# HG changeset patch
# User wenzelm
# Date 1414264592 -7200
# Node ID c6348a0621311e1cf9ec5c30b422cf3e98594f0f
# Parent  7305bad408b5132261388b8fbf690cab487c4de7
tuned whitespace;
more symbols;

diff -r 7305bad408b5 -r c6348a062131 src/HOL/Library/FuncSet.thy
--- a/src/HOL/Library/FuncSet.thy	Sat Oct 25 11:53:35 2014 +0200
+++ b/src/HOL/Library/FuncSet.thy	Sat Oct 25 21:16:32 2014 +0200
@@ -2,144 +2,143 @@
     Author:     Florian Kammueller and Lawrence C Paulson, Lukas Bulwahn
 *)
 
-header {* Pi and Function Sets *}
+header \<open>Pi and Function Sets\<close>
 
 theory FuncSet
 imports Hilbert_Choice Main
 begin
 
-definition
-  Pi :: "['a set, 'a => 'b set] => ('a => 'b) set" where
-  "Pi A B = {f. \<forall>x. x \<in> A --> f x \<in> B x}"
+definition Pi :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
+  where "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
 
-definition
-  extensional :: "'a set => ('a => 'b) set" where
-  "extensional A = {f. \<forall>x. x~:A --> f x = undefined}"
+definition extensional :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) set"
+  where "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}"
 
-definition
-  "restrict" :: "['a => 'b, 'a set] => ('a => 'b)" where
-  "restrict f A = (%x. if x \<in> A then f x else undefined)"
+definition "restrict" :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
+  where "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
 
-abbreviation
-  funcset :: "['a set, 'b set] => ('a => 'b) set"
-    (infixr "->" 60) where
-  "A -> B \<equiv> Pi A (%_. B)"
+abbreviation funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  (infixr "->" 60)
+  where "A -> B \<equiv> Pi A (\<lambda>_. B)"
 
 notation (xsymbols)
   funcset  (infixr "\<rightarrow>" 60)
 
 syntax
-  "_Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
-  "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
-
+  "_Pi"  :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3PI _:_./ _)" 10)
+  "_lam" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> 'b)"  ("(3%_:_./ _)" [0,0,3] 3)
 syntax (xsymbols)
-  "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
-  "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
-
+  "_Pi" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
+  "_lam" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
 syntax (HTML output)
-  "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
-  "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
-
+  "_Pi" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
+  "_lam" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
 translations
-  "PI x:A. B" \<rightleftharpoons> "CONST Pi A (%x. B)"
-  "%x:A. f" \<rightleftharpoons> "CONST restrict (%x. f) A"
+  "\<Pi> x\<in>A. B" \<rightleftharpoons> "CONST Pi A (\<lambda>x. B)"
+  "\<lambda>x\<in>A. f" \<rightleftharpoons> "CONST restrict (\<lambda>x. f) A"
 
-definition
-  "compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)" where
-  "compose A g f = (\<lambda>x\<in>A. g (f x))"
+definition "compose" :: "'a set \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c)"
+  where "compose A g f = (\<lambda>x\<in>A. g (f x))"
 
 
-subsection{*Basic Properties of @{term Pi}*}
+subsection \<open>Basic Properties of @{term Pi}\<close>
 
-lemma Pi_I[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
+lemma Pi_I[intro!]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B"
   by (simp add: Pi_def)
 
-lemma Pi_I'[simp]: "(!!x. x : A --> f x : B x) ==> f : Pi A B"
-by(simp add:Pi_def)
+lemma Pi_I'[simp]: "(\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B"
+  by (simp add:Pi_def)
 
-lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
+lemma funcsetI: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f \<in> A \<rightarrow> B"
   by (simp add: Pi_def)
 
-lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
+lemma Pi_mem: "f \<in> Pi A B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B x"
   by (simp add: Pi_def)
 
 lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
   unfolding Pi_def by auto
 
-lemma PiE [elim]:
-  "f : Pi A B ==> (f x : B x ==> Q) ==> (x ~: A ==> Q) ==> Q"
-by(auto simp: Pi_def)
+lemma PiE [elim]: "f \<in> Pi A B \<Longrightarrow> (f x \<in> B x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q"
+  by (auto simp: Pi_def)
 
-lemma Pi_cong:
-  "(\<And> w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi A B \<longleftrightarrow> g \<in> Pi A B"
+lemma Pi_cong: "(\<And>w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi A B \<longleftrightarrow> g \<in> Pi A B"
   by (auto simp: Pi_def)
 
 lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A"
   by auto
 
-lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
+lemma funcset_mem: "f \<in> A \<rightarrow> B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B"
   by (simp add: Pi_def)
 
-lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
+lemma funcset_image: "f \<in> A \<rightarrow> B \<Longrightarrow> f ` A \<subseteq> B"
   by auto
 
 lemma image_subset_iff_funcset: "F ` A \<subseteq> B \<longleftrightarrow> F \<in> A \<rightarrow> B"
   by auto
 
-lemma Pi_eq_empty[simp]: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B x = {})"
-apply (simp add: Pi_def, auto)
-txt{*Converse direction requires Axiom of Choice to exhibit a function
-picking an element from each non-empty @{term "B x"}*}
-apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
-apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
-done
+lemma Pi_eq_empty[simp]: "(\<Pi> x \<in> A. B x) = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})"
+  apply (simp add: Pi_def)
+  apply auto
+  txt \<open>Converse direction requires Axiom of Choice to exhibit a function
+  picking an element from each non-empty @{term "B x"}\<close>
+  apply (drule_tac x = "\<lambda>u. SOME y. y \<in> B u" in spec)
+  apply auto
+  apply (cut_tac P = "\<lambda>y. y \<in> B x" in some_eq_ex)
+  apply auto
+  done
 
 lemma Pi_empty [simp]: "Pi {} B = UNIV"
-by (simp add: Pi_def)
+  by (simp add: Pi_def)
 
 lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
   by auto
 
 lemma Pi_UN:
   fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
-  assumes "finite I" and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
+  assumes "finite I"
+    and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
   shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
 proof (intro set_eqI iffI)
-  fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
-  then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto
-  from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
+  fix f
+  assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
+  then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i"
+    by auto
+  from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)"
+    by auto
   obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
-    using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto
+    using \<open>finite I\<close> finite_nat_set_iff_bounded_le[of "n`I"] by auto
   have "f \<in> Pi I (A k)"
   proof (intro Pi_I)
-    fix i assume "i \<in> I"
+    fix i
+    assume "i \<in> I"
     from mono[OF this, of "n i" k] k[OF this] n[OF this]
     show "f i \<in> A k i" by auto
   qed
-  then show "f \<in> (\<Union>n. Pi I (A n))" by auto
+  then show "f \<in> (\<Union>n. Pi I (A n))"
+    by auto
 qed auto
 
-lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
-by (simp add: Pi_def)
+lemma Pi_UNIV [simp]: "A \<rightarrow> UNIV = UNIV"
+  by (simp add: Pi_def)
 
-text{*Covariance of Pi-sets in their second argument*}
-lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
-by auto
+text \<open>Covariance of Pi-sets in their second argument\<close>
+lemma Pi_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi A B \<subseteq> Pi A C"
+  by auto
 
-text{*Contravariance of Pi-sets in their first argument*}
-lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
-by auto
+text \<open>Contravariance of Pi-sets in their first argument\<close>
+lemma Pi_anti_mono: "A' \<subseteq> A \<Longrightarrow> Pi A B \<subseteq> Pi A' B"
+  by auto
 
 lemma prod_final:
-  assumes 1: "fst \<circ> f \<in> Pi A B" and 2: "snd \<circ> f \<in> Pi A C"
+  assumes 1: "fst \<circ> f \<in> Pi A B"
+    and 2: "snd \<circ> f \<in> Pi A C"
   shows "f \<in> (\<Pi> z \<in> A. B z \<times> C z)"
-proof (rule Pi_I) 
+proof (rule Pi_I)
   fix z
-  assume z: "z \<in> A" 
-  have "f z = (fst (f z), snd (f z))" 
+  assume z: "z \<in> A"
+  have "f z = (fst (f z), snd (f z))"
     by simp
-  also have "...  \<in> B z \<times> C z"
-    by (metis SigmaI PiE o_apply 1 2 z) 
+  also have "\<dots> \<in> B z \<times> C z"
+    by (metis SigmaI PiE o_apply 1 2 z)
   finally show "f z \<in> B z \<times> C z" .
 qed
 
