src/HOL/Binomial.thy
 author wenzelm Sat Nov 04 15:24:40 2017 +0100 (19 months ago) changeset 67003 49850a679c2c parent 66806 a4e82b58d833 child 67299 ba52a058942f permissions -rw-r--r--
more robust sorted_entries;
1 (*  Title:      HOL/Binomial.thy
2     Author:     Jacques D. Fleuriot
3     Author:     Lawrence C Paulson
5     Author:     Chaitanya Mangla
6     Author:     Manuel Eberl
7 *)
9 section \<open>Binomial Coefficients and Binomial Theorem\<close>
11 theory Binomial
12   imports Presburger Factorial
13 begin
15 subsection \<open>Binomial coefficients\<close>
17 text \<open>This development is based on the work of Andy Gordon and Florian Kammueller.\<close>
19 text \<open>Combinatorial definition\<close>
21 definition binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat"  (infixl "choose" 65)
22   where "n choose k = card {K\<in>Pow {0..<n}. card K = k}"
24 theorem n_subsets:
25   assumes "finite A"
26   shows "card {B. B \<subseteq> A \<and> card B = k} = card A choose k"
27 proof -
28   from assms obtain f where bij: "bij_betw f {0..<card A} A"
29     by (blast dest: ex_bij_betw_nat_finite)
30   then have [simp]: "card (f  C) = card C" if "C \<subseteq> {0..<card A}" for C
31     by (meson bij_betw_imp_inj_on bij_betw_subset card_image that)
32   from bij have "bij_betw (image f) (Pow {0..<card A}) (Pow A)"
33     by (rule bij_betw_Pow)
34   then have "inj_on (image f) (Pow {0..<card A})"
35     by (rule bij_betw_imp_inj_on)
36   moreover have "{K. K \<subseteq> {0..<card A} \<and> card K = k} \<subseteq> Pow {0..<card A}"
37     by auto
38   ultimately have "inj_on (image f) {K. K \<subseteq> {0..<card A} \<and> card K = k}"
39     by (rule inj_on_subset)
40   then have "card {K. K \<subseteq> {0..<card A} \<and> card K = k} =
41       card (image f  {K. K \<subseteq> {0..<card A} \<and> card K = k})" (is "_ = card ?C")
43   also have "?C = {K. K \<subseteq> f  {0..<card A} \<and> card K = k}"
44     by (auto elim!: subset_imageE)
45   also have "f  {0..<card A} = A"
46     by (meson bij bij_betw_def)
47   finally show ?thesis
49 qed
51 text \<open>Recursive characterization\<close>
53 lemma binomial_n_0 [simp, code]: "n choose 0 = 1"
54 proof -
55   have "{K \<in> Pow {0..<n}. card K = 0} = {{}}"
56     by (auto dest: finite_subset)
57   then show ?thesis
59 qed
61 lemma binomial_0_Suc [simp, code]: "0 choose Suc k = 0"
64 lemma binomial_Suc_Suc [simp, code]: "Suc n choose Suc k = (n choose k) + (n choose Suc k)"
65 proof -
66   let ?P = "\<lambda>n k. {K. K \<subseteq> {0..<n} \<and> card K = k}"
67   let ?Q = "?P (Suc n) (Suc k)"
68   have inj: "inj_on (insert n) (?P n k)"
69     by rule (auto; metis atLeastLessThan_iff insert_iff less_irrefl subsetCE)
70   have disjoint: "insert n  ?P n k \<inter> ?P n (Suc k) = {}"
71     by auto
72   have "?Q = {K\<in>?Q. n \<in> K} \<union> {K\<in>?Q. n \<notin> K}"
73     by auto
74   also have "{K\<in>?Q. n \<in> K} = insert n  ?P n k" (is "?A = ?B")
75   proof (rule set_eqI)
76     fix K
77     have K_finite: "finite K" if "K \<subseteq> insert n {0..<n}"
78       using that by (rule finite_subset) simp_all
79     have Suc_card_K: "Suc (card K - Suc 0) = card K" if "n \<in> K"
80       and "finite K"
81     proof -
82       from \<open>n \<in> K\<close> obtain L where "K = insert n L" and "n \<notin> L"
83         by (blast elim: Set.set_insert)
84       with that show ?thesis by (simp add: card_insert)
85     qed
86     show "K \<in> ?A \<longleftrightarrow> K \<in> ?B"
87       by (subst in_image_insert_iff)
88         (auto simp add: card_insert subset_eq_atLeast0_lessThan_finite
89           Diff_subset_conv K_finite Suc_card_K)
90   qed
91   also have "{K\<in>?Q. n \<notin> K} = ?P n (Suc k)"
92     by (auto simp add: atLeast0_lessThan_Suc)
93   finally show ?thesis using inj disjoint
94     by (simp add: binomial_def card_Un_disjoint card_image)
95 qed
97 lemma binomial_eq_0: "n < k \<Longrightarrow> n choose k = 0"
98   by (auto simp add: binomial_def dest: subset_eq_atLeast0_lessThan_card)
100 lemma zero_less_binomial: "k \<le> n \<Longrightarrow> n choose k > 0"
101   by (induct n k rule: diff_induct) simp_all
103 lemma binomial_eq_0_iff [simp]: "n choose k = 0 \<longleftrightarrow> n < k"
104   by (metis binomial_eq_0 less_numeral_extra(3) not_less zero_less_binomial)
106 lemma zero_less_binomial_iff [simp]: "n choose k > 0 \<longleftrightarrow> k \<le> n"
107   by (metis binomial_eq_0_iff not_less0 not_less zero_less_binomial)
109 lemma binomial_n_n [simp]: "n choose n = 1"
110   by (induct n) (simp_all add: binomial_eq_0)
112 lemma binomial_Suc_n [simp]: "Suc n choose n = Suc n"
113   by (induct n) simp_all
115 lemma binomial_1 [simp]: "n choose Suc 0 = n"
116   by (induct n) simp_all
118 lemma choose_reduce_nat:
119   "0 < n \<Longrightarrow> 0 < k \<Longrightarrow>
120     n choose k = ((n - 1) choose (k - 1)) + ((n - 1) choose k)"
121   using binomial_Suc_Suc [of "n - 1" "k - 1"] by simp
123 lemma Suc_times_binomial_eq: "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
124   apply (induct n arbitrary: k)
125    apply simp
126    apply arith
127   apply (case_tac k)
129   done
131 lemma binomial_le_pow2: "n choose k \<le> 2^n"
132   apply (induct n arbitrary: k)
133    apply (case_tac k)
134     apply simp_all
135   apply (case_tac k)
136    apply auto
138   done
140 text \<open>The absorption property.\<close>
141 lemma Suc_times_binomial: "Suc k * (Suc n choose Suc k) = Suc n * (n choose k)"
142   using Suc_times_binomial_eq by auto
144 text \<open>This is the well-known version of absorption, but it's harder to use
