src/Pure/term.ML
author clasohm
Thu Sep 16 12:20:38 1993 +0200 (1993-09-16)
changeset 0 a5a9c433f639
child 40 3f9f8395519e
permissions -rw-r--r--
Initial revision
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(*  Title: 	term.ML
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    ID:         $Id$
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    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   Cambridge University 1992
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*)
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(*Simply typed lambda-calculus: types, terms, and basic operations*)
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(*Indexnames can be quickly renamed by adding an offset to the integer part,
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  for resolution.*)
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type indexname = string*int;
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(* Types are classified by classes. *)
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type class = string;
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type sort  = class list;
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(* The sorts attached to TFrees and TVars specify the sort of that variable *)
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datatype typ = Type  of string * typ list
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             | TFree of string * sort
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	     | TVar  of indexname * sort;
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infixr 5 -->;
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fun S --> T = Type("fun",[S,T]);
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(*handy for multiple args: [T1,...,Tn]--->T  gives  T1-->(T2--> ... -->T)*)
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infixr --->;
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val op ---> = foldr (op -->);
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(*terms.  Bound variables are indicated by depth number.
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  Free variables, (scheme) variables and constants have names.
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  An term is "closed" if there every bound variable of level "lev"
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  is enclosed by at least "lev" abstractions. 
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  It is possible to create meaningless terms containing loose bound vars
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  or type mismatches.  But such terms are not allowed in rules. *)
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infix 9 $;  (*application binds tightly!*)
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datatype term = 
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    Const of string * typ
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  | Free  of string * typ 
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  | Var   of indexname * typ
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  | Bound of int
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  | Abs   of string*typ*term
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  | op $  of term*term;
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(*For errors involving type mismatches*)
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exception TYPE of string * typ list * term list;
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(*For system errors involving terms*)
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exception TERM of string * term list;
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(*Note variable naming conventions!
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    a,b,c: string
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    f,g,h: functions (including terms of function type)
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    i,j,m,n: int
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    t,u: term
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    v,w: indexnames
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    x,y: any
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    A,B,C: term (denoting formulae)
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    T,U: typ
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*)
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(** Discriminators **)
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fun is_Const (Const _) = true
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  | is_Const _ = false;
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fun is_Free (Free _) = true
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  | is_Free _ = false;
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fun is_Var (Var _) = true
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  | is_Var _ = false;
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fun is_TVar (TVar _) = true
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  | is_TVar _ = false;
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(** Destructors **)
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fun dest_Const (Const x) =  x
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  | dest_Const t = raise TERM("dest_Const", [t]);
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fun dest_Free (Free x) =  x
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  | dest_Free t = raise TERM("dest_Free", [t]);
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fun dest_Var (Var x) =  x
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  | dest_Var t = raise TERM("dest_Var", [t]);
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(* maps  [T1,...,Tn]--->T  to the list  [T1,T2,...,Tn]*)
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fun binder_types (Type("fun",[S,T])) = S :: binder_types T
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  | binder_types _   =  [];
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(* maps  [T1,...,Tn]--->T  to T*)
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fun body_type (Type("fun",[S,T])) = body_type T
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  | body_type T   =  T;
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(* maps  [T1,...,Tn]--->T  to   ([T1,T2,...,Tn], T)  *)
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fun strip_type T : typ list * typ =
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  (binder_types T, body_type T);
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(*Compute the type of the term, checking that combinations are well-typed
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  Ts = [T0,T1,...] holds types of bound variables 0, 1, ...*)
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fun type_of1 (Ts, Const (_,T)) = T
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  | type_of1 (Ts, Free  (_,T)) = T
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  | type_of1 (Ts, Bound i) = (nth_elem (i,Ts)  
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  	handle LIST _ => raise TYPE("type_of: bound variable", [], [Bound i]))
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  | type_of1 (Ts, Var (_,T)) = T
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  | type_of1 (Ts, Abs (_,T,body)) = T --> type_of1(T::Ts, body)
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  | type_of1 (Ts, f$u) = 
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      let val U = type_of1(Ts,u)
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          and T = type_of1(Ts,f)
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      in case T of
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	    Type("fun",[T1,T2]) =>
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	      if T1=U then T2  else raise TYPE
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	         ("type_of: type mismatch in application", [T1,U], [f$u])
