(*  Title:      HOL/Library/FSet.thy

     Author:     Ondrej Kuncar, TU Muenchen

     Author:     Cezary Kaliszyk and Christian Urban

     Author:     Andrei Popescu, TU Muenchen

 *)

 header {* Type of finite sets defined as a subtype of sets *}

 theory FSet

 imports Conditionally_Complete_Lattices

 begin

 subsection {* Definition of the type *}

 typedef 'a fset = "{A :: 'a set. finite A}"  morphisms fset Abs_fset

 by auto

 setup_lifting type_definition_fset

 subsection {* Basic operations and type class instantiations *}

 (* FIXME transfer and right_total vs. bi_total *)

 instantiation fset :: (finite) finite

 begin

 instance by default (transfer, simp)

 end

 instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"

 begin

 interpretation lifting_syntax .

 lift_definition bot_fset :: "'a fset" is "{}" parametric empty_transfer by simp 

 lift_definition less_eq_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" is subset_eq parametric subset_transfer 

   .

 definition less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" where "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"

 lemma less_fset_transfer[transfer_rule]:

   assumes [transfer_rule]: "bi_unique A" 

   shows "((pcr_fset A) ===> (pcr_fset A) ===> op =) op \<subset> op <"

   unfolding less_fset_def[abs_def] psubset_eq[abs_def] by transfer_prover

 lift_definition sup_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is union parametric union_transfer

   by simp

 lift_definition inf_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is inter parametric inter_transfer

   by simp

 lift_definition minus_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is minus parametric Diff_transfer

   by simp

 instance

 by default (transfer, auto)+

 end

 abbreviation fempty :: "'a fset" ("{||}") where "{||} \<equiv> bot"

 abbreviation fsubset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50) where "xs |\<subseteq>| ys \<equiv> xs \<le> ys"

 abbreviation fsubset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50) where "xs |\<subset>| ys \<equiv> xs < ys"

 abbreviation funion :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<union>|" 65) where "xs |\<union>| ys \<equiv> sup xs ys"

 abbreviation finter :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<inter>|" 65) where "xs |\<inter>| ys \<equiv> inf xs ys"

 abbreviation fminus :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|-|" 65) where "xs |-| ys \<equiv> minus xs ys"

 instantiation fset :: (equal) equal

 begin

 definition "HOL.equal A B \<longleftrightarrow> A |\<subseteq>| B \<and> B |\<subseteq>| A"

 instance by intro_classes (auto simp add: equal_fset_def)

 end 

 instantiation fset :: (type) conditionally_complete_lattice

 begin

 interpretation lifting_syntax .

 lemma right_total_Inf_fset_transfer:

   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"

   shows "(rel_set (rel_set A) ===> rel_set A) 

     (\<lambda>S. if finite (Inter S \<inter> Collect (Domainp A)) then Inter S \<inter> Collect (Domainp A) else {}) 

       (\<lambda>S. if finite (Inf S) then Inf S else {})"

     by transfer_prover

 lemma Inf_fset_transfer:

   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"

   shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>A. if finite (Inf A) then Inf A else {}) 

     (\<lambda>A. if finite (Inf A) then Inf A else {})"

   by transfer_prover

 lift_definition Inf_fset :: "'a fset set \<Rightarrow> 'a fset" is "\<lambda>A. if finite (Inf A) then Inf A else {}" 

 parametric right_total_Inf_fset_transfer Inf_fset_transfer by simp

 lemma Sup_fset_transfer:

   assumes [transfer_rule]: "bi_unique A"

   shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>A. if finite (Sup A) then Sup A else {})

   (\<lambda>A. if finite (Sup A) then Sup A else {})" by transfer_prover

 lift_definition Sup_fset :: "'a fset set \<Rightarrow> 'a fset" is "\<lambda>A. if finite (Sup A) then Sup A else {}"

 parametric Sup_fset_transfer by simp

 lemma finite_Sup: "\<exists>z. finite z \<and> (\<forall>a. a \<in> X \<longrightarrow> a \<le> z) \<Longrightarrow> finite (Sup X)"

 by (auto intro: finite_subset)

 lemma transfer_bdd_below[transfer_rule]: "(rel_set (pcr_fset op =) ===> op =) bdd_below bdd_below"

   by auto

 instance

 proof 

   fix x z :: "'a fset"

   fix X :: "'a fset set"

   {

     assume "x \<in> X" "bdd_below X" 

     then show "Inf X |\<subseteq>| x"  by transfer auto

   next

     assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> z |\<subseteq>| x)"

     then show "z |\<subseteq>| Inf X" by transfer (clarsimp, blast)

   next

     assume "x \<in> X" "bdd_above X"

     then obtain z where "x \<in> X" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)"

       by (auto simp: bdd_above_def)

     then show "x |\<subseteq>| Sup X"

       by transfer (auto intro!: finite_Sup)

   next

     assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)"

     then show "Sup X |\<subseteq>| z" by transfer (clarsimp, blast)

   }

 qed

 end

 instantiation fset :: (finite) complete_lattice 

 begin

 lift_definition top_fset :: "'a fset" is UNIV parametric right_total_UNIV_transfer UNIV_transfer by simp

 instance by default (transfer, auto)+

 end

 instantiation fset :: (finite) complete_boolean_algebra

 begin

 lift_definition uminus_fset :: "'a fset \<Rightarrow> 'a fset" is uminus 

   parametric right_total_Compl_transfer Compl_transfer by simp

 instance by (default, simp_all only: INF_def SUP_def) (transfer, simp add: Compl_partition Diff_eq)+

 end

 abbreviation fUNIV :: "'a::finite fset" where "fUNIV \<equiv> top"

 abbreviation fuminus :: "'a::finite fset \<Rightarrow> 'a fset" ("|-| _" [81] 80) where "|-| x \<equiv> uminus x"

 subsection {* Other operations *}

 lift_definition finsert :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is insert parametric Lifting_Set.insert_transfer

   by simp

 syntax

   "_insert_fset"     :: "args => 'a fset"  ("{|(_)|}")

 translations

   "{|x, xs|}" == "CONST finsert x {|xs|}"

   "{|x|}"     == "CONST finsert x {||}"

 lift_definition fmember :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<in>|" 50) is Set.member 

   parametric member_transfer .

 abbreviation notin_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50) where "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"

 context

 begin

 interpretation lifting_syntax .

 lift_definition ffilter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is Set.filter 

   parametric Lifting_Set.filter_transfer unfolding Set.filter_def by simp

 lift_definition fPow :: "'a fset \<Rightarrow> 'a fset fset" is Pow parametric Pow_transfer 

 by (simp add: finite_subset)

 lift_definition fcard :: "'a fset \<Rightarrow> nat" is card parametric card_transfer .

 lift_definition fimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset" (infixr "|`|" 90) is image 

   parametric image_transfer by simp

 lift_definition fthe_elem :: "'a fset \<Rightarrow> 'a" is the_elem .

 lift_definition fbind :: "'a fset \<Rightarrow> ('a \<Rightarrow> 'b fset) \<Rightarrow> 'b fset" is Set.bind parametric bind_transfer 

 by (simp add: Set.bind_def)

 lift_definition ffUnion :: "'a fset fset \<Rightarrow> 'a fset" is Union parametric Union_transfer by simp

 lift_definition fBall :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" is Ball parametric Ball_transfer .

 lift_definition fBex :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" is Bex parametric Bex_transfer .

 lift_definition ffold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" is Finite_Set.fold .

 subsection {* Transferred lemmas from Set.thy *}

 lemmas fset_eqI = set_eqI[Transfer.transferred]

 lemmas fset_eq_iff[no_atp] = set_eq_iff[Transfer.transferred]

 lemmas fBallI[intro!] = ballI[Transfer.transferred]

 lemmas fbspec[dest?] = bspec[Transfer.transferred]

 lemmas fBallE[elim] = ballE[Transfer.transferred]

 lemmas fBexI[intro] = bexI[Transfer.transferred]

 lemmas rev_fBexI[intro?] = rev_bexI[Transfer.transferred]

 lemmas fBexCI = bexCI[Transfer.transferred]

 lemmas fBexE[elim!] = bexE[Transfer.transferred]

 lemmas fBall_triv[simp] = ball_triv[Transfer.transferred]

 lemmas fBex_triv[simp] = bex_triv[Transfer.transferred]

 lemmas fBex_triv_one_point1[simp] = bex_triv_one_point1[Transfer.transferred]

 lemmas fBex_triv_one_point2[simp] = bex_triv_one_point2[Transfer.transferred]

 lemmas fBex_one_point1[simp] = bex_one_point1[Transfer.transferred]

 lemmas fBex_one_point2[simp] = bex_one_point2[Transfer.transferred]

 lemmas fBall_one_point1[simp] = ball_one_point1[Transfer.transferred]

 lemmas fBall_one_point2[simp] = ball_one_point2[Transfer.transferred]

 lemmas fBall_conj_distrib = ball_conj_distrib[Transfer.transferred]

 lemmas fBex_disj_distrib = bex_disj_distrib[Transfer.transferred]

 lemmas fBall_cong = ball_cong[Transfer.transferred]

 lemmas fBex_cong = bex_cong[Transfer.transferred]

 lemmas fsubsetI[intro!] = subsetI[Transfer.transferred]

 lemmas fsubsetD[elim, intro?] = subsetD[Transfer.transferred]

 lemmas rev_fsubsetD[no_atp,intro?] = rev_subsetD[Transfer.transferred]

 lemmas fsubsetCE[no_atp,elim] = subsetCE[Transfer.transferred]

 lemmas fsubset_eq[no_atp] = subset_eq[Transfer.transferred]

 lemmas contra_fsubsetD[no_atp] = contra_subsetD[Transfer.transferred]

 lemmas fsubset_refl = subset_refl[Transfer.transferred]

 lemmas fsubset_trans = subset_trans[Transfer.transferred]

 lemmas fset_rev_mp = set_rev_mp[Transfer.transferred]

 lemmas fset_mp = set_mp[Transfer.transferred]

 lemmas fsubset_not_fsubset_eq[code] = subset_not_subset_eq[Transfer.transferred]

 lemmas eq_fmem_trans = eq_mem_trans[Transfer.transferred]

 lemmas fsubset_antisym[intro!] = subset_antisym[Transfer.transferred]

 lemmas fequalityD1 = equalityD1[Transfer.transferred]

 lemmas fequalityD2 = equalityD2[Transfer.transferred]

 lemmas fequalityE = equalityE[Transfer.transferred]

 lemmas fequalityCE[elim] = equalityCE[Transfer.transferred]

 lemmas eqfset_imp_iff = eqset_imp_iff[Transfer.transferred]

 lemmas eqfelem_imp_iff = eqelem_imp_iff[Transfer.transferred]

 lemmas fempty_iff[simp] = empty_iff[Transfer.transferred]

 lemmas fempty_fsubsetI[iff] = empty_subsetI[Transfer.transferred]

 lemmas equalsffemptyI = equals0I[Transfer.transferred]

 lemmas equalsffemptyD = equals0D[Transfer.transferred]

 lemmas fBall_fempty[simp] = ball_empty[Transfer.transferred]

 lemmas fBex_fempty[simp] = bex_empty[Transfer.transferred]

 lemmas fPow_iff[iff] = Pow_iff[Transfer.transferred]

 lemmas fPowI = PowI[Transfer.transferred]

 lemmas fPowD = PowD[Transfer.transferred]

 lemmas fPow_bottom = Pow_bottom[Transfer.transferred]

 lemmas fPow_top = Pow_top[Transfer.transferred]

 lemmas fPow_not_fempty = Pow_not_empty[Transfer.transferred]

 lemmas finter_iff[simp] = Int_iff[Transfer.transferred]

 lemmas finterI[intro!] = IntI[Transfer.transferred]

 lemmas finterD1 = IntD1[Transfer.transferred]

 lemmas finterD2 = IntD2[Transfer.transferred]

 lemmas finterE[elim!] = IntE[Transfer.transferred]

 lemmas funion_iff[simp] = Un_iff[Transfer.transferred]

 lemmas funionI1[elim?] = UnI1[Transfer.transferred]

 lemmas funionI2[elim?] = UnI2[Transfer.transferred]

 lemmas funionCI[intro!] = UnCI[Transfer.transferred]

 lemmas funionE[elim!] = UnE[Transfer.transferred]

 lemmas fminus_iff[simp] = Diff_iff[Transfer.transferred]

 lemmas fminusI[intro!] = DiffI[Transfer.transferred]

 lemmas fminusD1 = DiffD1[Transfer.transferred]

 lemmas fminusD2 = DiffD2[Transfer.transferred]

 lemmas fminusE[elim!] = DiffE[Transfer.transferred]

 lemmas finsert_iff[simp] = insert_iff[Transfer.transferred]

 lemmas finsertI1 = insertI1[Transfer.transferred]

 lemmas finsertI2 = insertI2[Transfer.transferred]

 lemmas finsertE[elim!] = insertE[Transfer.transferred]

 lemmas finsertCI[intro!] = insertCI[Transfer.transferred]

 lemmas fsubset_finsert_iff = subset_insert_iff[Transfer.transferred]

 lemmas finsert_ident = insert_ident[Transfer.transferred]

 lemmas fsingletonI[intro!,no_atp] = singletonI[Transfer.transferred]

 lemmas fsingletonD[dest!,no_atp] = singletonD[Transfer.transferred]

 lemmas fsingleton_iff = singleton_iff[Transfer.transferred]

 lemmas fsingleton_inject[dest!] = singleton_inject[Transfer.transferred]

 lemmas fsingleton_finsert_inj_eq[iff,no_atp] = singleton_insert_inj_eq[Transfer.transferred]

 lemmas fsingleton_finsert_inj_eq'[iff,no_atp] = singleton_insert_inj_eq'[Transfer.transferred]

 lemmas fsubset_fsingletonD = subset_singletonD[Transfer.transferred]

 lemmas fminus_single_finsert = diff_single_insert[Transfer.transferred]

 lemmas fdoubleton_eq_iff = doubleton_eq_iff[Transfer.transferred]

 lemmas funion_fsingleton_iff = Un_singleton_iff[Transfer.transferred]

 lemmas fsingleton_funion_iff = singleton_Un_iff[Transfer.transferred]

 lemmas fimage_eqI[simp, intro] = image_eqI[Transfer.transferred]

 lemmas fimageI = imageI[Transfer.transferred]

 lemmas rev_fimage_eqI = rev_image_eqI[Transfer.transferred]

 lemmas fimageE[elim!] = imageE[Transfer.transferred]

 lemmas Compr_fimage_eq = Compr_image_eq[Transfer.transferred]

 lemmas fimage_funion = image_Un[Transfer.transferred]

 lemmas fimage_iff = image_iff[Transfer.transferred]

 lemmas fimage_fsubset_iff[no_atp] = image_subset_iff[Transfer.transferred]

 lemmas fimage_fsubsetI = image_subsetI[Transfer.transferred]

 lemmas fimage_ident[simp] = image_ident[Transfer.transferred]

 lemmas split_if_fmem1 = split_if_mem1[Transfer.transferred]

 lemmas split_if_fmem2 = split_if_mem2[Transfer.transferred]

 lemmas pfsubsetI[intro!,no_atp] = psubsetI[Transfer.transferred]

 lemmas pfsubsetE[elim!,no_atp] = psubsetE[Transfer.transferred]

 lemmas pfsubset_finsert_iff = psubset_insert_iff[Transfer.transferred]

 lemmas pfsubset_eq = psubset_eq[Transfer.transferred]

 lemmas pfsubset_imp_fsubset = psubset_imp_subset[Transfer.transferred]

 lemmas pfsubset_trans = psubset_trans[Transfer.transferred]

 lemmas pfsubsetD = psubsetD[Transfer.transferred]

 lemmas pfsubset_fsubset_trans = psubset_subset_trans[Transfer.transferred]

 lemmas fsubset_pfsubset_trans = subset_psubset_trans[Transfer.transferred]

 lemmas pfsubset_imp_ex_fmem = psubset_imp_ex_mem[Transfer.transferred]

 lemmas fimage_fPow_mono = image_Pow_mono[Transfer.transferred]

 lemmas fimage_fPow_surj = image_Pow_surj[Transfer.transferred]

 lemmas fsubset_finsertI = subset_insertI[Transfer.transferred]

 lemmas fsubset_finsertI2 = subset_insertI2[Transfer.transferred]

 lemmas fsubset_finsert = subset_insert[Transfer.transferred]

 lemmas funion_upper1 = Un_upper1[Transfer.transferred]

 lemmas funion_upper2 = Un_upper2[Transfer.transferred]

 lemmas funion_least = Un_least[Transfer.transferred]

 lemmas finter_lower1 = Int_lower1[Transfer.transferred]

 lemmas finter_lower2 = Int_lower2[Transfer.transferred]

 lemmas finter_greatest = Int_greatest[Transfer.transferred]

 lemmas fminus_fsubset = Diff_subset[Transfer.transferred]

 lemmas fminus_fsubset_conv = Diff_subset_conv[Transfer.transferred]

 lemmas fsubset_fempty[simp] = subset_empty[Transfer.transferred]

 lemmas not_pfsubset_fempty[iff] = not_psubset_empty[Transfer.transferred]

 lemmas finsert_is_funion = insert_is_Un[Transfer.transferred]

 lemmas finsert_not_fempty[simp] = insert_not_empty[Transfer.transferred]

 lemmas fempty_not_finsert = empty_not_insert[Transfer.transferred]

 lemmas finsert_absorb = insert_absorb[Transfer.transferred]

 lemmas finsert_absorb2[simp] = insert_absorb2[Transfer.transferred]

 lemmas finsert_commute = insert_commute[Transfer.transferred]

 lemmas finsert_fsubset[simp] = insert_subset[Transfer.transferred]

 lemmas finsert_inter_finsert[simp] = insert_inter_insert[Transfer.transferred]

 lemmas finsert_disjoint[simp,no_atp] = insert_disjoint[Transfer.transferred]

 lemmas disjoint_finsert[simp,no_atp] = disjoint_insert[Transfer.transferred]

 lemmas fimage_fempty[simp] = image_empty[Transfer.transferred]

 lemmas fimage_finsert[simp] = image_insert[Transfer.transferred]

 lemmas fimage_constant = image_constant[Transfer.transferred]

 lemmas fimage_constant_conv = image_constant_conv[Transfer.transferred]

 lemmas fimage_fimage = image_image[Transfer.transferred]

 lemmas finsert_fimage[simp] = insert_image[Transfer.transferred]

 lemmas fimage_is_fempty[iff] = image_is_empty[Transfer.transferred]

 lemmas fempty_is_fimage[iff] = empty_is_image[Transfer.transferred]

 lemmas fimage_cong = image_cong[Transfer.transferred]

 lemmas fimage_finter_fsubset = image_Int_subset[Transfer.transferred]

 lemmas fimage_fminus_fsubset = image_diff_subset[Transfer.transferred]

 lemmas finter_absorb = Int_absorb[Transfer.transferred]

 lemmas finter_left_absorb = Int_left_absorb[Transfer.transferred]

 lemmas finter_commute = Int_commute[Transfer.transferred]

 lemmas finter_left_commute = Int_left_commute[Transfer.transferred]

 lemmas finter_assoc = Int_assoc[Transfer.transferred]

 lemmas finter_ac = Int_ac[Transfer.transferred]

 lemmas finter_absorb1 = Int_absorb1[Transfer.transferred]

 lemmas finter_absorb2 = Int_absorb2[Transfer.transferred]

 lemmas finter_fempty_left = Int_empty_left[Transfer.transferred]

 lemmas finter_fempty_right = Int_empty_right[Transfer.transferred]

 lemmas disjoint_iff_fnot_equal = disjoint_iff_not_equal[Transfer.transferred]

 lemmas finter_funion_distrib = Int_Un_distrib[Transfer.transferred]

 lemmas finter_funion_distrib2 = Int_Un_distrib2[Transfer.transferred]

 lemmas finter_fsubset_iff[no_atp, simp] = Int_subset_iff[Transfer.transferred]

 lemmas funion_absorb = Un_absorb[Transfer.transferred]

 lemmas funion_left_absorb = Un_left_absorb[Transfer.transferred]

 lemmas funion_commute = Un_commute[Transfer.transferred]

 lemmas funion_left_commute = Un_left_commute[Transfer.transferred]

 lemmas funion_assoc = Un_assoc[Transfer.transferred]

 lemmas funion_ac = Un_ac[Transfer.transferred]

 lemmas funion_absorb1 = Un_absorb1[Transfer.transferred]

 lemmas funion_absorb2 = Un_absorb2[Transfer.transferred]

 lemmas funion_fempty_left = Un_empty_left[Transfer.transferred]

