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import OAI.Geometry.RecorderBoxes.Branches
namespace OAI
/-! Separated state-square placement of recorder controls. -/
namespace Solenoidal
namespace Bridge
variable {M : Machine}
def terminalControl : Recorder.Control M → Bool
| .S q => M.halt q
| _ => false
abbrev controlCount (M : Machine) : ℕ := Fintype.card (Recorder.Control M)
noncomputable def controlIndex (M : Machine) : Recorder.Control M ≃ Fin (controlCount M) :=
Fintype.equivFin _
theorem controlCount_pos (M : Machine) : 0 < controlCount M :=
Fintype.card_pos_iff.mpr ⟨.S M.initial⟩
noncomputable def kappa (M : Machine) : ℝ := 1 / (32 * (controlCount M : ℝ))
theorem kappa_pos (M : Machine) : 0 < kappa M :=
one_div_pos.mpr (mul_pos (by norm_num) (by exact_mod_cast controlCount_pos M))
theorem kappa_le (M : Machine) : kappa M ≤ 1 / 32 := by
apply one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 32)
have hN : (1 : ℝ) ≤ controlCount M := by exact_mod_cast controlCount_pos M
nlinarith
noncomputable def origin (s : Recorder.Control M) : ℝ :=
(if terminalControl s then 3 / 4 else 1 / 8) +
2 * ((controlIndex M s).val : ℝ) * kappa M
theorem state_extent (s : Recorder.Control M) :
(if terminalControl s then (3 / 4 : ℝ) else 1 / 8) ≤ origin s ∧
origin s + kappa M < (if terminalControl s then (13 / 16 : ℝ) else 3 / 16) := by
have hk := kappa_pos M
have h₀ : (0 : ℝ) ≤ (controlIndex M s).val := Nat.cast_nonneg _
have hN : (↑(controlIndex M s).val : ℝ) + 1 ≤ controlCount M := by
exact_mod_cast (controlIndex M s).isLt
have hprod : 2 * (controlCount M : ℝ) * kappa M = 1 / 16 := by
dsimp only [kappa]
have hn : (controlCount M : ℝ) ≠ 0 := by
exact_mod_cast (controlCount_pos M).ne'
field_simp
norm_num
have hscaled := mul_le_mul_of_nonneg_right hN hk.le
have hnonneg := mul_nonneg h₀ hk.le
dsimp only [origin]
cases terminalControl s <;> simp only [Bool.false_eq_true, ↓reduceIte] <;>
constructor <;> nlinarith
theorem state_gap {s t : Recorder.Control M} (hne : s ≠ t) :
(origin s + kappa M) + kappa M ≤ origin t ∨
(origin t + kappa M) + kappa M ≤ origin s := by
have hk := kappa_pos M
by_cases he : terminalControl s = terminalControl t
· have hidx : (controlIndex M s).val ≠ (controlIndex M t).val := by
intro hi
exact hne ((controlIndex M).injective (Fin.ext hi))
rcases lt_or_gt_of_ne hidx with hi | hi
· left
have hi' : (↑(controlIndex M s).val : ℝ) + 1 ≤ (controlIndex M t).val := by
exact_mod_cast hi
have hm := mul_le_mul_of_nonneg_right hi' hk.le
dsimp only [origin]
rw [he]
nlinarith
· right
have hi' : (↑(controlIndex M t).val : ℝ) + 1 ≤ (controlIndex M s).val := by
exact_mod_cast hi
have hm := mul_le_mul_of_nonneg_right hi' hk.le
dsimp only [origin]
rw [he]
nlinarith
· have hs := state_extent s
have ht := state_extent t
have hκ := kappa_le M
cases es : terminalControl s <;> cases et : terminalControl t
· exact False.elim (he (es.trans et.symm))
· simp only [es, Bool.false_eq_true, ↓reduceIte] at hs
simp only [et, ↓reduceIte] at ht
left; linarith
· simp only [es, ↓reduceIte] at hs
simp only [et, Bool.false_eq_true, ↓reduceIte] at ht
right; linarith
· exact False.elim (he (es.trans et.symm))
