the NatClass demo of the axclass tutorial;

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/FOL/ex/NatClass.ML	Fri Sep 01 14:26:26 1995 +0200
@@ -0,0 +1,62 @@
+(*  Title: 	FOL/ex/NatClass.ML
+    ID:         $Id$
+    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1992  University of Cambridge
+
+This is Nat.ML trivially modified to make it work with NatClass.thy.
+*)
+
+open NatClass;
+
+goal NatClass.thy "Suc(k) ~= (k::'a::nat)";
+by (res_inst_tac [("n","k")] induct 1);
+by (resolve_tac [notI] 1);
+by (eresolve_tac [Suc_neq_0] 1);
+by (resolve_tac [notI] 1);
+by (eresolve_tac [notE] 1);
+by (eresolve_tac [Suc_inject] 1);
+qed "Suc_n_not_n";
+
+
+goal NatClass.thy "(k+m)+n = k+(m+n)";
+prths ([induct] RL [topthm()]);  (*prints all 14 next states!*)
+by (resolve_tac [induct] 1);
+back();
+back();
+back();
+back();
+back();
+back();
+
+goalw NatClass.thy [add_def] "0+n = n";
+by (resolve_tac [rec_0] 1);
+qed "add_0";
+
+goalw NatClass.thy [add_def] "Suc(m)+n = Suc(m+n)";
+by (resolve_tac [rec_Suc] 1);
+qed "add_Suc";
+
+val add_ss = FOL_ss addsimps [add_0, add_Suc];
+
+goal NatClass.thy "(k+m)+n = k+(m+n)";
+by (res_inst_tac [("n","k")] induct 1);
+by (simp_tac add_ss 1);
+by (asm_simp_tac add_ss 1);
+qed "add_assoc";
+
+goal NatClass.thy "m+0 = m";
+by (res_inst_tac [("n","m")] induct 1);
+by (simp_tac add_ss 1);
+by (asm_simp_tac add_ss 1);
+qed "add_0_right";
+
+goal NatClass.thy "m+Suc(n) = Suc(m+n)";
+by (res_inst_tac [("n","m")] induct 1);
+by (ALLGOALS (asm_simp_tac add_ss));
+qed "add_Suc_right";
+
+val [prem] = goal NatClass.thy "(!!n. f(Suc(n)) = Suc(f(n))) ==> f(i+j) = i+f(j)";
+by (res_inst_tac [("n","i")] induct 1);
+by (simp_tac add_ss 1);
+by (asm_simp_tac (add_ss addsimps [prem]) 1);
+result();

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/FOL/ex/NatClass.thy	Fri Sep 01 14:26:26 1995 +0200
@@ -0,0 +1,34 @@
+(*  Title:      FOL/ex/NatClass.thy
+    ID:         $Id$
+    Author:     Markus Wenzel, TU Muenchen
+
+This is an abstract version of Nat.thy. Instead of axiomatizing a
+single type "nat" we define the class of all these types (up to
+isomorphism).
+
+Note: The "rec" operator had to be made 'monomorphic', because class
+axioms may not contain more than one type variable.
+*)
+
+NatClass = FOL +
+
+consts
+  "0"           :: "'a"                                 ("0")
+  Suc           :: "'a => 'a"
+  rec           :: "['a, 'a, ['a, 'a] => 'a] => 'a"
+
+axclass
+  nat < term
+  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))"
+
+consts
+  "+"           :: "['a::nat, 'a] => 'a"                (infixl 60)
+
+defs
+  add_def       "m + n == rec(m, n, %x y. Suc(y))"
+
+end