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

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
hm : m ≤ m𝓧
inst✝ : IsFiniteMeasure μ
hsf : SigmaFinite (ν.trim hm)
hμν : μ ≪ ν
this : SigmaFinite (Measure.map id ν)
h :
  (fun a ↦ ((Measure.map id μ).rnDeriv (Measure.map id ν) a).toReal) =ᵐ[Measure.map id ν]
    ν[fun a ↦ (μ.rnDeriv ν a).toReal | m]
⊢ (fun x ↦ ((Measure.map id μ).rnDeriv (Measure.map id ν) x).toReal) =ᵐ[Measure.map id ν]
    ν[fun x ↦ (μ.rnDeriv ν x).toReal | m]

Tactic:
convert! h

Goal state after:
case e'_2
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
hm : m ≤ m𝓧
inst✝ : IsFiniteMeasure μ
hsf : SigmaFinite (ν.trim hm)
hμν : μ ≪ ν
this : SigmaFinite (Measure.map id ν)
h :
  (fun a ↦ ((Measure.map id μ).rnDeriv (Measure.map id ν) a).toReal) =ᵐ[Measure.map id ν]
    ν[fun a ↦ (μ.rnDeriv ν a).toReal | m]
⊢ m = MeasurableSpace.comap id m

case e'_3
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
hm : m ≤ m𝓧
inst✝ : IsFiniteMeasure μ
hsf : SigmaFinite (ν.trim hm)
hμν : μ ≪ ν
this : SigmaFinite (Measure.map id ν)
h :
  (fun a ↦ ((Measure.map id μ).rnDeriv (Measure.map id ν) a).toReal) =ᵐ[Measure.map id ν]
    ν[fun a ↦ (μ.rnDeriv ν a).toReal | m]
e_2✝ : Measure 𝓧 = Measure 𝓧
⊢ m = MeasurableSpace.comap id m

case e'_5
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
hm : m ≤ m𝓧
inst✝ : IsFiniteMeasure μ
hsf : SigmaFinite (ν.trim hm)
hμν : μ ≪ ν
this : SigmaFinite (Measure.map id ν)
h :
  (fun a ↦ ((Measure.map id μ).rnDeriv (Measure.map id ν) a).toReal) =ᵐ[Measure.map id ν]
    ν[fun a ↦ (μ.rnDeriv ν a).toReal | m]
e_2✝ : Measure 𝓧 = Measure 𝓧
⊢ m = MeasurableSpace.comap id m
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: case refine_2
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
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
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ (fun x ↦ c * ((∂Measure.map g μ/∂Measure.map g ν) (g x)).toReal + c') ≤ᵐ[ν] fun x ↦
    f ((∂Measure.map g μ/∂Measure.map g ν) (g x)).toReal

pp: case refine_3
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
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
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ (fun x ↦ f ((∂Measure.map g μ/∂Measure.map g ν) (g x)).toReal) ≤ᵐ[ν] fun x ↦
    ν[fun x ↦ f ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨] x

pp: case refine_4
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
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
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ Integrable (fun x ↦ c * ((∂Measure.map g μ/∂Measure.map g ν) (g x)).toReal + c') ν

Tactic:
· exact ae_of_all _ (fun x ↦ h _ ENNReal.toReal_nonneg)

Goal state after:
case refine_3
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
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
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ (fun x ↦ f ((∂Measure.map g μ/∂Measure.map g ν) (g x)).toReal) ≤ᵐ[ν] fun x ↦
    ν[fun x ↦ f ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨] x

case refine_4
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
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
hf_cont : ContinuousOn f (Ici 0)
c c' : ℝ
h : ∀ (x : ℝ), 0 ≤ x → c * x + c' ≤ f x
⊢ Integrable (fun x ↦ c * ((∂Measure.map g μ/∂Measure.map g ν) (g x)).toReal + c') ν
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝ : Countable ι
X : ι → Ω → ℝ≥0∞
hX : ∀ (i : ι), AEMeasurable (X i) P
⊢ P⁻[∑' (i : ι), X i | mΩ] =ᵐ[P] ∑' (i : ι), P⁻[X i | mΩ]

