| import Mathlib |
| import OAI.Computability.MinUncut.Games.FinalCNFPattern |
| import OAI.Computability.MinUncut.Machines.RawInitialMachineFinish |
|
|
| namespace OAI |
|
|
| namespace MinUncutGames.Foundations.Complexity.FinalCNFMachine.Program |
|
|
| open Turing |
| open PCP.AlphabetTable |
|
|
| def headerFrame (input archive vertices darts rowIndex tail head accumulator : List Bool) : |
| Tape → List Bool |
| | .input => input |
| | .archive => archive |
| | .vertices => vertices |
| | .darts => darts |
| | .rowIndex => rowIndex |
| | .tail => tail |
| | .head => head |
| | .accumulator => accumulator |
| | _ => [] |
|
|
| def headerInputTapes (input : List Bool) : Tape → List Bool := |
| headerFrame input [] [] [] [] [] [] [] |
|
|
| def headerResultTapes (n m : Nat) (rowsBits : List Bool) : Tape → List Bool := |
| headerFrame rowsBits (encodeWords [n, m] ++ rowsBits) |
| (encodeWord n) (encodeWord m) (encodeWord 0) [] [] |
| (encodeWords [6 * n + 36864 * m, 40960 * m]).reverse |
|
|
| @[simp] theorem headerResultTapes_input (n m : Nat) (rowsBits : List Bool) : |
| headerResultTapes n m rowsBits .input = rowsBits := rfl |
|
|
| @[simp] theorem headerResultTapes_archive (n m : Nat) (rowsBits : List Bool) : |
| headerResultTapes n m rowsBits .archive = encodeWords [n, m] ++ rowsBits := rfl |
|
|
| @[simp] theorem headerResultTapes_accumulator (n m : Nat) (rowsBits : List Bool) : |
| headerResultTapes n m rowsBits .accumulator = |
| (encodeWords [6 * n + 36864 * m, 40960 * m]).reverse := rfl |
|
|
| @[simp] theorem headerResultTapes_rowIndex (n m : Nat) (rowsBits : List Bool) : |
| headerResultTapes n m rowsBits .rowIndex = encodeWord 0 := rfl |
|
|
| noncomputable def headerTimePolynomial : Polynomial Nat := |
| Polynomial.C 25 * Polynomial.X + Polynomial.C 52 |
|
|
| @[simp] theorem headerTimePolynomial_eval (length : Nat) : |
| headerTimePolynomial.eval length = 25 * length + 52 := by |
| simp [headerTimePolynomial] |
|
|
| def headerInTime (rowPlan : Plan) (n m : Nat) (rowsBits : List Bool) (ambient : Ambient) : |
| StateTransition.EvalsToInTime (TM2.step (program headerPlan rowPlan)) |
| ⟨some .copyFirst, ((ambient, ()), none), |
| headerInputTapes (encodeWords [n, m] ++ rowsBits)⟩ |
| (some ⟨some .guard, ((ambient, ()), none), headerResultTapes n m rowsBits⟩) |
| (headerTimePolynomial.eval (encodeWords [n, m] ++ rowsBits).length) := by |
| let word := encodeWords [n, m] ++ rowsBits |
| let bits := encodeWords [6 * n + 36864 * m, 40960 * m] |
| let b₀ := headerInputTapes word |
| let b₁ := headerFrame word word [] [] [] [] [] [] |
| let b₂ := headerFrame (encodeWord m ++ rowsBits) word (encodeWord n) [] [] [] [] [] |
| let b₃ := headerFrame rowsBits word (encodeWord n) (encodeWord m) [] [] [] [] |
| let b₄ := headerFrame rowsBits word (encodeWord n) (encodeWord m) |
| (encodeWord 0) (encodeWord 0) (encodeWord 0) [] |
| let b₅ := headerFrame rowsBits word (encodeWord n) (encodeWord m) |
| (encodeWord 0) (encodeWord 0) (encodeWord 0) bits.reverse |
| let b₆ := headerFrame rowsBits word (encodeWord n) (encodeWord m) |
| (encodeWord 0) [] (encodeWord 0) bits.reverse |
| let b₇ := headerResultTapes n m rowsBits |
| have h₀ : Function.update b₀ Tape.archive (b₀ .input ++ b₀ .archive) = b₁ := by |
| funext tape; cases tape <;> simp [b₀, b₁, headerInputTapes, headerFrame] |
| have h₁ : Hastad.SourceMachine.fieldTapes Tape.input Tape.vertices b₁ |
| (encodeWord m ++ rowsBits) (encodeWord n ++ b₁ .vertices) = b₂ := by |
| funext tape; cases tape <;> simp [Hastad.SourceMachine.fieldTapes, b₁, b₂, headerFrame] |
| have h₂ : Hastad.SourceMachine.fieldTapes Tape.input Tape.darts b₂ |
| rowsBits (encodeWord m ++ b₂ .darts) = b₃ := by |
| funext tape; cases tape <;> simp [Hastad.SourceMachine.fieldTapes, b₂, b₃, headerFrame] |
| have h₄ : Emitter.resultTapes Tape.accumulator b₄ |
| (Emitter.prefixBits (Emitter.listCommands headerPlan) (values n m 0 0 0) |
| ambient headerPlan.length) = b₅ := by |
| rw [Emitter.bits_listCommands, headerPlan_bits] |
| funext tape; cases tape <;> simp [Emitter.resultTapes, b₄, b₅, bits, headerFrame] |
| have h₅ : Function.update b₅ Tape.tail [] = b₆ := by |
| funext tape; cases tape <;> simp [b₅, b₆, headerFrame] |
| have h₆ : Function.update b₆ Tape.head [] = b₇ := by |
| funext tape; cases tape <;> simp [b₆, b₇, headerResultTapes, bits, word, headerFrame] |
| have hn : n ≤ word.length := by |
| simp only [word, List.length_append, encodeWords_length, List.sum_cons, |
| List.sum_nil, List.length_cons, List.length_nil] ; omega |
| have hm : m ≤ word.length := by |
| simp only [word, List.length_append, encodeWords_length, List.sum_cons, |
| List.sum_nil, List.length_cons, List.length_nil] ; omega |
| let p₀ := MachineCopy.copyInTime Tape.input Tape.archive Tape.scratch |
| (by decide) (by decide) (by decide) false .copyFirst .copySecond (some .startVertices) |
| (program headerPlan rowPlan) rfl rfl b₀ (by rfl) (ambient, ()) none |
| have p₀' : StateTransition.EvalsToInTime (TM2.step (program headerPlan rowPlan)) |
