Quelle  Groups_Big.thy

(*  Title:      HOL/Groups_Big.thy
    Author:     Tobias Nipkow
    Author:     Lawrence C Paulson
    Author:     Markus Wenzel
    Author:     Jeremy Avigad
*)

section java.lang.NullPointerException

theory Groups_Big java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
  imports Groups_Big
begingin

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

locale comm_monoid_set = comm_monoid
begin

subsubsection ‹Standard sum or product indexed by a finite set›

interpretation comp_fun_commute f
  by standard (simp add: fun_eq_iff left_commute)

interpretation comp?: comp_fun_commute "f ∘ g"
  by (fact comp_comp_fun_commute)

definition F :: "('b ==> 'a) ==> 'b set ==> 'a"
  where eq_fold: "F g A = Finite_Set.fold (f ∘ g) 1 A"

lemma infinite
  by (simp add: eq_fold)

lemma 
  by ( F :: (b\Rightarrow'a) \Rightarrow' set> '"

lemma insert [simp]: "finite A ==> x ∉ A ==>g) java.lang.NullPointerException: Cannot invoke "java.lang.CharSequence.toString()" because "replacement" is null
bysimp

lemma remove:
  assumes "finite A" and "x ∈
  shows "F g A = g x * F g (A - {x})"
proof -
  from ‹
     (aut dest: mk_disjoint_insert)
  moreover from ‹
 show ?ths by simp
qed

lemma insert_remove: "java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
 by (cases "x ∈ A") (simp_all add: remove insert_absorb)

  insert_if: "finite A ==> F g (insert x A) = (if x ∈ A then F g A else g x * F g A)"
 by (cases "x ∈ A") (simp_all add: insert_absorb)

  neutral: "∀
 by (induct A rule: infinite_finite_inducfrom \openx 🚫

  neutral_const [simp]: "F (λ: remo insert_)
 by (simp add: neutral)

  union_inter:
 assumes "finite A" and "finite B"
 shows "F g (A ∪ B) * F g (A ∩\Longrightarrow(if x \<in 
 ― ‹The reversed orientation looks more natural, but LOOPS as a simprule!›
 using assms
  (induct A)
 case empty
 then show ?case by simp
 
 case (insert x A)
 then show ?case
 by (auto simp: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
 

  union_inter_neutral:
 assumes "finite A" and "finite B"
 and "∀x ∈ A ∩ B. g x = 1"
 shows "F g (A ∪ B) = F g A x\in)(simp_aadd:in)
 using assms by (simp add: union_inter [symmetric] neutral)

  union_disjoint:
 assumes "finite A" and "finite B"
 assumes "A ∩ B = {}"
 shows "F g (A ∪ F g A = \1
 using assms by (simp add: union_inter_neutral)

  union_diff2:
 assumes "finite A" and "finite B"
 shows "F g (A ∪ B) = F g (A - B) * F g (B - A) * F g (A ∩ B)"
  -
 have "A ∪
 by auto
 with assms show ?thesis
 by simp (subst union_disjoint, auto)+
 

  subset_diff:
 assumes "B ⊆ \^A = \\>1"
 shows "F g A = F g (A - B) * F g B"
  -
 from assms have "finite (A - B)" by auto
 moreover from assms have "finite B" by (rule finite_subset)
 moreover from assms have "(A - B) ∩ B = {}" by aut
 ultimately have "F g (A - B ∪
 moreover from assms have "A ∪
 ultimately show ?thesis by simp
 

  Int_Diff:
 assumes "finite A"
java.lang.NullPointerException: Cannot invoke "java.lang.CharSequence.toString()" because "replacement" is null
 by (subst subset_diff [where B = "A - B"]) (auto simp: Diff_Diff_Int assms)

  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)

  not_neutral_contains_not_neutral:
java.lang.NullPointerException: Cannot invoke "java.lang.CharSequence.toString()" because "replacement" is null
 obtains a where "a ∈ A" and "g a ≠ assmsiA)
  -
 case em
 proof (induct A rule: infinite_finite_induct)
 case infinite
 then sho henshow?casby simp
 next
 case empty
 then show ?case by simp
 next
 case (insert a A)java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 show ?case by fastf
 qed
 with that show thesis by blast
 

  reindex:
 assumes "inj_on h A"
 shows "F g (h ` A) = F (g ∘ h) A"
  (cases "finite A")
 case True
 with assms show ?thesis
 by (simp add: eq_fold fold_image comp_assoc)
 
 case False
 with assms have "¬ finite (h ` A)" by (blast dest: finite_imageD)
 with False show ?thesis by simp
 

  cong [fundef_cong]:
 assumes "A = B"
 assumes g_h: "∧qed
 shows "F g A = F h B"
 using g_h unfolding ‹
 by (induct B rule: infinite_finite_induct) auto

  cong_simp [cong]:
 "[ A = B; ∧g x = \^
  (rule cong) (simp_all add: simp_implies_def)

  reindex_cong:
 assumes "inj_on l B"
 assumes "A = l ` B"
 assumes "∧x. x ∈ B ==> g (l x) = h x"
 shows "F g A = F h B"
 using assms by (simp add: reindex)

  image_eq:
 assumes "inj_on g A"
 shows "F (λx. x) (g ` A) = F g A"
 using assms reindex_cong by fastforce

  UNION_disjoint:
java.lang.NullPointerException: Cannot invoke "java.lang.CharSequence.toString()" because "replacement" is null
 and "∀i∈I. ∀j∈I. i ≠ j ⟶ A i ∩ A j = {}" using assms by (simp add: union [symmetric] ne)
 " A" and"fi B"
 using assms
  (induction rule: finite_induct)
 case (insert i I)
 then have "\<forallassumes
 by blast
java.lang.NullPointerException: Cannot invoke "java.lang.CharSequence.toString()" because "replacement" is null
 by blast
 with insert show ?case
 by (simp add: union_disjoint)
  auto

  Union_disjoint:
 assumes "∀A∈C. finite A" "∀ (simp a: un)
 shows "F g (∪
  (cases "finiC")
 case True
 from UNION_disjoint [OF this assms] show ?thesis by simp
 
 case False
 then show ?thesis by (auto dest: finite_UnionD intro: infinite)
 

  distrib: "F (λx. g x * h x) A = F g A
 by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)

  Sigma:
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
 shows "F (λx. F (g x) (B x)) A = F (case_prod g) (SIGMA x:A. B x)"
 unfolding Sigma_def
  (subst UNION_disjoint)
 show "F (λx. F (g x) (B x)) A = F (λx. F (λ(x, y). g x y) (∪y∈B x. {(x, y)})) A"
 proof (rule cong [OF refl])
 show "F (g x) (B x) = F (λ(x, y). g x y) (∪y∈B x. {(x, y)})"
 if "x ∈ A" for x
 using that assms by (simp add: UNION_disjoint)
 qed
  (use assms in auto)

  related:
 assumes Re: "R 1 1"
 and Rop: "∀x1 y1 x2 y2. R x1 x2 ∧ R y1 y2 ⟶ R (x1 * y1) (x2 * y2)"
 and fin: "finite S"
 and R_h_g: "∀x∈S. R (h x) (g x)"
 shows "R (F h S) (F g S)"
 using fin by (rule finite_subset_induct) (use assms in auto)

  mono_neutral_cong_left:
 assumes "finite T"
 and "S ⊆ T"
 and "∀i ∈ T - S. h i = 1"
 and "∧x. x ∈ S ==> g x = h x"
 shows "F g S = F h T"
 -
 have eq: "T = S ∪ (T - S)" using ‹S ⊆ T› by blast
 have d: "S ∩ (T - S) = {}" using ‹S ⊆ T› by blast
 from ‹finite T› ‹S ⊆ 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)])
 

  mono_neutral_cong_right:
 "finite T ==> S ⊆ T ==> ∀i ∈ T - S. g i = 1 ==> (∧x. x ∈ S ==> g x = h x) ==>
 F g T = F h S"
 by (auto intro!: mono_neutral_cong_left [symmetric])

  mono_neutral_left: "finite T ==> S ⊆ T ==> ∀i ∈ T - S. g i = 1 ==> F g S = F g T"
 by (blast intro: mono_neutral_cong_left)

  mono_neutral_right: "finite T ==> S ⊆ T ==> ∀i ∈ T - S. g i = 1 ==> F g T = F g S"
 by (blast intro!: mono_neutral_left [symmetric])

  mono_neutral_cong:
 assumes [simp]: "finite T" "finite S"
 and *: "∧i. i ∈ T - S ==> h i = 1" "∧i. i ∈ S - T ==> g i = 1"
 and gh: "∧x. x ∈ S ∩ T ==> g x = h x"
 shows "F g S = F h T"
 -
 have "F g S = F g (S ∩ T)"
    by(rule mono_neutral_right)(auto intro: *)
  also have "… = F h (S ∩ T)" using refl gh by(rule cong)
  also have "… = F h T"
    by(rule mono_neutral_left)(auto intro: *)
  finally show ?thesis .
qed

lemma reindex_bij_betw: "bij_betw h S T ==> F (λx. g (h x)) S = F g T"
  by (auto simp: bij_betw_def reindex)

