lean/OAI/Computability/MaxCut/Machines/MachineExpanderRowFrames.lean

import Mathlib
import OAI.Computability.MaxCut.Machines.MachineExpanderFamilyAffine

namespace OAI

/-!
# Concrete data and tape frames for an actual expander-row computation

Only the five actual row inputs are stored. Every address, table return and
finite control value below is computed from those inputs. The thirteen tape
frames use the existing phase-output functions. These are data identities and
size bounds, not assumed machine executions.
-/

namespace MaxCutGames.Foundations.Complexity.MachineExpanderRow

open PCP.ExpanderTables PCP.ExpanderRowControl PCP.ExpanderTableWords

structure RowData (v d : Nat) where
  oldTable : Table v (degree d)
  smallTable : Table (cloudSize d) d
  inputVertex : Fin v
  inputCloud : Fin (cloudSize d)
  inputPort : Fin (degree d)

def rowData {v d : Nat} (G : Table v (degree d)) (H : Table (cloudSize d) d)
    (vertex : Fin v) (cloud : Fin (cloudSize d)) (port : Fin (degree d)) : RowData v d :=
  ⟨G, H, vertex, cloud, port⟩

namespace RowData

variable {v d : Nat} (r : RowData v d)

def control0 : Control d := start r.smallTable r.inputCloud r.inputPort
def firstRow : Fin (v * degree d) := rowIndex v (degree d) (r.inputVertex, firstOffset r.control0)
def firstValue : Nat := (reverseIndex r.oldTable r.firstRow).val
def firstPair : Fin v × Fin (degree d) := lookup r.oldTable (r.inputVertex, firstOffset r.control0)
def firstVertex : Fin v := r.firstPair.1
def firstReturn : Fin (degree d) := r.firstPair.2
def control1 : Control d := receiveFirst r.control0 r.firstReturn
def secondRow : Fin (v * degree d) := rowIndex v (degree d) (r.firstVertex, secondOffset r.control1)
def secondValue : Nat := (reverseIndex r.oldTable r.secondRow).val
def secondPair : Fin v × Fin (degree d) := lookup r.oldTable (r.firstVertex, secondOffset r.control1)
def secondVertex : Fin v := r.secondPair.1
def secondReturn : Fin (degree d) := r.secondPair.2
def control2 : Control d := receiveSecond r.smallTable r.control1 r.secondReturn
def query1 : Nat := firstAddress r.inputVertex.val r.control0
def query2 : Nat := secondAddress r.firstVertex.val r.control1
def finalValue : Nat := outputAddress r.secondVertex.val r.control2
def tableLength : Nat := (encodeWords (rotationWords r.oldTable)).length

theorem query1_eq_firstRow : r.query1 = r.firstRow.val :=
  firstAddress_eq_rowIndex r.inputVertex r.control0

theorem query2_eq_secondRow : r.query2 = r.secondRow.val :=
  secondAddress_eq_rowIndex r.firstVertex r.control1

@[simp] theorem firstValue_div_degree : r.firstValue / degree d = r.firstVertex.val := rfl
@[simp] theorem firstValue_mod_degree : r.firstValue % degree d = r.firstReturn.val := rfl
@[simp] theorem secondValue_div_degree : r.secondValue / degree d = r.secondVertex.val := rfl
@[simp] theorem secondValue_mod_degree : r.secondValue % degree d = r.secondReturn.val := rfl

theorem firstReturn_eq_residue (positive : 0 < d) :
    r.firstReturn = MachineFixedDivMod.residue (degree d) (Nat.mul_pos positive positive)
      r.firstValue := Fin.ext rfl

theorem secondReturn_eq_residue (positive : 0 < d) :
    r.secondReturn = MachineFixedDivMod.residue (degree d) (Nat.mul_pos positive positive)
      r.secondValue := Fin.ext rfl

theorem receiveFirst_residue_eq_control1 (positive : 0 < d) :
    receiveFirst r.control0 (MachineFixedDivMod.residue (degree d)
      (Nat.mul_pos positive positive) r.firstValue) = r.control1 := by
  rw [← r.firstReturn_eq_residue positive]
  rfl

theorem receiveSecond_residue_eq_control2 (positive : 0 < d) :
    receiveSecond r.smallTable r.control1 (MachineFixedDivMod.residue (degree d)
      (Nat.mul_pos positive positive) r.secondValue) = r.control2 := by
  rw [← r.secondReturn_eq_residue positive]
  rfl

theorem firstSelected : (rotationWords r.oldTable)[r.query1]? = some r.firstValue := by
  rw [query1_eq_firstRow]
  exact rotationWords_getElem? r.oldTable r.firstRow

theorem secondSelected : (rotationWords r.oldTable)[r.query2]? = some r.secondValue := by
  rw [query2_eq_secondRow]
  exact rotationWords_getElem? r.oldTable r.secondRow

theorem evaluateRow_eq : evaluateRow r.oldTable r.smallTable r.inputVertex r.inputCloud r.inputPort =
    (r.secondVertex, r.control2) := rfl

theorem finalValue_eq_step_reverseIndex :
    r.finalValue = (reverseIndex (step r.oldTable r.smallTable)
      (rowIndex (v * cloudSize d) (degree d)
        (rowIndex v (cloudSize d) (r.inputVertex, r.inputCloud), r.inputPort))).val := by
  simpa only [evaluateRow_eq, finalValue] using outputAddress_eq_step_reverseIndex
    r.oldTable r.smallTable r.inputVertex r.inputCloud r.inputPort

theorem rows_le_tableLength : v * degree d ≤ r.tableLength := by
  simp only [tableLength, encodeWords_length, rotationWords_length]
  omega

theorem query1_le_tableLength : r.query1 ≤ r.tableLength := by
  rw [query1_eq_firstRow]
  exact r.firstRow.isLt.le.trans r.rows_le_tableLength

theorem query2_le_tableLength : r.query2 ≤ r.tableLength := by
  rw [query2_eq_secondRow]
  exact r.secondRow.isLt.le.trans r.rows_le_tableLength

