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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝¹ : Countable ι
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n ↦ f n a
h_sum : Summable fun n ↦ f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, (∑' (n : ι), f n) a b ∂κ a = ∑' (n : ι), ((κ.withDensity (f n)) a) s

pp: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝¹ : Countable ι
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n ↦ f n a
h_sum : Summable fun n ↦ f n
a : α
s : Set β
hs : MeasurableSet s
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))

Tactic:
swap

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝¹ : Countable ι
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n ↦ f n a
h_sum : Summable fun n ↦ f n
a : α
s : Set β
hs : MeasurableSet s
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))

α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝¹ : Countable ι
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n ↦ f n a
h_sum : Summable fun n ↦ f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, (∑' (n : ι), f n) a b ∂κ a = ∑' (n : ι), ((κ.withDensity (f n)) a) s
Module: Mathlib.Probability.Kernel.WithDensity

Goal state before:
pp: case e_f
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝¹ : Countable ι
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n ↦ f n a
h_sum : Summable fun n ↦ f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, (∑' (n : ι), f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b ↦ f n a b) b ∂κ a
n : ι
⊢ ∫⁻ (a_1 : β) in s, f n a a_1 ∂κ a = ((κ.withDensity (f n)) a) s

Tactic:
rw [Kernel.withDensity_apply' _ (hf n) a s]

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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ∞
hf : Measurable (Function.uncurry f)
⊢ IsSFiniteKernel (κ.withDensity f)

pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ∞
hf : ¬Measurable (Function.uncurry f)
⊢ IsSFiniteKernel (κ.withDensity f)

Tactic:
swap

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ∞
hf : ¬Measurable (Function.uncurry f)
⊢ IsSFiniteKernel (κ.withDensity f)

case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ∞
hf : Measurable (Function.uncurry f)
⊢ IsSFiniteKernel (κ.withDensity f)
Module: Mathlib.Probability.Kernel.WithDensity

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ∞
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b ↦ min (f a b) (↑n + 1) - min (f a b) ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
hf_eq_tsum : f = ∑' (n : ℕ), fs n
⊢ IsSFiniteKernel (Kernel.sum fun n ↦ κ.withDensity (fs n))

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✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ∞
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b ↦ min (f a b) (↑n + 1) - min (f a b) ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
hf_eq_tsum : f = ∑' (n : ℕ), fs n
⊢ ∀ (n : ℕ), Measurable (Function.uncurry (fs n))

Tactic:
swap

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ∞
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b ↦ min (f a b) (↑n + 1) - min (f a b) ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
hf_eq_tsum : f = ∑' (n : ℕ), fs n
⊢ ∀ (n : ℕ), Measurable (Function.uncurry (fs n))

case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ∞
hf : Measurable (Function.uncurry f)
fs : ℕ → α → β → ℝ≥0∞ := fun n a b ↦ min (f a b) (↑n + 1) - min (f a b) ↑n
h_le : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ ↑n
h_zero : ∀ (a : α) (b : β) (n : ℕ), ⌈(f a b).toReal⌉₊ ≤ n → fs n a b = 0
hf_eq_tsum : f = ∑' (n : ℕ), fs n
⊢ IsSFiniteKernel (Kernel.sum fun n ↦ κ.withDensity (fs n))
Module: Mathlib.Probability.Kernel.WithDensity

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
f : α → β → ℝ≥0
g : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
hg : Measurable (Function.uncurry g)
a : α
⊢ ((κ a).withDensity fun x ↦ ↑(f a x) * g a x) = ((κ.withDensity fun a x ↦ ↑(f a x)).withDensity g) a

pp: case hf
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
f : α → β → ℝ≥0
g : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
hg : Measurable (Function.uncurry g)
a : α
⊢ Measurable (Function.uncurry fun a x ↦ ↑(f a x) * g a x)

Tactic:
swap

Goal state after:
case hf
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
f : α → β → ℝ≥0
g : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
hg : Measurable (Function.uncurry g)
a : α
⊢ Measurable (Function.uncurry fun a x ↦ ↑(f a x) * g a x)

