src/HOL/Isar_Examples/Schroeder_Bernstein.thy
changeset 63980 f8e556c8ad6f
parent 63979 95c3ae4baba8
child 63981 6f7db4f8df4c
     1.1 --- a/src/HOL/Isar_Examples/Schroeder_Bernstein.thy	Sat Oct 01 17:16:35 2016 +0200
     1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.3 @@ -1,56 +0,0 @@
     1.4 -(*  Title:      HOL/Isar_Examples/Schroeder_Bernstein.thy
     1.5 -    Author:     Makarius
     1.6 -*)
     1.7 -
     1.8 -section \<open>Schröder-Bernstein Theorem\<close>
     1.9 -
    1.10 -theory Schroeder_Bernstein
    1.11 -  imports Main
    1.12 -begin
    1.13 -
    1.14 -text \<open>
    1.15 -  See also:
    1.16 -  \<^item> \<^file>\<open>$ISABELLE_HOME/src/HOL/ex/Set_Theory.thy\<close>
    1.17 -  \<^item> \<^url>\<open>http://planetmath.org/proofofschroederbernsteintheoremusingtarskiknastertheorem\<close>
    1.18 -  \<^item> Springer LNCS 828 (cover page)
    1.19 -\<close>
    1.20 -
    1.21 -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>
    1.22 -  for f :: \<open>'a \<Rightarrow> 'b\<close> and g :: \<open>'b \<Rightarrow> 'a\<close>
    1.23 -proof
    1.24 -  define A where \<open>A = lfp (\<lambda>X. - (g ` (- (f ` X))))\<close>
    1.25 -  define g' where \<open>g' = inv g\<close>
    1.26 -  let \<open>?h\<close> = \<open>\<lambda>z. if z \<in> A then f z else g' z\<close>
    1.27 -
    1.28 -  have \<open>A = - (g ` (- (f ` A)))\<close>
    1.29 -    unfolding A_def by (rule lfp_unfold) (blast intro: monoI)
    1.30 -  then have A_compl: \<open>- A = g ` (- (f ` A))\<close> by blast
    1.31 -  then have *: \<open>g' ` (- A) = - (f ` A)\<close>
    1.32 -    using g'_def \<open>inj g\<close> by auto
    1.33 -
    1.34 -  show \<open>inj ?h \<and> surj ?h\<close>
    1.35 -  proof
    1.36 -    from * show \<open>surj ?h\<close> by auto
    1.37 -    have \<open>inj_on f A\<close>
    1.38 -      using \<open>inj f\<close> by (rule subset_inj_on) blast
    1.39 -    moreover
    1.40 -    have \<open>inj_on g' (- A)\<close>
    1.41 -      unfolding g'_def
    1.42 -    proof (rule inj_on_inv_into)
    1.43 -      have \<open>g ` (- (f ` A)) \<subseteq> range g\<close> by blast
    1.44 -      then show \<open>- A \<subseteq> range g\<close> by (simp only: A_compl)
    1.45 -    qed
    1.46 -    moreover
    1.47 -    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
    1.48 -    proof -
    1.49 -      from a have fa: \<open>f a \<in> f ` A\<close> by (rule imageI)
    1.50 -      from b have \<open>g' b \<in> g' ` (- A)\<close> by (rule imageI)
    1.51 -      with * have \<open>g' b \<in> - (f ` A)\<close> by simp
    1.52 -      with eq fa show \<open>False\<close> by simp
    1.53 -    qed
    1.54 -    ultimately show \<open>inj ?h\<close>
    1.55 -      unfolding inj_on_def by (metis ComplI)
    1.56 -  qed
    1.57 -qed
    1.58 -
    1.59 -end