Tactic:
by
  by_cases hm : mΩ ≤ mΩ₀; swap
  · simp_rw [condLExp_of_not_le hm]; filter_upwards; simp
  by_cases hσ : SigmaFinite (P.trim hm); swap
  · simp_rw [condLExp_of_not_sigmaFinite hm hσ]; filter_upwards; simp
  refine (ae_eq_condLExp _ _ _ (by fun_prop) ?_).symm
  intro s hs
  simp only [ENNReal.tsum_apply]
  repeat rw [lintegral_tsum (by measurability)]
  congr with i
  exact setLIntegral_condLExp hm P (X i) hs

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
mΩ : MeasurableSpace Ω
s : Finset ι
X : ι → Ω → ℝ≥0∞
hX : ∀ (i : ι), AEMeasurable (X i) P
⊢ P⁻[∑ i ∈ s, X i | mΩ] =ᵐ[P] ∑ i ∈ s, P⁻[X i | mΩ]

Tactic:
convert! condLExp_tsum mΩ (fun i : s ↦ hX i)

Goal state after:
case e'_4
Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
mΩ : MeasurableSpace Ω
s : Finset ι
X : ι → Ω → ℝ≥0∞
hX : ∀ (i : ι), AEMeasurable (X i) P
⊢ ∑ i ∈ s, X i = ∑' (i : ↥s), X ↑i

case e'_5
Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
mΩ : MeasurableSpace Ω
s : Finset ι
X : ι → Ω → ℝ≥0∞
hX : ∀ (i : ι), AEMeasurable (X i) P
⊢ ∑ i ∈ s, P⁻[X i | mΩ] = ∑' (i : ↥s), P⁻[X ↑i | mΩ]
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
α : Type u_2
m mα : MeasurableSpace α
μ : Measure α
f : α → E
⊢ (fun x ↦ ‖μ[f | m] x‖) ≤ᵐ[μ] μ[fun x ↦ ‖f x‖ | m]

Tactic:
by
  by_cases! hm : ¬ m ≤ mα
  · simp [condExp_of_not_le hm]; aesop
  by_cases! hμm : ¬ SigmaFinite (μ.trim hm)
  · simp [condExp_of_not_sigmaFinite hm hμm]; aesop
  by_cases! hf_int : ¬ Integrable f μ
  · simp only [condExp_of_not_integrable hf_int, Pi.zero_apply, norm_zero]
    apply condExp_nonneg
    filter_upwards with a; positivity
  exact convexOn_univ_norm.map_condExp_le_univ hm continuous_norm.lowerSemicontinuous hf_int
    hf_int.norm

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

Goal state before:
pp: case intro
ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁶ : AddCommGroup V
inst✝⁵ : AffineSpace V P
inst✝⁴ : Ring k
inst✝³ : Module k V
b : AffineBasis ι k P
ι' : Type u_1
inst✝² : Fintype ι
inst✝¹ : Finite ι'
inst✝ : DecidableEq ι'
p : ι' → P
A : Matrix ι ι' k
hA : b.toMatrix p * A = 1
val✝ : Fintype ι'
w₁ : ι' → k
⊢ ∀ (w2 : ι' → k),
    ∑ i, w₁ i = 1 →
      ∑ i, w2 i = 1 →
        (Finset.affineCombination k Finset.univ p) w₁ = (Finset.affineCombination k Finset.univ p) w2 → w₁ = w2

