src/HOL/Binomial.thy
 author paulson Tue Apr 25 16:39:54 2017 +0100 (2017-04-25) changeset 65578 e4997c181cce parent 65552 f533820e7248 child 65581 baf96277ee76 permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
1 (*  Title:      HOL/Binomial.thy
2     Author:     Jacques D. Fleuriot
3     Author:     Lawrence C Paulson
4     Author:     Jeremy Avigad
5     Author:     Chaitanya Mangla
6     Author:     Manuel Eberl
7 *)
9 section \<open>Combinatorial Functions: Factorial Function, Rising Factorials, Binomial Coefficients and Binomial Theorem\<close>
11 theory Binomial
12   imports Pre_Main
13 begin
15 subsection \<open>Factorial\<close>
17 context semiring_char_0
18 begin
20 definition fact :: "nat \<Rightarrow> 'a"
21   where fact_prod: "fact n = of_nat (\<Prod>{1..n})"
23 lemma fact_prod_Suc: "fact n = of_nat (prod Suc {0..<n})"
24   by (cases n)
25     (simp_all add: fact_prod prod.atLeast_Suc_atMost_Suc_shift
26       atLeastLessThanSuc_atLeastAtMost)
28 lemma fact_prod_rev: "fact n = of_nat (\<Prod>i = 0..<n. n - i)"
29   using prod.atLeast_atMost_rev [of "\<lambda>i. i" 1 n]
30   by (cases n)
31     (simp_all add: fact_prod_Suc prod.atLeast_Suc_atMost_Suc_shift
32       atLeastLessThanSuc_atLeastAtMost)
34 lemma fact_0 [simp]: "fact 0 = 1"
35   by (simp add: fact_prod)
37 lemma fact_1 [simp]: "fact 1 = 1"
38   by (simp add: fact_prod)
40 lemma fact_Suc_0 [simp]: "fact (Suc 0) = 1"
41   by (simp add: fact_prod)
43 lemma fact_Suc [simp]: "fact (Suc n) = of_nat (Suc n) * fact n"
44   by (simp add: fact_prod atLeastAtMostSuc_conv algebra_simps)
46 lemma fact_2 [simp]: "fact 2 = 2"
47   by (simp add: numeral_2_eq_2)
49 lemma fact_split: "k \<le> n \<Longrightarrow> fact n = of_nat (prod Suc {n - k..<n}) * fact (n - k)"
50   by (simp add: fact_prod_Suc prod.union_disjoint [symmetric]
51     ivl_disj_un ac_simps of_nat_mult [symmetric])
53 end
55 lemma of_nat_fact [simp]: "of_nat (fact n) = fact n"
56   by (simp add: fact_prod)
58 lemma of_int_fact [simp]: "of_int (fact n) = fact n"
59   by (simp only: fact_prod of_int_of_nat_eq)
61 lemma fact_reduce: "n > 0 \<Longrightarrow> fact n = of_nat n * fact (n - 1)"
62   by (cases n) auto
64 lemma fact_nonzero [simp]: "fact n \<noteq> (0::'a::{semiring_char_0,semiring_no_zero_divisors})"
65   apply (induct n)
66   apply auto
67   using of_nat_eq_0_iff
68   apply fastforce
69   done
71 lemma fact_mono_nat: "m \<le> n \<Longrightarrow> fact m \<le> (fact n :: nat)"
72   by (induct n) (auto simp: le_Suc_eq)
74 lemma fact_in_Nats: "fact n \<in> \<nat>"
75   by (induct n) auto
77 lemma fact_in_Ints: "fact n \<in> \<int>"
78   by (induct n) auto
80 context
81   assumes "SORT_CONSTRAINT('a::linordered_semidom)"
82 begin
84 lemma fact_mono: "m \<le> n \<Longrightarrow> fact m \<le> (fact n :: 'a)"
85   by (metis of_nat_fact of_nat_le_iff fact_mono_nat)
87 lemma fact_ge_1 [simp]: "fact n \<ge> (1 :: 'a)"
88   by (metis le0 fact_0 fact_mono)
90 lemma fact_gt_zero [simp]: "fact n > (0 :: 'a)"
91   using fact_ge_1 less_le_trans zero_less_one by blast
93 lemma fact_ge_zero [simp]: "fact n \<ge> (0 :: 'a)"
94   by (simp add: less_imp_le)
96 lemma fact_not_neg [simp]: "\<not> fact n < (0 :: 'a)"
97   by (simp add: not_less_iff_gr_or_eq)
99 lemma fact_le_power: "fact n \<le> (of_nat (n^n) :: 'a)"
100 proof (induct n)
101   case 0
102   then show ?case by simp
103 next
104   case (Suc n)
105   then have *: "fact n \<le> (of_nat (Suc n ^ n) ::'a)"
106     by (rule order_trans) (simp add: power_mono del: of_nat_power)
107   have "fact (Suc n) = (of_nat (Suc n) * fact n ::'a)"
108     by (simp add: algebra_simps)
109   also have "\<dots> \<le> of_nat (Suc n) * of_nat (Suc n ^ n)"
110     by (simp add: * ordered_comm_semiring_class.comm_mult_left_mono del: of_nat_power)
111   also have "\<dots> \<le> of_nat (Suc n ^ Suc n)"
112     by (metis of_nat_mult order_refl power_Suc)
113   finally show ?case .
114 qed
116 end
118 text \<open>Note that @{term "fact 0 = fact 1"}\<close>
119 lemma fact_less_mono_nat: "0 < m \<Longrightarrow> m < n \<Longrightarrow> fact m < (fact n :: nat)"
120   by (induct n) (auto simp: less_Suc_eq)
122 lemma fact_less_mono: "0 < m \<Longrightarrow> m < n \<Longrightarrow> fact m < (fact n :: 'a::linordered_semidom)"
123   by (metis of_nat_fact of_nat_less_iff fact_less_mono_nat)
125 lemma fact_ge_Suc_0_nat [simp]: "fact n \<ge> Suc 0"
126   by (metis One_nat_def fact_ge_1)
128 lemma dvd_fact: "1 \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> m dvd fact n"
129   by (induct n) (auto simp: dvdI le_Suc_eq)
131 lemma fact_ge_self: "fact n \<ge> n"
132   by (cases "n = 0") (simp_all add: dvd_imp_le dvd_fact)
134 lemma fact_dvd: "n \<le> m \<Longrightarrow> fact n dvd (fact m :: 'a::{semiring_div,linordered_semidom})"
135   by (induct m) (auto simp: le_Suc_eq)
137 lemma fact_mod: "m \<le> n \<Longrightarrow> fact n mod (fact m :: 'a::{semiring_div,linordered_semidom}) = 0"
138   by (auto simp add: fact_dvd)
140 lemma fact_div_fact:
141   assumes "m \<ge> n"
142   shows "fact m div fact n = \<Prod>{n + 1..m}"
143 proof -
144   obtain d where "d = m - n"
145     by auto
146   with assms have "m = n + d"
147     by auto
148   have "fact (n + d) div (fact n) = \<Prod>{n + 1..n + d}"
149   proof (induct d)
150     case 0
151     show ?case by simp
152   next
153     case (Suc d')
154     have "fact (n + Suc d') div fact n = Suc (n + d') * fact (n + d') div fact n"
155       by simp
156     also from Suc.hyps have "\<dots> = Suc (n + d') * \<Prod>{n + 1..n + d'}"
157       unfolding div_mult1_eq[of _ "fact (n + d')"] by (simp add: fact_mod)
158     also have "\<dots> = \<Prod>{n + 1..n + Suc d'}"
159       by (simp add: atLeastAtMostSuc_conv)
160     finally show ?case .
161   qed
162   with \<open>m = n + d\<close> show ?thesis by simp
163 qed
165 lemma fact_num_eq_if: "fact m = (if m = 0 then 1 else of_nat m * fact (m - 1))"
166   by (cases m) auto
168 lemma fact_div_fact_le_pow:
169   assumes "r \<le> n"
170   shows "fact n div fact (n - r) \<le> n ^ r"
171 proof -
172   have "r \<le> n \<Longrightarrow> \<Prod>{n - r..n} = (n - r) * \<Prod>{Suc (n - r)..n}" for r
173     by (subst prod.insert[symmetric]) (auto simp: atLeastAtMost_insertL)
174   with assms show ?thesis
175     by (induct r rule: nat.induct) (auto simp add: fact_div_fact Suc_diff_Suc mult_le_mono)
176 qed
178 lemma fact_numeral: "fact (numeral k) = numeral k * fact (pred_numeral k)"
179   \<comment> \<open>Evaluation for specific numerals\<close>
180   by (metis fact_Suc numeral_eq_Suc of_nat_numeral)
183 subsection \<open>Binomial coefficients\<close>
185 text \<open>This development is based on the work of Andy Gordon and Florian Kammueller.\<close>
187 text \<open>Combinatorial definition\<close>
189 definition binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat"  (infixl "choose" 65)
190   where "n choose k = card {K\<in>Pow {0..<n}. card K = k}"
192 theorem n_subsets:
193   assumes "finite A"
194   shows "card {B. B \<subseteq> A \<and> card B = k} = card A choose k"
195 proof -
196   from assms obtain f where bij: "bij_betw f {0..<card A} A"
197     by (blast dest: ex_bij_betw_nat_finite)
198   then have [simp]: "card (f  C) = card C" if "C \<subseteq> {0..<card A}" for C
199     by (meson bij_betw_imp_inj_on bij_betw_subset card_image that)
200   from bij have "bij_betw (image f) (Pow {0..<card A}) (Pow A)"
201     by (rule bij_betw_Pow)
202   then have "inj_on (image f) (Pow {0..<card A})"
203     by (rule bij_betw_imp_inj_on)
204   moreover have "{K. K \<subseteq> {0..<card A} \<and> card K = k} \<subseteq> Pow {0..<card A}"
205     by auto
206   ultimately have "inj_on (image f) {K. K \<subseteq> {0..<card A} \<and> card K = k}"
207     by (rule inj_on_subset)
208   then have "card {K. K \<subseteq> {0..<card A} \<and> card K = k} =
209       card (image f  {K. K \<subseteq> {0..<card A} \<and> card K = k})" (is "_ = card ?C")
210     by (simp add: card_image)
211   also have "?C = {K. K \<subseteq> f  {0..<card A} \<and> card K = k}"
212     by (auto elim!: subset_imageE)
213   also have "f  {0..<card A} = A"
214     by (meson bij bij_betw_def)
215   finally show ?thesis
216     by (simp add: binomial_def)
217 qed
219 text \<open>Recursive characterization\<close>
221 lemma binomial_n_0 [simp, code]: "n choose 0 = 1"
222 proof -
223   have "{K \<in> Pow {0..<n}. card K = 0} = {{}}"
224     by (auto dest: finite_subset)
225   then show ?thesis
226     by (simp add: binomial_def)
227 qed
229 lemma binomial_0_Suc [simp, code]: "0 choose Suc k = 0"
230   by (simp add: binomial_def)
232 lemma binomial_Suc_Suc [simp, code]: "Suc n choose Suc k = (n choose k) + (n choose Suc k)"
233 proof -
234   let ?P = "\<lambda>n k. {K. K \<subseteq> {0..<n} \<and> card K = k}"
235   let ?Q = "?P (Suc n) (Suc k)"
236   have inj: "inj_on (insert n) (?P n k)"
237     by rule (auto; metis atLeastLessThan_iff insert_iff less_irrefl subsetCE)
238   have disjoint: "insert n  ?P n k \<inter> ?P n (Suc k) = {}"
239     by auto
240   have "?Q = {K\<in>?Q. n \<in> K} \<union> {K\<in>?Q. n \<notin> K}"
241     by auto
242   also have "{K\<in>?Q. n \<in> K} = insert n  ?P n k" (is "?A = ?B")
243   proof (rule set_eqI)
244     fix K
245     have K_finite: "finite K" if "K \<subseteq> insert n {0..<n}"
246       using that by (rule finite_subset) simp_all
247     have Suc_card_K: "Suc (card K - Suc 0) = card K" if "n \<in> K"
248       and "finite K"
249     proof -
250       from \<open>n \<in> K\<close> obtain L where "K = insert n L" and "n \<notin> L"
251         by (blast elim: Set.set_insert)
252       with that show ?thesis by (simp add: card_insert)
253     qed
254     show "K \<in> ?A \<longleftrightarrow> K \<in> ?B"
255       by (subst in_image_insert_iff)
