| import Mathlib |
| import OAI.Computability.MaxCut.Games.FinalCNFHeader |
|
|
| namespace OAI |
|
|
| /-! |
| Exact row enumeration and input/output stream identities for the concrete |
| final-CNF machine. Rows occur in stored order, including repeated rows. |
| -/ |
|
|
| namespace MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program |
|
|
| open PCP PCP.AlphabetTable FinalCNFTableAdapter |
|
|
| / |
| def remainingEvents (table : GraphTables.Table) (r : Nat) : List (Fin table.darts) := |
| (List.finRange table.darts).drop r |
|
|
| def processedEvents (table : GraphTables.Table) (r : Nat) : List (Fin table.darts) := |
| (List.finRange table.darts).take r |
|
|
| def inputStream (table : GraphTables.Table) (events : List (Fin table.darts)) : List Bool := |
| events.flatMap (fun e => encodeWords (GraphTables.rowWords table.rows[e])) |
|
|
| def outputStream (plan : Plan) (table : GraphTables.Table) |
| (events : List (Fin table.darts)) : List Bool := |
| events.flatMap (fun e => plan.flatMap (Emitter.commandBits (rowOperands table e) |
| (rowRelation table e))) |
|
|
| @[simp] theorem remainingEvents_zero (table : GraphTables.Table) : |
| remainingEvents table 0 = List.finRange table.darts := by |
| simp [remainingEvents] |
|
|
| @[simp] theorem remainingEvents_length (table : GraphTables.Table) (r : Nat) : |
| (remainingEvents table r).length = table.darts - r := by |
| simp [remainingEvents] |
|
|
| @[simp] theorem remainingEvents_done (table : GraphTables.Table) : |
| remainingEvents table table.darts = [] := by |
| simp [remainingEvents] |
|
|
| theorem remainingEvents_cons (table : GraphTables.Table) (r : Nat) (h : r < table.darts) : |
| remainingEvents table r = ⟨r, h⟩ :: remainingEvents table (r + 1) := by |
| unfold remainingEvents |
| rw [List.drop_eq_getElem_cons (l := List.finRange table.darts) (i := r) |
| (by simpa using h)] |
| simp |
|
|
| @[simp] theorem processedEvents_zero (table : GraphTables.Table) : |
| processedEvents table 0 = [] := by simp [processedEvents] |
|
|
| @[simp] theorem processedEvents_done (table : GraphTables.Table) : |
| processedEvents table table.darts = List.finRange table.darts := by |
| simp [processedEvents] |
|
|
| theorem processedEvents_succ (table : GraphTables.Table) (r : Nat) (h : r < table.darts) : |
| processedEvents table (r + 1) = processedEvents table r ++ [⟨r, h⟩] := by |
| unfold processedEvents |
| rw [List.take_succ_eq_append_getElem (l := List.finRange table.darts) (i := r) |
| (by simpa using h)] |
| simp |
|
|
| theorem processed_append_remaining (table : GraphTables.Table) (r : Nat) : |
| processedEvents table r ++ remainingEvents table r = List.finRange table.darts := by |
| exact List.take_append_drop r (List.finRange table.darts) |
|
|
| @[simp] theorem inputStream_nil (table : GraphTables.Table) : inputStream table [] = [] := rfl |
|
|
| @[simp] theorem inputStream_cons (table : GraphTables.Table) (e : Fin table.darts) |
| (events : List (Fin table.darts)) : |
| inputStream table (e :: events) = |
| encodeWords (GraphTables.rowWords table.rows[e]) ++ inputStream table events := rfl |
|
|
| @[simp] theorem inputStream_append (table : GraphTables.Table) |
| (first rest : List (Fin table.darts)) : |
| inputStream table (first ++ rest) = inputStream table first ++ inputStream table rest := by |
| simp only [inputStream, List.flatMap_append] |
|
|
| @[simp] theorem outputStream_nil (plan : Plan) (table : GraphTables.Table) : |
| outputStream plan table [] = [] := rfl |
|
|
| @[simp] theorem outputStream_cons (plan : Plan) (table : GraphTables.Table) |
| (e : Fin table.darts) (events : List (Fin table.darts)) : |
| outputStream plan table (e :: events) = |
| plan.flatMap (Emitter.commandBits (rowOperands table e) (rowRelation table e)) ++ |
| outputStream plan table events := rfl |
|
|
| @[simp] theorem outputStream_append (plan : Plan) (table : GraphTables.Table) |
| (first rest : List (Fin table.darts)) : |
| outputStream plan table (first ++ rest) = |
| outputStream plan table first ++ outputStream plan table rest := by |
| simp only [outputStream, List.flatMap_append] |
|
|
| theorem inputStream_remaining_cons (table : GraphTables.Table) (r : Nat) |
| (h : r < table.darts) : |
| inputStream table (remainingEvents table r) = |
