(* Title: HOL/Set.thy
Author: Tobias Nipkow
Author: Lawrence C Paulson
Author: Markus Wenzel
*)
section \<open>Set theory for higher-order logic\<close>
theory Set
imports Lattices
begin
subsection \<open>Sets as predicates\<close>
typedecl 'a set
axiomatization Collect :: "('a \ bool) \ 'a set" \ \comprehension\
and member :: "'a \ 'a set \ bool" \ \membership\
where mem_Collect_eq [iff, code_unfold]: "member a (Collect P) = P a"
and Collect_mem_eq [simp]: "Collect (\x. member x A) = A"
notation
member ("'(\')") and
member ("(_/ \ _)" [51, 51] 50)
abbreviation not_member
where "not_member x A \ \ (x \ A)" \ \non-membership\
notation
not_member ("'(\')") and
not_member ("(_/ \ _)" [51, 51] 50)
notation (ASCII)
member ("'(:')") and
member ("(_/ : _)" [51, 51] 50) and
not_member ("'(~:')") and
not_member ("(_/ ~: _)" [51, 51] 50)
text \<open>Set comprehensions\<close>
syntax
"_Coll" :: "pttrn \ bool \ 'a set" ("(1{_./ _})")
translations
"{x. P}" \<rightleftharpoons> "CONST Collect (\<lambda>x. P)"
syntax (ASCII)
"_Collect" :: "pttrn \ 'a set \ bool \ 'a set" ("(1{(_/: _)./ _})")
syntax
"_Collect" :: "pttrn \ 'a set \ bool \ 'a set" ("(1{(_/ \ _)./ _})")
translations
"{p:A. P}" \<rightharpoonup> "CONST Collect (\<lambda>p. p \<in> A \<and> P)"
lemma CollectI: "P a \ a \ {x. P x}"
by simp
lemma CollectD: "a \ {x. P x} \ P a"
by simp
lemma Collect_cong: "(\x. P x = Q x) \ {x. P x} = {x. Q x}"
by simp
text \<open>
Simproc for pulling \<open>x = t\<close> in \<open>{x. \<dots> \<and> x = t \<and> \<dots>}\<close>
to the front (and similarly for \<open>t = x\<close>):
\<close>
simproc_setup defined_Collect ("{x. P x \ Q x}") = \
fn _ => Quantifier1.rearrange_Collect
(fn ctxt =>
resolve_tac ctxt @{thms Collect_cong} 1 THEN
resolve_tac ctxt @{thms iffI} 1 THEN
ALLGOALS
(EVERY' [REPEAT_DETERM o eresolve_tac ctxt @{thms conjE},
DEPTH_SOLVE_1 o (assume_tac ctxt ORELSE' resolve_tac ctxt @{thms conjI})]))
\<close>
lemmas CollectE = CollectD [elim_format]
lemma set_eqI:
assumes "\x. x \ A \ x \ B"
shows "A = B"
proof -
from assms have "{x. x \ A} = {x. x \ B}"
by simp
then show ?thesis by simp
qed
lemma set_eq_iff: "A = B \ (\x. x \ A \ x \ B)"
by (auto intro:set_eqI)
lemma Collect_eqI:
assumes "\x. P x = Q x"
shows "Collect P = Collect Q"
using assms by (auto intro: set_eqI)
text \<open>Lifting of predicate class instances\<close>
instantiation set :: (type) boolean_algebra
begin
definition less_eq_set
where "A \ B \ (\x. member x A) \ (\x. member x B)"
definition less_set
where "A < B \ (\x. member x A) < (\x. member x B)"
definition inf_set
where "A \ B = Collect ((\x. member x A) \ (\x. member x B))"
definition sup_set
where "A \ B = Collect ((\x. member x A) \ (\x. member x B))"
definition bot_set
where "\ = Collect \"
definition top_set
where "\ = Collect \"
definition uminus_set
where "- A = Collect (- (\x. member x A))"
definition minus_set
where "A - B = Collect ((\x. member x A) - (\x. member x B))"
instance
by standard
(simp_all add: less_eq_set_def less_set_def inf_set_def sup_set_def
bot_set_def top_set_def uminus_set_def minus_set_def
less_le_not_le sup_inf_distrib1 diff_eq set_eqI fun_eq_iff
del: inf_apply sup_apply bot_apply top_apply minus_apply uminus_apply)
end
text \<open>Set enumerations\<close>
abbreviation empty :: "'a set" ("{}")
where "{} \ bot"
definition insert :: "'a \ 'a set \ 'a set"
where insert_compr: "insert a B = {x. x = a \ x \ B}"
syntax
"_Finset" :: "args \ 'a set" ("{(_)}")
translations
"{x, xs}" \<rightleftharpoons> "CONST insert x {xs}"
"{x}" \<rightleftharpoons> "CONST insert x {}"
subsection \<open>Subsets and bounded quantifiers\<close>
abbreviation subset :: "'a set \ 'a set \ bool"
where "subset \ less"
abbreviation subset_eq :: "'a set \ 'a set \ bool"
where "subset_eq \ less_eq"
notation
subset ("'(\')") and
subset ("(_/ \ _)" [51, 51] 50) and
subset_eq ("'(\')") and
subset_eq ("(_/ \ _)" [51, 51] 50)
abbreviation (input)
supset :: "'a set \ 'a set \ bool" where
"supset \ greater"
abbreviation (input)
supset_eq :: "'a set \ 'a set \ bool" where
"supset_eq \ greater_eq"
notation
supset ("'(\')") and
supset ("(_/ \ _)" [51, 51] 50) and
supset_eq ("'(\')") and
supset_eq ("(_/ \ _)" [51, 51] 50)
notation (ASCII output)
subset ("'(<')") and
subset ("(_/ < _)" [51, 51] 50) and
subset_eq ("'(<=')") and
subset_eq ("(_/ <= _)" [51, 51] 50)
definition Ball :: "'a set \ ('a \ bool) \ bool"
where "Ball A P \ (\x. x \ A \ P x)" \ \bounded universal quantifiers\
definition Bex :: "'a set \ ('a \ bool) \ bool"
where "Bex A P \ (\x. x \ A \ P x)" \ \bounded existential quantifiers\
syntax (ASCII)
"_Ball" :: "pttrn \ 'a set \ bool \ bool" ("(3ALL (_/:_)./ _)" [0, 0, 10] 10)
"_Bex" :: "pttrn \ 'a set \ bool \ bool" ("(3EX (_/:_)./ _)" [0, 0, 10] 10)
"_Bex1" :: "pttrn \ 'a set \ bool \ bool" ("(3EX! (_/:_)./ _)" [0, 0, 10] 10)
"_Bleast" :: "id \ 'a set \ bool \ 'a" ("(3LEAST (_/:_)./ _)" [0, 0, 10] 10)
syntax (input)
"_Ball" :: "pttrn \ 'a set \ bool \ bool" ("(3! (_/:_)./ _)" [0, 0, 10] 10)
"_Bex" :: "pttrn \ 'a set \ bool \ bool" ("(3? (_/:_)./ _)" [0, 0, 10] 10)
"_Bex1" :: "pttrn \ 'a set \ bool \ bool" ("(3?! (_/:_)./ _)" [0, 0, 10] 10)
syntax
"_Ball" :: "pttrn \ 'a set \ bool \ bool" ("(3\(_/\_)./ _)" [0, 0, 10] 10)
"_Bex" :: "pttrn \ 'a set \ bool \ bool" ("(3\(_/\_)./ _)" [0, 0, 10] 10)
"_Bex1" :: "pttrn \ 'a set \ bool \ bool" ("(3\!(_/\_)./ _)" [0, 0, 10] 10)
"_Bleast" :: "id \ 'a set \ bool \ 'a" ("(3LEAST(_/\_)./ _)" [0, 0, 10] 10)
translations
"\x\A. P" \ "CONST Ball A (\x. P)"
