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import OAI.Computability.MaxCut.PCP.PreprocessingRegularBounds
import OAI.Computability.MaxCut.Estimates.Basic
namespace OAI
namespace MaxCutGames.Foundations.Complexity.MachineRegularTable
open Turing MachineComposition
open MaxCutGames.Foundations.PCP
namespace Top
def inputData (t : GraphTables.Table) (output : List Bool) : Data where
table := GraphTables.tableBits t
globalIndex := []
owner := []
localRank := []
count := []
offset := []
darts := []
rotor := []
output := output
padding := []
level := []
def headerData (t : GraphTables.Table) (output : List Bool) : Data :=
{ inputData t output with darts := encodeWord t.darts }
def headerStacks (t : GraphTables.Table) : Header.Tape → List Bool :=
headerAux (encodeWord t.vertices) (encodeWord (Header.vertexTotal t))
(encodeWord (Header.dartTotal PreprocessingRegularTables.internalDegree t))
theorem headerTrace (H : BaseTable) (t : GraphTables.Table) (output : List Bool)
(counters : Fin 4 → List Bool) :
(advance (TM2.step (program H)))^[Header.totalTime PreprocessingRegularTables.internalDegree t output]
(some (cfg H (some (.header .init)) (inputData t output) (fun _ => []) counters)) =
some (cfg H (some .oldSeed)
(headerData t (output ++ Header.headerBits PreprocessingRegularTables.internalDegree t))
(headerStacks t) counters) := by
have raw := Header.trace PreprocessingRegularTables.internalDegree t output () none
have placed := Lift.trace headerTape headerView headerView_left headerView_right
Label.header (some .oldSeed) headerStates
(MachineRegularInternalRow.coreInitialState PreprocessingRegularTables.internalDegree
MachineRegularOriginalBody.degree_positive (), (MachineRegularFamily.readyState H, none))
(frame (inputData t output) (fun _ => []) counters)
(Header.program PreprocessingRegularTables.internalDegree) (program H) (fun _ => rfl)
_ _ _ raw
simpa only [Lift.configuration, MachineCloudPadding.Placement.label, headerPlacement,
headerAux_empty, cfg, readyState, MachineRegularOriginalBody.readyState, headerStates,
Equiv.coe_fn_mk, headerData, inputData, headerStacks] using placed
def oldData (t : GraphTables.Table) (index : Nat) (output : List Bool) : Data :=
{ headerData t output with globalIndex := encodeWord index }
theorem originalData_eq (t : GraphTables.Table) (e : Fin t.darts) (output : List Bool) :
MachineRegularOriginalBody.initialData t e output = oldData t e.val output := rfl
/-- The actual old-vertex body is transported with all outer counters and
header words untouched. No body-execution hypothesis occurs in this lemma. -/
theorem oldBodyTrace (H : BaseTable) (t : GraphTables.Table) (e : Fin t.darts)
(output : List Bool) (header : Header.Tape → List Bool) (counters : Fin 4 → List Bool) :
(advance (TM2.step (program H)))^[MachineRegularOriginalBody.totalSteps H t e output]
(some (cfg H (some (Label.oldBody MachineRegularOriginalBody.entry))
(oldData t e.val output) header counters)) =
some (cfg H (some .oldIncrement)
(oldData t e.val (output ++ MachineRegularOriginalBody.emittedBits H t e)) header counters) := by
have raw := MachineRegularOriginalBody.originalTrace H t e output
have placed := Lift.trace Sum.inl bodyView bodyView_left bodyView_right Label.oldBody
(some .oldIncrement) (Equiv.refl _) none
(frame (oldData t e.val output) header counters)
(MachineRegularOriginalBody.program H) (program H) (fun _ => rfl) _ _ _ raw
simpa only [Lift.configuration, MachineCloudPadding.Placement.label, bodyPlacement,
MachineRegularOriginalBody.cfg, originalData_eq, cfg, readyState, Equiv.refl_apply] using placed
def counterStacks (old owners dummy work : List Bool) : Fin 4 → List Bool :=
![old, owners, dummy, work]
@[simp] theorem frame_oldFuel (data : Data) (header : Header.Tape → List Bool)
(old owners dummy work : List Bool) :
frame data header (counterStacks old owners dummy work) oldFuel = old := rfl
@[simp] theorem frame_ownerFuel (data : Data) (header : Header.Tape → List Bool)
(old owners dummy work : List Bool) :
frame data header (counterStacks old owners dummy work) ownerFuel = owners := rfl
@[simp] theorem update_frame_oldFuel (data : Data) (header : Header.Tape → List Bool)
