(* 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 *}
typedecl 'a set
axiomatization Collect :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set" -- "comprehension"
and member :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" -- "membership"
where
mem_Collect_eq [iff, code_unfold]: "member a (Collect P) = P a"
and Collect_mem_eq [simp]: "Collect (\<lambda>x. member x A) = A"
notation
member ("op :") and
member ("(_/ : _)" [51, 51] 50)
abbreviation not_member where
"not_member x A \<equiv> ~ (x : A)" -- "non-membership"
notation
not_member ("op ~:") and
not_member ("(_/ ~: _)" [51, 51] 50)
notation (xsymbols)
member ("op \<in>") and
member ("(_/ \<in> _)" [51, 51] 50) and
not_member ("op \<notin>") and
not_member ("(_/ \<notin> _)" [51, 51] 50)
notation (HTML output)
member ("op \<in>") and
member ("(_/ \<in> _)" [51, 51] 50) and
not_member ("op \<notin>") and
not_member ("(_/ \<notin> _)" [51, 51] 50)
text {* Set comprehensions *}
syntax
"_Coll" :: "pttrn => bool => 'a set" ("(1{_./ _})")
translations
"{x. P}" == "CONST Collect (%x. P)"
syntax
"_Collect" :: "pttrn => 'a set => bool => 'a set" ("(1{_ :/ _./ _})")
syntax (xsymbols)
"_Collect" :: "pttrn => 'a set => bool => 'a set" ("(1{_ \<in>/ _./ _})")
translations
"{p:A. P}" => "CONST Collect (%p. p:A & P)"
lemma CollectI: "P a \<Longrightarrow> a \<in> {x. P x}"
by simp
lemma CollectD: "a \<in> {x. P x} \<Longrightarrow> P a"
by simp
lemma Collect_cong: "(\<And>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"}):
*}
simproc_setup defined_Collect ("{x. P x & Q x}") = {*
fn _ => Quantifier1.rearrange_Collect
(fn _ =>
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 @{thms conjI}]))
*}
lemmas CollectE = CollectD [elim_format]
lemma set_eqI:
assumes "\<And>x. x \<in> A \<longleftrightarrow> x \<in> B"
shows "A = B"
proof -
from assms have "{x. x \<in> A} = {x. x \<in> B}" by simp
then show ?thesis by simp
qed
lemma set_eq_iff:
"A = B \<longleftrightarrow> (\<forall>x. x \<in> A \<longleftrightarrow> x \<in> B)"
by (auto intro:set_eqI)
text {* Lifting of predicate class instances *}
instantiation set :: (type) boolean_algebra
begin
definition less_eq_set where
"A \<le> B \<longleftrightarrow> (\<lambda>x. member x A) \<le> (\<lambda>x. member x B)"
definition less_set where
"A < B \<longleftrightarrow> (\<lambda>x. member x A) < (\<lambda>x. member x B)"
definition inf_set where
"A \<sqinter> B = Collect ((\<lambda>x. member x A) \<sqinter> (\<lambda>x. member x B))"
definition sup_set where
"A \<squnion> B = Collect ((\<lambda>x. member x A) \<squnion> (\<lambda>x. member x B))"
definition bot_set where
"\<bottom> = Collect \<bottom>"
definition top_set where
"\<top> = Collect \<top>"
definition uminus_set where
"- A = Collect (- (\<lambda>x. member x A))"
definition minus_set where
"A - B = Collect ((\<lambda>x. member x A) - (\<lambda>x. member x B))"
instance proof
qed (simp_all add: less_eq_set_def less_set_def inf_set_def sup_set_def
bot_set_def top_set_def uminus_set_def minus_set_def
less_le_not_le inf_compl_bot sup_compl_top sup_inf_distrib1 diff_eq
set_eqI fun_eq_iff
del: inf_apply sup_apply bot_apply top_apply minus_apply uminus_apply)
end
text {* Set enumerations *}
abbreviation empty :: "'a set" ("{}") where
"{} \<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 ("(_/ < _)" [51, 51] 50) and
subset_eq ("op <=") and
subset_eq ("(_/ <= _)" [51, 51] 50)
notation (xsymbols)
subset ("op \<subset>") and
subset ("(_/ \<subset> _)" [51, 51] 50) and
subset_eq ("op \<subseteq>") and
subset_eq ("(_/ \<subseteq> _)" [51, 51] 50)
notation (HTML output)
subset ("op \<subset>") and
subset ("(_/ \<subset> _)" [51, 51] 50) and
subset_eq ("op \<subseteq>") and
subset_eq ("(_/ \<subseteq> _)" [51, 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> _)" [51, 51] 50) and
supset_eq ("op \<supseteq>") and
supset_eq ("(_/ \<supseteq> _)" [51, 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 All_binder = Mixfix.binder_name @{const_syntax All};
val Ex_binder = Mixfix.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', T) c n P =
if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
then Syntax.const c $ Syntax_Trans.mark_bound_body (v', T) $ n $ P
else raise Match;
fun tr' q = (q, fn _ =>
(fn [Const (@{syntax_const "_bound"}, _) $ Free (v, Type (@{type_name set}, _)),
Const (c, _) $
(Const (d, _) $ (Const (@{syntax_const "_bound"}, _) $ Free (v', T)) $ n) $ P] =>
(case AList.lookup (op =) trans (q, c, d) of
NONE => raise Match
| SOME l => mk v (v', T) l n P)
| _ => raise Match));
in
[tr' All_binder, tr' Ex_binder]
end
*}
text {*
\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 (Syntax_Trans.mk_binder_tr ("EX ", @{const_syntax Ex}));
fun nvars (Const (@{syntax_const "_idts"}, _) $ _ $ idts) = nvars idts + 1
| nvars _ = 1;
fun setcompr_tr ctxt [e, idts, b] =
let
val eq = Syntax.const @{const_syntax HOL.eq} $ Bound (nvars idts) $ e;
val P = Syntax.const @{const_syntax HOL.conj} $ eq $ b;
val exP = ex_tr ctxt [idts, P];
in Syntax.const @{const_syntax Collect} $ absdummy dummyT exP end;
in [(@{syntax_const "_Setcompr"}, setcompr_tr)] end;
*}
print_translation {*
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"},
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"}]
*} -- {* to avoid eta-contraction of body *}
print_translation {*
let
val ex_tr' = snd (Syntax_Trans.mk_binder_tr' (@{const_syntax Ex}, "DUMMY"));
fun setcompr_tr' ctxt [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' ctxt [abs]
in Syntax.const @{syntax_const "_Setcompr"} $ e $ idts $ Q end;
in
if check (P, 0) then tr' P
else
let
val (x as _ $ Free(xN, _), t) = Syntax_Trans.atomic_abs_tr' abs;
val M = Syntax.const @{syntax_const "_Coll"} $ x $ t;
in
case t of
Const (@{const_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;
*}
simproc_setup defined_Bex ("EX x:A. P x & Q x") = {*
fn _ => Quantifier1.rearrange_bex
(fn ctxt =>
unfold_tac ctxt @{thms Bex_def} THEN
Quantifier1.prove_one_point_ex_tac)
*}
simproc_setup defined_All ("ALL x:A. P x --> Q x") = {*
fn _ => Quantifier1.rearrange_ball
(fn ctxt =>
unfold_tac ctxt @{thms Ball_def} THEN
Quantifier1.prove_one_point_all_tac)
*}
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:
*}
setup {*
map_theory_claset (fn ctxt =>
ctxt addbefore ("bspec", fn _ => dtac @{thm bspec} THEN' assume_tac))
*}
ML {*
structure Simpdata =
struct
open Simpdata;
val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;
end;
open Simpdata;
*}
declaration {* fn _ =>
Simplifier.map_ss (Simplifier.set_mksimps (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
lemma ball_conj_distrib:
"(\<forall>x\<in>A. P x \<and> Q x) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>A. Q x))"
by blast
lemma bex_disj_distrib:
"(\<exists>x\<in>A. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>A. Q x))"
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"
by (simp add: less_eq_set_def le_fun_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, intro?]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B"
by (simp add: less_eq_set_def le_fun_def)
-- {* Rule in Modus Ponens style. *}
lemma rev_subsetD [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 [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
-- {* Classical elimination rule. *}
by (auto simp add: less_eq_set_def le_fun_def)
lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
by blast
lemma subset_refl: "A \<subseteq> A"
by (fact order_refl) (* already [iff] *)
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 subset_not_subset_eq [code]:
"A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"
by (fact less_le_not_le)
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 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_set_def bot_fun_def)
lemma empty_iff [simp]: "(c : {}) = False"
by (simp add: empty_def)
lemma emptyE [elim!]: "a : {} ==> P"
by simp
lemma empty_subsetI [iff]: "{} \<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_set_def top_fun_def)
lemma UNIV_I [simp]: "x : UNIV"
by (simp add: UNIV_def)
declare UNIV_I [intro] -- {* unsafe makes it less likely to cause problems *}
lemma UNIV_witness [intro?]: "EX x. x : UNIV"
by simp
lemma subset_UNIV: "A \<subseteq> UNIV"
by (fact top_greatest) (* already simp *)
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)
lemma empty_not_UNIV[simp]: "{} \<noteq> UNIV"
by blast
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: fun_Compl_def uminus_set_def)
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
by (simp add: fun_Compl_def uminus_set_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
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_set_def inf_fun_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_set_def sup_fun_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: 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: minus_set_def fun_diff_def)
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
by simp
lemma DiffD1: "c : A - B ==> c : A"
by simp
lemma DiffD2: "c : A - B ==> c : B ==> P"
by simp
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
by simp
lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast
lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"
by blast
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
lemma insert_eq_iff: assumes "a \<notin> A" "b \<notin> B"
shows "insert a A = insert b B \<longleftrightarrow>
(if a=b then A=B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)"
(is "?L \<longleftrightarrow> ?R")
proof
assume ?L
show ?R
proof cases
assume "a=b" with assms `?L` show ?R by (simp add: insert_ident)
next
assume "a\<noteq>b"
let ?C = "A - {b}"
have "A = insert b ?C \<and> b \<notin> ?C \<and> B = insert a ?C \<and> a \<notin> ?C"
using assms `?L` `a\<noteq>b` by auto
thus ?R using `a\<noteq>b` by auto
qed
next
assume ?R thus ?L by (auto split: if_splits)
qed
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 ==> 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)
lemma Un_singleton_iff:
"(A \<union> B = {x}) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"
by auto
lemma singleton_Un_iff:
"({x} = A \<union> B) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"
by auto
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: "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 Compr_image_eq:
"{x \<in> f ` A. P x} = f ` {x \<in> A. P (f x)}"
by auto
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 image_ident [simp]: "(%x. x) ` Y = Y"
by blast
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!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
by (unfold less_le) blast
lemma psubsetE [elim!]:
"[|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 (fact bot_unique)
lemma not_psubset_empty [iff]: "\<not> (A < {})"
by (fact not_less_bot) (* FIXME: already simp *)
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]
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]:
"(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) (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: "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)
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
text {* \medskip @{text range}. *}
lemma full_SetCompr_eq: "{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: "A \<inter> A = A"
by (fact inf_idem) (* already simp *)
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: "{} \<inter> B = {}"
by (fact inf_bot_left) (* already simp *)
lemma Int_empty_right: "A \<inter> {} = {}"
by (fact inf_bot_right) (* already simp *)
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: "UNIV \<inter> B = B"
by (fact inf_top_left) (* already simp *)
lemma Int_UNIV_right: "A \<inter> UNIV = A"
by (fact inf_top_right) (* already simp *)
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]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
by (fact inf_eq_top_iff) (* already simp *)
lemma Int_subset_iff [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: "A \<union> A = A"
by (fact sup_idem) (* already simp *)
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: "{} \<union> B = B"
by (fact sup_bot_left) (* already simp *)
lemma Un_empty_right: "A \<union> {} = A"
by (fact sup_bot_right) (* already simp *)
lemma Un_UNIV_left: "UNIV \<union> B = UNIV"
by (fact sup_top_left) (* already simp *)
lemma Un_UNIV_right: "A \<union> UNIV = UNIV"
by (fact sup_top_right) (* already simp *)
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) (* FIXME: already simp *)
lemma Un_subset_iff [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: "- (-A) = (A::'a set)"
by (fact double_compl) (* already simp *)
lemma Compl_Un: "-(A \<union> B) = (-A) \<inter> (-B)"
by (fact compl_sup) (* already simp *)
lemma Compl_Int: "-(A \<inter> B) = (-A) \<union> (-B)"
by (fact compl_inf) (* already simp *)
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: "-UNIV = {}"
by (fact compl_top_eq) (* already simp *)
lemma Compl_empty_eq: "-{} = UNIV"
by (fact compl_bot_eq) (* already simp *)
lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
by (fact compl_le_compl_iff) (* FIXME: already simp *)
lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"
by (fact compl_eq_compl_iff) (* FIXME: already simp *)
lemma Compl_insert: "- insert x A = (-A) - {x}"
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
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 UNIV_bool: "UNIV = {False, True}"
by (auto intro: bool_induct)
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}" for u])
lemma Pow_Compl: "Pow (- A) = {-B | B. A \<in> Pow B}"
by (blast intro: exI [where ?x = "- u" for u])
lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"
by blast
lemma Un_Pow_subset: "Pow A \<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: "(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 ball_simps [simp, no_atp]:
"\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"
"\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"
"\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"
"\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"
"\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"
"\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
"\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"
"\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"
"\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"
"\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"
by auto
lemma bex_simps [simp, no_atp]:
"\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"
"\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"
"\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"
"\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"
"\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"
"\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"
"\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
"\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"
by auto
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: "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 image_subset_iff_subset_vimage: "f ` A \<subseteq> B \<longleftrightarrow> A \<subseteq> f -` B"
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 vimage_ident [simp]: "(%x. x) -` Y = Y"
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
subsubsection {* Monad operation *}
definition bind :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
"bind A f = {x. \<exists>B \<in> f`A. x \<in> B}"
hide_const (open) bind
lemma bind_bind:
fixes A :: "'a set"
shows "Set.bind (Set.bind A B) C = Set.bind A (\<lambda>x. Set.bind (B x) C)"
by (auto simp add: bind_def)
lemma empty_bind [simp]:
"Set.bind {} f = {}"
by (simp add: bind_def)
lemma nonempty_bind_const:
"A \<noteq> {} \<Longrightarrow> Set.bind A (\<lambda>_. B) = B"
by (auto simp add: bind_def)
lemma bind_const: "Set.bind A (\<lambda>_. B) = (if A = {} then {} else B)"
by (auto simp add: bind_def)
subsubsection {* Operations for execution *}
definition is_empty :: "'a set \<Rightarrow> bool" where
[code_abbrev]: "is_empty A \<longleftrightarrow> A = {}"
hide_const (open) is_empty
definition remove :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set" where
[code_abbrev]: "remove x A = A - {x}"
hide_const (open) remove
lemma member_remove [simp]:
"x \<in> Set.remove y A \<longleftrightarrow> x \<in> A \<and> x \<noteq> y"
by (simp add: remove_def)
definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" where
[code_abbrev]: "filter P A = {a \<in> A. P a}"
hide_const (open) filter
lemma member_filter [simp]:
"x \<in> Set.filter P A \<longleftrightarrow> x \<in> A \<and> P x"
by (simp add: filter_def)
instantiation set :: (equal) equal
begin
definition
"HOL.equal A B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
instance proof
qed (auto simp add: equal_set_def)
end
text {* Misc *}
hide_const (open) member not_member
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 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