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import OAI.Computability.BinPacking.Search.SearchInitializeEncoding
namespace OAI
namespace BinPackingGap.SearchFinalizeMachine
open Turing BinPackingGames.Foundations.Complexity MachineComposition
open SearchSemantics SearchEncoding BinaryEncoding
open BinPackingGames.Reduction.MachineTransfer
inductive Tape
| input | binary | scratch | reversed | output
deriving DecidableEq
protected abbrev Tape.enumList : List Tape := [.input, .binary, .scratch, .reversed, .output]
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
| start
| convert (label : UnaryToBinaryMachine.Label)
| emitBinary | skipItem | skipLabel | items
| numerator | numeratorDigit | denominator | denominatorDigit
| fixed | option | name | nameDigit | reverse
deriving DecidableEq, Fintype
abbrev Register := Unit × Option Bool
def tapes (input binary scratch reversed output : List Bool) : Tape → List Bool
| .input => input
| .binary => binary
| .scratch => scratch
| .reversed => reversed
| .output => output
@[simp] theorem tapes_input (a b c d e : List Bool) : tapes a b c d e .input = a := rfl
@[simp] theorem tapes_binary (a b c d e : List Bool) : tapes a b c d e .binary = b := rfl
@[simp] theorem tapes_scratch (a b c d e : List Bool) : tapes a b c d e .scratch = c := rfl
@[simp] theorem tapes_reversed (a b c d e : List Bool) : tapes a b c d e .reversed = d := rfl
@[simp] theorem tapes_output (a b c d e : List Bool) : tapes a b c d e .output = e := rfl
@[simp] theorem update_input (a b c d e x : List Bool) :
Function.update (tapes a b c d e) .input x = tapes x b c d e := by
funext k; cases k <;> rfl
@[simp] theorem update_binary (a b c d e x : List Bool) :
Function.update (tapes a b c d e) .binary x = tapes a x c d e := by
funext k; cases k <;> rfl
@[simp] theorem update_scratch (a b c d e x : List Bool) :
Function.update (tapes a b c d e) .scratch x = tapes a b x d e := by
funext k; cases k <;> rfl
@[simp] theorem update_reversed (a b c d e x : List Bool) :
Function.update (tapes a b c d e) .reversed x = tapes a b c x e := by
funext k; cases k <;> rfl
@[simp] theorem update_output (a b c d e x : List Bool) :
Function.update (tapes a b c d e) .output x = tapes a b c d x := by
funext k; cases k <;> rfl
def go (next : Label) : TM2.Stmt (fun _ : Tape => Bool) Label Register :=
.load (fun _ => ((), none)) (.goto fun _ => next)
def reject : TM2.Stmt (fun _ : Tape => Bool) Label Register :=
.push .output (fun _ => false) (.load (fun _ => ((), none)) .halt)
def skipUnary (again next : Label) : TM2.Stmt (fun _ : Tape => Bool) Label Register :=
.pop .input (fun _ head => ((), head))
(.branch (fun state => state.2.getD false) (go again) (go next))
def skipDigit (next : Label) : TM2.Stmt (fun _ : Tape => Bool) Label Register :=
.pop .input (fun _ head => ((), head)) (go next)
def code : Label → TM2.Stmt (fun _ : Tape => Bool) Label Register
| .start => .pop .input (fun _ head => ((), head))
(.branch (fun state => state.2.getD false)
(.pop .input (fun _ head => ((), head))
(.branch (fun state => state.2.getD true) reject
(.push .reversed (fun _ => true) (go (.convert .scan)))))
reject)
| .convert label => UnaryToBinaryMachine.instruction .input .binary .scratch
Label.convert (some .emitBinary) label
| .emitBinary => .pop .binary (fun _ head => ((), head))
(.branch (fun state => state.2.isSome)
(.push .reversed (fun _ => true)
(.push .reversed (fun state => state.2.getD false) (go .emitBinary)))
(.push .reversed (fun _ => false) (go .skipItem)))
