lean/OAI/Computability/UniqueGames/Machines/MachineBinaryNameSearch.lean

import Mathlib.Tactic.Linarith
import OAI.Computability.UniqueGames.Machines.MachineBinaryNameCompare

namespace OAI

section

/-!
First-occurrence search over signed, framed binary names. The stream has no
clause markers. Every iteration discards one sign, scans and compares the
actual candidate payload, and increments a Boolean unary counter only after a
mismatch. The preserved key is a reversed payload, never a unary name.

The six explicitly distinct tapes are: stream, preserved key, candidate,
copied key, copy scratch, and unary occurrence counter. Successful search
consumes precisely the prefix through the first matching name, preserves the
unread suffix and all ambient tapes, and returns the three work tapes empty.
The time theorem follows from literal instruction equations and checked
comparison executions, with no supplied loop-body trace premise.
-/

namespace UniqueGamesTheorem.BinaryNameSearch

open Turing
open UniqueGamesTheorem.Foundations.Complexity
open MachineComposition
open UniqueGamesTheorem.Reduction.MachineTransfer

abbrev Token := Bool × List Bool

def tokenBits (token : Token) : List Bool :=
  token.1 :: BinaryNameMachine.frame token.2

def stream : List Token → List Bool
  | [] => []
  | token :: tokens => tokenBits token ++ stream tokens

def payloads (tokens : List Token) : List (List Bool) := tokens.map Prod.snd

def afterMatch : List Token → List Bool → List Token
  | [], _ => []
  | token :: tokens, key => if token.2 = key then tokens else afterMatch tokens key

def steps : List Token → List Bool → Nat
  | [], _ => 0
  | token :: tokens, key =>
      1 + BinaryNameCompare.steps token.2.length key.length +
        if token.2 = key then 0 else 1 + steps tokens key

def payloadSize (tokens : List Token) : Nat := (payloads tokens |>.map List.length).sum

@[simp] theorem stream_nil : stream [] = [] := rfl

@[simp] theorem stream_cons (token : Token) (tokens : List Token) :
    stream (token :: tokens) = tokenBits token ++ stream tokens := rfl

@[simp] theorem payloads_nil : payloads [] = [] := rfl

@[simp] theorem payloads_cons (token : Token) (tokens : List Token) :
    payloads (token :: tokens) = token.2 :: payloads tokens := rfl

@[simp] theorem payloadSize_nil : payloadSize [] = 0 := rfl

@[simp] theorem payloadSize_cons (token : Token) (tokens : List Token) :
    payloadSize (token :: tokens) = token.2.length + payloadSize tokens := rfl

@[simp] theorem stream_length (tokens : List Token) :
    (stream tokens).length = 2 * payloadSize tokens + 2 * tokens.length := by
  induction tokens with
  | nil => rfl
  | cons token tokens ih =>
      simp only [stream_cons, List.length_append, tokenBits, List.length_cons,
        BinaryNameMachine.frame_length, ih, payloadSize_cons]
      omega

/-- Work is linear in the scanned payloads plus one key-copy cost per token. -/
theorem steps_le (tokens : List Token) (key : List Bool) :
    steps tokens key ≤ 2 * payloadSize tokens + (3 * key.length + 6) * tokens.length := by
  induction tokens with
  | nil => simp [steps]
  | cons token tokens ih =>
      have h := BinaryNameCompare.steps_le token.2.length key.length
      simp only [steps, payloadSize_cons, List.length_cons]
      split <;> nlinarith

/-- This bound uses lengths of actual encoded words, not decoded name values. -/
theorem steps_le_stream (tokens : List Token) (key : List Bool) :
    steps tokens key ≤ (3 * key.length + 7) * (stream tokens).length := by
  have h := steps_le tokens key
  rw [stream_length]
  nlinarith

section Program

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

abbrev Alphabet (_ : K) := Bool
abbrev State (A : Type) := BinaryNameCompare.State A

abbrev clean (ambient : A) : State A := BinaryNameCompare.clean ambient

inductive Label
  | sign
  | comparison (label : BinaryNameCompare.Label)
  | increment
  deriving DecidableEq, Fintype

def compareTapes (tape : Fin 6 → K) : Fin 5 → K := fun i => tape i.castSucc

omit [DecidableEq K] in
theorem compareTapes_injective (tape : Fin 6 → K) (distinct : Function.Injective tape) :
    Function.Injective (compareTapes tape) := by
  intro i j h
  apply Fin.ext
  exact congrArg (fun i : Fin 6 => i.val) (distinct h)

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

def finish (exit : Option Λ) : TM2.Stmt (Alphabet (K := K)) Λ (State A) :=
  .load (fun state => clean state.1.1) (exitAt exit)

def instruction (tape : Fin 6 → K) (labels : Label → Λ)
    (foundExit missingExit malformedExit : Option Λ) :
    Label → TM2.Stmt (Alphabet (K := K)) Λ (State A)
  | .sign =>
      .pop (tape 0) (fun state head => (state.1, head))
        (.branch (fun state => state.2.isSome)
          (finish (some (labels (.comparison .scan)))) (finish missingExit))
  | .comparison l => BinaryNameCompare.instruction (compareTapes tape)
      (fun l => labels (.comparison l)) foundExit (some (labels .increment)) malformedExit l
  | .increment => .push (tape 5) (fun _ => true) (.goto fun _ => labels .sign)

@[simp] theorem stepAux_finish (exit : Option Λ) (state : State A) (base : K → List Bool) :
    TM2.stepAux (finish exit) state base = ⟨exit, clean state.1.1, base⟩ := by
  cases exit <;> rfl

private theorem update_source (source counter : K) (distinct : source ≠ counter)
    (base : K → List Bool) (input count replacement : List Bool) :
    Function.update (tapesAt source counter base input count) source replacement =
      tapesAt source counter base replacement count := by
  funext k
  by_cases hs : k = source
  · subst k; simp [tapesAt, distinct]
  · by_cases hc : k = counter
    · subst k; simp [tapesAt, Ne.symm distinct]
    · simp [tapesAt, hs, hc]

private theorem update_counter (source counter : K)
    (base : K → List Bool) (input count replacement : List Bool) :
    Function.update (tapesAt source counter base input count) counter replacement =
      tapesAt source counter base input replacement := by
  simp [tapesAt]

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

variable (tape : Fin 6 → K) (distinct : Function.Injective tape)
variable (labels : Label → Λ) (foundExit missingExit malformedExit : Option Λ)
variable (program : Λ → TM2.Stmt (Alphabet (K := K)) Λ (State A))
variable (atLabels : ∀ l, program (labels l) =
  instruction tape labels foundExit missingExit malformedExit l)
variable (base : K → List Bool) (ambient : A)

include distinct atLabels

theorem signStep (sign : Bool) (input counter : List Bool) :
    TM2.step program
      ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base (sign :: input) counter⟩ =
      some ⟨some (labels (.comparison .scan)), clean ambient,
        tapesAt (tape 0) (tape 5) base input counter⟩ := by
  have hd : tape 0 ≠ tape 5 := fun h => (by decide : (0 : Fin 6) ≠ 5) (distinct h)
  change some (TM2.stepAux (program (labels .sign)) _ _) = _
  rw [atLabels .sign]
  simp [instruction, TM2.stepAux, clean, BinaryNameCompare.clean, hd, update_source]

theorem emptyStep (counter : List Bool) :
    TM2.step program
      ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base [] counter⟩ =
      some ⟨missingExit, clean ambient, tapesAt (tape 0) (tape 5) base [] counter⟩ := by
  have hd : tape 0 ≠ tape 5 := fun h => (by decide : (0 : Fin 6) ≠ 5) (distinct h)
  change some (TM2.stepAux (program (labels .sign)) _ _) = _
  rw [atLabels .sign]
  simp [instruction, TM2.stepAux, clean, BinaryNameCompare.clean, hd, update_source]

omit distinct in
theorem incrementStep (input counter : List Bool) :
    TM2.step program
      ⟨some (labels .increment), clean ambient,
        tapesAt (tape 0) (tape 5) base input counter⟩ =
      some ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base input (true :: counter)⟩ := by
  change some (TM2.stepAux (program (labels .increment)) _ _) = _
  rw [atLabels .increment]
  simp [instruction, TM2.stepAux, update_counter]

/-- One sign is consumed, then the actual shared scan/copy/compare routine runs. -/
theorem tokenTrace (token : Token) (key suffix counter : List Bool)
    (canonical : BinaryNameMachine.canonical token.2 = true)
    (keyWord : base (tape 1) = key.reverse)
    (candidateEmpty : base (tape 2) = []) (copyEmpty : base (tape 3) = [])
    (scratchEmpty : base (tape 4) = []) :
    (advance (TM2.step program))^[1 + BinaryNameCompare.steps token.2.length key.length]
      (some ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base (tokenBits token ++ suffix) counter⟩) =
      some ⟨if token.2 = key then foundExit else some (labels .increment), clean ambient,
        tapesAt (tape 0) (tape 5) base suffix counter⟩ := by
  have hd (i j : Fin 6) (hne : i ≠ j) : tape i ≠ tape j := fun h => hne (distinct h)
  let initial := tapesAt (tape 0) (tape 5) base
    (BinaryNameMachine.frame token.2 ++ suffix) counter
  have first : (advance (TM2.step program))^[1]
      (some ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base (tokenBits token ++ suffix) counter⟩) =
      some ⟨some (labels (.comparison .scan)), clean ambient, initial⟩ := by
    simpa only [Function.iterate_one, advance_some, tokenBits, List.cons_append] using
      signStep tape distinct labels foundExit missingExit malformedExit program atLabels
        base ambient token.1 (BinaryNameMachine.frame token.2 ++ suffix) counter
  have second := BinaryNameCompare.compareTrace (compareTapes tape)
    (compareTapes_injective tape distinct) (fun l => labels (.comparison l))
    foundExit (some (labels .increment)) malformedExit program
    (fun l => atLabels (.comparison l)) initial token.2 suffix key.reverse canonical
    (by simp [initial, compareTapes, hd 0 5 (by decide)])
    (by simpa [initial, compareTapes, tapesAt,
      hd 1 0 (by decide), hd 1 5 (by decide)] using keyWord)
    (by simpa [initial, compareTapes, tapesAt, hd 2 0 (by decide), hd 2 5 (by decide)]
      using candidateEmpty)
    (by simpa [initial, compareTapes, tapesAt,
      hd 3 0 (by decide), hd 3 5 (by decide)] using copyEmpty)
    (by simpa [initial, compareTapes, tapesAt,
      hd 4 0 (by decide), hd 4 5 (by decide)] using scratchEmpty)
    ambient
  have heq : token.2.reverse = key.reverse ↔ token.2 = key := by
    constructor
    · intro h
      have := congrArg List.reverse h
      simpa only [List.reverse_reverse] using this
    · intro h; rw [h]
  simp only [List.length_reverse, heq] at second
  rw [show Function.update initial (compareTapes tape 0) suffix =
      tapesAt (tape 0) (tape 5) base suffix counter by
    exact update_source _ _ (hd 0 5 (by decide)) base _ counter suffix] at second
  exact joinTrace first second

