(* Title: FOL/ex/Nat.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Examples for the manuals.
Theory of the natural numbers: Peano's axioms, primitive recursion
*)
Nat = FOL +
types nat
arities nat :: term
consts "0" :: nat ("0")
Suc :: nat=>nat
rec :: [nat, 'a, [nat,'a]=>'a] => 'a
"+" :: [nat, nat] => nat (infixl 60)
rules induct "[| P(0); !!x. P(x) ==> P(Suc(x)) |] ==> P(n)"
Suc_inject "Suc(m)=Suc(n) ==> m=n"
Suc_neq_0 "Suc(m)=0 ==> R"
rec_0 "rec(0,a,f) = a"
rec_Suc "rec(Suc(m), a, f) = f(m, rec(m,a,f))"
add_def "m+n == rec(m, n, %x y. Suc(y))"
end