@@ -163,25 +162,27 @@
   apply (auto dest!: Pi_mem)
   done
 
-subsection{*Composition With a Restricted Domain: @{term compose}*}
+
+subsection \<open>Composition With a Restricted Domain: @{term compose}\<close>
 
-lemma funcset_compose:
-  "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
-by (simp add: Pi_def compose_def restrict_def)
+lemma funcset_compose: "f \<in> A \<rightarrow> B \<Longrightarrow> g \<in> B \<rightarrow> C \<Longrightarrow> compose A g f \<in> A \<rightarrow> C"
+  by (simp add: Pi_def compose_def restrict_def)
 
 lemma compose_assoc:
-    "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
-      ==> compose A h (compose A g f) = compose A (compose B h g) f"
-by (simp add: fun_eq_iff Pi_def compose_def restrict_def)
+  assumes "f \<in> A \<rightarrow> B"
+    and "g \<in> B \<rightarrow> C"
+    and "h \<in> C \<rightarrow> D"
+  shows "compose A h (compose A g f) = compose A (compose B h g) f"
+  using assms by (simp add: fun_eq_iff Pi_def compose_def restrict_def)
 
-lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
-by (simp add: compose_def restrict_def)
+lemma compose_eq: "x \<in> A \<Longrightarrow> compose A g f x = g (f x)"
+  by (simp add: compose_def restrict_def)
 
-lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
+lemma surj_compose: "f ` A = B \<Longrightarrow> g ` B = C \<Longrightarrow> compose A g f ` A = C"
   by (auto simp add: image_def compose_eq)
 
 
-subsection{*Bounded Abstraction: @{term restrict}*}
+subsection \<open>Bounded Abstraction: @{term restrict}\<close>
 
 lemma restrict_in_funcset: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> (\<lambda>x\<in>A. f x) \<in> A \<rightarrow> B"
   by (simp add: Pi_def restrict_def)
@@ -195,8 +196,7 @@
 lemma restrict_apply': "x \<in> A \<Longrightarrow> (\<lambda>y\<in>A. f y) x = f x"
   by simp
 
-lemma restrict_ext:
-    "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
+lemma restrict_ext: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
   by (simp add: fun_eq_iff Pi_def restrict_def)
 
 lemma restrict_UNIV: "restrict f UNIV = f"
@@ -205,12 +205,10 @@
 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
   by (simp add: inj_on_def restrict_def)
 
-lemma Id_compose:
-    "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
+lemma Id_compose: "f \<in> A \<rightarrow> B \<Longrightarrow> f \<in> extensional A \<Longrightarrow> compose A (\<lambda>y\<in>B. y) f = f"
   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
 
-lemma compose_Id:
-    "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
+lemma compose_Id: "g \<in> A \<rightarrow> B \<Longrightarrow> g \<in> extensional A \<Longrightarrow> compose A g (\<lambda>x\<in>A. x) = g"
   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
 
 lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
@@ -222,99 +220,99 @@
 lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
   by (auto simp: restrict_def)
 
-lemma restrict_upd[simp]:
-  "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
+lemma restrict_upd[simp]: "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
   by (auto simp: fun_eq_iff)
 
 lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
   by (auto simp: restrict_def Pi_def)
 
 
-subsection{*Bijections Between Sets*}
+subsection \<open>Bijections Between Sets\<close>
 
