src/Pure/type.ML

+(*  Title: 	Types and Sorts
+    ID:         $Id$
+    Author:	Tobias Nipkow & Lawrence C Paulson
+
+Maybe type classes should go in a separate module?
+*)
+
+
+signature TYPE =
+sig
+  structure Symtab:SYMTAB
+  type type_sig
+  val defaultS: type_sig -> sort
+  val extend: type_sig * (class * class list)list * sort *
+	      (string list * int)list *
+	      (string list * (sort list * class))list -> type_sig
+  val freeze: (indexname -> bool) -> term -> term
+  val freeze_vars: typ -> typ
+  val infer_types: type_sig * typ Symtab.table * (indexname -> typ option) *
+		   (indexname -> sort option) * (typ -> string) * typ * term
+		   -> term * (indexname*typ)list
+  val inst_term_tvars: type_sig * (indexname * typ)list -> term -> term
+  val logical_type: type_sig -> string -> bool
+  val merge: type_sig * type_sig -> type_sig
+  val thaw_vars: typ -> typ
+  val tsig0: type_sig
+  val type_errors: type_sig * (typ->string) -> typ * string list -> string list
+  val typ_instance: type_sig * typ * typ -> bool
+  val typ_match: type_sig -> (indexname*typ)list * (typ*typ) ->
+		 (indexname*typ)list
+  val unify: type_sig -> (typ*typ) * (indexname*typ)list -> (indexname*typ)list
+  val varifyT: typ -> typ
+  val varify: term * string list -> term
+  exception TUNIFY
+  exception TYPE_MATCH;
+end;
+
+functor TypeFun(structure Symtab:SYMTAB and Syntax:SYNTAX) : TYPE =
+struct
+structure Symtab = Symtab
+
+(* Miscellany *)
+
+val commas = space_implode ",";
+fun str_of_sort S = "{" ^ commas S ^ "}";
+fun str_of_dom dom = "(" ^ commas (map str_of_sort dom) ^ ")";
+fun str_of_decl(t,w,C) = t ^ ": " ^ str_of_dom w ^ C;
+
+
+(* Association list Manipulation  *)
+
+
+(* two-fold Association list lookup *)
+
+fun assoc2 (aal,(key1,key2)) = case assoc (aal,key1) of
+    Some (al) => assoc (al,key2)
+  | None => None;
+
+
+
+(**** TYPE CLASSES ****)
+
+type domain = sort list;
+type arity = domain * class;
+
+datatype type_sig =
+   TySg of {classes: class list,
+	    default: sort,
+	    subclass: (class * class list) list,
+	    args: (string * int) list,
+	    coreg: (string * (class * domain) list) list };
+
+(* classes: a list of all declared classes;
+   default: the default sort attached to all unconstrained TVars
+            occurring in a term to be type-inferred;
+   subclass: association list representation of subclass relationship;
+             (c,cs) is interpreted as "c is a proper subclass of all
+             elemenst of cs". Note that c itself is not a memeber of cs.
+   args: an association list of all declared types with the number of their
+         arguments
+   coreg: a two-fold association list of all type arities; (t,al) means that
+          type constructor t has the arities in al; an element (c,ss) of al
+          represents the arity (ss)c
+*)
+
+
+val tsig0 = TySg{classes = [],
+		 default = [],
+		 subclass = [],
+		 args = [],
+		 coreg = []};
+
+fun undcl_class (s) = error("Class " ^ s ^ " has not been declared");
+
+fun undcl_type(c) = "Undeclared type: " ^ c;
+fun undcl_type_err(c) = error(undcl_type(c));
+
+
+(* 'leq' checks the partial order on classes according to the
+   statements in the association list 'a' (i.e.'subclass')
+*)
+
+fun less a (C,D) = case assoc (a,C) of
+     Some(ss) => D mem ss
+   | None => undcl_class (C) ;
+
+fun leq a (C,D)  =  C = D orelse less a (C,D);
+
+
+fun defaultS(TySg{default,...}) = default;
+
+(* 'logical_type' checks if some type declaration t has as range
+   a class which is a subclass of "logic" *)
+
+fun logical_type(tsig as TySg{subclass,coreg,...}) t =
+  let fun is_log C = leq subclass (C,"logic")
+  in case assoc (coreg,t) of
+      Some(ars) => exists (is_log o #1) ars
