src/HOL/Set.thy
 author haftmann Tue, 20 Mar 2007 08:27:23 +0100 changeset 22478 110f7f6f8a5d parent 22455 f6f22aba2e0e child 22744 5cbe966d67a2 permissions -rw-r--r--
fixed typo

(*  Title:      HOL/Set.thy
ID:         $Id$
Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
*)

header {* Set theory for higher-order logic *}

theory Set
imports Lattices
begin

text {* A set in HOL is simply a predicate. *}

subsection {* Basic syntax *}

global

typedecl 'a set
arities set :: (type) type

consts
"{}"          :: "'a set"                             ("{}")
UNIV          :: "'a set"
insert        :: "'a => 'a set => 'a set"
Collect       :: "('a => bool) => 'a set"              -- "comprehension"
Int           :: "'a set => 'a set => 'a set"          (infixl 70)
Un            :: "'a set => 'a set => 'a set"          (infixl 65)
UNION         :: "'a set => ('a => 'b set) => 'b set"  -- "general union"
INTER         :: "'a set => ('a => 'b set) => 'b set"  -- "general intersection"
Union         :: "'a set set => 'a set"                -- "union of a set"
Inter         :: "'a set set => 'a set"                -- "intersection of a set"
Pow           :: "'a set => 'a set set"                -- "powerset"
Ball          :: "'a set => ('a => bool) => bool"      -- "bounded universal quantifiers"
Bex           :: "'a set => ('a => bool) => bool"      -- "bounded existential quantifiers"
Bex1          :: "'a set => ('a => bool) => bool"      -- "bounded unique existential quantifiers"
image         :: "('a => 'b) => 'a set => 'b set"      (infixr "" 90)
"op :"        :: "'a => 'a set => bool"                -- "membership"

notation
"op :"  ("op :") and
"op :"  ("(_/ : _)" [50, 51] 50)

local

subsection {* Additional concrete syntax *}

abbreviation
range :: "('a => 'b) => 'b set" where -- "of function"
"range f == f  UNIV"

abbreviation
"not_mem x A == ~ (x : A)" -- "non-membership"

notation
not_mem  ("op ~:") and
not_mem  ("(_/ ~: _)" [50, 51] 50)

notation (xsymbols)
"op Int"  (infixl "\<inter>" 70) and
"op Un"  (infixl "\<union>" 65) and
"op :"  ("op \<in>") and
"op :"  ("(_/ \<in> _)" [50, 51] 50) and
not_mem  ("op \<notin>") and
not_mem  ("(_/ \<notin> _)" [50, 51] 50) and
Union  ("\<Union>_" [90] 90) and
Inter  ("\<Inter>_" [90] 90)

notation (HTML output)
"op Int"  (infixl "\<inter>" 70) and
"op Un"  (infixl "\<union>" 65) and
"op :"  ("op \<in>") and
"op :"  ("(_/ \<in> _)" [50, 51] 50) and
not_mem  ("op \<notin>") and
not_mem  ("(_/ \<notin> _)" [50, 51] 50)

syntax
"@Finset"     :: "args => 'a set"                       ("{(_)}")
"@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")
"@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")
"@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ :/ _./ _})")
"@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
"@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
"@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 10] 10)
"@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 10] 10)
"_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 (HOL)
"_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)

translations
"{x, xs}"     == "insert x {xs}"
"{x}"         == "insert x {}"
"{x. P}"      == "Collect (%x. P)"
"{x:A. P}"    => "{x. x:A & P}"
"UN x y. B"   == "UN x. UN y. B"
"UN x. B"     == "UNION UNIV (%x. B)"
"UN x. B"     == "UN x:UNIV. B"
"INT x y. B"  == "INT x. INT y. B"
"INT x. B"    == "INTER UNIV (%x. B)"
"INT x. B"    == "INT x:UNIV. B"
"UN x:A. B"   == "UNION A (%x. B)"
"INT x:A. B"  == "INTER A (%x. B)"
"ALL x:A. P"  == "Ball A (%x. P)"
"EX x:A. P"   == "Bex A (%x. P)"
"EX! x:A. P"  == "Bex1 A (%x. P)"
"LEAST x:A. P" => "LEAST x. x:A & P"

syntax (xsymbols)
"_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
"_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
"_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
"_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10)

syntax (HTML output)
"_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
"_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
"_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)

syntax (xsymbols)
"@Collect"    :: "idt => 'a set => bool => 'a set"      ("(1{_ \<in>/ _./ _})")
"@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
"@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
"@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 10] 10)
"@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 10] 10)

syntax (latex output)
"@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
"@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
"@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)
"@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10)

text{*
Note the difference between ordinary xsymbol syntax of indexed
unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
former does not make the index expression a subscript of the
union/intersection symbol because this leads to problems with nested
subscripts in Proof General. *}

instance set :: (type) ord
subset_def:   "A <= B         == \<forall>x\<in>A. x \<in> B"
psubset_def:  "A < B          == (A::'a set) <= B & ~ A=B" ..

abbreviation
subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
"subset \<equiv> less"

abbreviation
subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
"subset_eq \<equiv> less_eq"

notation (output)
subset  ("op <") and
subset  ("(_/ < _)" [50, 51] 50) and
subset_eq  ("op <=") and
subset_eq  ("(_/ <= _)" [50, 51] 50)

notation (xsymbols)
subset  ("op \<subset>") and
subset  ("(_/ \<subset> _)" [50, 51] 50) and
subset_eq  ("op \<subseteq>") and
subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)

notation (HTML output)
subset  ("op \<subset>") and
subset  ("(_/ \<subset> _)" [50, 51] 50) and
subset_eq  ("op \<subseteq>") and
subset_eq  ("(_/ \<subseteq> _)" [50, 51] 50)

abbreviation (input)
supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
"supset \<equiv> greater"

abbreviation (input)
supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
"supset_eq \<equiv> greater_eq"

notation (xsymbols)
supset  ("op \<supset>") and
supset  ("(_/ \<supset> _)" [50, 51] 50) and
supset_eq  ("op \<supseteq>") and
supset_eq  ("(_/ \<supseteq> _)" [50, 51] 50)

subsubsection "Bounded quantifiers"

syntax (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 (xsymbols)
"_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
"_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
"_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
"_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
"_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)

syntax (HOL output)
"_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)

syntax (HTML output)
"_setlessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
"_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
"_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
"_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
"_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)

translations
"\<forall>A\<subset>B. P"   =>  "ALL A. A \<subset> B --> P"
"\<exists>A\<subset>B. P"   =>  "EX A. A \<subset> B & P"
"\<forall>A\<subseteq>B. P"   =>  "ALL A. A \<subseteq> B --> P"
"\<exists>A\<subseteq>B. P"   =>  "EX A. A \<subseteq> B & P"
"\<exists>!A\<subseteq>B. P"  =>  "EX! A. A \<subseteq> B & P"

print_translation {*
let
val Type (set_type, _) = @{typ "'a set"};
val All_binder = Syntax.binder_name @{const_syntax "All"};
val Ex_binder = Syntax.binder_name @{const_syntax "Ex"};
val impl = @{const_syntax "op -->"};
val conj = @{const_syntax "op &"};
val sbset = @{const_syntax "subset"};
val sbset_eq = @{const_syntax "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' c n P =
if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
then Syntax.const c $Syntax.mark_bound v'$ n $P else raise Match; fun tr' q = (q, fn [Const ("_bound", _)$ Free (v, Type (T, _)), Const (c, _) $(Const (d, _)$ (Const ("_bound", _) $Free (v', _))$ n) $P] => if T = (set_type) then case AList.lookup (op =) trans (q, c, d) of NONE => raise Match | SOME l => mk v v' l n P else raise Match | _ => raise Match); in [tr' All_binder, tr' Ex_binder] end *} text {* \medskip Translate between @{text "{e | x1...xn. P}"} and @{text "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is only translated if @{text "[0..n] subset bvs(e)"}. *} parse_translation {* let val ex_tr = snd (mk_binder_tr ("EX ", "Ex")); fun nvars (Const ("_idts", _)$ _ $idts) = nvars idts + 1 | nvars _ = 1; fun setcompr_tr [e, idts, b] = let val eq = Syntax.const "op ="$ Bound (nvars idts) $e; val P = Syntax.const "op &"$ eq $b; val exP = ex_tr [idts, P]; in Syntax.const "Collect"$ Term.absdummy (dummyT, exP) end;

in [("@SetCompr", setcompr_tr)] end;
*}

(* To avoid eta-contraction of body: *)
print_translation {*
let
fun btr' syn [A,Abs abs] =
let val (x,t) = atomic_abs_tr' abs
in Syntax.const syn $x$ A $t end in [("Ball", btr' "_Ball"),("Bex", btr' "_Bex"), ("UNION", btr' "@UNION"),("INTER", btr' "@INTER")] end *} print_translation {* let val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY")); fun setcompr_tr' [Abs (abs as (_, _, P))] = let fun check (Const ("Ex", _)$ Abs (_, _, P), n) = check (P, n + 1)
| check (Const ("op &", _) $(Const ("op =", _)$ Bound m $e)$ P, n) =
n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
((0 upto (n - 1)) subset add_loose_bnos (e, 0, []))
| check _ = false

fun tr' (_ $abs) = let val _$ idts $(_$ (_ $_$ e) $Q) = ex_tr' [abs] in Syntax.const "@SetCompr"$ e $idts$ Q end;
in if check (P, 0) then tr' P
else let val (x as _ $Free(xN,_), t) = atomic_abs_tr' abs val M = Syntax.const "@Coll"$ x $t in case t of Const("op &",_)$ (Const("op :",_) $(Const("_bound",_)$ Free(yN,_)) $A)$ P =>
if xN=yN then Syntax.const "@Collect" $x$ A \$ P else M
| _ => M
end
end;
in [("Collect", setcompr_tr')] end;
*}

subsection {* Rules and definitions *}

text {* Isomorphisms between predicates and sets. *}

axioms
mem_Collect_eq: "(a : {x. P(x)}) = P(a)"
Collect_mem_eq: "{x. x:A} = A"
finalconsts
Collect
"op :"

defs
Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"
Bex_def:      "Bex A P        == EX x. x:A & P(x)"
Bex1_def:     "Bex1 A P       == EX! x. x:A & P(x)"

instance set :: (type) minus
Compl_def:    "- A            == {x. ~x:A}"
set_diff_def: "A - B          == {x. x:A & ~x:B}" ..

defs
Un_def:       "A Un B         == {x. x:A | x:B}"
Int_def:      "A Int B        == {x. x:A & x:B}"
INTER_def:    "INTER A B      == {y. ALL x:A. y: B(x)}"
UNION_def:    "UNION A B      == {y. EX x:A. y: B(x)}"
Inter_def:    "Inter S        == (INT x:S. x)"
Union_def:    "Union S        == (UN x:S. x)"
Pow_def:      "Pow A          == {B. B <= A}"
empty_def:    "{}             == {x. False}"
UNIV_def:     "UNIV           == {x. True}"
insert_def:   "insert a B     == {x. x=a} Un B"
image_def:    "fA            == {y. EX x:A. y = f(x)}"

subsection {* Lemmas and proof tool setup *}

subsubsection {* Relating predicates and sets *}

declare mem_Collect_eq [iff]  Collect_mem_eq [simp]

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

lemmas CollectE = CollectD [elim_format]

subsubsection {* Bounded quantifiers *}

lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"

lemmas strip = impI allI ballI

lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"

lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
by (unfold Ball_def) blast

ML {* bind_thm ("rev_ballE", permute_prems 1 1 @{thm ballE}) *}

text {*
\medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
@{prop "a:A"}; creates assumption @{prop "P a"}.
*}

ML {*
fun ball_tac i = etac @{thm ballE} i THEN contr_tac (i + 1)
*}

text {*
Gives better instantiation for bound:
*}

ML_setup {*
change_claset (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1))
*}

lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
-- {* Normally the best argument order: @{prop "P x"} constrains the
choice of @{prop "x:A"}. *}
by (unfold Bex_def) blast

lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
-- {* The best argument order when there is only one @{prop "x:A"}. *}
by (unfold Bex_def) blast

lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
by (unfold Bex_def) blast

lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
by (unfold Bex_def) blast

lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
-- {* Trival rewrite rule. *}

lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
-- {* Dual form for existentials. *}

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

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

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

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

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

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

ML_setup {*
local
val unfold_bex_tac = unfold_tac @{thms "Bex_def"};
fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac;
val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;

val unfold_ball_tac = unfold_tac @{thms "Ball_def"};
fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac;
val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
in
val defBEX_regroup = Simplifier.simproc (the_context ())
"defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
val defBALL_regroup = Simplifier.simproc (the_context ())
"defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
end;

*}

subsubsection {* Congruence rules *}

lemma ball_cong:
"A = B ==> (!!x. x:B ==> P x = Q x) ==>
(ALL x:A. P x) = (ALL x:B. Q x)"

lemma strong_ball_cong [cong]:
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
(ALL x:A. P x) = (ALL x:B. Q x)"

lemma bex_cong:
"A = B ==> (!!x. x:B ==> P x = Q x) ==>
(EX x:A. P x) = (EX x:B. Q x)"
by (simp add: Bex_def cong: conj_cong)

lemma strong_bex_cong [cong]:
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
(EX x:A. P x) = (EX x:B. Q x)"
by (simp add: simp_implies_def Bex_def cong: conj_cong)

subsubsection {* Subsets *}

lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B"

text {*
\medskip Map the type @{text "'a set => anything"} to just @{typ
"'a set"}.
*}

lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
-- {* Rule in Modus Ponens style. *}
by (unfold subset_def) blast

declare subsetD [intro?] -- FIXME

lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
-- {* The same, with reversed premises for use with @{text erule} --
cf @{text rev_mp}. *}
by (rule subsetD)

declare rev_subsetD [intro?] -- FIXME

text {*
\medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
*}

ML {*
fun impOfSubs th = th RSN (2, @{thm rev_subsetD})
*}

lemma subsetCE [elim]: "A \<subseteq>  B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
-- {* Classical elimination rule. *}
by (unfold subset_def) blast

text {*
\medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and
creates the assumption @{prop "c \<in> B"}.
*}

ML {*
fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i
*}

lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
by blast

lemma subset_refl [simp,atp]: "A \<subseteq> A"
by fast

lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
by blast

subsubsection {* Equality *}

lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B"
apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals])
apply (rule Collect_mem_eq)
apply (rule Collect_mem_eq)
done

(* Due to Brian Huffman *)
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
by(auto intro:set_ext)

lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
-- {* Anti-symmetry of the subset relation. *}
by (iprover intro: set_ext subsetD)

lemmas equalityI [intro!] = subset_antisym

text {*
\medskip Equality rules from ZF set theory -- are they appropriate
here?
*}

lemma equalityD1: "A = B ==> A \<subseteq> B"

lemma equalityD2: "A = B ==> B \<subseteq> A"

text {*
\medskip Be careful when adding this to the claset as @{text
subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
\<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}!
*}

lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"

lemma equalityCE [elim]:
"A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> 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 universal set -- UNIV *}

lemma UNIV_I [simp]: "x : UNIV"

declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}

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

lemma subset_UNIV [simp]: "A \<subseteq> UNIV"
by (rule subsetI) (rule UNIV_I)

text {*
\medskip Eta-contracting these two rules (to remove @{text 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"

subsubsection {* The empty set *}

lemma empty_iff [simp]: "(c : {}) = False"

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

lemma empty_subsetI [iff]: "{} \<subseteq> A"
-- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
by blast

lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
by blast

lemma equals0D: "A = {} ==> a \<notin> A"
-- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
by blast

lemma ball_empty [simp]: "Ball {} P = True"

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

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

subsubsection {* The Powerset operator -- Pow *}

lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"

lemma PowI: "A \<subseteq> B ==> A \<in> Pow B"

lemma PowD: "A \<in> Pow B ==> A \<subseteq> B"

lemma Pow_bottom: "{} \<in> Pow B"
by simp

lemma Pow_top: "A \<in> Pow A"

subsubsection {* Set complement *}

lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
by (unfold Compl_def) blast

lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
by (unfold Compl_def) blast

text {*
\medskip This form, with negated conclusion, works well with the
Classical prover.  Negated assumptions behave like formulae on the
right side of the notional turnstile ... *}

lemma ComplD [dest!]: "c : -A ==> c~:A"
by (unfold Compl_def) blast

lemmas ComplE = ComplD [elim_format]

subsubsection {* Binary union -- Un *}

lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
by (unfold Un_def) blast

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

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

text {*
\medskip Classical introduction rule: no commitment to @{prop A} vs
@{prop B}.
*}

lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
by auto

lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
by (unfold Un_def) blast

subsubsection {* Binary intersection -- Int *}

lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
by (unfold Int_def) blast

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

lemma IntD1: "c : A Int B ==> c:A"
by simp

lemma IntD2: "c : A Int B ==> c:B"
by simp

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

subsubsection {* Set difference *}

lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
by (unfold set_diff_def) blast

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

subsubsection {* Augmenting a set -- insert *}

lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
by (unfold insert_def) 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"
by (unfold insert_def) blast

lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
-- {* Classical introduction rule. *}
by auto

lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
by auto

subsubsection {* Singletons, using insert *}

lemma singletonI [intro!]: "a : {a}"
-- {* Redundant? But unlike @{text 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 \<subseteq> {b})"
by blast

lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A \<subseteq> {b})"
by blast

lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}"
by fast

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} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B"
by blast

lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)"
by (blast elim: equalityE)

subsubsection {* Unions of families *}

text {*
@{term [source] "UN x:A. B x"} is @{term "Union (BA)"}.
*}

lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
by (unfold UNION_def) blast

lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
-- {* The order of the premises presupposes that @{term A} is rigid;
@{term b} may be flexible. *}
by auto

lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
by (unfold UNION_def) blast

lemma UN_cong [cong]:
"A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"

subsubsection {* Intersections of families *}

text {* @{term [source] "INT x:A. B x"} is @{term "Inter (BA)"}. *}

lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
by (unfold INTER_def) blast

lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
by (unfold INTER_def) blast

lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
by auto

lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
-- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
by (unfold INTER_def) blast

lemma INT_cong [cong]:
"A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"

subsubsection {* Union *}

lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)"
by (unfold Union_def) blast

lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
-- {* The order of the premises presupposes that @{term C} is rigid;
@{term A} may be flexible. *}
by auto

lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
by (unfold Union_def) blast

subsubsection {* Inter *}

lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)"
by (unfold Inter_def) blast

lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"

text {*
\medskip A destruct'' rule -- every @{term X} in @{term C}
contains @{term A} as an element, but @{prop "A:X"} can hold when
@{prop "X:C"} does not!  This rule is analogous to @{text spec}.
*}

lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
by auto

lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
-- {* Classical'' elimination rule -- does not require proving
@{prop "X:C"}. *}
by (unfold Inter_def) blast

text {*
\medskip Image of a set under a function.  Frequently @{term b} does
not have the syntactic form of @{term "f x"}.
*}

lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : fA"
by (unfold image_def) 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 : fA"
-- {* This version's more effective when we already have the
required @{term x}. *}
by (unfold image_def) blast

lemma imageE [elim!]:
"b : (%x. f x)A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
-- {* The eta-expansion gives variable-name preservation. *}
by (unfold image_def) blast

lemma image_Un: "f(A Un B) = fA Un fB"
by blast

lemma image_iff: "(z : fA) = (EX x:A. z = f x)"
by blast

lemma image_subset_iff: "(fA \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
-- {* This rewrite rule would confuse users if made default. *}
by blast

lemma subset_image_iff: "(B \<subseteq> fA) = (EX AA. AA \<subseteq> A & B = fAA)"
apply safe
prefer 2 apply fast
apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast)
done

lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> fA \<subseteq> B"
-- {* Replaces the three steps @{text subsetI}, @{text imageE},
@{text hypsubst}, but breaks too many existing proofs. *}
by blast

text {*
\medskip Range of a function -- just a translation for image!
*}

lemma range_eqI: "b = f x ==> b \<in> range f"
by simp

lemma rangeI: "f x \<in> range f"
by simp

lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P"
by blast

subsubsection {* Set reasoning tools *}

text {*
Rewrite rules for boolean case-splitting: faster than @{text
"split_if [split]"}.
*}

lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
by (rule split_if)

lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
by (rule split_if)

text {*
Split ifs on either side of the membership relation.  Not for @{text
"[simp]"} -- can cause goals to blow up!
*}

lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
by (rule split_if)

lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
by (rule split_if)

lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2

lemmas mem_simps =
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
-- {* Each of these has ALREADY been added @{text "[simp]"} above. *}

(*Would like to add these, but the existing code only searches for the
outer-level constant, which in this case is just "op :"; 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.
[("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]),
("op Int", [IntD1,IntD2]),
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
*)

ML_setup {*
val mksimps_pairs = [("Ball", @{thms bspec})] @ mksimps_pairs;
change_simpset (fn ss => ss setmksimps (mksimps mksimps_pairs));
*}

subsubsection {* The proper subset'' relation *}

lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
by (unfold psubset_def) blast

lemma psubsetE [elim!]:
"[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
by (unfold psubset_def) blast

lemma psubset_insert_iff:
"(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
by (auto simp add: psubset_def subset_insert_iff)

lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
by (simp only: psubset_def)

lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B"

lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
apply (unfold psubset_def)
apply (auto dest: subset_antisym)
done

lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
apply (unfold psubset_def)
apply (auto dest: subsetD)
done

lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"

lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C"

lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
by (unfold psubset_def) blast

lemma atomize_ball:
"(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
by (simp only: Ball_def atomize_all atomize_imp)

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

subsection {* Order on sets *}

instance set :: (type) order
by (intro_classes,
(assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)

subsection {* Further set-theory lemmas *}

subsubsection {* Derived rules involving subsets. *}

text {* @{text insert}. *}

lemma subset_insertI: "B \<subseteq> insert a B"
apply (rule subsetI)
apply (erule insertI2)
done

lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
by blast

lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
by blast

text {* \medskip Big Union -- least upper bound of a set. *}

lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
by (iprover intro: subsetI UnionI)

lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
by (iprover intro: subsetI elim: UnionE dest: subsetD)

text {* \medskip General union. *}

lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
by blast

lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
by (iprover intro: subsetI elim: UN_E dest: subsetD)

text {* \medskip Big Intersection -- greatest lower bound of a set. *}

lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
by blast

lemma Inter_subset:
"[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
by blast

lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
by (iprover intro: InterI subsetI dest: subsetD)

lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
by blast

lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
by (iprover intro: INT_I subsetI dest: subsetD)

text {* \medskip Finite Union -- the least upper bound of two sets. *}

lemma Un_upper1: "A \<subseteq> A \<union> B"
by blast

lemma Un_upper2: "B \<subseteq> A \<union> B"
by blast

lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
by blast

text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}

lemma Int_lower1: "A \<inter> B \<subseteq> A"
by blast

lemma Int_lower2: "A \<inter> B \<subseteq> B"
by blast

lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
by blast

text {* \medskip Set difference. *}

lemma Diff_subset: "A - B \<subseteq> A"
by blast

lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
by blast

text {* \medskip Monotonicity. *}

lemma mono_Un: "mono f ==> f A \<union> f B \<subseteq> f (A \<union> B)"

lemma mono_Int: "mono f ==> f (A \<inter> B) \<subseteq> f A \<inter> f B"

subsubsection {* Equalities involving union, intersection, inclusion, etc. *}

text {* @{text "{}"}. *}

lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
-- {* supersedes @{text "Collect_False_empty"} *}
by auto

lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
by blast

lemma not_psubset_empty [iff]: "\<not> (A < {})"
by (unfold psubset_def) blast

lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
by blast

lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
by blast

lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
by blast

lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}"
by blast

lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}"
by blast

lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}"
by blast

lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
by blast

lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
by blast

lemma Collect_ex_eq: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
by blast

lemma Collect_bex_eq: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
by blast

text {* \medskip @{text insert}. *}

lemma insert_is_Un: "insert a A = {a} Un A"
-- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *}
by blast

lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
by blast

lemmas empty_not_insert = insert_not_empty [symmetric, standard]
declare empty_not_insert [simp]

lemma insert_absorb: "a \<in> A ==> insert a A = A"
-- {* @{text "[simp]"} causes recursive calls when there are nested inserts *}
-- {* with \emph{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 \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
by blast

lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B"
-- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *}
apply (rule_tac x = "A - {a}" in exI, blast)
done

lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
by auto

lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
by blast

lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
by blast

lemma insert_disjoint[simp]:
"(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
"({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
by auto

lemma disjoint_insert[simp]:
"(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
"({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
by auto

text {* \medskip @{text image}. *}

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

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

lemma image_constant: "x \<in> A ==> (\<lambda>x. c)  A = {c}"
by auto

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

lemma image_image: "f  (g  A) = (\<lambda>x. f (g x))  A"
by blast

lemma insert_image [simp]: "x \<in> A ==> insert (f x) (fA) = fA"
by blast

lemma image_is_empty [iff]: "(fA = {}) = (A = {})"
by blast

lemma image_Collect: "f  {x. P x} = {f x | x. P x}"
-- {* NOT suitable as a default simprule: 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]:
"(\<lambda>x. if P x then f x else g x)  S
= (f  (S \<inter> {x. P x})) \<union> (g  (S \<inter> {x. \<not> P x}))"

lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> fM = gN"

text {* \medskip @{text range}. *}

lemma full_SetCompr_eq: "{u. \<exists>x. u = f x} = range f"
by auto

lemma range_composition [simp]: "range (\<lambda>x. f (g x)) = frange g"
by (subst image_image, simp)

text {* \medskip @{text Int} *}

lemma Int_absorb [simp]: "A \<inter> A = A"
by blast

lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
by blast

lemma Int_commute: "A \<inter> B = B \<inter> A"
by blast

lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
by blast

lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
by blast

lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
-- {* Intersection is an AC-operator *}

lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"
by blast

lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
by blast

lemma Int_empty_left [simp]: "{} \<inter> B = {}"
by blast

lemma Int_empty_right [simp]: "A \<inter> {} = {}"
by blast

lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"
by blast

lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
by blast

lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"
by blast

lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"
by blast

lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
by blast

lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
by blast

lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
by blast

lemma Int_UNIV [simp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
by blast

lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
by blast

lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"
by blast

text {* \medskip @{text Un}. *}

lemma Un_absorb [simp]: "A \<union> A = A"
by blast

lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
by blast

lemma Un_commute: "A \<union> B = B \<union> A"
by blast

lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
by blast

lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
by blast

lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
-- {* Union is an AC-operator *}

lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"
by blast

lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
by blast

lemma Un_empty_left [simp]: "{} \<union> B = B"
by blast

lemma Un_empty_right [simp]: "A \<union> {} = A"
by blast

lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"
by blast

lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"
by blast

lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
by blast

lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
by blast

lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
by blast

lemma Int_insert_left:
"(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
by auto

lemma Int_insert_right:
"A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
by auto

lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
by blast

lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
by blast

lemma Un_Int_crazy:
"(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
by blast

lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"
by blast

lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"
by blast

lemma Un_subset_iff [simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"
by blast

lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"
by blast

lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - B"
by blast

text {* \medskip Set complement *}

lemma Compl_disjoint [simp]: "A \<inter> -A = {}"
by blast

lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"
by blast

lemma Compl_partition: "A \<union> -A = UNIV"
by blast

lemma Compl_partition2: "-A \<union> A = UNIV"
by blast

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

lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"
by blast

lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"
by blast

lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
by blast

lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
by blast

lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"
by blast

lemma Un_Int_assoc_eq: "((A \<inter> B) \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"
-- {* Halmos, Naive Set Theory, page 16. *}
by blast

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

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

lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
by blast

lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"
by blast

text {* \medskip @{text Union}. *}

lemma Union_empty [simp]: "Union({}) = {}"
by blast

lemma Union_UNIV [simp]: "Union UNIV = UNIV"
by blast

lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
by blast

lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
by blast

lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
by blast

lemma Union_empty_conv [simp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
by blast

lemma empty_Union_conv [simp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
by blast

lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
by blast

text {* \medskip @{text Inter}. *}

lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
by blast

lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
by blast

lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
by blast

lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
by blast

lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
by blast

lemma Inter_UNIV_conv [simp]:
"(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
"(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
by blast+

text {*
\medskip @{text UN} and @{text INT}.

Basic identities: *}

lemma UN_empty [simp]: "(\<Union>x\<in>{}. B x) = {}"
by blast

lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
by blast

lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
by blast

lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
by auto

lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
by blast

lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
by blast

lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
by blast

lemma UN_Un: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
by blast

lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)"
by blast

lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
by blast

lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
by blast

lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
by blast

lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
by blast

lemma INT_insert_distrib:
"u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
by blast

lemma Union_image_eq [simp]: "\<Union>(BA) = (\<Union>x\<in>A. B x)"
by blast

lemma image_Union: "f  \<Union>S = (\<Union>x\<in>S. f  x)"
by blast

lemma Inter_image_eq [simp]: "\<Inter>(BA) = (\<Inter>x\<in>A. B x)"
by blast

lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
by auto

lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
by auto

lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
by blast

lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
-- {* Look: it has an \emph{existential} quantifier *}
by blast

lemma UNION_empty_conv[simp]:
"({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
"((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
by blast+

lemma INTER_UNIV_conv[simp]:
"(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
"((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
by blast+

text {* \medskip Distributive laws: *}

lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
by blast

lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
by blast

lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(AC) \<union> \<Union>(BC)"
-- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
-- {* Union of a family of unions *}
by blast

lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"
-- {* Equivalent version *}
by blast

lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
by blast

lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(AC) \<inter> \<Inter>(BC)"
by blast

lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"
-- {* Equivalent version *}
by blast

lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
-- {* Halmos, Naive Set Theory, page 35. *}
by blast

lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
by blast

lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
by blast

lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"
by blast

text {* \medskip 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 @{text Int}. *}

lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"
by blast

lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"
by blast

lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
by blast

lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
by blast

text {* \medskip Set difference. *}

lemma Diff_eq: "A - B = A \<inter> (-B)"
by blast

lemma Diff_eq_empty_iff [simp]: "(A - B = {}) = (A \<subseteq> B)"
by blast

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

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

lemma Diff_triv: "A \<inter> 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 \<notin> A ==> A - insert x B = A - B"
by blast

lemma Diff_insert: "A - insert a B = A - B - {a}"
-- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
by blast

lemma Diff_insert2: "A - insert a B = A - {a} - B"
-- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a 0"} *}
by blast

lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
by auto

lemma insert_Diff1 [simp]: "x \<in> 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 \<in> A ==> insert a (A - {a}) = A"
by blast

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

lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"
by blast

lemma Diff_partition: "A \<subseteq> B ==> A \<union> (B - A) = B"
by blast

lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"
by blast

lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"
by blast

lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"
by blast

lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"
by blast

lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"
by blast

lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"
by blast

lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"
by blast

lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"
by blast

lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"
by blast

lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"
by auto

lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"
by blast

text {* \medskip Quantification over type @{typ bool}. *}

lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"
by (cases x) auto

lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False"
by (auto intro: bool_induct)

lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True"
by (cases x) auto

lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False"
by (auto intro: bool_contrapos)

lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"

lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
by (auto intro: bool_contrapos)

lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
by (auto intro: bool_induct)

text {* \medskip @{text Pow} *}

lemma Pow_empty [simp]: "Pow {} = {{}}"

lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a  Pow A)"
by (blast intro: image_eqI [where ?x = "u - {a}", standard])

lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"
by (blast intro: exI [where ?x = "- u", standard])

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

lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"
by blast

lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
by blast

lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
by blast

lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
by blast

lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"
by blast

lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
by blast

text {* \medskip Miscellany. *}

lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"
by blast

lemma subset_iff: "(A \<subseteq> B) = (\<forall>t. t \<in> A --> t \<in> B)"
by blast

lemma subset_iff_psubset_eq: "(A \<subseteq> B) = ((A \<subset> B) | (A = B))"
by (unfold psubset_def) blast

lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) = (A = {})"
by blast

lemma ex_in_conv: "(\<exists>x. x \<in> A) = (A \<noteq> {})"
by blast

lemma distinct_lemma: "f x \<noteq> f y ==> x \<noteq> y"
by iprover

text {* \medskip Miniscoping: pushing in quantifiers and big Unions
and Intersections. *}

lemma UN_simps [simp]:
"!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
"!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
"!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
"!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
"!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
"!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
"!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
"!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
"!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
"!!A B f. (UN x:fA. B x)     = (UN a:A. B (f a))"
by auto

lemma INT_simps [simp]:
"!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
"!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
"!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
"!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
"!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
"!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
"!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
"!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
"!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
"!!A B f. (INT x:fA. B x)    = (INT a:A. B (f a))"
by auto

lemma ball_simps [simp]:
"!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
"!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
"!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
"!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
"!!P. (ALL x:{}. P x) = True"
"!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
"!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
"!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
"!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
"!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
"!!A P f. (ALL x:fA. P x) = (ALL x:A. P (f x))"
"!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
by auto

lemma bex_simps [simp]:
"!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
"!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
"!!P. (EX x:{}. P x) = False"
"!!P. (EX x:UNIV. P x) = (EX x. P x)"
"!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
"!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
"!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
"!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
"!!A P f. (EX x:fA. P x) = (EX x:A. P (f x))"
"!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
by auto

lemma ball_conj_distrib:
"(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
by blast

lemma bex_disj_distrib:
"(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
by blast

text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}

lemma UN_extend_simps:
"!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
"!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
"!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
"!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
"!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
"!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
"!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
"!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
"!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
"!!A B f. (UN a:A. B (f a)) = (UN x:fA. B x)"
by auto

lemma INT_extend_simps:
"!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
"!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
"!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
"!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
"!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
"!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
"!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
"!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
"!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
"!!A B f. (INT a:A. B (f a))    = (INT x:fA. B x)"
by auto

subsubsection {* Monotonicity of various operations *}

lemma image_mono: "A \<subseteq> B ==> fA \<subseteq> fB"
by blast

lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"
by blast

lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
by blast

lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
by blast

lemma UN_mono:
"A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
(\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
by (blast dest: subsetD)

lemma INT_anti_mono:
"B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
(\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
-- {* The last inclusion is POSITIVE! *}
by (blast dest: subsetD)

lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"
by blast

lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"
by blast

lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"
by blast

lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"
by blast

lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"
by blast

text {* \medskip Monotonicity of implications. *}

lemma in_mono: "A \<subseteq> B ==> x \<in> A --> x \<in> B"
apply (rule impI)
apply (erule subsetD, assumption)
done

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 ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"
by iprover

lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"
by iprover

lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"
by blast

lemma Int_Collect_mono:
"A \<subseteq> B ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> 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

lemma eq_to_mono2: "a = b ==> c = d ==> ~ b --> ~ d ==> ~ a --> ~ c"
by iprover

lemma Least_mono:
"mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
==> (LEAST y. y : f  S) = f (LEAST x. x : S)"
-- {* Courtesy of Stephan Merz *}
apply clarify
apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)
apply (rule LeastI2_order)
apply (auto elim: monoD intro!: order_antisym)
done

subsection {* Inverse image of a function *}

constdefs
vimage :: "('a => 'b) => 'b set => 'a set"    (infixr "-" 90)
"f - B == {x. f x : B}"

subsubsection {* Basic rules *}

lemma vimage_eq [simp]: "(a : f - B) = (f a : B)"
by (unfold vimage_def) blast

lemma vimage_singleton_eq: "(a : f - {b}) = (f a = b)"
by simp

lemma vimageI [intro]: "f a = b ==> b:B ==> a : f - B"
by (unfold vimage_def) blast

lemma vimageI2: "f a : A ==> a : f - A"
by (unfold vimage_def) fast

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

lemma vimageD: "a : f - A ==> f a : A"
by (unfold vimage_def) fast

subsubsection {* Equations *}

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

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

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

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

lemma vimage_Union: "f - (Union A) = (UN X:A. f - X)"
by blast

lemma vimage_UN: "f-(UN x:A. B x) = (UN x:A. f - B x)"
by blast

lemma vimage_INT: "f-(INT x:A. B x) = (INT x:A. f - B x)"
by blast

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}) Un (f-B)"
-- {* NOT suitable for rewriting because of the recurrence of @{term "{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_eq_UN: "f-B = (UN y: B. f-{y})"
-- {* NOT suitable for rewriting *}
by blast

lemma vimage_mono: "A \<subseteq> B ==> f - A \<subseteq> f - B"
-- {* monotonicity *}
by blast

subsection {* Getting the Contents of a Singleton Set *}

constdefs
contents :: "'a set => 'a"
"contents X == THE x. X = {x}"

lemma contents_eq [simp]: "contents {x} = x"

subsection {* Transitivity rules for calculational reasoning *}

lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
by (rule subsetD)

lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
by (rule subsetD)

lemmas basic_trans_rules [trans] =
order_trans_rules set_rev_mp set_mp

subsection {* Sets as lattice *}

instance set :: (type) distrib_lattice
inf_set_eq: "inf A B \<equiv> A \<inter> B"
sup_set_eq: "sup A B \<equiv> A \<union> B"
by intro_classes (auto simp add: inf_set_eq sup_set_eq)

subsection {* Basic ML bindings *}

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}
val UnE = @{thm UnE}
val UnI1 = @{thm UnI1}
val UnI2 = @{thm UnI2}
val ballE = @{thm ballE}
val ballI = @{thm ballI}
val bexCI = @{thm bexCI}
val bexE = @{thm bexE}
val bexI = @{thm bexI}
val bex_triv = @{thm bex_triv}
val bspec = @{thm bspec}
val contra_subsetD = @{thm contra_subsetD}
val distinct_lemma = @{thm distinct_lemma}
val eq_to_mono = @{thm eq_to_mono}
val eq_to_mono2 = @{thm eq_to_mono2}
val equalityCE = @{thm equalityCE}
val equalityD1 = @{thm equalityD1}
val equalityD2 = @{thm equalityD2}
val equalityE = @{thm equalityE}
val equalityI = @{thm equalityI}
val imageE = @{thm imageE}
val imageI = @{thm imageI}
val image_Un = @{thm image_Un}
val image_insert = @{thm image_insert}
val insert_commute = @{thm insert_commute}
val insert_iff = @{thm insert_iff}
val mem_Collect_eq = @{thm mem_Collect_eq}
val rangeE = @{thm rangeE}
val rangeI = @{thm rangeI}
val range_eqI = @{thm range_eqI}
val subsetCE = @{thm subsetCE}
val subsetD = @{thm subsetD}
val subsetI = @{thm subsetI}
val subset_refl = @{thm subset_refl}
val subset_trans = @{thm subset_trans}
val vimageD = @{thm vimageD}
val vimageE = @{thm vimageE}
val vimageI = @{thm vimageI}
val vimageI2 = @{thm vimageI2}
val vimage_Collect = @{thm vimage_Collect}
val vimage_Int = @{thm vimage_Int}
val vimage_Un = @{thm vimage_Un}
*}

end
`