import Mathlib
import OAI.Computability.MaxCut.Machines.MachineSubroutine
import OAI.Computability.MaxCut.Encoding.Encoding2

namespace OAI

namespace MaxCutGames.Reduction.SourceEquationLookup

open Turing
open MaxCutGames.Foundations.Complexity
open MaxCutGames.Foundations.Hastad
open MachineComposition SourceMachine SourceEncoding
open SourceClauseLookup (discardFieldsTrace)

inductive Tape
  | source | index | work | scratch | field (slot : Fin 4)
  deriving DecidableEq, Fintype

inductive Label
  | copyOut | copyBack | headerFirst | headerSecond | guard
  | skip (slot : Fin 4)
  | read (slot : Fin 5 × Bool)
  deriving DecidableEq, Fintype

def fieldDestination (r : Nat) : Tape := .field ⟨min r 3, by omega⟩
def readStart (r : Nat) : Label := .read (SourceFieldArray.boundedIndex 4 r, false)
def readLoop (r : Nat) : Label := .read (SourceFieldArray.boundedIndex 4 r, true)
def skipLabel (r : Nat) : Label := if h : r < 4 then .skip ⟨r, h⟩ else .guard
def headerLabel (r : Nat) : Label :=
  if r = 0 then .headerFirst else if r = 1 then .headerSecond else .guard

variable {σ : Type}

/-- Eight binary stacks and fixed finite control. Every loop scans one actual bit
or tests one actual counter bit. The arbitrary ambient state is preserved. -/
def program : Label → TM2.Stmt (fun _ : Tape => Bool) Label (σ × Option Bool)
  | .copyOut => Reduction.MachineTransfer.loopAt .source .scratch id false
      .copyOut (some .copyBack)
  | .copyBack => MachineCopy.forkLoop .scratch .source .work false
      .copyBack (some .headerFirst)
  | .headerFirst => MachineLookup.discard .work .headerFirst .headerSecond
  | .headerSecond => MachineLookup.discard .work .headerSecond .guard
  | .guard => MachineUnaryCounter.guard .index (skipLabel 0) (readStart 0)
  | .skip slot => MachineLookup.discard .work (.skip slot) (skipLabel (slot.val + 1))
  | .read slot =>
      if slot.1.val < 4 then
        if slot.2 then
          fieldLoop .work (fieldDestination slot.1.val) (.read (slot.1, true))
            (some (readStart (slot.1.val + 1)))
        else fieldStart (fieldDestination slot.1.val) (.read (slot.1, true))
      else SourceFieldArray.finish .work none

theorem atSkip (r : Nat) (hr : r < 4) :
    program (σ := σ) (skipLabel r) = MachineLookup.discard .work (skipLabel r) (skipLabel (r + 1)) := by
  simp [skipLabel, hr, program]

theorem atReadStart (r : Nat) (hr : r < 4) :
    program (σ := σ) (readStart r) = fieldStart (fieldDestination r) (readLoop r) := by
  simp [program, readStart, readLoop, SourceFieldArray.boundedIndex_val hr.le, hr]

theorem atReadLoop (r : Nat) (hr : r < 4) :
    program (σ := σ) (readLoop r) = fieldLoop .work (fieldDestination r) (readLoop r)
      (some (readStart (r + 1))) := by
  simp [program, readStart, readLoop, SourceFieldArray.boundedIndex_val hr.le, hr]

theorem atReadDone : program (σ := σ) (readStart 4) = SourceFieldArray.finish .work none := by
  simp [program, readStart, SourceFieldArray.boundedIndex]

