| import Mathlib.Algebra.BigOperators.Fin |
| import OAI.Computability.UniqueGames.Machines.MachineCompare |
| import OAI.Computability.UniqueGames.Machines.MachineCompositionLemmas |
| import OAI.Computability.UniqueGames.Machines.MachineCopy |
| import OAI.Computability.UniqueGames.Machines.MachineDrain |
| import OAI.Computability.UniqueGames.Machines.MachineLookupLemmas |
| import OAI.Computability.UniqueGames.PCP.InputLemmas |
|
|
| namespace OAI |
|
|
| /-! Actual finite-stack cleanup. Each selected stack is drained using the |
| checked pop loop, followed by one transition resetting the finite state. |
| Repeated selections are allowed; they only reduce the cost upper bound. -/ |
|
|
| namespace UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Cleanup |
|
|
| open Turing |
| open Complexity |
| open scoped BigOperators |
|
|
| variable {K Λ σ : Type} [DecidableEq K] |
|
|
| abbrev Alphabet (_ : K) := Bool |
|
|
| /-- Reset both the ambient finite state and the drain register before exiting. -/ |
| def finish (initial : σ) (exit : Option Λ) : |
| TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool) := |
| .load (fun _ => (initial, none)) |
| (match exit with | none => .halt | some label => .goto fun _ => label) |
|
|
| /-- A concrete instruction for each of the n drain phases and the final reset. -/ |
| def instruction {n : Nat} (selected : Fin n → K) (labels : Fin (n + 1) → Λ) |
| (initial : σ) (exit : Option Λ) (phase : Fin (n + 1)) : |
| TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool) := |
| if h : phase.val < n then |
| MachineDrain.drain (selected ⟨phase.val, h⟩) (labels phase) |
| (some (labels ⟨phase.val + 1, Nat.succ_lt_succ h⟩)) |
| else finish initial exit |
|
|
| omit [DecidableEq K] in |
| @[simp] theorem instruction_drain {n : Nat} (selected : Fin n → K) |
| (labels : Fin (n + 1) → Λ) (initial : σ) (exit : Option Λ) (i : Fin n) : |
| instruction selected labels initial exit i.castSucc = |
| MachineDrain.drain (selected i) (labels i.castSucc) (some (labels i.succ)) := by |
| simp [instruction, i.isLt] |
| rfl |
|
|
| omit [DecidableEq K] in |
| @[simp] theorem instruction_finish {n : Nat} (selected : Fin n → K) |
| (labels : Fin (n + 1) → Λ) (initial : σ) (exit : Option Λ) : |
| instruction selected labels initial exit (Fin.last n) = finish initial exit := by |
| simp [instruction] |
|
|
| /-- Actual tape family after the selected stacks have been drained in order. -/ |
| def clearTapes : {n : Nat} → (Fin n → K) → (K → List Bool) → K → List Bool |
| | 0, _, base => base |
| | n + 1, selected, base => |
| clearTapes (fun i : Fin n => selected i.succ) (Function.update base (selected 0) []) |
|
|
| /-- Exact number of transitions, including the final state-reset transition. -/ |
| def actualCost : {n : Nat} → (Fin n → K) → (K → List Bool) → Nat |
| | 0, _, _ => 1 |
| | n + 1, selected, base => |
| actualCost (fun i : Fin n => selected i.succ) (Function.update base (selected 0) []) + |
| ((base (selected 0)).length + 1) |
|
|
| theorem clearTapes_apply {n : Nat} (selected : Fin n → K) (base : K → List Bool) (k : K) : |
| clearTapes selected base k = if ∃ i, selected i = k then [] else base k := by |
| induction n generalizing base with |
| | zero => simp [clearTapes] |
| | succ n ih => |
| rw [clearTapes, ih] |
| by_cases first : selected 0 = k |
| · subst k |
| simp [Function.update] |
| · have notFirst : k ≠ selected 0 := Ne.symm first |
| by_cases later : ∃ i : Fin n, selected i.succ = k <;> |
| simp [Fin.exists_fin_succ, first, later, Function.update, notFirst] |
|
|
| @[simp] theorem clearTapes_selected {n : Nat} (selected : Fin n → K) |
| (base : K → List Bool) (i : Fin n) : clearTapes selected base (selected i) = [] := by |
| rw [clearTapes_apply, ite_eq_left ⟨i, rfl⟩] |
|
|
| theorem clearTapes_unselected {n : Nat} (selected : Fin n → K) |
| (base : K → List Bool) (k : K) (notSelected : ∀ i, selected i ≠ k) : |
| clearTapes selected base k = base k := by |
| rw [clearTapes_apply] |
| simp [notSelected] |
|
|
| theorem actualCost_le {n : Nat} (selected : Fin n → K) (base : K → List Bool) : |
| actualCost selected base ≤ (∑ i, (base (selected i)).length) + n + 1 := by |
| induction n generalizing base with |
| | zero => simp [actualCost] |
| | succ n ih => |
| have tailBound := ih (fun i : Fin n => selected i.succ) |
| (Function.update base (selected 0) []) |
| have sumBound : |
| (∑ i : Fin n, ((Function.update base (selected 0) []) (selected i.succ)).length) ≤ |
| ∑ i : Fin n, (base (selected i.succ)).length := by |
| apply Finset.sum_le_sum |
| intro i _ |
| by_cases h : selected i.succ = selected 0 <;> simp [h] |
| rw [actualCost, Fin.sum_univ_succ] |
| omega |
|
|
| /-- The trace is derived solely from the actual drain instructions and final |
| reset instruction. No execution of the entire cleanup is assumed. -/ |
| theorem cleanupTrace {n : Nat} (selected : Fin n → K) (labels : Fin (n + 1) → Λ) |
| (initial : σ) (exit : Option Λ) |
| (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)) |
| (atDrain : ∀ i : Fin n, program (labels i.castSucc) = |
| MachineDrain.drain (selected i) (labels i.castSucc) (some (labels i.succ))) |
| (atFinish : program (labels (Fin.last n)) = finish initial exit) |
| (base : K → List Bool) (ambient : σ) (register : Option Bool) : |
| (MachineComposition.advance (TM2.step program))^[actualCost selected base] |
| (some ⟨some (labels 0), (ambient, register), base⟩) = |
| some ⟨exit, (initial, none), clearTapes selected base⟩ := by |
| induction n generalizing base ambient register with |
| | zero => |
| change some (TM2.stepAux (program (labels 0)) (ambient, register) base) = _ |
| rw [show program (labels 0) = finish initial exit from atFinish] |
| cases exit <;> rfl |
| | succ n ih => |
| rw [actualCost, Function.iterate_add_apply] |
| have first := MachineDrain.drainTrace (selected 0) (labels (0 : Fin (n + 1)).castSucc) |
| (some (labels (0 : Fin (n + 1)).succ)) program (atDrain 0) |
| base (base (selected 0)) ambient register |
| simp only [Function.update_eq_self, Fin.castSucc_zero] at first |
