src/HOL/ex/set.thy
 changeset 13107 8743cc847224 parent 13058 ad6106d7b4bb child 14353 79f9fbef9106
```--- a/src/HOL/ex/set.thy	Tue May 07 14:27:07 2002 +0200
+++ b/src/HOL/ex/set.thy	Tue May 07 14:27:39 2002 +0200
@@ -2,176 +2,190 @@
ID:         \$Id\$
Author:     Tobias Nipkow and Lawrence C Paulson
+*)

-Set Theory examples: Cantor's Theorem, Schroeder-Berstein Theorem, etc.
-*)
+header {* Set Theory examples: Cantor's Theorem, Schr—der-Berstein Theorem, etc. *}

theory set = Main:

-text{*These two are cited in Benzmueller and Kohlhase's system description
-of LEO, CADE-15, 1998 (pages 139-143) as theorems LEO could not prove.*}
+text{*
+  These two are cited in Benzmueller and Kohlhase's system description
+  of LEO, CADE-15, 1998 (pages 139-143) as theorems LEO could not
+  prove.
+*}

-lemma "(X = Y Un Z) = (Y<=X & Z<=X & (ALL V. Y<=V & Z<=V --> X<=V))"
-by blast
-
-lemma "(X = Y Int Z) = (X<=Y & X<=Z & (ALL V. V<=Y & V<=Z --> V<=X))"
-by blast
+lemma "(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))"
+  by blast

-text{*trivial example of term synthesis: apparently hard for some provers!*}
-lemma "a ~= b ==> a:?X & b ~: ?X"
-by blast
+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 blast

-(** Examples for the Blast_tac paper **)
+text {*
+  Trivial example of term synthesis: apparently hard for some provers!
+*}

-text{*Union-image, called Un_Union_image on equalities.ML*}
-lemma "(UN x:C. f(x) Un g(x)) = Union(f`C)  Un  Union(g`C)"
-by blast
+lemma "a \<noteq> b \<Longrightarrow> a \<in> ?X \<and> b \<notin> ?X"
+  by blast
+
+
+subsection {* Examples for the @{text blast} paper *}

-text{*Inter-image, called Int_Inter_image on equalities.ML*}
-lemma "(INT x:C. f(x) Int g(x)) = Inter(f`C) Int Inter(g`C)"
-by blast
+lemma "(\<Union>x \<in> C. f x \<union> g x) = \<Union>(f ` C)  \<union>  \<Union>(g ` C)"
+  -- {* Union-image, called @{text Un_Union_image} in Main HOL *}
+  by blast

-text{*Singleton I.  Nice demonstration of blast_tac--and its limitations.
-For some unfathomable reason, UNIV_I increases the search space greatly*}
-lemma "!!S::'a set set. ALL x:S. ALL y:S. x<=y ==> EX z. S <= {z}"
-by (blast del: UNIV_I)
+lemma "(\<Inter>x \<in> C. f x \<inter> g x) = \<Inter>(f ` C) \<inter> \<Inter>(g ` C)"
+  -- {* Inter-image, called @{text Int_Inter_image} in Main HOL *}
+  by blast

-text{*Singleton II.  variant of the benchmark above*}
-lemma "ALL x:S. Union(S) <= x ==> EX z. S <= {z}"
-by (blast del: UNIV_I)
-
-text{* A unique fixpoint theorem --- fast/best/meson all fail *}
+lemma "\<And>S::'a set set. \<forall>x \<in> S. \<forall>y \<in> S. x \<subseteq> y \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
+  -- {* Singleton I.  Nice demonstration of @{text blast}--and its limitations. *}
+  -- {* For some unfathomable reason, @{text UNIV_I} increases the search space greatly. *}
+  by (blast del: UNIV_I)

