Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.DirectedFeedback.Machines.Label

namespace OAI

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section

namespace DFVSGames.Foundations.Complexity.MachineRegularExecutionBounds

open Turing MachineComposition PCP PCP.PreprocessingRegularTables PCP.PreprocessingCloudIndex
open PCP.PreprocessingMachineBounds MachineRegularInternalRow

theorem coreTimeBound_mono (q L R x v i k o m O L' R' x' v' i' k' o' m' O' : Nat)
    (hL : L ≤ L') (hR : R ≤ R') (hx : x ≤ x') (hv : v ≤ v') (hi : i ≤ i')
    (hk : k ≤ k') (ho : o ≤ o') (hm : m ≤ m') (hO : O ≤ O') :
    coreTimeBound q L R x v i k o m O ≤ coreTimeBound q L' R' x' v' i' k' o' m' O' := by
  unfold coreTimeBound
  gcongr

noncomputable def corePolynomial (q : Nat) : Polynomial Nat :=
  Polynomial.C (5*q+32) * rotorPolynomial q + 10 * Polynomial.X +
    2 * (Polynomial.C ExpanderFamily.growth * Polynomial.X) + 13 * Polynomial.X +
    5 * Polynomial.X + Polynomial.C (5*q+10) *
      (Polynomial.X + Polynomial.C ExpanderFamily.growth * Polynomial.X) +
    4 * (Polynomial.C ExpanderFamily.growth * Polynomial.X) +
    2 * regularPolynomial + Polynomial.C (8*q+41066)

theorem corePolynomial_eval (q L : Nat) :
    (corePolynomial q).eval L =
      coreTimeBound q L ((rotorPolynomial q).eval L)
        (ExpanderFamily.growth*L) L (ExpanderFamily.growth*L) L
        (ExpanderFamily.growth*L) L (regularPolynomial.eval L) := by
  simp only [corePolynomial, coreTimeBound, Polynomial.eval_add, Polynomial.eval_mul,
    Polynomial.eval_C, Polynomial.eval_X, Polynomial.eval_ofNat]
  omega

theorem internal_steps_le (q : Nat) (positive : 0 < q) (t : GraphTables.Table)
    (tables : ∀ v, ExpanderTables.Table (cloudSize t v + padding t v) q)
    (v : Fin t.vertices) (x : PaddedCloud t (padding t) v) (p : Fin q)
    (outputLength : Nat) (houtput : outputLength ≤ regularPolynomial.eval (inputLength t)) :
    coreSteps t (padding t) tables v x p outputLength ≤
      (corePolynomial q).eval (inputLength t) := by
  have hx : (vertexOrder t (padding t) x.val).val ≤ ExpanderFamily.growth * inputLength t :=
    (vertexOrder t (padding t) x.val).isLt.le.trans (regularVertices_le_input t)
  have hi : (paddedCloudRank t (padding t) v x).val ≤ ExpanderFamily.growth * inputLength t :=
    (paddedCloudRank t (padding t) v x).isLt.le.trans (cloudTotal_le_input t v)
  have hv : v.val ≤ inputLength t :=
    v.isLt.le.trans (GraphTables.vertices_le_tableBits_length t)
  have hm : t.darts ≤ inputLength t := GraphTables.darts_le_tableBits_length t
  rw [corePolynomial_eval]
  exact (coreSteps_le q positive t (padding t) tables v x p outputLength).trans
    (coreTimeBound_mono q _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ le_rfl
      (cloudRotorBits_le t v (tables v)) hx hv hi (cloudSize_le_input t v)
      (prefix_le_input t v.val) hm houtput)

noncomputable def familyPolynomial : Polynomial Nat :=
  MachinePaddedExpanderFamilyBounds.timePolynomial.comp (Polynomial.X + 1) +
    Polynomial.C (2 * ExpanderFamily.growth) * (Polynomial.X + 1) + 7

theorem family_budget_le (t : GraphTables.Table) (v : Fin t.vertices) :
    MachinePaddedExpanderFamilyBounds.timePolynomial.eval (cloudSize t v + 1) +
      2 * ExpanderFamily.growth * (cloudSize t v + 1) + 7 ≤
        familyPolynomial.eval (inputLength t) := by
  have h := Nat.add_le_add_right (cloudSize_le_input t v) 1
  have hp := natPolynomial_eval_mono MachinePaddedExpanderFamilyBounds.timePolynomial h
  have hm := Nat.mul_le_mul_left (2 * ExpanderFamily.growth) h
  simp only [familyPolynomial, Polynomial.eval_add, Polynomial.eval_comp,
    Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X, Polynomial.eval_one,
    Polynomial.eval_ofNat]
  omega

noncomputable def rowPolynomial (q : Nat) : Polynomial Nat :=
  Polynomial.C ExpanderFamily.growth * Polynomial.X +
    (Polynomial.C ExpanderFamily.growth * Polynomial.X) * Polynomial.C (q + 1) + 8192

theorem rowPolynomial_eval (q L : Nat) :
    (rowPolynomial q).eval L =
      ExpanderFamily.growth * L + (ExpanderFamily.growth * L) * (q + 1) + 8192 := by
  simp only [rowPolynomial, Polynomial.eval_add, Polynomial.eval_mul,
    Polynomial.eval_C, Polynomial.eval_X, Polynomial.eval_ofNat]

theorem actual_row_bound_le (q : Nat) (t : GraphTables.Table) :
    vertexCount t (padding t) + vertexCount t (padding t) * (q + 1) + 8192 ≤
      (rowPolynomial q).eval (inputLength t) := by
  rw [rowPolynomial_eval]
  have h := regularVertices_le_input t
  exact Nat.add_le_add_right (Nat.add_le_add h (Nat.mul_le_mul_right _ h)) _

noncomputable def blockCorePolynomial (q : Nat) : Polynomial Nat :=
  corePolynomial q + Polynomial.C (2*q) * rowPolynomial q

theorem coreTimeBound_add_output (q L R x v i k o m O A : Nat) :
    coreTimeBound q L R x v i k o m (O + A) =
      coreTimeBound q L R x v i k o m O + 2*A := by
  unfold coreTimeBound
  omega

