(* Title: Pure/type.ML
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 eq_tsig: tsig * tsig -> bool
val rep_tsig: tsig ->
{classes: Name_Space.T * Sorts.algebra,
default: sort,
types: decl Name_Space.table,
log_types: string list}
val empty_tsig: tsig
val class_space: tsig -> Name_Space.T
val class_alias: Name_Space.naming -> binding -> string -> tsig -> tsig
val intern_class: tsig -> xstring -> string
val extern_class: tsig -> string -> xstring
val defaultS: tsig -> sort
val logical_types: tsig -> string list
val eq_sort: tsig -> sort * sort -> bool
val subsort: tsig -> sort * sort -> bool
val of_sort: tsig -> typ * sort -> bool
val inter_sort: tsig -> sort * sort -> sort
val cert_class: tsig -> class -> class
val cert_sort: tsig -> sort -> sort
val minimize_sort: tsig -> sort -> sort
val witness_sorts: tsig -> (typ * sort) list -> sort list -> (typ * sort) list
type mode
val mode_default: mode
val mode_syntax: mode
val mode_abbrev: mode
val get_mode: Proof.context -> mode
val set_mode: mode -> Proof.context -> Proof.context
val restore_mode: Proof.context -> Proof.context -> Proof.context
val type_space: tsig -> Name_Space.T
val type_alias: Name_Space.naming -> binding -> string -> tsig -> tsig
val intern_type: tsig -> xstring -> string
val extern_type: tsig -> string -> xstring
val is_logtype: tsig -> string -> bool
val the_decl: tsig -> string -> decl
val cert_typ_mode: mode -> tsig -> typ -> typ
val cert_typ: tsig -> typ -> typ
val arity_number: tsig -> string -> int
val arity_sorts: Pretty.pp -> tsig -> string -> sort -> sort list
(*special treatment of type vars*)
val sort_of_atyp: typ -> sort
val strip_sorts: typ -> typ
val no_tvars: typ -> typ
val varify_global: (string * sort) list -> term -> ((string * sort) * indexname) list * term
val legacy_freeze_thaw_type: typ -> typ * (typ -> typ)
val legacy_freeze_type: typ -> typ
val legacy_freeze_thaw: term -> term * (term -> term)
val legacy_freeze: term -> term
(*matching and unification*)
exception TYPE_MATCH
type tyenv = (sort * typ) Vartab.table
val lookup: tyenv -> indexname * sort -> typ option
val devar: tyenv -> typ -> typ
val typ_match: tsig -> typ * typ -> tyenv -> tyenv
val typ_instance: tsig -> typ * typ -> bool
val raw_match: typ * typ -> tyenv -> tyenv
val raw_matches: typ list * typ list -> tyenv -> tyenv
val raw_instance: typ * typ -> bool
exception TUNIFY
val unify: tsig -> typ * typ -> tyenv * int -> tyenv * int
val raw_unify: typ * typ -> tyenv -> tyenv
val raw_unifys: typ list * typ list -> tyenv -> tyenv
val could_unify: typ * typ -> bool
val could_unifys: typ list * typ list -> bool
val eq_type: tyenv -> typ * typ -> bool
(*extend and merge type signatures*)
val add_class: Pretty.pp -> Name_Space.naming -> binding * class list -> tsig -> tsig
val hide_class: bool -> string -> tsig -> tsig
val set_defsort: sort -> tsig -> tsig
val add_type: Name_Space.naming -> binding * int -> tsig -> tsig
val add_abbrev: Name_Space.naming -> binding * string list * typ -> tsig -> tsig
val add_nonterminal: Name_Space.naming -> binding -> tsig -> tsig
val hide_type: bool -> string -> tsig -> tsig
val add_arity: Pretty.pp -> arity -> tsig -> tsig
val add_classrel: Pretty.pp -> class * class -> tsig -> tsig
val merge_tsig: 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;
(* type tsig *)
datatype tsig =
TSig of {
classes: Name_Space.T * Sorts.algebra, (*order-sorted algebra of type classes*)
default: sort, (*default sort on input*)
types: decl Name_Space.table, (*declared types*)
log_types: string list}; (*logical types sorted by number of arguments*)
fun eq_tsig
(TSig {classes = classes1, default = default1, types = types1, log_types = _},
TSig {classes = classes2, default = default2, types = types2, log_types = _}) =
