Section defs.
Variable A : Setoid.
Inductive is_subseq :
forall (m : Nat) (w : seq m A) (n : Nat) (v : seq n A), Prop :=
| is_subseq_empty :
forall (n : Nat) (v : seq n A) (w : seq 0 A), is_subseq w v
| is_subseq_of_tail :
forall (m : Nat) (w : seq m A) (n : Nat) (v : seq n A),
is_subseq w v -> forall a : A, is_subseq w (a;; v)
| is_subseq_cons :
forall (m : Nat) (w : seq m A) (n : Nat) (v : seq n A),
is_subseq w v -> forall a : A, is_subseq (a;; w) (a;; v)
| is_subseq_comp :
forall (m : Nat) (w w' : seq m A) (n : Nat) (v v' : seq n A),
is_subseq w' v' -> w =' w' in _ -> v =' v' in _ -> is_subseq w v.
intros.
induction H.
apply False_ind; auto with algebra.
case (IHis_subseq i).
intros.
destruct x.
exists (Build_finiteT (lt_n_S _ _ in_range_prf)).
apply Trans with (v (Build_finiteT in_range_prf)); auto with algebra.
destruct i; destruct index as [| n0].
exists (Build_finiteT (lt_O_Sn n)).
auto with algebra.
generalize (IHis_subseq (Build_finiteT (lt_S_n _ _ in_range_prf))); intro.
inversion_clear H0.
destruct x.
exists (Build_finiteT (lt_n_S _ _ in_range_prf0)).
apply Trans with (v (Build_finiteT in_range_prf0)); auto with algebra.
generalize (IHis_subseq i).
intros.
inversion_clear H2.
exists x.
apply Trans with (w' i); auto with algebra.
apply Trans with (v' x); auto with algebra.
Qed.

Lemma subseqs_are_not_longer :
forall (m n : Nat) (w : seq m A) (v : seq n A), is_subseq w v -> m <= n.