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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.RestrictedSeed
import OAI.Probability.CriticalPercolation.Exploration.DecisionHistory
import OAI.Probability.CriticalPercolation.Exploration.GrowthHistoryGeometry
import OAI.Probability.CriticalPercolation.Gluing.PinnedTest
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation.Corridor
open BondGraph Boundary BernoulliProduct
structure Scheme (V E : Type*) where
graph : BondGraph V E
coordinate : V → Point
j : ℝ
h : ℝ
seed : Set V
mark : Site → Set V
parameter : E → unitInterval
delta : ℝ
namespace Scheme
variable {V E : Type*} [Countable V] [Countable E] (P : Scheme V E)
def block (S : GrowthState) : Set E := revealed P.graph P.coordinate P.j P.h S
def edgeDomain (e : Edge) : Set E := corridorEdges P.graph P.coordinate P.j P.h e
def enlarged (n : ℕ) (S : GrowthState) : Set E := P.block (S.advance n false)
def hit (v : Site) (R : Set E) : Set (Set E) := P.graph.restrictedSeed P.seed (P.mark v) R
def incomingHit (n : ℕ) (S : GrowthState) : Set (Set E) := P.hit (slotSite n) (P.enlarged n S)
def nextHit (n : ℕ) (S : GrowthState) (b : Bool) : Set (Set E) :=
P.hit (target (slotSite n,b)) (P.enlarged n S∪P.edgeDomain (slotSite n,b))
def accepted (n : ℕ) (S : GrowthState) : Set (Set E) :=
(predictionFailure P.parameter (P.enlarged n S) (P.incomingHit n S)
(P.nextHit n S true) (P.nextHit n S false) P.delta)ᶜ
lemma accepted_measurable (n : ℕ) (S : GrowthState) :
MeasurableSet[blockSigma (P.enlarged n S)] (P.accepted n S) :=
(measurableSet_predictionFailure P.parameter (P.enlarged n S) _ _ _
(P.graph.measurableSet_restrictedSeed _ _ _) (P.graph.restrictedSeed_measurable _ _ _)
(P.graph.restrictedSeed_measurable _ _ _) P.delta).compl
omit [Countable V] [Countable E] in
lemma accepted_iff (n : ℕ) (S : GrowthState) (ω : Set E) :
ω∈P.accepted n S ↔ ω∈P.incomingHit n S ∧
∀ b : Bool, 1-P.delta ≤ (blockProbability P.parameter (P.enlarged n S) (P.nextHit n S b) ω).toReal := by
change (¬((ω∉P.incomingHit n S ∨ ω∈P.incomingHit n S ∧
(blockProbability P.parameter (P.enlarged n S) (P.nextHit n S true) ω).toReal<1-P.delta) ∨
ω∈P.incomingHit n S ∧
(blockProbability P.parameter (P.enlarged n S) (P.nextHit n S false) ω).toReal<1-P.delta)) ↔ _
constructor
· intro h
have ha : ω∈P.incomingHit n S := Classical.byContradiction (fun hn => h (Or.inl (Or.inl hn)))
refine ⟨ha,fun b => ?_⟩
cases b
· exact not_lt.mp (fun hb => h (Or.inr ⟨ha,hb⟩))
· exact not_lt.mp (fun hb => h (Or.inl (Or.inr ⟨ha,hb⟩)))
· rintro ⟨ha,hb⟩ (h|h)
· rcases h with h|h
· exact h ha
· exact not_lt_of_ge (hb true) h.2
· exact not_lt_of_ge (hb false) h.2
def rule : DecisionRule E GrowthState where
initial := GrowthState.initial
block := P.block
further := P.enlarged
contained n S := revealed_mono P.graph P.coordinate P.j P.h (S.seen_mono n false)
active n S := (incoming n S.good).isSome
accept := P.accepted
measurable := P.accepted_measurable
update := GrowthState.advance
block_update _ _ _ := rfl
lemma state_valid (n : ℕ) (w : DecisionCode n) : (P.rule.state n w).Valid n := by
induction n with
| zero => exact GrowthState.initial_valid
| succ n ih => exact GrowthState.advance_valid n _ w.2 (ih w.1)
lemma state_covered (n : ℕ) (w : DecisionCode n) : (P.rule.state n w).Covered := by
induction n with
| zero => exact GrowthState.initial_covered
| succ n ih => exact GrowthState.advance_covered n _ w.2 (ih w.1)
omit [Countable V] [Countable E] in
lemma block_enlarged (n : ℕ) (S : GrowthState) : P.block S⊆P.enlarged n S :=
revealed_mono P.graph P.coordinate P.j P.h (S.seen_mono n false)
omit [Countable V] [Countable E] in
lemma block_advance (n : ℕ) (S : GrowthState) (b : Bool) : P.block (S.advance n b)=P.enlarged n S := rfl
omit [Countable V] [Countable E] in
lemma enlarged_incoming (n : ℕ) (S : GrowthState) {e : Edge} (he : incoming n S.good=some e) :
P.enlarged n S=P.block S∪P.edgeDomain e :=
revealed_incoming P.graph P.coordinate P.j P.h n S false he
def rootEvent : Set (Set E) := {ω | ∀ b : Bool,
1-P.delta ≤ (blockProbability P.parameter (P.block GrowthState.initial)
(P.hit (target ((0,0),b)) (P.block GrowthState.initial∪P.edgeDomain ((0,0),b))) ω).toReal}
lemma rootEvent_measurable : MeasurableSet[blockSigma (P.block GrowthState.initial)] P.rootEvent := by
simp only [rootEvent,Set.ofPred_forall]
apply MeasurableSet.iInter
intro b
exact measurableSet_le measurable_const
(measurable_blockProbability P.parameter _ _ (P.graph.restrictedSeed_measurable _ _ _)).ennreal_toReal
lemma rootEvent_history : MeasurableSet[(P.rule.partition 0).historySigma] P.rootEvent := by
intro w
change MeasurableSet[blockSigma (P.block GrowthState.initial)] (P.rootEvent∩Set.univ)
simpa only [Set.inter_univ] using P.rootEvent_measurable
end Scheme
end CriticalPercolation.Corridor
end
end
end
end OAI
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