src/HOL/Library/FSet.thy

proper invariants

(*  Title:      HOL/Library/FSet.thy
    Author:     Ondrej Kuncar, TU Muenchen
    Author:     Cezary Kaliszyk and Christian Urban
    Author:     Andrei Popescu, TU Muenchen
*)

section \<open>Type of finite sets defined as a subtype of sets\<close>

theory FSet
imports Main Countable
begin

subsection \<open>Definition of the type\<close>

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

setup_lifting type_definition_fset


subsection \<open>Basic operations and type class instantiations\<close>

(* FIXME transfer and right_total vs. bi_total *)
instantiation fset :: (finite) finite
begin
instance by (standard; transfer; simp)
end

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

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]:
  includes lifting_syntax
  assumes [transfer_rule]: "bi_unique A"
  shows "((pcr_fset A) ===> (pcr_fset A) ===> (=)) (\<subset>) (<)"
  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 (standard; 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

context includes lifting_syntax
begin

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 (=)) ===> (=)) bdd_below bdd_below"
  by auto

end

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 (standard; 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 (standard; transfer) (simp_all add: Inf_Sup 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"

declare top_fset.rep_eq[simp]


subsection \<open>Other operations\<close>

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 .

lemma fmember_iff_member_fset: "x |\<in>| A \<longleftrightarrow> x \<in> fset A"
  by (rule fmember.rep_eq)

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

context includes lifting_syntax
begin

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 .

lift_definition fset_of_list :: "'a list \<Rightarrow> 'a fset" is set by (rule finite_set)

lift_definition sorted_list_of_fset :: "'a::linorder fset \<Rightarrow> 'a list" is sorted_list_of_set .

subsection \<open>Transferred lemmas from Set.thy\<close>

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[fundef_cong] = ball_cong[Transfer.transferred]
lemmas fBex_cong[fundef_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 = rev_subsetD[Transfer.transferred]
lemmas fset_mp = subsetD[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 if_split_fmem1 = if_split_mem1[Transfer.transferred]
lemmas if_split_fmem2 = if_split_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 ffunion_empty[simp] = Union_empty[Transfer.transferred]
lemmas ffunion_mono = Union_mono[Transfer.transferred]
lemmas ffunion_insert[simp] = Union_insert[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]
lemmas fset_of_list_simps[simp] = set_simps[Transfer.transferred]
lemmas fset_of_list_append[simp] = set_append[Transfer.transferred]
lemmas fset_of_list_rev[simp] = set_rev[Transfer.transferred]
lemmas fset_of_list_map[simp] = set_map[Transfer.transferred]


subsection \<open>Additional lemmas\<close>

subsubsection \<open>\<open>ffUnion\<close>\<close>

lemmas ffUnion_funion_distrib[simp] = Union_Un_distrib[Transfer.transferred]


subsubsection \<open>\<open>fbind\<close>\<close>

lemma fbind_cong[fundef_cong]: "A = B \<Longrightarrow> (\<And>x. x |\<in>| B \<Longrightarrow> f x = g x) \<Longrightarrow> fbind A f = fbind B g"
by transfer force


subsubsection \<open>\<open>fsingleton\<close>\<close>

lemmas fsingletonE = fsingletonD [elim_format]


subsubsection \<open>\<open>femepty\<close>\<close>

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 \<open>\<open>fset\<close>\<close>

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_iff_member_fset)

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 \<open>\<open>ffilter\<close>\<close>

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 \<and> \<not> P x \<and> Q x) \<Longrightarrow>
    ffilter P A |\<subset>| ffilter Q A"
  unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter)


subsubsection \<open>\<open>fset_of_list\<close>\<close>

lemma fset_of_list_filter[simp]:
  "fset_of_list (filter P xs) = ffilter P (fset_of_list xs)"
  by transfer (auto simp: Set.filter_def)

lemma fset_of_list_subset[intro]:
  "set xs \<subseteq> set ys \<Longrightarrow> fset_of_list xs |\<subseteq>| fset_of_list ys"
  by transfer simp

lemma fset_of_list_elem: "(x |\<in>| fset_of_list xs) \<longleftrightarrow> (x \<in> set xs)"
  by transfer simp


subsubsection \<open>\<open>finsert\<close>\<close>

(* 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 exI [where x = "A |-| {|a|}"]) blast

lemma finsert_eq_iff:
  assumes "a |\<notin>| A" and "b |\<notin>| B"
  shows "(finsert a A = finsert b B) =
    (if a = b then A = B else \<exists>C. A = finsert b C \<and> b |\<notin>| C \<and> B = finsert a C \<and> a |\<notin>| C)"
  using assms by transfer (force simp: insert_eq_iff)


subsubsection \<open>\<open>fimage\<close>\<close>

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)

lemma fimage_strict_mono:
  assumes "inj_on f (fset B)" and "A |\<subset>| B"
  shows "f |`| A |\<subset>| f |`| B"
  \<comment> \<open>TODO: Configure transfer framework to lift @{thm Fun.image_strict_mono}.\<close>
proof (rule pfsubsetI)
  from \<open>A |\<subset>| B\<close> have "A |\<subseteq>| B"
    by (rule pfsubset_imp_fsubset)
  thus "f |`| A |\<subseteq>| f |`| B"
    by (rule fimage_mono)
next
  from \<open>A |\<subset>| B\<close> have "A |\<subseteq>| B" and "A \<noteq> B"
    by (simp_all add: pfsubset_eq)

  have "fset A \<noteq> fset B"
    using \<open>A \<noteq> B\<close>
    by (simp add: fset_cong)
  hence "f ` fset A \<noteq> f ` fset B"
    using \<open>A |\<subseteq>| B\<close>
    by (simp add: inj_on_image_eq_iff[OF \<open>inj_on f (fset B)\<close>] less_eq_fset.rep_eq)
  hence "fset (f |`| A) \<noteq> fset (f |`| B)"
    by (simp add: fimage.rep_eq)
  thus "f |`| A \<noteq> f |`| B"
    by (simp add: fset_cong)
qed


subsubsection \<open>bounded quantification\<close>

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)+

lemma fBall_mono[mono]: "P \<le> Q \<Longrightarrow> fBall S P \<le> fBall S Q"
by auto

lemma fBex_mono[mono]: "P \<le> Q \<Longrightarrow> fBex S P \<le> fBex S Q"
by auto

end


subsubsection \<open>\<open>fcard\<close>\<close>

(* 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 fcard_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_remove)

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 \<open>\<open>sorted_list_of_fset\<close>\<close>

lemma sorted_list_of_fset_simps[simp]:
  "set (sorted_list_of_fset S) = fset S"
  "fset_of_list (sorted_list_of_fset S) = S"
by (transfer, simp)+


subsubsection \<open>\<open>ffold\<close>\<close>

(* 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[unfolded comp_fun_commute_def']
  by transfer (meson Finite_Set.fold_cong subset_UNIV)

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


subsubsection \<open>@{term fsubset}\<close>

lemma wfP_pfsubset: "wfP (|\<subset>|)"
proof (rule wfP_if_convertible_to_nat)
  show "\<And>x y. x |\<subset>| y \<Longrightarrow> fcard x < fcard y"
    by (rule pfsubset_fcard_mono)
qed


subsubsection \<open>Group operations\<close>

locale comm_monoid_fset = comm_monoid
begin

sublocale set: comm_monoid_set ..

lift_definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b fset \<Rightarrow> 'a" is set.F .

lemmas cong[fundef_cong] = set.cong[Transfer.transferred]

lemma cong_simp[cong]:
  "\<lbrakk> A = B;  \<And>x. x |\<in>| B =simp=> g x = h x \<rbrakk> \<Longrightarrow> F g A = F h B"
unfolding simp_implies_def by (auto cong: cong)

end

context comm_monoid_add begin

sublocale fsum: comm_monoid_fset plus 0
  rewrites "comm_monoid_set.F plus 0 = sum"
  defines fsum = fsum.F
proof -
  show "comm_monoid_fset (+) 0" by standard

  show "comm_monoid_set.F (+) 0 = sum" unfolding sum_def ..
qed

end


subsubsection \<open>Semilattice operations\<close>

locale semilattice_fset = semilattice
begin

sublocale set: semilattice_set ..

lift_definition F :: "'a fset \<Rightarrow> 'a" is set.F .

lemma eq_fold: "F (finsert x A) = ffold f x A"
  by transfer (rule set.eq_fold)

lemma singleton [simp]: "F {|x|} = x"
  by transfer (rule set.singleton)

lemma insert_not_elem: "x |\<notin>| A \<Longrightarrow> A \<noteq> {||} \<Longrightarrow> F (finsert x A) = x \<^bold>* F A"
  by transfer (rule set.insert_not_elem)

lemma in_idem: "x |\<in>| A \<Longrightarrow> x \<^bold>* F A = F A"
  by transfer (rule set.in_idem)

lemma insert [simp]: "A \<noteq> {||} \<Longrightarrow> F (finsert x A) = x \<^bold>* F A"
  by transfer (rule set.insert)

end

locale semilattice_order_fset = binary?: semilattice_order + semilattice_fset
begin

end


context linorder begin

sublocale fMin: semilattice_order_fset min less_eq less
  rewrites "semilattice_set.F min = Min"
  defines fMin = fMin.F
proof -
  show "semilattice_order_fset min (\<le>) (<)" by standard

  show "semilattice_set.F min = Min" unfolding Min_def ..
qed

sublocale fMax: semilattice_order_fset max greater_eq greater
  rewrites "semilattice_set.F max = Max"
  defines fMax = fMax.F
proof -
  show "semilattice_order_fset max (\<ge>) (>)"
    by standard

  show "semilattice_set.F max = Max"
    unfolding Max_def ..
qed

end

lemma mono_fMax_commute: "mono f \<Longrightarrow> A \<noteq> {||} \<Longrightarrow> f (fMax A) = fMax (f |`| A)"
  by transfer (rule mono_Max_commute)

lemma mono_fMin_commute: "mono f \<Longrightarrow> A \<noteq> {||} \<Longrightarrow> f (fMin A) = fMin (f |`| A)"
  by transfer (rule mono_Min_commute)

lemma fMax_in[simp]: "A \<noteq> {||} \<Longrightarrow> fMax A |\<in>| A"
  by transfer (rule Max_in)

lemma fMin_in[simp]: "A \<noteq> {||} \<Longrightarrow> fMin A |\<in>| A"
  by transfer (rule Min_in)

lemma fMax_ge[simp]: "x |\<in>| A \<Longrightarrow> x \<le> fMax A"
  by transfer (rule Max_ge)

lemma fMin_le[simp]: "x |\<in>| A \<Longrightarrow> fMin A \<le> x"
  by transfer (rule Min_le)

lemma fMax_eqI: "(\<And>y. y |\<in>| A \<Longrightarrow> y \<le> x) \<Longrightarrow> x |\<in>| A \<Longrightarrow> fMax A = x"
  by transfer (rule Max_eqI)

lemma fMin_eqI: "(\<And>y. y |\<in>| A \<Longrightarrow> x \<le> y) \<Longrightarrow> x |\<in>| A \<Longrightarrow> fMin A = x"
  by transfer (rule Min_eqI)

lemma fMax_finsert[simp]: "fMax (finsert x A) = (if A = {||} then x else max x (fMax A))"
  by transfer simp

lemma fMin_finsert[simp]: "fMin (finsert x A) = (if A = {||} then x else min x (fMin A))"
  by transfer simp

context linorder begin

lemma fset_linorder_max_induct[case_names fempty finsert]:
  assumes "P {||}"
  and     "\<And>x S. \<lbrakk>\<forall>y. y |\<in>| S \<longrightarrow> y < x; 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 fixing: less) (auto intro: finite_linorder_max_induct)
qed

lemma fset_linorder_min_induct[case_names fempty finsert]:
  assumes "P {||}"
  and     "\<And>x S. \<lbrakk>\<forall>y. y |\<in>| S \<longrightarrow> y > x; 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 fixing: less) (auto intro: finite_linorder_min_induct)
qed

end


subsection \<open>Choice in fsets\<close>

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 \<open>Induction and Cases rules for fsets\<close>

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 \<open>Setup for Lifting/Transfer\<close>

subsubsection \<open>Relator and predicator properties\<close>

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 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

subsubsection \<open>Transfer rules for the Transfer package\<close>

text \<open>Unconditional transfer rules\<close>

context includes lifting_syntax
begin

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 ===> (=)) ===> (=)) fBall fBall"
  unfolding rel_fset_alt_def rel_fun_def by blast

lemma fBex_transfer [transfer_rule]:
  "(rel_fset A ===> (A ===> (=)) ===> (=)) 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 ===> (=)) ===> rel_fset A ===> rel_fset B ===> (=))
    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"
  unfolding rel_fun_def
  using bind_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast

text \<open>Rules requiring bi-unique, bi-total or right-total relations\<close>

lemma fmember_transfer [transfer_rule]:
  assumes "bi_unique A"
  shows "(A ===> rel_fset A ===> (=)) (|\<in>|) (|\<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) (|-|) (|-|)"
  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 ===> (=)) (|\<subseteq>|) (|\<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"
  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 ===> (=)) ===> 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 ===> (=)) fcard fcard"
  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 \<open>BNF setup\<close>

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_Def.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage fst))\<inverse>\<inverse> OO
  BNF_Def.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage snd)) a b" (is "?L = ?R")
proof
  assume ?L
  define R' where "R' =
    the_inv fset (Collect (case_prod 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 \<open>?L\<close>]
      by (transfer, auto simp add: image_def Int_def split: prod.splits)
    from * show "b = fimage snd R'" using conjunct2[OF \<open>?L\<close>]
      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 (rule regularCard_natLeq)
    apply transfer apply (metis 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[unfolded OO_Grp_alt])
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 \<open>Size setup\<close>

context includes fset.lifting begin
lift_definition size_fset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a fset \<Rightarrow> nat" is "\<lambda>f. sum (Suc \<circ> f)" .
end

instantiation fset :: (type) size begin
definition size_fset where
  size_fset_overloaded_def: "size_fset = FSet.size_fset (\<lambda>_. 0)"
instance ..
end

lemmas size_fset_simps[simp] =
  size_fset_def[THEN meta_eq_to_obj_eq, THEN fun_cong, THEN fun_cong,
    unfolded map_fun_def comp_def id_apply]

lemmas size_fset_overloaded_simps[simp] =
  size_fset_simps[of "\<lambda>_. 0", unfolded add_0_left add_0_right,
    folded size_fset_overloaded_def]

lemma fset_size_o_map: "inj f \<Longrightarrow> size_fset g \<circ> fimage f = size_fset (g \<circ> f)"
  apply (subst fun_eq_iff)
  including fset.lifting by transfer (auto intro: sum.reindex_cong subset_inj_on)

setup \<open>
BNF_LFP_Size.register_size_global \<^type_name>\<open>fset\<close> \<^const_name>\<open>size_fset\<close>
  @{thm size_fset_overloaded_def} @{thms size_fset_simps size_fset_overloaded_simps}
  @{thms fset_size_o_map}
\<close>

lifting_update fset.lifting
lifting_forget fset.lifting

subsection \<open>Advanced relator customization\<close>

text \<open>Set vs. sum relators:\<close>

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


subsubsection \<open>Countability\<close>

lemma exists_fset_of_list: "\<exists>xs. fset_of_list xs = S"
including fset.lifting
by transfer (rule finite_list)

lemma fset_of_list_surj[simp, intro]: "surj fset_of_list"
proof -
  have "x \<in> range fset_of_list" for x :: "'a fset"
    unfolding image_iff
    using exists_fset_of_list by fastforce
  thus ?thesis by auto
qed

instance fset :: (countable) countable
proof
  obtain to_nat :: "'a list \<Rightarrow> nat" where "inj to_nat"
    by (metis ex_inj)
  moreover have "inj (inv fset_of_list)"
    using fset_of_list_surj by (rule surj_imp_inj_inv)
  ultimately have "inj (to_nat \<circ> inv fset_of_list)"
    by (rule inj_compose)
  thus "\<exists>to_nat::'a fset \<Rightarrow> nat. inj to_nat"
    by auto
qed


subsection \<open>Quickcheck setup\<close>

text \<open>Setup adapted from sets.\<close>

notation Quickcheck_Exhaustive.orelse (infixr "orelse" 55)

context
  includes term_syntax
begin

definition [code_unfold]:
"valterm_femptyset = Code_Evaluation.valtermify ({||} :: ('a :: typerep) fset)"

definition [code_unfold]:
"valtermify_finsert x s = Code_Evaluation.valtermify finsert {\<cdot>} (x :: ('a :: typerep * _)) {\<cdot>} s"

end

instantiation fset :: (exhaustive) exhaustive
begin

fun exhaustive_fset where
"exhaustive_fset f i = (if i = 0 then None else (f {||} orelse exhaustive_fset (\<lambda>A. f A orelse Quickcheck_Exhaustive.exhaustive (\<lambda>x. if x |\<in>| A then None else f (finsert x A)) (i - 1)) (i - 1)))"

instance ..

end

instantiation fset :: (full_exhaustive) full_exhaustive
begin

fun full_exhaustive_fset where
"full_exhaustive_fset f i = (if i = 0 then None else (f valterm_femptyset orelse full_exhaustive_fset (\<lambda>A. f A orelse Quickcheck_Exhaustive.full_exhaustive (\<lambda>x. if fst x |\<in>| fst A then None else f (valtermify_finsert x A)) (i - 1)) (i - 1)))"

instance ..

end

no_notation Quickcheck_Exhaustive.orelse (infixr "orelse" 55)

instantiation fset :: (random) random
begin

context
  includes state_combinator_syntax
begin

fun random_aux_fset :: "natural \<Rightarrow> natural \<Rightarrow> natural \<times> natural \<Rightarrow> ('a fset \<times> (unit \<Rightarrow> term)) \<times> natural \<times> natural" where
"random_aux_fset 0 j = Quickcheck_Random.collapse (Random.select_weight [(1, Pair valterm_femptyset)])" |
"random_aux_fset (Code_Numeral.Suc i) j =
  Quickcheck_Random.collapse (Random.select_weight
    [(1, Pair valterm_femptyset),
     (Code_Numeral.Suc i,
      Quickcheck_Random.random j \<circ>\<rightarrow> (\<lambda>x. random_aux_fset i j \<circ>\<rightarrow> (\<lambda>s. Pair (valtermify_finsert x s))))])"

lemma [code]:
  "random_aux_fset i j =
    Quickcheck_Random.collapse (Random.select_weight [(1, Pair valterm_femptyset),
      (i, Quickcheck_Random.random j \<circ>\<rightarrow> (\<lambda>x. random_aux_fset (i - 1) j \<circ>\<rightarrow> (\<lambda>s. Pair (valtermify_finsert x s))))])"
proof (induct i rule: natural.induct)
  case zero
  show ?case by (subst select_weight_drop_zero[symmetric]) (simp add: less_natural_def)
next
  case (Suc i)
  show ?case by (simp only: random_aux_fset.simps Suc_natural_minus_one)
qed

definition "random_fset i = random_aux_fset i i"

instance ..

end

end

end