src/FOLP/ex/Nat.thy
 changeset 0 a5a9c433f639 child 353 b5030aaca2ab
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/FOLP/ex/Nat.thy	Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,36 @@
+(*  Title: 	FOLP/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
+*)
+
+Nat = IFOLP +
+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)
+
+  (*Proof terms*)
+       nrec         :: "[nat,p,[nat,p]=>p]=>p"
+       ninj,nneq    :: "p=>p"
+       rec0, recSuc :: "p"
+
+rules
+  induct     "[| b:P(0); !!x u. u:P(x) ==> c(x,u):P(Suc(x)) \
+\             |] ==> nrec(n,b,c):P(n)"
+
+  Suc_inject "p:Suc(m)=Suc(n) ==> ninj(p) : m=n"
+  Suc_neq_0  "p:Suc(m)=0      ==> nneq(p) : R"
+  rec_0      "rec0 : rec(0,a,f) = a"
+  rec_Suc    "recSuc : rec(Suc(m), a, f) = f(m, rec(m,a,f))"
+  add_def    "m+n == rec(m, n, %x y. Suc(y))"
+
+  nrecB0     "b: A ==> nrec(0,b,c) = b : A"
+  nrecBSuc   "c(n,nrec(n,b,c)) : A ==> nrec(Suc(n),b,c) = c(n,nrec(n,b,c)) : A"
+end```