256         (auto simp add: card_insert subset_eq_atLeast0_lessThan_finite
257           Diff_subset_conv K_finite Suc_card_K)
258   qed
259   also have "{K\<in>?Q. n \<notin> K} = ?P n (Suc k)"
260     by (auto simp add: atLeast0_lessThan_Suc)
261   finally show ?thesis using inj disjoint
262     by (simp add: binomial_def card_Un_disjoint card_image)
263 qed
265 lemma binomial_eq_0: "n < k \<Longrightarrow> n choose k = 0"
266   by (auto simp add: binomial_def dest: subset_eq_atLeast0_lessThan_card)
268 lemma zero_less_binomial: "k \<le> n \<Longrightarrow> n choose k > 0"
269   by (induct n k rule: diff_induct) simp_all
271 lemma binomial_eq_0_iff [simp]: "n choose k = 0 \<longleftrightarrow> n < k"
272   by (metis binomial_eq_0 less_numeral_extra(3) not_less zero_less_binomial)
274 lemma zero_less_binomial_iff [simp]: "n choose k > 0 \<longleftrightarrow> k \<le> n"
275   by (metis binomial_eq_0_iff not_less0 not_less zero_less_binomial)
277 lemma binomial_n_n [simp]: "n choose n = 1"
278   by (induct n) (simp_all add: binomial_eq_0)
280 lemma binomial_Suc_n [simp]: "Suc n choose n = Suc n"
281   by (induct n) simp_all
283 lemma binomial_1 [simp]: "n choose Suc 0 = n"
284   by (induct n) simp_all
286 lemma choose_reduce_nat:
287   "0 < n \<Longrightarrow> 0 < k \<Longrightarrow>
288     n choose k = ((n - 1) choose (k - 1)) + ((n - 1) choose k)"
289   using binomial_Suc_Suc [of "n - 1" "k - 1"] by simp
291 lemma Suc_times_binomial_eq: "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
292   apply (induct n arbitrary: k)
293    apply simp
294    apply arith
295   apply (case_tac k)
297   done
299 lemma binomial_le_pow2: "n choose k \<le> 2^n"
300   apply (induct n arbitrary: k)
301    apply (case_tac k)
302     apply simp_all
303   apply (case_tac k)
304    apply auto
306   done
308 text \<open>The absorption property.\<close>
309 lemma Suc_times_binomial: "Suc k * (Suc n choose Suc k) = Suc n * (n choose k)"
310   using Suc_times_binomial_eq by auto
312 text \<open>This is the well-known version of absorption, but it's harder to use
313   because of the need to reason about division.\<close>
314 lemma binomial_Suc_Suc_eq_times: "(Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
315   by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right)
317 text \<open>Another version of absorption, with \<open>-1\<close> instead of \<open>Suc\<close>.\<close>
318 lemma times_binomial_minus1_eq: "0 < k \<Longrightarrow> k * (n choose k) = n * ((n - 1) choose (k - 1))"
319   using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 1"]
320   by (auto split: nat_diff_split)
323 subsection \<open>The binomial theorem (courtesy of Tobias Nipkow):\<close>
325 text \<open>Avigad's version, generalized to any commutative ring\<close>
326 theorem binomial_ring: "(a + b :: 'a::{comm_ring_1,power})^n =
327   (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))"
328 proof (induct n)
329   case 0
330   then show ?case by simp
331 next
332   case (Suc n)
333   have decomp: "{0..n+1} = {0} \<union> {n + 1} \<union> {1..n}"
334     by auto
335   have decomp2: "{0..n} = {0} \<union> {1..n}"
336     by auto
337   have "(a + b)^(n+1) = (a + b) * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n - k))"
338     using Suc.hyps by simp
339   also have "\<dots> = a * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k)) +
340       b * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
341     by (rule distrib_right)
342   also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) +
343       (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n - k + 1))"
344     by (auto simp add: sum_distrib_left ac_simps)
345   also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n + 1 - k)) +
346       (\<Sum>k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n + 1 - k))"
347     by (simp add:sum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps del: sum_cl_ivl_Suc)
348   also have "\<dots> = a^(n + 1) + b^(n + 1) +
349       (\<Sum>k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n + 1 - k)) +
350       (\<Sum>k=1..n. of_nat (n choose k) * a^k * b^(n + 1 - k))"
351     by (simp add: decomp2)
352   also have "\<dots> = a^(n + 1) + b^(n + 1) +
353       (\<Sum>k=1..n. of_nat (n + 1 choose k) * a^k * b^(n + 1 - k))"
354     by (auto simp add: field_simps sum.distrib [symmetric] choose_reduce_nat)
355   also have "\<dots> = (\<Sum>k=0..n+1. of_nat (n + 1 choose k) * a^k * b^(n + 1 - k))"
356     using decomp by (simp add: field_simps)
357   finally show ?case
358     by simp
359 qed
361 text \<open>Original version for the naturals.\<close>
362 corollary binomial: "(a + b :: nat)^n = (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n - k))"
363   using binomial_ring [of "int a" "int b" n]
364   by (simp only: of_nat_add [symmetric] of_nat_mult [symmetric] of_nat_power [symmetric]
365       of_nat_sum [symmetric] of_nat_eq_iff of_nat_id)
367 lemma binomial_fact_lemma: "k \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n"
368 proof (induct n arbitrary: k rule: nat_less_induct)
369   fix n k
370   assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) = fact m"
371   assume kn: "k \<le> n"
372   let ?ths = "fact k * fact (n - k) * (n choose k) = fact n"
373   consider "n = 0 \<or> k = 0 \<or> n = k" | m h where "n = Suc m" "k = Suc h" "h < m"
374     using kn by atomize_elim presburger
375   then show "fact k * fact (n - k) * (n choose k) = fact n"
376   proof cases
377     case 1
378     with kn show ?thesis by auto
379   next
380     case 2
381     note n = \<open>n = Suc m\<close>
382     note k = \<open>k = Suc h\<close>
383     note hm = \<open>h < m\<close>
384     have mn: "m < n"
385       using n by arith
386     have hm': "h \<le> m"
387       using hm by arith
388     have km: "k \<le> m"
389       using hm k n kn by arith
390     have "m - h = Suc (m - Suc h)"
391       using  k km hm by arith
392     with km k have "fact (m - h) = (m - h) * fact (m - k)"
393       by simp
394     with n k have "fact k * fact (n - k) * (n choose k) =
395         k * (fact h * fact (m - h) * (m choose h)) +
396         (m - h) * (fact k * fact (m - k) * (m choose k))"
397       by (simp add: field_simps)
398     also have "\<dots> = (k + (m - h)) * fact m"
399       using H[rule_format, OF mn hm'] H[rule_format, OF mn km]
400       by (simp add: field_simps)
401     finally show ?thesis
402       using k n km by simp
403   qed
404 qed
406 lemma binomial_fact':
407   assumes "k \<le> n"
408   shows "n choose k = fact n div (fact k * fact (n - k))"
409   using binomial_fact_lemma [OF assms]
410   by (metis fact_nonzero mult_eq_0_iff nonzero_mult_div_cancel_left)
412 lemma binomial_fact:
413   assumes kn: "k \<le> n"
414   shows "(of_nat (n choose k) :: 'a::field_char_0) = fact n / (fact k * fact (n - k))"
415   using binomial_fact_lemma[OF kn]
416   apply (simp add: field_simps)
417   apply (metis mult.commute of_nat_fact of_nat_mult)
418   done
420 lemma fact_binomial:
421   assumes "k \<le> n"
422   shows "fact k * of_nat (n choose k) = (fact n / fact (n - k) :: 'a::field_char_0)"
423   unfolding binomial_fact [OF assms] by (simp add: field_simps)
425 lemma choose_two: "n choose 2 = n * (n - 1) div 2"
426 proof (cases "n \<ge> 2")
427   case False
428   then have "n = 0 \<or> n = 1"
429     by auto
430   then show ?thesis by auto
431 next
432   case True
433   define m where "m = n - 2"
434   with True have "n = m + 2"
435     by simp
436   then have "fact n = n * (n - 1) * fact (n - 2)"
437     by (simp add: fact_prod_Suc atLeast0_lessThan_Suc algebra_simps)
438   with True show ?thesis
439     by (simp add: binomial_fact')
440 qed
442 lemma choose_row_sum: "(\<Sum>k=0..n. n choose k) = 2^n"
443   using binomial [of 1 "1" n] by (simp add: numeral_2_eq_2)
445 lemma sum_choose_lower: "(\<Sum>k=0..n. (r+k) choose k) = Suc (r+n) choose n"
446   by (induct n) auto
448 lemma sum_choose_upper: "(\<Sum>k=0..n. k choose m) = Suc n choose Suc m"
449   by (induct n) auto
451 lemma choose_alternating_sum:
452   "n > 0 \<Longrightarrow> (\<Sum>i\<le>n. (-1)^i * of_nat (n choose i)) = (0 :: 'a::comm_ring_1)"
453   using binomial_ring[of "-1 :: 'a" 1 n]
454   by (simp add: atLeast0AtMost mult_of_nat_commute zero_power)
456 lemma choose_even_sum:
457   assumes "n > 0"
458   shows "2 * (\<Sum>i\<le>n. if even i then of_nat (n choose i) else 0) = (2 ^ n :: 'a::comm_ring_1)"
459 proof -
460   have "2 ^ n = (\<Sum>i\<le>n. of_nat (n choose i)) + (\<Sum>i\<le>n. (-1) ^ i * of_nat (n choose i) :: 'a)"
461     using choose_row_sum[of n]
462     by (simp add: choose_alternating_sum assms atLeast0AtMost of_nat_sum[symmetric])
463   also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) + (-1) ^ i * of_nat (n choose i))"
464     by (simp add: sum.distrib)
465   also have "\<dots> = 2 * (\<Sum>i\<le>n. if even i then of_nat (n choose i) else 0)"
466     by (subst sum_distrib_left, intro sum.cong) simp_all
467   finally show ?thesis ..
468 qed
470 lemma choose_odd_sum:
471   assumes "n > 0"
472   shows "2 * (\<Sum>i\<le>n. if odd i then of_nat (n choose i) else 0) = (2 ^ n :: 'a::comm_ring_1)"
473 proof -
474   have "2 ^ n = (\<Sum>i\<le>n. of_nat (n choose i)) - (\<Sum>i\<le>n. (-1) ^ i * of_nat (n choose i) :: 'a)"
475     using choose_row_sum[of n]
476     by (simp add: choose_alternating_sum assms atLeast0AtMost of_nat_sum[symmetric])
477   also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) - (-1) ^ i * of_nat (n choose i))"
478     by (simp add: sum_subtractf)
479   also have "\<dots> = 2 * (\<Sum>i\<le>n. if odd i then of_nat (n choose i) else 0)"
480     by (subst sum_distrib_left, intro sum.cong) simp_all
481   finally show ?thesis ..