Tactic:
w₂

Goal state after:
case intro
ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁶ : AddCommGroup V
inst✝⁵ : AffineSpace V P
inst✝⁴ : Ring k
inst✝³ : Module k V
b : AffineBasis ι k P
ι' : Type u_1
inst✝² : Fintype ι
inst✝¹ : Finite ι'
inst✝ : DecidableEq ι'
p : ι' → P
A : Matrix ι ι' k
hA : b.toMatrix p * A = 1
val✝ : Fintype ι'
w₁ w₂ : ι' → k
⊢ ∑ i, w₁ i = 1 →
    ∑ i, w₂ i = 1 →
      (Finset.affineCombination k Finset.univ p) w₁ = (Finset.affineCombination k Finset.univ p) w₂ → w₁ = w₂
Module: Mathlib.LinearAlgebra.BilinearForm.IsometryEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
M₁ : Type u_3
M₂ : Type u_4
M₃ : Type u_5
M₄ : Type u_6
N : Type u_7
inst✝¹² : CommSemiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : AddCommMonoid M₁
inst✝⁹ : AddCommMonoid M₂
inst✝⁸ : AddCommMonoid M₃
inst✝⁷ : AddCommMonoid M₄
inst✝⁶ : AddCommMonoid N
inst✝⁵ : Module R M
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Module R M₃
inst✝¹ : Module R M₄
inst✝ : Module R N
B₁ : LinearMap.BilinForm R M₁
B₂ : LinearMap.BilinForm R M₂
B₃ : LinearMap.BilinForm R M₃
f : B₁.IsometryEquiv B₂
n✝ m✝ : M₂
⊢ (B₁ ((↑f.symm).toFun n✝)) ((↑f.symm).toFun m✝) = (B₂ n✝) m✝

Tactic:
← f.map_app

Goal state after:
R : Type u_1
M : Type u_2
M₁ : Type u_3
M₂ : Type u_4
M₃ : Type u_5
M₄ : Type u_6
N : Type u_7
inst✝¹² : CommSemiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : AddCommMonoid M₁
inst✝⁹ : AddCommMonoid M₂
inst✝⁸ : AddCommMonoid M₃
inst✝⁷ : AddCommMonoid M₄
inst✝⁶ : AddCommMonoid N
inst✝⁵ : Module R M
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Module R M₃
inst✝¹ : Module R M₄
inst✝ : Module R N
B₁ : LinearMap.BilinForm R M₁
B₂ : LinearMap.BilinForm R M₂
B₃ : LinearMap.BilinForm R M₃
f : B₁.IsometryEquiv B₂
n✝ m✝ : M₂
⊢ (B₂ (f ((↑f.symm).toFun n✝))) (f ((↑f.symm).toFun m✝)) = (B₂ n✝) m✝
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁷ : Ring k
inst✝⁶ : PartialOrder k
inst✝⁵ : IsOrderedAddMonoid k
inst✝⁴ : AddCommGroup V
inst✝³ : AffineSpace V P
inst✝² : Module k V
n : ℕ
inst✝¹ : NeZero n
s : Simplex k P n
i : Fin (n + 1)
inst✝ : ZeroLEOneClass k
w : Fin (n + 1) → k
hw : ∑ i, w i = 1
x✝ :
  (affineCombination k univ s.points) w ∈ s.closedInterior ∩ ↑((affineSpan k (s.points '' {i}ᶜ)).shift (s.points i) 0)
hp : ∀ (i : Fin (n + 1)), w i ∈ Set.Icc 0 1
hshift : w i = 1 - 0
⊢ w = Pi.single i 1

Tactic:
sub_zero

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁷ : Ring k
inst✝⁶ : PartialOrder k
inst✝⁵ : IsOrderedAddMonoid k
inst✝⁴ : AddCommGroup V
inst✝³ : AffineSpace V P
inst✝² : Module k V
n : ℕ
inst✝¹ : NeZero n
s : Simplex k P n
i : Fin (n + 1)
inst✝ : ZeroLEOneClass k
w : Fin (n + 1) → k
hw : ∑ i, w i = 1
x✝ :
  (affineCombination k univ s.points) w ∈ s.closedInterior ∩ ↑((affineSpan k (s.points '' {i}ᶜ)).shift (s.points i) 0)
hp : ∀ (i : Fin (n + 1)), w i ∈ Set.Icc 0 1
hshift : w i = 1
⊢ w = Pi.single i 1
Module: Mathlib.RingTheory.TensorProduct.IsBaseChangeHom

