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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 | import OAI.Computability.PerfectCompleteness.Machines.BinaryParsingLemmas
import OAI.Computability.PerfectCompleteness.Machines.EncodingLemmas
import OAI.Computability.PerfectCompleteness.Model
import OAI.Computability.PerfectCompleteness.Reduction.Value
import OAI.Computability.UniqueGames.Machines.MachineFiniteAlphabet
namespace OAI
section
namespace PerfectCompleteness
open Turing UniqueGamesTheorem.Foundations.Complexity
namespace BinaryGapReduction
variable {δ : ℚ} (reduction : BinaryGapReduction δ)
theorem nonempty (input : List Bool) : (reduction.construct input).edges ≠ [] :=
(reduction.construct input).nonempty
theorem exactly_two (input : List Bool)
(edge : Fin (reduction.construct input).edges.length) (label : Fin reduction.alphabet) :
((List.finRange (2 * reduction.alphabet)).filter
(fun a => decide ((reduction.construct input).edges[edge].projection.images[a] = label))).length = 2 :=
(reduction.construct input).edges[edge].projection.exactlyTwo label
theorem malformed_soundness (input : List Bool)
(malformed : BinaryEncoding.decodeFormula input = none) :
(reduction.construct input).value ≤ (δ : ℝ) := by
apply reduction.soundness input
rintro ⟨formula, decoded, _⟩
rw [malformed] at decoded
cases decoded
end BinaryGapReduction
end PerfectCompleteness
end
end OAI
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