(****************************** even/odd *********************************)

Inductive even (n : nat) : Prop :=
    even_intro : forall q : nat, n = 2 * q -> even n.(****************************** even/odd *********************************)

Inductive even (n : nat) : Prop :=
    even_intro : forall q : nat, n = 2 * q -> even n.Inductive odd (n : nat) : Prop :=
    odd_intro : forall q : nat, n = 2 * q + 1 -> odd n.Inductive odd (n : nat) : Prop :=
    odd_intro : forall q : nat, n = 2 * q + 1 -> odd n.

Lemma S_even_odd : forall a : nat, even a -> odd (S a).