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namespace OAI
namespace BinPackingGames.Foundations.Complexity.MachineExpanderRow
open Turing
open PCP.ExpanderTables PCP.ExpanderRowControl PCP.AlphabetTable
def emitSteps (n : Nat) : Nat := 3 * (n + 1) + 6
theorem affinePlan_bits {σ : Type} {bound : Nat} (coefficient : Nat)
(offset : σ → Fin bound) (n : Nat) (ambient : σ) :
Emitter.prefixBits (affinePlan coefficient offset) (fun _ : Fin 1 => n) ambient 3 =
encodeWord (coefficient * n + (offset ambient).val) := by
change Emitter.prefixBits
(Emitter.listCommands (Emitter.affineCommands [((0 : Fin 1), coefficient)] offset))
(fun _ : Fin 1 => n) ambient
(Emitter.affineCommands [((0 : Fin 1), coefficient)] offset).length = _
rw [Emitter.bits_listCommands, Emitter.affineCommands_bits]
simp [Emitter.affineValue]
theorem affinePlan_steps {σ : Type} {bound : Nat} (coefficient : Nat)
(offset : σ → Fin bound) (n : Nat) :
Emitter.prefixSteps (affinePlan coefficient offset) (fun _ : Fin 1 => n) 3 + 1 =
emitSteps n := by
simp [affinePlan, Emitter.prefixSteps, Emitter.commandAt, Emitter.listCommands,
Emitter.affineCommands, Emitter.commandSteps, emitSteps]
def emittedWord {K : Type} [DecidableEq K]
(target : K) (base : K → List Bool) (value : Nat) : K → List Bool :=
Function.update base target ((encodeWord value).reverse ++ base target)
@[simp] theorem emittedWord_target {K : Type} [DecidableEq K]
(target : K) (base : K → List Bool) (value : Nat) :
emittedWord target base value target = (encodeWord value).reverse ++ base target := by
simp [emittedWord]
theorem emittedWord_other {K : Type} [DecidableEq K]
(target : K) (base : K → List Bool) (value : Nat)
(tape : K) (different : tape ≠ target) :
emittedWord target base value tape = base tape := by
simp [emittedWord, different]
variable {ρ : Type} [Fintype ρ] {d : Nat}
@[simp] theorem program_firstEmit (positive : 0 < d) (H : Table (cloudSize d) d)
(label : Emitter.Label 3 (degree d)) :
program (ρ := ρ) positive H (.firstEmit label) =
Emitter.statement (firstPlan ρ d) (fun _ : Fin 1 => Tape.inputVertex)
.emitScratch .queryReverse Label.firstEmit (some Label.firstReverse) label := rfl
@[simp] theorem program_secondEmit (positive : 0 < d) (H : Table (cloudSize d) d)
(label : Emitter.Label 3 (degree d)) :
program (ρ := ρ) positive H (.secondEmit label) =
Emitter.statement (secondPlan ρ d) (fun _ : Fin 1 => Tape.quotientFirst)
.emitScratch .queryReverse Label.secondEmit (some Label.secondReverse) label := rfl
@[simp] theorem program_outputEmit (positive : 0 < d) (H : Table (cloudSize d) d)
(label : Emitter.Label 3 (rowFactor d)) :
program (ρ := ρ) positive H (.outputEmit label) =
Emitter.statement (outputPlan ρ d) (fun _ : Fin 1 => Tape.quotientSecond)
.emitScratch .output Label.outputEmit (some (Label.cleanup 0)) label := rfl
section Embedded
variable {K Λ : Type} [DecidableEq K]
theorem firstEmitTraceAt (positive : 0 < d) (H : Table (cloudSize d) d)
(ports : Tape → K) (portsInjective : Function.Injective ports)
(labels : Label d → Λ) (exit : Option Λ)
(target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
(atLabels : ∀ l, target (labels l) = statement positive H ports labels exit l)
(n : Nat) (base : K → List Bool)
(source : base (ports .inputVertex) = encodeWord n)
(scratch : base (ports .emitScratch) = [])
(ambient : Ambient ρ d × Fin (degree d)) :
(MachineComposition.advance (TM2.step target))^[emitSteps n]
(some ⟨some (labels (.firstEmit (Emitter.labelAt 3 _ 0 .entry))),
((ambient, ()), none), base⟩) =
some ⟨some (labels .firstReverse), ((ambient, ()), none),
emittedWord (ports .queryReverse) base (firstAddress n ambient.1.2)⟩ := by
have code : ∀ l, target (labels (.firstEmit l)) =
Emitter.statement (firstPlan ρ d) (fun _ : Fin 1 => ports .inputVertex)
(ports .emitScratch) (ports .queryReverse) (fun k => labels (.firstEmit k))
(some (labels .firstReverse)) l := by
intro l
rw [atLabels]
rfl
have run := Emitter.planTrace (firstPlan ρ d) (fun _ : Fin 1 => ports .inputVertex)
(ports .emitScratch) (ports .queryReverse)
(by intro i h; have h' := portsInjective h; cases h')
(by intro i h; have h' := portsInjective h; cases h')
(by intro h; have h' := portsInjective h; cases h')
(fun k => labels (.firstEmit k)) (some (labels .firstReverse)) target code
(fun _ : Fin 1 => n) base (fun _ => source) scratch ambient
simpa only [firstPlan, affinePlan_steps, affinePlan_bits, Emitter.resultTapes,
emittedWord, firstAddress] using run
theorem secondEmitTraceAt (positive : 0 < d) (H : Table (cloudSize d) d)
(ports : Tape → K) (portsInjective : Function.Injective ports)
(labels : Label d → Λ) (exit : Option Λ)
(target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
(atLabels : ∀ l, target (labels l) = statement positive H ports labels exit l)
(n : Nat) (base : K → List Bool)
(source : base (ports .quotientFirst) = encodeWord n)
(scratch : base (ports .emitScratch) = [])
(ambient : Ambient ρ d × Fin (degree d)) :
(MachineComposition.advance (TM2.step target))^[emitSteps n]
(some ⟨some (labels (.secondEmit (Emitter.labelAt 3 _ 0 .entry))),
((ambient, ()), none), base⟩) =
some ⟨some (labels .secondReverse), ((ambient, ()), none),
emittedWord (ports .queryReverse) base (secondAddress n ambient.1.2)⟩ := by
have code : ∀ l, target (labels (.secondEmit l)) =
Emitter.statement (secondPlan ρ d) (fun _ : Fin 1 => ports .quotientFirst)
(ports .emitScratch) (ports .queryReverse) (fun k => labels (.secondEmit k))
(some (labels .secondReverse)) l := by
intro l
rw [atLabels]
rfl
have run := Emitter.planTrace (secondPlan ρ d) (fun _ : Fin 1 => ports .quotientFirst)
(ports .emitScratch) (ports .queryReverse)
(by intro i h; have h' := portsInjective h; cases h')
(by intro i h; have h' := portsInjective h; cases h')
(by intro h; have h' := portsInjective h; cases h')
(fun k => labels (.secondEmit k)) (some (labels .secondReverse)) target code
(fun _ : Fin 1 => n) base (fun _ => source) scratch ambient
simpa only [secondPlan, affinePlan_steps, affinePlan_bits, Emitter.resultTapes,
emittedWord, secondAddress] using run
theorem outputEmitTraceAt (positive : 0 < d) (H : Table (cloudSize d) d)
(ports : Tape → K) (portsInjective : Function.Injective ports)
(labels : Label d → Λ) (exit : Option Λ)
(target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
(atLabels : ∀ l, target (labels l) = statement positive H ports labels exit l)
(n : Nat) (base : K → List Bool)
(source : base (ports .quotientSecond) = encodeWord n)
(scratch : base (ports .emitScratch) = [])
(ambient : Ambient ρ d × Fin (degree d)) :
(MachineComposition.advance (TM2.step target))^[emitSteps n]
(some ⟨some (labels (.outputEmit (Emitter.labelAt 3 _ 0 .entry))),
((ambient, ()), none), base⟩) =
some ⟨some (labels (.cleanup 0)), ((ambient, ()), none),
emittedWord (ports .output) base (outputAddress n ambient.1.2)⟩ := by
have code : ∀ l, target (labels (.outputEmit l)) =
Emitter.statement (outputPlan ρ d) (fun _ : Fin 1 => ports .quotientSecond)
(ports .emitScratch) (ports .output) (fun k => labels (.outputEmit k))
(some (labels (.cleanup 0))) l := by
intro l
rw [atLabels]
rfl
have run := Emitter.planTrace (outputPlan ρ d) (fun _ : Fin 1 => ports .quotientSecond)
(ports .emitScratch) (ports .output)
(by intro i h; have h' := portsInjective h; cases h')
(by intro i h; have h' := portsInjective h; cases h')
(by intro h; have h' := portsInjective h; cases h')
(fun k => labels (.outputEmit k)) (some (labels (.cleanup 0))) target code
(fun _ : Fin 1 => n) base (fun _ => source) scratch ambient
simpa only [outputPlan, affinePlan_steps, affinePlan_bits, Emitter.resultTapes,
emittedWord, outputAddress] using run
def firstEmitInTimeAt (positive : 0 < d) (H : Table (cloudSize d) d)
(ports : Tape → K) (portsInjective : Function.Injective ports)
(labels : Label d → Λ) (exit : Option Λ)
(target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
(atLabels : ∀ l, target (labels l) = statement positive H ports labels exit l)
(n : Nat) (base : K → List Bool)
(source : base (ports .inputVertex) = encodeWord n)
(scratch : base (ports .emitScratch) = [])
(ambient : Ambient ρ d × Fin (degree d)) :
StateTransition.EvalsToInTime (TM2.step target)
⟨some (labels (.firstEmit (Emitter.labelAt 3 _ 0 .entry))), ((ambient, ()), none), base⟩
(some ⟨some (labels .firstReverse), ((ambient, ()), none),
emittedWord (ports .queryReverse) base (firstAddress n ambient.1.2)⟩) (emitSteps n) where
steps := emitSteps n
evals_in_steps := firstEmitTraceAt positive H ports portsInjective labels exit target atLabels
n base source scratch ambient
steps_le_m := Nat.le_refl _
def secondEmitInTimeAt (positive : 0 < d) (H : Table (cloudSize d) d)
(ports : Tape → K) (portsInjective : Function.Injective ports)
(labels : Label d → Λ) (exit : Option Λ)
(target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
(atLabels : ∀ l, target (labels l) = statement positive H ports labels exit l)
(n : Nat) (base : K → List Bool)
(source : base (ports .quotientFirst) = encodeWord n)
(scratch : base (ports .emitScratch) = [])
(ambient : Ambient ρ d × Fin (degree d)) :
StateTransition.EvalsToInTime (TM2.step target)
⟨some (labels (.secondEmit (Emitter.labelAt 3 _ 0 .entry))), ((ambient, ()), none), base⟩
(some ⟨some (labels .secondReverse), ((ambient, ()), none),
emittedWord (ports .queryReverse) base (secondAddress n ambient.1.2)⟩) (emitSteps n) where
steps := emitSteps n
evals_in_steps := secondEmitTraceAt positive H ports portsInjective labels exit target atLabels
n base source scratch ambient
steps_le_m := Nat.le_refl _
def outputEmitInTimeAt (positive : 0 < d) (H : Table (cloudSize d) d)
(ports : Tape → K) (portsInjective : Function.Injective ports)
(labels : Label d → Λ) (exit : Option Λ)
(target : Λ → TM2.Stmt (fun _ : K => Bool) Λ (State ρ d))
(atLabels : ∀ l, target (labels l) = statement positive H ports labels exit l)
(n : Nat) (base : K → List Bool)
(source : base (ports .quotientSecond) = encodeWord n)
(scratch : base (ports .emitScratch) = [])
(ambient : Ambient ρ d × Fin (degree d)) :
StateTransition.EvalsToInTime (TM2.step target)
⟨some (labels (.outputEmit (Emitter.labelAt 3 _ 0 .entry))), ((ambient, ()), none), base⟩
(some ⟨some (labels (.cleanup 0)), ((ambient, ()), none),
emittedWord (ports .output) base (outputAddress n ambient.1.2)⟩) (emitSteps n) where
steps := emitSteps n
evals_in_steps := outputEmitTraceAt positive H ports portsInjective labels exit target atLabels
n base source scratch ambient
steps_le_m := Nat.le_refl _
end Embedded
theorem firstEmitTrace (positive : 0 < d) (H : Table (cloudSize d) d)
(n : Nat) (base : Tape → List Bool)
(source : base .inputVertex = encodeWord n) (scratch : base .emitScratch = [])
(ambient : Ambient ρ d × Fin (degree d)) :
(MachineComposition.advance (TM2.step (program positive H)))^[emitSteps n]
(some ⟨some (.firstEmit (Emitter.labelAt 3 _ 0 .entry)),
((ambient, ()), none), base⟩) =
some ⟨some .firstReverse, ((ambient, ()), none),
emittedWord .queryReverse base (firstAddress n ambient.1.2)⟩ := by
exact firstEmitTraceAt positive H id Function.injective_id id none (program positive H)
(fun _ => rfl) n base source scratch ambient
theorem secondEmitTrace (positive : 0 < d) (H : Table (cloudSize d) d)
(n : Nat) (base : Tape → List Bool)
(source : base .quotientFirst = encodeWord n) (scratch : base .emitScratch = [])
(ambient : Ambient ρ d × Fin (degree d)) :
(MachineComposition.advance (TM2.step (program positive H)))^[emitSteps n]
(some ⟨some (.secondEmit (Emitter.labelAt 3 _ 0 .entry)),
((ambient, ()), none), base⟩) =
some ⟨some .secondReverse, ((ambient, ()), none),
emittedWord .queryReverse base (secondAddress n ambient.1.2)⟩ := by
exact secondEmitTraceAt positive H id Function.injective_id id none (program positive H)
(fun _ => rfl) n base source scratch ambient
theorem outputEmitTrace (positive : 0 < d) (H : Table (cloudSize d) d)
(n : Nat) (base : Tape → List Bool)
(source : base .quotientSecond = encodeWord n) (scratch : base .emitScratch = [])
(ambient : Ambient ρ d × Fin (degree d)) :
(MachineComposition.advance (TM2.step (program positive H)))^[emitSteps n]
(some ⟨some (.outputEmit (Emitter.labelAt 3 _ 0 .entry)),
((ambient, ()), none), base⟩) =
some ⟨some (.cleanup 0), ((ambient, ()), none),
emittedWord .output base (outputAddress n ambient.1.2)⟩ := by
exact outputEmitTraceAt positive H id Function.injective_id id none (program positive H)
(fun _ => rfl) n base source scratch ambient
def firstEmitInTime (positive : 0 < d) (H : Table (cloudSize d) d)
(n : Nat) (base : Tape → List Bool)
(source : base .inputVertex = encodeWord n) (scratch : base .emitScratch = [])
(ambient : Ambient ρ d × Fin (degree d)) :
StateTransition.EvalsToInTime (TM2.step (program positive H))
⟨some (.firstEmit (Emitter.labelAt 3 _ 0 .entry)), ((ambient, ()), none), base⟩
(some ⟨some .firstReverse, ((ambient, ()), none),
emittedWord .queryReverse base (firstAddress n ambient.1.2)⟩) (emitSteps n) where
steps := emitSteps n
evals_in_steps := firstEmitTrace positive H n base source scratch ambient
steps_le_m := Nat.le_refl _
def secondEmitInTime (positive : 0 < d) (H : Table (cloudSize d) d)
(n : Nat) (base : Tape → List Bool)
(source : base .quotientFirst = encodeWord n) (scratch : base .emitScratch = [])
(ambient : Ambient ρ d × Fin (degree d)) :
StateTransition.EvalsToInTime (TM2.step (program positive H))
⟨some (.secondEmit (Emitter.labelAt 3 _ 0 .entry)), ((ambient, ()), none), base⟩
(some ⟨some .secondReverse, ((ambient, ()), none),
emittedWord .queryReverse base (secondAddress n ambient.1.2)⟩) (emitSteps n) where
steps := emitSteps n
evals_in_steps := secondEmitTrace positive H n base source scratch ambient
steps_le_m := Nat.le_refl _
def outputEmitInTime (positive : 0 < d) (H : Table (cloudSize d) d)
(n : Nat) (base : Tape → List Bool)
(source : base .quotientSecond = encodeWord n) (scratch : base .emitScratch = [])
(ambient : Ambient ρ d × Fin (degree d)) :
StateTransition.EvalsToInTime (TM2.step (program positive H))
⟨some (.outputEmit (Emitter.labelAt 3 _ 0 .entry)), ((ambient, ()), none), base⟩
(some ⟨some (.cleanup 0), ((ambient, ()), none),
emittedWord .output base (outputAddress n ambient.1.2)⟩) (emitSteps n) where
steps := emitSteps n
evals_in_steps := outputEmitTrace positive H n base source scratch ambient
steps_le_m := Nat.le_refl _
end BinPackingGames.Foundations.Complexity.MachineExpanderRow
end OAI
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