src/HOLCF/LowerPD.thy
author huffman
Thu Mar 27 00:27:16 2008 +0100 (2008-03-27)
changeset 26420 57a626f64875
parent 26407 562a1d615336
child 26806 40b411ec05aa
permissions -rw-r--r--
make preorder locale into a superclass of class po
     1 (*  Title:      HOLCF/LowerPD.thy
     2     ID:         $Id$
     3     Author:     Brian Huffman
     4 *)
     5 
     6 header {* Lower powerdomain *}
     7 
     8 theory LowerPD
     9 imports CompactBasis
    10 begin
    11 
    12 subsection {* Basis preorder *}
    13 
    14 definition
    15   lower_le :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> bool" (infix "\<le>\<flat>" 50) where
    16   "lower_le = (\<lambda>u v. \<forall>x\<in>Rep_pd_basis u. \<exists>y\<in>Rep_pd_basis v. x \<sqsubseteq> y)"
    17 
    18 lemma lower_le_refl [simp]: "t \<le>\<flat> t"
    19 unfolding lower_le_def by fast
    20 
    21 lemma lower_le_trans: "\<lbrakk>t \<le>\<flat> u; u \<le>\<flat> v\<rbrakk> \<Longrightarrow> t \<le>\<flat> v"
    22 unfolding lower_le_def
    23 apply (rule ballI)
    24 apply (drule (1) bspec, erule bexE)
    25 apply (drule (1) bspec, erule bexE)
    26 apply (erule rev_bexI)
    27 apply (erule (1) trans_less)
    28 done
    29 
    30 interpretation lower_le: preorder [lower_le]
    31 by (rule preorder.intro, rule lower_le_refl, rule lower_le_trans)
    32 
    33 lemma lower_le_minimal [simp]: "PDUnit compact_bot \<le>\<flat> t"
    34 unfolding lower_le_def Rep_PDUnit
    35 by (simp, rule Rep_pd_basis_nonempty [folded ex_in_conv])
    36 
    37 lemma PDUnit_lower_mono: "x \<sqsubseteq> y \<Longrightarrow> PDUnit x \<le>\<flat> PDUnit y"
    38 unfolding lower_le_def Rep_PDUnit by fast
    39 
    40 lemma PDPlus_lower_mono: "\<lbrakk>s \<le>\<flat> t; u \<le>\<flat> v\<rbrakk> \<Longrightarrow> PDPlus s u \<le>\<flat> PDPlus t v"
    41 unfolding lower_le_def Rep_PDPlus by fast
    42 
    43 lemma PDPlus_lower_less: "t \<le>\<flat> PDPlus t u"
    44 unfolding lower_le_def Rep_PDPlus by fast
    45 
    46 lemma lower_le_PDUnit_PDUnit_iff [simp]:
    47   "(PDUnit a \<le>\<flat> PDUnit b) = a \<sqsubseteq> b"
    48 unfolding lower_le_def Rep_PDUnit by fast
    49 
    50 lemma lower_le_PDUnit_PDPlus_iff:
    51   "(PDUnit a \<le>\<flat> PDPlus t u) = (PDUnit a \<le>\<flat> t \<or> PDUnit a \<le>\<flat> u)"
    52 unfolding lower_le_def Rep_PDPlus Rep_PDUnit by fast
    53 
    54 lemma lower_le_PDPlus_iff: "(PDPlus t u \<le>\<flat> v) = (t \<le>\<flat> v \<and> u \<le>\<flat> v)"
    55 unfolding lower_le_def Rep_PDPlus by fast
    56 
    57 lemma lower_le_induct [induct set: lower_le]:
    58   assumes le: "t \<le>\<flat> u"
    59   assumes 1: "\<And>a b. a \<sqsubseteq> b \<Longrightarrow> P (PDUnit a) (PDUnit b)"
    60   assumes 2: "\<And>t u a. P (PDUnit a) t \<Longrightarrow> P (PDUnit a) (PDPlus t u)"
    61   assumes 3: "\<And>t u v. \<lbrakk>P t v; P u v\<rbrakk> \<Longrightarrow> P (PDPlus t u) v"
    62   shows "P t u"
    63 using le
    64 apply (induct t arbitrary: u rule: pd_basis_induct)
    65 apply (erule rev_mp)
    66 apply (induct_tac u rule: pd_basis_induct)
    67 apply (simp add: 1)
    68 apply (simp add: lower_le_PDUnit_PDPlus_iff)
    69 apply (simp add: 2)
    70 apply (subst PDPlus_commute)
    71 apply (simp add: 2)
    72 apply (simp add: lower_le_PDPlus_iff 3)
    73 done
    74 
    75 lemma approx_pd_lower_mono1:
    76   "i \<le> j \<Longrightarrow> approx_pd i t \<le>\<flat> approx_pd j t"
    77 apply (induct t rule: pd_basis_induct)
    78 apply (simp add: compact_approx_mono1)
    79 apply (simp add: PDPlus_lower_mono)
    80 done
    81 
    82 lemma approx_pd_lower_le: "approx_pd i t \<le>\<flat> t"
    83 apply (induct t rule: pd_basis_induct)
    84 apply (simp add: compact_approx_le)
    85 apply (simp add: PDPlus_lower_mono)
    86 done
    87 
    88 lemma approx_pd_lower_mono:
    89   "t \<le>\<flat> u \<Longrightarrow> approx_pd n t \<le>\<flat> approx_pd n u"
    90 apply (erule lower_le_induct)
    91 apply (simp add: compact_approx_mono)
    92 apply (simp add: lower_le_PDUnit_PDPlus_iff)
    93 apply (simp add: lower_le_PDPlus_iff)
    94 done
    95 
    96 
    97 subsection {* Type definition *}
    98 
    99 cpodef (open) 'a lower_pd =
   100   "{S::'a::profinite pd_basis set. lower_le.ideal S}"
   101 apply (simp add: lower_le.adm_ideal)
   102 apply (fast intro: lower_le.ideal_principal)
   103 done
   104 
   105 lemma ideal_Rep_lower_pd: "lower_le.ideal (Rep_lower_pd x)"
   106 by (rule Rep_lower_pd [simplified])
   107 
   108 lemma Rep_lower_pd_mono: "x \<sqsubseteq> y \<Longrightarrow> Rep_lower_pd x \<subseteq> Rep_lower_pd y"
   109 unfolding less_lower_pd_def less_set_def .
   110 
   111 
   112 subsection {* Principal ideals *}
   113 
   114 definition
   115   lower_principal :: "'a pd_basis \<Rightarrow> 'a lower_pd" where
   116   "lower_principal t = Abs_lower_pd {u. u \<le>\<flat> t}"
   117 
   118 lemma Rep_lower_principal:
   119   "Rep_lower_pd (lower_principal t) = {u. u \<le>\<flat> t}"
   120 unfolding lower_principal_def
   121 apply (rule Abs_lower_pd_inverse [simplified])
   122 apply (rule lower_le.ideal_principal)
   123 done
   124 
   125 interpretation lower_pd:
   126   bifinite_basis [lower_le approx_pd lower_principal Rep_lower_pd]
   127 apply unfold_locales
   128 apply (rule approx_pd_lower_le)
   129 apply (rule approx_pd_idem)
   130 apply (erule approx_pd_lower_mono)
   131 apply (rule approx_pd_lower_mono1, simp)
   132 apply (rule finite_range_approx_pd)
   133 apply (rule ex_approx_pd_eq)
   134 apply (rule ideal_Rep_lower_pd)
   135 apply (rule cont_Rep_lower_pd)
   136 apply (rule Rep_lower_principal)
   137 apply (simp only: less_lower_pd_def less_set_def)
   138 done
   139 
   140 lemma lower_principal_less_iff [simp]:
   141   "(lower_principal t \<sqsubseteq> lower_principal u) = (t \<le>\<flat> u)"
   142 unfolding less_lower_pd_def Rep_lower_principal less_set_def
   143 by (fast intro: lower_le_refl elim: lower_le_trans)
   144 
   145 lemma lower_principal_mono:
   146   "t \<le>\<flat> u \<Longrightarrow> lower_principal t \<sqsubseteq> lower_principal u"
   147 by (rule lower_principal_less_iff [THEN iffD2])
   148 
   149 lemma compact_lower_principal: "compact (lower_principal t)"
   150 apply (rule compactI2)
   151 apply (simp add: less_lower_pd_def)
   152 apply (simp add: cont2contlubE [OF cont_Rep_lower_pd])
   153 apply (simp add: Rep_lower_principal set_cpo_simps)
   154 apply (simp add: subset_def)
   155 apply (drule spec, drule mp, rule lower_le_refl)
   156 apply (erule exE, rename_tac i)
   157 apply (rule_tac x=i in exI)
   158 apply clarify
   159 apply (erule (1) lower_le.idealD3 [OF ideal_Rep_lower_pd])
   160 done
   161 
   162 lemma lower_pd_minimal: "lower_principal (PDUnit compact_bot) \<sqsubseteq> ys"
   163 by (induct ys rule: lower_pd.principal_induct, simp, simp)
   164 
   165 instance lower_pd :: (bifinite) pcpo
   166 by (intro_classes, fast intro: lower_pd_minimal)
   167 
   168 lemma inst_lower_pd_pcpo: "\<bottom> = lower_principal (PDUnit compact_bot)"
   169 by (rule lower_pd_minimal [THEN UU_I, symmetric])
   170 
   171 
   172 subsection {* Approximation *}
   173 
   174 instance lower_pd :: (profinite) approx ..
