Proof with simpl; auto.
intro H1. destruct c... intros H2.
2: intros; left; unfold var_eq; auto.
2: intros; left; unfold var_eq; auto.
rewrite (@listd_sort_uniq_inter _ state), remove_trivial_atoms_equiv in H2.
rewrite (@listd_sort_uniq_un _ state). apply (H1 H2).
intros x y; rewrite <-pure_atom_cmp_eq; auto.
intros x y; rewrite <-pure_atom_cmp_eq; auto.
Qed.

Lemma rewrite_by_eqv s t atm st :
(s === t) st ->
pure_atom_denote atm st = pure_atom_denote (rewrite_by s t atm) st.