| rw [first] |
| exact ih (fun i : Fin n => selected i.succ) (fun i => labels i.succ) |
| (fun i => atDrain i.succ) atFinish (Function.update base (selected 0) []) ambient none |
|
|
| /-- Actual cleanup execution, bounded using the original selected-stack lengths. -/ |
| def cleanupInTime {n : Nat} (selected : Fin n → K) (labels : Fin (n + 1) → Λ) |
| (initial : σ) (exit : Option Λ) |
| (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)) |
| (atDrain : ∀ i : Fin n, program (labels i.castSucc) = |
| MachineDrain.drain (selected i) (labels i.castSucc) (some (labels i.succ))) |
| (atFinish : program (labels (Fin.last n)) = finish initial exit) |
| (base : K → List Bool) (ambient : σ) (register : Option Bool) : |
| StateTransition.EvalsToInTime (TM2.step program) |
| ⟨some (labels 0), (ambient, register), base⟩ |
| (some ⟨exit, (initial, none), clearTapes selected base⟩) |
| ((∑ i, (base (selected i)).length) + n + 1) where |
| steps := actualCost selected base |
| evals_in_steps := cleanupTrace selected labels initial exit program atDrain atFinish |
| base ambient register |
| steps_le_m := actualCost_le selected base |
|
|
| /-- Concrete n+1-label program, terminating after resetting the finite state. -/ |
| def program {n : Nat} (selected : Fin n → K) (initial : σ) : |
| Fin (n + 1) → TM2.Stmt (Alphabet (K := K)) (Fin (n + 1)) (σ × Option Bool) := |
| instruction selected id initial none |
|
|
| /-- A bundled finite machine. Its input/output tapes are designated explicitly. -/ |
| def machine [Fintype K] [Fintype σ] {n : Nat} (selected : Fin n → K) |
| (input output : K) (initial : σ) : FinTM2 where |
| K := K |
| k₀ := input |
| k₁ := output |
| Γ := Alphabet |
| Λ := Fin (n + 1) |
| main := 0 |
| σ := σ × Option Bool |
| initialState := (initial, none) |
| m := program selected initial |
|
|
| def machineInTime [Fintype K] [Fintype σ] {n : Nat} (selected : Fin n → K) |
| (input output : K) (initial : σ) (base : K → List Bool) |
| (ambient : σ) (register : Option Bool) : |
| StateTransition.EvalsToInTime (machine selected input output initial).step |
| ⟨some (0 : Fin (n + 1)), (ambient, register), base⟩ |
| (some ⟨none, (initial, none), clearTapes selected base⟩) |
| ((∑ i, (base (selected i)).length) + n + 1) := |
| cleanupInTime selected id initial none (program selected initial) |
| (fun i => instruction_drain selected id initial none i) |
| (instruction_finish selected id initial none) base ambient register |
|
|
| /-- Enumerating exactly the nonoutput stacks yields the standard final tape |
| family. Repeated selected stacks are permitted. -/ |
| theorem clearTapes_outputOnly {n : Nat} (selected : Fin n → K) |
| (output : K) (base : K → List Bool) |
| (excludesOutput : ∀ i, selected i ≠ output) |
| (coversOther : ∀ k, k ≠ output → ∃ i, selected i = k) : |
| clearTapes selected base = fun k => if k = output then base output else [] := by |
| funext k |
| by_cases h : k = output |
| · subst k |
| simp [clearTapes_unselected selected base output excludesOutput] |
| · rw [clearTapes_apply] |
| simp [h, coversOther k h] |
|
|
| /-- Full cleanup reaches `haltList`, with all nonoutput stacks empty and the |
| machine's actual initial finite state restored. -/ |
| def machineHaltInTime [Fintype K] [Fintype σ] {n : Nat} (selected : Fin n → K) |
| (input output : K) (initial : σ) (base : K → List Bool) |
| (ambient : σ) (register : Option Bool) |
| (excludesOutput : ∀ i, selected i ≠ output) |
| (coversOther : ∀ k, k ≠ output → ∃ i, selected i = k) : |
| StateTransition.EvalsToInTime (machine selected input output initial).step |
| ⟨some (0 : Fin (n + 1)), (ambient, register), base⟩ |
| (some (haltList (machine selected input output initial) (base output))) |
| ((∑ i, (base (selected i)).length) + n + 1) := by |
| have tapeEq := clearTapes_outputOnly selected output base excludesOutput coversOther |
| have haltEq : (⟨none, (initial, none), clearTapes selected base⟩ : |
| (machine selected input output initial).Cfg) = |
| haltList (machine selected input output initial) (base output) := by |
| congr 1 |
| rw [← haltEq] |
| exact machineInTime selected input output initial base ambient register |
| |
| end UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Cleanup |
| |
| namespace UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Lookup |
| |
| open Turing |
| open UniqueGamesTheorem.Foundations.Complexity |
| open MachineComposition GenericGraphTables |
| |
| variable {K Λ σ : Type} [DecidableEq K] |
| |
| abbrev Alphabet (_ : K) := Bool |
| |
| /-- A fixed finite chain skips Boolean unary words. A true head causes one |
| additional delimiter pop; a false head is already the complete zero word. |
| Only the one-bit head register is examined. -/ |
| def skipBits (source : K) : Nat → |
| TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool) → |
| TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool) |
| | 0, next => .load (fun state => (state.1, none)) next |
| | count + 1, next => |
| let rest := skipBits source count next |
| .pop source (fun state head => (state.1, head)) |
| (.branch (fun state => state.2.getD false) |
| (.pop source (fun state _ => (state.1, none)) rest) |
| (.load (fun state => (state.1, none)) rest)) |
| |
| theorem stepAux_skipBits (source : K) (bits : List Bool) |
| (next : TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)) |
| (base : K → List Bool) (suffix : List Bool) (ambient : σ) (register : Option Bool) : |
| TM2.stepAux (skipBits source bits.length next) (ambient, register) |
| (Function.update base source (encodeWords (bits.map bitWord) ++ suffix)) = |
| TM2.stepAux next (ambient, none) (Function.update base source suffix) := by |
| induction bits generalizing register with |
| | nil => simp [skipBits, encodeWords, TM2.stepAux] |
| | cons bit bits ih => |
| cases bit <;> |
| simpa only [List.length_cons, skipBits, List.map_cons, bitWord, Bool.false_eq_true, |
| ite_false, ite_true, encodeWords, encodeWord, List.replicate_zero, |
| List.replicate_succ, List.nil_append, List.cons_append, List.singleton_append, |