theorem firstValue_le_tableLength : r.firstValue ≤ r.tableLength :=
  (reverseIndex r.oldTable r.firstRow).isLt.le.trans r.rows_le_tableLength

theorem secondValue_le_tableLength : r.secondValue ≤ r.tableLength :=
  (reverseIndex r.oldTable r.secondRow).isLt.le.trans r.rows_le_tableLength

theorem inputVertex_le_tableLength : r.inputVertex.val ≤ r.tableLength := by
  have hq : 1 ≤ degree d := Nat.zero_lt_of_lt r.inputPort.isLt
  have hv : v ≤ v * degree d := by simpa using Nat.mul_le_mul_left v hq
  exact r.inputVertex.isLt.le.trans (hv.trans r.rows_le_tableLength)

theorem firstVertex_le_tableLength : r.firstVertex.val ≤ r.tableLength := by
  rw [← firstValue_div_degree]
  exact (Nat.div_le_self r.firstValue (degree d)).trans r.firstValue_le_tableLength

theorem secondVertex_le_tableLength : r.secondVertex.val ≤ r.tableLength := by
  rw [← secondValue_div_degree]
  exact (Nat.div_le_self r.secondValue (degree d)).trans r.secondValue_le_tableLength

theorem firstReturn_le_tableLength : r.firstReturn.val ≤ r.tableLength := by
  rw [← firstValue_mod_degree]
  exact (Nat.mod_le r.firstValue (degree d)).trans r.firstValue_le_tableLength

theorem secondReturn_le_tableLength : r.secondReturn.val ≤ r.tableLength := by
  rw [← secondValue_mod_degree]
  exact (Nat.mod_le r.secondValue (degree d)).trans r.secondValue_le_tableLength

end RowData

variable {v d : Nat}

/-- Complete explicit tape contents; all three scratch roles omitted here are empty. -/
def frameContents (r : RowData v d) (output : List Bool)
    (qr qi lo q1 q2 r1 r2 : List Bool) : Tape → List Bool
  | .inputVertex => encodeWord r.inputVertex.val
  | .table => encodeWords (rotationWords r.oldTable)
  | .output => output
  | .queryReverse => qr
  | .queryIndex => qi
  | .lookupOutput => lo
  | .quotientFirst => q1
  | .quotientSecond => q2
  | .remainderFirst => r1
  | .remainderSecond => r2
  | _ => []

def frame0 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  frameContents r output [] [] [] [] [] [] []
def frame1 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  initializedTapes id (frame0 r output)
def frame2 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  emittedWord .queryReverse (frame1 r output) r.query1
def frame3 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  Reduction.MachineTransfer.tapesAt .queryReverse .queryIndex (frame2 r output) []
    (encodeWord r.query1)
def frame4 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  lookupResultTapes id (frame3 r output) r.firstValue []
def frame5 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  divisionOutput d id .quotientFirst .remainderFirst (frame4 r output) r.firstValue [] [] []
def frame6 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  clearedQueryTapes id (frame5 r output)
def frame7 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  emittedWord .queryReverse (frame6 r output) r.query2
def frame8 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  Reduction.MachineTransfer.tapesAt .queryReverse .queryIndex (frame7 r output) []
    (encodeWord r.query2)
def frame9 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  lookupResultTapes id (frame8 r output) r.secondValue []
def frame10 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  divisionOutput d id .quotientSecond .remainderSecond (frame9 r output) r.secondValue [] [] []
def frame11 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  emittedWord .output (frame10 r output) r.finalValue
def frame12 (r : RowData v d) (output : List Bool) : Tape → List Bool :=
  cleanupTapes id (frame11 r output)

private theorem lookupResultTapes_id_eq_inline_MachineExpanderRowFrames (base : Tape → List Bool)
    (value : Nat) (suffix : List Bool) :
    lookupResultTapes id base value suffix =
      Function.update
        (Function.update (Function.update base .queryIndex (encodeWord 0 ++ suffix))
          .lookupWork [])
        .lookupOutput (encodeWord value ++ base .lookupOutput) := rfl

@[simp] theorem frame1_eq (r : RowData v d) (output : List Bool) :
    frame1 r output = frameContents r output [] [] []
      (encodeWord 0) (encodeWord 0) (encodeWord 0) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame1, frame0, frameContents, initializedTapes, encodeWord]

@[simp] theorem frame2_eq (r : RowData v d) (output : List Bool) :
    frame2 r output = frameContents r output (encodeWord r.query1).reverse [] []
      (encodeWord 0) (encodeWord 0) (encodeWord 0) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame2, frameContents, emittedWord]

@[simp] theorem frame3_eq (r : RowData v d) (output : List Bool) :
    frame3 r output = frameContents r output [] (encodeWord r.query1) []
      (encodeWord 0) (encodeWord 0) (encodeWord 0) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame3, frameContents, Reduction.MachineTransfer.tapesAt]

@[simp] theorem frame4_eq (r : RowData v d) (output : List Bool) :
    frame4 r output = frameContents r output [] (encodeWord 0) (encodeWord r.firstValue)
      (encodeWord 0) (encodeWord 0) (encodeWord 0) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame4, lookupResultTapes_id_eq_inline_MachineExpanderRowFrames, frameContents]

@[simp] theorem frame5_eq (r : RowData v d) (output : List Bool) :
    frame5 r output = frameContents r output [] (encodeWord 0) (encodeWord 0)
      (encodeWord r.firstVertex.val) (encodeWord 0)
      (encodeWord r.firstReturn.val) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame5, divisionOutput, MachineFixedDivMod.unaryTapes,
    MachineFixedDivMod.tapes, MachineCopy.forkTapes, frameContents]

@[simp] theorem frame6_eq (r : RowData v d) (output : List Bool) :
    frame6 r output = frameContents r output [] [] [] (encodeWord r.firstVertex.val)
      (encodeWord 0) (encodeWord r.firstReturn.val) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame6, clearedQueryTapes, frameContents, encodeWord]

