-- Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean
/-- For `κ' := map κ (Prod.map (id : β → β) e)`, the hypothesis `hη` is `fst κ' ⊗ₖ η = κ'`.
The conclusion of the lemma is `fst κ ⊗ₖ borelMarkovFromReal Ω η = comapRight (fst κ' ⊗ₖ η) _`. -/
lemma compProd_fst_borelMarkovFromReal_eq_comapRight_compProd
    (κ : Kernel α (β × Ω)) [IsSFiniteKernel κ] (η : Kernel (α × β) ℝ) [IsSFiniteKernel η]
    (hη : (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω)))) ⊗ₖ η
      = map κ (Prod.map (id : β → β) (embeddingReal Ω))) :
    fst κ ⊗ₖ borelMarkovFromReal Ω η
      = comapRight (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω))) ⊗ₖ η)
        (MeasurableEmbedding.id.prodMap (measurableEmbedding_embeddingReal Ω))

Proof state:
α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
e : Ω → ℝ := embeddingReal Ω
he : MeasurableEmbedding (embeddingReal Ω) := measurableEmbedding_embeddingReal Ω
κ' : Kernel α (β × ℝ) := κ.map (Prod.map id e)
hη' : κ'.fst ⊗ₖ η = κ'
h_prod_embed : MeasurableEmbedding (Prod.map id e)
h_fst : κ'.fst = κ.fst
a : α
t : Set (β × Ω)
ht : MeasurableSet t
⊢ (κ a) (Prod.map id e ⁻¹' (univ ×ˢ range e)ᶜ) = 0

case hs
α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
e : Ω → ℝ := embeddingReal Ω
he : MeasurableEmbedding (embeddingReal Ω) := measurableEmbedding_embeddingReal Ω
κ' : Kernel α (β × ℝ) := κ.map (Prod.map id e)
hη' : κ'.fst ⊗ₖ η = κ'
h_prod_embed : MeasurableEmbedding (Prod.map id e)
h_fst : κ'.fst = κ.fst
a : α
t : Set (β × Ω)
ht : MeasurableSet t
⊢ MeasurableSet (univ ×ˢ range e)ᶜ

Tactic:
swap

Result:
case hs
α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
e : Ω → ℝ := embeddingReal Ω
he : MeasurableEmbedding (embeddingReal Ω) := measurableEmbedding_embeddingReal Ω
κ' : Kernel α (β × ℝ) := κ.map (Prod.map id e)
hη' : κ'.fst ⊗ₖ η = κ'
h_prod_embed : MeasurableEmbedding (Prod.map id e)
h_fst : κ'.fst = κ.fst
a : α
t : Set (β × Ω)
ht : MeasurableSet t
⊢ MeasurableSet (univ ×ˢ range e)ᶜ

α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
e : Ω → ℝ := embeddingReal Ω
he : MeasurableEmbedding (embeddingReal Ω) := measurableEmbedding_embeddingReal Ω
κ' : Kernel α (β × ℝ) := κ.map (Prod.map id e)
hη' : κ'.fst ⊗ₖ η = κ'
h_prod_embed : MeasurableEmbedding (Prod.map id e)
h_fst : κ'.fst = κ.fst
a : α
t : Set (β × Ω)
ht : MeasurableSet t
⊢ (κ a) (Prod.map id e ⁻¹' (univ ×ˢ range e)ᶜ) = 0

Proof state:
case h.h
α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
e : Ω → ℝ := embeddingReal Ω
he : MeasurableEmbedding (embeddingReal Ω) := measurableEmbedding_embeddingReal Ω
κ' : Kernel α (β × ℝ) := κ.map (Prod.map id e)
hη' : κ'.fst ⊗ₖ η = κ'
h_prod_embed : MeasurableEmbedding (Prod.map id e)
a : α
u : Set β
a✝ : MeasurableSet u
⊢ (Measure.map (Prod.fst ∘ Prod.map id e) (κ a)) u = (Measure.map Prod.fst (κ a)) u

Tactic:
congr

Result:
no goals

Proof state:
case h
α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
e : Ω → ℝ := embeddingReal Ω
he : MeasurableEmbedding (embeddingReal Ω) := measurableEmbedding_embeddingReal Ω
κ' : Kernel α (β × ℝ) := κ.map (Prod.map id e)
hη' : κ'.fst ⊗ₖ η = κ'
h_prod_embed : MeasurableEmbedding (Prod.map id e)
h_fst : κ'.fst = κ.fst
a✝ : α
t : Set (β × Ω)
ht : MeasurableSet t
h_ae : ∀ᵐ (t : β) ∂κ.fst a✝, (a✝, t) ∈ {p | (η p) (range e)ᶜ = 0}
a : β
ha : (η (a✝, a)) (range e)ᶜ = 0
⊢ ((borelMarkovFromReal Ω η) (a✝, a)) {c | (a, c) ∈ t} = ((η.comapRight he) (a✝, a)) {c | (a, c) ∈ t}

Tactic:
rw [borelMarkovFromReal, comapRight_apply', comapRight_apply']

Result:
case h
α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
e : Ω → ℝ := embeddingReal Ω
he : MeasurableEmbedding (embeddingReal Ω) := measurableEmbedding_embeddingReal Ω
κ' : Kernel α (β × ℝ) := κ.map (Prod.map id e)
hη' : κ'.fst ⊗ₖ η = κ'
h_prod_embed : MeasurableEmbedding (Prod.map id e)
h_fst : κ'.fst = κ.fst
a✝ : α
t : Set (β × Ω)
ht : MeasurableSet t
h_ae : ∀ᵐ (t : β) ∂κ.fst a✝, (a✝, t) ∈ {p | (η p) (range e)ᶜ = 0}
a : β
ha : (η (a✝, a)) (range e)ᶜ = 0
⊢ ((piecewise ⋯ η (deterministic (fun x => Exists.choose ⋯) ⋯)) (a✝, a)) (embeddingReal Ω '' {c | (a, c) ∈ t}) =
    (η (a✝, a)) (embeddingReal Ω '' {c | (a, c) ∈ t})

case h.ht
α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
e : Ω → ℝ := embeddingReal Ω
he : MeasurableEmbedding (embeddingReal Ω) := measurableEmbedding_embeddingReal Ω
κ' : Kernel α (β × ℝ) := κ.map (Prod.map id e)
hη' : κ'.fst ⊗ₖ η = κ'
h_prod_embed : MeasurableEmbedding (Prod.map id e)
h_fst : κ'.fst = κ.fst
a✝ : α
t : Set (β × Ω)
ht : MeasurableSet t
h_ae : ∀ᵐ (t : β) ∂κ.fst a✝, (a✝, t) ∈ {p | (η p) (range e)ᶜ = 0}
a : β
ha : (η (a✝, a)) (range e)ᶜ = 0
⊢ MeasurableSet {c | (a, c) ∈ t}

case h.ht
α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
e : Ω → ℝ := embeddingReal Ω
he : MeasurableEmbedding (embeddingReal Ω) := measurableEmbedding_embeddingReal Ω
κ' : Kernel α (β × ℝ) := κ.map (Prod.map id e)
hη' : κ'.fst ⊗ₖ η = κ'
h_prod_embed : MeasurableEmbedding (Prod.map id e)
h_fst : κ'.fst = κ.fst
a✝ : α
t : Set (β × Ω)
ht : MeasurableSet t
h_ae : ∀ᵐ (t : β) ∂κ.fst a✝, (a✝, t) ∈ {p | (η p) (range e)ᶜ = 0}
a : β
ha : (η (a✝, a)) (range e)ᶜ = 0
⊢ MeasurableSet {c | (a, c) ∈ t}