| TM2.stepAux, Function.update_self, List.head?_cons, List.tail_cons, |
| Option.getD_some, Bool.cond_false, Bool.cond_true, Function.update_idem] using |
| ih none |
| |
| theorem stepAux_skipBits_fromTapes (source : K) (bits : List Bool) |
| (next : TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)) |
| (base : K → List Bool) (suffix : List Bool) |
| (hinput : base source = encodeWords (bits.map bitWord) ++ suffix) |
| (ambient : σ) (register : Option Bool) : |
| TM2.stepAux (skipBits source bits.length next) (ambient, register) base = |
| TM2.stepAux next (ambient, none) (Function.update base source suffix) := by |
| have h := stepAux_skipBits source bits next base suffix ambient register |
| have hbase : Function.update base source (encodeWords (bits.map bitWord) ++ suffix) = |
| base := by rw [← hinput]; exact Function.update_eq_self source base |
| rw [hbase] at h |
| exact h |
| |
| omit [DecidableEq K] in |
| theorem skipBits_pushBound (source : K) (count : Nat) |
| (next : TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)) : |
| Runtime.statementPushBound (skipBits source count next) = |
| Runtime.statementPushBound next := by |
| induction count with |
| | zero => rfl |
| | succ count ih => simp only [skipBits, Runtime.statementPushBound, ih, max_self] |
| |
| inductive Label |
| | guard | skipTail | skipReverse | skipPredicate | select | copy |
| deriving DecidableEq |
| |
| protected abbrev Label.enumList : List Label := [.guard, .skipTail, .skipReverse, .skipPredicate, |
| .select, .copy] |
| |
| protected theorem Label.enumList_getElem?_ctorIdx_eq (x : Label) : |
| Label.enumList[x.ctorIdx]? = some x := by |
| cases x <;> rfl |
| |
| protected theorem Label.enumList_nodup : Label.enumList.Nodup := by decide |
| |
| instance : Fintype Label where |
| elems := ⟨Label.enumList, Label.enumList_nodup⟩ |
| complete x := by cases x <;> decide |
| |
| /-- Six fixed control labels implement row selection. `q` fixes the finite |
| predicate parser; no data-dependent number is placed in the program syntax. -/ |
| def statement (q : Nat) (counter source destination : K) (labels : Label → Λ) |
| (exit : Option Λ) : Label → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool) |
| | .guard => MachineUnaryCounter.guard counter (labels .skipTail) (labels .select) |
| | .skipTail => MachineLookup.discard source (labels .skipTail) (labels .skipReverse) |
| | .skipReverse => MachineLookup.discard source (labels .skipReverse) (labels .skipPredicate) |
| | .skipPredicate => skipBits source (q * q) (.goto fun _ => labels .guard) |
| | .select => Hastad.SourceMachine.fieldStart destination (labels .copy) |
| | .copy => Hastad.SourceMachine.fieldLoop source destination (labels .copy) exit |
| |
| def program (q : Nat) (counter source destination : K) : |
| Label → TM2.Stmt (Alphabet (K := K)) Label (σ × Option Bool) := |
| statement q counter source destination id none |
| |
| def rowBits {q n m : Nat} (row : DartRow q n m) : List Bool := encodeWords (rowWords row) |
| |
| theorem rowBits_eq {q n m : Nat} (row : DartRow q n m) : |
| rowBits row = encodeWord row.tail.val ++ |
| (encodeWord row.reverseIndex.val ++ encodeWords (relationWords row.relation)) := by |
| simp [rowBits, rowWords, encodeWords] |
| |
| def rowsBits {q n m : Nat} (rows : List (DartRow q n m)) : List Bool := |
| encodeWords (rows.flatMap rowWords) |
| |
| @[simp] theorem rowsBits_nil {q n m : Nat} : rowsBits ([] : List (DartRow q n m)) = [] := rfl |
| |
| @[simp] theorem rowsBits_cons {q n m : Nat} (row : DartRow q n m) |
| (rows : List (DartRow q n m)) : rowsBits (row :: rows) = rowBits row ++ rowsBits rows := by |
| simp only [rowsBits, List.flatMap_cons, encodeWords_append, rowBits] |
| |
| def skipCost {q n m : Nat} (row : DartRow q n m) : Nat := |
| row.tail.val + row.reverseIndex.val + 4 |
| |
| def prefixCost {q n m : Nat} (rows : List (DartRow q n m)) : Nat := |
| (rows.map skipCost).sum |
| |
| def lookupCost {q n m : Nat} (prior : List (DartRow q n m)) (row : DartRow q n m) : Nat := |
| prefixCost prior + row.tail.val + 3 |
| |
| section Ambient |
| |
| variable {q n m : Nat} (counter source destination : K) (labels : Label → Λ) |
| (exit : Option Λ) |
| (P : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)) |
| (atStatement : ∀ label, P (labels label) = statement q counter source destination labels exit label) |
| |
| include atStatement |
| |
| /-- One skipped row: two actual unary scans followed by one finite predicate |
| statement. All registers are reset and every other tape is untouched. -/ |
| theorem skipRowTrace (base : K → List Bool) (row : DartRow q n m) (suffix : List Bool) |
| (ambient : σ) (register : Option Bool) : |
| (advance (TM2.step P))^[row.tail.val + row.reverseIndex.val + 3] |
| (some ⟨some (labels .skipTail), (ambient, register), |
| Function.update base source (rowBits row ++ suffix)⟩) = |
| some ⟨some (labels .guard), (ambient, none), Function.update base source suffix⟩ := by |
| have ht := MachineLookup.discardTrace_update source (labels .skipTail) (labels .skipReverse) |
| P (atStatement .skipTail) base row.tail.val |
| (encodeWord row.reverseIndex.val ++ (encodeWords (relationWords row.relation) ++ suffix)) |
| ambient register |
| have hr := MachineLookup.discardTrace_update source (labels .skipReverse) (labels .skipPredicate) |
| P (atStatement .skipReverse) base row.reverseIndex.val |
| (encodeWords (relationWords row.relation) ++ suffix) ambient none |
| have hp : TM2.step P |
| ⟨some (labels .skipPredicate), (ambient, none), Function.update base source |
| (encodeWords (relationWords row.relation) ++ suffix)⟩ = |
| some ⟨some (labels .guard), (ambient, none), Function.update base source suffix⟩ := by |
| change some (TM2.stepAux (P (labels .skipPredicate)) _ _) = _ |
| rw [atStatement .skipPredicate] |
| have h := congrArg some (stepAux_skipBits source row.relation.toList |
| (.goto fun _ => labels .guard) base suffix ambient none) |
| simpa only [statement, Vector.length_toList, relationWords, TM2.stepAux] using h |
| rw [rowBits_eq] |
| simp only [List.append_assoc] |
