lean/OAI/Computability/DirectedFeedback/Machines/MachinePaddedOverlay.lean

import Mathlib
import OAI.Computability.DirectedFeedback.Machines.MachineOverlayTable

namespace OAI

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namespace DFVSGames.Foundations.Complexity.MachinePaddedOverlay

open Turing MachineComposition MachineCloudPadding
open PCP

variable {d : Nat}

abbrev BaseTable := PreprocessingStageMaps.BaseTable
abbrev baseDegree := Expanders.baseDegree
abbrev overlayDegree := PreprocessingRegularTables.internalDegree
abbrev Tape := MachineVertexPadding.Tape ⊕ (MachineExpanderFamily.Tape ⊕ MachineOverlayTable.Tape)
abbrev Alphabet (_ : Tape) := Bool
abbrev Label (d : Nat) := MachineVertexPadding.Label d ⊕ (MachineExpanderFamily.Label baseDegree ⊕ MachineOverlayTable.Label d overlayDegree)
abbrev State (d : Nat) := MachineVertexPadding.State (MachineExpanderFamily.State Unit baseDegree × MachineOverlayTable.State Unit d overlayDegree)

theorem overlayDegree_positive : 0 < overlayDegree :=
  Nat.mul_pos MachineExpanderFamily.baseDegree_positive MachineExpanderFamily.baseDegree_positive

def paddingTape (k : MachineVertexPadding.Tape) : Tape := .inl k

def paddingView : Tape → Option MachineVertexPadding.Tape
  | .inl k => some k
  | .inr _ => none

@[simp] theorem paddingView_left (k : MachineVertexPadding.Tape) : paddingView (paddingTape k) = some k := rfl

theorem paddingView_right (j : Tape) (k : MachineVertexPadding.Tape)
    (h : paddingView j = some k) : paddingTape k = j := by
  cases j with
  | inl j => cases Option.some.inj h; rfl
  | inr j => simp [paddingView] at h

def familyTape (k : MachineExpanderFamily.Tape) : Tape :=
  if k = .inr .remainingLevel then .inl .level else .inr (.inl k)

def familyView : Tape → Option MachineExpanderFamily.Tape
  | .inl .level => some (.inr .remainingLevel)
  | .inr (.inl k) => if k = .inr .remainingLevel then none else some k
  | _ => none

@[simp] theorem familyView_left (k : MachineExpanderFamily.Tape) : familyView (familyTape k) = some k := by
  by_cases h : k = .inr .remainingLevel
  · subst k; rfl
  · simp [familyTape, familyView, h]

theorem familyView_right (j : Tape) (k : MachineExpanderFamily.Tape)
    (h : familyView j = some k) : familyTape k = j := by
  rcases j with j | j
  · cases j <;> simp_all [familyView, familyTape]
  · rcases j with j | j
    · by_cases hj : j = .inr .remainingLevel <;> simp_all [familyView, familyTape]
    · simp [familyView] at h

def overlayTape (k : MachineOverlayTable.Tape) : Tape :=
  if k = MachineOverlayTable.graphArchive then .inl .output
  else if k = MachineOverlayTable.expanderArchive then .inr (.inl MachineExpanderFamily.tableTape)
  else .inr (.inr k)

def overlayView : Tape → Option MachineOverlayTable.Tape
  | .inl .output => some MachineOverlayTable.graphArchive
  | .inr (.inl k) => if k = MachineExpanderFamily.tableTape then some MachineOverlayTable.expanderArchive else none
  | .inr (.inr k) =>
      if k = MachineOverlayTable.graphArchive ∨ k = MachineOverlayTable.expanderArchive then none else some k
  | _ => none

@[simp] theorem overlayView_left (k : MachineOverlayTable.Tape) : overlayView (overlayTape k) = some k := by
  by_cases hg : k = MachineOverlayTable.graphArchive
  · subst k; rfl
  · by_cases he : k = MachineOverlayTable.expanderArchive
    · subst k; rfl
    · simp [overlayTape, overlayView, hg, he]

theorem overlayView_right (j : Tape) (k : MachineOverlayTable.Tape)
    (h : overlayView j = some k) : overlayTape k = j := by
  rcases j with j | j
  · cases j <;> simp_all [overlayView, overlayTape, MachineOverlayTable.graphArchive]
  · rcases j with j | j
    · by_cases hj : j = MachineExpanderFamily.tableTape
      · subst j
        have hk : MachineOverlayTable.expanderArchive = k := by
          simpa [overlayView] using h
        subst k
        rfl
      · simp [overlayView, hj] at h
    · by_cases hg : j = MachineOverlayTable.graphArchive
      · simp [overlayView, hg] at h
      · by_cases he : j = MachineOverlayTable.expanderArchive
        · simp [overlayView, he] at h
        · simp_all [overlayView, overlayTape]

