| import Mathlib |
| import OAI.Computability.VertexCover.PCP.AlphabetRetraction |
| import OAI.Computability.VertexCover.PCP.FiniteGraph |
| import OAI.Computability.VertexCover.PCP.AmplificationIteration |
| import OAI.Computability.VertexCover.PCP.RoundGap |
| import OAI.Computability.VertexCover.PCP.RoundSize |
| import OAI.Computability.VertexCover.Fourier.SourceNonempty |
|
|
| namespace OAI |
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| noncomputable section |
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|
| namespace UniqueGames.Foundations.PCP.PCPIteration |
|
|
| open FiniteGraph |
|
|
| abbrev Label := AlphabetRetraction.Label64 |
| abbrev Graph := Bundle Label |
| abbrev Addresses := AmplificationRound.Addresses |
|
|
| def step (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) (G : Graph) : Graph := by |
| classical |
| exact Bundle.ofGraph (AmplificationRound.graph addresses complete G.graph) |
|
|
| def run (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) : Nat → Graph → Graph := |
| AmplificationIteration.run (step addresses complete) |
|
|
| def initial (F : Target.Formula) : Graph := Bundle.ofGraph (AlphabetRetraction.initial64 F) |
|
|
| def iterationCount (F : Target.Formula) : Nat := AmplificationIteration.rounds (initial F).size |
|
|
| def output (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (F : Target.Formula) : Graph := run addresses complete (iterationCount F) (initial F) |
|
|
| def polynomialDegree : Nat := AmplificationIteration.rounds RoundSize.sizeFactor |
|
|
| def finalClauseGap : ℚ := 1 / (40960 * (FinalConstants.walkLength : ℚ)) |
|
|
| theorem finalClauseGap_positive : 0 < finalClauseGap := by |
| apply one_div_pos.mpr |
| exact mul_pos (by norm_num) (by exact_mod_cast FinalConstants.walkLength_positive) |
|
|
| theorem sizeFactor_le_power : RoundSize.sizeFactor ≤ 2 ^ polynomialDegree := |
| (AmplificationIteration.rounds_large RoundSize.sizeFactor).le |
|
|
| theorem initial_satisfiable_iff (F : Target.Formula) : |
| (initial F).Satisfiable ↔ F.Satisfiable := |
| AlphabetRetraction.initial64_satisfiable_iff F |
|
|
| theorem initial_size_positive (F : Target.Formula) : 0 < (initial F).size := |
| (initial F).size_positive |
|
|
| theorem initial_size_bound (F : Target.Formula) : |
| (initial F).size ≤ 10 * F.clauses.length + 2 := by |
| change Fintype.card (InitialGraph.CompactVertex F) + |
| Fintype.card (InitialGraph.CompactDart F) ≤ _ |
| rw [InitialGraph.build_dart_count] |
| have h := InitialGraph.build_vertex_bound F |
| omega |
|
|
| theorem step_completeness (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (G : Graph) (satisfied : G.Satisfiable) : (step addresses complete G).Satisfiable := |
| AmplificationRound.completeness addresses complete G.graph satisfied |
|
|
| theorem step_gap (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) (G : Graph) : |
| min (2 * G.gap) FinalConstants.cap ≤ (step addresses complete G).gap := |
| RoundGap.amplifies addresses complete G.graph |
|
|
| theorem step_size (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) (G : Graph) : |
| (step addresses complete G).size ≤ RoundSize.sizeFactor * G.size := |
| RoundSize.output_total_le G.graph |
|
|
| theorem run_completeness (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (n : Nat) (G : Graph) (satisfied : G.Satisfiable) : |
| (run addresses complete n G).Satisfiable := |
| AmplificationIteration.run_preserves (step addresses complete) Bundle.Satisfiable |
| (step_completeness addresses complete) n G satisfied |
|
|
| theorem run_gap (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (n : Nat) (G : Graph) : |
| min (2 ^ n * G.gap) FinalConstants.cap ≤ (run addresses complete n G).gap := |
| AmplificationIteration.run_gap (step addresses complete) Bundle.gap |
| FinalConstants.cap FinalConstants.cap_positive.le (step_gap addresses complete) n G |
|
|
| theorem run_size (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (n : Nat) (G : Graph) : |
| (run addresses complete n G).size ≤ RoundSize.sizeFactor ^ n * G.size := |
| AmplificationIteration.run_size (step addresses complete) Bundle.size RoundSize.sizeFactor |
| (step_size addresses complete) n G |
|
|
| theorem output_completeness (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (F : Target.Formula) (satisfied : F.Satisfiable) : |
| (output addresses complete F).Satisfiable := |
| run_completeness addresses complete (iterationCount F) (initial F) |
| ((initial_satisfiable_iff F).mpr satisfied) |
|
|
| theorem output_gap (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (F : Target.Formula) (unsat : ¬ F.Satisfiable) : |
| FinalConstants.cap ≤ (output addresses complete F).gap := by |
| have hinitial : ¬ (initial F).Satisfiable := |
| fun h => unsat ((initial_satisfiable_iff F).mp h) |
| exact AmplificationIteration.run_reaches_cap (step addresses complete) Bundle.gap |
| FinalConstants.cap FinalConstants.cap_positive.le FinalConstants.cap_le_one |
| (step_gap addresses complete) (initial F).size (initial F) |
| (initial F).gap_nonnegative ((initial F).one_le_size_mul_gap hinitial) |
|
|
| theorem output_satisfiable_iff (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (F : Target.Formula) : (output addresses complete F).Satisfiable ↔ F.Satisfiable := by |
| constructor |
| · intro satisfied |
| by_contra unsat |
| have hgap := output_gap addresses complete F unsat |
| rw [(output addresses complete F).gap_eq_zero_iff.mpr satisfied] at hgap |
| exact (not_le_of_gt FinalConstants.cap_positive) hgap |
| · exact output_completeness addresses complete F |
|
|
| theorem output_size_polynomial (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (F : Target.Formula) : |
| (output addresses complete F).size ≤ |
| (2 * (initial F).size) ^ polynomialDegree * (initial F).size := |
| AmplificationIteration.run_size_polynomial (step addresses complete) Bundle.size |
| (initial_size_positive F) sizeFactor_le_power (step_size addresses complete) (initial F) |
|
|
| theorem output_size_clause_bound (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (F : Target.Formula) : |
| (output addresses complete F).size ≤ |
| (2 * (10 * F.clauses.length + 2)) ^ polynomialDegree * (10 * F.clauses.length + 2) := by |
| have h := initial_size_bound F |
| exact (output_size_polynomial addresses complete F).trans |
| (Nat.mul_le_mul (Nat.pow_le_pow_left (Nat.mul_le_mul_left 2 h) _) h) |
|
|
| theorem output_count_gap (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (F : Target.Formula) (unsat : ¬ F.Satisfiable) |
| (labeling : (output addresses complete F).Vertex → Label) : |
| Fintype.card (output addresses complete F).Dart ≤ |
| FinalConstants.walkLength * (output addresses complete F).rejectionCount labeling := by |
| have h := ((output addresses complete F).le_gap_iff FinalConstants.cap).mp |
| (output_gap addresses complete F unsat) labeling |
| have ht : (0 : ℝ) < FinalConstants.walkLength := by |
| exact_mod_cast FinalConstants.walkLength_positive |
| have h' : (Fintype.card (output addresses complete F).Dart : ℝ) / |
| FinalConstants.walkLength ≤ |
| ((output addresses complete F).rejectionCount labeling : ℝ) := by |
| simpa only [FinalConstants.cap, one_div_mul_eq_div] using h |
| have h'' := (div_le_iff₀ ht).mp h' |
| exact_mod_cast (show (Fintype.card (output addresses complete F).Dart : ℝ) ≤ |
| (FinalConstants.walkLength : ℝ) * |
| ((output addresses complete F).rejectionCount labeling : ℝ) by |
| simpa only [mul_comm] using h'') |
|
|
| section FinalCNF |
|
|
| variable (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (F : Target.Formula) {n m : Nat} |
| (vertices : (output addresses complete F).Vertex ≃ Fin n) |
| (darts : (output addresses complete F).Dart ≃ Fin m) |
|
|
| def finalCNF : Target.Formula := |