| rw [show row.tail.val + row.reverseIndex.val + 3 = |
| 1 + ((row.reverseIndex.val + 1) + (row.tail.val + 1)) by omega, |
| Function.iterate_add_apply, Function.iterate_add_apply, ht, hr, |
| Function.iterate_one, advance_some, hp] |
| |
| theorem skipRowTrace_fromTapes (base : K → List Bool) (row : DartRow q n m) |
| (suffix : List Bool) (hinput : base source = rowBits row ++ suffix) |
| (ambient : σ) (register : Option Bool) : |
| (advance (TM2.step P))^[row.tail.val + row.reverseIndex.val + 3] |
| (some ⟨some (labels .skipTail), (ambient, register), base⟩) = |
| some ⟨some (labels .guard), (ambient, none), Function.update base source suffix⟩ := by |
| have h := skipRowTrace counter source destination labels exit P atStatement |
| base row suffix ambient register |
| have hb : Function.update base source (rowBits row ++ suffix) = base := by |
| rw [← hinput] |
| exact Function.update_eq_self source base |
| rw [hb] at h |
| exact h |
| |
| theorem selectTrace (hsd : source ≠ destination) (base : K → List Bool) |
| (row : DartRow q n m) (suffix : List Bool) |
| (hinput : base source = rowBits row ++ suffix) (ambient : σ) (register : Option Bool) : |
| (advance (TM2.step P))^[row.tail.val + 2] |
| (some ⟨some (labels .select), (ambient, register), base⟩) = |
| some ⟨exit, (ambient, none), Hastad.SourceMachine.fieldTapes source destination base |
| (encodeWord row.reverseIndex.val ++ (encodeWords (relationWords row.relation) ++ suffix)) |
| (encodeWord row.tail.val ++ base destination)⟩ := by |
| let ht := Hastad.SourceMachine.fieldInTime source destination hsd |
| (labels .select) (labels .copy) exit P (atStatement .select) (atStatement .copy) |
| base row.tail.val |
| (encodeWord row.reverseIndex.val ++ (encodeWords (relationWords row.relation) ++ suffix)) |
| (by simpa only [rowBits_eq, List.append_assoc] using hinput) ambient register |
| have hsteps : ht.steps = row.tail.val + 2 := rfl |
| have htrace := ht.evals_in_steps |
| rw [hsteps] at htrace |
| exact htrace |
| |
| /-- Exact successful row lookup. The only data premise is the literal table |
| prefix; no lookup result or whole execution is postulated. -/ |
| theorem lookupTailTrace |
| (hcs : counter ≠ source) (hcd : counter ≠ destination) (hsd : source ≠ destination) |
| (base : K → List Bool) (prior : List (DartRow q n m)) (row : DartRow q n m) |
| (counterSuffix suffix output : List Bool) (ambient : σ) (register : Option Bool) : |
| (advance (TM2.step P))^[lookupCost prior row] |
| (some ⟨some (labels .guard), (ambient, register), |
| MachineLookup.tapes counter source destination base |
| (encodeWord prior.length ++ counterSuffix) |
| (rowsBits prior ++ (rowBits row ++ suffix)) output⟩) = |
| some ⟨exit, (ambient, none), |
| MachineLookup.tapes counter source destination base |
| (encodeWord 0 ++ counterSuffix) |
| (encodeWord row.reverseIndex.val ++ (encodeWords (relationWords row.relation) ++ suffix)) |
| (encodeWord row.tail.val ++ output)⟩ := by |
| induction prior generalizing register with |
| | nil => |
| simp only [lookupCost, prefixCost, List.map_nil, List.sum_nil, Nat.zero_add, |
| List.length_nil, rowsBits_nil, List.nil_append] |
| rw [show row.tail.val + 3 = (row.tail.val + 2) + 1 by omega, |
| Function.iterate_succ_apply] |
| have hg := MachineUnaryCounter.guardStep_zero counter (labels .guard) |
| (labels .skipTail) (labels .select) P (atStatement .guard) |
| (MachineLookup.tapes counter source destination base |
| (encodeWord 0 ++ counterSuffix) (rowBits row ++ suffix) output) |
| counterSuffix ambient register |
| have hcounter : MachineUnaryCounter.counterTapes counter |
| (MachineLookup.tapes counter source destination base |
| (encodeWord 0 ++ counterSuffix) (rowBits row ++ suffix) output) 0 counterSuffix = |
| MachineLookup.tapes counter source destination base |
| (encodeWord 0 ++ counterSuffix) (rowBits row ++ suffix) output := by |
| simp only [MachineUnaryCounter.counterTapes, MachineLookup.update_index _ _ _ hcs hcd] |
| rw [hcounter] at hg |
| simp only [advance_some] |
| rw [hg] |
| have hs := selectTrace counter source destination labels exit P atStatement hsd |
| (MachineLookup.tapes counter source destination base |
| (encodeWord 0 ++ counterSuffix) (rowBits row ++ suffix) output) |
| row suffix (by simp [hsd]) ambient none |
| simpa only [MachineLookup.tapes_destination, |
| MachineLookup.fieldTapes_eq _ _ _ hsd] using hs |
| | cons first prior ih => |
| simp only [lookupCost, prefixCost, List.map_cons, List.sum_cons, List.length_cons, |
| rowsBits_cons, List.append_assoc] |
| rw [show skipCost first + (prior.map skipCost).sum + row.tail.val + 3 = |
| ((prior.map skipCost).sum + row.tail.val + 3) + |
| (first.tail.val + first.reverseIndex.val + 3) + 1 by simp only [skipCost]; omega, |
| Function.iterate_succ_apply] |
| have hg := MachineUnaryCounter.guardStep_succ counter (labels .guard) |
| (labels .skipTail) (labels .select) P (atStatement .guard) |
| (MachineLookup.tapes counter source destination base |
| (encodeWord prior.length ++ counterSuffix) |
| (rowBits first ++ (rowsBits prior ++ (rowBits row ++ suffix))) output) |
| prior.length counterSuffix ambient register |
| simp only [MachineUnaryCounter.counterTapes, |
| MachineLookup.update_index _ _ _ hcs hcd] at hg |
| simp only [advance_some] |
| rw [hg, Function.iterate_add_apply] |
| have hs := skipRowTrace_fromTapes counter source destination labels exit P atStatement |
| (MachineLookup.tapes counter source destination base |
| (encodeWord prior.length ++ counterSuffix) |
| (rowBits first ++ (rowsBits prior ++ (rowBits row ++ suffix))) output) |
| first (rowsBits prior ++ (rowBits row ++ suffix)) (by simp [hsd]) ambient none |
| rw [hs] |
| simp only [MachineLookup.update_source _ _ _ hsd] |
| exact ih none |
| |
| end Ambient |
| |
| /-- The number of actual transitions is linear in the scanned encoding and |
| the unary row index. Fixed predicate statements contribute one transition. -/ |
| theorem lookupCost_le {q n m : Nat} (prior : List (DartRow q n m)) (row : DartRow q n m) : |