def paddingLabel (l : MachineVertexPadding.Label d) : Label d := .inl l
def familyLabel (l : MachineExpanderFamily.Label baseDegree) : Label d := .inr (.inl l)
def overlayLabel (l : MachineOverlayTable.Label d overlayDegree) : Label d := .inr (.inr l)
def main : Label d := paddingLabel (.splitCode .copyFirst)
def familyEntry : Label d := familyLabel (.inr .initialize)
def overlayEntry : Label d := overlayLabel (.inl .copyFirst)

def familyStates (d : Nat) :
    (MachineExpanderFamily.State Unit baseDegree × ((MachineOverlayTable.State Unit d overlayDegree × Bool) × Option Bool)) ≃
      State d where
  toFun s := (((s.1, s.2.1.1), s.2.1.2), s.2.2)
  invFun s := (s.1.1.1, ((s.1.1.2, s.1.2), s.2))
  left_inv := by rintro ⟨f, ⟨⟨o, b⟩, r⟩⟩; rfl
  right_inv := by rintro ⟨⟨⟨f, o⟩, b⟩, r⟩; rfl

def overlayStates (d : Nat) :
    (MachineOverlayTable.State Unit d overlayDegree × ((MachineExpanderFamily.State Unit baseDegree × Bool) × Option Bool)) ≃
      State d where
  toFun s := (((s.2.1.1, s.1), s.2.1.2), s.2.2)
  invFun s := (s.1.1.2, ((s.1.1.1, s.1.2), s.2))
  left_inv := by rintro ⟨o, ⟨⟨f, b⟩, r⟩⟩; rfl
  right_inv := by rintro ⟨⟨⟨f, o⟩, b⟩, r⟩; rfl

def familySource (H : BaseTable) (d : Nat) : MachineExpanderFamily.Label baseDegree →
    TM2.Stmt MachineExpanderFamily.BoolAlphabet (MachineExpanderFamily.Label baseDegree) (State d) :=
  MachineStateEquiv.program (familyStates d) (MachineStateFrame.frameProgram
    (MachineExpanderFamily.boolView MachineExpanderFamily.baseDegree_positive H MachineExpanderFamily.baseDegree_cloud_gt_one))

def overlaySource (d : Nat) (hd : 0 < d) : MachineOverlayTable.Label d overlayDegree →
    TM2.Stmt (fun _ : MachineOverlayTable.Tape => Bool) (MachineOverlayTable.Label d overlayDegree) (State d) :=
  MachineStateEquiv.program (overlayStates d) (MachineStateFrame.frameProgram
    (MachineOverlayTable.program d overlayDegree hd overlayDegree_positive))

def program (H : BaseTable) (d : Nat) (hd : 0 < d) :
    Label d → TM2.Stmt Alphabet (Label d) (State d)
  | .inl l => Placement.statement paddingTape paddingLabel (some familyEntry) (MachineVertexPadding.program d hd l)
  | .inr (.inl l) => Placement.statement familyTape familyLabel (some overlayEntry)
      (familySource H d l)
  | .inr (.inr l) => Placement.statement overlayTape overlayLabel none (overlaySource d hd l)

def clean (H : BaseTable) (d : Nat) (hd : 0 < d) : State d :=
  (((MachineExpanderFamily.initialState MachineExpanderFamily.baseDegree_positive H (),
    MachineOverlayTable.clean d overlayDegree hd overlayDegree_positive ()), false), none)

def memory (input padded level rotor count output : List Bool) : Tape → List Bool
  | .inl k => MachineVertexPadding.memory input padded [] [] [] level [] [] k
  | .inr (.inl (.inl (.inl .table))) => rotor
  | .inr (.inl (.inr .currentSize)) => count
  | .inr (.inl _) => []
  | .inr (.inr k) => if k = MachineOverlayTable.output then output else []

def padded {n d : Nat} (t : PortTables.Table n d) :=
  PreprocessingPaddingTables.pad t (PreprocessingLevels.le_paddedSize n)

def rotor (H : BaseTable) (n : Nat) : List Bool :=
  encodeWords (ExpanderTableWords.rotationWords
    (ExpanderTables.family H (PreprocessingLevels.boundedLevel n)))

def initialTapes {n d : Nat} (t : PortTables.Table n d) : Tape → List Bool :=
  memory (PortTables.tableBits t) [] [] [] [] []

def paddedTapes {n d : Nat} (t : PortTables.Table n d) : Tape → List Bool :=
  memory (PortTables.tableBits t) (PortTables.tableBits (padded t))
    (encodeWord (PreprocessingLevels.boundedLevel n)) [] [] []

