(* 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 \<Rightarrow> bool) \<Rightarrow> 'a set" \<comment> \<open>comprehension\<close>
and member :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" \<comment> \<open>membership\<close>
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 ("'(\<in>')") and
member ("(_/ \<in> _)" [51, 51] 50)
abbreviation not_member
where "not_member x A \<equiv> \<not> (x \<in> A)" \<comment> \<open>non-membership\<close>
notation
not_member ("'(\<notin>')") and
not_member ("(_/ \<notin> _)" [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 \<Rightarrow> bool \<Rightarrow> 'a set" ("(1{_./ _})")
translations
"{x. P}" \<rightleftharpoons> "CONST Collect (\<lambda>x. P)"
syntax (ASCII)
"_Collect" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a set" ("(1{(_/: _)./ _})")
syntax
"_Collect" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a set" ("(1{(_/ \<in> _)./ _})")
translations
"{p:A. P}" \<rightharpoonup> "CONST Collect (\<lambda>p. p \<in> A \<and> 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) \<Longrightarrow> {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 \<and> Q x}") = \<open>
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 "\<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)
lemma Collect_eqI:
assumes "\<And>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 \<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
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 "{} \<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 \<Rightarrow> '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 \<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
subset ("'(\<subset>')") and
subset ("(_/ \<subset> _)" [51, 51] 50) and
subset_eq ("'(\<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
supset ("'(\<supset>')") and
supset ("(_/ \<supset> _)" [51, 51] 50) and
supset_eq ("'(\<supseteq>')") and
supset_eq ("(_/ \<supseteq> _)" [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 \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
where "Ball A P \<longleftrightarrow> (\<forall>x. x \<in> A \<longrightarrow> P x)" \<comment> \<open>bounded universal quantifiers\<close>
definition Bex :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
where "Bex A P \<longleftrightarrow> (\<exists>x. x \<in> A \<and> P x)" \<comment> \<open>bounded existential quantifiers\<close>
syntax (ASCII)
"_Ball" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3ALL (_/:_)./ _)" [0, 0, 10] 10)
"_Bex" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3EX (_/:_)./ _)" [0, 0, 10] 10)
"_Bex1" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3EX! (_/:_)./ _)" [0, 0, 10] 10)
"_Bleast" :: "id \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a" ("(3LEAST (_/:_)./ _)" [0, 0, 10] 10)
syntax (input)
"_Ball" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3! (_/:_)./ _)" [0, 0, 10] 10)
"_Bex" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3? (_/:_)./ _)" [0, 0, 10] 10)
"_Bex1" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3?! (_/:_)./ _)" [0, 0, 10] 10)
syntax
"_Ball" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3\<forall>(_/\<in>_)./ _)" [0, 0, 10] 10)
"_Bex" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>(_/\<in>_)./ _)" [0, 0, 10] 10)
"_Bex1" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>!(_/\<in>_)./ _)" [0, 0, 10] 10)
"_Bleast" :: "id \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a" ("(3LEAST(_/\<in>_)./ _)" [0, 0, 10] 10)
translations
"\<forall>x\<in>A. P" \<rightleftharpoons> "CONST Ball A (\<lambda>x. P)"
"\<exists>x\<in>A. P" \<rightleftharpoons> "CONST Bex A (\<lambda>x. P)"
"\<exists>!x\<in>A. P" \<rightharpoonup> "\<exists>!x. x \<in> A \<and> P"
"LEAST x:A. P" \<rightharpoonup> "LEAST x. x \<in> A \<and> P"
syntax (ASCII output)
"_setlessAll" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3ALL _<_./ _)" [0, 0, 10] 10)
"_setlessEx" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3EX _<_./ _)" [0, 0, 10] 10)
"_setleAll" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10)
"_setleEx" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3EX _<=_./ _)" [0, 0, 10] 10)
"_setleEx1" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3EX! _<=_./ _)" [0, 0, 10] 10)
syntax
"_setlessAll" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10)
"_setlessEx" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10)
"_setleAll" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
"_setleEx" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
"_setleEx1" :: "[idt, 'a, bool] \<Rightarrow> bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
translations
"\<forall>A\<subset>B. P" \<rightharpoonup> "\<forall>A. A \<subset> B \<longrightarrow> P"
"\<exists>A\<subset>B. P" \<rightharpoonup> "\<exists>A. A \<subset> B \<and> P"
"\<forall>A\<subseteq>B. P" \<rightharpoonup> "\<forall>A. A \<subseteq> B \<longrightarrow> P"
"\<exists>A\<subseteq>B. P" \<rightharpoonup> "\<exists>A. A \<subseteq> B \<and> P"
"\<exists>!A\<subseteq>B. P" \<rightharpoonup> "\<exists>!A. A \<subseteq> B \<and> P"
print_translation \<open>
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 (=) 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 \<Rightarrow> idts \<Rightarrow> bool \<Rightarrow> 'a set" ("(1{_ |/_./ _})")
parse_translation \<open>
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;
\<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 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 (=) (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;
\<close>
simproc_setup defined_Bex ("\<exists>x\<in>A. P x \<and> Q x") = \<open>
fn _ => Quantifier1.rearrange_bex
(fn ctxt =>
unfold_tac ctxt @{thms Bex_def} THEN
Quantifier1.prove_one_point_ex_tac ctxt)
