Scheme Odd_alt_Even_alt_ind := Minimality for Odd_alt Sort Prop
with Even_alt_Odd_alt_ind := Minimality for Even_alt Sort Prop.

Lemma Odd_Even_ind (P Q : nat -> Prop) : (forall n, Even n -> Q n -> P (S n)) -> Q 0 -> (forall n, Odd n -> P n -> Q (S n)) -> forall n, Odd n -> P n.