--- a/Quot/Examples/IntEx2.thy	Tue Dec 08 23:32:54 2009 +0100
+++ b/Quot/Examples/IntEx2.thy	Wed Dec 09 00:03:18 2009 +0100
@@ -105,8 +105,6 @@
 
 end
 
-thm add_assoc
-
 lemma plus_raw_rsp[quot_respect]:
   shows "(op \<approx> ===> op \<approx> ===> op \<approx>) plus_raw plus_raw"
 by auto
@@ -171,42 +169,33 @@
   fix i j k :: int
   show "(i + j) + k = i + (j + k)"
     unfolding add_int_def
-    apply(tactic {* lift_tac @{context} @{thm plus_assoc_raw} 1 *})
-    done
+    by (lifting plus_assoc_raw)
   show "i + j = j + i" 
     unfolding add_int_def
-    apply(tactic {* lift_tac @{context} @{thm plus_sym_raw} 1 *})
-    done
+    by (lifting plus_sym_raw)
   show "0 + i = (i::int)"
     unfolding add_int_def Zero_int_def 
-    apply(tactic {* lift_tac @{context} @{thm plus_zero_raw} 1 *})
-    done
+    by (lifting plus_zero_raw)
   show "- i + i = 0"
     unfolding add_int_def minus_int_def Zero_int_def 
-    apply(tactic {* lift_tac @{context} @{thm plus_minus_zero_raw} 1 *})
-    done
+    by (lifting plus_minus_zero_raw)
   show "i - j = i + - j"
     by (simp add: diff_int_def)
   show "(i * j) * k = i * (j * k)"
     unfolding mult_int_def 
-    apply(tactic {* lift_tac @{context} @{thm mult_assoc_raw} 1 *})
-    done
+    by (lifting mult_assoc_raw)
   show "i * j = j * i"
     unfolding mult_int_def 
-    apply(tactic {* lift_tac @{context} @{thm mult_sym_raw} 1 *})
-    done
+    by (lifting mult_sym_raw)
   show "1 * i = i"
     unfolding mult_int_def One_int_def
-    apply(tactic {* lift_tac @{context} @{thm mult_one_raw} 1 *})
-    done
+    by (lifting mult_one_raw)
   show "(i + j) * k = i * k + j * k"
     unfolding mult_int_def add_int_def
-    apply(tactic {* lift_tac @{context} @{thm mult_plus_comm_raw} 1 *})
-    done
+    by (lifting mult_plus_comm_raw)
   show "0 \<noteq> (1::int)"
     unfolding Zero_int_def One_int_def
-    apply(tactic {* lift_tac @{context} @{thm one_zero_distinct} 1 *})
-    done
+    by (lifting one_zero_distinct)
 qed
 
 term of_nat
@@ -242,22 +231,18 @@
   fix i j k :: int
   show antisym: "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j"
     unfolding le_int_def
-    apply(tactic {* lift_tac @{context} @{thm le_antisym_raw} 1 *})
-    done
+    by (lifting le_antisym_raw)
   show "(i < j) = (i \<le> j \<and> \<not> j \<le> i)"
     by (auto simp add: less_int_def dest: antisym) 
   show "i \<le> i"
     unfolding le_int_def
-    apply(tactic {* lift_tac @{context} @{thm le_refl_raw} 1 *})
-    done
+    by (lifting le_refl_raw)
   show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
     unfolding le_int_def
-    apply(tactic {* lift_tac @{context} @{thm le_trans_raw} 1 *})
-    done
+    by (lifting le_trans_raw)
   show "i \<le> j \<or> j \<le> i"
     unfolding le_int_def
-    apply(tactic {* lift_tac @{context} @{thm le_cases_raw} 1 *})
-    done
+    by (lifting le_cases_raw)
 qed
 
 instantiation int :: distrib_lattice
@@ -285,8 +270,7 @@
   fix i j k :: int
   show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
     unfolding le_int_def add_int_def
-    apply(tactic {* lift_tac @{context} @{thm le_plus_raw} 1 *})
-    done
+    by (lifting le_plus_raw)
 qed
 
 lemma test:
@@ -303,8 +287,7 @@
   fix i j k :: int
   show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
     unfolding mult_int_def le_int_def less_int_def Zero_int_def
-    apply(tactic {* lift_tac @{context} @{thm test} 1 *})
-    done
+    by (lifting test)
   show "\<bar>i\<bar> = (if i < 0 then -i else i)"
     by (simp only: zabs_def)
   show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"