def familyTapes (H : BaseTable) {n d : Nat} (t : PortTables.Table n d) : Tape → List Bool :=
  memory (PortTables.tableBits t) (PortTables.tableBits (padded t))
    (encodeWord 0) (rotor H n) (encodeWord (PreprocessingLevels.paddedSize n)) []

def finalTapes (H : BaseTable) {n d : Nat} (t : PortTables.Table n d) : Tape → List Bool :=
  memory (PortTables.tableBits t) (PortTables.tableBits (padded t))
    (encodeWord 0) (rotor H n) (encodeWord (PreprocessingLevels.paddedSize n))
    (PortTables.tableBits (PreprocessingStageMaps.paddedOverlay H d ⟨n, t⟩).2)

private def placeExecution_inline_MachinePaddedOverlay {K Λ S : Type} [DecidableEq K]
    (tape : K → Tape) (view : Tape → Option K)
    (left : ∀ k, view (tape k) = some k)
    (right : ∀ j k, view j = some k → tape k = j)
    (labels : Λ → Label d) (exit : Option (Label d)) (extra : Tape → List Bool)
    (source : Λ → TM2.Stmt (fun _ : K => Bool) Λ S)
    (target : Label d → TM2.Stmt Alphabet (Label d) S)
    (atLabels : ∀ l, target (labels l) = Placement.statement tape labels exit (source l))
    {a b : TM2.Cfg (fun _ : K => Bool) Λ S} {budget : Nat}
    (run : StateTransition.EvalsToInTime (TM2.step source) a (some b) budget) :
    StateTransition.EvalsToInTime (TM2.step target)
      (Placement.configuration view labels exit extra a)
      (some (Placement.configuration view labels exit extra b)) budget :=
  liftExecutionInTime _ _ _
    (Placement.step_simulation tape view left right labels exit extra source target atLabels) run

theorem padding_initial_memory (input : List Bool) :
    Placement.tapes paddingView (MachineVertexPadding.memory input [] [] [] [] [] [] []) (fun _ => []) =
      memory input [] [] [] [] [] := by
  funext k
  rcases k with k | k
  · rfl
  · rcases k with k | k
    · rcases k with k | k
      · rcases k with k | k <;> cases k <;> rfl
      · cases k <;> rfl
    · simp [Placement.tapes, paddingView, memory]

theorem padding_final_memory (input output level : List Bool) :
    Placement.tapes paddingView (MachineVertexPadding.memory input output [] [] [] level [] []) (fun _ => []) =
      memory input output level [] [] [] := by
  funext k
  rcases k with k | k
  · rfl
  · rcases k with k | k
    · rcases k with k | k
      · rcases k with k | k <;> cases k <;> rfl
      · cases k <;> rfl
    · simp [Placement.tapes, paddingView, memory]

noncomputable def paddingInTime (H : BaseTable) {n d : Nat} (hd : 0 < d) (t : PortTables.Table n d) :
    StateTransition.EvalsToInTime (TM2.step (program H d hd))
      ⟨some main, clean H d hd, initialTapes t⟩
      (some ⟨some familyEntry, clean H d hd, paddedTapes t⟩)
      ((MachineVertexPadding.timePolynomial d).eval (PortTables.tableBits t).length) := by
  let ambient := (MachineExpanderFamily.initialState MachineExpanderFamily.baseDegree_positive H (),
    MachineOverlayTable.clean d overlayDegree hd overlayDegree_positive ())
  have run := placeExecution_inline_MachinePaddedOverlay paddingTape paddingView paddingView_left paddingView_right
    paddingLabel (some familyEntry) (fun _ => []) (MachineVertexPadding.program d hd) (program H d hd)
    (fun _ => rfl) (MachineVertexPadding.vertexPaddingInTime hd t ambient none)
  simpa only [Placement.configuration, Placement.label, padding_initial_memory,
    padding_final_memory, initialTapes, paddedTapes, padded, main, clean, ambient] using run

theorem family_initial_memory (input output : List Bool) (level : Nat) :
    Placement.tapes familyView
      (MachineExpanderFamily.toBoolTapes (MachineExpanderFamily.initialTapes level []))
      (memory input output (encodeWord level) [] [] []) =
      memory input output (encodeWord level) [] [] [] := by
  funext k
  rcases k with k | k
  · cases k <;> simp [Placement.tapes, familyView, memory,
      MachineVertexPadding.memory, MachineExpanderFamily.toBoolTapes,
      MachineExpanderFamily.toBoolWord, MachineExpanderFamily.initialTapes,
      MachineEmbedding.tapes]
  · rcases k with k | k
    · rcases k with k | k
      · rcases k with k | k <;> cases k <;>
          simp [Placement.tapes, familyView, memory, MachineExpanderFamily.toBoolTapes,
            MachineExpanderFamily.toBoolWord, MachineExpanderFamily.initialTapes,
            MachineExpanderFamily.tableFrame, MachineEmbedding.tapes]
      · cases k <;>
          simp [Placement.tapes, familyView, memory, MachineExpanderFamily.toBoolTapes,
            MachineExpanderFamily.toBoolWord, MachineExpanderFamily.initialTapes,
            MachineEmbedding.tapes]
    · rfl

