src/HOL/HOLCF/Ssum.thy
 author wenzelm Sun Nov 02 17:16:01 2014 +0100 (2014-11-02) changeset 58880 0baae4311a9f parent 49759 ecf1d5d87e5e child 61378 3e04c9ca001a permissions -rw-r--r--
     1 (*  Title:      HOL/HOLCF/Ssum.thy

     2     Author:     Franz Regensburger

     3     Author:     Brian Huffman

     4 *)

     5

     6 section {* The type of strict sums *}

     7

     8 theory Ssum

     9 imports Tr

    10 begin

    11

    12 default_sort pcpo

    13

    14 subsection {* Definition of strict sum type *}

    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 (infixr "++" 10) = "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 (xsymbols)

    29   ssum  ("(_ \<oplus>/ _)" [21, 20] 20)

    30 type_notation (HTML output)

    31   ssum  ("(_ \<oplus>/ _)" [21, 20] 20)

    32

    33

    34 subsection {* Definitions of constructors *}

    35

    36 definition

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

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

    39

    40 definition

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

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

    43

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

    45 by (simp add: ssum_def seq_conv_if)

    46

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

    48 by (simp add: ssum_def seq_conv_if)

    49

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

    51 by (simp add: sinl_def cont_Abs_ssum Abs_ssum_inverse sinl_ssum)

    52

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

    54 by (simp add: sinr_def cont_Abs_ssum Abs_ssum_inverse sinr_ssum)

    55

    56 lemmas Rep_ssum_simps =

    57   Rep_ssum_inject [symmetric] below_ssum_def

    58   prod_eq_iff below_prod_def

    59   Rep_ssum_strict Rep_ssum_sinl Rep_ssum_sinr

    60

    61 subsection {* Properties of \emph{sinl} and \emph{sinr} *}

    62

    63 text {* Ordering *}

    64

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

    66 by (simp add: Rep_ssum_simps seq_conv_if)

    67

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

    69 by (simp add: Rep_ssum_simps seq_conv_if)

    70

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

    72 by (simp add: Rep_ssum_simps seq_conv_if)

    73

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

    75 by (simp add: Rep_ssum_simps seq_conv_if)

    76

    77 text {* Equality *}

    78

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

    80 by (simp add: po_eq_conv)

    81

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

    83 by (simp add: po_eq_conv)

    84

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

    86 by (subst po_eq_conv, simp)

    87

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

    89 by (subst po_eq_conv, simp)

    90

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

    92 by (rule sinl_eq [THEN iffD1])

    93

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

    95 by (rule sinr_eq [THEN iffD1])

    96

    97 text {* Strictness *}

    98

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

   100 by (simp add: Rep_ssum_simps)

   101

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

   103 by (simp add: Rep_ssum_simps)

   104

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

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

   107

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

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

   110

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

   112 by simp

   113

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

   115 by simp

   116

   117 text {* Compactness *}

   118

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

   120 by (rule compact_ssum, simp add: Rep_ssum_sinl)

   121

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

   123 by (rule compact_ssum, simp add: Rep_ssum_sinr)

   124

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

   126 unfolding compact_def

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

   128

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

   130 unfolding compact_def

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

   132

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

   134 by (safe elim!: compact_sinl compact_sinlD)

   135

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

   137 by (safe elim!: compact_sinr compact_sinrD)

   138

   139 subsection {* Case analysis *}

   140

   141 lemma ssumE [case_names bottom sinl sinr, cases type: ssum]:

   142   obtains "p = \<bottom>"

   143   | x where "p = sinl\<cdot>x" and "x \<noteq> \<bottom>"

   144   | y where "p = sinr\<cdot>y" and "y \<noteq> \<bottom>"

   145 using Rep_ssum [of p] by (auto simp add: ssum_def Rep_ssum_simps)

   146

   147 lemma ssum_induct [case_names bottom sinl sinr, induct type: ssum]:

   148   "\<lbrakk>P \<bottom>;

   149    \<And>x. x \<noteq> \<bottom> \<Longrightarrow> P (sinl\<cdot>x);

   150    \<And>y. y \<noteq> \<bottom> \<Longrightarrow> P (sinr\<cdot>y)\<rbrakk> \<Longrightarrow> P x"

   151 by (cases x, simp_all)

   152

   153 lemma ssumE2 [case_names sinl sinr]:

   154   "\<lbrakk>\<And>x. p = sinl\<cdot>x \<Longrightarrow> Q; \<And>y. p = sinr\<cdot>y \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"

   155 by (cases p, simp only: sinl_strict [symmetric], simp, simp)

   156

   157 lemma below_sinlD: "p \<sqsubseteq> sinl\<cdot>x \<Longrightarrow> \<exists>y. p = sinl\<cdot>y \<and> y \<sqsubseteq> x"

   158 by (cases p, rule_tac x="\<bottom>" in exI, simp_all)

   159

   160 lemma below_sinrD: "p \<sqsubseteq> sinr\<cdot>x \<Longrightarrow> \<exists>y. p = sinr\<cdot>y \<and> y \<sqsubseteq> x"

   161 by (cases p, rule_tac x="\<bottom>" in exI, simp_all)

   162

   163 subsection {* Case analysis combinator *}

   164

   165 definition

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

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

   168

   169 translations

   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"

   171   "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"

   172

   173 translations

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

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

   176

   177 lemma beta_sscase:

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

   179 unfolding sscase_def by (simp add: cont_Rep_ssum)

   180

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

   182 unfolding beta_sscase by (simp add: Rep_ssum_strict)

   183

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

   185 unfolding beta_sscase by (simp add: Rep_ssum_sinl)

   186

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

   188 unfolding beta_sscase by (simp add: Rep_ssum_sinr)

   189

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

   191 by (cases z, simp_all)

   192

   193 subsection {* Strict sum preserves flatness *}

   194

   195 instance ssum :: (flat, flat) flat

   196 apply (intro_classes, clarify)

   197 apply (case_tac x, simp)

   198 apply (case_tac y, simp_all add: flat_below_iff)

   199 apply (case_tac y, simp_all add: flat_below_iff)

   200 done

   201

   202 end