Goal state before:
pp: case tmul
R : Type u_1
inst✝¹⁰ : CommSemiring R
S : Type u_2
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra R S
M : Type u_3
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
N : Type u_4
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
P : Type u_5
inst✝³ : AddCommMonoid P
inst✝² : Module R P
inst✝¹ : Module S P
inst✝ : IsScalarTower R S P
ε : N →ₗ[R] P
s t : S
f : M →ₗ[R] N
⊢ (TensorProduct.lift { toFun := fun s ↦ s • compRight R ε, map_add' := ⋯, map_smul' := ⋯ }) (s • t ⊗ₜ[R] f) =
    s • (TensorProduct.lift { toFun := fun s ↦ s • compRight R ε, map_add' := ⋯, map_smul' := ⋯ }) (t ⊗ₜ[R] f)

Tactic:
simp [TensorProduct.smul_tmul', mul_smul]

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

Goal state before:
pp: case tmul
R : Type u_1
inst✝⁸ : CommSemiring R
V : Type u_2
inst✝⁷ : AddCommMonoid V
inst✝⁶ : Module R V
W : Type u_3
inst✝⁵ : AddCommMonoid W
inst✝⁴ : Module R W
A : Type u_4
inst✝³ : CommSemiring A
inst✝² : Algebra R A
inst✝¹ : Module A W
inst✝ : IsScalarTower R A W
j : V →ₗ[R] W
ibc : IsBaseChange A j
a b : A
f : Dual R V
⊢ (TensorProduct.lift { toFun := fun a ↦ a • ibc.toDual, map_add' := ⋯, map_smul' := ⋯ }).toFun (a • b ⊗ₜ[R] f) =
    (RingHom.id A) a •
      (TensorProduct.lift { toFun := fun a ↦ a • ibc.toDual, map_add' := ⋯, map_smul' := ⋯ }).toFun (b ⊗ₜ[R] f)

Tactic:
simp [TensorProduct.smul_tmul', mul_smul]

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

Goal state before:
pp: R : Type u_1
inst✝¹⁷ : CommSemiring R
S : Type u_2
inst✝¹⁶ : CommSemiring S
inst✝¹⁵ : Algebra R S
M : Type u_3
inst✝¹⁴ : AddCommMonoid M
inst✝¹³ : Module R M
N : Type u_4
inst✝¹² : AddCommMonoid N
inst✝¹¹ : Module R N
P : Type u_5
inst✝¹⁰ : AddCommMonoid P
inst✝⁹ : Module R P
Q : Type u_6
inst✝⁸ : AddCommMonoid Q
inst✝⁷ : Module R Q
inst✝⁶ : Module S P
inst✝⁵ : IsScalarTower R S P
inst✝⁴ : Module S Q
inst✝³ : IsScalarTower R S Q
α : M →ₗ[R] P
β : N →ₗ[R] Q
ibcM : IsBaseChange S α
ibcN : IsBaseChange S β
ι : Type u_7
θ : Type u_8
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : Finite θ
b : Basis ι R M
c : Basis θ R N
f : M →ₗ[R] N
⊢ (toMatrix (basis b ibcM) (basis c ibcN)) ((ibcM.linearMapLeftRightHom β) f) = ((toMatrix b c) f).map ⇑(algebraMap R S)

Tactic:
by
  ext i j
  simp only [toMatrix_apply, Matrix.map_apply, basis_apply,
    linearMapLeftRightHom_comp_apply, basis_repr_comp_apply]

