src/HOL/Groups_Big.thy

 (*  Title:      HOL/Groups_Big.thy
-    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
-                with contributions by Jeremy Avigad
+    Author:     Tobias Nipkow
+    Author:     Lawrence C Paulson
+    Author:     Markus Wenzel
+    Author:     Jeremy Avigad
 *)
 
 section \<open>Big sum and product over finite (non-empty) sets\<close>
 
 theory Groups_Big
-imports Finite_Set Power
+  imports Finite_Set Power
 begin
 
 subsection \<open>Generic monoid operation over a set\<close>
 
 locale comm_monoid_set = comm_monoid

 
 interpretation comp?: comp_fun_commute "f \<circ> g"
   by (fact comp_comp_fun_commute)
 
 definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
-where
-  eq_fold: "F g A = Finite_Set.fold (f \<circ> g) \<^bold>1 A"
-
-lemma infinite [simp]:
-  "\<not> finite A \<Longrightarrow> F g A = \<^bold>1"
+  where eq_fold: "F g A = Finite_Set.fold (f \<circ> g) \<^bold>1 A"
+
+lemma infinite [simp]: "\<not> finite A \<Longrightarrow> F g A = \<^bold>1"
   by (simp add: eq_fold)
 
-lemma empty [simp]:
-  "F g {} = \<^bold>1"
+lemma empty [simp]: "F g {} = \<^bold>1"
   by (simp add: eq_fold)
 
-lemma insert [simp]:
-  assumes "finite A" and "x \<notin> A"
-  shows "F g (insert x A) = g x \<^bold>* F g A"
-  using assms by (simp add: eq_fold)
+lemma insert [simp]: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> F g (insert x A) = g x \<^bold>* F g A"
+  by (simp add: eq_fold)
 
 lemma remove:
   assumes "finite A" and "x \<in> A"
   shows "F g A = g x \<^bold>* F g (A - {x})"
 proof -
-  from \<open>x \<in> A\<close> obtain B where A: "A = insert x B" and "x \<notin> B"
+  from \<open>x \<in> A\<close> obtain B where B: "A = insert x B" and "x \<notin> B"
     by (auto dest: mk_disjoint_insert)
-  moreover from \<open>finite A\<close> A have "finite B" by simp
+  moreover from \<open>finite A\<close> B have "finite B" by simp
   ultimately show ?thesis by simp
 qed
 
-lemma insert_remove:
-  assumes "finite A"
-  shows "F g (insert x A) = g x \<^bold>* F g (A - {x})"
-  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
-
-lemma neutral:
-  assumes "\<forall>x\<in>A. g x = \<^bold>1"
-  shows "F g A = \<^bold>1"
-  using assms by (induct A rule: infinite_finite_induct) simp_all
-
-lemma neutral_const [simp]:
-  "F (\<lambda>_. \<^bold>1) A = \<^bold>1"
+lemma insert_remove: "finite A \<Longrightarrow> F g (insert x A) = g x \<^bold>* F g (A - {x})"
+  by (cases "x \<in> A") (simp_all add: remove insert_absorb)
+
+lemma neutral: "\<forall>x\<in>A. g x = \<^bold>1 \<Longrightarrow> F g A = \<^bold>1"
+  by (induct A rule: infinite_finite_induct) simp_all
+
+lemma neutral_const [simp]: "F (\<lambda>_. \<^bold>1) A = \<^bold>1"
   by (simp add: neutral)
 
 lemma union_inter:
   assumes "finite A" and "finite B"
   shows "F g (A \<union> B) \<^bold>* F g (A \<inter> B) = F g A \<^bold>* F g B"
   \<comment> \<open>The reversed orientation looks more natural, but LOOPS as a simprule!\<close>
-using assms proof (induct A)
-  case empty then show ?case by simp
-next
-  case (insert x A) then show ?case
-    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
+  using assms
+proof (induct A)
+  case empty
+  then show ?case by simp
+next
+  case (insert x A)
+  then show ?case
+    by (auto simp: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
 qed
 
 corollary union_inter_neutral:
   assumes "finite A" and "finite B"
-  and I0: "\<forall>x \<in> A \<inter> B. g x = \<^bold>1"
+    and "\<forall>x \<in> A \<inter> B. g x = \<^bold>1"
   shows "F g (A \<union> B) = F g A \<^bold>* F g B"
   using assms by (simp add: union_inter [symmetric] neutral)
 
 corollary union_disjoint:
   assumes "finite A" and "finite B"

   assumes "finite A" and "finite B"
   shows "F g (A \<union> B) = F g (A - B) \<^bold>* F g (B - A) \<^bold>* F g (A \<inter> B)"
 proof -
   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
     by auto
-  with assms show ?thesis by simp (subst union_disjoint, auto)+
+  with assms show ?thesis
+    by simp (subst union_disjoint, auto)+
 qed
 
 lemma subset_diff:
   assumes "B \<subseteq> A" and "finite A"
   shows "F g A = F g (A - B) \<^bold>* F g B"

   assumes "F g A \<noteq> \<^bold>1"
   obtains a where "a \<in> A" and "g a \<noteq> \<^bold>1"
 proof -
   from assms have "\<exists>a\<in>A. g a \<noteq> \<^bold>1"
   proof (induct A rule: infinite_finite_induct)
+    case infinite
+    then show ?case by simp
+  next
+    case empty
+    then show ?case by simp
+  next
     case (insert a A)
-    then show ?case by simp (rule, simp)
-  qed simp_all
+    then show ?case by fastforce
+  qed
   with that show thesis by blast
 qed
 
 lemma reindex:
   assumes "inj_on h A"
   shows "F g (h ` A) = F (g \<circ> h) A"
 proof (cases "finite A")
   case True
-  with assms show ?thesis by (simp add: eq_fold fold_image comp_assoc)
-next
-  case False with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
+  with assms show ?thesis
+    by (simp add: eq_fold fold_image comp_assoc)
+next
+  case False
+  with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
   with False show ?thesis by simp
 qed
 
 lemma cong [fundef_cong]:
   assumes "A = B"

   by (induct B rule: infinite_finite_induct) auto
 
 lemma strong_cong [cong]:
   assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
   shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
-  by (rule cong) (insert assms, simp_all add: simp_implies_def)
+  by (rule cong) (use assms in \<open>simp_all add: simp_implies_def\<close>)
 
 lemma reindex_cong:
   assumes "inj_on l B"
   assumes "A = l ` B"
   assumes "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x"
   shows "F g A = F h B"
   using assms by (simp add: reindex)
 
 lemma UNION_disjoint:
   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
-  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
+    and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
   shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
-apply (insert assms)
-apply (induct rule: finite_induct)
-apply simp
-apply atomize
-apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
- prefer 2 apply blast
-apply (subgoal_tac "A x Int UNION Fa A = {}")
- prefer 2 apply blast
-apply (simp add: union_disjoint)
-done
+  apply (insert assms)
+  apply (induct rule: finite_induct)
+   apply simp
+  apply atomize
+  apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
+   prefer 2 apply blast
+  apply (subgoal_tac "A x \<inter> UNION Fa A = {}")
+   prefer 2 apply blast
+  apply (simp add: union_disjoint)
+  done
 
 lemma Union_disjoint:
   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
   shows "F g (\<Union>C) = (F \<circ> F) g C"
-proof cases
-  assume "finite C"
-  from UNION_disjoint [OF this assms]
-  show ?thesis by simp
-qed (auto dest: finite_UnionD intro: infinite)
-
-lemma distrib:
-  "F (\<lambda>x. g x \<^bold>* h x) A = F g A \<^bold>* F h A"
+proof (cases "finite C")
+  case True
+  from UNION_disjoint [OF this assms] show ?thesis by simp
+next
+  case False
+  then show ?thesis by (auto dest: finite_UnionD intro: infinite)
+qed
+
+lemma distrib: "F (\<lambda>x. g x \<^bold>* h x) A = F g A \<^bold>* F h A"
   by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)
 