482 qed
484 lemma choose_row_sum': "(\<Sum>k\<le>n. (n choose k)) = 2 ^ n"
485   using choose_row_sum[of n] by (simp add: atLeast0AtMost)
487 text\<open>NW diagonal sum property\<close>
488 lemma sum_choose_diagonal:
489   assumes "m \<le> n"
490   shows "(\<Sum>k=0..m. (n - k) choose (m - k)) = Suc n choose m"
491 proof -
492   have "(\<Sum>k=0..m. (n-k) choose (m - k)) = (\<Sum>k=0..m. (n - m + k) choose k)"
493     using sum.atLeast_atMost_rev [of "\<lambda>k. (n - k) choose (m - k)" 0 m] assms
494       by simp
495   also have "\<dots> = Suc (n - m + m) choose m"
496     by (rule sum_choose_lower)
497   also have "\<dots> = Suc n choose m"
498     using assms by simp
499   finally show ?thesis .
500 qed
503 subsection \<open>Pochhammer's symbol: generalized rising factorial\<close>
505 text \<open>See \<^url>\<open>http://en.wikipedia.org/wiki/Pochhammer_symbol\<close>.\<close>
507 context comm_semiring_1
508 begin
510 definition pochhammer :: "'a \<Rightarrow> nat \<Rightarrow> 'a"
511   where pochhammer_prod: "pochhammer a n = prod (\<lambda>i. a + of_nat i) {0..<n}"
513 lemma pochhammer_prod_rev: "pochhammer a n = prod (\<lambda>i. a + of_nat (n - i)) {1..n}"
514   using prod.atLeast_lessThan_rev_at_least_Suc_atMost [of "\<lambda>i. a + of_nat i" 0 n]
515   by (simp add: pochhammer_prod)
517 lemma pochhammer_Suc_prod: "pochhammer a (Suc n) = prod (\<lambda>i. a + of_nat i) {0..n}"
518   by (simp add: pochhammer_prod atLeastLessThanSuc_atLeastAtMost)
520 lemma pochhammer_Suc_prod_rev: "pochhammer a (Suc n) = prod (\<lambda>i. a + of_nat (n - i)) {0..n}"
521   by (simp add: pochhammer_prod_rev prod.atLeast_Suc_atMost_Suc_shift)
523 lemma pochhammer_0 [simp]: "pochhammer a 0 = 1"
524   by (simp add: pochhammer_prod)
526 lemma pochhammer_1 [simp]: "pochhammer a 1 = a"
527   by (simp add: pochhammer_prod lessThan_Suc)
529 lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a"
530   by (simp add: pochhammer_prod lessThan_Suc)
532 lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
533   by (simp add: pochhammer_prod atLeast0_lessThan_Suc ac_simps)
535 end
537 lemma pochhammer_nonneg:
538   fixes x :: "'a :: linordered_semidom"
539   shows "x > 0 \<Longrightarrow> pochhammer x n \<ge> 0"
540   by (induction n) (auto simp: pochhammer_Suc intro!: mult_nonneg_nonneg add_nonneg_nonneg)
542 lemma pochhammer_pos:
543   fixes x :: "'a :: linordered_semidom"
544   shows "x > 0 \<Longrightarrow> pochhammer x n > 0"
545   by (induction n) (auto simp: pochhammer_Suc intro!: mult_pos_pos add_pos_nonneg)
547 lemma pochhammer_of_nat: "pochhammer (of_nat x) n = of_nat (pochhammer x n)"
548   by (simp add: pochhammer_prod)
550 lemma pochhammer_of_int: "pochhammer (of_int x) n = of_int (pochhammer x n)"
551   by (simp add: pochhammer_prod)
553 lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n"
554   by (simp add: pochhammer_prod prod.atLeast0_lessThan_Suc_shift ac_simps)
556 lemma pochhammer_rec': "pochhammer z (Suc n) = (z + of_nat n) * pochhammer z n"
557   by (simp add: pochhammer_prod prod.atLeast0_lessThan_Suc ac_simps)
559 lemma pochhammer_fact: "fact n = pochhammer 1 n"
560   by (simp add: pochhammer_prod fact_prod_Suc)
562 lemma pochhammer_of_nat_eq_0_lemma: "k > n \<Longrightarrow> pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
563   by (auto simp add: pochhammer_prod)
565 lemma pochhammer_of_nat_eq_0_lemma':
566   assumes kn: "k \<le> n"
567   shows "pochhammer (- (of_nat n :: 'a::{idom,ring_char_0})) k \<noteq> 0"
568 proof (cases k)
569   case 0
570   then show ?thesis by simp
571 next
572   case (Suc h)
573   then show ?thesis
574     apply (simp add: pochhammer_Suc_prod)
575     using Suc kn
576     apply (auto simp add: algebra_simps)
577     done
578 qed
580 lemma pochhammer_of_nat_eq_0_iff:
581   "pochhammer (- (of_nat n :: 'a::{idom,ring_char_0})) k = 0 \<longleftrightarrow> k > n"
582   (is "?l = ?r")
583   using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a]
584     pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a]
585   by (auto simp add: not_le[symmetric])
587 lemma pochhammer_eq_0_iff: "pochhammer a n = (0::'a::field_char_0) \<longleftrightarrow> (\<exists>k < n. a = - of_nat k)"
588   by (auto simp add: pochhammer_prod eq_neg_iff_add_eq_0)
590 lemma pochhammer_eq_0_mono:
591   "pochhammer a n = (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a m = 0"
592   unfolding pochhammer_eq_0_iff by auto
594 lemma pochhammer_neq_0_mono:
595   "pochhammer a m \<noteq> (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a n \<noteq> 0"
596   unfolding pochhammer_eq_0_iff by auto
598 lemma pochhammer_minus:
599   "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k"
600 proof (cases k)
601   case 0
602   then show ?thesis by simp
603 next
604   case (Suc h)
605   have eq: "((- 1) ^ Suc h :: 'a) = (\<Prod>i = 0..h. - 1)"
606     using prod_constant [where A="{0.. h}" and y="- 1 :: 'a"]
607     by auto
608   with Suc show ?thesis
609     using pochhammer_Suc_prod_rev [of "b - of_nat k + 1"]
610     by (auto simp add: pochhammer_Suc_prod prod.distrib [symmetric] eq of_nat_diff)
611 qed
613 lemma pochhammer_minus':
614   "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
615   apply (simp only: pochhammer_minus [where b = b])
616   apply (simp only: mult.assoc [symmetric])
617   apply (simp only: power_add [symmetric])
618   apply simp
619   done
621 lemma pochhammer_same: "pochhammer (- of_nat n) n =
622     ((- 1) ^ n :: 'a::{semiring_char_0,comm_ring_1,semiring_no_zero_divisors}) * fact n"
623   unfolding pochhammer_minus
624   by (simp add: of_nat_diff pochhammer_fact)
626 lemma pochhammer_product': "pochhammer z (n + m) = pochhammer z n * pochhammer (z + of_nat n) m"
627 proof (induct n arbitrary: z)
628   case 0
629   then show ?case by simp
630 next
631   case (Suc n z)
632   have "pochhammer z (Suc n) * pochhammer (z + of_nat (Suc n)) m =
633       z * (pochhammer (z + 1) n * pochhammer (z + 1 + of_nat n) m)"
634     by (simp add: pochhammer_rec ac_simps)
635   also note Suc[symmetric]
636   also have "z * pochhammer (z + 1) (n + m) = pochhammer z (Suc (n + m))"
637     by (subst pochhammer_rec) simp
638   finally show ?case
639     by simp
640 qed
642 lemma pochhammer_product:
643   "m \<le> n \<Longrightarrow> pochhammer z n = pochhammer z m * pochhammer (z + of_nat m) (n - m)"
644   using pochhammer_product'[of z m "n - m"] by simp
646 lemma pochhammer_times_pochhammer_half:
647   fixes z :: "'a::field_char_0"
648   shows "pochhammer z (Suc n) * pochhammer (z + 1/2) (Suc n) = (\<Prod>k=0..2*n+1. z + of_nat k / 2)"
649 proof (induct n)
650   case 0
651   then show ?case
652     by (simp add: atLeast0_atMost_Suc)
653 next
654   case (Suc n)
655   define n' where "n' = Suc n"
656   have "pochhammer z (Suc n') * pochhammer (z + 1 / 2) (Suc n') =
657       (pochhammer z n' * pochhammer (z + 1 / 2) n') * ((z + of_nat n') * (z + 1/2 + of_nat n'))"
658     (is "_ = _ * ?A")
659     by (simp_all add: pochhammer_rec' mult_ac)
660   also have "?A = (z + of_nat (Suc (2 * n + 1)) / 2) * (z + of_nat (Suc (Suc (2 * n + 1))) / 2)"
661     (is "_ = ?B")
662     by (simp add: field_simps n'_def)
663   also note Suc[folded n'_def]
664   also have "(\<Prod>k=0..2 * n + 1. z + of_nat k / 2) * ?B = (\<Prod>k=0..2 * Suc n + 1. z + of_nat k / 2)"
665     by (simp add: atLeast0_atMost_Suc)
666   finally show ?case
667     by (simp add: n'_def)
668 qed
670 lemma pochhammer_double:
671   fixes z :: "'a::field_char_0"
672   shows "pochhammer (2 * z) (2 * n) = of_nat (2^(2*n)) * pochhammer z n * pochhammer (z+1/2) n"
673 proof (induct n)
674   case 0
675   then show ?case by simp
676 next
677   case (Suc n)
678   have "pochhammer (2 * z) (2 * (Suc n)) = pochhammer (2 * z) (2 * n) *
679       (2 * (z + of_nat n)) * (2 * (z + of_nat n) + 1)"
680     by (simp add: pochhammer_rec' ac_simps)
681   also note Suc
682   also have "of_nat (2 ^ (2 * n)) * pochhammer z n * pochhammer (z + 1/2) n *
683         (2 * (z + of_nat n)) * (2 * (z + of_nat n) + 1) =
684       of_nat (2 ^ (2 * (Suc n))) * pochhammer z (Suc n) * pochhammer (z + 1/2) (Suc n)"
685     by (simp add: field_simps pochhammer_rec')
686   finally show ?case .
687 qed
689 lemma fact_double:
690   "fact (2 * n) = (2 ^ (2 * n) * pochhammer (1 / 2) n * fact n :: 'a::field_char_0)"
691   using pochhammer_double[of "1/2::'a" n] by (simp add: pochhammer_fact)
693 lemma pochhammer_absorb_comp: "(r - of_nat k) * pochhammer (- r) k = r * pochhammer (-r + 1) k"
694   (is "?lhs = ?rhs")
695   for r :: "'a::comm_ring_1"
696 proof -
697   have "?lhs = - pochhammer (- r) (Suc k)"
698     by (subst pochhammer_rec') (simp add: algebra_simps)
699   also have "\<dots> = ?rhs"
700     by (subst pochhammer_rec) simp
701   finally show ?thesis .
