Section Monad.
Definition state := nat.
Definition monad (A:Type) := state → state * A.
Definition Meval {A:Type} (m:monad A) st := m st.
Definition Meq {A:Type} (m:monad A) := ∀ st, st = fst (Meval m st).
Definition Mlt {A:Type} (m:monad A) := ∀ st, st < fst (Meval m st).
Definition Mle {A:Type} (m:monad A) := ∀ st, st ≤ fst (Meval m st).

Lemma mono_mono_Mbind_mono {A B:Type} (ma:monad A) (f:A → monad B):
Mmono ma → Mmono_one f → Mmono (ma >>= f).