| .skipItem => skipUnary .skipItem .skipLabel
| .skipLabel => skipUnary .skipLabel .items
| .items => skipUnary .numerator .fixed
| .numerator => skipUnary .numeratorDigit .denominator
| .numeratorDigit => skipDigit .numerator
| .denominator => skipUnary .denominatorDigit .items
| .denominatorDigit => skipDigit .denominator
| .fixed => .pop .input (fun _ head => ((), head))
(.push .reversed (fun state => state.2.getD false)
(.branch (fun state => state.2.getD false) (go .option) (go .reverse)))
| .option => .pop .input (fun _ head => ((), head))
(.branch (fun state => state.2.getD false) (go .name)
(.push .reversed (fun _ => false) (go .fixed)))
| .name => .pop .input (fun _ head => ((), head))
(.push .reversed (fun state => state.2.getD false)
(.branch (fun state => state.2.getD false) (go .nameDigit) (go .fixed)))
| .nameDigit => .pop .input (fun _ head => ((), head))
(.push .reversed (fun state => state.2.getD false) (go .name))
| .reverse => loopAt .reversed .output id false .reverse none
def program : MachineCanonicalOutput.Program Tape Label Register where
input := .input
output := .output
main := .start
initial := ((), none)
code := code
abbrev machine := MachineCanonicalOutput.sourceMachine program
def cfg (label : Option Label) (input binary scratch reversed output : List Bool)
(register : Option Bool := none) : machine.Cfg :=
⟨label, ((), register), tapes input binary scratch reversed output⟩
private theorem oneStepTrace {a b : machine.Cfg} (h : machine.step a = some b) :
(advance machine.step)^[1] (some a) = some b := by
simpa only [Function.iterate_one, advance_some] using h
private theorem transferNext (dst : Tape) : nextAt dst code = advance machine.step := by
funext state
rfl
private theorem joinTrace {X : Type*} {f : X → X} {a b c : X} {n m : Nat}
(first : f^[n] a = b) (second : f^[m] b = c) : f^[n + m] a = c := by
rw [Nat.add_comm, Function.iterate_add_apply, first, second]
theorem skipStep (again next current : Label) (h : code current = skipUnary again next)
(bit : Bool) (a b c d e : List Bool) :
machine.step (cfg (some current) (bit :: a) b c d e) =
some (cfg (some (if bit then again else next)) a b c d e) := by
change some (TM2.stepAux (code current) _ _) = _
rw [h]
cases bit <;> simp [skipUnary, go, TM2.stepAux, cfg]
all_goals rfl
theorem digitStep (next current : Label) (h : code current = skipDigit next)
(bit : Bool) (a b c d e : List Bool) :
machine.step (cfg (some current) (bit :: a) b c d e) =
some (cfg (some next) a b c d e) := by
change some (TM2.stepAux (code current) _ _) = _
rw [h]
simp [skipDigit, go, TM2.stepAux, cfg]
all_goals rfl
theorem skipUnaryTrace (current next : Label) (h : code current = skipUnary current next)
(n : Nat) (a b c d e : List Bool) :
(advance machine.step)^[n + 1]
(some (cfg (some current) (encodeWord n ++ a) b c d e)) =
some (cfg (some next) a b c d e) := by
induction n with
| zero =>
simpa only [encodeWord, List.replicate_zero, List.nil_append, List.singleton_append,
Nat.zero_add, Function.iterate_one, advance_some, Bool.false_eq_true, ↓reduceIte]
using skipStep current next current h false a b c d e
| succ n ih =>
rw [Function.iterate_succ_apply]
simp only [encodeWord, List.replicate_succ, List.cons_append, advance_some]
rw [skipStep current next current h true]
simpa only [encodeWord, Bool.true_eq, ↓reduceIte] using ih
theorem skipNameTrace (current digit next : Label)
(hc : code current = skipUnary digit next) (hd : code digit = skipDigit current)
(bits a b c d e : List Bool) :
