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).
Definition Mdo {A B:Type} (ma:monad A) (mb:monad B) := ma >>= (λ _, mb).
Infix ">>" := Mdo (at level 15, left associativity) : monad_scope. (* level at random… *)
(*Definition Mmake {A B:Type} (f:A→B) x:= Mreturn (f x).*)
Definition Mupdate (f:nat→nat) := (λ st, ((f st),tt)):monad unit.
Definition Mmap2 {A B C:Type} (f:A→B→C) (ma:monad A) (mb:monad B):=
(λ s0, let (s1,a) := ma s0 in let (s2,b) := mb s1 in (s2,(f a b))):monad C.

Lemma lt_le_Mmap2_lt {A B C:Type} (f:A→B→C) (ma:monad A) (mb:monad B):
Mlt ma → Mle mb → Mlt (Mmap2 f ma mb).