File size: 2,572 Bytes
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 | import OAI.Probability.FirstPassage.ContourLocalization
namespace OAI
namespace PlanarFPP
open Finset
namespace SquareLoop
variable {m : ℕ}
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}
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)
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
noncomputable instance enclosingLoopFintype (n : ℕ) : Fintype (EnclosingLoop n) :=
Fintype.ofInjective (enclosingLoopCode n) (enclosingLoopCode_injective n)
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)
end PlanarFPP
end OAI
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