openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Reduction /FixedTargetValue.lean
| import OAI.Computability.PerfectCompleteness.Reduction.FixedTarget | |
| namespace OAI | |
| namespace PerfectCompleteness.FixedTargetValue | |
| open UniqueGamesTheorem.Foundations Target | |
| open FixedParameters FixedRows FixedPreliminaryGame | |
| open scoped Classical | |
| noncomputable section | |
| private theorem packed_value_le (formula : Formula) [NeZero formula.clauses.length] | |
| (nonempty : formula.clauses ≠ []) | |
| (distinct : ∀ clause ∈ formula.clauses, ∀ i j : Fin 3, | |
| clause[i].variableIndex = clause[j].variableIndex → i = j) | |
| (branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat) | |
| (hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k) | |
| (hrows : ∀ k, 0 < rows (k + 1)) (δ : ℚ) (hδ : 0 < δ) : | |
| (NormalizedTarget.construct (branch := branch) (n := n) (t := t) | |
| rows repeats hn hbranch hrows δ ⟨formula, nonempty, distinct⟩).value ≤ | |
| PreliminaryOutput.value | |
| (fun index : Fin formula.clauses.length => | |
| ({ clause := PCP.clauseAt formula index | |
| distinct := fun i j same => distinct _ (List.getElem_mem index.isLt) i j same } : | |
| SourceClause.NormalizedClause formula.variables)) | |
| branch n t rows repeats hn hbranch hrows + (δ : ℝ) / 3 := by | |
| have clauses_eq : | |
| NormalizedSourceInput.clauses ⟨formula, nonempty, distinct⟩ = | |
| (fun index : Fin formula.clauses.length => | |
| ({ clause := PCP.clauseAt formula index | |
| distinct := fun i j same => distinct _ (List.getElem_mem index.isLt) i j same } : | |
| SourceClause.NormalizedClause formula.variables)) := by | |
| funext index | |
| rfl | |
| have native : | |
| (NormalizedTarget.construct (branch := branch) (n := n) (t := t) | |
| rows repeats hn hbranch hrows δ ⟨formula, nonempty, distinct⟩).value ≤ | |
| PreliminaryOutput.value (NormalizedSourceInput.clauses ⟨formula, nonempty, distinct⟩) | |
| branch n t rows repeats hn hbranch hrows + (δ : ℝ) / 3 := by | |
| simpa only [PreliminaryOutput.value] using | |
| (NormalizedTarget.value_le (branch := branch) (n := n) (t := t) | |
| rows repeats hn hbranch hrows δ ⟨formula, nonempty, distinct⟩ hδ) | |
| rw [clauses_eq] at native | |
| exact native | |
| variable {δ : ℚ} {hδ : 0 < δ} (p : Parameters δ hδ) | |
| theorem value_le (input : List Bool) : | |
| (FixedTarget.construct p input).value ≤ | |
| FixedPreliminaryGame.value p input + (δ : ℝ) / 3 := | |
| packed_value_le (PCPSource.normalizedFormula (BinaryLanguage.totalRename input)) | |
| (PCPSource.normalizedFormula_nonempty (BinaryLanguage.totalRename input)) | |
| (PCPSource.normalizedFormula_distinct (BinaryLanguage.totalRename input)) | |
| (branch p) p.plan.depth (sourceLength p.plan hδ) (rows p.plan) (repeats p.plan) | |
| (depth_pos p) (fun k _ => branch_pos p k) (fun k => rows_pos p (k + 1)) δ hδ | |
| end | |
| end PerfectCompleteness.FixedTargetValue | |
| end OAI | |