src/HOL/HOLCF/Ssum.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 62175 8ffc4d0e652d child 67312 0d25e02759b7 permissions -rw-r--r--
tuned proofs;
     1 (*  Title:      HOL/HOLCF/Ssum.thy

     2     Author:     Franz Regensburger

     3     Author:     Brian Huffman

     4 *)

     5

     6 section \<open>The type of strict sums\<close>

     7

     8 theory Ssum

     9 imports Tr

    10 begin

    11

    12 default_sort pcpo

    13

    14 subsection \<open>Definition of strict sum type\<close>

    15

    16 definition

    17   "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

    22 pcpodef ('a, 'b) ssum  ("(_ \<oplus>/ _)" [21, 20] 20) = "ssum :: (tr \<times> 'a \<times> 'b) set"

    23   unfolding ssum_def by simp_all

    24

    25 instance ssum :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin

    26 by (rule typedef_chfin [OF type_definition_ssum below_ssum_def])

    27

    28 type_notation (ASCII)

    29   ssum  (infixr "++" 10)

    30

    31

    32 subsection \<open>Definitions of constructors\<close>

    33

    34 definition

    35   sinl :: "'a \<rightarrow> ('a ++ 'b)" where

    36   "sinl = (\<Lambda> a. Abs_ssum (seq\<cdot>a\<cdot>TT, a, \<bottom>))"

    37

    38 definition

    39   sinr :: "'b \<rightarrow> ('a ++ 'b)" where

    40   "sinr = (\<Lambda> b. Abs_ssum (seq\<cdot>b\<cdot>FF, \<bottom>, b))"

    41

    42 lemma sinl_ssum: "(seq\<cdot>a\<cdot>TT, a, \<bottom>) \<in> ssum"

    43 by (simp add: ssum_def seq_conv_if)

    44

    45 lemma sinr_ssum: "(seq\<cdot>b\<cdot>FF, \<bottom>, b) \<in> ssum"

    46 by (simp add: ssum_def seq_conv_if)

    47

    48 lemma Rep_ssum_sinl: "Rep_ssum (sinl\<cdot>a) = (seq\<cdot>a\<cdot>TT, a, \<bottom>)"

    49 by (simp add: sinl_def cont_Abs_ssum Abs_ssum_inverse sinl_ssum)

    50

    51 lemma Rep_ssum_sinr: "Rep_ssum (sinr\<cdot>b) = (seq\<cdot>b\<cdot>FF, \<bottom>, b)"

    52 by (simp add: sinr_def cont_Abs_ssum Abs_ssum_inverse sinr_ssum)

    53

    54 lemmas Rep_ssum_simps =

    55   Rep_ssum_inject [symmetric] below_ssum_def

    56   prod_eq_iff below_prod_def

    57   Rep_ssum_strict Rep_ssum_sinl Rep_ssum_sinr

    58

    59 subsection \<open>Properties of \emph{sinl} and \emph{sinr}\<close>

    60

    61 text \<open>Ordering\<close>

    62

    63 lemma sinl_below [simp]: "(sinl\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x \<sqsubseteq> y)"

    64 by (simp add: Rep_ssum_simps seq_conv_if)

    65

    66 lemma sinr_below [simp]: "(sinr\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x \<sqsubseteq> y)"

    67 by (simp add: Rep_ssum_simps seq_conv_if)

    68

    69 lemma sinl_below_sinr [simp]: "(sinl\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x = \<bottom>)"

    70 by (simp add: Rep_ssum_simps seq_conv_if)

    71

    72 lemma sinr_below_sinl [simp]: "(sinr\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x = \<bottom>)"

    73 by (simp add: Rep_ssum_simps seq_conv_if)

    74

    75 text \<open>Equality\<close>

    76

    77 lemma sinl_eq [simp]: "(sinl\<cdot>x = sinl\<cdot>y) = (x = y)"

    78 by (simp add: po_eq_conv)

    79

    80 lemma sinr_eq [simp]: "(sinr\<cdot>x = sinr\<cdot>y) = (x = y)"

    81 by (simp add: po_eq_conv)

    82

    83 lemma sinl_eq_sinr [simp]: "(sinl\<cdot>x = sinr\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)"

    84 by (subst po_eq_conv, simp)

    85

    86 lemma sinr_eq_sinl [simp]: "(sinr\<cdot>x = sinl\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)"

    87 by (subst po_eq_conv, simp)

    88

    89 lemma sinl_inject: "sinl\<cdot>x = sinl\<cdot>y \<Longrightarrow> x = y"

    90 by (rule sinl_eq [THEN iffD1])

    91

    92 lemma sinr_inject: "sinr\<cdot>x = sinr\<cdot>y \<Longrightarrow> x = y"

    93 by (rule sinr_eq [THEN iffD1])

    94

    95 text \<open>Strictness\<close>

    96

    97 lemma sinl_strict [simp]: "sinl\<cdot>\<bottom> = \<bottom>"

    98 by (simp add: Rep_ssum_simps)

    99

   100 lemma sinr_strict [simp]: "sinr\<cdot>\<bottom> = \<bottom>"

   101 by (simp add: Rep_ssum_simps)

   102

   103 lemma sinl_bottom_iff [simp]: "(sinl\<cdot>x = \<bottom>) = (x = \<bottom>)"

   104 using sinl_eq [of "x" "\<bottom>"] by simp

   105

   106 lemma sinr_bottom_iff [simp]: "(sinr\<cdot>x = \<bottom>) = (x = \<bottom>)"

   107 using sinr_eq [of "x" "\<bottom>"] by simp

   108

   109 lemma sinl_defined: "x \<noteq> \<bottom> \<Longrightarrow> sinl\<cdot>x \<noteq> \<bottom>"

   110 by simp

   111

   112 lemma sinr_defined: "x \<noteq> \<bottom> \<Longrightarrow> sinr\<cdot>x \<noteq> \<bottom>"

   113 by simp

   114

   115 text \<open>Compactness\<close>

   116

   117 lemma compact_sinl: "compact x \<Longrightarrow> compact (sinl\<cdot>x)"

   118 by (rule compact_ssum, simp add: Rep_ssum_sinl)

   119

   120 lemma compact_sinr: "compact x \<Longrightarrow> compact (sinr\<cdot>x)"

   121 by (rule compact_ssum, simp add: Rep_ssum_sinr)

   122

   123 lemma compact_sinlD: "compact (sinl\<cdot>x) \<Longrightarrow> compact x"

   124 unfolding compact_def

   125 by (drule adm_subst [OF cont_Rep_cfun2 [where f=sinl]], simp)

   126

   127 lemma compact_sinrD: "compact (sinr\<cdot>x) \<Longrightarrow> compact x"

   128 unfolding compact_def

   129 by (drule adm_subst [OF cont_Rep_cfun2 [where f=sinr]], simp)

   130

   131 lemma compact_sinl_iff [simp]: "compact (sinl\<cdot>x) = compact x"

   132 by (safe elim!: compact_sinl compact_sinlD)

   133

   134 lemma compact_sinr_iff [simp]: "compact (sinr\<cdot>x) = compact x"

   135 by (safe elim!: compact_sinr compact_sinrD)

   136

   137 subsection \<open>Case analysis\<close>

   138

   139 lemma 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

   145 lemma 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

   151 lemma 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

   155 lemma 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

   158 lemma 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 subsection \<open>Case analysis combinator\<close>

   162

   163 definition

   164   sscase :: "('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a ++ 'b) \<rightarrow> 'c" where

   165   "sscase = (\<Lambda> f g s. (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y) (Rep_ssum s))"

   166

   167 translations

   168   "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"

   169   "case s of (XCONST sinl :: 'a)\<cdot>x \<Rightarrow> t1 | XCONST sinr\<cdot>y \<Rightarrow> t2" => "CONST sscase\<cdot>(\<Lambda> x. t1)\<cdot>(\<Lambda> y. t2)\<cdot>s"

   170

   171 translations

   172   "\<Lambda>(XCONST sinl\<cdot>x). t" == "CONST sscase\<cdot>(\<Lambda> x. t)\<cdot>\<bottom>"

   173   "\<Lambda>(XCONST sinr\<cdot>y). t" == "CONST sscase\<cdot>\<bottom>\<cdot>(\<Lambda> y. t)"

   174

   175 lemma beta_sscase:

   176   "sscase\<cdot>f\<cdot>g\<cdot>s = (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y) (Rep_ssum s)"

   177 unfolding sscase_def by (simp add: cont_Rep_ssum)

   178

   179 lemma sscase1 [simp]: "sscase\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>"

   180 unfolding beta_sscase by (simp add: Rep_ssum_strict)

   181

   182 lemma sscase2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = f\<cdot>x"

   183 unfolding beta_sscase by (simp add: Rep_ssum_sinl)

   184

   185 lemma sscase3 [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>y) = g\<cdot>y"

   186 unfolding beta_sscase by (simp add: Rep_ssum_sinr)

   187

   188 lemma sscase4 [simp]: "sscase\<cdot>sinl\<cdot>sinr\<cdot>z = z"

   189 by (cases z, simp_all)

   190

   191 subsection \<open>Strict sum preserves flatness\<close>

   192

   193 instance 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

   200 end