mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case e_f.e_a.hs
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n ↦ truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n ↦ ∑ i ∈ range n, Y i
hS : S = fun n ↦ ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n ↦ ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
N i : ℕ
⊢ ∀ i_1 ∈ range (u i), MemLp (Y i_1) 2 ℙ

Tactic:
intro j

Goal state after:
case e_f.e_a.hs
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n ↦ truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n ↦ ∑ i ∈ range n, Y i
hS : S = fun n ↦ ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n ↦ ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
N i j : ℕ
⊢ j ∈ range (u i) → MemLp (Y j) 2 ℙ
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : Semiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : V →ₗ[A] W
x✝ : G
⊢ ∀ (x : V × W), (LinearMap.fst A V W ∘ₗ (ρ.prod σ) x✝) x = (ρ x✝ ∘ₗ LinearMap.fst A V W) x

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : Semiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : V →ₗ[A] W
x✝ : G
⊢ ∀ (x : V × W), (LinearMap.snd A V W ∘ₗ (ρ.prod σ) x✝) x = (σ x✝ ∘ₗ LinearMap.snd A V W) x

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : Semiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : V →ₗ[A] W
φ : ρ.Equiv σ
g : G
⊢ ↑(LinearEquiv.refl A V) ∘ₗ ρ g = ρ g ∘ₗ ↑(LinearEquiv.refl A V)

Tactic:
simp

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case refine_2
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n ↦ truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n ↦ ∑ i ∈ range n, Y i
hS : S = fun n ↦ ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n ↦ ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
N : ℕ
⊢ ∀ a ∈ (range (u (N - 1))).sigma fun a ↦ {i ∈ range N | a < u i}, ⟨a.snd, a.fst⟩ ∈ (range N).sigma fun a ↦ range (u a)

Tactic:
simp

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case refine_3
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n ↦ truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n ↦ ∑ i ∈ range n, Y i
hS : S = fun n ↦ ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n ↦ ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
N : ℕ
⊢ ∀ a ∈ (range N).sigma fun a ↦ range (u a), ⟨⟨a.snd, a.fst⟩.snd, ⟨a.snd, a.fst⟩.fst⟩ = a

Tactic:
simp

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case refine_4
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n ↦ truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n ↦ ∑ i ∈ range n, Y i
hS : S = fun n ↦ ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n ↦ ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
N : ℕ
⊢ ∀ a ∈ (range (u (N - 1))).sigma fun a ↦ {i ∈ range N | a < u i}, ⟨⟨a.snd, a.fst⟩.snd, ⟨a.snd, a.fst⟩.fst⟩ = a

Tactic:
simp

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case refine_5
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n ↦ truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n ↦ ∑ i ∈ range n, Y i
hS : S = fun n ↦ ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n ↦ ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
N : ℕ
⊢ ∀ a ∈ (range N).sigma fun a ↦ range (u a),
    (↑(u a.fst) ^ 2)⁻¹ * Var[Y a.snd; ℙ] = (↑(u ⟨a.snd, a.fst⟩.snd) ^ 2)⁻¹ * Var[Y ⟨a.snd, a.fst⟩.fst; ℙ]

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : Monoid G
inst✝³ : AddCommMonoid V
inst✝² : AddCommMonoid W
inst✝¹ : Module A V
inst✝ : Module A W
ρ : Representation A G V
σ : Representation A G W
φ : ρ.Equiv σ
x✝ : V
⊢ (φ.trans φ.symm) x✝ = (refl ρ) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : Monoid G
inst✝³ : AddCommMonoid V
inst✝² : AddCommMonoid W
inst✝¹ : Module A V
inst✝ : Module A W
ρ : Representation A G V
σ : Representation A G W
φ : ρ.Equiv σ
x✝ : W
⊢ (φ.symm.trans φ) x✝ = (refl σ) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : V →ₗ[A] W
g₁ g₂ : ρ.IntertwiningMap σ
x✝ : ρ.asModule
⊢ { toFun := ⇑(g₁ + g₂).toLinearMap, map_add' := ⋯, map_smul' := ⋯ } x✝ =
    ({ toFun := ⇑g₁.toLinearMap, map_add' := ⋯, map_smul' := ⋯ } +
        { toFun := ⇑g₂.toLinearMap, map_add' := ⋯, map_smul' := ⋯ })
      x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : V →ₗ[A] W
t : A
g : ρ.IntertwiningMap σ
x✝ : ρ.asModule
⊢ { toFun := ⇑(t • g).toLinearMap, map_add' := ⋯, map_smul' := ⋯ } x✝ =
    ((RingHom.id A) t • { toFun := ⇑g.toLinearMap, map_add' := ⋯, map_smul' := ⋯ }) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f✝ : V →ₗ[A] W
f : ρ.asModule →ₗ[A[G]] σ.asModule
a : A
v : V
⊢ f.toFun (a • v) = (RingHom.id A) a • f.toFun v