| lookupCost prior row ≤ (rowsBits prior ++ rowBits row).length + |
| 2 * (encodeWord prior.length).length + 1 := by |
| have hp : prefixCost prior ≤ (rowsBits prior).length + 2 * prior.length := by |
| induction prior with |
| | nil => simp [prefixCost] |
| | cons first prior ih => |
| have hb : skipCost first ≤ (rowBits first).length + 2 := by |
| simp only [skipCost, rowBits_eq, List.length_append, encodeWord_length] |
| omega |
| simp only [prefixCost, List.map_cons, List.sum_cons, rowsBits_cons, |
| List.length_append, List.length_cons] at * |
| omega |
| have hr : row.tail.val + 1 ≤ (rowBits row).length := by |
| simp only [rowBits_eq, List.length_append, encodeWord_length] |
| omega |
| simp only [lookupCost, List.length_append, encodeWord_length] |
| omega |
| |
| /-- The exact row-selection trace packaged with its linear transition budget. -/ |
| def lookupTailInTime {q n m : Nat} |
| (counter source destination : K) (labels : Label → Λ) (exit : Option Λ) |
| (P : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)) |
| (atStatement : ∀ label, P (labels label) = statement q counter source destination labels exit label) |
| (hcs : counter ≠ source) (hcd : counter ≠ destination) (hsd : source ≠ destination) |
| (base : K → List Bool) (prior : List (DartRow q n m)) (row : DartRow q n m) |
| (counterSuffix suffix output : List Bool) (ambient : σ) (register : Option Bool) : |
| StateTransition.EvalsToInTime (TM2.step P) |
| ⟨some (labels .guard), (ambient, register), |
| MachineLookup.tapes counter source destination base |
| (encodeWord prior.length ++ counterSuffix) |
| (rowsBits prior ++ (rowBits row ++ suffix)) output⟩ |
| (some ⟨exit, (ambient, none), |
| MachineLookup.tapes counter source destination base |
| (encodeWord 0 ++ counterSuffix) |
| (encodeWord row.reverseIndex.val ++ (encodeWords (relationWords row.relation) ++ suffix)) |
| (encodeWord row.tail.val ++ output)⟩) |
| ((rowsBits prior ++ rowBits row).length + 2 * (encodeWord prior.length).length + 1) where |
| steps := lookupCost prior row |
| evals_in_steps := lookupTailTrace counter source destination labels exit P atStatement |
| hcs hcd hsd base prior row counterSuffix suffix output ambient register |
| steps_le_m := lookupCost_le prior row |
| |
| /-- The input's actual stored row vector determines the selected row. This |
| specialization derives the literal lookup prefix from the checked codec. -/ |
| theorem tableTailTrace {q : Nat} |
| (counter source destination : K) (labels : Label → Λ) (exit : Option Λ) |
| (P : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)) |
| (atStatement : ∀ label, P (labels label) = statement q counter source destination labels exit label) |
| (hcs : counter ≠ source) (hcd : counter ≠ destination) (hsd : source ≠ destination) |
| (base : K → List Bool) (table : Table q) (e : Fin table.darts) |
| (counterSuffix suffix output : List Bool) (ambient : σ) (register : Option Bool) : |
| (advance (TM2.step P))^[lookupCost ((rowList table).take e.val) table.rows[e]] |
| (some ⟨some (labels .guard), (ambient, register), |
| MachineLookup.tapes counter source destination base |
| (encodeWord e.val ++ counterSuffix) (rowsBits (rowList table) ++ suffix) output⟩) = |
| some ⟨exit, (ambient, none), |
| MachineLookup.tapes counter source destination base |
| (encodeWord 0 ++ counterSuffix) |
| (encodeWord table.rows[e].reverseIndex.val ++ |
| (encodeWords (relationWords table.rows[e].relation) ++ |
| (rowsBits ((rowList table).drop (e.val + 1)) ++ suffix))) |
| (encodeWord table.rows[e].tail.val ++ output)⟩ := by |
| have h := lookupTailTrace counter source destination labels exit P atStatement |
| hcs hcd hsd base ((rowList table).take e.val) table.rows[e] counterSuffix |
| (rowsBits ((rowList table).drop (e.val + 1)) ++ suffix) output ambient register |
| have hlength : ((rowList table).take e.val).length = e.val := by |
| simp only [List.length_take, rowList_length, Nat.min_eq_left (Nat.le_of_lt e.isLt)] |
| have hsplit : rowsBits (rowList table) ++ suffix = |
| rowsBits ((rowList table).take e.val) ++ |
| (rowBits table.rows[e] ++ (rowsBits ((rowList table).drop (e.val + 1)) ++ suffix)) := by |
| have hrow : table.rows[e.val] = table.rows[e] := rfl |
| simpa only [rowsBits, rowBits, rowList, Vector.getElem_toList, hrow] using |
| Input.rowsBits_append_split (rowList table) e.val (by simp) suffix |
| rw [hlength, ← hsplit] at h |
| exact h |
|
|
| /-- Two real delimiter-scanning loops remove the header of the copied table |
| before the row counter starts. The protected original table can occupy any |
| other tape in `base` and is unchanged. -/ |
| theorem fullTableTailTrace {q : Nat} |
| (counter source destination : K) (labels : Label → Λ) (exit : Option Λ) |
| (headerVertices headerDarts : Λ) |
| (P : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)) |
| (atStatement : ∀ label, P (labels label) = statement q counter source destination labels exit label) |
| (atVertices : P headerVertices = MachineLookup.discard source headerVertices headerDarts) |
| (atDarts : P headerDarts = MachineLookup.discard source headerDarts (labels .guard)) |
| (hcs : counter ≠ source) (hcd : counter ≠ destination) (hsd : source ≠ destination) |
| (base : K → List Bool) (table : Table q) (e : Fin table.darts) |
| (counterSuffix suffix output : List Bool) (ambient : σ) (register : Option Bool) : |
| (advance (TM2.step P))^[lookupCost ((rowList table).take e.val) table.rows[e] + |
| (table.darts + 1) + (table.vertices + 1)] |
| (some ⟨some headerVertices, (ambient, register), |
| MachineLookup.tapes counter source destination base |
| (encodeWord e.val ++ counterSuffix) (tableBits table ++ suffix) output⟩) = |
| some ⟨exit, (ambient, none), |
| MachineLookup.tapes counter source destination base |
| (encodeWord 0 ++ counterSuffix) |
| (encodeWord table.rows[e].reverseIndex.val ++ |
| (encodeWords (relationWords table.rows[e].relation) ++ |
| (rowsBits ((rowList table).drop (e.val + 1)) ++ suffix))) |
| (encodeWord table.rows[e].tail.val ++ output)⟩ := by |
| have hv := MachineLookup.discardTrace source headerVertices headerDarts P atVertices |