| encodeWords (GraphTables.rowWords table.rows[(⟨r, h⟩ : Fin table.darts)]) ++ |
| inputStream table (remainingEvents table (r + 1)) := by |
| rw [remainingEvents_cons table r h, inputStream_cons] |
|
|
| theorem outputStream_remaining_cons (plan : Plan) (table : GraphTables.Table) (r : Nat) |
| (h : r < table.darts) : |
| outputStream plan table (remainingEvents table r) = |
| plan.flatMap (Emitter.commandBits (rowOperands table ⟨r, h⟩) |
| (rowRelation table ⟨r, h⟩)) ++ |
| outputStream plan table (remainingEvents table (r + 1)) := by |
| rw [remainingEvents_cons table r h, outputStream_cons] |
|
|
| theorem outputStream_processed_succ (plan : Plan) (table : GraphTables.Table) (r : Nat) |
| (h : r < table.darts) : |
| outputStream plan table (processedEvents table (r + 1)) = |
| outputStream plan table (processedEvents table r) ++ |
| plan.flatMap (Emitter.commandBits (rowOperands table ⟨r, h⟩) |
| (rowRelation table ⟨r, h⟩)) := by |
| rw [processedEvents_succ table r h, outputStream_append] |
| simp |
|
|
| theorem inputStream_split (table : GraphTables.Table) (r : Nat) : |
| inputStream table (List.finRange table.darts) = |
| inputStream table (processedEvents table r) ++ |
| inputStream table (remainingEvents table r) := by |
| rw [← inputStream_append, processed_append_remaining] |
|
|
| theorem outputStream_split (plan : Plan) (table : GraphTables.Table) (r : Nat) : |
| outputStream plan table (List.finRange table.darts) = |
| outputStream plan table (processedEvents table r) ++ |
| outputStream plan table (remainingEvents table r) := by |
| rw [← outputStream_append, processed_append_remaining] |
|
|
| private theorem encodeWords_flatMap_rows_inline_FinalCNFStream {α : Type} (rows : List α) (words : α → List Nat) : |
| encodeWords (rows.flatMap words) = rows.flatMap (fun row => encodeWords (words row)) := by |
| induction rows with |
| | nil => rfl |
| | cons row rows ih => simp only [List.flatMap_cons, encodeWords_append, ih] |
|
|
| theorem rowList_eq_finRange_map (table : GraphTables.Table) : |
| GraphTables.rowList table = (List.finRange table.darts).map (fun e => table.rows[e]) := by |
| rw [← List.ofFn_eq_map, ← Vector.toList_ofFn] |
| change table.rows.toList = (Vector.ofFn (fun i => table.rows[i.val])).toList |
| rw [Vector.ofFn_getElem] |
|
|
| / |
| theorem inputStream_all (table : GraphTables.Table) : |
| inputStream table (List.finRange table.darts) = tableRowsBits (genericTable table) := by |
| rw [tableRowsBits, genericTable_rowList] |
| erw [List.flatMap_map] |
| change inputStream table (List.finRange table.darts) = |
| encodeWords ((GraphTables.rowList table).flatMap |
| (fun row => GenericGraphTables.rowWords (genericRow row))) |
| simp only [genericRow_words] |
| rw [rowList_eq_finRange_map, List.flatMap_map, encodeWords_flatMap_rows_inline_FinalCNFStream] |
| rfl |
|
|
| theorem inputStream_remaining_zero (table : GraphTables.Table) : |
| inputStream table (remainingEvents table 0) = tableRowsBits (genericTable table) := by |
| rw [remainingEvents_zero, inputStream_all] |
|
|
| / |
| theorem outputStream_accumulator_succ (plan : Plan) (table : GraphTables.Table) |
| (r : Nat) (h : r < table.darts) (header : List Bool) : |
| (plan.flatMap (Emitter.commandBits (rowOperands table ⟨r, h⟩) |
| (rowRelation table ⟨r, h⟩))).reverse ++ |
| (header ++ outputStream plan table (processedEvents table r)).reverse = |
| (header ++ outputStream plan table (processedEvents table (r + 1))).reverse := by |
| rw [outputStream_processed_succ plan table r h] |
| simp only [List.reverse_append, List.append_assoc] |
|
|
| end MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program |
|
|
| /-! The actual bounded loop over every stored row of the final CNF converter. -/ |
|
|
| namespace MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program |
|
|
| open Turing PCP PCP.AlphabetTable |
|
|
| structure LoopInvariant (table : GraphTables.Table) (r : Nat) (base : Tape → List Bool) : Prop where |
| input : base .input = inputStream table (remainingEvents table r) |
| archive : base .archive = GraphTables.tableBits table |
| vertices : base .vertices = encodeWord table.vertices |
| darts : base .darts = encodeWord table.darts |
| index : base .rowIndex = encodeWord r |
