Theory Countable_Set_Type

theory Countable_Set_Type
imports Countable_Set Cardinal_Notations
```(*  Title:      HOL/Library/Countable_Set_Type.thy
Author:     Andrei Popescu, TU Muenchen
Author:     Andreas Lochbihler, ETH Zurich

Type of (at most) countable sets.
*)

section ‹Type of (at Most) Countable Sets›

theory Countable_Set_Type
imports Countable_Set Cardinal_Notations HOL.Conditionally_Complete_Lattices
begin

subsection‹Cardinal stuff›

lemma countable_card_of_nat: "countable A ⟷ |A| ≤o |UNIV::nat set|"
unfolding countable_def card_of_ordLeq[symmetric] by auto

lemma countable_card_le_natLeq: "countable A ⟷ |A| ≤o natLeq"
unfolding countable_card_of_nat using card_of_nat ordLeq_ordIso_trans ordIso_symmetric by blast

lemma countable_or_card_of:
assumes "countable A"
shows "(finite A ∧ |A| <o |UNIV::nat set| ) ∨
(infinite A  ∧ |A| =o |UNIV::nat set| )"
by (metis assms countable_card_of_nat infinite_iff_card_of_nat ordIso_iff_ordLeq
ordLeq_iff_ordLess_or_ordIso)

lemma countable_cases_card_of[elim]:
assumes "countable A"
obtains (Fin) "finite A" "|A| <o |UNIV::nat set|"
| (Inf) "infinite A" "|A| =o |UNIV::nat set|"
using assms countable_or_card_of by blast

lemma countable_or:
"countable A ⟹ (∃ f::'a⇒nat. finite A ∧ inj_on f A) ∨ (∃ f::'a⇒nat. infinite A ∧ bij_betw f A UNIV)"
by (elim countable_enum_cases) fastforce+

lemma countable_cases[elim]:
assumes "countable A"
obtains (Fin) f :: "'a⇒nat" where "finite A" "inj_on f A"
| (Inf) f :: "'a⇒nat" where "infinite A" "bij_betw f A UNIV"
using assms countable_or by metis

lemma countable_ordLeq:
assumes "|A| ≤o |B|" and "countable B"
shows "countable A"
using assms unfolding countable_card_of_nat by(rule ordLeq_transitive)

lemma countable_ordLess:
assumes AB: "|A| <o |B|" and B: "countable B"
shows "countable A"
using countable_ordLeq[OF ordLess_imp_ordLeq[OF AB] B] .

subsection ‹The type of countable sets›

typedef 'a cset = "{A :: 'a set. countable A}" morphisms rcset acset
by (rule exI[of _ "{}"]) simp

setup_lifting type_definition_cset

declare
rcset_inverse[simp]
acset_inverse[Transfer.transferred, unfolded mem_Collect_eq, simp]
acset_inject[Transfer.transferred, unfolded mem_Collect_eq, simp]
rcset[Transfer.transferred, unfolded mem_Collect_eq, simp]

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

lift_definition bot_cset :: "'a cset" is "{}" parametric empty_transfer by simp

lift_definition less_eq_cset :: "'a cset ⇒ 'a cset ⇒ bool"
is subset_eq parametric subset_transfer .

definition less_cset :: "'a cset ⇒ 'a cset ⇒ bool"
where "xs < ys ≡ xs ≤ ys ∧ xs ≠ (ys::'a cset)"

lemma less_cset_transfer[transfer_rule]:
includes lifting_syntax
assumes [transfer_rule]: "bi_unique A"
shows "((pcr_cset A) ===> (pcr_cset A) ===> op =) op ⊂ op <"
unfolding less_cset_def[abs_def] psubset_eq[abs_def] by transfer_prover

lift_definition sup_cset :: "'a cset ⇒ 'a cset ⇒ 'a cset"
is union parametric union_transfer by simp

lift_definition inf_cset :: "'a cset ⇒ 'a cset ⇒ 'a cset"
is inter parametric inter_transfer by simp

lift_definition minus_cset :: "'a cset ⇒ 'a cset ⇒ 'a cset"
is minus parametric Diff_transfer by simp

instance by standard (transfer; auto)+

end

abbreviation cempty :: "'a cset" where "cempty ≡ bot"
abbreviation csubset_eq :: "'a cset ⇒ 'a cset ⇒ bool" where "csubset_eq xs ys ≡ xs ≤ ys"
abbreviation csubset :: "'a cset ⇒ 'a cset ⇒ bool" where "csubset xs ys ≡ xs < ys"
abbreviation cUn :: "'a cset ⇒ 'a cset ⇒ 'a cset" where "cUn xs ys ≡ sup xs ys"
abbreviation cInt :: "'a cset ⇒ 'a cset ⇒ 'a cset" where "cInt xs ys ≡ inf xs ys"
abbreviation cDiff :: "'a cset ⇒ 'a cset ⇒ 'a cset" where "cDiff xs ys ≡ minus xs ys"

lift_definition cin :: "'a ⇒ 'a cset ⇒ bool" is "op ∈" parametric member_transfer
.