| ⟨some .copyFirst, ((ambient, ()), none), b₀⟩ |
| (some ⟨some .startVertices, ((ambient, ()), none), b₁⟩) (2 * (word.length + 1)) := by |
| have h := p₀ |
| rw [h₀] at h |
| simpa only [show b₀ .input = word from rfl] using h |
| let p₁ := Hastad.SourceMachine.fieldInTime Tape.input Tape.vertices (by decide) |
| .startVertices .readVertices (some .startDarts) (program headerPlan rowPlan) |
| rfl rfl b₁ n (encodeWord m ++ rowsBits) |
| (by simp [b₁, headerFrame, word, encodeWords, List.append_assoc]) (ambient, ()) none |
| have p₁' : StateTransition.EvalsToInTime (TM2.step (program headerPlan rowPlan)) |
| ⟨some .startVertices, ((ambient, ()), none), b₁⟩ |
| (some ⟨some .startDarts, ((ambient, ()), none), b₂⟩) (n + 2) := by |
| simpa only [h₁] using p₁ |
| let p₂ := Hastad.SourceMachine.fieldInTime Tape.input Tape.darts (by decide) |
| .startDarts .readDarts (some .seedIndex) (program headerPlan rowPlan) |
| rfl rfl b₂ m rowsBits rfl (ambient, ()) none |
| have p₂' : StateTransition.EvalsToInTime (TM2.step (program headerPlan rowPlan)) |
| ⟨some .startDarts, ((ambient, ()), none), b₂⟩ |
| (some ⟨some .seedIndex, ((ambient, ()), none), b₃⟩) (m + 2) := by |
| simpa only [h₂] using p₂ |
| let p₃ : StateTransition.EvalsToInTime (TM2.step (program headerPlan rowPlan)) |
| ⟨some .seedIndex, ((ambient, ()), none), b₃⟩ |
| (some ⟨some (.header (Emitter.labelAt headerPlan.length 36864 0 .entry)), |
| ((ambient, ()), none), b₄⟩) 1 := { |
| steps := 1 |
| evals_in_steps := by |
| simp only [Function.iterate_one] |
| change TM2.step (program headerPlan rowPlan) |
| ⟨some .seedIndex, ((ambient, ()), none), b₃⟩ = _ |
| have ht : Function.update (Function.update (Function.update b₃ Tape.rowIndex [false]) |
| Tape.tail [false]) Tape.head [false] = b₄ := by |
| funext tape; cases tape <;> simp [b₃, b₄, headerFrame, encodeWord] |
| simp only [TM2.step, program, TM2.stepAux] |
| apply congrArg some |
| exact congrArg (TM2.Cfg.mk _ _) ht |
| steps_le_m := Nat.le_refl _ } |
| let p₄ := Emitter.planInTime (Emitter.listCommands headerPlan) source Tape.scratch |
| Tape.accumulator source_ne_scratch source_ne_accumulator (by decide) |
| Label.header (some .headerClearTail) (program headerPlan rowPlan) (fun _ => rfl) |
| (values n m 0 0 0) b₄ |
| (by intro i; fin_cases i <;> rfl) rfl ambient word.length |
| (by |
| intro i |
| fin_cases i |
| · exact hn |
| · exact hm |
| · exact Nat.zero_le _ |
| · exact Nat.zero_le _ |
| · exact Nat.zero_le _) |
| have p₄' : StateTransition.EvalsToInTime (TM2.step (program headerPlan rowPlan)) |
| ⟨some (.header (Emitter.labelAt headerPlan.length 36864 0 .entry)), |
| ((ambient, ()), none), b₄⟩ |
| (some ⟨some .headerClearTail, ((ambient, ()), none), b₅⟩) |
| (7 * (3 * (word.length + 1) + 3) + 1) := by |
| have h := p₄ |
| rw [h₄] at h |
| simpa only [headerPlan_length] using h |
| let p₅ := MachineLookup.discardInTime Tape.tail .headerClearTail .headerClearHead |
| (program headerPlan rowPlan) rfl b₅ 0 [] (by simp [b₅, headerFrame]) (ambient, ()) none |
| have p₅' : StateTransition.EvalsToInTime (TM2.step (program headerPlan rowPlan)) |
| ⟨some .headerClearTail, ((ambient, ()), none), b₅⟩ |
| (some ⟨some .headerClearHead, ((ambient, ()), none), b₆⟩) 1 := by |
| simpa only [h₅] using p₅ |
| let p₆ := MachineLookup.discardInTime Tape.head .headerClearHead .guard |
| (program headerPlan rowPlan) rfl b₆ 0 [] (by simp [b₆, headerFrame]) (ambient, ()) none |
| have p₆' : StateTransition.EvalsToInTime (TM2.step (program headerPlan rowPlan)) |
| ⟨some .headerClearHead, ((ambient, ()), none), b₆⟩ |
| (some ⟨some .guard, ((ambient, ()), none), b₇⟩) 1 := by |
| simpa only [h₆] using p₆ |
| let p₀₁ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ p₀' p₁' |
| let p₀₁₂ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ p₀₁ p₂' |
| let p₀₁₂₃ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ p₀₁₂ p₃ |
| let p₀₁₂₃₄ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ p₀₁₂₃ p₄' |
| let p₀₁₂₃₄₅ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ p₀₁₂₃₄ p₅' |
| let p := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ p₀₁₂₃₄₅ p₆' |
| exact { |
| toEvalsTo := p.toEvalsTo |
| steps_le_m := by |
| have h := p.steps_le_m |
| change _ ≤ headerTimePolynomial.eval word.length |
| rw [headerTimePolynomial_eval] |
| omega } |
|
|
| def tableRowsBits (table : PCP.GenericGraphTables.Table 64) : List Bool := |
| encodeWords ((PCP.GenericGraphTables.rowList table).flatMap PCP.GenericGraphTables.rowWords) |
|
|
| theorem tableBits_header_rows (table : PCP.GenericGraphTables.Table 64) : |
| PCP.GenericGraphTables.tableBits table = |
| encodeWords [table.vertices, table.darts] ++ tableRowsBits table := by |
| simp only [PCP.GenericGraphTables.tableBits, PCP.GenericGraphTables.tableWords, |
| encodeWords_append, tableRowsBits] |
|
|
| def tableHeaderInTime (rowPlan : Plan) (table : PCP.GenericGraphTables.Table 64) |
| (ambient : Ambient) : |
