(* Title: Pure/type_infer.ML
ID: $Id$
Author: Stefan Berghofer and Markus Wenzel, TU Muenchen
Type inference.
*)
signature TYPE_INFER =
sig
val infer_types: (term -> Pretty.T) -> (typ -> Pretty.T)
-> (string -> typ option) -> Sorts.classrel -> Sorts.arities
-> string list -> bool -> (indexname -> bool) -> term list -> typ list
-> term list * typ list * (indexname * typ) list
val appl_error: (term -> Pretty.T) -> (typ -> Pretty.T)
-> string -> term -> typ -> term -> typ -> string list
end;
structure TypeInfer: TYPE_INFER =
struct
(** term encodings **)
(*
Flavours of term encodings:
parse trees (type term):
A very complicated structure produced by the syntax module's
read functions. Encodes types and sorts as terms; may contain
explicit constraints and partial typing information (where
dummyT serves as wildcard).
Parse trees are INTERNAL! Users should never encounter them,
except in parse / print translation functions.
raw terms (type term):
Provide the user interface to type inferences. They may contain
partial type information (dummyT is wildcard) or explicit type
constraints (introduced via constrain: term -> typ -> term).
The type inference function also lets users specify a certain
subset of TVars to be treated as non-rigid inference parameters.
preterms (type preterm):
The internal representation for type inference.
well-typed term (type term):
Fully typed lambda terms to be accepted by appropriate
certification functions.
*)
(** pretyps and preterms **)
(*links to parameters may get instantiated, anything else is rigid*)
datatype pretyp =
PType of string * pretyp list |
PTFree of string * sort |
PTVar of indexname * sort |
Param of sort |
Link of pretyp ref;
datatype preterm =
PConst of string * pretyp |
PFree of string * pretyp |
PVar of indexname * pretyp |
PBound of int |
PAbs of string * pretyp * preterm |
PAppl of preterm * preterm |
Constraint of preterm * pretyp;
(* utils *)
val mk_param = Link o ref o Param;
fun deref (T as Link (ref (Param _))) = T
| deref (Link (ref T)) = deref T
| deref T = T;
fun foldl_pretyps f (x, PConst (_, T)) = f (x, T)
| foldl_pretyps f (x, PFree (_, T)) = f (x, T)
| foldl_pretyps f (x, PVar (_, T)) = f (x, T)
| foldl_pretyps _ (x, PBound _) = x
| foldl_pretyps f (x, PAbs (_, T, t)) = foldl_pretyps f (f (x, T), t)
| foldl_pretyps f (x, PAppl (t, u)) = foldl_pretyps f (foldl_pretyps f (x, t), u)
| foldl_pretyps f (x, Constraint (t, T)) = f (foldl_pretyps f (x, t), T);
(** raw typs/terms to pretyps/preterms **)
(* pretyp(s)_of *)
fun pretyp_of is_param (params, typ) =
let
fun add_parms (ps, TVar (xi as (x, _), S)) =
if is_param xi andalso is_none (assoc (ps, xi))
then (xi, mk_param S) :: ps else ps
| add_parms (ps, TFree _) = ps
| add_parms (ps, Type (_, Ts)) = foldl add_parms (ps, Ts);
val params' = add_parms (params, typ);
fun pre_of (TVar (v as (xi, _))) =
(case assoc (params', xi) of
None => PTVar v
| Some p => p)
| pre_of (TFree v) = PTFree v
| pre_of (T as Type (a, Ts)) =
if T = dummyT then mk_param []
else PType (a, map pre_of Ts);
in (params', pre_of typ) end;
fun pretyps_of is_param = foldl_map (pretyp_of is_param);
(* preterm(s)_of *)
fun preterm_of const_type is_param ((vparams, params), tm) =
let
fun add_vparm (ps, xi) =
if is_none (assoc (ps, xi)) then
(xi, mk_param []) :: ps
else ps;
fun add_vparms (ps, Var (xi, _)) = add_vparm (ps, xi)
| add_vparms (ps, Free (x, _)) = add_vparm (ps, (x, ~1))
| add_vparms (ps, Abs (_, _, t)) = add_vparms (ps, t)
| add_vparms (ps, t $ u) = add_vparms (add_vparms (ps, t), u)
| add_vparms (ps, _) = ps;
val vparams' = add_vparms (vparams, tm);
fun var_param xi = the (assoc (vparams', xi));
val preT_of = pretyp_of is_param;
fun constrain (ps, t) T =
if T = dummyT then (ps, t)
else
let val (ps', T') = preT_of (ps, T) in
