src/HOL/Groups_Big.thy
 author wenzelm Sun Dec 27 22:07:17 2015 +0100 (2015-12-27) changeset 61943 7fba644ed827 parent 61799 4cf66f21b764 child 61944 5d06ecfdb472 permissions -rw-r--r--
discontinued ASCII replacement syntax <*>;
1 (*  Title:      HOL/Groups_Big.thy
2     Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
3                 with contributions by Jeremy Avigad
4 *)
6 section \<open>Big sum and product over finite (non-empty) sets\<close>
8 theory Groups_Big
9 imports Finite_Set
10 begin
12 subsection \<open>Generic monoid operation over a set\<close>
14 no_notation times (infixl "*" 70)
15 no_notation Groups.one ("1")
17 locale comm_monoid_set = comm_monoid
18 begin
20 interpretation comp_fun_commute f
21   by standard (simp add: fun_eq_iff left_commute)
23 interpretation comp?: comp_fun_commute "f \<circ> g"
24   by (fact comp_comp_fun_commute)
26 definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
27 where
28   eq_fold: "F g A = Finite_Set.fold (f \<circ> g) 1 A"
30 lemma infinite [simp]:
31   "\<not> finite A \<Longrightarrow> F g A = 1"
32   by (simp add: eq_fold)
34 lemma empty [simp]:
35   "F g {} = 1"
36   by (simp add: eq_fold)
38 lemma insert [simp]:
39   assumes "finite A" and "x \<notin> A"
40   shows "F g (insert x A) = g x * F g A"
41   using assms by (simp add: eq_fold)
43 lemma remove:
44   assumes "finite A" and "x \<in> A"
45   shows "F g A = g x * F g (A - {x})"
46 proof -
47   from \<open>x \<in> A\<close> obtain B where A: "A = insert x B" and "x \<notin> B"
48     by (auto dest: mk_disjoint_insert)
49   moreover from \<open>finite A\<close> A have "finite B" by simp
50   ultimately show ?thesis by simp
51 qed
53 lemma insert_remove:
54   assumes "finite A"
55   shows "F g (insert x A) = g x * F g (A - {x})"
56   using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
58 lemma neutral:
59   assumes "\<forall>x\<in>A. g x = 1"
60   shows "F g A = 1"
61   using assms by (induct A rule: infinite_finite_induct) simp_all
63 lemma neutral_const [simp]:
64   "F (\<lambda>_. 1) A = 1"
65   by (simp add: neutral)
67 lemma union_inter:
68   assumes "finite A" and "finite B"
69   shows "F g (A \<union> B) * F g (A \<inter> B) = F g A * F g B"
70   \<comment> \<open>The reversed orientation looks more natural, but LOOPS as a simprule!\<close>
71 using assms proof (induct A)
72   case empty then show ?case by simp
73 next
74   case (insert x A) then show ?case
75     by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
76 qed
78 corollary union_inter_neutral:
79   assumes "finite A" and "finite B"
80   and I0: "\<forall>x \<in> A \<inter> B. g x = 1"
81   shows "F g (A \<union> B) = F g A * F g B"
82   using assms by (simp add: union_inter [symmetric] neutral)
84 corollary union_disjoint:
85   assumes "finite A" and "finite B"
86   assumes "A \<inter> B = {}"
87   shows "F g (A \<union> B) = F g A * F g B"
88   using assms by (simp add: union_inter_neutral)
90 lemma union_diff2:
91   assumes "finite A" and "finite B"
92   shows "F g (A \<union> B) = F g (A - B) * F g (B - A) * F g (A \<inter> B)"
93 proof -
94   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
95     by auto
96   with assms show ?thesis by simp (subst union_disjoint, auto)+
97 qed
99 lemma subset_diff:
100   assumes "B \<subseteq> A" and "finite A"
101   shows "F g A = F g (A - B) * F g B"
102 proof -
103   from assms have "finite (A - B)" by auto
104   moreover from assms have "finite B" by (rule finite_subset)
105   moreover from assms have "(A - B) \<inter> B = {}" by auto
106   ultimately have "F g (A - B \<union> B) = F g (A - B) * F g B" by (rule union_disjoint)
107   moreover from assms have "A \<union> B = A" by auto
108   ultimately show ?thesis by simp
109 qed
111 lemma setdiff_irrelevant:
112   assumes "finite A"
113   shows "F g (A - {x. g x = z}) = F g A"
114   using assms by (induct A) (simp_all add: insert_Diff_if)
116 lemma not_neutral_contains_not_neutral:
117   assumes "F g A \<noteq> z"
118   obtains a where "a \<in> A" and "g a \<noteq> z"
119 proof -
120   from assms have "\<exists>a\<in>A. g a \<noteq> z"
121   proof (induct A rule: infinite_finite_induct)
122     case (insert a A)
123     then show ?case by simp (rule, simp)
124   qed simp_all
125   with that show thesis by blast
126 qed
128 lemma reindex:
129   assumes "inj_on h A"
130   shows "F g (h ` A) = F (g \<circ> h) A"
131 proof (cases "finite A")
132   case True
133   with assms show ?thesis by (simp add: eq_fold fold_image comp_assoc)
134 next
135   case False with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
136   with False show ?thesis by simp
137 qed
139 lemma cong:
140   assumes "A = B"
141   assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
142   shows "F g A = F h B"
143   using g_h unfolding \<open>A = B\<close>
144   by (induct B rule: infinite_finite_induct) auto
146 lemma strong_cong [cong]:
147   assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
148   shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
149   by (rule cong) (insert assms, simp_all add: simp_implies_def)
151 lemma reindex_cong:
152   assumes "inj_on l B"
153   assumes "A = l ` B"
154   assumes "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x"
155   shows "F g A = F h B"
156   using assms by (simp add: reindex)
158 lemma UNION_disjoint:
159   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
160   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
161   shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
162 apply (insert assms)
163 apply (induct rule: finite_induct)
164 apply simp
165 apply atomize
166 apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
167  prefer 2 apply blast
168 apply (subgoal_tac "A x Int UNION Fa A = {}")
169  prefer 2 apply blast
170 apply (simp add: union_disjoint)
171 done
173 lemma Union_disjoint:
174   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
175   shows "F g (Union C) = (F \<circ> F) g C"
176 proof cases
177   assume "finite C"
178   from UNION_disjoint [OF this assms]
179   show ?thesis by simp
180 qed (auto dest: finite_UnionD intro: infinite)
182 lemma distrib:
183   "F (\<lambda>x. g x * h x) A = F g A * F h A"
184   using assms by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)
186 lemma Sigma:
187   "finite A \<Longrightarrow> \<forall>x\<in>A. finite (B x) \<Longrightarrow> F (\<lambda>x. F (g x) (B x)) A = F (case_prod g) (SIGMA x:A. B x)"
188 apply (subst Sigma_def)
189 apply (subst UNION_disjoint, assumption, simp)
190  apply blast
191 apply (rule cong)
192 apply rule
193 apply (simp add: fun_eq_iff)
194 apply (subst UNION_disjoint, simp, simp)
195  apply blast
196 apply (simp add: comp_def)
197 done
199 lemma related:
200   assumes Re: "R 1 1"
201   and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
202   and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
203   shows "R (F h S) (F g S)"
204   using fS by (rule finite_subset_induct) (insert assms, auto)
206 lemma mono_neutral_cong_left:
207   assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1"
208   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
209 proof-
210   have eq: "T = S \<union> (T - S)" using \<open>S \<subseteq> T\<close> by blast
211   have d: "S \<inter> (T - S) = {}" using \<open>S \<subseteq> T\<close> by blast
212   from \<open>finite T\<close> \<open>S \<subseteq> T\<close> have f: "finite S" "finite (T - S)"
213     by (auto intro: finite_subset)
214   show ?thesis using assms(4)
215     by (simp add: union_disjoint [OF f d, unfolded eq [symmetric]] neutral [OF assms(3)])
216 qed
218 lemma mono_neutral_cong_right:
219   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1; \<And>x. x \<in> S \<Longrightarrow> g x = h x \<rbrakk>
220    \<Longrightarrow> F g T = F h S"
221   by (auto intro!: mono_neutral_cong_left [symmetric])
223 lemma mono_neutral_left:
224   "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g S = F g T"
225   by (blast intro: mono_neutral_cong_left)
227 lemma mono_neutral_right:
228   "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S"
229   by (blast intro!: mono_neutral_left [symmetric])
231 lemma reindex_bij_betw: "bij_betw h S T \<Longrightarrow> F (\<lambda>x. g (h x)) S = F g T"
232   by (auto simp: bij_betw_def reindex)
234 lemma reindex_bij_witness:
235   assumes witness:
236     "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"
237     "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"
238     "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"
239     "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"
240   assumes eq:
241     "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
242   shows "F g S = F h T"
243 proof -
244   have "bij_betw j S T"
245     using bij_betw_byWitness[where A=S and f=j and f'=i and A'=T] witness by auto
246   moreover have "F g S = F (\<lambda>x. h (j x)) S"
247     by (intro cong) (auto simp: eq)
248   ultimately show ?thesis
249     by (simp add: reindex_bij_betw)
250 qed
252 lemma reindex_bij_betw_not_neutral:
253   assumes fin: "finite S'" "finite T'"
254   assumes bij: "bij_betw h (S - S') (T - T')"
255   assumes nn:
256     "\<And>a. a \<in> S' \<Longrightarrow> g (h a) = z"
257     "\<And>b. b \<in> T' \<Longrightarrow> g b = z"
258   shows "F (\<lambda>x. g (h x)) S = F g T"
259 proof -
260   have [simp]: "finite S \<longleftrightarrow> finite T"
261     using bij_betw_finite[OF bij] fin by auto
263   show ?thesis
264   proof cases
265     assume "finite S"
266     with nn have "F (\<lambda>x. g (h x)) S = F (\<lambda>x. g (h x)) (S - S')"
267       by (intro mono_neutral_cong_right) auto
268     also have "\<dots> = F g (T - T')"
269       using bij by (rule reindex_bij_betw)
270     also have "\<dots> = F g T"
271       using nn \<open>finite S\<close> by (intro mono_neutral_cong_left) auto
272     finally show ?thesis .
273   qed simp
274 qed
276 lemma reindex_nontrivial:
277   assumes "finite A"
278   and nz: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> h x = h y \<Longrightarrow> g (h x) = 1"
279   shows "F g (h ` A) = F (g \<circ> h) A"
280 proof (subst reindex_bij_betw_not_neutral [symmetric])
281   show "bij_betw h (A - {x \<in> A. (g \<circ> h) x = 1}) (h ` A - h ` {x \<in> A. (g \<circ> h) x = 1})"
282     using nz by (auto intro!: inj_onI simp: bij_betw_def)
283 qed (insert \<open>finite A\<close>, auto)
285 lemma reindex_bij_witness_not_neutral:
286   assumes fin: "finite S'" "finite T'"
287   assumes witness:
288     "\<And>a. a \<in> S - S' \<Longrightarrow> i (j a) = a"
289     "\<And>a. a \<in> S - S' \<Longrightarrow> j a \<in> T - T'"
290     "\<And>b. b \<in> T - T' \<Longrightarrow> j (i b) = b"
291     "\<And>b. b \<in> T - T' \<Longrightarrow> i b \<in> S - S'"
292   assumes nn:
293     "\<And>a. a \<in> S' \<Longrightarrow> g a = z"
294     "\<And>b. b \<in> T' \<Longrightarrow> h b = z"
295   assumes eq:
296     "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
297   shows "F g S = F h T"
298 proof -
299   have bij: "bij_betw j (S - (S' \<inter> S)) (T - (T' \<inter> T))"
300     using witness by (intro bij_betw_byWitness[where f'=i]) auto
301   have F_eq: "F g S = F (\<lambda>x. h (j x)) S"
302     by (intro cong) (auto simp: eq)
303   show ?thesis
304     unfolding F_eq using fin nn eq
305     by (intro reindex_bij_betw_not_neutral[OF _ _ bij]) auto
306 qed
308 lemma delta:
309   assumes fS: "finite S"
310   shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)"
311 proof-
312   let ?f = "(\<lambda>k. if k=a then b k else 1)"
313   { assume a: "a \<notin> S"
314     hence "\<forall>k\<in>S. ?f k = 1" by simp
315     hence ?thesis  using a by simp }
316   moreover
317   { assume a: "a \<in> S"
318     let ?A = "S - {a}"
319     let ?B = "{a}"
320     have eq: "S = ?A \<union> ?B" using a by blast
321     have dj: "?A \<inter> ?B = {}" by simp
322     from fS have fAB: "finite ?A" "finite ?B" by auto
323     have "F ?f S = F ?f ?A * F ?f ?B"
324       using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
325       by simp
326     then have ?thesis using a by simp }
327   ultimately show ?thesis by blast
328 qed
330 lemma delta':
331   assumes fS: "finite S"
332   shows "F (\<lambda>k. if a = k then b k else 1) S = (if a \<in> S then b a else 1)"
333   using delta [OF fS, of a b, symmetric] by (auto intro: cong)
335 lemma If_cases:
336   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
337   assumes fA: "finite A"
338   shows "F (\<lambda>x. if P x then h x else g x) A =
339     F h (A \<inter> {x. P x}) * F g (A \<inter> - {x. P x})"
340 proof -
341   have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}"
342           "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
343     by blast+
344   from fA
345   have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
346   let ?g = "\<lambda>x. if P x then h x else g x"
347   from union_disjoint [OF f a(2), of ?g] a(1)
348   show ?thesis
349     by (subst (1 2) cong) simp_all
350 qed
352 lemma cartesian_product:
353    "F (\<lambda>x. F (g x) B) A = F (case_prod g) (A \<times> B)"
354 apply (rule sym)
355 apply (cases "finite A")
356  apply (cases "finite B")
357   apply (simp add: Sigma)
358  apply (cases "A={}", simp)
359  apply simp
360 apply (auto intro: infinite dest: finite_cartesian_productD2)
361 apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
362 done
364 lemma inter_restrict:
365   assumes "finite A"
366   shows "F g (A \<inter> B) = F (\<lambda>x. if x \<in> B then g x else 1) A"
367 proof -
368   let ?g = "\<lambda>x. if x \<in> A \<inter> B then g x else 1"
369   have "\<forall>i\<in>A - A \<inter> B. (if i \<in> A \<inter> B then g i else 1) = 1"
370    by simp
371   moreover have "A \<inter> B \<subseteq> A" by blast
372   ultimately have "F ?g (A \<inter> B) = F ?g A" using \<open>finite A\<close>
373     by (intro mono_neutral_left) auto
374   then show ?thesis by simp
375 qed
377 lemma inter_filter:
378   "finite A \<Longrightarrow> F g {x \<in> A. P x} = F (\<lambda>x. if P x then g x else 1) A"
379   by (simp add: inter_restrict [symmetric, of A "{x. P x}" g, simplified mem_Collect_eq] Int_def)
381 lemma Union_comp:
382   assumes "\<forall>A \<in> B. finite A"
383     and "\<And>A1 A2 x. A1 \<in> B \<Longrightarrow> A2 \<in> B  \<Longrightarrow> A1 \<noteq> A2 \<Longrightarrow> x \<in> A1 \<Longrightarrow> x \<in> A2 \<Longrightarrow> g x = 1"
384   shows "F g (\<Union>B) = (F \<circ> F) g B"
385 using assms proof (induct B rule: infinite_finite_induct)
386   case (infinite A)
387   then have "\<not> finite (\<Union>A)" by (blast dest: finite_UnionD)
388   with infinite show ?case by simp
389 next
390   case empty then show ?case by simp
391 next
392   case (insert A B)
393   then have "finite A" "finite B" "finite (\<Union>B)" "A \<notin> B"
394     and "\<forall>x\<in>A \<inter> \<Union>B. g x = 1"
395     and H: "F g (\<Union>B) = (F o F) g B" by auto
396   then have "F g (A \<union> \<Union>B) = F g A * F g (\<Union>B)"
397     by (simp add: union_inter_neutral)
398   with \<open>finite B\<close> \<open>A \<notin> B\<close> show ?case
399     by (simp add: H)
400 qed
402 lemma commute:
403   "F (\<lambda>i. F (g i) B) A = F (\<lambda>j. F (\<lambda>i. g i j) A) B"
404   unfolding cartesian_product
405   by (rule reindex_bij_witness [where i = "\<lambda>(i, j). (j, i)" and j = "\<lambda>(i, j). (j, i)"]) auto
407 lemma commute_restrict:
408   "finite A \<Longrightarrow> finite B \<Longrightarrow>
409     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"
410   by (simp add: inter_filter) (rule commute)
412 lemma Plus:
413   fixes A :: "'b set" and B :: "'c set"
414   assumes fin: "finite A" "finite B"
415   shows "F g (A <+> B) = F (g \<circ> Inl) A * F (g \<circ> Inr) B"
416 proof -
417   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
418   moreover from fin have "finite (Inl ` A :: ('b + 'c) set)" "finite (Inr ` B :: ('b + 'c) set)"
419     by auto
420   moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('b + 'c) set)" by auto
421   moreover have "inj_on (Inl :: 'b \<Rightarrow> 'b + 'c) A" "inj_on (Inr :: 'c \<Rightarrow> 'b + 'c) B"
422     by (auto intro: inj_onI)
423   ultimately show ?thesis using fin
424     by (simp add: union_disjoint reindex)
425 qed
427 lemma same_carrier:
428   assumes "finite C"
429   assumes subset: "A \<subseteq> C" "B \<subseteq> C"
430   assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = 1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = 1"
431   shows "F g A = F h B \<longleftrightarrow> F g C = F h C"
432 proof -
433   from \<open>finite C\<close> subset have
434     "finite A" and "finite B" and "finite (C - A)" and "finite (C - B)"
435     by (auto elim: finite_subset)
436   from subset have [simp]: "A - (C - A) = A" by auto
437   from subset have [simp]: "B - (C - B) = B" by auto
438   from subset have "C = A \<union> (C - A)" by auto
439   then have "F g C = F g (A \<union> (C - A))" by simp
440   also have "\<dots> = F g (A - (C - A)) * F g (C - A - A) * F g (A \<inter> (C - A))"
441     using \<open>finite A\<close> \<open>finite (C - A)\<close> by (simp only: union_diff2)
442   finally have P: "F g C = F g A" using trivial by simp
443   from subset have "C = B \<union> (C - B)" by auto
444   then have "F h C = F h (B \<union> (C - B))" by simp
445   also have "\<dots> = F h (B - (C - B)) * F h (C - B - B) * F h (B \<inter> (C - B))"
446     using \<open>finite B\<close> \<open>finite (C - B)\<close> by (simp only: union_diff2)
447   finally have Q: "F h C = F h B" using trivial by simp
448   from P Q show ?thesis by simp
449 qed
451 lemma same_carrierI:
452   assumes "finite C"
453   assumes subset: "A \<subseteq> C" "B \<subseteq> C"
454   assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = 1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = 1"
455   assumes "F g C = F h C"
456   shows "F g A = F h B"
457   using assms same_carrier [of C A B] by simp
459 end
461 notation times (infixl "*" 70)
462 notation Groups.one ("1")
465 subsection \<open>Generalized summation over a set\<close>
468 begin
470 sublocale setsum: comm_monoid_set plus 0
471 defines
472   setsum = setsum.F ..
474 abbreviation
475   Setsum ("\<Sum>_"  999) where
476   "\<Sum>A \<equiv> setsum (%x. x) A"
478 end
480 text\<open>Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
481 written \<open>\<Sum>x\<in>A. e\<close>.\<close>
483 syntax
484   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(3SUM _:_./ _)" [0, 51, 10] 10)
485 syntax (xsymbols)
486   "_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add"    ("(2\<Sum>_\<in>_./ _)" [0, 51, 10] 10)
488 translations \<comment> \<open>Beware of argument permutation!\<close>
489   "SUM i:A. b" == "CONST setsum (%i. b) A"
490   "\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A"
492 text\<open>Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
493  \<open>\<Sum>x|P. e\<close>.\<close>
495 syntax
496   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
497 syntax (xsymbols)
498   "_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(2\<Sum>_ | (_)./ _)" [0,0,10] 10)
500 translations
501   "SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
502   "\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
504 print_translation \<open>
505 let
506   fun setsum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) \$ Abs (y, Ty, P)] =
507         if x <> y then raise Match
508         else
509           let
510             val x' = Syntax_Trans.mark_bound_body (x, Tx);
511             val t' = subst_bound (x', t);
512             val P' = subst_bound (x', P);
513           in
514             Syntax.const @{syntax_const "_qsetsum"} \$ Syntax_Trans.mark_bound_abs (x, Tx) \$ P' \$ t'
515           end
516     | setsum_tr' _ = raise Match;
517 in [(@{const_syntax setsum}, K setsum_tr')] end
518 \<close>
520 text \<open>TODO generalization candidates\<close>
522 lemma setsum_image_gen:
523   assumes fS: "finite S"
524   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
525 proof-
526   { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
527   hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
528     by simp
529   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
530     by (rule setsum.commute_restrict [OF fS finite_imageI [OF fS]])
531   finally show ?thesis .
532 qed
535 subsubsection \<open>Properties in more restricted classes of structures\<close>
537 lemma setsum_Un: "finite A ==> finite B ==>
538   (setsum f (A Un B) :: 'a :: ab_group_add) =
539    setsum f A + setsum f B - setsum f (A Int B)"
540 by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
542 lemma setsum_Un2:
543   assumes "finite (A \<union> B)"
544   shows "setsum f (A \<union> B) = setsum f (A - B) + setsum f (B - A) + setsum f (A \<inter> B)"
545 proof -
546   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
547     by auto
548   with assms show ?thesis by simp (subst setsum.union_disjoint, auto)+
549 qed
551 lemma setsum_diff1: "finite A \<Longrightarrow>
552   (setsum f (A - {a}) :: ('a::ab_group_add)) =
553   (if a:A then setsum f A - f a else setsum f A)"
554 by (erule finite_induct) (auto simp add: insert_Diff_if)
556 lemma setsum_diff:
557   assumes le: "finite A" "B \<subseteq> A"
558   shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
559 proof -
560   from le have finiteB: "finite B" using finite_subset by auto
561   show ?thesis using finiteB le
562   proof induct
563     case empty
564     thus ?case by auto
565   next
566     case (insert x F)
567     thus ?case using le finiteB
568       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
569   qed
570 qed
572 lemma setsum_mono:
573   assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, ordered_ab_semigroup_add}))"
574   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
575 proof (cases "finite K")
576   case True
577   thus ?thesis using le
578   proof induct
579     case empty
580     thus ?case by simp
581   next
582     case insert
583     thus ?case using add_mono by fastforce
584   qed
585 next
586   case False then show ?thesis by simp
587 qed
589 lemma setsum_strict_mono:
590   fixes f :: "'a \<Rightarrow> 'b::{ordered_cancel_ab_semigroup_add,comm_monoid_add}"
591   assumes "finite A"  "A \<noteq> {}"
592     and "!!x. x:A \<Longrightarrow> f x < g x"
593   shows "setsum f A < setsum g A"
594   using assms
595 proof (induct rule: finite_ne_induct)
596   case singleton thus ?case by simp
597 next
598   case insert thus ?case by (auto simp: add_strict_mono)
599 qed
601 lemma setsum_strict_mono_ex1:
602 fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add, ordered_cancel_ab_semigroup_add}"
603 assumes "finite A" and "ALL x:A. f x \<le> g x" and "EX a:A. f a < g a"
604 shows "setsum f A < setsum g A"
605 proof-
606   from assms(3) obtain a where a: "a:A" "f a < g a" by blast
607   have "setsum f A = setsum f ((A-{a}) \<union> {a})"
608     by(simp add:insert_absorb[OF \<open>a:A\<close>])
609   also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
610     using \<open>finite A\<close> by(subst setsum.union_disjoint) auto
611   also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
612     by(rule setsum_mono)(simp add: assms(2))
613   also have "setsum f {a} < setsum g {a}" using a by simp
614   also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
615     using \<open>finite A\<close> by(subst setsum.union_disjoint[symmetric]) auto
616   also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF \<open>a:A\<close>])
617   finally show ?thesis by (auto simp add: add_right_mono add_strict_left_mono)
618 qed
620 lemma setsum_negf: "(\<Sum>x\<in>A. - f x::'a::ab_group_add) = - (\<Sum>x\<in>A. f x)"
621 proof (cases "finite A")
622   case True thus ?thesis by (induct set: finite) auto
623 next
624   case False thus ?thesis by simp
625 qed
627 lemma setsum_subtractf: "(\<Sum>x\<in>A. f x - g x::'a::ab_group_add) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
628   using setsum.distrib [of f "- g" A] by (simp add: setsum_negf)
630 lemma setsum_subtractf_nat:
631   "(\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x) \<Longrightarrow> (\<Sum>x\<in>A. f x - g x::nat) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
632   by (induction A rule: infinite_finite_induct) (auto simp: setsum_mono)
634 lemma setsum_nonneg:
635   assumes nn: "\<forall>x\<in>A. (0::'a::{ordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
636   shows "0 \<le> setsum f A"
637 proof (cases "finite A")
638   case True thus ?thesis using nn
639   proof induct
640     case empty then show ?case by simp
641   next
642     case (insert x F)
643     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
644     with insert show ?case by simp
645   qed
646 next
647   case False thus ?thesis by simp
648 qed
650 lemma setsum_nonpos:
651   assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{ordered_ab_semigroup_add,comm_monoid_add})"
652   shows "setsum f A \<le> 0"
653 proof (cases "finite A")
654   case True thus ?thesis using np
655   proof induct
656     case empty then show ?case by simp
657   next
658     case (insert x F)
659     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
660     with insert show ?case by simp
661   qed
662 next
663   case False thus ?thesis by simp
664 qed
666 lemma setsum_nonneg_leq_bound:
667   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
668   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
669   shows "f i \<le> B"
670 proof -
671   have "0 \<le> (\<Sum> i \<in> s - {i}. f i)" and "0 \<le> f i"
672     using assms by (auto intro!: setsum_nonneg)
673   moreover
674   have "(\<Sum> i \<in> s - {i}. f i) + f i = B"
675     using assms by (simp add: setsum_diff1)
676   ultimately show ?thesis by auto
677 qed
679 lemma setsum_nonneg_0:
680   fixes f :: "'a \<Rightarrow> 'b::{ordered_ab_group_add}"
681   assumes "finite s" and pos: "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
682   and "(\<Sum> i \<in> s. f i) = 0" and i: "i \<in> s"
683   shows "f i = 0"
684   using setsum_nonneg_leq_bound[OF assms] pos[OF i] by auto
686 lemma setsum_mono2:
687 fixes f :: "'a \<Rightarrow> 'b :: ordered_comm_monoid_add"
688 assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
689 shows "setsum f A \<le> setsum f B"
690 proof -
691   have "setsum f A \<le> setsum f A + setsum f (B-A)"
692     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
693   also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
694     by (simp add: setsum.union_disjoint del:Un_Diff_cancel)
695   also have "A \<union> (B-A) = B" using sub by blast
696   finally show ?thesis .
697 qed
699 lemma setsum_le_included:
700   fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"
701   assumes "finite s" "finite t"
702   and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"
703   shows "setsum f s \<le> setsum g t"
704 proof -
705   have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
706   proof (rule setsum_mono)
707     fix y assume "y \<in> s"
708     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
709     with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
710       using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
711       by (auto intro!: setsum_mono2)
712   qed
713   also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
714     using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
715   also have "... \<le> setsum g t"
716     using assms by (auto simp: setsum_image_gen[symmetric])
717   finally show ?thesis .
718 qed
720 lemma setsum_mono3: "finite B ==> A <= B ==>
721     ALL x: B - A.
723         setsum f A <= setsum f B"
724   apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
725   apply (erule ssubst)
726   apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
727   apply simp
728   apply (rule add_left_mono)
729   apply (erule setsum_nonneg)
730   apply (subst setsum.union_disjoint [THEN sym])
731   apply (erule finite_subset, assumption)
732   apply (rule finite_subset)
733   prefer 2
734   apply assumption
735   apply (auto simp add: sup_absorb2)
736 done
738 lemma setsum_right_distrib:
739   fixes f :: "'a => ('b::semiring_0)"
740   shows "r * setsum f A = setsum (%n. r * f n) A"
741 proof (cases "finite A")
742   case True
743   thus ?thesis
744   proof induct
745     case empty thus ?case by simp
746   next
747     case (insert x A) thus ?case by (simp add: distrib_left)
748   qed
749 next
750   case False thus ?thesis by simp
751 qed
753 lemma setsum_left_distrib:
754   "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
755 proof (cases "finite A")
756   case True
757   then show ?thesis
758   proof induct
759     case empty thus ?case by simp
760   next
761     case (insert x A) thus ?case by (simp add: distrib_right)
762   qed
763 next
764   case False thus ?thesis by simp
765 qed
767 lemma setsum_divide_distrib:
768   "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
769 proof (cases "finite A")
770   case True
771   then show ?thesis
772   proof induct
773     case empty thus ?case by simp
774   next
775     case (insert x A) thus ?case by (simp add: add_divide_distrib)
776   qed
777 next
778   case False thus ?thesis by simp
779 qed
781 lemma setsum_abs[iff]:
782   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
783   shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
784 proof (cases "finite A")
785   case True
786   thus ?thesis
787   proof induct
788     case empty thus ?case by simp
789   next
790     case (insert x A)
791     thus ?case by (auto intro: abs_triangle_ineq order_trans)
792   qed
793 next
794   case False thus ?thesis by simp
795 qed
797 lemma setsum_abs_ge_zero[iff]:
798   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
799   shows "0 \<le> setsum (%i. abs(f i)) A"
800   by (simp add: setsum_nonneg)
802 lemma abs_setsum_abs[simp]:
803   fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
804   shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
805 proof (cases "finite A")
806   case True
807   thus ?thesis
808   proof induct
809     case empty thus ?case by simp
810   next
811     case (insert a A)
812     hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
813     also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
814     also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
815       by (simp del: abs_of_nonneg)
816     also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
817     finally show ?case .
818   qed
819 next
820   case False thus ?thesis by simp
821 qed
823 lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
824   shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
825   unfolding setsum.remove [OF assms] by auto
827 lemma setsum_product:
828   fixes f :: "'a => ('b::semiring_0)"
829   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
830   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum.commute)
832 lemma setsum_mult_setsum_if_inj:
833 fixes f :: "'a => ('b::semiring_0)"
834 shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
835   setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
836 by(auto simp: setsum_product setsum.cartesian_product
837         intro!:  setsum.reindex_cong[symmetric])
839 lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
840 apply (case_tac "finite A")
841  prefer 2 apply simp
842 apply (erule rev_mp)
843 apply (erule finite_induct, auto)
844 done
846 lemma setsum_eq_0_iff [simp]:
847   "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
848   by (induct set: finite) auto
850 lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
851   setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
852 apply(erule finite_induct)
854 done
856 lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
858 lemma setsum_Un_nat: "finite A ==> finite B ==>
859   (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
860   \<comment> \<open>For the natural numbers, we have subtraction.\<close>
861 by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
863 lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
864   (if a:A then setsum f A - f a else setsum f A)"
865 apply (case_tac "finite A")
866  prefer 2 apply simp
867 apply (erule finite_induct)
868  apply (auto simp add: insert_Diff_if)
869 apply (drule_tac a = a in mk_disjoint_insert, auto)
870 done
872 lemma setsum_diff_nat:
873 assumes "finite B" and "B \<subseteq> A"
874 shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
875 using assms
876 proof induct
877   show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
878 next
879   fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
880     and xFinA: "insert x F \<subseteq> A"
881     and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
882   from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
883   from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
884     by (simp add: setsum_diff1_nat)
885   from xFinA have "F \<subseteq> A" by simp
886   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
887   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
888     by simp
889   from xnotinF have "A - insert x F = (A - F) - {x}" by auto
890   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
891     by simp
892   from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
893   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
894     by simp
895   thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
896 qed
898 lemma setsum_comp_morphism:
899   assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
900   shows "setsum (h \<circ> g) A = h (setsum g A)"
901 proof (cases "finite A")
902   case False then show ?thesis by (simp add: assms)
903 next
904   case True then show ?thesis by (induct A) (simp_all add: assms)
905 qed
907 lemma (in comm_semiring_1) dvd_setsum:
908   "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd setsum f A"
909   by (induct A rule: infinite_finite_induct) simp_all
911 lemma setsum_pos2:
912     assumes "finite I" "i \<in> I" "0 < f i" "(\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i)"
913       shows "(0::'a::{ordered_ab_group_add,comm_monoid_add}) < setsum f I"
914 proof -
915   have "0 \<le> setsum f (I-{i})"
916     using assms by (simp add: setsum_nonneg)
917   also have "... < setsum f (I-{i}) + f i"
918     using assms by auto
919   also have "... = setsum f I"
920     using assms by (simp add: setsum.remove)
921   finally show ?thesis .
922 qed
924 lemma setsum_cong_Suc:
925   assumes "0 \<notin> A" "\<And>x. Suc x \<in> A \<Longrightarrow> f (Suc x) = g (Suc x)"
926   shows   "setsum f A = setsum g A"
927 proof (rule setsum.cong)
928   fix x assume "x \<in> A"
929   with assms(1) show "f x = g x" by (cases x) (auto intro!: assms(2))
930 qed simp_all
933 subsubsection \<open>Cardinality as special case of @{const setsum}\<close>
935 lemma card_eq_setsum:
936   "card A = setsum (\<lambda>x. 1) A"
937 proof -
938   have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
939     by (simp add: fun_eq_iff)
940   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) = Finite_Set.fold (\<lambda>_. Suc)"
941     by (rule arg_cong)
942   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
943     by (blast intro: fun_cong)
944   then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
945 qed
947 lemma setsum_constant [simp]:
948   "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
949 apply (cases "finite A")
950 apply (erule finite_induct)
951 apply (auto simp add: algebra_simps)
952 done
954 lemma setsum_Suc: "setsum (\<lambda>x. Suc(f x)) A = setsum f A + card A"
955   using setsum.distrib[of f "\<lambda>_. 1" A]
956   by simp
958 lemma setsum_bounded_above:
959   assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_ab_semigroup_add})"
960   shows "setsum f A \<le> of_nat (card A) * K"
961 proof (cases "finite A")
962   case True
963   thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
964 next
965   case False thus ?thesis by simp
966 qed
968 lemma setsum_bounded_above_strict:
969   assumes "\<And>i. i\<in>A \<Longrightarrow> f i < (K::'a::{ordered_cancel_ab_semigroup_add,semiring_1})"
970           "card A > 0"
971   shows "setsum f A < of_nat (card A) * K"
972 using assms setsum_strict_mono[where A=A and g = "%x. K"]
973 by (simp add: card_gt_0_iff)
975 lemma setsum_bounded_below:
976   assumes le: "\<And>i. i\<in>A \<Longrightarrow> (K::'a::{semiring_1, ordered_ab_semigroup_add}) \<le> f i"
977   shows "of_nat (card A) * K \<le> setsum f A"
978 proof (cases "finite A")
979   case True
980   thus ?thesis using le setsum_mono[where K=A and f = "%x. K"] by simp
981 next
982   case False thus ?thesis by simp
983 qed
985 lemma card_UN_disjoint:
986   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
987     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
988   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
989 proof -
990   have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
991   with assms show ?thesis by (simp add: card_eq_setsum setsum.UNION_disjoint del: setsum_constant)
992 qed
994 lemma card_Union_disjoint:
995   "finite C ==> (ALL A:C. finite A) ==>
996    (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
997    ==> card (Union C) = setsum card C"
998 apply (frule card_UN_disjoint [of C id])
999 apply simp_all
1000 done
1002 lemma setsum_multicount_gen:
1003   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
1004   shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
1005 proof-
1006   have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
1007   also have "\<dots> = ?r" unfolding setsum.commute_restrict [OF assms(1-2)]
1008     using assms(3) by auto
1009   finally show ?thesis .
1010 qed
1012 lemma setsum_multicount:
1013   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
1014   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
1015 proof-
1016   have "?l = setsum (\<lambda>i. k) T" by (rule setsum_multicount_gen) (auto simp: assms)
1017   also have "\<dots> = ?r" by (simp add: mult.commute)
1018   finally show ?thesis by auto
1019 qed
1021 lemma (in ordered_comm_monoid_add) setsum_pos:
1022   "finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> 0 < f i) \<Longrightarrow> 0 < setsum f I"
1023   by (induct I rule: finite_ne_induct) (auto intro: add_pos_pos)
1026 subsubsection \<open>Cardinality of products\<close>
1028 lemma card_SigmaI [simp]:
1029   "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
1030   \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
1031 by(simp add: card_eq_setsum setsum.Sigma del:setsum_constant)
1033 (*
1034 lemma SigmaI_insert: "y \<notin> A ==>
1035   (SIGMA x:(insert y A). B x) = (({y} \<times> (B y)) \<union> (SIGMA x: A. B x))"
1036   by auto
1037 *)
1039 lemma card_cartesian_product: "card (A \<times> B) = card(A) * card(B)"
1040   by (cases "finite A \<and> finite B")
1041     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
1043 lemma card_cartesian_product_singleton:  "card({x} \<times> A) = card(A)"
1044 by (simp add: card_cartesian_product)
1047 subsection \<open>Generalized product over a set\<close>
1049 context comm_monoid_mult
1050 begin
1052 sublocale setprod: comm_monoid_set times 1
1053 defines
1054   setprod = setprod.F ..
1056 abbreviation
1057   Setprod ("\<Prod>_"  999) where
1058   "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
1060 end
1062 syntax
1063   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(4PROD _:_./ _)" [0, 51, 10] 10)
1064 syntax (xsymbols)
1065   "_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"  ("(2\<Prod>_\<in>_./ _)" [0, 51, 10] 10)
1067 translations \<comment> \<open>Beware of argument permutation!\<close>
1068   "PROD i:A. b" == "CONST setprod (%i. b) A"
1069   "\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A"
1071 text\<open>Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
1072  \<open>\<Prod>x|P. e\<close>.\<close>
1074 syntax
1075   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(4PROD _ |/ _./ _)" [0,0,10] 10)
1076 syntax (xsymbols)
1077   "_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(2\<Prod>_ | (_)./ _)" [0,0,10] 10)
1079 translations
1080   "PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
1081   "\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
1083 context comm_monoid_mult
1084 begin
1086 lemma setprod_dvd_setprod:
1087   "(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> setprod f A dvd setprod g A"
1088 proof (induct A rule: infinite_finite_induct)
1089   case infinite then show ?case by (auto intro: dvdI)
1090 next
1091   case empty then show ?case by (auto intro: dvdI)
1092 next
1093   case (insert a A) then
1094   have "f a dvd g a" and "setprod f A dvd setprod g A" by simp_all
1095   then obtain r s where "g a = f a * r" and "setprod g A = setprod f A * s" by (auto elim!: dvdE)
1096   then have "g a * setprod g A = f a * setprod f A * (r * s)" by (simp add: ac_simps)
1097   with insert.hyps show ?case by (auto intro: dvdI)
1098 qed
1100 lemma setprod_dvd_setprod_subset:
1101   "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> setprod f A dvd setprod f B"
1102   by (auto simp add: setprod.subset_diff ac_simps intro: dvdI)
1104 end
1107 subsubsection \<open>Properties in more restricted classes of structures\<close>
1109 context comm_semiring_1
1110 begin
1112 lemma dvd_setprod_eqI [intro]:
1113   assumes "finite A" and "a \<in> A" and "b = f a"
1114   shows "b dvd setprod f A"
1115 proof -
1116   from \<open>finite A\<close> have "setprod f (insert a (A - {a})) = f a * setprod f (A - {a})"
1117     by (intro setprod.insert) auto
1118   also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A" by blast
1119   finally have "setprod f A = f a * setprod f (A - {a})" .
1120   with \<open>b = f a\<close> show ?thesis by simp
1121 qed
1123 lemma dvd_setprodI [intro]:
1124   assumes "finite A" and "a \<in> A"
1125   shows "f a dvd setprod f A"
1126   using assms by auto
1128 lemma setprod_zero:
1129   assumes "finite A" and "\<exists>a\<in>A. f a = 0"
1130   shows "setprod f A = 0"
1131 using assms proof (induct A)
1132   case empty then show ?case by simp
1133 next
1134   case (insert a A)
1135   then have "f a = 0 \<or> (\<exists>a\<in>A. f a = 0)" by simp
1136   then have "f a * setprod f A = 0" by rule (simp_all add: insert)
1137   with insert show ?case by simp
1138 qed
1140 lemma setprod_dvd_setprod_subset2:
1141   assumes "finite B" and "A \<subseteq> B" and "\<And>a. a \<in> A \<Longrightarrow> f a dvd g a"
1142   shows "setprod f A dvd setprod g B"
1143 proof -
1144   from assms have "setprod f A dvd setprod g A"
1145     by (auto intro: setprod_dvd_setprod)
1146   moreover from assms have "setprod g A dvd setprod g B"
1147     by (auto intro: setprod_dvd_setprod_subset)
1148   ultimately show ?thesis by (rule dvd_trans)
1149 qed
1151 end
1153 lemma setprod_zero_iff [simp]:
1154   assumes "finite A"
1155   shows "setprod f A = (0::'a::semidom) \<longleftrightarrow> (\<exists>a\<in>A. f a = 0)"
1156   using assms by (induct A) (auto simp: no_zero_divisors)
1158 lemma (in semidom_divide) setprod_diff1:
1159   assumes "finite A" and "f a \<noteq> 0"
1160   shows "setprod f (A - {a}) = (if a \<in> A then setprod f A div f a else setprod f A)"
1161 proof (cases "a \<notin> A")
1162   case True then show ?thesis by simp
1163 next
1164   case False with assms show ?thesis
1165   proof (induct A rule: finite_induct)
1166     case empty then show ?case by simp
1167   next
1168     case (insert b B)
1169     then show ?case
1170     proof (cases "a = b")
1171       case True with insert show ?thesis by simp
1172     next
1173       case False with insert have "a \<in> B" by simp
1174       def C \<equiv> "B - {a}"
1175       with \<open>finite B\<close> \<open>a \<in> B\<close>
1176         have *: "B = insert a C" "finite C" "a \<notin> C" by auto
1177       with insert show ?thesis by (auto simp add: insert_commute ac_simps)
1178     qed
1179   qed
1180 qed
1182 lemma (in field) setprod_inversef:
1183   "finite A \<Longrightarrow> setprod (inverse \<circ> f) A = inverse (setprod f A)"
1184   by (induct A rule: finite_induct) simp_all
1186 lemma (in field) setprod_dividef:
1187   "finite A \<Longrightarrow> (\<Prod>x\<in>A. f x / g x) = setprod f A / setprod g A"
1188   using setprod_inversef [of A g] by (simp add: divide_inverse setprod.distrib)
1190 lemma setprod_Un:
1191   fixes f :: "'b \<Rightarrow> 'a :: field"
1192   assumes "finite A" and "finite B"
1193   and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0"
1194   shows "setprod f (A \<union> B) = setprod f A * setprod f B / setprod f (A \<inter> B)"
1195 proof -
1196   from assms have "setprod f A * setprod f B = setprod f (A \<union> B) * setprod f (A \<inter> B)"
1197     by (simp add: setprod.union_inter [symmetric, of A B])
1198   with assms show ?thesis by simp
1199 qed
1201 lemma (in linordered_semidom) setprod_nonneg:
1202   "(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> setprod f A"
1203   by (induct A rule: infinite_finite_induct) simp_all
1205 lemma (in linordered_semidom) setprod_pos:
1206   "(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < setprod f A"
1207   by (induct A rule: infinite_finite_induct) simp_all
1209 lemma (in linordered_semidom) setprod_mono:
1210   assumes "\<forall>i\<in>A. 0 \<le> f i \<and> f i \<le> g i"
1211   shows "setprod f A \<le> setprod g A"
1212   using assms by (induct A rule: infinite_finite_induct)
1213     (auto intro!: setprod_nonneg mult_mono)
1215 lemma (in linordered_semidom) setprod_mono_strict:
1216     assumes"finite A" "\<forall>i\<in>A. 0 \<le> f i \<and> f i < g i" "A \<noteq> {}"
1217     shows "setprod f A < setprod g A"
1218 using assms
1219 apply (induct A rule: finite_induct)
1220 apply (simp add: )
1221 apply (force intro: mult_strict_mono' setprod_nonneg)
1222 done
1224 lemma (in linordered_field) abs_setprod:
1225   "\<bar>setprod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
1226   by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult)
1228 lemma setprod_eq_1_iff [simp]:
1229   "finite A \<Longrightarrow> setprod f A = 1 \<longleftrightarrow> (\<forall>a\<in>A. f a = (1::nat))"
1230   by (induct A rule: finite_induct) simp_all
1232 lemma setprod_pos_nat_iff [simp]:
1233   "finite A \<Longrightarrow> setprod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > (0::nat))"
1234   using setprod_zero_iff by (simp del:neq0_conv add:neq0_conv [symmetric])
1236 lemma setsum_nonneg_eq_0_iff:
1237   fixes f :: "'a \<Rightarrow> 'b::ordered_ab_group_add"
1238   shows "\<lbrakk>finite A; \<forall>x\<in>A. 0 \<le> f x\<rbrakk> \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
1239   apply (induct set: finite, simp)