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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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367 |
let fun subst (Abs(a,T,body)) = Abs(a, T, subst body)
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368 |
| subst (f$t') = subst f $ subst t'
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|
369 |
| subst t = (case assoc(instl,t) of
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370 |
Some u => u | None => t)
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|
371 |
in subst t end;
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|
372 |
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373 |
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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375 |
| subst(T as TFree _) = T
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376 |
| subst(T as TVar(ixn,_)) =
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(case assoc(iTs,ixn) of None => T | Some(U) => U)
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378 |
in subst T end;
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379 |
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380 |
val subst_TVars = map_term_types o typ_subst_TVars;
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381 |
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|
382 |
fun subst_Vars itms t = if null itms then t else
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383 |
let fun subst(v as Var(ixn,_)) =
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384 |
(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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|
386 |
| subst(f$t) = subst f $ subst t
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387 |
| subst(t) = t
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|
388 |
in subst t end;
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|
389 |
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390 |
fun subst_vars(iTs,itms) = if null iTs then subst_Vars itms else
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|
391 |
let fun subst(Const(a,T)) = Const(a,typ_subst_TVars iTs T)
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392 |
| subst(Free(a,T)) = Free(a,typ_subst_TVars iTs T)
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|
393 |
| subst(v as Var(ixn,T)) = (case assoc(itms,ixn) of
|
|
394 |
None => Var(ixn,typ_subst_TVars iTs T)
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395 |
| Some t => t)
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|
396 |
| subst(b as Bound _) = b
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|
397 |
| subst(Abs(a,T,t)) = Abs(a,typ_subst_TVars iTs T,subst t)
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|
398 |
| subst(f$t) = subst f $ subst t
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|
399 |
in subst end;
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|
400 |
|
|
401 |
|
|
402 |
(*Computing the maximum index of a typ*)
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|
403 |
fun maxidx_of_typ(Type(_,Ts)) =
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|
404 |
if Ts=[] then ~1 else max(map maxidx_of_typ Ts)
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|
405 |
| maxidx_of_typ(TFree _) = ~1
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|
406 |
| maxidx_of_typ(TVar((_,i),_)) = i;
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|
407 |
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|
408 |
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|
409 |
(*Computing the maximum index of a term*)
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|
410 |
fun maxidx_of_term (Const(_,T)) = maxidx_of_typ T
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|
411 |
| maxidx_of_term (Bound _) = ~1
|
|
412 |
| maxidx_of_term (Free(_,T)) = maxidx_of_typ T
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|
413 |
| maxidx_of_term (Var ((_,i), T)) = max[i, maxidx_of_typ T]
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|
414 |
| maxidx_of_term (Abs (_,T,body)) = max[maxidx_of_term body, maxidx_of_typ T]
|
|
415 |
| maxidx_of_term (f$t) = max [maxidx_of_term f, maxidx_of_term t];
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|
416 |
|
|
417 |
|
|
418 |
(* Increment the index of all Poly's in T by k *)
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|
419 |
fun incr_tvar k (Type(a,Ts)) = Type(a, map (incr_tvar k) Ts)
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|
420 |
| incr_tvar k (T as TFree _) = T
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|
421 |
| incr_tvar k (TVar((a,i),rs)) = TVar((a,i+k),rs);
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|
422 |
|
|
423 |
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|
424 |
(**** Syntax-related declarations ****)
|
|
425 |
|
|
426 |
|
|
427 |
(*Dummy type for parsing. Will be replaced during type inference. *)
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|
428 |
val dummyT = Type("dummy",[]);
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|
429 |
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|
430 |
(*scan a numeral of the given radix, normally 10*)
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|
431 |
fun scan_radixint (radix: int, cs) : int * string list =
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|
432 |
let val zero = ord"0"
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|
433 |
val limit = zero+radix
|
|
434 |
fun scan (num,[]) = (num,[])
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|
435 |
| scan (num, c::cs) =
|
|
436 |
if zero <= ord c andalso ord c < limit
|
|
437 |
then scan(radix*num + ord c - zero, cs)
|
|
438 |
else (num, c::cs)
|
|
439 |
in scan(0,cs) end;
|
|
440 |
|
|
441 |
fun scan_int cs = scan_radixint(10,cs);
|
|
442 |
|
|
443 |
|
|
444 |
(*** Printing ***)
|
|
445 |
|
|
446 |
|
|
447 |
(*Makes a variant of the name c distinct from the names in bs.
|
|
448 |
First attaches the suffix "a" and then increments this. *)
|
|
449 |
fun variant bs c : string =
|
|
450 |
let fun vary2 c = if (c mem bs) then vary2 (bump_string c) else c
|
|
451 |
fun vary1 c = if (c mem bs) then vary2 (c ^ "a") else c
|
|
452 |
in vary1 (if c="" then "u" else c) end;
|
|
453 |
|
|
454 |
(*Create variants of the list of names, with priority to the first ones*)
|
|
455 |
fun variantlist ([], used) = []
|
|
456 |
| variantlist(b::bs, used) =
|
|
457 |
let val b' = variant used b
|
|
458 |
in b' :: variantlist (bs, b'::used) end;
|
|
459 |
|
|
460 |
(** TFrees and TVars **)
|
|
461 |
|
|
462 |
(*maps (bs,v) to v'::bs this reverses the identifiers bs*)
|
|
463 |
fun add_new_id (bs, c) : string list = variant bs c :: bs;
|
|
464 |
|
|
465 |
(*Accumulates the names in the term, suppressing duplicates.
|
|
466 |
Includes Frees and Consts. For choosing unambiguous bound var names.*)
|
|
467 |
fun add_term_names (Const(a,_), bs) = a ins bs
|
|
468 |
| add_term_names (Free(a,_), bs) = a ins bs
|
|
469 |
| add_term_names (f$u, bs) = add_term_names (f, add_term_names(u, bs))
|
|
470 |
| add_term_names (Abs(_,_,t), bs) = add_term_names(t,bs)
|
|
471 |
| add_term_names (_, bs) = bs;
|
|
472 |
|
|
473 |
(*Accumulates the TVars in a type, suppressing duplicates. *)
|
|
474 |
fun add_typ_tvars(Type(_,Ts),vs) = foldr add_typ_tvars (Ts,vs)
|
|
475 |
| add_typ_tvars(TFree(_),vs) = vs
|
|
476 |
| add_typ_tvars(TVar(v),vs) = v ins vs;
|
|
477 |
|
|
478 |
(*Accumulates the TFrees in a type, suppressing duplicates. *)
|
|
479 |
fun add_typ_tfree_names(Type(_,Ts),fs) = foldr add_typ_tfree_names (Ts,fs)
|
|
480 |
| add_typ_tfree_names(TFree(f,_),fs) = f ins fs
|
|
481 |
| add_typ_tfree_names(TVar(_),fs) = fs;
|
|
482 |
|
|
483 |
fun add_typ_tfrees(Type(_,Ts),fs) = foldr add_typ_tfrees (Ts,fs)
|
|
484 |
| add_typ_tfrees(TFree(f),fs) = f ins fs
|
|
485 |
| add_typ_tfrees(TVar(_),fs) = fs;
|
|
486 |
|
|
487 |
(*Accumulates the TVars in a term, suppressing duplicates. *)
|
|
488 |
val add_term_tvars = it_term_types add_typ_tvars;
|
|
489 |
val add_term_tvar_ixns = (map #1) o (it_term_types add_typ_tvars);
|
|
490 |
|
|
491 |
(*Accumulates the TFrees in a term, suppressing duplicates. *)
|
|
492 |
val add_term_tfrees = it_term_types add_typ_tfrees;
|
|
493 |
val add_term_tfree_names = it_term_types add_typ_tfree_names;
|
|
494 |
|
|
495 |
(*Non-list versions*)
|
|
496 |
fun typ_tfrees T = add_typ_tfrees(T,[]);
|
|
497 |
fun typ_tvars T = add_typ_tvars(T,[]);
|
|
498 |
fun term_tfrees t = add_term_tfrees(t,[]);
|
|
499 |
fun term_tvars t = add_term_tvars(t,[]);
|
|
500 |
|
|
501 |
(** Frees and Vars **)
|
|
502 |
|
|
503 |
(*a partial ordering (not reflexive) for atomic terms*)
|
|
504 |
fun atless (Const (a,_), Const (b,_)) = a<b
|
|
505 |
| atless (Free (a,_), Free (b,_)) = a<b
|
|
506 |
| atless (Var(v,_), Var(w,_)) = xless(v,w)
|
|
507 |
| atless (Bound i, Bound j) = i<j
|
|
508 |
| atless _ = false;
|
|
509 |
|
|
510 |
(*insert atomic term into partially sorted list, suppressing duplicates (?)*)
|
|
511 |
fun insert_aterm (t,us) =
|
|
512 |
let fun inserta [] = [t]
|
|
513 |
| inserta (us as u::us') =
|
|
514 |
if atless(t,u) then t::us
|
|
515 |
else if t=u then us (*duplicate*)
|
|
516 |
else u :: inserta(us')
|
|
517 |
in inserta us end;
|
|
518 |
|
|
519 |
(*Accumulates the Vars in the term, suppressing duplicates*)
|
|
520 |
fun add_term_vars (t, vars: term list) = case t of
|
|
521 |
Var _ => insert_aterm(t,vars)
|
|
522 |
| Abs (_,_,body) => add_term_vars(body,vars)
|
|
523 |
| f$t => add_term_vars (f, add_term_vars(t, vars))
|
|
524 |
| _ => vars;
|
|
525 |
|
|
526 |
fun term_vars t = add_term_vars(t,[]);
|
|
527 |
|
|
528 |
(*Accumulates the Frees in the term, suppressing duplicates*)
|
|
529 |
fun add_term_frees (t, frees: term list) = case t of
|
|
530 |
Free _ => insert_aterm(t,frees)
|
|
531 |
| Abs (_,_,body) => add_term_frees(body,frees)
|
|
532 |
| f$t => add_term_frees (f, add_term_frees(t, frees))
|
|
533 |
| _ => frees;
|
|
534 |
|
|
535 |
fun term_frees t = add_term_frees(t,[]);
|
|
536 |
|
|
537 |
(*Given an abstraction over P, replaces the bound variable by a Free variable
|
|
538 |
having a unique name. *)
|
|
539 |
fun variant_abs (a,T,P) =
|
|
540 |
let val b = variant (add_term_names(P,[])) a
|
|
541 |
in (b, subst_bounds ([Free(b,T)], P)) end;
|
|
542 |
|
|
543 |
(* renames and reverses the strings in vars away from names *)
|
|
544 |
fun rename_aTs names vars : (string*typ)list =
|
|
545 |
let fun rename_aT (vars,(a,T)) =
|
|
546 |
(variant (map #1 vars @ names) a, T) :: vars
|
|
547 |
in foldl rename_aT ([],vars) end;
|
|
548 |
|
|
549 |
fun rename_wrt_term t = rename_aTs (add_term_names(t,[]));
|