| import Mathlib |
| 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 |
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|
| open Target Complexity |
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| 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) |
|
|
| @ theorem variable_count (table : GraphTables.Table) : |
| (output table).«variables» = table.vertices * 6 + table.darts * 36864 := |
| FinalBooleanVerifier.cnf_variable_count _ (Equiv.refl _) (Equiv.refl _) |
|
|
| @ 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 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 |
| 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 |
|
|
| 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 |
| omega |
| have hc : (output table).clauses.length ≤ 40960 * (GraphTables.tableBits table).length := by |
| rw |
| 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 |
| exact (formulaBits_length_le (output table)).trans (Nat.add_le_add hsum hproduct) |
|
|
| end UniqueGames.Foundations.PCP.FinalTableFormula |
| end |
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| end |
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| end OAI |
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