Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.DirectedFeedback.Machines.MachineSubdivisionLoop

namespace OAI


namespace DFVSGames.Foundations.PCP.NameCompaction

open DFVSGames.Foundations.Target

def clauseNames {n : Nat} (c : Clause n) : List (Fin n) :=
  [(c)[0].variableIndex, (c)[1].variableIndex, (c)[2].variableIndex]

def sourceNames (F : Formula) : List (Fin F.«variables») :=
  F.clauses.flatMap clauseNames

def activeNames (F : Formula) : List (Fin F.«variables») :=
  (sourceNames F).eraseDups

theorem literal_mem_active (F : Formula) (c : Clause F.«variables»)
    (hc : c ∈ F.clauses) (i : Fin 3) :
    (c)[i].variableIndex ∈ activeNames F := by
  simp only [activeNames, List.mem_eraseDups]
  apply List.mem_flatMap.mpr
  refine ⟨c, hc, ?_⟩
  have hi : i = 0 ∨ i = 1 ∨ i = 2 := by omega
  rcases hi with rfl | rfl | rfl <;> simp [clauseNames]

def compactIndex (F : Formula) (v : Fin F.«variables»)
    (hv : v ∈ activeNames F) : Fin (activeNames F).length :=
  ⟨(activeNames F).idxOf v, List.idxOf_lt_length_iff.mpr hv⟩

def decodeName (F : Formula) (v : Fin (activeNames F).length) : Fin F.«variables» :=
  (activeNames F)[v.val]

def compactLiteral (F : Formula) (l : Literal F.«variables»)
    (hl : l.variableIndex ∈ activeNames F) : Literal (activeNames F).length :=
  ⟨compactIndex F l.variableIndex hl, l.positive⟩

def compactClause (F : Formula) (c : Clause F.«variables»)
    (hc : c ∈ F.clauses) : Clause (activeNames F).length :=
  #v[compactLiteral F (c)[0] (literal_mem_active F c hc 0),
    compactLiteral F (c)[1] (literal_mem_active F c hc 1),
    compactLiteral F (c)[2] (literal_mem_active F c hc 2)]

def compactClauses (F : Formula) : List (Clause (activeNames F).length) :=
  F.clauses.attach.map (fun c => compactClause F c.val c.property)

def compact (F : Formula) : Formula where
  «variables» := (activeNames F).length
  clauses := compactClauses F

def restrictAssignment (F : Formula) (A : Fin F.«variables» → Bool) :
    Fin (activeNames F).length → Bool := fun v => A (decodeName F v)

def extendAssignment (F : Formula) (B : Fin (activeNames F).length → Bool)
    (v : Fin F.«variables») : Bool :=
  if hv : v ∈ activeNames F then B (compactIndex F v hv) else false

@[simp] theorem decode_compactIndex (F : Formula) (v : Fin F.«variables»)
    (hv : v ∈ activeNames F) : decodeName F (compactIndex F v hv) = v := by
  exact List.getElem_idxOf (List.idxOf_lt_length_iff.mpr hv)

@[simp] theorem eval_compact_restrict (F : Formula) (c : Clause F.«variables»)
    (hc : c ∈ F.clauses) (A : Fin F.«variables» → Bool) :
    (compactClause F c hc).eval (restrictAssignment F A) = c.eval A := by
  simp [compactClause, compactLiteral, Clause.eval, Literal.eval,
    restrictAssignment, decode_compactIndex]

@[simp] theorem eval_compact_extend (F : Formula) (c : Clause F.«variables»)
    (hc : c ∈ F.clauses) (B : Fin (activeNames F).length → Bool) :
    (compactClause F c hc).eval B = c.eval (extendAssignment F B) := by
  have h₀ : (c)[0].variableIndex ∈ activeNames F := literal_mem_active F c hc 0
  have h₁ : (c)[1].variableIndex ∈ activeNames F := literal_mem_active F c hc 1
  have h₂ : (c)[2].variableIndex ∈ activeNames F := literal_mem_active F c hc 2
  simp [compactClause, compactLiteral, Clause.eval, Literal.eval, extendAssignment,
    h₀, h₁, h₂]
  rfl

theorem map_attach_val {α β : Type} (xs : List α) (f : α → β) :
    xs.attach.map (fun x => f x.val) = xs.map f := by
  exact List.attach_map_val

def evaluationList (F : Formula) (A : Fin F.«variables» → Bool) : List Bool :=
  F.clauses.map (fun c => c.eval A)

def satisfiedCount (F : Formula) (A : Fin F.«variables» → Bool) : Nat :=
  (evaluationList F A).count true

def failedCount (F : Formula) (A : Fin F.«variables» → Bool) : Nat :=
  (evaluationList F A).count false

