(* Title: ZF/ex/Bin.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Arithmetic on binary integers.
*)
Bin = Integ + Univ +
consts
bin_rec :: "[i, i, i, [i,i,i]=>i] => i"
integ_of_bin :: "i=>i"
bin_succ :: "i=>i"
bin_pred :: "i=>i"
bin_minus :: "i=>i"
bin_add,bin_mult :: "[i,i]=>i"
bin :: "i"
datatype
"bin" = Plus
| Minus
| "$$" ("w: bin", "b: bool") (infixl 60)
type_intrs "[bool_into_univ]"
rules
bin_rec_def
"bin_rec(z,a,b,h) == \
\ Vrec(z, %z g. bin_case(a, b, %w x. h(w, x, g`w), z))"
integ_of_bin_def
"integ_of_bin(w) == bin_rec(w, $#0, $~($#1), %w x r. $#x $+ r $+ r)"
bin_succ_def
"bin_succ(w0) == bin_rec(w0, Plus$$1, Plus, %w x r. cond(x, r$$0, w$$1))"
bin_pred_def
"bin_pred(w0) == \
\ bin_rec(w0, Minus, Minus$$0, %w x r. cond(x, w$$0, r$$1))"
bin_minus_def
"bin_minus(w0) == \
\ bin_rec(w0, Plus, Plus$$1, %w x r. cond(x, bin_pred(r$$0), r$$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, v$$x, bin_pred(v$$x), \
\ %w y s. (r`cond(x and y, bin_succ(w), w)) \
\ $$ (x xor y))) ` w0"
bin_mult_def
"bin_mult(v0,w) == \
\ bin_rec(v0, Plus, bin_minus(w), \
\ %v x r. cond(x, bin_add(r$$0,w), r$$0))"
end