(*  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"

proof (cases "finite A")

  case True

  then have "\<And>C. C \<subseteq> A \<longrightarrow> (\<forall>x\<in>C. g x = h x) \<longrightarrow> F g C = F h C"

  proof induct

    case empty then show ?case by simp

  next

    case (insert x F) then show ?case apply -

    apply (simp add: subset_insert_iff, clarify)

    apply (subgoal_tac "finite C")

      prefer 2 apply (blast dest: finite_subset [rotated])

    apply (subgoal_tac "C = insert x (C - {x})")

      prefer 2 apply blast

    apply (erule ssubst)

    apply (simp add: Ball_def del: insert_Diff_single)

    done

  qed

  with `A = B` g_h show ?thesis by simp

next

  case False

  with `A = B` show ?thesis by simp

qed

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 eq_general:

  assumes h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y" 

  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"

  shows "F f1 S = F f2 S'"

proof-

  from h f12 have hS: "h ` S = S'" by blast

  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"

    from f12 h H  have "x = y" by auto }

  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast

  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto 

  from hS have "F f2 S' = F f2 (h ` S)" by simp

  also have "\<dots> = F (f2 o h) S" using reindex [OF hinj, of f2] .

  also have "\<dots> = F f1 S " using th cong [of _ _ "f2 o h" f1]

    by blast

  finally show ?thesis ..

qed

lemma eq_general_reverses:

  assumes kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"

  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"

  shows "F j S = F g T"

  (* metis solves it, but not yet available here *)

  apply (rule eq_general [of T S h g j])

  apply (rule ballI)

  apply (frule kh)

  apply (rule ex1I[])

  apply blast

  apply clarsimp

  apply (drule hk) apply simp

  apply (rule sym)

  apply (erule conjunct1[OF conjunct2[OF hk]])

  apply (rule ballI)

  apply (drule hk)

  apply blast

  done

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 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 ==> 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"

using nz proof (induct rule: finite_induct [OF fS])

  case 1 thus ?case by simp

next

  case (2 x F) 

  { assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto

    then obtain y where y: "y \<in> F" "f x = f y" by auto 

    from "2.hyps" y have xy: "x \<noteq> y" by auto

    from "2.prems" [of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp

    have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto

    also have "\<dots> = setsum (h o f) (insert x F)" 

      unfolding setsum.insert[OF `finite F` `x\<notin>F`]

      using h0

      apply (simp cong del: setsum.strong_cong)

      apply (rule "2.hyps"(3))

      apply (rule_tac y="y" in  "2.prems")

      apply simp_all

      done

    finally have ?case . }

  moreover

  { assume fxF: "f x \<notin> f ` F"

    have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" 

      using fxF "2.hyps" by simp 

    also have "\<dots> = setsum (h o f) (insert x F)"

      unfolding setsum.insert[OF `finite F` `x\<notin>F`]

      apply (simp cong del: setsum.strong_cong)

      apply (rule cong [OF refl [of "op + (h (f x))"]])

      apply (rule "2.hyps"(3))

      apply (rule_tac y="y" in  "2.prems")

      apply simp_all

      done

    finally have ?case . }

  ultimately show ?case by blast

qed

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)"

proof (simp add: setsum_cartesian_product)

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

    (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"

    (is "?s = _")

    apply (simp add: setsum.reindex [where h = "%(i, j). (j, i)"] swap_inj_on)

    apply (simp add: split_def)

    done

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

    (is "_ = ?t")

    apply (simp add: swap_product)

    done

  finally show "?s = ?t" .

qed

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)

lemma setsum_eq_general_reverses:

  assumes fS: "finite S" and fT: "finite T"

  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"

  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"

  shows "setsum f S = setsum g T"

  using kh hk by (fact setsum.eq_general_reverses)

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} *}