Definition ad_monotonic (pf:ad -> ad) :=
    forall a a':ad, Nless a a' = true -> Nless (pf a) (pf a') = true.

Lemma ad_comp_double_plus_un_monotonic : forall pf:ad -> ad, ad_monotonic pf -> ad_monotonic (fun a0:ad => pf (Ndouble_plus_one a0)).