Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.MaxCut.PCP.PoweringOpinionTables

namespace OAI

/-!
Serialization boundaries and direct row lookup for the shared executable
fixed-alphabet graph codec. The alphabet is a fixed parameter. Rows remain in
their stored order, including repeated darts and loops. Word offsets are fixed;
bit offsets are represented by actual encoded prefixes because unary fields
have data-dependent lengths. These are data identities, not runtime certificates.
-/

namespace MaxCutGames.Foundations.PCP.AlphabetTable.Input

open MaxCutGames.Foundations.Complexity

export GenericGraphTables (Label RelationTable DartRow Rows Valid Table
  relationIndex relationIndex_val relationAt relationOf relationAt_relationOf
  reverseAt acceptsAt rowList rowList_length semantics semantics_reverse
  semantics_tail semantics_accepts ofGraph semantics_ofGraph ofEnumeratedGraph
  bitWord relationWords relationWords_length rowWords rowWords_length
  rowsWords_length tableWords tableWords_length tableBits encoding
  parseRelation parseRow parseRows decodeTableWords decodeTableBits
  parseRelation_encoded parseRow_encoded parseRows_encoded
  decodeTableWords_encoded decodeTableBits_encoded tableBits_injective
  relationBits_length_le rowBits_length_le rowsBits_length_le tableWords_length_le_bits
  tableBits_length_le vertices_le_tableBits_length darts_le_tableBits_length)

variable {q n m : Nat}

abbrev rowWidth (q : Nat) : Nat := q * q + 2

/-- Offset measured in natural-word fields, including the two header fields. -/
def rowOffset (q e : Nat) : Nat := 2 + rowWidth q * e

/-- Exact decomposition before, at, and after a selected stored row. -/
theorem rowsWords_split (rows : List (DartRow q n m)) (e : Nat)
    (he : e < rows.length) :
    rows.flatMap rowWords = (rows.take e).flatMap rowWords ++
      (rowWords rows[e] ++ (rows.drop (e + 1)).flatMap rowWords) := by
  have hs : rows.take e ++ rows[e] :: rows.drop (e + 1) = rows := by
    rw [List.getElem_cons_drop he, List.take_append_drop]
  have hw := congrArg (fun rs : List (DartRow q n m) => rs.flatMap rowWords) hs
  simpa only [List.flatMap_append, List.flatMap_cons, List.append_assoc] using hw.symm

/-- The unary serialization keeps exactly the same row boundaries, with
separate encoded prefixes rather than an assumed constant bit offset. -/
theorem rowsBits_split (rows : List (DartRow q n m)) (e : Nat)
    (he : e < rows.length) :
    encodeWords (rows.flatMap rowWords) =
      encodeWords ((rows.take e).flatMap rowWords) ++
        (encodeWords (rowWords rows[e]) ++
          encodeWords ((rows.drop (e + 1)).flatMap rowWords)) := by
  rw [rowsWords_split rows e he]
  simp only [encodeWords_append]

/-- A selected row can be followed by an arbitrary caller-owned suffix. -/
theorem rowsBits_append_split (rows : List (DartRow q n m)) (e : Nat)
    (he : e < rows.length) (suffix : List Bool) :
    encodeWords (rows.flatMap rowWords) ++ suffix =
      encodeWords ((rows.take e).flatMap rowWords) ++
        (encodeWords (rowWords rows[e]) ++
          (encodeWords ((rows.drop (e + 1)).flatMap rowWords) ++ suffix)) := by
  rw [rowsBits_split rows e he]
  simp only [List.append_assoc]

def prefixWords (table : Table q) (e : Fin table.darts) : List Nat :=
  [table.vertices, table.darts] ++ ((rowList table).take e.val).flatMap rowWords

def suffixWords (table : Table q) (e : Fin table.darts) : List Nat :=
  ((rowList table).drop (e.val + 1)).flatMap rowWords

theorem prefixWords_length (table : Table q) (e : Fin table.darts) :
    (prefixWords table e).length = rowOffset q e.val := by
  simp [prefixWords, rowOffset, rowWidth, Nat.mul_comm, Nat.add_comm]
  omega

theorem tableWords_at_row (table : Table q) (e : Fin table.darts) :
    tableWords table = prefixWords table e ++
      (rowWords table.rows[e] ++ suffixWords table e) := by
  have hs := rowsWords_split (rowList table) e.val (by simpa only [rowList_length] using e.isLt)
  simp only [rowList, Vector.getElem_toList] at hs
  unfold tableWords prefixWords suffixWords rowList
  rw [hs]
  simp only [List.append_assoc, Fin.getElem_fin]

theorem tableBits_at_row (table : Table q) (e : Fin table.darts) :
    tableBits table = encodeWords (prefixWords table e) ++
      (encodeWords (rowWords table.rows[e]) ++ encodeWords (suffixWords table e)) := by
  rw [tableBits, tableWords_at_row table e]
  simp only [encodeWords_append]