theorem family_final_memory (H : BaseTable) (input output : List Bool) (level : Nat) :
    Placement.tapes familyView
      (MachineExpanderFamily.toBoolTapes (MachineExpanderFamily.familyTapes H level []))
      (memory input output (encodeWord level) [] [] []) =
      memory input output (encodeWord 0)
        (encodeWords (ExpanderTableWords.rotationWords (ExpanderTables.family H level)))
        (encodeWord (ExpanderTables.vertexCount overlayDegree level)) [] := by
  funext k
  rcases k with k | k
  · cases k <;> simp [Placement.tapes, familyView, memory,
      MachineVertexPadding.memory, MachineExpanderFamily.toBoolTapes,
      MachineExpanderFamily.toBoolWord, MachineExpanderFamily.familyTapes,
      MachineExpanderFamily.boundaryTapes, MachineExpanderFamily.extraFrame,
      MachineEmbedding.tapes]
  · rcases k with k | k
    · rcases k with k | k
      · rcases k with k | k <;> cases k <;>
          simp [Placement.tapes, familyView, memory, MachineExpanderFamily.toBoolTapes,
            MachineExpanderFamily.toBoolWord, MachineExpanderFamily.familyTapes,
            MachineExpanderFamily.boundaryTapes, MachineExpanderFamily.tableFrame,
            MachineEmbedding.tapes]
      · cases k <;>
          simp [Placement.tapes, familyView, memory, MachineExpanderFamily.toBoolTapes,
            MachineExpanderFamily.toBoolWord, MachineExpanderFamily.familyTapes,
            MachineExpanderFamily.boundaryTapes, MachineExpanderFamily.extraFrame,
            MachineEmbedding.tapes, overlayDegree, PreprocessingRegularTables.internalDegree,
            ExpanderRowControl.degree]
    · simp [Placement.tapes, familyView, memory]

noncomputable def familyInTime (H : BaseTable) {n d : Nat} (hd : 0 < d)
    (t : PortTables.Table n d) :
    StateTransition.EvalsToInTime (TM2.step (program H d hd))
      ⟨some familyEntry, clean H d hd, paddedTapes t⟩
      (some ⟨some overlayEntry, clean H d hd, familyTapes H t⟩)
      (MachineExpanderFamilyBounds.inputCoefficient * (n + 1)^5) := by
  let f := MachineExpanderFamily.initialState MachineExpanderFamily.baseDegree_positive H ()
  let o := MachineOverlayTable.clean d overlayDegree hd overlayDegree_positive ()
  have caller : MachineExpanderFamily.caller f = () := Subsingleton.elim _ _
  have castRun := MachineAlphabetTransport.successfulExecutionInTime
    MachineExpanderFamily.alphabet_eq
    (MachineExpanderFamily.program MachineExpanderFamily.baseDegree_positive H
      MachineExpanderFamily.baseDegree_cloud_gt_one)
    (MachineExpanderFamily.paddedFamilyInTime H n f [])
  have boolRun : StateTransition.EvalsToInTime
      (TM2.step (MachineExpanderFamily.boolView MachineExpanderFamily.baseDegree_positive H
        MachineExpanderFamily.baseDegree_cloud_gt_one))
      ⟨some (.inr .initialize), f, MachineExpanderFamily.toBoolTapes
        (MachineExpanderFamily.initialTapes (PreprocessingLevels.boundedLevel n) [])⟩
      (some ⟨none, f, MachineExpanderFamily.toBoolTapes
        (MachineExpanderFamily.familyTapes H (PreprocessingLevels.boundedLevel n) [])⟩)
      (MachineExpanderFamilyBounds.inputCoefficient * (n + 1)^5) := by
    simpa only [MachineExpanderFamily.boolView, MachineExpanderFamily.configuration_toBool,
      caller] using castRun
  have framed := MachineStateFrame.frameExecution
    (MachineExpanderFamily.boolView MachineExpanderFamily.baseDegree_positive H
      MachineExpanderFamily.baseDegree_cloud_gt_one) ((o, false), (none : Option Bool)) boolRun
  have transported := MachineStateEquiv.execution (familyStates d)
    (MachineStateFrame.frameProgram
      (MachineExpanderFamily.boolView MachineExpanderFamily.baseDegree_positive H
        MachineExpanderFamily.baseDegree_cloud_gt_one)) framed
  have sourceRun : StateTransition.EvalsToInTime (TM2.step (familySource H d))
      ⟨some (.inr .initialize), clean H d hd, MachineExpanderFamily.toBoolTapes
        (MachineExpanderFamily.initialTapes (PreprocessingLevels.boundedLevel n) [])⟩
      (some ⟨none, clean H d hd, MachineExpanderFamily.toBoolTapes
        (MachineExpanderFamily.familyTapes H (PreprocessingLevels.boundedLevel n) [])⟩)
      (MachineExpanderFamilyBounds.inputCoefficient * (n + 1)^5) := by
    simpa only [familySource, MachineStateEquiv.configuration,
      MachineStateFrame.frameConfiguration, Option.map_some, familyStates,
      Equiv.coe_fn_mk, clean, f, o]
      using transported
  have run := placeExecution_inline_MachinePaddedOverlay familyTape familyView familyView_left familyView_right
    familyLabel (some overlayEntry) (paddedTapes t) (familySource H d) (program H d hd)
    (fun _ => rfl) sourceRun
  have finishRaw := family_final_memory H (PortTables.tableBits t)
    (PortTables.tableBits (padded t)) (PreprocessingLevels.boundedLevel n)
  have hsize : ExpanderTables.vertexCount overlayDegree (PreprocessingLevels.boundedLevel n) =
      PreprocessingLevels.paddedSize n := PreprocessingLevels.table_vertexCount_eq_paddedSize n
  have count : memory (PortTables.tableBits t) (PortTables.tableBits (padded t))
      (encodeWord 0) (rotor H n)
      (encodeWord (ExpanderTables.vertexCount overlayDegree (PreprocessingLevels.boundedLevel n))) [] =
      familyTapes H t :=
    congrArg (fun size => memory (PortTables.tableBits t) (PortTables.tableBits (padded t))
      (encodeWord 0) (rotor H n) (encodeWord size) []) hsize
  have finish := finishRaw.trans count
  simpa only [Placement.configuration, Placement.label, paddedTapes,
    family_initial_memory, finish, familyEntry] using run

theorem rotationWords_resize {n m q : Nat} (h : n = m) (H : ExpanderTables.Table n q) :
    ExpanderTableWords.rotationWords (PreprocessingRegularTables.resizeTable h H) =
      ExpanderTableWords.rotationWords H := by
  cases h
  rfl

theorem rotor_familyAt (H : BaseTable) (n : Nat) :
    MachineOverlayTable.rawExpander (PreprocessingStageMaps.familyAt H n) = rotor H n := by
  unfold MachineOverlayTable.rawExpander PreprocessingStageMaps.familyAt rotor
  exact congrArg encodeWords
    (rotationWords_resize (PreprocessingLevels.table_vertexCount_eq_paddedSize n)
      (ExpanderTables.family H (PreprocessingLevels.boundedLevel n)))

theorem overlay_memory (input graph raw count output : List Bool) :
    Placement.tapes overlayView
      (MachineOverlayTable.memory graph raw [] [] [] [] [] [] [] output)
      (memory input graph (encodeWord 0) raw count []) =
      memory input graph (encodeWord 0) raw count output := by
  funext k
  rcases k with k | k
  · cases k <;> simp [Placement.tapes, overlayView, memory, MachineVertexPadding.memory,
      MachineOverlayTable.memory, MachineOverlayTable.graphArchive]
  · rcases k with k | k
    · rcases k with k | k
      · rcases k with k | k <;> cases k <;>
          simp [Placement.tapes, overlayView, memory, MachineExpanderFamily.tableTape,
            MachineOverlayTable.memory, MachineOverlayTable.expanderArchive]
      · cases k <;>
          simp [Placement.tapes, overlayView, memory, MachineExpanderFamily.tableTape,
            ]
    · rcases k with k | k <;> fin_cases k <;>
        simp [Placement.tapes, overlayView, memory, MachineOverlayTable.memory,
          MachineOverlayTable.graphArchive, MachineOverlayTable.expanderArchive,
          MachineOverlayTable.output]

noncomputable def overlayInTime (H : BaseTable) {n d : Nat} (hd : 0 < d)
    (t : PortTables.Table n d) :
    StateTransition.EvalsToInTime (TM2.step (program H d hd))
      ⟨some overlayEntry, clean H d hd, familyTapes H t⟩
      (some ⟨none, clean H d hd, finalTapes H t⟩)
      ((MachineOverlayTable.timePolynomial d overlayDegree).eval
        ((PortTables.tableBits (padded t)).length + (rotor H n).length)) := by
  let f := MachineExpanderFamily.initialState MachineExpanderFamily.baseDegree_positive H ()
  have original := MachineOverlayTable.tableInTime d overlayDegree hd overlayDegree_positive
    (padded t) (PreprocessingStageMaps.familyAt H n) ()
  have framed := MachineStateFrame.frameExecution
    (MachineOverlayTable.program d overlayDegree hd overlayDegree_positive)
    ((f, false), (none : Option Bool))
    original
  have transported := MachineStateEquiv.execution (overlayStates d)
    (MachineStateFrame.frameProgram
      (MachineOverlayTable.program d overlayDegree hd overlayDegree_positive)) framed
  have sourceRun : StateTransition.EvalsToInTime (TM2.step (overlaySource d hd))
      ⟨some (.inl .copyFirst), clean H d hd,
        MachineOverlayTable.initialTapes (PortTables.tableBits (padded t)) (rotor H n)⟩
      (some ⟨none, clean H d hd,
        MachineOverlayTable.memory (PortTables.tableBits (padded t)) (rotor H n)
          [] [] [] [] [] [] []
          (PortTables.tableBits (PreprocessingStageMaps.paddedOverlay H d ⟨n, t⟩).2)⟩)
      ((MachineOverlayTable.timePolynomial d overlayDegree).eval
        ((PortTables.tableBits (padded t)).length + (rotor H n).length)) := by
    simpa only [overlaySource, MachineStateEquiv.configuration,
      MachineStateFrame.frameConfiguration, Option.map_some, overlayStates, Equiv.coe_fn_mk, clean, f,
      rotor_familyAt, PreprocessingStageMaps.paddedOverlay, PreprocessingStageMaps.padding,
      padded] using transported
  have run := placeExecution_inline_MachinePaddedOverlay overlayTape overlayView overlayView_left overlayView_right
    overlayLabel none (familyTapes H t) (overlaySource d hd) (program H d hd)
    (fun _ => rfl) sourceRun
  simpa only [Placement.configuration, Placement.label, familyTapes, finalTapes,
    MachineOverlayTable.initialTapes_memory, overlay_memory, overlayEntry] using run

noncomputable def sizePolynomial : Polynomial Nat :=
  Polynomial.C ExpanderFamily.growth * (Polynomial.X + 1)

noncomputable def archivePolynomial (d : Nat) : Polynomial Nat :=
  let s := sizePolynomial
  s + s * Polynomial.C d + 2 +
    (s * Polynomial.C d) * (s + s * Polynomial.C d + 8192) +
    (s * Polynomial.C overlayDegree) * (s * Polynomial.C overlayDegree + 1)

theorem archive_length_le (H : BaseTable) {n d : Nat} (t : PortTables.Table n d) :
    (PortTables.tableBits (padded t)).length + (rotor H n).length ≤
      (archivePolynomial d).eval (PortTables.tableBits t).length := by
  let m := PreprocessingLevels.paddedSize n
  let s := ExpanderFamily.growth * ((PortTables.tableBits t).length + 1)
  have hm : m ≤ s := MachineVertexPadding.power_length_bound t
  have hmd : m*d ≤ s*d := Nat.mul_le_mul_right d hm
  have hmq : m*overlayDegree ≤ s*overlayDegree := Nat.mul_le_mul_right overlayDegree hm
  have hg := GraphTables.tableBits_length_le (PortTables.graphTable (padded t))
  change (PortTables.tableBits (padded t)).length ≤
    m + m*d + 2 + m*d*(m + m*d + 8192) at hg
  have hg' := hg.trans (Nat.add_le_add
    (Nat.add_le_add_right (Nat.add_le_add hm hmd) 2)
    (Nat.mul_le_mul hmd (Nat.add_le_add_right (Nat.add_le_add hm hmd) 8192)))
  have hr := ExpanderTableWords.encode_rotationWords_length_le
    (PreprocessingStageMaps.familyAt H n)
  change (MachineOverlayTable.rawExpander (PreprocessingStageMaps.familyAt H n)).length ≤
    (m*overlayDegree)*(m*overlayDegree+1) at hr
  rw [rotor_familyAt] at hr
  have hr' := hr.trans (Nat.mul_le_mul hmq (Nat.add_le_add_right hmq 1))
  have h := Nat.add_le_add hg' hr'
  simpa only [archivePolynomial, sizePolynomial, Polynomial.eval_add, Polynomial.eval_mul,
    Polynomial.eval_C, Polynomial.eval_X, Polynomial.eval_one, Polynomial.eval_ofNat,
    s] using h

noncomputable def familyPolynomial : Polynomial Nat :=
  Polynomial.C MachineExpanderFamilyBounds.inputCoefficient * (Polynomial.X + 1)^5

noncomputable def timePolynomial (d : Nat) : Polynomial Nat :=
  MachineVertexPadding.timePolynomial d + familyPolynomial +
    (MachineOverlayTable.timePolynomial d overlayDegree).comp (archivePolynomial d)

noncomputable def execution (H : BaseTable) {n d : Nat} (hd : 0 < d)
    (t : PortTables.Table n d) :
    StateTransition.EvalsToInTime (TM2.step (program H d hd))
      ⟨some main, clean H d hd, initialTapes t⟩
      (some ⟨none, clean H d hd, finalTapes H t⟩)
      ((timePolynomial d).eval (PortTables.tableBits t).length) := by
  let first := paddingInTime H hd t
  let second := familyInTime H hd t
  let third := overlayInTime H hd t
  let firstTwo := StateTransition.EvalsToInTime.trans (TM2.step (program H d hd))
    _ _ _ _ _ first second
  let all := StateTransition.EvalsToInTime.trans (TM2.step (program H d hd))
    _ _ _ _ _ firstTwo third
  refine { toEvalsTo := all.toEvalsTo, steps_le_m := all.steps_le_m.trans ?_ }
  have hn := PortTables.vertices_le_tableBits_length t
  have hf := Nat.mul_le_mul_left MachineExpanderFamilyBounds.inputCoefficient
    (Nat.pow_le_pow_left (Nat.add_le_add_right hn 1) 5)
  have ho := natPolynomial_eval_mono (MachineOverlayTable.timePolynomial d overlayDegree)
    (archive_length_le H t)
  have h := Nat.add_le_add
    (Nat.add_le_add (Nat.le_refl ((MachineVertexPadding.timePolynomial d).eval
      (PortTables.tableBits t).length)) hf) ho
  simpa only [timePolynomial, familyPolynomial, Polynomial.eval_add, Polynomial.eval_comp,
    Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_pow, Polynomial.eval_X,
    Polynomial.eval_one, Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using h

def machine (H : BaseTable) (d : Nat) (hd : 0 < d) : FinTM2 where
  K := Tape
  k₀ := .inl .original
  k₁ := .inr (.inr MachineOverlayTable.output)
  Γ := Alphabet
  Λ := Label d
  main := main
  σ := State d
  initialState := clean H d hd
  m := program H d hd

theorem alphabet_finite (H : BaseTable) (d : Nat) (hd : 0 < d) :
    ∀ k, Finite ((machine H d hd).Γ k) := by
  intro k
  change Finite Bool
  infer_instance

theorem initial_configuration (H : BaseTable) {n d : Nat} (hd : 0 < d)
    (t : PortTables.Table n d) :
    initList (machine H d hd) (PortTables.tableBits t) =
      ⟨some main, clean H d hd, initialTapes t⟩ := by
  have ht : (initList (machine H d hd) (PortTables.tableBits t)).stk = initialTapes t := by
    funext k
    rcases k with k | k
    · cases k <;> simp [initList, machine, initialTapes, memory, MachineVertexPadding.memory]
      rfl
    · rcases k with k | k
      · rcases k with k | k
        · rcases k with k | k <;> cases k <;>
            simp [initList, machine, initialTapes, memory]
        · cases k <;> simp [initList, machine, initialTapes, memory]
      · simp [initList, machine, initialTapes, memory]
  exact congrArg (TM2.Cfg.mk _ _) ht

theorem final_output (H : BaseTable) {n d : Nat} (hd : 0 < d)
    (t : PortTables.Table n d) :
    finalTapes H t (machine H d hd).k₁ =
      PortTables.inputBits (PreprocessingStageMaps.paddedOverlay H d ⟨n, t⟩) := by
  simp [finalTapes, machine, memory, PortTables.inputBits]

theorem original_preserved (H : BaseTable) {n d : Nat} (t : PortTables.Table n d) :
    finalTapes H t (.inl .original) = PortTables.tableBits t := rfl

end DFVSGames.Foundations.Complexity.MachinePaddedOverlay
end

section

namespace DFVSGames.Foundations.Complexity.MachinePaddedOverlayRuntime

open Turing
open PCP
open MachinePaddedOverlay (BaseTable Tape Label State)

def rawProgram (H : BaseTable) (d : Nat) (hd : 0 < d) :
    MachineCanonicalOutput.Program Tape (Label d) (State d) where
  input := .inl .original
  output := .inr (.inr MachineOverlayTable.output)
  main := MachinePaddedOverlay.main
  initial := MachinePaddedOverlay.clean H d hd
  code := MachinePaddedOverlay.program H d hd

theorem sourceMachine_eq (H : BaseTable) (d : Nat) (hd : 0 < d) :
    MachineCanonicalOutput.sourceMachine (rawProgram H d hd) =
      MachinePaddedOverlay.machine H d hd := rfl

noncomputable def cleanupTapes : List Tape :=
  (Finset.univ.erase (.inr (.inr MachineOverlayTable.output) : Tape)).toList

theorem cleanup_complete (H : BaseTable) (d : Nat) (hd : 0 < d) (k : Tape) :
    k ∈ cleanupTapes ↔ k ≠ (rawProgram H d hd).output := by
  simp [cleanupTapes, rawProgram]

theorem initial_configuration (H : BaseTable) (d : Nat) (hd : 0 < d)
    (a : PortTables.Input d) :
    initList (MachineCanonicalOutput.sourceMachine (rawProgram H d hd))
        (PortTables.inputBits a) =
      ⟨some MachinePaddedOverlay.main, MachinePaddedOverlay.clean H d hd,
        MachinePaddedOverlay.initialTapes a.2⟩ := by
  change initList (MachinePaddedOverlay.machine H d hd) (PortTables.tableBits a.2) = _
  exact MachinePaddedOverlay.initial_configuration H hd a.2

noncomputable def terminalRun (H : BaseTable) (d : Nat) (hd : 0 < d)
    (a : PortTables.Input d) :
    MachineCanonicalOutput.TerminalRun (rawProgram H d hd) (PortTables.inputBits a)
      (PortTables.inputBits (PreprocessingStageMaps.paddedOverlay H d a))
      ((MachinePaddedOverlay.timePolynomial d).eval (PortTables.inputBits a).length) where
  state := MachinePaddedOverlay.clean H d hd
  tapes := MachinePaddedOverlay.finalTapes H a.2
  execution := by
    rw [initial_configuration]
    exact MachinePaddedOverlay.execution H hd a.2
  output_eq := MachinePaddedOverlay.final_output H hd a.2

noncomputable def computableInPolyTime (H : BaseTable) (d : Nat) (hd : 0 < d) :
    TM2ComputableInPolyTime (PortTables.inputBits (ports := d))
      (PortTables.inputBits (ports := d + MachinePaddedOverlay.overlayDegree))
      (PreprocessingStageMaps.paddedOverlay H d) :=
  MachineCanonicalOutput.computableInPolyTime (rawProgram H d hd) cleanupTapes
    (cleanup_complete H d hd) PortTables.inputBits PortTables.inputBits
    (PreprocessingStageMaps.paddedOverlay H d) (MachinePaddedOverlay.timePolynomial d)
    (terminalRun H d hd)

theorem finite_alphabet (H : BaseTable) (d : Nat) (hd : 0 < d) :
    ∀ k, Finite ((computableInPolyTime H d hd).tm.Γ k) :=
  MachineCanonicalOutput.computableInPolyTime_finite_alphabet (rawProgram H d hd) cleanupTapes
    (cleanup_complete H d hd) PortTables.inputBits PortTables.inputBits
    (PreprocessingStageMaps.paddedOverlay H d) (MachinePaddedOverlay.timePolynomial d)
    (terminalRun H d hd)

end DFVSGames.Foundations.Complexity.MachinePaddedOverlayRuntime
end

section

namespace DFVSGames.Foundations.Complexity.MachineLazyCodec
open PCP.GraphTables PCP.PreprocessingLazyWords

theorem rowsBits_ofFn (v e count : Nat) :
    MachineDummyRows.rowsBits v e count =
      (List.ofFn fun p : Fin count => MachineDummyRows.rowBits v (e + p.val)).flatten := by
  induction count generalizing e with
  | zero => rfl
  | succ count ih =>
    rw [MachineDummyRows.rowsBits, List.ofFn_succ, List.flatten_cons]
    simp only [Fin.val_zero, Nat.add_zero]
    apply congrArg (MachineDummyRows.rowBits v e ++ ·)
    rw [ih]
    apply congrArg List.flatten
    congr 1
    funext p
    simp only [Fin.val_succ]
    congr 1
    omega

private theorem encodeWords_flatMap_inline_MachineLazyCodec {α : Type*} (items : List α) (words : α → List Nat) :
    encodeWords (items.flatMap words) = items.flatMap (fun x => encodeWords (words x)) := by
  induction items with
  | nil => rfl
  | cons item items ih => simp only [List.flatMap_cons, encodeWords_append, ih]

theorem rowBits_stay {n d : Nat} (v : Fin n) (p : Fin d) :
    MachineDummyRows.rowBits v.val (2 * d * v.val + p.val) =
      encodeWords (rowWords (stayRow d v p)) := by
  have h := MachineDummyRows.rowBits_eq_graph_row v (stayRow d v p).reverseIndex
  rw [stayRow_reverse_val] at h
  exact h

theorem rowsBits_stay {n d : Nat} (v : Fin n) :
    MachineDummyRows.rowsBits v.val (2 * d * v.val) d =
      encodeWords ((List.ofFn (stayRow d v)).flatMap rowWords) := by
  rw [rowsBits_ofFn, encodeWords_flatMap_inline_MachineLazyCodec]
  simp only [List.flatMap_def, List.map_ofFn]
  apply congrArg List.flatten
  congr 1
  funext p
  exact rowBits_stay v p

end DFVSGames.Foundations.Complexity.MachineLazyCodec
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