
diff -r a962f349c8c9 -r a95e7432d86c src/HOL/Finite_Set.thy
--- a/src/HOL/Finite_Set.thy	Wed Jul 06 14:09:13 2016 +0200
+++ b/src/HOL/Finite_Set.thy	Wed Jul 06 20:19:51 2016 +0200
@@ -16,9 +16,9 @@
 begin
 
 inductive finite :: "'a set \<Rightarrow> bool"
-  where
-    emptyI [simp, intro!]: "finite {}"
-  | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
+where
+  emptyI [simp, intro!]: "finite {}"
+| insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
 
 end
 
@@ -32,14 +32,16 @@
   assumes "P {}"
     and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
   shows "P F"
-using \<open>finite F\<close>
+  using \<open>finite F\<close>
 proof induct
   show "P {}" by fact
-  fix x F assume F: "finite F" and P: "P F"
+next
+  fix x F
+  assume F: "finite F" and P: "P F"
   show "P (insert x F)"
   proof cases
     assume "x \<in> F"
-    hence "insert x F = F" by (rule insert_absorb)
+    then have "insert x F = F" by (rule insert_absorb)
     with P show ?thesis by (simp only:)
   next
     assume "x \<notin> F"
@@ -49,13 +51,15 @@
 
 lemma infinite_finite_induct [case_names infinite empty insert]:
   assumes infinite: "\<And>A. \<not> finite A \<Longrightarrow> P A"
-  assumes empty: "P {}"
-  assumes insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
+    and empty: "P {}"
+    and insert: "\<And>x F. finite F \<Longrightarrow> x \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert x F)"
   shows "P A"
 proof (cases "finite A")
-  case False with infinite show ?thesis .
+  case False
+  with infinite show ?thesis .
 next
-  case True then show ?thesis by (induct A) (fact empty insert)+
+  case True
+  then show ?thesis by (induct A) (fact empty insert)+
 qed
 
 
@@ -71,16 +75,18 @@
 
 text \<open>A finite choice principle. Does not need the SOME choice operator.\<close>
 
-lemma finite_set_choice:
-  "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
+lemma finite_set_choice: "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
 proof (induct rule: finite_induct)
-  case empty then show ?case by simp
+  case empty
+  then show ?case by simp
 next
   case (insert a A)
-  then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
-  show ?case (is "EX f. ?P f")
+  then obtain f b where f: "\<forall>x\<in>A. P x (f x)" and ab: "P a b"
+    by auto
+  show ?case (is "\<exists>f. ?P f")
   proof
-    show "?P(%x. if x = a then b else f x)" using f ab by auto
+    show "?P (\<lambda>x. if x = a then b else f x)"
+      using f ab by auto
   qed
 qed
 
@@ -88,100 +94,101 @@
 subsubsection \<open>Finite sets are the images of initial segments of natural numbers\<close>
 
 lemma finite_imp_nat_seg_image_inj_on:
-  assumes "finite A" 
+  assumes "finite A"
   shows "\<exists>(n::nat) f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
-using assms
+  using assms
 proof induct
   case empty
   show ?case
   proof
-    show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}" by simp 
+    show "\<exists>f. {} = f ` {i::nat. i < 0} \<and> inj_on f {i. i < 0}"
+      by simp
   qed
 next
   case (insert a A)
   have notinA: "a \<notin> A" by fact
-  from insert.hyps obtain n f
-    where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
-  hence "insert a A = f(n:=a) ` {i. i < Suc n}"
-        "inj_on (f(n:=a)) {i. i < Suc n}" using notinA
-    by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
-  thus ?case by blast
+  from insert.hyps obtain n f where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}"
+    by blast
+  then have "insert a A = f(n:=a) ` {i. i < Suc n}" and "inj_on (f(n:=a)) {i. i < Suc n}"
+    using notinA by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
+  then show ?case by blast
 qed
 
-lemma nat_seg_image_imp_finite:
-  "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
+lemma nat_seg_image_imp_finite: "A = f ` {i::nat. i < n} \<Longrightarrow> finite A"
 proof (induct n arbitrary: A)
-  case 0 thus ?case by simp
+  case 0
+  then show ?case by simp
 next
   case (Suc n)
   let ?B = "f ` {i. i < n}"
-  have finB: "finite ?B" by(rule Suc.hyps[OF refl])
+  have finB: "finite ?B" by (rule Suc.hyps[OF refl])
   show ?case
-  proof cases
-    assume "\<exists>k<n. f n = f k"
-    hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
-    thus ?thesis using finB by simp
+  proof (cases "\<exists>k<n. f n = f k")
+    case True
+    then have "A = ?B"
+      using Suc.prems by (auto simp:less_Suc_eq)
+    then show ?thesis
+      using finB by simp
   next
-    assume "\<not>(\<exists> k<n. f n = f k)"
-    hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
-    thus ?thesis using finB by simp
+    case False
+    then have "A = insert (f n) ?B"
+      using Suc.prems by (auto simp:less_Suc_eq)
+    then show ?thesis using finB by simp
   qed
 qed
 
-lemma finite_conv_nat_seg_image:
-  "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
+lemma finite_conv_nat_seg_image: "finite A \<longleftrightarrow> (\<exists>(n::nat) f. A = f ` {i::nat. i < n})"
   by (blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
 
 lemma finite_imp_inj_to_nat_seg:
   assumes "finite A"
   shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
 proof -
-  from finite_imp_nat_seg_image_inj_on[OF \<open>finite A\<close>]
+  from finite_imp_nat_seg_image_inj_on [OF \<open>finite A\<close>]
   obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
-    by (auto simp:bij_betw_def)
+    by (auto simp: bij_betw_def)
   let ?f = "the_inv_into {i. i<n} f"
-  have "inj_on ?f A & ?f ` A = {i. i<n}"
+  have "inj_on ?f A \<and> ?f ` A = {i. i<n}"
     by (fold bij_betw_def) (rule bij_betw_the_inv_into[OF bij])
-  thus ?thesis by blast
+  then show ?thesis by blast
 qed
 
-lemma finite_Collect_less_nat [iff]:
-  "finite {n::nat. n < k}"
+lemma finite_Collect_less_nat [iff]: "finite {n::nat. n < k}"
   by (fastforce simp: finite_conv_nat_seg_image)
 
-lemma finite_Collect_le_nat [iff]:
-  "finite {n::nat. n \<le> k}"
+lemma finite_Collect_le_nat [iff]: "finite {n::nat. n \<le> k}"
   by (simp add: le_eq_less_or_eq Collect_disj_eq)
 
 
 subsubsection \<open>Finiteness and common set operations\<close>
 
-lemma rev_finite_subset:
-  "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
+lemma rev_finite_subset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> finite A"
 proof (induct arbitrary: A rule: finite_induct)
   case empty
   then show ?case by simp
 next
   case (insert x F A)
-  have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})" by fact+
+  have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F \<Longrightarrow> finite (A - {x})"
+    by fact+
   show "finite A"
   proof cases
     assume x: "x \<in> A"
     with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
     with r have "finite (A - {x})" .
-    hence "finite (insert x (A - {x}))" ..
-    also have "insert x (A - {x}) = A" using x by (rule insert_Diff)
+    then have "finite (insert x (A - {x}))" ..
+    also have "insert x (A - {x}) = A"
+      using x by (rule insert_Diff)
     finally show ?thesis .
   next
     show ?thesis when "A \<subseteq> F"
       using that by fact
     assume "x \<notin> A"
-    with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
+    with A show "A \<subseteq> F"
+      by (simp add: subset_insert_iff)
   qed
 qed
 
-lemma finite_subset:
-  "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
+lemma finite_subset: "A \<subseteq> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
   by (rule rev_finite_subset)
 
 lemma finite_UnI:
@@ -189,8 +196,7 @@
   shows "finite (F \<union> G)"
   using assms by induct simp_all
 
-lemma finite_Un [iff]:
-  "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
+lemma finite_Un [iff]: "finite (F \<union> G) \<longleftrightarrow> finite F \<and> finite G"
   by (blast intro: finite_UnI finite_subset [of _ "F \<union> G"])
 
 lemma finite_insert [simp]: "finite (insert a A) \<longleftrightarrow> finite A"
@@ -200,8 +206,7 @@
   then show ?thesis by simp
 qed
 
-lemma finite_Int [simp, intro]:
-  "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
+lemma finite_Int [simp, intro]: "finite F \<or> finite G \<Longrightarrow> finite (F \<inter> G)"
   by (blast intro: finite_subset)
 
 lemma finite_Collect_conjI [simp, intro]:
@@ -212,112 +217,110 @@
   "finite {x. P x \<or> Q x} \<longleftrightarrow> finite {x. P x} \<and> finite {x. Q x}"
   by (simp add: Collect_disj_eq)
 
-lemma finite_Diff [simp, intro]:
-  "finite A \<Longrightarrow> finite (A - B)"
+lemma finite_Diff [simp, intro]: "finite A \<Longrightarrow> finite (A - B)"
   by (rule finite_subset, rule Diff_subset)
 
 lemma finite_Diff2 [simp]:
   assumes "finite B"
   shows "finite (A - B) \<longleftrightarrow> finite A"
 proof -
-  have "finite A \<longleftrightarrow> finite((A - B) \<union> (A \<inter> B))" by (simp add: Un_Diff_Int)
-  also have "\<dots> \<longleftrightarrow> finite (A - B)" using \<open>finite B\<close> by simp
+  have "finite A \<longleftrightarrow> finite ((A - B) \<union> (A \<inter> B))"
+    by (simp add: Un_Diff_Int)
+  also have "\<dots> \<longleftrightarrow> finite (A - B)"
+    using \<open>finite B\<close> by simp
   finally show ?thesis ..
 qed
 
-lemma finite_Diff_insert [iff]:
-  "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
+lemma finite_Diff_insert [iff]: "finite (A - insert a B) \<longleftrightarrow> finite (A - B)"
 proof -
   have "finite (A - B) \<longleftrightarrow> finite (A - B - {a})" by simp
   moreover have "A - insert a B = A - B - {a}" by auto
   ultimately show ?thesis by simp
 qed
 
-lemma finite_compl[simp]:
+lemma finite_compl [simp]:
   "finite (A :: 'a set) \<Longrightarrow> finite (- A) \<longleftrightarrow> finite (UNIV :: 'a set)"
   by (simp add: Compl_eq_Diff_UNIV)
 
-lemma finite_Collect_not[simp]:
+lemma finite_Collect_not [simp]:
   "finite {x :: 'a. P x} \<Longrightarrow> finite {x. \<not> P x} \<longleftrightarrow> finite (UNIV :: 'a set)"
   by (simp add: Collect_neg_eq)
 
 lemma finite_Union [simp, intro]:
-  "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite(\<Union>A)"
+  "finite A \<Longrightarrow> (\<And>M. M \<in> A \<Longrightarrow> finite M) \<Longrightarrow> finite (\<Union>A)"
   by (induct rule: finite_induct) simp_all
 
 lemma finite_UN_I [intro]:
   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (\<Union>a\<in>A. B a)"
   by (induct rule: finite_induct) simp_all
 
-lemma finite_UN [simp]:
-  "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
+lemma finite_UN [simp]: "finite A \<Longrightarrow> finite (UNION A B) \<longleftrightarrow> (\<forall>x\<in>A. finite (B x))"
   by (blast intro: finite_subset)
 
-lemma finite_Inter [intro]:
-  "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
+lemma finite_Inter [intro]: "\<exists>A\<in>M. finite A \<Longrightarrow> finite (\<Inter>M)"
   by (blast intro: Inter_lower finite_subset)
 
-lemma finite_INT [intro]:
-  "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
+lemma finite_INT [intro]: "\<exists>x\<in>I. finite (A x) \<Longrightarrow> finite (\<Inter>x\<in>I. A x)"
   by (blast intro: INT_lower finite_subset)
 
-lemma finite_imageI [simp, intro]:
-  "finite F \<Longrightarrow> finite (h ` F)"
+lemma finite_imageI [simp, intro]: "finite F \<Longrightarrow> finite (h ` F)"
   by (induct rule: finite_induct) simp_all
 
-lemma finite_image_set [simp]:
-  "finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
+lemma finite_image_set [simp]: "finite {x. P x} \<Longrightarrow> finite {f x |x. P x}"
   by (simp add: image_Collect [symmetric])
 
 lemma finite_image_set2:
-  "finite {x. P x} \<Longrightarrow> finite {y. Q y} \<Longrightarrow> finite {f x y | x y. P x \<and> Q y}"
+  "finite {x. P x} \<Longrightarrow> finite {y. Q y} \<Longrightarrow> finite {f x y |x y. P x \<and> Q y}"
   by (rule finite_subset [where B = "\<Union>x \<in> {x. P x}. \<Union>y \<in> {y. Q y}. {f x y}"]) auto
 
 lemma finite_imageD:
   assumes "finite (f ` A)" and "inj_on f A"
   shows "finite A"
-using assms
+  using assms
 proof (induct "f ` A" arbitrary: A)
-  case empty then show ?case by simp
+  case empty
+  then show ?case by simp
 next
   case (insert x B)
-  then have B_A: "insert x B = f ` A" by simp
-  then obtain y where "x = f y" and "y \<in> A" by blast
-  from B_A \<open>x \<notin> B\<close> have "B = f ` A - {x}" by blast
-  with B_A \<open>x \<notin> B\<close> \<open>x = f y\<close> \<open>inj_on f A\<close> \<open>y \<in> A\<close> have "B = f ` (A - {y})" 
+  then have B_A: "insert x B = f ` A"
+    by simp
+  then obtain y where "x = f y" and "y \<in> A"
+    by blast
+  from B_A \<open>x \<notin> B\<close> have "B = f ` A - {x}"
+    by blast
+  with B_A \<open>x \<notin> B\<close> \<open>x = f y\<close> \<open>inj_on f A\<close> \<open>y \<in> A\<close> have "B = f ` (A - {y})"
     by (simp add: inj_on_image_set_diff Set.Diff_subset)
-  moreover from \<open>inj_on f A\<close> have "inj_on f (A - {y})" by (rule inj_on_diff)
-  ultimately have "finite (A - {y})" by (rule insert.hyps)
-  then show "finite A" by simp
+  moreover from \<open>inj_on f A\<close> have "inj_on f (A - {y})"
+    by (rule inj_on_diff)
+  ultimately have "finite (A - {y})"
+    by (rule insert.hyps)
+  then show "finite A"
+    by simp
 qed
 
-lemma finite_image_iff:
-  assumes "inj_on f A"
-  shows "finite (f ` A) \<longleftrightarrow> finite A"
-using assms finite_imageD by blast
+lemma finite_image_iff: "inj_on f A \<Longrightarrow> finite (f ` A) \<longleftrightarrow> finite A"
+  using finite_imageD by blast
 
-lemma finite_surj:
-  "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
+lemma finite_surj: "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> finite B"
   by (erule finite_subset) (rule finite_imageI)
 
-lemma finite_range_imageI:
-  "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
+lemma finite_range_imageI: "finite (range g) \<Longrightarrow> finite (range (\<lambda>x. f (g x)))"
   by (drule finite_imageI) (simp add: range_composition)
 
 lemma finite_subset_image:
   assumes "finite B"
   shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
-using assms
+  using assms
 proof induct
-  case empty then show ?case by simp
+  case empty
+  then show ?case by simp
 next
-  case insert then show ?case
-    by (clarsimp simp del: image_insert simp add: image_insert [symmetric])
-       blast
+  case insert
+  then show ?case
+    by (clarsimp simp del: image_insert simp add: image_insert [symmetric]) blast
 qed
 
-lemma finite_vimage_IntI:
-  "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
+lemma finite_vimage_IntI: "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
   apply (induct rule: finite_induct)
    apply simp_all
   apply (subst vimage_insert)
@@ -334,15 +337,14 @@
     by (simp only: * assms finite_UN_I)
 qed
 
-lemma finite_vimageI:
-  "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
+lemma finite_vimageI: "finite F \<Longrightarrow> inj h \<Longrightarrow> finite (h -` F)"
   using finite_vimage_IntI[of F h UNIV] by auto
 
-lemma finite_vimageD': "\<lbrakk> finite (f -` A); A \<subseteq> range f \<rbrakk> \<Longrightarrow> finite A"
-by(auto simp add: subset_image_iff intro: finite_subset[rotated])
+lemma finite_vimageD': "finite (f -` A) \<Longrightarrow> A \<subseteq> range f \<Longrightarrow> finite A"
+  by (auto simp add: subset_image_iff intro: finite_subset[rotated])
 
-lemma finite_vimageD: "\<lbrakk> finite (h -` F); surj h \<rbrakk> \<Longrightarrow> finite F"
-by(auto dest: finite_vimageD')
+lemma finite_vimageD: "finite (h -` F) \<Longrightarrow> surj h \<Longrightarrow> finite F"
+  by (auto dest: finite_vimageD')
 
 lemma finite_vimage_iff: "bij h \<Longrightarrow> finite (h -` F) \<longleftrightarrow> finite F"
   unfolding bij_def by (auto elim: finite_vimageD finite_vimageI)
@@ -359,30 +361,36 @@
   assumes "finite {y. P y}"
   shows "finite {x. \<exists>y. P y \<and> Q x y} \<longleftrightarrow> (\<forall>y. P y \<longrightarrow> finite {x. Q x y})"
 proof -
-  have "{x. EX y. P y & Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})" by auto
-  with assms show ?thesis by simp
+  have "{x. \<exists>y. P y \<and> Q x y} = (\<Union>y\<in>{y. P y}. {x. Q x y})"
+    by auto
+  with assms show ?thesis
+    by simp
 qed
 
-lemma finite_Plus:
-  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
+lemma finite_Plus: "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A <+> B)"
   by (simp add: Plus_def)
 
-lemma finite_PlusD: 
+lemma finite_PlusD:
   fixes A :: "'a set" and B :: "'b set"
   assumes fin: "finite (A <+> B)"
   shows "finite A" "finite B"
 proof -
-  have "Inl ` A \<subseteq> A <+> B" by auto
-  then have "finite (Inl ` A :: ('a + 'b) set)" using fin by (rule finite_subset)
-  then show "finite A" by (rule finite_imageD) (auto intro: inj_onI)
+  have "Inl ` A \<subseteq> A <+> B"
+    by auto
+  then have "finite (Inl ` A :: ('a + 'b) set)"
+    using fin by (rule finite_subset)
+  then show "finite A"
+    by (rule finite_imageD) (auto intro: inj_onI)
 next
-  have "Inr ` B \<subseteq> A <+> B" by auto
-  then have "finite (Inr ` B :: ('a + 'b) set)" using fin by (rule finite_subset)
-  then show "finite B" by (rule finite_imageD) (auto intro: inj_onI)
+  have "Inr ` B \<subseteq> A <+> B"
+    by auto
+  then have "finite (Inr ` B :: ('a + 'b) set)"
+    using fin by (rule finite_subset)
+  then show "finite B"
+    by (rule finite_imageD) (auto intro: inj_onI)
 qed
 
-lemma finite_Plus_iff [simp]:
-  "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
+lemma finite_Plus_iff [simp]: "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
   by (auto intro: finite_PlusD finite_Plus)
 
 lemma finite_Plus_UNIV_iff [simp]:
@@ -390,21 +398,22 @@
   by (subst UNIV_Plus_UNIV [symmetric]) (rule finite_Plus_iff)
 
 lemma finite_SigmaI [simp, intro]:
-  "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) ==> finite (SIGMA a:A. B a)"
-  by (unfold Sigma_def) blast
+  "finite A \<Longrightarrow> (\<And>a. a\<in>A \<Longrightarrow> finite (B a)) \<Longrightarrow> finite (SIGMA a:A. B a)"
+  unfolding Sigma_def by blast
 
 lemma finite_SigmaI2:
   assumes "finite {x\<in>A. B x \<noteq> {}}"
   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)"
   shows "finite (Sigma A B)"
 proof -
-  from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)" by auto
-  also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B" by auto
+  from assms have "finite (Sigma {x\<in>A. B x \<noteq> {}} B)"
+    by auto
+  also have "Sigma {x:A. B x \<noteq> {}} B = Sigma A B"
+    by auto
   finally show ?thesis .
 qed
 
-lemma finite_cartesian_product:
-  "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
+lemma finite_cartesian_product: "finite A \<Longrightarrow> finite B \<Longrightarrow> finite (A \<times> B)"
   by (rule finite_SigmaI)
 
 lemma finite_Prod_UNIV:
@@ -417,10 +426,12 @@
 proof -
   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
     by (auto simp add: finite_conv_nat_seg_image)
-  then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}" by simp
+  then have "fst ` (A \<times> B) = fst ` f ` {i::nat. i < n}"
+    by simp
   with \<open>B \<noteq> {}\<close> have "A = (fst \<circ> f) ` {i::nat. i < n}"
     by (simp add: image_comp)
-  then have "\<exists>n f. A = f ` {i::nat. i < n}" by blast
+  then have "\<exists>n f. A = f ` {i::nat. i < n}"
+    by blast
   then show ?thesis
     by (auto simp add: finite_conv_nat_seg_image)
 qed
@@ -431,10 +442,12 @@
 proof -
   from assms obtain n f where "A \<times> B = f ` {i::nat. i < n}"
     by (auto simp add: finite_conv_nat_seg_image)
-  then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}" by simp
+  then have "snd ` (A \<times> B) = snd ` f ` {i::nat. i < n}"
+    by simp
   with \<open>A \<noteq> {}\<close> have "B = (snd \<circ> f) ` {i::nat. i < n}"
     by (simp add: image_comp)
-  then have "\<exists>n f. B = f ` {i::nat. i < n}" by blast
+  then have "\<exists>n f. B = f ` {i::nat. i < n}"
+    by blast
   then show ?thesis
     by (auto simp add: finite_conv_nat_seg_image)
 qed
@@ -443,48 +456,52 @@
   "finite (A \<times> B) \<longleftrightarrow> (A = {} \<or> B = {} \<or> (finite A \<and> finite B))"
   by (auto dest: finite_cartesian_productD1 finite_cartesian_productD2 finite_cartesian_product)
 
-lemma finite_prod: 
+lemma finite_prod:
   "finite (UNIV :: ('a \<times> 'b) set) \<longleftrightarrow> finite (UNIV :: 'a set) \<and> finite (UNIV :: 'b set)"
   using finite_cartesian_product_iff[of UNIV UNIV] by simp
 
-lemma finite_Pow_iff [iff]:
-  "finite (Pow A) \<longleftrightarrow> finite A"
+lemma finite_Pow_iff [iff]: "finite (Pow A) \<longleftrightarrow> finite A"
 proof
   assume "finite (Pow A)"
-  then have "finite ((%x. {x}) ` A)" by (blast intro: finite_subset)
-  then show "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
+  then have "finite ((\<lambda>x. {x}) ` A)"
+    by (blast intro: finite_subset)
+  then show "finite A"
+    by (rule finite_imageD [unfolded inj_on_def]) simp
 next
   assume "finite A"
   then show "finite (Pow A)"
     by induct (simp_all add: Pow_insert)
 qed
 
-corollary finite_Collect_subsets [simp, intro]:
-  "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
+corollary finite_Collect_subsets [simp, intro]: "finite A \<Longrightarrow> finite {B. B \<subseteq> A}"
   by (simp add: Pow_def [symmetric])
 
 lemma finite_set: "finite (UNIV :: 'a set set) \<longleftrightarrow> finite (UNIV :: 'a set)"
-by(simp only: finite_Pow_iff Pow_UNIV[symmetric])
+  by (simp only: finite_Pow_iff Pow_UNIV[symmetric])
 
-lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
+lemma finite_UnionD: "finite (\<Union>A) \<Longrightarrow> finite A"
   by (blast intro: finite_subset [OF subset_Pow_Union])
 
-lemma finite_set_of_finite_funs: assumes "finite A" "finite B"
-shows "finite{f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}" (is "finite ?S")
-proof-
+lemma finite_set_of_finite_funs:
+  assumes "finite A" "finite B"
+  shows "finite {f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B) \<and> (x \<notin> A \<longrightarrow> f x = d)}" (is "finite ?S")
+proof -
   let ?F = "\<lambda>f. {(a,b). a \<in> A \<and> b = f a}"
-  have "?F ` ?S \<subseteq> Pow(A \<times> B)" by auto
-  from finite_subset[OF this] assms have 1: "finite (?F ` ?S)" by simp
+  have "?F ` ?S \<subseteq> Pow(A \<times> B)"
+    by auto
+  from finite_subset[OF this] assms have 1: "finite (?F ` ?S)"
+    by simp
   have 2: "inj_on ?F ?S"
-    by(fastforce simp add: inj_on_def set_eq_iff fun_eq_iff)
-  show ?thesis by(rule finite_imageD[OF 1 2])
+    by (fastforce simp add: inj_on_def set_eq_iff fun_eq_iff)
+  show ?thesis
+    by (rule finite_imageD [OF 1 2])
 qed
 
 lemma not_finite_existsD:
   assumes "\<not> finite {a. P a}"
   shows "\<exists>a. P a"
 proof (rule classical)
-  assume "\<not> (\<exists>a. P a)"
+  assume "\<not> ?thesis"
   with assms show ?thesis by auto
 qed
 
