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(* Title: Pure/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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Simply typed lambda-calculus: types, terms, and basic operations
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*)
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infix 9 $;
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infixr 5 -->;
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infixr --->;
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infix aconv;
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structure Term =
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struct
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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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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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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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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
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("type_of: function type is expected in application",
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[T,U], [f$u])
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end;
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fun type_of t : typ = type_of1 ([],t);
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(*Determines the type of a term, with minimal checking*)
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fun fastype_of1 (Ts, f$u) =
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(case fastype_of1 (Ts,f) of
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Type("fun",[_,T]) => T
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| _ => raise TERM("fastype_of: expected function type", [f$u]))
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lcp@61
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| fastype_of1 (_, Const (_,T)) = T
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lcp@61
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| fastype_of1 (_, Free (_,T)) = T
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lcp@61
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| fastype_of1 (Ts, Bound i) = (nth_elem(i,Ts)
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handle LIST _ => raise TERM("fastype_of: Bound", [Bound i]))
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lcp@61
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| fastype_of1 (_, Var (_,T)) = T
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lcp@61
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| fastype_of1 (Ts, Abs (_,T,u)) = T --> fastype_of1 (T::Ts, u);
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lcp@61
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lcp@61
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fun fastype_of t : typ = fastype_of1 ([],t);
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(* maps (x1,...,xn)t to t *)
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fun strip_abs_body (Abs(_,_,t)) = strip_abs_body t
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| strip_abs_body u = u;
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(* maps (x1,...,xn)t to [x1, ..., xn] *)
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fun strip_abs_vars (Abs(a,T,t)) = (a,T) :: strip_abs_vars t
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| strip_abs_vars u = [] : (string*typ) list;
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fun strip_qnt_body qnt =
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let fun strip(tm as Const(c,_)$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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clasohm@0
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nipkow@949
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fun map_type_tvar f (Type(a,Ts)) = Type(a, map (map_type_tvar f) Ts)
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nipkow@949
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| map_type_tvar f (T as TFree _) = T
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nipkow@949
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| map_type_tvar f (TVar x) = f x;
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nipkow@949
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nipkow@949
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fun map_type_tfree f (Type(a,Ts)) = Type(a, map (map_type_tfree f) Ts)
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nipkow@949
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| map_type_tfree f (TFree x) = f x
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nipkow@949
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| map_type_tfree f (T as TVar _) = T;
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nipkow@949
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clasohm@0
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(* apply a function to all types in a term *)
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clasohm@0
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fun map_term_types f =
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clasohm@0
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let fun map(Const(a,T)) = Const(a, f T)
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clasohm@0
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| map(Free(a,T)) = Free(a, f T)
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clasohm@0
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| map(Var(v,T)) = Var(v, f T)
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clasohm@0
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| map(t as Bound _) = t
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clasohm@0
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| map(Abs(a,T,t)) = Abs(a, f T, map t)
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clasohm@0
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| map(f$t) = map f $ map t;
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clasohm@0
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in map end;
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clasohm@0
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clasohm@0
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(* iterate a function over all types in a term *)
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clasohm@0
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fun it_term_types f =
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clasohm@0
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let fun iter(Const(_,T), a) = f(T,a)
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clasohm@0
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| iter(Free(_,T), a) = f(T,a)
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clasohm@0
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| iter(Var(_,T), a) = f(T,a)
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clasohm@0
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| iter(Abs(_,T,t), a) = iter(t,f(T,a))
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clasohm@0
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| iter(f$u, a) = iter(f, iter(u, a))
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clasohm@0
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| iter(Bound _, a) = a
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clasohm@0
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in iter end
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clasohm@0
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clasohm@0
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clasohm@0
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(** Connectives of higher order logic **)
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clasohm@0
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wenzelm@375
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val logicC: class = "logic";
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wenzelm@375
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val logicS: sort = [logicC];
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wenzelm@375
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wenzelm@375
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fun itselfT ty = Type ("itself", [ty]);
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wenzelm@375
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val a_itselfT = itselfT (TFree ("'a", logicS));
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wenzelm@375
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clasohm@0
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val propT : typ = Type("prop",[]);
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clasohm@0
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clasohm@0
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val implies = Const("==>", propT-->propT-->propT);
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clasohm@0
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clasohm@0
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fun all T = Const("all", (T-->propT)-->propT);
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clasohm@0
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clasohm@0
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fun equals T = Const("==", T-->T-->propT);
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clasohm@0
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clasohm@0
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fun flexpair T = Const("=?=", T-->T-->propT);
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clasohm@0
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clasohm@0
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(* maps !!x1...xn. t to t *)
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clasohm@0
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fun strip_all_body (Const("all",_)$Abs(_,_,t)) = strip_all_body t
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clasohm@0
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| strip_all_body t = t;
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clasohm@0
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clasohm@0
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(* maps !!x1...xn. t to [x1, ..., xn] *)
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clasohm@0
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fun strip_all_vars (Const("all",_)$Abs(a,T,t)) =
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clasohm@1460
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(a,T) :: strip_all_vars t
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clasohm@0
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| strip_all_vars t = [] : (string*typ) list;
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clasohm@0
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clasohm@0
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(*increments a term's non-local bound variables
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clasohm@0
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required when moving a term within abstractions
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clasohm@0
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inc is increment for bound variables
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clasohm@0
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lev is level at which a bound variable is considered 'loose'*)
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clasohm@0
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fun incr_bv (inc, lev, u as Bound i) = if i>=lev then Bound(i+inc) else u
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clasohm@0
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| incr_bv (inc, lev, Abs(a,T,body)) =
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clasohm@1460
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Abs(a, T, incr_bv(inc,lev+1,body))
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clasohm@0
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| incr_bv (inc, lev, f$t) =
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clasohm@0
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incr_bv(inc,lev,f) $ incr_bv(inc,lev,t)
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clasohm@0
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| incr_bv (inc, lev, u) = u;
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clasohm@0
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clasohm@0
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fun incr_boundvars 0 t = t
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clasohm@0
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| incr_boundvars inc t = incr_bv(inc,0,t);
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clasohm@0
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clasohm@0
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clasohm@0
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(*Accumulate all 'loose' bound vars referring to level 'lev' or beyond.
