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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.RelayProbability
import OAI.Probability.CriticalPercolation.GraphGeometry.CoordinateScheme
import OAI.Probability.CriticalPercolation.Criticality.GrowthEstimate
import OAI.Probability.CriticalPercolation.ProductMeasure.SourceMissLaw
import OAI.Probability.CriticalPercolation.Gluing.ProspectiveCorridor
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation.Corridor.CoordinateData
open BondGraph Boundary BernoulliProduct
variable {V E : Type*} [Countable V] [Countable E] (C : CoordinateData V E)
lemma active_slot_positive (n : ℕ) (S : GrowthState) (hv : S.Valid n)
(ha : (incoming n S.good).isSome) : 0<n := by
obtain ⟨e,he⟩ := Option.isSome_iff_exists.mp ha
have ht := parent_slot_lt e (hv.2.1 _ (incoming_source he)).1
rw [incoming_target he,gridSlot_slotSite] at ht
omega
theorem prospective_estimate (h : C.Bounds)
(hmove : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real
(C.graph.coordinateMove C.time C.height C.m C.e y s))
(n : ℕ) (S : GrowthState) (hv : S.Valid n) (_hc : S.Covered)
(ha : (incoming n S.good).isSome) (ω : Set E) (b : Bool) :
(blockProbability C.scheme.parameter (C.scheme.block S)
(C.scheme.incomingHit n S\C.scheme.nextHit n S b) ω).toReal≤(C.L+1)*C.kappa := by
let v := slotSite n
let F := C.scheme.enlarged n S
let U := F∪C.scheme.edgeDomain (v,b)
have hdom := C.domain_eq v b
have hDU : C.graph.inducedEdges (relayDomain (localTime C.time C.T v)
(localHeight C.height C.T v) C.J C.T (sign b))⊆U := by
rw [← hdom]
exact Set.subset_union_right
have hfresh : Disjoint (C.scheme.block S)
(C.graph.inducedEdges (relayDomain (localTime C.time C.T v)
(localHeight C.height C.T v) C.J C.T (sign b))) := by
rw [← hdom]
exact outgoing_unrevealed C.graph (normalized C.time C.height C.T) (C.J/C.T) (2520/C.T)
(C.layout h).1 (C.layout h).2.1 (C.layout h).2.2 n S hv (active_slot_positive n S hv ha) rfl
have hmoves : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real
(C.graph.coordinateMove (localTime C.time C.T v) (localHeight C.height C.T v)
C.m C.e y s) := by
intro y s hs
rw [coordinateMove_translate]
exact hmove y s hs
have hP : 1≤C.T := by
have hh : (416 : ℝ)≤C.L := by exact_mod_cast C.L_large h
dsimp [T]; linarith
have hh := C.graph.coordinate_pinned_corridor_failure h.connected h.locallyFinite
C.scheme.seed (C.graph.vertexBall_finite h.locallyFinite C.origin C.m) C.scheme.parameter
(C.scheme.block S) F U ω Set.subset_union_left
(localTime C.time C.T v) (localHeight C.height C.T v) (sign b) C.e C.rho C.J C.T
C.L C.R C.m (sign_abs b) h.e_pos h.rho_nonneg h.J_lower (C.L_large h) h.e_small h.rho_small
hP (C.source_behind h n (active_slot_positive n S hv ha))
(fun {_ _} hh => C.local_time_step h v hh) (fun {_ _} hh => C.local_height_step h v hh) h.R_gt C.pc C.p h.pc_pos h.pc_lt
h.p_lower h.p_upper (fun _ _ => rfl) hDU hfresh C.D C.k C.M C.VM h.degree h.M_pos
h.M_bound h.VM_bound C.kappa C.eta h.kappa_pos h.kappa_le h.eta_nonneg h.eta_small
h.k_large h.R_large hmoves
change (blockProbability C.scheme.parameter (C.scheme.block S)
(C.graph.restrictedSeed C.scheme.seed (relayMark (localTime C.time C.T v) (localHeight C.height C.T v) 0 (sign b) C.rho) F\
C.graph.restrictedSeed C.scheme.seed (relayMark (localTime C.time C.T v) (localHeight C.height C.T v) C.T (sign b) C.rho) U) ω).toReal≤_ at hh
rw [mark_initial,mark_target] at hh
exact hh
lemma root_corridor_estimate (h : C.Bounds)
(hmove : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real
(C.graph.coordinateMove C.time C.height C.m C.e y s)) (b : Bool) :
(BernoulliProduct.law C.scheme.parameter).real
(C.scheme.hit (target ((0,0),b))
