| import OAI.Probability.FirstPassage.BoundaryGeometry |
| import Mathlib.Algebra.BigOperators.Group.Finset.Basic |
| import Mathlib.Algebra.BigOperators.Group.Finset.Sigma |
|
|
| namespace OAI |
|
|
| namespace PlanarFPP |
| open Finset |
|
|
| def SquareSide.rayCross (e : SquareSide) : ℤ := |
| match e.2 with |
| | .north => if e.1.2 = 0 ∧ 0 ≤ e.1.1 then 1 else 0 |
| | .south => if e.1.2 = 0 ∧ 0 < e.1.1 then -1 else 0 |
| | _ => 0 |
|
|
| def SquareSide.opposite (e : SquareSide) : SquareSide := |
| (e.1 + e.2.outward, e.2.reverse) |
|
|
| @[simp] theorem SquareSide.opposite_opposite (e : SquareSide) : |
| e.opposite.opposite = e := by |
| rcases e with ⟨⟨x,y⟩,d⟩ |
| cases d <;> simp [opposite, Step.outward, Step.right, Step.reverse, Step.delta] |
|
|
| theorem SquareSide.opposite_ne_self (e : SquareSide) : e.opposite ≠ e := by |
| intro h |
| have hh := congrArg Prod.snd h |
| cases hd : e.2 <;> simp [opposite, hd, Step.reverse] at hh |
|
|
| @[simp] theorem SquareSide.rayCross_opposite (e : SquareSide) : |
| e.opposite.rayCross = -e.rayCross := by |
| rcases e with ⟨⟨x,y⟩,d⟩ |
| cases d <;> simp [opposite, rayCross, Step.outward, Step.right, |
| Step.reverse, Step.delta] <;> split_ifs <;> omega |
|
|
| theorem square_rayCross_sum (q : Vertex) : |
| ∑ d : Step, SquareSide.rayCross (q,d) = if q = 0 then 1 else 0 := by |
| have hu : (univ : Finset Step) = {Step.east,Step.west,Step.north,Step.south} := rfl |
| rw [hu] |
| rcases q with ⟨x,y⟩ |
| simp [SquareSide.rayCross, Prod.ext_iff] |
| split_ifs <;> omega |
|
|
| def internalSides (S : Finset Vertex) : Finset SquareSide := |
| (S ×ˢ univ).filter (fun e => e.1 + e.2.outward ∈ S) |
|
|
| theorem internalSides_opposite (S : Finset Vertex) (e : SquareSide) |
| (he : e ∈ internalSides S) : e.opposite ∈ internalSides S := by |
| simp only [internalSides, mem_filter, mem_product, mem_univ, and_true] at he ⊢ |
| rcases e with ⟨⟨x,y⟩,d⟩ |
| cases d <;> simp_all [SquareSide.opposite, Step.outward, |
| Step.right, Step.reverse, Step.delta, Prod.mk_add_mk, add_assoc] |
|
|
| theorem internalSides_rayCross_sum (S : Finset Vertex) : |
| ∑ e ∈ internalSides S, e.rayCross = 0 := by |
| refine sum_involution (fun e _ => e.opposite) ?_ ?_ (internalSides_opposite S) ?_ |
| · intro e he |
| simp |
| · intro e he hh |
| exact e.opposite_ne_self |
| · intro e he |
| exact e.opposite_opposite |
|
|
| theorem boundarySides_rayCross_sum (S : Finset Vertex) : |
| ∑ e ∈ boundarySides S, e.rayCross = if (0 : Vertex) ∈ S then 1 else 0 := by |
| classical |
| have hsplit := sum_filter_add_sum_filter_not (S ×ˢ (univ : Finset Step)) |
| (fun e : SquareSide => e.1 + e.2.outward ∈ S) SquareSide.rayCross |
| change (∑ e ∈ internalSides S, e.rayCross) + |
| (∑ e ∈ boundarySides S, e.rayCross) = _ at hsplit |
| rw [internalSides_rayCross_sum, zero_add] at hsplit |
| rw [hsplit, sum_product] |
| change (∑ q ∈ S, ∑ d : Step, SquareSide.rayCross (q,d)) = _ |
| simp_rw [square_rayCross_sum] |
| by_cases h0 : (0 : Vertex) ∈ S |
| · rw [ite_eq_left h0] |
| calc |
| _ = (if (0 : Vertex) = 0 then (1 : ℤ) else 0) := by |
| apply sum_eq_single 0 |
| · intro b hb hne |
| exact ite_eq_right hne |
| · exact fun hh => False.elim (hh h0) |
| _ = 1 := ite_eq_left rfl |
| · rw [ite_eq_right h0] |
| apply sum_eq_zero |
| intro q hq |
| apply ite_eq_right |
| intro hh |
| exact h0 (hh ▸ hq) |
|
|
| end PlanarFPP |
|
|
| end OAI |
|
|
