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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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | import OAI.Probability.FirstPassage.ContourCounts
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Finset.Prod
namespace OAI
namespace PlanarFPP
open Finset
instance : Fintype Step where
elems := {Step.east, Step.west, Step.north, Step.south}
complete d := by cases d <;> simp
def Step.left : Step → Step
| .east => .north | .north => .west | .west => .south | .south => .east
def Step.right : Step → Step
| .east => .south | .south => .west | .west => .north | .north => .east
def Step.outward (d : Step) : Vertex := d.right.delta
def Step.corner : Step → Vertex
| .east => (0,0) | .north => (1,0) | .west => (1,1) | .south => (0,1)
abbrev SquareSide := Vertex × Step
def SquareSide.tail (e : SquareSide) : Vertex := e.1 + e.2.corner
def SquareSide.head (e : SquareSide) : Vertex := e.tail + e.2.delta
def IsBoundarySide (S : Finset Vertex) (e : SquareSide) : Prop :=
e.1 ∈ S ∧ e.1 + e.2.outward ∉ S
instance (S : Finset Vertex) (e : SquareSide) : Decidable (IsBoundarySide S e) :=
inferInstanceAs (Decidable (_ ∧ _))
def boundarySides (S : Finset Vertex) : Finset SquareSide :=
(S ×ˢ univ).filter (fun e => e.1 + e.2.outward ∉ S)
@[simp] theorem mem_boundarySides (S : Finset Vertex) (e : SquareSide) :
e ∈ boundarySides S ↔ IsBoundarySide S e := by
simp [boundarySides, IsBoundarySide]
def boundaryNext (S : Finset Vertex) (e : SquareSide) : SquareSide :=
let q := e.1
let d := e.2
let r := q + d.left.outward
let t := r + d.outward
if r ∉ S then (q,d.left) else if t ∉ S then (r,d) else (t,d.right)
def boundaryPrev (S : Finset Vertex) (e : SquareSide) : SquareSide :=
let q := e.1
let d := e.2
let r := q + d.right.outward
let t := r + d.outward
if r ∉ S then (q,d.right) else if t ∉ S then (r,d) else (t,d.left)
theorem boundaryNext_boundary (S : Finset Vertex) (e : SquareSide)
(he : IsBoundarySide S e) : IsBoundarySide S (boundaryNext S e) := by
rcases e with ⟨⟨x,y⟩,d⟩
unfold boundaryNext
dsimp only
split
· cases d <;> simp_all [IsBoundarySide, Step.left, Step.outward, Step.right,
Step.delta]
· split
· cases d <;> simp_all [IsBoundarySide, Step.left, Step.outward, Step.right,
Step.delta]
· cases d <;>
simp_all [IsBoundarySide, Step.left, Step.outward, Step.right,
Step.delta, Prod.mk_add_mk, add_assoc]
theorem boundaryNext_tail (S : Finset Vertex) (e : SquareSide) :
(boundaryNext S e).tail = e.head := by
rcases e with ⟨⟨x,y⟩,d⟩
unfold boundaryNext
dsimp only
split
· cases d <;> simp [SquareSide.tail, SquareSide.head, Step.corner,
Step.left, Step.delta, Prod.mk_add_mk]
· split <;> cases d <;>
simp [SquareSide.tail, SquareSide.head, Step.corner, Step.left,
Step.right, Step.outward, Step.delta, Prod.mk_add_mk]
theorem boundaryPrev_next (S : Finset Vertex) (e : SquareSide)
(he : IsBoundarySide S e) : boundaryPrev S (boundaryNext S e) = e := by
rcases e with ⟨⟨x,y⟩,d⟩
unfold boundaryNext
dsimp only
split
· cases d <;>
simp_all [boundaryPrev, IsBoundarySide, Step.left, Step.right,
Step.outward, Step.delta, Prod.mk_add_mk]
· split <;> cases d <;>
simp_all [boundaryPrev, IsBoundarySide, Step.left, Step.right,
Step.outward, Step.delta, Prod.mk_add_mk, add_assoc]
abbrev BoundarySide (S : Finset Vertex) := {e // e ∈ boundarySides S}
def boundarySuccessor (S : Finset Vertex) (e : BoundarySide S) : BoundarySide S :=
⟨boundaryNext S e.val, (mem_boundarySides S _).mpr
(boundaryNext_boundary S e.val ((mem_boundarySides S _).mp e.property))⟩
theorem boundarySuccessor_injective (S : Finset Vertex) :
Function.Injective (boundarySuccessor S) := by
intro e f h
apply Subtype.ext
have hh := congrArg (fun a : BoundarySide S => boundaryPrev S a.val) h
simpa only [boundarySuccessor, boundaryPrev_next S e.val
((mem_boundarySides S _).mp e.property), boundaryPrev_next S f.val
((mem_boundarySides S _).mp f.property)] using hh
noncomputable def boundaryPerm (S : Finset Vertex) : Equiv.Perm (BoundarySide S) :=
Equiv.ofBijective (boundarySuccessor S)
⟨boundarySuccessor_injective S, Finite.surjective_of_injective
(boundarySuccessor_injective S)⟩
@[simp] theorem boundaryPerm_tail (S : Finset Vertex) (e : BoundarySide S) :
(boundaryPerm S e).val.tail = e.val.head := boundaryNext_tail S e.val
theorem boundaryPerm_ne_self (S : Finset Vertex) (e : BoundarySide S) :
boundaryPerm S e ≠ e := by
intro h
have hh := boundaryPerm_tail S e
rw [h] at hh
have hz : e.val.2.delta = 0 := by
have : e.val.tail + e.val.2.delta = e.val.tail + 0 := by
simpa [SquareSide.head] using hh.symm
exact add_left_cancel this
cases hd : e.val.2 <;> simp [hd, Step.delta] at hz
theorem boundary_dualEdge_injective (S : Finset Vertex) (e f : SquareSide)
(he : IsBoundarySide S e) (hf : IsBoundarySide S f)
(hh : e.2.edge e.tail = f.2.edge f.tail) : e = f := by
rcases e with ⟨⟨x,y⟩,d⟩
rcases f with ⟨⟨u,v⟩,a⟩
cases d <;> cases a <;>
simp_all [IsBoundarySide, SquareSide.tail, Step.edge, Step.corner,
Step.outward, Step.right, Step.delta, Prod.mk_add_mk, Prod.ext_iff]
end PlanarFPP
end OAI
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