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b317718 | 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 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.Probability.GammaPassage.SquareLoops
import OAI.Probability.GammaPassage.OrientedSelection
namespace OAI
/-! A loop with nonzero ray-crossing count starts within O(m) of zero.
This supplies the location factor in the source's contour enumeration. -/
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]
split_ifs <;> omega
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]
omega
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
/-- With the source's length m, every enclosing loop has a permitted starting
point in a deterministic box of cardinality (2m+3)^2. -/
theorem start_mem_box (c : SquareLoop m) (hm : 0 < m) (hc : c.winding ≠ 0) :
(c.side ⟨0,hm⟩).tail ∈ orientedSelectionBox (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 [orientedSelectionBox, mem_product, mem_Icc]
constructor <;> constructor <;> omega
end SquareLoop
end PlanarFPP
end OAI
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