src/HOL/Library/FSet.thy
author wenzelm
Wed Jun 22 10:09:20 2016 +0200 (2016-06-22)
changeset 63343 fb5d8a50c641
parent 63331 247eac9758dd
child 63622 7fb02cee1cba
permissions -rw-r--r--
bundle lifting_syntax;
     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 section \<open>Type of finite sets defined as a subtype of sets\<close>
     8 
     9 theory FSet
    10 imports Main
    11 begin
    12 
    13 subsection \<open>Definition of the type\<close>
    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 \<open>Basic operations and type class instantiations\<close>
    22 
    23 (* FIXME transfer and right_total vs. bi_total *)
    24 instantiation fset :: (finite) finite
    25 begin
    26 instance by (standard; transfer; simp)
    27 end
    28 
    29 instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
    30 begin
    31 
    32 lift_definition bot_fset :: "'a fset" is "{}" parametric empty_transfer by simp
    33 
    34 lift_definition less_eq_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" is subset_eq parametric subset_transfer
    35   .
    36 
    37 definition less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" where "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"
    38 
    39 lemma less_fset_transfer[transfer_rule]:
    40   includes lifting_syntax
    41   assumes [transfer_rule]: "bi_unique A"
    42   shows "((pcr_fset A) ===> (pcr_fset A) ===> op =) op \<subset> op <"
    43   unfolding less_fset_def[abs_def] psubset_eq[abs_def] by transfer_prover
    44 
    45 
    46 lift_definition sup_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is union parametric union_transfer
    47   by simp
    48 
    49 lift_definition inf_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is inter parametric inter_transfer
    50   by simp
    51 
    52 lift_definition minus_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is minus parametric Diff_transfer
    53   by simp
    54 
    55 instance
    56   by (standard; transfer; auto)+
    57 
    58 end
    59 
    60 abbreviation fempty :: "'a fset" ("{||}") where "{||} \<equiv> bot"
    61 abbreviation fsubset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50) where "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
    62 abbreviation fsubset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50) where "xs |\<subset>| ys \<equiv> xs < ys"
    63 abbreviation funion :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<union>|" 65) where "xs |\<union>| ys \<equiv> sup xs ys"
    64 abbreviation finter :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<inter>|" 65) where "xs |\<inter>| ys \<equiv> inf xs ys"
    65 abbreviation fminus :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|-|" 65) where "xs |-| ys \<equiv> minus xs ys"
    66 
    67 instantiation fset :: (equal) equal
    68 begin
    69 definition "HOL.equal A B \<longleftrightarrow> A |\<subseteq>| B \<and> B |\<subseteq>| A"
    70 instance by intro_classes (auto simp add: equal_fset_def)
    71 end
    72 
    73 instantiation fset :: (type) conditionally_complete_lattice
    74 begin
    75 
    76 context includes lifting_syntax
    77 begin
    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 end
   110 
   111 instance
   112 proof
   113   fix x z :: "'a fset"
   114   fix X :: "'a fset set"
   115   {
   116     assume "x \<in> X" "bdd_below X"
   117     then show "Inf X |\<subseteq>| x" by transfer auto
   118   next
   119     assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> z |\<subseteq>| x)"
   120     then show "z |\<subseteq>| Inf X" by transfer (clarsimp, blast)
   121   next
   122     assume "x \<in> X" "bdd_above X"
   123     then obtain z where "x \<in> X" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)"
   124       by (auto simp: bdd_above_def)
   125     then show "x |\<subseteq>| Sup X"
   126       by transfer (auto intro!: finite_Sup)
   127   next
   128     assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)"
   129     then show "Sup X |\<subseteq>| z" by transfer (clarsimp, blast)
   130   }
   131 qed
   132 end
   133 
   134 instantiation fset :: (finite) complete_lattice
   135 begin
   136 
   137 lift_definition top_fset :: "'a fset" is UNIV parametric right_total_UNIV_transfer UNIV_transfer
   138   by simp
   139 
   140 instance
   141   by (standard; transfer; auto)
   142 
   143 end
   144 
   145 instantiation fset :: (finite) complete_boolean_algebra
   146 begin
   147 
   148 lift_definition uminus_fset :: "'a fset \<Rightarrow> 'a fset" is uminus
   149   parametric right_total_Compl_transfer Compl_transfer by simp
   150 
   151 instance
   152   by (standard; transfer) (simp_all add: Diff_eq)
   153 
   154 end
   155 
   156 abbreviation fUNIV :: "'a::finite fset" where "fUNIV \<equiv> top"
   157 abbreviation fuminus :: "'a::finite fset \<Rightarrow> 'a fset" ("|-| _" [81] 80) where "|-| x \<equiv> uminus x"
   158 
   159 declare top_fset.rep_eq[simp]
   160 
   161 
   162 subsection \<open>Other operations\<close>
   163 
   164 lift_definition finsert :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is insert parametric Lifting_Set.insert_transfer
   165   by simp
   166 
   167 syntax
   168   "_insert_fset"     :: "args => 'a fset"  ("{|(_)|}")
   169 
   170 translations
   171   "{|x, xs|}" == "CONST finsert x {|xs|}"
   172   "{|x|}"     == "CONST finsert x {||}"
   173 
   174 lift_definition fmember :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<in>|" 50) is Set.member
   175   parametric member_transfer .
   176 
   177 abbreviation notin_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50) where "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
   178 
   179 context includes lifting_syntax
   180 begin
   181 
   182 lift_definition ffilter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is Set.filter
   183   parametric Lifting_Set.filter_transfer unfolding Set.filter_def by simp
   184 
   185 lift_definition fPow :: "'a fset \<Rightarrow> 'a fset fset" is Pow parametric Pow_transfer
   186 by (simp add: finite_subset)
   187 
   188 lift_definition fcard :: "'a fset \<Rightarrow> nat" is card parametric card_transfer .
   189 
   190 lift_definition fimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset" (infixr "|`|" 90) is image
   191   parametric image_transfer by simp
   192 
   193 lift_definition fthe_elem :: "'a fset \<Rightarrow> 'a" is the_elem .
   194 
   195 lift_definition fbind :: "'a fset \<Rightarrow> ('a \<Rightarrow> 'b fset) \<Rightarrow> 'b fset" is Set.bind parametric bind_transfer
   196 by (simp add: Set.bind_def)
   197 
   198 lift_definition ffUnion :: "'a fset fset \<Rightarrow> 'a fset" is Union parametric Union_transfer by simp
   199 
   200 lift_definition fBall :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" is Ball parametric Ball_transfer .
   201 lift_definition fBex :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" is Bex parametric Bex_transfer .
   202 
   203 lift_definition ffold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" is Finite_Set.fold .
   204 
   205 
   206 subsection \<open>Transferred lemmas from Set.thy\<close>
   207 
   208 lemmas fset_eqI = set_eqI[Transfer.transferred]
   209 lemmas fset_eq_iff[no_atp] = set_eq_iff[Transfer.transferred]
   210 lemmas fBallI[intro!] = ballI[Transfer.transferred]
   211 lemmas fbspec[dest?] = bspec[Transfer.transferred]
   212 lemmas fBallE[elim] = ballE[Transfer.transferred]
   213 lemmas fBexI[intro] = bexI[Transfer.transferred]
   214 lemmas rev_fBexI[intro?] = rev_bexI[Transfer.transferred]
   215 lemmas fBexCI = bexCI[Transfer.transferred]
   216 lemmas fBexE[elim!] = bexE[Transfer.transferred]
   217 lemmas fBall_triv[simp] = ball_triv[Transfer.transferred]
   218 lemmas fBex_triv[simp] = bex_triv[Transfer.transferred]
   219 lemmas fBex_triv_one_point1[simp] = bex_triv_one_point1[Transfer.transferred]
   220 lemmas fBex_triv_one_point2[simp] = bex_triv_one_point2[Transfer.transferred]
   221 lemmas fBex_one_point1[simp] = bex_one_point1[Transfer.transferred]
   222 lemmas fBex_one_point2[simp] = bex_one_point2[Transfer.transferred]
   223 lemmas fBall_one_point1[simp] = ball_one_point1[Transfer.transferred]
   224 lemmas fBall_one_point2[simp] = ball_one_point2[Transfer.transferred]
   225 lemmas fBall_conj_distrib = ball_conj_distrib[Transfer.transferred]
   226 lemmas fBex_disj_distrib = bex_disj_distrib[Transfer.transferred]
   227 lemmas fBall_cong = ball_cong[Transfer.transferred]
   228 lemmas fBex_cong = bex_cong[Transfer.transferred]
   229 lemmas fsubsetI[intro!] = subsetI[Transfer.transferred]
   230 lemmas fsubsetD[elim, intro?] = subsetD[Transfer.transferred]
   231 lemmas rev_fsubsetD[no_atp,intro?] = rev_subsetD[Transfer.transferred]
   232 lemmas fsubsetCE[no_atp,elim] = subsetCE[Transfer.transferred]
   233 lemmas fsubset_eq[no_atp] = subset_eq[Transfer.transferred]
   234 lemmas contra_fsubsetD[no_atp] = contra_subsetD[Transfer.transferred]
   235 lemmas fsubset_refl = subset_refl[Transfer.transferred]
   236 lemmas fsubset_trans = subset_trans[Transfer.transferred]
   237 lemmas fset_rev_mp = set_rev_mp[Transfer.transferred]
   238 lemmas fset_mp = set_mp[Transfer.transferred]
   239 lemmas fsubset_not_fsubset_eq[code] = subset_not_subset_eq[Transfer.transferred]
   240 lemmas eq_fmem_trans = eq_mem_trans[Transfer.transferred]
   241 lemmas fsubset_antisym[intro!] = subset_antisym[Transfer.transferred]
   242 lemmas fequalityD1 = equalityD1[Transfer.transferred]
   243 lemmas fequalityD2 = equalityD2[Transfer.transferred]
   244 lemmas fequalityE = equalityE[Transfer.transferred]
   245 lemmas fequalityCE[elim] = equalityCE[Transfer.transferred]
   246 lemmas eqfset_imp_iff = eqset_imp_iff[Transfer.transferred]
   247 lemmas eqfelem_imp_iff = eqelem_imp_iff[Transfer.transferred]
   248 lemmas fempty_iff[simp] = empty_iff[Transfer.transferred]
   249 lemmas fempty_fsubsetI[iff] = empty_subsetI[Transfer.transferred]
   250 lemmas equalsffemptyI = equals0I[Transfer.transferred]
   251 lemmas equalsffemptyD = equals0D[Transfer.transferred]
   252 lemmas fBall_fempty[simp] = ball_empty[Transfer.transferred]
   253 lemmas fBex_fempty[simp] = bex_empty[Transfer.transferred]
   254 lemmas fPow_iff[iff] = Pow_iff[Transfer.transferred]
   255 lemmas fPowI = PowI[Transfer.transferred]
   256 lemmas fPowD = PowD[Transfer.transferred]
   257 lemmas fPow_bottom = Pow_bottom[Transfer.transferred]
   258 lemmas fPow_top = Pow_top[Transfer.transferred]
   259 lemmas fPow_not_fempty = Pow_not_empty[Transfer.transferred]
   260 lemmas finter_iff[simp] = Int_iff[Transfer.transferred]
   261 lemmas finterI[intro!] = IntI[Transfer.transferred]
   262 lemmas finterD1 = IntD1[Transfer.transferred]
   263 lemmas finterD2 = IntD2[Transfer.transferred]
   264 lemmas finterE[elim!] = IntE[Transfer.transferred]
   265 lemmas funion_iff[simp] = Un_iff[Transfer.transferred]
   266 lemmas funionI1[elim?] = UnI1[Transfer.transferred]
   267 lemmas funionI2[elim?] = UnI2[Transfer.transferred]
   268 lemmas funionCI[intro!] = UnCI[Transfer.transferred]
   269 lemmas funionE[elim!] = UnE[Transfer.transferred]
   270 lemmas fminus_iff[simp] = Diff_iff[Transfer.transferred]
   271 lemmas fminusI[intro!] = DiffI[Transfer.transferred]
   272 lemmas fminusD1 = DiffD1[Transfer.transferred]
   273 lemmas fminusD2 = DiffD2[Transfer.transferred]
   274 lemmas fminusE[elim!] = DiffE[Transfer.transferred]
   275 lemmas finsert_iff[simp] = insert_iff[Transfer.transferred]
   276 lemmas finsertI1 = insertI1[Transfer.transferred]
   277 lemmas finsertI2 = insertI2[Transfer.transferred]
   278 lemmas finsertE[elim!] = insertE[Transfer.transferred]
   279 lemmas finsertCI[intro!] = insertCI[Transfer.transferred]
   280 lemmas fsubset_finsert_iff = subset_insert_iff[Transfer.transferred]
   281 lemmas finsert_ident = insert_ident[Transfer.transferred]
   282 lemmas fsingletonI[intro!,no_atp] = singletonI[Transfer.transferred]
   283 lemmas fsingletonD[dest!,no_atp] = singletonD[Transfer.transferred]
   284 lemmas fsingleton_iff = singleton_iff[Transfer.transferred]
   285 lemmas fsingleton_inject[dest!] = singleton_inject[Transfer.transferred]
   286 lemmas fsingleton_finsert_inj_eq[iff,no_atp] = singleton_insert_inj_eq[Transfer.transferred]
   287 lemmas fsingleton_finsert_inj_eq'[iff,no_atp] = singleton_insert_inj_eq'[Transfer.transferred]
   288 lemmas fsubset_fsingletonD = subset_singletonD[Transfer.transferred]
   289 lemmas fminus_single_finsert = Diff_single_insert[Transfer.transferred]
   290 lemmas fdoubleton_eq_iff = doubleton_eq_iff[Transfer.transferred]
   291 lemmas funion_fsingleton_iff = Un_singleton_iff[Transfer.transferred]
   292 lemmas fsingleton_funion_iff = singleton_Un_iff[Transfer.transferred]
   293 lemmas fimage_eqI[simp, intro] = image_eqI[Transfer.transferred]
   294 lemmas fimageI = imageI[Transfer.transferred]
   295 lemmas rev_fimage_eqI = rev_image_eqI[Transfer.transferred]
   296 lemmas fimageE[elim!] = imageE[Transfer.transferred]
   297 lemmas Compr_fimage_eq = Compr_image_eq[Transfer.transferred]
   298 lemmas fimage_funion = image_Un[Transfer.transferred]
   299 lemmas fimage_iff = image_iff[Transfer.transferred]
   300 lemmas fimage_fsubset_iff[no_atp] = image_subset_iff[Transfer.transferred]
   301 lemmas fimage_fsubsetI = image_subsetI[Transfer.transferred]
   302 lemmas fimage_ident[simp] = image_ident[Transfer.transferred]
   303 lemmas if_split_fmem1 = if_split_mem1[Transfer.transferred]
   304 lemmas if_split_fmem2 = if_split_mem2[Transfer.transferred]
   305 lemmas pfsubsetI[intro!,no_atp] = psubsetI[Transfer.transferred]
   306 lemmas pfsubsetE[elim!,no_atp] = psubsetE[Transfer.transferred]
   307 lemmas pfsubset_finsert_iff = psubset_insert_iff[Transfer.transferred]
   308 lemmas pfsubset_eq = psubset_eq[Transfer.transferred]
   309 lemmas pfsubset_imp_fsubset = psubset_imp_subset[Transfer.transferred]
   310 lemmas pfsubset_trans = psubset_trans[Transfer.transferred]
   311 lemmas pfsubsetD = psubsetD[Transfer.transferred]
   312 lemmas pfsubset_fsubset_trans = psubset_subset_trans[Transfer.transferred]
   313 lemmas fsubset_pfsubset_trans = subset_psubset_trans[Transfer.transferred]
   314 lemmas pfsubset_imp_ex_fmem = psubset_imp_ex_mem[Transfer.transferred]
   315 lemmas fimage_fPow_mono = image_Pow_mono[Transfer.transferred]
   316 lemmas fimage_fPow_surj = image_Pow_surj[Transfer.transferred]
   317 lemmas fsubset_finsertI = subset_insertI[Transfer.transferred]
   318 lemmas fsubset_finsertI2 = subset_insertI2[Transfer.transferred]
   319 lemmas fsubset_finsert = subset_insert[Transfer.transferred]
   320 lemmas funion_upper1 = Un_upper1[Transfer.transferred]
   321 lemmas funion_upper2 = Un_upper2[Transfer.transferred]
   322 lemmas funion_least = Un_least[Transfer.transferred]
   323 lemmas finter_lower1 = Int_lower1[Transfer.transferred]
   324 lemmas finter_lower2 = Int_lower2[Transfer.transferred]
   325 lemmas finter_greatest = Int_greatest[Transfer.transferred]
   326 lemmas fminus_fsubset = Diff_subset[Transfer.transferred]
   327 lemmas fminus_fsubset_conv = Diff_subset_conv[Transfer.transferred]
   328 lemmas fsubset_fempty[simp] = subset_empty[Transfer.transferred]
   329 lemmas not_pfsubset_fempty[iff] = not_psubset_empty[Transfer.transferred]
   330 lemmas finsert_is_funion = insert_is_Un[Transfer.transferred]
   331 lemmas finsert_not_fempty[simp] = insert_not_empty[Transfer.transferred]
   332 lemmas fempty_not_finsert = empty_not_insert[Transfer.transferred]
   333 lemmas finsert_absorb = insert_absorb[Transfer.transferred]
   334 lemmas finsert_absorb2[simp] = insert_absorb2[Transfer.transferred]
   335 lemmas finsert_commute = insert_commute[Transfer.transferred]
   336 lemmas finsert_fsubset[simp] = insert_subset[Transfer.transferred]
   337 lemmas finsert_inter_finsert[simp] = insert_inter_insert[Transfer.transferred]
   338 lemmas finsert_disjoint[simp,no_atp] = insert_disjoint[Transfer.transferred]
   339 lemmas disjoint_finsert[simp,no_atp] = disjoint_insert[Transfer.transferred]
   340 lemmas fimage_fempty[simp] = image_empty[Transfer.transferred]
   341 lemmas fimage_finsert[simp] = image_insert[Transfer.transferred]
   342 lemmas fimage_constant = image_constant[Transfer.transferred]
   343 lemmas fimage_constant_conv = image_constant_conv[Transfer.transferred]
   344 lemmas fimage_fimage = image_image[Transfer.transferred]
   345 lemmas finsert_fimage[simp] = insert_image[Transfer.transferred]
   346 lemmas fimage_is_fempty[iff] = image_is_empty[Transfer.transferred]
   347 lemmas fempty_is_fimage[iff] = empty_is_image[Transfer.transferred]
   348 lemmas fimage_cong = image_cong[Transfer.transferred]
   349 lemmas fimage_finter_fsubset = image_Int_subset[Transfer.transferred]
   350 lemmas fimage_fminus_fsubset = image_diff_subset[Transfer.transferred]
   351 lemmas finter_absorb = Int_absorb[Transfer.transferred]
   352 lemmas finter_left_absorb = Int_left_absorb[Transfer.transferred]
   353 lemmas finter_commute = Int_commute[Transfer.transferred]
   354 lemmas finter_left_commute = Int_left_commute[Transfer.transferred]
   355 lemmas finter_assoc = Int_assoc[Transfer.transferred]
   356 lemmas finter_ac = Int_ac[Transfer.transferred]
   357 lemmas finter_absorb1 = Int_absorb1[Transfer.transferred]
   358 lemmas finter_absorb2 = Int_absorb2[Transfer.transferred]
   359 lemmas finter_fempty_left = Int_empty_left[Transfer.transferred]
   360 lemmas finter_fempty_right = Int_empty_right[Transfer.transferred]
   361 lemmas disjoint_iff_fnot_equal = disjoint_iff_not_equal[Transfer.transferred]
   362 lemmas finter_funion_distrib = Int_Un_distrib[Transfer.transferred]
   363 lemmas finter_funion_distrib2 = Int_Un_distrib2[Transfer.transferred]
   364 lemmas finter_fsubset_iff[no_atp, simp] = Int_subset_iff[Transfer.transferred]
   365 lemmas funion_absorb = Un_absorb[Transfer.transferred]
   366 lemmas funion_left_absorb = Un_left_absorb[Transfer.transferred]
   367 lemmas funion_commute = Un_commute[Transfer.transferred]
   368 lemmas funion_left_commute = Un_left_commute[Transfer.transferred]
   369 lemmas funion_assoc = Un_assoc[Transfer.transferred]
   370 lemmas funion_ac = Un_ac[Transfer.transferred]
   371 lemmas funion_absorb1 = Un_absorb1[Transfer.transferred]
   372 lemmas funion_absorb2 = Un_absorb2[Transfer.transferred]
   373 lemmas funion_fempty_left = Un_empty_left[Transfer.transferred]
   374 lemmas funion_fempty_right = Un_empty_right[Transfer.transferred]
   375 lemmas funion_finsert_left[simp] = Un_insert_left[Transfer.transferred]
   376 lemmas funion_finsert_right[simp] = Un_insert_right[Transfer.transferred]
   377 lemmas finter_finsert_left = Int_insert_left[Transfer.transferred]
   378 lemmas finter_finsert_left_ifffempty[simp] = Int_insert_left_if0[Transfer.transferred]
   379 lemmas finter_finsert_left_if1[simp] = Int_insert_left_if1[Transfer.transferred]
   380 lemmas finter_finsert_right = Int_insert_right[Transfer.transferred]
   381 lemmas finter_finsert_right_ifffempty[simp] = Int_insert_right_if0[Transfer.transferred]
   382 lemmas finter_finsert_right_if1[simp] = Int_insert_right_if1[Transfer.transferred]
   383 lemmas funion_finter_distrib = Un_Int_distrib[Transfer.transferred]
   384 lemmas funion_finter_distrib2 = Un_Int_distrib2[Transfer.transferred]
   385 lemmas funion_finter_crazy = Un_Int_crazy[Transfer.transferred]
   386 lemmas fsubset_funion_eq = subset_Un_eq[Transfer.transferred]
   387 lemmas funion_fempty[iff] = Un_empty[Transfer.transferred]
   388 lemmas funion_fsubset_iff[no_atp, simp] = Un_subset_iff[Transfer.transferred]
   389 lemmas funion_fminus_finter = Un_Diff_Int[Transfer.transferred]
   390 lemmas fminus_finter2 = Diff_Int2[Transfer.transferred]
   391 lemmas funion_finter_assoc_eq = Un_Int_assoc_eq[Transfer.transferred]
   392 lemmas fBall_funion = ball_Un[Transfer.transferred]
