1(* Title: HOL/HOLCF/Ssum.thy 2 Author: Franz Regensburger 3 Author: Brian Huffman 4*) 5 6section \<open>The type of strict sums\<close> 7 8theory Ssum 9 imports Tr 10begin 11 12default_sort pcpo 13 14 15subsection \<open>Definition of strict sum type\<close> 16 17definition "ssum = 18 {p :: tr \<times> ('a \<times> 'b). p = \<bottom> \<or> 19 (fst p = TT \<and> fst (snd p) \<noteq> \<bottom> \<and> snd (snd p) = \<bottom>) \<or> 20 (fst p = FF \<and> fst (snd p) = \<bottom> \<and> snd (snd p) \<noteq> \<bottom>)}" 21 22pcpodef ('a, 'b) ssum ("(_ \<oplus>/ _)" [21, 20] 20) = "ssum :: (tr \<times> 'a \<times> 'b) set" 23 by (simp_all add: ssum_def) 24 25instance ssum :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin 26 by (rule typedef_chfin [OF type_definition_ssum below_ssum_def]) 27 28type_notation (ASCII) 29 ssum (infixr "++" 10) 30 31 32subsection \<open>Definitions of constructors\<close> 33 34definition sinl :: "'a \<rightarrow> ('a ++ 'b)" 35 where "sinl = (\<Lambda> a. Abs_ssum (seq\<cdot>a\<cdot>TT, a, \<bottom>))" 36 37definition sinr :: "'b \<rightarrow> ('a ++ 'b)" 38 where "sinr = (\<Lambda> b. Abs_ssum (seq\<cdot>b\<cdot>FF, \<bottom>, b))" 39 40lemma sinl_ssum: "(seq\<cdot>a\<cdot>TT, a, \<bottom>) \<in> ssum" 41 by (simp add: ssum_def seq_conv_if) 42 43lemma sinr_ssum: "(seq\<cdot>b\<cdot>FF, \<bottom>, b) \<in> ssum" 44 by (simp add: ssum_def seq_conv_if) 45 46lemma Rep_ssum_sinl: "Rep_ssum (sinl\<cdot>a) = (seq\<cdot>a\<cdot>TT, a, \<bottom>)" 47 by (simp add: sinl_def cont_Abs_ssum Abs_ssum_inverse sinl_ssum) 48 49lemma Rep_ssum_sinr: "Rep_ssum (sinr\<cdot>b) = (seq\<cdot>b\<cdot>FF, \<bottom>, b)" 50 by (simp add: sinr_def cont_Abs_ssum Abs_ssum_inverse sinr_ssum) 51 52lemmas Rep_ssum_simps = 53 Rep_ssum_inject [symmetric] below_ssum_def 54 prod_eq_iff below_prod_def 55 Rep_ssum_strict Rep_ssum_sinl Rep_ssum_sinr 56 57 58subsection \<open>Properties of \emph{sinl} and \emph{sinr}\<close> 59 60text \<open>Ordering\<close> 61 62lemma sinl_below [simp]: "sinl\<cdot>x \<sqsubseteq> sinl\<cdot>y \<longleftrightarrow> x \<sqsubseteq> y" 63 by (simp add: Rep_ssum_simps seq_conv_if) 64 65lemma sinr_below [simp]: "sinr\<cdot>x \<sqsubseteq> sinr\<cdot>y \<longleftrightarrow> x \<sqsubseteq> y" 66 by (simp add: Rep_ssum_simps seq_conv_if) 67 68lemma sinl_below_sinr [simp]: "sinl\<cdot>x \<sqsubseteq> sinr\<cdot>y \<longleftrightarrow> x = \<bottom>" 69 by (simp add: Rep_ssum_simps seq_conv_if) 70 71lemma sinr_below_sinl [simp]: "sinr\<cdot>x \<sqsubseteq> sinl\<cdot>y \<longleftrightarrow> x = \<bottom>" 72 by (simp add: Rep_ssum_simps seq_conv_if) 73 74text \<open>Equality\<close> 75 76lemma sinl_eq [simp]: "sinl\<cdot>x = sinl\<cdot>y \<longleftrightarrow> x = y" 77 by (simp add: po_eq_conv) 78 79lemma sinr_eq [simp]: "sinr\<cdot>x = sinr\<cdot>y \<longleftrightarrow> x = y" 80 by (simp add: po_eq_conv) 81 82lemma sinl_eq_sinr [simp]: "sinl\<cdot>x = sinr\<cdot>y \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>" 83 by (subst po_eq_conv) simp 84 85lemma sinr_eq_sinl [simp]: "sinr\<cdot>x = sinl\<cdot>y \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>" 86 by (subst po_eq_conv) simp 87 88lemma sinl_inject: "sinl\<cdot>x = sinl\<cdot>y \<Longrightarrow> x = y" 89 by (rule sinl_eq [THEN iffD1]) 90 91lemma sinr_inject: "sinr\<cdot>x = sinr\<cdot>y \<Longrightarrow> x = y" 92 by (rule sinr_eq [THEN iffD1]) 93 94text \<open>Strictness\<close> 95 96lemma sinl_strict [simp]: "sinl\<cdot>\<bottom> = \<bottom>" 97 by (simp add: Rep_ssum_simps) 98 99lemma sinr_strict [simp]: "sinr\<cdot>\<bottom> = \<bottom>" 100 by (simp add: Rep_ssum_simps) 101 102lemma sinl_bottom_iff [simp]: "sinl\<cdot>x = \<bottom> \<longleftrightarrow> x = \<bottom>" 103 using sinl_eq [of "x" "\<bottom>"] by simp 104 105lemma sinr_bottom_iff [simp]: "sinr\<cdot>x = \<bottom> \<longleftrightarrow> x = \<bottom>" 106 using sinr_eq [of "x" "\<bottom>"] by simp 107 108lemma sinl_defined: "x \<noteq> \<bottom> \<Longrightarrow> sinl\<cdot>x \<noteq> \<bottom>" 109 by simp 110 111lemma sinr_defined: "x \<noteq> \<bottom> \<Longrightarrow> sinr\<cdot>x \<noteq> \<bottom>" 112 by simp 113 114text \<open>Compactness\<close> 115 116lemma compact_sinl: "compact x \<Longrightarrow> compact (sinl\<cdot>x)" 117 by (rule compact_ssum) (simp add: Rep_ssum_sinl) 118 119lemma compact_sinr: "compact x \<Longrightarrow> compact (sinr\<cdot>x)" 120 by (rule compact_ssum) (simp add: Rep_ssum_sinr) 121 122lemma compact_sinlD: "compact (sinl\<cdot>x) \<Longrightarrow> compact x" 123 unfolding compact_def 124 by (drule adm_subst [OF cont_Rep_cfun2 [where f=sinl]], simp) 125 126lemma compact_sinrD: "compact (sinr\<cdot>x) \<Longrightarrow> compact x" 127 unfolding compact_def 128 by (drule adm_subst [OF cont_Rep_cfun2 [where f=sinr]], simp) 129 130lemma compact_sinl_iff [simp]: "compact (sinl\<cdot>x) = compact x" 131 by (safe elim!: compact_sinl compact_sinlD) 132 133lemma compact_sinr_iff [simp]: "compact (sinr\<cdot>x) = compact x" 134 by (safe elim!: compact_sinr compact_sinrD) 135 136 137subsection \<open>Case analysis\<close> 138 139lemma ssumE [case_names bottom sinl sinr, cases type: ssum]: 140 obtains "p = \<bottom>" 141 | x where "p = sinl\<cdot>x" and "x \<noteq> \<bottom>" 142 | y where "p = sinr\<cdot>y" and "y \<noteq> \<bottom>" 143 using Rep_ssum [of p] by (auto simp add: ssum_def Rep_ssum_simps) 144 145lemma ssum_induct [case_names bottom sinl sinr, induct type: ssum]: 146 "\<lbrakk>P \<bottom>; 147 \<And>x. x \<noteq> \<bottom> \<Longrightarrow> P (sinl\<cdot>x); 148 \<And>y. y \<noteq> \<bottom> \<Longrightarrow> P (sinr\<cdot>y)\<rbrakk> \<Longrightarrow> P x" 149 by (cases x) simp_all 150 151lemma ssumE2 [case_names sinl sinr]: 152 "\<lbrakk>\<And>x. p = sinl\<cdot>x \<Longrightarrow> Q; \<And>y. p = sinr\<cdot>y \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" 153 by (cases p, simp only: sinl_strict [symmetric], simp, simp) 154 155lemma below_sinlD: "p \<sqsubseteq> sinl\<cdot>x \<Longrightarrow> \<exists>y. p = sinl\<cdot>y \<and> y \<sqsubseteq> x" 156 by (cases p, rule_tac x="\<bottom>" in exI, simp_all) 157 158lemma below_sinrD: "p \<sqsubseteq> sinr\<cdot>x \<Longrightarrow> \<exists>y. p = sinr\<cdot>y \<and> y \<sqsubseteq> x" 159 by (cases p, rule_tac x="\<bottom>" in exI, simp_all) 160 161 162subsection \<open>Case analysis combinator\<close> 163 164definition sscase :: "('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a ++ 'b) \<rightarrow> 'c" 165 where "sscase = (\<Lambda> f g s. (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y) (Rep_ssum s))" 166 167translations 168 "case s of XCONST sinl\<cdot>x \<Rightarrow> t1 | XCONST sinr\<cdot>y \<Rightarrow> t2" \<rightleftharpoons> "CONST sscase\<cdot>(\<Lambda> x. t1)\<cdot>(\<Lambda> y. t2)\<cdot>s" 169 "case s of (XCONST sinl :: 'a)\<cdot>x \<Rightarrow> t1 | XCONST sinr\<cdot>y \<Rightarrow> t2" \<rightharpoonup> "CONST sscase\<cdot>(\<Lambda> x. t1)\<cdot>(\<Lambda> y. t2)\<cdot>s" 170 171translations 172 "\<Lambda>(XCONST sinl\<cdot>x). t" \<rightleftharpoons> "CONST sscase\<cdot>(\<Lambda> x. t)\<cdot>\<bottom>" 173 "\<Lambda>(XCONST sinr\<cdot>y). t" \<rightleftharpoons> "CONST sscase\<cdot>\<bottom>\<cdot>(\<Lambda> y. t)" 174 175lemma beta_sscase: "sscase\<cdot>f\<cdot>g\<cdot>s = (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y) (Rep_ssum s)" 176 by (simp add: sscase_def cont_Rep_ssum) 177 178lemma sscase1 [simp]: "sscase\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>" 179 by (simp add: beta_sscase Rep_ssum_strict) 180 181lemma sscase2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = f\<cdot>x" 182 by (simp add: beta_sscase Rep_ssum_sinl) 183 184lemma sscase3 [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>y) = g\<cdot>y" 185 by (simp add: beta_sscase Rep_ssum_sinr) 186 187lemma sscase4 [simp]: "sscase\<cdot>sinl\<cdot>sinr\<cdot>z = z" 188 by (cases z) simp_all 189 190 191subsection \<open>Strict sum preserves flatness\<close> 192 193instance ssum :: (flat, flat) flat 194 apply (intro_classes, clarify) 195 apply (case_tac x, simp) 196 apply (case_tac y, simp_all add: flat_below_iff) 197 apply (case_tac y, simp_all add: flat_below_iff) 198 done 199 200end 201