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import OAI.Computability.UniqueGames.Machines.MachineDrain
import OAI.Computability.UniqueGames.Machines.MachineLemmas
import OAI.Computability.UniqueGames.PCP.OverlayLemmas
import OAI.Computability.UniqueGames.Reduction.MachineSubstitution
namespace OAI
namespace UniqueGamesTheorem.Foundations.Complexity.MachineCeilingPower
open Turing
open MachineComposition
open UniqueGamesTheorem.Foundations.PCP
inductive Tape
| input | work | power | saved | level | fuel | spare | product
deriving DecidableEq
protected abbrev Tape.enumList : List Tape := [.input, .work, .power, .saved, .level, .fuel,
.spare, .product]
protected theorem Tape.enumList_getElem?_ctorIdx_eq (x : Tape) :
Tape.enumList[x.ctorIdx]? = some x := by
cases x <;> rfl
protected theorem Tape.enumList_nodup : Tape.enumList.Nodup := by decide
instance : Fintype Tape where
elems := ⟨Tape.enumList, Tape.enumList_nodup⟩
complete x := by cases x <;> decide
inductive Label
| init | fuelFirst | fuelSecond | guard | copyFirst | copySecond
| compare | restore | decide | scale | transfer | cleanup
deriving DecidableEq
protected abbrev Label.enumList : List Label := [.init, .fuelFirst, .fuelSecond, .guard,
.copyFirst, .copySecond, .compare, .restore, .decide, .scale, .transfer, .cleanup]
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
abbrev Alphabet (_ : Tape) := Bool
abbrev State (σ : Type) := (σ × Bool) × Option Bool
def memory (input work power saved level fuel spare product : List Bool) : Tape → List Bool
| .input => input
| .work => work
| .power => power
| .saved => saved
| .level => level
| .fuel => fuel
| .spare => spare
| .product => product
@[simp] theorem update_work (a b c d e f g h x : List Bool) :
Function.update (memory a b c d e f g h) .work x = memory a x c d e f g h := by
funext k; cases k <;> rfl
@[simp] theorem update_power (a b c d e f g h x : List Bool) :
Function.update (memory a b c d e f g h) .power x = memory a b x d e f g h := by
funext k; cases k <;> rfl
@[simp] theorem update_saved (a b c d e f g h x : List Bool) :
Function.update (memory a b c d e f g h) .saved x = memory a b c x e f g h := by
funext k; cases k <;> rfl
@[simp] theorem update_level (a b c d e f g h x : List Bool) :
Function.update (memory a b c d e f g h) .level x = memory a b c d x f g h := by
funext k; cases k <;> rfl
@[simp] theorem update_fuel (a b c d e f g h x : List Bool) :
Function.update (memory a b c d e f g h) .fuel x = memory a b c d e x g h := by
funext k; cases k <;> rfl
@[simp] theorem update_product (a b c d e f g h x : List Bool) :
Function.update (memory a b c d e f g h) .product x = memory a b c d e f g x := by
funext k; cases k <;> rfl
variable {σ : Type}
def compareLoop : TM2.Stmt Alphabet Label (State σ) :=
.pop .work (fun state head => (state.1, head))
(.branch (fun state => state.2.getD false)
(.peek .power (fun state head => (state.1, head))
(.branch (fun state => state.2.getD false)
(.pop .power (fun state _ => (state.1, none))
(.push .saved (fun _ => true) (.goto fun _ => .compare)))
(.load (fun state => ((state.1.1, true), none)) (.goto fun _ => .compare))))
(.load (fun state => (state.1, none)) (.goto fun _ => .restore)))
/-- Each input tally emits the fixed block of `g` tallies. The delimiter
starts the new power and the level is incremented by an actual push. -/
def scaleLoop (g : Nat) : TM2.Stmt Alphabet Label (State σ) :=
.pop .power (fun state head => (state.1, head))
(.branch (fun state => state.2.getD false)
(Reduction.MachineSubstitution.pushWord .product (List.replicate g true)
(.goto fun _ => .scale))
(.push .power (fun _ => false)
(.push .level (fun _ => true)
(.load (fun state => (state.1, none)) (.goto fun _ => .transfer)))))
def program (g : Nat) : Label → TM2.Stmt Alphabet Label (State σ)
| .init => .push .power (fun _ => false) (.push .power (fun _ => true)
(.push .level (fun _ => false) (.goto fun _ => .fuelFirst)))
| .fuelFirst => Reduction.MachineTransfer.loopAt
.input .spare id false .fuelFirst (some .fuelSecond)
| .fuelSecond => MachineCopy.forkLoop
.spare .input .fuel false .fuelSecond (some .guard)
| .guard => MachineUnaryCounter.guard .fuel .copyFirst .cleanup
| .copyFirst => Reduction.MachineTransfer.loopAt
.input .spare id false .copyFirst (some .copySecond)
| .copySecond => MachineCopy.forkLoop
.spare .input .work false .copySecond (some .compare)
| .compare => compareLoop
| .restore => Reduction.MachineTransfer.loopAt
.saved .power id false .restore (some .decide)
| .decide => .branch (fun state => state.1.2)
(.load (fun state => ((state.1.1, false), none)) (.goto fun _ => .scale))
(.load (fun state => ((state.1.1, false), none)) (.goto fun _ => .cleanup))
| .scale => scaleLoop g
| .transfer => Reduction.MachineTransfer.loopAt
.product .power id false .transfer (some .guard)
| .cleanup => MachineDrain.drain .fuel .cleanup none
def overflow (flag : Bool) (x p : Nat) : Bool := flag || decide (p < x)
theorem compareStep_zero (g p : Nat) (input saved level fuel spare product : List Bool)
(ambient : σ) (flag : Bool) (register : Option Bool) :
TM2.step (program g)
⟨some .compare, ((ambient, flag), register),
memory input (encodeWord 0) (encodeWord p) saved level fuel spare product⟩ =
some ⟨some .restore, ((ambient, flag), none),
memory input [] (encodeWord p) saved level fuel spare product⟩ := by
change some (TM2.stepAux compareLoop _ _) = _
simp [compareLoop, TM2.stepAux, memory, encodeWord]
theorem compareStep_succ_zero (g x : Nat) (input saved level fuel spare product : List Bool)
(ambient : σ) (flag : Bool) (register : Option Bool) :
TM2.step (program g)
⟨some .compare, ((ambient, flag), register),
memory input (encodeWord (x + 1)) (encodeWord 0) saved level fuel spare product⟩ =
some ⟨some .compare, ((ambient, true), none),
memory input (encodeWord x) (encodeWord 0) saved level fuel spare product⟩ := by
change some (TM2.stepAux compareLoop _ _) = _
simp [compareLoop, TM2.stepAux, memory, encodeWord, List.replicate_succ]
theorem compareStep_succ_succ (g x p : Nat) (input saved level fuel spare product : List Bool)
(ambient : σ) (flag : Bool) (register : Option Bool) :
TM2.step (program g)
⟨some .compare, ((ambient, flag), register),
memory input (encodeWord (x + 1)) (encodeWord (p + 1)) saved level fuel spare product⟩ =
some ⟨some .compare, ((ambient, flag), none),
memory input (encodeWord x) (encodeWord p) (true :: saved) level fuel spare product⟩ := by
change some (TM2.stepAux compareLoop _ _) = _
simp [compareLoop, TM2.stepAux, memory, encodeWord, List.replicate_succ]
/-- The comparison is performed by actual finite-head transitions. -/
theorem compareTrace (g x p : Nat) (input saved level fuel spare product : List Bool)
(ambient : σ) (flag : Bool) (register : Option Bool) :
(advance (TM2.step (program g)))^[x + 1]
(some ⟨some .compare, ((ambient, flag), register),
memory input (encodeWord x) (encodeWord p) saved level fuel spare product⟩) =
some ⟨some .restore, ((ambient, overflow flag x p), none),
memory input [] (encodeWord (p - x))
(List.replicate (min x p) true ++ saved) level fuel spare product⟩ := by
induction x generalizing p saved flag register with
| zero =>
simpa [overflow] using compareStep_zero g p input saved level fuel spare product ambient flag register
| succ x ih =>
rw [Function.iterate_succ_apply]
change (advance (TM2.step (program g)))^[x + 1]
(TM2.step (program g) ⟨some .compare, ((ambient, flag), register),
memory input (encodeWord (x + 1)) (encodeWord p) saved level fuel spare product⟩) = _
cases p with
| zero =>
rw [compareStep_succ_zero]
simpa [overflow] using ih 0 saved true none
| succ p =>
rw [compareStep_succ_succ]
simpa [overflow, Nat.succ_min_succ, List.replicate_succ', List.append_assoc] using
ih p (true :: saved) flag none
theorem replicate_encodeWord (a b : Nat) :
List.replicate a true ++ encodeWord b = encodeWord (a + b) := by
simp only [encodeWord, ← List.append_assoc, List.replicate_append_replicate]
theorem compareRestoredTrace (g k p : Nat) (level fuel : List Bool)
(ambient : σ) (register : Option Bool) :
(advance (TM2.step (program g)))^[k + min k p + 2]
(some ⟨some .compare, ((ambient, false), register),
memory (encodeWord k) (encodeWord k) (encodeWord p) [] level fuel [] []⟩) =
some ⟨some .decide, ((ambient, decide (p < k)), none),
memory (encodeWord k) [] (encodeWord p) [] level fuel [] []⟩ := by
have hc := compareTrace g k p (encodeWord k) [] level fuel [] [] ambient false register
simp only [overflow, Bool.false_or, List.append_nil] at hc
let mid := memory (encodeWord k) [] (encodeWord (p - k))
(List.replicate (min k p) true) level fuel [] []
have hr := Reduction.MachineTransfer.transferAt_fromTapes Tape.saved Tape.power (by decide)
id false Label.restore (some Label.decide) (program g) rfl mid (ambient, decide (p < k)) none
have hsum : min k p + (p - k) = p := by omega
simp only [mid, memory, List.length_replicate, List.reverse_replicate, List.map_id,
Reduction.MachineTransfer.tapesAt, update_saved, update_power] at hr
rw [replicate_encodeWord, hsum] at hr
rw [show k + min k p + 2 = (min k p + 1) + (k + 1) by omega,
Function.iterate_add_apply, hc]
exact hr
private theorem appendTrace {α : Type*} (f : α → α) {a b c : α} {s t : Nat}
(hs : f^[s] a = b) (ht : f^[t] b = c) : f^[s + t] a = c := by
rw [Nat.add_comm s t, Function.iterate_add_apply, hs, ht]
theorem scaleStep_succ (g p b : Nat) (input level fuel : List Bool)
(ambient : σ) (register : Option Bool) :
TM2.step (program g)
⟨some .scale, ((ambient, false), register),
memory input [] (encodeWord (p + 1)) [] level fuel [] (List.replicate b true)⟩ =
some ⟨some .scale, ((ambient, false), some true),
memory input [] (encodeWord p) [] level fuel [] (List.replicate (g + b) true)⟩ := by
change some (TM2.stepAux (scaleLoop g) _ _) = _
simp [scaleLoop, TM2.stepAux, memory, encodeWord, List.replicate_succ,
Reduction.MachineSubstitution.stepAux_pushWord]
theorem scaleStep_zero (g b : Nat) (input level fuel : List Bool)
(ambient : σ) (register : Option Bool) :
TM2.step (program g)
⟨some .scale, ((ambient, false), register),
memory input [] (encodeWord 0) [] level fuel [] (List.replicate b true)⟩ =
some ⟨some .transfer, ((ambient, false), none),
memory input [] (encodeWord 0) [] (true :: level) fuel [] (List.replicate b true)⟩ := by
change some (TM2.stepAux (scaleLoop g) _ _) = _
simp [scaleLoop, TM2.stepAux, memory, encodeWord]
theorem scaleScanTrace (g p b : Nat) (input level fuel : List Bool)
(ambient : σ) (register : Option Bool) :
(advance (TM2.step (program g)))^[p + 1]
(some ⟨some .scale, ((ambient, false), register),
