diff --git a/src/HOL/Groups_Big.thy b/src/HOL/Groups_Big.thy
--- a/src/HOL/Groups_Big.thy
+++ b/src/HOL/Groups_Big.thy
@@ -1,1634 +1,1639 @@
 (*  Title:      HOL/Groups_Big.thy
     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 Power
 begin
 
 subsection \<open>Generic monoid operation over a set\<close>
 
 locale comm_monoid_set = comm_monoid
 begin
 
 subsubsection \<open>Standard sum or product indexed by a finite set\<close>
 
 interpretation comp_fun_commute f
   by standard (simp add: fun_eq_iff left_commute)
 
 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"
   by (simp add: eq_fold)
 
 lemma empty [simp]: "F g {} = \<^bold>1"
   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 B: "A = insert x B" and "x \<notin> B"
     by (auto dest: mk_disjoint_insert)
   moreover from \<open>finite A\<close> B have "finite B" by simp
   ultimately show ?thesis by simp
 qed
 
 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 insert_if: "finite A \<Longrightarrow> F g (insert x A) = (if x \<in> A then F g A else g x \<^bold>* F g A)"
   by (cases "x \<in> A") (simp_all add: 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: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
 qed
 
 corollary union_inter_neutral:
   assumes "finite A" and "finite B"
     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 "A \<inter> B = {}"
   shows "F g (A \<union> B) = F g A \<^bold>* F g B"
   using assms by (simp add: union_inter_neutral)
 
 lemma union_diff2:
   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)+
 qed
 
 lemma subset_diff:
   assumes "B \<subseteq> A" and "finite A"
   shows "F g A = F g (A - B) \<^bold>* F g B"
 proof -
   from assms have "finite (A - B)" by auto
   moreover from assms have "finite B" by (rule finite_subset)
   moreover from assms have "(A - B) \<inter> B = {}" by auto
   ultimately have "F g (A - B \<union> B) = F g (A - B) \<^bold>* F g B" by (rule union_disjoint)
   moreover from assms have "A \<union> B = A" by auto
   ultimately show ?thesis by simp
 qed
 
 lemma Int_Diff:
   assumes "finite A"
   shows "F g A = F g (A \<inter> B) \<^bold>* F g (A - B)"
   by (subst subset_diff [where B = "A - B"]) (auto simp:  Diff_Diff_Int assms)
 
 lemma setdiff_irrelevant:
   assumes "finite A"
   shows "F g (A - {x. g x = z}) = F g A"
   using assms by (induct A) (simp_all add: insert_Diff_if)
 
 lemma not_neutral_contains_not_neutral:
   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 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 False show ?thesis by simp
 qed
 
 lemma cong [fundef_cong]:
   assumes "A = B"
   assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
   shows "F g A = F h B"
   using g_h unfolding \<open>A = B\<close>
   by (induct B rule: infinite_finite_induct) auto
 
 lemma cong_simp [cong]:
   "\<lbrakk> A = B;  \<And>x. x \<in> B =simp=> g x = h x \<rbrakk> \<Longrightarrow> F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
 by (rule cong) (simp_all add: simp_implies_def)
 
 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 image_eq:
+  assumes "inj_on g A"  
+  shows "F (\<lambda>x. x) (g ` A) = F g A"
+  using assms reindex_cong by fastforce
+
 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 = {}"
   shows "F g (\<Union>(A ` I)) = F (\<lambda>x. F g (A x)) I"
   using assms
 proof (induction rule: finite_induct)
   case (insert i I)
   then have "\<forall>j\<in>I. j \<noteq> i"
     by blast
   with insert.prems have "A i \<inter> \<Union>(A ` I) = {}"
     by blast
   with insert show ?case
     by (simp add: union_disjoint)
 qed auto
 
 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 "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:
   assumes "finite A" "\<forall>x\<in>A. finite (B x)"
   shows "F (\<lambda>x. F (g x) (B x)) A = F (case_prod g) (SIGMA x:A. B x)"
   unfolding Sigma_def
 proof (subst UNION_disjoint)
   show "F (\<lambda>x. F (g x) (B x)) A = F (\<lambda>x. F (\<lambda>(x, y). g x y) (\<Union>y\<in>B x. {(x, y)})) A"
   proof (rule cong [OF refl])
     show "F (g x) (B x) = F (\<lambda>(x, y). g x y) (\<Union>y\<in>B x. {(x, y)})"
       if "x \<in> A" for x
       using that assms by (simp add: UNION_disjoint)
   qed
 qed (use assms in auto)
 
 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 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 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"
 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:
   "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: "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: "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 mono_neutral_cong:
   assumes [simp]: "finite T" "finite S"
     and *: "\<And>i. i \<in> T - S \<Longrightarrow> h i = \<^bold>1" "\<And>i. i \<in> S - T \<Longrightarrow> g i = \<^bold>1"
     and gh: "\<And>x. x \<in> S \<inter> T \<Longrightarrow> g x = h x"
  shows "F g S = F h T"
 proof-
   have "F g S = F g (S \<inter> T)"
     by(rule mono_neutral_right)(auto intro: *)
   also have "\<dots> = F h (S \<inter> T)" using refl gh by(rule cong)
   also have "\<dots> = F h T"
     by(rule mono_neutral_left)(auto intro: *)
   finally show ?thesis .
 qed
 
 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)
 
 lemma reindex_bij_witness:
   assumes witness:
     "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"
     "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"
     "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"
     "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"
   assumes eq:
     "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
   shows "F g S = F h T"
 proof -
   have "bij_betw j S T"
     using bij_betw_byWitness[where A=S and f=j and f'=i and A'=T] witness by auto
   moreover have "F g S = F (\<lambda>x. h (j x)) S"
     by (intro cong) (auto simp: eq)
   ultimately show ?thesis
     by (simp add: reindex_bij_betw)
 qed
 
 lemma reindex_bij_betw_not_neutral:
   assumes fin: "finite S'" "finite T'"
   assumes bij: "bij_betw h (S - S') (T - T')"
   assumes nn:
     "\<And>a. a \<in> S' \<Longrightarrow> g (h a) = z"
     "\<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 "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 .
   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"
   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 (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"
     "\<And>a. a \<in> S - S' \<Longrightarrow> j a \<in> T - T'"
     "\<And>b. b \<in> T - T' \<Longrightarrow> j (i b) = b"
     "\<And>b. b \<in> T - T' \<Longrightarrow> i b \<in> S - S'"
   assumes nn:
     "\<And>a. a \<in> S' \<Longrightarrow> g a = z"
     "\<And>b. b \<in> T' \<Longrightarrow> h b = z"
   assumes eq:
     "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
   shows "F g S = F h T"
 proof -
   have bij: "bij_betw j (S - (S' \<inter> S)) (T - (T' \<inter> T))"
     using witness by (intro bij_betw_byWitness[where f'=i]) auto
   have F_eq: "F g S = F (\<lambda>x. h (j x)) S"
     by (intro cong) (auto simp: eq)
   show ?thesis
     unfolding F_eq using fin nn eq
     by (intro reindex_bij_betw_not_neutral[OF _ _ bij]) auto
 qed
 
 lemma delta_remove:
   assumes fS: "finite S"
   shows "F (\<lambda>k. if k = a then b k else c k) S = (if a \<in> S then b a \<^bold>* F c (S-{a}) else F c (S-{a}))"
 proof -
   let ?f = "(\<lambda>k. if k = a then b k else c k)"
   show ?thesis
   proof (cases "a \<in> S")
     case False
     then have "\<forall>k\<in>S. ?f k = c k" by simp
     with False show ?thesis by simp
   next
     case True
     let ?A = "S - {a}"
     let ?B = "{a}"
     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
     with True show ?thesis
       using comm_monoid_set.remove comm_monoid_set_axioms fS by fastforce
   qed
 qed
 
 lemma delta [simp]:
   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)"
   by (simp add: delta_remove [OF assms])
 
 lemma delta' [simp]:
   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 fin, of a b, symmetric] by (auto intro: cong)
 
 lemma If_cases:
   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
   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 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
     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)"
 proof (cases "A = {} \<or> B = {}")
   case True
   then show ?thesis
     by auto
 next
   case False
   then have "A \<noteq> {}" "B \<noteq> {}" by auto
   show ?thesis
   proof (cases "finite A \<and> finite B")
     case True
     then show ?thesis
       by (simp add: Sigma)
   next
     case False
     then consider "infinite A" | "infinite B" by auto
     then have "infinite (A \<times> B)"
       by cases (use \<open>A \<noteq> {}\<close> \<open>B \<noteq> {}\<close> in \<open>auto dest: finite_cartesian_productD1 finite_cartesian_productD2\<close>)
     then show ?thesis
       using False by auto
   qed
 qed
 
