Move towards standard functions.
(* Title: Pure/type.ML
ID: $Id$
Author: Tobias Nipkow, Lawrence C Paulson, and Markus Wenzel
Type signatures and certified types, special treatment of type vars,
matching and unification of types, extend and merge type signatures.
*)
signature TYPE =
sig
(*type signatures and certified types*)
datatype decl =
LogicalType of int |
Abbreviation of string list * typ * bool |
Nonterminal
type tsig
val rep_tsig: tsig ->
{classes: Sorts.classes,
default: sort,
types: (decl * stamp) Symtab.table,
arities: Sorts.arities,
log_types: string list,
witness: (typ * sort) option}
val empty_tsig: tsig
val classes: tsig -> class list
val defaultS: tsig -> sort
val logical_types: tsig -> string list
val universal_witness: tsig -> (typ * sort) option
val eq_sort: tsig -> sort * sort -> bool
val subsort: tsig -> sort * sort -> bool
val of_sort: tsig -> typ * sort -> bool
val cert_class: tsig -> class -> class
val cert_sort: tsig -> sort -> sort
val witness_sorts: tsig -> sort list -> sort list -> (typ * sort) list
val cert_typ: tsig -> typ -> typ
val cert_typ_syntax: tsig -> typ -> typ
val cert_typ_raw: tsig -> typ -> typ
(*special treatment of type vars*)
val strip_sorts: typ -> typ
val no_tvars: typ -> typ
val varifyT: typ -> typ
val unvarifyT: typ -> typ
val varify: term * string list -> term * (string * indexname) list
val freeze_thaw_type : typ -> typ * (typ -> typ)
val freeze_thaw : term -> term * (term -> term)
(*matching and unification*)
val inst_typ_tvars: Pretty.pp -> tsig -> (indexname * typ) list -> typ -> typ
val inst_term_tvars: Pretty.pp -> tsig -> (indexname * typ) list -> term -> term
exception TYPE_MATCH
val typ_match: tsig -> typ Vartab.table * (typ * typ) -> typ Vartab.table
val typ_instance: tsig -> typ * typ -> bool
exception TUNIFY
val unify: tsig -> typ Vartab.table * int -> typ * typ -> typ Vartab.table * int
val raw_unify: typ * typ -> bool
(*extend and merge type signatures*)
val add_classes: Pretty.pp -> (class * class list) list -> tsig -> tsig
val add_classrel: Pretty.pp -> (class * class) list -> tsig -> tsig
val set_defsort: sort -> tsig -> tsig
val add_types: (string * int) list -> tsig -> tsig
val add_abbrs: (string * string list * typ) list -> tsig -> tsig
val add_nonterminals: string list -> tsig -> tsig
val add_arities: Pretty.pp -> arity list -> tsig -> tsig
val merge_tsigs: Pretty.pp -> tsig * tsig -> tsig
end;
structure Type: TYPE =
struct
(** type signatures and certified types **)
(* type declarations *)
datatype decl =
LogicalType of int |
Abbreviation of string list * typ * bool |
Nonterminal;
fun str_of_decl (LogicalType _) = "logical type constructor"
| str_of_decl (Abbreviation _) = "type abbreviation"
| str_of_decl Nonterminal = "syntactic type";
(* type tsig *)
datatype tsig =
TSig of {
classes: Sorts.classes, (*declared classes with proper subclass relation*)
default: sort, (*default sort on input*)
types: (decl * stamp) Symtab.table, (*declared types*)
arities: Sorts.arities, (*image specification of types wrt. sorts*)
log_types: string list, (*logical types sorted by number of arguments*)
witness: (typ * sort) option}; (*witness for non-emptiness of strictest sort*)
fun rep_tsig (TSig comps) = comps;
fun make_tsig (classes, default, types, arities, log_types, witness) =
TSig {classes = classes, default = default, types = types, arities = arities,
log_types = log_types, witness = witness};
fun map_tsig f (TSig {classes, default, types, arities, log_types, witness}) =
make_tsig (f (classes, default, types, arities, log_types, witness));
fun build_tsig (classes, default, types, arities) =
let
