mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E'
⊢ ↑↑↑((condExpL2 E' 𝕜 hm) (indicatorConstLp 2 hs hμs x)) =ᵐ[μ] fun a ↦
    ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x

Tactic:
indicatorConstLp_eq_toSpanSingleton_compLp hs hμs x

Goal state after:
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E'
⊢ ↑↑↑((condExpL2 E' 𝕜 hm) ((toSpanSingleton ℝ x).compLp (indicatorConstLp 2 hs hμs 1))) =ᵐ[μ] fun a ↦
    ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x
Module: Mathlib.InformationTheory.KullbackLeibler.ChainRule

Goal state before:
pp: case neg
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η
h_int : ¬Integrable (llr μ ν) μ ∨ ¬Integrable (llr (μ ⊗ₘ κ) (μ ⊗ₘ η)) (μ ⊗ₘ κ)
⊢ ∞ = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)

Tactic:
cases h_int with
    | inl h => simp [h]
    | inr h => simp [h]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
φY : ↥Y → ℝ := φ ∘ ⇑Y.subtypeL
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
hfY_int : Integrable fY μ
lem2 : μ[f | m] =ᵐ[μ] ⇑Y.subtypeL ∘ μ[fY | m]
⊢ φ ∘ μ[f | m] ≤ᵐ[μ] μ[φ ∘ f | m]

Tactic:
have lem3 : φ ∘ f =ᵐ[μ] φY ∘ fY := by filter_upwards [lem1] with a ha; simp [φY, ha, fX]

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
φY : ↥Y → ℝ := φ ∘ ⇑Y.subtypeL
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
hfY_int : Integrable fY μ
lem2 : μ[f | m] =ᵐ[μ] ⇑Y.subtypeL ∘ μ[fY | m]
lem3 : φ ∘ f =ᵐ[μ] φY ∘ fY
⊢ φ ∘ μ[f | m] ≤ᵐ[μ] μ[φ ∘ f | m]
Module: Mathlib.RingTheory.TensorProduct.IsBaseChangeFree

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
V : Type u_3
inst✝⁵ : AddCommMonoid V
inst✝⁴ : Module R V
W : Type u_4
inst✝³ : AddCommMonoid W
inst✝² : Module R W
inst✝¹ : Module S W
inst✝ : IsScalarTower R S W
ι : Type u_5
ε : V →ₗ[R] W
b : Module.Basis ι R V
ibc : IsBaseChange S ε
v : V
⊢ (basis b ibc).repr (ε v) = (Finsupp.mapRange.linearMap (Algebra.linearMap R S)) (b.repr v)

Tactic:
by
  ext i
  simp [basis_repr_comp_apply]

Goal state after:
Module: Mathlib.LinearAlgebra.Alternating.DomCoprod

Goal state before:
pp: case refine_1
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
m₁ m₂ : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x✝ : ιa ⊕ ιb → Mᵢ
⊢ ((m₁ + m₂).domCoprod n) x✝ = (m₁.domCoprod n + m₂.domCoprod n) x✝

Tactic:
simp only [domCoprod_apply, add_apply, smul_apply, ← Finset.sum_add_distrib,
        Finset.smul_sum, _root_.sum_apply, domCoprod.summand]

Goal state after:
case refine_1
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
m₁ m₂ : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x✝ : ιa ⊕ ιb → Mᵢ
⊢ ∑ i, (Quotient.liftOn' i (fun σ ↦ Perm.sign σ • MultilinearMap.domDomCongr σ ((↑(m₁ + m₂)).domCoprod ↑n)) ⋯) x✝ =
    ∑ x,
      ((Quotient.liftOn' x (fun σ ↦ Perm.sign σ • MultilinearMap.domDomCongr σ ((↑m₁).domCoprod ↑n)) ⋯) x✝ +
        (Quotient.liftOn' x (fun σ ↦ Perm.sign σ • MultilinearMap.domDomCongr σ ((↑m₂).domCoprod ↑n)) ⋯) x✝)
Module: Mathlib.LinearAlgebra.Alternating.DomCoprod

Goal state before:
pp: case refine_2
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
c : R'
m : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x✝ : ιa ⊕ ιb → Mᵢ
⊢ ((c • m).domCoprod n) x✝ = (c • m.domCoprod n) x✝