memory input [] (encodeWord p) [] level fuel [] (List.replicate b true)⟩) =
some ⟨some .transfer, ((ambient, false), none),
memory input [] (encodeWord 0) [] (true :: level) fuel [] (List.replicate (g * p + b) true)⟩ := by
induction p generalizing b register with
| zero => simpa using scaleStep_zero g b input level fuel ambient register
| succ p ih =>
rw [Function.iterate_succ_apply]
change (advance (TM2.step (program g)))^[p + 1]
(TM2.step (program g) ⟨some .scale, ((ambient, false), register),
memory input [] (encodeWord (p + 1)) [] level fuel [] (List.replicate b true)⟩) = _
rw [scaleStep_succ, ih]
have h : g * p + (g + b) = g * (p + 1) + b := by rw [Nat.mul_succ]; omega
rw [h]
/-- The fixed-coefficient unary multiplication is an actual scan and transfer;
the level increment is one explicit push in the final scan transition. -/
theorem scaleTrace (g k p e : Nat) (fuel : List Bool)
(ambient : σ) (register : Option Bool) :
(advance (TM2.step (program g)))^[p + g * p + 2]
(some ⟨some .scale, ((ambient, false), register),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) fuel [] []⟩) =
some ⟨some .guard, ((ambient, false), none),
memory (encodeWord k) [] (encodeWord (p * g)) [] (encodeWord (e + 1)) fuel [] []⟩ := by
have hs := scaleScanTrace g p 0 (encodeWord k) (encodeWord e) fuel ambient register
simp only [List.replicate_zero, Nat.add_zero] at hs
let mid := memory (encodeWord k) [] (encodeWord 0) [] (true :: encodeWord e) fuel []
(List.replicate (g * p) true)
have ht := Reduction.MachineTransfer.transferAt_fromTapes Tape.product Tape.power (by decide)
id false Label.transfer (some Label.guard) (program g) rfl mid (ambient, false) none
simp only [mid, memory, List.length_replicate, List.reverse_replicate, List.map_id,
Reduction.MachineTransfer.tapesAt, update_product, update_power,
replicate_encodeWord, Nat.add_zero] at ht
have he : true :: encodeWord e = encodeWord (e + 1) := by simp [encodeWord, List.replicate_succ]
rw [he, Nat.mul_comm g p] at ht
rw [he, Nat.mul_comm g p] at hs
have h := appendTrace (advance (TM2.step (program g))) hs ht
simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm, Nat.mul_comm] using h
theorem cleanupTrace (g k p e f : Nat) (ambient : σ) (register : Option Bool) :
(advance (TM2.step (program g)))^[f + 2]
(some ⟨some .cleanup, ((ambient, false), register),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord f) [] []⟩) =
some ⟨none, ((ambient, false), none),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) [] [] []⟩ := by
have h := MachineDrain.drainTrace Tape.fuel Label.cleanup none (program g) rfl
(memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) [] [] [])
(encodeWord f) (ambient, false) register
simpa only [encodeWord_length, update_fuel, Nat.add_assoc] using h
theorem guardStep_succ (g k p e f : Nat) (ambient : σ) (register : Option Bool) :
TM2.step (program g)
⟨some .guard, ((ambient, false), register),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord (f + 1)) [] []⟩ =
some ⟨some .copyFirst, ((ambient, false), none),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord f) [] []⟩ := by
have h := MachineUnaryCounter.guardStep_succ Tape.fuel Label.guard Label.copyFirst Label.cleanup
(program g) rfl (memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) [] [] [])
f [] (ambient, false) register
simpa only [MachineUnaryCounter.counterTapes, List.append_nil, update_fuel] using h
theorem guardStep_zero (g k p e : Nat) (ambient : σ) (register : Option Bool) :
TM2.step (program g)
