mathlib-initiative/mathlib-tactics huggingface dataset:
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
f : ρ.IntertwiningMap σ
g : G
⊢ ∀ ⦃v : V⦄, v ∈ f.ker → (ρ g) v ∈ f.ker

Tactic:
by

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
ι : Type u_6
s : Finset ι
f : ι → ρ.IntertwiningMap σ
⊢ (∑ i ∈ s, f i).toLinearMap = ∑ i ∈ s, (f i).toLinearMap

Tactic:
by

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
ι : Type u_6
s : Finset ι
f : ι → ρ.IntertwiningMap σ
v : V
⊢ (∑ i ∈ s, f i) v = ∑ i ∈ s, (f i) v

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
h_indep : iIndepFun X μ
c : ι → ℝ≥0
h_meas : ∀ (i : ι), AEMeasurable (X i) μ
s : Finset ι
h_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ
⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ s, X i ω) (∑ i ∈ s, c i) μ

Tactic:
by

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✝¹³ : Semiring A
inst✝¹² : Monoid G
inst✝¹¹ : AddCommMonoid V✝
inst✝¹⁰ : AddCommMonoid W✝
inst✝⁹ : AddCommMonoid U
inst✝⁸ : Module A V✝
inst✝⁷ : Module A W✝
inst✝⁶ : Module A U
ρ✝ : Representation A G V✝
σ✝ : Representation A G W✝
τ✝ : Representation A G U
f✝¹ : V✝ →ₗ[A] W✝
V : Type u_6
W : Type u_7
P : Type u_8
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommGroup W
inst✝³ : AddCommGroup P
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A P
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G P
f✝ : V →ₗ[A] W
f : ρ.IntertwiningMap σ
⊢ ∀ (g : G), (-f.toLinearMap) ∘ₗ ρ g = σ g ∘ₗ (-f.toLinearMap)

Tactic:
by

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✝¹³ : Semiring A
inst✝¹² : Monoid G
inst✝¹¹ : AddCommMonoid V✝
inst✝¹⁰ : AddCommMonoid W✝
inst✝⁹ : AddCommMonoid U
inst✝⁸ : Module A V✝
inst✝⁷ : Module A W✝
inst✝⁶ : Module A U
ρ✝ : Representation A G V✝
σ✝ : Representation A G W✝
τ✝ : Representation A G U
f✝¹ : V✝ →ₗ[A] W✝
V : Type u_6
W : Type u_7
P : Type u_8
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommGroup W
inst✝³ : AddCommGroup P
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A P
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G P
f✝ : V →ₗ[A] W
f g : ρ.IntertwiningMap σ
⊢ ∀ (g_1 : G), (f.toLinearMap - g.toLinearMap) ∘ₗ ρ g_1 = σ g_1 ∘ₗ (f.toLinearMap - g.toLinearMap)

Tactic:
by

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✝¹³ : Semiring A
inst✝¹² : Monoid G
inst✝¹¹ : AddCommMonoid V✝
inst✝¹⁰ : AddCommMonoid W✝
inst✝⁹ : AddCommMonoid U
inst✝⁸ : Module A V✝
inst✝⁷ : Module A W✝
inst✝⁶ : Module A U
ρ✝ : Representation A G V✝
σ✝ : Representation A G W✝
τ✝ : Representation A G U
f✝¹ : V✝ →ₗ[A] W✝
V : Type u_6
W : Type u_7
P : Type u_8
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommGroup W
inst✝³ : AddCommGroup P
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A P
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G P
f✝ : V →ₗ[A] W
z : ℤ
f : ρ.IntertwiningMap σ
⊢ ∀ (g : G), (z • f.toLinearMap) ∘ₗ ρ g = σ g ∘ₗ (z • f.toLinearMap)

Tactic:
by

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
⊢ ∀ (g : G), LinearMap.id ∘ₗ ρ g = ρ g ∘ₗ LinearMap.id

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
h_indep : iIndepFun X μ
c : ι → ℝ≥0
s : Finset ι
h_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ
⊢ HasSubgaussianMGF (fun ω ↦ ∑ i, X (↑i) ω) (∑ i, c ↑i) μ

Tactic:
by

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
f : σ.IntertwiningMap τ
g : ρ.IntertwiningMap σ
⊢ ∀ (g_1 : G), (f.toLinearMap ∘ₗ g.toLinearMap) ∘ₗ ρ g_1 = τ g_1 ∘ₗ f.toLinearMap ∘ₗ g.toLinearMap

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
h_indep : iIndepFun X μ
c : ι → ℝ≥0
s : Finset ι
h_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ
⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ s, X i ω) (∑ i ∈ s, c i) μ

Tactic:
by

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
f₁ f₂ : σ.IntertwiningMap τ
g : ρ.IntertwiningMap σ
⊢ (f₁ + f₂).comp g = f₁.comp g + f₂.comp g

