdjointPair_iff_eq_of_nondegenerate
theorem isAdjointPair_iff_eq_of_nondegenerate (B : BilinForm K V) (b : B.Nondegenerate)
    (ψ φ : V →ₗ[K] V) : IsAdjointPair B B ψ φ ↔ ψ = B.leftAdjointOfNondegenerate b φ 
Arguments: {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] (B : LinearMap.BilinForm K V) (b : B.Nondegenerate) (ψ φ : V →ₗ[K] V)
Goal: B.IsAdjointPair B ψ φ ↔ ψ = B.leftAdjointOfNondegenerate b φ
-- AffineEquiv.map_vadd'
-- Porting note(#5171): this linter isn't ported yet.
-- @[nolint has_nonempty_instance]
structure AffineEquiv (k P₁ P₂ : Type*) {V₁ V₂ : Type*} [Ring k] [AddCommGroup V₁] [Module k V₁]
  [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] extends P₁ ≃ P₂ where
  linear : V₁ ≃ₗ[k] V₂
  map_vadd' : ∀ (p : P₁) (v : V₁), toEquiv (v +ᵥ p) = linear v +ᵥ toEquiv p
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_4} {V₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (self : P₁ ≃ᵃ[k] P₂) (p : P₁) (v : V₁)
Goal: self.toEquiv (v +ᵥ p) = self.linear v +ᵥ self.toEquiv p
-- AffineEquiv.toAffineMap_mk
@[simp]
theorem toAffineMap_mk (f : P₁ ≃ P₂) (f' : V₁ ≃ₗ[k] V₂) (h) :
    toAffineMap (mk f f' h) = ⟨f, f', h⟩ 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (f : P₁ ≃ P₂) (f' : V₁ ≃ₗ[k] V₂) (h : ∀ (p : P₁) (v : V₁), f (v +ᵥ p) = f' v +ᵥ f p)
Goal: ↑{ toEquiv := f, linear := f', map_vadd' := h } = { toFun := ⇑f, linear := ↑f', map_vadd' := h }
-- AffineEquiv.linear_toAffineMap
@[simp]
theorem linear_toAffineMap (e : P₁ ≃ᵃ[k] P₂) : e.toAffineMap.linear = e.linear 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AffineSpace V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (e : P₁ ≃ᵃ[k] P₂)
Goal: (↑e).linear = ↑e.linear