src/HOL/Library/FSet.thy
author blanchet
Thu Mar 06 14:57:14 2014 +0100 (2014-03-06)
changeset 55938 f20d1db5aa3c
parent 55933 12ee2c407dad
child 55943 5c2df04e97d1
permissions -rw-r--r--
renamed 'set_rel' to 'rel_set'
     1 (*  Title:      HOL/Library/FSet.thy
     2     Author:     Ondrej Kuncar, TU Muenchen
     3     Author:     Cezary Kaliszyk and Christian Urban
     4     Author:     Andrei Popescu, TU Muenchen
     5 *)
     6 
     7 header {* Type of finite sets defined as a subtype of sets *}
     8 
     9 theory FSet
    10 imports Conditionally_Complete_Lattices
    11 begin
    12 
    13 subsection {* Definition of the type *}
    14 
    15 typedef 'a fset = "{A :: 'a set. finite A}"  morphisms fset Abs_fset
    16 by auto
    17 
    18 setup_lifting type_definition_fset
    19 
    20 
    21 subsection {* Basic operations and type class instantiations *}
    22 
    23 (* FIXME transfer and right_total vs. bi_total *)
    24 instantiation fset :: (finite) finite
    25 begin
    26 instance by default (transfer, simp)
    27 end
    28 
    29 instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
    30 begin
    31 
    32 interpretation lifting_syntax .
    33 
    34 lift_definition bot_fset :: "'a fset" is "{}" parametric empty_transfer by simp 
    35 
    36 lift_definition less_eq_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" is subset_eq parametric subset_transfer 
    37   .
    38 
    39 definition less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" where "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"
    40 
    41 lemma less_fset_transfer[transfer_rule]:
    42   assumes [transfer_rule]: "bi_unique A" 
    43   shows "((pcr_fset A) ===> (pcr_fset A) ===> op =) op \<subset> op <"
    44   unfolding less_fset_def[abs_def] psubset_eq[abs_def] by transfer_prover
    45   
    46 
    47 lift_definition sup_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is union parametric union_transfer
    48   by simp
    49 
    50 lift_definition inf_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is inter parametric inter_transfer
    51   by simp
    52 
    53 lift_definition minus_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is minus parametric Diff_transfer
    54   by simp
    55 
    56 instance
    57 by default (transfer, auto)+
    58 
    59 end
    60 
    61 abbreviation fempty :: "'a fset" ("{||}") where "{||} \<equiv> bot"
    62 abbreviation fsubset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50) where "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
    63 abbreviation fsubset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50) where "xs |\<subset>| ys \<equiv> xs < ys"
    64 abbreviation funion :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<union>|" 65) where "xs |\<union>| ys \<equiv> sup xs ys"
    65 abbreviation finter :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<inter>|" 65) where "xs |\<inter>| ys \<equiv> inf xs ys"
    66 abbreviation fminus :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|-|" 65) where "xs |-| ys \<equiv> minus xs ys"
    67 
    68 instantiation fset :: (equal) equal
    69 begin
    70 definition "HOL.equal A B \<longleftrightarrow> A |\<subseteq>| B \<and> B |\<subseteq>| A"
    71 instance by intro_classes (auto simp add: equal_fset_def)
    72 end 
    73 
    74 instantiation fset :: (type) conditionally_complete_lattice
    75 begin
    76 
    77 interpretation lifting_syntax .
    78 
    79 lemma right_total_Inf_fset_transfer:
    80   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
    81   shows "(rel_set (rel_set A) ===> rel_set A) 
    82     (\<lambda>S. if finite (Inter S \<inter> Collect (Domainp A)) then Inter S \<inter> Collect (Domainp A) else {}) 
    83       (\<lambda>S. if finite (Inf S) then Inf S else {})"
    84     by transfer_prover
    85 
    86 lemma Inf_fset_transfer:
    87   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
    88   shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>A. if finite (Inf A) then Inf A else {}) 
    89     (\<lambda>A. if finite (Inf A) then Inf A else {})"
    90   by transfer_prover
    91 
    92 lift_definition Inf_fset :: "'a fset set \<Rightarrow> 'a fset" is "\<lambda>A. if finite (Inf A) then Inf A else {}" 
    93 parametric right_total_Inf_fset_transfer Inf_fset_transfer by simp
    94 
    95 lemma Sup_fset_transfer:
    96   assumes [transfer_rule]: "bi_unique A"
    97   shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>A. if finite (Sup A) then Sup A else {})
    98   (\<lambda>A. if finite (Sup A) then Sup A else {})" by transfer_prover
    99 
   100 lift_definition Sup_fset :: "'a fset set \<Rightarrow> 'a fset" is "\<lambda>A. if finite (Sup A) then Sup A else {}"
   101 parametric Sup_fset_transfer by simp
   102 
   103 lemma finite_Sup: "\<exists>z. finite z \<and> (\<forall>a. a \<in> X \<longrightarrow> a \<le> z) \<Longrightarrow> finite (Sup X)"
   104 by (auto intro: finite_subset)
   105 
   106 lemma transfer_bdd_below[transfer_rule]: "(rel_set (pcr_fset op =) ===> op =) bdd_below bdd_below"
   107   by auto
   108 
   109 instance
   110 proof 
   111   fix x z :: "'a fset"
   112   fix X :: "'a fset set"
   113   {
   114     assume "x \<in> X" "bdd_below X" 
   115     then show "Inf X |\<subseteq>| x"  by transfer auto
   116   next
   117     assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> z |\<subseteq>| x)"
   118     then show "z |\<subseteq>| Inf X" by transfer (clarsimp, blast)
   119   next
   120     assume "x \<in> X" "bdd_above X"
   121     then obtain z where "x \<in> X" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)"
   122       by (auto simp: bdd_above_def)
   123     then show "x |\<subseteq>| Sup X"
   124       by transfer (auto intro!: finite_Sup)
   125   next
   126     assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)"
   127     then show "Sup X |\<subseteq>| z" by transfer (clarsimp, blast)
   128   }
   129 qed
   130 end
   131 
   132 instantiation fset :: (finite) complete_lattice 
   133 begin
   134 
   135 lift_definition top_fset :: "'a fset" is UNIV parametric right_total_UNIV_transfer UNIV_transfer by simp
   136 
   137 instance by default (transfer, auto)+
   138 end
   139 
   140 instantiation fset :: (finite) complete_boolean_algebra
   141 begin
   142 
   143 lift_definition uminus_fset :: "'a fset \<Rightarrow> 'a fset" is uminus 
   144   parametric right_total_Compl_transfer Compl_transfer by simp
   145 
   146 instance by (default, simp_all only: INF_def SUP_def) (transfer, auto)+
   147 
   148 end
   149 
   150 abbreviation fUNIV :: "'a::finite fset" where "fUNIV \<equiv> top"
   151 abbreviation fuminus :: "'a::finite fset \<Rightarrow> 'a fset" ("|-| _" [81] 80) where "|-| x \<equiv> uminus x"
   152 
   153 
   154 subsection {* Other operations *}
   155 
   156 lift_definition finsert :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is insert parametric Lifting_Set.insert_transfer
   157   by simp
   158 
   159 syntax
   160   "_insert_fset"     :: "args => 'a fset"  ("{|(_)|}")
   161 
   162 translations
   163   "{|x, xs|}" == "CONST finsert x {|xs|}"
   164   "{|x|}"     == "CONST finsert x {||}"
   165 
   166 lift_definition fmember :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<in>|" 50) is Set.member 
   167   parametric member_transfer .
   168 
   169 abbreviation notin_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50) where "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
   170 
   171 context
   172 begin
   173 
   174 interpretation lifting_syntax .
   175 
   176 lift_definition ffilter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is Set.filter 
   177   parametric Lifting_Set.filter_transfer unfolding Set.filter_def by simp
   178 
   179 lift_definition fPow :: "'a fset \<Rightarrow> 'a fset fset" is Pow parametric Pow_transfer 
   180 by (simp add: finite_subset)
   181 
   182 lift_definition fcard :: "'a fset \<Rightarrow> nat" is card parametric card_transfer .
   183 
   184 lift_definition fimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset" (infixr "|`|" 90) is image 
   185   parametric image_transfer by simp
   186 
   187 lift_definition fthe_elem :: "'a fset \<Rightarrow> 'a" is the_elem .
   188 
   189 lift_definition fbind :: "'a fset \<Rightarrow> ('a \<Rightarrow> 'b fset) \<Rightarrow> 'b fset" is Set.bind parametric bind_transfer 
   190 by (simp add: Set.bind_def)
   191 
   192 lift_definition ffUnion :: "'a fset fset \<Rightarrow> 'a fset" is Union parametric Union_transfer by simp
   193 
   194 lift_definition fBall :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" is Ball parametric Ball_transfer .
   195 lift_definition fBex :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" is Bex parametric Bex_transfer .
   196 
   197 lift_definition ffold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" is Finite_Set.fold .
   198 
   199 
   200 subsection {* Transferred lemmas from Set.thy *}
   201 
   202 lemmas fset_eqI = set_eqI[Transfer.transferred]
   203 lemmas fset_eq_iff[no_atp] = set_eq_iff[Transfer.transferred]
   204 lemmas fBallI[intro!] = ballI[Transfer.transferred]
   205 lemmas fbspec[dest?] = bspec[Transfer.transferred]
   206 lemmas fBallE[elim] = ballE[Transfer.transferred]
   207 lemmas fBexI[intro] = bexI[Transfer.transferred]
   208 lemmas rev_fBexI[intro?] = rev_bexI[Transfer.transferred]
   209 lemmas fBexCI = bexCI[Transfer.transferred]
   210 lemmas fBexE[elim!] = bexE[Transfer.transferred]
   211 lemmas fBall_triv[simp] = ball_triv[Transfer.transferred]
   212 lemmas fBex_triv[simp] = bex_triv[Transfer.transferred]
   213 lemmas fBex_triv_one_point1[simp] = bex_triv_one_point1[Transfer.transferred]
   214 lemmas fBex_triv_one_point2[simp] = bex_triv_one_point2[Transfer.transferred]
   215 lemmas fBex_one_point1[simp] = bex_one_point1[Transfer.transferred]
   216 lemmas fBex_one_point2[simp] = bex_one_point2[Transfer.transferred]
   217 lemmas fBall_one_point1[simp] = ball_one_point1[Transfer.transferred]
   218 lemmas fBall_one_point2[simp] = ball_one_point2[Transfer.transferred]
   219 lemmas fBall_conj_distrib = ball_conj_distrib[Transfer.transferred]
   220 lemmas fBex_disj_distrib = bex_disj_distrib[Transfer.transferred]
   221 lemmas fBall_cong = ball_cong[Transfer.transferred]
   222 lemmas fBex_cong = bex_cong[Transfer.transferred]
   223 lemmas fsubsetI[intro!] = subsetI[Transfer.transferred]
   224 lemmas fsubsetD[elim, intro?] = subsetD[Transfer.transferred]
   225 lemmas rev_fsubsetD[no_atp,intro?] = rev_subsetD[Transfer.transferred]
   226 lemmas fsubsetCE[no_atp,elim] = subsetCE[Transfer.transferred]
   227 lemmas fsubset_eq[no_atp] = subset_eq[Transfer.transferred]
   228 lemmas contra_fsubsetD[no_atp] = contra_subsetD[Transfer.transferred]
   229 lemmas fsubset_refl = subset_refl[Transfer.transferred]
   230 lemmas fsubset_trans = subset_trans[Transfer.transferred]
   231 lemmas fset_rev_mp = set_rev_mp[Transfer.transferred]
   232 lemmas fset_mp = set_mp[Transfer.transferred]
   233 lemmas fsubset_not_fsubset_eq[code] = subset_not_subset_eq[Transfer.transferred]
   234 lemmas eq_fmem_trans = eq_mem_trans[Transfer.transferred]
   235 lemmas fsubset_antisym[intro!] = subset_antisym[Transfer.transferred]
   236 lemmas fequalityD1 = equalityD1[Transfer.transferred]
   237 lemmas fequalityD2 = equalityD2[Transfer.transferred]
   238 lemmas fequalityE = equalityE[Transfer.transferred]
   239 lemmas fequalityCE[elim] = equalityCE[Transfer.transferred]
   240 lemmas eqfset_imp_iff = eqset_imp_iff[Transfer.transferred]
   241 lemmas eqfelem_imp_iff = eqelem_imp_iff[Transfer.transferred]
   242 lemmas fempty_iff[simp] = empty_iff[Transfer.transferred]
   243 lemmas fempty_fsubsetI[iff] = empty_subsetI[Transfer.transferred]
   244 lemmas equalsffemptyI = equals0I[Transfer.transferred]
   245 lemmas equalsffemptyD = equals0D[Transfer.transferred]
   246 lemmas fBall_fempty[simp] = ball_empty[Transfer.transferred]
   247 lemmas fBex_fempty[simp] = bex_empty[Transfer.transferred]
   248 lemmas fPow_iff[iff] = Pow_iff[Transfer.transferred]
   249 lemmas fPowI = PowI[Transfer.transferred]
   250 lemmas fPowD = PowD[Transfer.transferred]
   251 lemmas fPow_bottom = Pow_bottom[Transfer.transferred]
   252 lemmas fPow_top = Pow_top[Transfer.transferred]
   253 lemmas fPow_not_fempty = Pow_not_empty[Transfer.transferred]
   254 lemmas finter_iff[simp] = Int_iff[Transfer.transferred]
   255 lemmas finterI[intro!] = IntI[Transfer.transferred]
   256 lemmas finterD1 = IntD1[Transfer.transferred]
   257 lemmas finterD2 = IntD2[Transfer.transferred]
   258 lemmas finterE[elim!] = IntE[Transfer.transferred]
   259 lemmas funion_iff[simp] = Un_iff[Transfer.transferred]
   260 lemmas funionI1[elim?] = UnI1[Transfer.transferred]
   261 lemmas funionI2[elim?] = UnI2[Transfer.transferred]
   262 lemmas funionCI[intro!] = UnCI[Transfer.transferred]
   263 lemmas funionE[elim!] = UnE[Transfer.transferred]
   264 lemmas fminus_iff[simp] = Diff_iff[Transfer.transferred]
   265 lemmas fminusI[intro!] = DiffI[Transfer.transferred]
   266 lemmas fminusD1 = DiffD1[Transfer.transferred]
   267 lemmas fminusD2 = DiffD2[Transfer.transferred]
   268 lemmas fminusE[elim!] = DiffE[Transfer.transferred]
   269 lemmas finsert_iff[simp] = insert_iff[Transfer.transferred]
   270 lemmas finsertI1 = insertI1[Transfer.transferred]
   271 lemmas finsertI2 = insertI2[Transfer.transferred]
   272 lemmas finsertE[elim!] = insertE[Transfer.transferred]
   273 lemmas finsertCI[intro!] = insertCI[Transfer.transferred]
   274 lemmas fsubset_finsert_iff = subset_insert_iff[Transfer.transferred]
   275 lemmas finsert_ident = insert_ident[Transfer.transferred]
   276 lemmas fsingletonI[intro!,no_atp] = singletonI[Transfer.transferred]
   277 lemmas fsingletonD[dest!,no_atp] = singletonD[Transfer.transferred]
   278 lemmas fsingleton_iff = singleton_iff[Transfer.transferred]
   279 lemmas fsingleton_inject[dest!] = singleton_inject[Transfer.transferred]
   280 lemmas fsingleton_finsert_inj_eq[iff,no_atp] = singleton_insert_inj_eq[Transfer.transferred]
   281 lemmas fsingleton_finsert_inj_eq'[iff,no_atp] = singleton_insert_inj_eq'[Transfer.transferred]
   282 lemmas fsubset_fsingletonD = subset_singletonD[Transfer.transferred]
   283 lemmas fminus_single_finsert = diff_single_insert[Transfer.transferred]
   284 lemmas fdoubleton_eq_iff = doubleton_eq_iff[Transfer.transferred]
   285 lemmas funion_fsingleton_iff = Un_singleton_iff[Transfer.transferred]
   286 lemmas fsingleton_funion_iff = singleton_Un_iff[Transfer.transferred]
   287 lemmas fimage_eqI[simp, intro] = image_eqI[Transfer.transferred]
   288 lemmas fimageI = imageI[Transfer.transferred]
   289 lemmas rev_fimage_eqI = rev_image_eqI[Transfer.transferred]
   290 lemmas fimageE[elim!] = imageE[Transfer.transferred]
   291 lemmas Compr_fimage_eq = Compr_image_eq[Transfer.transferred]
   292 lemmas fimage_funion = image_Un[Transfer.transferred]
   293 lemmas fimage_iff = image_iff[Transfer.transferred]
   294 lemmas fimage_fsubset_iff[no_atp] = image_subset_iff[Transfer.transferred]
   295 lemmas fimage_fsubsetI = image_subsetI[Transfer.transferred]
   296 lemmas fimage_ident[simp] = image_ident[Transfer.transferred]
   297 lemmas split_if_fmem1 = split_if_mem1[Transfer.transferred]
   298 lemmas split_if_fmem2 = split_if_mem2[Transfer.transferred]
   299 lemmas pfsubsetI[intro!,no_atp] = psubsetI[Transfer.transferred]
   300 lemmas pfsubsetE[elim!,no_atp] = psubsetE[Transfer.transferred]
   301 lemmas pfsubset_finsert_iff = psubset_insert_iff[Transfer.transferred]
   302 lemmas pfsubset_eq = psubset_eq[Transfer.transferred]
   303 lemmas pfsubset_imp_fsubset = psubset_imp_subset[Transfer.transferred]
   304 lemmas pfsubset_trans = psubset_trans[Transfer.transferred]
   305 lemmas pfsubsetD = psubsetD[Transfer.transferred]
   306 lemmas pfsubset_fsubset_trans = psubset_subset_trans[Transfer.transferred]
   307 lemmas fsubset_pfsubset_trans = subset_psubset_trans[Transfer.transferred]
   308 lemmas pfsubset_imp_ex_fmem = psubset_imp_ex_mem[Transfer.transferred]
   309 lemmas fimage_fPow_mono = image_Pow_mono[Transfer.transferred]
   310 lemmas fimage_fPow_surj = image_Pow_surj[Transfer.transferred]
   311 lemmas fsubset_finsertI = subset_insertI[Transfer.transferred]
   312 lemmas fsubset_finsertI2 = subset_insertI2[Transfer.transferred]
   313 lemmas fsubset_finsert = subset_insert[Transfer.transferred]
   314 lemmas funion_upper1 = Un_upper1[Transfer.transferred]
   315 lemmas funion_upper2 = Un_upper2[Transfer.transferred]
   316 lemmas funion_least = Un_least[Transfer.transferred]
   317 lemmas finter_lower1 = Int_lower1[Transfer.transferred]
   318 lemmas finter_lower2 = Int_lower2[Transfer.transferred]
   319 lemmas finter_greatest = Int_greatest[Transfer.transferred]
   320 lemmas fminus_fsubset = Diff_subset[Transfer.transferred]
   321 lemmas fminus_fsubset_conv = Diff_subset_conv[Transfer.transferred]
   322 lemmas fsubset_fempty[simp] = subset_empty[Transfer.transferred]
   323 lemmas not_pfsubset_fempty[iff] = not_psubset_empty[Transfer.transferred]
   324 lemmas finsert_is_funion = insert_is_Un[Transfer.transferred]
   325 lemmas finsert_not_fempty[simp] = insert_not_empty[Transfer.transferred]
   326 lemmas fempty_not_finsert = empty_not_insert[Transfer.transferred]
   327 lemmas finsert_absorb = insert_absorb[Transfer.transferred]
   328 lemmas finsert_absorb2[simp] = insert_absorb2[Transfer.transferred]
   329 lemmas finsert_commute = insert_commute[Transfer.transferred]
   330 lemmas finsert_fsubset[simp] = insert_subset[Transfer.transferred]
   331 lemmas finsert_inter_finsert[simp] = insert_inter_insert[Transfer.transferred]
   332 lemmas finsert_disjoint[simp,no_atp] = insert_disjoint[Transfer.transferred]
   333 lemmas disjoint_finsert[simp,no_atp] = disjoint_insert[Transfer.transferred]
   334 lemmas fimage_fempty[simp] = image_empty[Transfer.transferred]
   335 lemmas fimage_finsert[simp] = image_insert[Transfer.transferred]
   336 lemmas fimage_constant = image_constant[Transfer.transferred]
   337 lemmas fimage_constant_conv = image_constant_conv[Transfer.transferred]
   338 lemmas fimage_fimage = image_image[Transfer.transferred]
   339 lemmas finsert_fimage[simp] = insert_image[Transfer.transferred]
   340 lemmas fimage_is_fempty[iff] = image_is_empty[Transfer.transferred]
   341 lemmas fempty_is_fimage[iff] = empty_is_image[Transfer.transferred]
   342 lemmas fimage_cong = image_cong[Transfer.transferred]
   343 lemmas fimage_finter_fsubset = image_Int_subset[Transfer.transferred]
   344 lemmas fimage_fminus_fsubset = image_diff_subset[Transfer.transferred]
   345 lemmas finter_absorb = Int_absorb[Transfer.transferred]
   346 lemmas finter_left_absorb = Int_left_absorb[Transfer.transferred]
   347 lemmas finter_commute = Int_commute[Transfer.transferred]
   348 lemmas finter_left_commute = Int_left_commute[Transfer.transferred]
   349 lemmas finter_assoc = Int_assoc[Transfer.transferred]
   350 lemmas finter_ac = Int_ac[Transfer.transferred]
   351 lemmas finter_absorb1 = Int_absorb1[Transfer.transferred]
   352 lemmas finter_absorb2 = Int_absorb2[Transfer.transferred]
   353 lemmas finter_fempty_left = Int_empty_left[Transfer.transferred]
   354 lemmas finter_fempty_right = Int_empty_right[Transfer.transferred]
   355 lemmas disjoint_iff_fnot_equal = disjoint_iff_not_equal[Transfer.transferred]
   356 lemmas finter_funion_distrib = Int_Un_distrib[Transfer.transferred]
   357 lemmas finter_funion_distrib2 = Int_Un_distrib2[Transfer.transferred]
   358 lemmas finter_fsubset_iff[no_atp, simp] = Int_subset_iff[Transfer.transferred]
   359 lemmas funion_absorb = Un_absorb[Transfer.transferred]
   360 lemmas funion_left_absorb = Un_left_absorb[Transfer.transferred]
   361 lemmas funion_commute = Un_commute[Transfer.transferred]
   362 lemmas funion_left_commute = Un_left_commute[Transfer.transferred]
   363 lemmas funion_assoc = Un_assoc[Transfer.transferred]
   364 lemmas funion_ac = Un_ac[Transfer.transferred]
   365 lemmas funion_absorb1 = Un_absorb1[Transfer.transferred]
   366 lemmas funion_absorb2 = Un_absorb2[Transfer.transferred]
   367 lemmas funion_fempty_left = Un_empty_left[Transfer.transferred]
   368 lemmas funion_fempty_right = Un_empty_right[Transfer.transferred]
   369 lemmas funion_finsert_left[simp] = Un_insert_left[Transfer.transferred]
   370 lemmas funion_finsert_right[simp] = Un_insert_right[Transfer.transferred]
   371 lemmas finter_finsert_left = Int_insert_left[Transfer.transferred]
   372 lemmas finter_finsert_left_ifffempty[simp] = Int_insert_left_if0[Transfer.transferred]
   373 lemmas finter_finsert_left_if1[simp] = Int_insert_left_if1[Transfer.transferred]
   374 lemmas finter_finsert_right = Int_insert_right[Transfer.transferred]
   375 lemmas finter_finsert_right_ifffempty[simp] = Int_insert_right_if0[Transfer.transferred]
   376 lemmas finter_finsert_right_if1[simp] = Int_insert_right_if1[Transfer.transferred]
   377 lemmas funion_finter_distrib = Un_Int_distrib[Transfer.transferred]
   378 lemmas funion_finter_distrib2 = Un_Int_distrib2[Transfer.transferred]
   379 lemmas funion_finter_crazy = Un_Int_crazy[Transfer.transferred]
   380 lemmas fsubset_funion_eq = subset_Un_eq[Transfer.transferred]
   381 lemmas funion_fempty[iff] = Un_empty[Transfer.transferred]
   382 lemmas funion_fsubset_iff[no_atp, simp] = Un_subset_iff[Transfer.transferred]
   383 lemmas funion_fminus_finter = Un_Diff_Int[Transfer.transferred]
   384 lemmas fminus_finter2 = Diff_Int2[Transfer.transferred]
   385 lemmas funion_finter_assoc_eq = Un_Int_assoc_eq[Transfer.transferred]
   386 lemmas fBall_funion = ball_Un[Transfer.transferred]
   387 lemmas fBex_funion = bex_Un[Transfer.transferred]
   388 lemmas fminus_eq_fempty_iff[simp,no_atp] = Diff_eq_empty_iff[Transfer.transferred]
   389 lemmas fminus_cancel[simp] = Diff_cancel[Transfer.transferred]
   390 lemmas fminus_idemp[simp] = Diff_idemp[Transfer.transferred]
   391 lemmas fminus_triv = Diff_triv[Transfer.transferred]
   392 lemmas fempty_fminus[simp] = empty_Diff[Transfer.transferred]
   393 lemmas fminus_fempty[simp] = Diff_empty[Transfer.transferred]
   394 lemmas fminus_finsertffempty[simp,no_atp] = Diff_insert0[Transfer.transferred]
   395 lemmas fminus_finsert = Diff_insert[Transfer.transferred]
   396 lemmas fminus_finsert2 = Diff_insert2[Transfer.transferred]
   397 lemmas finsert_fminus_if = insert_Diff_if[Transfer.transferred]
   398 lemmas finsert_fminus1[simp] = insert_Diff1[Transfer.transferred]
   399 lemmas finsert_fminus_single[simp] = insert_Diff_single[Transfer.transferred]
   400 lemmas finsert_fminus = insert_Diff[Transfer.transferred]
   401 lemmas fminus_finsert_absorb = Diff_insert_absorb[Transfer.transferred]
   402 lemmas fminus_disjoint[simp] = Diff_disjoint[Transfer.transferred]
   403 lemmas fminus_partition = Diff_partition[Transfer.transferred]
   404 lemmas double_fminus = double_diff[Transfer.transferred]
   405 lemmas funion_fminus_cancel[simp] = Un_Diff_cancel[Transfer.transferred]
   406 lemmas funion_fminus_cancel2[simp] = Un_Diff_cancel2[Transfer.transferred]
   407 lemmas fminus_funion = Diff_Un[Transfer.transferred]
   408 lemmas fminus_finter = Diff_Int[Transfer.transferred]
   409 lemmas funion_fminus = Un_Diff[Transfer.transferred]
   410 lemmas finter_fminus = Int_Diff[Transfer.transferred]
   411 lemmas fminus_finter_distrib = Diff_Int_distrib[Transfer.transferred]
   412 lemmas fminus_finter_distrib2 = Diff_Int_distrib2[Transfer.transferred]
   413 lemmas fUNIV_bool[no_atp] = UNIV_bool[Transfer.transferred]
   414 lemmas fPow_fempty[simp] = Pow_empty[Transfer.transferred]
   415 lemmas fPow_finsert = Pow_insert[Transfer.transferred]
   416 lemmas funion_fPow_fsubset = Un_Pow_subset[Transfer.transferred]
   417 lemmas fPow_finter_eq[simp] = Pow_Int_eq[Transfer.transferred]
   418 lemmas fset_eq_fsubset = set_eq_subset[Transfer.transferred]
   419 lemmas fsubset_iff[no_atp] = subset_iff[Transfer.transferred]
   420 lemmas fsubset_iff_pfsubset_eq = subset_iff_psubset_eq[Transfer.transferred]
   421 lemmas all_not_fin_conv[simp] = all_not_in_conv[Transfer.transferred]
   422 lemmas ex_fin_conv = ex_in_conv[Transfer.transferred]
   423 lemmas fimage_mono = image_mono[Transfer.transferred]
   424 lemmas fPow_mono = Pow_mono[Transfer.transferred]
   425 lemmas finsert_mono = insert_mono[Transfer.transferred]
   426 lemmas funion_mono = Un_mono[Transfer.transferred]
   427 lemmas finter_mono = Int_mono[Transfer.transferred]
   428 lemmas fminus_mono = Diff_mono[Transfer.transferred]
   429 lemmas fin_mono = in_mono[Transfer.transferred]
   430 lemmas fthe_felem_eq[simp] = the_elem_eq[Transfer.transferred]
   431 lemmas fLeast_mono = Least_mono[Transfer.transferred]
   432 lemmas fbind_fbind = bind_bind[Transfer.transferred]
   433 lemmas fempty_fbind[simp] = empty_bind[Transfer.transferred]
   434 lemmas nonfempty_fbind_const = nonempty_bind_const[Transfer.transferred]
   435 lemmas fbind_const = bind_const[Transfer.transferred]
   436 lemmas ffmember_filter[simp] = member_filter[Transfer.transferred]
   437 lemmas fequalityI = equalityI[Transfer.transferred]
   438 
   439 
   440 subsection {* Additional lemmas*}
   441 
   442 subsubsection {* @{text fsingleton} *}
   443 
   444 lemmas fsingletonE = fsingletonD [elim_format]
   445 
   446 
   447 subsubsection {* @{text femepty} *}
   448 
   449 lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {||}"
   450 by transfer auto
   451 
   452 (* FIXME, transferred doesn't work here *)
   453 lemma femptyE [elim!]: "a |\<in>| {||} \<Longrightarrow> P"
   454   by simp
   455 
   456 
   457 subsubsection {* @{text fset} *}
   458 
   459 lemmas fset_simps[simp] = bot_fset.rep_eq finsert.rep_eq
   460 
   461 lemma finite_fset [simp]: 
   462   shows "finite (fset S)"
   463   by transfer simp
   464 
   465 lemmas fset_cong = fset_inject
   466 
   467 lemma filter_fset [simp]:
   468   shows "fset (ffilter P xs) = Collect P \<inter> fset xs"
   469   by transfer auto
   470 
   471 lemma notin_fset: "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S" by (simp add: fmember.rep_eq)
   472 
   473 lemmas inter_fset[simp] = inf_fset.rep_eq
   474 
   475 lemmas union_fset[simp] = sup_fset.rep_eq
   476 
   477 lemmas minus_fset[simp] = minus_fset.rep_eq
   478 
   479 
   480 subsubsection {* @{text filter_fset} *}
   481 
   482 lemma subset_ffilter: 
   483   "ffilter P A |\<subseteq>| ffilter Q A = (\<forall> x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)"
   484   by transfer auto
   485 
   486 lemma eq_ffilter: 
   487   "(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)"
   488   by transfer auto
   489 
   490 lemma pfsubset_ffilter:
   491   "(\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| A & \<not> P x & Q x) \<Longrightarrow> 
   492     ffilter P A |\<subset>| ffilter Q A"
   493   unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter)
   494 
   495 
   496 subsubsection {* @{text finsert} *}
   497 
   498 (* FIXME, transferred doesn't work here *)
   499 lemma set_finsert:
   500   assumes "x |\<in>| A"
   501   obtains B where "A = finsert x B" and "x |\<notin>| B"
   502 using assms by transfer (metis Set.set_insert finite_insert)
   503 
   504 lemma mk_disjoint_finsert: "a |\<in>| A \<Longrightarrow> \<exists>B. A = finsert a B \<and> a |\<notin>| B"
   505   by (rule_tac x = "A |-| {|a|}" in exI, blast)
   506 
   507 
   508 subsubsection {* @{text fimage} *}
   509 
   510 lemma subset_fimage_iff: "(B |\<subseteq>| f|`|A) = (\<exists> AA. AA |\<subseteq>| A \<and> B = f|`|AA)"
   511 by transfer (metis mem_Collect_eq rev_finite_subset subset_image_iff)
   512 
   513 
   514 subsubsection {* bounded quantification *}
   515 