\<close>
simproc_setup defined_All ("\<forall>x\<in>A. P x \<longrightarrow> Q x") = \<open>
fn _ => Quantifier1.rearrange_ball
(fn ctxt =>
unfold_tac ctxt @{thms Ball_def} THEN
Quantifier1.prove_one_point_all_tac ctxt)
\<close>
lemma ballI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> P x) \<Longrightarrow> \<forall>x\<in>A. P x"
by (simp add: Ball_def)
lemmas strip = impI allI ballI
lemma bspec [dest?]: "\<forall>x\<in>A. P x \<Longrightarrow> x \<in> A \<Longrightarrow> 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 Ball}, @{thms bspec})] @ mksimps_pairs;
end;
open Simpdata;
\<close>
declaration \<open>fn _ => Simplifier.map_ss (Simplifier.set_mksimps (mksimps mksimps_pairs))\<close>
lemma ballE [elim]: "\<forall>x\<in>A. P x \<Longrightarrow> (P x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q"
unfolding Ball_def by blast
lemma bexI [intro]: "P x \<Longrightarrow> x \<in> A \<Longrightarrow> \<exists>x\<in>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 \<in> A \<Longrightarrow> P x \<Longrightarrow> \<exists>x\<in>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: "(\<forall>x\<in>A. \<not> P x \<Longrightarrow> P a) \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>x\<in>A. P x"
unfolding Bex_def by blast
lemma bexE [elim!]: "\<exists>x\<in>A. P x \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P x \<Longrightarrow> Q) \<Longrightarrow> Q"
unfolding Bex_def by blast
lemma ball_triv [simp]: "(\<forall>x\<in>A. P) \<longleftrightarrow> ((\<exists>x. x \<in> A) \<longrightarrow> P)"
\<comment> \<open>Trival rewrite rule.\<close>
by (simp add: Ball_def)
lemma bex_triv [simp]: "(\<exists>x\<in>A. P) \<longleftrightarrow> ((\<exists>x. x \<in> A) \<and> P)"
\<comment> \<open>Dual form for existentials.\<close>
by (simp add: Bex_def)
lemma bex_triv_one_point1 [simp]: "(\<exists>x\<in>A. x = a) \<longleftrightarrow> a \<in> A"
by blast
lemma bex_triv_one_point2 [simp]: "(\<exists>x\<in>A. a = x) \<longleftrightarrow> a \<in> A"
by blast
lemma bex_one_point1 [simp]: "(\<exists>x\<in>A. x = a \<and> P x) \<longleftrightarrow> a \<in> A \<and> P a"
by blast
lemma bex_one_point2 [simp]: "(\<exists>x\<in>A. a = x \<and> P x) \<longleftrightarrow> a \<in> A \<and> P a"
by blast
lemma ball_one_point1 [simp]: "(\<forall>x\<in>A. x = a \<longrightarrow> P x) \<longleftrightarrow> (a \<in> A \<longrightarrow> P a)"
by blast
lemma ball_one_point2 [simp]: "(\<forall>x\<in>A. a = x \<longrightarrow> P x) \<longleftrightarrow> (a \<in> A \<longrightarrow> 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 \<open>Congruence rules\<close>
lemma ball_cong:
"A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow>
(\<forall>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>B. Q x)"
by (simp add: Ball_def)
lemma strong_ball_cong [cong]:
"A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> P x \<longleftrightarrow> Q x) \<Longrightarrow>
(\<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 \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> P x \<longleftrightarrow> Q x) \<Longrightarrow>
(\<exists>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>B. Q x)"
by (simp add: Bex_def cong: conj_cong)
lemma strong_bex_cong [cong]:
"A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> P x \<longleftrightarrow> Q x) \<Longrightarrow>
(\<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: "(\<exists>!x\<in>X. P x) \<longleftrightarrow> (\<exists>x\<in>X. P x) \<and> (\<forall>x\<in>X. \<forall>y\<in>X. P x \<longrightarrow> P y \<longrightarrow> x = y)"
by auto
subsection \<open>Basic operations\<close>
subsubsection \<open>Subsets\<close>
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 \<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 \<subseteq> B \<Longrightarrow> c \<in> A \<Longrightarrow> c \<in> B"
by (simp add: less_eq_set_def le_fun_def)
\<comment> \<open>Rule in Modus Ponens style.\<close>
lemma rev_subsetD [intro?]: "c \<in> A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> c \<in> B"
\<comment> \<open>The same, with reversed premises for use with @{method erule} -- cf. @{thm rev_mp}.\<close>
by (rule subsetD)
lemma subsetCE [elim]: "A \<subseteq> B \<Longrightarrow> (c \<notin> A \<Longrightarrow> P) \<Longrightarrow> (c \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
\<comment> \<open>Classical elimination rule.\<close>
by (auto simp add: less_eq_set_def le_fun_def)
lemma subset_eq: "A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"
by blast
lemma contra_subsetD: "A \<subseteq> B \<Longrightarrow> c \<notin> B \<Longrightarrow> c \<notin> A"
by blast
lemma subset_refl: "A \<subseteq> A"
by (fact order_refl) (* already [iff] *)
lemma subset_trans: "A \<subseteq> B \<Longrightarrow> B \<subseteq> C \<Longrightarrow> A \<subseteq> C"
by (fact order_trans)
lemma set_rev_mp: "x \<in> A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> x \<in> B"
by (rule subsetD)
lemma set_mp: "A \<subseteq> B \<Longrightarrow> x \<in> A \<Longrightarrow> x \<in> 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 \<Longrightarrow> b \<in> A \<Longrightarrow> a \<in> A"
by simp
lemmas basic_trans_rules [trans] =
order_trans_rules set_rev_mp set_mp eq_mem_trans
subsubsection \<open>Equality\<close>
lemma subset_antisym [intro!]: "A \<subseteq> B \<Longrightarrow> B \<subseteq> A \<Longrightarrow> 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 \<Longrightarrow> A \<subseteq> B"
by simp
lemma equalityD2: "A = B \<Longrightarrow> B \<subseteq> A"
by simp
text \<open>
\<^medskip>
Be careful when adding this to the claset as \<open>subset_empty\<close> is in the
simpset: @{prop "A = {}"} goes to @{prop "{} \<subseteq> A"} and @{prop "A \<subseteq> {}"}
and then back to @{prop "A = {}"}!
