lean/OAI/Computability/UniqueGames/Machines/MachineCloudPadding.lean

import OAI.Computability.UniqueGames.Machines.MachineCeilingPower
import OAI.Computability.UniqueGames.Machines.MachineCloudCount

namespace OAI

/-!
# Actual cloud counting, geometric padding, and unary subtraction

One finite program places the checked cloud counter and geometric search on
shared tapes. Its final phase physically subtracts the preserved cloud size
from the computed power. A head test handles the empty cloud separately.
-/

namespace UniqueGamesTheorem.Foundations.Complexity.MachineCloudPadding

open Turing
open MachineComposition
open UniqueGamesTheorem.Foundations.PCP

/- Static placement of homogeneous tapes, with an explicit finite partial
inverse. Complement tapes are preserved by every simulated transition. -/
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

protected abbrev Tape.enumList : List Tape := [.table, .query, .count, .work, .scratch, .spare,
  .power, .level, .fuel, .product]

protected theorem Tape.enumList_getElem?_ctorIdx_eq (x : Tape) :
    Tape.enumList[x.ctorIdx]? = some x := by
  cases x <;> rfl

protected theorem Tape.enumList_nodup : Tape.enumList.Nodup := by decide

instance : Fintype Tape where
  elems := ⟨Tape.enumList, Tape.enumList_nodup⟩
  complete x := by cases x <;> decide

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 {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

/-- Physical unary subtraction preserves the complete count word by moving
its consumed tallies to scratch and then transferring them back. -/
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 _ 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)

/-- Count zero is handled by a head test. Positive counts execute the exact
power search and subtraction traces on the shared count tape. -/
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 _ hb (appendTrace _ 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)

/-- One complete concrete program returns the cloud size, geometric level,
and required number of padding vertices, preserving input and query. -/
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 _ (appendTrace _ 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

/-- The bound is quadratic in the combined actual graph/query tape length.
All transitions are supplied by the concrete program and its phase traces. -/
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 UniqueGamesTheorem.Foundations.Complexity.MachineCloudPadding

end OAI