α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
f : α → β → ℝ≥0
g : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
hg : Measurable (Function.uncurry g)
a : α
⊢ ((κ a).withDensity fun x ↦ ↑(f a x) * g a x) = ((κ.withDensity fun a x ↦ ↑(f a x)).withDensity g) a
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
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (b : γ) in s, ↑(κ.rnDerivAux (κ + η) a b).toNNReal ∂(κ + η) a = (κ a) s

pp: case hf
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
s : Set γ
hs : MeasurableSet s
⊢ Measurable (Function.uncurry fun a x ↦ ↑(κ.rnDerivAux (κ + η) a x).toNNReal)

Tactic:
swap

Goal state after:
case hf
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
s : Set γ
hs : MeasurableSet s
⊢ Measurable (Function.uncurry fun a x ↦ ↑(κ.rnDerivAux (κ + η) a x).toNNReal)

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
s : Set γ
hs : MeasurableSet s
⊢ ∫⁻ (b : γ) in s, ↑(κ.rnDerivAux (κ + η) a b).toNNReal ∂(κ + η) a = (κ a) s
Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: case inr
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b : ℕ
hab : a ≤ b
⊢ partialTraj κ a b = (Kernel.id ×ₖ (partialTraj κ a b).map (restrict₂ ⋯)).map (_root_.IicProdIoc a b)

Tactic:
induction b, hab using Nat.le_induction with

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
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hsm : MeasurableSet s
hs : s ⊆ κ.mutuallySingularSetSlice η a
hs' : ∀ x ∈ s, 1 ≤ κ.rnDerivAux (κ + η) a x
⊢ ∫⁻ (b : γ) in s,
      ↑(κ.rnDerivAux (κ + η) a b).toNNReal - ↑(1 - κ.rnDerivAux (κ + η) a b).toNNReal * κ.rnDeriv η a b ∂(κ + η) a =
    (κ a) s

pp: case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hsm : MeasurableSet s
hs : s ⊆ κ.mutuallySingularSetSlice η a
hs' : ∀ x ∈ s, 1 ≤ κ.rnDerivAux (κ + η) a x
⊢ Measurable
    (Function.uncurry fun a x ↦
      ↑(κ.rnDerivAux (κ + η) a x).toNNReal - ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal * κ.rnDeriv η a x)

Tactic:
swap

Goal state after:
case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hsm : MeasurableSet s
hs : s ⊆ κ.mutuallySingularSetSlice η a
hs' : ∀ x ∈ s, 1 ≤ κ.rnDerivAux (κ + η) a x
⊢ Measurable
    (Function.uncurry fun a x ↦
      ↑(κ.rnDerivAux (κ + η) a x).toNNReal - ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal * κ.rnDeriv η a x)

α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hsm : MeasurableSet s
hs : s ⊆ κ.mutuallySingularSetSlice η a
hs' : ∀ x ∈ s, 1 ≤ κ.rnDerivAux (κ + η) a x
⊢ ∫⁻ (b : γ) in s,
      ↑(κ.rnDerivAux (κ + η) a b).toNNReal - ↑(1 - κ.rnDerivAux (κ + η) a b).toNNReal * κ.rnDeriv η a b ∂(κ + η) a =
    (κ a) s
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
pp: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
hκ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
⊢ ∫⁻ (x : γ), κ.rnDeriv η a x ∂η a ≤ κ.bound

pp: case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
hκ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
⊢ Measurable (Function.uncurry (κ.rnDeriv η))

Tactic:
swap

Goal state after:
case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
hκ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
⊢ Measurable (Function.uncurry (κ.rnDeriv η))

α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
hκ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
⊢ ∫⁻ (x : γ), κ.rnDeriv η a x ∂η a ≤ κ.bound
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
pp: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : κ a ≪ η a
⊢ (η a).withDensity (κ.rnDeriv η a) = κ a

pp: case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : κ a ≪ η a
⊢ Measurable (Function.uncurry (κ.rnDeriv η))

Tactic:
swap

Goal state after:
case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : κ a ≪ η a
⊢ Measurable (Function.uncurry (κ.rnDeriv η))

α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : κ a ≪ η a
⊢ (η a).withDensity (κ.rnDeriv η a) = κ a
Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a b : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
c : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
hab : a ≤ b
⊢ lmarginalPartialTraj κ b c f = f

Tactic:
ext x
  rw [lmarginalPartialTraj_eq_lintegral_map mf]
  refine @lintegral_eq_const _ _ _ ?_ _ _ (ae_of_all _ fun y ↦ hf fun i hi ↦ ?_)
  · refine @IsMarkovKernel.isProbabilityMeasure _ _ _ _ _ ?_ _
    exact IsMarkovKernel.map _ (by fun_prop)
  · simp_all only [coe_Iic, Set.mem_Iic, Function.updateFinset, mem_Ioc, dite_eq_right_iff]
    lia

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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : SFinite μ
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : ¬SFinite μ
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

Tactic:
swap

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : ¬SFinite μ
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : SFinite μ
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η
Module: Mathlib.Probability.Kernel.Composition.AbsolutelyContinuous

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : SFinite μ
hν : SFinite ν
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : SFinite μ
hν : ¬SFinite ν
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

Tactic:
swap

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : SFinite μ
hν : ¬SFinite ν
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : SFinite μ
hν : SFinite ν
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
this : ⇑Kernel.id =ᵐ[μ] ⇑(Kernel.id†μ)
⊢ ⇑(Kernel.id†μ) =ᵐ[μ] ⇑Kernel.id

Tactic:
filter_upwards [this] with a ha using ha.symm

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

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : MeasurableSpace.CountablyGenerated 𝓧
f : Ω → 𝓧
hf : Measurable f
⊢ ⇑(Kernel.id ∘ₖ Kernel.deterministic f hf ∥ₖ (Kernel.deterministic f hf ∘ₖ Kernel.id)) ∘ₘ ⇑(Kernel.copy Ω) ∘ₘ μ =
    ⇑(Kernel.deterministic f hf ∥ₖ Kernel.deterministic f hf) ∘ₘ ⇑(Kernel.copy Ω) ∘ₘ μ

Tactic:
Kernel.id_comp,

Goal state after:
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : MeasurableSpace.CountablyGenerated 𝓧
f : Ω → 𝓧
hf : Measurable f
⊢ ⇑(Kernel.deterministic f hf ∥ₖ (Kernel.deterministic f hf ∘ₖ Kernel.id)) ∘ₘ ⇑(Kernel.copy Ω) ∘ₘ μ =
    ⇑(Kernel.deterministic f hf ∥ₖ Kernel.deterministic f hf) ∘ₘ ⇑(Kernel.copy Ω) ∘ₘ μ
Module: Mathlib.Probability.Decision.Risk.Basic

Goal state before:
pp: Θ : Type u_1
𝓧 : Type u_2
𝓨 : Type u_4
mΘ : MeasurableSpace Θ
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
ℓ : Θ → 𝓨 → ℝ≥0∞
hl : Measurable (uncurry ℓ)
μ : Measure 𝓧
inst✝¹ : SFinite μ
π : Measure Θ
inst✝ : SFinite π
hl_pos : μ Set.univ = ∞ → ⨅ y, ∫⁻ (θ : Θ), ℓ θ y ∂π = 0 → ∃ y, ∫⁻ (θ : Θ), ℓ θ y ∂π = 0
h_zero : μ = 0 → Nonempty 𝓨
⊢ ⨅ y, ∫⁻ (θ : Θ), ℓ θ y * μ Set.univ ∂π ≤ bayesRisk ℓ (Kernel.const Θ μ) π

Tactic:
simp_rw [bayesRisk, le_iInf_iff]

Goal state after:
Θ : Type u_1
𝓧 : Type u_2
𝓨 : Type u_4
mΘ : MeasurableSpace Θ
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
ℓ : Θ → 𝓨 → ℝ≥0∞
hl : Measurable (uncurry ℓ)
μ : Measure 𝓧
inst✝¹ : SFinite μ
π : Measure Θ
inst✝ : SFinite π
hl_pos : μ Set.univ = ∞ → ⨅ y, ∫⁻ (θ : Θ), ℓ θ y ∂π = 0 → ∃ y, ∫⁻ (θ : Θ), ℓ θ y ∂π = 0
h_zero : μ = 0 → Nonempty 𝓨
⊢ ∀ (i : Kernel 𝓧 𝓨), IsMarkovKernel i → ⨅ y, ∫⁻ (θ : Θ), ℓ θ y * μ Set.univ ∂π ≤ avgRisk ℓ (Kernel.const Θ μ) i π
Module: Mathlib.Probability.Decision.Risk.Basic