| (MachineLookup.tapes counter source destination base |
| (encodeWord e.val ++ counterSuffix) (tableBits table ++ suffix) output) |
| table.vertices (encodeWord table.darts ++ (rowsBits (rowList table) ++ suffix)) |
| (by simp only [MachineLookup.tapes_source _ _ _ hsd, Input.tableBits_header, |
| rowsBits, List.append_assoc]) ambient register |
| rw [Function.iterate_add_apply, hv] |
| simp only [MachineLookup.update_source _ _ _ hsd] |
| have hd := MachineLookup.discardTrace source headerDarts (labels .guard) P atDarts |
| (MachineLookup.tapes counter source destination base |
| (encodeWord e.val ++ counterSuffix) |
| (encodeWord table.darts ++ (rowsBits (rowList table) ++ suffix)) output) |
| table.darts (rowsBits (rowList table) ++ suffix) (by simp [hsd]) ambient none |
| rw [Function.iterate_add_apply, hd] |
| simp only [MachineLookup.update_source _ _ _ hsd] |
| exact tableTailTrace counter source destination labels exit P atStatement hcs hcd hsd |
| base table e counterSuffix suffix output ambient none |
|
|
| theorem lookup_frame (counter source destination other : K) |
| (hc : other ≠ counter) (hs : other ≠ source) (hd : other ≠ destination) |
| (base : K → List Bool) (counterBits input output : List Bool) : |
| MachineLookup.tapes counter source destination base counterBits input output other = base other := |
| MachineLookup.tapes_other counter source destination other hc hs hd base counterBits input output |
|
|
| def tableTailCost {q : Nat} (table : Table q) (e : Fin table.darts) : Nat := |
| lookupCost ((rowList table).take e.val) table.rows[e] + |
| (table.darts + 1) + (table.vertices + 1) |
|
|
| theorem tableTailCost_le {q : Nat} (table : Table q) (e : Fin table.darts) : |
| tableTailCost table e ≤ (tableBits table).length + 2 * (encodeWord e.val).length + 1 := by |
| have htake : ((rowList table).take e.val).length = e.val := by |
| simp only [List.length_take, rowList_length, Nat.min_eq_left (Nat.le_of_lt e.isLt)] |
| have hrow : table.rows[e.val] = table.rows[e] := rfl |
| have hsplit : rowsBits (rowList table) = rowsBits ((rowList table).take e.val) ++ |
| (rowBits table.rows[e] ++ rowsBits ((rowList table).drop (e.val + 1))) := by |
| simpa only [rowsBits, rowBits, rowList, Vector.getElem_toList, hrow] using |
| Input.rowsBits_split (rowList table) e.val (by simp) |
| have hp : (rowsBits ((rowList table).take e.val) ++ rowBits table.rows[e]).length ≤ |
| (rowsBits (rowList table)).length := by |
| rw [hsplit] |
| simp only [List.length_append] |
| omega |
| have hb := lookupCost_le ((rowList table).take e.val) table.rows[e] |
| rw [htake] at hb |
| have hh := congrArg List.length (Input.tableBits_header table) |
| change (tableBits table).length = |
| (encodeWord table.vertices ++ (encodeWord table.darts ++ rowsBits (rowList table))).length at hh |
| simp only [List.length_append, encodeWord_length] at hh |
| unfold tableTailCost |
| omega |
|
|
| def headIndex {q : Nat} (table : Table q) (e : Fin table.darts) : Fin table.darts := |
| table.rows[e].reverseIndex |
|
|
| def headValue {q : Nat} (table : Table q) (e : Fin table.darts) : Nat := |
| table.rows[headIndex table e].tail.val |
|
|
| /-- The looked-up endpoint is the head of the actual graph semantics, including |
| paired loop darts whose reverse index differs from their own index. -/ |
| theorem headValue_eq_semantics {q : Nat} (table : Table q) (e : Fin table.darts) : |
| headValue table e = ((semantics table).head e).val := rfl |
|
|
| abbrev HeadTapes (K : Type) := Fin 6 → K |
|
|
| def headLookupTapes {q : Nat} (tape : HeadTapes K) (base : K → List Bool) |
| (table : Table q) (e : Fin table.darts) : K → List Bool := |
| MachineLookup.tapes (tape 3) (tape 2) (tape 4) base |
| (encodeWord 0 ++ base (tape 3)) |
| (encodeWord table.rows[headIndex table e].reverseIndex.val ++ |
| (encodeWords (relationWords table.rows[headIndex table e].relation) ++ |
| (rowsBits ((rowList table).drop ((headIndex table e).val + 1)) ++ base (tape 2)))) |
| (encodeWord (headValue table e)) |
|
|
| def headLookupCost {q : Nat} (table : Table q) (e : Fin table.darts) : Nat := |
| 2 * ((tableBits table).length + 1) + |
| 2 * ((encodeWord (headIndex table e).val).length + 1) + tableTailCost table (headIndex table e) |
|
|
| theorem headLookupCost_le {q : Nat} (table : Table q) (e : Fin table.darts) : |
| headLookupCost table e ≤ 7 * (tableBits table).length + 5 := by |
| have hi : (encodeWord (headIndex table e).val).length ≤ (tableBits table).length := by |
| rw [encodeWord_length] |
| exact (Nat.succ_le_of_lt (headIndex table e).isLt).trans (darts_le_tableBits_length table) |
| have hl := tableTailCost_le table (headIndex table e) |
| unfold headLookupCost |
| omega |
|
|
| noncomputable def headTimePolynomial : Polynomial Nat := |
| Polynomial.C 7 * Polynomial.X + Polynomial.C 5 |
|
|
| def headLookupInTime {q : Nat} |
| (tape : HeadTapes K) (hinj : Function.Injective tape) |
| (tableFirst tableSecond indexFirst indexSecond headerVertices headerDarts : Λ) |
| (labels : Label → Λ) (exit : Option Λ) |
| (P : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)) |
| (atTableFirst : P tableFirst = Reduction.MachineTransfer.loopAt |
| (tape 0) (tape 5) id false tableFirst (some tableSecond)) |
| (atTableSecond : P tableSecond = MachineCopy.forkLoop |
| (tape 5) (tape 0) (tape 2) false tableSecond (some indexFirst)) |
| (atIndexFirst : P indexFirst = Reduction.MachineTransfer.loopAt |
| (tape 1) (tape 5) id false indexFirst (some indexSecond)) |
| (atIndexSecond : P indexSecond = MachineCopy.forkLoop |
| (tape 5) (tape 1) (tape 3) false indexSecond (some headerVertices)) |
| (atVertices : P headerVertices = MachineLookup.discard (tape 2) headerVertices headerDarts) |
| (atDarts : P headerDarts = MachineLookup.discard (tape 2) headerDarts (labels .guard)) |
| (atStatement : ∀ label, |
| P (labels label) = statement q (tape 3) (tape 2) (tape 4) labels exit label) |
| (base : K → List Bool) (table : Table q) (e : Fin table.darts) |
| (hTable : base (tape 0) = tableBits table) |
