--- a/src/Sequents/LK/Nat.thy Mon Nov 20 21:23:12 2006 +0100
+++ b/src/Sequents/LK/Nat.thy Mon Nov 20 23:47:10 2006 +0100
@@ -27,6 +27,52 @@
rec_Suc: "|- rec(Suc(m), a, f) = f(m, rec(m,a,f))"
add_def: "m+n == rec(m, n, %x y. Suc(y))"
-ML {* use_legacy_bindings (the_context ()) *}
+
+declare Suc_neq_0 [simp]
+
+lemma Suc_inject_rule: "$H, $G, m = n |- $E \<Longrightarrow> $H, Suc(m) = Suc(n), $G |- $E"
+ by (rule L_of_imp [OF Suc_inject])
+
+lemma Suc_n_not_n: "|- Suc(k) ~= k"
+ apply (rule_tac n = k in induct)
+ apply (tactic {* simp_tac (LK_ss addsimps [thm "Suc_neq_0"]) 1 *})
+ apply (tactic {* fast_tac (LK_pack add_safes [thm "Suc_inject_rule"]) 1 *})
+ done
+
+lemma add_0: "|- 0+n = n"
+ apply (unfold add_def)
+ apply (rule rec_0)
+ done
+
+lemma add_Suc: "|- Suc(m)+n = Suc(m+n)"
+ apply (unfold add_def)
+ apply (rule rec_Suc)
+ done
+
+declare add_0 [simp] add_Suc [simp]
+
+lemma add_assoc: "|- (k+m)+n = k+(m+n)"
+ apply (rule_tac n = "k" in induct)
+ apply (tactic {* simp_tac (LK_ss addsimps [thm "add_0"]) 1 *})
+ apply (tactic {* simp_tac (LK_ss addsimps [thm "add_Suc"]) 1 *})
+ done
+
+lemma add_0_right: "|- m+0 = m"
+ apply (rule_tac n = "m" in induct)
+ apply (tactic {* simp_tac (LK_ss addsimps [thm "add_0"]) 1 *})
+ apply (tactic {* simp_tac (LK_ss addsimps [thm "add_Suc"]) 1 *})
+ done
+
+lemma add_Suc_right: "|- m+Suc(n) = Suc(m+n)"
+ apply (rule_tac n = "m" in induct)
+ apply (tactic {* simp_tac (LK_ss addsimps [thm "add_0"]) 1 *})
+ apply (tactic {* simp_tac (LK_ss addsimps [thm "add_Suc"]) 1 *})
+ done
+
+lemma "(!!n. |- f(Suc(n)) = Suc(f(n))) ==> |- f(i+j) = i+f(j)"
+ apply (rule_tac n = "i" in induct)
+ apply (tactic {* simp_tac (LK_ss addsimps [thm "add_0"]) 1 *})
+ apply (tactic {* asm_simp_tac (LK_ss addsimps [thm "add_Suc"]) 1 *})
+ done
end