openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Reduction /NormalizedTarget.lean
| import OAI.Computability.PerfectCompleteness.Construction.SourceQuestionOrderLemmas | |
| import OAI.Computability.PerfectCompleteness.Machines.KeyMetadataMachine | |
| import OAI.Computability.PerfectCompleteness.Machines.SignedTupleCleanupMachineLemmas | |
| import OAI.Computability.PerfectCompleteness.Machines.SourceOccurrenceProducerLemmas | |
| import OAI.Computability.PerfectCompleteness.Reduction.ExactPreliminaryTarget | |
| namespace OAI | |
| namespace PerfectCompleteness.NormalizedTarget | |
| open UniqueGamesTheorem.Foundations.Complexity | |
| open scoped Classical | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n t : Nat} | |
| def inputBits (input : NormalizedSourceInput.Input) : List Bool := | |
| NormalizedSourceInput.bits input | |
| @[simp] theorem inputBits_eq (input : NormalizedSourceInput.Input) : | |
| inputBits input = SourceOccurrenceEncoding.bits input.formula := rfl | |
| theorem decode_inputBits (input : NormalizedSourceInput.Input) : | |
| SourceOccurrenceEncoding.decode (inputBits input) = some input.formula := | |
| NormalizedSourceInput.decode_bits input | |
| theorem inputBits_injective : Function.Injective inputBits := | |
| NormalizedSourceInput.bits_injective | |
| abbrev SourceTuple (input : NormalizedSourceInput.Input) := | |
| Fin (TreeCanonical.locationCount branch n t) → | |
| Fin (NormalizedSourceInput.clauseCount input) | |
| def tupleQuestion (input : NormalizedSourceInput.Input) | |
| (indices : SourceTuple (branch := branch) (n := n) (t := t) input) : | |
| PreliminarySampler.Questions branch n t (NormalizedSourceInput.clauseCount input) := | |
| SourceQuestionOrder.tupleToQuestion indices | |
| @[simp] theorem tupleQuestion_digits (input : NormalizedSourceInput.Input) | |
| (indices : SourceTuple (branch := branch) (n := n) (t := t) input) : | |
| SourceQuestionOrder.questionToTuple (tupleQuestion input indices) = indices := | |
| SourceQuestionOrder.questionToTuple_tupleToQuestion indices | |
| def sourceShapes (input : NormalizedSourceInput.Input) | |
| (indices : SourceTuple (branch := branch) (n := n) (t := t) input) : | |
| Fin (TreeCanonical.locationCount branch n t) → CanonicalKeyShape.Shape := | |
| fun position => SourceClauseReaderMachine.clauseShape | |
| (SourceTupleData.selectedClause input.formula indices position) | |
| def parsedVariableIDs (input : NormalizedSourceInput.Input) | |
| (indices : SourceTuple (branch := branch) (n := n) (t := t) input) | |
| (position : Fin (TreeCanonical.locationCount branch n t)) (slot : Fin 3) : Nat := | |
| (SourceTupleData.selectedClause input.formula indices position)[slot].variableIndex.val | |
| @[simp] theorem parsedVariableIDs_eq (input : NormalizedSourceInput.Input) | |
| (indices : SourceTuple (branch := branch) (n := n) (t := t) input) | |
| (position : Fin (TreeCanonical.locationCount branch n t)) (slot : Fin 3) : | |
| parsedVariableIDs input indices position slot = | |
| ((NormalizedSourceInput.clauses input (indices position)).variable slot).val := rfl | |
| theorem sourceSigns_eq (input : NormalizedSourceInput.Input) | |
| (indices : SourceTuple (branch := branch) (n := n) (t := t) input) : | |
| SourceTupleScheduleMachine.sourceSigns input.formula indices = | |
| MetadataFreeSampler.sourceSigns (NormalizedSourceInput.clauses input) | |
| (tupleQuestion input indices) := by | |
| funext leaf coordinate | |
| rfl | |
| theorem signTuple_eq (input : NormalizedSourceInput.Input) | |
| (indices : SourceTuple (branch := branch) (n := n) (t := t) input) : | |
| SourceTupleScheduleMachine.signTuple (sourceShapes input indices) = | |
| MetadataFreeSampler.sourceSigns (NormalizedSourceInput.clauses input) | |
| (tupleQuestion input indices) := | |
| sourceSigns_eq input indices | |
| theorem numberedSlots_eq (input : NormalizedSourceInput.Input) | |
| (indices : SourceTuple (branch := branch) (n := n) (t := t) input) | |
| (position : Fin (TreeCanonical.locationCount branch n t)) : | |
| TreeCanonical.numberedSlots | |
| (MetadataFreeSampler.sourceSlots (NormalizedSourceInput.clauses input) | |
| (tupleQuestion input indices)) position = | |
| KeyMetadataMachine.actualSlot (sourceShapes input indices position) | |
| (indices position).val (parsedVariableIDs input indices position) := by | |
| change SourceKeys.slot (NormalizedSourceInput.clauses input) | |
| (.clause (SourceQuestionOrder.questionToTuple | |
| (SourceQuestionOrder.tupleToQuestion | |
| (branch := branch) (n := n) (t := t) indices) position)) = _ | |
| simp only [SourceQuestionOrder.questionToTuple_tupleToQuestion]; rfl | |
| theorem sourceTable_eq (input : NormalizedSourceInput.Input) | |
| (indices : SourceTuple (branch := branch) (n := n) (t := t) input) | |
| (prepared : Fin (TreeCanonical.locationCount branch n t) → Bool) : | |
