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f61e547 | 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 | import OAI.Probability.FirstPassage.SquareLoops
import OAI.Probability.FirstPassage.OrientedCompactness
import OAI.Probability.FirstPassage.SeparatedSelection
namespace OAI
namespace PlanarFPP
open Finset
theorem exists_enclosing_squareLoop (S : Finset Vertex) (h0 : (0 : Vertex) ∈ S) :
∃ m : ℕ, 0 < m ∧ ∃ c : SquareLoop m,
c.NoRepeatedSides ∧ c.winding ≠ 0 ∧ ∀ i, IsBoundarySide S (c.side i) := by
classical
obtain ⟨x,hn,hpos,_,hnext,hw⟩ := exists_boundary_cycle S h0
let es := (boundaryPerm S).toList x
have hlen : 0 < es.length := hpos
have : NeZero es.length := ⟨Nat.ne_of_gt hlen⟩
let c : SquareLoop es.length :=
{ side := fun i => (es.get i).val
follows := by
intro i
have hh := hnext (es.get i) (List.get_mem es i)
change (es.next es[i.val] (List.get_mem es i)).val.tail = es[i.val].val.head at hh
rw [List.next_getElem es hn i.val i.isLt] at hh
simpa only [List.get_eq_getElem, finRotate_apply, Fin.val_add,
Fin.val_one', Nat.add_mod_mod] using hh }
refine ⟨es.length,hlen,c,?_,?_,?_⟩
· exact Subtype.val_injective.comp hn.injective_get
· change (∑ i : Fin es.length, (es.get i).val.rayCross) ≠ 0
have hh : (∑ i : Fin es.length, (es.get i).val.rayCross) =
∑ e ∈ es.toFinset, e.val.rayCross := by
rw [List.sum_toFinset _ hn, ← Fin.sum_ofFn]
have hmap : List.map (fun e : BoundarySide S => e.val.rayCross) (List.ofFn es.get) =
List.ofFn (fun i => (es.get i).val.rayCross) := List.map_ofFn
simpa only [List.ofFn_get] using congrArg List.sum hmap.symm
rw [hh]
exact hw
· intro i
exact (mem_boundarySides S _).mp (es.get i).property
theorem exists_enclosing_squareLoop_unoriented (S : Finset Vertex) (h0 : (0 : Vertex) ∈ S) :
∃ m : ℕ, 0 < m ∧ ∃ c : SquareLoop m,
c.winding ≠ 0 ∧ (∀ i, IsBoundarySide S (c.side i)) ∧
Function.Injective (fun i => (c.side i).2.edge (c.side i).tail) := by
obtain ⟨m,hm,c,hc,hw,hb⟩ := exists_enclosing_squareLoop S h0
refine ⟨m,hm,c,hw,hb,?_⟩
intro i j hij
exact hc (boundary_dualEdge_injective S _ _ (hb i) (hb j) hij)
theorem SquareLoop.badSites_bad {m : ℕ} (c : SquareLoop m)
(S : Finset Vertex) (G : Vertex → Prop)
(hb : ∀ i, IsBoundarySide S (c.side i))
(hforward : ∀ q ∈ S, G q → q+(1,0) ∈ S ∧ q+(0,1) ∈ S) :
∀ q ∈ c.badSites, ¬G q := by
intro q hq hg
obtain ⟨i,hi,rfl⟩ := mem_image.mp hq
have hf := hforward _ (hb i).1 hg
have hout := (hb i).2
rcases (mem_filter.mp hi).2 with hd | hd
· exact hout (by simpa [hd,Step.outward,Step.right,Step.delta] using hf.1)
· exact hout (by simpa [hd,Step.outward,Step.right,Step.delta] using hf.2)
theorem no_infinite_path_enclosing_contour (G : Vertex → Prop) (k : ℕ)
(h : ¬infiniteOrientedPath G 0) :
∃ m : ℕ, 0 < m ∧ ∃ c : SquareLoop m,
c.NoRepeatedSides ∧ c.winding ≠ 0 ∧
(∀ q ∈ c.badSites, ¬G q) ∧
∃ T : Finset Vertex, T ⊆ c.badSites ∧
Set.Pairwise (↑T : Set Vertex) (fun p q => ¬latticeNear k p q) ∧
m ≤ 8*(2*k+1)^2*T.card := by
classical
let hf := orientedReachable_finite_of_no_infinite_path G 0 h
let S := hf.toFinset
have hS (q : Vertex) : q ∈ S ↔ q ∈ orientedReachable G 0 := hf.mem_toFinset
have h0 : (0 : Vertex) ∈ S := (hS 0).mpr (orientedReachable_origin G 0)
obtain ⟨m,hm,c,hc,hw,hb⟩ := exists_enclosing_squareLoop S h0
refine ⟨m,hm,c,hc,hw,?_,?_⟩
· apply c.badSites_bad S G hb
intro q hq hg
have hh := orientedReachable_forward G 0 q ((hS q).mp hq) hg
exact ⟨(hS _).mpr hh.1,(hS _).mpr hh.2⟩
· obtain ⟨T,hT,hpair,hcard⟩ := lattice_separated_selection c.badSites k
refine ⟨T,hT,hpair,?_⟩
have hh := c.length_le_eight_badSites hc
nlinarith
end PlanarFPP
end OAI
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