(old owners dummy work replacement : List Bool) :
Function.update (frame data header (counterStacks old owners dummy work)) oldFuel replacement =
frame data header (counterStacks replacement owners dummy work) := by
funext j
cases j with
| inl j => rfl
| inr j => cases j with
| inl j => rfl
| inr i => fin_cases i <;> rfl
@[simp] theorem update_frame_ownerFuel (data : Data) (header : Header.Tape → List Bool)
(old owners dummy work replacement : List Bool) :
Function.update (frame data header (counterStacks old owners dummy work)) ownerFuel replacement =
frame data header (counterStacks old replacement dummy work) := by
funext j
cases j with
| inl j => rfl
| inr j => cases j with
| inl j => rfl
| inr i => fin_cases i <;> rfl
@[simp] theorem update_frame_global (data : Data) (header : Header.Tape → List Bool)
(counters : Fin 4 → List Bool) (replacement : List Bool) :
Function.update (frame data header counters) (coreTape 1) replacement =
frame { data with globalIndex := replacement } header counters := by
funext j
cases j with
| inl j => cases j with
| inl i => fin_cases i <;> rfl
| inr e => cases e with
| inl e => cases e <;> rfl
| inr e => rfl
| inr j => cases j <;> rfl
@[simp] theorem update_frame_owner (data : Data) (header : Header.Tape → List Bool)
(counters : Fin 4 → List Bool) (replacement : List Bool) :
Function.update (frame data header counters) (coreTape 2) replacement =
frame { data with owner := replacement } header counters := by
funext j
cases j with
| inl j => cases j with
| inl i => fin_cases i <;> rfl
| inr e => cases e with
| inl e => cases e <;> rfl
| inr e => rfl
| inr j => cases j <;> rfl
theorem oldGuardStep (H : BaseTable) (data : Data) (header : Header.Tape → List Bool)
(remaining : Nat) :
(advance (TM2.step (program H)))^[1]
(some (cfg H (some .oldGuard) data header
(counterStacks (encodeWord (remaining + 1)) [] [] []))) =
some (cfg H (some (Label.oldBody MachineRegularOriginalBody.entry)) data header
(counterStacks (encodeWord remaining) [] [] [])) := by
change some (TM2.stepAux (Fuel.guard oldFuel (Label.oldBody MachineRegularOriginalBody.entry)
(some Label.ownerSeed)) (readyState H) _) = _
simp only [Fuel.guard, TM2.stepAux, frame_oldFuel, encodeWord, List.replicate_succ,
List.cons_append, List.head?_cons, List.tail_cons, Option.getD_some,
Bool.cond_true, update_frame_oldFuel]
rfl
theorem oldGuardExit (H : BaseTable) (data : Data) (header : Header.Tape → List Bool) :
(advance (TM2.step (program H)))^[1]
(some (cfg H (some .oldGuard) data header (counterStacks (encodeWord 0) [] [] []))) =
some (cfg H (some Label.ownerSeed) data header (counterStacks [] [] [] [])) := by
change some (TM2.stepAux (Fuel.guard oldFuel (Label.oldBody MachineRegularOriginalBody.entry)
(some Label.ownerSeed)) (readyState H) _) = _
simp only [Fuel.guard, TM2.stepAux, frame_oldFuel, encodeWord, List.replicate_zero,
List.nil_append, List.head?_cons, List.tail_cons, Option.getD_some,
Bool.cond_false, update_frame_oldFuel]
rfl
theorem oldIncrementStep (H : BaseTable) (t : GraphTables.Table) (index : Nat)
(output : List Bool) (header : Header.Tape → List Bool) (counters : Fin 4 → List Bool) :
(advance (TM2.step (program H)))^[1]
(some (cfg H (some .oldIncrement) (oldData t index output) header counters)) =
some (cfg H (some .oldGuard) (oldData t (index + 1) output) header counters) := by
change some (TM2.stepAux (Fuel.increment (coreTape 1) Label.oldGuard)
(readyState H) (frame (oldData t index output) header counters)) = _
simp only [Fuel.increment, TM2.stepAux, frame_core]
change some (⟨some Label.oldGuard, readyState H,
Function.update (frame (oldData t index output) header counters) (coreTape 1)
(true :: encodeWord index)⟩ : TM2.Cfg (fun _ : Tape => Bool) Label State) = _
rw [update_frame_global]
simp only [cfg, oldData, encodeWord, List.replicate_succ, List.cons_append]
def oldPrefix (H : BaseTable) (t : GraphTables.Table) (k : Nat) : List Bool :=
PreprocessingRegularLoopWords.blocksPrefix (MachineRegularOriginalBody.emittedBits H t) k
noncomputable def oldElapsed (H : BaseTable) (t : GraphTables.Table) (output : List Bool) : Nat → Nat
| 0 => 0
| k + 1 => oldElapsed H t output k +