   516 lemma bex_simps [simp, no_atp]:
   517   "\<And>A P Q. fBex A (\<lambda>x. P x \<and> Q) = (fBex A P \<and> Q)" 
   518   "\<And>A P Q. fBex A (\<lambda>x. P \<and> Q x) = (P \<and> fBex A Q)"
   519   "\<And>P. fBex {||} P = False" 
   520   "\<And>a B P. fBex (finsert a B) P = (P a \<or> fBex B P)"
   521   "\<And>A P f. fBex (f |`| A) P = fBex A (\<lambda>x. P (f x))"
   522   "\<And>A P. (\<not> fBex A P) = fBall A (\<lambda>x. \<not> P x)"
   523 by auto
   524 
   525 lemma ball_simps [simp, no_atp]:
   526   "\<And>A P Q. fBall A (\<lambda>x. P x \<or> Q) = (fBall A P \<or> Q)"
   527   "\<And>A P Q. fBall A (\<lambda>x. P \<or> Q x) = (P \<or> fBall A Q)"
   528   "\<And>A P Q. fBall A (\<lambda>x. P \<longrightarrow> Q x) = (P \<longrightarrow> fBall A Q)"
   529   "\<And>A P Q. fBall A (\<lambda>x. P x \<longrightarrow> Q) = (fBex A P \<longrightarrow> Q)"
   530   "\<And>P. fBall {||} P = True"
   531   "\<And>a B P. fBall (finsert a B) P = (P a \<and> fBall B P)"
   532   "\<And>A P f. fBall (f |`| A) P = fBall A (\<lambda>x. P (f x))"
   533   "\<And>A P. (\<not> fBall A P) = fBex A (\<lambda>x. \<not> P x)"
   534 by auto
   535 
   536 lemma atomize_fBall:
   537     "(\<And>x. x |\<in>| A ==> P x) == Trueprop (fBall A (\<lambda>x. P x))"
   538 apply (simp only: atomize_all atomize_imp)
   539 apply (rule equal_intr_rule)
   540 by (transfer, simp)+
   541 
   542 end
   543 
   544 
   545 subsubsection {* @{text fcard} *}
   546 
   547 (* FIXME: improve transferred to handle bounded meta quantification *)
   548 
   549 lemma fcard_fempty:
   550   "fcard {||} = 0"
   551   by transfer (rule card_empty)
   552 
   553 lemma fcard_finsert_disjoint:
   554   "x |\<notin>| A \<Longrightarrow> fcard (finsert x A) = Suc (fcard A)"
   555   by transfer (rule card_insert_disjoint)
   556 
   557 lemma fcard_finsert_if:
   558   "fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))"
   559   by transfer (rule card_insert_if)
   560 
   561 lemma card_0_eq [simp, no_atp]:
   562   "fcard A = 0 \<longleftrightarrow> A = {||}"
   563   by transfer (rule card_0_eq)
   564 
   565 lemma fcard_Suc_fminus1:
   566   "x |\<in>| A \<Longrightarrow> Suc (fcard (A |-| {|x|})) = fcard A"
   567   by transfer (rule card_Suc_Diff1)
   568 
   569 lemma fcard_fminus_fsingleton:
   570   "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) = fcard A - 1"
   571   by transfer (rule card_Diff_singleton)
   572 
   573 lemma fcard_fminus_fsingleton_if:
   574   "fcard (A |-| {|x|}) = (if x |\<in>| A then fcard A - 1 else fcard A)"
   575   by transfer (rule card_Diff_singleton_if)
   576 
   577 lemma fcard_fminus_finsert[simp]:
   578   assumes "a |\<in>| A" and "a |\<notin>| B"
   579   shows "fcard (A |-| finsert a B) = fcard (A |-| B) - 1"
   580 using assms by transfer (rule card_Diff_insert)
   581 
   582 lemma fcard_finsert: "fcard (finsert x A) = Suc (fcard (A |-| {|x|}))"
   583 by transfer (rule card_insert)
   584 
   585 lemma fcard_finsert_le: "fcard A \<le> fcard (finsert x A)"
   586 by transfer (rule card_insert_le)
   587 
   588 lemma fcard_mono:
   589   "A |\<subseteq>| B \<Longrightarrow> fcard A \<le> fcard B"
   590 by transfer (rule card_mono)
   591 
   592 lemma fcard_seteq: "A |\<subseteq>| B \<Longrightarrow> fcard B \<le> fcard A \<Longrightarrow> A = B"
   593 by transfer (rule card_seteq)
   594 
   595 lemma pfsubset_fcard_mono: "A |\<subset>| B \<Longrightarrow> fcard A < fcard B"
   596 by transfer (rule psubset_card_mono)
   597 
   598 lemma fcard_funion_finter: 
   599   "fcard A + fcard B = fcard (A |\<union>| B) + fcard (A |\<inter>| B)"
   600 by transfer (rule card_Un_Int)
   601 
   602 lemma fcard_funion_disjoint:
   603   "A |\<inter>| B = {||} \<Longrightarrow> fcard (A |\<union>| B) = fcard A + fcard B"
   604 by transfer (rule card_Un_disjoint)
   605 
   606 lemma fcard_funion_fsubset:
   607   "B |\<subseteq>| A \<Longrightarrow> fcard (A |-| B) = fcard A - fcard B"
   608 by transfer (rule card_Diff_subset)
   609 
   610 lemma diff_fcard_le_fcard_fminus:
   611   "fcard A - fcard B \<le> fcard(A |-| B)"
   612 by transfer (rule diff_card_le_card_Diff)
   613 
   614 lemma fcard_fminus1_less: "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) < fcard A"
   615 by transfer (rule card_Diff1_less)
   616 
   617 lemma fcard_fminus2_less:
   618   "x |\<in>| A \<Longrightarrow> y |\<in>| A \<Longrightarrow> fcard (A |-| {|x|} |-| {|y|}) < fcard A"
   619 by transfer (rule card_Diff2_less)
   620 
   621 lemma fcard_fminus1_le: "fcard (A |-| {|x|}) \<le> fcard A"
   622 by transfer (rule card_Diff1_le)
   623 
   624 lemma fcard_pfsubset: "A |\<subseteq>| B \<Longrightarrow> fcard A < fcard B \<Longrightarrow> A < B"
   625 by transfer (rule card_psubset)
   626 
   627 
   628 subsubsection {* @{text ffold} *}
   629 
   630 (* FIXME: improve transferred to handle bounded meta quantification *)
   631 
   632 context comp_fun_commute
   633 begin
   634   lemmas ffold_empty[simp] = fold_empty[Transfer.transferred]
   635 
   636   lemma ffold_finsert [simp]:
   637     assumes "x |\<notin>| A"
   638     shows "ffold f z (finsert x A) = f x (ffold f z A)"
   639     using assms by (transfer fixing: f) (rule fold_insert)
   640 
   641   lemma ffold_fun_left_comm:
   642     "f x (ffold f z A) = ffold f (f x z) A"
   643     by (transfer fixing: f) (rule fold_fun_left_comm)
   644 
   645   lemma ffold_finsert2:
   646     "x |\<notin>| A \<Longrightarrow> ffold f z (finsert x A)  = ffold f (f x z) A"
   647     by (transfer fixing: f) (rule fold_insert2)
   648 
   649   lemma ffold_rec:
   650     assumes "x |\<in>| A"
   651     shows "ffold f z A = f x (ffold f z (A |-| {|x|}))"
   652     using assms by (transfer fixing: f) (rule fold_rec)
   653   
   654   lemma ffold_finsert_fremove:
   655     "ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
   656      by (transfer fixing: f) (rule fold_insert_remove)
   657 end
   658 
   659 lemma ffold_fimage:
   660   assumes "inj_on g (fset A)"
   661   shows "ffold f z (g |`| A) = ffold (f \<circ> g) z A"
   662 using assms by transfer' (rule fold_image)
   663 
   664 lemma ffold_cong:
   665   assumes "comp_fun_commute f" "comp_fun_commute g"
   666   "\<And>x. x |\<in>| A \<Longrightarrow> f x = g x"
   667     and "s = t" and "A = B"
   668   shows "ffold f s A = ffold g t B"
   669 using assms by transfer (metis Finite_Set.fold_cong)
   670 
   671 context comp_fun_idem
   672 begin
   673 
   674   lemma ffold_finsert_idem:
   675     "ffold f z (finsert x A)  = f x (ffold f z A)"
   676     by (transfer fixing: f) (rule fold_insert_idem)
   677   
   678   declare ffold_finsert [simp del] ffold_finsert_idem [simp]
   679   
   680   lemma ffold_finsert_idem2:
   681     "ffold f z (finsert x A) = ffold f (f x z) A"
   682     by (transfer fixing: f) (rule fold_insert_idem2)
   683 
   684 end
   685 
   686 
   687 subsection {* Choice in fsets *}
   688 
   689 lemma fset_choice: 
   690   assumes "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
   691   shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
   692   using assms by transfer metis
   693 
   694 
   695 subsection {* Induction and Cases rules for fsets *}
   696 
   697 lemma fset_exhaust [case_names empty insert, cases type: fset]:
   698   assumes fempty_case: "S = {||} \<Longrightarrow> P" 
   699   and     finsert_case: "\<And>x S'. S = finsert x S' \<Longrightarrow> P"
   700   shows "P"
   701   using assms by transfer blast
   702 
   703 lemma fset_induct [case_names empty insert]:
   704   assumes fempty_case: "P {||}"
   705   and     finsert_case: "\<And>x S. P S \<Longrightarrow> P (finsert x S)"
   706   shows "P S"
   707 proof -
   708   (* FIXME transfer and right_total vs. bi_total *)
   709   note Domainp_forall_transfer[transfer_rule]
   710   show ?thesis
   711   using assms by transfer (auto intro: finite_induct)
   712 qed
   713 
   714 lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
   715   assumes empty_fset_case: "P {||}"
   716   and     insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
   717   shows "P S"
   718 proof -
   719   (* FIXME transfer and right_total vs. bi_total *)
   720   note Domainp_forall_transfer[transfer_rule]
   721   show ?thesis
   722   using assms by transfer (auto intro: finite_induct)
   723 qed
   724 
   725 lemma fset_card_induct:
   726   assumes empty_fset_case: "P {||}"
   727   and     card_fset_Suc_case: "\<And>S T. Suc (fcard S) = (fcard T) \<Longrightarrow> P S \<Longrightarrow> P T"
   728   shows "P S"
   729 proof (induct S)
   730   case empty
   731   show "P {||}" by (rule empty_fset_case)
   732 next
   733   case (insert x S)
   734   have h: "P S" by fact
   735   have "x |\<notin>| S" by fact
   736   then have "Suc (fcard S) = fcard (finsert x S)" 
   737     by transfer auto
   738   then show "P (finsert x S)" 
   739     using h card_fset_Suc_case by simp
   740 qed
   741 
   742 lemma fset_strong_cases:
   743   obtains "xs = {||}"
   744     | ys x where "x |\<notin>| ys" and "xs = finsert x ys"
   745 by transfer blast
   746 
   747 lemma fset_induct2:
   748   "P {||} {||} \<Longrightarrow>
   749   (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>
   750   (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>
   751   (\<And>x xs y ys. \<lbrakk>P xs ys; x |\<notin>| xs; y |\<notin>| ys\<rbrakk> \<Longrightarrow> P (finsert x xs) (finsert y ys)) \<Longrightarrow>
   752   P xsa ysa"
   753   apply (induct xsa arbitrary: ysa)
   754   apply (induct_tac x rule: fset_induct_stronger)
   755   apply simp_all
   756   apply (induct_tac xa rule: fset_induct_stronger)
   757   apply simp_all
   758   done
   759 
   760 
   761 subsection {* Setup for Lifting/Transfer *}
   762 
   763 subsubsection {* Relator and predicator properties *}
   764 
   765 lift_definition rel_fset :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is rel_set
   766 parametric rel_set_transfer .
   767 
   768 lemma rel_fset_alt_def: "rel_fset R = (\<lambda>A B. (\<forall>x.\<exists>y. x|\<in>|A \<longrightarrow> y|\<in>|B \<and> R x y) 
   769   \<and> (\<forall>y. \<exists>x. y|\<in>|B \<longrightarrow> x|\<in>|A \<and> R x y))"
   770 apply (rule ext)+
   771 apply transfer'
   772 apply (subst rel_set_def[unfolded fun_eq_iff]) 
   773 by blast
   774 
   775 lemma rel_fset_conversep: "rel_fset (conversep R) = conversep (rel_fset R)"
   776   unfolding rel_fset_alt_def by auto
   777 
   778 lemmas rel_fset_eq [relator_eq] = rel_set_eq[Transfer.transferred]
   779 
   780 lemma rel_fset_mono[relator_mono]: "A \<le> B \<Longrightarrow> rel_fset A \<le> rel_fset B"
   781 unfolding rel_fset_alt_def by blast
   782 
   783 lemma finite_rel_set:
   784   assumes fin: "finite X" "finite Z"
   785   assumes R_S: "rel_set (R OO S) X Z"
   786   shows "\<exists>Y. finite Y \<and> rel_set R X Y \<and> rel_set S Y Z"
   787 proof -
   788   obtain f where f: "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)"
   789   apply atomize_elim
   790   apply (subst bchoice_iff[symmetric])
   791   using R_S[unfolded rel_set_def OO_def] by blast
   792   
   793   obtain g where g: "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R  x (g z))"
   794   apply atomize_elim
   795   apply (subst bchoice_iff[symmetric])
   796   using R_S[unfolded rel_set_def OO_def] by blast
   797   
   798   let ?Y = "f ` X \<union> g ` Z"
   799   have "finite ?Y" by (simp add: fin)
   800   moreover have "rel_set R X ?Y"
   801     unfolding rel_set_def
   802     using f g by clarsimp blast
   803   moreover have "rel_set S ?Y Z"
   804     unfolding rel_set_def
   805     using f g by clarsimp blast
   806   ultimately show ?thesis by metis
   807 qed
   808 
   809 lemma rel_fset_OO[relator_distr]: "rel_fset R OO rel_fset S = rel_fset (R OO S)"
   810 apply (rule ext)+
   811 by transfer (auto intro: finite_rel_set rel_set_OO[unfolded fun_eq_iff, rule_format, THEN iffD1])
   812 
   813 lemma Domainp_fset[relator_domain]:
   814   assumes "Domainp T = P"
   815   shows "Domainp (rel_fset T) = (\<lambda>A. fBall A P)"
   816 proof -
   817   from assms obtain f where f: "\<forall>x\<in>Collect P. T x (f x)"
   818     unfolding Domainp_iff[abs_def]
   819     apply atomize_elim
   820     by (subst bchoice_iff[symmetric]) auto
   821   from assms f show ?thesis
   822     unfolding fun_eq_iff rel_fset_alt_def Domainp_iff
   823     apply clarify
   824     apply (rule iffI)
   825       apply blast
   826     by (rename_tac A, rule_tac x="f |`| A" in exI, blast)
   827 qed
   828 
   829 lemma right_total_rel_fset[transfer_rule]: "right_total A \<Longrightarrow> right_total (rel_fset A)"
   830 unfolding right_total_def 
   831 apply transfer
   832 apply (subst(asm) choice_iff)
   833 apply clarsimp
   834 apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
   835 by (auto simp add: rel_set_def)
   836 
   837 lemma left_total_rel_fset[reflexivity_rule]: "left_total A \<Longrightarrow> left_total (rel_fset A)"
   838 unfolding left_total_def 
   839 apply transfer
   840 apply (subst(asm) choice_iff)
   841 apply clarsimp
   842 apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
   843 by (auto simp add: rel_set_def)
   844 
   845 lemmas right_unique_rel_fset[transfer_rule] = right_unique_rel_set[Transfer.transferred]
   846 lemmas left_unique_rel_fset[reflexivity_rule] = left_unique_rel_set[Transfer.transferred]
   847 
   848 thm right_unique_rel_fset left_unique_rel_fset
   849 
   850 lemma bi_unique_rel_fset[transfer_rule]: "bi_unique A \<Longrightarrow> bi_unique (rel_fset A)"
   851 by (auto intro: right_unique_rel_fset left_unique_rel_fset iff: bi_unique_iff)
   852 
   853 lemma bi_total_rel_fset[transfer_rule]: "bi_total A \<Longrightarrow> bi_total (rel_fset A)"
   854 by (auto intro: right_total_rel_fset left_total_rel_fset iff: bi_total_iff)
   855 
   856 lemmas fset_invariant_commute [invariant_commute] = set_invariant_commute[Transfer.transferred]
   857 
   858 
   859 subsubsection {* Quotient theorem for the Lifting package *}
   860 
   861 lemma Quotient_fset_map[quot_map]:
   862   assumes "Quotient R Abs Rep T"
   863   shows "Quotient (rel_fset R) (fimage Abs) (fimage Rep) (rel_fset T)"
   864   using assms unfolding Quotient_alt_def4
   865   by (simp add: rel_fset_OO[symmetric] rel_fset_conversep) (simp add: rel_fset_alt_def, blast)
   866 
   867 
   868 subsubsection {* Transfer rules for the Transfer package *}
   869 
   870 text {* Unconditional transfer rules *}
   871 
   872 context
   873 begin
   874 
   875 interpretation lifting_syntax .
   876 
   877 lemmas fempty_transfer [transfer_rule] = empty_transfer[Transfer.transferred]
   878 
   879 lemma finsert_transfer [transfer_rule]:
   880   "(A ===> rel_fset A ===> rel_fset A) finsert finsert"
   881   unfolding fun_rel_def rel_fset_alt_def by blast
   882 
   883 lemma funion_transfer [transfer_rule]:
   884   "(rel_fset A ===> rel_fset A ===> rel_fset A) funion funion"
   885   unfolding fun_rel_def rel_fset_alt_def by blast
   886 
   887 lemma ffUnion_transfer [transfer_rule]:
   888   "(rel_fset (rel_fset A) ===> rel_fset A) ffUnion ffUnion"
   889   unfolding fun_rel_def rel_fset_alt_def by transfer (simp, fast)
   890 
   891 lemma fimage_transfer [transfer_rule]:
   892   "((A ===> B) ===> rel_fset A ===> rel_fset B) fimage fimage"
   893   unfolding fun_rel_def rel_fset_alt_def by simp blast
   894 
   895 lemma fBall_transfer [transfer_rule]:
   896   "(rel_fset A ===> (A ===> op =) ===> op =) fBall fBall"
   897   unfolding rel_fset_alt_def fun_rel_def by blast
   898 
   899 lemma fBex_transfer [transfer_rule]:
   900   "(rel_fset A ===> (A ===> op =) ===> op =) fBex fBex"
   901   unfolding rel_fset_alt_def fun_rel_def by blast
   902 
   903 (* FIXME transfer doesn't work here *)
   904 lemma fPow_transfer [transfer_rule]:
   905   "(rel_fset A ===> rel_fset (rel_fset A)) fPow fPow"
   906   unfolding fun_rel_def
   907   using Pow_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred]
   908   by blast
   909 
   910 lemma rel_fset_transfer [transfer_rule]:
   911   "((A ===> B ===> op =) ===> rel_fset A ===> rel_fset B ===> op =)
   912     rel_fset rel_fset"
   913   unfolding fun_rel_def
   914   using rel_set_transfer[unfolded fun_rel_def,rule_format, Transfer.transferred, where A = A and B = B]
   915   by simp
   916 
   917 lemma bind_transfer [transfer_rule]:
   918   "(rel_fset A ===> (A ===> rel_fset B) ===> rel_fset B) fbind fbind"
   919   using assms unfolding fun_rel_def
   920   using bind_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
   921 
   922 text {* Rules requiring bi-unique, bi-total or right-total relations *}
   923 
   924 lemma fmember_transfer [transfer_rule]:
   925   assumes "bi_unique A"
   926   shows "(A ===> rel_fset A ===> op =) (op |\<in>|) (op |\<in>|)"
   927   using assms unfolding fun_rel_def rel_fset_alt_def bi_unique_def by metis
   928 
   929 lemma finter_transfer [transfer_rule]:
   930   assumes "bi_unique A"
   931   shows "(rel_fset A ===> rel_fset A ===> rel_fset A) finter finter"
   932   using assms unfolding fun_rel_def
   933   using inter_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
   934 
   935 lemma fminus_transfer [transfer_rule]:
   936   assumes "bi_unique A"
   937   shows "(rel_fset A ===> rel_fset A ===> rel_fset A) (op |-|) (op |-|)"
   938   using assms unfolding fun_rel_def
   939   using Diff_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
   940 
   941 lemma fsubset_transfer [transfer_rule]:
   942   assumes "bi_unique A"
   943   shows "(rel_fset A ===> rel_fset A ===> op =) (op |\<subseteq>|) (op |\<subseteq>|)"
   944   using assms unfolding fun_rel_def
   945   using subset_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
   946 
   947 lemma fSup_transfer [transfer_rule]:
   948   "bi_unique A \<Longrightarrow> (rel_set (rel_fset A) ===> rel_fset A) Sup Sup"
   949   using assms unfolding fun_rel_def
   950   apply clarify
   951   apply transfer'
   952   using Sup_fset_transfer[unfolded fun_rel_def] by blast
   953 
   954 (* FIXME: add right_total_fInf_transfer *)
   955 
   956 lemma fInf_transfer [transfer_rule]:
   957   assumes "bi_unique A" and "bi_total A"
   958   shows "(rel_set (rel_fset A) ===> rel_fset A) Inf Inf"
   959   using assms unfolding fun_rel_def
   960   apply clarify
   961   apply transfer'
   962   using Inf_fset_transfer[unfolded fun_rel_def] by blast
   963 
   964 lemma ffilter_transfer [transfer_rule]:
   965   assumes "bi_unique A"
   966   shows "((A ===> op=) ===> rel_fset A ===> rel_fset A) ffilter ffilter"
   967   using assms unfolding fun_rel_def
   968   using Lifting_Set.filter_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
   969 
   970 lemma card_transfer [transfer_rule]:
   971   "bi_unique A \<Longrightarrow> (rel_fset A ===> op =) fcard fcard"
   972   using assms unfolding fun_rel_def
   973   using card_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
   974 
   975 end
   976 
   977 lifting_update fset.lifting
   978 lifting_forget fset.lifting
   979 
   980 
   981 subsection {* BNF setup *}
   982 
   983 context
   984 includes fset.lifting
   985 begin
   986 
   987 lemma rel_fset_alt:
   988   "rel_fset R a b \<longleftrightarrow> (\<forall>t \<in> fset a. \<exists>u \<in> fset b. R t u) \<and> (\<forall>t \<in> fset b. \<exists>u \<in> fset a. R u t)"
   989 by transfer (simp add: rel_set_def)
   990 
   991 lemma fset_to_fset: "finite A \<Longrightarrow> fset (the_inv fset A) = A"
   992 apply (rule f_the_inv_into_f[unfolded inj_on_def])
   993 apply (simp add: fset_inject)
   994 apply (rule range_eqI Abs_fset_inverse[symmetric] CollectI)+
   995 .
   996 
   997 lemma rel_fset_aux:
   998 "(\<forall>t \<in> fset a. \<exists>u \<in> fset b. R t u) \<and> (\<forall>u \<in> fset b. \<exists>t \<in> fset a. R t u) \<longleftrightarrow>
   999  ((BNF_Util.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage fst))\<inverse>\<inverse> OO
  1000   BNF_Util.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage snd)) a b" (is "?L = ?R")
  1001 proof
  1002   assume ?L
  1003   def R' \<equiv> "the_inv fset (Collect (split R) \<inter> (fset a \<times> fset b))" (is "the_inv fset ?L'")
  1004   have "finite ?L'" by (intro finite_Int[OF disjI2] finite_cartesian_product) (transfer, simp)+
  1005   hence *: "fset R' = ?L'" unfolding R'_def by (intro fset_to_fset)
  1006   show ?R unfolding Grp_def relcompp.simps conversep.simps
  1007   proof (intro CollectI case_prodI exI[of _ a] exI[of _ b] exI[of _ R'] conjI refl)
  1008     from * show "a = fimage fst R'" using conjunct1[OF `?L`]
  1009       by (transfer, auto simp add: image_def Int_def split: prod.splits)
  1010     from * show "b = fimage snd R'" using conjunct2[OF `?L`]
  1011       by (transfer, auto simp add: image_def Int_def split: prod.splits)
  1012   qed (auto simp add: *)
  1013 next
  1014   assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps
  1015   apply (simp add: subset_eq Ball_def)
  1016   apply (rule conjI)
  1017   apply (transfer, clarsimp, metis snd_conv)
  1018   by (transfer, clarsimp, metis fst_conv)
  1019 qed
  1020 
  1021 bnf "'a fset"
  1022   map: fimage
  1023   sets: fset 
  1024   bd: natLeq
  1025   wits: "{||}"
  1026   rel: rel_fset
  1027 apply -
  1028           apply transfer' apply simp
  1029          apply transfer' apply force
  1030         apply transfer apply force
  1031        apply transfer' apply force
  1032       apply (rule natLeq_card_order)
  1033      apply (rule natLeq_cinfinite)
  1034     apply transfer apply (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq)
  1035    apply (fastforce simp: rel_fset_alt)
  1036  apply (simp add: Grp_def relcompp.simps conversep.simps fun_eq_iff rel_fset_alt rel_fset_aux) 
  1037 apply transfer apply simp
  1038 done
  1039 
  1040 lemma rel_fset_fset: "rel_set \<chi> (fset A1) (fset A2) = rel_fset \<chi> A1 A2"
  1041   by transfer (rule refl)
  1042 
  1043 end
  1044 
  1045 lemmas [simp] = fset.map_comp fset.map_id fset.set_map
  1046 
  1047 
  1048 subsection {* Advanced relator customization *}
  1049 
  1050 (* Set vs. sum relators: *)
  1051 
  1052 lemma rel_set_sum_rel[simp]: 
  1053 "rel_set (sum_rel \<chi> \<phi>) A1 A2 \<longleftrightarrow> 
  1054  rel_set \<chi> (Inl -` A1) (Inl -` A2) \<and> rel_set \<phi> (Inr -` A1) (Inr -` A2)"
  1055 (is "?L \<longleftrightarrow> ?Rl \<and> ?Rr")
  1056 proof safe
  1057   assume L: "?L"
  1058   show ?Rl unfolding rel_set_def Bex_def vimage_eq proof safe
  1059     fix l1 assume "Inl l1 \<in> A1"
  1060     then obtain a2 where a2: "a2 \<in> A2" and "sum_rel \<chi> \<phi> (Inl l1) a2"
  1061     using L unfolding rel_set_def by auto
  1062     then obtain l2 where "a2 = Inl l2 \<and> \<chi> l1 l2" by (cases a2, auto)
  1063     thus "\<exists> l2. Inl l2 \<in> A2 \<and> \<chi> l1 l2" using a2 by auto
  1064   next
  1065     fix l2 assume "Inl l2 \<in> A2"
  1066     then obtain a1 where a1: "a1 \<in> A1" and "sum_rel \<chi> \<phi> a1 (Inl l2)"
  1067     using L unfolding rel_set_def by auto
  1068     then obtain l1 where "a1 = Inl l1 \<and> \<chi> l1 l2" by (cases a1, auto)
  1069     thus "\<exists> l1. Inl l1 \<in> A1 \<and> \<chi> l1 l2" using a1 by auto
  1070   qed
  1071   show ?Rr unfolding rel_set_def Bex_def vimage_eq proof safe
  1072     fix r1 assume "Inr r1 \<in> A1"
  1073     then obtain a2 where a2: "a2 \<in> A2" and "sum_rel \<chi> \<phi> (Inr r1) a2"
  1074     using L unfolding rel_set_def by auto
  1075     then obtain r2 where "a2 = Inr r2 \<and> \<phi> r1 r2" by (cases a2, auto)
  1076     thus "\<exists> r2. Inr r2 \<in> A2 \<and> \<phi> r1 r2" using a2 by auto
  1077   next
  1078     fix r2 assume "Inr r2 \<in> A2"
  1079     then obtain a1 where a1: "a1 \<in> A1" and "sum_rel \<chi> \<phi> a1 (Inr r2)"
  1080     using L unfolding rel_set_def by auto
  1081     then obtain r1 where "a1 = Inr r1 \<and> \<phi> r1 r2" by (cases a1, auto)
  1082     thus "\<exists> r1. Inr r1 \<in> A1 \<and> \<phi> r1 r2" using a1 by auto
  1083   qed
  1084 next
  1085   assume Rl: "?Rl" and Rr: "?Rr"
  1086   show ?L unfolding rel_set_def Bex_def vimage_eq proof safe
  1087     fix a1 assume a1: "a1 \<in> A1"
  1088     show "\<exists> a2. a2 \<in> A2 \<and> sum_rel \<chi> \<phi> a1 a2"
  1089     proof(cases a1)
  1090       case (Inl l1) then obtain l2 where "Inl l2 \<in> A2 \<and> \<chi> l1 l2"
  1091       using Rl a1 unfolding rel_set_def by blast
  1092       thus ?thesis unfolding Inl by auto
  1093     next
  1094       case (Inr r1) then obtain r2 where "Inr r2 \<in> A2 \<and> \<phi> r1 r2"
  1095       using Rr a1 unfolding rel_set_def by blast
  1096       thus ?thesis unfolding Inr by auto
  1097     qed
  1098   next
  1099     fix a2 assume a2: "a2 \<in> A2"
  1100     show "\<exists> a1. a1 \<in> A1 \<and> sum_rel \<chi> \<phi> a1 a2"
  1101     proof(cases a2)
  1102       case (Inl l2) then obtain l1 where "Inl l1 \<in> A1 \<and> \<chi> l1 l2"
  1103       using Rl a2 unfolding rel_set_def by blast
  1104       thus ?thesis unfolding Inl by auto
  1105     next
  1106       case (Inr r2) then obtain r1 where "Inr r1 \<in> A1 \<and> \<phi> r1 r2"
  1107       using Rr a2 unfolding rel_set_def by blast
  1108       thus ?thesis unfolding Inr by auto
  1109     qed
  1110   qed
  1111 qed
  1112 
  1113 end