src/HOL/MetisExamples/set.thy

 (*  Title:      HOL/MetisExamples/set.thy
-    ID:         $Id$
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
 
 Testing the metis method
 *)
 

 assume 2: "(\<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X \<or> Y \<subseteq> x) \<or> X \<noteq> Y \<union> Z"
 assume 3: "(\<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X \<or> Z \<subseteq> x) \<or> X \<noteq> Y \<union> Z"
 assume 4: "(\<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X \<or> \<not> X \<subseteq> x) \<or> X \<noteq> Y \<union> Z"
 assume 5: "\<And>V. ((\<not> Y \<subseteq> V \<or> \<not> Z \<subseteq> V) \<or> X \<subseteq> V) \<or> X = Y \<union> Z"
 have 6: "sup Y Z = X \<or> Y \<subseteq> X"
-  by (metis 0 sup_set_eq)
+  by (metis 0)
 have 7: "sup Y Z = X \<or> Z \<subseteq> X"
-  by (metis 1 sup_set_eq)
+  by (metis 1)
 have 8: "\<And>X3. sup Y Z = X \<or> X \<subseteq> X3 \<or> \<not> Y \<subseteq> X3 \<or> \<not> Z \<subseteq> X3"
-  by (metis 5 sup_set_eq)
+  by (metis 5)
 have 9: "Y \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X"
-  by (metis 2 sup_set_eq)
+  by (metis 2)
 have 10: "Z \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X"
-  by (metis 3 sup_set_eq)
+  by (metis 3)
 have 11: "sup Y Z \<noteq> X \<or> \<not> X \<subseteq> x \<or> \<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X"
-  by (metis 4 sup_set_eq)
+  by (metis 4)
 have 12: "Z \<subseteq> X"
-  by (metis Un_upper2 sup_set_eq 7)
+  by (metis Un_upper2 7)
 have 13: "\<And>X3. sup Y Z = X \<or> X \<subseteq> sup X3 Z \<or> \<not> Y \<subseteq> sup X3 Z"
-  by (metis 8 Un_upper2 sup_set_eq)
+  by (metis 8 Un_upper2)
 have 14: "Y \<subseteq> X"
-  by (metis Un_upper1 sup_set_eq 6)
+  by (metis Un_upper1 6)
 have 15: "Z \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X"
   by (metis 10 12)
 have 16: "Y \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X"
   by (metis 9 12)
 have 17: "sup Y Z \<noteq> X \<or> \<not> X \<subseteq> x \<or> \<not> Y \<subseteq> X"

 have 19: "Z \<subseteq> x \<or> sup Y Z \<noteq> X"
   by (metis 15 14)
 have 20: "Y \<subseteq> x \<or> sup Y Z \<noteq> X"
   by (metis 16 14)
 have 21: "sup Y Z = X \<or> X \<subseteq> sup Y Z"
-  by (metis 13 Un_upper1 sup_set_eq)
+  by (metis 13 Un_upper1)
 have 22: "sup Y Z = X \<or> \<not> sup Y Z \<subseteq> X"
   by (metis equalityI 21)
 have 23: "sup Y Z = X \<or> \<not> Z \<subseteq> X \<or> \<not> Y \<subseteq> X"
-  by (metis 22 Un_least sup_set_eq)
+  by (metis 22 Un_least)
 have 24: "sup Y Z = X \<or> \<not> Y \<subseteq> X"
   by (metis 23 12)
 have 25: "sup Y Z = X"
   by (metis 24 14)
 have 26: "\<And>X3. X \<subseteq> X3 \<or> \<not> Z \<subseteq> X3 \<or> \<not> Y \<subseteq> X3"
-  by (metis Un_least sup_set_eq 25)
+  by (metis Un_least 25)
 have 27: "Y \<subseteq> x"
   by (metis 20 25)
 have 28: "Z \<subseteq> x"
   by (metis 19 25)
 have 29: "\<not> X \<subseteq> x"

 assume 2: "(\<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X \<or> Y \<subseteq> x) \<or> X \<noteq> Y \<union> Z"
 assume 3: "(\<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X \<or> Z \<subseteq> x) \<or> X \<noteq> Y \<union> Z"
 assume 4: "(\<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X \<or> \<not> X \<subseteq> x) \<or> X \<noteq> Y \<union> Z"
 assume 5: "\<And>V. ((\<not> Y \<subseteq> V \<or> \<not> Z \<subseteq> V) \<or> X \<subseteq> V) \<or> X = Y \<union> Z"
 have 6: "sup Y Z = X \<or> Y \<subseteq> X"
-  by (metis 0 sup_set_eq)
+  by (metis 0)
 have 7: "Y \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X"
-  by (metis 2 sup_set_eq)
+  by (metis 2)
 have 8: "sup Y Z \<noteq> X \<or> \<not> X \<subseteq> x \<or> \<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X"
-  by (metis 4 sup_set_eq)
+  by (metis 4)
 have 9: "\<And>X3. sup Y Z = X \<or> X \<subseteq> sup X3 Z \<or> \<not> Y \<subseteq> sup X3 Z"
-  by (metis 5 sup_set_eq Un_upper2 sup_set_eq)
+  by (metis 5 Un_upper2)
 have 10: "Z \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X"
-  by (metis 3 sup_set_eq Un_upper2 sup_set_eq sup_set_eq)
+  by (metis 3 Un_upper2)
 have 11: "sup Y Z \<noteq> X \<or> \<not> X \<subseteq> x \<or> \<not> Y \<subseteq> X"
-  by (metis 8 Un_upper2 sup_set_eq sup_set_eq)
+  by (metis 8 Un_upper2)
 have 12: "Z \<subseteq> x \<or> sup Y Z \<noteq> X"
-  by (metis 10 Un_upper1 sup_set_eq)
+  by (metis 10 Un_upper1)
 have 13: "sup Y Z = X \<or> X \<subseteq> sup Y Z"
-  by (metis 9 Un_upper1 sup_set_eq)
+  by (metis 9 Un_upper1)
 have 14: "sup Y Z = X \<or> \<not> Z \<subseteq> X \<or> \<not> Y \<subseteq> X"
-  by (metis equalityI 13 Un_least sup_set_eq)
+  by (metis equalityI 13 Un_least)
 have 15: "sup Y Z = X"
-  by (metis 14 sup_set_eq 1 sup_set_eq sup_set_eq 6)
+  by (metis 14 1 6)
 have 16: "Y \<subseteq> x"
-  by (metis 7 Un_upper2 sup_set_eq sup_set_eq Un_upper1 sup_set_eq 15)
+  by (metis 7 Un_upper2 Un_upper1 15)
 have 17: "\<not> X \<subseteq> x"
-  by (metis 11 Un_upper1 sup_set_eq 15)
+  by (metis 11 Un_upper1 15)
 have 18: "X \<subseteq> x"
-  by (metis Un_least sup_set_eq 15 12 15 16)
+  by (metis Un_least 15 12 15 16)
 show "False"
   by (metis 18 17)
 qed
 
 declare [[sledgehammer_modulus = 3]]

 assume 2: "(\<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X \<or> Y \<subseteq> x) \<or> X \<noteq> Y \<union> Z"
 assume 3: "(\<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X \<or> Z \<subseteq> x) \<or> X \<noteq> Y \<union> Z"
 assume 4: "(\<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X \<or> \<not> X \<subseteq> x) \<or> X \<noteq> Y \<union> Z"
 assume 5: "\<And>V. ((\<not> Y \<subseteq> V \<or> \<not> Z \<subseteq> V) \<or> X \<subseteq> V) \<or> X = Y \<union> Z"
 have 6: "Z \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X"
-  by (metis 3 sup_set_eq)
+  by (metis 3)
 have 7: "\<And>X3. sup Y Z = X \<or> X \<subseteq> sup X3 Z \<or> \<not> Y \<subseteq> sup X3 Z"
-  by (metis 5 sup_set_eq Un_upper2 sup_set_eq)
+  by (metis 5 Un_upper2)
 have 8: "Y \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X"
-  by (metis 2 sup_set_eq Un_upper2 sup_set_eq sup_set_eq)
+  by (metis 2 Un_upper2)
 have 9: "Z \<subseteq> x \<or> sup Y Z \<noteq> X"
-  by (metis 6 Un_upper2 sup_set_eq sup_set_eq Un_upper1 sup_set_eq sup_set_eq)
+  by (metis 6 Un_upper2 Un_upper1)
 have 10: "sup Y Z = X \<or> \<not> sup Y Z \<subseteq> X"
-  by (metis equalityI 7 Un_upper1 sup_set_eq)
+  by (metis equalityI 7 Un_upper1)
 have 11: "sup Y Z = X"
-  by (metis 10 Un_least sup_set_eq sup_set_eq 1 sup_set_eq sup_set_eq 0 sup_set_eq)
+  by (metis 10 Un_least 1 0)
 have 12: "Z \<subseteq> x"
   by (metis 9 11)
 have 13: "X \<subseteq> x"
-  by (metis Un_least sup_set_eq 11 12 8 Un_upper1 sup_set_eq sup_set_eq 11)
-show "False"
-  by (metis 13 4 sup_set_eq Un_upper2 sup_set_eq sup_set_eq Un_upper1 sup_set_eq sup_set_eq 11)
+  by (metis Un_least 11 12 8 Un_upper1 11)
+show "False"
+  by (metis 13 4 Un_upper2 Un_upper1 11)
 qed
 