 lemma Sigma:
   "finite A \<Longrightarrow> \<forall>x\<in>A. finite (B x) \<Longrightarrow> F (\<lambda>x. F (g x) (B x)) A = F (case_prod g) (SIGMA x:A. B x)"
-apply (subst Sigma_def)
-apply (subst UNION_disjoint, assumption, simp)
- apply blast
-apply (rule cong)
-apply rule
-apply (simp add: fun_eq_iff)
-apply (subst UNION_disjoint, simp, simp)
- apply blast
-apply (simp add: comp_def)
-done
+  apply (subst Sigma_def)
+  apply (subst UNION_disjoint)
+     apply assumption
+    apply simp
+   apply blast
+  apply (rule cong)
+   apply rule
+  apply (simp add: fun_eq_iff)
+  apply (subst UNION_disjoint)
+     apply simp
+    apply simp
+   apply blast
+  apply (simp add: comp_def)
+  done
 
 lemma related:
   assumes Re: "R \<^bold>1 \<^bold>1"
-  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 \<^bold>* y1) (x2 \<^bold>* y2)"
-  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
+    and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 \<^bold>* y1) (x2 \<^bold>* y2)"
+    and fin: "finite S"
+    and R_h_g: "\<forall>x\<in>S. R (h x) (g x)"
   shows "R (F h S) (F g S)"
-  using fS by (rule finite_subset_induct) (insert assms, auto)
+  using fin by (rule finite_subset_induct) (use assms in auto)
 
 lemma mono_neutral_cong_left:
-  assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = \<^bold>1"
-  and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
+  assumes "finite T"
+    and "S \<subseteq> T"
+    and "\<forall>i \<in> T - S. h i = \<^bold>1"
+    and "\<And>x. x \<in> S \<Longrightarrow> g x = h x"
+  shows "F g S = F h T"
 proof-
   have eq: "T = S \<union> (T - S)" using \<open>S \<subseteq> T\<close> by blast
   have d: "S \<inter> (T - S) = {}" using \<open>S \<subseteq> T\<close> by blast
   from \<open>finite T\<close> \<open>S \<subseteq> T\<close> have f: "finite S" "finite (T - S)"
     by (auto intro: finite_subset)
   show ?thesis using assms(4)
     by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
 qed
 
 lemma mono_neutral_cong_right:
-  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = \<^bold>1; \<And>x. x \<in> S \<Longrightarrow> g x = h x \<rbrakk>
-   \<Longrightarrow> F g T = F h S"
+  "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> g x = h x) \<Longrightarrow>
+    F g T = F h S"
   by (auto intro!: mono_neutral_cong_left [symmetric])
 
-lemma mono_neutral_left:
-  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = \<^bold>1 \<rbrakk> \<Longrightarrow> F g S = F g T"
+lemma mono_neutral_left: "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> F g S = F g T"
   by (blast intro: mono_neutral_cong_left)
 
-lemma mono_neutral_right:
-  "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = \<^bold>1 \<rbrakk> \<Longrightarrow> F g T = F g S"
+lemma mono_neutral_right: "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> F g T = F g S"
   by (blast intro!: mono_neutral_left [symmetric])
 
 lemma reindex_bij_betw: "bij_betw h S T \<Longrightarrow> F (\<lambda>x. g (h x)) S = F g T"
   by (auto simp: bij_betw_def reindex)
 

     "\<And>b. b \<in> T' \<Longrightarrow> g b = z"
   shows "F (\<lambda>x. g (h x)) S = F g T"
 proof -
   have [simp]: "finite S \<longleftrightarrow> finite T"
     using bij_betw_finite[OF bij] fin by auto
-
   show ?thesis
-  proof cases
-    assume "finite S"
+  proof (cases "finite S")
+    case True
     with nn have "F (\<lambda>x. g (h x)) S = F (\<lambda>x. g (h x)) (S - S')"
       by (intro mono_neutral_cong_right) auto
     also have "\<dots> = F g (T - T')"
       using bij by (rule reindex_bij_betw)
     also have "\<dots> = F g T"
       using nn \<open>finite S\<close> by (intro mono_neutral_cong_left) auto
     finally show ?thesis .
-  qed simp
+  next
+    case False
+    then show ?thesis by simp
+  qed
 qed
 
 lemma reindex_nontrivial:
   assumes "finite A"
-  and nz: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> h x = h y \<Longrightarrow> g (h x) = \<^bold>1"
+    and nz: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> h x = h y \<Longrightarrow> g (h x) = \<^bold>1"
   shows "F g (h ` A) = F (g \<circ> h) A"
 proof (subst reindex_bij_betw_not_neutral [symmetric])
   show "bij_betw h (A - {x \<in> A. (g \<circ> h) x = \<^bold>1}) (h ` A - h ` {x \<in> A. (g \<circ> h) x = \<^bold>1})"
     using nz by (auto intro!: inj_onI simp: bij_betw_def)
-qed (insert \<open>finite A\<close>, auto)
+qed (use \<open>finite A\<close> in auto)
 
 lemma reindex_bij_witness_not_neutral:
   assumes fin: "finite S'" "finite T'"
   assumes witness:
     "\<And>a. a \<in> S - S' \<Longrightarrow> i (j a) = a"

 qed
 
 lemma delta:
   assumes fS: "finite S"
   shows "F (\<lambda>k. if k = a then b k else \<^bold>1) S = (if a \<in> S then b a else \<^bold>1)"
-proof-
-  let ?f = "(\<lambda>k. if k=a then b k else \<^bold>1)"
-  { assume a: "a \<notin> S"
-    hence "\<forall>k\<in>S. ?f k = \<^bold>1" by simp
-    hence ?thesis  using a by simp }
-  moreover
-  { assume a: "a \<in> S"
+proof -
+  let ?f = "(\<lambda>k. if k = a then b k else \<^bold>1)"
+  show ?thesis
+  proof (cases "a \<in> S")
+    case False
+    then have "\<forall>k\<in>S. ?f k = \<^bold>1" by simp
+    with False show ?thesis by simp
+  next
+    case True
     let ?A = "S - {a}"
     let ?B = "{a}"
-    have eq: "S = ?A \<union> ?B" using a by blast
+    from True have eq: "S = ?A \<union> ?B" by blast
     have dj: "?A \<inter> ?B = {}" by simp
     from fS have fAB: "finite ?A" "finite ?B" by auto
     have "F ?f S = F ?f ?A \<^bold>* F ?f ?B"
-      using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
-      by simp
-    then have ?thesis using a by simp }
-  ultimately show ?thesis by blast
+      using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]] by simp
+    with True show ?thesis by simp
+  qed
 qed
 
 lemma delta':
-  assumes fS: "finite S"
+  assumes fin: "finite S"
   shows "F (\<lambda>k. if a = k then b k else \<^bold>1) S = (if a \<in> S then b a else \<^bold>1)"
-  using delta [OF fS, of a b, symmetric] by (auto intro: cong)
+  using delta [OF fin, of a b, symmetric] by (auto intro: cong)
 
 lemma If_cases:
   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
-  assumes fA: "finite A"
-  shows "F (\<lambda>x. if P x then h x else g x) A =
-    F h (A \<inter> {x. P x}) \<^bold>* F g (A \<inter> - {x. P x})"
-proof -
-  have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}"
-          "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
+  assumes fin: "finite A"
+  shows "F (\<lambda>x. if P x then h x else g x) A = F h (A \<inter> {x. P x}) \<^bold>* F g (A \<inter> - {x. P x})"
+proof -
+  have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}" "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
     by blast+
-  from fA
-  have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
+  from fin have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
   let ?g = "\<lambda>x. if P x then h x else g x"
-  from union_disjoint [OF f a(2), of ?g] a(1)
-  show ?thesis
+  from union_disjoint [OF f a(2), of ?g] a(1) show ?thesis
     by (subst (1 2) cong) simp_all
 qed
 