Tactic:
simp only [domCoprod_apply, add_apply, smul_apply, ← Finset.sum_add_distrib,
        Finset.smul_sum, _root_.sum_apply, domCoprod.summand]

Goal state after:
case refine_2
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
c : R'
m : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x✝ : ιa ⊕ ιb → Mᵢ
⊢ ∑ i, (Quotient.liftOn' i (fun σ ↦ Perm.sign σ • MultilinearMap.domDomCongr σ ((↑(c • m)).domCoprod ↑n)) ⋯) x✝ =
    ∑ x, c • (Quotient.liftOn' x (fun σ ↦ Perm.sign σ • MultilinearMap.domDomCongr σ ((↑m).domCoprod ↑n)) ⋯) x✝
Module: Mathlib.LinearAlgebra.Alternating.DomCoprod

Goal state before:
pp: case refine_3
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
m : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n₁ n₂ : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x✝ : ιa ⊕ ιb → Mᵢ
⊢ (m.domCoprod (n₁ + n₂)) x✝ = (m.domCoprod n₁ + m.domCoprod n₂) x✝

Tactic:
simp only [domCoprod_apply, add_apply, smul_apply, ← Finset.sum_add_distrib,
        Finset.smul_sum, _root_.sum_apply, domCoprod.summand]

Goal state after:
case refine_3
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
m : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n₁ n₂ : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x✝ : ιa ⊕ ιb → Mᵢ
⊢ ∑ i, (Quotient.liftOn' i (fun σ ↦ Perm.sign σ • MultilinearMap.domDomCongr σ ((↑m).domCoprod ↑(n₁ + n₂))) ⋯) x✝ =
    ∑ x,
      ((Quotient.liftOn' x (fun σ ↦ Perm.sign σ • MultilinearMap.domDomCongr σ ((↑m).domCoprod ↑n₁)) ⋯) x✝ +
        (Quotient.liftOn' x (fun σ ↦ Perm.sign σ • MultilinearMap.domDomCongr σ ((↑m).domCoprod ↑n₂)) ⋯) x✝)
Module: Mathlib.LinearAlgebra.Alternating.DomCoprod

Goal state before:
pp: case refine_4
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
c : R'
m : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x✝ : ιa ⊕ ιb → Mᵢ
⊢ (m.domCoprod (c • n)) x✝ = (c • m.domCoprod n) x✝

Tactic:
simp only [domCoprod_apply, add_apply, smul_apply, ← Finset.sum_add_distrib,
        Finset.smul_sum, _root_.sum_apply, domCoprod.summand]

Goal state after:
case refine_4
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
c : R'
m : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x✝ : ιa ⊕ ιb → Mᵢ
⊢ ∑ i, (Quotient.liftOn' i (fun σ ↦ Perm.sign σ • MultilinearMap.domDomCongr σ ((↑m).domCoprod ↑(c • n))) ⋯) x✝ =
    ∑ x, c • (Quotient.liftOn' x (fun σ ↦ Perm.sign σ • MultilinearMap.domDomCongr σ ((↑m).domCoprod ↑n)) ⋯) x✝
Module: Mathlib.LinearAlgebra.TensorProduct.Opposite

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra R B
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
e₁ : Aᵐᵒᵖ ⊗[R] Bᵐᵒᵖ ≃ₗ[S] (A ⊗[R] B)ᵐᵒᵖ :=
  TensorProduct.AlgebraTensorModule.congr (opLinearEquiv S).symm (opLinearEquiv R).symm ≪≫ₗ opLinearEquiv S
e₂ : A ⊗[R] B ≃ₗ[S] (Aᵐᵒᵖ ⊗[R] Bᵐᵒᵖ)ᵐᵒᵖ :=
  TensorProduct.AlgebraTensorModule.congr (opLinearEquiv S) (opLinearEquiv R) ≪≫ₗ opLinearEquiv S
a₁ a₂ : Aᵐᵒᵖ
b₁ b₂ : Bᵐᵒᵖ
⊢ unop (↑e₁ ((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂))) = unop (↑e₁ (a₁ ⊗ₜ[R] b₁) * ↑e₁ (a₂ ⊗ₜ[R] b₂))

