Proof with simpl; auto.
intros H1 H2. destruct c... destruct gamma, delta...
destruct p... destruct delta... destruct d... intro H3.
simpl in H1; spec H1; auto.
inversion H1. 2: inversion H. clear H1.
subst priority0.
induction gamma. simpl in H2; spec H2; auto. induction delta...
inversion H2. left. apply rewrite_by_sound... right. apply IHdelta...
destruct H3 as [H3 H4]. apply IHgamma... intro H5. simpl in H2; spec H2.
split... rewrite <-rewrite_by_eqv in H3... auto.
Qed.

(* Specifications: *)
Lemma clause_denote_foldl (f: clause -> clause -> clause) l c0 s :
listd clause_denote inter TT l s -> clause_denote c0 s ->
(forall c d, clause_denote c s -> clause_denote d s ->
clause_denote (f c d) s) ->
clause_denote (fold_left f l c0) s.