theorem evaluationList_restrict (F : Formula) (A : Fin F.«variables» → Bool) :
    evaluationList (compact F) (restrictAssignment F A) = evaluationList F A := by
  simp only [evaluationList, compact, compactClauses, List.map_map]
  simpa only [Function.comp_def, eval_compact_restrict] using
    map_attach_val F.clauses (fun c => c.eval A)

theorem evaluationList_extend (F : Formula) (B : Fin (compact F).«variables» → Bool) :
    evaluationList (compact F) B = evaluationList F (extendAssignment F B) := by
  simp only [evaluationList, compact, compactClauses, List.map_map]
  change F.clauses.attach.map (fun c => (compactClause F c.val c.property).eval B) = _
  calc
    _ = F.clauses.attach.map (fun c => c.val.eval (extendAssignment F B)) := by
      apply List.map_congr_left
      intro c _
      exact eval_compact_extend F c.val c.property B
    _ = _ := map_attach_val F.clauses (fun c => c.eval (extendAssignment F B))

theorem satisfiedCount_restrict (F : Formula) (A : Fin F.«variables» → Bool) :
    satisfiedCount (compact F) (restrictAssignment F A) = satisfiedCount F A := by
  simp only [satisfiedCount, evaluationList_restrict]

theorem failedCount_restrict (F : Formula) (A : Fin F.«variables» → Bool) :
    failedCount (compact F) (restrictAssignment F A) = failedCount F A := by
  simp only [failedCount, evaluationList_restrict]

theorem satisfiedCount_extend (F : Formula) (B : Fin (compact F).«variables» → Bool) :
    satisfiedCount (compact F) B = satisfiedCount F (extendAssignment F B) := by
  simp only [satisfiedCount, evaluationList_extend]

theorem failedCount_extend (F : Formula) (B : Fin (compact F).«variables» → Bool) :
    failedCount (compact F) B = failedCount F (extendAssignment F B) := by
  simp only [failedCount, evaluationList_extend]

theorem clauseNames_flat_length {n : Nat} (cs : List (Clause n)) :
    (cs.flatMap clauseNames).length = 3 * cs.length := by
  induction cs with
  | nil => simp
  | cons c cs ih =>
    simp only [List.flatMap_cons, List.length_append, clauseNames,
      List.length_cons, List.length_nil, ih]
    omega

@[simp] theorem compact_clause_count (F : Formula) :
    (compact F).clauses.length = F.clauses.length := by
  simp [compact, compactClauses]

theorem length_eraseDups_le {α : Type} [BEq α] (xs : List α) :
    xs.eraseDups.length ≤ xs.length := by
  match xs with
  | [] => simp
  | a :: rest =>
    rw [List.eraseDups_cons]
    have filtered := List.length_filter_le (fun b => !(b == a)) rest
    have ih := length_eraseDups_le (rest.filter (fun b => !(b == a)))
    simp only [List.length_cons]
    omega
termination_by xs.length
decreasing_by
  exact Nat.lt_succ_of_le (List.length_filter_le _ _)

theorem compact_active_bound (F : Formula) :
    (compact F).«variables» ≤ 3 * F.clauses.length := by
  have h := length_eraseDups_le (sourceNames F)
  simpa only [sourceNames, clauseNames_flat_length, compact, activeNames] using h

theorem compact_completeness (F : Formula) (hs : F.Satisfiable) :
    (compact F).Satisfiable := by
  rcases hs with ⟨A, hA⟩
  refine ⟨restrictAssignment F A, ?_⟩
  intro d hd
  change d ∈ F.clauses.attach.map (fun c => compactClause F c.val c.property) at hd
  rcases List.mem_map.mp hd with ⟨c, _, rfl⟩
  exact (eval_compact_restrict F c.val c.property A).trans (hA c.val c.property)

theorem compact_reflects (F : Formula) (hs : (compact F).Satisfiable) :
    F.Satisfiable := by
  rcases hs with ⟨B, hB⟩
  refine ⟨extendAssignment F B, ?_⟩
  intro c hc
  have hm : compactClause F c hc ∈ (compact F).clauses := by
    change _ ∈ F.clauses.attach.map (fun d => compactClause F d.val d.property)
    exact List.mem_map.mpr ⟨⟨c, hc⟩, by simp, rfl⟩
  rw [← eval_compact_extend F c hc B]
  exact hB _ hm

theorem compact_satisfiable_iff (F : Formula) :
    (compact F).Satisfiable ↔ F.Satisfiable :=
  ⟨compact_reflects F, compact_completeness F⟩

theorem compact_preserves_gap (F : Formula) (a b : Nat)
    (gap : ∀ A, a * F.clauses.length ≤ b * failedCount F A)
    (B : Fin (compact F).«variables» → Bool) :
    a * (compact F).clauses.length ≤ b * failedCount (compact F) B := by
  rw [compact_clause_count, failedCount_extend]
  exact gap (extendAssignment F B)