 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
   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
   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"
   shows "F g (\<Union>B) = (F \<circ> F) g B"
   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
 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 \<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 swap: "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 swap_restrict:
   "finite A \<Longrightarrow> finite B \<Longrightarrow>
     F (\<lambda>x. F (g x) {y. y \<in> B \<and> R x y}) A = F (\<lambda>y. F (\<lambda>x. g x y) {x. x \<in> A \<and> R x y}) B"
   by (simp add: inter_filter) (rule swap)
 
 lemma image_gen:
   assumes fin: "finite S"
   shows "F h S = F (\<lambda>y. F h {x. x \<in> S \<and> g x = y}) (g ` S)"
 proof -
   have "{y. y\<in> g`S \<and> g x = y} = {g x}" if "x \<in> S" for x
     using that by auto
   then have "F h S = F (\<lambda>x. F (\<lambda>y. h x) {y. y\<in> g`S \<and> g x = y}) S"
     by simp
   also have "\<dots> = F (\<lambda>y. F h {x. x \<in> S \<and> g x = y}) (g ` S)"
     by (rule swap_restrict [OF fin finite_imageI [OF fin]])
   finally show ?thesis .
 qed
 
 lemma group:
   assumes fS: "finite S" and fT: "finite T" and fST: "g ` S \<subseteq> T"
   shows "F (\<lambda>y. F h {x. x \<in> S \<and> g x = y}) T = F h S"
   unfolding image_gen[OF fS, of h g]
   by (auto intro: neutral mono_neutral_right[OF fT fST])
 
 lemma Plus:
   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)" "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 -
   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 *: "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 "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"
   assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = \<^bold>1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = \<^bold>1"
   assumes "F g C = F h C"
   shows "F g A = F h B"
   using assms same_carrier [of C A B] by simp
 
 lemma eq_general:
   assumes B: "\<And>y. y \<in> B \<Longrightarrow> \<exists>!x. x \<in> A \<and> h x = y" and A: "\<And>x. x \<in> A \<Longrightarrow> h x \<in> B \<and> \<gamma>(h x) = \<phi> x"
   shows "F \<phi> A = F \<gamma> B"
 proof -
   have eq: "B = h ` A"
     by (auto dest: assms)
   have h: "inj_on h A"
     using assms by (blast intro: inj_onI)
   have "F \<phi> A = F (\<gamma> \<circ> h) A"
     using A by auto
   also have "\<dots> = F \<gamma> B"
     by (simp add: eq reindex h)
   finally show ?thesis .
 qed
 
 lemma eq_general_inverses:
   assumes B: "\<And>y. y \<in> B \<Longrightarrow> k y \<in> A \<and> h(k y) = y" and A: "\<And>x. x \<in> A \<Longrightarrow> h x \<in> B \<and> k(h x) = x \<and> \<gamma>(h x) = \<phi> x"
   shows "F \<phi> A = F \<gamma> B"
   by (rule eq_general [where h=h]) (force intro: dest: A B)+
 
 subsubsection \<open>HOL Light variant: sum/product indexed by the non-neutral subset\<close>
 text \<open>NB only a subset of the properties above are proved\<close>
 
 definition G :: "['b \<Rightarrow> 'a,'b set] \<Rightarrow> 'a"
   where "G p I \<equiv> if finite {x \<in> I. p x \<noteq> \<^bold>1} then F p {x \<in> I. p x \<noteq> \<^bold>1} else \<^bold>1"
 
 lemma finite_Collect_op:
   shows "\<lbrakk>finite {i \<in> I. x i \<noteq> \<^bold>1}; finite {i \<in> I. y i \<noteq> \<^bold>1}\<rbrakk> \<Longrightarrow> finite {i \<in> I. x i \<^bold>* y i \<noteq> \<^bold>1}"
   apply (rule finite_subset [where B = "{i \<in> I. x i \<noteq> \<^bold>1} \<union> {i \<in> I. y i \<noteq> \<^bold>1}"]) 
   using left_neutral by force+
 
 lemma empty' [simp]: "G p {} = \<^bold>1"
   by (auto simp: G_def)
 
 lemma eq_sum [simp]: "finite I \<Longrightarrow> G p I = F p I"
   by (auto simp: G_def intro: mono_neutral_cong_left)
 
 lemma insert' [simp]:
   assumes "finite {x \<in> I. p x \<noteq> \<^bold>1}"
   shows "G p (insert i I) = (if i \<in> I then G p I else p i \<^bold>* G p I)"
 proof -
   have "{x. x = i \<and> p x \<noteq> \<^bold>1 \<or> x \<in> I \<and> p x \<noteq> \<^bold>1} = (if p i = \<^bold>1 then {x \<in> I. p x \<noteq> \<^bold>1} else insert i {x \<in> I. p x \<noteq> \<^bold>1})"
     by auto
   then show ?thesis
     using assms by (simp add: G_def conj_disj_distribR insert_absorb)
 qed
 
 lemma distrib_triv':
   assumes "finite I"
   shows "G (\<lambda>i. g i \<^bold>* h i) I = G g I \<^bold>* G h I"
   by (simp add: assms local.distrib)
 
 lemma non_neutral': "G g {x \<in> I. g x \<noteq> \<^bold>1} = G g I"
   by (simp add: G_def)
 
 lemma distrib':
   assumes "finite {x \<in> I. g x \<noteq> \<^bold>1}" "finite {x \<in> I. h x \<noteq> \<^bold>1}"
   shows "G (\<lambda>i. g i \<^bold>* h i) I = G g I \<^bold>* G h I"
 proof -
   have "a \<^bold>* a \<noteq> a \<Longrightarrow> a \<noteq> \<^bold>1" for a
     by auto
   then have "G (\<lambda>i. g i \<^bold>* h i) I = G (\<lambda>i. g i \<^bold>* h i) ({i \<in> I. g i \<noteq> \<^bold>1} \<union> {i \<in> I. h i \<noteq> \<^bold>1})"
     using assms  by (force simp: G_def finite_Collect_op intro!: mono_neutral_cong)
   also have "\<dots> = G g I \<^bold>* G h I"
   proof -
     have "F g ({i \<in> I. g i \<noteq> \<^bold>1} \<union> {i \<in> I. h i \<noteq> \<^bold>1}) = G g I"
          "F h ({i \<in> I. g i \<noteq> \<^bold>1} \<union> {i \<in> I. h i \<noteq> \<^bold>1}) = G h I"
       by (auto simp: G_def assms intro: mono_neutral_right)
     then show ?thesis
       using assms by (simp add: distrib)
   qed
   finally show ?thesis .
 qed
 
 lemma cong':
   assumes "A = B"
   assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
   shows "G g A = G h B"
   using assms by (auto simp: G_def cong: conj_cong intro: cong)
 
 
 lemma mono_neutral_cong_left':
   assumes "S \<subseteq> T"
     and "\<And>i. i \<in> T - S \<Longrightarrow> h i = \<^bold>1"
     and "\<And>x. x \<in> S \<Longrightarrow> g x = h x"
   shows "G g S = G h T"
 proof -
   have *: "{x \<in> S. g x \<noteq> \<^bold>1} = {x \<in> T. h x \<noteq> \<^bold>1}"
     using assms by (metis DiffI subset_eq) 
   then have "finite {x \<in> S. g x \<noteq> \<^bold>1} = finite {x \<in> T. h x \<noteq> \<^bold>1}"
     by simp
   then show ?thesis
     using assms by (auto simp add: G_def * intro: cong)
 qed
 
 lemma mono_neutral_cong_right':
   "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>
     G g T = G h S"
   by (auto intro!: mono_neutral_cong_left' [symmetric])
 
 lemma mono_neutral_left': "S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> G g S = G g T"
   by (blast intro: mono_neutral_cong_left')
 
 lemma mono_neutral_right': "S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> G g T = G g S"
   by (blast intro!: mono_neutral_left' [symmetric])
 
 end
 
 
 subsection \<open>Generalized summation over a set\<close>
 
 context comm_monoid_add
 begin
 
 sublocale sum: comm_monoid_set plus 0
   defines sum = sum.F and sum' = sum.G ..
 
 abbreviation Sum ("\<Sum>")
   where "\<Sum> \<equiv> sum (\<lambda>x. x)"
 
 end
 
 text \<open>Now: lots of fancy syntax. First, \<^term>\<open>sum (\<lambda>x. e) A\<close> is written \<open>\<Sum>x\<in>A. e\<close>.\<close>
 
 syntax (ASCII)
   "_sum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::comm_monoid_add"  ("(3SUM (_/:_)./ _)" [0, 51, 10] 10)
 syntax
   "_sum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::comm_monoid_add"  ("(2\<Sum>(_/\<in>_)./ _)" [0, 51, 10] 10)
 translations \<comment> \<open>Beware of argument permutation!\<close>
   "\<Sum>i\<in>A. b" \<rightleftharpoons> "CONST sum (\<lambda>i. b) A"
 
 text \<open>Instead of \<^term>\<open>\<Sum>x\<in>{x. P}. e\<close> we introduce the shorter \<open>\<Sum>x|P. e\<close>.\<close>
 
 syntax (ASCII)
   "_qsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(3SUM _ |/ _./ _)" [0, 0, 10] 10)
 syntax
   "_qsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Sum>_ | (_)./ _)" [0, 0, 10] 10)
 translations
   "\<Sum>x|P. t" => "CONST sum (\<lambda>x. t) {x. P}"
 
 print_translation \<open>
 let
   fun sum_tr' [Abs (x, Tx, t), Const (\<^const_syntax>\<open>Collect\<close>, _) $ Abs (y, Ty, P)] =
         if x <> y then raise Match
         else
           let
             val x' = Syntax_Trans.mark_bound_body (x, Tx);
             val t' = subst_bound (x', t);
             val P' = subst_bound (x', P);
           in
             Syntax.const \<^syntax_const>\<open>_qsum\<close> $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'
           end
     | sum_tr' _ = raise Match;
 in [(\<^const_syntax>\<open>sum\<close>, K sum_tr')] end
 \<close>
 
 
 subsubsection \<open>Properties in more restricted classes of structures\<close>
 
 lemma sum_Un:
   "finite A \<Longrightarrow> finite B \<Longrightarrow> sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
   for f :: "'b \<Rightarrow> 'a::ab_group_add"
   by (subst sum.union_inter [symmetric]) (auto simp add: algebra_simps)
 
 lemma sum_Un2:
   assumes "finite (A \<union> B)"
   shows "sum f (A \<union> B) = sum f (A - B) + sum f (B - A) + sum 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 sum.union_disjoint, auto)+
 qed
 
 lemma sum_diff1:
   fixes f :: "'b \<Rightarrow> 'a::ab_group_add"
   assumes "finite A"
   shows "sum f (A - {a}) = (if a \<in> A then sum f A - f a else sum f A)"
   using assms by induct (auto simp: insert_Diff_if)
 
 lemma sum_diff:
   fixes f :: "'b \<Rightarrow> 'a::ab_group_add"
   assumes "finite A" "B \<subseteq> A"
   shows "sum f (A - B) = sum f A - sum 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 simp
   next
     case (insert x F)
     with \<open>finite A\<close> \<open>finite B\<close> show ?case
       by (simp add: Diff_insert[where a=x and B=F] sum_diff1 insert_absorb)
   qed
 qed
 
