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namespace OAI
namespace BinPackingGames.Foundations.Complexity.MachineFixedDivMod
open Turing
open Reduction.MachineSubstitution (pushWord stepAux_pushWord statementPushBound_pushWord)
variable {K Λ σ : Type} [DecidableEq K]
abbrev Alphabet (_ : K) := Bool
abbrev State (σ : Type) (d : Nat) := (σ × Fin d) × Option Bool
def residue (d : Nat) (positive : 0 < d) (n : Nat) : Fin d :=
⟨n % d, Nat.mod_lt n positive⟩
def nextResidue (d : Nat) (positive : 0 < d) (r : Fin d) : Fin d :=
residue d positive (r.val + 1)
theorem nextResidue_residue (d : Nat) (positive : 0 < d) (n : Nat) :
nextResidue d positive (residue d positive n) = residue d positive (n + 1) := by
apply Fin.ext
change (n % d + 1) % d = (n + 1) % d
simp only [Nat.add_mod, Nat.mod_mod]
def scanLoop (d : Nat) (positive : 0 < d) (source quotient : K)
(scanLabel : Λ) (emitterLabel : Fin d → Λ) :
TM2.Stmt (Alphabet (K := K)) Λ (State σ d) :=
.pop source (fun state head => (state.1, head))
(.branch (fun state => state.2.getD false)
(.branch (fun state => decide ((nextResidue d positive state.1.2).val = 0))
(.push quotient (fun _ => true)
(.load (fun state => ((state.1.1, nextResidue d positive state.1.2), state.2))
(.goto fun _ => scanLabel)))
(.load (fun state => ((state.1.1, nextResidue d positive state.1.2), state.2))
(.goto fun _ => scanLabel)))
(.push source (fun _ => false) (.goto fun state => emitterLabel state.1.2)))
def emitter (d : Nat) (positive : 0 < d) (remainder : K) (exit : Option Λ)
(r : Fin d) : TM2.Stmt (Alphabet (K := K)) Λ (State σ d) :=
pushWord remainder (List.replicate r.val true)
(.load (fun state => ((state.1.1, residue d positive 0), none))
(Reduction.MachineTransfer.exitAt remainder exit))
omit [DecidableEq K] in
theorem scanPushBound (d : Nat) (positive : 0 < d) (source quotient : K)
(scanLabel : Λ) (emitterLabel : Fin d → Λ) :
Runtime.statementPushBound (scanLoop (σ := σ) d positive source quotient scanLabel
emitterLabel) = 1 := rfl
omit [DecidableEq K] in
theorem emitterPushBound (d : Nat) (positive : 0 < d) (remainder : K)
(exit : Option Λ) (r : Fin d) :
Runtime.statementPushBound (emitter (σ := σ) d positive remainder exit r) = r.val := by
cases exit <;>
simp [emitter, statementPushBound_pushWord, Runtime.statementPushBound,
Reduction.MachineTransfer.exitAt]
abbrev tapes (source quotient remainder : K) (base : K → List Bool)
(input quotientWord remainderWord : List Bool) : K → List Bool :=
MachineCopy.forkTapes source quotient remainder base input quotientWord remainderWord
def unaryTapes (source quotient remainder : K) (base : K → List Bool)
(n q r : Nat) (sourceSuffix quotientSuffix remainderSuffix : List Bool) : K → List Bool :=
tapes source quotient remainder base (encodeWord n ++ sourceSuffix)
(encodeWord q ++ quotientSuffix) (encodeWord r ++ remainderSuffix)
@[simp] theorem unaryTapes_source (source quotient remainder : K)
(sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
(base : K → List Bool) (n q r : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool) :
unaryTapes source quotient remainder base n q r sourceSuffix quotientSuffix
remainderSuffix source = encodeWord n ++ sourceSuffix := by
simp [unaryTapes, sourceQuotient, sourceRemainder]
@[simp] theorem unaryTapes_quotient (source quotient remainder : K)
(quotientRemainder : quotient ≠ remainder) (base : K → List Bool) (n q r : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool) :
unaryTapes source quotient remainder base n q r sourceSuffix quotientSuffix
remainderSuffix quotient = encodeWord q ++ quotientSuffix := by
simp [unaryTapes, quotientRemainder]
@[simp] theorem unaryTapes_remainder (source quotient remainder : K)
(base : K → List Bool) (n q r : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool) :
unaryTapes source quotient remainder base n q r sourceSuffix quotientSuffix
remainderSuffix remainder = encodeWord r ++ remainderSuffix := by
simp [unaryTapes]
theorem unaryTapes_other (source quotient remainder other : K)
(notSource : other ≠ source) (notQuotient : other ≠ quotient)
(notRemainder : other ≠ remainder) (base : K → List Bool) (n q r : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool) :
unaryTapes source quotient remainder base n q r sourceSuffix quotientSuffix
remainderSuffix other = base other := by
simp [unaryTapes, tapes, MachineCopy.forkTapes, notSource, notQuotient, notRemainder]
private theorem update_source (source quotient remainder : K)
(sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
(quotientRemainder : quotient ≠ remainder) (base : K → List Bool)
(input quotientWord remainderWord replacement : List Bool) :
Function.update (tapes source quotient remainder base input quotientWord remainderWord)
source replacement = tapes source quotient remainder base replacement quotientWord remainderWord := by
funext k
