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title.tex
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(* Title: HOL/Groups_Big.thy
Author: Tobias Nipkow
Author: Lawrence C Paulson
Author: Markus Wenzel
Author: Jeremy Avigad
*)
section \<open>Big sum and product over finite (non-empty) sets\<close>
theory Groups_Big
imports Power
begin
subsection \<open>Generic monoid operation over a set\<close>
locale comm_monoid_set = comm_monoid
begin
subsubsection \<open>Standard sum or product indexed by a finite set\<close>
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) \<^bold>1 A"
lemma infinite [simp]: "\ finite A \ F g A = \<^bold>1"
by (simp add: eq_fold)
lemma empty [simp]: "F g {} = \<^bold>1"
by (simp add: eq_fold)
lemma insert [simp]: "finite A \ x \ A \ F g (insert x A) = g x \<^bold>* F g A"
by (simp add: eq_fold)
lemma remove:
assumes "finite A" and "x \ A"
shows "F g A = g x \<^bold>* F g (A - {x})"
proof -
from \<open>x \<in> A\<close> obtain B where B: "A = insert x B" and "x \<notin> B"
by (auto dest: mk_disjoint_insert)
moreover from \<open>finite A\<close> B have "finite B" by simp
ultimately show ?thesis by simp
qed
lemma insert_remove: "finite A \ F g (insert x A) = g x \<^bold>* 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 \<^bold>* F g A)"
by (cases "x \ A") (simp_all add: insert_absorb)
lemma neutral: "\x\A. g x = \<^bold>1 \ F g A = \<^bold>1"
by (induct A rule: infinite_finite_induct) simp_all
lemma neutral_const [simp]: "F (\_. \<^bold>1) A = \<^bold>1"
by (simp add: neutral)
lemma union_inter:
assumes "finite A" and "finite B"
shows "F g (A \ B) \<^bold>* F g (A \ B) = F g A \<^bold>* F g B"
\<comment> \<open>The reversed orientation looks more natural, but LOOPS as a simprule!\<close>
using assms
proof (induct A)
case empty
then show ?case by simp
next
case (insert x A)
then show ?case
by (auto simp: 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 = \<^bold>1"
shows "F g (A \ B) = F g A \<^bold>* 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 \<^bold>* 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) \<^bold>* F g (B - A) \<^bold>* 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) \<^bold>* 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) \<^bold>* 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) \<^bold>* 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 \ \<^bold>1"
obtains a where "a \ A" and "g a \ \<^bold>1"
proof -
from assms have "\a\A. g a \ \<^bold>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 \<open>A = B\<close>
by (induct B rule: infinite_finite_induct) auto
lemma cong_simp [cong]:
"\ A = B; \x. x \ B =simp=> g x = h x \ \ F (\x. g x) A = F (\x. 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 (\x. 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 (\x. 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 (\x. g x \<^bold>* h x) A = F g A \<^bold>* 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 (\x. F (g x) (B x)) A = F (case_prod g) (SIGMA x:A. B x)"
unfolding Sigma_def
proof (subst UNION_disjoint)
show "F (\x. F (g x) (B x)) A = F (\x. F (\(x, y). g x y) (\y\B x. {(x, y)})) A"
proof (rule cong [OF refl])
show "F (g x) (B x) = F (\(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 \<^bold>1 \<^bold>1"
and Rop: "\x1 y1 x2 y2. R x1 x2 \ R y1 y2 \ R (x1 \<^bold>* y1) (x2 \<^bold>* 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 = \<^bold>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 \<open>finite T\<close> \<open>S \<subseteq> T\<close> have f: "finite S" "finite (T - S)"
by (auto intro: finite_subset)
show ?thesis using assms(4)
by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
qed