| StateTransition.EvalsToInTime (TM2.step (program headerPlan rowPlan)) |
| ⟨some .copyFirst, ((ambient, ()), none), |
| headerInputTapes (PCP.GenericGraphTables.tableBits table)⟩ |
| (some ⟨some .guard, ((ambient, ()), none), |
| headerResultTapes table.vertices table.darts (tableRowsBits table)⟩) |
| (headerTimePolynomial.eval (PCP.GenericGraphTables.tableBits table).length) := by |
| rw [tableBits_header_rows] |
| exact headerInTime rowPlan table.vertices table.darts (tableRowsBits table) ambient |
|
|
| theorem headerInput_configuration (rowPlan : Plan) (input : List Bool) : |
| initList (machine headerPlan rowPlan) input = |
| ⟨some .copyFirst, ((((), fun _ => false), ()), none), headerInputTapes input⟩ := by |
| have ht : (initList (machine headerPlan rowPlan) input).stk = headerInputTapes input := by |
| funext tape |
| cases tape <;> simp [initList, machine, headerInputTapes, headerFrame] |
| all_goals rfl |
| exact congrArg (TM2.Cfg.mk _ _) ht |
|
|
| def initializedTableHeaderInTime (rowPlan : Plan) (table : PCP.GenericGraphTables.Table 64) : |
| StateTransition.EvalsToInTime (machine headerPlan rowPlan).step |
| (initList (machine headerPlan rowPlan) (PCP.GenericGraphTables.tableBits table)) |
| (some ⟨some .guard, ((((), fun _ => false), ()), none), |
| headerResultTapes table.vertices table.darts (tableRowsBits table)⟩) |
| (headerTimePolynomial.eval (PCP.GenericGraphTables.tableBits table).length) := by |
| rw [headerInput_configuration] |
| exact tableHeaderInTime rowPlan table ((), fun _ => false) |
|
|
| open Turing PCP PCP.AlphabetTable |
|
|
| def endpointTapes (base : Tape → List Bool) (tail reverse : Nat) (rest : List Bool) := |
| Hastad.SourceMachine.fieldTapes Tape.input Tape.reverseIndex |
| (Hastad.SourceMachine.fieldTapes Tape.input Tape.tail base |
| (encodeWord reverse ++ rest) (encodeWord tail ++ base .tail)) |
| rest (encodeWord reverse ++ base .reverseIndex) |
|
|
| def endpointsInTime (headerPlan plan : Plan) (base : Tape → List Bool) |
| (tail reverse : Nat) (rest : List Bool) |
| (hinput : base .input = encodeWord tail ++ encodeWord reverse ++ rest) |
| (ambient : Ambient) : |
| StateTransition.EvalsToInTime (TM2.step (program headerPlan plan)) |
| ⟨some .startTail, ((ambient, ()), none), base⟩ |
| (some ⟨some .headTableFirst, ((ambient, ()), none), |
| endpointTapes base tail reverse rest⟩) (tail + reverse + 4) := by |
| let first := Hastad.SourceMachine.fieldTapes Tape.input Tape.tail base |
| (encodeWord reverse ++ rest) (encodeWord tail ++ base .tail) |
| let run₁ := Hastad.SourceMachine.fieldInTime Tape.input Tape.tail (by decide) |
| .startTail .readTail (some .startReverse) (program headerPlan plan) rfl rfl |
| base tail (encodeWord reverse ++ rest) (by simpa only [List.append_assoc] using hinput) |
| (ambient, ()) none |
| have input₁ : first .input = encodeWord reverse ++ rest := by |
| simp [first, Hastad.SourceMachine.fieldTapes] |
| have reverse₁ : first .reverseIndex = base .reverseIndex := by |
| simp [first, Hastad.SourceMachine.fieldTapes] |
| let run₂ := Hastad.SourceMachine.fieldInTime Tape.input Tape.reverseIndex (by decide) |
| .startReverse .readReverse (some .headTableFirst) (program headerPlan plan) rfl rfl |
| first reverse rest input₁ (ambient, ()) none |
| have joined := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ run₁ run₂ |
| simpa only [first, reverse₁, endpointTapes, Nat.add_assoc, |
| Nat.add_left_comm, Nat.add_comm] using joined |
|
|
| def relationInTime (headerPlan plan : Plan) (base : Tape → List Bool) |
| (relation : GraphTables.RelationTable) (rest : List Bool) |
| (hinput : base .input = encodeWords (GraphTables.relationWords relation) ++ rest) |
| (ambient : Ambient) : |
| StateTransition.EvalsToInTime (TM2.step (program headerPlan plan)) |
| ⟨some .readRelation, ((ambient, ()), none), base⟩ |
| (some ⟨some (.row (Emitter.labelAt plan.length 36864 0 .entry)), |
| ((((), fun i => relation[i]), ()), none), Function.update base .input rest⟩) 1 where |
| steps := 1 |
| evals_in_steps := by |
| change some (TM2.stepAux (program headerPlan plan .readRelation) _ _) = _ |
| rw [program, stepAux_readRelationAt _ _ _ _ _ _ _ _ _ hinput] |
| steps_le_m := Nat.le_refl _ |
|
|
| def emitInTime (headerPlan plan : Plan) (base : Tape → List Bool) |
| (operands : Fin 5 → Nat) (hoperands : ∀ i, base (source i) = encodeWord (operands i)) |
| (hscratch : base .scratch = []) (ambient : Ambient) |
| (N : Nat) (hbounded : ∀ i, operands i ≤ N) : |
| StateTransition.EvalsToInTime (TM2.step (program headerPlan plan)) |
| ⟨some (.row (Emitter.labelAt plan.length 36864 0 .entry)), |
| ((ambient, ()), none), base⟩ |
| (some ⟨some .clearTail, ((ambient, ()), none), |
| Function.update base .accumulator |
| ((plan.flatMap (Emitter.commandBits operands ambient)).reverse ++ base .accumulator)⟩) |
| (plan.length * (3 * (N + 1) + 3) + 1) := by |
| have run := Emitter.planInTime (Emitter.listCommands plan) |
| source Tape.scratch Tape.accumulator source_ne_scratch source_ne_accumulator (by decide) |
