mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.GroupTheory.SpecificGroups.ZGroup

Goal state before:
pp: case refine_1
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
hG : IsZGroup G
⊢ ∃ N H φ x, IsCyclic ↥H ∧ IsCyclic ↥N ∧ (Nat.card ↥N).Coprime (Nat.card ↥H)

pp: case refine_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
⊢ (∃ N H φ x, IsCyclic ↥H ∧ IsCyclic ↥N ∧ (Nat.card ↥N).Coprime (Nat.card ↥H)) → IsZGroup G

Tactic:
·

Goal state after:
case refine_1
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
hG : IsZGroup G
⊢ ∃ N H φ x, IsCyclic ↥H ∧ IsCyclic ↥N ∧ (Nat.card ↥N).Coprime (Nat.card ↥H)
Module: Mathlib.GroupTheory.SpecificGroups.ZGroup

Goal state before:
pp: case refine_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
⊢ (∃ N H φ x, IsCyclic ↥H ∧ IsCyclic ↥N ∧ (Nat.card ↥N).Coprime (Nat.card ↥H)) → IsZGroup G

Tactic:
·

Goal state after:
case refine_2
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
⊢ (∃ N H φ x, IsCyclic ↥H ∧ IsCyclic ↥N ∧ (Nat.card ↥N).Coprime (Nat.card ↥H)) → IsZGroup G
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 z : (i : ι) → β i
⊢ hammingDist x y ≤ hammingDist x z + hammingDist y z

Tactic:
by
  rw [hammingDist_comm y]
  exact hammingDist_triangle _ _ _

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

Goal state before:
pp: case pos
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
n : ℕ
h✝ : n = 0
⊢ IsFiniteKernel κ

pp: case neg
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
n : ℕ
h✝ : ¬n = 0
⊢ IsFiniteKernel 0

Tactic:
·

Goal state after:
case pos
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
n : ℕ
h✝ : n = 0
⊢ IsFiniteKernel κ
Module: Mathlib.Probability.Kernel.Defs

Goal state before:
pp: case neg
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
n : ℕ
h✝ : ¬n = 0
⊢ IsFiniteKernel 0

Tactic:
·

Goal state after:
case neg
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
n : ℕ
h✝ : ¬n = 0
⊢ IsFiniteKernel 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 (swap α β)

Tactic:
rw [swap]; infer_instance

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

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 : Kernel α β
hμβ : IsZeroOrProbabilityMeasure 0
⊢ IsZeroOrMarkovKernel (const α 0)

pp: case inr
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
hμβ : IsZeroOrProbabilityMeasure μβ
h : IsProbabilityMeasure μβ
⊢ IsZeroOrMarkovKernel (const α μβ)

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 : Kernel α β
hμβ : IsZeroOrProbabilityMeasure 0
⊢ IsZeroOrMarkovKernel (const α 0)
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: case inr
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
hμβ : IsZeroOrProbabilityMeasure μβ
h : IsProbabilityMeasure μβ
⊢ IsZeroOrMarkovKernel (const α μβ)

Tactic:
·

Goal state after:
case inr
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
hμβ : IsZeroOrProbabilityMeasure μβ
h : IsProbabilityMeasure μβ
⊢ IsZeroOrMarkovKernel (const α μβ)
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 (Kernel.sum κs)

pp: case inr
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✝ : Infinite ι
⊢ IsSFiniteKernel (Kernel.sum κs)

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 (Kernel.sum κs)
Module: Mathlib.InformationTheory.Hamming

Goal state before:
pp: case mp
α : Type u_1
ι : Type u_2
β : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → DecidableEq (β i)
s : Set (Hamming β × Hamming β)
⊢ s ∈ uniformity (Hamming β) → ∃ ε > 0, ∀ {a b : Hamming β}, ↑(hammingDist (ofHamming a) (ofHamming b)) < ε → (a, b) ∈ s

pp: case mpr
α : Type u_1
ι : Type u_2
β : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → DecidableEq (β i)
s : Set (Hamming β × Hamming β)
⊢ (∃ ε > 0, ∀ {a b : Hamming β}, ↑(hammingDist (ofHamming a) (ofHamming b)) < ε → (a, b) ∈ s) →
    s ∈ uniformity (Hamming β)

