(* Title: HOLCF/Cprod.thy ID: $Id: Cprod.thy,v 1.22 2007/10/21 12:21:48 wenzelm Exp $ Author: Franz Regensburger Partial ordering for cartesian product of HOL products. *) header {* The cpo of cartesian products *} theory Cprod imports Cfun begin defaultsort cpo subsection {* Type @{typ unit} is a pcpo *} instance unit :: sq_ord .. defs (overloaded) less_unit_def [simp]: "x \<sqsubseteq> (y::unit) ≡ True" instance unit :: po by intro_classes simp_all instance unit :: cpo by intro_classes (simp add: is_lub_def is_ub_def) instance unit :: pcpo by intro_classes simp definition unit_when :: "'a -> unit -> 'a" where "unit_when = (Λ a _. a)" translations "Λ(). t" == "CONST unit_when·t" lemma unit_when [simp]: "unit_when·a·u = a" by (simp add: unit_when_def) subsection {* Product type is a partial order *} instance "*" :: (sq_ord, sq_ord) sq_ord .. defs (overloaded) less_cprod_def: "(op \<sqsubseteq>) ≡ λp1 p2. (fst p1 \<sqsubseteq> fst p2 ∧ snd p1 \<sqsubseteq> snd p2)" lemma refl_less_cprod: "(p::'a * 'b) \<sqsubseteq> p" by (simp add: less_cprod_def) lemma antisym_less_cprod: "[|(p1::'a * 'b) \<sqsubseteq> p2; p2 \<sqsubseteq> p1|] ==> p1 = p2" apply (unfold less_cprod_def) apply (rule injective_fst_snd) apply (fast intro: antisym_less) apply (fast intro: antisym_less) done lemma trans_less_cprod: "[|(p1::'a*'b) \<sqsubseteq> p2; p2 \<sqsubseteq> p3|] ==> p1 \<sqsubseteq> p3" apply (unfold less_cprod_def) apply (fast intro: trans_less) done instance "*" :: (cpo, cpo) po by intro_classes (assumption | rule refl_less_cprod antisym_less_cprod trans_less_cprod)+ subsection {* Monotonicity of @{text "(_,_)"}, @{term fst}, @{term snd} *} text {* Pair @{text "(_,_)"} is monotone in both arguments *} lemma monofun_pair1: "monofun (λx. (x, y))" by (simp add: monofun_def less_cprod_def) lemma monofun_pair2: "monofun (λy. (x, y))" by (simp add: monofun_def less_cprod_def) lemma monofun_pair: "[|x1 \<sqsubseteq> x2; y1 \<sqsubseteq> y2|] ==> (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: "chain S ==> 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 (erule monofun_fst [THEN ch2ch_monofun]) apply (rule is_ub_thelub) apply (erule monofun_snd [THEN ch2ch_monofun]) apply (rule_tac t = "u" in surjective_pairing [THEN ssubst]) apply (rule monofun_pair) apply (rule is_lub_thelub) apply (erule monofun_fst [THEN ch2ch_monofun]) apply (erule monofun_fst [THEN ub2ub_monofun]) apply (rule is_lub_thelub) apply (erule monofun_snd [THEN ch2ch_monofun]) apply (erule monofun_snd [THEN ub2ub_monofun]) done lemma thelub_cprod: "chain S ==> lub (range S) = (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))" by (rule lub_cprod [THEN thelubI]) lemma cpo_cprod: "chain (S::nat => 'a::cpo * 'b::cpo) ==> ∃x. range S <<| x" by (rule exI, erule lub_cprod) instance "*" :: (cpo, cpo) cpo by intro_classes (rule cpo_cprod) subsection {* Product type is pointed *} lemma minimal_cprod: "(⊥, ⊥) \<sqsubseteq> p" by (simp add: less_cprod_def) lemma least_cprod: "EX x::'a::pcpo * 'b::pcpo. ALL y. x \<sqsubseteq> y" apply (rule_tac x = "(⊥, ⊥)" in exI) apply (rule minimal_cprod [THEN allI]) done instance "*" :: (pcpo, pcpo) pcpo by intro_classes (rule least_cprod) text {* for compatibility with old HOLCF-Version *} lemma inst_cprod_pcpo: "UU = (UU,UU)" by (rule minimal_cprod [THEN UU_I, symmetric]) subsection {* Continuity of @{text "(_,_)"}, @{term fst}, @{term snd} *} lemma contlub_pair1: "contlub (λx. (x, y))" apply (rule contlubI) apply (subst thelub_cprod) apply (erule monofun_pair1 [THEN ch2ch_monofun]) apply simp done lemma contlub_pair2: "contlub (λy. (x, y))" apply (rule contlubI) apply (subst thelub_cprod) apply (erule monofun_pair2 [THEN ch2ch_monofun]) apply simp done lemma cont_pair1: "cont (λx. (x, y))" apply (rule monocontlub2cont) apply (rule monofun_pair1) apply (rule contlub_pair1) done lemma cont_pair2: "cont (λy. (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 -> 'b -> ('a * 'b)" -- {* continuous pairing *} where "cpair = (Λ x y. (x, y))" definition cfst :: "('a * 'b) -> 'a" where "cfst = (Λ p. fst p)" definition csnd :: "('a * 'b) -> 'b" where "csnd = (Λ p. snd p)" definition csplit :: "('a -> 'b -> 'c) -> ('a * 'b) -> 'c" where "csplit = (Λ f p. f·(cfst·p)·(csnd·p))" syntax "_ctuple" :: "['a, args] => 'a * 'b" ("(1<_,/ _>)") syntax (xsymbols) "_ctuple" :: "['a, args] => 'a * 'b" ("(1〈_,/ _〉)") translations "〈x, y, z〉" == "〈x, 〈y, z〉〉" "〈x, y〉" == "CONST cpair·x·y" translations "Λ(CONST cpair·x·y). t" == "CONST csplit·(Λ 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 inject_cpair: "<a,b> = <aa,ba> ==> a = aa ∧ b = ba" by (simp add: cpair_eq_pair) lemma cpair_eq [iff]: "(<a, b> = <a', b'>) = (a = a' ∧ b = b')" by (simp add: cpair_eq_pair) lemma cpair_less [iff]: "(<a, b> \<sqsubseteq> <a', b'>) = (a \<sqsubseteq> a' ∧ b \<sqsubseteq> b')" by (simp add: cpair_eq_pair less_cprod_def) lemma cpair_defined_iff [iff]: "(<x, y> = ⊥) = (x = ⊥ ∧ y = ⊥)" by (simp add: inst_cprod_pcpo cpair_eq_pair) lemma cpair_strict: "<⊥, ⊥> = ⊥" by simp lemma inst_cprod_pcpo2: "⊥ = <⊥, ⊥>" by (rule cpair_strict [symmetric]) lemma defined_cpair_rev: "<a,b> = ⊥ ==> a = ⊥ ∧ b = ⊥" by simp lemma Exh_Cprod2: "∃a b. z = <a, b>" by (simp add: cpair_eq_pair) lemma cprodE: "[|!!x y. p = <x, y> ==> Q|] ==> Q" by (cut_tac Exh_Cprod2, auto) lemma cfst_cpair [simp]: "cfst·<x, y> = x" by (simp add: cpair_eq_pair cfst_def cont_fst) lemma csnd_cpair [simp]: "csnd·<x, y> = y" by (simp add: cpair_eq_pair csnd_def cont_snd) lemma cfst_strict [simp]: "cfst·⊥ = ⊥" by (simp add: inst_cprod_pcpo2) lemma csnd_strict [simp]: "csnd·⊥ = ⊥" by (simp add: inst_cprod_pcpo2) lemma surjective_pairing_Cprod2: "<cfst·p, csnd·p> = p" apply (unfold cfst_def csnd_def) apply (simp add: cont_fst cont_snd cpair_eq_pair) done lemma less_cprod: "x \<sqsubseteq> y = (cfst·x \<sqsubseteq> cfst·y ∧ csnd·x \<sqsubseteq> csnd·y)" by (simp add: less_cprod_def cfst_def csnd_def cont_fst cont_snd) lemma