Section count.
Context {X: Type} (p: X -> bool).
Fixpoint count (l: list X): nat :=
match l with
| nil => 0
| h :: t => if p h then S (count t) else count t
end.
Proof with auto.
induction l...
simpl.
destruct (p a)...
intros.
rewrite IHl...
Qed.
Proof with auto. induction l... simpl. intros. rewrite H... Qed.
Proof with auto with arith.
induction l...
simpl.
destruct (p a)...
Qed.
Proof with auto with arith.
induction x...
simpl.
destruct (f a).
simpl.
destruct (p a)...
destruct (p a)...
Qed.
Hint Resolve count_le.

Lemma count_lt v l: In v l -> p v = false -> count l < length l.