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clasohm@0
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(Bound 0) is loose at level 0 *)
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clasohm@0
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266 |
fun add_loose_bnos (Bound i, lev, js) =
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paulson@2176
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if i<lev then js else (i-lev) ins_int js
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clasohm@0
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| add_loose_bnos (Abs (_,_,t), lev, js) = add_loose_bnos (t, lev+1, js)
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clasohm@0
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| add_loose_bnos (f$t, lev, js) =
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clasohm@1460
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add_loose_bnos (f, lev, add_loose_bnos (t, lev, js))
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clasohm@0
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| add_loose_bnos (_, _, js) = js;
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clasohm@0
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clasohm@0
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fun loose_bnos t = add_loose_bnos (t, 0, []);
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clasohm@0
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clasohm@0
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(* loose_bvar(t,k) iff t contains a 'loose' bound variable referring to
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clasohm@0
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level k or beyond. *)
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clasohm@0
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fun loose_bvar(Bound i,k) = i >= k
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clasohm@0
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| loose_bvar(f$t, k) = loose_bvar(f,k) orelse loose_bvar(t,k)
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clasohm@0
|
279 |
| loose_bvar(Abs(_,_,t),k) = loose_bvar(t,k+1)
|
clasohm@0
|
280 |
| loose_bvar _ = false;
|
clasohm@0
|
281 |
|
nipkow@2792
|
282 |
fun loose_bvar1(Bound i,k) = i = k
|
nipkow@2792
|
283 |
| loose_bvar1(f$t, k) = loose_bvar1(f,k) orelse loose_bvar1(t,k)
|
nipkow@2792
|
284 |
| loose_bvar1(Abs(_,_,t),k) = loose_bvar1(t,k+1)
|
nipkow@2792
|
285 |
| loose_bvar1 _ = false;
|
clasohm@0
|
286 |
|
clasohm@0
|
287 |
(*Substitute arguments for loose bound variables.
|
clasohm@0
|
288 |
Beta-reduction of arg(n-1)...arg0 into t replacing (Bound i) with (argi).
|
clasohm@0
|
289 |
Note that for ((x,y)c)(a,b), the bound vars in c are x=1 and y=0
|
clasohm@1460
|
290 |
and the appropriate call is subst_bounds([b,a], c) .
|
clasohm@0
|
291 |
Loose bound variables >=n are reduced by "n" to
|
clasohm@0
|
292 |
compensate for the disappearance of lambdas.
|
clasohm@0
|
293 |
*)
|
clasohm@0
|
294 |
fun subst_bounds (args: term list, t) : term =
|
clasohm@0
|
295 |
let val n = length args;
|
clasohm@0
|
296 |
fun subst (t as Bound i, lev) =
|
paulson@2580
|
297 |
(if i<lev then t (*var is locally bound*)
|
paulson@2580
|
298 |
else incr_boundvars lev (List.nth(args, i-lev))
|
paulson@2580
|
299 |
handle Subscript => Bound(i-n) (*loose: change it*))
|
clasohm@1460
|
300 |
| subst (Abs(a,T,body), lev) = Abs(a, T, subst(body,lev+1))
|
clasohm@1460
|
301 |
| subst (f$t, lev) = subst(f,lev) $ subst(t,lev)
|
clasohm@1460
|
302 |
| subst (t,lev) = t
|
clasohm@0
|
303 |
in case args of [] => t | _ => subst (t,0) end;
|
clasohm@0
|
304 |
|
paulson@2192
|
305 |
(*Special case: one argument*)
|
paulson@2192
|
306 |
fun subst_bound (arg, t) : term =
|
paulson@2192
|
307 |
let fun subst (t as Bound i, lev) =
|
paulson@2192
|
308 |
if i<lev then t (*var is locally bound*)
|
paulson@2192
|
309 |
else if i=lev then incr_boundvars lev arg
|
paulson@2192
|
310 |
else Bound(i-1) (*loose: change it*)
|
paulson@2192
|
311 |
| subst (Abs(a,T,body), lev) = Abs(a, T, subst(body,lev+1))
|
paulson@2192
|
312 |
| subst (f$t, lev) = subst(f,lev) $ subst(t,lev)
|
paulson@2192
|
313 |
| subst (t,lev) = t
|
paulson@2192
|
314 |
in subst (t,0) end;
|
paulson@2192
|
315 |
|
clasohm@0
|
316 |
(*beta-reduce if possible, else form application*)
|
paulson@2192
|
317 |
fun betapply (Abs(_,_,t), u) = subst_bound (u,t)
|
clasohm@0
|
318 |
| betapply (f,u) = f$u;
|
clasohm@0
|
319 |
|
paulson@2192
|
320 |
(** Equality, membership and insertion of indexnames (string*ints) **)
|
paulson@2192
|
321 |
|
paulson@2192
|
322 |
(*optimized equality test for indexnames. Yields a huge gain under Poly/ML*)
|
wenzelm@2959
|
323 |
fun eq_ix ((a, i):indexname, (a',i'):indexname) = (a=a') andalso i=i';
|
paulson@2192
|
324 |
|
paulson@2192
|
325 |
(*membership in a list, optimized version for indexnames*)
|
wenzelm@2959
|
326 |
fun mem_ix (_, []) = false
|
paulson@2192
|
327 |
| mem_ix (x, y :: ys) = eq_ix(x,y) orelse mem_ix (x, ys);
|
paulson@2192
|
328 |
|
paulson@2192
|
329 |
(*insertion into list, optimized version for indexnames*)
|
paulson@2192
|
330 |
fun ins_ix (x,xs) = if mem_ix (x, xs) then xs else x :: xs;
|
paulson@2192
|
331 |
|
clasohm@0
|
332 |
(*Tests whether 2 terms are alpha-convertible and have same type.
|
clasohm@0
|
333 |
Note that constants and Vars may have more than one type.*)
|
clasohm@0
|
334 |
fun (Const(a,T)) aconv (Const(b,U)) = a=b andalso T=U
|
paulson@2752
|
335 |
| (Free(a,T)) aconv (Free(b,U)) = a=b andalso T=U
|
paulson@2752
|
336 |
| (Var(v,T)) aconv (Var(w,U)) = eq_ix(v,w) andalso T=U
|
paulson@2752
|
337 |
| (Bound i) aconv (Bound j) = i=j
|
clasohm@0
|
338 |
| (Abs(_,T,t)) aconv (Abs(_,U,u)) = t aconv u andalso T=U
|
paulson@2752
|
339 |
| (f$t) aconv (g$u) = (f aconv g) andalso (t aconv u)
|
clasohm@0
|
340 |
| _ aconv _ = false;
|
clasohm@0
|
341 |
|
paulson@2176
|
342 |
(** Membership, insertion, union for terms **)
|
paulson@2176
|
343 |
|
paulson@2176
|
344 |
fun mem_term (_, []) = false
|
paulson@2176
|
345 |
| mem_term (t, t'::ts) = t aconv t' orelse mem_term(t,ts);
|
paulson@2176
|
346 |
|
paulson@2182
|
347 |
fun subset_term ([], ys) = true
|
paulson@2182
|
348 |
| subset_term (x :: xs, ys) = mem_term (x, ys) andalso subset_term(xs, ys);
|
paulson@2182
|
349 |
|
paulson@2182
|
350 |
fun eq_set_term (xs, ys) =
|
paulson@2182
|
351 |
xs = ys orelse (subset_term (xs, ys) andalso subset_term (ys, xs));
|
paulson@2182
|
352 |
|
paulson@2176
|
353 |
fun ins_term(t,ts) = if mem_term(t,ts) then ts else t :: ts;
|
paulson@2176
|
354 |
|
paulson@2176
|
355 |
fun union_term (xs, []) = xs
|
paulson@2176
|
356 |
| union_term ([], ys) = ys
|
paulson@2176
|
357 |
| union_term ((x :: xs), ys) = union_term (xs, ins_term (x, ys));
|
paulson@2176
|
358 |
|
paulson@2176
|
359 |
(** Equality, membership and insertion of sorts (string lists) **)
|
paulson@2176
|
360 |
|
paulson@2176
|
361 |
fun eq_sort ([]:sort, []) = true
|
paulson@2176
|
362 |
| eq_sort ((s::ss), (t::ts)) = s=t andalso eq_sort(ss,ts)
|
paulson@2176
|
363 |
| eq_sort (_, _) = false;
|
paulson@2176
|
364 |
|
paulson@2176
|
365 |
fun mem_sort (_:sort, []) = false
|
paulson@2176
|
366 |
| mem_sort (S, S'::Ss) = eq_sort (S, S') orelse mem_sort(S,Ss);
|
paulson@2176
|
367 |
|
paulson@2176
|
368 |
fun ins_sort(S,Ss) = if mem_sort(S,Ss) then Ss else S :: Ss;
|
paulson@2176
|
369 |
|
paulson@2176
|
370 |
fun union_sort (xs, []) = xs
|
paulson@2176
|
371 |
| union_sort ([], ys) = ys
|
paulson@2176
|
372 |
| union_sort ((x :: xs), ys) = union_sort (xs, ins_sort (x, ys));
|
paulson@2176
|
373 |
|
paulson@2182
|
374 |
fun subset_sort ([], ys) = true
|
paulson@2182
|
375 |
| subset_sort (x :: xs, ys) = mem_sort (x, ys) andalso subset_sort(xs, ys);
|
paulson@2182
|
376 |
|
paulson@2182
|
377 |
fun eq_set_sort (xs, ys) =
|
paulson@2182
|
378 |
xs = ys orelse (subset_sort (xs, ys) andalso subset_sort (ys, xs));
|
paulson@2182
|
379 |
|
clasohm@0
|
380 |
(*are two term lists alpha-convertible in corresponding elements?*)
|
clasohm@0
|
381 |
fun aconvs ([],[]) = true
|
clasohm@0
|
382 |
| aconvs (t::ts, u::us) = t aconv u andalso aconvs(ts,us)
|
clasohm@0
|
383 |
| aconvs _ = false;
|
clasohm@0
|
384 |
|
clasohm@0
|
385 |
(*A fast unification filter: true unless the two terms cannot be unified.
|
clasohm@0
|
386 |
Terms must be NORMAL. Treats all Vars as distinct. *)
|
clasohm@0
|
387 |
fun could_unify (t,u) =
|
clasohm@0
|
388 |
let fun matchrands (f$t, g$u) = could_unify(t,u) andalso matchrands(f,g)
|
clasohm@1460
|
389 |
| matchrands _ = true
|
clasohm@0
|
390 |
in case (head_of t , head_of u) of
|
clasohm@1460
|
391 |
(_, Var _) => true
|
clasohm@0
|
392 |
| (Var _, _) => true
|
clasohm@0
|
393 |
| (Const(a,_), Const(b,_)) => a=b andalso matchrands(t,u)
|
clasohm@0
|
394 |
| (Free(a,_), Free(b,_)) => a=b andalso matchrands(t,u)
|
clasohm@0
|
395 |
| (Bound i, Bound j) => i=j andalso matchrands(t,u)
|
clasohm@0
|
396 |
| (Abs _, _) => true (*because of possible eta equality*)
|
clasohm@0
|
397 |
| (_, Abs _) => true
|
clasohm@0
|
398 |
| _ => false
|
clasohm@0
|
399 |
end;
|
clasohm@0
|
400 |
|
clasohm@0
|
401 |
(*Substitute new for free occurrences of old in a term*)
|
clasohm@0
|
402 |
fun subst_free [] = (fn t=>t)
|
clasohm@0
|
403 |
| subst_free pairs =
|
clasohm@0
|
404 |
let fun substf u =
|
clasohm@1460
|
405 |
case gen_assoc (op aconv) (pairs, u) of
|
clasohm@1460
|
406 |
Some u' => u'
|
clasohm@1460
|
407 |
| None => (case u of Abs(a,T,t) => Abs(a, T, substf t)
|
clasohm@1460
|
408 |
| t$u' => substf t $ substf u'
|
clasohm@1460
|
409 |
| _ => u)
|
clasohm@0
|
410 |
in substf end;
|
clasohm@0
|
411 |
|
clasohm@0
|
412 |
(*a total, irreflexive ordering on index names*)
|
clasohm@0
|
413 |
fun xless ((a,i), (b,j): indexname) = i<j orelse (i=j andalso a<b);
|
clasohm@0
|
414 |
|
clasohm@0
|
415 |
|
clasohm@0
|
416 |
(*Abstraction of the term "body" over its occurrences of v,
|
clasohm@0
|
417 |
which must contain no loose bound variables.
|
clasohm@0
|
418 |
The resulting term is ready to become the body of an Abs.*)
|
clasohm@0
|
419 |
fun abstract_over (v,body) =
|
clasohm@0
|
420 |
let fun abst (lev,u) = if (v aconv u) then (Bound lev) else
|
clasohm@0
|
421 |
(case u of
|
clasohm@0
|
422 |
Abs(a,T,t) => Abs(a, T, abst(lev+1, t))
|
clasohm@1460
|
423 |
| f$rand => abst(lev,f) $ abst(lev,rand)
|
clasohm@1460
|
424 |
| _ => u)
|
clasohm@0
|
425 |
in abst(0,body) end;
|
clasohm@0
|
426 |
|
clasohm@0
|
427 |
|
clasohm@0
|
428 |
(*Form an abstraction over a free variable.*)
|
clasohm@0
|
429 |
fun absfree (a,T,body) = Abs(a, T, abstract_over (Free(a,T), body));
|
clasohm@0
|
430 |
|
clasohm@0
|
431 |
(*Abstraction over a list of free variables*)
|
clasohm@0
|
432 |
fun list_abs_free ([ ] , t) = t
|
clasohm@0
|
433 |
| list_abs_free ((a,T)::vars, t) =
|
clasohm@0
|
434 |
absfree(a, T, list_abs_free(vars,t));
|
clasohm@0
|
435 |
|
clasohm@0
|
436 |
(*Quantification over a list of free variables*)
|
clasohm@0
|
437 |
fun list_all_free ([], t: term) = t
|
clasohm@0
|
438 |
| list_all_free ((a,T)::vars, t) =
|
clasohm@0
|
439 |
(all T) $ (absfree(a, T, list_all_free(vars,t)));
|
clasohm@0
|
440 |
|
clasohm@0
|
441 |
(*Quantification over a list of variables (already bound in body) *)
|
clasohm@0
|
442 |
fun list_all ([], t) = t
|
clasohm@0
|
443 |
| list_all ((a,T)::vars, t) =
|
clasohm@0
|
444 |
(all T) $ (Abs(a, T, list_all(vars,t)));
|
clasohm@0
|
445 |
|
clasohm@0
|
446 |
(*Replace the ATOMIC term ti by ui; instl = [(t1,u1), ..., (tn,un)].
