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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : Measurable f
a : α
s : Set β
hs : MeasurableSet s
⊢ ({ toFun := fun a ↦ Measure.dirac (f a), measurable' := ⋯ } a) s = s.indicator (fun x ↦ 1) (f a)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : Measurable f
a : α
s : Set β
hs : MeasurableSet s
⊢ ({ toFun := fun a ↦ Measure.dirac (f a), measurable' := ⋯ } a) s = s.indicator (fun x ↦ 1) (f a)
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
f : α → β
hf : Measurable f
a : α
⊢ IsProbabilityMeasure (Measure.dirac (f a))

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
f : α → β
hf : Measurable f
a : α
⊢ IsProbabilityMeasure (Measure.dirac (f a))
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
f : α → β
hf : Measurable f
a : α
⊢ IsProbabilityMeasure ((deterministic f hf) a)

Tactic:
rw [deterministic_apply hf]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
f : α → β
hf : Measurable f
a : α
⊢ IsProbabilityMeasure (Measure.dirac (f a))
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
g : α → β
a : α
hg : Measurable g
hf : Measurable f
⊢ f (g a) = f (g a)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
g : α → β
a : α
hg : Measurable g
inst✝ : MeasurableSingletonClass β
⊢ f (g a) = f (g a)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
g : α → β
a : α
hg : Measurable g
hf : Measurable f
s : Set β
hs : MeasurableSet s
inst✝ : Decidable (g a ∈ s)
⊢ (if g a ∈ s then f (g a) else 0) = if g a ∈ s then f (g a) else 0

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable ι
κ : ι → Kernel α β
a : α
s : Set β
hs : MeasurableSet s
⊢ ∑' (i : ι), ((κ i) a) s = ∑' (n : ι), ((κ n) a) s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable ι
a : α
s : Set β
hs : MeasurableSet s
⊢ ∑' (n : ι), (0 a) s = (0 a) s

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable ι
a : α
s : Set β
hs : MeasurableSet s
⊢ ∑' (n : ι), (0 a) s = (0 a) s
Module: Mathlib.Probability.Kernel.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Countable ι
κ : ι → ι → Kernel α β
a : α
s : Set β
a✝ : MeasurableSet s
⊢ (Measure.sum fun m ↦ Measure.sum fun n ↦ (κ n m) a) s = (Measure.sum fun n ↦ Measure.sum fun n_1 ↦ (κ n_1 n) a) s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
g : α → β
a : α
hg : Measurable g
inst✝¹ : MeasurableSingletonClass β
s : Set β
inst✝ : Decidable (g a ∈ s)
⊢ (if g a ∈ s then f (g a) else 0) = if g a ∈ s then f (g a) else 0

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
h : IsFiniteKernel κ
a : α
s : Set β
hs : MeasurableSet s
⊢ (κ a) s = ∑' (n : ℕ), ((if n = 0 then κ else 0) a) s

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
h : IsFiniteKernel κ
a : α
s : Set β
hs : MeasurableSet s
⊢ (κ a) s = ∑' (n : ℕ), ((if n = 0 then κ else 0) a) s
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
⊢ IsMarkovKernel (deterministic id ⋯)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
⊢ IsMarkovKernel (deterministic id ⋯)
Module: Mathlib.InformationTheory.Hamming

Goal state before:
pp: ι : Type u_2
β : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → DecidableEq (β i)
x y : (i : ι) → β i
H : x = y
⊢ hammingDist y y = 0

Tactic:
]

Goal state after:
ι : Type u_2
β : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → DecidableEq (β i)
x y : (i : ι) → β i
H : x = y
⊢ hammingDist y y = 0
Module: Mathlib.Probability.Kernel.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
h : IsFiniteKernel κ
a : α
s : Set β
hs : MeasurableSet s
this : (fun i ↦ ((if i = 0 then κ else 0) a) s) = fun i ↦ if i = 0 then (κ a) s else 0
⊢ (κ a) s = (κ a) s

Tactic:
]

Goal state after:
Module: Mathlib.InformationTheory.Hamming

Goal state before:
pp: ι : Type u_2
β : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → DecidableEq (β i)
x y : (i : ι) → β i
⊢ x = y ↔ x = y

Tactic:
]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
a : α
⊢ Measure.dirac a = Measure.dirac a

Tactic:
]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
a : α
⊢ Kernel.id a = Measure.dirac a

Tactic:
rw [Kernel.id, deterministic_apply, id_def]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
h : IsSFiniteKernel κ
a : α
⊢ κ a = κ a

Tactic:
]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
f : α → ℝ≥0∞
hf : Measurable f
a : α
⊢ f a = f a

Tactic:
]

Goal state after:
Module: Mathlib.InformationTheory.Hamming

Goal state before:
pp: ι : Type u_2
β : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → DecidableEq (β i)
x y : (i : ι) → β i
⊢ hammingDist x y = 0 ↔ hammingDist x y = 0

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ η✝ κ η : Kernel α β
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
⊢ κ + η = κ + η

Tactic:
]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
f : α → ℝ≥0∞
a : α
⊢ f a = f a

Tactic:
]

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

Goal state before:
pp: case empty
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κs : ι → Kernel α β
h : ∀ i ∈ ∅, IsSFiniteKernel (κs i)
⊢ IsSFiniteKernel 0

Tactic:
]

Goal state after:
case empty
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κs : ι → Kernel α β
h : ∀ i ∈ ∅, IsSFiniteKernel (κs i)
⊢ IsSFiniteKernel 0
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
⊢ IsMarkovKernel (deterministic Function.diag ⋯)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
⊢ IsMarkovKernel (deterministic Function.diag ⋯)
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
⊢ IsMarkovKernel (deterministic (fun x ↦ PUnit.unit) ⋯)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
⊢ IsMarkovKernel (deterministic (fun x ↦ PUnit.unit) ⋯)
Module: Mathlib.Probability.Kernel.Defs

Goal state before:
pp: case insert
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κs : ι → Kernel α β
i : ι
I : Finset ι
hi_notMem_I : i ∉ I
h_ind : (∀ i ∈ I, IsSFiniteKernel (κs i)) → IsSFiniteKernel (∑ i ∈ I, κs i)
h : ∀ i_1 ∈ insert i I, IsSFiniteKernel (κs i_1)
⊢ IsSFiniteKernel (κs i + ∑ x ∈ I, κs x)

Tactic:
]

Goal state after:
case insert
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κs : ι → Kernel α β
i : ι
I : Finset ι
hi_notMem_I : i ∉ I
h_ind : (∀ i ∈ I, IsSFiniteKernel (κs i)) → IsSFiniteKernel (∑ i ∈ I, κs i)
h : ∀ i_1 ∈ insert i I, IsSFiniteKernel (κs i_1)
⊢ IsSFiniteKernel (κs i + ∑ x ∈ I, κs x)
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
⊢ IsMarkovKernel (deterministic Prod.swap ⋯)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
⊢ IsMarkovKernel (deterministic Prod.swap ⋯)
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
ab : α × β
⊢ Measure.dirac ab.swap = Measure.dirac ab.swap

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
ab : α × β
s : Set (β × α)
hs : MeasurableSet s
⊢ s.indicator 1 ab.swap = s.indicator 1 ab.swap

Tactic:
]

Goal state after:
Module: Mathlib.InformationTheory.Hamming

Goal state before:
pp: ι : Type u_2
β : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → DecidableEq (β i)
inst✝ : (i : ι) → Zero (β i)
x : (i : ι) → β i
⊢ hammingNorm x = hammingNorm x

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μβ : Measure β
inst✝ : SFinite μβ
⊢ const α μβ = const α μβ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Denumerable ι
κs : ι → Kernel α β
hκs : ∀ (n : ι), IsSFiniteKernel (κs n)
e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm
hκ_eq : Kernel.sum κs = Kernel.sum fun n ↦ Kernel.sum (κs n).seq
a : α
s : Set β
hs : MeasurableSet s
⊢ ((Kernel.sum fun n ↦ Kernel.sum (κs n).seq) a) s = ((Kernel.sum fun n ↦ (κs (e n).1).seq (e n).2) a) s

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Denumerable ι
κs : ι → Kernel α β
hκs : ∀ (n : ι), IsSFiniteKernel (κs n)
e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm
hκ_eq : Kernel.sum κs = Kernel.sum fun n ↦ Kernel.sum (κs n).seq
a : α
s : Set β
hs : MeasurableSet s
⊢ ((Kernel.sum fun n ↦ Kernel.sum (κs n).seq) a) s = ((Kernel.sum fun n ↦ (κs (e n).1).seq (e n).2) a) s
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
μ : Measure β
a : α
⊢ ∫⁻ (x : β), f x ∂μ = ∫⁻ (x : β), f x ∂μ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
μ : Measure β
a : α
s : Set β
⊢ ∫⁻ (x : β) in s, f x ∂μ = ∫⁻ (x : β) in s, f x ∂μ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : Denumerable ι
κs : ι → Kernel α β
hκs : ∀ (n : ι), IsSFiniteKernel (κs n)
e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm
hκ_eq : Kernel.sum κs = Kernel.sum fun n ↦ Kernel.sum (κs n).seq
a : α
s : Set β
hs : MeasurableSet s
⊢ ∑' (i : ι) (m : ℕ), (((κs i).seq m) a) s = ∑' (b : ι) (c : ℕ), (((κs (b, c).1).seq (b, c).2) a) s

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
s t : Set β
κ : Kernel α β
hs : MeasurableSet s
a : α
ht : MeasurableSet t
⊢ (κ a) (t ∩ s) = (κ a) (t ∩ s)

Tactic:
]

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

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝ : Countable ι
κs : ι → Kernel α β
hκs : ∀ (n : ι), IsSFiniteKernel (κs n)
val✝ : Fintype ι
⊢ IsSFiniteKernel (∑ i, κs i)

Tactic:
]

Goal state after:
case inl
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝ : Countable ι
κs : ι → Kernel α β
hκs : ∀ (n : ι), IsSFiniteKernel (κs n)
val✝ : Fintype ι
⊢ IsSFiniteKernel (∑ i, κs i)
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
a : α
⊢ κ a = κ a

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
s : Set β
κ : Kernel α β
hs : MeasurableSet s
a : α
f : β → ℝ≥0∞
⊢ ∫⁻ (b : β) in s, f b ∂κ a = ∫⁻ (b : β) in s, f b ∂κ a

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
s : Set β
κ : Kernel α β
hs : MeasurableSet s
a : α
f : β → ℝ≥0∞
t : Set β
⊢ ∫⁻ (b : β) in t ∩ s, f b ∂κ a = ∫⁻ (b : β) in t ∩ s, f b ∂κ a

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
s t : Set β
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hs : MeasurableSet s
a : α
⊢ (κ a) (Set.univ ∩ s) ≤ κ.bound

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
s t : Set β
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hs : MeasurableSet s
a : α
⊢ (κ a) (Set.univ ∩ s) ≤ κ.bound
Module: Mathlib.Probability.Kernel.MeasurableLIntegral

Goal state before:
pp: case basic
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t₁ : Set α
ht₁ : MeasurableSet t₁
t₂ : Set β
ht₂ : MeasurableSet t₂
h_eq_ite : (fun a ↦ (κ a) (if a ∈ t₁ then t₂ else ∅)) = fun a ↦ if a ∈ t₁ then (κ a) t₂ else 0
⊢ Measurable fun a ↦ if a ∈ t₁ then (κ a) t₂ else 0

Tactic:
]

Goal state after:
case basic
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t₁ : Set α
ht₁ : MeasurableSet t₁
t₂ : Set β
ht₂ : MeasurableSet t₂
h_eq_ite : (fun a ↦ (κ a) (if a ∈ t₁ then t₂ else ∅)) = fun a ↦ if a ∈ t₁ then (κ a) t₂ else 0
⊢ Measurable fun a ↦ if a ∈ t₁ then (κ a) t₂ else 0
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
hf : MeasurableEmbedding f
t : Set γ
ht : MeasurableSet t
a : α
⊢ (κ a) (f '' t) = (κ a) (f '' t)

pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
hf : MeasurableEmbedding f
t : Set γ
ht : MeasurableSet t
a : α
⊢ ∀ (s : Set γ), MeasurableSet s → MeasurableSet (f '' s)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
hf : MeasurableEmbedding f
t : Set γ
ht : MeasurableSet t
a : α
⊢ ∀ (s : Set γ), MeasurableSet s → MeasurableSet (f '' s)
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
hf : MeasurableEmbedding f
t : Set γ
ht : MeasurableSet t
this : (fun a ↦ (Measure.comap f (κ a)) t) = fun a ↦ (κ a) (f '' t)
⊢ Measurable fun a ↦ (κ a) (f '' t)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
hf : MeasurableEmbedding f
t : Set γ
ht : MeasurableSet t
this : (fun a ↦ (Measure.comap f (κ a)) t) = fun a ↦ (κ a) (f '' t)
⊢ Measurable fun a ↦ (κ a) (f '' t)
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
hf : MeasurableEmbedding f
a : α
t : Set γ
ht : MeasurableSet t
⊢ (κ a) (f '' t) = (κ a) (f '' t)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
a✝ : α
s✝ : Set β
hs : MeasurableSet s✝
⊢ (κ a✝) (id '' s✝) = (κ a✝) s✝

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
a✝ : α
s✝ : Set β
hs : MeasurableSet s✝
⊢ (κ a✝) (id '' s✝) = (κ a✝) s✝
Module: Mathlib.Probability.Kernel.MeasurableLIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t✝ : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t : Set (α × β)
htm : MeasurableSet t
iht : Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' t)
h_eq_sdiff : ∀ (a : α), Prod.mk a ⁻¹' tᶜ = univ \ Prod.mk a ⁻¹' t
a : α
⊢ (κ a) univ - (κ a) (Prod.mk a ⁻¹' t) = (κ a) univ - (κ a) (Prod.mk a ⁻¹' t)

pp: case h₂
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t✝ : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t : Set (α × β)
htm : MeasurableSet t
iht : Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' t)
h_eq_sdiff : ∀ (a : α), Prod.mk a ⁻¹' tᶜ = univ \ Prod.mk a ⁻¹' t
a : α
⊢ NullMeasurableSet (Prod.mk a ⁻¹' t) (κ a)

pp: case h_fin
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t✝ : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t : Set (α × β)
htm : MeasurableSet t
iht : Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' t)
h_eq_sdiff : ∀ (a : α), Prod.mk a ⁻¹' tᶜ = univ \ Prod.mk a ⁻¹' t
a : α
⊢ (κ a) (Prod.mk a ⁻¹' t) ≠ ∞

Tactic:
]

Goal state after:
case h₂
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t✝ : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t : Set (α × β)
htm : MeasurableSet t
iht : Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' t)
h_eq_sdiff : ∀ (a : α), Prod.mk a ⁻¹' tᶜ = univ \ Prod.mk a ⁻¹' t
a : α
⊢ NullMeasurableSet (Prod.mk a ⁻¹' t) (κ a)

case h_fin
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t✝ : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t : Set (α × β)
htm : MeasurableSet t
iht : Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' t)
h_eq_sdiff : ∀ (a : α), Prod.mk a ⁻¹' tᶜ = univ \ Prod.mk a ⁻¹' t
a : α
⊢ (κ a) (Prod.mk a ⁻¹' t) ≠ ∞
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
hf : MeasurableEmbedding f
hκ : ∀ (a : α), (κ a) (Set.range f) = 1
a : α
⊢ (κ a) (f '' Set.univ) = 1

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
hf : MeasurableEmbedding f
hκ : ∀ (a : α), (κ a) (Set.range f) = 1
a : α
⊢ (κ a) (f '' Set.univ) = 1
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf : MeasurableEmbedding f
a : α
⊢ (κ a) (f '' Set.univ) ≤ κ.bound

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf : MeasurableEmbedding f
a : α
⊢ (κ a) (f '' Set.univ) ≤ κ.bound
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
hf : MeasurableEmbedding f
a : α
⊢ (κ.comapRight hf) a = Measure.sum fun n ↦ ((κ.seq n).comapRight hf) a

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
hf : MeasurableEmbedding f
a : α
⊢ (κ.comapRight hf) a = Measure.sum fun n ↦ ((κ.seq n).comapRight hf) a
Module: Mathlib.Probability.Kernel.MeasurableLIntegral

Goal state before:
pp: case compl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t✝ : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t : Set (α × β)
htm : MeasurableSet t
iht : Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' t)
h_eq_sdiff : ∀ (a : α), Prod.mk a ⁻¹' tᶜ = univ \ Prod.mk a ⁻¹' t
this : (fun a ↦ (κ a) (univ \ Prod.mk a ⁻¹' t)) = fun a ↦ (κ a) univ - (κ a) (Prod.mk a ⁻¹' t)
⊢ Measurable fun a ↦ (κ a) univ - (κ a) (Prod.mk a ⁻¹' t)

Tactic:
]

Goal state after:
case compl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t✝ : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t : Set (α × β)
htm : MeasurableSet t
iht : Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' t)
h_eq_sdiff : ∀ (a : α), Prod.mk a ⁻¹' tᶜ = univ \ Prod.mk a ⁻¹' t
this : (fun a ↦ (κ a) (univ \ Prod.mk a ⁻¹' t)) = fun a ↦ (κ a) univ - (κ a) (Prod.mk a ⁻¹' t)
⊢ Measurable fun a ↦ (κ a) univ - (κ a) (Prod.mk a ⁻¹' t)
Module: Mathlib.Probability.Kernel.Composition.Comp

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
η : Kernel β γ
κ : Kernel α β
a : α
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (a : β), (η a) s ∂κ a = ∫⁻ (b : β), (η b) s ∂κ a

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Comp

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
κ : Kernel α β
η : Kernel β γ
a : α
⊢ ∫⁻ (b : β), (η b) Set.univ ∂κ a ≤ (κ a) Set.univ * η.bound

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
a : α
⊢ ∫⁻ (b : β), (η b) Set.univ ∂κ a ≤ (κ a) Set.univ * η.bound
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
hf : MeasurableEmbedding f
a : α
t : Set γ
ht : MeasurableSet t
⊢ ∑' (i : ℕ), (Measure.comap f ((κ.seq i) a)) t = ∑' (i : ℕ), ((κ.seq i) a) (f '' t)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
γ : Type u_4
mγ : MeasurableSpace γ
f : γ → β
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
hf : MeasurableEmbedding f
a : α
t : Set γ
ht : MeasurableSet t
⊢ ∑' (i : ℕ), (Measure.comap f ((κ.seq i) a)) t = ∑' (i : ℕ), ((κ.seq i) a) (f '' t)