|
0
|
1 |
(* Title: ZF/arith.thy
|
|
|
2 |
ID: $Id$
|
|
|
3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
|
|
|
4 |
Copyright 1992 University of Cambridge
|
|
|
5 |
|
|
|
6 |
Arithmetic operators and their definitions
|
|
|
7 |
*)
|
|
|
8 |
|
|
|
9 |
Arith = Epsilon +
|
|
|
10 |
consts
|
|
|
11 |
rec :: "[i, i, [i,i]=>i]=>i"
|
|
|
12 |
"#*" :: "[i,i]=>i" (infixl 70)
|
|
|
13 |
div :: "[i,i]=>i" (infixl 70)
|
|
|
14 |
mod :: "[i,i]=>i" (infixl 70)
|
|
|
15 |
"#+" :: "[i,i]=>i" (infixl 65)
|
|
|
16 |
"#-" :: "[i,i]=>i" (infixl 65)
|
|
|
17 |
|
|
|
18 |
rules
|
|
|
19 |
rec_def "rec(k,a,b) == transrec(k, %n f. nat_case(n, a, %m. b(m, f`m)))"
|
|
|
20 |
|
|
|
21 |
add_def "m#+n == rec(m, n, %u v.succ(v))"
|
|
|
22 |
diff_def "m#-n == rec(n, m, %u v. rec(v, 0, %x y.x))"
|
|
|
23 |
mult_def "m#*n == rec(m, 0, %u v. n #+ v)"
|
|
|
24 |
mod_def "m mod n == transrec(m, %j f. if(j:n, j, f`(j#-n)))"
|
|
|
25 |
div_def "m div n == transrec(m, %j f. if(j:n, 0, succ(f`(j#-n))))"
|
|
|
26 |
end
|