(** [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.

(* Specifications: *)
Lemma env_equiv_refl:
forall e, env_equiv e e.
Lemma env_equiv_refl':
forall e1 e2, e1 = e2 -> env_equiv e1 e2.
Lemma env_equiv_sym:
forall e1 e2, env_equiv e1 e2 -> env_equiv e2 e1.
Lemma env_equiv_trans:
forall e1 e2 e3, env_equiv e1 e2 -> env_equiv e2 e3 -> env_equiv e1 e3.
Lemma exec_par_env_equiv:
forall m e1 e2,
move_no_temp m ->
env_equiv e1 e2 ->
env_equiv (exec_par m e1) (exec_par m e2).