Update

src/subsequences.v

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+Require Import Nat.
+Require Import PeanoNat.
+Require Import List.
+Import ListNotations.
+
+
+Definition subsequence (l s : list nat) :=
+  exists (l1: list nat) (l2 : list (list nat)),
+    length s = length l2
+    /\ l = l1 ++ flat_map (fun e => (fst e) :: (snd e)) (combine s l2).
+
+Definition subsequence2 (l s : list nat) :=
+  exists (t: list bool),
+    length t = length l /\ s = map snd (filter fst (combine t l)).
+
+Theorem subsequence_eq_def : forall l s, subsequence l s <-> subsequence2 l s.
+Proof.
+  intro l. induction l.
+  (* first part of the induction *)
+  intro s. unfold subsequence. unfold subsequence2.
+  split. exists nil. split. reflexivity. simpl.
+  destruct H. destruct H. destruct H.
+  assert (x = nil). destruct x. reflexivity. simpl in H0.
+  apply nil_cons in H0. contradiction H0. rewrite H1 in H0.
+  simpl in H0. assert (combine s x0 = nil).
+  destruct (combine s x0). reflexivity. simpl in H0.
+  apply nil_cons in H0. contradiction H0.
+  destruct x0. destruct s. reflexivity. simpl in H.
+  apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
+  destruct s. reflexivity. simpl in H2.
+  symmetry in H2. apply nil_cons in H2. contradiction H2.
+  exists nil. exists nil. destruct s. simpl. easy.
+  destruct H. destruct H.
+  assert (x = nil). destruct x. reflexivity. simpl in H.
+  apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
+  rewrite H1 in H0. simpl in H0.
+  symmetry in H0. apply nil_cons in H0. contradiction H0.
+  (* second part of the induction *)
+  intro s. destruct s. unfold subsequence. unfold subsequence2. split.
+  exists (repeat false (S (length l))). rewrite repeat_length.
+  split. easy. simpl.
+  assert (forall u,
+    (nil: list nat)
+       = map snd (filter fst (combine (repeat false (length u)) u))).
+  intro u. induction u. reflexivity. simpl. assumption. apply H0.
+  exists (a::l). exists (nil). simpl. split; try rewrite app_nil_r; reflexivity.
+
+
+
+
+  destruct H. destruct H. destruct H.
+  assert (x0 = nil). symmetry in H. apply length_zero_iff_nil in H.
+  assumption. rewrite H1 in H0. simpl in H0.
+
+
+
+
+
+
+
+
+Example test1: subsequence [1;2;3;4;5] [1;3;5].
+Proof.
+  unfold subsequence.
+  exists [].
+  exists [[2];[4];[]]. simpl. easy.
+Qed.
+
+Example test2: subsequence [1;2;3;4;5] [2;4].
+Proof.
+  unfold subsequence.
+  exists [1].
+  exists [[3];[5]]. simpl. easy.
+Qed.