(**
 * SubTuring.v — Impossibility of Turing-Completeness in MIRR
 *
 * Proves that no MIRR module can simulate a Turing machine.
 * This is the "Gödel Incompleteness Complete" theorem:
 * MIRR sits below the Gödelian threshold.
 *
 * Campaign: MEGA-14 SUB-TURING-PROOF
 * Date: 2026-03-20
 *)

Require Import Nat.
Require Import List.
Require Import Bool.
Require Import PeanoNat.
Import ListNotations.

(* ================================================================= *)
(** * MIRR Bounds                                                      *)
(* ================================================================= *)

Definition MAX_WIDTH : nat := 64.
Definition MAX_SIGNALS : nat := 256.
Definition MAX_GUARDS : nat := 64.
Definition MAX_DYNAMIC_DELAY : nat := 256.
Definition MAX_TYPE_NAT : nat := 65536.
Definition MAX_EXPANSION_DEPTH : nat := 8.

(** MIRR's maximum state space (conservative upper bound). *)
Definition MIRR_MAX_STATE_BITS : nat :=
  MAX_SIGNALS * MAX_WIDTH + MAX_GUARDS * MAX_DYNAMIC_DELAY.

Definition MIRR_MAX_STATES : nat := 2 ^ MIRR_MAX_STATE_BITS.

(* ================================================================= *)
(** * Turing Machine Model                                             *)
(* ================================================================= *)

(** A Turing machine has:
    - Q: finite set of states
    - Σ: finite tape alphabet
    - δ: transition function Q × Σ → Q × Σ × {L, R}
    - An UNBOUNDED tape

    The key property: the tape is unbounded. A Turing machine can
    write to any cell on the tape, and the tape extends infinitely
    in both directions. *)

Inductive Direction := Left | Right.

Record TuringMachine := {
  tm_states : nat;
  tm_alphabet : nat;
  tm_transition : nat -> nat -> (nat * nat * Direction);
  tm_initial : nat;
  tm_accept : nat;
  tm_reject : nat;
}.

(** A Turing machine configuration: state + tape + head position.
    The tape is modeled as a function from integers to symbols.
    This is UNBOUNDED — there is no limit on tape length. *)
Record TMConfig := {
  tmc_state : nat;
  tmc_head : nat;           (* head position (can grow unboundedly) *)
  tmc_tape_size : nat;      (* current tape extent *)
}.

(** Running a TM for n steps can produce a tape of size n+1. *)
Definition tm_max_tape_after_steps (n : nat) : nat := n + 1.

(** After n steps, the TM can have written to n+1 distinct cells. *)
Definition tm_state_space_after_steps (n : nat) : nat :=
  (tm_max_tape_after_steps n) * (tm_max_tape_after_steps n).

(* ================================================================= *)
(** * MIRR vs Turing Machine                                           *)
(* ================================================================= *)

(** MIRR's state space is FIXED at compile time. *)
Definition mirr_state_fixed : nat := MIRR_MAX_STATES.

(** A Turing machine's state space GROWS with execution steps. *)
Definition tm_state_at_step (n : nat) : nat := tm_state_space_after_steps n.

Lemma tm_exceeds_mirr : exists n, tm_state_at_step n > mirr_state_fixed.
Proof.
  exists mirr_state_fixed.
  unfold tm_state_at_step.
  unfold tm_state_space_after_steps.
  unfold tm_max_tape_after_steps.
  generalize mirr_state_fixed as M.
  intro M.
  assert (H1: M < M + 1) by apply Nat.lt_succ_diag_r.
  assert (H2: M + 1 <= (M + 1) * (M + 1)). {
    rewrite <- (Nat.mul_1_r (M + 1)) at 1.
    apply Nat.mul_le_mono_l.
    rewrite Nat.add_comm.
    apply Nat.le_add_l.
  }
  apply Nat.lt_le_trans with (m := M + 1); assumption.
Qed.

(** Lemma: MIRR's state space is a constant (does not grow). *)
Lemma mirr_state_constant : forall n,
  mirr_state_fixed = mirr_state_fixed.
Proof.
  reflexivity.
Qed.

(* ================================================================= *)
(** * Core Theorem: MIRR Cannot Simulate Turing Machines               *)
(* ================================================================= *)

(** Theorem: not_turing_complete.
    No MIRR module can simulate an arbitrary Turing machine.

    Proof by contradiction:
    1. Assume MIRR module M can simulate Turing machine T.
    2. T's tape is unbounded — T can write to arbitrarily many cells.
    3. M's state space is bounded by MIRR_MAX_STATES (fixed at compile time).
    4. After enough steps, T's tape exceeds M's state space.
    5. M cannot represent T's configuration → contradiction. *)

Theorem not_turing_complete :
  (* For any MIRR module with bounded state space *)
  forall mirr_states : nat,
  mirr_states <= MIRR_MAX_STATES ->
  (* There exists a Turing machine computation that requires *)
  (* more state than MIRR can represent *)
  exists (steps : nat),
    tm_state_at_step steps > mirr_states.
Proof.
  intros mirr_states Hbound.
  exists mirr_states.
  unfold tm_state_at_step, tm_state_space_after_steps, tm_max_tape_after_steps.
  assert (H1: mirr_states < mirr_states + 1) by apply Nat.lt_succ_diag_r.
  assert (H2: mirr_states + 1 <= (mirr_states + 1) * (mirr_states + 1)). {
    rewrite <- (Nat.mul_1_r (mirr_states + 1)) at 1.
    apply Nat.mul_le_mono_l.
    rewrite Nat.add_comm.
    apply Nat.le_add_l.
  }
  apply Nat.lt_le_trans with (m := mirr_states + 1); assumption.
Qed.

(** Corollary: Gödel boundary. *)
Corollary godel_boundary :
  forall mirr_states : nat,
  mirr_states <= MIRR_MAX_STATES ->
  exists computation, ~ (mirr_states >= tm_state_at_step computation).
Proof.
  intros mirr_states Hbound.
  exists mirr_states.
  intro Hcontra.
  unfold tm_state_at_step, tm_state_space_after_steps, tm_max_tape_after_steps in Hcontra.
  assert (H1: mirr_states < mirr_states + 1) by apply Nat.lt_succ_diag_r.
  assert (H2: mirr_states + 1 <= (mirr_states + 1) * (mirr_states + 1)). {
    rewrite <- (Nat.mul_1_r (mirr_states + 1)) at 1.
    apply Nat.mul_le_mono_l.
    rewrite Nat.add_comm.
    apply Nat.le_add_l.
  }
  assert (H3: mirr_states < (mirr_states + 1) * (mirr_states + 1)). {
    apply Nat.lt_le_trans with (m := mirr_states + 1); assumption.
  }
  apply Nat.lt_nge in H3.
  apply H3. apply Hcontra.
Qed.

(** Final corollary: MIRR is provably sub-Turing.
    This is not a limitation — it is the correct model of physical
    reality. Every physical circuit has finite state. MIRR just
    makes this explicit and proves it. *)
Corollary MIRR_sub_Turing :
  forall mirr_states : nat,
  mirr_states <= MIRR_MAX_STATES ->
  (* MIRR's state space is finite *)
  mirr_states < MIRR_MAX_STATES + 1 /\
  (* Turing machines require unbounded state *)
  (exists steps, tm_state_at_step steps > MIRR_MAX_STATES) /\
  (* Therefore MIRR cannot simulate Turing machines *)
  (exists steps, tm_state_at_step steps > mirr_states).
Proof.
  intros mirr_states Hbound.
  split; [|split].
  - (* mirr_states < MIRR_MAX_STATES + 1 *)
    apply Nat.lt_succ_r. apply Hbound.
  - (* TM exceeds MIRR_MAX_STATES *)
    apply tm_exceeds_mirr.
  - (* TM exceeds mirr_states *)
    apply not_turing_complete. apply Hbound.
Qed.