src/HOL/Set.thy

bot comes before top, inf before sup etc.

+
−(*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel *)
+
−
+
−header {* Set theory for higher-order logic *}
+
−
+
−theory Set
+
−imports Lattices
+
−begin
+
−
+
−subsection {* Sets as predicates *}
+
−
+
−types 'a set = "'a \<Rightarrow> bool"
+
−
+
−definition Collect :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set" where -- "comprehension"
+
−  "Collect P = P"
+
−
+
−definition member :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" where -- "membership"
+
−  mem_def: "member x A = A x"
+
−
+
−notation
+
−  member  ("op :") and
+
−  member  ("(_/ : _)" [50, 51] 50)
+
−
+
−abbreviation not_member where
+
−  "not_member x A \<equiv> ~ (x : A)" -- "non-membership"
+
−
+
−notation
+
−  not_member  ("op ~:") and
+
−  not_member  ("(_/ ~: _)" [50, 51] 50)
+
−
+
−notation (xsymbols)
+
−  member      ("op \<in>") and
+
−  member      ("(_/ \<in> _)" [50, 51] 50) and
+
−  not_member  ("op \<notin>") and
+
−  not_member  ("(_/ \<notin> _)" [50, 51] 50)
+
−
+
−notation (HTML output)
+
−  member      ("op \<in>") and
+
−  member      ("(_/ \<in> _)" [50, 51] 50) and
+
−  not_member  ("op \<notin>") and
+
−  not_member  ("(_/ \<notin> _)" [50, 51] 50)
+
−
+
−text {* Set comprehensions *}
+
−
+
−syntax
+
−  "_Coll" :: "pttrn => bool => 'a set"    ("(1{_./ _})")
+
−translations
+
−  "{x. P}" == "CONST Collect (%x. P)"
+
−
+
−syntax
+
−  "_Collect" :: "idt => 'a set => bool => 'a set"    ("(1{_ :/ _./ _})")
+
−syntax (xsymbols)
+
−  "_Collect" :: "idt => 'a set => bool => 'a set"    ("(1{_ \<in>/ _./ _})")
+
−translations
+
−  "{x:A. P}" => "{x. x:A & P}"
+
−
+
−lemma mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"
+
−  by (simp add: Collect_def mem_def)
+
−
+
−lemma Collect_mem_eq [simp]: "{x. x:A} = A"
+
−  by (simp add: Collect_def mem_def)
+
−
+
−lemma CollectI: "P(a) ==> a : {x. P(x)}"
+
−  by simp
+
−
+
−lemma CollectD: "a : {x. P(x)} ==> P(a)"
+
−  by simp
+
−
+
−lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
+
−  by simp
+
−
+
−text {*
+
−Simproc for pulling @{text "x=t"} in @{text "{x. \<dots> & x=t & \<dots>}"}
+
−to the front (and similarly for @{text "t=x"}):
+
−*}
+
−
+
−setup {*
+
−let
+
−  val Coll_perm_tac = rtac @{thm Collect_cong} 1 THEN rtac @{thm iffI} 1 THEN
+
−    ALLGOALS(EVERY'[REPEAT_DETERM o (etac @{thm conjE}),
+
−                    DEPTH_SOLVE_1 o (ares_tac [@{thm conjI}])])
+
−  val defColl_regroup = Simplifier.simproc_global @{theory}
+
−    "defined Collect" ["{x. P x & Q x}"]
+
−    (Quantifier1.rearrange_Coll Coll_perm_tac)
+
−in
+
−  Simplifier.map_simpset (fn ss => ss addsimprocs [defColl_regroup])
+
−end
+
−*}
+
−
+
−lemmas CollectE = CollectD [elim_format]
+
−
+
−text {* Set enumerations *}
+
−
+
−abbreviation empty :: "'a set" ("{}") where
+
−  "{} \<equiv> bot"
+
−
+
−definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where
+
−  insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
+
−
+
−syntax
+
−  "_Finset" :: "args => 'a set"    ("{(_)}")
+
−translations
+
−  "{x, xs}" == "CONST insert x {xs}"
+
−  "{x}" == "CONST insert x {}"
+
−
+
−
+
−subsection {* Subsets and bounded quantifiers *}
+
−
+
−abbreviation
+
−  subset :: "'a set \<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)
+
−
+
−definition Ball :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
+
−  "Ball A P \<longleftrightarrow> (\<forall>x. x \<in> A \<longrightarrow> P x)"   -- "bounded universal quantifiers"
+
−
+
−definition Bex :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
+
−  "Bex A P \<longleftrightarrow> (\<exists>x. x \<in> A \<and> P x)"   -- "bounded existential quantifiers"
+
−
+
−syntax
+
−  "_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)
+
−
+
−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)
+
−
+
−translations
+
−  "ALL x:A. P" == "CONST Ball A (%x. P)"
+
−  "EX x:A. P" == "CONST Bex A (%x. P)"
+
−  "EX! x:A. P" => "EX! x. x:A & P"
+
−  "LEAST x:A. P" => "LEAST x. x:A & P"
+
−
+
−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"};   (* FIXME 'a => bool (!?!) *)
+
−  val All_binder = Syntax.binder_name @{const_syntax All};
+
−  val Ex_binder = Syntax.binder_name @{const_syntax Ex};
+
−  val impl = @{const_syntax HOL.implies};
+
−  val conj = @{const_syntax HOL.conj};
+
−  val sbset = @{const_syntax subset};
+
−  val sbset_eq = @{const_syntax subset_eq};
+
−
+
−  val trans =
+
−   [((All_binder, impl, sbset), @{syntax_const "_setlessAll"}),
+
−    ((All_binder, impl, sbset_eq), @{syntax_const "_setleAll"}),
+
−    ((Ex_binder, conj, sbset), @{syntax_const "_setlessEx"}),
+
−    ((Ex_binder, conj, sbset_eq), @{syntax_const "_setleEx"})];
+
−
+
−  fun mk v v' 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 (@{syntax_const "_bound"}, _) $ Free (v, Type (T, _)),
+
−            Const (c, _) $
+
−              (Const (d, _) $ (Const (@{syntax_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)"}.
+
−*}
+
−
+
−syntax
+
−  "_Setcompr" :: "'a => idts => bool => 'a set"    ("(1{_ |/_./ _})")
+
−
+
−parse_translation {*
+
−  let
+
−    val ex_tr = snd (mk_binder_tr ("EX ", @{const_syntax Ex}));
+
−
+
−    fun nvars (Const (@{syntax_const "_idts"}, _) $ _ $ idts) = nvars idts + 1
+
−      | nvars _ = 1;
+
−
+
−    fun setcompr_tr [e, idts, b] =
+
−      let
+
−        val eq = Syntax.const @{const_syntax HOL.eq} $ Bound (nvars idts) $ e;
+
−        val P = Syntax.const @{const_syntax HOL.conj} $ eq $ b;
+
−        val exP = ex_tr [idts, P];
+
−      in Syntax.const @{const_syntax Collect} $ Term.absdummy (dummyT, exP) end;
+
−
+
−  in [(@{syntax_const "_Setcompr"}, setcompr_tr)] end;
+
−*}
+
−
+
−print_translation {*
+
− [Syntax.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"},
+
−  Syntax.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"}]
+
−*} -- {* to avoid eta-contraction of body *}
+
−
+
−print_translation {*
+
−let
+
−  val ex_tr' = snd (mk_binder_tr' (@{const_syntax Ex}, "DUMMY"));
+
−
+
−  fun setcompr_tr' [Abs (abs as (_, _, P))] =
+
−    let
+
−      fun check (Const (@{const_syntax Ex}, _) $ Abs (_, _, P), n) = check (P, n + 1)
+
−        | check (Const (@{const_syntax HOL.conj}, _) $
+
−              (Const (@{const_syntax HOL.eq}, _) $ Bound m $ e) $ P, n) =
+
−            n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
+
−            subset (op =) (0 upto (n - 1), add_loose_bnos (e, 0, []))
+
−        | check _ = false;
+
−
+
−        fun tr' (_ $ abs) =
+
−          let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
+
−          in Syntax.const @{syntax_const "_Setcompr"} $ e $ idts $ Q end;
+
−    in
+
−      if check (P, 0) then tr' P
+
−      else
+
−        let
+
−          val (x as _ $ Free(xN, _), t) = atomic_abs_tr' abs;
+
−          val M = Syntax.const @{syntax_const "_Coll"} $ x $ t;
+
−        in
+
−          case t of
+
−            Const (@{const_syntax HOL.conj}, _) $
+
−              (Const (@{const_syntax Set.member}, _) $
+
−                (Const (@{syntax_const "_bound"}, _) $ Free (yN, _)) $ A) $ P =>
+
−            if xN = yN then Syntax.const @{syntax_const "_Collect"} $ x $ A $ P else M
+
−          | _ => M
+
−        end
+
−    end;
+
−  in [(@{const_syntax Collect}, setcompr_tr')] end;
+
−*}
+
−
+
−setup {*
+
−let
+
−  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;
+
−  val defBEX_regroup = Simplifier.simproc_global @{theory}
+
−    "defined BEX" ["EX x:A. P x & Q x"] rearrange_bex;
+
−  val defBALL_regroup = Simplifier.simproc_global @{theory}
+
−    "defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball;
+
−in
+
−  Simplifier.map_simpset (fn ss => ss addsimprocs [defBALL_regroup, defBEX_regroup])
+
−end
+
−*}
+
−
+
−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)
+
−
+
−text {*
+
−  Gives better instantiation for bound:
+
−*}
+
−
+
−declaration {* fn _ =>
+
−  Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1))
+
−*}
+
−
+
−ML {*
+
−structure Simpdata =
+
−struct
+
−
+
−open Simpdata;
+
−
+
−val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;
+
−
+
−end;
+
−
+
−open Simpdata;
+
−*}
+
−
+
−declaration {* fn _ =>
+
−  Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs))
+
−*}
+
−
+
−lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
+
−  by (unfold Ball_def) blast
+
−
+
−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
+
−
+
−
+
−text {* Congruence rules *}
+
−
+
−lemma ball_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 strong_ball_cong [cong]:
+
−  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
+
−    (ALL x:A. P x) = (ALL x:B. Q x)"
+
−  by (simp add: simp_implies_def Ball_def)
+
−
+
−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)
+
−
+
−
+
−subsection {* Basic operations *}
+
−
+
−subsubsection {* Subsets *}
+
−
+
−lemma subsetI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B) \<Longrightarrow> A \<subseteq> B"
+
−  unfolding mem_def by (rule le_funI, rule le_boolI)
+
−
+
−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, intro?]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
+
−  unfolding mem_def by (erule le_funE, erule le_boolE)
+
−  -- {* Rule in Modus Ponens style. *}
+
−
+
−lemma rev_subsetD [no_atp,intro?]: "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)
+
−
+
−text {*
+
−  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
+
−*}
+
−
+
−lemma subsetCE [no_atp,elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
+
−  -- {* Classical elimination rule. *}
+
−  unfolding mem_def by (blast dest: le_funE le_boolE)
+
−
+
−lemma subset_eq [no_atp]: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast
+
−
+
−lemma contra_subsetD [no_atp]: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
+
−  by blast
+
−
+
−lemma subset_refl [simp]: "A \<subseteq> A"
+
−  by (fact order_refl)
+
−
+
−lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C"
+
−  by (fact order_trans)
+
−
+
−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)
+
−
+
−lemma eq_mem_trans: "a=b ==> b \<in> A ==> a \<in> A"
+
−  by simp
+
−
+
−lemmas basic_trans_rules [trans] =
+
−  order_trans_rules set_rev_mp set_mp eq_mem_trans
+
−
+
−
+
−subsubsection {* Equality *}
+
−
+
−lemma set_eqI: 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 set_eq_iff [no_atp]: "(A = B) = (ALL x. (x:A) = (x:B))"
+
−by(auto intro:set_eqI)
+
−
+
−lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
+
−  -- {* Anti-symmetry of the subset relation. *}
+
−  by (iprover intro: set_eqI subsetD)
+
−
+
−text {*
+
−  \medskip Equality rules from ZF set theory -- are they appropriate
+
−  here?
+
−*}
+
−
+
−lemma equalityD1: "A = B ==> A \<subseteq> B"
+
−  by simp
+
−
+
−lemma equalityD2: "A = B ==> B \<subseteq> A"
+
−  by simp
+
−
+
−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
+
−
+
−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 empty set *}
+
−
+
−lemma empty_def:
+
−  "{} = {x. False}"
+
−  by (simp add: bot_fun_def bot_bool_def Collect_def)
+
−
+
−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: @{text "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)
+
−
+
−
+
−subsubsection {* The universal set -- UNIV *}
+
−
+
−abbreviation UNIV :: "'a set" where
+
−  "UNIV \<equiv> top"
+
−
+
−lemma UNIV_def:
+
−  "UNIV = {x. True}"
+
−  by (simp add: top_fun_def top_bool_def Collect_def)
+
−
+
−lemma UNIV_I [simp]: "x : UNIV"
+
−  by (simp add: UNIV_def)
+
−
+
−declare UNIV_I [intro]  -- {* unsafe makes it less likely to cause problems *}
+
−
+
−lemma UNIV_witness [intro?]: "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"
+
−  by (simp add: Ball_def)
+
−
+
−lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
+
−  by (simp add: Bex_def)
+
−
+
−lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"
+
−  by auto
+
−
+
−lemma UNIV_not_empty [iff]: "UNIV ~= {}"
+
−  by (blast elim: equalityE)
+
−
+
−
+
−subsubsection {* The Powerset operator -- Pow *}
+
−
+
−definition Pow :: "'a set => 'a set set" where
+
−  Pow_def: "Pow A = {B. B \<le> A}"
+
−
+
−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
+
−
+
−lemma Pow_not_empty: "Pow A \<noteq> {}"
+
−  using Pow_top by blast
+
−
+
−
+
−subsubsection {* Set complement *}
+
−
+
−lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
+
−  by (simp add: mem_def fun_Compl_def bool_Compl_def)
+
−
+
−lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
+
−  by (unfold mem_def fun_Compl_def bool_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 (simp add: mem_def fun_Compl_def bool_Compl_def)
+
−
+
−lemmas ComplE = ComplD [elim_format]
+
−
+
−lemma Compl_eq: "- A = {x. ~ x : A}" by blast
+
−
+
−
+
−subsubsection {* Binary intersection *}
+
−
+
−abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
+
−  "op Int \<equiv> inf"
+
−
+
−notation (xsymbols)
+
−  inter  (infixl "\<inter>" 70)
+
−
+
−notation (HTML output)
+
−  inter  (infixl "\<inter>" 70)
+
−
+
−lemma Int_def:
+
−  "A \<inter> B = {x. x \<in> A \<and> x \<in> B}"
+
−  by (simp add: inf_fun_def inf_bool_def Collect_def mem_def)
+
−
+
−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
+
−
+
−lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
+
−  by (fact mono_inf)
+
−
+
−
+
−subsubsection {* Binary union *}
+
−
+
−abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where
+
−  "union \<equiv> sup"
+
−
+
−notation (xsymbols)
+
−  union  (infixl "\<union>" 65)
+
−
+
−notation (HTML output)
+
−  union  (infixl "\<union>" 65)
+
−
+
−lemma Un_def:
+
−  "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
+
−  by (simp add: sup_fun_def sup_bool_def Collect_def mem_def)
+
−
+
−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
+
−
+
−lemma insert_def: "insert a B = {x. x = a} \<union> B"
+
−  by (simp add: Collect_def mem_def insert_compr Un_def)
+
−
+
−lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
+
−  by (fact mono_sup)
+
−
+
−
+
−subsubsection {* Set difference *}
+
−
+
−lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
+
−  by (simp add: mem_def fun_diff_def bool_diff_def)
+
−
+
−lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
+
−  by simp
+
−
+
−lemma DiffD1: "c : A - B ==> c : A"
+
−  by simp
+
−
+
−lemma DiffD2: "c : A - B ==> c : B ==> P"
+
−  by simp
+
−
+
−lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
+
−  by simp
+
−
+
−lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast
+
−
+
−lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"
+
−by blast
+
−
+
−
+
−subsubsection {* Augmenting a set -- @{const 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
+
−
+
−lemma set_insert:
+
−  assumes "x \<in> A"
+
−  obtains B where "A = insert x B" and "x \<notin> B"
+
−proof
+
−  from assms show "A = insert x (A - {x})" by blast
+
−next
+
−  show "x \<notin> A - {x}" by blast
+
−qed
+
−
+
−lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)"
+
−by auto
+
−
+
−subsubsection {* Singletons, using insert *}
+
−
+
−lemma singletonI [intro!,no_atp]: "a : {a}"
+
−    -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
+
−  by (rule insertI1)
+
−
+
−lemma singletonD [dest!,no_atp]: "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,no_atp]:
+
−     "({b} = insert a A) = (a = b & A \<subseteq> {b})"
+
−  by blast
+
−
+
−lemma singleton_insert_inj_eq' [iff,no_atp]:
+
−     "(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 {* Image of a set under a function *}
+
−
+
−text {*
+
−  Frequently @{term b} does not have the syntactic form of @{term "f x"}.
+
−*}
+
−
+
−definition image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) where
+
−  image_def [no_atp]: "f ` A = {y. EX x:A. y = f(x)}"
+
−
+
−abbreviation
+
−  range :: "('a => 'b) => 'b set" where -- "of function"
+
−  "range f == f ` UNIV"
+
−
+
−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 [no_atp]: "(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 {* Some rules with @{text "if"} *}
+
−
+
−text{* Elimination of @{text"{x. \<dots> & x=t & \<dots>}"}. *}
+
−
+
−lemma Collect_conv_if: "{x. x=a & P x} = (if P a then {a} else {})"
+
−  by auto
+
−
+
−lemma Collect_conv_if2: "{x. a=x & P x} = (if P a then {a} else {})"
+
−  by auto
+
−
+
−text {*
+
−  Rewrite rules for boolean case-splitting: faster than @{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 [where P="%S. a : S"])
+
−
+
−lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
+
−
+
−(*Would like to add these, but the existing code only searches for the
+
−  outer-level constant, which in this case is just Set.member; we instead need
+
−  to use term-nets to associate patterns with rules.  Also, if a rule fails to
+
−  apply, then the formula should be kept.
+
−  [("uminus", Compl_iff RS iffD1), ("minus", [Diff_iff RS iffD1]),
+
−   ("Int", [IntD1,IntD2]),
+
−   ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
+
− *)
+
−
+
−
+
−subsection {* Further operations and lemmas *}
+
−
+
−subsubsection {* The ``proper subset'' relation *}
+
−
+
−lemma psubsetI [intro!,no_atp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
+
−  by (unfold less_le) blast
+
−
+
−lemma psubsetE [elim!,no_atp]: 
+
−    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
+
−  by (unfold less_le) 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: less_le subset_insert_iff)
+
−
+
−lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)"
+
−  by (simp only: less_le)
+
−
+
−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 less_le)
+
−apply (auto dest: subset_antisym)
+
−done
+
−
+
−lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
+
−apply (unfold less_le)
+
−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 less_le) 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
+
−
+
−lemma image_Pow_mono:
+
−  assumes "f ` A \<le> B"
+
−  shows "(image f) ` (Pow A) \<le> Pow B"
+
−using assms by blast
+
−
+
−lemma image_Pow_surj:
+
−  assumes "f ` A = B"
+
−  shows "(image f) ` (Pow A) = Pow B"
+
−using assms unfolding Pow_def proof(auto)
+
−  fix Y assume *: "Y \<le> f ` A"
+
−  obtain X where X_def: "X = {x \<in> A. f x \<in> Y}" by blast
+
−  have "f ` X = Y \<and> X \<le> A" unfolding X_def using * by auto
+
−  thus "Y \<in> (image f) ` {X. X \<le> A}" by blast
+
−qed
+
−
+
−subsubsection {* Derived rules involving subsets. *}
+
−
+
−text {* @{text insert}. *}
+
−
+
−lemma subset_insertI: "B \<subseteq> insert a B"
+
−  by (rule subsetI) (erule insertI2)
+
−
+
−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 Finite Union -- the least upper bound of two sets. *}
+
−
+
−lemma Un_upper1: "A \<subseteq> A \<union> B"
+
−  by (fact sup_ge1)
+
−
+
−lemma Un_upper2: "B \<subseteq> A \<union> B"
+
−  by (fact sup_ge2)
+
−
+
−lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
+
−  by (fact sup_least)
+
−
+
−
+
−text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
+
−
+
−lemma Int_lower1: "A \<inter> B \<subseteq> A"
+
−  by (fact inf_le1)
+
−
+
−lemma Int_lower2: "A \<inter> B \<subseteq> B"
+
−  by (fact inf_le2)
+
−
+
−lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
+
−  by (fact inf_greatest)
+
−
+
−
+
−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
+
−
+
−
+
−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 less_le) 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
+
−
+
−
+
−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 insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
+
−  by blast
+
−
+
−lemma insert_disjoint [simp,no_atp]:
+
− "(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,no_atp]:
+
− "(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) (f`B)"
+
−  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) (f`A) = f`A"
+
−by blast
+
−
+
−lemma image_is_empty [iff]: "(f`A = {}) = (A = {})"
+
−by blast
+
−
+
−lemma empty_is_image[iff]: "({} = f ` A) = (A = {})"
+
−by blast
+
−
+
−
+
−lemma image_Collect [no_atp]: "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}))"
+
−  by (auto simp add: image_def)
+
−
+
−lemma image_cong: "M = N ==> (!!x. x \<in> N ==> f x = g x) ==> f`M = g`N"
+
−  by (simp add: image_def)
+
−
+
−
+
−text {* \medskip @{text range}. *}
+
−
+
−lemma full_SetCompr_eq [no_atp]: "{u. \<exists>x. u = f x} = range f"
+
−  by auto
+
−
+
−lemma range_composition: "range (\<lambda>x. f (g x)) = f`range g"
+
−by (subst image_image, simp)
+
−
+
−
+
−text {* \medskip @{text Int} *}
+
−
+
−lemma Int_absorb [simp]: "A \<inter> A = A"
+
−  by (fact inf_idem)
+
−
+
−lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
+
−  by (fact inf_left_idem)
+
−
+
−lemma Int_commute: "A \<inter> B = B \<inter> A"
+
−  by (fact inf_commute)
+
−
+
−lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
+
−  by (fact inf_left_commute)
+
−
+
−lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
+
−  by (fact inf_assoc)
+
−
+
−lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
+
−  -- {* Intersection is an AC-operator *}
+
−
+
−lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"
+
−  by (fact inf_absorb2)
+
−
+
−lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
+
−  by (fact inf_absorb1)
+
−
+
−lemma Int_empty_left [simp]: "{} \<inter> B = {}"
+
−  by (fact inf_bot_left)
+
−
+
−lemma Int_empty_right [simp]: "A \<inter> {} = {}"
+
−  by (fact inf_bot_right)
+
−
+
−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 (fact inf_top_left)
+
−
+
−lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"
+
−  by (fact inf_top_right)
+
−
+
−lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
+
−  by (fact inf_sup_distrib1)
+
−
+
−lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
+
−  by (fact inf_sup_distrib2)
+
−
+
−lemma Int_UNIV [simp,no_atp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
+
−  by (fact inf_eq_top_iff)
+
−
+
−lemma Int_subset_iff [no_atp, simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
+
−  by (fact le_inf_iff)
+
−
+
−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 (fact sup_idem)
+
−
+
−lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
+
−  by (fact sup_left_idem)
+
−
+
−lemma Un_commute: "A \<union> B = B \<union> A"
+
−  by (fact sup_commute)
+
−
+
−lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
+
−  by (fact sup_left_commute)
+
−
+
−lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
+
−  by (fact sup_assoc)
+
−
+
−lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
+
−  -- {* Union is an AC-operator *}
+
−
+
−lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"
+
−  by (fact sup_absorb2)
+
−
+
−lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
+
−  by (fact sup_absorb1)
+
−
+
−lemma Un_empty_left [simp]: "{} \<union> B = B"
+
−  by (fact sup_bot_left)
+
−
+
−lemma Un_empty_right [simp]: "A \<union> {} = A"
+
−  by (fact sup_bot_right)
+
−
+
−lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"
+
−  by (fact sup_top_left)
+
−
+
−lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"
+
−  by (fact sup_top_right)
+
−
+
−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_left_if0[simp]:
+
−    "a \<notin> C \<Longrightarrow> (insert a B) Int C = B \<inter> C"
+
−  by auto
+
−
+
−lemma Int_insert_left_if1[simp]:
+
−    "a \<in> C \<Longrightarrow> (insert a B) Int C = insert a (B Int 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 Int_insert_right_if0[simp]:
+
−    "a \<notin> A \<Longrightarrow> A Int (insert a B) = A Int B"
+
−  by auto
+
−
+
−lemma Int_insert_right_if1[simp]:
+
−    "a \<in> A \<Longrightarrow> A Int (insert a B) = insert a (A Int B)"
+
−  by auto
+
−
+
−lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
+
−  by (fact sup_inf_distrib1)
+
−
+
−lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
+
−  by (fact sup_inf_distrib2)
+
−
+
−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 (fact le_iff_sup)
+
−
+
−lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"
+
−  by (fact sup_eq_bot_iff)
+
−
+
−lemma Un_subset_iff [no_atp, simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"
+
−  by (fact le_sup_iff)
+
−
+
−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 (fact inf_compl_bot)
+
−
+
−lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"
+
−  by (fact compl_inf_bot)
+
−
+
−lemma Compl_partition: "A \<union> -A = UNIV"
+
−  by (fact sup_compl_top)
+
−
+
−lemma Compl_partition2: "-A \<union> A = UNIV"
+
−  by (fact compl_sup_top)
+
−
+
−lemma double_complement [simp]: "- (-A) = (A::'a set)"
+
−  by (fact double_compl)
+
−
+
−lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"
+
−  by (fact compl_sup)
+
−
+
−lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"
+
−  by (fact compl_inf)
+
−
+
−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 (fact compl_top_eq)
+
−
+
−lemma Compl_empty_eq [simp]: "-{} = UNIV"
+
−  by (fact compl_bot_eq)
+
−
+
−lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
+
−  by (fact compl_le_compl_iff)
+
−
+
−lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"
+
−  by (fact compl_eq_compl_iff)
+
−
+
−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
+
−
+
−
+
−text {* \medskip Set difference. *}
+
−
+
−lemma Diff_eq: "A - B = A \<inter> (-B)"
+
−  by blast
+
−
+
−lemma Diff_eq_empty_iff [simp,no_atp]: "(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,no_atp]: "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)
+
−
+
−text {* \medskip @{text Pow} *}
+
−
+
−lemma Pow_empty [simp]: "Pow {} = {{}}"
+
−  by (auto simp add: Pow_def)
+
−
+
−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 Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"
+
−  by blast
+
−
+
−
+
−text {* \medskip Miscellany. *}
+
−
+
−lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"
+
−  by blast
+
−
+
−lemma subset_iff [no_atp]: "(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 less_le) 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
+
−
+
−
+
−subsubsection {* Monotonicity of various operations *}
+
−
+
−lemma image_mono: "A \<subseteq> B ==> f`A \<subseteq> f`B"
+
−  by blast
+
−
+
−lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"
+
−  by blast
+
−
+
−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 (fact sup_mono)
+
−
+
−lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"
+
−  by (fact inf_mono)
+
−
+
−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 (fact compl_mono)
+
−
+
−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 not_mono: "Q --> P ==> ~ P --> ~ Q"
+
−  by iprover
+
−
+
−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
+
−
+
−
+
−subsubsection {* Inverse image of a function *}
+
−
+
−definition vimage :: "('a => 'b) => 'b set => 'a set" (infixr "-`" 90) where
+
−  "f -` B == {x. f x : B}"
+
−
+
−lemma vimage_eq [simp]: "(a : f -` B) = (f a : B)"
+
−  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
+
−
+
−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_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_mono: "A \<subseteq> B ==> f -` A \<subseteq> f -` B"
+
−  -- {* monotonicity *}
+
−  by blast
+
−
+
−lemma vimage_image_eq [no_atp]: "f -` (f ` A) = {y. EX x:A. f x = f y}"
+
−by (blast intro: sym)
+
−
+
−lemma image_vimage_subset: "f ` (f -` A) <= A"
+
−by blast
+
−
+
−lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int range f"
+
−by blast
+
−
+
−lemma vimage_const [simp]: "((\<lambda>x. c) -` A) = (if c \<in> A then UNIV else {})"
+
−  by auto
+
−
+
−lemma vimage_if [simp]: "((\<lambda>x. if x \<in> B then c else d) -` A) = 
+
−   (if c \<in> A then (if d \<in> A then UNIV else B)
+
−    else if d \<in> A then -B else {})"  
+
−  by (auto simp add: vimage_def) 
+
−
+
−lemma vimage_inter_cong:
+
−  "(\<And> w. w \<in> S \<Longrightarrow> f w = g w) \<Longrightarrow> f -` y \<inter> S = g -` y \<inter> S"
+
−  by auto
+
−
+
−lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B"
+
−by blast
+
−
+
−lemma image_diff_subset: "f`A - f`B <= f`(A - B)"
+
−by blast
+
−
+
−
+
−subsubsection {* Getting the Contents of a Singleton Set *}
+
−
+
−definition the_elem :: "'a set \<Rightarrow> 'a" where
+
−  "the_elem X = (THE x. X = {x})"
+
−
+
−lemma the_elem_eq [simp]: "the_elem {x} = x"
+
−  by (simp add: the_elem_def)
+
−
+
−
+
−subsubsection {* Least value operator *}
+
−
+
−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 {* Misc *}
+
−
+
−text {* Rudimentary code generation *}
+
−
+
−lemma insert_code [code]: "insert y A x \<longleftrightarrow> y = x \<or> A x"
+
−  by (auto simp add: insert_compr Collect_def mem_def)
+
−
+
−lemma vimage_code [code]: "(f -` A) x = A (f x)"
+
−  by (simp add: vimage_def Collect_def mem_def)
+
−
+
−hide_const (open) member
+
−
+
−text {* Misc theorem and ML bindings *}
+
−
+
−lemmas equalityI = subset_antisym
+
−
+
−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 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