repeat split; repeat intro.
destruct x; simpl in H...
destruct x; destruct y; destruct z; simpl in *...
do_comp expr_cspec e e1...
do_comp expr_cspec e1 e3...
do_comp expr_cspec e1 e3...
rewrite <- (comp_trans _ expr_cspec _ _ _ e5 e6)...
destruct x; destruct y.
simpl.
do_comp expr_cspec e e1...
do_comp expr_cspec e0 e2; econstructor; eauto; simpl...
econstructor; simpl...
econstructor; simpl...
Qed.
Hint Resolve pure_atom_cspec.

Lemma pure_atom_cmp_eq a b : a = b <-> Eq = pure_atom_cmp a b.
Hint Resolve pure_atom_cmp_eq.