| 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 |
|
|
