# HG changeset patch
# User paulson
# Date 1058785327 -7200
# Node ID 3a73850c6c7d4f439932d23ff3e97a020c74b9e6
# Parent  fb9c392644a15e3ca05a5941d1b4b825785cb167
Tidied some examples

diff -r fb9c392644a1 -r 3a73850c6c7d src/ZF/OrderArith.thy
--- a/src/ZF/OrderArith.thy	Mon Jul 21 10:58:16 2003 +0200
+++ b/src/ZF/OrderArith.thy	Mon Jul 21 13:02:07 2003 +0200
@@ -543,6 +543,27 @@
 apply (frule ok, assumption+, blast) 
 done
 
+subsubsection{*Bijections involving Powersets*}
+
+lemma Pow_sum_bij:
+    "(\<lambda>Z \<in> Pow(A+B). <{x \<in> A. Inl(x) \<in> Z}, {y \<in> B. Inr(y) \<in> Z}>)  
+     \<in> bij(Pow(A+B), Pow(A)*Pow(B))"
+apply (rule_tac d = "%<X,Y>. {Inl (x). x \<in> X} Un {Inr (y). y \<in> Y}" 
+       in lam_bijective)
+apply force+
+done
+
+text{*As a special case, we have @{term "bij(Pow(A*B), A -> Pow(B))"} *}
+lemma Pow_Sigma_bij:
+    "(\<lambda>r \<in> Pow(Sigma(A,B)). \<lambda>x \<in> A. r``{x})  
+     \<in> bij(Pow(Sigma(A,B)), \<Pi>x \<in> A. Pow(B(x)))"
+apply (rule_tac d = "%f. \<Union>x \<in> A. \<Union>y \<in> f`x. {<x,y>}" in lam_bijective)
+apply (blast intro: lam_type)
+apply (blast dest: apply_type, simp_all)
+apply fast (*strange, but blast can't do it*)
+apply (rule fun_extension, auto)
+by blast
+
 
 ML {*
 val measure_def = thm "measure_def";
diff -r fb9c392644a1 -r 3a73850c6c7d src/ZF/ex/misc.thy
--- a/src/ZF/ex/misc.thy	Mon Jul 21 10:58:16 2003 +0200
+++ b/src/ZF/ex/misc.thy	Mon Jul 21 13:02:07 2003 +0200
@@ -3,32 +3,37 @@
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1993  University of Cambridge
 
-Miscellaneous examples for Zermelo-Fraenkel Set Theory 
 Composition of homomorphisms, Pastre's examples, ...
 *)
 
+header{*Miscellaneous ZF Examples*}
+
 theory misc = Main:
 
-
+subsection{*Various Small Problems*}
 
-(*These two are cited in Benzmueller and Kohlhase's system description of LEO,
-  CADE-15, 1998 (page 139-143) as theorems LEO could not prove.*)
+text{*A weird property of ordered pairs.*}
+lemma "b\<noteq>c ==> <a,b> Int <a,c> = <a,a>"
+by (simp add: Pair_def Int_cons_left Int_cons_right doubleton_eq_iff, blast)
+
+text{*These two are cited in Benzmueller and Kohlhase's system description of
+ LEO, CADE-15, 1998 (page 139-143) as theorems LEO could not prove.*}
 lemma "(X = Y Un Z) <-> (Y \<subseteq> X & Z \<subseteq> X & (\<forall>V. Y \<subseteq> V & Z \<subseteq> V --> X \<subseteq> V))"
 by (blast intro!: equalityI)
 
-(*the dual of the previous one*)
+text{*the dual of the previous one}
 lemma "(X = Y Int Z) <-> (X \<subseteq> Y & X \<subseteq> Z & (\<forall>V. V \<subseteq> Y & V \<subseteq> Z --> V \<subseteq> X))"
 by (blast intro!: equalityI)
 
-(*trivial example of term synthesis: apparently hard for some provers!*)
+text{*trivial example of term synthesis: apparently hard for some provers!}
 lemma "a \<noteq> b ==> a:?X & b \<notin> ?X"
 by blast
 
-(*Nice Blast_tac benchmark.  Proved in 0.3s; old tactics can't manage it!*)
+text{*Nice Blast_tac benchmark.  Proved in 0.3s; old tactics can't manage it!}
 lemma "\<forall>x \<in> S. \<forall>y \<in> S. x \<subseteq> y ==> \<exists>z. S \<subseteq> {z}"
 by blast
 
