Theory FSet

(*  Title:      HOL/Library/FSet.thy
    Author:     Ondrej Kuncar, TU Muenchen
    Author:     Cezary Kaliszyk and Christian Urban
    Author:     Andrei Popescu, TU Muenchen
    Author:     Martin Desharnais, MPI-INF Saarbruecken

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

theory FSet
imports Main Countable

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
instance by (standard; transfer; simp)

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

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

  by (standard; transfer; auto)+


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
definition "HOL.equal A B  A |⊆| B  B |⊆| A"
instance by intro_classes (auto simp add: equal_fset_def)

instantiation fset :: (type) conditionally_complete_lattice

context includes lifting_syntax

lemma right_total_Inf_fset_transfer:
  assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
  shows "(rel_set (rel_set A) ===> rel_set A)
    (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


  fix x z :: "'a fset"
  fix X :: "'a fset set"
    assume "x  X" "bdd_below X"
    then show "Inf X |⊆| x" by transfer auto
    assume "X  {}" "(x. x  X  z |⊆| x)"
    then show "z |⊆| Inf X" by transfer (clarsimp, blast)
    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)
    assume "X  {}" "(x. x  X  x |⊆| z)"
    then show "Sup X |⊆| z" by transfer (clarsimp, blast)

instantiation fset :: (finite) complete_lattice

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

  by (standard; transfer; auto)


instantiation fset :: (finite) complete_boolean_algebra

lift_definition uminus_fset :: "'a fset  'a fset" is uminus
  parametric right_total_Compl_transfer Compl_transfer by simp

  by (standard; transfer) (simp_all add: Inf_Sup Diff_eq)

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

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

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

abbreviation fmember :: "'a  'a fset  bool" (infix "|∈|" 50) where
  "x |∈| X  x  fset X"

abbreviation not_fmember :: "'a  'a fset  bool" (infix "|∉|" 50) where
  "x |∉| X  x  fset X"


qualified abbreviation Ball :: "'a fset  ('a  bool)  bool" where
  "Ball X  Set.Ball (fset X)"

alias fBall = FSet.Ball

qualified abbreviation Bex :: "'a fset  ('a  bool)  bool" where
  "Bex X  Set.Bex (fset X)"

alias fBex = FSet.Bex


context includes lifting_syntax

lemma fmember_transfer0[transfer_rule]:
  assumes [transfer_rule]: "bi_unique A"
  shows "(A ===> pcr_fset A ===> (=)) (∈) (|∈|)"
  by transfer_prover

lemma fBall_transfer0[transfer_rule]:
  assumes [transfer_rule]: "bi_unique A"
  shows "(pcr_fset A ===> (A ===> (=)) ===> (=)) (Ball) (fBall)"
  by transfer_prover

lemma fBex_transfer0[transfer_rule]:
  assumes [transfer_rule]: "bi_unique A"
  shows "(pcr_fset A ===> (A ===> (=)) ===> (=)) (Bex) (fBex)"
  by transfer_prover

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
by (simp add: finite_subset)

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
by (simp add: Set.bind_def)

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

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›

lemma fset_eqI: "(x. (x |∈| A) = (x |∈| B))  A = B"
  by (rule set_eqI[Transfer.transferred])

lemma fset_eq_iff[no_atp]: "(A = B) = (x. (x |∈| A) = (x |∈| B))"
  by (rule set_eq_iff[Transfer.transferred])

lemma fBallI[no_atp]: "(x. x |∈| A  P x)  fBall A P"
  by (rule ballI[Transfer.transferred])

lemma fbspec[no_atp]: "fBall A P  x |∈| A  P x"
  by (rule bspec[Transfer.transferred])

lemma fBallE[no_atp]: "fBall A P  (P x  Q)  (x |∉| A  Q)  Q"
  by (rule ballE[Transfer.transferred])

lemma fBexI[no_atp]: "P x  x |∈| A  fBex A P"
  by (rule bexI[Transfer.transferred])

lemma rev_fBexI[no_atp]: "x |∈| A  P x  fBex A P"
  by (rule rev_bexI[Transfer.transferred])

lemma fBexCI[no_atp]: "(fBall A (λx. ¬ P x)  P a)  a |∈| A  fBex A P"
  by (rule bexCI[Transfer.transferred])

lemma fBexE[no_atp]: "fBex A P  (x. x |∈| A  P x  Q)  Q"
  by (rule bexE[Transfer.transferred])

lemma fBall_triv[no_atp]: "fBall A (λx. P) = ((x. x |∈| A)  P)"
  by (rule ball_triv[Transfer.transferred])

lemma fBex_triv[no_atp]: "fBex A (λx. P) = ((x. x |∈| A)  P)"
  by (rule bex_triv[Transfer.transferred])

lemma fBex_triv_one_point1[no_atp]: "fBex A (λx. x = a) = (a |∈| A)"
  by (rule bex_triv_one_point1[Transfer.transferred])

lemma fBex_triv_one_point2[no_atp]: "fBex A ((=) a) = (a |∈| A)"
  by (rule bex_triv_one_point2[Transfer.transferred])

lemma fBex_one_point1[no_atp]: "fBex A (λx. x = a  P x) = (a |∈| A  P a)"
  by (rule bex_one_point1[Transfer.transferred])

lemma fBex_one_point2[no_atp]: "fBex A (λx. a = x  P x) = (a |∈| A  P a)"
  by (rule bex_one_point2[Transfer.transferred])

lemma fBall_one_point1[no_atp]: "fBall A (λx. x = a  P x) = (a |∈| A  P a)"
  by (rule ball_one_point1[Transfer.transferred])