\<close>
lemma equalityE: "A = B \<Longrightarrow> (A \<subseteq> B \<Longrightarrow> B \<subseteq> A \<Longrightarrow> P) \<Longrightarrow> P"
by simp
lemma equalityCE [elim]: "A = B \<Longrightarrow> (c \<in> A \<Longrightarrow> c \<in> B \<Longrightarrow> P) \<Longrightarrow> (c \<notin> A \<Longrightarrow> c \<notin> B \<Longrightarrow> P) \<Longrightarrow> P"
by blast
lemma eqset_imp_iff: "A = B \<Longrightarrow> x \<in> A \<longleftrightarrow> x \<in> B"
by simp
lemma eqelem_imp_iff: "x = y \<Longrightarrow> x \<in> A \<longleftrightarrow> y \<in> 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 \<in> {} \<longleftrightarrow> False"
by (simp add: empty_def)
lemma emptyE [elim!]: "a \<in> {} \<Longrightarrow> P"
by simp
lemma empty_subsetI [iff]: "{} \<subseteq> A"
\<comment> \<open>One effect is to delete the ASSUMPTION @{prop "{} \<subseteq> A"}\<close>
by blast
lemma equals0I: "(\<And>y. y \<in> A \<Longrightarrow> False) \<Longrightarrow> A = {}"
by blast
lemma equals0D: "A = {} \<Longrightarrow> a \<notin> A"
\<comment> \<open>Use for reasoning about disjointness: \<open>A \<inter> B = {}\<close>\<close>
by blast
lemma ball_empty [simp]: "Ball {} P \<longleftrightarrow> True"
by (simp add: Ball_def)
lemma bex_empty [simp]: "Bex {} P \<longleftrightarrow> False"
by (simp add: Bex_def)
subsubsection \<open>The universal set -- UNIV\<close>
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 \<in> 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?]: "\<exists>x. x \<in> UNIV"
by simp
lemma subset_UNIV: "A \<subseteq> 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 \<longleftrightarrow> All P"
by (simp add: Ball_def)
lemma bex_UNIV [simp]: "Bex UNIV P \<longleftrightarrow> 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 \<noteq> {}"
by (blast elim: equalityE)
lemma empty_not_UNIV[simp]: "{} \<noteq> UNIV"
by blast
subsubsection \<open>The Powerset operator -- Pow\<close>
definition Pow :: "'a set \<Rightarrow> 'a set set"
where Pow_def: "Pow A = {B. B \<subseteq> A}"
lemma Pow_iff [iff]: "A \<in> Pow B \<longleftrightarrow> A \<subseteq> B"
by (simp add: Pow_def)
lemma PowI: "A \<subseteq> B \<Longrightarrow> A \<in> Pow B"
by (simp add: Pow_def)
lemma PowD: "A \<in> Pow B \<Longrightarrow> 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 \<open>Set complement\<close>
lemma Compl_iff [simp]: "c \<in> - A \<longleftrightarrow> c \<notin> A"
by (simp add: fun_Compl_def uminus_set_def)
lemma ComplI [intro!]: "(c \<in> A \<Longrightarrow> False) \<Longrightarrow> c \<in> - 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 \<in> - A \<Longrightarrow> c \<notin> A"
by simp
lemmas ComplE = ComplD [elim_format]
lemma Compl_eq: "- A = {x. \<not> x \<in> A}"
by blast
subsubsection \<open>Binary intersection\<close>
abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<inter>" 70)
where "(\<inter>) \<equiv> inf"
notation (ASCII)
inter (infixl "Int" 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 \<in> A \<inter> B \<longleftrightarrow> c \<in> A \<and> c \<in> B"
unfolding Int_def by blast
lemma IntI [intro!]: "c \<in> A \<Longrightarrow> c \<in> B \<Longrightarrow> c \<in> A \<inter> B"
by simp
lemma IntD1: "c \<in> A \<inter> B \<Longrightarrow> c \<in> A"
by simp
lemma IntD2: "c \<in> A \<inter> B \<Longrightarrow> c \<in> B"
by simp
lemma IntE [elim!]: "c \<in> A \<inter> B \<Longrightarrow> (c \<in> A \<Longrightarrow> c \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
by simp
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
by (fact mono_inf)
subsubsection \<open>Binary union\<close>
abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<union>" 65)
where "union \<equiv> sup"
notation (ASCII)
union (infixl "Un" 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 \<in> A \<union> B \<longleftrightarrow> c \<in> A \<or> c \<in> B"
unfolding Un_def by blast
lemma UnI1 [elim?]: "c \<in> A \<Longrightarrow> c \<in> A \<union> B"
by simp
lemma UnI2 [elim?]: "c \<in> B \<Longrightarrow> c \<in> A \<union> B"
by simp
text \<open>\<^medskip> Classical introduction rule: no commitment to \<open>A\<close> vs. \<open>B\<close>.\<close>
lemma UnCI [intro!]: "(c \<notin> B \<Longrightarrow> c \<in> A) \<Longrightarrow> c \<in> A \<union> B"
by auto
lemma UnE [elim!]: "c \<in> A \<union> B \<Longrightarrow> (c \<in> A \<Longrightarrow> P) \<Longrightarrow> (c \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
unfolding Un_def by 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 \<open>Set difference\<close>
lemma Diff_iff [simp]: "c \<in> A - B \<longleftrightarrow> c \<in> A \<and> c \<notin> B"
by (simp add: minus_set_def fun_diff_def)
lemma DiffI [intro!]: "c \<in> A \<Longrightarrow> c \<notin> B \<Longrightarrow> c \<in> A - B"
by simp
lemma DiffD1: "c \<in> A - B \<Longrightarrow> c \<in> A"
by simp
lemma DiffD2: "c \<in> A - B \<Longrightarrow> c \<in> B \<Longrightarrow> P"
by simp
lemma DiffE [elim!]: "c \<in> A - B \<Longrightarrow> (c \<in> A \<Longrightarrow> c \<notin> B \<Longrightarrow> P) \<Longrightarrow> P"
by simp
lemma set_diff_eq: "A - B = {x. x \<in> A \<and> x \<notin> B}"
by blast
lemma Compl_eq_Diff_UNIV: "- A = (UNIV - A)"
by blast
subsubsection \<open>Augmenting a set -- @{const insert}\<close>
