| import OAI.Probability.FirstPassage.SquareLoops |
| import OAI.Probability.FirstPassage.SpatialModulus |
|
|
| namespace OAI |
|
|
| namespace PlanarFPP |
| open Finset |
|
|
| namespace SquareSide |
|
|
| def axisCross (e : SquareSide) : ℤ := |
| match e.2 with |
| | .north => if e.1.2 = 0 then 1 else 0 |
| | .south => if e.1.2 = 0 then -1 else 0 |
| | _ => 0 |
|
|
| def aboveAxis (p : Vertex) : ℤ := if 0 < p.2 then 1 else 0 |
|
|
| theorem axisCross_eq (e : SquareSide) : |
| e.axisCross = aboveAxis e.head - aboveAxis e.tail := by |
| rcases e with ⟨⟨x,y⟩,d⟩ |
| cases d <;> simp [axisCross, aboveAxis, head, tail, Step.corner, Step.delta] <;> |
| split_ifs <;> omega |
|
|
| theorem rayCross_eq_axisCross (e : SquareSide) (hx : 0 < e.tail.1) : |
| e.rayCross = e.axisCross := by |
| rcases e with ⟨⟨x,y⟩,d⟩ |
| cases d <;> simp_all [rayCross, axisCross, tail, Step.corner] |
|
|
| theorem rayCross_ne_zero_location (e : SquareSide) (he : e.rayCross ≠ 0) : |
| 0 < e.tail.1 ∧ 0 ≤ e.tail.2 ∧ e.tail.2 ≤ 1 := by |
| rcases e with ⟨⟨x,y⟩,d⟩ |
| cases d <;> simp_all [rayCross, tail, Step.corner] |
|
|
| end SquareSide |
|
|
| namespace SquareLoop |
| variable {m : ℕ} |
|
|
| theorem sum_axisCross (c : SquareLoop m) : ∑ i, (c.side i).axisCross = 0 := by |
| simp_rw [SquareSide.axisCross_eq, ← c.follows] |
| rw [sum_sub_distrib, Equiv.sum_comp (finRotate m) |
| (fun i => SquareSide.aboveAxis (c.side i).tail), sub_self] |
|
|
| theorem winding_locations (c : SquareLoop m) (hc : c.winding ≠ 0) : |
| (∃ i, (c.side i).tail.1 ≤ 0) ∧ |
| (∃ i, 0 < (c.side i).tail.1 ∧ 0 ≤ (c.side i).tail.2 ∧ (c.side i).tail.2 ≤ 1) := by |
| constructor |
| · by_contra hn |
| push Not at hn |
| apply hc |
| change (∑ i, (c.side i).rayCross) = 0 |
| simp_rw [SquareSide.rayCross_eq_axisCross _ (hn _)] |
| exact c.sum_axisCross |
| · have he : ∃ i, (c.side i).rayCross ≠ 0 := by |
| by_contra hn |
| push Not at hn |
| exact hc (sum_eq_zero (fun i _ => hn i)) |
| obtain ⟨i,hi⟩ := he |
| exact ⟨i,SquareSide.rayCross_ne_zero_location _ hi⟩ |
|
|
| theorem tail_bounds (c : SquareLoop m) (hm : 0 < m) (i : Fin m) : |
| (c.side ⟨0,hm⟩).tail.1 - i.val ≤ (c.side i).tail.1 ∧ |
| (c.side i).tail.1 ≤ (c.side ⟨0,hm⟩).tail.1 + i.val ∧ |
| (c.side ⟨0,hm⟩).tail.2 - i.val ≤ (c.side i).tail.2 ∧ |
| (c.side i).tail.2 ≤ (c.side ⟨0,hm⟩).tail.2 + i.val := by |
| have : NeZero m := ⟨Nat.ne_of_gt hm⟩ |
| obtain ⟨i,hi⟩ := i |
| induction i with |
| | zero => simp |
| | succ n ih => |
| have hn : n < m := by omega |
| have hh := c.follows ⟨n,hn⟩ |
| have hr : finRotate m ⟨n,hn⟩ = ⟨n+1,hi⟩ := by |
| apply Fin.ext |
| simp only [finRotate_apply, Fin.val_add, Fin.val_one', Nat.add_mod_mod] |
| exact Nat.mod_eq_of_lt hi |
| rw [hr, SquareSide.head] at hh |
| have hx := congrArg Prod.fst hh |
| have hy := congrArg Prod.snd hh |
| have ihh := ih hn |
| cases hd : (c.side ⟨n,hn⟩).2 <;> |
| simp [hd,Step.delta] at hx hy <;> dsimp only at * <;> omega |
|
|
| theorem start_mem_box (c : SquareLoop m) (hm : 0 < m) (hc : c.winding ≠ 0) : |
| (c.side ⟨0,hm⟩).tail ∈ latticeBox (m+1) := by |
| obtain ⟨⟨i,hi⟩,⟨j,hj0,hj1,hj2⟩⟩ := c.winding_locations hc |
| have hbi := c.tail_bounds hm i |
| have hbj := c.tail_bounds hm j |
| have hi' := i.isLt |
| have hj' := j.isLt |
| simp only [latticeBox, mem_product, mem_Icc] |
| constructor <;> constructor <;> omega |
|
|
| end SquareLoop |
|
|
| end PlanarFPP |
|
|
| end OAI |
|
|
