src/HOL/Library/FSet.thy
author blanchet
Fri Jan 24 11:51:45 2014 +0100 (2014-01-24)
changeset 55129 26bd1cba3ab5
parent 54258 adfc759263ab
child 55414 eab03e9cee8a
permissions -rw-r--r--
killed 'More_BNFs' by moving its various bits where they (now) belong
     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   by simp
    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 "(set_rel (set_rel A) ===> set_rel 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 "(set_rel (set_rel A) ===> set_rel 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 "(set_rel (set_rel A) ===> set_rel 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]: "(set_rel (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 by simp
   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 lemma compose_rel_to_Domainp:
   180   assumes "left_unique R"
   181   assumes "(R ===> op=) P P'"
   182   shows "(R OO Lifting.invariant P' OO R\<inverse>\<inverse>) x y \<longleftrightarrow> Domainp R x \<and> P x \<and> x = y"
   183 using assms unfolding OO_def conversep_iff Domainp_iff left_unique_def fun_rel_def invariant_def
   184 by blast
   185 
   186 lift_definition fPow :: "'a fset \<Rightarrow> 'a fset fset" is Pow parametric Pow_transfer 
   187 by (subst compose_rel_to_Domainp [OF _ finite_transfer]) (auto intro: transfer_raw finite_subset 
   188   simp add: fset.pcr_cr_eq[symmetric] Domainp_set fset.domain_eq)
   189 
   190 lift_definition fcard :: "'a fset \<Rightarrow> nat" is card parametric card_transfer by simp
   191 
   192 lift_definition fimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset" (infixr "|`|" 90) is image 
   193   parametric image_transfer by simp
   194 
   195 lift_definition fthe_elem :: "'a fset \<Rightarrow> 'a" is the_elem ..
   196 
   197 (* FIXME why is not invariant here unfolded ? *)
   198 lift_definition fbind :: "'a fset \<Rightarrow> ('a \<Rightarrow> 'b fset) \<Rightarrow> 'b fset" is Set.bind parametric bind_transfer
   199 unfolding invariant_def Set.bind_def by clarsimp metis
   200 
   201 lift_definition ffUnion :: "'a fset fset \<Rightarrow> 'a fset" is Union parametric Union_transfer
   202 by (subst(asm) compose_rel_to_Domainp [OF _ finite_transfer])
   203   (auto intro: transfer_raw simp add: fset.pcr_cr_eq[symmetric] Domainp_set fset.domain_eq invariant_def)
   204 
   205 lift_definition fBall :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" is Ball parametric Ball_transfer ..
   206 lift_definition fBex :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" is Bex parametric Bex_transfer ..
   207 
   208 lift_definition ffold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" is Finite_Set.fold ..
   209 
   210 
   211 subsection {* Transferred lemmas from Set.thy *}
   212 
   213 lemmas fset_eqI = set_eqI[Transfer.transferred]
   214 lemmas fset_eq_iff[no_atp] = set_eq_iff[Transfer.transferred]
   215 lemmas fBallI[intro!] = ballI[Transfer.transferred]
   216 lemmas fbspec[dest?] = bspec[Transfer.transferred]
   217 lemmas fBallE[elim] = ballE[Transfer.transferred]
   218 lemmas fBexI[intro] = bexI[Transfer.transferred]
   219 lemmas rev_fBexI[intro?] = rev_bexI[Transfer.transferred]
   220 lemmas fBexCI = bexCI[Transfer.transferred]
   221 lemmas fBexE[elim!] = bexE[Transfer.transferred]
   222 lemmas fBall_triv[simp] = ball_triv[Transfer.transferred]
   223 lemmas fBex_triv[simp] = bex_triv[Transfer.transferred]
   224 lemmas fBex_triv_one_point1[simp] = bex_triv_one_point1[Transfer.transferred]
   225 lemmas fBex_triv_one_point2[simp] = bex_triv_one_point2[Transfer.transferred]
   226 lemmas fBex_one_point1[simp] = bex_one_point1[Transfer.transferred]
   227 lemmas fBex_one_point2[simp] = bex_one_point2[Transfer.transferred]
   228 lemmas fBall_one_point1[simp] = ball_one_point1[Transfer.transferred]
   229 lemmas fBall_one_point2[simp] = ball_one_point2[Transfer.transferred]
   230 lemmas fBall_conj_distrib = ball_conj_distrib[Transfer.transferred]
   231 lemmas fBex_disj_distrib = bex_disj_distrib[Transfer.transferred]
   232 lemmas fBall_cong = ball_cong[Transfer.transferred]
   233 lemmas fBex_cong = bex_cong[Transfer.transferred]
   234 lemmas fsubsetI[intro!] = subsetI[Transfer.transferred]
   235 lemmas fsubsetD[elim, intro?] = subsetD[Transfer.transferred]
   236 lemmas rev_fsubsetD[no_atp,intro?] = rev_subsetD[Transfer.transferred]
   237 lemmas fsubsetCE[no_atp,elim] = subsetCE[Transfer.transferred]
   238 lemmas fsubset_eq[no_atp] = subset_eq[Transfer.transferred]
   239 lemmas contra_fsubsetD[no_atp] = contra_subsetD[Transfer.transferred]
   240 lemmas fsubset_refl = subset_refl[Transfer.transferred]
   241 lemmas fsubset_trans = subset_trans[Transfer.transferred]
   242 lemmas fset_rev_mp = set_rev_mp[Transfer.transferred]
   243 lemmas fset_mp = set_mp[Transfer.transferred]
   244 lemmas fsubset_not_fsubset_eq[code] = subset_not_subset_eq[Transfer.transferred]
   245 lemmas eq_fmem_trans = eq_mem_trans[Transfer.transferred]
   246 lemmas fsubset_antisym[intro!] = subset_antisym[Transfer.transferred]
   247 lemmas fequalityD1 = equalityD1[Transfer.transferred]
   248 lemmas fequalityD2 = equalityD2[Transfer.transferred]
   249 lemmas fequalityE = equalityE[Transfer.transferred]
   250 lemmas fequalityCE[elim] = equalityCE[Transfer.transferred]
   251 lemmas eqfset_imp_iff = eqset_imp_iff[Transfer.transferred]
   252 lemmas eqfelem_imp_iff = eqelem_imp_iff[Transfer.transferred]
   253 lemmas fempty_iff[simp] = empty_iff[Transfer.transferred]
   254 lemmas fempty_fsubsetI[iff] = empty_subsetI[Transfer.transferred]
   255 lemmas equalsffemptyI = equals0I[Transfer.transferred]
   256 lemmas equalsffemptyD = equals0D[Transfer.transferred]
   257 lemmas fBall_fempty[simp] = ball_empty[Transfer.transferred]
   258 lemmas fBex_fempty[simp] = bex_empty[Transfer.transferred]
   259 lemmas fPow_iff[iff] = Pow_iff[Transfer.transferred]
   260 lemmas fPowI = PowI[Transfer.transferred]
   261 lemmas fPowD = PowD[Transfer.transferred]
