--- a/src/HOLCF/Cfun.thy Fri May 08 19:28:11 2009 +0100
+++ b/src/HOLCF/Cfun.thy Fri May 08 16:19:51 2009 -0700
@@ -105,19 +105,19 @@
by (rule typedef_finite_po [OF type_definition_CFun])
instance "->" :: (finite_po, chfin) chfin
-by (rule typedef_chfin [OF type_definition_CFun less_CFun_def])
+by (rule typedef_chfin [OF type_definition_CFun below_CFun_def])
instance "->" :: (cpo, discrete_cpo) discrete_cpo
-by intro_classes (simp add: less_CFun_def Rep_CFun_inject)
+by intro_classes (simp add: below_CFun_def Rep_CFun_inject)
instance "->" :: (cpo, pcpo) pcpo
-by (rule typedef_pcpo [OF type_definition_CFun less_CFun_def UU_CFun])
+by (rule typedef_pcpo [OF type_definition_CFun below_CFun_def UU_CFun])
lemmas Rep_CFun_strict =
- typedef_Rep_strict [OF type_definition_CFun less_CFun_def UU_CFun]
+ typedef_Rep_strict [OF type_definition_CFun below_CFun_def UU_CFun]
lemmas Abs_CFun_strict =
- typedef_Abs_strict [OF type_definition_CFun less_CFun_def UU_CFun]
+ typedef_Abs_strict [OF type_definition_CFun below_CFun_def UU_CFun]
text {* function application is strict in its first argument *}
@@ -153,11 +153,11 @@
text {* Extensionality wrt. ordering for continuous functions *}
-lemma expand_cfun_less: "f \<sqsubseteq> g = (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)"
-by (simp add: less_CFun_def expand_fun_less)
+lemma expand_cfun_below: "f \<sqsubseteq> g = (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)"
+by (simp add: below_CFun_def expand_fun_below)
-lemma less_cfun_ext: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g"
-by (simp add: expand_cfun_less)
+lemma below_cfun_ext: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g"
+by (simp add: expand_cfun_below)
text {* Congruence for continuous function application *}
@@ -205,13 +205,13 @@
text {* monotonicity of application *}
lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x"
-by (simp add: expand_cfun_less)
+by (simp add: expand_cfun_below)
lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y"
by (rule monofun_Rep_CFun2 [THEN monofunE])
lemma monofun_cfun: "\<lbrakk>f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>y"
-by (rule trans_less [OF monofun_cfun_fun monofun_cfun_arg])
+by (rule below_trans [OF monofun_cfun_fun monofun_cfun_arg])
text {* ch2ch - rules for the type @{typ "'a -> 'b"} *}
@@ -230,7 +230,7 @@
lemma ch2ch_LAM [simp]:
"\<lbrakk>\<And>x. chain (\<lambda>i. S i x); \<And>i. cont (\<lambda>x. S i x)\<rbrakk> \<Longrightarrow> chain (\<lambda>i. \<Lambda> x. S i x)"
-by (simp add: chain_def expand_cfun_less)
+by (simp add: chain_def expand_cfun_below)
text {* contlub, cont properties of @{term Rep_CFun} in both arguments *}
@@ -316,7 +316,7 @@
lemma cont2mono_LAM:
"\<lbrakk>\<And>x. cont (\<lambda>y. f x y); \<And>y. monofun (\<lambda>x. f x y)\<rbrakk>
\<Longrightarrow> monofun (\<lambda>x. \<Lambda> y. f x y)"
- unfolding monofun_def expand_cfun_less by simp
+ unfolding monofun_def expand_cfun_below by simp
text {* cont2cont Lemma for @{term "%x. LAM y. f x y"} *}
@@ -365,7 +365,7 @@
lemma semi_monofun_Abs_CFun:
"\<lbrakk>cont f; cont g; f \<sqsubseteq> g\<rbrakk> \<Longrightarrow> Abs_CFun f \<sqsubseteq> Abs_CFun g"
-by (simp add: less_CFun_def Abs_CFun_inverse2)
+by (simp add: below_CFun_def Abs_CFun_inverse2)
text {* some lemmata for functions with flat/chfin domain/range types *}
@@ -401,7 +401,7 @@
apply simp
done
-lemma injection_less:
+lemma injection_below:
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x \<sqsubseteq> g\<cdot>y) = (x \<sqsubseteq> y)"
apply (rule iffI)
apply (drule_tac f=f in monofun_cfun_arg)