(* 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 = LK +
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 "[| $H |- $E, P(0), $F;
!!x. $H |- $E, P(x) --> P(Suc(x)), $F |] ==> $H |- $E, P(n), $F"
Suc_inject "|- Suc(m)=Suc(n) --> m=n"
Suc_neq_0 "|- Suc(m) ~= 0"
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