| (hReverse : base (tape 1) = encodeWord (headIndex table e).val) |
| (hHead : base (tape 4) = []) (hScratch : base (tape 5) = []) |
| (ambient : σ) (register : Option Bool) : |
| StateTransition.EvalsToInTime (TM2.step P) |
| ⟨some tableFirst, (ambient, register), base⟩ |
| (some ⟨exit, (ambient, none), headLookupTapes tape base table e⟩) |
| (headTimePolynomial.eval (tableBits table).length) := by |
| have hne (i j : Fin 6) (h : i ≠ j) : tape i ≠ tape j := fun heq => h (hinj heq) |
| let t₁ := Function.update base (tape 2) (tableBits table ++ base (tape 2)) |
| let t₂ := Function.update t₁ (tape 3) (encodeWord (headIndex table e).val ++ base (tape 3)) |
| have run₁ : StateTransition.EvalsToInTime (TM2.step P) |
| ⟨some tableFirst, (ambient, register), base⟩ |
| (some ⟨some indexFirst, (ambient, none), t₁⟩) |
| (2 * ((tableBits table).length + 1)) := by |
| simpa only [hTable, t₁] using MachineCopy.copyInTime (tape 0) (tape 2) (tape 5) |
| (hne 0 2 (by decide)) (hne 0 5 (by decide)) (hne 2 5 (by decide)) false |
| tableFirst tableSecond (some indexFirst) P atTableFirst atTableSecond |
| base hScratch ambient register |
| have hsrc : t₁ (tape 1) = encodeWord (headIndex table e).val := by |
| simp only [t₁, Function.update_of_ne (hne 1 2 (by decide)), hReverse] |
| have hdst : t₁ (tape 3) = base (tape 3) := by |
| simp only [t₁, Function.update_of_ne (hne 3 2 (by decide))] |
| have hscratch₁ : t₁ (tape 5) = [] := by |
| simp only [t₁, Function.update_of_ne (hne 5 2 (by decide)), hScratch] |
| have run₂ : StateTransition.EvalsToInTime (TM2.step P) |
| ⟨some indexFirst, (ambient, none), t₁⟩ |
| (some ⟨some headerVertices, (ambient, none), t₂⟩) |
| (2 * ((encodeWord (headIndex table e).val).length + 1)) := by |
| simpa only [hsrc, hdst, t₂] using MachineCopy.copyInTime (tape 1) (tape 3) (tape 5) |
| (hne 1 3 (by decide)) (hne 1 5 (by decide)) (hne 3 5 (by decide)) false |
| indexFirst indexSecond (some headerVertices) P atIndexFirst atIndexSecond |
| t₁ hscratch₁ ambient none |
| have ht₂ : t₂ = MachineLookup.tapes (tape 3) (tape 2) (tape 4) base |
| (encodeWord (headIndex table e).val ++ base (tape 3)) |
| (tableBits table ++ base (tape 2)) [] := by |
| funext k |
| by_cases hi : k = tape 3 |
| · subst k |
| simp [t₂, MachineLookup.tapes, hne 3 2 (by decide), hne 3 4 (by decide)] |
| · by_cases hs : k = tape 2 |
| · subst k |
| simp [t₂, t₁, MachineLookup.tapes, hne 2 3 (by decide), hne 2 4 (by decide)] |
| · by_cases hh : k = tape 4 |
| · subst k |
| simp [t₂, t₁, MachineLookup.tapes, hne 4 3 (by decide), |
| hne 4 2 (by decide), hHead] |
| · simp [t₂, t₁, MachineLookup.tapes, hi, hs, hh] |
| let run₃ : StateTransition.EvalsToInTime (TM2.step P) |
| ⟨some headerVertices, (ambient, none), t₂⟩ |
| (some ⟨exit, (ambient, none), headLookupTapes tape base table e⟩) |
| (tableTailCost table (headIndex table e)) := { |
| steps := tableTailCost table (headIndex table e) |
| evals_in_steps := by |
| change (advance (TM2.step P))^[tableTailCost table (headIndex table e)] |
| (some ⟨some headerVertices, (ambient, none), t₂⟩) = |
| some ⟨exit, (ambient, none), headLookupTapes tape base table e⟩ |
| rw [ht₂] |
| simpa only [headLookupTapes, headValue, tableTailCost, List.append_nil] using |
| fullTableTailTrace (tape 3) (tape 2) (tape 4) labels exit |
| headerVertices headerDarts P atStatement atVertices atDarts |
| (hne 3 2 (by decide)) (hne 3 4 (by decide)) (hne 2 4 (by decide)) |
| base table (headIndex table e) (base (tape 3)) (base (tape 2)) [] ambient none |
| steps_le_m := Nat.le_refl _ } |
| let run₁₂ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ run₁ run₂ |
| let run := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ run₁₂ run₃ |
| refine { toEvalsTo := run.toEvalsTo, steps_le_m := ?_ } |
| have ht := run.steps_le_m |
| have hb := headLookupCost_le table e |
| simp only [headTimePolynomial, Polynomial.eval_add, Polynomial.eval_mul, |
| Polynomial.eval_C, Polynomial.eval_X] |
| unfold headLookupCost at hb |
| omega |
|
|
| theorem headLookupTapes_head {q : Nat} (tape : HeadTapes K) (base : K → List Bool) |
| (table : Table q) (e : Fin table.darts) : |
| headLookupTapes tape base table e (tape 4) = encodeWord (headValue table e) := by |
| simp [headLookupTapes] |
|
|
| theorem headLookupTapes_frame {q : Nat} (tape : HeadTapes K) (base : K → List Bool) |
| (table : Table q) (e : Fin table.darts) (other : K) |
| (hc : other ≠ tape 3) (hs : other ≠ tape 2) (hh : other ≠ tape 4) : |
| headLookupTapes tape base table e other = base other := |
| lookup_frame (tape 3) (tape 2) (tape 4) other hc hs hh base _ _ _ |
|
|
| def readRowTapes {q n m : Nat} (source tail reverse : K) (base : K → List Bool) |
| (row : DartRow q n m) (suffix : List Bool) : K → List Bool := |
| Function.update (Function.update (Function.update base source suffix) |
| tail (encodeWord row.tail.val)) reverse (encodeWord row.reverseIndex.val) |
|
|
| theorem readRowTapes_source {q n m : Nat} (source tail reverse : K) |
| (hst : source ≠ tail) (hsr : source ≠ reverse) |
| (base : K → List Bool) (row : DartRow q n m) (suffix : List Bool) : |
| readRowTapes source tail reverse base row suffix source = suffix := by |
| simp [readRowTapes, hst, hsr] |
|
|
| theorem readRowTapes_tail {q n m : Nat} (source tail reverse : K) |
| (htr : tail ≠ reverse) (base : K → List Bool) |
| (row : DartRow q n m) (suffix : List Bool) : |
| readRowTapes source tail reverse base row suffix tail = encodeWord row.tail.val := by |
| simp [readRowTapes, htr] |
|
|
| theorem readRowTapes_reverse {q n m : Nat} (source tail reverse : K) |
| (base : K → List Bool) (row : DartRow q n m) (suffix : List Bool) : |
| readRowTapes source tail reverse base row suffix reverse = |
| encodeWord row.reverseIndex.val := by |
| simp [readRowTapes] |
|
|
| theorem readRowTapes_frame {q n m : Nat} (source tail reverse other : K) |
| (hs : other ≠ source) (ht : other ≠ tail) (hr : other ≠ reverse) |
| (base : K → List Bool) (row : DartRow q n m) (suffix : List Bool) : |
| readRowTapes source tail reverse base row suffix other = base other := by |
| simp [readRowTapes, hs, ht, hr] |
