# Theory Groups_Big

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

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

theory Groups_Big
imports Power Equiv_Relations
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"

lemma empty [simp]: "F g {} = ❙1"

lemma insert [simp]: "finite A ⟹ x ∉ A ⟹ F g (insert x A) = g x ❙* F g A"

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: "∀x∈A. 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"

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 "∃a∈A. 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 "∀i∈I. finite (A i)"
and "∀i∈I. ∀j∈I. 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 "∀j∈I. j ≠ i"
by blast
with insert.prems have "A i ∩ ⋃(A ` I) = {}"
by blast
with insert show ?case
qed auto

lemma Union_disjoint:
assumes "∀A∈C. finite A" "∀A∈C. ∀B∈C. 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" "∀x∈A. 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) (⋃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
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: "∀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)

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
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
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 "∀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)"
with ‹finite B› ‹A ∉ B› show ?case
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"

lemma non_neutral': "G g {x ∈ I. g x ≠ ❙1} = G g I"

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›

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 ‹∑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!›
"∑i∈A. b" ⇌ "CONST sum (λi. b) A"

text ‹Instead of \<^term>‹∑x∈{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_syntax>‹Collect›, _) \$ Abs (y, Ty, P)] =
if x <> y then raise Match
else
let
val x' = Syntax_Trans.mark_bound_body (x, Tx);
val t' = subst_bound (x', t);
val P' = subst_bound (x', P);
in
Syntax.const \<^syntax_const>‹_qsum› \$ Syntax_Trans.mark_bound_abs (x, Tx) \$ P' \$ t'
end
| sum_tr' _ = raise Match;
in [(\<^const_syntax>‹sum›, 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

(*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)

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])

"(⋀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)

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
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})"
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)

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)"

lemma sum_nonneg_0:
"finite s ⟹ (⋀i. i ∈ s ⟹ f i ≥ 0) ⟹ (∑ i ∈ s. f i) = 0 ⟹ i ∈ s ⟹ f i = 0"

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)"
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

"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
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"

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))"

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"
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

"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)

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
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)"
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
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:
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:
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"]

lemma sum_bounded_below:
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¦"
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)"
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
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 = 0" if asm: "y ∈ R - g ` S" for y
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 ∈ C. f (g x))"
using eqv finite disjoint
by (simp flip: sum.Union_disjoint[simplified] add: Union_quotient)
also have "... = (∑y ∈ g ` S. ∑x ∈ ?cls y. f (g x))"
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 ∈ R. of_nat (card (?cls y)) * f y)"
using rest_0 by (simp add: sum.subset_diff[OF ‹g ` S ⊆ R› ‹finite R›])
finally show ?thesis
qed

subsubsection ‹Cardinality of products›

lemma card_SigmaI [simp]:
"finite A ⟹ ∀a∈A. finite (B a) ⟹ card (SIGMA x: A. B x) = (∑a∈A. 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"

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!›
"∏i∈A. b" == "CONST prod (λi. b) A"

text ‹Instead of \<^term>‹∏x∈{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)"
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 "∃a∈A. 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 ∨ (∃a∈A. f a = 0)" by simp
then have "f a * prod f A = 0" by (rule disjE) (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 ⟷ (∃a∈A. f a = 0)"
using assms by (induct A) (a```