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

Goal state before:
pp: Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
inst✝⁵ : (s : S) → TopologicalSpace (E s)
inst✝⁴ : (s : S) → MeasurableSpace (E s)
inst✝³ : ∀ (s : S), BorelSpace (E s)
inst✝² : ∀ (s : S), HasOuterApproxClosed (E s)
X : (s : S) → Ω → E s
inst✝¹ : Fintype S
inst✝ : IsProbabilityMeasure P
A : Set Ω
mA : NullMeasurableSet A P
mX : ∀ (s : S), AEMeasurable (X s) P
h : ∀ (f : (s : S) → E s →ᵇ ℝ), ∫ (ω : Ω) in A, ∏ s, (f s) (X s ω) ∂P = P.real A * ∫ (ω : Ω), ∏ s, (f s) (X s ω) ∂P
f : ℝ →ᵇ ℝ
g : (t : S) → E t →ᵇ ℝ
h1 :
  ∀ (ω : Ω),
    f (A.indicator 1 ω) * ∏ s, (g s) (X s ω) =
      A.indicator (fun ω ↦ f 1 * ∏ s, (g s) (X s ω)) ω + f 0 * ∏ s, (g s) (X s ω) -
        A.indicator (fun ω ↦ f 0 * ∏ s, (g s) (X s ω)) ω
h2 : ∀ (ω : Ω), f (A.indicator 1 ω) = A.indicator (fun x ↦ f 1) ω + Aᶜ.indicator (fun x ↦ f 0) ω
hg : ∀ {c : ℝ}, Integrable (fun ω ↦ c * ∏ s, (g s) (X s ω)) P
⊢ ∫ (x : Ω), (⇑f ∘ A.indicator 1 * ∏ t, ⇑(g t) ∘ X t) x ∂P =
    (∫ (x : Ω), (⇑f ∘ A.indicator 1) x ∂P) * ∫ (x : Ω), (∏ t, ⇑(g t) ∘ X t) x ∂P

Tactic:
simp_rw [Pi.mul_apply, Finset.prod_apply, Function.comp_apply, h1, h2]

Goal state after:
Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
inst✝⁵ : (s : S) → TopologicalSpace (E s)
inst✝⁴ : (s : S) → MeasurableSpace (E s)
inst✝³ : ∀ (s : S), BorelSpace (E s)
inst✝² : ∀ (s : S), HasOuterApproxClosed (E s)
X : (s : S) → Ω → E s
inst✝¹ : Fintype S
inst✝ : IsProbabilityMeasure P
A : Set Ω
mA : NullMeasurableSet A P
mX : ∀ (s : S), AEMeasurable (X s) P
h : ∀ (f : (s : S) → E s →ᵇ ℝ), ∫ (ω : Ω) in A, ∏ s, (f s) (X s ω) ∂P = P.real A * ∫ (ω : Ω), ∏ s, (f s) (X s ω) ∂P
f : ℝ →ᵇ ℝ
g : (t : S) → E t →ᵇ ℝ
h1 :
  ∀ (ω : Ω),
    f (A.indicator 1 ω) * ∏ s, (g s) (X s ω) =
      A.indicator (fun ω ↦ f 1 * ∏ s, (g s) (X s ω)) ω + f 0 * ∏ s, (g s) (X s ω) -
        A.indicator (fun ω ↦ f 0 * ∏ s, (g s) (X s ω)) ω
h2 : ∀ (ω : Ω), f (A.indicator 1 ω) = A.indicator (fun x ↦ f 1) ω + Aᶜ.indicator (fun x ↦ f 0) ω
hg : ∀ {c : ℝ}, Integrable (fun ω ↦ c * ∏ s, (g s) (X s ω)) P
⊢ ∫ (x : Ω),
      A.indicator (fun ω ↦ f 1 * ∏ s, (g s) (X s ω)) x + f 0 * ∏ x_1, (g x_1) (X x_1 x) -
        A.indicator (fun ω ↦ f 0 * ∏ s, (g s) (X s ω)) x ∂P =
    (∫ (x : Ω), A.indicator (fun x ↦ f 1) x + Aᶜ.indicator (fun x ↦ f 0) x ∂P) * ∫ (x : Ω), ∏ x_1, (g x_1) (X x_1 x) ∂P
Module: Mathlib.Probability.Independence.BoundedContinuousFunction

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
G : Type u_6
inst✝⁴ : TopologicalSpace G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
inst✝¹ : HasOuterApproxClosed G
Z : Ω → G
inst✝ : IsProbabilityMeasure P
A : Set Ω
mA : NullMeasurableSet A P
mZ : AEMeasurable Z P
h : ∀ (f : G →ᵇ ℝ), ∫ (ω : Ω) in A, f (Z ω) ∂P = P.real A * ∫ (ω : Ω), f (Z ω) ∂P
⊢ A.indicator 1 ⟂ᵢ[P] Z

Tactic:
suffices (A.indicator (1 : Ω → ℝ)) ⟂ᵢ[P] (fun ω (_ : Unit) ↦ Z ω) from
    this.comp (measurable_id) (measurable_pi_apply ())
  refine indicator_indepFun_pi_of_prod_bcf mA (fun _ ↦ mZ) fun f ↦ ?_
  convert! h (f ()) <;> simp

Goal state after:
Module: Mathlib.Probability.ProbabilityMassFunction.Binomial

Goal state before:
pp: k b : ℕ
hb : k ≤ b
x : ℝ≥0
h : x ≤ 1
eq0 : k % (b + 1) = k
eq1 : 1 - ↑x = ENNReal.ofReal (1 - ↑x)
this : 1 - ↑x ≥ 0
⊢ ENNReal.ofReal (↑(b.choose k) * ↑x ^ k * (1 - ↑x) ^ (b - k)) = (binomial x h b) (Fin.ofNat (b + 1) k)

Tactic:
Fin.ofNat_eq_cast,

Goal state after:
k b : ℕ
hb : k ≤ b
x : ℝ≥0
h : x ≤ 1
eq0 : k % (b + 1) = k
eq1 : 1 - ↑x = ENNReal.ofReal (1 - ↑x)
this : 1 - ↑x ≥ 0
⊢ ENNReal.ofReal (↑(b.choose k) * ↑x ^ k * (1 - ↑x) ^ (b - k)) = (binomial x h b) ↑k
Module: Mathlib.Probability.Process.LocalProperty

Goal state before:
pp: ι : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : LinearOrder ι
𝓕 : Filtration ι mΩ
X : ι → Ω → E
p q : (ι → Ω → E) → Prop
inst✝³ : OrderBot ι
inst✝² : Zero E
inst✝¹ : TopologicalSpace ι
inst✝ : OrderTopology ι
hp : IsStable 𝓕 p
hq : IsStable 𝓕 q
⊢ Locally (fun Y ↦ p Y ∧ q Y) 𝓕 X P ↔ Locally p 𝓕 X P ∧ Locally q 𝓕 X P

Tactic:
refine ⟨Locally.of_and, fun ⟨hpX, hqX⟩ ↦
    ⟨_, hpX.isLocalizingSequence_localSeq.min hqX.isLocalizingSequence_localSeq, fun n ↦ ?_⟩⟩

Goal state after:
ι : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : LinearOrder ι
𝓕 : Filtration ι mΩ
X : ι → Ω → E
p q : (ι → Ω → E) → Prop
inst✝³ : OrderBot ι
inst✝² : Zero E
inst✝¹ : TopologicalSpace ι
inst✝ : OrderTopology ι
hp : IsStable 𝓕 p
hq : IsStable 𝓕 q
x✝ : Locally p 𝓕 X P ∧ Locally q 𝓕 X P
hpX : Locally p 𝓕 X P
hqX : Locally q 𝓕 X P
n : ℕ
⊢ (fun Y ↦ p Y ∧ q Y)
    (stoppedProcess (fun i ↦ {ω | ⊥ < (hpX.localSeq ⊓ hqX.localSeq) n ω}.indicator (X i))
      ((hpX.localSeq ⊓ hqX.localSeq) n))
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K i : ℕ
x✝ : i ∈ range K
⊢ ⋃ N, {ω | X ω ∈ Set.Ioc ↑i ↑N} ⊆ {ω | X ω ∈ Set.Ioi ↑i}

Tactic:
simp +contextual only [Set.mem_Ioc, Set.mem_Ioi,
          Set.iUnion_subset_iff, Set.ofPred_subset_ofPred, imp_true_iff]

Goal state after:
Module: Mathlib.RepresentationTheory.Action

Goal state before:
pp: k : Type u
G : Type v
V : Type u'
W : Type v'
inst✝⁵ : Monoid G
inst✝⁴ : Semiring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddCommGroup W
inst✝ : Module k W
σ : Representation k G V
ρ : Representation k G W
X Y Z : Action (Type w) G
g : G
⊢ ↑(MonoidAlgebra.uniqueLinearEquiv k PUnit.{w + 1}).symm ∘ₗ (trivial k G k) g =
    (linearize k G (𝟙_ (Action (Type w) G))) g ∘ₗ ↑(MonoidAlgebra.uniqueLinearEquiv k PUnit.{w + 1}).symm

Tactic:
ext1

Goal state after:
k : Type u
G : Type v
V : Type u'
W : Type v'
inst✝⁵ : Monoid G
inst✝⁴ : Semiring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddCommGroup W
inst✝ : Module k W
σ : Representation k G V
ρ : Representation k G W
X Y Z : Action (Type w) G
g : G
⊢ (↑(MonoidAlgebra.uniqueLinearEquiv k PUnit.{w + 1}).symm ∘ₗ (trivial k G k) g) 1 =
    ((linearize k G (𝟙_ (Action (Type w) G))) g ∘ₗ ↑(MonoidAlgebra.uniqueLinearEquiv k PUnit.{w + 1}).symm) 1
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case h
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
a✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
⊢ ∫ (x : ℝ) in Set.Ioc ↑k ↑(k + 1), 2 / (↑k + 1) * x ^ 2 ∂ρ ≤ ∫ (x : ℝ) in ↑k..↑(k + 1), 2 * x ∂ρ

Tactic:
intervalIntegral.integral_of_le Ik

Goal state after:
case h
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
a✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
⊢ ∫ (x : ℝ) in Set.Ioc ↑k ↑(k + 1), 2 / (↑k + 1) * x ^ 2 ∂ρ ≤ ∫ (x : ℝ) in Set.Ioc ↑k ↑(k + 1), 2 * x ∂ρ
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:
ext1

Goal state after:
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).toLinearMap = (f₁.comp g + f₂.comp g).toLinearMap
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:
ext1

Goal state after:
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)).toLinearMap = (f.comp g₁ + f.comp g₂).toLinearMap
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case 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 j : ℕ
a✝ : j ∈ range (u (N - 1))
⊢ (∑ i ∈ range N with j < u i, (↑(u i) ^ 2)⁻¹) * Var[Y j; ℙ] ≤ c ^ 5 * (c - 1)⁻¹ ^ 3 / ↑j ^ 2 * Var[Y j; ℙ]

Tactic:
rcases eq_zero_or_pos j with (rfl | hj)

Goal state after:
case h.inl
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✝ : 0 ∈ range (u (N - 1))
⊢ (∑ i ∈ range N with 0 < u i, (↑(u i) ^ 2)⁻¹) * Var[Y 0; ℙ] ≤ c ^ 5 * (c - 1)⁻¹ ^ 3 / ↑0 ^ 2 * Var[Y 0; ℙ]

case h.inr
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 j : ℕ
a✝ : j ∈ range (u (N - 1))
hj : 0 < j
⊢ (∑ i ∈ range N with j < u i, (↑(u i) ^ 2)⁻¹) * Var[Y j; ℙ] ≤ c ^ 5 * (c - 1)⁻¹ ^ 3 / ↑j ^ 2 * Var[Y j; ℙ]
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 := ⋯
S : ℕ → Ω → ℝ := ⋯
hS : S = fun n ↦ ∑ i ∈ range n, Y i
u : ℕ → 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 := ⋯
I2 : ∀ (N : ℕ), ∑ i ∈ range N, (↑(u i) ^ 2)⁻¹ * Var[S (u i); ℙ] ≤ C
N : ℕ
⊢ ∑ i ∈ range N, Var[S (u i); ℙ] / (↑(u i) * ε) ^ 2 ≤ ε⁻¹ ^ 2 * C

Tactic:
div_eq_inv_mul,

Goal state after:
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)
I2 : ∀ (N : ℕ), ∑ i ∈ range N, (↑(u i) ^ 2)⁻¹ * Var[S (u i); ℙ] ≤ C
N : ℕ
⊢ ∑ x ∈ range N, ((↑(u x) * ε) ^ 2)⁻¹ * Var[S (u x); ℙ] ≤ ε⁻¹ ^ 2 * C
Module: Mathlib.RepresentationTheory.FDRep

Goal state before:
pp: R k : Type u
G : Type v
inst✝² : CommRing R
inst✝¹ : Field k
inst✝ : Monoid G
X Y : FDRep R G
f : (forget₂ (FDRep R G) (Rep R G)).obj X ⟶ (forget₂ (FDRep R G) (Rep R G)).obj Y
g : G
⊢ X.ρ g ≫ InducedCategory.homMk (ModuleCat.ofHom (Rep.Hom.hom f).toLinearMap) =
    InducedCategory.homMk (ModuleCat.ofHom (Rep.Hom.hom f).toLinearMap) ≫ Y.ρ g

Tactic:
ext1

Goal state after:
R k : Type u
G : Type v
inst✝² : CommRing R
inst✝¹ : Field k
inst✝ : Monoid G
X Y : FDRep R G
f : (forget₂ (FDRep R G) (Rep R G)).obj X ⟶ (forget₂ (FDRep R G) (Rep R G)).obj Y
g : G
⊢ ModuleCat.Hom.hom (X.ρ g ≫ InducedCategory.homMk (ModuleCat.ofHom (Rep.Hom.hom f).toLinearMap)).hom =
    ModuleCat.Hom.hom (InducedCategory.homMk (ModuleCat.ofHom (Rep.Hom.hom f).toLinearMap) ≫ Y.ρ g).hom
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case e'_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)
I2 : ∀ (N : ℕ), ∑ i ∈ range N, (↑(u i) ^ 2)⁻¹ * Var[S (u i); ℙ] ≤ C
I3 : ∀ (N : ℕ), ∑ i ∈ range N, ℙ {ω | ↑(u i) * ε ≤ |S (u i) ω - ∫ (a : Ω), S (u i) a|} ≤ ENNReal.ofReal (ε⁻¹ ^ 2 * C)
I4 : ∑' (i : ℕ), ℙ {ω | ↑(u i) * ε ≤ |S (u i) ω - ∫ (a : Ω), S (u i) a|} < ∞
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, |∑ c ∈ range (u n), Y c ω - ∫ (a : Ω), ∑ c ∈ range (u n), Y c a| < ε * ↑(u n)
x✝¹ : ℕ
x✝ : Ω
⊢ (range ⌊c ^ x✝¹⌋₊).sum Y x✝ = ∑ c ∈ range (u x✝¹), Y c x✝

Tactic:
simp only [Y, u, Finset.sum_apply]

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: case mp
k : Type u
inst✝¹ : CommRing k
G : Type v
inst✝ : Group G
X Y : Rep k G
f : ↑X →ₗ[k] ↑Y
g : G
h : Y.ρ g ∘ₗ f ∘ₗ X.ρ g⁻¹ = f
⊢ Y.ρ g ∘ₗ f ∘ₗ X.ρ 1 = Y.ρ g ∘ₗ f

Tactic:
map_one,

Goal state after:
case mp
k : Type u
inst✝¹ : CommRing k
G : Type v
inst✝ : Group G
X Y : Rep k G
f : ↑X →ₗ[k] ↑Y
g : G
h : Y.ρ g ∘ₗ f ∘ₗ X.ρ g⁻¹ = f
⊢ Y.ρ g ∘ₗ f ∘ₗ 1 = Y.ρ g ∘ₗ f
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: case mpr
k : Type u
inst✝¹ : CommRing k
G : Type v
inst✝ : Group G
X Y : Rep k G
f : ↑X →ₗ[k] ↑Y
g : G
h : f ∘ₗ X.ρ g = Y.ρ g ∘ₗ f
⊢ f ∘ₗ X.ρ 1 = f

Tactic:
map_one,

Goal state after:
case mpr
k : Type u
inst✝¹ : CommRing k
G : Type v
inst✝ : Group G
X Y : Rep k G
f : ↑X →ₗ[k] ↑Y
g : G
h : f ∘ₗ X.ρ g = Y.ρ g ∘ₗ f
⊢ f ∘ₗ 1 = f
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case refine_2
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : IsProbabilityMeasure μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSpace E
X : ℕ → Ω → E
h' : Measurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
φ : SimpleFunc E E
⊢ ∀ ⦃f g : SimpleFunc E E⦄,
    Disjoint (Function.support ⇑f) (Function.support ⇑g) →
      (∀ᵐ (ω : Ω) ∂μ,
          Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, f (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑f ∘ X 0) x ∂μ))) →
        (∀ᵐ (ω : Ω) ∂μ,
            Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, g (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑g ∘ X 0) x ∂μ))) →
          ∀ᵐ (ω : Ω) ∂μ,
            Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, (f + g) (X i ω)) atTop
              (𝓝 (∫ (x : Ω), (⇑(f + g) ∘ X 0) x ∂μ))

Tactic:
rintro φ ψ - hφ hψ
    filter_upwards [hφ, hψ] with ω hωφ hωψ
    convert! hωφ.add hωψ using 1
    · simp [sum_add_distrib]
    · congr 1
      rw [← integral_add]
      · rfl
      · exact (φ.comp (X 0) h').integrable_of_isFiniteMeasure
      · exact (ψ.comp (X 0) h').integrable_of_isFiniteMeasure

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁵ : IsProbabilityMeasure μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
h' : StronglyMeasurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
s : Set E := Set.range (X 0) ∪ {0}
zero_s : 0 ∈ s
this✝ : SeparableSpace ↑s
this : Nonempty ↑s
φ : ℕ → SimpleFunc E E := SimpleFunc.nearestPt fun k ↦ Nat.casesOn k 0 (Subtype.val ∘ denseSeq ↑s)
Y : ℕ → ℕ → Ω → E := fun k i ↦ ⇑(φ k) ∘ X i
A : ∀ᵐ (ω : Ω) ∂μ, ∀ (k : ℕ), Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω) atTop (𝓝 (∫ (x : Ω), Y k 0 x ∂μ))
B :
  ∀ᵐ (ω : Ω) ∂μ,
    ∀ (k : ℕ),
      Tendsto (fun n ↦ (∑ i ∈ Finset.range n, ‖(X i - Y k i) ω‖) / ↑n) atTop
        (𝓝 (∫ (x : Ω), (fun ω ↦ ‖(X 0 - Y k 0) ω‖) x ∂μ))
ω : Ω
hω : ∀ (k : ℕ), Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω) atTop (𝓝 (∫ (x : Ω), Y k 0 x ∂μ))
h'ω :
  ∀ (k : ℕ),
    Tendsto (fun n ↦ (∑ i ∈ Finset.range n, ‖(X i - Y k i) ω‖) / ↑n) atTop (𝓝 (∫ (x : Ω), ‖(X 0 - Y k 0) x‖ ∂μ))
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : 0 < δ
hδ : δ + δ + δ < ε
⊢ ∃ k, ∫ (ω : Ω), ‖(X 0 - Y k 0) ω‖ ∂μ < δ

