--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Metis_Examples/set.thy Tue Oct 20 19:52:04 2009 +0200
@@ -0,0 +1,283 @@
+(* Title: HOL/Metis_Examples/set.thy
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+
+Testing the metis method.
+*)
+
+theory set
+imports Main
+begin
+
+lemma "EX x X. ALL y. EX z Z. (~P(y,y) | P(x,x) | ~S(z,x)) &
+ (S(x,y) | ~S(y,z) | Q(Z,Z)) &
+ (Q(X,y) | ~Q(y,Z) | S(X,X))"
+by metis
+(*??But metis can't prove the single-step version...*)
+
+
+
+lemma "P(n::nat) ==> ~P(0) ==> n ~= 0"
+by metis
+
+declare [[sledgehammer_modulus = 1]]
+
+
+(*multiple versions of this example*)
+lemma (*equal_union: *)
+ "(X = Y \<union> Z) =
+ (Y \<subseteq> X \<and> Z \<subseteq> X \<and> (\<forall>V. Y \<subseteq> V \<and> Z \<subseteq> V \<longrightarrow> X \<subseteq> V))"
+proof (neg_clausify)
+fix x
+assume 0: "Y \<subseteq> X \<or> X = Y \<union> Z"
+assume 1: "Z \<subseteq> X \<or> X = Y \<union> Z"
+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)
+have 7: "sup Y Z = X \<or> Z \<subseteq> X"
+ 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)
+have 9: "Y \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X"
+ 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)
+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)
+have 12: "Z \<subseteq> X"
+ 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)
+have 14: "Y \<subseteq> X"
+ 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"
+ by (metis 11 12)
+have 18: "sup Y Z \<noteq> X \<or> \<not> X \<subseteq> x"
+ by (metis 17 14)
+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)
+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)
+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 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"
+ by (metis 18 25)
+have 30: "X \<subseteq> x \<or> \<not> Y \<subseteq> x"
+ by (metis 26 28)
+have 31: "X \<subseteq> x"
+ by (metis 30 27)
+show "False"
+ by (metis 31 29)
+qed
+
+declare [[sledgehammer_modulus = 2]]
+
+lemma (*equal_union: *)
+ "(X = Y \<union> Z) =
+ (Y \<subseteq> X \<and> Z \<subseteq> X \<and> (\<forall>V. Y \<subseteq> V \<and> Z \<subseteq> V \<longrightarrow> X \<subseteq> V))"
+proof (neg_clausify)
+fix x
+assume 0: "Y \<subseteq> X \<or> X = Y \<union> Z"
+assume 1: "Z \<subseteq> X \<or> X = Y \<union> Z"
+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)
+have 7: "Y \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X \<or> \<not> Z \<subseteq> X"
+ 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)
+have 9: "\<And>X3. sup Y Z = X \<or> X \<subseteq> sup X3 Z \<or> \<not> Y \<subseteq> sup X3 Z"
+ by (metis 5 Un_upper2)
+have 10: "Z \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X"
+ 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)
+have 12: "Z \<subseteq> x \<or> sup Y Z \<noteq> X"
+ by (metis 10 Un_upper1)
+have 13: "sup Y Z = X \<or> X \<subseteq> sup Y Z"
+ 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)
+have 15: "sup Y Z = X"
+ by (metis 14 1 6)
+have 16: "Y \<subseteq> x"
+ by (metis 7 Un_upper2 Un_upper1 15)
+have 17: "\<not> X \<subseteq> x"
+ by (metis 11 Un_upper1 15)
+have 18: "X \<subseteq> x"
+ by (metis Un_least 15 12 15 16)
+show "False"
+ by (metis 18 17)
+qed
+
+declare [[sledgehammer_modulus = 3]]
+
+lemma (*equal_union: *)
+ "(X = Y \<union> Z) =
+ (Y \<subseteq> X \<and> Z \<subseteq> X \<and> (\<forall>V. Y \<subseteq> V \<and> Z \<subseteq> V \<longrightarrow> X \<subseteq> V))"
+proof (neg_clausify)
+fix x
+assume 0: "Y \<subseteq> X \<or> X = Y \<union> Z"
+assume 1: "Z \<subseteq> X \<or> X = Y \<union> Z"
+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)
+have 7: "\<And>X3. sup Y Z = X \<or> X \<subseteq> sup X3 Z \<or> \<not> Y \<subseteq> sup X3 Z"
+ by (metis 5 Un_upper2)
+have 8: "Y \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X"
+ by (metis 2 Un_upper2)
+have 9: "Z \<subseteq> x \<or> sup Y Z \<noteq> X"
+ 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)
+have 11: "sup Y Z = X"
