(* Title: HOLCF/Cprod.thy
ID: $Id$
Author: Franz Regensburger
Partial ordering for cartesian product of HOL products.
*)
header {* The cpo of cartesian products *}
theory Cprod
imports Bifinite
begin
defaultsort cpo
subsection {* Type @{typ unit} is a pcpo *}
instantiation unit :: po
begin
definition
less_unit_def [simp]: "x \<sqsubseteq> (y::unit) \<equiv> True"
instance
by intro_classes simp_all
end
instance unit :: finite_po ..
instance unit :: pcpo
by intro_classes simp
definition
unit_when :: "'a \<rightarrow> unit \<rightarrow> 'a" where
"unit_when = (\<Lambda> a _. a)"
translations
"\<Lambda>(). t" == "CONST unit_when\<cdot>t"
lemma unit_when [simp]: "unit_when\<cdot>a\<cdot>u = a"
by (simp add: unit_when_def)
subsection {* Product type is a partial order *}
instantiation "*" :: (po, po) po
begin
definition
less_cprod_def: "(op \<sqsubseteq>) \<equiv> \<lambda>p1 p2. (fst p1 \<sqsubseteq> fst p2 \<and> snd p1 \<sqsubseteq> snd p2)"
instance
proof
fix x :: "'a \<times> 'b"
show "x \<sqsubseteq> x"
unfolding less_cprod_def by simp
next
fix x y :: "'a \<times> 'b"
assume "x \<sqsubseteq> y" "y \<sqsubseteq> x" thus "x = y"
unfolding less_cprod_def Pair_fst_snd_eq
by (fast intro: antisym_less)
next
fix x y z :: "'a \<times> 'b"
assume "x \<sqsubseteq> y" "y \<sqsubseteq> z" thus "x \<sqsubseteq> z"
unfolding less_cprod_def
by (fast intro: trans_less)
qed
end
subsection {* Monotonicity of @{text "(_,_)"}, @{term fst}, @{term snd} *}
text {* Pair @{text "(_,_)"} is monotone in both arguments *}
lemma monofun_pair1: "monofun (\<lambda>x. (x, y))"
by (simp add: monofun_def less_cprod_def)
lemma monofun_pair2: "monofun (\<lambda>y. (x, y))"
by (simp add: monofun_def less_cprod_def)
lemma monofun_pair:
"\<lbrakk>x1 \<sqsubseteq> x2; y1 \<sqsubseteq> y2\<rbrakk> \<Longrightarrow> (x1, y1) \<sqsubseteq> (x2, y2)"
by (simp add: less_cprod_def)
text {* @{term fst} and @{term snd} are monotone *}
lemma monofun_fst: "monofun fst"
by (simp add: monofun_def less_cprod_def)
lemma monofun_snd: "monofun snd"
by (simp add: monofun_def less_cprod_def)
subsection {* Product type is a cpo *}
lemma lub_cprod:
fixes S :: "nat \<Rightarrow> ('a::cpo \<times> 'b::cpo)"
assumes S: "chain S"
shows "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
apply (rule is_lubI)
apply (rule ub_rangeI)
apply (rule_tac t = "S i" in surjective_pairing [THEN ssubst])
apply (rule monofun_pair)
apply (rule is_ub_thelub)
apply (rule ch2ch_monofun [OF monofun_fst S])
apply (rule is_ub_thelub)
apply (rule ch2ch_monofun [OF monofun_snd S])
apply (rule_tac t = "u" in surjective_pairing [THEN ssubst])
apply (rule monofun_pair)
apply (rule is_lub_thelub)
apply (rule ch2ch_monofun [OF monofun_fst S])
apply (erule monofun_fst [THEN ub2ub_monofun])
apply (rule is_lub_thelub)
apply (rule ch2ch_monofun [OF monofun_snd S])
apply (erule monofun_snd [THEN ub2ub_monofun])
done
lemma thelub_cprod:
"chain (S::nat \<Rightarrow> 'a::cpo \<times> 'b::cpo)
\<Longrightarrow> lub (range S) = (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
by (rule lub_cprod [THEN thelubI])
instance "*" :: (cpo, cpo) cpo
proof
fix S :: "nat \<Rightarrow> ('a \<times> 'b)"
assume "chain S"
hence "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
by (rule lub_cprod)
thus "\<exists>x. range S <<| x" ..
qed
instance "*" :: (finite_po, finite_po) finite_po ..
