Theory Groups_Big

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

section Big sum and product over finite (non-empty) sets

theory Groups_Big
  imports Power
begin

subsection Generic monoid operation over a set

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 [simp]: "¬ finite A  F g A = 1"
  by (simp add: eq_fold)

lemma empty [simp]: "F g {} = 1"
  by (simp add: eq_fold)

lemma insert [simp]: "finite A  x  A  F g (insert x A) = g x * F g A"
  by (simp add: eq_fold)

lemma remove:
  assumes "finite A" and "x  A"
  shows "F g A = g x * F g (A - {x})"
proof -
  from x  A obtain B where B: "A = insert x B" and "x  B"
    by (auto dest: mk_disjoint_insert)
  moreover from finite A B have "finite B" by simp
  ultimately show ?thesis by simp
qed

lemma insert_remove: "finite A  F g (insert x A) = g x * F g (A - {x})"
  by (cases "x  A") (simp_all add: remove insert_absorb)

lemma 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)

lemma neutral: "xA. g x = 1  F g A = 1"
  by (induct A rule: infinite_finite_induct) simp_all

lemma neutral_const [simp]: "F (λ_. 1) A = 1"
  by (simp add: neutral)

lemma union_inter:
  assumes "finite A" and "finite B"
  shows "F g (A  B) * F g (A  B) = F g A * F g B"
  ― ‹The reversed orientation looks more natural, but LOOPS as a simprule!
  using assms
proof (induct A)
  case empty
  then show ?case by simp
next
  case (insert x A)
  then show ?case
    by (auto simp: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
qed

corollary union_inter_neutral:
  assumes "finite A" and "finite B"
    and "x  A  B. g x = 1"
  shows "F g (A  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  B = {}"
  shows "F g (A  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  B) = F g (A - B) * F g (B - A) * F g (A  B)"
proof -
  have "A  B = A - B  (B - A)  A  B"
    by auto
  with assms show ?thesis
    by simp (subst union_disjoint, auto)+
qed

lemma subset_diff:
  assumes "B  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)  B = {}" by auto
  ultimately have "F g (A - B  B) = F g (A - B) * F g B" by (rule union_disjoint)
  moreover from assms have "A  B = A" by auto
  ultimately show ?thesis by simp
qed

lemma Int_Diff:
  assumes "finite A"
  shows "F g A = F g (A  B) * F g (A - B)"
  by (subst subset_diff [where B = "A - B"]) (auto simp:  Diff_Diff_Int assms)

lemma setdiff_irrelevant:
  assumes "finite A"
  shows "F g (A - {x. g x = z}) = F g A"
  using assms by (induct A) (simp_all add: insert_Diff_if)

lemma not_neutral_contains_not_neutral:
  assumes "F g A  1"
  obtains a where "a  A" and "g a  1"
proof -
  from assms have "aA. g a  1"
  proof (induct A rule: infinite_finite_induct)
    case infinite
    then show ?case by simp
  next
    case empty
    then show ?case by simp
  next
    case (insert a A)
    then show ?case by fastforce
  qed
  with that show thesis by blast
qed

lemma reindex:
  assumes "inj_on h A"
  shows "F g (h ` A) = F (g  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 "¬ finite (h ` A)" by (blast dest: finite_imageD)
  with False show ?thesis by simp
qed

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

lemma cong_simp [cong]:
  " A = B;  x. x  B =simp=> g x = h x   F (λx. g x) A = F (λx. h x) B"
by (rule cong) (simp_all add: simp_implies_def)

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

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

lemma UNION_disjoint:
  assumes "finite I" and "iI. finite (A i)"
    and "iI. jI. i  j  A i  A j = {}"
  shows "F g ((A ` I)) = F (λx. F g (A x)) I"
  using assms
proof (induction rule: finite_induct)
  case (insert i I)
  then have "jI. j  i"
    by blast
  with insert.prems have "A i  (A ` I) = {}"
    by blast
  with insert show ?case
    by (simp add: union_disjoint)
qed auto

lemma Union_disjoint:
  assumes "AC. finite A" "AC. BC. A  B  A  B = {}"
  shows "F g (C) = (F  F) g C"
proof (cases "finite C")
  case True
  from UNION_disjoint [OF this assms] show ?thesis by simp
next
  case False
  then show ?thesis by (auto dest: finite_UnionD intro: infinite)
qed

