--- a/src/HOL/Isar_Examples/Schroeder_Bernstein.thy Sat Oct 01 15:21:43 2016 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,56 +0,0 @@
-(* Title: HOL/Isar_Examples/Schroeder_Bernstein.thy
- Author: Makarius
-*)
-
-section \<open>Schröder-Bernstein Theorem\<close>
-
-theory Schroeder_Bernstein
- imports Main
-begin
-
-text \<open>
- See also:
- \<^item> \<^file>\<open>$ISABELLE_HOME/src/HOL/ex/Set_Theory.thy\<close>
- \<^item> \<^url>\<open>http://planetmath.org/proofofschroederbernsteintheoremusingtarskiknastertheorem\<close>
- \<^item> Springer LNCS 828 (cover page)
-\<close>
-
-theorem Schroeder_Bernstein: \<open>\<exists>h :: 'a \<Rightarrow> 'b. inj h \<and> surj h\<close> if \<open>inj f\<close> and \<open>inj g\<close>
- for f :: \<open>'a \<Rightarrow> 'b\<close> and g :: \<open>'b \<Rightarrow> 'a\<close>
-proof
- define A where \<open>A = lfp (\<lambda>X. - (g ` (- (f ` X))))\<close>
- define g' where \<open>g' = inv g\<close>
- let \<open>?h\<close> = \<open>\<lambda>z. if z \<in> A then f z else g' z\<close>
-
- have \<open>A = - (g ` (- (f ` A)))\<close>
- unfolding A_def by (rule lfp_unfold) (blast intro: monoI)
- then have A_compl: \<open>- A = g ` (- (f ` A))\<close> by blast
- then have *: \<open>g' ` (- A) = - (f ` A)\<close>
- using g'_def \<open>inj g\<close> by auto
-
- show \<open>inj ?h \<and> surj ?h\<close>
- proof
- from * show \<open>surj ?h\<close> by auto
- have \<open>inj_on f A\<close>
- using \<open>inj f\<close> by (rule subset_inj_on) blast
- moreover
- have \<open>inj_on g' (- A)\<close>
- unfolding g'_def
- proof (rule inj_on_inv_into)
- have \<open>g ` (- (f ` A)) \<subseteq> range g\<close> by blast
- then show \<open>- A \<subseteq> range g\<close> by (simp only: A_compl)
- qed
- moreover
- have \<open>False\<close> if eq: \<open>f a = g' b\<close> and a: \<open>a \<in> A\<close> and b: \<open>b \<in> - A\<close> for a b
- proof -
- from a have fa: \<open>f a \<in> f ` A\<close> by (rule imageI)
- from b have \<open>g' b \<in> g' ` (- A)\<close> by (rule imageI)
- with * have \<open>g' b \<in> - (f ` A)\<close> by simp
- with eq fa show \<open>False\<close> by simp
- qed
- ultimately show \<open>inj ?h\<close>
- unfolding inj_on_def by (metis ComplI)
- qed
-qed
-
-end