lemma mono_neutral_cong_right:
"finite T \ S \ T \ \i \ T - S. g i = \<^bold>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 = \<^bold>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 = \<^bold>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 = \<^bold>1" "\i. i \ S - T \ g i = \<^bold>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 (\x. 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 (\x. h (j x)) S"
by (intro cong) (auto simp: eq)
ultimately show ?thesis
by (simp add: reindex_bij_betw)
qed
lemma reindex_bij_betw_not_neutral:
assumes fin: "finite S'" "finite T'"
assumes bij: "bij_betw h (S - S') (T - T')"
assumes nn:
"\a. a \ S' \ g (h a) = z"
"\b. b \ T' \ g b = z"
shows "F (\x. 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 (\x. g (h x)) S = F (\x. 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 \<open>finite S\<close> 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) = \<^bold>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 = \<^bold>1}) (h ` A - h ` {x \ A. (g \ h) x = \<^bold>1})"
using nz by (auto intro!: inj_onI simp: bij_betw_def)
qed (use \<open>finite A\<close> 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 (\x. h (j x)) S"
by (intro cong) (auto simp: eq)
show ?thesis
unfolding F_eq using fin nn eq
by (intro reindex_bij_betw_not_neutral[OF _ _ bij]) auto
qed
lemma delta_remove:
assumes fS: "finite S"
shows "F (\k. if k = a then b k else c k) S = (if a \ S then b a \<^bold>* F c (S-{a}) else F c (S-{a}))"
proof -
let ?f = "(\k. 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 \<^bold>* 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 (\k. if k = a then b k else \<^bold>1) S = (if a \ S then b a else \<^bold>1)"
by (simp add: delta_remove [OF assms])
lemma delta' [simp]:
assumes fin: "finite S"
shows "F (\k. if a = k then b k else \<^bold>1) S = (if a \ S then b a else \<^bold>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 (\x. if P x then h x else g x) A = F h (A \ {x. P x}) \<^bold>* 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 = "\x. if P x then h x else g x"
from union_disjoint [OF f a(2), of ?g] a(1) show ?thesis
by (subst (1 2) cong) simp_all
qed
lemma cartesian_product: "F (\x. 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 \<open>A \<noteq> {}\<close> \<open>B \<noteq> {}\<close> in \<open>auto dest: finite_cartesian_productD1 finite_cartesian_productD2\<close>)
then show ?thesis
using False by auto
qed
qed
lemma inter_restrict:
assumes "finite A"
shows "F g (A \ B) = F (\x. if x \ B then g x else \<^bold>1) A"
proof -
let ?g = "\x. if x \ A \ B then g x else \<^bold>1"
have "\i\A - A \ B. (if i \ A \ B then g i else \<^bold>1) = \<^bold>1" by simp
moreover have "A \ B \ A" by blast
ultimately have "F ?g (A \ B) = F ?g A"
using \<open>finite A\<close> by (intro mono_neutral_left) auto
then show ?thesis by simp
qed
lemma inter_filter:
"finite A \ F g {x \ A. P x} = F (\x. if P x then g x else \<^bold>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 = \<^bold>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 = \<^bold>1"
and H: "F g (\B) = (F \ F) g B" by auto
then have "F g (A \ \B) = F g A \<^bold>* F g (\B)"
by (simp add: union_inter_neutral)
with \<open>finite B\<close> \<open>A \<notin> B\<close> show ?case
by (simp add: H)
qed
lemma swap: "F (\i. F (g i) B) A = F (\j. F (\i. g i j) A) B"
unfolding cartesian_product
by (rule reindex_bij_witness [where i = "\(i, j). (j, i)" and j = "\(i, j). (j, i)"]) auto
lemma swap_restrict:
"finite A \ finite B \
F (\<lambda>x. F (g x) {y. y \<in> B \<and> R x y}) A = F (\<lambda>y. F (\<lambda>x. g x y) {x. x \<in> A \<and> R x y}) B"
by (simp add: inter_filter) (rule swap)
lemma image_gen:
assumes fin: "finite S"
shows "F h S = F (\y. 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 (\x. F (\y. h x) {y. y\ g`S \ g x = y}) S"
by simp
also have "\ = F (\y. 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 (\y. 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 \<^bold>* 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 = \<^bold>1" "\b. b \ C - B \ h b = \<^bold>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 \<open>finite C\<close> 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)) \<^bold>* F g (C - A - A) \<^bold>* F g (A \ (C - A))"
using \<open>finite A\<close> \<open>finite (C - A)\<close> 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)) \<^bold>* F h (C - B - B) \<^bold>* F h (B \ (C - B))"
using \<open>finite B\<close> \<open>finite (C - B)\<close> 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 = \<^bold>1" "\b. b \ C - B \ h b = \<^bold>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 \ \(h x) = \ x"
shows "F \ A = F \ 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 \ A = F (\ \ h) A"
using A by auto
also have "\ = F \ 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 \ \(h x) = \ x"
shows "F \ A = F \ B"
by (rule eq_general [where h=h]) (force intro: dest: A B)+
subsubsection \<open>HOL Light variant: sum/product indexed by the non-neutral subset\<close>
text \<open>NB only a subset of the properties above are proved\<close>
definition G :: "['b \ 'a,'b set] \ 'a"
where "G p I \ if finite {x \ I. p x \ \<^bold>1} then F p {x \ I. p x \ \<^bold>1} else \<^bold>1"
lemma finite_Collect_op:
shows "\finite {i \ I. x i \ \<^bold>1}; finite {i \ I. y i \ \<^bold>1}\ \ finite {i \ I. x i \<^bold>* y i \ \<^bold>1}"
apply (rule finite_subset [where B = "{i \ I. x i \ \<^bold>1} \ {i \ I. y i \ \<^bold>1}"])
using left_neutral by force+
lemma empty' [simp]: "G p {} = \<^bold>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 \ \<^bold>1}"
shows "G p (insert i I) = (if i \ I then G p I else p i \<^bold>* G p I)"
proof -
have "{x. x = i \ p x \ \<^bold>1 \ x \ I \ p x \ \<^bold>1} = (if p i = \<^bold>1 then {x \ I. p x \ \<^bold>1} else insert i {x \ I. p x \ \<^bold>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 (\i. g i \<^bold>* h i) I = G g I \<^bold>* G h I"
by (simp add: assms local.distrib)
lemma non_neutral': "G g {x \ I. g x \ \<^bold>1} = G g I"
by (simp add: G_def)
lemma distrib':
assumes "finite {x \ I. g x \ \<^bold>1}" "finite {x \ I. h x \ \<^bold>1}"
shows "G (\i. g i \<^bold>* h i) I = G g I \<^bold>* G h I"
proof -
have "a \<^bold>* a \ a \ a \ \<^bold>1" for a
by auto
then have "G (\i. g i \<^bold>* h i) I = G (\i. g i \<^bold>* h i) ({i \ I. g i \ \<^bold>1} \ {i \ I. h i \ \<^bold>1})"
using assms by (force simp: G_def finite_Collect_op intro!: mono_neutral_cong)
also have "\ = G g I \<^bold>* G h I"
proof -
have "F g ({i \ I. g i \ \<^bold>1} \ {i \ I. h i \ \<^bold>1}) = G g I"
"F h ({i \ I. g i \ \<^bold>1} \ {i \ I. h i \ \<^bold>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 = \<^bold>1"
and "\x. x \ S \ g x = h x"
shows "G g S = G h T"
proof -
have *: "{x \ S. g x \ \<^bold>1} = {x \ T. h x \ \<^bold>1}"
using assms by (metis DiffI subset_eq)
then have "finite {x \ S. g x \ \<^bold>1} = finite {x \ T. h x \ \<^bold>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 = \<^bold>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 = \<^bold>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 = \<^bold>1 \ G g T = G g S"
by (blast intro!: mono_neutral_left' [symmetric])
end
subsection \<open>Generalized summation over a set\<close>
context comm_monoid_add
begin
sublocale sum: comm_monoid_set plus 0
defines sum = sum.F and sum' = sum.G ..