(advance machine.step)^[(BinPackingCompleteness.BinaryEncoding.frame bits).length]
(some (cfg (some current) (BinPackingCompleteness.BinaryEncoding.frame bits ++ a) b c d e)) =
some (cfg (some next) a b c d e) := by
induction bits with
| nil =>
simpa only [BinPackingCompleteness.BinaryEncoding.frame, List.length_singleton,
List.singleton_append, Function.iterate_one, advance_some,
Bool.false_eq_true, ↓reduceIte] using skipStep digit next current hc false a b c d e
| cons bit bits ih =>
have first := skipStep digit next current hc true
(bit :: (BinPackingCompleteness.BinaryEncoding.frame bits ++ a)) b c d e
have second := digitStep current digit hd bit
(BinPackingCompleteness.BinaryEncoding.frame bits ++ a) b c d e
simp only [↓reduceIte] at first
have full := joinTrace (joinTrace
(oneStepTrace first)
(oneStepTrace second)) ih
simpa only [BinPackingCompleteness.BinaryEncoding.frame, List.length_cons,
List.cons_append, Nat.add_comm, Nat.add_assoc] using full
theorem skipItemsTrace (r : RawInstance) (a b c d e : List Bool) :
(advance machine.step)^[(rawInstanceBits r).length]
(some (cfg (some .items) (rawInstanceBits r ++ a) b c d e)) =
some (cfg (some .fixed) a b c d e) := by
induction r with
| nil =>
simpa only [rawInstanceBits, listBits, List.length_singleton, List.singleton_append,
Function.iterate_one, advance_some, Bool.false_eq_true, ↓reduceIte]
using skipStep .numerator .fixed .items rfl false a b c d e
| cons q r ih =>
have first := skipStep .numerator .fixed .items rfl true
(natBits q.1 ++ natBits q.2 ++ rawInstanceBits r ++ a) b c d e
have second := skipNameTrace .numerator .numeratorDigit .denominator rfl rfl q.1.bits
(natBits q.2 ++ rawInstanceBits r ++ a) b c d e
have third := skipNameTrace .denominator .denominatorDigit .items rfl rfl q.2.bits
(rawInstanceBits r ++ a) b c d e
simp only [↓reduceIte] at first
simp only [natBits, BinPackingCompleteness.BinaryEncoding.nameBits,
List.append_assoc] at first second third
have full := joinTrace (joinTrace (joinTrace
(oneStepTrace first) second) third) ih
have clock : 1 + (BinPackingCompleteness.BinaryEncoding.frame q.1.bits).length +
(BinPackingCompleteness.BinaryEncoding.frame q.2.bits).length +
(rawInstanceBits r).length = (rawInstanceBits (q :: r)).length := by
simp only [rawInstanceBits, listBits, pairBits, natBits,
BinPackingCompleteness.BinaryEncoding.nameBits, List.length_cons, List.length_append]
omega
rw [clock] at full
simpa only [rawInstanceBits, listBits, pairBits, natBits,
BinPackingCompleteness.BinaryEncoding.nameBits, List.length_cons,
List.length_append, List.cons_append, List.append_assoc, Nat.add_assoc] using full
theorem emitBinaryStep_nil (a c d e : List Bool) :
machine.step (cfg (some .emitBinary) a [] c d e) =
some (cfg (some .skipItem) a [] c (false :: d) e) := by
change some (TM2.stepAux (code .emitBinary) _ _) = _
simp [code, go, TM2.stepAux, cfg]
all_goals rfl
theorem emitBinaryStep_cons (bit : Bool) (a b c d e : List Bool) :
machine.step (cfg (some .emitBinary) a (bit :: b) c d e) =
some (cfg (some .emitBinary) a b c (bit :: true :: d) e) := by
change some (TM2.stepAux (code .emitBinary) _ _) = _
simp [code, go, TM2.stepAux, cfg]
all_goals rfl
theorem emitBinaryTrace (a bits c d e : List Bool) :
(advance machine.step)^[bits.length + 1]
(some (cfg (some .emitBinary) a bits c d e)) =
some (cfg (some .skipItem) a [] c
((BinPackingCompleteness.BinaryEncoding.frame bits).reverse ++ d) e) := by
induction bits generalizing d with
| nil => simpa [advance, BinPackingCompleteness.BinaryEncoding.frame] using!
