Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.DirectedFeedback.Encoding.ExpanderTableWords

namespace OAI

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namespace DFVSGames.Foundations.Complexity.MachineExpanderRow

open Turing
open PCP.ExpanderTables PCP.ExpanderRowControl

private theorem controlStatement_inverse_inline_MachineExpanderRowDivision {K Λ σ τ : Type} {Γ : K → Type}
    (e : σ ≃ τ) (q : TM2.Stmt Γ Λ σ) :
    MachineControl.statement id e.symm (MachineControl.statement id e q) = q := by
  induction q <;>
    simp_all only [MachineControl.statement, Equiv.apply_symm_apply,
      Equiv.symm_apply_apply, id_eq]

private theorem transportedDivision_inline_MachineExpanderRowDivision {K Λ σ τ : Type} [DecidableEq K] [Fintype σ]
    (q : Nat) (positive : 0 < q) (source quotient remainder : K)
    (sourceQuotient : source ≠ quotient) (sourceRemainder : source ≠ remainder)
    (quotientRemainder : quotient ≠ remainder)
    (states : MachineFixedDivMod.State σ q ≃ τ)
    (scanLabel : Λ) (emitterLabel : Fin q → Λ) (exit : Option Λ)
    (finish : σ → Fin q → σ)
    (target : Λ → TM2.Stmt (fun _ : K => Bool) Λ τ)
    (atScan : target scanLabel = MachineControl.statement id states
      (MachineFixedDivMod.scanLoop q positive source quotient scanLabel emitterLabel))
    (atEmitter : ∀ r, target (emitterLabel r) = MachineControl.statement id states
      (MachineFixedDivMod.emitterWithFinish q 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 target))^[n + 2]
      (some (MachineControl.configuration id states
        ⟨some scanLabel, ((ambient, MachineFixedDivMod.residue q positive 0), register), base⟩)) =
      some (MachineControl.configuration id states
        ⟨exit, ((finish ambient (MachineFixedDivMod.residue q positive n),
          MachineFixedDivMod.residue q positive 0), none),
          MachineFixedDivMod.unaryTapes source quotient remainder base 0 (n / q) (n % q)
            sourceSuffix quotientSuffix remainderSuffix⟩) := by
  let sourceProgram := MachineControl.program (Equiv.refl Λ) states.symm target
  have sourceScan : sourceProgram scanLabel =
      MachineFixedDivMod.scanLoop q positive source quotient scanLabel emitterLabel := by
    change MachineControl.statement id states.symm (target scanLabel) = _
    rw [atScan]
    exact controlStatement_inverse_inline_MachineExpanderRowDivision states _
  have sourceEmit : ∀ r, sourceProgram (emitterLabel r) =
      MachineFixedDivMod.emitterWithFinish q positive remainder exit finish r := by
    intro r
    change MachineControl.statement id states.symm (target (emitterLabel r)) = _
    rw [atEmitter]
    exact controlStatement_inverse_inline_MachineExpanderRowDivision states _
  have htrace := MachineFixedDivMod.divModFromTapesWithFinish q positive source quotient
    remainder sourceQuotient sourceRemainder quotientRemainder scanLabel emitterLabel exit
    finish sourceProgram sourceScan sourceEmit base n sourceSuffix quotientSuffix remainderSuffix
    sourceInput quotientInput remainderInput ambient register
  have roundtrip : MachineControl.program (Equiv.refl Λ) states sourceProgram = target := by
    funext label
    change MachineControl.statement id states
      (MachineControl.statement id states.symm (target label)) = target label
    exact controlStatement_inverse_inline_MachineExpanderRowDivision states.symm _
  have simulation : ∀ a b, TM2.step sourceProgram a = some b →
      TM2.step target (MachineControl.configuration id states a) =
        some (MachineControl.configuration id states b) := by
    intro a b hab
    have h := MachineControl.step_simulation (Equiv.refl Λ) states sourceProgram a
    rw [roundtrip, hab] at h
    exact h
  exact MachineComposition.liftSuccessfulTrace (TM2.step sourceProgram) (TM2.step target)
    (MachineControl.configuration id states) simulation (n + 2) _ _ htrace

variable {K Λ ρ : Type} [DecidableEq K] [Fintype ρ]

@[simp] theorem divisionStates_apply (ρ : Type) (d : Nat)
    (s : MachineFixedDivMod.State (Ambient ρ d × Unit) (degree d)) :
    divisionStates ρ d s = (((s.1.1.1, s.1.2), ()), s.2) := rfl

def divisionState {d : Nat} (positive : 0 < d) (ambient : ρ) (control : Control d)
    (register : Option Bool) : State ρ d :=
  ((((ambient, control),
    MachineFixedDivMod.residue (degree d) (Nat.mul_pos positive positive) 0), ()), register)

def divisionOutput (d : Nat) (ports : Tape → K) (quotient remainder : Tape)
    (base : K → List Bool) (n : Nat)
    (sourceSuffix quotientSuffix remainderSuffix : List Bool) : K → List Bool :=
  MachineFixedDivMod.unaryTapes (ports .lookupOutput) (ports quotient) (ports remainder)
    base 0 (n / degree d) (n % degree d) sourceSuffix quotientSuffix remainderSuffix

theorem firstDivisionTrace {d : Nat} (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))
    (atProgram : ∀ label, target (labels label) = statement positive H ports labels exit label)
    (base : K → List Bool) (n : Nat)
    (sourceSuffix quotientSuffix remainderSuffix : List Bool)
    (sourceInput : base (ports .lookupOutput) = encodeWord n ++ sourceSuffix)
    (quotientInput : base (ports .quotientFirst) = encodeWord 0 ++ quotientSuffix)
    (remainderInput : base (ports .remainderFirst) = encodeWord 0 ++ remainderSuffix)
    (ambient : ρ) (control : Control d) (register : Option Bool) :
    (MachineComposition.advance (TM2.step target))^[n + 2]
      (some ⟨some (labels .firstScan), divisionState positive ambient control register, base⟩) =
      some ⟨some (labels .clearQuery), divisionState positive ambient
        (receiveFirst control (MachineFixedDivMod.residue (degree d)
          (Nat.mul_pos positive positive) n)) none,
        divisionOutput d ports .quotientFirst .remainderFirst base n
          sourceSuffix quotientSuffix remainderSuffix⟩ := by
  have hsq : ports .lookupOutput ≠ ports .quotientFirst :=
    fun h => Tape.noConfusion (portsInjective h)
  have hsr : ports .lookupOutput ≠ ports .remainderFirst :=
    fun h => Tape.noConfusion (portsInjective h)
  have hqr : ports .quotientFirst ≠ ports .remainderFirst :=
    fun h => Tape.noConfusion (portsInjective h)
  have hs : target (labels .firstScan) = MachineControl.statement id (divisionStates ρ d)
      (MachineFixedDivMod.scanLoop (degree d) (Nat.mul_pos positive positive)
        (ports .lookupOutput) (ports .quotientFirst) (labels .firstScan)
        (fun r => labels (.firstResidue r))) := atProgram .firstScan
  have he : ∀ r, target (labels (.firstResidue r)) =
      MachineControl.statement id (divisionStates ρ d)
        (MachineFixedDivMod.emitterWithFinish (degree d) (Nat.mul_pos positive positive)
          (ports .remainderFirst) (some (labels .clearQuery)) firstFinish r) :=
    fun r => atProgram (.firstResidue r)
  have h := transportedDivision_inline_MachineExpanderRowDivision (degree d) (Nat.mul_pos positive positive)
    (ports .lookupOutput) (ports .quotientFirst) (ports .remainderFirst) hsq hsr hqr
    (divisionStates ρ d) (labels .firstScan) (fun r => labels (.firstResidue r))
    (some (labels .clearQuery)) firstFinish target hs he base n sourceSuffix quotientSuffix
    remainderSuffix sourceInput quotientInput remainderInput ((ambient, control), ()) register
  simpa only [MachineControl.configuration, divisionStates_apply, firstFinish, divisionState,
    divisionOutput, Option.map_some, id_eq] using h

theorem secondDivisionTrace {d : Nat} (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))
    (atProgram : ∀ label, target (labels label) = statement positive H ports labels exit label)
    (base : K → List Bool) (n : Nat)
    (sourceSuffix quotientSuffix remainderSuffix : List Bool)
    (sourceInput : base (ports .lookupOutput) = encodeWord n ++ sourceSuffix)
    (quotientInput : base (ports .quotientSecond) = encodeWord 0 ++ quotientSuffix)
    (remainderInput : base (ports .remainderSecond) = encodeWord 0 ++ remainderSuffix)
    (ambient : ρ) (control : Control d) (register : Option Bool) :
    (MachineComposition.advance (TM2.step target))^[n + 2]
      (some ⟨some (labels .secondScan), divisionState positive ambient control register, base⟩) =
      some ⟨some (labels (.outputEmit (PCP.AlphabetTable.Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient
          (receiveSecond H control (MachineFixedDivMod.residue (degree d)
            (Nat.mul_pos positive positive) n)) none,
        divisionOutput d ports .quotientSecond .remainderSecond base n
          sourceSuffix quotientSuffix remainderSuffix⟩ := by
  have hsq : ports .lookupOutput ≠ ports .quotientSecond :=
    fun h => Tape.noConfusion (portsInjective h)
  have hsr : ports .lookupOutput ≠ ports .remainderSecond :=
    fun h => Tape.noConfusion (portsInjective h)
  have hqr : ports .quotientSecond ≠ ports .remainderSecond :=
    fun h => Tape.noConfusion (portsInjective h)
  have hs : target (labels .secondScan) = MachineControl.statement id (divisionStates ρ d)
      (MachineFixedDivMod.scanLoop (degree d) (Nat.mul_pos positive positive)
        (ports .lookupOutput) (ports .quotientSecond) (labels .secondScan)
        (fun r => labels (.secondResidue r))) := atProgram .secondScan
  have he : ∀ r, target (labels (.secondResidue r)) =
      MachineControl.statement id (divisionStates ρ d)
        (MachineFixedDivMod.emitterWithFinish (degree d) (Nat.mul_pos positive positive)
          (ports .remainderSecond)
          (some (labels (.outputEmit (PCP.AlphabetTable.Emitter.labelAt 3 _ 0 .entry))))
          (secondFinish H) r) := fun r => atProgram (.secondResidue r)
  have h := transportedDivision_inline_MachineExpanderRowDivision (degree d) (Nat.mul_pos positive positive)
    (ports .lookupOutput) (ports .quotientSecond) (ports .remainderSecond) hsq hsr hqr
    (divisionStates ρ d) (labels .secondScan) (fun r => labels (.secondResidue r))
    (some (labels (.outputEmit (PCP.AlphabetTable.Emitter.labelAt 3 _ 0 .entry))))
    (secondFinish H) target hs he base n sourceSuffix quotientSuffix remainderSuffix
    sourceInput quotientInput remainderInput ((ambient, control), ()) register
  simpa only [MachineControl.configuration, divisionStates_apply, secondFinish, divisionState,
    divisionOutput, Option.map_some, id_eq] using h

@[simp] theorem divisionOutput_source (d : Nat) (ports : Tape → K)
    (portsInjective : Function.Injective ports) (quotient remainder : Tape)
    (notSourceQuotient : Tape.lookupOutput ≠ quotient)
    (notSourceRemainder : Tape.lookupOutput ≠ remainder)
    (base : K → List Bool) (n : Nat)
    (sourceSuffix quotientSuffix remainderSuffix : List Bool) :
    divisionOutput d ports quotient remainder base n sourceSuffix quotientSuffix remainderSuffix
      (ports .lookupOutput) = encodeWord 0 ++ sourceSuffix := by
  exact MachineFixedDivMod.unaryTapes_source _ _ _
    (fun h => notSourceQuotient (portsInjective h))
    (fun h => notSourceRemainder (portsInjective h)) _ _ _ _ _ _ _