lemma fBall_one_point2[no_atp]: "fBall A (λx. a = x  P x) = (a |∈| A  P a)"
  by (rule ball_one_point2[Transfer.transferred])

lemma fBall_conj_distrib: "fBall A (λx. P x  Q x) = (fBall A P  fBall A Q)"
  by (rule ball_conj_distrib[Transfer.transferred])

lemma fBex_disj_distrib: "fBex A (λx. P x  Q x) = (fBex A P  fBex A Q)"
  by (rule bex_disj_distrib[Transfer.transferred])

lemma fBall_cong[fundef_cong]: "A = B  (x. x |∈| B  P x = Q x)  fBall A P = fBall B Q"
  by (rule ball_cong[Transfer.transferred])

lemma fBex_cong[fundef_cong]: "A = B  (x. x |∈| B  P x = Q x)  fBex A P = fBex B Q"
  by (rule bex_cong[Transfer.transferred])

lemma fsubsetI[intro!]: "(x. x |∈| A  x |∈| B)  A |⊆| B"
  by (rule subsetI[Transfer.transferred])

lemma fsubsetD[elim, intro?]: "A |⊆| B  c |∈| A  c |∈| B"
  by (rule subsetD[Transfer.transferred])

lemma rev_fsubsetD[no_atp,intro?]: "c |∈| A  A |⊆| B  c |∈| B"
  by (rule rev_subsetD[Transfer.transferred])

lemma fsubsetCE[no_atp,elim]: "A |⊆| B  (c |∉| A  P)  (c |∈| B  P)  P"
  by (rule subsetCE[Transfer.transferred])

lemma fsubset_eq[no_atp]: "(A |⊆| B) = fBall A (λx. x |∈| B)"
  by (rule subset_eq[Transfer.transferred])

lemma contra_fsubsetD[no_atp]: "A |⊆| B  c |∉| B  c |∉| A"
  by (rule contra_subsetD[Transfer.transferred])

lemma fsubset_refl: "A |⊆| A"
  by (rule subset_refl[Transfer.transferred])

lemma fsubset_trans: "A |⊆| B  B |⊆| C  A |⊆| C"
  by (rule subset_trans[Transfer.transferred])

lemma fset_rev_mp: "c |∈| A  A |⊆| B  c |∈| B"
  by (rule rev_subsetD[Transfer.transferred])

lemma fset_mp: "A |⊆| B  c |∈| A  c |∈| B"
  by (rule subsetD[Transfer.transferred])

lemma fsubset_not_fsubset_eq[code]: "(A |⊂| B) = (A |⊆| B  ¬ B |⊆| A)"
  by (rule subset_not_subset_eq[Transfer.transferred])

lemma eq_fmem_trans: "a = b  b |∈| A  a |∈| A"
  by (rule eq_mem_trans[Transfer.transferred])

lemma fsubset_antisym[intro!]: "A |⊆| B  B |⊆| A  A = B"
  by (rule subset_antisym[Transfer.transferred])

lemma fequalityD1: "A = B  A |⊆| B"
  by (rule equalityD1[Transfer.transferred])

lemma fequalityD2: "A = B  B |⊆| A"
  by (rule equalityD2[Transfer.transferred])

lemma fequalityE: "A = B  (A |⊆| B  B |⊆| A  P)  P"
  by (rule equalityE[Transfer.transferred])

lemma fequalityCE[elim]:
  "A = B  (c |∈| A  c |∈| B  P)  (c |∉| A  c |∉| B  P)  P"
  by (rule equalityCE[Transfer.transferred])

lemma eqfset_imp_iff: "A = B  (x |∈| A) = (x |∈| B)"
  by (rule eqset_imp_iff[Transfer.transferred])

lemma eqfelem_imp_iff: "x = y  (x |∈| A) = (y |∈| A)"
  by (rule eqelem_imp_iff[Transfer.transferred])

lemma fempty_iff[simp]: "(c |∈| {||}) = False"
  by (rule empty_iff[Transfer.transferred])

lemma fempty_fsubsetI[iff]: "{||} |⊆| x"
  by (rule empty_subsetI[Transfer.transferred])

lemma equalsffemptyI: "(y. y |∈| A  False)  A = {||}"
  by (rule equals0I[Transfer.transferred])

lemma equalsffemptyD: "A = {||}  a |∉| A"
  by (rule equals0D[Transfer.transferred])

lemma fBall_fempty[simp]: "fBall {||} P = True"
  by (rule ball_empty[Transfer.transferred])

lemma fBex_fempty[simp]: "fBex {||} P = False"
  by (rule bex_empty[Transfer.transferred])

lemma fPow_iff[iff]: "(A |∈| fPow B) = (A |⊆| B)"
  by (rule Pow_iff[Transfer.transferred])

lemma fPowI: "A |⊆| B  A |∈| fPow B"
  by (rule PowI[Transfer.transferred])

lemma fPowD: "A |∈| fPow B  A |⊆| B"
  by (rule PowD[Transfer.transferred])

lemma fPow_bottom: "{||} |∈| fPow B"
  by (rule Pow_bottom[Transfer.transferred])

lemma fPow_top: "A |∈| fPow A"
  by (rule Pow_top[Transfer.transferred])

lemma fPow_not_fempty: "fPow A  {||}"
  by (rule Pow_not_empty[Transfer.transferred])

lemma finter_iff[simp]: "(c |∈| A |∩| B) = (c |∈| A  c |∈| B)"
  by (rule Int_iff[Transfer.transferred])

lemma finterI[intro!]: "c |∈| A  c |∈| B  c |∈| A |∩| B"
  by (rule IntI[Transfer.transferred])

lemma finterD1: "c |∈| A |∩| B  c |∈| A"
  by (rule IntD1[Transfer.transferred])

lemma finterD2: "c |∈| A |∩| B  c |∈| B"
  by (rule IntD2[Transfer.transferred])

lemma finterE[elim!]: "c |∈| A |∩| B  (c |∈| A  c |∈| B  P)  P"
  by (rule IntE[Transfer.transferred])

lemma funion_iff[simp]: "(c |∈| A |∪| B) = (c |∈| A  c |∈| B)"
  by (rule Un_iff[Transfer.transferred])

lemma funionI1[elim?]: "c |∈| A  c |∈| A |∪| B"
  by (rule UnI1[Transfer.transferred])

lemma funionI2[elim?]: "c |∈| B  c |∈| A |∪| B"
  by (rule UnI2[Transfer.transferred])

lemma funionCI[intro!]: "(c |∉| B  c |∈| A)  c |∈| A |∪| B"
  by (rule UnCI[Transfer.transferred])

lemma funionE[elim!]: "c |∈| A |∪| B  (c |∈| A  P)  (c |∈| B  P)  P"
  by (rule UnE[Transfer.transferred])

lemma fminus_iff[simp]: "(c |∈| A |-| B) = (c |∈| A  c |∉| B)"
  by (rule Diff_iff[Transfer.transferred])

lemma fminusI[intro!]: "c |∈| A  c |∉| B  c |∈| A |-| B"
  by (rule DiffI[Transfer.transferred])

lemma fminusD1: "c |∈| A |-| B  c |∈| A"
  by (rule DiffD1[Transfer.transferred])

lemma fminusD2: "c |∈| A |-| B  c |∈| B  P"
  by (rule DiffD2[Transfer.transferred])

lemma fminusE[elim!]: "c |∈| A |-| B  (c |∈| A  c |∉| B  P)  P"
  by (rule DiffE[Transfer.transferred])

lemma finsert_iff[simp]: "(a |∈| finsert b A) = (a = b  a |∈| A)"
  by (rule insert_iff[Transfer.transferred])

lemma finsertI1: "a |∈| finsert a B"
  by (rule insertI1[Transfer.transferred])

lemma finsertI2: "a |∈| B  a |∈| finsert b B"
  by (rule insertI2[Transfer.transferred])

lemma finsertE[elim!]: "a |∈| finsert b A  (a = b  P)  (a |∈| A  P)  P"
  by (rule insertE[Transfer.transferred])

lemma finsertCI[intro!]: "(a |∉| B  a = b)  a |∈| finsert b B"
  by (rule insertCI[Transfer.transferred])

lemma fsubset_finsert_iff:
  "(A |⊆| finsert x B) = (if x |∈| A then A |-| {|x|} |⊆| B else A |⊆| B)"
  by (rule subset_insert_iff[Transfer.transferred])

lemma finsert_ident: "x |∉| A  x |∉| B  (finsert x A = finsert x B) = (A = B)"
  by (rule insert_ident[Transfer.transferred])

lemma fsingletonI[intro!,no_atp]: "a |∈| {|a|}"
  by (rule singletonI[Transfer.transferred])

lemma fsingletonD[dest!,no_atp]: "b |∈| {|a|}  b = a"
  by (rule singletonD[Transfer.transferred])

lemma fsingleton_iff: "(b |∈| {|a|}) = (b = a)"
  by (rule singleton_iff[Transfer.transferred])

lemma fsingleton_inject[dest!]: "{|a|} = {|b|}  a = b"
  by (rule singleton_inject[Transfer.transferred])

lemma fsingleton_finsert_inj_eq[iff,no_atp]: "({|b|} = finsert a A) = (a = b  A |⊆| {|b|})"
  by (rule singleton_insert_inj_eq[Transfer.transferred])

lemma fsingleton_finsert_inj_eq'[iff,no_atp]: "(finsert a A = {|b|}) = (a = b  A |⊆| {|b|})"
  by (rule singleton_insert_inj_eq'[Transfer.transferred])

lemma fsubset_fsingletonD: "A |⊆| {|x|}  A = {||}  A = {|x|}"
  by (rule subset_singletonD[Transfer.transferred])

lemma fminus_single_finsert: "A |-| {|x|} |⊆| B  A |⊆| finsert x B"
  by (rule Diff_single_insert[Transfer.transferred])

lemma fdoubleton_eq_iff: "({|a, b|} = {|c, d|}) = (a = c  b = d  a = d  b = c)"
  by (rule doubleton_eq_iff[Transfer.transferred])

lemma funion_fsingleton_iff:
  "(A |∪| B = {|x|}) = (A = {||}  B = {|x|}  A = {|x|}  B = {||}  A = {|x|}  B = {|x|})"
  by (rule Un_singleton_iff[Transfer.transferred])

lemma fsingleton_funion_iff:
  "({|x|} = A |∪| B) = (A = {||}  B = {|x|}  A = {|x|}  B = {||}  A = {|x|}  B = {|x|})"
  by (rule singleton_Un_iff[Transfer.transferred])

lemma fimage_eqI[simp, intro]: "b = f x  x |∈| A  b |∈| f |`| A"
  by (rule image_eqI[Transfer.transferred])

lemma fimageI: "x |∈| A  f x |∈| f |`| A"
  by (rule imageI[Transfer.transferred])

lemma rev_fimage_eqI: "x |∈| A  b = f x  b |∈| f |`| A"
  by (rule rev_image_eqI[Transfer.transferred])

lemma fimageE[elim!]: "b |∈| f |`| A  (x. b = f x  x |∈| A  thesis)  thesis"
  by (rule imageE[Transfer.transferred])

lemma Compr_fimage_eq: "{x. x |∈| f |`| A  P x} = f ` {x. x |∈| A  P (f x)}"
  by (rule Compr_image_eq[Transfer.transferred])

lemma fimage_funion: "f |`| (A |∪| B) = f |`| A |∪| f |`| B"
  by (rule image_Un[Transfer.transferred])

lemma fimage_iff: "(z |∈| f |`| A) = fBex A (λx. z = f x)"
  by (rule image_iff[Transfer.transferred])

lemma fimage_fsubset_iff[no_atp]: "(f |`| A |⊆| B) = fBall A (λx. f x |∈| B)"
  by (rule image_subset_iff[Transfer.transferred])

lemma fimage_fsubsetI: "(x. x |∈| A  f x |∈| B)  f |`| A |⊆| B"
  by (rule image_subsetI[Transfer.transferred])

lemma fimage_ident[simp]: "(λx. x) |`| Y = Y"
  by (rule image_ident[Transfer.transferred])

lemma if_split_fmem1: "((if Q then x else y) |∈| b) = ((Q  x |∈| b)  (¬ Q  y |∈| b))"
  by (rule if_split_mem1[Transfer.transferred])

lemma if_split_fmem2: "(a |∈| (if Q then x else y)) = ((Q  a |∈| x)  (¬ Q  a |∈| y))"
  by (rule if_split_mem2[Transfer.transferred])

lemma pfsubsetI[intro!,no_atp]: "A |⊆| B  A  B  A |⊂| B"
  by (rule psubsetI[Transfer.transferred])

lemma pfsubsetE[elim!,no_atp]: "A |⊂| B  (A |⊆| B  ¬ B |⊆| A  R)  R"
  by (rule psubsetE[Transfer.transferred])

lemma pfsubset_finsert_iff:
  "(A |⊂| finsert x B) =
    (if x |∈| B then A |⊂| B else if x |∈| A then A |-| {|x|} |⊂| B else A |⊆| B)"
  by (rule psubset_insert_iff[Transfer.transferred])

lemma pfsubset_eq: "(A |⊂| B) = (A |⊆| B  A  B)"
  by (rule psubset_eq[Transfer.transferred])

lemma pfsubset_imp_fsubset: "A |⊂| B  A |⊆| B"
  by (rule psubset_imp_subset[Transfer.transferred])

lemma pfsubset_trans: "A |⊂| B  B |⊂| C  A |⊂| C"
  by (rule psubset_trans[Transfer.transferred])

lemma pfsubsetD: "A |⊂| B  c |∈| A  c |∈| B"
  by (rule psubsetD[Transfer.transferred])

lemma pfsubset_fsubset_trans: "A |⊂| B  B |⊆| C  A |⊂| C"
  by (rule psubset_subset_trans[Transfer.transferred])

lemma fsubset_pfsubset_trans: "A |⊆| B  B |⊂| C  A |⊂| C"
  by (rule subset_psubset_trans[Transfer.transferred])

lemma pfsubset_imp_ex_fmem: "A |⊂| B  b. b |∈| B |-| A"
  by (rule psubset_imp_ex_mem[Transfer.transferred])

lemma fimage_fPow_mono: "f |`| A |⊆| B  (|`|) f |`| fPow A |⊆| fPow B"
  by (rule image_Pow_mono[Transfer.transferred])

lemma fimage_fPow_surj: "f |`| A = B  (|`|) f |`| fPow A = fPow B"
  by (rule image_Pow_surj[Transfer.transferred])

lemma fsubset_finsertI: "B |⊆| finsert a B"
  by (rule subset_insertI[Transfer.transferred])

lemma fsubset_finsertI2: "A |⊆| B  A |⊆| finsert b B"
  by (rule subset_insertI2[Transfer.transferred])

lemma fsubset_finsert: "x |∉| A  (A |⊆| finsert x B) = (A |⊆| B)"
  by (rule subset_insert[Transfer.transferred])

lemma funion_upper1: "A |⊆| A |∪| B"
  by (rule Un_upper1[Transfer.transferred])

lemma funion_upper2: "B |⊆| A |∪| B"
  by (rule Un_upper2[Transfer.transferred])

lemma funion_least: "A |⊆| C  B |⊆| C  A |∪| B |⊆| C"
  by (rule Un_least[Transfer.transferred])

lemma finter_lower1: "A |∩| B |⊆| A"
  by (rule Int_lower1[Transfer.transferred])

lemma finter_lower2: "A |∩| B |⊆| B"
  by (rule Int_lower2[Transfer.transferred])

lemma finter_greatest: "C |⊆| A  C |⊆| B  C |⊆| A |∩| B"
  by (rule Int_greatest[Transfer.transferred])

lemma fminus_fsubset: "A |-| B |⊆| A"
  by (rule Diff_subset[Transfer.transferred])

lemma fminus_fsubset_conv: "(A |-| B |⊆| C) = (A |⊆| B |∪| C)"
  by (rule Diff_subset_conv[Transfer.transferred])

lemma fsubset_fempty[simp]: "(A |⊆| {||}) = (A = {||})"
  by (rule subset_empty[Transfer.transferred])

lemma not_pfsubset_fempty[iff]: "¬ A |⊂| {||}"
  by (rule not_psubset_empty[Transfer.transferred])

lemma finsert_is_funion: "finsert a A = {|a|} |∪| A"
  by (rule insert_is_Un[Transfer.transferred])

lemma finsert_not_fempty[simp]: "finsert a A  {||}"
  by (rule insert_not_empty[Transfer.transferred])

lemma fempty_not_finsert: "{||}  finsert a A"
  by (rule empty_not_insert[Transfer.transferred])

lemma finsert_absorb: "a |∈| A  finsert a A = A"
  by (rule insert_absorb[Transfer.transferred])

