(* Title: Pure/type_infer.ML
Author: Stefan Berghofer and Markus Wenzel, TU Muenchen
Representation of type-inference problems. Simple type inference.
*)
signature TYPE_INFER =
sig
val is_param: indexname -> bool
val is_paramT: typ -> bool
val param_maxidx: term -> int -> int
val param_maxidx_of: term list -> int
val param: int -> string * sort -> typ
val mk_param: int -> sort -> typ
val anyT: sort -> typ
val paramify_vars: typ -> typ
val paramify_dummies: typ -> int -> typ * int
val deref: typ Vartab.table -> typ -> typ
val finish: Proof.context -> typ Vartab.table -> typ list * term list -> typ list * term list
val fixate: Proof.context -> term list -> term list
val prepare: Proof.context -> (string -> typ option) -> (string * int -> typ option) ->
term list -> int * term list
val infer_types: Proof.context -> (string -> typ option) -> (indexname -> typ option) ->
term list -> term list
end;
structure Type_Infer: TYPE_INFER =
struct
(** type parameters and constraints **)
(* type inference parameters -- may get instantiated *)
fun is_param (x, _: int) = String.isPrefix "?" x;
fun is_paramT (TVar (xi, _)) = is_param xi
| is_paramT _ = false;
val param_maxidx =
(Term.fold_types o Term.fold_atyps)
(fn (TVar (xi as (_, i), _)) => if is_param xi then Integer.max i else I | _ => I);
fun param_maxidx_of ts = fold param_maxidx ts ~1;
fun param i (x, S) = TVar (("?" ^ x, i), S);
fun mk_param i S = TVar (("?'a", i), S);
(* pre-stage parameters *)
fun anyT S = TFree ("'_dummy_", 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;
(** prepare types/terms: create inference parameters **)
(* prepare_typ *)
fun prepare_typ typ params_idx =
let
val (params', idx) = fold_atyps
(fn TVar (xi, S) =>
(fn ps_idx as (ps, idx) =>
if is_param xi andalso not (Vartab.defined ps xi)
then (Vartab.update (xi, mk_param idx S) ps, idx + 1) else ps_idx)
| _ => I) typ params_idx;
fun prepare (T as Type (a, Ts)) idx =
if T = dummyT then (mk_param idx [], idx + 1)
else
let val (Ts', idx') = fold_map prepare Ts idx
in (Type (a, Ts'), idx') end
| prepare (T as TVar (xi, _)) idx =
(case Vartab.lookup params' xi of
NONE => T
| SOME p => p, idx)
| prepare (TFree ("'_dummy_", S)) idx = (mk_param idx S, idx + 1)
| prepare (T as TFree _) idx = (T, idx);
val (typ', idx') = prepare typ idx;
in (typ', (params', idx')) end;
(* prepare_term *)
fun prepare_term ctxt const_type tm (vparams, params, idx) =
let
fun add_vparm xi (ps_idx as (ps, idx)) =
if not (Vartab.defined ps xi) then
(Vartab.update (xi, mk_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);
fun polyT_of T idx = apsnd snd (prepare_typ (paramify_vars T) (Vartab.empty, idx));
fun constraint T t ps =
if T = dummyT then (t, ps)
else
let val (T', ps') = prepare_typ T ps
in (Type.constraint T' t, ps') end;
fun prepare (Const ("_type_constraint_", T) $ t) ps_idx =
let
fun err () =
error ("Malformed internal type constraint: " ^ Syntax.string_of_typ ctxt T);
val A = (case T of Type ("fun", [A, B]) => if A = B then A else err () | _ => err ());
val (A', ps_idx') = prepare_typ A ps_idx;
val (t', ps_idx'') = prepare t ps_idx';
in (Const ("_type_constraint_", A' --> A') $ t', ps_idx'') end
| prepare (Const (c, T)) (ps, idx) =
