Update

src/subsequences.v

@@ -427,6 +427,80 @@ Proof.
   inversion I; inversion J; reflexivity.
   rewrite H4; try rewrite H1; try rewrite H; reflexivity.
 Qed.
+
+
+Lemma subsequence_length {X: Type} :
+  forall (u v: list X), subsequence u v -> length v <= length u.
+Proof.
+  intros u v. generalize u. induction v; intro u0; intro H. apply Nat.le_0_l.
+  apply subsequence_eq_def_1 in H. apply subsequence_eq_def_2 in H.
+  destruct H. destruct H. destruct H. apply subsequence_eq_def_3 in H0.
+  apply IHv in H0. simpl. apply Nat.le_succ_l. rewrite H.
+  rewrite app_length. apply Nat.lt_lt_add_l.
+  rewrite <- Nat.le_succ_l. simpl. rewrite <- Nat.succ_le_mono.
+  assumption.
+Qed.
+
+
+Theorem subsequence_id {X: Type} :
+  forall (u v: list X),
+  subsequence u v -> subsequence v u -> u = v.
+Proof.
+  intro u. induction u; intro v; intros H I;
+  apply subsequence_eq_def_1 in H; apply subsequence_eq_def_1 in I.
+
+  destruct v. reflexivity.
+  apply subsequence2_nil_cons_r in H. contradiction H.
+  destruct v. apply subsequence2_nil_cons_r in I. contradiction I.
+
+  destruct H. destruct H. destruct I. destruct H1.
+  destruct x0. apply O_S in H. contradiction H.
+  destruct x1. apply O_S in H1. contradiction H1.
+  destruct b; destruct b0.
+
+  assert (subsequence2 u v). exists x0.
+  split; inversion H; inversion H0; reflexivity.
+  assert (subsequence2 v u). exists x1.
+  split; inversion H1; inversion H2; reflexivity.
+  apply subsequence_eq_def_2 in H3; apply subsequence_eq_def_3 in H3.
+  apply subsequence_eq_def_2 in H4; apply subsequence_eq_def_3 in H4.
+  apply IHu in H4. rewrite H4. inversion H0. reflexivity. assumption.
+
+  assert (subsequence2 u v). exists x0.
+  split; inversion H; inversion H0; reflexivity.
+  assert (subsequence2 v (a::u)). exists x1.
+  split; inversion H1; inversion H2; reflexivity.
+  assert (subsequence u (a::u)). apply subsequence_trans with (l2 := v);
+  apply subsequence_eq_def_3; apply subsequence_eq_def_2; assumption.
+  apply subsequence_length in H5. apply Nat.nle_succ_diag_l in H5.
+  contradiction H5.
+
+  assert (subsequence2 u (x::v)). exists x0.
+  split; inversion H; inversion H0; reflexivity.
+  assert (subsequence2 v u). exists x1.
+  split; inversion H1; inversion H2; reflexivity.
+  assert (subsequence v (x::v)). apply subsequence_trans with (l2 := u);
+  apply subsequence_eq_def_3; apply subsequence_eq_def_2; assumption.
+  apply subsequence_length in H5. apply Nat.nle_succ_diag_l in H5.
+  contradiction H5.
+
+  assert (subsequence2 u (x::v)). exists x0.
+  split; inversion H; inversion H0; reflexivity.
+  assert (subsequence2 v (a::u)). exists x1.
+  split; inversion H1; inversion H2; reflexivity.
+  apply subsequence2_cons_r in H3. apply subsequence2_cons_r in H4.
+  apply subsequence_eq_def_2 in H3. apply subsequence_eq_def_2 in H4.
+  apply subsequence_eq_def_3 in H3. apply subsequence_eq_def_3 in H4.
+  apply IHu in H4. rewrite H4 in H0. rewrite H4 in H.
+  simpl in H0. assert (subsequence2 v (x::v)). exists x0.
+  split. inversion H. reflexivity. assumption.
+  apply subsequence_eq_def_2 in H5. apply subsequence_eq_def_3 in H5.
+  apply subsequence_length in H5. apply Nat.nle_succ_diag_l in H5.
+  contradiction H5. assumption.
+Qed.
+
+
+
  
 
 (** * Tests