145   because of the need to reason about division.\<close>
146 lemma binomial_Suc_Suc_eq_times: "(Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
147   by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right)
149 text \<open>Another version of absorption, with \<open>-1\<close> instead of \<open>Suc\<close>.\<close>
150 lemma times_binomial_minus1_eq: "0 < k \<Longrightarrow> k * (n choose k) = n * ((n - 1) choose (k - 1))"
151   using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 1"]
152   by (auto split: nat_diff_split)
155 subsection \<open>The binomial theorem (courtesy of Tobias Nipkow):\<close>
157 text \<open>Avigad's version, generalized to any commutative ring\<close>
158 theorem binomial_ring: "(a + b :: 'a::{comm_ring_1,power})^n =
159   (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))"
160 proof (induct n)
161   case 0
162   then show ?case by simp
163 next
164   case (Suc n)
165   have decomp: "{0..n+1} = {0} \<union> {n + 1} \<union> {1..n}"
166     by auto
167   have decomp2: "{0..n} = {0} \<union> {1..n}"
168     by auto
169   have "(a + b)^(n+1) = (a + b) * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n - k))"
170     using Suc.hyps by simp
171   also have "\<dots> = a * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k)) +
172       b * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
173     by (rule distrib_right)
174   also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) +
175       (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n - k + 1))"
176     by (auto simp add: sum_distrib_left ac_simps)
177   also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n + 1 - k)) +
178       (\<Sum>k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n + 1 - k))"
179     by (simp add:sum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps del: sum_cl_ivl_Suc)
180   also have "\<dots> = a^(n + 1) + b^(n + 1) +
181       (\<Sum>k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n + 1 - k)) +
182       (\<Sum>k=1..n. of_nat (n choose k) * a^k * b^(n + 1 - k))"
184   also have "\<dots> = a^(n + 1) + b^(n + 1) +
185       (\<Sum>k=1..n. of_nat (n + 1 choose k) * a^k * b^(n + 1 - k))"
186     by (auto simp add: field_simps sum.distrib [symmetric] choose_reduce_nat)
187   also have "\<dots> = (\<Sum>k=0..n+1. of_nat (n + 1 choose k) * a^k * b^(n + 1 - k))"
188     using decomp by (simp add: field_simps)
189   finally show ?case
190     by simp
191 qed
193 text \<open>Original version for the naturals.\<close>
194 corollary binomial: "(a + b :: nat)^n = (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n - k))"
195   using binomial_ring [of "int a" "int b" n]
196   by (simp only: of_nat_add [symmetric] of_nat_mult [symmetric] of_nat_power [symmetric]
197       of_nat_sum [symmetric] of_nat_eq_iff of_nat_id)
199 lemma binomial_fact_lemma: "k \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n"
200 proof (induct n arbitrary: k rule: nat_less_induct)
201   fix n k
202   assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) = fact m"
203   assume kn: "k \<le> n"
204   let ?ths = "fact k * fact (n - k) * (n choose k) = fact n"
205   consider "n = 0 \<or> k = 0 \<or> n = k" | m h where "n = Suc m" "k = Suc h" "h < m"
206     using kn by atomize_elim presburger
207   then show "fact k * fact (n - k) * (n choose k) = fact n"
208   proof cases
209     case 1
210     with kn show ?thesis by auto
211   next
212     case 2
213     note n = \<open>n = Suc m\<close>
214     note k = \<open>k = Suc h\<close>
215     note hm = \<open>h < m\<close>
216     have mn: "m < n"
217       using n by arith
218     have hm': "h \<le> m"
219       using hm by arith
220     have km: "k \<le> m"
221       using hm k n kn by arith
222     have "m - h = Suc (m - Suc h)"
223       using  k km hm by arith
224     with km k have "fact (m - h) = (m - h) * fact (m - k)"
225       by simp
226     with n k have "fact k * fact (n - k) * (n choose k) =
227         k * (fact h * fact (m - h) * (m choose h)) +
228         (m - h) * (fact k * fact (m - k) * (m choose k))"
230     also have "\<dots> = (k + (m - h)) * fact m"
231       using H[rule_format, OF mn hm'] H[rule_format, OF mn km]
233     finally show ?thesis
234       using k n km by simp
235   qed
236 qed
238 lemma binomial_fact':
239   assumes "k \<le> n"
240   shows "n choose k = fact n div (fact k * fact (n - k))"
241   using binomial_fact_lemma [OF assms]
242   by (metis fact_nonzero mult_eq_0_iff nonzero_mult_div_cancel_left)
244 lemma binomial_fact:
245   assumes kn: "k \<le> n"
246   shows "(of_nat (n choose k) :: 'a::field_char_0) = fact n / (fact k * fact (n - k))"
247   using binomial_fact_lemma[OF kn]
249   apply (metis mult.commute of_nat_fact of_nat_mult)
250   done
252 lemma fact_binomial:
253   assumes "k \<le> n"
254   shows "fact k * of_nat (n choose k) = (fact n / fact (n - k) :: 'a::field_char_0)"
255   unfolding binomial_fact [OF assms] by (simp add: field_simps)
257 lemma choose_two: "n choose 2 = n * (n - 1) div 2"
258 proof (cases "n \<ge> 2")
259   case False
260   then have "n = 0 \<or> n = 1"
261     by auto
262   then show ?thesis by auto
263 next
264   case True
265   define m where "m = n - 2"
266   with True have "n = m + 2"
267     by simp
268   then have "fact n = n * (n - 1) * fact (n - 2)"
269     by (simp add: fact_prod_Suc atLeast0_lessThan_Suc algebra_simps)
270   with True show ?thesis
272 qed
274 lemma choose_row_sum: "(\<Sum>k=0..n. n choose k) = 2^n"
275   using binomial [of 1 "1" n] by (simp add: numeral_2_eq_2)
277 lemma sum_choose_lower: "(\<Sum>k=0..n. (r+k) choose k) = Suc (r+n) choose n"
278   by (induct n) auto
280 lemma sum_choose_upper: "(\<Sum>k=0..n. k choose m) = Suc n choose Suc m"
281   by (induct n) auto
283 lemma choose_alternating_sum:
284   "n > 0 \<Longrightarrow> (\<Sum>i\<le>n. (-1)^i * of_nat (n choose i)) = (0 :: 'a::comm_ring_1)"
285   using binomial_ring[of "-1 :: 'a" 1 n]
286   by (simp add: atLeast0AtMost mult_of_nat_commute zero_power)
288 lemma choose_even_sum:
289   assumes "n > 0"
290   shows "2 * (\<Sum>i\<le>n. if even i then of_nat (n choose i) else 0) = (2 ^ n :: 'a::comm_ring_1)"
291 proof -
292   have "2 ^ n = (\<Sum>i\<le>n. of_nat (n choose i)) + (\<Sum>i\<le>n. (-1) ^ i * of_nat (n choose i) :: 'a)"
293     using choose_row_sum[of n]
294     by (simp add: choose_alternating_sum assms atLeast0AtMost of_nat_sum[symmetric])
295   also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) + (-1) ^ i * of_nat (n choose i))"
297   also have "\<dots> = 2 * (\<Sum>i\<le>n. if even i then of_nat (n choose i) else 0)"
298     by (subst sum_distrib_left, intro sum.cong) simp_all
299   finally show ?thesis ..
300 qed
302 lemma choose_odd_sum:
303   assumes "n > 0"
304   shows "2 * (\<Sum>i\<le>n. if odd i then of_nat (n choose i) else 0) = (2 ^ n :: 'a::comm_ring_1)"
305 proof -
306   have "2 ^ n = (\<Sum>i\<le>n. of_nat (n choose i)) - (\<Sum>i\<le>n. (-1) ^ i * of_nat (n choose i) :: 'a)"
307     using choose_row_sum[of n]
308     by (simp add: choose_alternating_sum assms atLeast0AtMost of_nat_sum[symmetric])
309   also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) - (-1) ^ i * of_nat (n choose i))"
311   also have "\<dots> = 2 * (\<Sum>i\<le>n. if odd i then of_nat (n choose i) else 0)"
312     by (subst sum_distrib_left, intro sum.cong) simp_all
313   finally show ?thesis ..
314 qed
316 lemma choose_row_sum': "(\<Sum>k\<le>n. (n choose k)) = 2 ^ n"
317   using choose_row_sum[of n] by (simp add: atLeast0AtMost)
319 text\<open>NW diagonal sum property\<close>
320 lemma sum_choose_diagonal:
321   assumes "m \<le> n"
322   shows "(\<Sum>k=0..m. (n - k) choose (m - k)) = Suc n choose m"
323 proof -
324   have "(\<Sum>k=0..m. (n-k) choose (m - k)) = (\<Sum>k=0..m. (n - m + k) choose k)"
325     using sum.atLeast_atMost_rev [of "\<lambda>k. (n - k) choose (m - k)" 0 m] assms
326       by simp
327   also have "\<dots> = Suc (n - m + m) choose m"
328     by (rule sum_choose_lower)
329   also have "\<dots> = Suc n choose m"
330     using assms by simp
331   finally show ?thesis .
332 qed
335 subsection \<open>Generalized binomial coefficients\<close>
337 definition gbinomial :: "'a::{semidom_divide,semiring_char_0} \<Rightarrow> nat \<Rightarrow> 'a"  (infixl "gchoose" 65)
338   where gbinomial_prod_rev: "a gchoose n = prod (\<lambda>i. a - of_nat i) {0..<n} div fact n"
340 lemma gbinomial_0 [simp]:
341   "a gchoose 0 = 1"
342   "0 gchoose (Suc n) = 0"
343   by (simp_all add: gbinomial_prod_rev prod.atLeast0_lessThan_Suc_shift)
345 lemma gbinomial_Suc: "a gchoose (Suc k) = prod (\<lambda>i. a - of_nat i) {0..k} div fact (Suc k)"
346   by (simp add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)
348 lemma gbinomial_mult_fact: "fact n * (a gchoose n) = (\<Prod>i = 0..<n. a - of_nat i)"
349   for a :: "'a::field_char_0"
350   by (simp_all add: gbinomial_prod_rev field_simps)
352 lemma gbinomial_mult_fact': "(a gchoose n) * fact n = (\<Prod>i = 0..<n. a - of_nat i)"
353   for a :: "'a::field_char_0"
354   using gbinomial_mult_fact [of n a] by (simp add: ac_simps)
356 lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / fact n"
357   for a :: "'a::field_char_0"
358   by (cases n)
361        power_mult_distrib [symmetric] atLeastLessThanSuc_atLeastAtMost
362        prod.atLeast_Suc_atMost_Suc_shift of_nat_diff)
364 lemma gbinomial_pochhammer': "s gchoose n = pochhammer (s - of_nat n + 1) n / fact n"
365   for s :: "'a::field_char_0"
366 proof -
367   have "s gchoose n = ((-1)^n * (-1)^n) * pochhammer (s - of_nat n + 1) n / fact n"
368     by (simp add: gbinomial_pochhammer pochhammer_minus mult_ac)
369   also have "(-1 :: 'a)^n * (-1)^n = 1"
370     by (subst power_add [symmetric]) simp
371   finally show ?thesis
372     by simp
373 qed
375 lemma gbinomial_binomial: "n gchoose k = n choose k"
376 proof (cases "k \<le> n")
377   case False
378   then have "n < k"
380   then have "0 \<in> (op - n)  {0..<k}"
381     by auto
382   then have "prod (op - n) {0..<k} = 0"
383     by (auto intro: prod_zero)
384   with \<open>n < k\<close> show ?thesis
385     by (simp add: binomial_eq_0 gbinomial_prod_rev prod_zero)
386 next
387   case True
388   from True have *: "prod (op - n) {0..<k} = \<Prod>{Suc (n - k)..n}"
389     by (intro prod.reindex_bij_witness[of _ "\<lambda>i. n - i" "\<lambda>i. n - i"]) auto
390   from True have "n choose k = fact n div (fact k * fact (n - k))"
391     by (rule binomial_fact')
392   with * show ?thesis
393     by (simp add: gbinomial_prod_rev mult.commute [of "fact k"] div_mult2_eq fact_div_fact)
394 qed
396 lemma of_nat_gbinomial: "of_nat (n gchoose k) = (of_nat n gchoose k :: 'a::field_char_0)"
397 proof (cases "k \<le> n")
398   case False
399   then show ?thesis
400     by (simp add: not_le gbinomial_binomial binomial_eq_0 gbinomial_prod_rev)
401 next
402   case True
403   define m where "m = n - k"
404   with True have n: "n = m + k"
405     by arith
406   from n have "fact n = ((\<Prod>i = 0..<m + k. of_nat (m + k - i) ):: 'a)"
408   also have "\<dots> = ((\<Prod>i\<in>{0..<k} \<union> {k..<m + k}. of_nat (m + k - i)) :: 'a)"
