lean/OAI/Computability/DirectedFeedback/Machines/MachineCloudPadding.lean

import Mathlib
import OAI.Computability.DirectedFeedback.Encoding.PreprocessingRegularTables
import OAI.Computability.DirectedFeedback.Machines.MachineCloudCount

namespace OAI

namespace DFVSGames.Foundations.Complexity.MachineCloudPadding

open Turing
open MachineComposition
open DFVSGames.Foundations.PCP

namespace Placement

variable {K K' Λ Λ' σ : Type}

def tapes (view : K' → Option K) (source : K → List Bool)
    (extra : K' → List Bool) : K' → List Bool :=
  fun j => match view j with
    | some k => source k
    | none => extra j

def label (labels : Λ → Λ') (exit : Option Λ') : Option Λ → Option Λ'
  | some l => some (labels l)
  | none => exit

def configuration (view : K' → Option K) (labels : Λ → Λ') (exit : Option Λ')
    (extra : K' → List Bool) (c : TM2.Cfg (fun _ : K => Bool) Λ σ) :
    TM2.Cfg (fun _ : K' => Bool) Λ' σ :=
  ⟨label labels exit c.l, c.var, tapes view c.stk extra⟩

def statement (tape : K → K') (labels : Λ → Λ') (exit : Option Λ') :
    TM2.Stmt (fun _ : K => Bool) Λ σ → TM2.Stmt (fun _ : K' => Bool) Λ' σ
  | .push k f next => .push (tape k) f (statement tape labels exit next)
  | .peek k f next => .peek (tape k) f (statement tape labels exit next)
  | .pop k f next => .pop (tape k) f (statement tape labels exit next)
  | .load f next => .load f (statement tape labels exit next)
  | .branch f yes no => .branch f (statement tape labels exit yes) (statement tape labels exit no)
  | .goto f => .goto (fun state => labels (f state))
  | .halt => match exit with
    | some l => .goto (fun _ => l)
    | none => .halt

variable [DecidableEq K] [DecidableEq K']

omit [DecidableEq K] [DecidableEq K'] in
theorem tapes_apply (tape : K → K') (view : K' → Option K)
    (left : ∀ k, view (tape k) = some k)
    (source : K → List Bool) (extra : K' → List Bool) (k : K) :
    tapes view source extra (tape k) = source k := by
  simp only [tapes, left]

theorem tapes_update (tape : K → K') (view : K' → Option K)
    (left : ∀ k, view (tape k) = some k)
    (right : ∀ j k, view j = some k → tape k = j)
    (source : K → List Bool) (extra : K' → List Bool) (k : K) (w : List Bool) :
    tapes view (Function.update source k w) extra =
      Function.update (tapes view source extra) (tape k) w := by
  funext j
  by_cases h : j = tape k
  · subst j
    simp [tapes, left]
  · cases hv : view j with
    | none => simp [tapes, hv, h]
    | some i =>
      have hi : i ≠ k := by
        intro hi
        subst i
        exact h (right j k hv).symm
      simp [tapes, hv, h, hi]

theorem stepAux_simulation (tape : K → K') (view : K' → Option K)
    (left : ∀ k, view (tape k) = some k)
    (right : ∀ j k, view j = some k → tape k = j)
    (labels : Λ → Λ') (exit : Option Λ') (extra : K' → List Bool)
    (q : TM2.Stmt (fun _ : K => Bool) Λ σ) (state : σ) (source : K → List Bool) :
    TM2.stepAux (statement tape labels exit q) state (tapes view source extra) =
      configuration view labels exit extra (TM2.stepAux q state source) := by
  induction q generalizing state source with
  | push k f next ih =>
    simp only [statement, TM2.stepAux, tapes_apply tape view left]
    rw [← tapes_update tape view left right]
    exact ih state (Function.update source k (f state :: source k))
  | peek k f next ih =>
    simpa only [statement, TM2.stepAux, tapes_apply tape view left] using
      ih (f state (source k).head?) source
  | pop k f next ih =>
    simp only [statement, TM2.stepAux, tapes_apply tape view left]
    rw [← tapes_update tape view left right]
    exact ih (f state (source k).head?) (Function.update source k (source k).tail)
  | load f next ih => simpa only [statement, TM2.stepAux] using ih (f state) source
  | branch f yes no ihYes ihNo =>
    cases h : f state with
    | false => simpa only [statement, TM2.stepAux, h, Bool.cond_false] using ihNo state source
    | true => simpa only [statement, TM2.stepAux, h, Bool.cond_true] using ihYes state source
  | goto f => rfl
  | halt => cases exit <;> rfl

theorem step_simulation (tape : K → K') (view : K' → Option K)
    (left : ∀ k, view (tape k) = some k)
    (right : ∀ j k, view j = some k → tape k = j)
    (labels : Λ → Λ') (exit : Option Λ') (extra : K' → List Bool)
    (source : Λ → TM2.Stmt (fun _ : K => Bool) Λ σ)
    (target : Λ' → TM2.Stmt (fun _ : K' => Bool) Λ' σ)
    (atLabels : ∀ l, target (labels l) = statement tape labels exit (source l))
    (a b : TM2.Cfg (fun _ : K => Bool) Λ σ) (h : TM2.step source a = some b) :
    TM2.step target (configuration view labels exit extra a) =
      some (configuration view labels exit extra b) := by
  cases a with
  | mk l state sourceTapes =>
    cases l with
    | none => simp [TM2.step] at h
    | some l =>
      have hb : TM2.stepAux (source l) state sourceTapes = b := Option.some.inj h
      rw [← hb]
      change some (TM2.stepAux (target (labels l)) state (tapes view sourceTapes extra)) = _
      rw [atLabels, stepAux_simulation tape view left right]

theorem trace (tape : K → K') (view : K' → Option K)
    (left : ∀ k, view (tape k) = some k)
    (right : ∀ j k, view j = some k → tape k = j)
    (labels : Λ → Λ') (exit : Option Λ') (extra : K' → List Bool)
    (source : Λ → TM2.Stmt (fun _ : K => Bool) Λ σ)
    (target : Λ' → TM2.Stmt (fun _ : K' => Bool) Λ' σ)
    (atLabels : ∀ l, target (labels l) = statement tape labels exit (source l))
    (n : Nat) (a b : TM2.Cfg (fun _ : K => Bool) Λ σ)
    (run : (advance (TM2.step source))^[n] (some a) = some b) :
    (advance (TM2.step target))^[n]
      (some (configuration view labels exit extra a)) =
      some (configuration view labels exit extra b) :=
  liftSuccessfulTrace _ _ _
    (step_simulation tape view left right labels exit extra source target atLabels) n a b run

end Placement

inductive Tape
  | table | query | count | work | scratch | spare | power | level | fuel | product
  deriving DecidableEq

instance : Fintype Tape := derive_fintype% Tape

inductive Label
  | init | countCode (l : MachineCloudCount.Label)
  | branch | powerCode (l : MachineCeilingPower.Label)
  | subtract | restore
  deriving DecidableEq, Fintype

abbrev Alphabet (_ : Tape) := Bool
abbrev State (σ : Type) := (σ × Bool) × Option Bool

def memory (table query count work scratch spare power level fuel product : List Bool) :
    Tape → List Bool
  | .table => table
  | .query => query
  | .count => count
  | .work => work
  | .scratch => scratch
  | .spare => spare
  | .power => power
  | .level => level
  | .fuel => fuel
  | .product => product