lift_definition cinsert :: "'a ⇒ 'a cset ⇒ 'a cset" is insert parametric Lifting_Set.insert_transfer
by (rule countable_insert)
abbreviation csingle :: "'a ⇒ 'a cset" where "csingle x ≡ cinsert x cempty"
lift_definition cimage :: "('a ⇒ 'b) ⇒ 'a cset ⇒ 'b cset" is "op `" parametric image_transfer
by (rule countable_image)
lift_definition cBall :: "'a cset ⇒ ('a ⇒ bool) ⇒ bool" is Ball parametric Ball_transfer .
lift_definition cBex :: "'a cset ⇒ ('a ⇒ bool) ⇒ bool" is Bex parametric Bex_transfer .
lift_definition cUNION :: "'a cset ⇒ ('a ⇒ 'b cset) ⇒ 'b cset"
is "UNION" parametric UNION_transfer by simp
definition cUnion :: "'a cset cset ⇒ 'a cset" where "cUnion A = cUNION A id"

lemma Union_conv_UNION: "⋃A = UNION A id"
by auto

lemma cUnion_transfer [transfer_rule]:
"rel_fun (pcr_cset (pcr_cset A)) (pcr_cset A) Union cUnion"
proof -
have "rel_fun (pcr_cset (pcr_cset A)) (pcr_cset A) (λA. UNION A id) (λA. cUNION A id)"
by transfer_prover
then show ?thesis by (simp add: cUnion_def [symmetric])
qed

lemmas cset_eqI = set_eqI[Transfer.transferred]
lemmas cset_eq_iff[no_atp] = set_eq_iff[Transfer.transferred]
lemmas cBallI[intro!] = ballI[Transfer.transferred]
lemmas cbspec[dest?] = bspec[Transfer.transferred]
lemmas cBallE[elim] = ballE[Transfer.transferred]
lemmas cBexI[intro] = bexI[Transfer.transferred]
lemmas rev_cBexI[intro?] = rev_bexI[Transfer.transferred]
lemmas cBexCI = bexCI[Transfer.transferred]
lemmas cBexE[elim!] = bexE[Transfer.transferred]
lemmas cBall_triv[simp] = ball_triv[Transfer.transferred]
lemmas cBex_triv[simp] = bex_triv[Transfer.transferred]
lemmas cBex_triv_one_point1[simp] = bex_triv_one_point1[Transfer.transferred]
lemmas cBex_triv_one_point2[simp] = bex_triv_one_point2[Transfer.transferred]
lemmas cBex_one_point1[simp] = bex_one_point1[Transfer.transferred]
lemmas cBex_one_point2[simp] = bex_one_point2[Transfer.transferred]
lemmas cBall_one_point1[simp] = ball_one_point1[Transfer.transferred]
lemmas cBall_one_point2[simp] = ball_one_point2[Transfer.transferred]
lemmas cBall_conj_distrib = ball_conj_distrib[Transfer.transferred]
lemmas cBex_disj_distrib = bex_disj_distrib[Transfer.transferred]
lemmas cBall_cong = ball_cong[Transfer.transferred]
lemmas cBex_cong = bex_cong[Transfer.transferred]
lemmas csubsetI[intro!] = subsetI[Transfer.transferred]
lemmas csubsetD[elim, intro?] = subsetD[Transfer.transferred]
lemmas rev_csubsetD[no_atp,intro?] = rev_subsetD[Transfer.transferred]
lemmas csubsetCE[no_atp,elim] = subsetCE[Transfer.transferred]
lemmas csubset_eq[no_atp] = subset_eq[Transfer.transferred]
lemmas contra_csubsetD[no_atp] = contra_subsetD[Transfer.transferred]
lemmas csubset_refl = subset_refl[Transfer.transferred]
lemmas csubset_trans = subset_trans[Transfer.transferred]
lemmas cset_rev_mp = set_rev_mp[Transfer.transferred]
lemmas cset_mp = set_mp[Transfer.transferred]
lemmas csubset_not_fsubset_eq[code] = subset_not_subset_eq[Transfer.transferred]
lemmas eq_cmem_trans = eq_mem_trans[Transfer.transferred]
lemmas csubset_antisym[intro!] = subset_antisym[Transfer.transferred]
lemmas cequalityD1 = equalityD1[Transfer.transferred]
lemmas cequalityD2 = equalityD2[Transfer.transferred]
lemmas cequalityE = equalityE[Transfer.transferred]
lemmas cequalityCE[elim] = equalityCE[Transfer.transferred]
lemmas eqcset_imp_iff = eqset_imp_iff[Transfer.transferred]
lemmas eqcelem_imp_iff = eqelem_imp_iff[Transfer.transferred]
lemmas cempty_iff[simp] = empty_iff[Transfer.transferred]
lemmas cempty_fsubsetI[iff] = empty_subsetI[Transfer.transferred]
lemmas equals_cemptyI = equals0I[Transfer.transferred]
lemmas equals_cemptyD = equals0D[Transfer.transferred]
lemmas cBall_cempty[simp] = ball_empty[Transfer.transferred]
lemmas cBex_cempty[simp] = bex_empty[Transfer.transferred]
lemmas cInt_iff[simp] = Int_iff[Transfer.transferred]
lemmas cIntI[intro!] = IntI[Transfer.transferred]
lemmas cIntD1 = IntD1[Transfer.transferred]
lemmas cIntD2 = IntD2[Transfer.transferred]
lemmas cIntE[elim!] = IntE[Transfer.transferred]
lemmas cUn_iff[simp] = Un_iff[Transfer.transferred]
lemmas cUnI1[elim?] = UnI1[Transfer.transferred]
lemmas cUnI2[elim?] = UnI2[Transfer.transferred]
lemmas cUnCI[intro!] = UnCI[Transfer.transferred]
lemmas cuUnE[elim!] = UnE[Transfer.transferred]
lemmas cDiff_iff[simp] = Diff_iff[Transfer.transferred]
lemmas cDiffI[intro!] = DiffI[Transfer.transferred]
lemmas cDiffD1 = DiffD1[Transfer.transferred]
lemmas cDiffD2 = DiffD2[Transfer.transferred]
lemmas cDiffE[elim!] = DiffE[Transfer.transferred]
lemmas cinsert_iff[simp] = insert_iff[Transfer.transferred]
lemmas cinsertI1 = insertI1[Transfer.transferred]
lemmas cinsertI2 = insertI2[Transfer.transferred]
lemmas cinsertE[elim!] = insertE[Transfer.transferred]
lemmas cinsertCI[intro!] = insertCI[Transfer.transferred]
lemmas csubset_cinsert_iff = subset_insert_iff[Transfer.transferred]
lemmas cinsert_ident = insert_ident[Transfer.transferred]
lemmas csingletonI[intro!,no_atp] = singletonI[Transfer.transferred]
lemmas csingletonD[dest!,no_atp] = singletonD[Transfer.transferred]
lemmas fsingletonE = csingletonD [elim_format]
lemmas csingleton_iff = singleton_iff[Transfer.transferred]
lemmas csingleton_inject[dest!] = singleton_inject[Transfer.transferred]
lemmas csingleton_finsert_inj_eq[iff,no_atp] = singleton_insert_inj_eq[Transfer.transferred]
lemmas csingleton_finsert_inj_eq'[iff,no_atp] = singleton_insert_inj_eq'[Transfer.transferred]
lemmas csubset_csingletonD = subset_singletonD[Transfer.transferred]
lemmas cDiff_single_cinsert = Diff_single_insert[Transfer.transferred]
lemmas cdoubleton_eq_iff = doubleton_eq_iff[Transfer.transferred]
lemmas cUn_csingleton_iff = Un_singleton_iff[Transfer.transferred]
lemmas csingleton_cUn_iff = singleton_Un_iff[Transfer.transferred]
lemmas cimage_eqI[simp, intro] = image_eqI[Transfer.transferred]
lemmas cimageI = imageI[Transfer.transferred]
lemmas rev_cimage_eqI = rev_image_eqI[Transfer.transferred]
lemmas cimageE[elim!] = imageE[Transfer.transferred]
lemmas Compr_cimage_eq = Compr_image_eq[Transfer.transferred]
lemmas cimage_cUn = image_Un[Transfer.transferred]
lemmas cimage_iff = image_iff[Transfer.transferred]
lemmas cimage_csubset_iff[no_atp] = image_subset_iff[Transfer.transferred]
lemmas cimage_csubsetI = image_subsetI[Transfer.transferred]
lemmas cimage_ident[simp] = image_ident[Transfer.transferred]
lemmas if_split_cin1 = if_split_mem1[Transfer.transferred]
lemmas if_split_cin2 = if_split_mem2[Transfer.transferred]
lemmas cpsubsetI[intro!,no_atp] = psubsetI[Transfer.transferred]
lemmas cpsubsetE[elim!,no_atp] = psubsetE[Transfer.transferred]
lemmas cpsubset_finsert_iff = psubset_insert_iff[Transfer.transferred]
lemmas cpsubset_eq = psubset_eq[Transfer.transferred]
lemmas cpsubset_imp_fsubset = psubset_imp_subset[Transfer.transferred]
lemmas cpsubset_trans = psubset_trans[Transfer.transferred]
lemmas cpsubsetD = psubsetD[Transfer.transferred]
lemmas cpsubset_csubset_trans = psubset_subset_trans[Transfer.transferred]
lemmas csubset_cpsubset_trans = subset_psubset_trans[Transfer.transferred]
lemmas cpsubset_imp_ex_fmem = psubset_imp_ex_mem[Transfer.transferred]
lemmas csubset_cinsertI = subset_insertI[Transfer.transferred]
lemmas csubset_cinsertI2 = subset_insertI2[Transfer.transferred]
lemmas csubset_cinsert = subset_insert[Transfer.transferred]
lemmas cUn_upper1 = Un_upper1[Transfer.transferred]
lemmas cUn_upper2 = Un_upper2[Transfer.transferred]
lemmas cUn_least = Un_least[Transfer.transferred]
lemmas cInt_lower1 = Int_lower1[Transfer.transferred]
lemmas cInt_lower2 = Int_lower2[Transfer.transferred]
lemmas cInt_greatest = Int_greatest[Transfer.transferred]
lemmas cDiff_csubset = Diff_subset[Transfer.transferred]
lemmas cDiff_csubset_conv = Diff_subset_conv[Transfer.transferred]
lemmas csubset_cempty[simp] = subset_empty[Transfer.transferred]
lemmas not_cpsubset_cempty[iff] = not_psubset_empty[Transfer.transferred]
lemmas cinsert_is_cUn = insert_is_Un[Transfer.transferred]
lemmas cinsert_not_cempty[simp] = insert_not_empty[Transfer.transferred]
lemmas cempty_not_cinsert = empty_not_insert[Transfer.transferred]
lemmas cinsert_absorb = insert_absorb[Transfer.transferred]
lemmas cinsert_absorb2[simp] = insert_absorb2[Transfer.transferred]
lemmas cinsert_commute = insert_commute[Transfer.transferred]
lemmas cinsert_csubset[simp] = insert_subset[Transfer.transferred]
lemmas cinsert_cinter_cinsert[simp] = insert_inter_insert[Transfer.transferred]
lemmas cinsert_disjoint[simp,no_atp] = insert_disjoint[Transfer.transferred]
lemmas disjoint_cinsert[simp,no_atp] = disjoint_insert[Transfer.transferred]
lemmas cimage_cempty[simp] = image_empty[Transfer.transferred]
lemmas cimage_cinsert[simp] = image_insert[Transfer.transferred]
lemmas cimage_constant = image_constant[Transfer.transferred]
lemmas cimage_constant_conv = image_constant_conv[Transfer.transferred]
lemmas cimage_cimage = image_image[Transfer.transferred]
lemmas cinsert_cimage[simp] = insert_image[Transfer.transferred]
lemmas cimage_is_cempty[iff] = image_is_empty[Transfer.transferred]
lemmas cempty_is_cimage[iff] = empty_is_image[Transfer.transferred]
lemmas cimage_cong = image_cong[Transfer.transferred]
lemmas cimage_cInt_csubset = image_Int_subset[Transfer.transferred]
lemmas cimage_cDiff_csubset = image_diff_subset[Transfer.transferred]
lemmas cInt_absorb = Int_absorb[Transfer.transferred]
lemmas cInt_left_absorb = Int_left_absorb[Transfer.transferred]
lemmas cInt_commute = Int_commute[Transfer.transferred]
lemmas cInt_left_commute = Int_left_commute[Transfer.transferred]
lemmas cInt_assoc = Int_assoc[Transfer.transferred]
lemmas cInt_ac = Int_ac[Transfer.transferred]
lemmas cInt_absorb1 = Int_absorb1[Transfer.transferred]
lemmas cInt_absorb2 = Int_absorb2[Transfer.transferred]
lemmas cInt_cempty_left = Int_empty_left[Transfer.transferred]
lemmas cInt_cempty_right = Int_empty_right[Transfer.transferred]
lemmas disjoint_iff_cnot_equal = disjoint_iff_not_equal[Transfer.transferred]
lemmas cInt_cUn_distrib = Int_Un_distrib[Transfer.transferred]
lemmas cInt_cUn_distrib2 = Int_Un_distrib2[Transfer.transferred]
lemmas cInt_csubset_iff[no_atp, simp] = Int_subset_iff[Transfer.transferred]
lemmas cUn_absorb = Un_absorb[Transfer.transferred]
lemmas cUn_left_absorb = Un_left_absorb[Transfer.transferred]
lemmas cUn_commute = Un_commute[Transfer.transferred]
lemmas cUn_left_commute = Un_left_commute[Transfer.transferred]
lemmas cUn_assoc = Un_assoc[Transfer.transferred]
lemmas cUn_ac = Un_ac[Transfer.transferred]
lemmas cUn_absorb1 = Un_absorb1[Transfer.transferred]
