Update

src/subsequences2.v

@@ -172,19 +172,19 @@ Proof.
   apply IHl. destruct H0.
 
   rewrite <- Subsequence_cons_eq with (a := a) in s0.
-  apply Subsequence_cons_r in s0. rewrite Subsequence_bools in s0.
-  left. rewrite Subsequence_bools. assumption.
+  apply Subsequence_cons_r in s0. apply Subsequence_bools in s0.
+  left. apply Subsequence_bools. assumption.
 
   destruct s. left. apply Subsequence_nil_r.
   assert ({x=a}+{x<>a}). apply H. destruct H0. rewrite e.
   destruct IHl with (s := s); [ left | right ];
   rewrite Subsequence_cons_eq. assumption. assumption.
 
-  right. intro I. rewrite Subsequence_bools in I.
+  right. intro I. apply Subsequence_bools in I.
   destruct I. destruct H0. destruct x0.
   symmetry in H1. apply nil_cons in H1. contradiction H1.
   destruct b. inversion H1. rewrite H3 in n0. easy.
-  apply n. rewrite Subsequence_bools. exists x0; split; inversion H0; easy.
+  apply n. apply Subsequence_bools. exists x0; split; inversion H0; easy.
 Qed.
 
 
@@ -433,24 +433,22 @@ Proof.
 Qed.
 
 
-
-
-
-Theorem subsequence_map {X Y: Type} :
+Theorem Subsequence_map {X Y: Type} :
   forall (u v: list X) (f: X -> Y),
-    subsequence u v -> subsequence (map f u) (map f v).
+    Subsequence u v -> Subsequence (map f u) (map f v).
 Proof.
   intros u v. generalize u. induction v; intros u0 f; intro H.
-  apply subsequence_nil_r. apply subsequence_eq_def_3.
-  apply subsequence_eq_def_1 in H. apply subsequence_eq_def_2 in H.
-  destruct H. destruct H. destruct H.
+  apply Subsequence_nil_r. destruct H. destruct H. destruct H.
   exists (map f x). exists (map f x0). split.
   rewrite <- map_cons. rewrite <- map_app. rewrite H. reflexivity.
-  apply subsequence_eq_def_2. apply subsequence_eq_def_1.
-  apply IHv. apply subsequence_eq_def_3. assumption.
+  apply IHv. assumption.
 Qed.
 
 
+
+
+
+
 Theorem subsequence_double_cons {X: Type} :
   forall (u v: list X) a, subsequence (a::u) (a::v) <-> subsequence u v.
 Proof.