| Label.row (some .clearTail) (program headerPlan plan) (fun _ => rfl) |
| operands base hoperands hscratch ambient N hbounded |
| simpa only [Emitter.bits_listCommands, Emitter.resultTapes] using run |
|
|
| def finishTapes (base : Tape → List Bool) (row : Nat) : Tape → List Bool := |
| Function.update |
| (Function.update (Function.update (Function.update base Tape.tail []) Tape.head []) |
| Tape.reverseIndex []) Tape.rowIndex (encodeWord (row + 1)) |
|
|
| def finishInTime (headerPlan plan : Plan) (base : Tape → List Bool) |
| (tail head reverse row : Nat) |
| (htail : base .tail = encodeWord tail) (hhead : base .head = encodeWord head) |
| (hreverse : base .reverseIndex = encodeWord reverse) |
| (hrow : base .rowIndex = encodeWord row) (ambient : Ambient) : |
| StateTransition.EvalsToInTime (TM2.step (program headerPlan plan)) |
| ⟨some .clearTail, ((ambient, ()), none), base⟩ |
| (some ⟨some .guard, ((ambient, ()), none), finishTapes base row⟩) |
| (tail + head + reverse + 4) := by |
| let t₁ := Function.update base Tape.tail [] |
| let t₂ := Function.update t₁ Tape.head [] |
| let t₃ := Function.update t₂ Tape.reverseIndex [] |
| let run₁ := MachineLookup.discardInTime Tape.tail .clearTail .clearHead |
| (program headerPlan plan) rfl base tail [] (by simpa using htail) (ambient, ()) none |
| have hhead₁ : t₁ .head = encodeWord head := by simp [t₁, hhead] |
| let run₂ := MachineLookup.discardInTime Tape.head .clearHead .clearReverse |
| (program headerPlan plan) rfl t₁ head [] (by simpa using hhead₁) (ambient, ()) none |
| have hreverse₂ : t₂ .reverseIndex = encodeWord reverse := by simp [t₂, t₁, hreverse] |
| let run₃ := MachineLookup.discardInTime Tape.reverseIndex .clearReverse .nextRow |
| (program headerPlan plan) rfl t₂ reverse [] (by simpa using hreverse₂) (ambient, ()) none |
| let run₄ : StateTransition.EvalsToInTime (TM2.step (program headerPlan plan)) |
| ⟨some .nextRow, ((ambient, ()), none), t₃⟩ |
| (some ⟨some .guard, ((ambient, ()), none), finishTapes base row⟩) 1 := { |
| steps := 1 |
| evals_in_steps := by |
| change some (TM2.stepAux (program headerPlan plan .nextRow) _ _) = _ |
| simp only [program, TM2.stepAux] |
| congr 2 |
| simp [t₃, t₂, t₁, finishTapes, hrow, encodeWord, List.replicate_succ] |
| steps_le_m := Nat.le_refl _ } |
| let r₁₂ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ run₁ run₂ |
| let r₁₂₃ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ r₁₂ run₃ |
| have run := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ r₁₂₃ run₄ |
| simpa only [Nat.add_assoc, Nat.add_left_comm, Nat.add_comm] using run |
|
|
| end MinUncutGames.Foundations.Complexity.FinalCNFMachine.Program |
|
|
| namespace MinUncutGames.Foundations.Complexity.FinalCNFTableAdapter |
|
|
| open PCP |
|
|
| def genericRow {n m : Nat} (row : GraphTables.DartRow n m) : |
| GenericGraphTables.DartRow 64 n m := |
| ⟨row.tail, row.reverseIndex, row.relation⟩ |
|
|
| @[simp] theorem genericRow_tail {n m : Nat} (row : GraphTables.DartRow n m) : |
| (genericRow row).tail = row.tail := rfl |
|
|
| @[simp] theorem genericRow_reverseIndex {n m : Nat} (row : GraphTables.DartRow n m) : |
| (genericRow row).reverseIndex = row.reverseIndex := rfl |
|
|
| @[simp] theorem genericRow_relation {n m : Nat} (row : GraphTables.DartRow n m) : |
| (genericRow row).relation = row.relation := rfl |
|
|
| def genericRows {n m : Nat} (rows : GraphTables.Rows n m) : |
| GenericGraphTables.Rows 64 n m := rows.map genericRow |
|
|
| @[simp] theorem genericRows_get {n m : Nat} (rows : GraphTables.Rows n m) (e : Fin m) : |
| (genericRows rows)[e] = genericRow rows[e] := by |
| simp [genericRows] |
|
|
| @[simp] theorem genericRows_reverseAt {n m : Nat} (rows : GraphTables.Rows n m) |
| (e : Fin m) : |
| GenericGraphTables.reverseAt (genericRows rows) e = GraphTables.reverseAt rows e := by |
| change (genericRows rows)[e].reverseIndex = rows[e].reverseIndex |
| rw [genericRows_get] |
| rfl |
|
|
| @[simp] theorem genericRows_acceptsAt {n m : Nat} (rows : GraphTables.Rows n m) |
| (e : Fin m) (a b : Fin 64) : |
| GenericGraphTables.acceptsAt (genericRows rows) e a b = |
| GraphTables.acceptsAt rows e a b := by |
| simp only [GenericGraphTables.acceptsAt, genericRows_get, genericRow_relation] |
| rfl |
|
|
| theorem genericRows_valid {n m : Nat} (rows : GraphTables.Rows n m) |
| (valid : GraphTables.Valid rows) : GenericGraphTables.Valid (genericRows rows) := by |
| constructor |
| · intro e |
| simpa only [genericRows_reverseAt] using valid.1 e |
| · intro e a b |
| simpa only [genericRows_reverseAt, genericRows_acceptsAt] using valid.2 e a b |
|
|
| def genericTable (table : GraphTables.Table) : GenericGraphTables.Table 64 where |
| vertices := table.vertices |
| darts := table.darts |
| rows := genericRows table.rows |
| valid := genericRows_valid table.rows table.valid |
|
|
| @[simp] theorem genericTable_vertices (table : GraphTables.Table) : |
| (genericTable table).vertices = table.vertices := rfl |