@[simp] theorem update_count (a b c d e f g h i j x : List Bool) :
    Function.update (memory a b c d e f g h i j) .count x = memory a b x d e f g h i j := by
  funext k; cases k <;> rfl
@[simp] theorem update_scratch (a b c d e f g h i j x : List Bool) :
    Function.update (memory a b c d e f g h i j) .scratch x = memory a b c d x f g h i j := by
  funext k; cases k <;> rfl
@[simp] theorem update_power (a b c d e f g h i j x : List Bool) :
    Function.update (memory a b c d e f g h i j) .power x = memory a b c d e f x h i j := by
  funext k; cases k <;> rfl

@[simp] theorem update_level (a b c d e f g h i j x : List Bool) :
    Function.update (memory a b c d e f g h i j) .level x = memory a b c d e f g x i j := by
  funext k; cases k <;> rfl

def countTape : MachineCloudCount.Tape → Tape
  | .original => .table
  | .target => .query
  | .count => .count
  | .work => .work
  | .scratch => .scratch
  | .spare => .spare

def countView : Tape → Option MachineCloudCount.Tape
  | .table => some .original
  | .query => some .target
  | .count => some .count
  | .work => some .work
  | .scratch => some .scratch
  | .spare => some .spare
  | _ => none

theorem countView_left (k : MachineCloudCount.Tape) : countView (countTape k) = some k := by
  cases k <;> rfl
theorem countView_right (j : Tape) (k : MachineCloudCount.Tape)
    (h : countView j = some k) : countTape k = j := by
  cases j <;> cases k <;> simp_all [countView, countTape]

def powerTape : MachineCeilingPower.Tape → Tape
  | .input => .count
  | .work => .work
  | .power => .power
  | .saved => .scratch
  | .level => .level
  | .fuel => .fuel
  | .spare => .spare
  | .product => .product

def powerView : Tape → Option MachineCeilingPower.Tape
  | .count => some .input
  | .work => some .work
  | .power => some .power
  | .scratch => some .saved
  | .level => some .level
  | .fuel => some .fuel
  | .spare => some .spare
  | .product => some .product
  | _ => none

theorem powerView_left (k : MachineCeilingPower.Tape) : powerView (powerTape k) = some k := by
  cases k <;> rfl
theorem powerView_right (j : Tape) (k : MachineCeilingPower.Tape)
    (h : powerView j = some k) : powerTape k = j := by
  cases j <;> cases k <;> simp_all [powerView, powerTape]

@[simp] theorem countMemory (a b c d e f g h i j : List Bool) :
    Placement.tapes countView (MachineCloudCount.memory a d b e c f)
      (memory a b c d e f g h i j) = memory a b c d e f g h i j := by
  funext k; cases k <;> rfl

@[simp] theorem powerMemory (a b c d e f g h i j : List Bool) :
    Placement.tapes powerView (MachineCeilingPower.memory c d g e h i f j)
      (memory a b c d e f g h i j) = memory a b c d e f g h i j := by
  funext k; cases k <;> rfl

theorem countTapes (a b c d e f : List Bool) (extra : Tape → List Bool) :
    Placement.tapes countView (MachineCloudCount.memory a d b e c f) extra =
      memory a b c d e f (extra .power) (extra .level) (extra .fuel) (extra .product) := by
  funext k; cases k <;> rfl

theorem powerTapes (c d g e h i f j : List Bool) (extra : Tape → List Bool) :
    Placement.tapes powerView (MachineCeilingPower.memory c d g e h i f j) extra =
      memory (extra .table) (extra .query) c d e f g h i j := by
  funext k; cases k <;> rfl

variable {σ : Type}

def subtractLoop : TM2.Stmt Alphabet Label (State σ) :=
  .pop .count (fun state head => (state.1, head))
    (.branch (fun state => state.2.getD false)
      (.pop .power (fun state _ => (state.1, none))
        (.push .scratch (fun _ => true) (.goto fun _ => .subtract)))
      (.push .count (fun _ => false)
        (.load (fun state => (state.1, none)) (.goto fun _ => .restore))))

def program : Label → TM2.Stmt Alphabet Label (State σ)
  | .init => .push .count (fun _ => false) (.goto fun _ => .countCode .copyFirst)
  | .countCode l => Placement.statement countTape Label.countCode (some .branch)
      (MachineCloudCount.program l)
  | .branch => .peek .count (fun state head => (state.1, head))
      (.branch (fun state => state.2.getD false)
        (.load (fun state => (state.1, none)) (.goto fun _ => .powerCode .init))
        (.push .power (fun _ => false) (.push .level (fun _ => false)
          (.load (fun state => (state.1, none)) .halt))))
  | .powerCode l => Placement.statement powerTape Label.powerCode (some .subtract)
      (MachineCeilingPower.program ExpanderFamily.growth l)
  | .subtract => subtractLoop
  | .restore => Reduction.MachineTransfer.loopAt
      .scratch .count id false .restore none

private theorem appendTrace_inline_MachineCloudPadding {α : Type} (f : α → α) {a b : Nat} {x y z : α}
    (hs : f^[a] x = y) (ht : f^[b] y = z) : f^[a + b] x = z := by
  rw [Nat.add_comm a b, Function.iterate_add_apply, hs, ht]

theorem countTrace (t : GraphTables.Table) (v : Fin t.vertices)
    (suffix : List Bool) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[MachineCloudCount.totalSteps t.vertices t.darts v.val
      (MachineCloudCount.tableRows t)]
      (some ⟨some (.countCode .copyFirst), ((ambient, false), register),
        memory (GraphTables.tableBits t) (encodeWord v.val ++ suffix)
          (encodeWord 0) [] [] [] [] [] [] []⟩) =
      some ⟨some .branch, ((ambient, false), none),
        memory (GraphTables.tableBits t) (encodeWord v.val ++ suffix)
          (encodeWord (PreprocessingCloudIndex.cloudSize t v)) [] [] [] [] [] [] []⟩ := by
  have h := Placement.trace countTape countView countView_left countView_right
    Label.countCode (some Label.branch) (fun _ => [])
    (MachineCloudCount.program (σ := σ)) program (fun _ => rfl) _ _ _
    (MachineCloudCount.cloudCountTrace t v suffix [] ambient register)
  simpa only [Placement.configuration, Placement.label, countTapes, List.append_nil] using h

theorem powerTrace (table query : List Bool) (k : Nat) (ambient : σ)
    (register : Option Bool) :
    (advance (TM2.step program))^[MachineCeilingPower.totalTime ExpanderFamily.growth k]
      (some ⟨some (.powerCode .init), ((ambient, false), register),
        memory table query (encodeWord k) [] [] [] [] [] [] []⟩) =
      some ⟨some .subtract, ((ambient, false), none),
        memory table query (encodeWord k) [] [] []
          (encodeWord (PreprocessingLevels.paddedSize k))
          (encodeWord (PreprocessingLevels.boundedLevel k)) [] []⟩ := by
  have h := Placement.trace powerTape powerView powerView_left powerView_right
    Label.powerCode (some Label.subtract) (memory table query [] [] [] [] [] [] [] [])
    (MachineCeilingPower.program ExpanderFamily.growth (σ := σ)) program (fun _ => rfl)
    _ _ _ (MachineCeilingPower.paddingTrace k ambient register)
  simpa only [Placement.configuration, Placement.label, powerTapes, memory] using h

theorem subtractStep_zero (table query level saved : List Bool) (p : Nat)
    (ambient : σ) (register : Option Bool) :
    TM2.step program
      ⟨some .subtract, ((ambient, false), register),
        memory table query (encodeWord 0) [] saved [] (encodeWord p) level [] []⟩ =
      some ⟨some .restore, ((ambient, false), none),
        memory table query (encodeWord 0) [] saved [] (encodeWord p) level [] []⟩ := by
  change some (TM2.stepAux subtractLoop _ _) = _
  simp [subtractLoop, TM2.stepAux, memory, encodeWord]

theorem subtractStep_succ (table query level saved : List Bool) (k p : Nat)
    (ambient : σ) (register : Option Bool) :
    TM2.step program
      ⟨some .subtract, ((ambient, false), register),
        memory table query (encodeWord (k + 1)) [] saved [] (encodeWord (p + 1)) level [] []⟩ =
      some ⟨some .subtract, ((ambient, false), none),
        memory table query (encodeWord k) [] (true :: saved) [] (encodeWord p) level [] []⟩ := by
  change some (TM2.stepAux subtractLoop _ _) = _
  simp [subtractLoop, TM2.stepAux, memory, encodeWord, List.replicate_succ]

theorem subtractScanTrace (table query level : List Bool) (k p : Nat)
    (saved : List Bool) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[k + 1]
      (some ⟨some .subtract, ((ambient, false), register),
        memory table query (encodeWord k) [] saved [] (encodeWord (k + p)) level [] []⟩) =
      some ⟨some .restore, ((ambient, false), none),
        memory table query (encodeWord 0) [] (List.replicate k true ++ saved) []
          (encodeWord p) level [] []⟩ := by
  induction k generalizing saved register with
  | zero => simpa only [Nat.zero_add, Function.iterate_one, advance_some,
      List.replicate_zero, List.nil_append] using
      subtractStep_zero table query level saved p ambient register
  | succ k ih =>
    rw [Function.iterate_succ_apply]
    simp only [Nat.succ_add]
    change (advance (TM2.step program))^[k + 1]
      (TM2.step program ⟨some .subtract, ((ambient, false), register),
        memory table query (encodeWord (k + 1)) [] saved []
          (encodeWord ((k + p) + 1)) level [] []⟩) = _
    rw [subtractStep_succ]
    simpa only [List.replicate_succ', List.append_assoc, List.singleton_append] using
      ih (true :: saved) none

theorem subtractTrace (table query level : List Bool) (k p : Nat) (hkp : k ≤ p)
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[2 * k + 2]
      (some ⟨some .subtract, ((ambient, false), register),
        memory table query (encodeWord k) [] [] [] (encodeWord p) level [] []⟩) =
      some ⟨none, ((ambient, false), none),
        memory table query (encodeWord k) [] [] [] (encodeWord (p - k)) level [] []⟩ := by
  have hs := subtractScanTrace table query level k (p - k) [] ambient register
  rw [show k + (p - k) = p by omega] at hs
  simp only [List.append_nil] at hs
  let mid := memory table query (encodeWord 0) [] (List.replicate k true) []
    (encodeWord (p - k)) level [] []
  have hr := Reduction.MachineTransfer.transferAt_fromTapes Tape.scratch Tape.count (by decide)
    id false Label.restore none program rfl mid (ambient, false) none
  simp only [mid, memory, List.length_replicate, List.reverse_replicate, List.map_id,
    Reduction.MachineTransfer.tapesAt, update_scratch, update_count] at hr
  rw [MachineCeilingPower.replicate_encodeWord, Nat.add_zero] at hr
  have h := appendTrace_inline_MachineCloudPadding _ hs hr
  simpa only [show (k + 1) + (k + 1) = 2 * k + 2 by omega, Nat.add_zero] using h

theorem branchStep_zero (table query : List Bool) (ambient : σ) (register : Option Bool) :
    TM2.step program
      ⟨some .branch, ((ambient, false), register),
        memory table query (encodeWord 0) [] [] [] [] [] [] []⟩ =
      some ⟨none, ((ambient, false), none),
        memory table query (encodeWord 0) [] [] [] (encodeWord 0) (encodeWord 0) [] []⟩ := by
  simp [TM2.step, program, TM2.stepAux, memory, encodeWord]

theorem branchStep_succ (table query : List Bool) (k : Nat) (ambient : σ)
    (register : Option Bool) :
    TM2.step program
      ⟨some .branch, ((ambient, false), register),
        memory table query (encodeWord (k + 1)) [] [] [] [] [] [] []⟩ =
      some ⟨some (.powerCode .init), ((ambient, false), none),
        memory table query (encodeWord (k + 1)) [] [] [] [] [] [] []⟩ := by
  simp [TM2.step, program, TM2.stepAux, memory, encodeWord, List.replicate_succ]

def padding (k : Nat) : Nat := PreprocessingLevels.cloudPaddedSize k - k

def postTime (k : Nat) : Nat :=
  1 + if k = 0 then 0 else MachineCeilingPower.totalTime ExpanderFamily.growth k + (2 * k + 2)

theorem postTrace (table query : List Bool) (k : Nat) (ambient : σ)
    (register : Option Bool) :
    (advance (TM2.step program))^[postTime k]
      (some ⟨some .branch, ((ambient, false), register),
        memory table query (encodeWord k) [] [] [] [] [] [] []⟩) =
      some ⟨none, ((ambient, false), none),
        memory table query (encodeWord k) [] [] [] (encodeWord (padding k))
          (encodeWord (PreprocessingLevels.boundedLevel k)) [] []⟩ := by
  cases k with
  | zero =>
    simpa only [postTime, ite_eq_left rfl, ite_true, Nat.add_zero, Function.iterate_one, advance_some,
      padding, PreprocessingLevels.cloudPaddedSize_zero, Nat.sub_zero,
      PreprocessingLevels.boundedLevel_zero] using branchStep_zero table query ambient register
  | succ k =>
    have hb : (advance (TM2.step program))^[1]
        (some ⟨some .branch, ((ambient, false), register),
          memory table query (encodeWord (k + 1)) [] [] [] [] [] [] []⟩) =
        some ⟨some (.powerCode .init), ((ambient, false), none),
          memory table query (encodeWord (k + 1)) [] [] [] [] [] [] []⟩ :=
      branchStep_succ table query k ambient register
    have hp := powerTrace table query (k + 1) ambient none
    have hs := subtractTrace table query (encodeWord (PreprocessingLevels.boundedLevel (k + 1)))
      (k + 1) (PreprocessingLevels.paddedSize (k + 1))
      (PreprocessingLevels.le_paddedSize (k + 1)) ambient none
    have h := appendTrace_inline_MachineCloudPadding _ hb (appendTrace_inline_MachineCloudPadding _ hp hs)
    simpa only [postTime, Nat.succ_ne_zero, ite_false, padding,
      PreprocessingLevels.cloudPaddedSize_of_pos (Nat.succ_pos k)] using h

theorem initStep (table query : List Bool) (ambient : σ) (register : Option Bool) :
    TM2.step program
      ⟨some .init, ((ambient, false), register), memory table query [] [] [] [] [] [] [] []⟩ =
      some ⟨some (.countCode .copyFirst), ((ambient, false), register),
        memory table query (encodeWord 0) [] [] [] [] [] [] []⟩ := by
  simp [TM2.step, program, TM2.stepAux, memory, encodeWord]

def totalTime (t : GraphTables.Table) (v : Fin t.vertices) : Nat :=
  (1 + MachineCloudCount.totalSteps t.vertices t.darts v.val (MachineCloudCount.tableRows t)) +
    postTime (PreprocessingCloudIndex.cloudSize t v)

theorem cloudPaddingTrace (t : GraphTables.Table) (v : Fin t.vertices)
    (suffix : List Bool) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[totalTime t v]
      (some ⟨some .init, ((ambient, false), register),
        memory (GraphTables.tableBits t) (encodeWord v.val ++ suffix) [] [] [] [] [] [] [] []⟩) =
      some ⟨none, ((ambient, false), none),
        memory (GraphTables.tableBits t) (encodeWord v.val ++ suffix)
          (encodeWord (PreprocessingCloudIndex.cloudSize t v)) [] [] []
          (encodeWord (padding (PreprocessingCloudIndex.cloudSize t v)))
          (encodeWord (PreprocessingLevels.boundedLevel (PreprocessingCloudIndex.cloudSize t v)))
          [] []⟩ := by
  have hi : (advance (TM2.step program))^[1]
      (some ⟨some .init, ((ambient, false), register),
        memory (GraphTables.tableBits t) (encodeWord v.val ++ suffix) [] [] [] [] [] [] [] []⟩) =
      some ⟨some (.countCode .copyFirst), ((ambient, false), register),
        memory (GraphTables.tableBits t) (encodeWord v.val ++ suffix)
          (encodeWord 0) [] [] [] [] [] [] []⟩ :=
    initStep _ _ ambient register
  exact appendTrace_inline_MachineCloudPadding _ (appendTrace_inline_MachineCloudPadding _ hi (countTrace t v suffix ambient register))
    (postTrace _ _ (PreprocessingCloudIndex.cloudSize t v) ambient none)

def inputLength (t : GraphTables.Table) (v : Fin t.vertices) (suffix : List Bool) : Nat :=
  (GraphTables.tableBits t).length + (encodeWord v.val ++ suffix).length

theorem count_length_le (t : GraphTables.Table) (v : Fin t.vertices) (suffix : List Bool) :
    PreprocessingCloudIndex.cloudSize t v + 1 ≤ inputLength t v suffix := by
  have h := (PreprocessingCloudIndex.cloudSize_le_darts t v).trans
    (GraphTables.darts_le_tableBits_length t)
  simp only [inputLength, List.length_append, encodeWord_length]
  omega

theorem table_length_le (t : GraphTables.Table) (v : Fin t.vertices) (suffix : List Bool) :
    (GraphTables.tableBits t).length ≤ inputLength t v suffix := by
  unfold inputLength
  omega

theorem postTime_le (k : Nat) :
    postTime k ≤ 1 + MachineCeilingPower.timeBound ExpanderFamily.growth k + (2 * k + 2) := by
  have h := MachineCeilingPower.totalTime_le ExpanderFamily.growth k
  unfold postTime
  split_ifs <;> omega

noncomputable def timePolynomial : Polynomial Nat :=
  Polynomial.C (ExpanderFamily.growth + 8) * Polynomial.X ^ 2 + 25 * Polynomial.X + 10

theorem totalTime_le (t : GraphTables.Table) (v : Fin t.vertices) (suffix : List Bool) :
    totalTime t v ≤ timePolynomial.eval (inputLength t v suffix) := by
  have hc := MachineCloudCount.totalSteps_le t.vertices t.darts v.val (MachineCloudCount.tableRows t)
  rw [MachineCloudCount.tableRows_input] at hc
  have hp := postTime_le (PreprocessingCloudIndex.cloudSize t v)
  have hk := count_length_le t v suffix
  have ht := table_length_le t v suffix
  have hsq := Nat.mul_le_mul_left (ExpanderFamily.growth + 8)
    (Nat.pow_le_pow_left hk 2)
  simp only [timePolynomial, Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C,
    Polynomial.eval_pow, Polynomial.eval_X, Polynomial.eval_ofNat]
  unfold MachineCeilingPower.timeBound at hp
  unfold totalTime
  omega

def cloudPaddingInTime (t : GraphTables.Table) (v : Fin t.vertices)
    (suffix : List Bool) (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some .init, ((ambient, false), register),
        memory (GraphTables.tableBits t) (encodeWord v.val ++ suffix) [] [] [] [] [] [] [] []⟩
      (some ⟨none, ((ambient, false), none),
        memory (GraphTables.tableBits t) (encodeWord v.val ++ suffix)
          (encodeWord (PreprocessingCloudIndex.cloudSize t v)) [] [] []
          (encodeWord (padding (PreprocessingCloudIndex.cloudSize t v)))
          (encodeWord (PreprocessingLevels.boundedLevel (PreprocessingCloudIndex.cloudSize t v)))
          [] []⟩)
      (timePolynomial.eval (inputLength t v suffix)) where
  steps := totalTime t v
  evals_in_steps := cloudPaddingTrace t v suffix ambient register
  steps_le_m := totalTime_le t v suffix

theorem count_add_padding (k : Nat) : k + padding k = PreprocessingLevels.cloudPaddedSize k := by
  have h := (PreprocessingLevels.cloudPaddedSize_bounds k).1
  unfold padding
  omega

end DFVSGames.Foundations.Complexity.MachineCloudPadding

namespace DFVSGames.Foundations.PCP.PreprocessingPaddingOffsets

open PreprocessingRegularTables
open scoped BigOperators

private theorem idxOf_map_injective_inline_PreprocessingPaddingOffsets {A B : Type*} [DecidableEq A] [DecidableEq B]
    (f : A → B) (hf : Function.Injective f) (xs : List A) (a : A) :
    (xs.map f).idxOf (f a) = xs.idxOf a := by
  induction xs with
  | nil => rfl
  | cons x xs ih =>
    by_cases h : x = a
    · subst x; simp
    · simp [List.idxOf_cons_ne, h, hf.ne h, ih]

theorem sigma_idxOf {A : Type*} [DecidableEq A] (sizes : A → Nat)
    (xs : List A) (a : A) (ha : a ∈ xs) (j : Fin (sizes a)) :
    (xs.sigma (fun x => List.finRange (sizes x))).idxOf ⟨a, j⟩ =
      ((xs.take (xs.idxOf a)).map sizes).sum + j.val := by
  induction xs with
  | nil => simp at ha
  | cons x xs ih =>
    rw [List.sigma_cons]
    by_cases h : x = a
    · subst x
      have hm : (⟨a, j⟩ : Σ x, Fin (sizes x)) ∈
          (List.finRange (sizes a)).map (Sigma.mk a) := by
        exact List.mem_map.mpr ⟨j, List.mem_finRange j, rfl⟩
      rw [List.idxOf_append_of_mem hm,
        idxOf_map_injective_inline_PreprocessingPaddingOffsets (Sigma.mk a) (fun x y h => by cases h; rfl),
        List.idxOf_finRange]
      simp
    · have hm : (⟨a, j⟩ : Σ x, Fin (sizes x)) ∉
          (List.finRange (sizes x)).map (Sigma.mk x) := by
        intro hm
        rcases List.mem_map.mp hm with ⟨k, _, hk⟩
        exact h (congrArg Sigma.fst hk)
      rw [List.idxOf_append_of_notMem hm, List.length_map, List.length_finRange,
        ih (List.mem_of_ne_of_mem (Ne.symm h) ha), List.idxOf_cons_ne xs h]
      simp only [List.take_succ_cons, List.map_cons, List.sum_cons]
      omega

def offset {n : Nat} (padding : Fin n → Nat) (k : Nat) : Nat :=
  (((List.finRange n).take k).map padding).sum

theorem sigma_length {A : Type*} (sizes : A → Nat) (xs : List A) :
    (xs.sigma (fun x => List.finRange (sizes x))).length = (xs.map sizes).sum := by
  induction xs with
  | nil => rfl
  | cons x xs ih => simp [List.sigma_cons, ih]

theorem paddingOrder_val {n : Nat} (padding : Fin n → Nat)
    (v : Fin n) (j : Fin (padding v)) :
    (PreprocessingRegularTables.paddingOrder padding ⟨v, j⟩).val =
      offset padding v.val + j.val := by
  change (paddingList padding).idxOf ⟨v, j⟩ = _
  have h := sigma_idxOf padding (List.finRange n) v (List.mem_finRange v) j
  simpa only [List.idxOf_finRange, paddingList, offset] using h

theorem vertexOrder_dummy_val (t : GraphTables.Table)
    (padding : Fin t.vertices → Nat) (v : Fin t.vertices) (j : Fin (padding v)) :
    (vertexOrder t padding (Sum.inr ⟨v, j⟩)).val =
      t.darts + offset padding v.val + j.val := by
  rw [vertexOrder_dummy, paddingOrder_val]
  omega

@[simp] theorem offset_zero {n : Nat} (padding : Fin n → Nat) : offset padding 0 = 0 := by
  simp [offset]

theorem offset_all {n : Nat} (padding : Fin n → Nat) :
    offset padding n = ∑ v, padding v := by
  have ht : (List.finRange n).take n = List.finRange n := by simp
  unfold offset
  rw [ht, ← sigma_length padding (List.finRange n)]
  exact paddingList_length padding

end DFVSGames.Foundations.PCP.PreprocessingPaddingOffsets

namespace DFVSGames.Foundations.Complexity.MachineCloudPrefix

open Turing MachineComposition
open DFVSGames.Foundations.PCP
open scoped BigOperators

inductive Tape
  | inner (k : MachineCloudPadding.Tape)
  | bound | remaining | total
  deriving DecidableEq, Fintype

inductive Label
  | init | copyFirst | copySecond | guard | done
  | padding (l : MachineCloudPadding.Label)
  | add | clearCount | clearLevel | increment
  deriving DecidableEq, Fintype

abbrev Alphabet (_ : Tape) := Bool
abbrev State (σ : Type) := (σ × Bool) × Option Bool

def memory (inner : MachineCloudPadding.Tape → List Bool) (bound remaining total : List Bool) :
    Tape → List Bool
  | .inner k => inner k
  | .bound => bound
  | .remaining => remaining
  | .total => total

def frame (table query count power level bound remaining total : List Bool) : Tape → List Bool :=
  memory (MachineCloudPadding.memory table query count [] [] [] power level [] []) bound remaining total

@[simp] theorem frame_query (a b c d e f g h : List Bool) :
    frame a b c d e f g h (.inner .query) = b := rfl
@[simp] theorem frame_count (a b c d e f g h : List Bool) :
    frame a b c d e f g h (.inner .count) = c := rfl
@[simp] theorem frame_power (a b c d e f g h : List Bool) :
    frame a b c d e f g h (.inner .power) = d := rfl
@[simp] theorem frame_level (a b c d e f g h : List Bool) :
    frame a b c d e f g h (.inner .level) = e := rfl
@[simp] theorem frame_bound (a b c d e f g h : List Bool) :
    frame a b c d e f g h .bound = f := rfl
@[simp] theorem frame_remaining (a b c d e f g h : List Bool) :
    frame a b c d e f g h .remaining = g := rfl
@[simp] theorem frame_total (a b c d e f g h : List Bool) :
    frame a b c d e f g h .total = h := rfl

@[simp] theorem update_frame_query (a b c d e f g h x : List Bool) :
    Function.update (frame a b c d e f g h) (.inner .query) x =
      frame a x c d e f g h := by
  funext k; cases k with
  | inner k => cases k <;> rfl
  | bound => rfl
  | remaining => rfl
  | total => rfl
@[simp] theorem update_frame_count (a b c d e f g h x : List Bool) :
    Function.update (frame a b c d e f g h) (.inner .count) x =
      frame a b x d e f g h := by
  funext k; cases k with
  | inner k => cases k <;> rfl
  | bound => rfl
  | remaining => rfl
  | total => rfl
@[simp] theorem update_frame_power (a b c d e f g h x : List Bool) :
    Function.update (frame a b c d e f g h) (.inner .power) x =
      frame a b c x e f g h := by
  funext k; cases k with
  | inner k => cases k <;> rfl
  | bound => rfl
  | remaining => rfl
  | total => rfl
@[simp] theorem update_frame_level (a b c d e f g h x : List Bool) :
    Function.update (frame a b c d e f g h) (.inner .level) x =
      frame a b c d x f g h := by
  funext k; cases k with
  | inner k => cases k <;> rfl
  | bound => rfl
  | remaining => rfl
  | total => rfl
@[simp] theorem update_frame_remaining (a b c d e f g h x : List Bool) :
    Function.update (frame a b c d e f g h) .remaining x =
      frame a b c d e f x h := by
  funext k; cases k <;> rfl
@[simp] theorem update_frame_total (a b c d e f g h x : List Bool) :
    Function.update (frame a b c d e f g h) .total x =
      frame a b c d e f g x := by
  funext k; cases k <;> rfl

def paddingView : Tape → Option MachineCloudPadding.Tape
  | .inner k => some k
  | _ => none

theorem paddingView_left (k : MachineCloudPadding.Tape) : paddingView (.inner k) = some k := rfl

theorem paddingView_right (j : Tape) (k : MachineCloudPadding.Tape)
    (h : paddingView j = some k) : Tape.inner k = j := by
  cases j <;> simp_all [paddingView]

theorem placedTapes (inner : MachineCloudPadding.Tape → List Bool) (extra : Tape → List Bool) :
    MachineCloudPadding.Placement.tapes paddingView inner extra =
      memory inner (extra .bound) (extra .remaining) (extra .total) := by
  funext j; cases j <;> rfl

variable {σ : Type}

def program : Label → TM2.Stmt Alphabet Label (State σ)
  | .init => .push (.inner .query) (fun _ => false)
      (.push .total (fun _ => false) (.goto fun _ => .copyFirst))
  | .copyFirst => Reduction.MachineTransfer.loopAt
      .bound (.inner .scratch) id false .copyFirst (some .copySecond)
  | .copySecond => MachineCopy.forkLoop
      (.inner .scratch) .bound .remaining false .copySecond (some .guard)
  | .guard => MachineUnaryCounter.guard .remaining (.padding .init) .done
  | .done => .halt
  | .padding l => MachineCloudPadding.Placement.statement Tape.inner Label.padding (some .add) (MachineCloudPadding.program l)
  | .add => MachineUnaryAddAt.loop (.inner .power) .total .add (some .clearCount)
  | .clearCount => MachineLookup.discard (.inner .count) .clearCount .clearLevel
  | .clearLevel => MachineLookup.discard (.inner .level) .clearLevel .increment
  | .increment => .pop (.inner .power) (fun state _ => (state.1, none))
      (.push (.inner .query) (fun _ => true) (.goto fun _ => .guard))

private theorem appendTrace_inline_MachineCloudPrefix {α : Type} (f : α → α) {a b : Nat} {x y z : α}
    (hs : f^[a] x = y) (ht : f^[b] y = z) : f^[a + b] x = z := by
  rw [Nat.add_comm a b, Function.iterate_add_apply, hs, ht]

theorem paddingTrace (t : GraphTables.Table) (v : Fin t.vertices)
    (bound remaining total : List Bool) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[MachineCloudPadding.totalTime t v]
      (some ⟨some (.padding .init), ((ambient, false), register),
        frame (GraphTables.tableBits t) (encodeWord v.val) [] [] [] bound remaining total⟩) =
      some ⟨some .add, ((ambient, false), none),
        frame (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)))
          bound remaining total⟩ := by
  have h := MachineCloudPadding.Placement.trace Tape.inner paddingView paddingView_left paddingView_right
    Label.padding (some Label.add) (frame [] [] [] [] [] bound remaining total)
    (MachineCloudPadding.program (σ := σ)) program (fun _ => rfl) _ _ _
    (MachineCloudPadding.cloudPaddingTrace t v [] ambient register)
  simpa only [MachineCloudPadding.Placement.configuration, MachineCloudPadding.Placement.label, placedTapes,
    frame_bound, frame_remaining, frame_total, List.append_nil, frame, memory] using h

theorem incrementStep (table bound remaining total : List Bool) (v : Nat)
    (ambient : σ) (register : Option Bool) :
    TM2.step (program (σ := σ))
      ⟨some .increment, ((ambient, false), register),
        frame table (encodeWord v) [] (encodeWord 0) [] bound remaining total⟩ =
      some ⟨some .guard, ((ambient, false), none),
        frame table (encodeWord (v + 1)) [] [] [] bound remaining total⟩ := by
  simp [TM2.step, program, TM2.stepAux, encodeWord, List.replicate_succ]

def bodyTime (t : GraphTables.Table) (v : Fin t.vertices) : Nat :=
  MachineCloudPadding.totalTime t v + (MachineCloudPadding.padding (PreprocessingCloudIndex.cloudSize t v) + 1) +
    (PreprocessingCloudIndex.cloudSize t v + 1) +
      (PreprocessingLevels.boundedLevel (PreprocessingCloudIndex.cloudSize t v) + 1) + 1

theorem bodyTrace (t : GraphTables.Table) (v : Fin t.vertices)
    (bound remaining : List Bool) (acc : Nat) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[bodyTime t v]
      (some ⟨some (.padding .init), ((ambient, false), register),
        frame (GraphTables.tableBits t) (encodeWord v.val) [] [] []
          bound remaining (encodeWord acc)⟩) =
      some ⟨some .guard, ((ambient, false), none),
        frame (GraphTables.tableBits t) (encodeWord (v.val + 1)) [] [] []
          bound remaining (encodeWord (MachineCloudPadding.padding (PreprocessingCloudIndex.cloudSize t v) + acc))⟩ := by
  let k := PreprocessingCloudIndex.cloudSize t v
  let p := MachineCloudPadding.padding k
  let l := PreprocessingLevels.boundedLevel k
  have hp := paddingTrace t v bound remaining (encodeWord acc) ambient register
  have ha := MachineUnaryAddAt.addFromTapes (Tape.inner .power) Tape.total (by decide)
    Label.add (some Label.clearCount) program rfl
    (frame (GraphTables.tableBits t) (encodeWord v.val) (encodeWord k)
      (encodeWord p) (encodeWord l) bound remaining (encodeWord acc))
    p acc [] [] (by simp) (by simp) (ambient, false) none
  simp only [MachineUnaryAddAt.unaryTapes, Reduction.MachineTransfer.tapesAt,
    List.append_nil, update_frame_power, update_frame_total] at ha
  have hc := MachineLookup.discardTrace (Tape.inner .count) Label.clearCount Label.clearLevel
    program rfl
    (frame (GraphTables.tableBits t) (encodeWord v.val) (encodeWord k)
      (encodeWord 0) (encodeWord l) bound remaining (encodeWord (p + acc)))
    k [] (by simp) (ambient, false) none
  simp only [update_frame_count] at hc
  have hl := MachineLookup.discardTrace (Tape.inner .level) Label.clearLevel Label.increment
    program rfl
    (frame (GraphTables.tableBits t) (encodeWord v.val) []
      (encodeWord 0) (encodeWord l) bound remaining (encodeWord (p + acc)))
    l [] (by simp) (ambient, false) none
  simp only [update_frame_level] at hl
  have hi : (advance (TM2.step program))^[1]
      (some ⟨some .increment, ((ambient, false), none),
        frame (GraphTables.tableBits t) (encodeWord v.val) [] (encodeWord 0) []
          bound remaining (encodeWord (p + acc))⟩) =
      some ⟨some .guard, ((ambient, false), none),
        frame (GraphTables.tableBits t) (encodeWord (v.val + 1)) [] [] []
          bound remaining (encodeWord (p + acc))⟩ :=
    incrementStep _ _ _ _ _ ambient none
  exact appendTrace_inline_MachineCloudPrefix _ (appendTrace_inline_MachineCloudPrefix _ (appendTrace_inline_MachineCloudPrefix _ (appendTrace_inline_MachineCloudPrefix _ hp ha) hc) hl) hi

theorem guardStep_succ (table bound suffix total : List Bool) (v n : Nat)
    (ambient : σ) (register : Option Bool) :
    TM2.step (program (σ := σ))
      ⟨some .guard, ((ambient, false), register),
        frame table (encodeWord v) [] [] [] bound (encodeWord (n + 1) ++ suffix) total⟩ =
      some ⟨some (.padding .init), ((ambient, false), none),
        frame table (encodeWord v) [] [] [] bound (encodeWord n ++ suffix) total⟩ := by
  simp [TM2.step, program, MachineUnaryCounter.guard, TM2.stepAux, encodeWord,
    List.replicate_succ]

theorem guardStep_zero (table bound suffix total : List Bool) (v : Nat)
    (ambient : σ) (register : Option Bool) :
    TM2.step (program (σ := σ))
      ⟨some .guard, ((ambient, false), register),
        frame table (encodeWord v) [] [] [] bound (encodeWord 0 ++ suffix) total⟩ =
      some ⟨some .done, ((ambient, false), none),
        frame table (encodeWord v) [] [] [] bound (encodeWord 0 ++ suffix) total⟩ := rfl

theorem zeroTrace (table bound suffix total : List Bool) (v : Nat)
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step (program (σ := σ))))^[2]
      (some ⟨some .guard, ((ambient, false), register),
        frame table (encodeWord v) [] [] [] bound (encodeWord 0 ++ suffix) total⟩) =
      some ⟨none, ((ambient, false), none),
        frame table (encodeWord v) [] [] [] bound (encodeWord 0 ++ suffix) total⟩ := by
  rw [Function.iterate_succ_apply]
  change advance (TM2.step (program (σ := σ)))
    (TM2.step program ⟨some .guard, ((ambient, false), register),
      frame table (encodeWord v) [] [] [] bound (encodeWord 0 ++ suffix) total⟩) = _
  rw [guardStep_zero]
  rfl

def amount (t : GraphTables.Table) (v : Nat) : Nat :=
  if h : v < t.vertices then MachineCloudPadding.padding (PreprocessingCloudIndex.cloudSize t ⟨v, h⟩) else 0

def cycleTime (t : GraphTables.Table) (v : Nat) : Nat :=
  if h : v < t.vertices then 1 + bodyTime t ⟨v, h⟩ else 0

def sumFrom (t : GraphTables.Table) (v : Nat) : Nat → Nat
  | 0 => 0
  | n + 1 => amount t v + sumFrom t (v + 1) n

def loopTime (t : GraphTables.Table) (v : Nat) : Nat → Nat
  | 0 => 2
  | n + 1 => cycleTime t v + loopTime t (v + 1) n

theorem loopTrace (t : GraphTables.Table) (n v : Nat) (hvn : v + n ≤ t.vertices)
    (bound suffix : List Bool) (acc : Nat) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[loopTime t v n]
      (some ⟨some .guard, ((ambient, false), register),
        frame (GraphTables.tableBits t) (encodeWord v) [] [] []
          bound (encodeWord n ++ suffix) (encodeWord acc)⟩) =
      some ⟨none, ((ambient, false), none),
        frame (GraphTables.tableBits t) (encodeWord (v + n)) [] [] []
          bound (encodeWord 0 ++ suffix) (encodeWord (sumFrom t v n + acc))⟩ := by
  induction n generalizing v acc register with
  | zero =>
      simpa only [loopTime, sumFrom, Nat.add_zero, Nat.zero_add] using
        zeroTrace _ bound suffix (encodeWord acc) v ambient register
  | succ n ih =>
      have hv : v < t.vertices := by omega
      have hnext : v + 1 + n ≤ t.vertices := by omega
      have hg : (advance (TM2.step program))^[1]
          (some ⟨some .guard, ((ambient, false), register),
            frame (GraphTables.tableBits t) (encodeWord v) [] [] []
              bound (encodeWord (n + 1) ++ suffix) (encodeWord acc)⟩) =
          some ⟨some (.padding .init), ((ambient, false), none),
            frame (GraphTables.tableBits t) (encodeWord v) [] [] []
              bound (encodeWord n ++ suffix) (encodeWord acc)⟩ :=
        guardStep_succ _ bound suffix (encodeWord acc) v n ambient register
      have hb := bodyTrace t ⟨v, hv⟩ bound (encodeWord n ++ suffix) acc ambient none
      have ht := ih (v + 1) hnext
        (MachineCloudPadding.padding (PreprocessingCloudIndex.cloudSize t ⟨v, hv⟩) + acc) none
      have hall := appendTrace_inline_MachineCloudPrefix _ (appendTrace_inline_MachineCloudPrefix _ hg hb) ht
      simpa only [loopTime, cycleTime, dite_eq_left hv, sumFrom, amount,
        Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using hall

theorem sumFrom_list (t : GraphTables.Table) (xs : List (Fin t.vertices)) (v : Nat)
    (hseq : ∀ i (hi : i < xs.length), (xs[i]).val = v + i) :
    sumFrom t v xs.length = (xs.map (PreprocessingRegularTables.padding t)).sum := by
  induction xs generalizing v with
  | nil => simp [sumFrom]
  | cons x xs ih =>
      have hx : x.val = v := by
        have hhead := hseq 0 (by simp)
        simp only [List.getElem_cons_zero, Nat.add_zero] at hhead
        exact hhead
      have hv : v < t.vertices := by simpa only [hx] using x.isLt
      have he : (⟨v, hv⟩ : Fin t.vertices) = x := Fin.ext hx.symm
      have hnext : ∀ i (hi : i < xs.length), (xs[i]).val = v + 1 + i := by
        intro i hi
        have h := hseq (i + 1) (by simpa using Nat.succ_lt_succ hi)
        simpa only [List.getElem_cons_succ, Nat.add_assoc, Nat.add_comm,
          Nat.add_left_comm] using h
      simp only [List.length_cons, sumFrom, amount, dite_eq_left hv, he,
        List.map_cons, List.sum_cons]
      rw [ih (v + 1) hnext]
      rfl

theorem sumFrom_eq_offset (t : GraphTables.Table) (b : Nat) (hb : b ≤ t.vertices) :
    sumFrom t 0 b =
      PreprocessingPaddingOffsets.offset (PreprocessingRegularTables.padding t) b := by
  have h := sumFrom_list t ((List.finRange t.vertices).take b) 0
    (by intro i hi; simp only [List.getElem_take, List.getElem_finRange, Fin.val_cast,
      Nat.zero_add])
  simpa only [List.length_take, List.length_finRange, Nat.min_eq_left hb,
    PreprocessingPaddingOffsets.offset] using h

theorem initStep (table bound : List Bool) (ambient : σ) (register : Option Bool) :
    TM2.step (program (σ := σ))
      ⟨some .init, ((ambient, false), register), frame table [] [] [] [] bound [] []⟩ =
      some ⟨some .copyFirst, ((ambient, false), register),
        frame table (encodeWord 0) [] [] [] bound [] (encodeWord 0)⟩ := by
  simp [TM2.step, program, TM2.stepAux, encodeWord]

def totalTime (t : GraphTables.Table) (b : Nat) (suffix : List Bool) : Nat :=
  (1 + 2 * ((encodeWord b ++ suffix).length + 1)) + loopTime t 0 b

theorem prefixTrace (t : GraphTables.Table) (b : Nat) (hb : b ≤ t.vertices)
    (suffix : List Bool) (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[totalTime t b suffix]
      (some ⟨some .init, ((ambient, false), register),
        frame (GraphTables.tableBits t) [] [] [] [] (encodeWord b ++ suffix) [] []⟩) =
      some ⟨none, ((ambient, false), none),
        frame (GraphTables.tableBits t) (encodeWord b) [] [] []
          (encodeWord b ++ suffix) (encodeWord 0 ++ suffix)
          (encodeWord (PreprocessingPaddingOffsets.offset
            (PreprocessingRegularTables.padding t) b))⟩ := by
  have hi : (advance (TM2.step program))^[1]
      (some ⟨some .init, ((ambient, false), register),
        frame (GraphTables.tableBits t) [] [] [] [] (encodeWord b ++ suffix) [] []⟩) =
      some ⟨some .copyFirst, ((ambient, false), register),
        frame (GraphTables.tableBits t) (encodeWord 0) [] [] []
          (encodeWord b ++ suffix) [] (encodeWord 0)⟩ :=
    initStep _ _ ambient register
  have hc := MachineCopy.copyTrace Tape.bound Tape.remaining (Tape.inner .scratch)
    (by decide) (by decide) (by decide) false Label.copyFirst Label.copySecond
    (some Label.guard) program rfl rfl
    (frame (GraphTables.tableBits t) (encodeWord 0) [] [] []
      (encodeWord b ++ suffix) [] (encodeWord 0)) rfl (ambient, false) register
  simp only [frame_bound, frame_remaining, List.append_nil, update_frame_remaining] at hc
  have hl := loopTrace t b 0 (by simpa only [Nat.zero_add] using hb)
    (encodeWord b ++ suffix) suffix 0 ambient none
  simp only [Nat.zero_add, Nat.add_zero, sumFrom_eq_offset t b hb] at hl
  exact appendTrace_inline_MachineCloudPrefix _ (appendTrace_inline_MachineCloudPrefix _ hi hc) hl

def cycleBudget (L : Nat) : Nat :=
  (ExpanderFamily.growth + 8) * (2 * L) ^ 2 + 25 * (2 * L) + 10 +
    (ExpanderFamily.growth + 2) * L + 5

theorem bodyTime_le (t : GraphTables.Table) (v : Fin t.vertices) (L : Nat)
    (hL : (GraphTables.tableBits t).length ≤ L) :
    1 + bodyTime t v ≤ cycleBudget L := by
  have hn := GraphTables.vertices_le_tableBits_length t
  have hv := v.isLt
  have hinput : MachineCloudPadding.inputLength t v [] ≤ 2 * L := by
    simp only [MachineCloudPadding.inputLength, List.append_nil, encodeWord_length]
    omega
  have hp := (MachineCloudPadding.totalTime_le t v []).trans
    (natPolynomial_eval_mono MachineCloudPadding.timePolynomial hinput)
  simp only [MachineCloudPadding.timePolynomial, Polynomial.eval_add, Polynomial.eval_mul,
    Polynomial.eval_C, Polynomial.eval_pow, Polynomial.eval_X, Polynomial.eval_ofNat] at hp
  have hk : PreprocessingCloudIndex.cloudSize t v ≤ L :=
    (PreprocessingCloudIndex.cloudSize_le_darts t v).trans
      ((GraphTables.darts_le_tableBits_length t).trans hL)
  have hl := (PreprocessingLevels.boundedLevel_le_input
    (PreprocessingCloudIndex.cloudSize t v)).trans hk
  have hpad : MachineCloudPadding.padding (PreprocessingCloudIndex.cloudSize t v) ≤
      ExpanderFamily.growth * L := by
    have hsize := (PreprocessingLevels.cloudPaddedSize_bounds
      (PreprocessingCloudIndex.cloudSize t v)).2
    have hmul := Nat.mul_le_mul_left ExpanderFamily.growth hk
    unfold MachineCloudPadding.padding
    omega
  unfold bodyTime cycleBudget
  simp only [Nat.add_mul] at hp ⊢
  omega

theorem loopTime_le (t : GraphTables.Table) (n v : Nat) (hvn : v + n ≤ t.vertices)
    (L : Nat) (hL : (GraphTables.tableBits t).length ≤ L) :
    loopTime t v n ≤ n * cycleBudget L + 2 := by
  induction n generalizing v with
  | zero => simp only [loopTime, Nat.zero_mul, Nat.zero_add, le_refl]
  | succ n ih =>
      have hv : v < t.vertices := by omega
      have hnext : v + 1 + n ≤ t.vertices := by omega
      have hbody := bodyTime_le t ⟨v, hv⟩ L hL
      have htail := ih (v + 1) hnext
      simp only [loopTime, cycleTime, dite_eq_left hv, Nat.succ_mul]
      omega

def inputLength (t : GraphTables.Table) (b : Nat) (suffix : List Bool) : Nat :=
  (GraphTables.tableBits t).length + (encodeWord b ++ suffix).length

noncomputable def timePolynomial : Polynomial Nat :=
  Polynomial.X *
      (Polynomial.C (ExpanderFamily.growth + 8) * (2 * Polynomial.X) ^ 2 +
        25 * (2 * Polynomial.X) + 10 +
          Polynomial.C (ExpanderFamily.growth + 2) * Polynomial.X + 5) +
    2 * (Polynomial.X + 1) + 3

theorem totalTime_le (t : GraphTables.Table) (b : Nat) (hb : b ≤ t.vertices)
    (suffix : List Bool) :
    totalTime t b suffix ≤ timePolynomial.eval (inputLength t b suffix) := by
  let L := inputLength t b suffix
  have htable : (GraphTables.tableBits t).length ≤ L := by
    unfold L inputLength
    omega
  have hbound : (encodeWord b ++ suffix).length ≤ L := by
    unfold L inputLength
    omega
  have hbL : b ≤ L := hb.trans ((GraphTables.vertices_le_tableBits_length t).trans htable)
  have hloop := loopTime_le t b 0 (by simpa only [Nat.zero_add] using hb) L htable
  have hmul := Nat.mul_le_mul_right (cycleBudget L) hbL
  have htotal : totalTime t b suffix ≤ L * cycleBudget L + 2 * (L + 1) + 3 := by
    unfold totalTime
    omega
  simpa only [timePolynomial, Polynomial.eval_add, Polynomial.eval_mul,
    Polynomial.eval_C, Polynomial.eval_pow, Polynomial.eval_X, Polynomial.eval_ofNat,
    Polynomial.eval_one, cycleBudget, L] using htotal

def prefixInTime (t : GraphTables.Table) (b : Nat) (hb : b ≤ t.vertices)
    (suffix : List Bool) (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some .init, ((ambient, false), register),
        frame (GraphTables.tableBits t) [] [] [] [] (encodeWord b ++ suffix) [] []⟩
      (some ⟨none, ((ambient, false), none),
        frame (GraphTables.tableBits t) (encodeWord b) [] [] []
          (encodeWord b ++ suffix) (encodeWord 0 ++ suffix)
          (encodeWord (PreprocessingPaddingOffsets.offset
            (PreprocessingRegularTables.padding t) b))⟩)
      (timePolynomial.eval (inputLength t b suffix)) where
  steps := totalTime t b suffix
  evals_in_steps := prefixTrace t b hb suffix ambient register
  steps_le_m := totalTime_le t b hb suffix

theorem all_padding_sum (t : GraphTables.Table) :
    sumFrom t 0 t.vertices = ∑ v, PreprocessingRegularTables.padding t v := by
  rw [sumFrom_eq_offset t t.vertices (Nat.le_refl _), PreprocessingPaddingOffsets.offset_all]

theorem paddingOrder_eq_prefix (t : GraphTables.Table) (v : Fin t.vertices)
    (j : Fin (PreprocessingRegularTables.padding t v)) :
    (PreprocessingRegularTables.paddingOrder (PreprocessingRegularTables.padding t) ⟨v, j⟩).val =
      sumFrom t 0 v.val + j.val := by
  rw [PreprocessingPaddingOffsets.paddingOrder_val,
    sumFrom_eq_offset t v.val v.isLt.le]

end DFVSGames.Foundations.Complexity.MachineCloudPrefix

end OAI