| tail : base .tail = [] |
| head : base .head = [] |
| reverse : base .reverseIndex = [] |
| scratch : base .scratch = [] |
|
|
| structure LoopRun (headerPlan plan : Plan) (table : GraphTables.Table) (r remaining : Nat) |
| (base : Tape → List Bool) (ambient : Ambient) where |
| finalAmbient : Ambient |
| finalTapes : Tape → List Bool |
| execution : StateTransition.EvalsToInTime (TM2.step (program headerPlan plan)) |
| ⟨some .guard, ((ambient, ()), none), base⟩ |
| (some ⟨some .reverseOutput, ((finalAmbient, ()), none), finalTapes⟩) |
| (remaining * (rowTime plan (GraphTables.tableBits table).length + 1) + 1) |
| accumulator : finalTapes .accumulator = |
| (outputStream plan table (remainingEvents table r)).reverse ++ base .accumulator |
| output : finalTapes .output = base .output |
|
|
| theorem rowInvariant_succ (plan : Plan) (table : GraphTables.Table) (r : Nat) |
| (hr : r < table.darts) (base : Tape → List Bool) (invariant : LoopInvariant table r base) : |
| LoopInvariant table (r + 1) |
| (rowResultTapes plan base table ⟨r, hr⟩ |
| (inputStream table (remainingEvents table (r + 1)))) := by |
| constructor |
| · exact rowResult_input _ _ _ _ _ |
| · exact (rowResult_frame _ _ _ _ _ .archive (by decide) (by decide) (by decide) |
| (by decide) (by decide) (by decide) (by decide) (by decide)).trans invariant.archive |
| · exact (rowResult_frame _ _ _ _ _ .vertices (by decide) (by decide) (by decide) |
| (by decide) (by decide) (by decide) (by decide) (by decide)).trans invariant.vertices |
| · exact (rowResult_frame _ _ _ _ _ .darts (by decide) (by decide) (by decide) |
| (by decide) (by decide) (by decide) (by decide) (by decide)).trans invariant.darts |
| · exact rowResult_index _ _ _ _ _ |
| · exact rowResult_tail _ _ _ _ _ |
| · exact rowResult_head _ _ _ _ _ |
| · exact rowResult_reverse _ _ _ _ _ |
| · exact (rowResult_frame _ _ _ _ _ .scratch (by decide) (by decide) (by decide) |
| (by decide) (by decide) (by decide) (by decide) (by decide)).trans invariant.scratch |
|
|
| theorem rowWords_nonempty (table : GraphTables.Table) (e : Fin table.darts) : |
| encodeWords (GraphTables.rowWords table.rows[e]) ≠ [] := by |
| simp [GraphTables.rowWords, encodeWords, encodeWord] |
|
|
| / |
| input suffix. Every iteration invokes the checked row program. -/ |
| noncomputable def loopInTime (headerPlan plan : Plan) (table : GraphTables.Table) |
| (remaining : Nat) : |
| ∀ (r : Nat) (_hcount : r + remaining = table.darts) (base : Tape → List Bool) |
| (ambient : Ambient), LoopInvariant table r base → |
| LoopRun headerPlan plan table r remaining base ambient := by |
| induction remaining with |
| | zero => |
| intro r hcount base ambient invariant |
| have hr : r = table.darts := by omega |
| have hinput : base .input = [] := by |
| simpa only [hr, remainingEvents_done, inputStream_nil] using invariant.input |
| refine { |
| finalAmbient := ambient |
| finalTapes := base |
| execution := ?_ |
| accumulator := ?_ |
| output := rfl } |
| · simpa only [Nat.zero_mul, Nat.zero_add] using |
| guardEndInTime headerPlan plan base ambient hinput |
| · simp only [hr, remainingEvents_done, outputStream_nil, List.reverse_nil, List.nil_append] |
| | succ remaining ih => |
| intro r hcount base ambient invariant |
| have hr : r < table.darts := by omega |
| let e : Fin table.darts := ⟨r, hr⟩ |
| let rest := inputStream table (remainingEvents table (r + 1)) |
| let next := rowResultTapes plan base table e rest |
| have hinput : base .input = encodeWords (GraphTables.rowWords table.rows[e]) ++ rest := by |
| rw [invariant.input, inputStream_remaining_cons table r hr] |
| have hnonempty : base .input ≠ [] := by |
| rw [hinput] |
| exact List.append_ne_nil_of_left_ne_nil (rowWords_nonempty table e) rest |
| let guardRun := guardRowInTime headerPlan plan base ambient hnonempty |
| let rowRun := rowInTime headerPlan plan base table e rest hinput invariant.archive |
| invariant.vertices invariant.darts invariant.index invariant.tail invariant.head |
| invariant.reverse invariant.scratch ambient |
| let first := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ guardRun rowRun |
| let later := ih (r + 1) (by omega) next (rowRelation table e) |
| (rowInvariant_succ plan table r hr base invariant) |