(case const_type c of
SOME U =>
let val (U', idx') = polyT_of U idx
in constraint T (Const (c, U')) (ps, idx') end
| NONE => error ("Undeclared constant: " ^ quote c))
| prepare (Var (xi, Type ("_polymorphic_", [T]))) (ps, idx) =
let val (T', idx') = polyT_of T idx
in (Var (xi, T'), (ps, idx')) end
| prepare (Var (xi, T)) ps_idx = constraint T (Var (xi, var_param xi)) ps_idx
| prepare (Free (x, T)) ps_idx = constraint T (Free (x, var_param (x, ~1))) ps_idx
| prepare (Bound i) ps_idx = (Bound i, ps_idx)
| prepare (Abs (x, T, t)) ps_idx =
let
val (T', ps_idx') = prepare_typ T ps_idx;
val (t', ps_idx'') = prepare t ps_idx';
in (Abs (x, T', t'), ps_idx'') end
| prepare (t $ u) ps_idx =
let
val (t', ps_idx') = prepare t ps_idx;
val (u', ps_idx'') = prepare u ps_idx';
in (t' $ u', ps_idx'') end;
val (tm', (params', idx'')) = prepare tm (params, idx');
in (tm', (vparams', params', idx'')) end;
(** results **)
(* dereferenced views *)
fun deref tye (T as TVar (xi, _)) =
(case Vartab.lookup tye xi of
NONE => T
| SOME U => deref tye U)
| deref _ T = T;
fun add_parms tye T =
(case deref tye T of
Type (_, Ts) => fold (add_parms tye) Ts
| TVar (xi, _) => if is_param xi then insert (op =) xi else I
| _ => I);
fun add_names tye T =
(case deref tye T of
Type (_, Ts) => fold (add_names tye) Ts
| TFree (x, _) => Name.declare x
| TVar ((x, i), _) => if is_param (x, i) then I else Name.declare x);
(* finish -- standardize remaining parameters *)
fun finish ctxt tye (Ts, ts) =
let
val used =
(fold o fold_types) (add_names tye) ts (fold (add_names tye) Ts (Variable.names_of ctxt));
val parms = rev ((fold o fold_types) (add_parms tye) ts (fold (add_parms tye) Ts []));
val names = Name.invents used ("?" ^ Name.aT) (length parms);
val tab = Vartab.make (parms ~~ names);
fun finish_typ T =
(case deref tye T of
Type (a, Ts) => Type (a, map finish_typ Ts)
| U as TFree _ => U
| U as TVar (xi, S) =>
(case Vartab.lookup tab xi of
NONE => U
| SOME a => TVar ((a, 0), S)));
in (map finish_typ Ts, map (Type.strip_constraints o Term.map_types finish_typ) ts) end;
(* fixate -- introduce fresh type variables *)
fun fixate ctxt 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 used = (fold o fold_types) Term.declare_typ_names ts (Variable.names_of ctxt);
val (inst, _) = fold_rev subst_param (fold Term.add_tvars ts []) ([], used);
in (map o map_types) (Term_Subst.instantiateT inst) ts end;
(** order-sorted unification of types **)
exception NO_UNIFIER of string * typ Vartab.table;
fun unify ctxt =
let
val thy = Proof_Context.theory_of ctxt;
val arity_sorts = Type.arity_sorts (Context.pretty ctxt) (Sign.tsig_of thy);
(* adjust sorts of parameters *)
fun not_of_sort x S' S =
"Variable " ^ x ^ "::" ^ Syntax.string_of_sort ctxt S' ^ " not of sort " ^
Syntax.string_of_sort ctxt S;
fun meet (_, []) tye_idx = tye_idx
| meet (Type (a, Ts), S) (tye_idx as (tye, _)) =
meets (Ts, arity_sorts a S handle ERROR msg => raise NO_UNIFIER (msg, tye)) tye_idx
| meet (TFree (x, S'), S) (tye_idx as (tye, _)) =
if Sign.subsort thy (S', S) then tye_idx
else raise NO_UNIFIER (not_of_sort x S' S, tye)
| meet (TVar (xi, S'), S) (tye_idx as (tye, idx)) =
if Sign.subsort thy (S', S) then tye_idx
else if is_param xi then
(Vartab.update_new (xi, mk_param idx (Sign.inter_sort thy (S', S))) tye, idx + 1)
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 assignment *)
fun occurs_check tye xi (TVar (xi', _)) =
if xi = xi' then raise NO_UNIFIER ("Occurs check!", tye)
else
(case Vartab.lookup tye xi' of
NONE => ()
| SOME T => occurs_check tye xi T)
| occurs_check tye xi (Type (_, Ts)) = List.app (occurs_check tye xi) Ts
| occurs_check _ _ _ = ();
fun assign xi (T as TVar (xi', _)) S env =
if xi = xi' then env
else env |> meet (T, S) |>> Vartab.update_new (xi, T)
| assign xi T S (env as (tye, _)) =
(occurs_check tye xi T; env |> meet (T, S) |>> Vartab.update_new (xi, T));
(* unification *)
fun show_tycon (a, Ts) =
quote (Syntax.string_of_typ ctxt (Type (a, replicate (length Ts) dummyT)));
fun unif (T1, T2) (env as (tye, _)) =
(case pairself (`is_paramT o deref tye) (T1, T2) of
((true, TVar (xi, S)), (_, T)) => assign xi T S env
| ((_, T), (true, TVar (xi, S))) => assign xi T S env
| ((_, Type (a, Ts)), (_, Type (b, Us))) =>
if a <> b then
raise NO_UNIFIER
("Clash of types " ^ show_tycon (a, Ts) ^ " and " ^ show_tycon (b, Us), tye)
else fold unif (Ts ~~ Us) env
| ((_, T), (_, U)) => if T = U then env else raise NO_UNIFIER ("", tye));
in unif end;
(** simple type inference **)
(* infer *)
fun infer ctxt =
let
(* errors *)
fun prep_output tye bs ts Ts =
let
val (Ts_bTs', ts') = finish ctxt tye (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_Trans.mark_boundT xs, t) end;
in (map prep ts', Ts') end;
fun err_loose i = error ("Loose bound variable: B." ^ string_of_int i);
fun unif_failed msg =
"Type unification failed" ^ (if msg = "" then "" else ": " ^ msg) ^ "\n\n";
fun err_appl msg tye bs t T u U =
let val ([t', u'], [T', U']) = prep_output tye bs [t, u] [T, U]
in error (unif_failed msg ^ Type.appl_error ctxt t' T' u' U' ^ "\n") end;
(* main *)
fun inf _ (Const (_, T)) tye_idx = (T, tye_idx)
| inf _ (Free (_, T)) tye_idx = (T, tye_idx)
| inf _ (Var (_, T)) tye_idx = (T, tye_idx)
| inf bs (Bound i) tye_idx =
(snd (nth bs i handle Subscript => err_loose i), tye_idx)
| inf bs (Abs (x, T, t)) tye_idx =
let val (U, tye_idx') = inf ((x, T) :: bs) t tye_idx
in (T --> U, tye_idx') end
| inf bs (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 = mk_param idx [];
val tye_idx'' = unify ctxt (U --> V, T) (tye, idx + 1)
handle NO_UNIFIER (msg, tye') => err_appl msg tye' bs t T u U;
in (V, tye_idx'') end;
in inf [] end;
(* main interfaces *)
fun prepare ctxt const_type var_type raw_ts =
let
val get_type = the_default dummyT o var_type;
val constrain_vars = Term.map_aterms
(fn Free (x, T) => Type.constraint T (Free (x, get_type (x, ~1)))
| Var (xi, T) => Type.constraint T (Var (xi, get_type xi))
| t => t);
val ts = burrow_types (Syntax.check_typs ctxt) raw_ts;
val idx = param_maxidx_of ts + 1;
val (ts', (_, _, idx')) =
fold_map (prepare_term ctxt const_type o constrain_vars) ts
(Vartab.empty, Vartab.empty, idx);
in (idx', ts') end;
fun infer_types ctxt const_type var_type raw_ts =
let
val (idx, ts) = prepare ctxt const_type var_type raw_ts;
val (tye, _) = fold (snd oo infer ctxt) ts (Vartab.empty, idx);
val (_, ts') = finish ctxt tye ([], ts);
in ts' end;
end;