Tactic:
with_unfolding_all rfl

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Opposite

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra R B
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
e₁ : Aᵐᵒᵖ ⊗[R] Bᵐᵒᵖ ≃ₗ[S] (A ⊗[R] B)ᵐᵒᵖ :=
  TensorProduct.AlgebraTensorModule.congr (opLinearEquiv S).symm (opLinearEquiv R).symm ≪≫ₗ opLinearEquiv S
e₂ : A ⊗[R] B ≃ₗ[S] (Aᵐᵒᵖ ⊗[R] Bᵐᵒᵖ)ᵐᵒᵖ :=
  TensorProduct.AlgebraTensorModule.congr (opLinearEquiv S) (opLinearEquiv R) ≪≫ₗ opLinearEquiv S
a₁ a₂ : A
b₁ b₂ : B
⊢ unop (↑e₂ ((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂))) = unop (↑e₂ (a₁ ⊗ₜ[R] b₁) * ↑e₂ (a₂ ⊗ₜ[R] b₂))

Tactic:
with_unfolding_all rfl

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Contraction

Goal state before:
pp: R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
d✝ d' : Module.Dual R M
c : R
d : Module.Dual R M
x : CliffordAlgebra Q
⊢ (foldr' Q (contractLeftAux Q (c • d)) ⋯ 0) x = (RingHom.id R) c • (foldr' Q (contractLeftAux Q d) ⋯ 0) x

Tactic:
RingHom.id_apply

Goal state after:
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
d✝ d' : Module.Dual R M
c : R
d : Module.Dual R M
x : CliffordAlgebra Q
⊢ (foldr' Q (contractLeftAux Q (c • d)) ⋯ 0) x = c • (foldr' Q (contractLeftAux Q d) ⋯ 0) x
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Contraction

Goal state before:
pp: R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
d : Module.Dual R M
r : R
b : CliffordAlgebra Q
⊢ (contractRight (r • b)) d = (algebraMap R (CliffordAlgebra Q)) r * (contractRight b) d

Tactic:
LinearMap.map_smul₂,

Goal state after:
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
d : Module.Dual R M
r : R
b : CliffordAlgebra Q
⊢ r • (contractRight b) d = (algebraMap R (CliffordAlgebra Q)) r * (contractRight b) d
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Contraction

Goal state before:
pp: R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
d : Module.Dual R M
r : R
⊢ reverse ((contractLeft d) (reverse ((algebraMap R (CliffordAlgebra Q)) r))) = 0

Tactic:
reverse.commutes,

Goal state after:
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
d : Module.Dual R M
r : R
⊢ reverse ((contractLeft d) ((algebraMap R (CliffordAlgebra Q)) r)) = 0
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
i : ι × ι
a : ℬ i.1
b : 𝒜 i.2
i✝ : ι × ι
⊢ ((gradedCommAux R 𝒜 ℬ) ((gradedCommAux R ℬ 𝒜) ((lof R (ι × ι) (fun i ↦ ℬ i.1 ⊗[R] 𝒜 i.2) i) (a ⊗ₜ[R] b)))) i✝ =
    ((lof R (ι × ι) (fun i ↦ ℬ i.1 ⊗[R] 𝒜 i.2) i) (a ⊗ₜ[R] b)) i✝

Tactic:
gradedCommAux_lof_tmul,

Goal state after:
R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
i : ι × ι
a : ℬ i.1
b : 𝒜 i.2
i✝ : ι × ι
⊢ ((gradedCommAux R 𝒜 ℬ) ((-1) ^ (i.2 * i.1) • (lof R (ι × ι) (fun i ↦ 𝒜 i.1 ⊗[R] ℬ i.2) (i.2, i.1)) (b ⊗ₜ[R] a))) i✝ =
    ((lof R (ι × ι) (fun i ↦ ℬ i.1 ⊗[R] 𝒜 i.2) i) (a ⊗ₜ[R] b)) i✝
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
i : ι × ι
a : ℬ i.1
b : 𝒜 i.2
i✝ : ι × ι
⊢ ((-1) ^ (i.2 * i.1) • (gradedCommAux R 𝒜 ℬ) ((lof R (ι × ι) (fun i ↦ 𝒜 i.1 ⊗[R] ℬ i.2) (i.2, i.1)) (b ⊗ₜ[R] a))) i✝ =
    ((lof R (ι × ι) (fun i ↦ ℬ i.1 ⊗[R] 𝒜 i.2) i) (a ⊗ₜ[R] b)) i✝

Tactic:
gradedCommAux_lof_tmul,

Goal state after:
R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
i : ι × ι
a : ℬ i.1
b : 𝒜 i.2
i✝ : ι × ι
⊢ ((-1) ^ (i.2 * i.1) • (-1) ^ (i.1 * i.2) • (lof R (ι × ι) (fun i ↦ ℬ i.1 ⊗[R] 𝒜 i.2) (i.1, i.2)) (a ⊗ₜ[R] b)) i✝ =
    ((lof R (ι × ι) (fun i ↦ ℬ i.1 ⊗[R] 𝒜 i.2) i) (a ⊗ₜ[R] b)) i✝
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
i j : ι
a : 𝒜 i
b : ℬ j
⊢ (TensorProduct.directSum R R ℬ 𝒜).symm
      ((gradedCommAux R 𝒜 ℬ) ((lof R (ι × ι) (fun i ↦ 𝒜 i.1 ⊗[R] ℬ i.2) (i, j)) (a ⊗ₜ[R] b))) =
    (-1) ^ (j * i) • (lof R ι ℬ j) b ⊗ₜ[R] (lof R ι 𝒜 i) a

Tactic:
gradedCommAux_lof_tmul,

Goal state after:
R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
i j : ι
a : 𝒜 i
b : ℬ j
⊢ (TensorProduct.directSum R R ℬ 𝒜).symm
      ((-1) ^ (j * i) • (lof R (ι × ι) (fun i ↦ ℬ i.1 ⊗[R] 𝒜 i.2) (j, i)) (b ⊗ₜ[R] a)) =
    (-1) ^ (j * i) • (lof R ι ℬ j) b ⊗ₜ[R] (lof R ι 𝒜 i) a
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
x : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i
r : R
⊢ ((gradedMul R 𝒜 ℬ) x) ((algebraMap R (DirectSum ι 𝒜)) r ⊗ₜ[R] 1) = r • x

Tactic:
by
  suffices (gradedMul R 𝒜 ℬ).flip (algebraMap R _ r ⊗ₜ 1) = DistribSMul.toLinearMap R _ r by
    exact DFunLike.congr_fun this x
  ext
  dsimp
  erw [tmul_of_gradedMul_of_tmul]
  rw [mul_zero, uzpow_zero, one_smul, smul_tmul',
      mul_one, _root_.Algebra.smul_def, Algebra.commutes]
  rfl

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
⊢ 𝒜 ᵍ⊗[R] ℬ →ₗ[R] 𝒜 ᵍ⊗[R] ℬ →ₗ[R] 𝒜 ᵍ⊗[R] ℬ

Tactic:
letI fAB1 := auxEquiv R 𝒜 ℬ
  have := ((gradedMul R (𝒜 ·) (ℬ ·)).compl₁₂ fAB1.toLinearMap fAB1.toLinearMap).compr₂
    fAB1.symm.toLinearMap
  exact this

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
x y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
mA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=
  gradedMul R 𝒜 ℬ
ixa : ι
xa : 𝒜 ixa
ixb : ι
xb : ℬ ixb
iya : ι
ya : 𝒜 iya
iyb : ι
yb : ℬ iyb
iza : ι
za : 𝒜 iza
izb : ι
zb : ℬ izb
⊢ ((gradedMul R 𝒜 ℬ)
        (↑↑((-1) ^ (ixb * iya)) •
          ((lof R ι 𝒜 ixa) xa * (lof R ι 𝒜 iya) ya) ⊗ₜ[R] ((lof R ι ℬ ixb) xb * (lof R ι ℬ iyb) yb)))
      ((lof R ι 𝒜 iza) za ⊗ₜ[R] (lof R ι ℬ izb) zb) =
    ((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 ixa) xa ⊗ₜ[R] (lof R ι ℬ ixb) xb))
      (↑↑((-1) ^ (iyb * iza)) •
        ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za) ⊗ₜ[R] ((lof R ι ℬ iyb) yb * (lof R ι ℬ izb) zb))