Tactic:
·

Goal state after:
case mp
α : Type u_1
ι : Type u_2
β : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → DecidableEq (β i)
s : Set (Hamming β × Hamming β)
⊢ s ∈ uniformity (Hamming β) → ∃ ε > 0, ∀ {a b : Hamming β}, ↑(hammingDist (ofHamming a) (ofHamming b)) < ε → (a, b) ∈ s
Module: Mathlib.InformationTheory.Hamming

Goal state before:
pp: case mpr
α : Type u_1
ι : Type u_2
β : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → DecidableEq (β i)
s : Set (Hamming β × Hamming β)
⊢ (∃ ε > 0, ∀ {a b : Hamming β}, ↑(hammingDist (ofHamming a) (ofHamming b)) < ε → (a, b) ∈ s) →
    s ∈ uniformity (Hamming β)

Tactic:
·

Goal state after:
case mpr
α : Type u_1
ι : Type u_2
β : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → DecidableEq (β i)
s : Set (Hamming β × Hamming β)
⊢ (∃ ε > 0, ∀ {a b : Hamming β}, ↑(hammingDist (ofHamming a) (ofHamming b)) < ε → (a, b) ∈ s) →
    s ∈ uniformity (Hamming β)
Module: Mathlib.Probability.Kernel.MeasurableLIntegral

Goal state before:
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)
Module: Mathlib.Probability.Kernel.MeasurableLIntegral

Goal state before:
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 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.InformationTheory.Coding.KraftMcMillan

Goal state before:
pp: case left
α : Type u_1
S : Finset (List α)
r : ℕ
w : Fin r → ↥S
h0 : ∀ (c : ↥S), ↑c ≠ []
⊢ r ≤ ∑ i, (↑(w i)).length

pp: case right
α : Type u_1
S : Finset (List α)
r : ℕ
w : Fin r → ↥S
h0 : ∀ (c : ↥S), ↑c ≠ []
⊢ ∑ i, (↑(w i)).length ≤ r * S.sup List.length

Tactic:
·

Goal state after:
case left
α : Type u_1
S : Finset (List α)
r : ℕ
w : Fin r → ↥S
h0 : ∀ (c : ↥S), ↑c ≠ []
⊢ r ≤ ∑ i, (↑(w i)).length
Module: Mathlib.Probability.Kernel.MeasurableLIntegral

Goal state before:
pp: case hn
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)
f : ℕ → Set (α × β)
h_disj : Pairwise (Disjoint on f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf : ∀ (i : ℕ), Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' f i)
a : α
⊢ Pairwise (Disjoint on fun i ↦ Prod.mk a ⁻¹' f i)

pp: case h
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)
f : ℕ → Set (α × β)
h_disj : Pairwise (Disjoint on f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf : ∀ (i : ℕ), Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' f i)
a : α
⊢ ∀ (i : ℕ), MeasurableSet (Prod.mk a ⁻¹' f i)

Tactic:
·

Goal state after:
case hn
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)
f : ℕ → Set (α × β)
h_disj : Pairwise (Disjoint on f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf : ∀ (i : ℕ), Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' f i)
a : α
⊢ Pairwise (Disjoint on fun i ↦ Prod.mk a ⁻¹' f i)
Module: Mathlib.Probability.Kernel.MeasurableLIntegral

Goal state before:
pp: case h
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)
f : ℕ → Set (α × β)
h_disj : Pairwise (Disjoint on f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf : ∀ (i : ℕ), Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' f i)
a : α
⊢ ∀ (i : ℕ), MeasurableSet (Prod.mk a ⁻¹' f i)

Tactic:
·

Goal state after:
case h
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)
f : ℕ → Set (α × β)
h_disj : Pairwise (Disjoint on f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf : ∀ (i : ℕ), Measurable fun a ↦ (κ a) (Prod.mk a ⁻¹' f i)
a : α
⊢ ∀ (i : ℕ), MeasurableSet (Prod.mk a ⁻¹' f i)
Module: Mathlib.InformationTheory.Coding.KraftMcMillan

