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import OAI.Computability.VertexCover.Encoding.FinalCNFPattern
import OAI.Computability.VertexCover.PCP.GraphTableComplexity
import OAI.Computability.VertexCover.PCP.PCPIteration
namespace OAI
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namespace UniqueGames.Foundations.PCP.FinalTableFormula
open Target Complexity
def output (table : GraphTables.Table) : Formula :=
FinalBooleanVerifier.cnf (FinalCNFPattern.tableGraph table) (Equiv.refl _) (Equiv.refl _)
theorem satisfiable_iff (table : GraphTables.Table) :
(output table).Satisfiable ↔ (GraphTables.semantics table).Satisfiable :=
(FinalBooleanVerifier.cnf_satisfiable_iff (FinalCNFPattern.tableGraph table)
(Equiv.refl _) (Equiv.refl _)).trans
((GraphTables.semantics table).satisfiable_reindex (Equiv.refl _) (Equiv.refl _)
FinalCNFPattern.labelEquiv.symm)
@[simp] theorem variable_count (table : GraphTables.Table) :
(output table).«variables» = table.vertices * 6 + table.darts * 36864 :=
FinalBooleanVerifier.cnf_variable_count _ (Equiv.refl _) (Equiv.refl _)
@[simp] theorem clause_count (table : GraphTables.Table) :
(output table).clauses.length = table.darts * 40960 :=
FinalBooleanVerifier.cnf_clause_count _ (Equiv.refl _) (Equiv.refl _)
theorem nonempty (table : GraphTables.Table) (hd : 0 < table.darts) :
(output table).clauses ≠ [] :=
FinalBooleanVerifier.cnf_nonempty _ (Equiv.refl _) (Equiv.refl _) hd
theorem tableGraph_rejectionCount (table : GraphTables.Table)
(labeling : Fin table.vertices → FinalBooleanVerifier.Label) :
(FinalCNFPattern.tableGraph table).rejectionCount labeling =
(GraphTables.semantics table).rejectionCount
(fun v => FinalCNFPattern.labelEquiv (labeling v)) := by
have h := (GraphTables.semantics table).reindex_rejectionCount
(Equiv.refl _) (Equiv.refl _) FinalCNFPattern.labelEquiv.symm
(fun v => FinalCNFPattern.labelEquiv (labeling v))
simpa [FinalCNFPattern.tableGraph, ConstraintGraph.labelingEquiv] using h
theorem tableGraph_count_lower (table : GraphTables.Table) (walkLength : Nat)
(hgap : ∀ labeling : Fin table.vertices → GraphTables.Label,
table.darts ≤ walkLength * (GraphTables.semantics table).rejectionCount labeling)
(labeling : Fin table.vertices → FinalBooleanVerifier.Label) :
Fintype.card (Fin table.darts) ≤
walkLength * (FinalCNFPattern.tableGraph table).rejectionCount labeling := by
rw [Fintype.card_fin, tableGraph_rejectionCount]
exact hgap _
theorem clauseGap (table : GraphTables.Table) (hd : 0 < table.darts)
(hgap : ∀ labeling : Fin table.vertices → GraphTables.Label,
table.darts ≤ FinalConstants.walkLength *
(GraphTables.semantics table).rejectionCount labeling) :
Hastad.SourceGap.ClauseGap (output table) PCPIteration.finalClauseGap := by
let : Nonempty (Fin table.darts) := ⟨⟨0, hd⟩⟩
exact Hastad.SourceNonempty.cnf_clauseGap_unit (FinalCNFPattern.tableGraph table)
(Equiv.refl _) (Equiv.refl _) FinalConstants.walkLength
FinalConstants.walkLength_positive (tableGraph_count_lower table _ hgap)
theorem total_size_le (table : GraphTables.Table) :
(output table).«variables» + (output table).clauses.length ≤
77824 * (table.vertices + table.darts) := by
have hv := variable_count table
have hc := clause_count table
omega
noncomputable def sizePolynomial : Polynomial Nat :=
Polynomial.C 77824 * Polynomial.X + Polynomial.C 2 +
(Polynomial.C 40960 * Polynomial.X) *
(Polynomial.C 3 * (Polynomial.C 36864 * Polynomial.X + Polynomial.C 2))
theorem sizePolynomial_eval (N : Nat) :
sizePolynomial.eval N = 77824 * N + 2 + (40960 * N) * (3 * (36864 * N + 2)) := by
simp [sizePolynomial]
theorem formulaBits_length_le_polynomial (table : GraphTables.Table) :
(formulaBits (output table)).length ≤
sizePolynomial.eval (GraphTables.tableBits table).length := by
have hinput := GraphTableComplexity.size_add_two_le_bits table
have hv : (output table).«variables» ≤ 36864 * (GraphTables.tableBits table).length := by
rw [variable_count]
omega
have hc : (output table).clauses.length ≤ 40960 * (GraphTables.tableBits table).length := by
rw [clause_count]
omega
have hsum : (output table).«variables» + (output table).clauses.length + 2 ≤
77824 * (GraphTables.tableBits table).length + 2 := by omega
have hproduct := Nat.mul_le_mul hc (Nat.mul_le_mul_left 3 (Nat.add_le_add_right hv 2))
rw [sizePolynomial_eval]
exact (formulaBits_length_le (output table)).trans (Nat.add_le_add hsum hproduct)
end UniqueGames.Foundations.PCP.FinalTableFormula
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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