openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Machines /BinaryRenameWords.lean
| import OAI.Computability.PerfectCompleteness.Machines.BinaryNameSearch | |
| import OAI.Computability.PerfectCompleteness.Machines.BinaryOccurrenceRename | |
| import OAI.Computability.PerfectCompleteness.Machines.BinaryParsingLemmas | |
| import OAI.Computability.PerfectCompleteness.Machines.BinaryTokenMachine | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.BinaryRenameWords | |
| open BinaryFormula UniqueGamesTheorem.Foundations | |
| def clauseLiterals (c : Clause) : List Literal := [(c)[0], (c)[1], (c)[2]] | |
| def literals (clauses : List Clause) : List Literal := clauses.flatMap clauseLiterals | |
| def token (literal : Literal) : BinaryNameSearch.Token := | |
| (literal.positive, literal.name.bits) | |
| def tokens (clauses : List Clause) : List BinaryNameSearch.Token := | |
| (literals clauses).map token | |
| @[simp] theorem literals_nil : literals [] = [] := rfl | |
| @[simp] theorem literals_cons (c : Clause) (cs : List Clause) : | |
| literals (c :: cs) = (c)[0] :: (c)[1] :: (c)[2] :: literals cs := rfl | |
| @[simp] theorem literals_length (cs : List Clause) : | |
| (literals cs).length = 3 * cs.length := by | |
| induction cs with | |
| | nil => rfl | |
| | cons c cs ih => simp only [literals_cons, List.length_cons, ih]; omega | |
| @[simp] theorem tokens_length (cs : List Clause) : | |
| (tokens cs).length = 3 * cs.length := by simp [tokens] | |
| theorem literals_names (F : Formula) : | |
| (literals F.clauses).map Literal.name = sourceNames F := by | |
| simp only [literals, sourceNames, List.map_flatMap] | |
| rfl | |
| theorem tokens_payloads (F : Formula) : | |
| BinaryNameSearch.payloads (tokens F.clauses) = (sourceNames F).map Nat.bits := by | |
| rw [← literals_names] | |
| simp only [BinaryNameSearch.payloads, tokens, List.map_map, token, Function.comp_def] | |
| theorem token_bits (literal : Literal) : | |
| BinaryNameSearch.tokenBits (token literal) = BinaryEncoding.literalBits literal := by | |
| simp only [BinaryNameSearch.tokenBits, token, BinaryEncoding.literalBits, | |
| BinaryEncoding.nameBits, BinaryParsing.frame_eq] | |
| theorem tokens_stream (cs : List Clause) : | |
| BinaryNameSearch.stream (tokens cs) = BinaryTokenMachine.tokens cs := by | |
| induction cs with | |
| | nil => rfl | |
| | cons c cs ih => | |
| simp only [tokens, literals_cons, List.map_cons, BinaryNameSearch.stream_cons, | |
| token_bits, BinaryTokenMachine.tokens_cons, BinaryEncoding.clauseBits] | |
| simpa only [tokens, List.append_assoc] using | |
| congrArg (fun tail => BinaryEncoding.literalBits (c)[0] ++ | |
| BinaryEncoding.literalBits (c)[1] ++ BinaryEncoding.literalBits (c)[2] ++ tail) ih | |
| theorem tokens_canonical (cs : List Clause) : | |
| ∀ t ∈ tokens cs, BinaryNameMachine.canonical t.2 = true := by | |
| intro t mem | |
| obtain ⟨literal, _, rfl⟩ := List.mem_map.mp mem | |
| exact BinaryParsing.canonical_nat_bits literal.name | |
| theorem payload_index (F : Formula) (name : Nat) : | |
| (BinaryNameSearch.payloads (tokens F.clauses)).idxOf name.bits = | |
| (sourceNames F).idxOf name := by | |
| rw [tokens_payloads, BinaryOccurrenceRename.idxOf_binary_payloads] | |
| theorem token_payload_mem (cs : List Clause) (literal : Literal) | |
