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92416a2 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import Mathlib
import OAI.Computability.VertexCover.PCP.PoweringWalks
namespace OAI
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
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