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

lemma Sigma:
  assumes "finite A" "xA. finite (B x)"
  shows "F (λx. F (g x) (B x)) A = F (case_prod g) (SIGMA x:A. B x)"
  unfolding Sigma_def
proof (subst UNION_disjoint)
  show "F (λx. F (g x) (B x)) A = F (λx. F (λ(x, y). g x y) (yB x. {(x, y)})) A"
  proof (rule cong [OF refl])
    show "F (g x) (B x) = F (λ(x, y). g x y) (yB x. {(x, y)})"
      if "x  A" for x
      using that assms by (simp add: UNION_disjoint)
  qed
qed (use assms in auto)

lemma 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: "xS. R (h x) (g x)"
  shows "R (F h S) (F g S)"
  using fin by (rule finite_subset_induct) (use assms in auto)

lemma mono_neutral_cong_left:
  assumes "finite T"
    and "S  T"
    and "i  T - S. h i = 1"
    and "x. x  S  g x = h x"
  shows "F g S = F h T"
proof-
  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)])
qed

lemma 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])

lemma 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)

lemma 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])

lemma 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"
proof-
  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 "kS. ?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 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 "iA - 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 "xA  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  1}  {i  I. h i  1}) = G g I"
         "F h ({i  I. g i  1}  {i  I. h i  1}) = G h I"
      by (auto simp: G_def assms intro: mono_neutral_right)
    then show ?thesis
      using assms by (simp add: distrib)
  qed
  finally show ?thesis .
qed

lemma cong':
  assumes "A = B"
  assumes g_h: "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, termsum (λx. e) A is written ∑x∈A. e›.

syntax (ASCII)
  "_sum" :: "pttrn  'a set  'b  'b::comm_monoid_add"  ("(3SUM (_/:_)./ _)" [0, 51, 10] 10)
syntax
  "_sum" :: "pttrn  'a set  'b  'b::comm_monoid_add"  ("(2(_/_)./ _)" [0, 51, 10] 10)
translations ― ‹Beware of argument permutation!
  "iA. b"  "CONST sum (λi. b) A"

text Instead of termx{x. P}. e we introduce the shorter ∑x|P. e›.

syntax (ASCII)
  "_qsum" :: "pttrn  bool  'a  'a"  ("(3SUM _ |/ _./ _)" [0, 0, 10] 10)
syntax
  "_qsum" :: "pttrn  bool  'a  'a"  ("(2_ | (_)./ _)" [0, 0, 10] 10)
translations
  "x|P. t" => "CONST sum (λx. t) {x. P}"

print_translation 
let
  fun sum_tr' [Abs (x, Tx, t), Const (const_syntaxCollect, _) $ 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‹_qsum› $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'
          end
    | sum_tr' _ = raise Match;
in [(const_syntaxsum, K sum_tr')] end



subsubsection Properties in more restricted classes of structures

lemma 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)

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

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 induct (auto simp: insert_Diff_if)

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"
proof -
  from assms(2,1) have "finite B" by (rule finite_subset)
  from this B  A
  show ?thesis
  proof induct
    case empty
    thus ?case by simp
  next
    case (insert x F)
    with finite A finite B show ?case
      by (simp add: Diff_insert[where a=x and B=F] sum_diff1 insert_absorb)
  qed
qed

lemma sum_diff1'_aux:
  fixes f :: "'a  '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  'b::ab_group_add"
  assumes "finite {i  I. f i  0}"
  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. iK  f i  g i)  (iK. f i)  (iK. g i)"
  by (induct K rule: infinite_finite_induct) (use add_mono in auto)

lemma (in strict_ordered_comm_monoid_add) sum_strict_mono:
  assumes "finite A" "A  {}"
    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 "xA. f x  g x"
    and "aA. 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 "iI. 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 "... = (xA. 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: "(xA. - f x) = - (xA. f x)"
  for f :: "'b  'a::ab_group_add"
  by (induct A rule: infinite_finite_induct) auto

lemma sum_subtractf: "(xA. f x - g x) = (xA. f x) - (xA. 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)  (xA. f x - g x) = (xA. f x) - (xA. 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  (xA. 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 "yt. 0  g y" "(xs. yt. i y = x  f x  g y)"
  shows "sum f s  sum g t"
proof -
  have "sum f s  sum (λy. sum g {x. xt  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. xt  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) = (aF. 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 = (nA. r * f n)"
  by (induct A rule: infinite_finite_induct) (simp_all add: algebra_simps)

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

end

lemma sum_divide_distrib: "sum f A / r = (nA. 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]: "¦aA. ¦f a¦¦ = (aA. ¦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 "¦ainsert a A. ¦f a¦¦ = ¦¦f a¦ + (aA. ¦f a¦)¦" by simp
  also from insert have " = ¦¦f a¦ + ¦aA. ¦f a¦¦¦" by simp
  also have " = ¦f a¦ + ¦aA. ¦f a¦¦" by (simp del: abs_of_nonneg)
  also from insert have " = (ainsert 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 = (iA. jB. 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  aA. 0 < f a"
  by (induct A rule: infinite_finite_induct) auto

lemma sum_eq_Suc0_iff:
  "finite A  sum f A = Suc 0  (aA. f a = Suc 0  (bA. 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
  then show ?case
    apply (auto simp: insert_Diff_if)
    apply (drule mk_disjoint_insert)
    apply auto
    done
qed

lemma sum_diff_nat:
  fixes f :: "'a  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 constsum

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. iA  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. iA  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. iA  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. iA  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 "¦(iI. a i * x i) - b¦  δ"
proof -
  have [simp]: "(iI. c * x i) = c" for c
    by (simp flip: sum_distrib_left 1)
  then have "¦(iI. a i * x i) - b¦ = ¦iI. (a i - b) * x i¦"
    by (simp add: sum_subtractf left_diff_distrib)
  also have "  (iI. ¦(a i - b) * x i¦)"
    using abs_abs abs_of_nonneg by blast
  also have "  (iI. ¦(a i - b)¦ * x i)"
    by (simp add: abs_mult 0)
  also have "  (iI. 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 "iI. finite (A i)"
    and "iI. jI. i  j  A i  A j = {}"
  shows "card ((A ` I)) = (iI. card(A i))"
proof -
  have "(iI. card (A i)) = (iI. xA 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:
  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_UN_le:
  assumes "finite I"
  shows "card(iI. A i)  (iI. card(A i))"
  using assms
proof induction
  case (insert i I)
  then show ?case
    using card_Un_le nat_add_left_cancel_le by (force intro: order_trans) 
qed auto

lemma sum_multicount_gen:
  assumes "finite s" "finite t" "jt. (card {is. R i j} = k j)"
  shows "sum (λi. (card {jt. R i j})) s = sum k t"
    (is "?l = ?r")
proof-
  have "?l = sum (λi. sum (λx.1) {jt. 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" "jT. (card {iS. R i j} = k)"
  shows "sum (λi. card {jT. 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


subsubsection Cardinality of products

lemma card_SigmaI [simp]:
  "finite A  aA. finite (B a)  card (SIGMA x: A. B x) = (aA. card (B a))"
  by (simp add: card_eq_sum sum.Sigma del: sum_constant)

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

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

lemma card_cartesian_product_singleton:  "card ({x} × A) = card A"
  by (simp add: card_cartesian_product)


subsection Generalized product over a set

context comm_monoid_mult
begin

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

abbreviation Prod ("_" [1000] 999)
  where "A  prod (λx. x) A"

end

syntax (ASCII)
  "_prod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(4PROD (_/:_)./ _)" [0, 51, 10] 10)
syntax
  "_prod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(2(_/_)./ _)" [0, 51, 10] 10)
translations ― ‹Beware of argument permutation!
  "iA. b" == "CONST prod (λi. b) A"

text Instead of termx{x. P}. e we introduce the shorter ∏x|P. e›.

syntax (ASCII)
  "_qprod" :: "pttrn  bool  'a  'a"  ("(4PROD _ |/ _./ _)" [0, 0, 10] 10)
syntax
  "_qprod" :: "pttrn  bool  'a  'a"  ("(2_ | (_)./ _)" [0, 0, 10] 10)
translations
  "x|P. t" => "CONST prod (λx. t) {x. P}"

context comm_monoid_mult
begin

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

lemma prod_dvd_prod_subset: "finite B  A  B  prod f A dvd prod f B"
  by (auto simp add: prod.subset_diff ac_simps intro: dvdI)

end


subsubsection Properties in more restricted classes of structures

context linordered_nonzero_semiring
begin

lemma prod_ge_1: "(x. x  A  1  f x)  1  prod f A"
proof (induct A rule: infinite_finite_induct)
  case infinite
  then show ?case by simp
next
  case empty
  then show ?case by simp
next
  case (insert x F)
  have "1 * 1  f x * prod f F"
    by (rule mult_mono') (use insert in auto)
  with insert show ?case by simp
qed

lemma prod_le_1:
  fixes f :: "'b  'a"
  assumes "x. x  A  0  f x  f x  1"
  shows "prod f A  1"
    using assms
proof (induct A rule: infinite_finite_induct)
  case infinite
  then show ?case by simp
next
  case empty
  then show ?case by simp
next
  case (insert x F)
  then show ?case by (force simp: mult.commute intro: dest: mult_le_one)
qed

