(* Title: CTT/arith
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
Arithmetic operators and their definitions
Proves about elementary arithmetic: addition, multiplication, etc.
Tests definitions and simplifier.
*)
Arith = CTT +
consts "#+","-","|-|" :: "[i,i]=>i" (infixr 65)
"#*",div,mod :: "[i,i]=>i" (infixr 70)
rules
add_def "a#+b == rec(a, b, %u v. succ(v))"
diff_def "a-b == rec(b, a, %u v. rec(v, 0, %x y. x))"
absdiff_def "a|-|b == (a-b) #+ (b-a)"
mult_def "a#*b == rec(a, 0, %u v. b #+ v)"
mod_def "a mod b == rec(a, 0, %u v. rec(succ(v) |-| b, 0, %x y. succ(v)))"
div_def "a div b == rec(a, 0, %u v. rec(succ(u) mod b, succ(v), %x y. v))"
end