theorem compact_encoding_bound (F : Formula) :
    (Complexity.formulaBits (compact F)).length ≤
      9 * F.clauses.length * F.clauses.length + 10 * F.clauses.length + 2 := by
  have enc := Complexity.formulaBits_length_le (compact F)
  rw [compact_clause_count] at enc
  have active := compact_active_bound F
  have h₁ := Nat.add_le_add_right active (F.clauses.length + 2)
  have h₂ := Nat.mul_le_mul_left F.clauses.length
    (Nat.mul_le_mul_left 3 (Nat.add_le_add_right active 2))
  have total := Nat.add_le_add h₁ h₂
  have polynomial : 3 * F.clauses.length + (F.clauses.length + 2) +
      F.clauses.length * (3 * (3 * F.clauses.length + 2)) =
      9 * F.clauses.length * F.clauses.length + 10 * F.clauses.length + 2 := by
    simp [Nat.mul_add, Nat.mul_comm, Nat.mul_left_comm]
    omega
  rw [polynomial] at total
  omega

theorem failureCount_evaluations (F : Formula) (A : Fin F.«variables» → Bool)
    (indices : List (Fin F.clauses.length)) :
    failureCount F A indices =
      (indices.map (fun i => (clauseAt F i).eval A)).count false := by
  induction indices with
  | nil => simp [failureCount]
  | cons i rest ih =>
    cases h : (clauseAt F i).eval A <;>
      simp [failureCount, clauseFailure, h, ih, Nat.add_comm]

theorem evaluations_allIndices (F : Formula) (A : Fin F.«variables» → Bool) :
    ((allIndices F).map (fun i => (clauseAt F i).eval A)) = evaluationList F A := by
  calc
    _ = (List.ofFn (fun i : Fin F.clauses.length => F.clauses[i.val])).map
        (fun c => c.eval A) := by
      simp only [allIndices, List.finRange, clauseAt, List.map_ofFn, Function.comp_def]
    _ = _ := by rw [List.ofFn_getElem]; rfl

theorem verifier_failureCount (F : Formula) (A : Fin F.«variables» → Bool) :
    failureCount F A (allIndices F) = failedCount F A := by
  rw [failureCount_evaluations, evaluations_allIndices]
  rfl

theorem compact_verifier_soundness (F : Formula) (a b : Nat)
    (gap : ∀ A, a * F.clauses.length ≤ b * failedCount F A)
    (alice : AliceStrategy (compact F)) (bob : BobStrategy (compact F)) :
    b * ((allEvents (compact F)).filter (accepts (compact F) alice bob)).length +
        a * F.clauses.length ≤ b * (3 * F.clauses.length) := by
  have sourceGap : ∀ A : BobStrategy (compact F),
      a * (compact F).clauses.length ≤
        b * failureCount (compact F) A (allIndices (compact F)) := by
    intro A
    rw [verifier_failureCount]
    exact compact_preserves_gap F a b gap A
  have h := verifier_event_soundness (compact F) a b sourceGap alice bob
  simpa only [length_allEvents, compact_clause_count] using h

end DFVSGames.Foundations.PCP.NameCompaction


namespace DFVSGames.BinaryFormula

open DFVSGames.Foundations

def sourceNames (F : Formula) : List Nat :=
  F.clauses.flatMap clauseNames

def activeNames (F : Formula) : List Nat :=
  (sourceNames F).eraseDups

theorem literal_mem_active (F : Formula) (c : Clause)
    (hc : c ∈ F.clauses) (i : Fin 3) : (c)[i].name ∈ activeNames F := by
  simp only [activeNames, List.mem_eraseDups]
  apply List.mem_flatMap.mpr
  refine ⟨c, hc, ?_⟩
  have hi : i = 0 ∨ i = 1 ∨ i = 2 := by omega
  rcases hi with rfl | rfl | rfl <;> simp [clauseNames]

def compactIndex (F : Formula) (v : Nat)
    (hv : v ∈ activeNames F) : Fin (activeNames F).length :=
  ⟨(activeNames F).idxOf v, List.idxOf_lt_length_iff.mpr hv⟩

def decodeName (F : Formula) (v : Fin (activeNames F).length) : Nat :=
  (activeNames F)[v.val]

def compactLiteral (F : Formula) (l : Literal)
    (hl : l.name ∈ activeNames F) : Target.Literal (activeNames F).length :=
  ⟨compactIndex F l.name hl, l.positive⟩

def compactClause (F : Formula) (c : Clause)
    (hc : c ∈ F.clauses) : Target.Clause (activeNames F).length :=
  #v[compactLiteral F (c)[0] (literal_mem_active F c hc 0),
    compactLiteral F (c)[1] (literal_mem_active F c hc 1),
    compactLiteral F (c)[2] (literal_mem_active F c hc 2)]

def compactClauses (F : Formula) : List (Target.Clause (activeNames F).length) :=
  F.clauses.attach.map (fun c => compactClause F c.val c.property)

def dense (F : Formula) : Target.Formula where
  «variables» := (activeNames F).length
  clauses := compactClauses F

def restrictAssignment (F : Formula) (A : Nat → Bool) :
    Fin (activeNames F).length → Bool := fun v => A (decodeName F v)

def extendAssignment (F : Formula) (B : Fin (activeNames F).length → Bool)
    (v : Nat) : Bool :=
  if hv : v ∈ activeNames F then B (compactIndex F v hv) else false

@[simp] theorem decode_compactIndex (F : Formula) (v : Nat)
    (hv : v ∈ activeNames F) : decodeName F (compactIndex F v hv) = v := by
  exact List.getElem_idxOf (List.idxOf_lt_length_iff.mpr hv)

theorem compactIndex_injective (F : Formula) (u v : Nat)
    (hu : u ∈ activeNames F) (hv : v ∈ activeNames F)
    (h : compactIndex F u hu = compactIndex F v hv) : u = v := by
  have decoded := congrArg (decodeName F) h
  simpa only [decode_compactIndex] using decoded

@[simp] theorem compactLiteral_positive (F : Formula) (l : Literal)
    (hl : l.name ∈ activeNames F) : (compactLiteral F l hl).positive = l.positive := rfl

@[simp] theorem eval_compact_restrict (F : Formula) (c : Clause)
    (hc : c ∈ F.clauses) (A : Nat → Bool) :
    (compactClause F c hc).eval (restrictAssignment F A) = c.eval A := by
  simp [compactClause, compactLiteral, Target.Clause.eval, Target.Literal.eval,
    Clause.eval, Literal.eval, restrictAssignment, decode_compactIndex]

@[simp] theorem eval_compact_extend (F : Formula) (c : Clause)
    (hc : c ∈ F.clauses) (B : Fin (activeNames F).length → Bool) :
    (compactClause F c hc).eval B = c.eval (extendAssignment F B) := by
  have h₀ : (c)[0].name ∈ activeNames F := literal_mem_active F c hc 0
  have h₁ : (c)[1].name ∈ activeNames F := literal_mem_active F c hc 1
  have h₂ : (c)[2].name ∈ activeNames F := literal_mem_active F c hc 2
  simp [compactClause, compactLiteral, Target.Clause.eval, Target.Literal.eval,
    Clause.eval, Literal.eval, extendAssignment, h₀, h₁, h₂] ; rfl

def evaluationList (F : Formula) (A : Nat → Bool) : List Bool :=
  F.clauses.map (fun c => c.eval A)

theorem evaluationList_restrict (F : Formula) (A : Nat → Bool) :
    PCP.NameCompaction.evaluationList (dense F) (restrictAssignment F A) =
      evaluationList F A := by
  simp only [PCP.NameCompaction.evaluationList, evaluationList, dense, compactClauses,
    List.map_map]
  simpa only [Function.comp_def, eval_compact_restrict] using
    PCP.NameCompaction.map_attach_val F.clauses (fun c => c.eval A)

theorem evaluationList_extend (F : Formula) (B : Fin (dense F).«variables» → Bool) :
    PCP.NameCompaction.evaluationList (dense F) B =
      evaluationList F (extendAssignment F B) := by
  simp only [PCP.NameCompaction.evaluationList, evaluationList, dense, compactClauses,
    List.map_map]
  change F.clauses.attach.map (fun c => (compactClause F c.val c.property).eval B) = _
  calc
    _ = F.clauses.attach.map (fun c => c.val.eval (extendAssignment F B)) := by
      apply List.map_congr_left
      intro c _
      exact eval_compact_extend F c.val c.property B
    _ = _ := PCP.NameCompaction.map_attach_val F.clauses
      (fun c => c.eval (extendAssignment F B))

def failedCount (F : Formula) (A : Nat → Bool) : Nat :=
  (evaluationList F A).count false

theorem failedCount_restrict (F : Formula) (A : Nat → Bool) :
    PCP.NameCompaction.failedCount (dense F) (restrictAssignment F A) = failedCount F A := by
  simp only [PCP.NameCompaction.failedCount, failedCount, evaluationList_restrict]

theorem failedCount_extend (F : Formula) (B : Fin (dense F).«variables» → Bool) :
    PCP.NameCompaction.failedCount (dense F) B = failedCount F (extendAssignment F B) := by
  simp only [PCP.NameCompaction.failedCount, failedCount, evaluationList_extend]

theorem clauseNames_flat_length (cs : List Clause) :
    (cs.flatMap clauseNames).length = 3 * cs.length := by
  induction cs with
  | nil => simp
  | cons c cs ih =>
    simp only [List.flatMap_cons, List.length_append, clauseNames,
      List.length_cons, List.length_nil, ih]
    omega

@[simp] theorem dense_clause_count (F : Formula) :
    (dense F).clauses.length = F.clauses.length := by
  simp [dense, compactClauses]

theorem dense_variable_bound (F : Formula) :
    (dense F).«variables» ≤ 3 * F.clauses.length := by
  have h := PCP.NameCompaction.length_eraseDups_le (sourceNames F)
  simpa only [sourceNames, clauseNames_flat_length, dense, activeNames] using h

theorem dense_completeness (F : Formula) (hs : F.Satisfiable) :
    (dense F).Satisfiable := by
  rcases hs with ⟨A, hA⟩
  refine ⟨restrictAssignment F A, ?_⟩
  intro d hd
  change d ∈ F.clauses.attach.map (fun c => compactClause F c.val c.property) at hd
  rcases List.mem_map.mp hd with ⟨c, _, rfl⟩
  exact (eval_compact_restrict F c.val c.property A).trans (hA c.val c.property)

theorem dense_reflects (F : Formula) (hs : (dense F).Satisfiable) : F.Satisfiable := by
  rcases hs with ⟨B, hB⟩
  refine ⟨extendAssignment F B, ?_⟩
  intro c hc
  have hm : compactClause F c hc ∈ (dense F).clauses := by
    change _ ∈ F.clauses.attach.map (fun d => compactClause F d.val d.property)
    exact List.mem_map.mpr ⟨⟨c, hc⟩, by simp, rfl⟩
  rw [← eval_compact_extend F c hc B]
  exact hB _ hm

theorem dense_satisfiable_iff (F : Formula) :
    (dense F).Satisfiable ↔ F.Satisfiable :=
  ⟨dense_reflects F, dense_completeness F⟩

theorem dense_preserves_gap (F : Formula) (a b : Nat)
    (gap : ∀ A, a * F.clauses.length ≤ b * failedCount F A)
    (B : Fin (dense F).«variables» → Bool) :
    a * (dense F).clauses.length ≤ b * PCP.NameCompaction.failedCount (dense F) B := by
  rw [dense_clause_count, failedCount_extend]
  exact gap (extendAssignment F B)

theorem dense_encoding_bound (F : Formula) :
    (Complexity.formulaBits (dense F)).length ≤
      9 * F.clauses.length * F.clauses.length + 10 * F.clauses.length + 2 := by
  have enc := Complexity.formulaBits_length_le (dense F)
  rw [dense_clause_count] at enc
  have active := dense_variable_bound F
  have h₁ := Nat.add_le_add_right active (F.clauses.length + 2)
  have h₂ := Nat.mul_le_mul_left F.clauses.length
    (Nat.mul_le_mul_left 3 (Nat.add_le_add_right active 2))
  have total := Nat.add_le_add h₁ h₂
  have polynomial : 3 * F.clauses.length + (F.clauses.length + 2) +
      F.clauses.length * (3 * (3 * F.clauses.length + 2)) =
      9 * F.clauses.length * F.clauses.length + 10 * F.clauses.length + 2 := by
    ring
  rw [polynomial] at total
  omega

end DFVSGames.BinaryFormula


namespace DFVSGames.BinaryEncoding

open BinaryFormula

theorem dense_encoding_bound (formula : Formula) :
    (DFVSGames.Foundations.Complexity.formulaBits (dense formula)).length ≤
      9 * (formulaBits formula).length * (formulaBits formula).length +
      10 * (formulaBits formula).length + 2 := by
  have h := BinaryFormula.dense_encoding_bound formula
  have hm : formula.clauses.length ≤ (formulaBits formula).length :=
    (clauses_length_lt_bits formula.clauses).le
  have hsq := Nat.mul_self_le_mul_self hm
  nlinarith

end DFVSGames.BinaryEncoding


namespace DFVSGames.Foundations.Complexity.MachineCompare

open Turing
open DFVSGames.Reduction.MachineTransfer

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

abbrev Alphabet (K : Type) : K → Type := fun _ => Bool
abbrev State (σ : Type) := σ × Bool × Option Bool × Option Bool

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

def finish (exit : Option Λ) : TM2.Stmt (Alphabet K) Λ (State σ) :=
  .load (fun state => (state.1, false, none, none)) (exitAt exit)

def loop (left right : K) (loopLabel : Λ) (equalExit differentExit : Option Λ) :
    TM2.Stmt (Alphabet K) Λ (State σ) :=
  .pop left (fun state head => (state.1, state.2.1, head, state.2.2.2))
    (.pop right (fun state head => (state.1, state.2.1, state.2.2.1, head))
      (.branch (fun state => state.2.2.1.isNone && state.2.2.2.isNone)
        (.branch (fun state => state.2.1) (finish differentExit) (finish equalExit))
        (.load (fun state =>
          (state.1, state.2.1 || decide (state.2.2.1 ≠ state.2.2.2), none, none))
          (.goto fun _ => loopLabel))))

def mismatch (flag : Bool) (leftWord rightWord : List Bool) : Bool :=
  flag || decide (leftWord ≠ rightWord)

@[simp] theorem mismatch_nil_nil (flag : Bool) : mismatch flag [] [] = flag := by
  simp [mismatch]

@[simp] theorem mismatch_nil_cons (flag head : Bool) (tail : List Bool) :
    mismatch flag [] (head :: tail) = true := by simp [mismatch]

@[simp] theorem mismatch_cons_nil (flag head : Bool) (tail : List Bool) :
    mismatch flag (head :: tail) [] = true := by simp [mismatch]

@[simp] theorem mismatch_true (leftWord rightWord : List Bool) :
    mismatch true leftWord rightWord = true := by simp [mismatch]

theorem mismatch_cons (flag leftHead rightHead : Bool) (leftTail rightTail : List Bool) :
    mismatch (flag || decide (leftHead ≠ rightHead)) leftTail rightTail =
      mismatch flag (leftHead :: leftTail) (rightHead :: rightTail) := by
  cases flag <;> cases leftHead <;> cases rightHead <;> simp [mismatch]

@[simp] theorem stepAux_exitAt (exit : Option Λ) (state : State σ)
    (tapes : K → List Bool) :
    TM2.stepAux (exitAt exit) state tapes = ⟨exit, state, tapes⟩ := by
  cases exit <;> rfl

@[simp] theorem stepAux_finish (exit : Option Λ) (state : State σ)
    (tapes : K → List Bool) :
    TM2.stepAux (finish exit) state tapes =
      ⟨exit, (state.1, false, none, none), tapes⟩ := by
  simp [finish, TM2.stepAux]

private theorem update_left_inline_MachineCompare (left right : K) (distinct : left ≠ right)
    (base : K → List Bool) (leftWord rightWord replacement : List Bool) :
    Function.update (tapesAt left right base leftWord rightWord) left replacement =
      tapesAt left right base replacement rightWord := by
  funext k
  by_cases hl : k = left
  · subst k
    simp [tapesAt, distinct]
  · by_cases hr : k = right
    · subst k
      simp [tapesAt, Ne.symm distinct]
    · simp [tapesAt, hl, hr]

private theorem update_right_inline_MachineCompare (left right : K) (base : K → List Bool)
    (leftWord rightWord replacement : List Bool) :
    Function.update (tapesAt left right base leftWord rightWord) right replacement =
      tapesAt left right base leftWord replacement := by
  simp [tapesAt]

theorem step_empty (left right : K) (distinct : left ≠ right)
    (loopLabel : Λ) (equalExit differentExit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet K) Λ (State σ))
    (atLoop : program loopLabel = loop left right loopLabel equalExit differentExit)
    (base : K → List Bool) (ambient : σ) (flag : Bool)
    (leftRegister rightRegister : Option Bool) :
    TM2.step program
      ⟨some loopLabel, (ambient, flag, leftRegister, rightRegister),
        tapesAt left right base [] []⟩ =
      some ⟨if flag then differentExit else equalExit, (ambient, false, none, none),
        tapesAt left right base [] []⟩ := by
  change some (TM2.stepAux (program loopLabel) (ambient, flag, leftRegister, rightRegister)
    (tapesAt left right base [] [])) = _
  rw [atLoop]
  cases flag <;>
    simp [loop, TM2.stepAux, distinct, update_left_inline_MachineCompare, update_right_inline_MachineCompare]

theorem step_nil_cons (left right : K) (distinct : left ≠ right)
    (loopLabel : Λ) (equalExit differentExit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet K) Λ (State σ))
    (atLoop : program loopLabel = loop left right loopLabel equalExit differentExit)
    (base : K → List Bool) (head : Bool) (tail : List Bool) (ambient : σ) (flag : Bool)
    (leftRegister rightRegister : Option Bool) :
    TM2.step program
      ⟨some loopLabel, (ambient, flag, leftRegister, rightRegister),
        tapesAt left right base [] (head :: tail)⟩ =
      some ⟨some loopLabel, (ambient, true, none, none), tapesAt left right base [] tail⟩ := by
  change some (TM2.stepAux (program loopLabel) (ambient, flag, leftRegister, rightRegister)
    (tapesAt left right base [] (head :: tail))) = _
  rw [atLoop]
  simp [loop, TM2.stepAux, distinct, update_left_inline_MachineCompare, update_right_inline_MachineCompare]

