(** * Reasoning about monadic computations *)

(** The tactics below simplify hypotheses of the form [f ... = OK x s i]
  where [f] is a monadic computation.  For instance, the hypothesis
  [(do x <- a; b) s = OK y s' i] will generate the additional witnesses
  [x], [s1], [i1], [i'] and the additional hypotheses
  [a s = OK x s1 i1] and [b x s1 = OK y s' i'], reflecting the fact that
  both monadic computations [a] and [b] succeeded.
*)

Remark bind_inversion:
  forall (A B: Type) (f: mon A) (g: A -> mon B)
         (y: B) (s1 s3: state) (i: state_incr s1 s3),
  bind f g s1 = OK y s3 i ->
  exists x, exists s2, exists i1, exists i2,
  f s1 = OK x s2 i1 /\ g x s2 = OK y s3 i2.Remark bind2_inversion:
  forall (A B C: Type) (f: mon (A*B)) (g: A -> B -> mon C)
         (z: C) (s1 s3: state) (i: state_incr s1 s3),
  bind2 f g s1 = OK z s3 i ->
  exists x, exists y, exists s2, exists i1, exists i2,
  f s1 = OK (x, y) s2 i1 /\ g x y s2 = OK z s3 i2.Ltac monadInv1 H :=
  match type of H with
  | (OK _ _ _ = OK _ _ _) =>
      inversion H; clear H; try subst
  | (Error _ _ = OK _ _ _) =>
      discriminate
  | (ret _ _ = OK _ _ _) =>
      inversion H; clear H; try subst
  | (error _ _ = OK _ _ _) =>
      discriminate
  | (bind ?F ?G ?S = OK ?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)))))))
  | (bind2 ?F ?G ?S = OK ?X ?S' ?I) =>
      let x1 := fresh "x" in (
      let x2 := 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 (bind2_inversion _ _ _ F G X S S' I H) as [x1 [x2 [s [i1 [i2 [EQ1 EQ2]]]]]];
      clear H;
      try (monadInv1 EQ2))))))))
  end.Ltac monadInv H :=
  match type of H with
  | (ret _ _ = OK _ _ _) => monadInv1 H
  | (error _ _ = OK _ _ _) => monadInv1 H
  | (bind ?F ?G ?S = OK ?X ?S' ?I) => monadInv1 H
  | (bind2 ?F ?G ?S = OK ?X ?S' ?I) => monadInv1 H
  | (?F _ _ _ _ _ _ _ _ = OK _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ _ _ _ = OK _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ _ _ = OK _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ _ = OK _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ _ = OK _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ _ = OK _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ _ = OK _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  | (?F _ = OK _ _ _) =>
      ((progress simpl in H) || unfold F in H); monadInv1 H
  end.(** The following tactic saturates the hypotheses with
  [state_incr] properties that follow by transitivity from
  the known hypotheses. *)

Ltac saturateTrans :=
  match goal with
  | H1: state_incr ?x ?y, H2: state_incr ?y ?z |- _ =>
      match goal with
      | H: state_incr x z |- _  =>
         fail 1
      | _ =>
         let i := fresh "INCR" in
         (generalize (state_incr_trans x y z H1 H2); intro i;
          saturateTrans)
      end
  | _ => idtac
  end.(** * Validity and freshness of registers *)

(** An RTL pseudo-register is valid in a given state if it was created
  earlier, that is, it is less than the next fresh register of the state.
  Otherwise, the pseudo-register is said to be fresh. *)

Definition reg_valid (r: reg) (s: state) : Prop :=
  Plt r s.(st_nextreg).Definition reg_fresh (r: reg) (s: state) : Prop :=
  ~(Plt r s.(st_nextreg)).

Lemma reg_fresh_decr: forall r s1 s2, state_incr s1 s2 -> reg_fresh r s2 -> reg_fresh r s1.