| let run := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ first later.execution |
| refine { |
| finalAmbient := later.finalAmbient |
| finalTapes := later.finalTapes |
| execution := { toEvalsTo := run.toEvalsTo, steps_le_m := ?_ } |
| accumulator := ?_ |
| output := ?_ } |
| · have hb := run.steps_le_m |
| rw [Nat.succ_mul] |
| omega |
| · rw [later.accumulator, outputStream_remaining_cons plan table r hr] |
| simp only [next, rowResult_accumulator, List.reverse_append, List.append_assoc] |
| rfl |
| · rw [later.output] |
| exact rowResult_frame _ _ _ _ _ .output (by decide) (by decide) (by decide) |
| (by decide) (by decide) (by decide) (by decide) (by decide) |
|
|
| end MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program |
|
|
| /-! |
| The complete actual output-producing trace of the final CNF machine. Its |
| terminal configuration still contains private work tapes before the |
| canonical halt required by the machine interface. |
| -/ |
|
|
| namespace MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program |
|
|
| open Turing PCP PCP.AlphabetTable FinalCNFTableAdapter |
|
|
| def emittedBytes (plan : Plan) (table : GraphTables.Table) : List Bool := |
| encodeWords [6 * table.vertices + 36864 * table.darts, 40960 * table.darts] ++ |
| outputStream plan table (List.finRange table.darts) |
|
|
| noncomputable def rawBudget (plan : Plan) (table : GraphTables.Table) : Nat := |
| headerTimePolynomial.eval (GraphTables.tableBits table).length + |
| table.darts * (rowTime plan (GraphTables.tableBits table).length + 1) + 1 + |
| (emittedBytes plan table).length + 1 |
|
|
| structure RawRun (plan : Plan) (table : GraphTables.Table) where |
| finalAmbient : Ambient |
| finalTapes : Tape → List Bool |
| execution : StateTransition.EvalsToInTime (machine headerPlan plan).step |
| (initList (machine headerPlan plan) (GraphTables.tableBits table)) |
| (some ⟨none, ((finalAmbient, ()), none), finalTapes⟩) (rawBudget plan table) |
| output : finalTapes .output = emittedBytes plan table |
|
|
| theorem header_loopInvariant (_plan : Plan) (table : GraphTables.Table) : |
| LoopInvariant table 0 (headerResultTapes table.vertices table.darts |
| (tableRowsBits (genericTable table))) := by |
| constructor |
| · rw [headerResultTapes_input, inputStream_remaining_zero] |
| · rw [headerResultTapes_archive] |
| exact (tableBits_header_rows (genericTable table)).symm.trans (genericTable_tableBits table) |
| · rfl |
| · rfl |
| · exact headerResultTapes_rowIndex _ _ _ |
| · rfl |
| · rfl |
| · rfl |
| · rfl |
|
|
| / |
| row, final guard, and reversal are all actual transitions of this one program. -/ |
| noncomputable def rawRun (plan : Plan) (table : GraphTables.Table) : RawRun plan table := by |
| let base := headerResultTapes table.vertices table.darts (tableRowsBits (genericTable table)) |
| let ambient : Ambient := ((), fun _ => false) |
| have headerRun : StateTransition.EvalsToInTime (machine headerPlan plan).step |
| (initList (machine headerPlan plan) (GraphTables.tableBits table)) |
| (some ⟨some .guard, ((ambient, ()), none), base⟩) |
| (headerTimePolynomial.eval (GraphTables.tableBits table).length) := by |
| simpa only [genericTable_tableBits, genericTable_vertices, genericTable_darts, base, ambient] |
| using initializedTableHeaderInTime plan (genericTable table) |
| let loop := loopInTime headerPlan plan table table.darts 0 (by omega) |
| base ambient (header_loopInvariant plan table) |
| have acc : loop.finalTapes .accumulator = (emittedBytes plan table).reverse := by |
| rw [loop.accumulator] |
| simp only [remainingEvents_zero, base, headerResultTapes_accumulator, emittedBytes, |
| List.reverse_append] |
| have out : loop.finalTapes .output = [] := by |
| rw [loop.output] |
| rfl |
| let finalTapes := Reduction.MachineTransfer.tapesAt Tape.accumulator Tape.output |
| loop.finalTapes [] (emittedBytes plan table) |
| have finish : StateTransition.EvalsToInTime (machine headerPlan plan).step |
| ⟨some .reverseOutput, ((loop.finalAmbient, ()), none), loop.finalTapes⟩ |
| (some ⟨none, ((loop.finalAmbient, ()), none), finalTapes⟩) |
| ((emittedBytes plan table).length + 1) := by |
| have run := Reduction.MachineTransfer.transferAtInTime Tape.accumulator Tape.output |