end

context comm_semiring_1
begin

lemma dvd_prod_eqI [intro]:
  assumes "finite A" and "a  A" and "b = f a"
  shows "b dvd prod f A"
proof -
  from finite A have "prod f (insert a (A - {a})) = f a * prod f (A - {a})"
    by (intro prod.insert) auto
  also from a  A have "insert a (A - {a}) = A"
    by blast
  finally have "prod f A = f a * prod f (A - {a})" .
  with b = f a show ?thesis
    by simp
qed

lemma dvd_prodI [intro]: "finite A  a  A  f a dvd prod f A"
  by auto

lemma prod_zero:
  assumes "finite A" and "aA. f a = 0"
  shows "prod f A = 0"
  using assms
proof (induct A)
  case empty
  then show ?case by simp
next
  case (insert a A)
  then have "f a = 0  (aA. f a = 0)" by simp
  then have "f a * prod f A = 0" by rule (simp_all add: insert)
  with insert show ?case by simp
qed

lemma prod_dvd_prod_subset2:
  assumes "finite B" and "A  B" and "a. a  A  f a dvd g a"
  shows "prod f A dvd prod g B"
proof -
  from assms have "prod f A dvd prod g A"
    by (auto intro: prod_dvd_prod)
  moreover from assms have "prod g A dvd prod g B"
    by (auto intro: prod_dvd_prod_subset)
  ultimately show ?thesis by (rule dvd_trans)
qed

end

lemma (in semidom) prod_zero_iff [simp]:
  fixes f :: "'b  'a"
  assumes "finite A"
  shows "prod f A = 0  (aA. f a = 0)"
  using assms by (induct A) (auto simp: no_zero_divisors)

lemma (in semidom_divide) prod_diff1:
  assumes "finite A" and "f a  0"
  shows "prod f (A - {a}) = (if a  A then prod f A div f a else prod f A)"
proof (cases "a  A")
  case True
  then show ?thesis by simp
next
  case False
  with assms show ?thesis
  proof induct
    case empty
    then show ?case by simp
  next
    case (insert b B)
    then show ?case
    proof (cases "a = b")
      case True
      with insert show ?thesis by simp
    next
      case False
      with insert have "a  B" by simp
      define C where "C = B - {a}"
      with finite B a  B have "B = insert a C" "finite C" "a  C"
        by auto
      with insert show ?thesis
        by (auto simp add: insert_commute ac_simps)
    qed
  qed
qed

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

lemma sum_zero_power' [simp]:
  "(iA. c i * 0^i / d i) = (if finite A  0  A then c 0 / d 0 else 0)"
  for c :: "nat  'a::field"
  using sum_zero_power [of "λi. c i / d i" A] by auto

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

lemma (in field) prod_dividef: "(xA. f x / g x) = prod f A / prod g A"
  using prod_inversef [of g A] by (simp add: divide_inverse prod.distrib)

lemma prod_Un:
  fixes f :: "'b  'a :: field"
  assumes "finite A" and "finite B"
    and "xA  B. f x  0"
  shows "prod f (A  B) = prod f A * prod f B / prod f (A  B)"
proof -
  from assms have "prod f A * prod f B = prod f (A  B) * prod f (A  B)"
    by (simp add: prod.union_inter [symmetric, of A B])
  with assms show ?thesis
    by simp
qed

context linordered_semidom
begin

lemma prod_nonneg: "(aA. 0  f a)  0  prod f A"
  by (induct A rule: infinite_finite_induct) simp_all

lemma prod_pos: "(aA. 0 < f a)  0 < prod f A"
  by (induct A rule: infinite_finite_induct) simp_all

lemma prod_mono:
  "(i. i  A  0  f i  f i  g i)  prod f A  prod g A"
  by (induct A rule: infinite_finite_induct) (force intro!: prod_nonneg mult_mono)+

lemma prod_mono_strict:
  assumes "finite A" "i. i  A  0  f i  f i < g i" "A  {}"
  shows "prod f A < prod g A"
  using assms
proof (induct A rule: finite_induct)
  case empty
  then show ?case by simp
next
  case insert
  then show ?case by (force intro: mult_strict_mono' prod_nonneg)
qed

lemma prod_le_power:
  assumes A: "i. i  A  0  f i  f i  n" "card A  k" and "n  1"
  shows "prod f A  n ^ k"
  using A
proof (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  n * n ^ k'"
    using insert n  1 k' by (intro prod_nonneg mult_mono; force)
  then show ?case 
    by (auto simp: k = Suc k' insert.hyps)
qed (use n  1 in auto)

end

lemma prod_mono2:
  fixes f :: "'a  'b :: linordered_idom"
  assumes fin: "finite B"
    and sub: "A  B"
    and nn: "