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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 | import Mathlib
import OAI.Geometry.RecorderBoxes.Encoding
import OAI.Geometry.RecorderBoxes.InitialCode
import OAI.Analysis.RecorderRadix.Scales
namespace OAI
/-! Rational endpoints and the initial physical coordinate. -/
namespace Solenoidal
namespace Bridge
variable {M : Machine}
def castPoint (p : ℚ × ℚ) : ℝ × ℝ := ((p.1 : ℝ), (p.2 : ℝ))
def Rectangle.RationalEndpoints (P : Rectangle) : Prop :=
∃ l u : ℚ × ℚ, P = ⟨castPoint l, castPoint u⟩
theorem source_rational {m : ℕ} (a l : Fin (m + 1)) :
(sourceBox a l).RationalEndpoints := by
refine ⟨(Radix.rationalPrefix l 0, Radix.rationalPrefix a 0),
(Radix.rationalPrefix l 1, Radix.rationalPrefix a 1), ?_⟩
simp [sourceBox, castPoint]
theorem target_rational {m : ℕ} (b l : Fin (m + 1)) (d : Move) :
(targetBox b l d).RationalEndpoints := by
cases d with
| stay => exact source_rational b l
| right =>
refine ⟨(Radix.rationalPrefix b (Radix.rationalPrefix l 0), 0),
(Radix.rationalPrefix b (Radix.rationalPrefix l 1), 1), ?_⟩
simp [targetBox, castPoint]
| left =>
refine ⟨(0, Radix.rationalPrefix l (Radix.rationalPrefix b 0)),
(1, Radix.rationalPrefix l (Radix.rationalPrefix b 1)), ?_⟩
simp [targetBox, castPoint]
def rationalKappa (M : Machine) : ℚ := 1 / (32 * (controlCount M : ℚ))
noncomputable def rationalOrigin (s : Recorder.Control M) : ℚ :=
(if terminalControl s then 3 / 4 else 1 / 8) +
2 * ((controlIndex M s).val : ℚ) * rationalKappa M
noncomputable def rationalPlace (s : Recorder.Control M) (p : ℚ × ℚ) : ℚ × ℚ :=
(rationalOrigin s + rationalKappa M * p.1, 3 / 8 + rationalKappa M * p.2)
@[simp] theorem rationalKappa_coe (M : Machine) : (rationalKappa M : ℝ) = kappa M := by
simp [rationalKappa, kappa]
@[simp] theorem rationalOrigin_coe (s : Recorder.Control M) :
(rationalOrigin s : ℝ) = origin s := by
cases h : terminalControl s <;> simp [rationalOrigin, origin, h]
@[simp] theorem rationalPlace_cast (s : Recorder.Control M) (p : ℚ × ℚ) :
castPoint (rationalPlace s p) = place s (castPoint p) := by
apply Prod.ext <;> simp [castPoint, rationalPlace, place]
theorem physical_rational (s : Recorder.Control M) {P : Rectangle}
(hp : P.RationalEndpoints) : (physicalBox s P).RationalEndpoints := by
obtain ⟨l, u, rfl⟩ := hp
exact ⟨rationalPlace s l, rationalPlace s u, by simp [physicalBox]⟩
theorem rectangle_rational {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1)) (i : Branch M) :
(sourceRectangle e i).RationalEndpoints ∧ (targetRectangle e i).RationalEndpoints :=
⟨physical_rational i.src (source_rational _ _),
physical_rational i.dst (target_rational _ _ _)⟩
noncomputable def initialPhysicalRational {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1))
(w : List M.Symbol) : ℚ × ℚ :=
rationalPlace (.S M.initial) (initialRadixRational e w)
theorem initial_physical_is_rational {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1))
(w : List M.Symbol) :
configCode e (Recorder.initial w 2) = castPoint (initialPhysicalRational e w) := by
change place (.S M.initial) (Radix.tapeCode (fun k => e ((Recorder.initial w 2).tape k)) 0) = _
rw [initial_radix_is_rational]
exact (rationalPlace_cast (.S M.initial) (initialRadixRational e w)).symm
end Bridge
end Solenoidal
end OAI
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