(* Title: Pure/type_infer.ML
ID: $Id$
Author: Stefan Berghofer and Markus Wenzel, TU Muenchen
Type inference.
*)
signature TYPE_INFER =
sig
val anyT: sort -> typ
val logicT: typ
val mixfixT: Syntax.mixfix -> typ
val polymorphicT: typ -> typ
val appl_error: Pretty.pp -> string -> term -> typ -> term -> typ -> string list
val constrain: term -> typ -> term
val param: int -> string * sort -> typ
val paramify_dummies: typ -> int -> typ * int
val get_sort: Type.tsig -> (indexname -> sort option) -> (sort -> sort)
-> (indexname * sort) list -> indexname -> sort
val infer_types: Pretty.pp
-> Type.tsig -> (string -> typ option) -> (indexname -> typ option)
-> (indexname -> sort option) -> (string -> string) -> (typ -> typ)
-> (sort -> sort) -> string list -> bool -> typ list -> term list
-> term list * (indexname * typ) 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
dummies serve as wildcards).
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 (dummies are wildcards) 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 fold_pretyps f (PConst (_, T)) x = f T x
| fold_pretyps f (PFree (_, T)) x = f T x
| fold_pretyps f (PVar (_, T)) x = f T x
| fold_pretyps _ (PBound _) x = x
| fold_pretyps f (PAbs (_, T, t)) x = fold_pretyps f t (f T x)
| fold_pretyps f (PAppl (t, u)) x = fold_pretyps f u (fold_pretyps f t x)
| fold_pretyps f (Constraint (t, T)) x = f T (fold_pretyps f t x);
(** raw typs/terms to pretyps/preterms **)
(* pretyp(s)_of *)
fun anyT S = TFree ("'_dummy_", S);
val logicT = anyT [];
fun mixfixT (Binder _) = (logicT --> logicT) --> logicT
| mixfixT mx = replicate (Syntax.mixfix_args mx) logicT ---> logicT;
(*indicate polymorphic Vars*)
fun polymorphicT T = Type ("_polymorphic_", [T]);
fun pretyp_of is_param (params, typ) =
let
fun add_parms (TVar (xi as (x, _), S)) ps =
if is_param xi andalso not (AList.defined (op =) ps xi)
then (xi, mk_param S) :: ps else ps
| add_parms (TFree _) ps = ps
| add_parms (Type (_, Ts)) ps = fold add_parms Ts ps;
val params' = add_parms typ params;
fun pre_of (TVar (v as (xi, _))) =
(case AList.lookup (op =) params' xi of
NONE => PTVar v
| SOME p => p)
| pre_of (TFree ("'_dummy_", S)) = mk_param S
| 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 not (AList.defined (op =) ps xi) then
(xi, mk_param []) :: ps
else ps;
fun add_vparms (ps, Var (xi, Type ("_polymorphic_", _))) = ps
| 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 o AList.lookup (op =) 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, Var (xi, Type ("_polymorphic_", [T]))) =
(ps, PVar (xi, snd (pretyp_of (K true) ([], T))))
| pre_of (ps, Var (xi, T)) = constrain (ps, PVar (xi, var_param xi)) T
| pre_of (ps, Free (x, T)) = constrain (ps, PFree (x, var_param (x, ~1))) 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 (PType (_, Ts)) rs = fold add_parmsT Ts rs
| add_parmsT (Link (r as ref (Param _))) rs = r ins rs
| add_parmsT (Link (ref T)) rs = add_parmsT T rs
| add_parmsT _ rs = rs;
val add_parms = fold_pretyps add_parmsT;
(* add_names *)
fun add_namesT (PType (_, Ts)) xs = fold add_namesT Ts xs
| add_namesT (PTFree (x, _)) xs = x ins_string xs
| add_namesT (PTVar ((x, _), _)) xs = x ins_string xs
| add_namesT (Link (ref T)) xs = add_namesT T xs
| add_namesT (Param _) xs = xs;
val add_names = fold_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' = fold add_names ts (fold add_namesT Ts used);
val parms = rev (fold add_parms ts (fold add_parmsT Ts []));
val names = Term.invent_names used' (prfx ^ "'a") (length parms);
in
ListPair.app 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 pp classes arities =
let
(* adjust sorts of parameters *)
fun not_in_sort x S' S =
"Variable " ^ x ^ "::" ^ Pretty.string_of_sort pp S' ^ " not of sort " ^
Pretty.string_of_sort pp S;
fun no_domain (a, c) = "No way to get " ^ Pretty.string_of_arity pp (a, [], [c]);
fun meet (_, []) = ()
| meet (Link (r as (ref (Param S'))), S) =
if Sorts.sort_le classes (S', S) then ()
else r := mk_param (Sorts.inter_sort classes (S', S))
| meet (Link (ref T), S) = meet (T, S)
| meet (PType (a, Ts), S) =
ListPair.app meet (Ts, Sorts.mg_domain (classes, arities) a S
handle Sorts.DOMAIN ac => raise NO_UNIFIER (no_domain ac))
| meet (PTFree (x, S'), S) =
if Sorts.sort_le classes (S', S) then ()
else raise NO_UNIFIER (not_in_sort x S' S)
| meet (PTVar (xi, S'), S) =
if Sorts.sort_le classes (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)) = List.app (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 (meet (T', S); r := T')
| T' => (occurs_check r T'; meet (T', S); r := T'));
(* 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 ListPair.app unif (Ts, Us)
| unif (T, U) = if T = U then () else raise NO_UNIFIER "";
in unif end;
(** type inference **)
fun appl_error pp why t T u U =
["Type error in application: " ^ why,
"",
Pretty.string_of (Pretty.block
[Pretty.str "Operator:", Pretty.brk 2, Pretty.term pp t,
Pretty.str " ::", Pretty.brk 1, Pretty.typ pp T]),
Pretty.string_of (Pretty.block
[Pretty.str "Operand:", Pretty.brk 3, Pretty.term pp u,
Pretty.str " ::", Pretty.brk 1, Pretty.typ pp U]),
""];
(* infer *) (*DESTRUCTIVE*)
fun infer pp classes 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 (Ts', Ts'') = chop (length Ts) Ts_bTs';
val xs = map Free (map fst bs ~~ Ts'');
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 pp 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, Pretty.term pp t',
Pretty.str " ::", Pretty.brk 1, Pretty.typ pp T']),
Pretty.string_of (Pretty.block
[Pretty.str "Type:", Pretty.brk 2, Pretty.typ pp U']), ""];
in raise TYPE (text, [T', U'], [t']) end;
(* main *)
val unif = unify pp classes arities;
fun inf _ (PConst (_, T)) = T
| inf _ (PFree (_, T)) = T
| inf _ (PVar (_, T)) = T
| inf bs (PBound i) = snd (List.nth (bs, i) handle Subscript => 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;
(* basic_infer_types *)
fun basic_infer_types pp const_type classes 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 _ = List.app (fn t => (infer pp classes 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 = List.mapPartial 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;
(** type inference **)
(* user constraints *)
fun constrain t T =
if T = dummyT then t
else Const ("_type_constraint_", T) $ t;
(* user parameters *)
fun is_param (x, _) = size x > 0 andalso ord x = ord "?";
fun param i (x, S) = TVar (("?" ^ x, i), S);
val paramify_dummies =
let
fun dummy S maxidx = (param (maxidx + 1) ("'dummy", S), maxidx + 1);
fun paramify (TFree ("'_dummy_", S)) maxidx = dummy S maxidx
| paramify (Type ("dummy", _)) maxidx = dummy [] maxidx
| paramify (Type (a, Ts)) maxidx =
let val (Ts', maxidx') = fold_map paramify Ts maxidx
in (Type (a, Ts'), maxidx') end
| paramify T maxidx = (T, maxidx);
in paramify end;
(* decode sort constraints *)
fun get_sort tsig def_sort map_sort raw_env =
let
fun eq ((xi: indexname, S), (xi', S')) =
xi = xi' andalso Type.eq_sort tsig (S, S');
val env = distinct eq (map (apsnd map_sort) raw_env);
val _ = (case duplicates (eq_fst (op =)) env of [] => ()
| dups => error ("Inconsistent sort constraints for type variable(s) "
^ commas_quote (map (Syntax.string_of_vname' o fst) dups)));
fun get xi =
(case (AList.lookup (op =) env xi, def_sort xi) of
(NONE, NONE) => Type.defaultS tsig
| (NONE, SOME S) => S
| (SOME S, NONE) => S
| (SOME S, SOME S') =>
if Type.eq_sort tsig (S, S') then S'
else error ("Sort constraint inconsistent with default for type variable " ^
quote (Syntax.string_of_vname' xi)));
in get end;
(* decode_types -- transform parse tree into raw term *)
fun decode_types tsig is_const def_type def_sort map_const map_type map_sort tm =
let
fun get_type xi = the_default dummyT (def_type xi);
fun is_free x = is_some (def_type (x, ~1));
val raw_env = Syntax.raw_term_sorts tm;
val sort_of = get_sort tsig def_sort map_sort raw_env;
val certT = Type.cert_typ tsig o map_type;
fun decodeT t = certT (Syntax.typ_of_term sort_of map_sort t);
fun decode (Const ("_constrain", _) $ t $ typ) =
constrain (decode t) (decodeT typ)
| decode (Const ("_constrainAbs", _) $ (Abs (x, T, t)) $ typ) =
if T = dummyT then Abs (x, decodeT typ, decode t)
else constrain (Abs (x, certT T, decode t)) (decodeT typ --> dummyT)
| decode (Abs (x, T, t)) = Abs (x, certT T, decode t)
| decode (t $ u) = decode t $ decode u
| decode (Const (x, T)) =
let val c = (case try (unprefix Syntax.constN) x of SOME c => c | NONE => map_const x)
in Const (c, certT T) end
| decode (Free (x, T)) =
let val c = map_const x in
if not (is_free x) andalso (is_const c orelse NameSpace.is_qualified c) then
Const (c, certT T)
else if T = dummyT then Free (x, get_type (x, ~1))
else constrain (Free (x, certT T)) (get_type (x, ~1))
end
| decode (Var (xi, T)) =
if T = dummyT then Var (xi, get_type xi)
else constrain (Var (xi, certT T)) (get_type xi)
| decode (t as Bound _) = t;
in decode tm end;
(* infer_types *)
(*Given [T1,...,Tn] and [t1,...,tn], ensure that the type of ti
unifies with Ti (for i=1,...,n).
tsig: type signature
const_type: name mapping and signature lookup
def_type: partial map from indexnames to types (constrains Frees and Vars)
def_sort: partial map from indexnames to sorts (constrains TFrees and TVars)
used: list of already used type variables
freeze: if true then generated parameters are turned into TFrees, else TVars*)
fun infer_types pp tsig const_type def_type def_sort
map_const map_type map_sort used freeze pat_Ts raw_ts =
let
val {classes, arities, ...} = Type.rep_tsig tsig;
val pat_Ts' = map (Type.cert_typ tsig) pat_Ts;
val is_const = is_some o const_type;
val raw_ts' =
map (decode_types tsig is_const def_type def_sort map_const map_type map_sort) raw_ts;
val (ts, Ts, unifier) = basic_infer_types pp const_type
(#2 classes) arities used freeze is_param raw_ts' pat_Ts';
in (ts, unifier) end;
end;