"\x\A. P" \ "CONST Bex A (\x. P)"
"\!x\A. P" \ "\!x. x \ A \ P"
"LEAST x:A. P" \<rightharpoonup> "LEAST x. x \<in> A \<and> P"
syntax (ASCII output)
"_setlessAll" :: "[idt, 'a, bool] \ bool" ("(3ALL _<_./ _)" [0, 0, 10] 10)
"_setlessEx" :: "[idt, 'a, bool] \ bool" ("(3EX _<_./ _)" [0, 0, 10] 10)
"_setleAll" :: "[idt, 'a, bool] \ bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10)
"_setleEx" :: "[idt, 'a, bool] \ bool" ("(3EX _<=_./ _)" [0, 0, 10] 10)
"_setleEx1" :: "[idt, 'a, bool] \ bool" ("(3EX! _<=_./ _)" [0, 0, 10] 10)
syntax
"_setlessAll" :: "[idt, 'a, bool] \ bool" ("(3\_\_./ _)" [0, 0, 10] 10)
"_setlessEx" :: "[idt, 'a, bool] \ bool" ("(3\_\_./ _)" [0, 0, 10] 10)
"_setleAll" :: "[idt, 'a, bool] \ bool" ("(3\_\_./ _)" [0, 0, 10] 10)
"_setleEx" :: "[idt, 'a, bool] \ bool" ("(3\_\_./ _)" [0, 0, 10] 10)
"_setleEx1" :: "[idt, 'a, bool] \ bool" ("(3\!_\_./ _)" [0, 0, 10] 10)
translations
"\A\B. P" \ "\A. A \ B \ P"
"\A\B. P" \ "\A. A \ B \ P"
"\A\B. P" \ "\A. A \ B \ P"
"\A\B. P" \ "\A. A \ B \ P"
"\!A\B. P" \ "\!A. A \ B \ P"
print_translation \<open>
let
val All_binder = Mixfix.binder_name \<^const_syntax>\<open>All\<close>;
val Ex_binder = Mixfix.binder_name \<^const_syntax>\<open>Ex\<close>;
val impl = \<^const_syntax>\<open>HOL.implies\<close>;
val conj = \<^const_syntax>\<open>HOL.conj\<close>;
val sbset = \<^const_syntax>\<open>subset\<close>;
val sbset_eq = \<^const_syntax>\<open>subset_eq\<close>;
val trans =
[((All_binder, impl, sbset), \<^syntax_const>\<open>_setlessAll\<close>),
((All_binder, impl, sbset_eq), \<^syntax_const>\<open>_setleAll\<close>),
((Ex_binder, conj, sbset), \<^syntax_const>\<open>_setlessEx\<close>),
((Ex_binder, conj, sbset_eq), \<^syntax_const>\<open>_setleEx\<close>)];
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>\<open>_bound\<close>, _) $ Free (v, Type (\<^type_name>\<open>set\<close>, _)),
Const (c, _) $
(Const (d, _) $ (Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v', T)) $ n) $ P] =>
(case AList.lookup (=) trans (q, c, d) of
NONE => raise Match
| SOME l => mk v (v', T) l n P)
| _ => raise Match));
in
[tr' All_binder, tr' Ex_binder]
end
\<close>
text \<open>
\<^medskip>
Translate between \<open>{e | x1\<dots>xn. P}\<close> and \<open>{u. \<exists>x1\<dots>xn. u = e \<and> P}\<close>;
\<open>{y. \<exists>x1\<dots>xn. y = e \<and> P}\<close> is only translated if \<open>[0..n] \<subseteq> bvs e\<close>.
\<close>
syntax
"_Setcompr" :: "'a \ idts \ bool \ 'a set" ("(1{_ |/_./ _})")
parse_translation \<open>
let
val ex_tr = snd (Syntax_Trans.mk_binder_tr ("EX ", \<^const_syntax>\<open>Ex\<close>));
fun nvars (Const (\<^syntax_const>\<open>_idts\<close>, _) $ _ $ idts) = nvars idts + 1
| nvars _ = 1;
fun setcompr_tr ctxt [e, idts, b] =
let
val eq = Syntax.const \<^const_syntax>\<open>HOL.eq\<close> $ Bound (nvars idts) $ e;
val P = Syntax.const \<^const_syntax>\<open>HOL.conj\<close> $ eq $ b;
val exP = ex_tr ctxt [idts, P];
in Syntax.const \<^const_syntax>\<open>Collect\<close> $ absdummy dummyT exP end;
in [(\<^syntax_const>\<open>_Setcompr\<close>, setcompr_tr)] end
\<close>
print_translation \<open>
[Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\Ball\ \<^syntax_const>\_Ball\,
Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\Bex\ \<^syntax_const>\_Bex\]
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
print_translation \<open>
let
val ex_tr' = snd (Syntax_Trans.mk_binder_tr' (\<^const_syntax>\<open>Ex\<close>, "DUMMY"));
fun setcompr_tr' ctxt [Abs (abs as (_, _, P))] =
let
fun check (Const (\<^const_syntax>\<open>Ex\<close>, _) $ Abs (_, _, P), n) = check (P, n + 1)
| check (Const (\<^const_syntax>\<open>HOL.conj\<close>, _) $
(Const (\<^const_syntax>\<open>HOL.eq\<close>, _) $ Bound m $ e) $ P, n) =
n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
subset (=) (0 upto (n - 1), add_loose_bnos (e, 0, []))
| check _ = false;
fun tr' (_ $ abs) =
let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' ctxt [abs]
in Syntax.const \<^syntax_const>\<open>_Setcompr\<close> $ 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>\<open>_Coll\<close> $ x $ t;
in
case t of
Const (\<^const_syntax>\<open>HOL.conj\<close>, _) $
(Const (\<^const_syntax>\<open>Set.member\<close>, _) $
(Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (yN, _)) $ A) $ P =>
if xN = yN then Syntax.const \<^syntax_const>\<open>_Collect\<close> $ x $ A $ P else M
| _ => M
end
end;
in [(\<^const_syntax>\<open>Collect\<close>, setcompr_tr')] end
\<close>
simproc_setup defined_Bex ("\x\A. P x \ Q x") = \
fn _ => Quantifier1.rearrange_Bex
(fn ctxt => unfold_tac ctxt @{thms Bex_def})
\<close>
simproc_setup defined_All ("\x\A. P x \ Q x") = \
fn _ => Quantifier1.rearrange_Ball
(fn ctxt => unfold_tac ctxt @{thms Ball_def})
\<close>
lemma ballI [intro!]: "(\x. x \ A \ P x) \ \x\A. P x"
by (simp add: Ball_def)
lemmas strip = impI allI ballI
lemma bspec [dest?]: "\x\A. P x \ x \ A \ P x"
by (simp add: Ball_def)
text \<open>Gives better instantiation for bound:\<close>
setup \<open>
map_theory_claset (fn ctxt =>
ctxt addbefore ("bspec", fn ctxt' => dresolve_tac ctxt' @{thms bspec} THEN' assume_tac ctxt'))
\<close>
ML \<open>
structure Simpdata =
struct
open Simpdata;
val mksimps_pairs = [(\<^const_name>\<open>Ball\<close>, @{thms bspec})] @ mksimps_pairs;
end;
open Simpdata;
\<close>
declaration \<open>fn _ => Simplifier.map_ss (Simplifier.set_mksimps (mksimps mksimps_pairs))\<close>
lemma ballE [elim]: "\x\A. P x \ (P x \ Q) \ (x \ A \ Q) \ Q"