 lemma sum_diff1'_aux:
   fixes f :: "'a \<Rightarrow> 'b::ab_group_add"
   assumes "finite F" "{i \<in> I. f i \<noteq> 0} \<subseteq> F"
   shows "sum' f (I - {i}) = (if i \<in> I then sum' f I - f i else sum' f I)"
   using assms
 proof induct
   case (insert x F)
   have 1: "finite {x \<in> I. f x \<noteq> 0} \<Longrightarrow> finite {x \<in> I. x \<noteq> i \<and> f x \<noteq> 0}"
     by (erule rev_finite_subset) auto
   have 2: "finite {x \<in> I. x \<noteq> i \<and> f x \<noteq> 0} \<Longrightarrow> finite {x \<in> I. f x \<noteq> 0}"
     apply (drule finite_insert [THEN iffD2])
     by (erule rev_finite_subset) auto
   have 3: "finite {i \<in> I. f i \<noteq> 0}"
     using finite_subset insert by blast
   show ?case
     using insert sum_diff1 [of "{i \<in> I. f i \<noteq> 0}" f i]
     by (auto simp: sum.G_def 1 2 3 set_diff_eq conj_ac)
 qed (simp add: sum.G_def)
 
 lemma sum_diff1':
   fixes f :: "'a \<Rightarrow> 'b::ab_group_add"
   assumes "finite {i \<in> I. f i \<noteq> 0}"
   shows "sum' f (I - {i}) = (if i \<in> I then sum' f I - f i else sum' f I)"
   by (rule sum_diff1'_aux [OF assms order_refl])
 
 lemma (in ordered_comm_monoid_add) sum_mono:
   "(\<And>i. i\<in>K \<Longrightarrow> f i \<le> g i) \<Longrightarrow> (\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
   by (induct K rule: infinite_finite_induct) (use add_mono in auto)
 
 lemma (in strict_ordered_comm_monoid_add) sum_strict_mono:
   assumes "finite A" "A \<noteq> {}"
     and "\<And>x. x \<in> A \<Longrightarrow> f x < g x"
   shows "sum f A < sum g A"
   using assms
 proof (induct rule: finite_ne_induct)
   case singleton
   then show ?case by simp
 next
   case insert
   then show ?case by (auto simp: add_strict_mono)
 qed
 
 lemma sum_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"
   shows "sum f A < sum g A"
 proof-
   from assms(3) obtain a where a: "a \<in> A" "f a < g a" by blast
   have "sum f A = sum f ((A - {a}) \<union> {a})"
     by(simp add: insert_absorb[OF \<open>a \<in> A\<close>])
   also have "\<dots> = sum f (A - {a}) + sum f {a}"
     using \<open>finite A\<close> by(subst sum.union_disjoint) auto
   also have "sum f (A - {a}) \<le> sum g (A - {a})"
     by (rule sum_mono) (simp add: assms(2))
   also from a have "sum f {a} < sum g {a}" by simp
   also have "sum g (A - {a}) + sum g {a} = sum g((A - {a}) \<union> {a})"
     using \<open>finite A\<close> by (subst sum.union_disjoint[symmetric]) auto
   also have "\<dots> = sum 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 sum_mono_inv:
   fixes f g :: "'i \<Rightarrow> 'a :: ordered_cancel_comm_monoid_add"
   assumes eq: "sum f I = sum 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 "\<not> ?thesis"
   with le[OF i] have "f i < g i" by simp
   with i have "\<exists>i\<in>I. f i < g i" ..
   from sum_strict_mono_ex1[OF I _ this] le have "sum f I < sum g I"
     by blast
   with eq show False by simp
 qed
 
 lemma member_le_sum:
   fixes f :: "_ \<Rightarrow> 'b::{semiring_1, ordered_comm_monoid_add}"
   assumes "i \<in> A"
     and le: "\<And>x. x \<in> A - {i} \<Longrightarrow> 0 \<le> f x"
     and "finite A"
   shows "f i \<le> sum f A"
 proof -
   have "f i \<le> sum f (A \<inter> {i})"
     by (simp add: assms)
   also have "... = (\<Sum>x\<in>A. if x \<in> {i} then f x else 0)"
     using assms sum.inter_restrict by blast
   also have "... \<le> sum f A"
     apply (rule sum_mono)
     apply (auto simp: le)
     done
   finally show ?thesis .
 qed
 
 lemma sum_negf: "(\<Sum>x\<in>A. - f x) = - (\<Sum>x\<in>A. f x)"
   for f :: "'b \<Rightarrow> 'a::ab_group_add"
   by (induct A rule: infinite_finite_induct) auto
 
 lemma sum_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 sum.distrib [of f "- g" A] by (simp add: sum_negf)
 
 lemma sum_subtractf_nat:
   "(\<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: sum_mono)
 
 context ordered_comm_monoid_add
 begin
 
 lemma sum_nonneg: "(\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x) \<Longrightarrow> 0 \<le> sum f A"
 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 x F)
   then have "0 + 0 \<le> f x + sum f F" by (blast intro: add_mono)
   with insert show ?case by simp
 qed
 
 lemma sum_nonpos: "(\<And>x. x \<in> A \<Longrightarrow> f x \<le> 0) \<Longrightarrow> sum f A \<le> 0"
 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 x F)
   then have "f x + sum f F \<le> 0 + 0" by (blast intro: add_mono)
   with insert show ?case by simp
 qed
 
 lemma sum_nonneg_eq_0_iff:
   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x) \<Longrightarrow> sum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
   by (induct set: finite) (simp_all add: add_nonneg_eq_0_iff sum_nonneg)
 
 lemma sum_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: sum_nonneg_eq_0_iff)
 
 lemma sum_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 -
   from assms have "f i \<le> f i + (\<Sum>i \<in> s - {i}. f i)"
     by (intro add_increasing2 sum_nonneg) auto
   also have "\<dots> = B"
     using sum.remove[of s i f] assms by simp
   finally show ?thesis by auto
 qed
 
 lemma sum_mono2:
   assumes fin: "finite B"
     and sub: "A \<subseteq> B"
     and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
   shows "sum f A \<le> sum f B"
 proof -
   have "sum f A \<le> sum f A + sum f (B-A)"
     by (auto intro: add_increasing2 [OF sum_nonneg] nn)
   also from fin finite_subset[OF sub fin] have "\<dots> = sum f (A \<union> (B-A))"
     by (simp add: sum.union_disjoint del: Un_Diff_cancel)
   also from sub have "A \<union> (B-A) = B" by blast
   finally show ?thesis .
 qed
 
 lemma sum_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 "sum f s \<le> sum g t"
 proof -
   have "sum f s \<le> sum (\<lambda>y. sum g {x. x\<in>t \<and> i x = y}) s"
   proof (rule sum_mono)
     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> sum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
       using order_trans[of "?A (i z)" "sum g {z}" "?B (i z)", intro]
       by (auto intro!: sum_mono2)
   qed
   also have "\<dots> \<le> sum (\<lambda>y. sum g {x. x\<in>t \<and> i x = y}) (i ` t)"
     using assms(2-4) by (auto intro!: sum_mono2 sum_nonneg)
   also have "\<dots> \<le> sum g t"
     using assms by (auto simp: sum.image_gen[symmetric])
   finally show ?thesis .
 qed
 
 end
 
 lemma (in canonically_ordered_monoid_add) sum_eq_0_iff [simp]:
   "finite F \<Longrightarrow> (sum f F = 0) = (\<forall>a\<in>F. f a = 0)"
   by (intro ballI sum_nonneg_eq_0_iff zero_le)
 
 context semiring_0
 begin
 
 lemma sum_distrib_left: "r * sum f A = (\<Sum>n\<in>A. r * f n)"
   by (induct A rule: infinite_finite_induct) (simp_all add: algebra_simps)
 
 lemma sum_distrib_right: "sum f A * r = (\<Sum>n\<in>A. f n * r)"
   by (induct A rule: infinite_finite_induct) (simp_all add: algebra_simps)
 
 end
 
 lemma sum_divide_distrib: "sum f A / r = (\<Sum>n\<in>A. f n / r)"
   for r :: "'a::field"
 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
   then show ?case by (simp add: add_divide_distrib)
 qed
 
 lemma sum_abs[iff]: "\<bar>sum f A\<bar> \<le> sum (\<lambda>i. \<bar>f i\<bar>) A"
   for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
 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
   then show ?case by (auto intro: abs_triangle_ineq order_trans)
 qed
 
 lemma sum_abs_ge_zero[iff]: "0 \<le> sum (\<lambda>i. \<bar>f i\<bar>) A"
   for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
   by (simp add: sum_nonneg)
 
 lemma abs_sum_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 (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 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
 
 lemma sum_product:
   fixes f :: "'a \<Rightarrow> 'b::semiring_0"
   shows "sum f A * sum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
   by (simp add: sum_distrib_left sum_distrib_right) (rule sum.swap)
 
 lemma sum_mult_sum_if_inj:
   fixes f :: "'a \<Rightarrow> 'b::semiring_0"
   shows "inj_on (\<lambda>(a, b). f a * g b) (A \<times> B) \<Longrightarrow>
     sum f A * sum g B = sum id {f a * g b |a b. a \<in> A \<and> b \<in> B}"
   by(auto simp: sum_product sum.cartesian_product intro!: sum.reindex_cong[symmetric])
 
 lemma sum_SucD: "sum f A = Suc n \<Longrightarrow> \<exists>a\<in>A. 0 < f a"
   by (induct A rule: infinite_finite_induct) auto
 
 lemma sum_eq_Suc0_iff:
   "finite A \<Longrightarrow> sum 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))"
   by (induct A rule: finite_induct) (auto simp add: add_is_1)
 
 lemmas sum_eq_1_iff = sum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
 
 lemma sum_Un_nat:
   "finite A \<Longrightarrow> finite B \<Longrightarrow> sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
   for f :: "'a \<Rightarrow> nat"
   \<comment> \<open>For the natural numbers, we have subtraction.\<close>
   by (subst sum.union_inter [symmetric]) (auto simp: algebra_simps)
 
 lemma sum_diff1_nat: "sum f (A - {a}) = (if a \<in> A then sum f A - f a else sum f A)"
   for f :: "'a \<Rightarrow> nat"
 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
   then show ?case
     apply (auto simp: insert_Diff_if)
     apply (drule mk_disjoint_insert)
     apply auto
     done
 qed
 