⟨some .guard, ((ambient, false), register),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord 0) [] []⟩ =
some ⟨some .cleanup, ((ambient, false), none),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord 0) [] []⟩ := by
have h := MachineUnaryCounter.guardStep_zero Tape.fuel Label.guard Label.copyFirst Label.cleanup
(program g) rfl (memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) [] [] [])
[] (ambient, false) register
simpa only [MachineUnaryCounter.counterTapes, List.append_nil, update_fuel] using h
theorem decideStep (g k p e f : Nat) (ambient : σ) :
TM2.step (program g)
⟨some .decide, ((ambient, decide (p < k)), none),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord f) [] []⟩ =
some ⟨some (if k ≤ p then Label.cleanup else Label.scale), ((ambient, false), none),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord f) [] []⟩ := by
by_cases h : k ≤ p
· have hn : ¬p < k := by omega
simp [TM2.step, program, TM2.stepAux, h, hn]
· have hn : p < k := by omega
simp [TM2.step, program, TM2.stepAux, h, hn]
def prefixTime (k p : Nat) : Nat := (1 + 2 * (k + 2)) + (k + min k p + 2) + 1
/-- The loop prefix is an actual guard, preserving copy, comparison,
restoration and finite decision, all in the same fixed program. -/
theorem prefixTrace (g k p e f : Nat) (ambient : σ) (register : Option Bool) :
(advance (TM2.step (program g)))^[prefixTime k p]
(some ⟨some .guard, ((ambient, false), register),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord (f + 1)) [] []⟩) =
some ⟨some (if k ≤ p then Label.cleanup else Label.scale), ((ambient, false), none),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord f) [] []⟩ := by
have hg : (advance (TM2.step (program g)))^[1]
(some ⟨some .guard, ((ambient, false), register),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord (f + 1)) [] []⟩) =
some ⟨some .copyFirst, ((ambient, false), none),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord f) [] []⟩ :=
guardStep_succ g k p e f ambient register
have hc := MachineCopy.copyTrace Tape.input Tape.work Tape.spare
(by decide) (by decide) (by decide) false Label.copyFirst Label.copySecond
(some Label.compare) (program g) rfl rfl
(memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord f) [] [])
rfl (ambient, false) none
simp only [memory, encodeWord_length, List.append_nil, update_work] at hc
have hcmp := compareRestoredTrace g k p (encodeWord e) (encodeWord f) ambient none
have hd : (advance (TM2.step (program g)))^[1]
(some ⟨some .decide, ((ambient, decide (p < k)), none),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord f) [] []⟩) =
some ⟨some (if k ≤ p then Label.cleanup else Label.scale), ((ambient, false), none),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord f) [] []⟩ :=
decideStep g k p e f ambient
exact appendTrace _ (appendTrace _ (appendTrace _ hg hc) hcmp) hd
def coreTime (g k : Nat) : Nat → Nat → Nat
| 0, _ => 3
| f + 1, p => prefixTime k p +
if k ≤ p then f + 2 else (p + g * p + 2) + coreTime g k f (p * g)
/-- The fixed machine realizes the exact previously checked bounded loop.
Every initial fuel tally is either used or cleared before halting. -/
theorem coreTrace (g k f e p : Nat) (ambient : σ) (register : Option Bool) :
(advance (TM2.step (program g)))^[coreTime g k f p]
(some ⟨some .guard, ((ambient, false), register),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord f) [] []⟩) =
some ⟨none, ((ambient, false), none),
memory (encodeWord k) []
(encodeWord (PreprocessingLevels.levelLoop g k f e p).2) []
