lean/OAI/Computability/DirectedFeedback/Machines/MachineExpanderTableLoop.lean

import Mathlib
import OAI.Computability.DirectedFeedback.Machines.MachineExpanderRow

namespace OAI

section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section

section

namespace DFVSGames.Foundations.Complexity.MachineExpanderTable

open Turing MachineComposition
open PCP.ExpanderTables PCP.ExpanderRowControl PCP.ExpanderTableWords
open PCP.ExpanderTableEnumeration

def oldTableWord {v d : Nat} (G : Table v (degree d)) : List Bool :=
  encodeWords (rotationWords G)

def accumulatedVertices {v d : Nat} (G : Table v (degree d))
    (H : Table (cloudSize d) d) (current : Nat) : List Bool :=
  accumulate (priorVertices G H current) []

@[simp] theorem accumulatedVertices_zero {v d : Nat} (G : Table v (degree d))
    (H : Table (cloudSize d) d) : accumulatedVertices G H 0 = [] := by
  simp [accumulatedVertices, accumulate]

theorem accumulatedVertices_succ {v d : Nat} (G : Table v (degree d))
    (H : Table (cloudSize d) d) (current : Nat) (valid : current < v) :
    accumulatedVertices G H (current + 1) =
      accumulate (vertexWords G H ⟨current, valid⟩) (accumulatedVertices G H current) := by
  simp only [accumulatedVertices, priorVertices_succ G H current valid, accumulate_append]

theorem accumulatedVertices_full {v d : Nat} (G : Table v (degree d))
    (H : Table (cloudSize d) d) : accumulatedVertices G H v =
      (encodeWords (rotationWords (step G H))).reverse := by
  simp [accumulatedVertices, accumulate_generatedWords]

def vertexBodyBudget {v d : Nat} (G : Table v (degree d)) : Nat :=
  rowFactor d * (80 * ((oldTableWord G).length + 1) + 2)

def vertexLoopBudget {v d : Nat} (G : Table v (degree d)) (remaining : Nat) : Nat :=
  remaining * (vertexBodyBudget G + 1) + 1

def vertexLoopTapes {v d : Nat} (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (current remaining : Nat) (suffix : List Bool) : ∀ tape, List (Alphabet tape) :=
  boundaryTapes current remaining (oldTableWord G) (accumulatedVertices G H current) suffix []

structure VertexLoopRun {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (current remaining : Nat) (state : State ρ d) (suffix : List Bool) where
  finalState : State ρ d
  caller_preserved : caller finalState = caller state
  position_zero : finalState.2 = zeroPosition positive
  execution : StateTransition.EvalsToInTime (TM2.step (program positive H))
    ⟨some (.inr .vertexGuard), state, vertexLoopTapes G H current remaining suffix⟩
    (some ⟨some (.inr .reverseOutput), finalState, vertexLoopTapes G H v 0 suffix⟩)
    (vertexLoopBudget G remaining)

private def boundarySingleStep_inline_MachineExpanderTableLoop {S : Type*} (f : S → Option S) (a b : S)
    (h : f a = some b) : StateTransition.EvalsToInTime f a (some b) 1 where
  steps := 1
  evals_in_steps := by change f a = some b; exact h
  steps_le_m := Nat.le_refl _

def vertexLoopInTime {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (current remaining : Nat) (state : State ρ d) (suffix : List Bool)
    (count : current + remaining = v) (position : state.2 = zeroPosition positive) :
    VertexLoopRun positive G H current remaining state suffix := by
  induction remaining generalizing current state with
  | zero =>
      have hc : current = v := by omega
      subst current
      refine ⟨clearRegister state, caller_clearRegister state, ?_, ?_⟩
      · simpa only [clearRegister] using position
      · simpa only [vertexLoopBudget, Nat.zero_mul, Nat.zero_add, vertexLoopTapes] using
          boundarySingleStep_inline_MachineExpanderTableLoop _ _ _ (vertexGuard_boundary_zeroStep positive H v (oldTableWord G)
            (accumulatedVertices G H v) suffix [] state)
  | succ remaining ih =>
      have valid : current < v := by omega
      let vertex : Fin v := ⟨current, valid⟩
      let body := vertexInTime positive G H vertex (clearRegister state)
        (by simpa only [clearRegister] using position) remaining
        (accumulatedVertices G H current) suffix []
      have nextCount : current + 1 + remaining = v := by omega
      let rest := ih (current + 1) body.finalState nextCount body.position_zero
      have output_eq : accumulate (vertexWords G H vertex)
          (accumulatedVertices G H current) = accumulatedVertices G H (current + 1) :=
        (accumulatedVertices_succ G H current valid).symm
      let guard := boundarySingleStep_inline_MachineExpanderTableLoop _ _ _
        (vertexGuard_boundary_succStep positive H current remaining (oldTableWord G)
          (accumulatedVertices G H current) suffix [] state)
      have bodyRun : StateTransition.EvalsToInTime (TM2.step (program positive H))
          ⟨some (.inr .prepareRow), clearRegister state,
            boundaryTapes current remaining (oldTableWord G)
              (accumulatedVertices G H current) suffix []⟩
          (some ⟨some (.inr .vertexGuard), body.finalState,
            vertexLoopTapes G H (current + 1) remaining suffix⟩)
          (vertexBodyBudget G) := by
        simpa only [vertexLoopTapes, vertexBodyBudget, oldTableWord, output_eq] using body.execution
      let first := StateTransition.EvalsToInTime.trans (TM2.step (program positive H))
        _ _ _ _ _ guard bodyRun
      let all := StateTransition.EvalsToInTime.trans (TM2.step (program positive H))
        _ _ _ _ _ first rest.execution
      refine ⟨rest.finalState, ?_, rest.position_zero, ?_⟩
      · exact rest.caller_preserved.trans
          (body.caller_preserved.trans (caller_clearRegister state))
      · refine { toEvalsTo := all.toEvalsTo, steps_le_m := ?_ }
        have bound := all.steps_le_m
        simp only [vertexLoopBudget, Nat.succ_mul] at bound ⊢
        omega

end DFVSGames.Foundations.Complexity.MachineExpanderTable
end

section

namespace DFVSGames.Foundations.Complexity.MachineExpanderTable

open Turing
open MachineComposition
open PCP.ExpanderTables PCP.ExpanderRowControl

variable {ρ : Type} {d : Nat}

@[simp] theorem clearRegister_position (state : State ρ d) :
    (clearRegister state).2 = state.2 := rfl

@[simp] theorem clearRegister_register (state : State ρ d) :
    (clearRegister state).1.2 = none := rfl

@[simp] theorem clearRegister_idempotent (state : State ρ d) :
    clearRegister (clearRegister state) = clearRegister state := rfl

theorem boundaryTapes_reverse_update (vertex remaining : Nat)
    (oldTable output countSuffix result nextOutput nextResult : List Bool) :
    Function.update
      (Function.update (boundaryTapes vertex remaining oldTable output countSuffix result)
        (.inl .output) nextOutput)
      (.inr .result) nextResult =
      boundaryTapes vertex remaining oldTable nextOutput countSuffix nextResult := by
  funext tape
  rcases tape with row | extra
  · cases row <;>
      simp [boundaryTapes, rowTapes, MachineEmbedding.tapes, Function.update]
  · cases extra <;>
      simp [boundaryTapes, extraTapes, MachineEmbedding.tapes, Function.update]

variable [Fintype ρ]

theorem reverseOutputTrace (positive : 0 < d) (H : Table (cloudSize d) d)
    (vertex remaining : Nat) (oldTable output countSuffix result : List Bool)
    (state : State ρ d) :
    (advance (TM2.step (program positive H)))^[output.length + 1]
      (some ⟨some (.inr .reverseOutput), state,
        boundaryTapes vertex remaining oldTable output countSuffix result⟩) =
      some ⟨some (.inr .done), clearRegister state,
        boundaryTapes vertex remaining oldTable [] countSuffix (output.reverse ++ result)⟩ := by
  induction output generalizing result state with
  | nil =>
      simpa only [List.length_nil, Nat.zero_add, Function.iterate_one, advance_some,
        List.reverse_nil, List.nil_append] using
        reverseOutput_nilStep positive H
          (boundaryTapes vertex remaining oldTable [] countSuffix result) state rfl
  | cons symbol output ih =>
      rw [List.length_cons, Function.iterate_succ_apply]
      simp only [advance_some]
      rw [reverseOutput_consStep positive H
        (boundaryTapes vertex remaining oldTable (symbol :: output) countSuffix result)
        state symbol output rfl]
      change (advance (TM2.step (program positive H)))^[output.length + 1]
        (some ⟨some (.inr .reverseOutput), ((state.1.1, some symbol), state.2),
          Function.update
            (Function.update
              (boundaryTapes vertex remaining oldTable (symbol :: output) countSuffix result)
              (.inl .output) output)
            (.inr .result) (symbol :: result)⟩) = _
      rw [boundaryTapes_reverse_update]
      simpa only [clearRegister, List.reverse_cons, List.append_assoc,
        List.singleton_append] using
        ih (symbol :: result) ((state.1.1, some symbol), state.2)

theorem reverseOutputHaltTrace (positive : 0 < d) (H : Table (cloudSize d) d)
    (vertex remaining : Nat) (oldTable output countSuffix result : List Bool)
    (state : State ρ d) :
    (advance (TM2.step (program positive H)))^[output.length + 2]
      (some ⟨some (.inr .reverseOutput), state,
        boundaryTapes vertex remaining oldTable output countSuffix result⟩) =
      some ⟨none, clearRegister state,
        boundaryTapes vertex remaining oldTable [] countSuffix (output.reverse ++ result)⟩ := by
  rw [show output.length + 2 = (output.length + 1) + 1 by omega,
    Function.iterate_succ_apply',
    reverseOutputTrace positive H vertex remaining oldTable output countSuffix result state]
  exact doneStep positive H
    (boundaryTapes vertex remaining oldTable [] countSuffix (output.reverse ++ result))
    (clearRegister state)

def reverseOutputInTime (positive : 0 < d) (H : Table (cloudSize d) d)
    (vertex remaining : Nat) (oldTable output countSuffix result : List Bool)
    (state : State ρ d) :
    StateTransition.EvalsToInTime (TM2.step (program positive H))
      ⟨some (.inr .reverseOutput), state,
        boundaryTapes vertex remaining oldTable output countSuffix result⟩
      (some ⟨some (.inr .done), clearRegister state,
        boundaryTapes vertex remaining oldTable [] countSuffix (output.reverse ++ result)⟩)
      (output.length + 1) where
  steps := output.length + 1
  evals_in_steps := reverseOutputTrace positive H vertex remaining oldTable output countSuffix result state
  steps_le_m := Nat.le_refl _

def reverseOutputHaltInTime (positive : 0 < d) (H : Table (cloudSize d) d)
    (vertex remaining : Nat) (oldTable output countSuffix result : List Bool)
    (state : State ρ d) :
    StateTransition.EvalsToInTime (TM2.step (program positive H))
      ⟨some (.inr .reverseOutput), state,
        boundaryTapes vertex remaining oldTable output countSuffix result⟩
      (some ⟨none, clearRegister state,
        boundaryTapes vertex remaining oldTable [] countSuffix (output.reverse ++ result)⟩)
      (output.length + 2) where
  steps := output.length + 2
  evals_in_steps := reverseOutputHaltTrace positive H vertex remaining oldTable output countSuffix result state
  steps_le_m := Nat.le_refl _

end DFVSGames.Foundations.Complexity.MachineExpanderTable
end

section

namespace DFVSGames.Foundations.Complexity.MachineExpanderTable

open Turing MachineComposition
open PCP.ExpanderTables PCP.ExpanderRowControl PCP.ExpanderTableWords
open PCP.ExpanderTableEnumeration

def completeBudget {v d : Nat} (G : Table v (degree d)) (H : Table (cloudSize d) d) : Nat :=
  1 + vertexLoopBudget G v + ((encodeWords (rotationWords (step G H))).length + 2)

def timeCoefficient (d : Nat) : Nat :=
  rowFactor d * (rowFactor d + 1) + 82 * rowFactor d + 5

noncomputable def timePolynomial (d : Nat) : Polynomial Nat :=
  Polynomial.C (timeCoefficient d) * (Polynomial.X + 1) ^ 2

theorem vertices_le_word_length {v d : Nat} (positive : 0 < d)
    (G : Table v (degree d)) : v ≤ (oldTableWord G).length := by
  have hq : 1 ≤ degree d := Nat.mul_pos positive positive
  have hv : v ≤ v * degree d := by simpa using Nat.mul_le_mul_left v hq
  have hr : v * degree d ≤ (oldTableWord G).length := by
    simp only [oldTableWord, encodeWords_length, rotationWords_length]
    omega
  exact hv.trans hr

theorem completeBudget_le {v d : Nat} (positive : 0 < d)
    (G : Table v (degree d)) (H : Table (cloudSize d) d) :
    completeBudget G H ≤ (timePolynomial d).eval (oldTableWord G).length := by
  let L := (oldTableWord G).length
  let M := rowFactor d
  have vBound : v ≤ L + 1 := (vertices_le_word_length positive G).trans (Nat.le_succ L)
  have emitBound : 80 * (L + 1) + 2 ≤ 82 * (L + 1) := by omega
  have bodyBound : vertexBodyBudget G + 1 ≤ (82 * M + 1) * (L + 1) := by
    have mulBound := Nat.mul_le_mul_left M emitBound
    change M * (80 * (L + 1) + 2) + 1 ≤ _
    nlinarith
  have iterations := Nat.mul_le_mul vBound bodyBound
  have loopBound : v * (vertexBodyBudget G + 1) ≤ (82 * M + 1) * (L + 1)^2 := by
    calc
      _ ≤ (L + 1) * ((82 * M + 1) * (L + 1)) := iterations
      _ = _ := by ring
  have rowBound : v * M ≤ M * (L + 1) := by
    simpa only [Nat.mul_comm v M] using Nat.mul_le_mul_left M vBound
  have rowSuccBound : v * M + 1 ≤ (M + 1) * (L + 1) := by nlinarith
  have productBound := Nat.mul_le_mul rowBound rowSuccBound
  have lengthBound : (encodeWords (rotationWords (step G H))).length ≤
      M * (M + 1) * (L + 1)^2 := by
    calc
      _ ≤ (v * M) * (v * M + 1) := by
        simpa only [generatedWords_eq_rotationWords] using encode_generatedWords_length_le G H
      _ ≤ (M * (L + 1)) * ((M + 1) * (L + 1)) := productBound
      _ = _ := by ring
  have oneBound : 1 ≤ (L + 1)^2 := by nlinarith
  have combined := Nat.add_le_add loopBound lengthBound
  simp only [timePolynomial, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_pow,
    Polynomial.eval_add, Polynomial.eval_X, Polynomial.eval_one]
  change completeBudget G H ≤ (M * (M + 1) + 82 * M + 5) * (L + 1)^2
  unfold completeBudget vertexLoopBudget
  nlinarith

structure TableRun {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (state : State ρ d) (suffix : List Bool) where
  finalState : State ρ d
  caller_preserved : caller finalState = caller state
  position_zero : finalState.2 = zeroPosition positive
  execution : StateTransition.EvalsToInTime (TM2.step (program positive H))
    ⟨some (.inr .initialize), state, initialTapes v (oldTableWord G) suffix⟩
    (some ⟨none, finalState,
      finalTapes v (oldTableWord G) (encodeWords (rotationWords (step G H))) suffix⟩)
    ((timePolynomial d).eval (oldTableWord G).length)

private def initializeInTime_inline_MachineExpanderTable {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (state : State ρ d) (suffix : List Bool) :
    StateTransition.EvalsToInTime (TM2.step (program positive H))
      ⟨some (.inr .initialize), state, initialTapes v (oldTableWord G) suffix⟩
      (some ⟨some (.inr .vertexGuard), initialState positive H (caller state),
        vertexLoopTapes G H 0 v suffix⟩) 1 where
  steps := 1
  evals_in_steps := by
    change TM2.step (program (ρ := ρ) positive H)
      ⟨some (.inr .initialize), state, initialTapes v (oldTableWord G) suffix⟩ =
      some ⟨some (.inr .vertexGuard), initialState positive H (caller state),
        vertexLoopTapes G H 0 v suffix⟩
    simpa only [vertexLoopTapes, accumulatedVertices_zero] using
      initialize_boundaryStep positive H v (oldTableWord G) suffix state
  steps_le_m := Nat.le_refl _

def tableInTime {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (state : State ρ d) (suffix : List Bool) : TableRun positive G H state suffix := by
  let loop := vertexLoopInTime positive G H 0 v
    (initialState positive H (caller state)) suffix (by omega) rfl
  let initialRun := initializeInTime_inline_MachineExpanderTable positive G H state suffix
  have finish : StateTransition.EvalsToInTime (TM2.step (program positive H))
      ⟨some (.inr .reverseOutput), loop.finalState, vertexLoopTapes G H v 0 suffix⟩
      (some ⟨none, clearRegister loop.finalState,
        finalTapes v (oldTableWord G) (encodeWords (rotationWords (step G H))) suffix⟩)
      ((encodeWords (rotationWords (step G H))).length + 2) := by
    simpa only [vertexLoopTapes, finalTapes, accumulatedVertices_full, List.reverse_reverse,
      List.append_nil, List.length_reverse] using
      reverseOutputHaltInTime positive H v 0 (oldTableWord G)
        (accumulatedVertices G H v) suffix [] loop.finalState
  let first := StateTransition.EvalsToInTime.trans (TM2.step (program positive H))
    _ _ _ _ _ initialRun loop.execution
  let all := StateTransition.EvalsToInTime.trans (TM2.step (program positive H))
    _ _ _ _ _ first finish
  refine ⟨clearRegister loop.finalState, ?_, ?_, ?_⟩
  · exact (caller_clearRegister loop.finalState).trans loop.caller_preserved
  · simpa only [clearRegister] using loop.position_zero
  · refine { toEvalsTo := all.toEvalsTo, steps_le_m := ?_ }
    have bound := all.steps_le_m
    have polynomialBound := completeBudget_le positive G H
    unfold completeBudget at polynomialBound
    omega

end DFVSGames.Foundations.Complexity.MachineExpanderTable
end

section

namespace DFVSGames.Foundations.Complexity.MachineExpanderFamily

open PCP.ExpanderTables PCP.ExpanderRowControl PCP.ExpanderTableWords

def cycleWords (remaining : Nat) (word current vertex count result suffix : List Bool) :
    Tape → List Bool
  | .inl (.inl .table) => word
  | .inl (.inl .inputVertex) => vertex
  | .inl (.inr .vertexCount) => count
  | .inl (.inr .result) => result
  | .inr .remainingLevel => encodeWord remaining ++ suffix
  | .inr .currentSize => current
  | _ => []

def cycleTapes (remaining : Nat) (word current vertex count result suffix : List Bool) :
    (tape : Tape) → List (Alphabet tape) :=
  fromBoolTapes (cycleWords remaining word current vertex count result suffix)

@[simp] theorem toBool_cycleTapes (remaining : Nat)
    (word current vertex count result suffix : List Bool) :
    toBoolTapes (cycleTapes remaining word current vertex count result suffix) =
      cycleWords remaining word current vertex count result suffix := toBool_fromBool _

theorem boundaryTapes_eq_cycleTapes (remaining current : Nat) (word suffix : List Bool) :
    boundaryTapes remaining current word suffix =
      cycleTapes remaining word (encodeWord current) [] [] [] suffix := by
  funext tape
  rcases tape with tape | tape
  · rcases tape with row | extra
    · cases row <;> rfl
    · cases extra <;> rfl
  · cases tape <;> rfl

def copyFrame (remaining current : Nat) (word suffix : List Bool) :
    (tape : Tape) → List (Alphabet tape) :=
  MachineEmbedding.tapes (MachineExpanderTable.initialTapes current word [])
    (extraFrame remaining current suffix)

def returnFrame (remaining current : Nat) (oldWord newWord suffix : List Bool) :
    (tape : Tape) → List (Alphabet tape) :=
  MachineEmbedding.tapes (MachineExpanderTable.finalTapes current oldWord newWord [])
    (extraFrame remaining current suffix)

def installedFrame (remaining current : Nat) (newWord suffix : List Bool) :
    (tape : Tape) → List (Alphabet tape) :=
  cycleTapes remaining newWord [] (encodeWord current) (encodeWord 0) [] suffix

def multipliedFrame (d remaining current : Nat) (newWord suffix : List Bool) :
    (tape : Tape) → List (Alphabet tape) :=
  cycleTapes remaining newWord (encodeWord (cloudSize d * current))
    (encodeWord current) (encodeWord 0) [] suffix

theorem copyFrame_eq_cycleTapes (remaining current : Nat) (word suffix : List Bool) :
    copyFrame remaining current word suffix =
      cycleTapes remaining word (encodeWord current) [] (encodeWord current) [] suffix := by
  funext tape
  rcases tape with tape | tape
  · rcases tape with row | extra
    · cases row <;> rfl
    · cases extra <;> simp [copyFrame, MachineEmbedding.tapes,
        MachineExpanderTable.initialTapes, MachineExpanderTable.extraTapes,
        cycleTapes, fromBoolTapes, boolWord, cycleWords]
  · cases tape <;> rfl

theorem returnFrame_eq_cycleTapes (remaining current : Nat)
    (oldWord newWord suffix : List Bool) :
    returnFrame remaining current oldWord newWord suffix =
      cycleTapes remaining oldWord (encodeWord current) (encodeWord current)
        (encodeWord 0) newWord suffix := by
  funext tape
  rcases tape with tape | tape
  · rcases tape with row | extra
    · cases row <;> rfl
    · cases extra <;> simp [returnFrame, MachineEmbedding.tapes,
        MachineExpanderTable.finalTapes, MachineExpanderTable.boundaryTapes,
        MachineExpanderTable.extraTapes, cycleTapes, fromBoolTapes, boolWord, cycleWords]
  · cases tape <;> rfl

theorem copyFrame_eq_update (remaining current : Nat) (word suffix : List Bool) :
    Function.update (boundaryTapes remaining current word suffix)
      vertexCountTape (encodeWord current) = copyFrame remaining current word suffix := by
  rw [boundaryTapes_eq_cycleTapes, copyFrame_eq_cycleTapes]
  funext tape
  rcases tape with tape | tape
  · rcases tape with row | extra
    · cases row <;> rfl
    · cases extra <;> rfl
  · cases tape <;> rfl

theorem multipliedFrame_eq_update (d remaining current : Nat) (word suffix : List Bool) :
    Function.update (installedFrame remaining current word suffix)
      (.inr .currentSize) (encodeWord (cloudSize d * current)) =
      multipliedFrame d remaining current word suffix := by
  funext tape
  rcases tape with tape | tape
  · rcases tape with row | extra
    · cases row <;> rfl
    · cases extra <;> rfl
  · cases tape <;> rfl

end DFVSGames.Foundations.Complexity.MachineExpanderFamily
end

section

namespace DFVSGames.Foundations.Complexity.MachineExpanderFamilyBounds

open PCP.ExpanderTables PCP.ExpanderRowControl PCP.ExpanderTableWords

noncomputable def resizeCost {v d : Nat} (G : Table v (degree d))
    (H : Table (cloudSize d) d) : Nat :=
  (MachineExpanderTable.timePolynomial d).eval (MachineExpanderTable.oldTableWord G).length +
    6 * v + (MachineExpanderTable.oldTableWord G).length +
    2 * (encodeWords (rotationWords (step G H))).length + 15

def resizeCoefficient (d : Nat) : Nat :=
  MachineExpanderTable.timeCoefficient d * (degree d + 1) ^ 4 +
    3 * (degree d + 1) ^ 2 + 21

theorem wordLength_add_one_le {v q N : Nat} (G : Table v q) (hv : v ≤ N) :
    (encodeWords (rotationWords G)).length + 1 ≤ (q + 1)^2 * (N + 1)^2 := by
  have hlength := encode_rotationWords_length_le G
  have hrow : v * q + 1 ≤ (q + 1) * (N + 1) := by
    have h := Nat.mul_le_mul_right q hv
    nlinarith
  calc
    _ ≤ (v * q + 1)^2 := by nlinarith
    _ ≤ ((q + 1) * (N + 1))^2 := Nat.pow_le_pow_left hrow 2
    _ = _ := by ring

theorem resizeCost_le {v d : Nat} (G : Table v (degree d))
    (H : Table (cloudSize d) d) (N : Nat) (hv : v ≤ N)
    (hnext : v * cloudSize d ≤ N) :
    resizeCost G H ≤ resizeCoefficient d * (N + 1)^4 := by
  let oldL := (MachineExpanderTable.oldTableWord G).length
  let newL := (encodeWords (rotationWords (step G H))).length
  have oldBound : oldL + 1 ≤ (degree d + 1)^2 * (N + 1)^2 :=
    wordLength_add_one_le G hv
  have newBound : newL + 1 ≤ (degree d + 1)^2 * (N + 1)^2 :=
    wordLength_add_one_le (step G H) hnext
  have timeBound : (MachineExpanderTable.timePolynomial d).eval oldL ≤
      MachineExpanderTable.timeCoefficient d * (degree d + 1)^4 * (N + 1)^4 := by
    simp only [MachineExpanderTable.timePolynomial, Polynomial.eval_mul,
      Polynomial.eval_C, Polynomial.eval_pow, Polynomial.eval_add,
      Polynomial.eval_X, Polynomial.eval_one]
    calc
      _ ≤ MachineExpanderTable.timeCoefficient d *
          ((degree d + 1)^2 * (N + 1)^2)^2 :=
        Nat.mul_le_mul_left _ (Nat.pow_le_pow_left oldBound 2)
      _ = _ := by ring
  have pow24 : (N + 1)^2 ≤ (N + 1)^4 :=
    Nat.pow_le_pow_right (Nat.succ_pos N) (by decide)
  have overheadBound : oldL + 2 * newL ≤
      3 * (degree d + 1)^2 * (N + 1)^4 := by
    have h := Nat.mul_le_mul_left ((degree d + 1)^2) pow24
    calc
      _ ≤ 3 * ((degree d + 1)^2 * (N + 1)^2) := by omega
      _ ≤ 3 * ((degree d + 1)^2 * (N + 1)^4) := Nat.mul_le_mul_left 3 h
      _ = _ := by ring
  have npow : N + 1 ≤ (N + 1)^4 := by
    simpa only [pow_one] using
      Nat.pow_le_pow_right (Nat.succ_pos N) (show 1 ≤ 4 by decide)
  have linearBound : 6 * v + 15 ≤ 21 * (N + 1)^4 := by omega
  change (MachineExpanderTable.timePolynomial d).eval oldL + 6 * v + oldL +
    2 * newL + 15 ≤ _
  unfold resizeCoefficient
  nlinarith only [timeBound, overheadBound, linearBound]

theorem vertexCount_monotone {q : Nat} (positive : 0 < q) :
    Monotone (vertexCount q) := by
  intro m n hmn
  rw [vertexCount_eq, vertexCount_eq]
  exact Nat.pow_le_pow_right (Nat.mul_pos positive positive) hmn

theorem level_le_vertexCount {q : Nat} (growth : 1 < q * q) (level : Nat) :
    level ≤ vertexCount q level := by
  have positive : 0 < q := by nlinarith
  induction level with
  | zero => exact Nat.zero_le _
  | succ level ih =>
    have hpos := vertexCount_positive positive level
    have hmul := Nat.mul_le_mul_left (vertexCount q level)
      (show 2 ≤ q * q by omega)
    change level + 1 ≤ vertexCount q level * (q * q)
    nlinarith

noncomputable def resizeSum {d : Nat} (H : Table (cloudSize d) d) (level : Nat) : Nat :=
  ∑ i ∈ Finset.range level, resizeCost (family H i) H

@[simp] theorem resizeSum_zero {d : Nat} (H : Table (cloudSize d) d) :
    resizeSum H 0 = 0 := by simp [resizeSum]

theorem resizeSum_succ {d : Nat} (H : Table (cloudSize d) d) (level : Nat) :
    resizeSum H (level + 1) = resizeSum H level + resizeCost (family H level) H := by
  simp only [resizeSum, Finset.sum_range_succ]

theorem resizeSum_add {d : Nat} (H : Table (cloudSize d) d) (start count : Nat) :
    resizeSum H (start + count) = resizeSum H start +
      ∑ i ∈ Finset.range count, resizeCost (family H (start + i)) H :=
  Finset.sum_range_add (fun i => resizeCost (family H i) H) start count

noncomputable def familyBudget {d : Nat} (H : Table (cloudSize d) d) (level : Nat) : Nat :=
  resizeSum H level + level + 3

@[simp] theorem familyBudget_zero {d : Nat} (H : Table (cloudSize d) d) :
    familyBudget H 0 = 3 := by simp [familyBudget]

theorem familyBudget_succ {d : Nat} (H : Table (cloudSize d) d) (level : Nat) :
    familyBudget H (level + 1) =
      familyBudget H level + resizeCost (family H level) H + 1 := by
  simp only [familyBudget, resizeSum_succ]
  omega

theorem resizeSum_le {d : Nat} (H : Table (cloudSize d) d)
    (growth : 1 < cloudSize d) (level : Nat) :
    resizeSum H level ≤ resizeCoefficient d * (vertexCount (degree d) level + 1)^5 := by
  let N := vertexCount (degree d) level
  have positive : 0 < degree d := by
    have h : 1 < degree d * degree d := growth
    nlinarith
  have hmono := vertexCount_monotone positive
  have hlevel : level ≤ N := level_le_vertexCount growth level
  have eachBound : ∀ i ∈ Finset.range level,
      resizeCost (family H i) H ≤ resizeCoefficient d * (N + 1)^4 := by
    intro i hi
    have hi' : i < level := Finset.mem_range.mp hi
    apply resizeCost_le (family H i) H N
    · exact hmono (by omega)
    · exact hmono (show i + 1 ≤ level by omega)
  have sumBound : resizeSum H level ≤ level * (resizeCoefficient d * (N + 1)^4) := by
    calc
      _ ≤ ∑ _i ∈ Finset.range level, resizeCoefficient d * (N + 1)^4 :=
        Finset.sum_le_sum eachBound
      _ = _ := by simp
  calc
    _ ≤ level * (resizeCoefficient d * (N + 1)^4) := sumBound
    _ ≤ (N + 1) * (resizeCoefficient d * (N + 1)^4) :=
      Nat.mul_le_mul_right _ (hlevel.trans (Nat.le_succ N))
    _ = _ := by ring

theorem familyBudget_le {d : Nat} (H : Table (cloudSize d) d)
    (growth : 1 < cloudSize d) (level : Nat) :
    familyBudget H level ≤
      (resizeCoefficient d + 4) * (vertexCount (degree d) level + 1)^5 := by
  have hsum := resizeSum_le H growth level
  have hlevel := level_le_vertexCount growth level
  have hpow : vertexCount (degree d) level + 1 ≤
      (vertexCount (degree d) level + 1)^5 := by
    simpa only [pow_one] using Nat.pow_le_pow_right
      (Nat.succ_pos (vertexCount (degree d) level)) (show 1 ≤ 5 by decide)
  have overhead : level + 3 ≤ 4 * (vertexCount (degree d) level + 1)^5 := by omega
  unfold familyBudget
  nlinarith only [hsum, overhead]

theorem paddedSize_le_succ_input (k : Nat) :
    PCP.PreprocessingLevels.paddedSize k ≤ PCP.ExpanderFamily.growth * (k + 1) := by
  by_cases hk : k = 0
  · subst k
    have h := PCP.ExpanderFamily.growth_gt_one
    simp only [PCP.PreprocessingLevels.paddedSize_zero, Nat.zero_add, Nat.mul_one]
    omega
  · exact (PCP.PreprocessingLevels.paddedSize_bounds (Nat.pos_of_ne_zero hk)).2.trans
      (Nat.mul_le_mul_left _ (Nat.le_succ k))

def inputCoefficient : Nat :=
  (resizeCoefficient PCP.Expanders.baseDegree + 4) * (PCP.ExpanderFamily.growth + 1)^5

theorem familyBudget_at_boundedLevel_le
    (H : Table (cloudSize PCP.Expanders.baseDegree) PCP.Expanders.baseDegree) (k : Nat) :
    familyBudget H (PCP.PreprocessingLevels.boundedLevel k) ≤ inputCoefficient * (k + 1)^5 := by
  have hg : cloudSize PCP.Expanders.baseDegree = PCP.ExpanderFamily.growth := by
    unfold cloudSize degree PCP.ExpanderFamily.growth
    ring
  have growth : 1 < cloudSize PCP.Expanders.baseDegree := by
    rw [hg]
    exact PCP.ExpanderFamily.growth_gt_one
  have bound := familyBudget_le H growth (PCP.PreprocessingLevels.boundedLevel k)
  rw [PCP.PreprocessingLevels.table_vertexCount_eq_paddedSize] at bound
  have sizeBound : PCP.PreprocessingLevels.paddedSize k + 1 ≤
      (PCP.ExpanderFamily.growth + 1) * (k + 1) := by
    have h := paddedSize_le_succ_input k
    nlinarith
  calc
    _ ≤ (resizeCoefficient PCP.Expanders.baseDegree + 4) *
        (PCP.PreprocessingLevels.paddedSize k + 1)^5 := bound
    _ ≤ (resizeCoefficient PCP.Expanders.baseDegree + 4) *
        ((PCP.ExpanderFamily.growth + 1) * (k + 1))^5 :=
      Nat.mul_le_mul_left _ (Nat.pow_le_pow_left sizeBound 5)
    _ = _ := by unfold inputCoefficient; ring

end DFVSGames.Foundations.Complexity.MachineExpanderFamilyBounds

end

section

namespace DFVSGames.Foundations.Complexity.MachineExpanderFamily

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

theorem installed_returnFrame (remaining current : Nat) (oldWord newWord suffix : List Bool) :
    fromBoolTapes (installedTapes (toBoolTapes (returnFrame remaining current oldWord newWord suffix))
      newWord) = installedFrame remaining current newWord suffix := by
  rw [returnFrame_eq_cycleTapes, toBool_cycleTapes]
  funext tape
  rcases tape with tape | tape
  · rcases tape with row | extra
    · cases row <;> rfl
    · cases extra <;> rfl
  · cases tape <;> rfl

theorem cleaned_multipliedFrame (d remaining current : Nat) (newWord suffix : List Bool) :
    fromBoolTapes (cleanedCounterTapes
      (toBoolTapes (multipliedFrame d remaining current newWord suffix))) =
      boundaryTapes remaining (current * cloudSize d) newWord suffix := by
  rw [boundaryTapes_eq_cycleTapes]
  unfold multipliedFrame
  rw [toBool_cycleTapes]
  have multiply : cloudSize d * current = current * cloudSize d := Nat.mul_comm _ _
  rw [multiply]
  funext tape
  rcases tape with tape | tape
  · rcases tape with row | extra
    · cases row <;> rfl
    · cases extra <;> rfl
  · cases tape <;> rfl

structure ResizeRun {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (H : Table (cloudSize d) d) (growth : 1 < cloudSize d)
    (G : Table v (degree d)) (remaining : Nat) (state : State ρ d) (suffix : List Bool) where
  finalState : State ρ d
  caller_preserved : caller finalState = caller state
  execution : StateTransition.EvalsToInTime (TM2.step (program positive H growth))
    ⟨some (.inr (.affine .copyCount .seed)), state,
      boundaryTapes remaining v (MachineExpanderTable.oldTableWord G) suffix⟩
    (some ⟨some (.inr .levelGuard), finalState,
      boundaryTapes remaining (v * cloudSize d)
        (encodeWords (rotationWords (step G H))) suffix⟩)
    (resizeCost G H)

noncomputable def resizeInTime {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (H : Table (cloudSize d) d) (growth : 1 < cloudSize d)
    (G : Table v (degree d)) (remaining : Nat) (state : State ρ d) (suffix : List Bool) :
    ResizeRun positive H growth G remaining state suffix := by
  let oldWord := MachineExpanderTable.oldTableWord G
  let newWord := encodeWords (rotationWords (step G H))
  have copy := copyCountInTime positive H growth
    (boundaryTapes remaining v oldWord suffix) v rfl rfl rfl state
  rw [copyFrame_eq_update] at copy
  let called := MachineExpanderTable.tableInTime positive G H (clearRegister state).1 []
  let returned : State ρ d := (called.finalState, ())
  have eta : ((clearRegister state).1, ()) = clearRegister state := by
    rcases state with ⟨state, extra⟩
    cases extra
    rfl
  have call : StateTransition.EvalsToInTime (TM2.step (program positive H growth))
      ⟨some (.inl (.inr .initialize)), clearRegister state,
        copyFrame remaining v oldWord suffix⟩
      (some ⟨some (.inr .clearOldTable), returned,
        returnFrame remaining v oldWord newWord suffix⟩)
      ((MachineExpanderTable.timePolynomial d).eval oldWord.length) := by
    simpa only [MachineEmbedding.configuration, MachineEmbedding.label, tableReturn, eta,
      copyFrame, returnFrame, oldWord, newWord, returned] using
      tableExecution positive H growth (extraFrame remaining v suffix) called.execution
  have install := installTableInTime positive H growth
    (returnFrame remaining v oldWord newWord suffix) newWord
    (by rw [returnFrame_eq_cycleTapes, toBool_cycleTapes]; rfl)
    (by rw [returnFrame_eq_cycleTapes, toBool_cycleTapes]; rfl) returned
  rw [installed_returnFrame] at install
  have installRun : StateTransition.EvalsToInTime (TM2.step (program positive H growth))
      ⟨some (.inr .clearOldTable), returned, returnFrame remaining v oldWord newWord suffix⟩
      (some ⟨some (.inr (.affine .multiplySize .seed)), clearRegister returned,
        installedFrame remaining v newWord suffix⟩)
      (oldWord.length + 2 * newWord.length + (v + 1) + 4) := by
    simpa only [returnFrame_eq_cycleTapes, toBool_cycleTapes, cycleWords, tableTape,
      encodeWord_length] using install
  have multiply := multiplySizeInTime positive H growth
    (installedFrame remaining v newWord suffix) v rfl rfl rfl (clearRegister returned)
  rw [clearRegister_idempotent, multipliedFrame_eq_update] at multiply
  have clean := cleanupCountersInTime positive H growth
    (multipliedFrame d remaining v newWord suffix) (clearRegister returned)
  rw [clearRegister_idempotent, cleaned_multipliedFrame] at clean
  have cleanRun : StateTransition.EvalsToInTime (TM2.step (program positive H growth))
      ⟨some (.inr .drainInputVertex), clearRegister returned,
        multipliedFrame d remaining v newWord suffix⟩
      (some ⟨some (.inr .levelGuard), clearRegister returned,
        boundaryTapes remaining (v * cloudSize d) newWord suffix⟩) (v + 4) := by
    simpa only [multipliedFrame, toBool_cycleTapes, cycleWords, inputVertexTape,
      vertexCountTape, encodeWord_length, Nat.zero_add, Nat.add_assoc] using clean
  let a := StateTransition.EvalsToInTime.trans (TM2.step (program positive H growth))
    _ _ _ _ _ copy call
  let b := StateTransition.EvalsToInTime.trans (TM2.step (program positive H growth))
    _ _ _ _ _ a installRun
  let c := StateTransition.EvalsToInTime.trans (TM2.step (program positive H growth))
    _ _ _ _ _ b multiply
  let all := StateTransition.EvalsToInTime.trans (TM2.step (program positive H growth))
    _ _ _ _ _ c cleanRun
  refine ⟨clearRegister returned, ?_, ?_⟩
  · exact (caller_clearRegister returned).trans
      (called.caller_preserved.trans (caller_clearRegister state))
  · refine { toEvalsTo := all.toEvalsTo, steps_le_m := ?_ }
    have bound := all.steps_le_m
    unfold resizeCost
    change _ ≤ (MachineExpanderTable.timePolynomial d).eval oldWord.length +
      6 * v + oldWord.length + 2 * newWord.length + 15
    omega

end DFVSGames.Foundations.Complexity.MachineExpanderFamily
end

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

end OAI