(* 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 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) ❙1A" 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: "∀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" 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 "∃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 by (simp add: union_disjoint) 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 : 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 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 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 = ❙1then {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 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) 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. 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 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 = 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 by (simp add: eq_commute) 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" by (simp add: card_cartesian_product) subsection ‹Generalized product over a set› context comm_monoid_mult begin 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)" 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 "∃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