702 qed
705 subsection \<open>Generalized binomial coefficients\<close>
707 definition gbinomial :: "'a::{semidom_divide,semiring_char_0} \<Rightarrow> nat \<Rightarrow> 'a"  (infixl "gchoose" 65)
708   where gbinomial_prod_rev: "a gchoose n = prod (\<lambda>i. a - of_nat i) {0..<n} div fact n"
710 lemma gbinomial_0 [simp]:
711   "a gchoose 0 = 1"
712   "0 gchoose (Suc n) = 0"
713   by (simp_all add: gbinomial_prod_rev prod.atLeast0_lessThan_Suc_shift)
715 lemma gbinomial_Suc: "a gchoose (Suc k) = prod (\<lambda>i. a - of_nat i) {0..k} div fact (Suc k)"
716   by (simp add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)
718 lemma gbinomial_mult_fact: "fact n * (a gchoose n) = (\<Prod>i = 0..<n. a - of_nat i)"
719   for a :: "'a::field_char_0"
720   by (simp_all add: gbinomial_prod_rev field_simps)
722 lemma gbinomial_mult_fact': "(a gchoose n) * fact n = (\<Prod>i = 0..<n. a - of_nat i)"
723   for a :: "'a::field_char_0"
724   using gbinomial_mult_fact [of n a] by (simp add: ac_simps)
726 lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / fact n"
727   for a :: "'a::field_char_0"
728   by (cases n)
729     (simp_all add: pochhammer_minus,
730      simp_all add: gbinomial_prod_rev pochhammer_prod_rev
731        power_mult_distrib [symmetric] atLeastLessThanSuc_atLeastAtMost
732        prod.atLeast_Suc_atMost_Suc_shift of_nat_diff)
734 lemma gbinomial_pochhammer': "s gchoose n = pochhammer (s - of_nat n + 1) n / fact n"
735   for s :: "'a::field_char_0"
736 proof -
737   have "s gchoose n = ((-1)^n * (-1)^n) * pochhammer (s - of_nat n + 1) n / fact n"
738     by (simp add: gbinomial_pochhammer pochhammer_minus mult_ac)
739   also have "(-1 :: 'a)^n * (-1)^n = 1"
740     by (subst power_add [symmetric]) simp
741   finally show ?thesis
742     by simp
743 qed
745 lemma gbinomial_binomial: "n gchoose k = n choose k"
746 proof (cases "k \<le> n")
747   case False
748   then have "n < k"
749     by (simp add: not_le)
750   then have "0 \<in> (op - n)  {0..<k}"
751     by auto
752   then have "prod (op - n) {0..<k} = 0"
753     by (auto intro: prod_zero)
754   with \<open>n < k\<close> show ?thesis
755     by (simp add: binomial_eq_0 gbinomial_prod_rev prod_zero)
756 next
757   case True
758   from True have *: "prod (op - n) {0..<k} = \<Prod>{Suc (n - k)..n}"
759     by (intro prod.reindex_bij_witness[of _ "\<lambda>i. n - i" "\<lambda>i. n - i"]) auto
760   from True have "n choose k = fact n div (fact k * fact (n - k))"
761     by (rule binomial_fact')
762   with * show ?thesis
763     by (simp add: gbinomial_prod_rev mult.commute [of "fact k"] div_mult2_eq fact_div_fact)
764 qed
766 lemma of_nat_gbinomial: "of_nat (n gchoose k) = (of_nat n gchoose k :: 'a::field_char_0)"
767 proof (cases "k \<le> n")
768   case False
769   then show ?thesis
770     by (simp add: not_le gbinomial_binomial binomial_eq_0 gbinomial_prod_rev)
771 next
772   case True
773   define m where "m = n - k"
774   with True have n: "n = m + k"
775     by arith
776   from n have "fact n = ((\<Prod>i = 0..<m + k. of_nat (m + k - i) ):: 'a)"
777     by (simp add: fact_prod_rev)
778   also have "\<dots> = ((\<Prod>i\<in>{0..<k} \<union> {k..<m + k}. of_nat (m + k - i)) :: 'a)"
779     by (simp add: ivl_disj_un)
780   finally have "fact n = (fact m * (\<Prod>i = 0..<k. of_nat m + of_nat k - of_nat i) :: 'a)"
781     using prod_shift_bounds_nat_ivl [of "\<lambda>i. of_nat (m + k - i) :: 'a" 0 k m]
782     by (simp add: fact_prod_rev [of m] prod.union_disjoint of_nat_diff)
783   then have "fact n / fact (n - k) = ((\<Prod>i = 0..<k. of_nat n - of_nat i) :: 'a)"
784     by (simp add: n)
785   with True have "fact k * of_nat (n gchoose k) = (fact k * (of_nat n gchoose k) :: 'a)"
786     by (simp only: gbinomial_mult_fact [of k "of_nat n"] gbinomial_binomial [of n k] fact_binomial)
787   then show ?thesis
788     by simp
789 qed
791 lemma binomial_gbinomial: "of_nat (n choose k) = (of_nat n gchoose k :: 'a::field_char_0)"
792   by (simp add: gbinomial_binomial [symmetric] of_nat_gbinomial)
794 setup
795   \<open>Sign.add_const_constraint (@{const_name gbinomial}, SOME @{typ "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a"})\<close>
797 lemma gbinomial_1[simp]: "a gchoose 1 = a"
798   by (simp add: gbinomial_prod_rev lessThan_Suc)
800 lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a"
801   by (simp add: gbinomial_prod_rev lessThan_Suc)
803 lemma gbinomial_mult_1:
804   fixes a :: "'a::field_char_0"
805   shows "a * (a gchoose n) = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
806   (is "?l = ?r")
807 proof -
808   have "?r = ((- 1) ^n * pochhammer (- a) n / (fact n)) * (of_nat n - (- a + of_nat n))"
809     apply (simp only: gbinomial_pochhammer pochhammer_Suc right_diff_distrib power_Suc)
810     apply (simp del: of_nat_Suc fact_Suc)
811     apply (auto simp add: field_simps simp del: of_nat_Suc)
812     done
813   also have "\<dots> = ?l"
814     by (simp add: field_simps gbinomial_pochhammer)
815   finally show ?thesis ..
816 qed
818 lemma gbinomial_mult_1':
819   "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
820   for a :: "'a::field_char_0"
821   by (simp add: mult.commute gbinomial_mult_1)
823 lemma gbinomial_Suc_Suc: "(a + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
824   for a :: "'a::field_char_0"
825 proof (cases k)
826   case 0
827   then show ?thesis by simp
828 next
829   case (Suc h)
830   have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)"
831     apply (rule prod.reindex_cong [where l = Suc])
832       using Suc
833       apply (auto simp add: image_Suc_atMost)
834     done
835   have "fact (Suc k) * (a gchoose k + (a gchoose (Suc k))) =
836       (a gchoose Suc h) * (fact (Suc (Suc h))) +
837       (a gchoose Suc (Suc h)) * (fact (Suc (Suc h)))"
838     by (simp add: Suc field_simps del: fact_Suc)
839   also have "\<dots> =
840     (a gchoose Suc h) * of_nat (Suc (Suc h) * fact (Suc h)) + (\<Prod>i=0..Suc h. a - of_nat i)"
841     apply (simp del: fact_Suc add: gbinomial_mult_fact field_simps mult.left_commute [of _ "2"])
842     apply (simp del: fact_Suc add: fact_Suc [of "Suc h"] field_simps gbinomial_mult_fact
843       mult.left_commute [of _ "2"] atLeastLessThanSuc_atLeastAtMost)
844     done
845   also have "\<dots> =
846     (fact (Suc h) * (a gchoose Suc h)) * of_nat (Suc (Suc h)) + (\<Prod>i=0..Suc h. a - of_nat i)"
847     by (simp only: fact_Suc mult.commute mult.left_commute of_nat_fact of_nat_id of_nat_mult)
848   also have "\<dots> =
849     of_nat (Suc (Suc h)) * (\<Prod>i=0..h. a - of_nat i) + (\<Prod>i=0..Suc h. a - of_nat i)"
850     unfolding gbinomial_mult_fact atLeastLessThanSuc_atLeastAtMost by auto
851   also have "\<dots> =
852     (\<Prod>i=0..Suc h. a - of_nat i) + (of_nat h * (\<Prod>i=0..h. a - of_nat i) + 2 * (\<Prod>i=0..h. a - of_nat i))"
853     by (simp add: field_simps)
854   also have "\<dots> =
855     ((a gchoose Suc h) * (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{0..Suc h}. a - of_nat i)"
856     unfolding gbinomial_mult_fact'
857     by (simp add: comm_semiring_class.distrib field_simps Suc atLeastLessThanSuc_atLeastAtMost)
858   also have "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 1)"
859     unfolding gbinomial_mult_fact' atLeast0_atMost_Suc
860     by (simp add: field_simps Suc atLeastLessThanSuc_atLeastAtMost)
861   also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)"
862     using eq0
863     by (simp add: Suc prod.atLeast0_atMost_Suc_shift)
864   also have "\<dots> = (fact (Suc k)) * ((a + 1) gchoose (Suc k))"
865     by (simp only: gbinomial_mult_fact atLeastLessThanSuc_atLeastAtMost)
866   finally show ?thesis
867     using fact_nonzero [of "Suc k"] by auto
868 qed
870 lemma gbinomial_reduce_nat: "0 < k \<Longrightarrow> a gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)"
871   for a :: "'a::field_char_0"
872   by (metis Suc_pred' diff_add_cancel gbinomial_Suc_Suc)
874 lemma gchoose_row_sum_weighted:
875   "(\<Sum>k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))"
876   for r :: "'a::field_char_0"
877   by (induct m) (simp_all add: field_simps distrib gbinomial_mult_1)
879 lemma binomial_symmetric:
880   assumes kn: "k \<le> n"
881   shows "n choose k = n choose (n - k)"
882 proof -
883   have kn': "n - k \<le> n"
884     using kn by arith
885   from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']
886   have "fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))"
887     by simp
888   then show ?thesis
889     using kn by simp
890 qed
892 lemma choose_rising_sum:
893   "(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))"
894   "(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose m)"
895 proof -
896   show "(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))"
897     by (induct m) simp_all
898   also have "\<dots> = (n + m + 1) choose m"
899     by (subst binomial_symmetric) simp_all
900   finally show "(\<Sum>j\<le>m. ((n + j) choose n)) = (n + m + 1) choose m" .
901 qed
903 lemma choose_linear_sum: "(\<Sum>i\<le>n. i * (n choose i)) = n * 2 ^ (n - 1)"
904 proof (cases n)
905   case 0
906   then show ?thesis by simp
907 next
908   case (Suc m)
909   have "(\<Sum>i\<le>n. i * (n choose i)) = (\<Sum>i\<le>Suc m. i * (Suc m choose i))"
910     by (simp add: Suc)
911   also have "\<dots> = Suc m * 2 ^ m"
912     by (simp only: sum_atMost_Suc_shift Suc_times_binomial sum_distrib_left[symmetric])
913        (simp add: choose_row_sum')
914   finally show ?thesis
915     using Suc by simp
916 qed
918 lemma choose_alternating_linear_sum:
919   assumes "n \<noteq> 1"
920   shows "(\<Sum>i\<le>n. (-1)^i * of_nat i * of_nat (n choose i) :: 'a::comm_ring_1) = 0"
921 proof (cases n)
922   case 0
923   then show ?thesis by simp
924 next
925   case (Suc m)
926   with assms have "m > 0"
927     by simp
928   have "(\<Sum>i\<le>n. (-1) ^ i * of_nat i * of_nat (n choose i) :: 'a) =
929       (\<Sum>i\<le>Suc m. (-1) ^ i * of_nat i * of_nat (Suc m choose i))"
930     by (simp add: Suc)
931   also have "\<dots> = (\<Sum>i\<le>m. (-1) ^ (Suc i) * of_nat (Suc i * (Suc m choose Suc i)))"
932     by (simp only: sum_atMost_Suc_shift sum_distrib_left[symmetric] mult_ac of_nat_mult) simp
933   also have "\<dots> = - of_nat (Suc m) * (\<Sum>i\<le>m. (-1) ^ i * of_nat (m choose i))"
934     by (subst sum_distrib_left, rule sum.cong[OF refl], subst Suc_times_binomial)
935        (simp add: algebra_simps)
936   also have "(\<Sum>i\<le>m. (-1 :: 'a) ^ i * of_nat ((m choose i))) = 0"
937     using choose_alternating_sum[OF \<open>m > 0\<close>] by simp
938   finally show ?thesis
939     by simp
940 qed
942 lemma vandermonde: "(\<Sum>k\<le>r. (m choose k) * (n choose (r - k))) = (m + n) choose r"
943 proof (induct n arbitrary: r)
944   case 0
945   have "(\<Sum>k\<le>r. (m choose k) * (0 choose (r - k))) = (\<Sum>k\<le>r. if k = r then (m choose k) else 0)"
946     by (intro sum.cong) simp_all
947   also have "\<dots> = m choose r"
948     by (simp add: sum.delta)
949   finally show ?case
950     by simp
951 next
952   case (Suc n r)
953   show ?case
954     by (cases r) (simp_all add: Suc [symmetric] algebra_simps sum.distrib Suc_diff_le)
955 qed
957 lemma choose_square_sum: "(\<Sum>k\<le>n. (n choose k)^2) = ((2*n) choose n)"
958   using vandermonde[of n n n]
959   by (simp add: power2_eq_square mult_2 binomial_symmetric [symmetric])
961 lemma pochhammer_binomial_sum:
962   fixes a b :: "'a::comm_ring_1"
963   shows "pochhammer (a + b) n = (\<Sum>k\<le>n. of_nat (n choose k) * pochhammer a k * pochhammer b (n - k))"
964 proof (induction n arbitrary: a b)
965   case 0
966   then show ?case by simp
967 next
968   case (Suc n a b)
969   have "(\<Sum>k\<le>Suc n. of_nat (Suc n choose k) * pochhammer a k * pochhammer b (Suc n - k)) =
970       (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
971       ((\<Sum>i\<le>n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
972       pochhammer b (Suc n))"
973     by (subst sum_atMost_Suc_shift) (simp add: ring_distribs sum.distrib)
974   also have "(\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) =
975       a * pochhammer ((a + 1) + b) n"
976     by (subst Suc) (simp add: sum_distrib_left pochhammer_rec mult_ac)
977   also have "(\<Sum>i\<le>n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
978         pochhammer b (Suc n) =
979       (\<Sum>i=0..Suc n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))"
980     apply (subst sum_head_Suc)
981     apply simp
982     apply (subst sum_shift_bounds_cl_Suc_ivl)
983     apply (simp add: atLeast0AtMost)
984     done
985   also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))"
986     using Suc by (intro sum.mono_neutral_right) (auto simp: not_le binomial_eq_0)
987   also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc (n - i)))"
988     by (intro sum.cong) (simp_all add: Suc_diff_le)
989   also have "\<dots> = b * pochhammer (a + (b + 1)) n"
990     by (subst Suc) (simp add: sum_distrib_left mult_ac pochhammer_rec)
991   also have "a * pochhammer ((a + 1) + b) n + b * pochhammer (a + (b + 1)) n =
992       pochhammer (a + b) (Suc n)"
993     by (simp add: pochhammer_rec algebra_simps)
994   finally show ?case ..
995 qed
997 text \<open>Contributed by Manuel Eberl, generalised by LCP.
998   Alternative definition of the binomial coefficient as @{term "\<Prod>i<k. (n - i) / (k - i)"}.\<close>
999 lemma gbinomial_altdef_of_nat: "x gchoose k = (\<Prod>i = 0..<k. (x - of_nat i) / of_nat (k - i) :: 'a)"
1000   for k :: nat and x :: "'a::field_char_0"
1001   by (simp add: prod_dividef gbinomial_prod_rev fact_prod_rev)
1003 lemma gbinomial_ge_n_over_k_pow_k:
1004   fixes k :: nat
1005     and x :: "'a::linordered_field"
1006   assumes "of_nat k \<le> x"
1007   shows "(x / of_nat k :: 'a) ^ k \<le> x gchoose k"
1008 proof -
1009   have x: "0 \<le> x"
1010     using assms of_nat_0_le_iff order_trans by blast
1011   have "(x / of_nat k :: 'a) ^ k = (\<Prod>i = 0..<k. x / of_nat k :: 'a)"
1012     by (simp add: prod_constant)
1013   also have "\<dots> \<le> x gchoose k" (* FIXME *)
1014     unfolding gbinomial_altdef_of_nat
1015     apply (safe intro!: prod_mono)
1016     apply simp_all
1017     prefer 2
1018     subgoal premises for i
1019     proof -
1020       from assms have "x * of_nat i \<ge> of_nat (i * k)"
1021         by (metis mult.commute mult_le_cancel_right of_nat_less_0_iff of_nat_mult)
1022       then have "x * of_nat k - x * of_nat i \<le> x * of_nat k - of_nat (i * k)"
1023         by arith
1024       then have "x * of_nat (k - i) \<le> (x - of_nat i) * of_nat k"
1025         using \<open>i < k\<close> by (simp add: algebra_simps zero_less_mult_iff of_nat_diff)
1026       then have "x * of_nat (k - i) \<le> (x - of_nat i) * (of_nat k :: 'a)"
1027         by (simp only: of_nat_mult[symmetric] of_nat_le_iff)
1028       with assms show ?thesis
1029         using \<open>i < k\<close> by (simp add: field_simps)
1030     qed
1031     apply (simp add: x zero_le_divide_iff)
1032     done
1033   finally show ?thesis .
1034 qed
1036 lemma gbinomial_negated_upper: "(a gchoose b) = (-1) ^ b * ((of_nat b - a - 1) gchoose b)"
1037   by (simp add: gbinomial_pochhammer pochhammer_minus algebra_simps)
1039 lemma gbinomial_minus: "((-a) gchoose b) = (-1) ^ b * ((a + of_nat b - 1) gchoose b)"
1040   by (subst gbinomial_negated_upper) (simp add: add_ac)
1042 lemma Suc_times_gbinomial: "of_nat (Suc b) * ((a + 1) gchoose (Suc b)) = (a + 1) * (a gchoose b)"
1043 proof (cases b)
1044   case 0
1045   then show ?thesis by simp
1046 next
1047   case (Suc b)
1048   then have "((a + 1) gchoose (Suc (Suc b))) = (\<Prod>i = 0..Suc b. a + (1 - of_nat i)) / fact (b + 2)"
1049     by (simp add: field_simps gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)
1050   also have "(\<Prod>i = 0..Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i = 0..b. a - of_nat i)"
1051     by (simp add: prod.atLeast0_atMost_Suc_shift)
1052   also have "\<dots> / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)"
1053     by (simp_all add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)
1054   finally show ?thesis by (simp add: Suc field_simps del: of_nat_Suc)
1055 qed
1057 lemma gbinomial_factors: "((a + 1) gchoose (Suc b)) = (a + 1) / of_nat (Suc b) * (a gchoose b)"
1058 proof (cases b)
1059   case 0
1060   then show ?thesis by simp
1061 next
1062   case (Suc b)
1063   then have "((a + 1) gchoose (Suc (Suc b))) = (\<Prod>i = 0 .. Suc b. a + (1 - of_nat i)) / fact (b + 2)"
1064     by (simp add: field_simps gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)
1065   also have "(\<Prod>i = 0 .. Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i = 0..b. a - of_nat i)"
1066     by (simp add: prod.atLeast0_atMost_Suc_shift)
1067   also have "\<dots> / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)"
1068     by (simp_all add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)
1069   finally show ?thesis
1070     by (simp add: Suc)
1071 qed
1073 lemma gbinomial_rec: "((r + 1) gchoose (Suc k)) = (r gchoose k) * ((r + 1) / of_nat (Suc k))"
1074   using gbinomial_mult_1[of r k]
1075   by (subst gbinomial_Suc_Suc) (simp add: field_simps del: of_nat_Suc, simp add: algebra_simps)
1077 lemma gbinomial_of_nat_symmetric: "k \<le> n \<Longrightarrow> (of_nat n) gchoose k = (of_nat n) gchoose (n - k)"
1078   using binomial_symmetric[of k n] by (simp add: binomial_gbinomial [symmetric])
1081 text \<open>The absorption identity (equation 5.5 \cite[p.~157]{GKP}):
1082 $1083 {r \choose k} = \frac{r}{k}{r - 1 \choose k - 1},\quad \textnormal{integer } k \neq 0. 1084$\<close>
1085 lemma gbinomial_absorption': "k > 0 \<Longrightarrow> r gchoose k = (r / of_nat k) * (r - 1 gchoose (k - 1))"
1086   using gbinomial_rec[of "r - 1" "k - 1"]
1087   by (simp_all add: gbinomial_rec field_simps del: of_nat_Suc)
1089 text \<open>The absorption identity is written in the following form to avoid
1090 division by $k$ (the lower index) and therefore remove the $k \neq 0$
1091 restriction\cite[p.~157]{GKP}:
1092 $1093 k{r \choose k} = r{r - 1 \choose k - 1}, \quad \textnormal{integer } k. 1094$\<close>
1095 lemma gbinomial_absorption: "of_nat (Suc k) * (r gchoose Suc k) = r * ((r - 1) gchoose k)"
1096   using gbinomial_absorption'[of "Suc k" r] by (simp add: field_simps del: of_nat_Suc)
1098 text \<open>The absorption identity for natural number binomial coefficients:\<close>
1099 lemma binomial_absorption: "Suc k * (n choose Suc k) = n * ((n - 1) choose k)"
1100   by (cases n) (simp_all add: binomial_eq_0 Suc_times_binomial del: binomial_Suc_Suc mult_Suc)
1102 text \<open>The absorption companion identity for natural number coefficients,
1103   following the proof by GKP \cite[p.~157]{GKP}:\<close>
1104 lemma binomial_absorb_comp: "(n - k) * (n choose k) = n * ((n - 1) choose k)"
1105   (is "?lhs = ?rhs")
1106 proof (cases "n \<le> k")
1107   case True
1108   then show ?thesis by auto
1109 next
1110   case False
1111   then have "?rhs = Suc ((n - 1) - k) * (n choose Suc ((n - 1) - k))"
1112     using binomial_symmetric[of k "n - 1"] binomial_absorption[of "(n - 1) - k" n]
1113     by simp
1114   also have "Suc ((n - 1) - k) = n - k"
1115     using False by simp
1116   also have "n choose \<dots> = n choose k"
1117     using False by (intro binomial_symmetric [symmetric]) simp_all
1118   finally show ?thesis ..
