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import OAI.Computability.VertexCover.PCP.FinalBooleanVerifier
import OAI.Computability.VertexCover.Fourier.SourceGap
namespace OAI
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namespace UniqueGames.Foundations.Hastad.SourceNonempty
open Target PCP
theorem real_clause_gap_of_count (F : Formula) (a b : ℕ) (hb : 0 < b)
(hgap : ∀ assignment : Fin F.«variables» → Bool,
a * F.clauses.length ≤ b * NameCompaction.failedCount F assignment)
(assignment : Fin F.«variables» → Bool) :
(a : ℝ) / (b : ℝ) * F.clauses.length ≤
(failureCount F assignment (allIndices F) : ℝ) := by
rw [NameCompaction.verifier_failureCount]
have hreal : (a : ℝ) * F.clauses.length ≤
(b : ℝ) * (NameCompaction.failedCount F assignment : ℝ) := by
exact_mod_cast hgap assignment
rw [div_mul_eq_mul_div]
apply (div_le_iff₀ (Nat.cast_pos.mpr hb)).mpr
simpa only [mul_comm] using hreal
theorem clauseGap_of_count (F : Formula) (a b : ℕ) (hne : F.clauses ≠ [])
(hb : 0 < b)
(hgap : ∀ assignment : Fin F.«variables» → Bool,
a * F.clauses.length ≤ b * NameCompaction.failedCount F assignment) :
SourceGap.ClauseGap F ((a : ℚ) / (b : ℚ)) := by
refine ⟨hne, fun assignment => ?_⟩
simpa only [Rat.cast_div, Rat.cast_natCast] using
real_clause_gap_of_count F a b hb hgap assignment
variable {V E : Type*} {n m : ℕ}
theorem cnf_nonempty_of_nonempty_darts [Fintype E] [Nonempty E]
(G : ConstraintGraph V E FinalBooleanVerifier.Label)
(vertices : V ≃ Fin n) (edges : E ≃ Fin m) :
(FinalBooleanVerifier.cnf G vertices edges).clauses ≠ [] := by
have hcard : Fintype.card E = m := by
simpa only [Fintype.card_fin] using Fintype.card_congr edges
have hm : 0 < m := hcard ▸ Fintype.card_pos
exact FinalBooleanVerifier.cnf_nonempty G vertices edges hm
theorem cnf_clauseGap [Fintype E] [Nonempty E]
(G : ConstraintGraph V E FinalBooleanVerifier.Label)
(vertices : V ≃ Fin n) (edges : E ≃ Fin m) (a b : ℕ) (hb : 0 < b)
(hgap : ∀ labeling : V → FinalBooleanVerifier.Label,
a * Fintype.card E ≤ b * G.rejectionCount labeling) :
SourceGap.ClauseGap (FinalBooleanVerifier.cnf G vertices edges)
((a : ℚ) / ((b : ℚ) * 40960)) := by
have h := clauseGap_of_count (FinalBooleanVerifier.cnf G vertices edges) a (b * 40960)
(cnf_nonempty_of_nonempty_darts G vertices edges)
(Nat.mul_pos hb (by decide)) (FinalBooleanVerifier.cnf_gap G vertices edges a b hgap)
simpa only [Nat.cast_mul, Nat.cast_ofNat] using h
theorem cnf_clauseGap_unit [Fintype E] [Nonempty E]
(G : ConstraintGraph V E FinalBooleanVerifier.Label)
(vertices : V ≃ Fin n) (edges : E ≃ Fin m) (walkLength : ℕ)
(hwalk : 0 < walkLength)
(hgap : ∀ labeling : V → FinalBooleanVerifier.Label,
Fintype.card E ≤ walkLength * G.rejectionCount labeling) :
SourceGap.ClauseGap (FinalBooleanVerifier.cnf G vertices edges)
(1 / (40960 * (walkLength : ℚ))) := by
have h := cnf_clauseGap G vertices edges 1 walkLength hwalk
(by simpa only [one_mul] using hgap)
simpa only [Nat.cast_one, mul_comm] using h
end UniqueGames.Foundations.Hastad.SourceNonempty
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
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end
end
end OAI
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