Goal state before:
pp: Θ : Type u_1
𝓧 : Type u_2
𝓨 : Type u_4
mΘ : MeasurableSpace Θ
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
ℓ : Θ → 𝓨 → ℝ≥0∞
P : Kernel Θ 𝓧
κ : Kernel 𝓧 𝓨
π : Measure Θ
C : ℝ≥0
hℓC : ∀ (θ : Θ) (y : 𝓨), ℓ θ y ≤ ↑C
⊢ ∫⁻ (θ : Θ), ↑C * ∫⁻ (x : 𝓧), (κ x) Set.univ ∂P θ ∂π ≤ ∫⁻ (θ : Θ), ↑C * ∫⁻ (x : 𝓧), κ.bound ∂P θ ∂π

Tactic:
gcongr with θ x
    exact Kernel.measure_le_bound κ x Set.univ

Goal state after:
Module: Mathlib.Probability.Decision.BayesEstimator

Goal state before:
pp: Θ : Type u_1
𝓧 : Type u_2
𝓨 : Type u_3
mΘ : MeasurableSpace Θ
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
ℓ : Θ → 𝓨 → ℝ≥0∞
inst✝⁴ : StandardBorelSpace Θ
inst✝³ : Nonempty Θ
hl : Measurable (Function.uncurry ℓ)
P : Kernel Θ 𝓧
inst✝² : IsFiniteKernel P
κ : Kernel 𝓧 𝓨
inst✝¹ : IsSFiniteKernel κ
π : Measure Θ
inst✝ : IsFiniteMeasure π
⊢ ⇑(Kernel.id ∘ₖ P†π ×ₖ (κ ∘ₖ Kernel.id)) ∘ₘ ⇑P ∘ₘ π = ⇑(P†π ×ₖ κ) ∘ₘ ⇑P ∘ₘ π

Tactic:
Kernel.id_comp,

Goal state after:
Θ : Type u_1
𝓧 : Type u_2
𝓨 : Type u_3
mΘ : MeasurableSpace Θ
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
ℓ : Θ → 𝓨 → ℝ≥0∞
inst✝⁴ : StandardBorelSpace Θ
inst✝³ : Nonempty Θ
hl : Measurable (Function.uncurry ℓ)
P : Kernel Θ 𝓧
inst✝² : IsFiniteKernel P
κ : Kernel 𝓧 𝓨
inst✝¹ : IsSFiniteKernel κ
π : Measure Θ
inst✝ : IsFiniteMeasure π
⊢ ⇑(P†π ×ₖ (κ ∘ₖ Kernel.id)) ∘ₘ ⇑P ∘ₘ π = ⇑(P†π ×ₖ κ) ∘ₘ ⇑P ∘ₘ π
Module: Mathlib.Probability.Decision.Risk.Countable

Goal state before:
pp: Θ : Type u_1
𝓧 : Type u_3
𝓨 : Type u_5
mΘ : MeasurableSpace Θ
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
ℓ : Θ → 𝓨 → ℝ≥0∞
P : Kernel Θ 𝓧
π : Measure Θ
inst✝¹ : Countable 𝓨
inst✝ : MeasurableSingletonClass 𝓨
hℓ : Measurable ℓ
⊢ bayesRisk ℓ P π = ⨅ κ, ⨅ (_ : IsMarkovKernel κ), ∑' (y : 𝓨), ∫⁻ (θ : Θ), ℓ θ y * (⇑κ ∘ₘ P θ) {y} ∂π

Tactic:
simp [bayesRisk, avgRisk_countable' hℓ]

Goal state after:
Module: Mathlib.Probability.Distributions.Cauchy

Goal state before:
pp: case pos
x₀ : ℝ
γ : ℝ≥0
x : ℝ
h : γ = 0
⊢ π⁻¹ * ↑γ * ((x - x₀) ^ 2 + ↑γ ^ 2)⁻¹ = π⁻¹ * ↑γ⁻¹ * (1 + ((x - x₀) / ↑γ) ^ 2)⁻¹

Tactic:
simp [h]

