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[Merged by Bors] - chore(probability/kernel/composition): swap the order of the arguments of prod_mk_left #18929

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[Merged by Bors] - chore(probability/kernel/composition): swap the order of the arguments of prod_mk_left #18929

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20 changes: 10 additions & 10 deletions src/probability/kernel/composition.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -461,32 +461,32 @@ open_locale probability_theory
section fst_snd

/-- Define a `kernel (γ × α) β` from a `kernel α β` by taking the comap of the projection. -/
def prod_mk_left (κ : kernel α β) (γ : Type*) [measurable_space γ] : kernel (γ × α) β :=
def prod_mk_left (γ : Type*) [measurable_space γ] (κ : kernel α β) : kernel (γ × α) β :=
comap κ prod.snd measurable_snd

variables {γ : Type*} {mγ : measurable_space γ} {f : β → γ} {g : γ → α}

include mγ

lemma prod_mk_left_apply (κ : kernel α β) (ca : γ × α) :
prod_mk_left κ γ ca = κ ca.snd := rfl
prod_mk_left γ κ ca = κ ca.snd := rfl

lemma prod_mk_left_apply' (κ : kernel α β) (ca : γ × α) (s : set β) :
prod_mk_left κ γ ca s = κ ca.snd s := rfl
prod_mk_left γ κ ca s = κ ca.snd s := rfl

lemma lintegral_prod_mk_left (κ : kernel α β) (ca : γ × α) (g : β → ℝ≥0∞) :
∫⁻ b, g b ∂(prod_mk_left κ γ ca) = ∫⁻ b, g b ∂(κ ca.snd) := rfl
∫⁻ b, g b ∂(prod_mk_left γ κ ca) = ∫⁻ b, g b ∂(κ ca.snd) := rfl

instance is_markov_kernel.prod_mk_left (κ : kernel α β) [is_markov_kernel κ] :
is_markov_kernel (prod_mk_left κ γ) :=
is_markov_kernel (prod_mk_left γ κ) :=
by { rw prod_mk_left, apply_instance, }

instance is_finite_kernel.prod_mk_left (κ : kernel α β) [is_finite_kernel κ] :
is_finite_kernel (prod_mk_left κ γ) :=
is_finite_kernel (prod_mk_left γ κ) :=
by { rw prod_mk_left, apply_instance, }

instance is_s_finite_kernel.prod_mk_left (κ : kernel α β) [is_s_finite_kernel κ] :
is_s_finite_kernel (prod_mk_left κ γ) :=
is_s_finite_kernel (prod_mk_left γ κ) :=
by { rw prod_mk_left, apply_instance, }

/-- Define a `kernel (β × α) γ` from a `kernel (α × β) γ` by taking the comap of `prod.swap`. -/
Expand Down Expand Up @@ -612,7 +612,7 @@ include mγ
noncomputable
def comp (η : kernel β γ) [is_s_finite_kernel η] (κ : kernel α β) [is_s_finite_kernel κ] :
kernel α γ :=
snd (κ ⊗ₖ prod_mk_left η α)
snd (κ ⊗ₖ prod_mk_left α η)

localized "infix (name := kernel.comp) ` ∘ₖ `:100 := probability_theory.kernel.comp" in
probability_theory
Expand All @@ -631,7 +631,7 @@ lemma lintegral_comp (η : kernel β γ) [is_s_finite_kernel η] (κ : kernel α
∫⁻ c, g c ∂((η ∘ₖ κ) a) = ∫⁻ b, ∫⁻ c, g c ∂(η b) ∂(κ a) :=
begin
rw [comp, lintegral_snd _ _ hg],
change ∫⁻ bc, (λ a b, g b) bc.fst bc.snd ∂((κ ⊗ₖ prod_mk_left η α) a)
change ∫⁻ bc, (λ a b, g b) bc.fst bc.snd ∂((κ ⊗ₖ prod_mk_left α η) a)
= ∫⁻ b, ∫⁻ c, g c ∂(η b) ∂(κ a),
exact lintegral_comp_prod _ _ _ (hg.comp measurable_snd),
end
Expand Down Expand Up @@ -690,7 +690,7 @@ include mγ
noncomputable
def prod (κ : kernel α β) [is_s_finite_kernel κ] (η : kernel α γ) [is_s_finite_kernel η] :
kernel α (β × γ) :=
κ ⊗ₖ (swap_left (prod_mk_left η β))
κ ⊗ₖ (swap_left (prod_mk_left β η))

lemma prod_apply (κ : kernel α β) [is_s_finite_kernel κ] (η : kernel α γ) [is_s_finite_kernel η]
(a : α) {s : set (β × γ)} (hs : measurable_set s) :
Expand Down