| import Mathlib |
| import OAI.Probability.CriticalPercolation.Exploration.ClosedFresh |
|
|
| namespace OAI |
|
|
| section |
| section |
| open Set MeasureTheory ProbabilityTheory |
| open Set |
| open scoped BigOperators |
| open scoped Classical |
|
|
| namespace CriticalPercolation.SearchTree |
| open scoped Classical |
| open FiniteProduct LazyDFS |
|
|
| variable {E R : Type*} [Fintype E] |
|
|
| theorem terminalProbability_eq_indicator (T : SearchTree E R) (F : Set E → Prop) |
| (H : R → Prop) (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1) |
| (hT : ∀ ω, density w ω ≠ 0 → (F ω ↔ H (T.run (fun e => decide (ω e))))) |
| (ω : E → Prop) (hω : density w ω ≠ 0) : |
| T.terminalProbability F w ω = if F ω then 1 else 0 := by |
| induction T generalizing w with |
| | done r => |
| change prob w F = _ |
| calc |
| prob w F = prob w (fun _ => H r) := prob_congr_support w F _ hT |
| _ = if H r then 1 else 0 := expect_const w hw _ |
| _ = if F ω then 1 else 0 := by rw [hT ω hω]; rfl |
| | ask e l r hl hr => |
| have he := coordinate_ne_zero_of_density w ω hω e |
| by_cases hb : ω e |
| · have hb' : ω e = True := propext (iff_true_intro hb) |
| have he' : w e True ≠ 0 := by simpa only [hb'] using he |
| have hTr : ∀ η, density (pin w e True) η ≠ 0 → |
| (F η ↔ H (r.run (fun j => decide (η j)))) := by |
| intro η hη |
| have hbit := density_pin_support w e True η hη |
| have hh := hT η (density_pin_old_support w e True η he' hη) |
| simpa only [run,hbit,decide_true,Bool.true_eq,ite_true] using hh |
| change (if ω e then r.terminalProbability F (pin w e True) ω |
| else l.terminalProbability F (pin w e False) ω) = _ |
| rw [ite_eq_left hb] |
| exact hr (pin w e True) (pin_normalized w hw e True) hTr |
| (density_pin_of_support w ω e True hω hb') |
| · have hb' : ω e = False := propext (iff_false_intro hb) |
| have he' : w e False ≠ 0 := by simpa only [hb'] using he |
| have hTl : ∀ η, density (pin w e False) η ≠ 0 → |
| (F η ↔ H (l.run (fun j => decide (η j)))) := by |
| intro η hη |
| have hbit := density_pin_support w e False η hη |
| have hh := hT η (density_pin_old_support w e False η he' hη) |
| simpa only [run,hbit,decide_false,Bool.false_eq_true,ite_false] using hh |
| change (if ω e then r.terminalProbability F (pin w e True) ω |
| else l.terminalProbability F (pin w e False) ω) = _ |
| rw [ite_eq_right hb] |
| exact hl (pin w e False) (pin_normalized w hw e False) hTl |
| (density_pin_of_support w ω e False hω hb') |
|
|
| variable [DecidableEq E] |
|
|
| theorem contact_normalization (T : SearchTree E R) (used : Finset E) (hvalid : T.Valid used) |
| (p : ℝ) (C : Set E) (F : Set E → Prop) (H : R → Prop) (w v : E → Prop → ℝ) |
| (hw : ∀ e, ∑ b, w e b = 1) (hv : ∀ e, ∑ b, v e b = 1) |
| (hf : Fresh w p used) (hc : ClosedFresh v p C used) |
| (hT : ∀ ω, density w ω ≠ 0 → (F ω ↔ H (T.run (fun e => decide (ω e))))) |
| (hac : ∀ ω, density v ω ≠ 0 → density w ω ≠ 0) |
| (hfail : ∀ ω, density v ω ≠ 0 → ¬F ω) : |
| expect v (T.contactScore p C F w) = prob w F := by |
| have ht : expect v (T.terminalProbability F w) = 0 := by |
| calc |
| expect v (T.terminalProbability F w) = expect v (fun _ => 0) := by |
| apply expect_congr_support |
| intro ω hω |
| rw [terminalProbability_eq_indicator T F H w hw hT ω (hac ω hω),ite_eq_right (hfail ω hω)] |
| _ = 0 := expect_zero v |
| have hm := contact_mean_terminal T used hvalid p C F w v hw hv hf hc |
| rw [ht,add_zero] at hm |
| exact hm |
|
|
| end CriticalPercolation.SearchTree |
|
|
| namespace CriticalPercolation.FiniteProduct |
| open scoped Classical |
|
|
| variable {E : Type*} |
|
|
| noncomputable def forceClosed (w : E → Prop → ℝ) (C : Set E) (e : E) (b : Prop) : ℝ := |
| if e ∈ C then (if b then 0 else 1) else w e b |
|
|
| variable [Fintype E] |
| omit [Fintype E] in |
| lemma forceClosed_normalized (w : E → Prop → ℝ) (C : Set E) |
| (hw : ∀ e, ∑ b, w e b = 1) : ∀ e, ∑ b, forceClosed w C e b = 1 := by |
| intro e |
| by_cases he : e ∈ C |
| · simp [forceClosed,he,Fintype.univ_Prop] |
| · simpa only [forceClosed,ite_eq_right he] using hw e |
|
|
| omit [Fintype E] in |
| lemma forceClosed_nonneg (w : E → Prop → ℝ) (C : Set E) |
| (hn : ∀ e b, 0 ≤ w e b) : ∀ e b, 0 ≤ forceClosed w C e b := by |
| intro e b |
| unfold forceClosed |
| split_ifs <;> first | positivity | exact hn e b |
|
|
| lemma forceClosed_support (w : E → Prop → ℝ) (C : Set E) (ω : E → Prop) |
| (hω : density (forceClosed w C) ω ≠ 0) : ∀ e ∈ C, ¬ω e := by |
| intro e he hopen |
| have h := coordinate_ne_zero_of_density (forceClosed w C) ω hω e |
| simp only [forceClosed,ite_eq_left he,ite_eq_left hopen,ne_eq,not_true_eq_false] at h |
|
|
| omit [Fintype E] in |
| lemma homogeneous_normalized (w : E → Prop → ℝ) (p : ℝ) |
| (h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p) : |
| ∀ e, ∑ b, w e b = 1 := by |
| intro e |
| simp [Fintype.univ_Prop,h0,h1] |
|
|
| omit [Fintype E] in |
| lemma homogeneous_positive (w : E → Prop → ℝ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) |
| (h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p) : |
| ∀ e b, 0 < w e b := by |
| intro e b |
| by_cases hb : b |
| · have : b = True := propext (iff_true_intro hb) |
| rw [this,h1]; exact hp.1 |
| · have : b = False := propext (iff_false_intro hb) |
| rw [this,h0]; exact sub_pos.mpr hp.2 |
|
|
| lemma homogeneous_support (w : E → Prop → ℝ) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) |
| (h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p) (ω : E → Prop) : |
| density w ω ≠ 0 := by |
| apply Finset.prod_ne_zero_iff.mpr |
| intro e _ |
| exact (homogeneous_positive w hp h0 h1 e (ω e)).ne' |
| |
| end CriticalPercolation.FiniteProduct |
| |
| end |
| end |
| |
| end OAI |
| |