lemma reindex_bij_witness:
  assumes witness:
    "∧a. a ∈ S ==> i (j a) = a"
    "∧a. a ∈ S ==> j a ∈ T"
    "∧b. b ∈ T ==> j (i b) = b"
    "∧b. b ∈ T ==> i b ∈ S"
  assumes eq:
    "∧a. a ∈ S ==> 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 (λx. 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:
    "∧a. a ∈ S' ==> g (h a) = z"
    "∧b. b ∈ T' ==> g b = z"
  shows "F (λx. g (h x)) S = F g T"
proof -
  have [simp]: "finite S ⟷ finite T"
    using bij_betw_finite[OF bij] fin by auto
  show ?thesis
  proof (cases "finite S")
    case True
    with nn have "F (λx. g (h x)) S = F (λx. g (h x)) (S - S')"
      by (intro mono_neutral_cong_right) auto
    also have "… = F g (T - T')"
      using bij by (rule reindex_bij_betw)
    also have "… = F g T"
      using nn ‹finite S› by (intro mono_neutral_cong_left) auto
    finally show ?thesis .
  next
    case False
    then show ?thesis by simp
  qed
qed

lemma reindex_nontrivial:
  assumes "finite A"
    and nz: "∧x y. x ∈ A ==> y ∈ A ==> x ≠ y ==> h x = h y ==> g (h x) = 1"
  shows "F g (h ` A) = F (g ∘ h) A"
proof (subst reindex_bij_betw_not_neutral [symmetric])
  show "bij_betw h (A - {x ∈ A. (g ∘ h) x = 1}) (h ` A - h ` {x ∈ A. (g ∘ h) x = 1})"
    using nz by (auto intro!: inj_onI simp: bij_betw_def)
qed (use ‹finite A› in auto)

lemma reindex_bij_witness_not_neutral:
  assumes fin: "finite S'" "finite T'"
  assumes witness:
    "∧a. a ∈ S - S' ==> i (j a) = a"
    "∧a. a ∈ S - S' ==> j a ∈ T - T'"
    "∧b. b ∈ T - T' ==> j (i b) = b"
    "∧b. b ∈ T - T' ==> i b ∈ S - S'"
  assumes nn:
    "∧a. a ∈ S' ==> g a = z"
    "∧b. b ∈ T' ==> h b = z"
  assumes eq:
    "∧a. a ∈ S ==> h (j a) = g a"
  shows "F g S = F h T"
proof -
  have bij: "bij_betw j (S - (S' ∩ S)) (T - (T' ∩ T))"
    using witness by (intro bij_betw_byWitness[where f'=i]) auto
  have F_eq: "F g S = F (λx. h (j x)) S"
    by (intro cong) (auto simp: eq)
  show ?thesis
    unfolding F_eq using fin nn eq
    by (intro reindex_bij_betw_not_neutral[OF _ _ bij]) auto
qed

lemma delta_remove:
  assumes fS: "finite S"
  shows "F (λk. if k = a then b k else c k) S = (if a ∈ S then b a * F c (S-{a}) else F c (S-{a}))"
proof -
  let ?f = "(λk. if k = a then b k else c k)"
  show ?thesis
  proof (cases "a ∈ S")
    case False
    then have "∀k∈S. ?f k = c k" by simp
    with False show ?thesis by simp
  next
    case True
    let ?A = "S - {a}"
    let ?B = "{a}"
    from True have eq: "S = ?A ∪ ?B" by blast
    have dj: "?A ∩ ?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
    with True show ?thesis
      using comm_monoid_set.remove comm_monoid_set_axioms fS by fastforce
  qed
qed

lemma delta [simp]:
  assumes fS: "finite S"
  shows "F (λk. if k = a then b k else 1) S = (if a ∈ S then b a else 1)"
  by (simp add: delta_remove [OF assms])

lemma delta' [simp]:
  assumes fin: "finite S"
  shows "F (λk. if a = k then b k else 1) S = (if a ∈ S then b a else 1)"
  using delta [OF fin, of a b, symmetric] by (auto intro: cong)

lemma If_cases:
  fixes P :: "'b ==> bool" and g h :: "'b ==> 'a"
  assumes fin: "finite A"
  shows "F (λx. if P x then h x else g x) A = F h (A ∩ {x. P x}) * F g (A ∩ - {x. P x})"
proof -
  have a: "A = A ∩ {x. P x} ∪ A ∩ -{x. P x}" "(A ∩ {x. P x}) ∩ (A ∩ -{x. P x}) = {}"
    by blast+
  from fin have f: "finite (A ∩ {x. P x})" "finite (A ∩ -{x. P x})" by auto
  let ?g = "λx. 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 (λx. F (g x) B) A = F (case_prod g) (A × B)"
proof (cases "A = {} ∨ B = {}")
  case True
  then show ?thesis
    by auto
next
  case False
  then have "A ≠ {}" "B ≠ {}" by auto
  show ?thesis
  proof (cases "finite A ∧ finite B")
    case True
    then show ?thesis
      by (simp add: Sigma)
  next
    case False
    then consider "infinite A" | "infinite B" by auto
    then have "infinite (A × B)"
      by cases (use ‹A ≠ {}› ‹B ≠ {}› in ‹auto dest: finite_cartesian_productD1 finite_cartesian_productD2›)
    then show ?thesis
      using False by auto
  qed
qed

lemma cartesian_product':
  "F g (A × B) = F (λx. F (λy. g (x,y)) B) A"
  unfolding cartesian_product by simp


lemma inter_restrict:
  assumes "finite A"
  shows "F g (A ∩ B) = F (λx. if x ∈ B then g x else 1) A"
proof -
  let ?g = "λx. if x ∈ A ∩ B then g x else 1"
  have "∀i∈A - A ∩ B. (if i ∈ A ∩ B then g i else 1) = 1" by simp
  moreover have "A ∩ B ⊆ A" by blast
  ultimately have "F ?g (A ∩ B) = F ?g A"
    using ‹finite A› by (intro mono_neutral_left) auto
  then show ?thesis by simp
qed

lemma inter_filter:
  "finite A ==> F g {x ∈ A. P x} = F (λx. 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 "∀A ∈ B. finite A"
    and "∧A1 A2 x. A1 ∈ B ==> A2 ∈ B ==> A1 ≠ A2 ==> x ∈ A1 ==> x ∈ A2 ==> g x = 1"
  shows "F g (∪B) = (F ∘ F) g B"
  using assms
proof (induct B rule: infinite_finite_induct)
  case (infinite A)
  then have "¬ finite (∪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 (∪B)" "A ∉ B"
    and "∀x∈A ∩ ∪B. g x = 1"
    and H: "F g (∪B) = (F ∘ F) g B" by auto
  then have "F g (A ∪ ∪B) = F g A * F g (∪B)"
    by (simp add: union_inter_neutral)
  with ‹finite B› ‹A ∉ B› show ?case
    by (simp add: H)
qed

lemma swap: "F (λi. F (g i) B) A = F (λj. F (λi. g i j) A) B"
  unfolding cartesian_product
  by (rule reindex_bij_witness [where i = "λ(i, j). (j, i)" and j = "λ(i, j). (j, i)"]) auto

lemma swap_restrict:
  "finite A ==> finite B ==>
    F (λx. F (g x) {y. y ∈ B ∧ R x y}) A = F (λy. F (λx. g x y) {x. x ∈ A ∧ R x y}) B"
  by (simp add: inter_filter) (rule swap)

lemma image_gen:
  assumes fin: "finite S"
  shows "F h S = F (λy. F h {x. x ∈ S ∧ g x = y}) (g ` S)"
proof -
  have "{y. y∈ g`S ∧ g x = y} = {g x}" if "x ∈ S" for x
    using that by auto
  then have "F h S = F (λx. F (λy. h x) {y. y∈ g`S ∧ g x = y}) S"
    by simp
  also have "… = F (λy. F h {x. x ∈ S ∧ g x = y}) (g ` S)"
    by (rule swap_restrict [OF fin finite_imageI [OF fin]])
  finally show ?thesis .
qed

lemma group:
  assumes fS: "finite S" and fT: "finite T" and fST: "g ` S ⊆ T"
  shows "F (λy. F h {x. x ∈ S ∧ g x = y}) T = F h S"
  unfolding image_gen[OF fS, of h g]
  by (auto intro: neutral mono_neutral_right[OF fT fST])

lemma Plus:
  fixes A :: "'b set" and B :: "'c set"
  assumes fin: "finite A" "finite B"
  shows "F g (A <+> B) = F (g ∘ Inl) A * F (g ∘ Inr) B"
proof -
  have "A <+> B = Inl ` A ∪ Inr ` B" by auto
  moreover from fin have "finite (Inl ` A)" "finite (Inr ` B)" by auto
  moreover have "Inl ` A ∩ Inr ` B = {}" by auto
  moreover have "inj_on Inl A" "inj_on Inr B" by (auto intro: inj_onI)
  ultimately show ?thesis
    using fin by (simp add: union_disjoint reindex)
qed

