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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 | import OAI.Computability.PerfectCompleteness.Machines.FixedBinaryGapReduction
import OAI.Computability.PerfectCompleteness.Reduction.FixedPreliminarySoundness
import OAI.Computability.PerfectCompleteness.Reduction.PreliminaryOutputLemmas
namespace OAI
section
namespace PerfectCompleteness.Theorem11
noncomputable section
theorem preliminary_value_le {δ : ℚ} {hδ : 0 < δ}
(parameters : FixedParameters.Parameters δ hδ) (input : List Bool)
(unsat : ¬BinaryLanguage.language input) :
FixedPreliminaryGame.value parameters input ≤ InitialParameters.epsilon δ :=
FixedPreliminaryValue.value_le_of_no_large_strategy parameters input
(InitialParameters.epsilon δ) (fun strategy hsuccess =>
FixedPreliminarySoundness.no_large_strategy parameters input unsat strategy hsuccess)
theorem exists_reduction (δ : ℚ) (positive : 0 < δ) (less_than_one : δ < 1) :
Nonempty (BinaryGapReduction δ) := by
obtain ⟨parameters⟩ := FixedParameters.exists_parameters positive less_than_one
exact ⟨FixedBinaryGapReduction.ofPreliminarySoundness parameters
(preliminary_value_le parameters)⟩
end
end PerfectCompleteness.Theorem11
end
end OAI
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