src/HOL/HOLCF/Cpodef.thy
author wenzelm
Thu Mar 14 16:55:06 2019 +0100 (5 weeks ago)
changeset 69913 ca515cf61651
parent 69605 a96320074298
permissions -rw-r--r--
more specific keyword kinds;
     1 (*  Title:      HOL/HOLCF/Cpodef.thy
     2     Author:     Brian Huffman
     3 *)
     4 
     5 section \<open>Subtypes of pcpos\<close>
     6 
     7 theory Cpodef
     8   imports Adm
     9   keywords "pcpodef" "cpodef" :: thy_goal_defn
    10 begin
    11 
    12 subsection \<open>Proving a subtype is a partial order\<close>
    13 
    14 text \<open>
    15   A subtype of a partial order is itself a partial order,
    16   if the ordering is defined in the standard way.
    17 \<close>
    18 
    19 setup \<open>Sign.add_const_constraint (\<^const_name>\<open>Porder.below\<close>, NONE)\<close>
    20 
    21 theorem typedef_po:
    22   fixes Abs :: "'a::po \<Rightarrow> 'b::type"
    23   assumes type: "type_definition Rep Abs A"
    24     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    25   shows "OFCLASS('b, po_class)"
    26   apply (intro_classes, unfold below)
    27     apply (rule below_refl)
    28    apply (erule (1) below_trans)
    29   apply (rule type_definition.Rep_inject [OF type, THEN iffD1])
    30   apply (erule (1) below_antisym)
    31   done
    32 
    33 setup \<open>Sign.add_const_constraint (\<^const_name>\<open>Porder.below\<close>, SOME \<^typ>\<open>'a::below \<Rightarrow> 'a::below \<Rightarrow> bool\<close>)\<close>
    34 
    35 
    36 subsection \<open>Proving a subtype is finite\<close>
    37 
    38 lemma typedef_finite_UNIV:
    39   fixes Abs :: "'a::type \<Rightarrow> 'b::type"
    40   assumes type: "type_definition Rep Abs A"
    41   shows "finite A \<Longrightarrow> finite (UNIV :: 'b set)"
    42 proof -
    43   assume "finite A"
    44   then have "finite (Abs ` A)"
    45     by (rule finite_imageI)
    46   then show "finite (UNIV :: 'b set)"
    47     by (simp only: type_definition.Abs_image [OF type])
    48 qed
    49 
    50 
    51 subsection \<open>Proving a subtype is chain-finite\<close>
    52 
    53 lemma ch2ch_Rep:
    54   assumes below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    55   shows "chain S \<Longrightarrow> chain (\<lambda>i. Rep (S i))"
    56   unfolding chain_def below .
    57 
    58 theorem typedef_chfin:
    59   fixes Abs :: "'a::chfin \<Rightarrow> 'b::po"
    60   assumes type: "type_definition Rep Abs A"
    61     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    62   shows "OFCLASS('b, chfin_class)"
    63   apply intro_classes
    64   apply (drule ch2ch_Rep [OF below])
    65   apply (drule chfin)
    66   apply (unfold max_in_chain_def)
    67   apply (simp add: type_definition.Rep_inject [OF type])
    68   done
    69 
    70 
    71 subsection \<open>Proving a subtype is complete\<close>
    72 
    73 text \<open>
    74   A subtype of a cpo is itself a cpo if the ordering is
    75   defined in the standard way, and the defining subset
    76   is closed with respect to limits of chains.  A set is
    77   closed if and only if membership in the set is an
    78   admissible predicate.