lemma insert_iff [simp]: "a \<in> insert b A \<longleftrightarrow> a = b \<or> a \<in> A"
unfolding insert_def by blast
lemma insertI1: "a \<in> insert a B"
by simp
lemma insertI2: "a \<in> B \<Longrightarrow> a \<in> insert b B"
by simp
lemma insertE [elim!]: "a \<in> insert b A \<Longrightarrow> (a = b \<Longrightarrow> P) \<Longrightarrow> (a \<in> A \<Longrightarrow> P) \<Longrightarrow> P"
unfolding insert_def by blast
lemma insertCI [intro!]: "(a \<notin> B \<Longrightarrow> a = b) \<Longrightarrow> a \<in> insert b B"
\<comment> \<open>Classical introduction rule.\<close>
by auto
lemma subset_insert_iff: "A \<subseteq> insert x B \<longleftrightarrow> (if x \<in> 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
show "A = insert x (A - {x})" using assms by blast
show "x \<notin> A - {x}" by blast
qed
lemma insert_ident: "x \<notin> A \<Longrightarrow> x \<notin> B \<Longrightarrow> insert x A = insert x B \<longleftrightarrow> 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
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 \<and> b \<notin> ?C \<and> B = insert a ?C \<and> a \<notin> ?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 \<in> {a}"
\<comment> \<open>Redundant? But unlike \<open>insertCI\<close>, it proves the subgoal immediately!\<close>
by (rule insertI1)
lemma singletonD [dest!]: "b \<in> {a} \<Longrightarrow> b = a"
by blast
lemmas singletonE = singletonD [elim_format]
lemma singleton_iff: "b \<in> {a} \<longleftrightarrow> b = a"
by blast
lemma singleton_inject [dest!]: "{a} = {b} \<Longrightarrow> a = b"
by blast
lemma singleton_insert_inj_eq [iff]: "{b} = insert a A \<longleftrightarrow> a = b \<and> A \<subseteq> {b}"
by blast
lemma singleton_insert_inj_eq' [iff]: "insert a A = {b} \<longleftrightarrow> a = b \<and> A \<subseteq> {b}"
by blast
lemma subset_singletonD: "A \<subseteq> {x} \<Longrightarrow> A = {} \<or> A = {x}"
by fast
lemma subset_singleton_iff: "X \<subseteq> {a} \<longleftrightarrow> X = {} \<or> X = {a}"
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} \<subseteq> B \<Longrightarrow> A \<subseteq> insert x B"
by blast
lemma subset_Diff_insert: "A \<subseteq> B - insert x C \<longleftrightarrow> A \<subseteq> B - C \<and> x \<notin> A"
by blast
lemma doubleton_eq_iff: "{a, b} = {c, d} \<longleftrightarrow> a = c \<and> b = d \<or> a = d \<and> b = c"
by (blast elim: equalityE)
lemma Un_singleton_iff: "A \<union> B = {x} \<longleftrightarrow> 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 \<longleftrightarrow> A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> 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 \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "`" 90)
where "f ` A = {y. \<exists>x\<in>A. y = f x}"
lemma image_eqI [simp, intro]: "b = f x \<Longrightarrow> x \<in> A \<Longrightarrow> b \<in> f ` A"
unfolding image_def by blast
lemma imageI: "x \<in> A \<Longrightarrow> f x \<in> f ` A"
by (rule image_eqI) (rule refl)
lemma rev_image_eqI: "x \<in> A \<Longrightarrow> b = f x \<Longrightarrow> b \<in> 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 \<in> (\<lambda>x. f x) ` A" \<comment> \<open>The eta-expansion gives variable-name preservation.\<close>
obtains x where "b = f x" and "x \<in> A"
using assms unfolding image_def by 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 \<union> B) = f ` A \<union> f ` B"
by blast
lemma image_iff: "z \<in> f ` A \<longleftrightarrow> (\<exists>x\<in>A. z = f x)"
by blast
lemma image_subsetI: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` A \<subseteq> 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 \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. f x \<in> B)"
\<comment> \<open>This rewrite rule would confuse users if made default.\<close>
by blast
lemma subset_imageE:
assumes "B \<subseteq> f ` A"
obtains C where "C \<subseteq> A" and "B = f ` C"
proof -
from assms have "B = f ` {a \<in> A. f a \<in> B}" by fast
moreover have "{a \<in> A. f a \<in> B} \<subseteq> A" by blast
ultimately show thesis by (blast intro: that)
qed
lemma subset_image_iff: "B \<subseteq> f ` A \<longleftrightarrow> (\<exists>AA\<subseteq>A. B = f ` AA)"
by (blast elim: subset_imageE)
lemma image_ident [simp]: "(\<lambda>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 \<in> A \<Longrightarrow> (\<lambda>x. c) ` A = {c}"
by auto
lemma image_constant_conv: "(\<lambda>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 \<Longrightarrow> insert (f x) (f ` A) = f ` A"
by blast
lemma image_is_empty [iff]: "f ` A = {} \<longleftrightarrow> A = {}"
by blast
lemma empty_is_image [iff]: "{} = f ` A \<longleftrightarrow> 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]:
"(\<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
lemma image_cong: "\<lbrakk> M = N; \<And>x. x \<in> N \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow> f ` M = g ` N"
by (simp add: image_def)
lemma image_cong_strong: "\<lbrakk> M = N; \<And>x. x \<in> N =simp=> f x = g x\<rbrakk> \<Longrightarrow> f ` M = g ` N"
by (simp add: image_def simp_implies_def)
lemma image_Int_subset: "f ` (A \<inter> B) \<subseteq> f ` A \<inter> f ` B"
by blast
lemma image_diff_subset: "f ` A - f ` B \<subseteq> f ` (A - B)"
by blast
lemma Setcompr_eq_image: "{f x |x. x \<in> A} = f ` A"