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : ℕ → Ω → ℝ
h_indep : iIndepFun X μ
c : ℝ≥0
n : ℕ
h_subG : ∀ i < n, HasSubgaussianMGF (X i) c μ
ε : ℝ
hε : 0 ≤ ε
⊢ ∀ i ∈ Finset.range n, HasSubgaussianMGF (X i) c μ

Tactic:
by

Goal state after:
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

pp: case e_f.e_a.h
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
⊢ (↑(range (u i))).Pairwise fun i j ↦ Y i ⟂ᵢ Y j

Tactic:
· intro j _
          exact (hident j).aestronglyMeasurable_fst.memLp_truncation

Goal state after:
case e_f.e_a.h
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
⊢ (↑(range (u i))).Pairwise fun i j ↦ Y i ⟂ᵢ Y j
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 : σ.IntertwiningMap τ
g₁ g₂ : ρ.IntertwiningMap σ
⊢ f.comp (g₁ + g₂) = f.comp g₁ + f.comp g₂

Tactic:
by

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.fst A V W ∘ₗ (ρ.prod σ) x✝) x = (ρ x✝ ∘ₗ LinearMap.fst A V W) x

Tactic:
by

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:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : ℕ → Ω → ℝ
h_indep : iIndepFun X μ
c : ℝ≥0
n : ℕ
h_subG : ∀ i < n, HasSubgaussianMGF (X i) c μ
ε : ℝ
hε : 0 ≤ ε
⊢ μ.real {ω | ε ≤ ∑ i ∈ Finset.range n, X i ω} ≤ rexp (-ε ^ 2 / (2 * ↑n * ↑c))

Tactic:
by

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
f : ρ.IntertwiningMap σ
g : ρ.IntertwiningMap τ
x✝ : G
⊢ ∀ (x : V), (f.prod g.toLinearMap ∘ₗ ρ x✝) x = ((σ.prod τ) x✝ ∘ₗ f.prod g.toLinearMap) x

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) cX μ
hY : HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ) cY μ
hindep : X ⟂ᵢ[μ] Y
h_le : ∫ (x : Ω), Y x ∂μ ≤ ∫ (x : Ω), X x ∂μ
⊢ μ.real {ω | X ω ≤ Y ω} =
    μ.real {ω | ∫ (x : Ω), X x ∂μ - ∫ (x : Ω), Y x ∂μ ≤ Y ω - ∫ (x : Ω), Y x ∂μ - (X ω - ∫ (x : Ω), X x ∂μ)}

Tactic:
by

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 ψ : σ.Equiv ρ
h : φ.toLinearEquiv = ψ.toLinearEquiv
⊢ φ = ψ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: 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
⊢ ∑ i ∈ range N, (↑(u i) ^ 2)⁻¹ * Var[S (u i); ℙ] = ∑ i ∈ range N, (↑(u i) ^ 2)⁻¹ * ∑ j ∈ range (u i), Var[Y j; ℙ]

Tactic:
by

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✝ φ ψ : ρ.Equiv σ
h1 : ⇑φ.toLinearEquiv = ⇑ψ.toLinearEquiv
h2 : φ.invFun = ψ.invFun
⊢ φ = ψ

Tactic:
by

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 ψ : ρ.Equiv σ
h : ⇑φ = ⇑ψ
⊢ φ = ψ

Tactic:
by

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:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) cX μ
hY : HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ) cY μ
hindep : X ⟂ᵢ[μ] Y
h_le : ∫ (x : Ω), Y x ∂μ ≤ ∫ (x : Ω), X x ∂μ
⊢ Measurable fun x ↦ x - ∫ (x : Ω), Y x ∂μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) cX μ
hY : HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ) cY μ
hindep : X ⟂ᵢ[μ] Y
h_le : ∫ (x : Ω), Y x ∂μ ≤ ∫ (x : Ω), X x ∂μ
⊢ Measurable fun x ↦ x - ∫ (x : Ω), X x ∂μ

Tactic:
by

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✝ φ : ρ.Equiv σ
g : G
⊢ ↑φ.toLinearEquiv.symm ∘ₗ σ g = ρ g ∘ₗ ↑φ.toLinearEquiv.symm

Tactic:
by

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
e : V ≃ₗ[A] W
he : ∀ (g : G), ↑e ∘ₗ ρ g = σ g ∘ₗ ↑e
g : G
⊢ ↑e.symm ∘ₗ σ g = ρ g ∘ₗ ↑e.symm

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) cX μ
hY : HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ) cY μ
hindep : X ⟂ᵢ[μ] Y
h_le : ∫ (x : Ω), Y x ∂μ ≤ ∫ (x : Ω), X x ∂μ
⊢ μ.real {ω | ∫ (x : Ω), X x ∂μ - ∫ (x : Ω), Y x ∂μ ≤ Y ω - ∫ (x : Ω), Y x ∂μ - (X ω - ∫ (x : Ω), X x ∂μ)} ≤
    rexp (-(∫ (x : Ω), Y x ∂μ - ∫ (x : Ω), X x ∂μ) ^ 2 / (2 * (↑cX + ↑cY)))