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	  | _ => raise TYPE ("type_of: Rator must have function type",
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	                        [T,U], [f$u])
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      end;
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fun type_of t : typ = type_of1 ([],t);
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(*Determines the type of a term, with minimal checking*)
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fun fastype_of(Ts, f$u) = (case fastype_of(Ts,f) of
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	Type("fun",[_,T]) => T
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	| _ => raise TERM("fastype_of: expected function type", [f$u]))
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  | fastype_of(_, Const (_,T)) = T
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  | fastype_of(_, Free (_,T)) = T
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  | fastype_of(Ts, Bound i) = (nth_elem(i,Ts)
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  	 handle LIST _ => raise TERM("fastype_of: Bound", [Bound i]))
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  | fastype_of(_, Var (_,T)) = T 
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  | fastype_of(Ts, Abs (_,T,u)) = T --> fastype_of(T::Ts, u);
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(* maps  (x1,...,xn)t   to   t  *)
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fun strip_abs_body (Abs(_,_,t))  =  strip_abs_body t  
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  | strip_abs_body u  =  u;
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(* maps  (x1,...,xn)t   to   [x1, ..., xn]  *)
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fun strip_abs_vars (Abs(a,T,t))  =  (a,T) :: strip_abs_vars t 
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  | strip_abs_vars u  =  [] : (string*typ) list;
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fun strip_qnt_body qnt =
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let fun strip(tm as Const(c,_)$Abs(_,_,t)) = if c=qnt then strip t else tm
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      | strip t = t
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in strip end;
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fun strip_qnt_vars qnt =
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let fun strip(Const(c,_)$Abs(a,T,t)) = if c=qnt then (a,T)::strip t else []
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      | strip t  =  [] : (string*typ) list
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in strip end;
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(* maps   (f, [t1,...,tn])  to  f(t1,...,tn) *)
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val list_comb : term * term list -> term = foldl (op $);
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(* maps   f(t1,...,tn)  to  (f, [t1,...,tn]) ; naturally tail-recursive*)
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fun strip_comb u : term * term list = 
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    let fun stripc (f$t, ts) = stripc (f, t::ts)
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        |   stripc  x =  x 
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    in  stripc(u,[])  end;
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(* maps   f(t1,...,tn)  to  f , which is never a combination *)
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fun head_of (f$t) = head_of f
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  | head_of u = u;
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(*Number of atoms and abstractions in a term*)
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fun size_of_term (Abs (_,_,body)) = 1 + size_of_term body
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  | size_of_term (f$t) = size_of_term f  +  size_of_term t
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  | size_of_term _ = 1;
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(* apply a function to all types in a term *)
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fun map_term_types f =
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let fun map(Const(a,T)) = Const(a, f T)
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      | map(Free(a,T)) = Free(a, f T)
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      | map(Var(v,T)) = Var(v, f T)
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      | map(t as Bound _)  = t
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      | map(Abs(a,T,t)) = Abs(a, f T, map t)
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      | map(f$t) = map f $ map t;
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in map end;
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(* iterate a function over all types in a term *)
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fun it_term_types f =
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let fun iter(Const(_,T), a) = f(T,a)
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      | iter(Free(_,T), a) = f(T,a)
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      | iter(Var(_,T), a) = f(T,a)
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      | iter(Abs(_,T,t), a) = iter(t,f(T,a))
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      | iter(f$u, a) = iter(f, iter(u, a))
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      | iter(Bound _, a) = a
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in iter end
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(** Connectives of higher order logic **)
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val propT : typ = Type("prop",[]);
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val implies = Const("==>", propT-->propT-->propT);
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fun all T = Const("all", (T-->propT)-->propT);
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fun equals T = Const("==", T-->T-->propT);
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fun flexpair T = Const("=?=", T-->T-->propT);
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(* maps  !!x1...xn. t   to   t  *)
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fun strip_all_body (Const("all",_)$Abs(_,_,t))  =  strip_all_body t  
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  | strip_all_body t  =  t;
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(* maps  !!x1...xn. t   to   [x1, ..., xn]  *)
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fun strip_all_vars (Const("all",_)$Abs(a,T,t))  =
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		(a,T) :: strip_all_vars t 
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  | strip_all_vars t  =  [] : (string*typ) list;
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(*increments a term's non-local bound variables
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  required when moving a term within abstractions
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     inc is  increment for bound variables
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     lev is  level at which a bound variable is considered 'loose'*)
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fun incr_bv (inc, lev, u as Bound i) = if i>=lev then Bound(i+inc) else u 
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  | incr_bv (inc, lev, Abs(a,T,body)) =
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	Abs(a, T, incr_bv(inc,lev+1,body))
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  | incr_bv (inc, lev, f$t) = 
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      incr_bv(inc,lev,f) $ incr_bv(inc,lev,t)
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  | incr_bv (inc, lev, u) = u;
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fun incr_boundvars  0  t = t
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  | incr_boundvars inc t = incr_bv(inc,0,t);
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(*Accumulate all 'loose' bound vars referring to level 'lev' or beyond.
