Quelle  Set.thy

(*  Title:      HOL/Set.thy
  Author: Tobias Nipkow
  Author: Lawrence C Paulson
  Author: Markus Wenzel
*)

section ‹Set theory for higher-order logic›

theory Set
  imports Lattices Boolean_Algebras
begin

subsection ‹Sets as predicates›

typedecl 'a set

axiomatization Collect :: "('a ==> bool) ==> 'a set" 🍋 ‹comprehension›
  and member :: "'a ==> 'a set ==> bool" 🍋 ‹membership›
  where mem_Collect_eq [iff, code_unfold]: "member a (Collect P) = P a"
    and Collect_mem_eq [simp, code_unfold]: "Collect (λx. member x A) = A"

notation
  member  (‹'(∈')›) and
  member  (‹(‹notation=‹infix ∈›\›_/ ∈ _)› [51, 51] 50)

abbreviation not_member
  where "not_member x A ≡ ¬ (x ∈ A)" 🍋 ‹non-membership›
notation
  not_member  (‹'(∉')›) and
  not_member  (‹(‹notation=‹infix ∉›\›_/ ∉ _)› [51, 51] 50)

open_bundle member_ASCII_syntax
begin
notation (ASCII)
  member  (‹'(:')›) and
  member  (‹(‹notation=‹infix :›\›_/ : _)› [51, 51] 50) and
  not_member  (‹'(~:')›) and
  not_member  (‹(‹notation=‹infix ~:›\›_/ ~: _)› [51, 51] 50)
end


text ‹Set comprehensions›

syntax
  "_Coll" :: "pttrn ==> bool ==> 'a set"    (‹(‹indent=1 notation=‹mixfix set comprehension›\›{_./ _})›)
syntax_consts
  "_Coll" ⇌ Collect
translations
  "{x. P}" ⇌ "CONST Collect (λx. P)"

syntax (ASCII)
  "_Collect" :: "pttrn ==> 'a set ==> bool ==> 'a set"  (‹(‹indent=1 notation=‹mixfix set comprehension›\›{(_/: _)./ _})›)
syntax
  "_Collect" :: "pttrn ==> 'a set ==> bool ==> 'a set"  (‹(‹indent=1 notation=‹mixfix set comprehension›\›{(_/ ∈ _)./ _})›)
translations
  "{p:A. P}" ⇀ "CONST Collect (λp. p ∈ A ∧ P)"

ML ‹
  fun Collect_binder_tr' c [Abs (x, T, t), Const (🍋‹Collect›, _) $ Abs (y, _, P)] =
  if x = y then
  let
  val x' = Syntax_Trans.mark_bound_body (x, T);
  val t' = subst_bound (x', t);
  val P' = subst_bound (x', P);
  in Syntax.const c $ Syntax_Trans.mark_bound_abs (x, T) $ P' $ t' end
  else raise Match
  | Collect_binder_tr' _ _ = raise Match
 ›

lemma CollectI: "P a ==> a ∈ {x. P x}"
  by simp

lemma CollectD: "a ∈ {x. P x} ==> P a"
  by simp

lemma Collect_cong: "(∧x. P x = Q x) ==> {x. P x} = {x. Q x}"
  by simp

text ‹
  Simproc for pulling ‹x = t› in ‹{x. … ∧ x = t ∧ …}›
  to the front (and similarly for ‹t = x›):
 ›

simproc_setup defined_Collect ("{x. P x ∧ Q x}") = ‹
  K (Quantifier1.rearrange_Collect
  (fn ctxt =>
  resolve_tac ctxt @{thms Collect_cong} 1 THEN
  resolve_tac ctxt @{thms iffI} 1 THEN
  ALLGOALS
  (EVERY' [REPEAT_DETERM o eresolve_tac ctxt @{thms conjE},
  DEPTH_SOLVE_1 o (assume_tac ctxt ORELSE' resolve_tac ctxt @{thms conjI})])))
 ›

lemmas CollectE = CollectD [elim_format]

lemma set_eqI:
  assumes "∧x. x ∈ A ⟷ x ∈ B"
  shows "A = B"
proof -
  from assms have "{x. x ∈ A} = {x. x ∈ B}"
    by simp
  then show ?thesis by simp
qed

lemma set_eq_iff: "A = B ⟷ (∀x. x ∈ A ⟷ x ∈ B)"
  by (auto intro:set_eqI)

lemma Collect_eqI:
  assumes "∧x. P x = Q x"
  shows "Collect P = Collect Q"
  using assms by (auto intro: set_eqI)

text ‹Lifting of predicate class instances›

instantiation set :: (type) boolean_algebra
begin

definition less_eq_set
  where "A ≤ B ⟷ (λx. member x A) ≤ (λx. member x B)"

definition less_set
  where "A < B ⟷ (λx. member x A) < (λx. member x B)"

definition inf_set
  where "A ⊓ B = Collect ((λx. member x A) ⊓ (λx. member x B))"

definition sup_set
  where "A ⊔ B = Collect ((λx. member x A) ⊔ (λx. member x B))"

definition bot_set
  where "⊥ = Collect ⊥"

definition top_set
  where "⊤ = Collect ⊤"

definition uminus_set
  where "- A = Collect (- (λx. member x A))"

definition minus_set
  where "A - B = Collect ((λx. member x A) - (λx. member x B))"

instance
  by standard
    (simp_all add: less_eq_set_def less_set_def inf_set_def sup_set_def
      bot_set_def top_set_def uminus_set_def minus_set_def
      less_le_not_le sup_inf_distrib1 diff_eq set_eqI fun_eq_iff
      del: inf_apply sup_apply bot_apply top_apply minus_apply uminus_apply)

end

text ‹Set enumerations›

abbreviation empty :: "'a set" (‹{}›)
  where "{} ≡ bot"

definition insert :: "'a ==> 'a set ==> 'a set"
  where insert_compr: "insert a B = {x. x = a ∨ x ∈ B}"

open_bundle set_enumeration_syntax
begin

syntax
  "_Finset" :: "args ==> 'a set"  (‹(‹indent=1 notation=‹mixfix set enumeration›\›{_})›)
syntax_consts
  "_Finset" ⇌ insert
translations
  "{x, xs}" ⇌ "CONST insert x {xs}"
  "{x}" ⇌ "CONST insert x {}"

end


subsection ‹Subsets and bounded quantifiers›

abbreviation subset :: "'a set ==> 'a set ==> bool"
  where "subset ≡ less"

abbreviation subset_eq :: "'a set ==> 'a set ==> bool"
  where "subset_eq ≡ less_eq"

notation
  subset  (‹'(⊂')›) and
  subset  (‹(‹notation=‹infix ⊂›\›_/ ⊂ _)› [51, 51] 50) and
  subset_eq  (‹'(⊆')›) and
  subset_eq  (‹(‹notation=‹infix ⊆›\›_/ ⊆ _)› [51, 51] 50)

abbreviation (input)
  supset :: "'a set ==> 'a set ==> bool" where
  "supset ≡ greater"

abbreviation (input)
  supset_eq :: "'a set ==> 'a set ==> bool" where
  "supset_eq ≡ greater_eq"

notation
  supset  (‹'(🪙')›) and
  supset  (‹(‹notation=‹infix 🪙›\›_/ 🪙 _)› [51, 51] 50) and
  supset_eq  (‹'(🪙')›) and
  supset_eq  (‹(‹notation=‹infix 🪙›\›_/ 🪙 _)› [51, 51] 50)

notation (ASCII output)
  subset  (‹'(🚫›) and
  subset  (‹(‹notation=‹infix 🚫close>›_/ 🚫)› [51, 51] 50) and
  subset_eq (‹'(🚫)›) and
  subset_eq (‹(‹notation=‹infix 🚫›\›_/ 🚫_)› [51, 51] 50)
 
 definition Ball :: "'a set ==> ('a ==> bool) ==> bool"
  where "Ball A P ⟷ (∀x. x ∈ A ⟶ P x)" 🍋 ‹bounded universal quantifiers›
 
 definition Bex :: "'a set ==> ('a ==> bool) ==> bool"
  where "Bex A P ⟷ (∃x. x ∈ A ∧ P x)" 🍋 ‹bounded existential quantifiers›
 
 syntax (ASCII)
  "_Ball" :: "pttrn ==> 'a set ==> bool ==> bool" (‹(‹indent=3 notation=‹binder ALL :›\›ALL (_/:_)./ _)› [0, 0, 10] 10)
  "_Bex" :: "pttrn ==> 'a set ==> bool ==> bool" (‹(‹indent=3 notation=‹binder EX :›\›EX (_/:_)./ _)› [0, 0, 10] 10)
  "_Bex1" :: "pttrn ==> 'a set ==> bool ==> bool" (‹(‹indent=3 notation=‹binder EX! :›\›EX! (_/:_)./ _)› [0, 0, 10] 10)
  "_Bleast" :: "id ==> 'a set ==> bool ==> 'a" (‹(‹indent=3 notation=‹binder LEAST :›\›LEAST (_/:_)./ _)› [0, 0, 10] 10)
 
 syntax (input)
  "_Ball" :: "pttrn ==> 'a set ==> bool ==> bool" (‹(‹indent=3 notation=‹binder ! :›\›! (_/:_)./ _)› [0, 0, 10] 10)
  "_Bex" :: "pttrn ==> 'a set ==> bool ==> bool" (‹(‹indent=3 notation=‹binder ? :›\›? (_/:_)./ _)› [0, 0, 10] 10)
  "_Bex1" :: "pttrn ==> 'a set ==> bool ==> bool" (‹(‹indent=3 notation=‹binder ?! :›\›?! (_/:_)./ _)› [0, 0, 10] 10)
 
 syntax
  "_Ball" :: "pttrn ==> 'a set ==> bool ==> bool" (‹(‹indent=3 notation=‹binder ∀›\›\∀(_/∈_)./ _)› [0, 0, 10] 10)
  "_Bex" :: "pttrn ==> 'a set ==> bool ==> bool" (‹(‹indent=3 notation=‹binder ∃›\›\∃(_/∈_)./ _)› [0, 0, 10] 10)
  "_Bex1" :: "pttrn ==> 'a set ==> bool ==> bool" (‹(‹indent=3 notation=‹binder ∃!›\›\∃!(_/∈_)./ _)› [0, 0, 10] 10)
  "_Bleast" :: "id ==> 'a set ==> bool ==> 'a" (‹(‹indent=3 notation=‹binder LEAST›\›LEAST(_/∈_)./ _)› [0, 0, 10] 10)
 
 syntax_consts
  "_Ball" ⇌ Ball and
  "_Bex" ⇌ Bex and
  "_Bex1" ⇌ Ex1 and
  "_Bleast" ⇌ Least
 
 translations
  "∀x∈A. P" ⇌ "CONST Ball A (λx. P)"
  "∃x∈A. P" ⇌ "CONST Bex A (λx. P)"
  "∃!x∈A. P" ⇀ "∃!x. x ∈ A ∧ P"
  "LEAST x:A. P" ⇀ "LEAST x. x ∈ A ∧ P"
 
 syntax (ASCII output)
  "_setlessAll" :: "[idt, 'a, bool] ==> bool" (‹(‹indent=3 notation=‹binder ALL›\›ALL _🚫/ _)› [0, 0, 10] 10)
  "_setlessEx" :: "[idt, 'a, bool] ==> bool" (‹(‹indent=3 notation=‹binder EX›\›EX _🚫/ _)› [0, 0, 10] 10)
  "_setleAll" :: "[idt, 'a, bool] ==> bool" (‹(‹indent=3 notation=‹binder ALL›\›ALL _🚫./ _)› [0, 0, 10] 10)
  "_setleEx" :: "[idt, 'a, bool] ==> bool" (‹(‹indent=3 notation=‹binder EX›\›EX _🚫./ _)› [0, 0, 10] 10)
  "_setleEx1" :: "[idt, 'a, bool] ==> bool" (‹(‹indent=3 notation=‹binder EX!›\›EX! _🚫./ _)› [0, 0, 10] 10)
 
 syntax
  "_setlessAll" :: "[idt, 'a, bool] ==> bool" (‹(‹indent=3 notation=‹binder ∀›\›\∀_⊂_./ _)› [0, 0, 10] 10)
  "_setlessEx" :: "[idt, 'a, bool] ==> bool" (‹(‹indent=3 notation=‹binder ∃›\›\∃_⊂_./ _)› [0, 0, 10] 10)
  "_setleAll" :: "[idt, 'a, bool] ==> bool" (‹(‹indent=3 notation=‹binder ∀›\›\∀_⊆_./ _)› [0, 0, 10] 10)
  "_setleEx" :: "[idt, 'a, bool] ==> bool" (‹(‹indent=3 notation=‹binder ∃›\›\∃_⊆_./ _)› [0, 0, 10] 10)
  "_setleEx1" :: "[idt, 'a, bool] ==> bool" (‹(‹indent=3 notation=‹binder ∃!›\›\∃!_⊆_./ _)› [0, 0, 10] 10)
 
 syntax_consts
  "_setlessAll" "_setleAll" ⇌ All and
  "_setlessEx" "_setleEx" ⇌ Ex and
  "_setleEx1" ⇌ Ex1
 
 translations
  "∀A⊂B. P" ⇀ "∀A. A ⊂ B ⟶ P"
  "∃A⊂B. P" ⇀ "∃A. A ⊂ B ∧ P"
  "∀A⊆B. P" ⇀ "∀A. A ⊆ B ⟶ P"
  "∃A⊆B. P" ⇀ "∃A. A ⊆ B ∧ P"
  "∃!A⊆B. P" ⇀ "∃!A. A ⊆ B ∧ P"
 
 print_translation ‹
  let
  val All_binder = Mixfix.binder_name 🍋‹All›;
  val Ex_binder = Mixfix.binder_name 🍋‹Ex›;
  val impl = 🍋‹HOL.implies›;
  val conj = 🍋‹HOL.conj›;
  val sbset = 🍋‹subset›;
  val sbset_eq = 🍋‹subset_eq›;
 
  val trans =
  [((All_binder, impl, sbset), 🍋‹_setlessAll›),
  ((All_binder, impl, sbset_eq), 🍋‹_setleAll›),
  ((Ex_binder, conj, sbset), 🍋‹_setlessEx›),
  ((Ex_binder, conj, sbset_eq), 🍋‹_setleEx›)];
 
  fun mk v (v', T) c n P =
  if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
  then Syntax.const c $ Syntax_Trans.mark_bound_body (v', T) $ n $ P
  else raise Match;
 
  fun tr' q = (q, fn _ =>
  (fn [Const (🍋‹_bound›, _) $ Free (v, 🍋‹set _›),
  Const (c, _) $
  (Const (d, _) $ (Const (🍋‹_bound›, _) $ Free (v', T)) $ n) $ P] =>
  (case AList.lookup (=) trans (q, c, d) of
  NONE => raise Match
  | SOME l => mk v (v', T) l n P)
  | _ => raise Match));
  in
  [tr' All_binder, tr' Ex_binder]
  end
 ›
 
 
 text ‹
  🪙
  Translate between ‹{e | x1…xn. P}› and ‹{u. ∃x1…xn. u = e ∧ P}›;
  ‹{y. ∃x1…xn. y = e ∧ P}› is only translated if ‹[0..n] ⊆ bvs e›.
 ›
 
 syntax
  "_Setcompr" :: "'a ==> idts ==> bool ==> 'a set"
  (‹(‹indent=1 notation=‹mixfix set comprehension›\›{_ |/_./ _})›)
 syntax_consts
  "_Setcompr" ⇌ Collect
 
 parse_translation ‹
  let
  val ex_tr = snd (Syntax_Trans.mk_binder_tr ("EX ", 🍋‹Ex›));
 
  fun nvars (Const (🍋‹_idts›, _) $ _ $ idts) = nvars idts + 1
  | nvars _ = 1;
 
  fun setcompr_tr ctxt [e, idts, b] =
  let
  val eq = Syntax.const 🍋‹HOL.eq› $ Bound (nvars idts) $ e;
  val P = Syntax.const 🍋‹HOL.conj› $ eq $ b;
  val exP = ex_tr ctxt [idts, P];
  in Syntax.const 🍋‹Collect› $ absdummy dummyT exP end;
 
  in [(🍋‹_Setcompr›, setcompr_tr)] end
 ›
 
 typed_print_translation ‹
  [(🍋‹Ball›, Syntax_Trans.preserve_binder_abs2_tr' 🍋‹_Ball›),
  (🍋‹Bex›, Syntax_Trans.preserve_binder_abs2_tr' 🍋‹_Bex›)]
 ›
 
 print_translation ‹
 let
  val ex_tr' = snd (Syntax_Trans.mk_binder_tr' (🍋‹Ex›, "DUMMY"));
 
  fun setcompr_tr' ctxt [Abs (abs as (_, _, P))] =
  let
  fun check (Const (🍋‹Ex›, _) $ Abs (_, _, P), n) = check (P, n + 1)
  | check (Const (🍋‹HOL.conj›, _) $
  (Const (🍋‹HOL.eq›, _) $ Bound m $ e) $ P, n) =
  n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
  subset (=) (0 upto (n - 1), add_loose_bnos (e, 0, []))
  | check _ = false;
 
  fun tr' (_ $ abs) =
  let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' ctxt [abs]
  in Syntax.const 🍋‹_Setcompr› $ e $ idts $ Q end;
  in
  if check (P, 0) then tr' P
  else
  let
  val (x as _ $ Free(xN, _), t) = Syntax_Trans.atomic_abs_tr' ctxt abs;
  val M = Syntax.const 🍋‹_Coll› $ x $ t;
  in
  case t of
  Const (🍋‹HOL.conj›, _) $
  (Const (🍋‹Set.member›, _) $
  (Const (🍋‹_bound›, _) $ Free (yN, _)) $ A) $ P =>
  if xN = yN then Syntax.const 🍋‹_Collect› $ x $ A $ P else M
  | _ => M
  end
  end;
  in [(🍋‹Collect›, setcompr_tr')] end
 ›
 
 simproc_setup defined_Bex ("∃x∈A. P x ∧ Q x") = ‹
  K (Quantifier1.rearrange_Bex (fn ctxt => unfold_tac ctxt @{thms Bex_def}))
 ›
 
 simproc_setup defined_All ("∀x∈A. P x ⟶ Q x") = ‹
  K (Quantifier1.rearrange_Ball (fn ctxt => unfold_tac ctxt @{thms Ball_def}))
 ›
 
 lemma ballI [intro!]: "(∧x. x ∈ A ==> P x) ==> ∀x∈A. P x"
  by (simp add: Ball_def)
 
 lemmas strip = impI allI ballI
 
 lemma bspec [dest?]: "∀x∈A. P x ==> x ∈ A ==> P x"
  by (simp add: Ball_def)
 
 text ‹Gives better instantiation for bound:›
 setup ‹
  map_theory_claset (fn ctxt =>
  ctxt addbefore ("bspec", fn ctxt' => dresolve_tac ctxt' @{thms bspec} THEN' assume_tac ctxt'))
 ›
 
 ML ‹
 structure Simpdata =
 struct
  open Simpdata;
  val mksimps_pairs = [(🍋‹Ball›, @{thms bspec})] @ mksimps_pairs;
 end;
 
 open Simpdata;
 ›
 
 declaration ‹fn _ => Simplifier.map_ss (Simplifier.set_mksimps (mksimps mksimps_pairs))›
 
 lemma ballE [elim]: "∀x∈A. P x ==> (P x ==> Q) ==> (x ∉ A ==> Q) ==> Q"
  unfolding Ball_def by blast
 
 lemma bexI [intro]: "P x ==> x ∈ A ==> ∃x∈A. P x"
  🍋 ‹Normally the best argument order: ‹P x› constrains the choice of ‹x ∈ A›.›
  unfolding Bex_def by blast
 
 lemma rev_bexI [intro?]: "x ∈ A ==> P x ==> ∃x∈A. P x"
  🍋 ‹The best argument order when there is only one ‹x ∈ A›.›
  unfolding Bex_def by blast
 
 lemma bexCI: "(∀x∈A. ¬ P x ==> P a) ==> a ∈ A ==> ∃x∈A. P x"
  unfolding Bex_def by blast
 
 lemma bexE [elim!]: "∃x∈A. P x ==> (∧x. x ∈ A ==> P x ==> Q) ==> Q"
  unfolding Bex_def by blast
 
 lemma ball_triv [simp]: "(∀x∈A. P) ⟷ ((∃x. x ∈ A) ⟶ P)"
  🍋 ‹trivial rewrite rule.›
  by (simp add: Ball_def)
 
 lemma bex_triv [simp]: "(∃x∈A. P) ⟷ ((∃x. x ∈ A) ∧ P)"
  🍋 ‹Dual form for existentials.›
  by (simp add: Bex_def)
 
 lemma bex_triv_one_point1 [simp]: "(∃x∈A. x = a) ⟷ a ∈ A"
  by blast
 
 lemma bex_triv_one_point2 [simp]: "(∃x∈A. a = x) ⟷ a ∈ A"
  by blast
 
 lemma bex_one_point1 [simp]: "(∃x∈A. x = a ∧ P x) ⟷ a ∈ A ∧ P a"
  by blast
 
 lemma bex_one_point2 [simp]: "(∃x∈A. a = x ∧ P x) ⟷ a ∈ A ∧ P a"
  by blast
 
 lemma ball_one_point1 [simp]: "(∀x∈A. x = a ⟶ P x) ⟷ (a ∈ A ⟶ P a)"
  by blast
 
 lemma ball_one_point2 [simp]: "(∀x∈A. a = x ⟶ P x) ⟷ (a ∈ A ⟶ P a)"
  by blast
 
 lemma ball_conj_distrib: "(∀x∈A. P x ∧ Q x) ⟷ (∀x∈A. P x) ∧ (∀x∈A. Q x)"
  by blast
 
 lemma bex_disj_distrib: "(∃x∈A. P x ∨ Q x) ⟷ (∃x∈A. P x) ∨ (∃x∈A. Q x)"
  by blast
 
 
 text ‹Congruence rules›
 
 lemma ball_cong:
  "[ A = B; ∧x. x ∈ B ==> P x ⟷ Q x ] ==>
  (∀x∈A. P x) ⟷ (∀x∈B. Q x)"
 by (simp add: Ball_def)
 
 lemma ball_cong_simp [cong]:
  "[ A = B; ∧x. x ∈ B =simp=> P x ⟷ Q x ] ==>
  (∀x∈A. P x) ⟷ (∀x∈B. Q x)"
 by (simp add: simp_implies_def Ball_def)
 
 lemma bex_cong:
  "[ A = B; ∧x. x ∈ B ==> P x ⟷ Q x ] ==>
  (∃x∈A. P x) ⟷ (∃x∈B. Q x)"
 by (simp add: Bex_def cong: conj_cong)
 
 lemma bex_cong_simp [cong]:
  "[ A = B; ∧x. x ∈ B =simp=> P x ⟷ Q x ] ==>
  (∃x∈A. P x) ⟷ (∃x∈B. Q x)"
 by (simp add: simp_implies_def Bex_def cong: conj_cong)
 
 lemma bex1_def: "(∃!x∈X. P x) ⟷ (∃x∈X. P x) ∧ (∀x∈X. ∀y∈X. P x ⟶ P y ⟶ x = y)"
  by auto
 
 
 subsection ‹Basic operations›
 
 subsubsection ‹Subsets›
 
 lemma subsetI [intro!]: "(∧x. x ∈ A ==> x ∈ B) ==> A ⊆ B"
  by (simp add: less_eq_set_def le_fun_def)
 
 text ‹
  🪙
  Map the type ‹'a set ==> anything› to just ‹'a›; for overloading constants
  whose first argument has type ‹'a set›.
 ›
 
 lemma subsetD [elim, intro?]: "A ⊆ B ==> c ∈ A ==> c ∈ B"
  by (simp add: less_eq_set_def le_fun_def)
  🍋 ‹Rule in Modus Ponens style.›
 
 lemma rev_subsetD [intro?,no_atp]: "c ∈ A ==> A ⊆ B ==> c ∈ B"
  🍋 ‹The same, with reversed premises for use with @{method erule} -- cf. @{thm rev_mp}.›
  by (rule subsetD)
 
 lemma subsetCE [elim,no_atp]: "A ⊆ B ==> (c ∉ A ==> P) ==> (c ∈ B ==> P) ==> P"
  🍋 ‹Classical elimination rule.›
  by (auto simp add: less_eq_set_def le_fun_def)
 
 lemma subset_eq: "A ⊆ B ⟷ (∀x∈A. x ∈ B)"
  by blast
 
 lemma contra_subsetD [no_atp]: "A ⊆ B ==> c ∉ B ==> c ∉ A"
  by blast
 
 lemma subset_refl: "A ⊆ A"
  by (fact order_refl) (* already [iff] *)

lemma subset_trans: "A ⊆ B ==> B ⊆ C ==> A ⊆ C"
  by (fact order_trans)

lemma subset_not_subset_eq [code]: "A ⊂ B ⟷ A ⊆ B ∧ ¬ B ⊆ A"
  by (fact less_le_not_le)

lemma eq_mem_trans: "a = b ==> b ∈ A ==> a ∈ A"
  by simp

lemmas basic_trans_rules [trans] =
  order_trans_rules rev_subsetD subsetD eq_mem_trans


subsubsection ‹Equality›

lemma subset_antisym [intro!]: "A ⊆ B ==> B ⊆ A ==> A = B"
  🍋 ‹Anti-symmetry of the subset relation.›
  by (iprover intro: set_eqI subsetD)

text ‹🪙 Equality rules from ZF set theory -- are they appropriate here?›

lemma equalityD1: "A = B ==> A ⊆ B"
  by simp

lemma equalityD2: "A = B ==> B ⊆ A"
  by simp

text ‹
  🪙
  Be careful when adding this to the claset as ‹subset_empty› is in the
  simpset: 🍋‹A = {}› goes to 🍋‹{} ⊆ A› and 🍋‹A ⊆ {}›
  and then back to 🍋‹A = {}›!
 ›

lemma equalityE: "A = B ==> (A ⊆ B ==> B ⊆ A ==> P) ==> P"
  by simp

lemma equalityCE [elim]: "A = B ==> (c ∈ A ==> c ∈ B ==> P) ==> (c ∉ A ==> c ∉ B ==> P) ==> P"
  by blast

lemma eqset_imp_iff: "A = B ==> x ∈ A ⟷ x ∈ B"
  by simp

lemma eqelem_imp_iff: "x = y ==> x ∈ A ⟷ y ∈ A"
  by simp


subsubsection ‹The empty set›

lemma empty_def: "{} = {x. False}"
  by (simp add: bot_set_def bot_fun_def)

lemma empty_iff [simp]: "c ∈ {} ⟷ False"
  by (simp add: empty_def)

lemma emptyE [elim!]: "a ∈ {} ==> P"
  by simp

lemma empty_subsetI [iff]: "{} ⊆ A"
  🍋 ‹One effect is to delete the ASSUMPTION 🍋‹{} ⊆ A›\›
  by blast

lemma equals0I: "(∧y. y ∈ A ==> False) ==> A = {}"
  by blast

lemma equals0D: "A = {} ==> a ∉ A"
  🍋 ‹Use for reasoning about disjointness: ‹A ∩ B = {}›\›
  by blast

lemma ball_empty [simp]: "Ball {} P ⟷ True"
  by (simp add: Ball_def)

lemma bex_empty [simp]: "Bex {} P ⟷ False"
  by (simp add: Bex_def)


subsubsection ‹The universal set -- UNIV›

abbreviation UNIV :: "'a set"
  where "UNIV ≡ top"

lemma UNIV_def: "UNIV = {x. True}"
  by (simp add: top_set_def top_fun_def)

lemma UNIV_I [simp]: "x ∈ UNIV"
  by (simp add: UNIV_def)

declare UNIV_I [intro]  🍋 ‹unsafe makes it less likely to cause problems›

lemma UNIV_witness [intro?]: "∃x. x ∈ UNIV"
  by simp

lemma subset_UNIV: "A ⊆ UNIV"
  by (fact top_greatest) (* already simp *)

text ‹
  🪙
  Eta-contracting these two rules (to remove ‹P›)
  to be ignored because of their interaction with congruence rules.
›

lemma ball_UNIV [simp]: "Ball UNIV P ⟷ All P"
  by (simp add: Ball_def)

lemma bex_UNIV [simp]: "Bex UNIV P ⟷ Ex P"
  by (simp add: Bex_def)

lemma UNIV_eq_I: "(∧x. x ∈ A) ==> UNIV = A"
  by auto

lemma UNIV_not_empty [iff]: "UNIV ≠ {}"
  by (blast elim: equalityE)

lemma empty_not_UNIV[simp]: "{} ≠ UNIV"
  by blast


subsubsection ‹The Powerset operator -- Pow›

definition Pow :: "'a set ==> 'a set set"
  where Pow_def: "Pow A = {B. B ⊆ A}"

lemma Pow_iff [iff]: "A ∈ Pow B ⟷ A ⊆ B"
  by (simp add: Pow_def)

lemma PowI: "A ⊆ B ==> A ∈ Pow B"
  by (simp add: Pow_def)

lemma PowD: "A ∈ Pow B ==> A ⊆ B"
  by (simp add: Pow_def)

lemma Pow_bottom: "{} ∈ Pow B"
  by simp

lemma Pow_top: "A ∈ Pow A"
  by simp

lemma Pow_not_empty: "Pow A ≠ {}"
  using Pow_top by blast


subsubsection ‹Set complement›

lemma Compl_iff [simp]: "c ∈ - A ⟷ c ∉ A"
  by (simp add: fun_Compl_def uminus_set_def)

lemma ComplI [intro!]: "(c ∈ A ==> False) ==> c ∈ - A"
  by (simp add: fun_Compl_def uminus_set_def) blast

text ‹
  🪙
  This form, with negated conclusion, works well with the Classical prover.
  Negated assumptions behave like formulae on the right side of the
  notional turnstile \dots
 ›

lemma ComplD [dest!]: "c ∈ - A ==> c ∉ A"
  by simp

lemmas ComplE = ComplD [elim_format]

lemma Compl_eq: "- A = {x. ¬ x ∈ A}"
  by blast


subsubsection ‹Binary intersection›

abbreviation inter :: "'a set ==> 'a set ==> 'a set"  (infixl ‹∩› 70)
  where "(∩) ≡ inf"

notation (ASCII)
  inter  (infixl ‹Int› 70)

lemma Int_def: "A ∩ B = {x. x ∈ A ∧ x ∈ B}"
  by (simp add: inf_set_def inf_fun_def)

lemma Int_iff [simp]: "c ∈ A ∩ B ⟷ c ∈ A ∧ c ∈ B"
  unfolding Int_def by blast

lemma IntI [intro!]: "c ∈ A ==> c ∈ B ==> c ∈ A ∩ B"
  by simp

lemma IntD1: "c ∈ A ∩ B ==> c ∈ A"
  by simp

lemma IntD2: "c ∈ A ∩ B ==> c ∈ B"
  by simp

lemma IntE [elim!]: "c ∈ A ∩ B ==> (c ∈ A ==> c ∈ B ==> P) ==> P"
  by simp


subsubsection ‹Binary union›

abbreviation union :: "'a set ==> 'a set ==> 'a set"  (infixl ‹∪› 65)
  where "union ≡ sup"

notation (ASCII)
  union  (infixl ‹Un› 65)

lemma Un_def: "A ∪ B = {x. x ∈ A ∨ x ∈ B}"
  by (simp add: sup_set_def sup_fun_def)

lemma Un_iff [simp]: "c ∈ A ∪ B ⟷ c ∈ A ∨ c ∈ B"
  unfolding Un_def by blast

lemma UnI1 [elim?]: "c ∈ A ==> c ∈ A ∪ B"
  by simp

lemma UnI2 [elim?]: "c ∈ B ==> c ∈ A ∪ B"
  by simp

text ‹🪙 Classical introduction rule: no commitment to ‹A› vs. ‹B›.›
lemma UnCI [intro!]: "(c ∉ B ==> c ∈ A) ==> c ∈ A ∪ B"
  by auto

lemma UnE [elim!]: "c ∈ A ∪ B ==> (c ∈ A ==> P) ==> (c ∈ B ==> P) ==> P"
  unfolding Un_def by blast

lemma insert_def: "insert a B = {x. x = a} ∪ B"
  by (simp add: insert_compr Un_def)


subsubsection ‹Set difference›

lemma Diff_iff [simp]: "c ∈ A - B ⟷ c ∈ A ∧ c ∉ B"
  by (simp add: minus_set_def fun_diff_def)

lemma DiffI [intro!]: "c ∈ A ==> c ∉ B ==> c ∈ A - B"
  by simp

lemma DiffD1: "c ∈ A - B ==> c ∈ A"
  by simp

lemma DiffD2: "c ∈ A - B ==> c ∈ B ==> P"
  by simp

lemma DiffE [elim!]: "c ∈ A - B ==> (c ∈ A ==> c ∉ B ==> P) ==> P"
  by simp

lemma set_diff_eq: "A - B = {x. x ∈ A ∧ x ∉ B}"
  by blast

lemma Compl_eq_Diff_UNIV: "- A = (UNIV - A)"
  by blast

abbreviation sym_diff :: "'a set ==> 'a set ==> 'a set" where
  "sym_diff A B ≡ ((A - B) ∪ (B-A))"


subsubsection ‹Augmenting a set -- 🍋‹insert›\
lemma insert_iff [simp]: "a ∈ insert b A ⟷ a = b ∨ a ∈ A"
  unfolding insert_def by blast

lemma insertI1: "a ∈ insert a B"
  by simp

lemma insertI2: "a ∈ B ==> a ∈ insert b B"
  by simp

lemma insertE [elim!]: "a ∈ insert b A ==> (a = b ==> P) ==> (a ∈ A ==> P) ==> P"
  unfolding insert_def by blast

lemma insertCI [intro!]: "(a ∉ B ==> a = b) ==> a ∈ insert b B"
  🍋 ‹Classical introduction rule.›
  by auto

lemma subset_insert_iff: "A ⊆ insert x B ⟷ (if x ∈ A then A - {x} ⊆ B else A ⊆ B)"
  by auto

lemma set_insert:
  assumes "x ∈ A"
  obtains B where "A = insert x B" and "x ∉ B"
proof
  show "A = insert x (A - {x})" using assms by blast
  show "x ∉ A - {x}" by blast
qed

lemma insert_ident: "x ∉ A ==> x ∉ B ==> insert x A = insert x B ⟷ A = B"
  by auto

lemma insert_eq_iff:
  assumes "a ∉ A" "b ∉ B"
  shows "insert a A = insert b B ⟷
    (if a = b then A = B else ∃C. A = insert b C ∧ b ∉ C ∧ B = insert a C ∧ a ∉ C)"
    (is "?L ⟷ ?R")
proof
  show ?R if ?L
  proof (cases "a = b")
    case True
    with assms ‹?L› show ?R
      by (simp add: insert_ident)
  next
    case False
    let ?C = "A - {b}"
    have "A = insert b ?C ∧ b ∉ ?C ∧ B = insert a ?C ∧ a ∉ ?C"
      using assms ‹?L› ‹a ≠ b› by auto
    then show ?R using ‹a ≠ b› by auto
  qed
  show ?L if ?R
    using that by (auto split: if_splits)
qed

lemma insert_UNIV[simp]: "insert x UNIV = UNIV"
  by auto


subsubsection ‹Singletons, using insert›

lemma singletonI [intro!]: "a ∈ {a}"
  🍋 ‹Redundant? But unlike ‹insertCI›, it proves the subgoal immediately!›
  by (rule insertI1)

lemma singletonD [dest!]: "b ∈ {a} ==> b = a"
  by blast

lemmas singletonE = singletonD [elim_format]

lemma singleton_iff: "b ∈ {a} ⟷ b = a"
  by blast

lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
  by blast

lemma singleton_insert_inj_eq [iff]: "{b} = insert a A ⟷ a = b ∧ A ⊆ {b}"
  by blast

lemma singleton_insert_inj_eq' [iff]: "insert a A = {b} ⟷ a = b ∧ A ⊆ {b}"
  by blast

lemma subset_singletonD: "A ⊆ {x} ==> A = {} ∨ A = {x}"
  by fast

lemma subset_singleton_iff: "X ⊆ {a} ⟷ X = {} ∨ X = {a}"
  by blast

lemma subset_singleton_iff_Uniq: "(∃a. A ⊆ {a}) ⟷ (∃🪙≤🪙1x. x ∈ A)"
  unfolding Uniq_def by blast

lemma singleton_conv [simp]: "{x. x = a} = {a}"
  by blast

lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
  by blast

lemma Diff_single_insert: "A - {x} ⊆ B ==> A ⊆ insert x B"
  by blast

lemma subset_Diff_insert: "A ⊆ B - insert x C ⟷ A ⊆ B - C ∧ x ∉ A"
  by blast

lemma doubleton_eq_iff: "{a, b} = {c, d} ⟷ a = c ∧ b = d ∨ a = d ∧ b = c"
  by (blast elim: equalityE)

lemma Un_singleton_iff: "A ∪ B = {x} ⟷ A = {} ∧ B = {x} ∨ A = {x} ∧ B = {} ∨ A = {x} ∧ B = {x}"
  by auto

lemma singleton_Un_iff: "{x} = A ∪ B ⟷ A = {} ∧ B = {x} ∨ A = {x} ∧ B = {} ∨ A = {x} ∧ B = {x}"
  by auto


subsubsection ‹Image of a set under a function›

text ‹Frequently ‹b› does not have the syntactic form of ‹f x›.›

definition image :: "('a ==> 'b) ==> 'a set ==> 'b set"    (infixr ‹`› 90)
  where "f ` A = {y. ∃x∈A. y = f x}"

lemma image_eqI [simp, intro]: "b = f x ==> x ∈ A ==> b ∈ f ` A"
  unfolding image_def by blast

lemma imageI: "x ∈ A ==> f x ∈ f ` A"
  by (rule image_eqI) (rule refl)

lemma rev_image_eqI: "x ∈ A ==> b = f x ==> b ∈ f ` A"
  🍋 ‹This version's more effective when we already have the required ‹x›.›
  by (rule image_eqI)

lemma imageE [elim!]:
  assumes "b ∈ (λx. f x) ` A"  🍋 ‹The eta-expansion gives variable-name preservation.›
  obtains x where "b = f x" and "x ∈ A"
  using assms unfolding image_def by blast

lemma Compr_image_eq: "{x ∈ f ` A. P x} = f ` {x ∈ A. P (f x)}"
  by auto

lemma image_Un: "f ` (A ∪ B) = f ` A ∪ f ` B"
  by blast

lemma image_iff: "z ∈ f ` A ⟷ (∃x∈A. z = f x)"
  by blast

lemma image_subsetI: "(∧x. x ∈ A ==> f x ∈ B) ==> f ` A ⊆ B"
  🍋 ‹Replaces the three steps ‹subsetI›, ‹imageE›,
  ‹hypsubst›, but breaks too many existing proofs.›
  by blast

lemma image_subset_iff: "f ` A ⊆ B ⟷ (∀x∈A. f x ∈ B)"
  🍋 ‹This rewrite rule would confuse users if made default.›
  by blast

lemma subset_imageE:
  assumes "B ⊆ f ` A"
  obtains C where "C ⊆ A" and "B = f ` C"
proof -
  from assms have "B = f ` {a ∈ A. f a ∈ B}" by fast
  moreover have "{a ∈ A. f a ∈ B} ⊆ A" by blast
  ultimately show thesis by (blast intro: that)
qed

lemma subset_image_iff: "B ⊆ f ` A ⟷ (∃AA⊆A. B = f ` AA)"
  by (blast elim: subset_imageE)

lemma image_ident [simp]: "(λx. x) ` Y = Y"
  by blast

lemma image_empty [simp]: "f ` {} = {}"
  by blast

lemma image_insert [simp]: "f ` insert a B = insert (f a) (f ` B)"
  by blast

lemma image_constant: "x ∈ A ==> (λx. c) ` A = {c}"
  by auto

lemma image_constant_conv: "(λx. c) ` A = (if A = {} then {} else {c})"
  by auto

lemma image_image: "f ` (g ` A) = (λx. f (g x)) ` A"
  by blast

lemma insert_image [simp]: "x ∈ A ==> insert (f x) (f ` A) = f ` A"
  by blast

lemma image_is_empty [iff]: "f ` A = {} ⟷ A = {}"
  by blast

lemma empty_is_image [iff]: "{} = f ` A ⟷ A = {}"
  by blast

lemma image_Collect: "f ` {x. P x} = {f x | x. P x}"
  🍋 ‹NOT suitable as a default simp rule: the RHS isn't simpler than the LHS,
  with its implicit quantifier and conjunction. Also image enjoys better
  equational properties than does the RHS.›
  by blast

lemma if_image_distrib [simp]:
  "(λx. if P x then f x else g x) ` S = f ` (S ∩ {x. P x}) ∪ g ` (S ∩ {x. ¬ P x})"
  by auto

lemma image_cong:
  "f ` M = g ` N" if "M = N" "∧x. x ∈ N ==> f x = g x"
  using that by (simp add: image_def)

lemma image_cong_simp [cong]:
  "f ` M = g ` N" if "M = N" "∧x. x ∈ N =simp=> f x = g x"
  using that image_cong [of M N f g] by (simp add: simp_implies_def)

lemma image_Int_subset: "f ` (A ∩ B) ⊆ f ` A ∩ f ` B"
  by blast

lemma image_diff_subset: "f ` A - f ` B ⊆ f ` (A - B)"
  by blast

lemma Setcompr_eq_image: "{f x |x. x ∈ A} = f ` A"
  by blast

lemma setcompr_eq_image: "{f x |x. P x} = f ` {x. P x}"
  by auto

lemma ball_imageD: "∀x∈f ` A. P x ==> ∀x∈A. P (f x)"
  by simp

lemma bex_imageD: "∃x∈f ` A. P x ==> ∃x∈A. P (f x)"
  by auto

lemma image_add_0 [simp]: "(+) (0::'a::comm_monoid_add) ` S = S"
  by auto

theorem Cantors_theorem: "∄f. f ` A = Pow A"
proof
  assume "∃f. f ` A = Pow A"
  then obtain f where f: "f ` A = Pow A" ..
  let ?X = "{a ∈ A. a ∉ f a}"
  have "?X ∈ Pow A" by blast
  then have "?X ∈ f ` A" by (simp only: f)
  then obtain x where "x ∈ A" and "f x = ?X" by blast
  then show False by blast
qed