unfolding Ball_def by blast
lemma bexI [intro]: "P x \ x \ A \ \x\A. P x"
\<comment> \<open>Normally the best argument order: \<open>P x\<close> constrains the choice of \<open>x \<in> A\<close>.\<close>
unfolding Bex_def by blast
lemma rev_bexI [intro?]: "x \ A \ P x \ \x\A. P x"
\<comment> \<open>The best argument order when there is only one \<open>x \<in> A\<close>.\<close>
unfolding Bex_def by blast
lemma bexCI: "(\x\A. \ P x \ P a) \ a \ A \ \x\A. P x"
unfolding Bex_def by blast
lemma bexE [elim!]: "\x\A. P x \ (\x. x \ A \ P x \ Q) \ Q"
unfolding Bex_def by blast
lemma ball_triv [simp]: "(\x\A. P) \ ((\x. x \ A) \ P)"
\<comment> \<open>trivial rewrite rule.\<close>
by (simp add: Ball_def)
lemma bex_triv [simp]: "(\x\A. P) \ ((\x. x \ A) \ P)"
\<comment> \<open>Dual form for existentials.\<close>
by (simp add: Bex_def)
lemma bex_triv_one_point1 [simp]: "(\x\A. x = a) \ a \ A"
by blast
lemma bex_triv_one_point2 [simp]: "(\x\A. a = x) \ a \ A"
by blast
lemma bex_one_point1 [simp]: "(\x\A. x = a \ P x) \ a \ A \ P a"
by blast
lemma bex_one_point2 [simp]: "(\x\A. a = x \ P x) \ a \ A \ P a"
by blast
lemma ball_one_point1 [simp]: "(\x\A. x = a \ P x) \ (a \ A \ P a)"
by blast
lemma ball_one_point2 [simp]: "(\x\A. a = x \ P x) \ (a \ A \ P a)"
by blast
lemma ball_conj_distrib: "(\x\A. P x \ Q x) \ (\x\A. P x) \ (\x\A. Q x)"
by blast
lemma bex_disj_distrib: "(\x\A. P x \ Q x) \ (\x\A. P x) \ (\x\A. Q x)"
by blast
text \<open>Congruence rules\<close>
lemma ball_cong:
"\ A = B; \x. x \ B \ P x \ Q x \ \
(\<forall>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>B. Q x)"
by (simp add: Ball_def)
lemma ball_cong_simp [cong]:
"\ A = B; \x. x \ B =simp=> P x \ Q x \ \
(\<forall>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>B. Q x)"
by (simp add: simp_implies_def Ball_def)
lemma bex_cong:
"\ A = B; \x. x \ B \ P x \ Q x \ \
(\<exists>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>B. Q x)"
by (simp add: Bex_def cong: conj_cong)
lemma bex_cong_simp [cong]:
"\ A = B; \x. x \ B =simp=> P x \ Q x \ \
(\<exists>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>B. Q x)"
by (simp add: simp_implies_def Bex_def cong: conj_cong)
lemma bex1_def: "(\!x\X. P x) \ (\x\X. P x) \ (\x\X. \y\X. P x \ P y \ x = y)"
by auto
subsection \<open>Basic operations\<close>
subsubsection \<open>Subsets\<close>
lemma subsetI [intro!]: "(\x. x \ A \ x \ B) \ A \ B"
by (simp add: less_eq_set_def le_fun_def)
text \<open>
\<^medskip>
Map the type \<open>'a set \<Rightarrow> anything\<close> to just \<open>'a\<close>; for overloading constants
whose first argument has type \<open>'a set\<close>.
\<close>
lemma subsetD [elim, intro?]: "A \ B \ c \ A \ c \ B"
by (simp add: less_eq_set_def le_fun_def)
\<comment> \<open>Rule in Modus Ponens style.\<close>
lemma rev_subsetD [intro?,no_atp]: "c \ A \ A \ B \ c \ B"
\<comment> \<open>The same, with reversed premises for use with @{method erule} -- cf. @{thm rev_mp}.\<close>
by (rule subsetD)
lemma subsetCE [elim,no_atp]: "A \ B \ (c \ A \ P) \ (c \ B \ P) \ P"
\<comment> \<open>Classical elimination rule.\<close>
by (auto simp add: less_eq_set_def le_fun_def)
lemma subset_eq: "A \ B \ (\x\A. x \ B)"
by blast
lemma contra_subsetD [no_atp]: "A \ B \ c \ B \ c \ A"
by blast
lemma subset_refl: "A \ A"
by (fact order_refl) (* already [iff] *)
lemma subset_trans: "A \ B \ B \ C \ A \ C"
by (fact order_trans)
lemma subset_not_subset_eq [code]: "A \ B \ A \ B \ \ B \ A"
by (fact less_le_not_le)
lemma eq_mem_trans: "a = b \ b \ A \ a \ A"
by simp
lemmas basic_trans_rules [trans] =
order_trans_rules rev_subsetD subsetD eq_mem_trans
subsubsection \<open>Equality\<close>
lemma subset_antisym [intro!]: "A \ B \ B \ A \ A = B"
\<comment> \<open>Anti-symmetry of the subset relation.\<close>
by (iprover intro: set_eqI subsetD)
text \<open>\<^medskip> Equality rules from ZF set theory -- are they appropriate here?\<close>
lemma equalityD1: "A = B \ A \ B"
by simp
lemma equalityD2: "A = B \ B \ A"
by simp
text \<open>
\<^medskip>
Be careful when adding this to the claset as \<open>subset_empty\<close> is in the
simpset: \<^prop>\<open>A = {}\<close> goes to \<^prop>\<open>{} \<subseteq> A\<close> and \<^prop>\<open>A \<subseteq> {}\<close>
and then back to \<^prop>\<open>A = {}\<close>!
\<close>
lemma equalityE: "A = B \ (A \ B \ B \ A \ P) \ P"
by simp
lemma equalityCE [elim]: "A = B \ (c \ A \ c \ B \ P) \ (c \ A \ c \ B \ P) \ P"
by blast
lemma eqset_imp_iff: "A = B \ x \ A \ x \ B"
by simp
lemma eqelem_imp_iff: "x = y \ x \ A \ y \ A"
by simp
subsubsection \<open>The empty set\<close>
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]: "{} \ A"
\<comment> \<open>One effect is to delete the ASSUMPTION \<^prop>\<open>{} \<subseteq> A\<close>\<close>
by blast
lemma equals0I: "(\y. y \ A \ False) \ A = {}"
by blast
lemma equals0D: "A = {} \ a \ A"
\<comment> \<open>Use for reasoning about disjointness: \<open>A \<inter> B = {}\<close>\<close>
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 \<open>The universal set -- UNIV\<close>
abbreviation UNIV :: "'a set"
where "UNIV \ 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] \<comment> \<open>unsafe makes it less likely to cause problems\<close>
lemma UNIV_witness [intro?]: "\x. x \ UNIV"
by simp
lemma subset_UNIV: "A \ UNIV"
by (fact top_greatest) (* already simp *)
text \<open>
\<^medskip>
Eta-contracting these two rules (to remove \<open>P\<close>) causes them
to be ignored because of their interaction with congruence rules.
