# Theory FSet

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

section ‹Type of finite sets defined as a subtype of sets›

theory FSet
imports Main Countable
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 (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 ⇒ 'a fset ⇒ bool" is subset_eq parametric subset_transfer
.

definition less_fset :: "'a fset ⇒ 'a fset ⇒ bool" where "xs < ys ≡ xs ≤ ys ∧ xs ≠ (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) ===> (=)) (⊂) (<)"
unfolding less_fset_def[abs_def] psubset_eq[abs_def] by transfer_prover

lift_definition sup_fset :: "'a fset ⇒ 'a fset ⇒ 'a fset" is union parametric union_transfer
by simp

lift_definition inf_fset :: "'a fset ⇒ 'a fset ⇒ 'a fset" is inter parametric inter_transfer
by simp

lift_definition minus_fset :: "'a fset ⇒ 'a fset ⇒ 'a fset" is minus parametric Diff_transfer
by simp

instance
by (standard; transfer; auto)+

end

abbreviation fempty :: "'a fset" ("{||}") where "{||} ≡ bot"
abbreviation fsubset_eq :: "'a fset ⇒ 'a fset ⇒ bool" (infix "|⊆|" 50) where "xs |⊆| ys ≡ xs ≤ ys"
abbreviation fsubset :: "'a fset ⇒ 'a fset ⇒ bool" (infix "|⊂|" 50) where "xs |⊂| ys ≡ xs < ys"
abbreviation funion :: "'a fset ⇒ 'a fset ⇒ 'a fset" (infixl "|∪|" 65) where "xs |∪| ys ≡ sup xs ys"
abbreviation finter :: "'a fset ⇒ 'a fset ⇒ 'a fset" (infixl "|∩|" 65) where "xs |∩| ys ≡ inf xs ys"
abbreviation fminus :: "'a fset ⇒ 'a fset ⇒ 'a fset" (infixl "|-|" 65) where "xs |-| ys ≡ minus xs ys"

instantiation fset :: (equal) equal
begin
definition "HOL.equal A B ⟷ A |⊆| B ∧ B |⊆| 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)
(λS. if finite (⋂S ∩ Collect (Domainp A)) then ⋂S ∩ Collect (Domainp A) else {})
(λS. 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) (λA. if finite (Inf A) then Inf A else {})
(λA. if finite (Inf A) then Inf A else {})"
by transfer_prover

lift_definition Inf_fset :: "'a fset set ⇒ 'a fset" is "λA. 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) (λA. if finite (Sup A) then Sup A else {})
(λA. if finite (Sup A) then Sup A else {})" by transfer_prover

lift_definition Sup_fset :: "'a fset set ⇒ 'a fset" is "λA. if finite (Sup A) then Sup A else {}"
parametric Sup_fset_transfer by simp

lemma finite_Sup: "∃z. finite z ∧ (∀a. a ∈ X ⟶ a ≤ z) ⟹ 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 ∈ X" "bdd_below X"
then show "Inf X |⊆| x" by transfer auto
next
assume "X ≠ {}" "(⋀x. x ∈ X ⟹ z |⊆| x)"
then show "z |⊆| Inf X" by transfer (clarsimp, blast)
next
assume "x ∈ X" "bdd_above X"
then obtain z where "x ∈ X" "(⋀x. x ∈ X ⟹ x |⊆| z)"
by (auto simp: bdd_above_def)
then show "x |⊆| Sup X"
by transfer (auto intro!: finite_Sup)
next
assume "X ≠ {}" "(⋀x. x ∈ X ⟹ x |⊆| z)"
then show "Sup X |⊆| 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 ⇒ '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 ≡ top"
abbreviation fuminus :: "'a::finite fset ⇒ 'a fset" ("|-| _" [81] 80) where "|-| x ≡ uminus x"

declare top_fset.rep_eq[simp]

subsection ‹Other operations›

lift_definition finsert :: "'a ⇒ 'a fset ⇒ '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 ⇒ 'a fset ⇒ bool" (infix "|∈|" 50) is Set.member
parametric member_transfer .

abbreviation notin_fset :: "'a ⇒ 'a fset ⇒ bool" (infix "|∉|" 50) where "x |∉| S ≡ ¬ (x |∈| S)"

context includes lifting_syntax
begin

lift_definition ffilter :: "('a ⇒ bool) ⇒ 'a fset ⇒ 'a fset" is Set.filter
parametric Lifting_Set.filter_transfer unfolding Set.filter_def by simp

lift_definition fPow :: "'a fset ⇒ 'a fset fset" is Pow parametric Pow_transfer

lift_definition fcard :: "'a fset ⇒ nat" is card parametric card_transfer .

lift_definition fimage :: "('a ⇒ 'b) ⇒ 'a fset ⇒ 'b fset" (infixr "|`|" 90) is image
parametric image_transfer by simp

lift_definition fthe_elem :: "'a fset ⇒ 'a" is the_elem .

lift_definition fbind :: "'a fset ⇒ ('a ⇒ 'b fset) ⇒ 'b fset" is Set.bind parametric bind_transfer

lift_definition ffUnion :: "'a fset fset ⇒ 'a fset" is Union parametric Union_transfer by simp

lift_definition fBall :: "'a fset ⇒ ('a ⇒ bool) ⇒ bool" is Ball parametric Ball_transfer .
lift_definition fBex :: "'a fset ⇒ ('a ⇒ bool) ⇒ bool" is Bex parametric Bex_transfer .

lift_definition ffold :: "('a ⇒ 'b ⇒ 'b) ⇒ 'b ⇒ 'a fset ⇒ 'b" is Finite_Set.fold .

lift_definition fset_of_list :: "'a list ⇒ 'a fset" is set by (rule finite_set)

lift_definition sorted_list_of_fset :: "'a::linorder fset ⇒ 'a list" is sorted_list_of_set .

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[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 = 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 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]

subsubsection ‹‹ffUnion››

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

subsubsection ‹‹fbind››

lemma fbind_cong[fundef_cong]: "A = B ⟹ (⋀x. x |∈| B ⟹ f x = g x) ⟹ fbind A f = fbind B g"
by transfer force

subsubsection ‹‹fsingleton››

lemmas fsingletonE = fsingletonD [elim_format]

subsubsection ‹‹femepty››

lemma fempty_ffilter[simp]: "ffilter (λ_. False) A = {||}"
by transfer auto

(* FIXME, transferred doesn't work here *)
lemma femptyE [elim!]: "a |∈| {||} ⟹ P"
by simp

subsubsection ‹‹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 ∩ fset xs"
by transfer auto

lemma notin_fset: "x |∉| S ⟷ x ∉ 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 ‹‹ffilter››

lemma subset_ffilter:
"ffilter P A |⊆| ffilter Q A = (∀ x. x |∈| A ⟶ P x ⟶ Q x)"
by transfer auto

lemma eq_ffilter:
"(ffilter P A = ffilter Q A) = (∀x. x |∈| A ⟶ P x = Q x)"
by transfer auto

lemma pfsubset_ffilter:
"(⋀x. x |∈| A ⟹ P x ⟹ Q x) ⟹ (x |∈| A ∧ ¬ P x ∧ Q x) ⟹
ffilter P A |⊂| ffilter Q A"
unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter)

subsubsection ‹‹fset_of_list››

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 ⊆ set ys ⟹ fset_of_list xs |⊆| fset_of_list ys"
by transfer simp

lemma fset_of_list_elem: "(x |∈| fset_of_list xs) ⟷ (x ∈ set xs)"
by transfer simp

subsubsection ‹‹finsert››

(* FIXME, transferred doesn't work here *)
lemma set_finsert:
assumes "x |∈| A"
obtains B where "A = finsert x B" and "x |∉| B"
using assms by transfer (metis Set.set_insert finite_insert)

lemma mk_disjoint_finsert: "a |∈| A ⟹ ∃B. A = finsert a B ∧ a |∉| B"
by (rule exI [where x = "A |-| {|a|}"]) blast

lemma finsert_eq_iff:
assumes "a |∉| A" and "b |∉| B"
shows "(finsert a A = finsert b B) =
(if a = b then A = B else ∃C. A = finsert b C ∧ b |∉| C ∧ B = finsert a C ∧ a |∉| C)"
using assms by transfer (force simp: insert_eq_iff)

subsubsection ‹‹fimage››

lemma subset_fimage_iff: "(B |⊆| f|`|A) = (∃ AA. AA |⊆| A ∧ B = f|`|AA)"
by transfer (metis mem_Collect_eq rev_finite_subset subset_image_iff)

subsubsection ‹bounded quantification›

lemma bex_simps [simp, no_atp]:
"⋀A P Q. fBex A (λx. P x ∧ Q) = (fBex A P ∧ Q)"
"⋀A P Q. fBex A (λx. P ∧ Q x) = (P ∧ fBex A Q)"
"⋀P. fBex {||} P = False"
"⋀a B P. fBex (finsert a B) P = (P a ∨ fBex B P)"
"⋀A P f. fBex (f |`| A) P = fBex A (λx. P (f x))"
"⋀A P. (¬ fBex A P) = fBall A (λx. ¬ P x)"
by auto

lemma ball_simps [simp, no_atp]:
"⋀A P Q. fBall A (λx. P x ∨ Q) = (fBall A P ∨ Q)"
"⋀A P Q. fBall A (λx. P ∨ Q x) = (P ∨ fBall A Q)"
"⋀A P Q. fBall A (λx. P ⟶ Q x) = (P ⟶ fBall A Q)"
"⋀A P Q. fBall A (λx. P x ⟶ Q) = (fBex A P ⟶ Q)"
"⋀P. fBall {||} P = True"
"⋀a B P. fBall (finsert a B) P = (P a ∧ fBall B P)"
"⋀A P f. fBall (f |`| A) P = fBall A (λx. P (f x))"
"⋀A P. (¬ fBall A P) = fBex A (λx. ¬ P x)"
by auto

lemma atomize_fBall:
"(⋀x. x |∈| A ==> P x) == Trueprop (fBall A (λx. P x))"
apply (simp only: atomize_all atomize_imp)
apply (rule equal_intr_rule)
by (transfer, simp)+

lemma fBall_mono[mono]: "P ≤ Q ⟹ fBall S P ≤ fBall S Q"
by auto

lemma fBex_mono[mono]: "P ≤ Q ⟹ fBex S P ≤ fBex S Q"
by auto

end

subsubsection ‹‹fcard››

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

lemma fcard_fempty:
"fcard {||} = 0"
by transfer (rule card_empty)

lemma fcard_finsert_disjoint:
"x |∉| A ⟹ fcard (finsert x A) = Suc (fcard A)"
by transfer (rule card_insert_disjoint)

lemma fcard_finsert_if:
"fcard (finsert x A) = (if x |∈| A then fcard A else Suc (fcard A))"
by transfer (rule card_insert_if)

lemma fcard_0_eq [simp, no_atp]:
"fcard A = 0 ⟷ A = {||}"
by transfer (rule card_0_eq)

lemma fcard_Suc_fminus1:
"x |∈| A ⟹ Suc (fcard (A |-| {|x|})) = fcard A"
by transfer (rule card_Suc_Diff1)

lemma fcard_fminus_fsingleton:
"x |∈| A ⟹ fcard (A |-| {|x|}) = fcard A - 1"
by transfer (rule card_Diff_singleton)

lemma fcard_fminus_fsingleton_if:
"fcard (A |-| {|x|}) = (if x |∈| A then fcard A - 1 else fcard A)"
by transfer (rule card_Diff_singleton_if)

lemma fcard_fminus_finsert[simp]:
assumes "a |∈| A" and "a |∉| 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 ≤ fcard (finsert x A)"
by transfer (rule card_insert_le)

lemma fcard_mono:
"A |⊆| B ⟹ fcard A ≤ fcard B"
by transfer (rule card_mono)

lemma fcard_seteq: "A |⊆| B ⟹ fcard B ≤ fcard A ⟹ A = B"
by transfer (rule card_seteq)

lemma pfsubset_fcard_mono: "A |⊂| B ⟹ fcard A < fcard B"
by transfer (rule psubset_card_mono)

lemma fcard_funion_finter:
"fcard A + fcard B = fcard (A |∪| B) + fcard (A |∩| B)"
by transfer (rule card_Un_Int)

lemma fcard_funion_disjoint:
"A |∩| B = {||} ⟹ fcard (A |∪| B) = fcard A + fcard B"
by transfer (rule card_Un_disjoint)

lemma fcard_funion_fsubset:
"B |⊆| A ⟹ fcard (A |-| B) = fcard A - fcard B"
by transfer (rule card_Diff_subset)

lemma diff_fcard_le_fcard_fminus:
"fcard A - fcard B ≤ fcard(A |-| B)"
by transfer (rule diff_card_le_card_Diff)

lemma fcard_fminus1_less: "x |∈| A ⟹ fcard (A |-| {|x|}) < fcard A"
by transfer (rule card_Diff1_less)

lemma fcard_fminus2_less:
"x |∈| A ⟹ y |∈| A ⟹ fcard (A |-| {|x|} |-| {|y|}) < fcard A"
by transfer (rule card_Diff2_less)

lemma fcard_fminus1_le: "fcard (A |-| {|x|}) ≤ fcard A"
by transfer (rule card_Diff1_le)

lemma fcard_pfsubset: "A |⊆| B ⟹ fcard A < fcard B ⟹ A < B"
by transfer (rule card_psubset)

subsubsection ‹‹sorted_list_of_fset››

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 ‹‹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 |∉| 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 |∉| A ⟹ ffold f z (finsert x A) = ffold f (f x z) A"
by (transfer fixing: f) (rule fold_insert2)

lemma ffold_rec:
assumes "x |∈| 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 ∘ g) z A"
using assms by transfer' (rule fold_image)

lemma ffold_cong:
assumes "comp_fun_commute f" "comp_fun_commute g"
"⋀x. x |∈| A ⟹ 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

subsubsection ‹Group operations›

locale comm_monoid_fset = comm_monoid
begin

sublocale set: comm_monoid_set ..

lift_definition F :: "('b ⇒ 'a) ⇒ 'b fset ⇒ 'a" is set.F .

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

lemma strong_cong[cong]:
assumes "A = B" "⋀x. x |∈| B =simp=> g x = h x"
shows "F g A = F h B"
using assms unfolding simp_implies_def by (auto cong: cong)

end

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 ‹Semilattice operations›

locale semilattice_fset = semilattice
begin

sublocale set: semilattice_set ..

lift_definition F :: "'a fset ⇒ '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 |∉| A ⟹ A ≠ {||} ⟹ F (finsert x A) = x ❙* F A"
by transfer (rule set.insert_not_elem)

lemma in_idem: "x |∈| A ⟹ x ❙* F A = F A"
by transfer (rule set.in_idem)

lemma insert [simp]: "A ≠ {||} ⟹ F (finsert x A) = x ❙* 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 (≤) (<)" 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 (≥) (>)"
by standard

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

end

lemma mono_fMax_commute: "mono f ⟹ A ≠ {||} ⟹ f (fMax A) = fMax (f |`| A)"
by transfer (rule mono_Max_commute)

lemma mono_fMin_commute: "mono f ⟹ A ≠ {||} ⟹ f (fMin A) = fMin (f |`| A)"
by transfer (rule mono_Min_commute)

lemma fMax_in[simp]: "A ≠ {||} ⟹ fMax A |∈| A"
by transfer (rule Max_in)

lemma fMin_in[simp]: "A ≠ {||} ⟹ fMin A |∈| A"
by transfer (rule Min_in)

lemma fMax_ge[simp]: "x |∈| A ⟹ x ≤ fMax A"
by transfer (rule Max_ge)

lemma fMin_le[simp]: "x |∈| A ⟹ fMin A ≤ x"
by transfer (rule Min_le)

lemma fMax_eqI: "(⋀y. y |∈| A ⟹ y ≤ x) ⟹ x |∈| A ⟹ fMax A = x"
by transfer (rule Max_eqI)

lemma fMin_eqI: "(⋀y. y |∈| A ⟹ x ≤ y) ⟹ x |∈| A ⟹ 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     "⋀x S. ⟦∀y. y |∈| S ⟶ y < x; P S⟧ ⟹ 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     "⋀x S. ⟦∀y. y |∈| S ⟶ y > x; P S⟧ ⟹ 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 ‹Choice in fsets›

lemma fset_choice:
assumes "∀x. x |∈| A ⟶ (∃y. P x y)"
shows "∃f. ∀x. x |∈| A ⟶ 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 = {||} ⟹ P"
and     finsert_case: "⋀x S'. S = finsert x S' ⟹ P"
shows "P"
using assms by transfer blast

lemma fset_induct [case_names empty insert]:
assumes fempty_case: "P {||}"
and     finsert_case: "⋀x S. P S ⟹ 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: "⋀x S. ⟦x |∉| S; P S⟧ ⟹ 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: "⋀S T. Suc (fcard S) = (fcard T) ⟹ P S ⟹ 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 |∉| 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 |∉| ys" and "xs = finsert x ys"
by transfer blast

lemma fset_induct2:
"P {||} {||} ⟹
(⋀x xs. x |∉| xs ⟹ P (finsert x xs) {||}) ⟹
(⋀y ys. y |∉| ys ⟹ P {||} (finsert y ys)) ⟹
(⋀x xs y ys. ⟦P xs ys; x |∉| xs; y |∉| ys⟧ ⟹ P (finsert x xs) (finsert y ys)) ⟹
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 ⇒ 'b ⇒ bool) ⇒ 'a fset ⇒ 'b fset ⇒ bool" is rel_set
parametric rel_set_transfer .

lemma rel_fset_alt_def: "rel_fset R = (λA B. (∀x.∃y. x|∈|A ⟶ y|∈|B ∧ R x y)
∧ (∀y. ∃x. y|∈|B ⟶ x|∈|A ∧ 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 "∃Y. finite Y ∧ rel_set R X Y ∧ rel_set S Y Z"
proof -
obtain f where f: "∀x∈X. R x (f x) ∧ (∃z∈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: "∀z∈Z. S (g z) z ∧ (∃x∈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 ∪ 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 ‹Transfer rules for the Transfer package›

text ‹Unconditional transfer rules›

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 ‹Rules requiring bi-unique, bi-total or right-total relations›

lemma fmember_transfer [transfer_rule]:
assumes "bi_unique A"
shows "(A ===> rel_fset A ===> (=)) (|∈|) (|∈|)"
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 ===> (=)) (|⊆|) (|⊆|)"
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 ⟹ (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

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 ⟹ (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 ‹BNF setup›

context
includes fset.lifting
begin

lemma rel_fset_alt:
"rel_fset R a b ⟷ (∀t ∈ fset a. ∃u ∈ fset b. R t u) ∧ (∀t ∈ fset b. ∃u ∈ fset a. R u t)"

lemma fset_to_fset: "finite A ⟹ fset (the_inv fset A) = A"
apply (rule f_the_inv_into_f[unfolded inj_on_def])
apply (rule range_eqI Abs_fset_inverse[symmetric] CollectI)+
.

lemma rel_fset_aux:
"(∀t ∈ fset a. ∃u ∈ fset b. R t u) ∧ (∀u ∈ fset b. ∃t ∈ fset a. R t u) ⟷
((BNF_Def.Grp {a. fset a ⊆ {(a, b). R a b}} (fimage fst))¯¯ OO
BNF_Def.Grp {a. fset a ⊆ {(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) ∩ (fset a × 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)
next
assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps
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[unfolded OO_Grp_alt])
apply transfer apply simp
done

lemma rel_fset_fset: "rel_set χ (fset A1) (fset A2) = rel_fset χ A1 A2"
by transfer (rule refl)

end

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

subsection ‹Size setup›

context includes fset.lifting begin
lift_definition size_fset :: "('a ⇒ nat) ⇒ 'a fset ⇒ nat" is "λf. sum (Suc ∘ f)" .
end

instantiation fset :: (type) size begin
definition size_fset where
size_fset_overloaded_def: "size_fset = FSet.size_fset (λ_. 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]

lemma fset_size_o_map: "inj f ⟹ size_fset g ∘ fimage f = size_fset (g ∘ f)"
apply (subst fun_eq_iff)
including fset.lifting by transfer (auto intro: sum.reindex_cong subset_inj_on)

setup ‹
BNF_LFP_Size.register_size_global @{type_name fset} @{const_name size_fset}
@{thms fset_size_o_map}
›

lifting_update fset.lifting
lifting_forget fset.lifting

text ‹Set vs. sum relators:›

lemma rel_set_rel_sum[simp]:
"rel_set (rel_sum χ φ) A1 A2 ⟷
rel_set χ (Inl -` A1) (Inl -` A2) ∧ rel_set φ (Inr -` A1) (Inr -` A2)"
(is "?L ⟷ ?Rl ∧ ?Rr")
proof safe
assume L: "?L"
show ?Rl unfolding rel_set_def Bex_def vimage_eq proof safe
fix l1 assume "Inl l1 ∈ A1"
then obtain a2 where a2: "a2 ∈ A2" and "rel_sum χ φ (Inl l1) a2"
using L unfolding rel_set_def by auto
then obtain l2 where "a2 = Inl l2 ∧ χ l1 l2" by (cases a2, auto)
thus "∃ l2. Inl l2 ∈ A2 ∧ χ l1 l2" using a2 by auto
next
fix l2 assume "Inl l2 ∈ A2"
then obtain a1 where a1: "a1 ∈ A1" and "rel_sum χ φ a1 (Inl l2)"
using L unfolding rel_set_def by auto
then obtain l1 where "a1 = Inl l1 ∧ χ l1 l2" by (cases a1, auto)
thus "∃ l1. Inl l1 ∈ A1 ∧ χ l1 l2" using a1 by auto
qed
show ?Rr unfolding rel_set_def Bex_def vimage_eq proof safe
fix r1 assume "Inr r1 ∈ A1"
then obtain a2 where a2: "a2 ∈ A2" and "rel_sum χ φ (Inr r1) a2"
using L unfolding rel_set_def by auto
then obtain r2 where "a2 = Inr r2 ∧ φ r1 r2" by (cases a2, auto)
thus "∃ r2. Inr r2 ∈ A2 ∧ φ r1 r2" using a2 by auto
next
fix r2 assume "Inr r2 ∈ A2"
then obtain a1 where a1: "a1 ∈ A1" and "rel_sum χ φ a1 (Inr r2)"
using L unfolding rel_set_def by auto
then obtain r1 where "a1 = Inr r1 ∧ φ r1 r2" by (cases a1, auto)
thus "∃ r1. Inr r1 ∈ A1 ∧ φ 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 ∈ A1"
show "∃ a2. a2 ∈ A2 ∧ rel_sum χ φ a1 a2"
proof(cases a1)
case (Inl l1) then obtain l2 where "Inl l2 ∈ A2 ∧ χ 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 ∈ A2 ∧ φ r1 r2"
using Rr a1 unfolding rel_set_def by blast
thus ?thesis unfolding Inr by auto
qed
next
fix a2 assume a2: "a2 ∈ A2"
show "∃ a1. a1 ∈ A1 ∧ rel_sum χ φ a1 a2"
proof(cases a2)
case (Inl l2) then obtain l1 where "Inl l1 ∈ A1 ∧ χ 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 ∈ A1 ∧ φ r1 r2"
using Rr a2 unfolding rel_set_def by blast
thus ?thesis unfolding Inr by auto
qed
qed
qed

subsubsection ‹Countability›

lemma exists_fset_of_list: "∃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 ∈ 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 ⇒ 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 ∘ inv fset_of_list)"
by (rule inj_comp)
thus "∃to_nat::'a fset ⇒ nat. inj to_nat"
by auto
qed

subsection ‹Quickcheck setup›

notation Quickcheck_Exhaustive.orelse (infixr "orelse" 55)

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

definition (in term_syntax) [code_unfold]:
"valtermify_finsert x s = Code_Evaluation.valtermify finsert {⋅} (x :: ('a :: typerep * _)) {⋅} s"

instantiation fset :: (exhaustive) exhaustive
begin

fun exhaustive_fset where
"exhaustive_fset f i = (if i = 0 then None else (f {||} orelse exhaustive_fset (λA. f A orelse Quickcheck_Exhaustive.exhaustive (λx. if x |∈| 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 (λA. f A orelse Quickcheck_Exhaustive.full_exhaustive (λx. if fst x |∈| fst A then None else f (valtermify_finsert x A)) (i - 1)) (i - 1)))"

instance ..

end

no_notation Quickcheck_Exhaustive.orelse (infixr "orelse" 55)

notation scomp (infixl "∘→" 60)

instantiation fset :: (random) random
begin

fun random_aux_fset :: "natural ⇒ natural ⇒ natural × natural ⇒ ('a fset × (unit ⇒ term)) × natural × 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 ∘→ (λx. random_aux_fset i j ∘→ (λs. 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 ∘→ (λx. random_aux_fset (i - 1) j ∘→ (λs. 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

no_notation scomp (infixl "∘→" 60)

end
```