   393 lemmas fBex_funion = bex_Un[Transfer.transferred]
   394 lemmas fminus_eq_fempty_iff[simp,no_atp] = Diff_eq_empty_iff[Transfer.transferred]
   395 lemmas fminus_cancel[simp] = Diff_cancel[Transfer.transferred]
   396 lemmas fminus_idemp[simp] = Diff_idemp[Transfer.transferred]
   397 lemmas fminus_triv = Diff_triv[Transfer.transferred]
   398 lemmas fempty_fminus[simp] = empty_Diff[Transfer.transferred]
   399 lemmas fminus_fempty[simp] = Diff_empty[Transfer.transferred]
   400 lemmas fminus_finsertffempty[simp,no_atp] = Diff_insert0[Transfer.transferred]
   401 lemmas fminus_finsert = Diff_insert[Transfer.transferred]
   402 lemmas fminus_finsert2 = Diff_insert2[Transfer.transferred]
   403 lemmas finsert_fminus_if = insert_Diff_if[Transfer.transferred]
   404 lemmas finsert_fminus1[simp] = insert_Diff1[Transfer.transferred]
   405 lemmas finsert_fminus_single[simp] = insert_Diff_single[Transfer.transferred]
   406 lemmas finsert_fminus = insert_Diff[Transfer.transferred]
   407 lemmas fminus_finsert_absorb = Diff_insert_absorb[Transfer.transferred]
   408 lemmas fminus_disjoint[simp] = Diff_disjoint[Transfer.transferred]
   409 lemmas fminus_partition = Diff_partition[Transfer.transferred]
   410 lemmas double_fminus = double_diff[Transfer.transferred]
   411 lemmas funion_fminus_cancel[simp] = Un_Diff_cancel[Transfer.transferred]
   412 lemmas funion_fminus_cancel2[simp] = Un_Diff_cancel2[Transfer.transferred]
   413 lemmas fminus_funion = Diff_Un[Transfer.transferred]
   414 lemmas fminus_finter = Diff_Int[Transfer.transferred]
   415 lemmas funion_fminus = Un_Diff[Transfer.transferred]
   416 lemmas finter_fminus = Int_Diff[Transfer.transferred]
   417 lemmas fminus_finter_distrib = Diff_Int_distrib[Transfer.transferred]
   418 lemmas fminus_finter_distrib2 = Diff_Int_distrib2[Transfer.transferred]
   419 lemmas fUNIV_bool[no_atp] = UNIV_bool[Transfer.transferred]
   420 lemmas fPow_fempty[simp] = Pow_empty[Transfer.transferred]
   421 lemmas fPow_finsert = Pow_insert[Transfer.transferred]
   422 lemmas funion_fPow_fsubset = Un_Pow_subset[Transfer.transferred]
   423 lemmas fPow_finter_eq[simp] = Pow_Int_eq[Transfer.transferred]
   424 lemmas fset_eq_fsubset = set_eq_subset[Transfer.transferred]
   425 lemmas fsubset_iff[no_atp] = subset_iff[Transfer.transferred]
   426 lemmas fsubset_iff_pfsubset_eq = subset_iff_psubset_eq[Transfer.transferred]
   427 lemmas all_not_fin_conv[simp] = all_not_in_conv[Transfer.transferred]
   428 lemmas ex_fin_conv = ex_in_conv[Transfer.transferred]
   429 lemmas fimage_mono = image_mono[Transfer.transferred]
   430 lemmas fPow_mono = Pow_mono[Transfer.transferred]
   431 lemmas finsert_mono = insert_mono[Transfer.transferred]
   432 lemmas funion_mono = Un_mono[Transfer.transferred]
   433 lemmas finter_mono = Int_mono[Transfer.transferred]
   434 lemmas fminus_mono = Diff_mono[Transfer.transferred]
   435 lemmas fin_mono = in_mono[Transfer.transferred]
   436 lemmas fthe_felem_eq[simp] = the_elem_eq[Transfer.transferred]
   437 lemmas fLeast_mono = Least_mono[Transfer.transferred]
   438 lemmas fbind_fbind = bind_bind[Transfer.transferred]
   439 lemmas fempty_fbind[simp] = empty_bind[Transfer.transferred]
   440 lemmas nonfempty_fbind_const = nonempty_bind_const[Transfer.transferred]
   441 lemmas fbind_const = bind_const[Transfer.transferred]
   442 lemmas ffmember_filter[simp] = member_filter[Transfer.transferred]
   443 lemmas fequalityI = equalityI[Transfer.transferred]
   444 
   445 
   446 subsection \<open>Additional lemmas\<close>
   447 
   448 subsubsection \<open>\<open>fsingleton\<close>\<close>
   449 
   450 lemmas fsingletonE = fsingletonD [elim_format]
   451 
   452 
   453 subsubsection \<open>\<open>femepty\<close>\<close>
   454 
   455 lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {||}"
   456 by transfer auto
   457 
   458 (* FIXME, transferred doesn't work here *)
   459 lemma femptyE [elim!]: "a |\<in>| {||} \<Longrightarrow> P"
   460   by simp
   461 
   462 
   463 subsubsection \<open>\<open>fset\<close>\<close>
   464 
   465 lemmas fset_simps[simp] = bot_fset.rep_eq finsert.rep_eq
   466 
   467 lemma finite_fset [simp]:
   468   shows "finite (fset S)"
   469   by transfer simp
   470 
   471 lemmas fset_cong = fset_inject
   472 
   473 lemma filter_fset [simp]:
   474   shows "fset (ffilter P xs) = Collect P \<inter> fset xs"
   475   by transfer auto
   476 
   477 lemma notin_fset: "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S" by (simp add: fmember.rep_eq)
   478 
   479 lemmas inter_fset[simp] = inf_fset.rep_eq
   480 
   481 lemmas union_fset[simp] = sup_fset.rep_eq
   482 
   483 lemmas minus_fset[simp] = minus_fset.rep_eq
   484 
   485 
   486 subsubsection \<open>\<open>filter_fset\<close>\<close>
   487 
   488 lemma subset_ffilter:
   489   "ffilter P A |\<subseteq>| ffilter Q A = (\<forall> x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)"
   490   by transfer auto
   491 
   492 lemma eq_ffilter:
   493   "(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)"
   494   by transfer auto
   495 
   496 lemma pfsubset_ffilter:
   497   "(\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| A & \<not> P x & Q x) \<Longrightarrow>
   498     ffilter P A |\<subset>| ffilter Q A"
   499   unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter)
   500 
   501 
   502 subsubsection \<open>\<open>finsert\<close>\<close>
   503 
   504 (* FIXME, transferred doesn't work here *)
   505 lemma set_finsert:
   506   assumes "x |\<in>| A"
   507   obtains B where "A = finsert x B" and "x |\<notin>| B"
   508 using assms by transfer (metis Set.set_insert finite_insert)
   509 
   510 lemma mk_disjoint_finsert: "a |\<in>| A \<Longrightarrow> \<exists>B. A = finsert a B \<and> a |\<notin>| B"
   511   by (rule_tac x = "A |-| {|a|}" in exI, blast)
   512 
   513 
   514 subsubsection \<open>\<open>fimage\<close>\<close>
   515 
   516 lemma subset_fimage_iff: "(B |\<subseteq>| f|`|A) = (\<exists> AA. AA |\<subseteq>| A \<and> B = f|`|AA)"
   517 by transfer (metis mem_Collect_eq rev_finite_subset subset_image_iff)
   518 
   519 
   520 subsubsection \<open>bounded quantification\<close>
   521 
   522 lemma bex_simps [simp, no_atp]:
   523   "\<And>A P Q. fBex A (\<lambda>x. P x \<and> Q) = (fBex A P \<and> Q)"
   524   "\<And>A P Q. fBex A (\<lambda>x. P \<and> Q x) = (P \<and> fBex A Q)"
   525   "\<And>P. fBex {||} P = False"
   526   "\<And>a B P. fBex (finsert a B) P = (P a \<or> fBex B P)"
   527   "\<And>A P f. fBex (f |`| A) P = fBex A (\<lambda>x. P (f x))"
   528   "\<And>A P. (\<not> fBex A P) = fBall A (\<lambda>x. \<not> P x)"
   529 by auto
   530 
   531 lemma ball_simps [simp, no_atp]:
   532   "\<And>A P Q. fBall A (\<lambda>x. P x \<or> Q) = (fBall A P \<or> Q)"
   533   "\<And>A P Q. fBall A (\<lambda>x. P \<or> Q x) = (P \<or> fBall A Q)"
   534   "\<And>A P Q. fBall A (\<lambda>x. P \<longrightarrow> Q x) = (P \<longrightarrow> fBall A Q)"
   535   "\<And>A P Q. fBall A (\<lambda>x. P x \<longrightarrow> Q) = (fBex A P \<longrightarrow> Q)"
   536   "\<And>P. fBall {||} P = True"
   537   "\<And>a B P. fBall (finsert a B) P = (P a \<and> fBall B P)"
   538   "\<And>A P f. fBall (f |`| A) P = fBall A (\<lambda>x. P (f x))"
   539   "\<And>A P. (\<not> fBall A P) = fBex A (\<lambda>x. \<not> P x)"
   540 by auto
   541 
   542 lemma atomize_fBall:
   543     "(\<And>x. x |\<in>| A ==> P x) == Trueprop (fBall A (\<lambda>x. P x))"
   544 apply (simp only: atomize_all atomize_imp)
   545 apply (rule equal_intr_rule)
   546 by (transfer, simp)+
   547 
   548 end
   549 
   550 
   551 subsubsection \<open>\<open>fcard\<close>\<close>
   552 
   553 (* FIXME: improve transferred to handle bounded meta quantification *)
   554 
   555 lemma fcard_fempty:
   556   "fcard {||} = 0"
   557   by transfer (rule card_empty)
   558 
   559 lemma fcard_finsert_disjoint:
   560   "x |\<notin>| A \<Longrightarrow> fcard (finsert x A) = Suc (fcard A)"
   561   by transfer (rule card_insert_disjoint)
   562 
   563 lemma fcard_finsert_if:
   564   "fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))"
   565   by transfer (rule card_insert_if)
   566 
   567 lemma card_0_eq [simp, no_atp]:
   568   "fcard A = 0 \<longleftrightarrow> A = {||}"
   569   by transfer (rule card_0_eq)
   570 
   571 lemma fcard_Suc_fminus1:
   572   "x |\<in>| A \<Longrightarrow> Suc (fcard (A |-| {|x|})) = fcard A"
   573   by transfer (rule card_Suc_Diff1)
   574 
   575 lemma fcard_fminus_fsingleton:
   576   "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) = fcard A - 1"
   577   by transfer (rule card_Diff_singleton)
   578 
   579 lemma fcard_fminus_fsingleton_if:
   580   "fcard (A |-| {|x|}) = (if x |\<in>| A then fcard A - 1 else fcard A)"
   581   by transfer (rule card_Diff_singleton_if)
   582 
   583 lemma fcard_fminus_finsert[simp]:
   584   assumes "a |\<in>| A" and "a |\<notin>| B"
   585   shows "fcard (A |-| finsert a B) = fcard (A |-| B) - 1"
   586 using assms by transfer (rule card_Diff_insert)
   587 
   588 lemma fcard_finsert: "fcard (finsert x A) = Suc (fcard (A |-| {|x|}))"
   589 by transfer (rule card_insert)
   590 
   591 lemma fcard_finsert_le: "fcard A \<le> fcard (finsert x A)"
   592 by transfer (rule card_insert_le)
   593 
   594 lemma fcard_mono:
   595   "A |\<subseteq>| B \<Longrightarrow> fcard A \<le> fcard B"
   596 by transfer (rule card_mono)
   597 
   598 lemma fcard_seteq: "A |\<subseteq>| B \<Longrightarrow> fcard B \<le> fcard A \<Longrightarrow> A = B"
   599 by transfer (rule card_seteq)
   600 
   601 lemma pfsubset_fcard_mono: "A |\<subset>| B \<Longrightarrow> fcard A < fcard B"
   602 by transfer (rule psubset_card_mono)
   603 
   604 lemma fcard_funion_finter:
   605   "fcard A + fcard B = fcard (A |\<union>| B) + fcard (A |\<inter>| B)"
   606 by transfer (rule card_Un_Int)
   607 
   608 lemma fcard_funion_disjoint:
   609   "A |\<inter>| B = {||} \<Longrightarrow> fcard (A |\<union>| B) = fcard A + fcard B"
   610 by transfer (rule card_Un_disjoint)
   611 
   612 lemma fcard_funion_fsubset:
   613   "B |\<subseteq>| A \<Longrightarrow> fcard (A |-| B) = fcard A - fcard B"
   614 by transfer (rule card_Diff_subset)
   615 
   616 lemma diff_fcard_le_fcard_fminus:
   617   "fcard A - fcard B \<le> fcard(A |-| B)"
   618 by transfer (rule diff_card_le_card_Diff)
   619 
   620 lemma fcard_fminus1_less: "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) < fcard A"
   621 by transfer (rule card_Diff1_less)
   622 
   623 lemma fcard_fminus2_less:
   624   "x |\<in>| A \<Longrightarrow> y |\<in>| A \<Longrightarrow> fcard (A |-| {|x|} |-| {|y|}) < fcard A"
   625 by transfer (rule card_Diff2_less)
   626 
   627 lemma fcard_fminus1_le: "fcard (A |-| {|x|}) \<le> fcard A"
   628 by transfer (rule card_Diff1_le)
   629 
   630 lemma fcard_pfsubset: "A |\<subseteq>| B \<Longrightarrow> fcard A < fcard B \<Longrightarrow> A < B"
   631 by transfer (rule card_psubset)
   632 
   633 
   634 subsubsection \<open>\<open>ffold\<close>\<close>
   635 
   636 (* FIXME: improve transferred to handle bounded meta quantification *)
   637 
   638 context comp_fun_commute
   639 begin
   640   lemmas ffold_empty[simp] = fold_empty[Transfer.transferred]
   641 
   642   lemma ffold_finsert [simp]:
   643     assumes "x |\<notin>| A"
   644     shows "ffold f z (finsert x A) = f x (ffold f z A)"
   645     using assms by (transfer fixing: f) (rule fold_insert)
   646 
   647   lemma ffold_fun_left_comm:
   648     "f x (ffold f z A) = ffold f (f x z) A"
   649     by (transfer fixing: f) (rule fold_fun_left_comm)
   650 
   651   lemma ffold_finsert2:
   652     "x |\<notin>| A \<Longrightarrow> ffold f z (finsert x A) = ffold f (f x z) A"
   653     by (transfer fixing: f) (rule fold_insert2)
   654 
   655   lemma ffold_rec:
   656     assumes "x |\<in>| A"
   657     shows "ffold f z A = f x (ffold f z (A |-| {|x|}))"
   658     using assms by (transfer fixing: f) (rule fold_rec)
   659 
   660   lemma ffold_finsert_fremove:
   661     "ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
   662      by (transfer fixing: f) (rule fold_insert_remove)
   663 end
   664 
   665 lemma ffold_fimage:
   666   assumes "inj_on g (fset A)"
   667   shows "ffold f z (g |`| A) = ffold (f \<circ> g) z A"
   668 using assms by transfer' (rule fold_image)
   669 
   670 lemma ffold_cong:
   671   assumes "comp_fun_commute f" "comp_fun_commute g"
   672   "\<And>x. x |\<in>| A \<Longrightarrow> f x = g x"
   673     and "s = t" and "A = B"
   674   shows "ffold f s A = ffold g t B"
   675 using assms by transfer (metis Finite_Set.fold_cong)
   676 
   677 context comp_fun_idem
   678 begin
   679 
   680   lemma ffold_finsert_idem:
   681     "ffold f z (finsert x A) = f x (ffold f z A)"
   682     by (transfer fixing: f) (rule fold_insert_idem)
   683 
   684   declare ffold_finsert [simp del] ffold_finsert_idem [simp]
   685 
   686   lemma ffold_finsert_idem2:
   687     "ffold f z (finsert x A) = ffold f (f x z) A"
   688     by (transfer fixing: f) (rule fold_insert_idem2)
   689 
   690 end
   691 
   692 
   693 subsection \<open>Choice in fsets\<close>
   694 
   695 lemma fset_choice:
   696   assumes "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
   697   shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
   698   using assms by transfer metis
   699 
   700 
   701 subsection \<open>Induction and Cases rules for fsets\<close>
   702 
   703 lemma fset_exhaust [case_names empty insert, cases type: fset]:
   704   assumes fempty_case: "S = {||} \<Longrightarrow> P"
   705   and     finsert_case: "\<And>x S'. S = finsert x S' \<Longrightarrow> P"
   706   shows "P"
   707   using assms by transfer blast
   708 
   709 lemma fset_induct [case_names empty insert]:
   710   assumes fempty_case: "P {||}"
   711   and     finsert_case: "\<And>x S. P S \<Longrightarrow> P (finsert x S)"
   712   shows "P S"
   713 proof -
   714   (* FIXME transfer and right_total vs. bi_total *)
   715   note Domainp_forall_transfer[transfer_rule]
   716   show ?thesis
   717   using assms by transfer (auto intro: finite_induct)
   718 qed
   719 
   720 lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
   721   assumes empty_fset_case: "P {||}"
   722   and     insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
   723   shows "P S"
   724 proof -
   725   (* FIXME transfer and right_total vs. bi_total *)
   726   note Domainp_forall_transfer[transfer_rule]
   727   show ?thesis
   728   using assms by transfer (auto intro: finite_induct)
   729 qed
   730 
   731 lemma fset_card_induct:
   732   assumes empty_fset_case: "P {||}"
   733   and     card_fset_Suc_case: "\<And>S T. Suc (fcard S) = (fcard T) \<Longrightarrow> P S \<Longrightarrow> P T"
   734   shows "P S"
   735 proof (induct S)
   736   case empty
   737   show "P {||}" by (rule empty_fset_case)
   738 next
   739   case (insert x S)
   740   have h: "P S" by fact
   741   have "x |\<notin>| S" by fact
   742   then have "Suc (fcard S) = fcard (finsert x S)"
   743     by transfer auto
   744   then show "P (finsert x S)"
   745     using h card_fset_Suc_case by simp
   746 qed
   747 
   748 lemma fset_strong_cases:
   749   obtains "xs = {||}"
   750     | ys x where "x |\<notin>| ys" and "xs = finsert x ys"
   751 by transfer blast
   752 
   753 lemma fset_induct2:
   754   "P {||} {||} \<Longrightarrow>
   755   (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>
   756   (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>
   757   (\<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>
   758   P xsa ysa"
   759   apply (induct xsa arbitrary: ysa)
   760   apply (induct_tac x rule: fset_induct_stronger)
   761   apply simp_all
   762   apply (induct_tac xa rule: fset_induct_stronger)
   763   apply simp_all
   764   done
   765 
   766 
   767 subsection \<open>Setup for Lifting/Transfer\<close>
   768 
   769 subsubsection \<open>Relator and predicator properties\<close>
   770 
   771 lift_definition rel_fset :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is rel_set
   772 parametric rel_set_transfer .
   773 
   774 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)
   775   \<and> (\<forall>y. \<exists>x. y|\<in>|B \<longrightarrow> x|\<in>|A \<and> R x y))"
   776 apply (rule ext)+
   777 apply transfer'
   778 apply (subst rel_set_def[unfolded fun_eq_iff])
   779 by blast
   780 
   781 lemma finite_rel_set:
   782   assumes fin: "finite X" "finite Z"
   783   assumes R_S: "rel_set (R OO S) X Z"
   784   shows "\<exists>Y. finite Y \<and> rel_set R X Y \<and> rel_set S Y Z"
   785 proof -
   786   obtain f where f: "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)"
   787   apply atomize_elim
   788   apply (subst bchoice_iff[symmetric])
   789   using R_S[unfolded rel_set_def OO_def] by blast
   790 
   791   obtain g where g: "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R x (g z))"
   792   apply atomize_elim
   793   apply (subst bchoice_iff[symmetric])
   794   using R_S[unfolded rel_set_def OO_def] by blast
   795 
   796   let ?Y = "f ` X \<union> g ` Z"
   797   have "finite ?Y" by (simp add: fin)
   798   moreover have "rel_set R X ?Y"
   799     unfolding rel_set_def
   800     using f g by clarsimp blast
   801   moreover have "rel_set S ?Y Z"
   802     unfolding rel_set_def
   803     using f g by clarsimp blast
   804   ultimately show ?thesis by metis
   805 qed
   806 
   807 subsubsection \<open>Transfer rules for the Transfer package\<close>
   808 
   809 text \<open>Unconditional transfer rules\<close>
   810 
   811 context includes lifting_syntax
   812 begin
   813 
   814 lemmas fempty_transfer [transfer_rule] = empty_transfer[Transfer.transferred]
   815 
   816 lemma finsert_transfer [transfer_rule]:
   817   "(A ===> rel_fset A ===> rel_fset A) finsert finsert"
   818   unfolding rel_fun_def rel_fset_alt_def by blast
   819 
   820 lemma funion_transfer [transfer_rule]:
   821   "(rel_fset A ===> rel_fset A ===> rel_fset A) funion funion"
   822   unfolding rel_fun_def rel_fset_alt_def by blast
   823 
   824 lemma ffUnion_transfer [transfer_rule]:
   825   "(rel_fset (rel_fset A) ===> rel_fset A) ffUnion ffUnion"
   826   unfolding rel_fun_def rel_fset_alt_def by transfer (simp, fast)
   827 
   828 lemma fimage_transfer [transfer_rule]:
   829   "((A ===> B) ===> rel_fset A ===> rel_fset B) fimage fimage"
   830   unfolding rel_fun_def rel_fset_alt_def by simp blast
   831 
   832 lemma fBall_transfer [transfer_rule]:
   833   "(rel_fset A ===> (A ===> op =) ===> op =) fBall fBall"
   834   unfolding rel_fset_alt_def rel_fun_def by blast
   835 
   836 lemma fBex_transfer [transfer_rule]:
   837   "(rel_fset A ===> (A ===> op =) ===> op =) fBex fBex"
   838   unfolding rel_fset_alt_def rel_fun_def by blast
   839 
   840 (* FIXME transfer doesn't work here *)
   841 lemma fPow_transfer [transfer_rule]:
   842   "(rel_fset A ===> rel_fset (rel_fset A)) fPow fPow"
   843   unfolding rel_fun_def
   844   using Pow_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred]
   845   by blast