   262 lemmas fPow_bottom = Pow_bottom[Transfer.transferred]
   263 lemmas fPow_top = Pow_top[Transfer.transferred]
   264 lemmas fPow_not_fempty = Pow_not_empty[Transfer.transferred]
   265 lemmas finter_iff[simp] = Int_iff[Transfer.transferred]
   266 lemmas finterI[intro!] = IntI[Transfer.transferred]
   267 lemmas finterD1 = IntD1[Transfer.transferred]
   268 lemmas finterD2 = IntD2[Transfer.transferred]
   269 lemmas finterE[elim!] = IntE[Transfer.transferred]
   270 lemmas funion_iff[simp] = Un_iff[Transfer.transferred]
   271 lemmas funionI1[elim?] = UnI1[Transfer.transferred]
   272 lemmas funionI2[elim?] = UnI2[Transfer.transferred]
   273 lemmas funionCI[intro!] = UnCI[Transfer.transferred]
   274 lemmas funionE[elim!] = UnE[Transfer.transferred]
   275 lemmas fminus_iff[simp] = Diff_iff[Transfer.transferred]
   276 lemmas fminusI[intro!] = DiffI[Transfer.transferred]
   277 lemmas fminusD1 = DiffD1[Transfer.transferred]
   278 lemmas fminusD2 = DiffD2[Transfer.transferred]
   279 lemmas fminusE[elim!] = DiffE[Transfer.transferred]
   280 lemmas finsert_iff[simp] = insert_iff[Transfer.transferred]
   281 lemmas finsertI1 = insertI1[Transfer.transferred]
   282 lemmas finsertI2 = insertI2[Transfer.transferred]
   283 lemmas finsertE[elim!] = insertE[Transfer.transferred]
   284 lemmas finsertCI[intro!] = insertCI[Transfer.transferred]
   285 lemmas fsubset_finsert_iff = subset_insert_iff[Transfer.transferred]
   286 lemmas finsert_ident = insert_ident[Transfer.transferred]
   287 lemmas fsingletonI[intro!,no_atp] = singletonI[Transfer.transferred]
   288 lemmas fsingletonD[dest!,no_atp] = singletonD[Transfer.transferred]
   289 lemmas fsingleton_iff = singleton_iff[Transfer.transferred]
   290 lemmas fsingleton_inject[dest!] = singleton_inject[Transfer.transferred]
   291 lemmas fsingleton_finsert_inj_eq[iff,no_atp] = singleton_insert_inj_eq[Transfer.transferred]
   292 lemmas fsingleton_finsert_inj_eq'[iff,no_atp] = singleton_insert_inj_eq'[Transfer.transferred]
   293 lemmas fsubset_fsingletonD = subset_singletonD[Transfer.transferred]
   294 lemmas fminus_single_finsert = diff_single_insert[Transfer.transferred]
   295 lemmas fdoubleton_eq_iff = doubleton_eq_iff[Transfer.transferred]
   296 lemmas funion_fsingleton_iff = Un_singleton_iff[Transfer.transferred]
   297 lemmas fsingleton_funion_iff = singleton_Un_iff[Transfer.transferred]
   298 lemmas fimage_eqI[simp, intro] = image_eqI[Transfer.transferred]
   299 lemmas fimageI = imageI[Transfer.transferred]
   300 lemmas rev_fimage_eqI = rev_image_eqI[Transfer.transferred]
   301 lemmas fimageE[elim!] = imageE[Transfer.transferred]
   302 lemmas Compr_fimage_eq = Compr_image_eq[Transfer.transferred]
   303 lemmas fimage_funion = image_Un[Transfer.transferred]
   304 lemmas fimage_iff = image_iff[Transfer.transferred]
   305 lemmas fimage_fsubset_iff[no_atp] = image_subset_iff[Transfer.transferred]
   306 lemmas fimage_fsubsetI = image_subsetI[Transfer.transferred]
   307 lemmas fimage_ident[simp] = image_ident[Transfer.transferred]
   308 lemmas split_if_fmem1 = split_if_mem1[Transfer.transferred]
   309 lemmas split_if_fmem2 = split_if_mem2[Transfer.transferred]
   310 lemmas pfsubsetI[intro!,no_atp] = psubsetI[Transfer.transferred]
   311 lemmas pfsubsetE[elim!,no_atp] = psubsetE[Transfer.transferred]
   312 lemmas pfsubset_finsert_iff = psubset_insert_iff[Transfer.transferred]
   313 lemmas pfsubset_eq = psubset_eq[Transfer.transferred]
   314 lemmas pfsubset_imp_fsubset = psubset_imp_subset[Transfer.transferred]
   315 lemmas pfsubset_trans = psubset_trans[Transfer.transferred]
   316 lemmas pfsubsetD = psubsetD[Transfer.transferred]
   317 lemmas pfsubset_fsubset_trans = psubset_subset_trans[Transfer.transferred]
   318 lemmas fsubset_pfsubset_trans = subset_psubset_trans[Transfer.transferred]
   319 lemmas pfsubset_imp_ex_fmem = psubset_imp_ex_mem[Transfer.transferred]
   320 lemmas fimage_fPow_mono = image_Pow_mono[Transfer.transferred]
   321 lemmas fimage_fPow_surj = image_Pow_surj[Transfer.transferred]
   322 lemmas fsubset_finsertI = subset_insertI[Transfer.transferred]
   323 lemmas fsubset_finsertI2 = subset_insertI2[Transfer.transferred]
   324 lemmas fsubset_finsert = subset_insert[Transfer.transferred]
   325 lemmas funion_upper1 = Un_upper1[Transfer.transferred]
   326 lemmas funion_upper2 = Un_upper2[Transfer.transferred]
   327 lemmas funion_least = Un_least[Transfer.transferred]
   328 lemmas finter_lower1 = Int_lower1[Transfer.transferred]
   329 lemmas finter_lower2 = Int_lower2[Transfer.transferred]
   330 lemmas finter_greatest = Int_greatest[Transfer.transferred]
   331 lemmas fminus_fsubset = Diff_subset[Transfer.transferred]
   332 lemmas fminus_fsubset_conv = Diff_subset_conv[Transfer.transferred]
   333 lemmas fsubset_fempty[simp] = subset_empty[Transfer.transferred]
   334 lemmas not_pfsubset_fempty[iff] = not_psubset_empty[Transfer.transferred]
   335 lemmas finsert_is_funion = insert_is_Un[Transfer.transferred]
   336 lemmas finsert_not_fempty[simp] = insert_not_empty[Transfer.transferred]
   337 lemmas fempty_not_finsert = empty_not_insert[Transfer.transferred]
   338 lemmas finsert_absorb = insert_absorb[Transfer.transferred]
   339 lemmas finsert_absorb2[simp] = insert_absorb2[Transfer.transferred]
   340 lemmas finsert_commute = insert_commute[Transfer.transferred]
   341 lemmas finsert_fsubset[simp] = insert_subset[Transfer.transferred]
   342 lemmas finsert_inter_finsert[simp] = insert_inter_insert[Transfer.transferred]
   343 lemmas finsert_disjoint[simp,no_atp] = insert_disjoint[Transfer.transferred]
   344 lemmas disjoint_finsert[simp,no_atp] = disjoint_insert[Transfer.transferred]
   345 lemmas fimage_fempty[simp] = image_empty[Transfer.transferred]
   346 lemmas fimage_finsert[simp] = image_insert[Transfer.transferred]
   347 lemmas fimage_constant = image_constant[Transfer.transferred]
   348 lemmas fimage_constant_conv = image_constant_conv[Transfer.transferred]
   349 lemmas fimage_fimage = image_image[Transfer.transferred]
   350 lemmas finsert_fimage[simp] = insert_image[Transfer.transferred]
   351 lemmas fimage_is_fempty[iff] = image_is_empty[Transfer.transferred]
   352 lemmas fempty_is_fimage[iff] = empty_is_image[Transfer.transferred]
   353 lemmas fimage_cong = image_cong[Transfer.transferred]
   354 lemmas fimage_finter_fsubset = image_Int_subset[Transfer.transferred]
   355 lemmas fimage_fminus_fsubset = image_diff_subset[Transfer.transferred]
   356 lemmas finter_absorb = Int_absorb[Transfer.transferred]
   357 lemmas finter_left_absorb = Int_left_absorb[Transfer.transferred]
   358 lemmas finter_commute = Int_commute[Transfer.transferred]
   359 lemmas finter_left_commute = Int_left_commute[Transfer.transferred]
   360 lemmas finter_assoc = Int_assoc[Transfer.transferred]
   361 lemmas finter_ac = Int_ac[Transfer.transferred]
   362 lemmas finter_absorb1 = Int_absorb1[Transfer.transferred]
   363 lemmas finter_absorb2 = Int_absorb2[Transfer.transferred]
   364 lemmas finter_fempty_left = Int_empty_left[Transfer.transferred]
   365 lemmas finter_fempty_right = Int_empty_right[Transfer.transferred]
   366 lemmas disjoint_iff_fnot_equal = disjoint_iff_not_equal[Transfer.transferred]
   367 lemmas finter_funion_distrib = Int_Un_distrib[Transfer.transferred]
   368 lemmas finter_funion_distrib2 = Int_Un_distrib2[Transfer.transferred]
   369 lemmas finter_fsubset_iff[no_atp, simp] = Int_subset_iff[Transfer.transferred]
   370 lemmas funion_absorb = Un_absorb[Transfer.transferred]
   371 lemmas funion_left_absorb = Un_left_absorb[Transfer.transferred]
   372 lemmas funion_commute = Un_commute[Transfer.transferred]
   373 lemmas funion_left_commute = Un_left_commute[Transfer.transferred]
   374 lemmas funion_assoc = Un_assoc[Transfer.transferred]
   375 lemmas funion_ac = Un_ac[Transfer.transferred]
   376 lemmas funion_absorb1 = Un_absorb1[Transfer.transferred]
   377 lemmas funion_absorb2 = Un_absorb2[Transfer.transferred]
   378 lemmas funion_fempty_left = Un_empty_left[Transfer.transferred]
   379 lemmas funion_fempty_right = Un_empty_right[Transfer.transferred]
   380 lemmas funion_finsert_left[simp] = Un_insert_left[Transfer.transferred]
   381 lemmas funion_finsert_right[simp] = Un_insert_right[Transfer.transferred]
   382 lemmas finter_finsert_left = Int_insert_left[Transfer.transferred]
   383 lemmas finter_finsert_left_ifffempty[simp] = Int_insert_left_if0[Transfer.transferred]
   384 lemmas finter_finsert_left_if1[simp] = Int_insert_left_if1[Transfer.transferred]
   385 lemmas finter_finsert_right = Int_insert_right[Transfer.transferred]
   386 lemmas finter_finsert_right_ifffempty[simp] = Int_insert_right_if0[Transfer.transferred]
   387 lemmas finter_finsert_right_if1[simp] = Int_insert_right_if1[Transfer.transferred]
   388 lemmas funion_finter_distrib = Un_Int_distrib[Transfer.transferred]
   389 lemmas funion_finter_distrib2 = Un_Int_distrib2[Transfer.transferred]
   390 lemmas funion_finter_crazy = Un_Int_crazy[Transfer.transferred]
   391 lemmas fsubset_funion_eq = subset_Un_eq[Transfer.transferred]
   392 lemmas funion_fempty[iff] = Un_empty[Transfer.transferred]
   393 lemmas funion_fsubset_iff[no_atp, simp] = Un_subset_iff[Transfer.transferred]
   394 lemmas funion_fminus_finter = Un_Diff_Int[Transfer.transferred]
   395 lemmas fminus_finter2 = Diff_Int2[Transfer.transferred]
   396 lemmas funion_finter_assoc_eq = Un_Int_assoc_eq[Transfer.transferred]
   397 lemmas fBall_funion = ball_Un[Transfer.transferred]
   398 lemmas fBex_funion = bex_Un[Transfer.transferred]
   399 lemmas fminus_eq_fempty_iff[simp,no_atp] = Diff_eq_empty_iff[Transfer.transferred]
   400 lemmas fminus_cancel[simp] = Diff_cancel[Transfer.transferred]
   401 lemmas fminus_idemp[simp] = Diff_idemp[Transfer.transferred]
   402 lemmas fminus_triv = Diff_triv[Transfer.transferred]
   403 lemmas fempty_fminus[simp] = empty_Diff[Transfer.transferred]
   404 lemmas fminus_fempty[simp] = Diff_empty[Transfer.transferred]
   405 lemmas fminus_finsertffempty[simp,no_atp] = Diff_insert0[Transfer.transferred]
   406 lemmas fminus_finsert = Diff_insert[Transfer.transferred]
   407 lemmas fminus_finsert2 = Diff_insert2[Transfer.transferred]
   408 lemmas finsert_fminus_if = insert_Diff_if[Transfer.transferred]
   409 lemmas finsert_fminus1[simp] = insert_Diff1[Transfer.transferred]
   410 lemmas finsert_fminus_single[simp] = insert_Diff_single[Transfer.transferred]
   411 lemmas finsert_fminus = insert_Diff[Transfer.transferred]
   412 lemmas fminus_finsert_absorb = Diff_insert_absorb[Transfer.transferred]
   413 lemmas fminus_disjoint[simp] = Diff_disjoint[Transfer.transferred]
   414 lemmas fminus_partition = Diff_partition[Transfer.transferred]
   415 lemmas double_fminus = double_diff[Transfer.transferred]
   416 lemmas funion_fminus_cancel[simp] = Un_Diff_cancel[Transfer.transferred]
   417 lemmas funion_fminus_cancel2[simp] = Un_Diff_cancel2[Transfer.transferred]
   418 lemmas fminus_funion = Diff_Un[Transfer.transferred]
   419 lemmas fminus_finter = Diff_Int[Transfer.transferred]
   420 lemmas funion_fminus = Un_Diff[Transfer.transferred]
   421 lemmas finter_fminus = Int_Diff[Transfer.transferred]
   422 lemmas fminus_finter_distrib = Diff_Int_distrib[Transfer.transferred]
   423 lemmas fminus_finter_distrib2 = Diff_Int_distrib2[Transfer.transferred]
   424 lemmas fUNIV_bool[no_atp] = UNIV_bool[Transfer.transferred]
   425 lemmas fPow_fempty[simp] = Pow_empty[Transfer.transferred]
   426 lemmas fPow_finsert = Pow_insert[Transfer.transferred]
   427 lemmas funion_fPow_fsubset = Un_Pow_subset[Transfer.transferred]
   428 lemmas fPow_finter_eq[simp] = Pow_Int_eq[Transfer.transferred]
   429 lemmas fset_eq_fsubset = set_eq_subset[Transfer.transferred]
   430 lemmas fsubset_iff[no_atp] = subset_iff[Transfer.transferred]
   431 lemmas fsubset_iff_pfsubset_eq = subset_iff_psubset_eq[Transfer.transferred]
   432 lemmas all_not_fin_conv[simp] = all_not_in_conv[Transfer.transferred]
   433 lemmas ex_fin_conv = ex_in_conv[Transfer.transferred]
