src/HOLCF/Ssum.thy
 author huffman Mon May 11 08:28:09 2009 -0700 (2009-05-11) changeset 31095 b79d140f6d0b parent 31076 99fe356cbbc2 child 31115 7d6416f0d1e0 permissions -rw-r--r--
simplify fixrec proofs for mutually-recursive definitions; generate better fixpoint induction rules
```     1 (*  Title:      HOLCF/Ssum.thy
```
```     2     Author:     Franz Regensburger and Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* The type of strict sums *}
```
```     6
```
```     7 theory Ssum
```
```     8 imports Cprod Tr
```
```     9 begin
```
```    10
```
```    11 defaultsort pcpo
```
```    12
```
```    13 subsection {* Definition of strict sum type *}
```
```    14
```
```    15 pcpodef (Ssum)  ('a, 'b) "++" (infixr "++" 10) =
```
```    16   "{p :: tr \<times> ('a \<times> 'b).
```
```    17     (cfst\<cdot>p \<sqsubseteq> TT \<longleftrightarrow> csnd\<cdot>(csnd\<cdot>p) = \<bottom>) \<and>
```
```    18     (cfst\<cdot>p \<sqsubseteq> FF \<longleftrightarrow> cfst\<cdot>(csnd\<cdot>p) = \<bottom>)}"
```
```    19 by simp_all
```
```    20
```
```    21 instance "++" :: ("{finite_po,pcpo}", "{finite_po,pcpo}") finite_po
```
```    22 by (rule typedef_finite_po [OF type_definition_Ssum])
```
```    23
```
```    24 instance "++" :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin
```
```    25 by (rule typedef_chfin [OF type_definition_Ssum below_Ssum_def])
```
```    26
```
```    27 syntax (xsymbols)
```
```    28   "++"		:: "[type, type] => type"	("(_ \<oplus>/ _)" [21, 20] 20)
```
```    29 syntax (HTML output)
```
```    30   "++"		:: "[type, type] => type"	("(_ \<oplus>/ _)" [21, 20] 20)
```
```    31
```
```    32 subsection {* Definitions of constructors *}
```
```    33
```
```    34 definition
```
```    35   sinl :: "'a \<rightarrow> ('a ++ 'b)" where
```
```    36   "sinl = (\<Lambda> a. Abs_Ssum <strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>>)"
```
```    37
```
```    38 definition
```
```    39   sinr :: "'b \<rightarrow> ('a ++ 'b)" where
```
```    40   "sinr = (\<Lambda> b. Abs_Ssum <strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b>)"
```
```    41
```
```    42 lemma sinl_Ssum: "<strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>> \<in> Ssum"
```
```    43 by (simp add: Ssum_def strictify_conv_if)
```
```    44
```
```    45 lemma sinr_Ssum: "<strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b> \<in> Ssum"
```
```    46 by (simp add: Ssum_def strictify_conv_if)
```
```    47
```
```    48 lemma sinl_Abs_Ssum: "sinl\<cdot>a = Abs_Ssum <strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>>"
```
```    49 by (unfold sinl_def, simp add: cont_Abs_Ssum sinl_Ssum)
```
```    50
```
```    51 lemma sinr_Abs_Ssum: "sinr\<cdot>b = Abs_Ssum <strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b>"
```
```    52 by (unfold sinr_def, simp add: cont_Abs_Ssum sinr_Ssum)
```
```    53
```
```    54 lemma Rep_Ssum_sinl: "Rep_Ssum (sinl\<cdot>a) = <strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>>"
```
```    55 by (simp add: sinl_Abs_Ssum Abs_Ssum_inverse sinl_Ssum)
```
```    56
```
```    57 lemma Rep_Ssum_sinr: "Rep_Ssum (sinr\<cdot>b) = <strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b>"
```
```    58 by (simp add: sinr_Abs_Ssum Abs_Ssum_inverse sinr_Ssum)
```
```    59
```
```    60 subsection {* Properties of @{term sinl} and @{term sinr} *}
```
```    61
```
```    62 text {* Ordering *}
```
```    63
```
```    64 lemma sinl_below [simp]: "(sinl\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x \<sqsubseteq> y)"
```
```    65 by (simp add: below_Ssum_def Rep_Ssum_sinl strictify_conv_if)
```
```    66
```
```    67 lemma sinr_below [simp]: "(sinr\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x \<sqsubseteq> y)"
```
```    68 by (simp add: below_Ssum_def Rep_Ssum_sinr strictify_conv_if)
```
```    69
```
```    70 lemma sinl_below_sinr [simp]: "(sinl\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x = \<bottom>)"
```
```    71 by (simp add: below_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strictify_conv_if)
```
```    72
```
```    73 lemma sinr_below_sinl [simp]: "(sinr\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x = \<bottom>)"
```
```    74 by (simp add: below_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strictify_conv_if)
```
```    75
```
```    76 text {* Equality *}
```
```    77
```
```    78 lemma sinl_eq [simp]: "(sinl\<cdot>x = sinl\<cdot>y) = (x = y)"
```
```    79 by (simp add: po_eq_conv)
```
```    80
```
```    81 lemma sinr_eq [simp]: "(sinr\<cdot>x = sinr\<cdot>y) = (x = y)"
```
```    82 by (simp add: po_eq_conv)
```
```    83
```
```    84 lemma sinl_eq_sinr [simp]: "(sinl\<cdot>x = sinr\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)"
```
```    85 by (subst po_eq_conv, simp)
```
```    86
```
```    87 lemma sinr_eq_sinl [simp]: "(sinr\<cdot>x = sinl\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)"
```
```    88 by (subst po_eq_conv, simp)
```
```    89
```
```    90 lemma sinl_inject: "sinl\<cdot>x = sinl\<cdot>y \<Longrightarrow> x = y"
```
```    91 by (rule sinl_eq [THEN iffD1])
```
```    92
```
```    93 lemma sinr_inject: "sinr\<cdot>x = sinr\<cdot>y \<Longrightarrow> x = y"
```
```    94 by (rule sinr_eq [THEN iffD1])
```
```    95
```
```    96 text {* Strictness *}
```
```    97
```
```    98 lemma sinl_strict [simp]: "sinl\<cdot>\<bottom> = \<bottom>"
```
```    99 by (simp add: sinl_Abs_Ssum Abs_Ssum_strict)
```
```   100
```
```   101 lemma sinr_strict [simp]: "sinr\<cdot>\<bottom> = \<bottom>"
```
```   102 by (simp add: sinr_Abs_Ssum Abs_Ssum_strict)
```
```   103
```
```   104 lemma sinl_defined_iff [simp]: "(sinl\<cdot>x = \<bottom>) = (x = \<bottom>)"
```
```   105 by (cut_tac sinl_eq [of "x" "\<bottom>"], simp)
```
```   106
```
```   107 lemma sinr_defined_iff [simp]: "(sinr\<cdot>x = \<bottom>) = (x = \<bottom>)"
```
```   108 by (cut_tac sinr_eq [of "x" "\<bottom>"], simp)
```
```   109
```
```   110 lemma sinl_defined [intro!]: "x \<noteq> \<bottom> \<Longrightarrow> sinl\<cdot>x \<noteq> \<bottom>"
```
```   111 by simp
```
```   112
```
```   113 lemma sinr_defined [intro!]: "x \<noteq> \<bottom> \<Longrightarrow> sinr\<cdot>x \<noteq> \<bottom>"
```
```   114 by simp
```
```   115
```
```   116 text {* Compactness *}
```
```   117
```
```   118 lemma compact_sinl: "compact x \<Longrightarrow> compact (sinl\<cdot>x)"
```
```   119 by (rule compact_Ssum, simp add: Rep_Ssum_sinl strictify_conv_if)
```
```   120
```
```   121 lemma compact_sinr: "compact x \<Longrightarrow> compact (sinr\<cdot>x)"
```
```   122 by (rule compact_Ssum, simp add: Rep_Ssum_sinr strictify_conv_if)
```
```   123
```
```   124 lemma compact_sinlD: "compact (sinl\<cdot>x) \<Longrightarrow> compact x"
```
```   125 unfolding compact_def
```