theorem block_core_bound_le (q : Nat) (t : GraphTables.Table)
    (tables : ∀ v, ExpanderTables.Table (cloudSize t v + padding t v) q)
    (v : Fin t.vertices) (x : PaddedCloud t (padding t) v)
    (initialOutput : Nat) (houtput : initialOutput ≤ regularPolynomial.eval (inputLength t)) :
    coreTimeBound q (GraphTables.tableBits t).length
      (encodeWords (ExpanderTableWords.rotationWords (tables v))).length
      (vertexOrder t (padding t) x.val).val v.val (paddedCloudRank t (padding t) v x).val
      (cloudSize t v) (PreprocessingPaddingOffsets.offset (padding t) v.val) t.darts
      (initialOutput + q *
        (vertexCount t (padding t) + vertexCount t (padding t) * (q + 1) + 8192)) ≤
      (blockCorePolynomial q).eval (inputLength t) := by
  have hx : (vertexOrder t (padding t) x.val).val ≤ ExpanderFamily.growth * inputLength t :=
    (vertexOrder t (padding t) x.val).isLt.le.trans (regularVertices_le_input t)
  have hi : (paddedCloudRank t (padding t) v x).val ≤ ExpanderFamily.growth * inputLength t :=
    (paddedCloudRank t (padding t) v x).isLt.le.trans (cloudTotal_le_input t v)
  have hv : v.val ≤ inputLength t :=
    v.isLt.le.trans (GraphTables.vertices_le_tableBits_length t)
  have hm : t.darts ≤ inputLength t := GraphTables.darts_le_tableBits_length t
  have hr := Nat.mul_le_mul_left q (actual_row_bound_le q t)
  have hO := Nat.add_le_add houtput hr
  have h := coreTimeBound_mono q (inputLength t) _ _ _ _ _ _ _ _
    (inputLength t) _ _ _ _ _ _ _ _ le_rfl
    (cloudRotorBits_le t v (tables v)) hx hv hi (cloudSize_le_input t v)
    (prefix_le_input t v.val) hm hO
  simp only [coreTimeBound_add_output] at h ⊢
  rw [← corePolynomial_eval] at h
  simpa only [inputLength, blockCorePolynomial, Polynomial.eval_add, Polynomial.eval_mul,
    Polynomial.eval_C, Nat.mul_assoc] using h

end DFVSGames.Foundations.Complexity.MachineRegularExecutionBounds
end

section

namespace DFVSGames.Foundations.Complexity.MachineRegularOriginalBodyBounds

open Turing MachineComposition PCP PreprocessingCloudIndex PreprocessingRegularTables
open PreprocessingMachineBounds MachineRegularOriginalBody

noncomputable def wide : Polynomial Nat := Polynomial.C (ExpanderFamily.growth + 1) * Polynomial.X
noncomputable def rowPolynomial (q : Nat) : Polynomial Nat :=
  wide + wide * Polynomial.C (q + 1) + 8192
noncomputable def outputPolynomial (q : Nat) : Polynomial Nat :=
  wide + Polynomial.C q * rowPolynomial q
noncomputable def corePolynomial (q : Nat) : Polynomial Nat :=
  Polynomial.C (5*q+32) * rotorPolynomial q + 10*wide + 2*wide + 13*wide + 5*wide +
    Polynomial.C (5*q+10)*(wide+wide) + 4*wide + 2*outputPolynomial q +
      Polynomial.C (8*q+41066)
noncomputable def blockPolynomial (q : Nat) : Polynomial Nat :=
  Polynomial.C q * corePolynomial q + Polynomial.C (5*q+20)*wide +
    2*outputPolynomial q + Polynomial.C (5*q+40995)
noncomputable def cleanupPolynomial (q : Nat) : Polynomial Nat := 6*wide + rotorPolynomial q + 13
noncomputable def timePolynomial : Polynomial Nat :=
  MachineRegularMetadata.timePolynomial + MachineRegularExecutionBounds.familyPolynomial +
    blockPolynomial internalDegree + cleanupPolynomial internalDegree

def bodySize (t : GraphTables.Table) (output : List Bool) : Nat :=
  inputLength t + output.length

theorem blockPolynomial_eval (q L : Nat) :
    (blockPolynomial q).eval L =
      MachineRegularVertexBlock.originalTimeBound q
        ((ExpanderFamily.growth+1)*L) ((rotorPolynomial q).eval L)
        ((ExpanderFamily.growth+1)*L) ((ExpanderFamily.growth+1)*L)
        ((ExpanderFamily.growth+1)*L) ((ExpanderFamily.growth+1)*L)
        ((ExpanderFamily.growth+1)*L) ((ExpanderFamily.growth+1)*L)
        ((ExpanderFamily.growth+1)*L) ((ExpanderFamily.growth+1)*L) := by
  simp only [blockPolynomial, corePolynomial, outputPolynomial, rowPolynomial, wide,
    Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X,
    Polynomial.eval_ofNat, MachineRegularVertexBlock.originalTimeBound,
    MachineRegularVertexBlock.portTimeBound, MachineRegularInternalRow.coreTimeBound,
    MachineRegularOriginalClean.timeBound, MachineRegularVertexBlock.rowSizeBound]
  ring

theorem originalTimeBound_le (q L R x v i k o m O V W R' : Nat)
    (hL : L ≤ W) (hR : R ≤ R') (hx : x ≤ W) (hv : v ≤ W)
    (hi : i ≤ W) (hk : k ≤ W) (ho : o ≤ W) (hm : m ≤ W)
    (hO : O ≤ W) (hV : V ≤ W) :
    MachineRegularVertexBlock.originalTimeBound q L R x v i k o m O V ≤
      MachineRegularVertexBlock.originalTimeBound q W R' W W W W W W W W := by
  unfold MachineRegularVertexBlock.originalTimeBound MachineRegularVertexBlock.portTimeBound
    MachineRegularInternalRow.coreTimeBound MachineRegularOriginalClean.timeBound
    MachineRegularVertexBlock.rowSizeBound
  gcongr

theorem wide_le (t : GraphTables.Table) (output : List Bool) :
    inputLength t ≤ (ExpanderFamily.growth+1)*bodySize t output ∧
    ExpanderFamily.growth*inputLength t ≤ (ExpanderFamily.growth+1)*bodySize t output ∧
    output.length ≤ (ExpanderFamily.growth+1)*bodySize t output := by
  have hN : inputLength t ≤ bodySize t output := Nat.le_add_right _ _
  have hO : output.length ≤ bodySize t output := Nat.le_add_left _ _
  have hg := Nat.mul_le_mul_left ExpanderFamily.growth hN
  rw [Nat.add_mul, Nat.one_mul]
  omega

theorem vertexSteps_le (H : PreprocessingRegularTables.BaseTable) (t : GraphTables.Table) (e : Fin t.darts)
    (output : List Bool) :
    MachineRegularOriginalBody.vertexSteps H t e output ≤
      (blockPolynomial internalDegree).eval (bodySize t output) := by
  let v := t.rows[e].tail
  let W := (ExpanderFamily.growth+1)*bodySize t output
  have bounds := wide_le t output
  have hx := e.isLt.le.trans (GraphTables.darts_le_tableBits_length t)
  have hv := v.isLt.le.trans (GraphTables.vertices_le_tableBits_length t)
  have hk := cloudSize_le_input t v
  have hi := (cloudRank t v (MachineRegularMetadata.originalMember t e)).isLt.le.trans hk
  have hm := GraphTables.darts_le_tableBits_length t
  have ho := prefix_le_input t v.val
  have hV := regularVertices_le_input t
  have hR := (cloudRotorBits_le t v (familyCloudTable H t v)).trans
    (natPolynomial_eval_mono (rotorPolynomial internalDegree)
      (Nat.le_add_right (inputLength t) output.length))
  have raw := MachineRegularVertexBlock.originalSteps_le internalDegree degree_positive
    t (padding t) (familyCloudTable H t) e output.length
  have bound := originalTimeBound_le internalDegree _ _ _ _ _ _ _ _ _ _ W _
    bounds.1 hR (hx.trans bounds.1) (hv.trans bounds.1) (hi.trans bounds.1)
    (hk.trans bounds.1) (ho.trans bounds.2.1) (hm.trans bounds.1) bounds.2.2
    (hV.trans bounds.2.1)
  rw [blockPolynomial_eval]
  exact raw.trans bound