Goal state after:
Module: Mathlib.LinearAlgebra.FixedSubmodule

Goal state before:
pp: case a
R : Type u_1
inst✝² : Semiring R
V : Type u_3
inst✝¹ : AddCommMonoid V
inst✝ : Module R V
e : V ≃ₗ[R] V
W : Submodule R V
hW : W ≤ (↑e).fixedSubmodule
x : V
hx : x ∈ W
⊢ x ∈ (↑e).fixedSubmodule

Tactic:
exact hW hx

Goal state after:
Module: Mathlib.LinearAlgebra.Center

Goal state before:
pp: case inr
R : Type u_1
V : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
ι : Type u_3
inst✝ : Nontrivial ι
b : Basis ι R V
f : V →ₗ[R] V
hV : Nontrivial V
h_allEq : ∀ (i j : ι), (b.coord i) (f (b i)) = (b.coord j) (f (b j))
hcomm : ∀ (i : ι) (r : R), r • f (b i) = (b.coord i) (f (b i)) • r • b i
i : ι := Classical.ofNonempty
x : V
⊢ f ((linearCombination R ⇑b) (b.repr x)) = (⟨(b.coord i) (f (b i)), ⋯⟩ • 1) ((linearCombination R ⇑b) (b.repr x))

Tactic:
linearCombination_apply,

Goal state after:
case inr
R : Type u_1
V : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
ι : Type u_3
inst✝ : Nontrivial ι
b : Basis ι R V
f : V →ₗ[R] V
hV : Nontrivial V
h_allEq : ∀ (i j : ι), (b.coord i) (f (b i)) = (b.coord j) (f (b j))
hcomm : ∀ (i : ι) (r : R), r • f (b i) = (b.coord i) (f (b i)) • r • b i
i : ι := Classical.ofNonempty
x : V
⊢ f ((b.repr x).sum fun i a ↦ a • b i) = (⟨(b.coord i) (f (b i)), ⋯⟩ • 1) ((b.repr x).sum fun i a ↦ a • b i)
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup V
inst✝ : Module R V
f : Dual R V
v : V
h : f v = 0
x : V
⊢ (LinearMap.transvection f (-v)) ((LinearMap.transvection f v) x) = x

Tactic:
simp [comp_of_left_eq_apply h]

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

Goal state before:
pp: V : Type u_2
inst✝² : AddCommGroup V
K : Type u_3
inst✝¹ : DivisionRing K
inst✝ : Module K V
e : V ≃ₗ[K] V
u : V →ₗ[K] V := ↑e - LinearMap.id
hu : u + LinearMap.id = ↑e
hr : 1 ≤ Module.rank K ↥u.range
b : Basis Unit K ↥u.range
x : V
this : ↑(u.rangeRestrict x) = u x
⊢ ↑(u.rangeRestrict x) = (b.repr (u.rangeRestrict x)) default • ↑(b default)

Tactic:
← Submodule.coe_smul,

Goal state after:
V : Type u_2
inst✝² : AddCommGroup V
K : Type u_3
inst✝¹ : DivisionRing K
inst✝ : Module K V
e : V ≃ₗ[K] V
u : V →ₗ[K] V := ↑e - LinearMap.id
hu : u + LinearMap.id = ↑e
hr : 1 ≤ Module.rank K ↥u.range
b : Basis Unit K ↥u.range
x : V
this : ↑(u.rangeRestrict x) = u x
⊢ ↑(u.rangeRestrict x) = ↑((b.repr (u.rangeRestrict x)) default • b default)
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: case tmul
R : Type u_3
V : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid V
inst✝⁶ : Module R V
A : Type u_5
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
W : Type u_6
inst✝³ : AddCommMonoid W
inst✝² : Module R W
inst✝¹ : Module A W
inst✝ : IsScalarTower R A W
ε : V →ₗ[R] W
ibc : IsBaseChange A ε
f : Dual R V
v x : V
⊢ (ibc.endHom (LinearMap.transvection f v)) (ε x) = (LinearMap.transvection (ibc.toDual f) (ε v)) (ε x)