Tactic:
simp

Goal state after:
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: case zero
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : StandardBorelSpace Ω
Y : ℕ → Ω → ℝ
cY : ℕ → ℝ≥0
ℱ : Filtration ℕ mΩ
inst✝ : IsZeroOrProbabilityMeasure μ
h_adapted : StronglyAdapted ℱ Y
h0 : HasSubgaussianMGF (Y 0) (cY 0) μ
h_subG : ∀ i < 0 - 1, HasCondSubgaussianMGF (↑ℱ i) ⋯ (Y (i + 1)) (cY (i + 1)) μ
⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ Finset.range 0, Y i ω) (∑ i ∈ Finset.range 0, cY i) μ

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f✝ : V →ₗ[A] W
a : A
f g : ρ.IntertwiningMap ρ
x✝ : V
⊢ (f * a • g).toLinearMap x✝ = (a • (f * g)).toLinearMap x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : V →ₗ[A] W
x✝ : V
⊢ (centralAlgebraMul ρ ⋯).toLinearMap x✝ = (toLinearMap 1) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : V →ₗ[A] W
x✝² x✝¹ : ↥(Submonoid.center A[G])
x✝ : V
⊢ (centralAlgebraMul ρ ⋯).toLinearMap x✝ = (centralAlgebraMul ρ ⋯ * centralAlgebraMul ρ ⋯).toLinearMap x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁹ : CommSemiring A
inst✝⁸ : Monoid G
inst✝⁷ : AddCommMonoid V
inst✝⁶ : AddCommMonoid W
inst✝⁵ : AddCommMonoid U
inst✝⁴ : Module A V
inst✝³ : Module A W
inst✝² : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
P : Type u_6
inst✝¹ : AddCommMonoid P
inst✝ : Module A P
π : Representation A G P
f : ρ.IntertwiningMap σ
a : A
g : τ.IntertwiningMap π
⊢ f.tensor (a • g) = a • f.tensor g

Tactic:
by ext; simp [TensorProduct.tmul_smul]