theorem clearSteps_frame (data : Data) :
    clearSteps (frame data) 7 = data.owner.length + data.localRank.length +
      data.count.length + data.offset.length + data.rotor.length + data.padding.length +
      data.level.length + 7 := by
  simp [clearSteps, clearMemory, clearTape, frame, MachineRegularMetadata.frame]
  omega

theorem cleanupSteps_le (H : PreprocessingRegularTables.BaseTable) (t : GraphTables.Table) (e : Fin t.darts)
    (output : List Bool) :
    clearSteps (frame (emittedData H t e output)) 7 ≤
      (cleanupPolynomial internalDegree).eval (bodySize t output) := by
  let v := t.rows[e].tail
  have bounds := wide_le t output
  have hv := v.isLt.le.trans (GraphTables.vertices_le_tableBits_length t)
  have hk := cloudSize_le_input t v
  have hi := (cloudRank t v (MachineRegularMetadata.originalMember t e)).isLt.le.trans hk
  have ho := prefix_le_input t v.val
  have hp := cloudTotal_le_input t v
  have hl := level_le_input t v
  have hrotor := PreprocessingFamilyBridge.familyRotor_eq_familyCloudTable H t v
    (PreprocessingFamilyBridge.cloudSize_pos_of_dart t e)
  change MachineRegularFamily.rotor H (cloudSize t v) = _ at hrotor
  have hR := (cloudRotorBits_le t v (familyCloudTable H t v)).trans
    (natPolynomial_eval_mono (rotorPolynomial internalDegree)
      (Nat.le_add_right (inputLength t) output.length))
  rw [← hrotor] at hR
  rw [clearSteps_frame]
  simp only [emittedData, familyData, metadataData, MachineRegularMetadata.originalData,
    MachineRegularMetadata.cloudData, MachineRegularMetadata.prefixData,
    MachineRegularMetadata.rankData, MachineRegularMetadata.ownerData, initialData,
    encodeWord_length, cleanupPolynomial, wide, Polynomial.eval_add, Polynomial.eval_mul,
    Polynomial.eval_C, Polynomial.eval_X, Polynomial.eval_ofNat]
  change _ ≤ 6*((ExpanderFamily.growth+1)*bodySize t output) +
    (rotorPolynomial internalDegree).eval (bodySize t output)+13
  change cloudSize t v + MachineCloudPadding.padding (cloudSize t v) ≤ _ at hp
  dsimp only [v, bodySize, inputLength] at *
  omega

theorem totalSteps_le (H : PreprocessingRegularTables.BaseTable) (t : GraphTables.Table) (e : Fin t.darts)
    (output : List Bool) :
    totalSteps H t e output ≤ timePolynomial.eval (bodySize t output) := by
  have hsize : inputLength t ≤ bodySize t output := Nat.le_add_right _ _
  have hm := (MachineRegularMetadata.totalTime_le t e).trans
    (natPolynomial_eval_mono MachineRegularMetadata.timePolynomial hsize)
  have hf := (familyExecution H t e output).steps_le_m
  have hfb := MachineRegularExecutionBounds.family_budget_le t t.rows[e].tail
  simp only [MachineRegularFamily.timePolynomial, encodeWord_length,
    Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X,
    Polynomial.eval_ofNat] at hf
  have hf' := (hf.trans hfb).trans
    (natPolynomial_eval_mono MachineRegularExecutionBounds.familyPolynomial hsize)
  have hv := vertexSteps_le H t e output
  have hc := cleanupSteps_le H t e output
  simp only [totalSteps, familySteps, timePolynomial, Polynomial.eval_add]
  omega

noncomputable def originalInTime (H : PreprocessingRegularTables.BaseTable) (t : GraphTables.Table) (e : Fin t.darts)
    (output : List Bool) :
    StateTransition.EvalsToInTime (TM2.step (program H))
      (cfg H (some entry) (initialData t e output))
      (some (cfg H none (initialData t e (output ++ emittedBits H t e))))
      (timePolynomial.eval (bodySize t output)) where
  steps := totalSteps H t e output
  evals_in_steps := originalTrace H t e output
  steps_le_m := totalSteps_le H t e output

end DFVSGames.Foundations.Complexity.MachineRegularOriginalBodyBounds
end

section

namespace DFVSGames.Foundations.Complexity.MachineRegularOwnerBodyBounds

open Turing MachineComposition PCP PreprocessingCloudIndex PreprocessingRegularTables
open PreprocessingMachineBounds PreprocessingRegularLoopWords

def bodySize (t : GraphTables.Table) (output : List Bool) : Nat :=
  inputLength t + output.length

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

noncomputable def rowPolynomial (q : Nat) : Polynomial Nat :=
  wide + wide * Polynomial.C (q + 1) + 8192

noncomputable def outputPolynomial (q : Nat) : Polynomial Nat :=
  wide + regularPolynomial + Polynomial.C q * rowPolynomial q

noncomputable def corePolynomial (q : Nat) : Polynomial Nat :=
  Polynomial.C (5*q+32) * rotorPolynomial q + 10*wide + 2*wide + 13*wide + 5*wide +
    Polynomial.C (5*q+10)*(wide+wide) + 4*wide + 2*outputPolynomial q +
      Polynomial.C (8*q+41066)

noncomputable def blockPolynomial (q : Nat) : Polynomial Nat :=
  Polynomial.C q * corePolynomial q + Polynomial.C (5*q+11)*wide +
    2*outputPolynomial q + Polynomial.C (5*q+40986)

theorem blockPolynomial_eval (q L : Nat) :
    (blockPolynomial q).eval L =
      MachineRegularVertexBlock.dummyTimeBound q
        ((ExpanderFamily.growth+1)*L) ((rotorPolynomial q).eval L)
        ((ExpanderFamily.growth+1)*L) ((ExpanderFamily.growth+1)*L)
        ((ExpanderFamily.growth+1)*L) ((ExpanderFamily.growth+1)*L)
        ((ExpanderFamily.growth+1)*L) ((ExpanderFamily.growth+1)*L)
        (((ExpanderFamily.growth+1)*L) + regularPolynomial.eval L)
        ((ExpanderFamily.growth+1)*L) := by
  simp only [blockPolynomial, corePolynomial, outputPolynomial, rowPolynomial, wide,
    Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X,
    Polynomial.eval_ofNat, MachineRegularVertexBlock.dummyTimeBound,
    MachineRegularVertexBlock.portTimeBound, MachineRegularInternalRow.coreTimeBound,
    MachineRegularDummyRow.timeBound, MachineRegularVertexBlock.rowSizeBound]
  simp only [Nat.add_assoc]