if h : k < t.darts then
1 + MachineRegularOriginalBody.totalSteps H t ⟨k,h⟩ (output ++ oldPrefix H t k) + 1
else 0
private theorem joinTrace_inline_MachineRegularTable {A : Type*} {f : A → A} {m n : Nat} {a b c : A}
(first : f^[m] a = b) (second : f^[n] b = c) : f^[m + n] a = c := by
rw [Nat.add_comm m n, Function.iterate_add_apply, first, second]
/-- Each actual fuel/body/increment segment appends exactly the next original
vertex block. This induction is over a specification index, not a register. -/
theorem oldPrefixTrace (H : BaseTable) (t : GraphTables.Table) (output : List Bool)
(header : Header.Tape → List Bool) (k : Nat) : k ≤ t.darts →
(advance (TM2.step (program H)))^[oldElapsed H t output k]
(some (cfg H (some .oldGuard) (oldData t 0 output) header
(counterStacks (encodeWord t.darts) [] [] []))) =
some (cfg H (some .oldGuard) (oldData t k (output ++ oldPrefix H t k)) header
(counterStacks (encodeWord (t.darts-k)) [] [] [])) := by
induction k with
| zero =>
intro _
simp [oldElapsed, oldPrefix]
| succ k ih =>
intro hk
have hlt : k < t.darts := by omega
have hremaining : t.darts-k = (t.darts-(k+1))+1 := by omega
have previous := ih (by omega)
have guard := oldGuardStep H (oldData t k (output ++ oldPrefix H t k)) header
(t.darts-(k+1))
rw [← hremaining] at guard
have body := oldBodyTrace H t ⟨k,hlt⟩ (output ++ oldPrefix H t k) header
(counterStacks (encodeWord (t.darts-(k+1))) [] [] [])
have increment := oldIncrementStep H t k
((output ++ oldPrefix H t k) ++ MachineRegularOriginalBody.emittedBits H t ⟨k,hlt⟩)
header (counterStacks (encodeWord (t.darts-(k+1))) [] [] [])
have combined := joinTrace_inline_MachineRegularTable (joinTrace_inline_MachineRegularTable (joinTrace_inline_MachineRegularTable previous guard) body) increment
have bits := PreprocessingRegularLoopWords.blocksPrefix_succ
(MachineRegularOriginalBody.emittedBits H t) k hlt
change oldPrefix H t (k+1) = oldPrefix H t k ++
MachineRegularOriginalBody.emittedBits H t ⟨k,hlt⟩ at bits
simpa only [oldElapsed, dite_eq_left hlt, bits, List.append_assoc, Nat.add_assoc] using combined
theorem oldLoopTrace (H : BaseTable) (t : GraphTables.Table) (output : List Bool)
(header : Header.Tape → List Bool) :
(advance (TM2.step (program H)))^[oldElapsed H t output t.darts + 1]
(some (cfg H (some .oldGuard) (oldData t 0 output) header
(counterStacks (encodeWord t.darts) [] [] []))) =
some (cfg H (some Label.ownerSeed)
(oldData t t.darts (output ++ oldPrefix H t t.darts)) header
(counterStacks [] [] [] [])) := by
have prefixRun := oldPrefixTrace H t output header t.darts (Nat.le_refl _)
simp only [Nat.sub_self] at prefixRun
exact joinTrace_inline_MachineRegularTable prefixRun (oldGuardExit H _ header)
@[simp] theorem frame_scratch (data : Data) (header : Header.Tape → List Bool)
(old owners dummy work : List Bool) :
frame data header (counterStacks old owners dummy work) scratch = work := rfl
@[simp] theorem frame_darts (data : Data) (header : Header.Tape → List Bool)
(counters : Fin 4 → List Bool) : frame data header counters (coreTape 6) = data.darts := rfl
@[simp] theorem frame_header_n (data : Data) (n N R : List Bool)
(counters : Fin 4 → List Bool) :
frame data (headerAux n N R) counters (headerTape Header.nTape) = n := rfl
theorem oldSeedStep (H : BaseTable) (t : GraphTables.Table) (output : List Bool)
(header : Header.Tape → List Bool) (counters : Fin 4 → List Bool) :
(advance (TM2.step (program H)))^[1]
(some (cfg H (some .oldSeed) (headerData t output) header counters)) =
some (cfg H (some (.copyFirst .old)) (oldData t 0 output) header counters) := by
change some (TM2.stepAux (.push (coreTape 1) (fun _ => false)
(.goto fun _ => Label.copyFirst CopyPhase.old)) (readyState H) (frame (headerData t output) header counters)) = _
change some (⟨some (Label.copyFirst CopyPhase.old), readyState H,
Function.update (frame (headerData t output) header counters) (coreTape 1) [false]⟩ :
TM2.Cfg (fun _ : Tape => Bool) Label State) = _
rw [update_frame_global]
rfl
theorem copyOldTrace (H : BaseTable) (t : GraphTables.Table) (output : List Bool)
(header : Header.Tape → List Bool) :
(advance (TM2.step (program H)))^[2*(t.darts+2)]
