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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Gluing.ContourSites
namespace OAI
section
section
noncomputable section
open Set Function
open scoped Classical BigOperators
namespace CriticalPercolation.Boundary
def wordDir {n : ℕ} (w : Fin n → Dir) (i : ℕ) : Dir :=
if h : i<n then w ⟨i,h⟩ else if h₀ : 0<n then w ⟨0,h₀⟩ else 0
def wordEdge {n : ℕ} (p : Grid) (w : Fin n → Dir) (i : ℕ) : Edge :=
(p+∑ j∈Finset.range i, step (wordDir w j),wordDir w i)
lemma wordEdge_chain {n : ℕ} (p : Grid) (w : Fin n → Dir) (i : ℕ) :
head (wordEdge p w i)=tail (wordEdge p w (i+1)) := by
simp [wordEdge,head,tail,Finset.sum_range_succ,add_assoc]
lemma wordEdge_zero {n : ℕ} (p : Grid) (w : Fin n → Dir) :
(wordEdge p w 0).1=p := by simp [wordEdge]
lemma wordEdge_reconstruct {n : ℕ} (hn : 0<n) (e : ℕ → Edge)
(hchain : ∀ i, head (e i)=tail (e (i+1))) (hclose : e n=e 0) :
∀ i≤n, wordEdge (e 0).1 (fun j : Fin n => (e j).2) i=e i := by
let w : Fin n → Dir := fun j => (e j).2
have hd (i : ℕ) (hi : i<n) : wordDir w i=(e i).2 := by simp [wordDir,hi,w]
have hv : ∀ i≤n, (wordEdge (e 0).1 w i).1=(e i).1 := by
intro i hi
induction i with
| zero => exact wordEdge_zero _ _
| succ i ih =>
have hh : i<n := by omega
have ih' := ih (by omega)
have hc := wordEdge_chain (e 0).1 w i
change (wordEdge (e 0).1 w i).1+step (wordDir w i)=
(wordEdge (e 0).1 w (i+1)).1 at hc
rw [← hc,ih',hd i hh]
exact hchain i
intro i hi
apply Prod.ext (hv i hi)
change wordDir w i=(e i).2
by_cases hlt : i<n
· exact hd i hlt
· have hie : i=n := by omega
subst i
simp [wordDir,hn,hclose,w]
lemma card_words (n : ℕ) : Fintype.card (Fin n → Dir)=4^n := by
simp [Dir]
def wordValid {n : ℕ} (p : Grid) (w : Fin n → Dir) : Prop :=
wordEdge p w n=wordEdge p w 0 ∧ Set.InjOn (wordEdge p w) (Set.Iio n)
lemma wordValid_reconstruct {n : ℕ} (hn : 0<n) (e : ℕ → Edge)
(hchain : ∀ i, head (e i)=tail (e (i+1))) (hclose : e n=e 0)
(hinj : Set.InjOn e (Set.Iio n)) :
wordValid (e 0).1 (fun j : Fin n => (e j).2) := by
have hr := wordEdge_reconstruct hn e hchain hclose
refine ⟨?_,?_⟩
· rw [hr n le_rfl,hr 0 (Nat.zero_le n),hclose]
· intro i hi j hj he
rw [hr i hi.le,hr j hj.le] at he
exact hinj hi hj he
lemma outsideSites_congr {n : ℕ} {e f : ℕ → Edge}
(h : ∀ i<n, e i=f i) : outsideSites e n=outsideSites f n := by
have hf : forwardIndices e n=forwardIndices f n := by
apply Finset.filter_congr
intro i hi
rw [h i (Finset.mem_range.mp hi)]
unfold outsideSites
rw [hf]
apply Finset.image_congr
intro i hi
dsimp only
rw [h i (Finset.mem_range.mp (Finset.mem_filter.mp hi).1)]
theorem finite_good_contour (S : Set Grid) (hS : S.Finite) (hroot : (0,0)∈S)
(hquad : ∀ v∈S, 0≤v.1 ∧ 0≤v.2) :
∃ n : ℕ, ∃ w : Fin (n+1) → Dir,
wordValid (0,1) w ∧
(∀ z∈outsideSites (wordEdge (0,1) w) (n+1),
z∉S ∧ ∃ y∈S, z=y+(1,0) ∨ z=y+(0,1)) := by
obtain ⟨l,hl,e,h0,hend,hmem,hchain,hinj⟩ := squareBoundary_closed_trail hS
((0,1),3) (west_root_edge_mem S hroot hquad)
obtain ⟨n,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hl)
have he : e (n+1)=e 0 := hend.trans h0.symm
have hp : (e 0).1=(0,1) := congrArg Prod.fst h0
refine ⟨n,(fun j : Fin (n+1) => (e j).2),?_,?_⟩
· simpa only [hp] using wordValid_reconstruct hl e hchain he hinj
· have hr := wordEdge_reconstruct hl e hchain he
rw [hp] at hr
rw [outsideSites_congr (fun i hi => hr i hi.le)]
exact outsideSites_failed_children S e (n+1) hmem
end CriticalPercolation.Boundary
end
end
end
end OAI
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