+ 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 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]]
+
+lemma (*equal_union: *)
+ "(X = Y \<union> Z) =
+ (Y \<subseteq> X \<and> Z \<subseteq> X \<and> (\<forall>V. Y \<subseteq> V \<and> Z \<subseteq> V \<longrightarrow> X \<subseteq> V))"
+proof (neg_clausify)
+fix x
+assume 0: "Y \<subseteq> X \<or> X = Y \<union> Z"
+assume 1: "Z \<subseteq> X \<or> X = Y \<union> Z"
+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)
+have 7: "Z \<subseteq> x \<or> sup Y Z \<noteq> X \<or> \<not> Y \<subseteq> X"
+ by (metis 3 Un_upper2)
+have 8: "Z \<subseteq> x \<or> sup Y Z \<noteq> X"
+ 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 Un_upper2 Un_upper1 Un_least)
+have 10: "Y \<subseteq> x"
+ 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 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
+
+declare [[ atp_problem_prefix = "set__equal_union" ]]
+lemma (*equal_union: *)
+ "(X = Y \<union> Z) =
+ (Y \<subseteq> X \<and> Z \<subseteq> X \<and> (\<forall>V. Y \<subseteq> V \<and> Z \<subseteq> V \<longrightarrow> X \<subseteq> V))"
+(*One shot proof: hand-reduced. Metis can't do the full proof any more.*)
+by (metis Un_least Un_upper1 Un_upper2 set_eq_subset)
+
+
+declare [[ atp_problem_prefix = "set__equal_inter" ]]
+lemma "(X = Y \<inter> Z) =
+ (X \<subseteq> Y \<and> X \<subseteq> Z \<and> (\<forall>V. V \<subseteq> Y \<and> V \<subseteq> Z \<longrightarrow> V \<subseteq> X))"
+by (metis Int_greatest Int_lower1 Int_lower2 set_eq_subset)
+
+declare [[ atp_problem_prefix = "set__fixedpoint" ]]
+lemma fixedpoint:
+ "\<exists>!x. f (g x) = x \<Longrightarrow> \<exists>!y. g (f y) = y"
+by metis
+
+lemma (*fixedpoint:*)
+ "\<exists>!x. f (g x) = x \<Longrightarrow> \<exists>!y. g (f y) = y"
+proof (neg_clausify)
+fix x xa
+assume 0: "f (g x) = x"
+assume 1: "\<And>y. y = x \<or> f (g y) \<noteq> y"
+assume 2: "\<And>x. g (f (xa x)) = xa x \<or> g (f x) \<noteq> x"
+assume 3: "\<And>x. g (f x) \<noteq> x \<or> xa x \<noteq> x"
+have 4: "\<And>X1. g (f X1) \<noteq> X1 \<or> g x \<noteq> X1"
+ by (metis 3 1 2)
+show "False"
+ by (metis 4 0)
+qed
+
+declare [[ atp_problem_prefix = "set__singleton_example" ]]
+lemma (*singleton_example_2:*)
+ "\<forall>x \<in> S. \<Union>S \<subseteq> x \<Longrightarrow> \<exists>z. S \<subseteq> {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_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}"
+assume 1: "\<And>x. x \<notin> S \<or> \<Union>S \<subseteq> x"
+have 2: "\<And>X3. X3 = \<Union>S \<or> \<not> X3 \<subseteq> \<Union>S \<or> X3 \<notin> S"
+ by (metis set_eq_subset 1)
+have 3: "\<And>X3. S \<subseteq> insert (\<Union>S) X3"
+ by (metis insert_iff Set.subsetI Union_upper 2 Set.subsetI)
+show "False"
+ by (metis 3 0)
+qed
+
+
+
+text {*
+ From W. W. Bledsoe and Guohui Feng, SET-VAR. JAR 11 (3), 1993, pages
+ 293-314.
+*}
+
+declare [[ atp_problem_prefix = "set__Bledsoe_Fung" ]]
+(*Notes: 1, the numbering doesn't completely agree with the paper.
+2, we must rename set variables to avoid type clashes.*)
+lemma "\<exists>B. (\<forall>x \<in> B. x \<le> (0::int))"
+ "D \<in> F \<Longrightarrow> \<exists>G. \<forall>A \<in> G. \<exists>B \<in> F. A \<subseteq> B"
+ "P a \<Longrightarrow> \<exists>A. (\<forall>x \<in> A. P x) \<and> (\<exists>y. y \<in> A)"
+ "a < b \<and> b < (c::int) \<Longrightarrow> \<exists>B. a \<notin> B \<and> b \<in> B \<and> c \<notin> B"
+ "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
+ "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
+ "\<exists>A. a \<notin> A"
+ "(\<forall>C. (0, 0) \<in> C \<and> (\<forall>x y. (x, y) \<in> C \<longrightarrow> (Suc x, Suc y) \<in> C) \<longrightarrow> (n, m) \<in> C) \<and> Q n \<longrightarrow> Q m"
+apply (metis atMost_iff)
+apply (metis emptyE)
+apply (metis insert_iff singletonE)
+apply (metis insertCI singletonE zless_le)
+apply (metis Collect_def Collect_mem_eq)
+apply (metis Collect_def Collect_mem_eq)
+apply (metis DiffE)
+apply (metis pair_in_Id_conv)
+done
+
+end