author wenzelm Sun, 19 Jun 2016 22:51:42 +0200 changeset 63316 dff40165618c parent 63315 67c38b9ea2fb child 63321 0c89eef79701 child 63322 bc1f17d45e91
misc tuning and modernization;
 src/HOL/Set.thy file | annotate | diff | comparison | revisions
--- a/src/HOL/Set.thy	Sun Jun 19 17:40:51 2016 +0200
+++ b/src/HOL/Set.thy	Sun Jun 19 22:51:42 2016 +0200
@@ -35,7 +35,7 @@
text \<open>Set comprehensions\<close>

syntax
-  "_Coll" :: "pttrn => bool => 'a set"    ("(1{_./ _})")
+  "_Coll" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a set"    ("(1{_./ _})")
translations
"{x. P}" \<rightleftharpoons> "CONST Collect (\<lambda>x. P)"

@@ -52,15 +52,15 @@
lemma CollectD: "a \<in> {x. P x} \<Longrightarrow> P a"
by simp

-lemma Collect_cong: "(\<And>x. P x = Q x) ==> {x. P x} = {x. Q x}"
+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> & x=t & \<dots>}\<close>
-to the front (and similarly for \<open>t=x\<close>):
+  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}") = \<open>
+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
@@ -80,8 +80,7 @@
then show ?thesis by simp
qed

-lemma set_eq_iff:
-  "A = B \<longleftrightarrow> (\<forall>x. x \<in> A \<longleftrightarrow> x \<in> B)"
+lemma set_eq_iff: "A = B \<longleftrightarrow> (\<forall>x. x \<in> A \<longleftrightarrow> x \<in> B)"
by (auto intro:set_eqI)

text \<open>Lifting of predicate class instances\<close>
@@ -89,52 +88,52 @@
instantiation set :: (type) boolean_algebra
begin

-definition less_eq_set where
-  "A \<le> B \<longleftrightarrow> (\<lambda>x. member x A) \<le> (\<lambda>x. member x B)"
-
-definition less_set where
-  "A < B \<longleftrightarrow> (\<lambda>x. member x A) < (\<lambda>x. member x B)"
-
-definition inf_set where
-  "A \<sqinter> B = Collect ((\<lambda>x. member x A) \<sqinter> (\<lambda>x. member x B))"
-
-definition sup_set where
-  "A \<squnion> B = Collect ((\<lambda>x. member x A) \<squnion> (\<lambda>x. member x B))"
-
-definition bot_set where
-  "\<bottom> = Collect \<bottom>"
-
-definition top_set where
-  "\<top> = Collect \<top>"
-
-definition uminus_set where
-  "- A = Collect (- (\<lambda>x. member x A))"
-
-definition minus_set where
-  "A - B = Collect ((\<lambda>x. member x A) - (\<lambda>x. member x B))"
-
-instance proof
-qed (simp_all add: less_eq_set_def less_set_def inf_set_def sup_set_def
-  bot_set_def top_set_def uminus_set_def minus_set_def
-  less_le_not_le inf_compl_bot sup_compl_top sup_inf_distrib1 diff_eq
-  set_eqI fun_eq_iff
-  del: inf_apply sup_apply bot_apply top_apply minus_apply uminus_apply)
+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}"
+abbreviation empty :: "'a set" ("{}")
+  where "{} \<equiv> bot"
+
+definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
+  where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"

syntax
-  "_Finset" :: "args => 'a set"    ("{(_)}")
+  "_Finset" :: "args \<Rightarrow> 'a set"    ("{(_)}")
translations
-  "{x, xs}" == "CONST insert x {xs}"
-  "{x}" == "CONST insert x {}"
+  "{x, xs}" \<rightleftharpoons> "CONST insert x {xs}"
+  "{x}" \<rightleftharpoons> "CONST insert x {}"

subsection \<open>Subsets and bounded quantifiers\<close>
@@ -171,28 +170,28 @@
subset_eq  ("op <=") 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> "bounded universal quantifiers"
-
-definition Bex :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
-  "Bex A P \<longleftrightarrow> (\<exists>x. x \<in> A \<and> P x)"   \<comment> "bounded existential quantifiers"
+definition Ball :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
+  where "Ball A P \<longleftrightarrow> (\<forall>x. x \<in> A \<longrightarrow> P x)"   \<comment> "bounded universal quantifiers"
+
+definition Bex :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
+  where "Bex A P \<longleftrightarrow> (\<exists>x. x \<in> A \<and> P x)"   \<comment> "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)
+  "_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 => '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)
+  "_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 => 'a set => bool => bool"      ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
-  "_Bex"        :: "pttrn => 'a set => bool => bool"      ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
-  "_Bex1"       :: "pttrn => 'a set => bool => bool"      ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10)
-  "_Bleast"     :: "id => 'a set => bool => 'a"           ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10)
+  "_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)"
@@ -201,25 +200,25 @@
"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)
+  "_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] => bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
-  "_setlessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
-  "_setleAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
-  "_setleEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
-  "_setleEx1"   :: "[idt, 'a, bool] => bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
+  "_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"
+ "\<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
@@ -256,12 +255,13 @@

text \<open>
-  \medskip Translate between \<open>{e | x1...xn. P}\<close> and \<open>{u. EX x1..xn. u = e & P}\<close>; \<open>{y. EX x1..xn. y = e & P}\<close> is
-  only translated if \<open>[0..n] subset bvs(e)\<close>.
+  \<^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{_ |/_./ _})")
+  "_Setcompr" :: "'a \<Rightarrow> idts \<Rightarrow> bool \<Rightarrow> 'a set"    ("(1{_ |/_./ _})")

parse_translation \<open>
let
@@ -319,31 +319,29 @@
in [(@{const_syntax Collect}, setcompr_tr')] end;
\<close>

