lean/OAI/Computability/UniqueGames/Reduction/MachineTemplateAddress.lean

import OAI.Computability.UniqueGames.Machines.MachineDrainManyLemmas
import OAI.Computability.UniqueGames.Machines.MachineRadixStepLemmas
import OAI.Computability.UniqueGames.PCP.SourceFieldArrayLemmas
import OAI.Computability.UniqueGames.Reduction.EncodingLemmas
import OAI.Computability.UniqueGames.Reduction.MachineFieldTemplate

namespace OAI

section

/-! A fixed canonical-word template followed by actual radix-address execution.
The program is determined by the fixed tokens and tape layout. Numeric field
values enter only the tape invariants and execution theorem, never the labels. -/

namespace UniqueGamesTheorem.Reduction.MachineTemplateAddress

open Turing
open UniqueGamesTheorem.Foundations.Complexity
open UniqueGamesTheorem.Foundations.Hastad

inductive Tape (q width : Nat)
  | field (i : Fin q)
  | radix | output | reversed | forward | copyScratch
  | accA | accB | counter | hornerScratch
  | digit (i : Fin width)
  deriving DecidableEq, Fintype

abbrev State (σ : Type) := (σ × Unit) × Option Bool
abbrev Alphabet {K : Type} (_ : K) := Bool

variable {q width : Nat} {K Λ σ : Type} [DecidableEq K]

def hornerLayout : MachineHorner.Layout width ↪ Tape q width where
  toFun
    | .inl 0 => .radix
    | .inl 1 => .accA
    | .inl 2 => .accB
    | .inl 3 => .output
    | .inl 4 => .counter
    | .inl 5 => .hornerScratch
    | .inr i => .digit i
  inj' := by
    rintro (a | a) (b | b) h
    · fin_cases a <;> fin_cases b <;> simp_all
    · fin_cases a <;> simp_all
    · fin_cases b <;> simp_all
    · exact congrArg Sum.inr (Tape.digit.inj h)

def hornerSlots (slots : Tape q width ↪ K) : MachineHorner.Layout width ↪ K :=
  hornerLayout.trans slots

def chosen (slots : Tape q width ↪ K) : List K := List.ofFn (fun i => slots (.digit i))

omit [DecidableEq K] in
@[simp] theorem mem_chosen (slots : Tape q width ↪ K) (tape : K) :
    tape ∈ chosen slots ↔ ∃ i, slots (.digit i) = tape := by
  simp [chosen, List.mem_ofFn]

abbrev Label (m width : Nat) (cleanup : List K) :=
  MachineFieldTemplate.Label m ⊕ Unit ⊕ (Fin (width + 1) × Bool) ⊕
    MachineHorner.Label width ⊕ MachineDrainMany.Label cleanup

def emitLabel {m : Nat} {cleanup : List K} (l : MachineFieldTemplate.Label m) :
    Label m width cleanup := .inl l
def reverseLabel {m : Nat} {cleanup : List K} : Label m width cleanup := .inr (.inl ())
def parseLabel {m : Nat} {cleanup : List K} (l : Fin (width + 1) × Bool) :
    Label m width cleanup := .inr (.inr (.inl l))
def hornerLabel {m : Nat} {cleanup : List K} (l : MachineHorner.Label width) :
    Label m width cleanup := .inr (.inr (.inr (.inl l)))
def cleanupLabel {m : Nat} {cleanup : List K} (l : MachineDrainMany.Label cleanup) :
    Label m width cleanup := .inr (.inr (.inr (.inr l)))

def parseDestination (slots : Tape q width ↪ K) (i : Nat) : K :=
  if h : i < width then slots (.digit ⟨i, h⟩) else slots .forward

omit [DecidableEq K] in
@[simp] theorem parseDestination_below (slots : Tape q width ↪ K) (i : Nat) (hi : i < width) :
    parseDestination slots i = slots (.digit ⟨i, hi⟩) := by simp [parseDestination, hi]