lemma same_carrier:
  assumes "finite C"
  assumes subset: "A ⊆ C" "B ⊆ C"
  assumes trivial: "∧a. a ∈ C - A ==> g a = 1" "∧b. b ∈ C - B ==> h b = 1"
  shows "F g A = F h B ⟷ F g C = F h C"
proof -
  have "finite A" and "finite B" and "finite (C - A)" and "finite (C - B)"
    using ‹finite C› subset by (auto elim: finite_subset)
  from subset have [simp]: "A - (C - A) = A" by auto
  from subset have [simp]: "B - (C - B) = B" by auto
  from subset have "C = A ∪ (C - A)" by auto
  then have "F g C = F g (A ∪ (C - A))" by simp
  also have "… = F g (A - (C - A)) * F g (C - A - A) * F g (A ∩ (C - A))"
    using ‹finite A› ‹finite (C - A)› by (simp only: union_diff2)
  finally have *: "F g C = F g A" using trivial by simp
  from subset have "C = B ∪ (C - B)" by auto
  then have "F h C = F h (B ∪ (C - B))" by simp
  also have "… = F h (B - (C - B)) * F h (C - B - B) * F h (B ∩ (C - B))"
    using ‹finite B› ‹finite (C - B)› by (simp only: union_diff2)
  finally have "F h C = F h B"
    using trivial by simp
  with * show ?thesis by simp
qed

lemma same_carrierI:
  assumes "finite C"
  assumes subset: "A ⊆ C" "B ⊆ C"
  assumes trivial: "∧a. a ∈ C - A ==> g a = 1" "∧b. b ∈ C - B ==> 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

lemma eq_general:
  assumes B: "∧y. y ∈ B ==> ∃!x. x ∈ A ∧ h x = y" and A: "∧x. x ∈ A ==> h x ∈ B ∧ γ(h x) = φ x"
  shows "F φ A = F γ B"
proof -
  have eq: "B = h ` A"
    by (auto dest: assms)
  have h: "inj_on h A"
    using assms by (blast intro: inj_onI)
  have "F φ A = F (γ ∘ h) A"
    using A by auto
  also have "… = F γ B"
    by (simp add: eq reindex h)
  finally show ?thesis .
qed

lemma eq_general_inverses:
  assumes B: "∧y. y ∈ B ==> k y ∈ A ∧ h(k y) = y" and A: "∧x. x ∈ A ==> h x ∈ B ∧ k(h x) = x ∧ γ(h x) = φ x"
  shows "F φ A = F γ B"
  by (rule eq_general [where h=h]) (force intro: dest: A B)+

subsubsection ‹HOL Light variant: sum/product indexed by the non-neutral subset›
text ‹NB only a subset of the properties above are proved›

definition G :: "['b ==> 'a,'b set] ==> 'a"
  where "G p I ≡ if finite {x ∈ I. p x ≠ 1} then F p {x ∈ I. p x ≠ 1} else 1"

lemma finite_Collect_op:
  shows "[finite {i ∈ I. x i ≠ 1}; finite {i ∈ I. y i ≠ 1}] ==> finite {i ∈ I. x i * y i ≠ 1}"
  apply (rule finite_subset [where B = "{i ∈ I. x i ≠ 1} ∪ {i ∈ I. y i ≠ 1}"]) 
  using left_neutral by force+

lemma empty' [simp]: "G p {} = 1"
  by (auto simp: G_def)

lemma eq_sum [simp]: "finite I ==> G p I = F p I"
  by (auto simp: G_def intro: mono_neutral_cong_left)

lemma insert' [simp]:
  assumes "finite {x ∈ I. p x ≠ 1}"
  shows "G p (insert i I) = (if i ∈ I then G p I else p i * G p I)"
proof -
  have "{x. x = i ∧ p x ≠ 1 ∨ x ∈ I ∧ p x ≠ 1} = (if p i = 1 then {x ∈ I. p x ≠ 1} else insert i {x ∈ I. p x ≠ 1})"
    by auto
  then show ?thesis
    using assms by (simp add: G_def conj_disj_distribR insert_absorb)
qed

lemma distrib_triv':
  assumes "finite I"
  shows "G (λi. g i * h i) I = G g I * G h I"
  by (simp add: assms local.distrib)

lemma non_neutral': "G g {x ∈ I. g x ≠ 1} = G g I"
  by (simp add: G_def)

lemma distrib':
  assumes "finite {x ∈ I. g x ≠ 1}" "finite {x ∈ I. h x ≠ 1}"
  shows "G (λi. g i * h i) I = G g I * G h I"
proof -
  have "a * a ≠ a ==> a ≠ 1" for a
    by auto
  then have "G (λi. g i * h i) I = G (λi. g i * h i) ({i ∈ I. g i ≠ 1} ∪ {i ∈ I. h i ≠ 1})"
    using assms  by (force simp: G_def finite_Collect_op intro!: mono_neutral_cong)
  also have "… = G g I * G h I"
  proof -
    have "F g ({i ∈ I. g i ≠ -
          h (i \in> {i ∈bol
      by (auto simp: G_def assms intro: mono_neutral_right)
    then show ?thesis
      using assms by (simp add: distrib)
  qed
  finally show ?thesis .
qed

lemma cong':
  assumes "A = B"
  assumes g_h: "∧x. x ∈ B ==> g x = h x"
  shows "G g A = G h B"
  using assms by (auto simp: G_def cong: conj_cong intro: cong)


lemma mono_neutral_cong_left':
  assumes "S ⊆ T"
    and "∧i. i ∈ T - S ==> h i = 1"
    and "∧x. x ∈ S ==> g x = h x"
  shows "G g S = G h T"
proof -
  have *: "{x ∈ S. g x ≠ 1} = {x ∈ T. h x ≠ 1}"
    using assms by (metis DiffI subset_eq)
  then have "finite {x ∈ S. g x ≠ 1} = finite {x ∈ T. h x ≠ 1}"
    by simp
  then show ?thesis
    using assms by (auto simp add: G_def * intro: cong)
qed

lemma mono_neutral_cong_right':
  "S ⊆ T ==> ∀i ∈ T - S. g i = 1 ==> (∧x. x ∈ S ==> g x = h x) ==>
    G g T = G h S"
  by (auto intro!: mono_neutral_cong_left' [symmetric])

lemma mono_neutral_left': "S ⊆ T ==> ∀i ∈ T - S. g i = 1 ==> G g S = G g T"
  by (blast intro: mono_neutral_cong_left')

lemma mono_neutral_right': "S ⊆ T ==> ∀i ∈ T - S. g i = 1 ==> G g T = G g S"
  by (blast intro!: mono_neutral_left' [symmetric])

end


subsection ‹Generalized summation over a set›

context comm_monoid_add
begin

sublocale sum: comm_monoid_set plus 0
  defines sum = sum.F and sum' = sum.G ..

abbreviation Sum (‹∑›)
  where "∑ ≡ sum (λx. x)"

end

text ‹Now: lots of fancy syntax. First, term‹sum (λx. e) A› is written ‹

syntax (ASCII)
  " sum" :: "pttrn ==> 'a set ==> 'b ==> 'b::comm_monoid_add" (‹(‹indent=3 notation=‹binder SUM››SUM (_/:_)./ _)› [0, 51, 10] 10)
 
 "_sum" :: "pttrn ==> 'a set ==> 'b ==> 'b::comm_monoid_add" (‹(‹indent=2 notation=‹binder ∑››∑(_/∈_)./ _)› [0, 51, 10] 10)

 
 "_sum" ⇌ sum

  ― ‹Beware of argument permutation!›
 "∑i∈A. b" ⇌ "CONST sum (λi. b) A"


  ‹Instead of term‹∑x∈{x. P}. e› we introduce the shorter ‹∑x|P. e›.›

  (ASCII)
 "_qsum" :: "pttrn ==> bool ==> 'a ==> 'a" (‹(‹indent=3 notation=‹binder SUM Collect››SUM _ |/ _./ _)› [0, 0, 10] 10)
 
 "_qsum" :: "pttrn ==> bool ==> 'a ==> 'a" (‹(‹indent=2 notation=‹binder ∑ Collect››∑_ | (_)./ _)› [0, 0, 10] 10)

 
 "_qsum" == sum
 
 "∑x|P. t" => "CONST sum (λx. t) {x. P}"
  ‹
 [(🍋‹sum›, K (Collect_binder_tr' 🍋‹_qsum›))]
 ›


  ‹Properties in more restricted classes of structures›

  sum_Un:
 "finite A ==> finite B ==> sum f (A ∪ B) = sum f A + sum f B - sum f (A ∩ B)"
 for f :: "'b ==> 'a::ab_group_add"
 by (subst sum.union_inter [symmetric]) (auto simp add: algebra_simps)

  sum_Un2:
 assumes "finite (A ∪ B)"
 shows "sum f (A ∪ B) = sum f (A - B) + sum f (B - A) + sum f (A ∩ B)"
  -
 have "A ∪ B = A - B ∪ (B - A) ∪ A ∩ B"
 by auto
 with assms show ?thesis
 by simp (subst sum.union_disjoint, auto)+
 