 lemmas funion_fempty_right = Un_empty_right[Transfer.transferred]

 lemmas funion_finsert_left[simp] = Un_insert_left[Transfer.transferred]

 lemmas funion_finsert_right[simp] = Un_insert_right[Transfer.transferred]

 lemmas finter_finsert_left = Int_insert_left[Transfer.transferred]

 lemmas finter_finsert_left_ifffempty[simp] = Int_insert_left_if0[Transfer.transferred]

 lemmas finter_finsert_left_if1[simp] = Int_insert_left_if1[Transfer.transferred]

 lemmas finter_finsert_right = Int_insert_right[Transfer.transferred]

 lemmas finter_finsert_right_ifffempty[simp] = Int_insert_right_if0[Transfer.transferred]

 lemmas finter_finsert_right_if1[simp] = Int_insert_right_if1[Transfer.transferred]

 lemmas funion_finter_distrib = Un_Int_distrib[Transfer.transferred]

 lemmas funion_finter_distrib2 = Un_Int_distrib2[Transfer.transferred]

 lemmas funion_finter_crazy = Un_Int_crazy[Transfer.transferred]

 lemmas fsubset_funion_eq = subset_Un_eq[Transfer.transferred]

 lemmas funion_fempty[iff] = Un_empty[Transfer.transferred]

 lemmas funion_fsubset_iff[no_atp, simp] = Un_subset_iff[Transfer.transferred]

 lemmas funion_fminus_finter = Un_Diff_Int[Transfer.transferred]

 lemmas fminus_finter2 = Diff_Int2[Transfer.transferred]

 lemmas funion_finter_assoc_eq = Un_Int_assoc_eq[Transfer.transferred]

 lemmas fBall_funion = ball_Un[Transfer.transferred]

 lemmas fBex_funion = bex_Un[Transfer.transferred]

 lemmas fminus_eq_fempty_iff[simp,no_atp] = Diff_eq_empty_iff[Transfer.transferred]

 lemmas fminus_cancel[simp] = Diff_cancel[Transfer.transferred]

 lemmas fminus_idemp[simp] = Diff_idemp[Transfer.transferred]

 lemmas fminus_triv = Diff_triv[Transfer.transferred]

 lemmas fempty_fminus[simp] = empty_Diff[Transfer.transferred]

 lemmas fminus_fempty[simp] = Diff_empty[Transfer.transferred]

 lemmas fminus_finsertffempty[simp,no_atp] = Diff_insert0[Transfer.transferred]

 lemmas fminus_finsert = Diff_insert[Transfer.transferred]

 lemmas fminus_finsert2 = Diff_insert2[Transfer.transferred]

 lemmas finsert_fminus_if = insert_Diff_if[Transfer.transferred]

 lemmas finsert_fminus1[simp] = insert_Diff1[Transfer.transferred]

 lemmas finsert_fminus_single[simp] = insert_Diff_single[Transfer.transferred]

 lemmas finsert_fminus = insert_Diff[Transfer.transferred]

 lemmas fminus_finsert_absorb = Diff_insert_absorb[Transfer.transferred]

 lemmas fminus_disjoint[simp] = Diff_disjoint[Transfer.transferred]

 lemmas fminus_partition = Diff_partition[Transfer.transferred]

 lemmas double_fminus = double_diff[Transfer.transferred]

 lemmas funion_fminus_cancel[simp] = Un_Diff_cancel[Transfer.transferred]

 lemmas funion_fminus_cancel2[simp] = Un_Diff_cancel2[Transfer.transferred]

 lemmas fminus_funion = Diff_Un[Transfer.transferred]

 lemmas fminus_finter = Diff_Int[Transfer.transferred]

 lemmas funion_fminus = Un_Diff[Transfer.transferred]

 lemmas finter_fminus = Int_Diff[Transfer.transferred]

 lemmas fminus_finter_distrib = Diff_Int_distrib[Transfer.transferred]

 lemmas fminus_finter_distrib2 = Diff_Int_distrib2[Transfer.transferred]

 lemmas fUNIV_bool[no_atp] = UNIV_bool[Transfer.transferred]

 lemmas fPow_fempty[simp] = Pow_empty[Transfer.transferred]

 lemmas fPow_finsert = Pow_insert[Transfer.transferred]

 lemmas funion_fPow_fsubset = Un_Pow_subset[Transfer.transferred]

 lemmas fPow_finter_eq[simp] = Pow_Int_eq[Transfer.transferred]

 lemmas fset_eq_fsubset = set_eq_subset[Transfer.transferred]

 lemmas fsubset_iff[no_atp] = subset_iff[Transfer.transferred]

 lemmas fsubset_iff_pfsubset_eq = subset_iff_psubset_eq[Transfer.transferred]

 lemmas all_not_fin_conv[simp] = all_not_in_conv[Transfer.transferred]

 lemmas ex_fin_conv = ex_in_conv[Transfer.transferred]

 lemmas fimage_mono = image_mono[Transfer.transferred]

 lemmas fPow_mono = Pow_mono[Transfer.transferred]

 lemmas finsert_mono = insert_mono[Transfer.transferred]

 lemmas funion_mono = Un_mono[Transfer.transferred]

 lemmas finter_mono = Int_mono[Transfer.transferred]

 lemmas fminus_mono = Diff_mono[Transfer.transferred]

 lemmas fin_mono = in_mono[Transfer.transferred]

 lemmas fthe_felem_eq[simp] = the_elem_eq[Transfer.transferred]

 lemmas fLeast_mono = Least_mono[Transfer.transferred]

 lemmas fbind_fbind = bind_bind[Transfer.transferred]

 lemmas fempty_fbind[simp] = empty_bind[Transfer.transferred]

 lemmas nonfempty_fbind_const = nonempty_bind_const[Transfer.transferred]

 lemmas fbind_const = bind_const[Transfer.transferred]

 lemmas ffmember_filter[simp] = member_filter[Transfer.transferred]

 lemmas fequalityI = equalityI[Transfer.transferred]

 subsection {* Additional lemmas*}

 subsubsection {* @{text fsingleton} *}

 lemmas fsingletonE = fsingletonD [elim_format]

 subsubsection {* @{text femepty} *}

 lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {||}"

 by transfer auto

 (* FIXME, transferred doesn't work here *)

 lemma femptyE [elim!]: "a |\<in>| {||} \<Longrightarrow> P"

   by simp

 subsubsection {* @{text fset} *}

 lemmas fset_simps[simp] = bot_fset.rep_eq finsert.rep_eq

 lemma finite_fset [simp]: 

   shows "finite (fset S)"

   by transfer simp

 lemmas fset_cong = fset_inject

 lemma filter_fset [simp]:

   shows "fset (ffilter P xs) = Collect P \<inter> fset xs"

   by transfer auto

 lemma notin_fset: "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S" by (simp add: fmember.rep_eq)

 lemmas inter_fset[simp] = inf_fset.rep_eq

 lemmas union_fset[simp] = sup_fset.rep_eq

 lemmas minus_fset[simp] = minus_fset.rep_eq

 subsubsection {* @{text filter_fset} *}

 lemma subset_ffilter: 

   "ffilter P A |\<subseteq>| ffilter Q A = (\<forall> x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)"

   by transfer auto

 lemma eq_ffilter: 

   "(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)"

   by transfer auto

 lemma pfsubset_ffilter:

   "(\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| A & \<not> P x & Q x) \<Longrightarrow> 

     ffilter P A |\<subset>| ffilter Q A"

   unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter)

 subsubsection {* @{text finsert} *}

 (* FIXME, transferred doesn't work here *)

 lemma set_finsert:

   assumes "x |\<in>| A"

   obtains B where "A = finsert x B" and "x |\<notin>| B"

 using assms by transfer (metis Set.set_insert finite_insert)

 lemma mk_disjoint_finsert: "a |\<in>| A \<Longrightarrow> \<exists>B. A = finsert a B \<and> a |\<notin>| B"

   by (rule_tac x = "A |-| {|a|}" in exI, blast)

 subsubsection {* @{text fimage} *}

 lemma subset_fimage_iff: "(B |\<subseteq>| f|`|A) = (\<exists> AA. AA |\<subseteq>| A \<and> B = f|`|AA)"

 by transfer (metis mem_Collect_eq rev_finite_subset subset_image_iff)

 subsubsection {* bounded quantification *}

 lemma bex_simps [simp, no_atp]:

   "\<And>A P Q. fBex A (\<lambda>x. P x \<and> Q) = (fBex A P \<and> Q)" 

   "\<And>A P Q. fBex A (\<lambda>x. P \<and> Q x) = (P \<and> fBex A Q)"

   "\<And>P. fBex {||} P = False" 

   "\<And>a B P. fBex (finsert a B) P = (P a \<or> fBex B P)"

   "\<And>A P f. fBex (f |`| A) P = fBex A (\<lambda>x. P (f x))"

   "\<And>A P. (\<not> fBex A P) = fBall A (\<lambda>x. \<not> P x)"

 by auto

 lemma ball_simps [simp, no_atp]:

   "\<And>A P Q. fBall A (\<lambda>x. P x \<or> Q) = (fBall A P \<or> Q)"

   "\<And>A P Q. fBall A (\<lambda>x. P \<or> Q x) = (P \<or> fBall A Q)"

   "\<And>A P Q. fBall A (\<lambda>x. P \<longrightarrow> Q x) = (P \<longrightarrow> fBall A Q)"

   "\<And>A P Q. fBall A (\<lambda>x. P x \<longrightarrow> Q) = (fBex A P \<longrightarrow> Q)"

   "\<And>P. fBall {||} P = True"

   "\<And>a B P. fBall (finsert a B) P = (P a \<and> fBall B P)"

   "\<And>A P f. fBall (f |`| A) P = fBall A (\<lambda>x. P (f x))"

   "\<And>A P. (\<not> fBall A P) = fBex A (\<lambda>x. \<not> P x)"

 by auto

 lemma atomize_fBall:

     "(\<And>x. x |\<in>| A ==> P x) == Trueprop (fBall A (\<lambda>x. P x))"

 apply (simp only: atomize_all atomize_imp)

 apply (rule equal_intr_rule)

 by (transfer, simp)+

 end

 subsubsection {* @{text fcard} *}

 (* FIXME: improve transferred to handle bounded meta quantification *)

 lemma fcard_fempty:

   "fcard {||} = 0"

   by transfer (rule card_empty)

 lemma fcard_finsert_disjoint:

   "x |\<notin>| A \<Longrightarrow> fcard (finsert x A) = Suc (fcard A)"

   by transfer (rule card_insert_disjoint)

 lemma fcard_finsert_if:

   "fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))"

   by transfer (rule card_insert_if)

 lemma card_0_eq [simp, no_atp]:

   "fcard A = 0 \<longleftrightarrow> A = {||}"

   by transfer (rule card_0_eq)

 lemma fcard_Suc_fminus1:

   "x |\<in>| A \<Longrightarrow> Suc (fcard (A |-| {|x|})) = fcard A"

   by transfer (rule card_Suc_Diff1)

 lemma fcard_fminus_fsingleton:

   "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) = fcard A - 1"

   by transfer (rule card_Diff_singleton)

 lemma fcard_fminus_fsingleton_if:

   "fcard (A |-| {|x|}) = (if x |\<in>| A then fcard A - 1 else fcard A)"

   by transfer (rule card_Diff_singleton_if)

 lemma fcard_fminus_finsert[simp]:

   assumes "a |\<in>| A" and "a |\<notin>| B"

   shows "fcard (A |-| finsert a B) = fcard (A |-| B) - 1"

 using assms by transfer (rule card_Diff_insert)

 lemma fcard_finsert: "fcard (finsert x A) = Suc (fcard (A |-| {|x|}))"

 by transfer (rule card_insert)

 lemma fcard_finsert_le: "fcard A \<le> fcard (finsert x A)"

 by transfer (rule card_insert_le)

 lemma fcard_mono:

   "A |\<subseteq>| B \<Longrightarrow> fcard A \<le> fcard B"

 by transfer (rule card_mono)

 lemma fcard_seteq: "A |\<subseteq>| B \<Longrightarrow> fcard B \<le> fcard A \<Longrightarrow> A = B"

 by transfer (rule card_seteq)

 lemma pfsubset_fcard_mono: "A |\<subset>| B \<Longrightarrow> fcard A < fcard B"

 by transfer (rule psubset_card_mono)

 lemma fcard_funion_finter: 

   "fcard A + fcard B = fcard (A |\<union>| B) + fcard (A |\<inter>| B)"

 by transfer (rule card_Un_Int)

 lemma fcard_funion_disjoint:

   "A |\<inter>| B = {||} \<Longrightarrow> fcard (A |\<union>| B) = fcard A + fcard B"

 by transfer (rule card_Un_disjoint)

 lemma fcard_funion_fsubset:

   "B |\<subseteq>| A \<Longrightarrow> fcard (A |-| B) = fcard A - fcard B"

 by transfer (rule card_Diff_subset)

 lemma diff_fcard_le_fcard_fminus:

   "fcard A - fcard B \<le> fcard(A |-| B)"

 by transfer (rule diff_card_le_card_Diff)

 lemma fcard_fminus1_less: "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) < fcard A"

 by transfer (rule card_Diff1_less)

 lemma fcard_fminus2_less:

   "x |\<in>| A \<Longrightarrow> y |\<in>| A \<Longrightarrow> fcard (A |-| {|x|} |-| {|y|}) < fcard A"

 by transfer (rule card_Diff2_less)

 lemma fcard_fminus1_le: "fcard (A |-| {|x|}) \<le> fcard A"

 by transfer (rule card_Diff1_le)

 lemma fcard_pfsubset: "A |\<subseteq>| B \<Longrightarrow> fcard A < fcard B \<Longrightarrow> A < B"

 by transfer (rule card_psubset)

 subsubsection {* @{text ffold} *}

 (* FIXME: improve transferred to handle bounded meta quantification *)

 context comp_fun_commute

 begin

   lemmas ffold_empty[simp] = fold_empty[Transfer.transferred]

   lemma ffold_finsert [simp]:

     assumes "x |\<notin>| A"

     shows "ffold f z (finsert x A) = f x (ffold f z A)"

     using assms by (transfer fixing: f) (rule fold_insert)

   lemma ffold_fun_left_comm:

     "f x (ffold f z A) = ffold f (f x z) A"

     by (transfer fixing: f) (rule fold_fun_left_comm)

   lemma ffold_finsert2:

     "x |\<notin>| A \<Longrightarrow> ffold f z (finsert x A)  = ffold f (f x z) A"

     by (transfer fixing: f) (rule fold_insert2)

   lemma ffold_rec:

     assumes "x |\<in>| A"

     shows "ffold f z A = f x (ffold f z (A |-| {|x|}))"

     using assms by (transfer fixing: f) (rule fold_rec)

   lemma ffold_finsert_fremove:

     "ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"

      by (transfer fixing: f) (rule fold_insert_remove)

 end

 lemma ffold_fimage:

   assumes "inj_on g (fset A)"

   shows "ffold f z (g |`| A) = ffold (f \<circ> g) z A"

 using assms by transfer' (rule fold_image)

 lemma ffold_cong:

   assumes "comp_fun_commute f" "comp_fun_commute g"

   "\<And>x. x |\<in>| A \<Longrightarrow> f x = g x"

     and "s = t" and "A = B"

   shows "ffold f s A = ffold g t B"

 using assms by transfer (metis Finite_Set.fold_cong)

 context comp_fun_idem

 begin

   lemma ffold_finsert_idem:

     "ffold f z (finsert x A)  = f x (ffold f z A)"

     by (transfer fixing: f) (rule fold_insert_idem)

   declare ffold_finsert [simp del] ffold_finsert_idem [simp]

   lemma ffold_finsert_idem2:

     "ffold f z (finsert x A) = ffold f (f x z) A"

     by (transfer fixing: f) (rule fold_insert_idem2)

 end

 subsection {* Choice in fsets *}

 lemma fset_choice: 

   assumes "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"

   shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"

   using assms by transfer metis

 subsection {* Induction and Cases rules for fsets *}

 lemma fset_exhaust [case_names empty insert, cases type: fset]:

   assumes fempty_case: "S = {||} \<Longrightarrow> P" 

   and     finsert_case: "\<And>x S'. S = finsert x S' \<Longrightarrow> P"

   shows "P"

   using assms by transfer blast

 lemma fset_induct [case_names empty insert]:

   assumes fempty_case: "P {||}"

   and     finsert_case: "\<And>x S. P S \<Longrightarrow> P (finsert x S)"

   shows "P S"

 proof -

   (* FIXME transfer and right_total vs. bi_total *)

   note Domainp_forall_transfer[transfer_rule]

   show ?thesis

   using assms by transfer (auto intro: finite_induct)

 qed

 lemma fset_induct_stronger [case_names empty insert, induct type: fset]:

   assumes empty_fset_case: "P {||}"

   and     insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"

   shows "P S"

 proof -

   (* FIXME transfer and right_total vs. bi_total *)

   note Domainp_forall_transfer[transfer_rule]

   show ?thesis

   using assms by transfer (auto intro: finite_induct)

 qed

 lemma fset_card_induct:

   assumes empty_fset_case: "P {||}"

   and     card_fset_Suc_case: "\<And>S T. Suc (fcard S) = (fcard T) \<Longrightarrow> P S \<Longrightarrow> P T"

   shows "P S"

 proof (induct S)

   case empty

   show "P {||}" by (rule empty_fset_case)

 next

   case (insert x S)

   have h: "P S" by fact

   have "x |\<notin>| S" by fact

   then have "Suc (fcard S) = fcard (finsert x S)" 

     by transfer auto

   then show "P (finsert x S)" 

     using h card_fset_Suc_case by simp

 qed

 lemma fset_strong_cases:

   obtains "xs = {||}"

     | ys x where "x |\<notin>| ys" and "xs = finsert x ys"

 by transfer blast

 lemma fset_induct2:

   "P {||} {||} \<Longrightarrow>

   (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>

   (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>

   (\<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>

   P xsa ysa"

   apply (induct xsa arbitrary: ysa)

   apply (induct_tac x rule: fset_induct_stronger)

   apply simp_all

   apply (induct_tac xa rule: fset_induct_stronger)

   apply simp_all

   done

 subsection {* Setup for Lifting/Transfer *}

 subsubsection {* Relator and predicator properties *}

 lift_definition rel_fset :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is rel_set

 parametric rel_set_transfer .

 lemma rel_fset_alt_def: "rel_fset R = (\<lambda>A B. (\<forall>x.\<exists>y. x|\<in>|A \<longrightarrow> y|\<in>|B \<and> R x y) 

   \<and> (\<forall>y. \<exists>x. y|\<in>|B \<longrightarrow> x|\<in>|A \<and> R x y))"

 apply (rule ext)+

 apply transfer'

 apply (subst rel_set_def[unfolded fun_eq_iff]) 

 by blast

 lemma rel_fset_conversep: "rel_fset (conversep R) = conversep (rel_fset R)"

   unfolding rel_fset_alt_def by auto

 lemmas rel_fset_eq [relator_eq] = rel_set_eq[Transfer.transferred]

 lemma rel_fset_mono[relator_mono]: "A \<le> B \<Longrightarrow> rel_fset A \<le> rel_fset B"

 unfolding rel_fset_alt_def by blast

 lemma finite_rel_set:

   assumes fin: "finite X" "finite Z"

   assumes R_S: "rel_set (R OO S) X Z"

   shows "\<exists>Y. finite Y \<and> rel_set R X Y \<and> rel_set S Y Z"

 proof -

   obtain f where f: "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)"

   apply atomize_elim

   apply (subst bchoice_iff[symmetric])

   using R_S[unfolded rel_set_def OO_def] by blast

   obtain g where g: "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R  x (g z))"

   apply atomize_elim

   apply (subst bchoice_iff[symmetric])

   using R_S[unfolded rel_set_def OO_def] by blast

   let ?Y = "f ` X \<union> g ` Z"

   have "finite ?Y" by (simp add: fin)

   moreover have "rel_set R X ?Y"

     unfolding rel_set_def

     using f g by clarsimp blast

   moreover have "rel_set S ?Y Z"

     unfolding rel_set_def

     using f g by clarsimp blast

   ultimately show ?thesis by metis

 qed

 lemma rel_fset_OO[relator_distr]: "rel_fset R OO rel_fset S = rel_fset (R OO S)"

 apply (rule ext)+

 by transfer (auto intro: finite_rel_set rel_set_OO[unfolded fun_eq_iff, rule_format, THEN iffD1])

 lemma Domainp_fset[relator_domain]: "Domainp (rel_fset T) = (\<lambda>A. fBall A (Domainp T))"

 proof -

   obtain f where f: "\<forall>x\<in>Collect (Domainp T). T x (f x)"

     unfolding Domainp_iff[abs_def]

     apply atomize_elim

     by (subst bchoice_iff[symmetric]) (auto iff: bchoice_iff[symmetric])

   from f show ?thesis

     unfolding fun_eq_iff rel_fset_alt_def Domainp_iff

     apply clarify

     apply (rule iffI)

       apply blast

     by (rename_tac A, rule_tac x="f |`| A" in exI, blast)

 qed

 lemma right_total_rel_fset[transfer_rule]: "right_total A \<Longrightarrow> right_total (rel_fset A)"

 unfolding right_total_def 

 apply transfer

 apply (subst(asm) choice_iff)

 apply clarsimp

 apply (rename_tac A f y, rule_tac x = "f ` y" in exI)

 by (auto simp add: rel_set_def)

 lemma left_total_rel_fset[transfer_rule]: "left_total A \<Longrightarrow> left_total (rel_fset A)"

 unfolding left_total_def 

 apply transfer

 apply (subst(asm) choice_iff)

 apply clarsimp

 apply (rename_tac A f y, rule_tac x = "f ` y" in exI)

 by (auto simp add: rel_set_def)

 lemmas right_unique_rel_fset[transfer_rule] = right_unique_rel_set[Transfer.transferred]

 lemmas left_unique_rel_fset[transfer_rule] = left_unique_rel_set[Transfer.transferred]

 thm right_unique_rel_fset left_unique_rel_fset

 lemma bi_unique_rel_fset[transfer_rule]: "bi_unique A \<Longrightarrow> bi_unique (rel_fset A)"

 by (auto intro: right_unique_rel_fset left_unique_rel_fset iff: bi_unique_alt_def)

 lemma bi_total_rel_fset[transfer_rule]: "bi_total A \<Longrightarrow> bi_total (rel_fset A)"

 by (auto intro: right_total_rel_fset left_total_rel_fset iff: bi_total_alt_def)

 lemmas fset_relator_eq_onp [relator_eq_onp] = set_relator_eq_onp[Transfer.transferred]

 subsubsection {* Quotient theorem for the Lifting package *}

 lemma Quotient_fset_map[quot_map]:

   assumes "Quotient R Abs Rep T"

   shows "Quotient (rel_fset R) (fimage Abs) (fimage Rep) (rel_fset T)"

   using assms unfolding Quotient_alt_def4

   by (simp add: rel_fset_OO[symmetric] rel_fset_conversep) (simp add: rel_fset_alt_def, blast)

 subsubsection {* Transfer rules for the Transfer package *}

 text {* Unconditional transfer rules *}

 context

 begin

 interpretation lifting_syntax .

 lemmas fempty_transfer [transfer_rule] = empty_transfer[Transfer.transferred]

 lemma finsert_transfer [transfer_rule]:

   "(A ===> rel_fset A ===> rel_fset A) finsert finsert"

   unfolding rel_fun_def rel_fset_alt_def by blast

 lemma funion_transfer [transfer_rule]:

   "(rel_fset A ===> rel_fset A ===> rel_fset A) funion funion"

   unfolding rel_fun_def rel_fset_alt_def by blast

 lemma ffUnion_transfer [transfer_rule]:

   "(rel_fset (rel_fset A) ===> rel_fset A) ffUnion ffUnion"

   unfolding rel_fun_def rel_fset_alt_def by transfer (simp, fast)

 lemma fimage_transfer [transfer_rule]:

   "((A ===> B) ===> rel_fset A ===> rel_fset B) fimage fimage"

   unfolding rel_fun_def rel_fset_alt_def by simp blast

 lemma fBall_transfer [transfer_rule]:

   "(rel_fset A ===> (A ===> op =) ===> op =) fBall fBall"

   unfolding rel_fset_alt_def rel_fun_def by blast

 lemma fBex_transfer [transfer_rule]:

   "(rel_fset A ===> (A ===> op =) ===> op =) fBex fBex"

   unfolding rel_fset_alt_def rel_fun_def by blast

 (* FIXME transfer doesn't work here *)

 lemma fPow_transfer [transfer_rule]:

   "(rel_fset A ===> rel_fset (rel_fset A)) fPow fPow"

   unfolding rel_fun_def

   using Pow_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred]

   by blast

 lemma rel_fset_transfer [transfer_rule]:

   "((A ===> B ===> op =) ===> rel_fset A ===> rel_fset B ===> op =)

     rel_fset rel_fset"

   unfolding rel_fun_def

   using rel_set_transfer[unfolded rel_fun_def,rule_format, Transfer.transferred, where A = A and B = B]

   by simp

 lemma bind_transfer [transfer_rule]:

   "(rel_fset A ===> (A ===> rel_fset B) ===> rel_fset B) fbind fbind"

   using assms unfolding rel_fun_def

   using bind_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast

 text {* Rules requiring bi-unique, bi-total or right-total relations *}

 lemma fmember_transfer [transfer_rule]:

   assumes "bi_unique A"

   shows "(A ===> rel_fset A ===> op =) (op |\<in>|) (op |\<in>|)"

   using assms unfolding rel_fun_def rel_fset_alt_def bi_unique_def by metis

 lemma finter_transfer [transfer_rule]:

   assumes "bi_unique A"

   shows "(rel_fset A ===> rel_fset A ===> rel_fset A) finter finter"

   using assms unfolding rel_fun_def

   using inter_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast

 lemma fminus_transfer [transfer_rule]:

   assumes "bi_unique A"

   shows "(rel_fset A ===> rel_fset A ===> rel_fset A) (op |-|) (op |-|)"

   using assms unfolding rel_fun_def

   using Diff_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast

 lemma fsubset_transfer [transfer_rule]:

   assumes "bi_unique A"

   shows "(rel_fset A ===> rel_fset A ===> op =) (op |\<subseteq>|) (op |\<subseteq>|)"

   using assms unfolding rel_fun_def

   using subset_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast

 lemma fSup_transfer [transfer_rule]:

   "bi_unique A \<Longrightarrow> (rel_set (rel_fset A) ===> rel_fset A) Sup Sup"

   using assms unfolding rel_fun_def

   apply clarify

   apply transfer'

   using Sup_fset_transfer[unfolded rel_fun_def] by blast

 (* FIXME: add right_total_fInf_transfer *)

 lemma fInf_transfer [transfer_rule]:

   assumes "bi_unique A" and "bi_total A"

   shows "(rel_set (rel_fset A) ===> rel_fset A) Inf Inf"

   using assms unfolding rel_fun_def

   apply clarify

   apply transfer'

   using Inf_fset_transfer[unfolded rel_fun_def] by blast

 lemma ffilter_transfer [transfer_rule]:

   assumes "bi_unique A"

   shows "((A ===> op=) ===> rel_fset A ===> rel_fset A) ffilter ffilter"

   using assms unfolding rel_fun_def

   using Lifting_Set.filter_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast

 lemma card_transfer [transfer_rule]:

   "bi_unique A \<Longrightarrow> (rel_fset A ===> op =) fcard fcard"

   using assms unfolding rel_fun_def

   using card_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast

 end

 lifting_update fset.lifting

 lifting_forget fset.lifting

 subsection {* BNF setup *}

 context

 includes fset.lifting

 begin

 lemma rel_fset_alt:

   "rel_fset R a b \<longleftrightarrow> (\<forall>t \<in> fset a. \<exists>u \<in> fset b. R t u) \<and> (\<forall>t \<in> fset b. \<exists>u \<in> fset a. R u t)"

 by transfer (simp add: rel_set_def)

 lemma fset_to_fset: "finite A \<Longrightarrow> fset (the_inv fset A) = A"

 apply (rule f_the_inv_into_f[unfolded inj_on_def])

 apply (simp add: fset_inject)

 apply (rule range_eqI Abs_fset_inverse[symmetric] CollectI)+

 .

 lemma rel_fset_aux:

 "(\<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>

  ((BNF_Util.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage fst))\<inverse>\<inverse> OO

   BNF_Util.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage snd)) a b" (is "?L = ?R")

 proof

   assume ?L

   def R' \<equiv> "the_inv fset (Collect (split R) \<inter> (fset a \<times> fset b))" (is "the_inv fset ?L'")

   have "finite ?L'" by (intro finite_Int[OF disjI2] finite_cartesian_product) (transfer, simp)+

   hence *: "fset R' = ?L'" unfolding R'_def by (intro fset_to_fset)

   show ?R unfolding Grp_def relcompp.simps conversep.simps

   proof (intro CollectI case_prodI exI[of _ a] exI[of _ b] exI[of _ R'] conjI refl)

     from * show "a = fimage fst R'" using conjunct1[OF `?L`]

       by (transfer, auto simp add: image_def Int_def split: prod.splits)

     from * show "b = fimage snd R'" using conjunct2[OF `?L`]

       by (transfer, auto simp add: image_def Int_def split: prod.splits)

   qed (auto simp add: *)

 next

   assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps

   apply (simp add: subset_eq Ball_def)

   apply (rule conjI)

   apply (transfer, clarsimp, metis snd_conv)

   by (transfer, clarsimp, metis fst_conv)

 qed

 bnf "'a fset"

   map: fimage

   sets: fset 

   bd: natLeq

   wits: "{||}"

   rel: rel_fset

 apply -

           apply transfer' apply simp

          apply transfer' apply force

         apply transfer apply force

        apply transfer' apply force

       apply (rule natLeq_card_order)

      apply (rule natLeq_cinfinite)

     apply transfer apply (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq)

    apply (fastforce simp: rel_fset_alt)

  apply (simp add: Grp_def relcompp.simps conversep.simps fun_eq_iff rel_fset_alt rel_fset_aux) 

 apply transfer apply simp

 done

 lemma rel_fset_fset: "rel_set \<chi> (fset A1) (fset A2) = rel_fset \<chi> A1 A2"

   by transfer (rule refl)

 end

 lemmas [simp] = fset.map_comp fset.map_id fset.set_map

 subsection {* Advanced relator customization *}

 (* Set vs. sum relators: *)

 lemma rel_set_rel_sum[simp]: 

 "rel_set (rel_sum \<chi> \<phi>) A1 A2 \<longleftrightarrow> 

  rel_set \<chi> (Inl -` A1) (Inl -` A2) \<and> rel_set \<phi> (Inr -` A1) (Inr -` A2)"

 (is "?L \<longleftrightarrow> ?Rl \<and> ?Rr")

 proof safe

   assume L: "?L"

   show ?Rl unfolding rel_set_def Bex_def vimage_eq proof safe

     fix l1 assume "Inl l1 \<in> A1"

     then obtain a2 where a2: "a2 \<in> A2" and "rel_sum \<chi> \<phi> (Inl l1) a2"

     using L unfolding rel_set_def by auto

     then obtain l2 where "a2 = Inl l2 \<and> \<chi> l1 l2" by (cases a2, auto)

     thus "\<exists> l2. Inl l2 \<in> A2 \<and> \<chi> l1 l2" using a2 by auto

   next

     fix l2 assume "Inl l2 \<in> A2"

     then obtain a1 where a1: "a1 \<in> A1" and "rel_sum \<chi> \<phi> a1 (Inl l2)"

     using L unfolding rel_set_def by auto

     then obtain l1 where "a1 = Inl l1 \<and> \<chi> l1 l2" by (cases a1, auto)

     thus "\<exists> l1. Inl l1 \<in> A1 \<and> \<chi> l1 l2" using a1 by auto

   qed

   show ?Rr unfolding rel_set_def Bex_def vimage_eq proof safe

     fix r1 assume "Inr r1 \<in> A1"

     then obtain a2 where a2: "a2 \<in> A2" and "rel_sum \<chi> \<phi> (Inr r1) a2"

     using L unfolding rel_set_def by auto

     then obtain r2 where "a2 = Inr r2 \<and> \<phi> r1 r2" by (cases a2, auto)

     thus "\<exists> r2. Inr r2 \<in> A2 \<and> \<phi> r1 r2" using a2 by auto

   next

     fix r2 assume "Inr r2 \<in> A2"

     then obtain a1 where a1: "a1 \<in> A1" and "rel_sum \<chi> \<phi> a1 (Inr r2)"

     using L unfolding rel_set_def by auto

     then obtain r1 where "a1 = Inr r1 \<and> \<phi> r1 r2" by (cases a1, auto)

     thus "\<exists> r1. Inr r1 \<in> A1 \<and> \<phi> r1 r2" using a1 by auto

   qed

 next

   assume Rl: "?Rl" and Rr: "?Rr"

   show ?L unfolding rel_set_def Bex_def vimage_eq proof safe

     fix a1 assume a1: "a1 \<in> A1"

     show "\<exists> a2. a2 \<in> A2 \<and> rel_sum \<chi> \<phi> a1 a2"

     proof(cases a1)

       case (Inl l1) then obtain l2 where "Inl l2 \<in> A2 \<and> \<chi> l1 l2"

       using Rl a1 unfolding rel_set_def by blast

       thus ?thesis unfolding Inl by auto

     next

       case (Inr r1) then obtain r2 where "Inr r2 \<in> A2 \<and> \<phi> r1 r2"

       using Rr a1 unfolding rel_set_def by blast

       thus ?thesis unfolding Inr by auto

     qed

   next

     fix a2 assume a2: "a2 \<in> A2"

     show "\<exists> a1. a1 \<in> A1 \<and> rel_sum \<chi> \<phi> a1 a2"

     proof(cases a2)

       case (Inl l2) then obtain l1 where "Inl l1 \<in> A1 \<and> \<chi> l1 l2"

       using Rl a2 unfolding rel_set_def by blast

       thus ?thesis unfolding Inl by auto

     next

       case (Inr r2) then obtain r1 where "Inr r1 \<in> A1 \<and> \<phi> r1 r2"

       using Rr a2 unfolding rel_set_def by blast

       thus ?thesis unfolding Inr by auto

     qed

   qed

 qed

 end