--- a/src/HOL/Isar_Examples/Schroeder_Bernstein.thy Sat Jun 11 16:22:42 2016 +0200
+++ b/src/HOL/Isar_Examples/Schroeder_Bernstein.thy Sat Jun 11 20:54:31 2016 +0200
@@ -16,42 +16,42 @@
\<close>
theorem Schroeder_Bernstein:
- fixes f :: "'a \<Rightarrow> 'b"
- and g :: "'b \<Rightarrow> 'a"
- assumes "inj f" and "inj g"
- shows "\<exists>h :: 'a \<Rightarrow> 'b. inj h \<and> surj h"
+ 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>
proof
- define A where "A = lfp (\<lambda>X. - (g ` (- (f ` X))))"
- define g' where "g' = inv g"
- let ?h = "\<lambda>z. if z \<in> A then f z else g' z"
+ 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>
- have "A = - (g ` (- (f ` A)))"
+ have \<open>A = - (g ` (- (f ` A)))\<close>
unfolding A_def by (rule lfp_unfold) (blast intro: monoI)
- then have A_compl: "- A = g ` (- (f ` A))" by blast
- then have *: "g' ` (- A) = - (f ` A)"
+ 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 "inj ?h \<and> surj ?h"
+ show \<open>inj ?h \<and> surj ?h\<close>
proof
- from * show "surj ?h" by auto
- have "inj_on f A"
+ 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 "inj_on g' (- A)"
+ have \<open>inj_on g' (- A)\<close>
unfolding g'_def
proof (rule inj_on_inv_into)
- have "g ` (- (f ` A)) \<subseteq> range g" by blast
- then show "- A \<subseteq> range g" by (simp only: A_compl)
+ 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 False if eq: "f a = g' b" and a: "a \<in> A" and b: "b \<in> - A" for a b
+ 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
proof -
- from a have fa: "f a \<in> f ` A" by (rule imageI)
- from b have "g' b \<in> g' ` (- A)" by (rule imageI)
- with * have "g' b \<in> - (f ` A)" by simp
+ 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
qed
- ultimately show "inj ?h"
+ ultimately show \<open>inj ?h\<close>
unfolding inj_on_def by (metis ComplI)
qed
qed