(** ** Working with the state monad *)

Remark bind_inversion:
  forall (A B: Type) (f: mon A) (g: A -> mon B)
         (y: B) (s1 s3: state) (i: sincr s1 s3),
  bind f g s1 = R y s3 i ->
  exists x, exists s2, exists i1, exists i2,
  f s1 = R x s2 i1 /\ g x s2 = R y s3 i2.Ltac monadInv1 H :=
  match type of H with
  | (R _ _ _ = R _ _ _) =>
      inversion H; clear H; try subst
  | (ret _ _ = R _ _ _) =>
      inversion H; clear H; try subst
  | (bind ?F ?G ?S = R ?X ?S' ?I) =>
      let x := fresh "x" in (
      let s := fresh "s" in (
      let i1 := fresh "INCR" in (
      let i2 := fresh "INCR" in (
      let EQ1 := fresh "EQ" in (
      let EQ2 := fresh "EQ" in (
      destruct (bind_inversion _ _ F G X S S' I H) as [x [s [i1 [i2 [EQ1 EQ2]]]]];
      clear H;
      try (monadInv1 EQ2)))))))
  end.Ltac monadInv H :=
  match type of H with
  | (ret _ _ = R _ _ _) => monadInv1 H
  | (bind ?F ?G ?S = R ?X ?S' ?I) => monadInv1 H
  | (?F _ _ _ _ _ _ _ _ = R _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ _ _ _ = R _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ _ _ = R _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ _ = R _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ = R _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ = R _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ = R _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ = R _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  end.Remark mlist_iter2_fold:
  forall (A B: Type) (f: A -> B -> mon unit) l s,
  exists i,
  mlist_iter2 f l s =
  R tt (fold_left (fun a p => match f (fst p) (snd p) a with R _ s2 _ => s2 end) l s) i.

(* Specifications: *)
Lemma ptree_mfold_spec: forall (A: Type) (f: positive -> A -> mon unit) t s x s' i, ptree_mfold f t s = R x s' i -> exists i', mlist_iter2 f (PTree.elements t) s = R tt s' i'.