 lemma sum_diff_nat:
   fixes f :: "'a \<Rightarrow> nat"
   assumes "finite B" and "B \<subseteq> A"
   shows "sum f (A - B) = sum f A - sum f B"
   using assms
 proof induct
   case empty
   then show ?case by simp
 next
   case (insert x F)
   note IH = \<open>F \<subseteq> A \<Longrightarrow> sum f (A - F) = sum f A - sum 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: "sum f ((A - F) - {x}) = sum f (A - F) - f x"
     by (simp add: sum_diff1_nat)
   from \<open>insert x F \<subseteq> A\<close> have "F \<subseteq> A" by simp
   with IH have "sum f (A - F) = sum f A - sum f F" by simp
   with A have B: "sum f ((A - F) - {x}) = sum f A - sum f F - f x"
     by simp
   from \<open>x \<notin> F\<close> have "A - insert x F = (A - F) - {x}" by auto
   with B have C: "sum f (A - insert x F) = sum f A - sum f F - f x"
     by simp
   from \<open>finite F\<close> \<open>x \<notin> F\<close> have "sum f (insert x F) = sum f F + f x"
     by simp
   with C have "sum f (A - insert x F) = sum f A - sum f (insert x F)"
     by simp
   then show ?case by simp
 qed
 
 lemma sum_comp_morphism:
   "h 0 = 0 \<Longrightarrow> (\<And>x y. h (x + y) = h x + h y) \<Longrightarrow> sum (h \<circ> g) A = h (sum g A)"
   by (induct A rule: infinite_finite_induct) simp_all
 
 lemma (in comm_semiring_1) dvd_sum: "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd sum f A"
   by (induct A rule: infinite_finite_induct) simp_all
 
 lemma (in ordered_comm_monoid_add) sum_pos:
   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> 0 < f i) \<Longrightarrow> 0 < sum f I"
   by (induct I rule: finite_ne_induct) (auto intro: add_pos_pos)
 
 lemma (in ordered_comm_monoid_add) sum_pos2:
   assumes I: "finite I" "i \<in> I" "0 < f i" "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
   shows "0 < sum f I"
 proof -
   have "0 < f i + sum f (I - {i})"
     using assms by (intro add_pos_nonneg sum_nonneg) auto
   also have "\<dots> = sum f I"
     using assms by (simp add: sum.remove)
   finally show ?thesis .
 qed
 
 lemma sum_cong_Suc:
   assumes "0 \<notin> A" "\<And>x. Suc x \<in> A \<Longrightarrow> f (Suc x) = g (Suc x)"
   shows "sum f A = sum g A"
 proof (rule sum.cong)
   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>\<open>sum\<close>\<close>
 
 lemma card_eq_sum: "card A = sum (\<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 sum.eq_fold)
 qed
 
 context semiring_1
 begin
 
 lemma sum_constant [simp]:
   "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
   by (induct A rule: infinite_finite_induct) (simp_all add: algebra_simps)
 
 end
 
 lemma sum_Suc: "sum (\<lambda>x. Suc(f x)) A = sum f A + card A"
   using sum.distrib[of f "\<lambda>_. 1" A] by simp
 
 lemma sum_bounded_above:
   fixes K :: "'a::{semiring_1,ordered_comm_monoid_add}"
   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> K"
   shows "sum f A \<le> of_nat (card A) * K"
 proof (cases "finite A")
   case True
   then show ?thesis
     using le sum_mono[where K=A and g = "\<lambda>x. K"] by simp
 next
   case False
   then show ?thesis by simp
 qed
 
 lemma sum_bounded_above_divide:
   fixes K :: "'a::linordered_field"
   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> K / of_nat (card A)" and fin: "finite A" "A \<noteq> {}"
   shows "sum f A \<le> K"
   using sum_bounded_above [of A f "K / of_nat (card A)", OF le] fin by simp
 
 lemma sum_bounded_above_strict:
   fixes K :: "'a::{ordered_cancel_comm_monoid_add,semiring_1}"
   assumes "\<And>i. i\<in>A \<Longrightarrow> f i < K" "card A > 0"
   shows "sum f A < of_nat (card A) * K"
   using assms sum_strict_mono[where A=A and g = "\<lambda>x. K"]
   by (simp add: card_gt_0_iff)
 
 lemma sum_bounded_below:
   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> sum f A"
 proof (cases "finite A")
   case True
   then show ?thesis
     using le sum_mono[where K=A and f = "\<lambda>x. K"] by simp
 next
   case False
   then show ?thesis by simp
 qed
 
 lemma convex_sum_bound_le:
   fixes x :: "'a \<Rightarrow> 'b::linordered_idom"
   assumes 0: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> x i" and 1: "sum x I = 1"
       and \<delta>: "\<And>i. i \<in> I \<Longrightarrow> \<bar>a i - b\<bar> \<le> \<delta>"
     shows "\<bar>(\<Sum>i\<in>I. a i * x i) - b\<bar> \<le> \<delta>"
 proof -
   have [simp]: "(\<Sum>i\<in>I. c * x i) = c" for c
     by (simp flip: sum_distrib_left 1)
   then have "\<bar>(\<Sum>i\<in>I. a i * x i) - b\<bar> = \<bar>\<Sum>i\<in>I. (a i - b) * x i\<bar>"
     by (simp add: sum_subtractf left_diff_distrib)
   also have "\<dots> \<le> (\<Sum>i\<in>I. \<bar>(a i - b) * x i\<bar>)"
     using abs_abs abs_of_nonneg by blast
   also have "\<dots> \<le> (\<Sum>i\<in>I. \<bar>(a i - b)\<bar> * x i)"
     by (simp add: abs_mult 0)
   also have "\<dots> \<le> (\<Sum>i\<in>I. \<delta> * x i)"
     by (rule sum_mono) (use \<delta> "0" mult_right_mono in blast)
   also have "\<dots> = \<delta>"
     by simp
   finally show ?thesis .
 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>(A ` I)) = (\<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_sum sum.UNION_disjoint del: sum_constant)
 qed
 
 lemma card_Union_disjoint:
   assumes "pairwise disjnt C" and fin: "\<And>A. A \<in> C \<Longrightarrow> finite A"
   shows "card (\<Union>C) = sum card C"
 proof (cases "finite C")
   case True
   then show ?thesis
     using card_UN_disjoint [OF True, of "\<lambda>x. x"] assms
     by (simp add: disjnt_def fin pairwise_def)
 next
   case False
   then show ?thesis
     using assms card_eq_0_iff finite_UnionD by fastforce
 qed
 
 lemma card_Union_le_sum_card:
   fixes U :: "'a set set"
   assumes "\<forall>u \<in> U. finite u"
   shows "card (\<Union>U) \<le> sum card U"
 proof (cases "finite U")
   case False
   then show "card (\<Union>U) \<le> sum card U"
     using card_eq_0_iff finite_UnionD by auto
 next
   case True
   then show "card (\<Union>U) \<le> sum card U"
   proof (induct U rule: finite_induct)
     case empty
     then show ?case by auto
   next
     case (insert x F)
     then have "card(\<Union>(insert x F)) \<le> card(x) + card (\<Union>F)" using card_Un_le by auto
     also have "... \<le> card(x) + sum card F" using insert.hyps by auto
     also have "... = sum card (insert x F)" using sum.insert_if and insert.hyps by auto
     finally show ?case .
   qed
 qed
 
 lemma card_UN_le:
   assumes "finite I"
   shows "card(\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. card(A i))"
   using assms
 proof induction
   case (insert i I)
   then show ?case
     using card_Un_le nat_add_left_cancel_le by (force intro: order_trans) 
 qed auto
 
 lemma sum_multicount_gen:
   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
   shows "sum (\<lambda>i. (card {j\<in>t. R i j})) s = sum k t"
     (is "?l = ?r")
 proof-
   have "?l = sum (\<lambda>i. sum (\<lambda>x.1) {j\<in>t. R i j}) s"
     by auto
   also have "\<dots> = ?r"
     unfolding sum.swap_restrict [OF assms(1-2)]
     using assms(3) by auto
   finally show ?thesis .
 qed
 
 lemma sum_multicount:
   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
   shows "sum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
 proof-
   have "?l = sum (\<lambda>i. k) T"
     by (rule sum_multicount_gen) (auto simp: assms)
   also have "\<dots> = ?r" by (simp add: mult.commute)
   finally show ?thesis by auto
 qed
 
 lemma sum_card_image:
   assumes "finite A"
   assumes "pairwise (\<lambda>s t. disjnt (f s) (f t)) A"
   shows "sum card (f ` A) = sum (\<lambda>a. card (f a)) A"
 using assms
 proof (induct A)
   case (insert a A)
   show ?case
   proof cases
     assume "f a = {}"
     with insert show ?case
       by (subst sum.mono_neutral_right[where S="f ` A"]) (auto simp: pairwise_insert)
   next
     assume "f a \<noteq> {}"
     then have "sum card (insert (f a) (f ` A)) = card (f a) + sum card (f ` A)"
       using insert
       by (subst sum.insert) (auto simp: pairwise_insert)
     with insert show ?case by (simp add: pairwise_insert)
   qed
 qed simp
 
 
 subsubsection \<open>Cardinality of products\<close>
 
 lemma card_SigmaI [simp]:
   "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_sum sum.Sigma del: sum_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"
   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)
 
 
 subsection \<open>Generalized product over a set\<close>
 
 context comm_monoid_mult
 begin
 
 sublocale prod: comm_monoid_set times 1
   defines prod = prod.F and prod' = prod.G ..
 