+    | None => undcl_type_err(t)
+  end;
+
+
+(* 'sortorder' checks the ordering on sets of classes,i.e. on sorts:
+   S1 <= S2 ,iff for every class C2 in S2 there exists a class C1 in S1
+   with C1 <= C2 (according to an association list 'a')
+*)
+
+fun sortorder a (S1,S2) =
+  forall  (fn C2 => exists  (fn C1 => leq a (C1,C2))  S1)  S2;
+
+
+(* 'inj' inserts a new class C into a given class set S (i.e.sort) only if
+  there exists no class in S which is <= C;
+  the resulting set is minimal if S was minimal
+*)
+
+fun inj a (C,S) =
+  let fun inj1 [] = [C]
+        | inj1 (D::T) = if leq a (D,C) then D::T
+                        else if leq a (C,D) then inj1 T
+                             else D::(inj1 T)
+  in inj1 S end;
+
+
+(* 'union_sort' forms the minimal union set of two sorts S1 and S2
+   under the assumption that S2 is minimal *)
+
+fun union_sort a = foldr (inj a);
+
+
+(* 'elementwise_union' forms elementwise the minimal union set of two
+   sort lists under the assumption that the two lists have the same length
+*) 
+
+fun elementwise_union a (Ss1,Ss2) = map (union_sort a) (Ss1~~Ss2);
+   
+
+(* 'lew' checks for two sort lists the ordering for all corresponding list
+   elements (i.e. sorts) *)
+
+fun lew a (w1,w2) = forall (sortorder a)  (w1~~w2);
+
+ 
+(* 'is_min' checks if a class C is minimal in a given sort S under the 
+   assumption that S contains C *) 
+
+fun is_min a S C = not (exists (fn (D) => less a (D,C)) S);
+
+
+(* 'min_sort' reduces a sort to its minimal classes *)
+
+fun min_sort a S = distinct(filter (is_min a S) S);
+
+
+(* 'min_domain' minimizes the domain sorts of type declarationsl;
+   the function will be applied on the type declarations in extensions *) 
+
+fun min_domain subclass =
+  let fun one_min (f,(doms,ran)) = (f, (map (min_sort subclass) doms, ran))
+  in map one_min end;
+
+
+(* 'min_filter' filters a list 'ars' consisting of arities (domain * class)
+   and gives back a list of those range classes whose domains meet the 
+   predicate 'pred' *)
+   
+fun min_filter a pred ars =
+  let fun filt ([],l) = l
+        | filt ((c,x)::xs,l) = if pred(x) then filt (xs,inj a (c,l))
+                               else filt (xs,l)
+  in filt (ars,[]) end;
+
+
+(* 'cod_above' filters all arities whose domains are elementwise >= than
+   a given domain 'w' and gives back a list of the corresponding range 
+   classes *)
+
+fun cod_above (a,w,ars) = min_filter a (fn w' => lew a (w,w')) ars;
+
+
+(* 'least_sort' returns for a given type its maximum sort:
+   - type variables, free types: the sort brought with
+   - type constructors: recursive determination of the maximum sort of the
+                    arguments if the type is declared in 'coreg' of the 
+                    given type signature  *) 
+
+fun least_sort (tsig as TySg{subclass,coreg,...}) =
+  let fun ls(T as Type(a,Ts)) =
+	  let val ars = case assoc (coreg,a) of Some(ars) => ars
+			      | None => raise TYPE(undcl_type a,[T],[]);
+	  in cod_above(subclass,map ls Ts,ars) end
+	| ls(TFree(a,S)) = S
+	| ls(TVar(a,S)) = S
+  in ls end;
+
+
+fun check_has_sort(tsig as TySg{subclass,coreg,...},T,S) =
+  if sortorder subclass ((least_sort tsig T),S) then ()
+  else raise TYPE("Type not of sort " ^ (str_of_sort S),[T],[])
+
+
+(*Instantiation of type variables in types *)
+fun inst_typ_tvars(tsig,tye) =
+  let fun inst(Type(a,Ts)) = Type(a, map inst Ts)
+	| inst(T as TFree _) = T
+	| inst(T as TVar(v,S)) = (case assoc(tye,v) of
+		None => T | Some(U) => (check_has_sort(tsig,U,S); U))
+  in inst end;
+
+(*Instantiation of type variables in terms *)
+fun inst_term_tvars(tsig,tye) = map_term_types (inst_typ_tvars(tsig,tye));
+