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   (Bound 0) is loose at level 0 *)
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fun add_loose_bnos (Bound i, lev, js) = 
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	if i<lev then js  else  (i-lev) :: js
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  | add_loose_bnos (Abs (_,_,t), lev, js) = add_loose_bnos (t, lev+1, js)
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  | add_loose_bnos (f$t, lev, js) =
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	add_loose_bnos (f, lev, add_loose_bnos (t, lev, js)) 
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  | add_loose_bnos (_, _, js) = js;
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fun loose_bnos t = add_loose_bnos (t, 0, []);
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(* loose_bvar(t,k) iff t contains a 'loose' bound variable referring to
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   level k or beyond. *)
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fun loose_bvar(Bound i,k) = i >= k
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  | loose_bvar(f$t, k) = loose_bvar(f,k) orelse loose_bvar(t,k)
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  | loose_bvar(Abs(_,_,t),k) = loose_bvar(t,k+1)
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  | loose_bvar _ = false;
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(*Substitute arguments for loose bound variables.
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  Beta-reduction of arg(n-1)...arg0 into t replacing (Bound i) with (argi).
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  Note that for ((x,y)c)(a,b), the bound vars in c are x=1 and y=0
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	and the appropriate call is  subst_bounds([b,a], c) .
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  Loose bound variables >=n are reduced by "n" to
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     compensate for the disappearance of lambdas.
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*)
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fun subst_bounds (args: term list, t) : term = 
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  let val n = length args;
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      fun subst (t as Bound i, lev) =
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 	    if i<lev then  t    (*var is locally bound*)
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	    else  (case (drop (i-lev,args)) of
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		  []     => Bound(i-n)  (*loose: change it*)
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	        | arg::_ => incr_boundvars lev arg)
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	| subst (Abs(a,T,body), lev) = Abs(a, T,  subst(body,lev+1))
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	| subst (f$t, lev) =  subst(f,lev)  $  subst(t,lev)
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	| subst (t,lev) = t
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  in   case args of [] => t  | _ => subst (t,0)  end;
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(*beta-reduce if possible, else form application*)
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fun betapply (Abs(_,_,t), u) = subst_bounds([u],t)
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  | betapply (f,u) = f$u;
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(*Tests whether 2 terms are alpha-convertible and have same type.
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  Note that constants and Vars may have more than one type.*)
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infix aconv;
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fun (Const(a,T)) aconv (Const(b,U)) = a=b  andalso  T=U
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  | (Free(a,T)) aconv (Free(b,U)) = a=b  andalso  T=U
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  | (Var(v,T)) aconv (Var(w,U)) =   v=w  andalso  T=U
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  | (Bound i) aconv (Bound j)  =   i=j
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  | (Abs(_,T,t)) aconv (Abs(_,U,u)) = t aconv u  andalso  T=U
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  | (f$t) aconv (g$u) = (f aconv g) andalso (t aconv u)
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  | _ aconv _  =  false;
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(*are two term lists alpha-convertible in corresponding elements?*)
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fun aconvs ([],[]) = true
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  | aconvs (t::ts, u::us) = t aconv u andalso aconvs(ts,us)
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  | aconvs _ = false;
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(*A fast unification filter: true unless the two terms cannot be unified. 
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  Terms must be NORMAL.  Treats all Vars as distinct. *)
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fun could_unify (t,u) =
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  let fun matchrands (f$t, g$u) = could_unify(t,u) andalso  matchrands(f,g)
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	| matchrands _ = true
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  in case (head_of t , head_of u) of
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	(_, Var _) => true
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      | (Var _, _) => true
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      | (Const(a,_), Const(b,_)) =>  a=b andalso matchrands(t,u)
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      | (Free(a,_), Free(b,_)) =>  a=b andalso matchrands(t,u)
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      | (Bound i, Bound j) =>  i=j andalso matchrands(t,u)
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      | (Abs _, _) =>  true   (*because of possible eta equality*)
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      | (_, Abs _) =>  true
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      | _ => false
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  end;
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(*Substitute new for free occurrences of old in a term*)
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fun subst_free [] = (fn t=>t)
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  | subst_free pairs =
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      let fun substf u = 
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	    case gen_assoc (op aconv) (pairs, u) of
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		Some u' => u'
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	      | None => (case u of Abs(a,T,t) => Abs(a, T, substf t)
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				 | t$u' => substf t $ substf u'
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				 | _ => u)
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      in  substf  end;
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(*a total, irreflexive ordering on index names*)
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fun xless ((a,i), (b,j): indexname) = i<j  orelse  (i=j andalso a<b);
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(*Abstraction of the term "body" over its occurrences of v, 
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    which must contain no loose bound variables.
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  The resulting term is ready to become the body of an Abs.*)
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fun abstract_over (v,body) =
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  let fun abst (lev,u) = if (v aconv u) then (Bound lev) else
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      (case u of
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          Abs(a,T,t) => Abs(a, T, abst(lev+1, t))
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	| f$rand => abst(lev,f) $ abst(lev,rand)
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	| _ => u)
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  in  abst(0,body)  end;
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(*Form an abstraction over a free variable.*)
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fun absfree (a,T,body) = Abs(a, T, abstract_over (Free(a,T), body));
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(*Abstraction over a list of free variables*)
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fun list_abs_free ([ ] ,     t) = t
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  | list_abs_free ((a,T)::vars, t) = 
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      absfree(a, T, list_abs_free(vars,t));
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(*Quantification over a list of free variables*)
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fun list_all_free ([], t: term) = t
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  | list_all_free ((a,T)::vars, t) = 
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        (all T) $ (absfree(a, T, list_all_free(vars,t)));
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(*Quantification over a list of variables (already bound in body) *)
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fun list_all ([], t) = t
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  | list_all ((a,T)::vars, t) = 
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        (all T) $ (Abs(a, T, list_all(vars,t)));
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(*Replace the ATOMIC term ti by ui;    instl = [(t1,u1), ..., (tn,un)]. 
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  A simultaneous substitution:  [ (a,b), (b,a) ] swaps a and b.  *)
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fun subst_atomic [] t = t : term
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  | subst_atomic (instl: (term*term) list) t =
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      let fun subst (Abs(a,T,body)) = Abs(a, T, subst body)
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	    | subst (f$t') = subst f $ subst t'
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	    | subst t = (case assoc(instl,t) of
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		           Some u => u  |  None => t)
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      in  subst t  end;
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fun typ_subst_TVars iTs T = if null iTs then T else
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  let fun subst(Type(a,Ts)) = Type(a, map subst Ts)
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	| subst(T as TFree _) = T
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	| subst(T as TVar(ixn,_)) =
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            (case assoc(iTs,ixn) of None => T | Some(U) => U)
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  in subst T end;
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val subst_TVars = map_term_types o typ_subst_TVars;
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fun subst_Vars itms t = if null itms then t else
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  let fun subst(v as Var(ixn,_)) =
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            (case assoc(itms,ixn) of None => v | Some t => t)
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        | subst(Abs(a,T,t)) = Abs(a,T,subst t)
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        | subst(f$t) = subst f $ subst t
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        | subst(t) = t
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  in subst t end;
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fun subst_vars(iTs,itms) = if null iTs then subst_Vars itms else
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  let fun subst(Const(a,T)) = Const(a,typ_subst_TVars iTs T)
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        | subst(Free(a,T)) = Free(a,typ_subst_TVars iTs T)
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        | subst(v as Var(ixn,T)) = (case assoc(itms,ixn) of
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            None   => Var(ixn,typ_subst_TVars iTs T)
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          | Some t => t)
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        | subst(b as Bound _) = b
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        | subst(Abs(a,T,t)) = Abs(a,typ_subst_TVars iTs T,subst t)
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        | subst(f$t) = subst f $ subst t
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  in subst end;
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(*Computing the maximum index of a typ*)
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fun maxidx_of_typ(Type(_,Ts)) =
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	if Ts=[] then ~1 else max(map maxidx_of_typ Ts)
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  | maxidx_of_typ(TFree _) = ~1
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  | maxidx_of_typ(TVar((_,i),_)) = i;
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(*Computing the maximum index of a term*)
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fun maxidx_of_term (Const(_,T)) = maxidx_of_typ T
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  | maxidx_of_term (Bound _) = ~1
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  | maxidx_of_term (Free(_,T)) = maxidx_of_typ T
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  | maxidx_of_term (Var ((_,i), T)) = max[i, maxidx_of_typ T]
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  | maxidx_of_term (Abs (_,T,body)) = max[maxidx_of_term body, maxidx_of_typ T]
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  | maxidx_of_term (f$t) = max [maxidx_of_term f,  maxidx_of_term t];
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(* Increment the index of all Poly's in T by k *)
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fun incr_tvar k (Type(a,Ts)) = Type(a, map (incr_tvar k) Ts)
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  | incr_tvar k (T as TFree _) = T
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  | incr_tvar k (TVar((a,i),rs)) = TVar((a,i+k),rs);
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(**** Syntax-related declarations ****)
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(*Dummy type for parsing.  Will be replaced during type inference. *)
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val dummyT = Type("dummy",[]);
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(*scan a numeral of the given radix, normally 10*)
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fun scan_radixint (radix: int, cs) : int * string list =
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  let val zero = ord"0"
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      val limit = zero+radix
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      fun scan (num,[]) = (num,[])
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	| scan (num, c::cs) =
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	      if  zero <= ord c  andalso  ord c < limit
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	      then scan(radix*num + ord c - zero, cs)
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	      else (num, c::cs)
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  in  scan(0,cs)  end;
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fun scan_int cs = scan_radixint(10,cs);
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(*** Printing ***)
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(*Makes a variant of the name c distinct from the names in bs.
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  First attaches the suffix "a" and then increments this. *)
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fun variant bs c : string =
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  let fun vary2 c = if (c mem bs) then  vary2 (bump_string c)  else  c
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      fun vary1 c = if (c mem bs) then  vary2 (c ^ "a")  else  c
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  in  vary1 (if c="" then "u" else c)  end;
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   454
(*Create variants of the list of names, with priority to the first ones*)
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fun variantlist ([], used) = []
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  | variantlist(b::bs, used) = 
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      let val b' = variant used b
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      in  b' :: variantlist (bs, b'::used)  end;
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   460
(** TFrees and TVars **)
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   461
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(*maps  (bs,v)  to   v'::bs    this reverses the identifiers bs*)
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fun add_new_id (bs, c) : string list =  variant bs c  ::  bs;
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   465
(*Accumulates the names in the term, suppressing duplicates.
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  Includes Frees and Consts.  For choosing unambiguous bound var names.*)
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fun add_term_names (Const(a,_), bs) = a ins bs
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  | add_term_names (Free(a,_), bs) = a ins bs
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   469
  | add_term_names (f$u, bs) = add_term_names (f, add_term_names(u, bs))
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   470
  | add_term_names (Abs(_,_,t), bs) = add_term_names(t,bs)
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   471
  | add_term_names (_, bs) = bs;
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   472
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   473
(*Accumulates the TVars in a type, suppressing duplicates. *)
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fun add_typ_tvars(Type(_,Ts),vs) = foldr add_typ_tvars (Ts,vs)
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  | add_typ_tvars(TFree(_),vs) = vs
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   476
  | add_typ_tvars(TVar(v),vs) = v ins vs;
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   477
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   478
(*Accumulates the TFrees in a type, suppressing duplicates. *)
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   479
fun add_typ_tfree_names(Type(_,Ts),fs) = foldr add_typ_tfree_names (Ts,fs)
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   480
  | add_typ_tfree_names(TFree(f,_),fs) = f ins fs
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   481
  | add_typ_tfree_names(TVar(_),fs) = fs;
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   482
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   483
fun add_typ_tfrees(Type(_,Ts),fs) = foldr add_typ_tfrees (Ts,fs)
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   484
  | add_typ_tfrees(TFree(f),fs) = f ins fs
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   485
  | add_typ_tfrees(TVar(_),fs) = fs;
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   486
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   487
(*Accumulates the TVars in a term, suppressing duplicates. *)
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   488
val add_term_tvars = it_term_types add_typ_tvars;
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   489
val add_term_tvar_ixns = (map #1) o (it_term_types add_typ_tvars);
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   490
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   491
(*Accumulates the TFrees in a term, suppressing duplicates. *)
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   492
val add_term_tfrees = it_term_types add_typ_tfrees;
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   493
val add_term_tfree_names = it_term_types add_typ_tfree_names;
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   494
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   495
(*Non-list versions*)
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   496
fun typ_tfrees T = add_typ_tfrees(T,[]);
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   497
fun typ_tvars T = add_typ_tvars(T,[]);
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   498
fun term_tfrees t = add_term_tfrees(t,[]);
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   499
fun term_tvars t = add_term_tvars(t,[]);
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   500
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   501
(** Frees and Vars **)
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   502
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   503
(*a partial ordering (not reflexive) for atomic terms*)
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   504
fun atless (Const (a,_), Const (b,_))  =  a<b
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   505
  | atless (Free (a,_), Free (b,_)) =  a<b
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   506
  | atless (Var(v,_), Var(w,_))  =  xless(v,w)
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   507
  | atless (Bound i, Bound j)  =   i<j
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   508
  | atless _  =  false;
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   509
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   510
(*insert atomic term into partially sorted list, suppressing duplicates (?)*)
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   511
fun insert_aterm (t,us) =
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   512
  let fun inserta [] = [t]
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   513
        | inserta (us as u::us') = 
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   514
	      if atless(t,u) then t::us
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   515
	      else if t=u then us (*duplicate*)
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   516
	      else u :: inserta(us')
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   517
  in  inserta us  end;
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   518
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   519
(*Accumulates the Vars in the term, suppressing duplicates*)
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   520
fun add_term_vars (t, vars: term list) = case t of
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   521
    Var   _ => insert_aterm(t,vars)
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   522
  | Abs (_,_,body) => add_term_vars(body,vars)
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   523
  | f$t =>  add_term_vars (f, add_term_vars(t, vars))
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   524
  | _ => vars;
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   525
clasohm@0
   526
fun term_vars t = add_term_vars(t,[]);
clasohm@0
   527
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   528
(*Accumulates the Frees in the term, suppressing duplicates*)
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   529
fun add_term_frees (t, frees: term list) = case t of
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   530
    Free   _ => insert_aterm(t,frees)
clasohm@0
   531
  | Abs (_,_,body) => add_term_frees(body,frees)
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   532
  | f$t =>  add_term_frees (f, add_term_frees(t, frees))
clasohm@0
   533
  | _ => frees;
clasohm@0
   534
clasohm@0
   535
fun term_frees t = add_term_frees(t,[]);
clasohm@0
   536
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   537
(*Given an abstraction over P, replaces the bound variable by a Free variable
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   538
  having a unique name. *)
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   539
fun variant_abs (a,T,P) =
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   540
  let val b = variant (add_term_names(P,[])) a
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   541
  in  (b,  subst_bounds ([Free(b,T)], P))  end;
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   542
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   543
(* renames and reverses the strings in vars away from names *)
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   544
fun rename_aTs names vars : (string*typ)list =
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   545
  let fun rename_aT (vars,(a,T)) =
clasohm@0
   546
		(variant (map #1 vars @ names) a, T) :: vars
clasohm@0
   547
  in foldl rename_aT ([],vars) end;
clasohm@0
   548
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   549
fun rename_wrt_term t = rename_aTs (add_term_names(t,[]));