@[simp] theorem divisionOutput_quotient (d : Nat) (ports : Tape → K)
    (portsInjective : Function.Injective ports) (quotient remainder : Tape)
    (distinct : quotient ≠ remainder) (base : K → List Bool) (n : Nat)
    (sourceSuffix quotientSuffix remainderSuffix : List Bool) :
    divisionOutput d ports quotient remainder base n sourceSuffix quotientSuffix remainderSuffix
      (ports quotient) = encodeWord (n / degree d) ++ quotientSuffix := by
  exact MachineFixedDivMod.unaryTapes_quotient _ _ _
    (fun h => distinct (portsInjective h)) _ _ _ _ _ _ _

@[simp] theorem divisionOutput_remainder (d : Nat) (ports : Tape → K)
    (quotient remainder : Tape) (base : K → List Bool) (n : Nat)
    (sourceSuffix quotientSuffix remainderSuffix : List Bool) :
    divisionOutput d ports quotient remainder base n sourceSuffix quotientSuffix remainderSuffix
      (ports remainder) = encodeWord (n % degree d) ++ remainderSuffix := by
  exact MachineFixedDivMod.unaryTapes_remainder _ _ _ _ _ _ _ _ _ _

theorem divisionOutput_other (d : Nat) (ports : Tape → K) (quotient remainder : Tape)
    (base : K → List Bool) (n : Nat)
    (sourceSuffix quotientSuffix remainderSuffix : List Bool) (other : K)
    (notSource : other ≠ ports .lookupOutput) (notQuotient : other ≠ ports quotient)
    (notRemainder : other ≠ ports remainder) :
    divisionOutput d ports quotient remainder base n sourceSuffix quotientSuffix remainderSuffix
      other = base other := by
  exact MachineFixedDivMod.unaryTapes_other _ _ _ other notSource notQuotient notRemainder
    _ _ _ _ _ _ _

def firstDivisionInTime {d : Nat} (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))
    (atProgram : ∀ label, target (labels label) = statement positive H ports labels exit label)
    (base : K → List Bool) (n : Nat)
    (sourceSuffix quotientSuffix remainderSuffix : List Bool)
    (sourceInput : base (ports .lookupOutput) = encodeWord n ++ sourceSuffix)
    (quotientInput : base (ports .quotientFirst) = encodeWord 0 ++ quotientSuffix)
    (remainderInput : base (ports .remainderFirst) = encodeWord 0 ++ remainderSuffix)
    (ambient : ρ) (control : Control d) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step target)
      ⟨some (labels .firstScan), divisionState positive ambient control register, base⟩
      (some ⟨some (labels .clearQuery), divisionState positive ambient
        (receiveFirst control (MachineFixedDivMod.residue (degree d)
          (Nat.mul_pos positive positive) n)) none,
        divisionOutput d ports .quotientFirst .remainderFirst base n
          sourceSuffix quotientSuffix remainderSuffix⟩) (n + 2) where
  steps := n + 2
  evals_in_steps := firstDivisionTrace positive H ports portsInjective labels exit target atProgram
    base n sourceSuffix quotientSuffix remainderSuffix sourceInput quotientInput remainderInput
    ambient control register
  steps_le_m := Nat.le_refl _

def secondDivisionInTime {d : Nat} (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))
    (atProgram : ∀ label, target (labels label) = statement positive H ports labels exit label)
    (base : K → List Bool) (n : Nat)
    (sourceSuffix quotientSuffix remainderSuffix : List Bool)
    (sourceInput : base (ports .lookupOutput) = encodeWord n ++ sourceSuffix)
    (quotientInput : base (ports .quotientSecond) = encodeWord 0 ++ quotientSuffix)
    (remainderInput : base (ports .remainderSecond) = encodeWord 0 ++ remainderSuffix)
    (ambient : ρ) (control : Control d) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step target)
      ⟨some (labels .secondScan), divisionState positive ambient control register, base⟩
      (some ⟨some (labels (.outputEmit (PCP.AlphabetTable.Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient
          (receiveSecond H control (MachineFixedDivMod.residue (degree d)
            (Nat.mul_pos positive positive) n)) none,
        divisionOutput d ports .quotientSecond .remainderSecond base n
          sourceSuffix quotientSuffix remainderSuffix⟩) (n + 2) where
  steps := n + 2
  evals_in_steps := secondDivisionTrace positive H ports portsInjective labels exit target atProgram
    base n sourceSuffix quotientSuffix remainderSuffix sourceInput quotientInput remainderInput
    ambient control register
  steps_le_m := Nat.le_refl _

end DFVSGames.Foundations.Complexity.MachineExpanderRow

end

section

namespace DFVSGames.Foundations.Complexity.MachineExpanderTable

open Turing
open PCP.ExpanderTables PCP.ExpanderRowControl

inductive ExtraTape
  | vertexCount | result
  deriving DecidableEq

instance : Fintype ExtraTape := derive_fintype% ExtraTape

abbrev Tape := MachineExpanderRow.Tape ⊕ ExtraTape

abbrev Alphabet : Tape → Type :=
  MachineEmbedding.Alphabet (fun _ : MachineExpanderRow.Tape => Bool)
    (fun _ : ExtraTape => Bool)

instance alphabetFintype (tape : Tape) : Fintype (Alphabet tape) := by
  cases tape <;> exact inferInstanceAs (Fintype Bool)

inductive OuterLabel
  | initialize | vertexGuard | prepareRow | afterRow | reverseOutput | done
  deriving DecidableEq

instance : Fintype OuterLabel := derive_fintype% OuterLabel

abbrev Label (d : Nat) := MachineExpanderRow.Label d ⊕ OuterLabel
abbrev Position (d : Nat) := Fin (rowFactor d)
abbrev State (ρ : Type) (d : Nat) := MachineExpanderRow.State ρ d × Position d
abbrev RowStatePrefix (ρ : Type) (d : Nat) :=
  (MachineExpanderRow.Ambient ρ d × Fin (degree d)) × Unit

def guardStates (ρ : Type) (d : Nat) :
    ((RowStatePrefix ρ d × Position d) × Option Bool) ≃ State ρ d where
  toFun s := ((s.1.1, s.2), s.1.2)
  invFun s := ((s.1.1, s.2), s.1.2)
  left_inv := by rintro ⟨⟨a, p⟩, b⟩; rfl
  right_inv := by rintro ⟨⟨a, b⟩, p⟩; rfl

@[simp] theorem guardStates_apply (ρ : Type) (d : Nat)
    (s : (RowStatePrefix ρ d × Position d) × Option Bool) :
    guardStates ρ d s = ((s.1.1, s.2), s.1.2) := rfl

@[simp] theorem guardStates_symm_apply (ρ : Type) (d : Nat) (s : State ρ d) :
    (guardStates ρ d).symm s = ((s.1.1, s.2), s.1.2) := rfl

theorem rowFactor_pos {d : Nat} (positive : 0 < d) : 0 < rowFactor d := by
  have hq : 0 < degree d := Nat.mul_pos positive positive
  exact Nat.mul_pos (Nat.mul_pos hq hq) hq

def zeroPosition {d : Nat} (positive : 0 < d) : Position d :=
  ⟨0, rowFactor_pos positive⟩

def nextPosition {d : Nat} (positive : 0 < d) (p : Position d) : Position d :=
  ⟨(p.val + 1) % rowFactor d, Nat.mod_lt _ (rowFactor_pos positive)⟩

def positionPair {d : Nat} (p : Position d) : Fin (cloudSize d) × Fin (degree d) :=
  (rowIndex (cloudSize d) (degree d)).symm p

def caller {ρ : Type} {d : Nat} (s : State ρ d) : ρ := s.1.1.1.1.1

def boundaryState {ρ : Type} {d : Nat} (positive : 0 < d)
    (H : Table (cloudSize d) d) (ambient : ρ) (p : Position d) : State ρ d :=
  (MachineExpanderRow.divisionState positive ambient
    (start H (positionPair p).1 (positionPair p).2) none, p)

def initialState {ρ : Type} {d : Nat} (positive : 0 < d)
    (H : Table (cloudSize d) d) (ambient : ρ) : State ρ d :=
  boundaryState positive H ambient (zeroPosition positive)

def prepareState {ρ : Type} {d : Nat} (positive : 0 < d)
    (H : Table (cloudSize d) d) (s : State ρ d) : State ρ d :=
  boundaryState positive H (caller s) s.2

def resetState {ρ : Type} {d : Nat} (positive : 0 < d)
    (H : Table (cloudSize d) d) (s : State ρ d) : State ρ d :=
  initialState positive H (caller s)

def clearRegister {ρ : Type} {d : Nat} (s : State ρ d) : State ρ d :=
  ((s.1.1, none), s.2)

def advancePositionState {ρ : Type} {d : Nat} (positive : 0 < d)
    (s : State ρ d) : State ρ d := (s.1, nextPosition positive s.2)

def resetPositionState {ρ : Type} {d : Nat} (positive : 0 < d)
    (s : State ρ d) : State ρ d := (s.1, zeroPosition positive)

def vertexGuardCore {ρ : Type} {d : Nat} :
    TM2.Stmt Alphabet (Label d) ((RowStatePrefix ρ d × Position d) × Option Bool) :=
  .peek (.inr .vertexCount) (fun s head => (s.1, head))
    (.branch (fun s => s.2.getD false)
      (.pop (.inr .vertexCount) (fun s _ => (s.1, none))
        (.goto fun _ => .inr .prepareRow))
      (.load (fun s => (s.1, none)) (.goto fun _ => .inr .reverseOutput)))

variable {ρ : Type} [Fintype ρ]

def outerStatement {d : Nat} (positive : 0 < d) (H : Table (cloudSize d) d) :
    OuterLabel → TM2.Stmt Alphabet (Label d) (State ρ d)
  | .initialize =>
    .push (.inl .inputVertex) (fun _ => false)
      (.load (resetState positive H) (.goto fun _ => .inr .vertexGuard))
  | .vertexGuard => MachineControl.statement id (guardStates ρ d) vertexGuardCore
  | .prepareRow =>
    .load (prepareState positive H) (.goto fun _ => .inl .initialize)
  | .afterRow =>
    .branch (fun s => decide (s.2.val + 1 < rowFactor d))
      (.load (advancePositionState positive) (.goto fun _ => .inr .prepareRow))
      (.push (.inl .inputVertex) (fun _ => true)
        (.load (resetPositionState positive) (.goto fun _ => .inr .vertexGuard)))
  | .reverseOutput =>
    MachineControl.statement id (guardStates ρ d)
      (Reduction.MachineTransfer.loopAt (Γ := Alphabet) (Λ := Label d)
        (σ := RowStatePrefix ρ d × Position d) (.inl .output) (.inr .result) id false
        (.inr .reverseOutput) (some (.inr .done)))
  | .done => .halt

def rowReturn (d : Nat) : Option (Label d) := some (.inr .afterRow)

def program {d : Nat} (positive : 0 < d) (H : Table (cloudSize d) d) :
    Label d → TM2.Stmt Alphabet (Label d) (State ρ d) :=
  MachineEmbedding.program (rowReturn d) (MachineExpanderRow.program positive H)
    (outerStatement positive H)

@[simp] theorem program_row {d : Nat} (positive : 0 < d)
    (H : Table (cloudSize d) d) (label : MachineExpanderRow.Label d) :
    program (ρ := ρ) positive H (.inl label) =
      MachineEmbedding.statement (rowReturn d)
        (MachineExpanderRow.program positive H label) := rfl

@[simp] theorem program_outer {d : Nat} (positive : 0 < d)
    (H : Table (cloudSize d) d) (label : OuterLabel) :
    program (ρ := ρ) positive H (.inr label) = outerStatement positive H label := rfl

def rowExecution {d : Nat} (positive : 0 < d) (H : Table (cloudSize d) d)
    (position : Position d) (extra : ExtraTape → List Bool)
    {a b : TM2.Cfg (fun _ : MachineExpanderRow.Tape => Bool)
      (MachineExpanderRow.Label d) (MachineExpanderRow.State ρ d)} {budget : Nat}
    (execution : StateTransition.EvalsToInTime
      (TM2.step (MachineExpanderRow.program positive H)) a (some b) budget) :
    StateTransition.EvalsToInTime (TM2.step (program positive H))
      (MachineEmbedding.configuration (rowReturn d) position extra a)
      (some (MachineEmbedding.configuration (rowReturn d) position extra b)) budget :=
  MachineComposition.embeddedExecution (rowReturn d) position extra
    (MachineExpanderRow.program positive H) (outerStatement positive H) execution

@[simp] theorem rowExecution_steps {d : Nat} (positive : 0 < d)
    (H : Table (cloudSize d) d) (position : Position d) (extra : ExtraTape → List Bool)
    {a b : TM2.Cfg (fun _ : MachineExpanderRow.Tape => Bool)
      (MachineExpanderRow.Label d) (MachineExpanderRow.State ρ d)} {budget : Nat}
    (execution : StateTransition.EvalsToInTime
      (TM2.step (MachineExpanderRow.program positive H)) a (some b) budget) :
    (rowExecution positive H position extra execution).steps = execution.steps := rfl

def rowTapes (vertex : Nat) (oldTable output : List Bool) : MachineExpanderRow.Tape → List Bool
  | .inputVertex => encodeWord vertex
  | .table => oldTable
  | .output => output
  | _ => []

def extraTapes (remaining : Nat) (countSuffix result : List Bool) : ExtraTape → List Bool
  | .vertexCount => encodeWord remaining ++ countSuffix
  | .result => result

def boundaryTapes (vertex remaining : Nat) (oldTable output countSuffix result : List Bool) :
    (tape : Tape) → List (Alphabet tape) :=
  MachineEmbedding.tapes (rowTapes vertex oldTable output) (extraTapes remaining countSuffix result)

def initialTapes (vertices : Nat) (oldTable countSuffix : List Bool) :
    (tape : Tape) → List (Alphabet tape) :=
  MachineEmbedding.tapes
    (fun tape => match tape with | .table => oldTable | _ => [])
    (extraTapes vertices countSuffix [])

def finalTapes (vertices : Nat) (oldTable output countSuffix : List Bool) :
    (tape : Tape) → List (Alphabet tape) :=
  boundaryTapes vertices 0 oldTable [] countSuffix output

end DFVSGames.Foundations.Complexity.MachineExpanderTable
end

section

namespace DFVSGames.Foundations.Complexity.MachineAlphabetTransport

open Turing.TM2

variable {K Λ σ : Type} {Γ Δ : K → Type}

def tapes (h : Γ = Δ) (source : ∀ k, List (Γ k)) : ∀ k, List (Δ k) := h ▸ source

def statement (h : Γ = Δ) (source : Stmt Γ Λ σ) : Stmt Δ Λ σ := h ▸ source

def configuration (h : Γ = Δ) (source : Cfg Γ Λ σ) : Cfg Δ Λ σ := h ▸ source

def program (h : Γ = Δ) (source : Λ → Stmt Γ Λ σ) : Λ → Stmt Δ Λ σ :=
  fun label => statement h (source label)

abbrev castStatement (h : Γ = Δ) (source : Stmt Γ Λ σ) : Stmt Δ Λ σ := statement h source
abbrev castConfiguration (h : Γ = Δ) (source : Cfg Γ Λ σ) : Cfg Δ Λ σ := configuration h source
abbrev castProgram (h : Γ = Δ) (source : Λ → Stmt Γ Λ σ) : Λ → Stmt Δ Λ σ := program h source

@[simp] theorem tapes_rfl (source : ∀ k, List (Γ k)) : tapes rfl source = source := rfl
@[simp] theorem statement_rfl (source : Stmt Γ Λ σ) : statement rfl source = source := rfl
@[simp] theorem configuration_rfl (source : Cfg Γ Λ σ) : configuration rfl source = source := rfl
@[simp] theorem program_rfl (source : Λ → Stmt Γ Λ σ) : program rfl source = source := rfl

@[simp] theorem program_apply (h : Γ = Δ) (source : Λ → Stmt Γ Λ σ) (label : Λ) :
    program h source label = statement h (source label) := rfl

@[simp] theorem statement_roundtrip (h : Γ = Δ) (source : Stmt Δ Λ σ) :
    statement h (statement h.symm source) = source := by cases h; rfl

@[simp] theorem statement_symm_roundtrip (h : Γ = Δ) (source : Stmt Γ Λ σ) :
    statement h.symm (statement h source) = source := by cases h; rfl

@[simp] theorem configuration_roundtrip (h : Γ = Δ) (source : Cfg Δ Λ σ) :
    configuration h (configuration h.symm source) = source := by cases h; rfl

@[simp] theorem configuration_symm_roundtrip (h : Γ = Δ) (source : Cfg Γ Λ σ) :
    configuration h.symm (configuration h source) = source := by cases h; rfl

@[simp] theorem program_roundtrip (h : Γ = Δ) (source : Λ → Stmt Δ Λ σ) :
    program h (program h.symm source) = source := by cases h; rfl

@[simp] theorem program_symm_roundtrip (h : Γ = Δ) (source : Λ → Stmt Γ Λ σ) :
    program h.symm (program h source) = source := by cases h; rfl

@[simp] theorem tapes_roundtrip (h : Γ = Δ) (source : ∀ k, List (Δ k)) :
    tapes h (tapes h.symm source) = source := by cases h; rfl

@[simp] theorem tapes_symm_roundtrip (h : Γ = Δ) (source : ∀ k, List (Γ k)) :
    tapes h.symm (tapes h source) = source := by cases h; rfl

@[simp] theorem configuration_label (h : Γ = Δ) (source : Cfg Γ Λ σ) :
    (configuration h source).l = source.l := by cases h; rfl

@[simp] theorem configuration_state (h : Γ = Δ) (source : Cfg Γ Λ σ) :
    (configuration h source).var = source.var := by cases h; rfl

@[simp] theorem configuration_tapes (h : Γ = Δ) (source : Cfg Γ Λ σ) :
    (configuration h source).stk = tapes h source.stk := by cases h; rfl

@[simp] theorem configuration_mk (h : Γ = Δ) (label : Option Λ) (state : σ)
    (source : ∀ k, List (Γ k)) :
    configuration h ⟨label, state, source⟩ = ⟨label, state, tapes h source⟩ := by cases h; rfl

@[simp] theorem map_configuration_rfl (source : Option (Cfg Γ Λ σ)) :
    source.map (configuration (rfl : Γ = Γ)) = source := by cases source <;> rfl

variable [DecidableEq K]

theorem stepAux_simulation (h : Γ = Δ) (source : Stmt Γ Λ σ) (state : σ)
    (sourceTapes : ∀ k, List (Γ k)) :
    stepAux (statement h source) state (tapes h sourceTapes) =
      configuration h (stepAux source state sourceTapes) := by cases h; rfl

theorem step_simulation (h : Γ = Δ) (source : Λ → Stmt Γ Λ σ) (start : Cfg Γ Λ σ) :
    step (program h source) (configuration h start) =
      (step source start).map (configuration h) := by
  cases h
  simp only [program_rfl, configuration_rfl, map_configuration_rfl]

theorem trace_transport (h : Γ = Δ) (source : Λ → Stmt Γ Λ σ)
    (n : Nat) (start : Option (Cfg Γ Λ σ)) :
    (MachineComposition.advance (step (program h source)))^[n]
      (start.map (configuration h)) =
      ((MachineComposition.advance (step source))^[n] start).map (configuration h) := by
  cases h
  simp only [program_rfl, map_configuration_rfl]

theorem successfulTrace (h : Γ = Δ) (source : Λ → Stmt Γ Λ σ)
    (n : Nat) (start finish : Cfg Γ Λ σ)
    (trace : (MachineComposition.advance (step source))^[n] (some start) = some finish) :
    (MachineComposition.advance (step (program h source)))^[n]
      (some (configuration h start)) = some (configuration h finish) := by
  have transported := trace_transport h source n (some start)
  simpa only [Option.map_some, trace] using transported

def executionInTime (h : Γ = Δ) (source : Λ → Stmt Γ Λ σ)
    {start : Cfg Γ Λ σ} {finish : Option (Cfg Γ Λ σ)} {budget : Nat}
    (execution : StateTransition.EvalsToInTime (step source) start finish budget) :
    StateTransition.EvalsToInTime (step (program h source)) (configuration h start)
      (finish.map (configuration h)) budget where
  steps := execution.steps
  evals_in_steps := by
    cases h
    simpa only [program_rfl, configuration_rfl, map_configuration_rfl]
      using execution.evals_in_steps
  steps_le_m := execution.steps_le_m

@[simp] theorem executionInTime_steps (h : Γ = Δ) (source : Λ → Stmt Γ Λ σ)
    {start : Cfg Γ Λ σ} {finish : Option (Cfg Γ Λ σ)} {budget : Nat}
    (execution : StateTransition.EvalsToInTime (step source) start finish budget) :
    (executionInTime h source execution).steps = execution.steps := rfl

def successfulExecutionInTime (h : Γ = Δ) (source : Λ → Stmt Γ Λ σ)
    {start finish : Cfg Γ Λ σ} {budget : Nat}
    (execution : StateTransition.EvalsToInTime (step source) start (some finish) budget) :
    StateTransition.EvalsToInTime (step (program h source)) (configuration h start)
      (some (configuration h finish)) budget := executionInTime h source execution

@[simp] theorem successfulExecutionInTime_steps (h : Γ = Δ) (source : Λ → Stmt Γ Λ σ)
    {start finish : Cfg Γ Λ σ} {budget : Nat}
    (execution : StateTransition.EvalsToInTime (step source) start (some finish) budget) :
    (successfulExecutionInTime h source execution).steps = execution.steps := rfl

end DFVSGames.Foundations.Complexity.MachineAlphabetTransport

end

section

namespace DFVSGames.Foundations.Complexity.MachineExpanderFamily

open Turing
open PCP.ExpanderTables PCP.ExpanderRowControl PCP.ExpanderTableWords

inductive ExtraTape
  | remainingLevel | currentSize | unaryScratch | tableReverse
  deriving DecidableEq

instance : Fintype ExtraTape := derive_fintype% ExtraTape

abbrev Tape := MachineExpanderTable.Tape ⊕ ExtraTape
abbrev Alphabet : Tape → Type :=
  MachineEmbedding.Alphabet MachineExpanderTable.Alphabet (fun _ : ExtraTape => Bool)
abbrev BoolAlphabet (_ : Tape) := Bool

theorem alphabet_eq : Alphabet = BoolAlphabet := by
  funext tape
  cases tape with
  | inl tape => cases tape <;> rfl
  | inr tape => rfl

instance alphabetFintype (tape : Tape) : Fintype (Alphabet tape) := by
  rw [alphabet_eq]
  exact inferInstanceAs (Fintype Bool)

inductive AffinePhase
  | copyCount | multiplySize
  deriving DecidableEq

instance : Fintype AffinePhase := derive_fintype% AffinePhase

inductive AffineLabel
  | seed | scan | restore
  deriving DecidableEq

instance : Fintype AffineLabel := derive_fintype% AffineLabel

inductive OuterLabel
  | initialize
  | levelGuard
  | affine (phase : AffinePhase) (label : AffineLabel)
  | clearOldTable | reverseResult | reverseTable | clearCurrentSize
  | drainInputVertex | drainVertexCount | done
  deriving DecidableEq, Fintype

abbrev Label (d : Nat) := MachineExpanderTable.Label d ⊕ OuterLabel
abbrev State (ρ : Type) (d : Nat) := MachineExpanderTable.State ρ d × Unit
abbrev RegisterAmbient (ρ : Type) (d : Nat) :=
  (MachineExpanderTable.RowStatePrefix ρ d × MachineExpanderTable.Position d) × Unit

def registerStates (ρ : Type) (d : Nat) :
    (RegisterAmbient ρ d × Option Bool) ≃ State ρ d where
  toFun s := (((s.1.1.1, s.2), s.1.1.2), s.1.2)
  invFun s := (((s.1.1.1, s.1.2), s.2), s.1.1.2)
  left_inv := by rintro ⟨⟨⟨a, p⟩, u⟩, b⟩; rfl
  right_inv := by rintro ⟨⟨⟨a, b⟩, p⟩, u⟩; rfl

@[simp] theorem registerStates_apply (ρ : Type) (d : Nat)
    (s : RegisterAmbient ρ d × Option Bool) :
    registerStates ρ d s = (((s.1.1.1, s.2), s.1.1.2), s.1.2) := rfl

@[simp] theorem registerStates_symm_apply (ρ : Type) (d : Nat) (s : State ρ d) :
    (registerStates ρ d).symm s = (((s.1.1.1, s.1.2), s.2), s.1.1.2) := rfl

def caller {ρ : Type} {d : Nat} (s : State ρ d) : ρ :=
  MachineExpanderTable.caller s.1

def initialState {ρ : Type} {d : Nat} (positive : 0 < d)
    (H : Table (cloudSize d) d) (ambient : ρ) : State ρ d :=
  (MachineExpanderTable.initialState positive H ambient, ())

def normalizeState {ρ : Type} {d : Nat} (positive : 0 < d)
    (H : Table (cloudSize d) d) (s : State ρ d) : State ρ d :=
  initialState positive H (caller s)

def initialEncoding (d : Nat) : List Bool := encodeWords (rotationWords (initial d))

theorem initialEncoding_eq_family {d : Nat} (H : Table (cloudSize d) d) :
    initialEncoding d = encodeWords (rotationWords (family H 0)) := rfl

def tableTape : Tape := .inl (.inl .table)
def inputVertexTape : Tape := .inl (.inl .inputVertex)
def vertexCountTape : Tape := .inl (.inr .vertexCount)
def resultTape : Tape := .inl (.inr .result)

def affineSource : AffinePhase → Tape
  | .copyCount => .inr .currentSize
  | .multiplySize => inputVertexTape

def affineDestination : AffinePhase → Tape
  | .copyCount => vertexCountTape
  | .multiplySize => .inr .currentSize

def affineCoefficient (d : Nat) : AffinePhase → Nat
  | .multiplySize => cloudSize d
  | .copyCount => 1

def affineExit (d : Nat) : AffinePhase → Label d
  | .copyCount => .inl (.inr .initialize)
  | .multiplySize => .inr .drainInputVertex

def affineStatement {ρ : Type} {d : Nat} (phase : AffinePhase) :
    AffineLabel → TM2.Stmt BoolAlphabet (Label d) (State ρ d)
  | .seed =>
    MachineControl.statement id (registerStates ρ d)
      (MachineUnaryAffineAt.seed (Λ := Label d) (σ := RegisterAmbient ρ d)
        (affineDestination phase) 0 (.inr (.affine phase .scan)))
  | .scan =>
    MachineControl.statement id (registerStates ρ d)
      (MachineUnaryAffineAt.scan (Λ := Label d) (σ := RegisterAmbient ρ d)
        (affineSource phase) (.inr .unaryScratch)
        (affineDestination phase) (affineCoefficient d phase)
        (.inr (.affine phase .scan)) (.inr (.affine phase .restore)))
  | .restore =>
    MachineControl.statement id (registerStates ρ d)
      (Reduction.MachineTransfer.loopAt (Γ := BoolAlphabet) (Λ := Label d)
        (σ := RegisterAmbient ρ d) (.inr .unaryScratch) (affineSource phase) id false
        (.inr (.affine phase .restore)) (some (affineExit d phase)))

def drainStatement {ρ : Type} {d : Nat} (tape : Tape) (again next : OuterLabel) :
    TM2.Stmt BoolAlphabet (Label d) (State ρ d) :=
  MachineControl.statement id (registerStates ρ d)
    (MachineDrain.drain (Λ := Label d) (σ := RegisterAmbient ρ d)
      tape (.inr again) (some (.inr next)))

variable {ρ : Type} [Fintype ρ]

def boolOuterStatement {d : Nat} (positive : 0 < d) (H : Table (cloudSize d) d) :
    OuterLabel → TM2.Stmt BoolAlphabet (Label d) (State ρ d)
  | .initialize =>
    Reduction.MachineSubstitution.pushWord tableTape (initialEncoding d).reverse
      (Reduction.MachineSubstitution.pushWord (.inr .currentSize) (encodeWord 1).reverse
        (.load (normalizeState positive H)
          (.goto fun _ => .inr .levelGuard)))
  | .levelGuard =>
    MachineControl.statement id (registerStates ρ d)
      (MachineUnaryCounter.guard (K := Tape) (Λ := Label d) (σ := RegisterAmbient ρ d)
        (.inr .remainingLevel) (.inr (.affine .copyCount .seed)) (.inr .done))
  | .affine phase label => affineStatement phase label
  | .clearOldTable => drainStatement tableTape .clearOldTable .reverseResult
  | .reverseResult =>
    MachineControl.statement id (registerStates ρ d)
      (Reduction.MachineTransfer.loopAt (Γ := BoolAlphabet) (Λ := Label d)
        (σ := RegisterAmbient ρ d) resultTape (.inr .tableReverse) id false
        (.inr .reverseResult) (some (.inr .reverseTable)))
  | .reverseTable =>
    MachineControl.statement id (registerStates ρ d)
      (Reduction.MachineTransfer.loopAt (Γ := BoolAlphabet) (Λ := Label d)
        (σ := RegisterAmbient ρ d) (.inr .tableReverse) tableTape id false
        (.inr .reverseTable) (some (.inr .clearCurrentSize)))
  | .clearCurrentSize =>
    drainStatement (.inr .currentSize) .clearCurrentSize (.affine .multiplySize .seed)
  | .drainInputVertex => drainStatement inputVertexTape .drainInputVertex .drainVertexCount
  | .drainVertexCount => drainStatement vertexCountTape .drainVertexCount .levelGuard
  | .done => .load (normalizeState positive H) .halt

def outerStatement {d : Nat} (positive : 0 < d) (H : Table (cloudSize d) d)
    (label : OuterLabel) : TM2.Stmt Alphabet (Label d) (State ρ d) :=
  MachineAlphabetTransport.statement alphabet_eq.symm (boolOuterStatement positive H label)

def tableReturn (d : Nat) : Option (Label d) := some (.inr .clearOldTable)

def program {d : Nat} (positive : 0 < d) (H : Table (cloudSize d) d)
    (_growth : 1 < cloudSize d) : Label d → TM2.Stmt Alphabet (Label d) (State ρ d) :=
  MachineEmbedding.program (tableReturn d) (MachineExpanderTable.program positive H)
    (outerStatement positive H)

@[simp] theorem program_table {d : Nat} (positive : 0 < d)
    (H : Table (cloudSize d) d) (growth : 1 < cloudSize d)
    (label : MachineExpanderTable.Label d) :
    program (ρ := ρ) positive H growth (.inl label) =
      MachineEmbedding.statement (tableReturn d)
        (MachineExpanderTable.program positive H label) := rfl

@[simp] theorem program_outer {d : Nat} (positive : 0 < d)
    (H : Table (cloudSize d) d) (growth : 1 < cloudSize d) (label : OuterLabel) :
    program (ρ := ρ) positive H growth (.inr label) = outerStatement positive H label := rfl

def boolView {d : Nat} (positive : 0 < d) (H : Table (cloudSize d) d)
    (growth : 1 < cloudSize d) : Label d → TM2.Stmt BoolAlphabet (Label d) (State ρ d) :=
  MachineAlphabetTransport.program alphabet_eq (program positive H growth)

@[simp] theorem boolView_outer {d : Nat} (positive : 0 < d)
    (H : Table (cloudSize d) d) (growth : 1 < cloudSize d) (label : OuterLabel) :
    boolView (ρ := ρ) positive H growth (.inr label) =
      boolOuterStatement positive H label := by
  change MachineAlphabetTransport.statement alphabet_eq
    (MachineAlphabetTransport.statement alphabet_eq.symm
      (boolOuterStatement positive H label)) = _
  exact MachineAlphabetTransport.statement_roundtrip alphabet_eq _

def tableExecution {d : Nat} (positive : 0 < d) (H : Table (cloudSize d) d)
    (growth : 1 < cloudSize d) (extra : ExtraTape → List Bool)
    {a b : TM2.Cfg MachineExpanderTable.Alphabet (MachineExpanderTable.Label d)
      (MachineExpanderTable.State ρ d)} {budget : Nat}
    (execution : StateTransition.EvalsToInTime
      (TM2.step (MachineExpanderTable.program positive H)) a (some b) budget) :
    StateTransition.EvalsToInTime (TM2.step (program positive H growth))
      (MachineEmbedding.configuration (tableReturn d) () extra a)
      (some (MachineEmbedding.configuration (tableReturn d) () extra b)) budget :=
  MachineComposition.embeddedExecution (tableReturn d) () extra
    (MachineExpanderTable.program positive H) (outerStatement positive H) execution

@[simp] theorem tableExecution_steps {d : Nat} (positive : 0 < d)
    (H : Table (cloudSize d) d) (growth : 1 < cloudSize d)
    (extra : ExtraTape → List Bool)
    {a b : TM2.Cfg MachineExpanderTable.Alphabet (MachineExpanderTable.Label d)
      (MachineExpanderTable.State ρ d)} {budget : Nat}
    (execution : StateTransition.EvalsToInTime
      (TM2.step (MachineExpanderTable.program positive H)) a (some b) budget) :
    (tableExecution positive H growth extra execution).steps = execution.steps := rfl

def tableFrame (word : List Bool) : (tape : MachineExpanderTable.Tape) →
    List (MachineExpanderTable.Alphabet tape)
  | .inl .table => word
  | .inl _ => []
  | .inr _ => []

def extraFrame (remaining current : Nat) (levelSuffix : List Bool) : ExtraTape → List Bool
  | .remainingLevel => encodeWord remaining ++ levelSuffix
  | .currentSize => encodeWord current
  | _ => []

def boundaryTapes (remaining current : Nat) (word levelSuffix : List Bool) :
    (tape : Tape) → List (Alphabet tape) :=
  MachineEmbedding.tapes (tableFrame word) (extraFrame remaining current levelSuffix)

def initialTapes (level : Nat) (levelSuffix : List Bool) :
    (tape : Tape) → List (Alphabet tape) :=
  MachineEmbedding.tapes (tableFrame [])
    (fun tape => match tape with
      | .remainingLevel => encodeWord level ++ levelSuffix
      | _ => [])

def familyTapes {d : Nat} (H : Table (cloudSize d) d) (level : Nat)
    (levelSuffix : List Bool) : (tape : Tape) → List (Alphabet tape) :=
  boundaryTapes 0 (vertexCount (degree d) level)
    (encodeWords (rotationWords (family H level))) levelSuffix

end DFVSGames.Foundations.Complexity.MachineExpanderFamily
end

end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end

end OAI