| import OAI.Probability.FirstPassage.ContourLocalization |
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| namespace OAI |
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| namespace PlanarFPP |
| open Finset |
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| namespace SquareLoop |
| variable {m : ℕ} |
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| theorem eq_of_start_direction (c d : SquareLoop m) (hm : 0 < m) |
| (h0 : (c.side ⟨0,hm⟩).tail = (d.side ⟨0,hm⟩).tail) |
| (hd : ∀ i, (c.side i).2 = (d.side i).2) : c = d := by |
| have : NeZero m := ⟨Nat.ne_of_gt hm⟩ |
| have hp (i : Fin m) : (c.side i).tail = (d.side i).tail := by |
| obtain ⟨n,hn⟩ := i |
| induction n with |
| | zero => exact h0 |
| | succ n ih => |
| have hn' : n < m := by omega |
| have hr : finRotate m ⟨n,hn'⟩ = ⟨n+1,hn⟩ := by |
| apply Fin.ext |
| simp only [finRotate_apply, Fin.val_add, Fin.val_one', Nat.add_mod_mod] |
| exact Nat.mod_eq_of_lt hn |
| have hc := c.follows ⟨n,hn'⟩ |
| have hh := d.follows ⟨n,hn'⟩ |
| rw [hr] at hc hh |
| rw [hc,hh,SquareSide.head,SquareSide.head,ih hn',hd] |
| have he : c.side = d.side := by |
| funext i |
| apply Prod.ext |
| · have hh := hp i |
| change (c.side i).1 + (c.side i).2.corner = |
| (d.side i).1 + (d.side i).2.corner at hh |
| rw [hd] at hh |
| exact add_right_cancel hh |
| · exact hd i |
| cases c |
| cases d |
| cases he |
| rfl |
|
|
| end SquareLoop |
|
|
| abbrev EnclosingLoop (n : ℕ) := |
| {c : SquareLoop (n+1) // c.winding ≠ 0 ∧ c.NoRepeatedSides} |
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| def enclosingLoopCode (n : ℕ) (c : EnclosingLoop n) : |
| {p // p ∈ latticeBox (n+2)} × (Fin (n+1) → Step) := |
| (⟨(c.val.side ⟨0,by omega⟩).tail,c.val.start_mem_box (by omega) c.property.1⟩, |
| fun i => (c.val.side i).2) |
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| theorem enclosingLoopCode_injective (n : ℕ) : Function.Injective (enclosingLoopCode n) := by |
| intro c d h |
| apply Subtype.ext |
| apply SquareLoop.eq_of_start_direction c.val d.val (by omega) |
| · exact congrArg (fun x => x.1.val) h |
| · intro i |
| exact congrArg (fun x => x.2 i) h |
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| noncomputable instance enclosingLoopFintype (n : ℕ) : Fintype (EnclosingLoop n) := |
| Fintype.ofInjective (enclosingLoopCode n) (enclosingLoopCode_injective n) |
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| theorem enclosingLoop_card (n : ℕ) : |
| Fintype.card (EnclosingLoop n) ≤ 9*(n+2)^2*4^(n+1) := by |
| have hh := Fintype.card_le_of_injective _ (enclosingLoopCode_injective n) |
| have hf : Fintype.card Step = 4 := by decide |
| simp only [Fintype.card_prod, Fintype.card_coe, latticeBox_card, |
| Fintype.card_fun, Fintype.card_fin, hf] at hh |
| have hsq : (2*(n+2)+1)^2 ≤ 9*(n+2)^2 := by |
| calc |
| _ ≤ (3*(n+2))^2 := Nat.pow_le_pow_left (by omega) 2 |
| _ = _ := by ring |
| exact hh.trans (Nat.mul_le_mul_right _ hsq) |
|
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| end PlanarFPP |
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| end OAI |
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