openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Exploration /SearchMarkedContactBound.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Exploration.ClosedPivotWeight | |
| namespace OAI | |
| section | |
| section | |
| open Set MeasureTheory ProbabilityTheory | |
| open Set | |
| open scoped BigOperators | |
| open scoped Classical | |
| namespace CriticalPercolation.LazyDFS | |
| open scoped Classical | |
| open FiniteProduct SearchTree | |
| variable {V E : Type*} [Fintype E] [Fintype V] [DecidableEq V] [DecidableEq E] | |
| theorem search_marked_contact_bound (G : BondGraph V E) (B : Set V) | |
| (order : List (Dart V E)) (horder : GoodDarts G order) (ho : Exhaustive G order) | |
| (a : V) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) (w : E → Prop → ℝ) | |
| (h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p) : | |
| let F := fun ω : Set E => ∃ b ∈ B, (G.openGraph ω).Reachable a b | |
| let T := search G order a B | |
| expect w (fun ω => ∑ e, T.closedPivotWeight p F w ω e * | |
| (1 + T.prefixScore p F w (Function.update ω e True) e)) ≤ 4 := by | |
| let F := fun ω : Set E => ∃ b ∈ B, (G.openGraph ω).Reachable a b | |
| let T := search G order a B | |
| have hv : T.Valid ∅ := compile_valid G order B _ _ _ | |
| have hF : ∀ s t, s ⊆ t → F s → F t := by | |
| rintro s t hst ⟨b,hb,hab⟩ | |
| exact ⟨b,hb,hab.mono (G.openGraph_mono hst)⟩ | |
| have hm := search_target_pivotal_moments G B order horder ho a hp w h0 h1 | |
| have hb := marked_prefix_bound T ∅ hv hp F hF w (homogeneous_normalized w p h0 h1) | |
| (fun e b => (homogeneous_positive w hp h0 h1 e b).le) | |
| (fun e _ => ⟨h0 e,h1 e⟩) h0 h1 | |
| exact hb.trans (by linarith [hm.1,hm.2]) | |
| end CriticalPercolation.LazyDFS | |
| end | |
| end | |
| end OAI | |