subsection {* Product type is pointed *}
lemma minimal_cprod: "(\<bottom>, \<bottom>) \<sqsubseteq> p"
by (simp add: less_cprod_def)
instance "*" :: (pcpo, pcpo) pcpo
by intro_classes (fast intro: minimal_cprod)
lemma inst_cprod_pcpo: "\<bottom> = (\<bottom>, \<bottom>)"
by (rule minimal_cprod [THEN UU_I, symmetric])
subsection {* Continuity of @{text "(_,_)"}, @{term fst}, @{term snd} *}
lemma contlub_pair1: "contlub (\<lambda>x. (x, y))"
apply (rule contlubI)
apply (subst thelub_cprod)
apply (erule monofun_pair1 [THEN ch2ch_monofun])
apply simp
done
lemma contlub_pair2: "contlub (\<lambda>y. (x, y))"
apply (rule contlubI)
apply (subst thelub_cprod)
apply (erule monofun_pair2 [THEN ch2ch_monofun])
apply simp
done
lemma cont_pair1: "cont (\<lambda>x. (x, y))"
apply (rule monocontlub2cont)
apply (rule monofun_pair1)
apply (rule contlub_pair1)
done
lemma cont_pair2: "cont (\<lambda>y. (x, y))"
apply (rule monocontlub2cont)
apply (rule monofun_pair2)
apply (rule contlub_pair2)
done
lemma contlub_fst: "contlub fst"
apply (rule contlubI)
apply (simp add: thelub_cprod)
done
lemma contlub_snd: "contlub snd"
apply (rule contlubI)
apply (simp add: thelub_cprod)
done
lemma cont_fst: "cont fst"
apply (rule monocontlub2cont)
apply (rule monofun_fst)
apply (rule contlub_fst)
done
lemma cont_snd: "cont snd"
apply (rule monocontlub2cont)
apply (rule monofun_snd)
apply (rule contlub_snd)
done
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)"
definition
csplit :: "('a \<rightarrow> 'b \<rightarrow> 'c) \<rightarrow> ('a * 'b) \<rightarrow> 'c" where
"csplit = (\<Lambda> f p. f\<cdot>(cfst\<cdot>p)\<cdot>(csnd\<cdot>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)"
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_less [iff]: "(<a, b> \<sqsubseteq> <a', b'>) = (a \<sqsubseteq> a' \<and> b \<sqsubseteq> b')"
by (simp add: cpair_eq_pair less_cprod_def)
lemma cpair_defined_iff [iff]: "(<x, y> = \<bottom>) = (x = \<bottom> \<and> y = \<bottom>)"
by (simp add: inst_cprod_pcpo 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 cont_fst)
lemma csnd_cpair [simp]: "csnd\<cdot><x, y> = y"
by (simp add: cpair_eq_pair csnd_def cont_snd)
lemma cfst_strict [simp]: "cfst\<cdot>\<bottom> = \<bottom>"
unfolding inst_cprod_pcpo2 by (rule cfst_cpair)
lemma csnd_strict [simp]: "csnd\<cdot>\<bottom> = \<bottom>"
unfolding inst_cprod_pcpo2 by (rule csnd_cpair)
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 less_cprod: "x \<sqsubseteq> y = (cfst\<cdot>x \<sqsubseteq> cfst\<cdot>y \<and> csnd\<cdot>x \<sqsubseteq> csnd\<cdot>y)"
by (simp add: less_cprod_def cfst_def csnd_def cont_fst cont_snd)
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 less_cprod)
lemma cfst_less_iff: "cfst\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> <y, csnd\<cdot>x>"
by (simp add: less_cprod)
lemma csnd_less_iff: "csnd\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> <cfst\<cdot>x, y>"
by (simp add: less_cprod)
lemma compact_cfst: "compact x \<Longrightarrow> compact (cfst\<cdot>x)"
by (rule compactI, simp add: cfst_less_iff)
lemma compact_csnd: "compact x \<Longrightarrow> compact (csnd\<cdot>x)"
by (rule compactI, simp add: csnd_less_iff)
lemma compact_cpair: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact <x, y>"
by (rule compactI, simp add: less_cprod)
lemma compact_cpair_iff [simp]: "compact <x, y> = (compact x \<and> compact y)"
apply (safe intro!: compact_cpair)
apply (drule compact_cfst, simp)
apply (drule compact_csnd, simp)
done
instance "*" :: (chfin, chfin) chfin
apply (intro_classes, clarify)
apply (erule compact_imp_max_in_chain)
apply (rule_tac p="\<Squnion>i. Y i" in cprodE, simp)
done
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 cont_fst cont_snd)
apply (erule lub_cprod)
done
lemma thelub_cprod2:
"chain S \<Longrightarrow> lub (range S) = <\<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)
lemma csplit3 [simp]: "csplit\<cdot>cpair\<cdot>z = z"
by (simp add: csplit_def cpair_cfst_csnd)
lemmas Cprod_rews = cfst_cpair csnd_cpair csplit2
subsection {* Product type is a bifinite domain *}
instance "*" :: (bifinite_cpo, bifinite_cpo) approx ..
defs (overloaded)
approx_cprod_def:
"approx \<equiv> \<lambda>n. \<Lambda>\<langle>x, y\<rangle>. \<langle>approx n\<cdot>x, approx n\<cdot>y\<rangle>"
instance "*" :: (bifinite_cpo, bifinite_cpo) bifinite_cpo
proof
fix i :: nat and x :: "'a \<times> 'b"
show "chain (\<lambda>i. approx i\<cdot>x)"
unfolding approx_cprod_def by simp
show "(\<Squnion>i. approx i\<cdot>x) = x"
unfolding approx_cprod_def
by (simp add: lub_distribs eta_cfun)
show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
unfolding approx_cprod_def csplit_def by simp
have "{x::'a \<times> 'b. approx i\<cdot>x = x} \<subseteq>
{x::'a. approx i\<cdot>x = x} \<times> {x::'b. approx i\<cdot>x = x}"
unfolding approx_cprod_def
by (clarsimp simp add: pair_eq_cpair)
thus "finite {x::'a \<times> 'b. approx i\<cdot>x = x}"
by (rule finite_subset,
intro finite_cartesian_product finite_fixes_approx)
qed
instance "*" :: (bifinite, bifinite) bifinite ..
lemma approx_cpair [simp]:
"approx i\<cdot>\<langle>x, y\<rangle> = \<langle>approx i\<cdot>x, approx i\<cdot>y\<rangle>"
unfolding approx_cprod_def by simp
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