emitBinaryStep_nil a c d e
| cons bit bits ih =>
rw [List.length_cons, Function.iterate_succ_apply]
simp only [advance_some, emitBinaryStep_cons]
simpa only [BinPackingCompleteness.BinaryEncoding.frame, List.reverse_cons,
List.append_assoc, List.cons_append, List.nil_append, List.singleton_append] using ih (bit :: true :: d)
theorem fixedStep (bit : Bool) (a b c d e : List Bool) :
machine.step (cfg (some .fixed) (bit :: a) b c d e) =
some (cfg (some (if bit then .option else .reverse)) a b c (bit :: d) e) := by
change some (TM2.stepAux (code .fixed) _ _) = _
cases bit <;> simp [code, go, TM2.stepAux, cfg]
all_goals rfl
theorem optionStep (bit : Bool) (a b c d e : List Bool) :
machine.step (cfg (some .option) (bit :: a) b c d e) =
some (cfg (some (if bit then .name else .fixed)) a b c
(if bit then d else false :: d) e) := by
change some (TM2.stepAux (code .option) _ _) = _
cases bit <;> simp [code, go, TM2.stepAux, cfg]
all_goals rfl
theorem nameStep (bit : Bool) (a b c d e : List Bool) :
machine.step (cfg (some .name) (bit :: a) b c d e) =
some (cfg (some (if bit then .nameDigit else .fixed)) a b c (bit :: d) e) := by
change some (TM2.stepAux (code .name) _ _) = _
cases bit <;> simp [code, go, TM2.stepAux, cfg]
all_goals rfl
theorem nameDigitStep (bit : Bool) (a b c d e : List Bool) :
machine.step (cfg (some .nameDigit) (bit :: a) b c d e) =
some (cfg (some .name) a b c (bit :: d) e) := by
change some (TM2.stepAux (code .nameDigit) _ _) = _
simp [code, go, TM2.stepAux, cfg]
all_goals rfl
theorem nameTrace (bits a b c d e : List Bool) :
(advance machine.step)^[(BinPackingCompleteness.BinaryEncoding.frame bits).length]
(some (cfg (some .name) (BinPackingCompleteness.BinaryEncoding.frame bits ++ a) b c d e)) =
some (cfg (some .fixed) a b c
((BinPackingCompleteness.BinaryEncoding.frame bits).reverse ++ d) e) := by
induction bits generalizing d with
| nil =>
simpa [BinPackingCompleteness.BinaryEncoding.frame, advance] using! nameStep false a b c d e
| cons bit bits ih =>
have first := nameStep true (bit :: (BinPackingCompleteness.BinaryEncoding.frame bits ++ a))
b c d e
have second := nameDigitStep bit (BinPackingCompleteness.BinaryEncoding.frame bits ++ a)
b c (true :: d) e
simp only [↓reduceIte] at first
have full := joinTrace (joinTrace
(oneStepTrace first)
(oneStepTrace second)) (ih (bit :: true :: d))
simpa only [BinPackingCompleteness.BinaryEncoding.frame, List.length_cons,
List.cons_append, List.reverse_cons, List.append_assoc, List.singleton_append,
List.nil_append, Nat.add_comm, Nat.add_assoc] using full
theorem fixedTrace (fixed : List (Option Nat)) (a b c d e : List Bool) :
(advance machine.step)^[(fixedBits fixed).length]
(some (cfg (some .fixed) (fixedBits fixed ++ a) b c d e)) =
some (cfg (some .reverse) a b c ((assignmentBits (labels fixed)).reverse ++ d) e) := by
induction fixed generalizing d with
| nil =>
simpa [fixedBits, listBits, labels, assignmentBits, advance] using!
fixedStep false a b c d e
| cons value fixed ih =>
cases value with
| none =>
have first := fixedStep true (false :: (fixedBits fixed ++ a)) b c d e
have second := optionStep false (fixedBits fixed ++ a) b c (true :: d) e
simp only [Bool.false_eq_true, ↓reduceIte] at first second
have full := joinTrace (joinTrace
(oneStepTrace first)
(oneStepTrace second)) (ih (false :: true :: d))
have clock : 1 + 1 + (fixedBits fixed).length =
(fixedBits (none :: fixed)).length := by
simp only [fixedBits, listBits, optionBits, List.length_cons,
List.length_append, List.length_nil]
omega
rw [clock] at full
simpa only [fixedBits, listBits, optionBits, labels, List.map_cons,
Option.getD_none, assignmentBits, natBits,
BinPackingCompleteness.BinaryEncoding.nameBits, Nat.zero_bits,
BinPackingCompleteness.BinaryEncoding.frame, List.reverse_cons,
List.reverse_append, List.reverse_nil, List.cons_append,
List.nil_append, List.append_nil, List.append_assoc] using full
| some n =>
have first := fixedStep true (true :: (natBits n ++ fixedBits fixed ++ a)) b c d e
have second := optionStep true (natBits n ++ fixedBits fixed ++ a) b c (true :: d) e
have third := nameTrace n.bits (fixedBits fixed ++ a) b c (true :: d) e
simp only [↓reduceIte] at first second
simp only [natBits, BinPackingCompleteness.BinaryEncoding.nameBits,
List.append_assoc] at first second third
have full := joinTrace (joinTrace (joinTrace
(oneStepTrace first)
(oneStepTrace second)) third)
(ih ((natBits n).reverse ++ true :: d))
have clock : 1 + 1 + (BinPackingCompleteness.BinaryEncoding.frame n.bits).length +
(fixedBits fixed).length = (fixedBits (some n :: fixed)).length := by
simp only [fixedBits, listBits, optionBits, natBits,
BinPackingCompleteness.BinaryEncoding.nameBits, List.length_cons, List.length_append]
omega
rw [clock] at full
simpa only [fixedBits, listBits, optionBits, labels, List.map_cons,
Option.getD_some, assignmentBits, natBits, BinPackingCompleteness.BinaryEncoding.nameBits,
List.length_cons, List.length_append, List.cons_append, List.reverse_cons,
List.reverse_append, List.append_assoc, List.singleton_append,
List.nil_append, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using full
theorem convertTrace (n : Nat) (a d e : List Bool) :
(advance machine.step)^[UnaryToBinaryMachine.steps n 0]
(some (cfg (some (.convert .scan)) (encodeWord n ++ a) [] [] d e)) =
some (cfg (some .emitBinary) a n.bits [] d e) := by
have h := UnaryToBinaryMachine.convertTrace
(K := Tape) (A := Unit) .input .binary .scratch (by decide) (by decide) (by decide)
Label.convert (some .emitBinary) code (fun _ => rfl)
(tapes [] [] [] d e) () n 0 a
have step_eq : TM2.step code = machine.step := rfl
rw [step_eq] at h
simpa [UnaryToBinaryMachine.tapes, cfg] using! h
theorem reverseTrace (a b c d e : List Bool) :
(advance machine.step)^[d.length + 1]
(some (cfg (some .reverse) a b c d e)) =
some (cfg none a b c [] (d.reverse ++ e)) := by
have h := transferAt_fromTapes (Γ := fun _ : Tape => Bool) .reversed .output
(by decide) id false .reverse none code rfl (tapes a b c d e) () none
simpa only [transferNext, cfg, tapes_reversed, tapes_output, tapesAt,
update_reversed, update_output, List.map_id_fun, id_eq] using! h
theorem startDone (a : List Bool) :
machine.step (cfg (some .start) (true :: false :: a) [] [] [] []) =
some (cfg (some (.convert .scan)) a [] [] [true] []) := by
change some (TM2.stepAux (code .start) _ _) = _
simp [code, go, TM2.stepAux, cfg]
all_goals rfl
theorem startFalse (a : List Bool) :
machine.step (cfg (some .start) (false :: a) [] [] [] []) =
some (cfg none a [] [] [] [false]) := by
change some (TM2.stepAux (code .start) _ _) = _
simp [code, reject, TM2.stepAux, cfg]
all_goals rfl
theorem startInvalid (a : List Bool) :
machine.step (cfg (some .start) (true :: true :: a) [] [] [] []) =
some (cfg none a [] [] [] [false]) := by
change some (TM2.stepAux (code .start) _ _) = _
simp [code, reject, TM2.stepAux, cfg]
all_goals rfl
def bodyBits (s : State) : List Bool :=
encodeWord s.bins ++ encodeWord s.item ++ encodeWord s.label ++
rawInstanceBits s.items ++ fixedBits s.fixed
def resultBits (bins : Nat) (fixed : List (Option Nat)) : List Bool :=
packingResultBits (some ⟨bins, labels fixed⟩)
def doneSteps (s : State) : Nat :=
1 + UnaryToBinaryMachine.steps s.bins 0 + (s.bins.bits.length + 1) +
(s.item + 1) + (s.label + 1) + (rawInstanceBits s.items).length +
(fixedBits s.fixed).length + ((resultBits s.bins s.fixed).length + 1)
theorem doneTrace (s : State) :
(advance machine.step)^[doneSteps s]
(some (cfg (some .start) (true :: false :: bodyBits s) [] [] [] [])) =
some (cfg none [] [] [] [] (resultBits s.bins s.fixed)) := by
let rest := encodeWord s.item ++ (encodeWord s.label ++
(rawInstanceBits s.items ++ fixedBits s.fixed))
let reverseHeader := (natBits s.bins).reverse ++ [true]
have first := startDone (bodyBits s)
have second := convertTrace s.bins rest [true] []
have third := emitBinaryTrace rest s.bins.bits [] [true] []
have fourth := skipUnaryTrace .skipItem .skipLabel rfl s.item
(encodeWord s.label ++ (rawInstanceBits s.items ++ fixedBits s.fixed))
[] [] reverseHeader []
have fifth := skipUnaryTrace .skipLabel .items rfl s.label
(rawInstanceBits s.items ++ fixedBits s.fixed) [] [] reverseHeader []
have sixth := skipItemsTrace s.items (fixedBits s.fixed) [] [] reverseHeader []
have seventh := fixedTrace s.fixed [] [] [] reverseHeader []
simp only [List.append_nil] at seventh
have finalWord :
((assignmentBits (labels s.fixed)).reverse ++ reverseHeader).reverse =
resultBits s.bins s.fixed := by
simp only [reverseHeader, resultBits, packingResultBits, optionBits, rawPackingBits,
List.reverse_append, List.reverse_reverse, List.reverse_singleton,
List.cons_append, List.nil_append]
have finalLength :
((assignmentBits (labels s.fixed)).reverse ++ reverseHeader).length =
(resultBits s.bins s.fixed).length := by
simpa only [List.length_reverse] using congrArg List.length finalWord
have eighth := reverseTrace [] [] []
((assignmentBits (labels s.fixed)).reverse ++ reverseHeader) []
rw [finalLength, finalWord] at eighth
simp only [List.append_nil] at eighth
have headerEq :
(BinPackingCompleteness.BinaryEncoding.frame s.bins.bits).reverse ++ [true] = reverseHeader := rfl
rw [headerEq] at third
have firstSecond := joinTrace
(oneStepTrace first)
(show (advance machine.step)^[UnaryToBinaryMachine.steps s.bins 0]
(some (cfg (some (.convert .scan)) (bodyBits s) [] [] [true] [])) = _ from by
simpa only [bodyBits, rest, List.append_assoc] using second)
have full := joinTrace (joinTrace (joinTrace (joinTrace (joinTrace
(joinTrace firstSecond third) fourth) fifth) sixth) seventh) eighth
simpa only [doneSteps] using full
theorem labelsBits_le (fixed : List (Option Nat)) :
(assignmentBits (labels fixed)).length ≤ (fixedBits fixed).length := by
induction fixed with
| nil => simp [assignmentBits, labels, fixedBits, listBits]
| cons value fixed ih =>
cases value <;>
simp only [assignmentBits, labels, List.map_cons, fixedBits, listBits,
optionBits, Option.getD_none, Option.getD_some, List.length_cons,
List.length_append] at ih ⊢
· simpa using ih
· omega
theorem resultBits_length_le (bins : Nat) (fixed : List (Option Nat)) :
(resultBits bins fixed).length ≤ 2 * bins + (fixedBits fixed).length + 2 := by
have hf := labelsBits_le fixed
have hn : bins.size ≤ bins := Nat.size_le.mpr bins.lt_two_pow_self
simp only [resultBits, packingResultBits, optionBits, rawPackingBits,
List.length_cons, List.length_append, natBits_length]
omega
noncomputable def rawTime : Polynomial Nat := 10 * (Polynomial.X + 1) ^ 2
theorem doneSteps_le (s : State) : doneSteps s ≤ rawTime.eval (stateBits s).length := by
have ht := UnaryToBinaryMachine.steps_le s.bins 0
have hr := resultBits_length_le s.bins s.fixed
have hn : s.bins.bits.length ≤ s.bins := by
rw [Nat.size_eq_bits_len]
exact Nat.size_le.mpr s.bins.lt_two_pow_self
have hlen := stateBits_length s
have hbins : s.bins ≤ (stateBits s).length := by omega
have hsq : s.bins * s.bins ≤ (stateBits s).length * (stateBits s).length :=
Nat.mul_le_mul hbins hbins
simp only [doneSteps, rawTime, Polynomial.eval_mul, Polynomial.eval_ofNat,
Polynomial.eval_pow, Polynomial.eval_add, Polynomial.eval_X, Polynomial.eval_one]
simp only [Nat.zero_add] at ht
nlinarith
theorem init_eq (input : List Bool) :
initList machine input = cfg (some .start) input [] [] [] [] := by
unfold initList cfg
congr 1
funext k
cases k <;> rfl
private def rejectedRun (s : State) (rest : List Bool)
(h : machine.step (cfg (some .start) (stateBits s) [] [] [] []) =
some (cfg none rest [] [] [] [false]))
(hout : packingResultBits (SearchSemantics.output s) = [false]) :
MachineCanonicalOutput.TerminalRun program (stateBits s)
(packingResultBits (SearchSemantics.output s)) (rawTime.eval (stateBits s).length) where
state := ((), none)
tapes := tapes rest [] [] [] [false]
execution := {
steps := 1
evals_in_steps := by
change machine.step (initList machine (stateBits s)) = _
rw [init_eq]
exact h
steps_le_m := by
simp only [rawTime, Polynomial.eval_mul, Polynomial.eval_ofNat,
Polynomial.eval_pow, Polynomial.eval_add, Polynomial.eval_X, Polynomial.eval_one]
have positive : 1 ≤ (stateBits s).length + 1 := by omega
have square := Nat.mul_le_mul positive positive
nlinarith only [square] }
output_eq := hout.symm
def rawRun (s : State) :
MachineCanonicalOutput.TerminalRun program (stateBits s)
(packingResultBits (SearchSemantics.output s)) (rawTime.eval (stateBits s).length) := by
cases hp : s.phase with
| find =>
apply rejectedRun s (false :: bodyBits s)
· simpa only [stateBits, hp, phaseBits, bodyBits, List.cons_append,
List.nil_append, List.append_assoc] using startFalse (false :: bodyBits s)
· simp [SearchSemantics.output, hp, packingResultBits, optionBits]
| fill =>
apply rejectedRun s (true :: bodyBits s)
· simpa only [stateBits, hp, phaseBits, bodyBits, List.cons_append,
List.nil_append, List.append_assoc] using startFalse (true :: bodyBits s)
· simp [SearchSemantics.output, hp, packingResultBits, optionBits]
| invalid =>
apply rejectedRun s (bodyBits s)
· simpa only [stateBits, hp, phaseBits, bodyBits, List.cons_append,
List.nil_append, List.append_assoc] using startInvalid (bodyBits s)
· simp [SearchSemantics.output, hp, packingResultBits, optionBits]
| done =>
refine {
state := ((), none)
tapes := tapes [] [] [] [] (resultBits s.bins s.fixed)
execution := {
steps := doneSteps s
evals_in_steps := ?_
steps_le_m := doneSteps_le s }
output_eq := ?_ }
· change (advance machine.step)^[doneSteps s]
(some (initList machine (stateBits s))) =
some (cfg none [] [] [] [] (resultBits s.bins s.fixed))
rw [init_eq]
simpa only [stateBits, hp, phaseBits, bodyBits,
List.cons_append, List.nil_append, List.append_assoc] using doneTrace s
· change resultBits s.bins s.fixed = packingResultBits (SearchSemantics.output s)
simp only [resultBits, SearchSemantics.output, hp]
def cleanupTapes : List Tape := [.input, .binary, .scratch, .reversed]
theorem cleanup_complete (k : Tape) : k ∈ cleanupTapes ↔ k ≠ program.output := by
cases k <;> simp [cleanupTapes, program]
noncomputable def computation :
TM2ComputableInPolyTime stateBits packingResultBits SearchSemantics.output :=
MachineCanonicalOutput.computableInPolyTime program cleanupTapes cleanup_complete
stateBits packingResultBits SearchSemantics.output rawTime rawRun
theorem finiteAlphabet : MachineFiniteAlphabet.FiniteAlphabet computation.tm :=
MachineCanonicalOutput.computableInPolyTime_finite_alphabet program cleanupTapes
cleanup_complete stateBits packingResultBits SearchSemantics.output rawTime rawRun
end BinPackingGap.SearchFinalizeMachine
end OAI
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