| FinalBooleanVerifier.cnf (output addresses complete F).graph vertices darts |
|
|
| theorem finalCNF_satisfiable_iff : |
| (finalCNF addresses complete F vertices darts).Satisfiable ↔ F.Satisfiable := |
| (FinalBooleanVerifier.cnf_satisfiable_iff _ vertices darts).trans |
| (output_satisfiable_iff addresses complete F) |
|
|
| theorem finalCNF_clause_count : |
| (finalCNF addresses complete F vertices darts).clauses.length = m * 40960 := |
| FinalBooleanVerifier.cnf_clause_count _ vertices darts |
|
|
| theorem finalCNF_variable_count : |
| (finalCNF addresses complete F vertices darts).«variables» = n * 6 + m * 36864 := |
| FinalBooleanVerifier.cnf_variable_count _ vertices darts |
|
|
| theorem finalCNF_nonempty : (finalCNF addresses complete F vertices darts).clauses ≠ [] := by |
| have hm : Fintype.card (output addresses complete F).Dart = m := by |
| simpa only [Fintype.card_fin] using Fintype.card_congr darts |
| have hpositive : 0 < m := hm ▸ (Fintype.card_pos : |
| 0 < Fintype.card (output addresses complete F).Dart) |
| exact FinalBooleanVerifier.cnf_nonempty _ vertices darts hpositive |
|
|
| theorem finalCNF_gap (unsat : ¬ F.Satisfiable) |
| (assignment : Fin (finalCNF addresses complete F vertices darts).«variables» → Bool) : |
| (finalCNF addresses complete F vertices darts).clauses.length ≤ |
| (FinalConstants.walkLength * 40960) * |
| NameCompaction.failedCount (finalCNF addresses complete F vertices darts) assignment := by |
| have source (labeling : (output addresses complete F).Vertex → Label) : |
| 1 * Fintype.card (output addresses complete F).Dart ≤ |
| FinalConstants.walkLength * (output addresses complete F).graph.rejectionCount labeling := by |
| simpa only [one_mul, Bundle.rejectionCount] using output_count_gap addresses complete F unsat labeling |
| simpa only [one_mul, finalCNF] using FinalBooleanVerifier.cnf_gap |
| (output addresses complete F).graph vertices darts 1 FinalConstants.walkLength source assignment |
|
|
| theorem finalCNF_clauseGap (unsat : ¬ F.Satisfiable) : |
| Hastad.SourceGap.ClauseGap (finalCNF addresses complete F vertices darts) finalClauseGap := |
| Hastad.SourceNonempty.cnf_clauseGap_unit (output addresses complete F).graph |
| vertices darts FinalConstants.walkLength FinalConstants.walkLength_positive |
| (output_count_gap addresses complete F unsat) |
|
|
| theorem finalCNF_total_size : |
| (finalCNF addresses complete F vertices darts).«variables» + |
| (finalCNF addresses complete F vertices darts).clauses.length ≤ |
| 77824 * (output addresses complete F).size := by |
| have hn : Fintype.card (output addresses complete F).Vertex = n := by |
| simpa only [Fintype.card_fin] using Fintype.card_congr vertices |
| have hm : Fintype.card (output addresses complete F).Dart = m := by |
| simpa only [Fintype.card_fin] using Fintype.card_congr darts |
| have hsize : (output addresses complete F).size = n + m := congrArg₂ Nat.add hn hm |
| calc |
| _ = n * 6 + m * 36864 + m * 40960 := |
| congrArg₂ Nat.add (finalCNF_variable_count addresses complete F vertices darts) |
| (finalCNF_clause_count addresses complete F vertices darts) |
| _ ≤ 77824 * (n + m) := by omega |
| _ = 77824 * (output addresses complete F).size := |
| congrArg (fun k => 77824 * k) hsize.symm |
|
|
| theorem finalCNF_polynomial_size : |
| (finalCNF addresses complete F vertices darts).«variables» + |
| (finalCNF addresses complete F vertices darts).clauses.length ≤ |
| 77824 * ((2 * (10 * F.clauses.length + 2)) ^ polynomialDegree * |
| (10 * F.clauses.length + 2)) := |
| (finalCNF_total_size addresses complete F vertices darts).trans |
| (Nat.mul_le_mul_left _ (output_size_clause_bound addresses complete F)) |
|
|
| end FinalCNF |
|
|
| end UniqueGames.Foundations.PCP.PCPIteration |
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| end |
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|
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| end OAI |
|
|