(C.scheme.block GrowthState.initial∪C.scheme.edgeDomain ((0,0),b)))ᶜ≤C.eta+C.L*C.kappa := by
let U := C.scheme.block GrowthState.initial∪C.scheme.edgeDomain ((0,0),b)
have hDU : C.graph.inducedEdges (relayDomain (localTime C.time C.T (0,0))
(localHeight C.height C.T (0,0)) C.J C.T (sign b))⊆U := by
rw [← C.domain_eq]
exact Set.subset_union_right
have hmoves : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real
(C.graph.coordinateMove (localTime C.time C.T (0,0)) (localHeight C.height C.T (0,0))
C.m C.e y s) := by
intro y s hs
rw [coordinateMove_translate]
exact hmove y s hs
have hh := C.graph.original_root_corridor_miss h.connected h.locallyFinite C.origin
(localTime C.time C.T (0,0)) (localHeight C.height C.T (0,0)) (sign b) C.e C.rho C.J
C.L C.R C.m U hDU (sign_abs b) h.e_pos h.rho_nonneg h.J_lower (C.L_large h) h.e_small h.rho_small
(by simp [localTime,timeCenter,h.time_origin]) (by simp [localHeight,heightCenter,h.height_origin])
(fun {_ _} hh => C.local_time_step h (0,0) hh) (fun {_ _} hh => C.local_height_step h (0,0) hh) h.R_gt C.pc C.p h.pc_pos h.pc_lt
h.p_lower h.p_upper C.D C.k C.M C.VM h.degree h.M_pos h.M_bound h.VM_bound
C.kappa C.eta h.kappa_pos h.kappa_le h.eta_nonneg h.eta_small h.k_large h.R_large hmoves
change (BernoulliProduct.law (fun _ : E => C.p)).real
(C.graph.restrictedSeed (C.graph.vertexBall C.origin C.m)
(relayMark (localTime C.time C.T (0,0)) (localHeight C.height C.T (0,0)) C.T (sign b) C.rho) U)ᶜ≤_ at hh
rw [mark_target] at hh
exact hh
lemma root_positive (h : C.Bounds)
(hmove : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real
(C.graph.coordinateMove C.time C.height C.m C.e y s)) :
0<(BernoulliProduct.law C.scheme.parameter).real C.scheme.rootEvent := by
have he : C.scheme.rootEvent=twoPredictions C.scheme.parameter (C.scheme.block GrowthState.initial)
(C.scheme.hit (target ((0,0),true))
(C.scheme.block GrowthState.initial∪C.scheme.edgeDomain ((0,0),true)))
(C.scheme.hit (target ((0,0),false))
(C.scheme.block GrowthState.initial∪C.scheme.edgeDomain ((0,0),false))) C.delta := by
ext ω
change (∀ b : Bool, 1-C.delta≤(blockProbability C.scheme.parameter
(C.scheme.block GrowthState.initial) (C.scheme.hit (target ((0,0),b))
(C.scheme.block GrowthState.initial∪C.scheme.edgeDomain ((0,0),b))) ω).toReal) ↔ (_ ∧ _)
constructor
· intro hh; exact ⟨hh true,hh false⟩
· rintro ⟨ht,hf⟩ b; cases b <;> assumption
rw [he]
apply twoPredictions_positive _ _ _ _
(C.graph.restrictedSeed_measurable _ _ _) (C.graph.restrictedSeed_measurable _ _ _)
C.delta (C.eta+C.L*C.kappa) h.delta_pos h.root_error
(C.root_corridor_estimate h hmove true) (C.root_corridor_estimate h hmove false)
theorem moves_force_percolation (h : C.Bounds)
(hmove : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real
(C.graph.coordinateMove C.time C.height C.m C.e y s)) :
0<C.graph.law C.p C.graph.percolates := by
let hf := C.graph.vertexBall_finite h.locallyFinite C.origin C.m
exact C.scheme.exploration_percolates (C.layout h).1 (C.layout h).2.1 (C.layout h).2.2
hf.toFinset (by simp only [Set.Finite.coe_toFinset]; rfl) (C.root_seed h) (C.marks_disjoint h)
C.p rfl (C.root_positive h hmove) h.delta_pos ((C.L+1)*C.kappa) C.contour
h.contour_nonneg h.contour_small (mul_nonneg (by positivity) h.kappa_pos.le) h.test_error (C.prospective_estimate h hmove)
theorem moves_impossible_below_criticality (h : C.Bounds)
(hp : C.p<C.graph.criticalProbability)
(hmove : ∀ y s, |s|=1 → 1-C.eta≤(BernoulliProduct.law (fun _ : E => C.p)).real
(C.graph.coordinateMove C.time C.height C.m C.e y s)) : False :=
(not_le_of_gt hp) (C.graph.criticalProbability_le_of_pos (C.moves_force_percolation h hmove))
end CriticalPercolation.Corridor.CoordinateData
end
end
end
end OAI
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