(some (cfg H (some (.copyFirst .old)) (oldData t 0 output) header
(counterStacks [] [] [] []))) =
some (cfg H (some .oldGuard) (oldData t 0 output) header
(counterStacks (encodeWord t.darts) [] [] [])) := by
have h := MachineCopy.copyTrace (coreTape 6) oldFuel scratch
(by decide) (by decide) (by decide) false (.copyFirst .old) (.copySecond .old)
(some .oldGuard) (program H) rfl rfl
(frame (oldData t 0 output) header (counterStacks [] [] [] [])) rfl
(MachineRegularOriginalBody.readyState H) none
simpa only [frame_darts, oldData, headerData, inputData, encodeWord_length,
frame_oldFuel, List.append_nil, update_frame_oldFuel, cfg, readyState, Nat.add_assoc] using h
theorem oldStageTrace (H : BaseTable) (t : GraphTables.Table) (output : List Bool)
(header : Header.Tape → List Bool) :
(advance (TM2.step (program H)))^[1+2*(t.darts+2)+(oldElapsed H t output t.darts+1)]
(some (cfg H (some .oldSeed) (headerData t output) header (counterStacks [] [] [] []))) =
some (cfg H (some Label.ownerSeed)
(oldData t t.darts (output ++ oldPrefix H t t.darts)) header
(counterStacks [] [] [] [])) :=
joinTrace_inline_MachineRegularTable (joinTrace_inline_MachineRegularTable (oldSeedStep H t output header _) (copyOldTrace H t output header))
(oldLoopTrace H t output header)
@[simp] theorem update_frame_offset (data : Data) (header : Header.Tape → List Bool)
(counters : Fin 4 → List Bool) (replacement : List Bool) :
Function.update (frame data header counters) (coreTape 5) replacement =
frame { data with offset := replacement } header counters := by
funext j
cases j with
| inl j => cases j with
| inl i => fin_cases i <;> rfl
| inr e => cases e with
| inl e => cases e <;> rfl
| inr e => rfl
| inr j => cases j <;> rfl
def ownerData (t : GraphTables.Table) (index : Nat) (output : List Bool) : Data :=
{ headerData t output with
globalIndex := encodeWord (t.darts + PreprocessingPaddingOffsets.offset
(PreprocessingRegularTables.padding t) index)
owner := encodeWord index
offset := encodeWord (PreprocessingPaddingOffsets.offset
(PreprocessingRegularTables.padding t) index) }
theorem ownerSeedStep (H : BaseTable) (t : GraphTables.Table) (output : List Bool)
(header : Header.Tape → List Bool) (counters : Fin 4 → List Bool) :
(advance (TM2.step (program H)))^[1]
(some (cfg H (some Label.ownerSeed) (oldData t t.darts output) header counters)) =
some (cfg H (some (.copyFirst .owner)) (ownerData t 0 output) header counters) := by
change some (TM2.stepAux
(.push (coreTape 2) (fun _ => false) (.push (coreTape 5) (fun _ => false)
(.goto fun _ => Label.copyFirst CopyPhase.owner))) (readyState H)
(frame (oldData t t.darts output) header counters)) = _
simp only [TM2.stepAux, frame_core]
change some (⟨some (Label.copyFirst CopyPhase.owner), readyState H,
Function.update (Function.update (frame (oldData t t.darts output) header counters)
(coreTape 2) [false]) (coreTape 5) [false]⟩ :
TM2.Cfg (fun _ : Tape => Bool) Label State) = _
rw [update_frame_owner, update_frame_offset]
simp only [cfg, ownerData, oldData, PreprocessingPaddingOffsets.offset_zero, Nat.add_zero,
encodeWord, List.replicate_zero, List.nil_append]
theorem copyOwnerTrace (H : BaseTable) (t : GraphTables.Table) (output : List Bool) :
(advance (TM2.step (program H)))^[2*(t.vertices+2)]
(some (cfg H (some (.copyFirst .owner)) (ownerData t 0 output) (headerStacks t)
(counterStacks [] [] [] []))) =
some (cfg H (some .ownerGuard) (ownerData t 0 output) (headerStacks t)
(counterStacks [] (encodeWord t.vertices) [] [])) := by
have h := MachineCopy.copyTrace (headerTape Header.nTape) ownerFuel scratch
(by decide) (by decide) (by decide) false (.copyFirst .owner) (.copySecond .owner)
(some .ownerGuard) (program H) rfl rfl
(frame (ownerData t 0 output) (headerStacks t) (counterStacks [] [] [] [])) rfl
(MachineRegularOriginalBody.readyState H) none
simpa only [headerStacks, frame_header_n, encodeWord_length, frame_ownerFuel,
List.append_nil, update_frame_ownerFuel, cfg, readyState, Nat.add_assoc] using h
theorem ownerPlacement (data extra : Data) (header : Header.Tape → List Bool)
(old owners : List Bool) :
MachineCloudPadding.Placement.tapes ownerView (MachineRegularOwnerBody.frame data)
