src/HOL/Library/FSet.thy
 author haftmann Fri Nov 01 18:51:14 2013 +0100 (2013-11-01) changeset 54230 b1d955791529 parent 54014 21dac9a60f0c child 54258 adfc759263ab permissions -rw-r--r--
more simplification rules on unary and binary minus
```     1 (*  Title:      HOL/Library/FSet.thy
```
```     2     Author:     Ondrej Kuncar, TU Muenchen
```
```     3     Author:     Cezary Kaliszyk and Christian Urban
```
```     4 *)
```
```     5
```
```     6 header {* Type of finite sets defined as a subtype of sets *}
```
```     7
```
```     8 theory FSet
```
```     9 imports Main Conditionally_Complete_Lattices
```
```    10 begin
```
```    11
```
```    12 subsection {* Definition of the type *}
```
```    13
```
```    14 typedef 'a fset = "{A :: 'a set. finite A}"  morphisms fset Abs_fset
```
```    15 by auto
```
```    16
```
```    17 setup_lifting type_definition_fset
```
```    18
```
```    19 subsection {* Basic operations and type class instantiations *}
```
```    20
```
```    21 (* FIXME transfer and right_total vs. bi_total *)
```
```    22 instantiation fset :: (finite) finite
```
```    23 begin
```
```    24 instance by default (transfer, simp)
```
```    25 end
```
```    26
```
```    27 instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
```
```    28 begin
```
```    29
```
```    30 interpretation lifting_syntax .
```
```    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   by simp
```
```    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   assumes [transfer_rule]: "bi_unique A"
```
```    41   shows "((pcr_fset A) ===> (pcr_fset A) ===> op =) op \<subset> op <"
```
```    42   unfolding less_fset_def[abs_def] psubset_eq[abs_def] by transfer_prover
```
```    43
```
```    44
```
```    45 lift_definition sup_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is union parametric union_transfer
```
```    46   by simp
```
```    47
```
```    48 lift_definition inf_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is inter parametric inter_transfer
```
```    49   by simp
```
```    50
```
```    51 lift_definition minus_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is minus parametric Diff_transfer
```
```    52   by simp
```
```    53
```
```    54 instance
```
```    55 by default (transfer, auto)+
```
```    56
```
```    57 end
```
```    58
```
```    59 abbreviation fempty :: "'a fset" ("{||}") where "{||} \<equiv> bot"
```
```    60 abbreviation fsubset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50) where "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
```
```    61 abbreviation fsubset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50) where "xs |\<subset>| ys \<equiv> xs < ys"
```
```    62 abbreviation funion :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<union>|" 65) where "xs |\<union>| ys \<equiv> sup xs ys"
```
```    63 abbreviation finter :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<inter>|" 65) where "xs |\<inter>| ys \<equiv> inf xs ys"
```
```    64 abbreviation fminus :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|-|" 65) where "xs |-| ys \<equiv> minus xs ys"
```
```    65
```
```    66 instantiation fset :: (equal) equal
```
```    67 begin
```
```    68 definition "HOL.equal A B \<longleftrightarrow> A |\<subseteq>| B \<and> B |\<subseteq>| A"
```
```    69 instance by intro_classes (auto simp add: equal_fset_def)
```
```    70 end
```
```    71
```
```    72 instantiation fset :: (type) conditionally_complete_lattice
```
```    73 begin
```
```    74
```
```    75 interpretation lifting_syntax .
```
```    76
```
```    77 lemma right_total_Inf_fset_transfer:
```
```    78   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
```
```    79   shows "(set_rel (set_rel A) ===> set_rel A)
```
```    80     (\<lambda>S. if finite (Inter S \<inter> Collect (Domainp A)) then Inter S \<inter> Collect (Domainp A) else {})
```
```    81       (\<lambda>S. if finite (Inf S) then Inf S else {})"
```
```    82     by transfer_prover
```
```    83
```
```    84 lemma Inf_fset_transfer:
```
```    85   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
```
```    86   shows "(set_rel (set_rel A) ===> set_rel A) (\<lambda>A. if finite (Inf A) then Inf A else {})
```
```    87     (\<lambda>A. if finite (Inf A) then Inf A else {})"
```
```    88   by transfer_prover
```
```    89
```
```    90 lift_definition Inf_fset :: "'a fset set \<Rightarrow> 'a fset" is "\<lambda>A. if finite (Inf A) then Inf A else {}"
```
```    91 parametric right_total_Inf_fset_transfer Inf_fset_transfer by simp
```
```    92
```
```    93 lemma Sup_fset_transfer:
```
```    94   assumes [transfer_rule]: "bi_unique A"
```
```    95   shows "(set_rel (set_rel A) ===> set_rel A) (\<lambda>A. if finite (Sup A) then Sup A else {})
```
```    96   (\<lambda>A. if finite (Sup A) then Sup A else {})" by transfer_prover
```
```    97
```
```    98 lift_definition Sup_fset :: "'a fset set \<Rightarrow> 'a fset" is "\<lambda>A. if finite (Sup A) then Sup A else {}"
```
```    99 parametric Sup_fset_transfer by simp
```
```   100
```
```   101 lemma finite_Sup: "\<exists>z. finite z \<and> (\<forall>a. a \<in> X \<longrightarrow> a \<le> z) \<Longrightarrow> finite (Sup X)"
```
```   102 by (auto intro: finite_subset)
```
```   103
```
```   104 instance
```
```   105 proof
```
```   106   fix x z :: "'a fset"
```
```   107   fix X :: "'a fset set"
```
```   108   {
```
```   109     assume "x \<in> X" "(\<And>a. a \<in> X \<Longrightarrow> z |\<subseteq>| a)"
```
```   110     then show "Inf X |\<subseteq>| x"  by transfer auto
```
```   111   next
```
```   112     assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> z |\<subseteq>| x)"
```
```   113     then show "z |\<subseteq>| Inf X" by transfer (clarsimp, blast)
```
```   114   next
```
```   115     assume "x \<in> X" "(\<And>a. a \<in> X \<Longrightarrow> a |\<subseteq>| z)"
```
```   116     then show "x |\<subseteq>| Sup X" by transfer (auto intro!: finite_Sup)
```
```   117   next
```
```   118     assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)"
```
```   119     then show "Sup X |\<subseteq>| z" by transfer (clarsimp, blast)
```
```   120   }
```
```   121 qed
```
```   122 end
```
```   123
```
```   124 instantiation fset :: (finite) complete_lattice
```
```   125 begin
```
```   126
```
```   127 lift_definition top_fset :: "'a fset" is UNIV parametric right_total_UNIV_transfer UNIV_transfer by simp
```
```   128
```
```   129 instance by default (transfer, auto)+
```
```   130 end
```
```   131
```
```   132 instantiation fset :: (finite) complete_boolean_algebra
```
```   133 begin
```
```   134
```
```   135 lift_definition uminus_fset :: "'a fset \<Rightarrow> 'a fset" is uminus
```
```   136   parametric right_total_Compl_transfer Compl_transfer by simp
```
```   137
```
```   138 instance by (default, simp_all only: INF_def SUP_def) (transfer, auto)+
```
```   139
```
```   140 end
```
```   141
```
```   142 abbreviation fUNIV :: "'a::finite fset" where "fUNIV \<equiv> top"
```
```   143 abbreviation fuminus :: "'a::finite fset \<Rightarrow> 'a fset" ("|-| _" [81] 80) where "|-| x \<equiv> uminus x"
```
```   144
```
```   145 subsection {* Other operations *}
```
```   146
```
```   147 lift_definition finsert :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is insert parametric Lifting_Set.insert_transfer
```
```   148   by simp
```
```   149
```
```   150 syntax
```
```   151   "_insert_fset"     :: "args => 'a fset"  ("{|(_)|}")
```
```   152
```
```   153 translations
```
```   154   "{|x, xs|}" == "CONST finsert x {|xs|}"
```
```   155   "{|x|}"     == "CONST finsert x {||}"
```
```   156
```
```   157 lift_definition fmember :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<in>|" 50) is Set.member
```
```   158   parametric member_transfer by simp
```
```   159
```
```   160 abbreviation notin_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50) where "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
```
```   161
```
```   162 context
```
```   163 begin
```
```   164 interpretation lifting_syntax .
```
```   165
```
```   166 lift_definition ffilter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is Set.filter
```
```   167   parametric Lifting_Set.filter_transfer unfolding Set.filter_def by simp
```
```   168
```
```   169 lemma compose_rel_to_Domainp:
```
```   170   assumes "left_unique R"
```
```   171   assumes "(R ===> op=) P P'"
```
```   172   shows "(R OO Lifting.invariant P' OO R\<inverse>\<inverse>) x y \<longleftrightarrow> Domainp R x \<and> P x \<and> x = y"
```
```   173 using assms unfolding OO_def conversep_iff Domainp_iff left_unique_def fun_rel_def invariant_def
```
```   174 by blast
```
```   175
```
```   176 lift_definition fPow :: "'a fset \<Rightarrow> 'a fset fset" is Pow parametric Pow_transfer
```
```   177 by (subst compose_rel_to_Domainp [OF _ finite_transfer]) (auto intro: transfer_raw finite_subset
```
```   178   simp add: fset.pcr_cr_eq[symmetric] Domainp_set fset.domain_eq)
```
```   179
```
```   180 lift_definition fcard :: "'a fset \<Rightarrow> nat" is card parametric card_transfer by simp
```
```   181
```
```   182 lift_definition fimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset" (infixr "|`|" 90) is image
```
```   183   parametric image_transfer by simp
```
```   184
```
```   185 lift_definition fthe_elem :: "'a fset \<Rightarrow> 'a" is the_elem ..
```
```   186
```
```   187 (* FIXME why is not invariant here unfolded ? *)
```
```   188 lift_definition fbind :: "'a fset \<Rightarrow> ('a \<Rightarrow> 'b fset) \<Rightarrow> 'b fset" is Set.bind parametric bind_transfer
```
```   189 unfolding invariant_def Set.bind_def by clarsimp metis
```
```   190
```
```   191 lift_definition ffUnion :: "'a fset fset \<Rightarrow> 'a fset" is Union parametric Union_transfer
```
```   192 by (subst(asm) compose_rel_to_Domainp [OF _ finite_transfer])
```
```   193   (auto intro: transfer_raw simp add: fset.pcr_cr_eq[symmetric] Domainp_set fset.domain_eq invariant_def)
```
```   194
```
```   195 lift_definition fBall :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" is Ball parametric Ball_transfer ..
```
```   196 lift_definition fBex :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" is Bex parametric Bex_transfer ..
```
```   197
```
```   198 lift_definition ffold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" is Finite_Set.fold ..
```
```   199
```
```   200 subsection {* Transferred lemmas from Set.thy *}
```
```   201
```
```   202 lemmas fset_eqI = set_eqI[Transfer.transferred]
```
```   203 lemmas fset_eq_iff[no_atp] = set_eq_iff[Transfer.transferred]
```
```   204 lemmas fBallI[intro!] = ballI[Transfer.transferred]
```
```   205 lemmas fbspec[dest?] = bspec[Transfer.transferred]
```
```   206 lemmas fBallE[elim] = ballE[Transfer.transferred]
```
```   207 lemmas fBexI[intro] = bexI[Transfer.transferred]
```
```   208 lemmas rev_fBexI[intro?] = rev_bexI[Transfer.transferred]
```
```   209 lemmas fBexCI = bexCI[Transfer.transferred]
```
```   210 lemmas fBexE[elim!] = bexE[Transfer.transferred]
```
```   211 lemmas fBall_triv[simp] = ball_triv[Transfer.transferred]
```
```   212 lemmas fBex_triv[simp] = bex_triv[Transfer.transferred]
```
```   213 lemmas fBex_triv_one_point1[simp] = bex_triv_one_point1[Transfer.transferred]
```
```   214 lemmas fBex_triv_one_point2[simp] = bex_triv_one_point2[Transfer.transferred]
```
```   215 lemmas fBex_one_point1[simp] = bex_one_point1[Transfer.transferred]
```
```   216 lemmas fBex_one_point2[simp] = bex_one_point2[Transfer.transferred]
```
```   217 lemmas fBall_one_point1[simp] = ball_one_point1[Transfer.transferred]
```
```   218 lemmas fBall_one_point2[simp] = ball_one_point2[Transfer.transferred]
```
```   219 lemmas fBall_conj_distrib = ball_conj_distrib[Transfer.transferred]
```
```   220 lemmas fBex_disj_distrib = bex_disj_distrib[Transfer.transferred]
```
```   221 lemmas fBall_cong = ball_cong[Transfer.transferred]
```
```   222 lemmas fBex_cong = bex_cong[Transfer.transferred]
```
```   223 lemmas fsubsetI[intro!] = subsetI[Transfer.transferred]
```
```   224 lemmas fsubsetD[elim, intro?] = subsetD[Transfer.transferred]
```
```   225 lemmas rev_fsubsetD[no_atp,intro?] = rev_subsetD[Transfer.transferred]
```
```   226 lemmas fsubsetCE[no_atp,elim] = subsetCE[Transfer.transferred]
```
```   227 lemmas fsubset_eq[no_atp] = subset_eq[Transfer.transferred]
```
```   228 lemmas contra_fsubsetD[no_atp] = contra_subsetD[Transfer.transferred]
```
```   229 lemmas fsubset_refl = subset_refl[Transfer.transferred]
```
```   230 lemmas fsubset_trans = subset_trans[Transfer.transferred]
```
```   231 lemmas fset_rev_mp = set_rev_mp[Transfer.transferred]
```
```   232 lemmas fset_mp = set_mp[Transfer.transferred]
```
```   233 lemmas fsubset_not_fsubset_eq[code] = subset_not_subset_eq[Transfer.transferred]
```
```   234 lemmas eq_fmem_trans = eq_mem_trans[Transfer.transferred]
```
```   235 lemmas fsubset_antisym[intro!] = subset_antisym[Transfer.transferred]
```
```   236 lemmas fequalityD1 = equalityD1[Transfer.transferred]
```
```   237 lemmas fequalityD2 = equalityD2[Transfer.transferred]
```
```   238 lemmas fequalityE = equalityE[Transfer.transferred]
```
```   239 lemmas fequalityCE[elim] = equalityCE[Transfer.transferred]
```
```   240 lemmas eqfset_imp_iff = eqset_imp_iff[Transfer.transferred]
```
```   241 lemmas eqfelem_imp_iff = eqelem_imp_iff[Transfer.transferred]
```
```   242 lemmas fempty_iff[simp] = empty_iff[Transfer.transferred]
```
```   243 lemmas fempty_fsubsetI[iff] = empty_subsetI[Transfer.transferred]
```
```   244 lemmas equalsffemptyI = equals0I[Transfer.transferred]
```
```   245 lemmas equalsffemptyD = equals0D[Transfer.transferred]
```
```   246 lemmas fBall_fempty[simp] = ball_empty[Transfer.transferred]
```
```   247 lemmas fBex_fempty[simp] = bex_empty[Transfer.transferred]
```
```   248 lemmas fPow_iff[iff] = Pow_iff[Transfer.transferred]
```
```   249 lemmas fPowI = PowI[Transfer.transferred]
```
```   250 lemmas fPowD = PowD[Transfer.transferred]
```
```   251 lemmas fPow_bottom = Pow_bottom[Transfer.transferred]
```
```   252 lemmas fPow_top = Pow_top[Transfer.transferred]
```
```   253 lemmas fPow_not_fempty = Pow_not_empty[Transfer.transferred]
```
```   254 lemmas finter_iff[simp] = Int_iff[Transfer.transferred]
```
```   255 lemmas finterI[intro!] = IntI[Transfer.transferred]
```
```   256 lemmas finterD1 = IntD1[Transfer.transferred]
```
```   257 lemmas finterD2 = IntD2[Transfer.transferred]
```
```   258 lemmas finterE[elim!] = IntE[Transfer.transferred]
```
```   259 lemmas funion_iff[simp] = Un_iff[Transfer.transferred]
```
```   260 lemmas funionI1[elim?] = UnI1[Transfer.transferred]
```
```   261 lemmas funionI2[elim?] = UnI2[Transfer.transferred]
```
```   262 lemmas funionCI[intro!] = UnCI[Transfer.transferred]
```
```   263 lemmas funionE[elim!] = UnE[Transfer.transferred]
```
```   264 lemmas fminus_iff[simp] = Diff_iff[Transfer.transferred]
```
```   265 lemmas fminusI[intro!] = DiffI[Transfer.transferred]
```
```   266 lemmas fminusD1 = DiffD1[Transfer.transferred]
```
```   267 lemmas fminusD2 = DiffD2[Transfer.transferred]
```
```   268 lemmas fminusE[elim!] = DiffE[Transfer.transferred]
```
```   269 lemmas finsert_iff[simp] = insert_iff[Transfer.transferred]
```
```   270 lemmas finsertI1 = insertI1[Transfer.transferred]
```
```   271 lemmas finsertI2 = insertI2[Transfer.transferred]
```
```   272 lemmas finsertE[elim!] = insertE[Transfer.transferred]
```
```   273 lemmas finsertCI[intro!] = insertCI[Transfer.transferred]
```
```   274 lemmas fsubset_finsert_iff = subset_insert_iff[Transfer.transferred]
```
```   275 lemmas finsert_ident = insert_ident[Transfer.transferred]
```
```   276 lemmas fsingletonI[intro!,no_atp] = singletonI[Transfer.transferred]
```
```   277 lemmas fsingletonD[dest!,no_atp] = singletonD[Transfer.transferred]
```
```   278 lemmas fsingleton_iff = singleton_iff[Transfer.transferred]
```
```   279 lemmas fsingleton_inject[dest!] = singleton_inject[Transfer.transferred]
```
```   280 lemmas fsingleton_finsert_inj_eq[iff,no_atp] = singleton_insert_inj_eq[Transfer.transferred]
```
```   281 lemmas fsingleton_finsert_inj_eq'[iff,no_atp] = singleton_insert_inj_eq'[Transfer.transferred]
```
```   282 lemmas fsubset_fsingletonD = subset_singletonD[Transfer.transferred]
```
```   283 lemmas fminus_single_finsert = diff_single_insert[Transfer.transferred]
```
```   284 lemmas fdoubleton_eq_iff = doubleton_eq_iff[Transfer.transferred]
```
```   285 lemmas funion_fsingleton_iff = Un_singleton_iff[Transfer.transferred]
```
```   286 lemmas fsingleton_funion_iff = singleton_Un_iff[Transfer.transferred]
```
```   287 lemmas fimage_eqI[simp, intro] = image_eqI[Transfer.transferred]
```
```   288 lemmas fimageI = imageI[Transfer.transferred]
```
```   289 lemmas rev_fimage_eqI = rev_image_eqI[Transfer.transferred]
```
```   290 lemmas fimageE[elim!] = imageE[Transfer.transferred]
```
```   291 lemmas Compr_fimage_eq = Compr_image_eq[Transfer.transferred]
```
```   292 lemmas fimage_funion = image_Un[Transfer.transferred]
```
```   293 lemmas fimage_iff = image_iff[Transfer.transferred]
```
```   294 lemmas fimage_fsubset_iff[no_atp] = image_subset_iff[Transfer.transferred]
```
```   295 lemmas fimage_fsubsetI = image_subsetI[Transfer.transferred]
```
```   296 lemmas fimage_ident[simp] = image_ident[Transfer.transferred]
```
```   297 lemmas split_if_fmem1 = split_if_mem1[Transfer.transferred]
```
```   298 lemmas split_if_fmem2 = split_if_mem2[Transfer.transferred]
```
```   299 lemmas pfsubsetI[intro!,no_atp] = psubsetI[Transfer.transferred]
```
```   300 lemmas pfsubsetE[elim!,no_atp] = psubsetE[Transfer.transferred]
```
```   301 lemmas pfsubset_finsert_iff = psubset_insert_iff[Transfer.transferred]
```
```   302 lemmas pfsubset_eq = psubset_eq[Transfer.transferred]
```
```   303 lemmas pfsubset_imp_fsubset = psubset_imp_subset[Transfer.transferred]
```
```   304 lemmas pfsubset_trans = psubset_trans[Transfer.transferred]
```
```   305 lemmas pfsubsetD = psubsetD[Transfer.transferred]
```
```   306 lemmas pfsubset_fsubset_trans = psubset_subset_trans[Transfer.transferred]
```
```   307 lemmas fsubset_pfsubset_trans = subset_psubset_trans[Transfer.transferred]
```
```   308 lemmas pfsubset_imp_ex_fmem = psubset_imp_ex_mem[Transfer.transferred]
```
```   309 lemmas fimage_fPow_mono = image_Pow_mono[Transfer.transferred]
```
```   310 lemmas fimage_fPow_surj = image_Pow_surj[Transfer.transferred]
```
```   311 lemmas fsubset_finsertI = subset_insertI[Transfer.transferred]
```
```   312 lemmas fsubset_finsertI2 = subset_insertI2[Transfer.transferred]
```
```   313 lemmas fsubset_finsert = subset_insert[Transfer.transferred]
```
```   314 lemmas funion_upper1 = Un_upper1[Transfer.transferred]
```
```   315 lemmas funion_upper2 = Un_upper2[Transfer.transferred]
```
```   316 lemmas funion_least = Un_least[Transfer.transferred]
```
```   317 lemmas finter_lower1 = Int_lower1[Transfer.transferred]
```
```   318 lemmas finter_lower2 = Int_lower2[Transfer.transferred]
```
```   319 lemmas finter_greatest = Int_greatest[Transfer.transferred]
```
```   320 lemmas fminus_fsubset = Diff_subset[Transfer.transferred]
```
```   321 lemmas fminus_fsubset_conv = Diff_subset_conv[Transfer.transferred]
```
```   322 lemmas fsubset_fempty[simp] = subset_empty[Transfer.transferred]
```
```   323 lemmas not_pfsubset_fempty[iff] = not_psubset_empty[Transfer.transferred]
```
```   324 lemmas finsert_is_funion = insert_is_Un[Transfer.transferred]
```
```   325 lemmas finsert_not_fempty[simp] = insert_not_empty[Transfer.transferred]
```
```   326 lemmas fempty_not_finsert = empty_not_insert[Transfer.transferred]
```
```   327 lemmas finsert_absorb = insert_absorb[Transfer.transferred]
```
```   328 lemmas finsert_absorb2[simp] = insert_absorb2[Transfer.transferred]
```
```   329 lemmas finsert_commute = insert_commute[Transfer.transferred]
```
```   330 lemmas finsert_fsubset[simp] = insert_subset[Transfer.transferred]
```
```   331 lemmas finsert_inter_finsert[simp] = insert_inter_insert[Transfer.transferred]
```
```   332 lemmas finsert_disjoint[simp,no_atp] = insert_disjoint[Transfer.transferred]
```
```   333 lemmas disjoint_finsert[simp,no_atp] = disjoint_insert[Transfer.transferred]
```
```   334 lemmas fimage_fempty[simp] = image_empty[Transfer.transferred]
```
```   335 lemmas fimage_finsert[simp] = image_insert[Transfer.transferred]
```
```   336 lemmas fimage_constant = image_constant[Transfer.transferred]
```
```   337 lemmas fimage_constant_conv = image_constant_conv[Transfer.transferred]
```
```   338 lemmas fimage_fimage = image_image[Transfer.transferred]
```
```   339 lemmas finsert_fimage[simp] = insert_image[Transfer.transferred]
```
```   340 lemmas fimage_is_fempty[iff] = image_is_empty[Transfer.transferred]
```
```   341 lemmas fempty_is_fimage[iff] = empty_is_image[Transfer.transferred]
```
```   342 lemmas fimage_cong = image_cong[Transfer.transferred]
```
```   343 lemmas fimage_finter_fsubset = image_Int_subset[Transfer.transferred]
```
```   344 lemmas fimage_fminus_fsubset = image_diff_subset[Transfer.transferred]
```
```   345 lemmas finter_absorb = Int_absorb[Transfer.transferred]
```
```   346 lemmas finter_left_absorb = Int_left_absorb[Transfer.transferred]
```
```   347 lemmas finter_commute = Int_commute[Transfer.transferred]
```
```   348 lemmas finter_left_commute = Int_left_commute[Transfer.transferred]
```
```   349 lemmas finter_assoc = Int_assoc[Transfer.transferred]
```
```   350 lemmas finter_ac = Int_ac[Transfer.transferred]
```
```   351 lemmas finter_absorb1 = Int_absorb1[Transfer.transferred]
```
```   352 lemmas finter_absorb2 = Int_absorb2[Transfer.transferred]
```
```   353 lemmas finter_fempty_left = Int_empty_left[Transfer.transferred]
```
```   354 lemmas finter_fempty_right = Int_empty_right[Transfer.transferred]
```
```   355 lemmas disjoint_iff_fnot_equal = disjoint_iff_not_equal[Transfer.transferred]
```
```   356 lemmas finter_funion_distrib = Int_Un_distrib[Transfer.transferred]
```
```   357 lemmas finter_funion_distrib2 = Int_Un_distrib2[Transfer.transferred]
```
```   358 lemmas finter_fsubset_iff[no_atp, simp] = Int_subset_iff[Transfer.transferred]
```
```   359 lemmas funion_absorb = Un_absorb[Transfer.transferred]
```
```   360 lemmas funion_left_absorb = Un_left_absorb[Transfer.transferred]
```
```   361 lemmas funion_commute = Un_commute[Transfer.transferred]
```
```   362 lemmas funion_left_commute = Un_left_commute[Transfer.transferred]
```
```   363 lemmas funion_assoc = Un_assoc[Transfer.transferred]
```
```   364 lemmas funion_ac = Un_ac[Transfer.transferred]
```
```   365 lemmas funion_absorb1 = Un_absorb1[Transfer.transferred]
```
```   366 lemmas funion_absorb2 = Un_absorb2[Transfer.transferred]
```
```   367 lemmas funion_fempty_left = Un_empty_left[Transfer.transferred]
```
```   368 lemmas funion_fempty_right = Un_empty_right[Transfer.transferred]
```
```   369 lemmas funion_finsert_left[simp] = Un_insert_left[Transfer.transferred]
```
```   370 lemmas funion_finsert_right[simp] = Un_insert_right[Transfer.transferred]
```
```   371 lemmas finter_finsert_left = Int_insert_left[Transfer.transferred]
```
```   372 lemmas finter_finsert_left_ifffempty[simp] = Int_insert_left_if0[Transfer.transferred]
```
```   373 lemmas finter_finsert_left_if1[simp] = Int_insert_left_if1[Transfer.transferred]
```
```   374 lemmas finter_finsert_right = Int_insert_right[Transfer.transferred]
```
```   375 lemmas finter_finsert_right_ifffempty[simp] = Int_insert_right_if0[Transfer.transferred]
```
```   376 lemmas finter_finsert_right_if1[simp] = Int_insert_right_if1[Transfer.transferred]
```
```   377 lemmas funion_finter_distrib = Un_Int_distrib[Transfer.transferred]
```
```   378 lemmas funion_finter_distrib2 = Un_Int_distrib2[Transfer.transferred]
```
```   379 lemmas funion_finter_crazy = Un_Int_crazy[Transfer.transferred]
```
```   380 lemmas fsubset_funion_eq = subset_Un_eq[Transfer.transferred]
```
```   381 lemmas funion_fempty[iff] = Un_empty[Transfer.transferred]
```
```   382 lemmas funion_fsubset_iff[no_atp, simp] = Un_subset_iff[Transfer.transferred]
```
```   383 lemmas funion_fminus_finter = Un_Diff_Int[Transfer.transferred]
```
```   384 lemmas fminus_finter2 = Diff_Int2[Transfer.transferred]
```
```   385 lemmas funion_finter_assoc_eq = Un_Int_assoc_eq[Transfer.transferred]
```
```   386 lemmas fBall_funion = ball_Un[Transfer.transferred]
```
```   387 lemmas fBex_funion = bex_Un[Transfer.transferred]
```
```   388 lemmas fminus_eq_fempty_iff[simp,no_atp] = Diff_eq_empty_iff[Transfer.transferred]
```
```   389 lemmas fminus_cancel[simp] = Diff_cancel[Transfer.transferred]
```
```   390 lemmas fminus_idemp[simp] = Diff_idemp[Transfer.transferred]
```
```   391 lemmas fminus_triv = Diff_triv[Transfer.transferred]
```
```   392 lemmas fempty_fminus[simp] = empty_Diff[Transfer.transferred]
```
```   393 lemmas fminus_fempty[simp] = Diff_empty[Transfer.transferred]
```
```   394 lemmas fminus_finsertffempty[simp,no_atp] = Diff_insert0[Transfer.transferred]
```
```   395 lemmas fminus_finsert = Diff_insert[Transfer.transferred]
```
```   396 lemmas fminus_finsert2 = Diff_insert2[Transfer.transferred]
```
```   397 lemmas finsert_fminus_if = insert_Diff_if[Transfer.transferred]
```
```   398 lemmas finsert_fminus1[simp] = insert_Diff1[Transfer.transferred]
```
```   399 lemmas finsert_fminus_single[simp] = insert_Diff_single[Transfer.transferred]
```
```   400 lemmas finsert_fminus = insert_Diff[Transfer.transferred]
```
```   401 lemmas fminus_finsert_absorb = Diff_insert_absorb[Transfer.transferred]
```
```   402 lemmas fminus_disjoint[simp] = Diff_disjoint[Transfer.transferred]
```
```   403 lemmas fminus_partition = Diff_partition[Transfer.transferred]
```
```   404 lemmas double_fminus = double_diff[Transfer.transferred]
```
```   405 lemmas funion_fminus_cancel[simp] = Un_Diff_cancel[Transfer.transferred]
```
```   406 lemmas funion_fminus_cancel2[simp] = Un_Diff_cancel2[Transfer.transferred]
```
```   407 lemmas fminus_funion = Diff_Un[Transfer.transferred]
```
```   408 lemmas fminus_finter = Diff_Int[Transfer.transferred]
```
```   409 lemmas funion_fminus = Un_Diff[Transfer.transferred]
```
```   410 lemmas finter_fminus = Int_Diff[Transfer.transferred]
```
```   411 lemmas fminus_finter_distrib = Diff_Int_distrib[Transfer.transferred]
```
```   412 lemmas fminus_finter_distrib2 = Diff_Int_distrib2[Transfer.transferred]
```
```   413 lemmas fUNIV_bool[no_atp] = UNIV_bool[Transfer.transferred]
```
```   414 lemmas fPow_fempty[simp] = Pow_empty[Transfer.transferred]
```
```   415 lemmas fPow_finsert = Pow_insert[Transfer.transferred]
```
```   416 lemmas funion_fPow_fsubset = Un_Pow_subset[Transfer.transferred]
```
```   417 lemmas fPow_finter_eq[simp] = Pow_Int_eq[Transfer.transferred]
```
```   418 lemmas fset_eq_fsubset = set_eq_subset[Transfer.transferred]
```
```   419 lemmas fsubset_iff[no_atp] = subset_iff[Transfer.transferred]
```
```   420 lemmas fsubset_iff_pfsubset_eq = subset_iff_psubset_eq[Transfer.transferred]
```
```   421 lemmas all_not_fin_conv[simp] = all_not_in_conv[Transfer.transferred]
```
```   422 lemmas ex_fin_conv = ex_in_conv[Transfer.transferred]
```
```   423 lemmas fimage_mono = image_mono[Transfer.transferred]
```
```   424 lemmas fPow_mono = Pow_mono[Transfer.transferred]
```
```   425 lemmas finsert_mono = insert_mono[Transfer.transferred]
```
```   426 lemmas funion_mono = Un_mono[Transfer.transferred]
```
```   427 lemmas finter_mono = Int_mono[Transfer.transferred]
```
```   428 lemmas fminus_mono = Diff_mono[Transfer.transferred]
```
```   429 lemmas fin_mono = in_mono[Transfer.transferred]
```
```   430 lemmas fthe_felem_eq[simp] = the_elem_eq[Transfer.transferred]
```
```   431 lemmas fLeast_mono = Least_mono[Transfer.transferred]
```
```   432 lemmas fbind_fbind = bind_bind[Transfer.transferred]
```
```   433 lemmas fempty_fbind[simp] = empty_bind[Transfer.transferred]
```
```   434 lemmas nonfempty_fbind_const = nonempty_bind_const[Transfer.transferred]
```
```   435 lemmas fbind_const = bind_const[Transfer.transferred]
```
```   436 lemmas ffmember_filter[simp] = member_filter[Transfer.transferred]
```
```   437 lemmas fequalityI = equalityI[Transfer.transferred]
```
```   438
```
```   439 subsection {* Additional lemmas*}
```
```   440
```
```   441 subsubsection {* @{text fsingleton} *}
```
```   442
```
```   443 lemmas fsingletonE = fsingletonD [elim_format]
```
```   444
```
```   445 subsubsection {* @{text femepty} *}
```
```   446
```
```   447 lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {||}"
```
```   448 by transfer auto
```
```   449
```
```   450 (* FIXME, transferred doesn't work here *)
```
```   451 lemma femptyE [elim!]: "a |\<in>| {||} \<Longrightarrow> P"
```
```   452   by simp
```
```   453
```
```   454 subsubsection {* @{text fset} *}
```
```   455
```
```   456 lemmas fset_simps[simp] = bot_fset.rep_eq finsert.rep_eq
```
```   457
```
```   458 lemma finite_fset [simp]:
```
```   459   shows "finite (fset S)"
```
```   460   by transfer simp
```
```   461
```
```   462 lemmas fset_cong = fset_inject
```
```   463
```
```   464 lemma filter_fset [simp]:
```
```   465   shows "fset (ffilter P xs) = Collect P \<inter> fset xs"
```
```   466   by transfer auto
```
```   467
```
```   468 lemma notin_fset: "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S" by (simp add: fmember.rep_eq)
```
```   469
```
```   470 lemmas inter_fset[simp] = inf_fset.rep_eq
```
```   471
```
```   472 lemmas union_fset[simp] = sup_fset.rep_eq
```
```   473
```
```   474 lemmas minus_fset[simp] = minus_fset.rep_eq
```
```   475
```
```   476 subsubsection {* @{text filter_fset} *}
```
```   477
```
```   478 lemma subset_ffilter:
```
```   479   "ffilter P A |\<subseteq>| ffilter Q A = (\<forall> x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)"
```
```   480   by transfer auto
```
```   481
```
```   482 lemma eq_ffilter:
```
```   483   "(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)"
```
```   484   by transfer auto
```
```   485
```
```   486 lemma pfsubset_ffilter:
```
```   487   "(\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| A & \<not> P x & Q x) \<Longrightarrow>
```
```   488     ffilter P A |\<subset>| ffilter Q A"
```
```   489   unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter)
```
```   490
```
```   491 subsubsection {* @{text finsert} *}
```
```   492
```
```   493 (* FIXME, transferred doesn't work here *)
```
```   494 lemma set_finsert:
```
```   495   assumes "x |\<in>| A"
```
```   496   obtains B where "A = finsert x B" and "x |\<notin>| B"
```
```   497 using assms by transfer (metis Set.set_insert finite_insert)
```
```   498
```
```   499 lemma mk_disjoint_finsert: "a |\<in>| A \<Longrightarrow> \<exists>B. A = finsert a B \<and> a |\<notin>| B"
```
```   500   by (rule_tac x = "A |-| {|a|}" in exI, blast)
```
```   501
```
```   502 subsubsection {* @{text fimage} *}
```
```   503
```
```   504 lemma subset_fimage_iff: "(B |\<subseteq>| f|`|A) = (\<exists> AA. AA |\<subseteq>| A \<and> B = f|`|AA)"
```
```   505 by transfer (metis mem_Collect_eq rev_finite_subset subset_image_iff)
```
```   506
```
```   507 subsubsection {* bounded quantification *}
```
```   508
```
```   509 lemma bex_simps [simp, no_atp]:
```
```   510   "\<And>A P Q. fBex A (\<lambda>x. P x \<and> Q) = (fBex A P \<and> Q)"
```
```   511   "\<And>A P Q. fBex A (\<lambda>x. P \<and> Q x) = (P \<and> fBex A Q)"
```
```   512   "\<And>P. fBex {||} P = False"
```
```   513   "\<And>a B P. fBex (finsert a B) P = (P a \<or> fBex B P)"
```
```   514   "\<And>A P f. fBex (f |`| A) P = fBex A (\<lambda>x. P (f x))"
```
```   515   "\<And>A P. (\<not> fBex A P) = fBall A (\<lambda>x. \<not> P x)"
```
```   516 by auto
```
```   517
```
```   518 lemma ball_simps [simp, no_atp]:
```
```   519   "\<And>A P Q. fBall A (\<lambda>x. P x \<or> Q) = (fBall A P \<or> Q)"
```
```   520   "\<And>A P Q. fBall A (\<lambda>x. P \<or> Q x) = (P \<or> fBall A Q)"
```
```   521   "\<And>A P Q. fBall A (\<lambda>x. P \<longrightarrow> Q x) = (P \<longrightarrow> fBall A Q)"
```
```   522   "\<And>A P Q. fBall A (\<lambda>x. P x \<longrightarrow> Q) = (fBex A P \<longrightarrow> Q)"
```
```   523   "\<And>P. fBall {||} P = True"
```
```   524   "\<And>a B P. fBall (finsert a B) P = (P a \<and> fBall B P)"
```
```   525   "\<And>A P f. fBall (f |`| A) P = fBall A (\<lambda>x. P (f x))"
```
```   526   "\<And>A P. (\<not> fBall A P) = fBex A (\<lambda>x. \<not> P x)"
```
```   527 by auto
```
```   528
```
```   529 lemma atomize_fBall:
```
```   530     "(\<And>x. x |\<in>| A ==> P x) == Trueprop (fBall A (\<lambda>x. P x))"
```
```   531 apply (simp only: atomize_all atomize_imp)
```
```   532 apply (rule equal_intr_rule)
```
```   533 by (transfer, simp)+
```
```   534
```
```   535 end
```
```   536
```
```   537 subsubsection {* @{text fcard} *}
```
```   538
```
```   539 (* FIXME: improve transferred to handle bounded meta quantification *)
```
```   540
```
```   541 lemma fcard_fempty:
```
```   542   "fcard {||} = 0"
```
```   543   by transfer (rule card_empty)
```
```   544
```
```   545 lemma fcard_finsert_disjoint:
```
```   546   "x |\<notin>| A \<Longrightarrow> fcard (finsert x A) = Suc (fcard A)"
```
```   547   by transfer (rule card_insert_disjoint)
```
```   548
```
```   549 lemma fcard_finsert_if:
```
```   550   "fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))"
```
```   551   by transfer (rule card_insert_if)
```
```   552
```
```   553 lemma card_0_eq [simp, no_atp]:
```
```   554   "fcard A = 0 \<longleftrightarrow> A = {||}"
```
```   555   by transfer (rule card_0_eq)
```
```   556
```
```   557 lemma fcard_Suc_fminus1:
```
```   558   "x |\<in>| A \<Longrightarrow> Suc (fcard (A |-| {|x|})) = fcard A"
```
```   559   by transfer (rule card_Suc_Diff1)
```
```   560
```
```   561 lemma fcard_fminus_fsingleton:
```
```   562   "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) = fcard A - 1"
```
```   563   by transfer (rule card_Diff_singleton)
```
```   564
```
```   565 lemma fcard_fminus_fsingleton_if:
```
```   566   "fcard (A |-| {|x|}) = (if x |\<in>| A then fcard A - 1 else fcard A)"
```
```   567   by transfer (rule card_Diff_singleton_if)
```
```   568
```
```   569 lemma fcard_fminus_finsert[simp]:
```
```   570   assumes "a |\<in>| A" and "a |\<notin>| B"
```
```   571   shows "fcard (A |-| finsert a B) = fcard (A |-| B) - 1"
```
```   572 using assms by transfer (rule card_Diff_insert)
```
```   573
```
```   574 lemma fcard_finsert: "fcard (finsert x A) = Suc (fcard (A |-| {|x|}))"
```
```   575 by transfer (rule card_insert)
```
```   576
```
```   577 lemma fcard_finsert_le: "fcard A \<le> fcard (finsert x A)"
```
```   578 by transfer (rule card_insert_le)
```
```   579
```
```   580 lemma fcard_mono:
```
```   581   "A |\<subseteq>| B \<Longrightarrow> fcard A \<le> fcard B"
```
```   582 by transfer (rule card_mono)
```
```   583
```
```   584 lemma fcard_seteq: "A |\<subseteq>| B \<Longrightarrow> fcard B \<le> fcard A \<Longrightarrow> A = B"
```
```   585 by transfer (rule card_seteq)
```
```   586
```
```   587 lemma pfsubset_fcard_mono: "A |\<subset>| B \<Longrightarrow> fcard A < fcard B"
```
```   588 by transfer (rule psubset_card_mono)
```
```   589
```
```   590 lemma fcard_funion_finter:
```
```   591   "fcard A + fcard B = fcard (A |\<union>| B) + fcard (A |\<inter>| B)"
```
```   592 by transfer (rule card_Un_Int)
```
```   593
```
```   594 lemma fcard_funion_disjoint:
```
```   595   "A |\<inter>| B = {||} \<Longrightarrow> fcard (A |\<union>| B) = fcard A + fcard B"
```
```   596 by transfer (rule card_Un_disjoint)
```
```   597
```
```   598 lemma fcard_funion_fsubset:
```
```   599   "B |\<subseteq>| A \<Longrightarrow> fcard (A |-| B) = fcard A - fcard B"
```
```   600 by transfer (rule card_Diff_subset)
```
```   601
```
```   602 lemma diff_fcard_le_fcard_fminus:
```
```   603   "fcard A - fcard B \<le> fcard(A |-| B)"
```
```   604 by transfer (rule diff_card_le_card_Diff)
```
```   605
```
```   606 lemma fcard_fminus1_less: "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) < fcard A"
```
```   607 by transfer (rule card_Diff1_less)
```
```   608
```
```   609 lemma fcard_fminus2_less:
```
```   610   "x |\<in>| A \<Longrightarrow> y |\<in>| A \<Longrightarrow> fcard (A |-| {|x|} |-| {|y|}) < fcard A"
```
```   611 by transfer (rule card_Diff2_less)
```
```   612
```
```   613 lemma fcard_fminus1_le: "fcard (A |-| {|x|}) \<le> fcard A"
```
```   614 by transfer (rule card_Diff1_le)
```
```   615
```
```   616 lemma fcard_pfsubset: "A |\<subseteq>| B \<Longrightarrow> fcard A < fcard B \<Longrightarrow> A < B"
```
```   617 by transfer (rule card_psubset)
```
```   618
```
```   619 subsubsection {* @{text ffold} *}
```
```   620
```
```   621 (* FIXME: improve transferred to handle bounded meta quantification *)
```
```   622
```
```   623 context comp_fun_commute
```
```   624 begin
```
```   625   lemmas ffold_empty[simp] = fold_empty[Transfer.transferred]
```
```   626
```
```   627   lemma ffold_finsert [simp]:
```
```   628     assumes "x |\<notin>| A"
```
```   629     shows "ffold f z (finsert x A) = f x (ffold f z A)"
```
```   630     using assms by (transfer fixing: f) (rule fold_insert)
```
```   631
```
```   632   lemma ffold_fun_left_comm:
```
```   633     "f x (ffold f z A) = ffold f (f x z) A"
```
```   634     by (transfer fixing: f) (rule fold_fun_left_comm)
```
```   635
```
```   636   lemma ffold_finsert2:
```
```   637     "x |\<notin>| A \<Longrightarrow> ffold f z (finsert x A)  = ffold f (f x z) A"
```
```   638     by (transfer fixing: f) (rule fold_insert2)
```
```   639
```
```   640   lemma ffold_rec:
```
```   641     assumes "x |\<in>| A"
```
```   642     shows "ffold f z A = f x (ffold f z (A |-| {|x|}))"
```
```   643     using assms by (transfer fixing: f) (rule fold_rec)
```
```   644
```
```   645   lemma ffold_finsert_fremove:
```
```   646     "ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
```
```   647      by (transfer fixing: f) (rule fold_insert_remove)
```
```   648 end
```
```   649
```
```   650 lemma ffold_fimage:
```
```   651   assumes "inj_on g (fset A)"
```
```   652   shows "ffold f z (g |`| A) = ffold (f \<circ> g) z A"
```
```   653 using assms by transfer' (rule fold_image)
```
```   654
```
```   655 lemma ffold_cong:
```
```   656   assumes "comp_fun_commute f" "comp_fun_commute g"
```
```   657   "\<And>x. x |\<in>| A \<Longrightarrow> f x = g x"
```
```   658     and "s = t" and "A = B"
```
```   659   shows "ffold f s A = ffold g t B"
```
```   660 using assms by transfer (metis Finite_Set.fold_cong)
```
```   661
```
```   662 context comp_fun_idem
```
```   663 begin
```
```   664
```
```   665   lemma ffold_finsert_idem:
```
```   666     "ffold f z (finsert x A)  = f x (ffold f z A)"
```
```   667     by (transfer fixing: f) (rule fold_insert_idem)
```
```   668
```
```   669   declare ffold_finsert [simp del] ffold_finsert_idem [simp]
```
```   670
```
```   671   lemma ffold_finsert_idem2:
```
```   672     "ffold f z (finsert x A) = ffold f (f x z) A"
```
```   673     by (transfer fixing: f) (rule fold_insert_idem2)
```
```   674
```
```   675 end
```
```   676
```
```   677 subsection {* Choice in fsets *}
```
```   678
```
```   679 lemma fset_choice:
```
```   680   assumes "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
```
```   681   shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
```
```   682   using assms by transfer metis
```
```   683
```
```   684 subsection {* Induction and Cases rules for fsets *}
```
```   685
```
```   686 lemma fset_exhaust [case_names empty insert, cases type: fset]:
```
```   687   assumes fempty_case: "S = {||} \<Longrightarrow> P"
```
```   688   and     finsert_case: "\<And>x S'. S = finsert x S' \<Longrightarrow> P"
```
```   689   shows "P"
```
```   690   using assms by transfer blast
```
```   691
```
```   692 lemma fset_induct [case_names empty insert]:
```
```   693   assumes fempty_case: "P {||}"
```
```   694   and     finsert_case: "\<And>x S. P S \<Longrightarrow> P (finsert x S)"
```
```   695   shows "P S"
```
```   696 proof -
```
```   697   (* FIXME transfer and right_total vs. bi_total *)
```
```   698   note Domainp_forall_transfer[transfer_rule]
```
```   699   show ?thesis
```
```   700   using assms by transfer (auto intro: finite_induct)
```
```   701 qed
```
```   702
```
```   703 lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
```
```   704   assumes empty_fset_case: "P {||}"
```
```   705   and     insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
```
```   706   shows "P S"
```
```   707 proof -
```
```   708   (* FIXME transfer and right_total vs. bi_total *)
```
```   709   note Domainp_forall_transfer[transfer_rule]
```
```   710   show ?thesis
```
```   711   using assms by transfer (auto intro: finite_induct)
```
```   712 qed
```
```   713
```
```   714 lemma fset_card_induct:
```
```   715   assumes empty_fset_case: "P {||}"
```
```   716   and     card_fset_Suc_case: "\<And>S T. Suc (fcard S) = (fcard T) \<Longrightarrow> P S \<Longrightarrow> P T"
```
```   717   shows "P S"
```
```   718 proof (induct S)
```
```   719   case empty
```
```   720   show "P {||}" by (rule empty_fset_case)
```
```   721 next
```
```   722   case (insert x S)
```
```   723   have h: "P S" by fact
```
```   724   have "x |\<notin>| S" by fact
```
```   725   then have "Suc (fcard S) = fcard (finsert x S)"
```
```   726     by transfer auto
```
```   727   then show "P (finsert x S)"
```
```   728     using h card_fset_Suc_case by simp
```
```   729 qed
```
```   730
```
```   731 lemma fset_strong_cases:
```
```   732   obtains "xs = {||}"
```
```   733     | ys x where "x |\<notin>| ys" and "xs = finsert x ys"
```
```   734 by transfer blast
```
```   735
```
```   736 lemma fset_induct2:
```
```   737   "P {||} {||} \<Longrightarrow>
```
```   738   (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>
```
```   739   (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>
```
```   740   (\<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>
```
```   741   P xsa ysa"
```
```   742   apply (induct xsa arbitrary: ysa)
```
```   743   apply (induct_tac x rule: fset_induct_stronger)
```
```   744   apply simp_all
```
```   745   apply (induct_tac xa rule: fset_induct_stronger)
```
```   746   apply simp_all
```
```   747   done
```
```   748
```
```   749 subsection {* Setup for Lifting/Transfer *}
```
```   750
```
```   751 subsubsection {* Relator and predicator properties *}
```
```   752
```
```   753 lift_definition fset_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is set_rel
```
```   754 parametric set_rel_transfer ..
```
```   755
```
```   756 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)
```
```   757   \<and> (\<forall>y. \<exists>x. y|\<in>|B \<longrightarrow> x|\<in>|A \<and> R x y))"
```
```   758 apply (rule ext)+
```
```   759 apply transfer'
```
```   760 apply (subst set_rel_def[unfolded fun_eq_iff])
```
```   761 by blast
```
```   762
```
```   763 lemma fset_rel_conversep: "fset_rel (conversep R) = conversep (fset_rel R)"
```
```   764   unfolding fset_rel_alt_def by auto
```
```   765
```
```   766 lemmas fset_rel_eq [relator_eq] = set_rel_eq[Transfer.transferred]
```
```   767
```
```   768 lemma fset_rel_mono[relator_mono]: "A \<le> B \<Longrightarrow> fset_rel A \<le> fset_rel B"
```
```   769 unfolding fset_rel_alt_def by blast
```
```   770
```
```   771 lemma finite_set_rel:
```
```   772   assumes fin: "finite X" "finite Z"
```
```   773   assumes R_S: "set_rel (R OO S) X Z"
```
```   774   shows "\<exists>Y. finite Y \<and> set_rel R X Y \<and> set_rel S Y Z"
```
```   775 proof -
```
```   776   obtain f where f: "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)"
```
```   777   apply atomize_elim
```
```   778   apply (subst bchoice_iff[symmetric])
```
```   779   using R_S[unfolded set_rel_def OO_def] by blast
```
```   780
```
```   781   obtain g where g: "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R  x (g z))"
```
```   782   apply atomize_elim
```
```   783   apply (subst bchoice_iff[symmetric])
```
```   784   using R_S[unfolded set_rel_def OO_def] by blast
```
```   785
```
```   786   let ?Y = "f ` X \<union> g ` Z"
```
```   787   have "finite ?Y" by (simp add: fin)
```
```   788   moreover have "set_rel R X ?Y"
```
```   789     unfolding set_rel_def
```
```   790     using f g by clarsimp blast
```
```   791   moreover have "set_rel S ?Y Z"
```
```   792     unfolding set_rel_def
```
```   793     using f g by clarsimp blast
```
```   794   ultimately show ?thesis by metis
```
```   795 qed
```
```   796
```
```   797 lemma fset_rel_OO[relator_distr]: "fset_rel R OO fset_rel S = fset_rel (R OO S)"
```
```   798 apply (rule ext)+
```
```   799 by transfer (auto intro: finite_set_rel set_rel_OO[unfolded fun_eq_iff, rule_format, THEN iffD1])
```
```   800
```
```   801 lemma Domainp_fset[relator_domain]:
```
```   802   assumes "Domainp T = P"
```
```   803   shows "Domainp (fset_rel T) = (\<lambda>A. fBall A P)"
```
```   804 proof -
```
```   805   from assms obtain f where f: "\<forall>x\<in>Collect P. T x (f x)"
```
```   806     unfolding Domainp_iff[abs_def]
```
```   807     apply atomize_elim
```
```   808     by (subst bchoice_iff[symmetric]) auto
```
```   809   from assms f show ?thesis
```
```   810     unfolding fun_eq_iff fset_rel_alt_def Domainp_iff
```
```   811     apply clarify
```
```   812     apply (rule iffI)
```
```   813       apply blast
```
```   814     by (rename_tac A, rule_tac x="f |`| A" in exI, blast)
```
```   815 qed
```
```   816
```
```   817 lemmas reflp_fset_rel[reflexivity_rule] = reflp_set_rel[Transfer.transferred]
```
```   818
```
```   819 lemma right_total_fset_rel[transfer_rule]: "right_total A \<Longrightarrow> right_total (fset_rel A)"
```
```   820 unfolding right_total_def
```
```   821 apply transfer
```
```   822 apply (subst(asm) choice_iff)
```
```   823 apply clarsimp
```
```   824 apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
```
```   825 by (auto simp add: set_rel_def)
```
```   826
```
```   827 lemma left_total_fset_rel[reflexivity_rule]: "left_total A \<Longrightarrow> left_total (fset_rel A)"
```
```   828 unfolding left_total_def
```
```   829 apply transfer
```
```   830 apply (subst(asm) choice_iff)
```
```   831 apply clarsimp
```
```   832 apply (rename_tac A f y, rule_tac x = "f ` y" in exI)
```
```   833 by (auto simp add: set_rel_def)
```
```   834
```
```   835 lemmas right_unique_fset_rel[transfer_rule] = right_unique_set_rel[Transfer.transferred]
```
```   836 lemmas left_unique_fset_rel[reflexivity_rule] = left_unique_set_rel[Transfer.transferred]
```
```   837
```
```   838 thm right_unique_fset_rel left_unique_fset_rel
```
```   839
```
```   840 lemma bi_unique_fset_rel[transfer_rule]: "bi_unique A \<Longrightarrow> bi_unique (fset_rel A)"
```
```   841 by (auto intro: right_unique_fset_rel left_unique_fset_rel iff: bi_unique_iff)
```
```   842
```
```   843 lemma bi_total_fset_rel[transfer_rule]: "bi_total A \<Longrightarrow> bi_total (fset_rel A)"
```
```   844 by (auto intro: right_total_fset_rel left_total_fset_rel iff: bi_total_iff)
```
```   845
```
```   846 lemmas fset_invariant_commute [invariant_commute] = set_invariant_commute[Transfer.transferred]
```
```   847
```
```   848 subsubsection {* Quotient theorem for the Lifting package *}
```
```   849
```
```   850 lemma Quotient_fset_map[quot_map]:
```
```   851   assumes "Quotient R Abs Rep T"
```
```   852   shows "Quotient (fset_rel R) (fimage Abs) (fimage Rep) (fset_rel T)"
```
```   853   using assms unfolding Quotient_alt_def4
```
```   854   by (simp add: fset_rel_OO[symmetric] fset_rel_conversep) (simp add: fset_rel_alt_def, blast)
```
```   855
```
```   856 subsubsection {* Transfer rules for the Transfer package *}
```
```   857
```
```   858 text {* Unconditional transfer rules *}
```
```   859
```
```   860 context
```
```   861 begin
```
```   862
```
```   863 interpretation lifting_syntax .
```
```   864
```
```   865 lemmas fempty_transfer [transfer_rule] = empty_transfer[Transfer.transferred]
```
```   866
```
```   867 lemma finsert_transfer [transfer_rule]:
```
```   868   "(A ===> fset_rel A ===> fset_rel A) finsert finsert"
```
```   869   unfolding fun_rel_def fset_rel_alt_def by blast
```
```   870
```
```   871 lemma funion_transfer [transfer_rule]:
```
```   872   "(fset_rel A ===> fset_rel A ===> fset_rel A) funion funion"
```
```   873   unfolding fun_rel_def fset_rel_alt_def by blast
```
```   874
```
```   875 lemma ffUnion_transfer [transfer_rule]:
```
```   876   "(fset_rel (fset_rel A) ===> fset_rel A) ffUnion ffUnion"
```
```   877   unfolding fun_rel_def fset_rel_alt_def by transfer (simp, fast)
```
```   878
```
```   879 lemma fimage_transfer [transfer_rule]:
```
```   880   "((A ===> B) ===> fset_rel A ===> fset_rel B) fimage fimage"
```
```   881   unfolding fun_rel_def fset_rel_alt_def by simp blast
```
```   882
```
```   883 lemma fBall_transfer [transfer_rule]:
```
```   884   "(fset_rel A ===> (A ===> op =) ===> op =) fBall fBall"
```
```   885   unfolding fset_rel_alt_def fun_rel_def by blast
```
```   886
```
```   887 lemma fBex_transfer [transfer_rule]:
```
```   888   "(fset_rel A ===> (A ===> op =) ===> op =) fBex fBex"
```
```   889   unfolding fset_rel_alt_def fun_rel_def by blast
```
```   890
```
```   891 (* FIXME transfer doesn't work here *)
```
```   892 lemma fPow_transfer [transfer_rule]:
```
```   893   "(fset_rel A ===> fset_rel (fset_rel A)) fPow fPow"
```
```   894   unfolding fun_rel_def
```
```   895   using Pow_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred]
```
```   896   by blast
```
```   897
```
```   898 lemma fset_rel_transfer [transfer_rule]:
```
```   899   "((A ===> B ===> op =) ===> fset_rel A ===> fset_rel B ===> op =)
```
```   900     fset_rel fset_rel"
```
```   901   unfolding fun_rel_def
```
```   902   using set_rel_transfer[unfolded fun_rel_def,rule_format, Transfer.transferred, where A = A and B = B]
```
```   903   by simp
```
```   904
```
```   905 lemma bind_transfer [transfer_rule]:
```
```   906   "(fset_rel A ===> (A ===> fset_rel B) ===> fset_rel B) fbind fbind"
```
```   907   using assms unfolding fun_rel_def
```
```   908   using bind_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
```
```   909
```
```   910 text {* Rules requiring bi-unique, bi-total or right-total relations *}
```
```   911
```
```   912 lemma fmember_transfer [transfer_rule]:
```
```   913   assumes "bi_unique A"
```
```   914   shows "(A ===> fset_rel A ===> op =) (op |\<in>|) (op |\<in>|)"
```
```   915   using assms unfolding fun_rel_def fset_rel_alt_def bi_unique_def by metis
```
```   916
```
```   917 lemma finter_transfer [transfer_rule]:
```
```   918   assumes "bi_unique A"
```
```   919   shows "(fset_rel A ===> fset_rel A ===> fset_rel A) finter finter"
```
```   920   using assms unfolding fun_rel_def
```
```   921   using inter_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
```
```   922
```
```   923 lemma fminus_transfer [transfer_rule]:
```
```   924   assumes "bi_unique A"
```
```   925   shows "(fset_rel A ===> fset_rel A ===> fset_rel A) (op |-|) (op |-|)"
```
```   926   using assms unfolding fun_rel_def
```
```   927   using Diff_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
```
```   928
```
```   929 lemma fsubset_transfer [transfer_rule]:
```
```   930   assumes "bi_unique A"
```
```   931   shows "(fset_rel A ===> fset_rel A ===> op =) (op |\<subseteq>|) (op |\<subseteq>|)"
```
```   932   using assms unfolding fun_rel_def
```
```   933   using subset_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
```
```   934
```
```   935 lemma fSup_transfer [transfer_rule]:
```
```   936   "bi_unique A \<Longrightarrow> (set_rel (fset_rel A) ===> fset_rel A) Sup Sup"
```
```   937   using assms unfolding fun_rel_def
```
```   938   apply clarify
```
```   939   apply transfer'
```
```   940   using Sup_fset_transfer[unfolded fun_rel_def] by blast
```
```   941
```
```   942 (* FIXME: add right_total_fInf_transfer *)
```
```   943
```
```   944 lemma fInf_transfer [transfer_rule]:
```
```   945   assumes "bi_unique A" and "bi_total A"
```
```   946   shows "(set_rel (fset_rel A) ===> fset_rel A) Inf Inf"
```
```   947   using assms unfolding fun_rel_def
```
```   948   apply clarify
```
```   949   apply transfer'
```
```   950   using Inf_fset_transfer[unfolded fun_rel_def] by blast
```
```   951
```
```   952 lemma ffilter_transfer [transfer_rule]:
```
```   953   assumes "bi_unique A"
```
```   954   shows "((A ===> op=) ===> fset_rel A ===> fset_rel A) ffilter ffilter"
```
```   955   using assms unfolding fun_rel_def
```
```   956   using Lifting_Set.filter_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
```
```   957
```
```   958 lemma card_transfer [transfer_rule]:
```
```   959   "bi_unique A \<Longrightarrow> (fset_rel A ===> op =) fcard fcard"
```
```   960   using assms unfolding fun_rel_def
```
```   961   using card_transfer[unfolded fun_rel_def, rule_format, Transfer.transferred] by blast
```
```   962
```
```   963 end
```
```   964
```
```   965 lifting_update fset.lifting
```
```   966 lifting_forget fset.lifting
```
```   967
```
```   968 end
```