   175 
   176 defs (overloaded)
   177   approx_lower_pd_def:
   178     "approx \<equiv> (\<lambda>n. lower_pd.basis_fun (\<lambda>t. lower_principal (approx_pd n t)))"
   179 
   180 lemma approx_lower_principal [simp]:
   181   "approx n\<cdot>(lower_principal t) = lower_principal (approx_pd n t)"
   182 unfolding approx_lower_pd_def
   183 apply (rule lower_pd.basis_fun_principal)
   184 apply (erule lower_principal_mono [OF approx_pd_lower_mono])
   185 done
   186 
   187 lemma chain_approx_lower_pd:
   188   "chain (approx :: nat \<Rightarrow> 'a lower_pd \<rightarrow> 'a lower_pd)"
   189 unfolding approx_lower_pd_def
   190 by (rule lower_pd.chain_basis_fun_take)
   191 
   192 lemma lub_approx_lower_pd:
   193   "(\<Squnion>i. approx i\<cdot>xs) = (xs::'a lower_pd)"
   194 unfolding approx_lower_pd_def
   195 by (rule lower_pd.lub_basis_fun_take)
   196 
   197 lemma approx_lower_pd_idem:
   198   "approx n\<cdot>(approx n\<cdot>xs) = approx n\<cdot>(xs::'a lower_pd)"
   199 apply (induct xs rule: lower_pd.principal_induct, simp)
   200 apply (simp add: approx_pd_idem)
   201 done
   202 
   203 lemma approx_eq_lower_principal:
   204   "\<exists>t\<in>Rep_lower_pd xs. approx n\<cdot>xs = lower_principal (approx_pd n t)"
   205 unfolding approx_lower_pd_def
   206 by (rule lower_pd.basis_fun_take_eq_principal)
   207 
   208 lemma finite_fixes_approx_lower_pd:
   209   "finite {xs::'a lower_pd. approx n\<cdot>xs = xs}"
   210 unfolding approx_lower_pd_def
   211 by (rule lower_pd.finite_fixes_basis_fun_take)
   212 
   213 instance lower_pd :: (profinite) profinite
   214 apply intro_classes
   215 apply (simp add: chain_approx_lower_pd)
   216 apply (rule lub_approx_lower_pd)
   217 apply (rule approx_lower_pd_idem)
   218 apply (rule finite_fixes_approx_lower_pd)
   219 done
   220 
   221 instance lower_pd :: (bifinite) bifinite ..
   222 
   223 lemma compact_imp_lower_principal:
   224   "compact xs \<Longrightarrow> \<exists>t. xs = lower_principal t"
   225 apply (drule bifinite_compact_eq_approx)
   226 apply (erule exE)
   227 apply (erule subst)
   228 apply (cut_tac n=i and xs=xs in approx_eq_lower_principal)
   229 apply fast
   230 done
   231 
   232 lemma lower_principal_induct:
   233   "\<lbrakk>adm P; \<And>t. P (lower_principal t)\<rbrakk> \<Longrightarrow> P xs"
   234 apply (erule approx_induct, rename_tac xs)
   235 apply (cut_tac n=n and xs=xs in approx_eq_lower_principal)
   236 apply (clarify, simp)
   237 done
   238 
   239 lemma lower_principal_induct2:
   240   "\<lbrakk>\<And>ys. adm (\<lambda>xs. P xs ys); \<And>xs. adm (\<lambda>ys. P xs ys);
   241     \<And>t u. P (lower_principal t) (lower_principal u)\<rbrakk> \<Longrightarrow> P xs ys"
   242 apply (rule_tac x=ys in spec)
   243 apply (rule_tac xs=xs in lower_principal_induct, simp)
   244 apply (rule allI, rename_tac ys)
   245 apply (rule_tac xs=ys in lower_principal_induct, simp)
   246 apply simp
   247 done
   248 
   249 
   250 subsection {* Monadic unit *}
   251 
   252 definition
   253   lower_unit :: "'a \<rightarrow> 'a lower_pd" where
   254   "lower_unit = compact_basis.basis_fun (\<lambda>a. lower_principal (PDUnit a))"
   255 
   256 lemma lower_unit_Rep_compact_basis [simp]:
   257   "lower_unit\<cdot>(Rep_compact_basis a) = lower_principal (PDUnit a)"
   258 unfolding lower_unit_def
   259 apply (rule compact_basis.basis_fun_principal)
   260 apply (rule lower_principal_mono)
   261 apply (erule PDUnit_lower_mono)
   262 done
   263 
   264 lemma lower_unit_strict [simp]: "lower_unit\<cdot>\<bottom> = \<bottom>"
   265 unfolding inst_lower_pd_pcpo Rep_compact_bot [symmetric] by simp
   266 
   267 lemma approx_lower_unit [simp]:
   268   "approx n\<cdot>(lower_unit\<cdot>x) = lower_unit\<cdot>(approx n\<cdot>x)"
   269 apply (induct x rule: compact_basis_induct, simp)
   270 apply (simp add: approx_Rep_compact_basis)
   271 done
   272 
   273 lemma lower_unit_less_iff [simp]:
   274   "(lower_unit\<cdot>x \<sqsubseteq> lower_unit\<cdot>y) = (x \<sqsubseteq> y)"
   275  apply (rule iffI)
   276   apply (rule bifinite_less_ext)
   277   apply (drule_tac f="approx i" in monofun_cfun_arg, simp)
   278   apply (cut_tac x="approx i\<cdot>x" in compact_imp_Rep_compact_basis, simp)
   279   apply (cut_tac x="approx i\<cdot>y" in compact_imp_Rep_compact_basis, simp)
   280   apply (clarify, simp add: compact_le_def)
   281  apply (erule monofun_cfun_arg)
   282 done
   283 
   284 lemma lower_unit_eq_iff [simp]:
   285   "(lower_unit\<cdot>x = lower_unit\<cdot>y) = (x = y)"
   286 unfolding po_eq_conv by simp
   287 
   288 lemma lower_unit_strict_iff [simp]: "(lower_unit\<cdot>x = \<bottom>) = (x = \<bottom>)"
   289 unfolding lower_unit_strict [symmetric] by (rule lower_unit_eq_iff)
   290 
   291 lemma compact_lower_unit_iff [simp]:
   292   "compact (lower_unit\<cdot>x) = compact x"
   293 unfolding bifinite_compact_iff by simp
   294 
   295 
   296 subsection {* Monadic plus *}
   297 
   298 definition
   299   lower_plus :: "'a lower_pd \<rightarrow> 'a lower_pd \<rightarrow> 'a lower_pd" where
   300   "lower_plus = lower_pd.basis_fun (\<lambda>t. lower_pd.basis_fun (\<lambda>u.
   301       lower_principal (PDPlus t u)))"
   302 
   303 abbreviation
   304   lower_add :: "'a lower_pd \<Rightarrow> 'a lower_pd \<Rightarrow> 'a lower_pd"
   305     (infixl "+\<flat>" 65) where
   306   "xs +\<flat> ys == lower_plus\<cdot>xs\<cdot>ys"
   307 
   308 lemma lower_plus_principal [simp]:
   309   "lower_plus\<cdot>(lower_principal t)\<cdot>(lower_principal u) =
   310    lower_principal (PDPlus t u)"
   311 unfolding lower_plus_def
   312 by (simp add: lower_pd.basis_fun_principal
   313     lower_pd.basis_fun_mono PDPlus_lower_mono)
   314 
   315 lemma approx_lower_plus [simp]:
   316   "approx n\<cdot>(lower_plus\<cdot>xs\<cdot>ys) = lower_plus\<cdot>(approx n\<cdot>xs)\<cdot>(approx n\<cdot>ys)"
   317 by (induct xs ys rule: lower_principal_induct2, simp, simp, simp)
   318 
   319 lemma lower_plus_commute: "lower_plus\<cdot>xs\<cdot>ys = lower_plus\<cdot>ys\<cdot>xs"
   320 apply (induct xs ys rule: lower_principal_induct2, simp, simp)
   321 apply (simp add: PDPlus_commute)
   322 done
   323 
   324 lemma lower_plus_assoc:
   325   "lower_plus\<cdot>(lower_plus\<cdot>xs\<cdot>ys)\<cdot>zs = lower_plus\<cdot>xs\<cdot>(lower_plus\<cdot>ys\<cdot>zs)"
   326 apply (induct xs ys arbitrary: zs rule: lower_principal_induct2, simp, simp)
   327 apply (rule_tac xs=zs in lower_principal_induct, simp)
   328 apply (simp add: PDPlus_assoc)
   329 done
   330 
   331 lemma lower_plus_absorb: "lower_plus\<cdot>xs\<cdot>xs = xs"
   332 apply (induct xs rule: lower_principal_induct, simp)
   333 apply (simp add: PDPlus_absorb)
   334 done
   335 
   336 lemma lower_plus_less1: "xs \<sqsubseteq> lower_plus\<cdot>xs\<cdot>ys"
   337 apply (induct xs ys rule: lower_principal_induct2, simp, simp)
   338 apply (simp add: PDPlus_lower_less)
   339 done
   340 
   341 lemma lower_plus_less2: "ys \<sqsubseteq> lower_plus\<cdot>xs\<cdot>ys"
   342 by (subst lower_plus_commute, rule lower_plus_less1)
   343 
   344 lemma lower_plus_least: "\<lbrakk>xs \<sqsubseteq> zs; ys \<sqsubseteq> zs\<rbrakk> \<Longrightarrow> lower_plus\<cdot>xs\<cdot>ys \<sqsubseteq> zs"
   345 apply (subst lower_plus_absorb [of zs, symmetric])
   346 apply (erule (1) monofun_cfun [OF monofun_cfun_arg])
   347 done
   348 
   349 lemma lower_plus_less_iff:
   350   "(lower_plus\<cdot>xs\<cdot>ys \<sqsubseteq> zs) = (xs \<sqsubseteq> zs \<and> ys \<sqsubseteq> zs)"
   351 apply safe
   352 apply (erule trans_less [OF lower_plus_less1])
   353 apply (erule trans_less [OF lower_plus_less2])
   354 apply (erule (1) lower_plus_least)
   355 done
   356 
   357 lemma lower_plus_strict_iff [simp]:
   358   "(lower_plus\<cdot>xs\<cdot>ys = \<bottom>) = (xs = \<bottom> \<and> ys = \<bottom>)"
   359 apply safe
   360 apply (rule UU_I, erule subst, rule lower_plus_less1)
   361 apply (rule UU_I, erule subst, rule lower_plus_less2)
   362 apply (rule lower_plus_absorb)
   363 done
   364 
   365 lemma lower_plus_strict1 [simp]: "lower_plus\<cdot>\<bottom>\<cdot>ys = ys"
   366 apply (rule antisym_less [OF _ lower_plus_less2])
   367 apply (simp add: lower_plus_least)
   368 done
   369 
   370 lemma lower_plus_strict2 [simp]: "lower_plus\<cdot>xs\<cdot>\<bottom> = xs"
   371 apply (rule antisym_less [OF _ lower_plus_less1])
   372 apply (simp add: lower_plus_least)
   373 done
   374 
   375 lemma lower_unit_less_plus_iff:
   376   "(lower_unit\<cdot>x \<sqsubseteq> lower_plus\<cdot>ys\<cdot>zs) =
   377     (lower_unit\<cdot>x \<sqsubseteq> ys \<or> lower_unit\<cdot>x \<sqsubseteq> zs)"
   378  apply (rule iffI)
   379   apply (subgoal_tac
