| import OAI.Probability.GammaPassage.ContourCounts |
| import Mathlib.Data.Fintype.Prod |
| import Mathlib.Data.Finset.Prod |
|
|
| namespace OAI |
|
|
| /-! The uncancelled, counterclockwise square sides in |
| crossings.tex:cr:oriented-paths. A dual vertex `p` represents the actual |
| point `p - (1/2,1/2)`, so the square at the primal site `q` has lower-left |
| corner `q` in these integer coordinates. -/ |
| 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] |
|
|
| /-- Follow the first uncancelled side counterclockwise around the arriving |
| corner. The two tests inspect the other two squares meeting that corner. -/ |
| 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 |
|
|
| /-- Cancellation of shared sides leaves a permutation of the remaining sides; |
| its cycles are directed closed walks with no repeated directed side. -/ |
| 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 |
|
|
| /-- An unoriented dual edge cannot occur twice among uncancelled sides: |
| its two possible orientations have opposite occupied and unoccupied cells. -/ |
| 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 |
|
|