Goal state after:
Module: Mathlib.Probability.Distributions.Gamma

Goal state before:
pp: a r : ℝ
ha : 0 < a
hr : 0 < r
⊢ ∫⁻ (x : ℝ) in Iio 0, 0 ∂volume = 0

Tactic:
lintegral_zero

Goal state after:
a r : ℝ
ha : 0 < a
hr : 0 < r
⊢ 0 = 0
Module: Mathlib.Probability.ProbabilityMassFunction.Basic

Goal state before:
pp: case pos
α : Type u_1
p : PMF α
a b : α
h : b = a
⊢ p a ≤ p b

Tactic:
simp [h]

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

Goal state before:
pp: case neg
α : Type u_1
p : PMF α
a b : α
h : ¬b = a
⊢ 0 ≤ p b

Tactic:
simp [h]

Goal state after:
Module: Mathlib.Probability.Distributions.Geometric

Goal state before:
pp: p : ↑unitInterval
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℕ → E
hp : p ≠ 0
⊢ (Summable fun i ↦ (ENNReal.ofReal ((1 - ↑p) ^ i * ↑p)).toReal * ‖f i‖) ↔ Summable fun n ↦ (1 - ↑p) ^ n * ↑p * ‖f n‖

Tactic:
congrm Summable (fun n ↦ ?_ * _)

Goal state after:
p : ↑unitInterval
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℕ → E
hp : p ≠ 0
n : ℕ
⊢ (ENNReal.ofReal ((1 - ↑p) ^ n * ↑p)).toReal = (1 - ↑p) ^ n * ↑p
Module: Mathlib.Probability.Distributions.Poisson.Basic

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
r : ℝ≥0
f : ℕ → E
⊢ (Summable fun i ↦ (ENNReal.ofReal (rexp (-↑r) * ↑r ^ i / ↑i !)).toReal * ‖f i‖) ↔
    Summable fun n ↦ rexp (-↑r) * ↑r ^ n / ↑n ! * ‖f n‖

Tactic:
congrm Summable (fun n ↦ ?_ * _)

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
r : ℝ≥0
f : ℕ → E
n : ℕ
⊢ (ENNReal.ofReal (rexp (-↑r) * ↑r ^ n / ↑n !)).toReal = rexp (-↑r) * ↑r ^ n / ↑n !
Module: Mathlib.Probability.ProbabilityMassFunction.Monad

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
f : (a' : α) → a' ∈ (pure a).support → PMF β
b : β
a' : α
⊢ ((if a' = a then 1 else 0) * if h : (if a' = a then 1 else 0) = 0 then 0 else (f a' h) b) =
    if a' = a then (f a ⋯) b else 0

Tactic:
by_cases h : a' = a

Goal state after:
case pos
α : Type u_1
β : Type u_2
a : α
f : (a' : α) → a' ∈ (pure a).support → PMF β
b : β
a' : α
h : a' = a
⊢ ((if a' = a then 1 else 0) * if h : (if a' = a then 1 else 0) = 0 then 0 else (f a' h) b) =
    if a' = a then (f a ⋯) b else 0

case neg
α : Type u_1
β : Type u_2
a : α
f : (a' : α) → a' ∈ (pure a).support → PMF β
b : β
a' : α
h : ¬a' = a
⊢ ((if a' = a then 1 else 0) * if h : (if a' = a then 1 else 0) = 0 then 0 else (f a' h) b) =
    if a' = a then (f a ⋯) b else 0
Module: Mathlib.Probability.ProbabilityMassFunction.Monad

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
a : α
f : (a' : α) → a' ∈ (pure a).support → PMF β
b : β
a' : α
h : a' = a
⊢ ((if a' = a then 1 else 0) * if h : (if a' = a then 1 else 0) = 0 then 0 else (f a' h) b) =
    if a' = a then (f a ⋯) b else 0

Tactic:
simp [h]

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

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
a : α
f : (a' : α) → a' ∈ (pure a).support → PMF β
b : β
a' : α
h : ¬a' = a
⊢ ((if a' = a then 1 else 0) * if h : (if a' = a then 1 else 0) = 0 then 0 else (f a' h) b) =
    if a' = a then (f a ⋯) b else 0

Tactic:
simp [h]

Goal state after: