1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/HOLCF/Ssum.thy Sat Nov 27 16:08:10 2010 -0800
1.3 @@ -0,0 +1,198 @@
1.4 +(* Title: HOLCF/Ssum.thy
1.5 + Author: Franz Regensburger
1.6 + Author: Brian Huffman
1.7 +*)
1.8 +
1.9 +header {* The type of strict sums *}
1.10 +
1.11 +theory Ssum
1.12 +imports Tr
1.13 +begin
1.14 +
1.15 +default_sort pcpo
1.16 +
1.17 +subsection {* Definition of strict sum type *}
1.18 +
1.19 +pcpodef ('a, 'b) ssum (infixr "++" 10) =
1.20 + "{p :: tr \<times> ('a \<times> 'b). p = \<bottom> \<or>
1.21 + (fst p = TT \<and> fst (snd p) \<noteq> \<bottom> \<and> snd (snd p) = \<bottom>) \<or>
1.22 + (fst p = FF \<and> fst (snd p) = \<bottom> \<and> snd (snd p) \<noteq> \<bottom>) }"
1.23 +by simp_all
1.24 +
1.25 +instance ssum :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin
1.26 +by (rule typedef_chfin [OF type_definition_ssum below_ssum_def])
1.27 +
1.28 +type_notation (xsymbols)
1.29 + ssum ("(_ \<oplus>/ _)" [21, 20] 20)
1.30 +type_notation (HTML output)
1.31 + ssum ("(_ \<oplus>/ _)" [21, 20] 20)
1.32 +
1.33 +
1.34 +subsection {* Definitions of constructors *}
1.35 +
1.36 +definition
1.37 + sinl :: "'a \<rightarrow> ('a ++ 'b)" where
1.38 + "sinl = (\<Lambda> a. Abs_ssum (seq\<cdot>a\<cdot>TT, a, \<bottom>))"
1.39 +
1.40 +definition
1.41 + sinr :: "'b \<rightarrow> ('a ++ 'b)" where
1.42 + "sinr = (\<Lambda> b. Abs_ssum (seq\<cdot>b\<cdot>FF, \<bottom>, b))"
1.43 +
1.44 +lemma sinl_ssum: "(seq\<cdot>a\<cdot>TT, a, \<bottom>) \<in> ssum"
1.45 +by (simp add: ssum_def seq_conv_if)
1.46 +
1.47 +lemma sinr_ssum: "(seq\<cdot>b\<cdot>FF, \<bottom>, b) \<in> ssum"
1.48 +by (simp add: ssum_def seq_conv_if)
1.49 +
1.50 +lemma Rep_ssum_sinl: "Rep_ssum (sinl\<cdot>a) = (seq\<cdot>a\<cdot>TT, a, \<bottom>)"
1.51 +by (simp add: sinl_def cont_Abs_ssum Abs_ssum_inverse sinl_ssum)
1.52 +
1.53 +lemma Rep_ssum_sinr: "Rep_ssum (sinr\<cdot>b) = (seq\<cdot>b\<cdot>FF, \<bottom>, b)"
1.54 +by (simp add: sinr_def cont_Abs_ssum Abs_ssum_inverse sinr_ssum)
1.55 +
1.56 +lemmas Rep_ssum_simps =
1.57 + Rep_ssum_inject [symmetric] below_ssum_def
1.58 + Pair_fst_snd_eq below_prod_def
1.59 + Rep_ssum_strict Rep_ssum_sinl Rep_ssum_sinr
1.60 +
1.61 +subsection {* Properties of \emph{sinl} and \emph{sinr} *}
1.62 +
1.63 +text {* Ordering *}
1.64 +
1.65 +lemma sinl_below [simp]: "(sinl\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x \<sqsubseteq> y)"
1.66 +by (simp add: Rep_ssum_simps seq_conv_if)
1.67 +
1.68 +lemma sinr_below [simp]: "(sinr\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x \<sqsubseteq> y)"
1.69 +by (simp add: Rep_ssum_simps seq_conv_if)
1.70 +
1.71 +lemma sinl_below_sinr [simp]: "(sinl\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x = \<bottom>)"
1.72 +by (simp add: Rep_ssum_simps seq_conv_if)
1.73 +
1.74 +lemma sinr_below_sinl [simp]: "(sinr\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x = \<bottom>)"