 abbreviation Prod ("\<Prod>_" [1000] 999)
   where "\<Prod>A \<equiv> prod (\<lambda>x. x) A"
 
 end
 
 syntax (ASCII)
   "_prod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(4PROD (_/:_)./ _)" [0, 51, 10] 10)
 syntax
   "_prod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(2\<Prod>(_/\<in>_)./ _)" [0, 51, 10] 10)
 translations \<comment> \<open>Beware of argument permutation!\<close>
   "\<Prod>i\<in>A. b" == "CONST prod (\<lambda>i. b) A"
 
 text \<open>Instead of \<^term>\<open>\<Prod>x\<in>{x. P}. e\<close> we introduce the shorter \<open>\<Prod>x|P. e\<close>.\<close>
 
 syntax (ASCII)
   "_qprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(4PROD _ |/ _./ _)" [0, 0, 10] 10)
 syntax
   "_qprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a"  ("(2\<Prod>_ | (_)./ _)" [0, 0, 10] 10)
 translations
   "\<Prod>x|P. t" => "CONST prod (\<lambda>x. t) {x. P}"
 
 context comm_monoid_mult
 begin
 
 lemma prod_dvd_prod: "(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> prod f A dvd prod 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 "prod f A dvd prod g A"
     by simp_all
   then obtain r s where "g a = f a * r" and "prod g A = prod f A * s"
     by (auto elim!: dvdE)
   then have "g a * prod g A = f a * prod f A * (r * s)"
     by (simp add: ac_simps)
   with insert.hyps show ?case
     by (auto intro: dvdI)
 qed
 
 lemma prod_dvd_prod_subset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> prod f A dvd prod f B"
   by (auto simp add: prod.subset_diff ac_simps intro: dvdI)
 
 end
 
 
 subsubsection \<open>Properties in more restricted classes of structures\<close>
 
 context linordered_nonzero_semiring
 begin
 
 lemma prod_ge_1: "(\<And>x. x \<in> A \<Longrightarrow> 1 \<le> f x) \<Longrightarrow> 1 \<le> prod f A"
 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 x F)
   have "1 * 1 \<le> f x * prod f F"
     by (rule mult_mono') (use insert in auto)
   with insert show ?case by simp
 qed
 
 lemma prod_le_1:
   fixes f :: "'b \<Rightarrow> 'a"
   assumes "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x \<and> f x \<le> 1"
   shows "prod f A \<le> 1"
     using assms
 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 x F)
   then show ?case by (force simp: mult.commute intro: dest: mult_le_one)
 qed
 
 end
 
 context comm_semiring_1
 begin
 
 lemma dvd_prod_eqI [intro]:
   assumes "finite A" and "a \<in> A" and "b = f a"
   shows "b dvd prod f A"
 proof -
   from \<open>finite A\<close> have "prod f (insert a (A - {a})) = f a * prod f (A - {a})"
     by (intro prod.insert) auto
   also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A"
     by blast
   finally have "prod f A = f a * prod f (A - {a})" .
   with \<open>b = f a\<close> show ?thesis
     by simp
 qed
 
 lemma dvd_prodI [intro]: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> f a dvd prod f A"
   by auto
 
 lemma prod_zero:
   assumes "finite A" and "\<exists>a\<in>A. f a = 0"
   shows "prod f A = 0"
   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 * prod f A = 0" by rule (simp_all add: insert)
   with insert show ?case by simp
 qed
 
 lemma prod_dvd_prod_subset2:
   assumes "finite B" and "A \<subseteq> B" and "\<And>a. a \<in> A \<Longrightarrow> f a dvd g a"
   shows "prod f A dvd prod g B"
 proof -
   from assms have "prod f A dvd prod g A"
     by (auto intro: prod_dvd_prod)
   moreover from assms have "prod g A dvd prod g B"
     by (auto intro: prod_dvd_prod_subset)
   ultimately show ?thesis by (rule dvd_trans)
 qed
 
 end
 
 lemma (in semidom) prod_zero_iff [simp]:
   fixes f :: "'b \<Rightarrow> 'a"
   assumes "finite A"
   shows "prod 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) prod_diff1:
   assumes "finite A" and "f a \<noteq> 0"
   shows "prod f (A - {a}) = (if a \<in> A then prod f A div f a else prod f A)"
 proof (cases "a \<notin> A")
   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
     next
       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)
     qed
   qed
 qed
 
 lemma sum_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 sum_zero_power' [simp]:
   "(\<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 sum_zero_power [of "\<lambda>i. c i / d i" A] by auto
 
 lemma (in field) prod_inversef: "prod (inverse \<circ> f) A = inverse (prod f A)"
  proof (cases "finite A")
    case True
    then show ?thesis
      by (induct A rule: finite_induct) simp_all
  next
    case False
    then show ?thesis
      by auto
  qed
 
 lemma (in field) prod_dividef: "(\<Prod>x\<in>A. f x / g x) = prod f A / prod g A"
   using prod_inversef [of g A] by (simp add: divide_inverse prod.distrib)
 
 lemma prod_Un:
   fixes f :: "'b \<Rightarrow> 'a :: field"
   assumes "finite A" and "finite B"
     and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0"
   shows "prod f (A \<union> B) = prod f A * prod f B / prod f (A \<inter> B)"
 proof -
   from assms have "prod f A * prod f B = prod f (A \<union> B) * prod f (A \<inter> B)"
     by (simp add: prod.union_inter [symmetric, of A B])
   with assms show ?thesis
     by simp
 qed
 
 context linordered_semidom
 begin
 
 lemma prod_nonneg: "(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> prod f A"
   by (induct A rule: infinite_finite_induct) simp_all
 
 lemma prod_pos: "(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < prod f A"
   by (induct A rule: infinite_finite_induct) simp_all
 
 lemma prod_mono:
   "(\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i \<and> f i \<le> g i) \<Longrightarrow> prod f A \<le> prod g A"
   by (induct A rule: infinite_finite_induct) (force intro!: prod_nonneg mult_mono)+
 
 lemma prod_mono_strict:
   assumes "finite A" "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i \<and> f i < g i" "A \<noteq> {}"
   shows "prod f A < prod 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' prod_nonneg)
 qed
 
 end
 
 lemma prod_mono2:
   fixes f :: "'a \<Rightarrow> 'b :: linordered_idom"
   assumes fin: "finite B"
     and sub: "A \<subseteq> B"
     and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 1 \<le> f b"
     and A: "\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a"
   shows "prod f A \<le> prod f B"
 proof -
   have "prod f A \<le> prod f A * prod f (B-A)"
     by (metis prod_ge_1 A mult_le_cancel_left1 nn not_less prod_nonneg)
   also from fin finite_subset[OF sub fin] have "\<dots> = prod f (A \<union> (B-A))"
     by (simp add: prod.union_disjoint del: Un_Diff_cancel)
   also from sub have "A \<union> (B-A) = B" by blast
   finally show ?thesis .
 qed
 
 lemma less_1_prod:
   fixes f :: "'a \<Rightarrow> 'b::linordered_idom"
   shows "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> 1 < f i) \<Longrightarrow> 1 < prod f I"
   by (induct I rule: finite_ne_induct) (auto intro: less_1_mult)
 
 lemma less_1_prod2:
   fixes f :: "'a \<Rightarrow> 'b::linordered_idom"
   assumes I: "finite I" "i \<in> I" "1 < f i" "\<And>i. i \<in> I \<Longrightarrow> 1 \<le> f i"
   shows "1 < prod f I"
 proof -
   have "1 < f i * prod f (I - {i})"
     using assms
     by (meson DiffD1 leI less_1_mult less_le_trans mult_le_cancel_left1 prod_ge_1)
   also have "\<dots> = prod f I"
     using assms by (simp add: prod.remove)
   finally show ?thesis .
 qed
 
 lemma (in linordered_field) abs_prod: "\<bar>prod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
   by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult)
 
 lemma prod_eq_1_iff [simp]: "finite A \<Longrightarrow> prod 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 prod_pos_nat_iff [simp]: "finite A \<Longrightarrow> prod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > 0)"
   for f :: "'a \<Rightarrow> nat"
   using prod_zero_iff by (simp del: neq0_conv add: zero_less_iff_neq_zero)
 
 lemma prod_constant [simp]: "(\<Prod>x\<in> A. y) = y ^ card A"
   for y :: "'a::comm_monoid_mult"
   by (induct A rule: infinite_finite_induct) simp_all
 
 lemma prod_power_distrib: "prod f A ^ n = prod (\<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_sum: "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 prod_gen_delta:
   fixes b :: "'b \<Rightarrow> 'a::comm_monoid_mult"
   assumes fin: "finite S"
   shows "prod (\<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)"
   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: prod_constant)
   next
     case True
     let ?A = "S - {a}"
     let ?B = "{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: "prod ?f ?A = prod (\<lambda>i. c) ?A"
       by (rule prod.cong) auto
     from fin True have card_A: "card ?A = card S - 1" by auto
     have f_A1: "prod ?f ?A = c ^ card ?A"
       unfolding f_A0 by (rule prod_constant)
     have "prod ?f ?A * prod ?f ?B = prod ?f S"
       using prod.union_disjoint[OF fin' disjoint, of ?f, unfolded eq[symmetric]]
       by simp
     with True card_A show ?thesis
       by (simp add: f_A1 field_simps cong add: prod.cong cong del: if_weak_cong)
   qed
 qed
 
 lemma sum_image_le:
   fixes g :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"
   assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> g(f i)"
     shows "sum g (f ` I) \<le> sum (g \<circ> f) I"
   using assms
 proof induction
   case empty
   then show ?case by auto
 next
   case (insert x F)
   from insertI1 have "0 \<le> g (f x)" by (rule insert)
   hence 1: "sum g (f ` F) \<le> g (f x) + sum g (f ` F)" using add_increasing by blast
   have 2: "sum g (f ` F) \<le> sum (g \<circ> f) F" using insert by blast
   have "sum g (f ` insert x F) = sum g (insert (f x) (f ` F))" by simp
   also have "\<dots> \<le> g (f x) + sum g (f ` F)" by (simp add: 1 insert sum.insert_if)
   also from 2 have "\<dots> \<le> g (f x) + sum (g \<circ> f) F" by (rule add_left_mono)
   also from insert(1, 2) have "\<dots> = sum (g \<circ> f) (insert x F)" by (simp add: sum.insert_if)
   finally show ?case .
 qed
 
 end
diff --git a/src/HOL/Library/Infinite_Set.thy b/src/HOL/Library/Infinite_Set.thy
--- a/src/HOL/Library/Infinite_Set.thy
+++ b/src/HOL/Library/Infinite_Set.thy
@@ -1,588 +1,590 @@
 (*  Title:      HOL/Library/Infinite_Set.thy
     Author:     Stephan Merz
 *)
 
 section \<open>Infinite Sets and Related Concepts\<close>
 
 theory Infinite_Set
   imports Main
 begin
 
 subsection \<open>The set of natural numbers is infinite\<close>
 
 lemma infinite_nat_iff_unbounded_le: "infinite S \<longleftrightarrow> (\<forall>m. \<exists>n\<ge>m. n \<in> S)"
   for S :: "nat set"
   using frequently_cofinite[of "\<lambda>x. x \<in> S"]
   by (simp add: cofinite_eq_sequentially frequently_def eventually_sequentially)
 
 lemma infinite_nat_iff_unbounded: "infinite S \<longleftrightarrow> (\<forall>m. \<exists>n>m. n \<in> S)"
   for S :: "nat set"
   using frequently_cofinite[of "\<lambda>x. x \<in> S"]
   by (simp add: cofinite_eq_sequentially frequently_def eventually_at_top_dense)
 
 lemma finite_nat_iff_bounded: "finite S \<longleftrightarrow> (\<exists>k. S \<subseteq> {..<k})"
   for S :: "nat set"
   using infinite_nat_iff_unbounded_le[of S] by (simp add: subset_eq) (metis not_le)
 
 lemma finite_nat_iff_bounded_le: "finite S \<longleftrightarrow> (\<exists>k. S \<subseteq> {.. k})"
   for S :: "nat set"
   using infinite_nat_iff_unbounded[of S] by (simp add: subset_eq) (metis not_le)
 
 lemma finite_nat_bounded: "finite S \<Longrightarrow> \<exists>k. S \<subseteq> {..<k}"
   for S :: "nat set"
   by (simp add: finite_nat_iff_bounded)
 
 
 text \<open>
   For a set of natural numbers to be infinite, it is enough to know
   that for any number larger than some \<open>k\<close>, there is some larger
   number that is an element of the set.
 \<close>
 
 lemma unbounded_k_infinite: "\<forall>m>k. \<exists>n>m. n \<in> S \<Longrightarrow> infinite (S::nat set)"
   apply (clarsimp simp add: finite_nat_set_iff_bounded)
   apply (drule_tac x="Suc (max m k)" in spec)
   using less_Suc_eq apply fastforce
   done
 
 lemma nat_not_finite: "finite (UNIV::nat set) \<Longrightarrow> R"
   by simp
 
 lemma range_inj_infinite:
   fixes f :: "nat \<Rightarrow> 'a"
   assumes "inj f"
   shows "infinite (range f)"
 proof
   assume "finite (range f)"
   from this assms have "finite (UNIV::nat set)"
     by (rule finite_imageD)
   then show False by simp
 qed
 
 
 subsection \<open>The set of integers is also infinite\<close>
 
 lemma infinite_int_iff_infinite_nat_abs: "infinite S \<longleftrightarrow> infinite ((nat \<circ> abs) ` S)"
   for S :: "int set"
 proof (unfold Not_eq_iff, rule iffI)
   assume "finite ((nat \<circ> abs) ` S)"
   then have "finite (nat ` (abs ` S))"
     by (simp add: image_image cong: image_cong)
   moreover have "inj_on nat (abs ` S)"
     by (rule inj_onI) auto
   ultimately have "finite (abs ` S)"
     by (rule finite_imageD)
   then show "finite S"
     by (rule finite_image_absD)
 qed simp
 
 proposition infinite_int_iff_unbounded_le: "infinite S \<longleftrightarrow> (\<forall>m. \<exists>n. \<bar>n\<bar> \<ge> m \<and> n \<in> S)"
   for S :: "int set"
   by (simp add: infinite_int_iff_infinite_nat_abs infinite_nat_iff_unbounded_le o_def image_def)
     (metis abs_ge_zero nat_le_eq_zle le_nat_iff)
 
 proposition infinite_int_iff_unbounded: "infinite S \<longleftrightarrow> (\<forall>m. \<exists>n. \<bar>n\<bar> > m \<and> n \<in> S)"
   for S :: "int set"
   by (simp add: infinite_int_iff_infinite_nat_abs infinite_nat_iff_unbounded o_def image_def)
     (metis (full_types) nat_le_iff nat_mono not_le)
 
 proposition finite_int_iff_bounded: "finite S \<longleftrightarrow> (\<exists>k. abs ` S \<subseteq> {..<k})"
   for S :: "int set"
   using infinite_int_iff_unbounded_le[of S] by (simp add: subset_eq) (metis not_le)
 
 proposition finite_int_iff_bounded_le: "finite S \<longleftrightarrow> (\<exists>k. abs ` S \<subseteq> {.. k})"
   for S :: "int set"
   using infinite_int_iff_unbounded[of S] by (simp add: subset_eq) (metis not_le)
 
 
 subsection \<open>Infinitely Many and Almost All\<close>
 
 text \<open>
   We often need to reason about the existence of infinitely many
   (resp., all but finitely many) objects satisfying some predicate, so
   we introduce corresponding binders and their proof rules.
 \<close>
 
 lemma not_INFM [simp]: "\<not> (INFM x. P x) \<longleftrightarrow> (MOST x. \<not> P x)"
   by (rule not_frequently)
 
 lemma not_MOST [simp]: "\<not> (MOST x. P x) \<longleftrightarrow> (INFM x. \<not> P x)"
   by (rule not_eventually)
 
 lemma INFM_const [simp]: "(INFM x::'a. P) \<longleftrightarrow> P \<and> infinite (UNIV::'a set)"
   by (simp add: frequently_const_iff)
 
 lemma MOST_const [simp]: "(MOST x::'a. P) \<longleftrightarrow> P \<or> finite (UNIV::'a set)"
   by (simp add: eventually_const_iff)
 
 lemma INFM_imp_distrib: "(INFM x. P x \<longrightarrow> Q x) \<longleftrightarrow> ((MOST x. P x) \<longrightarrow> (INFM x. Q x))"
   by (rule frequently_imp_iff)
 
 lemma MOST_imp_iff: "MOST x. P x \<Longrightarrow> (MOST x. P x \<longrightarrow> Q x) \<longleftrightarrow> (MOST x. Q x)"
   by (auto intro: eventually_rev_mp eventually_mono)
 
 lemma INFM_conjI: "INFM x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> INFM x. P x \<and> Q x"
   by (rule frequently_rev_mp[of P]) (auto elim: eventually_mono)
 
 
 text \<open>Properties of quantifiers with injective functions.\<close>
 
 lemma INFM_inj: "INFM x. P (f x) \<Longrightarrow> inj f \<Longrightarrow> INFM x. P x"
   using finite_vimageI[of "{x. P x}" f] by (auto simp: frequently_cofinite)
 
 lemma MOST_inj: "MOST x. P x \<Longrightarrow> inj f \<Longrightarrow> MOST x. P (f x)"
   using finite_vimageI[of "{x. \<not> P x}" f] by (auto simp: eventually_cofinite)
 
 
 text \<open>Properties of quantifiers with singletons.\<close>
 
 lemma not_INFM_eq [simp]:
   "\<not> (INFM x. x = a)"
   "\<not> (INFM x. a = x)"
   unfolding frequently_cofinite by simp_all
 
 lemma MOST_neq [simp]:
   "MOST x. x \<noteq> a"
   "MOST x. a \<noteq> x"
   unfolding eventually_cofinite by simp_all
 
 lemma INFM_neq [simp]:
   "(INFM x::'a. x \<noteq> a) \<longleftrightarrow> infinite (UNIV::'a set)"
   "(INFM x::'a. a \<noteq> x) \<longleftrightarrow> infinite (UNIV::'a set)"
   unfolding frequently_cofinite by simp_all
 
 lemma MOST_eq [simp]:
   "(MOST x::'a. x = a) \<longleftrightarrow> finite (UNIV::'a set)"
   "(MOST x::'a. a = x) \<longleftrightarrow> finite (UNIV::'a set)"
   unfolding eventually_cofinite by simp_all
 
 lemma MOST_eq_imp:
   "MOST x. x = a \<longrightarrow> P x"
   "MOST x. a = x \<longrightarrow> P x"
   unfolding eventually_cofinite by simp_all
 
 
 text \<open>Properties of quantifiers over the naturals.\<close>
 
 lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P n) \<longleftrightarrow> (\<exists>m. \<forall>n>m. P n)"
   for P :: "nat \<Rightarrow> bool"
   by (auto simp add: eventually_cofinite finite_nat_iff_bounded_le subset_eq simp flip: not_le)
 
 lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P n) \<longleftrightarrow> (\<exists>m. \<forall>n\<ge>m. P n)"
   for P :: "nat \<Rightarrow> bool"
   by (auto simp add: eventually_cofinite finite_nat_iff_bounded subset_eq simp flip: not_le)
 
 lemma INFM_nat: "(\<exists>\<^sub>\<infinity>n. P n) \<longleftrightarrow> (\<forall>m. \<exists>n>m. P n)"
   for P :: "nat \<Rightarrow> bool"
   by (simp add: frequently_cofinite infinite_nat_iff_unbounded)
 
 lemma INFM_nat_le: "(\<exists>\<^sub>\<infinity>n. P n) \<longleftrightarrow> (\<forall>m. \<exists>n\<ge>m. P n)"
   for P :: "nat \<Rightarrow> bool"
   by (simp add: frequently_cofinite infinite_nat_iff_unbounded_le)
 
 lemma MOST_INFM: "infinite (UNIV::'a set) \<Longrightarrow> MOST x::'a. P x \<Longrightarrow> INFM x::'a. P x"
   by (simp add: eventually_frequently)
 
 lemma MOST_Suc_iff: "(MOST n. P (Suc n)) \<longleftrightarrow> (MOST n. P n)"
   by (simp add: cofinite_eq_sequentially)
 
 lemma MOST_SucI: "MOST n. P n \<Longrightarrow> MOST n. P (Suc n)"
   and MOST_SucD: "MOST n. P (Suc n) \<Longrightarrow> MOST n. P n"
   by (simp_all add: MOST_Suc_iff)
 
 lemma MOST_ge_nat: "MOST n::nat. m \<le> n"
   by (simp add: cofinite_eq_sequentially)
 
 \<comment> \<open>legacy names\<close>
 lemma Inf_many_def: "Inf_many P \<longleftrightarrow> infinite {x. P x}" by (fact frequently_cofinite)
 lemma Alm_all_def: "Alm_all P \<longleftrightarrow> \<not> (INFM x. \<not> P x)" by simp
 lemma INFM_iff_infinite: "(INFM x. P x) \<longleftrightarrow> infinite {x. P x}" by (fact frequently_cofinite)
 lemma MOST_iff_cofinite: "(MOST x. P x) \<longleftrightarrow> finite {x. \<not> P x}" by (fact eventually_cofinite)
 lemma INFM_EX: "(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)" by (fact frequently_ex)
 lemma ALL_MOST: "\<forall>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x" by (fact always_eventually)
 lemma INFM_mono: "\<exists>\<^sub>\<infinity>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> \<exists>\<^sub>\<infinity>x. Q x" by (fact frequently_elim1)
 lemma MOST_mono: "\<forall>\<^sub>\<infinity>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x" by (fact eventually_mono)
 lemma INFM_disj_distrib: "(\<exists>\<^sub>\<infinity>x. P x \<or> Q x) \<longleftrightarrow> (\<exists>\<^sub>\<infinity>x. P x) \<or> (\<exists>\<^sub>\<infinity>x. Q x)" by (fact frequently_disj_iff)
 lemma MOST_rev_mp: "\<forall>\<^sub>\<infinity>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x \<longrightarrow> Q x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x" by (fact eventually_rev_mp)
 lemma MOST_conj_distrib: "(\<forall>\<^sub>\<infinity>x. P x \<and> Q x) \<longleftrightarrow> (\<forall>\<^sub>\<infinity>x. P x) \<and> (\<forall>\<^sub>\<infinity>x. Q x)" by (fact eventually_conj_iff)
 lemma MOST_conjI: "MOST x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> MOST x. P x \<and> Q x" by (fact eventually_conj)
 lemma INFM_finite_Bex_distrib: "finite A \<Longrightarrow> (INFM y. \<exists>x\<in>A. P x y) \<longleftrightarrow> (\<exists>x\<in>A. INFM y. P x y)" by (fact frequently_bex_finite_distrib)
 lemma MOST_finite_Ball_distrib: "finite A \<Longrightarrow> (MOST y. \<forall>x\<in>A. P x y) \<longleftrightarrow> (\<forall>x\<in>A. MOST y. P x y)" by (fact eventually_ball_finite_distrib)
 lemma INFM_E: "INFM x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> thesis) \<Longrightarrow> thesis" by (fact frequentlyE)
 lemma MOST_I: "(\<And>x. P x) \<Longrightarrow> MOST x. P x" by (rule eventuallyI)
 lemmas MOST_iff_finiteNeg = MOST_iff_cofinite
 
 
 subsection \<open>Enumeration of an Infinite Set\<close>
 
 text \<open>The set's element type must be wellordered (e.g. the natural numbers).\<close>
 
 text \<open>
   Could be generalized to
     \<^prop>\<open>enumerate' S n = (SOME t. t \<in> s \<and> finite {s\<in>S. s < t} \<and> card {s\<in>S. s < t} = n)\<close>.
 \<close>
 
 primrec (in wellorder) enumerate :: "'a set \<Rightarrow> nat \<Rightarrow> 'a"
   where
     enumerate_0: "enumerate S 0 = (LEAST n. n \<in> S)"
   | enumerate_Suc: "enumerate S (Suc n) = enumerate (S - {LEAST n. n \<in> S}) n"
 
 lemma enumerate_Suc': "enumerate S (Suc n) = enumerate (S - {enumerate S 0}) n"
   by simp
 
 lemma enumerate_in_set: "infinite S \<Longrightarrow> enumerate S n \<in> S"
 proof (induct n arbitrary: S)
   case 0
   then show ?case
     by (fastforce intro: LeastI dest!: infinite_imp_nonempty)
 next
   case (Suc n)
   then show ?case
     by simp (metis DiffE infinite_remove)
 qed
 
 declare enumerate_0 [simp del] enumerate_Suc [simp del]
 
 lemma enumerate_step: "infinite S \<Longrightarrow> enumerate S n < enumerate S (Suc n)"
 proof (induction n arbitrary: S)
   case 0
   then have "enumerate S 0 \<le> enumerate S (Suc 0)"
     by (simp add: enumerate_0 Least_le enumerate_in_set)
   moreover have "enumerate (S - {enumerate S 0}) 0 \<in> S - {enumerate S 0}"
     by (meson "0.prems" enumerate_in_set infinite_remove)
   then have "enumerate S 0 \<noteq> enumerate (S - {enumerate S 0}) 0"
     by auto
   ultimately show ?case
     by (simp add: enumerate_Suc')
 next
   case (Suc n)
   then show ?case 
     by (simp add: enumerate_Suc')
 qed
 
 lemma enumerate_mono: "m < n \<Longrightarrow> infinite S \<Longrightarrow> enumerate S m < enumerate S n"
   by (induct m n rule: less_Suc_induct) (auto intro: enumerate_step)
 
 lemma le_enumerate:
   assumes S: "infinite S"
   shows "n \<le> enumerate S n"
   using S
 proof (induct n)
   case 0
   then show ?case by simp
 next
   case (Suc n)
   then have "n \<le> enumerate S n" by simp
   also note enumerate_mono[of n "Suc n", OF _ \<open>infinite S\<close>]
   finally show ?case by simp
 qed
 
 lemma infinite_enumerate:
   assumes fS: "infinite S"
   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. r n \<in> S)"
   unfolding strict_mono_def
   using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
 
 lemma enumerate_Suc'':
   fixes S :: "'a::wellorder set"
   assumes "infinite S"
   shows "enumerate S (Suc n) = (LEAST s. s \<in> S \<and> enumerate S n < s)"
   using assms
 proof (induct n arbitrary: S)
   case 0
   then have "\<forall>s \<in> S. enumerate S 0 \<le> s"
     by (auto simp: enumerate.simps intro: Least_le)
   then show ?case
     unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"]
     by (intro arg_cong[where f = Least] ext) auto
 next
   case (Suc n S)
   show ?case
     using enumerate_mono[OF zero_less_Suc \<open>infinite S\<close>, of n] \<open>infinite S\<close>
     apply (subst (1 2) enumerate_Suc')
     apply (subst Suc)
      apply (use \<open>infinite S\<close> in simp)
     apply (intro arg_cong[where f = Least] ext)
     apply (auto simp flip: enumerate_Suc')
     done
 qed
 
 lemma enumerate_Ex:
   fixes S :: "nat set"
   assumes S: "infinite S"
     and s: "s \<in> S"
   shows "\<exists>n. enumerate S n = s"
   using s
 proof (induct s rule: less_induct)
   case (less s)
   show ?case
   proof (cases "\<exists>y\<in>S. y < s")
     case True
     let ?y = "Max {s'\<in>S. s' < s}"
     from True have y: "\<And>x. ?y < x \<longleftrightarrow> (\<forall>s'\<in>S. s' < s \<longrightarrow> s' < x)"
       by (subst Max_less_iff) auto
     then have y_in: "?y \<in> {s'\<in>S. s' < s}"
       by (intro Max_in) auto
     with less.hyps[of ?y] obtain n where "enumerate S n = ?y"
       by auto
     with S have "enumerate S (Suc n) = s"
       by (auto simp: y less enumerate_Suc'' intro!: Least_equality)
     then show ?thesis by auto
   next
     case False
     then have "\<forall>t\<in>S. s \<le> t" by auto
     with \<open>s \<in> S\<close> show ?thesis
       by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0)
   qed
 qed
 
 lemma inj_enumerate:
   fixes S :: "'a::wellorder set"
   assumes S: "infinite S"
   shows "inj (enumerate S)"
   unfolding inj_on_def
 proof clarsimp
   show "\<And>x y. enumerate S x = enumerate S y \<Longrightarrow> x = y"
     by (metis neq_iff enumerate_mono[OF _ \<open>infinite S\<close>]) 
 qed
 
 text \<open>To generalise this, we'd need a condition that all initial segments were finite\<close>
 lemma bij_enumerate:
   fixes S :: "nat set"
   assumes S: "infinite S"
   shows "bij_betw (enumerate S) UNIV S"
 proof -
   have "\<forall>s \<in> S. \<exists>i. enumerate S i = s"
     using enumerate_Ex[OF S] by auto
   moreover note \<open>infinite S\<close> inj_enumerate
   ultimately show ?thesis
     unfolding bij_betw_def by (auto intro: enumerate_in_set)
 qed
 
 text \<open>A pair of weird and wonderful lemmas from HOL Light.\<close>
 lemma finite_transitivity_chain:
   assumes "finite A"
     and R: "\<And>x. \<not> R x x" "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"
     and A: "\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> A \<and> R x y"
   shows "A = {}"
   using \<open>finite A\<close> A
 proof (induct A)
   case empty
   then show ?case by simp
 next
   case (insert a A)
   have False
     using R(1)[of a] R(2)[of _ a] insert(3,4) by blast   
   thus ?case ..
 qed
 
 corollary Union_maximal_sets:
   assumes "finite \<F>"
   shows "\<Union>{T \<in> \<F>. \<forall>U\<in>\<F>. \<not> T \<subset> U} = \<Union>\<F>"
     (is "?lhs = ?rhs")
 proof
   show "?lhs \<subseteq> ?rhs" by force
   show "?rhs \<subseteq> ?lhs"
   proof (rule Union_subsetI)
     fix S
     assume "S \<in> \<F>"
     have "{T \<in> \<F>. S \<subseteq> T} = {}"
       if "\<not> (\<exists>y. y \<in> {T \<in> \<F>. \<forall>U\<in>\<F>. \<not> T \<subset> U} \<and> S \<subseteq> y)"
     proof -
       have \<section>: "\<And>x. x \<in> \<F> \<and> S \<subseteq> x \<Longrightarrow> \<exists>y. y \<in> \<F> \<and> S \<subseteq> y \<and> x \<subset> y"
         using that by (blast intro: dual_order.trans psubset_imp_subset)
       show ?thesis
       proof (rule finite_transitivity_chain [of _ "\<lambda>T U. S \<subseteq> T \<and> T \<subset> U"])
       qed (use assms in \<open>auto intro: \<section>\<close>)
     qed
     with \<open>S \<in> \<F>\<close> show "\<exists>y. y \<in> {T \<in> \<F>. \<forall>U\<in>\<F>. \<not> T \<subset> U} \<and> S \<subseteq> y"
       by blast
   qed
 qed
 
 subsection \<open>Properties of @{term enumerate} on finite sets\<close>
 
 lemma finite_enumerate_in_set: "\<lbrakk>finite S; n < card S\<rbrakk> \<Longrightarrow> enumerate S n \<in> S"
 proof (induction n arbitrary: S)
   case 0
   then show ?case
     by (metis all_not_in_conv card_empty enumerate.simps(1) not_less0 wellorder_Least_lemma(1))
 next
   case (Suc n)
   show ?case
     using Suc.prems Suc.IH [of "S - {LEAST n. n \<in> S}"]
     apply (simp add: enumerate.simps)
     by (metis Diff_empty Diff_insert0 Suc_lessD card.remove less_Suc_eq)
 qed
 
 lemma finite_enumerate_step: "\<lbrakk>finite S; Suc n < card S\<rbrakk> \<Longrightarrow> enumerate S n < enumerate S (Suc n)"
 proof (induction n arbitrary: S)
   case 0
   then have "enumerate S 0 \<le> enumerate S (Suc 0)"
     by (simp add: Least_le enumerate.simps(1) finite_enumerate_in_set)
   moreover have "enumerate (S - {enumerate S 0}) 0 \<in> S - {enumerate S 0}"
     by (metis 0 Suc_lessD Suc_less_eq card_Suc_Diff1 enumerate_in_set finite_enumerate_in_set)
   then have "enumerate S 0 \<noteq> enumerate (S - {enumerate S 0}) 0"
     by auto
   ultimately show ?case
     by (simp add: enumerate_Suc')
 next
   case (Suc n)
   then show ?case
     by (simp add: enumerate_Suc' finite_enumerate_in_set)
 qed
 
 lemma finite_enumerate_mono: "\<lbrakk>m < n; finite S; n < card S\<rbrakk> \<Longrightarrow> enumerate S m < enumerate S n"
   by (induct m n rule: less_Suc_induct) (auto intro: finite_enumerate_step)
 
 
 lemma finite_le_enumerate:
   assumes "finite S" "n < card S"
   shows "n \<le> enumerate S n"
   using assms
 proof (induction n)
   case 0
   then show ?case by simp
 next
   case (Suc n)
   then have "n \<le> enumerate S n" by simp
   also note finite_enumerate_mono[of n "Suc n", OF _ \<open>finite S\<close>]
   finally show ?case
     using Suc.prems(2) Suc_leI by blast
 qed
 
 lemma finite_enumerate:
   assumes fS: "finite S"
   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono_on r {..<card S} \<and> (\<forall>n<card S. r n \<in> S)"
   unfolding strict_mono_def
   using finite_enumerate_in_set[OF fS] finite_enumerate_mono[of _ _ S] fS
   by (metis lessThan_iff strict_mono_on_def)
 
 lemma finite_enumerate_Suc'':
   fixes S :: "'a::wellorder set"
   assumes "finite S" "Suc n < card S"
   shows "enumerate S (Suc n) = (LEAST s. s \<in> S \<and> enumerate S n < s)"
   using assms
 proof (induction n arbitrary: S)
   case 0
   then have "\<forall>s \<in> S. enumerate S 0 \<le> s"
     by (auto simp: enumerate.simps intro: Least_le)
   then show ?case
     unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"]
     by (metis Diff_iff dual_order.strict_iff_order singletonD singletonI)
 next
   case (Suc n S)
   then have "Suc n < card (S - {enumerate S 0})"
     using Suc.prems(2) finite_enumerate_in_set by force
   then show ?case
     apply (subst (1 2) enumerate_Suc')
     apply (simp add: Suc)
     apply (intro arg_cong[where f = Least] HOL.ext)
     using finite_enumerate_mono[OF zero_less_Suc \<open>finite S\<close>, of n] Suc.prems
     by (auto simp flip: enumerate_Suc')
 qed
 
 lemma finite_enumerate_initial_segment:
   fixes S :: "'a::wellorder set"
-  assumes "finite S" "s \<in> S" and n: "n < card (S \<inter> {..<s})"
+  assumes "finite S" and n: "n < card (S \<inter> {..<s})"
   shows "enumerate (S \<inter> {..<s}) n = enumerate S n"
   using n
 proof (induction n)
   case 0
   have "(LEAST n. n \<in> S \<and> n < s) = (LEAST n. n \<in> S)"
   proof (rule Least_equality)
     have "\<exists>t. t \<in> S \<and> t < s"
       by (metis "0" card_gt_0_iff disjoint_iff_not_equal lessThan_iff)
     then show "(LEAST n. n \<in> S) \<in> S \<and> (LEAST n. n \<in> S) < s"
       by (meson LeastI Least_le le_less_trans)
   qed (simp add: Least_le)
   then show ?case
     by (auto simp: enumerate_0)
 next
   case (Suc n)
   then have less_card: "Suc n < card S"
     by (meson assms(1) card_mono inf_sup_ord(1) leD le_less_linear order.trans)
-  obtain t where t: "t \<in> {s \<in> S. enumerate S n < s}"
+  obtain T where T: "T \<in> {s \<in> S. enumerate S n < s}"
     by (metis Infinite_Set.enumerate_step enumerate_in_set finite_enumerate_in_set finite_enumerate_step less_card mem_Collect_eq)
-  have "(LEAST x. x \<in> S \<and> x < s \<and> enumerate S n < x) = (LEAST s. s \<in> S \<and> enumerate S n < s)"
+  have "(LEAST x. x \<in> S \<and> x < s \<and> enumerate S n < x) = (LEAST x. x \<in> S \<and> enumerate S n < x)"
        (is "_ = ?r")
   proof (intro Least_equality conjI)
     show "?r \<in> S"
-      by (metis (mono_tags, lifting) LeastI mem_Collect_eq t)
-    show "?r < s"
-      using not_less_Least [of _ "\<lambda>t. t \<in> S \<and> enumerate S n < t"] Suc assms 
-      by (metis (no_types, lifting) Int_Collect Suc_lessD finite_Int finite_enumerate_in_set finite_enumerate_step lessThan_def linorder_cases)
+      by (metis (mono_tags, lifting) LeastI mem_Collect_eq T)
+    have "\<not> s \<le> ?r"
+      using not_less_Least [of _ "\<lambda>x. x \<in> S \<and> enumerate S n < x"] Suc assms
+      by (metis (mono_tags, lifting) Int_Collect Suc_lessD finite_Int finite_enumerate_in_set finite_enumerate_step lessThan_def less_le_trans)
+    then show "?r < s"
+      by auto
     show "enumerate S n < ?r"
-      by (metis (no_types, lifting) LeastI mem_Collect_eq t)
+      by (metis (no_types, lifting) LeastI mem_Collect_eq T)
   qed (auto simp: Least_le)
   then show ?case
     using Suc assms by (simp add: finite_enumerate_Suc'' less_card)
 qed
 
 lemma finite_enumerate_Ex:
   fixes S :: "'a::wellorder set"
   assumes S: "finite S"
     and s: "s \<in> S"
   shows "\<exists>n<card S. enumerate S n = s"
   using s S
 proof (induction s arbitrary: S rule: less_induct)
   case (less s)
   show ?case
   proof (cases "\<exists>y\<in>S. y < s")
     case True
     let ?T = "S \<inter> {..<s}"
     have "finite ?T"
       using less.prems(2) by blast
     have TS: "card ?T < card S"
       using less.prems by (blast intro: psubset_card_mono [OF \<open>finite S\<close>])
     from True have y: "\<And>x. Max ?T < x \<longleftrightarrow> (\<forall>s'\<in>S. s' < s \<longrightarrow> s' < x)"
       by (subst Max_less_iff) (auto simp: \<open>finite ?T\<close>)
     then have y_in: "Max ?T \<in> {s'\<in>S. s' < s}"
       using Max_in \<open>finite ?T\<close> by fastforce
     with less.IH[of "Max ?T" ?T] obtain n where n: "enumerate ?T n = Max ?T" "n < card ?T"
       using \<open>finite ?T\<close> by blast
     then have "Suc n < card S"
       using TS less_trans_Suc by blast
     with S n have "enumerate S (Suc n) = s"
       by (subst finite_enumerate_Suc'') (auto simp: y finite_enumerate_initial_segment less finite_enumerate_Suc'' intro!: Least_equality)
     then show ?thesis
       using \<open>Suc n < card S\<close> by blast
   next
     case False
     then have "\<forall>t\<in>S. s \<le> t" by auto
     moreover have "0 < card S"
       using card_0_eq less.prems by blast
     ultimately show ?thesis
       using \<open>s \<in> S\<close>
       by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0)
   qed
 qed
 
 lemma finite_bij_enumerate:
   fixes S :: "'a::wellorder set"
   assumes S: "finite S"
   shows "bij_betw (enumerate S) {..<card S} S"
 proof -
   have "\<And>n m. \<lbrakk>n \<noteq> m; n < card S; m < card S\<rbrakk> \<Longrightarrow> enumerate S n \<noteq> enumerate S m"
     using finite_enumerate_mono[OF _ \<open>finite S\<close>] by (auto simp: neq_iff)
   then have "inj_on (enumerate S) {..<card S}"
     by (auto simp: inj_on_def)
   moreover have "\<forall>s \<in> S. \<exists>i<card S. enumerate S i = s"
     using finite_enumerate_Ex[OF S] by auto
   moreover note \<open>finite S\<close>
   ultimately show ?thesis
     unfolding bij_betw_def by (auto intro: finite_enumerate_in_set)
 qed
 
 lemma ex_bij_betw_strict_mono_card:
   fixes M :: "'a::wellorder set"
   assumes "finite M" 
   obtains h where "bij_betw h {..<card M} M" and "strict_mono_on h {..<card M}"
 proof
   show "bij_betw (enumerate M) {..<card M} M"
     by (simp add: assms finite_bij_enumerate)
   show "strict_mono_on (enumerate M) {..<card M}"
     by (simp add: assms finite_enumerate_mono strict_mono_on_def)
 qed
 
 end