-lemma "EX! x. f(g(x))=x ==> EX! y. g(f(y))=y"
-apply (erule ex1E, rule ex1I, erule arg_cong)
-apply (rule subst, assumption, erule allE, rule arg_cong, erule mp)
-apply (erule arg_cong)
-done
+lemma "\<forall>x \<in> S. \<Union>S \<subseteq> x \<Longrightarrow> \<exists>z. S \<subseteq> {z}"
+  -- {*Singleton II.  Variant of the benchmark above. *}
+  by (blast del: UNIV_I)
+
+lemma "\<exists>!x. f (g x) = x \<Longrightarrow> \<exists>!y. g (f y) = y"
+  -- {* A unique fixpoint theorem --- @{text fast}/@{text best}/@{text meson} all fail. *}
+  apply (erule ex1E, rule ex1I, erule arg_cong)
+  apply (rule subst, assumption, erule allE, rule arg_cong, erule mp)
+  apply (erule arg_cong)
+  done

-text{* Cantor's Theorem: There is no surjection from a set to its powerset. *}
+subsection {* Cantor's Theorem: There is no surjection from a set to its powerset *}

-text{*requires best-first search because it is undirectional*}
-lemma cantor1: "~ (EX f:: 'a=>'a set. ALL S. EX x. f(x) = S)"
-by best
+lemma cantor1: "\<not> (\<exists>f:: 'a \<Rightarrow> 'a set. \<forall>S. \<exists>x. f x = S)"
+  -- {* Requires best-first search because it is undirectional. *}
+  by best

-text{*This form displays the diagonal term*}
-lemma "ALL f:: 'a=>'a set. ALL x. f(x) ~= ?S(f)"
-by best
+lemma "\<forall>f:: 'a \<Rightarrow> 'a set. \<forall>x. f x \<noteq> ?S f"
+  -- {*This form displays the diagonal term. *}
+  by best

-text{*This form exploits the set constructs*}
-lemma "?S ~: range(f :: 'a=>'a set)"
-by (rule notI, erule rangeE, best)
+lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
+  -- {* This form exploits the set constructs. *}
+  by (rule notI, erule rangeE, best)

-text{*Or just this!*}
-lemma "?S ~: range(f :: 'a=>'a set)"
-by best
+lemma "?S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
+  -- {* Or just this! *}
+  by best
+

-text{* The Schroeder-Berstein Theorem *}
+subsection {* The Schr—der-Berstein Theorem *}

-lemma disj_lemma: "[| -(f`X) = g`(-X);  f(a)=g(b);  a:X |] ==> b:X"
-by blast
+lemma disj_lemma: "- (f ` X) = g ` (-X) \<Longrightarrow> f a = g b \<Longrightarrow> a \<in> X \<Longrightarrow> b \<in> X"
+  by blast

lemma surj_if_then_else:
-     "-(f`X) = g`(-X) ==> surj(%z. if z:X then f(z) else g(z))"
+  "-(f ` X) = g ` (-X) \<Longrightarrow> surj (\<lambda>z. if z \<in> X then f z else g z)"
+  by (simp add: surj_def) blast

-lemma bij_if_then_else:
-     "[| inj_on f X;  inj_on g (-X);  -(f`X) = g`(-X);
-         h = (%z. if z:X then f(z) else g(z)) |]
-      ==> inj(h) & surj(h)"
-apply (unfold inj_on_def)
-apply (blast dest: disj_lemma sym)
-done
+lemma bij_if_then_else:
+  "inj_on f X \<Longrightarrow> inj_on g (-X) \<Longrightarrow> -(f ` X) = g ` (-X) \<Longrightarrow>
+    h = (\<lambda>z. if z \<in> X then f z else g z) \<Longrightarrow> inj h \<and> surj h"
+  apply (unfold inj_on_def)
+  apply (blast dest: disj_lemma sym)
+  done

-lemma decomposition: "EX X. X = - (g`(- (f`X)))"
-apply (rule exI)
-apply (rule lfp_unfold)
-apply (rule monoI, blast)
-done
+lemma decomposition: "\<exists>X. X = - (g ` (- (f ` X)))"
+  apply (rule exI)
+  apply (rule lfp_unfold)
+  apply (rule monoI, blast)
+  done

-text{*Schroeder-Bernstein Theorem*}
-lemma "[| inj (f:: 'a=>'b);  inj (g:: 'b=>'a) |]
-       ==> EX h:: 'a=>'b. inj(h) & surj(h)"
-apply (rule decomposition [THEN exE])
-apply (rule exI)
-apply (rule bij_if_then_else)
-   apply (rule_tac  refl)
-  apply (rule_tac  inj_on_inv)
-  apply (erule subset_inj_on [OF subset_UNIV])
-  txt{*tricky variable instantiations!*}
- apply (erule ssubst, subst double_complement)
- apply (rule subsetI, erule imageE, erule ssubst, rule rangeI)
-apply (erule ssubst, subst double_complement, erule inv_image_comp [symmetric])
-done
+theorem Schroeder_Bernstein:
+  "inj (f :: 'a \<Rightarrow> 'b) \<Longrightarrow> inj (g :: 'b \<Rightarrow> 'a)
+    \<Longrightarrow> \<exists>h:: 'a \<Rightarrow> 'b. inj h \<and> surj h"
+  apply (rule decomposition [THEN exE])
+  apply (rule exI)
+  apply (rule bij_if_then_else)
+     apply (rule_tac  refl)
+    apply (rule_tac  inj_on_inv)
+    apply (erule subset_inj_on [OF subset_UNIV])
+   txt {* Tricky variable instantiations! *}
+   apply (erule ssubst, subst double_complement)
+   apply (rule subsetI, erule imageE, erule ssubst, rule rangeI)
+  apply (erule ssubst, subst double_complement, erule inv_image_comp [symmetric])
+  done

-text{*Set variable instantiation examples from
-W. W. Bledsoe and Guohui Feng, SET-VAR.
-JAR 11 (3), 1993, pages 293-314.
+subsection {* Set variable instantiation examples *}

-Isabelle can prove the easy examples without any special mechanisms, but it
-can't prove the hard ones.
+text {*
+  From W. W. Bledsoe and Guohui Feng, SET-VAR. JAR 11 (3), 1993, pages
+  293-314.
+
+  Isabelle can prove the easy examples without any special mechanisms,
+  but it can't prove the hard ones.
*}

-text{*Example 1, page 295.*}
-lemma "(EX A. (ALL x:A. x <= (0::int)))"
-by force
+lemma "\<exists>A. (\<forall>x \<in> A. x \<le> (0::int))"
+  -- {* Example 1, page 295. *}
+  by force

-text{*Example 2*}
-lemma "D : F --> (EX G. (ALL A:G. EX B:F. A <= B))";
-by force
+lemma "D \<in> F \<Longrightarrow> \<exists>G. \<forall>A \<in> G. \<exists>B \<in> F. A \<subseteq> B"
+  -- {* Example 2. *}
+  by force

-text{*Example 3*}
-lemma "P(a) --> (EX A. (ALL x:A. P(x)) & (EX y. y:A))";
-by force
+lemma "P a \<Longrightarrow> \<exists>A. (\<forall>x \<in> A. P x) \<and> (\<exists>y. y \<in> A)"
+  -- {* Example 3. *}
+  by force

-text{*Example 4*}
-lemma "a<b & b<(c::int) --> (EX A. a~:A & b:A & c~: A)"
-by force
+lemma "a < b \<and> b < (c::int) \<Longrightarrow> \<exists>A. a \<notin> A \<and> b \<in> A \<and> c \<notin> A"
+  -- {* Example 4. *}
+  by force

-text{*Example 5, page 298.*}