|
|
| @[simp] theorem genericTable_darts (table : GraphTables.Table) : |
| (genericTable table).darts = table.darts := rfl |
|
|
| @[simp] theorem genericTable_row (table : GraphTables.Table) (e : Fin table.darts) : |
| (genericTable table).rows[e] = genericRow table.rows[e] := genericRows_get _ _ |
|
|
| @[simp] theorem genericTable_tail (table : GraphTables.Table) (e : Fin table.darts) : |
| (genericTable table).rows[e].tail = table.rows[e].tail := by |
| rw [genericTable_row] |
| rfl |
|
|
| @[simp] theorem genericTable_reverseIndex (table : GraphTables.Table) (e : Fin table.darts) : |
| (genericTable table).rows[e].reverseIndex = table.rows[e].reverseIndex := by |
| rw [genericTable_row] |
| rfl |
|
|
| @[simp] theorem genericTable_relation (table : GraphTables.Table) (e : Fin table.darts) : |
| (genericTable table).rows[e].relation = table.rows[e].relation := by |
| rw [genericTable_row] |
| rfl |
|
|
| @[simp] theorem genericRow_words {n m : Nat} (row : GraphTables.DartRow n m) : |
| GenericGraphTables.rowWords (genericRow row) = GraphTables.rowWords row := rfl |
|
|
| theorem genericTable_rowList (table : GraphTables.Table) : |
| GenericGraphTables.rowList (genericTable table) = |
| (GraphTables.rowList table).map genericRow := by |
| simp only [GenericGraphTables.rowList, GraphTables.rowList, genericTable, |
| genericRows, Vector.toList_map] |
|
|
| @[simp] theorem genericTable_tableWords (table : GraphTables.Table) : |
| GenericGraphTables.tableWords (genericTable table) = GraphTables.tableWords table := by |
| simp only [GenericGraphTables.tableWords, GraphTables.tableWords, |
| genericTable_vertices, genericTable_darts, genericTable_rowList, List.flatMap_map, |
| genericRow_words] |
|
|
| @[simp] theorem genericTable_tableBits (table : GraphTables.Table) : |
| GenericGraphTables.tableBits (genericTable table) = GraphTables.tableBits table := by |
| simp only [GenericGraphTables.tableBits, GraphTables.tableBits, genericTable_tableWords] |
|
|
| @[simp] theorem genericTable_semantics_tail (table : GraphTables.Table) (e : Fin table.darts) : |
| (GenericGraphTables.semantics (genericTable table)).tail e = |
| (GraphTables.semantics table).tail e := genericTable_tail table e |
|
|
| @[simp] theorem genericTable_semantics_reverse (table : GraphTables.Table) |
| (e : Fin table.darts) : |
| (GenericGraphTables.semantics (genericTable table)).reverse e = |
| (GraphTables.semantics table).reverse e := genericTable_reverseIndex table e |
|
|
| @[simp] theorem genericTable_semantics_head (table : GraphTables.Table) (e : Fin table.darts) : |
| (GenericGraphTables.semantics (genericTable table)).head e = |
| (GraphTables.semantics table).head e := by |
| simp only [ConstraintGraph.head, genericTable_semantics_reverse, genericTable_semantics_tail] |
| rfl |
|
|
| @[simp] theorem genericTable_semantics_accepts (table : GraphTables.Table) |
| (e : Fin table.darts) (a b : Fin 64) : |
| (GenericGraphTables.semantics (genericTable table)).accepts e a b = |
| (GraphTables.semantics table).accepts e a b := genericRows_acceptsAt _ _ _ _ |
|
|
| @[simp] theorem genericTable_headIndex (table : GraphTables.Table) (e : Fin table.darts) : |
| AlphabetTable.Lookup.headIndex (genericTable table) e = table.rows[e].reverseIndex := |
| genericTable_reverseIndex table e |
|
|
| @[simp] theorem genericTable_headValue (table : GraphTables.Table) (e : Fin table.darts) : |
| AlphabetTable.Lookup.headValue (genericTable table) e = |
| ((GraphTables.semantics table).head e).val := by |
| exact (AlphabetTable.Lookup.headValue_eq_semantics (genericTable table) e).trans |
| (congrArg Fin.val (genericTable_semantics_head table e)) |
|
|
| theorem genericTable_headValue_tableGraph (table : GraphTables.Table) (e : Fin table.darts) : |
| AlphabetTable.Lookup.headValue (genericTable table) e = |
| ((FinalCNFPattern.tableGraph table).head e).val := by |
| rw [genericTable_headValue] |
| rfl |
|
|
| theorem genericTable_pattern_relation (table : GraphTables.Table) (e : Fin table.darts) |
| (p : VerifierToCNF.PatternIndex 12) : |
| (genericTable table).rows[e].relation[FinalCNFPattern.patternRelationIndex p] = |
| (FinalCNFPattern.tableVerifier table).accepts e (VerifierToCNF.patternAt 12 p) := by |
| rw [genericTable_relation] |
| exact FinalCNFPattern.pattern_lookup_eq_verifier_accepts table e p |
|
|
| end MinUncutGames.Foundations.Complexity.FinalCNFTableAdapter |
|
|
| namespace MinUncutGames.Foundations.Complexity.FinalCNFMachine.Program |
|
|
| open Turing PCP PCP.AlphabetTable FinalCNFTableAdapter |
|
|
| @[simp] theorem headRoles_zero : headRoles 0 = Tape.archive := rfl |
| @[simp] theorem headRoles_one : headRoles 1 = Tape.reverseIndex := rfl |
| @[simp] theorem headRoles_two : headRoles 2 = Tape.scanWork := rfl |
| @[simp] theorem headRoles_three : headRoles 3 = Tape.indexWork := rfl |
| @[simp] theorem headRoles_four : headRoles 4 = Tape.head := rfl |
| @[simp] theorem headRoles_five : headRoles 5 = Tape.scratch := rfl |
|
|
