(* Title: Pure/type_infer.ML
Author: Stefan Berghofer and Markus Wenzel, TU Muenchen
Simple type inference.
*)
signature TYPE_INFER =
sig
val anyT: sort -> typ
val polymorphicT: typ -> typ
val constrain: typ -> term -> term
val is_param: indexname -> bool
val param: int -> string * sort -> typ
val paramify_vars: typ -> typ
val paramify_dummies: typ -> int -> typ * int
val fixate_params: Name.context -> term list -> term list
val appl_error: Pretty.pp -> string -> term -> typ -> term -> typ -> string list
val infer_types: Pretty.pp -> Type.tsig -> (typ list -> typ list) ->
(string -> typ option) -> (indexname -> typ option) -> Name.context -> int ->
term list -> term list
end;
structure TypeInfer: TYPE_INFER =
struct
(** type parameters and constraints **)
fun anyT S = TFree ("'_dummy_", S);
(*indicate polymorphic Vars*)
fun polymorphicT T = Type ("_polymorphic_", [T]);
val constrain = Syntax.type_constraint;
(* user parameters *)
fun is_param (x, _: int) = String.isPrefix "?" x;
fun param i (x, S) = TVar (("?" ^ x, i), S);
val paramify_vars =
Same.commit
(Term_Subst.map_atypsT_same
(fn TVar ((x, i), S) => (param i (x, S)) | _ => raise Same.SAME));
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;
fun fixate_params name_context ts =
let
fun subst_param (xi, S) (inst, used) =
if is_param xi then
let
val [a] = Name.invents used Name.aT 1;
val used' = Name.declare a used;
in (((xi, S), TFree (a, S)) :: inst, used') end
else (inst, used);
val name_context' = (fold o fold_types) Term.declare_typ_names ts name_context;
val (inst, _) = fold_rev subst_param (fold Term.add_tvars ts []) ([], name_context');
in (map o map_types) (Term_Subst.instantiateT inst) ts end;
(** pretyps and preterms **)
(*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 int * sort;
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 *)
fun deref tye (T as Param (i, S)) =
(case Inttab.lookup tye i of
NONE => T
| SOME U => deref tye U)
| deref tye 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_para typ params_idx =
let
val (params', idx) = fold_atyps
(fn TVar (xi as (x, _), S) =>
(fn ps_idx as (ps, idx) =>
if is_para xi andalso not (Vartab.defined ps xi)
then (Vartab.update (xi, Param (idx, S)) ps, idx + 1) else ps_idx)
| _ => I) typ params_idx;
fun pre_of (TVar (v as (xi, _))) idx =
(case Vartab.lookup params' xi of
NONE => PTVar v
| SOME p => p, idx)
| pre_of (TFree ("'_dummy_", S)) idx = (Param (idx, S), idx + 1)
| pre_of (TFree v) idx = (PTFree v, idx)
| pre_of (T as Type (a, Ts)) idx =
if T = dummyT then (Param (idx, []), idx + 1)
else
let val (Ts', idx') = fold_map pre_of Ts idx
in (PType (a, Ts'), idx') end;
val (ptyp, idx') = pre_of typ idx;
in (ptyp, (params', idx')) end;
(* preterm_of *)
fun preterm_of const_type is_para tm (vparams, params, idx) =
let
fun add_vparm xi (ps_idx as (ps, idx)) =
if not (Vartab.defined ps xi) then
(Vartab.update (xi, Param (idx, [])) ps, idx + 1)
else ps_idx;
val (vparams', idx') = fold_aterms
(fn Var (_, Type ("_polymorphic_", _)) => I
| Var (xi, _) => add_vparm xi
| Free (x, _) => add_vparm (x, ~1)
| _ => I)
tm (vparams, idx);
fun var_param xi = the (Vartab.lookup vparams' xi);
val preT_of = pretyp_of is_para;
fun polyT_of T idx = apsnd snd (pretyp_of (K true) T (Vartab.empty, idx));
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, idx) =
(case const_type c of
SOME U =>
let val (pU, idx') = polyT_of U idx
in constraint T (PConst (c, pU)) (ps, idx') end
| NONE => raise TYPE ("No such constant: " ^ quote c, [], []))
| pre_of (Var (xi, Type ("_polymorphic_", [T]))) (ps, idx) =
let val (pT, idx') = polyT_of T idx
in (PVar (xi, pT), (ps, idx')) end
| pre_of (Var (xi, T)) ps_idx = constraint T (PVar (xi, var_param xi)) ps_idx
| pre_of (Free (x, T)) ps_idx = constraint T (PFree (x, var_param (x, ~1))) ps_idx
| pre_of (Const ("_type_constraint_", Type ("fun", [T, _])) $ t) ps_idx =
pre_of t ps_idx |-> constraint T
| pre_of (Bound i) ps_idx = (PBound i, ps_idx)
| pre_of (Abs (x, T, t)) ps_idx =
let
val (T', ps_idx') = preT_of T ps_idx;
val (t', ps_idx'') = pre_of t ps_idx';
in (PAbs (x, T', t'), ps_idx'') end
| pre_of (t $ u) ps_idx =
let
val (t', ps_idx') = pre_of t ps_idx;
val (u', ps_idx'') = pre_of u ps_idx';
in (PAppl (t', u'), ps_idx'') end;
val (tm', (params', idx'')) = pre_of tm (params, idx');
in (tm', (vparams', params', idx'')) end;
(** pretyps/terms to typs/terms **)
(* add_parms *)
fun add_parmsT tye T =
(case deref tye T of
PType (_, Ts) => fold (add_parmsT tye) Ts
| Param (i, _) => insert (op =) i
| _ => I);
fun add_parms tye = fold_pretyps (add_parmsT tye);
(* add_names *)
fun add_namesT tye T =
(case deref tye T of
PType (_, Ts) => fold (add_namesT tye) Ts
| PTFree (x, _) => Name.declare x
| PTVar ((x, _), _) => Name.declare x
| Param _ => I);
fun add_names tye = fold_pretyps (add_namesT tye);
(* simple_typ/term_of *)
fun simple_typ_of tye f T =
(case deref tye T of
PType (a, Ts) => Type (a, map (simple_typ_of tye f) Ts)
| PTFree v => TFree v
| PTVar v => TVar v
| Param (i, S) => TVar (f i, S));
(*convert types, drop constraints*)
fun simple_term_of tye f (PConst (c, T)) = Const (c, simple_typ_of tye f T)
| simple_term_of tye f (PFree (x, T)) = Free (x, simple_typ_of tye f T)
| simple_term_of tye f (PVar (xi, T)) = Var (xi, simple_typ_of tye f T)
| simple_term_of tye f (PBound i) = Bound i
| simple_term_of tye f (PAbs (x, T, t)) =
Abs (x, simple_typ_of tye f T, simple_term_of tye f t)
| simple_term_of tye f (PAppl (t, u)) =
simple_term_of tye f t $ simple_term_of tye f u
| simple_term_of tye f (Constraint (t, _)) = simple_term_of tye f t;
(* typs_terms_of *)
fun typs_terms_of tye used maxidx (Ts, ts) =
let
val used' = fold (add_names tye) ts (fold (add_namesT tye) Ts used);
val parms = rev (fold (add_parms tye) ts (fold (add_parmsT tye) Ts []));
val names = Name.invents used' ("?" ^ Name.aT) (length parms);
val tab = Inttab.make (parms ~~ names);
fun f i = (the (Inttab.lookup tab i), maxidx + 1);
in (map (simple_typ_of tye f) Ts, map (simple_term_of tye f) ts) end;
(** order-sorted unification of types **)
exception NO_UNIFIER of string * pretyp Inttab.table;
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 (_, []) tye_idx = tye_idx
| meet (Param (i, S'), S) (tye_idx as (tye, idx)) =
if Type.subsort tsig (S', S) then tye_idx
else (Inttab.update_new (i,
Param (idx, Type.inter_sort tsig (S', S))) tye, idx + 1)
| meet (PType (a, Ts), S) (tye_idx as (tye, _)) =
meets (Ts, Type.arity_sorts pp tsig a S
handle ERROR msg => raise NO_UNIFIER (msg, tye)) tye_idx
| meet (PTFree (x, S'), S) (tye_idx as (tye, _)) =
if Type.subsort tsig (S', S) then tye_idx
else raise NO_UNIFIER (not_of_sort x S' S, tye)
| meet (PTVar (xi, S'), S) (tye_idx as (tye, _)) =
if Type.subsort tsig (S', S) then tye_idx
else raise NO_UNIFIER (not_of_sort (Term.string_of_vname xi) S' S, tye)
and meets (T :: Ts, S :: Ss) (tye_idx as (tye, _)) =
meets (Ts, Ss) (meet (deref tye T, S) tye_idx)
| meets _ tye_idx = tye_idx;
(* occurs check and assigment *)
fun occurs_check tye i (Param (i', S)) =
if i = i' then raise NO_UNIFIER ("Occurs check!", tye)
else
(case Inttab.lookup tye i' of
NONE => ()
| SOME T => occurs_check tye i T)
| occurs_check tye i (PType (_, Ts)) = List.app (occurs_check tye i) Ts
| occurs_check _ _ _ = ();
fun assign i (T as Param (i', _)) S (tye_idx as (tye, idx)) =
if i = i' then tye_idx
else meet (T, S) (Inttab.update_new (i, T) tye, idx)
| assign i T S (tye, idx) =
(occurs_check tye i T; meet (T, S) (Inttab.update_new (i, T) tye, idx));
(* unification *)
fun unif (T1, T2) (tye_idx as (tye, idx)) =
(case (deref tye T1, deref tye T2) of
(Param (i, S), T) => assign i T S tye_idx
| (T, Param (i, S)) => assign i T S tye_idx
| (PType (a, Ts), PType (b, Us)) =>
if a <> b then
raise NO_UNIFIER ("Clash of types " ^ quote a ^ " and " ^ quote b, tye)
else fold unif (Ts ~~ Us) tye_idx
| (T, U) => if T = U then tye_idx else raise NO_UNIFIER ("", tye));
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 *)
fun infer pp tsig =
let
(* errors *)
fun unif_failed msg =
"Type unification failed" ^ (if msg = "" then "" else ": " ^ msg) ^ "\n";
fun prep_output tye bs ts Ts =
let
val (Ts_bTs', ts') = typs_terms_of tye Name.context ~1 (Ts @ map snd bs, ts);
val (Ts', Ts'') = chop (length Ts) Ts_bTs';
fun prep t =
let val xs = rev (Term.variant_frees t (rev (map fst bs ~~ Ts'')))
in Term.subst_bounds (map Syntax.mark_boundT xs, t) end;
in (map prep ts', Ts') end;
fun err_loose i =
raise TYPE ("Loose bound variable: B." ^ string_of_int i, [], []);
fun err_appl msg tye bs t T u U =
let
val ([t', u'], [T', U']) = prep_output tye 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 tye bs t T U =
let
val ([t'], [T', U']) = prep_output tye 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)) tye_idx = (T, tye_idx)
| inf _ (PFree (_, T)) tye_idx = (T, tye_idx)
| inf _ (PVar (_, T)) tye_idx = (T, tye_idx)
| inf bs (PBound i) tye_idx =
(snd (nth bs i handle Subscript => err_loose i), tye_idx)
| inf bs (PAbs (x, T, t)) tye_idx =
let val (U, tye_idx') = inf ((x, T) :: bs) t tye_idx
in (PType ("fun", [T, U]), tye_idx') end
| inf bs (PAppl (t, u)) tye_idx =
let
val (T, tye_idx') = inf bs t tye_idx;
val (U, (tye, idx)) = inf bs u tye_idx';
val V = Param (idx, []);
val U_to_V = PType ("fun", [U, V]);
val tye_idx'' = unif (U_to_V, T) (tye, idx + 1)
handle NO_UNIFIER (msg, tye') => err_appl msg tye' bs t T u U;
in (V, tye_idx'') end
| inf bs (Constraint (t, U)) tye_idx =
let val (T, tye_idx') = inf bs t tye_idx in
(T,
unif (T, U) tye_idx'
handle NO_UNIFIER (msg, tye) => err_constraint msg tye bs t T U)
end;
in inf [] end;
(* infer_types *)
fun infer_types pp tsig check_typs const_type var_type used maxidx raw_ts =
let
(*constrain vars*)
val get_type = the_default dummyT o var_type;
val constrain_vars = Term.map_aterms
(fn Free (x, T) => constrain T (Free (x, get_type (x, ~1)))
| Var (xi, T) => constrain T (Var (xi, get_type xi))
| t => t);
(*convert to preterms*)
val ts = burrow_types check_typs raw_ts;
val (ts', (_, _, idx)) =
fold_map (preterm_of const_type is_param o constrain_vars) ts
(Vartab.empty, Vartab.empty, 0);
(*do type inference*)
val (tye, _) = fold (snd oo infer pp tsig) ts' (Inttab.empty, idx);
in #2 (typs_terms_of tye used maxidx ([], ts')) end;
end;