(*  Title:      HOL/Groups_Big.thy

     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel

                 with contributions by Jeremy Avigad

 *)

 header {* Big sum and product over finite (non-empty) sets *}

 theory Groups_Big

 imports Finite_Set

 begin

 subsection {* Generic monoid operation over a set *}

 no_notation times (infixl "*" 70)

 no_notation Groups.one ("1")

 locale comm_monoid_set = comm_monoid

 begin

 interpretation comp_fun_commute f

   by default (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) 1 A"

 lemma infinite [simp]:

   "\<not> finite A \<Longrightarrow> F g A = 1"

   by (simp add: eq_fold)

 lemma empty [simp]:

   "F g {} = 1"

   by (simp add: eq_fold)

 lemma insert [simp]:

   assumes "finite A" and "x \<notin> A"

   shows "F g (insert x A) = g x * F g A"

   using assms by (simp add: eq_fold)

 lemma remove:

   assumes "finite A" and "x \<in> A"

   shows "F g A = g x * F g (A - {x})"

 proof -

   from `x \<in> A` obtain B where A: "A = insert x B" and "x \<notin> B"

     by (auto dest: mk_disjoint_insert)

   moreover from `finite A` A 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 * 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 = 1"

   shows "F g A = 1"

   using assms by (induct A rule: infinite_finite_induct) simp_all

 lemma neutral_const [simp]:

   "F (\<lambda>_. 1) A = 1"

   by (simp add: neutral)

 lemma union_inter:

   assumes "finite A" and "finite B"

   shows "F g (A \<union> B) * F g (A \<inter> B) = F g A * F g B"

   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}

 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)

 qed

 corollary union_inter_neutral:

   assumes "finite A" and "finite B"

   and I0: "\<forall>x \<in> A \<inter> B. g x = 1"

   shows "F g (A \<union> B) = F g A * 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 * F g B"

   using assms by (simp add: union_inter_neutral)

 lemma subset_diff:

   assumes "B \<subseteq> A" and "finite A"

   shows "F g A = F g (A - B) * 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) * F g B" by (rule union_disjoint)

   moreover from assms have "A \<union> B = A" by auto

   ultimately show ?thesis by simp

 qed

 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> z"

   obtains a where "a \<in> A" and "g a \<noteq> z"

 proof -

   from assms have "\<exists>a\<in>A. g a \<noteq> z"

   proof (induct A rule: infinite_finite_induct)

     case (insert a A)

     then show ?case by simp (rule, simp)

   qed simp_all

   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:

   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 `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)

 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 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

 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 (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 * h x) A = F g A * F h A"

   using assms 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 (split 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

 lemma related: 

   assumes Re: "R 1 1" 

   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" 

   and fS: "finite S" and Rfg: "\<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)

 lemma mono_neutral_cong_left:

   assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 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 `S \<subseteq> T` by blast

   have d: "S \<inter> (T - S) = {}" using `S \<subseteq> T` by blast

   from `finite T` `S \<subseteq> T` 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 = 1; \<And>x. x \<in> S \<Longrightarrow> g x = h x \<rbrakk>

    \<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 = 1 \<rbrakk> \<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 = 1 \<rbrakk> \<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)

 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

     assume "finite S"

     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 `finite S` by (intro mono_neutral_cong_left) auto

     finally show ?thesis .

   qed simp

 qed

 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: 

   assumes fS: "finite S"

   shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)"

 proof-

   let ?f = "(\<lambda>k. if k=a then b k else 1)"

   { assume a: "a \<notin> S"

     hence "\<forall>k\<in>S. ?f k = 1" by simp

     hence ?thesis  using a by simp }

   moreover

   { assume a: "a \<in> S"

     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 "F ?f S = F ?f ?A * 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

 qed

 lemma delta': 

   assumes fS: "finite S"

   shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"

   using delta [OF fS, 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}) * 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

   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 (split g) (A <*> 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

 end

 notation times (infixl "*" 70)

 notation Groups.one ("1")

 subsection {* Generalized summation over a set *}

 context comm_monoid_add

 begin

 definition setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"

 where

   "setsum = comm_monoid_set.F plus 0"

 sublocale setsum!: comm_monoid_set plus 0

 where

   "comm_monoid_set.F plus 0 = setsum"

 proof -

   show "comm_monoid_set plus 0" ..