410   finally have "fact n = (fact m * (\<Prod>i = 0..<k. of_nat m + of_nat k - of_nat i) :: 'a)"
411     using prod_shift_bounds_nat_ivl [of "\<lambda>i. of_nat (m + k - i) :: 'a" 0 k m]
412     by (simp add: fact_prod_rev [of m] prod.union_disjoint of_nat_diff)
413   then have "fact n / fact (n - k) = ((\<Prod>i = 0..<k. of_nat n - of_nat i) :: 'a)"
415   with True have "fact k * of_nat (n gchoose k) = (fact k * (of_nat n gchoose k) :: 'a)"
416     by (simp only: gbinomial_mult_fact [of k "of_nat n"] gbinomial_binomial [of n k] fact_binomial)
417   then show ?thesis
418     by simp
419 qed
421 lemma binomial_gbinomial: "of_nat (n choose k) = (of_nat n gchoose k :: 'a::field_char_0)"
422   by (simp add: gbinomial_binomial [symmetric] of_nat_gbinomial)
424 setup
425   \<open>Sign.add_const_constraint (@{const_name gbinomial}, SOME @{typ "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a"})\<close>
427 lemma gbinomial_1[simp]: "a gchoose 1 = a"
428   by (simp add: gbinomial_prod_rev lessThan_Suc)
430 lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a"
431   by (simp add: gbinomial_prod_rev lessThan_Suc)
433 lemma gbinomial_mult_1:
434   fixes a :: "'a::field_char_0"
435   shows "a * (a gchoose n) = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
436   (is "?l = ?r")
437 proof -
438   have "?r = ((- 1) ^n * pochhammer (- a) n / (fact n)) * (of_nat n - (- a + of_nat n))"
439     apply (simp only: gbinomial_pochhammer pochhammer_Suc right_diff_distrib power_Suc)
440     apply (simp del: of_nat_Suc fact_Suc)
441     apply (auto simp add: field_simps simp del: of_nat_Suc)
442     done
443   also have "\<dots> = ?l"
444     by (simp add: field_simps gbinomial_pochhammer)
445   finally show ?thesis ..
446 qed
448 lemma gbinomial_mult_1':
449   "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
450   for a :: "'a::field_char_0"
451   by (simp add: mult.commute gbinomial_mult_1)
453 lemma gbinomial_Suc_Suc: "(a + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
454   for a :: "'a::field_char_0"
455 proof (cases k)
456   case 0
457   then show ?thesis by simp
458 next
459   case (Suc h)
460   have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)"
461     apply (rule prod.reindex_cong [where l = Suc])
462       using Suc
463       apply (auto simp add: image_Suc_atMost)
464     done
465   have "fact (Suc k) * (a gchoose k + (a gchoose (Suc k))) =
466       (a gchoose Suc h) * (fact (Suc (Suc h))) +
467       (a gchoose Suc (Suc h)) * (fact (Suc (Suc h)))"
468     by (simp add: Suc field_simps del: fact_Suc)
469   also have "\<dots> =
470     (a gchoose Suc h) * of_nat (Suc (Suc h) * fact (Suc h)) + (\<Prod>i=0..Suc h. a - of_nat i)"
471     apply (simp del: fact_Suc add: gbinomial_mult_fact field_simps mult.left_commute [of _ "2"])
472     apply (simp del: fact_Suc add: fact_Suc [of "Suc h"] field_simps gbinomial_mult_fact
473       mult.left_commute [of _ "2"] atLeastLessThanSuc_atLeastAtMost)
474     done
475   also have "\<dots> =
476     (fact (Suc h) * (a gchoose Suc h)) * of_nat (Suc (Suc h)) + (\<Prod>i=0..Suc h. a - of_nat i)"
477     by (simp only: fact_Suc mult.commute mult.left_commute of_nat_fact of_nat_id of_nat_mult)
478   also have "\<dots> =
479     of_nat (Suc (Suc h)) * (\<Prod>i=0..h. a - of_nat i) + (\<Prod>i=0..Suc h. a - of_nat i)"
480     unfolding gbinomial_mult_fact atLeastLessThanSuc_atLeastAtMost by auto
481   also have "\<dots> =
482     (\<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))"
484   also have "\<dots> =
485     ((a gchoose Suc h) * (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{0..Suc h}. a - of_nat i)"
486     unfolding gbinomial_mult_fact'
487     by (simp add: comm_semiring_class.distrib field_simps Suc atLeastLessThanSuc_atLeastAtMost)
488   also have "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 1)"
489     unfolding gbinomial_mult_fact' atLeast0_atMost_Suc
490     by (simp add: field_simps Suc atLeastLessThanSuc_atLeastAtMost)
491   also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)"
492     using eq0
493     by (simp add: Suc prod.atLeast0_atMost_Suc_shift)
494   also have "\<dots> = (fact (Suc k)) * ((a + 1) gchoose (Suc k))"
495     by (simp only: gbinomial_mult_fact atLeastLessThanSuc_atLeastAtMost)
496   finally show ?thesis
497     using fact_nonzero [of "Suc k"] by auto
498 qed
500 lemma gbinomial_reduce_nat: "0 < k \<Longrightarrow> a gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)"
501   for a :: "'a::field_char_0"
502   by (metis Suc_pred' diff_add_cancel gbinomial_Suc_Suc)
504 lemma gchoose_row_sum_weighted:
505   "(\<Sum>k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))"
506   for r :: "'a::field_char_0"
507   by (induct m) (simp_all add: field_simps distrib gbinomial_mult_1)
509 lemma binomial_symmetric:
510   assumes kn: "k \<le> n"
511   shows "n choose k = n choose (n - k)"
512 proof -
513   have kn': "n - k \<le> n"
514     using kn by arith
515   from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']
516   have "fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))"
517     by simp
518   then show ?thesis
519     using kn by simp
520 qed
522 lemma choose_rising_sum:
523   "(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))"
524   "(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose m)"
525 proof -
526   show "(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))"
527     by (induct m) simp_all
528   also have "\<dots> = (n + m + 1) choose m"
529     by (subst binomial_symmetric) simp_all
530   finally show "(\<Sum>j\<le>m. ((n + j) choose n)) = (n + m + 1) choose m" .