-(*variant of the benchmark above*)
+text{*variant of the benchmark above}
 lemma "\<forall>x \<in> S. Union(S) \<subseteq> x ==> \<exists>z. S \<subseteq> {z}"
 by blast
 
@@ -39,16 +44,19 @@
 lemma "(\<forall>F. {x} \<in> F --> {y} \<in> F) --> (\<forall>A. x \<in> A --> y \<in> A)"
 by best
 
+text{*A characterization of functions suggested by Tobias Nipkow*}
+lemma "r \<in> domain(r)->B  <->  r \<subseteq> domain(r)*B & (\<forall>X. r `` (r -`` X) \<subseteq> X)"
+by (unfold Pi_def function_def, best)
 
-(*** Composition of homomorphisms is a homomorphism ***)
 
-(*Given as a challenge problem in
+subsection{*Composition of homomorphisms is a Homomorphism*}
+
+text{*Given as a challenge problem in
   R. Boyer et al.,
   Set Theory in First-Order Logic: Clauses for G\"odel's Axioms,
-  JAR 2 (1986), 287-327 
-*)
+  JAR 2 (1986), 287-327 *}
 
-(*collecting the relevant lemmas*)
+text{*collecting the relevant lemmas}
 declare comp_fun [simp] SigmaI [simp] apply_funtype [simp]
 
 (*Force helps prove conditions of rewrites such as comp_fun_apply, since
@@ -60,7 +68,7 @@
        (K O J) \<in> hom(A,f,C,h)"
 by force
 
-(*Another version, with meta-level rewriting*)
+text{*Another version, with meta-level rewriting}
 lemma "(!! A f B g. hom(A,f,B,g) ==  
            {H \<in> A->B. f \<in> A*A->A & g \<in> B*B->B &  
                      (\<forall>x \<in> A. \<forall>y \<in> A. H`(f`<x,y>) = g`<H`x,H`y>)}) 
@@ -68,17 +76,11 @@
 by force
 
 
-
-(** A characterization of functions suggested by Tobias Nipkow **)
-
-lemma "r \<in> domain(r)->B  <->  r \<subseteq> domain(r)*B & (\<forall>X. r `` (r -`` X) \<subseteq> X)"
-by (unfold Pi_def function_def, best)
+subsection{*Pastre's Examples*}
 
-(**** From D Pastre.  Automatic theorem proving in set theory. 
-         Artificial Intelligence, 10:1--27, 1978.
-
-      Previously, these were done using ML code, but blast manages fine.
-****)
+text{*D Pastre.  Automatic theorem proving in set theory. 
+        Artificial Intelligence, 10:1--27, 1978.
+Previously, these were done using ML code, but blast manages fine.*}
 
 lemmas compIs [intro] = comp_surj comp_inj comp_fun [intro]
 lemmas compDs [dest] =  comp_mem_injD1 comp_mem_surjD1 
@@ -120,26 +122,5 @@
 by (unfold bij_def, blast)
 
 
-(** Yet another example... **)
-
-lemma Pow_sum_bij:
-    "(\<lambda>Z \<in> Pow(A+B). <{x \<in> A. Inl(x) \<in> Z}, {y \<in> B. Inr(y) \<in> Z}>)  
-     \<in> bij(Pow(A+B), Pow(A)*Pow(B))"
-apply (rule_tac d = "%<X,Y>. {Inl (x). x \<in> X} Un {Inr (y). y \<in> Y}" 
-       in lam_bijective)
-apply force+
-done
-
-(*As a special case, we have  bij(Pow(A*B), A -> Pow B)  *)
-lemma Pow_Sigma_bij:
-    "(\<lambda>r \<in> Pow(Sigma(A,B)). \<lambda>x \<in> A. r``{x})  
-     \<in> bij(Pow(Sigma(A,B)), \<Pi>x \<in> A. Pow(B(x)))"
-apply (rule_tac d = "%f. \<Union>x \<in> A. \<Union>y \<in> f`x. {<x,y>}" in lam_bijective)
-apply (blast intro: lam_type)
-apply (blast dest: apply_type, simp_all)
-apply fast (*strange, but blast can't do it*)
-apply (rule fun_extension, auto)
-by blast
-
 end