mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this : IsFiniteKernel κ
hu_meas : MeasurableSet u
x : γ
h0 : (ν a) (countablePartitionSet n x) = 0
⊢ (κ.fst a) (countablePartitionSet n x) = 0

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this : IsFiniteKernel κ
hu_meas : MeasurableSet u
x : γ
h0 : (ν a) (countablePartitionSet n x) = 0
⊢ (κ.fst a) (countablePartitionSet n x) = 0
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this✝ : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u, (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ∂ν a =
    ∫⁻ (x : γ) in u, (κ a) (u ×ˢ s) / (ν a) u ∂ν a
h0 : (ν a) u = 0
⊢ (κ.fst a) u = 0

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this✝ : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u, (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ∂ν a =
    ∫⁻ (x : γ) in u, (κ a) (u ×ˢ s) / (ν a) u ∂ν a
h0 : (ν a) u = 0
⊢ (κ.fst a) u = 0
Module: Mathlib.Probability.Kernel.Disintegration.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
κ : Kernel α (β × Ω)
κCond✝ : Kernel (α × β) Ω
inst✝ : Countable α
κCond : α → Kernel β Ω
h_atom : ∀ (x y : α), x ∈ measurableAtom y → κCond x = κCond y
x y : α
hx : β
hy : y ∈ measurableAtom x
⊢ (κCond (y, hx).1) (y, hx).2 = (κCond (x, hx).1) (x, hx).2

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
κ : Kernel α (β × Ω)
κCond✝ : Kernel (α × β) Ω
inst✝ : Countable α
κCond : α → Kernel β Ω
h_atom : ∀ (x y : α), x ∈ measurableAtom y → κCond x = κCond y
x y : α
hx : β
hy : y ∈ measurableAtom x
⊢ (κCond y) hx = (κCond x) hx
Module: Mathlib.Probability.Kernel.Disintegration.Integral

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
inst✝¹ : CountableOrCountablyGenerated α β
κ : Kernel α (β × Ω)
inst✝ : IsFiniteKernel κ
a : α
s : Set β
hs : MeasurableSet s
t : Set Ω
ht : MeasurableSet t
this : (κ a) (s ×ˢ t) = ((κ.fst ⊗ₖ κ.condKernel) a) (s ×ˢ t)
⊢ ∫⁻ (b : β) in s, (κ.condKernel (a, b)) t ∂κ.fst a = (κ a) (s ×ˢ t)

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
inst✝¹ : CountableOrCountablyGenerated α β
κ : Kernel α (β × Ω)
inst✝ : IsFiniteKernel κ
a : α
s : Set β
hs : MeasurableSet s
t : Set Ω
ht : MeasurableSet t
this : (κ a) (s ×ˢ t) = ((κ.fst ⊗ₖ κ.condKernel) a) (s ×ˢ t)
⊢ ∫⁻ (b : β) in s, (κ.condKernel (a, b)) t ∂κ.fst a = ((κ.fst ⊗ₖ κ.condKernel) a) (s ×ˢ t)
Module: Mathlib.Probability.Kernel.Disintegration.Integral

Goal state before:
pp: β : Type u_1
Ω : Type u_2
mβ : MeasurableSpace β
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
ρ : Measure (β × Ω)
inst✝ : IsFiniteMeasure ρ
s : Set β
hs : MeasurableSet s
t : Set Ω
ht : MeasurableSet t
this : ρ (s ×ˢ t) = (ρ.fst ⊗ₘ ρ.condKernel) (s ×ˢ t)
⊢ ∫⁻ (b : β) in s, (ρ.condKernel b) t ∂ρ.fst = ρ (s ×ˢ t)

Tactic:
simpa using

Goal state after:
β : Type u_1
Ω : Type u_2
mβ : MeasurableSpace β
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
ρ : Measure (β × Ω)
inst✝ : IsFiniteMeasure ρ
s : Set β
hs : MeasurableSet s
t : Set Ω
ht : MeasurableSet t
this : ρ (s ×ˢ t) = (ρ.fst ⊗ₘ ρ.condKernel) (s ×ˢ t)
⊢ ∫⁻ (b : β) in s, (ρ.condKernel b) t ∂ρ.fst = (ρ.fst ⊗ₘ ρ.condKernel) (s ×ˢ t)
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: case neg.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 : ¬(ν a) (countablePartitionSet n x) = 0
this :
  Tendsto (fun m ↦ (κ a) (countablePartitionSet n x ×ˢ seq m)) atTop
    (𝓝 ((κ a) (⋂ n_1, countablePartitionSet n x ×ˢ seq n_1)))
⊢ Tendsto (fun m ↦ (κ a) (countablePartitionSet n x ×ˢ seq m)) atTop (𝓝 ((κ a) (countablePartitionSet n x ×ˢ ∅)))

Tactic:
simpa using

Goal state after:
case neg.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 : ¬(ν a) (countablePartitionSet n x) = 0
this :
  Tendsto (fun m ↦ (κ a) (countablePartitionSet n x ×ˢ seq m)) atTop
    (𝓝 ((κ a) (⋂ n_1, countablePartitionSet n x ×ˢ seq n_1)))
⊢ Tendsto (fun m ↦ (κ a) (countablePartitionSet n x ×ˢ seq m)) atTop (𝓝 ((κ a) (countablePartitionSet n x ×ˢ ∅)))
Module: Mathlib.Probability.Kernel.IonescuTulcea.Maps

Goal state before:
pp: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : LocallyFiniteOrder ι
inst✝¹ : DecidableLE ι
X : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (X i)
a b c : ι
i : ↥(Ioc a c)
h : ↑i ≤ b
⊢ Measurable fun c_1 ↦ IocProdIoc a b c c_1 i

Tactic:
simpa using

Goal state after:
case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : LocallyFiniteOrder ι
inst✝¹ : DecidableLE ι
X : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (X i)
a b c : ι
i : ↥(Ioc a c)
h : ↑i ≤ b
⊢ Measurable fun c_1 ↦ c_1.1 ⟨↑i, ⋯⟩
Module: Mathlib.Probability.Kernel.IonescuTulcea.Maps

Goal state before:
pp: case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : LocallyFiniteOrder ι
inst✝¹ : DecidableLE ι
X : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (X i)
a b c : ι
i : ↥(Ioc a c)
h : ¬↑i ≤ b
⊢ Measurable fun c_1 ↦ IocProdIoc a b c c_1 i

Tactic:
simpa using

Goal state after:
case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : LocallyFiniteOrder ι
inst✝¹ : DecidableLE ι
X : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (X i)
a b c : ι
i : ↥(Ioc a c)
h : ¬↑i ≤ b
⊢ Measurable fun c_1 ↦ c_1.2 ⟨↑i, ⋯⟩
Module: Mathlib.Probability.Kernel.IonescuTulcea.Maps

Goal state before:
pp: case pos
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrder ι
inst✝² : DecidableLE ι
X : ι → Type u_2
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : (i : ι) → MeasurableSpace (X i)
m n : ι
i : ↥(Iic n)
h : ↑i ≤ m
⊢ Measurable fun c ↦ IicProdIoc m n c i

Tactic:
simpa using

Goal state after:
case pos
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrder ι
inst✝² : DecidableLE ι
X : ι → Type u_2
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : (i : ι) → MeasurableSpace (X i)
m n : ι
i : ↥(Iic n)
h : ↑i ≤ m
⊢ Measurable fun c ↦ c.1 ⟨↑i, ⋯⟩
Module: Mathlib.Probability.Kernel.IonescuTulcea.Maps

Goal state before:
pp: case neg
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrder ι
inst✝² : DecidableLE ι
X : ι → Type u_2
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : (i : ι) → MeasurableSpace (X i)
m n : ι
i : ↥(Iic n)
h : ¬↑i ≤ m
⊢ Measurable fun c ↦ IicProdIoc m n c i

Tactic:
simpa using

Goal state after:
case neg
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrder ι
inst✝² : DecidableLE ι
X : ι → Type u_2
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : (i : ι) → MeasurableSpace (X i)
m n : ι
i : ↥(Iic n)
h : ¬↑i ≤ m
⊢ Measurable fun c ↦ c.2 ⟨↑i, ⋯⟩
Module: Mathlib.Probability.Kernel.IonescuTulcea.Maps

Goal state before:
pp: case pos
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrder ι
inst✝² : DecidableLE ι
X : ι → Type u_2
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : (i : ι) → MeasurableSpace (X i)
a b : ι
hab : a ≤ b
x : ↥(Iic b)
h : ↑x ≤ a
⊢ Measurable fun c ↦
    { toFun := fun x i ↦ if h : ↑i ≤ a then x.1 ⟨↑i, ⋯⟩ else x.2 ⟨↑i, ⋯⟩,
        invFun := fun x ↦ (fun i ↦ x ⟨↑i, ⋯⟩, fun i ↦ x ⟨↑i, ⋯⟩), left_inv := ⋯, right_inv := ⋯ }
      c x

Tactic:
simpa using

Goal state after:
case pos
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrder ι
inst✝² : DecidableLE ι
X : ι → Type u_2
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : (i : ι) → MeasurableSpace (X i)
a b : ι
hab : a ≤ b
x : ↥(Iic b)
h : ↑x ≤ a
⊢ Measurable fun c ↦ c.1 ⟨↑x, ⋯⟩
Module: Mathlib.Probability.Kernel.IonescuTulcea.Maps

Goal state before:
pp: case neg
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrder ι
inst✝² : DecidableLE ι
X : ι → Type u_2
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : (i : ι) → MeasurableSpace (X i)
a b : ι
hab : a ≤ b
x : ↥(Iic b)
h : ¬↑x ≤ a
⊢ Measurable fun c ↦
    { toFun := fun x i ↦ if h : ↑i ≤ a then x.1 ⟨↑i, ⋯⟩ else x.2 ⟨↑i, ⋯⟩,
        invFun := fun x ↦ (fun i ↦ x ⟨↑i, ⋯⟩, fun i ↦ x ⟨↑i, ⋯⟩), left_inv := ⋯, right_inv := ⋯ }
      c x

Tactic:
simpa using

Goal state after:
case neg
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrder ι
inst✝² : DecidableLE ι
X : ι → Type u_2
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : (i : ι) → MeasurableSpace (X i)
a b : ι
hab : a ≤ b
x : ↥(Iic b)
h : ¬↑x ≤ a
⊢ Measurable fun c ↦ c.2 ⟨↑x, ⋯⟩
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: case refine_1.refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
x : ℝ≥0∞
hx : x ∈ Iio ((κ a) univ + 1)
⊢ x < (κ a) univ + 1

Tactic:
simpa using

Goal state after:
case refine_1.refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
x : ℝ≥0∞
hx : x ∈ Iio ((κ a) univ + 1)
⊢ x < (κ a) univ + 1
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal 0
h : Tendsto (fun m ↦ (κ a) (univ ×ˢ seq m)) atTop (𝓝 ((κ a) (⋂ n, (fun m ↦ univ ×ˢ seq m) n)))
⊢ Tendsto (fun m ↦ (κ a) (univ ×ˢ seq m)) atTop (𝓝 0)

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal 0
h : Tendsto (fun m ↦ (κ a) (univ ×ˢ seq m)) atTop (𝓝 ((κ a) (⋂ n, (fun m ↦ univ ×ˢ seq m) n)))
⊢ Tendsto (fun m ↦ (κ a) (univ ×ˢ seq m)) atTop (𝓝 0)
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
hx : ¬(if (κ.fst a) (countablePartitionSet n x) = 0 then 0 else 1) = 1
⊢ (κ.fst a) (countablePartitionSet n x) = 0

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
hx : ¬(if (κ.fst a) (countablePartitionSet n x) = 0 then 0 else 1) = 1
⊢ (κ.fst a) (countablePartitionSet n x) = 0
Module: Mathlib.Probability.Distributions.SetBernoulli

Goal state before:
pp: ι : Type u_1
inst✝ : Countable ι
u : Set ι
hu : u.Finite
p : ↑I
k : ℕ
this : {s | s ⊆ u ∧ s.ncard ∈ {k}}.Finite
h1 : ∀ s ∈ this.toFinset, setBer(u, p).real {s} = ↑p ^ k * (1 - ↑p) ^ (u.ncard - k)
⊢ ↑(#this.toFinset) * (↑p ^ k * (1 - ↑p) ^ (u.ncard - k)) = ↑(u.ncard.choose k) * (↑p ^ k * (1 - ↑p) ^ (u.ncard - k))

Tactic:
← Set.ncard_eq_toFinset_card _ _

Goal state after:
ι : Type u_1
inst✝ : Countable ι
u : Set ι
hu : u.Finite
p : ↑I
k : ℕ
this : {s | s ⊆ u ∧ s.ncard ∈ {k}}.Finite
h1 : ∀ s ∈ this.toFinset, setBer(u, p).real {s} = ↑p ^ k * (1 - ↑p) ^ (u.ncard - k)
⊢ ↑{s | s ⊆ u ∧ s.ncard ∈ {k}}.ncard * (↑p ^ k * (1 - ↑p) ^ (u.ncard - k)) =
    ↑(u.ncard.choose k) * (↑p ^ k * (1 - ↑p) ^ (u.ncard - k))
Module: Mathlib.Probability.Distributions.SetBernoulli

Goal state before:
pp: case pos
ι : Type u_1
p : ↑I
inst✝ : Countable ι
s : Set (Set ι)
hs : MeasurableSet s
h : ∅ ∈ s
⊢ setBer(∅, p) {s_1 | s_1 ∈ s ∧ s_1 ⊆ ∅} = (dirac ∅) s

Tactic:
have : {t | t ∈ s ∧ t ⊆ ∅} = {∅} := by grind
    simp_all

Goal state after:
Module: Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs

Goal state before:
pp: V : Type u_1
p : ↑I
inst✝ : Countable V
S : Set (Sym2 V)
hS : S ⊆ Sym2.diagSetᶜ
⊢ (fromEdgeSet S).edgeSet = S

Tactic:
simpa using

Goal state after:
V : Type u_1
p : ↑I
inst✝ : Countable V
S : Set (Sym2 V)
hS : S ⊆ Sym2.diagSetᶜ
⊢ Disjoint S Sym2.diagSet
Module: Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs

Goal state before:
pp: V : Type u_1
p : ↑I
inst✝ : Countable V
s : Set (Sym2 V)
hs : s ⊆ Sym2.diagSetᶜ
⊢ (fromEdgeSet s).edgeSet = s

Tactic:
simpa using

Goal state after:
V : Type u_1
p : ↑I
inst✝ : Countable V
s : Set (Sym2 V)
hs : s ⊆ Sym2.diagSetᶜ
⊢ Disjoint s Sym2.diagSet
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: case inl
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
μ : (n : ℕ) → Measure (X n)
hμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)
a b : ℕ
hab : a ≤ b
s : (i : ↥(Iic b)) → Set (X ↑i)
ms : ∀ (i : ↥(Iic b)), MeasurableSet (s i)
⊢ ∏ x ∈ Iic a ∪ Ioc a b, Function.extend Subtype.val (fun i ↦ (μ ↑i) (s i)) 1 x =
    (∏ x ≤ a, Function.extend Subtype.val (fun i ↦ (μ ↑i) (frestrictLe₂ hab s i)) 1 x) *
      ∏ x ∈ Ioc a b, Function.extend Subtype.val (fun i ↦ (μ ↑i) (restrict₂ ⋯ s i)) 1 x

Tactic:
prod_union (Iic_disjoint_Ioc le_rfl)

Goal state after:
case inl
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
μ : (n : ℕ) → Measure (X n)
hμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)
a b : ℕ
hab : a ≤ b
s : (i : ↥(Iic b)) → Set (X ↑i)
ms : ∀ (i : ↥(Iic b)), MeasurableSet (s i)
⊢ (∏ x ≤ a, Function.extend Subtype.val (fun i ↦ (μ ↑i) (s i)) 1 x) *
      ∏ x ∈ Ioc a b, Function.extend Subtype.val (fun i ↦ (μ ↑i) (s i)) 1 x =
    (∏ x ≤ a, Function.extend Subtype.val (fun i ↦ (μ ↑i) (frestrictLe₂ hab s i)) 1 x) *
      ∏ x ∈ Ioc a b, Function.extend Subtype.val (fun i ↦ (μ ↑i) (restrict₂ ⋯ s i)) 1 x
Module: Mathlib.Probability.Kernel.WithDensity

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
hf : Measurable (Function.uncurry f)
a : α
g : β → ℝ≥0∞
hg : Measurable g
⊢ ∫⁻ (a_1 : β), (f a * g) a_1 ∂κ a = ∫⁻ (b : β), f a b * g b ∂κ a

Tactic:
simp_rw [Pi.mul_apply]

Goal state after:
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
f : ℕ → ((n : ℕ) → X n) → ℝ≥0∞
a : ℕ → ℕ
hcte : ∀ (n : ℕ), DependsOn (f n) ↑(Iic (a n))
mf : ∀ (n : ℕ), Measurable (f n)
bound : ℝ≥0∞
fin_bound : bound ≠ ∞
le_bound : ∀ (n : ℕ) (x : (n : ℕ) → X n), f n x ≤ bound
k : ℕ
anti : ∀ (x : (n : ℕ) → X n), Antitone fun n ↦ lmarginalPartialTraj κ (k + 1) (a n) (f n) x
l : ((n : ℕ) → X n) → ℝ≥0∞
htendsto : ∀ (x : (n : ℕ) → X n), Tendsto (fun n ↦ lmarginalPartialTraj κ (k + 1) (a n) (f n) x) atTop (𝓝 (l x))
ε : ℝ≥0∞
y : (i : ↥(Iic k)) → X ↑i
hpos : ∀ (x : (i : ℕ) → X i) (n : ℕ), ε ≤ lmarginalPartialTraj κ k (a n) (f n) (updateFinset x (Iic k) y)
x✝ : ∀ (n : ℕ), Nonempty (X n)
F : ℕ → ((n : ℕ) → X n) → ℝ≥0∞ := fun n ↦ lmarginalPartialTraj κ (k + 1) (a n) (f n)
tendstoF : ∀ (x : (n : ℕ) → X n), Tendsto (fun x_1 ↦ F x_1 x) atTop (𝓝 (l x))
f_eq : ∀ (x : (n : ℕ) → X n) (n : ℕ), lmarginalPartialTraj κ k (a n) (f n) x = lmarginalPartialTraj κ k (k + 1) (F n) x
n : ℕ
x : (n : ℕ) → X n
⊢ F n x ≤ bound

Tactic:
simpa using

Goal state after:
X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
f : ℕ → ((n : ℕ) → X n) → ℝ≥0∞
a : ℕ → ℕ
hcte : ∀ (n : ℕ), DependsOn (f n) ↑(Iic (a n))
mf : ∀ (n : ℕ), Measurable (f n)
bound : ℝ≥0∞
fin_bound : bound ≠ ∞
le_bound : ∀ (n : ℕ) (x : (n : ℕ) → X n), f n x ≤ bound
k : ℕ
anti : ∀ (x : (n : ℕ) → X n), Antitone fun n ↦ lmarginalPartialTraj κ (k + 1) (a n) (f n) x
l : ((n : ℕ) → X n) → ℝ≥0∞
htendsto : ∀ (x : (n : ℕ) → X n), Tendsto (fun n ↦ lmarginalPartialTraj κ (k + 1) (a n) (f n) x) atTop (𝓝 (l x))
ε : ℝ≥0∞
y : (i : ↥(Iic k)) → X ↑i
hpos : ∀ (x : (i : ℕ) → X i) (n : ℕ), ε ≤ lmarginalPartialTraj κ k (a n) (f n) (updateFinset x (Iic k) y)
x✝ : ∀ (n : ℕ), Nonempty (X n)
F : ℕ → ((n : ℕ) → X n) → ℝ≥0∞ := fun n ↦ lmarginalPartialTraj κ (k + 1) (a n) (f n)
tendstoF : ∀ (x : (n : ℕ) → X n), Tendsto (fun x_1 ↦ F x_1 x) atTop (𝓝 (l x))
f_eq : ∀ (x : (n : ℕ) → X n) (n : ℕ), lmarginalPartialTraj κ k (a n) (f n) x = lmarginalPartialTraj κ k (k + 1) (F n) x
n : ℕ
x : (n : ℕ) → X n
⊢ ∫⁻ (z : (i : ↥(Iic (a n))) → X ↑i),
      f n (updateFinset x (Iic (a n)) z) ∂(partialTraj κ (k + 1) (a n)) (frestrictLe (k + 1) x) ≤
    bound
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
A : ℕ → Set ((n : ℕ) → X n)
A_mem : ∀ (n : ℕ), A n ∈ measurableCylinders X
A_anti : Antitone A
A_inter : ⋂ n, A n = ∅
p : ℕ
x₀ : (i : ↥(Iic p)) → X ↑i
x✝ : ∀ (n : ℕ), Nonempty (X n)
n : ℕ
⊢ ∃ a S, MeasurableSet S ∧ A n = cylinder (Iic a) S

Tactic:
simpa using

Goal state after:
X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
A : ℕ → Set ((n : ℕ) → X n)
A_mem : ∀ (n : ℕ), A n ∈ measurableCylinders X
A_anti : Antitone A
A_inter : ⋂ n, A n = ∅
p : ℕ
x₀ : (i : ↥(Iic p)) → X ↑i
x✝ : ∀ (n : ℕ), Nonempty (X n)
n : ℕ
⊢ ∃ a S, MeasurableSet S ∧ A n = cylinder (Iic a) S
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
μ : (n : ℕ) → Measure (X n)
hμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)
a b : ℕ
hba : b ≤ a
⊢ IsEmpty ↥(Ioc a b)

Tactic:
simpa using

Goal state after:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
μ : (n : ℕ) → Measure (X n)
hμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)
a b : ℕ
hba : b ≤ a
⊢ IsEmpty { x // False }
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
μ : (n : ℕ) → Measure (X n)
hμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)
A : Set ((n : ℕ) → X n)
hA : A ∈ measurableCylinders X
⊢ ∃ s S, MeasurableSet S ∧ A = cylinder s S

Tactic:
simpa using

Goal state after:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
μ : (n : ℕ) → Measure (X n)
hμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)
A : Set ((n : ℕ) → X n)
hA : A ∈ measurableCylinders X
⊢ ∃ s S, MeasurableSet S ∧ A = cylinder s S
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
pp: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
this :
  (((κ + η).withDensity fun a x ↦ ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal) a) {x | 1 ≤ κ.rnDerivAux (κ + η) a x} = 0
⊢ (η a) (κ.mutuallySingularSetSlice η a) = 0

Tactic:
by
    rwa [withDensity_one_sub_rnDerivAux κ η] at this

Goal state after:
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
pp: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
a : α
x : γ
⊢ κ.rnDeriv η a x = ∞ ↔ x ∈ κ.mutuallySingularSetSlice η a

Tactic:
rw [rnDeriv_eq_top_iff, mutuallySingularSet, mutuallySingularSetSlice, mem_ofPred, mem_ofPred]

Goal state after: