(*  Title:      HOL/Set.thy

    ID:         $Id$

    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel

    License:    GPL (GNU GENERAL PUBLIC LICENSE)

*)

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

theory Set = HOL:

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"

  image         :: "('a => 'b) => 'a set => 'b set"      (infixr "`" 90)

syntax

  "op :"        :: "'a => 'a set => bool"                ("op :")

consts

  "op :"        :: "'a => 'a set => bool"                ("(_/ : _)" [50, 51] 50)  -- "membership"

local

instance set :: (type) ord ..

instance set :: (type) minus ..

subsection {* Additional concrete syntax *}

syntax

  range         :: "('a => 'b) => 'b set"             -- "of function"

  "op ~:"       :: "'a => 'a set => bool"                 ("op ~:")  -- "non-membership"

  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ ~: _)" [50, 51] 50)

  "@Finset"     :: "args => 'a set"                       ("{(_)}")

  "@Coll"       :: "pttrn => bool => 'a set"              ("(1{_./ _})")

  "@SetCompr"   :: "'a => idts => bool => 'a set"         ("(1{_ |/_./ _})")

  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" 10)

  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" 10)

  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" 10)

  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" 10)

  "_Ball"       :: "pttrn => 'a set => bool => bool"      ("(3ALL _:_./ _)" [0, 0, 10] 10)

  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3EX _:_./ _)" [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)

translations

  "range f"     == "f`UNIV"

  "x ~: y"      == "~ (x : y)"

  "{x, xs}"     == "insert x {xs}"

  "{x}"         == "insert x {}"

  "{x. P}"      == "Collect (%x. 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)"

syntax (output)

  "_setle"      :: "'a set => 'a set => bool"             ("op <=")

  "_setle"      :: "'a set => 'a set => bool"             ("(_/ <= _)" [50, 51] 50)

  "_setless"    :: "'a set => 'a set => bool"             ("op <")

  "_setless"    :: "'a set => 'a set => bool"             ("(_/ < _)" [50, 51] 50)

syntax (xsymbols)

  "_setle"      :: "'a set => 'a set => bool"             ("op \<subseteq>")

  "_setle"      :: "'a set => 'a set => bool"             ("(_/ \<subseteq> _)" [50, 51] 50)

  "_setless"    :: "'a set => 'a set => bool"             ("op \<subset>")

  "_setless"    :: "'a set => 'a set => bool"             ("(_/ \<subset> _)" [50, 51] 50)

  "op Int"      :: "'a set => 'a set => 'a set"           (infixl "\<inter>" 70)

  "op Un"       :: "'a set => 'a set => 'a set"           (infixl "\<union>" 65)

  "op :"        :: "'a => 'a set => bool"                 ("op \<in>")

  "op :"        :: "'a => 'a set => bool"                 ("(_/ \<in> _)" [50, 51] 50)

  "op ~:"       :: "'a => 'a set => bool"                 ("op \<notin>")

  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ \<notin> _)" [50, 51] 50)

  Union         :: "'a set set => 'a set"                 ("\<Union>_" [90] 90)

  Inter         :: "'a set set => 'a set"                 ("\<Inter>_" [90] 90)

  "_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)

syntax (HTML output)

  "_setle"      :: "'a set => 'a set => bool"             ("op \<subseteq>")

  "_setle"      :: "'a set => 'a set => bool"             ("(_/ \<subseteq> _)" [50, 51] 50)

  "_setless"    :: "'a set => 'a set => bool"             ("op \<subset>")

  "_setless"    :: "'a set => 'a set => bool"             ("(_/ \<subset> _)" [50, 51] 50)

  "op Int"      :: "'a set => 'a set => 'a set"           (infixl "\<inter>" 70)

  "op Un"       :: "'a set => 'a set => 'a set"           (infixl "\<union>" 65)

  "op :"        :: "'a => 'a set => bool"                 ("op \<in>")

  "op :"        :: "'a => 'a set => bool"                 ("(_/ \<in> _)" [50, 51] 50)

  "op ~:"       :: "'a => 'a set => bool"                 ("op \<notin>")

  "op ~:"       :: "'a => 'a set => bool"                 ("(_/ \<notin> _)" [50, 51] 50)

  "_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)

syntax (input)

  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" 10)

  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" 10)

  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" 10)

  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" 10)

syntax (xsymbols)

  "@UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>\<^bsub>_\<^esub>/ _)" 10)

  "@INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>\<^bsub>_\<^esub>/ _)" 10)

  "@UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>\<^bsub>_\<in>_\<^esub>/ _)" 10)

  "@INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>\<^bsub>_\<in>_\<^esub>/ _)" 10)

translations

  "op \<subseteq>" => "op <= :: _ set => _ set => bool"

  "op \<subset>" => "op <  :: _ set => _ set => bool"

typed_print_translation {*

  let

    fun le_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts =

          list_comb (Syntax.const "_setle", ts)

      | le_tr' _ _ _ = raise Match;

    fun less_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts =

          list_comb (Syntax.const "_setless", ts)

      | less_tr' _ _ _ = raise Match;

  in [("op <=", le_tr'), ("op <", less_tr')] 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" $ Abs ("", 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,t) = atomic_abs_tr' abs

            in Syntax.const "@Coll" $ x $ t end

    end;

  in [("Collect", setcompr_tr')] end;

*}

subsection {* Rules and definitions *}

text {* Isomorphisms between predicates and sets. *}

axioms

  mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"

  Collect_mem_eq [simp]: "{x. x:A} = A"

defs

  Ball_def:     "Ball A P       == ALL x. x:A --> P(x)"

  Bex_def:      "Bex A P        == EX x. x:A & P(x)"

defs (overloaded)

  subset_def:   "A <= B         == ALL x:A. x:B"

  psubset_def:  "A < B          == (A::'a set) <= B & ~ A=B"

  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:    "f`A            == {y. EX x:A. y = f(x)}"

subsection {* Lemmas and proof tool setup *}

subsubsection {* Relating predicates and sets *}

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"

  by (simp add: Ball_def)

lemmas strip = impI allI ballI

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

  by (simp add: Ball_def)

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 {*

  local val ballE = thm "ballE"

  in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end;

*}

text {*

  Gives better instantiation for bound:

*}

ML_setup {*

  claset_ref() := claset() 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. *}

  by (simp add: Ball_def)

lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"

  -- {* Dual form for existentials. *}

  by (simp add: Bex_def)

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 Ball_def = thm "Ball_def";

    val Bex_def = thm "Bex_def";

    val prove_bex_tac =

      rewrite_goals_tac [Bex_def] THEN Quantifier1.prove_one_point_ex_tac;

    val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;

    val prove_ball_tac =

      rewrite_goals_tac [Ball_def] THEN Quantifier1.prove_one_point_all_tac;

    val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;

  in

    val defBEX_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))

      "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;

    val defBALL_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))

      "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;

  end;

  Addsimprocs [defBALL_regroup, defBEX_regroup];

*}

subsubsection {* Congruence rules *}

lemma ball_cong [cong]:

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

    (ALL x:A. P x) = (ALL x:B. Q x)"

  by (simp add: Ball_def)

lemma bex_cong [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)

subsubsection {* Subsets *}

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

  by (simp add: subset_def)

text {*

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

  'a}; for overloading constants whose first argument has type @{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 {*

  local val rev_subsetD = thm "rev_subsetD"

  in fun impOfSubs th = th RSN (2, rev_subsetD) end;

*}

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 {*

  local val subsetCE = thm "subsetCE"

  in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end;

*}

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

  by blast

lemma subset_refl: "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

lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"

  -- {* Anti-symmetry of the subset relation. *}

  by (rules 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"

  by (simp add: subset_refl)

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

  by (simp add: subset_refl)

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"

  by (simp add: subset_refl)

lemma equalityCE [elim]:

    "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"

  by blast

text {*

  \medskip Lemma for creating induction formulae -- for "pattern

  matching" on @{text p}.  To make the induction hypotheses usable,

  apply @{text spec} or @{text bspec} to put universal quantifiers over the free

  variables in @{text p}.

*}

lemma setup_induction: "p:A ==> (!!z. z:A ==> p = z --> R) ==> R"

  by simp

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"

  by (simp add: UNIV_def)

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: "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"

  by (simp add: Ball_def)

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

  by (simp add: Bex_def)

subsubsection {* The empty set *}

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

  by (simp add: empty_def)

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"

  by (simp add: Ball_def)

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

  by (simp add: Bex_def)

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)"

  by (simp add: Pow_def)

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

  by (simp add: Pow_def)

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

  by (simp add: Pow_def)

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

  by simp

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

  by (simp add: subset_refl)

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: "c : -A ==> c~:A"

  by (unfold Compl_def) blast

lemmas ComplE [elim!] = 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: "b : {a} ==> b = a"

  by blast

lemmas singletonE [elim!] = 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

subsubsection {* Unions of families *}

text {*

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

*}

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)"

  by (simp add: UNION_def)

subsubsection {* Intersections of families *}

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

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)"

  by (simp add: INTER_def)

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"

  by (simp add: Inter_def)

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 : f`A"

  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 : f`A"

  -- {* 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) = f`A Un f`B"

  by blast

lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"

  by blast

lemma image_subset_iff: "(f`A \<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> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)"

  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) ==> f`A \<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.

  [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]),

   ("op Int", [IntD1,IntD2]),

   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]

 *)

ML_setup {*

  val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs;

  simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);

*}

declare subset_UNIV [simp] subset_refl [simp]

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"

  by (simp add: psubset_eq)

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"

  by (auto simp add: psubset_eq)

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

  by (auto simp add: psubset_eq)

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)

declare atomize_ball [symmetric, rulify]

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 (rules intro: subsetI UnionI)

lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"

  by (rules 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 (rules 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 (rules 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 (rules 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