\<close>
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: "(\x. x \ A) \ UNIV = A"
by auto
lemma UNIV_not_empty [iff]: "UNIV \ {}"
by (blast elim: equalityE)
lemma empty_not_UNIV[simp]: "{} \ UNIV"
by blast
subsubsection \<open>The Powerset operator -- Pow\<close>
definition Pow :: "'a set \ 'a set set"
where Pow_def: "Pow A = {B. B \ A}"
lemma Pow_iff [iff]: "A \ Pow B \ A \ B"
by (simp add: Pow_def)
lemma PowI: "A \ B \ A \ Pow B"
by (simp add: Pow_def)
lemma PowD: "A \ Pow B \ A \ B"
by (simp add: Pow_def)
lemma Pow_bottom: "{} \ Pow B"
by simp
lemma Pow_top: "A \ Pow A"
by simp
lemma Pow_not_empty: "Pow A \ {}"
using Pow_top by blast
subsubsection \<open>Set complement\<close>
lemma Compl_iff [simp]: "c \ - A \ c \ A"
by (simp add: fun_Compl_def uminus_set_def)
lemma ComplI [intro!]: "(c \ A \ False) \ c \ - A"
by (simp add: fun_Compl_def uminus_set_def) blast
text \<open>
\<^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 \dots
\<close>
lemma ComplD [dest!]: "c \ - A \ c \ A"
by simp
lemmas ComplE = ComplD [elim_format]
lemma Compl_eq: "- A = {x. \ x \ A}"
by blast
subsubsection \<open>Binary intersection\<close>
abbreviation inter :: "'a set \ 'a set \ 'a set" (infixl "\" 70)
where "(\) \ inf"
notation (ASCII)
inter (infixl "Int" 70)
lemma Int_def: "A \ B = {x. x \ A \ x \ B}"
by (simp add: inf_set_def inf_fun_def)
lemma Int_iff [simp]: "c \ A \ B \ c \ A \ c \ B"
unfolding Int_def by blast
lemma IntI [intro!]: "c \ A \ c \ B \ c \ A \ B"
by simp
lemma IntD1: "c \ A \ B \ c \ A"
by simp
lemma IntD2: "c \ A \ B \ c \ B"
by simp
lemma IntE [elim!]: "c \ A \ B \ (c \ A \ c \ B \ P) \ P"
by simp
lemma mono_Int: "mono f \ f (A \ B) \ f A \ f B"
by (fact mono_inf)
subsubsection \<open>Binary union\<close>
abbreviation union :: "'a set \ 'a set \ 'a set" (infixl "\" 65)
where "union \ sup"
notation (ASCII)
union (infixl "Un" 65)
lemma Un_def: "A \ B = {x. x \ A \ x \ B}"
by (simp add: sup_set_def sup_fun_def)
lemma Un_iff [simp]: "c \ A \ B \ c \ A \ c \ B"
unfolding Un_def by blast
lemma UnI1 [elim?]: "c \ A \ c \ A \ B"
by simp
lemma UnI2 [elim?]: "c \ B \ c \ A \ B"
by simp
text \<open>\<^medskip> Classical introduction rule: no commitment to \<open>A\<close> vs. \<open>B\<close>.\<close>
lemma UnCI [intro!]: "(c \ B \ c \ A) \ c \ A \ B"
by auto
lemma UnE [elim!]: "c \ A \ B \ (c \ A \ P) \ (c \ B \ P) \ P"
unfolding Un_def by blast
lemma insert_def: "insert a B = {x. x = a} \ B"
by (simp add: insert_compr Un_def)
lemma mono_Un: "mono f \ f A \ f B \ f (A \ B)"
by (fact mono_sup)
subsubsection \<open>Set difference\<close>
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 \<open>Augmenting a set -- \<^const>\<open>insert\<close>\<close>
lemma insert_iff [simp]: "a \ insert b A \ a = b \ a \ A"
unfolding insert_def by blast
lemma insertI1: "a \ insert a B"
by simp
lemma insertI2: "a \ B \ a \ insert b B"
by simp
lemma insertE [elim!]: "a \ insert b A \ (a = b \ P) \ (a \ A \ P) \ P"
unfolding insert_def by blast
lemma insertCI [intro!]: "(a \ B \ a = b) \ a \ insert b B"
\<comment> \<open>Classical introduction rule.\<close>
by auto
lemma subset_insert_iff: "A \ insert x B \ (if x \ A then A - {x} \ B else A \ B)"
by auto
lemma set_insert:
assumes "x \ A"
obtains B where "A = insert x B" and "x \ B"
proof
show "A = insert x (A - {x})" using assms by blast
show "x \ A - {x}" by blast
qed
lemma insert_ident: "x \ A \ x \ B \ insert x A = insert x B \ A = B"
by auto
lemma insert_eq_iff:
assumes "a \ A" "b \ B"
shows "insert a A = insert b B \
(if a = b then A = B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)"
(is "?L \ ?R")
proof
show ?R if ?L
proof (cases "a = b")
case True
with assms \<open>?L\<close> show ?R
by (simp add: insert_ident)
next
case False
let ?C = "A - {b}"
have "A = insert b ?C \ b \ ?C \ B = insert a ?C \ a \ ?C"
using assms \<open>?L\<close> \<open>a \<noteq> b\<close> by auto
then show ?R using \<open>a \<noteq> b\<close> by auto
qed
show ?L if ?R
using that by (auto split: if_splits)
qed
lemma insert_UNIV: "insert x UNIV = UNIV"
by auto
subsubsection \<open>Singletons, using insert\<close>
lemma singletonI [intro!]: "a \ {a}"
\<comment> \<open>Redundant? But unlike \<open>insertCI\<close>, it proves the subgoal immediately!\<close>
by (rule insertI1)
lemma singletonD [dest!]: "b \ {a} \ b = a"
by blast
lemmas singletonE = singletonD [elim_format]
lemma singleton_iff: "b \ {a} \ b = a"
by blast
lemma singleton_inject [dest!]: "{a} = {b} \ a = b"
by blast
lemma singleton_insert_inj_eq [iff]: "{b} = insert a A \ a = b \ A \ {b}"
by blast
lemma singleton_insert_inj_eq' [iff]: "insert a A = {b} \ a = b \ A \ {b}"
by blast
lemma subset_singletonD: "A \ {x} \ A = {} \ A = {x}"
by fast
lemma subset_singleton_iff: "X \ {a} \ X = {} \ X = {a}"
by blast
lemma subset_singleton_iff_Uniq: "(\a. A \ {a}) \ (\\<^sub>\\<^sub>1x. x \ A)"
unfolding Uniq_def by blast
lemma singleton_conv [simp]: "{x. x = a} = {a}"
by blast
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
by blast
lemma Diff_single_insert: "A - {x} \ B \ A \ insert x B"
by blast
lemma subset_Diff_insert: "A \ B - insert x C \ A \ B - C \ x \ A"
by blast
lemma doubleton_eq_iff: "{a, b} = {c, d} \ a = c \ b = d \ a = d \ b = c"
by (blast elim: equalityE)
lemma Un_singleton_iff: "A \ B = {x} \ A = {} \ B = {x} \ A = {x} \ B = {} \ A = {x} \ B = {x}"
by auto
lemma singleton_Un_iff: "{x} = A \ B \ A = {} \ B = {x} \ A = {x} \ B = {} \ A = {x} \ B = {x}"
by auto
subsubsection \<open>Image of a set under a function\<close>
text \<open>Frequently \<open>b\<close> does not have the syntactic form of \<open>f x\<close>.\<close>
definition image :: "('a \ 'b) \ 'a set \ 'b set" (infixr "`" 90)
where "f ` A = {y. \x\A. y = f x}"
lemma image_eqI [simp, intro]: "b = f x \ x \ A \ b \ f ` A"
unfolding image_def by blast
lemma imageI: "x \ A \ f x \ f ` A"