fun add_log_type (ts, (c, (LogicalType n, _))) = (c, n) :: ts
| add_log_type (ts, _) = ts;
val log_types =
Symtab.foldl add_log_type ([], types)
|> Library.sort (Library.int_ord o pairself #2) |> map #1;
val witness =
(case Sorts.witness_sorts (classes, arities) log_types [] [Graph.keys classes] of
[w] => SOME w | _ => NONE);
in make_tsig (classes, default, types, arities, log_types, witness) end;
fun change_tsig f (TSig {classes, default, types, arities, log_types = _, witness = _}) =
build_tsig (f (classes, default, types, arities));
val empty_tsig = build_tsig (Graph.empty, [], Symtab.empty, Symtab.empty);
(* classes and sorts *)
fun classes (TSig {classes = C, ...}) = Graph.keys C;
fun defaultS (TSig {default, ...}) = default;
fun logical_types (TSig {log_types, ...}) = log_types;
fun universal_witness (TSig {witness, ...}) = witness;
fun eq_sort (TSig {classes, ...}) = Sorts.sort_eq classes;
fun subsort (TSig {classes, ...}) = Sorts.sort_le classes;
fun of_sort (TSig {classes, arities, ...}) = Sorts.of_sort (classes, arities);
fun cert_class (TSig {classes, ...}) = Sorts.certify_class classes;
fun cert_sort (TSig {classes, ...}) = Sorts.certify_sort classes;
fun witness_sorts (tsig as TSig {classes, arities, log_types, ...}) =
Sorts.witness_sorts (classes, arities) log_types;
(* certified types *)
fun bad_nargs t = "Bad number of arguments for type constructor: " ^ quote t;
fun undecl_type c = "Undeclared type constructor: " ^ quote c;
local
fun inst_typ env (Type (c, Ts)) = Type (c, map (inst_typ env) Ts)
| inst_typ env (T as TFree (x, _)) = getOpt (Library.assoc_string (env, x),T)
| inst_typ _ T = T;
fun certify_typ normalize syntax tsig ty =
let
val TSig {classes, types, ...} = tsig;
fun err msg = raise TYPE (msg, [ty], []);
val check_syntax =
if syntax then K ()
else fn c => err ("Illegal occurrence of syntactic type: " ^ quote c);
fun cert (T as Type (c, Ts)) =
let
val Ts' = map cert Ts;
fun nargs n = if length Ts <> n then err (bad_nargs c) else ();
in
(case Symtab.lookup (types, c) of
SOME (LogicalType n, _) => (nargs n; Type (c, Ts'))
| SOME (Abbreviation (vs, U, syn), _) => (nargs (length vs);
if syn then check_syntax c else ();
if normalize then inst_typ (vs ~~ Ts') U
else Type (c, Ts'))
| SOME (Nonterminal, _) => (nargs 0; check_syntax c; T)
| NONE => err (undecl_type c))
end
| cert (TFree (x, S)) = TFree (x, Sorts.certify_sort classes S)
| cert (TVar (xi as (_, i), S)) =
if i < 0 then
err ("Malformed type variable: " ^ quote (Term.string_of_vname xi))
else TVar (xi, Sorts.certify_sort classes S);
val ty' = cert ty;
in if ty = ty' then ty else ty' end; (*avoid copying of already normal type*)
in
val cert_typ = certify_typ true false;
val cert_typ_syntax = certify_typ true true;
val cert_typ_raw = certify_typ false true;
end;
(** special treatment of type vars **)
(* strip_sorts *)
fun strip_sorts (Type (a, Ts)) = Type (a, map strip_sorts Ts)
| strip_sorts (TFree (x, _)) = TFree (x, [])
| strip_sorts (TVar (xi, _)) = TVar (xi, []);
(* no_tvars *)
fun no_tvars T =
(case typ_tvars T of [] => T
| vs => raise TYPE ("Illegal schematic type variable(s): " ^
commas_quote (map (Term.string_of_vname o #1) vs), [T], []));
(* varify, unvarify *)
val varifyT = map_type_tfree (fn (a, S) => TVar ((a, 0), S));
fun unvarifyT (Type (a, Ts)) = Type (a, map unvarifyT Ts)
| unvarifyT (TVar ((a, 0), S)) = TFree (a, S)
| unvarifyT T = T;
fun varify (t, fixed) =
let
val fs = add_term_tfree_names (t, []) \\ fixed;
val ixns = add_term_tvar_ixns (t, []);
val fmap = fs ~~ map (rpair 0) (variantlist (fs, map #1 ixns))
fun thaw (f as (a, S)) =
(case assoc (fmap, a) of
NONE => TFree f
| SOME b => TVar (b, S));
in (map_term_types (map_type_tfree thaw) t, fmap) end;
(* freeze_thaw: freeze TVars in a term; return the "thaw" inverse *)
local
fun new_name (ix, (pairs,used)) =
let val v = variant used (string_of_indexname ix)
in ((ix,v)::pairs, v::used) end;
fun freeze_one alist (ix,sort) =
TFree (valOf (assoc (alist, ix)), sort)
handle Option =>
raise TYPE ("Failure during freezing of ?" ^ string_of_indexname ix, [], []);
fun thaw_one alist (a,sort) = TVar (valOf (assoc (alist,a)), sort)
handle Option => TFree(a, sort);
in
(*this sort of code could replace unvarifyT*)
fun freeze_thaw_type T =
let
val used = add_typ_tfree_names (T, [])
and tvars = map #1 (add_typ_tvars (T, []));
val (alist, _) = Library.foldr new_name (tvars, ([], used));
in (map_type_tvar (freeze_one alist) T, map_type_tfree (thaw_one (map swap alist))) end;
fun freeze_thaw t =
let
val used = it_term_types add_typ_tfree_names (t, [])
and tvars = map #1 (it_term_types add_typ_tvars (t, []));
val (alist, _) = Library.foldr new_name (tvars, ([], used));
in
(case alist of
[] => (t, fn x => x) (*nothing to do!*)
| _ => (map_term_types (map_type_tvar (freeze_one alist)) t,
map_term_types (map_type_tfree (thaw_one (map swap alist)))))
end;
end;
(** matching and unification of types **)
(* instantiation *)
fun inst_typ_tvars pp tsig tye =
let
fun inst (var as (v, S)) =
(case assoc_string_int (tye, v) of
SOME U =>
if of_sort tsig (U, S) then U
else raise TYPE ("Type not of sort " ^ Pretty.string_of_sort pp S, [U], [])
| NONE => TVar var);
in map_type_tvar inst end;
fun inst_term_tvars _ _ [] t = t
| inst_term_tvars pp tsig tye t = map_term_types (inst_typ_tvars pp tsig tye) t;
(* matching *)
exception TYPE_MATCH;
fun typ_match tsig =
let
fun match (subs, (TVar (v, S), T)) =
(case Vartab.lookup (subs, v) of
NONE =>
if of_sort tsig (T, S) then Vartab.update ((v, T), subs)
else raise TYPE_MATCH
| SOME U => if U = T then subs else raise TYPE_MATCH)
| match (subs, (Type (a, Ts), Type (b, Us))) =
if a <> b then raise TYPE_MATCH
else Library.foldl match (subs, Ts ~~ Us)
| match (subs, (TFree x, TFree y)) =
if x = y then subs else raise TYPE_MATCH
| match _ = raise TYPE_MATCH;
in match end;
fun typ_instance tsig (T, U) =
(typ_match tsig (Vartab.empty, (U, T)); true) handle TYPE_MATCH => false;
(* unification *)
exception TUNIFY;
(*occurs_check*)
fun occurs v tye =
let
fun occ (Type (_, Ts)) = exists occ Ts
| occ (TFree _) = false
| occ (TVar (w, _)) =
eq_ix (v, w) orelse
(case Vartab.lookup (tye, w) of
NONE => false
| SOME U => occ U);
in occ end;
(*chase variable assignments; if devar returns a type var then it must be unassigned*)
fun devar (T as TVar (v, _), tye) =
(case Vartab.lookup (tye, v) of
SOME U => devar (U, tye)
| NONE => T)
| devar (T, tye) = T;
fun unify (tsig as TSig {classes, arities, ...}) (tyenv, maxidx) TU =
let
val tyvar_count = ref maxidx;
fun gen_tyvar S = TVar (("'a", inc tyvar_count), S);
fun mg_domain a S =
Sorts.mg_domain (classes, arities) a S handle Sorts.DOMAIN _ => raise TUNIFY;
fun meet ((_, []), tye) = tye
| meet ((TVar (xi, S'), S), tye) =
if Sorts.sort_le classes (S', S) then tye
else Vartab.update_new ((xi,
gen_tyvar (Sorts.inter_sort classes (S', S))), tye)
| meet ((TFree (_, S'), S), tye) =
if Sorts.sort_le classes (S', S) then tye
else raise TUNIFY
| meet ((Type (a, Ts), S), tye) = meets ((Ts, mg_domain a S), tye)
and meets (([], []), tye) = tye
| meets ((T :: Ts, S :: Ss), tye) =
meets ((Ts, Ss), meet ((devar (T, tye), S), tye))
| meets _ = sys_error "meets";
fun unif ((ty1, ty2), tye) =
(case (devar (ty1, tye), devar (ty2, tye)) of
(T as TVar (v, S1), U as TVar (w, S2)) =>
if eq_ix (v, w) then tye
else if Sorts.sort_le classes (S1, S2) then