```   126 by (drule adm_subst [OF cont_Rep_CFun2 [where f=sinl]], simp)
```
```   127
```
```   128 lemma compact_sinrD: "compact (sinr\<cdot>x) \<Longrightarrow> compact x"
```
```   129 unfolding compact_def
```
```   130 by (drule adm_subst [OF cont_Rep_CFun2 [where f=sinr]], simp)
```
```   131
```
```   132 lemma compact_sinl_iff [simp]: "compact (sinl\<cdot>x) = compact x"
```
```   133 by (safe elim!: compact_sinl compact_sinlD)
```
```   134
```
```   135 lemma compact_sinr_iff [simp]: "compact (sinr\<cdot>x) = compact x"
```
```   136 by (safe elim!: compact_sinr compact_sinrD)
```
```   137
```
```   138 subsection {* Case analysis *}
```
```   139
```
```   140 lemma Exh_Ssum:
```
```   141   "z = \<bottom> \<or> (\<exists>a. z = sinl\<cdot>a \<and> a \<noteq> \<bottom>) \<or> (\<exists>b. z = sinr\<cdot>b \<and> b \<noteq> \<bottom>)"
```
```   142 apply (rule_tac x=z in Abs_Ssum_induct)
```
```   143 apply (rule_tac p=y in cprodE, rename_tac t x)
```
```   144 apply (rule_tac p=x in cprodE, rename_tac a b)
```
```   145 apply (rule_tac p=t in trE)
```
```   146 apply (rule disjI1)
```
```   147 apply (simp add: Ssum_def cpair_strict Abs_Ssum_strict)
```
```   148 apply (rule disjI2, rule disjI1, rule_tac x=a in exI)
```
```   149 apply (simp add: sinl_Abs_Ssum Ssum_def)
```
```   150 apply (rule disjI2, rule disjI2, rule_tac x=b in exI)
```
```   151 apply (simp add: sinr_Abs_Ssum Ssum_def)
```
```   152 done
```
```   153
```
```   154 lemma ssumE [cases type: ++]:
```
```   155   "\<lbrakk>p = \<bottom> \<Longrightarrow> Q;
```
```   156    \<And>x. \<lbrakk>p = sinl\<cdot>x; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q;
```
```   157    \<And>y. \<lbrakk>p = sinr\<cdot>y; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
```
```   158 by (cut_tac z=p in Exh_Ssum, auto)
```
```   159
```
```   160 lemma ssum_induct [induct type: ++]:
```
```   161   "\<lbrakk>P \<bottom>;
```
```   162    \<And>x. x \<noteq> \<bottom> \<Longrightarrow> P (sinl\<cdot>x);
```
```   163    \<And>y. y \<noteq> \<bottom> \<Longrightarrow> P (sinr\<cdot>y)\<rbrakk> \<Longrightarrow> P x"
```
```   164 by (cases x, simp_all)
```
```   165
```
```   166 lemma ssumE2:
```
```   167   "\<lbrakk>\<And>x. p = sinl\<cdot>x \<Longrightarrow> Q; \<And>y. p = sinr\<cdot>y \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
```
```   168 by (cases p, simp only: sinl_strict [symmetric], simp, simp)
```
```   169
```
```   170 lemma below_sinlD: "p \<sqsubseteq> sinl\<cdot>x \<Longrightarrow> \<exists>y. p = sinl\<cdot>y \<and> y \<sqsubseteq> x"
```
```   171 by (cases p, rule_tac x="\<bottom>" in exI, simp_all)
```
```   172
```
```   173 lemma below_sinrD: "p \<sqsubseteq> sinr\<cdot>x \<Longrightarrow> \<exists>y. p = sinr\<cdot>y \<and> y \<sqsubseteq> x"
```
```   174 by (cases p, rule_tac x="\<bottom>" in exI, simp_all)
```
```   175
```
```   176 subsection {* Case analysis combinator *}
```
```   177
```
```   178 definition
```
```   179   sscase :: "('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a ++ 'b) \<rightarrow> 'c" where
```
```   180   "sscase = (\<Lambda> f g s. (\<Lambda><t, x, y>. If t then f\<cdot>x else g\<cdot>y fi)\<cdot>(Rep_Ssum s))"
```
```   181
```
```   182 translations
```
```   183   "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"
```
```   184
```
```   185 translations
```
```   186   "\<Lambda>(XCONST sinl\<cdot>x). t" == "CONST sscase\<cdot>(\<Lambda> x. t)\<cdot>\<bottom>"
```
```   187   "\<Lambda>(XCONST sinr\<cdot>y). t" == "CONST sscase\<cdot>\<bottom>\<cdot>(\<Lambda> y. t)"
```
```   188
```
```   189 lemma beta_sscase:
```
```   190   "sscase\<cdot>f\<cdot>g\<cdot>s = (\<Lambda><t, x, y>. If t then f\<cdot>x else g\<cdot>y fi)\<cdot>(Rep_Ssum s)"
```
```   191 unfolding sscase_def by (simp add: cont_Rep_Ssum cont2cont_LAM)
```
```   192
```
```   193 lemma sscase1 [simp]: "sscase\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>"
```
```   194 unfolding beta_sscase by (simp add: Rep_Ssum_strict)
```
```   195
```
```   196 lemma sscase2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = f\<cdot>x"
```
```   197 unfolding beta_sscase by (simp add: Rep_Ssum_sinl)
```
```   198
```
```   199 lemma sscase3 [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>y) = g\<cdot>y"
```
```   200 unfolding beta_sscase by (simp add: Rep_Ssum_sinr)
```
```   201
```
```   202 lemma sscase4 [simp]: "sscase\<cdot>sinl\<cdot>sinr\<cdot>z = z"
```
```   203 by (cases z, simp_all)
```
```   204
```
```   205 subsection {* Strict sum preserves flatness *}
```
```   206
```
```   207 instance "++" :: (flat, flat) flat
```
```   208 apply (intro_classes, clarify)
```
```   209 apply (rule_tac p=x in ssumE, simp)
```
```   210 apply (rule_tac p=y in ssumE, simp_all add: flat_below_iff)
```
```   211 apply (rule_tac p=y in ssumE, simp_all add: flat_below_iff)
```
```   212 done
```
```   213
```
```   214 subsection {* Strict sum is a bifinite domain *}
```
```   215
```
```   216 instantiation "++" :: (bifinite, bifinite) bifinite
```
```   217 begin
```
```   218
```
```   219 definition
```
```   220   approx_ssum_def:
```
```   221     "approx = (\<lambda>n. sscase\<cdot>(\<Lambda> x. sinl\<cdot>(approx n\<cdot>x))\<cdot>(\<Lambda> y. sinr\<cdot>(approx n\<cdot>y)))"
```
```   222
```
```   223 lemma approx_sinl [simp]: "approx i\<cdot>(sinl\<cdot>x) = sinl\<cdot>(approx i\<cdot>x)"
```
```   224 unfolding approx_ssum_def by (cases "x = \<bottom>") simp_all
```
```   225
```
```   226 lemma approx_sinr [simp]: "approx i\<cdot>(sinr\<cdot>x) = sinr\<cdot>(approx i\<cdot>x)"
```
```   227 unfolding approx_ssum_def by (cases "x = \<bottom>") simp_all
```
```   228
```
```   229 instance proof
```
```   230   fix i :: nat and x :: "'a \<oplus> 'b"
```
```   231   show "chain (approx :: nat \<Rightarrow> 'a \<oplus> 'b \<rightarrow> 'a \<oplus> 'b)"
```
```   232     unfolding approx_ssum_def by simp
```
```   233   show "(\<Squnion>i. approx i\<cdot>x) = x"
```
```   234     unfolding approx_ssum_def
```
```   235     by (simp add: lub_distribs eta_cfun)
```
```   236   show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
```
```   237     by (cases x, simp add: approx_ssum_def, simp, simp)
```
```   238   have "{x::'a \<oplus> 'b. approx i\<cdot>x = x} \<subseteq>
```
```   239         (\<lambda>x. sinl\<cdot>x) ` {x. approx i\<cdot>x = x} \<union>
```
```   240         (\<lambda>x. sinr\<cdot>x) ` {x. approx i\<cdot>x = x}"
```
```   241     by (rule subsetI, case_tac x rule: ssumE2, simp, simp)
```
```   242   thus "finite {x::'a \<oplus> 'b. approx i\<cdot>x = x}"
```
```   243     by (rule finite_subset,
```
```   244         intro finite_UnI finite_imageI finite_fixes_approx)
```
```   245 qed
```
```   246
```
```   247 end
```
```   248
```
```   249 end
```