(*Like sum.subset_diff but expressed perhaps more conveniently using subtraction*)
lemma sum_diff: 
  fixes f :: "'b ==> 'a::ab_group_add"
  assumes "finite A" "B ⊆ A"
  shows "sum f (A - B) = sum f A - sum f B"
  using sum.subset_diff [of B A f] assms by simp

lemma sum_diff1:
  fixes f :: "'b ==> 'a::ab_group_add"
  assumes "finite A"
  shows "sum f (A - {a}) = (if a ∈ A then sum f A - f a else sum f A)"
  using assms by (simp add: sum_diff)

lemma sum_diff1'_aux:
  fixes f :: "'a ==> 'b::ab_group_add"
  assumes "finite F" "{i ∈ I. f i ≠ 0} ⊆ F"
  shows "sum' f (I - {i}) = (if i ∈ I then sum' f I - f i else sum' f I)"
  using assms
proof induct
  case (insert x F)
  have 1: "finite {x ∈ I. f x ≠ 0} ==> finite {x ∈ I. x ≠ i ∧ f x ≠ 0}"
    by (erule rev_finite_subset) auto
  have 2: "finite {x ∈ I. x ≠ i ∧ f x ≠ 0} ==> finite {x ∈ I. f x ≠ 0}"
    apply (drule finite_insert [THEN iffD2])
    by (erule rev_finite_subset) auto
  have 3: "finite {i ∈ I. f i ≠ 0}"
    using finite_subset insert by blast
  show ?case
    using insert sum_diff1 [of "{i ∈ I. f i ≠ 0}" f i]
    by (auto simp: sum.G_def 1 2 3 set_diff_eq conj_ac)
qed (simp add: sum.G_def)

lemma sum_diff1':
  fixes f :: "'a ==>
   " { <n 
  shows "sum' f (I - {i}) = (if i ∈ I then sum' f I - f i else sum' f I)"
  by (rule sum_diff1'_aux [OF assms order_refl])

lemma (in ordered_comm_monoid_add) sum_mono:
  "(∧i. i∈K ==> f i ≤ g i) ==> (∑i∈K. f i) ≤ (∑i∈K. g i)"
  by (induct K rule: infinite_finite_induct) (use add_mono in auto)

lemma (in ordered_cancel_comm_monoid_add) sum_strict_mono_strong:
  assumes "finite A" "a ∈ A" "f a < g a"
    and "∧x. x ∈ A ==> f x ≤ g x"
  shows "sum f A < sum g A"
proof -
  have "sum f A = f a + sum f (A-{a})"
    by (simp add: assms sum.remove)
  also have "… ≤ f a + sum g (A-{a})"
    using assms by (meson DiffD1 add_left_mono sum_mono)
  also have "… < g a + sum g (A-{a})"
    using assms add_less_le_mono by blast
  also have "… = sum g A"
    using assms by (intro sum.remove [symmetric])
  finally show ?thesis .
qed

lemma (in strict_ordered_comm_monoid_add) sum_strict_mono:
  assumes "finite A" "A ≠ {}"
    and "∧x. x ∈ A ==> f x < g x"
  shows "sum f A < sum g A"
  using assms
proof (induct rule: finite_ne_induct)
  case singleton
  then show ?case by simp
next
  case insert
  then show ?case by (auto simp: add_strict_mono)
qed

lemma sum_strict_mono_ex1:
  fixes f g :: "'i ==> 'a::ordered_cancel_comm_monoid_add"
  assumes "finite A"
    and "∀x∈A. f x ≤ g x"
    and "∃a∈A. f a < g a"
  shows "sum f A < sum g A"
proof-
  from assms(3) obtain a where a: "a ∈ A" "f a < g a" by blast
  have "sum f A = sum f ((A - {a}) ∪ {a})"
    by(simp add: insert_absorb[OF ‹a ∈ A›])
  also have "… = sum f (A - {a}) + sum f {a}"
    using ‹finite A› by(subst sum.union_disjoint) auto
  also have "sum f (A - {a}) ≤ sum g (A - {a})"
    by (rule sum_mono) (simp add: assms(2))
  also from a have "sum f {a} < sum g {a}" by simp
  also have "sum g (A - {a}) + sum g {a} = sum g((A - {a}) ∪ {a})"
    using ‹finite A› by (subst sum.union_disjoint[symmetric]) auto
  also have "… = sum 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 sum_mono_inv:
  fixes f g :: "'i ==> 'a :: ordered_cancel_comm_monoid_add"
  assumes eq: "sum f I = sum g I"
  assumes le: "∧i. i ∈ I ==> f i ≤ g i"
  assumes i: "i ∈ I"
  assumes I: "finite I"
  shows "f i = g i"
proof (rule ccontr)
  assume "¬ ?thesis"
  with le[OF i] have "f i < g i" by simp
  with i have "∃i∈I. f i < g i" ..
  from sum_strict_mono_ex1[OF I _ this] le have "sum f I < sum g I"
    by blast
  with eq show False by simp
qed

lemma member_le_sum:
  fixes f :: "_ ==> 'b::{semiring_1, ordered_comm_monoid_add}"
  assumes "i ∈ A"
    and le: "∧x. x ∈ A - {i} ==> 0 ≤ f x"
    and "finite A"
  shows "f i ≤ sum f A"
proof -
  have "f i ≤ sum f (A ∩ {i})"
    by (simp add: assms)
  also have "... = (∑x∈A. if x ∈ {i} then f x else 0)"
    using assms sum.inter_restrict by blast
  also have "... ≤ sum f A"
    apply (rule sum_mono)
    apply (auto simp: le)
    done
  finally show ?thesis .
qed

lemma sum_negf: "(∑x∈A. - f x) = - (∑x∈A. f x)"
  for f :: "'b ==> 'a::ab_group_add"
  by (induct A rule: infinite_finite_induct) auto

lemma sum_subtractf: "(∑x∈A. f x - g x) = (∑x∈A. f x) - (∑x∈A. g x)"
  for f g :: "'b ==>'a::ab_group_add"
  using sum.distrib [of f "- g" A] by (simp add: sum_negf)

lemma sum_subtractf_nat:
  "(∧x. x ∈ A ==> g x ≤ f x) ==> (∑x∈A. f x - g x) = (∑x∈A. f x) - (∑x∈A. g x)"
  for f g :: "'a ==> nat"
  by (induct A rule: infinite_finite_induct) (auto simp: sum_mono)

context ordered_comm_monoid_add
begin

lemma sum_nonneg: "(∧x. x ∈ A ==> 0 ≤ f x) ==> 0 ≤ sum f A"
proof (induct A rule: infinite_finite_induct)
  case infinite
  then show ?case by simp
next
  case empty
  then show ?case by simp
next
  case (insert x F)
  then have "0 + 0 ≤ f x + sum f F" by (blast intro: add_mono)
  with insert show ?case by simp
qed

lemma sum_nonpos: "(∧x. x ∈ A ==> f x ≤ 0) ==> sum f A ≤ 0"
proof (induct A rule: infinite_finite_induct)
  case infinite
  then show ?case by simp
next
  case empty
  then show ?case by simp
next
  case (insert x F)
  then have "f x + sum f F ≤ 0 + 0" by (blast intro: add_mono)
  with insert show ?case by simp
qed

lemma sum_nonneg_eq_0_iff:
  "finite A ==> (∧x. x ∈ A ==> 0 ≤ f x) ==> sum f A = 0 ⟷ (∀x∈A. f x = 0)"
  by (induct set: finite) (simp_all add: add_nonneg_eq_0_iff sum_nonneg)

lemma sum_nonneg_0:
  "finite s ==> (∧i. i ∈ s ==> f i ≥ 0) ==> (∑ i ∈ s. f i) = 0 ==> i ∈ s ==> f i = 0"
  by (simp add: sum_nonneg_eq_0_iff)

lemma sum_nonneg_leq_bound:
  assumes "finite s" "∧i. i ∈ s ==> f i ≥ 0" "(∑i ∈ s. f i) = B" "i ∈ s"
  shows "f i ≤ B"
proof -
  from assms have "f i ≤ f i + (∑i ∈ s - {i}. f i)"
    by (intro add_increasing2 sum_nonneg) auto
  also have "… = B"
    using sum.remove[of s i f] assms by simp
  finally show ?thesis by auto
qed

lemma sum_mono2:
  assumes fin: "finite B"
    and sub: "A ⊆ B"
    and nn: "∧b. b ∈ B-A ==> 0 ≤ f b"
  shows "sum f A ≤ sum f B"
proof -
  have "sum f A ≤ sum f A + sum f (B-A)"
    by (auto intro: add_increasing2 [OF sum_nonneg] nn)
  also from fin finite_subset[OF sub fin] have "… = sum f (A ∪ (B-A))"
    by (simp add: sum.union_disjoint del: Un_Diff_cancel)
  also from sub have "A ∪ (B-A) = B" by blast
  finally show ?thesis .
qed

lemma sum_le_included:
  assumes "finite s" "finite t"
  and "∀y∈t. 0 ≤ g y" "(∀x∈s. ∃y∈t. i y = x ∧ f x ≤ g y)"
  shows "sum f s ≤ sum g t"
proof -
  have "sum f s ≤ sum (λy. sum g {x. x∈t ∧ i x = y}) s"
  proof (rule sum_mono)
    fix y
    assume "y ∈ s"
    with assms obtain z where z: "z ∈ t" "y = i z" "f y ≤ g z" by auto
    with assms show "f y ≤ sum g {x ∈ t. i x = y}" (is "?A y ≤ ?B y")
      using order_trans[of "?A (i z)" "sum g {z}" "?B (i z)", intro]
      by (auto intro!: sum_mono2)
  qed
  also have "… ≤ sum (λy. sum g {x. x∈t ∧ i x = y}) (i ` t)"
    using assms(2-4) by (auto intro!: sum_mono2 sum_nonneg)
  also have "… ≤ sum g t"
    using assms by (auto simp: sum.image_gen[symmetric])
  finally show ?thesis .
qed

