Definition quotient_euclide (a b:nat)(H:(b<>O)) := let (q,_) := (euclide a b H) in q.Definition remainder_euclide (a b:nat)(H:(b<>O)) := let (_,e0) := (euclide a b H) in let (r,_) := e0 in r.

Lemma dec_impl_lt_dec : forall (P:nat->Prop),(forall (n:nat),{(P n)}+{~(P n)})->(forall (m:nat),{n:nat | (n<m)/\(P(n))}+{(forall (n:nat),(n<m)->~(P n))}).