| (by decide) id false .reverseOutput none (program headerPlan plan) rfl |
| loop.finalTapes (loop.finalAmbient, ()) none |
| simpa only [acc, out, List.reverse_reverse, List.length_reverse, List.map_id, |
| List.append_nil, finalTapes, machine, FinTM2.step] using! run |
| let first := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ headerRun loop.execution |
| let run := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ first finish |
| exact { |
| finalAmbient := loop.finalAmbient |
| finalTapes := finalTapes |
| execution := { |
| toEvalsTo := run.toEvalsTo |
| steps_le_m := by |
| have hb := run.steps_le_m |
| unfold rawBudget |
| omega } |
| output := by simp [finalTapes, Reduction.MachineTransfer.tapesAt] } |
|
|
| end MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program |
|
|
| /-! |
| Fixed literal templates for the twelve-query converter. All control data have |
| constant size: twelve query references, nine auxiliary references, ten clauses, |
| and three literal slots. Pattern bits remain symbolic throughout the proof. |
|
|
| These templates specify the actual serialized rows. Their machine execution |
| is proved separately in FinalCNFRow and FinalCNFComputation. |
| -/ |
|
|
| namespace MaxCutGames.Foundations.Complexity.FinalCNFTemplate |
|
|
| open Target PCP |
|
|
| inductive Reference where |
| | query : Fin 12 → Reference |
| | auxiliary : Fin 9 → Reference |
| deriving DecidableEq |
|
|
| structure LiteralTemplate where |
| reference : Reference |
| positive : Bool |
| deriving DecidableEq |
|
|
| abbrev ClauseTemplate := Vector LiteralTemplate 3 |
|
|
| def queryLiteral (bits : Fin 12 → Bool) (i : Fin 12) : LiteralTemplate := |
| ⟨.query i, !(bits i)⟩ |
|
|
| def auxiliaryLiteral (j : Fin 9) (positive : Bool) : LiteralTemplate := |
| ⟨.auxiliary j, positive⟩ |
|
|
| def tautologyTemplate : ClauseTemplate := |
| #v[⟨.query 0, true⟩, ⟨.query 0, false⟩, ⟨.query 0, true⟩] |
|
|
| / |
| def chainTemplates (bits : Fin 12 → Bool) : Vector ClauseTemplate 10 := |
| #v[ |
| #v[queryLiteral bits 0, queryLiteral bits 1, auxiliaryLiteral 0 true], |
| #v[auxiliaryLiteral 0 false, queryLiteral bits 2, auxiliaryLiteral 1 true], |
| #v[auxiliaryLiteral 1 false, queryLiteral bits 3, auxiliaryLiteral 2 true], |
| #v[auxiliaryLiteral 2 false, queryLiteral bits 4, auxiliaryLiteral 3 true], |
| #v[auxiliaryLiteral 3 false, queryLiteral bits 5, auxiliaryLiteral 4 true], |
| #v[auxiliaryLiteral 4 false, queryLiteral bits 6, auxiliaryLiteral 5 true], |
| #v[auxiliaryLiteral 5 false, queryLiteral bits 7, auxiliaryLiteral 6 true], |
| #v[auxiliaryLiteral 6 false, queryLiteral bits 8, auxiliaryLiteral 7 true], |
| #v[auxiliaryLiteral 7 false, queryLiteral bits 9, auxiliaryLiteral 8 true], |
| #v[auxiliaryLiteral 8 false, queryLiteral bits 10, queryLiteral bits 11]] |
|
|
| def blockTemplates (accepted : Bool) (bits : Fin 12 → Bool) : Vector ClauseTemplate 10 := |
| if accepted then Vector.replicate 10 tautologyTemplate else chainTemplates bits |
|
|
| def templates (accepted : Bool) (bits : Fin 12 → Bool) : List ClauseTemplate := |
| (blockTemplates accepted bits).toList |
|
|
| / |
| def literalAt (accepted : Bool) (bits : Fin 12 → Bool) |
| (clause : Fin 10) (slot : Fin 3) : LiteralTemplate := |
| (blockTemplates accepted bits)[clause][slot] |
|
|
| @[simp] theorem templates_length (accepted : Bool) (bits : Fin 12 → Bool) : |
| (templates accepted bits).length = 10 := Vector.length_toList |
|
|
| theorem templates_true (bits : Fin 12 → Bool) : |
| templates true bits = List.replicate 10 tautologyTemplate := by rfl |
|
|
| theorem templates_false (bits : Fin 12 → Bool) : |
| templates false bits = (chainTemplates bits).toList := by rfl |
|
|
| def literalTemplates (accepted : Bool) (bits : Fin 12 → Bool) : List LiteralTemplate := |
| (templates accepted bits).flatMap fun c => [(c)[0], (c)[1], (c)[2]] |
|
|
| theorem literalTemplates_length (accepted : Bool) (bits : Fin 12 → Bool) : |
| (literalTemplates accepted bits).length = 30 := by |
| unfold literalTemplates |
| rw [VerifierToCNF.length_flatMap_constant _ _ 3 (by intro c hc; rfl), templates_length] |
|
|
| def literalWords (index : Reference → Nat) (literal : LiteralTemplate) : List Nat := |
| [index literal.reference, if literal.positive then 1 else 0] |
|
|