lemmas cUn_absorb2 = Un_absorb2[Transfer.transferred]
lemmas cUn_cempty_left = Un_empty_left[Transfer.transferred]
lemmas cUn_cempty_right = Un_empty_right[Transfer.transferred]
lemmas cUn_cinsert_left[simp] = Un_insert_left[Transfer.transferred]
lemmas cUn_cinsert_right[simp] = Un_insert_right[Transfer.transferred]
lemmas cInt_cinsert_left = Int_insert_left[Transfer.transferred]
lemmas cInt_cinsert_left_if0[simp] = Int_insert_left_if0[Transfer.transferred]
lemmas cInt_cinsert_left_if1[simp] = Int_insert_left_if1[Transfer.transferred]
lemmas cInt_cinsert_right = Int_insert_right[Transfer.transferred]
lemmas cInt_cinsert_right_if0[simp] = Int_insert_right_if0[Transfer.transferred]
lemmas cInt_cinsert_right_if1[simp] = Int_insert_right_if1[Transfer.transferred]
lemmas cUn_cInt_distrib = Un_Int_distrib[Transfer.transferred]
lemmas cUn_cInt_distrib2 = Un_Int_distrib2[Transfer.transferred]
lemmas cUn_cInt_crazy = Un_Int_crazy[Transfer.transferred]
lemmas csubset_cUn_eq = subset_Un_eq[Transfer.transferred]
lemmas cUn_cempty[iff] = Un_empty[Transfer.transferred]
lemmas cUn_csubset_iff[no_atp, simp] = Un_subset_iff[Transfer.transferred]
lemmas cUn_cDiff_cInt = Un_Diff_Int[Transfer.transferred]
lemmas cDiff_cInt2 = Diff_Int2[Transfer.transferred]
lemmas cUn_cInt_assoc_eq = Un_Int_assoc_eq[Transfer.transferred]
lemmas cBall_cUn = ball_Un[Transfer.transferred]
lemmas cBex_cUn = bex_Un[Transfer.transferred]
lemmas cDiff_eq_cempty_iff[simp,no_atp] = Diff_eq_empty_iff[Transfer.transferred]
lemmas cDiff_cancel[simp] = Diff_cancel[Transfer.transferred]
lemmas cDiff_idemp[simp] = Diff_idemp[Transfer.transferred]
lemmas cDiff_triv = Diff_triv[Transfer.transferred]
lemmas cempty_cDiff[simp] = empty_Diff[Transfer.transferred]
lemmas cDiff_cempty[simp] = Diff_empty[Transfer.transferred]
lemmas cDiff_cinsert0[simp,no_atp] = Diff_insert0[Transfer.transferred]
lemmas cDiff_cinsert = Diff_insert[Transfer.transferred]
lemmas cDiff_cinsert2 = Diff_insert2[Transfer.transferred]
lemmas cinsert_cDiff_if = insert_Diff_if[Transfer.transferred]
lemmas cinsert_cDiff1[simp] = insert_Diff1[Transfer.transferred]
lemmas cinsert_cDiff_single[simp] = insert_Diff_single[Transfer.transferred]
lemmas cinsert_cDiff = insert_Diff[Transfer.transferred]
lemmas cDiff_cinsert_absorb = Diff_insert_absorb[Transfer.transferred]
lemmas cDiff_disjoint[simp] = Diff_disjoint[Transfer.transferred]
lemmas cDiff_partition = Diff_partition[Transfer.transferred]
lemmas double_cDiff = double_diff[Transfer.transferred]
lemmas cUn_cDiff_cancel[simp] = Un_Diff_cancel[Transfer.transferred]
lemmas cUn_cDiff_cancel2[simp] = Un_Diff_cancel2[Transfer.transferred]
lemmas cDiff_cUn = Diff_Un[Transfer.transferred]
lemmas cDiff_cInt = Diff_Int[Transfer.transferred]
lemmas cUn_cDiff = Un_Diff[Transfer.transferred]
lemmas cInt_cDiff = Int_Diff[Transfer.transferred]
lemmas cDiff_cInt_distrib = Diff_Int_distrib[Transfer.transferred]
lemmas cDiff_cInt_distrib2 = Diff_Int_distrib2[Transfer.transferred]
lemmas cset_eq_csubset = set_eq_subset[Transfer.transferred]
lemmas csubset_iff[no_atp] = subset_iff[Transfer.transferred]
lemmas csubset_iff_pfsubset_eq = subset_iff_psubset_eq[Transfer.transferred]
lemmas all_not_cin_conv[simp] = all_not_in_conv[Transfer.transferred]
lemmas ex_cin_conv = ex_in_conv[Transfer.transferred]
lemmas cimage_mono = image_mono[Transfer.transferred]
lemmas cinsert_mono = insert_mono[Transfer.transferred]
lemmas cunion_mono = Un_mono[Transfer.transferred]
lemmas cinter_mono = Int_mono[Transfer.transferred]
lemmas cminus_mono = Diff_mono[Transfer.transferred]
lemmas cin_mono = in_mono[Transfer.transferred]
lemmas cLeast_mono = Least_mono[Transfer.transferred]
lemmas cequalityI = equalityI[Transfer.transferred]
lemmas cUN_iff [simp] = UN_iff[Transfer.transferred]
lemmas cUN_I [intro] = UN_I[Transfer.transferred]
lemmas cUN_E [elim!] = UN_E[Transfer.transferred]
lemmas cUN_upper = UN_upper[Transfer.transferred]
lemmas cUN_least = UN_least[Transfer.transferred]
lemmas cUN_cinsert_distrib = UN_insert_distrib[Transfer.transferred]
lemmas cUN_empty [simp] = UN_empty[Transfer.transferred]
lemmas cUN_empty2 [simp] = UN_empty2[Transfer.transferred]
lemmas cUN_absorb = UN_absorb[Transfer.transferred]
lemmas cUN_cinsert [simp] = UN_insert[Transfer.transferred]
lemmas cUN_cUn [simp] = UN_Un[Transfer.transferred]
lemmas cUN_cUN_flatten = UN_UN_flatten[Transfer.transferred]
lemmas cUN_csubset_iff = UN_subset_iff[Transfer.transferred]
lemmas cUN_constant [simp] = UN_constant[Transfer.transferred]
lemmas cimage_cUnion = image_Union[Transfer.transferred]
lemmas cUNION_cempty_conv [simp] = UNION_empty_conv[Transfer.transferred]
lemmas cBall_cUN = ball_UN[Transfer.transferred]
lemmas cBex_cUN = bex_UN[Transfer.transferred]
lemmas cUn_eq_cUN = Un_eq_UN[Transfer.transferred]
lemmas cUN_mono = UN_mono[Transfer.transferred]
lemmas cimage_cUN = image_UN[Transfer.transferred]
lemmas cUN_csingleton [simp] = UN_singleton[Transfer.transferred]

subsubsection ‹‹cempty››

lemma cemptyE [elim!]: "cin a cempty ⟹ P" by simp

subsubsection ‹‹cinsert››

lemma countable_insert_iff: "countable (insert x A) ⟷ countable A"
by (metis Diff_eq_empty_iff countable_empty countable_insert subset_insertI uncountable_minus_countable)

lemma set_cinsert:
assumes "cin x A"
obtains B where "A = cinsert x B" and "¬ cin x B"
using assms by transfer(erule Set.set_insert, simp add: countable_insert_iff)

lemma mk_disjoint_cinsert: "cin a A ⟹ ∃B. A = cinsert a B ∧ ¬ cin a B"
by (rule exI[where x = "cDiff A (csingle a)"]) blast

subsubsection ‹‹cimage››

lemma subset_cimage_iff: "csubset_eq B (cimage f A) ⟷ (∃AA. csubset_eq AA A ∧ B = cimage f AA)"
by transfer (metis countable_subset image_mono mem_Collect_eq subset_imageE)

subsubsection ‹bounded quantification›

lemma cBex_simps [simp, no_atp]:
"⋀A P Q. cBex A (λx. P x ∧ Q) = (cBex A P ∧ Q)"
"⋀A P Q. cBex A (λx. P ∧ Q x) = (P ∧ cBex A Q)"
"⋀P. cBex cempty P = False"
"⋀a B P. cBex (cinsert a B) P = (P a ∨ cBex B P)"
"⋀A P f. cBex (cimage f A) P = cBex A (λx. P (f x))"
"⋀A P. (¬ cBex A P) = cBall A (λx. ¬ P x)"
by auto

lemma cBall_simps [simp, no_atp]:
"⋀A P Q. cBall A (λx. P x ∨ Q) = (cBall A P ∨ Q)"
"⋀A P Q. cBall A (λx. P ∨ Q x) = (P ∨ cBall A Q)"
"⋀A P Q. cBall A (λx. P ⟶ Q x) = (P ⟶ cBall A Q)"
"⋀A P Q. cBall A (λx. P x ⟶ Q) = (cBex A P ⟶ Q)"
"⋀P. cBall cempty P = True"
"⋀a B P. cBall (cinsert a B) P = (P a ∧ cBall B P)"
"⋀A P f. cBall (cimage f A) P = cBall A (λx. P (f x))"
"⋀A P. (¬ cBall A P) = cBex A (λx. ¬ P x)"
by auto

lemma atomize_cBall:
"(⋀x. cin x A ==> P x) == Trueprop (cBall A (λx. P x))"
apply (simp only: atomize_all atomize_imp)
apply (rule equal_intr_rule)
by (transfer, simp)+

subsubsection ‹@{const cUnion}›

lemma cUNION_cimage: "cUNION (cimage f A) g = cUNION A (g ∘ f)"
including cset.lifting by transfer auto

subsection ‹Setup for Lifting/Transfer›

subsubsection ‹Relator and predicator properties›

lift_definition rel_cset :: "('a ⇒ 'b ⇒ bool) ⇒ 'a cset ⇒ 'b cset ⇒ bool"
is rel_set parametric rel_set_transfer .

lemma rel_cset_alt_def:
"rel_cset R a b ⟷
(∀t ∈ rcset a. ∃u ∈ rcset b. R t u) ∧
(∀t ∈ rcset b. ∃u ∈ rcset a. R u t)"

lemma rel_cset_iff:
"rel_cset R a b ⟷
(∀t. cin t a ⟶ (∃u. cin u b ∧ R t u)) ∧
(∀t. cin t b ⟶ (∃u. cin u a ∧ R u t))"

lemma rel_cset_cUNION:
"⟦ rel_cset Q A B; rel_fun Q (rel_cset R) f g ⟧
⟹ rel_cset R (cUNION A f) (cUNION B g)"
unfolding rel_fun_def by transfer(erule rel_set_UNION, simp add: rel_fun_def)

lemma rel_cset_csingle_iff [simp]: "rel_cset R (csingle x) (csingle y) ⟷ R x y"

subsubsection ‹Transfer rules for the Transfer package›

text ‹Unconditional transfer rules›

context includes lifting_syntax
begin

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

lemma cinsert_parametric [transfer_rule]:
"(A ===> rel_cset A ===> rel_cset A) cinsert cinsert"
unfolding rel_fun_def rel_cset_iff by blast

lemma cUn_parametric [transfer_rule]:
"(rel_cset A ===> rel_cset A ===> rel_cset A) cUn cUn"
unfolding rel_fun_def rel_cset_iff by blast

lemma cUnion_parametric [transfer_rule]:
"(rel_cset (rel_cset A) ===> rel_cset A) cUnion cUnion"
unfolding rel_fun_def
by transfer (auto simp: rel_set_def, metis+)

lemma cimage_parametric [transfer_rule]:
"((A ===> B) ===> rel_cset A ===> rel_cset B) cimage cimage"
unfolding rel_fun_def rel_cset_iff by blast

lemma cBall_parametric [transfer_rule]:
"(rel_cset A ===> (A ===> op =) ===> op =) cBall cBall"
unfolding rel_cset_iff rel_fun_def by blast

lemma cBex_parametric [transfer_rule]:
"(rel_cset A ===> (A ===> op =) ===> op =) cBex cBex"
unfolding rel_cset_iff rel_fun_def by blast

lemma rel_cset_parametric [transfer_rule]:
"((A ===> B ===> op =) ===> rel_cset A ===> rel_cset B ===> op =) rel_cset rel_cset"
unfolding rel_fun_def
using rel_set_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred, where A = A and B = B]
by simp

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

lemma cin_parametric [transfer_rule]:
"bi_unique A ⟹ (A ===> rel_cset A ===> op =) cin cin"
unfolding rel_fun_def rel_cset_iff bi_unique_def by metis

lemma cInt_parametric [transfer_rule]:
"bi_unique A ⟹ (rel_cset A ===> rel_cset A ===> rel_cset A) cInt cInt"
unfolding rel_fun_def
using inter_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred]
by blast

lemma cDiff_parametric [transfer_rule]:
"bi_unique A ⟹ (rel_cset A ===> rel_cset A ===> rel_cset A) cDiff cDiff"
unfolding rel_fun_def
using Diff_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast

lemma csubset_parametric [transfer_rule]:
"bi_unique A ⟹ (rel_cset A ===> rel_cset A ===> op =) csubset_eq csubset_eq"
unfolding rel_fun_def
using subset_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast

end

lifting_update cset.lifting
lifting_forget cset.lifting

subsection ‹Registration as BNF›

lemma card_of_countable_sets_range:
fixes A :: "'a set"
shows "|{X. X ⊆ A ∧ countable X ∧ X ≠ {}}| ≤o |{f::nat ⇒ 'a. range f ⊆ A}|"
apply(rule card_of_ordLeqI[of from_nat_into]) using inj_on_from_nat_into
unfolding inj_on_def by auto

lemma card_of_countable_sets_Func:
"|{X. X ⊆ A ∧ countable X ∧ X ≠ {}}| ≤o |A| ^c natLeq"
using card_of_countable_sets_range card_of_Func_UNIV[THEN ordIso_symmetric]
unfolding cexp_def Field_natLeq Field_card_of
by (rule ordLeq_ordIso_trans)

lemma ordLeq_countable_subsets:
"|A| ≤o |{X. X ⊆ A ∧ countable X}|"
apply (rule card_of_ordLeqI[of "λ a. {a}"]) unfolding inj_on_def by auto

lemma finite_countable_subset:
"finite {X. X ⊆ A ∧ countable X} ⟷ finite A"
apply (rule iffI)
apply (erule contrapos_pp)
apply (rule card_of_ordLeq_infinite)
apply (rule ordLeq_countable_subsets)
apply assumption
apply (rule finite_Collect_conjI)
apply (rule disjI1)
apply (erule finite_Collect_subsets)
done

lemma rcset_to_rcset: "countable A ⟹ rcset (the_inv rcset A) = A"
including cset.lifting
apply (rule f_the_inv_into_f[unfolded inj_on_def image_iff])
apply transfer' apply simp
apply transfer' apply simp
done

lemma Collect_Int_Times: "{(x, y). R x y} ∩ A × B = {(x, y). R x y ∧ x ∈ A ∧ y ∈ B}"
by auto

lemma rel_cset_aux:
"(∀t ∈ rcset a. ∃u ∈ rcset b. R t u) ∧ (∀t ∈ rcset b. ∃u ∈ rcset a. R u t) ⟷
((BNF_Def.Grp {x. rcset x ⊆ {(a, b). R a b}} (cimage fst))¯¯ OO
BNF_Def.Grp {x. rcset x ⊆ {(a, b). R a b}} (cimage snd)) a b" (is "?L = ?R")
proof
assume ?L
define R' where "R' = the_inv rcset (Collect (case_prod R) ∩ (rcset a × rcset b))"
(is "_ = the_inv rcset ?L'")
have L: "countable ?L'" by auto
hence *: "rcset R' = ?L'" unfolding R'_def by (intro rcset_to_rcset)
thus ?R unfolding Grp_def relcompp.simps conversep.simps including cset.lifting
proof (intro CollectI case_prodI exI[of _ a] exI[of _ b] exI[of _ R'] conjI refl)
from * ‹?L› show "a = cimage fst R'" by transfer (auto simp: image_def Collect_Int_Times)
from * ‹?L› show "b = cimage snd R'" by transfer (auto simp: image_def Collect_Int_Times)
qed simp_all
next
assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps
by (simp add: subset_eq Ball_def)(transfer, auto simp add: cimage.rep_eq, metis snd_conv, metis fst_conv)
qed

bnf "'a cset"
map: cimage
sets: rcset
bd: natLeq
wits: "cempty"
rel: rel_cset
proof -
show "cimage id = id" by auto
next
fix f g show "cimage (g ∘ f) = cimage g ∘ cimage f" by fastforce
next
fix C f g assume eq: "⋀a. a ∈ rcset C ⟹ f a = g a"
thus "cimage f C = cimage g C" including cset.lifting by transfer force
next
fix f show "rcset ∘ cimage f = op ` f ∘ rcset" including cset.lifting by transfer' fastforce
next
show "card_order natLeq" by (rule natLeq_card_order)
next
show "cinfinite natLeq" by (rule natLeq_cinfinite)
next
fix C show "|rcset C| ≤o natLeq"
including cset.lifting by transfer (unfold countable_card_le_natLeq)
next
fix R S
show "rel_cset R OO rel_cset S ≤ rel_cset (R OO S)"
unfolding rel_cset_alt_def[abs_def] by fast
next
fix R
show "rel_cset R = (λx y. ∃z. rcset z ⊆ {(x, y). R x y} ∧
cimage fst z = x ∧ cimage snd z = y)"
unfolding rel_cset_alt_def[abs_def] rel_cset_aux[unfolded OO_Grp_alt] by simp