|
|
| def readRowCost {q n m : Nat} (row : DartRow q n m) : Nat := |
| row.tail.val + row.reverseIndex.val + 5 |
|
|
| theorem readRowCost_le {q n m : Nat} (row : DartRow q n m) : |
| readRowCost row ≤ (rowBits row).length + 3 := by |
| rw [rowBits_eq] |
| simp only [List.length_append, encodeWord_length] |
| unfold readRowCost |
| omega |
|
|
| theorem readRowCost_table_le {q : Nat} (table : Table q) (e : Fin table.darts) : |
| readRowCost table.rows[e] ≤ 2 * (tableBits table).length + 5 := by |
| have ht := (Nat.le_of_lt table.rows[e].tail.isLt).trans |
| (vertices_le_tableBits_length table) |
| have hr := (Nat.le_of_lt table.rows[e].reverseIndex.isLt).trans |
| (darts_le_tableBits_length table) |
| unfold readRowCost |
| omega |
|
|
| /-- Read both unary fields using actual copy loops, then execute the checked |
| fixed predicate parser. Only the two field destinations must initially be |
| empty; the cursor suffix and every unrelated tape are retained exactly. -/ |
| def readRowInTime {q n m : Nat} (source tail reverse : K) |
| (hst : source ≠ tail) (hsr : source ≠ reverse) (htr : tail ≠ reverse) |
| (tailStart tailLoop reverseStart reverseLoop readLabel exitLabel : Λ) |
| (P : Λ → TM2.Stmt (Alphabet (K := K)) Λ (ReadRelation.State σ q)) |
| (atTailStart : P tailStart = Hastad.SourceMachine.fieldStart tail tailLoop) |
| (atTailLoop : P tailLoop = |
| Hastad.SourceMachine.fieldLoop source tail tailLoop (some reverseStart)) |
| (atReverseStart : P reverseStart = Hastad.SourceMachine.fieldStart reverse reverseLoop) |
| (atReverseLoop : P reverseLoop = |
| Hastad.SourceMachine.fieldLoop source reverse reverseLoop (some readLabel)) |
| (atRead : P readLabel = ReadRelation.parser source exitLabel) |
| (base : K → List Bool) (row : DartRow q n m) (suffix : List Bool) |
| (hinput : base source = rowBits row ++ suffix) |
| (hTail : base tail = []) (hReverse : base reverse = []) |
| (ambient : σ) (initial : RelationTable q) (register : Option Bool) : |
| StateTransition.EvalsToInTime (TM2.step P) |
| ⟨some tailStart, ((ambient, initial), register), base⟩ |
| (some ⟨some exitLabel, ((ambient, row.relation), none), |
| readRowTapes source tail reverse base row suffix⟩) |
| (readRowCost row) := by |
| let relationInput := encodeWords (relationWords row.relation) ++ suffix |
| let reverseInput := encodeWord row.reverseIndex.val ++ relationInput |
| let t₁ := Hastad.SourceMachine.fieldTapes source tail base reverseInput |
| (encodeWord row.tail.val) |
| let t₂ := Hastad.SourceMachine.fieldTapes source reverse t₁ relationInput |
| (encodeWord row.reverseIndex.val) |
| have hinput₁ : base source = encodeWord row.tail.val ++ reverseInput := by |
| simpa only [rowBits_eq, List.append_assoc, reverseInput, relationInput] using hinput |
| have run₁ : StateTransition.EvalsToInTime (TM2.step P) |
| ⟨some tailStart, ((ambient, initial), register), base⟩ |
| (some ⟨some reverseStart, ((ambient, initial), none), t₁⟩) |
| (row.tail.val + 2) := by |
| simpa only [hTail, List.append_nil, t₁] using |
| Hastad.SourceMachine.fieldInTime source tail hst tailStart tailLoop |
| (some reverseStart) P atTailStart atTailLoop base row.tail.val reverseInput |
| hinput₁ (ambient, initial) register |
| have hinput₂ : t₁ source = encodeWord row.reverseIndex.val ++ relationInput := by |
| simp only [t₁, Hastad.SourceMachine.fieldTapes_source source tail hst, reverseInput] |
| have hreverse₁ : t₁ reverse = [] := by |
| simp [t₁, Hastad.SourceMachine.fieldTapes, Ne.symm hsr, Ne.symm htr, hReverse] |
| have run₂ : StateTransition.EvalsToInTime (TM2.step P) |
| ⟨some reverseStart, ((ambient, initial), none), t₁⟩ |
| (some ⟨some readLabel, ((ambient, initial), none), t₂⟩) |
| (row.reverseIndex.val + 2) := by |
| simpa only [hreverse₁, List.append_nil, t₂] using |
| Hastad.SourceMachine.fieldInTime source reverse hsr reverseStart reverseLoop |
| (some readLabel) P atReverseStart atReverseLoop t₁ row.reverseIndex.val relationInput |
| hinput₂ (ambient, initial) none |
| have hinput₃ : t₂ source = encodeWords (relationWords row.relation) ++ suffix := by |
| simp only [t₂, Hastad.SourceMachine.fieldTapes_source source reverse hsr, relationInput] |
| have hrestore : Function.update t₂ source |
| (encodeWords (relationWords row.relation) ++ suffix) = t₂ := by |
| rw [← hinput₃] |
| exact Function.update_eq_self source t₂ |
| have hfinal : Function.update t₂ source suffix = |
| readRowTapes source tail reverse base row suffix := by |
| funext k |
| by_cases hs : k = source |
| · subst k |
| simp [readRowTapes, hst, hsr] |
| · by_cases ht : k = tail |
| · subst k |
| simp [t₂, t₁, Hastad.SourceMachine.fieldTapes, readRowTapes, hs, htr] |
| · by_cases hr : k = reverse |
| · subst k |
| simp [t₂, Hastad.SourceMachine.fieldTapes, readRowTapes, hs] |
| · simp [t₂, t₁, Hastad.SourceMachine.fieldTapes, readRowTapes, hs, ht, hr] |
| have run₃ : StateTransition.EvalsToInTime (TM2.step P) |
| ⟨some readLabel, ((ambient, initial), none), t₂⟩ |
| (some ⟨some exitLabel, ((ambient, row.relation), none), |
| readRowTapes source tail reverse base row suffix⟩) 1 := { |
| steps := 1 |
| evals_in_steps := by |
| change (advance (TM2.step P))^[1] _ = _ |
| simpa only [hrestore, hfinal] using ReadRelation.parserTrace source readLabel exitLabel |
| P atRead t₂ row.relation suffix ambient initial none |
| steps_le_m := Nat.le_refl _ } |
| let run₁₂ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ run₁ run₂ |
| let run := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ run₁₂ run₃ |
| refine { toEvalsTo := run.toEvalsTo, steps_le_m := ?_ } |
| have ht := run.steps_le_m |
| unfold readRowCost |
| omega |
|
|
| /-- A concrete finite machine instance, with input-dependent unary values |
| located only on its three tapes. -/ |
| def machine (q : Nat) : FinTM2 where |
| K := Fin 3 |
| k₀ := 1 |
| k₁ := 2 |
| Γ _ := Bool |
| Λ := Label |
| main := .guard |
| σ := Unit × Option Bool |
| initialState := ((), none) |
| m := program q 0 1 2 |
|
|
| end UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Lookup |
|
|
| end OAI |
|
|