    79 \<close>
    80 
    81 lemma typedef_is_lubI:
    82   assumes below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    83   shows "range (\<lambda>i. Rep (S i)) <<| Rep x \<Longrightarrow> range S <<| x"
    84   by (simp add: is_lub_def is_ub_def below)
    85 
    86 lemma Abs_inverse_lub_Rep:
    87   fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
    88   assumes type: "type_definition Rep Abs A"
    89     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
    90     and adm:  "adm (\<lambda>x. x \<in> A)"
    91   shows "chain S \<Longrightarrow> Rep (Abs (\<Squnion>i. Rep (S i))) = (\<Squnion>i. Rep (S i))"
    92   apply (rule type_definition.Abs_inverse [OF type])
    93   apply (erule admD [OF adm ch2ch_Rep [OF below]])
    94   apply (rule type_definition.Rep [OF type])
    95   done
    96 
    97 theorem typedef_is_lub:
    98   fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
    99   assumes type: "type_definition Rep Abs A"
   100     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   101     and adm: "adm (\<lambda>x. x \<in> A)"
   102   assumes S: "chain S"
   103   shows "range S <<| Abs (\<Squnion>i. Rep (S i))"
   104 proof -
   105   from S have "chain (\<lambda>i. Rep (S i))"
   106     by (rule ch2ch_Rep [OF below])
   107   then have "range (\<lambda>i. Rep (S i)) <<| (\<Squnion>i. Rep (S i))"
   108     by (rule cpo_lubI)
   109   then have "range (\<lambda>i. Rep (S i)) <<| Rep (Abs (\<Squnion>i. Rep (S i)))"
   110     by (simp only: Abs_inverse_lub_Rep [OF type below adm S])
   111   then show "range S <<| Abs (\<Squnion>i. Rep (S i))"
   112     by (rule typedef_is_lubI [OF below])
   113 qed
   114 
   115 lemmas typedef_lub = typedef_is_lub [THEN lub_eqI]
   116 
   117 theorem typedef_cpo:
   118   fixes Abs :: "'a::cpo \<Rightarrow> 'b::po"
   119   assumes type: "type_definition Rep Abs A"
   120     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   121     and adm: "adm (\<lambda>x. x \<in> A)"
   122   shows "OFCLASS('b, cpo_class)"
   123 proof
   124   fix S :: "nat \<Rightarrow> 'b"
   125   assume "chain S"
   126   then have "range S <<| Abs (\<Squnion>i. Rep (S i))"
   127     by (rule typedef_is_lub [OF type below adm])
   128   then show "\<exists>x. range S <<| x" ..
   129 qed
   130 
   131 
   132 subsubsection \<open>Continuity of \emph{Rep} and \emph{Abs}\<close>
   133 
   134 text \<open>For any sub-cpo, the \<^term>\<open>Rep\<close> function is continuous.\<close>
   135 
   136 theorem typedef_cont_Rep:
   137   fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
   138   assumes type: "type_definition Rep Abs A"
   139     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   140     and adm: "adm (\<lambda>x. x \<in> A)"
   141   shows "cont (\<lambda>x. f x) \<Longrightarrow> cont (\<lambda>x. Rep (f x))"
   142   apply (erule cont_apply [OF _ _ cont_const])
   143   apply (rule contI)
   144   apply (simp only: typedef_lub [OF type below adm])
   145   apply (simp only: Abs_inverse_lub_Rep [OF type below adm])
   146   apply (rule cpo_lubI)
   147   apply (erule ch2ch_Rep [OF below])
   148   done
   149 
   150 text \<open>
   151   For a sub-cpo, we can make the \<^term>\<open>Abs\<close> function continuous
   152   only if we restrict its domain to the defining subset by
   153   composing it with another continuous function.
   154 \<close>
   155 
   156 theorem typedef_cont_Abs:
   157   fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
   158   fixes f :: "'c::cpo \<Rightarrow> 'a::cpo"
   159   assumes type: "type_definition Rep Abs A"
   160     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   161     and adm: "adm (\<lambda>x. x \<in> A)" (* not used *)
   162     and f_in_A: "\<And>x. f x \<in> A"
   163   shows "cont f \<Longrightarrow> cont (\<lambda>x. Abs (f x))"
   164   unfolding cont_def is_lub_def is_ub_def ball_simps below
   165   by (simp add: type_definition.Abs_inverse [OF type f_in_A])
   166 
   167 
   168 subsection \<open>Proving subtype elements are compact\<close>
   169 
   170 theorem typedef_compact:
   171   fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
   172   assumes type: "type_definition Rep Abs A"
   173     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   174     and adm: "adm (\<lambda>x. x \<in> A)"
   175   shows "compact (Rep k) \<Longrightarrow> compact k"
   176 proof (unfold compact_def)
   177   have cont_Rep: "cont Rep"
   178     by (rule typedef_cont_Rep [OF type below adm cont_id])
   179   assume "adm (\<lambda>x. Rep k \<notsqsubseteq> x)"
   180   with cont_Rep have "adm (\<lambda>x. Rep k \<notsqsubseteq> Rep x)" by (rule adm_subst)
   181   then show "adm (\<lambda>x. k \<notsqsubseteq> x)" by (unfold below)
   182 qed
   183 
   184 
   185 subsection \<open>Proving a subtype is pointed\<close>
   186 
   187 text \<open>
   188   A subtype of a cpo has a least element if and only if
   189   the defining subset has a least element.
   190 \<close>
   191 
   192 theorem typedef_pcpo_generic:
   193   fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo"
   194   assumes type: "type_definition Rep Abs A"
   195     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   196     and z_in_A: "z \<in> A"
   197     and z_least: "\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x"
   198   shows "OFCLASS('b, pcpo_class)"
   199   apply (intro_classes)
   200   apply (rule_tac x="Abs z" in exI, rule allI)
   201   apply (unfold below)
   202   apply (subst type_definition.Abs_inverse [OF type z_in_A])
   203   apply (rule z_least [OF type_definition.Rep [OF type]])
   204   done
   205 
   206 text \<open>
   207   As a special case, a subtype of a pcpo has a least element
   208   if the defining subset contains \<^term>\<open>\<bottom>\<close>.
   209 \<close>
   210 
   211 theorem typedef_pcpo:
   212   fixes Abs :: "'a::pcpo \<Rightarrow> 'b::cpo"
   213   assumes type: "type_definition Rep Abs A"
   214     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   215     and bottom_in_A: "\<bottom> \<in> A"
   216   shows "OFCLASS('b, pcpo_class)"
   217   by (rule typedef_pcpo_generic [OF type below bottom_in_A], rule minimal)
   218 
   219 
   220 subsubsection \<open>Strictness of \emph{Rep} and \emph{Abs}\<close>
   221 
   222 text \<open>
   223   For a sub-pcpo where \<^term>\<open>\<bottom>\<close> is a member of the defining
   224   subset, \<^term>\<open>Rep\<close> and \<^term>\<open>Abs\<close> are both strict.
   225 \<close>
   226 
   227 theorem typedef_Abs_strict:
   228   assumes type: "type_definition Rep Abs A"
   229     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   230     and bottom_in_A: "\<bottom> \<in> A"
   231   shows "Abs \<bottom> = \<bottom>"
   232   apply (rule bottomI, unfold below)
   233   apply (simp add: type_definition.Abs_inverse [OF type bottom_in_A])
   234   done
   235 
   236 theorem typedef_Rep_strict:
   237   assumes type: "type_definition Rep Abs A"
   238     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   239     and bottom_in_A: "\<bottom> \<in> A"
   240   shows "Rep \<bottom> = \<bottom>"
   241   apply (rule typedef_Abs_strict [OF type below bottom_in_A, THEN subst])
   242   apply (rule type_definition.Abs_inverse [OF type bottom_in_A])
   243   done
   244 
   245 theorem typedef_Abs_bottom_iff:
   246   assumes type: "type_definition Rep Abs A"
   247     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   248     and bottom_in_A: "\<bottom> \<in> A"
   249   shows "x \<in> A \<Longrightarrow> (Abs x = \<bottom>) = (x = \<bottom>)"
   250   apply (rule typedef_Abs_strict [OF type below bottom_in_A, THEN subst])
   251   apply (simp add: type_definition.Abs_inject [OF type] bottom_in_A)
   252   done
   253 
   254 theorem typedef_Rep_bottom_iff:
   255   assumes type: "type_definition Rep Abs A"
   256     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   257     and bottom_in_A: "\<bottom> \<in> A"
   258   shows "(Rep x = \<bottom>) = (x = \<bottom>)"
   259   apply (rule typedef_Rep_strict [OF type below bottom_in_A, THEN subst])
   260   apply (simp add: type_definition.Rep_inject [OF type])
   261   done
   262 
   263 
   264 subsection \<open>Proving a subtype is flat\<close>
   265 
   266 theorem typedef_flat:
   267   fixes Abs :: "'a::flat \<Rightarrow> 'b::pcpo"
   268   assumes type: "type_definition Rep Abs A"
   269     and below: "(\<sqsubseteq>) \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
   270     and bottom_in_A: "\<bottom> \<in> A"
   271   shows "OFCLASS('b, flat_class)"
   272   apply (intro_classes)
   273   apply (unfold below)
   274   apply (simp add: type_definition.Rep_inject [OF type, symmetric])
   275   apply (simp add: typedef_Rep_strict [OF type below bottom_in_A])
   276   apply (simp add: ax_flat)
   277   done
   278 
   279 
   280 subsection \<open>HOLCF type definition package\<close>
   281 
   282 ML_file \<open>Tools/cpodef.ML\<close>
   283 
   284 end