src/Sequents/LK/Nat.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Sequents/LK/Nat.ML	Tue Jul 27 18:50:14 1999 +0200
@@ -0,0 +1,51 @@
+(*  Title:      FOL/ex/Nat.ML
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1992  University of Cambridge
+
+Proofs about the natural numbers.
+*)
+
+Addsimps [Suc_neq_0];
+
+Add_safes [Suc_inject RS L_of_imp];
+
+Goal "|- Suc(k) ~= k";
+by (res_inst_tac [("n","k")] induct 1);
+by (Simp_tac 1);
+by (Fast_tac 1);
+qed "Suc_n_not_n";
+
+Goalw [add_def] "|- 0+n = n";
+by (rtac rec_0 1);
+qed "add_0";
+
+Goalw [add_def] "|- Suc(m)+n = Suc(m+n)";
+by (rtac rec_Suc 1);
+qed "add_Suc";
+
+Addsimps [add_0, add_Suc];
+
+Goal "|- (k+m)+n = k+(m+n)";
+by (res_inst_tac [("n","k")] induct 1);
+by (Simp_tac 1);
+by (Asm_simp_tac 1);
+qed "add_assoc";
+
+Goal "|- m+0 = m";
+by (res_inst_tac [("n","m")] induct 1);
+by (Simp_tac 1);
+by (Asm_simp_tac 1);
+qed "add_0_right";
+
+Goal "|- m+Suc(n) = Suc(m+n)";
+by (res_inst_tac [("n","m")] induct 1);
+by (ALLGOALS (Asm_simp_tac));
+qed "add_Suc_right";
+
+(*Example used in Reference Manual, Doc/Ref/simplifier.tex*)
+val [prem] = Goal "(!!n. |- f(Suc(n)) = Suc(f(n))) ==> |- f(i+j) = i+f(j)";
+by (res_inst_tac [("n","i")] induct 1);
+by (Simp_tac 1);
+by (simp_tac (simpset() addsimps [prem]) 1);
+result();