@[simp] theorem frame7_eq (r : RowData v d) (output : List Bool) :
    frame7 r output = frameContents r output (encodeWord r.query2).reverse [] []
      (encodeWord r.firstVertex.val) (encodeWord 0)
      (encodeWord r.firstReturn.val) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame7, emittedWord, frameContents]

@[simp] theorem frame8_eq (r : RowData v d) (output : List Bool) :
    frame8 r output = frameContents r output [] (encodeWord r.query2) []
      (encodeWord r.firstVertex.val) (encodeWord 0)
      (encodeWord r.firstReturn.val) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame8, Reduction.MachineTransfer.tapesAt, frameContents]

@[simp] theorem frame9_eq (r : RowData v d) (output : List Bool) :
    frame9 r output = frameContents r output [] (encodeWord 0) (encodeWord r.secondValue)
      (encodeWord r.firstVertex.val) (encodeWord 0)
      (encodeWord r.firstReturn.val) (encodeWord 0) := by
  funext tape
  cases tape <;> simp [frame9, lookupResultTapes_id_eq_inline_MachineExpanderRowFrames, frameContents]

@[simp] theorem frame10_eq (r : RowData v d) (output : List Bool) :
    frame10 r output = frameContents r output [] (encodeWord 0) (encodeWord 0)
      (encodeWord r.firstVertex.val) (encodeWord r.secondVertex.val)
      (encodeWord r.firstReturn.val) (encodeWord r.secondReturn.val) := by
  funext tape
  cases tape <;> simp [frame10, divisionOutput, MachineFixedDivMod.unaryTapes,
    MachineFixedDivMod.tapes, MachineCopy.forkTapes, frameContents]

@[simp] theorem frame11_eq (r : RowData v d) (output : List Bool) :
    frame11 r output = frameContents r ((encodeWord r.finalValue).reverse ++ output)
      [] (encodeWord 0) (encodeWord 0)
      (encodeWord r.firstVertex.val) (encodeWord r.secondVertex.val)
      (encodeWord r.firstReturn.val) (encodeWord r.secondReturn.val) := by
  funext tape
  cases tape <;> simp [frame11, emittedWord, frameContents]

@[simp] theorem frame12_eq (r : RowData v d) (output : List Bool) :
    frame12 r output = frameContents r ((encodeWord r.finalValue).reverse ++ output)
      [] [] [] [] [] [] [] := by
  funext tape
  cases tape <;> simp [frame12, cleanupTapes, frameContents]

theorem final_frame_eq (r : RowData v d) (output : List Bool) :
    frame12 r output = emittedWord .output (frame0 r output) r.finalValue := by
  funext tape
  cases tape <;> simp [frame0, emittedWord, frameContents]

/-! Direct input facts for the individual checked phase traces. -/

@[simp] theorem frame0_inputVertex (r : RowData v d) (output : List Bool) :
    frame0 r output .inputVertex = encodeWord r.inputVertex.val := rfl

@[simp] theorem frame0_table (r : RowData v d) (output : List Bool) :
    frame0 r output .table = encodeWords (rotationWords r.oldTable) := rfl

@[simp] theorem frame0_output (r : RowData v d) (output : List Bool) :
    frame0 r output .output = output := rfl

@[simp] theorem frame1_inputVertex (r : RowData v d) (output : List Bool) :
    frame1 r output .inputVertex = encodeWord r.inputVertex.val := by simp [frameContents]

@[simp] theorem frame1_emitScratch (r : RowData v d) (output : List Bool) :
    frame1 r output .emitScratch = [] := by simp [frameContents]

@[simp] theorem frame2_queryReverse (r : RowData v d) (output : List Bool) :
    frame2 r output .queryReverse = (encodeWord r.query1).reverse := by simp [frameContents]

@[simp] theorem frame2_queryIndex (r : RowData v d) (output : List Bool) :
    frame2 r output .queryIndex = [] := by simp [frameContents]

@[simp] theorem frame3_table (r : RowData v d) (output : List Bool) :
    frame3 r output .table = encodeWords (rotationWords r.oldTable) := by simp [frameContents]

@[simp] theorem frame3_lookupRestore (r : RowData v d) (output : List Bool) :
    frame3 r output .lookupRestore = [] := by simp [frameContents]

@[simp] theorem frame3_lookupWork (r : RowData v d) (output : List Bool) :
    frame3 r output .lookupWork = [] := by simp [frameContents]

@[simp] theorem frame3_lookupOutput (r : RowData v d) (output : List Bool) :
    frame3 r output .lookupOutput = [] := by simp [frameContents]

@[simp] theorem frame3_queryIndex (r : RowData v d) (output : List Bool) :
    frame3 r output .queryIndex = encodeWord r.query1 := by simp [frameContents]

@[simp] theorem frame4_lookupOutput (r : RowData v d) (output : List Bool) :
    frame4 r output .lookupOutput = encodeWord r.firstValue := by simp [frameContents]

@[simp] theorem frame4_quotientFirst (r : RowData v d) (output : List Bool) :
    frame4 r output .quotientFirst = encodeWord 0 := by simp [frameContents]

@[simp] theorem frame4_remainderFirst (r : RowData v d) (output : List Bool) :
    frame4 r output .remainderFirst = encodeWord 0 := by simp [frameContents]

@[simp] theorem frame6_quotientFirst (r : RowData v d) (output : List Bool) :
    frame6 r output .quotientFirst = encodeWord r.firstVertex.val := by simp [frameContents]

@[simp] theorem frame6_emitScratch (r : RowData v d) (output : List Bool) :
    frame6 r output .emitScratch = [] := by simp [frameContents]

@[simp] theorem frame7_queryReverse (r : RowData v d) (output : List Bool) :
    frame7 r output .queryReverse = (encodeWord r.query2).reverse := by simp [frameContents]

@[simp] theorem frame7_queryIndex (r : RowData v d) (output : List Bool) :
    frame7 r output .queryIndex = [] := by simp [frameContents]

@[simp] theorem frame8_table (r : RowData v d) (output : List Bool) :
    frame8 r output .table = encodeWords (rotationWords r.oldTable) := by simp [frameContents]

@[simp] theorem frame8_lookupRestore (r : RowData v d) (output : List Bool) :
    frame8 r output .lookupRestore = [] := by simp [frameContents]

@[simp] theorem frame8_lookupWork (r : RowData v d) (output : List Bool) :
    frame8 r output .lookupWork = [] := by simp [frameContents]

@[simp] theorem frame8_lookupOutput (r : RowData v d) (output : List Bool) :
    frame8 r output .lookupOutput = [] := by simp [frameContents]

@[simp] theorem frame8_queryIndex (r : RowData v d) (output : List Bool) :
    frame8 r output .queryIndex = encodeWord r.query2 := by simp [frameContents]

@[simp] theorem frame9_lookupOutput (r : RowData v d) (output : List Bool) :
    frame9 r output .lookupOutput = encodeWord r.secondValue := by simp [frameContents]

@[simp] theorem frame9_quotientSecond (r : RowData v d) (output : List Bool) :
    frame9 r output .quotientSecond = encodeWord 0 := by simp [frameContents]

@[simp] theorem frame9_remainderSecond (r : RowData v d) (output : List Bool) :
    frame9 r output .remainderSecond = encodeWord 0 := by simp [frameContents]

@[simp] theorem frame10_quotientSecond (r : RowData v d) (output : List Bool) :
    frame10 r output .quotientSecond = encodeWord r.secondVertex.val := by simp [frameContents]

@[simp] theorem frame10_emitScratch (r : RowData v d) (output : List Bool) :
    frame10 r output .emitScratch = [] := by simp [frameContents]

@[simp] theorem frame10_output (r : RowData v d) (output : List Bool) :
    frame10 r output .output = output := by simp [frameContents]

theorem final_dirty_word_length_le (r : RowData v d) (output : List Bool) (i : Fin 6) :
    ((frame11 r output) (dirtyTape i)).length ≤ r.tableLength + 1 := by
  have h1 := r.firstVertex_le_tableLength
  have h2 := r.secondVertex_le_tableLength
  have h3 := r.firstReturn_le_tableLength
  have h4 := r.secondReturn_le_tableLength
  fin_cases i <;> simp only [frame11_eq, dirtyTape, frameContents, encodeWord_length] <;> omega

theorem cleanupSteps_le_tableLength (r : RowData v d) (output : List Bool) :
    cleanupSteps id (frame11 r output) ≤ 6 * (r.tableLength + 1) + 6 := by
  have h1 := r.firstVertex_le_tableLength
  have h2 := r.secondVertex_le_tableLength
  have h3 := r.firstReturn_le_tableLength
  have h4 := r.secondReturn_le_tableLength
  simp only [cleanupSteps, frame11_eq, id_eq, frameContents, encodeWord_length]
  omega

end MaxCutGames.Foundations.Complexity.MachineExpanderRow

/-!
# Complete execution of one expander-table row

The actual finite program performs both stored-table lookups, splits their
results by its fixed degree, emits the resulting square/zigzag reverse index,
and clears its work tapes. Its transition bound is linear in the unary input
table length. The output accumulator and finite caller state are arbitrary.
-/

namespace MaxCutGames.Foundations.Complexity.MachineExpanderRow

open Turing MachineComposition
open PCP.ExpanderTables PCP.ExpanderRowControl PCP.ExpanderTableWords PCP.AlphabetTable

private theorem concatenate_inline_MachineExpanderRow {A : Type*} {f : A → A} {m n : Nat} {a b c : A}
    (first : f^[m] a = b) (second : f^[n] b = c) : f^[m + n] a = c := by
  rw [Nat.add_comm m n, Function.iterate_add_apply, first, second]

/-- The actual number of transitions, retaining both concrete lookup costs. -/
def rowSteps {v d : Nat} (r : RowData v d) (output : List Bool) : Nat :=
  1 + emitSteps r.inputVertex.val + (r.query1 + 2) +
    MachinePreservingLookupClean.steps (rotationWords r.oldTable) r.firstRow.val +
    (r.firstValue + 2) + 1 + emitSteps r.firstVertex.val + (r.query2 + 2) +
    MachinePreservingLookupClean.steps (rotationWords r.oldTable) r.secondRow.val +
    (r.secondValue + 2) + emitSteps r.secondVertex.val +
    (cleanupSteps id (frame11 r output) + 1)

section Execution

variable {v d : Nat} {ρ Λ : Type} [Fintype ρ]
    (positive : 0 < d) (r : RowData v d) (output : List Bool) (ambient : ρ)
    (labels : Label d → Λ) (exit : Option Λ)
    (target : Λ → TM2.Stmt (fun _ : Tape => Bool) Λ (State ρ d))
    (code : ∀ l, target (labels l) = statement positive r.smallTable id labels exit l)

