mathlib-initiative/mathlib-tactics huggingface dataset:
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] fun ω x ↦ Z ω

Tactic:
refine indicator_indepFun_pi_of_prod_bcf mA (fun _ ↦ mZ) fun f ↦ ?_

Goal state after:
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
f : Unit → G →ᵇ ℝ
⊢ ∫ (ω : Ω) in A, ∏ s, (f s) (Z ω) ∂P = P.real A * ∫ (ω : Ω), ∏ s, (f s) (Z ω) ∂P
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
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
this :
  ∀ (p q : (ι → Ω → E) → Prop),
    IsStable 𝓕 p →
      IsStable 𝓕 q →
        ∀ (hpX : Locally p 𝓕 X P) (hqX : Locally q 𝓕 X P),
          p
            (stoppedProcess (fun i ↦ {ω | ⊥ < (hpX.localSeq ⊓ hqX.localSeq) n ω}.indicator (X i))
              ((hpX.localSeq ⊓ hqX.localSeq) n))
⊢ q
    (stoppedProcess (fun i ↦ {ω | ⊥ < (hqX.localSeq ⊓ hpX.localSeq) n ω}.indicator (X i))
      ((hqX.localSeq ⊓ hpX.localSeq) n))

Tactic:
exact this q p hq hp hqX hpX

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: α : Type u_1
f : α → ℝ
⊢ truncation f 0 = 0

Tactic:
simp [truncation]

Goal state after:
α : Type u_1
f : α → ℝ
⊢ (fun x ↦ 0) ∘ f = 0
Module: Mathlib.Probability.Process.LocalProperty

Goal state before:
pp: ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : ConditionallyCompleteLinearOrderBot ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
𝓕 : Filtration ι mΩ
inst✝¹ : SecondCountableTopology ι
inst✝ : IsFiniteMeasure P
τ : ℕ → Ω → WithTop ι
σ : ℕ → ℕ → Ω → WithTop ι
hτ : IsLocalizingSequence 𝓕 τ P
hσ : ∀ (n : ℕ), IsLocalizingSequence 𝓕 (σ n) P
T : ℕ → ι
hT : Tendsto T atTop atTop
n : ℕ
hn : ∀ (k : ℕ), (1 / 2) ^ n < P {ω | σ n k ω < min (τ n ω) ↑(T n)}
this : (1 / 2) ^ n ≤ P (⋂ k, {ω | σ n k ω < min (τ n ω) ↑(T n)})
ω : Ω
hTop : ∀ (i : ι), ∃ a, ∀ (b : ℕ), a ≤ b → ↑i < σ n b ω
hmem : ∀ (i : ℕ), σ n i ω < min (τ n ω) ↑(T n)
N : ℕ
hN : ↑(T n) < σ n N ω
⊢ False

Tactic:
specialize hmem N

Goal state after:
ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : ConditionallyCompleteLinearOrderBot ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
𝓕 : Filtration ι mΩ
inst✝¹ : SecondCountableTopology ι
inst✝ : IsFiniteMeasure P
τ : ℕ → Ω → WithTop ι
σ : ℕ → ℕ → Ω → WithTop ι
hτ : IsLocalizingSequence 𝓕 τ P
hσ : ∀ (n : ℕ), IsLocalizingSequence 𝓕 (σ n) P
T : ℕ → ι
hT : Tendsto T atTop atTop
n : ℕ
hn : ∀ (k : ℕ), (1 / 2) ^ n < P {ω | σ n k ω < min (τ n ω) ↑(T n)}
this : (1 / 2) ^ n ≤ P (⋂ k, {ω | σ n k ω < min (τ n ω) ↑(T n)})
ω : Ω
hTop : ∀ (i : ι), ∃ a, ∀ (b : ℕ), a ≤ b → ↑i < σ n b ω
N : ℕ
hmem : σ n N ω < min (τ n ω) ↑(T n)
hN : ↑(T n) < σ n N ω
⊢ False
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEStronglyMeasurable f μ
A : ℝ
n : ℕ
hn : n ≠ 0
h'f : 0 ≤ f
M : MeasurableSet (Set.Ioc 0 A)
M' : MeasurableSet (Set.Ioc A 0)
⊢ ∫ (x : α), (fun z ↦ (Set.Ioc 0 A).indicator id z ^ n) (f x) ∂μ = ∫ (y : ℝ) in 0..A, y ^ n ∂Measure.map f μ

Tactic:
rcases le_or_gt 0 A with (hA | hA)

Goal state after:
case inl
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEStronglyMeasurable f μ
A : ℝ
n : ℕ
hn : n ≠ 0
h'f : 0 ≤ f
M : MeasurableSet (Set.Ioc 0 A)
M' : MeasurableSet (Set.Ioc A 0)
hA : 0 ≤ A
⊢ ∫ (x : α), (fun z ↦ (Set.Ioc 0 A).indicator id z ^ n) (f x) ∂μ = ∫ (y : ℝ) in 0..A, y ^ n ∂Measure.map f μ

case inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEStronglyMeasurable f μ
A : ℝ
n : ℕ
hn : n ≠ 0
h'f : 0 ≤ f
M : MeasurableSet (Set.Ioc 0 A)
M' : MeasurableSet (Set.Ioc A 0)
hA : A < 0
⊢ ∫ (x : α), (fun z ↦ (Set.Ioc 0 A).indicator id z ^ n) (f x) ∂μ = ∫ (y : ℝ) in 0..A, y ^ n ∂Measure.map f μ
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case h₂
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K N : ℕ
hKN : K ≤ N
ρ : Measure ℝ := Measure.map X ℙ
this : IsProbabilityMeasure ρ
⊢ ∫ (x : ℝ) in Set.Ioc 0 ↑N, 1 ∂ρ ≤ 1

Tactic:
simp only [integral_const, measureReal_restrict_apply', measurableSet_Ioc, Set.univ_inter,
          smul_eq_mul, mul_one]

Goal state after:
case h₂
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K N : ℕ
hKN : K ≤ N
ρ : Measure ℝ := Measure.map X ℙ
this : IsProbabilityMeasure ρ
⊢ ρ.real (Set.Ioc 0 ↑N) ≤ 1
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
Ω'' : Type u_3
mΩ'' : MeasurableSpace Ω''
Y : Ω'' → ℝ
cY : ℝ≥0
inst✝¹ : SFinite ν
η : Kernel (Ω' × Ω) Ω''
inst✝ : IsZeroOrMarkovKernel η
hX : HasSubgaussianMGF X c κ ν
hY : HasSubgaussianMGF Y cY η (ν ⊗ₘ κ)
hκ : IsSFiniteKernel κ
q : ℚ
ω' : Ω'
hX_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
hX_int : Integrable (fun y ↦ rexp (↑q * X y)) (κ ω')
hY_mgf : ∀ᵐ (b : Ω) ∂κ ω', ∀ (t : ℝ), mgf Y (η (ω', b)) t ≤ rexp (↑cY * t ^ 2 / 2)
⊢ Integrable (fun ω ↦ rexp (↑q * (X ω.1 + Y ω.2))) ((κ ⊗ₖ η) ω') →
    mgf (fun p ↦ X p.1 + Y p.2) ((κ ⊗ₖ η) ω') ↑q ≤ rexp (↑(c + cY) * ↑q ^ 2 / 2)

Tactic:
h_int_mul

Goal state after:
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
Ω'' : Type u_3
mΩ'' : MeasurableSpace Ω''
Y : Ω'' → ℝ
cY : ℝ≥0
inst✝¹ : SFinite ν
η : Kernel (Ω' × Ω) Ω''
inst✝ : IsZeroOrMarkovKernel η
hX : HasSubgaussianMGF X c κ ν
hY : HasSubgaussianMGF Y cY η (ν ⊗ₘ κ)
hκ : IsSFiniteKernel κ
q : ℚ
ω' : Ω'
hX_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
hX_int : Integrable (fun y ↦ rexp (↑q * X y)) (κ ω')
hY_mgf : ∀ᵐ (b : Ω) ∂κ ω', ∀ (t : ℝ), mgf Y (η (ω', b)) t ≤ rexp (↑cY * t ^ 2 / 2)
h_int_mul : Integrable (fun ω ↦ rexp (↑q * (X ω.1 + Y ω.2))) ((κ ⊗ₖ η) ω')
⊢ mgf (fun p ↦ X p.1 + Y p.2) ((κ ⊗ₖ η) ω') ↑q ≤ rexp (↑(c + cY) * ↑q ^ 2 / 2)
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 : 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
⊢ ∑ j ∈ range K, (↑j ^ 2)⁻¹ * ∫ (x : ℝ) in 0..↑j, x ^ 2 ∂ρ =
    ∑ j ∈ range K, (↑j ^ 2)⁻¹ * ∑ k ∈ range j, ∫ (x : ℝ) in ↑k..↑(k + 1), x ^ 2 ∂ρ

Tactic:
congr 1 with j

Goal state after:
case e_f
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
j : ℕ
⊢ (↑j ^ 2)⁻¹ * ∫ (x : ℝ) in 0..↑j, x ^ 2 ∂ρ = (↑j ^ 2)⁻¹ * ∑ k ∈ range j, ∫ (x : ℝ) in ↑k..↑(k + 1), x ^ 2 ∂ρ
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case h.refine_1
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)
⊢ IntegrableOn (fun x ↦ 2 / (↑k + 1) * x ^ 2) (Set.Ioc ↑k ↑(k + 1)) ρ

Tactic:
apply Continuous.integrableOn_Ioc (by fun_prop)

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case h.refine_2
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)
⊢ IntegrableOn (HMul.hMul 2) (Set.Ioc ↑k ↑(k + 1)) ρ

Tactic:
apply Continuous.integrableOn_Ioc (by fun_prop)

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
K : ℕ
⊢ ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ ∑ j ∈ range K, (↑j ^ 2)⁻¹ * ∫ (a : Ω), (truncation (X 0) ↑j ^ 2) a

Tactic:
gcongr with j

Goal state after:
case hbc
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
K j : ℕ
a✝ : j ∈ range K
⊢ Var[Y j; ℙ] ≤ ∫ (a : Ω), (truncation (X 0) ↑j ^ 2) a
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:
simp [f.isIntertwining, g.isIntertwining]

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 : σ.IntertwiningMap τ
g : ρ.IntertwiningMap σ
γ : G
v : V
⊢ (f.toLinearMap ∘ₗ g.toLinearMap) ((ρ γ) v) = (τ γ) ((f.toLinearMap ∘ₗ g.toLinearMap) v)

Tactic:
simp [f.isIntertwining, g.isIntertwining]

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
g₁ g₂ : τ.IntertwiningMap π
⊢ f.tensor (g₁ + g₂) = f.tensor g₁ + f.tensor g₂

Tactic:
ext; simp [TensorProduct.tmul_add]

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

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : CommRing k
inst✝ : Monoid G
V : Rep k G
r : k
x : ↑V
⊢ (RestrictScalars.addEquiv k k[G] V.ρ.asModule).symm (V.ρ.asModuleEquiv.symm (r • x)) =
    (RingHom.id k) r • unitIsoAddEquiv.toFun x

Tactic:
simp only [Representation.asModuleEquiv_symm_map_smul]

Goal state after:
k : Type u
G : Type v
inst✝¹ : CommRing k
inst✝ : Monoid G
V : Rep k G
r : k
x : ↑V
⊢ (RestrictScalars.addEquiv k k[G] V.ρ.asModule).symm ((algebraMap k k[G]) r • V.ρ.asModuleEquiv.symm x) =
    (RingHom.id k) r • unitIsoAddEquiv.toFun x
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⟩
    exact {
      exact := by rwa [ShortComplex.exact_map_iff_of_faithful]
      mono_f := by simpa [mono_iff_injective] using! mono_f
      epi_g := by simpa [epi_iff_surjective] using! epi_g
    }

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: 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
c : E
s : Set E
hs : MeasurableSet s
F : E → ℝ := s.indicator 1
F_meas : Measurable F
Y : ℕ → Ω → ℝ := fun n ↦ F ∘ X n
this : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y i ω) atTop (𝓝 (∫ (x : Ω), Y 0 x ∂μ))
ω : Ω
hω : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y i ω) atTop (𝓝 (∫ (x : Ω), Y 0 x ∂μ))
⊢ Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, s.indicator (Function.const E c) (X i ω)) atTop
    (𝓝 (∫ (x : Ω), s.indicator (Function.const E c) (X 0 x) ∂μ))

Tactic:
have I : indicator s (Function.const E c) = (fun x ↦ (indicator s (1 : E → ℝ) x) • c) := by
      ext
      rw [← indicator_smul_const_apply]
      congr! 1
      ext
      simp

Goal state after:
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
c : E
s : Set E
hs : MeasurableSet s
F : E → ℝ := s.indicator 1
F_meas : Measurable F
Y : ℕ → Ω → ℝ := fun n ↦ F ∘ X n
this : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y i ω) atTop (𝓝 (∫ (x : Ω), Y 0 x ∂μ))
ω : Ω
hω : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y i ω) atTop (𝓝 (∫ (x : Ω), Y 0 x ∂μ))
I : s.indicator (Function.const E c) = fun x ↦ s.indicator 1 x • c
⊢ Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, s.indicator (Function.const E c) (X i ω)) atTop
    (𝓝 (∫ (x : Ω), s.indicator (Function.const E c) (X 0 x) ∂μ))
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 : res φ (leftRegular k H) ⟶ A
g : G
h : H
⊢ (fun h ↦ (Hom.hom f).toLinearMap (MonoidAlgebra.single h 1)) (φ g * h) =
    (A.ρ g) ((fun h ↦ (Hom.hom f).toLinearMap (MonoidAlgebra.single h 1)) h)

Tactic:
by
    simp only [res_obj_V, res_obj_ρ, Representation.IntertwiningMap.toLinearMap_apply]
    have := by simpa using (hom_comm_apply f g (.single h 1)).symm
    rw [← this]

Goal state after:
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 A B : Rep k G
⊢ ofHom (Representation.IntertwiningMap.inl k A.ρ B.ρ) ≫ ofHom (Representation.IntertwiningMap.snd k A.ρ B.ρ) = 0

Tactic:
ext1; simp [zero_hom]

Goal state after:
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 A B : Rep k G
⊢ ofHom (Representation.IntertwiningMap.inr k A.ρ B.ρ) ≫ ofHom (Representation.IntertwiningMap.fst k A.ρ B.ρ) = 0

Tactic:
ext1; simp [zero_hom]

Goal state after:
Module: Mathlib.RepresentationTheory.FiniteIndex

Goal state before:
pp: case inl
k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g : G
a : ↑A
s s₁ : ↥S
⊢ (if h : (QuotientGroup.rightRel S) (↑s * s₁ • g) g then A.ρ ⟨↑s * s₁ • g * g⁻¹, ⋯⟩ else 0) a =
    (A.ρ s) ((if h : (QuotientGroup.rightRel S) (s₁ • g) g then A.ρ ⟨s₁ • g * g⁻¹, ⋯⟩ else 0) a)

Tactic:
rw [dif_pos ⟨s * s₁, mul_assoc ..⟩, dif_pos ⟨s₁, rfl⟩]

Goal state after:
case inl
k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g : G
a : ↑A
s s₁ : ↥S
⊢ (A.ρ ⟨↑s * s₁ • g * g⁻¹, ⋯⟩) a = (A.ρ s) ((A.ρ ⟨s₁ • g * g⁻¹, ⋯⟩) a)
Module: Mathlib.RepresentationTheory.FiniteIndex

Goal state before:
pp: k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g₁ g₂ : G
a : ↑A
s : ↥S
⊢ (A.indToCoindAux (↑s * g₁)) ((A.ρ s) a) g₂ = (A.indToCoindAux g₁) a g₂

Tactic:
rcases em ((QuotientGroup.rightRel S).r g₂ g₁) with ⟨s₁, rfl⟩ | h
  · simp only [indToCoindAux, LinearMap.pi_apply]
    rw [dif_pos ⟨s₁ * s⁻¹, by simp [S.1.smul_def, smul_eq_mul, mul_assoc]⟩, dif_pos ⟨s₁, rfl⟩,
      ← Module.End.mul_apply, ← map_mul]
    congr
    simp [Subtype.ext_iff, S.1.smul_def, mul_assoc]
  · rw [indToCoindAux_of_not_rel (h := h), indToCoindAux_of_not_rel]
    exact mt (fun ⟨s₁, hs₁⟩ => ⟨s₁ * s, by simp_all [S.1.smul_def, mul_assoc]⟩) h

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality

Goal state before:
pp: k : Type u
G : Type v
inst✝⁴ : Ring k
inst✝³ : TopologicalSpace k
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
X : TopRep k G
n : ℕ
⊢ cocyclesMap (ContinuousMonoidHom.id G) (𝟙 X) n = 𝟙 (cocycles X n)

Tactic:
by
  simp [cocyclesMap]

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.ContCohomology.LowDegree

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝⁴ : Ring k
inst✝³ : Group G
inst✝² : TopologicalSpace k
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
X : TopRep k G
x : ↑X
h : ContinuousMap.const G x ∈ (X.resolution'.X 0).ρ.invariants
⊢ ⟨ContinuousMap.const G x, h⟩ ∈ (↑(TopModuleCat.Hom.hom (X.homogeneousCochains.d 0 1))).ker

Tactic:
by
  rw [LinearMap.mem_ker, Subtype.ext_iff, ContinuousLinearMap.coe_coe,
    homogeneousCochains.d_apply]
  simp [d_succ, hom_sub, ContIntertwiningMap.sub_apply, d_zero]

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.Hilbert90

Goal state before:
pp: case h
K L : Type
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
g : Gal(L/K)
hg : ∀ (x : Gal(L/K)), x ∈ Subgroup.zpowers g
x : L
hx : (Algebra.norm K) x = 1
H : ∀ (x : L), (Algebra.norm K) x = 1 → ∃ y, g ↑y / ↑y = x
hxinv : (Algebra.norm K) x⁻¹ = 1
y : Lˣ
hy : g ↑y = x⁻¹ * ↑y
⊢ ↑y / g ↑y = x

Tactic:
rw [hy]

Goal state after:
case h
K L : Type
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
g : Gal(L/K)
hg : ∀ (x : Gal(L/K)), x ∈ Subgroup.zpowers g
x : L
hx : (Algebra.norm K) x = 1
H : ∀ (x : L), (Algebra.norm K) x = 1 → ∃ y, g ↑y / ↑y = x
hxinv : (Algebra.norm K) x⁻¹ = 1
y : Lˣ
hy : g ↑y = x⁻¹ * ↑y
⊢ ↑y / (x⁻¹ * ↑y) = x
Module: Mathlib.RepresentationTheory.Homological.Resolution

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Monoid G
x : Fin 1 → G
⊢ (ModuleCat.Hom.hom ((forget₂ToModuleCatHomotopyEquiv k G).hom.f 0) ∘ₗ MonoidAlgebra.lsingle x) 1 =
    (ModuleCat.Hom.hom ((forget₂ (Rep k G) (ModuleCat k)).map (ε k G)) ∘ₗ MonoidAlgebra.lsingle x) 1

Tactic:
simp [forget₂ToModuleCatHomotopyEquiv, HomotopyEquiv.ofIso, HomotopyEquiv.trans,
    SimplicialObject.Augmented.ExtraDegeneracy.homotopyEquiv, ChainComplex.single₀_map_f_zero,
    AlgebraicTopology.AlternatingFaceMapComplex.ε_app_f_zero, compForgetAugmentedIso, eqToIso.inv,
    HomologicalComplex.eqToHom_f, compForgetAugmented, compForgetAugmented.toModule, ε,
    SimplicialObject.augment, Unique.eq_default (terminal.from _), MonoidAlgebra.coeff_single,
    Finsupp.single_apply, if_pos (Subsingleton.elim _ _)]

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality

Goal state before:
pp: k G H : Type u
inst✝² : CommRing k
inst✝¹ : Group G
inst✝ : Group H
A : Rep k H
B : Rep k G
f : G →* H
φ : res f A ⟶ B
n : ℕ
⊢ cochainsMap₁ f φ ≫ (shortComplexH1 B).g = (shortComplexH1 A).g ≫ cochainsMap₂ f φ

Tactic:
by
    ext x
    funext g
    simpa [shortComplexH1, d₁₂, cochainsMap₁, cochainsMap₂] using (hom_comm_apply φ _ _).symm

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Ideal

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
x : MvPolynomial σ R
s : Set (σ →₀ ℕ)
xi : σ →₀ ℕ
hxi : xi ∈ x.support
⊢ (∃ si ∈ s, si ≤ xi) ↔ ∃ si ∈ s, coeff xi x = 0 ∨ si ≤ xi

Tactic:
mem_support_iff.mp hxi,

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
x : MvPolynomial σ R
s : Set (σ →₀ ℕ)
xi : σ →₀ ℕ
hxi : xi ∈ x.support
⊢ (∃ si ∈ s, si ≤ xi) ↔ ∃ si ∈ s, False ∨ si ≤ xi
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
e : σ ↪ τ
t : τ
h : t ∉ Set.range ⇑e
⊢ ¬↑(single t 1).support ⊆ Set.range ⇑e

Tactic:
support_single _ (by simp),

Goal state after:
σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
e : σ ↪ τ
t : τ
h : t ∉ Set.range ⇑e
⊢ ¬↑{t} ⊆ Set.range ⇑e
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
f : G →₀ ↑A
h : f ∈ boundaries₁ A
⊢ f ∈ cycles₁ A

Tactic:
rcases h with ⟨x, rfl⟩
  exact d₂₁_apply_mem_cycles₁ x

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: case mp
G : Type u_1
A : Type u_2
inst✝² : Group G
inst✝¹ : AddCommGroup A
inst✝ : DistribMulAction G A
x : G →₀ A
⊢ IsBoundary₁ x → ∃ y, (y.sum fun g a ↦ single g.2 a - single (g.1 * g.2) (g.1 • a) + single g.1 (g.1 • a)) = x

Tactic:
rintro ⟨y, hy⟩
    use y.sum (fun g a => single g (g.1⁻¹ • a))
    simp_all [sum_sum_index]

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: case mp
G : Type u_1
A : Type u_2
inst✝² : Group G
inst✝¹ : AddCommGroup A
inst✝ : DistribMulAction G A
x : G × G →₀ A
⊢ IsBoundary₂ x →
    ∃ y,
      (y.sum fun g a ↦
          single (g.2.1, g.2.2) a - single (g.1 * g.2.1, g.2.2) (g.1 • a) + single (g.1, g.2.1 * g.2.2) (g.1 • a) -
            single (g.1, g.2.1) (g.1 • a)) =
        x

Tactic:
rintro ⟨y, hy⟩
    use y.sum (fun g a => single g (g.1⁻¹ • a))
    simp_all [sum_sum_index]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Equiv

Goal state before:
pp: case refine_1
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
p q : MvPowerSeries (Option σ) R
k : ℕ
x : σ →₀ ℕ
⊢ ∀ a ∈ antidiagonal (optionElim k x),
    ⟨(a.1 none, a.2 none), (a.1.some, a.2.some)⟩ ∈ (antidiagonal k).sigma fun a ↦ antidiagonal x

Tactic:
intros; simp_all [Finsupp.ext_iff]

Goal state after:
Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology

Goal state before:
pp: case mp
A : Type u_2
inst✝² : CommRing A
inst✝¹ : TopologicalSpace A
inst✝ : IsTopologicalRing A
⊢ ((∀ (n : ℕ), IsOpen[inst✝¹] ↑(⊥ ^ n)) ∧ ∀ s ∈ 𝓝 0, ∃ n, ↑(⊥ ^ n) ⊆ s) → DiscreteTopology A

Tactic:
rintro ⟨h, _h'⟩
    rw [discreteTopology_iff_isOpen_singleton_zero]
    simpa using h 1

Goal state after: