--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/Integ/Bin.thy Tue Sep 22 13:49:22 1998 +0200
@@ -0,0 +1,172 @@
+(* Title: ZF/ex/Bin.thy
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1994 University of Cambridge
+
+Arithmetic on binary integers.
+
+ The sign Pls stands for an infinite string of leading 0's.
+ The sign Min stands for an infinite string of leading 1's.
+
+A number can have multiple representations, namely leading 0's with sign
+Pls and leading 1's with sign Min. See twos-compl.ML/int_of_binary for
+the numerical interpretation.
+
+The representation expects that (m mod 2) is 0 or 1, even if m is negative;
+For instance, ~5 div 2 = ~3 and ~5 mod 2 = 1; thus ~5 = (~3)*2 + 1
+
+Division is not defined yet!
+*)
+
+Bin = Integ + Datatype +
+
+consts
+ bin_rec :: [i, i, i, [i,i,i]=>i] => i
+ integ_of :: i=>i
+ NCons :: [i,i]=>i
+ bin_succ :: i=>i
+ bin_pred :: i=>i
+ bin_minus :: i=>i
+ bin_add :: [i,i]=>i
+ bin_mult :: [i,i]=>i
+
+ bin :: i
+
+syntax
+ "_Int" :: xnum => i ("_")
+
+datatype
+ "bin" = Pls
+ | Min
+ | Cons ("w: bin", "b: bool")
+ type_intrs "[bool_into_univ]"
+
+
+defs
+
+ bin_rec_def
+ "bin_rec(z,a,b,h) ==
+ Vrec(z, %z g. bin_case(a, b, %w x. h(w, x, g`w), z))"
+
+ integ_of_def
+ "integ_of(w) == bin_rec(w, $#0, $~($#1), %w x r. $#x $+ r $+ r)"
+
+ (** recall that cond(1,b,c)=b and cond(0,b,c)=0 **)
+
+ (*NCons adds a bit, suppressing leading 0s and 1s*)
+ NCons_def
+ "NCons(w,b) ==
+ bin_case(cond(b,Cons(w,b),w), cond(b,w,Cons(w,b)),
+ %w' x'. Cons(w,b), w)"
+
+ bin_succ_def
+ "bin_succ(w0) ==
+ bin_rec(w0, Cons(Pls,1), Pls,
+ %w x r. cond(x, Cons(r,0), NCons(w,1)))"
+
+ bin_pred_def
+ "bin_pred(w0) ==
+ bin_rec(w0, Min, Cons(Min,0),
+ %w x r. cond(x, NCons(w,0), Cons(r,1)))"
+
+ bin_minus_def
+ "bin_minus(w0) ==
+ bin_rec(w0, Pls, Cons(Pls,1),
+ %w x r. cond(x, bin_pred(NCons(r,0)), Cons(r,0)))"
+
+ bin_add_def
+ "bin_add(v0,w0) ==
+ bin_rec(v0,
+ lam w:bin. w,
+ lam w:bin. bin_pred(w),
+ %v x r. lam w1:bin.
+ bin_rec(w1, Cons(v,x), bin_pred(Cons(v,x)),
+ %w y s. NCons(r`cond(x and y, bin_succ(w), w),
+ x xor y))) ` w0"
+
+ bin_mult_def
+ "bin_mult(v0,w) ==
+ bin_rec(v0, Pls, bin_minus(w),
+ %v x r. cond(x, bin_add(NCons(r,0),w), NCons(r,0)))"
+end
+
+
+ML
+
+(** Concrete syntax for integers **)
+
+local
+ open Syntax;
+
+ (* Bits *)
+
+ fun mk_bit 0 = const "0"
+ | mk_bit 1 = const "succ" $ const "0"
+ | mk_bit _ = sys_error "mk_bit";
+
+ fun dest_bit (Const ("0", _)) = 0
+ | dest_bit (Const ("succ", _) $ Const ("0", _)) = 1
+ | dest_bit _ = raise Match;
+
+
+ (* Bit strings *) (*we try to handle superfluous leading digits nicely*)
+
+ fun prefix_len _ [] = 0
+ | prefix_len pred (x :: xs) =
+ if pred x then 1 + prefix_len pred xs else 0;
+
+ fun mk_bin str =
+ let
+ val (sign, digs) =
+ (case Symbol.explode str of
+ "#" :: "-" :: cs => (~1, cs)
+ | "#" :: cs => (1, cs)
+ | _ => raise ERROR);
+
+ val zs = prefix_len (equal "0") digs;
+
+ fun bin_of 0 = []
+ | bin_of ~1 = [~1]
+ | bin_of n = (n mod 2) :: bin_of (n div 2);
+
+ fun term_of [] = const "Pls"
+ | term_of [~1] = const "Min"
+ | term_of (b :: bs) = const "Cons" $ term_of bs $ mk_bit b;
+ in
+ term_of (bin_of (sign * (#1 (read_int digs))))
+ end;
+
+ fun dest_bin tm =
+ let
+ fun bin_of (Const ("Pls", _)) = []
+ | bin_of (Const ("Min", _)) = [~1]
+ | bin_of (Const ("Cons", _) $ bs $ b) = dest_bit b :: bin_of bs
+ | bin_of _ = raise Match;
+
+ fun int_of [] = 0
+ | int_of (b :: bs) = b + 2 * int_of bs;
+
+ val rev_digs = bin_of tm;
+ val (sign, zs) =
+ (case rev rev_digs of
+ ~1 :: bs => ("-", prefix_len (equal 1) bs)
+ | bs => ("", prefix_len (equal 0) bs));
+ val num = string_of_int (abs (int_of rev_digs));
+ in
+ "#" ^ sign ^ implode (replicate zs "0") ^ num
+ end;
+
+
+ (* translation of integer constant tokens to and from binary *)
+
+ fun int_tr (*"_Int"*) [t as Free (str, _)] =
+ (const "integ_of" $
+ (mk_bin str handle ERROR => raise TERM ("int_tr", [t])))
+ | int_tr (*"_Int"*) ts = raise TERM ("int_tr", ts);
+
+ fun int_tr' (*"integ_of"*) [t] = const "_Int" $ free (dest_bin t)
+ | int_tr' (*"integ_of"*) _ = raise Match;
+in
+ val parse_translation = [("_Int", int_tr)];
+ val print_translation = [("integ_of", int_tr')];
+end;