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695bd70 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import Mathlib
import OAI.Analysis.RecorderRadix.Tape
namespace OAI
/-! Filled prefix rectangles and normalized separation gaps. -/
namespace Solenoidal
namespace Bridge
@[ext] structure Rectangle where
lower : ℝ × ℝ
upper : ℝ × ℝ
def Rectangle.carrier (P : Rectangle) : Set (ℝ × ℝ) :=
Set.Icc P.lower.1 P.upper.1 ×ˢ Set.Icc P.lower.2 P.upper.2
def Rectangle.SeparatedBy (δ : ℝ) (P Q : Rectangle) : Prop :=
(P.upper.1 + δ ≤ Q.lower.1 ∨ Q.upper.1 + δ ≤ P.lower.1) ∨
(P.upper.2 + δ ≤ Q.lower.2 ∨ Q.upper.2 + δ ≤ P.lower.2)
def Rectangle.InUnit (P : Rectangle) : Prop :=
0 ≤ P.lower.1 ∧ P.lower.1 ≤ P.upper.1 ∧ P.upper.1 ≤ 1 ∧
0 ≤ P.lower.2 ∧ P.lower.2 ≤ P.upper.2 ∧ P.upper.2 ≤ 1
theorem Rectangle.gap_mono {δ ε : ℝ} {P Q : Rectangle} (h : P.SeparatedBy ε Q)
(hδε : δ ≤ ε) : P.SeparatedBy δ Q := by
rcases h with (h | h) | (h | h)
· exact .inl (.inl (le_trans (by linarith) h))
· exact .inl (.inr (le_trans (by linarith) h))
· exact .inr (.inl (le_trans (by linarith) h))
· exact .inr (.inr (le_trans (by linarith) h))
theorem Rectangle.disjoint_of_gap {δ : ℝ} {P Q : Rectangle} (hδ : 0 < δ)
(h : P.SeparatedBy δ Q) : Disjoint P.carrier Q.carrier := by
apply Set.disjoint_left.mpr
intro p hp hq
change (P.lower.1 ≤ p.1 ∧ p.1 ≤ P.upper.1) ∧
(P.lower.2 ≤ p.2 ∧ p.2 ≤ P.upper.2) at hp
change (Q.lower.1 ≤ p.1 ∧ p.1 ≤ Q.upper.1) ∧
(Q.lower.2 ≤ p.2 ∧ p.2 ≤ Q.upper.2) at hq
rcases h with (h | h) | (h | h) <;> linarith [hp.1.1, hp.1.2, hp.2.1, hp.2.2,
hq.1.1, hq.1.2, hq.2.1, hq.2.2]
noncomputable def sourceBox {m : ℕ} (a l : Fin (m + 1)) : Rectangle :=
⟨(Radix.pushDigit l 0, Radix.pushDigit a 0),
(Radix.pushDigit l 1, Radix.pushDigit a 1)⟩
noncomputable def targetBox {m : ℕ} (b l : Fin (m + 1)) : Move → Rectangle
| .right => ⟨(Radix.pushDigit b (Radix.pushDigit l 0), 0),
(Radix.pushDigit b (Radix.pushDigit l 1), 1)⟩
| .stay => sourceBox b l
| .left => ⟨(0, Radix.pushDigit l (Radix.pushDigit b 0)),
(1, Radix.pushDigit l (Radix.pushDigit b 1))⟩
@[simp] theorem sourceBox_carrier {m : ℕ} (a l : Fin (m + 1)) :
(sourceBox a l).carrier = Radix.source a l := rfl
@[simp] theorem targetBox_carrier {m : ℕ} (b l : Fin (m + 1)) (d : Move) :
(targetBox b l d).carrier = Radix.target b l d := by
cases d <;> rfl
theorem first_bounds {m : ℕ} (a : Fin (m + 1)) :
0 ≤ Radix.pushDigit a 0 ∧ Radix.pushDigit a 0 ≤ Radix.pushDigit a 1 ∧
Radix.pushDigit a 1 ≤ 1 :=
⟨Radix.lower_nonneg a, (Radix.prefix_le_prefix_iff a).mpr (by norm_num),
Radix.upper_le_one a⟩
theorem two_bounds {m : ℕ} (a b : Fin (m + 1)) :
0 ≤ Radix.pushDigit a (Radix.pushDigit b 0) ∧
Radix.pushDigit a (Radix.pushDigit b 0) ≤ Radix.pushDigit a (Radix.pushDigit b 1) ∧
Radix.pushDigit a (Radix.pushDigit b 1) ≤ 1 := by
have ha := first_bounds a
have hb := first_bounds b
exact ⟨ha.1.trans ((Radix.prefix_le_prefix_iff a).mpr hb.1),
(Radix.prefix_le_prefix_iff a).mpr hb.2.1,
((Radix.prefix_le_prefix_iff a).mpr hb.2.2).trans ha.2.2⟩
theorem source_in_unit {m : ℕ} (a l : Fin (m + 1)) : (sourceBox a l).InUnit :=
⟨(first_bounds l).1, (first_bounds l).2.1, (first_bounds l).2.2,
(first_bounds a).1, (first_bounds a).2.1, (first_bounds a).2.2⟩
theorem target_in_unit {m : ℕ} (b l : Fin (m + 1)) (d : Move) :
(targetBox b l d).InUnit := by
cases d with
| stay => exact source_in_unit b l
| left => exact ⟨le_refl _, by norm_num [targetBox], le_refl _,
(two_bounds l b).1, (two_bounds l b).2.1, (two_bounds l b).2.2⟩
| right => exact ⟨(two_bounds b l).1, (two_bounds b l).2.1, (two_bounds b l).2.2,
le_refl _, by norm_num [targetBox], le_refl _⟩
theorem inverse_square_le_inverse (m : ℕ) : 1 / Radix.base m ^ 2 ≤ 1 / Radix.base m := by
apply one_div_le_one_div_of_le (Radix.base_pos m)
have := Radix.base_three_le m
nlinarith
theorem normalized_source_gap {m : ℕ} {a l a' l' : Fin (m + 1)}
(hne : (a, l) ≠ (a', l')) :
(sourceBox a l).SeparatedBy (1 / Radix.base m) (sourceBox a' l') := by
by_cases hl : l = l'
· have ha : a ≠ a' := fun he => hne (Prod.ext he hl)
exact .inr (Radix.first_separated ha)
· exact .inl (Radix.first_separated hl)
theorem normalized_target_gap {m : ℕ} {b l b' l' : Fin (m + 1)}
(hne : (b, l) ≠ (b', l')) (d : Move) :
(targetBox b l d).SeparatedBy (1 / Radix.base m ^ 2) (targetBox b' l' d) := by
cases d with
| right => exact .inl (Radix.two_separated hne)
| left =>
exact .inr (Radix.two_separated (by
intro he
exact hne (Prod.ext (congrArg Prod.snd he) (congrArg Prod.fst he))))
| stay => exact Rectangle.gap_mono (normalized_source_gap hne) (inverse_square_le_inverse m)
end Bridge
end Solenoidal
end OAI
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