File size: 3,733 Bytes
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 | import OAI.Computability.PerfectCompleteness.Construction.LocalCompletionOutput
import OAI.Computability.PerfectCompleteness.Machines.LocalCompletionMachineLemmas
namespace OAI
section
namespace PerfectCompleteness.LocalCompletionSerializedOutput
open Turing
open UniqueGamesTheorem.Foundations.Complexity
open CompletionSoundness CompletionSoundness.LegalProjectionGame
open LocalCompletionFamily
open scoped Classical
noncomputable section
variable {m l r q : Nat} {L : Fin l → Type*} {R : Fin r → Type*}
[∀ x, Fintype (L x)]
(G : LegalProjectionGame (Fin m) (Fin l) (Fin r) L R)
(legalL : ∀ x, L x ↪ Fin (2 * q)) (legalR : ∀ y, R y ↪ Fin q)
(defaultL : ∀ x, L x) (hq : 0 < q)
(hsmall : ∀ e b, Fintype.card {a : L (G.left e) // G.projection e a = b} ≤ 2)
private theorem projection_eq_of_images {a b : ProjectionTable q}
(h : a.images = b.images) : a = b := by
cases a
cases b
cases h
rfl
def entry (e : Fin m)
(seed : Fin ((LocalCompletionGame.family G legalL legalR defaultL hq hsmall).size e)) :
Edge l r q :=
CompletionRounding.entries (LocalCompletionGame.family G legalL legalR defaultL hq hsmall)
(CompletionRounding.enumeration
(LocalCompletionGame.family G legalL legalR defaultL hq hsmall) ⟨e, seed⟩)
@[simp] theorem entry_left (e : Fin m)
(seed : Fin ((LocalCompletionGame.family G legalL legalR defaultL hq hsmall).size e)) :
(entry G legalL legalR defaultL hq hsmall e seed).left = G.left e := by
simp only [entry, CompletionRounding.entries, Equiv.symm_apply_apply]
@[simp] theorem entry_right (e : Fin m)
(seed : Fin ((LocalCompletionGame.family G legalL legalR defaultL hq hsmall).size e)) :
(entry G legalL legalR defaultL hq hsmall e seed).right = G.right e := by
simp only [entry, CompletionRounding.entries, Equiv.symm_apply_apply]
theorem entry_projection (e : Fin m)
(seed : Fin ((LocalCompletionGame.family G legalL legalR defaultL hq hsmall).size e)) :
(entry G legalL legalR defaultL hq hsmall e seed).projection =
LocalCompletionSerialization.projection
(LocalCompletionGame.result_correct G legalL legalR defaultL hq hsmall e) seed := by
apply projection_eq_of_images
let F := LocalCompletionGame.family G legalL legalR defaultL hq hsmall
have h := congrArg (fun es : CompletionRounding.CompletedOccurrence F =>
Vector.ofFn (F.map es.1 es.2))
((CompletionRounding.enumeration F).symm_apply_apply ⟨e, seed⟩)
exact h
def local_machine_output (e : Fin m) :
TM2OutputsInTime (LocalCompletionSerialization.computation hq).tm
(LocalCompletionMachine.inputBits (LocalCompletionGame.input G legalL legalR defaultL e))
(some (LocalCompletionSerialization.payloadBits
(LocalCompletionGame.result G legalL legalR defaultL hq e))) 1 :=
LocalCompletionSerialization.machine_output hq _
theorem local_machine_output_decodes (e : Fin m) :
decodeWords (LocalCompletionSerialization.payloadBits
(LocalCompletionGame.result G legalL legalR defaultL hq e)) >>=
LocalCompletionSerialization.parsePayload q =
some ((List.finRange
((LocalCompletionGame.family G legalL legalR defaultL hq hsmall).size e)).map
(fun seed => (entry G legalL legalR defaultL hq hsmall e seed).projection)) := by
rw [LocalCompletionSerialization.decode_payloadBits, Option.bind_eq_bind, Option.bind_some]
rw [LocalCompletionSerialization.parse_payload
(LocalCompletionGame.result_correct G legalL legalR defaultL hq hsmall e)]
congr 1
apply List.map_congr_left
intro seed _
exact (entry_projection G legalL legalR defaultL hq hsmall e seed).symm
end
end PerfectCompleteness.LocalCompletionSerializedOutput
end
end OAI
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