@@ -496,11 +513,13 @@
   assumes "\<And>x. P {x}"
     and "\<And>x F. finite F \<Longrightarrow> F \<noteq> {} \<Longrightarrow> x \<notin> F \<Longrightarrow> P F  \<Longrightarrow> P (insert x F)"
   shows "P F"
-using assms
+  using assms
 proof induct
-  case empty then show ?case by simp
+  case empty
+  then show ?case by simp
 next
-  case (insert x F) then show ?case by cases auto
+  case (insert x F)
+  then show ?case by cases auto
 qed
 
 lemma finite_subset_induct [consumes 2, case_names empty insert]:
@@ -508,13 +527,12 @@
   assumes empty: "P {}"
     and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
   shows "P F"
-using \<open>finite F\<close> \<open>F \<subseteq> A\<close>
+  using \<open>finite F\<close> \<open>F \<subseteq> A\<close>
 proof induct
   show "P {}" by fact
 next
   fix x F
-  assume "finite F" and "x \<notin> F" and
-    P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
+  assume "finite F" and "x \<notin> F" and P: "F \<subseteq> A \<Longrightarrow> P F" and i: "insert x F \<subseteq> A"
   show "P (insert x F)"
   proof (rule insert)
     from i show "x \<in> A" by blast
@@ -531,11 +549,10 @@
     and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
   shows "P {}"
 proof -
-  have "\<And>B. B \<subseteq> A \<Longrightarrow> P (A - B)"
+  have "P (A - B)" if "B \<subseteq> A" for B :: "'a set"
   proof -
-    fix B :: "'a set"
-    assume "B \<subseteq> A"
-    with \<open>finite A\<close> have "finite B" by (rule rev_finite_subset)
+    from \<open>finite A\<close> that have "finite B"
+      by (rule rev_finite_subset)
     from this \<open>B \<subseteq> A\<close> show "P (A - B)"
     proof induct
       case empty
@@ -544,11 +561,15 @@
       case (insert b B)
       have "P (A - B - {b})"
       proof (rule remove)
-        from \<open>finite A\<close> show "finite (A - B)" by induct auto
-        from insert show "b \<in> A - B" by simp
-        from insert show "P (A - B)" by simp
+        from \<open>finite A\<close> show "finite (A - B)"
+          by induct auto
+        from insert show "b \<in> A - B"
+          by simp
+        from insert show "P (A - B)"
+          by simp
       qed
-      also have "A - B - {b} = A - insert b B" by (rule Diff_insert [symmetric])
+      also have "A - B - {b} = A - insert b B"
+        by (rule Diff_insert [symmetric])
       finally show ?case .
     qed
   qed
@@ -558,11 +579,13 @@
 
 lemma finite_update_induct [consumes 1, case_names const update]:
   assumes finite: "finite {a. f a \<noteq> c}"
-  assumes const: "P (\<lambda>a. c)"
-  assumes update: "\<And>a b f. finite {a. f a \<noteq> c} \<Longrightarrow> f a = c \<Longrightarrow> b \<noteq> c \<Longrightarrow> P f \<Longrightarrow> P (f(a := b))"
+    and const: "P (\<lambda>a. c)"
+    and update: "\<And>a b f. finite {a. f a \<noteq> c} \<Longrightarrow> f a = c \<Longrightarrow> b \<noteq> c \<Longrightarrow> P f \<Longrightarrow> P (f(a := b))"
   shows "P f"
-using finite proof (induct "{a. f a \<noteq> c}" arbitrary: f)
-  case empty with const show ?case by simp
+  using finite
+proof (induct "{a. f a \<noteq> c}" arbitrary: f)
+  case empty
+  with const show ?case by simp
 next
   case (insert a A)
   then have "A = {a'. (f(a := c)) a' \<noteq> c}" and "f a \<noteq> c"
@@ -573,7 +596,8 @@
     by simp
   from insert \<open>A = {a'. (f(a := c)) a' \<noteq> c}\<close> have "P (f(a := c))"
     by simp
-  with \<open>finite {a'. (f(a := c)) a' \<noteq> c}\<close> \<open>(f(a := c)) a = c\<close> \<open>f a \<noteq> c\<close> have "P ((f(a := c))(a := f a))"
+  with \<open>finite {a'. (f(a := c)) a' \<noteq> c}\<close> \<open>(f(a := c)) a = c\<close> \<open>f a \<noteq> c\<close>
+  have "P ((f(a := c))(a := f a))"
     by (rule update)
   then show ?case by simp
 qed
@@ -581,8 +605,7 @@
 
 subsection \<open>Class \<open>finite\<close>\<close>
 
-class finite =
-  assumes finite_UNIV: "finite (UNIV :: 'a set)"
+class finite = assumes finite_UNIV: "finite (UNIV :: 'a set)"
 begin
 
 lemma finite [simp]: "finite (A :: 'a set)"
@@ -596,20 +619,22 @@
 instance prod :: (finite, finite) finite
   by standard (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
 
-lemma inj_graph: "inj (%f. {(x, y). y = f x})"
-  by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
+lemma inj_graph: "inj (\<lambda>f. {(x, y). y = f x})"
+  by (rule inj_onI) (auto simp add: set_eq_iff fun_eq_iff)
 
 instance "fun" :: (finite, finite) finite
 proof
-  show "finite (UNIV :: ('a => 'b) set)"
+  show "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
   proof (rule finite_imageD)
-    let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
-    have "range ?graph \<subseteq> Pow UNIV" by simp
+    let ?graph = "\<lambda>f::'a \<Rightarrow> 'b. {(x, y). y = f x}"
+    have "range ?graph \<subseteq> Pow UNIV"
+      by simp
     moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
       by (simp only: finite_Pow_iff finite)
     ultimately show "finite (range ?graph)"
       by (rule finite_subset)
-    show "inj ?graph" by (rule inj_graph)
+    show "inj ?graph"
+      by (rule inj_graph)
   qed
 qed
 
@@ -629,8 +654,8 @@
 subsection \<open>A basic fold functional for finite sets\<close>
 
 text \<open>The intended behaviour is
-\<open>fold f z {x\<^sub>1, ..., x\<^sub>n} = f x\<^sub>1 (\<dots> (f x\<^sub>n z)\<dots>)\<close>
-if \<open>f\<close> is ``left-commutative'':
+  \<open>fold f z {x\<^sub>1, \<dots>, x\<^sub>n} = f x\<^sub>1 (\<dots> (f x\<^sub>n z)\<dots>)\<close>
+  if \<open>f\<close> is ``left-commutative'':
 \<close>
 
 locale comp_fun_commute =
@@ -641,34 +666,35 @@
 lemma fun_left_comm: "f y (f x z) = f x (f y z)"
   using comp_fun_commute by (simp add: fun_eq_iff)
 
-lemma commute_left_comp:
-  "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
+lemma commute_left_comp: "f y \<circ> (f x \<circ> g) = f x \<circ> (f y \<circ> g)"
   by (simp add: o_assoc comp_fun_commute)
 
 end
 
 inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
-for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
-  emptyI [intro]: "fold_graph f z {} z" |
-  insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
-      \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
+  for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b
+where
+  emptyI [intro]: "fold_graph f z {} z"
+| insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
 
 inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
 
-definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
-  "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"
+definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
+  where "fold f z A = (if finite A then (THE y. fold_graph f z A y) else z)"
 
-text\<open>A tempting alternative for the definiens is
-@{term "if finite A then THE y. fold_graph f z A y else e"}.
-It allows the removal of finiteness assumptions from the theorems
-\<open>fold_comm\<close>, \<open>fold_reindex\<close> and \<open>fold_distrib\<close>.
-The proofs become ugly. It is not worth the effort. (???)\<close>
+text \<open>
+  A tempting alternative for the definiens is
+  @{term "if finite A then THE y. fold_graph f z A y else e"}.
+  It allows the removal of finiteness assumptions from the theorems
+  \<open>fold_comm\<close>, \<open>fold_reindex\<close> and \<open>fold_distrib\<close>.
+  The proofs become ugly. It is not worth the effort. (???)
+\<close>
 
 lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
-by (induct rule: finite_induct) auto
+  by (induct rule: finite_induct) auto
 
 
-subsubsection\<open>From @{const fold_graph} to @{term fold}\<close>
+subsubsection \<open>From @{const fold_graph} to @{term fold}\<close>
 
 context comp_fun_commute
 begin
@@ -681,11 +707,16 @@
 lemma fold_graph_insertE_aux:
   "fold_graph f z A y \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>y'. y = f a y' \<and> fold_graph f z (A - {a}) y'"
 proof (induct set: fold_graph)
-  case (insertI x A y) show ?case
+  case emptyI
+  then show ?case by simp
+next
+  case (insertI x A y)
+  show ?case
   proof (cases "x = a")
-    assume "x = a" with insertI show ?case by auto
+    case True
+    with insertI show ?thesis by auto
   next
-    assume "x \<noteq> a"
+    case False
     then obtain y' where y: "y = f a y'" and y': "fold_graph f z (A - {a}) y'"
       using insertI by auto
     have "f x y = f a (f x y')"
@@ -693,86 +724,96 @@
     moreover have "fold_graph f z (insert x A - {a}) (f x y')"
       using y' and \<open>x \<noteq> a\<close> and \<open>x \<notin> A\<close>
       by (simp add: insert_Diff_if fold_graph.insertI)
-    ultimately show ?case by fast
+    ultimately show ?thesis
+      by fast
   qed
-qed simp
+qed
 
 lemma fold_graph_insertE:
   assumes "fold_graph f z (insert x A) v" and "x \<notin> A"
   obtains y where "v = f x y" and "fold_graph f z A y"
-using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
+  using assms by (auto dest: fold_graph_insertE_aux [OF _ insertI1])
 
-lemma fold_graph_determ:
-  "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
+lemma fold_graph_determ: "fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
 proof (induct arbitrary: y set: fold_graph)
+  case emptyI
+  then show ?case by fast
+next
   case (insertI x A y v)
   from \<open>fold_graph f z (insert x A) v\<close> and \<open>x \<notin> A\<close>
   obtain y' where "v = f x y'" and "fold_graph f z A y'"
     by (rule fold_graph_insertE)
-  from \<open>fold_graph f z A y'\<close> have "y' = y" by (rule insertI)
-  with \<open>v = f x y'\<close> show "v = f x y" by simp
-qed fast
+  from \<open>fold_graph f z A y'\<close> have "y' = y"
+    by (rule insertI)
+  with \<open>v = f x y'\<close> show "v = f x y"
+    by simp
+qed
 
-lemma fold_equality:
-  "fold_graph f z A y \<Longrightarrow> fold f z A = y"
+lemma fold_equality: "fold_graph f z A y \<Longrightarrow> fold f z A = y"
   by (cases "finite A") (auto simp add: fold_def intro: fold_graph_determ dest: fold_graph_finite)
 
 lemma fold_graph_fold:
   assumes "finite A"
   shows "fold_graph f z A (fold f z A)"
 proof -
-  from assms have "\<exists>x. fold_graph f z A x" by (rule finite_imp_fold_graph)
+  from assms have "\<exists>x. fold_graph f z A x"
+    by (rule finite_imp_fold_graph)
   moreover note fold_graph_determ
-  ultimately have "\<exists>!x. fold_graph f z A x" by (rule ex_ex1I)
-  then have "fold_graph f z A (The (fold_graph f z A))" by (rule theI')
-  with assms show ?thesis by (simp add: fold_def)
+  ultimately have "\<exists>!x. fold_graph f z A x"
+    by (rule ex_ex1I)
+  then have "fold_graph f z A (The (fold_graph f z A))"
+    by (rule theI')
+  with assms show ?thesis
+    by (simp add: fold_def)
 qed
 
 text \<open>The base case for \<open>fold\<close>:\<close>
 
-lemma (in -) fold_infinite [simp]:
-  assumes "\<not> finite A"
-  shows "fold f z A = z"
-  using assms by (auto simp add: fold_def)
+lemma (in -) fold_infinite [simp]: "\<not> finite A \<Longrightarrow> fold f z A = z"
+  by (auto simp: fold_def)
 
-lemma (in -) fold_empty [simp]:
-  "fold f z {} = z"
-  by (auto simp add: fold_def)
+lemma (in -) fold_empty [simp]: "fold f z {} = z"
+  by (auto simp: fold_def)
 
-text\<open>The various recursion equations for @{const fold}:\<close>
+text \<open>The various recursion equations for @{const fold}:\<close>
 
 lemma fold_insert [simp]:
   assumes "finite A" and "x \<notin> A"
   shows "fold f z (insert x A) = f x (fold f z A)"
 proof (rule fold_equality)
   fix z
-  from \<open>finite A\<close> have "fold_graph f z A (fold f z A)" by (rule fold_graph_fold)
-  with \<open>x \<notin> A\<close> have "fold_graph f z (insert x A) (f x (fold f z A))" by (rule fold_graph.insertI)
-  then show "fold_graph f z (insert x A) (f x (fold f z A))" by simp
+  from \<open>finite A\<close> have "fold_graph f z A (fold f z A)"
+    by (rule fold_graph_fold)
+  with \<open>x \<notin> A\<close> have "fold_graph f z (insert x A) (f x (fold f z A))"
+    by (rule fold_graph.insertI)
+  then show "fold_graph f z (insert x A) (f x (fold f z A))"
+    by simp
 qed
 
 declare (in -) empty_fold_graphE [rule del] fold_graph.intros [rule del]
   \<comment> \<open>No more proofs involve these.\<close>
 
-lemma fold_fun_left_comm:
-  "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
+lemma fold_fun_left_comm: "finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
 proof (induct rule: finite_induct)
-  case empty then show ?case by simp
+  case empty
+  then show ?case by simp
 next
-  case (insert y A) then show ?case
+  case insert
+  then show ?case
     by (simp add: fun_left_comm [of x])
 qed
 
-lemma fold_insert2:
-  "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A)  = fold f (f x z) A"
+lemma fold_insert2: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A)  = fold f (f x z) A"
   by (simp add: fold_fun_left_comm)
 
 lemma fold_rec:
   assumes "finite A" and "x \<in> A"
   shows "fold f z A = f x (fold f z (A - {x}))"
 proof -
-  have A: "A = insert x (A - {x})" using \<open>x \<in> A\<close> by blast
-  then have "fold f z A = fold f z (insert x (A - {x}))" by simp
+  have A: "A = insert x (A - {x})"
+    using \<open>x \<in> A\<close> by blast
+  then have "fold f z A = fold f z (insert x (A - {x}))"
+    by simp
   also have "\<dots> = f x (fold f z (A - {x}))"
     by (rule fold_insert) (simp add: \<open>finite A\<close>)+
   finally show ?thesis .
@@ -782,27 +823,32 @@
   assumes "finite A"
   shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
 proof -
-  from \<open>finite A\<close> have "finite (insert x A)" by auto
-  moreover have "x \<in> insert x A" by auto
+  from \<open>finite A\<close> have "finite (insert x A)"
+    by auto
+  moreover have "x \<in> insert x A"
+    by auto
   ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
     by (rule fold_rec)
-  then show ?thesis by simp
+  then show ?thesis
+    by simp
 qed
 
 lemma fold_set_union_disj:
   assumes "finite A" "finite B" "A \<inter> B = {}"
   shows "Finite_Set.fold f z (A \<union> B) = Finite_Set.fold f (Finite_Set.fold f z A) B"
-using assms(2,1,3) by induction simp_all
+  using assms(2,1,3) by induct simp_all
 
 end
 
-text\<open>Other properties of @{const fold}:\<close>
+text \<open>Other properties of @{const fold}:\<close>
 
 lemma fold_image:
   assumes "inj_on g A"
   shows "fold f z (g ` A) = fold (f \<circ> g) z A"
 proof (cases "finite A")
-  case False with assms show ?thesis by (auto dest: finite_imageD simp add: fold_def)
+  case False
+  with assms show ?thesis
+    by (auto dest: finite_imageD simp add: fold_def)
 next
   case True
   have "fold_graph f z (g ` A) = fold_graph (f \<circ> g) z A"
@@ -810,48 +856,63 @@
     fix w
     show "fold_graph f z (g ` A) w \<longleftrightarrow> fold_graph (f \<circ> g) z A w" (is "?P \<longleftrightarrow> ?Q")
     proof
-      assume ?P then show ?Q using assms
+      assume ?P
+      then show ?Q
+        using assms
       proof (induct "g ` A" w arbitrary: A)
-        case emptyI then show ?case by (auto intro: fold_graph.emptyI)
+        case emptyI
+        then show ?case by (auto intro: fold_graph.emptyI)
       next
         case (insertI x A r B)
-        from \<open>inj_on g B\<close> \<open>x \<notin> A\<close> \<open>insert x A = image g B\<close> obtain x' A' where
-          "x' \<notin> A'" and [simp]: "B = insert x' A'" "x = g x'" "A = g ` A'"
+        from \<open>inj_on g B\<close> \<open>x \<notin> A\<close> \<open>insert x A = image g B\<close> obtain x' A'
+          where "x' \<notin> A'" and [simp]: "B = insert x' A'" "x = g x'" "A = g ` A'"
           by (rule inj_img_insertE)
-        from insertI.prems have "fold_graph (f o g) z A' r"
+        from insertI.prems have "fold_graph (f \<circ> g) z A' r"
           by (auto intro: insertI.hyps)
         with \<open>x' \<notin> A'\<close> have "fold_graph (f \<circ> g) z (insert x' A') ((f \<circ> g) x' r)"
           by (rule fold_graph.insertI)
-        then show ?case by simp
+        then show ?case
+          by simp
       qed
     next
-      assume ?Q then show ?P using assms
+      assume ?Q
+      then show ?P
+        using assms
       proof induct
-        case emptyI thus ?case by (auto intro: fold_graph.emptyI)
+        case emptyI
+        then show ?case
+          by (auto intro: fold_graph.emptyI)
       next
         case (insertI x A r)
-        from \<open>x \<notin> A\<close> insertI.prems have "g x \<notin> g ` A" by auto
-        moreover from insertI have "fold_graph f z (g ` A) r" by simp
+        from \<open>x \<notin> A\<close> insertI.prems have "g x \<notin> g ` A"
+          by auto
+        moreover from insertI have "fold_graph f z (g ` A) r"
+          by simp
         ultimately have "fold_graph f z (insert (g x) (g ` A)) (f (g x) r)"
           by (rule fold_graph.insertI)
-        then show ?case by simp
+        then show ?case
+          by simp
       qed
     qed
   qed
-  with True assms show ?thesis by (auto simp add: fold_def)
+  with True assms show ?thesis
+    by (auto simp add: fold_def)
 qed
 
 lemma fold_cong:
   assumes "comp_fun_commute f" "comp_fun_commute g"
-  assumes "finite A" and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
+    and "finite A"
+    and cong: "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
     and "s = t" and "A = B"
   shows "fold f s A = fold g t B"
 proof -
-  have "fold f s A = fold g s A"  
-  using \<open>finite A\<close> cong proof (induct A)
-    case empty then show ?case by simp
+  have "fold f s A = fold g s A"
+    using \<open>finite A\<close> cong
+  proof (induct A)
+    case empty
+    then show ?case by simp
   next
-    case (insert x A)
+    case insert
     interpret f: comp_fun_commute f by (fact \<open>comp_fun_commute f\<close>)
     interpret g: comp_fun_commute g by (fact \<open>comp_fun_commute g\<close>)
     from insert show ?case by simp
@@ -874,16 +935,19 @@
   shows "fold f z (insert x A)  = f x (fold f z A)"
 proof cases
   assume "x \<in> A"
-  then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
-  then show ?thesis using assms by (simp add: comp_fun_idem fun_left_idem)
+  then obtain B where "A = insert x B" and "x \<notin> B"
+    by (rule set_insert)
+  then show ?thesis
+    using assms by (simp add: comp_fun_idem fun_left_idem)
 next
-  assume "x \<notin> A" then show ?thesis using assms by simp
+  assume "x \<notin> A"
+  then show ?thesis
+    using assms by simp
 qed
 
 declare fold_insert [simp del] fold_insert_idem [simp]
 
-lemma fold_insert_idem2:
-  "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
+lemma fold_insert_idem2: "finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
   by (simp add: fold_fun_left_comm)
 
 end
@@ -891,50 +955,54 @@
 
 subsubsection \<open>Liftings to \<open>comp_fun_commute\<close> etc.\<close>
 
-lemma (in comp_fun_commute) comp_comp_fun_commute:
-  "comp_fun_commute (f \<circ> g)"
-proof
-qed (simp_all add: comp_fun_commute)
+lemma (in comp_fun_commute) comp_comp_fun_commute: "comp_fun_commute (f \<circ> g)"
+  by standard (simp_all add: comp_fun_commute)
 
-lemma (in comp_fun_idem) comp_comp_fun_idem:
-  "comp_fun_idem (f \<circ> g)"
+lemma (in comp_fun_idem) comp_comp_fun_idem: "comp_fun_idem (f \<circ> g)"
   by (rule comp_fun_idem.intro, rule comp_comp_fun_commute, unfold_locales)
     (simp_all add: comp_fun_idem)
 
-lemma (in comp_fun_commute) comp_fun_commute_funpow:
-  "comp_fun_commute (\<lambda>x. f x ^^ g x)"
+lemma (in comp_fun_commute) comp_fun_commute_funpow: "comp_fun_commute (\<lambda>x. f x ^^ g x)"
 proof
-  fix y x
-  show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y"
+  show "f y ^^ g y \<circ> f x ^^ g x = f x ^^ g x \<circ> f y ^^ g y" for x y
   proof (cases "x = y")
-    case True then show ?thesis by simp
+    case True
+    then show ?thesis by simp
   next
-    case False show ?thesis
+    case False
+    show ?thesis
     proof (induct "g x" arbitrary: g)
-      case 0 then show ?case by simp
+      case 0
+      then show ?case by simp
     next
       case (Suc n g)
       have hyp1: "f y ^^ g y \<circ> f x = f x \<circ> f y ^^ g y"
       proof (induct "g y" arbitrary: g)
-        case 0 then show ?case by simp
+        case 0
+        then show ?case by simp
       next
         case (Suc n g)
         define h where "h z = g z - 1" for z
-        with Suc have "n = h y" by simp
+        with Suc have "n = h y"
+          by simp
         with Suc have hyp: "f y ^^ h y \<circ> f x = f x \<circ> f y ^^ h y"
           by auto
-        from Suc h_def have "g y = Suc (h y)" by simp
-        then show ?case by (simp add: comp_assoc hyp)
-          (simp add: o_assoc comp_fun_commute)
+        from Suc h_def have "g y = Suc (h y)"
+          by simp
+        then show ?case
+          by (simp add: comp_assoc hyp) (simp add: o_assoc comp_fun_commute)
       qed
       define h where "h z = (if z = x then g x - 1 else g z)" for z
-      with Suc have "n = h x" by simp
+      with Suc have "n = h x"
+        by simp
       with Suc have "f y ^^ h y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ h y"
         by auto
-      with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y" by simp
-      from Suc h_def have "g x = Suc (h x)" by simp
-      then show ?case by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2)
-        (simp add: comp_assoc hyp1)
+      with False h_def have hyp2: "f y ^^ g y \<circ> f x ^^ h x = f x ^^ h x \<circ> f y ^^ g y"
+        by simp
+      from Suc h_def have "g x = Suc (h x)"
+        by simp
+      then show ?case
+        by (simp del: funpow.simps add: funpow_Suc_right o_assoc hyp2) (simp add: comp_assoc hyp1)
     qed
   qed
 qed
@@ -942,51 +1010,45 @@
 
 subsubsection \<open>Expressing set operations via @{const fold}\<close>
 
-lemma comp_fun_commute_const:
-  "comp_fun_commute (\<lambda>_. f)"
-proof
-qed rule
+lemma comp_fun_commute_const: "comp_fun_commute (\<lambda>_. f)"
+  by standard rule
 
-lemma comp_fun_idem_insert:
-  "comp_fun_idem insert"
-proof
-qed auto
+lemma comp_fun_idem_insert: "comp_fun_idem insert"
+  by standard auto
 
-lemma comp_fun_idem_remove:
-  "comp_fun_idem Set.remove"
-proof
-qed auto
+lemma comp_fun_idem_remove: "comp_fun_idem Set.remove"
+  by standard auto
 
-lemma (in semilattice_inf) comp_fun_idem_inf:
-  "comp_fun_idem inf"
-proof
-qed (auto simp add: inf_left_commute)
+lemma (in semilattice_inf) comp_fun_idem_inf: "comp_fun_idem inf"
+  by standard (auto simp add: inf_left_commute)
 
-lemma (in semilattice_sup) comp_fun_idem_sup:
-  "comp_fun_idem sup"
-proof
-qed (auto simp add: sup_left_commute)
+lemma (in semilattice_sup) comp_fun_idem_sup: "comp_fun_idem sup"
+  by standard (auto simp add: sup_left_commute)
 
 lemma union_fold_insert:
   assumes "finite A"
   shows "A \<union> B = fold insert B A"
 proof -
-  interpret comp_fun_idem insert by (fact comp_fun_idem_insert)
-  from \<open>finite A\<close> show ?thesis by (induct A arbitrary: B) simp_all
+  interpret comp_fun_idem insert
+    by (fact comp_fun_idem_insert)
+  from \<open>finite A\<close> show ?thesis
+    by (induct A arbitrary: B) simp_all
 qed
 
 lemma minus_fold_remove:
   assumes "finite A"
   shows "B - A = fold Set.remove B A"
 proof -
-  interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove)
-  from \<open>finite A\<close> have "fold Set.remove B A = B - A" by (induct A arbitrary: B) auto
+  interpret comp_fun_idem Set.remove
+    by (fact comp_fun_idem_remove)
+  from \<open>finite A\<close> have "fold Set.remove B A = B - A"
+    by (induct A arbitrary: B) auto
   then show ?thesis ..
 qed
 
 lemma comp_fun_commute_filter_fold:
   "comp_fun_commute (\<lambda>x A'. if P x then Set.insert x A' else A')"
-proof - 
+proof -
   interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
   show ?thesis by standard (auto simp: fun_eq_iff)
 qed
@@ -994,77 +1056,79 @@
 lemma Set_filter_fold:
   assumes "finite A"
   shows "Set.filter P A = fold (\<lambda>x A'. if P x then Set.insert x A' else A') {} A"
-using assms
-by (induct A) 
-  (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
+  using assms
+  by induct
+    (auto simp add: Set.filter_def comp_fun_commute.fold_insert[OF comp_fun_commute_filter_fold])
 
-lemma inter_Set_filter:     
+lemma inter_Set_filter:
   assumes "finite B"
   shows "A \<inter> B = Set.filter (\<lambda>x. x \<in> A) B"
-using assms 
-by (induct B) (auto simp: Set.filter_def)
+  using assms
+  by induct (auto simp: Set.filter_def)
 
 lemma image_fold_insert:
   assumes "finite A"
   shows "image f A = fold (\<lambda>k A. Set.insert (f k) A) {} A"
-using assms
 proof -
-  interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A" by standard auto
-  show ?thesis using assms by (induct A) auto
+  interpret comp_fun_commute "\<lambda>k A. Set.insert (f k) A"
+    by standard auto
+  show ?thesis
+    using assms by (induct A) auto
 qed
 
 lemma Ball_fold:
   assumes "finite A"
   shows "Ball A P = fold (\<lambda>k s. s \<and> P k) True A"
-using assms
 proof -
-  interpret comp_fun_commute "\<lambda>k s. s \<and> P k" by standard auto
-  show ?thesis using assms by (induct A) auto
+  interpret comp_fun_commute "\<lambda>k s. s \<and> P k"
+    by standard auto
+  show ?thesis
+    using assms by (induct A) auto
 qed
 
 lemma Bex_fold:
   assumes "finite A"
   shows "Bex A P = fold (\<lambda>k s. s \<or> P k) False A"
-using assms
 proof -
-  interpret comp_fun_commute "\<lambda>k s. s \<or> P k" by standard auto
-  show ?thesis using assms by (induct A) auto
+  interpret comp_fun_commute "\<lambda>k s. s \<or> P k"
+    by standard auto
+  show ?thesis
+    using assms by (induct A) auto
 qed
 
-lemma comp_fun_commute_Pow_fold: 
-  "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)" 
+lemma comp_fun_commute_Pow_fold: "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)"
   by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
 
 lemma Pow_fold:
   assumes "finite A"
   shows "Pow A = fold (\<lambda>x A. A \<union> Set.insert x ` A) {{}} A"
-using assms
 proof -
-  interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A" by (rule comp_fun_commute_Pow_fold)
-  show ?thesis using assms by (induct A) (auto simp: Pow_insert)
+  interpret comp_fun_commute "\<lambda>x A. A \<union> Set.insert x ` A"
+    by (rule comp_fun_commute_Pow_fold)
+  show ?thesis
+    using assms by (induct A) (auto simp: Pow_insert)
 qed
 
 lemma fold_union_pair:
   assumes "finite B"
   shows "(\<Union>y\<in>B. {(x, y)}) \<union> A = fold (\<lambda>y. Set.insert (x, y)) A B"
 proof -
-  interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)" by standard auto
-  show ?thesis using assms  by (induct B arbitrary: A) simp_all
+  interpret comp_fun_commute "\<lambda>y. Set.insert (x, y)"
+    by standard auto
+  show ?thesis
+    using assms by (induct arbitrary: A) simp_all
 qed
 
-lemma comp_fun_commute_product_fold: 
-  assumes "finite B"
-  shows "comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)" 
-  by standard (auto simp: fold_union_pair[symmetric] assms)
+lemma comp_fun_commute_product_fold:
+  "finite B \<Longrightarrow> comp_fun_commute (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B)"
+  by standard (auto simp: fold_union_pair [symmetric])
 
 lemma product_fold:
-  assumes "finite A"
-  assumes "finite B"
+  assumes "finite A" "finite B"
   shows "A \<times> B = fold (\<lambda>x z. fold (\<lambda>y. Set.insert (x, y)) z B) {} A"
-using assms unfolding Sigma_def 
-by (induct A) 
-  (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
-
+  using assms unfolding Sigma_def
+  by (induct A)
+    (simp_all add: comp_fun_commute.fold_insert[OF comp_fun_commute_product_fold] fold_union_pair)
 
 context complete_lattice
 begin
@@ -1073,61 +1137,55 @@
   assumes "finite A"
   shows "inf (Inf A) B = fold inf B A"
 proof -
-  interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
-  from \<open>finite A\<close> fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
-    (simp_all add: inf_commute fun_eq_iff)
+  interpret comp_fun_idem inf
+    by (fact comp_fun_idem_inf)
+  from \<open>finite A\<close> fold_fun_left_comm show ?thesis
+    by (induct A arbitrary: B) (simp_all add: inf_commute fun_eq_iff)
 qed
 
 lemma sup_Sup_fold_sup:
   assumes "finite A"
   shows "sup (Sup A) B = fold sup B A"
 proof -
-  interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
-  from \<open>finite A\<close> fold_fun_left_comm show ?thesis by (induct A arbitrary: B)
-    (simp_all add: sup_commute fun_eq_iff)
+  interpret comp_fun_idem sup
+    by (fact comp_fun_idem_sup)
+  from \<open>finite A\<close> fold_fun_left_comm show ?thesis
+    by (induct A arbitrary: B) (simp_all add: sup_commute fun_eq_iff)
 qed
 
-lemma Inf_fold_inf:
-  assumes "finite A"
-  shows "Inf A = fold inf top A"
-  using assms inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
+lemma Inf_fold_inf: "finite A \<Longrightarrow> Inf A = fold inf top A"
+  using inf_Inf_fold_inf [of A top] by (simp add: inf_absorb2)
 
-lemma Sup_fold_sup:
-  assumes "finite A"
-  shows "Sup A = fold sup bot A"
-  using assms sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
+lemma Sup_fold_sup: "finite A \<Longrightarrow> Sup A = fold sup bot A"
+  using sup_Sup_fold_sup [of A bot] by (simp add: sup_absorb2)
 
 lemma inf_INF_fold_inf:
   assumes "finite A"
-  shows "inf B (INFIMUM A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold") 
-proof (rule sym)
+  shows "inf B (INFIMUM A f) = fold (inf \<circ> f) B A" (is "?inf = ?fold")
+proof -
   interpret comp_fun_idem inf by (fact comp_fun_idem_inf)
   interpret comp_fun_idem "inf \<circ> f" by (fact comp_comp_fun_idem)
-  from \<open>finite A\<close> show "?fold = ?inf"
-    by (induct A arbitrary: B)
-      (simp_all add: inf_left_commute)
+  from \<open>finite A\<close> have "?fold = ?inf"
+    by (induct A arbitrary: B) (simp_all add: inf_left_commute)
+  then show ?thesis ..
 qed
 
 lemma sup_SUP_fold_sup:
   assumes "finite A"
-  shows "sup B (SUPREMUM A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold") 
-proof (rule sym)
+  shows "sup B (SUPREMUM A f) = fold (sup \<circ> f) B A" (is "?sup = ?fold")
+proof -
   interpret comp_fun_idem sup by (fact comp_fun_idem_sup)
   interpret comp_fun_idem "sup \<circ> f" by (fact comp_comp_fun_idem)
-  from \<open>finite A\<close> show "?fold = ?sup"
-    by (induct A arbitrary: B)
-      (simp_all add: sup_left_commute)
+  from \<open>finite A\<close> have "?fold = ?sup"
+    by (induct A arbitrary: B) (simp_all add: sup_left_commute)
+  then show ?thesis ..
 qed
 
-lemma INF_fold_inf:
-  assumes "finite A"
-  shows "INFIMUM A f = fold (inf \<circ> f) top A"
-  using assms inf_INF_fold_inf [of A top] by simp
+lemma INF_fold_inf: "finite A \<Longrightarrow> INFIMUM A f = fold (inf \<circ> f) top A"
+  using inf_INF_fold_inf [of A top] by simp
 
-lemma SUP_fold_sup:
-  assumes "finite A"
-  shows "SUPREMUM A f = fold (sup \<circ> f) bot A"
-  using assms sup_SUP_fold_sup [of A bot] by simp
+lemma SUP_fold_sup: "finite A \<Longrightarrow> SUPREMUM A f = fold (sup \<circ> f) bot A"
+  using sup_SUP_fold_sup [of A bot] by simp
 
 end
 
@@ -1146,15 +1204,14 @@
   by standard (insert comp_fun_commute, simp add: fun_eq_iff)
 
 definition F :: "'a set \<Rightarrow> 'b"
-where
-  eq_fold: "F A = fold f z A"
+  where eq_fold: "F A = fold f z A"
 
 lemma empty [simp]:"F {} = z"
   by (simp add: eq_fold)
 
 lemma infinite [simp]: "\<not> finite A \<Longrightarrow> F A = z"
   by (simp add: eq_fold)
- 
+
 lemma insert [simp]:
   assumes "finite A" and "x \<notin> A"
   shows "F (insert x A) = f x (F A)"
@@ -1163,7 +1220,7 @@
   have "fold f z (insert x A) = f x (fold f z A)" by simp
   with \<open>finite A\<close> show ?thesis by (simp add: eq_fold fun_eq_iff)
 qed
- 
+
 lemma remove:
   assumes "finite A" and "x \<in> A"
   shows "F A = f x (F (A - {x}))"
@@ -1174,10 +1231,8 @@
   ultimately show ?thesis by simp
 qed
 
-lemma insert_remove:
-  assumes "finite A"
-  shows "F (insert x A) = f x (F (A - {x}))"
-  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
+lemma insert_remove: "finite A \<Longrightarrow> F (insert x A) = f x (F (A - {x}))"
+  by (cases "x \<in> A") (simp_all add: remove insert_absorb)
 
 end
 
@@ -1209,7 +1264,7 @@
 
 text \<open>
   The traditional definition
-  @{prop "card A \<equiv> LEAST n. EX f. A = {f i | i. i < n}"}
+  @{prop "card A \<equiv> LEAST n. \<exists>f. A = {f i |i. i < n}"}
   is ugly to work with.
   But now that we have @{const fold} things are easy:
 \<close>
@@ -1218,60 +1273,49 @@
   defines card = "folding.F (\<lambda>_. Suc) 0"
   by standard rule
 
-lemma card_infinite:
-  "\<not> finite A \<Longrightarrow> card A = 0"
+lemma card_infinite: "\<not> finite A \<Longrightarrow> card A = 0"
   by (fact card.infinite)
 
-lemma card_empty:
-  "card {} = 0"
+lemma card_empty: "card {} = 0"
   by (fact card.empty)
 
-lemma card_insert_disjoint:
-  "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
+lemma card_insert_disjoint: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> card (insert x A) = Suc (card A)"
   by (fact card.insert)
 
-lemma card_insert_if:
-  "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
+lemma card_insert_if: "finite A \<Longrightarrow> card (insert x A) = (if x \<in> A then card A else Suc (card A))"
   by auto (simp add: card.insert_remove card.remove)
 
-lemma card_ge_0_finite:
-  "card A > 0 \<Longrightarrow> finite A"
+lemma card_ge_0_finite: "card A > 0 \<Longrightarrow> finite A"
   by (rule ccontr) simp
 
-lemma card_0_eq [simp]:
-  "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
+lemma card_0_eq [simp]: "finite A \<Longrightarrow> card A = 0 \<longleftrightarrow> A = {}"
   by (auto dest: mk_disjoint_insert)
 
-lemma finite_UNIV_card_ge_0:
-  "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
+lemma finite_UNIV_card_ge_0: "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
   by (rule ccontr) simp
 
-lemma card_eq_0_iff:
-  "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
+lemma card_eq_0_iff: "card A = 0 \<longleftrightarrow> A = {} \<or> \<not> finite A"
   by auto
 
-lemma card_range_greater_zero:
-  "finite (range f) \<Longrightarrow> card (range f) > 0"
+lemma card_range_greater_zero: "finite (range f) \<Longrightarrow> card (range f) > 0"
   by (rule ccontr) (simp add: card_eq_0_iff)
 
-lemma card_gt_0_iff:
-  "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
-  by (simp add: neq0_conv [symmetric] card_eq_0_iff) 
+lemma card_gt_0_iff: "0 < card A \<longleftrightarrow> A \<noteq> {} \<and> finite A"
+  by (simp add: neq0_conv [symmetric] card_eq_0_iff)
 
-lemma card_Suc_Diff1:
-  "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
-apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
-apply(simp del:insert_Diff_single)
-done
+lemma card_Suc_Diff1: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
+  apply (rule insert_Diff [THEN subst, where t = A])
+  apply assumption
+  apply (simp del: insert_Diff_single)
+  done
 
-lemma card_insert_le_m1: "n>0 \<Longrightarrow> card y \<le> n-1 \<Longrightarrow> card (insert x y) \<le> n"
+lemma card_insert_le_m1: "n > 0 \<Longrightarrow> card y \<le> n - 1 \<Longrightarrow> card (insert x y) \<le> n"
   apply (cases "finite y")
   apply (cases "x \<in> y")
   apply (auto simp: insert_absorb)
   done
 
-lemma card_Diff_singleton:
-  "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"
+lemma card_Diff_singleton: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"
   by (simp add: card_Suc_Diff1 [symmetric])
 
 lemma card_Diff_singleton_if:
@@ -1282,124 +1326,137 @@
   assumes "finite A" and "a \<in> A" and "a \<notin> B"
   shows "card (A - insert a B) = card (A - B) - 1"
 proof -
-  have "A - insert a B = (A - B) - {a}" using assms by blast
-  then show ?thesis using assms by(simp add: card_Diff_singleton)
+  have "A - insert a B = (A - B) - {a}"
+    using assms by blast
+  then show ?thesis
+    using assms by (simp add: card_Diff_singleton)
 qed
 
-lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
+lemma card_insert: "finite A \<Longrightarrow> card (insert x A) = Suc (card (A - {x}))"
   by (fact card.insert_remove)
 
-lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
-by (simp add: card_insert_if)
+lemma card_insert_le: "finite A \<Longrightarrow> card A \<le> card (insert x A)"
+  by (simp add: card_insert_if)
 
-lemma card_Collect_less_nat[simp]: "card{i::nat. i < n} = n"
-by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
+lemma card_Collect_less_nat[simp]: "card {i::nat. i < n} = n"
+  by (induct n) (simp_all add:less_Suc_eq Collect_disj_eq)
 
-lemma card_Collect_le_nat[simp]: "card{i::nat. i <= n} = Suc n"
-using card_Collect_less_nat[of "Suc n"] by(simp add: less_Suc_eq_le)
+lemma card_Collect_le_nat[simp]: "card {i::nat. i \<le> n} = Suc n"
+  using card_Collect_less_nat[of "Suc n"] by (simp add: less_Suc_eq_le)
 
 lemma card_mono:
   assumes "finite B" and "A \<subseteq> B"
   shows "card A \<le> card B"
 proof -
-  from assms have "finite A" by (auto intro: finite_subset)
-  then show ?thesis using assms proof (induct A arbitrary: B)
-    case empty then show ?case by simp
+  from assms have "finite A"
+    by (auto intro: finite_subset)
+  then show ?thesis
+    using assms
+  proof (induct A arbitrary: B)
+    case empty
+    then show ?case by simp
   next
     case (insert x A)
-    then have "x \<in> B" by simp
-    from insert have "A \<subseteq> B - {x}" and "finite (B - {x})" by auto
-    with insert.hyps have "card A \<le> card (B - {x})" by auto
-    with \<open>finite A\<close> \<open>x \<notin> A\<close> \<open>finite B\<close> \<open>x \<in> B\<close> show ?case by simp (simp only: card.remove)
+    then have "x \<in> B"
+      by simp
+    from insert have "A \<subseteq> B - {x}" and "finite (B - {x})"
+      by auto
+    with insert.hyps have "card A \<le> card (B - {x})"
+      by auto
+    with \<open>finite A\<close> \<open>x \<notin> A\<close> \<open>finite B\<close> \<open>x \<in> B\<close> show ?case
+      by simp (simp only: card.remove)
   qed
 qed
 
-lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
-apply (induct rule: finite_induct)
-apply simp
-apply clarify
-apply (subgoal_tac "finite A & A - {x} <= F")
- prefer 2 apply (blast intro: finite_subset, atomize)
-apply (drule_tac x = "A - {x}" in spec)
-apply (simp add: card_Diff_singleton_if split add: if_split_asm)
-apply (case_tac "card A", auto)
-done
+lemma card_seteq: "finite B \<Longrightarrow> (\<And>A. A \<subseteq> B \<Longrightarrow> card B \<le> card A \<Longrightarrow> A = B)"
+  apply (induct rule: finite_induct)
+  apply simp
+  apply clarify
+  apply (subgoal_tac "finite A \<and> A - {x} \<subseteq> F")
+   prefer 2 apply (blast intro: finite_subset, atomize)
+  apply (drule_tac x = "A - {x}" in spec)
+  apply (simp add: card_Diff_singleton_if split add: if_split_asm)
+  apply (case_tac "card A", auto)
+  done
 
-lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
-apply (simp add: psubset_eq linorder_not_le [symmetric])
-apply (blast dest: card_seteq)
-done
+lemma psubset_card_mono: "finite B \<Longrightarrow> A < B \<Longrightarrow> card A < card B"
+  apply (simp add: psubset_eq linorder_not_le [symmetric])
+  apply (blast dest: card_seteq)
+  done
 
 lemma card_Un_Int:
-  assumes "finite A" and "finite B"
+  assumes "finite A" "finite B"
   shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
-using assms proof (induct A)
-  case empty then show ?case by simp
+  using assms
+proof (induct A)
+  case empty
+  then show ?case by simp
 next
- case (insert x A) then show ?case
+  case insert
+  then show ?case
     by (auto simp add: insert_absorb Int_insert_left)
 qed
 
-lemma card_Un_disjoint:
-  assumes "finite A" and "finite B"
-  assumes "A \<inter> B = {}"
-  shows "card (A \<union> B) = card A + card B"
-using assms card_Un_Int [of A B] by simp
+lemma card_Un_disjoint: "finite A \<Longrightarrow> finite B \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> card (A \<union> B) = card A + card B"
+  using card_Un_Int [of A B] by simp
 
 lemma card_Un_le: "card (A \<union> B) \<le> card A + card B"
-apply(cases "finite A")
- apply(cases "finite B")
-  using le_iff_add card_Un_Int apply blast
- apply simp
-apply simp
-done
+  apply (cases "finite A")
+   apply (cases "finite B")
+    using le_iff_add card_Un_Int apply blast
+   apply simp
+  apply simp
+  done
 
 lemma card_Diff_subset:
-  assumes "finite B" and "B \<subseteq> A"
+  assumes "finite B"
+    and "B \<subseteq> A"
   shows "card (A - B) = card A - card B"
 proof (cases "finite A")
-  case False with assms show ?thesis by simp
+  case False
+  with assms show ?thesis
+    by simp
 next
-  case True with assms show ?thesis by (induct B arbitrary: A) simp_all
+  case True
+  with assms show ?thesis
+    by (induct B arbitrary: A) simp_all
 qed
 
 lemma card_Diff_subset_Int:
-  assumes AB: "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
+  assumes "finite (A \<inter> B)"
+  shows "card (A - B) = card A - card (A \<inter> B)"
 proof -
   have "A - B = A - A \<inter> B" by auto
-  thus ?thesis
-    by (simp add: card_Diff_subset AB) 
+  with assms show ?thesis
+    by (simp add: card_Diff_subset)
 qed
 
 lemma diff_card_le_card_Diff:
-assumes "finite B" shows "card A - card B \<le> card(A - B)"
-proof-
+  assumes "finite B"
+  shows "card A - card B \<le> card (A - B)"
+proof -
   have "card A - card B \<le> card A - card (A \<inter> B)"
     using card_mono[OF assms Int_lower2, of A] by arith
-  also have "\<dots> = card(A-B)" using assms by(simp add: card_Diff_subset_Int)
+  also have "\<dots> = card (A - B)"
+    using assms by (simp add: card_Diff_subset_Int)
   finally show ?thesis .
 qed
 
-lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
-apply (rule Suc_less_SucD)
-apply (simp add: card_Suc_Diff1 del:card_Diff_insert)
-done
+lemma card_Diff1_less: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) < card A"
+  by (rule Suc_less_SucD) (simp add: card_Suc_Diff1 del: card_Diff_insert)
 
-lemma card_Diff2_less:
-  "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
-apply (case_tac "x = y")
- apply (simp add: card_Diff1_less del:card_Diff_insert)
-apply (rule less_trans)
- prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
-done
+lemma card_Diff2_less: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> card (A - {x} - {y}) < card A"
+  apply (cases "x = y")
+   apply (simp add: card_Diff1_less del:card_Diff_insert)
+  apply (rule less_trans)
+   prefer 2 apply (auto intro!: card_Diff1_less simp del: card_Diff_insert)
+  done
 
-lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
-apply (case_tac "x : A")
- apply (simp_all add: card_Diff1_less less_imp_le)
-done
+lemma card_Diff1_le: "finite A \<Longrightarrow> card (A - {x}) \<le> card A"
+  by (cases "x \<in> A") (simp_all add: card_Diff1_less less_imp_le)
 
-lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
-by (erule psubsetI, blast)
+lemma card_psubset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> card A < card B \<Longrightarrow> A < B"
+  by (erule psubsetI) blast
 
 lemma card_le_inj:
   assumes fA: "finite A"
@@ -1413,7 +1470,7 @@
 next
   case (insert x s t)
   then show ?case
-  proof (induct rule: finite_induct[OF "insert.prems"(1)])
+  proof (induct rule: finite_induct [OF insert.prems(1)])
     case 1
     then show ?case by simp
   next
@@ -1454,41 +1511,43 @@
 qed
 
 lemma insert_partition:
-  "\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
-  \<Longrightarrow> x \<inter> \<Union>F = {}"
-by auto
+  "x \<notin> F \<Longrightarrow> \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<Longrightarrow> x \<inter> \<Union>F = {}"
+  by auto
 
-lemma finite_psubset_induct[consumes 1, case_names psubset]:
-  assumes fin: "finite A" 
-  and     major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A" 
+lemma finite_psubset_induct [consumes 1, case_names psubset]:
+  assumes finite: "finite A"
+    and major: "\<And>A. finite A \<Longrightarrow> (\<And>B. B \<subset> A \<Longrightarrow> P B) \<Longrightarrow> P A"
   shows "P A"
-using fin
+  using finite
 proof (induct A taking: card rule: measure_induct_rule)
   case (less A)
   have fin: "finite A" by fact
-  have ih: "\<And>B. \<lbrakk>card B < card A; finite B\<rbrakk> \<Longrightarrow> P B" by fact
-  { fix B 
-    assume asm: "B \<subset> A"
-    from asm have "card B < card A" using psubset_card_mono fin by blast
+  have ih: "card B < card A \<Longrightarrow> finite B \<Longrightarrow> P B" for B by fact
+  have "P B" if "B \<subset> A" for B
+  proof -
+    from that have "card B < card A"
+      using psubset_card_mono fin by blast
     moreover
-    from asm have "B \<subseteq> A" by auto
-    then have "finite B" using fin finite_subset by blast
-    ultimately 
-    have "P B" using ih by simp
-  }
+    from that have "B \<subseteq> A"
+      by auto
+    then have "finite B"
+      using fin finite_subset by blast
+    ultimately show ?thesis using ih by simp
+  qed
   with fin show "P A" using major by blast
 qed
 
-lemma finite_induct_select[consumes 1, case_names empty select]:
+lemma finite_induct_select [consumes 1, case_names empty select]:
   assumes "finite S"
-  assumes "P {}"
-  assumes select: "\<And>T. T \<subset> S \<Longrightarrow> P T \<Longrightarrow> \<exists>s\<in>S - T. P (insert s T)"
+    and "P {}"
+    and select: "\<And>T. T \<subset> S \<Longrightarrow> P T \<Longrightarrow> \<exists>s\<in>S - T. P (insert s T)"
   shows "P S"
 proof -
   have "0 \<le> card S" by simp
   then have "\<exists>T \<subseteq> S. card T = card S \<and> P T"
   proof (induct rule: dec_induct)
-    case base with \<open>P {}\<close> show ?case
+    case base with \<open>P {}\<close>
+    show ?case
       by (intro exI[of _ "{}"]) auto
   next
     case (step n)
@@ -1506,24 +1565,27 @@
 qed
 
 lemma remove_induct [case_names empty infinite remove]:
-  assumes empty: "P ({} :: 'a set)" and infinite: "\<not>finite B \<Longrightarrow> P B"
-      and remove: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
+  assumes empty: "P ({} :: 'a set)"
+    and infinite: "\<not> finite B \<Longrightarrow> P B"
+    and remove: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
   shows "P B"
 proof (cases "finite B")
   assume "\<not>finite B"
-  thus ?thesis by (rule infinite)
+  then show ?thesis by (rule infinite)
 next
   define A where "A = B"
   assume "finite B"
-  hence "finite A" "A \<subseteq> B" by (simp_all add: A_def)
-  thus "P A"
-  proof (induction "card A" arbitrary: A)
+  then have "finite A" "A \<subseteq> B"
+    by (simp_all add: A_def)
+  then show "P A"
+  proof (induct "card A" arbitrary: A)
     case 0
-    hence "A = {}" by auto
+    then have "A = {}" by auto
     with empty show ?case by simp
   next
     case (Suc n A)
-    from \<open>A \<subseteq> B\<close> and \<open>finite B\<close> have "finite A" by (rule finite_subset)
+    from \<open>A \<subseteq> B\<close> and \<open>finite B\<close> have "finite A"
+      by (rule finite_subset)
     moreover from Suc.hyps have "A \<noteq> {}" by auto
     moreover note \<open>A \<subseteq> B\<close>
     moreover have "P (A - {x})" if x: "x \<in> A" for x
@@ -1533,92 +1595,99 @@
 qed
 
 lemma finite_remove_induct [consumes 1, case_names empty remove]:
-  assumes finite: "finite B" and empty: "P ({} :: 'a set)" 
-      and rm: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
+  fixes P :: "'a set \<Rightarrow> bool"
+  assumes finite: "finite B"
+    and empty: "P {}"
+    and rm: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
   defines "B' \<equiv> B"
-  shows   "P B'"
-  by (induction B' rule: remove_induct) (simp_all add: assms)
+  shows "P B'"
+  by (induct B' rule: remove_induct) (simp_all add: assms)
 
 
-text\<open>main cardinality theorem\<close>
+text \<open>Main cardinality theorem.\<close>
 lemma card_partition [rule_format]:
-  "finite C ==>
-     finite (\<Union>C) -->
-     (\<forall>c\<in>C. card c = k) -->
-     (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
-     k * card(C) = card (\<Union>C)"
-apply (erule finite_induct, simp)
-apply (simp add: card_Un_disjoint insert_partition 
-       finite_subset [of _ "\<Union>(insert x F)"])
-done
+  "finite C \<Longrightarrow> finite (\<Union>C) \<Longrightarrow> (\<forall>c\<in>C. card c = k) \<Longrightarrow>
+    (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}) \<Longrightarrow>
+    k * card C = card (\<Union>C)"
+  apply (induct rule: finite_induct)
+  apply simp
+  apply (simp add: card_Un_disjoint insert_partition finite_subset [of _ "\<Union>(insert x F)"])
+  done
 
 lemma card_eq_UNIV_imp_eq_UNIV:
   assumes fin: "finite (UNIV :: 'a set)"
-  and card: "card A = card (UNIV :: 'a set)"
+    and card: "card A = card (UNIV :: 'a set)"
   shows "A = (UNIV :: 'a set)"
 proof
   show "A \<subseteq> UNIV" by simp
   show "UNIV \<subseteq> A"
   proof
-    fix x
-    show "x \<in> A"
+    show "x \<in> A" for x
     proof (rule ccontr)
       assume "x \<notin> A"
       then have "A \<subset> UNIV" by auto
-      with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
+      with fin have "card A < card (UNIV :: 'a set)"
+        by (fact psubset_card_mono)
       with card show False by simp
     qed
   qed
 qed
 
-text\<open>The form of a finite set of given cardinality\<close>
+text \<open>The form of a finite set of given cardinality\<close>
 
 lemma card_eq_SucD:
-assumes "card A = Suc k"
-shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
+  assumes "card A = Suc k"
+  shows "\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {})"
 proof -
-  have fin: "finite A" using assms by (auto intro: ccontr)
-  moreover have "card A \<noteq> 0" using assms by auto
-  ultimately obtain b where b: "b \<in> A" by auto
+  have fin: "finite A"
+    using assms by (auto intro: ccontr)
+  moreover have "card A \<noteq> 0"
+    using assms by auto
+  ultimately obtain b where b: "b \<in> A"
+    by auto
   show ?thesis
   proof (intro exI conjI)
-    show "A = insert b (A-{b})" using b by blast
-    show "b \<notin> A - {b}" by blast
+    show "A = insert b (A - {b})"
+      using b by blast
+    show "b \<notin> A - {b}"
+      by blast
     show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
       using assms b fin by(fastforce dest:mk_disjoint_insert)+
   qed
 qed
 
 lemma card_Suc_eq:
-  "(card A = Suc k) =
-   (\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
- apply(auto elim!: card_eq_SucD)
- apply(subst card.insert)
- apply(auto simp add: intro:ccontr)
- done
+  "card A = Suc k \<longleftrightarrow>
+    (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))"
+  apply (auto elim!: card_eq_SucD)
+  apply (subst card.insert)
+  apply (auto simp add: intro:ccontr)
+  done
 
 lemma card_1_singletonE:
-    assumes "card A = 1" obtains x where "A = {x}"
+  assumes "card A = 1"
+  obtains x where "A = {x}"
   using assms by (auto simp: card_Suc_eq)
 
 lemma is_singleton_altdef: "is_singleton A \<longleftrightarrow> card A = 1"
   unfolding is_singleton_def
   by (auto elim!: card_1_singletonE is_singletonE simp del: One_nat_def)
 
-lemma card_le_Suc_iff: "finite A \<Longrightarrow>
-  Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
-by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
-  dest: subset_singletonD split: nat.splits if_splits)
+lemma card_le_Suc_iff:
+  "finite A \<Longrightarrow> Suc n \<le> card A = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
+  by (fastforce simp: card_Suc_eq less_eq_nat.simps(2) insert_eq_iff
+    dest: subset_singletonD split: nat.splits if_splits)
 
 lemma finite_fun_UNIVD2:
   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
   shows "finite (UNIV :: 'b set)"
 proof -
-  from fin have "\<And>arbitrary. finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
+  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))" for arbitrary
     by (rule finite_imageI)
-  moreover have "\<And>arbitrary. UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
+  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)" for arbitrary
     by (rule UNIV_eq_I) auto
-  ultimately show "finite (UNIV :: 'b set)" by simp
+  ultimately show "finite (UNIV :: 'b set)"
+    by simp
 qed
 
 lemma card_UNIV_unit [simp]: "card (UNIV :: unit set) = 1"
@@ -1628,164 +1697,176 @@
   assumes "\<not> finite A"
   shows "\<exists>B. finite B \<and> card B = n \<and> B \<subseteq> A"
 proof (induction n)
-  case 0 show ?case by (intro exI[of _ "{}"]) auto
-next 
+  case 0
+  show ?case by (intro exI[of _ "{}"]) auto
+next
   case (Suc n)
-  then guess B .. note B = this
+  then obtain B where B: "finite B \<and> card B = n \<and> B \<subseteq> A" ..
   with \<open>\<not> finite A\<close> have "A \<noteq> B" by auto
   with B have "B \<subset> A" by auto
-  hence "\<exists>x. x \<in> A - B" by (elim psubset_imp_ex_mem)
-  then guess x .. note x = this
+  then have "\<exists>x. x \<in> A - B"
+    by (elim psubset_imp_ex_mem)
+  then obtain x where x: "x \<in> A - B" ..
   with B have "finite (insert x B) \<and> card (insert x B) = Suc n \<and> insert x B \<subseteq> A"
     by auto
-  thus "\<exists>B. finite B \<and> card B = Suc n \<and> B \<subseteq> A" ..
+  then show "\<exists>B. finite B \<and> card B = Suc n \<and> B \<subseteq> A" ..
 qed
 
+
 subsubsection \<open>Cardinality of image\<close>
 
-lemma card_image_le: "finite A ==> card (f ` A) \<le> card A"
+lemma card_image_le: "finite A \<Longrightarrow> card (f ` A) \<le> card A"
   by (induct rule: finite_induct) (simp_all add: le_SucI card_insert_if)
 
 lemma card_image:
   assumes "inj_on f A"
   shows "card (f ` A) = card A"
 proof (cases "finite A")
-  case True then show ?thesis using assms by (induct A) simp_all
+  case True
+  then show ?thesis
+    using assms by (induct A) simp_all
 next
-  case False then have "\<not> finite (f ` A)" using assms by (auto dest: finite_imageD)
-  with False show ?thesis by simp
+  case False
+  then have "\<not> finite (f ` A)"
+    using assms by (auto dest: finite_imageD)
+  with False show ?thesis
+    by simp
 qed
 
 lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
-by(auto simp: card_image bij_betw_def)
+  by(auto simp: card_image bij_betw_def)
 
-lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
-by (simp add: card_seteq card_image)
+lemma endo_inj_surj: "finite A \<Longrightarrow> f ` A \<subseteq> A \<Longrightarrow> inj_on f A \<Longrightarrow> f ` A = A"
+  by (simp add: card_seteq card_image)
 
 lemma eq_card_imp_inj_on:
-  assumes "finite A" "card(f ` A) = card A" shows "inj_on f A"
-using assms
+  assumes "finite A" "card(f ` A) = card A"
+  shows "inj_on f A"
+  using assms
 proof (induct rule:finite_induct)
-  case empty show ?case by simp
+  case empty
+  show ?case by simp
 next
   case (insert x A)
-  then show ?case using card_image_le [of A f]
-    by (simp add: card_insert_if split: if_splits)
+  then show ?case
+    using card_image_le [of A f] by (simp add: card_insert_if split: if_splits)
 qed
 
-lemma inj_on_iff_eq_card: "finite A \<Longrightarrow> inj_on f A \<longleftrightarrow> card(f ` A) = card A"
+lemma inj_on_iff_eq_card: "finite A \<Longrightarrow> inj_on f A \<longleftrightarrow> card (f ` A) = card A"
   by (blast intro: card_image eq_card_imp_inj_on)
 
 lemma card_inj_on_le:
-  assumes "inj_on f A" "f ` A \<subseteq> B" "finite B" shows "card A \<le> card B"
+  assumes "inj_on f A" "f ` A \<subseteq> B" "finite B"
+  shows "card A \<le> card B"
 proof -
-  have "finite A" using assms
-    by (blast intro: finite_imageD dest: finite_subset)
-  then show ?thesis using assms 
-   by (force intro: card_mono simp: card_image [symmetric])
+  have "finite A"
+    using assms by (blast intro: finite_imageD dest: finite_subset)
+  then show ?thesis
+    using assms by (force intro: card_mono simp: card_image [symmetric])
 qed
 
 lemma surj_card_le: "finite A \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> card B \<le> card A"
   by (blast intro: card_image_le card_mono le_trans)
 
 lemma card_bij_eq:
-  "[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
-     finite A; finite B |] ==> card A = card B"
-by (auto intro: le_antisym card_inj_on_le)
+  "inj_on f A \<Longrightarrow> f ` A \<subseteq> B \<Longrightarrow> inj_on g B \<Longrightarrow> g ` B \<subseteq> A \<Longrightarrow> finite A \<Longrightarrow> finite B
+    \<Longrightarrow> card A = card B"
+  by (auto intro: le_antisym card_inj_on_le)
+
+lemma bij_betw_finite: "bij_betw f A B \<Longrightarrow> finite A \<longleftrightarrow> finite B"
+  unfolding bij_betw_def using finite_imageD [of f A] by auto
 
-lemma bij_betw_finite:
-  assumes "bij_betw f A B"
-  shows "finite A \<longleftrightarrow> finite B"
-using assms unfolding bij_betw_def
-using finite_imageD[of f A] by auto
+lemma inj_on_finite: "inj_on f A \<Longrightarrow> f ` A \<le> B \<Longrightarrow> finite B \<Longrightarrow> finite A"
+  using finite_imageD finite_subset by blast
 
-lemma inj_on_finite:
-assumes "inj_on f A" "f ` A \<le> B" "finite B"
-shows "finite A"
-using assms finite_imageD finite_subset by blast
+lemma card_vimage_inj: "inj f \<Longrightarrow> A \<subseteq> range f \<Longrightarrow> card (f -` A) = card A"
+  by (auto 4 3 simp: subset_image_iff inj_vimage_image_eq
+      intro: card_image[symmetric, OF subset_inj_on])
 
-lemma card_vimage_inj: "\<lbrakk> inj f; A \<subseteq> range f \<rbrakk> \<Longrightarrow> card (f -` A) = card A"
-by(auto 4 3 simp add: subset_image_iff inj_vimage_image_eq intro: card_image[symmetric, OF subset_inj_on])
 
 subsubsection \<open>Pigeonhole Principles\<close>
 
-lemma pigeonhole: "card A > card(f ` A) \<Longrightarrow> ~ inj_on f A "
-by (auto dest: card_image less_irrefl_nat)
+lemma pigeonhole: "card A > card (f ` A) \<Longrightarrow> \<not> inj_on f A "
+  by (auto dest: card_image less_irrefl_nat)
 
 lemma pigeonhole_infinite:
-assumes  "~ finite A" and "finite(f`A)"
-shows "EX a0:A. ~finite{a:A. f a = f a0}"
-proof -
-  have "finite(f`A) \<Longrightarrow> ~ finite A \<Longrightarrow> EX a0:A. ~finite{a:A. f a = f a0}"
-  proof(induct "f`A" arbitrary: A rule: finite_induct)
-    case empty thus ?case by simp
+  assumes "\<not> finite A" and "finite (f`A)"
+  shows "\<exists>a0\<in>A. \<not> finite {a\<in>A. f a = f a0}"
+  using assms(2,1)
+proof (induct "f`A" arbitrary: A rule: finite_induct)
+  case empty
+  then show ?case by simp
+next
+  case (insert b F)
+  show ?case
+  proof (cases "finite {a\<in>A. f a = b}")
+    case True
+    with \<open>\<not> finite A\<close> have "\<not> finite (A - {a\<in>A. f a = b})"
+      by simp
+    also have "A - {a\<in>A. f a = b} = {a\<in>A. f a \<noteq> b}"
+      by blast
+    finally have "\<not> finite {a\<in>A. f a \<noteq> b}" .
+    from insert(3)[OF _ this] insert(2,4) show ?thesis
+      by simp (blast intro: rev_finite_subset)
   next
-    case (insert b F)
-    show ?case
-    proof cases
-      assume "finite{a:A. f a = b}"
-      hence "~ finite(A - {a:A. f a = b})" using \<open>\<not> finite A\<close> by simp
-      also have "A - {a:A. f a = b} = {a:A. f a \<noteq> b}" by blast
-      finally have "~ finite({a:A. f a \<noteq> b})" .
-      from insert(3)[OF _ this]
-      show ?thesis using insert(2,4) by simp (blast intro: rev_finite_subset)
-    next
-      assume 1: "~finite{a:A. f a = b}"
-      hence "{a \<in> A. f a = b} \<noteq> {}" by force
-      thus ?thesis using 1 by blast
-    qed
+    case False
+    then have "{a \<in> A. f a = b} \<noteq> {}" by force
+    with False show ?thesis by blast
   qed
-  from this[OF assms(2,1)] show ?thesis .
 qed
 
 lemma pigeonhole_infinite_rel:
-assumes "~finite A" and "finite B" and "ALL a:A. EX b:B. R a b"
-shows "EX b:B. ~finite{a:A. R a b}"
+  assumes "\<not> finite A"
+    and "finite B"
+    and "\<forall>a\<in>A. \<exists>b\<in>B. R a b"
+  shows "\<exists>b\<in>B. \<not> finite {a:A. R a b}"
 proof -
-   let ?F = "%a. {b:B. R a b}"
-   from finite_Pow_iff[THEN iffD2, OF \<open>finite B\<close>]
-   have "finite(?F ` A)" by(blast intro: rev_finite_subset)
-   from pigeonhole_infinite[where f = ?F, OF assms(1) this]
-   obtain a0 where "a0\<in>A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
-   obtain b0 where "b0 : B" and "R a0 b0" using \<open>a0:A\<close> assms(3) by blast
-   { assume "finite{a:A. R a b0}"
-     then have "finite {a\<in>A. ?F a = ?F a0}"
-       using \<open>b0 : B\<close> \<open>R a0 b0\<close> by(blast intro: rev_finite_subset)
-   }
-   with 1 \<open>b0 : B\<close> show ?thesis by blast
+  let ?F = "\<lambda>a. {b\<in>B. R a b}"
+  from finite_Pow_iff[THEN iffD2, OF \<open>finite B\<close>] have "finite (?F ` A)"
+    by (blast intro: rev_finite_subset)
+  from pigeonhole_infinite [where f = ?F, OF assms(1) this]
+  obtain a0 where "a0 \<in> A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
+  obtain b0 where "b0 \<in> B" and "R a0 b0"
+    using \<open>a0 \<in> A\<close> assms(3) by blast
+  have "finite {a\<in>A. ?F a = ?F a0}" if "finite{a:A. R a b0}"
+    using \<open>b0 \<in> B\<close> \<open>R a0 b0\<close> that by (blast intro: rev_finite_subset)
+  with 1 \<open>b0 : B\<close> show ?thesis
+    by blast
 qed
 
 
 subsubsection \<open>Cardinality of sums\<close>
 
 lemma card_Plus:
-  assumes "finite A" and "finite B"
+  assumes "finite A" "finite B"
   shows "card (A <+> B) = card A + card B"
 proof -
   have "Inl`A \<inter> Inr`B = {}" by fast
   with assms show ?thesis
-    unfolding Plus_def
-    by (simp add: card_Un_disjoint card_image)
+    by (simp add: Plus_def card_Un_disjoint card_image)
 qed
 
 lemma card_Plus_conv_if:
   "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
   by (auto simp add: card_Plus)
 
-text \<open>Relates to equivalence classes.  Based on a theorem of F. Kamm\"uller.\<close>
+text \<open>Relates to equivalence classes.  Based on a theorem of F. Kammüller.\<close>
 
 lemma dvd_partition:
-  assumes f: "finite (\<Union>C)" and "\<forall>c\<in>C. k dvd card c" "\<forall>c1\<in>C. \<forall>c2\<in>C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}"
-    shows "k dvd card (\<Union>C)"
+  assumes f: "finite (\<Union>C)"
+    and "\<forall>c\<in>C. k dvd card c" "\<forall>c1\<in>C. \<forall>c2\<in>C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}"
+  shows "k dvd card (\<Union>C)"
 proof -
-  have "finite C" 
+  have "finite C"
     by (rule finite_UnionD [OF f])
-  then show ?thesis using assms
+  then show ?thesis
+    using assms
   proof (induct rule: finite_induct)
-    case empty show ?case by simp
+    case empty
+    show ?case by simp
   next
-    case (insert c C)
-    then show ?case 
+    case insert
+    then show ?case
       apply simp
       apply (subst card_Un_disjoint)
       apply (auto simp add: disjoint_eq_subset_Compl)
@@ -1793,34 +1874,33 @@
   qed
 qed
 
+
 subsubsection \<open>Relating injectivity and surjectivity\<close>
 
-lemma finite_surj_inj: assumes "finite A" "A \<subseteq> f ` A" shows "inj_on f A"
+lemma finite_surj_inj:
+  assumes "finite A" "A \<subseteq> f ` A"
+  shows "inj_on f A"
 proof -
-  have "f ` A = A" 
+  have "f ` A = A"
     by (rule card_seteq [THEN sym]) (auto simp add: assms card_image_le)
   then show ?thesis using assms
     by (simp add: eq_card_imp_inj_on)
 qed
 
-lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
-shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
-by (blast intro: finite_surj_inj subset_UNIV)
+lemma finite_UNIV_surj_inj: "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f" for f :: "'a \<Rightarrow> 'a"
+  by (blast intro: finite_surj_inj subset_UNIV)
 
-lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
-shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
-by(fastforce simp:surj_def dest!: endo_inj_surj)
+lemma finite_UNIV_inj_surj: "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f" for f :: "'a \<Rightarrow> 'a"
+  by (fastforce simp:surj_def dest!: endo_inj_surj)
 
-corollary infinite_UNIV_nat [iff]:
-  "\<not> finite (UNIV :: nat set)"
+corollary infinite_UNIV_nat [iff]: "\<not> finite (UNIV :: nat set)"
 proof
   assume "finite (UNIV :: nat set)"
-  with finite_UNIV_inj_surj [of Suc]
-  show False by simp (blast dest: Suc_neq_Zero surjD)
+  with finite_UNIV_inj_surj [of Suc] show False
+    by simp (blast dest: Suc_neq_Zero surjD)
 qed
 
-lemma infinite_UNIV_char_0:
-  "\<not> finite (UNIV :: 'a::semiring_char_0 set)"
+lemma infinite_UNIV_char_0: "\<not> finite (UNIV :: 'a::semiring_char_0 set)"
 proof
   assume "finite (UNIV :: 'a set)"
   with subset_UNIV have "finite (range of_nat :: 'a set)"
@@ -1836,7 +1916,7 @@
 hide_const (open) Finite_Set.fold
 
 
-subsection "Infinite Sets"
+subsection \<open>Infinite Sets\<close>
 
 text \<open>
   Some elementary facts about infinite sets, mostly by Stephan Merz.
@@ -1859,19 +1939,18 @@
   by simp
 
 lemma Diff_infinite_finite:
-  assumes T: "finite T" and S: "infinite S"
+  assumes "finite T" "infinite S"
   shows "infinite (S - T)"
-  using T
+  using \<open>finite T\<close>
 proof induct
-  from S
-  show "infinite (S - {})" by auto
+  from \<open>infinite S\<close> show "infinite (S - {})"
+    by auto
 next
   fix T x
   assume ih: "infinite (S - T)"
   have "S - (insert x T) = (S - T) - {x}"
     by (rule Diff_insert)
-  with ih
-  show "infinite (S - (insert x T))"
+  with ih show "infinite (S - (insert x T))"
     by (simp add: infinite_remove)
 qed
 
@@ -1882,21 +1961,23 @@
   by simp
 
 lemma infinite_super:
-  assumes T: "S \<subseteq> T" and S: "infinite S"
+  assumes "S \<subseteq> T"
+    and "infinite S"
   shows "infinite T"
 proof
   assume "finite T"
-  with T have "finite S" by (simp add: finite_subset)
-  with S show False by simp
+  with \<open>S \<subseteq> T\<close> have "finite S" by (simp add: finite_subset)
+  with \<open>infinite S\<close> show False by simp
 qed
 
 proposition infinite_coinduct [consumes 1, case_names infinite]:
   assumes "X A"
-  and step: "\<And>A. X A \<Longrightarrow> \<exists>x\<in>A. X (A - {x}) \<or> infinite (A - {x})"
+    and step: "\<And>A. X A \<Longrightarrow> \<exists>x\<in>A. X (A - {x}) \<or> infinite (A - {x})"
   shows "infinite A"
 proof
   assume "finite A"
-  then show False using \<open>X A\<close>
+  then show False
+    using \<open>X A\<close>
   proof (induction rule: finite_psubset_induct)
     case (psubset A)
     then obtain x where "x \<in> A" "X (A - {x}) \<or> infinite (A - {x})"
@@ -1906,7 +1987,8 @@
     show False
       apply (rule psubset.IH [where B = "A - {x}"])
       using \<open>x \<in> A\<close> apply blast
-      by (simp add: \<open>X (A - {x})\<close>)
+      apply (simp add: \<open>X (A - {x})\<close>)
+      done
   qed
 qed
 
@@ -1918,14 +2000,14 @@
 \<close>
 
 lemma inf_img_fin_dom':
-  assumes img: "finite (f ` A)" and dom: "infinite A"
+  assumes img: "finite (f ` A)"
+    and dom: "infinite A"
   shows "\<exists>y \<in> f ` A. infinite (f -` {y} \<inter> A)"
 proof (rule ccontr)
   have "A \<subseteq> (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by auto
-  moreover
-  assume "\<not> ?thesis"
+  moreover assume "\<not> ?thesis"
   with img have "finite (\<Union>y\<in>f ` A. f -` {y} \<inter> A)" by blast
-  ultimately have "finite A" by(rule finite_subset)
+  ultimately have "finite A" by (rule finite_subset)
   with dom show False by contradiction
 qed
 
@@ -1937,16 +2019,15 @@
 lemma inf_img_fin_dom:
   assumes img: "finite (f`A)" and dom: "infinite A"
   shows "\<exists>y \<in> f`A. infinite (f -` {y})"
-using inf_img_fin_dom'[OF assms] by auto
+  using inf_img_fin_dom'[OF assms] by auto
 
 lemma inf_img_fin_domE:
   assumes "finite (f`A)" and "infinite A"
   obtains y where "y \<in> f`A" and "infinite (f -` {y})"
   using assms by (blast dest: inf_img_fin_dom)
 
-proposition finite_image_absD:
-    fixes S :: "'a::linordered_ring set"
-    shows "finite (abs ` S) \<Longrightarrow> finite S"
+proposition finite_image_absD: "finite (abs ` S) \<Longrightarrow> finite S"
+  for S :: "'a::linordered_ring set"
   by (rule ccontr) (auto simp: abs_eq_iff vimage_def dest: inf_img_fin_dom)
 
 end




