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9b6e014 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 | import OAI.Probability.ClassicalON.LatticeRadius
namespace OAI
universe uK
noncomputable section
open Set
open scoped Classical
namespace ClassicalON
variable {K : Type uK}
def SeparatedCenters (N : ℤ) (z : K → Site) : Prop := ∀ i j,i≠j → 4*N<siteRadius (z i) (z j)
def centerLabel (N : ℤ) (z : K → Site) (v : Site) : Option K :=
if h : ∃ i,v∈annulusSites N (z i) then some h.choose else none
def centerPins (N : ℤ) (z : K → Site) : Set Site := {v | ∃ i,v∈annulusBoundary N (z i)}
theorem annulus_unique (N : ℤ) (z : K → Site) (hz : SeparatedCenters N z)
(v : Site) {i j : K} (hi : v∈annulusSites N (z i)) (hj : v∈annulusSites N (z j)) : i=j := by
by_contra h
exact Set.disjoint_left.mp (annuli_disjoint (hz i j h)) hi hj
theorem centerLabel_some_iff (N : ℤ) (z : K → Site) (hz : SeparatedCenters N z) (v : Site) (i : K) :
centerLabel N z v=some i ↔ v∈annulusSites N (z i) := by
unfold centerLabel
split_ifs with h
· constructor
· intro he
have hi := Option.some.inj he
exact hi ▸ h.choose_spec
· intro hv
congr 1
exact annulus_unique N z hz v h.choose_spec hv
· constructor
· intro he; cases he
· intro hv; exact False.elim (h ⟨i,hv⟩)
theorem centerLabel_none_iff (N : ℤ) (z : K → Site) (v : Site) :
centerLabel N z v=none ↔ ∀ i,v∉annulusSites N (z i) := by
simp only [centerLabel]
split_ifs with h
· constructor
· intro he; cases he
· intro hh; exact False.elim (hh h.choose h.choose_spec)
· constructor
· intro _ i hi; exact h ⟨i,hi⟩
· intro _; rfl
theorem annulusBoundary_subset (N : ℤ) (hN : 0≤N) (z : Site) :
annulusBoundary N z⊆annulusSites N z := by
intro v hv
change siteRadius z v=N ∨ siteRadius z v=2*N at hv
change N≤ siteRadius z v ∧ siteRadius z v≤2*N
omega
theorem local_centerPins_iff (N : ℤ) (hN : 0≤N) (z : K → Site) (hz : SeparatedCenters N z)
(v : Site) (i : K) (hi : v∈annulusSites N (z i)) :
v∈centerPins N z ↔ v∈annulusBoundary N (z i) := by
constructor
· rintro ⟨j,hj⟩
have hij := annulus_unique N z hz v hi (annulusBoundary_subset N hN _ hj)
exact hij ▸ hj
· exact fun h => ⟨i,h⟩
namespace LatticeGraph
def annulusVertexLabel (G : LatticeGraph) (N : ℤ) (z : K → Site) : G.vertices → Option K :=
fun v => centerLabel N z v.val
def annulusEdgeLabel (G : LatticeGraph) (N : ℤ) (z : K → Site) (e : G.edges) : Option K :=
if h : ∃ i,e.val.1.val∈annulusSites N (z i) ∧ e.val.2.val∈annulusSites N (z i) then some h.choose else none
def allAnnulusPins (G : LatticeGraph) (N : ℤ) (z : K → Site) : Set G.vertices :=
{v | v.val∈centerPins N z}
theorem annulusEdgeLabel_some_iff (G : LatticeGraph) (N : ℤ) (z : K → Site)
(hz : SeparatedCenters N z) (e : G.edges) (i : K) :
G.annulusEdgeLabel N z e=some i ↔
e.val.1.val∈annulusSites N (z i) ∧ e.val.2.val∈annulusSites N (z i) := by
unfold annulusEdgeLabel
split_ifs with h
· constructor
· intro he
exact Option.some.inj he ▸ h.choose_spec
· intro hv
congr 1
exact annulus_unique N z hz _ h.choose_spec.1 hv.1
· constructor
· intro he; cases he
· intro hv; exact False.elim (h ⟨i,hv⟩)
theorem annulus_vertex_edge_label (G : LatticeGraph) (N : ℤ) (_hN : 0≤N) (z : K → Site)
(hz : SeparatedCenters N z) (e : G.edges) (v w : G.vertices)
(he : (v=e.val.1 ∧ w=e.val.2) ∨ (v=e.val.2 ∧ w=e.val.1))
(hv : v∉G.allAnnulusPins N z) : G.annulusVertexLabel N z v=G.annulusEdgeLabel N z e := by
have hstep : PositiveNeighbor v.val w.val ∨ PositiveNeighbor w.val v.val := by
rcases he with ⟨rfl,rfl⟩ | ⟨rfl,rfl⟩
· exact Or.inl (G.nearest e.val e.property)
· exact Or.inr (G.nearest e.val e.property)
have hvp : v.val∉centerPins N z := hv
cases hl : G.annulusVertexLabel N z v with
| none =>
cases hel : G.annulusEdgeLabel N z e with
| none => rfl
| some i =>
have hh := (G.annulusEdgeLabel_some_iff N z hz e i).mp hel
have hh' : v.val∈annulusSites N (z i) := by rcases he with ⟨rfl,_⟩ | ⟨rfl,_⟩; exact hh.1; exact hh.2
have := (centerLabel_some_iff N z hz v.val i).mpr hh'
change G.annulusVertexLabel N z v=some i at this
rw [hl] at this
cases this
| some i =>
have hi := (centerLabel_some_iff N z hz v.val i).mp hl
have hvi : v.val∈annulusInterior N (z i) :=
(annulusSites_not_boundary N (z i) v.val).mp ⟨hi,fun h => hvp ⟨i,h⟩⟩
have hwi := neighbor_in_annulus hvi hstep
symm
apply (G.annulusEdgeLabel_some_iff N z hz e i).mpr
rcases he with ⟨rfl,rfl⟩ | ⟨rfl,rfl⟩
· exact ⟨hi,hwi⟩
· exact ⟨hwi,hi⟩
theorem annulus_left_label (G : LatticeGraph) (N : ℤ) (hN : 0≤N) (z : K → Site)
(hz : SeparatedCenters N z) (e : G.edges) (hv : e.val.1∉G.allAnnulusPins N z) :
G.annulusVertexLabel N z e.val.1=G.annulusEdgeLabel N z e :=
G.annulus_vertex_edge_label N hN z hz e e.val.1 e.val.2 (Or.inl ⟨rfl,rfl⟩) hv
theorem annulus_right_label (G : LatticeGraph) (N : ℤ) (hN : 0≤N) (z : K → Site)
(hz : SeparatedCenters N z) (e : G.edges) (hv : e.val.2∉G.allAnnulusPins N z) :
G.annulusVertexLabel N z e.val.2=G.annulusEdgeLabel N z e :=
G.annulus_vertex_edge_label N hN z hz e e.val.2 e.val.1 (Or.inr ⟨rfl,rfl⟩) hv
end LatticeGraph
end ClassicalON
end
end OAI
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