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) cX μ
hY : HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ) cY μ
hindep : X ⟂ᵢ[μ] Y
h_le : ∫ (x : Ω), Y x ∂μ ≤ ∫ (x : Ω), X x ∂μ
⊢ μ.real {ω | X ω ≤ Y ω} ≤ rexp (-(∫ (x : Ω), Y x ∂μ - ∫ (x : Ω), X x ∂μ) ^ 2 / (2 * (↑cX + ↑cY)))

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
inst✝ : IsProbabilityMeasure μ
a b t : ℝ
hm : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hc : ∫ (x : Ω), X x ∂μ = 0
ht : 0 < t
hi : ∀ (u : ℝ), Integrable (fun ω ↦ rexp (u * X ω)) μ
⊢ Set.Icc 0 t ⊆ interior (integrableExpSet X μ)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: 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
⊢ ∑ i ∈ range N, (↑(u i) ^ 2)⁻¹ * ∑ j ∈ range (u i), Var[Y j; ℙ] =
    ∑ j ∈ range (u (N - 1)), (∑ i ∈ range N with j < u i, (↑(u i) ^ 2)⁻¹) * Var[Y j; ℙ]

Tactic:
by

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✝ φ : ρ.Equiv σ
ψ : σ.Equiv τ
g : G
⊢ ↑(φ.toLinearEquiv ≪≫ₗ ψ.toLinearEquiv) ∘ₗ ρ g = τ g ∘ₗ ↑(φ.toLinearEquiv ≪≫ₗ ψ.toLinearEquiv)

Tactic:
by

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
⊢ φ.trans φ.symm = refl ρ

Tactic:
by

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
⊢ φ.symm.trans φ = refl σ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
inst✝ : IsProbabilityMeasure μ
a b t : ℝ
hm : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hc : ∫ (x : Ω), X x ∂μ = 0
ht : 0 < t
hi : ∀ (u : ℝ), Integrable (fun ω ↦ rexp (u * X ω)) μ
hs : Set.Icc 0 t ⊆ interior (integrableExpSet X μ)
u : ℝ
h1 : u ∈ Set.Ioo 0 t
h2 : cgf X μ t = iteratedDeriv 2 (cgf X μ) u * t ^ 2 / 2
⊢ iteratedDeriv 2 (cgf X μ) u = Var[X; μ.tilted fun x ↦ u * X x]

Tactic:
by

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
g : G
φ : ρ.Equiv σ
⊢ φ.toLinearEquiv.conj (ρ g) = σ g

Tactic:
by

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 : ρ.IntertwiningMap σ
⊢ ∀ (g : G), (a • f.toLinearMap) ∘ₗ ρ g = σ g ∘ₗ (a • f.toLinearMap)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
inst✝ : IsProbabilityMeasure μ
a b t : ℝ
hm : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hc : ∫ (x : Ω), X x ∂μ = 0
ht : 0 < t
hi : ∀ (u : ℝ), Integrable (fun ω ↦ rexp (u * X ω)) μ
hs : Set.Icc 0 t ⊆ interior (integrableExpSet X μ)
u : ℝ
h1 : u ∈ Set.Ioo 0 t
h2 : cgf X μ t = iteratedDeriv 2 (cgf X μ) u * t ^ 2 / 2
⊢ Var[X; μ.tilted fun x ↦ u * X x] ≤ ((b - a) / 2) ^ 2

Tactic:
by

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.IntertwiningMap σ
m : A[G]
v : ρ.asModule
⊢ f.toLinearMap (m • v) = (RingHom.id A[G]) m • f.toLinearMap v

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
inst✝ : IsProbabilityMeasure μ
a b t : ℝ
hm : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hc : ∫ (x : Ω), X x ∂μ = 0
ht : 0 < t
hi : ∀ (u : ℝ), Integrable (fun ω ↦ rexp (u * X ω)) μ
hs : Set.Icc 0 t ⊆ interior (integrableExpSet X μ)
u : ℝ
h1 : u ∈ Set.Ioo 0 t
h2 : cgf X μ t = iteratedDeriv 2 (cgf X μ) u * t ^ 2 / 2
⊢ ((b - a) / 2) ^ 2 = (↑‖b - a‖₊ / 2) ^ 2

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
inst✝ : IsProbabilityMeasure μ
a b t : ℝ
hm : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hc : ∫ (x : Ω), X x ∂μ = 0
ht : 0 < t
⊢ mgf X μ t ≤ rexp ((↑‖b - a‖₊ / 2) ^ 2 * t ^ 2 / 2)

Tactic:
by

Goal state after: