| import Mathlib |
| import OAI.Computability.MinUncut.Games.HintSimulation |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace MinUncut.Outer |
| attribute [local instance] Classical.propDecidable |
| variable {I X : Type*} [Fintype I] [Fintype X] [Nonempty X] |
|
|
| lemma expect_le_of_active_sections (A : I → Prop) (f : (I → X) → ℝ) (C : ℝ) |
| (h : ∀ bg : I → X, (𝔼 u : {i // A i} → X, f (patch A bg u)) ≤ C) : |
| (𝔼 w : I → X, f w) ≤ C := by |
| classical |
| let bg (v : {i // ¬A i} → X) : I → X := |
| fun i => if hi : ¬A i then v ⟨i,hi⟩ else Classical.ofNonempty |
| let e := Equiv.piEquivPiSubtypeProd A (fun _ => X) |
| have he : (𝔼 w : I → X, f w) = |
| 𝔼 z : ({i // A i} → X) × ({i // ¬A i} → X), f (patch A (bg z.2) z.1) := by |
| apply Fintype.expect_equiv e |
| intro w |
| congr 1 |
| funext i |
| by_cases hi : A i <;> simp [patch,bg,e,Equiv.piEquivPiSubtypeProd,hi] |
| rw [he,OuterSmoothness.expect_pair,Finset.expect_comm] |
| calc |
| _ ≤ 𝔼 _ : {i // ¬A i} → X, C := Finset.expect_le_expect (fun v _ => h (bg v)) |
| _ = C := Fintype.expect_const _ |
|
|
| end MinUncut.Outer |
|
|
| end |
| end |
|
|
| end OAI |
|
|