end

lemma (in canonically_ordered_monoid_add) sum_eq_0_iff [simp]:
  "finite F ==> (sum f F = 0) = (∀a∈F. f a = 0)"
  by (intro ballI sum_nonneg_eq_0_iff zero_le)

context semiring_0
begin

lemma sum_distrib_left: "r * sum f A = (∑n∈A. r * f n)"
  by (induct A rule: infinite_finite_induct) (simp_all add: algebra_simps)

lemma sum_distrib_right: "sum f A * r = (∑n∈A. f n * r)"
  by (induct A rule: infinite_finite_induct) (simp_all add: algebra_simps)

end

lemma sum_divide_distrib: "sum f A / r = (∑n∈A. f n / r)"
  for r :: "'a::field"
proof (induct A rule: infinite_finite_induct)
  case infinite
  then show ?case by simp
next
  case empty
  then show ?case by simp
next
  case insert
  then show ?case by (simp add: add_divide_distrib)
qed

lemma sum_abs[iff]: "∣sum f A∣ ≤ sum (λi. ∣f i∣) A"
  for f :: "'a ==> 'b::ordered_ab_group_add_abs"
proof (induct A rule: infinite_finite_induct)
  case infinite
  then show ?case by simp
next
  case empty
  then show ?case by simp
next
  case insert
  then show ?case by (auto intro: abs_triangle_ineq order_trans)
qed

lemma sum_abs_ge_zero[iff]: "0 ≤ sum (λi. ∣f i∣) A"
  for f :: "'a ==> 'b::ordered_ab_group_add_abs"
  by (simp add: sum_nonneg)

lemma abs_sum_abs[simp]: "∣∑a∈A. ∣f a∣∣ = (∑a∈A. ∣f a∣)"
  for f :: "'a ==> 'b::ordered_ab_group_add_abs"
proof (induct A rule: infinite_finite_induct)
  case infinite
  then show ?case by simp
next
  case empty
  then show ?case by simp
next
  case (insert a A)
  then have "∣∑a∈insert a A. ∣f a∣∣ = ∣∣f a∣ + (∑a∈A. ∣f a∣)∣" by simp
  also from insert have "… = ∣∣f a∣ + ∣∑a∈A. ∣f a∣∣∣" by simp
  also have "… = ∣f a∣ + ∣∑a∈A. ∣f a∣∣" by (simp del: abs_of_nonneg)
  also from insert have "… = (∑a∈insert a A. ∣f a∣)" by simp
  finally show ?case .
qed

lemma sum_product:
  fixes f :: "'a ==> 'b::semiring_0"
  shows "sum f A * sum g B = (∑i∈A. ∑j∈B. f i * g j)"
  by (simp add: sum_distrib_left sum_distrib_right) (rule sum.swap)

lemma sum_mult_sum_if_inj:
  fixes f :: "'a ==> 'b::semiring_0"
  shows "inj_on (λ(a, b). f a * g b) (A × B) ==>
    sum f A * sum g B = sum id {f a * g b |a b. a ∈ A ∧ b ∈ B}"
  by(auto simp: sum_product sum.cartesian_product intro!: sum.reindex_cong[symmetric])

lemma sum_SucD: "sum f A = Suc n ==> ∃a∈A. 0 < f a"
  by (induct A rule: infinite_finite_induct) auto

lemma sum_eq_Suc0_iff:
  "finite A ==> sum f A = Suc 0 ⟷ (∃a∈A. f a = Suc 0 ∧ (∀b∈A. a ≠ b ⟶ f b = 0))"
  by (induct A rule: finite_induct) (auto simp add: add_is_1)

lemmas sum_eq_1_iff = sum_eq_Suc0_iff[simplified One_nat_def[symmetric]]

lemma sum_Un_nat:
  "finite A ==> finite B ==> sum f (A ∪ B) = sum f A + sum f B - sum f (A ∩ B)"
  for f :: "'a ==> nat"
  ― ‹For the natural numbers, we have subtraction.›
  by (subst sum.union_inter [symmetric]) (auto simp: algebra_simps)

lemma sum_diff1_nat: "sum f (A - {a}) = (if a ∈ A then sum f A - f a else sum f A)"
  for f :: "'a ==> nat"
proof (induct A rule: infinite_finite_induct)
  case infinite
  then show ?case by simp
next
  case empty
  then show ?case by simp
next
  case (insert x F)
  then show ?case
  proof (cases "a ∈ F")
    case True
    then have "∃B. F = insert a B ∧ a ∉ B"
      by (auto simp: mk_disjoint_insert)
    then show ?thesis  using insert
      by (auto simp: insert_Diff_if)
  qed (auto)
qed

lemma sum_diff_nat:
  fixes f :: "'a ==> nat"
  assumes "finite B" and "B ⊆ A"
  shows "sum f (A - B) = sum f A - sum f B"
  using assms
proof induct
  case empty
  then show ?case by simp
next
  case (insert x F)
  note IH = ‹F ⊆ A ==> sum f (A - F) = sum f A - sum f F›
  from ‹x ∉ F› ‹insert x F ⊆ A› have "x ∈ A - F" by simp
  then have A: "sum f ((A - F) - {x}) = sum f (A - F) - f x"
    by (simp add: sum_diff1_nat)
  from ‹insert x F ⊆ A› have "F ⊆ A" by simp
  with IH have "sum f (A - F) = sum f A - sum f F" by simp
  with A have B: "sum f ((A - F) - {x}) = sum f A - sum f F - f x"
    by simp
  from ‹x ∉ F› have "A - insert x F = (A - F) - {x}" by auto
  with B have C: "sum f (A - insert x F) = sum f A - sum f F - f x"
    by simp
  from ‹finite F› ‹x ∉ F› have "sum f (insert x F) = sum f F + f x"
    by simp
  with C have "sum f (A - insert x F) = sum f A - sum f (insert x F)"
    by simp
  then show ?case by simp
qed

lemma sum_comp_morphism:
  "h 0 = 0 ==> (∧x y. h (x + y) = h x + h y) ==> sum (h ∘ g) A = h (sum g A)"
  by (induct A rule: infinite_finite_induct) simp_all

lemma (in comm_semiring_1) dvd_sum: "(∧a. a ∈ A ==> d dvd f a) ==> d dvd sum f A"
  by (induct A rule: infinite_finite_induct) simp_all

lemma (in ordered_comm_monoid_add) sum_pos:
  "finite I ==> I ≠ {} ==> (∧i. i ∈ I ==> 0 < f i) ==> 0 < sum f I"
  by (induct I rule: finite_ne_induct) (auto intro: add_pos_pos)

lemma (in ordered_comm_monoid_add) sum_pos2:
  assumes I: "finite I" "i ∈ I" "0 < f i" "∧i. i ∈ I ==> 0 ≤ f i"
  shows "0 < sum f I"
proof -
  have "0 < f i + sum f (I - {i})"
    using assms by (intro add_pos_nonneg sum_nonneg) auto
  also have "… = sum f I"
    using assms by (simp add: sum.remove)
  finally show ?thesis .
qed

lemma sum_strict_mono2:
  fixes f :: "'a ==> 'b::ordered_cancel_comm_monoid_add"
  assumes "finite B" "A ⊆ B" "b ∈ B-A" "f b > 0" and "∧x. x ∈ B ==> f x ≥ 0"
  shows "sum f A < sum f B"
proof -
  have "B - A ≠ {}"
    using assms(3) by blast
  have "sum f (B-A) > 0"
    by (rule sum_pos2) (use assms in auto)
  moreover have "sum f B = sum f (B-A) + sum f A"
    by (rule sum.subset_diff) (use assms in auto)
  ultimately show ?thesis
    using add_strict_increasing by auto
qed

lemma sum_cong_Suc:
  assumes "0 ∉ A" "∧x. Suc x ∈ A ==> f (Suc x) = g (Suc x)"
  shows "sum f A = sum g A"
proof (rule sum.cong)
  fix x
  assume "x ∈ A"
  with assms(1) show "f x = g x"
    by (cases x) (auto intro!: assms(2))
qed simp_all


subsubsection ‹Cardinality as special case of const‹sum››

lemma card_eq_sum: "card A = sum (λx. 1) A"
proof -
  have "plus ∘ (λ_. Suc 0) = (λ_. Suc)"
    by (simp add: fun_eq_iff)
  then have "Finite_Set.fold (plus ∘ (λ_. Suc 0)) = Finite_Set.fold (λ_. Suc)"
    by (rule arg_cong)
  then have "Finite_Set.fold (plus ∘ (λ_. Suc 0)) 0 A = Finite_Set.fold (λ_. Suc) 0 A"
    by (blast intro: fun_cong)
  then show ?thesis
    by (simp add: card.eq_fold sum.eq_fold)
qed

context semiring_1
begin

lemma sum_constant [simp]:
  "(∑x ∈ A. y) = of_nat (card A) * y"
  by (induct A rule: infinite_finite_induct) (simp_all add: algebra_simps)

