| import Mathlib |
| import OAI.Combinatorics.IndependentSets.Reduction.Bits |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace IndependentSetsGames.Foundations.PCP.CayleySampling |
|
|
| open scoped BigOperators |
| open Finset |
|
|
| variable {V D : Type*} |
|
|
| abbrev FalseFiber (χ : V → Bool) := {x : V // χ x = false} |
|
|
| def falseRepresentative (χ : V → Bool) (flip : V → V) |
| (hχ : ∀ x, χ (flip x) = !(χ x)) (x : V) : FalseFiber χ := |
| if h : χ x = false then ⟨x, h⟩ |
| else ⟨flip x, by cases hx : χ x <;> simp_all⟩ |
|
|
| def booleanSplitEquiv (χ : V → Bool) (flip : V → V) |
| (hflip : Function.Involutive flip) (hχ : ∀ x, χ (flip x) = !(χ x)) : |
| V ≃ Bool × FalseFiber χ where |
| toFun x := (χ x, falseRepresentative χ flip hχ x) |
| invFun p := if p.1 then flip p.2.val else p.2.val |
| left_inv x := by |
| cases hx : χ x <;> simp [falseRepresentative, hx] |
| exact hflip x |
| right_inv p := by |
| rcases p with ⟨b, x⟩ |
| apply Prod.ext |
| · cases b <;> simp [hχ, x.property] |
| · cases b <;> apply Subtype.ext <;> |
| simp [falseRepresentative, hχ, x.property] |
| exact hflip x.val |
|
|
| @[simp] theorem booleanSplitEquiv_fst (χ : V → Bool) (flip : V → V) |
| (hflip : Function.Involutive flip) (hχ : ∀ x, χ (flip x) = !(χ x)) (x : V) : |
| (booleanSplitEquiv χ flip hflip hχ x).1 = χ x := rfl |
|
|
| def sampleSplitEquiv (χ : V → Bool) (flip : V → V) |
| (hflip : Function.Involutive flip) (hχ : ∀ x, χ (flip x) = !(χ x)) : |
| (D → V) ≃ (D → Bool) × (D → FalseFiber χ) := |
| (Equiv.piCongrRight fun _ : D => booleanSplitEquiv χ flip hflip hχ).trans |
| (Equiv.arrowProdEquivProdArrow D (fun _ => Bool) (fun _ => FalseFiber χ)) |
|
|
| @[simp] theorem sampleSplitEquiv_fst (χ : V → Bool) (flip : V → V) |
| (hflip : Function.Involutive flip) (hχ : ∀ x, χ (flip x) = !(χ x)) (g : D → V) : |
| (sampleSplitEquiv χ flip hflip hχ g).1 = fun d => χ (g d) := rfl |
|
|
| theorem card_boolean_split [Fintype V] (χ : V → Bool) (flip : V → V) |
| (hflip : Function.Involutive flip) (hχ : ∀ x, χ (flip x) = !(χ x)) : |
| Fintype.card V = 2 * Fintype.card (FalseFiber χ) := by |
| simpa using Fintype.card_congr (booleanSplitEquiv χ flip hflip hχ) |
|
|
| theorem expect_boolean_samples [Fintype V] [Nonempty V] [Fintype D] [DecidableEq D] |
| (χ : V → Bool) (flip : V → V) (hflip : Function.Involutive flip) |
| (hχ : ∀ x, χ (flip x) = !(χ x)) (F : (D → Bool) → ℝ) : |
| (𝔼 g : D → V, F (fun d => χ (g d))) = 𝔼 b : D → Bool, F b := by |
| classical |
| have : Nonempty (FalseFiber χ) := |
| ⟨falseRepresentative χ flip hχ (Classical.choice ‹Nonempty V›)⟩ |
| calc |
| (𝔼 g : D → V, F (fun d => χ (g d))) = |
| 𝔼 p : (D → Bool) × (D → FalseFiber χ), F p.1 := by |
| apply Fintype.expect_equiv (sampleSplitEquiv χ flip hflip hχ) |
| intro g |
| rfl |
| _ = 𝔼 b : D → Bool, F b := by |
| rw [← Finset.univ_product_univ, Finset.expect_product] |
| simp only [Fintype.expect_const] |
|
|
| end IndependentSetsGames.Foundations.PCP.CayleySampling |
| end |
|
|
| end OAI |
|
|