   then interpret setsum!: comm_monoid_set plus 0 .

   from setsum_def show "comm_monoid_set.F plus 0 = setsum" by rule

 qed

 abbreviation

   Setsum ("\<Sum>_" [1000] 999) where

   "\<Sum>A \<equiv> setsum (%x. x) A"

 end

 text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is

 written @{text"\<Sum>x\<in>A. e"}. *}

 syntax

   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_. _)" [0, 51, 10] 10)

 syntax (xsymbols)

   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)

 syntax (HTML output)

   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)

 translations -- {* Beware of argument permutation! *}

   "SUM i:A. b" == "CONST setsum (%i. b) A"

   "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"

 text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter

  @{text"\<Sum>x|P. e"}. *}

 syntax

   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)

 syntax (xsymbols)

   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)

 syntax (HTML output)

   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)

 translations

   "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"

   "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"

 print_translation {*

 let

   fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) $ 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 "_qsetsum"} $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'

           end

     | setsum_tr' _ = raise Match;

 in [(@{const_syntax setsum}, K setsum_tr')] end

 *}

 text {* TODO These are candidates for generalization *}

 context comm_monoid_add

 begin

 lemma setsum_reindex_id: 

   "inj_on f B \<Longrightarrow> setsum f B = setsum id (f ` B)"

   by (simp add: setsum.reindex)

 lemma setsum_reindex_nonzero:

   assumes fS: "finite S"

   and nz: "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"

   shows "setsum h (f ` S) = setsum (h \<circ> f) S"

 proof (subst setsum.reindex_bij_betw_not_neutral[symmetric])

   show "bij_betw f (S - {x\<in>S. h (f x) = 0}) (f`S - f`{x\<in>S. h (f x) = 0})"

     using nz by (auto intro!: inj_onI simp: bij_betw_def)

 qed (insert fS, auto)

 lemma setsum_cong2:

   "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"

   by (auto intro: setsum.cong)

 lemma setsum_reindex_cong:

    "[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] 

     ==> setsum h B = setsum g A"

   by (simp add: setsum.reindex)

 lemma setsum_restrict_set:

   assumes fA: "finite A"

   shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"

 proof-

   from fA have fab: "finite (A \<inter> B)" by auto

   have aba: "A \<inter> B \<subseteq> A" by blast

   let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"

   from setsum.mono_neutral_left [OF fA aba, of ?g]

   show ?thesis by simp

 qed

 lemma setsum_Union_disjoint:

   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"

   shows "setsum f (Union C) = setsum (setsum f) C"

   using assms by (fact setsum.Union_disjoint)

 lemma setsum_cartesian_product:

   "(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)"

   by (fact setsum.cartesian_product)

 lemma setsum_UNION_zero:

   assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"

   and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"

   shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"

   using fSS f0

 proof(induct rule: finite_induct[OF fS])

   case 1 thus ?case by simp

 next

   case (2 T F)

   then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F" 

     and H: "setsum f (\<Union> F) = setsum (setsum f) F" by auto

   from fTF have fUF: "finite (\<Union>F)" by auto

   from "2.prems" TF fTF

   show ?case 

     by (auto simp add: H [symmetric] intro: setsum.union_inter_neutral [OF fTF(1) fUF, of f])

 qed

 text {* Commuting outer and inner summation *}

 lemma setsum_commute:

   "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"

   unfolding setsum_cartesian_product

   by (rule setsum.reindex_bij_witness[where i="\<lambda>(i, j). (j, i)" and j="\<lambda>(i, j). (j, i)"]) auto

 lemma setsum_Plus:

   fixes A :: "'a set" and B :: "'b set"

   assumes fin: "finite A" "finite B"

   shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"

 proof -

   have "A <+> B = Inl ` A \<union> Inr ` B" by auto

   moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"

     by auto

   moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto

   moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)

   ultimately show ?thesis using fin by(simp add: setsum.union_disjoint setsum.reindex)

 qed

 end

 text {* TODO These are legacy *}

 lemma setsum_empty:

   "setsum f {} = 0"

   by (fact setsum.empty)

 lemma setsum_insert:

   "finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"

   by (fact setsum.insert)

 lemma setsum_infinite:

   "~ finite A ==> setsum f A = 0"

   by (fact setsum.infinite)

 lemma setsum_reindex:

   "inj_on f B \<Longrightarrow> setsum h (f ` B) = setsum (h \<circ> f) B"

   by (fact setsum.reindex)

 lemma setsum_cong:

   "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"

   by (fact setsum.cong)

 lemma strong_setsum_cong:

   "A = B ==> (!!x. x:B =simp=> f x = g x)

    ==> setsum (%x. f x) A = setsum (%x. g x) B"

   by (fact setsum.strong_cong)

 lemmas setsum_0 = setsum.neutral_const

 lemmas setsum_0' = setsum.neutral

 lemma setsum_Un_Int: "finite A ==> finite B ==>

   setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"

   -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}

   by (fact setsum.union_inter)

 lemma setsum_Un_disjoint: "finite A ==> finite B

   ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"

   by (fact setsum.union_disjoint)

 lemma setsum_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>

     setsum f A = setsum f (A - B) + setsum f B"

   by (fact setsum.subset_diff)

 lemma setsum_mono_zero_left: 

   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 0 \<rbrakk> \<Longrightarrow> setsum f S = setsum f T"

   by (fact setsum.mono_neutral_left)

 lemmas setsum_mono_zero_right = setsum.mono_neutral_right

 lemma setsum_mono_zero_cong_left: 

   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 0; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>

   \<Longrightarrow> setsum f S = setsum g T"

   by (fact setsum.mono_neutral_cong_left)

 lemmas setsum_mono_zero_cong_right = setsum.mono_neutral_cong_right

 lemma setsum_delta: "finite S \<Longrightarrow>

   setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"

   by (fact setsum.delta)

 lemma setsum_delta': "finite S \<Longrightarrow>

   setsum (\<lambda>k. if a = k then b k else 0) S = (if a\<in> S then b a else 0)"

   by (fact setsum.delta')

 lemma setsum_cases:

   assumes "finite A"

   shows "setsum (\<lambda>x. if P x then f x else g x) A =

          setsum f (A \<inter> {x. P x}) + setsum g (A \<inter> - {x. P x})"

   using assms by (fact setsum.If_cases)

 (*But we can't get rid of finite I. If infinite, although the rhs is 0, 

   the lhs need not be, since UNION I A could still be finite.*)

 lemma setsum_UN_disjoint:

   assumes "finite I" and "ALL i:I. finite (A i)"

     and "ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}"

   shows "setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"

   using assms by (fact setsum.UNION_disjoint)

 (*But we can't get rid of finite A. If infinite, although the lhs is 0, 

   the rhs need not be, since SIGMA A B could still be finite.*)

 lemma setsum_Sigma:

   assumes "finite A" and  "ALL x:A. finite (B x)"

   shows "(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)"

   using assms by (fact setsum.Sigma)

 lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"

   by (fact setsum.distrib)

 lemma setsum_Un_zero:  

   "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 0 \<rbrakk> \<Longrightarrow>

   setsum f (S \<union> T) = setsum f S + setsum f T"

   by (fact setsum.union_inter_neutral)

 subsubsection {* Properties in more restricted classes of structures *}

 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_Un_Int [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_Un_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)

 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

   proof induct

     case empty

     thus ?case by auto

   next

     case (insert x F)

     thus ?case using le finiteB 

       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)

   qed

 qed

 lemma setsum_mono:

   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"

   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"

 proof (cases "finite K")

   case True

   thus ?thesis using le

   proof induct

     case empty

     thus ?case by simp

   next

     case insert

     thus ?case using add_mono by fastforce

   qed

 next

   case False then show ?thesis by simp

 qed

 lemma setsum_strict_mono:

   fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"

   assumes "finite A"  "A \<noteq> {}"

     and "!!x. x: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)

 qed

 lemma setsum_strict_mono_ex1:

 fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"

 assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a: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 `a:A`])

   also have "\<dots> = setsum f (A-{a}) + setsum f {a}"

     using `finite A` by(subst setsum_Un_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 `finite A` by(subst setsum_Un_disjoint[symmetric]) auto

   also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF `a:A`])

   finally show ?thesis by (auto simp add: add_right_mono add_strict_left_mono)

 qed

 lemma setsum_negf:

   "setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"

 proof (cases "finite A")

   case True thus ?thesis by (induct set: finite) auto

 next

   case False thus ?thesis by simp

 qed

 lemma setsum_subtractf:

   "setsum (%x. ((f x)::'a::ab_group_add) - g x) A =

     setsum f A - setsum g A"

   using setsum_addf [of f "- g" A] by (simp add: setsum_negf)

 lemma setsum_nonneg:

   assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"

   shows "0 \<le> setsum f A"

 proof (cases "finite A")

   case True thus ?thesis using nn

   proof induct

     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 setsum_nonpos:

   assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"

   shows "setsum f A \<le> 0"

 proof (cases "finite A")

   case True thus ?thesis using np

   proof induct

     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 setsum_nonneg_leq_bound:

   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"

   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 "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"

     using assms by (auto intro!: setsum_nonneg)

   moreover

   have "(\<Sum> i \<in> s - {i}. f i) + f i = B"

     using assms by (simp add: setsum_diff1)

   ultimately show ?thesis by auto

 qed

 lemma setsum_nonneg_0:

   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"

   assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"

   and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"

   shows "f i = 0"

   using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto

 lemma setsum_mono2:

 fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"

 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_Un_disjoint del:Un_Diff_cancel)

   also have "A \<union> (B-A) = B" using sub by blast

   finally show ?thesis .

 qed

 lemma setsum_mono3: "finite B ==> A <= B ==> 

     ALL x: B - A. 

       0 <= ((f x)::'a::{comm_monoid_add,ordered_ab_semigroup_add}) ==>

         setsum f A <= setsum f B"

   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")

   apply (erule ssubst)

   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")

   apply simp

   apply (rule add_left_mono)

   apply (erule setsum_nonneg)

   apply (subst setsum_Un_disjoint [THEN sym])

   apply (erule finite_subset, assumption)

   apply (rule finite_subset)

   prefer 2

   apply assumption

   apply (auto simp add: sup_absorb2)

 done

 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)"

 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)"

 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 "abs (setsum f A) \<le> setsum (%i. abs(f i)) 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 (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. abs(f i)) 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 auto

   qed

 next

   case False thus ?thesis by simp

 qed

 lemma abs_setsum_abs[simp]: 

   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"

   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"

 proof (cases "finite A")

   case True

   thus ?thesis

   proof induct

     case empty thus ?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

     finally show ?case .

   qed

 next

   case False thus ?thesis by simp

 qed

 lemma setsum_diff1'[rule_format]:

   "finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"

 apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])

 apply (auto simp add: insert_Diff_if add_ac)

 done

 lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"

   shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"

 unfolding setsum_diff1'[OF assms] by auto

 lemma setsum_product:

   fixes f :: "'a => ('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_0_iff [simp]:

   "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"

   by (induct set: finite) auto

 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

 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)"

   -- {* For the natural numbers, we have subtraction. *}

 by (subst setsum_Un_Int [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

 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

 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"

     by (simp add: setsum_diff1_nat)

   from xFinA 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

   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

   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

 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

 subsubsection {* Cardinality as special case of @{const setsum} *}

 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

 lemma setsum_bounded:

   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"

   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

 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_UN_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

 subsubsection {* Cardinality of products *}

 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)

 (*

 lemma SigmaI_insert: "y \<notin> A ==>

   (SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"

   by auto

 *)

 lemma card_cartesian_product: "card (A <*> 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} <*> A) = card(A)"

 by (simp add: card_cartesian_product)

 subsection {* Generalized product over a set *}

 context comm_monoid_mult

 begin

 definition setprod :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"

 where

   "setprod = comm_monoid_set.F times 1"

 sublocale setprod!: comm_monoid_set times 1

 where

   "comm_monoid_set.F times 1 = setprod"

 proof -

   show "comm_monoid_set times 1" ..

   then interpret setprod!: comm_monoid_set times 1 .

   from setprod_def show "comm_monoid_set.F times 1 = setprod" by rule

 qed

 abbreviation

   Setprod ("\<Prod>_" [1000] 999) where

   "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"

 end

 syntax

   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3PROD _:_. _)" [0, 51, 10] 10)

 syntax (xsymbols)

   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)

 syntax (HTML output)

   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)

 translations -- {* Beware of argument permutation! *}

   "PROD i:A. b" == "CONST setprod (%i. b) A" 

   "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A" 

 text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter

  @{text"\<Prod>x|P. e"}. *}

 syntax

   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)

 syntax (xsymbols)