531 qed
533 lemma choose_linear_sum: "(\<Sum>i\<le>n. i * (n choose i)) = n * 2 ^ (n - 1)"
534 proof (cases n)
535   case 0
536   then show ?thesis by simp
537 next
538   case (Suc m)
539   have "(\<Sum>i\<le>n. i * (n choose i)) = (\<Sum>i\<le>Suc m. i * (Suc m choose i))"
541   also have "\<dots> = Suc m * 2 ^ m"
542     by (simp only: sum_atMost_Suc_shift Suc_times_binomial sum_distrib_left[symmetric])
544   finally show ?thesis
545     using Suc by simp
546 qed
548 lemma choose_alternating_linear_sum:
549   assumes "n \<noteq> 1"
550   shows "(\<Sum>i\<le>n. (-1)^i * of_nat i * of_nat (n choose i) :: 'a::comm_ring_1) = 0"
551 proof (cases n)
552   case 0
553   then show ?thesis by simp
554 next
555   case (Suc m)
556   with assms have "m > 0"
557     by simp
558   have "(\<Sum>i\<le>n. (-1) ^ i * of_nat i * of_nat (n choose i) :: 'a) =
559       (\<Sum>i\<le>Suc m. (-1) ^ i * of_nat i * of_nat (Suc m choose i))"
561   also have "\<dots> = (\<Sum>i\<le>m. (-1) ^ (Suc i) * of_nat (Suc i * (Suc m choose Suc i)))"
562     by (simp only: sum_atMost_Suc_shift sum_distrib_left[symmetric] mult_ac of_nat_mult) simp
563   also have "\<dots> = - of_nat (Suc m) * (\<Sum>i\<le>m. (-1) ^ i * of_nat (m choose i))"
564     by (subst sum_distrib_left, rule sum.cong[OF refl], subst Suc_times_binomial)
566   also have "(\<Sum>i\<le>m. (-1 :: 'a) ^ i * of_nat ((m choose i))) = 0"
567     using choose_alternating_sum[OF \<open>m > 0\<close>] by simp
568   finally show ?thesis
569     by simp
570 qed
572 lemma vandermonde: "(\<Sum>k\<le>r. (m choose k) * (n choose (r - k))) = (m + n) choose r"
573 proof (induct n arbitrary: r)
574   case 0
575   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)"
576     by (intro sum.cong) simp_all
577   also have "\<dots> = m choose r"
579   finally show ?case
580     by simp
581 next
582   case (Suc n r)
583   show ?case
584     by (cases r) (simp_all add: Suc [symmetric] algebra_simps sum.distrib Suc_diff_le)
585 qed
587 lemma choose_square_sum: "(\<Sum>k\<le>n. (n choose k)^2) = ((2*n) choose n)"
588   using vandermonde[of n n n]
589   by (simp add: power2_eq_square mult_2 binomial_symmetric [symmetric])
591 lemma pochhammer_binomial_sum:
592   fixes a b :: "'a::comm_ring_1"
593   shows "pochhammer (a + b) n = (\<Sum>k\<le>n. of_nat (n choose k) * pochhammer a k * pochhammer b (n - k))"
594 proof (induction n arbitrary: a b)
595   case 0
596   then show ?case by simp
597 next
598   case (Suc n a b)
599   have "(\<Sum>k\<le>Suc n. of_nat (Suc n choose k) * pochhammer a k * pochhammer b (Suc n - k)) =
600       (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
601       ((\<Sum>i\<le>n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
602       pochhammer b (Suc n))"
603     by (subst sum_atMost_Suc_shift) (simp add: ring_distribs sum.distrib)
604   also have "(\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) =
605       a * pochhammer ((a + 1) + b) n"
606     by (subst Suc) (simp add: sum_distrib_left pochhammer_rec mult_ac)
607   also have "(\<Sum>i\<le>n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
608         pochhammer b (Suc n) =
609       (\<Sum>i=0..Suc n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))"
611     apply simp
612     apply (subst sum_shift_bounds_cl_Suc_ivl)
614     done
615   also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))"
616     using Suc by (intro sum.mono_neutral_right) (auto simp: not_le binomial_eq_0)
617   also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc (n - i)))"
618     by (intro sum.cong) (simp_all add: Suc_diff_le)
619   also have "\<dots> = b * pochhammer (a + (b + 1)) n"
620     by (subst Suc) (simp add: sum_distrib_left mult_ac pochhammer_rec)
621   also have "a * pochhammer ((a + 1) + b) n + b * pochhammer (a + (b + 1)) n =
622       pochhammer (a + b) (Suc n)"
623     by (simp add: pochhammer_rec algebra_simps)
624   finally show ?case ..
