(*  Title:      HOL/Groups_Big.thy

    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel

                with contributions by Jeremy Avigad

*)

section \<open>Big sum and product over finite (non-empty) sets\<close>

theory Groups_Big

imports Finite_Set

begin

subsection \<open>Generic monoid operation over a set\<close>

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 \<open>x \<in> A\<close> obtain B where A: "A = insert x B" and "x \<notin> B"

    by (auto dest: mk_disjoint_insert)

  moreover from \<open>finite A\<close> A have "finite B" by simp

  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"

  -- \<open>The reversed orientation looks more natural, but LOOPS as a simprule!\<close>

using assms proof (induct A)

  case empty then show ?case by simp

next

  case (insert x A) then show ?case

    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)

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 union_diff2:

  assumes "finite A" and "finite B"

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

  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 reindex_cong:

  assumes "inj_on l B"

  assumes "A = l ` B"

  assumes "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x"

  shows "F g A = F h B"

  using assms by (simp add: reindex)

lemma UNION_disjoint:

  assumes "finite I" and "\<forall>i\<in>I. finite (A i)"

  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"

  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 \<circ> F) g C"

proof cases

  assume "finite C"

  from UNION_disjoint [OF this assms]

  show ?thesis by simp

qed (auto dest: finite_UnionD intro: infinite)

lemma distrib:

  "F (\<lambda>x. g x * 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 (case_prod g) (SIGMA x:A. B x)"

apply (subst Sigma_def)

apply (subst UNION_disjoint, assumption, simp)

 apply blast

apply (rule cong)

apply rule

apply (simp add: fun_eq_iff)

apply (subst UNION_disjoint, simp, simp)

 apply blast

apply (simp add: comp_def)

done

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 \<open>S \<subseteq> T\<close> by blast

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

  from \<open>finite T\<close> \<open>S \<subseteq> T\<close> have f: "finite S" "finite (T - S)"

    by (auto intro: finite_subset)

  show ?thesis using assms(4)

    by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])

qed

lemma mono_neutral_cong_right:

  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 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 \<open>finite S\<close> by (intro mono_neutral_cong_left) auto

    finally show ?thesis .

  qed simp

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) = 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 = 1}) (h ` A - h ` {x \<in> A. (g \<circ> h) x = 1})"

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

qed (insert \<open>finite A\<close>, 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: 

  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 (case_prod 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

lemma inter_restrict:

  assumes "finite A"

  shows "F g (A \<inter> B) = F (\<lambda>x. if x \<in> B then g x else 1) A"

proof -

  let ?g = "\<lambda>x. if x \<in> A \<inter> B then g x else 1"

  have "\<forall>i\<in>A - A \<inter> B. (if i \<in> A \<inter> B then g i else 1) = 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 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 = 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 = 1"

    and H: "F g (\<Union>B) = (F o F) g B" by auto

  then have "F g (A \<union> \<Union>B) = F g A * F g (\<Union>B)"

    by (simp add: union_inter_neutral)

  with \<open>finite B\<close> \<open>A \<notin> B\<close> show ?case

    by (simp add: H)

qed

lemma commute:

  "F (\<lambda>i. F (g i) B) A = F (\<lambda>j. F (\<lambda>i. g i j) A) B"

  unfolding cartesian_product

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

lemma commute_restrict:

  "finite A \<Longrightarrow> finite B \<Longrightarrow>

    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 commute)

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 * F (g \<circ> Inr) B"

proof -

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

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

    by auto

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

  moreover have "inj_on (Inl :: 'b \<Rightarrow> 'b + 'c) A" "inj_on (Inr :: 'c \<Rightarrow> 'b + 'c) B"

    by (auto intro: inj_onI)

  ultimately show ?thesis using fin

    by (simp add: union_disjoint reindex)

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 = 1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = 1"

  shows "F g A = F h B \<longleftrightarrow> F g C = F h C"

proof -

  from \<open>finite C\<close> subset have

    "finite A" and "finite B" and "finite (C - A)" and "finite (C - B)"

    by (auto elim: finite_subset)

  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)) * F g (C - A - A) * F g (A \<inter> (C - A))"

    using \<open>finite A\<close> \<open>finite (C - A)\<close> by (simp only: union_diff2)

  finally have P: "F g C = F g A" using trivial by simp

  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)) * F h (C - B - B) * F h (B \<inter> (C - B))"

    using \<open>finite B\<close> \<open>finite (C - B)\<close> by (simp only: union_diff2)

  finally have Q: "F h C = F h B" using trivial by simp

  from P Q show ?thesis by simp

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 = 1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = 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

end

notation times (infixl "*" 70)

notation Groups.one ("1")

subsection \<open>Generalized summation over a set\<close>

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

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

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"    ("(2\<Sum>_\<in>_./ _)" [0, 51, 10] 10)

syntax (HTML output)