| def clauseWords (index : Reference → Nat) (clause : ClauseTemplate) : List Nat := |
| literalWords index clause[0] ++ literalWords index clause[1] ++ literalWords index clause[2] |
|
|
| def words (accepted : Bool) (bits : Fin 12 → Bool) (index : Reference → Nat) : List Nat := |
| (templates accepted bits).flatMap (clauseWords index) |
|
|
| theorem words_length (accepted : Bool) (bits : Fin 12 → Bool) (index : Reference → Nat) : |
| (words accepted bits index).length = 60 := by |
| unfold words |
| rw [VerifierToCNF.length_flatMap_constant _ _ 6 |
| (by intro c hc; simp [clauseWords, literalWords]), templates_length] |
|
|
| variable (V : VerifierToCNF.FiniteVerifier 12) (e : Fin V.events) |
| (p : VerifierToCNF.PatternIndex 12) |
|
|
| def resolve : Reference → Fin (VerifierToCNF.outputVariables V) |
| | .query i => VerifierToCNF.oldIndex V (V.query e i) |
| | .auxiliary j => VerifierToCNF.freshIndex V e p j |
|
|
| theorem resolve_query_value (i : Fin 12) : |
| (resolve V e p (.query i)).val = (V.query e i).val := by |
| simp only [resolve, VerifierToCNF.oldIndex, Fin.val_castAdd] |
|
|
| theorem resolve_auxiliary_value (j : Fin 9) : |
| (resolve V e p (.auxiliary j)).val = |
| V.«variables» + ((e.val * 4096 + p.val) * 9 + j.val) := by |
| change (VerifierToCNF.freshIndex V e p j).val = _ |
| exact (VerifierToCNF.freshIndex_value V e p j).trans |
| (congrArg (fun k : Nat => V.«variables» + ((e.val * k + p.val) * 9 + j.val)) |
| (show (VerifierToCNF.patterns 12).length = 4096 from VerifierToCNF.patterns_length 12)) |
|
|
| def instantiateLiteral (literal : LiteralTemplate) : Literal (VerifierToCNF.outputVariables V) := |
| ⟨resolve V e p literal.reference, literal.positive⟩ |
|
|
| def instantiateClause (clause : ClauseTemplate) : Clause (VerifierToCNF.outputVariables V) := |
| #v[instantiateLiteral V e p clause[0], instantiateLiteral V e p clause[1], |
| instantiateLiteral V e p clause[2]] |
|
|
| / |
| theorem splitLong_twelve {n : Nat} (ls : Fin 12 → Literal n) (ys : Fin 9 → Fin n) : |
| VerifierToCNF.splitLong (List.ofFn ls) (List.ofFn ys) = |
| [#v[ls 0, ls 1, ⟨ys 0, true⟩], |
| #v[⟨ys 0, false⟩, ls 2, ⟨ys 1, true⟩], |
| #v[⟨ys 1, false⟩, ls 3, ⟨ys 2, true⟩], |
| #v[⟨ys 2, false⟩, ls 4, ⟨ys 3, true⟩], |
| #v[⟨ys 3, false⟩, ls 5, ⟨ys 4, true⟩], |
| #v[⟨ys 4, false⟩, ls 6, ⟨ys 5, true⟩], |
| #v[⟨ys 5, false⟩, ls 7, ⟨ys 6, true⟩], |
| #v[⟨ys 6, false⟩, ls 8, ⟨ys 7, true⟩], |
| #v[⟨ys 7, false⟩, ls 9, ⟨ys 8, true⟩], |
| #v[⟨ys 8, false⟩, ls 10, ls 11]] := by |
| simp only [List.ofFn_succ, List.ofFn_zero, VerifierToCNF.splitLong] |
| rfl |
|
|
| / |
| theorem instantiate_templates : |
| (templates (V.accepts e (VerifierToCNF.patternAt 12 p)) |
| (VerifierToCNF.patternAt 12 p)).map (instantiateClause V e p) = |
| VerifierToCNF.block V (by decide) e p := by |
| cases h : V.accepts e (VerifierToCNF.patternAt 12 p) with |
| | false => |
| simp only [h, templates_false, VerifierToCNF.block, Bool.false_eq_true, ↓reduceIte, |
| VerifierToCNF.forbiddenClause, VerifierToCNF.auxiliaryNames] |
| rw [splitLong_twelve] |
| rfl |
| | true => |
| simp only [h, templates_true, VerifierToCNF.block, ↓reduceIte, List.map_replicate] |
| rfl |
|
|
| theorem instantiate_templates_of_acceptance (accepted : Bool) |
| (h : V.accepts e (VerifierToCNF.patternAt 12 p) = accepted) : |
| (templates accepted (VerifierToCNF.patternAt 12 p)).map (instantiateClause V e p) = |
| VerifierToCNF.block V (by decide) e p := by |
| rw [← h] |
| exact instantiate_templates V e p |
|
|
| theorem instantiateClause_words (clause : ClauseTemplate) : |
| Complexity.clauseWords (instantiateClause V e p clause) = |
| clauseWords (fun r => (resolve V e p r).val) clause := rfl |
|
|
| / |
| theorem words_eq_block : |
| words (V.accepts e (VerifierToCNF.patternAt 12 p)) (VerifierToCNF.patternAt 12 p) |
| (fun r => (resolve V e p r).val) = |
| (VerifierToCNF.block V (by decide) e p).flatMap Complexity.clauseWords := by |
| rw [← instantiate_templates V e p] |
| simp only [List.flatMap_map, instantiateClause_words] |
| rfl |
|
|
| theorem block_words_length : |
| ((VerifierToCNF.block V (by decide) e p).flatMap Complexity.clauseWords).length = 60 := by |
| rw [← words_eq_block V e p] |
| exact words_length _ _ _ |
|
|
| end MaxCutGames.Foundations.Complexity.FinalCNFTemplate |
|
|
| /-! |
| The fixed finite emission plan for one directed graph row. All choices based |
| on the relation are functions of its 4096-bit finite register. Vertex counts, |
| endpoint names and row indices are read by preserving unary scans. |
| -/ |
|
|
| namespace MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program |
|
|
| open PCP PCP.AlphabetTable FinalCNFTemplate |
|
|
| def referenceValue (vertices tail head row pattern : Nat) : Reference → Nat |
| | .query i => if i.val < 6 then 6 * tail + i.val else 6 * head + (i.val - 6) |
| | .auxiliary j => 6 * vertices + 36864 * row + (9 * pattern + j.val) |
|
|
| def auxiliaryFlag : Reference → Bool |
| | .query _ => false |
| | .auxiliary _ => true |
|
|
| def tailFlag : Reference → Bool |
| | .query i => decide (i.val < 6) |
| | .auxiliary _ => false |
|
|
| def headFlag : Reference → Bool |
| | .query i => decide (6 ≤ i.val) |
| | .auxiliary _ => false |
|
|
| theorem pattern_lt (p : VerifierToCNF.PatternIndex 12) : p.val < 4096 := by |
| have hlen : (VerifierToCNF.patterns 12).length = 4096 := VerifierToCNF.patterns_length 12 |
| exact lt_of_lt_of_eq p.isLt hlen |
|
|
| def referenceOffset (p : VerifierToCNF.PatternIndex 12) : Reference → Fin 36864 |
| | .query i => ⟨if i.val < 6 then i.val else i.val - 6, by split <;> omega⟩ |
| | .auxiliary j => ⟨9 * p.val + j.val, by have hp := pattern_lt p; omega⟩ |
|
|
| / |
| def literalPlan (p : VerifierToCNF.PatternIndex 12) (literal : Ambient → LiteralTemplate) : Plan := |
| [.scaledIf 0 6 (fun state => auxiliaryFlag (literal state).reference), |
| .scaledIf 2 6 (fun state => tailFlag (literal state).reference), |
| .scaledIf 3 6 (fun state => headFlag (literal state).reference), |
| .scaledIf 4 36864 (fun state => auxiliaryFlag (literal state).reference), |
| .bounded (fun state => referenceOffset p (literal state).reference), |
| .literal [false], |
| .bit (fun state => (literal state).positive)] |
|
|
| theorem literalPlan_length (p : VerifierToCNF.PatternIndex 12) |
| (literal : Ambient → LiteralTemplate) : (literalPlan p literal).length = 7 := rfl |
|
|
| theorem literalPlan_bits (p : VerifierToCNF.PatternIndex 12) |
| (literal : Ambient → LiteralTemplate) (vertices darts tail head row : Nat) |
| (ambient : Ambient) : |
| (literalPlan p literal).flatMap |
| (Emitter.commandBits (values vertices darts tail head row) ambient) = |
| encodeWords (FinalCNFTemplate.literalWords |
| (referenceValue vertices tail head row p.val) (literal ambient)) := by |
| rcases h : literal ambient with ⟨reference, positive⟩ |
| cases reference with |
| | query i => |
| by_cases hi : i.val < 6 |
| · simp [literalPlan, Emitter.commandBits, h, auxiliaryFlag, tailFlag, headFlag, |
| referenceOffset, referenceValue, FinalCNFTemplate.literalWords, values, |
| hi, show ¬ 6 ≤ i.val by omega, encodeWords, encodeWord, List.replicate_add, |
| List.append_assoc, -List.replicate_append_replicate] |
| rfl |
| · simp [literalPlan, Emitter.commandBits, h, auxiliaryFlag, tailFlag, headFlag, |
| referenceOffset, referenceValue, FinalCNFTemplate.literalWords, values, |
| hi, show 6 ≤ i.val by omega, encodeWords, encodeWord, List.replicate_add, |
| List.append_assoc, -List.replicate_append_replicate] |
| rfl |
| | auxiliary j => |
| simp [literalPlan, Emitter.commandBits, h, auxiliaryFlag, tailFlag, headFlag, |
| referenceOffset, referenceValue, FinalCNFTemplate.literalWords, values, |
| encodeWords, encodeWord, List.replicate_add, List.append_assoc, |
| -List.replicate_append_replicate] |
| rfl |
|
|
| def patternLiteral (p : VerifierToCNF.PatternIndex 12) (clause : Fin 10) (slot : Fin 3) |
| (ambient : Ambient) : LiteralTemplate := |
| literalAt (ambient.2 (FinalCNFPattern.patternRelationIndex p)) |
| (VerifierToCNF.patternAt 12 p) clause slot |
|
|
| def patternPlan (p : VerifierToCNF.PatternIndex 12) : Plan := |
| (List.finRange 10).flatMap fun clause => |
| (List.finRange 3).flatMap fun slot => literalPlan p (patternLiteral p clause slot) |
|
|
| def rowPlan : Plan := |
| (List.finRange (VerifierToCNF.patterns 12).length).flatMap patternPlan |
|
|
| theorem patternPlan_length (p : VerifierToCNF.PatternIndex 12) : |
| (patternPlan p).length = 210 := by |
| unfold patternPlan |