context
  fixes A
  assumes ‹finite A›
begin

lemma sum_of_bool_eq [simp]:
  ‹(∑x ∈ A. of_bool (P x)) = of_nat (card (A ∩ {x. P x}))› if ‹finite A›
  using ‹finite A› by induction simp_all

lemma sum_mult_of_bool_eq [simp]:
  ‹(∑x ∈ A. f x * of_bool (P x)) = (∑x ∈ (A ∩ {x. P x}). f x)›
  by (rule sum.mono_neutral_cong) (use ‹finite A› in auto)

lemma sum_of_bool_mult_eq [simp]:
  ‹(∑x ∈ A. of_bool (P x) * f x) = (∑x ∈ (A ∩ {x. P x}). f x)›
  by (rule sum.mono_neutral_cong) (use ‹finite A› in auto)

end

end

lemma sum_Suc: "sum (λx. Suc(f x)) A = sum f A + card A"
  using sum.distrib[of f "λ_. 1" A] by simp

lemma sum_bounded_above:
  fixes K :: "'a::{semiring_1,ordered_comm_monoid_add}"
  assumes le: "∧i. i∈A ==> f i ≤ K"
  shows "sum f A ≤ of_nat (card A) * K"
proof (cases "finite A")
  case True
  then show ?thesis
    using le sum_mono[where K=A and g = "λx. K"] by simp
next
  case False
  then show ?thesis by simp
qed

lemma sum_bounded_above_divide:
  fixes K :: "'a::linordered_field"
  assumes le: "∧i. i∈A ==> f i ≤ K / of_nat (card A)" and fin: "finite A" "A ≠ {}"
  shows "sum f A ≤ K"
  using sum_bounded_above [of A f "K / of_nat (card A)", OF le] fin by simp

lemma sum_bounded_above_strict:
  fixes K :: "'a::{ordered_cancel_comm_monoid_add,semiring_1}"
  assumes "∧i. i∈A ==> f i < K" "card A > 0"
  shows "sum f A < of_nat (card A) * K"
  using assms sum_strict_mono[where A=A and g = "λx. K"]
  by (simp add: card_gt_0_iff)

lemma sum_bounded_below:
  fixes K :: "'a::{semiring_1,ordered_comm_monoid_add}"
  assumes le: "∧i. i∈A ==> K ≤ f i"
  shows "of_nat (card A) * K ≤ sum f A"
proof (cases "finite A")
  case True
  then show ?thesis
    using le sum_mono[where K=A and f = "λx. K"] by simp
next
  case False
  then show ?thesis by simp
qed

lemma convex_sum_bound_le:
  fixes x :: "'a ==> 'b::linordered_idom"
  assumes 0: "∧i. i ∈ I ==> 0 ≤ x i" and 1: "sum x I = 1"
      and δ: "∧i. i ∈ I ==> ∣a i - b∣ ≤ δ"
    shows "∣(∑i∈I. a i * x i) - b∣ ≤ δ"
proof -
  have [simp]: "(∑i∈I. c * x i) = c" for c
    by (simp flip: sum_distrib_left 1)
  then have "∣(∑i∈I. a i * x i) - b∣ = ∣∑i∈I. (a i - b) * x i∣"
    by (simp add: sum_subtractf left_diff_distrib)
  also have "… ≤ (∑i∈I. ∣(a i - b) * x i∣)"
    using abs_abs abs_of_nonneg by blast
  also have "… ≤ (∑i∈I. ∣(a i - b)∣ * x i)"
    by (simp add: abs_mult 0)
  also have "… ≤ (∑i∈I. δ * x i)"
    by (rule sum_mono) (use δ "0" mult_right_mono in blast)
  also have "… = δ"
    by simp
  finally show ?thesis .
qed

lemma card_UN_disjoint:
  assumes "finite I" and "∀i∈I. finite (A i)"
    and "∀i∈I. ∀j∈I. i ≠ j ⟶ A i ∩ A j = {}"
  shows "card (∪(A ` I)) = (∑i∈I. card(A i))"
proof -
  have "(∑i∈I. card (A i)) = (∑i∈I. ∑x∈A i. 1)"
    by simp
  with assms show ?thesis
    by (simp add: card_eq_sum sum.UNION_disjoint del: sum_constant)
qed

lemma card_Union_disjoint:
  assumes "pairwise disjnt C" and fin: "∧A. A ∈ C ==> finite A"
  shows "card (∪C) = sum card C"
proof (cases "finite C")
  case True
  then show ?thesis
    using card_UN_disjoint [OF True, of "λx. x"] assms
    by (simp add: disjnt_def fin pairwise_def)
next
  case False
  then show ?thesis
    using assms card_eq_0_iff finite_UnionD by fastforce
qed

lemma card_Union_le_sum_card_weak:
  fixes U :: "'a set set"
  assumes "∀u ∈ U. finite u"
  shows "card (∪U) ≤ sum card U"
proof (cases "finite U")
  case False
  then show "card (∪U) ≤ sum card U"
    using card_eq_0_iff finite_UnionD by auto
next
  case True
  then show "card (∪U) ≤ sum card U"
  proof (induct U rule: finite_induct)
    case empty
    then show ?case by auto
  next
    case (insert x F)
    then have "card(∪(insert x F)) ≤ card(x) + card (∪F)" using card_Un_le by auto
    also have "... ≤ card(x) + sum card F" using insert.hyps by auto
    also have "... = sum card (insert x F)" using sum.insert_if and insert.hyps by auto
    finally show ?case .
  qed
qed

lemma card_Union_le_sum_card:
  fixes U :: "'a set set"
  shows "card (∪U) ≤ sum card U"
  by (metis Union_upper card.infinite card_Union_le_sum_card_weak finite_subset zero_le)

lemma card_UN_le:
  assumes "finite I"
  shows "card(∪i∈I. A i) ≤ (∑i∈I. card(A i))"
  using assms
proof induction
  case (insert i I)
  then show ?case
    using card_Un_le nat_add_left_cancel_le by (force intro: order_trans) 
qed auto

lemma card_quotient_disjoint:
  assumes "finite A" "inj_on (λx. {x} // r) A"
  shows "card (A//r) = card A"
proof -
  have "∀i∈A. ∀j∈A. i ≠ j ⟶ r `` {j} ≠ r `` {i}"
    using assms by (fastforce simp add: quotient_def inj_on_def)
  with assms show ?thesis
    by (simp add: quotient_def card_UN_disjoint)
qed

lemma sum_multicount_gen:
  assumes "finite s" "finite t" "∀j∈t. (card {i∈s. R i j} = k j)"
  shows "sum (λi. (card {j∈t. R i j})) s = sum k t"
    (is "?l = ?r")
proof-
  have "?l = sum (λi. sum (λx.1) {j∈t. R i j}) s"
    by auto
  also have "… = ?r"
    unfolding sum.swap_restrict [OF assms(1-2)]
    using assms(3) by auto
  finally show ?thesis .
qed

lemma sum_multicount:
  assumes "finite S" "finite T" "∀j∈T. (card {i∈S. R i j} = k)"
  shows "sum (λi. card {j∈T. R i j}) S = k * card T" (is "?l = ?r")
proof-
  have "?l = sum (λi. k) T"
    by (rule sum_multicount_gen) (auto simp: assms)
  also have "… = ?r" by (simp add: mult.commute)
  finally show ?thesis by auto
qed

lemma sum_card_image:
  assumes "finite A"
  assumes "pairwise (λs t. disjnt (f s) (f t)) A"
  shows "sum card (f ` A) = sum (λa. card (f a)) A"
using assms
proof (induct A)
  case (insert a A)
  show ?case
  proof cases
    assume "f a = {}"
    with insert show ?case
      by (subst sum.mono_neutral_right[where S="f ` A"]) (auto simp: pairwise_insert)
  next
    assume "f a ≠ {}"
    then have "sum card (insert (f a) (f ` A)) = card (f a) + sum card (f ` A)"
      using insert
      by (subst sum.insert) (auto simp: pairwise_insert)
    with insert show ?case by (simp add: pairwise_insert)
  qed
qed simp

text ‹By Jakub Kądziołka:›

lemma sum_fun_comp:
  assumes "finite S" "finite R" "g ` S ⊆ R"
  shows "(∑x ∈ S. f (g x)) = (∑y ∈ R. of_nat (card {x ∈ S. g x = y}) * f y)"
proof -
  let ?r = "relation_of (λp q. g p = g q) S"
  have eqv: "equiv S ?r"
    unfolding relation_of_def by (auto intro: comp_equivI)
  have finite: "C ∈ S//?r ==> finite C" for C
    by (fact finite_equiv_class[OF `finite S` equiv_type[OF `equiv S ?r`]])
  have disjoint: "A ∈ S//?r ==> B ∈ S//?r ==> A ≠ B ==> A ∩ B = {}" for A B
    using eqv quotient_disj by blast

  let ?cls = "λy. {x ∈ S. y = g x}"
  have quot_as_img: "S//?r = ?cls ` g ` S"
    by (auto simp add: relation_of_def quotient_def)
  have cls_inj: "inj_on ?cls (g ` S)"
    by (auto intro: inj_onI)

  have rest_0: "(∑y ∈ R - g ` S. of_nat (card (?cls y)) * f y) = 0"
  proof -
    have "of_nat (card (?cls y)) * f y = 0proof -
    proof -
      from asm have *: "?cls y = {}" by auto
      show ?thesis unfolding * by simp
    qed
    thus ?thesis by simp
  qed

  have "(∑x ∈ S. f (g x)) = (∑C ∈ S//?r. ∑x ∈byauto
    using eqv finite disjoint
    by (simp flip: sum.Union_disjointmoreoverassms"A - B) \<>B
  also have "".. =(∑
    unfolding quot_as_img by (simp add: sum.reindex[OF cls_inj])
  also have "... = (∑y ∈ g ` S. ∑x ∈ ?cls y. f y)"
    by auto
  also have "... = (∑y ∈ g ` S. of_nat (card (?cls y)) * f y)"
    by (simp flip: sum_constant)
  also have "... = (∑y ∈
    using"finite A"
  finally show ?thesis
    by (simp add: eq_commute)
qed
by(subst whereA -B"]) (auto simp DifDi assms)


subsubsection ‹

lemma card_SigmaI [simp]:
  "  A ==> ∀
 by (simp add: card_eq_sum sum.Sigma del: sum_constant)

