Section Transport.
(* Assume that P reaches a halting configuration from (0, 1) after NP steps, show that M is uniformly bounded. *)
Variable NP : nat. (* number of steps  *)
Variable NP_spec : CM.halting P (Nat.iter NP (CM.step P) cm_start).
Definition cm_end := Nat.iter NP (CM.step P) cm_start. (* final configuration *)
Definition CP : nat := CM.value cm_end. (* final counter value *)
Definition maybe_goto_1_time := (goto_time CP TP + NP * step_time CP TP + 2).
Definition maybe_index_try_stop_time :=
(NP * step_time CP TP + 2*maybe_goto_1_time + 1 + ((G + CP + 2) * (2 * goto_time CP TP + 3))).
Arguments maybe_first_goto_1 T {l r T' n X Y c Z}.
Definition maybe_increase_time := ((G + CP + 2) * (2 * maybe_goto_1_time + 3)).
Arguments maybe_increase l r {i j n} k m.
Definition maybe_index_try_step_time := 2*maybe_goto_1_time + maybe_increase_time +2.
Definition maybe_index_try_futile_time := maybe_index_try_stop_time + 3.
Definition maybe_bounded_try_time := TP + maybe_index_try_stop_time + 1.
Definition maybe_index_yes_futile_time := 2*maybe_goto_1_time + maybe_increase_time + maybe_index_try_stop_time + 5.
(* Grow Lemmas: lX0^mr ->> lX0^(m+1)r *)
Definition maybe_increase_no_grow_time := maybe_goto_1_time + maybe_index_try_stop_time + 3.
Definition maybe_index_yes_grow_time := maybe_index_yes_futile_time + maybe_increase_time + 3 * maybe_goto_1_time + maybe_index_try_stop_time + 4.
Definition maybe_index_try_grow_time := maybe_index_try_futile_time + (length P) * maybe_index_try_step_time + maybe_index_yes_grow_time + maybe_goto_1_time + 1.
Definition maybe_increase_yes_grow_time := maybe_index_try_grow_time + maybe_index_yes_futile_time + 2.
Definition maybe_index_no_grow_time := maybe_index_try_grow_time + maybe_goto_1_time + maybe_index_try_stop_time + 3.
Definition maybe_bounded_try_grow_time := maybe_bounded_try_time+1.
Definition maybe_basic_state_grow_time :=
maybe_index_try_stop_time + 2 +
maybe_bounded_try_grow_time +
maybe_index_try_grow_time + maybe_index_yes_grow_time + maybe_index_no_grow_time +
maybe_increase_yes_grow_time + maybe_increase_no_grow_time.

Lemma maybe_basic_state_grow l r m X : exists '(l', r', n', X', Y'),
maybe_reachable maybe_basic_state_grow_time (l, §0^m ++ r, basic_state X) (l', §0^(m+1) ++ r', goto n' X' Y').