theorem step_cons_nil (left right : K) (distinct : left ≠ right)
    (loopLabel : Λ) (equalExit differentExit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet K) Λ (State σ))
    (atLoop : program loopLabel = loop left right loopLabel equalExit differentExit)
    (base : K → List Bool) (head : Bool) (tail : List Bool) (ambient : σ) (flag : Bool)
    (leftRegister rightRegister : Option Bool) :
    TM2.step program
      ⟨some loopLabel, (ambient, flag, leftRegister, rightRegister),
        tapesAt left right base (head :: tail) []⟩ =
      some ⟨some loopLabel, (ambient, true, none, none), tapesAt left right base tail []⟩ := by
  change some (TM2.stepAux (program loopLabel) (ambient, flag, leftRegister, rightRegister)
    (tapesAt left right base (head :: tail) [])) = _
  rw [atLoop]
  simp [loop, TM2.stepAux, distinct, update_left_inline_MachineCompare, update_right_inline_MachineCompare]

theorem step_cons_cons (left right : K) (distinct : left ≠ right)
    (loopLabel : Λ) (equalExit differentExit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet K) Λ (State σ))
    (atLoop : program loopLabel = loop left right loopLabel equalExit differentExit)
    (base : K → List Bool) (leftHead rightHead : Bool) (leftTail rightTail : List Bool)
    (ambient : σ) (flag : Bool) (leftRegister rightRegister : Option Bool) :
    TM2.step program
      ⟨some loopLabel, (ambient, flag, leftRegister, rightRegister),
        tapesAt left right base (leftHead :: leftTail) (rightHead :: rightTail)⟩ =
      some ⟨some loopLabel,
        (ambient, flag || decide (leftHead ≠ rightHead), none, none),
        tapesAt left right base leftTail rightTail⟩ := by
  change some (TM2.stepAux (program loopLabel) (ambient, flag, leftRegister, rightRegister)
    (tapesAt left right base (leftHead :: leftTail) (rightHead :: rightTail))) = _
  rw [atLoop]
  simp [loop, TM2.stepAux, distinct, update_left_inline_MachineCompare, update_right_inline_MachineCompare]

private theorem trace_left_empty_inline_MachineCompare (left right : K) (distinct : left ≠ right)
    (loopLabel : Λ) (equalExit differentExit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet K) Λ (State σ))
    (atLoop : program loopLabel = loop left right loopLabel equalExit differentExit)
    (base : K → List Bool) (rightWord : List Bool) (ambient : σ) (flag : Bool)
    (leftRegister rightRegister : Option Bool) :
    (MachineComposition.advance (TM2.step program))^[rightWord.length + 1]
      (some ⟨some loopLabel, (ambient, flag, leftRegister, rightRegister),
        tapesAt left right base [] rightWord⟩) =
      some ⟨if mismatch flag [] rightWord then differentExit else equalExit,
        (ambient, false, none, none), tapesAt left right base [] []⟩ := by
  induction rightWord generalizing flag leftRegister rightRegister with
  | nil =>
    simpa only [List.length_nil, Nat.zero_add, Function.iterate_one,
      MachineComposition.advance_some, mismatch_nil_nil] using
      step_empty left right distinct loopLabel equalExit differentExit program atLoop base
        ambient flag leftRegister rightRegister
  | cons head tail ih =>
    rw [List.length_cons, Function.iterate_succ_apply]
    change (MachineComposition.advance (TM2.step program))^[tail.length + 1]
      (TM2.step program ⟨some loopLabel, (ambient, flag, leftRegister, rightRegister),
        tapesAt left right base [] (head :: tail)⟩) = _
    rw [step_nil_cons left right distinct loopLabel equalExit differentExit program atLoop]
    simpa only [mismatch_true, mismatch_nil_cons] using ih true none none

