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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | 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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