| import Mathlib |
| import OAI.Computability.VertexCover.PCP.PoweringWalks |
|
|
| namespace OAI |
|
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| |
|
|
| namespace UniqueGames.Foundations.PCP.PoweringLabels |
|
|
| open PoweringWalks |
| open scoped BigOperators |
|
|
| variable {V D A : Type*} |
|
|
| abbrev PortWords (D : Type*) (t : Nat) := (n : Fin (t + 1)) × (Fin n.val → D) |
| |
| instance finitePortWords [Finite D] (t : Nat) : Finite (PortWords D t) := by |
| let := Fintype.ofFinite D |
| exact Finite.of_fintype _ |
| |
| def Ball (G : PortGraph V D) (t : Nat) (v : V) := |
| {u : V // ∃ n, n ≤ t ∧ ∃ p : Fin n → D, wordEnd G n v p = u} |
| |
| def wordToBall (G : PortGraph V D) (t : Nat) (v : V) (w : PortWords D t) : Ball G t v := |
| ⟨wordEnd G w.1.val v w.2, w.1.val, Nat.le_of_lt_succ w.1.isLt, w.2, rfl⟩ |
| |
| theorem wordToBall_surjective (G : PortGraph V D) (t : Nat) (v : V) : |
| Function.Surjective (wordToBall G t v) := by |
| intro u |
| obtain ⟨n, hn, p, hp⟩ := u.property |
| refine ⟨⟨⟨n, Nat.lt_succ_of_le hn⟩, p⟩, ?_⟩ |
| exact Subtype.ext hp |
| |
| instance finiteBall [Finite D] (G : PortGraph V D) (t : Nat) (v : V) : |
| Finite (Ball G t v) := |
| Finite.of_surjective (wordToBall G t v) (wordToBall_surjective G t v) |
| |
| theorem card_portWords [Finite D] (t : Nat) : |
| Nat.card (PortWords D t) = ∑ n : Fin (t + 1), Nat.card D ^ n.val := by |
| simp [PortWords, Nat.card_sigma, Nat.card_fun] |
| |
| theorem card_ball_le [Finite D] (G : PortGraph V D) (t : Nat) (v : V) : |
| Nat.card (Ball G t v) ≤ ∑ n : Fin (t + 1), Nat.card D ^ n.val := by |
| calc |
| _ ≤ Nat.card (PortWords D t) := Nat.card_le_card_of_surjective |
| (wordToBall G t v) (wordToBall_surjective G t v) |
| _ = _ := card_portWords t |
| |
| structure AddressSelector (G : PortGraph V D) (t : Nat) (v : V) where |
| address : Ball G t v → PortWords D t |
| correct : ∀ u, wordToBall G t v (address u) = u |
|
|
| noncomputable def classicalSelector (G : PortGraph V D) (t : Nat) (v : V) : |
| AddressSelector G t v where |
| address u := Classical.choose (wordToBall_surjective G t v u) |
| correct u := Classical.choose_spec (wordToBall_surjective G t v u) |
|
|
| def scanWitness {A : Type*} (p : A → Prop) [DecidablePred p] : |
| (xs : List A) → (∃ a ∈ xs, p a) → {a // p a} |
| | [], h => False.elim (by simp at h) |
| | a :: xs, h => |
| if ha : p a then ⟨a, ha⟩ else |
| scanWitness p xs (by |
| obtain ⟨b, hb, hp⟩ := h |
| rcases List.mem_cons.mp hb with he | hm |
| · exact False.elim (ha (he ▸ hp)) |
| · exact ⟨b, hm, hp⟩) |
|
|
| def listSelector [DecidableEq V] (G : PortGraph V D) (t : Nat) (v : V) |
| (addresses : List (PortWords D t)) (complete : ∀ w, w ∈ addresses) : |
| AddressSelector G t v where |
| address u := (scanWitness (fun w => (wordToBall G t v w).val = u.val) addresses (by |
| obtain ⟨w, hw⟩ := wordToBall_surjective G t v u |
| exact ⟨w, complete w, congrArg Subtype.val hw⟩)).val |
| correct u := Subtype.ext (scanWitness |
| (fun w => (wordToBall G t v w).val = u.val) addresses (by |
| obtain ⟨w, hw⟩ := wordToBall_surjective G t v u |
| exact ⟨w, complete w, congrArg Subtype.val hw⟩)).property |
|
|
| abbrev PaddedLabel (D : Type*) (t : Nat) (A : Type*) := PortWords D t → A |
|
|
| def encode (G : PortGraph V D) (t : Nat) (v : V) (ℓ : Ball G t v → A) : |
| PaddedLabel D t A := fun w => ℓ (wordToBall G t v w) |
|
|
| def decode {G : PortGraph V D} {t : Nat} {v : V} (S : AddressSelector G t v) |
| (a : PaddedLabel D t A) : Ball G t v → A := fun u => a (S.address u) |
|
|
| theorem decode_encode {G : PortGraph V D} {t : Nat} {v : V} |
| (S : AddressSelector G t v) (ℓ : Ball G t v → A) : |
| decode S (encode G t v ℓ) = ℓ := by |
| funext u |
| change ℓ (wordToBall G t v (S.address u)) = ℓ u |
| rw [S.correct] |
|
|
| theorem decode_surjective {G : PortGraph V D} {t : Nat} {v : V} |
| (S : AddressSelector G t v) : Function.Surjective (decode (A := A) S) := |
| fun ℓ => ⟨encode G t v ℓ, decode_encode S ℓ⟩ |
|
|
| theorem encode_injective (G : PortGraph V D) (t : Nat) (v : V) : |
| Function.Injective (encode (A := A) G t v) := by |
| intro a b h |
| funext u |
| obtain ⟨w, rfl⟩ := wordToBall_surjective G t v u |
| exact congrFun h w |
|
|
| theorem decoded_word_collision {G : PortGraph V D} {t : Nat} {v : V} |
| (S : AddressSelector G t v) (a : PaddedLabel D t A) (w z : PortWords D t) |
| (h : (wordToBall G t v w).val = (wordToBall G t v z).val) : |
| decode S a (wordToBall G t v w) = decode S a (wordToBall G t v z) := |
| congrArg (decode S a) (Subtype.ext h) |
|
|
| theorem card_paddedLabel [Finite D] [Finite A] (t : Nat) : |
| Nat.card (PaddedLabel D t A) = |
| Nat.card A ^ (∑ n : Fin (t + 1), Nat.card D ^ n.val) := by |
| rw [Nat.card_fun, card_portWords] |
|
|
| end UniqueGames.Foundations.PCP.PoweringLabels |
|
|
| end OAI |
|
|