Goal state before:
pp: case right
α : Type u_1
S : Finset (List α)
r : ℕ
w : Fin r → ↥S
h0 : ∀ (c : ↥S), ↑c ≠ []
⊢ ∑ i, (↑(w i)).length ≤ r * S.sup List.length

Tactic:
·

Goal state after:
case right
α : Type u_1
S : Finset (List α)
r : ℕ
w : Fin r → ↥S
h0 : ∀ (c : ↥S), ↑c ≠ []
⊢ ∑ i, (↑(w i)).length ≤ r * S.sup List.length
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
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
⊢ ∃ s, MeasurableSet s ∧ μ s = 0 ∧ ∀ a ∉ s, IsProbabilityMeasure (κ a)

Tactic:
refine ⟨toMeasurable μ {a | ¬ IsProbabilityMeasure (κ a)}, measurableSet_toMeasurable _ _,
      by simpa [measure_toMeasurable] using! h, ?_⟩

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
⊢ ∀ a ∉ toMeasurable μ {a | ¬IsProbabilityMeasure (κ a)}, IsProbabilityMeasure (κ a)
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
a : α
ha : a ∈ sᶜ
⊢ ⇑κ =ᵐ[μ] ⇑(piecewise s_meas (const α (κ a)) κ)

pp: case refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
a : α
ha : a ∈ sᶜ
⊢ IsMarkovKernel (piecewise s_meas (const α (κ a)) κ)

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
a : α
ha : a ∈ sᶜ
⊢ ⇑κ =ᵐ[μ] ⇑(piecewise s_meas (const α (κ a)) κ)
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
a : α
ha : a ∈ sᶜ
⊢ IsMarkovKernel (piecewise s_meas (const α (κ a)) κ)

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
a : α
ha : a ∈ sᶜ
⊢ IsMarkovKernel (piecewise s_meas (const α (κ a)) κ)
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ s
⊢ IsProbabilityMeasure ((piecewise s_meas (const α (κ a)) κ) b)

pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
a : α
ha : a ∈ sᶜ
b : α
hb : b ∉ s
⊢ IsProbabilityMeasure ((piecewise s_meas (const α (κ a)) κ) b)

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ s
⊢ IsProbabilityMeasure ((piecewise s_meas (const α (κ a)) κ) b)
Module: Mathlib.Probability.Kernel.Basic

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
a : α
ha : a ∈ sᶜ
b : α
hb : b ∉ s
⊢ IsProbabilityMeasure ((piecewise s_meas (const α (κ a)) κ) b)

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
μ : Measure α
h : ∀ᵐ (a : α) ∂μ, IsProbabilityMeasure (κ a)
h' : μ ≠ 0
s : Set α
s_meas : MeasurableSet s
μs : μ s = 0
hs : ∀ a ∉ s, IsProbabilityMeasure (κ a)
a : α
ha : a ∈ sᶜ
b : α
hb : b ∉ s
⊢ IsProbabilityMeasure ((piecewise s_meas (const α (κ a)) κ) b)
Module: Mathlib.Probability.Kernel.MeasurableLIntegral

Goal state before:
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∞
hf : Measurable (uncurry f)
F : ℕ → SimpleFunc (α × β) ℝ≥0∞ := SimpleFunc.eapprox (uncurry f)
h : ∀ (a : α) (b : β), ⨆ n, (F n) (a, b) = f a b
a : α
⊢ ∀ (n : ℕ), Measurable fun b ↦ (F n) (a, b)

pp: case h_mono
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∞
hf : Measurable (uncurry f)
F : ℕ → SimpleFunc (α × β) ℝ≥0∞ := SimpleFunc.eapprox (uncurry f)
h : ∀ (a : α) (b : β), ⨆ n, (F n) (a, b) = f a b
a : α
⊢ Monotone fun n b ↦ (F n) (a, b)

Tactic:
·

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∞
hf : Measurable (uncurry f)
F : ℕ → SimpleFunc (α × β) ℝ≥0∞ := SimpleFunc.eapprox (uncurry f)
h : ∀ (a : α) (b : β), ⨆ n, (F n) (a, b) = f a b
a : α
⊢ ∀ (n : ℕ), Measurable fun b ↦ (F n) (a, b)
Module: Mathlib.Probability.Kernel.MeasurableLIntegral

