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92416a2 | 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 | import Mathlib
import OAI.Computability.VertexCover.PCP.AssignmentTester
import OAI.Computability.VertexCover.PCP.CodeComposition
namespace OAI
noncomputable section
namespace UniqueGames.Foundations.PCP.AlphabetReduction
open UniqueGames.Foundations.Hastad
open CodeComposition
variable {A : Type*} [Fintype A] [DecidableEq A] [Nonempty A]
abbrev LegalPair (accepts : A → A → Bool) :=
{p : A × A // accepts p.1 p.2 = true}
abbrev InputCoordinate (A : Type*) := Bool × Cube A
def pairEncoding (accepts : A → A → Bool) (p : LegalPair accepts) :
InputCoordinate A → Bool :=
pairWord (codeword p.1.1) (codeword p.1.2)
def edgeTesterReject (accepts : A → A → Bool) (left right : Cube A → Bool)
(F : Cube (LegalPair accepts) → Bool) : ℝ :=
AssignmentTester.explicitTesterReject (pairEncoding accepts) (pairWord left right) F
theorem rejected_edge_tester_bound (accepts : A → A → Bool)
(left right : Cube A → Bool)
(F : Cube (LegalPair accepts) → Bool)
(hbad : accepts (nearest left) (nearest right) = false) :
1 / 2048 ≤ edgeTesterReject accepts left right F := by
have hfar (p : LegalPair accepts) :
(1 / 8 : ℝ) ≤ AssignmentTester.distance (pairWord left right) (pairEncoding accepts p) := by
have h := rejected_pair_far accepts left right hbad p.1.1 p.1.2 p.2
convert h using 1
rfl
have h := AssignmentTester.explicit_proximity_soundness (pairEncoding accepts)
(pairWord left right) F (δ := 1 / 8) (by norm_num) (by norm_num) hfar
norm_num at h
exact h
omit [Nonempty A] in
theorem honest_edge_tester (accepts : A → A → Bool) (a b : A)
(hab : accepts a b = true) :
edgeTesterReject accepts (codeword a) (codeword b)
(fun f => f ⟨(a, b), hab⟩) = 0 := by
unfold edgeTesterReject
rw [AssignmentTester.explicitTesterReject_eq]
exact AssignmentTester.tester_perfect_completeness
(pairEncoding accepts) ⟨(a, b), hab⟩
end UniqueGames.Foundations.PCP.AlphabetReduction
end
end OAI
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