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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 | import Mathlib
import OAI.Geometry.RecorderBoxes.Affine
import OAI.Computability.SolenoidalRecorder.Halting
namespace OAI
/-! Configuration codes, unique affine branches and terminal observation strips. -/
namespace Solenoidal
namespace Bridge
variable {M : Machine}
noncomputable def configCode {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1))
(c : Recorder.Config M) : ℝ × ℝ :=
place c.control (Radix.tapeCode (fun k => e (c.tape k)) c.head)
theorem step_has_affine_branch {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1))
{c c' : Recorder.Config M} (hc : Recorder.Step c c') :
∃ i : Branch M, configCode e c ∈ (sourceRectangle e i).carrier ∧
affine e i (configCode e c) = configCode e c' := by
obtain ⟨s, b, d, hr, rfl⟩ := hc
let i : Branch M := ⟨c.control, c.tape c.head, c.tape (c.head - 1), s, b, d, hr⟩
refine ⟨i, ?_, ?_⟩
· rw [sourceRectangle, physicalBox_carrier]
refine ⟨Radix.tapeCode (fun k => e (c.tape k)) c.head, ?_, rfl⟩
exact Radix.tapeCode_mem_source (fun k => e (c.tape k)) c.head
· have ht : (fun k => e (Function.update c.tape c.head b k)) =
Function.update (fun k => e (c.tape k)) c.head (e b) := by
funext k
by_cases hk : k = c.head
· subst k; simp
· simp [hk]
dsimp only [affine, configCode, i]
rw [unplace_place, Radix.update_tapeCode (fun k => e (c.tape k)) c.head (e b) d]
simp only [Recorder.successor, ht]
theorem step_has_unique_affine_branch {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1))
{c c' : Recorder.Config M} (hc : Recorder.Step c c') :
∃! i : Branch M, configCode e c ∈ (sourceRectangle e i).carrier ∧
affine e i (configCode e c) = configCode e c' := by
obtain ⟨i, hi, ht⟩ := step_has_affine_branch e hc
refine ⟨i, ⟨hi, ht⟩, ?_⟩
intro j hj
by_contra hne
have hgap := source_rectangles_separated e hne
have hg : 0 < kappa M / Radix.base m ^ 2 :=
div_pos (kappa_pos M) (pow_pos (Radix.base_pos m) _)
exact Set.disjoint_left.mp (Rectangle.disjoint_of_gap hg hgap) hj.1 hi
theorem configCode_bounds {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1))
(c : Recorder.Config M) :
origin c.control ≤ (configCode e c).1 ∧
(configCode e c).1 ≤ origin c.control + kappa M ∧
(3 / 8 : ℝ) ≤ (configCode e c).2 ∧
(configCode e c).2 ≤ 3 / 8 + kappa M := by
have hL := Radix.code_bounds (Radix.leftStack (fun k => e (c.tape k)) c.head)
have hR := Radix.code_bounds (Radix.rightStack (fun k => e (c.tape k)) c.head)
have hL₀ := mul_nonneg (kappa_pos M).le hL.1
have hR₀ := mul_nonneg (kappa_pos M).le hR.1
have hL₁ := mul_le_mul_of_nonneg_left hL.2 (kappa_pos M).le
have hR₁ := mul_le_mul_of_nonneg_left hR.2 (kappa_pos M).le
dsimp [configCode, Radix.tapeCode, place]
constructor
· linarith
constructor
· linarith
constructor <;> linarith
theorem terminalControl_iff (c : Recorder.Config M) :
terminalControl c.control = true ↔ Recorder.Terminal c := by
cases h : c.control <;> simp [terminalControl, Recorder.Terminal, h]
theorem terminal_code_range {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1))
{c : Recorder.Config M} (hc : Recorder.Terminal c) :
(3 / 4 : ℝ) ≤ (configCode e c).1 ∧ (configCode e c).1 < 13 / 16 := by
have h := (terminalControl_iff c).mpr hc
have hs := state_extent c.control
simp only [h, ↓reduceIte] at hs
have hb := configCode_bounds e c
exact ⟨hs.1.trans hb.1, hb.2.1.trans_lt hs.2⟩
theorem nonterminal_code_range {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1))
{c : Recorder.Config M} (hc : ¬Recorder.Terminal c) :
(1 / 8 : ℝ) ≤ (configCode e c).1 ∧ (configCode e c).1 < 3 / 16 := by
have hf : terminalControl c.control = false := by
cases he : terminalControl c.control with
| false => rfl
| true => exact False.elim (hc ((terminalControl_iff c).mp he))
have hs := state_extent c.control
simp only [hf, Bool.false_eq_true, ↓reduceIte] at hs
have hb := configCode_bounds e c
exact ⟨hs.1.trans hb.1, hb.2.1.trans_lt hs.2⟩
end Bridge
end Solenoidal
end OAI
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