/-- Successful search returns exactly the first occurrence index. All labels
are literal instruction equations, so this theorem supplies its own body runs. -/
theorem searchTrace (tokens : List Token) (key suffix counter : List Bool)
    (canonical : ∀ token ∈ tokens, BinaryNameMachine.canonical token.2 = true)
    (present : key ∈ payloads tokens)
    (keyWord : base (tape 1) = key.reverse)
    (candidateEmpty : base (tape 2) = []) (copyEmpty : base (tape 3) = [])
    (scratchEmpty : base (tape 4) = []) :
    (advance (TM2.step program))^[steps tokens key]
      (some ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base (stream tokens ++ suffix) counter⟩) =
      some ⟨foundExit, clean ambient,
        tapesAt (tape 0) (tape 5) base (stream (afterMatch tokens key) ++ suffix)
          (List.replicate ((payloads tokens).idxOf key) true ++ counter)⟩ := by
  induction tokens generalizing counter with
  | nil => simp at present
  | cons token tokens ih =>
      have headCanonical := canonical token (by simp)
      have tailCanonical : ∀ t ∈ tokens, BinaryNameMachine.canonical t.2 = true :=
        fun t ht => canonical t (by simp [ht])
      have first := tokenTrace tape distinct labels foundExit missingExit malformedExit
        program atLabels base ambient token key (stream tokens ++ suffix) counter
        headCanonical keyWord candidateEmpty copyEmpty scratchEmpty
      by_cases same : token.2 = key
      · simpa only [steps, same, ite_true, Nat.add_zero, stream_cons,
          List.append_assoc, afterMatch, payloads_cons, List.idxOf_cons_self,
          List.replicate_zero, List.nil_append] using first
      · have remaining : key ∈ payloads tokens := by
          rcases List.mem_cons.mp present with equal | remaining
          · exact False.elim (same equal.symm)
          · exact remaining
        have increment : (advance (TM2.step program))^[1]
            (some ⟨some (labels .increment), clean ambient,
              tapesAt (tape 0) (tape 5) base (stream tokens ++ suffix) counter⟩) =
            some ⟨some (labels .sign), clean ambient,
              tapesAt (tape 0) (tape 5) base (stream tokens ++ suffix) (true :: counter)⟩ := by
          simpa only [Function.iterate_one, advance_some] using
            incrementStep tape labels foundExit missingExit malformedExit program
              atLabels base ambient (stream tokens ++ suffix) counter
        have rest := ih (true :: counter) tailCanonical remaining
        simp only [same, ite_false] at first
        have full := joinTrace first (joinTrace increment rest)
        have counterEq : List.replicate ((payloads tokens).idxOf key) true ++
            (true :: counter) =
            List.replicate ((payloads tokens).idxOf key + 1) true ++ counter := by
          rw [List.replicate_succ']
          simp only [List.append_assoc, List.singleton_append]
        have indexEq : (token.2 :: payloads tokens).idxOf key =
            (payloads tokens).idxOf key + 1 := by
          exact List.idxOf_cons_ne (payloads tokens) same
        simpa only [steps, same, ite_false, stream_cons, List.append_assoc, afterMatch,
          payloads_cons, indexEq, counterEq] using full

/-- Explicit polynomial bound for the actual successful run. -/
def searchInTime (tokens : List Token) (key suffix counter : List Bool)
    (canonical : ∀ token ∈ tokens, BinaryNameMachine.canonical token.2 = true)
    (present : key ∈ payloads tokens)
    (keyWord : base (tape 1) = key.reverse)
    (candidateEmpty : base (tape 2) = []) (copyEmpty : base (tape 3) = [])
    (scratchEmpty : base (tape 4) = []) :
    StateTransition.EvalsToInTime (TM2.step program)
      ⟨some (labels .sign), clean ambient,
        tapesAt (tape 0) (tape 5) base (stream tokens ++ suffix) counter⟩
      (some ⟨foundExit, clean ambient,
        tapesAt (tape 0) (tape 5) base (stream (afterMatch tokens key) ++ suffix)
          (List.replicate ((payloads tokens).idxOf key) true ++ counter)⟩)
      ((3 * key.length + 7) * (stream tokens).length) where
  steps := steps tokens key
  evals_in_steps := searchTrace tape distinct labels foundExit missingExit malformedExit
    program atLabels base ambient tokens key suffix counter canonical present keyWord
      candidateEmpty copyEmpty scratchEmpty
  steps_le_m := steps_le_stream tokens key

end Program

/-- Concrete finite-control instantiation; the counter is its designated output. -/
def machine : FinTM2 where
  K := Fin 6
  k₀ := 0
  k₁ := 5
  Γ _ := Bool
  Λ := Label ⊕ Fin 3
  main := .inl .sign
  σ := State Unit
  initialState := clean ()
  m label := match label with
    | .inl l => instruction id Sum.inl (some (.inr 0)) (some (.inr 1)) (some (.inr 2)) l
    | .inr _ => .halt

theorem machine_finiteAlphabet (k : machine.K) : Finite (machine.Γ k) := by
  change Finite Bool
  infer_instance

end UniqueGamesTheorem.BinaryNameSearch

end

end OAI