   434 lemmas fimage_mono = image_mono[Transfer.transferred]
   435 lemmas fPow_mono = Pow_mono[Transfer.transferred]
   436 lemmas finsert_mono = insert_mono[Transfer.transferred]
   437 lemmas funion_mono = Un_mono[Transfer.transferred]
   438 lemmas finter_mono = Int_mono[Transfer.transferred]
   439 lemmas fminus_mono = Diff_mono[Transfer.transferred]
   440 lemmas fin_mono = in_mono[Transfer.transferred]
   441 lemmas fthe_felem_eq[simp] = the_elem_eq[Transfer.transferred]
   442 lemmas fLeast_mono = Least_mono[Transfer.transferred]
   443 lemmas fbind_fbind = bind_bind[Transfer.transferred]
   444 lemmas fempty_fbind[simp] = empty_bind[Transfer.transferred]
   445 lemmas nonfempty_fbind_const = nonempty_bind_const[Transfer.transferred]
   446 lemmas fbind_const = bind_const[Transfer.transferred]
   447 lemmas ffmember_filter[simp] = member_filter[Transfer.transferred]
   448 lemmas fequalityI = equalityI[Transfer.transferred]
   449 
   450 
   451 subsection {* Additional lemmas*}
   452 
   453 subsubsection {* @{text fsingleton} *}
   454 
   455 lemmas fsingletonE = fsingletonD [elim_format]
   456 
   457 
   458 subsubsection {* @{text femepty} *}
   459 
   460 lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {||}"
   461 by transfer auto
   462 
   463 (* FIXME, transferred doesn't work here *)
   464 lemma femptyE [elim!]: "a |\<in>| {||} \<Longrightarrow> P"
   465   by simp
   466 
   467 
   468 subsubsection {* @{text fset} *}
   469 
   470 lemmas fset_simps[simp] = bot_fset.rep_eq finsert.rep_eq
   471 
   472 lemma finite_fset [simp]: 
   473   shows "finite (fset S)"
   474   by transfer simp
   475 
   476 lemmas fset_cong = fset_inject
   477 
   478 lemma filter_fset [simp]:
   479   shows "fset (ffilter P xs) = Collect P \<inter> fset xs"
   480   by transfer auto
   481 
   482 lemma notin_fset: "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S" by (simp add: fmember.rep_eq)
   483 
   484 lemmas inter_fset[simp] = inf_fset.rep_eq
   485 
   486 lemmas union_fset[simp] = sup_fset.rep_eq
   487 
   488 lemmas minus_fset[simp] = minus_fset.rep_eq
   489 
   490 
   491 subsubsection {* @{text filter_fset} *}
   492 
   493 lemma subset_ffilter: 
   494   "ffilter P A |\<subseteq>| ffilter Q A = (\<forall> x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)"
   495   by transfer auto
   496 
   497 lemma eq_ffilter: 
   498   "(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)"
   499   by transfer auto
   500 
   501 lemma pfsubset_ffilter:
   502   "(\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| A & \<not> P x & Q x) \<Longrightarrow> 
   503     ffilter P A |\<subset>| ffilter Q A"
   504   unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter)
   505 
   506 
   507 subsubsection {* @{text finsert} *}
   508 
   509 (* FIXME, transferred doesn't work here *)
   510 lemma set_finsert:
   511   assumes "x |\<in>| A"
   512   obtains B where "A = finsert x B" and "x |\<notin>| B"
   513 using assms by transfer (metis Set.set_insert finite_insert)
   514 
   515 lemma mk_disjoint_finsert: "a |\<in>| A \<Longrightarrow> \<exists>B. A = finsert a B \<and> a |\<notin>| B"
   516   by (rule_tac x = "A |-| {|a|}" in exI, blast)
   517 
   518 
   519 subsubsection {* @{text fimage} *}
   520 
   521 lemma subset_fimage_iff: "(B |\<subseteq>| f|`|A) = (\<exists> AA. AA |\<subseteq>| A \<and> B = f|`|AA)"
   522 by transfer (metis mem_Collect_eq rev_finite_subset subset_image_iff)
   523 
   524 
   525 subsubsection {* bounded quantification *}
   526 
   527 lemma bex_simps [simp, no_atp]:
   528   "\<And>A P Q. fBex A (\<lambda>x. P x \<and> Q) = (fBex A P \<and> Q)" 
   529   "\<And>A P Q. fBex A (\<lambda>x. P \<and> Q x) = (P \<and> fBex A Q)"
   530   "\<And>P. fBex {||} P = False" 
   531   "\<And>a B P. fBex (finsert a B) P = (P a \<or> fBex B P)"
   532   "\<And>A P f. fBex (f |`| A) P = fBex A (\<lambda>x. P (f x))"
   533   "\<And>A P. (\<not> fBex A P) = fBall A (\<lambda>x. \<not> P x)"
   534 by auto
   535 
   536 lemma ball_simps [simp, no_atp]:
   537   "\<And>A P Q. fBall A (\<lambda>x. P x \<or> Q) = (fBall A P \<or> Q)"
   538   "\<And>A P Q. fBall A (\<lambda>x. P \<or> Q x) = (P \<or> fBall A Q)"
   539   "\<And>A P Q. fBall A (\<lambda>x. P \<longrightarrow> Q x) = (P \<longrightarrow> fBall A Q)"
   540   "\<And>A P Q. fBall A (\<lambda>x. P x \<longrightarrow> Q) = (fBex A P \<longrightarrow> Q)"
   541   "\<And>P. fBall {||} P = True"
   542   "\<And>a B P. fBall (finsert a B) P = (P a \<and> fBall B P)"
   543   "\<And>A P f. fBall (f |`| A) P = fBall A (\<lambda>x. P (f x))"
   544   "\<And>A P. (\<not> fBall A P) = fBex A (\<lambda>x. \<not> P x)"
   545 by auto
   546 
   547 lemma atomize_fBall:
   548     "(\<And>x. x |\<in>| A ==> P x) == Trueprop (fBall A (\<lambda>x. P x))"
   549 apply (simp only: atomize_all atomize_imp)
   550 apply (rule equal_intr_rule)
   551 by (transfer, simp)+
   552 
   553 end
   554 
   555 
   556 subsubsection {* @{text fcard} *}
   557 
   558 (* FIXME: improve transferred to handle bounded meta quantification *)
   559 
   560 lemma fcard_fempty:
   561   "fcard {||} = 0"
   562   by transfer (rule card_empty)
   563 
   564 lemma fcard_finsert_disjoint:
   565   "x |\<notin>| A \<Longrightarrow> fcard (finsert x A) = Suc (fcard A)"
   566   by transfer (rule card_insert_disjoint)
   567 
   568 lemma fcard_finsert_if:
   569   "fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))"
   570   by transfer (rule card_insert_if)
   571 
   572 lemma card_0_eq [simp, no_atp]:
   573   "fcard A = 0 \<longleftrightarrow> A = {||}"
   574   by transfer (rule card_0_eq)
   575 
   576 lemma fcard_Suc_fminus1:
   577   "x |\<in>| A \<Longrightarrow> Suc (fcard (A |-| {|x|})) = fcard A"
   578   by transfer (rule card_Suc_Diff1)
   579 
   580 lemma fcard_fminus_fsingleton:
   581   "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) = fcard A - 1"
   582   by transfer (rule card_Diff_singleton)
   583 
   584 lemma fcard_fminus_fsingleton_if:
   585   "fcard (A |-| {|x|}) = (if x |\<in>| A then fcard A - 1 else fcard A)"
   586   by transfer (rule card_Diff_singleton_if)
   587 
   588 lemma fcard_fminus_finsert[simp]:
   589   assumes "a |\<in>| A" and "a |\<notin>| B"
   590   shows "fcard (A |-| finsert a B) = fcard (A |-| B) - 1"
   591 using assms by transfer (rule card_Diff_insert)
   592 
   593 lemma fcard_finsert: "fcard (finsert x A) = Suc (fcard (A |-| {|x|}))"
   594 by transfer (rule card_insert)
   595 
   596 lemma fcard_finsert_le: "fcard A \<le> fcard (finsert x A)"
   597 by transfer (rule card_insert_le)
   598 
   599 lemma fcard_mono:
   600   "A |\<subseteq>| B \<Longrightarrow> fcard A \<le> fcard B"
   601 by transfer (rule card_mono)
   602 
   603 lemma fcard_seteq: "A |\<subseteq>| B \<Longrightarrow> fcard B \<le> fcard A \<Longrightarrow> A = B"
   604 by transfer (rule card_seteq)
   605 
   606 lemma pfsubset_fcard_mono: "A |\<subset>| B \<Longrightarrow> fcard A < fcard B"
   607 by transfer (rule psubset_card_mono)
   608 
   609 lemma fcard_funion_finter: 
   610   "fcard A + fcard B = fcard (A |\<union>| B) + fcard (A |\<inter>| B)"
   611 by transfer (rule card_Un_Int)
   612 
   613 lemma fcard_funion_disjoint:
   614   "A |\<inter>| B = {||} \<Longrightarrow> fcard (A |\<union>| B) = fcard A + fcard B"
   615 by transfer (rule card_Un_disjoint)
   616 
   617 lemma fcard_funion_fsubset:
   618   "B |\<subseteq>| A \<Longrightarrow> fcard (A |-| B) = fcard A - fcard B"
   619 by transfer (rule card_Diff_subset)
   620 
   621 lemma diff_fcard_le_fcard_fminus:
   622   "fcard A - fcard B \<le> fcard(A |-| B)"
   623 by transfer (rule diff_card_le_card_Diff)
   624 
   625 lemma fcard_fminus1_less: "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) < fcard A"
   626 by transfer (rule card_Diff1_less)
   627 
   628 lemma fcard_fminus2_less:
   629   "x |\<in>| A \<Longrightarrow> y |\<in>| A \<Longrightarrow> fcard (A |-| {|x|} |-| {|y|}) < fcard A"
   630 by transfer (rule card_Diff2_less)
   631 
   632 lemma fcard_fminus1_le: "fcard (A |-| {|x|}) \<le> fcard A"
   633 by transfer (rule card_Diff1_le)
   634 
   635 lemma fcard_pfsubset: "A |\<subseteq>| B \<Longrightarrow> fcard A < fcard B \<Longrightarrow> A < B"
   636 by transfer (rule card_psubset)
   637 
   638 
   639 subsubsection {* @{text ffold} *}
   640 
   641 (* FIXME: improve transferred to handle bounded meta quantification *)
   642 
   643 context comp_fun_commute
   644 begin
   645   lemmas ffold_empty[simp] = fold_empty[Transfer.transferred]
   646 
   647   lemma ffold_finsert [simp]:
   648     assumes "x |\<notin>| A"
   649     shows "ffold f z (finsert x A) = f x (ffold f z A)"
   650     using assms by (transfer fixing: f) (rule fold_insert)
   651 
   652   lemma ffold_fun_left_comm:
   653     "f x (ffold f z A) = ffold f (f x z) A"
   654     by (transfer fixing: f) (rule fold_fun_left_comm)
   655 
   656   lemma ffold_finsert2:
   657     "x |\<notin>| A \<Longrightarrow> ffold f z (finsert x A)  = ffold f (f x z) A"
   658     by (transfer fixing: f) (rule fold_insert2)
   659 
   660   lemma ffold_rec:
   661     assumes "x |\<in>| A"
   662     shows "ffold f z A = f x (ffold f z (A |-| {|x|}))"
   663     using assms by (transfer fixing: f) (rule fold_rec)
   664   
   665   lemma ffold_finsert_fremove:
   666     "ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
   667      by (transfer fixing: f) (rule fold_insert_remove)
   668 end
   669 
   670 lemma ffold_fimage:
   671   assumes "inj_on g (fset A)"
   672   shows "ffold f z (g |`| A) = ffold (f \<circ> g) z A"
   673 using assms by transfer' (rule fold_image)
   674 
   675 lemma ffold_cong:
   676   assumes "comp_fun_commute f" "comp_fun_commute g"
   677   "\<And>x. x |\<in>| A \<Longrightarrow> f x = g x"
   678     and "s = t" and "A = B"
   679   shows "ffold f s A = ffold g t B"
   680 using assms by transfer (metis Finite_Set.fold_cong)
   681 
   682 context comp_fun_idem
   683 begin
   684 
   685   lemma ffold_finsert_idem:
   686     "ffold f z (finsert x A)  = f x (ffold f z A)"
   687     by (transfer fixing: f) (rule fold_insert_idem)
   688   
   689   declare ffold_finsert [simp del] ffold_finsert_idem [simp]
   690   
   691   lemma ffold_finsert_idem2:
   692     "ffold f z (finsert x A) = ffold f (f x z) A"
   693     by (transfer fixing: f) (rule fold_insert_idem2)
   694 
   695 end
   696 
   697 
   698 subsection {* Choice in fsets *}
   699 
   700 lemma fset_choice: 
   701   assumes "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
   702   shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
   703   using assms by transfer metis
   704 
   705 
   706 subsection {* Induction and Cases rules for fsets *}
   707 
   708 lemma fset_exhaust [case_names empty insert, cases type: fset]:
   709   assumes fempty_case: "S = {||} \<Longrightarrow> P" 
   710   and     finsert_case: "\<And>x S'. S = finsert x S' \<Longrightarrow> P"
   711   shows "P"
   712   using assms by transfer blast
   713 
   714 lemma fset_induct [case_names empty insert]:
   715   assumes fempty_case: "P {||}"
   716   and     finsert_case: "\<And>x S. P S \<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_induct_stronger [case_names empty insert, induct type: fset]:
   726   assumes empty_fset_case: "P {||}"
   727   and     insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
   728   shows "P S"
   729 proof -
   730   (* FIXME transfer and right_total vs. bi_total *)
   731   note Domainp_forall_transfer[transfer_rule]
   732   show ?thesis
   733   using assms by transfer (auto intro: finite_induct)
   734 qed
   735 
   736 lemma fset_card_induct:
   737   assumes empty_fset_case: "P {||}"
   738   and     card_fset_Suc_case: "\<And>S T. Suc (fcard S) = (fcard T) \<Longrightarrow> P S \<Longrightarrow> P T"
   739   shows "P S"
   740 proof (induct S)
   741   case empty
   742   show "P {||}" by (rule empty_fset_case)
   743 next
   744   case (insert x S)
   745   have h: "P S" by fact
   746   have "x |\<notin>| S" by fact
   747   then have "Suc (fcard S) = fcard (finsert x S)" 
   748     by transfer auto
   749   then show "P (finsert x S)" 
   750     using h card_fset_Suc_case by simp
   751 qed
   752 
   753 lemma fset_strong_cases:
   754   obtains "xs = {||}"
   755     | ys x where "x |\<notin>| ys" and "xs = finsert x ys"
   756 by transfer blast
   757 
   758 lemma fset_induct2:
   759   "P {||} {||} \<Longrightarrow>
   760   (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>
   761   (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>
   762   (\<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>
   763   P xsa ysa"
   764   apply (induct xsa arbitrary: ysa)
   765   apply (induct_tac x rule: fset_induct_stronger)
   766   apply simp_all
   767   apply (induct_tac xa rule: fset_induct_stronger)
   768   apply simp_all
   769   done
   770 
   771 
   772 subsection {* Setup for Lifting/Transfer *}
   773 
   774 subsubsection {* Relator and predicator properties *}
   775 
   776 lift_definition fset_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is set_rel
   777 parametric set_rel_transfer ..
   778 
   779 lemma fset_rel_alt_def: "fset_rel R = (\<lambda>A B. (\<forall>x.\<exists>y. x|\<in>|A \<longrightarrow> y|\<in>|B \<and> R x y) 
   780   \<and> (\<forall>y. \<exists>x. y|\<in>|B \<longrightarrow> x|\<in>|A \<and> R x y))"
   781 apply (rule ext)+
   782 apply transfer'
   783 apply (subst set_rel_def[unfolded fun_eq_iff]) 
   784 by blast
   785 
   786 lemma fset_rel_conversep: "fset_rel (conversep R) = conversep (fset_rel R)"
   787   unfolding fset_rel_alt_def by auto
   788 
   789 lemmas fset_rel_eq [relator_eq] = set_rel_eq[Transfer.transferred]
   790 
   791 lemma fset_rel_mono[relator_mono]: "A \<le> B \<Longrightarrow> fset_rel A \<le> fset_rel B"
   792 unfolding fset_rel_alt_def by blast
   793 
   794 lemma finite_set_rel:
   795   assumes fin: "finite X" "finite Z"
   796   assumes R_S: "set_rel (R OO S) X Z"
   797   shows "\<exists>Y. finite Y \<and> set_rel R X Y \<and> set_rel S Y Z"
   798 proof -
   799   obtain f where f: "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)"
   800   apply atomize_elim
   801   apply (subst bchoice_iff[symmetric])
   802   using R_S[unfolded set_rel_def OO_def] by blast
   803   
   804   obtain g where g: "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R  x (g z))"
   805   apply atomize_elim
   806   apply (subst bchoice_iff[symmetric])
   807   using R_S[unfolded set_rel_def OO_def] by blast
   808   
   809   let ?Y = "f ` X \<union> g ` Z"
   810   have "finite ?Y" by (simp add: fin)
   811   moreover have "set_rel R X ?Y"
   812     unfolding set_rel_def
   813     using f g by clarsimp blast
   814   moreover have "set_rel S ?Y Z"
   815     unfolding set_rel_def
   816     using f g by clarsimp blast
   817   ultimately show ?thesis by metis
   818 qed
   819 
   820 lemma fset_rel_OO[relator_distr]: "fset_rel R OO fset_rel S = fset_rel (R OO S)"
   821 apply (rule ext)+
   822 by transfer (auto intro: finite_set_rel set_rel_OO[unfolded fun_eq_iff, rule_format, THEN iffD1])
   823 
   824 lemma Domainp_fset[relator_domain]:
   825   assumes "Domainp T = P"
   826   shows "Domainp (fset_rel T) = (\<lambda>A. fBall A P)"
   827 proof -
   828   from assms obtain f where f: "\<forall>x\<in>Collect P. T x (f x)"
   829     unfolding Domainp_iff[abs_def]
   830     apply atomize_elim
   831     by (subst bchoice_iff[symmetric]) auto
   832   from assms f show ?thesis
   833     unfolding fun_eq_iff fset_rel_alt_def Domainp_iff
   834     apply clarify
   835     apply (rule iffI)
   836       apply blast
   837     by (rename_tac A, rule_tac x="f |`| A" in exI, blast)
   838 qed
   839 
   840 lemmas reflp_fset_rel[reflexivity_rule] = reflp_set_rel[Transfer.transferred]
   841 
   842 lemma right_total_fset_rel[transfer_rule]: "right_total A \<Longrightarrow> right_total (fset_rel A)"
   843 unfolding right_total_def 
   844 apply transfer
   845 apply (subst(asm) choice_iff)
   846 apply clarsimp
   847 apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
   848 by (auto simp add: set_rel_def)
   849 
   850 lemma left_total_fset_rel[reflexivity_rule]: "left_total A \<Longrightarrow> left_total (fset_rel A)"
   851 unfolding left_total_def 
   852 apply transfer
   853 apply (subst(asm) choice_iff)
   854 apply clarsimp
   855 apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
   856 by (auto simp add: set_rel_def)
   857 
   858 lemmas right_unique_fset_rel[transfer_rule] = right_unique_set_rel[Transfer.transferred]
   859 lemmas left_unique_fset_rel[reflexivity_rule] = left_unique_set_rel[Transfer.transferred]
   860 
   861 thm right_unique_fset_rel left_unique_fset_rel
   862 
   863 lemma bi_unique_fset_rel[transfer_rule]: "bi_unique A \<Longrightarrow> bi_unique (fset_rel A)"
   864 by (auto intro: right_unique_fset_rel left_unique_fset_rel iff: bi_unique_iff)
   865 
   866 lemma bi_total_fset_rel[transfer_rule]: "bi_total A \<Longrightarrow> bi_total (fset_rel A)"
   867 by (auto intro: right_total_fset_rel left_total_fset_rel iff: bi_total_iff)
   868 
   869 lemmas fset_invariant_commute [invariant_commute] = set_invariant_commute[Transfer.transferred]
   870 
   871 
   872 subsubsection {* Quotient theorem for the Lifting package *}
   873 
   874 lemma Quotient_fset_map[quot_map]:
   875   assumes "Quotient R Abs Rep T"
   876   shows "Quotient (fset_rel R) (fimage Abs) (fimage Rep) (fset_rel T)"
   877   using assms unfolding Quotient_alt_def4
   878   by (simp add: fset_rel_OO[symmetric] fset_rel_conversep) (simp add: fset_rel_alt_def, blast)
   879 
   880 
   881 subsubsection {* Transfer rules for the Transfer package *}
   882 
   883 text {* Unconditional transfer rules *}
   884 
   885 context
   886 begin
   887 
   888 interpretation lifting_syntax .
   889 
   890 lemmas fempty_transfer [transfer_rule] = empty_transfer[Transfer.transferred]
   891 
   892 lemma finsert_transfer [transfer_rule]:
   893   "(A ===> fset_rel A ===> fset_rel A) finsert finsert"
   894   unfolding fun_rel_def fset_rel_alt_def by blast
   895 
   896 lemma funion_transfer [transfer_rule]:
   897   "(fset_rel A ===> fset_rel A ===> fset_rel A) funion funion"
   898   unfolding fun_rel_def fset_rel_alt_def by blast
   899 
   900 lemma ffUnion_transfer [transfer_rule]:
   901   "(fset_rel (fset_rel A) ===> fset_rel A) ffUnion ffUnion"
   902   unfolding fun_rel_def fset_rel_alt_def by transfer (simp, fast)
   903 
   904 lemma fimage_transfer [transfer_rule]:
   905   "((A ===> B) ===> fset_rel A ===> fset_rel B) fimage fimage"
   906   unfolding fun_rel_def fset_rel_alt_def by simp blast
   907 
   908 lemma fBall_transfer [transfer_rule]:
   909   "(fset_rel A ===> (A ===> op =) ===> op =) fBall fBall"
   910   unfolding fset_rel_alt_def fun_rel_def by blast
   911 
   912 lemma fBex_transfer [transfer_rule]:
   913   "(fset_rel A ===> (A ===> op =) ===> op =) fBex fBex"
   914   unfolding fset_rel_alt_def fun_rel_def by blast
   915 
   916 (* FIXME transfer doesn't work here *)
   917 lemma fPow_transfer [transfer_rule]:
   918   "(fset_rel A ===> fset_rel (fset_rel A)) fPow fPow"
   919   unfolding fun_rel_def
   920   using Pow_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred]
   921   by blast
   922 
   923 lemma fset_rel_transfer [transfer_rule]:
   924   "((A ===> B ===> op =) ===> fset_rel A ===> fset_rel B ===> op =)
   925     fset_rel fset_rel"
   926   unfolding fun_rel_def
   927   using set_rel_transfer[unfolded fun_rel_def,rule_format, Transfer.transferred, where A = A and B = B]
   928   by simp
   929 
   930 lemma bind_transfer [transfer_rule]:
   931   "(fset_rel A ===> (A ===> fset_rel B) ===> fset_rel B) fbind fbind"
   932   using assms unfolding fun_rel_def
   933   using bind_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
   934 
   935 text {* Rules requiring bi-unique, bi-total or right-total relations *}
   936 
   937 lemma fmember_transfer [transfer_rule]:
   938   assumes "bi_unique A"
   939   shows "(A ===> fset_rel A ===> op =) (op |\<in>|) (op |\<in>|)"
   940   using assms unfolding fun_rel_def fset_rel_alt_def bi_unique_def by metis
   941 
   942 lemma finter_transfer [transfer_rule]:
   943   assumes "bi_unique A"
   944   shows "(fset_rel A ===> fset_rel A ===> fset_rel A) finter finter"
   945   using assms unfolding fun_rel_def
   946   using inter_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
   947 
   948 lemma fminus_transfer [transfer_rule]:
   949   assumes "bi_unique A"
   950   shows "(fset_rel A ===> fset_rel A ===> fset_rel A) (op |-|) (op |-|)"
   951   using assms unfolding fun_rel_def
   952   using Diff_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
   953 
   954 lemma fsubset_transfer [transfer_rule]:
   955   assumes "bi_unique A"
   956   shows "(fset_rel A ===> fset_rel A ===> op =) (op |\<subseteq>|) (op |\<subseteq>|)"
   957   using assms unfolding fun_rel_def
   958   using subset_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
   959 
   960 lemma fSup_transfer [transfer_rule]:
   961   "bi_unique A \<Longrightarrow> (set_rel (fset_rel A) ===> fset_rel A) Sup Sup"
   962   using assms unfolding fun_rel_def
   963   apply clarify
   964   apply transfer'
   965   using Sup_fset_transfer[unfolded fun_rel_def] by blast
   966 
   967 (* FIXME: add right_total_fInf_transfer *)
   968 
   969 lemma fInf_transfer [transfer_rule]:
   970   assumes "bi_unique A" and "bi_total A"
   971   shows "(set_rel (fset_rel A) ===> fset_rel A) Inf Inf"
   972   using assms unfolding fun_rel_def
   973   apply clarify
   974   apply transfer'
   975   using Inf_fset_transfer[unfolded fun_rel_def] by blast
   976 
   977 lemma ffilter_transfer [transfer_rule]:
   978   assumes "bi_unique A"
   979   shows "((A ===> op=) ===> fset_rel A ===> fset_rel A) ffilter ffilter"
   980   using assms unfolding fun_rel_def
   981   using Lifting_Set.filter_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
   982 
   983 lemma card_transfer [transfer_rule]:
   984   "bi_unique A \<Longrightarrow> (fset_rel A ===> op =) fcard fcard"
   985   using assms unfolding fun_rel_def
   986   using card_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
   987 
   988 end
   989 
   990 lifting_update fset.lifting
   991 lifting_forget fset.lifting
   992 
   993 
   994 subsection {* BNF setup *}
   995 
   996 context
   997 includes fset.lifting
   998 begin
   999 
  1000 lemma fset_rel_alt:
  1001   "fset_rel 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)"
  1002 by transfer (simp add: set_rel_def)
  1003 
  1004 lemma fset_to_fset: "finite A \<Longrightarrow> fset (the_inv fset A) = A"
  1005 apply (rule f_the_inv_into_f[unfolded inj_on_def])
  1006 apply (simp add: fset_inject)
  1007 apply (rule range_eqI Abs_fset_inverse[symmetric] CollectI)+
  1008 .
  1009 
  1010 lemma fset_rel_aux:
  1011 "(\<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>
  1012  ((BNF_Util.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage fst))\<inverse>\<inverse> OO
  1013   BNF_Util.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage snd)) a b" (is "?L = ?R")
  1014 proof
  1015   assume ?L
  1016   def R' \<equiv> "the_inv fset (Collect (split R) \<inter> (fset a \<times> fset b))" (is "the_inv fset ?L'")
  1017   have "finite ?L'" by (intro finite_Int[OF disjI2] finite_cartesian_product) (transfer, simp)+
  1018   hence *: "fset R' = ?L'" unfolding R'_def by (intro fset_to_fset)
  1019   show ?R unfolding Grp_def relcompp.simps conversep.simps
  1020   proof (intro CollectI prod_caseI exI[of _ a] exI[of _ b] exI[of _ R'] conjI refl)
  1021     from * show "a = fimage fst R'" using conjunct1[OF `?L`]
  1022       by (transfer, auto simp add: image_def Int_def split: prod.splits)
  1023     from * show "b = fimage snd R'" using conjunct2[OF `?L`]
  1024       by (transfer, auto simp add: image_def Int_def split: prod.splits)
  1025   qed (auto simp add: *)
  1026 next
  1027   assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps
  1028   apply (simp add: subset_eq Ball_def)
  1029   apply (rule conjI)
  1030   apply (transfer, clarsimp, metis snd_conv)
  1031   by (transfer, clarsimp, metis fst_conv)
  1032 qed
  1033 
  1034 bnf "'a fset"
  1035   map: fimage
  1036   sets: fset 
  1037   bd: natLeq
  1038   wits: "{||}"
  1039   rel: fset_rel
  1040 apply -
  1041           apply transfer' apply simp
  1042          apply transfer' apply force
  1043         apply transfer apply force
  1044        apply transfer' apply force
  1045       apply (rule natLeq_card_order)
  1046      apply (rule natLeq_cinfinite)
  1047     apply transfer apply (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq)
  1048    apply (fastforce simp: fset_rel_alt)
  1049  apply (simp add: Grp_def relcompp.simps conversep.simps fun_eq_iff fset_rel_alt fset_rel_aux) 
  1050 apply transfer apply simp
  1051 done
  1052 
  1053 lemma fset_rel_fset: "set_rel \<chi> (fset A1) (fset A2) = fset_rel \<chi> A1 A2"
  1054   by transfer (rule refl)
  1055 
  1056 end
  1057 
  1058 lemmas [simp] = fset.map_comp fset.map_id fset.set_map
  1059 
  1060 
  1061 subsection {* Advanced relator customization *}
  1062 
  1063 (* Set vs. sum relators: *)
  1064 
  1065 lemma set_rel_sum_rel[simp]: 
  1066 "set_rel (sum_rel \<chi> \<phi>) A1 A2 \<longleftrightarrow> 
  1067  set_rel \<chi> (Inl -` A1) (Inl -` A2) \<and> set_rel \<phi> (Inr -` A1) (Inr -` A2)"
  1068 (is "?L \<longleftrightarrow> ?Rl \<and> ?Rr")
  1069 proof safe
  1070   assume L: "?L"
  1071   show ?Rl unfolding set_rel_def Bex_def vimage_eq proof safe
  1072     fix l1 assume "Inl l1 \<in> A1"
  1073     then obtain a2 where a2: "a2 \<in> A2" and "sum_rel \<chi> \<phi> (Inl l1) a2"
  1074     using L unfolding set_rel_def by auto
  1075     then obtain l2 where "a2 = Inl l2 \<and> \<chi> l1 l2" by (cases a2, auto)
  1076     thus "\<exists> l2. Inl l2 \<in> A2 \<and> \<chi> l1 l2" using a2 by auto
  1077   next
  1078     fix l2 assume "Inl l2 \<in> A2"
  1079     then obtain a1 where a1: "a1 \<in> A1" and "sum_rel \<chi> \<phi> a1 (Inl l2)"
  1080     using L unfolding set_rel_def by auto
  1081     then obtain l1 where "a1 = Inl l1 \<and> \<chi> l1 l2" by (cases a1, auto)
  1082     thus "\<exists> l1. Inl l1 \<in> A1 \<and> \<chi> l1 l2" using a1 by auto
  1083   qed
  1084   show ?Rr unfolding set_rel_def Bex_def vimage_eq proof safe
  1085     fix r1 assume "Inr r1 \<in> A1"
  1086     then obtain a2 where a2: "a2 \<in> A2" and "sum_rel \<chi> \<phi> (Inr r1) a2"
  1087     using L unfolding set_rel_def by auto
  1088     then obtain r2 where "a2 = Inr r2 \<and> \<phi> r1 r2" by (cases a2, auto)
  1089     thus "\<exists> r2. Inr r2 \<in> A2 \<and> \<phi> r1 r2" using a2 by auto
  1090   next
  1091     fix r2 assume "Inr r2 \<in> A2"
  1092     then obtain a1 where a1: "a1 \<in> A1" and "sum_rel \<chi> \<phi> a1 (Inr r2)"
  1093     using L unfolding set_rel_def by auto
  1094     then obtain r1 where "a1 = Inr r1 \<and> \<phi> r1 r2" by (cases a1, auto)
  1095     thus "\<exists> r1. Inr r1 \<in> A1 \<and> \<phi> r1 r2" using a1 by auto
  1096   qed
  1097 next
  1098   assume Rl: "?Rl" and Rr: "?Rr"
  1099   show ?L unfolding set_rel_def Bex_def vimage_eq proof safe
  1100     fix a1 assume a1: "a1 \<in> A1"
  1101     show "\<exists> a2. a2 \<in> A2 \<and> sum_rel \<chi> \<phi> a1 a2"
  1102     proof(cases a1)
  1103       case (Inl l1) then obtain l2 where "Inl l2 \<in> A2 \<and> \<chi> l1 l2"
  1104       using Rl a1 unfolding set_rel_def by blast
  1105       thus ?thesis unfolding Inl by auto
  1106     next
  1107       case (Inr r1) then obtain r2 where "Inr r2 \<in> A2 \<and> \<phi> r1 r2"
  1108       using Rr a1 unfolding set_rel_def by blast
  1109       thus ?thesis unfolding Inr by auto
  1110     qed
  1111   next
  1112     fix a2 assume a2: "a2 \<in> A2"
  1113     show "\<exists> a1. a1 \<in> A1 \<and> sum_rel \<chi> \<phi> a1 a2"
  1114     proof(cases a2)
  1115       case (Inl l2) then obtain l1 where "Inl l1 \<in> A1 \<and> \<chi> l1 l2"
  1116       using Rl a2 unfolding set_rel_def by blast
  1117       thus ?thesis unfolding Inl by auto
  1118     next
  1119       case (Inr r2) then obtain r1 where "Inr r1 \<in> A1 \<and> \<phi> r1 r2"
  1120       using Rr a2 unfolding set_rel_def by blast
  1121       thus ?thesis unfolding Inr by auto
  1122     qed
  1123   qed
  1124 qed
  1125 
  1126 end