-lemma cartesian_product:
-   "F (\<lambda>x. F (g x) B) A = F (case_prod g) (A \<times> B)"
-apply (rule sym)
-apply (cases "finite A")
- apply (cases "finite B")
-  apply (simp add: Sigma)
- apply (cases "A={}", simp)
- apply simp
-apply (auto intro: infinite dest: finite_cartesian_productD2)
-apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
-done
+lemma cartesian_product: "F (\<lambda>x. F (g x) B) A = F (case_prod g) (A \<times> B)"
+  apply (rule sym)
+  apply (cases "finite A")
+   apply (cases "finite B")
+    apply (simp add: Sigma)
+   apply (cases "A = {}")
+    apply simp
+   apply simp
+   apply (auto intro: infinite dest: finite_cartesian_productD2)
+  apply (cases "B = {}")
+   apply (auto intro: infinite dest: finite_cartesian_productD1)
+  done
 
 lemma inter_restrict:
   assumes "finite A"
   shows "F g (A \<inter> B) = F (\<lambda>x. if x \<in> B then g x else \<^bold>1) A"
 proof -
   let ?g = "\<lambda>x. if x \<in> A \<inter> B then g x else \<^bold>1"
-  have "\<forall>i\<in>A - A \<inter> B. (if i \<in> A \<inter> B then g i else \<^bold>1) = \<^bold>1"
-   by simp
+  have "\<forall>i\<in>A - A \<inter> B. (if i \<in> A \<inter> B then g i else \<^bold>1) = \<^bold>1" by simp
   moreover have "A \<inter> B \<subseteq> A" by blast
-  ultimately have "F ?g (A \<inter> B) = F ?g A" using \<open>finite A\<close>
-    by (intro mono_neutral_left) auto
+  ultimately have "F ?g (A \<inter> B) = F ?g A"
+    using \<open>finite A\<close> by (intro mono_neutral_left) auto
   then show ?thesis by simp
 qed
 
 lemma inter_filter:
   "finite A \<Longrightarrow> F g {x \<in> A. P x} = F (\<lambda>x. if P x then g x else \<^bold>1) A"
   by (simp add: inter_restrict [symmetric, of A "{x. P x}" g, simplified mem_Collect_eq] Int_def)
 
 lemma Union_comp:
   assumes "\<forall>A \<in> B. finite A"
-    and "\<And>A1 A2 x. A1 \<in> B \<Longrightarrow> A2 \<in> B  \<Longrightarrow> A1 \<noteq> A2 \<Longrightarrow> x \<in> A1 \<Longrightarrow> x \<in> A2 \<Longrightarrow> g x = \<^bold>1"
+    and "\<And>A1 A2 x. A1 \<in> B \<Longrightarrow> A2 \<in> B \<Longrightarrow> A1 \<noteq> A2 \<Longrightarrow> x \<in> A1 \<Longrightarrow> x \<in> A2 \<Longrightarrow> g x = \<^bold>1"
   shows "F g (\<Union>B) = (F \<circ> F) g B"
-using assms proof (induct B rule: infinite_finite_induct)
+  using assms
+proof (induct B rule: infinite_finite_induct)
   case (infinite A)
   then have "\<not> finite (\<Union>A)" by (blast dest: finite_UnionD)
   with infinite show ?case by simp
 next
-  case empty then show ?case by simp
+  case empty
+  then show ?case by simp
 next
   case (insert A B)
   then have "finite A" "finite B" "finite (\<Union>B)" "A \<notin> B"
     and "\<forall>x\<in>A \<inter> \<Union>B. g x = \<^bold>1"
-    and H: "F g (\<Union>B) = (F o F) g B" by auto
+    and H: "F g (\<Union>B) = (F \<circ> F) g B" by auto
   then have "F g (A \<union> \<Union>B) = F g A \<^bold>* F g (\<Union>B)"
     by (simp add: union_inter_neutral)
   with \<open>finite B\<close> \<open>A \<notin> B\<close> show ?case
     by (simp add: H)
 qed
 
-lemma commute:
-  "F (\<lambda>i. F (g i) B) A = F (\<lambda>j. F (\<lambda>i. g i j) A) B"
+lemma commute: "F (\<lambda>i. F (g i) B) A = F (\<lambda>j. F (\<lambda>i. g i j) A) B"
   unfolding cartesian_product
   by (rule reindex_bij_witness [where i = "\<lambda>(i, j). (j, i)" and j = "\<lambda>(i, j). (j, i)"]) auto
 
 lemma commute_restrict:
   "finite A \<Longrightarrow> finite B \<Longrightarrow>

   fixes A :: "'b set" and B :: "'c set"
   assumes fin: "finite A" "finite B"
   shows "F g (A <+> B) = F (g \<circ> Inl) A \<^bold>* F (g \<circ> Inr) B"
 proof -
   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
-  moreover from fin have "finite (Inl ` A :: ('b + 'c) set)" "finite (Inr ` B :: ('b + 'c) set)"
-    by auto
-  moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('b + 'c) set)" by auto
-  moreover have "inj_on (Inl :: 'b \<Rightarrow> 'b + 'c) A" "inj_on (Inr :: 'c \<Rightarrow> 'b + 'c) B"
-    by (auto intro: inj_onI)
-  ultimately show ?thesis using fin
-    by (simp add: union_disjoint reindex)
+  moreover from fin have "finite (Inl ` A)" "finite (Inr ` B)" by auto
+  moreover have "Inl ` A \<inter> Inr ` B = {}" by auto
+  moreover have "inj_on Inl A" "inj_on Inr B" by (auto intro: inj_onI)
+  ultimately show ?thesis
+    using fin by (simp add: union_disjoint reindex)
 qed
 
 lemma same_carrier:
   assumes "finite C"
   assumes subset: "A \<subseteq> C" "B \<subseteq> C"
   assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = \<^bold>1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = \<^bold>1"
   shows "F g A = F h B \<longleftrightarrow> F g C = F h C"
 proof -
-  from \<open>finite C\<close> subset have
-    "finite A" and "finite B" and "finite (C - A)" and "finite (C - B)"
-    by (auto elim: finite_subset)
+  have "finite A" and "finite B" and "finite (C - A)" and "finite (C - B)"
+    using \<open>finite C\<close> subset by (auto elim: finite_subset)
   from subset have [simp]: "A - (C - A) = A" by auto
   from subset have [simp]: "B - (C - B) = B" by auto
   from subset have "C = A \<union> (C - A)" by auto
   then have "F g C = F g (A \<union> (C - A))" by simp
   also have "\<dots> = F g (A - (C - A)) \<^bold>* F g (C - A - A) \<^bold>* F g (A \<inter> (C - A))"
     using \<open>finite A\<close> \<open>finite (C - A)\<close> by (simp only: union_diff2)
-  finally have P: "F g C = F g A" using trivial by simp
+  finally have *: "F g C = F g A" using trivial by simp
   from subset have "C = B \<union> (C - B)" by auto
   then have "F h C = F h (B \<union> (C - B))" by simp
   also have "\<dots> = F h (B - (C - B)) \<^bold>* F h (C - B - B) \<^bold>* F h (B \<inter> (C - B))"
     using \<open>finite B\<close> \<open>finite (C - B)\<close> by (simp only: union_diff2)
-  finally have Q: "F h C = F h B" using trivial by simp
-  from P Q show ?thesis by simp
+  finally have "F h C = F h B"
+    using trivial by simp
+  with * show ?thesis by simp
 qed
 
 lemma same_carrierI:
   assumes "finite C"
   assumes subset: "A \<subseteq> C" "B \<subseteq> C"

 
 context comm_monoid_add
 begin
 
 sublocale setsum: comm_monoid_set plus 0
-defines
-  setsum = setsum.F ..
+  defines setsum = setsum.F ..
 
 abbreviation Setsum ("\<Sum>_" [1000] 999)
   where "\<Sum>A \<equiv> setsum (\<lambda>x. x) A"
 
 end

           end
     | setsum_tr' _ = raise Match;
 in [(@{const_syntax setsum}, K setsum_tr')] end
 \<close>
 
-text \<open>TODO generalization candidates\<close>
+(* TODO generalization candidates *)
 
 lemma (in comm_monoid_add) setsum_image_gen:
-  assumes fS: "finite S"
+  assumes fin: "finite S"
   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
-proof-
-  { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
-  hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
+proof -
+  have "{y. y\<in> f`S \<and> f x = y} = {f x}" if "x \<in> S" for x
+    using that by auto
+  then have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
     by simp
   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
-    by (rule setsum.commute_restrict [OF fS finite_imageI [OF fS]])
+    by (rule setsum.commute_restrict [OF fin finite_imageI [OF fin]])
   finally show ?thesis .
 qed
 
 
 subsubsection \<open>Properties in more restricted classes of structures\<close>
 
-lemma setsum_Un: "finite A ==> finite B ==>
-  (setsum f (A Un B) :: 'a :: ab_group_add) =
-   setsum f A + setsum f B - setsum f (A Int B)"
-by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
+lemma setsum_Un:
+  "finite A \<Longrightarrow> finite B \<Longrightarrow> setsum f (A \<union> B) = setsum f A + setsum f B - setsum f (A \<inter> B)"
+  for f :: "'b \<Rightarrow> 'a::ab_group_add"
+  by (subst setsum.union_inter [symmetric]) (auto simp add: algebra_simps)
 
 lemma setsum_Un2:
   assumes "finite (A \<union> B)"
   shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
 proof -
   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
     by auto
-  with assms show ?thesis by simp (subst setsum.union_disjoint, auto)+
-qed
-
-lemma setsum_diff1: "finite A \<Longrightarrow>
-  (setsum f (A - {a}) :: ('a::ab_group_add)) =
-  (if a:A then setsum f A - f a else setsum f A)"
-by (erule finite_induct) (auto simp add: insert_Diff_if)
+  with assms show ?thesis
+    by simp (subst setsum.union_disjoint, auto)+
+qed
+
+lemma setsum_diff1:
+  fixes f :: "'b \<Rightarrow> 'a::ab_group_add"
+  assumes "finite A"
+  shows "setsum f (A - {a}) = (if a \<in> A then setsum f A - f a else setsum f A)"
+  using assms by induct (auto simp: insert_Diff_if)
 
 lemma setsum_diff:
-  assumes le: "finite A" "B \<subseteq> A"
-  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
-proof -
-  from le have finiteB: "finite B" using finite_subset by auto
-  show ?thesis using finiteB le
+  fixes f :: "'b \<Rightarrow> 'a::ab_group_add"
+  assumes "finite A" "B \<subseteq> A"
+  shows "setsum f (A - B) = setsum f A - setsum f B"
+proof -
+  from assms(2,1) have "finite B" by (rule finite_subset)
+  from this \<open>B \<subseteq> A\<close>
+  show ?thesis
   proof induct
     case empty
-    thus ?case by auto
+    thus ?case by simp
   next
     case (insert x F)
-    thus ?case using le finiteB
+    with \<open>finite A\<close> \<open>finite B\<close> show ?case
       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
   qed
 qed
 
 lemma (in ordered_comm_monoid_add) setsum_mono:
   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f i \<le> g i"
   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
 proof (cases "finite K")
   case True
-  thus ?thesis using le
+  from this le show ?thesis
   proof induct
     case empty
-    thus ?case by simp
+    then show ?case by simp
   next
     case insert
-    thus ?case using add_mono by fastforce
-  qed
-next
-  case False then show ?thesis by simp
+    then show ?case using add_mono by fastforce
+  qed
+next
+  case False
+  then show ?thesis by simp
 qed
 
 lemma (in strict_ordered_comm_monoid_add) setsum_strict_mono:
-  assumes "finite A"  "A \<noteq> {}" and "\<And>x. x \<in> A \<Longrightarrow> f x < g x"
+  assumes "finite A" "A \<noteq> {}"
+    and "\<And>x. x \<in> A \<Longrightarrow> f x < g x"
   shows "setsum f A < setsum g A"
   using assms
 proof (induct rule: finite_ne_induct)
-  case singleton thus ?case by simp
-next
-  case insert thus ?case by (auto simp: add_strict_mono)
+  case singleton
+  then show ?case by simp
+next
+  case insert
+  then show ?case by (auto simp: add_strict_mono)
 qed
 
 lemma setsum_strict_mono_ex1:
   fixes f g :: "'i \<Rightarrow> 'a::ordered_cancel_comm_monoid_add"
-  assumes "finite A" and "\<forall>x\<in>A. f x \<le> g x" and "\<exists>a\<in>A. f a < g a"
+  assumes "finite A"
+    and "\<forall>x\<in>A. f x \<le> g x"
+    and "\<exists>a\<in>A. f a < g a"
   shows "setsum f A < setsum g A"
 proof-
-  from assms(3) obtain a where a: "a:A" "f a < g a" by blast
-  have "setsum f A = setsum f ((A-{a}) \<union> {a})"
-    by(simp add:insert_absorb[OF \<open>a:A\<close>])
-  also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
+  from assms(3) obtain a where a: "a \<in> A" "f a < g a" by blast
+  have "setsum f A = setsum f ((A - {a}) \<union> {a})"
+    by(simp add: insert_absorb[OF \<open>a \<in> A\<close>])
+  also have "\<dots> = setsum f (A - {a}) + setsum f {a}"
     using \<open>finite A\<close> by(subst setsum.union_disjoint) auto
-  also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
-    by(rule setsum_mono)(simp add: assms(2))
-  also have "setsum f {a} < setsum g {a}" using a by simp
-  also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
-    using \<open>finite A\<close> by(subst setsum.union_disjoint[symmetric]) auto
-  also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF \<open>a:A\<close>])
-  finally show ?thesis by (auto simp add: add_right_mono add_strict_left_mono)
+  also have "setsum f (A - {a}) \<le> setsum g (A - {a})"
+    by (rule setsum_mono) (simp add: assms(2))
+  also from a have "setsum f {a} < setsum g {a}" by simp
+  also have "setsum g (A - {a}) + setsum g {a} = setsum g((A - {a}) \<union> {a})"
+    using \<open>finite A\<close> by (subst setsum.union_disjoint[symmetric]) auto
+  also have "\<dots> = setsum g A" by (simp add: insert_absorb[OF \<open>a \<in> A\<close>])
+  finally show ?thesis
+    by (auto simp add: add_right_mono add_strict_left_mono)
 qed
 
 lemma setsum_mono_inv:
   fixes f g :: "'i \<Rightarrow> 'a :: ordered_cancel_comm_monoid_add"
   assumes eq: "setsum f I = setsum g I"
   assumes le: "\<And>i. i \<in> I \<Longrightarrow> f i \<le> g i"
   assumes i: "i \<in> I"
   assumes I: "finite I"
   shows "f i = g i"
-proof(rule ccontr)
-  assume "f i \<noteq> g i"
+proof (rule ccontr)
+  assume "\<not> ?thesis"
   with le[OF i] have "f i < g i" by simp
-  hence "\<exists>i\<in>I. f i < g i" using i ..
-  from setsum_strict_mono_ex1[OF I _ this] le have "setsum f I < setsum g I" by blast
+  with i have "\<exists>i\<in>I. f i < g i" ..
+  from setsum_strict_mono_ex1[OF I _ this] le have "setsum f I < setsum g I"
+    by blast
   with eq show False by simp
 qed
 
-lemma setsum_negf: "(\<Sum>x\<in>A. - f x::'a::ab_group_add) = - (\<Sum>x\<in>A. f x)"
-proof (cases "finite A")
-  case True thus ?thesis by (induct set: finite) auto
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_subtractf: "(\<Sum>x\<in>A. f x - g x::'a::ab_group_add) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
+lemma setsum_negf: "(\<Sum>x\<in>A. - f x) = - (\<Sum>x\<in>A. f x)"
+  for f :: "'b \<Rightarrow> 'a::ab_group_add"
+proof (cases "finite A")
+  case True
+  then show ?thesis by (induct set: finite) auto
+next
+  case False
+  then show ?thesis by simp
+qed
+
+lemma setsum_subtractf: "(\<Sum>x\<in>A. f x - g x) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
+  for f g :: "'b \<Rightarrow>'a::ab_group_add"
   using setsum.distrib [of f "- g" A] by (simp add: setsum_negf)
 
 lemma setsum_subtractf_nat:
-  "(\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x) \<Longrightarrow> (\<Sum>x\<in>A. f x - g x::nat) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
-  by (induction A rule: infinite_finite_induct) (auto simp: setsum_mono)
-
-lemma (in ordered_comm_monoid_add) setsum_nonneg:
+  "(\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x) \<Longrightarrow> (\<Sum>x\<in>A. f x - g x) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
+  for f g :: "'a \<Rightarrow> nat"
+  by (induct A rule: infinite_finite_induct) (auto simp: setsum_mono)
+
+context ordered_comm_monoid_add
+begin
+
+lemma setsum_nonneg:
   assumes nn: "\<forall>x\<in>A. 0 \<le> f x"
   shows "0 \<le> setsum f A"
 proof (cases "finite A")
-  case True thus ?thesis using nn
+  case True
+  then show ?thesis
+    using nn
   proof induct
-    case empty then show ?case by simp
+    case empty
+    then show ?case by simp
   next
     case (insert x F)
     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
     with insert show ?case by simp
   qed
 next
-  case False thus ?thesis by simp
-qed
-
-lemma (in ordered_comm_monoid_add) setsum_nonpos:
+  case False
+  then show ?thesis by simp
+qed
+
+lemma setsum_nonpos:
   assumes np: "\<forall>x\<in>A. f x \<le> 0"
   shows "setsum f A \<le> 0"
 proof (cases "finite A")
-  case True thus ?thesis using np
+  case True
+  then show ?thesis
+    using np
   proof induct
-    case empty then show ?case by simp
+    case empty
+    then show ?case by simp
   next
     case (insert x F)
     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
     with insert show ?case by simp
   qed
 next
   case False thus ?thesis by simp
 qed
 
-lemma (in ordered_comm_monoid_add) setsum_nonneg_eq_0_iff:
+lemma setsum_nonneg_eq_0_iff:
   "finite A \<Longrightarrow> \<forall>x\<in>A. 0 \<le> f x \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
-  by (induct set: finite, simp) (simp add: add_nonneg_eq_0_iff setsum_nonneg)
-
-lemma (in ordered_comm_monoid_add) setsum_nonneg_0:
+  by (induct set: finite) (simp_all add: add_nonneg_eq_0_iff setsum_nonneg)
+
+lemma setsum_nonneg_0:
   "finite s \<Longrightarrow> (\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0) \<Longrightarrow> (\<Sum> i \<in> s. f i) = 0 \<Longrightarrow> i \<in> s \<Longrightarrow> f i = 0"
   by (simp add: setsum_nonneg_eq_0_iff)
 
-lemma (in ordered_comm_monoid_add) setsum_nonneg_leq_bound:
+lemma setsum_nonneg_leq_bound:
   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
   shows "f i \<le> B"
 proof -
-  have "f i \<le> f i + (\<Sum>i \<in> s - {i}. f i)"
-    using assms by (intro add_increasing2 setsum_nonneg) auto
+  from assms have "f i \<le> f i + (\<Sum>i \<in> s - {i}. f i)"
+    by (intro add_increasing2 setsum_nonneg) auto
   also have "\<dots> = B"
     using setsum.remove[of s i f] assms by simp
   finally show ?thesis by auto
 qed
 
-lemma (in ordered_comm_monoid_add) setsum_mono2:
-  assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
+lemma setsum_mono2:
+  assumes fin: "finite B"
+    and sub: "A \<subseteq> B"
+    and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
   shows "setsum f A \<le> setsum f B"
 proof -
   have "setsum f A \<le> setsum f A + setsum f (B-A)"
     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
-  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
-    by (simp add: setsum.union_disjoint del:Un_Diff_cancel)
-  also have "A \<union> (B-A) = B" using sub by blast
+  also from fin finite_subset[OF sub fin] have "\<dots> = setsum f (A \<union> (B-A))"
+    by (simp add: setsum.union_disjoint del: Un_Diff_cancel)
+  also from sub have "A \<union> (B-A) = B" by blast
   finally show ?thesis .
 qed
 
-lemma (in ordered_comm_monoid_add) setsum_le_included:
+lemma setsum_le_included:
   assumes "finite s" "finite t"
   and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"
   shows "setsum f s \<le> setsum g t"
 proof -
   have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
   proof (rule setsum_mono)
-    fix y assume "y \<in> s"
+    fix y
+    assume "y \<in> s"
     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
     with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
       using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
       by (auto intro!: setsum_mono2)
   qed
-  also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
+  also have "\<dots> \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
     using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
-  also have "... \<le> setsum g t"
+  also have "\<dots> \<le> setsum g t"
     using assms by (auto simp: setsum_image_gen[symmetric])
   finally show ?thesis .
 qed
 
-lemma (in ordered_comm_monoid_add) setsum_mono3:
-  "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> \<forall>x\<in>B - A. 0 \<le> f x \<Longrightarrow> setsum f A \<le> setsum f B"
+lemma setsum_mono3: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> \<forall>x\<in>B - A. 0 \<le> f x \<Longrightarrow> setsum f A \<le> setsum f B"
   by (rule setsum_mono2) auto
+
+end
 
 lemma (in canonically_ordered_monoid_add) setsum_eq_0_iff [simp]:
   "finite F \<Longrightarrow> (setsum f F = 0) = (\<forall>a\<in>F. f a = 0)"
   by (intro ballI setsum_nonneg_eq_0_iff zero_le)
 
 lemma setsum_right_distrib:
-  fixes f :: "'a => ('b::semiring_0)"
-  shows "r * setsum f A = setsum (%n. r * f n) A"
-proof (cases "finite A")
-  case True
-  thus ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A) thus ?case by (simp add: distrib_left)
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_left_distrib:
-  "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
+  fixes f :: "'a \<Rightarrow> 'b::semiring_0"
+  shows "r * setsum f A = setsum (\<lambda>n. r * f n) A"
 proof (cases "finite A")
   case True
   then show ?thesis
   proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A) thus ?case by (simp add: distrib_right)
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_divide_distrib:
-  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
+    case empty
+    then show ?case by simp
+  next
+    case insert
+    then show ?case by (simp add: distrib_left)
+  qed
+next
+  case False
+  then show ?thesis by simp
+qed
+
+lemma setsum_left_distrib: "setsum f A * r = (\<Sum>n\<in>A. f n * r)"
+  for r :: "'a::semiring_0"
 proof (cases "finite A")
   case True
   then show ?thesis
   proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A) thus ?case by (simp add: add_divide_distrib)
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_abs[iff]:
-  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
-  shows "\<bar>setsum f A\<bar> \<le> setsum (%i. \<bar>f i\<bar>) A"
-proof (cases "finite A")
-  case True
-  thus ?thesis
+    case empty
+    then show ?case by simp
+  next
+    case insert
+    then show ?case by (simp add: distrib_right)
+  qed
+next
+  case False
+  then show ?thesis by simp
+qed
+
+lemma setsum_divide_distrib: "setsum f A / r = (\<Sum>n\<in>A. f n / r)"
+  for r :: "'a::field"
+proof (cases "finite A")
+  case True
+  then show ?thesis
   proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A)
-    thus ?case by (auto intro: abs_triangle_ineq order_trans)
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_abs_ge_zero[iff]:
-  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
-  shows "0 \<le> setsum (%i. \<bar>f i\<bar>) A"
+    case empty
+    then show ?case by simp
+  next
+    case insert
+    then show ?case by (simp add: add_divide_distrib)
+  qed
+next
+  case False
+  then show ?thesis by simp
+qed
+
+lemma setsum_abs[iff]: "\<bar>setsum f A\<bar> \<le> setsum (\<lambda>i. \<bar>f i\<bar>) A"
+  for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
+proof (cases "finite A")
+  case True
+  then show ?thesis
+  proof induct
+    case empty
+    then show ?case by simp
+  next
+    case insert
+    then show ?case by (auto intro: abs_triangle_ineq order_trans)
+  qed
+next
+  case False
+  then show ?thesis by simp
+qed
+
+lemma setsum_abs_ge_zero[iff]: "0 \<le> setsum (\<lambda>i. \<bar>f i\<bar>) A"
+  for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
   by (simp add: setsum_nonneg)
 
-lemma abs_setsum_abs[simp]:
-  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
-  shows "\<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar> = (\<Sum>a\<in>A. \<bar>f a\<bar>)"
-proof (cases "finite A")
-  case True
-  thus ?thesis
+lemma abs_setsum_abs[simp]: "\<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar> = (\<Sum>a\<in>A. \<bar>f a\<bar>)"
+  for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
+proof (cases "finite A")
+  case True
+  then show ?thesis
   proof induct
-    case empty thus ?case by simp
+    case empty
+    then show ?case by simp
   next
     case (insert a A)
-    hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
-    also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
-    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
-      by (simp del: abs_of_nonneg)
-    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
+    then have "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
+    also from insert have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>" by simp
+    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>" by (simp del: abs_of_nonneg)
+    also from insert have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" by simp
     finally show ?case .
   qed
 next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
-  shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
+  case False
+  then show ?thesis by simp
+qed
+
+lemma setsum_diff1_ring:
+  fixes f :: "'b \<Rightarrow> 'a::ring"
+  assumes "finite A" "a \<in> A"
+  shows "setsum f (A - {a}) = setsum f A - (f a)"
   unfolding setsum.remove [OF assms] by auto
 
 lemma setsum_product:
-  fixes f :: "'a => ('b::semiring_0)"
+  fixes f :: "'a \<Rightarrow> 'b::semiring_0"
   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum.commute)
 
 lemma setsum_mult_setsum_if_inj:
-fixes f :: "'a => ('b::semiring_0)"
-shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
-  setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
-by(auto simp: setsum_product setsum.cartesian_product
-        intro!:  setsum.reindex_cong[symmetric])
-
-lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
-apply (case_tac "finite A")
- prefer 2 apply simp
-apply (erule rev_mp)
-apply (erule finite_induct, auto)
-done
-
-lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
-  setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
-apply(erule finite_induct)
-apply (auto simp add:add_is_1)
-done
+  fixes f :: "'a \<Rightarrow> 'b::semiring_0"
+  shows "inj_on (\<lambda>(a, b). f a * g b) (A \<times> B) \<Longrightarrow>
+    setsum f A * setsum g B = setsum id {f a * g b |a b. a \<in> A \<and> b \<in> B}"
+  by(auto simp: setsum_product setsum.cartesian_product intro!: setsum.reindex_cong[symmetric])
+
+lemma setsum_SucD:
+  assumes "setsum f A = Suc n"
+  shows "\<exists>a\<in>A. 0 < f a"
+proof (cases "finite A")
+  case True
+  from this assms show ?thesis by induct auto
+next
+  case False
+  with assms show ?thesis by simp
+qed
+
+lemma setsum_eq_Suc0_iff:
+  assumes "finite A"
+  shows "setsum f A = Suc 0 \<longleftrightarrow> (\<exists>a\<in>A. f a = Suc 0 \<and> (\<forall>b\<in>A. a \<noteq> b \<longrightarrow> f b = 0))"
+  using assms by induct (auto simp add:add_is_1)
 
 lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
 
-lemma setsum_Un_nat: "finite A ==> finite B ==>
-  (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
+lemma setsum_Un_nat:
+  "finite A \<Longrightarrow> finite B \<Longrightarrow> setsum f (A \<union> B) = setsum f A + setsum f B - setsum f (A \<inter> B)"
+  for f :: "'a \<Rightarrow> nat"
   \<comment> \<open>For the natural numbers, we have subtraction.\<close>
-by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
-
-lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
-  (if a:A then setsum f A - f a else setsum f A)"
-apply (case_tac "finite A")
- prefer 2 apply simp
-apply (erule finite_induct)
- apply (auto simp add: insert_Diff_if)
-apply (drule_tac a = a in mk_disjoint_insert, auto)
-done
+  by (subst setsum.union_inter [symmetric]) (auto simp: algebra_simps)
+
+lemma setsum_diff1_nat: "setsum f (A - {a}) = (if a \<in> A then setsum f A - f a else setsum f A)"
+  for f :: "'a \<Rightarrow> nat"
+proof (cases "finite A")
+  case True
+  then show ?thesis
+    apply induct
+     apply (auto simp: insert_Diff_if)
+    apply (drule mk_disjoint_insert)
+    apply auto
+    done
+next
+  case False
+  then show ?thesis by simp
+qed
 
 lemma setsum_diff_nat:
-assumes "finite B" and "B \<subseteq> A"
-shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
-using assms
+  fixes f :: "'a \<Rightarrow> nat"
+  assumes "finite B" and "B \<subseteq> A"
+  shows "setsum f (A - B) = setsum f A - setsum f B"
+  using assms
 proof induct
-  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
-next
-  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
-    and xFinA: "insert x F \<subseteq> A"
-    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
-  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
-  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
+  case empty
+  then show ?case by simp
+next
+  case (insert x F)
+  note IH = \<open>F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F\<close>
+  from \<open>x \<notin> F\<close> \<open>insert x F \<subseteq> A\<close> have "x \<in> A - F" by simp
+  then have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
     by (simp add: setsum_diff1_nat)
-  from xFinA have "F \<subseteq> A" by simp
+  from \<open>insert x F \<subseteq> A\<close> have "F \<subseteq> A" by simp
   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
     by simp
-  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
+  from \<open>x \<notin> F\<close> have "A - insert x F = (A - F) - {x}" by auto
   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
     by simp
-  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
+  from \<open>finite F\<close> \<open>x \<notin> F\<close> have "setsum f (insert x F) = setsum f F + f x"
+    by simp
   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
     by simp
-  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
+  then show ?case by simp
 qed
 
 lemma setsum_comp_morphism:
   assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
   shows "setsum (h \<circ> g) A = h (setsum g A)"
 proof (cases "finite A")
-  case False then show ?thesis by (simp add: assms)
-next
-  case True then show ?thesis by (induct A) (simp_all add: assms)
-qed
-
-lemma (in comm_semiring_1) dvd_setsum:
-  "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd setsum f A"
+  case False
+  then show ?thesis by (simp add: assms)
+next
+  case True
+  then show ?thesis by (induct A) (simp_all add: assms)
+qed
+
+lemma (in comm_semiring_1) dvd_setsum: "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd setsum f A"
   by (induct A rule: infinite_finite_induct) simp_all
 
 lemma (in ordered_comm_monoid_add) setsum_pos:
   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> 0 < f i) \<Longrightarrow> 0 < setsum f I"
   by (induct I rule: finite_ne_induct) (auto intro: add_pos_pos)

   finally show ?thesis .
 qed
 
 lemma setsum_cong_Suc:
   assumes "0 \<notin> A" "\<And>x. Suc x \<in> A \<Longrightarrow> f (Suc x) = g (Suc x)"
-  shows   "setsum f A = setsum g A"
+  shows "setsum f A = setsum g A"
 proof (rule setsum.cong)
-  fix x assume "x \<in> A"
-  with assms(1) show "f x = g x" by (cases x) (auto intro!: assms(2))
+  fix x
+  assume "x \<in> A"
+  with assms(1) show "f x = g x"
+    by (cases x) (auto intro!: assms(2))
 qed simp_all
 
 
 subsubsection \<open>Cardinality as special case of @{const setsum}\<close>
 
-lemma card_eq_setsum:
-  "card A = setsum (\<lambda>x. 1) A"
+lemma card_eq_setsum: "card A = setsum (\<lambda>x. 1) A"
 proof -
   have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
     by (simp add: fun_eq_iff)
   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
     by (rule arg_cong)
   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
     by (blast intro: fun_cong)
-  then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
-qed
-
-lemma setsum_constant [simp]:
-  "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
-apply (cases "finite A")
-apply (erule finite_induct)
-apply (auto simp add: algebra_simps)
-done
+  then show ?thesis
+    by (simp add: card.eq_fold setsum.eq_fold)
+qed
+
+lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
+proof (cases "finite A")
+  case True
+  then show ?thesis by induct (auto simp: algebra_simps)
+next
+  case False
+  then show ?thesis by simp
+qed
 
 lemma setsum_Suc: "setsum (\<lambda>x. Suc(f x)) A = setsum f A + card A"
-  using setsum.distrib[of f "\<lambda>_. 1" A]
-  by simp
+  using setsum.distrib[of f "\<lambda>_. 1" A] by simp
 
 lemma setsum_bounded_above:
-  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_comm_monoid_add})"
+  fixes K :: "'a::{semiring_1,ordered_comm_monoid_add}"
+  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> K"
   shows "setsum f A \<le> of_nat (card A) * K"
 proof (cases "finite A")
   case True
-  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
-next
-  case False thus ?thesis by simp
+  then show ?thesis
+    using le setsum_mono[where K=A and g = "\<lambda>x. K"] by simp
+next
+  case False
+  then show ?thesis by simp
 qed
 
 lemma setsum_bounded_above_strict:
-  assumes "\<And>i. i\<in>A \<Longrightarrow> f i < (K::'a::{ordered_cancel_comm_monoid_add,semiring_1})"
-          "card A > 0"
+  fixes K :: "'a::{ordered_cancel_comm_monoid_add,semiring_1}"
+  assumes "\<And>i. i\<in>A \<Longrightarrow> f i < K" "card A > 0"
   shows "setsum f A < of_nat (card A) * K"
-using assms setsum_strict_mono[where A=A and g = "%x. K"]
-by (simp add: card_gt_0_iff)
+  using assms setsum_strict_mono[where A=A and g = "\<lambda>x. K"]
+  by (simp add: card_gt_0_iff)
 
 lemma setsum_bounded_below:
-  assumes le: "\<And>i. i\<in>A \<Longrightarrow> (K::'a::{semiring_1, ordered_comm_monoid_add}) \<le> f i"
+  fixes K :: "'a::{semiring_1,ordered_comm_monoid_add}"
+  assumes le: "\<And>i. i\<in>A \<Longrightarrow> K \<le> f i"
   shows "of_nat (card A) * K \<le> setsum f A"
 proof (cases "finite A")
   case True
-  thus ?thesis using le setsum_mono[where K=A and f = "%x. K"] by simp
-next
-  case False thus ?thesis by simp
+  then show ?thesis
+    using le setsum_mono[where K=A and f = "%x. K"] by simp
+next
+  case False
+  then show ?thesis by simp
 qed
 
 lemma card_UN_disjoint:
   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
 proof -
-  have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
-  with assms show ?thesis by (simp add: card_eq_setsum setsum.UNION_disjoint del: setsum_constant)
+  have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)"
+    by simp
+  with assms show ?thesis
+    by (simp add: card_eq_setsum setsum.UNION_disjoint del: setsum_constant)
 qed
 
 lemma card_Union_disjoint:
-  "finite C ==> (ALL A:C. finite A) ==>
-   (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
-   ==> card (\<Union>C) = setsum card C"
-apply (frule card_UN_disjoint [of C id])
-apply simp_all
-done
+  "finite C \<Longrightarrow> \<forall>A\<in>C. finite A \<Longrightarrow> \<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {} \<Longrightarrow>
+    card (\<Union>C) = setsum card C"
+  by (frule card_UN_disjoint [of C id]) simp_all
 
 lemma setsum_multicount_gen:
   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
-  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
+  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t"
+    (is "?l = ?r")
 proof-
-  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
-  also have "\<dots> = ?r" unfolding setsum.commute_restrict [OF assms(1-2)]
+  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s"
+    by auto
+  also have "\<dots> = ?r"
+    unfolding setsum.commute_restrict [OF assms(1-2)]
     using assms(3) by auto
   finally show ?thesis .
 qed
 
 lemma setsum_multicount:
   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
 proof-
-  have "?l = setsum (\<lambda>i. k) T" by (rule setsum_multicount_gen) (auto simp: assms)
+  have "?l = setsum (\<lambda>i. k) T"
+    by (rule setsum_multicount_gen) (auto simp: assms)
   also have "\<dots> = ?r" by (simp add: mult.commute)
   finally show ?thesis by auto
 qed
 
+
 subsubsection \<open>Cardinality of products\<close>
 
 lemma card_SigmaI [simp]:
-  "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
-  \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
-by(simp add: card_eq_setsum setsum.Sigma del:setsum_constant)
+  "finite A \<Longrightarrow> \<forall>a\<in>A. finite (B a) \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
+  by (simp add: card_eq_setsum setsum.Sigma del: setsum_constant)
 
 (*
 lemma SigmaI_insert: "y \<notin> A ==>
   (SIGMA x:(insert y A). B x) = (({y} \<times> (B y)) \<union> (SIGMA x: A. B x))"
   by auto
 *)
 
-lemma card_cartesian_product: "card (A \<times> B) = card(A) * card(B)"
+lemma card_cartesian_product: "card (A \<times> B) = card A * card B"
   by (cases "finite A \<and> finite B")
     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
 
-lemma card_cartesian_product_singleton:  "card({x} \<times> A) = card(A)"
-by (simp add: card_cartesian_product)
+lemma card_cartesian_product_singleton:  "card ({x} \<times> A) = card A"
+  by (simp add: card_cartesian_product)
 
 
 subsection \<open>Generalized product over a set\<close>
 
 context comm_monoid_mult
 begin
 
 sublocale setprod: comm_monoid_set times 1
-defines
-  setprod = setprod.F ..
-
-abbreviation
-  Setprod ("\<Prod>_" [1000] 999) where
-  "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
+  defines setprod = setprod.F ..
+
+abbreviation Setprod ("\<Prod>_" [1000] 999)
+  where "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
 
 end
 
 syntax (ASCII)
   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(4PROD _:_./ _)" [0, 51, 10] 10)

   "\<Prod>x|P. t" => "CONST setprod (\<lambda>x. t) {x. P}"
 
 context comm_monoid_mult
 begin
 
-lemma setprod_dvd_setprod:
-  "(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> setprod f A dvd setprod g A"
+lemma setprod_dvd_setprod: "(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> setprod f A dvd setprod g A"
 proof (induct A rule: infinite_finite_induct)
-  case infinite then show ?case by (auto intro: dvdI)
-next
-  case empty then show ?case by (auto intro: dvdI)
-next
-  case (insert a A) then
-  have "f a dvd g a" and "setprod f A dvd setprod g A" by simp_all
-  then obtain r s where "g a = f a * r" and "setprod g A = setprod f A * s" by (auto elim!: dvdE)
-  then have "g a * setprod g A = f a * setprod f A * (r * s)" by (simp add: ac_simps)
-  with insert.hyps show ?case by (auto intro: dvdI)
-qed
-
-lemma setprod_dvd_setprod_subset:
-  "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> setprod f A dvd setprod f B"
+  case infinite
+  then show ?case by (auto intro: dvdI)
+next
+  case empty
+  then show ?case by (auto intro: dvdI)
+next
+  case (insert a A)
+  then have "f a dvd g a" and "setprod f A dvd setprod g A"
+    by simp_all
+  then obtain r s where "g a = f a * r" and "setprod g A = setprod f A * s"
+    by (auto elim!: dvdE)
+  then have "g a * setprod g A = f a * setprod f A * (r * s)"
+    by (simp add: ac_simps)
+  with insert.hyps show ?case
+    by (auto intro: dvdI)
+qed
+
+lemma setprod_dvd_setprod_subset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> setprod f A dvd setprod f B"
   by (auto simp add: setprod.subset_diff ac_simps intro: dvdI)
 
 end
 
 

   assumes "finite A" and "a \<in> A" and "b = f a"
   shows "b dvd setprod f A"
 proof -
   from \<open>finite A\<close> have "setprod f (insert a (A - {a})) = f a * setprod f (A - {a})"
     by (intro setprod.insert) auto
-  also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A" by blast
+  also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A"
+    by blast
   finally have "setprod f A = f a * setprod f (A - {a})" .
-  with \<open>b = f a\<close> show ?thesis by simp
-qed
-
-lemma dvd_setprodI [intro]:
-  assumes "finite A" and "a \<in> A"
-  shows "f a dvd setprod f A"
-  using assms by auto
+  with \<open>b = f a\<close> show ?thesis
+    by simp
+qed
+
+lemma dvd_setprodI [intro]: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> f a dvd setprod f A"
+  by auto
 
 lemma setprod_zero:
   assumes "finite A" and "\<exists>a\<in>A. f a = 0"
   shows "setprod f A = 0"
-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 a A)
   then have "f a = 0 \<or> (\<exists>a\<in>A. f a = 0)" by simp
   then have "f a * setprod f A = 0" by rule (simp_all add: insert)
   with insert show ?case by simp

 qed
 
 end
 
 lemma setprod_zero_iff [simp]:
+  fixes f :: "'b \<Rightarrow> 'a::semidom"
   assumes "finite A"
-  shows "setprod f A = (0::'a::semidom) \<longleftrightarrow> (\<exists>a\<in>A. f a = 0)"
+  shows "setprod f A = 0 \<longleftrightarrow> (\<exists>a\<in>A. f a = 0)"
   using assms by (induct A) (auto simp: no_zero_divisors)
 
 lemma (in semidom_divide) setprod_diff1:
   assumes "finite A" and "f a \<noteq> 0"
   shows "setprod f (A - {a}) = (if a \<in> A then setprod f A div f a else setprod f A)"
 proof (cases "a \<notin> A")
-  case True then show ?thesis by simp
-next
-  case False with assms show ?thesis
-  proof (induct A rule: finite_induct)
-    case empty then show ?case by simp
+  case True
+  then show ?thesis by simp
+next
+  case False
+  with assms show ?thesis
+  proof induct
+    case empty
+    then show ?case by simp
   next
     case (insert b B)
     then show ?case
     proof (cases "a = b")
-      case True with insert show ?thesis by simp
+      case True
+      with insert show ?thesis by simp
     next
-      case False with insert have "a \<in> B" by simp
+      case False
+      with insert have "a \<in> B" by simp
       define C where "C = B - {a}"
-      with \<open>finite B\<close> \<open>a \<in> B\<close>
-        have *: "B = insert a C" "finite C" "a \<notin> C" by auto
-      with insert show ?thesis by (auto simp add: insert_commute ac_simps)
+      with \<open>finite B\<close> \<open>a \<in> B\<close> have "B = insert a C" "finite C" "a \<notin> C"
+        by auto
+      with insert show ?thesis
+        by (auto simp add: insert_commute ac_simps)
     qed
   qed
 qed
 
-lemma setsum_zero_power [simp]:
-  fixes c :: "nat \<Rightarrow> 'a::division_ring"
-  shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
-apply (cases "finite A")
-  by (induction A rule: finite_induct) auto
+lemma setsum_zero_power [simp]: "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
+  for c :: "nat \<Rightarrow> 'a::division_ring"
+  by (induct A rule: infinite_finite_induct) auto
 
 lemma setsum_zero_power' [simp]:
-  fixes c :: "nat \<Rightarrow> 'a::field"
-  shows "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
-  using setsum_zero_power [of "\<lambda>i. c i / d i" A]
-  by auto
+  "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
+  for c :: "nat \<Rightarrow> 'a::field"
+  using setsum_zero_power [of "\<lambda>i. c i / d i" A] by auto
 
 lemma (in field) setprod_inversef:
   "finite A \<Longrightarrow> setprod (inverse \<circ> f) A = inverse (setprod f A)"
   by (induct A rule: finite_induct) simp_all
 
-lemma (in field) setprod_dividef:
-  "finite A \<Longrightarrow> (\<Prod>x\<in>A. f x / g x) = setprod f A / setprod g A"
+lemma (in field) setprod_dividef: "finite A \<Longrightarrow> (\<Prod>x\<in>A. f x / g x) = setprod f A / setprod g A"
   using setprod_inversef [of A g] by (simp add: divide_inverse setprod.distrib)
 
 lemma setprod_Un:
   fixes f :: "'b \<Rightarrow> 'a :: field"
   assumes "finite A" and "finite B"
-  and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0"
+    and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0"
   shows "setprod f (A \<union> B) = setprod f A * setprod f B / setprod f (A \<inter> B)"
 proof -
   from assms have "setprod f A * setprod f B = setprod f (A \<union> B) * setprod f (A \<inter> B)"
     by (simp add: setprod.union_inter [symmetric, of A B])
-  with assms show ?thesis by simp
-qed
-
-lemma (in linordered_semidom) setprod_nonneg:
-  "(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> setprod f A"
+  with assms show ?thesis
+    by simp
+qed
+
+lemma (in linordered_semidom) setprod_nonneg: "(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> setprod f A"
   by (induct A rule: infinite_finite_induct) simp_all
 
-lemma (in linordered_semidom) setprod_pos:
-  "(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < setprod f A"
+lemma (in linordered_semidom) setprod_pos: "(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < setprod f A"
   by (induct A rule: infinite_finite_induct) simp_all
 
 lemma (in linordered_semidom) setprod_mono:
   "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i \<Longrightarrow> setprod f A \<le> setprod g A"
   by (induct A rule: infinite_finite_induct) (auto intro!: setprod_nonneg mult_mono)
 
 lemma (in linordered_semidom) setprod_mono_strict:
-    assumes"finite A" "\<forall>i\<in>A. 0 \<le> f i \<and> f i < g i" "A \<noteq> {}"
-    shows "setprod f A < setprod g A"
-using assms
-apply (induct A rule: finite_induct)
-apply (simp add: )
-apply (force intro: mult_strict_mono' setprod_nonneg)
-done
-
-lemma (in linordered_field) abs_setprod:
-  "\<bar>setprod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
+  assumes "finite A" "\<forall>i\<in>A. 0 \<le> f i \<and> f i < g i" "A \<noteq> {}"
+  shows "setprod f A < setprod g A"
+  using assms
+proof (induct A rule: finite_induct)
+  case empty
+  then show ?case by simp
+next
+  case insert
+  then show ?case by (force intro: mult_strict_mono' setprod_nonneg)
+qed
+
+lemma (in linordered_field) abs_setprod: "\<bar>setprod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
   by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult)
 
-lemma setprod_eq_1_iff [simp]:
-  "finite A \<Longrightarrow> setprod f A = 1 \<longleftrightarrow> (\<forall>a\<in>A. f a = (1::nat))"
+lemma setprod_eq_1_iff [simp]: "finite A \<Longrightarrow> setprod f A = 1 \<longleftrightarrow> (\<forall>a\<in>A. f a = 1)"
+  for f :: "'a \<Rightarrow> nat"
   by (induct A rule: finite_induct) simp_all
 
-lemma setprod_pos_nat_iff [simp]:
-  "finite A \<Longrightarrow> setprod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > (0::nat))"
+lemma setprod_pos_nat_iff [simp]: "finite A \<Longrightarrow> setprod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > 0)"
+  for f :: "'a \<Rightarrow> nat"
   using setprod_zero_iff by (simp del: neq0_conv add: zero_less_iff_neq_zero)
 
-lemma setprod_constant:
-  "(\<Prod>x\<in> A. (y::'a::comm_monoid_mult)) = y ^ card A"
+lemma setprod_constant: "(\<Prod>x\<in> A. y) = y ^ card A"
+  for y :: "'a::comm_monoid_mult"
   by (induct A rule: infinite_finite_induct) simp_all
 
-lemma setprod_power_distrib:
-  fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
-  shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
-proof (cases "finite A")
-  case True then show ?thesis
-    by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
-next
-  case False then show ?thesis
-    by simp
-qed
-
-lemma power_setsum:
-  "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
+lemma setprod_power_distrib: "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
+  for f :: "'a \<Rightarrow> 'b::comm_semiring_1"
+  by (induct A rule: infinite_finite_induct) (auto simp add: power_mult_distrib)
+
+lemma power_setsum: "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
   by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
 
 lemma setprod_gen_delta:
-  assumes fS: "finite S"
-  shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::comm_monoid_mult) * c^ (card S - 1) else c^ card S)"
-proof-
+  fixes b :: "'b \<Rightarrow> 'a::comm_monoid_mult"
+  assumes fin: "finite S"
+  shows "setprod (\<lambda>k. if k = a then b k else c) S =
+    (if a \<in> S then b a * c ^ (card S - 1) else c ^ card S)"
+proof -
   let ?f = "(\<lambda>k. if k=a then b k else c)"
-  {assume a: "a \<notin> S"
-    hence "\<forall> k\<in> S. ?f k = c" by simp
-    hence ?thesis using a setprod_constant by simp }
-  moreover
-  {assume a: "a \<in> S"
+  show ?thesis
+  proof (cases "a \<in> S")
+    case False
+    then have "\<forall> k\<in> S. ?f k = c" by simp
+    with False show ?thesis by (simp add: setprod_constant)
+  next
+    case True
     let ?A = "S - {a}"
     let ?B = "{a}"
-    have eq: "S = ?A \<union> ?B" using a by blast
-    have dj: "?A \<inter> ?B = {}" by simp
-    from fS have fAB: "finite ?A" "finite ?B" by auto
-    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
+    from True have eq: "S = ?A \<union> ?B" by blast
+    have disjoint: "?A \<inter> ?B = {}" by simp
+    from fin have fin': "finite ?A" "finite ?B" by auto
+    have f_A0: "setprod ?f ?A = setprod (\<lambda>i. c) ?A"
       by (rule setprod.cong) auto
-    have cA: "card ?A = card S - 1" using fS a by auto
-    have fA1: "setprod ?f ?A = c ^ card ?A"
-      unfolding fA0 by (rule setprod_constant)
+    from fin True have card_A: "card ?A = card S - 1" by auto
+    have f_A1: "setprod ?f ?A = c ^ card ?A"
+      unfolding f_A0 by (rule setprod_constant)
     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
-      using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
+      using setprod.union_disjoint[OF fin' disjoint, of ?f, unfolded eq[symmetric]]
       by simp
-    then have ?thesis using a cA
-      by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
-  ultimately show ?thesis by blast
+    with True card_A show ?thesis
+      by (simp add: f_A1 field_simps cong add: setprod.cong cong del: if_weak_cong)
+  qed
 qed
 
 end