/-- Fixed finite instruction assembly, with arbitrary placement and continuation. -/
def instruction (tokens : List (MachineFieldTemplate.Token q)) (slots : Tape q width ↪ K)
    (place : Label tokens.length width (chosen slots) → Λ) (exit : Option Λ) :
    Label tokens.length width (chosen slots) → TM2.Stmt (Alphabet (K := K)) Λ (State σ)
  | .inl l => MachineFieldTemplate.instruction tokens (fun j => slots (.field j))
      (slots .copyScratch) (slots .reversed) (fun l => place (emitLabel l))
      (some (place reverseLabel)) l
  | .inr (.inl _) => MachineTransfer.loopAt (slots .reversed) (slots .forward) id false
      (place reverseLabel) (some (place (parseLabel (SourceFieldArray.startLabel width 0))))
  | .inr (.inr (.inl l)) =>
      if h : l.1.val < width then
        if l.2 then
          SourceMachine.fieldLoop (slots .forward) (slots (.digit ⟨l.1.val, h⟩))
            (place (parseLabel (l.1, true)))
            (some (place (parseLabel (SourceFieldArray.startLabel width (l.1.val + 1)))))
        else SourceMachine.fieldStart (slots (.digit ⟨l.1.val, h⟩))
          (place (parseLabel (l.1, true)))
      else SourceFieldArray.finish (slots .forward) (some (place (hornerLabel .start)))
  | .inr (.inr (.inr (.inl l))) => MachineHorner.statement (hornerSlots slots)
      (fun l => place (hornerLabel l))
      (MachineDrainMany.entry (chosen slots) (fun l => place (cleanupLabel l)) exit) l
  | .inr (.inr (.inr (.inr l))) => MachineDrainMany.instruction (chosen slots)
      (fun l => place (cleanupLabel l)) exit l

def program (tokens : List (MachineFieldTemplate.Token q)) (slots : Tape q width ↪ K)
    (exit : Option (Label tokens.length width (chosen slots))) :
    Label tokens.length width (chosen slots) →
      TM2.Stmt (Alphabet (K := K)) (Label tokens.length width (chosen slots)) (State σ) :=
  instruction tokens slots id exit

structure Clean (slots : Tape q width ↪ K) (base : K → List Bool) : Prop where
  reversed : base (slots .reversed) = []
  forward : base (slots .forward) = []
  copyScratch : base (slots .copyScratch) = []
  accA : base (slots .accA) = []
  accB : base (slots .accB) = []
  counter : base (slots .counter) = []
  hornerScratch : base (slots .hornerScratch) = []
  digit : ∀ j, base (slots (.digit j)) = []

def digits (values : List Nat) (i : Nat) : Nat := values[i]?.getD 0

def forwardTapes (slots : Tape q width ↪ K) (base : K → List Bool) (values : List Nat) :
    K → List Bool := Function.update base (slots .forward) (encodeWords values)

def parsedTapes (slots : Tape q width ↪ K) (base : K → List Bool) (values : List Nat) :
    K → List Bool :=
  SourceFieldArray.sequenceTapes (slots .forward) (parseDestination slots)
    (forwardTapes slots base values) 0 values []

def resultTapes (slots : Tape q width ↪ K) (base : K → List Bool) (value : Nat) :
    K → List Bool := Function.update base (slots .output) (encodeWord value ++ base (slots .output))

def afterHorner (slots : Tape q width ↪ K) (base : K → List Bool) (B : Nat) (values : List Nat) :
    K → List Bool :=
  MachineHorner.resultTapes (hornerSlots slots) (parsedTapes slots base values)
    (MachineHorner.value B (digits values) width)

def phaseSteps (tokens : List (MachineFieldTemplate.Token q)) (slots : Tape q width ↪ K)
    (base : K → List Bool) (B : Nat) (values : List Nat) : Nat :=
  MachineFieldTemplate.templateSteps tokens (fun j => base (slots (.field j))) +
    ((encodeWords values).length + 1) + (values.sum + 2 * values.length + 1) +
    MachineHorner.steps B (digits values) width +
    MachineDrainMany.steps (chosen slots) (afterHorner slots base B values)

