(* *********************************************************************)
(*                                                                     *)
(*              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 "basics/types.ma".
include "basics/logic.ma".
include "basics/list.ma".
include "common/PreIdentifiers.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[0] :=
  | MSG: String → errcode
  | CTX: ∀tag:String. identifier tag → errcode.

definition errmsg: Type[0] ≝ list errcode.

definition msg : String → errmsg ≝ λs. [MSG s].

(* * * 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. *)

inductive res (A: Type[0]) : Type[0] ≝
| OK: A → res A
| Error: errmsg → res A.

(*Implicit Arguments Error [A].*)

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

definition bind ≝ λA,B:Type[0]. λf: res A. λg: A → res B.
  match f with
  [ OK x ⇒ g x
  | Error msg ⇒ Error ? msg
  ].

definition bind2 ≝ λA,B,C: Type[0]. λf: res (A × B). λg: A → B → res C.
  match f with
  [ OK v ⇒ match v with [ pair x y ⇒ g x y ]
  | Error msg => Error ? msg
  ].

definition bind3 ≝ λA,B,C,D: Type[0]. λf: res (A × B × C). λg: A → B → C → res D.
  match f with
  [ OK v ⇒ match v with [ pair xy z ⇒ match xy with [ pair x y ⇒ g x y z ] ]
  | 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 Prod bind" 'bind2 e f = (bind2 ??? e f).
notation > "vbox('do' 〈ident v1, ident v2, ident v3〉 ← e; break e')" with precedence 40 for @{'bind3 ${e} (λ${ident v1}.λ${ident v2}.λ${ident v3}.${e'})}.
notation > "vbox('do' 〈ident v1 : ty1, ident v2 : ty2, ident v3 : ty3〉 ← e; break e')" with precedence 40 for @{'bind3 ${e} (λ${ident v1} : ${ty1}.λ${ident v2} : ${ty2}.λ${ident v3} : ${ty3}.${e'})}.
notation < "vbox('do' \nbsp 〈ident v1, ident v2, ident v3〉 ← e; break e')" with precedence 40 for @{'bind3 ${e} (λ${ident v1}.λ${ident v2}.λ${ident v3}.${e'})}.
notation < "vbox('do' \nbsp 〈ident v1 : ty1, ident v2 : ty2, ident v3 : ty3〉 ← e; break e')" with precedence 40 for @{'bind3 ${e} (λ${ident v1} : ${ty1}.λ${ident v2} : ${ty2}.λ${ident v3} : ${ty3}.${e'})}.
interpretation "error monad triple bind" 'bind3 e f = (bind3 ???? 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.
*)
lemma bind_inversion:
  ∀A,B: Type[0]. ∀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 cases f; 
[ #a #e %{a} whd in e:(??%?); /2/;
| #m #H whd in H:(??%?); destruct (H);
] qed.

lemma bind2_inversion:
  ∀A,B,C: Type[0]. ∀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 cases f;
[ #ab cases ab; #a #b #e %{a} %{b} whd in e:(??%?); /2/;
| #m #H whd in H:(??%?); destruct
] qed. 

(*
Open Local Scope error_monad_scope.

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

let rec mmap (A, B: Type[0]) (f: A → res B) (l: list A) on l : res (list B) ≝
  match l with
  [ nil ⇒ OK ? []
  | cons hd tl ⇒ do hd' ← f hd; do tl' ← mmap ?? f tl; OK ? (hd'::tl')
  ].

(*
lemma mmap_inversion:
  ∀A, B: Type[0]. ∀f: A -> res B. ∀l: list A. ∀l': list B.
  mmap A B 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.
*)

(* A monadic fold_left2 *)

axiom WrongLength: String.

let rec mfold_left2
  (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C → res A) (accu: res A)
  (left: list B) (right: list C) on left: res A ≝
  match left with
  [ nil ⇒
    match right with
    [ nil ⇒ accu
    | cons hd tl ⇒ Error ? (msg WrongLength)
    ]
  | cons hd tl ⇒
    match right with
    [ nil ⇒ Error ? (msg WrongLength)
    | cons hd' tl' ⇒
       do accu ← accu;
       mfold_left2 … f (f accu hd hd') tl tl'
    ]
  ].

(*
(** * 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.
*)


definition opt_to_res ≝ λA.λmsg.λv:option A. match v with [ None ⇒ Error A msg | Some v ⇒ OK A v ].
lemma opt_OK: ∀A,m,P,e.
  (∀v. e = Some ? v → P v) →
  match opt_to_res A m e with [ Error _ ⇒ True | OK v ⇒ P v ].
#A #m #P #e elim e; /2/;
qed.

definition res_to_opt : ∀A:Type[0]. res A → option A ≝
 λA.λv. match v with [ Error _ ⇒ None ? | OK v ⇒ Some … v].