abbreviation Sum ("\")
where "\ \ sum (\x. x)"
end
text \<open>Now: lots of fancy syntax. First, \<^term>\<open>sum (\<lambda>x. e) A\<close> is written \<open>\<Sum>x\<in>A. e\<close>.\<close>
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 \<comment> \<open>Beware of argument permutation!\<close>
"\i\A. b" \ "CONST sum (\i. b) A"
text \<open>Instead of \<^term>\<open>\<Sum>x\<in>{x. P}. e\<close> we introduce the shorter \<open>\<Sum>x|P. e\<close>.\<close>
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 (\x. t) {x. P}"
print_translation \<open>
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>\<open>_qsum\<close> $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'
end
| sum_tr' _ = raise Match;
in [(\<^const_syntax>\<open>sum\<close>, K sum_tr')] end
\<close>
subsubsection \<open>Properties in more restricted classes of structures\<close>
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 \<open>B \<subseteq> A\<close>
show ?thesis
proof induct
case empty
thus ?case by simp
next
case (insert x F)
with \<open>finite A\<close> \<open>finite B\<close> 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. 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 \<open>a \<in> A\<close>])
also have "\ = sum f (A - {a}) + sum f {a}"
using \<open>finite A\<close> 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 \<open>finite A\<close> 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 (\y. 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 (\y. 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 (\i. \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 (\i. \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 (\(a, b). f a * g b) (A \ B) \
sum f A * sum g B = sum id {f a * g b |a b. a \<in> A \<and> b \<in> 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"
\<comment> \<open>For the natural numbers, we have subtraction.\<close>
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 = \<open>F \<subseteq> A \<Longrightarrow> sum f (A - F) = sum f A - sum f F\<close>
from \<open>x \<notin> F\<close> \<open>insert x F \<subseteq> A\<close> have "x \<in> 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 \<open>insert x F \<subseteq> A\<close> have "F \<subseteq> 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 \<open>x \<notin> F\<close> 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 \<open>finite F\<close> \<open>x \<notin> F\<close> 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 \<open>Cardinality as special case of \<^const>\<open>sum\<close>\<close>
lemma card_eq_sum: "card A = sum (\x. 1) A"
proof -
have "plus \ (\_. Suc 0) = (\_. Suc)"
by (simp add: fun_eq_iff)
then have "Finite_Set.fold (plus \ (\_. Suc 0)) = Finite_Set.fold (\_. Suc)"
by (rule arg_cong)
then have "Finite_Set.fold (plus \ (\_. Suc 0)) 0 A = Finite_Set.fold (\_. 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)
end
lemma sum_Suc: "sum (\x. Suc(f x)) A = sum f A + card A"
using sum.distrib[of f "\_. 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 = "\x. 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 = "\x. 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 = "\x. 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 \<delta>: "\<And>i. i \<in> I \<Longrightarrow> \<bar>a i - b\<bar> \<le> \<delta>"
shows "\(\i\I. a i * x i) - b\ \ \"
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. \ * x i)"
by (rule sum_mono) (use \<delta> "0" mult_right_mono in blast)
also have "\ = \"
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 "\x. 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(\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 sum_multicount_gen:
assumes "finite s" "finite t" "\j\t. (card {i\s. R i j} = k j)"
shows "sum (\i. (card {j\t. R i j})) s = sum k t"
(is "?l = ?r")
proof-
have "?l = sum (\i. sum (\x.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 (\i. card {j\T. R i j}) S = k * card T" (is "?l = ?r")
proof-
have "?l = sum (\i. 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 (\s t. disjnt (f s) (f t)) A"
shows "sum card (f ` A) = sum (\a. 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 \<open>Cardinality of products\<close>
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 \<notin> A ==>
(SIGMA x:(insert y A). B x) = (({y} \<times> (B y)) \<union> (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 \<open>Generalized product over a set\<close>
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 (\x. 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 \<comment> \<open>Beware of argument permutation!\<close>
"\i\A. b" == "CONST prod (\i. b) A"
text \<open>Instead of \<^term>\<open>\<Prod>x\<in>{x. P}. e\<close> we introduce the shorter \<open>\<Prod>x|P. e\<close>.\<close>
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 (\x. 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 \<open>Properties in more restricted classes of structures\<close>
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 \<open>finite A\<close> have "prod f (insert a (A - {a})) = f a * prod f (A - {a})"
by (intro prod.insert) auto
also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A"
by blast
finally have "prod f A = f a * prod f (A - {a})" .
with \<open>b = f a\<close> 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 (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) (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 \<open>finite B\<close> \<open>a \<in> B\<close> have "B = insert a C" "finite C" "a \<notin> C"
by auto
with insert show ?thesis
by (auto simp add: insert_commute ac_simps)
qed
qed
qed
lemma sum_zero_power [simp]: "(\i\A. 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]:
"(\i\A. 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 "\i. 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: "(\x\A. 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 "\x\A \ 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: "(\a\A. 0 \ f a) \ 0 \ prod f A"
by (induct A rule: infinite_finite_induct) simp_all
lemma prod_pos: "(\a\A. 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
end
lemma prod_mono2:
fixes f :: "'a \ 'b :: linordered_idom"
assumes fin: "finite B"
and sub: "A \ B"
and nn: "\b. b \ B-A \ 1 \ f b"
and A: "\a. a \ A \ 0 \ f a"
shows "prod f A \ prod f B"
proof -
have "prod f A \ prod f A * prod f (B-A)"
by (metis prod_ge_1 A mult_le_cancel_left1 nn not_less prod_nonneg)
also from fin finite_subset[OF sub fin] have "\ = prod f (A \ (B-A))"
by (simp add: prod.union_disjoint del: Un_Diff_cancel)
also from sub have "A \ (B-A) = B" by blast
finally show ?thesis .