+exception TYPE_MATCH;
+
+(* Typ matching
+   typ_match(ts,s,(U,T)) = s' <=> s'(U)=T and s' is an extension of s *)
+fun typ_match tsig =
+let fun tm(subs, (TVar(v,S), T)) = (case assoc(subs,v) of
+		None => ( (v, (check_has_sort(tsig,T,S); T))::subs
+			handle TYPE _ => raise TYPE_MATCH )
+	      | Some(U) => if U=T then subs else raise TYPE_MATCH)
+      | tm(subs, (Type(a,Ts), Type(b,Us))) =
+	if a<>b then raise TYPE_MATCH
+	else foldl tm (subs, Ts~~Us)
+      | tm(subs, (TFree(x), TFree(y))) =
+	if x=y then subs else raise TYPE_MATCH
+      | tm _ = raise TYPE_MATCH
+in tm end;
+
+fun typ_instance(tsig,T,U) = let val x = typ_match tsig ([],(U,T)) in true end
+			     handle TYPE_MATCH => false;
+
+
+(* EXTENDING AND MERGIN TYPE SIGNATURES *)
+
+fun not_ident(s) = error("Must be an identifier: " ^ s);
+
+fun twice(a) = error("Type constructor " ^a^ " has already been declared.");
+
+(*Is the type valid? Accumulates error messages in "errs".*)
+fun type_errors (tsig as TySg{classes,subclass,args,...},
+                 string_of_typ) (T,errs) =
+let fun class_err([],errs) = errs
+     |  class_err(S::Ss,errs) = 
+          if S mem classes then class_err (Ss,errs)
+	  else class_err (Ss,("Class " ^ S ^ " has not been declared") :: errs)
+    fun errors(Type(c,Us), errs) =
+	let val errs' = foldr errors (Us,errs)
+	in case assoc(args,c) of
+	     None => (undcl_type c) :: errs
+	   | Some(n) => if n=length(Us) then errs'
+		        else ("Wrong number of arguments: " ^ c) :: errs
+	end
+      | errors(TFree(_,S), errs) = class_err(S,errs)
+      | errors(TVar(_,S), errs) = class_err(S,errs);
+in case errors(T,[]) of
+     [] => ((least_sort tsig T; errs)
+	    handle TYPE(_,[U],_) => ("Ill-formed type: " ^ string_of_typ U)
+				    :: errs)
+   | errs' => errs'@errs
+end;
+
+
+(* 'add_class' adds a new class to the list of all existing classes *) 
+
+fun add_class (classes,(s,_)) =
+  if s mem classes then error("Class " ^ s ^ " declared twice.")
+  else s::classes;
+
+(* 'add_subclass' adds a tuple consisiting of a new class (the new class
+   has already been inserted into the 'classes' list) and its
+   superclasses (they must be declared in 'classes' too) to the 'subclass' 
+   list of the given type signature; 
+   furthermore all inherited superclasses according to the superclasses 
+   brought with are inserted and there is a check that there are no
+   cycles (i.e. C <= D <= C, with C <> D); *)
+
+fun add_subclass classes (subclass,(s,ges)) =
+let fun upd (subclass,s') = if s' mem classes then
+        let val Some(ges') = assoc (subclass,s)
+        in case assoc (subclass,s') of
+             Some(sups) => if s mem sups
+                           then error(" Cycle :" ^ s^" <= "^ s'^" <= "^ s )
+                           else overwrite (subclass,(s,sups union ges'))
+           | None => subclass
+         end
+         else undcl_class(s')
+in foldl upd (subclass@[(s,ges)],ges) end;
+
+
+(* 'extend_classes' inserts all new classes into the corresponding
+   lists ('classes','subclass') if possible *)
+
+fun extend_classes (classes,subclass,newclasses) =
+  if newclasses = [] then (classes,subclass) else
+  let val classes' = foldl add_class (classes,newclasses);
+      val subclass' = foldl (add_subclass classes') (subclass,newclasses);
+  in (classes',subclass') end;
+
+(* Corregularity *)
+
+(* 'is_unique_decl' checks if there exists just one declaration t:(Ss)C *)
+
+fun is_unique_decl coreg (t,(s,w)) = case assoc2 (coreg,(t,s)) of
+      Some(w1) => if w = w1 then () else
+	error("There are two declarations\n" ^
+              str_of_decl(t,w,s) ^ " and\n" ^
+	      str_of_decl(t,w1,s) ^ "\n" ^
+              "with the same result class.")
+    | None => ();
+
+(* 'restr2' checks if there are two declarations t:(Ss1)C1 and t:(Ss2)C2
+   such that C1 >= C2 then Ss1 >= Ss2 (elementwise) *)
+
+fun subs (classes,subclass) C = 
+  let fun sub (rl,l) = if leq subclass (l,C) then l::rl else rl
+  in foldl sub ([],classes) end;
+
+fun coreg_err(t,(w1,C),(w2,D)) =
+    error("Declarations " ^ str_of_decl(t,w1,C) ^ " and "
+                          ^ str_of_decl(t,w2,D) ^ " are in conflict");
+
+fun restr2 classes (subclass,coreg) (t,(s,w)) =
+  let fun restr ([],test) = ()
+        | restr (s1::Ss,test) = (case assoc2 (coreg,(t,s1)) of
+              Some (dom) => if lew subclass (test (w,dom)) then restr (Ss,test)
+                            else coreg_err (t,(w,s),(dom,s1))
+            | None => restr (Ss,test))
+      fun forward (t,(s,w)) =
+        let val s_sups = case assoc (subclass,s) of
+                   Some(s_sups) => s_sups | None => undcl_class(s);
+        in restr (s_sups,I) end
+      fun backward (t,(s,w)) =
+        let val s_subs = subs (classes,subclass) s
+        in restr (s_subs,fn (x,y) => (y,x)) end
+  in (backward (t,(s,w)); forward (t,(s,w))) end;
+
+
+fun varying_decls(t) =
+    error("Type constructor "^t^" has varying number of arguments.");
+
+
+
+(* 'coregular' checks
+   - the two restriction conditions 'is_unique_decl' and 'restr2'
+   - if the classes in the new type declarations are known in the 
+     given type signature
+   - if one type constructor has always the same number of arguments;
+   if one type declaration has passed all checks it is inserted into 
+   the 'coreg' association list of the given type signatrure  *)
+
+fun coregular (classes,subclass,args) =
+  let fun ex C = if C mem classes then () else undcl_class(C);
+
+      fun addar(w,C) (coreg,t) = case assoc(args,t) of
+            Some(n) => if n <> length w then varying_decls(t) else
+                     (is_unique_decl coreg (t,(C,w));
+	              (seq o seq) ex w;
+	              restr2 classes (subclass,coreg) (t,(C,w));
+                      let val Some(ars) = assoc(coreg,t)
+                      in overwrite(coreg,(t,(C,w) ins ars)) end)
+          | None => undcl_type_err(t);
+
+      fun addts(coreg,(ts,ar)) = foldl (addar ar) (coreg,ts)
+
+  in addts end;
+
+
+(* 'close' extends the 'coreg' association list after all new type
+   declarations have been inserted successfully:
+   for every declaration t:(Ss)C , for all classses D with C <= D:
+      if there is no declaration t:(Ss')C' with C < C' and C' <= D
+      then insert the declaration t:(Ss)D into 'coreg'
+   this means, if there exists a declaration t:(Ss)C and there is
+   no declaration t:(Ss')D with C <=D then the declaration holds
+   for all range classes more general than C *)   
+   
+fun close (coreg,subclass) =
+  let fun check sl (l,(s,dom)) = case assoc (subclass,s) of
+          Some(sups) =>
+            let fun close_sup (l,sup) =
+                  if exists (fn s'' => less subclass (s,s'') andalso
+                                       leq subclass (s'',sup)) sl
+                  then l
+                  else (sup,dom)::l
+            in foldl close_sup (l,sups) end
+        | None => l;
+      fun ext (s,l) = (s, foldl (check (map #1 l)) (l,l));
+  in map ext coreg end;
+
+fun add_types(ac,(ts,n)) =
+  let fun add_type((args,coreg),t) = case assoc(args,t) of
+          Some _ => twice(t) | None => ((t,n)::args,(t,[])::coreg)
+  in if n<0
+     then error("Type constructor cannot have negative number of arguments")
+     else foldl add_type (ac,ts)
+  end;
+
+(* 'extend' takes the above described check- and extend-functions to
+   extend a given type signature with new classes and new type declarations *)
+
+fun extend (TySg{classes,default,subclass,args,coreg},
+            newclasses,newdefault,types,arities) =
+let val (classes',subclass') = extend_classes(classes,subclass,newclasses);
+    val (args',coreg') = foldl add_types ((args,coreg),types);
+    val old_coreg = map #1 coreg;
+    fun is_old(c) = if c mem old_coreg then () else undcl_type_err(c);
+    fun is_new(c) = if c mem old_coreg then twice(c) else ();
+    val coreg'' = foldl (coregular (classes',subclass',args'))
+                        (coreg',min_domain subclass' arities);
+    val coreg''' = close (coreg'',subclass');
+    val default' = if null newdefault then default else newdefault;
+in TySg{classes=classes', default=default',subclass=subclass', args=args',
+	coreg=coreg'''} end;
+
+
+(* 'assoc_union' merges two association lists if the contents associated
+   the keys are lists *)
+
+fun assoc_union (as1,[]) = as1
+  | assoc_union (as1,(key,l2)::as2) = case assoc (as1,key) of
+        Some(l1) => assoc_union (overwrite(as1,(key,l1 union l2)),as2)
+      | None => assoc_union ((key,l2)::as1,as2);
+
+
+fun trcl r =
+  let val r' = transitive_closure r
+  in if exists (op mem) r' then error("Cyclic class structure!") else r' end;
+
+
+(* 'merge_coreg' builds the union of two 'coreg' lists;
+   it only checks the two restriction conditions and inserts afterwards
+   all elements of the second list into the first one *) 
+
+fun merge_coreg classes subclass1 =
+  let fun test_ar classes (t,ars1) (coreg1,(s,w)) =
+        (is_unique_decl coreg1 (t,(s,w));
+	 restr2 classes (subclass1,coreg1) (t,(s,w));
+	 overwrite (coreg1,(t,(s,w) ins ars1)));
+
+      fun merge_c (coreg1,(c as (t,ars2))) = case assoc (coreg1,t) of
+          Some(ars1) => foldl (test_ar classes (t,ars1)) (coreg1,ars2)
+        | None => c::coreg1
+  in foldl merge_c end;
+
+fun merge_args(args,(t,n)) = case assoc(args,t) of
+      Some(m) => if m=n then args else varying_decls(t)
+    | None => (t,n)::args;
+
+(* 'merge' takes the above declared functions to merge two type signatures *)
+
+fun merge(TySg{classes=classes1,default=default1,subclass=subclass1,args=args1,
+           coreg=coreg1},
+	  TySg{classes=classes2,default=default2,subclass=subclass2,args=args2,
+           coreg=coreg2}) =
+  let val classes' = classes1 union classes2;
+      val subclass' = trcl (assoc_union (subclass1,subclass2));
+      val args' = foldl merge_args (args1,args2)
+      val coreg' = merge_coreg classes' subclass' (coreg1,coreg2);
+      val default' = min_sort subclass' (default1 @ default2)
+  in TySg{classes=classes' , default=default',subclass=subclass', args=args',
+	  coreg=coreg'} 
+  end;
+
+(**** TYPE INFERENCE ****)
+
+(*
+
+Input:
+- a 'raw' term which contains only dummy types and some explicit type
+  constraints encoded as terms.
+- the expected type of the term.
+
+Output:
+- the correctly typed term
+- the substitution needed to unify the actual type of the term with its
+  expected type; only the TVars in the expected type are included.
+
+During type inference all TVars in the term have negative index. This keeps
+them apart from normal TVars, which is essential, because at the end the type
+of the term is unified with the expected type, which contains normal TVars.
+
+1. Add initial type information to the term (add_types).
+   This freezes (freeze_vars) TVars in explicitly provided types (eg
+   constraints or defaults) by turning them into TFrees.
+2. Carry out type inference, possibly introducing new negative TVars.
+3. Unify actual and expected type.
+4. Turn all (negative) TVars into unique new TFrees (freeze).
+5. Thaw all TVars frozen in step 1 (thaw_vars).
+
+*)
+
+(*Raised if types are not unifiable*)
+exception TUNIFY;
+
+val tyvar_count = ref(~1);
+
+fun tyinit() = (tyvar_count := ~1);
+
+fun new_tvar_inx() = (tyvar_count := !tyvar_count-1; !tyvar_count)
+
+(*
+Generate new TVar.  Index is < ~1 to distinguish it from TVars generated from
+variable names (see id_type).  Name is arbitrary because index is new.
+*)
+
+fun gen_tyvar(S) = TVar(("'a", new_tvar_inx()),S);
+fun new_id_type(a) = TVar(("'"^a, new_tvar_inx()),[]);
+
+(*Occurs check: type variable occurs in type?*)
+fun occ v tye =
+  let fun occ(Type(_,Ts)) = exists occ Ts
+        | occ(TFree _) = false
+        | occ(TVar(w,_)) = v=w orelse
+                           (case assoc(tye,w) of
+                              None   => false
+                            | Some U => occ U);
+  in occ end;
+
+(*Chase variable assignments in tye.  
+  If devar (T,tye) returns a type var then it must be unassigned.*) 
+fun devar (T as TVar(v,_), tye) = (case  assoc(tye,v)  of
+          Some U =>  devar (U,tye)
+        | None   =>  T)
+  | devar (T,tye) = T;
+
+
+(* 'dom' returns for a type constructor t the list of those domains
+   which deliver a given range class C *)
+
+fun dom coreg t C = case assoc2 (coreg, (t,C)) of
+    Some(Ss) => Ss
+  | None => raise TUNIFY;
+
+
+(* 'Dom' returns the union of all domain lists of 'dom' for a given sort S
+   (i.e. a set of range classes ); the union is carried out elementwise
+   for the seperate sorts in the domains *)
+
+fun Dom (subclass,coreg) (t,S) =
+  let val domlist = map (dom coreg t) S;
+  in if null domlist then []
+     else foldl (elementwise_union subclass) (hd domlist,tl domlist)
+  end;
+
+
+fun W ((T,S),tsig as TySg{subclass,coreg,...},tye) =
+  let fun Wd ((T,S),tye) = W ((devar (T,tye),S),tsig,tye)
+      fun Wk(T as TVar(v,S')) = 
+	      if sortorder subclass (S',S) then tye
+	      else (v,gen_tyvar(union_sort subclass (S',S)))::tye
+	| Wk(T as TFree(v,S')) = if sortorder subclass (S',S) then tye
+				 else raise TUNIFY
+	| Wk(T as Type(f,Ts)) = 
+	   if null S then tye 
+	   else foldr Wd (Ts~~(Dom (subclass,coreg) (f,S)) ,tye)
+  in Wk(T) end;
+
+
+(* Order-sorted Unification of Types (U)  *)
+
+
+(* Precondition: both types are well-formed w.r.t. type constructor arities *)
+fun unify (tsig as TySg{subclass,coreg,...}) = 
+  let fun unif ((T,U),tye) =
+        case (devar(T,tye), devar(U,tye)) of
+	  (T as TVar(v,S1), U as TVar(w,S2)) =>
+             if v=w then tye else
+             if sortorder subclass (S1,S2) then (w,T)::tye else
+             if sortorder subclass (S2,S1) then (v,U)::tye
+             else let val nu = gen_tyvar (union_sort subclass (S1,S2))
+                  in (v,nu)::(w,nu)::tye end
+        | (T as TVar(v,S), U) =>
+             if occ v tye U then raise TUNIFY else W ((U,S),tsig,(v,U)::tye)
+        | (U, T as TVar (v,S)) =>
+             if occ v tye U then raise TUNIFY else W ((U,S),tsig,(v,U)::tye)
+        | (Type(a,Ts),Type(b,Us)) =>
+	     if a<>b then raise TUNIFY else foldr unif (Ts~~Us,tye)
+        | (T,U) => if T=U then tye else raise TUNIFY
+  in unif end;
+
+(*Instantiation of type variables in types*)
+(*Pre: instantiations obey restrictions! *)
+fun inst_typ tye =
+  let fun inst(Type(a,Ts)) = Type(a, map inst Ts)
+        | inst(T as TFree _) = T
+        | inst(T as TVar(v,_)) =
+            (case assoc(tye,v) of Some U => inst U | None => T)
+  in inst end;
+
+(*Type inference for polymorphic term*)
+fun infer tsig =
+  let fun inf(Ts, Const (_,T), tye) = (T,tye)
+        | inf(Ts, Free  (_,T), tye) = (T,tye)
+        | inf(Ts, Bound i, tye) = ((nth_elem(i,Ts) , tye)
+          handle LIST _=> raise TYPE ("loose bound variable", [], [Bound i]))
+        | inf(Ts, Var (_,T), tye) = (T,tye)
+        | inf(Ts, Abs (_,T,body), tye) = 
+	    let val (U,tye') = inf(T::Ts, body, tye) in  (T-->U, tye')  end
+        | inf(Ts, f$u, tye) =
+	    let val (U,tyeU) = inf(Ts, u, tye);
+	        val (T,tyeT) = inf(Ts, f, tyeU);
+                fun err s =
+                  raise TYPE(s, [inst_typ tyeT T, inst_typ tyeT U], [f$u])
+	    in case T of
+	         Type("fun",[T1,T2]) =>
+		   ( (T2, unify tsig ((T1,U), tyeT))
+                     handle TUNIFY => err"type mismatch in application" )
+	       | TVar _ => 
+                   let val T2 = gen_tyvar([])
+                   in (T2, unify tsig ((T, U-->T2), tyeT))
+                      handle TUNIFY => err"type mismatch in application"
+                   end
+               | _ => err"rator must have function type"
+           end
+  in inf end;
+
+fun freeze_vars(Type(a,Ts)) = Type(a,map freeze_vars Ts)
+  | freeze_vars(T as TFree _) = T
+  | freeze_vars(TVar(v,S)) = TFree(Syntax.string_of_vname v, S);
+
+(* Attach a type to a constant *)
+fun type_const (a,T) = Const(a, incr_tvar (new_tvar_inx()) T);
+
+(*Find type of ident.  If not in table then use ident's name for tyvar
+  to get consistent typing.*)
+fun type_of_ixn(types,ixn as (a,_)) =
+      case types ixn of Some T => freeze_vars T | None => TVar(("'"^a,~1),[]);
+
+fun constrain(term,T) = Const(Syntax.constrainC,T-->T) $ term;
+fun constrainAbs(Abs(a,_,body),T) = Abs(a,T,body);
+
+(*
+
+Attach types to a term.  Input is a "parse tree" containing dummy types.
+Type constraints are translated and checked for validity wrt tsig.
+TVars in constraints are frozen.
+
+The atoms in the resulting term satisfy the following spec:
+
+Const(a,T):
+  T is a renamed copy of the generic type of a; renaming decreases index of
+  all TVars by new_tvar_inx(), which is less than ~1. The index of all TVars
+  in the generic type must be 0 for this to work!
+
+Free(a,T), Var(ixn,T):
+  T is either the frozen default type of a or TVar(("'"^a,~1),[])
+
+Abs(a,T,_):
+  T is either a type constraint or TVar(("'"^a,i),[]), where i is generated
+  by new_tvar_inx(). Thus different abstractions can have the bound variables
+  of the same name but different types.
+
+*)
+
+fun add_types (tsig, const_tab, types, sorts, string_of_typ) =
+  let val S0 = defaultS tsig;
+      fun defS0 ixn = case sorts ixn of Some S => S | None => S0;
+      fun prepareT(typ) =
+	let val T = Syntax.typ_of_term defS0 typ;
+	    val T' = freeze_vars T
+	in case type_errors (tsig,string_of_typ) (T,[]) of
+	     [] => T'
+	   | errs => raise TYPE(cat_lines errs,[T],[])
+	end
+      fun add (Const(a,_)) =
+            (case Symtab.lookup(const_tab, a) of
+               Some T => type_const(a,T)
+             | None => raise TYPE ("No such constant: "^a, [], []))
+        | add (Bound i) = Bound i
+        | add (Free(a,_)) =
+            (case Symtab.lookup(const_tab, a) of
+               Some T => type_const(a,T)
+             | None => Free(a, type_of_ixn(types,(a,~1))))
+        | add (Var(ixn,_)) = Var(ixn, type_of_ixn(types,ixn))
+        | add (Abs(a,_,body)) = Abs(a, new_id_type a, add body)
+        | add ((f as Const(a,_)$t1)$t2) =
+	    if a=Syntax.constrainC then constrain(add t1,prepareT t2) else
+	    if a=Syntax.constrainAbsC then constrainAbs(add t1,prepareT t2)
+	    else add f $ add t2
+        | add (f$t) = add f $ add t
+  in  add  end;
+
+
+(* Post-Processing *)
+
+
+(*Instantiation of type variables in terms*)
+fun inst_types tye = map_term_types (inst_typ tye);
+
+(*Delete explicit constraints -- occurrences of "_constrain" *)
+fun unconstrain (Abs(a,T,t)) = Abs(a, T, unconstrain t)
+  | unconstrain ((f as Const(a,_)) $ t) =
+      if a=Syntax.constrainC then unconstrain t
+      else unconstrain f $ unconstrain t
+  | unconstrain (f$t) = unconstrain f $ unconstrain t
+  | unconstrain (t) = t;
+
+
+(* Turn all TVars which satisfy p into new TFrees *)
+fun freeze p t =
+  let val fs = add_term_tfree_names(t,[]);
+      val inxs = filter p (add_term_tvar_ixns(t,[]));
+      val vmap = inxs ~~ variantlist(map #1 inxs, fs);
+      fun free(Type(a,Ts)) = Type(a, map free Ts)
+        | free(T as TVar(v,S)) =
+            (case assoc(vmap,v) of None => T | Some(a) => TFree(a,S))
+        | free(T as TFree _) = T
+  in map_term_types free t end;
+
+(* Thaw all TVars that were frozen in freeze_vars *)
+fun thaw_vars(Type(a,Ts)) = Type(a, map thaw_vars Ts)
+  | thaw_vars(T as TFree(a,S)) = (case explode a of
+	  "?"::"'"::vn => let val ((b,i),_) = Syntax.scan_varname vn
+			  in TVar(("'"^b,i),S) end
+	| _ => T)
+  | thaw_vars(T) = T;
+
+
+fun restrict tye =
+  let fun clean(tye1, ((a,i),T)) =
+	if i < 0 then tye1 else ((a,i),inst_typ tye T) :: tye1
+  in foldl clean ([],tye) end
+
+
+(*Infer types for term t using tables. Check that t's type and T unify *)
+
+fun infer_term (tsig, const_tab, types, sorts, string_of_typ, T, t) =
+  let val u = add_types (tsig, const_tab, types, sorts, string_of_typ) t;
+      val (U,tye) = infer tsig ([], u, []);
+      val uu = unconstrain u;
+      val tye' = unify tsig ((T,U),tye) handle TUNIFY => raise TYPE
+	("Term does not have expected type", [T, U], [inst_types tye uu])
+      val Ttye = restrict tye' (* restriction to TVars in T *)
+      val all = Const("", Type("", map snd Ttye)) $ (inst_types tye' uu)
+        (* all is a dummy term which contains all exported TVars *)
+      val Const(_,Type(_,Ts)) $ u'' =
+            map_term_types thaw_vars (freeze (fn (_,i) => i<0) all)
+        (* turn all internally generated TVars into TFrees
+           and thaw all initially frozen TVars *)
+  in (u'', (map fst Ttye) ~~ Ts) end;
+
+fun infer_types args = (tyinit(); infer_term args);
+
+
+(* Turn TFrees into TVars to allow types & axioms to be written without "?" *)
+fun varifyT(Type(a,Ts)) = Type(a,map varifyT Ts)
+  | varifyT(TFree(a,S)) = TVar((a,0),S)
+  | varifyT(T) = T;
+
+(* Turn TFrees except those in fixed into new TVars *)
+fun varify(t,fixed) =
+  let val fs = add_term_tfree_names(t,[]) \\ fixed;
+      val ixns = add_term_tvar_ixns(t,[]);
+      val fmap = fs ~~ variantlist(fs, map #1 ixns)
+      fun thaw(Type(a,Ts)) = Type(a, map thaw Ts)
+        | thaw(T as TVar _) = T
+        | thaw(T as TFree(a,S)) =
+            (case assoc(fmap,a) of None => T | Some b => TVar((b,0),S))
+  in map_term_types thaw t end;
+
+
+end;