(frame extra header (counterStacks old owners [] [])) =
frame data header (counterStacks old owners [] []) := by
funext j
cases j with
| inl j => rfl
| inr j => cases j with
| inl j => rfl
| inr i => fin_cases i <;> rfl
theorem ownerInitialData_eq (t : GraphTables.Table) (v : Fin t.vertices) (output : List Bool) :
MachineRegularOwnerBody.initialData t v output = ownerData t v.val output := rfl
theorem ownerBodyTrace (H : BaseTable) (t : GraphTables.Table) (v : Fin t.vertices)
(output : List Bool) (header : Header.Tape → List Bool) (owners : List Bool) :
(advance (TM2.step (program H)))^[MachineRegularOwnerBody.totalSteps H t v output]
(some (cfg H (some (.ownerBody MachineRegularOwnerBody.entry))
(ownerData t v.val output) header (counterStacks [] owners [] []))) =
some (cfg H (some .ownerIncrement) (MachineRegularOwnerBody.finalData H t v output)
header (counterStacks [] owners [] [])) := by
have raw := MachineRegularOwnerBody.ownerTrace H t v output
have placed := MachineCloudPadding.Placement.trace ownerTape ownerView ownerView_left ownerView_right
Label.ownerBody (some .ownerIncrement)
(frame (ownerData t v.val output) header (counterStacks [] owners [] []))
(MachineRegularOwnerBody.program H) (program H) (fun _ => rfl) _ _ _ raw
simpa only [MachineCloudPadding.Placement.configuration, MachineCloudPadding.Placement.label,
ownerPlacement, MachineRegularOwnerBody.cfg, MachineRegularOwnerBody.readyState,
ownerInitialData_eq, cfg, readyState] using placed
theorem ownerGuardStep (H : BaseTable) (data : Data) (header : Header.Tape → List Bool)
(remaining : Nat) :
(advance (TM2.step (program H)))^[1]
(some (cfg H (some .ownerGuard) data header
(counterStacks [] (encodeWord (remaining+1)) [] []))) =
some (cfg H (some (.ownerBody MachineRegularOwnerBody.entry)) data header
(counterStacks [] (encodeWord remaining) [] [])) := by
change some (TM2.stepAux (Fuel.guard ownerFuel (.ownerBody MachineRegularOwnerBody.entry)
(clearEntry 0)) (readyState H) _) = _
simp only [Fuel.guard, TM2.stepAux, frame_ownerFuel, encodeWord, List.replicate_succ,
List.cons_append, List.head?_cons, List.tail_cons, Option.getD_some,
Bool.cond_true, update_frame_ownerFuel]
rfl
theorem ownerGuardExit (H : BaseTable) (data : Data) (header : Header.Tape → List Bool) :
(advance (TM2.step (program H)))^[1]
(some (cfg H (some .ownerGuard) data header (counterStacks [] (encodeWord 0) [] []))) =
some (cfg H (clearEntry 0) data header (counterStacks [] [] [] [])) := by
change some (TM2.stepAux (Fuel.guard ownerFuel (.ownerBody MachineRegularOwnerBody.entry)
(clearEntry 0)) (readyState H) _) = _
simp only [Fuel.guard, TM2.stepAux, frame_ownerFuel, encodeWord, List.replicate_zero,
List.nil_append, List.head?_cons, List.tail_cons, Option.getD_some,
Bool.cond_false, update_frame_ownerFuel]
rfl
theorem ownerIncrementStep (H : BaseTable) (t : GraphTables.Table) (v : Fin t.vertices)
(output : List Bool) (header : Header.Tape → List Bool) (counters : Fin 4 → List Bool) :
(advance (TM2.step (program H)))^[1]
(some (cfg H (some .ownerIncrement) (MachineRegularOwnerBody.finalData H t v output)
header counters)) =
some (cfg H (some .ownerGuard)
(ownerData t (v.val+1) (output ++ MachineRegularOwnerBody.ownerBits H t v)) header counters) := by
change some (TM2.stepAux (Fuel.increment (coreTape 2) Label.ownerGuard) (readyState H)
(frame (MachineRegularOwnerBody.finalData H t v output) header counters)) = _
simp only [Fuel.increment, TM2.stepAux, frame_core]
change some (⟨some Label.ownerGuard, readyState H,
Function.update (frame (MachineRegularOwnerBody.finalData H t v output) header counters)
(coreTape 2) (true :: encodeWord v.val)⟩ :
TM2.Cfg (fun _ : Tape => Bool) Label State) = _
rw [update_frame_owner]
have offset := PreprocessingRegularBounds.offset_succ (PreprocessingRegularTables.padding t) v.val v.isLt
simp only [cfg, ownerData, headerData, inputData, MachineRegularOwnerBody.finalData,
MachineRegularOwnerBody.initialData, offset, encodeWord, List.replicate_succ,
List.cons_append, Nat.add_assoc]
def ownerPrefix (H : BaseTable) (t : GraphTables.Table) (k : Nat) : List Bool :=
PreprocessingRegularLoopWords.blocksPrefix (MachineRegularOwnerBody.ownerBits H t) k