include code

/-- The complete execution follows the concrete program. The only code premise
places its instructions in a caller; every lookup is derived from `oldTable`. -/
theorem rowTraceAt :
    (advance (TM2.step target))^[rowSteps r output]
      (some ⟨some (labels .initialize), divisionState positive ambient r.control0 none,
        frame0 r output⟩) =
      some ⟨exit, divisionState positive ambient r.control2 none,
        emittedWord .output (frame0 r output) r.finalValue⟩ := by
  let zero := MachineFixedDivMod.residue (degree d) (Nat.mul_pos positive positive) 0
  have h0 : (advance (TM2.step target))^[1]
      (some ⟨some (labels .initialize), divisionState positive ambient r.control0 none,
        frame0 r output⟩) =
      some ⟨some (labels (.firstEmit (Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient r.control0 none, frame1 r output⟩ := by
    simpa only [Function.iterate_one, advance_some, divisionState, frame1] using
      initializeStepAt positive r.smallTable id Function.injective_id labels exit target code
        (frame0 r output) (divisionState positive ambient r.control0 none)
  have h1 := firstEmitTraceAt positive r.smallTable id Function.injective_id
    labels exit target code r.inputVertex.val (frame1 r output)
    (by simp) (by simp) ((ambient,r.control0),zero)
  change (advance (TM2.step target))^[emitSteps r.inputVertex.val]
      (some ⟨some (labels (.firstEmit (Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient r.control0 none, frame1 r output⟩) =
      some ⟨some (labels .firstReverse), divisionState positive ambient r.control0 none,
        frame2 r output⟩ at h1
  have h2 := reversePhaseTrace_unary false positive r.smallTable id Function.injective_id
    labels exit target code (frame2 r output) (((ambient,r.control0),zero),()) none
    r.query1 [] (by simp) (by simp)
  simp only [List.append_nil] at h2
  change (advance (TM2.step target))^[r.query1 + 2]
      (some ⟨some (labels .firstReverse), divisionState positive ambient r.control0 none,
        frame2 r output⟩) =
      some ⟨some (labels (.firstLookup (.run .copyFirst))),
        divisionState positive ambient r.control0 none, frame3 r output⟩ at h2
  have h3 := lookupPhaseTrace false positive r.smallTable id Function.injective_id
    labels exit target code (frame3 r output) r.oldTable
    (by simp) (by simp)
    (r.inputVertex, firstOffset r.control0) []
    (by simpa [frameContents, RowData.firstRow] using
      congrArg encodeWord r.query1_eq_firstRow)
    (by simp) (((ambient,r.control0),zero),()) none
  change (advance (TM2.step target))^[MachinePreservingLookupClean.steps
      (rotationWords r.oldTable) r.firstRow.val]
      (some ⟨some (labels (.firstLookup (.run .copyFirst))),
        divisionState positive ambient r.control0 none, frame3 r output⟩) =
      some ⟨some (labels .firstScan), divisionState positive ambient r.control0 none,
        frame4 r output⟩ at h3
  have h4 := firstDivisionTrace positive r.smallTable id Function.injective_id labels exit
    target code (frame4 r output) r.firstValue [] [] []
    (by simp) (by simp) (by simp)
    ambient r.control0 none
  rw [← r.firstReturn_eq_residue positive] at h4
  change (advance (TM2.step target))^[r.firstValue + 2]
      (some ⟨some (labels .firstScan), divisionState positive ambient r.control0 none,
        frame4 r output⟩) =
      some ⟨some (labels .clearQuery), divisionState positive ambient r.control1 none,
        frame5 r output⟩ at h4
  have h5 : (advance (TM2.step target))^[1]
      (some ⟨some (labels .clearQuery), divisionState positive ambient r.control1 none,
        frame5 r output⟩) =
      some ⟨some (labels (.secondEmit (Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient r.control1 none, frame6 r output⟩ := by
    simpa only [Function.iterate_one, advance_some, frame6] using
      clearQueryStepAt positive r.smallTable id Function.injective_id labels exit target code
        (frame5 r output) (divisionState positive ambient r.control1 none)
  have h6 := secondEmitTraceAt positive r.smallTable id Function.injective_id
    labels exit target code r.firstVertex.val (frame6 r output)
    (by simp) (by simp) ((ambient,r.control1),zero)
  change (advance (TM2.step target))^[emitSteps r.firstVertex.val]
      (some ⟨some (labels (.secondEmit (Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient r.control1 none, frame6 r output⟩) =
      some ⟨some (labels .secondReverse), divisionState positive ambient r.control1 none,
        frame7 r output⟩ at h6
  have h7 := reversePhaseTrace_unary true positive r.smallTable id Function.injective_id
    labels exit target code (frame7 r output) (((ambient,r.control1),zero),()) none
    r.query2 [] (by simp) (by simp)
  simp only [List.append_nil] at h7
  change (advance (TM2.step target))^[r.query2 + 2]
      (some ⟨some (labels .secondReverse), divisionState positive ambient r.control1 none,
        frame7 r output⟩) =
      some ⟨some (labels (.secondLookup (.run .copyFirst))),
        divisionState positive ambient r.control1 none, frame8 r output⟩ at h7
  have h8 := lookupPhaseTrace true positive r.smallTable id Function.injective_id
    labels exit target code (frame8 r output) r.oldTable
    (by simp) (by simp)
    (r.firstVertex, secondOffset r.control1) []
    (by simpa [frameContents, RowData.secondRow] using
      congrArg encodeWord r.query2_eq_secondRow)
    (by simp) (((ambient,r.control1),zero),()) none
  change (advance (TM2.step target))^[MachinePreservingLookupClean.steps
      (rotationWords r.oldTable) r.secondRow.val]
      (some ⟨some (labels (.secondLookup (.run .copyFirst))),
        divisionState positive ambient r.control1 none, frame8 r output⟩) =
      some ⟨some (labels .secondScan), divisionState positive ambient r.control1 none,
        frame9 r output⟩ at h8
  have h9 := secondDivisionTrace positive r.smallTable id Function.injective_id labels exit
    target code (frame9 r output) r.secondValue [] [] []
    (by simp) (by simp) (by simp)
    ambient r.control1 none
  rw [← r.secondReturn_eq_residue positive] at h9
  change (advance (TM2.step target))^[r.secondValue + 2]
      (some ⟨some (labels .secondScan), divisionState positive ambient r.control1 none,
        frame9 r output⟩) =
      some ⟨some (labels (.outputEmit (Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient r.control2 none, frame10 r output⟩ at h9
  have h10 := outputEmitTraceAt positive r.smallTable id Function.injective_id
    labels exit target code r.secondVertex.val (frame10 r output)
    (by simp) (by simp) ((ambient,r.control2),zero)
  change (advance (TM2.step target))^[emitSteps r.secondVertex.val]
      (some ⟨some (labels (.outputEmit (Emitter.labelAt 3 _ 0 .entry))),
        divisionState positive ambient r.control2 none, frame10 r output⟩) =
      some ⟨some (labels (.cleanup 0)), divisionState positive ambient r.control2 none,
        frame11 r output⟩ at h10
  have h11 := cleanupExitTraceAt positive r.smallTable id Function.injective_id
    labels exit target code (frame11 r output) (divisionState positive ambient r.control2 none)
  change (advance (TM2.step target))^[cleanupSteps id (frame11 r output) + 1]
      (some ⟨some (labels (.cleanup 0)), divisionState positive ambient r.control2 none,
        frame11 r output⟩) =
      some ⟨exit, divisionState positive ambient r.control2 none, frame12 r output⟩ at h11
  rw [final_frame_eq] at h11
  exact concatenate_inline_MachineExpanderRow (concatenate_inline_MachineExpanderRow (concatenate_inline_MachineExpanderRow (concatenate_inline_MachineExpanderRow (concatenate_inline_MachineExpanderRow
    (concatenate_inline_MachineExpanderRow (concatenate_inline_MachineExpanderRow (concatenate_inline_MachineExpanderRow (concatenate_inline_MachineExpanderRow (concatenate_inline_MachineExpanderRow
      (concatenate_inline_MachineExpanderRow h0 h1) h2) h3) h4) h5) h6) h7) h8) h9) h10) h11

end Execution

theorem rowSteps_le {v d : Nat} (r : RowData v d) (output : List Bool) :
    rowSteps r output ≤ 80 * (r.tableLength + 1) := by
  have lookup1 := MachinePreservingLookupClean.steps_le (rotationWords r.oldTable)
    r.firstRow.val r.firstValue (rotationWords_getElem? r.oldTable r.firstRow)
  have lookup2 := MachinePreservingLookupClean.steps_le (rotationWords r.oldTable)
    r.secondRow.val r.secondValue (rotationWords_getElem? r.oldTable r.secondRow)
  have h0 := r.inputVertex_le_tableLength
  have h1 := r.query1_le_tableLength
  have h2 := r.firstValue_le_tableLength
  have h3 := r.firstVertex_le_tableLength
  have h4 := r.query2_le_tableLength
  have h5 := r.secondValue_le_tableLength
  have h6 := r.secondVertex_le_tableLength
  have clean := cleanupSteps_le_tableLength r output
  change _ ≤ 6 * r.tableLength + 4 at lookup1 lookup2
  simp only [rowSteps, emitSteps]
  omega

/-- The actual row program has a linear transition bound, with all temporary
tapes empty at return. The bound holds for every valid finite input table. -/
def rowInTimeAt {v d : Nat} {ρ Λ : Type} [Fintype ρ]
    (positive : 0 < d) (r : RowData v d) (output : List Bool) (ambient : ρ)
    (labels : Label d → Λ) (exit : Option Λ)
    (target : Λ → TM2.Stmt (fun _ : Tape => Bool) Λ (State ρ d))
    (code : ∀ l, target (labels l) = statement positive r.smallTable id labels exit l) :
    StateTransition.EvalsToInTime (TM2.step target)
      ⟨some (labels .initialize), divisionState positive ambient r.control0 none,
        frame0 r output⟩
      (some ⟨exit, divisionState positive ambient r.control2 none,
        emittedWord .output (frame0 r output) r.finalValue⟩)
      (80 * (r.tableLength + 1)) where
  steps := rowSteps r output
  evals_in_steps := rowTraceAt positive r output ambient labels exit target code
  steps_le_m := rowSteps_le r output

/-- Specialization to the standalone finite row program. -/
def rowInTime {v d : Nat} {ρ : Type} [Fintype ρ]
    (positive : 0 < d) (r : RowData v d) (output : List Bool) (ambient : ρ) :
    StateTransition.EvalsToInTime (TM2.step (program positive r.smallTable))
      ⟨some .initialize, divisionState positive ambient r.control0 none, frame0 r output⟩
      (some ⟨none, divisionState positive ambient r.control2 none,
        emittedWord .output (frame0 r output) r.finalValue⟩)
      (80 * (r.tableLength + 1)) :=
  rowInTimeAt positive r output ambient id none (program positive r.smallTable) (fun _ => rfl)

end MaxCutGames.Foundations.Complexity.MachineExpanderRow

/-!
# Concrete boundary steps of the expander-table machine

Every execution theorem below reduces the actual common program statement.
States and dependent tape frames are arbitrary; only the relevant unary word
or finite-position branch is specified. No phase execution is a premise.
-/

namespace MaxCutGames.Foundations.Complexity.MachineExpanderTable

open Turing
open PCP.ExpanderTables PCP.ExpanderRowControl

variable {ρ : Type} {d : Nat}

@[simp] theorem caller_boundaryState (positive : 0 < d) (H : Table (cloudSize d) d)
    (ambient : ρ) (position : Position d) :
    caller (boundaryState positive H ambient position) = ambient := rfl

@[simp] theorem prepareState_boundaryState (positive : 0 < d) (H : Table (cloudSize d) d)
    (ambient : ρ) (position : Position d) :
    prepareState positive H (boundaryState positive H ambient position) =
      boundaryState positive H ambient position := rfl

@[simp] theorem clearRegister_boundaryState (positive : 0 < d) (H : Table (cloudSize d) d)
    (ambient : ρ) (position : Position d) :
    clearRegister (boundaryState positive H ambient position) =
      boundaryState positive H ambient position := rfl

@[simp] theorem caller_prepareState (positive : 0 < d) (H : Table (cloudSize d) d)
    (state : State ρ d) : caller (prepareState positive H state) = caller state := rfl

@[simp] theorem caller_resetState (positive : 0 < d) (H : Table (cloudSize d) d)
    (state : State ρ d) : caller (resetState positive H state) = caller state := rfl

@[simp] theorem caller_clearRegister (state : State ρ d) :
    caller (clearRegister state) = caller state := rfl

@[simp] theorem caller_advancePositionState (positive : 0 < d) (state : State ρ d) :
    caller (advancePositionState positive state) = caller state := rfl

@[simp] theorem caller_resetPositionState (positive : 0 < d) (state : State ρ d) :
    caller (resetPositionState positive state) = caller state := rfl

theorem nextPosition_val_of_lt (positive : 0 < d) (position : Position d)
    (more : position.val + 1 < rowFactor d) :
    (nextPosition positive position).val = position.val + 1 := Nat.mod_eq_of_lt more