   380     "adm (\<lambda>f. f\<cdot>(lower_unit\<cdot>x) \<sqsubseteq> f\<cdot>ys \<or> f\<cdot>(lower_unit\<cdot>x) \<sqsubseteq> f\<cdot>zs)")
   381    apply (drule admD, rule chain_approx)
   382     apply (drule_tac f="approx i" in monofun_cfun_arg)
   383     apply (cut_tac x="approx i\<cdot>x" in compact_imp_Rep_compact_basis, simp)
   384     apply (cut_tac xs="approx i\<cdot>ys" in compact_imp_lower_principal, simp)
   385     apply (cut_tac xs="approx i\<cdot>zs" in compact_imp_lower_principal, simp)
   386     apply (clarify, simp add: lower_le_PDUnit_PDPlus_iff)
   387    apply simp
   388   apply simp
   389  apply (erule disjE)
   390   apply (erule trans_less [OF _ lower_plus_less1])
   391  apply (erule trans_less [OF _ lower_plus_less2])
   392 done
   393 
   394 lemmas lower_pd_less_simps =
   395   lower_unit_less_iff
   396   lower_plus_less_iff
   397   lower_unit_less_plus_iff
   398 
   399 
   400 subsection {* Induction rules *}
   401 
   402 lemma lower_pd_induct1:
   403   assumes P: "adm P"
   404   assumes unit: "\<And>x. P (lower_unit\<cdot>x)"
   405   assumes insert:
   406     "\<And>x ys. \<lbrakk>P (lower_unit\<cdot>x); P ys\<rbrakk> \<Longrightarrow> P (lower_plus\<cdot>(lower_unit\<cdot>x)\<cdot>ys)"
   407   shows "P (xs::'a lower_pd)"
   408 apply (induct xs rule: lower_principal_induct, rule P)
   409 apply (induct_tac t rule: pd_basis_induct1)
   410 apply (simp only: lower_unit_Rep_compact_basis [symmetric])
   411 apply (rule unit)
   412 apply (simp only: lower_unit_Rep_compact_basis [symmetric]
   413                   lower_plus_principal [symmetric])
   414 apply (erule insert [OF unit])
   415 done
   416 
   417 lemma lower_pd_induct:
   418   assumes P: "adm P"
   419   assumes unit: "\<And>x. P (lower_unit\<cdot>x)"
   420   assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (lower_plus\<cdot>xs\<cdot>ys)"
   421   shows "P (xs::'a lower_pd)"
   422 apply (induct xs rule: lower_principal_induct, rule P)
   423 apply (induct_tac t rule: pd_basis_induct)
   424 apply (simp only: lower_unit_Rep_compact_basis [symmetric] unit)
   425 apply (simp only: lower_plus_principal [symmetric] plus)
   426 done
   427 
   428 
   429 subsection {* Monadic bind *}
   430 
   431 definition
   432   lower_bind_basis ::
   433   "'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where
   434   "lower_bind_basis = fold_pd
   435     (\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a))
   436     (\<lambda>x y. \<Lambda> f. lower_plus\<cdot>(x\<cdot>f)\<cdot>(y\<cdot>f))"
   437 
   438 lemma ACI_lower_bind: "ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. lower_plus\<cdot>(x\<cdot>f)\<cdot>(y\<cdot>f))"
   439 apply unfold_locales
   440 apply (simp add: lower_plus_assoc)
   441 apply (simp add: lower_plus_commute)
   442 apply (simp add: lower_plus_absorb eta_cfun)
   443 done
   444 
   445 lemma lower_bind_basis_simps [simp]:
   446   "lower_bind_basis (PDUnit a) =
   447     (\<Lambda> f. f\<cdot>(Rep_compact_basis a))"
   448   "lower_bind_basis (PDPlus t u) =
   449     (\<Lambda> f. lower_plus\<cdot>(lower_bind_basis t\<cdot>f)\<cdot>(lower_bind_basis u\<cdot>f))"
   450 unfolding lower_bind_basis_def
   451 apply -
   452 apply (rule ab_semigroup_idem_mult.fold_pd_PDUnit [OF ACI_lower_bind])
   453 apply (rule ab_semigroup_idem_mult.fold_pd_PDPlus [OF ACI_lower_bind])
   454 done
   455 
   456 lemma lower_bind_basis_mono:
   457   "t \<le>\<flat> u \<Longrightarrow> lower_bind_basis t \<sqsubseteq> lower_bind_basis u"
   458 unfolding expand_cfun_less
   459 apply (erule lower_le_induct, safe)
   460 apply (simp add: compact_le_def monofun_cfun)