625 qed
627 text \<open>Contributed by Manuel Eberl, generalised by LCP.
628   Alternative definition of the binomial coefficient as @{term "\<Prod>i<k. (n - i) / (k - i)"}.\<close>
629 lemma gbinomial_altdef_of_nat: "x gchoose k = (\<Prod>i = 0..<k. (x - of_nat i) / of_nat (k - i) :: 'a)"
630   for k :: nat and x :: "'a::field_char_0"
631   by (simp add: prod_dividef gbinomial_prod_rev fact_prod_rev)
633 lemma gbinomial_ge_n_over_k_pow_k:
634   fixes k :: nat
635     and x :: "'a::linordered_field"
636   assumes "of_nat k \<le> x"
637   shows "(x / of_nat k :: 'a) ^ k \<le> x gchoose k"
638 proof -
639   have x: "0 \<le> x"
640     using assms of_nat_0_le_iff order_trans by blast
641   have "(x / of_nat k :: 'a) ^ k = (\<Prod>i = 0..<k. x / of_nat k :: 'a)"
643   also have "\<dots> \<le> x gchoose k" (* FIXME *)
644     unfolding gbinomial_altdef_of_nat
645     apply (safe intro!: prod_mono)
646     apply simp_all
647     prefer 2
648     subgoal premises for i
649     proof -
650       from assms have "x * of_nat i \<ge> of_nat (i * k)"
651         by (metis mult.commute mult_le_cancel_right of_nat_less_0_iff of_nat_mult)
652       then have "x * of_nat k - x * of_nat i \<le> x * of_nat k - of_nat (i * k)"
653         by arith
654       then have "x * of_nat (k - i) \<le> (x - of_nat i) * of_nat k"
655         using \<open>i < k\<close> by (simp add: algebra_simps zero_less_mult_iff of_nat_diff)
656       then have "x * of_nat (k - i) \<le> (x - of_nat i) * (of_nat k :: 'a)"
657         by (simp only: of_nat_mult[symmetric] of_nat_le_iff)
658       with assms show ?thesis
659         using \<open>i < k\<close> by (simp add: field_simps)
660     qed
661     apply (simp add: x zero_le_divide_iff)
662     done
663   finally show ?thesis .
664 qed
666 lemma gbinomial_negated_upper: "(a gchoose b) = (-1) ^ b * ((of_nat b - a - 1) gchoose b)"
667   by (simp add: gbinomial_pochhammer pochhammer_minus algebra_simps)
669 lemma gbinomial_minus: "((-a) gchoose b) = (-1) ^ b * ((a + of_nat b - 1) gchoose b)"
672 lemma Suc_times_gbinomial: "of_nat (Suc b) * ((a + 1) gchoose (Suc b)) = (a + 1) * (a gchoose b)"
673 proof (cases b)
674   case 0
675   then show ?thesis by simp
676 next
677   case (Suc b)
678   then have "((a + 1) gchoose (Suc (Suc b))) = (\<Prod>i = 0..Suc b. a + (1 - of_nat i)) / fact (b + 2)"
679     by (simp add: field_simps gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)
680   also have "(\<Prod>i = 0..Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i = 0..b. a - of_nat i)"
682   also have "\<dots> / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)"
683     by (simp_all add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)
684   finally show ?thesis by (simp add: Suc field_simps del: of_nat_Suc)
685 qed
687 lemma gbinomial_factors: "((a + 1) gchoose (Suc b)) = (a + 1) / of_nat (Suc b) * (a gchoose b)"
688 proof (cases b)
689   case 0
690   then show ?thesis by simp
691 next
692   case (Suc b)
693   then have "((a + 1) gchoose (Suc (Suc b))) = (\<Prod>i = 0 .. Suc b. a + (1 - of_nat i)) / fact (b + 2)"
694     by (simp add: field_simps gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)
695   also have "(\<Prod>i = 0 .. Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i = 0..b. a - of_nat i)"
697   also have "\<dots> / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)"
698     by (simp_all add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)
699   finally show ?thesis
701 qed
703 lemma gbinomial_rec: "((r + 1) gchoose (Suc k)) = (r gchoose k) * ((r + 1) / of_nat (Suc k))"
704   using gbinomial_mult_1[of r k]
705   by (subst gbinomial_Suc_Suc) (simp add: field_simps del: of_nat_Suc, simp add: algebra_simps)
707 lemma gbinomial_of_nat_symmetric: "k \<le> n \<Longrightarrow> (of_nat n) gchoose k = (of_nat n) gchoose (n - k)"
708   using binomial_symmetric[of k n] by (simp add: binomial_gbinomial [symmetric])
711 text \<open>The absorption identity (equation 5.5 \cite[p.~157]{GKP}):
712 $713 {r \choose k} = \frac{r}{k}{r - 1 \choose k - 1},\quad \textnormal{integer } k \neq 0. 714$\<close>
715 lemma gbinomial_absorption': "k > 0 \<Longrightarrow> r gchoose k = (r / of_nat k) * (r - 1 gchoose (k - 1))"
716   using gbinomial_rec[of "r - 1" "k - 1"]
717   by (simp_all add: gbinomial_rec field_simps del: of_nat_Suc)
719 text \<open>The absorption identity is written in the following form to avoid
720 division by $k$ (the lower index) and therefore remove the $k \neq 0$
721 restriction\cite[p.~157]{GKP}:
722 $723 k{r \choose k} = r{r - 1 \choose k - 1}, \quad \textnormal{integer } k. 724$\<close>
725 lemma gbinomial_absorption: "of_nat (Suc k) * (r gchoose Suc k) = r * ((r - 1) gchoose k)"
726   using gbinomial_absorption'[of "Suc k" r] by (simp add: field_simps del: of_nat_Suc)
728 text \<open>The absorption identity for natural number binomial coefficients:\<close>
729 lemma binomial_absorption: "Suc k * (n choose Suc k) = n * ((n - 1) choose k)"
730   by (cases n) (simp_all add: binomial_eq_0 Suc_times_binomial del: binomial_Suc_Suc mult_Suc)
732 text \<open>The absorption companion identity for natural number coefficients,
733   following the proof by GKP \cite[p.~157]{GKP}:\<close>
734 lemma binomial_absorb_comp: "(n - k) * (n choose k) = n * ((n - 1) choose k)"
735   (is "?lhs = ?rhs")
736 proof (cases "n \<le> k")
737   case True
738   then show ?thesis by auto
739 next
740   case False
741   then have "?rhs = Suc ((n - 1) - k) * (n choose Suc ((n - 1) - k))"
742     using binomial_symmetric[of k "n - 1"] binomial_absorption[of "(n - 1) - k" n]
743     by simp
744   also have "Suc ((n - 1) - k) = n - k"
745     using False by simp
746   also have "n choose \<dots> = n choose k"
747     using False by (intro binomial_symmetric [symmetric]) simp_all
748   finally show ?thesis ..
749 qed
751 text \<open>The generalised absorption companion identity:\<close>
752 lemma gbinomial_absorb_comp: "(r - of_nat k) * (r gchoose k) = r * ((r - 1) gchoose k)"
753   using pochhammer_absorb_comp[of r k] by (simp add: gbinomial_pochhammer)
756   "r gchoose (Suc k) = ((r - 1) gchoose (Suc k)) + ((r - 1) gchoose k)"
757   using gbinomial_Suc_Suc[of "r - 1" k] by (simp add: algebra_simps)
760   "0 < n \<Longrightarrow> n choose (Suc k) = ((n - 1) choose (Suc k)) + ((n - 1) choose k)"
761   by (subst choose_reduce_nat) simp_all
763 text \<open>
764   Equation 5.9 of the reference material \cite[p.~159]{GKP} is a useful
765   summation formula, operating on both indices:
766   $767 \sum\limits_{k \leq n}{r + k \choose k} = {r + n + 1 \choose n}, 768 \quad \textnormal{integer } n. 769$
770 \<close>
771 lemma gbinomial_parallel_sum: "(\<Sum>k\<le>n. (r + of_nat k) gchoose k) = (r + of_nat n + 1) gchoose n"
772 proof (induct n)
773   case 0
774   then show ?case by simp
775 next
776   case (Suc m)
777   then show ?case
778     using gbinomial_Suc_Suc[of "(r + of_nat m + 1)" m]
780 qed
783 subsubsection \<open>Summation on the upper index\<close>
785 text \<open>
786   Another summation formula is equation 5.10 of the reference material \cite[p.~160]{GKP},
787   aptly named \emph{summation on the upper index}:$\sum_{0 \leq k \leq n} {k \choose m} = 788 {n + 1 \choose m + 1}, \quad \textnormal{integers } m, n \geq 0.$
789 \<close>
790 lemma gbinomial_sum_up_index:
791   "(\<Sum>k = 0..n. (of_nat k gchoose m) :: 'a::field_char_0) = (of_nat n + 1) gchoose (m + 1)"
792 proof (induct n)
793   case 0
794   show ?case
795     using gbinomial_Suc_Suc[of 0 m]
796     by (cases m) auto
797 next
798   case (Suc n)
799   then show ?case
800     using gbinomial_Suc_Suc[of "of_nat (Suc n) :: 'a" m]
802 qed
804 lemma gbinomial_index_swap:
805   "((-1) ^ m) * ((- (of_nat n) - 1) gchoose m) = ((-1) ^ n) * ((- (of_nat m) - 1) gchoose n)"
806   (is "?lhs = ?rhs")
807 proof -
808   have "?lhs = (of_nat (m + n) gchoose m)"
809     by (subst gbinomial_negated_upper) (simp add: power_mult_distrib [symmetric])
810   also have "\<dots> = (of_nat (m + n) gchoose n)"
811     by (subst gbinomial_of_nat_symmetric) simp_all
812   also have "\<dots> = ?rhs"
813     by (subst gbinomial_negated_upper) simp
814   finally show ?thesis .
815 qed
817 lemma gbinomial_sum_lower_neg: "(\<Sum>k\<le>m. (r gchoose k) * (- 1) ^ k) = (- 1) ^ m * (r - 1 gchoose m)"
818   (is "?lhs = ?rhs")
819 proof -
820   have "?lhs = (\<Sum>k\<le>m. -(r + 1) + of_nat k gchoose k)"
821     by (intro sum.cong[OF refl]) (subst gbinomial_negated_upper, simp add: power_mult_distrib)
822   also have "\<dots>  = - r + of_nat m gchoose m"
823     by (subst gbinomial_parallel_sum) simp
824   also have "\<dots> = ?rhs"
825     by (subst gbinomial_negated_upper) (simp add: power_mult_distrib)
826   finally show ?thesis .
827 qed
829 lemma gbinomial_partial_row_sum:
830   "(\<Sum>k\<le>m. (r gchoose k) * ((r / 2) - of_nat k)) = ((of_nat m + 1)/2) * (r gchoose (m + 1))"
831 proof (induct m)
832   case 0
833   then show ?case by simp
834 next
835   case (Suc mm)
836   then have "(\<Sum>k\<le>Suc mm. (r gchoose k) * (r / 2 - of_nat k)) =
837       (r - of_nat (Suc mm)) * (r gchoose Suc mm) / 2"
839   also have "\<dots> = r * (r - 1 gchoose Suc mm) / 2"
840     by (subst gbinomial_absorb_comp) (rule refl)
841   also have "\<dots> = (of_nat (Suc mm) + 1) / 2 * (r gchoose (Suc mm + 1))"
842     by (subst gbinomial_absorption [symmetric]) simp
843   finally show ?case .
