# ImpParserLexing and Parsing in Coq

(* \$Date: 2013-07-01 18:48:47 -0400 (Mon, 01 Jul 2013) \$ *)

The development of the Imp language in Imp.v completely ignores issues of concrete syntax — how an ascii string that a programmer might write gets translated into the abstract syntax trees defined by the datatypes aexp, bexp, and com. In this file we illustrate how the rest of the story can be filled in by building a simple lexical analyzer and parser using Coq's functional programming facilities.
This development is not intended to be understood in detail: the explanations are fairly terse and there are no exercises. The main point is simply to demonstrate that it can be done. You are invited to look through the code — most of it is not very complicated, though the parser relies on some "monadic" programming idioms that may require a little work to make out — but most readers will probably want to just skip down to the Examples section at the very end to get the punchline.

# Internals

Require Import SfLib.
Require Import Imp.

Require Import String.
Require Import Ascii.

Open Scope list_scope.

## Lexical Analysis

Definition isWhite (c : ascii) : bool :=
let n := nat_of_ascii c in
orb (orb (beq_nat n 32) (* space *)
(beq_nat n 9)) (* tab *)
(orb (beq_nat n 10) (* linefeed *)
(beq_nat n 13)). (* Carriage return. *)

Notation "x '<=?' y" := (ble_nat x y)
(at level 70, no associativity) : nat_scope.

Definition isLowerAlpha (c : ascii) : bool :=
let n := nat_of_ascii c in
andb (97 <=? n) (n <=? 122).

Definition isAlpha (c : ascii) : bool :=
let n := nat_of_ascii c in
orb (andb (65 <=? n) (n <=? 90))
(andb (97 <=? n) (n <=? 122)).

Definition isDigit (c : ascii) : bool :=
let n := nat_of_ascii c in
andb (48 <=? n) (n <=? 57).

Inductive chartype := white | alpha | digit | other.

Definition classifyChar (c : ascii) : chartype :=
if isWhite c then
white
else if isAlpha c then
alpha
else if isDigit c then
digit
else
other.

Fixpoint list_of_string (s : string) : list ascii :=
match s with
| EmptyString ⇒ []
| String c sc :: (list_of_string s)
end.

Fixpoint string_of_list (xs : list ascii) : string :=
fold_right String EmptyString xs.

Definition token := string.

Fixpoint tokenize_helper (cls : chartype) (acc xs : list ascii)
: list (list ascii) :=
let tk := match acc with [] ⇒ [] | _::_ ⇒ [rev acc] end in
match xs with
| [] ⇒ tk
| (x::xs') ⇒
match cls, classifyChar x, x with
| _, _, "(" ⇒ tk ++ ["("]::(tokenize_helper other [] xs')
| _, _, ")" ⇒ tk ++ [")"]::(tokenize_helper other [] xs')
| _, white, _tk ++ (tokenize_helper white [] xs')
| alpha,alpha,xtokenize_helper alpha (x::acc) xs'
| digit,digit,xtokenize_helper digit (x::acc) xs'
| other,other,xtokenize_helper other (x::acc) xs'
| _,tp,xtk ++ (tokenize_helper tp [x] xs')
end
end %char.

Definition tokenize (s : string) : list string :=
map string_of_list (tokenize_helper white [] (list_of_string s)).

Example tokenize_ex1 :
tokenize "abc12==3 223*(3+(a+c))" %string
= ["abc"; "12"; "=="; "3"; "223";
"×"; "("; "3"; "+"; "(";
"a"; "+"; "c"; ")"; ")"]%string.
Proof. reflexivity. Qed.

## Parsing

### Options with Errors

(* An option with error messages. *)
Inductive optionE (X:Type) : Type :=
| SomeE : X optionE X
| NoneE : string optionE X.

Implicit Arguments SomeE [[X]].
Implicit Arguments NoneE [[X]].

(* Some syntactic sugar to make writing nested match-expressions on
optionE more convenient. *)

Notation "'DO' ( x , y ) <== e1 ; e2"
:= (match e1 with
| SomeE (x,y) ⇒ e2
| NoneE errNoneE err
end)
(right associativity, at level 60).

Notation "'DO' ( x , y ) <-- e1 ; e2 'OR' e3"
:= (match e1 with
| SomeE (x,y) ⇒ e2
| NoneE erre3
end)
(right associativity, at level 60, e2 at next level).

### Symbol Table

(* Build a mapping from tokens to nats.  A real parser would do
this incrementally as it encountered new symbols, but passing
around the symbol table inside the parsing functions is a bit
inconvenient, so instead we do it as a first pass. *)

Fixpoint build_symtable (xs : list token) (n : nat) : (token nat) :=
match xs with
| [] ⇒ (fun sn)
| x::xs
if (forallb isLowerAlpha (list_of_string x))
then (fun sif string_dec s x then n else (build_symtable xs (S n) s))
else build_symtable xs n
end.

### Generic Combinators for Building Parsers

Open Scope string_scope.

