src/Sequents/LK/Nat.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Sequents/LK/Nat.thy	Tue Jul 27 18:50:14 1999 +0200
@@ -0,0 +1,27 @@
+(*  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