(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the GNU General Public License as published by  *)
(*  the Free Software Foundation, either version 2 of the License, or  *)
(*  (at your option) any later version.  This file is also distributed *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)

include "datatypes/pairs.ma".
include "Plogic/equality.ma".
include "Plogic/connectives.ma".

(* * Error reporting and the error monad. *)
(*
(** * Representation of error messages. *)

(** Compile-time errors produce an error message, represented in Coq
  as a list of either substrings or positive numbers encoding
  a source-level identifier (see module AST). *)

Inductive errcode: Type :=
  | MSG: string -> errcode
  | CTX: positive -> errcode.

Definition errmsg: Type := list errcode.

Definition msg (s: string) : errmsg := MSG s :: nil.
*)
(* * * The error monad *)

(* * Compilation functions that can fail have return type [res A].
  The return value is either [OK res] to indicate success,
  or [Error msg] to indicate failure. *)

ninductive res (A: Type) : Type ≝
| OK: A → res A
| Error: (* FIXME errmsg →*) res A.

(*Implicit Arguments Error [A].*)

(* * To automate the propagation of errors, we use a monadic style
  with the following [bind] operation. *)

ndefinition bind ≝ λA,B:Type. λf: res A. λg: A → res B.
  match f with
  [ OK x ⇒ g x
  | Error (*msg*) ⇒ Error ? (*msg*)
  ].

ndefinition bind2 ≝ λA,B,C: Type. λf: res (A × B). λg: A → B → res C.
  match f with
  [ OK v ⇒ match v with [ mk_pair x y ⇒ g x y ]
  | Error (*msg*) => Error ? (*msg*)
  ].

(* Not sure what level to use *)
notation > "vbox('do' ident v ← e; break e')" with precedence 40 for @{'bind ${e} (λ${ident v}.${e'})}.
notation > "vbox('do' ident v : ty ← e; break e')" with precedence 40 for @{'bind ${e} (λ${ident v} : ${ty}.${e'})}.
notation < "vbox('do' \nbsp ident v ← e; break e')" with precedence 40 for @{'bind ${e} (λ${ident v}.${e'})}.
notation < "vbox('do' \nbsp ident v : ty ← e; break e')" with precedence 40 for @{'bind ${e} (λ${ident v} : ${ty}.${e'})}.
interpretation "error monad bind" 'bind e f = (bind ?? e f).
notation > "vbox('do' 〈ident v1, ident v2〉 ← e; break e')" with precedence 40 for @{'bind2 ${e} (λ${ident v1}.λ${ident v2}.${e'})}.
notation > "vbox('do' 〈ident v1 : ty1, ident v2 : ty2〉 ← e; break e')" with precedence 40 for @{'bind2 ${e} (λ${ident v1} : ${ty1}.λ${ident v2} : ${ty2}.${e'})}.
notation < "vbox('do' \nbsp 〈ident v1, ident v2〉 ← e; break e')" with precedence 40 for @{'bind2 ${e} (λ${ident v1}.λ${ident v2}.${e'})}.
notation < "vbox('do' \nbsp 〈ident v1 : ty1, ident v2 : ty2〉 ← e; break e')" with precedence 40 for @{'bind2 ${e} (λ${ident v1} : ${ty1}.λ${ident v2} : ${ty2}.${e'})}.
interpretation "error monad pair bind" 'bind2 e f = (bind2 ??? e f).
(*interpretation "error monad ret" 'ret e = (ret ? e).
notation "'ret' e" non associative with precedence 45 for @{'ret ${e}}.*)

(*
(** The [do] notation, inspired by Haskell's, keeps the code readable. *)

Notation "'do' X <- A ; B" := (bind A (fun X => B))
 (at level 200, X ident, A at level 100, B at level 200)
 : error_monad_scope.

Notation "'do' ( X , Y ) <- A ; B" := (bind2 A (fun X Y => B))
 (at level 200, X ident, Y ident, A at level 100, B at level 200)
 : error_monad_scope.
*)
nremark bind_inversion:
  ∀A,B: Type. ∀f: res A. ∀g: A → res B. ∀y: B.
  bind ?? f g = OK ? y →
  ∃x. f = OK ? x ∧ g x = OK ? y.
#A B f g y; ncases f; 
##[ #a e; @a; nwhd in e:(??%?); /2/;
##| #H; nwhd in H:(??%?); ndestruct (H);
##] nqed.

nremark bind2_inversion:
  ∀A,B,C: Type. ∀f: res (A×B). ∀g: A → B → res C. ∀z: C.
  bind2 ??? f g = OK ? z →
  ∃x. ∃y. f = OK ? 〈x, y〉 ∧ g x y = OK ? z.
#A B C f g z; ncases f;
##[ #ab; ncases ab; #a b e; @a; @b; nwhd in e:(??%?); /2/;
##| #H; nwhd in H:(??%?); ndestruct
##] nqed. 

(*
Open Local Scope error_monad_scope.

(** This is the familiar monadic map iterator. *)

Fixpoint mmap (A B: Type) (f: A -> res B) (l: list A) {struct l} : res (list B) :=
  match l with
  | nil => OK nil
  | hd :: tl => do hd' <- f hd; do tl' <- mmap f tl; OK (hd' :: tl')
  end.

Remark mmap_inversion:
  forall (A B: Type) (f: A -> res B) (l: list A) (l': list B),
  mmap f l = OK l' ->
  list_forall2 (fun x y => f x = OK y) l l'.
Proof.
  induction l; simpl; intros.
  inversion_clear H. constructor.
  destruct (bind_inversion _ _ H) as [hd' [P Q]].
  destruct (bind_inversion _ _ Q) as [tl' [R S]].
  inversion_clear S.
  constructor. auto. auto. 
Qed.

(** * Reasoning over monadic computations *)

(** The [monadInv H] tactic below simplifies hypotheses of the form
<<
        H: (do x <- a; b) = OK res
>>
    By definition of the bind operation, both computations [a] and
    [b] must succeed for their composition to succeed.  The tactic
    therefore generates the following hypotheses:

         x: ...
        H1: a = OK x
        H2: b x = OK res
*)

Ltac monadInv1 H :=
  match type of H with
  | (OK _ = OK _) =>
      inversion H; clear H; try subst
  | (Error _ = OK _) =>
      discriminate
  | (bind ?F ?G = OK ?X) =>
      let x := fresh "x" in (
      let EQ1 := fresh "EQ" in (
      let EQ2 := fresh "EQ" in (
      destruct (bind_inversion F G H) as [x [EQ1 EQ2]];
      clear H;
      try (monadInv1 EQ2))))
  | (bind2 ?F ?G = OK ?X) =>
      let x1 := fresh "x" in (
      let x2 := fresh "x" in (
      let EQ1 := fresh "EQ" in (
      let EQ2 := fresh "EQ" in (
      destruct (bind2_inversion F G H) as [x1 [x2 [EQ1 EQ2]]];
      clear H;
      try (monadInv1 EQ2)))))
  | (mmap ?F ?L = OK ?M) =>
      generalize (mmap_inversion F L H); intro
  end.

Ltac monadInv H :=
  match type of H with
  | (OK _ = OK _) => monadInv1 H
  | (Error _ = OK _) => monadInv1 H
  | (bind ?F ?G = OK ?X) => monadInv1 H
  | (bind2 ?F ?G = OK ?X) => 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.
*)