|
clasohm@0
|
447 |
A simultaneous substitution: [ (a,b), (b,a) ] swaps a and b. *)
|
clasohm@0
|
448 |
fun subst_atomic [] t = t : term
|
clasohm@0
|
449 |
| subst_atomic (instl: (term*term) list) t =
|
clasohm@0
|
450 |
let fun subst (Abs(a,T,body)) = Abs(a, T, subst body)
|
clasohm@1460
|
451 |
| subst (f$t') = subst f $ subst t'
|
clasohm@1460
|
452 |
| subst t = (case assoc(instl,t) of
|
clasohm@1460
|
453 |
Some u => u | None => t)
|
clasohm@0
|
454 |
in subst t end;
|
clasohm@0
|
455 |
|
lcp@728
|
456 |
(*Substitute for type Vars in a type*)
|
clasohm@0
|
457 |
fun typ_subst_TVars iTs T = if null iTs then T else
|
clasohm@0
|
458 |
let fun subst(Type(a,Ts)) = Type(a, map subst Ts)
|
clasohm@1460
|
459 |
| subst(T as TFree _) = T
|
clasohm@1460
|
460 |
| subst(T as TVar(ixn,_)) =
|
clasohm@0
|
461 |
(case assoc(iTs,ixn) of None => T | Some(U) => U)
|
clasohm@0
|
462 |
in subst T end;
|
clasohm@0
|
463 |
|
lcp@728
|
464 |
(*Substitute for type Vars in a term*)
|
clasohm@0
|
465 |
val subst_TVars = map_term_types o typ_subst_TVars;
|
clasohm@0
|
466 |
|
lcp@728
|
467 |
(*Substitute for Vars in a term; see also envir/norm_term*)
|
clasohm@0
|
468 |
fun subst_Vars itms t = if null itms then t else
|
clasohm@0
|
469 |
let fun subst(v as Var(ixn,_)) =
|
clasohm@0
|
470 |
(case assoc(itms,ixn) of None => v | Some t => t)
|
clasohm@0
|
471 |
| subst(Abs(a,T,t)) = Abs(a,T,subst t)
|
clasohm@0
|
472 |
| subst(f$t) = subst f $ subst t
|
clasohm@0
|
473 |
| subst(t) = t
|
clasohm@0
|
474 |
in subst t end;
|
clasohm@0
|
475 |
|
lcp@728
|
476 |
(*Substitute for type/term Vars in a term; see also envir/norm_term*)
|
clasohm@0
|
477 |
fun subst_vars(iTs,itms) = if null iTs then subst_Vars itms else
|
clasohm@0
|
478 |
let fun subst(Const(a,T)) = Const(a,typ_subst_TVars iTs T)
|
clasohm@0
|
479 |
| subst(Free(a,T)) = Free(a,typ_subst_TVars iTs T)
|
clasohm@0
|
480 |
| subst(v as Var(ixn,T)) = (case assoc(itms,ixn) of
|
clasohm@0
|
481 |
None => Var(ixn,typ_subst_TVars iTs T)
|
clasohm@0
|
482 |
| Some t => t)
|
clasohm@0
|
483 |
| subst(b as Bound _) = b
|
clasohm@0
|
484 |
| subst(Abs(a,T,t)) = Abs(a,typ_subst_TVars iTs T,subst t)
|
clasohm@0
|
485 |
| subst(f$t) = subst f $ subst t
|
clasohm@0
|
486 |
in subst end;
|
clasohm@0
|
487 |
|
clasohm@0
|
488 |
|
clasohm@0
|
489 |
(*Computing the maximum index of a typ*)
|
paulson@2146
|
490 |
fun maxidx_of_typ(Type(_,Ts)) = maxidx_of_typs Ts
|
clasohm@0
|
491 |
| maxidx_of_typ(TFree _) = ~1
|
paulson@2146
|
492 |
| maxidx_of_typ(TVar((_,i),_)) = i
|
paulson@2146
|
493 |
and maxidx_of_typs [] = ~1
|
paulson@2146
|
494 |
| maxidx_of_typs (T::Ts) = Int.max(maxidx_of_typ T, maxidx_of_typs Ts);
|
clasohm@0
|
495 |
|
clasohm@0
|
496 |
|
clasohm@0
|
497 |
(*Computing the maximum index of a term*)
|
clasohm@0
|
498 |
fun maxidx_of_term (Const(_,T)) = maxidx_of_typ T
|
clasohm@0
|
499 |
| maxidx_of_term (Bound _) = ~1
|
clasohm@0
|
500 |
| maxidx_of_term (Free(_,T)) = maxidx_of_typ T
|
paulson@2146
|
501 |
| maxidx_of_term (Var ((_,i), T)) = Int.max(i, maxidx_of_typ T)
|
paulson@2146
|
502 |
| maxidx_of_term (Abs (_,T,u)) = Int.max(maxidx_of_term u, maxidx_of_typ T)
|
paulson@2146
|
503 |
| maxidx_of_term (f$t) = Int.max(maxidx_of_term f, maxidx_of_term t);
|
clasohm@0
|
504 |
|
clasohm@0
|
505 |
|
clasohm@0
|
506 |
(* Increment the index of all Poly's in T by k *)
|
nipkow@949
|
507 |
fun incr_tvar k = map_type_tvar (fn ((a,i),S) => TVar((a,i+k),S));
|
clasohm@0
|
508 |
|
clasohm@0
|
509 |
|
clasohm@0
|
510 |
(**** Syntax-related declarations ****)
|
clasohm@0
|
511 |
|
clasohm@0
|
512 |
|
clasohm@0
|
513 |
(*Dummy type for parsing. Will be replaced during type inference. *)
|
clasohm@0
|
514 |
val dummyT = Type("dummy",[]);
|
clasohm@0
|
515 |
|
clasohm@0
|
516 |
(*scan a numeral of the given radix, normally 10*)
|
clasohm@0
|
517 |
fun scan_radixint (radix: int, cs) : int * string list =
|
clasohm@0
|
518 |
let val zero = ord"0"
|
clasohm@0
|
519 |
val limit = zero+radix
|
clasohm@0
|
520 |
fun scan (num,[]) = (num,[])
|
clasohm@1460
|
521 |
| scan (num, c::cs) =
|
clasohm@1460
|
522 |
if zero <= ord c andalso ord c < limit
|
clasohm@1460
|
523 |
then scan(radix*num + ord c - zero, cs)
|
clasohm@1460
|
524 |
else (num, c::cs)
|
clasohm@0
|
525 |
in scan(0,cs) end;
|
clasohm@0
|
526 |
|
clasohm@0
|
527 |
fun scan_int cs = scan_radixint(10,cs);
|
clasohm@0
|
528 |
|
clasohm@0
|
529 |
|
clasohm@0
|
530 |
(*** Printing ***)
|
clasohm@0
|
531 |
|
clasohm@0
|
532 |
|
clasohm@0
|
533 |
(*Makes a variant of the name c distinct from the names in bs.
|
clasohm@0
|
534 |
First attaches the suffix "a" and then increments this. *)
|
clasohm@0
|
535 |
fun variant bs c : string =
|
paulson@2138
|
536 |
let fun vary2 c = if (c mem_string bs) then vary2 (bump_string c) else c
|
paulson@2138
|
537 |
fun vary1 c = if (c mem_string bs) then vary2 (c ^ "a") else c
|
clasohm@0
|
538 |
in vary1 (if c="" then "u" else c) end;
|
clasohm@0
|
539 |
|
clasohm@0
|
540 |
(*Create variants of the list of names, with priority to the first ones*)
|
clasohm@0
|
541 |
fun variantlist ([], used) = []
|
clasohm@0
|
542 |
| variantlist(b::bs, used) =
|
clasohm@0
|
543 |
let val b' = variant used b
|
clasohm@0
|
544 |
in b' :: variantlist (bs, b'::used) end;
|
clasohm@0
|
545 |
|
clasohm@0
|
546 |
(** TFrees and TVars **)
|
clasohm@0
|
547 |
|
clasohm@0
|
548 |
(*maps (bs,v) to v'::bs this reverses the identifiers bs*)
|
clasohm@0
|
549 |
fun add_new_id (bs, c) : string list = variant bs c :: bs;
|
clasohm@0
|
550 |
|
clasohm@0
|
551 |
(*Accumulates the names in the term, suppressing duplicates.
|
clasohm@0
|
552 |
Includes Frees and Consts. For choosing unambiguous bound var names.*)
|
paulson@2176
|
553 |
fun add_term_names (Const(a,_), bs) = a ins_string bs
|
paulson@2176
|
554 |
| add_term_names (Free(a,_), bs) = a ins_string bs
|
clasohm@0
|
555 |
| add_term_names (f$u, bs) = add_term_names (f, add_term_names(u, bs))
|
clasohm@0
|
556 |
| add_term_names (Abs(_,_,t), bs) = add_term_names(t,bs)
|
clasohm@0
|
557 |
| add_term_names (_, bs) = bs;
|
clasohm@0
|
558 |
|
clasohm@0
|
559 |
(*Accumulates the TVars in a type, suppressing duplicates. *)
|
clasohm@0
|
560 |
fun add_typ_tvars(Type(_,Ts),vs) = foldr add_typ_tvars (Ts,vs)
|
clasohm@0
|
561 |
| add_typ_tvars(TFree(_),vs) = vs
|
clasohm@0
|
562 |
| add_typ_tvars(TVar(v),vs) = v ins vs;
|
clasohm@0
|
563 |
|
clasohm@0
|
564 |
(*Accumulates the TFrees in a type, suppressing duplicates. *)
|
clasohm@0
|
565 |
fun add_typ_tfree_names(Type(_,Ts),fs) = foldr add_typ_tfree_names (Ts,fs)
|
paulson@2176
|
566 |
| add_typ_tfree_names(TFree(f,_),fs) = f ins_string fs
|
clasohm@0
|
567 |
| add_typ_tfree_names(TVar(_),fs) = fs;
|
clasohm@0
|
568 |
|
clasohm@0
|
569 |
fun add_typ_tfrees(Type(_,Ts),fs) = foldr add_typ_tfrees (Ts,fs)
|
clasohm@0
|
570 |
| add_typ_tfrees(TFree(f),fs) = f ins fs
|
clasohm@0
|
571 |
| add_typ_tfrees(TVar(_),fs) = fs;
|
clasohm@0
|
572 |
|
nipkow@949
|
573 |
fun add_typ_varnames(Type(_,Ts),nms) = foldr add_typ_varnames (Ts,nms)
|
paulson@2176
|
574 |
| add_typ_varnames(TFree(nm,_),nms) = nm ins_string nms
|
paulson@2176
|
575 |
| add_typ_varnames(TVar((nm,_),_),nms) = nm ins_string nms;
|
nipkow@949
|
576 |
|
clasohm@0
|
577 |
(*Accumulates the TVars in a term, suppressing duplicates. *)
|
clasohm@0
|
578 |
val add_term_tvars = it_term_types add_typ_tvars;
|
clasohm@0
|
579 |
|
clasohm@0
|
580 |
(*Accumulates the TFrees in a term, suppressing duplicates. *)
|
clasohm@0
|
581 |
val add_term_tfrees = it_term_types add_typ_tfrees;
|
clasohm@0
|
582 |
val add_term_tfree_names = it_term_types add_typ_tfree_names;
|
clasohm@0
|
583 |
|
nipkow@949
|
584 |
val add_term_tvarnames = it_term_types add_typ_varnames;
|
nipkow@949
|
585 |
|
clasohm@0
|
586 |
(*Non-list versions*)
|
clasohm@0
|
587 |
fun typ_tfrees T = add_typ_tfrees(T,[]);
|
clasohm@0
|
588 |
fun typ_tvars T = add_typ_tvars(T,[]);
|
clasohm@0
|
589 |
fun term_tfrees t = add_term_tfrees(t,[]);
|
clasohm@0
|
590 |
fun term_tvars t = add_term_tvars(t,[]);
|
clasohm@0
|
591 |
|
nipkow@949
|
592 |
(*special code to enforce left-to-right collection of TVar-indexnames*)
|
nipkow@949
|
593 |
|
nipkow@949
|
594 |
fun add_typ_ixns(ixns,Type(_,Ts)) = foldl add_typ_ixns (ixns,Ts)
|
paulson@2176
|
595 |
| add_typ_ixns(ixns,TVar(ixn,_)) = if mem_ix (ixn, ixns) then ixns
|
paulson@2176
|
596 |
else ixns@[ixn]
|
nipkow@949
|
597 |
| add_typ_ixns(ixns,TFree(_)) = ixns;
|
nipkow@949
|
598 |
|
nipkow@949
|
599 |
fun add_term_tvar_ixns(Const(_,T),ixns) = add_typ_ixns(ixns,T)
|
nipkow@949
|
600 |
| add_term_tvar_ixns(Free(_,T),ixns) = add_typ_ixns(ixns,T)
|
nipkow@949
|
601 |
| add_term_tvar_ixns(Var(_,T),ixns) = add_typ_ixns(ixns,T)
|
nipkow@949
|
602 |
| add_term_tvar_ixns(Bound _,ixns) = ixns
|
nipkow@949
|
603 |
| add_term_tvar_ixns(Abs(_,T,t),ixns) =
|
nipkow@949
|
604 |
add_term_tvar_ixns(t,add_typ_ixns(ixns,T))
|
nipkow@949
|
605 |
| add_term_tvar_ixns(f$t,ixns) =
|
nipkow@949
|
606 |
add_term_tvar_ixns(t,add_term_tvar_ixns(f,ixns));
|
nipkow@949
|
607 |
|
clasohm@0
|
608 |
(** Frees and Vars **)
|
clasohm@0
|
609 |
|
clasohm@0
|
610 |
(*a partial ordering (not reflexive) for atomic terms*)
|
clasohm@0
|
611 |
fun atless (Const (a,_), Const (b,_)) = a<b
|
clasohm@0
|
612 |
| atless (Free (a,_), Free (b,_)) = a<b
|
clasohm@0
|
613 |
| atless (Var(v,_), Var(w,_)) = xless(v,w)
|
clasohm@0
|
614 |
| atless (Bound i, Bound j) = i<j
|
clasohm@0
|
615 |
| atless _ = false;
|
clasohm@0
|
616 |
|
clasohm@0
|
617 |
(*insert atomic term into partially sorted list, suppressing duplicates (?)*)
|
clasohm@0
|
618 |
fun insert_aterm (t,us) =
|
clasohm@0
|
619 |
let fun inserta [] = [t]
|
clasohm@0
|
620 |
| inserta (us as u::us') =
|
clasohm@1460
|
621 |
if atless(t,u) then t::us
|
clasohm@1460
|
622 |
else if t=u then us (*duplicate*)
|
clasohm@1460
|
623 |
else u :: inserta(us')
|
clasohm@0
|
624 |
in inserta us end;
|
clasohm@0
|
625 |
|
clasohm@0
|
626 |
(*Accumulates the Vars in the term, suppressing duplicates*)
|
clasohm@0
|
627 |
fun add_term_vars (t, vars: term list) = case t of
|
clasohm@0
|
628 |
Var _ => insert_aterm(t,vars)
|
clasohm@0
|
629 |
| Abs (_,_,body) => add_term_vars(body,vars)
|
clasohm@0
|
630 |
| f$t => add_term_vars (f, add_term_vars(t, vars))
|
clasohm@0
|
631 |
| _ => vars;
|
clasohm@0
|
632 |
|
clasohm@0
|
633 |
fun term_vars t = add_term_vars(t,[]);
|
clasohm@0
|
634 |
|
clasohm@0
|
635 |
(*Accumulates the Frees in the term, suppressing duplicates*)
|
clasohm@0
|
636 |
fun add_term_frees (t, frees: term list) = case t of
|
clasohm@0
|
637 |
Free _ => insert_aterm(t,frees)
|
clasohm@0
|
638 |
| Abs (_,_,body) => add_term_frees(body,frees)
|
clasohm@0
|
639 |
| f$t => add_term_frees (f, add_term_frees(t, frees))
|
clasohm@0
|
640 |
| _ => frees;
|
clasohm@0
|
641 |
|
clasohm@0
|
642 |
fun term_frees t = add_term_frees(t,[]);
|
clasohm@0
|
643 |
|
clasohm@0
|
644 |
(*Given an abstraction over P, replaces the bound variable by a Free variable
|
clasohm@0
|
645 |
having a unique name. *)
|
clasohm@0
|
646 |
fun variant_abs (a,T,P) =
|
clasohm@0
|
647 |
let val b = variant (add_term_names(P,[])) a
|
paulson@2192
|
648 |
in (b, subst_bound (Free(b,T), P)) end;
|
clasohm@0
|
649 |
|
clasohm@0
|
650 |
(* renames and reverses the strings in vars away from names *)
|
clasohm@0
|
651 |
fun rename_aTs names vars : (string*typ)list =
|
clasohm@0
|
652 |
let fun rename_aT (vars,(a,T)) =
|
clasohm@1460
|
653 |
(variant (map #1 vars @ names) a, T) :: vars
|
clasohm@0
|
654 |
in foldl rename_aT ([],vars) end;
|
clasohm@0
|
655 |
|
clasohm@0
|
656 |
fun rename_wrt_term t = rename_aTs (add_term_names(t,[]));
|
clasohm@1364
|
657 |
|
paulson@1417
|
658 |
|
paulson@1417
|
659 |
|
paulson@1426
|
660 |
(*** Compression of terms and types by sharing common subtrees.
|
paulson@1426
|
661 |
Saves 50-75% on storage requirements. As it is fairly slow,
|
paulson@1426
|
662 |
it is called only for axioms, stored theorems, etc. ***)
|
paulson@1417
|
663 |
|
paulson@1417
|
664 |
(** Sharing of types **)
|
paulson@1417
|
665 |
|
paulson@1417
|
666 |
fun atomic_tag (Type (a,_)) = if a<>"fun" then a else raise Match
|
paulson@1417
|
667 |
| atomic_tag (TFree (a,_)) = a
|
paulson@1417
|
668 |
| atomic_tag (TVar ((a,_),_)) = a;
|
paulson@1417
|
669 |
|
paulson@1417
|
670 |
fun type_tag (Type("fun",[S,T])) = atomic_tag S ^ type_tag T
|
paulson@1417
|
671 |
| type_tag T = atomic_tag T;
|
paulson@1417
|
672 |
|
paulson@1417
|
673 |
val memo_types = ref (Symtab.null : typ list Symtab.table);
|
paulson@1417
|
674 |
|
paulson@1417
|
675 |
fun find_type (T, []: typ list) = None
|
paulson@1417
|
676 |
| find_type (T, T'::Ts) =
|
paulson@1417
|
677 |
if T=T' then Some T'
|
paulson@1417
|
678 |
else find_type (T, Ts);
|
paulson@1417
|
679 |
|
paulson@1417
|
680 |
fun compress_type T =
|
paulson@1417
|
681 |
let val tag = type_tag T
|
paulson@1417
|
682 |
val tylist = the (Symtab.lookup (!memo_types, tag))
|
clasohm@1460
|
683 |
handle _ => []
|
paulson@1417
|
684 |
in
|
paulson@1417
|
685 |
case find_type (T,tylist) of
|
clasohm@1460
|
686 |
Some T' => T'
|
paulson@1417
|
687 |
| None =>
|
clasohm@1460
|
688 |
let val T' =
|
clasohm@1460
|
689 |
case T of
|
clasohm@1460
|
690 |
Type (a,Ts) => Type (a, map compress_type Ts)
|
clasohm@1460
|
691 |
| _ => T
|
clasohm@1460
|
692 |
in memo_types := Symtab.update ((tag, T'::tylist), !memo_types);
|
clasohm@1460
|
693 |
T
|
clasohm@1460
|
694 |
end
|
paulson@1417
|
695 |
end
|
paulson@1417
|
696 |
handle Match =>
|
paulson@1417
|
697 |
let val Type (a,Ts) = T
|
paulson@1417
|
698 |
in Type (a, map compress_type Ts) end;
|
paulson@1417
|
699 |
|
paulson@1417
|
700 |
(** Sharing of atomic terms **)
|
paulson@1417
|
701 |
|
paulson@1417
|
702 |
val memo_terms = ref (Symtab.null : term list Symtab.table);
|
paulson@1417
|
703 |
|
paulson@1417
|
704 |
fun find_term (t, []: term list) = None
|
paulson@1417
|
705 |
| find_term (t, t'::ts) =
|
paulson@1417
|
706 |
if t=t' then Some t'
|
paulson@1417
|
707 |
else find_term (t, ts);
|
paulson@1417
|
708 |
|
paulson@1417
|
709 |
fun const_tag (Const (a,_)) = a
|
paulson@1417
|
710 |
| const_tag (Free (a,_)) = a
|
paulson@1417
|
711 |
| const_tag (Var ((a,i),_)) = a
|
paulson@1417
|
712 |
| const_tag (t as Bound _) = ".B.";
|
paulson@1417
|
713 |
|
paulson@1417
|
714 |
fun share_term (t $ u) = share_term t $ share_term u
|
paulson@1417
|
715 |
| share_term (Abs (a,T,u)) = Abs (a, T, share_term u)
|
paulson@1417
|
716 |
| share_term t =
|
paulson@1417
|
717 |
let val tag = const_tag t
|
clasohm@1460
|
718 |
val ts = the (Symtab.lookup (!memo_terms, tag))
|
clasohm@1460
|
719 |
handle _ => []
|
paulson@1417
|
720 |
in
|
clasohm@1460
|
721 |
case find_term (t,ts) of
|
clasohm@1460
|
722 |
Some t' => t'
|
clasohm@1460
|
723 |
| None => (memo_terms := Symtab.update ((tag, t::ts), !memo_terms);
|
clasohm@1460
|
724 |
t)
|
paulson@1417
|
725 |
end;
|
paulson@1417
|
726 |
|
paulson@1417
|
727 |
val compress_term = share_term o map_term_types compress_type;
|
paulson@1417
|
728 |
|
clasohm@1364
|
729 |
end;
|
clasohm@1364
|
730 |
|
clasohm@1364
|
731 |
open Term;
|