(** [env_equiv] is an equivalence relation between environments
  capturing the fact that two environments assign the same values to
  non-temporary registers, but can disagree on the values of temporary
  registers. *)

Definition env_equiv (e1 e2: env) : Prop :=
  forall r, is_not_temp r -> e1 r = e2 r.

Theorem transitions_correctness: forall mu tau, move_no_temp mu -> is_mill mu -> transitions (State mu nil nil) (State nil nil tau) -> forall e, env_equiv (exec_seq (List.rev tau) e) (exec_par mu e).