| import Mathlib |
| import OAI.Probability.CriticalPercolation.Exploration.CompileContactPrefixCongr |
| import OAI.Probability.CriticalPercolation.Entropy.BinaryMarks |
|
|
| namespace OAI |
|
|
| section |
| section |
| namespace CriticalPercolation.BinaryLaw |
| open scoped Classical |
| open FiniteProduct BondGraph SearchTree BinaryInterpolation InterpolationForest |
|
|
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def switched (n : ℕ) (h : History E n) (ω : Set E) (S : Finset ℕ) : Set E := |
| {e | ω e ∨ ∃ i ∈ S, nodeMark n h i = some e} |
|
|
| lemma switched_mono (n : ℕ) (h : History E n) (ω : Set E) (S : Finset ℕ) : |
| ω ⊆ switched n h ω S := fun _ he => Or.inl he |
|
|
| lemma switched_insert (n : ℕ) (h : History E n) (ω : Set E) (S : Finset ℕ) |
| (i : ℕ) (e : E) (hm : nodeMark n h i = some e) : |
| switched n h ω (insert i S) = Function.update (switched n h ω S) e True := by |
| apply Set.ext |
| intro f |
| change (ω f ∨ ∃ j ∈ insert i S, nodeMark n h j = some f) ↔ |
| (Function.update (switched n h ω S : E → Prop) e True) f |
| by_cases hfe : f=e |
| · subst f |
| erw [Function.update_self] |
| simp only [iff_true] |
| exact Or.inr ⟨i,Finset.mem_insert_self _ _,hm⟩ |
| · erw [Function.update_of_ne hfe] |
| constructor |
| · rintro (ho|⟨j,hj,hjm⟩) |
| · exact Or.inl ho |
| · rcases Finset.mem_insert.mp hj with rfl|hj |
| · exact (hfe (Option.some.inj (hjm.symm.trans hm))).elim |
| · exact Or.inr ⟨j,hj,hjm⟩ |
| · rintro (ho|⟨j,hj,hjm⟩) |
| · exact Or.inl ho |
| · exact Or.inr ⟨j,Finset.mem_insert_of_mem hj,hjm⟩ |
|
|
| def targets (d : ℕ) (x : ℕ → V) (n : ℕ) (h : History E n) (i : ℕ) : Set V := |
| {v | ∃ j < i, (node d x n h j).1=v} |
|
|
| lemma canonical_misses_targets (d : ℕ) (x : ℕ → V) (n : ℕ) (h : History E n) |
| (ω : Set E) (i : ℕ) (hin : i < n+2) (hc : Canonical G (anchors d x n h) ω i) : |
| ¬∃ b ∈ targets d x n h i, (G.openGraph ω).Reachable (node d x n h i).1 b := by |
| rintro ⟨b,⟨j,hji,hjb⟩,hr⟩ |
| exact hc j _ _ hji (node_getElem? d x n h j (hji.trans hin)) |
| (node_getElem? d x n h i hin) (hjb ▸ hr.symm) |
|
|
| section Contact |
| variable [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] |
| variable (order : List (LazyDFS.Dart V E)) (d : ℕ) (x : ℕ → V) |
| variable (p : ℝ) (w : E → Prop → ℝ) (r₀ : (E → Prop) → ℝ) |
|
|
| lemma earlier_not_incident (n : ℕ) (h : History E n) (ω : E → Prop) |
| (hr : density G order d x p w r₀ n h ω ≠ 0) |
| (i j : ℕ) (hin : i < n+2) (hji : j < i) (hj : 2 ≤ j) |
| (hc : Canonical G (anchors d x n h) ω i) (e : E) (hm : nodeMark n h j = some e) : |
| ¬∃ v ∈ G.ends e, (G.openGraph ω).Reachable (node d x n h i).1 v := by |
| obtain ⟨_,u,v,hev,hu,hv⟩ := node_mark_geometry G order d x p w r₀ n h ω hr j hj |
| (hji.trans hin) e hm |
| rintro ⟨z,hz,hiz⟩ |
| rw [hev,Sym2.mem_iff] at hz |