1119 qed
1121 text \<open>The generalised absorption companion identity:\<close>
1122 lemma gbinomial_absorb_comp: "(r - of_nat k) * (r gchoose k) = r * ((r - 1) gchoose k)"
1123   using pochhammer_absorb_comp[of r k] by (simp add: gbinomial_pochhammer)
1126   "r gchoose (Suc k) = ((r - 1) gchoose (Suc k)) + ((r - 1) gchoose k)"
1127   using gbinomial_Suc_Suc[of "r - 1" k] by (simp add: algebra_simps)
1130   "0 < n \<Longrightarrow> n choose (Suc k) = ((n - 1) choose (Suc k)) + ((n - 1) choose k)"
1131   by (subst choose_reduce_nat) simp_all
1133 text \<open>
1134   Equation 5.9 of the reference material \cite[p.~159]{GKP} is a useful
1135   summation formula, operating on both indices:
1136   $1137 \sum\limits_{k \leq n}{r + k \choose k} = {r + n + 1 \choose n}, 1138 \quad \textnormal{integer } n. 1139$
1140 \<close>
1141 lemma gbinomial_parallel_sum: "(\<Sum>k\<le>n. (r + of_nat k) gchoose k) = (r + of_nat n + 1) gchoose n"
1142 proof (induct n)
1143   case 0
1144   then show ?case by simp
1145 next
1146   case (Suc m)
1147   then show ?case
1148     using gbinomial_Suc_Suc[of "(r + of_nat m + 1)" m]
1150 qed
1153 subsubsection \<open>Summation on the upper index\<close>
1155 text \<open>
1156   Another summation formula is equation 5.10 of the reference material \cite[p.~160]{GKP},
1157   aptly named \emph{summation on the upper index}:$\sum_{0 \leq k \leq n} {k \choose m} = 1158 {n + 1 \choose m + 1}, \quad \textnormal{integers } m, n \geq 0.$
1159 \<close>
1160 lemma gbinomial_sum_up_index:
1161   "(\<Sum>k = 0..n. (of_nat k gchoose m) :: 'a::field_char_0) = (of_nat n + 1) gchoose (m + 1)"
1162 proof (induct n)
1163   case 0
1164   show ?case
1165     using gbinomial_Suc_Suc[of 0 m]
1166     by (cases m) auto
1167 next
1168   case (Suc n)
1169   then show ?case
1170     using gbinomial_Suc_Suc[of "of_nat (Suc n) :: 'a" m]
1172 qed
1174 lemma gbinomial_index_swap:
1175   "((-1) ^ m) * ((- (of_nat n) - 1) gchoose m) = ((-1) ^ n) * ((- (of_nat m) - 1) gchoose n)"
1176   (is "?lhs = ?rhs")
1177 proof -
1178   have "?lhs = (of_nat (m + n) gchoose m)"
1179     by (subst gbinomial_negated_upper) (simp add: power_mult_distrib [symmetric])
1180   also have "\<dots> = (of_nat (m + n) gchoose n)"
1181     by (subst gbinomial_of_nat_symmetric) simp_all
1182   also have "\<dots> = ?rhs"
1183     by (subst gbinomial_negated_upper) simp
1184   finally show ?thesis .
1185 qed
1187 lemma gbinomial_sum_lower_neg: "(\<Sum>k\<le>m. (r gchoose k) * (- 1) ^ k) = (- 1) ^ m * (r - 1 gchoose m)"
1188   (is "?lhs = ?rhs")
1189 proof -
1190   have "?lhs = (\<Sum>k\<le>m. -(r + 1) + of_nat k gchoose k)"
1191     by (intro sum.cong[OF refl]) (subst gbinomial_negated_upper, simp add: power_mult_distrib)
1192   also have "\<dots>  = - r + of_nat m gchoose m"
1193     by (subst gbinomial_parallel_sum) simp
1194   also have "\<dots> = ?rhs"
1195     by (subst gbinomial_negated_upper) (simp add: power_mult_distrib)
1196   finally show ?thesis .
1197 qed
1199 lemma gbinomial_partial_row_sum:
1200   "(\<Sum>k\<le>m. (r gchoose k) * ((r / 2) - of_nat k)) = ((of_nat m + 1)/2) * (r gchoose (m + 1))"
1201 proof (induct m)
1202   case 0
1203   then show ?case by simp
1204 next
1205   case (Suc mm)
1206   then have "(\<Sum>k\<le>Suc mm. (r gchoose k) * (r / 2 - of_nat k)) =
1207       (r - of_nat (Suc mm)) * (r gchoose Suc mm) / 2"
1208     by (simp add: field_simps)
1209   also have "\<dots> = r * (r - 1 gchoose Suc mm) / 2"
1210     by (subst gbinomial_absorb_comp) (rule refl)
1211   also have "\<dots> = (of_nat (Suc mm) + 1) / 2 * (r gchoose (Suc mm + 1))"
1212     by (subst gbinomial_absorption [symmetric]) simp
1213   finally show ?case .
1214 qed
1216 lemma sum_bounds_lt_plus1: "(\<Sum>k<mm. f (Suc k)) = (\<Sum>k=1..mm. f k)"
1217   by (induct mm) simp_all
1219 lemma gbinomial_partial_sum_poly:
1220   "(\<Sum>k\<le>m. (of_nat m + r gchoose k) * x^k * y^(m-k)) =
1221     (\<Sum>k\<le>m. (-r gchoose k) * (-x)^k * (x + y)^(m-k))"
1222   (is "?lhs m = ?rhs m")
1223 proof (induction m)
1224   case 0
1225   then show ?case by simp
1226 next
1227   case (Suc mm)
1228   define G where "G i k = (of_nat i + r gchoose k) * x^k * y^(i - k)" for i k
1229   define S where "S = ?lhs"
1230   have SG_def: "S = (\<lambda>i. (\<Sum>k\<le>i. (G i k)))"
1231     unfolding S_def G_def ..
1233   have "S (Suc mm) = G (Suc mm) 0 + (\<Sum>k=Suc 0..Suc mm. G (Suc mm) k)"
1234     using SG_def by (simp add: sum_head_Suc atLeast0AtMost [symmetric])
1235   also have "(\<Sum>k=Suc 0..Suc mm. G (Suc mm) k) = (\<Sum>k=0..mm. G (Suc mm) (Suc k))"
1236     by (subst sum_shift_bounds_cl_Suc_ivl) simp
1237   also have "\<dots> = (\<Sum>k=0..mm. ((of_nat mm + r gchoose (Suc k)) +
1238       (of_nat mm + r gchoose k)) * x^(Suc k) * y^(mm - k))"
1239     unfolding G_def by (subst gbinomial_addition_formula) simp
1240   also have "\<dots> = (\<Sum>k=0..mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k)) +
1241       (\<Sum>k=0..mm. (of_nat mm + r gchoose k) * x^(Suc k) * y^(mm - k))"
1242     by (subst sum.distrib [symmetric]) (simp add: algebra_simps)
1243   also have "(\<Sum>k=0..mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k)) =
1244       (\<Sum>k<Suc mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k))"
1245     by (simp only: atLeast0AtMost lessThan_Suc_atMost)
1246   also have "\<dots> = (\<Sum>k<mm. (of_nat mm + r gchoose Suc k) * x^(Suc k) * y^(mm-k)) +
1247       (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm)"
1248     (is "_ = ?A + ?B")
1249     by (subst sum_lessThan_Suc) simp
1250   also have "?A = (\<Sum>k=1..mm. (of_nat mm + r gchoose k) * x^k * y^(mm - k + 1))"
1251   proof (subst sum_bounds_lt_plus1 [symmetric], intro sum.cong[OF refl], clarify)
1252     fix k
1253     assume "k < mm"
1254     then have "mm - k = mm - Suc k + 1"
1255       by linarith
1256     then show "(of_nat mm + r gchoose Suc k) * x ^ Suc k * y ^ (mm - k) =
1257         (of_nat mm + r gchoose Suc k) * x ^ Suc k * y ^ (mm - Suc k + 1)"
1258       by (simp only:)
1259   qed
1260   also have "\<dots> + ?B = y * (\<Sum>k=1..mm. (G mm k)) + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm)"
1261     unfolding G_def by (subst sum_distrib_left) (simp add: algebra_simps)
1262   also have "(\<Sum>k=0..mm. (of_nat mm + r gchoose k) * x^(Suc k) * y^(mm - k)) = x * (S mm)"
1263     unfolding S_def by (subst sum_distrib_left) (simp add: atLeast0AtMost algebra_simps)
1264   also have "(G (Suc mm) 0) = y * (G mm 0)"
1265     by (simp add: G_def)
1266   finally have "S (Suc mm) =
1267       y * (G mm 0 + (\<Sum>k=1..mm. (G mm k))) + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm) + x * (S mm)"
1268     by (simp add: ring_distribs)
1269   also have "G mm 0 + (\<Sum>k=1..mm. (G mm k)) = S mm"
1270     by (simp add: sum_head_Suc[symmetric] SG_def atLeast0AtMost)
1271   finally have "S (Suc mm) = (x + y) * (S mm) + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm)"
1272     by (simp add: algebra_simps)
1273   also have "(of_nat mm + r gchoose (Suc mm)) = (-1) ^ (Suc mm) * (- r gchoose (Suc mm))"
1274     by (subst gbinomial_negated_upper) simp
1275   also have "(-1) ^ Suc mm * (- r gchoose Suc mm) * x ^ Suc mm =
1276       (- r gchoose (Suc mm)) * (-x) ^ Suc mm"
1277     by (simp add: power_minus[of x])
1278   also have "(x + y) * S mm + \<dots> = (x + y) * ?rhs mm + (- r gchoose (Suc mm)) * (- x)^Suc mm"
1279     unfolding S_def by (subst Suc.IH) simp
1280   also have "(x + y) * ?rhs mm = (\<Sum>n\<le>mm. ((- r gchoose n) * (- x) ^ n * (x + y) ^ (Suc mm - n)))"
1281     by (subst sum_distrib_left, rule sum.cong) (simp_all add: Suc_diff_le)
1282   also have "\<dots> + (-r gchoose (Suc mm)) * (-x)^Suc mm =
1283       (\<Sum>n\<le>Suc mm. (- r gchoose n) * (- x) ^ n * (x + y) ^ (Suc mm - n))"
1284     by simp
1285   finally show ?case
1286     by (simp only: S_def)
1287 qed
1289 lemma gbinomial_partial_sum_poly_xpos:
1290   "(\<Sum>k\<le>m. (of_nat m + r gchoose k) * x^k * y^(m-k)) =
1291      (\<Sum>k\<le>m. (of_nat k + r - 1 gchoose k) * x^k * (x + y)^(m-k))"
1292   apply (subst gbinomial_partial_sum_poly)
1293   apply (subst gbinomial_negated_upper)
1294   apply (intro sum.cong, rule refl)
1295   apply (simp add: power_mult_distrib [symmetric])
1296   done
1298 lemma binomial_r_part_sum: "(\<Sum>k\<le>m. (2 * m + 1 choose k)) = 2 ^ (2 * m)"
1299 proof -
1300   have "2 * 2^(2*m) = (\<Sum>k = 0..(2 * m + 1). (2 * m + 1 choose k))"
1301     using choose_row_sum[where n="2 * m + 1"] by simp
1302   also have "(\<Sum>k = 0..(2 * m + 1). (2 * m + 1 choose k)) =
1303       (\<Sum>k = 0..m. (2 * m + 1 choose k)) +
1304       (\<Sum>k = m+1..2*m+1. (2 * m + 1 choose k))"
1305     using sum_ub_add_nat[of 0 m "\<lambda>k. 2 * m + 1 choose k" "m+1"]
1306     by (simp add: mult_2)
1307   also have "(\<Sum>k = m+1..2*m+1. (2 * m + 1 choose k)) =
1308       (\<Sum>k = 0..m. (2 * m + 1 choose (k + (m + 1))))"
1309     by (subst sum_shift_bounds_cl_nat_ivl [symmetric]) (simp add: mult_2)
1310   also have "\<dots> = (\<Sum>k = 0..m. (2 * m + 1 choose (m - k)))"
1311     by (intro sum.cong[OF refl], subst binomial_symmetric) simp_all
1312   also have "\<dots> = (\<Sum>k = 0..m. (2 * m + 1 choose k))"
1313     using sum.atLeast_atMost_rev [of "\<lambda>k. 2 * m + 1 choose (m - k)" 0 m]
1314     by simp
1315   also have "\<dots> + \<dots> = 2 * \<dots>"
1316     by simp
1317   finally show ?thesis
1318     by (subst (asm) mult_cancel1) (simp add: atLeast0AtMost)
1319 qed
1321 lemma gbinomial_r_part_sum: "(\<Sum>k\<le>m. (2 * (of_nat m) + 1 gchoose k)) = 2 ^ (2 * m)"
1322   (is "?lhs = ?rhs")
1323 proof -
1324   have "?lhs = of_nat (\<Sum>k\<le>m. (2 * m + 1) choose k)"
1326   also have "\<dots> = of_nat (2 ^ (2 * m))"
1327     by (subst binomial_r_part_sum) (rule refl)
1328   finally show ?thesis by simp
1329 qed
1331 lemma gbinomial_sum_nat_pow2:
1332   "(\<Sum>k\<le>m. (of_nat (m + k) gchoose k :: 'a::field_char_0) / 2 ^ k) = 2 ^ m"
1333   (is "?lhs = ?rhs")
1334 proof -
1335   have "2 ^ m * 2 ^ m = (2 ^ (2*m) :: 'a)"
1336     by (induct m) simp_all
1337   also have "\<dots> = (\<Sum>k\<le>m. (2 * (of_nat m) + 1 gchoose k))"
1338     using gbinomial_r_part_sum ..
