| 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 |
|
|
