Inductive ex_minn_spec : nat -> Type :=
ExMinnSpec m of P m & (forall n, P n -> n >= m) : ex_minn_spec m.

Lemma eq_ex_maxn (P Q : pred nat) m n exP ubP exQ ubQ :
P =1 Q -> @ex_maxn P m exP ubP = @ex_maxn Q n exQ ubQ.