(* Title: Pure/type_infer.ML
ID: $Id$
Author: Stefan Berghofer and Markus Wenzel, TU Muenchen
Simple type inference.
*)
signature TYPE_INFER =
sig
val anyT: sort -> typ
val logicT: typ
val polymorphicT: typ -> typ
val constrain: term -> typ -> term
val param: int -> string * sort -> typ
val paramify_vars: typ -> typ
val paramify_dummies: typ -> int -> typ * int
val appl_error: Pretty.pp -> string -> term -> typ -> term -> typ -> string list
val infer_types: Pretty.pp -> Type.tsig ->
(string -> typ option) -> (indexname -> typ option) ->
Name.context -> bool -> (term * typ) list -> (term * typ) list * (indexname * typ) list
end;
structure TypeInfer: TYPE_INFER =
struct
(** type parameters and constraints **)
fun anyT S = TFree ("'_dummy_", S);
val logicT = anyT [];
(*indicate polymorphic Vars*)
fun polymorphicT T = Type ("_polymorphic_", [T]);
fun constrain t T =
if T = dummyT then t
else Const ("_type_constraint_", T --> T) $ t;
(* user parameters *)
fun is_param_default (x, _) = size x > 0 andalso ord x = ord "?";
fun param i (x, S) = TVar (("?" ^ x, i), S);
val paramify_vars = Term.map_atyps (fn TVar ((x, i), S) => param i (x, S) | T => T);
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;
(** 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_of *)
fun pretyp_of is_param typ params =
let
val params' = fold_atyps
(fn TVar (xi as (x, _), S) =>
(fn ps =>
if is_param xi andalso not (Vartab.defined ps xi)
then Vartab.update (xi, mk_param S) ps else ps)
| _ => I) typ params;
fun pre_of (TVar (v as (xi, _))) =
(case Vartab.lookup 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 (pre_of typ, params') end;
(* preterm_of *)
fun preterm_of const_type is_param tm (vparams, params) =
let
fun add_vparm xi ps =
if not (Vartab.defined ps xi) then
Vartab.update (xi, mk_param []) ps
else ps;
val vparams' = fold_aterms
(fn Var (_, Type ("_polymorphic_", _)) => I
| Var (xi, _) => add_vparm xi
| Free (x, _) => add_vparm (x, ~1)
| _ => I)
tm vparams;
fun var_param xi = the (Vartab.lookup vparams' xi);
val preT_of = pretyp_of is_param;
fun polyT_of T = fst (pretyp_of (K true) T Vartab.empty);
fun constraint T t ps =
if T = dummyT then (t, ps)
else
let val (T', ps') = preT_of T ps
in (Constraint (t, T'), ps') end;
fun pre_of (Const (c, T)) ps =
(case const_type c of
SOME U => constraint T (PConst (c, polyT_of U)) ps
| NONE => raise TYPE ("No such constant: " ^ quote c, [], []))
| pre_of (Var (xi, Type ("_polymorphic_", [T]))) ps = (PVar (xi, polyT_of T), ps)
| pre_of (Var (xi, T)) ps = constraint T (PVar (xi, var_param xi)) ps
| pre_of (Free (x, T)) ps = constraint T (PFree (x, var_param (x, ~1))) ps
| pre_of (Const ("_type_constraint_", Type ("fun", [T, _])) $ t) ps =
pre_of t ps |-> constraint T
| pre_of (Bound i) ps = (PBound i, ps)
| pre_of (Abs (x, T, t)) ps =
let
val (T', ps') = preT_of T ps;
val (t', ps'') = pre_of t ps';
in (PAbs (x, T', t'), ps'') end
| pre_of (t $ u) ps =
let
val (t', ps') = pre_of t ps;
val (u', ps'') = pre_of u ps';
in (PAppl (t', u'), ps'') end;
val (tm', params') = pre_of tm params;
in (tm', (vparams', params')) end;
(** 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 = insert (op =) r 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)) = fold add_namesT Ts
| add_namesT (PTFree (x, _)) = Name.declare x
| add_namesT (PTVar ((x, _), _)) = Name.declare x
| add_namesT (Link (ref T)) = add_namesT T
| add_namesT (Param _) = I;
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 = Name.invents 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 tsig =
let
(* adjust sorts of parameters *)
fun not_of_sort x S' S =
"Variable " ^ x ^ "::" ^ Pretty.string_of_sort pp S' ^ " not of sort " ^
Pretty.string_of_sort pp S;
fun meet (_, []) = ()
| meet (Link (r as (ref (Param S'))), S) =
if Type.subsort tsig (S', S) then ()
else r := mk_param (Type.inter_sort tsig (S', S))
| meet (Link (ref T), S) = meet (T, S)
| meet (PType (a, Ts), S) =
ListPair.app meet (Ts, Type.arity_sorts pp tsig a S
handle ERROR msg => raise NO_UNIFIER msg)
| meet (PTFree (x, S'), S) =
if Type.subsort tsig (S', S) then ()
else raise NO_UNIFIER (not_of_sort x S' S)
| meet (PTVar (xi, S'), S) =
if Type.subsort tsig (S', S) then ()
else raise NO_UNIFIER (not_of_sort (Term.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 **)
(* appl_error *)
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 tsig =
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 Name.context 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 tsig;
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;
(* infer_types *)
fun infer_types pp tsig const_type var_type used freeze args =
let
(*certify types*)
val certT = Type.cert_typ tsig;
val (raw_ts, raw_Ts) = split_list args;
val ts = map (Term.map_types certT) raw_ts;
val Ts = map certT raw_Ts;
(*constrain vars*)
val get_type = the_default dummyT o var_type;
val constrain_vars = Term.map_aterms
(fn Free (x, T) => constrain (Free (x, get_type (x, ~1))) T
| Var (xi, T) => constrain (Var (xi, get_type xi)) T
| t => t);
(*convert to preterms/typs*)
val (Ts', Tps) = fold_map (pretyp_of (K true)) Ts Vartab.empty;
val (ts', (vps, ps)) =
fold_map (preterm_of const_type is_param_default o constrain_vars) ts (Vartab.empty, Tps);
(*run type inference*)
val tTs' = ListPair.map Constraint (ts', Ts');
val _ = List.app (fn t => (infer pp tsig t; ())) tTs';
(*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');
(*collect result unifier*)
val redundant = fn (xi, TVar (yi, _)) => xi = yi | _ => false;
val env = filter_out redundant (map (apsnd simple_typ_of) (Vartab.dest Tps));
in (final_ts ~~ final_Ts, env) end;
end;