by (rule image_eqI) (rule refl)
lemma rev_image_eqI: "x \ A \ b = f x \ b \ f ` A"
\<comment> \<open>This version's more effective when we already have the required \<open>x\<close>.\<close>
by (rule image_eqI)
lemma imageE [elim!]:
assumes "b \ (\x. f x) ` A" \ \The eta-expansion gives variable-name preservation.\
obtains x where "b = f x" and "x \ A"
using assms unfolding image_def by blast
lemma Compr_image_eq: "{x \ f ` A. P x} = f ` {x \ A. P (f x)}"
by auto
lemma image_Un: "f ` (A \ B) = f ` A \ f ` B"
by blast
lemma image_iff: "z \ f ` A \ (\x\A. z = f x)"
by blast
lemma image_subsetI: "(\x. x \ A \ f x \ B) \ f ` A \ B"
\<comment> \<open>Replaces the three steps \<open>subsetI\<close>, \<open>imageE\<close>,
\<open>hypsubst\<close>, but breaks too many existing proofs.\<close>
by blast
lemma image_subset_iff: "f ` A \ B \ (\x\A. f x \ B)"
\<comment> \<open>This rewrite rule would confuse users if made default.\<close>
by blast
lemma subset_imageE:
assumes "B \ f ` A"
obtains C where "C \ A" and "B = f ` C"
proof -
from assms have "B = f ` {a \ A. f a \ B}" by fast
moreover have "{a \ A. f a \ B} \ A" by blast
ultimately show thesis by (blast intro: that)
qed
lemma subset_image_iff: "B \ f ` A \ (\AA\A. B = f ` AA)"
by (blast elim: subset_imageE)
lemma image_ident [simp]: "(\x. x) ` Y = Y"
by blast
lemma image_empty [simp]: "f ` {} = {}"
by blast
lemma image_insert [simp]: "f ` insert a B = insert (f a) (f ` B)"
by blast
lemma image_constant: "x \ A \ (\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) = (\x. f (g x)) ` A"
by blast
lemma insert_image [simp]: "x \ A \ insert (f x) (f ` A) = f ` A"
by blast
lemma image_is_empty [iff]: "f ` A = {} \ A = {}"
by blast
lemma empty_is_image [iff]: "{} = f ` A \ A = {}"
by blast
lemma image_Collect: "f ` {x. P x} = {f x | x. P x}"
\<comment> \<open>NOT suitable as a default simp rule: the RHS isn't simpler than the LHS,
with its implicit quantifier and conjunction. Also image enjoys better
equational properties than does the RHS.\<close>
by blast
lemma if_image_distrib [simp]:
"(\x. if P x then f x else g x) ` S = f ` (S \ {x. P x}) \ g ` (S \ {x. \ P x})"
by auto
lemma image_cong:
"f ` M = g ` N" if "M = N" "\x. x \ N \ f x = g x"
using that by (simp add: image_def)
lemma image_cong_simp [cong]:
"f ` M = g ` N" if "M = N" "\x. x \ N =simp=> f x = g x"
using that image_cong [of M N f g] by (simp add: simp_implies_def)
lemma image_Int_subset: "f ` (A \ B) \ f ` A \ f ` B"
by blast
lemma image_diff_subset: "f ` A - f ` B \ f ` (A - B)"
by blast
lemma Setcompr_eq_image: "{f x |x. x \ A} = f ` A"
by blast
lemma setcompr_eq_image: "{f x |x. P x} = f ` {x. P x}"
by auto
lemma ball_imageD: "\x\f ` A. P x \ \x\A. P (f x)"
by simp
lemma bex_imageD: "\x\f ` A. P x \ \x\A. P (f x)"
by auto
lemma image_add_0 [simp]: "(+) (0::'a::comm_monoid_add) ` S = S"
by auto
text \<open>\<^medskip> Range of a function -- just an abbreviation for image!\<close>
abbreviation range :: "('a \ 'b) \ 'b set" \ \of function\
where "range f \ f ` UNIV"
lemma range_eqI: "b = f x \ b \ range f"
by simp
lemma rangeI: "f x \ range f"
by simp
lemma rangeE [elim?]: "b \ range (\x. f x) \ (\x. b = f x \ P) \ P"
by (rule imageE)
lemma full_SetCompr_eq: "{u. \x. u = f x} = range f"
by auto
lemma range_composition: "range (\x. f (g x)) = f ` range g"
by auto
lemma range_constant [simp]: "range (\_. x) = {x}"
by (simp add: image_constant)
lemma range_eq_singletonD: "range f = {a} \ f x = a"
by auto
subsubsection \<open>Some rules with \<open>if\<close>\<close>
text \<open>Elimination of \<open>{x. \<dots> \<and> x = t \<and> \<dots>}\<close>.\<close>
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 \<open>
Rewrite rules for boolean case-splitting: faster than \<open>if_split [split]\<close>.
\<close>
lemma if_split_eq1: "(if Q then x else y) = b \ (Q \ x = b) \ (\ Q \ y = b)"
by (rule if_split)
lemma if_split_eq2: "a = (if Q then x else y) \ (Q \ a = x) \ (\ Q \ a = y)"
by (rule if_split)
text \<open>
Split ifs on either side of the membership relation.
Not for \<open>[simp]\<close> -- can cause goals to blow up!
\<close>
lemma if_split_mem1: "(if Q then x else y) \ b \ (Q \ x \ b) \ (\ Q \ y \ b)"
by (rule if_split)
lemma if_split_mem2: "(a \ (if Q then x else y)) \ (Q \ a \ x) \ (\ Q \ a \ y)"
by (rule if_split [where P = "\S. a \ S"])
lemmas split_ifs = if_bool_eq_conj if_split_eq1 if_split_eq2 if_split_mem1 if_split_mem2
(*Would like to add these, but the existing code only searches for the
outer-level constant, which in this case is just Set.member; we instead need
to use term-nets to associate patterns with rules. Also, if a rule fails to
apply, then the formula should be kept.
[("uminus", Compl_iff RS iffD1), ("minus", [Diff_iff RS iffD1]),
("Int", [IntD1,IntD2]),
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
*)
subsection \<open>Further operations and lemmas\<close>
subsubsection \<open>The ``proper subset'' relation\<close>
lemma psubsetI [intro!]: "A \ B \ A \ B \ A \ B"
unfolding less_le by blast
lemma psubsetE [elim!]: "A \ B \ (A \ B \ \ B \ A \ R) \ R"
unfolding less_le by blast
lemma psubset_insert_iff:
"A \ insert x B \ (if x \ B then A \ B else if x \ A then A - {x} \ B else A \ B)"
by (auto simp add: less_le subset_insert_iff)
lemma psubset_eq: "A \ B \ A \ B \ A \ B"
by (simp only: less_le)
lemma psubset_imp_subset: "A \ B \ A \ B"
by (simp add: psubset_eq)
lemma psubset_trans: "A \ B \ B \ C \ A \ C"
unfolding less_le by (auto dest: subset_antisym)
lemma psubsetD: "A \ B \ c \ A \ c \ B"
unfolding less_le by (auto dest: subsetD)
lemma psubset_subset_trans: "A \ B \ B \ C \ A \ C"
by (auto simp add: psubset_eq)
lemma subset_psubset_trans: "A \ B \ B \ C \ A \ C"
by (auto simp add: psubset_eq)
lemma psubset_imp_ex_mem: "A \ B \ \b. b \ B - A"
unfolding less_le by blast
lemma atomize_ball: "(\x. x \ A \ P x) \ Trueprop (\x\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: "f ` A \ B \ image f ` Pow A \ Pow B"
by blast
lemma image_Pow_surj: "f ` A = B \ image f ` Pow A = Pow B"
by (blast elim: subset_imageE)
subsubsection \<open>Derived rules involving subsets.\<close>
text \<open>\<open>insert\<close>.\<close>
lemma subset_insertI: "B \ insert a B"
by (rule subsetI) (erule insertI2)
lemma subset_insertI2: "A \ B \ A \ insert b B"
by blast
lemma subset_insert: "x \ A \ A \ insert x B \ A \ B"
by blast
text \<open>\<^medskip> Finite Union -- the least upper bound of two sets.\<close>
lemma Un_upper1: "A \ A \ B"
by (fact sup_ge1)
lemma Un_upper2: "B \ A \ B"
by (fact sup_ge2)
lemma Un_least: "A \ C \ B \ C \ A \ B \ C"
by (fact sup_least)
text \<open>\<^medskip> Finite Intersection -- the greatest lower bound of two sets.\<close>
lemma Int_lower1: "A \ B \ A"
by (fact inf_le1)
lemma Int_lower2: "A \ B \ B"
by (fact inf_le2)
lemma Int_greatest: "C \ A \ C \ B \ C \ A \ B"
by (fact inf_greatest)
text \<open>\<^medskip> Set difference.\<close>
lemma Diff_subset[simp]: "A - B \ A"
by blast
lemma Diff_subset_conv: "A - B \ C \ A \ B \ C"
by blast
subsubsection \<open>Equalities involving union, intersection, inclusion, etc.\<close>
text \<open>\<open>{}\<close>.\<close>
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
\<comment> \<open>supersedes \<open>Collect_False_empty\<close>\<close>
by auto
lemma subset_empty [simp]: "A \ {} \ A = {}"
by (fact bot_unique)
lemma not_psubset_empty [iff]: "\ (A < {})"
by (fact not_less_bot) (* FIXME: already simp *)
lemma Collect_subset [simp]: "{x\A. P x} \ A" by auto
lemma Collect_empty_eq [simp]: "Collect P = {} \ (\x. \ P x)"
by blast
lemma empty_Collect_eq [simp]: "{} = Collect P \ (\x. \ P x)"
by blast
lemma Collect_neg_eq: "{x. \ P x} = - {x. P x}"
by blast
lemma Collect_disj_eq: "{x. P x \ Q x} = {x. P x} \ {x. Q x}"
by blast
lemma Collect_imp_eq: "{x. P x \ Q x} = - {x. P x} \ {x. Q x}"
by blast
lemma Collect_conj_eq: "{x. P x \ Q x} = {x. P x} \ {x. Q x}"
by blast
lemma Collect_mono_iff: "Collect P \ Collect Q \ (\x. P x \ Q x)"
by blast
text \<open>\<^medskip> \<open>insert\<close>.\<close>
lemma insert_is_Un: "insert a A = {a} \ A"
\<comment> \<open>NOT SUITABLE FOR REWRITING since \<open>{a} \<equiv> insert a {}\<close>\<close>
by blast
lemma insert_not_empty [simp]: "insert a A \ {}"
and empty_not_insert [simp]: "{} \ insert a A"
by blast+
lemma insert_absorb: "a \ A \ insert a A = A"
\<comment> \<open>\<open>[simp]\<close> causes recursive calls when there are nested inserts\<close>
\<comment> \<open>with \<^emph>\<open>quadratic\<close> running time\<close>
by blast
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
by blast
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
by blast
lemma insert_subset [simp]: "insert x A \ B \ x \ B \ A \ B"
by blast
lemma mk_disjoint_insert: "a \ A \ \B. A = insert a B \ a \ B"
\<comment> \<open>use new \<open>B\<close> rather than \<open>A - {a}\<close> to avoid infinite unfolding\<close>
by (rule exI [where x = "A - {a}"]) blast
lemma insert_Collect: "insert a (Collect P) = {u. u \ a \ P u}"
by auto
lemma insert_inter_insert [simp]: "insert a A \ insert a B = insert a (A \ B)"
by blast
lemma insert_disjoint [simp]:
"insert a A \ B = {} \ a \ B \ A \ B = {}"
"{} = insert a A \ B \ a \ B \ {} = A \ B"
by auto
lemma disjoint_insert [simp]:
"B \ insert a A = {} \ a \ B \ B \ A = {}"
"{} = A \ insert b B \ b \ A \ {} = A \ B"
by auto
text \<open>\<^medskip> \<open>Int\<close>\<close>
lemma Int_absorb: "A \ A = A"
by (fact inf_idem) (* already simp *)
lemma Int_left_absorb: "A \ (A \ B) = A \ B"
by (fact inf_left_idem)
lemma Int_commute: "A \ B = B \ A"
by (fact inf_commute)
lemma Int_left_commute: "A \ (B \ C) = B \ (A \ C)"
by (fact inf_left_commute)
lemma Int_assoc: "(A \ B) \ C = A \ (B \ C)"
by (fact inf_assoc)
lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
\<comment> \<open>Intersection is an AC-operator\<close>
lemma Int_absorb1: "B \ A \ A \ B = B"
by (fact inf_absorb2)
lemma Int_absorb2: "A \ B \ A \ B = A"
by (fact inf_absorb1)
lemma Int_empty_left: "{} \ B = {}"
by (fact inf_bot_left) (* already simp *)
lemma Int_empty_right: "A \ {} = {}"
by (fact inf_bot_right) (* already simp *)
lemma disjoint_eq_subset_Compl: "A \ B = {} \ A \ - B"
by blast
lemma disjoint_iff: "A \ B = {} \ (\x. x\A \ x \ B)"
by blast
lemma disjoint_iff_not_equal: "A \ B = {} \ (\x\A. \y\B. x \ y)"
by blast
lemma Int_UNIV_left: "UNIV \ B = B"
by (fact inf_top_left) (* already simp *)
lemma Int_UNIV_right: "A \ UNIV = A"
by (fact inf_top_right) (* already simp *)
lemma Int_Un_distrib: "A \ (B \ C) = (A \ B) \ (A \ C)"
by (fact inf_sup_distrib1)
lemma Int_Un_distrib2: "(B \ C) \ A = (B \ A) \ (C \ A)"
by (fact inf_sup_distrib2)
lemma Int_UNIV [simp]: "A \ B = UNIV \ A = UNIV \ B = UNIV"
by (fact inf_eq_top_iff) (* already simp *)
lemma Int_subset_iff [simp]: "C \ A \ B \ C \ A \ C \ B"
by (fact le_inf_iff)
lemma Int_Collect: "x \ A \ {x. P x} \ x \ A \ P x"
by blast
text \<open>\<^medskip> \<open>Un\<close>.\<close>
lemma Un_absorb: "A \ A = A"
by (fact sup_idem) (* already simp *)
lemma Un_left_absorb: "A \ (A \ B) = A \ B"
by (fact sup_left_idem)
lemma Un_commute: "A \ B = B \ A"
by (fact sup_commute)
lemma Un_left_commute: "A \ (B \ C) = B \ (A \ C)"
by (fact sup_left_commute)
lemma Un_assoc: "(A \ B) \ C = A \ (B \ C)"
by (fact sup_assoc)
lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
\<comment> \<open>Union is an AC-operator\<close>
lemma Un_absorb1: "A \ B \ A \ B = B"
by (fact sup_absorb2)
lemma Un_absorb2: "B \ A \ A \ B = A"
by (fact sup_absorb1)
lemma Un_empty_left: "{} \ B = B"
by (fact sup_bot_left) (* already simp *)
lemma Un_empty_right: "A \ {} = A"
by (fact sup_bot_right) (* already simp *)
lemma Un_UNIV_left: "UNIV \ B = UNIV"
by (fact sup_top_left) (* already simp *)
lemma Un_UNIV_right: "A \ UNIV = UNIV"
by (fact sup_top_right) (* already simp *)
lemma Un_insert_left [simp]: "(insert a B) \ C = insert a (B \ C)"
by blast
lemma Un_insert_right [simp]: "A \ (insert a B) = insert a (A \ B)"
by blast
lemma Int_insert_left: "(insert a B) \ C = (if a \ C then insert a (B \ C) else B \ C)"
by auto
lemma Int_insert_left_if0 [simp]: "a \ C \ (insert a B) \ C = B \ C"
by auto
lemma Int_insert_left_if1 [simp]: "a \ C \ (insert a B) \ C = insert a (B \ C)"
by auto
lemma Int_insert_right: "A \ (insert a B) = (if a \ A then insert a (A \ B) else A \ B)"
by auto
lemma Int_insert_right_if0 [simp]: "a \ A \ A \ (insert a B) = A \ B"
by auto
lemma Int_insert_right_if1 [simp]: "a \ A \ A \ (insert a B) = insert a (A \ B)"
by auto
lemma Un_Int_distrib: "A \ (B \ C) = (A \ B) \ (A \ C)"
by (fact sup_inf_distrib1)
lemma Un_Int_distrib2: "(B \ C) \ A = (B \ A) \ (C \ A)"
by (fact sup_inf_distrib2)
lemma Un_Int_crazy: "(A \ B) \ (B \ C) \ (C \ A) = (A \ B) \ (B \ C) \ (C \ A)"
by blast
lemma subset_Un_eq: "A \ B \ A \ B = B"
by (fact le_iff_sup)
lemma Un_empty [iff]: "A \ B = {} \ A = {} \ B = {}"
by (fact sup_eq_bot_iff) (* FIXME: already simp *)
lemma Un_subset_iff [simp]: "A \ B \ C \ A \ C \ B \ C"
by (fact le_sup_iff)
lemma Un_Diff_Int: "(A - B) \ (A \ B) = A"
by blast
lemma Diff_Int2: "A \ C - B \ C = A \ C - B"
by blast
lemma subset_UnE:
assumes "C \ A \ B"
obtains A' B' where "A' \ A" "B' \ B" "C = A' \ B'"
proof
show "C \ A \ A" "C \ B \ B" "C = (C \ A) \ (C \ B)"
using assms by blast+
qed
lemma Un_Int_eq [simp]: "(S \ T) \ S = S" "(S \ T) \ T = T" "S \ (S \ T) = S" "T \ (S \ T) = T"
by auto
lemma Int_Un_eq [simp]: "(S \ T) \ S = S" "(S \ T) \ T = T" "S \ (S \ T) = S" "T \ (S \ T) = T"
by auto
text \<open>\<^medskip> Set complement\<close>
lemma Compl_disjoint [simp]: "A \ - A = {}"
by (fact inf_compl_bot)
lemma Compl_disjoint2 [simp]: "- A \ A = {}"
by (fact compl_inf_bot)
lemma Compl_partition: "A \ - A = UNIV"
by (fact sup_compl_top)
lemma Compl_partition2: "- A \ A = UNIV"
by (fact compl_sup_top)
lemma double_complement: "- (-A) = A" for A :: "'a set"
by (fact double_compl) (* already simp *)
lemma Compl_Un: "- (A \ B) = (- A) \ (- B)"
by (fact compl_sup) (* already simp *)
lemma Compl_Int: "- (A \ B) = (- A) \ (- B)"
by (fact compl_inf) (* already simp *)
lemma subset_Compl_self_eq: "A \ - A \ A = {}"
by blast
lemma Un_Int_assoc_eq: "(A \ B) \ C = A \ (B \ C) \ C \ A"
\<comment> \<open>Halmos, Naive Set Theory, page 16.\<close>
by blast
lemma Compl_UNIV_eq: "- UNIV = {}"
by (fact compl_top_eq) (* already simp *)
lemma Compl_empty_eq: "- {} = UNIV"
by (fact compl_bot_eq) (* already simp *)
lemma Compl_subset_Compl_iff [iff]: "- A \ - B \ B \ A"
by (fact compl_le_compl_iff) (* FIXME: already simp *)
lemma Compl_eq_Compl_iff [iff]: "- A = - B \ A = B"
for A B :: "'a set"
by (fact compl_eq_compl_iff) (* FIXME: already simp *)
lemma Compl_insert: "- insert x A = (- A) - {x}"
by blast
text \<open>\<^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 \<open>Int\<close>.
\<close>
lemma ball_Un: "(\x \ A \ B. P x) \ (\x\A. P x) \ (\x\B. P x)"
by blast
lemma bex_Un: "(\x \ A \ B. P x) \ (\x\A. P x) \ (\x\B. P x)"
by blast
text \<open>\<^medskip> Set difference.\<close>
lemma Diff_eq: "A - B = A \ (- B)"
by blast
lemma Diff_eq_empty_iff [simp]: "A - B = {} \ A \ B"
by blast
lemma Diff_cancel [simp]: "A - A = {}"
by blast
lemma Diff_idemp [simp]: "(A - B) - B = A - B"
for A B :: "'a set"
by blast
lemma Diff_triv: "A \ B = {} \ A - B = A"
by (blast elim: equalityE)
lemma empty_Diff [simp]: "{} - A = {}"
by blast
lemma Diff_empty [simp]: "A - {} = A"
by blast
lemma Diff_UNIV [simp]: "A - UNIV = {}"
by blast
lemma Diff_insert0 [simp]: "x \ A \ A - insert x B = A - B"
by blast
lemma Diff_insert: "A - insert a B = A - B - {a}"
\<comment> \<open>NOT SUITABLE FOR REWRITING since \<open>{a} \<equiv> insert a 0\<close>\<close>
by blast
lemma Diff_insert2: "A - insert a B = A - {a} - B"
\<comment> \<open>NOT SUITABLE FOR REWRITING since \<open>{a} \<equiv> insert a 0\<close>\<close>
by blast
lemma insert_Diff_if: "insert x A - B = (if x \ B then A - B else insert x (A - B))"
by auto
lemma insert_Diff1 [simp]: "x \ B \ insert x A - B = A - B"
by blast
lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"
by blast
lemma insert_Diff: "a \ A \ insert a (A - {a}) = A"
by blast
lemma Diff_insert_absorb: "x \ A \ (insert x A) - {x} = A"
by auto
lemma Diff_disjoint [simp]: "A \ (B - A) = {}"
by blast
lemma Diff_partition: "A \ B \ A \ (B - A) = B"
by blast
lemma double_diff: "A \ B \ B \ C \ B - (C - A) = A"
by blast
lemma Un_Diff_cancel [simp]: "A \ (B - A) = A \ B"
by blast
lemma Un_Diff_cancel2 [simp]: "(B - A) \ A = B \ A"
by blast
lemma Diff_Un: "A - (B \ C) = (A - B) \ (A - C)"
by blast
lemma Diff_Int: "A - (B \ C) = (A - B) \ (A - C)"
by blast
lemma Diff_Diff_Int: "A - (A - B) = A \ B"
by blast
lemma Un_Diff: "(A \ B) - C = (A - C) \ (B - C)"
by blast
lemma Int_Diff: "(A \ B) - C = A \ (B - C)"
by blast
lemma Diff_Int_distrib: "C \ (A - B) = (C \ A) - (C \ B)"
by blast
lemma Diff_Int_distrib2: "(A - B) \ C = (A \ C) - (B \ C)"
by blast
lemma Diff_Compl [simp]: "A - (- B) = A \ B"
by auto
lemma Compl_Diff_eq [simp]: "- (A - B) = - A \ B"
by blast
lemma subset_Compl_singleton [simp]: "A \ - {b} \ b \ A"
by blast
text \<open>\<^medskip> Quantification over type \<^typ>\<open>bool\<close>.\<close>
lemma bool_induct: "P True \ P False \ P x"
by (cases x) auto
lemma all_bool_eq: "(\b. P b) \ P True \ P False"
by (auto intro: bool_induct)
lemma bool_contrapos: "P x \ \ P False \ P True"
by (cases x) auto
lemma ex_bool_eq: "(\b. P b) \ P True \ P False"
by (auto intro: bool_contrapos)
lemma UNIV_bool: "UNIV = {False, True}"
by (auto intro: bool_induct)
text \<open>\<^medskip> \<open>Pow\<close>\<close>
lemma Pow_empty [simp]: "Pow {} = {{}}"
by (auto simp add: Pow_def)
lemma Pow_singleton_iff [simp]: "Pow X = {Y} \ X = {} \ Y = {}"
by blast (* somewhat slow *)
lemma Pow_insert: "Pow (insert a A) = Pow A \ (insert a ` Pow A)"
by (blast intro: image_eqI [where ?x = "u - {a}" for u])
lemma Pow_Compl: "Pow (- A) = {- B | B. A \ Pow B}"
by (blast intro: exI [where ?x = "- u" for u])
lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"
by blast
lemma Un_Pow_subset: "Pow A \ Pow B \ Pow (A \ B)"
by blast
lemma Pow_Int_eq [simp]: "Pow (A \ B) = Pow A \ Pow B"
by blast
text \<open>\<^medskip> Miscellany.\<close>
lemma set_eq_subset: "A = B \ A \ B \ B \ A"
by blast
lemma subset_iff: "A \ B \ (\t. t \ A \ t \ B)"
by blast
lemma subset_iff_psubset_eq: "A \ B \ A \ B \ A = B"
unfolding less_le by blast
lemma all_not_in_conv [simp]: "(\x. x \ A) \ A = {}"
by blast
lemma ex_in_conv: "(\x. x \ A) \ A \ {}"
by blast
lemma ball_simps [simp, no_atp]:
"\A P Q. (\x\A. P x \ Q) \ ((\x\A. P x) \ Q)"
"\A P Q. (\x\A. P \ Q x) \ (P \ (\x\A. Q x))"
"\A P Q. (\x\A. P \ Q x) \ (P \ (\x\A. Q x))"
"\A P Q. (\x\A. P x \ Q) \ ((\x\A. P x) \ Q)"
"\P. (\x\{}. P x) \ True"
"\P. (\x\UNIV. P x) \ (\x. P x)"
"\a B P. (\x\insert a B. P x) \ (P a \ (\x\B. P x))"
"\P Q. (\x\Collect Q. P x) \ (\x. Q x \ P x)"
"\A P f. (\x\f`A. P x) \ (\x\A. P (f x))"
"\A P. (\ (\x\A. P x)) \ (\x\A. \ P x)"
by auto
lemma bex_simps [simp, no_atp]:
"\A P Q. (\x\A. P x \ Q) \ ((\x\A. P x) \ Q)"
"\A P Q. (\x\A. P \ Q x) \ (P \ (\x\A. Q x))"
"\P. (\x\{}. P x) \ False"
"\P. (\x\UNIV. P x) \ (\x. P x)"
"\a B P. (\x\insert a B. P x) \ (P a \ (\x\B. P x))"
"\P Q. (\x\Collect Q. P x) \ (\x. Q x \ P x)"
"\A P f. (\x\f`A. P x) \ (\x\A. P (f x))"
"\A P. (\(\x\A. P x)) \ (\x\A. \ P x)"
by auto
lemma ex_image_cong_iff [simp, no_atp]:
"(\x. x\f`A) \ A \ {}" "(\x. x\f`A \ P x) \ (\x\A. P (f x))"
by auto
subsubsection \<open>Monotonicity of various operations\<close>
lemma image_mono: "A \ B \ f ` A \ f ` B"
by blast
lemma Pow_mono: "A \ B \ Pow A \ Pow B"
by blast
lemma insert_mono: "C \ D \ insert a C \ insert a D"
by blast
lemma Un_mono: "A \ C \ B \ D \ A \ B \ C \ D"
by (fact sup_mono)
lemma Int_mono: "A \ C \ B \ D \ A \ B \ C \ D"
by (fact inf_mono)
lemma Diff_mono: "A \ C \ D \ B \ A - B \ C - D"
by blast
lemma Compl_anti_mono: "A \ B \ - B \ - A"
by (fact compl_mono)
text \<open>\<^medskip> Monotonicity of implications.\<close>
lemma in_mono: "A \ B \ x \ A \ x \ B"
by (rule impI) (erule subsetD)
lemma conj_mono: "P1 \ Q1 \ P2 \ Q2 \ (P1 \ P2) \ (Q1 \ Q2)"
by iprover
lemma disj_mono: "P1 \ Q1 \ P2 \ Q2 \ (P1 \ P2) \ (Q1 \ Q2)"
by iprover
lemma imp_mono: "Q1 \ P1 \ P2 \ Q2 \ (P1 \ P2) \ (Q1 \ Q2)"
by iprover
lemma imp_refl: "P \ P" ..
lemma not_mono: "Q \ P \ \ P \ \ Q"
by iprover
lemma ex_mono: "(\x. P x \ Q x) \ (\x. P x) \ (\x. Q x)"
by iprover
lemma all_mono: "(\x. P x \ Q x) \ (\x. P x) \ (\x. Q x)"
by iprover
lemma Collect_mono: "(\x. P x \ Q x) \ Collect P \ Collect Q"
by blast
lemma Int_Collect_mono: "A \ B \ (\x. x \ A \ P x \ Q x) \ A \ Collect P \ B \ Collect Q"
by blast
lemmas basic_monos =
subset_refl imp_refl disj_mono conj_mono ex_mono Collect_mono in_mono
lemma eq_to_mono: "a = b \ c = d \ b \ d \ a \ c"
by iprover
subsubsection \<open>Inverse image of a function\<close>
definition vimage :: "('a \ 'b) \ 'b set \ 'a set" (infixr "-`" 90)
where "f -` B \ {x. f x \ B}"
lemma vimage_eq [simp]: "a \ f -` B \ f a \ B"
unfolding vimage_def by blast
lemma vimage_singleton_eq: "a \ f -` {b} \ f a = b"
by simp
lemma vimageI [intro]: "f a = b \ b \ B \ a \ f -` B"
unfolding vimage_def by blast
lemma vimageI2: "f a \ A \ a \ f -` A"
unfolding vimage_def by fast
lemma vimageE [elim!]: "a \ f -` B \ (\x. f a = x \ x \ B \ P) \ P"
unfolding vimage_def by blast
lemma vimageD: "a \ f -` A \ f a \ A"
unfolding vimage_def by fast
lemma vimage_empty [simp]: "f -` {} = {}"
by blast
lemma vimage_Compl: "f -` (- A) = - (f -` A)"
by blast
lemma vimage_Un [simp]: "f -` (A \ B) = (f -` A) \ (f -` B)"
by blast
lemma vimage_Int [simp]: "f -` (A \ B) = (f -` A) \ (f -` B)"
by fast
lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}"
by blast
lemma vimage_Collect: "(\x. P (f x) = Q x) \ f -` (Collect P) = Collect Q"
by blast
lemma vimage_insert: "f -` (insert a B) = (f -` {a}) \ (f -` B)"
\<comment> \<open>NOT suitable for rewriting because of the recurrence of \<open>{a}\<close>.\<close>
by blast
lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)"
by blast
lemma vimage_UNIV [simp]: "f -` UNIV = UNIV"
by blast
lemma vimage_mono: "A \ B \ f -` A \ f -` B"
\<comment> \<open>monotonicity\<close>
by blast
lemma vimage_image_eq: "f -` (f ` A) = {y. \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 \ range f"
by blast
lemma image_subset_iff_subset_vimage: "f ` A \ B \ A \ f -` B"
by blast
lemma vimage_const [simp]: "((\x. c) -` A) = (if c \ A then UNIV else {})"
by auto
lemma vimage_if [simp]: "((\x. if x \ 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: "(\ w. w \ S \ f w = g w) \ f -` y \ S = g -` y \ S"
by auto
lemma vimage_ident [simp]: "(\x. x) -` Y = Y"
by blast
subsubsection \<open>Singleton sets\<close>
definition is_singleton :: "'a set \ bool"
where "is_singleton A \ (\x. A = {x})"
lemma is_singletonI [simp, intro!]: "is_singleton {x}"
unfolding is_singleton_def by simp
lemma is_singletonI': "A \ {} \ (\x y. x \ A \ y \ A \ x = y) \ is_singleton A"
unfolding is_singleton_def by blast
lemma is_singletonE: "is_singleton A \ (\x. A = {x} \ P) \ P"
unfolding is_singleton_def by blast
subsubsection \<open>Getting the contents of a singleton set\<close>
definition the_elem :: "'a set \ 'a"
where "the_elem X = (THE x. X = {x})"
lemma the_elem_eq [simp]: "the_elem {x} = x"
--> --------------------
--> maximum size reached
--> --------------------
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