-lemma "P(f(b)) --> (EX s A. (ALL x:A. P(x)) & f(s):A)";
-by force
+lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
+  -- {*Example 5, page 298. *}
+  by force

-text{*Example 6*}
-lemma "P(f(b)) --> (EX s A. (ALL x:A. P(x)) & f(s):A)";
-by force
+lemma "P (f b) \<Longrightarrow> \<exists>s A. (\<forall>x \<in> A. P x) \<and> f s \<in> A"
+  -- {* Example 6. *}
+  by force

-text{*Example 7*}
-lemma "EX A. a ~: A"
-by force
+lemma "\<exists>A. a \<notin> A"
+  -- {* Example 7. *}
+  by force

-text{*Example 8*}
-lemma "(ALL u v. u < (0::int) --> u ~= abs v) --> (EX A::int set. (ALL y. abs y ~: A) & -2 : A)"
-by force  text{*not blast, which can't simplify -2<0*}
+lemma "(\<forall>u v. u < (0::int) \<longrightarrow> u \<noteq> abs v)
+    \<longrightarrow> (\<exists>A::int set. (\<forall>y. abs y \<notin> A) \<and> -2 \<in> A)"
+  -- {* Example 8. *}
+  by force  -- {* not @{text blast}, which can't simplify @{text "-2 < 0"} *}

-text{*Example 9 omitted (requires the reals)*}
+text {* Example 9 omitted (requires the reals). *}

-text{*The paper has no Example 10!*}
+text {* The paper has no Example 10! *}

-text{*Example 11: needs a hint*}
-lemma "(ALL A. 0:A & (ALL x:A. Suc(x):A) --> n:A) &
-       P(0) & (ALL x. P(x) --> P(Suc(x))) --> P(n)"
-apply clarify
-apply (drule_tac x="{x. P x}" in spec)
-by force
+lemma "(\<forall>A. 0 \<in> A \<and> (\<forall>x \<in> A. Suc x \<in> A) \<longrightarrow> n \<in> A) \<and>
+  P 0 \<and> (\<forall>x. P x \<longrightarrow> P (Suc x)) \<longrightarrow> P n"
+  -- {* Example 11: needs a hint. *}
+  apply clarify
+  apply (drule_tac x = "{x. P x}" in spec)
+  apply force
+  done

-text{*Example 12*}
-lemma "(ALL A. (0,0):A & (ALL x y. (x,y):A --> (Suc(x),Suc(y)):A) --> (n,m):A)
-       & P(n) --> P(m)"
-by auto
+lemma
+  "(\<forall>A. (0, 0) \<in> A \<and> (\<forall>x y. (x, y) \<in> A \<longrightarrow> (Suc x, Suc y) \<in> A) \<longrightarrow> (n, m) \<in> A)
+    \<and> P n \<longrightarrow> P m"
+  -- {* Example 12. *}
+  by auto

-text{*Example EO1: typo in article, and with the obvious fix it seems
-      to require arithmetic reasoning.*}
-lemma "(ALL x. (EX u. x=2*u) = (~(EX v. Suc x = 2*v))) -->
-       (EX A. ALL x. (x : A) = (Suc x ~: A))"
-apply clarify
-apply (rule_tac x="{x. EX u. x = 2*u}" in exI, auto)
-apply (case_tac v, auto)
-apply (drule_tac x="Suc v" and P="%x. ?a(x) ~= ?b(x)" in spec, force)
-done
+lemma
+  "(\<forall>x. (\<exists>u. x = 2 * u) = (\<not> (\<exists>v. Suc x = 2 * v))) \<longrightarrow>
+    (\<exists>A. \<forall>x. (x \<in> A) = (Suc x \<notin> A))"
+  -- {* Example EO1: typo in article, and with the obvious fix it seems
+      to require arithmetic reasoning. *}
+  apply clarify
+  apply (rule_tac x = "{x. \<exists>u. x = 2 * u}" in exI, auto)
+   apply (case_tac v, auto)
+  apply (drule_tac x = "Suc v" and P = "\<lambda>x. ?a x \<noteq> ?b x" in spec, force)
+  done

end```