(*  Title: 	FOL/ex/nat.thy

     ID:         $Id$

     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1992  University of Cambridge

 Examples for the manual "Introduction to Isabelle"

 Theory of the natural numbers: Peano's axioms, primitive recursion

 INCOMPATIBLE with nat2.thy, Nipkow's example

 *)

 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