@[simp] theorem resultTapes_output (slots : Tape q width ↪ K) (base : K → List Bool) (value : Nat) :
    resultTapes slots base value (slots .output) = encodeWord value ++ base (slots .output) := by
  simp [resultTapes]

theorem resultTapes_other (slots : Tape q width ↪ K) (base : K → List Bool) (value : Nat)
    (tape : K) (hne : tape ≠ slots .output) : resultTapes slots base value tape = base tape := by
  simp [resultTapes, hne]

/-- The numeric recurrence used by the checked Horner machine is the direct
canonical-address radix, in the original template-word order. -/
theorem hornerValue_eq_radix (B : Nat) (values : List Nat) :
    MachineHorner.value B (digits values) values.length = CanonicalAddress.radix B values := by
  rw [MachineHorner.value_eq_foldl, CanonicalAddress.radix_eq_foldl]
  congr 1
  apply List.ext_getElem
  · simp
  · intro i hi hj
    simp only [List.getElem_map, List.getElem_range, digits,
      List.getElem?_eq_getElem hj, Option.getD_some]

theorem parsedTapes_other (slots : Tape q width ↪ K) (base : K → List Bool)
    (values : List Nat) (hlen : values.length = width) (tape : K)
    (notForward : tape ≠ slots .forward) (notDigit : ∀ i, tape ≠ slots (.digit i)) :
    parsedTapes slots base values tape = base tape := by
  rw [parsedTapes, SourceFieldArray.sequenceTapes_other]
  · exact Function.update_of_ne notForward _ _
  · exact notForward
  · intro r hr
    simp only [Nat.zero_add, parseDestination_below slots r (by omega)]
    exact notDigit _

theorem parsedTapes_source (slots : Tape q width ↪ K) (base : K → List Bool)
    (values : List Nat) (hlen : values.length = width) :
    parsedTapes slots base values (slots .forward) = [] := by
  apply SourceFieldArray.sequenceTapes_source
  · intro r hr
    simp only [Nat.zero_add, parseDestination_below slots r (by omega)]
    exact slots.injective.ne (by simp)
  · simp [forwardTapes]

theorem parsedTapes_digit (slots : Tape q width ↪ K) (base : K → List Bool)
    (values : List Nat) (hlen : values.length = width) (clean : Clean slots base) (i : Fin width) :
    parsedTapes slots base values (slots (.digit i)) = encodeWord (digits values i.val) := by
  have hi : i.val < values.length := by omega
  have h := SourceFieldArray.sequenceTapes_selected (slots .forward) (parseDestination slots)
    (forwardTapes slots base values) 0 values [] i.val hi
    (by
      intro r hr
      simp only [Nat.zero_add, parseDestination_below slots r (by omega)]
      exact slots.injective.ne (by simp))
    (by
      intro r s hr hs h
      simp only [Nat.zero_add, parseDestination_below slots r (by omega),
        parseDestination_below slots s (by omega)] at h
      exact congrArg Fin.val (Tape.digit.inj (slots.injective h)))
  simpa [parsedTapes, parseDestination, i.isLt, forwardTapes, clean.digit,
    slots.injective.ne (show Tape.digit i ≠ Tape.forward by simp), digits,
    List.getElem?_eq_getElem hi] using h

