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.
(* pair is already defined as we want *)
(* cons is already defined as we want *)
Definition triple {A B C:Type} (x:A*B) (c:C) := (fst x,snd x,c):A*B*C.
Notation "ma >>+ mb" := (Mmap2 pair ma mb) (at level 15, left associativity):monad_scope.
Notation "mab >>++ mc" := (Mmap2 triple mab mc) (at level 15, left associativity):monad_scope.
Notation "mx >>:: mxs" :=  (Mmap2 cons mx mxs) (at level 15, left associativity):monad_scope.

Lemma Mpair_on_first {A B:Type} (ma:monad A) (mb:monad B) (P:A→Prop):
(∀ st, P (snd (ma st))) →
∀ st, P (fst (snd ((ma >>+ mb) st))).