  "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(2\<Sum>_\<in>_./ _)" [0, 51, 10] 10)

translations -- \<open>Beware of argument permutation!\<close>

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

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

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

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

syntax

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

syntax (xsymbols)

  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(2\<Sum>_ | (_)./ _)" [0,0,10] 10)

syntax (HTML output)

  "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(2\<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 \<open>

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

\<close>

text \<open>TODO generalization candidates\<close>

lemma setsum_image_gen:

  assumes fS: "finite S"

  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"

proof-

  { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }

  hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"

    by simp

  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"

    by (rule setsum.commute_restrict [OF fS finite_imageI [OF fS]])

  finally show ?thesis .

qed

subsubsection \<open>Properties in more restricted classes of structures\<close>

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

  (setsum f (A Un B) :: 'a :: ab_group_add) =

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

by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)

lemma setsum_Un2:

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

  shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"

proof -

  have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"

    by auto

  with assms show ?thesis by simp (subst setsum.union_disjoint, auto)+

qed

lemma setsum_diff1: "finite A \<Longrightarrow>

  (setsum f (A - {a}) :: ('a::ab_group_add)) =

  (if a:A then setsum f A - f a else setsum f A)"

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

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 \<open>a:A\<close>])

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

    using \<open>finite A\<close> by(subst setsum.union_disjoint) auto

  also have "setsum f (A-{a}) \<le> setsum g (A-{a})"

    by(rule setsum_mono)(simp add: assms(2))

  also have "setsum f {a} < setsum g {a}" using a by simp

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

    using \<open>finite A\<close> by(subst setsum.union_disjoint[symmetric]) auto

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

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

qed

lemma setsum_negf: "(\<Sum>x\<in>A. - f x::'a::ab_group_add) = - (\<Sum>x\<in>A. f x)"

proof (cases "finite A")

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

next

  case False thus ?thesis by simp

qed

lemma setsum_subtractf: "(\<Sum>x\<in>A. f x - g x::'a::ab_group_add) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"

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

lemma setsum_subtractf_nat:

  "(\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x) \<Longrightarrow> (\<Sum>x\<in>A. f x - g x::nat) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"

  by (induction A rule: infinite_finite_induct) (auto simp: setsum_mono)

lemma 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.union_disjoint del:Un_Diff_cancel)

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

  finally show ?thesis .

qed

lemma setsum_le_included:

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

  assumes "finite s" "finite t"

  and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"

  shows "setsum f s \<le> setsum g t"

proof -

  have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"

  proof (rule setsum_mono)

    fix y assume "y \<in> s"

    with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto

    with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")

      using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]

      by (auto intro!: setsum_mono2)

  qed

  also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"

    using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)

  also have "... \<le> setsum g t"

    using assms by (auto simp: setsum_image_gen[symmetric])

  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.union_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"

  by (simp add: setsum_nonneg)

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_ring: assumes "finite A" "a \<in> A"

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

  unfolding setsum.remove [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)"

  -- \<open>For the natural numbers, we have subtraction.\<close>

by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)

lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =

  (if a:A then setsum f A - f a else setsum f A)"

apply (case_tac "finite A")

 prefer 2 apply simp

apply (erule finite_induct)

 apply (auto simp add: insert_Diff_if)

apply (drule_tac a = a in mk_disjoint_insert, auto)

done

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

lemma (in comm_semiring_1) dvd_setsum:

  "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd setsum f A"

  by (induct A rule: infinite_finite_induct) simp_all

lemma setsum_pos2:

    assumes "finite I" "i \<in> I" "0 < f i" "(\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i)"

      shows "(0::'a::{ordered_ab_group_add,comm_monoid_add}) < setsum f I"

proof -

  have "0 \<le> setsum f (I-{i})"

    using assms by (simp add: setsum_nonneg)

  also have "... < setsum f (I-{i}) + f i"

    using assms by auto

  also have "... = setsum f I"

    using assms by (simp add: setsum.remove)

  finally show ?thesis .

qed

subsubsection \<open>Cardinality as special case of @{const setsum}\<close>

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_Suc: "setsum (\<lambda>x. Suc(f x)) A = setsum f A + card A"

  using setsum.distrib[of f "\<lambda>_. 1" A] 

  by simp

lemma setsum_bounded_above:

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

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

          "card A > 0"

  shows "setsum f A < of_nat (card A) * K"

using assms setsum_strict_mono[where A=A and g = "%x. K"]

by (simp add: card_gt_0_iff)

lemma setsum_bounded_below:

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

  shows "of_nat (card A) * K \<le> setsum f A"

proof (cases "finite A")

  case True

  thus ?thesis using le setsum_mono[where K=A and f = "%x. K"] by simp

next

  case False thus ?thesis by simp

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 -