File size: 2,238 Bytes
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 | import Mathlib
import OAI.Computability.VertexCover.PCP.PoweringLabels
namespace OAI
namespace UniqueGames.Foundations.PCP.PoweringAddresses
open PoweringWalks PoweringLabels
open scoped BigOperators
variable {V : Type*}
def allWords (d : Nat) : (n : Nat) → List (Fin n → Fin d)
| 0 => [Fin.elim0]
| n + 1 => (List.finRange d).flatMap fun a =>
(allWords d n).map fun p => Fin.cases a p
theorem mem_allWords (d : Nat) :
∀ n (p : Fin n → Fin d), p ∈ allWords d n := by
intro n
induction n with
| zero =>
intro p
simp only [allWords, List.mem_singleton]
funext i
exact Fin.elim0 i
| succ n ih =>
intro p
simp only [allWords, List.mem_flatMap, List.mem_map]
refine ⟨p 0, List.mem_finRange (p 0), (fun j => p j.succ), ih _, ?_⟩
funext j
exact Fin.cases rfl (fun _ => rfl) j
def allAddresses (d t : Nat) : List (PortWords (Fin d) t) :=
(List.finRange (t + 1)).flatMap fun n =>
(allWords d n.val).map fun p => Sigma.mk n p
theorem mem_allAddresses (d t : Nat) (w : PortWords (Fin d) t) :
w ∈ allAddresses d t := by
rcases w with ⟨n, p⟩
unfold allAddresses
apply List.mem_flatMap.mpr
refine ⟨n, List.mem_finRange n, ?_⟩
exact List.mem_map.mpr ⟨p, mem_allWords d n.val p, rfl⟩
theorem length_allWords (d : Nat) :
∀ n, (allWords d n).length = d ^ n := by
intro n
induction n with
| zero => rfl
| succ n ih =>
simp only [allWords, List.length_flatMap, List.length_map, ih,
List.map_const', List.length_finRange, List.sum_replicate_nat]
exact (Nat.mul_comm d (d ^ n)).trans (Nat.pow_succ d n).symm
theorem length_allAddresses (d t : Nat) :
(allAddresses d t).length = ∑ n : Fin (t + 1), d ^ n.val := by
unfold allAddresses
simp only [List.length_flatMap, List.length_map, length_allWords]
exact (Fin.sum_univ_def (fun n : Fin (t + 1) => d ^ n.val)).symm
def finitePortSelector {d : Nat} [DecidableEq V] (G : PortGraph V (Fin d))
(t : Nat) (v : V) : AddressSelector G t v :=
listSelector G t v (allAddresses d t) (mem_allAddresses d t)
end UniqueGames.Foundations.PCP.PoweringAddresses
end OAI
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