| import OAI.Combinatorics.GraphThreshold.Resampling |
| import Mathlib.Analysis.SpecialFunctions.Exp |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| /-! # Success and failure of a local resampling transition -/ |
|
|
| noncomputable section |
|
|
| namespace LeanBlast.SecondKahnKalai |
|
|
| variable {X B : Type*} [DecidableEq X] [Fintype B] |
|
|
| / |
| halved integral capacity. -/ |
| def LocalSuccess (ν : FiniteLaw B) (A : B → Finset X) (ρ : ℝ) (m : ℕ) |
| (W : Finset X) (c b : B) : Prop := |
| Real.exp (-10 * (m : ℝ)) ≤ resamplingZ ν A ρ (W ∪ A c) ∧ |
| (A b \ W).card ≤ m / 2 |
|
|
| / |
| def LocalFailure (ν : FiniteLaw B) (A : B → Finset X) (ρ : ℝ) (m : ℕ) |
| (W : Finset X) (c b : B) : Prop := |
| ¬ LocalSuccess ν A ρ m W c b |
|
|
| theorem localFailure_iff (ν : FiniteLaw B) (A : B → Finset X) (ρ : ℝ) (m : ℕ) |
| (W : Finset X) (c b : B) : |
| LocalFailure ν A ρ m W c b ↔ |
| resamplingZ ν A ρ (W ∪ A c) < Real.exp (-10 * (m : ℝ)) ∨ |
| m / 2 < (A b \ W).card := by |
| simp only [LocalFailure, LocalSuccess, not_and_or, not_le] |
|
|
| theorem localFailure_iff_twice (ν : FiniteLaw B) (A : B → Finset X) (ρ : ℝ) (m : ℕ) |
| (W : Finset X) (c b : B) : |
| LocalFailure ν A ρ m W c b ↔ |
| resamplingZ ν A ρ (W ∪ A c) < Real.exp (-10 * (m : ℝ)) ∨ |
| m < 2 * (A b \ W).card := by |
| rw [localFailure_iff] |
| apply or_congr Iff.rfl |
| omega |
|
|
| end LeanBlast.SecondKahnKalai |
|
|
| end |
|
|
| end OAI |
|
|