   461 apply (simp add: rev_trans_less [OF lower_plus_less1])
   462 apply (simp add: lower_plus_less_iff)
   463 done
   464 
   465 definition
   466   lower_bind :: "'a lower_pd \<rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where
   467   "lower_bind = lower_pd.basis_fun lower_bind_basis"
   468 
   469 lemma lower_bind_principal [simp]:
   470   "lower_bind\<cdot>(lower_principal t) = lower_bind_basis t"
   471 unfolding lower_bind_def
   472 apply (rule lower_pd.basis_fun_principal)
   473 apply (erule lower_bind_basis_mono)
   474 done
   475 
   476 lemma lower_bind_unit [simp]:
   477   "lower_bind\<cdot>(lower_unit\<cdot>x)\<cdot>f = f\<cdot>x"
   478 by (induct x rule: compact_basis_induct, simp, simp)
   479 
   480 lemma lower_bind_plus [simp]:
   481   "lower_bind\<cdot>(lower_plus\<cdot>xs\<cdot>ys)\<cdot>f =
   482    lower_plus\<cdot>(lower_bind\<cdot>xs\<cdot>f)\<cdot>(lower_bind\<cdot>ys\<cdot>f)"
   483 by (induct xs ys rule: lower_principal_induct2, simp, simp, simp)
   484 
   485 lemma lower_bind_strict [simp]: "lower_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>"
   486 unfolding lower_unit_strict [symmetric] by (rule lower_bind_unit)
   487 
   488 
   489 subsection {* Map and join *}
   490 
   491 definition
   492   lower_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a lower_pd \<rightarrow> 'b lower_pd" where
   493   "lower_map = (\<Lambda> f xs. lower_bind\<cdot>xs\<cdot>(\<Lambda> x. lower_unit\<cdot>(f\<cdot>x)))"
   494 
   495 definition
   496   lower_join :: "'a lower_pd lower_pd \<rightarrow> 'a lower_pd" where
   497   "lower_join = (\<Lambda> xss. lower_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))"
   498 
   499 lemma lower_map_unit [simp]:
   500   "lower_map\<cdot>f\<cdot>(lower_unit\<cdot>x) = lower_unit\<cdot>(f\<cdot>x)"
   501 unfolding lower_map_def by simp
   502 
   503 lemma lower_map_plus [simp]:
   504   "lower_map\<cdot>f\<cdot>(lower_plus\<cdot>xs\<cdot>ys) =
   505    lower_plus\<cdot>(lower_map\<cdot>f\<cdot>xs)\<cdot>(lower_map\<cdot>f\<cdot>ys)"
   506 unfolding lower_map_def by simp
   507 
   508 lemma lower_join_unit [simp]:
   509   "lower_join\<cdot>(lower_unit\<cdot>xs) = xs"
   510 unfolding lower_join_def by simp
   511 
   512 lemma lower_join_plus [simp]:
   513   "lower_join\<cdot>(lower_plus\<cdot>xss\<cdot>yss) =
   514    lower_plus\<cdot>(lower_join\<cdot>xss)\<cdot>(lower_join\<cdot>yss)"
   515 unfolding lower_join_def by simp
   516 
   517 lemma lower_map_ident: "lower_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs"
   518 by (induct xs rule: lower_pd_induct, simp_all)
   519 
   520 lemma lower_map_map:
   521   "lower_map\<cdot>f\<cdot>(lower_map\<cdot>g\<cdot>xs) = lower_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs"
   522 by (induct xs rule: lower_pd_induct, simp_all)
   523 
   524 lemma lower_join_map_unit:
   525   "lower_join\<cdot>(lower_map\<cdot>lower_unit\<cdot>xs) = xs"
   526 by (induct xs rule: lower_pd_induct, simp_all)
   527 
   528 lemma lower_join_map_join:
   529   "lower_join\<cdot>(lower_map\<cdot>lower_join\<cdot>xsss) = lower_join\<cdot>(lower_join\<cdot>xsss)"
   530 by (induct xsss rule: lower_pd_induct, simp_all)
   531 
   532 lemma lower_join_map_map:
   533   "lower_join\<cdot>(lower_map\<cdot>(lower_map\<cdot>f)\<cdot>xss) =
   534    lower_map\<cdot>f\<cdot>(lower_join\<cdot>xss)"
   535 by (induct xss rule: lower_pd_induct, simp_all)
   536 
   537 lemma lower_map_approx: "lower_map\<cdot>(approx n)\<cdot>xs = approx n\<cdot>xs"
   538 by (induct xs rule: lower_pd_induct, simp_all)
   539 
   540 end