theorem compareTrace_flag (left right : K) (distinct : left ≠ right)
    (loopLabel : Λ) (equalExit differentExit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet K) Λ (State σ))
    (atLoop : program loopLabel = loop left right loopLabel equalExit differentExit)
    (base : K → List Bool) (leftWord rightWord : List Bool) (ambient : σ) (flag : Bool)
    (leftRegister rightRegister : Option Bool) :
    (MachineComposition.advance (TM2.step program))^[max leftWord.length rightWord.length + 1]
      (some ⟨some loopLabel, (ambient, flag, leftRegister, rightRegister),
        tapesAt left right base leftWord rightWord⟩) =
      some ⟨if mismatch flag leftWord rightWord then differentExit else equalExit,
        (ambient, false, none, none), tapesAt left right base [] []⟩ := by
  induction leftWord generalizing rightWord flag leftRegister rightRegister with
  | nil =>
    simpa only [List.length_nil, Nat.zero_max] using
      trace_left_empty_inline_MachineCompare left right distinct loopLabel equalExit differentExit program atLoop
        base rightWord ambient flag leftRegister rightRegister
  | cons head tail ih =>
    cases rightWord with
    | nil =>
      rw [List.length_cons, List.length_nil, Nat.max_zero, Function.iterate_succ_apply]
      change (MachineComposition.advance (TM2.step program))^[tail.length + 1]
        (TM2.step program ⟨some loopLabel, (ambient, flag, leftRegister, rightRegister),
          tapesAt left right base (head :: tail) []⟩) = _
      rw [step_cons_nil left right distinct loopLabel equalExit differentExit program atLoop]
      simpa only [List.length_nil, Nat.max_zero, mismatch_true, mismatch_cons_nil] using
        ih [] true none none
    | cons rightHead rightTail =>
      rw [List.length_cons, List.length_cons, Nat.succ_max_succ, Function.iterate_succ_apply]
      change (MachineComposition.advance (TM2.step program))^[max tail.length rightTail.length + 1]
        (TM2.step program ⟨some loopLabel, (ambient, flag, leftRegister, rightRegister),
          tapesAt left right base (head :: tail) (rightHead :: rightTail)⟩) = _
      rw [step_cons_cons left right distinct loopLabel equalExit differentExit program atLoop]
      simpa only [mismatch_cons] using
        ih rightTail (flag || decide (head ≠ rightHead)) none none

theorem compareTrace (left right : K) (distinct : left ≠ right)
    (loopLabel : Λ) (equalExit differentExit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet K) Λ (State σ))
    (atLoop : program loopLabel = loop left right loopLabel equalExit differentExit)
    (base : K → List Bool) (leftWord rightWord : List Bool) (ambient : σ)
    (leftRegister rightRegister : Option Bool) :
    (MachineComposition.advance (TM2.step program))^[max leftWord.length rightWord.length + 1]
      (some ⟨some loopLabel, (ambient, false, leftRegister, rightRegister),
        tapesAt left right base leftWord rightWord⟩) =
      some ⟨if leftWord = rightWord then equalExit else differentExit,
        (ambient, false, none, none), tapesAt left right base [] []⟩ := by
  have h := compareTrace_flag left right distinct loopLabel equalExit differentExit program
    atLoop base leftWord rightWord ambient false leftRegister rightRegister
  by_cases heq : leftWord = rightWord <;> simpa [mismatch, heq] using h

theorem compareTrace_fromTapes (left right : K) (distinct : left ≠ right)
    (loopLabel : Λ) (equalExit differentExit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet K) Λ (State σ))
    (atLoop : program loopLabel = loop left right loopLabel equalExit differentExit)
    (base : K → List Bool) (ambient : σ) (leftRegister rightRegister : Option Bool) :
    (MachineComposition.advance (TM2.step program))^[max (base left).length (base right).length + 1]
      (some ⟨some loopLabel, (ambient, false, leftRegister, rightRegister), base⟩) =
      some ⟨if base left = base right then equalExit else differentExit,
        (ambient, false, none, none), tapesAt left right base [] []⟩ := by
  simpa only [tapesAt_self] using
    compareTrace left right distinct loopLabel equalExit differentExit program atLoop base
      (base left) (base right) ambient leftRegister rightRegister

theorem drained_other (left right k : K) (notLeft : k ≠ left) (notRight : k ≠ right)
    (base : K → List Bool) : tapesAt left right base [] [] k = base k :=
  tapesAt_other left right k notLeft notRight base [] []

def compareInTime (left right : K) (distinct : left ≠ right)
    (loopLabel : Λ) (equalExit differentExit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet K) Λ (State σ))
    (atLoop : program loopLabel = loop left right loopLabel equalExit differentExit)
    (base : K → List Bool) (ambient : σ) (leftRegister rightRegister : Option Bool) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some loopLabel, (ambient, false, leftRegister, rightRegister), base⟩
      (some ⟨if base left = base right then equalExit else differentExit,
        (ambient, false, none, none), tapesAt left right base [] []⟩)
      (max (base left).length (base right).length + 1) where
  steps := max (base left).length (base right).length + 1
  evals_in_steps := compareTrace_fromTapes left right distinct loopLabel equalExit differentExit
    program atLoop base ambient leftRegister rightRegister
  steps_le_m := Nat.le_refl _

@[simp] theorem compareInTime_steps (left right : K) (distinct : left ≠ right)
    (loopLabel : Λ) (equalExit differentExit : Option Λ)
    (program : Λ → TM2.Stmt (Alphabet K) Λ (State σ))
    (atLoop : program loopLabel = loop left right loopLabel equalExit differentExit)
    (base : K → List Bool) (ambient : σ) (leftRegister rightRegister : Option Bool) :
    (compareInTime left right distinct loopLabel equalExit differentExit program atLoop base
      ambient leftRegister rightRegister).steps =
        max (base left).length (base right).length + 1 := rfl

end DFVSGames.Foundations.Complexity.MachineCompare

end OAI