# Theory Set

(*  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]: "Collect (λx. member x A) = A"

notation
member  ("'(∈')") and
member  ("(_/  _)" [51, 51] 50)

abbreviation not_member
where "not_member x A  ¬ (x  A)" ― ‹non-membership
notation
not_member  ("'(∉')") and
not_member  ("(_/  _)" [51, 51] 50)

notation (ASCII)
member  ("'(:')") and
member  ("(_/ : _)" [51, 51] 50) and
not_member  ("'(~:')") and
not_member  ("(_/ ~: _)" [51, 51] 50)

text Set comprehensions

syntax
"_Coll" :: "pttrn  bool  'a set"    ("(1{_./ _})")
translations
"{x. P}"  "CONST Collect (λx. P)"

syntax (ASCII)
"_Collect" :: "pttrn  'a set  bool  'a set"  ("(1{(_/: _)./ _})")
syntax
"_Collect" :: "pttrn  'a set  bool  'a set"  ("(1{(_/  _)./ _})")
translations
"{p:A. P}"  "CONST Collect (λp. p  A  P)"

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}") =
fn _ => 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

definition top_set
where

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}"

syntax
"_Finset" :: "args  'a set"    ("{(_)}")
translations
"{x, xs}"  "CONST insert x {xs}"
"{x}"  "CONST insert x {}"

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  ("(_/  _)" [51, 51] 50) and
subset_eq  ("'(⊆')") and
subset_eq  ("(_/  _)" [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  ("(_/  _)" [51, 51] 50) and
supset_eq  ("'(⊇')") and
supset_eq  ("(_/  _)" [51, 51] 50)

notation (ASCII output)
subset  ("'(<')") and
subset  ("(_/ < _)" [51, 51] 50) and
subset_eq  ("'(<=')") and
subset_eq  ("(_/ <= _)" [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"      ("(3ALL (_/:_)./ _)" [0, 0, 10] 10)
"_Bex"        :: "pttrn  'a set  bool  bool"      ("(3EX (_/:_)./ _)" [0, 0, 10] 10)
"_Bex1"       :: "pttrn  'a set  bool  bool"      ("(3EX! (_/:_)./ _)" [0, 0, 10] 10)
"_Bleast"     :: "id  'a set  bool  'a"           ("(3LEAST (_/:_)./ _)" [0, 0, 10] 10)

syntax (input)
"_Ball"       :: "pttrn  'a set  bool  bool"      ("(3! (_/:_)./ _)" [0, 0, 10] 10)
"_Bex"        :: "pttrn  'a set  bool  bool"      ("(3? (_/:_)./ _)" [0, 0, 10] 10)
"_Bex1"       :: "pttrn  'a set  bool  bool"      ("(3?! (_/:_)./ _)" [0, 0, 10] 10)

syntax
"_Ball"       :: "pttrn  'a set  bool  bool"      ("(3(_/_)./ _)" [0, 0, 10] 10)
"_Bex"        :: "pttrn  'a set  bool  bool"      ("(3(_/_)./ _)" [0, 0, 10] 10)
"_Bex1"       :: "pttrn  'a set  bool  bool"      ("(3∃!(_/_)./ _)" [0, 0, 10] 10)
"_Bleast"     :: "id  'a set  bool  'a"           ("(3LEAST(_/_)./ _)" [0, 0, 10] 10)

translations
"xA. P"  "CONST Ball A (λx. P)"
"xA. P"  "CONST Bex A (λx. P)"
"∃!xA. P"  "∃!x. x  A  P"
"LEAST x:A. P"  "LEAST x. x  A  P"

syntax (ASCII output)
"_setlessAll" :: "[idt, 'a, bool]  bool"  ("(3ALL _<_./ _)"  [0, 0, 10] 10)
"_setlessEx"  :: "[idt, 'a, bool]  bool"  ("(3EX _<_./ _)"  [0, 0, 10] 10)
"_setleAll"   :: "[idt, 'a, bool]  bool"  ("(3ALL _<=_./ _)" [0, 0, 10] 10)
"_setleEx"    :: "[idt, 'a, bool]  bool"  ("(3EX _<=_./ _)" [0, 0, 10] 10)
"_setleEx1"   :: "[idt, 'a, bool]  bool"  ("(3EX! _<=_./ _)" [0, 0, 10] 10)

syntax
"_setlessAll" :: "[idt, 'a, bool]  bool"   ("(3__./ _)"  [0, 0, 10] 10)
"_setlessEx"  :: "[idt, 'a, bool]  bool"   ("(3__./ _)"  [0, 0, 10] 10)
"_setleAll"   :: "[idt, 'a, bool]  bool"   ("(3__./ _)" [0, 0, 10] 10)
"_setleEx"    :: "[idt, 'a, bool]  bool"   ("(3__./ _)" [0, 0, 10] 10)
"_setleEx1"   :: "[idt, 'a, bool]  bool"   ("(3∃!__./ _)" [0, 0, 10] 10)

translations
"AB. P"  "A. A  B  P"
"AB. P"  "A. A  B  P"
"AB. P"  "A. A  B  P"
"AB. P"  "A. A  B  P"
"∃!AB. P"  "∃!A. A  B  P"

print_translation
let
val All_binder = Mixfix.binder_name const_syntaxAll;
val Ex_binder = Mixfix.binder_name const_syntaxEx;
val impl = const_syntaxHOL.implies;
val conj = const_syntaxHOL.conj;
val sbset = const_syntaxsubset;
val sbset_eq = const_syntaxsubset_eq;

val trans =
[((All_binder, impl, sbset), syntax_const‹_setlessAll›),
((All_binder, impl, sbset_eq), syntax_const‹_setleAll›),
((Ex_binder, conj, sbset), syntax_const‹_setlessEx›),
((Ex_binder, conj, sbset_eq), syntax_const‹_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 (syntax_const‹_bound›, _) \$ Free (v, Type (type_nameset, _)),
Const (c, _) \$
(Const (d, _) \$ (Const (syntax_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"    ("(1{_ |/_./ _})")

parse_translation
let
val ex_tr = snd (Syntax_Trans.mk_binder_tr ("EX ", const_syntaxEx));

fun nvars (Const (syntax_const‹_idts›, _) \$ _ \$ idts) = nvars idts + 1
| nvars _ = 1;

fun setcompr_tr ctxt [e, idts, b] =
let
val eq = Syntax.const const_syntaxHOL.eq \$ Bound (nvars idts) \$ e;
val P = Syntax.const const_syntaxHOL.conj \$ eq \$ b;
val exP = ex_tr ctxt [idts, P];
in Syntax.const const_syntaxCollect \$ absdummy dummyT exP end;

in [(syntax_const‹_Setcompr›, setcompr_tr)] end

print_translation
[Syntax_Trans.preserve_binder_abs2_tr' const_syntaxBall syntax_const‹_Ball›,
Syntax_Trans.preserve_binder_abs2_tr' const_syntaxBex syntax_const‹_Bex›]
― ‹to avoid eta-contraction of body

print_translation
let
val ex_tr' = snd (Syntax_Trans.mk_binder_tr' (const_syntaxEx, "DUMMY"));

fun setcompr_tr' ctxt [Abs (abs as (_, _, P))] =
let
fun check (Const (const_syntaxEx, _) \$ Abs (_, _, P), n) = check (P, n + 1)
| check (Const (const_syntaxHOL.conj, _) \$
(Const (const_syntaxHOL.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 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' abs;
val M = Syntax.const syntax_const‹_Coll› \$ x \$ t;
in
case t of
Const (const_syntaxHOL.conj, _) \$
(Const (const_syntaxSet.member, _) \$
(Const (syntax_const‹_bound›, _) \$ Free (yN, _)) \$ A) \$ P =>
if xN = yN then Syntax.const syntax_const‹_Collect› \$ x \$ A \$ P else M
| _ => M
end
end;
in [(const_syntaxCollect, setcompr_tr')] end

simproc_setup defined_Bex ("xA. P x  Q x") =
fn _ => Quantifier1.rearrange_Bex
(fn ctxt => unfold_tac ctxt @{thms Bex_def})

simproc_setup defined_All ("xA. P x  Q x") =
fn _ => Quantifier1.rearrange_Ball
(fn ctxt => unfold_tac ctxt @{thms Ball_def})

lemma ballI [intro!]: "(x. x  A  P x)  xA. P x"

lemmas strip = impI allI ballI

lemma bspec [dest?]: "xA. P x  x  A  P x"

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 = [(const_nameBall, @{thms bspec})] @ mksimps_pairs;
end;

open Simpdata;

declaration fn _ => Simplifier.map_ss (Simplifier.set_mksimps (mksimps mksimps_pairs))

lemma ballE [elim]: "xA. P x  (P x  Q)  (x  A  Q)  Q"
unfolding Ball_def by blast

lemma bexI [intro]: "P x  x  A  xA. 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  xA. P x"
― ‹The best argument order when there is only one x ∈ A›.
unfolding Bex_def by blast

lemma bexCI: "(xA. ¬ P x  P a)  a  A  xA. P x"
unfolding Bex_def by blast

lemma bexE [elim!]: "xA. P x  (x. x  A  P x  Q)  Q"
unfolding Bex_def by blast

lemma ball_triv [simp]: "(xA. P)  ((x. x  A)  P)"
― ‹trivial rewrite rule.

lemma bex_triv [simp]: "(xA. P)  ((x. x  A)  P)"
― ‹Dual form for existentials.

lemma bex_triv_one_point1 [simp]: "(xA. x = a)  a  A"
by blast

lemma bex_triv_one_point2 [simp]: "(xA. a = x)  a  A"
by blast

lemma bex_one_point1 [simp]: "(xA. x = a  P x)  a  A  P a"
by blast

lemma bex_one_point2 [simp]: "(xA. a = x  P x)  a  A  P a"
by blast

lemma ball_one_point1 [simp]: "(xA. x = a  P x)  (a  A  P a)"
by blast

lemma ball_one_point2 [simp]: "(xA. a = x  P x)  (a  A  P a)"
by blast

lemma ball_conj_distrib: "(xA. P x  Q x)  (xA. P x)  (xA. Q x)"
by blast

lemma bex_disj_distrib: "(xA. P x  Q x)  (xA. P x)  (xA. Q x)"
by blast

text Congruence rules

lemma ball_cong:
" A = B;  x. x  B  P x  Q x
(xA. P x)  (xB. Q x)"

lemma ball_cong_simp [cong]:
" A = B;  x. x  B =simp=> P x  Q x
(xA. P x)  (xB. Q x)"

lemma bex_cong:
" A = B;  x. x  B  P x  Q x
(xA. P x)  (xB. 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
(xA. P x)  (xB. Q x)"
by (simp add: simp_implies_def Bex_def cong: conj_cong)

lemma bex1_def: "(∃!xX. P x)  (xX. P x)  (xX. yX. P x  P y  x = y)"
by auto

subsection Basic operations

subsubsection Subsets

lemma subsetI [intro!]: "(x. x  A  x  B)  A  B"

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"
― ‹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  (xA. 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: propA = {} goes to prop{}  A and propA  {}
and then back to propA = {}!

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}"

lemma empty_iff [simp]: "c  {}  False"

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

lemma empty_subsetI [iff]: "{}  A"
― ‹One effect is to delete the ASSUMPTION prop{}  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"

lemma bex_empty [simp]: "Bex {} P  False"

subsubsection The universal set -- UNIV

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

lemma UNIV_def: "UNIV = {x. True}"

lemma UNIV_I [simp]: "x  UNIV"

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›) causes them
to be ignored because of their interaction with congruence rules.

lemma ball_UNIV [simp]: "Ball UNIV P  All P"

lemma bex_UNIV [simp]: "Bex UNIV P  Ex P"

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"

lemma PowI: "A  B  A  Pow B"

lemma PowD: "A  Pow B  A  B"

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"

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}"

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

lemma mono_Int: "mono f  f (A  B)  f A  f B"
by (fact mono_inf)

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}"

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"

lemma mono_Un: "mono f  f A  f B  f (A  B)"
by (fact mono_sup)

subsubsection Set difference

lemma Diff_iff [simp]: "c  A - B  c  A  c  B"

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

subsubsection Augmenting a set -- constinsert

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
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:
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. xA. 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  (xA. 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  (xA. 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  (AAA. 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: "xf ` A. P x  xA. P (f x)"
by simp

lemma bex_imageD: "xf ` A. P x  xA. P (f x)"
by auto

by auto

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}"

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"

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"

lemma subset_psubset_trans: "A  B  B  C  A  C"

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 (xA. 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) (* FIXME: already simp *)

lemma Collect_subset [simp]: "{xA. 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_mono_iff: "Collect P  Collect Q  (x. P x  Q x)"
by blast

text  insert›.

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
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. xA  x  B)"
by blast

lemma disjoint_iff_not_equal: "A  B = {}  (xA. yB. 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 [simp]: "A  B = UNIV  A = UNIV  B = UNIV"
by (fact inf_eq_top_iff) (* already simp *)

lemma Int_subset_iff [simp]: "C  A  B  C  A  C  B"
by (fact le_inf_iff)

lemma Int_Collect: "x  A  {x. P x}  x  A  P x"
by blast

text  Un›.

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 [simp]: "A  B  C  A  C  B  C"
by (fact le_sup_iff)

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)  (xA. P x)  (xB. P x)"
by blast

lemma bex_Un: "(x  A  B. P x)  (xA. P x)  (xB. P x)"
by blast

text  Set difference.

lemma Diff_eq: "A - B = A  (- B)"
by blast

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

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 typbool.

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 {} = {{}}"

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]:
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. (xA. P x  Q)  ((xA. P x)  Q)"
"A P Q. (xA. P  Q x)  (P  (xA. Q x))"
"A P Q. (xA. P  Q x)  (P  (xA. Q x))"
"A P Q. (xA. P x  Q)  ((xA. P x)  Q)"
"P. (x{}. P x)  True"
"P. (xUNIV. P x)  (x. P x)"
"a B P. (xinsert a B. P x)  (P a  (xB. P x))"
"P Q. (xCollect Q. P x)  (x. Q x  P x)"
"A P f. (xf`A. P x)  (xA. P (f x))"
"A P. (¬ (xA. P x))  (xA. ¬ P x)"
by auto

lemma bex_simps [simp, no_atp]:
"A P Q. (xA. P x  Q)  ((xA. P x)  Q)"
"A P Q. (xA. P  Q x)  (P  (xA. Q x))"
"P. (x{}. P x)  False"
"P. (xUNIV. P x)  (x. P x)"
"a B P. (xinsert a B. P x)  (P a  (xB. P x))"
"P Q. (xCollect Q. P x)  (x. Q x  P x)"
"A P f. (xf`A. P x)  (xA. P (f x))"
"A P. (¬(xA. P x))  (xA. ¬ P x)"
by auto

lemma ex_image_cong_iff [simp, no_atp]:
"(x. xf`A)  A  {}" "(x. xf`A  P x)  (xA. 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}›.
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. xA. 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  (xA. 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 {})"

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

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"

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 = f x"
shows "the_elem (f ` A) = f x"
unfolding the_elem_def
proof (rule the1_equality)
from A  {} obtain y where "y  A" by auto
with * have "f x = f y" by simp
with y  A have "f x  f ` A" by blast
with * show "f ` A = {f x}" by auto
then show "∃!x. f ` A = {x}" by auto
qed

subsubsection Least value operator

lemma Least_mono: "mono f  xS. yS. x  y  (LEAST y. y  f ` S) = f (LEAST x. x  S)"
for f :: "'a::order  'b::order"
― ‹Courtesy of Stephan Merz
apply clarify
apply (erule_tac P = "λx. x  S" in LeastI2_order)
apply fast
apply (rule LeastI2_order)
apply (auto elim: monoD intro!: order_antisym)
done

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]:

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

definition is_empty :: "'a set  bool"
where [code_abbrev]: "is_empty A  A = {}"

hide_const (open) is_empty

definition remove :: "'a  'a set  'a set"
where [code_abbrev]: "remove x A = A - {x}"

hide_const (open) remove

lemma member_remove [simp]: "x  Set.remove y A  x  A  x  y"

definition filter :: "('a  bool)  'a set  'a set"
where [code_abbrev]: "filter P A = {a  A. P a}"

hide_const (open) filter

lemma member_filter [simp]: "x  Set.filter P A  x  A  P x"

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  (xS. yS-{x}. R x y)"

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 {}"

lemma pairwise_singleton [simp]: "pairwise P {A}"

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_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"

lemma disjnt_insert2 [simp]: "disjnt Y (insert a X)  a  Y  disjnt Y X"

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 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

ML
val Ball_def = @{thm Ball_def}
val Bex_def = @{thm Bex_def}
val CollectD = @{thm CollectD}
val CollectE = @{thm CollectE}
val CollectI = @{thm CollectI}
val Collect_conj_eq = @{thm Collect_conj_eq}
val Collect_mem_eq = @{thm Collect_mem_eq}
val IntD1 = @{thm IntD1}
val IntD2 = @{thm IntD2}
val IntE = @{thm IntE}
val IntI = @{thm IntI}
val Int_Collect = @{thm Int_Collect}
val UNIV_I = @{thm UNIV_I}
val UNIV_witness = @{thm UNIV_witness}