noncomputable def ownerElapsed (H : BaseTable) (t : GraphTables.Table) (output : List Bool) : Nat → Nat
| 0 => 0
| k+1 => ownerElapsed H t output k +
if h : k < t.vertices then
1 + MachineRegularOwnerBody.totalSteps H t ⟨k,h⟩ (output ++ ownerPrefix H t k) + 1
else 0
theorem ownerPrefixTrace (H : BaseTable) (t : GraphTables.Table) (output : List Bool)
(header : Header.Tape → List Bool) (k : Nat) : k ≤ t.vertices →
(advance (TM2.step (program H)))^[ownerElapsed H t output k]
(some (cfg H (some .ownerGuard) (ownerData t 0 output) header
(counterStacks [] (encodeWord t.vertices) [] []))) =
some (cfg H (some .ownerGuard) (ownerData t k (output ++ ownerPrefix H t k)) header
(counterStacks [] (encodeWord (t.vertices-k)) [] [])) := by
induction k with
| zero =>
intro _
simp [ownerElapsed, ownerPrefix]
| succ k ih =>
intro hk
have hlt : k < t.vertices := by omega
have hremaining : t.vertices-k = (t.vertices-(k+1))+1 := by omega
have previous := ih (by omega)
have guard := ownerGuardStep H (ownerData t k (output ++ ownerPrefix H t k)) header
(t.vertices-(k+1))
rw [← hremaining] at guard
have body := ownerBodyTrace H t ⟨k,hlt⟩ (output ++ ownerPrefix H t k) header
(encodeWord (t.vertices-(k+1)))
have increment := ownerIncrementStep H t ⟨k,hlt⟩ (output ++ ownerPrefix H t k) header
(counterStacks [] (encodeWord (t.vertices-(k+1))) [] [])
have combined := joinTrace_inline_MachineRegularTable (joinTrace_inline_MachineRegularTable (joinTrace_inline_MachineRegularTable previous guard) body) increment
have bits := PreprocessingRegularLoopWords.blocksPrefix_succ
(MachineRegularOwnerBody.ownerBits H t) k hlt
change ownerPrefix H t (k+1) = ownerPrefix H t k ++
MachineRegularOwnerBody.ownerBits H t ⟨k,hlt⟩ at bits
simpa only [ownerElapsed, dite_eq_left hlt, bits, List.append_assoc, Nat.add_assoc] using combined
theorem ownerLoopTrace (H : BaseTable) (t : GraphTables.Table) (output : List Bool)
(header : Header.Tape → List Bool) :
(advance (TM2.step (program H)))^[ownerElapsed H t output t.vertices+1]
(some (cfg H (some .ownerGuard) (ownerData t 0 output) header
(counterStacks [] (encodeWord t.vertices) [] []))) =
some (cfg H (clearEntry 0)
(ownerData t t.vertices (output ++ ownerPrefix H t t.vertices)) header
(counterStacks [] [] [] [])) := by
have prefixRun := ownerPrefixTrace H t output header t.vertices (Nat.le_refl _)
simp only [Nat.sub_self] at prefixRun
exact joinTrace_inline_MachineRegularTable prefixRun (ownerGuardExit H _ header)
theorem ownerStageTrace (H : BaseTable) (t : GraphTables.Table) (output : List Bool) :
(advance (TM2.step (program H)))^[1+2*(t.vertices+2)+(ownerElapsed H t output t.vertices+1)]
(some (cfg H (some Label.ownerSeed) (oldData t t.darts output) (headerStacks t)
(counterStacks [] [] [] []))) =
some (cfg H (clearEntry 0)
(ownerData t t.vertices (output ++ ownerPrefix H t t.vertices)) (headerStacks t)
(counterStacks [] [] [] [])) :=
joinTrace_inline_MachineRegularTable (joinTrace_inline_MachineRegularTable (ownerSeedStep H t output (headerStacks t) _) (copyOwnerTrace H t output))
(ownerLoopTrace H t output (headerStacks t))
theorem owner_output_eq_ownerPrefix (H : BaseTable) (t : GraphTables.Table) (k : Nat) :
(Header.headerBits PreprocessingRegularTables.internalDegree t ++ oldPrefix H t t.darts) ++
ownerPrefix H t k =
PreprocessingRegularBounds.ownerPrefix t (PreprocessingRegularTables.padding t)
(PreprocessingRegularTables.familyCloudTable H t) k := by
have oldBits : MachineRegularOriginalBody.emittedBits H t =
PreprocessingRegularWords.originalVertexBits t (PreprocessingRegularTables.padding t)
(PreprocessingRegularTables.familyCloudTable H t) := rfl
have ownerBits : MachineRegularOwnerBody.ownerBits H t =
PreprocessingRegularBounds.ownerBits t (PreprocessingRegularTables.padding t)
(PreprocessingRegularTables.familyCloudTable H t) := rfl
simp only [Header.headerBits, Header.vertexTotal, Header.dartTotal, oldPrefix,
ownerPrefix, PreprocessingRegularLoopWords.blocksPrefix_all,
PreprocessingRegularBounds.ownerPrefix, PreprocessingRegularBounds.header]
rw [oldBits, ownerBits]
def cleaned (base : Tape → List Bool) : Nat → Tape → List Bool
| 0 => base