| SourceTupleData.stacks input.formula indices prepared .table = inputBits input := rfl | |
| theorem digit_counter (input : NormalizedSourceInput.Input) | |
| (indices : SourceTuple (branch := branch) (n := n) (t := t) input) | |
| (prepared : Fin (TreeCanonical.locationCount branch n t) → Bool) | |
| (position : Fin (TreeCanonical.locationCount branch n t)) : | |
| SourceTupleData.stacks input.formula indices prepared (.digit position) = | |
| List.replicate (indices position).val true := rfl | |
| theorem variable_counter (input : NormalizedSourceInput.Input) | |
| (indices : SourceTuple (branch := branch) (n := n) (t := t) input) | |
| (position : Fin (TreeCanonical.locationCount branch n t)) (slot : Fin 3) : | |
| SourceTupleData.stacks input.formula indices (fun _ => true) (.variableName position slot) = | |
| List.replicate (parsedVariableIDs input indices position slot) true := rfl | |
| variable (rows repeats : Nat → Nat) | |
| (hn : 0 < n) (hbranch : ∀ k < n, 0 < branch k) | |
| (hrows : ∀ k, 0 < rows (k + 1)) (δ : ℚ) | |
| local notation "q" => FinitePreliminaryCompletion.alphabet branch n t δ | |
| def construct (input : NormalizedSourceInput.Input) : Instance q := | |
| ExactPreliminaryTarget.construct (t := t) (NormalizedSourceInput.clauses input) | |
| rows repeats hn hbranch hrows δ | |
| def outputBits (input : NormalizedSourceInput.Input) : List Bool := | |
| Encoding.gameBits (construct (t := t) rows repeats hn hbranch hrows δ input) | |
| @[simp] theorem construct_eq (input : NormalizedSourceInput.Input) : | |
| construct (t := t) rows repeats hn hbranch hrows δ input = | |
| ExactPreliminaryTarget.construct (t := t) (NormalizedSourceInput.clauses input) | |
| rows repeats hn hbranch hrows δ := rfl | |
| theorem edge_count (input : NormalizedSourceInput.Input) : | |
| (construct (t := t) rows repeats hn hbranch hrows δ input).edges.length = | |
| SignedMultiplicity.denominator branch n t rows repeats hn hbranch hrows q * | |
| (NormalizedSourceInput.clauseCount input) ^ TreeCanonical.locationCount branch n t := | |
| ExactPreliminaryTarget.edge_count (t := t) (NormalizedSourceInput.clauses input) | |
| rows repeats hn hbranch hrows δ | |
| local instance leftVertex_fintype (input : NormalizedSourceInput.Input) : | |
| Fintype (CanonicalGame.LeftVertex (FinitePreliminaryCompletion.blockFamily | |
| (NormalizedSourceInput.clauses input) branch n t rows repeats)) := Fintype.ofFinite _ | |
| local instance rightVertex_fintype (input : NormalizedSourceInput.Input) : | |
| Fintype (CanonicalGame.RightVertex (FinitePreliminaryCompletion.blockFamily | |
| (NormalizedSourceInput.clauses input) branch n t rows repeats)) := Fintype.ofFinite _ | |
| local instance leftLabel_fintype (input : NormalizedSourceInput.Input) | |
| (x : CanonicalGame.LeftVertex (FinitePreliminaryCompletion.blockFamily | |
| (NormalizedSourceInput.clauses input) branch n t rows repeats)) : | |
| Fintype (CanonicalGame.LeftLabel (FinitePreliminaryCompletion.blockFamily | |
| (NormalizedSourceInput.clauses input) branch n t rows repeats) x) := Fintype.ofFinite _ | |
| local instance rightLabel_fintype (input : NormalizedSourceInput.Input) | |
| (y : CanonicalGame.RightVertex (FinitePreliminaryCompletion.blockFamily | |
| (NormalizedSourceInput.clauses input) branch n t rows repeats)) : | |
| Fintype (CanonicalGame.RightLabel (FinitePreliminaryCompletion.blockFamily | |
| (NormalizedSourceInput.clauses input) branch n t rows repeats) y) := Fintype.ofFinite _ | |
| theorem value_le (input : NormalizedSourceInput.Input) (hδ : 0 < δ) : | |
| (construct (t := t) rows repeats hn hbranch hrows δ input).value ≤ | |
| (PreliminarySampler.game (NormalizedSourceInput.clauses input) | |
| branch n t rows repeats hn hbranch hrows).value + (δ : ℝ) / 3 := | |
| ExactPreliminaryTarget.value_le (t := t) (NormalizedSourceInput.clauses input) | |
| rows repeats hn hbranch hrows δ hδ | |
| theorem perfectlyComplete (input : NormalizedSourceInput.Input) | |
| (satisfiable : input.formula.Satisfiable) : | |
| PerfectlyComplete (construct (t := t) rows repeats hn hbranch hrows δ input) := by | |
| obtain ⟨assignment, satisfies⟩ := satisfiable | |
| apply ExactPreliminaryTarget.perfectlyComplete (t := t) | |
| (NormalizedSourceInput.clauses input) rows repeats hn hbranch hrows δ assignment | |
| intro occurrence | |
| exact satisfies _ (List.getElem_mem occurrence.isLt) | |
| theorem outputBits_length_le (input : NormalizedSourceInput.Input) : | |
| let N := (construct (t := t) rows repeats hn hbranch hrows δ input).edges.length | |
| (outputBits (t := t) rows repeats hn hbranch hrows δ input).length ≤ | |
| 2 * N + q + N + 4 + N * (2 * N + 2 * q * q + 2 * q + 2) := | |
| ExactPreliminaryTarget.encoding_length_le (t := t) (NormalizedSourceInput.clauses input) | |
| rows repeats hn hbranch hrows δ | |
| end | |
| end PerfectCompleteness.NormalizedTarget | |
| end OAI | |