(* Title: CCL/ex/Nat.thy
ID: $Id$
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
header {* Programs defined over the natural numbers *}
theory Nat
imports Wfd
begin
consts
not :: "i=>i"
"#+" :: "[i,i]=>i" (infixr 60)
"#*" :: "[i,i]=>i" (infixr 60)
"#-" :: "[i,i]=>i" (infixr 60)
"##" :: "[i,i]=>i" (infixr 60)
"#<" :: "[i,i]=>i" (infixr 60)
"#<=" :: "[i,i]=>i" (infixr 60)
ackermann :: "[i,i]=>i"
defs
not_def: "not(b) == if b then false else true"
add_def: "a #+ b == nrec(a,b,%x g. succ(g))"
mult_def: "a #* b == nrec(a,zero,%x g. b #+ g)"
sub_def: "a #- b == letrec sub x y be ncase(y,x,%yy. ncase(x,zero,%xx. sub(xx,yy)))
in sub(a,b)"
le_def: "a #<= b == letrec le x y be ncase(x,true,%xx. ncase(y,false,%yy. le(xx,yy)))
in le(a,b)"
lt_def: "a #< b == not(b #<= a)"
div_def: "a ## b == letrec div x y be if x #< y then zero else succ(div(x#-y,y))
in div(a,b)"
ack_def:
"ackermann(a,b) == letrec ack n m be ncase(n,succ(m),%x.
ncase(m,ack(x,succ(zero)),%y. ack(x,ack(succ(x),y))))
in ack(a,b)"
ML {* use_legacy_bindings (the_context ()) *}
end