1339   also have "\<dots> = (\<Sum>k\<le>m. (of_nat (m + k) gchoose k) * 2 ^ (m - k))"
1340     using gbinomial_partial_sum_poly_xpos[where x="1" and y="1" and r="of_nat m + 1" and m="m"]
1342   also have "\<dots> = 2 ^ m * (\<Sum>k\<le>m. (of_nat (m + k) gchoose k) / 2 ^ k)"
1343     by (subst sum_distrib_left) (simp add: algebra_simps power_diff)
1344   finally show ?thesis
1345     by (subst (asm) mult_left_cancel) simp_all
1346 qed
1348 lemma gbinomial_trinomial_revision:
1349   assumes "k \<le> m"
1350   shows "(r gchoose m) * (of_nat m gchoose k) = (r gchoose k) * (r - of_nat k gchoose (m - k))"
1351 proof -
1352   have "(r gchoose m) * (of_nat m gchoose k) = (r gchoose m) * fact m / (fact k * fact (m - k))"
1353     using assms by (simp add: binomial_gbinomial [symmetric] binomial_fact)
1354   also have "\<dots> = (r gchoose k) * (r - of_nat k gchoose (m - k))"
1355     using assms by (simp add: gbinomial_pochhammer power_diff pochhammer_product)
1356   finally show ?thesis .
1357 qed
1359 text \<open>Versions of the theorems above for the natural-number version of "choose"\<close>
1360 lemma binomial_altdef_of_nat:
1361   "k \<le> n \<Longrightarrow> of_nat (n choose k) = (\<Prod>i = 0..<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
1362   for n k :: nat and x :: "'a::field_char_0"
1363   by (simp add: gbinomial_altdef_of_nat binomial_gbinomial of_nat_diff)
1365 lemma binomial_ge_n_over_k_pow_k: "k \<le> n \<Longrightarrow> (of_nat n / of_nat k :: 'a) ^ k \<le> of_nat (n choose k)"
1366   for k n :: nat and x :: "'a::linordered_field"
1367   by (simp add: gbinomial_ge_n_over_k_pow_k binomial_gbinomial of_nat_diff)
1369 lemma binomial_le_pow:
1370   assumes "r \<le> n"
1371   shows "n choose r \<le> n ^ r"
1372 proof -
1373   have "n choose r \<le> fact n div fact (n - r)"
1374     using assms by (subst binomial_fact_lemma[symmetric]) auto
1375   with fact_div_fact_le_pow [OF assms] show ?thesis
1376     by auto
1377 qed
1379 lemma binomial_altdef_nat: "k \<le> n \<Longrightarrow> n choose k = fact n div (fact k * fact (n - k))"
1380   for k n :: nat
1381   by (subst binomial_fact_lemma [symmetric]) auto
1383 lemma choose_dvd:
1384   "k \<le> n \<Longrightarrow> fact k * fact (n - k) dvd (fact n :: 'a::{semiring_div,linordered_semidom})"
1385   unfolding dvd_def
1386   apply (rule exI [where x="of_nat (n choose k)"])
1387   using binomial_fact_lemma [of k n, THEN arg_cong [where f = of_nat and 'b='a]]
1388   apply auto
1389   done
1391 lemma fact_fact_dvd_fact:
1392   "fact k * fact n dvd (fact (k + n) :: 'a::{semiring_div,linordered_semidom})"
1395 lemma choose_mult_lemma:
1396   "((m + r + k) choose (m + k)) * ((m + k) choose k) = ((m + r + k) choose k) * ((m + r) choose m)"
1397   (is "?lhs = _")
1398 proof -
1399   have "?lhs =
1400       fact (m + r + k) div (fact (m + k) * fact (m + r - m)) * (fact (m + k) div (fact k * fact m))"
1401     by (simp add: binomial_altdef_nat)
1402   also have "\<dots> = fact (m + r + k) div (fact r * (fact k * fact m))"
1403     apply (subst div_mult_div_if_dvd)
1404     apply (auto simp: algebra_simps fact_fact_dvd_fact)
1406     done
1407   also have "\<dots> = (fact (m + r + k) * fact (m + r)) div (fact r * (fact k * fact m) * fact (m + r))"
1408     apply (subst div_mult_div_if_dvd [symmetric])
1409     apply (auto simp add: algebra_simps)
1410     apply (metis fact_fact_dvd_fact dvd_trans nat_mult_dvd_cancel_disj)
1411     done
1412   also have "\<dots> =
1413       (fact (m + r + k) div (fact k * fact (m + r)) * (fact (m + r) div (fact r * fact m)))"
1414     apply (subst div_mult_div_if_dvd)
1415     apply (auto simp: fact_fact_dvd_fact algebra_simps)
1416     done
1417   finally show ?thesis
1418     by (simp add: binomial_altdef_nat mult.commute)
1419 qed
1421 text \<open>The "Subset of a Subset" identity.\<close>
1422 lemma choose_mult:
1423   "k \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> (n choose m) * (m choose k) = (n choose k) * ((n - k) choose (m - k))"
1424   using choose_mult_lemma [of "m-k" "n-m" k] by simp
1427 subsection \<open>More on Binomial Coefficients\<close>
1429 lemma choose_one: "n choose 1 = n" for n :: nat
1430   by simp
1432 lemma card_UNION:
1433   assumes "finite A"
1434     and "\<forall>k \<in> A. finite k"
1435   shows "card (\<Union>A) = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * int (card (\<Inter>I)))"
1436   (is "?lhs = ?rhs")
1437 proof -
1438   have "?rhs = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * (\<Sum>_\<in>\<Inter>I. 1))"
1439     by simp
1440   also have "\<dots> = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (\<Sum>_\<in>\<Inter>I. (- 1) ^ (card I + 1)))"
1441     (is "_ = nat ?rhs")
1442     by (subst sum_distrib_left) simp
1443   also have "?rhs = (\<Sum>(I, _)\<in>Sigma {I. I \<subseteq> A \<and> I \<noteq> {}} Inter. (- 1) ^ (card I + 1))"
1444     using assms by (subst sum.Sigma) auto
1445   also have "\<dots> = (\<Sum>(x, I)\<in>(SIGMA x:UNIV. {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}). (- 1) ^ (card I + 1))"
1446     by (rule sum.reindex_cong [where l = "\<lambda>(x, y). (y, x)"]) (auto intro: inj_onI simp add: split_beta)
1447   also have "\<dots> = (\<Sum>(x, I)\<in>(SIGMA x:\<Union>A. {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}). (- 1) ^ (card I + 1))"
1448     using assms
1449     by (auto intro!: sum.mono_neutral_cong_right finite_SigmaI2 intro: finite_subset[where B="\<Union>A"])
1450   also have "\<dots> = (\<Sum>x\<in>\<Union>A. (\<Sum>I|I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I. (- 1) ^ (card I + 1)))"
1451     using assms by (subst sum.Sigma) auto
1452   also have "\<dots> = (\<Sum>_\<in>\<Union>A. 1)" (is "sum ?lhs _ = _")
1453   proof (rule sum.cong[OF refl])
1454     fix x
1455     assume x: "x \<in> \<Union>A"
1456     define K where "K = {X \<in> A. x \<in> X}"
1457     with \<open>finite A\<close> have K: "finite K"
1458       by auto
1459     let ?I = "\<lambda>i. {I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I}"
1460     have "inj_on snd (SIGMA i:{1..card A}. ?I i)"
1461       using assms by (auto intro!: inj_onI)
1462     moreover have [symmetric]: "snd  (SIGMA i:{1..card A}. ?I i) = {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}"
1463       using assms
1464       by (auto intro!: rev_image_eqI[where x="(card a, a)" for a]
1465         simp add: card_gt_0_iff[folded Suc_le_eq]
1466         dest: finite_subset intro: card_mono)
1467     ultimately have "?lhs x = (\<Sum>(i, I)\<in>(SIGMA i:{1..card A}. ?I i). (- 1) ^ (i + 1))"
1468       by (rule sum.reindex_cong [where l = snd]) fastforce
1469     also have "\<dots> = (\<Sum>i=1..card A. (\<Sum>I|I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. (- 1) ^ (i + 1)))"
1470       using assms by (subst sum.Sigma) auto
1471     also have "\<dots> = (\<Sum>i=1..card A. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1))"
1472       by (subst sum_distrib_left) simp
1473     also have "\<dots> = (\<Sum>i=1..card K. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> K \<and> card I = i. 1))"
1474       (is "_ = ?rhs")
1475     proof (rule sum.mono_neutral_cong_right[rule_format])
1476       show "finite {1..card A}"
1477         by simp
1478       show "{1..card K} \<subseteq> {1..card A}"
1479         using \<open>finite A\<close> by (auto simp add: K_def intro: card_mono)
1480     next
1481       fix i
1482       assume "i \<in> {1..card A} - {1..card K}"
1483       then have i: "i \<le> card A" "card K < i"
1484         by auto
1485       have "{I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I} = {I. I \<subseteq> K \<and> card I = i}"
1486         by (auto simp add: K_def)
1487       also have "\<dots> = {}"
1488         using \<open>finite A\<close> i by (auto simp add: K_def dest: card_mono[rotated 1])
1489       finally show "(- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1 :: int) = 0"
1490         by (simp only:) simp
1491     next
1492       fix i
1493       have "(\<Sum>I | I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1) = (\<Sum>I | I \<subseteq> K \<and> card I = i. 1 :: int)"
1494         (is "?lhs = ?rhs")
1495         by (rule sum.cong) (auto simp add: K_def)
1496       then show "(- 1) ^ (i + 1) * ?lhs = (- 1) ^ (i + 1) * ?rhs"
1497         by simp
1498     qed
1499     also have "{I. I \<subseteq> K \<and> card I = 0} = {{}}"
1500       using assms by (auto simp add: card_eq_0_iff K_def dest: finite_subset)
1501     then have "?rhs = (\<Sum>i = 0..card K. (- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> K \<and> card I = i. 1 :: int)) + 1"
1502       by (subst (2) sum_head_Suc) simp_all
1503     also have "\<dots> = (\<Sum>i = 0..card K. (- 1) * ((- 1) ^ i * int (card K choose i))) + 1"
1504       using K by (subst n_subsets[symmetric]) simp_all
1505     also have "\<dots> = - (\<Sum>i = 0..card K. (- 1) ^ i * int (card K choose i)) + 1"
1506       by (subst sum_distrib_left[symmetric]) simp
1507     also have "\<dots> =  - ((-1 + 1) ^ card K) + 1"
1508       by (subst binomial_ring) (simp add: ac_simps)
1509     also have "\<dots> = 1"
1510       using x K by (auto simp add: K_def card_gt_0_iff)
1511     finally show "?lhs x = 1" .
1512   qed
1513   also have "nat \<dots> = card (\<Union>A)"
1514     by simp
1515   finally show ?thesis ..
1516 qed
1518 text \<open>The number of nat lists of length \<open>m\<close> summing to \<open>N\<close> is @{term "(N + m - 1) choose N"}:\<close>
1519 lemma card_length_sum_list_rec:
1520   assumes "m \<ge> 1"
1521   shows "card {l::nat list. length l = m \<and> sum_list l = N} =
1522       card {l. length l = (m - 1) \<and> sum_list l = N} +
1523       card {l. length l = m \<and> sum_list l + 1 = N}"
1524     (is "card ?C = card ?A + card ?B")
1525 proof -
1526   let ?A' = "{l. length l = m \<and> sum_list l = N \<and> hd l = 0}"
1527   let ?B' = "{l. length l = m \<and> sum_list l = N \<and> hd l \<noteq> 0}"
1528   let ?f = "\<lambda>l. 0 # l"
1529   let ?g = "\<lambda>l. (hd l + 1) # tl l"
1530   have 1: "xs \<noteq> [] \<Longrightarrow> x = hd xs \<Longrightarrow> x # tl xs = xs" for x xs
1531     by simp
1532   have 2: "xs \<noteq> [] \<Longrightarrow> sum_list(tl xs) = sum_list xs - hd xs" for xs :: "nat list"
1533     by (auto simp add: neq_Nil_conv)
1534   have f: "bij_betw ?f ?A ?A'"
1535     apply (rule bij_betw_byWitness[where f' = tl])
1536     using assms
1537     apply (auto simp: 2 length_0_conv[symmetric] 1 simp del: length_0_conv)
1538     done
1539   have 3: "xs \<noteq> [] \<Longrightarrow> hd xs + (sum_list xs - hd xs) = sum_list xs" for xs :: "nat list"
1540     by (metis 1 sum_list_simps(2) 2)
1541   have g: "bij_betw ?g ?B ?B'"
1542     apply (rule bij_betw_byWitness[where f' = "\<lambda>l. (hd l - 1) # tl l"])
1543     using assms
1544     by (auto simp: 2 length_0_conv[symmetric] intro!: 3
1545         simp del: length_greater_0_conv length_0_conv)
1546   have fin: "finite {xs. size xs = M \<and> set xs \<subseteq> {0..<N}}" for M N :: nat
1547     using finite_lists_length_eq[OF finite_atLeastLessThan] conj_commute by auto
1548   have fin_A: "finite ?A" using fin[of _ "N+1"]
1549     by (intro finite_subset[where ?A = "?A" and ?B = "{xs. size xs = m - 1 \<and> set xs \<subseteq> {0..<N+1}}"])
1550       (auto simp: member_le_sum_list_nat less_Suc_eq_le)
1551   have fin_B: "finite ?B"
1552     by (intro finite_subset[where ?A = "?B" and ?B = "{xs. size xs = m \<and> set xs \<subseteq> {0..<N}}"])
1553       (auto simp: member_le_sum_list_nat less_Suc_eq_le fin)
1554   have uni: "?C = ?A' \<union> ?B'"
1555     by auto
1556   have disj: "?A' \<inter> ?B' = {}" by blast
1557   have "card ?C = card(?A' \<union> ?B')"
1558     using uni by simp
1559   also have "\<dots> = card ?A + card ?B"
1560     using card_Un_disjoint[OF _ _ disj] bij_betw_finite[OF f] bij_betw_finite[OF g]
1561       bij_betw_same_card[OF f] bij_betw_same_card[OF g] fin_A fin_B
1562     by presburger
1563   finally show ?thesis .
1564 qed
1566 lemma card_length_sum_list: "card {l::nat list. size l = m \<and> sum_list l = N} = (N + m - 1) choose N"
1567   \<comment> "by Holden Lee, tidied by Tobias Nipkow"
1568 proof (cases m)
1569   case 0
1570   then show ?thesis
1571     by (cases N) (auto cong: conj_cong)
1572 next
1573   case (Suc m')
1574   have m: "m \<ge> 1"
1575     by (simp add: Suc)
1576   then show ?thesis
1577   proof (induct "N + m - 1" arbitrary: N m)
1578     case 0  \<comment> "In the base case, the only solution is ."
1579     have [simp]: "{l::nat list. length l = Suc 0 \<and> (\<forall>n\<in>set l. n = 0)} = {}"
1580       by (auto simp: length_Suc_conv)
1581     have "m = 1 \<and> N = 0"
1582       using 0 by linarith
1583     then show ?case
1584       by simp
1585   next
1586     case (Suc k)
1587     have c1: "card {l::nat list. size l = (m - 1) \<and> sum_list l =  N} = (N + (m - 1) - 1) choose N"
1588     proof (cases "m = 1")
1589       case True
1590       with Suc.hyps have "N \<ge> 1"
1591         by auto
1592       with True show ?thesis
1593         by (simp add: binomial_eq_0)
1594     next
1595       case False
1596       then show ?thesis
1597         using Suc by fastforce
1598     qed
1599     from Suc have c2: "card {l::nat list. size l = m \<and> sum_list l + 1 = N} =
1600       (if N > 0 then ((N - 1) + m - 1) choose (N - 1) else 0)"
1601     proof -
1602       have *: "n > 0 \<Longrightarrow> Suc m = n \<longleftrightarrow> m = n - 1" for m n
1603         by arith
1604       from Suc have "N > 0 \<Longrightarrow>
1605         card {l::nat list. size l = m \<and> sum_list l + 1 = N} =
1606           ((N - 1) + m - 1) choose (N - 1)"
1607         by (simp add: *)
1608       then show ?thesis
1609         by auto
1610     qed
1611     from Suc.prems have "(card {l::nat list. size l = (m - 1) \<and> sum_list l = N} +
1612           card {l::nat list. size l = m \<and> sum_list l + 1 = N}) = (N + m - 1) choose N"
1613       by (auto simp: c1 c2 choose_reduce_nat[of "N + m - 1" N] simp del: One_nat_def)
1614     then show ?case
1615       using card_length_sum_list_rec[OF Suc.prems] by auto
1616   qed
1617 qed
1619 lemma card_disjoint_shuffle:
1620   assumes "set xs \<inter> set ys = {}"
1621   shows   "card (shuffle xs ys) = (length xs + length ys) choose length xs"
1622 using assms
1623 proof (induction xs ys rule: shuffle.induct)
1624   case (3 x xs y ys)
1625   have "shuffle (x # xs) (y # ys) = op # x  shuffle xs (y # ys) \<union> op # y  shuffle (x # xs) ys"
1626     by (rule shuffle.simps)
1627   also have "card \<dots> = card (op # x  shuffle xs (y # ys)) + card (op # y  shuffle (x # xs) ys)"
1628     by (rule card_Un_disjoint) (insert "3.prems", auto)
1629   also have "card (op # x  shuffle xs (y # ys)) = card (shuffle xs (y # ys))"
1630     by (rule card_image) auto
1631   also have "\<dots> = (length xs + length (y # ys)) choose length xs"
1632     using "3.prems" by (intro "3.IH") auto
1633   also have "card (op # y  shuffle (x # xs) ys) = card (shuffle (x # xs) ys)"
1634     by (rule card_image) auto
1635   also have "\<dots> = (length (x # xs) + length ys) choose length (x # xs)"
1636     using "3.prems" by (intro "3.IH") auto
1637   also have "length xs + length (y # ys) choose length xs + \<dots> =
1638                (length (x # xs) + length (y # ys)) choose length (x # xs)" by simp
1639   finally show ?case .
1640 qed auto
1642 lemma Suc_times_binomial_add: "Suc a * (Suc (a + b) choose Suc a) = Suc b * (Suc (a + b) choose a)"
1643   \<comment> \<open>by Lukas Bulwahn\<close>
1644 proof -
1645   have dvd: "Suc a * (fact a * fact b) dvd fact (Suc (a + b))" for a b
1646     using fact_fact_dvd_fact[of "Suc a" "b", where 'a=nat]
1647     by (simp only: fact_Suc add_Suc[symmetric] of_nat_id mult.assoc)
1648   have "Suc a * (fact (Suc (a + b)) div (Suc a * fact a * fact b)) =
1649       Suc a * fact (Suc (a + b)) div (Suc a * (fact a * fact b))"
1650     by (subst div_mult_swap[symmetric]; simp only: mult.assoc dvd)
1651   also have "\<dots> = Suc b * fact (Suc (a + b)) div (Suc b * (fact a * fact b))"
1652     by (simp only: div_mult_mult1)
1653   also have "\<dots> = Suc b * (fact (Suc (a + b)) div (Suc b * (fact a * fact b)))"
1654     using dvd[of b a] by (subst div_mult_swap[symmetric]; simp only: ac_simps dvd)
1655   finally show ?thesis
1656     by (subst (1 2) binomial_altdef_nat)
1657       (simp_all only: ac_simps diff_Suc_Suc Suc_diff_le diff_add_inverse fact_Suc of_nat_id)
1658 qed
1661 subsection \<open>Misc\<close>
1663 lemma fact_code [code]:
1664   "fact n = (of_nat (fold_atLeastAtMost_nat (op *) 2 n 1) :: 'a::semiring_char_0)"
1665 proof -
1666   have "fact n = (of_nat (\<Prod>{1..n}) :: 'a)"
1667     by (simp add: fact_prod)
1668   also have "\<Prod>{1..n} = \<Prod>{2..n}"
1669     by (intro prod.mono_neutral_right) auto
1670   also have "\<dots> = fold_atLeastAtMost_nat (op *) 2 n 1"
1671     by (simp add: prod_atLeastAtMost_code)
1672   finally show ?thesis .
1673 qed
1675 lemma pochhammer_code [code]:
1676   "pochhammer a n =
1677     (if n = 0 then 1
1678      else fold_atLeastAtMost_nat (\<lambda>n acc. (a + of_nat n) * acc) 0 (n - 1) 1)"
1679   by (cases n)
1680     (simp_all add: pochhammer_prod prod_atLeastAtMost_code [symmetric]
1681       atLeastLessThanSuc_atLeastAtMost)
1683 lemma gbinomial_code [code]:
1684   "a gchoose n =
1685     (if n = 0 then 1
1686      else fold_atLeastAtMost_nat (\<lambda>n acc. (a - of_nat n) * acc) 0 (n - 1) 1 / fact n)"
1687   by (cases n)
1688     (simp_all add: gbinomial_prod_rev prod_atLeastAtMost_code [symmetric]
1689       atLeastLessThanSuc_atLeastAtMost)
1691 (* FIXME *)
1692 (*TODO: This code equation breaks Scala code generation in HOL-Codegenerator_Test. We have to figure out why and how to prevent that. *)
1694 (*
1695 lemma binomial_code [code]:
1696   "(n choose k) =
1697       (if k > n then 0
1698        else if 2 * k > n then (n choose (n - k))
1699        else (fold_atLeastAtMost_nat (op * ) (n-k+1) n 1 div fact k))"
1700 proof -
1701   {
1702     assume "k \<le> n"
1703     then have "{1..n} = {1..n-k} \<union> {n-k+1..n}" by auto
1704     then have "(fact n :: nat) = fact (n-k) * \<Prod>{n-k+1..n}"
1705       by (simp add: prod.union_disjoint fact_altdef_nat)
1706   }
1707   then show ?thesis by (auto simp: binomial_altdef_nat mult_ac prod_atLeastAtMost_code)
1708 qed
1709 *)
1711 end