1.1 --- a/src/HOLCF/Product_Cpo.thy Sat Nov 27 14:34:54 2010 -0800
1.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
1.3 @@ -1,299 +0,0 @@
1.4 -(* Title: HOLCF/Product_Cpo.thy
1.5 - Author: Franz Regensburger
1.6 -*)
1.7 -
1.8 -header {* The cpo of cartesian products *}
1.9 -
1.10 -theory Product_Cpo
1.11 -imports Adm
1.12 -begin
1.13 -
1.14 -default_sort cpo
1.15 -
1.16 -subsection {* Unit type is a pcpo *}
1.17 -
1.18 -instantiation unit :: discrete_cpo
1.19 -begin
1.20 -
1.21 -definition
1.22 - below_unit_def [simp]: "x \<sqsubseteq> (y::unit) \<longleftrightarrow> True"
1.23 -
1.24 -instance proof
1.25 -qed simp
1.26 -
1.27 -end
1.28 -
1.29 -instance unit :: pcpo
1.30 -by intro_classes simp
1.31 -
1.32 -
1.33 -subsection {* Product type is a partial order *}
1.34 -
1.35 -instantiation prod :: (below, below) below
1.36 -begin
1.37 -
1.38 -definition
1.39 - below_prod_def: "(op \<sqsubseteq>) \<equiv> \<lambda>p1 p2. (fst p1 \<sqsubseteq> fst p2 \<and> snd p1 \<sqsubseteq> snd p2)"
1.40 -
1.41 -instance ..
1.42 -end
1.43 -
1.44 -instance prod :: (po, po) po
1.45 -proof
1.46 - fix x :: "'a \<times> 'b"
1.47 - show "x \<sqsubseteq> x"
1.48 - unfolding below_prod_def by simp
1.49 -next
1.50 - fix x y :: "'a \<times> 'b"
1.51 - assume "x \<sqsubseteq> y" "y \<sqsubseteq> x" thus "x = y"
1.52 - unfolding below_prod_def Pair_fst_snd_eq
1.53 - by (fast intro: below_antisym)
1.54 -next
1.55 - fix x y z :: "'a \<times> 'b"
1.56 - assume "x \<sqsubseteq> y" "y \<sqsubseteq> z" thus "x \<sqsubseteq> z"
1.57 - unfolding below_prod_def
1.58 - by (fast intro: below_trans)
1.59 -qed
1.60 -
1.61 -subsection {* Monotonicity of \emph{Pair}, \emph{fst}, \emph{snd} *}
1.62 -
1.63 -lemma prod_belowI: "\<lbrakk>fst p \<sqsubseteq> fst q; snd p \<sqsubseteq> snd q\<rbrakk> \<Longrightarrow> p \<sqsubseteq> q"
1.64 -unfolding below_prod_def by simp
1.65 -
1.66 -lemma Pair_below_iff [simp]: "(a, b) \<sqsubseteq> (c, d) \<longleftrightarrow> a \<sqsubseteq> c \<and> b \<sqsubseteq> d"
1.67 -unfolding below_prod_def by simp
1.68 -
1.69 -text {* Pair @{text "(_,_)"} is monotone in both arguments *}
1.70 -
1.71 -lemma monofun_pair1: "monofun (\<lambda>x. (x, y))"
1.72 -by (simp add: monofun_def)
1.73 -
1.74 -lemma monofun_pair2: "monofun (\<lambda>y. (x, y))"
1.75 -by (simp add: monofun_def)
1.76 -
1.77 -lemma monofun_pair:
1.78 - "\<lbrakk>x1 \<sqsubseteq> x2; y1 \<sqsubseteq> y2\<rbrakk> \<Longrightarrow> (x1, y1) \<sqsubseteq> (x2, y2)"
1.79 -by simp
1.80 -
1.81 -lemma ch2ch_Pair [simp]:
1.82 - "chain X \<Longrightarrow> chain Y \<Longrightarrow> chain (\<lambda>i. (X i, Y i))"
1.83 -by (rule chainI, simp add: chainE)
1.84 -
1.85 -text {* @{term fst} and @{term snd} are monotone *}
1.86 -
1.87 -lemma fst_monofun: "x \<sqsubseteq> y \<Longrightarrow> fst x \<sqsubseteq> fst y"
1.88 -unfolding below_prod_def by simp
1.89 -
1.90 -lemma snd_monofun: "x \<sqsubseteq> y \<Longrightarrow> snd x \<sqsubseteq> snd y"
1.91 -unfolding below_prod_def by simp
1.92 -
1.93 -lemma monofun_fst: "monofun fst"
1.94 -by (simp add: monofun_def below_prod_def)
1.95 -
1.96 -lemma monofun_snd: "monofun snd"
1.97 -by (simp add: monofun_def below_prod_def)
1.98 -
1.99 -lemmas ch2ch_fst [simp] = ch2ch_monofun [OF monofun_fst]
1.100 -
1.101 -lemmas ch2ch_snd [simp] = ch2ch_monofun [OF monofun_snd]
1.102 -
1.103 -lemma prod_chain_cases:
1.104 - assumes "chain Y"
1.105 - obtains A B
1.106 - where "chain A" and "chain B" and "Y = (\<lambda>i. (A i, B i))"
1.107 -proof
1.108 - from `chain Y` show "chain (\<lambda>i. fst (Y i))" by (rule ch2ch_fst)
1.109 - from `chain Y` show "chain (\<lambda>i. snd (Y i))" by (rule ch2ch_snd)
1.110 - show "Y = (\<lambda>i. (fst (Y i), snd (Y i)))" by simp
1.111 -qed
1.112 -
1.113 -subsection {* Product type is a cpo *}
1.114 -
1.115 -lemma is_lub_Pair:
1.116 - "\<lbrakk>range A <<| x; range B <<| y\<rbrakk> \<Longrightarrow> range (\<lambda>i. (A i, B i)) <<| (x, y)"
1.117 -unfolding is_lub_def is_ub_def ball_simps below_prod_def by simp
1.118 -
1.119 -lemma lub_Pair:
1.120 - "\<lbrakk>chain (A::nat \<Rightarrow> 'a::cpo); chain (B::nat \<Rightarrow> 'b::cpo)\<rbrakk>
1.121 - \<Longrightarrow> (\<Squnion>i. (A i, B i)) = (\<Squnion>i. A i, \<Squnion>i. B i)"
1.122 -by (fast intro: lub_eqI is_lub_Pair elim: thelubE)
1.123 -
1.124 -lemma is_lub_prod:
1.125 - fixes S :: "nat \<Rightarrow> ('a::cpo \<times> 'b::cpo)"
1.126 - assumes S: "chain S"
1.127 - shows "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
1.128 -using S by (auto elim: prod_chain_cases simp add: is_lub_Pair cpo_lubI)
1.129 -
1.130 -lemma lub_prod:
1.131 - "chain (S::nat \<Rightarrow> 'a::cpo \<times> 'b::cpo)
1.132 - \<Longrightarrow> (\<Squnion>i. S i) = (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
1.133 -by (rule is_lub_prod [THEN lub_eqI])
1.134 -
1.135 -instance prod :: (cpo, cpo) cpo
1.136 -proof
1.137 - fix S :: "nat \<Rightarrow> ('a \<times> 'b)"
1.138 - assume "chain S"
1.139 - hence "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
1.140 - by (rule is_lub_prod)
1.141 - thus "\<exists>x. range S <<| x" ..
1.142 -qed
1.143 -
1.144 -instance prod :: (discrete_cpo, discrete_cpo) discrete_cpo
1.145 -proof
1.146 - fix x y :: "'a \<times> 'b"
1.147 - show "x \<sqsubseteq> y \<longleftrightarrow> x = y"
1.148 - unfolding below_prod_def Pair_fst_snd_eq
1.149 - by simp
1.150 -qed
1.151 -
1.152 -subsection {* Product type is pointed *}
1.153 -
1.154 -lemma minimal_prod: "(\<bottom>, \<bottom>) \<sqsubseteq> p"
1.155 -by (simp add: below_prod_def)
1.156 -
1.157 -instance prod :: (pcpo, pcpo) pcpo
1.158 -by intro_classes (fast intro: minimal_prod)
1.159 -
1.160 -lemma inst_prod_pcpo: "\<bottom> = (\<bottom>, \<bottom>)"
1.161 -by (rule minimal_prod [THEN UU_I, symmetric])
1.162 -
1.163 -lemma Pair_bottom_iff [simp]: "(x, y) = \<bottom> \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
1.164 -unfolding inst_prod_pcpo by simp
1.165 -
1.166 -lemma fst_strict [simp]: "fst \<bottom> = \<bottom>"
1.167 -unfolding inst_prod_pcpo by (rule fst_conv)
1.168 -
1.169 -lemma snd_strict [simp]: "snd \<bottom> = \<bottom>"
1.170 -unfolding inst_prod_pcpo by (rule snd_conv)
1.171 -
1.172 -lemma Pair_strict [simp]: "(\<bottom>, \<bottom>) = \<bottom>"
1.173 -by simp
1.174 -
1.175 -lemma split_strict [simp]: "split f \<bottom> = f \<bottom> \<bottom>"
1.176 -unfolding split_def by simp
1.177 -
1.178 -subsection {* Continuity of \emph{Pair}, \emph{fst}, \emph{snd} *}
1.179 -
1.180 -lemma cont_pair1: "cont (\<lambda>x. (x, y))"
1.181 -apply (rule contI)
1.182 -apply (rule is_lub_Pair)
1.183 -apply (erule cpo_lubI)
1.184 -apply (rule is_lub_const)
1.185 -done
1.186 -
1.187 -lemma cont_pair2: "cont (\<lambda>y. (x, y))"
1.188 -apply (rule contI)
1.189 -apply (rule is_lub_Pair)
1.190 -apply (rule is_lub_const)
1.191 -apply (erule cpo_lubI)
1.192 -done
1.193 -
1.194 -lemma cont_fst: "cont fst"
1.195 -apply (rule contI)
1.196 -apply (simp add: lub_prod)
1.197 -apply (erule cpo_lubI [OF ch2ch_fst])
1.198 -done
1.199 -
1.200 -lemma cont_snd: "cont snd"
1.201 -apply (rule contI)
1.202 -apply (simp add: lub_prod)
1.203 -apply (erule cpo_lubI [OF ch2ch_snd])
1.204 -done
1.205 -
1.206 -lemma cont2cont_Pair [simp, cont2cont]:
1.207 - assumes f: "cont (\<lambda>x. f x)"
1.208 - assumes g: "cont (\<lambda>x. g x)"
1.209 - shows "cont (\<lambda>x. (f x, g x))"
1.210 -apply (rule cont_apply [OF f cont_pair1])
1.211 -apply (rule cont_apply [OF g cont_pair2])
1.212 -apply (rule cont_const)
1.213 -done
1.214 -
1.215 -lemmas cont2cont_fst [simp, cont2cont] = cont_compose [OF cont_fst]
1.216 -
1.217 -lemmas cont2cont_snd [simp, cont2cont] = cont_compose [OF cont_snd]
1.218 -
1.219 -lemma cont2cont_prod_case:
1.220 - assumes f1: "\<And>a b. cont (\<lambda>x. f x a b)"
1.221 - assumes f2: "\<And>x b. cont (\<lambda>a. f x a b)"
1.222 - assumes f3: "\<And>x a. cont (\<lambda>b. f x a b)"
1.223 - assumes g: "cont (\<lambda>x. g x)"
1.224 - shows "cont (\<lambda>x. case g x of (a, b) \<Rightarrow> f x a b)"
1.225 -unfolding split_def
1.226 -apply (rule cont_apply [OF g])
1.227 -apply (rule cont_apply [OF cont_fst f2])