| k+1 => if h : k < 7 then Function.update (cleaned base k) (clearTape ⟨k,h⟩) []
else cleaned base k
def cleanupTime (base : Tape → List Bool) : Nat → Nat
| 0 => 0
| k+1 => cleanupTime base k + if h : k < 7 then
((cleaned base k) (clearTape ⟨k,h⟩)).length+1 else 0
theorem drainTrace (H : BaseTable) (i : Fin 7) (base : Tape → List Bool) :
(advance (TM2.step (program H)))^[(base (clearTape i)).length+1]
(some ⟨some (.clear i), readyState H, base⟩) =
some ⟨clearEntry (i.val+1), readyState H, Function.update base (clearTape i) []⟩ := by
have h := MachineDrain.drainTrace (clearTape i) (.clear i) (clearEntry (i.val+1))
(program H) rfl base (base (clearTape i)) (MachineRegularOriginalBody.readyState H) none
simpa only [Function.update_eq_self, readyState] using h
theorem cleanupPrefixTrace (H : BaseTable) (base : Tape → List Bool) (k : Nat) (hk : k ≤ 7) :
(advance (TM2.step (program H)))^[cleanupTime base k]
(some ⟨clearEntry 0, readyState H, base⟩) =
some ⟨clearEntry k, readyState H, cleaned base k⟩ := by
induction k with
| zero => rfl
| succ k ih =>
have h : k < 7 := by omega
have first := ih (by omega)
have second := drainTrace H ⟨k,h⟩ (cleaned base k)
have atEntry : clearEntry k = some (.clear ⟨k,h⟩) := by simp only [clearEntry, dite_eq_left h]
rw [atEntry] at first
simpa only [cleanupTime, cleaned, dite_eq_left h] using joinTrace_inline_MachineRegularTable first second
theorem cleanupTrace (H : BaseTable) (base : Tape → List Bool) :
(advance (TM2.step (program H)))^[cleanupTime base 7]
(some ⟨clearEntry 0, readyState H, base⟩) =
some ⟨none, readyState H, cleaned base 7⟩ := by
simpa only [clearEntry, lt_self_iff_false, dite_false] using cleanupPrefixTrace H base 7 le_rfl
def finalStacks (t : GraphTables.Table) (output : List Bool) : Tape → List Bool :=
frame (ownerData t t.vertices output) (headerStacks t) (counterStacks [] [] [] [])
theorem cleaned_finalStacks (t : GraphTables.Table) (output : List Bool) :
cleaned (finalStacks t output) 7 =
frame (inputData t output) (fun _ => []) (counterStacks [] [] [] []) := by
funext j
cases j with
| inl j => cases j with
| inl i => fin_cases i <;> rfl
| inr e => cases e with
| inl e => cases e <;> rfl
| inr e => rfl
| inr j => cases j with
| inl k => cases k with
| inl k => cases k with
| inner k => cases k <;> rfl
| bound => rfl
| remaining => rfl
| total => rfl
| inr k => cases k <;> rfl
| inr i => fin_cases i <;> rfl
theorem cleaned_length_le (base : Tape → List Bool) (k : Nat) :
∀ j, (cleaned base k j).length ≤ (base j).length := by
induction k with
| zero => intro j; exact Nat.le_refl _
| succ k ih =>
intro j
simp only [cleaned]
split
· rename_i hk
by_cases hj : j = clearTape ⟨k,hk⟩
· simp [hj]
· simpa only [Function.update_of_ne hj] using ih j
· exact ih j
theorem cleanupTime_le (base : Tape → List Bool) (cap : Nat)
(bounded : ∀ i : Fin 7, (base (clearTape i)).length ≤ cap) (k : Nat) :
cleanupTime base k ≤ k*(cap+1) := by
induction k with
| zero => simp [cleanupTime]
| succ k ih =>
simp only [cleanupTime]
split
· rename_i hk
have hword := (cleaned_length_le base k (clearTape ⟨k,hk⟩)).trans (bounded ⟨k,hk⟩)
calc
cleanupTime base k + ((cleaned base k (clearTape ⟨k,hk⟩)).length+1) ≤
k*(cap+1)+(cap+1) := by omega
_ = (k+1)*(cap+1) := by simp only [Nat.add_mul, one_mul]
· exact ih.trans (Nat.mul_le_mul_right _ (by omega))
theorem cleanupTime_final_le (t : GraphTables.Table) (output : List Bool) :
cleanupTime (finalStacks t output) 7 ≤
(7*(ExpanderFamily.growth*(PreprocessingRegularTables.internalDegree+4)+4)) *
(PreprocessingMachineBounds.inputLength t+1) := by
let L := PreprocessingMachineBounds.inputLength t
let g := ExpanderFamily.growth
let q := PreprocessingRegularTables.internalDegree
let C := g*(q+4)+4
have hm : t.darts ≤ L := GraphTables.darts_le_tableBits_length t
have hn : t.vertices ≤ L := GraphTables.vertices_le_tableBits_length t
have hs : PreprocessingPaddingOffsets.offset (PreprocessingRegularTables.padding t) t.vertices ≤ g*L :=
PreprocessingMachineBounds.prefix_le_input t t.vertices
have hN : Header.vertexTotal t ≤ g*L := PreprocessingMachineBounds.regularVertices_le_input t
have hone : 1 ≤ C := by dsimp [C]; omega
have hg : g ≤ C := by
have h := Nat.mul_le_mul_left g (show 1 ≤ q+4 by omega)