Tactic:
simp_rw [Pi.sub_apply, norm_sub_rev (X 0 _)]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁵ : IsProbabilityMeasure μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
h' : StronglyMeasurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
s : Set E := Set.range (X 0) ∪ {0}
zero_s : 0 ∈ s
this✝ : SeparableSpace ↑s
this : Nonempty ↑s
φ : ℕ → SimpleFunc E E := SimpleFunc.nearestPt fun k ↦ Nat.casesOn k 0 (Subtype.val ∘ denseSeq ↑s)
Y : ℕ → ℕ → Ω → E := fun k i ↦ ⇑(φ k) ∘ X i
A : ∀ᵐ (ω : Ω) ∂μ, ∀ (k : ℕ), Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω) atTop (𝓝 (∫ (x : Ω), Y k 0 x ∂μ))
B :
  ∀ᵐ (ω : Ω) ∂μ,
    ∀ (k : ℕ),
      Tendsto (fun n ↦ (∑ i ∈ Finset.range n, ‖(X i - Y k i) ω‖) / ↑n) atTop
        (𝓝 (∫ (x : Ω), (fun ω ↦ ‖(X 0 - Y k 0) ω‖) x ∂μ))
ω : Ω
hω : ∀ (k : ℕ), Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω) atTop (𝓝 (∫ (x : Ω), Y k 0 x ∂μ))
h'ω :
  ∀ (k : ℕ),
    Tendsto (fun n ↦ (∑ i ∈ Finset.range n, ‖(X i - Y k i) ω‖) / ↑n) atTop (𝓝 (∫ (x : Ω), ‖(X 0 - Y k 0) x‖ ∂μ))
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : 0 < δ
hδ : δ + δ + δ < ε
⊢ ∃ k, ∫ (ω : Ω), ‖Y k 0 ω - X 0 ω‖ ∂μ < δ
Module: Mathlib.RepresentationTheory.Coinduced

Goal state before:
pp: k : Type u
G : Type v
H : Type w
inst✝² : CommRing k
inst✝¹ : Monoid G
inst✝ : Monoid H
φ : G →* H
A : Rep k G
f g : ↑(coind' φ A)
hfg : ∀ (h : H), (Hom.hom f).toLinearMap (MonoidAlgebra.single h 1) = (Hom.hom g).toLinearMap (MonoidAlgebra.single h 1)
⊢ Hom.hom f = Hom.hom g

Tactic:
ext1

Goal state after:
k : Type u
G : Type v
H : Type w
inst✝² : CommRing k
inst✝¹ : Monoid G
inst✝ : Monoid H
φ : G →* H
A : Rep k G
f g : ↑(coind' φ A)
hfg : ∀ (h : H), (Hom.hom f).toLinearMap (MonoidAlgebra.single h 1) = (Hom.hom g).toLinearMap (MonoidAlgebra.single h 1)
⊢ (Hom.hom f).toLinearMap = (Hom.hom g).toLinearMap
Module: Mathlib.RepresentationTheory.Continuous.TopRep