qed
lemma less_1_prod:
fixes f :: "'a \ 'b::linordered_idom"
shows "finite I \ I \ {} \ (\i. i \ I \ 1 < f i) \ 1 < prod f I"
by (induct I rule: finite_ne_induct) (auto intro: less_1_mult)
lemma less_1_prod2:
fixes f :: "'a \ 'b::linordered_idom"
assumes I: "finite I" "i \ I" "1 < f i" "\i. i \ I \ 1 \ f i"
shows "1 < prod f I"
proof -
have "1 < f i * prod f (I - {i})"
using assms
by (meson DiffD1 leI less_1_mult less_le_trans mult_le_cancel_left1 prod_ge_1)
also have "\ = prod f I"
using assms by (simp add: prod.remove)
finally show ?thesis .
qed
lemma (in linordered_field) abs_prod: "\prod f A\ = (\x\A. \f x\)"
by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult)
lemma prod_eq_1_iff [simp]: "finite A \ prod f A = 1 \ (\a\A. f a = 1)"
for f :: "'a \ nat"
by (induct A rule: finite_induct) simp_all
lemma prod_pos_nat_iff [simp]: "finite A \ prod f A > 0 \ (\a\A. f a > 0)"
for f :: "'a \ nat"
using prod_zero_iff by (simp del: neq0_conv add: zero_less_iff_neq_zero)
lemma prod_constant [simp]: "(\x\ A. y) = y ^ card A"
for y :: "'a::comm_monoid_mult"
by (induct A rule: infinite_finite_induct) simp_all
lemma prod_power_distrib: "prod f A ^ n = prod (\x. (f x) ^ n) A"
for f :: "'a \ 'b::comm_semiring_1"
by (induct A rule: infinite_finite_induct) (auto simp add: power_mult_distrib)
lemma power_sum: "c ^ (\a\A. f a) = (\a\A. c ^ f a)"
by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
lemma prod_gen_delta:
fixes b :: "'b \ 'a::comm_monoid_mult"
assumes fin: "finite S"
shows "prod (\k. if k = a then b k else c) S =
(if a \<in> S then b a * c ^ (card S - 1) else c ^ card S)"
proof -
let ?f = "(\k. if k=a then b k else c)"
show ?thesis
proof (cases "a \ S")
case False
then have "\ k\ S. ?f k = c" by simp
with False show ?thesis by (simp add: prod_constant)
next
case True
let ?A = "S - {a}"
let ?B = "{a}"
from True have eq: "S = ?A \ ?B" by blast
have disjoint: "?A \ ?B = {}" by simp
from fin have fin': "finite ?A" "finite ?B" by auto
have f_A0: "prod ?f ?A = prod (\i. c) ?A"
by (rule prod.cong) auto
from fin True have card_A: "card ?A = card S - 1" by auto
have f_A1: "prod ?f ?A = c ^ card ?A"
unfolding f_A0 by (rule prod_constant)
have "prod ?f ?A * prod ?f ?B = prod ?f S"
using prod.union_disjoint[OF fin' disjoint, of ?f, unfolded eq[symmetric]]
by simp
with True card_A show ?thesis
by (simp add: f_A1 field_simps cong add: prod.cong cong del: if_weak_cong)
qed
qed
lemma sum_image_le:
fixes g :: "'a \ 'b::ordered_comm_monoid_add"
assumes "finite I" "\i. i \ I \ 0 \ g(f i)"
shows "sum g (f ` I) \ sum (g \ f) I"
using assms
proof induction
case empty
then show ?case by auto
next
case (insert x F)
from insertI1 have "0 \ g (f x)" by (rule insert)
hence 1: "sum g (f ` F) \ g (f x) + sum g (f ` F)" using add_increasing by blast
have 2: "sum g (f ` F) \ sum (g \ f) F" using insert by blast
have "sum g (f ` insert x F) = sum g (insert (f x) (f ` F))" by simp
also have "\ \ g (f x) + sum g (f ` F)" by (simp add: 1 insert sum.insert_if)
also from 2 have "\ \ g (f x) + sum (g \ f) F" by (rule add_left_mono)
also from insert(1, 2) have "\ = sum (g \ f) (insert x F)" by (simp add: sum.insert_if)
finally show ?case .
qed
end
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