| 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 |
|
|