1.228 -apply (rule cont_apply [OF cont_snd f3])
1.229 -apply (rule cont_const)
1.230 -apply (rule f1)
1.231 -done
1.232 -
1.233 -lemma prod_contI:
1.234 - assumes f1: "\<And>y. cont (\<lambda>x. f (x, y))"
1.235 - assumes f2: "\<And>x. cont (\<lambda>y. f (x, y))"
1.236 - shows "cont f"
1.237 -proof -
1.238 - have "cont (\<lambda>(x, y). f (x, y))"
1.239 - by (intro cont2cont_prod_case f1 f2 cont2cont)
1.240 - thus "cont f"
1.241 - by (simp only: split_eta)
1.242 -qed
1.243 -
1.244 -lemma prod_cont_iff:
1.245 - "cont f \<longleftrightarrow> (\<forall>y. cont (\<lambda>x. f (x, y))) \<and> (\<forall>x. cont (\<lambda>y. f (x, y)))"
1.246 -apply safe
1.247 -apply (erule cont_compose [OF _ cont_pair1])
1.248 -apply (erule cont_compose [OF _ cont_pair2])
1.249 -apply (simp only: prod_contI)
1.250 -done
1.251 -
1.252 -lemma cont2cont_prod_case' [simp, cont2cont]:
1.253 - assumes f: "cont (\<lambda>p. f (fst p) (fst (snd p)) (snd (snd p)))"
1.254 - assumes g: "cont (\<lambda>x. g x)"
1.255 - shows "cont (\<lambda>x. prod_case (f x) (g x))"
1.256 -using assms by (simp add: cont2cont_prod_case prod_cont_iff)
1.257 -
1.258 -text {* The simple version (due to Joachim Breitner) is needed if
1.259 - either element type of the pair is not a cpo. *}
1.260 -
1.261 -lemma cont2cont_split_simple [simp, cont2cont]:
1.262 - assumes "\<And>a b. cont (\<lambda>x. f x a b)"
1.263 - shows "cont (\<lambda>x. case p of (a, b) \<Rightarrow> f x a b)"
1.264 -using assms by (cases p) auto
1.265 -
1.266 -text {* Admissibility of predicates on product types. *}
1.267 -
1.268 -lemma adm_prod_case [simp]:
1.269 - assumes "adm (\<lambda>x. P x (fst (f x)) (snd (f x)))"
1.270 - shows "adm (\<lambda>x. case f x of (a, b) \<Rightarrow> P x a b)"
1.271 -unfolding prod_case_beta using assms .
1.272 -
1.273 -subsection {* Compactness and chain-finiteness *}
1.274 -
1.275 -lemma fst_below_iff: "fst (x::'a \<times> 'b) \<sqsubseteq> y \<longleftrightarrow> x \<sqsubseteq> (y, snd x)"
1.276 -unfolding below_prod_def by simp
1.277 -
1.278 -lemma snd_below_iff: "snd (x::'a \<times> 'b) \<sqsubseteq> y \<longleftrightarrow> x \<sqsubseteq> (fst x, y)"
1.279 -unfolding below_prod_def by simp
1.280 -
1.281 -lemma compact_fst: "compact x \<Longrightarrow> compact (fst x)"
1.282 -by (rule compactI, simp add: fst_below_iff)
1.283 -
1.284 -lemma compact_snd: "compact x \<Longrightarrow> compact (snd x)"
1.285 -by (rule compactI, simp add: snd_below_iff)
1.286 -
1.287 -lemma compact_Pair: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact (x, y)"
1.288 -by (rule compactI, simp add: below_prod_def)
1.289 -
1.290 -lemma compact_Pair_iff [simp]: "compact (x, y) \<longleftrightarrow> compact x \<and> compact y"
1.291 -apply (safe intro!: compact_Pair)
1.292 -apply (drule compact_fst, simp)
1.293 -apply (drule compact_snd, simp)
1.294 -done
1.295 -
1.296 -instance prod :: (chfin, chfin) chfin
1.297 -apply intro_classes
1.298 -apply (erule compact_imp_max_in_chain)
1.299 -apply (case_tac "\<Squnion>i. Y i", simp)
1.300 -done
1.301 -
1.302 -end