Goal state before:
pp: case h_mono
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∞
hf : Measurable (uncurry f)
F : ℕ → SimpleFunc (α × β) ℝ≥0∞ := SimpleFunc.eapprox (uncurry f)
h : ∀ (a : α) (b : β), ⨆ n, (F n) (a, b) = f a b
a : α
⊢ Monotone fun n b ↦ (F n) (a, b)

Tactic:
·

Goal state after:
case h_mono
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∞
hf : Measurable (uncurry f)
F : ℕ → SimpleFunc (α × β) ℝ≥0∞ := SimpleFunc.eapprox (uncurry f)
h : ∀ (a : α) (b : β), ⨆ n, (F n) (a, b) = f a b
a : α
⊢ Monotone fun n b ↦ (F n) (a, b)
Module: Mathlib.Probability.Kernel.MeasurableLIntegral

Goal state before:
pp: case refine_1
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∞
hf : Measurable (uncurry f)
F : ℕ → SimpleFunc (α × β) ℝ≥0∞ := SimpleFunc.eapprox (uncurry f)
h : ∀ (a : α) (b : β), ⨆ n, (F n) (a, b) = f a b
this : ∀ (a : α), ∫⁻ (b : β), ⨆ n, (F n) (a, b) ∂κ a = ⨆ n, ∫⁻ (b : β), (F n) (a, b) ∂κ a
n : ℕ
⊢ ∀ (c : ℝ≥0∞) {s : Set (α × β)} (hs : MeasurableSet s),
    Measurable fun a ↦
      ∫⁻ (b : β), (SimpleFunc.piecewise s hs (SimpleFunc.const (α × β) c) (SimpleFunc.const (α × β) 0)) (a, b) ∂κ a

pp: case refine_2
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∞
hf : Measurable (uncurry f)
F : ℕ → SimpleFunc (α × β) ℝ≥0∞ := SimpleFunc.eapprox (uncurry f)
h : ∀ (a : α) (b : β), ⨆ n, (F n) (a, b) = f a b
this : ∀ (a : α), ∫⁻ (b : β), ⨆ n, (F n) (a, b) ∂κ a = ⨆ n, ∫⁻ (b : β), (F n) (a, b) ∂κ a
n : ℕ
⊢ ∀ ⦃f g : SimpleFunc (α × β) ℝ≥0∞⦄,
    Disjoint (support ⇑f) (support ⇑g) →
      (Measurable fun a ↦ ∫⁻ (b : β), f (a, b) ∂κ a) →
        (Measurable fun a ↦ ∫⁻ (b : β), g (a, b) ∂κ a) → Measurable fun a ↦ ∫⁻ (b : β), (f + g) (a, b) ∂κ a

Tactic:
·

Goal state after:
case refine_1
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∞
hf : Measurable (uncurry f)
F : ℕ → SimpleFunc (α × β) ℝ≥0∞ := SimpleFunc.eapprox (uncurry f)
h : ∀ (a : α) (b : β), ⨆ n, (F n) (a, b) = f a b
this : ∀ (a : α), ∫⁻ (b : β), ⨆ n, (F n) (a, b) ∂κ a = ⨆ n, ∫⁻ (b : β), (F n) (a, b) ∂κ a
n : ℕ
⊢ ∀ (c : ℝ≥0∞) {s : Set (α × β)} (hs : MeasurableSet s),
    Measurable fun a ↦
      ∫⁻ (b : β), (SimpleFunc.piecewise s hs (SimpleFunc.const (α × β) c) (SimpleFunc.const (α × β) 0)) (a, b) ∂κ a
Module: Mathlib.Probability.Kernel.MeasurableLIntegral