theorem dummyTimeBound_le (q L R x v i k o m O V W R' O' : Nat)
    (hL : L ≤ W) (hR : R ≤ R') (hx : x ≤ W) (hv : v ≤ W)
    (hi : i ≤ W) (hk : k ≤ W) (ho : o ≤ W) (hm : m ≤ W)
    (hO : O ≤ O') (hV : V ≤ W) :
    MachineRegularVertexBlock.dummyTimeBound q L R x v i k o m O V ≤
      MachineRegularVertexBlock.dummyTimeBound q W R' W W W W W W O' W := by
  unfold MachineRegularVertexBlock.dummyTimeBound MachineRegularVertexBlock.portTimeBound
    MachineRegularInternalRow.coreTimeBound MachineRegularDummyRow.timeBound
    MachineRegularVertexBlock.rowSizeBound
  gcongr

theorem wide_le (t : GraphTables.Table) (output : List Bool) :
    inputLength t ≤ (ExpanderFamily.growth+1)*bodySize t output ∧
    ExpanderFamily.growth*inputLength t ≤ (ExpanderFamily.growth+1)*bodySize t output ∧
    output.length ≤ (ExpanderFamily.growth+1)*bodySize t output := by
  have hN : inputLength t ≤ bodySize t output := Nat.le_add_right _ _
  have hO : output.length ≤ bodySize t output := Nat.le_add_left _ _
  have hg := Nat.mul_le_mul_left ExpanderFamily.growth hN
  rw [Nat.add_mul, Nat.one_mul]
  omega

theorem ownerPrefix_length_le (H : BaseTable) (t : GraphTables.Table)
    (v : Fin t.vertices) (n : Nat) :
    (blocksPrefix (PreprocessingRegularWords.dummyVertexBits t (padding t)
      (familyCloudTable H t) v) n).length ≤ regularPolynomial.eval (inputLength t) := by
  have h := PreprocessingRegularBounds.regular_dummyPrefix_le t H v n
  simp only [PreprocessingRegularBounds.dummyPrefix, List.length_append] at h
  omega

theorem growingOutput_length_le (H : BaseTable) (t : GraphTables.Table)
    (v : Fin t.vertices) (n : Nat) (output : List Bool) :
    (output ++ blocksPrefix (PreprocessingRegularWords.dummyVertexBits t (padding t)
      (familyCloudTable H t) v) n).length ≤
      (ExpanderFamily.growth+1)*bodySize t output + regularPolynomial.eval (bodySize t output) := by
  have h := (ownerPrefix_length_le H t v n).trans
    (natPolynomial_eval_mono regularPolynomial (Nat.le_add_right (inputLength t) output.length))
  have hO := (wide_le t output).2.2
  rw [List.length_append]
  exact Nat.add_le_add hO h

theorem dummySteps_le (H : BaseTable) (t : GraphTables.Table) (v : Fin t.vertices)
    (j : Fin (padding t v)) (n : Nat) (output : List Bool) :
    MachineRegularVertexBlock.dummySteps t (padding t) (familyCloudTable H t) v j
      (output ++ blocksPrefix (PreprocessingRegularWords.dummyVertexBits t (padding t)
        (familyCloudTable H t) v) n).length ≤
      (blockPolynomial internalDegree).eval (bodySize t output) := by
  let x := paddedNew t (padding t) v j
  let W := (ExpanderFamily.growth+1)*bodySize t output
  have bounds := wide_le t output
  have hx := (vertexOrder t (padding t) x.val).isLt.le.trans (regularVertices_le_input t)
  have hv := v.isLt.le.trans (GraphTables.vertices_le_tableBits_length t)
  have hi := (paddedCloudRank t (padding t) v x).isLt.le.trans (cloudTotal_le_input t v)
  have hk := cloudSize_le_input t v
  have hm := GraphTables.darts_le_tableBits_length t
  have ho := prefix_le_input t v.val
  have hV := regularVertices_le_input t
  have hR := (cloudRotorBits_le t v (familyCloudTable H t v)).trans
    (natPolynomial_eval_mono (rotorPolynomial internalDegree)
      (Nat.le_add_right (inputLength t) output.length))
  have hO := growingOutput_length_le H t v n output
  have raw := MachineRegularVertexBlock.dummySteps_le internalDegree
    MachineRegularOriginalBody.degree_positive t (padding t) (familyCloudTable H t) v j
      (output ++ blocksPrefix (PreprocessingRegularWords.dummyVertexBits t (padding t)
        (familyCloudTable H t) v) n).length
  have bound := dummyTimeBound_le internalDegree _ _ _ _ _ _ _ _ _ _ W _ _
    bounds.1 hR (hx.trans bounds.2.1) (hv.trans bounds.1) (hi.trans bounds.2.1)
    (hk.trans bounds.1) (ho.trans bounds.2.1) (hm.trans bounds.1) hO
    (hV.trans bounds.2.1)
  rw [blockPolynomial_eval]
  exact raw.trans bound

noncomputable def prePolynomial : Polynomial Nat :=
  MachineCloudPadding.timePolynomial.comp (2*Polynomial.X+1) + 4*wide + 6

noncomputable def loopPolynomial : Polynomial Nat :=
  wide * (blockPolynomial internalDegree + 2)

noncomputable def cleanupPolynomial : Polynomial Nat :=
  6*wide + rotorPolynomial internalDegree + 13

noncomputable def timePolynomial : Polynomial Nat :=
  prePolynomial + MachineRegularExecutionBounds.familyPolynomial + loopPolynomial +
    cleanupPolynomial + 2

theorem preSteps_le (t : GraphTables.Table) (v : Fin t.vertices) (output : List Bool) :
    MachineRegularOwnerBody.preSteps t v ≤ prePolynomial.eval (bodySize t output) := by
  have bounds := wide_le t output
  have hk := cloudSize_le_input t v
  have hd := cloudTotal_le_input t v
  have hv := v.isLt.le.trans (GraphTables.vertices_le_tableBits_length t)
  change v.val ≤ inputLength t at hv
  have hN : inputLength t ≤ bodySize t output := Nat.le_add_right _ _
  have hinput : MachineCloudPadding.inputLength t v [] ≤ 2*bodySize t output+1 := by
    simp only [MachineCloudPadding.inputLength, List.append_nil, encodeWord_length]
    change inputLength t + (v.val+1) ≤ _
    omega
  have hm := (MachineCloudPadding.totalTime_le t v []).trans
    (natPolynomial_eval_mono MachineCloudPadding.timePolynomial hinput)
  simp only [MachineRegularOwnerBody.preSteps, prePolynomial, wide, Polynomial.eval_add,
    Polynomial.eval_mul, Polynomial.eval_comp, Polynomial.eval_C, Polynomial.eval_X,
    Polynomial.eval_ofNat, Polynomial.eval_one]
  omega

theorem loopSteps_le (H : BaseTable) (t : GraphTables.Table) (v : Fin t.vertices)
    (output : List Bool) (n : Nat) (hn : n ≤ padding t v) :
    MachineRegularOwnerBody.loopSteps H t v output n hn ≤
      n * ((blockPolynomial internalDegree).eval (bodySize t output) + 2) := by
  induction n with
  | zero => simp only [MachineRegularOwnerBody.loopSteps, Nat.zero_mul, le_refl]
  | succ n ih =>
    have h : n < padding t v := by omega
    have first := ih (by omega)
    have second := dummySteps_le H t v ⟨n, h⟩ n output
    rw [MachineRegularOwnerBody.loopSteps, Nat.succ_mul]
    change MachineRegularOwnerBody.loopSteps H t v output n _ +
      (1 + MachineRegularVertexBlock.dummySteps t (padding t) (familyCloudTable H t) v ⟨n, h⟩
        (output ++ blocksPrefix (PreprocessingRegularWords.dummyVertexBits t (padding t)
          (familyCloudTable H t) v) n).length + 1) ≤ _
    omega

theorem allLoopSteps_le (H : BaseTable) (t : GraphTables.Table) (v : Fin t.vertices)
    (output : List Bool) :
    MachineRegularOwnerBody.loopSteps H t v output (padding t v) le_rfl ≤
      loopPolynomial.eval (bodySize t output) := by
  have hd := (cloudTotal_le_input t v).trans (wide_le t output).2.1
  have hp : padding t v ≤ (ExpanderFamily.growth+1)*bodySize t output := by omega
  have h := (loopSteps_le H t v output (padding t v) le_rfl).trans
    (Nat.mul_le_mul_right _ hp)
  simpa only [loopPolynomial, wide, Polynomial.eval_add, Polynomial.eval_mul,
    Polynomial.eval_C, Polynomial.eval_X, Polynomial.eval_ofNat] using h

theorem cleanupSteps_working (data : MachineRegularOwnerBody.Data)
    (fuel saved : List Bool) (d : Nat) :
    MachineRegularOwnerCleanup.steps (MachineRegularOwnerBody.working data fuel saved) d =
      2*(d+1) + data.localRank.length + data.count.length + data.rotor.length +
        data.padding.length + data.level.length + fuel.length + 6 := by
  change 2*(d+1) + ((data.localRank.length+1) + ((data.count.length+1) +
    ((data.rotor.length+1) + ((data.padding.length+1) + ((data.level.length+1) +
      ((fuel.length+1)+0)))))) = _
  omega

theorem cleanupCost_le (H : BaseTable) (t : GraphTables.Table) (v : Fin t.vertices)
    (output : List Bool) :
    MachineRegularOwnerBody.cleanupCost H t v output ≤
      cleanupPolynomial.eval (bodySize t output) := by
  have bounds := wide_le t output
  have hk := (cloudSize_le_input t v).trans bounds.1
  have htotal := (cloudTotal_le_input t v).trans bounds.2.1
  have hl := (level_le_input t v).trans bounds.1
  have hp : MachineCloudPadding.padding (cloudSize t v) = padding t v := rfl
  by_cases hd : padding t v = 0
  · rw [MachineRegularOwnerBody.cleanupCost, ite_eq_left hd, cleanupSteps_working]
    simp only [MachineRegularOwnerBody.seededData, MachineRegularOwnerBody.metadataData,
      MachineRegularMetadata.cloudData, MachineRegularOwnerBody.initialData,
      hp, encodeWord_length, List.length_nil, cleanupPolynomial, wide,
      Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X,
      Polynomial.eval_ofNat]
    omega
  · have hpositive : 0 < padding t v := by omega
    have hkpos := PreprocessingFamilyBridge.cloudSize_pos_of_dummy t v ⟨0, hpositive⟩
    have hrotor := PreprocessingFamilyBridge.familyRotor_eq_familyCloudTable H t v hkpos
    change MachineRegularFamily.rotor H (cloudSize t v) = _ at hrotor
    have hR := (cloudRotorBits_le t v (familyCloudTable H t v)).trans
      (natPolynomial_eval_mono (rotorPolynomial internalDegree)
        (Nat.le_add_right (inputLength t) output.length))
    rw [← hrotor] at hR
    change (MachineRegularFamily.rotor H (cloudSize t v)).length ≤
      (rotorPolynomial internalDegree).eval (bodySize t output) at hR
    rw [MachineRegularOwnerBody.cleanupCost, ite_eq_right hd,
      MachineRegularOwnerBody.loopFrame, cleanupSteps_working]
    simp only [MachineRegularOwnerBody.loopData, MachineRegularOwnerBody.vertexData,
      MachineRegularOwnerBody.seededData, MachineRegularOwnerBody.metadataData,
      MachineRegularMetadata.cloudData, MachineRegularOwnerBody.initialData,
      hp, encodeWord_length, Nat.sub_self, cleanupPolynomial, wide,
      Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X,
      Polynomial.eval_ofNat]
    omega

theorem ownerTime_le (H : BaseTable) (t : GraphTables.Table) (v : Fin t.vertices)
    (output : List Bool) :
    MachineRegularOwnerBody.ownerTime H t v output ≤ timePolynomial.eval (bodySize t output) := by
  have hp := preSteps_le t v output
  have hc := cleanupCost_le H t v output
  have hl := allLoopSteps_le H t v output
  have hf := (MachineRegularExecutionBounds.family_budget_le t v).trans
    (natPolynomial_eval_mono MachineRegularExecutionBounds.familyPolynomial
      (Nat.le_add_right (inputLength t) output.length))
  have hfamily : MachineRegularFamily.timePolynomial.eval (encodeWord (cloudSize t v)).length ≤
      MachineRegularExecutionBounds.familyPolynomial.eval (bodySize t output) := by
    simpa only [bodySize, MachineRegularFamily.timePolynomial, encodeWord_length, Polynomial.eval_add,
      Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X, Polynomial.eval_ofNat] using hf
  simp only [MachineRegularOwnerBody.ownerTime, timePolynomial, Polynomial.eval_add,
    Polynomial.eval_ofNat]
  split_ifs <;> omega

theorem totalSteps_le (H : BaseTable) (t : GraphTables.Table) (v : Fin t.vertices)
    (output : List Bool) :
    MachineRegularOwnerBody.totalSteps H t v output ≤ timePolynomial.eval (bodySize t output) :=
  (MachineRegularOwnerBody.totalSteps_le H t v output).trans (ownerTime_le H t v output)

noncomputable def ownerInTime (H : BaseTable) (t : GraphTables.Table) (v : Fin t.vertices)
    (output : List Bool) :
    StateTransition.EvalsToInTime (TM2.step (MachineRegularOwnerBody.program H))
      (MachineRegularOwnerBody.cfg H (some MachineRegularOwnerBody.entry)
        (MachineRegularOwnerBody.initialData t v output))
      (some (MachineRegularOwnerBody.cfg H none
        (MachineRegularOwnerBody.finalData H t v output)))
      (timePolynomial.eval (bodySize t output)) where
  steps := MachineRegularOwnerBody.totalSteps H t v output
  evals_in_steps := MachineRegularOwnerBody.ownerTrace H t v output
  steps_le_m := totalSteps_le H t v output

end DFVSGames.Foundations.Complexity.MachineRegularOwnerBodyBounds
end

section

namespace DFVSGames.Foundations.Complexity.MachineRegularTable.Top

open Turing MachineComposition PCP PreprocessingRegularTables
open PreprocessingRegularBounds PreprocessingMachineBounds

noncomputable def oldLoopPolynomial : Polynomial Nat :=
  Polynomial.X *
    (MachineRegularOriginalBodyBounds.timePolynomial.comp
      (Polynomial.X + regularPolynomial) + 2)

noncomputable def ownerLoopPolynomial : Polynomial Nat :=
  Polynomial.X *
    (MachineRegularOwnerBodyBounds.timePolynomial.comp
      (Polynomial.X + regularPolynomial) + 2)

noncomputable def timePolynomial : Polynomial Nat :=
  Header.timePolynomial internalDegree + oldLoopPolynomial + ownerLoopPolynomial +
    4 * Polynomial.X + 12 +
    Polynomial.C (7 * (ExpanderFamily.growth * (internalDegree + 4) + 4)) *
      (Polynomial.X + 1)

theorem oldElapsed_prefix_le (H : BaseTable) (t : GraphTables.Table) (k : Nat) :
    oldElapsed H t (Header.headerBits internalDegree t) k ≤
      k * (MachineRegularOriginalBodyBounds.timePolynomial.eval
        (inputLength t + regularPolynomial.eval (inputLength t)) + 2) := by
  induction k with
  | zero => simp [oldElapsed]
  | succ k ih =>
    simp only [oldElapsed]
    split_ifs with hk
    · have raw := MachineRegularOriginalBodyBounds.totalSteps_le H t ⟨k, hk⟩
        (Header.headerBits internalDegree t ++ oldPrefix H t k)
      have hprefix :
          (Header.headerBits internalDegree t ++ oldPrefix H t k).length ≤
            regularPolynomial.eval (inputLength t) := by
        exact regular_originalPrefix_le t H k
      have hcall := raw.trans (natPolynomial_eval_mono
        MachineRegularOriginalBodyBounds.timePolynomial (Nat.add_le_add_left hprefix _))
      rw [Nat.succ_mul]
      omega
    · rw [Nat.succ_mul]
      omega

theorem oldElapsed_le (H : BaseTable) (t : GraphTables.Table) :
    oldElapsed H t (Header.headerBits internalDegree t) t.darts ≤
      oldLoopPolynomial.eval (inputLength t) := by
  have bound := (oldElapsed_prefix_le H t t.darts).trans
    (Nat.mul_le_mul_right _ (GraphTables.darts_le_tableBits_length t))
  simpa only [oldLoopPolynomial, inputLength, Polynomial.eval_mul, Polynomial.eval_X,
    Polynomial.eval_add, Polynomial.eval_comp, Polynomial.eval_ofNat] using bound

theorem ownerElapsed_prefix_le (H : BaseTable) (t : GraphTables.Table) (k : Nat) :
    ownerElapsed H t
      (Header.headerBits internalDegree t ++ oldPrefix H t t.darts) k ≤
      k * (MachineRegularOwnerBodyBounds.timePolynomial.eval
        (inputLength t + regularPolynomial.eval (inputLength t)) + 2) := by
  induction k with
  | zero => simp [ownerElapsed]
  | succ k ih =>
    simp only [ownerElapsed]
    split_ifs with hk
    · have raw := MachineRegularOwnerBodyBounds.totalSteps_le H t ⟨k, hk⟩
        ((Header.headerBits internalDegree t ++ oldPrefix H t t.darts) ++ ownerPrefix H t k)
      have hprefix :
          ((Header.headerBits internalDegree t ++ oldPrefix H t t.darts) ++
            ownerPrefix H t k).length ≤ regularPolynomial.eval (inputLength t) := by
        rw [owner_output_eq_ownerPrefix]
        exact regular_ownerPrefix_le t H k
      have hcall := raw.trans (natPolynomial_eval_mono
        MachineRegularOwnerBodyBounds.timePolynomial (Nat.add_le_add_left hprefix _))
      rw [Nat.succ_mul]
      omega
    · rw [Nat.succ_mul]
      omega

theorem ownerElapsed_le (H : BaseTable) (t : GraphTables.Table) :
    ownerElapsed H t
      (Header.headerBits internalDegree t ++ oldPrefix H t t.darts) t.vertices ≤
      ownerLoopPolynomial.eval (inputLength t) := by
  have bound := (ownerElapsed_prefix_le H t t.vertices).trans
    (Nat.mul_le_mul_right _ (GraphTables.vertices_le_tableBits_length t))
  simpa only [ownerLoopPolynomial, inputLength, Polynomial.eval_mul, Polynomial.eval_X,
    Polynomial.eval_add, Polynomial.eval_comp, Polynomial.eval_ofNat] using bound

theorem totalTime_le (H : BaseTable) (t : GraphTables.Table) :
    totalTime H t ≤ timePolynomial.eval (inputLength t) := by
  have header := Header.totalTime_le internalDegree t []
  simp only [List.length_nil, Nat.add_zero] at header
  change Header.totalTime internalDegree t [] ≤
    (Header.timePolynomial internalDegree).eval (inputLength t) at header
  have old := oldElapsed_le H t
  have owners := ownerElapsed_le H t
  have cleanup := cleanupTime_final_le t (finalOutput H t)
  have hm : t.darts ≤ inputLength t := GraphTables.darts_le_tableBits_length t
  have hn : t.vertices ≤ inputLength t := GraphTables.vertices_le_tableBits_length t
  simp only [totalTime, headerOutput, oldOutput, timePolynomial,
    Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X,
    Polynomial.eval_ofNat, Polynomial.eval_one]
  omega

noncomputable def machineInTime (H : BaseTable) (t : GraphTables.Table) :
    StateTransition.EvalsToInTime (TM2.step (program H))
      (initList (machine H) (GraphTables.tableBits t))
      (some ⟨none, readyState H,
        Function.update (initList (machine H) (GraphTables.tableBits t)).stk
          (coreTape 8) (PortTables.tableBits (regularize H t))⟩)
      (timePolynomial.eval (inputLength t)) where
  steps := totalTime H t
  evals_in_steps := trace H t
  steps_le_m := totalTime_le H t

end DFVSGames.Foundations.Complexity.MachineRegularTable.Top
end

section

namespace DFVSGames.Foundations.Complexity.MachineCanonicalOutput

open Turing MachineComposition

structure Program (K Λ σ : Type) where
  input : K
  output : K
  main : Λ
  initial : σ
  code : Λ → TM2.Stmt (fun _ : K => Bool) Λ σ

section Structural

variable {K Λ Λ' σ : Type} [DecidableEq K]

def extendedLabel (labels : Λ → Λ') (haltTarget : Option Λ') : Option Λ → Option Λ'
  | none => haltTarget
  | some label => some (labels label)

def extendedCfg (labels : Λ → Λ') (haltTarget : Option Λ') (register : Option Bool)
    (cfg : TM2.Cfg (fun _ : K => Bool) Λ σ) :
    TM2.Cfg (fun _ : K => Bool) Λ' (σ × Option Bool) :=
  ⟨extendedLabel labels haltTarget cfg.l, (cfg.var, register), cfg.stk⟩

def extendedStmt (labels : Λ → Λ') (haltTarget : Option Λ') :
    TM2.Stmt (fun _ : K => Bool) Λ σ →
      TM2.Stmt (fun _ : K => Bool) Λ' (σ × Option Bool)
  | .push k f next => .push k (fun state => f state.1)
      (extendedStmt labels haltTarget next)
  | .peek k f next => .peek k (fun state symbol => (f state.1 symbol, state.2))
      (extendedStmt labels haltTarget next)
  | .pop k f next => .pop k (fun state symbol => (f state.1 symbol, state.2))
      (extendedStmt labels haltTarget next)
  | .load f next => .load (fun state => (f state.1, state.2))
      (extendedStmt labels haltTarget next)
  | .branch test yes no => .branch (fun state => test state.1)
      (extendedStmt labels haltTarget yes) (extendedStmt labels haltTarget no)
  | .goto label => .goto (fun state => labels (label state.1))
  | .halt => match haltTarget with
    | none => .halt
    | some label => .goto (fun _ => label)

theorem stepAux_extended (labels : Λ → Λ') (haltTarget : Option Λ')
    (stmt : TM2.Stmt (fun _ : K => Bool) Λ σ) (state : σ)
    (register : Option Bool) (tapes : K → List Bool) :
    TM2.stepAux (extendedStmt labels haltTarget stmt) (state, register) tapes =
      extendedCfg labels haltTarget register (TM2.stepAux stmt state tapes) := by
  induction stmt generalizing state tapes with
  | push k f next ih => exact ih state (Function.update tapes k (f state :: tapes k))
  | peek k f next ih => exact ih (f state (tapes k).head?) tapes
  | pop k f next ih =>
      exact ih (f state (tapes k).head?) (Function.update tapes k (tapes k).tail)
  | load f next ih => exact ih (f state) tapes
  | branch test yes no ihYes ihNo =>
      cases h : test state with
      | false =>
          simpa only [extendedStmt, TM2.stepAux, h, Bool.cond_false] using ihNo state tapes
      | true =>
          simpa only [extendedStmt, TM2.stepAux, h, Bool.cond_true] using ihYes state tapes
  | goto label => rfl
  | halt => cases haltTarget <;> rfl

end Structural

section Machine

variable {K Λ σ : Type} [DecidableEq K] [Fintype K] [Fintype Λ] [Fintype σ]

def sourceMachine (P : Program K Λ σ) : FinTM2 where
  K := K
  k₀ := P.input
  k₁ := P.output
  Γ _ := Bool
  Λ := Λ
  main := P.main
  σ := σ
  initialState := P.initial
  m := P.code

abbrev Label (Λ : Type) (chosen : List K) := Λ ⊕ (MachineDrainMany.Label chosen ⊕ Unit)

def cleanupLabel (chosen : List K) : MachineDrainMany.Label chosen → Label Λ chosen :=
  fun label => .inr (.inl label)

def resetLabel (chosen : List K) : Label Λ chosen := .inr (.inr ())

def cleanupEntry (chosen : List K) : Option (Label Λ chosen) :=
  MachineDrainMany.entry chosen (cleanupLabel chosen) (some (resetLabel chosen))

def completedProgram (P : Program K Λ σ) (chosen : List K) :
    Label Λ chosen → TM2.Stmt (fun _ : K => Bool) (Label Λ chosen) (σ × Option Bool)
  | .inl label => extendedStmt Sum.inl (cleanupEntry chosen) (P.code label)
  | .inr (.inl label) => MachineDrainMany.instruction chosen (cleanupLabel chosen)
      (some (resetLabel chosen)) label
  | .inr (.inr _) => .load (fun _ => (P.initial, none)) .halt

def completedMachine (P : Program K Λ σ) (chosen : List K) : FinTM2 where
  K := K
  k₀ := P.input
  k₁ := P.output
  Γ _ := Bool
  Λ := Label Λ chosen
  main := .inl P.main
  σ := σ × Option Bool
  initialState := (P.initial, none)
  m := completedProgram P chosen

theorem finiteAlphabet (P : Program K Λ σ) (chosen : List K) :
    MachineFiniteAlphabet.FiniteAlphabet (completedMachine P chosen) := by
  intro k
  exact inferInstanceAs (Finite Bool)

def embeddedCfg (P : Program K Λ σ) (chosen : List K)
    (cfg : (sourceMachine P).Cfg) : (completedMachine P chosen).Cfg :=
  extendedCfg Sum.inl (cleanupEntry chosen) none cfg

theorem step_simulation (P : Program K Λ σ) (chosen : List K)
    (a b : (sourceMachine P).Cfg) (step : (sourceMachine P).step a = some b) :
    (completedMachine P chosen).step (embeddedCfg P chosen a) =
      some (embeddedCfg P chosen b) := by
  cases a with
  | mk label state tapes =>
    cases label with
    | none => simp [FinTM2.step, TM2.step] at step
    | some label =>
      have hb : TM2.stepAux (P.code label) state tapes = b := Option.some.inj step
      rw [← hb]
      change some (TM2.stepAux
        (extendedStmt Sum.inl (cleanupEntry chosen) (P.code label)) (state, none) tapes) = _
      exact congrArg some
        (stepAux_extended Sum.inl (cleanupEntry chosen) (P.code label) state none tapes)

@[simp] theorem embedded_init (P : Program K Λ σ) (chosen : List K) (input : List Bool) :
    embeddedCfg P chosen (initList (sourceMachine P) input) =
      initList (completedMachine P chosen) input := rfl

omit [Fintype K] [Fintype Λ] [Fintype σ] in
theorem finalTapes_eq (P : Program K Λ σ) (chosen : List K)
    (complete : ∀ k, k ∈ chosen ↔ k ≠ P.output) (base : K → List Bool) :
    MachineDrainMany.finalTapes chosen base =
      MachineDrainMany.haltTapes P.output (base P.output) := by
  funext k
  rw [MachineDrainMany.finalTapes_apply]
  by_cases h : k = P.output
  · subst k
    simp [MachineDrainMany.haltTapes, complete]
  · simp [MachineDrainMany.haltTapes, complete, h]

theorem haltList_eq (P : Program K Λ σ) (chosen : List K) (output : List Bool) :
    haltList (completedMachine P chosen) output =
      ⟨none, (P.initial, none), MachineDrainMany.haltTapes P.output output⟩ := by
  congr 1

def cleanupExecution (P : Program K Λ σ) (chosen : List K)
    (complete : ∀ k, k ∈ chosen ↔ k ≠ P.output) (state : σ)
    (register : Option Bool) (base : K → List Bool) :
    StateTransition.EvalsToInTime (completedMachine P chosen).step
      ⟨cleanupEntry chosen, (state, register), base⟩
      (some (haltList (completedMachine P chosen) (base P.output)))
      (MachineDrainMany.steps chosen base + 1) := by
  let after : (completedMachine P chosen).Cfg :=
    ⟨some (resetLabel chosen), (state, MachineDrainMany.finalRegister chosen register),
      MachineDrainMany.finalTapes chosen base⟩
  have drains : StateTransition.EvalsToInTime (completedMachine P chosen).step
      ⟨cleanupEntry chosen, (state, register), base⟩ (some after)
      (MachineDrainMany.steps chosen base) := {
    steps := MachineDrainMany.steps chosen base
    evals_in_steps := MachineDrainMany.trace chosen (cleanupLabel chosen)
      (some (resetLabel chosen)) (completedProgram P chosen) (fun _ => rfl) base state register
    steps_le_m := le_rfl }
  have reset : StateTransition.EvalsToInTime (completedMachine P chosen).step
      after (some (haltList (completedMachine P chosen) (base P.output))) 1 := {
    steps := 1
    evals_in_steps := by
      change some (⟨none, (P.initial, none), MachineDrainMany.finalTapes chosen base⟩ :
          TM2.Cfg (fun _ : K => Bool) (Label Λ chosen) (σ × Option Bool)) =
        some (haltList (completedMachine P chosen) (base P.output))
      rw [finalTapes_eq P chosen complete, haltList_eq]
      rfl
    steps_le_m := le_rfl }
  simpa only [Nat.add_comm] using
    StateTransition.EvalsToInTime.trans _ (MachineDrainMany.steps chosen base) 1
      _ after _ drains reset

def outputsInTime (P : Program K Λ σ) (chosen : List K)
    (complete : ∀ k, k ∈ chosen ↔ k ≠ P.output)
    (input output : List Bool) (state : σ) (base : K → List Bool) (budget : Nat)
    (raw : StateTransition.EvalsToInTime (sourceMachine P).step
      (initList (sourceMachine P) input) (some ⟨none, state, base⟩) budget)
    (correctOutput : base P.output = output) :
    TM2OutputsInTime (completedMachine P chosen) input (some output)
      (budget + chosen.length *
        (input.length + budget * Runtime.programPushBound (sourceMachine P) + 1) + 1) := by
  let lifted := liftExecutionInTime (sourceMachine P).step
    (completedMachine P chosen).step (embeddedCfg P chosen) (step_simulation P chosen) raw
  have joined := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ lifted
    (cleanupExecution P chosen complete state none base)
  have stackBound : ∀ k, (base k).length ≤
      input.length + budget * Runtime.programPushBound (sourceMachine P) := by
    intro k
    have h := Runtime.executionSizeBound (sourceMachine P).step
      (fun cfg => (cfg.stk k).length) (Runtime.programPushBound (sourceMachine P))
      (Runtime.stepStackLength (sourceMachine P) k) raw
    exact h.trans (Nat.add_le_add_right (Runtime.initialStackLength (sourceMachine P) input k) _)
  have cleanupBound := MachineDrainMany.steps_le_uniform chosen base _ stackBound
  rw [embedded_init, correctOutput] at joined
  exact { toEvalsTo := joined.toEvalsTo
          steps_le_m := joined.steps_le_m.trans (by omega) }

noncomputable def completedTime (P : Program K Λ σ) (chosen : List K)
    (rawTime : Polynomial Nat) : Polynomial Nat :=
  rawTime + Polynomial.C chosen.length *
    (Polynomial.X + rawTime * Polynomial.C (Runtime.programPushBound (sourceMachine P)) + 1) + 1

theorem completedTime_eval (P : Program K Λ σ) (chosen : List K)
    (rawTime : Polynomial Nat) (n : Nat) :
    (completedTime P chosen rawTime).eval n =
      rawTime.eval n + chosen.length *
        (n + rawTime.eval n * Runtime.programPushBound (sourceMachine P) + 1) + 1 := by
  simp [completedTime]

structure TerminalRun (P : Program K Λ σ) (input output : List Bool) (budget : Nat) where
  state : σ
  tapes : K → List Bool
  execution : StateTransition.EvalsToInTime (sourceMachine P).step
    (initList (sourceMachine P) input) (some ⟨none, state, tapes⟩) budget
  output_eq : tapes P.output = output

noncomputable def computableInPolyTime {α β : Type}
    (P : Program K Λ σ) (chosen : List K)
    (complete : ∀ k, k ∈ chosen ↔ k ≠ P.output)
    (encodeIn : α → List Bool) (encodeOut : β → List Bool) (f : α → β)
    (rawTime : Polynomial Nat)
    (run : ∀ a, TerminalRun P (encodeIn a) (encodeOut (f a))
      (rawTime.eval (encodeIn a).length)) :
    TM2ComputableInPolyTime encodeIn encodeOut f where
  tm := completedMachine P chosen
  inputAlphabet := Equiv.refl Bool
  outputAlphabet := Equiv.refl Bool
  time := completedTime P chosen rawTime
  outputsFun a := by
    have execution : TM2OutputsInTime (completedMachine P chosen) (encodeIn a)
        (some (encodeOut (f a))) ((completedTime P chosen rawTime).eval (encodeIn a).length) := by
      simpa only [completedTime_eval] using
        outputsInTime P chosen complete _ _ (run a).state (run a).tapes _
          (run a).execution (run a).output_eq
    exact (congrArg₂ (fun input output : List Bool =>
      TM2OutputsInTime (completedMachine P chosen) input (some output)
        ((completedTime P chosen rawTime).eval (encodeIn a).length))
      (List.map_id (encodeIn a)) (List.map_id (encodeOut (f a)))).mpr execution

theorem computableInPolyTime_finite_alphabet {α β : Type}
    (P : Program K Λ σ) (chosen : List K)
    (complete : ∀ k, k ∈ chosen ↔ k ≠ P.output)
    (encodeIn : α → List Bool) (encodeOut : β → List Bool) (f : α → β)
    (rawTime : Polynomial Nat)
    (run : ∀ a, TerminalRun P (encodeIn a) (encodeOut (f a))
      (rawTime.eval (encodeIn a).length)) :
    MachineFiniteAlphabet.FiniteAlphabet
      (computableInPolyTime P chosen complete encodeIn encodeOut f rawTime run).tm :=
  finiteAlphabet P chosen

noncomputable def computableInPolyTimeOfEnumeration {α β : Type} {N : Nat}
    (P : Program K Λ σ) (enumeration : Fin N ≃ K)
    (encodeIn : α → List Bool) (encodeOut : β → List Bool) (f : α → β)
    (rawTime : Polynomial Nat)
    (run : ∀ a, TerminalRun P (encodeIn a) (encodeOut (f a))
      (rawTime.eval (encodeIn a).length)) :
    TM2ComputableInPolyTime encodeIn encodeOut f :=
  computableInPolyTime P (MachineDrainMany.workTapes enumeration P.output)
    (fun k => MachineDrainMany.mem_workTapes enumeration P.output k)
    encodeIn encodeOut f rawTime run

end Machine

end DFVSGames.Foundations.Complexity.MachineCanonicalOutput
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