| import Mathlib |
| import OAI.Geometry.RecorderBoxes.Boxes |
| import OAI.Computability.SolenoidalRecorder.Rules |
|
|
| namespace OAI |
|
|
| /-! Finite recorder branches and disjoint source and target rectangles. -/ |
|
|
| namespace Solenoidal |
| namespace Bridge |
| @[ext] structure Branch (M : Machine) where |
| src : Recorder.Control M |
| read : Recorder.Letter M |
| left : Recorder.Letter M |
| dst : Recorder.Control M |
| written : Recorder.Letter M |
| move : Move |
| allowed : Recorder.Rule M src read dst written move |
|
|
| variable {M : Machine} |
|
|
| |
| theorem branch_source_key {i j : Branch M} (hs : i.src = j.src) |
| (ha : i.read = j.read) (hl : i.left = j.left) : i = j := by |
| have hh := Recorder.rule_deterministic i.allowed j.allowed hs ha |
| exact Branch.ext hs ha hl hh.1 hh.2.1 hh.2.2 |
|
|
| |
| theorem branch_target_key {i j : Branch M} (hs : i.dst = j.dst) |
| (hb : i.written = j.written) (hl : i.left = j.left) : i = j := by |
| have hj : Recorder.Rule M j.src j.read i.dst i.written j.move := by |
| simpa only [hs, hb] using j.allowed |
| have hh := Recorder.local_predecessor_unique i.allowed hj |
| exact Branch.ext hh.1 hh.2.1 hl hs hb hh.2.2 |
|
|
| |
| instance : Finite (Branch M) := by |
| let key : Branch M → Recorder.Control M × Recorder.Letter M × Recorder.Letter M := |
| fun i => (i.src, i.read, i.left) |
| apply Finite.of_injective key |
| intro i j h |
| exact branch_source_key (congrArg Prod.fst h) |
| (congrArg (fun x => x.2.1) h) (congrArg (fun x => x.2.2) h) |
|
|
| |
| noncomputable def branchSource {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1)) |
| (i : Branch M) : Rectangle := sourceBox (e i.read) (e i.left) |
|
|
| noncomputable def branchTarget {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1)) |
| (i : Branch M) : Rectangle := targetBox (e i.written) (e i.left) i.move |
|
|
| theorem branch_source_gap {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1)) |
| {i j : Branch M} (hij : i ≠ j) (hs : i.src = j.src) : |
| (branchSource e i).SeparatedBy (1 / Radix.base m) (branchSource e j) := by |
| apply normalized_source_gap |
| intro he |
| exact hij (branch_source_key hs (e.injective (congrArg Prod.fst he)) |
| (e.injective (congrArg Prod.snd he))) |
|
|
| theorem branch_target_gap {m : ℕ} (e : Recorder.Letter M ≃ Fin (m + 1)) |
| {i j : Branch M} (hij : i ≠ j) (hs : i.dst = j.dst) : |
| (branchTarget e i).SeparatedBy (1 / Radix.base m ^ 2) (branchTarget e j) := by |
| have hd : i.move = j.move := (Recorder.fixed_incoming i.allowed).trans |
| ((congrArg (Recorder.incoming M) hs).trans (Recorder.fixed_incoming j.allowed).symm) |
| dsimp only [branchTarget] |
| rw [hd] |
| apply normalized_target_gap |
| intro he |
| exact hij (branch_target_key hs (e.injective (congrArg Prod.fst he)) |
| (e.injective (congrArg Prod.snd he))) |
| end Bridge |
| end Solenoidal |
|
|
| end OAI |
|
|