/-- The two header fields are physically consumed from the work copy. -/
theorem discardHeadersTrace (base : Tape → List Bool) (variableCount equations : Nat)
    (suffix : List Bool) (hinput : base .work = encodeWords [variableCount, equations] ++ suffix) (ambient : σ) :
    (advance (TM2.step program))^[variableCount + equations + 2]
      (some ⟨some .headerFirst, (ambient, none), base⟩) =
      some ⟨some .guard, (ambient, none), Function.update base .work suffix⟩ := by
  have ht := discardFieldsTrace .work headerLabel program base 0 [variableCount, equations]
    suffix (by
      intro r hr
      have he : r = 0 ∨ r = 1 := by simp at hr; omega
      rcases he with rfl | rfl <;> rfl) hinput ambient
  simpa [encodeWords_length, headerLabel] using ht

/-- The fixed skip body consumes four complete unary fields and returns to the
same counter guard, preserving the counter and every other tape. -/
theorem discardFourTrace (base : Tape → List Bool) (values : List Nat)
    (hlen : values.length = 4) (suffix : List Bool)
    (hinput : base .work = encodeWords values ++ suffix) (ambient : σ) :
    (advance (TM2.step program))^[(encodeWords values).length]
      (some ⟨some (skipLabel 0), (ambient, none), base⟩) =
      some ⟨some .guard, (ambient, none), Function.update base .work suffix⟩ := by
  have ht := discardFieldsTrace .work skipLabel program base 0 values suffix
    (by intro r hr; simpa only [Nat.zero_add] using atSkip r (by omega)) hinput ambient
  simpa [hlen, skipLabel] using ht

/-- Read the selected four fields using the already checked scanner. -/
theorem readFourTrace (base : Tape → List Bool) (values : List Nat)
    (hlen : values.length = 4) (suffix : List Bool)
    (hinput : base .work = encodeWords values ++ suffix) (ambient : σ) :
    (advance (TM2.step program))^[(encodeWords values).length + 5]
      (some ⟨some (readStart 0), (ambient, none), base⟩) =
      some ⟨none, (ambient, none),
        SourceFieldArray.sequenceTapes .work fieldDestination base 0 values suffix⟩ := by
  have ht := SourceFieldArray.sequenceTrace .work fieldDestination readStart readLoop none
    program base 0 values suffix (by intro r hr; simp [fieldDestination])
    (by intro r hr; simpa only [Nat.zero_add] using atReadStart r (by omega))
    (by intro r hr; simpa only [Nat.zero_add] using atReadLoop r (by omega))
    (by simpa [hlen] using atReadDone) hinput ambient none
  have htime : values.sum + 2 * values.length + 1 = (encodeWords values).length + 5 := by
    rw [encodeWords_length, hlen]
  simpa only [Nat.zero_add, htime] using ht

/-- Normal form for the changing counter and consumed work-copy prefix. -/
def scanTapes (base : Tape → List Bool) (index : Nat)
    (indexSuffix input : List Bool) : Tape → List Bool :=
  MachineUnaryCounter.counterTapes .index (Function.update base .work input) index indexSuffix

@[simp] theorem scanTapes_work (base : Tape → List Bool) (i : Nat)
    (indexSuffix input : List Bool) : scanTapes base i indexSuffix input .work = input := by
  simp [scanTapes, MachineUnaryCounter.counterTapes]

@[simp] theorem scanTapes_index (base : Tape → List Bool) (i : Nat)
    (indexSuffix input : List Bool) :
    scanTapes base i indexSuffix input .index = encodeWord i ++ indexSuffix := by
  simp [scanTapes, MachineUnaryCounter.counterTapes]

theorem scanTapes_other (base : Tape → List Bool) (i : Nat)
    (indexSuffix input : List Bool) (p : Tape) (hi : p ≠ .index) (hw : p ≠ .work) :
    scanTapes base i indexSuffix input p = base p := by
  simp [scanTapes, MachineUnaryCounter.counterTapes, hi, hw]

theorem update_scanTapes_work (base : Tape → List Bool) (i : Nat)
    (indexSuffix input replacement : List Bool) :
    Function.update (scanTapes base i indexSuffix input) .work replacement =
      scanTapes base i indexSuffix replacement := by
  funext p
  cases p <;> simp [scanTapes, MachineUnaryCounter.counterTapes]

/-- Each equation-index bit causes one genuine guard transition and exactly four
field-discard loops. The final zero guard enters the field reader. -/
theorem skipEquationsTrace {n : Nat} (base : Tape → List Bool)
    (prior : List (CloneGap.Equation (Fin n))) (indexSuffix suffix : List Bool)
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[(encodeWords (prior.flatMap equationWords)).length + prior.length + 1]
      (some ⟨some .guard, (ambient, register), scanTapes base prior.length indexSuffix
        (encodeWords (prior.flatMap equationWords) ++ suffix)⟩) =
      some ⟨some (readStart 0), (ambient, none), scanTapes base 0 indexSuffix suffix⟩ := by
  induction prior generalizing register with
  | nil =>
    simpa only [List.flatMap_nil, List.length_nil, encodeWords, List.nil_append,
      Nat.zero_add, Function.iterate_one, advance_some, scanTapes] using
      MachineUnaryCounter.guardStep_zero .index .guard (skipLabel 0) (readStart 0)
        program rfl (Function.update base .work suffix) indexSuffix ambient register
  | cons equation prior ih =>
    have hencoding : encodeWords ((equation :: prior).flatMap equationWords) ++ suffix =
        encodeWords (equationWords equation) ++ (encodeWords (prior.flatMap equationWords) ++ suffix) := by
      simp only [List.flatMap_cons, encodeWords_append, List.append_assoc]
    have htime : (encodeWords ((equation :: prior).flatMap equationWords)).length +
        (equation :: prior).length + 1 =
        ((encodeWords (prior.flatMap equationWords)).length + prior.length + 1 +
          (encodeWords (equationWords equation)).length) + 1 := by
      simp only [List.flatMap_cons, encodeWords_append, List.length_append, List.length_cons]
      omega
    rw [htime, Function.iterate_succ_apply]
    change (advance (TM2.step program))^[
      (encodeWords (prior.flatMap equationWords)).length + prior.length + 1 +
        (encodeWords (equationWords equation)).length]
      (TM2.step program ⟨some .guard, (ambient, register),
        MachineUnaryCounter.counterTapes .index
          (Function.update base .work (encodeWords ((equation :: prior).flatMap equationWords) ++ suffix))
          (prior.length + 1) indexSuffix⟩) = _
    rw [MachineUnaryCounter.guardStep_succ .index .guard (skipLabel 0) (readStart 0)
      program rfl]
    rw [Function.iterate_add_apply]
    have hd := discardFourTrace
      (scanTapes base prior.length indexSuffix
        (encodeWords ((equation :: prior).flatMap equationWords) ++ suffix))
      (equationWords equation) (equationWords_length equation)
      (encodeWords (prior.flatMap equationWords) ++ suffix) (by
        rw [scanTapes_work, hencoding]) ambient
    change (advance (TM2.step program))^[
      (encodeWords (prior.flatMap equationWords)).length + prior.length + 1]
      ((advance (TM2.step program))^[(encodeWords (equationWords equation)).length]
        (some ⟨some (skipLabel 0), (ambient, none), scanTapes base prior.length indexSuffix
          (encodeWords ((equation :: prior).flatMap equationWords) ++ suffix)⟩)) = _
    rw [hd, update_scanTapes_work]
    exact ih none

theorem fieldDestination_eq (j : Fin 4) : fieldDestination j.val = .field j := by
  unfold fieldDestination
  congr 1
  apply Fin.ext
  exact Nat.min_eq_left (by omega)

theorem fieldDestination_injective_below (r s : Nat) (hr : r < 4) (hs : s < 4)
    (he : fieldDestination r = fieldDestination s) : r = s := by
  have hv := congrArg (fun t : Tape => match t with | .field j => j.val | _ => 4) he
  simpa [fieldDestination, Nat.min_eq_left (show r ≤ 3 by omega),
    Nat.min_eq_left (show s ≤ 3 by omega)] using hv

theorem readFour_output_field (base : Tape → List Bool) (values : List Nat)
    (hlen : values.length = 4) (suffix : List Bool) (j : Fin 4) :
    SourceFieldArray.sequenceTapes .work fieldDestination base 0 values suffix (.field j) =
      encodeWord (values[j.val]'(by omega)) ++ base (.field j) := by
  have ht := SourceFieldArray.sequenceTapes_selected .work fieldDestination base 0 values
    suffix j.val (by omega) (by intro r hr; simp [fieldDestination])
    (by
      intro r s hr hs he
      apply fieldDestination_injective_below r s (by omega) (by omega)
      simpa only [Nat.zero_add] using he)
  simpa only [Nat.zero_add, fieldDestination_eq] using ht

theorem readFour_output_work (base : Tape → List Bool) (values : List Nat)
    (_hlen : values.length = 4) (suffix : List Bool)
    (hinput : base .work = encodeWords values ++ suffix) :
    SourceFieldArray.sequenceTapes .work fieldDestination base 0 values suffix .work = suffix := by
  apply SourceFieldArray.sequenceTapes_source _ _ _ _ _ _ _ hinput
  intro r hr
  simp [fieldDestination]

theorem readFour_output_frame (base : Tape → List Bool) (values : List Nat)
    (suffix : List Bool) (p : Tape) (hwork : p ≠ .work)
    (hfield : ∀ j : Fin 4, p ≠ .field j) :
    SourceFieldArray.sequenceTapes .work fieldDestination base 0 values suffix p = base p := by
  apply SourceFieldArray.sequenceTapes_other _ _ _ hwork
  intro r hr
  exact hfield _

def outputTapes {n : Nat} (base : Tape → List Bool) (equation : CloneGap.Equation (Fin n))
    (indexSuffix suffix : List Bool) : Tape → List Bool :=
  SourceFieldArray.sequenceTapes .work fieldDestination
    (scanTapes base 0 indexSuffix (encodeWords (equationWords equation) ++ suffix))
    0 (equationWords equation) suffix

def scanSteps {n : Nat} (variableCount declaredEquations : Nat)
    (prior : List (CloneGap.Equation (Fin n))) (equation : CloneGap.Equation (Fin n)) : Nat :=
  variableCount + declaredEquations + 2 +
    (encodeWords (prior.flatMap equationWords)).length + prior.length + 1 +
    (encodeWords (equationWords equation)).length + 5

theorem scanReadTrace {n : Nat} (base : Tape → List Bool) (variableCount declaredEquations : Nat)
    (prior : List (CloneGap.Equation (Fin n))) (equation : CloneGap.Equation (Fin n))
    (indexSuffix suffix : List Bool) (ambient : σ) :
    (advance (TM2.step program))^[scanSteps variableCount declaredEquations prior equation]
      (some ⟨some .headerFirst, (ambient, none), scanTapes base prior.length indexSuffix
        (encodeWords [variableCount, declaredEquations] ++
          (encodeWords (prior.flatMap equationWords) ++
            (encodeWords (equationWords equation) ++ suffix)))⟩) =
      some ⟨none, (ambient, none), outputTapes base equation indexSuffix suffix⟩ := by
  have hh := discardHeadersTrace
    (scanTapes base prior.length indexSuffix
      (encodeWords [variableCount, declaredEquations] ++
        (encodeWords (prior.flatMap equationWords) ++ (encodeWords (equationWords equation) ++ suffix))))
    variableCount declaredEquations
    (encodeWords (prior.flatMap equationWords) ++ (encodeWords (equationWords equation) ++ suffix))
    (by simp) ambient
  have hs := skipEquationsTrace base prior indexSuffix (encodeWords (equationWords equation) ++ suffix) ambient none
  have hr := readFourTrace
    (scanTapes base 0 indexSuffix (encodeWords (equationWords equation) ++ suffix))
    (equationWords equation) (equationWords_length equation) suffix (by simp) ambient
  rw [show scanSteps variableCount declaredEquations prior equation =
      ((encodeWords (equationWords equation)).length + 5 +
        ((encodeWords (prior.flatMap equationWords)).length + prior.length + 1)) +
        (variableCount + declaredEquations + 2) by unfold scanSteps; omega]
  rw [Function.iterate_add_apply, hh, update_scanTapes_work,
    Function.iterate_add_apply, hs]
  exact hr

def lookupSteps {n : Nat} (F : SourceEncoding.Input) (prior : List (CloneGap.Equation (Fin n)))
    (equation : CloneGap.Equation (Fin n)) : Nat :=
  2 * ((inputBits F).length + 1) + scanSteps F.«variables» F.equations.length prior equation

theorem lookupWithSplitTrace (F : SourceEncoding.Input) (prior : List (CloneGap.Equation (Fin F.«variables»)))
    (equation : CloneGap.Equation (Fin F.«variables»)) (after : List (CloneGap.Equation (Fin F.«variables»)))
    (hsplit : F.equations = prior ++ equation :: after)
    (base : Tape → List Bool) (indexSuffix : List Bool)
    (hsource : base .source = inputBits F)
    (hindex : base .index = encodeWord prior.length ++ indexSuffix)
    (hwork : base .work = []) (hscratch : base .scratch = [])
    (ambient : σ) (register : Option Bool) :
    (advance (TM2.step program))^[lookupSteps F prior equation]
      (some ⟨some .copyOut, (ambient, register), base⟩) =
      some ⟨none, (ambient, none),
        outputTapes base equation indexSuffix (encodeWords (after.flatMap equationWords))⟩ := by
  have hbits : inputBits F = encodeWords [F.«variables», F.equations.length] ++
      (encodeWords (prior.flatMap equationWords) ++
        (encodeWords (equationWords equation) ++ encodeWords (after.flatMap equationWords))) := by
    simp only [inputBits, inputWords, encodeWords_append]
    rw [hsplit]
    simp only [List.flatMap_append, List.flatMap_cons, encodeWords_append]
  have hc := MachineCopy.copyTrace .source .work .scratch (by decide) (by decide)
    (by decide) false .copyOut .copyBack (some .headerFirst) program rfl rfl
    base hscratch ambient register
  have htapes : Function.update base .work (base .source ++ base .work) =
      scanTapes base prior.length indexSuffix (inputBits F) := by
    funext p
    cases p <;> simp [scanTapes, MachineUnaryCounter.counterTapes, hsource, hindex, hwork]
  rw [htapes, hsource] at hc
  rw [lookupSteps, Nat.add_comm, Function.iterate_add_apply, hc, hbits]
  exact scanReadTrace base F.«variables» F.equations.length prior equation indexSuffix
    (encodeWords (after.flatMap equationWords)) ambient

theorem output_source {n : Nat} (base : Tape → List Bool) (equation : CloneGap.Equation (Fin n))
    (indexSuffix suffix : List Bool) :
    outputTapes base equation indexSuffix suffix .source = base .source := by
  rw [outputTapes, readFour_output_frame _ _ _ .source (by decide) (by intro j; simp)]
  exact scanTapes_other _ _ _ _ _ (by decide) (by decide)

theorem output_index {n : Nat} (base : Tape → List Bool) (equation : CloneGap.Equation (Fin n))
    (indexSuffix suffix : List Bool) :
    outputTapes base equation indexSuffix suffix .index = encodeWord 0 ++ indexSuffix := by
  rw [outputTapes, readFour_output_frame _ _ _ .index (by decide) (by intro j; simp),
    scanTapes_index]

theorem output_work {n : Nat} (base : Tape → List Bool) (equation : CloneGap.Equation (Fin n))
    (indexSuffix suffix : List Bool) :
    outputTapes base equation indexSuffix suffix .work = suffix := by
  apply readFour_output_work _ _ (equationWords_length equation) _
  simp

theorem output_field {n : Nat} (base : Tape → List Bool) (equation : CloneGap.Equation (Fin n))
    (indexSuffix suffix : List Bool) (j : Fin 4) :
    outputTapes base equation indexSuffix suffix (.field j) =
      encodeWord ((equationWords equation)[j.val]'(by simp)) ++ base (.field j) := by
  rw [outputTapes, readFour_output_field _ _ (equationWords_length equation)]
  rw [scanTapes_other _ _ _ _ (.field j) (by simp) (by simp)]

theorem output_first {n : Nat} (base : Tape → List Bool)
    (equation : CloneGap.Equation (Fin n)) (indexSuffix suffix : List Bool) :
    outputTapes base equation indexSuffix suffix (.field 0) =
      encodeWord equation.first.val ++ base (.field 0) := by
  simpa [equationWords] using output_field base equation indexSuffix suffix (0 : Fin 4)

theorem output_second {n : Nat} (base : Tape → List Bool)
    (equation : CloneGap.Equation (Fin n)) (indexSuffix suffix : List Bool) :
    outputTapes base equation indexSuffix suffix (.field 1) =
      encodeWord equation.second.val ++ base (.field 1) := by
  simpa [equationWords] using output_field base equation indexSuffix suffix (1 : Fin 4)

theorem output_third {n : Nat} (base : Tape → List Bool)
    (equation : CloneGap.Equation (Fin n)) (indexSuffix suffix : List Bool) :
    outputTapes base equation indexSuffix suffix (.field 2) =
      encodeWord equation.third.val ++ base (.field 2) := by
  simpa [equationWords] using output_field base equation indexSuffix suffix (2 : Fin 4)

theorem output_rhs {n : Nat} (base : Tape → List Bool)
    (equation : CloneGap.Equation (Fin n)) (indexSuffix suffix : List Bool) :
    outputTapes base equation indexSuffix suffix (.field 3) =
      encodeWord (if equation.rhs then 1 else 0) ++ base (.field 3) := by
  simpa [equationWords] using output_field base equation indexSuffix suffix (3 : Fin 4)

theorem output_frame {n : Nat} (base : Tape → List Bool) (equation : CloneGap.Equation (Fin n))
    (indexSuffix suffix : List Bool) (p : Tape) (hindex : p ≠ .index) (hwork : p ≠ .work)
    (hfield : ∀ j : Fin 4, p ≠ .field j) :
    outputTapes base equation indexSuffix suffix p = base p := by
  rw [outputTapes, readFour_output_frame _ _ _ p hwork hfield]
  exact scanTapes_other _ _ _ _ p hindex hwork

theorem equationWords_four {n : Nat} (equation : CloneGap.Equation (Fin n)) :
    equationWords equation =
      [equation.first.val, equation.second.val, equation.third.val,
       if equation.rhs then 1 else 0] := rfl

def fieldWords (tapes : Tape → List Bool) : List (List Bool) :=
  [tapes (.field 0), tapes (.field 1), tapes (.field 2), tapes (.field 3)]

theorem output_fieldWords {n : Nat} (base : Tape → List Bool) (equation : CloneGap.Equation (Fin n))
    (indexSuffix suffix : List Bool) (hempty : ∀ j : Fin 4, base (.field j) = []) :
    fieldWords (outputTapes base equation indexSuffix suffix) =
      (equationWords equation).map encodeWord := by
  simp [fieldWords, output_field, hempty, equationWords_four]

theorem lookupSteps_le_input (F : SourceEncoding.Input) (prior : List (CloneGap.Equation (Fin F.«variables»)))
    (equation : CloneGap.Equation (Fin F.«variables»)) (after : List (CloneGap.Equation (Fin F.«variables»)))
    (hsplit : F.equations = prior ++ equation :: after) :
    lookupSteps F prior equation ≤ 4 * (inputBits F).length + 8 := by
  have hlength : (inputBits F).length = F.«variables» + F.equations.length + 2 +
      (encodeWords (prior.flatMap equationWords)).length +
      (encodeWords (equationWords equation)).length +
      (encodeWords (after.flatMap equationWords)).length := by
    simp only [inputBits, inputWords, encodeWords_append, List.length_append,
      encodeWords, encodeWord_length, List.length_nil]
    rw [hsplit]
    simp only [List.flatMap_append, List.flatMap_cons, encodeWords_append, List.length_append]
    omega
  have hp : prior.length ≤ F.equations.length := by
    rw [hsplit, List.length_append, List.length_cons]
    omega
  unfold lookupSteps scanSteps
  omega

noncomputable def timePolynomial : Polynomial Nat := Polynomial.C 4 * Polynomial.X + Polynomial.C 8

theorem timePolynomial_eval (L : Nat) : timePolynomial.eval L = 4 * L + 8 := by
  simp [timePolynomial]

/-- A polynomial bound on the actual complete copy/skip/read execution. -/
def lookupWithSplitInTime (F : SourceEncoding.Input) (prior : List (CloneGap.Equation (Fin F.«variables»)))
    (equation : CloneGap.Equation (Fin F.«variables»)) (after : List (CloneGap.Equation (Fin F.«variables»)))
    (hsplit : F.equations = prior ++ equation :: after)
    (base : Tape → List Bool) (indexSuffix : List Bool)
    (hsource : base .source = inputBits F)
    (hindex : base .index = encodeWord prior.length ++ indexSuffix)
    (hwork : base .work = []) (hscratch : base .scratch = [])
    (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some .copyOut, (ambient, register), base⟩
      (some ⟨none, (ambient, none),
        outputTapes base equation indexSuffix (encodeWords (after.flatMap equationWords))⟩)
      (timePolynomial.eval (inputBits F).length) where
  steps := lookupSteps F prior equation
  evals_in_steps := lookupWithSplitTrace F prior equation after hsplit base indexSuffix
    hsource hindex hwork hscratch ambient register
  steps_le_m := by
    rw [timePolynomial_eval]
    exact lookupSteps_le_input F prior equation after hsplit

def lookupInTime (F : SourceEncoding.Input) (i : Fin F.equations.length)
    (base : Tape → List Bool) (indexSuffix : List Bool)
    (hsource : base .source = inputBits F)
    (hindex : base .index = encodeWord i.val ++ indexSuffix)
    (hwork : base .work = []) (hscratch : base .scratch = [])
    (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some .copyOut, (ambient, register), base⟩
      (some ⟨none, (ambient, none), outputTapes base F.equations[i.val] indexSuffix
        (encodeWords ((F.equations.drop (i.val + 1)).flatMap equationWords))⟩)
      (timePolynomial.eval (inputBits F).length) := by
  have hp : (F.equations.take i.val).length = i.val := by
    rw [List.length_take, Nat.min_eq_left i.isLt.le]
  have hs : F.equations = F.equations.take i.val ++ F.equations[i.val] :: F.equations.drop (i.val + 1) := by
    rw [List.getElem_cons_drop i.isLt, List.take_append_drop]
  apply lookupWithSplitInTime F (F.equations.take i.val) F.equations[i.val]
    (F.equations.drop (i.val + 1)) hs base indexSuffix hsource
    (by simpa only [hp] using hindex) hwork hscratch ambient register

def machine : FinTM2 where
  K := Tape
  k₀ := .source
  k₁ := .field 0
  Γ _ := Bool
  Λ := Label
  main := .copyOut
  σ := Unit × Option Bool
  initialState := ((), none)
  m := program (σ := Unit)

/-- The same proved run stated directly for the concrete finite machine. -/
def machineInTime (F : SourceEncoding.Input) (i : Fin F.equations.length)
    (base : Tape → List Bool) (indexSuffix : List Bool)
    (hsource : base .source = inputBits F)
    (hindex : base .index = encodeWord i.val ++ indexSuffix)
    (hwork : base .work = []) (hscratch : base .scratch = [])
    (register : Option Bool) :
    StateTransition.EvalsToInTime machine.step
      ⟨some .copyOut, ((), register), base⟩
      (some ⟨none, ((), none), outputTapes base F.equations[i.val] indexSuffix
        (encodeWords ((F.equations.drop (i.val + 1)).flatMap equationWords))⟩)
      (timePolynomial.eval (inputBits F).length) := by
  exact lookupInTime F i base indexSuffix hsource hindex hwork hscratch () register

/-- Same-tape placement into the caller's finite control. Choosing the ambient
type to be `τ × Unit` gives the shared state `(τ × Unit) × Option Bool` used by
the field-template machinery. The lookup's final transition enters `exit`. -/
noncomputable def placedLookupInTime {Λ' : Type}
    (labels : Label → Λ') (exit : Option Λ')
    (target : Λ' → TM2.Stmt (fun _ : Tape => Bool) Λ' (σ × Option Bool))
    (atLabels : ∀ l, target (labels l) =
      MachineSubroutine.statement labels exit (program (σ := σ) l))
    (input : SourceEncoding.Input) (i : Fin input.equations.length)
    (base : Tape → List Bool) (indexSuffix : List Bool)
    (hsource : base .source = inputBits input)
    (hindex : base .index = encodeWord i.val ++ indexSuffix)
    (hwork : base .work = []) (hscratch : base .scratch = [])
    (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step target)
      ⟨some (labels .copyOut), (ambient, register), base⟩
      (some ⟨exit, (ambient, none), outputTapes base input.equations[i.val] indexSuffix
        (encodeWords ((input.equations.drop (i.val + 1)).flatMap equationWords))⟩)
      (timePolynomial.eval (inputBits input).length) :=
  MachineSubroutine.execution labels exit program target atLabels
    (lookupInTime input i base indexSuffix hsource hindex hwork hscratch ambient register)

/-- An optional disjoint-stack placement preserves arbitrary extra tapes and
state. It reuses the same generic lookup and introduces no extra transitions. -/
noncomputable def embeddedLookupInTime {E Λextra τ : Type} [DecidableEq E]
    (continuation : Option (Label ⊕ Λextra)) (extraState : τ)
    (extraTapes : E → List Bool)
    (extra : Λextra → TM2.Stmt
      (MachineEmbedding.Alphabet (fun _ : Tape => Bool) (fun _ : E => Bool))
      (Label ⊕ Λextra) ((σ × Option Bool) × τ))
    (input : SourceEncoding.Input) (i : Fin input.equations.length)
    (base : Tape → List Bool) (indexSuffix : List Bool)
    (hsource : base .source = inputBits input)
    (hindex : base .index = encodeWord i.val ++ indexSuffix)
    (hwork : base .work = []) (hscratch : base .scratch = [])
    (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime
      (TM2.step (MachineEmbedding.program continuation program extra))
      (MachineEmbedding.configuration continuation extraState extraTapes
        ⟨some .copyOut, (ambient, register), base⟩)
      (some (MachineEmbedding.configuration continuation extraState extraTapes
        ⟨none, (ambient, none), outputTapes base input.equations[i.val] indexSuffix
          (encodeWords ((input.equations.drop (i.val + 1)).flatMap equationWords))⟩))
      (timePolynomial.eval (inputBits input).length) :=
  MachineComposition.embeddedExecution continuation extraState extraTapes program extra
    (lookupInTime input i base indexSuffix hsource hindex hwork hscratch ambient register)

end MaxCutGames.Reduction.SourceEquationLookup

end OAI