Goal state before:
pp: case refine_2
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∞
hf : Measurable (uncurry f)
F : ℕ → SimpleFunc (α × β) ℝ≥0∞ := SimpleFunc.eapprox (uncurry f)
h : ∀ (a : α) (b : β), ⨆ n, (F n) (a, b) = f a b
this : ∀ (a : α), ∫⁻ (b : β), ⨆ n, (F n) (a, b) ∂κ a = ⨆ n, ∫⁻ (b : β), (F n) (a, b) ∂κ a
n : ℕ
⊢ ∀ ⦃f g : SimpleFunc (α × β) ℝ≥0∞⦄,
    Disjoint (support ⇑f) (support ⇑g) →
      (Measurable fun a ↦ ∫⁻ (b : β), f (a, b) ∂κ a) →
        (Measurable fun a ↦ ∫⁻ (b : β), g (a, b) ∂κ a) → Measurable fun a ↦ ∫⁻ (b : β), (f + g) (a, b) ∂κ a

Tactic:
·

Goal state after:
case refine_2
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∞
hf : Measurable (uncurry f)
F : ℕ → SimpleFunc (α × β) ℝ≥0∞ := SimpleFunc.eapprox (uncurry f)
h : ∀ (a : α) (b : β), ⨆ n, (F n) (a, b) = f a b
this : ∀ (a : α), ∫⁻ (b : β), ⨆ n, (F n) (a, b) ∂κ a = ⨆ n, ∫⁻ (b : β), (F n) (a, b) ∂κ a
n : ℕ
⊢ ∀ ⦃f g : SimpleFunc (α × β) ℝ≥0∞⦄,
    Disjoint (support ⇑f) (support ⇑g) →
      (Measurable fun a ↦ ∫⁻ (b : β), f (a, b) ∂κ a) →
        (Measurable fun a ↦ ∫⁻ (b : β), g (a, b) ∂κ a) → Measurable fun a ↦ ∫⁻ (b : β), (f + g) (a, b) ∂κ a
Module: Mathlib.Probability.Kernel.Composition.MapComap

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
f : β → γ
a✝¹ : α
s✝ : Set γ
a✝ : MeasurableSet s✝
hf : Measurable f
⊢ ((map 0 f) a✝¹) s✝ = (0 a✝¹) s✝

pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
f : β → γ
a✝¹ : α
s✝ : Set γ
a✝ : MeasurableSet s✝
hf : ¬Measurable f
⊢ ((map 0 f) a✝¹) s✝ = (0 a✝¹) s✝

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
f : β → γ
a✝¹ : α
s✝ : Set γ
a✝ : MeasurableSet s✝
hf : Measurable f
⊢ ((map 0 f) a✝¹) s✝ = (0 a✝¹) s✝
Module: Mathlib.Probability.Kernel.Composition.ParallelComp

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
η : Kernel γ δ
h : IsSFiniteKernel η
⊢ 0 ∥ₖ η = 0

pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
η : Kernel γ δ
h : ¬IsSFiniteKernel η
⊢ 0 ∥ₖ η = 0

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
η : Kernel γ δ
h : IsSFiniteKernel η
⊢ 0 ∥ₖ η = 0
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
h : ¬SFinite μ
⊢ μ ⊗ₘ κ = 0

Tactic:
compProd,

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
h : ¬SFinite μ
⊢ (Kernel.const Unit μ ⊗ₖ Kernel.prodMkLeft Unit κ) () = 0
Module: Mathlib.Probability.Kernel.Composition.MapComap

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
f : β → γ
a✝¹ : α
s✝ : Set γ
a✝ : MeasurableSet s✝
hf : ¬Measurable f
⊢ ((map 0 f) a✝¹) s✝ = (0 a✝¹) s✝

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
f : β → γ
a✝¹ : α
s✝ : Set γ
a✝ : MeasurableSet s✝
hf : ¬Measurable f
⊢ ((map 0 f) a✝¹) s✝ = (0 a✝¹) s✝
Module: Mathlib.Probability.Kernel.Composition.ParallelComp

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
η : Kernel γ δ
h : ¬IsSFiniteKernel η
⊢ 0 ∥ₖ η = 0

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
η : Kernel γ δ
h : ¬IsSFiniteKernel η
⊢ 0 ∥ₖ η = 0
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
h : ¬IsSFiniteKernel κ
⊢ μ ⊗ₘ κ = 0

Tactic:
compProd,

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
h : ¬IsSFiniteKernel κ
⊢ (Kernel.const Unit μ ⊗ₖ Kernel.prodMkLeft Unit κ) () = 0