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

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ
hf : Integrable f μ
h'f : 0 ≤ᵐ[μ] f
hm : m ≤ m𝓧
hμ : SigmaFinite (μ.trim hm)
A : μ[fun x ↦ (ENNReal.ofReal (f x)).toReal | m] =ᵐ[μ] μ[f | m]
B : 0 ≤ᵐ[μ] μ[f | m]
g : 𝓧 → ℝ≥0∞ := fun x ↦ ENNReal.ofReal (f x)
I : ∫⁻ (x : 𝓧), g x ∂μ ≠ ∞
J : ∀ᵐ (x : 𝓧) ∂μ, μ⁻[g | m] x < ∞
a : 𝓧
ha : (μ⁻[g | m] a).toReal = μ[fun x ↦ (g x).toReal | m] a
h'a : 0 a ≤ f a
h''a : μ[fun x ↦ (ENNReal.ofReal (f x)).toReal | m] a = μ[f | m] a
h'''a : 0 a ≤ μ[f | m] a
C : μ⁻[g | m] a < ∞
⊢ μ⁻[fun x ↦ ENNReal.ofReal (f x) | m] a = ENNReal.ofReal (μ[f | m] a)

Tactic:
rw [← ENNReal.toReal_eq_toReal_iff' C.ne, ENNReal.toReal_ofReal h'''a, ha, h''a]

Goal state after:
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ
hf : Integrable f μ
h'f : 0 ≤ᵐ[μ] f
hm : m ≤ m𝓧
hμ : SigmaFinite (μ.trim hm)
A : μ[fun x ↦ (ENNReal.ofReal (f x)).toReal | m] =ᵐ[μ] μ[f | m]
B : 0 ≤ᵐ[μ] μ[f | m]
g : 𝓧 → ℝ≥0∞ := fun x ↦ ENNReal.ofReal (f x)
I : ∫⁻ (x : 𝓧), g x ∂μ ≠ ∞
J : ∀ᵐ (x : 𝓧) ∂μ, μ⁻[g | m] x < ∞
a : 𝓧
ha : (μ⁻[g | m] a).toReal = μ[fun x ↦ (g x).toReal | m] a
h'a : 0 a ≤ f a
h''a : μ[fun x ↦ (ENNReal.ofReal (f x)).toReal | m] a = μ[f | m] a
h'''a : 0 a ≤ μ[f | m] a
C : μ⁻[g | m] a < ∞
⊢ ENNReal.ofReal (μ[f | m] a) ≠ ∞
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
⊢ Integrable (fun x ↦ f (ν[fun x ↦ ((∂μ/∂ν) x).toReal | m] x)) ν

Tactic:
have h := integrable_comp_rnDeriv_trim hm hμν hf hf_cvx hf_cont_at h_int
  refine integrable_of_integrable_trim hm ((integrable_congr ?_).mp h)
  filter_upwards [toReal_rnDeriv_trim hm hμν] with a ha
  rw [ha]

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.IsBaseChangeFree

Goal state before:
pp: case refine_2
R : Type u_1
inst✝⁷ : CommSemiring R
V : Type u_2
inst✝⁶ : AddCommMonoid V
inst✝⁵ : Module R V
A : Type u_3
inst✝⁴ : CommSemiring A
inst✝³ : Algebra A R
inst✝² : Module A V
inst✝¹ : IsScalarTower A R V
ι : Type u_4
b : Module.Basis ι R V
inst✝ : Fintype ι
this : DecidableEq ι
j : R ⊗[A] (ι → A) ≃ₗ[R] ι → R := piScalarRight A R R ι
⊢ ∀ (x : ι → A),
    (LinearEquiv.ofBijective (Fintype.linearCombination R ⇑b ∘ₗ ↑j) ⋯) (1 ⊗ₜ[A] x) = (Fintype.linearCombination A ⇑b) x

Tactic:
· intro x
    -- simp? [Fintype.linearCombination_apply] says:
    simp only [LinearEquiv.ofBijective_apply, LinearMap.coe_comp, LinearEquiv.coe_coe,
      Function.comp_apply, Fintype.linearCombination_apply]
    congr
    ext i
    rw [TensorProduct.piScalarRight_apply, TensorProduct.piScalarRightHom_tmul]
    simp

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

Goal state before:
pp: R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
⊢ { f // ∀ (m : A ⊗[R] V), f m * f m = (algebraMap A (A ⊗[R] CliffordAlgebra Q)) ((QuadraticForm.baseChange A Q) m) }

Tactic:
refine ⟨TensorProduct.AlgebraTensorModule.map (LinearMap.id : A →ₗ[A] A) (ι Q), ?_⟩

Goal state after:
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
⊢ ∀ (m : A ⊗[R] V),
    (TensorProduct.AlgebraTensorModule.map LinearMap.id (ι Q)) m *
        (TensorProduct.AlgebraTensorModule.map LinearMap.id (ι Q)) m =
      (algebraMap A (A ⊗[R] CliffordAlgebra Q)) ((QuadraticForm.baseChange A Q) m)
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Equivs

Goal state before:
pp: c : CliffordAlgebra Q
this✝ : toComplex (involute ((ι Q) 1)) = (starRingEnd ℂ) (toComplex ((ι Q) 1))
this : toComplex.comp involute = (↑Complex.conjAe).comp toComplex
⊢ toComplex (involute c) = (starRingEnd ℂ) (toComplex c)

Tactic:
by
    exact AlgHom.congr_fun this c

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
Q : QuadraticForm R M
A : Type u_3
B : Type u_4
inst✝³ : Ring A
inst✝² : Ring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
f✝ f : EvenHom Q A
⊢ ↥(even Q) →ₗ[R] A

Tactic:
refine ?_ ∘ₗ (even Q).val.toLinearMap
  exact LinearMap.fst R _ _ ∘ₗ foldr Q (fFold f) (fFold_fFold f) (1, 0)

Goal state after:
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)
a✝ : ⨁ (i : ι), 𝒜 i
b : ℬ 0
i : ι
a : 𝒜 i
⊢ (gradedComm R 𝒜 ℬ) ((lof R ι 𝒜 i) a ⊗ₜ[R] (lof R ι ℬ 0) b) = (lof R ι ℬ 0) b ⊗ₜ[R] (lof R ι 𝒜 i) a

Tactic:
gradedComm_of_tmul_of,

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)
a✝ : ⨁ (i : ι), 𝒜 i
b : ℬ 0
i : ι
a : 𝒜 i
⊢ (-1) ^ (0 * i) • (lof R ι ℬ 0) b ⊗ₜ[R] (lof R ι 𝒜 i) a = (lof R ι ℬ 0) 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✝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)
a : 𝒜 0
b✝ : ⨁ (i : ι), ℬ i
i : ι
b : ℬ i
⊢ (gradedComm R 𝒜 ℬ) ((lof R ι 𝒜 0) a ⊗ₜ[R] (lof R ι ℬ i) b) = (lof R ι ℬ i) b ⊗ₜ[R] (lof R ι 𝒜 0) a

Tactic:
gradedComm_of_tmul_of,

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)
a : 𝒜 0
b✝ : ⨁ (i : ι), ℬ i
i : ι
b : ℬ i
⊢ (-1) ^ (i * 0) • (lof R ι ℬ i) b ⊗ₜ[R] (lof R ι 𝒜 0) a = (lof R ι ℬ i) b ⊗ₜ[R] (lof R ι 𝒜 0) a
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: case mul
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x y : CliffordAlgebra Q
hx : ↑((toEven Q) (reverse (involute x))) = reverse ↑((toEven Q) x)
hy : ↑((toEven Q) (reverse (involute y))) = reverse ↑((toEven Q) y)
⊢ ↑((toEven Q) (reverse (involute (x * y)))) = reverse ↑((toEven Q) (x * y))

Tactic:
| mul x y hx hy =>

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

Goal state before:
pp: case add
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
m₂ : CliffordAlgebra Q₂
i₁ i₂ : ZMod 2
hm₂ : m₂ ∈ evenOdd Q₂ i₂
x✝ y✝ : CliffordAlgebra Q₁
hx✝ : x✝ ∈ ⨆ i, (ι Q₁).range ^ ↑i
hy✝ : y✝ ∈ ⨆ i, (ι Q₁).range ^ ↑i
ihx : (map f₁) x✝ * (map f₂) m₂ = (-1) ^ (i₂ * i₁) • ((map f₂) m₂ * (map f₁) x✝)
ihy : (map f₁) y✝ * (map f₂) m₂ = (-1) ^ (i₂ * i₁) • ((map f₂) m₂ * (map f₁) y✝)
⊢ (map f₁) (x✝ + y✝) * (map f₂) m₂ = (-1) ^ (i₂ * i₁) • ((map f₂) m₂ * (map f₁) (x✝ + y✝))

Tactic:
rw [map_add, add_mul, mul_add, ihx, ihy, smul_add]

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

Goal state before:
pp: case mem.add
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
m₂ : CliffordAlgebra Q₂
i₂ : ZMod 2
hm₂ : m₂ ∈ evenOdd Q₂ i₂
m₁' : CliffordAlgebra Q₁
i₁n : ℕ
x✝ y✝ : CliffordAlgebra Q₁
i✝ : ℕ
hx✝ : x✝ ∈ (ι Q₁).range ^ i✝
hy✝ : y✝ ∈ (ι Q₁).range ^ i✝
ihx : (map f₁) x✝ * (map f₂) m₂ = (-1) ^ (i₂ * ↑i✝) • ((map f₂) m₂ * (map f₁) x✝)
ihy : (map f₁) y✝ * (map f₂) m₂ = (-1) ^ (i₂ * ↑i✝) • ((map f₂) m₂ * (map f₁) y✝)
⊢ (map f₁) (x✝ + y✝) * (map f₂) m₂ = (-1) ^ (i₂ * ↑i✝) • ((map f₂) m₂ * (map f₁) (x✝ + y✝))

Tactic:
rw [map_add, add_mul, mul_add, ihx, ihy, smul_add]

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

Goal state before:
pp: case mem.mem_mul.e_a.e_a.add
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
m₂ : CliffordAlgebra Q₂
i₂ : ZMod 2
m₁' : CliffordAlgebra Q₁
i₁n i : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i
v₁ : M₁
x✝ y✝ : CliffordAlgebra Q₂
hx✝ : x✝ ∈ ⨆ i, (ι Q₂).range ^ ↑i
hy✝ : y✝ ∈ ⨆ i, (ι Q₂).range ^ ↑i
ihx : (ι Qₙ) (f₁ v₁) * (map f₂) x✝ = (-1) ^ i₂ • ((map f₂) x✝ * (ι Qₙ) (f₁ v₁))
ihy : (ι Qₙ) (f₁ v₁) * (map f₂) y✝ = (-1) ^ i₂ • ((map f₂) y✝ * (ι Qₙ) (f₁ v₁))
⊢ (ι Qₙ) (f₁ v₁) * (map f₂) (x✝ + y✝) = (-1) ^ i₂ • ((map f₂) (x✝ + y✝) * (ι Qₙ) (f₁ v₁))

Tactic:
rw [map_add, add_mul, mul_add, ihx, ihy, smul_add]

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 : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
i₁ : ι
a₁ : 𝒜 i₁
j₁ : ι
b₁ : ℬ j₁
i₂ : ι
a₂ : 𝒜 i₂
j₂ : ι
b₂ : ℬ j₂
⊢ (gradedComm R 𝒜 ℬ)
      (((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 i₁) a₁ ⊗ₜ[R] (lof R ι ℬ j₁) b₁)) ((lof R ι 𝒜 i₂) a₂ ⊗ₜ[R] (lof R ι ℬ j₂) b₂)) =
    ((gradedMul R ℬ 𝒜) ((gradedComm R 𝒜 ℬ) ((lof R ι 𝒜 i₁) a₁ ⊗ₜ[R] (lof R ι ℬ j₁) b₁)))
      ((gradedComm R 𝒜 ℬ) ((lof R ι 𝒜 i₂) a₂ ⊗ₜ[R] (lof R ι ℬ j₂) b₂))

Tactic:
gradedComm_of_tmul_of,

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 : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
i₁ : ι
a₁ : 𝒜 i₁
j₁ : ι
b₁ : ℬ j₁
i₂ : ι
a₂ : 𝒜 i₂
j₂ : ι
b₂ : ℬ j₂
⊢ (gradedComm R 𝒜 ℬ)
      (((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 i₁) a₁ ⊗ₜ[R] (lof R ι ℬ j₁) b₁)) ((lof R ι 𝒜 i₂) a₂ ⊗ₜ[R] (lof R ι ℬ j₂) b₂)) =
    ((gradedMul R ℬ 𝒜) ((-1) ^ (j₁ * i₁) • (lof R ι ℬ j₁) b₁ ⊗ₜ[R] (lof R ι 𝒜 i₁) a₁))
      ((gradedComm R 𝒜 ℬ) ((lof R ι 𝒜 i₂) a₂ ⊗ₜ[R] (lof R ι ℬ j₂) b₂))
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 : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
i₁ : ι
a₁ : 𝒜 i₁
j₁ : ι
b₁ : ℬ j₁
i₂ : ι
a₂ : 𝒜 i₂
j₂ : ι
b₂ : ℬ j₂
⊢ (gradedComm R 𝒜 ℬ)
      (((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 i₁) a₁ ⊗ₜ[R] (lof R ι ℬ j₁) b₁)) ((lof R ι 𝒜 i₂) a₂ ⊗ₜ[R] (lof R ι ℬ j₂) b₂)) =
    ((gradedMul R ℬ 𝒜) ((-1) ^ (j₁ * i₁) • (lof R ι ℬ j₁) b₁ ⊗ₜ[R] (lof R ι 𝒜 i₁) a₁))
      ((gradedComm R 𝒜 ℬ) ((lof R ι 𝒜 i₂) a₂ ⊗ₜ[R] (lof R ι ℬ j₂) b₂))

Tactic:
gradedComm_of_tmul_of,

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 : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
i₁ : ι
a₁ : 𝒜 i₁
j₁ : ι
b₁ : ℬ j₁
i₂ : ι
a₂ : 𝒜 i₂
j₂ : ι
b₂ : ℬ j₂
⊢ (gradedComm R 𝒜 ℬ)
      (((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 i₁) a₁ ⊗ₜ[R] (lof R ι ℬ j₁) b₁)) ((lof R ι 𝒜 i₂) a₂ ⊗ₜ[R] (lof R ι ℬ j₂) b₂)) =
    ((gradedMul R ℬ 𝒜) ((-1) ^ (j₁ * i₁) • (lof R ι ℬ j₁) b₁ ⊗ₜ[R] (lof R ι 𝒜 i₁) a₁))
      ((-1) ^ (j₂ * i₂) • (lof R ι ℬ j₂) b₂ ⊗ₜ[R] (lof R ι 𝒜 i₂) a₂)
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Prod

Goal state before:
pp: case mem.mem_mul.e_a.e_a.mem.add
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
x✝ y✝ : CliffordAlgebra Q₂
i✝ : ℕ
hx✝ : x✝ ∈ (ι Q₂).range ^ i✝
hy✝ : y✝ ∈ (ι Q₂).range ^ i✝
ihx : (ι Qₙ) (f₁ v₁) * (map f₂) x✝ = (-1) ^ ↑i✝ • ((map f₂) x✝ * (ι Qₙ) (f₁ v₁))
ihy : (ι Qₙ) (f₁ v₁) * (map f₂) y✝ = (-1) ^ ↑i✝ • ((map f₂) y✝ * (ι Qₙ) (f₁ v₁))
⊢ (ι Qₙ) (f₁ v₁) * (map f₂) (x✝ + y✝) = (-1) ^ ↑i✝ • ((map f₂) (x✝ + y✝) * (ι Qₙ) (f₁ v₁))

Tactic:
rw [map_add, add_mul, mul_add, ihx, ihy, smul_add]

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 : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
i₁ : ι
a₁ : 𝒜 i₁
j₁ : ι
b₁ : ℬ j₁
i₂ : ι
a₂ : 𝒜 i₂
j₂ : ι
b₂ : ℬ j₂
⊢ (-1) ^ (j₁ * i₂) •
      (gradedComm R 𝒜 ℬ)
        ((lof R ι 𝒜 (i₁ + i₂)) (GradedMonoid.GMul.mul a₁ a₂) ⊗ₜ[R]
          (lof R ι ℬ (j₁ + j₂)) (GradedMonoid.GMul.mul b₁ b₂)) =
    (-1) ^ (j₂ * i₂) •
      (-1) ^ (j₁ * i₁) •
        ((gradedMul R ℬ 𝒜) ((lof R ι ℬ j₁) b₁ ⊗ₜ[R] (lof R ι 𝒜 i₁) a₁)) ((lof R ι ℬ j₂) b₂ ⊗ₜ[R] (lof R ι 𝒜 i₂) a₂)

Tactic:
gradedComm_of_tmul_of,

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 : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
i₁ : ι
a₁ : 𝒜 i₁
j₁ : ι
b₁ : ℬ j₁
i₂ : ι
a₂ : 𝒜 i₂
j₂ : ι
b₂ : ℬ j₂
⊢ (-1) ^ (j₁ * i₂) •
      (-1) ^ ((j₁ + j₂) * (i₁ + i₂)) •
        (lof R ι ℬ (j₁ + j₂)) (GradedMonoid.GMul.mul b₁ b₂) ⊗ₜ[R] (lof R ι 𝒜 (i₁ + i₂)) (GradedMonoid.GMul.mul a₁ a₂) =
    (-1) ^ (j₂ * i₂) •
      (-1) ^ (j₁ * i₁) •
        ((gradedMul R ℬ 𝒜) ((lof R ι ℬ j₁) b₁ ⊗ₜ[R] (lof R ι 𝒜 i₁) a₁)) ((lof R ι ℬ j₂) b₂ ⊗ₜ[R] (lof R ι 𝒜 i₂) a₂)
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : CliffordAlgebra Q
y : (CliffordAlgebra Q)ˣ
hy₂ : ↑y = x
_hx₂ : star x * x = 1
hx₃ : ↑y * star ↑y = 1
hy₁ : y ∈ lipschitzGroup Q
hy₃ : y * star y = 1
⊢ star y ∈ lipschitzGroup Q

Tactic:
apply_fun fun x => y⁻¹ * x at hy₃

Goal state after:
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : CliffordAlgebra Q
y : (CliffordAlgebra Q)ˣ
hy₂ : ↑y = x
_hx₂ : star x * x = 1
hx₃ : ↑y * star ↑y = 1
hy₁ : y ∈ lipschitzGroup Q
hy₃ : y⁻¹ * (y * star y) = y⁻¹ * 1
⊢ star y ∈ lipschitzGroup Q
Module: Mathlib.LinearAlgebra.Dimension.Torsion.Finite

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : StrongRankCondition R
inst✝ : Module.Finite R M
⊢ finrank R M = 0 ↔ IsTorsion R M

Tactic:
simp [← rank_eq_zero_iff_isTorsion (R := R), ← finrank_eq_rank]

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)
i j : ι
a : ↥(𝒜 i)
b : ↥(ℬ j)
⊢ (gradedComm R (fun x ↦ ↥(𝒜 x)) fun x ↦ ↥(ℬ x)) ((lof R ι (fun i ↦ ↥(𝒜 i)) i) a ⊗ₜ[R] (lof R ι (fun i ↦ ↥(ℬ i)) j) b) =
    (auxEquiv R ℬ 𝒜) ((-1) ^ (j * i) • ↑b ᵍ⊗ₜ[R] ↑a)

Tactic:
gradedComm_of_tmul_of,

Goal state after:
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)
i j : ι
a : ↥(𝒜 i)
b : ↥(ℬ j)
⊢ (-1) ^ (j * i) • (lof R ι (fun x ↦ ↥(ℬ x)) j) b ⊗ₜ[R] (lof R ι (fun x ↦ ↥(𝒜 x)) i) a =
    (auxEquiv R ℬ 𝒜) ((-1) ^ (j * i) • ↑b ᵍ⊗ₜ[R] ↑a)
Module: Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower

Goal state before:
pp: case refine_2
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n✝ n : ℕ
x₀ : M
x : Fin n → M
ih :
  (List.map (fun a ↦ GradedMonoid.mk 1 ((PiTensorProduct.tprod R) fun x_1 ↦ x a)) (List.finRange n)).prod =
    GradedMonoid.mk n ((PiTensorProduct.tprod R) x)
⊢ GradedMonoid.mk 1 ((PiTensorProduct.tprod R) fun x ↦ x₀) *
      (List.map (fun x_1 ↦ GradedMonoid.mk 1 ((PiTensorProduct.tprod R) fun x_2 ↦ x x_1)) (List.finRange n)).prod =
    GradedMonoid.mk (n + 1) ((PiTensorProduct.tprod R) (Fin.cons x₀ x))

Tactic:
ih,

Goal state after:
case refine_2
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n✝ n : ℕ
x₀ : M
x : Fin n → M
ih :
  (List.map (fun a ↦ GradedMonoid.mk 1 ((PiTensorProduct.tprod R) fun x_1 ↦ x a)) (List.finRange n)).prod =
    GradedMonoid.mk n ((PiTensorProduct.tprod R) x)
⊢ GradedMonoid.mk 1 ((PiTensorProduct.tprod R) fun x ↦ x₀) * GradedMonoid.mk n ((PiTensorProduct.tprod R) x) =
    GradedMonoid.mk (n + 1) ((PiTensorProduct.tprod R) (Fin.cons x₀ x))
Module: Mathlib.LinearAlgebra.ExteriorAlgebra.Grading

Goal state before:
pp: case mem_mul
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ℕ
x : ExteriorAlgebra R M
i✝ : ℕ
x✝ : ExteriorAlgebra R M
hx✝ : x✝ ∈ (ι R).range ^ i✝
ih : (liftι R M) x✝ = (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝) ⟨x✝, hx✝⟩
w✝ : M
⊢ (liftι R M) ((ι R) w✝) * (liftι R M) x✝ = (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝.succ) ⟨(ι R) w✝ * x✝, ⋯⟩

Tactic:
ih,

Goal state after:
case mem_mul
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
i : ℕ
x : ExteriorAlgebra R M
i✝ : ℕ
x✝ : ExteriorAlgebra R M
hx✝ : x✝ ∈ (ι R).range ^ i✝
ih : (liftι R M) x✝ = (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝) ⟨x✝, hx✝⟩
w✝ : M
⊢ (liftι R M) ((ι R) w✝) * (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝) ⟨x✝, hx✝⟩ =
    (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝.succ) ⟨(ι R) w✝ * x✝, ⋯⟩
Module: Mathlib.LinearAlgebra.FreeModule.ModN

Goal state before:
pp: G : Type u_1
H : Type u_2
M : Type u_3
inst✝¹ : AddCommGroup G
n : ℕ
inst✝ : AddMonoid M
φ : { φ // ∀ (g : G), n • φ g = 0 }
g : G
⊢ ((LinearMap.lsmul ℤ G) ↑n) g ∈ (↑φ).ker