| (mem : literal ∈ literals cs) : | |
| literal.name.bits ∈ BinaryNameSearch.payloads (tokens cs) := by | |
| apply List.mem_map.mpr | |
| refine ⟨token literal, ?_, rfl⟩ | |
| exact List.mem_map.mpr ⟨literal, mem, rfl⟩ | |
| theorem payloadSize_tokens (cs : List Clause) : | |
| BinaryNameSearch.payloadSize (tokens cs) = BinaryEncoding.namesBitSize cs := by | |
| induction cs with | |
| | nil => rfl | |
| | cons c cs ih => | |
| simp only [tokens, literals_cons, List.map_cons, | |
| BinaryNameSearch.payloadSize_cons, token, Nat.size_eq_bits_len, | |
| BinaryEncoding.namesBitSize_cons] at * | |
| simpa only [BinaryEncoding.clauseNameSize, BinaryFormula.clauseNames, | |
| List.map_cons, List.map_nil, List.sum_cons, List.sum_nil, | |
| Nat.add_zero, Nat.add_assoc] using congrArg | |
| (fun tail => (c)[0].name.size + ((c)[1].name.size + ((c)[2].name.size + tail))) ih | |
| def outputLiteral (names : List Nat) (literal : Literal) : List Bool := | |
| Complexity.encodeWord (names.idxOf literal.name) ++ | |
| Complexity.encodeWord (if literal.positive then 1 else 0) | |
| def body (F : Formula) : List Bool := | |
| (literals F.clauses).flatMap (outputLiteral (sourceNames F)) | |
| theorem encoded_clause (F : Formula) (c : Clause) (mem : c ∈ F.clauses) : | |
| Complexity.encodeWords (Complexity.clauseWords (BinaryOccurrenceRename.clause F c mem)) = | |
| (clauseLiterals c).flatMap (outputLiteral (sourceNames F)) := by | |
| simp [Complexity.clauseWords, Complexity.literalWords, Complexity.encodeWords, | |
| BinaryOccurrenceRename.clause, BinaryOccurrenceRename.literal, | |
| BinaryOccurrenceRename.nameIndex, clauseLiterals, outputLiteral, List.append_assoc] | |
| rfl | |
| private theorem encodeWords_flatMap {α : Type} (xs : List α) (f : α → List Nat) : | |
| Complexity.encodeWords (xs.flatMap f) = | |
| xs.flatMap (fun x => Complexity.encodeWords (f x)) := by | |
| induction xs with | |
| | nil => rfl | |
| | cons x xs ih => simp only [List.flatMap_cons, Complexity.encodeWords_append, ih] | |
| theorem encoded_body (F : Formula) : | |
| Complexity.encodeWords ((BinaryOccurrenceRename.renamed F).clauses.flatMap | |
| Complexity.clauseWords) = body F := by | |
| rw [encodeWords_flatMap] | |
| change (F.clauses.attach.map (fun c => BinaryOccurrenceRename.clause F c.val c.property)).flatMap | |
| (fun c => Complexity.encodeWords (Complexity.clauseWords c)) = body F | |
| rw [List.flatMap_map] | |
| simp only [encoded_clause] | |
| have h := congrArg (fun cs : List Clause => | |
| cs.flatMap (fun c => (clauseLiterals c).flatMap (outputLiteral (sourceNames F)))) | |
| (List.attach_map_subtype_val F.clauses) | |
| simpa only [List.flatMap_map, body, literals, List.flatMap_assoc] using h | |
| theorem encoded_renamed (F : Formula) : | |
| Complexity.formulaBits (BinaryOccurrenceRename.renamed F) = | |
| Complexity.encodeWord (3 * F.clauses.length) ++ | |
| Complexity.encodeWord F.clauses.length ++ body F := by | |
| simp only [Complexity.formulaBits, Complexity.formulaWords, Complexity.encodeWords_append, | |
| Complexity.encodeWords, List.append_nil, | |
| BinaryOccurrenceRename.renamed_variable_count, | |
| BinaryOccurrenceRename.renamed_clause_count, encoded_body, List.append_assoc] | |
| end PerfectCompleteness.BinaryRenameWords | |
| end | |
| end OAI | |