-simproc_setup defined_Bex ("EX x:A. P x & Q x") = \<open>
+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 ("ALL x:A. P x --> Q x") = \<open>
+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!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
+lemma ballI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> P x) \<Longrightarrow> \<forall>x\<in>A. P x"

lemmas strip = impI allI ballI

-lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
+lemma bspec [dest?]: "\<forall>x\<in>A. P x \<Longrightarrow> x \<in> A \<Longrightarrow> P x"

-text \<open>
-  Gives better instantiation for bound:
-\<close>
+text \<open>Gives better instantiation for bound:\<close>

setup \<open>
map_theory_claset (fn ctxt =>
@@ -353,98 +351,91 @@
ML \<open>
structure Simpdata =
struct
-
-open Simpdata;
-
-val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs;
-
+  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]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
-  by (unfold Ball_def) blast
-
-lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
-  \<comment> \<open>Normally the best argument order: @{prop "P x"} constrains the
-    choice of @{prop "x:A"}.\<close>
-  by (unfold Bex_def) blast
-
-lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x"
-  \<comment> \<open>The best argument order when there is only one @{prop "x:A"}.\<close>
-  by (unfold Bex_def) blast
-
-lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
-  by (unfold Bex_def) blast
-
-lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
-  by (unfold Bex_def) blast
-
-lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
+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>

-lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
+lemma bex_triv [simp]: "(\<exists>x\<in>A. P) \<longleftrightarrow> ((\<exists>x. x \<in> A) \<and> P)"
\<comment> \<open>Dual form for existentials.\<close>

-lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
+lemma bex_triv_one_point1 [simp]: "(\<exists>x\<in>A. x = a) \<longleftrightarrow> a \<in> A"
by blast

-lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
+lemma bex_triv_one_point2 [simp]: "(\<exists>x\<in>A. a = x) \<longleftrightarrow> a \<in> A"
by blast

-lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
+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]: "(EX x:A. a = x & P x) = (a:A & P a)"
+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]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
+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]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
+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))"
+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))"
+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 ==> (!!x. x:B ==> P x = Q x) ==>
-    (ALL x:A. P x) = (ALL x:B. Q x)"
+  "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)"

lemma strong_ball_cong [cong]:
-  "A = B ==> (!!x. x:B =simp=> P x = Q x) ==>
-    (ALL x:A. P x) = (ALL x:B. Q x)"
+  "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)"

lemma bex_cong:
-  "A = B ==> (!!x. x:B ==> P x = Q x) ==>
-    (EX x:A. P x) = (EX x:B. Q x)"
+  "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 ==> (!!x. x:B =simp=> P x = Q x) ==>
-    (EX x:A. P x) = (EX x:B. Q x)"
+  "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>
@@ -453,50 +444,45 @@

text \<open>
-  \medskip Map the type \<open>'a set => anything\<close> to just @{typ
-  "'a set"}.
+  \<^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 ==> c \<in> A ==> c \<in> B"
+lemma subsetD [elim, intro?]: "A \<subseteq> B \<Longrightarrow> c \<in> A \<Longrightarrow> c \<in> B"
\<comment> \<open>Rule in Modus Ponens style.\<close>

-lemma rev_subsetD [intro?]: "c \<in> A ==> A \<subseteq> B ==> c \<in> B"
-  \<comment> \<open>The same, with reversed premises for use with \<open>erule\<close> --
-      cf \<open>rev_mp\<close>.\<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 \<open>erule\<close> -- cf. \<open>rev_mp\<close>.\<close>
by (rule subsetD)

-text \<open>
-  \medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}.
-\<close>
-
-lemma subsetCE [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P"
+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 \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast
-
-lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A"
+lemma subset_eq: "A \<le> B = (\<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 ==> B \<subseteq> C ==> A \<subseteq> C"
+lemma subset_trans: "A \<subseteq> B \<Longrightarrow> B \<subseteq> C \<Longrightarrow> A \<subseteq> C"
by (fact order_trans)

-lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
+lemma set_rev_mp: "x \<in> A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> x \<in> B"
by (rule subsetD)

-lemma set_mp: "A \<subseteq> B ==> x:A ==> x:B"
+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"
+lemma subset_not_subset_eq [code]: "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"
by (fact less_le_not_le)

-lemma eq_mem_trans: "a=b ==> b \<in> A ==> a \<in> A"
+lemma eq_mem_trans: "a = b \<Longrightarrow> b \<in> A \<Longrightarrow> a \<in> A"
by simp

lemmas basic_trans_rules [trans] =
@@ -505,123 +491,120 @@

subsubsection \<open>Equality\<close>

-lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B"
+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 ==> A \<subseteq> B"
+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 ==> B \<subseteq> A"
+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 = {}"}!
+  \<^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 ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P"
+lemma equalityE: "A = B \<Longrightarrow> (A \<subseteq> B \<Longrightarrow> B \<subseteq> A \<Longrightarrow> P) \<Longrightarrow> P"
by simp

-lemma equalityCE [elim]:
-    "A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P"
+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 ==> (x : A) = (x : B)"
+lemma eqset_imp_iff: "A = B \<Longrightarrow> x \<in> A \<longleftrightarrow> x \<in> B"
by simp

-lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)"
+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}"
+lemma empty_def: "{} = {x. False}"

-lemma empty_iff [simp]: "(c : {}) = False"
+lemma empty_iff [simp]: "c \<in> {} \<longleftrightarrow> False"

-lemma emptyE [elim!]: "a : {} ==> P"
+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 "{} <= A"}\<close>
+  \<comment> \<open>One effect is to delete the ASSUMPTION @{prop "{} \<subseteq> A"}\<close>
by blast

-lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
+lemma equals0I: "(\<And>y. y \<in> A \<Longrightarrow> False) \<Longrightarrow> A = {}"
by blast

-lemma equals0D: "A = {} ==> a \<notin> A"
-    \<comment> \<open>Use for reasoning about disjointness: \<open>A Int B = {}\<close>\<close>
+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 = True"
+lemma ball_empty [simp]: "Ball {} P \<longleftrightarrow> True"

-lemma bex_empty [simp]: "Bex {} P = False"
+lemma bex_empty [simp]: "Bex {} P \<longleftrightarrow> False"

subsubsection \<open>The universal set -- UNIV\<close>

-abbreviation UNIV :: "'a set" where
-  "UNIV \<equiv> top"
-
-lemma UNIV_def:
-  "UNIV = {x. True}"
+abbreviation UNIV :: "'a set"
+  where "UNIV \<equiv> top"
+
+lemma UNIV_def: "UNIV = {x. True}"

-lemma UNIV_I [simp]: "x : UNIV"
+lemma UNIV_I [simp]: "x \<in> UNIV"

declare UNIV_I [intro]  \<comment> \<open>unsafe makes it less likely to cause problems\<close>

-lemma UNIV_witness [intro?]: "EX x. x : UNIV"
+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.
+  \<^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"
+lemma ball_UNIV [simp]: "Ball UNIV P \<longleftrightarrow> All P"

-lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
+lemma bex_UNIV [simp]: "Bex UNIV P \<longleftrightarrow> Ex P"

lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"
by auto

-lemma UNIV_not_empty [iff]: "UNIV ~= {}"
+lemma UNIV_not_empty [iff]: "UNIV \<noteq> {}"
by (blast elim: equalityE)

lemma empty_not_UNIV[simp]: "{} \<noteq> UNIV"
-by blast
+  by blast
+

subsubsection \<open>The Powerset operator -- Pow\<close>

-definition Pow :: "'a set => 'a set set" where
-  Pow_def: "Pow A = {B. B \<le> A}"
-
-lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)"
+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"

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

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

lemma Pow_bottom: "{} \<in> Pow B"
@@ -636,23 +619,25 @@

subsubsection \<open>Set complement\<close>

-lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)"
+lemma Compl_iff [simp]: "c \<in> - A \<longleftrightarrow> c \<notin> A"

-lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A"
+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 ...\<close>
-
-lemma ComplD [dest!]: "c : -A ==> c~:A"
+  \<^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. ~ x : A}"
+lemma Compl_eq: "- A = {x. \<not> x \<in> A}"
by blast

@@ -664,23 +649,22 @@
notation (ASCII)
inter  (infixl "Int" 70)

-lemma Int_def:
-  "A \<inter> B = {x. x \<in> A \<and> x \<in> B}"
+lemma Int_def: "A \<inter> B = {x. x \<in> A \<and> x \<in> B}"

-lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
-  by (unfold Int_def) blast
-
-lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
+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 : A Int B ==> c:A"
+lemma IntD1: "c \<in> A \<inter> B \<Longrightarrow> c \<in> A"
by simp

-lemma IntD2: "c : A Int B ==> c:B"
+lemma IntD2: "c \<in> A \<inter> B \<Longrightarrow> c \<in> B"
by simp

-lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
+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"
@@ -695,29 +679,25 @@
notation (ASCII)
union  (infixl "Un" 65)

-lemma Un_def:
-  "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
+lemma Un_def: "A \<union> B = {x. x \<in> A \<or> x \<in> B}"

-lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
-  by (unfold Un_def) blast
-
-lemma UnI1 [elim?]: "c:A ==> c : A Un B"
-  by simp
-
-lemma UnI2 [elim?]: "c:B ==> c : A Un B"
+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

-text \<open>
-  \medskip Classical introduction rule: no commitment to @{prop A} vs
-  @{prop B}.
-\<close>
-
-lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
+lemma UnI2 [elim?]: "c \<in> B \<Longrightarrow> c \<in> A \<union> B"
+  by simp
+
+text \<open>\<^medskip> Classical introduction rule: no commitment to @{prop A} vs @{prop B}.\<close>
+
+lemma UnCI [intro!]: "(c \<notin> B \<Longrightarrow> c \<in> A) \<Longrightarrow> c \<in> A \<union> B"
by auto

-lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
-  by (unfold Un_def) blast
+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"
@@ -728,109 +708,110 @@

subsubsection \<open>Set difference\<close>

-lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
+lemma Diff_iff [simp]: "c \<in> A - B \<longleftrightarrow> c \<in> A \<and> c \<notin> B"

-lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
+lemma DiffI [intro!]: "c \<in> A \<Longrightarrow> c \<notin> B \<Longrightarrow> c \<in> A - B"
by simp

-lemma DiffD1: "c : A - B ==> c : A"
+lemma DiffD1: "c \<in> A - B \<Longrightarrow> c \<in> A"
by simp

-lemma DiffD2: "c : A - B ==> c : B ==> P"
+lemma DiffD2: "c \<in> A - B \<Longrightarrow> c \<in> B \<Longrightarrow> P"
by simp

-lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
+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 : A & ~ x : B}" by blast
-
-lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"
-by blast
+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 : insert b A) = (a = b | a:A)"
-  by (unfold insert_def) blast
-
-lemma insertI1: "a : insert a B"
+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 : B ==> a : insert b B"
+lemma insertI2: "a \<in> B \<Longrightarrow> a \<in> insert b B"
by simp

-lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
-  by (unfold insert_def) blast
-
-lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
+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) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)"
+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
-  from assms show "A = insert x (A - {x})" by blast
-next
+  show "A = insert x (A - {x})" using assms by blast
show "x \<notin> A - {x}" by blast
qed

-lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)"
-by auto
-
-lemma insert_eq_iff: assumes "a \<notin> A" "b \<notin> B"
-shows "insert a A = insert b B \<longleftrightarrow>
-  (if a=b then A=B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)"
-  (is "?L \<longleftrightarrow> ?R")
+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
-  assume ?L
-  show ?R
-  proof cases
-    assume "a=b" with assms \<open>?L\<close> show ?R by (simp add: insert_ident)
+  show ?R if ?L
+  proof (cases "a = b")
+    case True
+    with assms \<open>?L\<close> show ?R
next
-    assume "a\<noteq>b"
+    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
-    thus ?R using \<open>a\<noteq>b\<close> by auto
+      using assms \<open>?L\<close> \<open>a \<noteq> b\<close> by auto
+    then show ?R using \<open>a \<noteq> b\<close> by auto
qed
-next
-  assume ?R thus ?L by (auto split: if_splits)
+  show ?L if ?R
+    using that by (auto split: if_splits)
qed

lemma insert_UNIV: "insert x UNIV = UNIV"
-by auto
+  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>
+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 : {a} ==> b = a"
+lemma singletonD [dest!]: "b \<in> {a} \<Longrightarrow> b = a"
by blast

lemmas singletonE = singletonD [elim_format]

-lemma singleton_iff: "(b : {a}) = (b = a)"
+lemma singleton_iff: "b \<in> {a} \<longleftrightarrow> b = a"
by blast

-lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
+lemma singleton_inject [dest!]: "{a} = {b} \<Longrightarrow> a = b"
by blast

-lemma singleton_insert_inj_eq [iff]:
-     "({b} = insert a A) = (a = b & A \<subseteq> {b})"
+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}) = (a = b & A \<subseteq> {b})"
+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} ==> A = {} | A = {x}"
+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}"
@@ -842,73 +823,59 @@
lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
by blast

-lemma Diff_single_insert: "A - {x} \<subseteq> B ==> A \<subseteq> insert x B"
+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"
+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}) = (a=c & b=d | a=d & b=c)"
+lemma doubleton_eq_iff: "{a, b} = {c, d} \<longleftrightarrow> a = c \<and> b = d \<or> a = d & b = c"
by (blast elim: equalityE)

-lemma Un_singleton_iff:
-  "(A \<union> B = {x}) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"
-by auto
-
-lemma singleton_Un_iff:
-  "({x} = A \<union> B) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"
-by auto
+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 @{term b} does not have the syntactic form of @{term "f x"}.
-\<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"
-  by (unfold image_def) blast
-
-lemma imageI:
-  "x \<in> A \<Longrightarrow> f x \<in> f  A"
+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 @{term x}.\<close>
+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>
+  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 by (unfold image_def) blast
-
-lemma Compr_image_eq:
-  "{x \<in> f  A. P x} = f  {x \<in> A. P (f x)}"
+  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"
+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)"
+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"
+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)"
+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

@@ -921,68 +888,53 @@
ultimately show thesis by (blast intro: that)
qed

-lemma subset_image_iff:
-  "B \<subseteq> f  A \<longleftrightarrow> (\<exists>AA\<subseteq>A. B = f  AA)"
+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"
+lemma image_ident [simp]: "(\<lambda>x. x)  Y = Y"
by blast

-lemma image_empty [simp]:
-  "f  {} = {}"
+lemma image_empty [simp]: "f  {} = {}"
by blast

-lemma image_insert [simp]:
-  "f  insert a B = insert (f a) (f  B)"
+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}"
+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})"
+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"
+lemma image_image: "f  (g  A) = (\<lambda>x. f (g x))  A"
by blast

-lemma insert_image [simp]:
-  "x \<in> A ==> insert (f x) (f  A) = f  A"
+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 = {}"
+lemma image_is_empty [iff]: "f  A = {} \<longleftrightarrow> A = {}"
by blast

-lemma empty_is_image [iff]:
-  "{} = f  A \<longleftrightarrow> A = {}"
+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 simprule: the RHS isn't simpler than the LHS,
+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}))"
+  "(\<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:
-  "M = N \<Longrightarrow> (\<And>x. x \<in> N \<Longrightarrow> f x = g x) \<Longrightarrow> f  M = g  N"
+lemma image_cong: "M = N \<Longrightarrow> (\<And>x. x \<in> N \<Longrightarrow> f x = g x) \<Longrightarrow> f  M = g  N"

-lemma image_Int_subset:
-  "f  (A \<inter> B) \<subseteq> f  A \<inter> f  B"
+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)"
+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"
@@ -991,78 +943,67 @@
lemma setcompr_eq_image: "{f x |x. P x} = f  {x. P x}"
by auto

