| import Mathlib |
| import OAI.Probability.CriticalPercolation.Criticality.GrowthProcess |
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|
| namespace OAI |
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| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped Classical ENNReal |
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| namespace CriticalPercolation.Corridor.Scheme |
| open BondGraph Boundary BernoulliProduct |
|
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| variable {V E : Type*} [Countable V] [Countable E] (P : Scheme V E) |
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| def Stable (n : ℕ) (S : GrowthState) (ω : Set E) : Prop := |
| (∀ v∈S.good, ω∈P.hit v (P.block S)) ∧ |
| (∀ e : Edge, source e∈S.good → n ≤ gridSlot (target e) → |
| 1-P.delta ≤ (blockProbability P.parameter (P.block S) |
| (P.hit (target e) (P.block S∪P.edgeDomain e)) ω).toReal) |
|
|
| omit [Countable V] [Countable E] in |
| lemma stable_initial (hroot : (P.seed∩P.mark (0,0)).Nonempty) |
| {ω : Set E} (hω : ω∈P.rootEvent) : P.Stable 0 GrowthState.initial ω := by |
| constructor |
| · intro v hv |
| have he : v=(0,0) := by simpa [GrowthState.initial] using hv |
| subst v |
| change ω∈P.graph.restrictedSeed P.seed (P.mark (0,0)) _ |
| rw [P.graph.restrictedSeed_self _ _ hroot] |
| trivial |
| · intro e he _hn |
| obtain ⟨v,b⟩ := e |
| have hv : v=(0,0) := by simpa [GrowthState.initial,source] using he |
| subst v |
| exact hω b |
|
|
| lemma stable_advance (hh : 0≤P.h) (hhj : P.h≤P.j) (hsep : 2*P.j+P.h<1) |
| (n : ℕ) (S : GrowthState) (b : Bool) (ω : Set E) |
| (hv : S.Valid n) (hc : S.Covered) (hs : P.Stable n S ω) |
| (ha : b=true → ω∈P.accepted n S) : P.Stable (n+1) (S.advance n b) ω := by |
| have hold_hit : ∀ v∈S.good, ω∈P.hit v (P.block (S.advance n b)) := by |
| intro v hv |
| exact P.graph.restrictedSeed_mono _ _ (P.block_enlarged n S) (hs.1 v hv) |
| have hold_pred : ∀ e : Edge, source e∈S.good → n+1 ≤ gridSlot (target e) → |
| 1-P.delta ≤ (blockProbability P.parameter (P.block (S.advance n b)) |
| (P.hit (target e) (P.block (S.advance n b)∪P.edgeDomain e)) ω).toReal := by |
| intro e hp ht |
| have hI := later_revealed_intersection P.graph P.coordinate P.j P.h hh hhj hsep |
| (n+1) S (S.advance n b) hc (S.advance_valid n b hv) hp ht |
| have hprob := P.graph.seed_prediction_persists P.parameter P.seed (P.mark (target e)) |
| (P.block S) (P.block (S.advance n b)) (P.edgeDomain e) ω (P.block_enlarged n S) hI |
| exact (hs.2 e hp (by omega)).trans |
| (ENNReal.toReal_mono (ne_of_lt (lt_of_le_of_lt (blockProbability_le_one _ _ _ _) ENNReal.one_lt_top)) hprob) |
| constructor |
| · intro v hv |
| change v∈(if (incoming n S.good).isSome ∧ b=true then insert (slotSite n) S.good else S.good) at hv |
| split_ifs at hv with hb |
| · rcases Finset.mem_insert.mp hv with rfl|hv |
| · exact (P.accepted_iff n S ω).mp (ha hb.2) |>.1 |
| · exact hold_hit v hv |
| · exact hold_hit v hv |
| · intro e hp ht |
| change source e∈(if (incoming n S.good).isSome ∧ b=true then insert (slotSite n) S.good else S.good) at hp |
| split_ifs at hp with hb |
| · rcases Finset.mem_insert.mp hp with he|hp |
| · have he' : e=(slotSite n,e.2) := Prod.ext he rfl |
| rw [he'] |
| exact (P.accepted_iff n S ω).mp (ha hb.2) |>.2 e.2 |
| · exact hold_pred e hp ht |
| · exact hold_pred e hp ht |
|
|
| theorem stable_on_cell (hh : 0≤P.h) (hhj : P.h≤P.j) (hsep : 2*P.j+P.h<1) |
| (hroot : (P.seed∩P.mark (0,0)).Nonempty) (n : ℕ) (w : DecisionCode n) |
| (ω : Set E) (hc : ω∈(P.rule.partition n).cell w) (hB : ω∈P.rootEvent) : |
| P.Stable n (P.rule.state n w) ω := by |
| induction n with |
| | zero => exact P.stable_initial hroot hB |
| | succ n ih => |
| rcases w with ⟨w,b⟩ |
| rw [P.rule.partition_cell_succ] at hc |
| apply P.stable_advance hh hhj hsep n (P.rule.state n w) b ω |
| (P.state_valid n w) (P.state_covered n w) (ih w hc.1) |
| intro hb |
| have hdecision := hc.2 |
| rw [hb] at hdecision |
| change (if ω∈P.accepted n (P.rule.state n w) then true else false)=true at hdecision |
| by_contra hn |
| simp only [ite_eq_right hn,Bool.false_eq_true] at hdecision |
|
|
| end CriticalPercolation.Corridor.Scheme |
| end |
| end |
| end |
|
|
| end OAI |
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|
