Section SeqFind.
Variable a : pred T.
Fixpoint find s := if s is x :: s' then if a x then 0 else (find s').+1 else 0.
Fixpoint filter s :=
if s is x :: s' then if a x then x :: filter s' else filter s' else [::].
Fixpoint count s := if s is x :: s' then a x + count s' else 0.
Fixpoint has s := if s is x :: s' then a x || has s' else false.
Fixpoint all s := if s is x :: s' then a x && all s' else true.
End SubPred.
(* Surgery: drop, take, rot, rotr.                                        *)
Fixpoint drop n s {struct s} :=
match s, n with
| _ :: s', n'.+1 => drop n' s'
| _, _ => s
end.

Lemma take_take i j : i <= j -> forall s, take i (take j s) = take i s.