   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)

 syntax (HTML output)

   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)

 translations

   "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"

   "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"

 text {* TODO These are candidates for generalization *}

 context comm_monoid_mult

 begin

 lemma setprod_reindex_id:

   "inj_on f B ==> setprod f B = setprod id (f ` B)"

   by (auto simp add: setprod.reindex)

 lemma setprod_reindex_cong:

   "inj_on f A ==> B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"

   by (frule setprod.reindex, simp)

 lemma strong_setprod_reindex_cong:

   "inj_on f A \<Longrightarrow> B = f ` A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x) \<Longrightarrow> setprod h B = setprod g A"

   by (subst setprod.reindex_bij_betw[symmetric, where h=f])

      (auto simp: bij_betw_def)

 lemma setprod_Union_disjoint:

   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}" 

   shows "setprod f (Union C) = setprod (setprod f) C"

   using assms by (fact setprod.Union_disjoint)

 text{*Here we can eliminate the finiteness assumptions, by cases.*}

 lemma setprod_cartesian_product:

   "(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"

   by (fact setprod.cartesian_product)

 lemma setprod_Un2:

   assumes "finite (A \<union> B)"

   shows "setprod f (A \<union> B) = setprod f (A - B) * setprod f (B - A) * setprod 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 setprod.union_disjoint, auto)+

 qed

 end

 text {* TODO These are legacy *}

 lemma setprod_empty: "setprod f {} = 1"

   by (fact setprod.empty)

 lemma setprod_insert: "[| finite A; a \<notin> A |] ==>

     setprod f (insert a A) = f a * setprod f A"

   by (fact setprod.insert)

 lemma setprod_infinite: "~ finite A ==> setprod f A = 1"

   by (fact setprod.infinite)

 lemma setprod_reindex:

   "inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"

   by (fact setprod.reindex)

 lemma setprod_cong:

   "A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"

   by (fact setprod.cong)

 lemma strong_setprod_cong:

   "A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"

   by (fact setprod.strong_cong)

 lemma setprod_Un_one:

   "\<lbrakk> finite S; finite T; \<forall>x \<in> S\<inter>T. f x = 1 \<rbrakk>

   \<Longrightarrow> setprod f (S \<union> T) = setprod f S  * setprod f T"

   by (fact setprod.union_inter_neutral)

 lemmas setprod_1 = setprod.neutral_const

 lemmas setprod_1' = setprod.neutral

 lemma setprod_Un_Int: "finite A ==> finite B

     ==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"

   by (fact setprod.union_inter)

 lemma setprod_Un_disjoint: "finite A ==> finite B

   ==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"

   by (fact setprod.union_disjoint)

 lemma setprod_subset_diff: "\<lbrakk> B \<subseteq> A; finite A \<rbrakk> \<Longrightarrow>

     setprod f A = setprod f (A - B) * setprod f B"

   by (fact setprod.subset_diff)

 lemma setprod_mono_one_left:

   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. f i = 1 \<rbrakk> \<Longrightarrow> setprod f S = setprod f T"

   by (fact setprod.mono_neutral_left)

 lemmas setprod_mono_one_right = setprod.mono_neutral_right

 lemma setprod_mono_one_cong_left: 

   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> f x = g x \<rbrakk>

   \<Longrightarrow> setprod f S = setprod g T"

   by (fact setprod.mono_neutral_cong_left)

 lemmas setprod_mono_one_cong_right = setprod.mono_neutral_cong_right

 lemma setprod_delta: "finite S \<Longrightarrow>

   setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"

   by (fact setprod.delta)

 lemma setprod_delta': "finite S \<Longrightarrow>

   setprod (\<lambda>k. if a = k then b k else 1) S = (if a\<in> S then b a else 1)"

   by (fact setprod.delta')

 lemma setprod_UN_disjoint:

     "finite I ==> (ALL i:I. finite (A i)) ==>

         (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>

       setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"

   by (fact setprod.UNION_disjoint)

 lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>

     (\<Prod>x\<in>A. (\<Prod>y\<in> B x. f x y)) =

     (\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"

   by (fact setprod.Sigma)

 lemma setprod_timesf: "setprod (\<lambda>x. f x * g x) A = setprod f A * setprod g A"

   by (fact setprod.distrib)

 subsubsection {* Properties in more restricted classes of structures *}

 lemma setprod_zero:

      "finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"

 apply (induct set: finite, force, clarsimp)

 apply (erule disjE, auto)

 done

 lemma setprod_zero_iff[simp]: "finite A ==> 

   (setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =

   (EX x: A. f x = 0)"

 by (erule finite_induct, auto simp:no_zero_divisors)

 lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>

   (setprod f (A Un B) :: 'a ::{field})

    = setprod f A * setprod f B / setprod f (A Int B)"

 by (subst setprod_Un_Int [symmetric], auto)

 lemma setprod_nonneg [rule_format]:

    "(ALL x: A. (0::'a::linordered_semidom) \<le> f x) --> 0 \<le> setprod f A"

 by (cases "finite A", induct set: finite, simp_all)

 lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::linordered_semidom) < f x)

   --> 0 < setprod f A"

 by (cases "finite A", induct set: finite, simp_all)

 lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>

   (setprod f (A - {a}) :: 'a :: {field}) =

   (if a:A then setprod f A / f a else setprod f A)"

   by (erule finite_induct) (auto simp add: insert_Diff_if)

 lemma setprod_inversef: 

   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"

   shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"

 by (erule finite_induct) auto

 lemma setprod_dividef:

   fixes f :: "'b \<Rightarrow> 'a::field_inverse_zero"

   shows "finite A

     ==> setprod (%x. f x / g x) A = setprod f A / setprod g A"

 apply (subgoal_tac

          "setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")

 apply (erule ssubst)

 apply (subst divide_inverse)

 apply (subst setprod_timesf)

 apply (subst setprod_inversef, assumption+, rule refl)

 apply (rule setprod_cong, rule refl)

 apply (subst divide_inverse, auto)

 done

 lemma setprod_dvd_setprod [rule_format]: 

     "(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"

   apply (cases "finite A")

   apply (induct set: finite)

   apply (auto simp add: dvd_def)

   apply (rule_tac x = "k * ka" in exI)

   apply (simp add: algebra_simps)

 done

 lemma setprod_dvd_setprod_subset:

   "finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"

   apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")

   apply (unfold dvd_def, blast)

   apply (subst setprod_Un_disjoint [symmetric])

   apply (auto elim: finite_subset intro: setprod_cong)

 done

 lemma setprod_dvd_setprod_subset2:

   "finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow> 

       setprod f A dvd setprod g B"

   apply (rule dvd_trans)

   apply (rule setprod_dvd_setprod, erule (1) bspec)

   apply (erule (1) setprod_dvd_setprod_subset)

 done

 lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow> 

     (f i ::'a::comm_semiring_1) dvd setprod f A"

 by (induct set: finite) (auto intro: dvd_mult)

 lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow> 

     (d::'a::comm_semiring_1) dvd (SUM x : A. f x)"

   apply (cases "finite A")

   apply (induct set: finite)

   apply auto

 done

 lemma setprod_mono:

   fixes f :: "'a \<Rightarrow> 'b\<Colon>linordered_semidom"

   assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"

   shows "setprod f A \<le> setprod g A"

 proof (cases "finite A")

   case True

   hence ?thesis "setprod f A \<ge> 0" using subset_refl[of A]

   proof (induct A rule: finite_subset_induct)

     case (insert a F)

     thus "setprod f (insert a F) \<le> setprod g (insert a F)" "0 \<le> setprod f (insert a F)"

       unfolding setprod_insert[OF insert(1,3)]

       using assms[rule_format,OF insert(2)] insert

       by (auto intro: mult_mono)

   qed auto

   thus ?thesis by simp

 qed auto

 lemma abs_setprod:

   fixes f :: "'a \<Rightarrow> 'b\<Colon>{linordered_field,abs}"

   shows "abs (setprod f A) = setprod (\<lambda>x. abs (f x)) A"

 proof (cases "finite A")

   case True thus ?thesis

     by induct (auto simp add: field_simps abs_mult)

 qed auto

 lemma setprod_eq_1_iff [simp]:

   "finite F ==> setprod f F = 1 \<longleftrightarrow> (ALL a:F. f a = (1::nat))"

   by (induct set: finite) auto

 lemma setprod_pos_nat:

   "finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"

 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])

 lemma setprod_pos_nat_iff[simp]:

   "finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"

 using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])

 end