text ‹🪙 Range of a function -- just an abbreviation for image!›

abbreviation range :: "('a ==> 'b) ==> 'b set"  🍋 ‹of function›
  where "range f ≡ f ` UNIV"

lemma range_eqI: "b = f x ==> b ∈ range f"
  by simp

lemma rangeI: "f x ∈ range f"
  by simp

lemma rangeE [elim?]: "b ∈ range (λx. f x) ==> (∧x. b = f x ==> P) ==> P"
  by (rule imageE)

lemma range_subsetD: "range f ⊆ B ==> f i ∈ B"
  by blast

lemma full_SetCompr_eq: "{u. ∃x. u = f x} = range f"
  by auto

lemma range_composition: "range (λx. f (g x)) = f ` range g"
  by auto

lemma range_constant [simp]: "range (λ_. x) = {x}"
  by (simp add: image_constant)

lemma range_eq_singletonD: "range f = {a} ==> f x = a"
  by auto


subsubsection ‹Some rules with ‹if›\›

text ‹Elimination of ‹{x. … ∧ x = t ∧ …}›.›

lemma Collect_conv_if: "{x. x = a ∧ P x} = (if P a then {a} else {})"
  by auto

lemma Collect_conv_if2: "{x. a = x ∧ P x} = (if P a then {a} else {})"
  by auto

text ‹
  Rewrite rules for boolean case-splitting: faster than ‹if_split [split]›.
 ›

lemma if_split_eq1: "(if Q then x else y) = b ⟷ (Q ⟶ x = b) ∧ (¬ Q ⟶ y = b)"
  by (rule if_split)

lemma if_split_eq2: "a = (if Q then x else y) ⟷ (Q ⟶ a = x) ∧ (¬ Q ⟶ a = y)"
  by (rule if_split)

text ‹
  Split ifs on either side of the membership relation.
  Not for ‹[simp]› -- can cause goals to blow up!
 ›

lemma if_split_mem1: "(if Q then x else y) ∈ b ⟷ (Q ⟶ x ∈ b) ∧ (¬ Q ⟶ y ∈ b)"
  by (rule if_split)

lemma if_split_mem2: "(a ∈ (if Q then x else y)) ⟷ (Q ⟶ a ∈ x) ∧ (¬ Q ⟶ a ∈ y)"
  by (rule if_split [where P = "λS. a ∈ S"])

lemmas split_ifs = if_bool_eq_conj if_split_eq1 if_split_eq2 if_split_mem1 if_split_mem2

(*Would like to add these, but the existing code only searches for the
  outer-level constant, which in this case is just Set.member; we instead need
  to use term-nets to associate patterns with rules. Also, if a rule fails to
  apply, then the formula should be kept.
  [("uminus", Compl_iff RS iffD1), ("minus", [Diff_iff RS iffD1]),
  ("Int", [IntD1,IntD2]),
  ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
 *)


subsection ‹Further operations and lemmas›

subsubsection ‹The ``proper subset'' relation›

lemma psubsetI [intro!]: "A ⊆ B ==> A ≠ B ==> A ⊂ B"
  unfolding less_le by blast

lemma psubsetE [elim!]: "A ⊂ B ==> (A ⊆ B ==> ¬ B ⊆ A ==> R) ==> R"
  unfolding less_le by blast

lemma psubset_insert_iff:
  "A ⊂ insert x B ⟷ (if x ∈ B then A ⊂ B else if x ∈ A then A - {x} ⊂ B else A ⊆ B)"
  by (auto simp add: less_le subset_insert_iff)

lemma psubset_eq: "A ⊂ B ⟷ A ⊆ B ∧ A ≠ B"
  by (simp only: less_le)

lemma psubset_imp_subset: "A ⊂ B ==> A ⊆ B"
  by (simp add: psubset_eq)

lemma psubset_trans: "A ⊂ B ==> B ⊂ C ==> A ⊂ C"
  unfolding less_le by (auto dest: subset_antisym)

lemma psubsetD: "A ⊂ B ==> c ∈ A ==> c ∈ B"
  unfolding less_le by (auto dest: subsetD)

lemma psubset_subset_trans: "A ⊂ B ==> B ⊆ C ==> A ⊂ C"
  by (auto simp add: psubset_eq)

lemma subset_psubset_trans: "A ⊆ B ==> B ⊂ C ==> A ⊂ C"
  by (auto simp add: psubset_eq)

lemma psubset_imp_ex_mem: "A ⊂ B ==> ∃b. b ∈ B - A"
  unfolding less_le by blast

lemma atomize_ball: "(∧x. x ∈ A ==> P x) ≡ Trueprop (∀x∈A. P x)"
  by (simp only: Ball_def atomize_all atomize_imp)

lemmas [symmetric, rulify] = atomize_ball
  and [symmetric, defn] = atomize_ball

lemma image_Pow_mono: "f ` A ⊆ B ==> image f ` Pow A ⊆ Pow B"
  by blast

lemma image_Pow_surj: "f ` A = B ==> image f ` Pow A = Pow B"
  by (blast elim: subset_imageE)


subsubsection ‹Derived rules involving subsets.›

text ‹‹insert›.›

lemma subset_insertI: "B ⊆ insert a B"
  by (rule subsetI) (erule insertI2)

lemma subset_insertI2: "A ⊆ B ==> A ⊆ insert b B"
  by blast

lemma subset_insert: "x ∉ A ==> A ⊆ insert x B ⟷ A ⊆ B"
  by blast


text ‹🪙 Finite Union -- the least upper bound of two sets.›

lemma Un_upper1: "A ⊆ A ∪ B"
  by (fact sup_ge1)

lemma Un_upper2: "B ⊆ A ∪ B"
  by (fact sup_ge2)

lemma Un_least: "A ⊆ C ==> B ⊆ C ==> A ∪ B ⊆ C"
  by (fact sup_least)


text ‹🪙 Finite Intersection -- the greatest lower bound of two sets.›

lemma Int_lower1: "A ∩ B ⊆ A"
  by (fact inf_le1)

lemma Int_lower2: "A ∩ B ⊆ B"
  by (fact inf_le2)

lemma Int_greatest: "C ⊆ A ==> C ⊆ B ==> C ⊆ A ∩ B"
  by (fact inf_greatest)


text ‹🪙 Set difference.›

lemma Diff_subset[simp]: "A - B ⊆ A"
  by blast

lemma Diff_subset_conv: "A - B ⊆ C ⟷ A ⊆ B ∪ C"
  by blast


subsubsection ‹Equalities involving union, intersection, inclusion, etc.›

text ‹‹{}›.›

lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
  🍋 ‹supersedes ‹Collect_False_empty›\›
  by auto

lemma subset_empty [simp]: "A ⊆ {} ⟷ A = {}"
  by (fact bot_unique)

lemma not_psubset_empty [iff]: "¬ (A < {})"
  by (fact not_less_bot) (*already simp *)

lemma Collect_subset [simp]: "{x∈A. P x} ⊆ A" by auto

lemma Collect_empty_eq [simp]: "Collect P = {} ⟷ (∀x. ¬ P x)"
  by blast

lemma empty_Collect_eq [simp]: "{} = Collect P ⟷ (∀x. ¬ P x)"
  by blast

lemma Collect_neg_eq: "{x. ¬ P x} = - {x. P x}"
  by blast

lemma Collect_disj_eq: "{x. P x ∨ Q x} = {x. P x} ∪ {x. Q x}"
  by blast

lemma Collect_imp_eq: "{x. P x ⟶ Q x} = - {x. P x} ∪ {x. Q x}"
  by blast

lemma Collect_conj_eq: "{x. P x ∧ Q x} = {x. P x} ∩ {x. Q x}"
  by blast

lemma Collect_conj_eq2: "{x ∈ A. P x ∧ Q x} = {x ∈ A. P x} ∩ {x ∈ A. Q x}"
  by blast

lemma Collect_mono_iff: "Collect P ⊆ Collect Q ⟷ (∀x. P x ⟶ Q x)"
  by blast


text ‹🪙 ‹insert›.\<close>

lemma insert_is_Un: "insert a A = {a} ∪ A"
  🍋 ‹NOT SUITABLE FOR REWRITING since ‹{a} ≡ insert a {}›\›
  by blast

lemma insert_not_empty [simp]: "insert a A ≠ {}"
  and empty_not_insert [simp]: "{} ≠ insert a A"
  by blast+

lemma insert_absorb: "a ∈ A ==> insert a A = A"
  🍋 ‹‹[simp]› causes recursive calls when there are nested inserts›
  🍋 ‹with 🪙‹quadratic› running time›
  by blast

lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
  by blast

lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
  by blast

lemma insert_subset [simp]: "insert x A ⊆ B ⟷ x ∈ B ∧ A ⊆ B"
  by blast

lemma mk_disjoint_insert: "a ∈ A ==> ∃B. A = insert a B ∧ a ∉ B"
  🍋 ‹use new ‹B› rather than ‹A - {a}› to avoid infinite unfolding›
  by (rule exI [where x = "A - {a}"]) blast

lemma insert_Collect: "insert a (Collect P) = {u. u ≠ a ⟶ P u}"
  by auto

lemma insert_inter_insert [simp]: "insert a A ∩ insert a B = insert a (A ∩ B)"
  by blast

lemma insert_disjoint [simp]:
  "insert a A ∩ B = {} ⟷ a ∉ B ∧ A ∩ B = {}"
  "{} = insert a A ∩ B ⟷ a ∉ B ∧ {} = A ∩ B"
  by auto

lemma disjoint_insert [simp]:
  "B ∩ insert a A = {} ⟷ a ∉ B ∧ B ∩ A = {}"
  "{} = A ∩ insert b B ⟷ b ∉ A ∧ {} = A ∩ B"
  by auto


text ‹🪙 ‹Int›\›

lemma Int_absorb: "A ∩ A = A"
  by (fact inf_idem) (* already simp *)

lemma Int_left_absorb: "A ∩ (A ∩ B) = A ∩ B"
  by (fact inf_left_idem)

lemma Int_commute: "A ∩ B = B ∩ A"
  by (fact inf_commute)

lemma Int_left_commute: "A ∩ (B ∩ C) = B ∩ (A ∩ C)"
  by (fact inf_left_commute)

lemma Int_assoc: "(A ∩ B) ∩ C = A ∩ (B ∩ C)"
  by (fact inf_assoc)

lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
  🍋 ‹Intersection is an AC-operator›

lemma Int_absorb1: "B ⊆ A ==> A ∩ B = B"
  by (fact inf_absorb2)

lemma Int_absorb2: "A ⊆ B ==> A ∩ B = A"
  by (fact inf_absorb1)

lemma Int_empty_left: "{} ∩ B = {}"
  by (fact inf_bot_left) (* already simp *)

lemma Int_empty_right: "A ∩ {} = {}"
  by (fact inf_bot_right) (* already simp *)

lemma disjoint_eq_subset_Compl: "A ∩ B = {} ⟷ A ⊆ - B"
  by blast

lemma disjoint_iff: "A ∩ B = {} ⟷ (∀x. x∈A ⟶ x ∉ B)"
  by blast

lemma disjoint_iff_not_equal: "A ∩ B = {} ⟷ (∀x∈A. ∀y∈B. x ≠ y)"
  by blast

lemma Int_UNIV_left: "UNIV ∩ B = B"
  by (fact inf_top_left) (* already simp *)

lemma Int_UNIV_right: "A ∩ UNIV = A"
  by (fact inf_top_right) (* already simp *)

lemma Int_Un_distrib: "A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)"
  by (fact inf_sup_distrib1)

lemma Int_Un_distrib2: "(B ∪ C) ∩ A = (B ∩ A) ∪ (C ∩ A)"
  by (fact inf_sup_distrib2)

lemma Int_UNIV: "A ∩ B = UNIV ⟷ A = UNIV ∧ B = UNIV"
  by (fact inf_eq_top_iff) (* already simp *)

lemma Int_subset_iff: "C ⊆ A ∩ B ⟷ C ⊆ A ∧ C ⊆ B"
  by (fact le_inf_iff) (* already simp *)

lemma Int_Collect: "x ∈ A ∩ {x. P x} ⟷ x ∈ A ∧ P x"
  by blast


text ‹🪙 ‹Un›.\<close>

lemma Un_absorb: "A ∪ A = A"
  by (fact sup_idem) (* already simp *)

lemma Un_left_absorb: "A ∪ (A ∪ B) = A ∪ B"
  by (fact sup_left_idem)

lemma Un_commute: "A ∪ B = B ∪ A"
  by (fact sup_commute)

lemma Un_left_commute: "A ∪ (B ∪ C) = B ∪ (A ∪ C)"
  by (fact sup_left_commute)

lemma Un_assoc: "(A ∪ B) ∪ C = A ∪ (B ∪ C)"
  by (fact sup_assoc)

lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
  🍋 ‹Union is an AC-operator›

lemma Un_absorb1: "A ⊆ B ==> A ∪ B = B"
  by (fact sup_absorb2)

lemma Un_absorb2: "B ⊆ A ==> A ∪ B = A"
  by (fact sup_absorb1)

lemma Un_empty_left: "{} ∪ B = B"
  by (fact sup_bot_left) (* already simp *)

lemma Un_empty_right: "A ∪ {} = A"
  by (fact sup_bot_right) (* already simp *)

lemma Un_UNIV_left: "UNIV ∪ B = UNIV"
  by (fact sup_top_left) (* already simp *)

lemma Un_UNIV_right: "A ∪ UNIV = UNIV"
  by (fact sup_top_right) (* already simp *)

lemma Un_insert_left [simp]: "(insert a B) ∪ C = insert a (B ∪ C)"
  by blast

lemma Un_insert_right [simp]: "A ∪ (insert a B) = insert a (A ∪ B)"
  by blast

lemma Int_insert_left: "(insert a B) ∩ C = (if a ∈ C then insert a (B ∩ C) else B ∩ C)"
  by auto

lemma Int_insert_left_if0 [simp]: "a ∉ C ==> (insert a B) ∩ C = B ∩ C"
  by auto

lemma Int_insert_left_if1 [simp]: "a ∈ C ==> (insert a B) ∩ C = insert a (B ∩ C)"
  by auto

lemma Int_insert_right: "A ∩ (insert a B) = (if a ∈ A then insert a (A ∩ B) else A ∩ B)"
  by auto

lemma Int_insert_right_if0 [simp]: "a ∉ A ==> A ∩ (insert a B) = A ∩ B"
  by auto

lemma Int_insert_right_if1 [simp]: "a ∈ A ==> A ∩ (insert a B) = insert a (A ∩ B)"
  by auto

lemma Un_Int_distrib: "A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)"
  by (fact sup_inf_distrib1)

lemma Un_Int_distrib2: "(B ∩ C) ∪ A = (B ∪ A) ∩ (C ∪ A)"
  by (fact sup_inf_distrib2)

lemma Un_Int_crazy: "(A ∩ B) ∪ (B ∩ C) ∪ (C ∩ A) = (A ∪ B) ∩ (B ∪ C) ∩ (C ∪ A)"
  by blast

lemma subset_Un_eq: "A ⊆ B ⟷ A ∪ B = B"
  by (fact le_iff_sup)

lemma Un_empty [iff]: "A ∪ B = {} ⟷ A = {} ∧ B = {}"
  by (fact sup_eq_bot_iff) (* FIXME: already simp *)

lemma Un_subset_iff: "A ∪ B ⊆ C ⟷ A ⊆ C ∧ B ⊆ C"
  by (fact le_sup_iff) (* already simp *)

lemma Un_Diff_Int: "(A - B) ∪ (A ∩ B) = A"
  by blast

lemma Diff_Int2: "A ∩ C - B ∩ C = A ∩ C - B"
  by blast

lemma subset_UnE:
  assumes "C ⊆ A ∪ B"
  obtains A' B' where "A' ⊆ A" "B' ⊆ B" "C = A' ∪ B'"
proof
  show "C ∩ A ⊆ A" "C ∩ B ⊆ B" "C = (C ∩ A) ∪ (C ∩ B)"
    using assms by blast+
qed

lemma Un_Int_eq [simp]: "(S ∪ T) ∩ S = S" "(S ∪ T) ∩ T = T" "S ∩ (S ∪ T) = S" "T ∩ (S ∪ T) = T"
  by auto

lemma Int_Un_eq [simp]: "(S ∩ T) ∪ S = S" "(S ∩ T) ∪ T = T" "S ∪ (S ∩ T) = S" "T ∪ (S ∩ T) = T"
  by auto

text ‹🪙 Set complement›

lemma Compl_disjoint [simp]: "A ∩ - A = {}"
  by (fact inf_compl_bot)

lemma Compl_disjoint2 [simp]: "- A ∩ A = {}"
  by (fact compl_inf_bot)

lemma Compl_partition: "A ∪ - A = UNIV"
  by (fact sup_compl_top)

lemma Compl_partition2: "- A ∪ A = UNIV"
  by (fact compl_sup_top)

lemma double_complement: "- (-A) = A" for A :: "'a set"
  by (fact double_compl) (* already simp *)

lemma Compl_Un: "- (A ∪ B) = (- A) ∩ (- B)"
  by (fact compl_sup) (* already simp *)

lemma Compl_Int: "- (A ∩ B) = (- A) ∪ (- B)"
  by (fact compl_inf) (* already simp *)

lemma subset_Compl_self_eq: "A ⊆ - A ⟷ A = {}"
  by blast

lemma Un_Int_assoc_eq: "(A ∩ B) ∪ C = A ∩ (B ∪ C) ⟷ C ⊆ A"
  🍋 ‹Halmos, Naive Set Theory, page 16.›
  by blast

lemma Compl_UNIV_eq: "- UNIV = {}"
  by (fact compl_top_eq) (* already simp *)

lemma Compl_empty_eq: "- {} = UNIV"
  by (fact compl_bot_eq) (* already simp *)

lemma Compl_subset_Compl_iff [iff]: "- A ⊆ - B ⟷ B ⊆ A"
  by (fact compl_le_compl_iff) (* FIXME: already simp *)

lemma Compl_eq_Compl_iff [iff]: "- A = - B ⟷ A = B"
  for A B :: "'a set"
  by (fact compl_eq_compl_iff) (* FIXME: already simp *)

lemma Compl_insert: "- insert x A = (- A) - {x}"
  by blast

text ‹🪙 Bounded quantifiers.
 
  The following are not added to the default simpset because
  (a) they duplicate the body and (b) there are no similar rules for ‹Int›.
›

lemma ball_Un: "(∀x ∈ A ∪ B. P x) ⟷ (∀x∈A. P x) ∧ (∀x∈B. P x)"
  by blast

lemma bex_Un: "(∃x ∈ A ∪ B. P x) ⟷ (∃x∈A. P x) ∨ (∃x∈B. P x)"
  by blast


text ‹🪙 Set difference.›

lemma Diff_eq: "A - B = A ∩ (- B)"
  by(rule boolean_algebra_class.diff_eq)

lemma Diff_eq_empty_iff: "A - B = {} ⟷ A ⊆ B"
  by(rule boolean_algebra_class.diff_shunt_var) (* already simp *)

lemma Diff_cancel [simp]: "A - A = {}"
  by blast

lemma Diff_idemp [simp]: "(A - B) - B = A - B"
  for A B :: "'a set"
  by blast

lemma Diff_triv: "A ∩ B = {} ==> A - B = A"
  by (blast elim: equalityE)

lemma empty_Diff [simp]: "{} - A = {}"
  by blast

lemma Diff_empty [simp]: "A - {} = A"
  by blast

lemma Diff_UNIV [simp]: "A - UNIV = {}"
  by blast

lemma Diff_insert0 [simp]: "x ∉ A ==> A - insert x B = A - B"
  by blast

lemma Diff_insert: "A - insert a B = A - B - {a}"
  🍋 ‹NOT SUITABLE FOR REWRITING since ‹{a} ≡ insert a 0›\   by blast

lemma Diff_insert2: "A - insert a B = A - {a} - B"
  🍋 ‹NOT SUITABLE FOR REWRITING since ‹{a} ≡ insert a 0›\   by blast

lemma insert_Diff_if: "insert x A - B = (if x ∈ B then A - B else insert x (A - B))"
  by auto

lemma insert_Diff1 [simp]: "x ∈ B ==> insert x A - B = A - B"
  by blast

lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"
  by blast

lemma insert_Diff: "a ∈ A ==> insert a (A - {a}) = A"
  by blast

lemma Diff_insert_absorb: "x ∉ A ==> (insert x A) - {x} = A"
  by auto

lemma Diff_disjoint [simp]: "A ∩ (B - A) = {}"
  by blast

lemma Diff_partition: "A ⊆ B ==> A ∪ (B - A) = B"
  by blast

lemma double_diff: "A ⊆ B ==> B ⊆ C ==> B - (C - A) = A"
  by blast

lemma Un_Diff_cancel [simp]: "A ∪ (B - A) = A ∪ B"
  by blast

lemma Un_Diff_cancel2 [simp]: "(B - A) ∪ A = B ∪ A"
  by blast

lemma Diff_Un: "A - (B ∪ C) = (A - B) ∩ (A - C)"
  by blast

lemma Diff_Int: "A - (B ∩ C) = (A - B) ∪ (A - C)"
  by blast

lemma Diff_Diff_Int: "A - (A - B) = A ∩ B"
  by blast

lemma Un_Diff: "(A ∪ B) - C = (A - C) ∪ (B - C)"
  by blast

lemma Int_Diff: "(A ∩ B) - C = A ∩ (B - C)"
  by blast

lemma Diff_Int_distrib: "C ∩ (A - B) = (C ∩ A) - (C ∩ B)"
  by blast

lemma Diff_Int_distrib2: "(A - B) ∩ C = (A ∩ C) - (B ∩ C)"
  by blast

lemma Diff_Compl [simp]: "A - (- B) = A ∩ B"
  by auto

lemma Compl_Diff_eq [simp]: "- (A - B) = - A ∪ B"
  by blast

lemma subset_Compl_singleton [simp]: "A ⊆ - {b} ⟷ b ∉ A"
  by blast

text ‹🪙 Quantification over type 🍋‹bool›.\<close>

lemma bool_induct: "P True ==> P False ==> P x"
  by (cases x) auto

lemma all_bool_eq: "(∀b. P b) ⟷ P True ∧ P False"
  by (auto intro: bool_induct)

lemma bool_contrapos: "P x ==> ¬ P False ==> P True"
  by (cases x) auto

lemma ex_bool_eq: "(∃b. P b) ⟷ P True ∨ P False"
  by (auto intro: bool_contrapos)

lemma UNIV_bool: "UNIV = {False, True}"
  by (auto intro: bool_induct)

text ‹🪙 ‹Pow›\
lemma Pow_empty [simp]: "Pow {} = {{}}"
  by (auto simp add: Pow_def)

lemma Pow_singleton_iff [simp]: "Pow X = {Y} ⟷ X = {} ∧ Y = {}"
  by blast  (* somewhat slow *)

lemma Pow_insert: "Pow (insert a A) = Pow A ∪ (insert a ` Pow A)"
  by (blast intro: image_eqI [where ?x = "u - {a}" for u])

lemma Pow_Compl: "Pow (- A) = {- B | B. A ∈ Pow B}"
  by (blast intro: exI [where ?x = "- u" for u])

lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"
  by blast

lemma Un_Pow_subset: "Pow A ∪ Pow B ⊆ Pow (A ∪ B)"
  by blast

lemma Pow_Int_eq [simp]: "Pow (A ∩ B) = Pow A ∩ Pow B"
  by blast


text ‹🪙 Miscellany.›

lemma Int_Diff_disjoint: "A ∩ B ∩ (A - B) = {}"
  by blast

lemma Int_Diff_Un: "A ∩ B ∪ (A - B) = A"
  by blast

lemma set_eq_subset: "A = B ⟷ A ⊆ B ∧ B ⊆ A"
  by blast

lemma subset_iff: "A ⊆ B ⟷ (∀t. t ∈ A ⟶ t ∈ B)"
  by blast

lemma subset_iff_psubset_eq: "A ⊆ B ⟷ A ⊂ B ∨ A = B"
  unfolding less_le by blast

lemma all_not_in_conv [simp]: "(∀x. x ∉ A) ⟷ A = {}"
  by blast

lemma ex_in_conv: "(∃x. x ∈ A) ⟷ A ≠ {}"
  by blast

lemma ball_simps [simp, no_atp]:
  "∧A P Q. (∀x∈A. P x ∨ Q) ⟷ ((∀x∈A. P x) ∨ Q)"
  "∧A P Q. (∀x∈A. P ∨ Q x) ⟷ (P ∨ (∀x∈A. Q x))"
  "∧A P Q. (∀x∈A. P ⟶ Q x) ⟷ (P ⟶ (∀x∈A. Q x))"
  "∧A P Q. (∀x∈A. P x ⟶ Q) ⟷ ((∃x∈A. P x) ⟶ Q)"
  "∧P. (∀x∈{}. P x) ⟷ True"
  "∧P. (∀x∈UNIV. P x) ⟷ (∀x. P x)"
  "∧a B P. (∀x∈insert a B. P x) ⟷ (P a ∧ (∀x∈B. P x))"
  "∧P Q. (∀x∈Collect Q. P x) ⟷ (∀x. Q x ⟶ P x)"
  "∧A P f. (∀x∈f`A. P x) ⟷ (∀x∈A. P (f x))"
  "∧A P. (¬ (∀x∈A. P x)) ⟷ (∃x∈A. ¬ P x)"
  by auto

lemma bex_simps [simp, no_atp]:
  "∧A P Q. (∃x∈A. P x ∧ Q) ⟷ ((∃x∈A. P x) ∧ Q)"
  "∧A P Q. (∃x∈A. P ∧ Q x) ⟷ (P ∧ (∃x∈A. Q x))"
  "∧P. (∃x∈{}. P x) ⟷ False"
  "∧P. (∃x∈UNIV. P x) ⟷ (∃x. P x)"
  "∧a B P. (∃x∈insert a B. P x) ⟷ (P a ∨ (∃x∈B. P x))"
  "∧P Q. (∃x∈Collect Q. P x) ⟷ (∃x. Q x ∧ P x)"
  "∧A P f. (∃x∈f`A. P x) ⟷ (∃x∈A. P (f x))"
  "∧A P. (¬(∃x∈A. P x)) ⟷ (∀x∈A. ¬ P x)"
  by auto

lemma ex_image_cong_iff [simp, no_atp]:
  "(∃x. x∈f`A) ⟷ A ≠ {}" "(∃x. x∈f`A ∧ P x) ⟷ (∃x∈A. P (f x))"
  by auto

subsubsection ‹Monotonicity of various operations›

lemma image_mono: "A ⊆ B ==> f ` A ⊆ f ` B"
  by blast

lemma Pow_mono: "A ⊆ B ==> Pow A ⊆ Pow B"
  by blast

lemma insert_mono: "C ⊆ D ==> insert a C ⊆ insert a D"
  by blast

lemma Un_mono: "A ⊆ C ==> B ⊆ D ==> A ∪ B ⊆ C ∪ D"
  by (fact sup_mono)

lemma Int_mono: "A ⊆ C ==> B ⊆ D ==> A ∩ B ⊆ C ∩ D"
  by (fact inf_mono)

lemma Diff_mono: "A ⊆ C ==> D ⊆ B ==> A - B ⊆ C - D"
  by blast

lemma Compl_anti_mono: "A ⊆ B ==> - B ⊆ - A"
  by (fact compl_mono)

text ‹🪙 Monotonicity of implications.›

lemma in_mono: "A ⊆ B ==> x ∈ A ⟶ x ∈ B"
  by (rule impI) (erule subsetD)

lemma conj_mono: "P1 ⟶ Q1 ==> P2 ⟶ Q2 ==> (P1 ∧ P2) ⟶ (Q1 ∧ Q2)"
  by iprover

lemma disj_mono: "P1 ⟶ Q1 ==> P2 ⟶ Q2 ==> (P1 ∨ P2) ⟶ (Q1 ∨ Q2)"
  by iprover

lemma imp_mono: "Q1 ⟶ P1 ==> P2 ⟶ Q2 ==> (P1 ⟶ P2) ⟶ (Q1 ⟶ Q2)"
  by iprover

lemma imp_refl: "P ⟶ P" ..

lemma not_mono: "Q ⟶ P ==> ¬ P ⟶ ¬ Q"
  by iprover

lemma ex_mono: "(∧x. P x ⟶ Q x) ==> (∃x. P x) ⟶ (∃x. Q x)"
  by iprover

lemma all_mono: "(∧x. P x ⟶ Q x) ==> (∀x. P x) ⟶ (∀x. Q x)"
  by iprover

lemma Collect_mono: "(∧x. P x ⟶ Q x) ==> Collect P ⊆ Collect Q"
  by blast

lemma Int_Collect_mono: "A ⊆ B ==> (∧x. x ∈ A ==> P x ⟶ Q x) ==> A ∩ Collect P ⊆ B ∩ Collect Q"
  by blast

lemmas basic_monos =
  subset_refl imp_refl disj_mono conj_mono ex_mono Collect_mono in_mono

lemma eq_to_mono: "a = b ==> c = d ==> b ⟶ d ==> a ⟶ c"
  by iprover


subsubsection ‹Inverse image of a function›

definition vimage :: "('a ==> 'b) ==> 'b set ==> 'a set"  (infixr ‹-`› 90)
  where "f -` B ≡ {x. f x ∈ B}"

lemma vimage_eq [simp]: "a ∈ f -` B ⟷ f a ∈ B"
  unfolding vimage_def by blast

lemma vimage_singleton_eq: "a ∈ f -` {b} ⟷ f a = b"
  by simp

lemma vimageI [intro]: "f a = b ==> b ∈ B ==> a ∈ f -` B"
  unfolding vimage_def by blast

lemma vimageI2: "f a ∈ A ==> a ∈ f -` A"
  unfolding vimage_def by fast

lemma vimageE [elim!]: "a ∈ f -` B ==> (∧x. f a = x ==> x ∈ B ==> P) ==> P"
  unfolding vimage_def by blast

lemma vimageD: "a ∈ f -` A ==> f a ∈ A"
  unfolding vimage_def by fast

lemma vimage_empty [simp]: "f -` {} = {}"
  by blast

lemma vimage_Compl: "f -` (- A) = - (f -` A)"
  by blast

lemma vimage_Un [simp]: "f -` (A ∪ B) = (f -` A) ∪ (f -` B)"
  by blast

lemma vimage_Int [simp]: "f -` (A ∩ B) = (f -` A) ∩ (f -` B)"
  by fast

lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}"
  by blast

lemma vimage_Collect: "(∧x. P (f x) = Q x) ==> f -` (Collect P) = Collect Q"
  by blast

lemma vimage_insert: "f -` (insert a B) = (f -` {a}) ∪ (f -` B)"
  🍋 ‹NOT suitable for rewriting because of the recurrence of ‹{a}›.\<close>
  by blast

lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)"
  by blast

lemma vimage_UNIV [simp]: "f -` UNIV = UNIV"
  by blast

lemma vimage_mono: "A ⊆ B ==> f -` A ⊆ f -` B"
  🍋 ‹monotonicity›
  by blast

lemma vimage_image_eq: "f -` (f ` A) = {y. ∃x∈A. f x = f y}"
  by (blast intro: sym)

lemma image_vimage_subset: "f ` (f -` A) ⊆ A"
  by blast

lemma image_vimage_eq [simp]: "f ` (f -` A) = A ∩ range f"
  by blast

lemma image_subset_iff_subset_vimage: "f ` A ⊆ B ⟷ A ⊆ f -` B"
  by blast

lemma subset_vimage_iff: "A ⊆ f -` B ⟷ (∀x∈A. f x ∈ B)"
  by auto

lemma vimage_const [simp]: "((λx. c) -` A) = (if c ∈ A then UNIV else {})"
  by auto

lemma vimage_if [simp]: "((λx. if x ∈ B then c else d) -` A) =
   (if c ∈ A then (if d ∈ A then UNIV else B)
    else if d ∈ A then - B else {})"
  by (auto simp add: vimage_def)

lemma vimage_inter_cong: "(∧ w. w ∈ S ==> f w = g w) ==> f -` y ∩ S = g -` y ∩ S"
  by auto

lemma vimage_ident [simp]: "(λx. x) -` Y = Y"
  by blast


subsubsection ‹Singleton sets›

definition is_singleton :: "'a set ==> bool"
  where "is_singleton A ⟷ (∃x. A = {x})"

lemma is_singletonI [simp, intro!]: "is_singleton {x}"
  unfolding is_singleton_def by simp

lemma is_singletonI': "A ≠ {} ==> (∧x y. x ∈ A ==> y ∈ A ==> x = y) ==> is_singleton A"
  unfolding is_singleton_def by blast

lemma is_singletonE: "is_singleton A ==> (∧x. A = {x} ==> P) ==> P"
  unfolding is_singleton_def by blast

lemma is_singleton_iff_ex1:
  ‹is_singleton A ⟷ (∃!x. x ∈ A)›
  by (auto simp add: is_singleton_def)


subsubsection ‹Getting the contents of a singleton set›

definition the_elem :: "'a set ==> 'a"
  where "the_elem X = (THE x. X = {x})"

lemma the_elem_eq [simp]: "the_elem {x} = x"
  by (simp add: the_elem_def)

lemma is_singleton_the_elem: "is_singleton A ⟷ A = {the_elem A}"
  by (auto simp: is_singleton_def)

lemma the_elem_image_unique:
  assumes "A ≠ {}"
    and *: "∧y. y ∈ A ==> f y = a"
  shows "the_elem (f ` A) = a"
  unfolding the_elem_def
proof (rule the1_equality)
  from ‹A ≠ {}› obtain y where "y ∈ A" by auto
  with * ‹y ∈ A› have "a ∈ f ` A" by blast
  with * show "f ` A = {a}" by auto
  then show "∃!x. f ` A = {x}" by auto
qed


subsubsection ‹Monad operation›

definition bind :: "'a set ==> ('a ==> 'b set) ==> 'b set"
  where "bind A f = {x. ∃B ∈ f`A. x ∈ B}"

hide_const (open) bind

lemma bind_bind: "Set.bind (Set.bind A B) C = Set.bind A (λx. Set.bind (B x) C)"
  for A :: "'a set"
  by (auto simp: bind_def)

lemma empty_bind [simp]: "Set.bind {} f = {}"
  by (simp add: bind_def)

lemma nonempty_bind_const: "A ≠ {} ==> Set.bind A (λ_. B) = B"
  by (auto simp: bind_def)

lemma bind_const: "Set.bind A (λ_. B) = (if A = {} then {} else B)"
  by (auto simp: bind_def)

lemma bind_singleton_conv_image: "Set.bind A (λx. {f x}) = f ` A"
  by (auto simp: bind_def)


subsubsection ‹Operations for execution›

text ‹
  Use those operations only for generating executable / efficient code.
  Otherwise use the RHSs directly.
 ›

context
begin

qualified definition is_empty :: "'a set ==> bool" 🍋 ‹only for code generation›
  where is_empty_iff [code_abbrev, simp]: "is_empty A ⟷ A = {}"

qualified definition remove :: "'a ==> 'a set ==> 'a set" 🍋 ‹only for code generation›
  where remove_eq [code_abbrev, simp]: "remove x A = A - {x}"

qualified definition filter :: "('a ==> bool) ==> 'a set ==> 'a set" 🍋 ‹only for code generation›
  where filter_eq [code_abbrev, simp]: "filter P A = {a ∈ A. P a}"

qualified definition can_select :: "('a ==> bool) ==> 'a set ==> bool" 🍋 ‹only for code generation›
  where can_select_iff [code_abbrev, simp]: "can_select P A = (∃!x∈A. P x)"

qualified lemma can_select_iff_is_singleton:
  ‹Set.can_select P A ⟷ is_singleton (Set.filter P A)›
  by (simp add: is_singleton_iff_ex1)

end

instantiation set :: (equal) equal
begin

definition "HOL.equal A B ⟷ A ⊆ B ∧ B ⊆ A"

instance
  by standard (auto simp add: equal_set_def)

end


text ‹Misc›

definition pairwise :: "('a ==> 'a ==> bool) ==> 'a set ==> bool"
  where "pairwise R S ⟷ (∀x ∈ S. ∀y ∈ S. x ≠ y ⟶ R x y)"

lemma pairwise_alt: "pairwise R S ⟷ (∀x∈S. ∀y∈S-{x}. R x y)"
by (auto simp add: pairwise_def)

lemma pairwise_trivial [simp]: "pairwise (λi j. j ≠ i) I"
  by (auto simp: pairwise_def)

lemma pairwiseI [intro?]:
  "pairwise R S" if "∧x y. x ∈ S ==> y ∈ S ==> x ≠ y ==> R x y"
  using that by (simp add: pairwise_def)

lemma pairwiseD:
  "R x y" and "R y x"
  if "pairwise R S" "x ∈ S" and "y ∈ S" and "x ≠ y"
  using that by (simp_all add: pairwise_def)

lemma pairwise_empty [simp]: "pairwise P {}"
  by (simp add: pairwise_def)

lemma pairwise_singleton [simp]: "pairwise P {A}"
  by (simp add: pairwise_def)

lemma pairwise_insert:
  "pairwise r (insert x s) ⟷ (∀y. y ∈ s ∧ y ≠ x ⟶ r x y ∧ r y x) ∧ pairwise r s"
  by (force simp: pairwise_def)

lemma pairwise_subset: "pairwise P S ==> T ⊆ S ==> pairwise P T"
  by (force simp: pairwise_def)

lemma pairwise_mono: "[pairwise P A; ∧x y. P x y ==> Q x y; B ⊆ A] ==> pairwise Q B"
  by (fastforce simp: pairwise_def)

lemma pairwise_imageI:
  "pairwise P (f ` A)"
  if "∧x y. x ∈ A ==> y ∈ A ==> x ≠ y ==> f x ≠ f y ==> P (f x) (f y)"
  using that by (auto intro: pairwiseI)

lemma pairwise_image: "pairwise r (f ` s) ⟷ pairwise (λx y. (f x ≠ f y) ⟶ r (f x) (f y)) s"
  by (force simp: pairwise_def)

definition disjnt :: "'a set ==> 'a set ==> bool"
  where "disjnt A B ⟷ A ∩ B = {}"

lemma disjnt_self_iff_empty [simp]: "disjnt S S ⟷ S = {}"
  by (auto simp: disjnt_def)

lemma disjnt_commute: "disjnt A B = disjnt B A"
  by (auto simp: disjnt_def)

lemma disjnt_iff: "disjnt A B ⟷ (∀x. ¬ (x ∈ A ∧ x ∈ B))"
  by (force simp: disjnt_def)

lemma disjnt_sym: "disjnt A B ==> disjnt B A"
  using disjnt_iff by blast

lemma disjnt_empty1 [simp]: "disjnt {} A" and disjnt_empty2 [simp]: "disjnt A {}"
  by (auto simp: disjnt_def)

lemma disjnt_insert1 [simp]: "disjnt (insert a X) Y ⟷ a ∉ Y ∧ disjnt X Y"
  by (simp add: disjnt_def)

lemma disjnt_insert2 [simp]: "disjnt Y (insert a X) ⟷ a ∉ Y ∧ disjnt Y X"
  by (simp add: disjnt_def)

lemma disjnt_subset1 : "[disjnt X Y; Z ⊆ X] ==> disjnt Z Y"
  by (auto simp: disjnt_def)

lemma disjnt_subset2 : "[disjnt X Y; Z ⊆ Y] ==> disjnt X Z"
  by (auto simp: disjnt_def)

lemma disjnt_Un1 [simp]: "disjnt (A ∪ B) C ⟷ disjnt A C ∧ disjnt B C"
  by (auto simp: disjnt_def)

lemma disjnt_Un2 [simp]: "disjnt C (A ∪ B) ⟷ disjnt C A ∧ disjnt C B"
  by (auto simp: disjnt_def)

lemma disjnt_Diff1: "disjnt (X-Y) (U-V)" and disjnt_Diff2: "disjnt (U-V) (X-Y)" if "X ⊆ V"
  using that by (auto simp: disjnt_def)

lemma disjoint_image_subset: "[pairwise disjnt A; ∧X. X ∈ A ==> f X ⊆ X] ==> pairwise disjnt (f `A)"
  unfolding disjnt_def pairwise_def by fast

lemma pairwise_disjnt_iff: "pairwise disjnt A ⟷ (∀x. ∃🪙≤🪙1 X. X ∈ A ∧ x ∈ X)"
  by (auto simp: Uniq_def disjnt_iff pairwise_def)

lemma disjnt_insert: 🍋‹contributor ‹Lars Hupel›\   ‹disjnt (insert x M) N› if ‹x ∉ N› ‹disjnt M N›
  using that by (simp add: disjnt_def)

lemma Int_emptyI: "(∧x. x ∈ A ==> x ∈ B ==> False) ==> A ∩ B = {}"
  by blast

lemma in_image_insert_iff:
  assumes "∧C. C ∈ B ==> x ∉ C"
  shows "A ∈ insert x ` B ⟷ x ∈ A ∧ A - {x} ∈ B" (is "?P ⟷ ?Q")
proof
  assume ?P then show ?Q
    using assms by auto
next
  assume ?Q
  then have "x ∈ A" and "A - {x} ∈ B"
    by simp_all
  from ‹A - {x} ∈ B› have "insert x (A - {x}) ∈ insert x ` B"
    by (rule imageI)
  also from ‹x ∈ A›
  have "insert x (A - {x}) = A"
    by auto
  finally show ?P .
qed

hide_const (open) member not_member

lemmas equalityI = subset_antisym
lemmas set_mp = subsetD
lemmas set_rev_mp = rev_subsetD

end