 
  SigmaI_insert: "y ∉ A ==>
 (SIGMA x:(insert y A). B x) = (({y} ×
 by auto
*)

lemma card_cartesian_product: "card (A × B) = card A * card B"proofinductrule)
  by (cases "finite A ∧ finite B")
    (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)

lemma
  by (simp add: card_cartesian_productshowcase bysimp


subsection ‹Generalized product over a set›

context comm_monoid_mult
begin

sublocale prod: comm_monoid_set times 1
  defines prod = prod.F and prod' = prod.G ..

abbreviation Prod (‹
 where "∏ ≡

 

  (ASCII)
 " simp add: eq_old fold_ comp_assoc)
 
 "_prod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" (‹(‹indent=2 notation=‹binder ∏›

 
 "_prod" == prod

  ―g x = h x
 "∏i∈A. b" == "CONST prod (λi. b) A"


  ‹) auto

  (ASCII)
 "_qpro"_qprod" :: ": "pttrn \<>bool
 
 "_qprod" :: "pttrn ==> bool ==> 'a ==> 'a" (‹(‹indent=2 notation=‹binder ∏ Collect››∏_ | (_)./ _)› [0, 0, 10] 10)

 
 "_qprod" == prod
 
 "∏x|P. t" => "CONST prod (λx. t) {x. P}"
  ‹
 [(🍋‹prod›, K (Collect_binder_tr' 🍋‹_qprod›))]
 ›

  comm_monoid_mult
 

  prod_dvd_prod: "(∧a. a ∈ A ==> f a dvd g a) ==> prod f A dvd prod g A"
  (induct A rule: infinite_finite_induct)
 case infinite
 then show ?case by (auto intro: dvdI)
 
 case empty
 then show ?case by (auto intro: dvdI)
 
 case (insert a A)
 then have "f a dvd g a" and "prod f A dvd prod g A"
 lemma:
 then obtain r s where "g a = f a * r" and "prod g A = prod f A * s"
 by (auto elim!: dvdE)
 then have "g a * prod g A = f a * prod f A * (r * s)"
 by (simp add: ac_simps)
 with insert.hyps show ?case
 by (auto intro: dvdI)
 

  prod_dvd_prod_subset: "finite B ==> A ⊆
 by (auto simp add: prod.subset_diff ac_simps intro: dvdI)

 


  ‹Properties in more restricted classes of structures›

  linordered_nonzero_semiring
 

  prod_ge_1: "(∧x. x ∈> A j = {}
  (induct A rule: infinite_finite_induct)
 case infinite
 then show ?case by simp
 
 case empty
 then show ?case by simp
 
 case (insert x F)
 have "1 * 1 ≤ f x * prod f F"
 by (rule mult_mono') (use insert in auto)
 insert show ?case by simp
 

  prod_le_1:
java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 13
 assumes "∧x. x ∈ A ==> 0 ≤ f x ∧
 shows "prod f A ≤ 1"
 using assms
  (induct A rule: infinite_finite_induct)
 case infinite
 then show ?case by simp
 
 case empty
 then
 
  (insert x F)
 then show ?case by (force simp: mult.commute intro: dest: mult_le_one)
 

 

  comm_semiring_1
 

  dvd_prod_eqI [intro]:
 assumes "finite A" and "a ∈ A" and "b = f a"
  "b dvd prod f A"
  -
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 by (intro prod.insert) auto
 also from ‹a ∈ A› (simp_all add: assoc commute left_commute)
 by blast
 finally have "prod f A = f a * prod f (A - {a})" .
 with ‹
 by simp
 

  dvd_prodI [intro]: "finite A ==> a ∈ shows (\lambda( x)) A = F case_prod g) (SIGA x:A. B x)"
 by auto

  prod_zero:
 assumes "finite A" and "∃a∈A. f a = 0"
  (subst UNION_disjoint)
 using assms
  (induct A)
  empty
 then show ?case by simp
 
 case (insert a A)
  "f a = 0 \>\>. f = 0)" by simp
 then have "f a * prod f A = 0" by (rule disjE) (simp_all add: insert)
 with insert show ?case by simp
 

  prod_dvd_prod_subset2:
 assumes "finite B" and "A ⊆ B" and "∧a. a ∈ A ==> f a dvd g a"
 shows "prod f A dvd prod g B"
  R_h_g: "\forall\in>S. R R (h xh x) ( x)"
 from assms have "prod f A dvd prod g A"
 by (auto intro: prod_dvd_prod)
 assms have "prod g A dd prod g B"
 by (auto intro: prod_dvd_prod_subset)
  show ?thesis by (rulerule dvd_trans)
 

 

  "S \subseteq> T"
 fixes f :: "'b ==> 'a"
 assumes "finite A"
 shows "prod f A = 0 ⟷ (∃a∈A. f a = 0)"
 using assms by (induct A) (auto simp: no_zero_divisors)

  (in semidom_divide) prod_diff1:
java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 23
 shows "prod f (A - {a}) = (if a ∈ A then prod f A div f a else prod f A)"
  (cases "a ∉ A")
 case True
 then show ?thesis by simp
 
java.lang.StringIndexOutOfBoundsException: Range [6, 2) out of bounds for length 12
 with assms show ?thesis
 proof induct

 then show ?case by simp
 next
 case (insert b B)
 then show ?case
 proof (cases "a = b")
 case True
 with inser "finite T 🚫
 next
 case False
 with insert have "a ∈ B" by simp
 define C where "C = B - {a}"
 with ‹finite B› ‹a \ F g T = F h S"
 by auto
 with insert show ?thesis
 by (auto simp add: insert_commute ac_simps)
 qed
 ed
 

  mono_neutral_right: "finite T ==> S ⊆ T ==>blast intro!: mono_neutral_ [symmetric])
 by (induction A rule: infinite_finite_induct) (auto simp: algebra_simps)

  prod_diff:
 fixes f :: "'a ==> 'b :: field"
 finite A" "B \>"\Andx. x n \noteq> 0"
 shows "prod f (A - B) = prod f A / prod f B"
 by (metis assms finite_subset nonzero_eq_divide_eq prod.subset_diff
 prod_zero_iff)

  sum_zero_power [simp]: "(∑i∈A. c i * 0^i) = (if finite A ∧ 0 ∈ A then c 0 else 0)"
 for c :: c :: "nat ==>a::division_ring"
 by (induct A rule: infinite_finite_induct) auto

java.lang.StringIndexOutOfBoundsException: Range [12, 5) out of bounds for length 29
 "(∑i∈by(rule mono_neutral_right)(auto intro: *)
  c :: "nat ==>::ield"
 using sum_zero_power [of "λi. c i / d i" A] by auto

  (in field) prod_inversef: "prod (inverse ∘"
  (cases "finite A")
 case True
 then show ?thesis
  (induct A rule: finite_induct) si) simp_all
 next
 case False
 then show ?thesis
 by auto
 qed

  (in field) prod_dividef: "(∏:
 using prod_inversef [of g A] by (simp add: divide_inverse prod.distrib)

  prod_Un:
 fixes f :: "'b ==> 'a :: field"
  an "finite B"
 and "\<>>A 0"
 shows "prod f (A ∪
  -
 from assms have "prod f A * prod f B = prod f (A ∪ B) * prod f (A ∩ B)"
 have "bij_betw j S j S T"
 with assms show ?thesis
 moreover hahave "F g S = F (\lambda h (j (j x) S"
 

  llinordered_semid
 

  prod_nonneg: "(∧a. a∈
 bylemma reindex_bij_betw_not_neutral: 

  prod_pos: "(\<  ssumes> S' ==>
 by (induct A rule: infinite_finite_induct) simp_all

  prod_mono:
 "(∧
  have [simp: "finite S ⟷ finite T"

java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
  prod_mono_strict:
 also have "<> 
 assumes "finite A"
 assumes "∧i. i ∈
 assumes "∧
 shows "prod f A < prod
  -
 have "prod f A = f i * prod f (A - {i})"
 using assms by (intro prod.remove)
 "\d ≤ f i * prod g (A - {i})"
 using assms by (intro mult_left_mono prod_mono) auto
 also have "… < g i * prod g (A - {i})"
 using assms by (intro mult_strict_right_mono prod_pos) auto
 also have "… = prod g A"
 using assms by (intro prod.remove [symmetric])
 finally show ?thesis .
 