/-- Removing precisely the two encoded header words exposes the stored rows. -/
theorem tableBits_header (table : Table q) :
    tableBits table = encodeWord table.vertices ++
      (encodeWord table.darts ++ encodeWords ((rowList table).flatMap rowWords)) := by
  simp [tableBits, tableWords, encodeWords]

theorem tableWords_drop_row (table : Table q) (e : Fin table.darts) :
    (tableWords table).drop (rowOffset q e.val) =
      rowWords table.rows[e] ++ suffixWords table e := by
  rw [tableWords_at_row table e]
  exact List.drop_left' (l₂ := rowWords table.rows[e] ++ suffixWords table e)
    (prefixWords_length table e)
theorem tableWords_take_row (table : Table q) (e : Fin table.darts) :
    ((tableWords table).drop (rowOffset q e.val)).take (rowWidth q) =
      rowWords table.rows[e] := by
  rw [tableWords_drop_row]
  exact List.take_left' (l₂ := suffixWords table e) (rowWords_length table.rows[e])

theorem tableWords_get_tail (table : Table q) (e : Fin table.darts) :
    (tableWords table)[rowOffset q e.val]? = some table.rows[e].tail.val := by
  have h := congrArg (fun words : List Nat => words[0]?) (tableWords_drop_row table e)
  simpa [List.getElem?_drop, rowWords] using h

theorem tableWords_get_reverse (table : Table q) (e : Fin table.darts) :
    (tableWords table)[rowOffset q e.val + 1]? = some table.rows[e].reverseIndex.val := by
  have h := congrArg (fun words : List Nat => words[1]?) (tableWords_drop_row table e)
  simpa [List.getElem?_drop, rowWords] using h

theorem tableWords_drop_relation (table : Table q) (e : Fin table.darts) :
    (tableWords table).drop (rowOffset q e.val + 2) =
      relationWords table.rows[e].relation ++ suffixWords table e := by
  rw [← List.drop_drop, tableWords_drop_row]
  simp [rowWords]

theorem relationWords_get (relation : RelationTable q) (j : Fin (q * q)) :
    (relationWords relation)[j.val]? = some (bitWord relation[j]) := by
  rw [List.getElem?_eq_getElem (by simpa only [relationWords_length] using j.isLt)]
  simp [relationWords]

theorem tableWords_get_relation (table : Table q) (e : Fin table.darts)
    (j : Fin (q * q)) :
    (tableWords table)[rowOffset q e.val + 2 + j.val]? =
      some (bitWord table.rows[e].relation[j]) := by
  have h := congrArg (fun words : List Nat => words[j.val]?)
    (tableWords_drop_relation table e)
  rw [List.getElem?_drop,
    List.getElem?_append_left (by simpa only [relationWords_length] using j.isLt), relationWords_get] at h
  exact h

theorem tableWords_get_predicate (table : Table q) (e : Fin table.darts)
    (a b : Label q) :
    (tableWords table)[rowOffset q e.val + 2 + (b.val + q * a.val)]? =
      some (bitWord ((semantics table).accepts e a b)) := by
  simpa only [relationIndex_val, semantics_accepts, relationAt] using
    tableWords_get_relation table e ((relationIndex q) (a, b))

theorem rowOffset_lt_length (table : Table q) (e : Fin table.darts) :
    rowOffset q e.val < (tableWords table).length := by
  rw [tableWords_length]
  have h := Nat.mul_le_mul_left (rowWidth q) (Nat.succ_le_of_lt e.isLt)
  rw [Nat.mul_succ] at h
  unfold rowOffset rowWidth at *
  omega

theorem relationOffset_lt_length (table : Table q) (e : Fin table.darts)
    (j : Fin (q * q)) :
    rowOffset q e.val + 2 + j.val < (tableWords table).length := by
  rw [tableWords_length]
  have h := Nat.mul_le_mul_left (rowWidth q) (Nat.succ_le_of_lt e.isLt)
  rw [Nat.mul_succ] at h
  have hj := j.isLt
  unfold rowOffset rowWidth at *
  omega

end MaxCutGames.Foundations.PCP.AlphabetTable.Input

namespace MaxCutGames.Foundations.PCP.AlphabetTable.Setup

open Turing
open MaxCutGames.Foundations.Complexity
open MaxCutGames.Foundations.Hastad

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

abbrev Alphabet (_ : K) := Bool

abbrev Ports (K : Type) := Fin 7 → K

inductive Label
  | copyFirst | copySecond | verticesStart | verticesLoop
  | dartsStart | dartsLoop | counterFirst | counterSecond | initialize
  deriving DecidableEq

protected abbrev Label.enumList : List Label := [.copyFirst, .copySecond, .verticesStart,
  .verticesLoop, .dartsStart, .dartsLoop, .counterFirst, .counterSecond, .initialize]

protected theorem Label.enumList_getElem?_ctorIdx_eq (x : Label) :
    Label.enumList[x.ctorIdx]? = some x := by
  cases x <;> rfl

protected theorem Label.enumList_nodup : Label.enumList.Nodup := by decide

instance : Fintype Label where
  elems := ⟨Label.enumList, Label.enumList_nodup⟩
  complete x := by cases x <;> decide

def statement (ports : Ports K) (labels : Label → Λ) (exit : Option Λ) :
    Label → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool)
  | .copyFirst => Reduction.MachineTransfer.loopAt (ports 0) (ports 5) id false
      (labels .copyFirst) (some (labels .copySecond))
  | .copySecond => MachineCopy.forkLoop (ports 5) (ports 0) (ports 1) false
      (labels .copySecond) (some (labels .verticesStart))
  | .verticesStart => SourceMachine.fieldStart (ports 2) (labels .verticesLoop)
  | .verticesLoop => SourceMachine.fieldLoop (ports 1) (ports 2)
      (labels .verticesLoop) (some (labels .dartsStart))
  | .dartsStart => SourceMachine.fieldStart (ports 3) (labels .dartsLoop)
  | .dartsLoop => SourceMachine.fieldLoop (ports 1) (ports 3)
      (labels .dartsLoop) (some (labels .counterFirst))
  | .counterFirst => Reduction.MachineTransfer.loopAt (ports 3) (ports 5) id false
      (labels .counterFirst) (some (labels .counterSecond))
  | .counterSecond => MachineCopy.forkLoop (ports 5) (ports 3) (ports 4) false
      (labels .counterSecond) (some (labels .initialize))
  | .initialize => .push (ports 6) (fun _ => false)
      (.load (fun state => (state.1, none)) (Reduction.MachineTransfer.exitAt (ports 6) exit))

def resultTapes (ports : Ports K) (base : K → List Bool)
    (n m : Nat) (rows : List Bool) : K → List Bool :=
  Function.update
    (Function.update
      (Function.update
        (Function.update (Function.update base (ports 1) rows)
          (ports 2) (encodeWord n)) (ports 3) (encodeWord m))
      (ports 4) (encodeWord m)) (ports 6) (encodeWord 0)

theorem resultTapes_other (ports : Ports K) (base : K → List Bool)
    (n m : Nat) (rows : List Bool) (tape : K)
    (h : ∀ i : Fin 7, i ≠ 0 → i ≠ 5 → tape ≠ ports i) :
    resultTapes ports base n m rows tape = base tape := by
  simp [resultTapes, h]

/-- Actual program placement is the only execution premise. Header values and
unread row bits specify the input, and every auxiliary role starts empty. -/
def setupInTime (ports : Ports K) (distinct : Function.Injective ports)
    (labels : Label → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (atLabels : ∀ label, program (labels label) = statement ports labels exit label)
    (base : K → List Bool) (n m : Nat) (rows : List Bool)
    (initial : ∀ i : Fin 7, base (ports i) =
      if i = 0 then encodeWord n ++ (encodeWord m ++ rows) else [])
    (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels .copyFirst), (ambient, register), base⟩
      (some ⟨exit, (ambient, none), resultTapes ports base n m rows⟩)
      (2 * (encodeWord n ++ (encodeWord m ++ rows)).length + n + 3 * m + 11) := by
  let input := encodeWord n ++ (encodeWord m ++ rows)
  let t₀ := Function.update base (ports 1) input
  let t₁ := SourceMachine.afterField (ports 1) (ports 2) t₀ n (encodeWord m ++ rows)
  let t₂ := SourceMachine.afterField (ports 1) (ports 3) t₁ m rows
  let t₃ := Function.update t₂ (ports 4) (encodeWord m)
  have hne {i j : Fin 7} (h : i ≠ j) : ports i ≠ ports j := fun e => h (distinct e)
  have run₀ := MachineCopy.copyInTime (ports 0) (ports 1) (ports 5)
    (hne (by decide)) (hne (by decide)) (hne (by decide)) false
    (labels .copyFirst) (labels .copySecond) (some (labels .verticesStart))
    program (atLabels .copyFirst) (atLabels .copySecond) base
    (by simp [initial]) ambient register
  have hb0 : base (ports 0) = input := by simpa [input] using initial 0
  have hb1 : base (ports 1) = [] := by simpa using initial 1
  rw [hb0, hb1, List.append_nil] at run₀
  change StateTransition.EvalsToInTime (TM2.step program)
    ⟨some (labels .copyFirst), (ambient, register), base⟩
    (some ⟨some (labels .verticesStart), (ambient, none), t₀⟩)
    (2 * (input.length + 1)) at run₀
  have run₁ := SourceMachine.fieldInTime (ports 1) (ports 2) (hne (by decide))
    (labels .verticesStart) (labels .verticesLoop) (some (labels .dartsStart))
    program (atLabels .verticesStart) (atLabels .verticesLoop)
    t₀ n (encodeWord m ++ rows) (by simp [t₀, input]) ambient none
  have run₂ := SourceMachine.fieldInTime (ports 1) (ports 3) (hne (by decide))
    (labels .dartsStart) (labels .dartsLoop) (some (labels .counterFirst))
    program (atLabels .dartsStart) (atLabels .dartsLoop)
    t₁ m rows (by simp [t₁, SourceMachine.afterField, distinct.eq_iff]) ambient none
  have h₂m : t₂ (ports 3) = encodeWord m := by
    simp [t₂, t₁, t₀, SourceMachine.afterField, SourceMachine.fieldTapes,
      distinct.eq_iff, initial]
  have h₂c : t₂ (ports 4) = [] := by
    simp [t₂, t₁, t₀, SourceMachine.afterField, SourceMachine.fieldTapes,
      distinct.eq_iff, initial]
  have h₂s : t₂ (ports 5) = [] := by
    simp [t₂, t₁, t₀, SourceMachine.afterField, SourceMachine.fieldTapes,
      distinct.eq_iff, initial]
  have run₃ := MachineCopy.copyInTime (ports 3) (ports 4) (ports 5)
    (hne (by decide)) (hne (by decide)) (hne (by decide)) false
    (labels .counterFirst) (labels .counterSecond) (some (labels .initialize))
    program (atLabels .counterFirst) (atLabels .counterSecond) t₂ h₂s ambient none
  rw [h₂m, h₂c, List.append_nil, encodeWord_length] at run₃
  have h₃e : t₃ (ports 6) = [] := by
    simp [t₃, t₂, t₁, t₀, SourceMachine.afterField, SourceMachine.fieldTapes,
      distinct.eq_iff, initial]
  have hresult : Function.update t₃ (ports 6) [false] =
      resultTapes ports base n m rows := by
    funext tape
    by_cases h1 : tape = ports 1
    · subst tape; simp [t₃, t₂, t₁, t₀, SourceMachine.afterField,
        SourceMachine.fieldTapes, resultTapes, distinct.eq_iff]
    · by_cases h2 : tape = ports 2
      · subst tape; simp [t₃, t₂, t₁, t₀, SourceMachine.afterField,
          SourceMachine.fieldTapes, resultTapes, distinct.eq_iff, initial]
      · by_cases h3 : tape = ports 3
        · subst tape; simp [t₃, t₂, t₁, t₀, SourceMachine.afterField,
            SourceMachine.fieldTapes, resultTapes, distinct.eq_iff, initial]
        · simp [t₃, t₂, t₁, t₀, SourceMachine.afterField, SourceMachine.fieldTapes,
            resultTapes, Function.update_apply, h1, h2, h3, encodeWord]
  have run₄ : StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels .initialize), (ambient, none), t₃⟩
      (some ⟨exit, (ambient, none), resultTapes ports base n m rows⟩) 1 := {
    steps := 1
    evals_in_steps := by
      change some (TM2.stepAux (program (labels .initialize)) (ambient, none) t₃) = _
      rw [atLabels]
      cases exit <;> simp [statement, TM2.stepAux, Reduction.MachineTransfer.exitAt,
        h₃e, hresult]
    steps_le_m := Nat.le_refl _ }
  let r₀₁ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ run₀ run₁
  let r₀₁₂ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ r₀₁ run₂
  let r₀₁₂₃ := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ r₀₁₂ run₃
  let run := StateTransition.EvalsToInTime.trans _ _ _ _ _ _ r₀₁₂₃ run₄
  exact {
    toEvalsTo := run.toEvalsTo
    steps_le_m := by
      have h := run.steps_le_m
      dsimp [input] at h
      omega }

/-- Specialization to the checked graph-table codec. The bound follows from
the actual copy and delimiter-scan traces above. -/
def tableSetupInTime {q : Nat} (ports : Ports K) (distinct : Function.Injective ports)
    (labels : Label → Λ) (exit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (σ × Option Bool))
    (atLabels : ∀ label, program (labels label) = statement ports labels exit label)
    (base : K → List Bool) (table : GenericGraphTables.Table q)
    (initial : ∀ i : Fin 7, base (ports i) =
      if i = 0 then GenericGraphTables.tableBits table else [])
    (ambient : σ) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels .copyFirst), (ambient, register), base⟩
      (some ⟨exit, (ambient, none), resultTapes ports base table.vertices table.darts
        (encodeWords ((GenericGraphTables.rowList table).flatMap GenericGraphTables.rowWords))⟩)
      (6 * (GenericGraphTables.tableBits table).length + 11) := by
  have hb := Input.tableBits_header table
  let run := setupInTime ports distinct labels exit program atLabels base
    table.vertices table.darts
    (encodeWords ((GenericGraphTables.rowList table).flatMap GenericGraphTables.rowWords))
    (by intro i; rw [initial i, hb]) ambient register
  have hn := GenericGraphTables.vertices_le_tableBits_length table
  have hm := GenericGraphTables.darts_le_tableBits_length table
  exact {
    toEvalsTo := run.toEvalsTo
    steps_le_m := by
      have h := run.steps_le_m
      have hlen := congrArg List.length hb
      omega }

def program (ports : Ports K) : Label → TM2.Stmt (Alphabet (K := K)) Label (σ × Option Bool) :=
  statement ports id none

/-- This stage itself has only seven tapes and nine control labels. -/
def machine : FinTM2 where
  K := Fin 7
  k₀ := 0
  k₁ := 1
  Γ _ := Bool
  Λ := Label
  main := .copyFirst
  σ := Unit × Option Bool
  initialState := ((), none)
  m := program id

end MaxCutGames.Foundations.PCP.AlphabetTable.Setup

/-!
An actual fixed-alphabet TM2 parser for the complete Boolean relation table.
The finite syntax contains one slot reader for each of the `q*q` coordinates.
Each reader consumes `[false]` or `[true,false]`, stores the bit, and clears the
head register. No parser outcome or execution trace is an assumption.
-/

namespace MaxCutGames.Foundations.PCP.AlphabetTable.ReadRelation

open Turing
open MaxCutGames.Foundations.Complexity
open GenericGraphTables

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

abbrev Alphabet (_ : K) := Bool
abbrev State (σ : Type) (q : Nat) := (σ × RelationTable q) × Option Bool

/-- The relation register is a finite tuple of bits, with no unbounded storage. -/
def relationEquiv (q : Nat) : RelationTable q ≃ (Fin (q * q) → Bool) where
  toFun relation i := relation[i.val]
  invFun := Vector.ofFn
  left_inv relation := Vector.ofFn_getElem
  right_inv bits := by funext i; simp

/-- Executable enumeration used when assembling the finite machine state. -/
@[instance_reducible] def relationFintype (q : Nat) : Fintype (RelationTable q) :=
  Fintype.ofEquiv (Fin (q * q) → Bool) (relationEquiv q).symm

@[instance_reducible] def stateFintype (σ : Type) (q : Nat) [Fintype σ] :
    Fintype (State σ q) := by
  letI := relationFintype q
  exact inferInstance

/-- Read one encoded Boolean into a fixed slot of the finite relation register. -/
def readSlot (source : K) (i : Fin (q * q))
    (next : TM2.Stmt (Alphabet (K := K)) Λ (State σ q)) :
    TM2.Stmt (Alphabet (K := K)) Λ (State σ q) :=
  .pop source
    (fun state head =>
      ((state.1.1, state.1.2.set i.val (head.getD false) i.isLt), head))
    (.branch (fun state => state.2.getD false)
      (.pop source (fun state _ => (state.1, none)) next)
      (.load (fun state => (state.1, none)) next))

theorem stepAux_readSlot (source : K) (i : Fin (q * q)) (bit : Bool)
    (next : TM2.Stmt (Alphabet (K := K)) Λ (State σ q))
    (base : K → List Bool) (suffix : List Bool) (ambient : σ)
    (initial : RelationTable q) (register : Option Bool) :
    TM2.stepAux (readSlot source i next) ((ambient, initial), register)
      (Function.update base source (encodeWord (bitWord bit) ++ suffix)) =
      TM2.stepAux next ((ambient, initial.set i.val bit i.isLt), none)
        (Function.update base source suffix) := by
  cases bit <;> simp [readSlot, TM2.stepAux, bitWord, encodeWord]

def readSlots (source : K) : List (Fin (q * q)) →
    TM2.Stmt (Alphabet (K := K)) Λ (State σ q) →
    TM2.Stmt (Alphabet (K := K)) Λ (State σ q)
  | [], next => .load (fun state => (state.1, none)) next
  | i :: indices, next => readSlot source i (readSlots source indices next)

def indexedWords (relation : RelationTable q) (indices : List (Fin (q * q))) : List Nat :=
  indices.map (fun i => bitWord relation[i.val])

def fillSlots (relation : RelationTable q) : List (Fin (q * q)) →
    RelationTable q → RelationTable q
  | [], initial => initial
  | i :: indices, initial =>
      fillSlots relation indices (initial.set i.val relation[i.val] i.isLt)

theorem fillSlots_preserves_correct (relation : RelationTable q)
    (indices : List (Fin (q * q))) (initial : RelationTable q) (i : Fin (q * q))
    (hcorrect : initial[i.val] = relation[i.val]) :
    (fillSlots relation indices initial)[i.val] = relation[i.val] := by
  induction indices generalizing initial with
  | nil => exact hcorrect
  | cons j indices ih =>
    apply ih
    by_cases hji : j.val = i.val
    · simp [hji]
    · simp [hji, hcorrect]

theorem fillSlots_mem (relation : RelationTable q) (indices : List (Fin (q * q)))
    (initial : RelationTable q) (i : Fin (q * q)) (hi : i ∈ indices) :
    (fillSlots relation indices initial)[i.val] = relation[i.val] := by
  induction indices generalizing initial with
  | nil => simp at hi
  | cons j indices ih =>
    change (fillSlots relation indices (initial.set j.val relation[j.val] j.isLt))[i.val] = _
    rcases List.mem_cons.mp hi with h | h
    · subst j
      apply fillSlots_preserves_correct
      simp
    · exact ih _ h

theorem fillSlots_finRange (relation initial : RelationTable q) :
    fillSlots relation (List.finRange (q * q)) initial = relation := by
  apply Vector.ext
  intro i hi
  exact fillSlots_mem relation (List.finRange (q * q)) initial ⟨i, hi⟩
    (List.mem_finRange _)

theorem indexedWords_finRange (relation : RelationTable q) :
    indexedWords relation (List.finRange (q * q)) = relationWords relation := by
  have h : (List.finRange (q * q)).map (fun i => relation[i.val]) = relation.toList := by
    rw [List.finRange, List.map_ofFn]
    change List.ofFn (fun i : Fin (q * q) => relation[i.val]) = relation.toList
    rw [← Vector.toList_ofFn, Vector.ofFn_getElem]
  calc
    _ = ((List.finRange (q * q)).map (fun i => relation[i.val])).map bitWord := by
      simp only [indexedWords, List.map_map, Function.comp_def]
    _ = relation.toList.map bitWord := congrArg (fun bits : List Bool => bits.map bitWord) h
    _ = _ := rfl

/-- Each stored register bit is obtained from its actual serialized field. -/
theorem stepAux_readSlots (source : K) (indices : List (Fin (q * q)))
    (relation : RelationTable q)
    (next : TM2.Stmt (Alphabet (K := K)) Λ (State σ q))
    (base : K → List Bool) (suffix : List Bool) (ambient : σ)
    (initial : RelationTable q) (register : Option Bool) :
    TM2.stepAux (readSlots source indices next) ((ambient, initial), register)
      (Function.update base source (encodeWords (indexedWords relation indices) ++ suffix)) =
      TM2.stepAux next ((ambient, fillSlots relation indices initial), none)
        (Function.update base source suffix) := by
  induction indices generalizing initial register with
  | nil => simp [readSlots, fillSlots, indexedWords, encodeWords, TM2.stepAux]
  | cons i indices ih =>
    simp only [readSlots, indexedWords, List.map_cons, encodeWords, List.append_assoc, fillSlots]
    rw [stepAux_readSlot]
    exact ih _ none