Definition parser (T : Type) :=
list token optionE (T × list token).

Fixpoint many_helper {T} (p : parser T) acc steps xs :=
match steps, p xs with
| 0, _NoneE "Too many recursive calls"
| _, NoneE _SomeE ((rev acc), xs)
| S steps', SomeE (t, xs') ⇒ many_helper p (t::acc) steps' xs'
end.

(* A (step-indexed) parser which expects zero or more ps *)
Fixpoint many {T} (p : parser T) (steps : nat) : parser (list T) :=
many_helper p [] steps.

(* A parser which expects a given token, followed by p *)
Definition firstExpect {T} (t : token) (p : parser T) : parser T :=
fun xsmatch xs with
| x::xs'if string_dec x t
then p xs'
else NoneE ("expected '" ++ t ++ "'.")
| [] ⇒ NoneE ("expected '" ++ t ++ "'.")
end.

(* A parser which expects a particular token *)
Definition expect (t : token) : parser unit :=
firstExpect t (fun xsSomeE(tt, xs)).

### A Recursive-Descent Parser for Imp

(* Identifiers *)
Definition parseIdentifier (symtable :stringnat) (xs : list token)
: optionE (id × list token) :=
match xs with
| [] ⇒ NoneE "Expected identifier"
| x::xs'
if forallb isLowerAlpha (list_of_string x) then
SomeE (Id (symtable x), xs')
else
NoneE ("Illegal identifier:'" ++ x ++ "'")
end.

(* Numbers *)
Definition parseNumber (xs : list token) : optionE (nat × list token) :=
match xs with
| [] ⇒ NoneE "Expected number"
| x::xs'
if forallb isDigit (list_of_string x) then
SomeE (fold_left (fun n d
10 × n + (nat_of_ascii d - nat_of_ascii "0"%char))
(list_of_string x)
0,
xs')
else
NoneE "Expected number"
end.

(* Parse arithmetic expressions *)
Fixpoint parsePrimaryExp (steps:nat) symtable (xs : list token)
: optionE (aexp × list token) :=
match steps with
| 0 ⇒ NoneE "Too many recursive calls"
| S steps'
DO (i, rest) <-- parseIdentifier symtable xs ;
SomeE (AId i, rest)
OR DO (n, rest) <-- parseNumber xs ;
SomeE (ANum n, rest)
OR (DO (e, rest) <== firstExpect "(" (parseSumExp steps' symtable) xs;
DO (u, rest') <== expect ")" rest ;
SomeE(e,rest'))
end
with parseProductExp (steps:nat) symtable (xs : list token) :=
match steps with
| 0 ⇒ NoneE "Too many recursive calls"
| S steps'
DO (e, rest) <==
parsePrimaryExp steps' symtable xs ;
DO (es, rest') <==
many (firstExpect "×" (parsePrimaryExp steps' symtable)) steps' rest;
SomeE (fold_left AMult es e, rest')
end
with parseSumExp (steps:nat) symtable (xs : list token) :=
match steps with
| 0 ⇒ NoneE "Too many recursive calls"
| S steps'
DO (e, rest) <==
parseProductExp steps' symtable xs ;
DO (es, rest') <==
many (fun xs
DO (e,rest') <--
firstExpect "+" (parseProductExp steps' symtable) xs;
SomeE ( (true, e), rest')
OR DO (e,rest') <==
firstExpect "-" (parseProductExp steps' symtable) xs;
SomeE ( (false, e), rest'))
steps' rest;
SomeE (fold_left (fun e0 term
match term with
(true, e) ⇒ APlus e0 e
| (false, e) ⇒ AMinus e0 e
end)
es e,
rest')
end.

Definition parseAExp := parseSumExp.

(* Parsing boolean expressions. *)
Fixpoint parseAtomicExp (steps:nat) (symtable : stringnat) (xs : list token) :=
match steps with
| 0 ⇒ NoneE "Too many recursive calls"
| S steps'
DO (u,rest) <-- expect "true" xs;
SomeE (BTrue,rest)
OR DO (u,rest) <-- expect "false" xs;
SomeE (BFalse,rest)
OR DO (e,rest) <-- firstExpect "not" (parseAtomicExp steps' symtable) xs;
SomeE (BNot e, rest)
OR DO (e,rest) <-- firstExpect "(" (parseConjunctionExp steps' symtable) xs;
(DO (u,rest') <== expect ")" rest; SomeE (e, rest'))
OR DO (e, rest) <== parseProductExp steps' symtable xs ;
(DO (e', rest') <--
firstExpect "==" (parseAExp steps' symtable) rest ;
SomeE (BEq e e', rest')
OR DO (e', rest') <--
firstExpect "≤" (parseAExp steps' symtable) rest ;
SomeE (BLe e e', rest')
OR
NoneE "Expected '==' or '≤' after arithmetic expression")
end
with parseConjunctionExp (steps:nat) (symtable : stringnat) (xs : list token) :=
match steps with
| 0 ⇒ NoneE "Too many recursive calls"
| S steps'
DO (e, rest) <==
parseAtomicExp steps' symtable xs ;
DO (es, rest') <==
many (firstExpect "&&" (parseAtomicExp steps' symtable)) steps' rest;
SomeE (fold_left BAnd es e, rest')
end.