by blast
lemma setcompr_eq_image: "{f x |x. P x} = f ` {x. P x}"
by auto
lemma ball_imageD: "\<forall>x\<in>f ` A. P x \<Longrightarrow> \<forall>x\<in>A. P (f x)"
by simp
lemma bex_imageD: "\<exists>x\<in>f ` A. P x \<Longrightarrow> \<exists>x\<in>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 \<Rightarrow> 'b) \<Rightarrow> 'b set" \<comment> \<open>of function\<close>
where "range f \<equiv> f ` UNIV"
lemma range_eqI: "b = f x \<Longrightarrow> 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) \<Longrightarrow> (\<And>x. b = f x \<Longrightarrow> P) \<Longrightarrow> P"
by (rule imageE)
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 auto
lemma range_constant [simp]: "range (\<lambda>_. x) = {x}"
by (simp add: image_constant)
lemma range_eq_singletonD: "range f = {a} \<Longrightarrow> 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 \<and> P x} = (if P a then {a} else {})"
by auto
lemma Collect_conv_if2: "{x. a = x \<and> 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 \<longleftrightarrow> (Q \<longrightarrow> x = b) \<and> (\<not> Q \<longrightarrow> y = b)"
by (rule if_split)
lemma if_split_eq2: "a = (if Q then x else y) \<longleftrightarrow> (Q \<longrightarrow> a = x) \<and> (\<not> Q \<longrightarrow> 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) \<in> b \<longleftrightarrow> (Q \<longrightarrow> x \<in> b) \<and> (\<not> Q \<longrightarrow> y \<in> b)"
by (rule if_split)
lemma if_split_mem2: "(a \<in> (if Q then x else y)) \<longleftrightarrow> (Q \<longrightarrow> a \<in> x) \<and> (\<not> Q \<longrightarrow> a \<in> y)"
by (rule if_split [where P = "\<lambda>S. a \<in> 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 \<subseteq> B \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<subset> B"
unfolding less_le by blast
lemma psubsetE [elim!]: "A \<subset> B \<Longrightarrow> (A \<subseteq> B \<Longrightarrow> \<not> B \<subseteq> A \<Longrightarrow> R) \<Longrightarrow> R"
unfolding less_le by blast
lemma psubset_insert_iff:
"A \<subset> insert x B \<longleftrightarrow> (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 \<longleftrightarrow> A \<subseteq> B \<and> A \<noteq> B"
by (simp only: less_le)
lemma psubset_imp_subset: "A \<subset> B \<Longrightarrow> A \<subseteq> B"
by (simp add: psubset_eq)
lemma psubset_trans: "A \<subset> B \<Longrightarrow> B \<subset> C \<Longrightarrow> A \<subset> C"
unfolding less_le by (auto dest: subset_antisym)
lemma psubsetD: "A \<subset> B \<Longrightarrow> c \<in> A \<Longrightarrow> c \<in> B"
unfolding less_le by (auto dest: subsetD)
lemma psubset_subset_trans: "A \<subset> B \<Longrightarrow> B \<subseteq> C \<Longrightarrow> A \<subset> C"
by (auto simp add: psubset_eq)
lemma subset_psubset_trans: "A \<subseteq> B \<Longrightarrow> B \<subset> C \<Longrightarrow> A \<subset> C"
by (auto simp add: psubset_eq)
lemma psubset_imp_ex_mem: "A \<subset> B \<Longrightarrow> \<exists>b. b \<in> B - A"
unfolding less_le by blast
lemma atomize_ball: "(\<And>x. x \<in> A \<Longrightarrow> P x) \<equiv> 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: "f ` A \<subseteq> B \<Longrightarrow> image f ` Pow A \<subseteq> Pow B"
by blast
lemma image_Pow_surj: "f ` A = B \<Longrightarrow> 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 \<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 \<Longrightarrow> A \<subseteq> insert x B \<longleftrightarrow> A \<subseteq> B"
by blast
text \<open>\<^medskip> Finite Union -- the least upper bound of two sets.\<close>
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 \<Longrightarrow> B \<subseteq> C \<Longrightarrow> A \<union> B \<subseteq> C"
by (fact sup_least)
text \<open>\<^medskip> Finite Intersection -- the greatest lower bound of two sets.\<close>
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 \<Longrightarrow> C \<subseteq> B \<Longrightarrow> C \<subseteq> A \<inter> B"
by (fact inf_greatest)
text \<open>\<^medskip> Set difference.\<close>
lemma Diff_subset: "A - B \<subseteq> A"
by blast
lemma Diff_subset_conv: "A - B \<subseteq> C \<longleftrightarrow> A \<subseteq> B \<union> 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 \<subseteq> {} \<longleftrightarrow> A = {}"
by (fact bot_unique)
lemma not_psubset_empty [iff]: "\<not> (A < {})"
by (fact not_less_bot) (* FIXME: already simp *)
lemma Collect_subset [simp]: "{x\<in>A. P x} \<subseteq> A" by auto
lemma Collect_empty_eq [simp]: "Collect P = {} \<longleftrightarrow> (\<forall>x. \<not> P x)"
by blast
lemma empty_Collect_eq [simp]: "{} = Collect P \<longleftrightarrow> (\<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 \<or> Q x} = {x. P x} \<union> {x. Q x}"
by blast
lemma Collect_imp_eq: "{x. P x \<longrightarrow> Q x} = - {x. P x} \<union> {x. Q x}"
by blast
lemma Collect_conj_eq: "{x. P x \<and> Q x} = {x. P x} \<inter> {x. Q x}"
by blast
lemma Collect_mono_iff: "Collect P \<subseteq> Collect Q \<longleftrightarrow> (\<forall>x. P x \<longrightarrow> Q x)"
by blast
text \<open>\<^medskip> \<open>insert\<close>.\<close>
lemma insert_is_Un: "insert a A = {a} \<union> 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 \<noteq> {}"
and empty_not_insert [simp]: "{} \<noteq> insert a A"
by blast+
lemma insert_absorb: "a \<in> A \<Longrightarrow> 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 \<subseteq> B \<longleftrightarrow> x \<in> B \<and> A \<subseteq> B"
by blast
lemma mk_disjoint_insert: "a \<in> A \<Longrightarrow> \<exists>B. A = insert a B \<and> a \<notin> 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 \<noteq> a \<longrightarrow> 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 = {} \<longleftrightarrow> a \<notin> B \<and> A \<inter> B = {}"
"{} = insert a A \<inter> B \<longleftrightarrow> a \<notin> B \<and> {} = A \<inter> B"
by auto
lemma disjoint_insert [simp]:
"B \<inter> insert a A = {} \<longleftrightarrow> a \<notin> B \<and> B \<inter> A = {}"
"{} = A \<inter> insert b B \<longleftrightarrow> b \<notin> A \<and> {} = A \<inter> B"
by auto
text \<open>\<^medskip> \<open>Int\<close>\<close>
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
\<comment> \<open>Intersection is an AC-operator\<close>
lemma Int_absorb1: "B \<subseteq> A \<Longrightarrow> A \<inter> B = B"
by (fact inf_absorb2)
lemma Int_absorb2: "A \<subseteq> B \<Longrightarrow> 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 = {} \<longleftrightarrow> A \<subseteq> - B"
by blast
lemma disjoint_iff_not_equal: "A \<inter> B = {} \<longleftrightarrow> (\<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 \<longleftrightarrow> A = UNIV \<and> B = UNIV"
by (fact inf_eq_top_iff) (* already simp *)
lemma Int_subset_iff [simp]: "C \<subseteq> A \<inter> B \<longleftrightarrow> C \<subseteq> A \<and> C \<subseteq> B"
by (fact le_inf_iff)
lemma Int_Collect: "x \<in> A \<inter> {x. P x} \<longleftrightarrow> x \<in> A \<and> P x"
by blast
text \<open>\<^medskip> \<open>Un\<close>.\<close>
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
\<comment> \<open>Union is an AC-operator\<close>
lemma Un_absorb1: "A \<subseteq> B \<Longrightarrow> A \<union> B = B"
by (fact sup_absorb2)
lemma Un_absorb2: "B \<subseteq> A \<Longrightarrow> 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) \<inter> 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) \<inter> C = B \<inter> C"
by auto
lemma Int_insert_left_if1 [simp]: "a \<in> C \<Longrightarrow> (insert a B) \<inter> C = insert a (B \<inter> C)"
by auto
lemma Int_insert_right: "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
by auto
lemma Int_insert_right_if0 [simp]: "a \<notin> A \<Longrightarrow> A \<inter> (insert a B) = A \<inter> B"
by auto
lemma Int_insert_right_if1 [simp]: "a \<in> A \<Longrightarrow> A \<inter> (insert a B) = insert a (A \<inter> 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 \<longleftrightarrow> A \<union> B = B"
by (fact le_iff_sup)
lemma Un_empty [iff]: "A \<union> B = {} \<longleftrightarrow> A = {} \<and> B = {}"
by (fact sup_eq_bot_iff) (* FIXME: already simp *)
lemma Un_subset_iff [simp]: "A \<union> B \<subseteq> C \<longleftrightarrow> A \<subseteq> C \<and> 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 \<open>\<^medskip> Set complement\<close>
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" for 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 \<longleftrightarrow> A = {}"
by blast
lemma Un_Int_assoc_eq: "(A \<inter> B) \<union> C = A \<inter> (B \<union> C) \<longleftrightarrow> C \<subseteq> 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 \<subseteq> - B \<longleftrightarrow> B \<subseteq> A"
by (fact compl_le_compl_iff) (* FIXME: already simp *)
lemma Compl_eq_Compl_iff [iff]: "- A = - B \<longleftrightarrow> 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: "(\<forall>x \<in> A \<union> B. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>B. P x)"
by blast
lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>B. P x)"
by blast
text \<open>\<^medskip> Set difference.\<close>
lemma Diff_eq: "A - B = A \<inter> (- B)"
by blast
lemma Diff_eq_empty_iff [simp]: "A - B = {} \<longleftrightarrow> A \<subseteq> 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 \<inter> B = {} \<Longrightarrow> 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 \<Longrightarrow> 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 \<in> B then A - B else insert x (A - B))"
by auto
lemma insert_Diff1 [simp]: "x \<in> B \<Longrightarrow> 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 \<Longrightarrow> insert a (A - {a}) = A"
by blast
lemma Diff_insert_absorb: "x \<notin> A \<Longrightarrow> (insert x A) - {x} = A"
by auto
lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"
by blast
lemma Diff_partition: "A \<subseteq> B \<Longrightarrow> A \<union> (B - A) = B"
by blast
lemma double_diff: "A \<subseteq> B \<Longrightarrow> B \<subseteq> C \<Longrightarrow> 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 Diff_Diff_Int: "A - (A - B) = A \<inter> B"
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
lemma subset_Compl_singleton [simp]: "A \<subseteq> - {b} \<longleftrightarrow> b \<notin> A"
by blast
text \<open>\<^medskip> Quantification over type @{typ bool}.\<close>
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 \<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} \<longleftrightarrow> X = {} \<and> Y = {}"
by blast (* somewhat slow *)
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 \<open>\<^medskip> Miscellany.\<close>
lemma set_eq_subset: "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
by blast
lemma subset_iff: "A \<subseteq> B \<longleftrightarrow> (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
by blast
lemma subset_iff_psubset_eq: "A \<subseteq> B \<longleftrightarrow> A \<subset> B \<or> A = B"
unfolding less_le by blast
lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) \<longleftrightarrow> A = {}"
by blast
lemma ex_in_conv: "(\<exists>x. x \<in> A) \<longleftrightarrow> 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 \<or> (\<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 \<open>Monotonicity of various operations\<close>
lemma image_mono: "A \<subseteq> B \<Longrightarrow> f ` A \<subseteq> f ` B"
by blast
lemma Pow_mono: "A \<subseteq> B \<Longrightarrow> Pow A \<subseteq> Pow B"
by blast
lemma insert_mono: "C \<subseteq> D \<Longrightarrow> insert a C \<subseteq> insert a D"
by blast
lemma Un_mono: "A \<subseteq> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> A \<union> B \<subseteq> C \<union> D"
by (fact sup_mono)
lemma Int_mono: "A \<subseteq> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> A \<inter> B \<subseteq> C \<inter> D"
by (fact inf_mono)
lemma Diff_mono: "A \<subseteq> C \<Longrightarrow> D \<subseteq> B \<Longrightarrow> A - B \<subseteq> C - D"
by blast
lemma Compl_anti_mono: "A \<subseteq> B \<Longrightarrow> - B \<subseteq> - A"
by (fact compl_mono)
text \<open>\<^medskip> Monotonicity of implications.\<close>
lemma in_mono: "A \<subseteq> B \<Longrightarrow> x \<in> A \<longrightarrow> x \<in> B"
by (rule impI) (erule subsetD)
lemma conj_mono: "P1 \<longrightarrow> Q1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<and> P2) \<longrightarrow> (Q1 \<and> Q2)"
by iprover
lemma disj_mono: "P1 \<longrightarrow> Q1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<or> P2) \<longrightarrow> (Q1 \<or> Q2)"
by iprover
lemma imp_mono: "Q1 \<longrightarrow> P1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<longrightarrow> P2) \<longrightarrow> (Q1 \<longrightarrow> Q2)"
by iprover
lemma imp_refl: "P \<longrightarrow> P" ..
lemma not_mono: "Q \<longrightarrow> P \<Longrightarrow> \<not> P \<longrightarrow> \<not> Q"
by iprover
lemma ex_mono: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> (\<exists>x. P x) \<longrightarrow> (\<exists>x. Q x)"
by iprover
lemma all_mono: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> (\<forall>x. P x) \<longrightarrow> (\<forall>x. Q x)"
by iprover
lemma Collect_mono: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> Collect P \<subseteq> Collect Q"
by blast
lemma Int_Collect_mono: "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P x \<longrightarrow> Q x) \<Longrightarrow> 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 \<Longrightarrow> c = d \<Longrightarrow> b \<longrightarrow> d \<Longrightarrow> a \<longrightarrow> c"
by iprover
subsubsection \<open>Inverse image of a function\<close>
definition vimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'a set" (infixr "-`" 90)
where "f -` B \<equiv> {x. f x \<in> B}"
lemma vimage_eq [simp]: "a \<in> f -` B \<longleftrightarrow> f a \<in> B"
unfolding vimage_def by blast
lemma vimage_singleton_eq: "a \<in> f -` {b} \<longleftrightarrow> f a = b"
by simp
lemma vimageI [intro]: "f a = b \<Longrightarrow> b \<in> B \<Longrightarrow> a \<in> f -` B"
unfolding vimage_def by blast
lemma vimageI2: "f a \<in> A \<Longrightarrow> a \<in> f -` A"
unfolding vimage_def by fast
lemma vimageE [elim!]: "a \<in> f -` B \<Longrightarrow> (\<And>x. f a = x \<Longrightarrow> x \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
unfolding vimage_def by blast
lemma vimageD: "a \<in> f -` A \<Longrightarrow> f a \<in> 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 \<union> B) = (f -` A) \<union> (f -` B)"
by blast
lemma vimage_Int [simp]: "f -` (A \<inter> B) = (f -` A) \<inter> (f -` B)"
by fast
lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}"
by blast
lemma vimage_Collect: "(\<And>x. P (f x) = Q x) \<Longrightarrow> f -` (Collect P) = Collect Q"
by blast
lemma vimage_insert: "f -` (insert a B) = (f -` {a}) \<union> (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 \<subseteq> B \<Longrightarrow> f -` A \<subseteq> f -` B"
\<comment> \<open>monotonicity\<close>
by blast
lemma vimage_image_eq: "f -` (f ` A) = {y. \<exists>x\<in>A. f x = f y}"
by (blast intro: sym)
lemma image_vimage_subset: "f ` (f -` A) \<subseteq> A"
by blast
lemma image_vimage_eq [simp]: "f ` (f -` A) = A \<inter> 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]: "(\<lambda>x. x) -` Y = Y"
by blast
subsubsection \<open>Singleton sets\<close>
definition is_singleton :: "'a set \<Rightarrow> bool"
where "is_singleton A \<longleftrightarrow> (\<exists>x. A = {x})"
lemma is_singletonI [simp, intro!]: "is_singleton {x}"
unfolding is_singleton_def by simp
lemma is_singletonI': "A \<noteq> {} \<Longrightarrow> (\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x = y) \<Longrightarrow> is_singleton A"
unfolding is_singleton_def by blast
lemma is_singletonE: "is_singleton A \<Longrightarrow> (\<And>x. A = {x} \<Longrightarrow> P) \<Longrightarrow> P"
unfolding is_singleton_def by blast
subsubsection \<open>Getting the contents of a singleton set\<close>
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)
lemma is_singleton_the_elem: "is_singleton A \<longleftrightarrow> A = {the_elem A}"
by (auto simp: is_singleton_def)
lemma the_elem_image_unique:
assumes "A \<noteq> {}"
and *: "\<And>y. y \<in> A \<Longrightarrow> f y = f x"
shows "the_elem (f ` A) = f x"
unfolding the_elem_def
proof (rule the1_equality)
from \<open>A \<noteq> {}\<close> obtain y where "y \<in> A" by auto
with * have "f x = f y" by simp
with \<open>y \<in> A\<close> have "f x \<in> f ` A" by blast
with * show "f ` A = {f x}" by auto
then show "\<exists>!x. f ` A = {x}" by auto
qed
subsubsection \<open>Least value operator\<close>
lemma Least_mono: "mono f \<Longrightarrow> \<exists>x\<in>S. \<forall>y\<in>S. x \<le> y \<Longrightarrow> (LEAST y. y \<in> f ` S) = f (LEAST x. x \<in> S)"
for f :: "'a::order \<Rightarrow> 'b::order"
\<comment> \<open>Courtesy of Stephan Merz\<close>
apply clarify
apply (erule_tac P = "\<lambda>x. x \<in> S" in LeastI2_order)
apply fast
apply (rule LeastI2_order)
apply (auto elim: monoD intro!: order_antisym)
done
subsubsection \<open>Monad operation\<close>
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: "Set.bind (Set.bind A B) C = Set.bind A (\<lambda>x. Set.bind (B x) C)"
for A :: "'a set"
by (auto simp: 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: bind_def)
lemma bind_const: "Set.bind A (\<lambda>_. B) = (if A = {} then {} else B)"
by (auto simp: bind_def)
lemma bind_singleton_conv_image: "Set.bind A (\<lambda>x. {f x}) = f ` A"
by (auto simp: bind_def)
subsubsection \<open>Operations for execution\<close>
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 by standard (auto simp add: equal_set_def)
end
text \<open>Misc\<close>
definition pairwise :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
where "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> S. x \<noteq> y \<longrightarrow> R x y)"
lemma pairwiseI [intro?]:
"pairwise R S" if "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> R x y"
using that by (simp add: pairwise_def)
lemma pairwiseD:
"R x y" and "R y x"
if "pairwise R S" "x \<in> S" and "y \<in> S" and "x \<noteq> y"
using that by (simp_all add: pairwise_def)
lemma pairwise_empty [simp]: "pairwise P {}"
by (simp add: pairwise_def)
lemma pairwise_singleton [simp]: "pairwise P {A}"
by (simp add: pairwise_def)
lemma pairwise_insert:
"pairwise r (insert x s) \<longleftrightarrow> (\<forall>y. y \<in> s \<and> y \<noteq> x \<longrightarrow> r x y \<and> r y x) \<and> pairwise r s"
by (force simp: pairwise_def)
lemma pairwise_subset: "pairwise P S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> pairwise P T"
by (force simp: pairwise_def)
lemma pairwise_mono: "\<lbrakk>pairwise P A; \<And>x y. P x y \<Longrightarrow> Q x y; B \<subseteq> A\<rbrakk> \<Longrightarrow> pairwise Q B"
by (fastforce simp: pairwise_def)
lemma pairwise_imageI:
"pairwise P (f ` A)"
if "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x \<noteq> f y \<Longrightarrow> P (f x) (f y)"
using that by (auto intro: pairwiseI)
lemma pairwise_image: "pairwise r (f ` s) \<longleftrightarrow> pairwise (\<lambda>x y. (f x \<noteq> f y) \<longrightarrow> r (f x) (f y)) s"
by (force simp: pairwise_def)
definition disjnt :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "disjnt A B \<longleftrightarrow> A \<inter> B = {}"
lemma disjnt_self_iff_empty [simp]: "disjnt S S \<longleftrightarrow> S = {}"
by (auto simp: disjnt_def)
lemma disjnt_iff: "disjnt A B \<longleftrightarrow> (\<forall>x. \<not> (x \<in> A \<and> x \<in> B))"
by (force simp: disjnt_def)
lemma disjnt_sym: "disjnt A B \<Longrightarrow> disjnt B A"
using disjnt_iff by blast
lemma disjnt_empty1 [simp]: "disjnt {} A" and disjnt_empty2 [simp]: "disjnt A {}"
by (auto simp: disjnt_def)
lemma disjnt_insert1 [simp]: "disjnt (insert a X) Y \<longleftrightarrow> a \<notin> Y \<and> disjnt X Y"
by (simp add: disjnt_def)
lemma disjnt_insert2 [simp]: "disjnt Y (insert a X) \<longleftrightarrow> a \<notin> Y \<and> disjnt Y X"
by (simp add: disjnt_def)
lemma disjnt_subset1 : "\<lbrakk>disjnt X Y; Z \<subseteq> X\<rbrakk> \<Longrightarrow> disjnt Z Y"
by (auto simp: disjnt_def)
lemma disjnt_subset2 : "\<lbrakk>disjnt X Y; Z \<subseteq> Y\<rbrakk> \<Longrightarrow> disjnt X Z"
by (auto simp: disjnt_def)
lemma disjoint_image_subset: "\<lbrakk>pairwise disjnt \<A>; \<And>X. X \<in> \<A> \<Longrightarrow> f X \<subseteq> X\<rbrakk> \<Longrightarrow> pairwise disjnt (f `\<A>)"
unfolding disjnt_def pairwise_def by fast
lemma Int_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B \<Longrightarrow> False) \<Longrightarrow> A \<inter> B = {}"
by blast
lemma in_image_insert_iff:
assumes "\<And>C. C \<in> B \<Longrightarrow> x \<notin> C"
shows "A \<in> insert x ` B \<longleftrightarrow> x \<in> A \<and> A - {x} \<in> B" (is "?P \<longleftrightarrow> ?Q")
proof
assume ?P then show ?Q
using assms by auto
next
assume ?Q
then have "x \<in> A" and "A - {x} \<in> B"
by simp_all
from \<open>A - {x} \<in> B\<close> have "insert x (A - {x}) \<in> insert x ` B"
by (rule imageI)
also from \<open>x \<in> A\<close>
have "insert x (A - {x}) = A"
by auto
finally show ?P .
qed
hide_const (open) member not_member
lemmas equalityI = subset_antisym
ML \<open>
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}
\<close>
end