| rw [VerifierToCNF.length_flatMap_constant _ _ 21] |
| · simp |
| · intro clause hclause |
| rw [VerifierToCNF.length_flatMap_constant _ _ 7] |
| · simp |
| · intro slot hslot |
| exact literalPlan_length _ _ |
|
|
| theorem rowPlan_length : rowPlan.length = 860160 := by |
| unfold rowPlan |
| rw [VerifierToCNF.length_flatMap_constant _ _ 210] |
| · rw [List.length_finRange, VerifierToCNF.patterns_length]; norm_num |
| · intro p hp |
| exact patternPlan_length p |
|
|
| theorem encodeWords_flatMap {α : Type*} (xs : List α) (words : α → List Nat) : |
| encodeWords (xs.flatMap words) = xs.flatMap (fun x => encodeWords (words x)) := by |
| induction xs with |
| | nil => rfl |
| | cons x xs ih => simp only [List.flatMap_cons, encodeWords_append, ih] |
|
|
| theorem patternPlan_bits (p : VerifierToCNF.PatternIndex 12) |
| (vertices darts tail head row : Nat) (ambient : Ambient) : |
| (patternPlan p).flatMap |
| (Emitter.commandBits (values vertices darts tail head row) ambient) = |
| encodeWords (FinalCNFTemplate.words |
| (ambient.2 (FinalCNFPattern.patternRelationIndex p)) |
| (VerifierToCNF.patternAt 12 p) (referenceValue vertices tail head row p.val)) := by |
| unfold patternPlan |
| simp only [List.flatMap_assoc] |
| simp_rw [literalPlan_bits] |
| have hclauses : |
| (List.finRange 10).map |
| (fun c => (blockTemplates (ambient.2 (FinalCNFPattern.patternRelationIndex p)) |
| (VerifierToCNF.patternAt 12 p))[c]) = |
| templates (ambient.2 (FinalCNFPattern.patternRelationIndex p)) |
| (VerifierToCNF.patternAt 12 p) := by |
| rw [← List.ofFn_eq_map] |
| change List.ofFn (fun c : Fin 10 => |
| (blockTemplates (ambient.2 (FinalCNFPattern.patternRelationIndex p)) |
| (VerifierToCNF.patternAt 12 p))[c.val]) = _ |
| rw [← Vector.toList_ofFn, Vector.ofFn_getElem] |
| rfl |
| unfold FinalCNFTemplate.words |
| rw [← hclauses, List.flatMap_map, encodeWords_flatMap] |
| apply List.flatMap_congr |
| intro clause hclause |
| simp only [List.finRange, List.ofFn_succ, List.ofFn_zero, List.flatMap_cons, |
| List.flatMap_nil, patternLiteral, literalAt, FinalCNFTemplate.clauseWords, |
| encodeWords_append, List.append_nil, List.append_assoc] |
| rfl |
|
|
| theorem referenceValue_eq_resolve (table : GraphTables.Table) (e : Fin table.darts) |
| (p : VerifierToCNF.PatternIndex 12) (reference : Reference) : |
| referenceValue table.vertices table.rows[e].tail.val |
| table.rows[table.rows[e].reverseIndex].tail.val e.val p.val reference = |
| (resolve (FinalCNFPattern.tableVerifier table) e p reference).val := by |
| cases reference with |
| | query i => |
| rw [resolve_query_value] |
| refine Fin.addCases (m := 6) (n := 6) (fun j => ?_) (fun j => ?_) i |
| · rw [FinalBooleanVerifier.query_tail] |
| simp only [referenceValue, Fin.val_castAdd, j.isLt, ite_true] |
| change 6 * table.rows[e].tail.val + j.val = j.val + 6 * table.rows[e].tail.val |
| omega |
| · rw [FinalBooleanVerifier.query_head] |
| simp only [referenceValue, Fin.val_natAdd] |
| split |
| · omega |
| · change 6 * table.rows[table.rows[e].reverseIndex].tail.val + |
| (6 + j.val - 6) = j.val + 6 * table.rows[table.rows[e].reverseIndex].tail.val |
| omega |
| | auxiliary j => |
| rw [resolve_auxiliary_value] |
| change 6 * table.vertices + 36864 * e.val + (9 * p.val + j.val) = |
| table.vertices * 6 + ((e.val * 4096 + p.val) * 9 + j.val) |
| omega |
|
|
| theorem rowPlan_bits (table : GraphTables.Table) (e : Fin table.darts) : |
| rowPlan.flatMap (Emitter.commandBits |
| (values table.vertices table.darts table.rows[e].tail.val |
| table.rows[table.rows[e].reverseIndex].tail.val e.val) |
| ((), fun i => table.rows[e].relation[i])) = |
| encodeWords ((VerifierToCNF.eventBlock (FinalCNFPattern.tableVerifier table) |
| (by decide) e).flatMap Complexity.clauseWords) := by |
| unfold rowPlan VerifierToCNF.eventBlock |
| simp only [List.flatMap_assoc] |
| rw [encodeWords_flatMap] |
| apply List.flatMap_congr |
| intro p hp |
| rw [patternPlan_bits] |
| dsimp only |
| rw [FinalCNFPattern.pattern_lookup_eq_verifier_accepts] |
| have href := funext (referenceValue_eq_resolve table e p) |
| rw [href] |
| exact congrArg encodeWords (words_eq_block (FinalCNFPattern.tableVerifier table) e p) |
|
|
| end MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program |
|
|
| end OAI |
|
|