-text{*The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of
-the theorems belong here, or need at least @{term Hilbert_Choice}.*}
+text \<open>The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of
+the theorems belong here, or need at least @{term Hilbert_Choice}.\<close>
 
 lemma bij_betwI:
-assumes "f \<in> A \<rightarrow> B" and "g \<in> B \<rightarrow> A"
-    and g_f: "\<And>x. x\<in>A \<Longrightarrow> g (f x) = x" and f_g: "\<And>y. y\<in>B \<Longrightarrow> f (g y) = y"
-shows "bij_betw f A B"
-unfolding bij_betw_def
+  assumes "f \<in> A \<rightarrow> B"
+    and "g \<in> B \<rightarrow> A"
+    and g_f: "\<And>x. x\<in>A \<Longrightarrow> g (f x) = x"
+    and f_g: "\<And>y. y\<in>B \<Longrightarrow> f (g y) = y"
+  shows "bij_betw f A B"
+  unfolding bij_betw_def
 proof
-  show "inj_on f A" by (metis g_f inj_on_def)
-next
-  have "f ` A \<subseteq> B" using `f \<in> A \<rightarrow> B` by auto
+  show "inj_on f A"
+    by (metis g_f inj_on_def)
+  have "f ` A \<subseteq> B"
+    using \<open>f \<in> A \<rightarrow> B\<close> by auto
   moreover
-  have "B \<subseteq> f ` A" by auto (metis Pi_mem `g \<in> B \<rightarrow> A` f_g image_iff)
-  ultimately show "f ` A = B" by blast
+  have "B \<subseteq> f ` A"
+    by auto (metis Pi_mem \<open>g \<in> B \<rightarrow> A\<close> f_g image_iff)
+  ultimately show "f ` A = B"
+    by blast
 qed
 
 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
-by (auto simp add: bij_betw_def)
+  by (auto simp add: bij_betw_def)
 
-lemma inj_on_compose:
-  "[| bij_betw f A B; inj_on g B |] ==> inj_on (compose A g f) A"
-by (auto simp add: bij_betw_def inj_on_def compose_eq)
+lemma inj_on_compose: "bij_betw f A B \<Longrightarrow> inj_on g B \<Longrightarrow> inj_on (compose A g f) A"
+  by (auto simp add: bij_betw_def inj_on_def compose_eq)
 
-lemma bij_betw_compose:
-  "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
-apply (simp add: bij_betw_def compose_eq inj_on_compose)
-apply (auto simp add: compose_def image_def)
-done
+lemma bij_betw_compose: "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (compose A g f) A C"
+  apply (simp add: bij_betw_def compose_eq inj_on_compose)
+  apply (auto simp add: compose_def image_def)
+  done
 
-lemma bij_betw_restrict_eq [simp]:
-  "bij_betw (restrict f A) A B = bij_betw f A B"
-by (simp add: bij_betw_def)
+lemma bij_betw_restrict_eq [simp]: "bij_betw (restrict f A) A B = bij_betw f A B"
+  by (simp add: bij_betw_def)
 
 
-subsection{*Extensionality*}
+subsection \<open>Extensionality\<close>
 
 lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
   unfolding extensional_def by auto
 
-lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = undefined"
-by (simp add: extensional_def)
+lemma extensional_arb: "f \<in> extensional A \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = undefined"
+  by (simp add: extensional_def)
 
 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
-by (simp add: restrict_def extensional_def)
+  by (simp add: restrict_def extensional_def)
 
 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
-by (simp add: compose_def)
+  by (simp add: compose_def)
 
 lemma extensionalityI:
-  "[| f \<in> extensional A; g \<in> extensional A;
-      !!x. x\<in>A ==> f x = g x |] ==> f = g"
-by (force simp add: fun_eq_iff extensional_def)
+  assumes "f \<in> extensional A"
+    and "g \<in> extensional A"
+    and "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
+  shows "f = g"
+  using assms by (force simp add: fun_eq_iff extensional_def)
 
 lemma extensional_restrict:  "f \<in> extensional A \<Longrightarrow> restrict f A = f"
-by(rule extensionalityI[OF restrict_extensional]) auto
+  by (rule extensionalityI[OF restrict_extensional]) auto
 
 lemma extensional_subset: "f \<in> extensional A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f \<in> extensional B"
   unfolding extensional_def by auto
 
-lemma inv_into_funcset: "f ` A = B ==> (\<lambda>x\<in>B. inv_into A f x) : B -> A"
-by (unfold inv_into_def) (fast intro: someI2)
+lemma inv_into_funcset: "f ` A = B \<Longrightarrow> (\<lambda>x\<in>B. inv_into A f x) \<in> B \<rightarrow> A"
+  by (unfold inv_into_def) (fast intro: someI2)
 
-lemma compose_inv_into_id:
-  "bij_betw f A B ==> compose A (\<lambda>y\<in>B. inv_into A f y) f = (\<lambda>x\<in>A. x)"
-apply (simp add: bij_betw_def compose_def)
-apply (rule restrict_ext, auto)
-done
+lemma compose_inv_into_id: "bij_betw f A B \<Longrightarrow> compose A (\<lambda>y\<in>B. inv_into A f y) f = (\<lambda>x\<in>A. x)"
+  apply (simp add: bij_betw_def compose_def)
+  apply (rule restrict_ext, auto)
+  done
 
-lemma compose_id_inv_into:
-  "f ` A = B ==> compose B f (\<lambda>y\<in>B. inv_into A f y) = (\<lambda>x\<in>B. x)"
-apply (simp add: compose_def)
-apply (rule restrict_ext)
-apply (simp add: f_inv_into_f)
-done
+lemma compose_id_inv_into: "f ` A = B \<Longrightarrow> compose B f (\<lambda>y\<in>B. inv_into A f y) = (\<lambda>x\<in>B. x)"
+  apply (simp add: compose_def)
+  apply (rule restrict_ext)
+  apply (simp add: f_inv_into_f)
+  done
 
 lemma extensional_insert[intro, simp]:
   assumes "a \<in> extensional (insert i I)"
   shows "a(i := b) \<in> extensional (insert i I)"
   using assms unfolding extensional_def by auto
 
-lemma extensional_Int[simp]:
-  "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
+lemma extensional_Int[simp]: "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
   unfolding extensional_def by auto
 
 lemma extensional_UNIV[simp]: "extensional UNIV = UNIV"
@@ -332,61 +330,66 @@
   unfolding extensional_def by auto
 
 
-subsection{*Cardinality*}
+subsection \<open>Cardinality\<close>
 
-lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
-by (rule card_inj_on_le) auto
+lemma card_inj: "f \<in> A \<rightarrow> B \<Longrightarrow> inj_on f A \<Longrightarrow> finite B \<Longrightarrow> card A \<le> card B"
+  by (rule card_inj_on_le) auto
 
 lemma card_bij:
-  "[|f \<in> A\<rightarrow>B; inj_on f A;
-     g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
-by (blast intro: card_inj order_antisym)
+  assumes "f \<in> A \<rightarrow> B" "inj_on f A"
+    and "g \<in> B \<rightarrow> A" "inj_on g B"
+    and "finite A" "finite B"
+  shows "card A = card B"
+  using assms by (blast intro: card_inj order_antisym)
 
-subsection {* Extensional Function Spaces *} 
+
+subsection \<open>Extensional Function Spaces\<close>
 
-definition PiE :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set" where
-  "PiE S T = Pi S T \<inter> extensional S"
+definition PiE :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
+  where "PiE S T = Pi S T \<inter> extensional S"
 
 abbreviation "Pi\<^sub>E A B \<equiv> PiE A B"
 
-syntax "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)
-
-syntax (xsymbols) "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
+syntax
+  "_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3PIE _:_./ _)" 10)
+syntax (xsymbols)
+  "_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
+syntax (HTML output)
+  "_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
+translations "\<Pi>\<^sub>E x\<in>A. B" \<rightleftharpoons> "CONST Pi\<^sub>E A (\<lambda>x. B)"
 
-syntax (HTML output) "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
-
-translations "PIE x:A. B" \<rightleftharpoons> "CONST Pi\<^sub>E A (%x. B)"
-
-abbreviation extensional_funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "->\<^sub>E" 60) where
-  "A ->\<^sub>E B \<equiv> (\<Pi>\<^sub>E i\<in>A. B)"
+abbreviation extensional_funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "->\<^sub>E" 60)
+  where "A ->\<^sub>E B \<equiv> (\<Pi>\<^sub>E i\<in>A. B)"
 
 notation (xsymbols)
   extensional_funcset  (infixr "\<rightarrow>\<^sub>E" 60)
 
-lemma extensional_funcset_def: "extensional_funcset S T = (S -> T) \<inter> extensional S"
+lemma extensional_funcset_def: "extensional_funcset S T = (S \<rightarrow> T) \<inter> extensional S"
   by (simp add: PiE_def)
 
-lemma PiE_empty_domain[simp]: "PiE {} T = {%x. undefined}"
+lemma PiE_empty_domain[simp]: "PiE {} T = {\<lambda>x. undefined}"
   unfolding PiE_def by simp
 
 lemma PiE_UNIV_domain: "PiE UNIV T = Pi UNIV T"
   unfolding PiE_def by simp
 
-lemma PiE_empty_range[simp]: "i \<in> I \<Longrightarrow> F i = {} \<Longrightarrow> (PIE i:I. F i) = {}"
+lemma PiE_empty_range[simp]: "i \<in> I \<Longrightarrow> F i = {} \<Longrightarrow> (\<Pi>\<^sub>E i\<in>I. F i) = {}"
   unfolding PiE_def by auto
 
-lemma PiE_eq_empty_iff:
-  "Pi\<^sub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
+lemma PiE_eq_empty_iff: "Pi\<^sub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
 proof
   assume "Pi\<^sub>E I F = {}"
   show "\<exists>i\<in>I. F i = {}"
   proof (rule ccontr)
     assume "\<not> ?thesis"
-    then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)" by auto
+    then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)"
+      by auto
     from choice[OF this]
     obtain f where " \<forall>x. (x \<in> I \<longrightarrow> f x \<in> F x) \<and> (x \<notin> I \<longrightarrow> f x = undefined)" ..
-    then have "f \<in> Pi\<^sub>E I F" by (auto simp: extensional_def PiE_def)
-    with `Pi\<^sub>E I F = {}` show False by auto
+    then have "f \<in> Pi\<^sub>E I F"
+      by (auto simp: extensional_def PiE_def)
+    with \<open>Pi\<^sub>E I F = {}\<close> show False
+      by auto
   qed
 qed (auto simp: PiE_def)
 
@@ -411,21 +414,24 @@
     with assms have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)"
       by (auto intro!: image_eqI[where x="(f x, f(x := undefined))"] intro: fun_upd_in_PiE PiE_mem)
   }
-  then show ?thesis using assms by (auto intro: PiE_fun_upd)
+  then show ?thesis
+    using assms by (auto intro: PiE_fun_upd)
 qed
 
-lemma PiE_Int: "(Pi\<^sub>E I A) \<inter> (Pi\<^sub>E I B) = Pi\<^sub>E I (\<lambda>x. A x \<inter> B x)"
+lemma PiE_Int: "Pi\<^sub>E I A \<inter> Pi\<^sub>E I B = Pi\<^sub>E I (\<lambda>x. A x \<inter> B x)"
   by (auto simp: PiE_def)
 
-lemma PiE_cong:
-  "(\<And>i. i\<in>I \<Longrightarrow> A i = B i) \<Longrightarrow> Pi\<^sub>E I A = Pi\<^sub>E I B"
+lemma PiE_cong: "(\<And>i. i\<in>I \<Longrightarrow> A i = B i) \<Longrightarrow> Pi\<^sub>E I A = Pi\<^sub>E I B"
   unfolding PiE_def by (auto simp: Pi_cong)
 
 lemma PiE_E [elim]:
-  "f \<in> PiE A B \<Longrightarrow> (x \<in> A \<Longrightarrow> f x \<in> B x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> f x = undefined \<Longrightarrow> Q) \<Longrightarrow> Q"
-by(auto simp: Pi_def PiE_def extensional_def)
+  assumes "f \<in> PiE A B"
+  obtains "x \<in> A" and "f x \<in> B x"
+    | "x \<notin> A" and "f x = undefined"
+  using assms by (auto simp: Pi_def PiE_def extensional_def)
 
-lemma PiE_I[intro!]: "(\<And>x. x \<in> A ==> f x \<in> B x) \<Longrightarrow> (\<And>x. x \<notin> A \<Longrightarrow> f x = undefined) \<Longrightarrow> f \<in> PiE A B"
+lemma PiE_I[intro!]:
+  "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> (\<And>x. x \<notin> A \<Longrightarrow> f x = undefined) \<Longrightarrow> f \<in> PiE A B"
   by (simp add: PiE_def extensional_def)
 
 lemma PiE_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> PiE A B \<subseteq> PiE A C"
@@ -442,24 +448,31 @@
 
 lemma PiE_eq_subset:
   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
-  assumes eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'" and "i \<in> I"
+    and eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
+    and "i \<in> I"
   shows "F i \<subseteq> F' i"
 proof
-  fix x assume "x \<in> F i"
-  with ne have "\<forall>j. \<exists>y. ((j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined))"
+  fix x
+  assume "x \<in> F i"
+  with ne have "\<forall>j. \<exists>y. (j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined)"
     by auto
   from choice[OF this] obtain f
     where f: " \<forall>j. (j \<in> I \<longrightarrow> f j \<in> F j \<and> (i = j \<longrightarrow> x = f j)) \<and> (j \<notin> I \<longrightarrow> f j = undefined)" ..
-  then have "f \<in> Pi\<^sub>E I F" by (auto simp: extensional_def PiE_def)
-  then have "f \<in> Pi\<^sub>E I F'" using assms by simp
-  then show "x \<in> F' i" using f `i \<in> I` by (auto simp: PiE_def)
+  then have "f \<in> Pi\<^sub>E I F"
+    by (auto simp: extensional_def PiE_def)
+  then have "f \<in> Pi\<^sub>E I F'"
+    using assms by simp
+  then show "x \<in> F' i"
+    using f \<open>i \<in> I\<close> by (auto simp: PiE_def)
 qed
 
 lemma PiE_eq_iff_not_empty:
   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
   shows "Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
 proof (intro iffI ballI)
-  fix i assume eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'" and i: "i \<in> I"
+  fix i
+  assume eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
+  assume i: "i \<in> I"
   show "F i = F' i"
     using PiE_eq_subset[of I F F', OF ne eq i]
     using PiE_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
@@ -473,15 +486,16 @@
   assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
   then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
     using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by auto
-  with PiE_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" by auto
+  with PiE_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i"
+    by auto
 next
   assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
   then show "Pi\<^sub>E I F = Pi\<^sub>E I F'"
     using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by (auto simp: PiE_def)
 qed
 
-lemma extensional_funcset_fun_upd_restricts_rangeI: 
-  "\<forall>y \<in> S. f x \<noteq> f y \<Longrightarrow> f : (insert x S) \<rightarrow>\<^sub>E T ==> f(x := undefined) : S \<rightarrow>\<^sub>E (T - {f x})"
+lemma extensional_funcset_fun_upd_restricts_rangeI:
+  "\<forall>y \<in> S. f x \<noteq> f y \<Longrightarrow> f \<in> (insert x S) \<rightarrow>\<^sub>E T \<Longrightarrow> f(x := undefined) \<in> S \<rightarrow>\<^sub>E (T - {f x})"
   unfolding extensional_funcset_def extensional_def
   apply auto
   apply (case_tac "x = xa")
@@ -490,66 +504,70 @@
 
 lemma extensional_funcset_fun_upd_extends_rangeI:
   assumes "a \<in> T" "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
-  shows "f(x := a) \<in> (insert x S) \<rightarrow>\<^sub>E  T"
+  shows "f(x := a) \<in> insert x S \<rightarrow>\<^sub>E  T"
   using assms unfolding extensional_funcset_def extensional_def by auto
 
-subsubsection {* Injective Extensional Function Spaces *}
+
+subsubsection \<open>Injective Extensional Function Spaces\<close>
 
 lemma extensional_funcset_fun_upd_inj_onI:
-  assumes "f \<in> S \<rightarrow>\<^sub>E (T - {a})" "inj_on f S"
+  assumes "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
+    and "inj_on f S"
   shows "inj_on (f(x := a)) S"
-  using assms unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
+  using assms
+  unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
 
 lemma extensional_funcset_extend_domain_inj_on_eq:
   assumes "x \<notin> S"
-  shows"{f. f \<in> (insert x S) \<rightarrow>\<^sub>E T \<and> inj_on f (insert x S)} =
-    (%(y, g). g(x:=y)) ` {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E (T - {y}) \<and> inj_on g S}"
-proof -
-  from assms show ?thesis
-    apply (auto del: PiE_I PiE_E)
-    apply (auto intro: extensional_funcset_fun_upd_inj_onI extensional_funcset_fun_upd_extends_rangeI del: PiE_I PiE_E)
-    apply (auto simp add: image_iff inj_on_def)
-    apply (rule_tac x="xa x" in exI)
-    apply (auto intro: PiE_mem del: PiE_I PiE_E)
-    apply (rule_tac x="xa(x := undefined)" in exI)
-    apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
-    apply (auto dest!: PiE_mem split: split_if_asm)
-    done
-qed
+  shows "{f. f \<in> (insert x S) \<rightarrow>\<^sub>E T \<and> inj_on f (insert x S)} =
+    (\<lambda>(y, g). g(x:=y)) ` {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E (T - {y}) \<and> inj_on g S}"
+  using assms
+  apply (auto del: PiE_I PiE_E)
+  apply (auto intro: extensional_funcset_fun_upd_inj_onI
+    extensional_funcset_fun_upd_extends_rangeI del: PiE_I PiE_E)
+  apply (auto simp add: image_iff inj_on_def)
+  apply (rule_tac x="xa x" in exI)
+  apply (auto intro: PiE_mem del: PiE_I PiE_E)
+  apply (rule_tac x="xa(x := undefined)" in exI)
+  apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
+  apply (auto dest!: PiE_mem split: split_if_asm)
+  done
 
 lemma extensional_funcset_extend_domain_inj_onI:
   assumes "x \<notin> S"
   shows "inj_on (\<lambda>(y, g). g(x := y)) {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E (T - {y}) \<and> inj_on g S}"
-proof -
-  from assms show ?thesis
-    apply (auto intro!: inj_onI)
-    apply (metis fun_upd_same)
-    by (metis assms PiE_arb fun_upd_triv fun_upd_upd)
-qed
-  
+  using assms
+  apply (auto intro!: inj_onI)
+  apply (metis fun_upd_same)
+  apply (metis assms PiE_arb fun_upd_triv fun_upd_upd)
+  done
 
-subsubsection {* Cardinality *}
 
-lemma finite_PiE: "finite S \<Longrightarrow> (\<And>i. i \<in> S \<Longrightarrow> finite (T i)) \<Longrightarrow> finite (PIE i : S. T i)"
+subsubsection \<open>Cardinality\<close>
+
+lemma finite_PiE: "finite S \<Longrightarrow> (\<And>i. i \<in> S \<Longrightarrow> finite (T i)) \<Longrightarrow> finite (\<Pi>\<^sub>E i \<in> S. T i)"
   by (induct S arbitrary: T rule: finite_induct) (simp_all add: PiE_insert_eq)
 
 lemma inj_combinator: "x \<notin> S \<Longrightarrow> inj_on (\<lambda>(y, g). g(x := y)) (T x \<times> Pi\<^sub>E S T)"
 proof (safe intro!: inj_onI ext)
-  fix f y g z assume "x \<notin> S" and fg: "f \<in> Pi\<^sub>E S T" "g \<in> Pi\<^sub>E S T"
+  fix f y g z
+  assume "x \<notin> S"
+  assume fg: "f \<in> Pi\<^sub>E S T" "g \<in> Pi\<^sub>E S T"
   assume "f(x := y) = g(x := z)"
   then have *: "\<And>i. (f(x := y)) i = (g(x := z)) i"
     unfolding fun_eq_iff by auto
   from this[of x] show "y = z" by simp
-  fix i from *[of i] `x \<notin> S` fg show "f i = g i"
+  fix i from *[of i] \<open>x \<notin> S\<close> fg show "f i = g i"
     by (auto split: split_if_asm simp: PiE_def extensional_def)
 qed
 
-lemma card_PiE:
-  "finite S \<Longrightarrow> card (PIE i : S. T i) = (\<Prod> i\<in>S. card (T i))"
+lemma card_PiE: "finite S \<Longrightarrow> card (\<Pi>\<^sub>E i \<in> S. T i) = (\<Prod> i\<in>S. card (T i))"
 proof (induct rule: finite_induct)
-  case empty then show ?case by auto
+  case empty
+  then show ?case by auto
 next
-  case (insert x S) then show ?case
+  case (insert x S)
+  then show ?case
     by (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product)
 qed