(ps', Constraint (t, T'))
end;
fun pre_of (ps, Const (c, T)) =
(case const_type c of
Some U => constrain (ps, PConst (c, snd (pretyp_of (K true) ([], U)))) T
| None => raise TYPE ("No such constant: " ^ quote c, [], []))
| pre_of (ps, Free (x, T)) = constrain (ps, PFree (x, var_param (x, ~1))) T
| pre_of (ps, Var (xi, T)) = constrain (ps, PVar (xi, var_param xi)) T
| pre_of (ps, Const ("_type_constraint_", T) $ t) = constrain (pre_of (ps, t)) T
| pre_of (ps, Bound i) = (ps, PBound i)
| pre_of (ps, Abs (x, T, t)) =
let
val (ps', T') = preT_of (ps, T);
val (ps'', t') = pre_of (ps', t);
in (ps'', PAbs (x, T', t')) end
| pre_of (ps, t $ u) =
let
val (ps', t') = pre_of (ps, t);
val (ps'', u') = pre_of (ps', u);
in (ps'', PAppl (t', u')) end;
val (params', tm') = pre_of (params, tm);
in
((vparams', params'), tm')
end;
fun preterms_of const_type is_param = foldl_map (preterm_of const_type is_param);
(** pretyps/terms to typs/terms **)
(* add_parms *)
fun add_parmsT (rs, PType (_, Ts)) = foldl add_parmsT (rs, Ts)
| add_parmsT (rs, Link (r as ref (Param _))) = r ins rs
| add_parmsT (rs, Link (ref T)) = add_parmsT (rs, T)
| add_parmsT (rs, _) = rs;
val add_parms = foldl_pretyps add_parmsT;
(* add_names *)
fun add_namesT (xs, PType (_, Ts)) = foldl add_namesT (xs, Ts)
| add_namesT (xs, PTFree (x, _)) = x ins xs
| add_namesT (xs, PTVar ((x, _), _)) = x ins xs
| add_namesT (xs, Link (ref T)) = add_namesT (xs, T)
| add_namesT (xs, Param _) = xs;
val add_names = foldl_pretyps add_namesT;
(* simple_typ/term_of *)
(*deref links, fail on params*)
fun simple_typ_of (PType (a, Ts)) = Type (a, map simple_typ_of Ts)
| simple_typ_of (PTFree v) = TFree v
| simple_typ_of (PTVar v) = TVar v
| simple_typ_of (Link (ref T)) = simple_typ_of T
| simple_typ_of (Param _) = sys_error "simple_typ_of: illegal Param";
(*convert types, drop constraints*)
fun simple_term_of (PConst (c, T)) = Const (c, simple_typ_of T)
| simple_term_of (PFree (x, T)) = Free (x, simple_typ_of T)
| simple_term_of (PVar (xi, T)) = Var (xi, simple_typ_of T)
| simple_term_of (PBound i) = Bound i
| simple_term_of (PAbs (x, T, t)) = Abs (x, simple_typ_of T, simple_term_of t)
| simple_term_of (PAppl (t, u)) = simple_term_of t $ simple_term_of u
| simple_term_of (Constraint (t, _)) = simple_term_of t;
(* typs_terms_of *) (*DESTRUCTIVE*)
fun typs_terms_of used mk_var prfx (Ts, ts) =
let
fun elim (r as ref (Param S), x) = r := mk_var (x, S)
| elim _ = ();
val used' = foldl add_names (foldl add_namesT (used, Ts), ts);
val parms = rev (foldl add_parms (foldl add_parmsT ([], Ts), ts));
val pre_names = replicate (length parms) (prfx ^ "'");
val names = variantlist (pre_names, prfx ^ "'" :: used');
in
seq2 elim (parms, names);
(map simple_typ_of Ts, map simple_term_of ts)
end;
(** order-sorted unification of types **) (*DESTRUCTIVE*)
exception NO_UNIFIER of string;
fun unify classrel arities =
let
(* adjust sorts of parameters *)
fun not_in_sort x S' S =
"Variable " ^ x ^ "::" ^ Sorts.str_of_sort S' ^ " not of sort " ^
Sorts.str_of_sort S ^ ".";
fun meet (_, []) = ()
| meet (Link (r as (ref (Param S'))), S) =
if Sorts.sort_le classrel (S', S) then ()
else r := mk_param (Sorts.inter_sort classrel (S', S))
| meet (Link (ref T), S) = meet (T, S)
| meet (PType (a, Ts), S) =
seq2 meet (Ts, Sorts.mg_domain classrel arities a S
handle TYPE (msg, _, _) => raise NO_UNIFIER msg)
| meet (PTFree (x, S'), S) =
if Sorts.sort_le classrel (S', S) then ()
else raise NO_UNIFIER (not_in_sort x S' S)
| meet (PTVar (xi, S'), S) =
if Sorts.sort_le classrel (S', S) then ()
else raise NO_UNIFIER (not_in_sort (Syntax.string_of_vname xi) S' S)
| meet (Param _, _) = sys_error "meet";
(* occurs check and assigment *)
fun occurs_check r (Link (r' as ref T)) =
if r = r' then raise NO_UNIFIER "Occurs check!"
else occurs_check r T
| occurs_check r (PType (_, Ts)) = seq (occurs_check r) Ts
| occurs_check _ _ = ();
fun assign r T S =
(case deref T of
T' as Link (r' as ref (Param _)) =>
if r = r' then () else (r := T'; meet (T', S))
| T' => (occurs_check r T'; r := T'; meet (T', S)));
(* unification *)
fun unif (Link (r as ref (Param S)), T) = assign r T S
| unif (T, Link (r as ref (Param S))) = assign r T S
| unif (Link (ref T), U) = unif (T, U)
| unif (T, Link (ref U)) = unif (T, U)
| unif (PType (a, Ts), PType (b, Us)) =
if a <> b then
raise NO_UNIFIER ("Clash of types " ^ quote a ^ " and " ^ quote b ^ ".")
else seq2 unif (Ts, Us)
| unif (T, U) = if T = U then () else raise NO_UNIFIER "";
in unif end;
(** type inference **)
fun appl_error prt prT why t T u U =
["Type error in application: " ^ why,
"",
Pretty.string_of
(Pretty.block [Pretty.str "Operator:", Pretty.brk 2, prt t,
Pretty.str " ::", Pretty.brk 1, prT T]),
Pretty.string_of
(Pretty.block [Pretty.str "Operand:", Pretty.brk 3, prt u,
Pretty.str " ::", Pretty.brk 1, prT U]),
""];
(* infer *) (*DESTRUCTIVE*)
fun infer prt prT classrel arities =
let
(* errors *)
fun unif_failed msg =
"Type unification failed" ^ (if msg = "" then "." else ": " ^ msg) ^ "\n";
fun prep_output bs ts Ts =
let
val (Ts_bTs', ts') = typs_terms_of [] PTFree "??" (Ts @ map snd bs, ts);
val len = length Ts;
val Ts' = take (len, Ts_bTs');
val xs = map Free (map fst bs ~~ drop (len, Ts_bTs'));
val ts'' = map (fn t => subst_bounds (xs, t)) ts';
in (ts'', Ts') end;
fun err_loose i =
raise TYPE ("Loose bound variable: B." ^ string_of_int i, [], []);
fun err_appl msg bs t T u U =
let
val ([t', u'], [T', U']) = prep_output bs [t, u] [T, U];
val why =
(case T' of
Type ("fun", _) => "Incompatible operand type."
| _ => "Operator not of function type.");
val text = unif_failed msg ^
cat_lines (appl_error prt prT why t' T' u' U');
in raise TYPE (text, [T', U'], [t', u']) end;
fun err_constraint msg bs t T U =
let
val ([t'], [T', U']) = prep_output bs [t] [T, U];
val text = cat_lines
[unif_failed msg,
"Cannot meet type constraint:", "",
Pretty.string_of
(Pretty.block [Pretty.str "Term:", Pretty.brk 2, prt t',
Pretty.str " ::", Pretty.brk 1, prT T']),
Pretty.string_of
(Pretty.block [Pretty.str "Type:", Pretty.brk 2, prT U']), ""];
in raise TYPE (text, [T', U'], [t']) end;
(* main *)
val unif = unify classrel arities;
fun inf _ (PConst (_, T)) = T
| inf _ (PFree (_, T)) = T
| inf _ (PVar (_, T)) = T
| inf bs (PBound i) = snd (nth_elem (i, bs) handle LIST _ => err_loose i)
| inf bs (PAbs (x, T, t)) = PType ("fun", [T, inf ((x, T) :: bs) t])
| inf bs (PAppl (t, u)) =
let
val T = inf bs t;
val U = inf bs u;
val V = mk_param [];
val U_to_V = PType ("fun", [U, V]);
val _ = unif (U_to_V, T) handle NO_UNIFIER msg => err_appl msg bs t T u U;
in V end
| inf bs (Constraint (t, U)) =
let val T = inf bs t in
unif (T, U) handle NO_UNIFIER msg => err_constraint msg bs t T U;
T
end;
in inf [] end;
(* infer_types *)
fun infer_types prt prT const_type classrel arities used freeze is_param ts Ts =
let
(*convert to preterms/typs*)
val (Tps, Ts') = pretyps_of (K true) ([], Ts);
val ((vps, ps), ts') = preterms_of const_type is_param (([], Tps), ts);
(*run type inference*)
val tTs' = ListPair.map Constraint (ts', Ts');
val _ = seq (fn t => (infer prt prT classrel arities t; ())) tTs';
(*collect result unifier*)
fun ch_var (xi, Link (r as ref (Param S))) = (r := PTVar (xi, S); None)
| ch_var xi_T = Some xi_T;
val env = mapfilter ch_var Tps;
(*convert back to terms/typs*)
val mk_var =
if freeze then PTFree
else (fn (x, S) => PTVar ((x, 0), S));
val (final_Ts, final_ts) = typs_terms_of used mk_var "" (Ts', ts');
val final_env = map (apsnd simple_typ_of) env;
in
(final_ts, final_Ts, final_env)
end;
end;