--- a/src/HOL/HOLCF/Cont.thy Wed Jan 13 23:02:28 2016 +0100
+++ b/src/HOL/HOLCF/Cont.thy Wed Jan 13 23:07:06 2016 +0100
@@ -3,23 +3,23 @@
Author: Brian Huffman
*)
-section {* Continuity and monotonicity *}
+section \<open>Continuity and monotonicity\<close>
theory Cont
imports Pcpo
begin
-text {*
+text \<open>
Now we change the default class! Form now on all untyped type variables are
of default class po
-*}
+\<close>
default_sort po
-subsection {* Definitions *}
+subsection \<open>Definitions\<close>
definition
- monofun :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" -- "monotonicity" where
+ monofun :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" \<comment> "monotonicity" where
"monofun f = (\<forall>x y. x \<sqsubseteq> y \<longrightarrow> f x \<sqsubseteq> f y)"
definition
@@ -44,9 +44,9 @@
by (simp add: monofun_def)
-subsection {* Equivalence of alternate definition *}
+subsection \<open>Equivalence of alternate definition\<close>
-text {* monotone functions map chains to chains *}
+text \<open>monotone functions map chains to chains\<close>
lemma ch2ch_monofun: "\<lbrakk>monofun f; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. f (Y i))"
apply (rule chainI)
@@ -54,7 +54,7 @@
apply (erule chainE)
done
-text {* monotone functions map upper bound to upper bounds *}
+text \<open>monotone functions map upper bound to upper bounds\<close>
lemma ub2ub_monofun:
"\<lbrakk>monofun f; range Y <| u\<rbrakk> \<Longrightarrow> range (\<lambda>i. f (Y i)) <| f u"
@@ -63,7 +63,7 @@
apply (erule ub_rangeD)
done
-text {* a lemma about binary chains *}
+text \<open>a lemma about binary chains\<close>
lemma binchain_cont:
"\<lbrakk>cont f; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> range (\<lambda>i::nat. f (if i = 0 then x else y)) <<| f y"
@@ -75,7 +75,7 @@
apply (erule is_lub_bin_chain [THEN lub_eqI])
done
-text {* continuity implies monotonicity *}
+text \<open>continuity implies monotonicity\<close>
lemma cont2mono: "cont f \<Longrightarrow> monofun f"
apply (rule monofunI)
@@ -88,7 +88,7 @@
lemmas ch2ch_cont = cont2mono [THEN ch2ch_monofun]
-text {* continuity implies preservation of lubs *}
+text \<open>continuity implies preservation of lubs\<close>
lemma cont2contlubE:
"\<lbrakk>cont f; chain Y\<rbrakk> \<Longrightarrow> f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i))"
@@ -118,26 +118,26 @@
by (rule thelubE)
qed
-subsection {* Collection of continuity rules *}
+subsection \<open>Collection of continuity rules\<close>
named_theorems cont2cont "continuity intro rule"
-subsection {* Continuity of basic functions *}
+subsection \<open>Continuity of basic functions\<close>
-text {* The identity function is continuous *}
+text \<open>The identity function is continuous\<close>
lemma cont_id [simp, cont2cont]: "cont (\<lambda>x. x)"
apply (rule contI)
apply (erule cpo_lubI)
done
-text {* constant functions are continuous *}
+text \<open>constant functions are continuous\<close>
lemma cont_const [simp, cont2cont]: "cont (\<lambda>x. c)"
using is_lub_const by (rule contI)
-text {* application of functions is continuous *}
+text \<open>application of functions is continuous\<close>
lemma cont_apply:
fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo" and t :: "'a \<Rightarrow> 'b"
@@ -165,7 +165,7 @@
"\<lbrakk>cont c; cont (\<lambda>x. f x)\<rbrakk> \<Longrightarrow> cont (\<lambda>x. c (f x))"
by (rule cont_apply [OF _ _ cont_const])
-text {* Least upper bounds preserve continuity *}
+text \<open>Least upper bounds preserve continuity\<close>
lemma cont2cont_lub [simp]:
assumes chain: "\<And>x. chain (\<lambda>i. F i x)" and cont: "\<And>i. cont (\<lambda>x. F i x)"
@@ -176,15 +176,15 @@
apply (simp add: diag_lub ch2ch_cont [OF cont] chain)
done
-text {* if-then-else is continuous *}
+text \<open>if-then-else is continuous\<close>
lemma cont_if [simp, cont2cont]:
"\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (\<lambda>x. if b then f x else g x)"
by (induct b) simp_all
-subsection {* Finite chains and flat pcpos *}
+subsection \<open>Finite chains and flat pcpos\<close>
-text {* Monotone functions map finite chains to finite chains. *}
+text \<open>Monotone functions map finite chains to finite chains.\<close>
lemma monofun_finch2finch:
"\<lbrakk>monofun f; finite_chain Y\<rbrakk> \<Longrightarrow> finite_chain (\<lambda>n. f (Y n))"
@@ -193,13 +193,13 @@
apply (force simp add: max_in_chain_def)
done
-text {* The same holds for continuous functions. *}
+text \<open>The same holds for continuous functions.\<close>
lemma cont_finch2finch:
"\<lbrakk>cont f; finite_chain Y\<rbrakk> \<Longrightarrow> finite_chain (\<lambda>n. f (Y n))"
by (rule cont2mono [THEN monofun_finch2finch])
-text {* All monotone functions with chain-finite domain are continuous. *}
+text \<open>All monotone functions with chain-finite domain are continuous.\<close>
lemma chfindom_monofun2cont: "monofun f \<Longrightarrow> cont (f::'a::chfin \<Rightarrow> 'b::cpo)"
apply (erule contI2)
@@ -210,7 +210,7 @@
apply (force simp add: max_in_chain_def)
done
-text {* All strict functions with flat domain are continuous. *}
+text \<open>All strict functions with flat domain are continuous.\<close>
lemma flatdom_strict2mono: "f \<bottom> = \<bottom> \<Longrightarrow> monofun (f::'a::flat \<Rightarrow> 'b::pcpo)"
apply (rule monofunI)
@@ -221,7 +221,7 @@
lemma flatdom_strict2cont: "f \<bottom> = \<bottom> \<Longrightarrow> cont (f::'a::flat \<Rightarrow> 'b::pcpo)"
by (rule flatdom_strict2mono [THEN chfindom_monofun2cont])
-text {* All functions with discrete domain are continuous. *}
+text \<open>All functions with discrete domain are continuous.\<close>
lemma cont_discrete_cpo [simp, cont2cont]: "cont (f::'a::discrete_cpo \<Rightarrow> 'b::cpo)"
apply (rule contI)