 (*Example included in TPHOLs paper*)
 
 declare [[sledgehammer_modulus = 4]]

 assume 2: "(\<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X \<or> Y \<subseteq> x) \<or> X \<noteq> Y \<union> Z"
 assume 3: "(\<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X \<or> Z \<subseteq> x) \<or> X \<noteq> Y \<union> Z"
 assume 4: "(\<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X \<or> \<not> X \<subseteq> x) \<or> X \<noteq> Y \<union> Z"
 assume 5: "\<And>V. ((\<not> Y \<subseteq> V \<or> \<not> Z \<subseteq> V) \<or> X \<subseteq> V) \<or> X = Y \<union> Z"
 have 6: "sup Y Z \<noteq> X \<or> \<not> X \<subseteq> x \<or> \<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X"
-  by (metis 4 sup_set_eq)
+  by (metis 4)
 have 7: "Z \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X"
-  by (metis 3 sup_set_eq Un_upper2 sup_set_eq sup_set_eq)
+  by (metis 3 Un_upper2)
 have 8: "Z \<subseteq> x \<or> sup Y Z \<noteq> X"
-  by (metis 7 Un_upper1 sup_set_eq sup_set_eq)
+  by (metis 7 Un_upper1)
 have 9: "sup Y Z = X \<or> \<not> Z \<subseteq> X \<or> \<not> Y \<subseteq> X"
-  by (metis equalityI 5 sup_set_eq Un_upper2 sup_set_eq Un_upper1 sup_set_eq Un_least sup_set_eq)
+  by (metis equalityI 5 Un_upper2 Un_upper1 Un_least)
 have 10: "Y \<subseteq> x"
-  by (metis 2 sup_set_eq Un_upper2 sup_set_eq 1 sup_set_eq Un_upper1 sup_set_eq 0 sup_set_eq 9 Un_upper2 sup_set_eq 1 sup_set_eq Un_upper1 sup_set_eq 0 sup_set_eq)
+  by (metis 2 Un_upper2 1 Un_upper1 0 9 Un_upper2 1 Un_upper1 0)
 have 11: "X \<subseteq> x"
-  by (metis Un_least sup_set_eq 9 Un_upper2 sup_set_eq 1 sup_set_eq Un_upper1 sup_set_eq 0 sup_set_eq 8 9 Un_upper2 sup_set_eq 1 sup_set_eq Un_upper1 sup_set_eq 0 sup_set_eq 10)
-show "False"
-  by (metis 11 6 Un_upper2 sup_set_eq 1 sup_set_eq Un_upper1 sup_set_eq 0 sup_set_eq 9 Un_upper2 sup_set_eq 1 sup_set_eq Un_upper1 sup_set_eq 0 sup_set_eq)
+  by (metis Un_least 9 Un_upper2 1 Un_upper1 0 8 9 Un_upper2 1 Un_upper1 0 10)
+show "False"
+  by (metis 11 6 Un_upper2 1 Un_upper1 0 9 Un_upper2 1 Un_upper1 0)
 qed 
 
 ML {*AtpWrapper.problem_name := "set__equal_union"*}
 lemma (*equal_union: *)
    "(X = Y \<union> Z) =

 by (metis Set.subsetI Union_upper insertCI set_eq_subset)
   --{*found by SPASS*}
 
 lemma (*singleton_example_2:*)
      "\<forall>x \<in> S. \<Union>S \<subseteq> x \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
-by (metis Set.subsetI Union_upper insert_code mem_def set_eq_subset)
+by (metis Set.subsetI Union_upper insert_iff set_eq_subset)
 
 lemma singleton_example_2:
      "\<forall>x \<in> S. \<Union>S \<subseteq> x \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
 proof (neg_clausify)
 assume 0: "\<And>x. \<not> S \<subseteq> {x}"