-lemma ball_imageD:
-  assumes "\<forall>x\<in>f  A. P x"
-  shows "\<forall>x\<in>A. P (f x)"
-  using assms by simp
-
-lemma bex_imageD:
-  assumes "\<exists>x\<in>f  A. P x"
-  shows "\<exists>x\<in>A. P (f x)"
-  using assms 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]: "op + (0::'a::comm_monoid_add)  S = S"
by auto

-text \<open>
-  \medskip Range of a function -- just a translation for image!
-\<close>
-
-abbreviation range :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set"
-where \<comment> "of function"
-  "range f \<equiv> f  UNIV"
-
-lemma range_eqI:
-  "b = f x \<Longrightarrow> b \<in> range f"
+text \<open>\<^medskip> Range of a function -- just an abbreviation for image!\<close>
+
+abbreviation range :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set"  \<comment> "of function"
+  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"
+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"
+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"
+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"
+lemma range_composition: "range (\<lambda>x. f (g x)) = f  range g"
by auto

subsubsection \<open>Some rules with \<open>if\<close>\<close>

-text\<open>Elimination of \<open>{x. \<dots> & x=t & \<dots>}\<close>.\<close>
-
-lemma Collect_conv_if: "{x. x=a & P x} = (if P a then {a} else {})"
+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 & P x} = (if P a then {a} else {})"
+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) = ((Q --> x = b) & (~ Q --> y = b))"
+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)) = ((Q --> a = x) & (~ Q --> a = y))"
+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!
+  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))"
+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 : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
-  by (rule if_split [where P="%S. a : S"])
+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

@@ -1080,58 +1021,48 @@

subsubsection \<open>The proper subset'' relation\<close>

-lemma psubsetI [intro!]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B"
-  by (unfold less_le) blast
-
-lemma psubsetE [elim!]:
-    "[|A \<subset> B;  [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R"
-  by (unfold less_le) blast
+lemma 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) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
+  "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) = (A \<subseteq> B & A \<noteq> B)"
+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 ==> A \<subseteq> B"
+lemma psubset_imp_subset: "A \<subset> B \<Longrightarrow> A \<subseteq> B"

-lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C"
-apply (unfold less_le)
-apply (auto dest: subset_antisym)
-done
-
-lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B"
-apply (unfold less_le)
-apply (auto dest: subsetD)
-done
-
-lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C"
+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"

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

-lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)"
-  by (unfold less_le) blast
-
-lemma atomize_ball:
-    "(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)"
+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:
-  assumes "f  A \<subseteq> B"
-  shows "image f  Pow A \<subseteq> Pow B"
-  using assms by blast
-
-lemma image_Pow_surj:
-  assumes "f  A = B"
-  shows "image f  Pow A = Pow B"
-  using assms by (blast elim: subset_imageE)
+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>
@@ -1144,11 +1075,11 @@
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
by blast

-lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)"
+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>
+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)
@@ -1156,11 +1087,11 @@
lemma Un_upper2: "B \<subseteq> A \<union> B"
by (fact sup_ge2)

-lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
+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>
+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)
@@ -1168,17 +1099,17 @@
lemma Int_lower2: "A \<inter> B \<subseteq> B"
by (fact inf_le2)

-lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
+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>
+text \<open>\<^medskip> Set difference.\<close>

lemma Diff_subset: "A - B \<subseteq> A"
by blast

-lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
-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>
@@ -1189,49 +1120,47 @@
\<comment> \<open>supersedes \<open>Collect_False_empty\<close>\<close>
by auto

-lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})"
+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_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)"
-by blast
-
-lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)"
-by blast
+lemma Collect_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 | Q x} = {x. P x} \<union> {x. Q x}"
+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 --> Q x} = -{x. P x} \<union> {x. Q x}"
+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 & Q x} = {x. P x} \<inter> {x. Q x}"
+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} Un A"
-  \<comment> \<open>NOT SUITABLE FOR REWRITING since \<open>{a} == insert a {}\<close>\<close>
+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> {}"
-  by blast
-
-lemmas empty_not_insert = insert_not_empty [symmetric]
-declare empty_not_insert [simp]
-
-lemma insert_absorb: "a \<in> A ==> insert a A = A"
+  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{quadratic} running time\<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"
@@ -1240,32 +1169,31 @@
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
by blast

-lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)"
+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 ==> \<exists>B. A = insert a B & a \<notin> B"
+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>
-  apply (rule_tac x = "A - {a}" in exI, blast)
-  done
-
-lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}"
+  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)"
+lemma insert_inter_insert [simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
by blast

lemma insert_disjoint [simp]:
- "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
- "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
+  "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 = {}) = (a \<notin> B \<and> B \<inter> A = {})"
- "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
+  "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>
+text \<open>\<^medskip> \<open>Int\<close>\<close>

lemma Int_absorb: "A \<inter> A = A"
by (fact inf_idem) (* already simp *)
@@ -1285,10 +1213,10 @@
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 ==> A \<inter> B = B"
+lemma Int_absorb1: "B \<subseteq> A \<Longrightarrow> A \<inter> B = B"
by (fact inf_absorb2)

-lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
+lemma Int_absorb2: "A \<subseteq> B \<Longrightarrow> A \<inter> B = A"
by (fact inf_absorb1)

lemma Int_empty_left: "{} \<inter> B = {}"
@@ -1297,10 +1225,10 @@
lemma Int_empty_right: "A \<inter> {} = {}"
by (fact inf_bot_right) (* already simp *)

-lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"
+lemma disjoint_eq_subset_Compl: "A \<inter> B = {} \<longleftrightarrow> A \<subseteq> - B"
by blast

-lemma disjoint_iff_not_equal: "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
+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"
@@ -1315,17 +1243,17 @@
lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
by (fact inf_sup_distrib2)

-lemma Int_UNIV [simp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
+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) = (C \<subseteq> A & C \<subseteq> B)"
+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}) = (x \<in> A & P x)"
+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>
+text \<open>\<^medskip> \<open>Un\<close>.\<close>

lemma Un_absorb: "A \<union> A = A"
by (fact sup_idem) (* already simp *)
@@ -1345,10 +1273,10 @@
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 ==> A \<union> B = B"
+lemma Un_absorb1: "A \<subseteq> B \<Longrightarrow> A \<union> B = B"
by (fact sup_absorb2)

-lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
+lemma Un_absorb2: "B \<subseteq> A \<Longrightarrow> A \<union> B = A"
by (fact sup_absorb1)

lemma Un_empty_left: "{} \<union> B = B"
@@ -1369,28 +1297,22 @@
lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
by blast

-lemma Int_insert_left:
-    "(insert a B) Int C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
+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) Int C = B \<inter> C"
+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) Int C = insert a (B Int C)"
+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)"
+lemma Int_insert_right: "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
by auto

-lemma Int_insert_right_if0[simp]:
-    "a \<notin> A \<Longrightarrow> A Int (insert a B) = A Int B"
+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 Int (insert a B) = insert a (A Int B)"
+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)"
@@ -1399,17 +1321,16 @@
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)"
+lemma Un_Int_crazy: "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
by blast

-lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"
+lemma subset_Un_eq: "A \<subseteq> B \<longleftrightarrow> A \<union> B = B"
by (fact le_iff_sup)

-lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"
+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) = (A \<subseteq> C & B \<subseteq> C)"
+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"
@@ -1419,78 +1340,79 @@
by blast

-text \<open>\medskip Set complement\<close>
-
-lemma Compl_disjoint [simp]: "A \<inter> -A = {}"
+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 = {}"
+lemma Compl_disjoint2 [simp]: "- A \<inter> A = {}"
by (fact compl_inf_bot)

-lemma Compl_partition: "A \<union> -A = UNIV"
+lemma Compl_partition: "A \<union> - A = UNIV"
by (fact sup_compl_top)

-lemma Compl_partition2: "-A \<union> A = UNIV"
+lemma Compl_partition2: "- A \<union> A = UNIV"
by (fact compl_sup_top)

-lemma double_complement: "- (-A) = (A::'a set)"
+lemma double_complement: "- (-A) = A" for A :: "'a set"
by (fact double_compl) (* already simp *)

-lemma Compl_Un: "-(A \<union> B) = (-A) \<inter> (-B)"
+lemma Compl_Un: "- (A \<union> B) = (- A) \<inter> (- B)"
by (fact compl_sup) (* already simp *)

-lemma Compl_Int: "-(A \<inter> B) = (-A) \<union> (-B)"
+lemma Compl_Int: "- (A \<inter> B) = (- A) \<union> (- B)"
by (fact compl_inf) (* already simp *)

-lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"
+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)) = (C \<subseteq> A)"
+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 = {}"
+lemma Compl_UNIV_eq: "- UNIV = {}"
by (fact compl_top_eq) (* already simp *)

-lemma Compl_empty_eq: "-{} = UNIV"
+lemma Compl_empty_eq: "- {} = UNIV"
by (fact compl_bot_eq) (* already simp *)

-lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
+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) = (A = (B::'a set))"
+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}"
+lemma Compl_insert: "- insert x A = (- A) - {x}"
by blast

-text \<open>\medskip Bounded quantifiers.
+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) = ((\<forall>x\<in>A. P x) & (\<forall>x\<in>B. P x))"
+  (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) = ((\<exists>x\<in>A. P x) | (\<exists>x\<in>B. P x))"
+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)"
+text \<open>\<^medskip> Set difference.\<close>
+
+lemma Diff_eq: "A - B = A \<inter> (- B)"
by blast

-lemma Diff_eq_empty_iff [simp]: "(A - B = {}) = (A \<subseteq> B)"
+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::'a set)"
-by blast
-
-lemma Diff_triv: "A \<inter> B = {} ==> A - B = A"
+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 = {}"
@@ -1502,39 +1424,39 @@
lemma Diff_UNIV [simp]: "A - UNIV = {}"
by blast

-lemma Diff_insert0 [simp]: "x \<notin> A ==> A - insert x B = A - B"
+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} == insert a 0\<close>\<close>
+  \<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} == insert a 0\<close>\<close>
+  \<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 ==> insert x A - B = A - B"
+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 ==> insert a (A - {a}) = A"
by blast

-lemma Diff_insert_absorb: "x \<notin> A ==> (insert x A) - {x} = A"
+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 ==> A \<union> (B - A) = B"
+lemma Diff_partition: "A \<subseteq> B \<Longrightarrow> A \<union> (B - A) = B"
by blast

-lemma double_diff: "A \<subseteq> B ==> B \<subseteq> C ==> B - (C - A) = A"
+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"
@@ -1567,13 +1489,13 @@
lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"
by auto

-lemma Compl_Diff_eq [simp]: "- (A - B) = -A \<union> B"
+lemma Compl_Diff_eq [simp]: "- (A - B) = - A \<union> B"
by blast

-lemma subset_Compl_singleton [simp]: "A \<subseteq> - {b} \<longleftrightarrow> (b \<notin> A)"
+lemma subset_Compl_singleton [simp]: "A \<subseteq> - {b} \<longleftrightarrow> b \<notin> A"
by blast

-text \<open>\medskip Quantification over type @{typ bool}.\<close>
+text \<open>\<^medskip> Quantification over type @{typ bool}.\<close>

lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"
by (cases x) auto
@@ -1590,7 +1512,7 @@
lemma UNIV_bool: "UNIV = {False, True}"
by (auto intro: bool_induct)

-text \<open>\medskip \<open>Pow\<close>\<close>
+text \<open>\<^medskip> \<open>Pow\<close>\<close>

lemma Pow_empty [simp]: "Pow {} = {{}}"
@@ -1601,7 +1523,7 @@
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}"
+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"
@@ -1614,21 +1536,21 @@
by blast

-text \<open>\medskip Miscellany.\<close>
-
-lemma set_eq_subset: "(A = B) = (A \<subseteq> B & B \<subseteq> A)"
+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) = (\<forall>t. t \<in> A --> t \<in> B)"
+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) = ((A \<subset> B) | (A = B))"
-  by (unfold less_le) blast
-
-lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) = (A = {})"
+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) = (A \<noteq> {})"
+lemma ex_in_conv: "(\<exists>x. x \<in> A) \<longleftrightarrow> A \<noteq> {}"
by blast

lemma ball_simps [simp, no_atp]:
@@ -1658,112 +1580,110 @@

subsubsection \<open>Monotonicity of various operations\<close>

-lemma image_mono: "A \<subseteq> B ==> fA \<subseteq> fB"
+lemma image_mono: "A \<subseteq> B \<Longrightarrow> f  A \<subseteq> f  B"
by blast

-lemma Pow_mono: "A \<subseteq> B ==> Pow A \<subseteq> Pow B"
+lemma Pow_mono: "A \<subseteq> B \<Longrightarrow> Pow A \<subseteq> Pow B"
by blast

-lemma insert_mono: "C \<subseteq> D ==> insert a C \<subseteq> insert a D"
+lemma insert_mono: "C \<subseteq> D \<Longrightarrow> insert a C \<subseteq> insert a D"
by blast

-lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"
+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 ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"
+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 ==> D \<subseteq> B ==> A - B \<subseteq> C - D"
+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 ==> -B \<subseteq> -A"
+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 ==> x \<in> A --> x \<in> B"
+text \<open>\<^medskip> Monotonicity of implications.\<close>
+
+lemma in_mono: "A \<subseteq> B \<Longrightarrow> x \<in> A \<longrightarrow> x \<in> B"
apply (rule impI)
apply (erule subsetD, assumption)
done

-lemma conj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 & P2) --> (Q1 & Q2)"
-  by iprover
-
-lemma disj_mono: "P1 --> Q1 ==> P2 --> Q2 ==> (P1 | P2) --> (Q1 | Q2)"
+lemma conj_mono: "P1 \<longrightarrow> Q1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<and> P2) \<longrightarrow> (Q1 \<and> Q2)"
by iprover

-lemma imp_mono: "Q1 --> P1 ==> P2 --> Q2 ==> (P1 --> P2) --> (Q1 --> Q2)"
+lemma disj_mono: "P1 \<longrightarrow> Q1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<or> P2) \<longrightarrow> (Q1 \<or> Q2)"
by iprover

-lemma imp_refl: "P --> P" ..
-
-lemma not_mono: "Q --> P ==> ~ P --> ~ Q"
+lemma imp_mono: "Q1 \<longrightarrow> P1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<longrightarrow> P2) \<longrightarrow> (Q1 \<longrightarrow> Q2)"
by iprover

-lemma ex_mono: "(!!x. P x --> Q x) ==> (EX x. P x) --> (EX x. Q x)"
+lemma imp_refl: "P \<longrightarrow> P" ..
+
+lemma not_mono: "Q \<longrightarrow> P \<Longrightarrow> \<not> P \<longrightarrow> \<not> Q"
by iprover

-lemma all_mono: "(!!x. P x --> Q x) ==> (ALL x. P x) --> (ALL x. Q x)"
+lemma ex_mono: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> (\<exists>x. P x) \<longrightarrow> (\<exists>x. Q x)"
by iprover

-lemma Collect_mono: "(!!x. P x --> Q x) ==> Collect P \<subseteq> Collect Q"
+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 ==> (!!x. x \<in> A ==> P x --> Q x) ==> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
+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 ==> c = d ==> b --> d ==> a --> c"
+  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 => 'b) => 'b set => 'a set" (infixr "-" 90) where
-  "f - B == {x. f x : B}"
-
-lemma vimage_eq [simp]: "(a : f - B) = (f a : B)"
-  by (unfold vimage_def) blast
-
-lemma vimage_singleton_eq: "(a : f - {b}) = (f a = b)"
+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 ==> b:B ==> a : f - B"
-  by (unfold vimage_def) blast
-
-lemma vimageI2: "f a : A ==> a : f - A"
-  by (unfold vimage_def) fast
-
-lemma vimageE [elim!]: "a: f - B ==> (!!x. f a = x ==> x:B ==> P) ==> P"
-  by (unfold vimage_def) blast
-
-lemma vimageD: "a : f - A ==> f a : A"
-  by (unfold vimage_def) fast
+lemma 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)"
+lemma vimage_Compl: "f - (- A) = - (f - A)"
by blast

-lemma vimage_Un [simp]: "f - (A Un B) = (f - A) Un (f - B)"
+lemma vimage_Un [simp]: "f - (A \<union> B) = (f - A) \<union> (f - B)"
by blast

-lemma vimage_Int [simp]: "f - (A Int B) = (f - A) Int (f - B)"
+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: "(!!x. P (f x) = Q x) ==> f - (Collect P) = Collect Q"
+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}) Un (f-B)"
-  \<comment> \<open>NOT suitable for rewriting because of the recurrence of @{term "{a}"}.\<close>
+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)"
@@ -1772,18 +1692,18 @@
lemma vimage_UNIV [simp]: "f - UNIV = UNIV"
by blast

-lemma vimage_mono: "A \<subseteq> B ==> f - A \<subseteq> f - B"
+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. EX x:A. f x = f y}"
-by (blast intro: sym)
-
-lemma image_vimage_subset: "f  (f - A) <= A"
-by blast
-
-lemma image_vimage_eq [simp]: "f  (f - A) = A Int range f"
-by blast
+lemma vimage_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
@@ -1793,21 +1713,20 @@

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 {})"
+    else if d \<in> A then - B else {})"

-lemma vimage_inter_cong:
-  "(\<And> w. w \<in> S \<Longrightarrow> f w = g w) \<Longrightarrow> f - y \<inter> S = g - y \<inter> S"
+lemma vimage_inter_cong: "(\<And> w. w \<in> S \<Longrightarrow> f w = g w) \<Longrightarrow> f - y \<inter> S = g - y \<inter> S"
by auto

-lemma vimage_ident [simp]: "(%x. x) - Y = Y"
+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})"
+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
@@ -1819,10 +1738,10 @@
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})"
+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"
@@ -1832,9 +1751,10 @@

lemma the_elem_image_unique:
assumes "A \<noteq> {}"
-  assumes *: "\<And>y. y \<in> A \<Longrightarrow> f y = f x"
+    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)
+  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
@@ -1845,12 +1765,12 @@

subsubsection \<open>Least value operator\<close>

-lemma Least_mono:
-  "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
-    ==> (LEAST y. y : f  S) = f (LEAST x. x : S)"
-    \<comment> \<open>Courtesy of Stephan Merz\<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 = "%x. x : S" in LeastI2_order, fast)
+  apply (erule_tac P = "\<lambda>x. x : S" in LeastI2_order)
+  apply fast
apply (rule LeastI2_order)
apply (auto elim: monoD intro!: order_antisym)
done
@@ -1858,22 +1778,18 @@

-definition bind :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
-  "bind A f = {x. \<exists>B \<in> fA. x \<in> B}"
+definition bind :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"
+  where "bind A f = {x. \<exists>B \<in> fA. x \<in> B}"

hide_const (open) bind

-lemma bind_bind:
-  fixes A :: "'a set"
-  shows "Set.bind (Set.bind A B) C = Set.bind A (\<lambda>x. Set.bind (B x) C)"
+lemma bind_bind: "Set.bind (Set.bind A B) C = Set.bind A (\<lambda>x. Set.bind (B x) C)" for A :: "'a set"

-lemma empty_bind [simp]:
-  "Set.bind {} f = {}"
+lemma empty_bind [simp]: "Set.bind {} f = {}"

-lemma nonempty_bind_const:
-  "A \<noteq> {} \<Longrightarrow> Set.bind A (\<lambda>_. B) = B"
+lemma nonempty_bind_const: "A \<noteq> {} \<Longrightarrow> Set.bind A (\<lambda>_. B) = B"

lemma bind_const: "Set.bind A (\<lambda>_. B) = (if A = {} then {} else B)"
@@ -1882,53 +1798,50 @@
lemma bind_singleton_conv_image: "Set.bind A (\<lambda>x. {f x}) = f  A"

+
subsubsection \<open>Operations for execution\<close>

-definition is_empty :: "'a set \<Rightarrow> bool" where
-  [code_abbrev]: "is_empty A \<longleftrightarrow> A = {}"
+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}"
+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"
+lemma member_remove [simp]: "x \<in> Set.remove y A \<longleftrightarrow> x \<in> A \<and> x \<noteq> y"

-definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" where
-  [code_abbrev]: "filter P A = {a \<in> A. P a}"
+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"
+lemma member_filter [simp]: "x \<in> Set.filter P A \<longleftrightarrow> x \<in> A \<and> P x"

instantiation set :: (equal) equal
begin

-definition
-  "HOL.equal A B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
-
-instance proof
+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 R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
-
-lemma pairwise_subset: "\<lbrakk>pairwise P S; T \<subseteq> S\<rbrakk> \<Longrightarrow> pairwise P T"
+definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x \<noteq> y \<longrightarrow> R x y)"
+
+lemma pairwise_subset: "pairwise P S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> pairwise P T"
by (force simp: pairwise_def)

-definition disjnt where "disjnt A B \<equiv> A \<inter> B = {}"
-
-lemma disjnt_iff: "disjnt A B \<longleftrightarrow> (\<forall>x. ~ (x \<in> A \<and> x \<in> B))"
+definition disjnt where "disjnt A B \<longleftrightarrow> A \<inter> B = {}"
+
+lemma disjnt_iff: "disjnt A B \<longleftrightarrow> (\<forall>x. \<not> (x \<in> A \<and> x \<in> B))"
by (force simp: disjnt_def)

lemma pairwise_empty [simp]: "pairwise P {}"
@@ -1938,12 +1851,11 @@

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_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)
+  "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_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)

lemma Int_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B \<Longrightarrow> False) \<Longrightarrow> A \<inter> B = {}"
by blast`