/-- Fixed finite program fragment; it can be inserted before any continuation. -/
def readRelation (source : K)
    (next : TM2.Stmt (Alphabet (K := K)) Λ (State σ q)) :
    TM2.Stmt (Alphabet (K := K)) Λ (State σ q) :=
  readSlots source (List.finRange (q * q)) next

theorem stepAux_readRelation (source : K) (relation : RelationTable q)
    (next : TM2.Stmt (Alphabet (K := K)) Λ (State σ q))
    (base : K → List Bool) (suffix : List Bool) (ambient : σ)
    (initial : RelationTable q) (register : Option Bool) :
    TM2.stepAux (readRelation source next) ((ambient, initial), register)
      (Function.update base source (encodeWords (relationWords relation) ++ suffix)) =
      TM2.stepAux next ((ambient, relation), none) (Function.update base source suffix) := by
  simpa only [readRelation, indexedWords_finRange, fillSlots_finRange] using
    stepAux_readSlots source (List.finRange (q * q)) relation next
      base suffix ambient initial register

theorem stepAux_readRelation_fromTapes (source : K) (relation : RelationTable q)
    (next : TM2.Stmt (Alphabet (K := K)) Λ (State σ q))
    (base : K → List Bool) (suffix : List Bool)
    (hinput : base source = encodeWords (relationWords relation) ++ suffix)
    (ambient : σ) (initial : RelationTable q) (register : Option Bool) :
    TM2.stepAux (readRelation source next) ((ambient, initial), register) base =
      TM2.stepAux next ((ambient, relation), none) (Function.update base source suffix) := by
  have h := stepAux_readRelation source relation next base suffix ambient initial register
  have hbase : Function.update base source (encodeWords (relationWords relation) ++ suffix) =
      base := by rw [← hinput]; exact Function.update_eq_self source base
  rw [hbase] at h
  exact h

omit [DecidableEq K] in
theorem readSlots_pushBound (source : K) (indices : List (Fin (q * q)))
    (next : TM2.Stmt (Alphabet (K := K)) Λ (State σ q)) :
    Runtime.statementPushBound (readSlots source indices next) =
      Runtime.statementPushBound next := by
  induction indices with
  | nil => rfl
  | cons i indices ih =>
    simp only [readSlots, readSlot, Runtime.statementPushBound, ih, max_self]

omit [DecidableEq K] in
theorem readRelation_pushBound (source : K)
    (next : TM2.Stmt (Alphabet (K := K)) Λ (State σ q)) :
    Runtime.statementPushBound (readRelation source next) =
      Runtime.statementPushBound next :=
  readSlots_pushBound source (List.finRange (q * q)) next

def parser (source : K) (exitLabel : Λ) :
    TM2.Stmt (Alphabet (K := K)) Λ (State σ q) :=
  readRelation source (.goto fun _ => exitLabel)

omit [DecidableEq K] in
theorem parser_pushBound (source : K) (exitLabel : Λ) :
    Runtime.statementPushBound (parser (σ := σ) (q := q) source exitLabel) = 0 := by
  rw [parser, readRelation_pushBound]
  rfl

theorem parserStep (source : K) (readLabel exitLabel : Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ q))
    (atRead : program readLabel = parser source exitLabel)
    (base : K → List Bool) (relation : RelationTable q) (suffix : List Bool)
    (ambient : σ) (initial : RelationTable q) (register : Option Bool) :
    TM2.step program
      ⟨some readLabel, ((ambient, initial), register),
        Function.update base source (encodeWords (relationWords relation) ++ suffix)⟩ =
      some ⟨some exitLabel, ((ambient, relation), none), Function.update base source suffix⟩ := by
  change some (TM2.stepAux (program readLabel) ((ambient, initial), register)
    (Function.update base source (encodeWords (relationWords relation) ++ suffix))) = _
  rw [atRead, parser, stepAux_readRelation]
  rfl

/-- One transition executes the fixed chain of `q*q` slot readers. -/
theorem parserTrace (source : K) (readLabel exitLabel : Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ q))
    (atRead : program readLabel = parser source exitLabel)
    (base : K → List Bool) (relation : RelationTable q) (suffix : List Bool)
    (ambient : σ) (initial : RelationTable q) (register : Option Bool) :
    (MachineComposition.advance (TM2.step program))^[1]
      (some ⟨some readLabel, ((ambient, initial), register),
        Function.update base source (encodeWords (relationWords relation) ++ suffix)⟩) =
      some ⟨some exitLabel, ((ambient, relation), none), Function.update base source suffix⟩ := by
  simpa only [Function.iterate_one, MachineComposition.advance_some] using
    parserStep source readLabel exitLabel program atRead base relation suffix ambient initial register

def parserInTime (source : K) (readLabel exitLabel : Λ)
    (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State σ q))
    (atRead : program readLabel = parser source exitLabel)
    (base : K → List Bool) (relation : RelationTable q) (suffix : List Bool)
    (ambient : σ) (initial : RelationTable q) (register : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some readLabel, ((ambient, initial), register),
        Function.update base source (encodeWords (relationWords relation) ++ suffix)⟩
      (some ⟨some exitLabel, ((ambient, relation), none), Function.update base source suffix⟩) 1 where
  steps := 1
  evals_in_steps := by
    change (MachineComposition.advance (TM2.step program))^[1] _ = _
    exact parserTrace source readLabel exitLabel program atRead base relation suffix ambient initial register
  steps_le_m := Nat.le_refl _

end MaxCutGames.Foundations.PCP.AlphabetTable.ReadRelation

/-!
# An actual finite machine for a fixed-size Boolean block map

The program parameters `N`, `M`, and `F` are fixed. The local register has
exactly `N` bits. A syntactic chain of `N` pops fills that register, and a
syntactic chain of `M` pushes writes `F` of the register. No instruction
applies `F` to an unbounded word. An arbitrary ambient state is preserved,
and the local register is reset before the optional continuation.

