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760f90e | 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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | import OAI.Computability.UniqueGames.Machines.FinalCNFPattern
import OAI.Computability.UniqueGames.PCP.InputLemmas
import OAI.Computability.UniqueGames.PCP.SourceGapLemmas
namespace OAI
section
/-! The exact table-to-formula converter used by the final CNF machine.
Its semantic count premise is explicit; amplification supplies that premise
elsewhere. The converter chooses no vertex or dart numbering. -/
namespace UniqueGamesTheorem.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
/-- Boolean labels and numeric labels have exactly the same rejected darts. -/
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
/-- A uniform numeric-label count bound transfers to the machine's Boolean-label graph. -/
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
/-- A fixed quadratic in the full serialized input length. -/
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 UniqueGamesTheorem.Foundations.PCP.FinalTableFormula
end
end OAI
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