   846 
   847 lemma rel_fset_transfer [transfer_rule]:
   848   "((A ===> B ===> op =) ===> rel_fset A ===> rel_fset B ===> op =)
   849     rel_fset rel_fset"
   850   unfolding rel_fun_def
   851   using rel_set_transfer[unfolded rel_fun_def,rule_format, Transfer.transferred, where A = A and B = B]
   852   by simp
   853 
   854 lemma bind_transfer [transfer_rule]:
   855   "(rel_fset A ===> (A ===> rel_fset B) ===> rel_fset B) fbind fbind"
   856   unfolding rel_fun_def
   857   using bind_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
   858 
   859 text \<open>Rules requiring bi-unique, bi-total or right-total relations\<close>
   860 
   861 lemma fmember_transfer [transfer_rule]:
   862   assumes "bi_unique A"
   863   shows "(A ===> rel_fset A ===> op =) (op |\<in>|) (op |\<in>|)"
   864   using assms unfolding rel_fun_def rel_fset_alt_def bi_unique_def by metis
   865 
   866 lemma finter_transfer [transfer_rule]:
   867   assumes "bi_unique A"
   868   shows "(rel_fset A ===> rel_fset A ===> rel_fset A) finter finter"
   869   using assms unfolding rel_fun_def
   870   using inter_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
   871 
   872 lemma fminus_transfer [transfer_rule]:
   873   assumes "bi_unique A"
   874   shows "(rel_fset A ===> rel_fset A ===> rel_fset A) (op |-|) (op |-|)"
   875   using assms unfolding rel_fun_def
   876   using Diff_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
   877 
   878 lemma fsubset_transfer [transfer_rule]:
   879   assumes "bi_unique A"
   880   shows "(rel_fset A ===> rel_fset A ===> op =) (op |\<subseteq>|) (op |\<subseteq>|)"
   881   using assms unfolding rel_fun_def
   882   using subset_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
   883 
   884 lemma fSup_transfer [transfer_rule]:
   885   "bi_unique A \<Longrightarrow> (rel_set (rel_fset A) ===> rel_fset A) Sup Sup"
   886   unfolding rel_fun_def
   887   apply clarify
   888   apply transfer'
   889   using Sup_fset_transfer[unfolded rel_fun_def] by blast
   890 
   891 (* FIXME: add right_total_fInf_transfer *)
   892 
   893 lemma fInf_transfer [transfer_rule]:
   894   assumes "bi_unique A" and "bi_total A"
   895   shows "(rel_set (rel_fset A) ===> rel_fset A) Inf Inf"
   896   using assms unfolding rel_fun_def
   897   apply clarify
   898   apply transfer'
   899   using Inf_fset_transfer[unfolded rel_fun_def] by blast
   900 
   901 lemma ffilter_transfer [transfer_rule]:
   902   assumes "bi_unique A"
   903   shows "((A ===> op=) ===> rel_fset A ===> rel_fset A) ffilter ffilter"
   904   using assms unfolding rel_fun_def
   905   using Lifting_Set.filter_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
   906 
   907 lemma card_transfer [transfer_rule]:
   908   "bi_unique A \<Longrightarrow> (rel_fset A ===> op =) fcard fcard"
   909   unfolding rel_fun_def
   910   using card_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
   911 
   912 end
   913 
   914 lifting_update fset.lifting
   915 lifting_forget fset.lifting
   916 
   917 
   918 subsection \<open>BNF setup\<close>
   919 
   920 context
   921 includes fset.lifting
   922 begin
   923 
   924 lemma rel_fset_alt:
   925   "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)"
   926 by transfer (simp add: rel_set_def)
   927 
   928 lemma fset_to_fset: "finite A \<Longrightarrow> fset (the_inv fset A) = A"
   929 apply (rule f_the_inv_into_f[unfolded inj_on_def])
   930 apply (simp add: fset_inject)
   931 apply (rule range_eqI Abs_fset_inverse[symmetric] CollectI)+
   932 .
   933 
   934 lemma rel_fset_aux:
   935 "(\<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>
   936  ((BNF_Def.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage fst))\<inverse>\<inverse> OO
   937   BNF_Def.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage snd)) a b" (is "?L = ?R")
   938 proof
   939   assume ?L
   940   define R' where "R' =
   941     the_inv fset (Collect (case_prod R) \<inter> (fset a \<times> fset b))" (is "_ = the_inv fset ?L'")
   942   have "finite ?L'" by (intro finite_Int[OF disjI2] finite_cartesian_product) (transfer, simp)+
   943   hence *: "fset R' = ?L'" unfolding R'_def by (intro fset_to_fset)
   944   show ?R unfolding Grp_def relcompp.simps conversep.simps
   945   proof (intro CollectI case_prodI exI[of _ a] exI[of _ b] exI[of _ R'] conjI refl)
   946     from * show "a = fimage fst R'" using conjunct1[OF \<open>?L\<close>]
   947       by (transfer, auto simp add: image_def Int_def split: prod.splits)
   948     from * show "b = fimage snd R'" using conjunct2[OF \<open>?L\<close>]
   949       by (transfer, auto simp add: image_def Int_def split: prod.splits)
   950   qed (auto simp add: *)
   951 next
   952   assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps
   953   apply (simp add: subset_eq Ball_def)
   954   apply (rule conjI)
   955   apply (transfer, clarsimp, metis snd_conv)
   956   by (transfer, clarsimp, metis fst_conv)
   957 qed
   958 
   959 bnf "'a fset"
   960   map: fimage
   961   sets: fset
   962   bd: natLeq
   963   wits: "{||}"
   964   rel: rel_fset
   965 apply -
   966           apply transfer' apply simp
   967          apply transfer' apply force
   968         apply transfer apply force
   969        apply transfer' apply force
   970       apply (rule natLeq_card_order)
   971      apply (rule natLeq_cinfinite)
   972     apply transfer apply (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq)
   973    apply (fastforce simp: rel_fset_alt)
   974  apply (simp add: Grp_def relcompp.simps conversep.simps fun_eq_iff rel_fset_alt
   975    rel_fset_aux[unfolded OO_Grp_alt])
   976 apply transfer apply simp
   977 done
   978 
   979 lemma rel_fset_fset: "rel_set \<chi> (fset A1) (fset A2) = rel_fset \<chi> A1 A2"
   980   by transfer (rule refl)
   981 
   982 end
   983 
   984 lemmas [simp] = fset.map_comp fset.map_id fset.set_map
   985 
   986 
   987 subsection \<open>Size setup\<close>
   988 
   989 context includes fset.lifting begin
   990 lift_definition size_fset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a fset \<Rightarrow> nat" is "\<lambda>f. setsum (Suc \<circ> f)" .
   991 end
   992 
   993 instantiation fset :: (type) size begin
   994 definition size_fset where
   995   size_fset_overloaded_def: "size_fset = FSet.size_fset (\<lambda>_. 0)"
   996 instance ..
   997 end
   998 
   999 lemmas size_fset_simps[simp] =
  1000   size_fset_def[THEN meta_eq_to_obj_eq, THEN fun_cong, THEN fun_cong,
  1001     unfolded map_fun_def comp_def id_apply]
  1002 
  1003 lemmas size_fset_overloaded_simps[simp] =
  1004   size_fset_simps[of "\<lambda>_. 0", unfolded add_0_left add_0_right,
  1005     folded size_fset_overloaded_def]
  1006 
  1007 lemma fset_size_o_map: "inj f \<Longrightarrow> size_fset g \<circ> fimage f = size_fset (g \<circ> f)"
  1008   apply (subst fun_eq_iff)
  1009   including fset.lifting by transfer (auto intro: setsum.reindex_cong subset_inj_on)
  1010 
  1011 setup \<open>
  1012 BNF_LFP_Size.register_size_global @{type_name fset} @{const_name size_fset}
  1013   @{thm size_fset_overloaded_def} @{thms size_fset_simps size_fset_overloaded_simps}
  1014   @{thms fset_size_o_map}
  1015 \<close>
  1016 
  1017 lifting_update fset.lifting
  1018 lifting_forget fset.lifting
  1019 
  1020 subsection \<open>Advanced relator customization\<close>
  1021 
  1022 (* Set vs. sum relators: *)
  1023 
  1024 lemma rel_set_rel_sum[simp]:
  1025 "rel_set (rel_sum \<chi> \<phi>) A1 A2 \<longleftrightarrow>
  1026  rel_set \<chi> (Inl -` A1) (Inl -` A2) \<and> rel_set \<phi> (Inr -` A1) (Inr -` A2)"
  1027 (is "?L \<longleftrightarrow> ?Rl \<and> ?Rr")
  1028 proof safe
  1029   assume L: "?L"
  1030   show ?Rl unfolding rel_set_def Bex_def vimage_eq proof safe
  1031     fix l1 assume "Inl l1 \<in> A1"
  1032     then obtain a2 where a2: "a2 \<in> A2" and "rel_sum \<chi> \<phi> (Inl l1) a2"
  1033     using L unfolding rel_set_def by auto
  1034     then obtain l2 where "a2 = Inl l2 \<and> \<chi> l1 l2" by (cases a2, auto)
  1035     thus "\<exists> l2. Inl l2 \<in> A2 \<and> \<chi> l1 l2" using a2 by auto
  1036   next
  1037     fix l2 assume "Inl l2 \<in> A2"
  1038     then obtain a1 where a1: "a1 \<in> A1" and "rel_sum \<chi> \<phi> a1 (Inl l2)"
  1039     using L unfolding rel_set_def by auto
  1040     then obtain l1 where "a1 = Inl l1 \<and> \<chi> l1 l2" by (cases a1, auto)
  1041     thus "\<exists> l1. Inl l1 \<in> A1 \<and> \<chi> l1 l2" using a1 by auto
  1042   qed
  1043   show ?Rr unfolding rel_set_def Bex_def vimage_eq proof safe
  1044     fix r1 assume "Inr r1 \<in> A1"
  1045     then obtain a2 where a2: "a2 \<in> A2" and "rel_sum \<chi> \<phi> (Inr r1) a2"
  1046     using L unfolding rel_set_def by auto
  1047     then obtain r2 where "a2 = Inr r2 \<and> \<phi> r1 r2" by (cases a2, auto)
  1048     thus "\<exists> r2. Inr r2 \<in> A2 \<and> \<phi> r1 r2" using a2 by auto
  1049   next
  1050     fix r2 assume "Inr r2 \<in> A2"
  1051     then obtain a1 where a1: "a1 \<in> A1" and "rel_sum \<chi> \<phi> a1 (Inr r2)"
  1052     using L unfolding rel_set_def by auto
  1053     then obtain r1 where "a1 = Inr r1 \<and> \<phi> r1 r2" by (cases a1, auto)
  1054     thus "\<exists> r1. Inr r1 \<in> A1 \<and> \<phi> r1 r2" using a1 by auto
  1055   qed
  1056 next
  1057   assume Rl: "?Rl" and Rr: "?Rr"
  1058   show ?L unfolding rel_set_def Bex_def vimage_eq proof safe
  1059     fix a1 assume a1: "a1 \<in> A1"
  1060     show "\<exists> a2. a2 \<in> A2 \<and> rel_sum \<chi> \<phi> a1 a2"
  1061     proof(cases a1)
  1062       case (Inl l1) then obtain l2 where "Inl l2 \<in> A2 \<and> \<chi> l1 l2"
  1063       using Rl a1 unfolding rel_set_def by blast
  1064       thus ?thesis unfolding Inl by auto
  1065     next
  1066       case (Inr r1) then obtain r2 where "Inr r2 \<in> A2 \<and> \<phi> r1 r2"
  1067       using Rr a1 unfolding rel_set_def by blast
  1068       thus ?thesis unfolding Inr by auto
  1069     qed
  1070   next
  1071     fix a2 assume a2: "a2 \<in> A2"
  1072     show "\<exists> a1. a1 \<in> A1 \<and> rel_sum \<chi> \<phi> a1 a2"
  1073     proof(cases a2)
  1074       case (Inl l2) then obtain l1 where "Inl l1 \<in> A1 \<and> \<chi> l1 l2"
  1075       using Rl a2 unfolding rel_set_def by blast
  1076       thus ?thesis unfolding Inl by auto
  1077     next
  1078       case (Inr r2) then obtain r1 where "Inr r1 \<in> A1 \<and> \<phi> r1 r2"
  1079       using Rr a2 unfolding rel_set_def by blast
  1080       thus ?thesis unfolding Inr by auto
  1081     qed
  1082   qed
  1083 qed
  1084 
  1085 
  1086 subsection \<open>Quickcheck setup\<close>
  1087 
  1088 text \<open>Setup adapted from sets.\<close>
  1089 
  1090 notation Quickcheck_Exhaustive.orelse (infixr "orelse" 55)
  1091 
  1092 definition (in term_syntax) [code_unfold]:
  1093 "valterm_femptyset = Code_Evaluation.valtermify ({||} :: ('a :: typerep) fset)"
  1094 
  1095 definition (in term_syntax) [code_unfold]:
  1096 "valtermify_finsert x s = Code_Evaluation.valtermify finsert {\<cdot>} (x :: ('a :: typerep * _)) {\<cdot>} s"
  1097 
  1098 instantiation fset :: (exhaustive) exhaustive
  1099 begin
  1100 
  1101 fun exhaustive_fset where
  1102 "exhaustive_fset f i = (if i = 0 then None else (f {||} orelse exhaustive_fset (\<lambda>A. f A orelse Quickcheck_Exhaustive.exhaustive (\<lambda>x. if x |\<in>| A then None else f (finsert x A)) (i - 1)) (i - 1)))"
  1103 
  1104 instance ..
  1105 
  1106 end
  1107 
  1108 instantiation fset :: (full_exhaustive) full_exhaustive
  1109 begin
  1110 
  1111 fun full_exhaustive_fset where
  1112 "full_exhaustive_fset f i = (if i = 0 then None else (f valterm_femptyset orelse full_exhaustive_fset (\<lambda>A. f A orelse Quickcheck_Exhaustive.full_exhaustive (\<lambda>x. if fst x |\<in>| fst A then None else f (valtermify_finsert x A)) (i - 1)) (i - 1)))"
  1113 
  1114 instance ..
  1115 
  1116 end
  1117 
  1118 no_notation Quickcheck_Exhaustive.orelse (infixr "orelse" 55)
  1119 
  1120 notation scomp (infixl "\<circ>\<rightarrow>" 60)
  1121 
  1122 instantiation fset :: (random) random
  1123 begin
  1124 
  1125 fun random_aux_fset :: "natural \<Rightarrow> natural \<Rightarrow> natural \<times> natural \<Rightarrow> ('a fset \<times> (unit \<Rightarrow> term)) \<times> natural \<times> natural" where
  1126 "random_aux_fset 0 j = Quickcheck_Random.collapse (Random.select_weight [(1, Pair valterm_femptyset)])" |
  1127 "random_aux_fset (Code_Numeral.Suc i) j =
  1128   Quickcheck_Random.collapse (Random.select_weight
  1129     [(1, Pair valterm_femptyset),
  1130      (Code_Numeral.Suc i,
  1131       Quickcheck_Random.random j \<circ>\<rightarrow> (\<lambda>x. random_aux_fset i j \<circ>\<rightarrow> (\<lambda>s. Pair (valtermify_finsert x s))))])"
  1132 
  1133 lemma [code]:
  1134   "random_aux_fset i j =
  1135     Quickcheck_Random.collapse (Random.select_weight [(1, Pair valterm_femptyset),
  1136       (i, Quickcheck_Random.random j \<circ>\<rightarrow> (\<lambda>x. random_aux_fset (i - 1) j \<circ>\<rightarrow> (\<lambda>s. Pair (valtermify_finsert x s))))])"
  1137 proof (induct i rule: natural.induct)
  1138   case zero
  1139   show ?case by (subst select_weight_drop_zero[symmetric]) (simp add: less_natural_def)
  1140 next
  1141   case (Suc i)
  1142   show ?case by (simp only: random_aux_fset.simps Suc_natural_minus_one)
  1143 qed
  1144 
  1145 definition "random_fset i = random_aux_fset i i"
  1146 
  1147 instance ..
  1148 
  1149 end
  1150 
  1151 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
  1152 
  1153 end