| import Mathlib.Data.Fintype.Powerset |
| import Mathlib.Basic.Real.Basic |
| import Mathlib.Algebra.BigOperators.Ring.Finset |
| import Mathlib.Combinatorics.SimpleGraph.Copy |
| import Mathlib.Analysis.SpecialFunctions.Log.Basic |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| universe u |
|
|
| namespace LeanBlast.SecondKahnKalai |
|
|
| def bernoulliWeight {X : Type u} [Fintype X] [DecidableEq X] |
| (p : ℝ) (W : Finset X) : ℝ := |
| ∏ x, if x ∈ W then p else 1 - p |
|
|
| def bernoulliExpectation {X : Type u} [Fintype X] [DecidableEq X] |
| (p : ℝ) (f : Finset X → ℝ) : ℝ := |
| ∑ W, bernoulliWeight p W * f W |
|
|
| def bernoulliProbability {X : Type u} [Fintype X] [DecidableEq X] |
| (p : ℝ) (P : Finset X → Prop) : ℝ := by |
| classical |
| exact ∑ W, if P W then bernoulliWeight p W else 0 |
|
|
| / |
| abbrev Edge (n : ℕ) := {e : Sym2 (Fin n) // ¬e.IsDiag} |
|
|
| / |
| def graphOfEdges {n : ℕ} (W : Finset (Edge n)) : SimpleGraph (Fin n) := |
| SimpleGraph.fromEdgeSet (Subtype.val '' (W : Set (Edge n))) |
|
|
| / |
| def edgeCount {V : Type u} (H : SimpleGraph V) : ℕ := Nat.card H.edgeSet |
|
|
| / |
| def expectedCopies {V : Type u} (n : ℕ) (p : ℝ) (H : SimpleGraph V) : ℝ := |
| bernoulliExpectation p (fun W : Finset (Edge n) => |
| ((graphOfEdges W).copyCount H : ℝ)) |
|
|
| / |
| def containmentProbability {V : Type u} (n : ℕ) (p : ℝ) |
| (H : SimpleGraph V) : ℝ := |
| bernoulliProbability p (fun W : Finset (Edge n) => |
| 0 < (graphOfEdges W).copyCount H) |
|
|
| def criticalThreshold {V : Type u} (n : ℕ) (H : SimpleGraph V) : ℝ := |
| sInf {p : ℝ | p ∈ Set.Icc 0 1 ∧ (1 / 2 : ℝ) ≤ containmentProbability n p H} |
|
|
| def expectationThreshold {V : Type u} (n : ℕ) (H : SimpleGraph V) : ℝ := |
| sInf {p : ℝ | p ∈ Set.Icc 0 1 ∧ |
| ∀ F : H.Subgraph, (1 / 2 : ℝ) ≤ expectedCopies n p F.coe} |
|
|
| def logTwo (x : ℝ) : ℝ := Real.log x / Real.log 2 |
|
|
| def SecondKahnKalaiBounds {V : Type u} (n : ℕ) (H : SimpleGraph V) : Prop := |
| criticalThreshold n H ≤ |
| min 1 (2048 * Real.exp 50 * expectationThreshold n H * |
| (1 + logTwo (edgeCount H : ℝ))) ∧ |
| criticalThreshold n H ≤ |
| 6144 * Real.exp 50 * expectationThreshold n H * logTwo (n : ℝ) |
|
|
| end LeanBlast.SecondKahnKalai |
|
|
| end |
|
|
| end OAI |
|
|
