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(** Error reporting and the error monad. *)
Require Import String.
Require Import Coqlib.
Close Scope string_scope.
Set Implicit Arguments.
(** * 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: Set :=
| MSG: string -> errcode
| CTX: positive -> errcode.
Definition errmsg: Set := 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. *)
Inductive res (A: Set) : Set :=
| 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 (A B: Set) (f: res A) (g: A -> res B) : res B :=
match f with
| OK x => g x
| Error msg => Error msg
end.
Definition bind2 (A B C: Set) (f: res (A * B)) (g: A -> B -> res C) : res C :=
match f with
| OK (x, y) => g x y
| Error msg => Error msg
end.
(** 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.
Remark bind_inversion:
forall (A B: Set) (f: res A) (g: A -> res B) (y: B),
bind f g = OK y ->
exists x, f = OK x /\ g x = OK y.
Proof.
intros until y. destruct f; simpl; intros.
exists a; auto.
discriminate.
Qed.
Remark bind2_inversion:
forall (A B C: Set) (f: res (A*B)) (g: A -> B -> res C) (z: C),
bind2 f g = OK z ->
exists x, exists y, f = OK (x, y) /\ g x y = OK z.
Proof.
intros until z. destruct f; simpl.
destruct p; simpl; intros. exists a; exists b; auto.
intros; discriminate.
Qed.
Open Local Scope error_monad_scope.
(** This is the familiar monadic map iterator. *)
Fixpoint mmap (A B: Set) (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: Set) (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.
|