/-- Replacing the count word leaves the entire row frame and result unchanged. -/
theorem boundaryTapes_update_count (vertex remaining next : Nat)
    (oldTable output countSuffix result : List Bool) :
    Function.update (boundaryTapes vertex remaining oldTable output countSuffix result)
      (.inr .vertexCount) (encodeWord next ++ countSuffix) =
      boundaryTapes vertex next oldTable output countSuffix result := by
  funext tape
  rcases tape with row | extra
  · cases row <;> simp [boundaryTapes, rowTapes, MachineEmbedding.tapes, Function.update]
  · cases extra <;> simp [boundaryTapes, extraTapes, MachineEmbedding.tapes, Function.update]

/-- The last row's one actual push increments the unary vertex word. -/
theorem boundaryTapes_increment_vertex (vertex remaining : Nat)
    (oldTable output countSuffix result : List Bool) :
    Function.update (boundaryTapes vertex remaining oldTable output countSuffix result)
      (.inl .inputVertex) (true :: encodeWord vertex) =
      boundaryTapes (vertex + 1) remaining oldTable output countSuffix result := by
  funext tape
  rcases tape with row | extra
  · cases row <;>
      simp [boundaryTapes, rowTapes, MachineEmbedding.tapes, Function.update,
        encodeWord, List.replicate_succ]
  · cases extra <;>
      simp [boundaryTapes, extraTapes, MachineEmbedding.tapes, Function.update]

theorem initialTapes_initialize (vertices : Nat) (oldTable countSuffix : List Bool) :
    Function.update (initialTapes vertices oldTable countSuffix) (.inl .inputVertex) [false] =
      boundaryTapes 0 vertices oldTable [] countSuffix [] := by
  funext tape
  rcases tape with row | extra
  · cases row <;>
      simp [initialTapes, boundaryTapes, rowTapes, MachineEmbedding.tapes,
        Function.update, encodeWord]
  · cases extra <;>
      simp [initialTapes, boundaryTapes, extraTapes, MachineEmbedding.tapes,
        Function.update]

variable [Fintype ρ]

theorem initializeStep (positive : 0 < d) (H : Table (cloudSize d) d)
    (base : ∀ tape, List (Alphabet tape)) (state : State ρ d) :
    TM2.step (program positive H) ⟨some (.inr .initialize), state, base⟩ =
      some ⟨some (.inr .vertexGuard), resetState positive H state,
        Function.update base (.inl .inputVertex) (false :: base (.inl .inputVertex))⟩ := by
  change some (TM2.stepAux (program positive H (.inr .initialize)) state base) = _
  rw [program_outer]
  rfl

theorem initialize_boundaryStep (positive : 0 < d) (H : Table (cloudSize d) d)
    (vertices : Nat) (oldTable countSuffix : List Bool) (state : State ρ d) :
    TM2.step (program positive H)
      ⟨some (.inr .initialize), state, initialTapes vertices oldTable countSuffix⟩ =
      some ⟨some (.inr .vertexGuard), initialState positive H (caller state),
        boundaryTapes 0 vertices oldTable [] countSuffix []⟩ := by
  have h := initializeStep positive H (initialTapes vertices oldTable countSuffix) state
  have input : initialTapes vertices oldTable countSuffix (.inl .inputVertex) = [] := rfl
  simp only [input, resetState] at h
  exact h.trans (congrArg
    (fun tapes : ∀ tape, List (Alphabet tape) =>
      (some ⟨some (.inr .vertexGuard), initialState positive H (caller state), tapes⟩ :
        Option (TM2.Cfg Alphabet (Label d) (State ρ d))))
    (initialTapes_initialize vertices oldTable countSuffix))

/-- A positive vertex counter is physically decremented; its suffix remains. -/
theorem vertexGuard_succStep (positive : 0 < d) (H : Table (cloudSize d) d)
    (base : ∀ tape, List (Alphabet tape)) (state : State ρ d)
    (remaining : Nat) (suffix : List Bool)
    (counter : base (.inr .vertexCount) = encodeWord (remaining + 1) ++ suffix) :
    TM2.step (program positive H) ⟨some (.inr .vertexGuard), state, base⟩ =
      some ⟨some (.inr .prepareRow), clearRegister state,
        Function.update base (.inr .vertexCount) (encodeWord remaining ++ suffix)⟩ := by
  change some (TM2.stepAux (program positive H (.inr .vertexGuard)) state base) = _
  rw [program_outer]
  simp [outerStatement, MachineControl.statement, vertexGuardCore, TM2.stepAux,
    guardStates, clearRegister, counter, encodeWord, List.replicate_succ]

/-- The zero delimiter is retained when the loop enters output reversal. -/
theorem vertexGuard_zeroStep (positive : 0 < d) (H : Table (cloudSize d) d)
    (base : ∀ tape, List (Alphabet tape)) (state : State ρ d)
    (suffix : List Bool)
    (counter : base (.inr .vertexCount) = encodeWord 0 ++ suffix) :
    TM2.step (program positive H) ⟨some (.inr .vertexGuard), state, base⟩ =
      some ⟨some (.inr .reverseOutput), clearRegister state, base⟩ := by
  change some (TM2.stepAux (program positive H (.inr .vertexGuard)) state base) = _
  rw [program_outer]
  simp [outerStatement, MachineControl.statement, vertexGuardCore, TM2.stepAux,
    guardStates, clearRegister, counter, encodeWord]

theorem vertexGuard_boundary_succStep (positive : 0 < d) (H : Table (cloudSize d) d)
    (vertex remaining : Nat) (oldTable output countSuffix result : List Bool)
    (state : State ρ d) :
    TM2.step (program positive H)
      ⟨some (.inr .vertexGuard), state,
        boundaryTapes vertex (remaining + 1) oldTable output countSuffix result⟩ =
      some ⟨some (.inr .prepareRow), clearRegister state,
        boundaryTapes vertex remaining oldTable output countSuffix result⟩ := by
  simpa only [boundaryTapes_update_count] using
    vertexGuard_succStep positive H
      (boundaryTapes vertex (remaining + 1) oldTable output countSuffix result)
      state remaining countSuffix rfl

theorem vertexGuard_boundary_zeroStep (positive : 0 < d) (H : Table (cloudSize d) d)
    (vertex : Nat) (oldTable output countSuffix result : List Bool) (state : State ρ d) :
    TM2.step (program positive H)
      ⟨some (.inr .vertexGuard), state,
        boundaryTapes vertex 0 oldTable output countSuffix result⟩ =
      some ⟨some (.inr .reverseOutput), clearRegister state,
        boundaryTapes vertex 0 oldTable output countSuffix result⟩ :=
  vertexGuard_zeroStep positive H
    (boundaryTapes vertex 0 oldTable output countSuffix result) state countSuffix rfl

