author | huffman |
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(* Title: HOLCF/Cprod.thy |
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ID: $Id$ |
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Author: Franz Regensburger |
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Partial ordering for cartesian product of HOL products. |
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*) |
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header {* The cpo of cartesian products *} |
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theory Cprod |
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imports Cfun |
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begin |
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defaultsort cpo |
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subsection {* Type @{typ unit} is a pcpo *} |
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instantiation unit :: sq_ord |
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begin |
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definition |
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less_unit_def [simp]: "x \<sqsubseteq> (y::unit) \<equiv> True" |
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instance .. |
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end |
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instance unit :: po |
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by intro_classes simp_all |
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instance unit :: dcpo |
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by intro_classes (simp add: is_lub_def is_ub_def) |
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instance unit :: pcpo |
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by intro_classes simp |
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definition |
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unit_when :: "'a \<rightarrow> unit \<rightarrow> 'a" where |
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"unit_when = (\<Lambda> a _. a)" |
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translations |
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"\<Lambda>(). t" == "CONST unit_when\<cdot>t" |
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lemma unit_when [simp]: "unit_when\<cdot>a\<cdot>u = a" |
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by (simp add: unit_when_def) |
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subsection {* Product type is a partial order *} |
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instantiation "*" :: (sq_ord, sq_ord) sq_ord |
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begin |
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definition |
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less_cprod_def: "(op \<sqsubseteq>) \<equiv> \<lambda>p1 p2. (fst p1 \<sqsubseteq> fst p2 \<and> snd p1 \<sqsubseteq> snd p2)" |
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instance .. |
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end |
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instance "*" :: (po, po) po |
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proof |
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fix x :: "'a \<times> 'b" |
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show "x \<sqsubseteq> x" |
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unfolding less_cprod_def by simp |
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next |
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fix x y :: "'a \<times> 'b" |
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assume "x \<sqsubseteq> y" "y \<sqsubseteq> x" thus "x = y" |
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unfolding less_cprod_def Pair_fst_snd_eq |
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by (fast intro: antisym_less) |
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next |
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fix x y z :: "'a \<times> 'b" |
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assume "x \<sqsubseteq> y" "y \<sqsubseteq> z" thus "x \<sqsubseteq> z" |
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unfolding less_cprod_def |
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by (fast intro: trans_less) |
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qed |
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subsection {* Monotonicity of @{text "(_,_)"}, @{term fst}, @{term snd} *} |
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text {* Pair @{text "(_,_)"} is monotone in both arguments *} |
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lemma monofun_pair1: "monofun (\<lambda>x. (x, y))" |
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by (simp add: monofun_def less_cprod_def) |
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lemma monofun_pair2: "monofun (\<lambda>y. (x, y))" |
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by (simp add: monofun_def less_cprod_def) |
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lemma monofun_pair: |
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"\<lbrakk>x1 \<sqsubseteq> x2; y1 \<sqsubseteq> y2\<rbrakk> \<Longrightarrow> (x1, y1) \<sqsubseteq> (x2, y2)" |
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by (simp add: less_cprod_def) |
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text {* @{term fst} and @{term snd} are monotone *} |
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lemma monofun_fst: "monofun fst" |
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by (simp add: monofun_def less_cprod_def) |
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lemma monofun_snd: "monofun snd" |
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by (simp add: monofun_def less_cprod_def) |
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subsection {* Product type is a cpo *} |
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lemma lub_cprod: |
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assumes S: "chain S" |
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shows "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))" |
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apply (rule is_lubI) |
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apply (rule ub_rangeI) |
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apply (rule_tac t = "S i" in surjective_pairing [THEN ssubst]) |
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apply (rule monofun_pair) |
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apply (rule is_ub_thelub) |
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apply (rule ch2ch_monofun [OF monofun_fst S]) |
