Update

src/subsequences.v

@@ -620,24 +620,6 @@ Proof.
 Qed.
 
 
-Theorem subsequence_filter_2 {X: Type} :
-  forall (u v: list X) f,
-    subsequence u v -> subsequence (filter f u) (filter f v).
-Proof.
-  (*
-  intro u. induction u; intros v f; intro H.
-  assert (v = nil). destruct v. reflexivity.
-  apply subsequence_eq_def_1 in H. apply subsequence_eq_def_2 in H.
-  destruct H. destruct H. destruct H.
-  destruct x0; apply nil_cons in H; contradiction.
-  rewrite H0. apply subsequence_nil_r.
-  simpl. destruct (f a).
-  *)
-  intros u v. generalize u. induction v; intros u0 f; intro H.
-  apply subsequence_nil_r. simpl. destruct (f a).
-  apply subsequence_cons_r.
-
-
 Theorem subsequence_incl {X: Type} :
   forall (u v: list X), subsequence u v -> incl v u.
 Proof.
@@ -696,6 +678,78 @@ Proof.
 Qed.
 
 
+Theorem subsequence_split_2 {X: Type} :
+  forall (u v w: list X),
+    subsequence u (v++w) 
+    -> (exists a b, u = a++b /\ (subsequence a v) /\ subsequence b w).
+Proof.
+  intros u v w. intro H. destruct H. destruct H. destruct H.
+  exists (x ++
+     flat_map (fun e : X * list X => fst e :: snd e) (combine v x0)).
+  exists (
+     flat_map (fun e : X * list X => fst e :: snd e)
+     (combine w (skipn (length v) x0))).
+   split. rewrite H0. rewrite <- app_assoc. apply app_inv_head_iff.
+   rewrite <- flat_map_app.
+
+   assert (L: forall (w y: list X) (z : list (list X)),
+     length z = length w + length y ->
+     combine (w ++ y) z = combine w z ++ combine y (skipn (length w) z)).
+   intros w0. induction w0; intros y z; intro I. reflexivity.
+   destruct z. simpl. rewrite combine_nil. reflexivity.
+   simpl. rewrite IHw0. reflexivity. inversion I. reflexivity.
+   rewrite L. reflexivity. rewrite app_length in H. rewrite H.
+   reflexivity. split.
+   exists x. exists (firstn (length v) x0). split.
+   rewrite firstn_length. rewrite <- H. rewrite app_length.
+   rewrite min_l. reflexivity. apply PeanoNat.Nat.le_add_r.
+   rewrite combine_firstn_l. reflexivity.
+
+   exists nil. exists (skipn (length v) x0). split.
+   rewrite app_length in H. rewrite skipn_length. rewrite <- H.
+   rewrite PeanoNat.Nat.add_sub_swap. rewrite PeanoNat.Nat.sub_diag.
+   reflexivity. reflexivity. reflexivity.
+ Qed.
+
+
+Theorem subsequence_filter_2 {X: Type} :
+  forall (u v: list X) f,
+    subsequence u v -> subsequence (filter f u) (filter f v).
+Proof.
+  (*
+  intro u. induction u; intros v f; intro H.
+  assert (v = nil). destruct v. reflexivity.
+  apply subsequence_eq_def_1 in H. apply subsequence_eq_def_2 in H.
+  destruct H. destruct H. destruct H.
+  destruct x0; apply nil_cons in H; contradiction.
+  rewrite H0. apply subsequence_nil_r.
+  simpl. destruct (f a).
+  *)
+  intros u v. generalize u. induction v; intros u0 f; intro H.
+  apply subsequence_nil_r.
+
+  assert (f a = true \/ f a = false).
+  destruct (f a); [ left | right ]; reflexivity. destruct H0.
+  simpl. rewrite H0.
+  replace (a::v) with ([a] ++ v) in H. apply subsequence_split in H.
+
+
+
+
+  destruct H. destruct H. destruct x0. destruct H.
+  apply PeanoNat.Nat.neq_succ_0 in H. contradiction.
+  destruct H. simpl in H0. rewrite H0.
+
+
+
+  simpl. destruct (f a).
+
+
+
+  apply subsequence_cons_r.
+
+
+
 
 
 (** * Tests
@@ -800,7 +854,6 @@ Proof.
   apply subsequence_cons_l. assumption.
 Qed.
 
-
 Theorem smerge_subsequence2 {X: Type}:
   (forall x y : X, {x = y} + {x <> y})
   -> forall (l s1 s2: list X), smerge l s1 s2 -> subsequence l s2.