--- a/src/HOL/HOLCF/Adm.thy Wed Jan 13 23:02:28 2016 +0100
+++ b/src/HOL/HOLCF/Adm.thy Wed Jan 13 23:07:06 2016 +0100
@@ -2,7 +2,7 @@
Author: Franz Regensburger and Brian Huffman
*)
-section {* Admissibility and compactness *}
+section \<open>Admissibility and compactness\<close>
theory Adm
imports Cont
@@ -10,7 +10,7 @@
default_sort cpo
-subsection {* Definitions *}
+subsection \<open>Definitions\<close>
definition
adm :: "('a::cpo \<Rightarrow> bool) \<Rightarrow> bool" where
@@ -29,14 +29,14 @@
lemma triv_admI: "\<forall>x. P x \<Longrightarrow> adm P"
by (rule admI, erule spec)
-subsection {* Admissibility on chain-finite types *}
+subsection \<open>Admissibility on chain-finite types\<close>
-text {* For chain-finite (easy) types every formula is admissible. *}
+text \<open>For chain-finite (easy) types every formula is admissible.\<close>
lemma adm_chfin [simp]: "adm (P::'a::chfin \<Rightarrow> bool)"
by (rule admI, frule chfin, auto simp add: maxinch_is_thelub)
-subsection {* Admissibility of special formulae and propagation *}
+subsection \<open>Admissibility of special formulae and propagation\<close>
lemma adm_const [simp]: "adm (\<lambda>x. t)"
by (rule admI, simp)
@@ -53,7 +53,7 @@
"(\<And>y. y \<in> A \<Longrightarrow> adm (\<lambda>x. P x y)) \<Longrightarrow> adm (\<lambda>x. \<forall>y\<in>A. P x y)"
by (fast intro: admI elim: admD)
-text {* Admissibility for disjunction is hard to prove. It requires 2 lemmas. *}
+text \<open>Admissibility for disjunction is hard to prove. It requires 2 lemmas.\<close>
lemma adm_disj_lemma1:
assumes adm: "adm P"
@@ -108,7 +108,7 @@
\<Longrightarrow> adm (\<lambda>x. P x = Q x)"
by (subst iff_conv_conj_imp, rule adm_conj)
-text {* admissibility and continuity *}
+text \<open>admissibility and continuity\<close>
lemma adm_below [simp]:
"\<lbrakk>cont (\<lambda>x. u x); cont (\<lambda>x. v x)\<rbrakk> \<Longrightarrow> adm (\<lambda>x. u x \<sqsubseteq> v x)"
@@ -124,7 +124,7 @@
lemma adm_not_below [simp]: "cont (\<lambda>x. t x) \<Longrightarrow> adm (\<lambda>x. t x \<notsqsubseteq> u)"
by (rule admI, simp add: cont2contlubE ch2ch_cont lub_below_iff)
-subsection {* Compactness *}
+subsection \<open>Compactness\<close>
definition
compact :: "'a::cpo \<Rightarrow> bool" where
@@ -161,7 +161,7 @@
apply (erule (1) below_trans [OF is_ub_thelub])
done
-text {* admissibility and compactness *}
+text \<open>admissibility and compactness\<close>
lemma adm_compact_not_below [simp]:
"\<lbrakk>compact k; cont (\<lambda>x. t x)\<rbrakk> \<Longrightarrow> adm (\<lambda>x. k \<notsqsubseteq> t x)"
@@ -178,7 +178,7 @@
lemma compact_bottom [simp, intro]: "compact \<bottom>"
by (rule compactI, simp)
-text {* Any upward-closed predicate is admissible. *}
+text \<open>Any upward-closed predicate is admissible.\<close>
lemma adm_upward:
assumes P: "\<And>x y. \<lbrakk>P x; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> P y"