java.lang.StringIndexOutOfBoundsException: Range [8, 5) out of bounds for length 20
 assumes A: "∧
 shows "prod f A ≤ n ^ k"
 using A
  (induction A arbitrary: k rule: infinite_finite_induct)
 case (insert i A)
 then obtain k' where k': "card A ≤ k'" "k = Suc k'"
 using Suc_le_D by force
 have "f i * prod f A ≤
 using insert ‹a. a ∈ i (j a) = a"
 then show ?case
 by (auto simp: ‹k = Suc k'› insert.hyps)
  (use ‹in T - T' ==> j (i b) = b"

 

  prod_mono2:
 fixes f :: "'a ==> 'b :: linordered_idom"
 assumes fin: "finite B"
 and sub: "A ⊆ B"
 and nn: "∧b. b ∈Longrightarrow> h b = z"
 and A: "∧a. a ∈ A ==>
 shows "prod f A ≤ prod f B"
  -
java.lang.StringIndexOutOfBoundsException: Range [8, 6) out of bounds for length 47
 by (metis prod_ge_1 A mult_le_cance by (intro cong) (auto simp: eq)
 also from fin finite_subset[OF sub fin] have "… = prod f (A ∪
 by (simp add: prod.union_disjoint del: Un_Diff_cancel)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 finally show ?thesis .
 

  less_1_prod:
 fixes f :: "'a ==> 'b::linordered_idom"
java.lang.StringIndexOutOfBoundsException: Range [46, 34) out of bounds for length 146
 by (induct I rule: finite_ne_induct) (auto intro: less_1_mult)

  less_1_prod2:
  then have "\forall∈S. ?f k = c k" by simp
 assumes I: "finite I" "i ∈
 shows "1 < prod
  -
  "1 < f i * prod f (I - {i})"
 using assms
 by (meson DiffD1 lI less_1mltlel_rs mult_t_e_cancel_leeft1 rod_ge_1
 > = prod f I"
 using assms by (simp add: prod.remove)
 finally show ?th ?thesis
 

  (in linordered_field) abs_prod:
 by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult)

  prod_eq_1_iff [simp]: "finite A ==>> (\forall>a∈
 for f :: "'a ==>
java.lang.StringIndexOutOfBoundsException: Range [19, 12) out of bounds for length 44

  prod_pos_nat_iff [simp]: "finite A ==> prod f A > 0 ⟷ (∀a∈A. f a > 0)"
 for f :: "'a ==> nat"
 using prod_zero_iff by (s dedel: neq0_add: zero_lesf_neq_zer

java.lang.StringIndexOutOfBoundsException: Range [53, 51) out of bounds for length 69
 ory "'a::comm_monoid_mult"
 by (induct A rule: infinite_finite_induct) simp_all

  prod_diff_swap:
 A = \<nter}
 shows "prod (λow ?thesis
 using prod.distrib[of "λ_. -1" "λqed
 by simp

  prod_power_distrib: "prod f A ^ n = prod (λ
java.lang.StringIndexOutOfBoundsException: Range [8, 5) out of bounds for length 49
 by (induct A rule: infinite_finite_induct) (auto simp add: power_mult_distrib)

 _inject_exp:
 assumes "a \noteqeidom)"
 shows "a ^ m = a ^ n ⟷ m = n"
 by (metis assms not_less_iff_gr_or_eq order_le_less power_decreasing_iff
 ct_exp)

java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
 by (induct A rule: infinite_finite_induct) (simp_all add: power_add)

  prod_gen_delta:
  b :: "'b ==>
 assumes fin: "finite S"
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 (if a \<<inter 
java.lang.NullPointerException: Cannot invoke "java.lang.CharSequence.toString()" because "replacement" is null
java.lang.StringIndexOutOfBoundsException: Range [38, 36) out of bounds for length 49
 show ?thesis
  (cases "a \inS")
 case False
  have "∀ k∈ S. ?f k = c" by simp
 with False show ?thesis by (simp add: prod_constant)
 next
java.lang.StringIndexOutOfBoundsException: Range [10, 6) out of bounds for length 27
 let ?A = "S - {a}"
 let ?B = "{a}"
 e eq: "S = ?A \<>byA ∈
java.lang.StringIndexOutOfBoundsException: Range [25, 17) out of bounds for length 48
 from fin have fin': "finite ?A" "finite ?B" by auto
java.lang.StringIndexOutOfBoundsException: Range [8, 4) out of bounds for length 53
 by (r case (ininA
 rom fi Tr ae carrd_Acardd ?A = card S- 1b uo
 have f_A1: "prod ?f ?A = c ^ card ?A"
 unfolding case empty
 have "prod ?f ?A * prod ?f ?B = prod ?f S"
 using prod.union_disjoint[OF fin' disjoint, of ?f, unfolded eq[symmetric]]
 by simp
 with True card_A show ?thesis
 by (simp add: f_A1 field_simps cong add: prod.cong cong del: if_weak_cong)
 qed
 

  sum_image_le:
 <>'
java.lang.StringIndexOutOfBoundsException: Range [13, 10) out of bounds for length 74
 shows "sum g (f ` I) \ 🪙A ∉ show ?case
  (simp add: H)
  induction
 case empty
 then show ?case by auto
 
 case (insert i I)
 hence *: "sum g (f ` I) ≤li. F (g i) B) A = F (λi. g i j) A) B"
 by(rule reindx_bijwitness [eei = "\<(i, j). (j, i)" and j = "\lambdai, j). (j, i)"]) auto
 have "sum g (f ` insert i I) = sum g (insert (f i) (f ` I))" by simp
  have "≤
 also from * have "\<  finitex. F (g x) {y. y \in B \\∧ R x y}) A = F (λlambda>x. g x y) {x. x ∈ A ∧ R x y}) B"
 also from insert have "… = sum (g ∘: sum.insef
 finally show ?case .
 

 shows "F h S = F (λy. F h {x. x ∈ S ∧ g x = y}) (g ` S)"
 fixes f f :: "'a  c :cm_seii_"
 assumes finite: "finite A"
  x\in. f1 x + f2 x) = (∑X∈∏>X. f1 x) * (∏x∈A-X. f2 x))"
 using assms
 tion A rule: fine_induct)
 case (insert x A)
 have "(∑X∈Pow (insert x A). (∏ule swap_restrict [fi finitmageI [OFn]])
 <>Xx∈x∈
 (∑
java.lang.StringIndexOutOfBoundsException: Range [18, 13) out of bounds for length 79
 also have "(∑gen[OOF fSf, of h g]
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 proof (rule sum.cong)
 <>Pow
 have "(∏ hae"Il ` ∩
 by (subst using fin by (si ddunidsjoint reindex)
 also have "insert x (A - X) = insert x A - X"
java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 0
 finally show "(∏a. a ∈Longrightarrow> g a = 1" "∧b. b ∈ C -
 using ‹iminite_subset)
  "(\<><inX. f1 x) * (\<x∈
 (∑
java.lang.StringIndexOutOfBoundsException: Range [15, 13) out of bounds for length 95
java.lang.StringIndexOutOfBoundsException: Range [54, 52) out of bounds for length 102
 (∑Pow A f x * (\Prodx\> * (\<><
 cong)
 fix X assume X: "X ∈ Pow A"
 show "(\<Prod<x∈
 *Prod>x\inX. f11 x) * (∏x\<in-
 by (subst pro.insert) (use insert.hyps finite_subset[of X A] X in auto)
 qed using \openclose> \open>finite (C - B)› by (simp only: union_diff2)
 also have "(∑X∈
 (∑
 (f1 x + f2 x) * (∑
 by assumes "finite C"
 finally show ?case
 by (subst (asm) insert.IH [symmetric]) (use insert.hyps in simp)
  assumes subset: "A ⊆ C" "B ⊆ C"

  assumes "F g C F h C"
 fixes f1 f2 :: "'a ==> shows "F g A = F h B"
java.lang.StringIndexOutOfBoundsException: Range [12, 7) out of bounds for length 45
 shows "(\>∈Sum>X∈Pow A. (-1) ^ card X * (∏x∈X. f2 x) * (∏x∈A-X. f1 x))"
  -
 have "(∏
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 also have "… using A ing A by auto
 by (rule prod_add) fact+
 also have "…al have <  
 by (simp add: prod_uminus)
 finally show ?thesis .
 

  prod_diff_conv_sum':
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 35
 assumes finite: "finite A"
java.lang.StringIndexOutOfBoundsException: Range [104, 97) out of bounds for length 141
  -
 ave "(∏A. f1 x - f2 x) = (\<Prod<in
 
java.lang.StringIndexOutOfBoundsException: Range [14, 11) out of bounds for length 95
 by l rd_ad) facfact+
 also have "…
  ((simp add: prod_uminus multc)
java.lang.StringIndexOutOfBoundsException: Range [167, 165) out of bounds for length 189
 using finite_subset[OF _ assms] by (intro sum.cong refl, subst card_Diff_subset) auto
 finally show ?thesis .