Vartab.update_new ((w, T), tye)
else if Sorts.sort_le classes (S2, S1) then
Vartab.update_new ((v, U), tye)
else
let val S = gen_tyvar (Sorts.inter_sort classes (S1, S2)) in
Vartab.update_new ((v, S), Vartab.update_new ((w, S), tye))
end
| (TVar (v, S), T) =>
if occurs v tye T then raise TUNIFY
else meet ((T, S), Vartab.update_new ((v, T), tye))
| (T, TVar (v, S)) =>
if occurs v tye T then raise TUNIFY
else meet ((T, S), Vartab.update_new ((v, T), tye))
| (Type (a, Ts), Type (b, Us)) =>
if a <> b then raise TUNIFY
else Library.foldr unif (Ts ~~ Us, tye)
| (T, U) => if T = U then tye else raise TUNIFY);
in (unif (TU, tyenv), ! tyvar_count) end;
(*purely structural unification *)
fun raw_unify (ty1, ty2) =
(unify empty_tsig (Vartab.empty, 0) (strip_sorts ty1, strip_sorts ty2); true)
handle TUNIFY => false;
(** extend and merge type signatures **)
(* arities *)
local
fun err_decl t decl = error ("Illegal " ^ str_of_decl decl ^ ": " ^ quote t);
fun for_classes _ NONE = ""
| for_classes pp (SOME (c1, c2)) =
" for classes " ^ Pretty.string_of_classrel pp [c1, c2];
fun err_conflict pp t cc (c, Ss) (c', Ss') =
error ("Conflict of type arities" ^ for_classes pp cc ^ ":\n " ^
Pretty.string_of_arity pp (t, Ss, [c]) ^ " and\n " ^
Pretty.string_of_arity pp (t, Ss', [c']));
fun coregular pp C t (c, Ss) ars =
let
fun conflict (c', Ss') =
if Sorts.class_le C (c, c') andalso not (Sorts.sorts_le C (Ss, Ss')) then
SOME ((c, c'), (c', Ss'))
else if Sorts.class_le C (c', c) andalso not (Sorts.sorts_le C (Ss', Ss)) then
SOME ((c', c), (c', Ss'))
else NONE;
in
(case Library.get_first conflict ars of
SOME ((c1, c2), (c', Ss')) => err_conflict pp t (SOME (c1, c2)) (c, Ss) (c', Ss')
| NONE => (c, Ss) :: ars)
end;
fun insert pp C t ((c, Ss), ars) =
(case assoc_string (ars, c) of
NONE => coregular pp C t (c, Ss) ars
| SOME Ss' =>
if Sorts.sorts_le C (Ss, Ss') then ars
else if Sorts.sorts_le C (Ss', Ss)
then coregular pp C t (c, Ss) (ars \ (c, Ss'))
else err_conflict pp t NONE (c, Ss) (c, Ss'));
fun complete C (c, Ss) = map (rpair Ss) (Graph.all_succs C [c]);
fun insert_arities pp classes (arities, (t, ars)) =
let val ars' =
Symtab.lookup_multi (arities, t)
|> curry (Library.foldr (insert pp classes t)) (List.concat (map (complete classes) ars))
in Symtab.update ((t, ars'), arities) end;
fun insert_table pp classes = Symtab.foldl (fn (arities, (t, ars)) =>
insert_arities pp classes (arities, (t, map (apsnd (map (Sorts.norm_sort classes))) ars)));
in
fun add_arities pp decls tsig = tsig |> change_tsig (fn (classes, default, types, arities) =>
let
fun prep (t, Ss, S) =
(case Symtab.lookup (types, t) of
SOME (LogicalType n, _) =>
if length Ss = n then
(t, map (cert_sort tsig) Ss, cert_sort tsig S)
handle TYPE (msg, _, _) => error msg
else error (bad_nargs t)
| SOME (decl, _) => err_decl t decl
| NONE => error (undecl_type t));
val ars = decls |> map ((fn (t, Ss, S) => (t, map (fn c => (c, Ss)) S)) o prep);
val arities' = Library.foldl (insert_arities pp classes) (arities, ars);
in (classes, default, types, arities') end);
fun rebuild_arities pp classes arities =
insert_table pp classes (Symtab.empty, arities);
fun merge_arities pp classes (arities1, arities2) =
insert_table pp classes (insert_table pp classes (Symtab.empty, arities1), arities2);
end;
(* classes *)
local
fun err_dup_classes cs =
error ("Duplicate declaration of class(es): " ^ commas_quote cs);
fun err_cyclic_classes pp css =
error (cat_lines (map (fn cs =>
"Cycle in class relation: " ^ Pretty.string_of_classrel pp cs) css));
fun add_class pp (c, cs) tsig = tsig |> change_tsig (fn (classes, default, types, arities) =>
let
val cs' = map (cert_class tsig) cs
handle TYPE (msg, _, _) => error msg;
val classes' = classes |> Graph.new_node (c, stamp ())
handle Graph.DUP d => err_dup_classes [d];
val classes'' = classes' |> fold Graph.add_edge_trans_acyclic (map (pair c) cs')
handle Graph.CYCLES css => err_cyclic_classes pp css;
in (classes'', default, types, arities) end);
in
val add_classes = fold o add_class;
fun add_classrel pp ps tsig = tsig |> change_tsig (fn (classes, default, types, arities) =>
let
val ps' = map (pairself (cert_class tsig)) ps
handle TYPE (msg, _, _) => error msg;
val classes' = classes |> fold Graph.add_edge_trans_acyclic ps'
handle Graph.CYCLES css => err_cyclic_classes pp css;
val default' = default |> Sorts.norm_sort classes';
val arities' = arities |> rebuild_arities pp classes';
in (classes', default', types, arities') end);
fun merge_classes pp CC = Graph.merge_trans_acyclic (op =) CC
handle Graph.DUPS cs => err_dup_classes cs
| Graph.CYCLES css => err_cyclic_classes pp css;
end;
(* default sort *)
fun set_defsort S tsig = tsig |> change_tsig (fn (classes, _, types, arities) =>
(classes, cert_sort tsig S handle TYPE (msg, _, _) => error msg, types, arities));
(* types *)
local
fun err_neg_args c =
error ("Negative number of arguments in type constructor declaration: " ^ quote c);
fun err_in_decls c decl decl' =
let val s = str_of_decl decl and s' = str_of_decl decl' in
if s = s' then error ("Duplicate declaration of " ^ s ^ ": " ^ quote c)
else error ("Conflict of " ^ s ^ " with " ^ s' ^ ": " ^ quote c)
end;
fun new_decl (c, decl) types =
(case Symtab.lookup (types, c) of
SOME (decl', _) => err_in_decls c decl decl'
| NONE => Symtab.update ((c, (decl, stamp ())), types));
fun the_decl types c = fst (valOf (Symtab.lookup (types, c)));
fun change_types f = change_tsig (fn (classes, default, types, arities) =>
(classes, default, f types, arities));
fun syntactic types (Type (c, Ts)) =
(case Symtab.lookup (types, c) of SOME (Nonterminal, _) => true | _ => false)
orelse exists (syntactic types) Ts
| syntactic _ _ = false;
fun add_abbr (a, vs, rhs) tsig = tsig |> change_types (fn types =>
let
fun err msg =
error (msg ^ "\nThe error(s) above occurred in type abbreviation: " ^ quote a);
val rhs' = strip_sorts (no_tvars (cert_typ_syntax tsig rhs))
handle TYPE (msg, _, _) => err msg;
in
(case duplicates vs of
[] => []
| dups => err ("Duplicate variables on lhs: " ^ commas_quote dups));
(case gen_rems (op =) (map (#1 o #1) (typ_tvars rhs'), vs) of
[] => []
| extras => err ("Extra variables on rhs: " ^ commas_quote extras));
types |> new_decl (a, Abbreviation (vs, rhs', syntactic types rhs'))
end);
in
fun add_types ps = change_types (fold new_decl (ps |> map (fn (c, n) =>
if n < 0 then err_neg_args c else (c, LogicalType n))));
val add_abbrs = fold add_abbr;
val add_nonterminals = change_types o fold new_decl o map (rpair Nonterminal);
fun merge_types (types1, types2) =
Symtab.merge Library.eq_snd (types1, types2) handle Symtab.DUPS (d :: _) =>
err_in_decls d (the_decl types1 d) (the_decl types2 d);
end;
(* merge type signatures *)
fun merge_tsigs pp (tsig1, tsig2) =
let
val (TSig {classes = classes1, default = default1, types = types1, arities = arities1,
log_types = _, witness = _}) = tsig1;
val (TSig {classes = classes2, default = default2, types = types2, arities = arities2,
log_types = _, witness = _}) = tsig2;
val classes' = merge_classes pp (classes1, classes2);
val default' = Sorts.inter_sort classes' (default1, default2);
val types' = merge_types (types1, types2);
val arities' = merge_arities pp classes' (arities1, arities2);
in build_tsig (classes', default', types', arities') end;
end;