Goal state before:
pp: k : Type u
G : Type v
X Y : Type w
inst✝¹² : TopologicalSpace k
inst✝¹¹ : Ring k
inst✝¹⁰ : Monoid G
inst✝⁹ : AddCommGroup X
inst✝⁸ : Module k X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : IsTopologicalAddGroup X
inst✝⁵ : ContinuousSMul k X
inst✝⁴ : AddCommGroup Y
inst✝³ : Module k Y
inst✝² : TopologicalSpace Y
inst✝¹ : IsTopologicalAddGroup Y
inst✝ : ContinuousSMul k Y
ρ : ContRepresentation k G X
σ : ContRepresentation k G Y
A B C : TopRep k G
X✝ Y✝ : TopRep k G
f : X✝ ⟶ Y✝
g : G
⊢ { V := TopModuleCat.of k ↑X✝, ρ := (TopModuleCat.of k ↑X✝).endRingEquiv.symm.toMonoidHom.comp ↑X✝.ρ }.ρ g ≫
      Hom.toTopModuleCatHom f =
    Hom.toTopModuleCatHom f ≫
      { V := TopModuleCat.of k ↑Y✝, ρ := (TopModuleCat.of k ↑Y✝).endRingEquiv.symm.toMonoidHom.comp ↑Y✝.ρ }.ρ g

Tactic:
ext1

Goal state after:
k : Type u
G : Type v
X Y : Type w
inst✝¹² : TopologicalSpace k
inst✝¹¹ : Ring k
inst✝¹⁰ : Monoid G
inst✝⁹ : AddCommGroup X
inst✝⁸ : Module k X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : IsTopologicalAddGroup X
inst✝⁵ : ContinuousSMul k X
inst✝⁴ : AddCommGroup Y
inst✝³ : Module k Y
inst✝² : TopologicalSpace Y
inst✝¹ : IsTopologicalAddGroup Y
inst✝ : ContinuousSMul k Y
ρ : ContRepresentation k G X
σ : ContRepresentation k G Y
A B C : TopRep k G
X✝ Y✝ : TopRep k G
f : X✝ ⟶ Y✝
g : G
⊢ ConcreteCategory.hom
      ({ V := TopModuleCat.of k ↑X✝, ρ := (TopModuleCat.of k ↑X✝).endRingEquiv.symm.toMonoidHom.comp ↑X✝.ρ }.ρ g ≫
        Hom.toTopModuleCatHom f) =
    ConcreteCategory.hom
      (Hom.toTopModuleCatHom f ≫
        { V := TopModuleCat.of k ↑Y✝, ρ := (TopModuleCat.of k ↑Y✝).endRingEquiv.symm.toMonoidHom.comp ↑Y✝.ρ }.ρ g)
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A B C : Rep k G
f₁ f₂ : A ⟶ B
g : B ⟶ C
⊢ (f₁ + f₂) ≫ g = f₁ ≫ g + f₂ ≫ g

Tactic:
ext1

Goal state after:
k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A B C : Rep k G
f₁ f₂ : A ⟶ B
g : B ⟶ C
⊢ Hom.hom ((f₁ + f₂) ≫ g) = Hom.hom (f₁ ≫ g + f₂ ≫ g)
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A B C : Rep k G
f : A ⟶ B
g₁ g₂ : B ⟶ C
⊢ f ≫ (g₁ + g₂) = f ≫ g₁ + f ≫ g₂

Tactic:
ext1

Goal state after:
k : Type u
G : Type v
inst✝¹ : Semiring k
inst✝ : Monoid G
A B C : Rep k G
f : A ⟶ B
g₁ g₂ : B ⟶ C
⊢ Hom.hom (f ≫ (g₁ + g₂)) = Hom.hom (f ≫ g₁ + f ≫ g₂)
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : Ring k
inst✝ : Monoid G
A B C X Y : Rep k G
f g : X ⟶ Y
⊢ (forget₂ (Rep k G) (ModuleCat k)).map (f + g) =
    (forget₂ (Rep k G) (ModuleCat k)).map f + (forget₂ (Rep k G) (ModuleCat k)).map g

Tactic:
ext1

Goal state after:
k : Type u
G : Type v
inst✝¹ : Ring k
inst✝ : Monoid G
A B C X Y : Rep k G
f g : X ⟶ Y
⊢ ModuleCat.Hom.hom ((forget₂ (Rep k G) (ModuleCat k)).map (f + g)) =
    ModuleCat.Hom.hom ((forget₂ (Rep k G) (ModuleCat k)).map f + (forget₂ (Rep k G) (ModuleCat k)).map g)