src/FOL/ex/Nat.thy

+(*  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 0
+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