/-- Preparing a row modifies finite control only, preserving every tape. -/
theorem prepareRowStep (positive : 0 < d) (H : Table (cloudSize d) d)
    (base : ∀ tape, List (Alphabet tape)) (state : State ρ d) :
    TM2.step (program positive H) ⟨some (.inr .prepareRow), state, base⟩ =
      some ⟨some (.inl .initialize), prepareState positive H state, base⟩ := by
  change some (TM2.stepAux (program positive H (.inr .prepareRow)) state base) = _
  rw [program_outer]
  rfl

/-- A nonfinal row advances the finite position without changing tapes. -/
theorem afterRow_moreStep (positive : 0 < d) (H : Table (cloudSize d) d)
    (base : ∀ tape, List (Alphabet tape)) (state : State ρ d)
    (more : state.2.val + 1 < rowFactor d) :
    TM2.step (program positive H) ⟨some (.inr .afterRow), state, base⟩ =
      some ⟨some (.inr .prepareRow), advancePositionState positive state, base⟩ := by
  change some (TM2.stepAux (program positive H (.inr .afterRow)) state base) = _
  rw [program_outer]
  simp [outerStatement, TM2.stepAux, more]

/-- The final row increments the tape vertex and resets the finite position. -/
theorem afterRow_lastStep (positive : 0 < d) (H : Table (cloudSize d) d)
    (base : ∀ tape, List (Alphabet tape)) (state : State ρ d)
    (last : ¬ state.2.val + 1 < rowFactor d) :
    TM2.step (program positive H) ⟨some (.inr .afterRow), state, base⟩ =
      some ⟨some (.inr .vertexGuard), resetPositionState positive state,
        Function.update base (.inl .inputVertex) (true :: base (.inl .inputVertex))⟩ := by
  change some (TM2.stepAux (program positive H (.inr .afterRow)) state base) = _
  rw [program_outer]
  simp [outerStatement, TM2.stepAux, last]

theorem afterRow_last_boundaryStep (positive : 0 < d) (H : Table (cloudSize d) d)
    (vertex remaining : Nat) (oldTable output countSuffix result : List Bool)
    (state : State ρ d) (last : ¬ state.2.val + 1 < rowFactor d) :
    TM2.step (program positive H)
      ⟨some (.inr .afterRow), state,
        boundaryTapes vertex remaining oldTable output countSuffix result⟩ =
      some ⟨some (.inr .vertexGuard), resetPositionState positive state,
        boundaryTapes (vertex + 1) remaining oldTable output countSuffix result⟩ := by
  have input : boundaryTapes vertex remaining oldTable output countSuffix result
      (.inl .inputVertex) = encodeWord vertex := rfl
  have h := afterRow_lastStep positive H
    (boundaryTapes vertex remaining oldTable output countSuffix result) state last
  simp only [input] at h
  exact h.trans (congrArg
    (fun tapes : ∀ tape, List (Alphabet tape) =>
      (some ⟨some (.inr .vertexGuard), resetPositionState positive state, tapes⟩ :
        Option (TM2.Cfg Alphabet (Label d) (State ρ d))))
    (boundaryTapes_increment_vertex vertex remaining oldTable output countSuffix result))

/-- One reversal step moves an actual symbol and preserves the finite position. -/
theorem reverseOutput_consStep (positive : 0 < d) (H : Table (cloudSize d) d)
    (base : ∀ tape, List (Alphabet tape)) (state : State ρ d)
    (symbol : Bool) (word : List Bool) (source : base (.inl .output) = symbol :: word) :
    TM2.step (program positive H) ⟨some (.inr .reverseOutput), state, base⟩ =
      some ⟨some (.inr .reverseOutput), ((state.1.1, some symbol), state.2),
        Function.update (Function.update base (.inl .output) word)
          (.inr .result) (symbol :: base (.inr .result))⟩ := by
  change some (TM2.stepAux (program positive H (.inr .reverseOutput)) state base) = _
  rw [program_outer]
  simp [outerStatement, MachineControl.statement, Reduction.MachineTransfer.loopAt,
    Reduction.MachineTransfer.exitAt, TM2.stepAux, guardStates, source]

theorem reverseOutput_nilStep (positive : 0 < d) (H : Table (cloudSize d) d)
    (base : ∀ tape, List (Alphabet tape)) (state : State ρ d)
    (source : base (.inl .output) = []) :
    TM2.step (program positive H) ⟨some (.inr .reverseOutput), state, base⟩ =
      some ⟨some (.inr .done), clearRegister state, base⟩ := by
  have same : Function.update base (.inl .output) [] = base := by
    simpa only [source] using Function.update_eq_self (.inl .output) base
  change some (TM2.stepAux (program positive H (.inr .reverseOutput)) state base) = _
  rw [program_outer]
  simp [outerStatement, MachineControl.statement, Reduction.MachineTransfer.loopAt,
    Reduction.MachineTransfer.exitAt, TM2.stepAux, guardStates, clearRegister, source, same]

theorem doneStep (positive : 0 < d) (H : Table (cloudSize d) d)
    (base : ∀ tape, List (Alphabet tape)) (state : State ρ d) :
    TM2.step (program positive H) ⟨some (.inr .done), state, base⟩ =
      some ⟨none, state, base⟩ := by
  change some (TM2.stepAux (program positive H (.inr .done)) state base) = _
  rw [program_outer]
  rfl

end MaxCutGames.Foundations.Complexity.MachineExpanderTable

end OAI