theorem parsedTapes_clean (slots : Tape q width ↪ K) (base : K → List Bool)
    (values : List Nat) (hlen : values.length = width) (clean : Clean slots base) :
    MachineHorner.Clean (hornerSlots slots) (parsedTapes slots base values) := by
  constructor
  · rw [show hornerSlots slots (.inl 1) = slots .accA from rfl,
      parsedTapes_other slots base values hlen _ (slots.injective.ne (by simp))
        (fun _ => slots.injective.ne (by simp))]
    exact clean.accA
  · rw [show hornerSlots slots (.inl 2) = slots .accB from rfl,
      parsedTapes_other slots base values hlen _ (slots.injective.ne (by simp))
        (fun _ => slots.injective.ne (by simp))]
    exact clean.accB
  · rw [show hornerSlots slots (.inl 4) = slots .counter from rfl,
      parsedTapes_other slots base values hlen _ (slots.injective.ne (by simp))
        (fun _ => slots.injective.ne (by simp))]
    exact clean.counter
  · rw [show hornerSlots slots (.inl 5) = slots .hornerScratch from rfl,
      parsedTapes_other slots base values hlen _ (slots.injective.ne (by simp))
        (fun _ => slots.injective.ne (by simp))]
    exact clean.hornerScratch

theorem cleanup_result (slots : Tape q width ↪ K) (base : K → List Bool)
    (B : Nat) (values : List Nat) (hlen : values.length = width) (clean : Clean slots base) :
    MachineDrainMany.finalTapes (chosen slots) (afterHorner slots base B values) =
      resultTapes slots base (CanonicalAddress.radix B values) := by
  have hv : MachineHorner.value B (digits values) width = CanonicalAddress.radix B values := by
    rw [← hlen, hornerValue_eq_radix]
  funext tape
  rw [MachineDrainMany.finalTapes_apply]
  by_cases hd : tape ∈ chosen slots
  · rw [ite_eq_left hd]
    obtain ⟨i, rfl⟩ := (mem_chosen slots tape).mp hd
    simp [resultTapes, slots.injective.ne (show Tape.digit i ≠ Tape.output by simp), clean.digit]
  · rw [ite_eq_right hd]
    have notDigit : ∀ i, tape ≠ slots (.digit i) := by
      intro i hi
      exact hd ((mem_chosen slots tape).mpr ⟨i, hi.symm⟩)
    by_cases ho : tape = slots .output
    · subst tape
      simp only [afterHorner, MachineHorner.resultTapes,
        show hornerSlots slots (.inl 3) = slots .output from rfl,
        Function.update_self, resultTapes_output, hv]
      rw [parsedTapes_other slots base values hlen _ (slots.injective.ne (by simp)) notDigit]
    · simp only [afterHorner, MachineHorner.resultTapes,
        show hornerSlots slots (.inl 3) = slots .output from rfl,
        resultTapes, Function.update_of_ne ho]
      by_cases hf : tape = slots .forward
      · subst tape
        rw [parsedTapes_source slots base values hlen, clean.forward]
      · exact parsedTapes_other slots base values hlen tape hf notDigit

section Execution

variable (tokens : List (MachineFieldTemplate.Token q)) (slots : Tape q width ↪ K)
variable (place : Label tokens.length width (chosen slots) → Λ) (exit : Option Λ)
variable (P : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
variable (atInstruction : ∀ l, P (place l) = instruction tokens slots place exit l)

include atInstruction

omit [DecidableEq K] in
theorem parse_atStart (r : Nat) (hr : r < width) :
    P (place (parseLabel (SourceFieldArray.startLabel width r))) =
      SourceMachine.fieldStart (parseDestination slots r)
        (place (parseLabel (SourceFieldArray.loopLabel width r))) := by
  rw [atInstruction]
  simp [instruction, parseLabel, SourceFieldArray.startLabel, SourceFieldArray.loopLabel,
    SourceFieldArray.boundedIndex_val hr.le, hr]

omit [DecidableEq K] in
theorem parse_atLoop (r : Nat) (hr : r < width) :
    P (place (parseLabel (SourceFieldArray.loopLabel width r))) =
      SourceMachine.fieldLoop (slots .forward) (parseDestination slots r)
        (place (parseLabel (SourceFieldArray.loopLabel width r)))
        (some (place (parseLabel (SourceFieldArray.startLabel width (r + 1))))) := by
  rw [atInstruction]
  simp [instruction, parseLabel, SourceFieldArray.startLabel, SourceFieldArray.loopLabel,
    SourceFieldArray.boundedIndex_val hr.le, hr]