TM2 executes a whole finite statement in one transition. The push bound
below separately records the exact `M` pushes in this statement.
-/

namespace MaxCutGames.Foundations.Complexity.MachineFixedBlockMap

open Turing

abbrev Buffer (N : Nat) := Fin N → Bool

def emptyBuffer (N : Nat) : Buffer N := fun _ => false

section Chains

variable {K Λ σ : Type} {N : Nat}

def readSlots (src : K) : List (Fin N) →
    TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N) →
    TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N)
  | [], next => next
  | i :: slots, next =>
      .pop src (fun state head =>
        (state.1, Function.update state.2 i (head.getD false)))
        (readSlots src slots next)

/-- Proof-level description of the finite register updates. -/
def fill (slots : List (Fin N)) (bits buffer : Buffer N) : Buffer N :=
  slots.foldl (fun buffer i => Function.update buffer i (bits i)) buffer

theorem fill_apply (slots : List (Fin N)) (bits buffer : Buffer N) (i : Fin N) :
    fill slots bits buffer i = if i ∈ slots then bits i else buffer i := by
  induction slots generalizing buffer with
  | nil => simp [fill]
  | cons j slots ih =>
      change fill slots bits (Function.update buffer j (bits j)) i = _
      rw [ih]
      by_cases hm : i ∈ slots
      · simp [hm]
      · by_cases he : i = j <;> simp [hm, he]

@[simp] theorem fill_all (bits buffer : Buffer N) :
    fill (List.ofFn id) bits buffer = bits := by
  funext i
  simp [fill_apply, List.mem_ofFn]

variable [DecidableEq K]

/-- The input symbols are physically popped; the remaining suffix is exact. -/
theorem stepAux_readSlots (src : K) (slots : List (Fin N))
    (next : TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N))
    (ambient : σ) (bits buffer : Buffer N) (tapes : K → List Bool)
    (suffix : List Bool) :
    TM2.stepAux (readSlots src slots next) (ambient, buffer)
        (Function.update tapes src (slots.map bits ++ suffix)) =
      TM2.stepAux next (ambient, fill slots bits buffer)
        (Function.update tapes src suffix) := by
  induction slots generalizing buffer tapes with
  | nil => simp [readSlots, fill]
  | cons i slots ih =>
      simpa only [readSlots, List.map_cons, List.cons_append, TM2.stepAux,
        Function.update_self, List.head?_cons, Option.getD_some, List.tail_cons,
        Function.update_idem, fill, List.foldl_cons] using
        ih (Function.update buffer i (bits i)) tapes

theorem stepAux_readAll (src : K)
    (next : TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N))
    (state : σ × Buffer N) (bits : Buffer N) (tapes : K → List Bool)
    (suffix : List Bool) (hinput : tapes src = List.ofFn bits ++ suffix) :
    TM2.stepAux (readSlots src (List.ofFn id) next) state tapes =
      TM2.stepAux next (state.1, bits) (Function.update tapes src suffix) := by
  have h := stepAux_readSlots src (List.ofFn id) next state.1 bits state.2 tapes suffix
  have hin : Function.update tapes src ((List.ofFn id).map bits ++ suffix) = tapes := by
    simpa only [List.map_ofFn, Function.comp_id, ← hinput] using
      Function.update_eq_self src tapes
  rw [hin, fill_all] at h
  exact h

omit [DecidableEq K] in
theorem statementPushBound_readSlots (src : K) (slots : List (Fin N))
    (next : TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N)) :
    Runtime.statementPushBound (readSlots src slots next) =
      Runtime.statementPushBound next := by
  induction slots with
  | nil => rfl
  | cons i slots ih => exact ih

/-- The slot list is a fixed part of the syntax; each instruction reads only
the finite internal state. Push order is reversed on the stack. -/
def writeSlots {M : Nat} (dst : K) (emit : σ → Buffer M) : List (Fin M) →
    TM2.Stmt (fun _ : K => Bool) Λ σ → TM2.Stmt (fun _ : K => Bool) Λ σ
  | [], next => next
  | i :: slots, next => .push dst (fun state => emit state i) (writeSlots dst emit slots next)

theorem stepAux_writeSlots {M : Nat} (dst : K) (emit : σ → Buffer M)
    (slots : List (Fin M)) (next : TM2.Stmt (fun _ : K => Bool) Λ σ)
    (state : σ) (tapes : K → List Bool) :
    TM2.stepAux (writeSlots dst emit slots next) state tapes =
      TM2.stepAux next state
        (Function.update tapes dst ((slots.map (emit state)).reverse ++ tapes dst)) := by
  induction slots generalizing tapes with
  | nil => simp [writeSlots]
  | cons i slots ih =>
      simp only [writeSlots, TM2.stepAux]
      rw [ih]
      simp only [Function.update_self, Function.update_idem, List.map_cons,
        List.reverse_cons, List.append_assoc, List.singleton_append]

omit [DecidableEq K] in
theorem statementPushBound_writeSlots {M : Nat} (dst : K) (emit : σ → Buffer M)
    (slots : List (Fin M)) (next : TM2.Stmt (fun _ : K => Bool) Λ σ) :
    Runtime.statementPushBound (writeSlots dst emit slots next) =
      slots.length + Runtime.statementPushBound next := by
  induction slots with
  | nil => simp [writeSlots]
  | cons i slots ih =>
      simp only [writeSlots, Runtime.statementPushBound, ih, List.length_cons]
      omega

end Chains

section Block

variable {K Λ σ : Type} {N M : Nat}

/-- Read a fixed block, write its fixed finite transformation, reset the local
register, and execute the next finite statement. -/
def blockStmt (src dst : K) (F : Buffer N → Buffer M)
    (next : TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N)) :
    TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N) :=
  readSlots src (List.ofFn id)
    (writeSlots dst (fun state => F state.2) (List.ofFn id).reverse
      (.load (fun state => (state.1, emptyBuffer N)) next))

def finishAt (exit : Option Λ) : TM2.Stmt (fun _ : K => Bool) Λ σ :=
  match exit with
  | none => .halt
  | some label => .goto fun _ => label

def blockMapAt (src dst : K) (F : Buffer N → Buffer M) (exit : Option Λ) :
    TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N) :=
  blockStmt src dst F (finishAt exit)

theorem statementPushBound_blockStmt (src dst : K) (F : Buffer N → Buffer M)
    (next : TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N)) :
    Runtime.statementPushBound (blockStmt src dst F next) =
      M + Runtime.statementPushBound next := by
  simp only [blockStmt, statementPushBound_readSlots, statementPushBound_writeSlots,
    Runtime.statementPushBound, List.length_reverse, List.length_ofFn]

theorem statementPushBound_blockMapAt (src dst : K) (F : Buffer N → Buffer M)
    (exit : Option Λ) :
    Runtime.statementPushBound (blockMapAt (σ := σ) src dst F exit) = M := by
  cases exit <;> simp [blockMapAt, statementPushBound_blockStmt, finishAt,
    Runtime.statementPushBound]

variable [DecidableEq K]

theorem stepAux_blockStmt (src dst : K) (F : Buffer N → Buffer M)
    (next : TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N))
    (hne : src ≠ dst) (bits : Buffer N) (suffix : List Bool)
    (state : σ × Buffer N) (tapes : K → List Bool)
    (hinput : tapes src = List.ofFn bits ++ suffix) :
    TM2.stepAux (blockStmt src dst F next) state tapes =
      TM2.stepAux next (state.1, emptyBuffer N)
        (Function.update (Function.update tapes src suffix) dst
          (List.ofFn (F bits) ++ tapes dst)) := by
  unfold blockStmt
  rw [stepAux_readAll src _ state bits tapes suffix hinput, stepAux_writeSlots]
  simp only [List.map_reverse, List.map_ofFn, Function.comp_id, List.reverse_reverse,
    Function.update_of_ne (Ne.symm hne), TM2.stepAux]

/-- The suffix and all unmentioned tapes are preserved, and the local register
is reset, in the actual transition semantics. -/
theorem stepAux_blockMapAt (src dst : K) (F : Buffer N → Buffer M)
    (exit : Option Λ) (hne : src ≠ dst) (bits : Buffer N) (suffix : List Bool)
    (state : σ × Buffer N) (tapes : K → List Bool)
    (hinput : tapes src = List.ofFn bits ++ suffix) :
    TM2.stepAux (blockMapAt src dst F exit) state tapes =
      { l := exit, var := (state.1, emptyBuffer N),
        stk := Function.update (Function.update tapes src suffix) dst
          (List.ofFn (F bits) ++ tapes dst) } := by
  rw [blockMapAt, stepAux_blockStmt src dst F _ hne bits suffix state tapes hinput]
  cases exit <;> rfl

/-- Placement in an arbitrary program gives one actual TM2 transition. -/
theorem step_blockMapAt (src dst : K) (F : Buffer N → Buffer M)
    (exit : Option Λ)
    (program : Λ → TM2.Stmt (fun _ : K => Bool) Λ (σ × Buffer N))
    (label : Λ) (hprogram : program label = blockMapAt src dst F exit)
    (hne : src ≠ dst) (bits : Buffer N) (suffix : List Bool)
    (state : σ × Buffer N) (tapes : K → List Bool)
    (hinput : tapes src = List.ofFn bits ++ suffix) :
    TM2.step program { l := some label, var := state, stk := tapes } =
      some { l := exit
             var := (state.1, emptyBuffer N)
             stk := Function.update (Function.update tapes src suffix) dst
               (List.ofFn (F bits) ++ tapes dst) } := by
  simp only [TM2.step, hprogram,
    stepAux_blockMapAt src dst F exit hne bits suffix state tapes hinput]

end Block

/-- The fixed block map is also a concrete machine with two Boolean tapes and
a finite register, independently of any ambient placement. -/
def machine {N M : Nat} (F : Buffer N → Buffer M) : FinTM2 where
  K := Bool
  k₀ := false
  k₁ := true
  Γ _ := Bool
  Λ := Unit
  main := ()
  σ := Unit × Buffer N
  initialState := ((), emptyBuffer N)
  m _ := blockMapAt false true F none

theorem machine_statementPushBound {N M : Nat} (F : Buffer N → Buffer M) :
    Runtime.statementPushBound ((machine F).m ()) = M :=
  statementPushBound_blockMapAt false true F none

end MaxCutGames.Foundations.Complexity.MachineFixedBlockMap

end OAI