Tactic:
LinearMap.map_smul₂,

Goal state after:
R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
x y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
mA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=
  gradedMul R 𝒜 ℬ
ixa : ι
xa : 𝒜 ixa
ixb : ι
xb : ℬ ixb
iya : ι
ya : 𝒜 iya
iyb : ι
yb : ℬ iyb
iza : ι
za : 𝒜 iza
izb : ι
zb : ℬ izb
⊢ ↑↑((-1) ^ (ixb * iya)) •
      ((gradedMul R 𝒜 ℬ) (((lof R ι 𝒜 ixa) xa * (lof R ι 𝒜 iya) ya) ⊗ₜ[R] ((lof R ι ℬ ixb) xb * (lof R ι ℬ iyb) yb)))
        ((lof R ι 𝒜 iza) za ⊗ₜ[R] (lof R ι ℬ izb) zb) =
    ((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 ixa) xa ⊗ₜ[R] (lof R ι ℬ ixb) xb))
      (↑↑((-1) ^ (iyb * iza)) •
        ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za) ⊗ₜ[R] ((lof R ι ℬ iyb) yb * (lof R ι ℬ izb) zb))
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Prod

Goal state before:
pp: case mem.mem_mul.e_a.e_a.mem
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : AddCommGroup N
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ (f₁ x) (f₂ y)
m₁ : CliffordAlgebra Q₁
i₂ : ZMod 2
m₁' : CliffordAlgebra Q₁
i₁n i : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i
v₁ : M₁
i₂' : { n // ↑n = i₂ }
m₂' : CliffordAlgebra Q₂
hm₂ : m₂' ∈ (ι Q₂).range ^ ↑i₂'
⊢ (ι Qₙ) (f₁ v₁) * (map f₂) m₂' = (-1) ^ i₂ • ((map f₂) m₂' * (ι Qₙ) (f₁ v₁))

Tactic:
obtain ⟨i₂n, rfl⟩ := i₂'

Goal state after:
case mem.mem_mul.e_a.e_a.mem
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : AddCommGroup N
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ (f₁ x) (f₂ y)
m₁ m₁' : CliffordAlgebra Q₁
i₁n i : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i
v₁ : M₁
m₂' : CliffordAlgebra Q₂
i₂n : ℕ
hm₂ : m₂' ∈ (ι Q₂).range ^ ↑⟨i₂n, ⋯⟩
⊢ (ι Qₙ) (f₁ v₁) * (map f₂) m₂' = (-1) ^ ↑i₂n • ((map f₂) m₂' * (ι Qₙ) (f₁ v₁))
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: case e_f
R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
a₁ a₂ : A
i : ι
⊢ (of R 𝒜 ℬ) ((a₁ * ↑(((decompose 𝒜) a₂) i)) ⊗ₜ[R] 1) =
    (of R 𝒜 ℬ) (a₁ ⊗ₜ[R] 1) * (of R 𝒜 ℬ) (↑(((decompose 𝒜) a₂) i) ⊗ₜ[R] 1)

Tactic:
rw [← SetLike.coe_gOne ℬ, tmul_coe_mul_coe_tmul, zero_mul, uzpow_zero, one_smul,
      SetLike.coe_gOne, one_mul]

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : DecidableEq ι
inst✝⁹ : CommRing R
inst✝⁸ : Ring A
inst✝⁷ : Ring B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝⁴ : GradedAlgebra 𝒜
inst✝³ : GradedAlgebra ℬ
inst✝² : Module ι (Additive ℤˣ)
C : Type ?u.46
inst✝¹ : Ring C
inst✝ : Algebra R C
x y : 𝒜 ᵍ⊗[R] ℬ
⊢ ((auxEquiv R 𝒜 ℬ ≪≫ₗ gradedComm R (fun i ↦ ↥(𝒜 i)) fun i ↦ ↥(ℬ i)) ≪≫ₗ (auxEquiv R ℬ 𝒜).symm) (x * y) =
    ((auxEquiv R 𝒜 ℬ ≪≫ₗ gradedComm R (fun i ↦ ↥(𝒜 i)) fun i ↦ ↥(ℬ i)) ≪≫ₗ (auxEquiv R ℬ 𝒜).symm) x *
      ((auxEquiv R 𝒜 ℬ ≪≫ₗ gradedComm R (fun i ↦ ↥(𝒜 i)) fun i ↦ ↥(ℬ i)) ≪≫ₗ (auxEquiv R ℬ 𝒜).symm) y

Tactic:
dsimp
      simp_rw [auxEquiv_mul, gradedComm_gradedMul, LinearEquiv.symm_apply_eq,
        ← gradedComm_gradedMul, auxEquiv_mul, LinearEquiv.apply_symm_apply, gradedComm_gradedMul]

Goal state after: