proposition: Lemma interp_check_distinct : forall ha diseq, check_distinct ha diseq = true -> interp_form_hatom ha = afold_left bool true andb (amap (Lit.interp rho) diseq).
proof: Proof. intros ha diseq; rewrite check_distinct_spec; intros [A [dist [H1 H2]]]; rewrite check_diseqs_spec in H2; destruct H2 as [H2 H3]; unfold Atom.interp_form_hatom, Atom.interp_bool, Atom.interp_hatom; rewrite Atom.t_interp_wf; auto with smtcoq_spl_op; rewrite H1; simpl; generalize (Atom.compute_interp_spec_rev t_i (get (Atom.t_interp t_i t_func t_atom)) A dist); case (Atom.compute_interp t_i (get (Atom.t_interp t_i t_func t_atom)) A nil); simpl. intros l H4; case_eq (distinct (Typ.i_eqb t_i A) (rev l)). rewrite distinct_spec; intro H5; symmetry; apply afold_left_andb_true; rewrite length_amap; intros i Hi; rewrite get_amap by exact Hi; destruct (H2 _ Hi) as [H9 [a [H10 [h1 [h2 [H6 [H7 H8]]]]]]]; unfold Lit.interp; replace (Lit.is_pos (diseq .[ i])) with false by (case_eq (Lit.is_pos (diseq .[ i])); auto with smtcoq_spl_op smtcoq_core); unfold Var.interp; rewrite Form.wf_interp_form; auto with smtcoq_spl_op smtcoq_core; rewrite H10; simpl; rewrite Atom.t_interp_wf; auto with smtcoq_spl_op smtcoq_core; rewrite H6; simpl; unfold Atom.apply_binop; unfold Atom.wt in wt_t_atom; unfold is_true in wt_t_atom; rewrite aforallbi_spec in wt_t_atom; assert (H11: a <? length t_atom). case_eq (a <? length t_atom); auto with smtcoq_spl_op smtcoq_core; intro H11; rewrite (get_outofbound _ _ _ H11) in H6; rewrite default_t_atom in H6; inversion H6. generalize (wt_t_atom _ H11); rewrite H6; simpl; rewrite !andb_true_iff; change (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) h1) A = true) with (is_true (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) h1) A)); change (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) h2) A = true) with (is_true (Typ.eqb (Atom.get_type' t_i (Atom.t_interp t_i t_func t_atom) h2) A)); rewrite !Typ.eqb_spec; intros [[_ H13] H12]; generalize (Atom.check_aux_interp_hatom _ t_func _ wf_t_atom h1); rewrite H13; intros [v1 HH1]; generalize (Atom.check_aux_interp_hatom _ t_func _ wf_t_atom h2); rewrite H12; intros [v2 HH2]; rewrite HH1, HH2; simpl; rewrite Typ.cast_refl; simpl; destruct H8 as [H8|H8]; [ |rewrite Typ.i_eqb_sym]; rewrite H5; auto with smtcoq_spl_op smtcoq_core; rewrite H4; [exists h2; exists h1|exists h1; exists h2]; auto with smtcoq_spl_op smtcoq_core. rewrite distinct_false_spec; intros [v2 [v1 [H5 H6]]]; rewrite H4 in H5; destruct H5 as [a [b [H5 [H7 H8]]]]; clear H4; change (Typ.i_eqb t_i A v2 v1 = true) with (is_true (Typ.i_eqb t_i A v2 v1)) in H6; rewrite Typ.i_eqb_spec in H6; subst v2; clear H2; destruct (H3 _ _ H5) as [i [H2 [H4 [hb [H6 [H9 H10]]]]]]; clear H3; symmetry; apply (afold_left_andb_false i); rewrite ?length_amap; auto with smtcoq_spl_op smtcoq_core; rewrite get_amap by assumption; unfold Lit.interp; replace (Lit.is_pos (diseq .[ i])) with false by (case_eq (Lit.is_pos (diseq .[ i])); auto with smtcoq_spl_op smtcoq_core); unfold Var.interp; rewrite Form.wf_interp_form; auto with smtcoq_spl_op smtcoq_core; rewrite H6; simpl; rewrite Atom.t_interp_wf; auto with smtcoq_spl_op smtcoq_core; destruct H10 as [H10|H10]; rewrite H10; simpl; rewrite H7, H8; simpl; rewrite Typ.cast_refl; simpl; replace (Typ.i_eqb t_i A v1 v1) with true; auto with smtcoq_spl_op smtcoq_core; symmetry; change (is_true (Typ.i_eqb t_i A v1 v1)); rewrite Typ.i_eqb_spec; auto with smtcoq_spl_op smtcoq_core. intros [a [H20 H21]]; assert (H4: ha <? length t_atom). case_eq (ha <? length t_atom); auto with smtcoq_spl_op smtcoq_core; intro Heq; generalize H1; rewrite get_outofbound; auto with smtcoq_spl_op smtcoq_core; rewrite default_t_atom; discriminate. unfold Atom.wt in wt_t_atom; unfold is_true in wt_t_atom; rewrite aforallbi_spec in wt_t_atom; generalize (wt_t_atom _ H4); rewrite H1; simpl; rewrite andb_true_iff, forallb_forall; intros [_ H5]; assert (H6 := H5 _ H20); generalize (Atom.check_aux_interp_hatom _ t_func _ wf_t_atom a); intros [va Ha]; rewrite Ha in H21; simpl in H21; elim H21; apply Typ.eqb_spec; auto with smtcoq_spl_op smtcoq_core. Qed.
imports: List PArray Bool Int63 ZMicromega Misc State SMT_terms