eq_cprod: "(x = y) = (cfst·x = cfst·y ∧ csnd·x = csnd·y)" by (auto simp add: po_eq_conv less_cprod) lemma compact_cpair [simp]: "[|compact x; compact y|] ==> compact <x, y>" by (rule compactI, simp add: less_cprod) lemma lub_cprod2: "chain S ==> range S <<| <\<Squnion>i. cfst·(S i), \<Squnion>i. csnd·(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 ==> lub (range S) = <\<Squnion>i. cfst·(S i), \<Squnion>i. csnd·(S i)>" by (rule lub_cprod2 [THEN thelubI]) lemma csplit1 [simp]: "csplit·f·⊥ = f·⊥·⊥" by (simp add: csplit_def) lemma csplit2 [simp]: "csplit·f·<x,y> = f·x·y" by (simp add: csplit_def) lemma csplit3 [simp]: "csplit·cpair·z = z" by (simp add: csplit_def surjective_pairing_Cprod2) lemmas Cprod_rews = cfst_cpair csnd_cpair csplit2 end
lemma unit_when:
(LAM (). a)·u = a
lemma refl_less_cprod:
p << p
lemma antisym_less_cprod:
[| p1.0 << p2.0; p2.0 << p1.0 |] ==> p1.0 = p2.0
lemma trans_less_cprod:
[| p1.0 << p2.0; p2.0 << p3.0 |] ==> p1.0 << p3.0
lemma monofun_pair1:
monofun (λx. (x, y))
lemma monofun_pair2:
monofun (Pair x)
lemma monofun_pair:
[| x1.0 << x2.0; y1.0 << y2.0 |] ==> (x1.0, y1.0) << (x2.0, y2.0)
lemma monofun_fst:
monofun fst
lemma monofun_snd:
monofun snd
lemma lub_cprod:
chain S ==> range S <<| (LUB i. fst (S i), LUB i. snd (S i))
lemma thelub_cprod:
chain S ==> Lub S = (LUB i. fst (S i), LUB i. snd (S i))
lemma cpo_cprod:
chain S ==> ∃x. range S <<| x
lemma minimal_cprod:
(UU, UU) << p
lemma least_cprod:
∃x. ∀y. x << y
lemma inst_cprod_pcpo:
UU = (UU, UU)
lemma contlub_pair1:
contlub (λx. (x, y))
lemma contlub_pair2:
contlub (Pair x)
lemma cont_pair1:
cont (λx. (x, y))
lemma cont_pair2:
cont (Pair x)
lemma contlub_fst:
contlub fst
lemma contlub_snd:
contlub snd
lemma cont_fst:
cont fst
lemma cont_snd:
cont snd
lemma cpair_eq_pair:
<x, y> = (x, y)
lemma inject_cpair:
<a, b> = <aa, ba> ==> a = aa ∧ b = ba
lemma cpair_eq:
(<a, b> = <a', b'>) = (a = a' ∧ b = b')
lemma cpair_less:
<a, b> << <a', b'> = (a << a' ∧ b << b')
lemma cpair_defined_iff:
(<x, y> = UU) = (x = UU ∧ y = UU)
lemma cpair_strict:
<UU, UU> = UU
lemma inst_cprod_pcpo2:
UU = <UU, UU>
lemma defined_cpair_rev:
<a, b> = UU ==> a = UU ∧ b = UU
lemma Exh_Cprod2:
∃a b. z = <a, b>
lemma cprodE:
(!!x y. p = <x, y> ==> Q) ==> Q
lemma cfst_cpair:
cfst·<x, y> = x
lemma csnd_cpair:
csnd·<x, y> = y
lemma cfst_strict:
cfst·UU = UU
lemma csnd_strict:
csnd·UU = UU
lemma surjective_pairing_Cprod2:
<cfst·p, csnd·p> = p
lemma less_cprod:
x << y = (cfst·x << cfst·y ∧ csnd·x << csnd·y)
lemma eq_cprod:
(x = y) = (cfst·x = cfst·y ∧ csnd·x = csnd·y)
lemma compact_cpair:
[| compact x; compact y |] ==> compact <x, y>
lemma lub_cprod2:
chain S ==> range S <<| <LUB i. cfst·(S i), LUB i. csnd·(S i)>
lemma thelub_cprod2:
chain S ==> Lub S = <LUB i. cfst·(S i), LUB i. csnd·(S i)>
lemma csplit1:
csplit·f·UU = f·UU·UU
lemma csplit2:
csplit·f·<x, y> = f·x·y
lemma csplit3:
csplit·cpair·z = z
lemma Cprod_rews:
cfst·<x, y> = x
csnd·<x, y> = y
csplit·f·<x, y> = f·x·y