| have hp := nodeParent_lt G d x ω n h j hj |
| rcases hz with rfl|rfl |
| · exact hc j _ _ hji (node_getElem? d x n h j (hji.trans hin)) |
| (node_getElem? d x n h i hin) (hu.trans hiz.symm) |
| · exact hc _ _ _ (hp.trans hji) (node_getElem? d x n h _ ((hp.trans hji).trans hin)) |
| (node_getElem? d x n h i hin) (hv.trans hiz.symm) |
|
|
| lemma switched_earlier_incidence (n : ℕ) (h : History E n) (ω : E → Prop) |
| (hr : density G order d x p w r₀ n h ω ≠ 0) |
| (i : ℕ) (hin : i < n+2) (hc : Canonical G (anchors d x n h) ω i) |
| (S : Finset ℕ) (hS : ∀ j ∈ S, 2 ≤ j ∧ j < i) : |
| ∀ e, (∃ v ∈ G.ends e, (G.openGraph ω).Reachable (node d x n h i).1 v) → |
| (ω e ↔ switched n h ω S e) := by |
| intro e he |
| constructor |
| · exact Or.inl |
| · rintro (ho|⟨j,hj,hjm⟩) |
| · exact ho |
| · exact (earlier_not_incident G order d x p w r₀ n h ω hr i j hin (hS j hj).2 |
| (hS j hj).1 hc e hjm he).elim |
|
|
| theorem switched_contact_reconstruction (n : ℕ) (h : History E n) (ω : E → Prop) |
| (hr : density G order d x p w r₀ n h ω ≠ 0) |
| (i : ℕ) (hi : 2 ≤ i) (hin : i < n+2) |
| (hc : Canonical G (anchors d x n h) ω i) (e : E) (hm : nodeMark n h i = some e) |
| (S : Finset ℕ) (hS : ∀ j ∈ S, 2 ≤ j ∧ j < i) (offset : ℝ) : |
| let B := targets d x n h i |
| let a := (node d x n h i).1 |
| let F := fun σ : Set E => ∃ b ∈ B, (G.openGraph σ).Reachable a b |
| let T := LazyDFS.search G order a B |
| T.queryDelta F w (Function.update ω e True) e = |
| T.queryDelta F w (switched n h ω (insert i S)) e ∧ |
| T.prefixScore p F w (Function.update ω e True) e = |
| T.prefixScore p F w (switched n h ω (insert i S)) e ∧ |
| T.reverseGuess p F w (Function.update ω e True) offset e = |
| T.reverseGuess p F w (switched n h ω (insert i S)) offset e := by |
| dsimp only |
| rw [switched_insert n h ω S i e hm] |
| obtain ⟨hclosed,u,v,hev,hu,hv⟩ := node_mark_geometry G order d x p w r₀ n h ω hr i hi hin e hm |
| have hb : (node d x n h (nodeParent G d x ω n h i)).1 ∈ targets d x n h i := |
| ⟨_,nodeParent_lt G d x ω n h i hi,rfl⟩ |
| have hup (σ : Set E) : |
| @Function.update E (fun _ => Prop) (fun a b => Classical.propDecidable (a=b)) σ e True = |
| Function.update σ e True := by |
| exact congrArg (fun d' : DecidableEq E => |
| @Function.update E (fun _ => Prop) d' σ e True) (Subsingleton.elim _ _) |
| simpa only [hup] using LazyDFS.search_contact_reconstruction G order ω (switched n h ω S) _ _ u v e |
| hev hclosed (canonical_misses_targets G d x n h ω i hin hc) hu |
| ⟨_,hb,hv.symm⟩ ⟨_,hb,hv.symm.mono (G.openGraph_mono (switched_mono n h ω S))⟩ |
| (switched_earlier_incidence G order d x p w r₀ n h ω hr i hin hc S hS) p w offset |
|
|
| end Contact |
| end CriticalPercolation.BinaryLaw |
| end |
| end |
|
|
| end OAI |
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