Tactic:
simpa using φ.property g

Goal state after:
Module: Mathlib.LinearAlgebra.FreeModule.ModN

Goal state before:
pp: case refine_1
G : Type u_1
H✝ : Type u_2
M : Type u_3
inst✝¹ : AddCommGroup G
n : ℕ
inst✝ : NeZero n
ι : Type u_4
b : Basis ι ℤ G
nG : Submodule ℤ G := ((LinearMap.lsmul ℤ G) ↑n).range
H : Type u_1 := G ⧸ nG
φ : G →ₗ[ℤ] H := nG.mkQ
mod : (ι →₀ ℤ) →ₗ[ℤ] ι →₀ ZMod n := mapRange.linearMap (Int.castAddHom (ZMod n)).toIntLinearMap
f : G →ₗ[ℤ] ι →₀ ℤ := ↑b.repr
hker : nG ≤ (mod ∘ₗ f).ker
g : H →ₗ[ℤ] ι →₀ ZMod n := nG.liftQ (mod ∘ₗ f) hker
x : G
hx : ∀ (b : ι), ↑n ∣ (f x) b
⊢ Submodule.Quotient.mk x = 0

Tactic:
Submodule.Quotient.mk_eq_zero

Goal state after:
case refine_1
G : Type u_1
H✝ : Type u_2
M : Type u_3
inst✝¹ : AddCommGroup G
n : ℕ
inst✝ : NeZero n
ι : Type u_4
b : Basis ι ℤ G
nG : Submodule ℤ G := ((LinearMap.lsmul ℤ G) ↑n).range
H : Type u_1 := G ⧸ nG
φ : G →ₗ[ℤ] H := nG.mkQ
mod : (ι →₀ ℤ) →ₗ[ℤ] ι →₀ ZMod n := mapRange.linearMap (Int.castAddHom (ZMod n)).toIntLinearMap
f : G →ₗ[ℤ] ι →₀ ℤ := ↑b.repr
hker : nG ≤ (mod ∘ₗ f).ker
g : H →ₗ[ℤ] ι →₀ ZMod n := nG.liftQ (mod ∘ₗ f) hker
x : G
hx : ∀ (b : ι), ↑n ∣ (f x) b
⊢ x ∈ nG
Module: Mathlib.LinearAlgebra.FreeProduct.Basic

Goal state before:
pp: I : Type u
inst✝³ : DecidableEq I
R : Type v
inst✝² : CommSemiring R
A : I → Type w
inst✝¹ : (i : I) → Semiring (A i)
inst✝ : (i : I) → Algebra R (A i)
x : ⨁ (i : I), A i
⊢ ↑((ι R) x) = (ι' R A) x

Tactic:
by
  aesop (add simp [ι', mkAlgHom])

Goal state after:
Module: Mathlib.LinearAlgebra.FreeProduct.Basic

Goal state before:
pp: I : Type u
inst✝⁵ : DecidableEq I
i : I
R : Type v
inst✝⁴ : CommSemiring R
A : I → Type w
inst✝³ : (i : I) → Semiring (A i)
inst✝² : (i : I) → Algebra R (A i)
B : Type w'
inst✝¹ : Semiring B
inst✝ : Algebra R B
maps : {i : I} → A i →ₐ[R] B
⊢ ((lift R A) fun {i} ↦ maps) ∘ₐ ι R A i = maps

Tactic:
by
  ext a
  simp [lift_apply, ι, ← ι_apply]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.AbsoluteValue

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Nontrivial R
inst✝⁴ : CommRing S
inst✝³ : LinearOrder S
inst✝² : IsStrictOrderedRing S
n : Type u_3
inst✝¹ : Fintype n
inst✝ : DecidableEq n
ι : Type u_4
s : Finset ι
A : ι → Matrix n n R
abv : AbsoluteValue R S
x : S
hx : ∀ (k : ι) (i j : n), abv (A k i j) ≤ x
i j : n
⊢ abv ((∑ k ∈ s, A k) i j) = abv (∑ k ∈ s, A k i j)

Tactic:
by simp only [sum_apply]

Goal state after: