lean/OAI/Computability/DirectedFeedback/Machines/LoopInit.lean

import Mathlib
import OAI.Computability.DirectedFeedback.Games.TestEmit
import OAI.Computability.DirectedFeedback.Games.SignaturePrepare

namespace OAI


namespace DFVSGames.Foundations.Hastad.SourceLoopInit

open Turing DFVSGames.Foundations.Complexity
open MachineComposition SourceMachine

inductive HeaderTape | variableCount | clauseCount
  deriving DecidableEq

instance : Fintype HeaderTape where
  elems := {.variableCount, .clauseCount}
  complete header := by cases header <;> simp

abbrev Tape (u : Nat) (Extra : Type) := SourceContextLoad.Tape u (HeaderTape ⊕ Extra)

inductive Label (u : Nat)
  | inputCopyOut | inputCopyBack | read (slot : Fin 3 × Bool) | clearWork
  | seed (j : Fin u) | digitCopyOut (j : Fin u) | digitCopyBack (j : Fin u)
  | trim (j : Fin u) | done
  deriving DecidableEq, Fintype

variable {u : Nat} {Extra : Type}

def variableHeader : Tape u Extra := .extra (.inl .variableCount)
def clauseHeader : Tape u Extra := .extra (.inl .clauseCount)
def headerDestination (r : Nat) : Tape u Extra := if r = 0 then variableHeader else clauseHeader
def readStart (r : Nat) : Label u := .read (SourceFieldArray.boundedIndex 2 r, false)
def readLoop (r : Nat) : Label u := .read (SourceFieldArray.boundedIndex 2 r, true)
def labelAt (r : Nat) : Label u := if h : r < u then .seed ⟨r, h⟩ else .done

def program : Label u → TM2.Stmt (fun _ : Tape u Extra => Bool) (Label u) (Unit × Option Bool)
  | .inputCopyOut => Reduction.MachineTransfer.loopAt .formula .scratch id false
      .inputCopyOut (some .inputCopyBack)
  | .inputCopyBack => MachineCopy.forkLoop .scratch .formula .work false
      .inputCopyBack (some (readStart 0))
  | .read slot => if slot.1.val < 2 then
      if slot.2 then fieldLoop .work (headerDestination slot.1.val) (.read (slot.1, true))
        (some (readStart (slot.1.val + 1)))
      else fieldStart (headerDestination slot.1.val) (.read (slot.1, true))
    else SourceFieldArray.finish .work (some .clearWork)
  | .clearWork => MachineDrain.drain .work .clearWork (some (labelAt 0))
  | .seed j => .push (.current j) (fun _ => false) (.goto fun _ => .digitCopyOut j)
  | .digitCopyOut j => Reduction.MachineTransfer.loopAt clauseHeader .copyScratch id false
      (.digitCopyOut j) (some (.digitCopyBack j))
  | .digitCopyBack j => MachineCopy.forkLoop .copyScratch clauseHeader (.remaining j) false
      (.digitCopyBack j) (some (.trim j))
  | .trim j => .pop (.remaining j) (fun state _ => state) (.goto fun _ => labelAt (j.val + 1))
  | .done => .halt

theorem atReadStart (r : Nat) (hr : r < 2) :
    program (u := u) (Extra := Extra) (readStart r) =
      fieldStart (headerDestination r) (readLoop r) := by
  simp [program, readStart, readLoop, SourceFieldArray.boundedIndex_val hr.le, hr]

theorem atReadLoop (r : Nat) (hr : r < 2) :
    program (u := u) (Extra := Extra) (readLoop r) = fieldLoop .work (headerDestination r)
      (readLoop r) (some (readStart (r + 1))) := by
  simp [program, readStart, readLoop, SourceFieldArray.boundedIndex_val hr.le, hr]

theorem atReadDone : program (u := u) (Extra := Extra) (readStart 2) =
    SourceFieldArray.finish .work (some .clearWork) := by
  simp [program, readStart]

variable [DecidableEq Extra]

def copiedInput (base : Tape u Extra → List Bool) : Tape u Extra → List Bool :=
  Function.update base .work (base .formula)

def readOutput (F : Target.Formula) (base : Tape u Extra → List Bool) : Tape u Extra → List Bool :=
  SourceFieldArray.sequenceTapes .work headerDestination (copiedInput base) 0
    [F.«variables», F.clauses.length] (encodeWords (F.clauses.flatMap clauseWords))

def headerOutput (F : Target.Formula) (base : Tape u Extra → List Bool) : Tape u Extra → List Bool :=
  Function.update (Function.update (Function.update base .work []) variableHeader
    (encodeWord F.«variables» ++ base variableHeader)) clauseHeader
    (encodeWord F.clauses.length ++ base clauseHeader)

theorem clear_readOutput (F : Target.Formula) (base : Tape u Extra → List Bool) :
    Function.update (readOutput F base) .work [] = headerOutput F base := by
  funext p
  cases p <;> simp [readOutput, headerOutput, SourceFieldArray.sequenceTapes,
    afterField, fieldTapes, copiedInput, headerDestination, variableHeader, clauseHeader]
  rename_i e
  cases e with
  | inl h => cases h <;> simp
  | inr e => simp

theorem readOutput_work (F : Target.Formula) (base : Tape u Extra → List Bool) :
    readOutput F base .work = encodeWords (F.clauses.flatMap clauseWords) := by
  simp [readOutput, SourceFieldArray.sequenceTapes, afterField, fieldTapes,
    headerDestination, variableHeader, clauseHeader, encodeWords]

theorem headerOutput_frame (F : Target.Formula) (base : Tape u Extra → List Bool)
    (p : Tape u Extra) (hw : p ≠ .work) (hn : p ≠ variableHeader) (hm : p ≠ clauseHeader) :
    headerOutput F base p = base p := by simp [headerOutput, hw, hn, hm]

def headerInTime (F : Target.Formula) (base : Tape u Extra → List Bool)
    (hformula : base .formula = formulaBits F) (hwork : base .work = [])
    (hscratch : base .scratch = []) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some .inputCopyOut, ((), register), base⟩
      (some ⟨some (labelAt 0), ((), none), headerOutput F base⟩)
      (3 * (formulaBits F).length + 6) := by
  have copying := MachineCopy.copyInTime .formula .work .scratch
    (by simp) (by simp) (by simp) false .inputCopyOut .inputCopyBack (some (readStart 0))
    program rfl rfl base hscratch () register
  have hc : Function.update base .work (base .formula ++ base .work) = copiedInput base := by
    simp [copiedInput, hwork]
  rw [hc] at copying
  have reading := SourceFieldArray.sequenceInTime .work headerDestination readStart readLoop
    (some .clearWork) program (copiedInput base) 0 [F.«variables», F.clauses.length]
    (encodeWords (F.clauses.flatMap clauseWords))
    (by intro r hr; simp [headerDestination, variableHeader, clauseHeader]; split <;> simp)
    (by intro r hr; simpa only [Nat.zero_add] using atReadStart (u := u) (Extra := Extra) r (by simpa using hr))
    (by intro r hr; simpa only [Nat.zero_add] using atReadLoop (u := u) (Extra := Extra) r (by simpa using hr))
    (by simpa using (atReadDone (u := u) (Extra := Extra)))
    (by simp [copiedInput, hformula, formulaBits, formulaWords, encodeWords, List.append_assoc]) () none
  have draining := MachineDrain.drainInTime .work .clearWork (some (labelAt 0))
    program rfl (readOutput F base) () none
  have joined := StateTransition.EvalsToInTime.trans (TM2.step program) _ _ _ _ _
    (StateTransition.EvalsToInTime.trans (TM2.step program) _ _ _ _ _ copying reading) draining
  refine { steps := joined.steps, evals_in_steps := ?_, steps_le_m := ?_ }
  · simpa only [clear_readOutput] using joined.evals_in_steps
  · apply Nat.le_trans joined.steps_le_m
    rw [readOutput_work, hformula]
    have he : (formulaBits F).length = F.«variables» + F.clauses.length + 2 +
        (encodeWords (F.clauses.flatMap clauseWords)).length := by
      simp [formulaBits, formulaWords, encodeWords, Nat.add_assoc]
      omega
    simp only [List.sum_cons, List.sum_nil, List.length_cons, List.length_nil]
    omega

def digitOutput (j : Fin u) (m : Nat) (base : Tape u Extra → List Bool) : Tape u Extra → List Bool :=
  Function.update (Function.update base (.current j) (encodeWord 0)) (.remaining j) (encodeWord (m - 1))

theorem digitOutput_frame (j : Fin u) (m : Nat) (base : Tape u Extra → List Bool)
    (p : Tape u Extra) (hc : p ≠ .current j) (hr : p ≠ .remaining j) :
    digitOutput j m base p = base p := by simp [digitOutput, hc, hr]

def digitInTime (j : Fin u) (m : Nat) (hm : 0 < m) (base : Tape u Extra → List Bool)
    (hcount : base clauseHeader = encodeWord m)
    (hcurrent : base (.current j) = []) (hremaining : base (.remaining j) = [])
    (hscratch : base .copyScratch = []) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (.seed j), ((), none), base⟩
      (some ⟨some (labelAt (j.val + 1)), ((), none), digitOutput j m base⟩)
      (2 * m + 6) := by
  let seeded := Function.update base (.current j) (encodeWord 0)
  have seedRun : StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (.seed j), ((), none), base⟩
      (some ⟨some (.digitCopyOut j), ((), none), seeded⟩) 1 := by
    refine { steps := 1, evals_in_steps := ?_, steps_le_m := Nat.le_refl _ }
    change TM2.step program ⟨some (.seed j), ((), none), base⟩ = _
    simp [program, TM2.step, TM2.stepAux, seeded, hcurrent, encodeWord]
  have copying := MachineCopy.copyInTime clauseHeader (.remaining j) .copyScratch
    (by simp [clauseHeader]) (by simp [clauseHeader]) (by simp)
    false (.digitCopyOut j) (.digitCopyBack j) (some (.trim j)) program rfl rfl seeded
    (by simpa [seeded] using hscratch) () none
  let copied := Function.update seeded (.remaining j) (encodeWord m)
  have he : Function.update seeded (.remaining j)
      (seeded clauseHeader ++ seeded (.remaining j)) = copied := by
    have hcount' : base (.extra (.inl .clauseCount)) = encodeWord m := hcount
    simp [copied, seeded, clauseHeader, hremaining, hcount']
  rw [he] at copying
  have trimRun : StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (.trim j), ((), none), copied⟩
      (some ⟨some (labelAt (j.val + 1)), ((), none), digitOutput j m base⟩) 1 := by
    refine { steps := 1, evals_in_steps := ?_, steps_le_m := Nat.le_refl _ }
    change TM2.step program ⟨some (.trim j), ((), none), copied⟩ = _
    have hn : m = (m - 1) + 1 := by omega
    have htail : (encodeWord m).tail = encodeWord (m - 1) := by
      conv_lhs => rw [hn]
      simp [encodeWord, List.replicate_succ]
    simp [program, TM2.step, TM2.stepAux, copied, seeded, digitOutput, htail]
  have joined := StateTransition.EvalsToInTime.trans (TM2.step program) _ _ _ _ _
    (StateTransition.EvalsToInTime.trans (TM2.step program) _ _ _ _ _ seedRun copying) trimRun
  refine { steps := joined.steps, evals_in_steps := joined.evals_in_steps, steps_le_m := ?_ }
  apply Nat.le_trans joined.steps_le_m
  have hs : seeded clauseHeader = encodeWord m := by simpa [seeded, clauseHeader] using hcount
  rw [hs, encodeWord_length]
  omega

def stageTapes (m : Nat) (base : Tape u Extra → List Bool) : Nat → Tape u Extra → List Bool
  | 0 => base
  | r + 1 => if h : r < u then digitOutput ⟨r, h⟩ m (stageTapes m base r)
    else stageTapes m base r

theorem stageTapes_frame (m : Nat) (base : Tape u Extra → List Bool) (r : Nat)
    (p : Tape u Extra) (hc : ∀ j, p ≠ .current j) (hr : ∀ j, p ≠ .remaining j) :
    stageTapes m base r p = base p := by
  induction r with
  | zero => rfl
  | succ r ih =>
    simp only [stageTapes]
    split
    · rw [digitOutput_frame _ _ _ _ (hc _) (hr _), ih]
    · exact ih

theorem stageTapes_current (m : Nat) (base : Tape u Extra → List Bool) (r : Nat) (j : Fin u) :
    stageTapes m base r (.current j) = if j.val < r then encodeWord 0 else base (.current j) := by
  induction r with
  | zero => simp [stageTapes]
  | succ r ih =>
    simp only [stageTapes]
    split
    next hr =>
      by_cases hj : j = (⟨r, hr⟩ : Fin u)
      · subst j; simp [digitOutput]
      · rw [digitOutput_frame _ _ _ _ (by simpa using hj) (by simp), ih]
        have hval : j.val ≠ r := by intro h; apply hj; exact Fin.ext h
        have he : j.val < r + 1 ↔ j.val < r := by omega
        simp only [he]
    next hr =>
      rw [ih]
      have hjr : j.val < r := by omega
      have hjr' : j.val < r + 1 := by omega
      simp only [hjr, hjr', ite_true]

theorem stageTapes_remaining (m : Nat) (base : Tape u Extra → List Bool) (r : Nat) (j : Fin u) :
    stageTapes m base r (.remaining j) =
      if j.val < r then encodeWord (m - 1) else base (.remaining j) := by
  induction r with
  | zero => simp [stageTapes]
  | succ r ih =>
    simp only [stageTapes]
    split
    next hr =>
      by_cases hj : j = (⟨r, hr⟩ : Fin u)
      · subst j; simp [digitOutput]
      · rw [digitOutput_frame _ _ _ _ (by simp) (by simpa using hj), ih]
        have hval : j.val ≠ r := by intro h; apply hj; exact Fin.ext h
        have he : j.val < r + 1 ↔ j.val < r := by omega
        simp only [he]
    next hr =>
      rw [ih]
      have hjr : j.val < r := by omega
      have hjr' : j.val < r + 1 := by omega
      simp only [hjr, hjr', ite_true]

def prefixInTime (m : Nat) (hm : 0 < m) (base : Tape u Extra → List Bool)
    (hcount : base clauseHeader = encodeWord m)
    (hcurrent : ∀ j, base (.current j) = []) (hremaining : ∀ j, base (.remaining j) = [])
    (hscratch : base .copyScratch = []) (r : Nat) (hr : r ≤ u) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labelAt 0), ((), none), base⟩
      (some ⟨some (labelAt r), ((), none), stageTapes m base r⟩)
      (r * (2 * m + 6)) := by
  induction r with
  | zero => exact { steps := 0, evals_in_steps := rfl, steps_le_m := by omega }
  | succ r ih =>
    have hru : r < u := by omega
    let j : Fin u := ⟨r, hru⟩
    let before := stageTapes m base r
    have one := digitInTime j m hm before
      (by dsimp only [before]; rw [stageTapes_frame _ _ _ _ (by simp [clauseHeader]) (by simp [clauseHeader])]; exact hcount)
      (by simp [before, stageTapes_current, j, hcurrent])
      (by simp [before, stageTapes_remaining, j, hremaining])
      (by dsimp only [before]; rw [stageTapes_frame _ _ _ _ (by simp) (by simp)]; exact hscratch)
    have one' : StateTransition.EvalsToInTime (TM2.step program)
        ⟨some (labelAt r), ((), none), before⟩
        (some ⟨some (labelAt (r + 1)), ((), none), stageTapes m base (r + 1)⟩)
        (2 * m + 6) := by
      simpa only [labelAt, dite_eq_left hru, j, stageTapes, before] using one
    have joined := StateTransition.EvalsToInTime.trans (TM2.step program) _ _ _ _ _
      (ih (by omega)) one'
    simpa only [Nat.add_mul, Nat.one_mul, Nat.add_comm] using joined

def digitsInTime (m : Nat) (hm : 0 < m) (base : Tape u Extra → List Bool)
    (hcount : base clauseHeader = encodeWord m)
    (hcurrent : ∀ j, base (.current j) = []) (hremaining : ∀ j, base (.remaining j) = [])
    (hscratch : base .copyScratch = []) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labelAt 0), ((), none), base⟩
      (some ⟨none, ((), none), stageTapes m base u⟩)
      (u * (2 * m + 6) + 1) := by
  have initialRun := prefixInTime m hm base hcount hcurrent hremaining hscratch u (Nat.le_refl _)
  have doneRun : StateTransition.EvalsToInTime (TM2.step (program (Extra := Extra)))
      ⟨some (labelAt u), ((), none), stageTapes m base u⟩
      (some ⟨none, ((), none), stageTapes m base u⟩) 1 := by
    refine { steps := 1, evals_in_steps := ?_, steps_le_m := Nat.le_refl _ }
    change TM2.step program ⟨some (labelAt u), ((), none), stageTapes m base u⟩ = _
    simp only [labelAt, Nat.lt_irrefl, ↓reduceDIte]
    rfl
  simpa only [Nat.add_comm] using
    StateTransition.EvalsToInTime.trans (TM2.step program) _ _ _ _ _ initialRun doneRun

def outputTapes (F : Target.Formula) (base : Tape u Extra → List Bool) : Tape u Extra → List Bool :=
  stageTapes F.clauses.length (headerOutput F base) u

def initializeInTime (F : Target.Formula) (hm : 0 < F.clauses.length)
    (base : Tape u Extra → List Bool) (hformula : base .formula = formulaBits F)
    (hwork : base .work = []) (hscratch : base .scratch = []) (hcopy : base .copyScratch = [])
    (hclauseHeader : base clauseHeader = [])
    (hcurrent : ∀ j, base (.current j) = []) (hremaining : ∀ j, base (.remaining j) = []) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some .inputCopyOut, ((), none), base⟩
      (some ⟨none, ((), none), outputTapes F base⟩)
      ((2 * u + 3) * (formulaBits F).length + 6 * u + 7) := by
  have firstRun := headerInTime F base hformula hwork hscratch none
  have hcount : headerOutput F base clauseHeader = encodeWord F.clauses.length := by
    simp [headerOutput, hclauseHeader]
  have secondRun := digitsInTime F.clauses.length hm (headerOutput F base) hcount
    (by intro j; rw [headerOutput_frame _ _ _ (by simp) (by simp [variableHeader])
      (by simp [clauseHeader])]; exact hcurrent j)
    (by intro j; rw [headerOutput_frame _ _ _ (by simp) (by simp [variableHeader])
      (by simp [clauseHeader])]; exact hremaining j)
    (by rw [headerOutput_frame _ _ _ (by simp) (by simp [variableHeader])
      (by simp [clauseHeader])]; exact hcopy)
  have joined := StateTransition.EvalsToInTime.trans (TM2.step program) _ _ _ _ _ firstRun secondRun
  refine { steps := joined.steps, evals_in_steps := joined.evals_in_steps, steps_le_m := ?_ }
  apply Nat.le_trans joined.steps_le_m
  have hL := SourceContextLoad.clauses_length_le_input F
  have hb : u * (2 * F.clauses.length + 6) ≤ u * (2 * (formulaBits F).length + 6) :=
    Nat.mul_le_mul_left u (by omega)
  have he : (2 * u + 3) * (formulaBits F).length + 6 * u + 7 =
      u * (2 * (formulaBits F).length + 6) + 3 * (formulaBits F).length + 7 := by
    simp [Nat.add_mul, Nat.mul_add, Nat.mul_comm, Nat.mul_assoc,
      Nat.add_assoc, Nat.add_comm, Nat.add_left_comm]
  rw [he]
  omega

@[simp] theorem output_current (F : Target.Formula) (base : Tape u Extra → List Bool) (j : Fin u) :
    outputTapes F base (.current j) = encodeWord 0 := by
  simp only [outputTapes, stageTapes_current, j.isLt, ite_true]

@[simp] theorem output_remaining (F : Target.Formula) (base : Tape u Extra → List Bool) (j : Fin u) :
    outputTapes F base (.remaining j) = encodeWord (F.clauses.length - 1) := by
  simp only [outputTapes, stageTapes_remaining, j.isLt, ite_true]

theorem output_frame (F : Target.Formula) (base : Tape u Extra → List Bool) (p : Tape u Extra)
    (hw : p ≠ .work) (hn : p ≠ variableHeader) (hm : p ≠ clauseHeader)
    (hc : ∀ j, p ≠ .current j) (hr : ∀ j, p ≠ .remaining j) :
    outputTapes F base p = base p := by
  rw [outputTapes, stageTapes_frame _ _ _ _ hc hr, headerOutput_frame _ _ _ hw hn hm]

@[simp] theorem output_work (F : Target.Formula) (base : Tape u Extra → List Bool) :
    outputTapes F base .work = [] := by
  rw [outputTapes, stageTapes_frame _ _ _ _ (by simp) (by simp)]
  simp [headerOutput, variableHeader, clauseHeader]

@[simp] theorem output_variableHeader (F : Target.Formula) (base : Tape u Extra → List Bool) :
    outputTapes F base variableHeader = encodeWord F.«variables» ++ base variableHeader := by
  rw [outputTapes, stageTapes_frame _ _ _ _ (by simp [variableHeader]) (by simp [variableHeader])]
  simp [headerOutput, variableHeader, clauseHeader]

@[simp] theorem output_clauseHeader (F : Target.Formula) (base : Tape u Extra → List Bool) :
    outputTapes F base clauseHeader = encodeWord F.clauses.length ++ base clauseHeader := by
  rw [outputTapes, stageTapes_frame _ _ _ _ (by simp [clauseHeader]) (by simp [clauseHeader])]
  simp [headerOutput]

theorem output_headers (F : Target.Formula) (base : Tape u Extra → List Bool)
    (hn : base variableHeader = []) (hm : base clauseHeader = []) :
    outputTapes F base variableHeader = encodeWord F.«variables» ∧
      outputTapes F base clauseHeader = encodeWord F.clauses.length := by
  simp only [output_variableHeader, output_clauseHeader, hn, hm, List.append_nil, and_self]

@[simp] theorem output_formula (F : Target.Formula) (base : Tape u Extra → List Bool) :
    outputTapes F base .formula = base .formula :=
  output_frame F base .formula (by simp) (by simp [variableHeader])
    (by simp [clauseHeader]) (by simp) (by simp)

noncomputable def timePolynomial (u : Nat) : Polynomial Nat :=
  Polynomial.C (2 * u + 3) * Polynomial.X + Polynomial.C (6 * u + 7)

theorem timePolynomial_eval (u L : Nat) :
    (timePolynomial u).eval L = (2 * u + 3) * L + 6 * u + 7 := by
  simp [timePolynomial, Nat.add_assoc]

def machine (u : Nat) (Extra : Type) [DecidableEq Extra] [Fintype Extra] : FinTM2 where
  K := Tape u Extra
  k₀ := .formula
  k₁ := clauseHeader
  Γ _ := Bool
  Λ := Label u
  main := .inputCopyOut
  σ := Unit × Option Bool
  initialState := ((), none)
  m := program

end DFVSGames.Foundations.Hastad.SourceLoopInit


namespace DFVSGames.Foundations.Hastad.SourceRankPhase

open Turing Complexity
open scoped BigOperators
open Target SourceContexts SourceOccurrences SourceAddressArithmetic

def rank {width : Nat} (radix : Nat) (digits : Fin width → Nat) : Nat :=
  ∑ i : Fin width, digits i * radix ^ i.val

def reverseDigits {width : Nat} (digits : Fin width → Nat) (i : Nat) : Nat :=
  if h : i < width then digits (Fin.rev ⟨i, h⟩) else 0

@[simp] theorem reverseDigits_apply {width : Nat} (digits : Fin width → Nat) (i : Fin width) :
    reverseDigits digits i.val = digits i.rev := by
  simp [reverseDigits, i.isLt]

theorem horner_value_sum (radix : Nat) (digits : Nat → Nat) (n : Nat) :
    MachineHorner.value radix digits n =
      ∑ i ∈ Finset.range n, digits i * radix ^ (n - 1 - i) := by
  induction n with
  | zero => simp [MachineHorner.value]
  | succ n ih =>
    rw [MachineHorner.value, ih, Finset.sum_range_succ, Finset.mul_sum]
    simp only [Nat.add_sub_cancel, Nat.sub_self, pow_zero, Nat.mul_one]
    congr 1
    apply Finset.sum_congr rfl
    intro i hi
    have hi' := Finset.mem_range.mp hi
    rw [show n - i = (n - 1 - i) + 1 by omega, pow_succ]
    ac_rfl

theorem horner_reverse_eq_rank {width : Nat} (radix : Nat) (digits : Fin width → Nat) :
    MachineHorner.value radix (reverseDigits digits) width = rank radix digits := by
  rw [horner_value_sum, Finset.sum_range]
  calc
    (∑ i : Fin width, reverseDigits digits i.val * radix ^ (width - 1 - i.val)) =
        ∑ i : Fin width, digits i.rev * radix ^ i.rev.val := by
      apply Finset.sum_congr rfl
      intro i _
      rw [reverseDigits_apply]
      have he : width - 1 - i.val = i.rev.val := by
        simp only [Fin.rev]
        omega
      rw [he]
    _ = rank radix digits := by
      simpa only [rank, Fin.revPerm_apply] using
        Equiv.sum_comp (Fin.revPerm (n := width)) (fun i => digits i * radix ^ i.val)

variable {K Λ σ : Type} {width : Nat}

def reverseSlots (slots : MachineHorner.Layout width ↪ K) : MachineHorner.Layout width ↪ K :=
  (Equiv.sumCongr (Equiv.refl (Fin 6)) (Fin.revPerm (n := width))).toEmbedding.trans slots

@[simp] theorem reverseSlots_control (slots : MachineHorner.Layout width ↪ K) (i : Fin 6) :
    reverseSlots slots (.inl i) = slots (.inl i) := rfl

@[simp] theorem reverseSlots_digit (slots : MachineHorner.Layout width ↪ K) (i : Fin width) :
    reverseSlots slots (.inr i) = slots (.inr i.rev) := rfl

variable [DecidableEq K]

omit [DecidableEq K] in
theorem clean_reverseSlots (slots : MachineHorner.Layout width ↪ K) (base : K → List Bool)
    (clean : MachineHorner.Clean slots base) : MachineHorner.Clean (reverseSlots slots) base where
  accA := by simpa only [reverseSlots_control] using clean.accA
  accB := by simpa only [reverseSlots_control] using clean.accB
  counter := by simpa only [reverseSlots_control] using clean.counter
  scratch := by simpa only [reverseSlots_control] using clean.scratch

def statement (slots : MachineHorner.Layout width ↪ K)
    (labels : MachineHorner.Label width → Λ) (exit : Option Λ) :
    MachineHorner.Label width → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ) :=
  MachineHorner.statement (reverseSlots slots) labels exit

def program (slots : MachineHorner.Layout width ↪ K) :
    MachineHorner.Label width → TM2.Stmt (fun _ : K => Bool)
      (MachineHorner.Label width) (MachineHorner.State σ) :=
  statement slots id none

theorem rankTrace (slots : MachineHorner.Layout width ↪ K)
    (labels : MachineHorner.Label width → Λ) (exit : Option Λ)
    (p : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
    (atLabels : ∀ label, p (labels label) = statement slots labels exit label)
    (base : K → List Bool) (radix : Nat) (digits : Fin width → Nat)
    (operandRadix : base (slots (.inl 0)) = encodeWord radix)
    (operandDigits : ∀ i, base (slots (.inr i)) = encodeWord (digits i))
    (clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool) :
    (MachineComposition.advance (TM2.step p))^[
      MachineHorner.steps radix (reverseDigits digits) width]
      (some ⟨some (labels .start), ((ambient, ()), register), base⟩) =
      some ⟨exit, ((ambient, ()), none), MachineHorner.resultTapes slots base (rank radix digits)⟩ := by
  have h := MachineHorner.hornerTrace (reverseSlots slots) labels exit p atLabels base radix
    (reverseDigits digits) operandRadix
    (fun i => by simpa only [reverseSlots_digit, reverseDigits_apply] using operandDigits i.rev)
    (clean_reverseSlots slots base clean) ambient register
  rw [horner_reverse_eq_rank] at h
  simpa only [MachineHorner.resultTapes, reverseSlots_control] using h

def rankInTime (slots : MachineHorner.Layout width ↪ K)
    (labels : MachineHorner.Label width → Λ) (exit : Option Λ)
    (p : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
    (atLabels : ∀ label, p (labels label) = statement slots labels exit label)
    (base : K → List Bool) (radix : Nat) (digits : Fin width → Nat)
    (operandRadix : base (slots (.inl 0)) = encodeWord radix)
    (operandDigits : ∀ i, base (slots (.inr i)) = encodeWord (digits i))
    (clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step p)
      ⟨some (labels .start), ((ambient, ()), register), base⟩
      (some ⟨exit, ((ambient, ()), none), MachineHorner.resultTapes slots base (rank radix digits)⟩)
      (MachineHorner.steps radix (reverseDigits digits) width) where
  steps := MachineHorner.steps radix (reverseDigits digits) width
  evals_in_steps := rankTrace slots labels exit p atLabels base radix digits operandRadix
    operandDigits clean ambient register
  steps_le_m := Nat.le_refl _

def rankInPolynomialTime (slots : MachineHorner.Layout width ↪ K)
    (labels : MachineHorner.Label width → Λ) (exit : Option Λ)
    (p : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
    (atLabels : ∀ label, p (labels label) = statement slots labels exit label)
    (base : K → List Bool) (radix : Nat) (digits : Fin width → Nat)
    (operandRadix : base (slots (.inl 0)) = encodeWord radix)
    (operandDigits : ∀ i, base (slots (.inr i)) = encodeWord (digits i))
    (clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool)
    (magnitude : Nat) (radixBound : radix ≤ magnitude) (digitBound : ∀ i, digits i ≤ magnitude) :
    StateTransition.EvalsToInTime (TM2.step p)
      ⟨some (labels .start), ((ambient, ()), register), base⟩
      (some ⟨exit, ((ambient, ()), none), MachineHorner.resultTapes slots base (rank radix digits)⟩)
      ((MachineHorner.timePolynomial width).eval magnitude) where
  steps := MachineHorner.steps radix (reverseDigits digits) width
  evals_in_steps := rankTrace slots labels exit p atLabels base radix digits operandRadix
    operandDigits clean ambient register
  steps_le_m := MachineHorner.steps_le_timePolynomial radix (reverseDigits digits) width magnitude
    radixBound (fun i hi => by simpa only [reverseDigits, dite_eq_left hi] using digitBound (Fin.rev ⟨i, hi⟩))

def variableRankInTime (F : Formula) (u : Nat) (v : VariableContext F u)
    (slots : MachineHorner.Layout u ↪ K) (labels : MachineHorner.Label u → Λ) (exit : Option Λ)
    (p : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
    (atLabels : ∀ label, p (labels label) = statement slots labels exit label)
    (base : K → List Bool) (operandRadix : base (slots (.inl 0)) = encodeWord F.«variables»)
    (operandDigits : ∀ i, base (slots (.inr i)) = encodeWord (v i).val)
    (clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step p)
      ⟨some (labels .start), ((ambient, ()), register), base⟩
      (some ⟨exit, ((ambient, ()), none),
        MachineHorner.resultTapes slots base (((variableEncoding F u).code v).val)⟩)
      ((MachineHorner.timePolynomial u).eval F.«variables») := by
  have h := rankInPolynomialTime slots labels exit p atLabels base F.«variables» (fun i => (v i).val)
    operandRadix operandDigits clean ambient register F.«variables» (Nat.le_refl _) (fun i => (v i).isLt.le)
  simpa only [rank, ← variable_rank F u v] using h

def clauseRankInTime (F : Formula) (u : Nat) (c : ClauseContext F u)
    (slots : MachineHorner.Layout u ↪ K) (labels : MachineHorner.Label u → Λ) (exit : Option Λ)
    (p : Λ → TM2.Stmt (fun _ : K => Bool) Λ (MachineHorner.State σ))
    (atLabels : ∀ label, p (labels label) = statement slots labels exit label)
    (base : K → List Bool) (operandRadix : base (slots (.inl 0)) = encodeWord F.clauses.length)
    (operandDigits : ∀ i, base (slots (.inr i)) = encodeWord (c i).val)
    (clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step p)
      ⟨some (labels .start), ((ambient, ()), register), base⟩
      (some ⟨exit, ((ambient, ()), none),
        MachineHorner.resultTapes slots base (((clauseEncoding F u).code c).val)⟩)
      ((MachineHorner.timePolynomial u).eval F.clauses.length) := by
  have h := rankInPolynomialTime slots labels exit p atLabels base F.clauses.length (fun i => (c i).val)
    operandRadix operandDigits clean ambient register F.clauses.length (Nat.le_refl _) (fun i => (c i).isLt.le)
  simpa only [rank, ← clause_rank F u c] using h

def programInTime (slots : MachineHorner.Layout width ↪ K)
    (base : K → List Bool) (radix : Nat) (digits : Fin width → Nat)
    (operandRadix : base (slots (.inl 0)) = encodeWord radix)
    (operandDigits : ∀ i, base (slots (.inr i)) = encodeWord (digits i))
    (clean : MachineHorner.Clean slots base) (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step (program (σ := σ) slots))
      ⟨some .start, ((ambient, ()), register), base⟩
      (some ⟨none, ((ambient, ()), none), MachineHorner.resultTapes slots base (rank radix digits)⟩)
      (MachineHorner.steps radix (reverseDigits digits) width) :=
  rankInTime slots id none (program slots) (fun _ => rfl) base radix digits operandRadix
    operandDigits clean ambient register

def machine (width : Nat) : FinTM2 where
  K := MachineHorner.Layout width
  k₀ := .inl 0
  k₁ := .inl 3
  Γ _ := Bool
  Λ := MachineHorner.Label width
  main := .start
  σ := MachineHorner.State Unit
  initialState := (((), ()), none)
  m := program (Function.Embedding.refl (MachineHorner.Layout width))

def machineInTime (width radix : Nat) (digits : Fin width → Nat)
    (base : MachineHorner.Layout width → List Bool)
    (operandRadix : base (.inl 0) = encodeWord radix)
    (operandDigits : ∀ i, base (.inr i) = encodeWord (digits i))
    (clean : MachineHorner.Clean (Function.Embedding.refl _) base) (register : Option Bool) :
    StateTransition.EvalsToInTime (machine width).step
      ⟨some (MachineHorner.Label.start), (((), ()), register), base⟩
      (some ⟨none, (((), ()), none),
        MachineHorner.resultTapes (Function.Embedding.refl _) base (rank radix digits)⟩)
      (MachineHorner.steps radix (reverseDigits digits) width) :=
  programInTime (Function.Embedding.refl _) base radix digits operandRadix operandDigits clean () register


variable {u : Nat} {Extra : Type}

def clauseSlots (controls : Fin 6 ↪ Extra) :
    MachineHorner.Layout u ↪ SourceContextLoad.Tape u Extra where
  toFun
    | .inl i => .extra (controls i)
    | .inr i => .current i
  inj' := by
    intro a b h
    cases a with
    | inl a =>
      cases b with
      | inl b => exact congrArg Sum.inl (controls.injective (SourceContextLoad.Tape.extra.inj h))
      | inr b => cases h
    | inr a =>
      cases b with
      | inl b => cases h
      | inr b => exact congrArg Sum.inr (SourceContextLoad.Tape.current.inj h)

def variableSlots (controls : Fin 6 ↪ Extra) (selected : SlotContext u) :
    MachineHorner.Layout u ↪ SourceContextLoad.Tape u Extra where
  toFun
    | .inl i => .extra (controls i)
    | .inr i => SourceContextLoad.variableField i (slotEncoding.code (selected i))
  inj' := by
    intro a b h
    cases a with
    | inl a =>
      cases b with
      | inl b => exact congrArg Sum.inl (controls.injective (SourceContextLoad.Tape.extra.inj h))
      | inr b => cases h
    | inr a =>
      cases b with
      | inl b => cases h
      | inr b => exact congrArg Sum.inr (SourceContextLoad.Tape.field.inj h).1

@[simp] theorem clauseSlots_control (controls : Fin 6 ↪ Extra) (i : Fin 6) :
    clauseSlots (u := u) controls (.inl i) = .extra (controls i) := rfl

@[simp] theorem clauseSlots_digit (controls : Fin 6 ↪ Extra) (i : Fin u) :
    clauseSlots controls (.inr i) = .current i := rfl

@[simp] theorem variableSlots_control (controls : Fin 6 ↪ Extra) (selected : SlotContext u) (i : Fin 6) :
    variableSlots controls selected (.inl i) = .extra (controls i) := rfl

@[simp] theorem variableSlots_digit (controls : Fin 6 ↪ Extra) (selected : SlotContext u) (i : Fin u) :
    variableSlots controls selected (.inr i) =
      SourceContextLoad.variableField i (slotEncoding.code (selected i)) := rfl

theorem selectedDigits_loaded (F : Formula) (c : ClauseContext F u) (selected : SlotContext u)
    (controls : Fin 6 ↪ Extra) (base : SourceContextLoad.Tape u Extra → List Bool)
    (empty : ∀ i, base (SourceContextLoad.variableField i (slotEncoding.code (selected i))) = [])
    (i : Fin u) :
    SourceContextLoad.stageTapes F c base u (variableSlots controls selected (.inr i)) =
      encodeWord (sampledVariables F c selected i).val := by
  rw [variableSlots_digit]
  dsimp only [slotEncoding] at empty ⊢
  rw [SourceContextLoad.output_variable, empty i, List.append_nil]
  congr 1
  cases hs : selected i <;> simp [sampledVariables, PCP.nameAt, PCP.clauseAt, hs] <;> rfl


end DFVSGames.Foundations.Hastad.SourceRankPhase


namespace DFVSGames.Foundations.Hastad.SourceRuntimeModel

open Turing Complexity SourceContexts SourceOccurrences SourceLocalSignature SourceProfileBridge

inductive BodyWork (Extra : Type)
  | leftBlock | dummy | bitCount | occurrenceCount
  | zero | rank | rightBase | leftBase
  | accA | accB | counter | arithScratch | coefficient
  | queryTemporary | queryScratch | accumulator
  | extra (value : Extra)
  deriving DecidableEq, Fintype

abbrev BodyExtra (Extra : Type) := SourceLoopInit.HeaderTape ⊕ BodyWork Extra
abbrev Arena (u : Nat) (Extra : Type) := SourceContextLoad.Tape u (BodyExtra Extra)
abbrev QueryState (u D : Nat) := SourceQueryLoop.State u D
abbrev ArithmeticState := MachineHorner.State Unit
abbrev State (u D : Nat) := SourceContextPrepare.State u (QueryState u D × ArithmeticState)
abbrev QueryMetadata (u : Nat) := SourceContextPrepare.LoadState ×
  (SourceSignaturePrepare.ProfileState u × (Signature u × ArithmeticState))
abbrev ArithmeticMetadata (u D : Nat) := SourceContextPrepare.LoadState ×
  (SourceSignaturePrepare.ProfileState u × (Signature u × QueryState u D))

variable {u D : Nat} {Extra : Type}

def workTape (role : BodyWork Extra) : Arena u Extra := .extra (.inr role)
def variableHeader : Arena u Extra := SourceLoopInit.variableHeader
def clauseHeader : Arena u Extra := SourceLoopInit.clauseHeader

def canonicalSignature (u : Nat) : Signature u :=
  ⟨fun _ => false, fun _ _ => false, fun _ => .first⟩

def canonicalState (initialQuery : SourceQueryLoop.Query u D) : State u D :=
  SourceContextPrepare.initialState (fun _ _ => false) (canonicalSignature u)
    (((canonicalSignature u, initialQuery), none), (((), ()), none))

def queryStateEquiv (u D : Nat) : QueryState u D × QueryMetadata u ≃ State u D where
  toFun x := (x.2.1, (x.2.2.1, (x.2.2.2.1, (x.1, x.2.2.2.2))))
  invFun x := (x.2.2.2.1, (x.1, (x.2.1, (x.2.2.1, x.2.2.2.2))))
  left_inv _ := rfl
  right_inv _ := rfl

def arithmeticStateEquiv (u D : Nat) : ArithmeticState × ArithmeticMetadata u D ≃ State u D where
  toFun x := (x.2.1, (x.2.2.1, (x.2.2.2.1, (x.2.2.2.2, x.1))))
  invFun x := (x.2.2.2.2, (x.1, (x.2.1, (x.2.2.1, x.2.2.2.1))))
  left_inv _ := rfl
  right_inv _ := rfl

def rankControls (header : SourceLoopInit.HeaderTape) : Fin 6 ↪ BodyExtra Extra where
  toFun
    | 0 => .inl header
    | 1 => .inr .accA
    | 2 => .inr .accB
    | 3 => .inr .rank
    | 4 => .inr .counter
    | 5 => .inr .arithScratch
  inj' := by intro a b h; fin_cases a <;> fin_cases b <;> simp_all

def clauseRankSlots : MachineHorner.Layout u ↪ Arena u Extra :=
  SourceRankPhase.clauseSlots (rankControls .clauseCount)

def variableRankSlots (selected : SlotContext u) : MachineHorner.Layout u ↪ Arena u Extra :=
  SourceRankPhase.variableSlots (rankControls .variableCount) selected

def baseSlots (right : Bool) : MachineHorner.Layout 2 ↪ Arena u Extra where
  toFun
    | .inl 0 => workTape .rank
    | .inl 1 => workTape .accA
    | .inl 2 => workTape .accB
    | .inl 3 => workTape (if right then .rightBase else .leftBase)
    | .inl 4 => workTape .counter
    | .inl 5 => workTape .arithScratch
    | .inr 0 => workTape .coefficient
    | .inr 1 => workTape (if right then .leftBlock else .zero)
  inj' := by
    intro a b h
    cases a with
    | inl a =>
      cases b with
      | inl b => cases right <;> fin_cases a <;> fin_cases b <;> simp_all [workTape]
      | inr b => cases right <;> fin_cases a <;> fin_cases b <;> simp_all [workTape]
    | inr a =>
      cases b with
      | inl b => cases right <;> fin_cases a <;> fin_cases b <;> simp_all [workTape]
      | inr b => cases right <;> fin_cases a <;> fin_cases b <;> simp_all [workTape]

def querySources : Fin 3 → Arena u Extra
  | 0 => workTape .leftBase
  | 1 => workTape .rightBase
  | 2 => workTape .dummy

def queryLayout : SourceTestAppend.Layout (Arena u Extra) where
  sources := querySources
  temporary := workTape .queryTemporary
  scratch := workTape .queryScratch
  accumulator := workTape .accumulator
  sourceTemporary side := by fin_cases side <;> simp [querySources, workTape]
  sourceScratch side := by fin_cases side <;> simp [querySources, workTape]
  sourceAccumulator side := by fin_cases side <;> simp [querySources, workTape]
  temporaryScratch := by simp [workTape]
  temporaryAccumulator := by simp [workTape]
  scratchAccumulator := by simp [workTape]

def perSlotCleanup : List (Arena u Extra) := [workTape .rank, workTape .leftBase]
def finalBaseCleanup : List (Arena u Extra) := [workTape .rank, workTape .rightBase, workTape .zero]

@[simp] theorem initializer_arena :
    SourceLoopInit.Tape u (BodyWork Extra) = Arena u Extra := rfl

end DFVSGames.Foundations.Hastad.SourceRuntimeModel

end OAI