dsimp [C]
omega
have hgp : g+1 ≤ C := by
have h := Nat.mul_le_mul_left g (show 1 ≤ q+4 by omega)
dsimp [C]
omega
have hgq : g*(q+1) ≤ C := by
have h := Nat.mul_le_mul_left g (show q+1 ≤ q+4 by omega)
dsimp [C]
omega
have hmC : t.darts ≤ C*L := hm.trans (by simpa using Nat.mul_le_mul_right L hone)
have hnC : t.vertices ≤ C*L := hn.trans (by simpa using Nat.mul_le_mul_right L hone)
have hsC : PreprocessingPaddingOffsets.offset (PreprocessingRegularTables.padding t) t.vertices ≤ C*L :=
hs.trans (Nat.mul_le_mul_right L hg)
have hNC : Header.vertexTotal t ≤ C*L := hN.trans (Nat.mul_le_mul_right L hg)
have hxC : t.darts +
PreprocessingPaddingOffsets.offset (PreprocessingRegularTables.padding t) t.vertices ≤ C*L := by
calc
_ ≤ L+g*L := Nat.add_le_add hm hs
_ = (g+1)*L := by simp only [Nat.add_mul, one_mul, Nat.add_comm]
_ ≤ C*L := Nat.mul_le_mul_right L hgp
have hRC : Header.dartTotal q t ≤ C*L := by
calc
_ ≤ (g*L)*(q+1) := Nat.mul_le_mul_right _ hN
_ = (g*(q+1))*L := by ac_rfl
_ ≤ C*L := Nat.mul_le_mul_right L hgq
have bounds : ∀ i : Fin 7, (finalStacks t output (clearTape i)).length ≤ C*L+1 := by
intro i
fin_cases i
· change (encodeWord _).length ≤ _
simpa only [encodeWord_length] using Nat.add_le_add_right hxC 1
· change (encodeWord _).length ≤ _
simpa only [encodeWord_length] using Nat.add_le_add_right hnC 1
· change (encodeWord _).length ≤ _
simpa only [encodeWord_length] using Nat.add_le_add_right hsC 1
· change (encodeWord _).length ≤ _
simpa only [encodeWord_length] using Nat.add_le_add_right hmC 1
· change (encodeWord _).length ≤ _
simpa only [encodeWord_length] using Nat.add_le_add_right hnC 1
· change (encodeWord _).length ≤ _
simpa only [encodeWord_length] using Nat.add_le_add_right hNC 1
· change (encodeWord _).length ≤ _
simpa only [encodeWord_length] using Nat.add_le_add_right hRC 1
have actual := cleanupTime_le (finalStacks t output) (C*L+1) bounds 7
have hC : 2 ≤ C := by dsimp [C]; omega
change cleanupTime (finalStacks t output) 7 ≤ (7*C)*(L+1)
nlinarith
def headerOutput (t : GraphTables.Table) : List Bool :=
Header.headerBits PreprocessingRegularTables.internalDegree t
def oldOutput (H : BaseTable) (t : GraphTables.Table) : List Bool :=
headerOutput t ++ oldPrefix H t t.darts
def finalOutput (H : BaseTable) (t : GraphTables.Table) : List Bool :=
oldOutput H t ++ ownerPrefix H t t.vertices
noncomputable def totalTime (H : BaseTable) (t : GraphTables.Table) : Nat :=
Header.totalTime PreprocessingRegularTables.internalDegree t [] +
(1+2*(t.darts+2)+(oldElapsed H t (headerOutput t) t.darts+1)) +
(1+2*(t.vertices+2)+(ownerElapsed H t (oldOutput H t) t.vertices+1)) +
cleanupTime (finalStacks t (finalOutput H t)) 7
theorem old_output_eq_originalPrefix (H : BaseTable) (t : GraphTables.Table) (k : Nat) :
Header.headerBits PreprocessingRegularTables.internalDegree t ++ oldPrefix H t k =
PreprocessingRegularBounds.originalPrefix t (PreprocessingRegularTables.padding t)
(PreprocessingRegularTables.familyCloudTable H t) k := rfl
theorem finalOutput_eq (H : BaseTable) (t : GraphTables.Table) :
finalOutput H t = PortTables.tableBits (PreprocessingRegularTables.regularize H t) := by
change finalOutput H t = PortTables.tableBits (PreprocessingRegularTables.ofCloudTables t
(PreprocessingRegularTables.padding t) (PreprocessingRegularTables.familyCloudTable H t))
rw [PreprocessingRegularWords.tableBits_ofCloudTables]
have oldBits : MachineRegularOriginalBody.emittedBits H t =
PreprocessingRegularWords.originalVertexBits t (PreprocessingRegularTables.padding t)
(PreprocessingRegularTables.familyCloudTable H t) := rfl
have ownerBits : MachineRegularOwnerBody.ownerBits H t =
(fun v => (List.ofFn (PreprocessingRegularWords.dummyVertexBits t
(PreprocessingRegularTables.padding t) (PreprocessingRegularTables.familyCloudTable H t) v)).flatten) := rfl
simp only [finalOutput, oldOutput, headerOutput, Header.headerBits, Header.vertexTotal,
Header.dartTotal, oldPrefix, ownerPrefix, PreprocessingRegularLoopWords.blocksPrefix_all,
List.append_assoc]
rw [oldBits, ownerBits]
end Top
end MaxCutGames.Foundations.Complexity.MachineRegularTable
end OAI
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