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dd14c26 | 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 | import OAI.Probability.LatticePassage.Lattice
import OAI.Probability.LatticePassage.CheapPath
namespace OAI
namespace CompanionFPP
open Finset
private def neighbors (x : Vertex) : Finset Vertex :=
{(x.1 + 1, x.2), (x.1 - 1, x.2), (x.1, x.2 + 1), (x.1, x.2 - 1)}
private theorem mem_neighbors {x y : Vertex} : y ∈ neighbors x ↔ lattice.Adj x y := by
rcases x with ⟨a,b⟩
rcases y with ⟨c,d⟩
change (c,d) ∈ neighbors (a,b) ↔ |a-c| + |b-d| = 1
simp only [neighbors, mem_insert, mem_singleton, Prod.mk.injEq]
rcases le_total 0 (a-c) with ha | ha <;>
rcases le_total 0 (b-d) with hb | hb
all_goals simp only [abs_of_nonneg, abs_of_nonpos, ha, hb]
all_goals omega
private theorem card_neighbors (x : Vertex) : (neighbors x).card = 4 := by
have h₁ : (x.1 + 1, x.2) ≠ (x.1 - 1, x.2) := by intro h; have := congrArg Prod.fst h; omega
have h₂ : (x.1 + 1, x.2) ≠ (x.1, x.2 + 1) := by intro h; have := congrArg Prod.fst h; omega
have h₃ : (x.1 + 1, x.2) ≠ (x.1, x.2 - 1) := by intro h; have := congrArg Prod.fst h; omega
have h₄ : (x.1 - 1, x.2) ≠ (x.1, x.2 + 1) := by intro h; have := congrArg Prod.fst h; omega
have h₅ : (x.1 - 1, x.2) ≠ (x.1, x.2 - 1) := by intro h; have := congrArg Prod.fst h; omega
have h₆ : (x.1, x.2 + 1) ≠ (x.1, x.2 - 1) := by intro h; have := congrArg Prod.snd h; omega
simp [neighbors, h₁,h₂,h₃,h₄,h₅,h₆]
private theorem card_neighbors_erase {prev x : Vertex} (h : lattice.Adj x prev) :
((neighbors x).erase prev).card = 3 := by
rw [card_erase_of_mem (mem_neighbors.mpr h), card_neighbors]
noncomputable def avoidingWalks (n : ℕ) (prev x : Vertex) :
Finset (Σ y : Vertex, lattice.Walk x y) := by
classical
exact match n with
| 0 => {⟨x, .nil⟩}
| n+1 => ((neighbors x).erase prev).attach.biUnion fun v =>
(avoidingWalks n x v).image fun q =>
⟨q.1, .cons (mem_neighbors.mp (mem_of_mem_erase v.property)) q.2⟩
theorem card_avoidingWalks (n : ℕ) {prev x : Vertex} (h : lattice.Adj x prev) :
(avoidingWalks n prev x).card ≤ 3^n := by
classical
induction n generalizing prev x with
| zero => simp [avoidingWalks]
| succ n ih =>
calc
_ ≤ ∑ v ∈ ((neighbors x).erase prev).attach,
((avoidingWalks n x v).image (fun q =>
(⟨q.1, .cons (mem_neighbors.mp (mem_of_mem_erase v.property)) q.2⟩ :
Σ y : Vertex, lattice.Walk x y))).card := card_biUnion_le
_ ≤ ∑ _v ∈ ((neighbors x).erase prev).attach, 3^n := by
apply sum_le_sum
intro v hv
exact card_image_le.trans (ih (mem_neighbors.mp (mem_of_mem_erase v.property)).symm)
_ = 3^(n+1) := by
simp [card_neighbors_erase h, pow_succ, Nat.mul_comm]
theorem trail_mem_avoiding {prev x y : Vertex} (p : lattice.Walk x y)
(hp : p.IsTrail) (hne : s(prev,x) ∉ p.edges) :
(⟨y,p⟩ : Σ y : Vertex, lattice.Walk x y) ∈ avoidingWalks p.length prev x := by
classical
induction p generalizing prev with
| nil => simp [avoidingWalks]
| @cons x v y h p ih =>
have ht : p.IsTrail := hp.of_cons
have hnot : s(x,v) ∉ p.edges := (List.nodup_cons.mp hp.edges_nodup).1
have hvne : v ≠ prev := by
intro heq
subst v
apply hne
simp [SimpleGraph.Walk.edges_cons, Sym2.eq_swap]
have hv : v ∈ (neighbors x).erase prev := mem_erase.mpr ⟨hvne, mem_neighbors.mpr h⟩
simp only [SimpleGraph.Walk.length_cons, avoidingWalks, mem_biUnion]
refine ⟨⟨v,hv⟩, mem_attach _ _, ?_⟩
exact mem_image.mpr ⟨⟨y,p⟩, ih ht hnot, rfl⟩
noncomputable def startingWalks (n : ℕ) (x : Vertex) :
Finset (Σ y : Vertex, lattice.Walk x y) := by
classical
exact match n with
| 0 => {⟨x, .nil⟩}
| n+1 => (neighbors x).attach.biUnion fun v =>
(avoidingWalks n x v).image fun q =>
⟨q.1, .cons (mem_neighbors.mp v.property) q.2⟩
theorem card_startingWalks (n : ℕ) (x : Vertex) :
(startingWalks (n+1) x).card ≤ 4 * 3^n := by
classical
calc
_ ≤ ∑ v ∈ (neighbors x).attach,
((avoidingWalks n x v).image (fun q =>
(⟨q.1, .cons (mem_neighbors.mp v.property) q.2⟩ :
Σ y : Vertex, lattice.Walk x y))).card := card_biUnion_le
_ ≤ ∑ _v ∈ (neighbors x).attach, 3^n := by
apply sum_le_sum
intro v hv
exact card_image_le.trans (card_avoidingWalks n (mem_neighbors.mp v.property).symm)
_ = 4 * 3^n := by simp [card_neighbors]
theorem trail_mem_starting {x y : Vertex} (p : lattice.Walk x y) (hp : p.IsTrail) :
(⟨y,p⟩ : Σ y : Vertex, lattice.Walk x y) ∈ startingWalks p.length x := by
classical
cases p with
| nil => simp [startingWalks]
| @cons x v y h p =>
have hnot : s(x,v) ∉ p.edges := (List.nodup_cons.mp hp.edges_nodup).1
simp only [SimpleGraph.Walk.length_cons, startingWalks, mem_biUnion]
refine ⟨⟨v,mem_neighbors.mpr h⟩, mem_attach _ _, ?_⟩
exact mem_image.mpr ⟨⟨y,p⟩, trail_mem_avoiding p hp.of_cons hnot, rfl⟩
end CompanionFPP
end OAI
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