by_cases hs : k = source
· subst k
simp [tapes, MachineCopy.forkTapes, sourceQuotient, sourceRemainder]
· by_cases hq : k = quotient
· subst k
simp [tapes, MachineCopy.forkTapes, Ne.symm sourceQuotient, quotientRemainder]
· by_cases hr : k = remainder
· subst k
simp [tapes, MachineCopy.forkTapes, Ne.symm sourceRemainder]
· simp [tapes, MachineCopy.forkTapes, hs, hq, hr]
private theorem update_quotient (source quotient remainder : K)
(quotientRemainder : quotient ≠ remainder) (base : K → List Bool)
(input quotientWord remainderWord replacement : List Bool) :
Function.update (tapes source quotient remainder base input quotientWord remainderWord)
quotient replacement = tapes source quotient remainder base input replacement remainderWord := by
funext k
by_cases hq : k = quotient
· subst k
simp [tapes, MachineCopy.forkTapes, quotientRemainder]
· by_cases hr : k = remainder
· subst k
simp [tapes, MachineCopy.forkTapes, Ne.symm quotientRemainder]
· simp [tapes, MachineCopy.forkTapes, hq, hr]
private theorem update_remainder (source quotient remainder : K) (base : K → List Bool)
(input quotientWord remainderWord replacement : List Bool) :
Function.update (tapes source quotient remainder base input quotientWord remainderWord)
remainder replacement = tapes source quotient remainder base input quotientWord replacement := by
simp [tapes, MachineCopy.forkTapes]
theorem scanAux_true (d : Nat) (positive : 0 < d) (source quotient remainder : K)
(sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
(quotientRemainder : quotient ≠ remainder) (scanLabel : Λ) (emitterLabel : Fin d → Λ)
(base : K → List Bool) (input quotientWord remainderWord : List Bool)
(ambient : σ) (r : Fin d) (register : Option Bool) :
TM2.stepAux (scanLoop d positive source quotient scanLabel emitterLabel)
((ambient, r), register)
(tapes source quotient remainder base (true :: input) quotientWord remainderWord) =
⟨some scanLabel, ((ambient, nextResidue d positive r), some true),
tapes source quotient remainder base input
(if (nextResidue d positive r).val = 0 then true :: quotientWord else quotientWord)
remainderWord⟩ := by
by_cases hcarry : (nextResidue d positive r).val = 0 <;>
simp [scanLoop, TM2.stepAux, sourceQuotient, sourceRemainder, quotientRemainder,
update_source, update_quotient, hcarry]
theorem scanAux_false (d : Nat) (positive : 0 < d) (source quotient remainder : K)
(sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
(quotientRemainder : quotient ≠ remainder) (scanLabel : Λ) (emitterLabel : Fin d → Λ)
(base : K → List Bool) (input quotientWord remainderWord : List Bool)
(ambient : σ) (r : Fin d) (register : Option Bool) :
TM2.stepAux (scanLoop d positive source quotient scanLabel emitterLabel)
((ambient, r), register)
(tapes source quotient remainder base (false :: input) quotientWord remainderWord) =
⟨some (emitterLabel r), ((ambient, r), some false),
tapes source quotient remainder base (false :: input) quotientWord remainderWord⟩ := by
simp [scanLoop, TM2.stepAux, sourceQuotient, sourceRemainder, quotientRemainder,
update_source]
theorem scanStep_succ (d : Nat) (positive : 0 < d) (source quotient remainder : K)
(sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
(quotientRemainder : quotient ≠ remainder) (scanLabel : Λ) (emitterLabel : Fin d → Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ d))
(atScan : program scanLabel = scanLoop d positive source quotient scanLabel emitterLabel)
(base : K → List Bool) (n k : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool)
(ambient : σ) (register : Option Bool) :
TM2.step program ⟨some scanLabel, ((ambient, residue d positive k), register),
unaryTapes source quotient remainder base (n + 1) (k / d) 0
sourceSuffix quotientSuffix remainderSuffix⟩ =
some ⟨some scanLabel, ((ambient, residue d positive (k + 1)), some true),
unaryTapes source quotient remainder base n ((k + 1) / d) 0
sourceSuffix quotientSuffix remainderSuffix⟩ := by
change some (TM2.stepAux (program scanLabel) ((ambient, residue d positive k), register)
(unaryTapes source quotient remainder base (n + 1) (k / d) 0
sourceSuffix quotientSuffix remainderSuffix)) = _
rw [atScan]
simp only [unaryTapes, encodeWord, List.replicate_succ, List.cons_append]
rw [scanAux_true d positive source quotient remainder sourceQuotient sourceRemainder
quotientRemainder, nextResidue_residue]
by_cases hmod : (k + 1) % d = 0
· rw [Nat.succ_div_of_mod_eq_zero hmod]
simp [residue, hmod, List.replicate_succ]
· rw [Nat.succ_div_of_mod_ne_zero hmod]
simp [residue, hmod]
theorem scanStep_zero (d : Nat) (positive : 0 < d) (source quotient remainder : K)
(sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
(quotientRemainder : quotient ≠ remainder) (scanLabel : Λ) (emitterLabel : Fin d → Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ d))
(atScan : program scanLabel = scanLoop d positive source quotient scanLabel emitterLabel)
(base : K → List Bool) (q : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool)
(ambient : σ) (r : Fin d) (register : Option Bool) :
TM2.step program ⟨some scanLabel, ((ambient, r), register),
unaryTapes source quotient remainder base 0 q 0
sourceSuffix quotientSuffix remainderSuffix⟩ =
some ⟨some (emitterLabel r), ((ambient, r), some false),
unaryTapes source quotient remainder base 0 q 0
sourceSuffix quotientSuffix remainderSuffix⟩ := by
change some (TM2.stepAux (program scanLabel) ((ambient, r), register)
(unaryTapes source quotient remainder base 0 q 0
sourceSuffix quotientSuffix remainderSuffix)) = _
rw [atScan]
simp only [unaryTapes, encodeWord, List.replicate_zero, List.nil_append,
List.singleton_append]
rw [scanAux_false d positive source quotient remainder sourceQuotient sourceRemainder
quotientRemainder]
theorem emitterStep (d : Nat) (positive : 0 < d) (source quotient remainder : K)
(emitterLabel : Fin d → Λ) (exit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ d))
(atEmitter : ∀ r, program (emitterLabel r) = emitter d positive remainder exit r)
(base : K → List Bool) (n q : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool)
(ambient : σ) (r : Fin d) (register : Option Bool) :
TM2.step program ⟨some (emitterLabel r), ((ambient, r), register),
unaryTapes source quotient remainder base n q 0
sourceSuffix quotientSuffix remainderSuffix⟩ =
some ⟨exit, ((ambient, residue d positive 0), none),
unaryTapes source quotient remainder base n q r.val
sourceSuffix quotientSuffix remainderSuffix⟩ := by
change some (TM2.stepAux (program (emitterLabel r)) ((ambient, r), register)
(unaryTapes source quotient remainder base n q 0
sourceSuffix quotientSuffix remainderSuffix)) = _
rw [atEmitter]
unfold emitter
rw [stepAux_pushWord]
simp only [unaryTapes, MachineCopy.forkTapes_right, List.reverse_replicate]
rw [update_remainder]
cases exit <;>
simp [TM2.stepAux, Reduction.MachineTransfer.exitAt, encodeWord, List.append_assoc]
theorem trace_fromCount (d : Nat) (positive : 0 < d) (source quotient remainder : K)
(sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
(quotientRemainder : quotient ≠ remainder) (scanLabel : Λ) (emitterLabel : Fin d → Λ)
(exit : Option Λ) (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ d))
(atScan : program scanLabel = scanLoop d positive source quotient scanLabel emitterLabel)
(atEmitter : ∀ r, program (emitterLabel r) = emitter d positive remainder exit r)
(base : K → List Bool) (n k : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool)
(ambient : σ) (register : Option Bool) :
(MachineComposition.advance (TM2.step program))^[n + 2]
(some ⟨some scanLabel, ((ambient, residue d positive k), register),
unaryTapes source quotient remainder base n (k / d) 0
sourceSuffix quotientSuffix remainderSuffix⟩) =
some ⟨exit, ((ambient, residue d positive 0), none),
unaryTapes source quotient remainder base 0 ((k + n) / d) ((k + n) % d)
sourceSuffix quotientSuffix remainderSuffix⟩ := by
induction n generalizing k register with
| zero =>
change (TM2.step program ⟨some scanLabel, ((ambient, residue d positive k), register),
unaryTapes source quotient remainder base 0 (k / d) 0
sourceSuffix quotientSuffix remainderSuffix⟩).bind (TM2.step program) = _
rw [scanStep_zero d positive source quotient remainder sourceQuotient sourceRemainder
quotientRemainder scanLabel emitterLabel program atScan, Option.bind_some]
simpa only [residue, Nat.add_zero] using
emitterStep d positive source quotient remainder emitterLabel exit program atEmitter
base 0 (k / d) sourceSuffix quotientSuffix remainderSuffix ambient
(residue d positive k) (some false)
| succ n ih =>
rw [Function.iterate_succ_apply]
change (MachineComposition.advance (TM2.step program))^[n + 2]
(TM2.step program ⟨some scanLabel, ((ambient, residue d positive k), register),
unaryTapes source quotient remainder base (n + 1) (k / d) 0
sourceSuffix quotientSuffix remainderSuffix⟩) = _
rw [scanStep_succ d positive source quotient remainder sourceQuotient sourceRemainder
quotientRemainder scanLabel emitterLabel program atScan]
simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using ih (k + 1) (some true)
theorem divModTrace (d : Nat) (positive : 0 < d) (source quotient remainder : K)
(sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
(quotientRemainder : quotient ≠ remainder) (scanLabel : Λ) (emitterLabel : Fin d → Λ)
(exit : Option Λ) (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ d))
(atScan : program scanLabel = scanLoop d positive source quotient scanLabel emitterLabel)
(atEmitter : ∀ r, program (emitterLabel r) = emitter d positive remainder exit r)
(base : K → List Bool) (n : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool)
(ambient : σ) (register : Option Bool) :
(MachineComposition.advance (TM2.step program))^[n + 2]
(some ⟨some scanLabel, ((ambient, residue d positive 0), register),
unaryTapes source quotient remainder base n 0 0
sourceSuffix quotientSuffix remainderSuffix⟩) =
some ⟨exit, ((ambient, residue d positive 0), none),
unaryTapes source quotient remainder base 0 (n / d) (n % d)
sourceSuffix quotientSuffix remainderSuffix⟩ := by
simpa only [Nat.zero_add, Nat.zero_div] using
trace_fromCount d positive source quotient remainder sourceQuotient sourceRemainder
quotientRemainder scanLabel emitterLabel exit program atScan atEmitter base n 0
sourceSuffix quotientSuffix remainderSuffix ambient register
theorem divModFromTapes (d : Nat) (positive : 0 < d) (source quotient remainder : K)
(sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
(quotientRemainder : quotient ≠ remainder) (scanLabel : Λ) (emitterLabel : Fin d → Λ)
(exit : Option Λ) (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ d))
(atScan : program scanLabel = scanLoop d positive source quotient scanLabel emitterLabel)
(atEmitter : ∀ r, program (emitterLabel r) = emitter d positive remainder exit r)
(base : K → List Bool) (n : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool)
(sourceInput : base source = encodeWord n ++ sourceSuffix)
(quotientInput : base quotient = encodeWord 0 ++ quotientSuffix)
(remainderInput : base remainder = encodeWord 0 ++ remainderSuffix)
(ambient : σ) (register : Option Bool) :
(MachineComposition.advance (TM2.step program))^[n + 2]
(some ⟨some scanLabel, ((ambient, residue d positive 0), register), base⟩) =
some ⟨exit, ((ambient, residue d positive 0), none),
unaryTapes source quotient remainder base 0 (n / d) (n % d)
sourceSuffix quotientSuffix remainderSuffix⟩ := by
have hbase : unaryTapes source quotient remainder base n 0 0 sourceSuffix quotientSuffix
remainderSuffix = base := by
simp only [unaryTapes, ← sourceInput, ← quotientInput, ← remainderInput,
MachineCopy.forkTapes_self]
have h := divModTrace d positive source quotient remainder sourceQuotient sourceRemainder
quotientRemainder scanLabel emitterLabel exit program atScan atEmitter base n
sourceSuffix quotientSuffix remainderSuffix ambient register
rw [hbase] at h
exact h
def divModInTime (d : Nat) (positive : 0 < d) (source quotient remainder : K)
(sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
(quotientRemainder : quotient ≠ remainder) (scanLabel : Λ) (emitterLabel : Fin d → Λ)
(exit : Option Λ) (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ d))
(atScan : program scanLabel = scanLoop d positive source quotient scanLabel emitterLabel)
(atEmitter : ∀ r, program (emitterLabel r) = emitter d positive remainder exit r)
(base : K → List Bool) (n : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool)
(sourceInput : base source = encodeWord n ++ sourceSuffix)
(quotientInput : base quotient = encodeWord 0 ++ quotientSuffix)
(remainderInput : base remainder = encodeWord 0 ++ remainderSuffix)
(ambient : σ) (register : Option Bool) :
StateTransition.EvalsToInTime (TM2.step program)
⟨some scanLabel, ((ambient, residue d positive 0), register), base⟩
(some ⟨exit, ((ambient, residue d positive 0), none),
unaryTapes source quotient remainder base 0 (n / d) (n % d)
sourceSuffix quotientSuffix remainderSuffix⟩) (n + 2) where
steps := n + 2
evals_in_steps := divModFromTapes d positive source quotient remainder sourceQuotient
sourceRemainder quotientRemainder scanLabel emitterLabel exit program atScan atEmitter
base n sourceSuffix quotientSuffix remainderSuffix sourceInput quotientInput remainderInput
ambient register
steps_le_m := Nat.le_refl _
theorem scanTrace_fromCount (d : Nat) (positive : 0 < d) (source quotient remainder : K)
(sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
(quotientRemainder : quotient ≠ remainder) (scanLabel : Λ) (emitterLabel : Fin d → Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ d))
(atScan : program scanLabel = scanLoop d positive source quotient scanLabel emitterLabel)
(base : K → List Bool) (n k : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool)
(ambient : σ) (register : Option Bool) :
(MachineComposition.advance (TM2.step program))^[n + 1]
(some ⟨some scanLabel, ((ambient, residue d positive k), register),
unaryTapes source quotient remainder base n (k / d) 0
sourceSuffix quotientSuffix remainderSuffix⟩) =
some ⟨some (emitterLabel (residue d positive (k + n))),
((ambient, residue d positive (k + n)), some false),
unaryTapes source quotient remainder base 0 ((k + n) / d) 0
sourceSuffix quotientSuffix remainderSuffix⟩ := by
induction n generalizing k register with
| zero =>
simpa only [Nat.zero_add, Nat.add_zero, Function.iterate_one,
MachineComposition.advance_some] using
scanStep_zero d positive source quotient remainder sourceQuotient sourceRemainder
quotientRemainder scanLabel emitterLabel program atScan base (k / d)
sourceSuffix quotientSuffix remainderSuffix ambient (residue d positive k) register
| succ n ih =>
rw [Function.iterate_succ_apply]
change (MachineComposition.advance (TM2.step program))^[n + 1]
(TM2.step program ⟨some scanLabel, ((ambient, residue d positive k), register),
unaryTapes source quotient remainder base (n + 1) (k / d) 0
sourceSuffix quotientSuffix remainderSuffix⟩) = _
rw [scanStep_succ d positive source quotient remainder sourceQuotient sourceRemainder
quotientRemainder scanLabel emitterLabel program atScan]
simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using ih (k + 1) (some true)
def emitterWithFinish [Fintype σ] (d : Nat) (positive : 0 < d) (remainder : K)
(exit : Option Λ) (finish : σ → Fin d → σ) (r : Fin d) :
TM2.Stmt (Alphabet (K := K)) Λ (State σ d) :=
pushWord remainder (List.replicate r.val true)
(.load (fun state => ((finish state.1.1 r, residue d positive 0), none))
(Reduction.MachineTransfer.exitAt remainder exit))
omit [DecidableEq K] in
theorem emitterWithFinishPushBound [Fintype σ] (d : Nat) (positive : 0 < d)
(remainder : K) (exit : Option Λ) (finish : σ → Fin d → σ) (r : Fin d) :
Runtime.statementPushBound (emitterWithFinish d positive remainder exit finish r) =
r.val := by
cases exit <;>
simp [emitterWithFinish, statementPushBound_pushWord, Runtime.statementPushBound,
Reduction.MachineTransfer.exitAt]
theorem emitterWithFinishStep [Fintype σ] (d : Nat) (positive : 0 < d)
(source quotient remainder : K) (emitterLabel : Fin d → Λ) (exit : Option Λ)
(finish : σ → Fin d → σ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ d))
(atEmitter : ∀ r, program (emitterLabel r) =
emitterWithFinish d positive remainder exit finish r)
(base : K → List Bool) (n q : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool)
(ambient : σ) (r : Fin d) (register : Option Bool) :
TM2.step program ⟨some (emitterLabel r), ((ambient, r), register),
unaryTapes source quotient remainder base n q 0
sourceSuffix quotientSuffix remainderSuffix⟩ =
some ⟨exit, ((finish ambient r, residue d positive 0), none),
unaryTapes source quotient remainder base n q r.val
sourceSuffix quotientSuffix remainderSuffix⟩ := by
change some (TM2.stepAux (program (emitterLabel r)) ((ambient, r), register)
(unaryTapes source quotient remainder base n q 0
sourceSuffix quotientSuffix remainderSuffix)) = _
rw [atEmitter]
unfold emitterWithFinish
rw [stepAux_pushWord]
simp only [unaryTapes, MachineCopy.forkTapes_right, List.reverse_replicate]
rw [update_remainder]
cases exit <;>
simp [TM2.stepAux, Reduction.MachineTransfer.exitAt, encodeWord, List.append_assoc]
theorem divModTraceWithFinish [Fintype σ] (d : Nat) (positive : 0 < d)
(source quotient remainder : K)
(sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
(quotientRemainder : quotient ≠ remainder) (scanLabel : Λ) (emitterLabel : Fin d → Λ)
(exit : Option Λ) (finish : σ → Fin d → σ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ d))
(atScan : program scanLabel = scanLoop d positive source quotient scanLabel emitterLabel)
(atEmitter : ∀ r, program (emitterLabel r) =
emitterWithFinish d positive remainder exit finish r)
(base : K → List Bool) (n : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool)
(ambient : σ) (register : Option Bool) :
(MachineComposition.advance (TM2.step program))^[n + 2]
(some ⟨some scanLabel, ((ambient, residue d positive 0), register),
unaryTapes source quotient remainder base n 0 0
sourceSuffix quotientSuffix remainderSuffix⟩) =
some ⟨exit, ((finish ambient (residue d positive n), residue d positive 0), none),
unaryTapes source quotient remainder base 0 (n / d) (n % d)
sourceSuffix quotientSuffix remainderSuffix⟩ := by
have hscan := scanTrace_fromCount d positive source quotient remainder
sourceQuotient sourceRemainder quotientRemainder scanLabel emitterLabel program atScan
base n 0 sourceSuffix quotientSuffix remainderSuffix ambient register
simp only [Nat.zero_add, Nat.zero_div] at hscan
rw [Function.iterate_succ_apply', hscan, MachineComposition.advance_some]
exact emitterWithFinishStep d positive source quotient remainder emitterLabel exit finish
program atEmitter base 0 (n / d) sourceSuffix quotientSuffix remainderSuffix ambient
(residue d positive n) (some false)
theorem divModFromTapesWithFinish [Fintype σ] (d : Nat) (positive : 0 < d)
(source quotient remainder : K)
(sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
(quotientRemainder : quotient ≠ remainder) (scanLabel : Λ) (emitterLabel : Fin d → Λ)
(exit : Option Λ) (finish : σ → Fin d → σ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ d))
(atScan : program scanLabel = scanLoop d positive source quotient scanLabel emitterLabel)
(atEmitter : ∀ r, program (emitterLabel r) =
emitterWithFinish d positive remainder exit finish r)
(base : K → List Bool) (n : Nat)
(sourceSuffix quotientSuffix remainderSuffix : List Bool)
(sourceInput : base source = encodeWord n ++ sourceSuffix)
(quotientInput : base quotient = encodeWord 0 ++ quotientSuffix)
(remainderInput : base remainder = encodeWord 0 ++ remainderSuffix)
(ambient : σ) (register : Option Bool) :
(MachineComposition.advance (TM2.step program))^[n + 2]
(some ⟨some scanLabel, ((ambient, residue d positive 0), register), base⟩) =
some ⟨exit, ((finish ambient (residue d positive n), residue d positive 0), none),
unaryTapes source quotient remainder base 0 (n / d) (n % d)
sourceSuffix quotientSuffix remainderSuffix⟩ := by
have hbase : unaryTapes source quotient remainder base n 0 0 sourceSuffix quotientSuffix
remainderSuffix = base := by
simp only [unaryTapes, ← sourceInput, ← quotientInput, ← remainderInput,
MachineCopy.forkTapes_self]
have h := divModTraceWithFinish d positive source quotient remainder sourceQuotient
sourceRemainder quotientRemainder scanLabel emitterLabel exit finish program atScan
atEmitter base n sourceSuffix quotientSuffix remainderSuffix ambient register
rw [hbase] at h
exact h
end BinPackingGames.Foundations.Complexity.MachineFixedDivMod
namespace BinPackingGames.Foundations.Complexity.MachinePortReindex
open Turing MachineComposition
open Reduction.MachineSubstitution (pushWord stepAux_pushWord)
variable {K Λ A : Type} [DecidableEq K]
abbrev Alphabet (_ : K) := Bool
abbrev State (A : Type) (d : Nat) := MachineFixedDivMod.State A d
def clean (d : Nat) (positive : 0 < d) (ambient : A) : State A d :=
((ambient, MachineFixedDivMod.residue d positive 0), none)
inductive Label (d : Nat)
| copySeed | copyScan | copyRestore | initialize | divide
| emit (r : Fin d)
| affineScan | affineRestore | drainQuotient | finish
deriving DecidableEq, Fintype
def exitAt {d : Nat} (exit : Option Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A d) :=
match exit with
| none => .halt
| some label => .goto fun _ => label
def emit (d : Nat) (positive : 0 < d) (output : K) (b : Nat)
(r : Fin d) (next : Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A d) :=
pushWord output (List.replicate (r.val + b) true)
(.load (fun s => clean d positive s.1.1) (.goto fun _ => next))
def instruction (d : Nat) (positive : 0 < d) (c b : Nat)
(tape : Fin 5 → K) (labels : Label d → Λ) (exit : Option Λ) :
Label d → TM2.Stmt (Alphabet (K := K)) Λ (State A d)
| .copySeed => MachineUnaryAffineAt.seed (tape 2) 0 (labels .copyScan)
| .copyScan => MachineUnaryAffineAt.scan (tape 0) (tape 1) (tape 2) 1
(labels .copyScan) (labels .copyRestore)
| .copyRestore => Reduction.MachineTransfer.loopAt (tape 1) (tape 0) id false
(labels .copyRestore) (some (labels .initialize))
| .initialize => .push (tape 3) (fun _ => false)
(.push (tape 4) (fun _ => false) (.goto fun _ => labels .divide))
| .divide => MachineFixedDivMod.scanLoop d positive (tape 2) (tape 3)
(labels .divide) (fun r => labels (.emit r))
| .emit r => emit d positive (tape 4) b r (labels .affineScan)
| .affineScan => MachineUnaryAffineAt.scan (tape 3) (tape 1) (tape 4) c
(labels .affineScan) (labels .affineRestore)
| .affineRestore => Reduction.MachineTransfer.loopAt (tape 1) (tape 3) id false
(labels .affineRestore) (some (labels .drainQuotient))
| .drainQuotient => MachineDrain.drain (tape 3) (labels .drainQuotient)
(some (labels .finish))
| .finish => .pop (tape 2) (fun s _ => clean d positive s.1.1) (exitAt exit)
def steps (d j : Nat) : Nat := 3 * j + 3 * (j / d) + 11
theorem steps_le (d j : Nat) : steps d j ≤ 6 * j + 11 := by
have h := Nat.div_le_self j d
unfold steps
omega
theorem steps_le_encodedLength (d j : Nat) :
steps d j ≤ 6 * (encodeWord j).length + 5 := by
have h := steps_le d j
rw [encodeWord_length]
omega
def value (d c b j : Nat) : Nat := c * (j / d) + (j % d) + b
private def frame (tape : Fin 5 → K) (base : K → List Bool)
(work quotient output : List Bool) : K → List Bool :=
MachineCopy.forkTapes (tape 2) (tape 3) (tape 4) base work quotient output
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 reindexTrace (d : Nat) (positive : 0 < d) (c b : Nat)
(tape : Fin 5 → K) (distinct : Function.Injective tape)
(labels : Label d → Λ) (exit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A d))
(atLabels : ∀ l, program (labels l) = instruction d positive c b tape labels exit l)
(base : K → List Bool) (j : Nat) (suffix : List Bool)
(sourceWord : base (tape 0) = encodeWord j ++ suffix)
(scratchEmpty : base (tape 1) = []) (workEmpty : base (tape 2) = [])
(quotientEmpty : base (tape 3) = []) (ambient : A) :
(advance (TM2.step program))^[steps d j]
(some ⟨some (labels .copySeed), clean d positive ambient, base⟩) =
some ⟨exit, clean d positive ambient,
Function.update base (tape 4) (encodeWord (value d c b j) ++ base (tape 4))⟩ := by
have hd (i k : Fin 5) (h : i ≠ k) : tape i ≠ tape k := fun e => h (distinct e)
let q := j / d
let r := MachineFixedDivMod.residue d positive j
let copied := Function.update base (tape 2) (encodeWord j)
let initialized := frame tape base (encodeWord j) (encodeWord 0)
(encodeWord 0 ++ base (tape 4))
let divided := frame tape base (encodeWord 0) (encodeWord q)
(encodeWord 0 ++ base (tape 4))
let emitted := frame tape base (encodeWord 0) (encodeWord q)
(encodeWord (r.val + b) ++ base (tape 4))
let computed := frame tape base (encodeWord 0) (encodeWord q)
(encodeWord (c * q + (r.val + b)) ++ base (tape 4))
let drained := frame tape base (encodeWord 0) []
(encodeWord (c * q + (r.val + b)) ++ base (tape 4))
have copyRun : (advance (TM2.step program))^[2 * (j + 1) + 1]
(some ⟨some (labels .copySeed), clean d positive ambient, base⟩) =
some ⟨some (labels .initialize), clean d positive ambient, copied⟩ := by
simpa only [clean, copied, Nat.one_mul, Nat.add_zero, workEmpty, List.append_nil] using
MachineUnaryAffineAt.seededAffineTrace (tape 0) (tape 1) (tape 2)
(hd 0 1 (by decide)) (hd 0 2 (by decide)) (hd 1 2 (by decide)) 1 0
(labels .copySeed) (labels .copyScan) (labels .copyRestore)
(some (labels .initialize)) program (atLabels .copySeed) (atLabels .copyScan)
(atLabels .copyRestore) base j suffix sourceWord scratchEmpty
(ambient, MachineFixedDivMod.residue d positive 0) none
have initRun : (advance (TM2.step program))^[1]
(some ⟨some (labels .initialize), clean d positive ambient, copied⟩) =
some ⟨some (labels .divide), clean d positive ambient, initialized⟩ := by
change some (TM2.stepAux (program (labels .initialize)) _ _) = _
rw [atLabels]
simp [instruction, TM2.stepAux, copied, initialized, frame, MachineCopy.forkTapes,
hd 3 2 (by decide), hd 4 2 (by decide), hd 4 3 (by decide), quotientEmpty, encodeWord]
have divisionInput : MachineFixedDivMod.unaryTapes (tape 2) (tape 3) (tape 4)
base j 0 0 [] [] (base (tape 4)) = initialized := by
simp only [MachineFixedDivMod.unaryTapes, MachineFixedDivMod.tapes,
List.append_nil, initialized, frame]
have divisionOutput : MachineFixedDivMod.unaryTapes (tape 2) (tape 3) (tape 4)
base 0 q 0 [] [] (base (tape 4)) = divided := by
simp only [MachineFixedDivMod.unaryTapes, MachineFixedDivMod.tapes,
List.append_nil, divided, frame]
have divideRun : (advance (TM2.step program))^[j + 1]
(some ⟨some (labels .divide), clean d positive ambient, initialized⟩) =
some ⟨some (labels (.emit r)), ((ambient, r), some false), divided⟩ := by
have h := MachineFixedDivMod.scanTrace_fromCount d positive (tape 2) (tape 3) (tape 4)
(hd 2 3 (by decide)) (hd 2 4 (by decide)) (hd 3 4 (by decide))
(labels .divide) (fun a => labels (.emit a)) program (atLabels .divide)
base j 0 [] [] (base (tape 4)) ambient none
simpa only [Nat.zero_div, Nat.zero_add, divisionInput, divisionOutput, clean, r, q] using h
have emitRun : (advance (TM2.step program))^[1]
(some ⟨some (labels (.emit r)), ((ambient, r), some false), divided⟩) =
some ⟨some (labels .affineScan), clean d positive ambient, emitted⟩ := by
change some (TM2.stepAux (program (labels (.emit r))) _ _) = _
rw [atLabels]
simp [instruction, emit, stepAux_pushWord, TM2.stepAux, divided, emitted, frame,
MachineCopy.forkTapes, List.reverse_replicate, encodeWord, List.append_assoc]
have affineInput : MachineUnaryAffineAt.tapes (tape 3) (tape 1) (tape 4)
emitted (encodeWord q ++ []) [] (encodeWord (r.val + b) ++ base (tape 4)) = emitted := by
have heq : emitted (tape 3) = encodeWord q := by
simp [emitted, frame, MachineCopy.forkTapes, hd 3 4 (by decide)]
have hempty : emitted (tape 1) = [] := by
simp [emitted, frame, MachineCopy.forkTapes, hd 1 2 (by decide),
hd 1 3 (by decide), hd 1 4 (by decide), scratchEmpty]
have hout : emitted (tape 4) = encodeWord (r.val + b) ++ base (tape 4) := by
simp [emitted, frame, MachineCopy.forkTapes]
rw [List.append_nil, ← heq, ← hempty, ← hout]
exact MachineCopy.forkTapes_self _ _ _ _
have affineOutput : MachineUnaryAffineAt.tapes (tape 3) (tape 1) (tape 4)
emitted (encodeWord q ++ []) [] (encodeWord (c * q + (r.val + b)) ++ base (tape 4)) =
computed := by
funext k
by_cases h1 : k = tape 1
· subst k
simp [MachineUnaryAffineAt.tapes, emitted, computed, frame, MachineCopy.forkTapes,
hd 1 2 (by decide), hd 1 3 (by decide), hd 1 4 (by decide), scratchEmpty]
· by_cases h3 : k = tape 3
· subst k
simp [MachineUnaryAffineAt.tapes, emitted, computed, frame, MachineCopy.forkTapes,
hd 3 1 (by decide), hd 3 4 (by decide)]
· by_cases h4 : k = tape 4
· subst k; simp [MachineUnaryAffineAt.tapes, computed, frame, MachineCopy.forkTapes]
· simp [MachineUnaryAffineAt.tapes, emitted, computed, frame, MachineCopy.forkTapes,
h1, h3, h4]
have affineRun : (advance (TM2.step program))^[2 * (q + 1)]
(some ⟨some (labels .affineScan), clean d positive ambient, emitted⟩) =
some ⟨some (labels .drainQuotient), clean d positive ambient, computed⟩ := by
have h := MachineUnaryAffineAt.affineTrace (tape 3) (tape 1) (tape 4)
(hd 3 1 (by decide)) (hd 3 4 (by decide)) (hd 1 4 (by decide)) c
(labels .affineScan) (labels .affineRestore) (some (labels .drainQuotient))
program (atLabels .affineScan) (atLabels .affineRestore) emitted q (r.val + b)
[] (base (tape 4)) (ambient, MachineFixedDivMod.residue d positive 0) none
simpa only [affineInput, affineOutput, clean] using h
have quotientWord : computed (tape 3) = encodeWord q := by
simp [computed, frame, MachineCopy.forkTapes, hd 3 4 (by decide)]
have drainOutput : Function.update computed (tape 3) [] = drained := by
funext k
by_cases h3 : k = tape 3
· subst k
simp [drained, frame, MachineCopy.forkTapes, hd 3 4 (by decide)]
· by_cases h4 : k = tape 4
· subst k
simp [computed, drained, frame, MachineCopy.forkTapes, hd 4 3 (by decide)]
· simp [computed, drained, frame, MachineCopy.forkTapes, h3, h4]
have drainRun : (advance (TM2.step program))^[q + 2]
(some ⟨some (labels .drainQuotient), clean d positive ambient, computed⟩) =
some ⟨some (labels .finish), clean d positive ambient, drained⟩ := by
have h := MachineDrain.drainTrace (tape 3) (labels .drainQuotient)
(some (labels .finish)) program (atLabels .drainQuotient)
computed (computed (tape 3)) (ambient, MachineFixedDivMod.residue d positive 0) none
rw [Function.update_eq_self, drainOutput, quotientWord, encodeWord_length] at h
exact h
have finishOutput : Function.update drained (tape 2) [] =
Function.update base (tape 4) (encodeWord (value d c b j) ++ base (tape 4)) := by
have hv : c * q + (r.val + b) = value d c b j := by
simp [value, q, r, MachineFixedDivMod.residue, Nat.add_assoc]
funext k
by_cases h2 : k = tape 2
· subst k
simp [hd 2 4 (by decide), workEmpty]
· by_cases h3 : k = tape 3
· subst k
simp [drained, frame, MachineCopy.forkTapes, hd 3 2 (by decide),
hd 3 4 (by decide), quotientEmpty]
· by_cases h4 : k = tape 4
· subst k
simp [drained, frame, MachineCopy.forkTapes, hd 4 2 (by decide), hv]
· simp [drained, frame, MachineCopy.forkTapes, h2, h3, h4]
have finishRun : (advance (TM2.step program))^[1]
(some ⟨some (labels .finish), clean d positive ambient, drained⟩) =
some ⟨exit, clean d positive ambient,
Function.update base (tape 4) (encodeWord (value d c b j) ++ base (tape 4))⟩ := by
have hw : drained (tape 2) = encodeWord 0 := by
simp [drained, frame, MachineCopy.forkTapes, hd 2 3 (by decide), hd 2 4 (by decide)]
change some (TM2.stepAux (program (labels .finish)) _ _) = _
rw [atLabels]
cases exit <;> simp [instruction, TM2.stepAux, exitAt, hw, encodeWord, finishOutput, clean]
have total := joinTrace (joinTrace (joinTrace (joinTrace (joinTrace
(joinTrace copyRun initRun) divideRun) emitRun) affineRun) drainRun) finishRun
have hsteps : steps d j =
(((((2 * (j + 1) + 1) + 1) + (j + 1)) + 1) + 2 * (q + 1)) + (q + 2) + 1 := by
dsimp [steps, q]
omega
rw [hsteps]
exact total
def reindexInTime (d : Nat) (positive : 0 < d) (c b : Nat)
(tape : Fin 5 → K) (distinct : Function.Injective tape)
(labels : Label d → Λ) (exit : Option Λ)
(program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A d))
(atLabels : ∀ l, program (labels l) = instruction d positive c b tape labels exit l)
(base : K → List Bool) (j : Nat) (suffix : List Bool)
(sourceWord : base (tape 0) = encodeWord j ++ suffix)
(scratchEmpty : base (tape 1) = []) (workEmpty : base (tape 2) = [])
(quotientEmpty : base (tape 3) = []) (ambient : A) :
StateTransition.EvalsToInTime (TM2.step program)
⟨some (labels .copySeed), clean d positive ambient, base⟩
(some ⟨exit, clean d positive ambient,
Function.update base (tape 4) (encodeWord (value d c b j) ++ base (tape 4))⟩)
(6 * j + 11) where
steps := steps d j
evals_in_steps := reindexTrace d positive c b tape distinct labels exit program atLabels
base j suffix sourceWord scratchEmpty workEmpty quotientEmpty ambient
steps_le_m := steps_le d j
def machine (d : Nat) (positive : 0 < d) (c b : Nat) : FinTM2 where
K := Fin 5
k₀ := 0
k₁ := 4
Γ _ := Bool
Λ := Label d
main := .copySeed
σ := State Unit d
initialState := clean d positive ()
m := instruction d positive c b id id none
def machineInTime (d : Nat) (positive : 0 < d) (c b : Nat)
(base : Fin 5 → List Bool) (j : Nat) (suffix : List Bool)
(sourceWord : base 0 = encodeWord j ++ suffix)
(scratchEmpty : base 1 = []) (workEmpty : base 2 = [])
(quotientEmpty : base 3 = []) :
StateTransition.EvalsToInTime (machine d positive c b).step
⟨some .copySeed, clean d positive (), base⟩
(some ⟨none, clean d positive (),
Function.update base (4 : Fin 5) (encodeWord (value d c b j) ++ base 4)⟩)
(6 * j + 11) :=
reindexInTime d positive c b id (fun _ _ h => h) id none
(instruction d positive c b id id none) (fun _ => rfl)
base j suffix sourceWord scratchEmpty workEmpty quotientEmpty ()
theorem lazy_value (d j : Nat) : value d (2 * d) d j =
2 * d * (j / d) + d + (j % d) := by
unfold value
omega
end BinPackingGames.Foundations.Complexity.MachinePortReindex
end OAI
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