author wenzelm Mon, 13 Jun 2016 22:42:38 +0200 changeset 63297 ce995deef4b0 parent 63296 3951a15a05d1 child 63298 70d144cead25 child 63299 71805faedeb2 child 63301 d3c87eb0bad2
tuned;
```--- a/src/HOL/Isar_Examples/Schroeder_Bernstein.thy	Mon Jun 13 17:39:52 2016 +0200
+++ b/src/HOL/Isar_Examples/Schroeder_Bernstein.thy	Mon Jun 13 22:42:38 2016 +0200
@@ -15,15 +15,12 @@
\<^item> Springer LNCS 828 (cover page)
\<close>

-theorem Schroeder_Bernstein:
-  fixes f :: \<open>'a \<Rightarrow> 'b\<close>
-    and g :: \<open>'b \<Rightarrow> 'a\<close>
-  assumes \<open>inj f\<close> and \<open>inj g\<close>
-  shows \<open>\<exists>h :: 'a \<Rightarrow> 'b. inj h \<and> surj h\<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 ?h = \<open>\<lambda>z. if z \<in> A then f z else g' z\<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)
@@ -44,12 +41,12 @@
then show \<open>- A \<subseteq> range g\<close> by (simp only: A_compl)
qed
moreover
-    have False 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
+    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 False 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)```