(encodeWord (PreprocessingLevels.levelLoop g k f e p).1) [] [] []⟩ := by
induction f generalizing e p register with
| zero =>
have hg : (advance (TM2.step (program g)))^[1]
(some ⟨some .guard, ((ambient, false), register),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord 0) [] []⟩) =
some ⟨some .cleanup, ((ambient, false), none),
memory (encodeWord k) [] (encodeWord p) [] (encodeWord e) (encodeWord 0) [] []⟩ :=
guardStep_zero g k p e ambient register
exact appendTrace _ hg (cleanupTrace g k p e 0 ambient none)
| succ f ih =>
have hpref := prefixTrace g k p e f ambient register
by_cases h : k ≤ p
· simp only [h, ite_true] at hpref
have hrun := appendTrace _ hpref (cleanupTrace g k p e f ambient none)
simpa only [coreTime, PreprocessingLevels.levelLoop, h, ite_true] using hrun
· simp only [h, ite_false] at hpref
have hscale := scaleTrace g k p e (encodeWord f) ambient none
have hrun := appendTrace _ hpref (appendTrace _ hscale (ih (e + 1) (p * g) none))
simpa only [coreTime, PreprocessingLevels.levelLoop, h, ite_false] using hrun
def roundBudget (g k : Nat) : Nat := (g + 8) * (k + 1) + 16
theorem prefixTime_le (k p : Nat) : prefixTime k p ≤ 4 * k + 8 := by
have h := Nat.min_le_left k p
unfold prefixTime
omega
theorem prefixTime_le_roundBudget (g k p : Nat) : prefixTime k p ≤ roundBudget g k := by
have h := prefixTime_le k p
unfold roundBudget
simp only [Nat.add_mul, Nat.mul_add, Nat.mul_one]
omega
theorem continuingTime_le (g k p : Nat) (h : p < k) :
prefixTime k p + (p + g * p + 2) ≤ roundBudget g k := by
have hprefix := prefixTime_le k p
have hmul := Nat.mul_le_mul_left g h.le
unfold roundBudget
simp only [Nat.add_mul, Nat.mul_add, Nat.mul_one]
omega
theorem coreTime_le (g k f p : Nat) :
coreTime g k f p ≤ (f + 1) * roundBudget g k := by
have hB : 3 ≤ roundBudget g k := by unfold roundBudget; omega
induction f generalizing p with
| zero => simpa only [coreTime, Nat.zero_add, Nat.one_mul] using hB
| succ f ih =>
rw [coreTime]
by_cases h : k ≤ p
· rw [ite_eq_left h]
have hp := prefixTime_le_roundBudget g k p
have hm := Nat.mul_le_mul_left (f + 1) hB
have hc : f + 2 ≤ (f + 1) * roundBudget g k := by omega
calc
prefixTime k p + (f + 2) ≤ roundBudget g k + (f + 1) * roundBudget g k :=
Nat.add_le_add hp hc
_ = (f + 1 + 1) * roundBudget g k := by ring
· rw [ite_eq_right h]
have hb := continuingTime_le g k p (by omega)
have hr := ih (p * g)
calc
prefixTime k p + ((p + g * p + 2) + coreTime g k f (p * g)) =
(prefixTime k p + (p + g * p + 2)) + coreTime g k f (p * g) := by omega
_ ≤ roundBudget g k + (f + 1) * roundBudget g k := Nat.add_le_add hb hr
_ = (f + 1 + 1) * roundBudget g k := by ring
theorem initStep (g k : Nat) (ambient : σ) (register : Option Bool) :
TM2.step (program g)
⟨some .init, ((ambient, false), register), memory (encodeWord k) [] [] [] [] [] [] []⟩ =
some ⟨some .fuelFirst, ((ambient, false), register),
memory (encodeWord k) [] (encodeWord 1) [] (encodeWord 0) [] [] []⟩ := by
change some (TM2.stepAux (program g .init) _ _) = _
simp [program, TM2.stepAux, memory, encodeWord]
def totalTime (g k : Nat) : Nat := 1 + 2 * (k + 2) + coreTime g k k 1
/-- Initialization writes one and zero, copies the actual input to the fuel
tape, and executes the proved loop. All auxiliary tapes are empty on exit. -/
theorem searchTrace (g k : Nat) (ambient : σ) (register : Option Bool) :
(advance (TM2.step (program g)))^[totalTime g k]
(some ⟨some .init, ((ambient, false), register), memory (encodeWord k) [] [] [] [] [] [] []⟩) =
some ⟨none, ((ambient, false), none),
memory (encodeWord k) []
(encodeWord (PreprocessingLevels.levelLoop g k k 0 1).2) []
(encodeWord (PreprocessingLevels.levelLoop g k k 0 1).1) [] [] []⟩ := by
have hi : (advance (TM2.step (program g)))^[1]
(some ⟨some .init, ((ambient, false), register), memory (encodeWord k) [] [] [] [] [] [] []⟩) =
some ⟨some .fuelFirst, ((ambient, false), register),
memory (encodeWord k) [] (encodeWord 1) [] (encodeWord 0) [] [] []⟩ :=
initStep g k ambient register
have hc := MachineCopy.copyTrace Tape.input Tape.fuel Tape.spare
(by decide) (by decide) (by decide) false Label.fuelFirst Label.fuelSecond
(some Label.guard) (program g) rfl rfl
(memory (encodeWord k) [] (encodeWord 1) [] (encodeWord 0) [] [] [])
rfl (ambient, false) register
simp only [memory, encodeWord_length, List.append_nil, update_fuel] at hc
exact appendTrace _ (appendTrace _ hi hc) (coreTrace g k k 0 1 ambient none)
def timeBound (g k : Nat) : Nat := (g + 8) * (k + 1) ^ 2 + 18 * (k + 1) + 3
theorem totalTime_le (g k : Nat) : totalTime g k ≤ timeBound g k := by
have h := coreTime_le g k k 1
calc
totalTime g k ≤ 1 + 2 * (k + 2) + (k + 1) * roundBudget g k :=
Nat.add_le_add_left h _
_ = timeBound g k := by unfold roundBudget timeBound; ring
/-- The bound is one fixed quadratic polynomial in the actual unary input
length. Its coefficient may depend on the fixed growth factor. -/
noncomputable def timePolynomial (g : Nat) : Polynomial Nat :=
Polynomial.C (g + 8) * Polynomial.X ^ 2 + 18 * Polynomial.X + 3
theorem timePolynomial_encoding (g k : Nat) :
(timePolynomial g).eval (encodeWord k).length = timeBound g k := by
simp [timePolynomial, timeBound, encodeWord_length]
/-- Instantiate the fixed program with the actual expander growth. This is
the exact executable result already identified with the ceiling logarithm. -/
theorem paddingTrace (k : Nat) (ambient : σ) (register : Option Bool) :
(advance (TM2.step (program ExpanderFamily.growth)))^[totalTime ExpanderFamily.growth k]
(some ⟨some .init, ((ambient, false), register), memory (encodeWord k) [] [] [] [] [] [] []⟩) =
some ⟨none, ((ambient, false), none),
memory (encodeWord k) [] (encodeWord (PreprocessingLevels.paddedSize k)) []
(encodeWord (PreprocessingLevels.boundedLevel k)) [] [] []⟩ := by
exact searchTrace ExpanderFamily.growth k ambient register
def paddingInTime (k : Nat) (ambient : σ) (register : Option Bool) :
StateTransition.EvalsToInTime (TM2.step (program ExpanderFamily.growth))
⟨some .init, ((ambient, false), register), memory (encodeWord k) [] [] [] [] [] [] []⟩
(some ⟨none, ((ambient, false), none),
memory (encodeWord k) [] (encodeWord (PreprocessingLevels.paddedSize k)) []
(encodeWord (PreprocessingLevels.boundedLevel k)) [] [] []⟩)
((timePolynomial ExpanderFamily.growth).eval (encodeWord k).length) where
steps := totalTime ExpanderFamily.growth k
evals_in_steps := paddingTrace k ambient register
steps_le_m := by rw [timePolynomial_encoding]; exact totalTime_le _ _
theorem paddingOutput_bounds (k : Nat) :
PreprocessingLevels.boundedLevel k ≤ k ∧
k ≤ PreprocessingLevels.paddedSize k ∧
PreprocessingLevels.paddedSize k ≤ ExpanderFamily.growth * (k + 1) := by
refine ⟨PreprocessingLevels.boundedLevel_le_input k, PreprocessingLevels.le_paddedSize k, ?_⟩
by_cases hk : k = 0
· subst k
simpa using ExpanderFamily.growth_gt_one.le
· exact (PreprocessingLevels.paddedSize_bounds (Nat.pos_of_ne_zero hk)).2.trans
(Nat.mul_le_mul_left ExpanderFamily.growth (Nat.le_succ k))
end UniqueGamesTheorem.Foundations.Complexity.MachineCeilingPower
end OAI
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