pointer_eq (classes1, classes2) andalso
default1 = default2 andalso
pointer_eq (types1, types2);
fun rep_tsig (TSig comps) = comps;
fun make_tsig (classes, default, types, log_types) =
TSig {classes = classes, default = default, types = types, log_types = log_types};
fun build_tsig (classes, default, types) =
let
val log_types =
Symtab.fold (fn (c, LogicalType n) => cons (c, n) | _ => I) (snd types) []
|> Library.sort (int_ord o pairself snd) |> map fst;
in make_tsig (classes, default, types, log_types) end;
fun map_tsig f (TSig {classes, default, types, log_types = _}) =
build_tsig (f (classes, default, types));
val empty_tsig =
build_tsig ((Name_Space.empty "class", Sorts.empty_algebra), [], Name_Space.empty_table "type");
(* classes and sorts *)
val class_space = #1 o #classes o rep_tsig;
fun class_alias naming binding name = map_tsig (fn ((space, classes), default, types) =>
((Name_Space.alias naming binding name space, classes), default, types));
val intern_class = Name_Space.intern o class_space;
val extern_class = Name_Space.extern o class_space;
fun defaultS (TSig {default, ...}) = default;
fun logical_types (TSig {log_types, ...}) = log_types;
fun eq_sort (TSig {classes, ...}) = Sorts.sort_eq (#2 classes);
fun subsort (TSig {classes, ...}) = Sorts.sort_le (#2 classes);
fun of_sort (TSig {classes, ...}) = Sorts.of_sort (#2 classes);
fun inter_sort (TSig {classes, ...}) = Sorts.inter_sort (#2 classes);
fun cert_class (TSig {classes, ...}) = Sorts.certify_class (#2 classes);
fun cert_sort (TSig {classes, ...}) = Sorts.certify_sort (#2 classes);
fun minimize_sort (TSig {classes, ...}) = Sorts.minimize_sort (#2 classes);
fun witness_sorts (TSig {classes, log_types, ...}) =
Sorts.witness_sorts (#2 classes) log_types;
(* certification mode *)
datatype mode = Mode of {normalize: bool, logical: bool};
val mode_default = Mode {normalize = true, logical = true};
val mode_syntax = Mode {normalize = true, logical = false};
val mode_abbrev = Mode {normalize = false, logical = false};
structure Mode = Proof_Data
(
type T = mode;
fun init _ = mode_default;
);
val get_mode = Mode.get;
fun set_mode mode = Mode.map (K mode);
fun restore_mode ctxt = set_mode (get_mode ctxt);
(* types *)
val type_space = #1 o #types o rep_tsig;
fun type_alias naming binding name = map_tsig (fn (classes, default, (space, types)) =>
(classes, default, (Name_Space.alias naming binding name space, types)));
val intern_type = Name_Space.intern o type_space;
val extern_type = Name_Space.extern o type_space;
val is_logtype = member (op =) o logical_types;
fun undecl_type c = "Undeclared type constructor: " ^ quote c;
fun lookup_type (TSig {types = (_, types), ...}) = Symtab.lookup types;
fun the_decl tsig c =
(case lookup_type tsig c of
NONE => error (undecl_type c)
| SOME decl => decl);
(* certified types *)
fun bad_nargs t = "Bad number of arguments for type constructor: " ^ quote t;
local
fun inst_typ env (Type (c, Ts)) = Type (c, map (inst_typ env) Ts)
| inst_typ env (T as TFree (x, _)) = the_default T (AList.lookup (op =) env x)
| inst_typ _ T = T;
in
fun cert_typ_mode (Mode {normalize, logical}) tsig ty =
let
fun err msg = raise TYPE (msg, [ty], []);
val check_logical =
if logical then fn c => err ("Illegal occurrence of syntactic type: " ^ quote c)
else fn _ => ();
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 the_decl tsig c of
LogicalType n => (nargs n; Type (c, Ts'))
| Abbreviation (vs, U, syn) =>
(nargs (length vs);
if syn then check_logical c else ();
if normalize then inst_typ (vs ~~ Ts') U
else Type (c, Ts'))
| Nonterminal => (nargs 0; check_logical c; T))
end
| cert (TFree (x, S)) = TFree (x, cert_sort tsig S)
| cert (TVar (xi as (_, i), S)) =
if i < 0 then
err ("Malformed type variable: " ^ quote (Term.string_of_vname xi))
else TVar (xi, cert_sort tsig S);
val ty' = cert ty;
in if ty = ty' then ty else ty' end; (*avoid copying of already normal type*)
val cert_typ = cert_typ_mode mode_default;
end;
(* type arities *)
fun arity_number tsig a =
(case lookup_type tsig a of
SOME (LogicalType n) => n
| _ => error (undecl_type a));
fun arity_sorts _ tsig a [] = replicate (arity_number tsig a) []
| arity_sorts pp (TSig {classes, ...}) a S = Sorts.mg_domain (#2 classes) a S
handle Sorts.CLASS_ERROR err => error (Sorts.class_error pp err);
(** special treatment of type vars **)
(* sort_of_atyp *)
fun sort_of_atyp (TFree (_, S)) = S
| sort_of_atyp (TVar (_, S)) = S
| sort_of_atyp T = raise TYPE ("sort_of_atyp", [T], []);
(* 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 Term.add_tvarsT T [] of [] => T
| vs => raise TYPE ("Illegal schematic type variable(s): " ^
commas_quote (map (Term.string_of_vname o #1) (rev vs)), [T], []));
(* varify_global *)
fun varify_global fixed t =
let
val fs = Term.fold_types (Term.fold_atyps
(fn TFree v => if member (op =) fixed v then I else insert (op =) v | _ => I)) t [];
val used = Name.context
|> fold_types (fold_atyps (fn TVar ((a, _), _) => Name.declare a | _ => I)) t;
val fmap = fs ~~ map (rpair 0) (#1 (Name.variants (map fst fs) used));
fun thaw (f as (_, S)) =
(case AList.lookup (op =) fmap f of
NONE => TFree f
| SOME xi => TVar (xi, S));
in (fmap, map_types (map_type_tfree thaw) t) end;
(* freeze_thaw: freeze TVars in a term; return the "thaw" inverse *)
local
fun new_name (ix, (pairs, used)) =
let val v = Name.variant used (string_of_indexname ix)
in ((ix, v) :: pairs, v :: used) end;
fun freeze_one alist (ix, sort) =
TFree (the (AList.lookup (op =) alist ix), sort)
handle Option =>
raise TYPE ("Failure during freezing of ?" ^ string_of_indexname ix, [], []);
fun thaw_one alist (a, sort) = TVar (the (AList.lookup (op =) alist a), sort)
handle Option => TFree (a, sort);
in
fun legacy_freeze_thaw_type T =
let
val used = OldTerm.add_typ_tfree_names (T, [])
and tvars = map #1 (OldTerm.add_typ_tvars (T, []));
val (alist, _) = List.foldr new_name ([], used) tvars;
in (map_type_tvar (freeze_one alist) T, map_type_tfree (thaw_one (map swap alist))) end;
val legacy_freeze_type = #1 o legacy_freeze_thaw_type;
fun legacy_freeze_thaw t =
let
val used = OldTerm.it_term_types OldTerm.add_typ_tfree_names (t, [])
and tvars = map #1 (OldTerm.it_term_types OldTerm.add_typ_tvars (t, []));
val (alist, _) = List.foldr new_name ([], used) tvars;
in
(case alist of
[] => (t, fn x => x) (*nothing to do!*)
| _ => (map_types (map_type_tvar (freeze_one alist)) t,
map_types (map_type_tfree (thaw_one (map swap alist)))))
end;
val legacy_freeze = #1 o legacy_freeze_thaw;
end;
(** matching and unification of types **)
type tyenv = (sort * typ) Vartab.table;
fun tvar_clash ixn S S' = raise TYPE ("Type variable " ^
quote (Term.string_of_vname ixn) ^ " has two distinct sorts",
[TVar (ixn, S), TVar (ixn, S')], []);
fun lookup tye (ixn, S) =
(case Vartab.lookup tye ixn of
NONE => NONE
| SOME (S', T) => if S = S' then SOME T else tvar_clash ixn S S');
(* matching *)
exception TYPE_MATCH;
fun typ_match tsig =
let
fun match (TVar (v, S), T) subs =
(case lookup subs (v, S) of
NONE =>
if of_sort tsig (T, S) then Vartab.update_new (v, (S, T)) subs
else raise TYPE_MATCH
| SOME U => if U = T then subs else raise TYPE_MATCH)
| match (Type (a, Ts), Type (b, Us)) subs =
if a <> b then raise TYPE_MATCH
else matches (Ts, Us) subs
| match (TFree x, TFree y) subs =
if x = y then subs else raise TYPE_MATCH
| match _ _ = raise TYPE_MATCH
and matches (T :: Ts, U :: Us) subs = matches (Ts, Us) (match (T, U) subs)
| matches _ subs = subs;
in match end;
fun typ_instance tsig (T, U) =
(typ_match tsig (U, T) Vartab.empty; true) handle TYPE_MATCH => false;
(*purely structural matching*)
fun raw_match (TVar (v, S), T) subs =
(case lookup subs (v, S) of
NONE => Vartab.update_new (v, (S, T)) subs
| SOME U => if U = T then subs else raise TYPE_MATCH)
| raw_match (Type (a, Ts), Type (b, Us)) subs =
if a <> b then raise TYPE_MATCH
else raw_matches (Ts, Us) subs
| raw_match (TFree x, TFree y) subs =
if x = y then subs else raise TYPE_MATCH
| raw_match _ _ = raise TYPE_MATCH
and raw_matches (T :: Ts, U :: Us) subs = raw_matches (Ts, Us) (raw_match (T, U) subs)
| raw_matches ([], []) subs = subs
| raw_matches _ _ = raise TYPE_MATCH;
fun raw_instance (T, U) =
(raw_match (U, T) Vartab.empty; 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, S)) =
Term.eq_ix (v, w) orelse
(case lookup tye (w, S) 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 tye (T as TVar v) =
(case lookup tye v of
SOME U => devar tye U
| NONE => T)
| devar _ T = T;
(*order-sorted unification*)
fun unify (TSig {classes = (_, classes), ...}) TU (tyenv, maxidx) =
let
val tyvar_count = Unsynchronized.ref maxidx;
fun gen_tyvar S = TVar ((Name.aT, Unsynchronized.inc tyvar_count), S);
fun mg_domain a S = Sorts.mg_domain classes a S
handle Sorts.CLASS_ERROR _ => 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, (S', 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 (T :: Ts, S :: Ss) tye = meets (Ts, Ss) (meet (devar tye T, S) tye)
| meets _ tye = tye;
fun unif (ty1, ty2) tye =
(case (devar tye ty1, devar tye ty2) of
(T as TVar (v, S1), U as TVar (w, S2)) =>
if Term.eq_ix (v, w) then
if S1 = S2 then tye else tvar_clash v S1 S2
else if Sorts.sort_le classes (S1, S2) then
Vartab.update_new (w, (S2, T)) tye
else if Sorts.sort_le classes (S2, S1) then
Vartab.update_new (v, (S1, U)) tye
else
let val S = gen_tyvar (Sorts.inter_sort classes (S1, S2)) in
Vartab.update_new (v, (S1, S)) (Vartab.update_new (w, (S2, S)) tye)
end
| (TVar (v, S), T) =>
if occurs v tye T then raise TUNIFY
else meet (T, S) (Vartab.update_new (v, (S, T)) tye)
| (T, TVar (v, S)) =>
if occurs v tye T then raise TUNIFY
else meet (T, S) (Vartab.update_new (v, (S, T)) tye)
| (Type (a, Ts), Type (b, Us)) =>
if a <> b then raise TUNIFY
else unifs (Ts, Us) tye
| (T, U) => if T = U then tye else raise TUNIFY)
and unifs (T :: Ts, U :: Us) tye = unifs (Ts, Us) (unif (T, U) tye)
| unifs _ tye = tye;
in (unif TU tyenv, ! tyvar_count) end;
(*purely structural unification*)
fun raw_unify (ty1, ty2) tye =
(case (devar tye ty1, devar tye ty2) of
(T as TVar (v, S1), TVar (w, S2)) =>
if Term.eq_ix (v, w) then
if S1 = S2 then tye else tvar_clash v S1 S2
else Vartab.update_new (w, (S2, T)) tye
| (TVar (v, S), T) =>
if occurs v tye T then raise TUNIFY
else Vartab.update_new (v, (S, T)) tye
| (T, TVar (v, S)) =>
if occurs v tye T then raise TUNIFY
else Vartab.update_new (v, (S, T)) tye
| (Type (a, Ts), Type (b, Us)) =>
if a <> b then raise TUNIFY
else raw_unifys (Ts, Us) tye
| (T, U) => if T = U then tye else raise TUNIFY)
and raw_unifys (T :: Ts, U :: Us) tye = raw_unifys (Ts, Us) (raw_unify (T, U) tye)
| raw_unifys ([], []) tye = tye
| raw_unifys _ _ = raise TUNIFY;
(*fast unification filter*)
fun could_unify (Type (a, Ts), Type (b, Us)) = a = b andalso could_unifys (Ts, Us)
| could_unify (TFree (a, _), TFree (b, _)) = a = b
| could_unify (TVar _, _) = true
| could_unify (_, TVar _) = true
| could_unify _ = false
and could_unifys (T :: Ts, U :: Us) = could_unify (T, U) andalso could_unifys (Ts, Us)
| could_unifys ([], []) = true
| could_unifys _ = false;
(*equality with respect to a type environment*)
fun equal_type tye (T, T') =
(case (devar tye T, devar tye T') of
(Type (s, Ts), Type (s', Ts')) =>
s = s' andalso ListPair.all (equal_type tye) (Ts, Ts')
| (U, U') => U = U');
fun eq_type tye =
if Vartab.is_empty tye then op = else equal_type tye;
(** extend and merge type signatures **)
(* classes *)
fun add_class pp naming (c, cs) tsig =
tsig |> map_tsig (fn ((space, classes), default, types) =>
let
val cs' = map (cert_class tsig) cs
handle TYPE (msg, _, _) => error msg;
val (c', space') = space |> Name_Space.declare true naming c;
val classes' = classes |> Sorts.add_class pp (c', cs');
in ((space', classes'), default, types) end);
fun hide_class fully c = map_tsig (fn ((space, classes), default, types) =>
((Name_Space.hide fully c space, classes), default, types));
(* arities *)
fun add_arity pp (t, Ss, S) tsig = tsig |> map_tsig (fn ((space, classes), default, types) =>
let
val _ =
(case lookup_type tsig t of
SOME (LogicalType n) => if length Ss <> n then error (bad_nargs t) else ()
| SOME _ => error ("Logical type constructor expected: " ^ quote t)
| NONE => error (undecl_type t));
val (Ss', S') = (map (cert_sort tsig) Ss, cert_sort tsig S)
handle TYPE (msg, _, _) => error msg;
val classes' = classes |> Sorts.add_arities pp ((t, map (fn c' => (c', Ss')) S'));
in ((space, classes'), default, types) end);
(* classrel *)
fun add_classrel pp rel tsig =
tsig |> map_tsig (fn ((space, classes), default, types) =>
let
val rel' = pairself (cert_class tsig) rel
handle TYPE (msg, _, _) => error msg;
val classes' = classes |> Sorts.add_classrel pp rel';
in ((space, classes'), default, types) end);
(* default sort *)
fun set_defsort S tsig = tsig |> map_tsig (fn (classes, _, types) =>
(classes, cert_sort tsig S handle TYPE (msg, _, _) => error msg, types));
(* types *)
local
fun new_decl naming (c, decl) types =
#2 (Name_Space.define true naming (c, decl) types);
fun map_types f = map_tsig (fn (classes, default, types) =>
let
val (space', tab') = f types;
val _ = Name_Space.intern space' "dummy" = "dummy" orelse
error "Illegal declaration of dummy type";
in (classes, default, (space', tab')) end);
fun syntactic types (Type (c, Ts)) =
(case Symtab.lookup types c of SOME Nonterminal => true | _ => false)
orelse exists (syntactic types) Ts
| syntactic _ _ = false;
in
fun add_type naming (c, n) =
if n < 0 then error ("Bad type constructor declaration " ^ quote (Binding.str_of c))
else map_types (new_decl naming (c, LogicalType n));
fun add_abbrev naming (a, vs, rhs) tsig = tsig |> map_types (fn types =>
let
fun err msg =
cat_error msg ("The error(s) above occurred in type abbreviation " ^
quote (Binding.str_of a));
val rhs' = strip_sorts (no_tvars (cert_typ_mode mode_syntax tsig rhs))
handle TYPE (msg, _, _) => err msg;
val _ =
(case duplicates (op =) vs of
[] => []
| dups => err ("Duplicate variables on lhs: " ^ commas_quote dups));
val _ =
(case subtract (op =) vs (map #1 (Term.add_tfreesT rhs' [])) of
[] => []
| extras => err ("Extra variables on rhs: " ^ commas_quote extras));
in types |> new_decl naming (a, Abbreviation (vs, rhs', syntactic (#2 types) rhs')) end);
fun add_nonterminal naming = map_types o new_decl naming o rpair Nonterminal;
end;
fun hide_type fully c = map_tsig (fn (classes, default, (space, types)) =>
(classes, default, (Name_Space.hide fully c space, types)));
(* merge type signatures *)
fun merge_tsig pp (tsig1, tsig2) =
let
val (TSig {classes = (space1, classes1), default = default1, types = types1,
log_types = _}) = tsig1;
val (TSig {classes = (space2, classes2), default = default2, types = types2,
log_types = _}) = tsig2;
val space' = Name_Space.merge (space1, space2);
val classes' = Sorts.merge_algebra pp (classes1, classes2);
val default' = Sorts.inter_sort classes' (default1, default2);
val types' = Name_Space.merge_tables (types1, types2);
in build_tsig ((space', classes'), default', types') end;
end;