| def rowOperands (table : GraphTables.Table) (e : Fin table.darts) : Fin 5 → Nat := |
| values table.vertices table.darts table.rows[e].tail.val |
| table.rows[table.rows[e].reverseIndex].tail.val e.val |
|
|
| def rowRelation (table : GraphTables.Table) (e : Fin table.darts) : Ambient := |
| ((), fun i => table.rows[e].relation[i]) |
|
|
| def rowRest (table : GraphTables.Table) (e : Fin table.darts) (rest : List Bool) : List Bool := |
| encodeWords (GraphTables.relationWords table.rows[e].relation) ++ rest |
|
|
| def preparedTapes (base : Tape → List Bool) (table : GraphTables.Table) |
| (e : Fin table.darts) (rest : List Bool) : Tape → List Bool := |
| Lookup.headLookupTapes headRoles |
| (endpointTapes base table.rows[e].tail.val table.rows[e].reverseIndex.val |
| (rowRest table e rest)) (genericTable table) e |
|
|
| theorem prepared_input (base : Tape → List Bool) (table : GraphTables.Table) |
| (e : Fin table.darts) (rest : List Bool) : |
| preparedTapes base table e rest .input = rowRest table e rest := by |
| simp [preparedTapes, Lookup.headLookupTapes, MachineLookup.tapes, endpointTapes, |
| Hastad.SourceMachine.fieldTapes] |
|
|
| theorem prepared_head (base : Tape → List Bool) (table : GraphTables.Table) |
| (e : Fin table.darts) (rest : List Bool) : |
| preparedTapes base table e rest .head = |
| encodeWord table.rows[table.rows[e].reverseIndex].tail.val := by |
| simp [preparedTapes, Lookup.headLookupTapes, MachineLookup.tapes, |
| genericTable_headValue, GraphTables.semantics, ConstraintGraph.head, |
| GraphTables.reverseAt] |
|
|
| theorem prepared_tail (base : Tape → List Bool) (table : GraphTables.Table) |
| (e : Fin table.darts) (rest : List Bool) : |
| preparedTapes base table e rest .tail = encodeWord table.rows[e].tail.val ++ base .tail := by |
| simp [preparedTapes, Lookup.headLookupTapes, MachineLookup.tapes, endpointTapes, |
| Hastad.SourceMachine.fieldTapes] |
|
|
| theorem prepared_reverse (base : Tape → List Bool) (table : GraphTables.Table) |
| (e : Fin table.darts) (rest : List Bool) : |
| preparedTapes base table e rest .reverseIndex = |
| encodeWord table.rows[e].reverseIndex.val ++ base .reverseIndex := by |
| simp [preparedTapes, Lookup.headLookupTapes, MachineLookup.tapes, endpointTapes, |
| Hastad.SourceMachine.fieldTapes] |
|
|
| theorem prepared_frame (base : Tape → List Bool) (table : GraphTables.Table) |
| (e : Fin table.darts) (rest : List Bool) (tape : Tape) |
| (hi : tape ≠ .input) (ht : tape ≠ .tail) (hr : tape ≠ .reverseIndex) |
| (hh : tape ≠ .head) (hs : tape ≠ .scanWork) (hx : tape ≠ .indexWork) : |
| preparedTapes base table e rest tape = base tape := by |
| simp [preparedTapes, Lookup.headLookupTapes, MachineLookup.tapes, endpointTapes, |
| Hastad.SourceMachine.fieldTapes, hi, ht, hr, hh, hs, hx] |
|
|
| def emittedTapes (plan : Plan) (base : Tape → List Bool) (table : GraphTables.Table) |
| (e : Fin table.darts) (rest : List Bool) : Tape → List Bool := |
| Function.update (Function.update (preparedTapes base table e rest) Tape.input rest) |
| Tape.accumulator ((plan.flatMap (Emitter.commandBits (rowOperands table e) |
| (rowRelation table e))).reverse ++ base .accumulator) |
|
|
| def rowResultTapes (plan : Plan) (base : Tape → List Bool) (table : GraphTables.Table) |
| (e : Fin table.darts) (rest : List Bool) : Tape → List Bool := |
| finishTapes (emittedTapes plan base table e rest) e.val |
|
|
| @[simp] theorem rowResult_input (plan : Plan) (base : Tape → List Bool) |
| (table : GraphTables.Table) (e : Fin table.darts) (rest : List Bool) : |
| rowResultTapes plan base table e rest .input = rest := by |
| simp [rowResultTapes, finishTapes, emittedTapes] |
|
|
| @[simp] theorem rowResult_tail (plan : Plan) (base : Tape → List Bool) |
| (table : GraphTables.Table) (e : Fin table.darts) (rest : List Bool) : |
| rowResultTapes plan base table e rest .tail = [] := by |
| simp [rowResultTapes, finishTapes] |
|
|
| @[simp] theorem rowResult_head (plan : Plan) (base : Tape → List Bool) |
| (table : GraphTables.Table) (e : Fin table.darts) (rest : List Bool) : |
| rowResultTapes plan base table e rest .head = [] := by |
| simp [rowResultTapes, finishTapes] |
|
|
| @[simp] theorem rowResult_reverse (plan : Plan) (base : Tape → List Bool) |
| (table : GraphTables.Table) (e : Fin table.darts) (rest : List Bool) : |
| rowResultTapes plan base table e rest .reverseIndex = [] := by |
| simp [rowResultTapes, finishTapes] |
|
|
| @[simp] theorem rowResult_index (plan : Plan) (base : Tape → List Bool) |
| (table : GraphTables.Table) (e : Fin table.darts) (rest : List Bool) : |
| rowResultTapes plan base table e rest .rowIndex = encodeWord (e.val + 1) := by |
| simp [rowResultTapes, finishTapes] |
|
|
| @[simp] theorem rowResult_accumulator (plan : Plan) (base : Tape → List Bool) |
| (table : GraphTables.Table) (e : Fin table.darts) (rest : List Bool) : |
| rowResultTapes plan base table e rest .accumulator = |
| (plan.flatMap (Emitter.commandBits (rowOperands table e) (rowRelation table e))).reverse ++ |
| base .accumulator := by |
| simp [rowResultTapes, finishTapes, emittedTapes] |
|
|
| theorem rowResult_frame (plan : Plan) (base : Tape → List Bool) |
| (table : GraphTables.Table) (e : Fin table.darts) (rest : List Bool) (tape : Tape) |
| (hi : tape ≠ .input) (ht : tape ≠ .tail) (hr : tape ≠ .reverseIndex) |
| (hh : tape ≠ .head) (hs : tape ≠ .scanWork) (hx : tape ≠ .indexWork) |
| (ha : tape ≠ .accumulator) (hc : tape ≠ .rowIndex) : |
| rowResultTapes plan base table e rest tape = base tape := by |
| simp only [rowResultTapes, finishTapes, emittedTapes, Function.update_of_ne hi, |
| Function.update_of_ne ht, Function.update_of_ne hr, Function.update_of_ne hh, |
| Function.update_of_ne ha, Function.update_of_ne hc] |
| exact prepared_frame base table e rest tape hi ht hr hh hs hx |
|
|
| def guardRowInTime (headerPlan plan : Plan) (base : Tape → List Bool) |
| (ambient : Ambient) (hinput : base .input ≠ []) : |
| StateTransition.EvalsToInTime (TM2.step (program headerPlan plan)) |
| ⟨some .guard, ((ambient, ()), none), base⟩ |
| (some ⟨some .startTail, ((ambient, ()), none), base⟩) 1 where |
| steps := 1 |
| evals_in_steps := by |
| change some (TM2.stepAux (program headerPlan plan .guard) _ _) = _ |
| cases hb : base .input with |
| | nil => exact False.elim (hinput hb) |
| | cons b bs => simp [program, guard, TM2.stepAux, hb] |
| steps_le_m := Nat.le_refl _ |
|
|
| def guardEndInTime (headerPlan plan : Plan) (base : Tape → List Bool) |
| (ambient : Ambient) (hinput : base .input = []) : |
| StateTransition.EvalsToInTime (TM2.step (program headerPlan plan)) |
| ⟨some .guard, ((ambient, ()), none), base⟩ |
| (some ⟨some .reverseOutput, ((ambient, ()), none), base⟩) 1 where |
| steps := 1 |
| evals_in_steps := by |
| change some (TM2.stepAux (program headerPlan plan .guard) _ _) = _ |
| simp [program, guard, TM2.stepAux, hinput] |
| steps_le_m := Nat.le_refl _ |
|
|
| theorem prepared_operands (base : Tape → List Bool) (table : GraphTables.Table) |
| (e : Fin table.darts) (rest : List Bool) |
| (hn : base .vertices = encodeWord table.vertices) |
| (hm : base .darts = encodeWord table.darts) |
| (hr : base .rowIndex = encodeWord e.val) |
| (ht : base .tail = []) : |
| ∀ i, (Function.update (preparedTapes base table e rest) Tape.input rest) (source i) = |
| encodeWord (rowOperands table e i) := by |
| intro i |
| fin_cases i |
| · simpa [source, rowOperands, values] using |
| prepared_frame base table e rest .vertices (by decide) (by decide) (by decide) |
| (by decide) (by decide) (by decide) |>.trans hn |
| · change (Function.update (preparedTapes base table e rest) Tape.input rest) Tape.darts = _ |
| simp only [Function.update_of_ne (show Tape.darts ≠ Tape.input by decide)] |
| exact (prepared_frame base table e rest .darts (by decide) (by decide) (by decide) |
| (by decide) (by decide) (by decide)).trans hm |
| · change (Function.update (preparedTapes base table e rest) Tape.input rest) Tape.tail = _ |
| simp only [Function.update_of_ne (show Tape.tail ≠ Tape.input by decide), prepared_tail, |
| ht, List.append_nil] |
| rfl |
| · change (Function.update (preparedTapes base table e rest) Tape.input rest) Tape.head = _ |
| simp only [Function.update_of_ne (show Tape.head ≠ Tape.input by decide), prepared_head] |
| rfl |
| · change (Function.update (preparedTapes base table e rest) Tape.input rest) Tape.rowIndex = _ |
| simp only [Function.update_of_ne (show Tape.rowIndex ≠ Tape.input by decide)] |
| exact (prepared_frame base table e rest .rowIndex (by decide) (by decide) (by decide) |
| (by decide) (by decide) (by decide)).trans hr |
|
|
| theorem rowOperands_bounded (table : GraphTables.Table) (e : Fin table.darts) : |
| ∀ i, rowOperands table e i ≤ (GraphTables.tableBits table).length := by |
| intro i |
| fin_cases i |
| · exact GraphTables.vertices_le_tableBits_length table |
| · exact GraphTables.darts_le_tableBits_length table |
| · exact table.rows[e].tail.isLt.le.trans (GraphTables.vertices_le_tableBits_length table) |
| · exact table.rows[table.rows[e].reverseIndex].tail.isLt.le.trans |
| (GraphTables.vertices_le_tableBits_length table) |
| · exact e.isLt.le.trans (GraphTables.darts_le_tableBits_length table) |
|
|
| def rowTime (plan : Plan) (N : Nat) : Nat := |
| plan.length * (3 * (N + 1) + 3) + 12 * N + 15 |
|
|
| noncomputable def rowInTime (headerPlan plan : Plan) (base : Tape → List Bool) |
| (table : GraphTables.Table) (e : Fin table.darts) (rest : List Bool) |
| (hinput : base .input = encodeWords (GraphTables.rowWords table.rows[e]) ++ rest) |
| (harchive : base .archive = GraphTables.tableBits table) |
| (hn : base .vertices = encodeWord table.vertices) |
| (hm : base .darts = encodeWord table.darts) |
| (hrow : base .rowIndex = encodeWord e.val) |
| (ht : base .tail = []) (hh : base .head = []) (hr : base .reverseIndex = []) |
| (hscratch : base .scratch = []) (ambient : Ambient) : |
| StateTransition.EvalsToInTime (TM2.step (program headerPlan plan)) |
| ⟨some .startTail, ((ambient, ()), none), base⟩ |
| (some ⟨some .guard, ((rowRelation table e, ()), none), |
| rowResultTapes plan base table e rest⟩) |
| (rowTime plan (GraphTables.tableBits table).length) := by |
| let parsed := endpointTapes base table.rows[e].tail.val table.rows[e].reverseIndex.val |
| (rowRest table e rest) |
| let ready := preparedTapes base table e rest |
| let read := Function.update ready Tape.input rest |
| let output := emittedTapes plan base table e rest |
| let run₁ := endpointsInTime headerPlan plan base table.rows[e].tail.val |
| table.rows[e].reverseIndex.val (rowRest table e rest) |
| (by simpa [GraphTables.rowWords, encodeWords, List.append_assoc, rowRest] using hinput) ambient |
| have parsed_archive : parsed .archive = GenericGraphTables.tableBits (genericTable table) := by |
| simpa [parsed, endpointTapes, Hastad.SourceMachine.fieldTapes] using harchive |
| have parsed_reverse : parsed .reverseIndex = |
| encodeWord (Lookup.headIndex (genericTable table) e).val := by |
| simp [parsed, endpointTapes, Hastad.SourceMachine.fieldTapes, hr] |
| have parsed_head : parsed .head = [] := by |
| simp [parsed, endpointTapes, Hastad.SourceMachine.fieldTapes, hh] |
| have parsed_scratch : parsed .scratch = [] := by |
| simp [parsed, endpointTapes, Hastad.SourceMachine.fieldTapes, hscratch] |
| let run₂ := headPhaseInTime headerPlan plan parsed (genericTable table) e |
| parsed_archive parsed_reverse parsed_head parsed_scratch ambient |
| let run₃ := relationInTime headerPlan plan ready table.rows[e].relation rest |
| (prepared_input base table e rest) ambient |
| have read_operands : ∀ i, read (source i) = encodeWord (rowOperands table e i) := |
| prepared_operands base table e rest hn hm hrow ht |
| have read_scratch : read .scratch = [] := by |
| simp only [read, Function.update_of_ne (show Tape.scratch ≠ Tape.input by decide)] |
| exact (prepared_frame base table e rest .scratch (by decide) (by decide) (by decide) |
| (by decide) (by decide) (by decide)).trans hscratch |
| have read_acc : read .accumulator = base .accumulator := by |
| simp only [read, Function.update_of_ne (show Tape.accumulator ≠ Tape.input by decide)] |
| exact prepared_frame base table e rest .accumulator (by decide) (by decide) (by decide) |
| (by decide) (by decide) (by decide) |
| have run₄ : StateTransition.EvalsToInTime (TM2.step (program headerPlan plan)) |
| ⟨some (.row (Emitter.labelAt plan.length 36864 0 .entry)), |
| ((rowRelation table e, ()), none), read⟩ |
| (some ⟨some .clearTail, ((rowRelation table e, ()), none), output⟩) |
| (plan.length * (3 * ((GraphTables.tableBits table).length + 1) + 3) + 1) := by |
| simpa only [read_acc, output, emittedTapes, read, ready] using |
| emitInTime headerPlan plan read (rowOperands table e) read_operands read_scratch |
| (rowRelation table e) (GraphTables.tableBits table).length (rowOperands_bounded table e) |
| have output_tail : output .tail = encodeWord table.rows[e].tail.val := by |
| simp [output, emittedTapes, prepared_tail, ht] |
| have output_head : output .head = encodeWord table.rows[table.rows[e].reverseIndex].tail.val := by |
| simp [output, emittedTapes, prepared_head] |
| have output_reverse : output .reverseIndex = encodeWord table.rows[e].reverseIndex.val := by |
| simp [output, emittedTapes, prepared_reverse, hr] |
| have output_row : output .rowIndex = encodeWord e.val := by |
| simp only [output, emittedTapes, |
| Function.update_of_ne (show Tape.rowIndex ≠ Tape.accumulator by decide), |
| Function.update_of_ne (show Tape.rowIndex ≠ Tape.input by decide)] |
| exact (prepared_frame base table e rest .rowIndex (by decide) (by decide) (by decide) |
| (by decide) (by decide) (by decide)).trans hrow |
| let run₅ := finishInTime headerPlan plan output table.rows[e].tail.val |
| table.rows[table.rows[e].reverseIndex].tail.val table.rows[e].reverseIndex.val e.val |
| output_tail output_head output_reverse output_row (rowRelation table e) |
| let r₁₂ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ run₁ run₂ |
| let r₁₂₃ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ r₁₂ run₃ |
| let r₁₂₃₄ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ r₁₂₃ run₄ |
| let run := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ r₁₂₃₄ run₅ |
| refine { toEvalsTo := run.toEvalsTo, steps_le_m := ?_ } |
| have hb := run.steps_le_m |
| have htbound := rowOperands_bounded table e 2 |
| have hhbound := rowOperands_bounded table e 3 |
| have hrbound := table.rows[e].reverseIndex.isLt.le.trans (GraphTables.darts_le_tableBits_length table) |
| change table.rows[e].tail.val ≤ _ at htbound |
| change table.rows[table.rows[e].reverseIndex].tail.val ≤ _ at hhbound |
| simp only [Lookup.headTimePolynomial, Polynomial.eval_add, Polynomial.eval_mul, |
| Polynomial.eval_C, Polynomial.eval_X, genericTable_tableBits] at hb |
| unfold rowTime |
| omega |
|
|
| end MinUncutGames.Foundations.Complexity.FinalCNFMachine.Program |
|
|
| end OAI |
|
|
