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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 | import Mathlib
import OAI.Analysis.RecorderRadix.Tape
import OAI.Computability.SolenoidalRecorder.Checkpoints
namespace OAI
/-! Exact rational radix coordinates of the initialized recorder. -/
namespace Solenoidal
namespace Bridge
variable {M : Machine}
def blankLetter (M : Machine) : Recorder.Letter M := ⟨M.blank, .empty, false⟩
theorem initial_tape_left (w : List M.Symbol) (k : ℤ) (hk : k < 0) :
(Recorder.initial w 2).tape k = blankLetter M := by
have hk₂ : k ≠ 2 := by omega
simp [Recorder.initial, Recorder.checkpoint, Recorder.tracks, Machine.inputTape,
Recorder.initialHistory, hk₂, not_le.mpr hk, blankLetter]
theorem initial_tape_right (w : List M.Symbol) (k : ℤ)
(hk : ((max w.length 3 : ℕ) : ℤ) ≤ k) :
(Recorder.initial w 2).tape k = blankLetter M := by
have hk₀ : 0 ≤ k := by omega
have hk₂ : k ≠ 2 := by omega
have hlen : w.length ≤ k.toNat := by omega
have hw : w[k.toNat]? = none := by simp [List.getElem?_eq_none hlen]
simp [Recorder.initial, Recorder.checkpoint, Recorder.tracks, Machine.inputTape,
Recorder.initialHistory, hk₂, hk₀, hw, blankLetter]
def initialRadixRational {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1))
(w : List M.Symbol) : ℚ × ℚ :=
let t : ℤ → Fin (m + 1) := fun k => e ((Recorder.initial w 2).tape k)
(Radix.finiteCode (Radix.leftStack t 0) (e (blankLetter M)) 0,
Radix.finiteCode (Radix.rightStack t 0) (e (blankLetter M)) (max w.length 3))
theorem initial_radix_is_rational {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1))
(w : List M.Symbol) :
Radix.tapeCode (fun k => e ((Recorder.initial w 2).tape k)) 0 =
(((initialRadixRational e w).1 : ℝ), ((initialRadixRational e w).2 : ℝ)) := by
apply Prod.ext
· apply Radix.code_is_finiteCode
intro j _
change e ((Recorder.initial w 2).tape (0 - ((j : ℤ) + 1))) = _
rw [initial_tape_left w _ (by omega)]
· apply Radix.code_is_finiteCode
intro j hj
change e ((Recorder.initial w 2).tape (0 + (j : ℤ))) = _
rw [initial_tape_right w _ (by omega)]
end Bridge
end Solenoidal
end OAI
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