Update

src/subsequences.v

@@ -584,6 +584,64 @@ Theorem subsequence_map {X Y: Type} :
 Qed.
 
 
+Theorem subsequence_double_cons {X: Type} :
+  forall (u v: list X) a, subsequence (a::u) (a::v) -> subsequence u v.
+Proof.
+  intros u v a. intro H. destruct H. destruct H. destruct H.
+  destruct x; destruct x0. symmetry in H0. apply nil_cons in H0.
+  contradiction. inversion H0.
+  exists l. exists x0. split. inversion H. reflexivity. reflexivity.
+  apply PeanoNat.Nat.neq_succ_0 in H. contradiction.
+  inversion H0. exists (x1++x::l). exists x0. split. inversion H.
+  reflexivity. rewrite <- app_assoc. reflexivity.
+Qed.
+
+
+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.
+  destruct x. apply O_S in H. contradiction. destruct b.
+  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.
+  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.
+  exists x. split. inversion H. reflexivity. apply H0.
+  apply subsequence_cons_l. apply IHu with (f := f). assumption.
+Qed.
+
+
+Theorem subsequence_filter {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_trans {X: Type} :
+  forall (l1 l2 l3: list X),
+  subsequence l1 l2 -> subsequence l2 l3 -> subsequence l1 l3.
+
+
+
+
+
 Theorem subsequence_incl {X: Type} :
   forall (u v: list X), subsequence u v -> incl v u.
 Proof.
@@ -610,14 +668,12 @@ Theorem subsequence_split {X: Type} :
     -> (exists a b, w = a++b /\ (subsequence u a) /\ subsequence v b).
 Proof.
   intros u v w. intro H.
-  apply subsequence_eq_def_1 in H.
-  destruct H. destruct H.
+  apply subsequence_eq_def_1 in H. destruct H. destruct H.
 
   assert (forall (b: list bool) (u v: list X),
     length b = length (u++v)
     -> exists x y, b = x++y /\ length x = length u /\ length y = length v).
-  intros b u0 v0. intro I.
-  rewrite app_length in I.
+  intros b u0 v0. intro I. rewrite app_length in I.
   exists (firstn (length u0) b). exists (skipn (length u0) b).
   split. symmetry. apply firstn_skipn. split.
   rewrite firstn_length_le. reflexivity. rewrite I.
@@ -633,16 +689,12 @@ Proof.
   assert (L: forall (g i: list bool) (h j: list X), length g = length h
     -> (combine g h) ++ (combine i j) = combine (g++i) (h++j)).
   intro g. induction g; intros i h j; intro I.
-  symmetry in I. apply length_zero_iff_nil in I. rewrite I.
-  reflexivity.
+  symmetry in I. apply length_zero_iff_nil in I. rewrite I. reflexivity.
   destruct h. apply PeanoNat.Nat.neq_succ_0 in I. contradiction.
   simpl. rewrite IHg. reflexivity. inversion I. reflexivity.
-  rewrite L. rewrite <- H. split. assumption. split.
-
-  apply subsequence_eq_def_3. apply subsequence_eq_def_2.
-  exists x0. split. assumption. reflexivity.
-  apply subsequence_eq_def_3. apply subsequence_eq_def_2.
-  exists x1. split. assumption. reflexivity.
+  rewrite L. rewrite <- H. split. assumption.
+  split; apply subsequence_eq_def_3; apply subsequence_eq_def_2;
+  [ exists x0 | exists x1]; split; try assumption; reflexivity.
 
   assumption.
 Qed.