--- a/src/HOLCF/Cprod.thy Mon Mar 22 23:33:23 2010 -0700
+++ b/src/HOLCF/Cprod.thy Mon Mar 22 23:34:23 2010 -0700
@@ -22,150 +22,22 @@
lemma unit_when [simp]: "unit_when\<cdot>a\<cdot>u = a"
by (simp add: unit_when_def)
-subsection {* Continuous versions of constants *}
-
-definition
- cpair :: "'a \<rightarrow> 'b \<rightarrow> ('a * 'b)" -- {* continuous pairing *} where
- "cpair = (\<Lambda> x y. (x, y))"
-
-definition
- cfst :: "('a * 'b) \<rightarrow> 'a" where
- "cfst = (\<Lambda> p. fst p)"
-
-definition
- csnd :: "('a * 'b) \<rightarrow> 'b" where
- "csnd = (\<Lambda> p. snd p)"
+subsection {* Continuous version of split function *}
definition
csplit :: "('a \<rightarrow> 'b \<rightarrow> 'c) \<rightarrow> ('a * 'b) \<rightarrow> 'c" where
"csplit = (\<Lambda> f p. f\<cdot>(fst p)\<cdot>(snd p))"
-syntax
- "_ctuple" :: "['a, args] \<Rightarrow> 'a * 'b" ("(1<_,/ _>)")
-
-syntax (xsymbols)
- "_ctuple" :: "['a, args] \<Rightarrow> 'a * 'b" ("(1\<langle>_,/ _\<rangle>)")
-
translations
- "\<langle>x, y, z\<rangle>" == "\<langle>x, \<langle>y, z\<rangle>\<rangle>"
- "\<langle>x, y\<rangle>" == "CONST cpair\<cdot>x\<cdot>y"
-
-translations
- "\<Lambda>(CONST cpair\<cdot>x\<cdot>y). t" == "CONST csplit\<cdot>(\<Lambda> x y. t)"
- "\<Lambda>(CONST Pair x y). t" => "CONST csplit\<cdot>(\<Lambda> x y. t)"
+ "\<Lambda>(CONST Pair x y). t" == "CONST csplit\<cdot>(\<Lambda> x y. t)"
subsection {* Convert all lemmas to the continuous versions *}
-lemma cpair_eq_pair: "<x, y> = (x, y)"
-by (simp add: cpair_def cont_pair1 cont_pair2)
-
-lemma pair_eq_cpair: "(x, y) = <x, y>"
-by (simp add: cpair_def cont_pair1 cont_pair2)
-
-lemma inject_cpair: "<a,b> = <aa,ba> \<Longrightarrow> a = aa \<and> b = ba"
-by (simp add: cpair_eq_pair)
-
-lemma cpair_eq [iff]: "(<a, b> = <a', b'>) = (a = a' \<and> b = b')"
-by (simp add: cpair_eq_pair)
-
-lemma cpair_below [iff]: "(<a, b> \<sqsubseteq> <a', b'>) = (a \<sqsubseteq> a' \<and> b \<sqsubseteq> b')"
-by (simp add: cpair_eq_pair)
-
-lemma cpair_defined_iff [iff]: "(<x, y> = \<bottom>) = (x = \<bottom> \<and> y = \<bottom>)"
-by (simp add: cpair_eq_pair)
-
-lemma cpair_strict [simp]: "\<langle>\<bottom>, \<bottom>\<rangle> = \<bottom>"
-by simp
-
-lemma inst_cprod_pcpo2: "\<bottom> = <\<bottom>, \<bottom>>"
-by (rule cpair_strict [symmetric])
-
-lemma defined_cpair_rev:
- "<a,b> = \<bottom> \<Longrightarrow> a = \<bottom> \<and> b = \<bottom>"
-by simp
-
-lemma Exh_Cprod2: "\<exists>a b. z = <a, b>"
-by (simp add: cpair_eq_pair)
-
-lemma cprodE: "\<lbrakk>\<And>x y. p = <x, y> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
-by (cut_tac Exh_Cprod2, auto)
-
-lemma cfst_cpair [simp]: "cfst\<cdot><x, y> = x"
-by (simp add: cpair_eq_pair cfst_def)
-
-lemma csnd_cpair [simp]: "csnd\<cdot><x, y> = y"
-by (simp add: cpair_eq_pair csnd_def)
-
-lemma cfst_strict [simp]: "cfst\<cdot>\<bottom> = \<bottom>"
-by (simp add: cfst_def)
-
-lemma csnd_strict [simp]: "csnd\<cdot>\<bottom> = \<bottom>"
-by (simp add: csnd_def)
-
-lemma cpair_cfst_csnd: "\<langle>cfst\<cdot>p, csnd\<cdot>p\<rangle> = p"
-by (cases p rule: cprodE, simp)
-
-lemmas surjective_pairing_Cprod2 = cpair_cfst_csnd
-
-lemma below_cprod: "x \<sqsubseteq> y = (cfst\<cdot>x \<sqsubseteq> cfst\<cdot>y \<and> csnd\<cdot>x \<sqsubseteq> csnd\<cdot>y)"
-by (simp add: below_prod_def cfst_def csnd_def)
-
-lemma eq_cprod: "(x = y) = (cfst\<cdot>x = cfst\<cdot>y \<and> csnd\<cdot>x = csnd\<cdot>y)"
-by (auto simp add: po_eq_conv below_cprod)
-
-lemma cfst_below_iff: "cfst\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> <y, csnd\<cdot>x>"
-by (simp add: below_cprod)
-
-lemma csnd_below_iff: "csnd\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> <cfst\<cdot>x, y>"
-by (simp add: below_cprod)
-
-lemma compact_cfst: "compact x \<Longrightarrow> compact (cfst\<cdot>x)"
-by (rule compactI, simp add: cfst_below_iff)
-
-lemma compact_csnd: "compact x \<Longrightarrow> compact (csnd\<cdot>x)"
-by (rule compactI, simp add: csnd_below_iff)
-
-lemma compact_cpair: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact <x, y>"
-by (simp add: cpair_eq_pair)
-
-lemma compact_cpair_iff [simp]: "compact <x, y> = (compact x \<and> compact y)"
-by (simp add: cpair_eq_pair)
-
-lemma lub_cprod2:
- "chain S \<Longrightarrow> range S <<| <\<Squnion>i. cfst\<cdot>(S i), \<Squnion>i. csnd\<cdot>(S i)>"
-apply (simp add: cpair_eq_pair cfst_def csnd_def)
-apply (erule lub_cprod)
-done
-
-lemma thelub_cprod2:
- "chain S \<Longrightarrow> (\<Squnion>i. S i) = <\<Squnion>i. cfst\<cdot>(S i), \<Squnion>i. csnd\<cdot>(S i)>"
-by (rule lub_cprod2 [THEN thelubI])
-
lemma csplit1 [simp]: "csplit\<cdot>f\<cdot>\<bottom> = f\<cdot>\<bottom>\<cdot>\<bottom>"
by (simp add: csplit_def)
-lemma csplit2 [simp]: "csplit\<cdot>f\<cdot><x,y> = f\<cdot>x\<cdot>y"
-by (simp add: csplit_def cpair_def)
-
-lemma csplit3 [simp]: "csplit\<cdot>cpair\<cdot>z = z"
-by (simp add: csplit_def cpair_def)
-
lemma csplit_Pair [simp]: "csplit\<cdot>f\<cdot>(x, y) = f\<cdot>x\<cdot>y"
by (simp add: csplit_def)
-lemmas Cprod_rews = cfst_cpair csnd_cpair csplit2
-
-subsection {* Product type is a bifinite domain *}
-
-lemma approx_cpair [simp]:
- "approx i\<cdot>\<langle>x, y\<rangle> = \<langle>approx i\<cdot>x, approx i\<cdot>y\<rangle>"
-by (simp add: cpair_eq_pair)
-
-lemma cfst_approx: "cfst\<cdot>(approx i\<cdot>p) = approx i\<cdot>(cfst\<cdot>p)"
-by (cases p rule: cprodE, simp)
-
-lemma csnd_approx: "csnd\<cdot>(approx i\<cdot>p) = approx i\<cdot>(csnd\<cdot>p)"
-by (cases p rule: cprodE, simp)
-
end