omit [DecidableEq K] in
theorem parse_atDone :
    P (place (parseLabel (SourceFieldArray.startLabel width width))) =
      SourceFieldArray.finish (slots .forward) (some (place (hornerLabel .start))) := by
  rw [atInstruction]
  simp [instruction, parseLabel, SourceFieldArray.startLabel]

theorem phaseTrace (base : K → List Bool) (B : Nat) (values : List Nat)
    (hlen : values.length = width) (clean : Clean slots base)
    (baseField : base (slots .radix) = encodeWord B)
    (wordCorrect : MachineFieldTemplate.templateOutput tokens (fun j => base (slots (.field j))) =
      encodeWords values) (ambient : σ) (register : Option Bool) :
    (MachineComposition.advance (TM2.step P))^[phaseSteps tokens slots base B values]
      (some ⟨some (place (emitLabel (MachineFieldTemplate.startAt tokens.length 0))),
        ((ambient, ()), register), base⟩) =
      some ⟨exit, ((ambient, ()), none), resultTapes slots base (CanonicalAddress.radix B values)⟩ := by
  let emitted := MachineFieldTemplate.outputTapes base (slots .reversed) (encodeWords values)
  have emitRun := MachineFieldTemplate.phaseTrace tokens (fun j => slots (.field j))
    (slots .copyScratch) (slots .reversed)
    (fun _ => slots.injective.ne (by simp)) (fun _ => slots.injective.ne (by simp))
    (slots.injective.ne (by simp)) (fun l => place (emitLabel l))
    (some (place reverseLabel)) P (fun l => atInstruction (emitLabel l))
    base clean.copyScratch ((ambient, ()), register)
  rw [wordCorrect] at emitRun
  change (MachineComposition.advance (TM2.step P))^[_] _ =
    some ⟨some (place reverseLabel), ((ambient, ()), none), emitted⟩ at emitRun
  have reverseRun := MachineTransfer.transferAt_fromTapes
    (Γ := fun _ : K => Bool) (σ := σ × Unit)
    (slots .reversed) (slots .forward) (slots.injective.ne (by simp)) id false
    (place reverseLabel) (some (place (parseLabel (SourceFieldArray.startLabel width 0))))
    P (atInstruction reverseLabel) emitted (ambient, ()) none
  have reverseResult : MachineTransfer.tapesAt (slots .reversed) (slots .forward) emitted []
      ((emitted (slots .reversed)).reverse.map id ++ emitted (slots .forward)) =
      forwardTapes slots base values := by
    funext tape
    by_cases hr : tape = slots .reversed
    · subst tape
      simp [emitted, MachineFieldTemplate.outputTapes, MachineTransfer.tapesAt, forwardTapes,
        slots.injective.ne (show Tape.reversed ≠ Tape.forward by simp), clean.reversed]
    · by_cases hf : tape = slots .forward
      · subst tape
        simp [emitted, MachineFieldTemplate.outputTapes, MachineTransfer.tapesAt, forwardTapes,
          slots.injective.ne (show Tape.forward ≠ Tape.reversed by simp),
          clean.reversed, clean.forward]
      · simp [emitted, MachineFieldTemplate.outputTapes, MachineTransfer.tapesAt,
          forwardTapes, hr, hf]
  rw [reverseResult] at reverseRun
  have reverseLength : (emitted (slots .reversed)).length + 1 = (encodeWords values).length + 1 := by
    simp [emitted, MachineFieldTemplate.outputTapes, clean.reversed]
  rw [reverseLength] at reverseRun
  change (MachineComposition.advance (TM2.step P))^[(encodeWords values).length + 1]
