File size: 2,717 Bytes
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 | 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
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