Tactic:
| tmul x =>

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 : Module.Dual R M
x : CliffordAlgebra Q
⊢ (contractLeft d) ((contractLeft d) x) = 0

Tactic:
induction x using CliffordAlgebra.left_induction with
  | algebraMap => simp_rw [contractLeft_algebraMap, map_zero]
  | add _ _ hx hy => rw [map_add, map_add, hx, hy, add_zero]
  | ι_mul _ _ hx =>
    rw [contractLeft_ι_mul, map_sub, contractLeft_ι_mul, hx, map_smul, mul_zero, sub_zero, sub_self]

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

Goal state before:
pp: case neg.hy
K : Type u_3
V : Type u_4
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : Dual K V
v : V
hfv : f v = 0
hv : ¬v = 0
hf : ¬f = 0
ι : Set V
b : Basis (↑ι) K V
i j : ↑ι
hi : i ≠ j
hj : v = b i ∧ f = b.coord j
this : Fintype ↑ι
x y : ↑ι
⊢ (Finsupp.single i ((Finsupp.single y 1) j)) x = Matrix.single i j 1 x y

Tactic:
by_cases h : i = x ∧ j = y
      · rw [h.1, h.2]; simp
      · rcases not_and_or.mp h with h' | h' <;>
          simp [Finsupp.single_eq_of_ne' h',
            Finsupp.single_eq_of_ne h',
            Matrix.single_apply_of_ne (h := h)]

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 Q' : QuadraticForm R M
B : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
⊢ (changeForm h) 1 = 1

Tactic:
simpa using changeForm_algebraMap h (1 : R)

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 Q' : QuadraticForm R M
B : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
m : M
⊢ (ι Q') m = (ι Q') m - 0

Tactic:
sub_zero

Goal state after:
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' : QuadraticForm R M
B : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
m : M
⊢ (ι Q') m = (ι Q') m
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Contraction

Goal state before:
pp: case ι_mul
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x✝ : CliffordAlgebra Q
m✝ : M
hx : (changeForm ⋯) x✝ = x✝
⊢ (ι Q) m✝ * x✝ - 0 = (ι Q) m✝ * x✝

Tactic:
sub_zero

Goal state after:
case ι_mul
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x✝ : CliffordAlgebra Q
m✝ : M
hx : (changeForm ⋯) x✝ = x✝
⊢ (ι Q) m✝ * x✝ = (ι Q) m✝ * x✝
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
⊢ (gradedMul R 𝒜 ℬ).compr₂ ↑(gradedComm R 𝒜 ℬ) = (gradedMul R ℬ 𝒜 ∘ₗ ↑(gradedComm R 𝒜 ℬ)).compl₂ ↑(gradedComm R 𝒜 ℬ)

Tactic:
ext i₁ a₁ j₁ b₁ i₂ a₂ j₂ b₂

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
⊢ ((AlgebraTensorModule.curry
              (((AlgebraTensorModule.curry ((gradedMul R 𝒜 ℬ).compr₂ ↑(gradedComm R 𝒜 ℬ)) ∘ₗ lof R ι 𝒜 i₁) a₁ ∘ₗ
                  lof R ι ℬ j₁)
                b₁) ∘ₗ
            lof R ι 𝒜 i₂)
          a₂ ∘ₗ
        lof R ι ℬ j₂)
      b₂ =
    ((AlgebraTensorModule.curry
              (((AlgebraTensorModule.curry ((gradedMul R ℬ 𝒜 ∘ₗ ↑(gradedComm R 𝒜 ℬ)).compl₂ ↑(gradedComm R 𝒜 ℬ)) ∘ₗ
                      lof R ι 𝒜 i₁)
                    a₁ ∘ₗ
                  lof R ι ℬ j₁)
                b₁) ∘ₗ
            lof R ι 𝒜 i₂)
          a₂ ∘ₗ
        lof R ι ℬ j₂)
      b₂
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Prod