    (some ⟨some (place reverseLabel), ((ambient, ()), none), emitted⟩) =
    some ⟨some (place (parseLabel (SourceFieldArray.startLabel width 0))),
      ((ambient, ()), none), forwardTapes slots base values⟩ at reverseRun
  have parseRun := SourceFieldArray.sequenceTrace (slots .forward) (parseDestination slots)
    (fun r => place (parseLabel (SourceFieldArray.startLabel width r)))
    (fun r => place (parseLabel (SourceFieldArray.loopLabel width r)))
    (some (place (hornerLabel .start))) P (forwardTapes slots base values) 0 values []
    (by
      intro r hr
      simp only [Nat.zero_add, parseDestination_below slots r (by omega)]
      exact slots.injective.ne (by simp))
    (by intro r hr; simpa using parse_atStart tokens slots place exit P atInstruction r (by omega))
    (by intro r hr; simpa using parse_atLoop tokens slots place exit P atInstruction r (by omega))
    (by simpa only [Nat.zero_add, hlen] using parse_atDone tokens slots place exit P atInstruction)
    (by simp [forwardTapes]) (ambient, ()) none
  change (MachineComposition.advance (TM2.step P))^[values.sum + 2 * values.length + 1]
    (some ⟨some (place (parseLabel (SourceFieldArray.startLabel width 0))),
      ((ambient, ()), none), forwardTapes slots base values⟩) =
    some ⟨some (place (hornerLabel .start)), ((ambient, ()), none),
      parsedTapes slots base values⟩ at parseRun
  have hbase : parsedTapes slots base values (hornerSlots slots (.inl 0)) = encodeWord B := by
    change parsedTapes slots base values (slots .radix) = _
    rw [parsedTapes_other slots base values hlen _ (slots.injective.ne (by simp))
      (fun _ => slots.injective.ne (by simp))]
    exact baseField
  have hornerRun := MachineHorner.hornerTrace (hornerSlots slots) (fun l => place (hornerLabel l))
    (MachineDrainMany.entry (chosen slots) (fun l => place (cleanupLabel l)) exit)
    P (fun l => atInstruction (hornerLabel l)) (parsedTapes slots base values) B (digits values)
    hbase (parsedTapes_digit slots base values hlen clean)
    (parsedTapes_clean slots base values hlen clean) ambient none
  have cleanupRun := MachineDrainMany.trace (chosen slots) (fun l => place (cleanupLabel l)) exit
    P (fun l => atInstruction (cleanupLabel l)) (afterHorner slots base B values) (ambient, ()) none
  rw [MachineDrainMany.finalRegister_none, cleanup_result slots base B values hlen clean] at cleanupRun
  rw [show phaseSteps tokens slots base B values =
    MachineDrainMany.steps (chosen slots) (afterHorner slots base B values) +
      (MachineHorner.steps B (digits values) width +
      ((values.sum + 2 * values.length + 1) +
      (((encodeWords values).length + 1) +
      MachineFieldTemplate.templateSteps tokens (fun j => base (slots (.field j)))))) by
        unfold phaseSteps; omega]
  rw [Function.iterate_add_apply]
  rw [Function.iterate_add_apply (m := MachineHorner.steps B (digits values) width)]
  rw [Function.iterate_add_apply (m := values.sum + 2 * values.length + 1)]
  rw [Function.iterate_add_apply (m := (encodeWords values).length + 1)]
  rw [emitRun, reverseRun, parseRun, hornerRun]
  exact cleanupRun

end Execution

/-- The emitted finite program discharges every instruction-placement premise. -/
theorem programTrace (tokens : List (MachineFieldTemplate.Token q)) (slots : Tape q width ↪ K)
    (exit : Option (Label tokens.length width (chosen slots)))
    (base : K → List Bool) (B : Nat) (values : List Nat) (hlen : values.length = width)
    (clean : Clean slots base) (baseField : base (slots .radix) = encodeWord B)
    (wordCorrect : MachineFieldTemplate.templateOutput tokens (fun j => base (slots (.field j))) =
      encodeWords values) (ambient : σ) (register : Option Bool) :
    (MachineComposition.advance (TM2.step (program tokens slots exit)))^[phaseSteps tokens slots base B values]
      (some ⟨some (emitLabel (MachineFieldTemplate.startAt tokens.length 0)),
        ((ambient, ()), register), base⟩) =
      some ⟨exit, ((ambient, ()), none), resultTapes slots base (CanonicalAddress.radix B values)⟩ :=
  phaseTrace tokens slots id exit (program tokens slots exit) (fun _ => rfl)
    base B values hlen clean baseField wordCorrect ambient register

/-- Timed witness with the exact sum of the five concrete phase counts. -/
def phaseInTime (tokens : List (MachineFieldTemplate.Token q)) (slots : Tape q width ↪ K)
    (place : Label tokens.length width (chosen slots) → Λ) (exit : Option Λ)
    (P : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
    (atInstruction : ∀ l, P (place l) = instruction tokens slots place exit l)
    (base : K → List Bool) (B : Nat) (values : List Nat) (hlen : values.length = width)
    (clean : Clean slots base) (baseField : base (slots .radix) = encodeWord B)
    (wordCorrect : MachineFieldTemplate.templateOutput tokens (fun j => base (slots (.field j))) =
      encodeWords values) (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step P)
      ⟨some (place (emitLabel (MachineFieldTemplate.startAt tokens.length 0))),
        ((ambient, ()), register), base⟩
      (some ⟨exit, ((ambient, ()), none), resultTapes slots base (CanonicalAddress.radix B values)⟩)
      (phaseSteps tokens slots base B values) where
  steps := phaseSteps tokens slots base B values
  evals_in_steps := phaseTrace tokens slots place exit P atInstruction
    base B values hlen clean baseField wordCorrect ambient register
  steps_le_m := Nat.le_refl _

omit [DecidableEq K] in
private theorem copiedLength_le (tokens : List (MachineFieldTemplate.Token q))
    (fields : Fin q → List Bool) (M : Nat) (bounded : ∀ j, (fields j).length ≤ M) :
    MachineFieldTemplate.copiedLength tokens fields ≤ tokens.length * M := by
  induction tokens with
  | nil => simp [MachineFieldTemplate.copiedLength]
  | cons token tokens ih =>
    cases token with
    | literal bits =>
      simp only [MachineFieldTemplate.copiedLength, List.map_cons, List.sum_cons,
        List.length_cons, Nat.zero_add] at *
      nlinarith
    | copy j =>
      simp only [MachineFieldTemplate.copiedLength, List.map_cons, List.sum_cons,
        List.length_cons] at *
      have h := bounded j
      nlinarith

omit [DecidableEq K] in
private theorem lengthSum_le (selected : List K) (base : K → List Bool) (bound : Nat)
    (bounded : ∀ tape ∈ selected, (base tape).length ≤ bound) :
    MachineDrainMany.lengthSum selected base ≤ selected.length * bound := by
  induction selected with
  | nil => simp [MachineDrainMany.lengthSum]
  | cons tape selected ih =>
    have ht := bounded tape (by simp)
    have hs := ih (fun t ht => bounded t (by simp [ht]))
    simp only [MachineDrainMany.lengthSum, List.map_cons, List.sum_cons,
      List.length_cons] at *
    nlinarith

/-- A fixed polynomial in a magnitude bounding input fields, the serialized
word list, the radix, and the digits. Width and token count are static. -/
noncomputable def timePolynomial (tokenCount width : Nat) : Polynomial Nat :=
  MachineHorner.timePolynomial width +
    Polynomial.C (3 * tokenCount + width + 2) * Polynomial.X +
    Polynomial.C (3 * tokenCount + 3 * width + 3)

omit [DecidableEq K] in
theorem timePolynomial_eval (tokenCount width M : Nat) :
    (timePolynomial tokenCount width).eval M =
      (MachineHorner.timePolynomial width).eval M +
      (3 * tokenCount + width + 2) * M + (3 * tokenCount + 3 * width + 3) := by
  simp [timePolynomial]

theorem phaseSteps_le_timePolynomial (tokens : List (MachineFieldTemplate.Token q))
    (slots : Tape q width ↪ K) (base : K → List Bool) (B : Nat) (values : List Nat)
    (hlen : values.length = width) (clean : Clean slots base) (M : Nat)
    (fieldsBound : ∀ j, (base (slots (.field j))).length ≤ M)
    (wordsBound : (encodeWords values).length ≤ M) (radixBound : B ≤ M)
    (digitBound : ∀ i, i < width → digits values i ≤ M) :
    phaseSteps tokens slots base B values ≤ (timePolynomial tokens.length width).eval M := by
  have hcopied := copiedLength_le tokens (fun j => base (slots (.field j))) M fieldsBound
  have htemplate := MachineFieldTemplate.templateSteps_le tokens (fun j => base (slots (.field j)))
  have hhorner := MachineHorner.steps_le_timePolynomial B (digits values) width M radixBound digitBound
  have hselected : ∀ tape ∈ chosen slots, (afterHorner slots base B values tape).length ≤ M + 1 := by
    intro tape ht
    obtain ⟨i, rfl⟩ := (mem_chosen slots tape).mp ht
    have heq : afterHorner slots base B values (slots (.digit i)) =
        encodeWord (digits values i.val) := by
      simp only [afterHorner, MachineHorner.resultTapes,
        show hornerSlots slots (.inl 3) = slots .output from rfl,
        Function.update_of_ne (slots.injective.ne (show Tape.digit i ≠ Tape.output by simp))]
      exact parsedTapes_digit slots base values hlen clean i
    rw [heq]
    simpa only [encodeWord, List.length_append, List.length_replicate, List.length_singleton] using
      Nat.add_le_add_right (digitBound i.val i.isLt) 1
  have hsum := lengthSum_le (chosen slots) (afterHorner slots base B values) (M + 1) hselected
  have hcleanup := MachineDrainMany.steps_le (chosen slots) (afterHorner slots base B values)
  have hchosen : (chosen slots).length = width := by simp [chosen]
  rw [hchosen] at hsum hcleanup
  have hparse := SourceFieldArray.sequence_time_eq_length values
  rw [hlen] at hparse
  rw [timePolynomial_eval]
  unfold phaseSteps
  rw [hlen]
  nlinarith

def phaseInPolynomialTime (tokens : List (MachineFieldTemplate.Token q))
    (slots : Tape q width ↪ K) (place : Label tokens.length width (chosen slots) → Λ)
    (exit : Option Λ) (P : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ))
    (atInstruction : ∀ l, P (place l) = instruction tokens slots place exit l)
    (base : K → List Bool) (B : Nat) (values : List Nat) (hlen : values.length = width)
    (clean : Clean slots base) (baseField : base (slots .radix) = encodeWord B)
    (wordCorrect : MachineFieldTemplate.templateOutput tokens (fun j => base (slots (.field j))) =
      encodeWords values) (ambient : σ) (register : Option Bool) (M : Nat)
    (fieldsBound : ∀ j, (base (slots (.field j))).length ≤ M)
    (wordsBound : (encodeWords values).length ≤ M) (radixBound : B ≤ M)
    (digitBound : ∀ i, i < width → digits values i ≤ M) :
    StateTransition.EvalsToInTime (TM2.step P)
      ⟨some (place (emitLabel (MachineFieldTemplate.startAt tokens.length 0))),
        ((ambient, ()), register), base⟩
      (some ⟨exit, ((ambient, ()), none), resultTapes slots base (CanonicalAddress.radix B values)⟩)
      ((timePolynomial tokens.length width).eval M) where
  steps := phaseSteps tokens slots base B values
  evals_in_steps := phaseTrace tokens slots place exit P atInstruction
    base B values hlen clean baseField wordCorrect ambient register
  steps_le_m := phaseSteps_le_timePolynomial tokens slots base B values hlen clean M
    fieldsBound wordsBound radixBound digitBound

end UniqueGamesTheorem.Reduction.MachineTemplateAddress

end

end OAI