lean/OAI/Computability/DirectedFeedback/Machines/MachineRegularMetadata.lean

import Mathlib
import OAI.Computability.DirectedFeedback.Machines.MachineTableRows

namespace OAI

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namespace DFVSGames.Foundations.Complexity.MachineRegularMetadata

open Turing MachineComposition
open DFVSGames.Foundations.PCP

inductive Extra | padding | level
  deriving DecidableEq

instance : Fintype Extra := derive_fintype% Extra

abbrev Tape := Fin 27 ⊕ Extra
abbrev Alphabet (_ : Tape) := Bool
abbrev State (σ : Type) := (σ × Bool) × Option Bool

inductive Label
  | owner (l : MachineAffineLookup.Label)
  | ownerClearQuery | ownerClearWork
  | copyXFirst | copyXSecond | copyOwnerFirst | copyOwnerSecond | rankInit
  | rank (l : MachineCloudRank.RankLabel)
  | rankClearQuery | rankClearWork | rankClearFuel
  | prefixCode (l : MachineCloudPrefix.Label)
  | prefixClearQuery | prefixClearFuel
  | cloud (l : MachineCloudPadding.Label)
  deriving DecidableEq, Fintype

def ownerTapes : Fin 5 → Tape := ![.inl 0, .inl 9, .inl 10, .inl 2, .inl 11]

def rankTape : MachineCloudCount.Tape → Tape
  | .original => .inl 0
  | .work => .inl 10
  | .target => .inl 9
  | .scratch => .inl 11
  | .count => .inl 3
  | .spare => .inl 12

def prefixTape : MachineCloudPrefix.Tape → Tape
  | .inner .table => .inl 0
  | .inner .query => .inl 9
  | .inner .count => .inl 10
  | .inner .work => .inl 11
  | .inner .scratch => .inl 12
  | .inner .spare => .inl 13
  | .inner .power => .inl 14
  | .inner .level => .inl 15
  | .inner .fuel => .inl 16
  | .inner .product => .inl 17
  | .bound => .inl 2
  | .remaining => .inl 18
  | .total => .inl 5

def cloudTape : MachineCloudPadding.Tape → Tape
  | .table => .inl 0
  | .query => .inl 2
  | .count => .inl 4
  | .work => .inl 9
  | .scratch => .inl 10
  | .spare => .inl 11
  | .power => .inr .padding
  | .level => .inr .level
  | .fuel => .inl 12
  | .product => .inl 13

variable {σ : Type}

def program : Label → TM2.Stmt Alphabet Label (State σ)
  | .owner l => MachineAffineLookup.instruction (.inl 1) ownerTapes 4098 2
      Label.owner (some .ownerClearQuery) l
  | .ownerClearQuery => MachineDrain.drain (.inl 9) .ownerClearQuery (some .ownerClearWork)
  | .ownerClearWork => MachineDrain.drain (.inl 10) .ownerClearWork (some .copyXFirst)
  | .copyXFirst => Reduction.MachineTransfer.loopAt (.inl 1) (.inl 10) id false
      .copyXFirst (some .copyXSecond)
  | .copyXSecond => MachineCopy.forkLoop (.inl 10) (.inl 1) (.inl 9) false
      .copyXSecond (some .copyOwnerFirst)
  | .copyOwnerFirst => Reduction.MachineTransfer.loopAt (.inl 2) (.inl 10) id false
      .copyOwnerFirst (some .copyOwnerSecond)
  | .copyOwnerSecond => MachineCopy.forkLoop (.inl 10) (.inl 2) (.inl 9) false
      .copyOwnerSecond (some .rankInit)
  | .rankInit => .push (.inl 3) (fun _ => false) (.goto fun _ => .rank .queryFirst)
  | .rank l => MachineCloudPadding.Placement.statement rankTape Label.rank
      (some .rankClearQuery) (MachineCloudRank.rankProgram l)
  | .rankClearQuery => MachineDrain.drain (.inl 9) .rankClearQuery (some .rankClearWork)
  | .rankClearWork => MachineDrain.drain (.inl 10) .rankClearWork (some .rankClearFuel)
  | .rankClearFuel => MachineDrain.drain (.inl 12) .rankClearFuel (some (.prefixCode .init))
  | .prefixCode l => MachineCloudPadding.Placement.statement prefixTape Label.prefixCode
      (some .prefixClearQuery) (MachineCloudPrefix.program l)
  | .prefixClearQuery => MachineDrain.drain (.inl 9) .prefixClearQuery (some .prefixClearFuel)
  | .prefixClearFuel => MachineDrain.drain (.inl 18) .prefixClearFuel (some (.cloud .init))
  | .cloud l => MachineCloudPadding.Placement.statement cloudTape Label.cloud none
      (MachineCloudPadding.program l)

def rankView : Tape → Option MachineCloudCount.Tape
  | .inl i => match i.val with
    | 0 => some .original
    | 10 => some .work
    | 9 => some .target
    | 11 => some .scratch
    | 3 => some .count
    | 12 => some .spare
    | _ => none
  | .inr _ => none

def prefixView : Tape → Option MachineCloudPrefix.Tape
  | .inl i => match i.val with
    | 0 => some (.inner .table)
    | 9 => some (.inner .query)
    | 10 => some (.inner .count)
    | 11 => some (.inner .work)
    | 12 => some (.inner .scratch)
    | 13 => some (.inner .spare)
    | 14 => some (.inner .power)
    | 15 => some (.inner .level)
    | 16 => some (.inner .fuel)
    | 17 => some (.inner .product)
    | 2 => some .bound
    | 18 => some .remaining
    | 5 => some .total
    | _ => none
  | .inr _ => none

def cloudView : Tape → Option MachineCloudPadding.Tape
  | .inl i => match i.val with
    | 0 => some .table
    | 2 => some .query
    | 4 => some .count
    | 9 => some .work
    | 10 => some .scratch
    | 11 => some .spare
    | 12 => some .fuel
    | 13 => some .product
    | _ => none
  | .inr .padding => some .power
  | .inr .level => some .level

theorem rankView_left (k : MachineCloudCount.Tape) : rankView (rankTape k) = some k := by
  cases k <;> rfl
theorem rankView_right (j : Tape) (k : MachineCloudCount.Tape)
    (h : rankView j = some k) : rankTape k = j := by
  cases j with
  | inl j => fin_cases j <;> simp_all [rankView, rankTape] <;> subst k <;> rfl
  | inr j => simp_all [rankView]

theorem prefixView_left (k : MachineCloudPrefix.Tape) : prefixView (prefixTape k) = some k := by
  cases k with
  | inner k => cases k <;> rfl
  | bound => rfl
  | remaining => rfl
  | total => rfl
theorem prefixView_right (j : Tape) (k : MachineCloudPrefix.Tape)
    (h : prefixView j = some k) : prefixTape k = j := by
  cases j with
  | inl j => fin_cases j <;> simp_all [prefixView, prefixTape] <;> subst k <;> rfl
  | inr j => simp_all [prefixView]

theorem cloudView_left (k : MachineCloudPadding.Tape) : cloudView (cloudTape k) = some k := by
  cases k <;> rfl
theorem cloudView_right (j : Tape) (k : MachineCloudPadding.Tape)
    (h : cloudView j = some k) : cloudTape k = j := by
  cases j with
  | inl j => fin_cases j <;> simp_all [cloudView, cloudTape] <;> subst k <;> rfl
  | inr j => cases j <;> simp_all [cloudView, cloudTape] <;> subst k <;> rfl

structure Data where
  table : List Bool
  globalIndex : List Bool
  owner : List Bool
  localRank : List Bool
  count : List Bool
  offset : List Bool
  darts : List Bool
  rotor : List Bool
  output : List Bool
  padding : List Bool
  level : List Bool

def frame (data : Data) : Tape → List Bool
  | .inl i => match i.val with
    | 0 => data.table
    | 1 => data.globalIndex
    | 2 => data.owner
    | 3 => data.localRank
    | 4 => data.count
    | 5 => data.offset
    | 6 => data.darts
    | 7 => data.rotor
    | 8 => data.output
    | _ => []
  | .inr .padding => data.padding
  | .inr .level => data.level

@[simp] theorem frame_index (data : Data) :
    frame data (.inl 1) = data.globalIndex := rfl

@[simp] theorem frame_query_empty (data : Data) :
    frame data (.inl 9) = [] := rfl

def cfg (label : Option Label) (data : Data) (ambient : σ) (register : Option Bool) :
    TM2.Cfg Alphabet Label (State σ) :=
  ⟨label, ((ambient, false), register), frame data⟩

def cloudData (t : GraphTables.Table) (v : Fin t.vertices) (data : Data) : Data :=
  { data with
    count := encodeWord (PreprocessingCloudIndex.cloudSize t v)
    padding := encodeWord (MachineCloudPadding.padding (PreprocessingCloudIndex.cloudSize t v))
    level := encodeWord (PreprocessingLevels.boundedLevel (PreprocessingCloudIndex.cloudSize t v)) }

theorem cloudTrace (t : GraphTables.Table) (v : Fin t.vertices) (data : Data)
    (htable : data.table = GraphTables.tableBits t) (howner : data.owner = encodeWord v.val)
    (hcount : data.count = []) (hpadding : data.padding = []) (hlevel : data.level = [])
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[MachineCloudPadding.totalTime t v]
      (some (cfg (some (.cloud .init)) data ambient register)) =
      some (cfg none (cloudData t v data) ambient none) := by
  have raw := MachineCloudPadding.cloudPaddingTrace t v [] ambient register
  simp only [List.append_nil] at raw
  have placed := MachineCloudPadding.Placement.trace cloudTape cloudView cloudView_left cloudView_right
    Label.cloud none (frame data) MachineCloudPadding.program program (fun _ => rfl)
    (MachineCloudPadding.totalTime t v) _ _ raw
  have hstart : MachineCloudPadding.Placement.tapes cloudView
      (MachineCloudPadding.memory (GraphTables.tableBits t) (encodeWord v.val)
        [] [] [] [] [] [] [] []) (frame data) = frame data := by
    funext j
    cases j with
    | inl j => fin_cases j <;>
        simp_all [MachineCloudPadding.Placement.tapes, cloudView, frame, MachineCloudPadding.memory]
    | inr j => cases j <;>
        simp_all [MachineCloudPadding.Placement.tapes, cloudView, frame, MachineCloudPadding.memory]
  have hfinish : MachineCloudPadding.Placement.tapes cloudView
      (MachineCloudPadding.memory (GraphTables.tableBits t) (encodeWord v.val)
        (encodeWord (PreprocessingCloudIndex.cloudSize t v)) [] [] []
        (encodeWord (MachineCloudPadding.padding (PreprocessingCloudIndex.cloudSize t v)))
        (encodeWord (PreprocessingLevels.boundedLevel (PreprocessingCloudIndex.cloudSize t v))) [] [])
      (frame data) = frame (cloudData t v data) := by
    funext j
    cases j with
    | inl j => fin_cases j <;>
        simp_all [MachineCloudPadding.Placement.tapes, cloudView, frame, cloudData, MachineCloudPadding.memory]
    | inr j => cases j <;>
        simp_all [MachineCloudPadding.Placement.tapes, cloudView, frame, cloudData, MachineCloudPadding.memory]
  simpa only [MachineCloudPadding.Placement.configuration, MachineCloudPadding.Placement.label,
    hstart, hfinish, cfg] using placed

private theorem join_trace_inline_MachineRegularMetadata {A : Type*} {f : A → A} {m n : Nat} {a b c : A}
    (first : f^[m] a = b) (second : f^[n] b = c) : f^[m + n] a = c := by
  rw [Nat.add_comm m n, Function.iterate_add_apply, first, second]

private theorem drainAt_inline_MachineRegularMetadata (source : Tape) (again : Label) (exit : Option Label)
    (code : program (σ := σ) again = MachineDrain.drain source again exit)
    (base : Tape → List Bool) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[(base source).length + 1]
      (some ⟨some again, ((ambient, false), register), base⟩) =
      some ⟨exit, ((ambient, false), none), Function.update base source []⟩ := by
  simpa only [Function.update_eq_self] using
    MachineDrain.drainTrace source again exit program code base (base source)
      (ambient, false) register

def prefixData (t : GraphTables.Table) (v : Fin t.vertices) (data : Data) : Data :=
  { data with offset := encodeWord (PreprocessingPaddingOffsets.offset
      (PreprocessingRegularTables.padding t) v.val) }

def prefixSteps (t : GraphTables.Table) (v : Fin t.vertices) : Nat :=
  MachineCloudPrefix.totalTime t v.val [] + (v.val + 2) + 2

theorem prefixTrace (t : GraphTables.Table) (v : Fin t.vertices) (data : Data)
    (htable : data.table = GraphTables.tableBits t) (howner : data.owner = encodeWord v.val)
    (hoffset : data.offset = []) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[prefixSteps t v]
      (some (cfg (some (.prefixCode .init)) data ambient register)) =
      some (cfg (some (.cloud .init)) (prefixData t v data) ambient none) := by
  have raw := MachineCloudPrefix.prefixTrace t v.val v.isLt.le [] ambient register
  simp only [List.append_nil] at raw
  let mid := MachineCloudPadding.Placement.tapes prefixView
    (MachineCloudPrefix.frame (GraphTables.tableBits t) (encodeWord v.val) [] [] []
      (encodeWord v.val) (encodeWord 0)
      (encodeWord (PreprocessingPaddingOffsets.offset
        (PreprocessingRegularTables.padding t) v.val))) (frame data)
  have placed := MachineCloudPadding.Placement.trace prefixTape prefixView prefixView_left
    prefixView_right Label.prefixCode (some .prefixClearQuery) (frame data)
    MachineCloudPrefix.program program (fun _ => rfl)
    (MachineCloudPrefix.totalTime t v.val []) _ _ raw
  have hstart : MachineCloudPadding.Placement.tapes prefixView
      (MachineCloudPrefix.frame (GraphTables.tableBits t) [] [] [] [] (encodeWord v.val) [] [])
      (frame data) = frame data := by
    funext j
    cases j with
    | inl j => fin_cases j <;>
        simp [htable, howner, hoffset, MachineCloudPadding.Placement.tapes, prefixView, frame,
          MachineCloudPrefix.frame, MachineCloudPrefix.memory, MachineCloudPadding.memory]
    | inr j => cases j <;> rfl
  have phase : (advance (TM2.step program))^[MachineCloudPrefix.totalTime t v.val []]
      (some (cfg (some (.prefixCode .init)) data ambient register)) =
      some ⟨some .prefixClearQuery, ((ambient, false), none), mid⟩ := by
    simpa only [MachineCloudPadding.Placement.configuration,
      MachineCloudPadding.Placement.label, hstart, cfg, mid] using placed
  have first := drainAt_inline_MachineRegularMetadata (.inl 9) .prefixClearQuery (some .prefixClearFuel) rfl mid ambient none
  have second := drainAt_inline_MachineRegularMetadata (.inl 18) .prefixClearFuel (some (.cloud .init)) rfl
    (Function.update mid (.inl 9) []) ambient none
  have hquery : (mid (.inl 9)).length + 1 = v.val + 2 := by
    simp [mid, MachineCloudPadding.Placement.tapes, prefixView, MachineCloudPrefix.frame, MachineCloudPrefix.memory,
      MachineCloudPadding.memory, encodeWord_length]
  have hfuel : ((Function.update mid (.inl 9) []) (.inl 18)).length + 1 = 2 := by
    simp [mid, MachineCloudPadding.Placement.tapes, prefixView, MachineCloudPrefix.frame, MachineCloudPrefix.memory,
      encodeWord_length]
  have hfinish : Function.update (Function.update mid (.inl 9) []) (.inl 18) [] =
      frame (prefixData t v data) := by
    funext j
    cases j with
    | inl j => fin_cases j <;>
        simp [htable, howner, mid, MachineCloudPadding.Placement.tapes, prefixView, frame, prefixData,
          MachineCloudPrefix.frame, MachineCloudPrefix.memory, MachineCloudPadding.memory]
    | inr j => cases j <;> rfl
  rw [hquery] at first
  rw [hfuel, hfinish] at second
  exact join_trace_inline_MachineRegularMetadata (join_trace_inline_MachineRegularMetadata phase first) second

theorem selected_owner (t : GraphTables.Table) (e : Fin t.darts) :
    (GraphTables.tableWords t)[4098 * e.val + 2]? = some t.rows[e].tail.val := by
  have h := congrArg (fun words : List Nat => words[0]?)
    (MachineRegularOriginalRow.tableWords_drop_row t e)
  simp only [List.getElem?_drop, Nat.add_zero] at h
  rw [show 2 + 4098 * e.val = 4098 * e.val + 2 by omega] at h
  simpa [GraphTables.rowWords] using h

def ownerData (t : GraphTables.Table) (e : Fin t.darts) (data : Data) : Data :=
  { data with owner := encodeWord t.rows[e].tail.val }

def ownerRemainder (t : GraphTables.Table) (e : Fin t.darts) : List Bool :=
  encodeWords ((GraphTables.tableWords t).drop (4098 * e.val + 2 + 1))

def ownerSteps (t : GraphTables.Table) (e : Fin t.darts) : Nat :=
  MachineAffineLookup.steps (GraphTables.tableWords t) e.val 4098 2 + 2 +
    ((ownerRemainder t e).length + 1)

theorem ownerTrace (t : GraphTables.Table) (e : Fin t.darts) (data : Data)
    (htable : data.table = GraphTables.tableBits t)
    (hindex : data.globalIndex = encodeWord e.val) (howner : data.owner = [])
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[ownerSteps t e]
      (some (cfg (some (.owner .seed)) data ambient register)) =
      some (cfg (some .copyXFirst) (ownerData t e data) ambient none) := by
  have distinct : Function.Injective ownerTapes := by
    intro i j h
    fin_cases i <;> fin_cases j <;> simp_all [ownerTapes]
  have outside : ∀ i, (.inl 1 : Tape) ≠ ownerTapes i := by
    intro i
    fin_cases i <;> decide
  let mid := MachineAffineLookup.finalTapes ownerTapes (frame data)
    (GraphTables.tableWords t) (4098 * e.val + 2) t.rows[e].tail.val
  have phase : (advance (TM2.step program))^[
      MachineAffineLookup.steps (GraphTables.tableWords t) e.val 4098 2]
      (some (cfg (some (.owner .seed)) data ambient register)) =
      some ⟨some .ownerClearQuery, ((ambient, false), none), mid⟩ := by
    exact MachineAffineLookup.affineLookupTrace (.inl 1) ownerTapes distinct outside 4098 2
      Label.owner (some .ownerClearQuery) program (fun _ => rfl) (frame data)
      (GraphTables.tableWords t) htable rfl e.val []
      (by change data.globalIndex = encodeWord e.val ++ [];
          simpa only [List.append_nil] using hindex)
      t.rows[e].tail.val (selected_owner t e) (ambient, false) register
  have first := drainAt_inline_MachineRegularMetadata (.inl 9) .ownerClearQuery (some .ownerClearWork) rfl mid ambient none
  have second := drainAt_inline_MachineRegularMetadata (.inl 10) .ownerClearWork (some .copyXFirst) rfl
    (Function.update mid (.inl 9) []) ambient none
  have hquery : (mid (.inl 9)).length + 1 = 2 := by
    simp [mid, MachineAffineLookup.finalTapes, MachinePreservingLookup.finalTapes,
      MachineLookup.tapes, ownerTapes, frame, encodeWord_length]
  have hwork : (Function.update mid (.inl 9) []) (.inl 10) = ownerRemainder t e := by
    simp [mid, MachineAffineLookup.finalTapes, MachinePreservingLookup.finalTapes,
      MachineLookup.tapes, ownerTapes, frame, ownerRemainder]
  have hfinish : Function.update (Function.update mid (.inl 9) []) (.inl 10) [] =
      frame (ownerData t e data) := by
    funext j
    cases j with
    | inl j => fin_cases j <;>
        simp [howner, mid, MachineAffineLookup.finalTapes, MachinePreservingLookup.finalTapes,
          MachineLookup.tapes, ownerTapes, frame, ownerData]
    | inr j => cases j <;> rfl
  rw [hquery] at first
  rw [hwork, hfinish] at second
  exact join_trace_inline_MachineRegularMetadata (join_trace_inline_MachineRegularMetadata phase first) second

def rankQuery (v x : Nat) : List Bool := MachineCloudRank.queryWord v x []

def queryFrame (data : Data) (v x : Nat) : Tape → List Bool :=
  Function.update (Function.update (frame data) (.inl 9) (rankQuery v x))
    (.inl 3) (encodeWord 0)

def querySteps (v x : Nat) : Nat := 2 * (x + 2) + 2 * (v + 2) + 1

theorem queryTrace (data : Data) (v x : Nat)
    (howner : data.owner = encodeWord v) (hindex : data.globalIndex = encodeWord x)
    (hrank : data.localRank = []) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[querySteps v x]
      (some (cfg (some .copyXFirst) data ambient register)) =
      some ⟨some (.rank .queryFirst), ((ambient, false), none), queryFrame data v x⟩ := by
  let first := Function.update (frame data) (.inl 9) data.globalIndex
  let second := Function.update first (.inl 9) (data.owner ++ data.globalIndex)
  have copyX := MachineCopy.copyTrace (.inl 1 : Tape) (.inl 9) (.inl 10)
    (by decide) (by decide) (by decide) false Label.copyXFirst Label.copyXSecond
    (some .copyOwnerFirst) program rfl rfl (frame data) rfl (ambient, false) register
  have copyOwner := MachineCopy.copyTrace (.inl 2 : Tape) (.inl 9) (.inl 10)
    (by decide) (by decide) (by decide) false Label.copyOwnerFirst Label.copyOwnerSecond
    (some .rankInit) program rfl rfl first (by simp [first, frame]) (ambient, false) none
  have firstRun : (advance (TM2.step program))^[2 * (x + 2)]
      (some (cfg (some .copyXFirst) data ambient register)) =
      some ⟨some .copyOwnerFirst, ((ambient, false), none), first⟩ := by
    simpa only [frame_index, frame_query_empty, hindex, encodeWord_length,
      List.append_nil, cfg, first] using copyX
  have secondRun : (advance (TM2.step program))^[2 * (v + 2)]
      (some ⟨some .copyOwnerFirst, ((ambient, false), none), first⟩) =
      some ⟨some .rankInit, ((ambient, false), none), second⟩ := by
    simpa [first, frame, howner, encodeWord_length, second] using copyOwner
  have init : (advance (TM2.step program))^[1]
      (some ⟨some .rankInit, ((ambient, false), none), second⟩) =
      some ⟨some (.rank .queryFirst), ((ambient, false), none), queryFrame data v x⟩ := by
    have ht : Function.update second (.inl 3) (false :: second (.inl 3)) =
        queryFrame data v x := by
      funext j
      cases j with
      | inl j => fin_cases j <;>
          simp_all [queryFrame, second, first, frame, rankQuery,
            MachineCloudRank.queryWord, encodeWord]
      | inr j => cases j <;> rfl
    simp only [Function.iterate_one, advance_some, TM2.step, program, TM2.stepAux,
      ht]

  exact join_trace_inline_MachineRegularMetadata (join_trace_inline_MachineRegularMetadata firstRun secondRun) init

def rankData (t : GraphTables.Table) (v : Fin t.vertices)
    (e : DegreeReplacement.Cloud (GraphTables.semantics t) v) (data : Data) : Data :=
  { data with localRank := encodeWord (PreprocessingCloudIndex.cloudRank t v e).val }

def rankRemainder (t : GraphTables.Table)
    (v : Fin t.vertices) (e : DegreeReplacement.Cloud (GraphTables.semantics t) v) : List Bool :=
  encodeWords (((MachineCloudCount.tableRows t).drop e.val.val).flatMap MachineCloudCount.rowWords)

def rankSteps (t : GraphTables.Table) (v : Fin t.vertices)
    (e : DegreeReplacement.Cloud (GraphTables.semantics t) v) : Nat :=
  querySteps v.val e.val.val +
    MachineCloudRank.rankSteps t.vertices t.darts v.val e.val.val (MachineCloudCount.tableRows t) [] +
    ((rankQuery v.val e.val.val).length + 1) + ((rankRemainder t v e).length + 1) + 2

theorem rankTrace (t : GraphTables.Table) (v : Fin t.vertices)
    (e : DegreeReplacement.Cloud (GraphTables.semantics t) v) (data : Data)
    (htable : data.table = GraphTables.tableBits t)
    (howner : data.owner = encodeWord v.val) (hindex : data.globalIndex = encodeWord e.val.val)
    (hrank : data.localRank = []) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[rankSteps t v e]
      (some (cfg (some .copyXFirst) data ambient register)) =
      some (cfg (some (.prefixCode .init)) (rankData t v e data) ambient none) := by
  have query := queryTrace data v.val e.val.val howner hindex hrank ambient register
  have raw := MachineCloudRank.cloudRankTrace t v e [] [] ambient none
  simp only [List.append_nil] at raw
  let mid := MachineCloudPadding.Placement.tapes rankView
    (MachineCloudCount.memory (GraphTables.tableBits t) (rankRemainder t v e)
      (rankQuery v.val e.val.val) []
      (encodeWord (PreprocessingCloudIndex.cloudRank t v e).val) (encodeWord 0)) (frame data)
  have placed := MachineCloudPadding.Placement.trace rankTape rankView rankView_left
    rankView_right Label.rank (some .rankClearQuery) (frame data)
    MachineCloudRank.rankProgram program (fun _ => rfl)
    (MachineCloudRank.rankSteps t.vertices t.darts v.val e.val.val (MachineCloudCount.tableRows t) [])
    _ _ raw
  have hstart : MachineCloudPadding.Placement.tapes rankView
      (MachineCloudCount.memory (GraphTables.tableBits t) []
        (rankQuery v.val e.val.val) [] (encodeWord 0) []) (frame data) =
      queryFrame data v.val e.val.val := by
    funext j
    cases j with
    | inl j => fin_cases j <;>
        simp_all [MachineCloudPadding.Placement.tapes, rankView, frame,
          queryFrame, MachineCloudCount.memory]
    | inr j => cases j <;> rfl
  simp only [rankQuery] at hstart
  have phase : (advance (TM2.step program))^[
      MachineCloudRank.rankSteps t.vertices t.darts v.val e.val.val (MachineCloudCount.tableRows t) []]
      (some ⟨some (.rank .queryFirst), ((ambient, false), none), queryFrame data v.val e.val.val⟩) =
      some ⟨some .rankClearQuery, ((ambient, false), none), mid⟩ := by
    simpa only [MachineCloudPadding.Placement.configuration, MachineCloudPadding.Placement.label,
      hstart, mid, rankQuery, rankRemainder] using placed
  have first := drainAt_inline_MachineRegularMetadata (.inl 9) .rankClearQuery (some .rankClearWork) rfl mid ambient none
  have second := drainAt_inline_MachineRegularMetadata (.inl 10) .rankClearWork (some .rankClearFuel) rfl
    (Function.update mid (.inl 9) []) ambient none
  have third := drainAt_inline_MachineRegularMetadata (.inl 12) .rankClearFuel (some (.prefixCode .init)) rfl
    (Function.update (Function.update mid (.inl 9) []) (.inl 10) []) ambient none
  have hquery : mid (.inl 9) = rankQuery v.val e.val.val := rfl
  have hwork : Function.update mid (.inl 9) [] (.inl 10) = rankRemainder t v e := by
    simp [mid, MachineCloudPadding.Placement.tapes, rankView, MachineCloudCount.memory]
  have hfuel : ((Function.update (Function.update mid (.inl 9) []) (.inl 10) [])
      (.inl 12)).length + 1 = 2 := by
    simp [mid, MachineCloudPadding.Placement.tapes, rankView, MachineCloudCount.memory,
      encodeWord_length]
  have hfinish : Function.update
      (Function.update (Function.update mid (.inl 9) []) (.inl 10) []) (.inl 12) [] =
      frame (rankData t v e data) := by
    funext j
    cases j with
    | inl j => fin_cases j <;>
        simp [htable, mid, MachineCloudPadding.Placement.tapes, rankView, frame, rankData,
          MachineCloudCount.memory]
    | inr j => cases j <;> rfl
  rw [hquery] at first
  rw [hwork] at second
  rw [hfuel, hfinish] at third
  exact join_trace_inline_MachineRegularMetadata (join_trace_inline_MachineRegularMetadata (join_trace_inline_MachineRegularMetadata (join_trace_inline_MachineRegularMetadata query phase) first) second) third

def originalMember (t : GraphTables.Table) (e : Fin t.darts) :
    DegreeReplacement.Cloud (GraphTables.semantics t) t.rows[e].tail := ⟨e, rfl⟩

def originalData (t : GraphTables.Table) (e : Fin t.darts) (data : Data) : Data :=
  cloudData t t.rows[e].tail (prefixData t t.rows[e].tail
    (rankData t t.rows[e].tail (originalMember t e) (ownerData t e data)))

def totalTime (t : GraphTables.Table) (e : Fin t.darts) : Nat :=
  ownerSteps t e + rankSteps t t.rows[e].tail (originalMember t e) +
    prefixSteps t t.rows[e].tail + MachineCloudPadding.totalTime t t.rows[e].tail

theorem originalTrace (t : GraphTables.Table) (e : Fin t.darts) (data : Data)
    (htable : data.table = GraphTables.tableBits t)
    (hindex : data.globalIndex = encodeWord e.val)
    (howner : data.owner = []) (hrank : data.localRank = []) (hcount : data.count = [])
    (hoffset : data.offset = []) (hpadding : data.padding = []) (hlevel : data.level = [])
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[totalTime t e]
      (some (cfg (some (.owner .seed)) data ambient register)) =
      some (cfg none (originalData t e data) ambient none) := by
  have own := ownerTrace t e data htable hindex howner ambient register
  have rank := rankTrace t t.rows[e].tail (originalMember t e) (ownerData t e data)
    htable rfl hindex hrank ambient none
  have pref := prefixTrace t t.rows[e].tail
    (rankData t t.rows[e].tail (originalMember t e) (ownerData t e data))
    htable rfl hoffset ambient none
  have cloud := cloudTrace t t.rows[e].tail
    (prefixData t t.rows[e].tail
      (rankData t t.rows[e].tail (originalMember t e) (ownerData t e data)))
    htable rfl hcount hpadding hlevel ambient none
  exact join_trace_inline_MachineRegularMetadata (join_trace_inline_MachineRegularMetadata (join_trace_inline_MachineRegularMetadata own rank) pref) cloud

private theorem encodeWords_drop_length_le_inline_MachineRegularMetadata (values : List Nat) (n : Nat) :
    (encodeWords (values.drop n)).length ≤ (encodeWords values).length := by
  have h := congrArg (fun words : List Nat => (encodeWords words).length)
    (List.take_append_drop n values)
  simp only [encodeWords_append, List.length_append] at h
  omega

theorem ownerRemainder_length_le (t : GraphTables.Table) (e : Fin t.darts) :
    (ownerRemainder t e).length ≤ (GraphTables.tableBits t).length :=
  encodeWords_drop_length_le_inline_MachineRegularMetadata _ _

theorem rankRemainder_length_le (t : GraphTables.Table) (v : Fin t.vertices)
    (e : DegreeReplacement.Cloud (GraphTables.semantics t) v) :
    (rankRemainder t v e).length ≤ (GraphTables.tableBits t).length := by
  have split := congrArg List.length
    (MachineCloudRank.encoded_rows_split (MachineCloudCount.tableRows t) e.val.val)
  have input := MachineCloudCount.inputWord_length t.vertices t.darts (MachineCloudCount.tableRows t)
  rw [MachineCloudCount.tableRows_input] at input
  simp only [List.length_append] at split
  unfold rankRemainder
  omega

noncomputable def timePolynomial : Polynomial Nat :=
  (MachineCloudPrefix.timePolynomial + MachineCloudPadding.timePolynomial).comp
      (2 * Polynomial.X) + 40 * (Polynomial.X + 1)

def overhead (N x v : Nat) : Nat :=
  (2*x + 5*N + 6) + 2 + (N+1) + 2*(x+2) + 2*(v+2) + 1 +
    (5*N + 3*(v+x+2) + 6) + (v+x+3) + (N+1) + 2 + (v+2) + 2

theorem overhead_le (N x v : Nat) (hx : x ≤ N) (hv : v ≤ N) :
    overhead N x v ≤ 40*(N+1) := by
  unfold overhead
  omega

theorem totalTime_le (t : GraphTables.Table) (e : Fin t.darts) :
    totalTime t e ≤ timePolynomial.eval (GraphTables.tableBits t).length := by
  let v := t.rows[e].tail
  let c := originalMember t e
  let N := (GraphTables.tableBits t).length
  have hn := GraphTables.vertices_le_tableBits_length t
  have hm := GraphTables.darts_le_tableBits_length t
  have hv : v.val < N := lt_of_lt_of_le v.isLt hn
  have hx : e.val ≤ N := e.isLt.le.trans hm
  have ownerBound := MachineAffineLookup.steps_le (GraphTables.tableWords t)
    e.val 4098 2 t.rows[e].tail.val (selected_owner t e)
  have ownerRest := ownerRemainder_length_le t e
  have rankBound := MachineCloudRank.rankSteps_le t.vertices t.darts v.val e.val
    (MachineCloudCount.tableRows t) []
  rw [MachineCloudCount.tableRows_input] at rankBound
  have rankRest := rankRemainder_length_le t v c
  have hpIn : MachineCloudPrefix.inputLength t v.val [] ≤ 2*N := by
    simp only [MachineCloudPrefix.inputLength, List.append_nil, encodeWord_length]
    omega
  have hcIn : MachineCloudPadding.inputLength t v [] ≤ 2*N := by
    simp only [MachineCloudPadding.inputLength, List.append_nil, encodeWord_length]
    omega
  have hp := (MachineCloudPrefix.totalTime_le t v.val v.isLt.le []).trans
    (natPolynomial_eval_mono MachineCloudPrefix.timePolynomial hpIn)
  have hc := (MachineCloudPadding.totalTime_le t v []).trans
    (natPolynomial_eval_mono MachineCloudPadding.timePolynomial hcIn)
  have ho := overhead_le N e.val v.val hx hv.le
  have hsmall : ownerSteps t e + rankSteps t v c + (v.val+2) + 2 ≤
      overhead N e.val v.val := by
    simp only [MachineCloudRank.queryWord, List.length_append, encodeWord_length,
      List.length_nil] at rankBound
    simp only [ownerSteps, rankSteps, querySteps, rankQuery, MachineCloudRank.queryWord,
      List.length_append, encodeWord_length, List.length_nil, overhead]
    change _ ≤ _ at ownerBound
    dsimp only [c, originalMember] at rankRest ⊢
    change MachineAffineLookup.steps (GraphTables.tableWords t) e.val 4098 2 ≤
      2*e.val+5*N+6 at ownerBound
    omega
  have htotal : totalTime t e = ownerSteps t e + rankSteps t v c +
      MachineCloudPrefix.totalTime t v.val [] + (v.val+2) + 2 +
        MachineCloudPadding.totalTime t v := by
    simp only [totalTime, prefixSteps, v, c]
    omega
  rw [htotal]
  simp only [timePolynomial, Polynomial.eval_add, Polynomial.eval_comp,
    Polynomial.eval_mul, Polynomial.eval_ofNat, Polynomial.eval_X, Polynomial.eval_one]
  change _ ≤ MachineCloudPrefix.timePolynomial.eval (2*N) +
    MachineCloudPadding.timePolynomial.eval (2*N) + 40*(N+1)
  omega

def originalInTime (t : GraphTables.Table) (e : Fin t.darts) (data : Data)
    (htable : data.table = GraphTables.tableBits t)
    (hindex : data.globalIndex = encodeWord e.val)
    (howner : data.owner = []) (hrank : data.localRank = []) (hcount : data.count = [])
    (hoffset : data.offset = []) (hpadding : data.padding = []) (hlevel : data.level = [])
    (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      (cfg (some (.owner .seed)) data ambient register)
      (some (cfg none (originalData t e data) ambient none))
      (timePolynomial.eval (GraphTables.tableBits t).length) where
  steps := totalTime t e
  evals_in_steps := originalTrace t e data htable hindex howner hrank hcount hoffset
    hpadding hlevel ambient register
  steps_le_m := totalTime_le t e

def cloudInTime (t : GraphTables.Table) (v : Fin t.vertices) (data : Data)
    (htable : data.table = GraphTables.tableBits t) (howner : data.owner = encodeWord v.val)
    (hcount : data.count = []) (hpadding : data.padding = []) (hlevel : data.level = [])
    (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      (cfg (some (.cloud .init)) data ambient register)
      (some (cfg none (cloudData t v data) ambient none))
      (MachineCloudPadding.timePolynomial.eval (MachineCloudPadding.inputLength t v [])) where
  steps := MachineCloudPadding.totalTime t v
  evals_in_steps := cloudTrace t v data htable howner hcount hpadding hlevel ambient register
  steps_le_m := MachineCloudPadding.totalTime_le t v []

end DFVSGames.Foundations.Complexity.MachineRegularMetadata
end

section

namespace DFVSGames.Foundations.PCP.ExpanderRowControl

open ExpanderTables

abbrev degree (d : Nat) : Nat := d * d
abbrev cloudSize (d : Nat) : Nat := degree d * degree d
abbrev rowFactor (d : Nat) : Nat := cloudSize d * degree d

structure Control (d : Nat) where
  firstH : Fin (cloudSize d) × Fin d
  lastInput : Fin d
  firstG : Fin (degree d)
  secondG : Fin (degree d)
  lastH : Fin (cloudSize d) × Fin d
  deriving DecidableEq, Fintype

def squarePorts {d : Nat} (s : Control d) : Fin (degree d) × Fin (degree d) :=
  (rowIndex (degree d) (degree d)).symm s.firstH.1

def start {d : Nat} (H : Table (cloudSize d) d)
    (cloud : Fin (cloudSize d)) (port : Fin (degree d)) : Control d :=
  let ports := (rowIndex d d).symm port
  let first := lookup H (cloud, ports.1)
  let square := (rowIndex (degree d) (degree d)).symm first.1
  { firstH := first
    lastInput := ports.2
    firstG := square.1
    secondG := square.2
    lastH := first }

def receiveFirst {d : Nat} (s : Control d) (returnedPort : Fin (degree d)) : Control d :=
  { s with firstG := returnedPort }

def receiveSecond {d : Nat} (H : Table (cloudSize d) d)
    (s : Control d) (returnedPort : Fin (degree d)) : Control d :=
  { s with
    secondG := returnedPort
    lastH := lookup H
      (rowIndex (degree d) (degree d) (returnedPort,s.firstG),s.lastInput) }

def firstOffset {d : Nat} (s : Control d) : Fin (degree d) := (squarePorts s).1
def secondOffset {d : Nat} (s : Control d) : Fin (degree d) := (squarePorts s).2

def outputPort {d : Nat} (s : Control d) : Fin (degree d) :=
  rowIndex d d (s.lastH.2,s.firstH.2)

def outputOffset {d : Nat} (s : Control d) : Fin (rowFactor d) :=
  rowIndex (cloudSize d) (degree d) (s.lastH.1,outputPort s)

def firstAddress {d : Nat} (vertex : Nat) (s : Control d) : Nat :=
  degree d * vertex + (firstOffset s).val

def secondAddress {d : Nat} (vertex : Nat) (s : Control d) : Nat :=
  degree d * vertex + (secondOffset s).val

def outputAddress {d : Nat} (vertex : Nat) (s : Control d) : Nat :=
  rowFactor d * vertex + (outputOffset s).val

@[simp] theorem firstOffset_receiveFirst {d : Nat} (s : Control d) (r : Fin (degree d)) :
    firstOffset (receiveFirst s r) = firstOffset s := rfl

@[simp] theorem secondOffset_receiveFirst {d : Nat} (s : Control d) (r : Fin (degree d)) :
    secondOffset (receiveFirst s r) = secondOffset s := rfl

@[simp] theorem receiveFirst_port {d : Nat} (s : Control d) (r : Fin (degree d)) :
    (receiveFirst s r).firstG = r := rfl

@[simp] theorem receiveSecond_port {d : Nat} (H : Table (cloudSize d) d)
    (s : Control d) (r : Fin (degree d)) : (receiveSecond H s r).secondG = r := rfl

theorem firstAddress_eq_rowIndex {v d : Nat} (vertex : Fin v) (s : Control d) :
    firstAddress vertex.val s =
      (rowIndex v (degree d) (vertex,firstOffset s)).val := by
  rw [rowIndex_val]
  exact Nat.add_comm _ _

theorem secondAddress_eq_rowIndex {v d : Nat} (vertex : Fin v) (s : Control d) :
    secondAddress vertex.val s =
      (rowIndex v (degree d) (vertex,secondOffset s)).val := by
  rw [rowIndex_val]
  exact Nat.add_comm _ _

theorem firstAddress_lt {v d : Nat} (vertex : Fin v) (s : Control d) :
    firstAddress vertex.val s < v * degree d := by
  rw [firstAddress_eq_rowIndex]
  exact (rowIndex v (degree d) (vertex,firstOffset s)).isLt

theorem secondAddress_lt {v d : Nat} (vertex : Fin v) (s : Control d) :
    secondAddress vertex.val s < v * degree d := by
  rw [secondAddress_eq_rowIndex]
  exact (rowIndex v (degree d) (vertex,secondOffset s)).isLt

theorem lookup_vertex_div {v q : Nat} (G : Table v q) (x : Fin v × Fin q) :
    (lookup G x).1.val = (reverseIndex G (rowIndex v q x)).val / q := rfl

theorem lookup_port_mod {v q : Nat} (G : Table v q) (x : Fin v × Fin q) :
    (lookup G x).2.val = (reverseIndex G (rowIndex v q x)).val % q := rfl

theorem lookup_port_eq_remainder {v q : Nat} (G : Table v q) (x : Fin v × Fin q)
    (r : Fin q) (hr : r.val = (reverseIndex G (rowIndex v q x)).val % q) :
    (lookup G x).2 = r := Fin.ext ((lookup_port_mod G x).trans hr.symm)

theorem rowIndex_lookup {v q : Nat} (G : Table v q) (x : Fin v × Fin q) :
    rowIndex v q (lookup G x) = reverseIndex G (rowIndex v q x) := by
  simp only [lookup, Equiv.apply_symm_apply]

def outputPair {v d : Nat} (vertex : Fin v) (s : Control d) :
    Fin (v * cloudSize d) × Fin (degree d) :=
  (rowIndex v (cloudSize d) (vertex,s.lastH.1),outputPort s)

theorem outputAddress_eq_rowIndex {v d : Nat} (vertex : Fin v) (s : Control d) :
    outputAddress vertex.val s =
      (rowIndex (v * cloudSize d) (degree d) (outputPair vertex s)).val := by
  simp only [outputAddress, outputOffset, outputPair, rowIndex_val, rowFactor]
  ring

theorem outputAddress_lt {v d : Nat} (vertex : Fin v) (s : Control d) :
    outputAddress vertex.val s < (v * cloudSize d) * degree d := by
  rw [outputAddress_eq_rowIndex]
  exact (rowIndex (v * cloudSize d) (degree d) (outputPair vertex s)).isLt

def evaluateRow {v d : Nat} (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (vertex : Fin v) (cloud : Fin (cloudSize d)) (port : Fin (degree d)) :
    Fin v × Control d :=
  let s₀ := start H cloud port
  let g₁ := lookup G (vertex,firstOffset s₀)
  let s₁ := receiveFirst s₀ g₁.2
  let g₂ := lookup G (g₁.1,secondOffset s₁)
  (g₂.1, receiveSecond H s₁ g₂.2)

theorem evaluateRow_eq_lookup_step {v d : Nat}
    (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (vertex : Fin v) (cloud : Fin (cloudSize d)) (port : Fin (degree d)) :
    outputPair (evaluateRow G H vertex cloud port).1 (evaluateRow G H vertex cloud port).2 =
      lookup (step G H) (rowIndex v (cloudSize d) (vertex,cloud),port) := by
  rw [lookup_step]
  simp only [evaluateRow, start, receiveFirst, receiveSecond, firstOffset, secondOffset,
    squarePorts, outputPair, outputPort, stepLookup, Equiv.symm_apply_apply]

theorem outputAddress_eq_step_row {v d : Nat}
    (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (vertex : Fin v) (cloud : Fin (cloudSize d)) (port : Fin (degree d)) :
    outputAddress (evaluateRow G H vertex cloud port).1.val (evaluateRow G H vertex cloud port).2 =
      (rowIndex (v * cloudSize d) (degree d)
        (lookup (step G H) (rowIndex v (cloudSize d) (vertex,cloud),port))).val := by
  rw [outputAddress_eq_rowIndex, evaluateRow_eq_lookup_step]

theorem outputAddress_eq_step_reverseIndex {v d : Nat}
    (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (vertex : Fin v) (cloud : Fin (cloudSize d)) (port : Fin (degree d)) :
    outputAddress (evaluateRow G H vertex cloud port).1.val (evaluateRow G H vertex cloud port).2 =
      (reverseIndex (step G H) (rowIndex (v * cloudSize d) (degree d)
        (rowIndex v (cloudSize d) (vertex,cloud),port))).val := by
  rw [outputAddress_eq_step_row, rowIndex_lookup]

end DFVSGames.Foundations.PCP.ExpanderRowControl
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