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁵ : CommSemiring A
inst✝⁴ : Monoid G
inst✝³ : AddCommMonoid V
inst✝² : AddCommMonoid W
inst✝¹ : Module A V
inst✝ : Module A W
ρ : Representation A G V
σ : Representation A G W
x✝¹ : V
x✝ : W
⊢ ((TensorProduct.AlgebraTensorModule.curry (lTensor ρ (id σ)).toLinearMap) x✝¹) x✝ =
    ((TensorProduct.AlgebraTensorModule.curry (id (ρ.tprod σ)).toLinearMap) x✝¹) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
x✝¹ : V
x✝ : W
⊢ ((TensorProduct.AlgebraTensorModule.curry (lTensor ρ 0).toLinearMap) x✝¹) x✝ =
    ((TensorProduct.AlgebraTensorModule.curry (toLinearMap 0)) x✝¹) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁵ : CommSemiring A
inst✝⁴ : Monoid G
inst✝³ : AddCommMonoid V
inst✝² : AddCommMonoid W
inst✝¹ : Module A V
inst✝ : Module A W
ρ : Representation A G V
σ : Representation A G W
x✝¹ : W
x✝ : V
⊢ ((TensorProduct.AlgebraTensorModule.curry (rTensor ρ (id σ)).toLinearMap) x✝¹) x✝ =
    ((TensorProduct.AlgebraTensorModule.curry (id (σ.tprod ρ)).toLinearMap) x✝¹) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
x✝¹ : W
x✝ : V
⊢ ((TensorProduct.AlgebraTensorModule.curry (rTensor ρ 0).toLinearMap) x✝¹) x✝ =
    ((TensorProduct.AlgebraTensorModule.curry (toLinearMap 0)) x✝¹) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁹ : CommSemiring A
inst✝⁸ : Monoid G
inst✝⁷ : AddCommMonoid V
inst✝⁶ : AddCommMonoid W
inst✝⁵ : AddCommMonoid U
inst✝⁴ : Module A V
inst✝³ : Module A W
inst✝² : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
Q : Type u_7
inst✝¹ : AddCommMonoid Q
inst✝ : Module A Q
υ : Representation A G Q
f : ρ.IntertwiningMap τ
g : σ.IntertwiningMap υ
x✝¹ : V
x✝ : W
⊢ ((TensorProduct.AlgebraTensorModule.curry ((rTensor υ f).comp (lTensor ρ g)).toLinearMap) x✝¹) x✝ =
    ((TensorProduct.AlgebraTensorModule.curry (f.tensor g).toLinearMap) x✝¹) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁹ : CommSemiring A
inst✝⁸ : Monoid G
inst✝⁷ : AddCommMonoid V
inst✝⁶ : AddCommMonoid W
inst✝⁵ : AddCommMonoid U
inst✝⁴ : Module A V
inst✝³ : Module A W
inst✝² : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
Q : Type u_7
inst✝¹ : AddCommMonoid Q
inst✝ : Module A Q
υ : Representation A G Q
f : ρ.IntertwiningMap τ
g : σ.IntertwiningMap υ
x✝¹ : V
x✝ : W
⊢ ((TensorProduct.AlgebraTensorModule.curry ((lTensor τ g).comp (rTensor σ f)).toLinearMap) x✝¹) x✝ =
    ((TensorProduct.AlgebraTensorModule.curry (f.tensor g).toLinearMap) x✝¹) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : V →ₗ[A] W
g : G
x✝¹ : V
x✝ : W
⊢ ((TensorProduct.AlgebraTensorModule.curry (↑(_root_.TensorProduct.comm A V W) ∘ₗ (ρ.tprod σ) g)) x✝¹) x✝ =
    ((TensorProduct.AlgebraTensorModule.curry ((σ.tprod ρ) g ∘ₗ ↑(_root_.TensorProduct.comm A V W))) x✝¹) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : σ.IntertwiningMap τ
x✝¹ : V
x✝ : W
⊢ ((TensorProduct.AlgebraTensorModule.curry ((↑(comm ρ τ)).comp (IntertwiningMap.lTensor ρ f)).toLinearMap) x✝¹) x✝ =
    ((TensorProduct.AlgebraTensorModule.curry ((IntertwiningMap.rTensor ρ f).comp ↑(comm ρ σ)).toLinearMap) x✝¹) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : σ.IntertwiningMap τ
x✝¹ : W
x✝ : V
⊢ ((TensorProduct.AlgebraTensorModule.curry ((↑(comm τ ρ)).comp (IntertwiningMap.rTensor ρ f)).toLinearMap) x✝¹) x✝ =
    ((TensorProduct.AlgebraTensorModule.curry ((IntertwiningMap.lTensor ρ f).comp ↑(comm σ ρ)).toLinearMap) x✝¹) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : V →ₗ[A] W
g : G
x✝² : V
x✝¹ : W
x✝ : U
⊢ (((TensorProduct.AlgebraTensorModule.curry
            (TensorProduct.AlgebraTensorModule.curry
              (↑(_root_.TensorProduct.assoc A V W U) ∘ₗ ((ρ.tprod σ).tprod τ) g)))
          x✝²)
        x✝¹)
      x✝ =
    (((TensorProduct.AlgebraTensorModule.curry
            (TensorProduct.AlgebraTensorModule.curry
              ((ρ.tprod (σ.tprod τ)) g ∘ₗ ↑(_root_.TensorProduct.assoc A V W U))))
          x✝²)
        x✝¹)
      x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : V →ₗ[A] W
g : G
x✝ : W
⊢ ((TensorProduct.AlgebraTensorModule.curry (↑(_root_.TensorProduct.rid A W) ∘ₗ (σ.tprod (trivial A G A)) g)) x✝) 1 =
    ((TensorProduct.AlgebraTensorModule.curry (σ g ∘ₗ ↑(_root_.TensorProduct.rid A W))) x✝) 1

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f : V →ₗ[A] W
g : G
x✝ : W
⊢ ((TensorProduct.AlgebraTensorModule.curry (↑(_root_.TensorProduct.lid A W) ∘ₗ ((trivial A G A).tprod σ) g)) 1) x✝ =
    ((TensorProduct.AlgebraTensorModule.curry (σ g ∘ₗ ↑(_root_.TensorProduct.lid A W))) 1) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Iso

