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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.LatticeGrowth
import OAI.Probability.CriticalPercolation.Gluing.CorridorGraph
namespace OAI
section
section
noncomputable section
open Set Function
open scoped Classical
namespace CriticalPercolation.Corridor
open BondGraph Boundary
variable {V E : Type*} (G : BondGraph V E) (ξ : V → Point) (j h : ℝ)
def revealed (S : GrowthState) : Set E :=
boxEdges G ξ j h (0,0)∪⋃ e∈S.seen, corridorEdges G ξ j h e
lemma revealed_mono {S T : GrowthState} (hST : S.seen⊆T.seen) :
revealed G ξ j h S⊆revealed G ξ j h T := by
apply Set.union_subset_union_right
intro b hb
obtain ⟨e,he,hb⟩ := Set.mem_iUnion₂.mp hb
exact Set.mem_iUnion₂.mpr ⟨e,hST he,hb⟩
lemma boxEdges_subset_corridor_source (e : Edge) :
boxEdges G ξ j h (source e)⊆corridorEdges G ξ j h e := by
intro b hb v hv
exact Or.inl (Or.inl (hb v hv))
lemma boxEdges_subset_corridor_target (e : Edge) :
boxEdges G ξ j h (target e)⊆corridorEdges G ξ j h e := by
intro b hb v hv
exact Or.inl (Or.inr (hb v hv))
lemma good_box_revealed {S : GrowthState} (hs : S.Covered) {v : Site} (hv : v∈S.good) :
boxEdges G ξ j h v⊆revealed G ξ j h S := by
rcases hs v hv with rfl|⟨e,he,rfl⟩
· exact Set.subset_union_left
· intro b hb
exact Or.inr (Set.mem_iUnion₂.mpr ⟨e,he,boxEdges_subset_corridor_target G ξ j h e hb⟩)
lemma revealed_initial : revealed G ξ j h GrowthState.initial=boxEdges G ξ j h (0,0) := by
simp [revealed,GrowthState.initial]
lemma revealed_advance (n : ℕ) (S : GrowthState) (b : Bool) :
revealed G ξ j h (S.advance n b)=revealed G ξ j h S∪
⋃ e∈(incoming n S.good).toFinset, corridorEdges G ξ j h e := by
ext e
constructor
· rintro (he|he)
· exact Or.inl (Or.inl he)
· obtain ⟨f,hf,he⟩ := Set.mem_iUnion₂.mp he
rcases Finset.mem_union.mp hf with hf|hf
· exact Or.inl (Or.inr (Set.mem_iUnion₂.mpr ⟨f,hf,he⟩))
· exact Or.inr (Set.mem_iUnion₂.mpr ⟨f,hf,he⟩)
· rintro ((he|he)|he)
· exact Or.inl he
· obtain ⟨f,hf,he⟩ := Set.mem_iUnion₂.mp he
exact Or.inr (Set.mem_iUnion₂.mpr ⟨f,Finset.mem_union.mpr (Or.inl hf),he⟩)
· obtain ⟨f,hf,he⟩ := Set.mem_iUnion₂.mp he
exact Or.inr (Set.mem_iUnion₂.mpr ⟨f,Finset.mem_union.mpr (Or.inr hf),he⟩)
lemma revealed_incoming (n : ℕ) (S : GrowthState) (b : Bool) {e : Edge}
(he : incoming n S.good=some e) :
revealed G ξ j h (S.advance n b)=revealed G ξ j h S∪corridorEdges G ξ j h e := by
rw [revealed_advance,he]
simp
lemma later_revealed_intersection (hh : 0≤h) (hhj : h≤j) (hsep : 2*j+h<1)
(n : ℕ) (S T : GrowthState) (hc : S.Covered) (hv : T.Valid n)
{e : Edge} (hp : source e∈S.good) (ht : n ≤ gridSlot (target e)) :
revealed G ξ j h T∩corridorEdges G ξ j h e⊆revealed G ξ j h S := by
intro b hb
rcases hb.1 with hb0|hbT
· exact Or.inl hb0
· have hh' := T.seen_intersection n hv ht
have hh'' := history_intersection_subset G ξ j h hh hhj hsep (↑T.seen) hh'.1 hh'.2
exact good_box_revealed G ξ j h hc hp (hh'' ⟨hbT,hb.2⟩)
lemma outgoing_unrevealed (hh : 0≤h) (hhj : h≤j) (hsep : 2*j+h<1)
(n : ℕ) (S : GrowthState) (hv : S.Valid n) (hn : 0<n)
{e : Edge} (he : source e=slotSite n) :
Disjoint (revealed G ξ j h S) (corridorEdges G ξ j h e) := by
apply Disjoint.sup_left
· apply edge_box_disjoint G ξ j h hh hhj hsep
intro hroot
have he1 := parent_slot_lt e (he ▸ slotSite_quadrant n)
have he2 : gridSlot (source e)=n := by rw [he,gridSlot_slotSite]
rcases hroot with hh|hh
· have he0 : gridSlot (source e)=0 := by rw [← hh]; rfl
omega
· have he0 : gridSlot (target e)=0 := by rw [← hh]; rfl
omega
· exact history_disjoint G ξ j h hh hhj hsep (↑S.seen)
(fun f hf => S.outgoing_nonincident n hv he hf)
end CriticalPercolation.Corridor
end
end
end
end OAI
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