Definition parseBExp := parseConjunctionExp.

(*
Eval compute in
(parseProductExp 100 (tokenize "x*y*(x*x)*x")).

Eval compute in
(parseDisjunctionExp 100 (tokenize "not((x==x||x*x<=(x*x)*x)&&x==x)")).
*)

(* Parsing commands *)
Fixpoint parseSimpleCommand (steps:nat) (symtable:stringnat) (xs : list token) :=
match steps with
| 0 ⇒ NoneE "Too many recursive calls"
| S steps'
DO (u, rest) <-- expect "SKIP" xs;
SomeE (SKIP, rest)
OR DO (e,rest) <--
firstExpect "IF" (parseBExp steps' symtable) xs;
DO (c,rest') <==
firstExpect "THEN" (parseSequencedCommand steps' symtable) rest;
DO (c',rest'') <==
firstExpect "ELSE" (parseSequencedCommand steps' symtable) rest';
DO (u,rest''') <==
expect "END" rest'';
SomeE(IFB e THEN c ELSE c' FI, rest''')
OR DO (e,rest) <--
firstExpect "WHILE" (parseBExp steps' symtable) xs;
DO (c,rest') <==
firstExpect "DO" (parseSequencedCommand steps' symtable) rest;
DO (u,rest'') <==
expect "END" rest';
SomeE(WHILE e DO c END, rest'')
OR DO (i, rest) <==
parseIdentifier symtable xs;
DO (e, rest') <==
firstExpect ":=" (parseAExp steps' symtable) rest;
SomeE(i ::= e, rest')
end

with parseSequencedCommand (steps:nat) (symtable:stringnat) (xs : list token) :=
match steps with
| 0 ⇒ NoneE "Too many recursive calls"
| S steps'
DO (c, rest) <==
parseSimpleCommand steps' symtable xs;
DO (c', rest') <--
firstExpect ";;" (parseSequencedCommand steps' symtable) rest;
SomeE(c ;; c', rest')
OR
SomeE(c, rest)
end.

Definition bignumber := 1000.

Definition parse (str : string) : optionE (com × list token) :=
let tokens := tokenize str in
parseSequencedCommand bignumber (build_symtable tokens 0) tokens.

# Examples

(*
Eval compute in parse "
IF x == y + 1 + 2 - y * 6 + 3 THEN
x := x * 1;;
y := 0
ELSE
SKIP
END  ".
====>
SomeE
(IFB BEq (AId (Id 0))
(APlus
(AMinus (APlus (APlus (AId (Id 1)) (ANum 1)) (ANum 2))
(AMult (AId (Id 1)) (ANum 6)))
(ANum 3))
THEN Id 0 ::= AMult (AId (Id 0)) (ANum 1);; Id 1 ::= ANum 0
ELSE SKIP FI, )
*)

(*
Eval compute in parse "
SKIP;;
z:=x*y*(x*x);;
WHILE x==x DO
IF z <= z*z && not x == 2 THEN
x := z;;
y := z
ELSE
SKIP
END;;
SKIP
END;;
x:=z  ".
====>
SomeE
(SKIP;;
Id 0 ::= AMult (AMult (AId (Id 1)) (AId (Id 2)))
(AMult (AId (Id 1)) (AId (Id 1)));;
WHILE BEq (AId (Id 1)) (AId (Id 1)) DO
IFB BAnd (BLe (AId (Id 0)) (AMult (AId (Id 0)) (AId (Id 0))))
(BNot (BEq (AId (Id 1)) (ANum 2)))
THEN Id 1 ::= AId (Id 0);; Id 2 ::= AId (Id 0)
ELSE SKIP FI;;
SKIP
END;;
Id 1 ::= AId (Id 0),

*)

(*
Eval compute in parse "
SKIP;;
z:=x*y*(x*x);;
WHILE x==x DO
IF z <= z*z && not x == 2 THEN
x := z;;
y := z
ELSE
SKIP
END;;
SKIP
END;;
x:=z  ".
=====>
SomeE
(SKIP;;
Id 0 ::= AMult (AMult (AId (Id 1)) (AId (Id 2)))
(AMult (AId (Id 1)) (AId (Id 1)));;
WHILE BEq (AId (Id 1)) (AId (Id 1)) DO
IFB BAnd (BLe (AId (Id 0)) (AMult (AId (Id 0)) (AId (Id 0))))
(BNot (BEq (AId (Id 1)) (ANum 2)))
THEN Id 1 ::= AId (Id 0);;
Id 2 ::= AId (Id 0)
ELSE SKIP
FI;;
SKIP
END;;
Id 1 ::= AId (Id 0),
).
*)