noncomputable def place (s : Recorder.Control M) (p : ℝ × ℝ) : ℝ × ℝ :=
(origin s + kappa M * p.1, 3 / 8 + kappa M * p.2)
noncomputable def unplace (s : Recorder.Control M) (p : ℝ × ℝ) : ℝ × ℝ :=
((p.1 - origin s) / kappa M, (p.2 - 3 / 8) / kappa M)
@[simp] theorem unplace_place (s : Recorder.Control M) (p : ℝ × ℝ) :
unplace s (place s p) = p := by
apply Prod.ext <;> dsimp [unplace, place] <;> field_simp [(kappa_pos M).ne'] <;> ring
@[simp] theorem place_unplace (s : Recorder.Control M) (p : ℝ × ℝ) :
place s (unplace s p) = p := by
apply Prod.ext <;> dsimp [unplace, place] <;> field_simp [(kappa_pos M).ne'] <;> ring
noncomputable def physicalBox (s : Recorder.Control M) (P : Rectangle) : Rectangle :=
⟨place s P.lower, place s P.upper⟩
theorem physicalBox_carrier (s : Recorder.Control M) (P : Rectangle) :
(physicalBox s P).carrier = place s '' P.carrier := by
ext q
constructor
· intro hq
refine ⟨unplace s q, ?_, place_unplace s q⟩
change (P.lower.1 ≤ (q.1 - origin s) / kappa M ∧
(q.1 - origin s) / kappa M ≤ P.upper.1) ∧
(P.lower.2 ≤ (q.2 - 3 / 8) / kappa M ∧
(q.2 - 3 / 8) / kappa M ≤ P.upper.2)
change (origin s + kappa M * P.lower.1 ≤ q.1 ∧
q.1 ≤ origin s + kappa M * P.upper.1) ∧
(3 / 8 + kappa M * P.lower.2 ≤ q.2 ∧
q.2 ≤ 3 / 8 + kappa M * P.upper.2) at hq
simp only [le_div_iff₀ (kappa_pos M), div_le_iff₀ (kappa_pos M)]
constructor <;> constructor <;> nlinarith [hq.1.1, hq.1.2, hq.2.1, hq.2.2]
· rintro ⟨p, hp, rfl⟩
change (origin s + kappa M * P.lower.1 ≤ origin s + kappa M * p.1 ∧
origin s + kappa M * p.1 ≤ origin s + kappa M * P.upper.1) ∧
(3 / 8 + kappa M * P.lower.2 ≤ 3 / 8 + kappa M * p.2 ∧
3 / 8 + kappa M * p.2 ≤ 3 / 8 + kappa M * P.upper.2)
have h₁ := mul_le_mul_of_nonneg_left hp.1.1 (kappa_pos M).le
have h₂ := mul_le_mul_of_nonneg_left hp.1.2 (kappa_pos M).le
have h₃ := mul_le_mul_of_nonneg_left hp.2.1 (kappa_pos M).le
have h₄ := mul_le_mul_of_nonneg_left hp.2.2 (kappa_pos M).le
constructor <;> constructor <;> linarith
theorem physical_gap_same (s : Recorder.Control M) {P Q : Rectangle} {δ : ℝ}
(h : P.SeparatedBy δ Q) :
(physicalBox s P).SeparatedBy (kappa M * δ) (physicalBox s Q) := by
rcases h with (h | h) | (h | h)
all_goals have hm := mul_le_mul_of_nonneg_left h (kappa_pos M).le
· exact .inl (.inl (by dsimp [physicalBox, place]; nlinarith))
· exact .inl (.inr (by dsimp [physicalBox, place]; nlinarith))
· exact .inr (.inl (by dsimp [physicalBox, place]; nlinarith))
· exact .inr (.inr (by dsimp [physicalBox, place]; nlinarith))
theorem physical_gap_different {s t : Recorder.Control M} {P Q : Rectangle}
(hs : s ≠ t) (hP : P.InUnit) (hQ : Q.InUnit) :
(physicalBox s P).SeparatedBy (kappa M) (physicalBox t Q) := by
have hk := kappa_pos M
have hpU := mul_le_mul_of_nonneg_left hP.2.2.1 hk.le
have hpL := mul_nonneg hk.le hP.1
have hqU := mul_le_mul_of_nonneg_left hQ.2.2.1 hk.le
have hqL := mul_nonneg hk.le hQ.1
rcases state_gap hs with hst | hst
· exact .inl (.inl (by dsimp [physicalBox, place]; nlinarith))
· exact .inl (.inr (by dsimp [physicalBox, place]; nlinarith))
end Bridge
end Solenoidal
end OAI
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