Goal state before:
pp: case mem.mem_mul.e_a.e_a.mem.mem_mul
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 i : ℕ
x₂ : CliffordAlgebra Q₂
_hx₂ : x₂ ∈ (ι Q₂).range ^ i
ih₂ : (ι Qₙ) (f₁ v₁) * (map f₂) x₂ = (-1) ^ ↑i • ((map f₂) x₂ * (ι Qₙ) (f₁ v₁))
v₂ : M₂
⊢ -(-1) ^ ↑i • ((ι Qₙ) (f₂ v₂) * (map f₂) x₂ * (ι Qₙ) (f₁ v₁)) =
    ((-1) ^ ↑i * (-1) ^ 1) • ((ι Qₙ) (f₂ v₂) * (map f₂) x₂ * (ι Qₙ) (f₁ v₁))

Tactic:
uzpow_one,

Goal state after:
case mem.mem_mul.e_a.e_a.mem.mem_mul
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 i : ℕ
x₂ : CliffordAlgebra Q₂
_hx₂ : x₂ ∈ (ι Q₂).range ^ i
ih₂ : (ι Qₙ) (f₁ v₁) * (map f₂) x₂ = (-1) ^ ↑i • ((map f₂) x₂ * (ι Qₙ) (f₁ v₁))
v₂ : M₂
⊢ -(-1) ^ ↑i • ((ι Qₙ) (f₂ v₂) * (map f₂) x₂ * (ι Qₙ) (f₁ v₁)) =
    ((-1) ^ ↑i * -1) • ((ι Qₙ) (f₂ v₂) * (map f₂) x₂ * (ι Qₙ) (f₁ v₁))
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Prod

Goal state before:
pp: 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
m : M₁ × M₂
⊢ ((ι Q₁) m.1 * (ι Q₁) m.1) ᵍ⊗ₜ[R] 1 + (ι Q₁) m.1 ᵍ⊗ₜ[R] (ι Q₂) m.2 +
      ((-1) ^ 1 • (ι Q₁) m.1 ᵍ⊗ₜ[R] ((ι Q₂) m.2 * 1) + 1 ᵍ⊗ₜ[R] ((ι Q₂) m.2 * (ι Q₂) m.2)) =
    (algebraMap R (CliffordAlgebra Q₁)) (Q₁ m.1) ᵍ⊗ₜ[R] 1 + (algebraMap R (CliffordAlgebra Q₁)) (Q₂ m.2) ᵍ⊗ₜ[R] 1

Tactic:
uzpow_one,

Goal state after:
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
m : M₁ × M₂
⊢ ((ι Q₁) m.1 * (ι Q₁) m.1) ᵍ⊗ₜ[R] 1 + (ι Q₁) m.1 ᵍ⊗ₜ[R] (ι Q₂) m.2 +
      (-1 • (ι Q₁) m.1 ᵍ⊗ₜ[R] ((ι Q₂) m.2 * 1) + 1 ᵍ⊗ₜ[R] ((ι Q₂) m.2 * (ι Q₂) m.2)) =
    (algebraMap R (CliffordAlgebra Q₁)) (Q₁ m.1) ᵍ⊗ₜ[R] 1 + (algebraMap R (CliffordAlgebra Q₁)) (Q₂ m.2) ᵍ⊗ₜ[R] 1
Module: Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n m : ℕ
h : n = m
x : ⨂[R]^n M
⊢ GradedMonoid.mk m ((PiTensorProduct.reindex R (fun x ↦ M) (Equiv.cast ⋯)) x) = GradedMonoid.mk n x

Tactic:
mk_reindex_cast h

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n m : ℕ
h : n = m
x : ⨂[R]^n M
⊢ GradedMonoid.mk n x = GradedMonoid.mk n x