Goal state before:
pp: case add
k : Type u_1
G : Type u_2
inst✝⁵ : CommRing k
inst✝⁴ : Monoid G
V : Type u_3
W : Type u_4
inst✝³ : AddCommGroup V
inst✝² : AddCommGroup W
inst✝¹ : Module k V
inst✝ : Module k W
ρ : G →* V →ₗ[k] V
σ : G →* W →ₗ[k] W
f : V →ₗ[k] W
w : ∀ (g : G), f ∘ₗ ρ g = σ g ∘ₗ f
r : k[G]
x : V
g h : k[G]
gw : f ((((lift k (V →ₗ[k] V) G) ρ) g) x) = (((lift k (W →ₗ[k] W) G) σ) g) (f x)
⊢ f ((((lift k (V →ₗ[k] V) G) ρ) h) x) = (((lift k (W →ₗ[k] W) G) σ) h) (f x) →
    f ((((lift k (V →ₗ[k] V) G) ρ) (g + h)) x) = (((lift k (W →ₗ[k] W) G) σ) (g + h)) (f x)

Tactic:
hw

Goal state after:
case add
k : Type u_1
G : Type u_2
inst✝⁵ : CommRing k
inst✝⁴ : Monoid G
V : Type u_3
W : Type u_4
inst✝³ : AddCommGroup V
inst✝² : AddCommGroup W
inst✝¹ : Module k V
inst✝ : Module k W
ρ : G →* V →ₗ[k] V
σ : G →* W →ₗ[k] W
f : V →ₗ[k] W
w : ∀ (g : G), f ∘ₗ ρ g = σ g ∘ₗ f
r : k[G]
x : V
g h : k[G]
gw : f ((((lift k (V →ₗ[k] V) G) ρ) g) x) = (((lift k (W →ₗ[k] W) G) σ) g) (f x)
hw : f ((((lift k (V →ₗ[k] V) G) ρ) h) x) = (((lift k (W →ₗ[k] W) G) σ) h) (f x)
⊢ f ((((lift k (V →ₗ[k] V) G) ρ) (g + h)) x) = (((lift k (W →ₗ[k] W) G) σ) (g + h)) (f x)
Module: Mathlib.RepresentationTheory.Rep.Res

Goal state before:
pp: k : Type u
inst✝² : Semiring k
G : Type v1
H : Type v2
inst✝¹ : Monoid G
inst✝ : Monoid H
f : H →* G
hf : Function.Surjective ⇑f
X Y : Rep k G
f' : (resFunctor f).obj X ⟶ (resFunctor f).obj Y
x✝ : ↑((resFunctor f).obj X)
⊢ (Hom.hom ((resFunctor f).map (liftHomOfSurj f hf f'))).toLinearMap x✝ = (Hom.hom f').toLinearMap x✝

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Res

Goal state before:
pp: case mpr
G : Type v1
H : Type v2
inst✝² : Monoid G
inst✝¹ : Monoid H
k : Type u
inst✝ : CommRing k
φ : H →* G
S : ShortComplex (Rep k G)
⊢ S.ShortExact → (S.map (resFunctor φ)).ShortExact

Tactic:
rintro @⟨_, mono_f, epi_g⟩

Goal state after:
case mpr
G : Type v1
H : Type v2
inst✝² : Monoid G
inst✝¹ : Monoid H
k : Type u
inst✝ : CommRing k
φ : H →* G
S : ShortComplex (Rep k G)
exact✝ : S.Exact
mono_f : Mono S.f
epi_g : Epi S.g
⊢ (S.map (resFunctor φ)).ShortExact
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case e'_7
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
ω : Ω
hω :
  (fun n ↦
      ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω -
        ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a) =o[atTop]
    fun n ↦ ↑⌊c ^ n⌋₊
A : Tendsto (fun n ↦ ⌊c ^ n⌋₊) atTop atTop
n : ℕ
⊢ ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ↑⌊c ^ n⌋₊ * ∫ (a : Ω), X 0 a =
    (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a) +
      ((fun n ↦ (∫ (a : Ω), (∑ i ∈ range n, truncation (X i) ↑i) a) - ↑n * ∫ (a : Ω), X 0 a) ∘ fun n ↦ ⌊c ^ n⌋₊) n

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: case zero
k : Type u_1
G : Type u_2
V : Type u_3
inst✝³ : CommRing k
inst✝² : Group G
inst✝¹ : AddCommGroup V
inst✝ : Module k V
ρ : Representation k G V
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
x : V
hx : (ρ g) x = x
⊢ (ρ ((fun x ↦ g ^ x) 0)) x = x

Tactic:
simp

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: case refine_1
k : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddCommGroup W
inst✝ : Module k W
ρ : Representation k G V
σ : Representation k G W
f : V →ₗ[k] W
γ : G
hf : σ γ ∘ₗ f ∘ₗ ρ γ⁻¹ = f
v : V
⊢ (σ γ ∘ₗ f ∘ₗ ρ γ⁻¹) ((ρ γ) v) = (σ γ) (f v)

Tactic:
simp

Goal state after: