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2f887db | 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 | import OAI.Computability.PerfectCompleteness.Machines.CanonicalKeyShapeMachineLemmas
import OAI.Computability.PerfectCompleteness.Reduction.ExactPreliminaryTarget
namespace OAI
section
namespace PerfectCompleteness.ExactTargetOnlineEncoding
open UniqueGamesTheorem.Foundations.Complexity
open scoped Classical
noncomputable section
private theorem encodeWords_flatMap {α : Type*} (xs : List α) (words : α → List Nat) :
encodeWords (xs.flatMap words) = xs.flatMap (fun x => encodeWords (words x)) := by
induction xs with
| nil => rfl
| cons x xs ih => simp only [List.flatMap_cons, encodeWords_append, ih]
variable {branch : Nat → Nat} {n t v m : Nat} [NeZero m]
(clauses : Fin m → SourceClause.NormalizedClause v) (rows repeats : Nat → Nat)
(hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k)
(hrows : ∀ k, 0 < rows (k + 1)) (δ : ℚ)
local notation "F" => FinitePreliminaryCompletion.blockFamily clauses branch n t rows repeats
local notation "C" => CompletedSignedLaw.Completed (t := t) clauses rows repeats hn hbranch hrows δ
local notation "order" => CompletedVisitOrder.completedOrder
(t := t) clauses rows repeats hn hbranch hrows δ
local notation "target" => ExactPreliminaryTarget.construct
(t := t) clauses rows repeats hn hbranch hrows δ
local notation "edge" => ExactPreliminaryTarget.entry
(t := t) clauses rows repeats hn hbranch hrows δ
def leftKeys (occurrences : List C) := occurrences.map (fun e => CanonicalGame.leftKey F e.1)
def rightKeys (occurrences : List C) := occurrences.map (fun e => CanonicalGame.rightKey F e.1)
theorem left_online (earlier suffix : List C) (e : C) (split : order = earlier ++ e :: suffix) :
CanonicalOnlineNames.onlineID (leftKeys (t := t) clauses rows repeats hn hbranch hrows δ earlier)
(CanonicalGame.leftKey F e.1) = (edge e).left.val := by
change _ = (BinaryNameSearch.payloads (CanonicalVertexNames.tokens
(((order).map Sigma.fst).map (CanonicalGame.leftKey F)))).idxOf
(CanonicalVertexNames.payload (CanonicalGame.leftKey F e.1))
rw [List.map_map, split, List.map_append, List.map_cons]
exact CanonicalOnlineNames.onlineID_eq_global _ _ _
theorem right_online (earlier suffix : List C) (e : C) (split : order = earlier ++ e :: suffix) :
CanonicalOnlineNames.onlineID (rightKeys (t := t) clauses rows repeats hn hbranch hrows δ earlier)
(CanonicalGame.rightKey F e.1) = (edge e).right.val := by
change _ = (BinaryNameSearch.payloads (CanonicalVertexNames.tokens
(((order).map Sigma.fst).map (CanonicalGame.rightKey F)))).idxOf
(CanonicalVertexNames.payload (CanonicalGame.rightKey F e.1))
rw [List.map_map, split, List.map_append, List.map_cons]
exact CanonicalOnlineNames.onlineID_eq_global _ _ _
def onlineWords (earlier : List C) (e : C) : List Nat :=
[CanonicalOnlineNames.onlineID
(leftKeys (t := t) clauses rows repeats hn hbranch hrows δ earlier)
(CanonicalGame.leftKey F e.1),
CanonicalOnlineNames.onlineID
(rightKeys (t := t) clauses rows repeats hn hbranch hrows δ earlier)
(CanonicalGame.rightKey F e.1)] ++ Encoding.tableWords (edge e).projection
theorem onlineWords_eq (earlier suffix : List C) (e : C)
(split : order = earlier ++ e :: suffix) :
onlineWords (t := t) clauses rows repeats hn hbranch hrows δ earlier e =
Encoding.edgeWords (edge e) := by
unfold onlineWords Encoding.edgeWords
rw [left_online (t := t) clauses rows repeats hn hbranch hrows δ earlier suffix e split,
right_online (t := t) clauses rows repeats hn hbranch hrows δ earlier suffix e split]
theorem reverse_edge_fields {l r q : Nat} (e : Edge l r q) (tail : List Bool) :
(encodeWords (Encoding.tableWords e.projection)).reverse ++
((encodeWord e.right.val).reverse ++ ((encodeWord e.left.val).reverse ++ tail)) =
(encodeWords (Encoding.edgeWords e)).reverse ++ tail := by
simp only [Encoding.edgeWords, encodeWords_append, encodeWords,
List.append_nil, List.reverse_append, List.append_assoc]
theorem target_gameBits :
Encoding.gameBits target =
encodeWords [(order).length, (order).length,
FinitePreliminaryCompletion.alphabet branch n t δ, (order).length] ++
(order).flatMap (fun e => encodeWords (Encoding.edgeWords (edge e))) := by
change encodeWords
([(CompletedVisitOrder.rawOrder (t := t) clauses rows repeats hn hbranch hrows δ).length,
(CompletedVisitOrder.rawOrder (t := t) clauses rows repeats hn hbranch hrows δ).length,
FinitePreliminaryCompletion.alphabet branch n t δ, ((order).map edge).length] ++
((order).map edge).flatMap Encoding.edgeWords) = _
rw [encodeWords_append, encodeWords_flatMap]
apply congrArg₂ List.append
· apply congrArg encodeWords
simp only [List.length_map]
simp only [CompletedVisitOrder.rawOrder, List.length_map]
· simp only [List.flatMap_map]
end
end PerfectCompleteness.ExactTargetOnlineEncoding
end
end OAI
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