Update

src/subsequences2.v

@@ -462,29 +462,32 @@ Proof.
 Qed.
 
 
-
-
-
-Theorem subsequence_filter {X: Type} :
-  forall (u v: list X) f, subsequence (filter f u) v -> subsequence u v.
+Theorem Subsequence_filter {X: Type} :
+  forall (u v: list X) f, Subsequence (filter f u) v -> Subsequence u v.
 Proof.
   intro u. induction u; intros v f; intro H. assumption.
   simpl in H. destruct (f a). assert (K := H).
-  apply subsequence_eq_def_1 in H. destruct H. destruct H.
+  apply Subsequence_bools in H.
+  destruct H. destruct H.
   destruct x. apply O_S in H. contradiction. destruct b.
-  destruct v. apply subsequence_nil_r. assert (x0 = a). inversion H0.
+  destruct v. apply Subsequence_nil_r. assert (x0 = a). inversion H0.
   reflexivity. rewrite H1.
-  apply subsequence_eq_def_3. exists nil. exists u. split. reflexivity.
-  apply subsequence_eq_def_2. apply subsequence_eq_def_1.
+  exists nil. exists u. split. reflexivity.
   apply IHu with (f := f). rewrite H1 in K.
-  apply subsequence_double_cons in K. assumption.
-  apply subsequence_cons_l. apply IHu with (f := f).
-  apply subsequence_eq_def_3. apply subsequence_eq_def_2.
+  apply Subsequence_double_cons in K. assumption.
+  apply Subsequence_cons_l. apply IHu with (f := f).
+  apply Subsequence_bools.
   exists x. split. inversion H. reflexivity. apply H0.
-  apply subsequence_cons_l. apply IHu with (f := f). assumption.
+  apply Subsequence_cons_l. apply IHu with (f := f). assumption.
 Qed.
 
 
+
+
+
+
+
+
 Theorem subsequence_incl {X: Type} :
   forall (u v: list X), subsequence u v -> incl v u.
 Proof.