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apply (rule is_ub_thelub) |
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apply (rule ch2ch_monofun [OF monofun_snd S]) |
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apply (rule_tac t = "u" in surjective_pairing [THEN ssubst]) |
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apply (rule monofun_pair) |
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apply (rule is_lub_thelub) |
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apply (rule ch2ch_monofun [OF monofun_fst S]) |
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apply (erule monofun_fst [THEN ub2ub_monofun]) |
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apply (rule is_lub_thelub) |
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apply (rule ch2ch_monofun [OF monofun_snd S]) |
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apply (erule monofun_snd [THEN ub2ub_monofun]) |
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done |
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lemma directed_lub_cprod: |
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fixes S :: "('a::dcpo \<times> 'b::dcpo) set" |
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assumes S: "directed S" |
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shows "S <<| (\<Squnion>x\<in>S. fst x, \<Squnion>x\<in>S. snd x)" |
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apply (rule is_lubI) |
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apply (rule is_ubI) |
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apply (rule_tac t=x in surjective_pairing [THEN ssubst]) |
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apply (rule monofun_pair) |
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apply (erule is_ub_thelub' [OF dir2dir_monofun [OF monofun_fst S] imageI]) |
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apply (erule is_ub_thelub' [OF dir2dir_monofun [OF monofun_snd S] imageI]) |
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apply (rule_tac t=u in surjective_pairing [THEN ssubst]) |
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apply (rule monofun_pair) |
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apply (rule is_lub_thelub') |
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apply (rule dir2dir_monofun [OF monofun_fst S]) |
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apply (erule ub2ub_monofun' [OF monofun_fst]) |
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apply (rule is_lub_thelub') |
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apply (rule dir2dir_monofun [OF monofun_snd S]) |
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apply (erule ub2ub_monofun' [OF monofun_snd]) |
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140 |
done |
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141 |
|
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142 |
lemma thelub_cprod: |
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"chain (S::nat \<Rightarrow> 'a::cpo \<times> 'b::cpo) |
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\<Longrightarrow> lub (range S) = (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))" |
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by (rule lub_cprod [THEN thelubI]) |
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|
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instance "*" :: (cpo, cpo) cpo |
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148 |
proof |
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fix S :: "nat \<Rightarrow> ('a \<times> 'b)" |
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assume "chain S" |
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hence "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))" |
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152 |
by (rule lub_cprod) |
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153 |
thus "\<exists>x. range S <<| x" .. |
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154 |
qed |
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155 |
|
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156 |
instance "*" :: (dcpo, dcpo) dcpo |
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157 |
proof |
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158 |
fix S :: "('a \<times> 'b) set" |
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assume "directed S" |
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hence "S <<| (\<Squnion>x\<in>S. fst x, \<Squnion>x\<in>S. snd x)" |
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161 |
by (rule directed_lub_cprod) |
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162 |
thus "\<exists>x. S <<| x" .. |
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163 |
qed |
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|
18289 | 165 |
subsection {* Product type is pointed *} |
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166 |
|
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167 |
lemma minimal_cprod: "(\<bottom>, \<bottom>) \<sqsubseteq> p" |
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by (simp add: less_cprod_def) |
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169 |
|
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170 |
lemma least_cprod: "EX x::'a::pcpo * 'b::pcpo. ALL y. x \<sqsubseteq> y" |
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171 |
apply (rule_tac x = "(\<bottom>, \<bottom>)" in exI) |
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172 |
apply (rule minimal_cprod [THEN allI]) |
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173 |
done |
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174 |
|
15609 | 175 |
instance "*" :: (pcpo, pcpo) pcpo |
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176 |
by intro_classes (rule least_cprod) |
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177 |
|
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text {* for compatibility with old HOLCF-Version *} |
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179 |
lemma inst_cprod_pcpo: "UU = (UU,UU)" |
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180 |
by (rule minimal_cprod [THEN UU_I, symmetric]) |
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181 |
|
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182 |
|
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183 |
subsection {* Continuity of @{text "(_,_)"}, @{term fst}, @{term snd} *} |
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184 |
|
16916 | 185 |
lemma contlub_pair1: "contlub (\<lambda>x. (x, y))" |
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186 |
apply (rule contlubI) |
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187 |
apply (subst thelub_cprod) |
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188 |
apply (erule monofun_pair1 [THEN ch2ch_monofun]) |
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189 |
apply simp |
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190 |
done |
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|
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192 |
lemma contlub_pair2: "contlub (\<lambda>y. (x, y))" |
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193 |
apply (rule contlubI) |
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194 |
apply (subst thelub_cprod) |
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195 |
apply (erule monofun_pair2 [THEN ch2ch_monofun]) |
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196 |
apply simp |
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197 |
done |
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198 |
|
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199 |
lemma cont_pair1: "cont (\<lambda>x. (x, y))" |
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200 |
apply (rule monocontlub2cont) |
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201 |
apply (rule monofun_pair1) |
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202 |
apply (rule contlub_pair1) |
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203 |
done |
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204 |
|
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205 |
lemma cont_pair2: "cont (\<lambda>y. (x, y))" |
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206 |
apply (rule monocontlub2cont) |
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207 |
apply (rule monofun_pair2) |
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208 |
apply (rule contlub_pair2) |
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|
209 |
done |
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210 |
|
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211 |
lemma contlub_fst: "contlub fst" |
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212 |
apply (rule contlubI) |
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213 |
apply (simp add: thelub_cprod) |
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214 |
done |
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215 |
|
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216 |
lemma contlub_snd: "contlub snd" |
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217 |
apply (rule contlubI) |
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218 |
apply (simp add: thelub_cprod) |
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219 |
done |
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220 |
|
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221 |
lemma cont_fst: "cont fst" |
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222 |
apply (rule monocontlub2cont) |
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223 |
apply (rule monofun_fst) |
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224 |
apply (rule contlub_fst) |
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225 |
done |
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226 |
|
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227 |
lemma cont_snd: "cont snd" |
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228 |
apply (rule monocontlub2cont) |
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229 |
apply (rule monofun_snd) |
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230 |
apply (rule contlub_snd) |
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231 |
done |
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232 |
|
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233 |
subsection {* Continuous versions of constants *} |
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234 |
|
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235 |
definition |
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236 |
cpair :: "'a \<rightarrow> 'b \<rightarrow> ('a * 'b)" -- {* continuous pairing *} where |
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237 |
"cpair = (\<Lambda> x y. (x, y))" |
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238 |
|
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239 |
definition |
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240 |
cfst :: "('a * 'b) \<rightarrow> 'a" where |
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241 |
"cfst = (\<Lambda> p. fst p)" |
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|
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|
243 |
definition |
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244 |
csnd :: "('a * 'b) \<rightarrow> 'b" where |
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245 |
"csnd = (\<Lambda> p. snd p)" |
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|
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247 |
definition |
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248 |
csplit :: "('a \<rightarrow> 'b \<rightarrow> 'c) \<rightarrow> ('a * 'b) \<rightarrow> 'c" where |
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249 |
"csplit = (\<Lambda> f p. f\<cdot>(cfst\<cdot>p)\<cdot>(csnd\<cdot>p))" |
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250 |
|
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251 |
syntax |
17834 | 252 |
"_ctuple" :: "['a, args] \<Rightarrow> 'a * 'b" ("(1<_,/ _>)") |
253 |
||
254 |
syntax (xsymbols) |
|
255 |
"_ctuple" :: "['a, args] \<Rightarrow> 'a * 'b" ("(1\<langle>_,/ _\<rangle>)") |
|
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256 |
|
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257 |
translations |
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258 |
"\<langle>x, y, z\<rangle>" == "\<langle>x, \<langle>y, z\<rangle>\<rangle>" |
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259 |
"\<langle>x, y\<rangle>" == "CONST cpair\<cdot>x\<cdot>y" |
17834 | 260 |
|
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261 |
translations |
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262 |
"\<Lambda>(CONST cpair\<cdot>x\<cdot>y). t" == "CONST csplit\<cdot>(\<Lambda> x y. t)" |
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263 |
|
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|
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subsection {* Convert all lemmas to the continuous versions *} |
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|
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lemma cpair_eq_pair: "<x, y> = (x, y)" |
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by (simp add: cpair_def cont_pair1 cont_pair2) |
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|
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lemma inject_cpair: "<a,b> = <aa,ba> \<Longrightarrow> a = aa \<and> b = ba" |