lemma finsert_absorb2[simp]: "finsert x (finsert x A) = finsert x A"
  by (rule insert_absorb2[Transfer.transferred])

lemma finsert_commute: "finsert x (finsert y A) = finsert y (finsert x A)"
  by (rule insert_commute[Transfer.transferred])

lemma finsert_fsubset[simp]: "(finsert x A |⊆| B) = (x |∈| B  A |⊆| B)"
  by (rule insert_subset[Transfer.transferred])

lemma finsert_inter_finsert[simp]: "finsert a A |∩| finsert a B = finsert a (A |∩| B)"
  by (rule insert_inter_insert[Transfer.transferred])

lemma finsert_disjoint[simp,no_atp]:
  "(finsert a A |∩| B = {||}) = (a |∉| B  A |∩| B = {||})"
  "({||} = finsert a A |∩| B) = (a |∉| B  {||} = A |∩| B)"
  by (rule insert_disjoint[Transfer.transferred])+

lemma disjoint_finsert[simp,no_atp]:
  "(B |∩| finsert a A = {||}) = (a |∉| B  B |∩| A = {||})"
  "({||} = A |∩| finsert b B) = (b |∉| A  {||} = A |∩| B)"
  by (rule disjoint_insert[Transfer.transferred])+

lemma fimage_fempty[simp]: "f |`| {||} = {||}"
  by (rule image_empty[Transfer.transferred])

lemma fimage_finsert[simp]: "f |`| finsert a B = finsert (f a) (f |`| B)"
  by (rule image_insert[Transfer.transferred])

lemma fimage_constant: "x |∈| A  (λx. c) |`| A = {|c|}"
  by (rule image_constant[Transfer.transferred])

lemma fimage_constant_conv: "(λx. c) |`| A = (if A = {||} then {||} else {|c|})"
  by (rule image_constant_conv[Transfer.transferred])

lemma fimage_fimage: "f |`| g |`| A = (λx. f (g x)) |`| A"
  by (rule image_image[Transfer.transferred])

lemma finsert_fimage[simp]: "x |∈| A  finsert (f x) (f |`| A) = f |`| A"
  by (rule insert_image[Transfer.transferred])

lemma fimage_is_fempty[iff]: "(f |`| A = {||}) = (A = {||})"
  by (rule image_is_empty[Transfer.transferred])

lemma fempty_is_fimage[iff]: "({||} = f |`| A) = (A = {||})"
  by (rule empty_is_image[Transfer.transferred])

lemma fimage_cong: "M = N  (x. x |∈| N  f x = g x)  f |`| M = g |`| N"
  by (rule image_cong[Transfer.transferred])

lemma fimage_finter_fsubset: "f |`| (A |∩| B) |⊆| f |`| A |∩| f |`| B"
  by (rule image_Int_subset[Transfer.transferred])

lemma fimage_fminus_fsubset: "f |`| A |-| f |`| B |⊆| f |`| (A |-| B)"
  by (rule image_diff_subset[Transfer.transferred])

lemma finter_absorb: "A |∩| A = A"
  by (rule Int_absorb[Transfer.transferred])

lemma finter_left_absorb: "A |∩| (A |∩| B) = A |∩| B"
  by (rule Int_left_absorb[Transfer.transferred])

lemma finter_commute: "A |∩| B = B |∩| A"
  by (rule Int_commute[Transfer.transferred])

lemma finter_left_commute: "A |∩| (B |∩| C) = B |∩| (A |∩| C)"
  by (rule Int_left_commute[Transfer.transferred])

lemma finter_assoc: "A |∩| B |∩| C = A |∩| (B |∩| C)"
  by (rule Int_assoc[Transfer.transferred])

lemma finter_ac:
  "A |∩| B |∩| C = A |∩| (B |∩| C)"
  "A |∩| (A |∩| B) = A |∩| B"
  "A |∩| B = B |∩| A"
  "A |∩| (B |∩| C) = B |∩| (A |∩| C)"
  by (rule Int_ac[Transfer.transferred])+

lemma finter_absorb1: "B |⊆| A  A |∩| B = B"
  by (rule Int_absorb1[Transfer.transferred])

lemma finter_absorb2: "A |⊆| B  A |∩| B = A"
  by (rule Int_absorb2[Transfer.transferred])

lemma finter_fempty_left: "{||} |∩| B = {||}"
  by (rule Int_empty_left[Transfer.transferred])

lemma finter_fempty_right: "A |∩| {||} = {||}"
  by (rule Int_empty_right[Transfer.transferred])

lemma disjoint_iff_fnot_equal: "(A |∩| B = {||}) = fBall A (λx. fBall B ((≠) x))"
  by (rule disjoint_iff_not_equal[Transfer.transferred])

lemma finter_funion_distrib: "A |∩| (B |∪| C) = A |∩| B |∪| (A |∩| C)"
  by (rule Int_Un_distrib[Transfer.transferred])

lemma finter_funion_distrib2: "B |∪| C |∩| A = B |∩| A |∪| (C |∩| A)"
  by (rule Int_Un_distrib2[Transfer.transferred])

lemma finter_fsubset_iff[no_atp, simp]: "(C |⊆| A |∩| B) = (C |⊆| A  C |⊆| B)"
  by (rule Int_subset_iff[Transfer.transferred])

lemma funion_absorb: "A |∪| A = A"
  by (rule Un_absorb[Transfer.transferred])

lemma funion_left_absorb: "A |∪| (A |∪| B) = A |∪| B"
  by (rule Un_left_absorb[Transfer.transferred])

lemma funion_commute: "A |∪| B = B |∪| A"
  by (rule Un_commute[Transfer.transferred])

lemma funion_left_commute: "A |∪| (B |∪| C) = B |∪| (A |∪| C)"
  by (rule Un_left_commute[Transfer.transferred])

lemma funion_assoc: "A |∪| B |∪| C = A |∪| (B |∪| C)"
  by (rule Un_assoc[Transfer.transferred])

lemma funion_ac:
  "A |∪| B |∪| C = A |∪| (B |∪| C)"
  "A |∪| (A |∪| B) = A |∪| B"
  "A |∪| B = B |∪| A"
  "A |∪| (B |∪| C) = B |∪| (A |∪| C)"
  by (rule Un_ac[Transfer.transferred])+

lemma funion_absorb1: "A |⊆| B  A |∪| B = B"
  by (rule Un_absorb1[Transfer.transferred])

lemma funion_absorb2: "B |⊆| A  A |∪| B = A"
  by (rule Un_absorb2[Transfer.transferred])

lemma funion_fempty_left: "{||} |∪| B = B"
  by (rule Un_empty_left[Transfer.transferred])

lemma funion_fempty_right: "A |∪| {||} = A"
  by (rule Un_empty_right[Transfer.transferred])

lemma funion_finsert_left[simp]: "finsert a B |∪| C = finsert a (B |∪| C)"
  by (rule Un_insert_left[Transfer.transferred])

lemma funion_finsert_right[simp]: "A |∪| finsert a B = finsert a (A |∪| B)"
  by (rule Un_insert_right[Transfer.transferred])

lemma finter_finsert_left: "finsert a B |∩| C = (if a |∈| C then finsert a (B |∩| C) else B |∩| C)"
  by (rule Int_insert_left[Transfer.transferred])

lemma finter_finsert_left_ifffempty[simp]: "a |∉| C  finsert a B |∩| C = B |∩| C"
  by (rule Int_insert_left_if0[Transfer.transferred])

lemma finter_finsert_left_if1[simp]: "a |∈| C  finsert a B |∩| C = finsert a (B |∩| C)"
  by (rule Int_insert_left_if1[Transfer.transferred])

lemma finter_finsert_right:
  "A |∩| finsert a B = (if a |∈| A then finsert a (A |∩| B) else A |∩| B)"
  by (rule Int_insert_right[Transfer.transferred])

lemma finter_finsert_right_ifffempty[simp]: "a |∉| A  A |∩| finsert a B = A |∩| B"
  by (rule Int_insert_right_if0[Transfer.transferred])

lemma finter_finsert_right_if1[simp]: "a |∈| A  A |∩| finsert a B = finsert a (A |∩| B)"
  by (rule Int_insert_right_if1[Transfer.transferred])

lemma funion_finter_distrib: "A |∪| (B |∩| C) = A |∪| B |∩| (A |∪| C)"
  by (rule Un_Int_distrib[Transfer.transferred])

lemma funion_finter_distrib2: "B |∩| C |∪| A = B |∪| A |∩| (C |∪| A)"
  by (rule Un_Int_distrib2[Transfer.transferred])

lemma funion_finter_crazy:
  "A |∩| B |∪| (B |∩| C) |∪| (C |∩| A) = A |∪| B |∩| (B |∪| C) |∩| (C |∪| A)"
  by (rule Un_Int_crazy[Transfer.transferred])

lemma fsubset_funion_eq: "(A |⊆| B) = (A |∪| B = B)"
  by (rule subset_Un_eq[Transfer.transferred])

lemma funion_fempty[iff]: "(A |∪| B = {||}) = (A = {||}  B = {||})"
  by (rule Un_empty[Transfer.transferred])

lemma funion_fsubset_iff[no_atp, simp]: "(A |∪| B |⊆| C) = (A |⊆| C  B |⊆| C)"
  by (rule Un_subset_iff[Transfer.transferred])

lemma funion_fminus_finter: "A |-| B |∪| (A |∩| B) = A"
  by (rule Un_Diff_Int[Transfer.transferred])

lemma ffunion_empty[simp]: "ffUnion {||} = {||}"
  by (rule Union_empty[Transfer.transferred])

lemma ffunion_mono: "A |⊆| B  ffUnion A |⊆| ffUnion B"
  by (rule Union_mono[Transfer.transferred])

lemma ffunion_insert[simp]: "ffUnion (finsert a B) = a |∪| ffUnion B"
  by (rule Union_insert[Transfer.transferred])

lemma fminus_finter2: "A |∩| C |-| (B |∩| C) = A |∩| C |-| B"
  by (rule Diff_Int2[Transfer.transferred])

lemma funion_finter_assoc_eq: "(A |∩| B |∪| C = A |∩| (B |∪| C)) = (C |⊆| A)"
  by (rule Un_Int_assoc_eq[Transfer.transferred])

lemma fBall_funion: "fBall (A |∪| B) P = (fBall A P  fBall B P)"
  by (rule ball_Un[Transfer.transferred])

lemma fBex_funion: "fBex (A |∪| B) P = (fBex A P  fBex B P)"
  by (rule bex_Un[Transfer.transferred])

lemma fminus_eq_fempty_iff[simp,no_atp]: "(A |-| B = {||}) = (A |⊆| B)"
  by (rule Diff_eq_empty_iff[Transfer.transferred])

lemma fminus_cancel[simp]: "A |-| A = {||}"
  by (rule Diff_cancel[Transfer.transferred])

lemma fminus_idemp[simp]: "A |-| B |-| B = A |-| B"
  by (rule Diff_idemp[Transfer.transferred])

lemma fminus_triv: "A |∩| B = {||}  A |-| B = A"
  by (rule Diff_triv[Transfer.transferred])

lemma fempty_fminus[simp]: "{||} |-| A = {||}"
  by (rule empty_Diff[Transfer.transferred])

lemma fminus_fempty[simp]: "A |-| {||} = A"
  by (rule Diff_empty[Transfer.transferred])

lemma fminus_finsertffempty[simp,no_atp]: "x |∉| A  A |-| finsert x B = A |-| B"
  by (rule Diff_insert0[Transfer.transferred])

lemma fminus_finsert: "A |-| finsert a B = A |-| B |-| {|a|}"
  by (rule Diff_insert[Transfer.transferred])

lemma fminus_finsert2: "A |-| finsert a B = A |-| {|a|} |-| B"
  by (rule Diff_insert2[Transfer.transferred])

lemma finsert_fminus_if: "finsert x A |-| B = (if x |∈| B then A |-| B else finsert x (A |-| B))"
  by (rule insert_Diff_if[Transfer.transferred])

lemma finsert_fminus1[simp]: "x |∈| B  finsert x A |-| B = A |-| B"
  by (rule insert_Diff1[Transfer.transferred])

lemma finsert_fminus_single[simp]: "finsert a (A |-| {|a|}) = finsert a A"
  by (rule insert_Diff_single[Transfer.transferred])

lemma finsert_fminus: "a |∈| A  finsert a (A |-| {|a|}) = A"
  by (rule insert_Diff[Transfer.transferred])

lemma fminus_finsert_absorb: "x |∉| A  finsert x A |-| {|x|} = A"
  by (rule Diff_insert_absorb[Transfer.transferred])

lemma fminus_disjoint[simp]: "A |∩| (B |-| A) = {||}"
  by (rule Diff_disjoint[Transfer.transferred])

lemma fminus_partition: "A |⊆| B  A |∪| (B |-| A) = B"
  by (rule Diff_partition[Transfer.transferred])

lemma double_fminus: "A |⊆| B  B |⊆| C  B |-| (C |-| A) = A"
  by (rule double_diff[Transfer.transferred])

lemma funion_fminus_cancel[simp]: "A |∪| (B |-| A) = A |∪| B"
  by (rule Un_Diff_cancel[Transfer.transferred])

lemma funion_fminus_cancel2[simp]: "B |-| A |∪| A = B |∪| A"
  by (rule Un_Diff_cancel2[Transfer.transferred])

lemma fminus_funion: "A |-| (B |∪| C) = A |-| B |∩| (A |-| C)"
  by (rule Diff_Un[Transfer.transferred])

lemma fminus_finter: "A |-| (B |∩| C) = A |-| B |∪| (A |-| C)"
  by (rule Diff_Int[Transfer.transferred])

lemma funion_fminus: "A |∪| B |-| C = A |-| C |∪| (B |-| C)"
  by (rule Un_Diff[Transfer.transferred])

lemma finter_fminus: "A |∩| B |-| C = A |∩| (B |-| C)"
  by (rule Int_Diff[Transfer.transferred])

lemma fminus_finter_distrib: "C |∩| (A |-| B) = C |∩| A |-| (C |∩| B)"
  by (rule Diff_Int_distrib[Transfer.transferred])

lemma fminus_finter_distrib2: "A |-| B |∩| C = A |∩| C |-| (B |∩| C)"
  by (rule Diff_Int_distrib2[Transfer.transferred])

lemma fUNIV_bool[no_atp]: "fUNIV = {|False, True|}"
  by (rule UNIV_bool[Transfer.transferred])

lemma fPow_fempty[simp]: "fPow {||} = {|{||}|}"
  by (rule Pow_empty[Transfer.transferred])

lemma fPow_finsert: "fPow (finsert a A) = fPow A |∪| finsert a |`| fPow A"
  by (rule Pow_insert[Transfer.transferred])

lemma funion_fPow_fsubset: "fPow A |∪| fPow B |⊆| fPow (A |∪| B)"
  by (rule Un_Pow_subset[Transfer.transferred])

lemma fPow_finter_eq[simp]: "fPow (A |∩| B) = fPow A |∩| fPow B"
  by (rule Pow_Int_eq[Transfer.transferred])

lemma fset_eq_fsubset: "(A = B) = (A |⊆| B  B |⊆| A)"
  by (rule set_eq_subset[Transfer.transferred])

lemma fsubset_iff[no_atp]: "(A |⊆| B) = (t. t |∈| A  t |∈| B)"
  by (rule subset_iff[Transfer.transferred])

lemma fsubset_iff_pfsubset_eq: "(A |⊆| B) = (A |⊂| B  A = B)"
  by (rule subset_iff_psubset_eq[Transfer.transferred])

lemma all_not_fin_conv[simp]: "(x. x |∉| A) = (A = {||})"
  by (rule all_not_in_conv[Transfer.transferred])

lemma ex_fin_conv: "(x. x |∈| A) = (A  {||})"
  by (rule ex_in_conv[Transfer.transferred])

lemma fimage_mono: "A |⊆| B  f |`| A |⊆| f |`| B"
  by (rule image_mono[Transfer.transferred])

lemma fPow_mono: "A |⊆| B  fPow A |⊆| fPow B"
  by (rule Pow_mono[Transfer.transferred])

lemma finsert_mono: "C |⊆| D  finsert a C |⊆| finsert a D"
  by (rule insert_mono[Transfer.transferred])

lemma funion_mono: "A |⊆| C  B |⊆| D  A |∪| B |⊆| C |∪| D"
  by (rule Un_mono[Transfer.transferred])

lemma finter_mono: "A |⊆| C  B |⊆| D  A |∩| B |⊆| C |∩| D"
  by (rule Int_mono[Transfer.transferred])

lemma fminus_mono: "A |⊆| C  D |⊆| B  A |-| B |⊆| C |-| D"
  by (rule Diff_mono[Transfer.transferred])

lemma fin_mono: "A |⊆| B  x |∈| A  x |∈| B"
  by (rule in_mono[Transfer.transferred])

lemma fthe_felem_eq[simp]: "fthe_elem {|x|} = x"
  by (rule the_elem_eq[Transfer.transferred])

lemma fLeast_mono:
  "mono f  fBex S (λx. fBall S ((≤) x))  (LEAST y. y |∈| f |`| S) = f (LEAST x. x |∈| S)"
  by (rule Least_mono[Transfer.transferred])

lemma fbind_fbind: "fbind (fbind A B) C = fbind A (λx. fbind (B x) C)"
  by (rule Set.bind_bind[Transfer.transferred])

lemma fempty_fbind[simp]: "fbind {||} f = {||}"
  by (rule empty_bind[Transfer.transferred])

lemma nonfempty_fbind_const: "A  {||}  fbind A (λ_. B) = B"
  by (rule nonempty_bind_const[Transfer.transferred])

lemma fbind_const: "fbind A (λ_. B) = (if A = {||} then {||} else B)"
  by (rule bind_const[Transfer.transferred])

lemma ffmember_filter[simp]: "(x |∈| ffilter P A) = (x |∈| A  P x)"
  by (rule member_filter[Transfer.transferred])

lemma fequalityI: "A |⊆| B  B |⊆| A  A = B"
  by (rule equalityI[Transfer.transferred])

lemma fset_of_list_simps[simp]:
  "fset_of_list [] = {||}"
  "fset_of_list (x21 # x22) = finsert x21 (fset_of_list x22)"
  by (rule set_simps[Transfer.transferred])+

lemma fset_of_list_append[simp]: "fset_of_list (xs @ ys) = fset_of_list xs |∪| fset_of_list ys"
  by (rule set_append[Transfer.transferred])

lemma fset_of_list_rev[simp]: "fset_of_list (rev xs) = fset_of_list xs"
  by (rule set_rev[Transfer.transferred])

lemma fset_of_list_map[simp]: "fset_of_list (map f xs) = f |`| fset_of_list xs"
  by (rule set_map[Transfer.transferred])

subsection ‹Additional lemmas›

subsubsection ffUnion›

lemma ffUnion_funion_distrib[simp]: "ffUnion (A |∪| B) = ffUnion A |∪| ffUnion B"
  by (rule 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›

lemma fsingletonE: " b |∈| {|a|}  (b = a  thesis)  thesis"
  by (rule 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›

lemma fset_simps[simp]:
  "fset {||} = {}"
  "fset (finsert x X) = insert x (fset X)"
  by (rule 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 inter_fset[simp]: "fset (A |∩| B) = fset A  fset B"
  by (rule inf_fset.rep_eq)

lemma union_fset[simp]: "fset (A |∪| B) = fset A  fset B"
  by (rule sup_fset.rep_eq)

lemma minus_fset[simp]: "fset (A |-| B) = fset A - fset B"
  by (rule 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)

lemma fimage_strict_mono:
  assumes "inj_on f (fset B)" and "A |⊂| B"
  shows "f |`| A |⊂| f |`| B"
  ― ‹TODO: Configure transfer framework to lift @{thm Fun.image_strict_mono}.›
proof (rule pfsubsetI)
  from A |⊂| B have "A |⊆| B"
    by (rule pfsubset_imp_fsubset)
  thus "f |`| A |⊆| f |`| B"
    by (rule fimage_mono)
  from A |⊂| B have "A |⊆| B" and "A  B"
    by (simp_all add: pfsubset_eq)

  have "fset A  fset B"
    using A  B
    by (simp add: fset_cong)
  hence "f ` fset A  f ` fset B"
    using A |⊆| B
    by (simp add: inj_on_image_eq_iff[OF inj_on f (fset B)] less_eq_fset.rep_eq)
  hence "fset (f |`| A)  fset (f |`| B)"
    by (simp add: fimage.rep_eq)
  thus "f |`| A  f |`| B"
    by (simp add: fset_cong)

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


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_remove)

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
  lemma ffold_empty[simp]: "ffold f z {||} = z"
    by (rule 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)

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

context comp_fun_idem

  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)


subsubsection @{term fsubset}

lemma wfP_pfsubset: "wfP (|⊂|)"
proof (rule wfP_if_convertible_to_nat)
  show "x y. x |⊂| y  fcard x < fcard y"
    by (rule pfsubset_fcard_mono)

subsubsection ‹Group operations›

locale comm_monoid_fset = comm_monoid

sublocale set: comm_monoid_set ..

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

lemma cong[fundef_cong]: "A = B  (x. x |∈| B  g x = h x)  F g A = F h B"
  by (rule set.cong[Transfer.transferred])

lemma cong_simp[cong]:
  " A = B;  x. x |∈| B =simp=> g x = h x   F g A = F h B"
unfolding simp_implies_def by (auto cong: cong)


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 ..


subsubsection ‹Semilattice operations›

locale semilattice_fset = semilattice

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)


locale semilattice_order_fset = binary?: semilattice_order + semilattice_fset


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 ..

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 ..


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)

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)


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)

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)

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)
  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

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

subsection ‹Lemmas depending on induction›

lemma ffUnion_fsubset_iff: "ffUnion A |⊆| B  fBall A (λx. x |⊆| B)"
  by (induction A) simp_all

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: "xX. R x (f x)  (zZ. 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: "zZ. S (g z) z  (xX. 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

subsubsection ‹Transfer rules for the Transfer package›

text ‹Unconditional transfer rules›

context includes lifting_syntax

lemma fempty_transfer [transfer_rule]:
  "rel_fset A {||} {||}"
  by (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