| import OAI.Combinatorics.Automata.RankLoss |
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| namespace OAI |
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| namespace OneWayLiveness |
| open scoped Classical |
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| namespace BRel |
| variable {H : Type*} |
| lemma singleton_context (a : BRel H) (i j : H) : |
| (∃ x y, (restrictedId {i} * a * restrictedId {j}).holds x y) ↔ a.holds i j := by |
| simp only [mul_restrictedId, restrictedId_mul, Set.mem_singleton_iff] |
| aesop |
|
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| lemma context_separates (u v : List (BRel H)) |
| (h : ∀ x y : List (BRel H), live (x ++ u ++ y) ↔ live (x ++ v ++ y)) : u.prod = v.prod := by |
| ext i j |
| have hh := h [restrictedId {i}] [restrictedId {j}] |
| simpa only [live, List.prod_append, List.prod_cons, List.prod_nil, mul_one, singleton_context] using hh |
| end BRel |
|
|
| /-- Any matching recognition of liveness induces the full relation divisor. |
| The recognizer predicate may in particular sandwich the fixed endmarkers |
| and inspect the designated test ports. -/ |
| theorem matching_recognition_degree {H A : Type} [Fintype H] [Fintype A] |
| (η : BRel H → Pairing (A ⊕ A)) (test : Pairing (A ⊕ A) → Prop) |
| (hη : ∀ w : List (BRel H), test (w.map η).prod ↔ BRel.live w) |
| (x y : H) (hxy : x ≠ y) : rankLoss (Fintype.card H) ≤ Fintype.card A := by |
| classical |
| let S : Submonoid (Pairing (A ⊕ A) × BRel H) := { |
| carrier := {p | ∃ w : List (BRel H), p = ((w.map η).prod, w.prod)} |
| one_mem' := ⟨[], by simp only [List.map_nil, List.prod_nil]; rfl⟩ |
| mul_mem' := by |
| rintro a b ⟨u, rfl⟩ ⟨v, rfl⟩ |
| exact ⟨u ++ v, by simp⟩ } |
| let ρ : S →* Pairing (A ⊕ A) := (MonoidHom.fst _ _).comp S.subtype |
| let φ : S →* BRel H := (MonoidHom.snd _ _).comp S.subtype |
| have hρ : Function.Injective ρ := by |
| intro a b hab |
| obtain ⟨u, hu⟩ := a.property |
| obtain ⟨v, hv⟩ := b.property |
| have hd : (u.map η).prod = (v.map η).prod := by |
| change a.val.1 = b.val.1 at hab |
| simpa only [hu, hv] using hab |
| have hr : u.prod = v.prod := by |
| apply BRel.context_separates |
| intro l r |
| rw [← hη, ← hη] |
| simp only [List.map_append, List.prod_append, hd] |
| apply Subtype.ext |
| rw [hu, hv, hd, hr] |
| have hφ : Function.Surjective φ := by |
| intro R |
| exact ⟨⟨(η R, R), [R], by simp only [List.map_cons, List.map_nil, List.prod_cons, List.prod_nil, mul_one]⟩, rfl⟩ |
| exact relation_divisor_degree ρ hρ φ hφ x y hxy |
|
|
| end OneWayLiveness |
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| end OAI |
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