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by (simp add: cpair_eq_pair) |
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|
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lemma cpair_eq [iff]: "(<a, b> = <a', b'>) = (a = a' \<and> b = b')" |
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by (simp add: cpair_eq_pair) |
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|
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lemma cpair_less [iff]: "(<a, b> \<sqsubseteq> <a', b'>) = (a \<sqsubseteq> a' \<and> b \<sqsubseteq> b')" |
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by (simp add: cpair_eq_pair less_cprod_def) |
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|
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lemma cpair_defined_iff [iff]: "(<x, y> = \<bottom>) = (x = \<bottom> \<and> y = \<bottom>)" |
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by (simp add: inst_cprod_pcpo cpair_eq_pair) |
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||
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lemma cpair_strict: "<\<bottom>, \<bottom>> = \<bottom>" |
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by simp |
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|
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lemma inst_cprod_pcpo2: "\<bottom> = <\<bottom>, \<bottom>>" |
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by (rule cpair_strict [symmetric]) |
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|
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lemma defined_cpair_rev: |
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"<a,b> = \<bottom> \<Longrightarrow> a = \<bottom> \<and> b = \<bottom>" |
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by simp |
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|
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lemma Exh_Cprod2: "\<exists>a b. z = <a, b>" |
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by (simp add: cpair_eq_pair) |
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|
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lemma cprodE: "\<lbrakk>\<And>x y. p = <x, y> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" |
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by (cut_tac Exh_Cprod2, auto) |
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|
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lemma cfst_cpair [simp]: "cfst\<cdot><x, y> = x" |
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by (simp add: cpair_eq_pair cfst_def cont_fst) |
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|
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lemma csnd_cpair [simp]: "csnd\<cdot><x, y> = y" |
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by (simp add: cpair_eq_pair csnd_def cont_snd) |
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|
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lemma cfst_strict [simp]: "cfst\<cdot>\<bottom> = \<bottom>" |
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by (simp add: inst_cprod_pcpo2) |
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|
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lemma csnd_strict [simp]: "csnd\<cdot>\<bottom> = \<bottom>" |
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by (simp add: inst_cprod_pcpo2) |
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|
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lemma surjective_pairing_Cprod2: "<cfst\<cdot>p, csnd\<cdot>p> = p" |
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apply (unfold cfst_def csnd_def) |
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apply (simp add: cont_fst cont_snd cpair_eq_pair) |
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done |
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|
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lemma less_cprod: "x \<sqsubseteq> y = (cfst\<cdot>x \<sqsubseteq> cfst\<cdot>y \<and> csnd\<cdot>x \<sqsubseteq> csnd\<cdot>y)" |
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by (simp add: less_cprod_def cfst_def csnd_def cont_fst cont_snd) |
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||
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lemma eq_cprod: "(x = y) = (cfst\<cdot>x = cfst\<cdot>y \<and> csnd\<cdot>x = csnd\<cdot>y)" |
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by (auto simp add: po_eq_conv less_cprod) |
|
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||
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lemma compact_cpair [simp]: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact <x, y>" |
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by (rule compactI, simp add: less_cprod) |
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||
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lemma lub_cprod2: |
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"chain S \<Longrightarrow> range S <<| <\<Squnion>i. cfst\<cdot>(S i), \<Squnion>i. csnd\<cdot>(S i)>" |
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apply (simp add: cpair_eq_pair cfst_def csnd_def cont_fst cont_snd) |
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apply (erule lub_cprod) |
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done |
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|
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lemma thelub_cprod2: |
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"chain S \<Longrightarrow> lub (range S) = <\<Squnion>i. cfst\<cdot>(S i), \<Squnion>i. csnd\<cdot>(S i)>" |
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by (rule lub_cprod2 [THEN thelubI]) |
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|
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lemma csplit1 [simp]: "csplit\<cdot>f\<cdot>\<bottom> = f\<cdot>\<bottom>\<cdot>\<bottom>" |
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by (simp add: csplit_def) |
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|
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lemma csplit2 [simp]: "csplit\<cdot>f\<cdot><x,y> = f\<cdot>x\<cdot>y" |
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by (simp add: csplit_def) |
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|
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lemma csplit3 [simp]: "csplit\<cdot>cpair\<cdot>z = z" |
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by (simp add: csplit_def surjective_pairing_Cprod2) |
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|
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lemmas Cprod_rews = cfst_cpair csnd_cpair csplit2 |
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|
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end |