Proof with
simpl; auto;
try solve [simpl; destruct delta as [|p]; simpl; auto; destruct p; simpl; auto].
intros A; destruct c as [gamma delta| |]...
destruct gamma... destruct delta... destruct p... destruct delta... destruct p...
(*simpl; destruct delta as [|p]; simpl; auto; destruct p... destruct delta as [|p]; simpl; auto; destruct p...*)
remember (expr_eq e e1) as b1; remember (expr_eq e0 e2) as b2.
destruct b1, b2...
if_tac... simpl; split; auto. intro Hyp.
rewrite (listd_insert_uniq_un pure_atom_denote FF). 2: apply pure_atom_cmp_eq.
rewrite (expr_eq_eq' _ _ Heqb1), (expr_eq_eq' _ _ Heqb2) in A |- *.
simpl in A; destruct A as [A|[A|A]]; [auto| | |right; auto].
solve [left; if_tac; simpl; solve[auto | apply var_eq_sym; auto]].
solve [left; if_tac; simpl; solve[auto | apply var_eq_sym; auto]].
replace (pure_atom_eq (Eqv e e0) (Eqv e1 e2)) with false; auto.
unfold pure_atom_eq; simpl.
unfold expr_eq in Heqb1. destruct (expr_cmp e e1); simpl; try congruence.
unfold expr_eq in Heqb2. destruct (expr_cmp e0 e2);simpl; try congruence.
replace (pure_atom_eq (Eqv e e0) (Eqv e1 e2)) with false; auto.
unfold pure_atom_eq; simpl.
unfold expr_eq in Heqb1. destruct (expr_cmp e e1); simpl; try congruence.
replace (pure_atom_eq (Eqv e e0) (Eqv e1 e2) ) with false; auto.
unfold pure_atom_eq; simpl.
unfold expr_eq in Heqb1. destruct (expr_cmp e e1); simpl; try congruence.
Qed.

(* Specifications: *)
Lemma equality_factoring_sound c l0 s :
clause_denote c s -> listd clause_denote inter TT l0 s ->
listd clause_denote inter TT (equality_factoring c l0) s.
Lemma sp_sound ce c d l0 s :
clause_denote c s -> clause_denote d s -> listd clause_denote inter TT l0 s ->
listd clause_denote inter TT (sp ce c d l0) s.