1.75 +by (simp add: Rep_ssum_simps seq_conv_if)
1.76 +
1.77 +text {* Equality *}
1.78 +
1.79 +lemma sinl_eq [simp]: "(sinl\<cdot>x = sinl\<cdot>y) = (x = y)"
1.80 +by (simp add: po_eq_conv)
1.81 +
1.82 +lemma sinr_eq [simp]: "(sinr\<cdot>x = sinr\<cdot>y) = (x = y)"
1.83 +by (simp add: po_eq_conv)
1.84 +
1.85 +lemma sinl_eq_sinr [simp]: "(sinl\<cdot>x = sinr\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)"
1.86 +by (subst po_eq_conv, simp)
1.87 +
1.88 +lemma sinr_eq_sinl [simp]: "(sinr\<cdot>x = sinl\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)"
1.89 +by (subst po_eq_conv, simp)
1.90 +
1.91 +lemma sinl_inject: "sinl\<cdot>x = sinl\<cdot>y \<Longrightarrow> x = y"
1.92 +by (rule sinl_eq [THEN iffD1])
1.93 +
1.94 +lemma sinr_inject: "sinr\<cdot>x = sinr\<cdot>y \<Longrightarrow> x = y"
1.95 +by (rule sinr_eq [THEN iffD1])
1.96 +
1.97 +text {* Strictness *}
1.98 +
1.99 +lemma sinl_strict [simp]: "sinl\<cdot>\<bottom> = \<bottom>"
1.100 +by (simp add: Rep_ssum_simps)
1.101 +
1.102 +lemma sinr_strict [simp]: "sinr\<cdot>\<bottom> = \<bottom>"
1.103 +by (simp add: Rep_ssum_simps)
1.104 +
1.105 +lemma sinl_bottom_iff [simp]: "(sinl\<cdot>x = \<bottom>) = (x = \<bottom>)"
1.106 +using sinl_eq [of "x" "\<bottom>"] by simp
1.107 +
1.108 +lemma sinr_bottom_iff [simp]: "(sinr\<cdot>x = \<bottom>) = (x = \<bottom>)"
1.109 +using sinr_eq [of "x" "\<bottom>"] by simp
1.110 +
1.111 +lemma sinl_defined: "x \<noteq> \<bottom> \<Longrightarrow> sinl\<cdot>x \<noteq> \<bottom>"
1.112 +by simp
1.113 +
1.114 +lemma sinr_defined: "x \<noteq> \<bottom> \<Longrightarrow> sinr\<cdot>x \<noteq> \<bottom>"
1.115 +by simp
1.116 +
1.117 +text {* Compactness *}
1.118 +
1.119 +lemma compact_sinl: "compact x \<Longrightarrow> compact (sinl\<cdot>x)"
1.120 +by (rule compact_ssum, simp add: Rep_ssum_sinl)
1.121 +
1.122 +lemma compact_sinr: "compact x \<Longrightarrow> compact (sinr\<cdot>x)"
1.123 +by (rule compact_ssum, simp add: Rep_ssum_sinr)
1.124 +
1.125 +lemma compact_sinlD: "compact (sinl\<cdot>x) \<Longrightarrow> compact x"
1.126 +unfolding compact_def
1.127 +by (drule adm_subst [OF cont_Rep_cfun2 [where f=sinl]], simp)
1.128 +
1.129 +lemma compact_sinrD: "compact (sinr\<cdot>x) \<Longrightarrow> compact x"
1.130 +unfolding compact_def
1.131 +by (drule adm_subst [OF cont_Rep_cfun2 [where f=sinr]], simp)
1.132 +
1.133 +lemma compact_sinl_iff [simp]: "compact (sinl\<cdot>x) = compact x"
1.134 +by (safe elim!: compact_sinl compact_sinlD)
1.135 +
1.136 +lemma compact_sinr_iff [simp]: "compact (sinr\<cdot>x) = compact x"
1.137 +by (safe elim!: compact_sinr compact_sinrD)
1.138 +
1.139 +subsection {* Case analysis *}
1.140 +
1.141 +lemma ssumE [case_names bottom sinl sinr, cases type: ssum]:
1.142 + obtains "p = \<bottom>"
1.143 + | x where "p = sinl\<cdot>x" and "x \<noteq> \<bottom>"
1.144 + | y where "p = sinr\<cdot>y" and "y \<noteq> \<bottom>"
1.145 +using Rep_ssum [of p] by (auto simp add: ssum_def Rep_ssum_simps)
1.146 +
1.147 +lemma ssum_induct [case_names bottom sinl sinr, induct type: ssum]:
1.148 + "\<lbrakk>P \<bottom>;
1.149 + \<And>x. x \<noteq> \<bottom> \<Longrightarrow> P (sinl\<cdot>x);
1.150 + \<And>y. y \<noteq> \<bottom> \<Longrightarrow> P (sinr\<cdot>y)\<rbrakk> \<Longrightarrow> P x"
1.151 +by (cases x, simp_all)
1.152 +
1.153 +lemma ssumE2 [case_names sinl sinr]:
1.154 + "\<lbrakk>\<And>x. p = sinl\<cdot>x \<Longrightarrow> Q; \<And>y. p = sinr\<cdot>y \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
1.155 +by (cases p, simp only: sinl_strict [symmetric], simp, simp)
1.156 +
1.157 +lemma below_sinlD: "p \<sqsubseteq> sinl\<cdot>x \<Longrightarrow> \<exists>y. p = sinl\<cdot>y \<and> y \<sqsubseteq> x"
1.158 +by (cases p, rule_tac x="\<bottom>" in exI, simp_all)
1.159 +
1.160 +lemma below_sinrD: "p \<sqsubseteq> sinr\<cdot>x \<Longrightarrow> \<exists>y. p = sinr\<cdot>y \<and> y \<sqsubseteq> x"
1.161 +by (cases p, rule_tac x="\<bottom>" in exI, simp_all)
1.162 +
1.163 +subsection {* Case analysis combinator *}
1.164 +
1.165 +definition
1.166 + sscase :: "('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a ++ 'b) \<rightarrow> 'c" where
1.167 + "sscase = (\<Lambda> f g s. (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y) (Rep_ssum s))"
1.168 +
1.169 +translations
1.170 + "case s of XCONST sinl\<cdot>x \<Rightarrow> t1 | XCONST sinr\<cdot>y \<Rightarrow> t2" == "CONST sscase\<cdot>(\<Lambda> x. t1)\<cdot>(\<Lambda> y. t2)\<cdot>s"
1.171 +
1.172 +translations
1.173 + "\<Lambda>(XCONST sinl\<cdot>x). t" == "CONST sscase\<cdot>(\<Lambda> x. t)\<cdot>\<bottom>"
1.174 + "\<Lambda>(XCONST sinr\<cdot>y). t" == "CONST sscase\<cdot>\<bottom>\<cdot>(\<Lambda> y. t)"
1.175 +
1.176 +lemma beta_sscase:
1.177 + "sscase\<cdot>f\<cdot>g\<cdot>s = (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y) (Rep_ssum s)"
1.178 +unfolding sscase_def by (simp add: cont_Rep_ssum [THEN cont_compose])
1.179 +
1.180 +lemma sscase1 [simp]: "sscase\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>"
1.181 +unfolding beta_sscase by (simp add: Rep_ssum_strict)
1.182 +
1.183 +lemma sscase2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = f\<cdot>x"
1.184 +unfolding beta_sscase by (simp add: Rep_ssum_sinl)
1.185 +
1.186 +lemma sscase3 [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>y) = g\<cdot>y"
1.187 +unfolding beta_sscase by (simp add: Rep_ssum_sinr)
1.188 +
1.189 +lemma sscase4 [simp]: "sscase\<cdot>sinl\<cdot>sinr\<cdot>z = z"
1.190 +by (cases z, simp_all)
1.191 +
1.192 +subsection {* Strict sum preserves flatness *}
1.193 +
1.194 +instance ssum :: (flat, flat) flat
1.195 +apply (intro_classes, clarify)
1.196 +apply (case_tac x, simp)
1.197 +apply (case_tac y, simp_all add: flat_below_iff)
1.198 +apply (case_tac y, simp_all add: flat_below_iff)
1.199 +done
1.200 +
1.201 +end