openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /FiniteInterpolation.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Entropy.BinaryEventBound | |
| import OAI.Probability.CriticalPercolation.Entropy.BinaryCostBound | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| namespace CriticalPercolation.BinaryLaw | |
| open scoped Classical | |
| open FiniteProduct BondGraph BinaryInterpolation ExactExposure | |
| variable {V E : Type*} [Fintype V] [Fintype E] | |
| local instance interpolationFinDec (n : ℕ) : DecidableEq (Fin n) := Classical.decEq _ | |
| def allDarts (G : BondGraph V E) : List (LazyDFS.Dart V E) := | |
| (Finset.univ.filter (fun a : LazyDFS.Dart V E => G.ends a.2.1 = s(a.1,a.2.2))).toList | |
| lemma allDarts_good (G : BondGraph V E) : LazyDFS.GoodDarts G (allDarts G) := by | |
| intro a ha | |
| simpa only [allDarts,Finset.mem_toList,Finset.mem_filter,Finset.mem_univ,true_and] using ha | |
| lemma allDarts_exhaustive (G : BondGraph V E) : LazyDFS.Exhaustive G (allDarts G) := by | |
| intro v e t h | |
| simp only [allDarts,Finset.mem_toList,Finset.mem_filter,Finset.mem_univ,true_and] | |
| exact h | |
| theorem finite_interpolation_index (G : BondGraph V E) (d : ℕ) (x : ℕ → V) | |
| (hd : 2≤d) {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) | |
| {c : ℝ} (hc : 0<c) | |
| (hconn : ∀ j k, j<k → k≤d → k-j<d → | |
| c ≤ prob w (fun ω => (G.openGraph ω).Reachable (x j) (x k))) | |
| (s : ℕ) (hδ : 0 < prob w (endpointEvent G (x 0) (x d) s)) : | |
| ∃ j<d, 0 < prob w (endpointEvent G (x j) (x (j+1)) s) ∧ | |
| -Real.log (prob w (endpointEvent G (x j) (x (j+1)) s)) ≤ | |
| (4*(16/(2*p*(1-p)))+6)*(Real.log (1/prob w (endpointEvent G (x 0) (x d) s)) + | |
| (Nat.clog 2 d)*(Real.log 8+Real.log (1/c))) + | |
| (2*(16/(2*p*(1-p)))+5)*(Nat.clog 2 d)*Real.log 4 := by | |
| let order := allDarts G | |
| let n := Nat.clog 2 d | |
| have hc1 : c≤1 := (hconn 0 1 (by omega) (by omega) (by omega)).trans | |
| (prob_le_one w (homogeneous_normalized w p h0 h1) | |
| (fun e b => (homogeneous_positive w hp h0 h1 e b).le) _) | |
| obtain ⟨D,hD⟩ := exists_descriptor G order d x p w s n | |
| (jointMass_normalized G order d x p w s n (allDarts_exhaustive G) hd hp h0 h1 hc hconn hδ) | |
| have hD0 : 0 < descriptorMass G order d x p w s n D := | |
| (zpow_pos (by norm_num : (0:ℝ)<8) _).trans_le hD | |
| have he := conditioned_event_bound G order d x p w s n D (allDarts_good G) | |
| (allDarts_exhaustive G) (by omega) hp h0 h1 hc hc1 | |
| (fun j hj hjd => hconn 0 j hj hjd.le (by simpa only [Nat.sub_zero] using hjd)) hδ hD0 | |
| let h := fromWords (E := E) n (fun j => (D j).1) (fun _ => none) | |
| have hadj := interval_adjacent (by omega : 0<d) (le_refl (Nat.clog 2 d)) h | |
| have hb := interval_bounds (by omega : 0<d) n h | |
| refine ⟨(interval d n h).1,by omega,?_⟩ | |
| change 0 < prob w (endpointEvent G (x (interval d n h).1) (x (interval d n h).2) s) ∧ _ at he | |
| rw [hadj] at he | |
| refine ⟨he.1,he.2.trans ?_⟩ | |
| apply add_le_add ?_ le_rfl | |
| apply mul_le_mul_of_nonneg_left (constructionCost_le G order d x p w s n D hc hδ hD) | |
| have : 0 < 2*p*(1-p) := mul_pos (mul_pos (by norm_num) hp.1) (sub_pos.mpr hp.2) | |
| positivity | |
| end CriticalPercolation.BinaryLaw | |
| end | |
| end | |
| end | |
| end OAI | |