844 qed
846 lemma sum_bounds_lt_plus1: "(\<Sum>k<mm. f (Suc k)) = (\<Sum>k=1..mm. f k)"
847   by (induct mm) simp_all
849 lemma gbinomial_partial_sum_poly:
850   "(\<Sum>k\<le>m. (of_nat m + r gchoose k) * x^k * y^(m-k)) =
851     (\<Sum>k\<le>m. (-r gchoose k) * (-x)^k * (x + y)^(m-k))"
852   (is "?lhs m = ?rhs m")
853 proof (induction m)
854   case 0
855   then show ?case by simp
856 next
857   case (Suc mm)
858   define G where "G i k = (of_nat i + r gchoose k) * x^k * y^(i - k)" for i k
859   define S where "S = ?lhs"
860   have SG_def: "S = (\<lambda>i. (\<Sum>k\<le>i. (G i k)))"
861     unfolding S_def G_def ..
863   have "S (Suc mm) = G (Suc mm) 0 + (\<Sum>k=Suc 0..Suc mm. G (Suc mm) k)"
865   also have "(\<Sum>k=Suc 0..Suc mm. G (Suc mm) k) = (\<Sum>k=0..mm. G (Suc mm) (Suc k))"
866     by (subst sum_shift_bounds_cl_Suc_ivl) simp
867   also have "\<dots> = (\<Sum>k=0..mm. ((of_nat mm + r gchoose (Suc k)) +
868       (of_nat mm + r gchoose k)) * x^(Suc k) * y^(mm - k))"
869     unfolding G_def by (subst gbinomial_addition_formula) simp
870   also have "\<dots> = (\<Sum>k=0..mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k)) +
871       (\<Sum>k=0..mm. (of_nat mm + r gchoose k) * x^(Suc k) * y^(mm - k))"
872     by (subst sum.distrib [symmetric]) (simp add: algebra_simps)
873   also have "(\<Sum>k=0..mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k)) =
874       (\<Sum>k<Suc mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k))"
875     by (simp only: atLeast0AtMost lessThan_Suc_atMost)
876   also have "\<dots> = (\<Sum>k<mm. (of_nat mm + r gchoose Suc k) * x^(Suc k) * y^(mm-k)) +
877       (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm)"
878     (is "_ = ?A + ?B")
879     by (subst sum_lessThan_Suc) simp
880   also have "?A = (\<Sum>k=1..mm. (of_nat mm + r gchoose k) * x^k * y^(mm - k + 1))"
881   proof (subst sum_bounds_lt_plus1 [symmetric], intro sum.cong[OF refl], clarify)
882     fix k
883     assume "k < mm"
884     then have "mm - k = mm - Suc k + 1"
885       by linarith
886     then show "(of_nat mm + r gchoose Suc k) * x ^ Suc k * y ^ (mm - k) =
887         (of_nat mm + r gchoose Suc k) * x ^ Suc k * y ^ (mm - Suc k + 1)"
888       by (simp only:)
889   qed
890   also have "\<dots> + ?B = y * (\<Sum>k=1..mm. (G mm k)) + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm)"
891     unfolding G_def by (subst sum_distrib_left) (simp add: algebra_simps)
892   also have "(\<Sum>k=0..mm. (of_nat mm + r gchoose k) * x^(Suc k) * y^(mm - k)) = x * (S mm)"
893     unfolding S_def by (subst sum_distrib_left) (simp add: atLeast0AtMost algebra_simps)
894   also have "(G (Suc mm) 0) = y * (G mm 0)"
896   finally have "S (Suc mm) =
897       y * (G mm 0 + (\<Sum>k=1..mm. (G mm k))) + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm) + x * (S mm)"
899   also have "G mm 0 + (\<Sum>k=1..mm. (G mm k)) = S mm"
901   finally have "S (Suc mm) = (x + y) * (S mm) + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm)"
903   also have "(of_nat mm + r gchoose (Suc mm)) = (-1) ^ (Suc mm) * (- r gchoose (Suc mm))"
904     by (subst gbinomial_negated_upper) simp
905   also have "(-1) ^ Suc mm * (- r gchoose Suc mm) * x ^ Suc mm =
906       (- r gchoose (Suc mm)) * (-x) ^ Suc mm"
907     by (simp add: power_minus[of x])
908   also have "(x + y) * S mm + \<dots> = (x + y) * ?rhs mm + (- r gchoose (Suc mm)) * (- x)^Suc mm"
909     unfolding S_def by (subst Suc.IH) simp
910   also have "(x + y) * ?rhs mm = (\<Sum>n\<le>mm. ((- r gchoose n) * (- x) ^ n * (x + y) ^ (Suc mm - n)))"
911     by (subst sum_distrib_left, rule sum.cong) (simp_all add: Suc_diff_le)
912   also have "\<dots> + (-r gchoose (Suc mm)) * (-x)^Suc mm =
913       (\<Sum>n\<le>Suc mm. (- r gchoose n) * (- x) ^ n * (x + y) ^ (Suc mm - n))"
914     by simp
915   finally show ?case
916     by (simp only: S_def)
917 qed
919 lemma gbinomial_partial_sum_poly_xpos:
920   "(\<Sum>k\<le>m. (of_nat m + r gchoose k) * x^k * y^(m-k)) =
921      (\<Sum>k\<le>m. (of_nat k + r - 1 gchoose k) * x^k * (x + y)^(m-k))"
922   apply (subst gbinomial_partial_sum_poly)
923   apply (subst gbinomial_negated_upper)
924   apply (intro sum.cong, rule refl)
925   apply (simp add: power_mult_distrib [symmetric])
926   done
928 lemma binomial_r_part_sum: "(\<Sum>k\<le>m. (2 * m + 1 choose k)) = 2 ^ (2 * m)"
929 proof -
930   have "2 * 2^(2*m) = (\<Sum>k = 0..(2 * m + 1). (2 * m + 1 choose k))"
931     using choose_row_sum[where n="2 * m + 1"] by simp
932   also have "(\<Sum>k = 0..(2 * m + 1). (2 * m + 1 choose k)) =
933       (\<Sum>k = 0..m. (2 * m + 1 choose k)) +
934       (\<Sum>k = m+1..2*m+1. (2 * m + 1 choose k))"
935     using sum_ub_add_nat[of 0 m "\<lambda>k. 2 * m + 1 choose k" "m+1"]
937   also have "(\<Sum>k = m+1..2*m+1. (2 * m + 1 choose k)) =
938       (\<Sum>k = 0..m. (2 * m + 1 choose (k + (m + 1))))"
939     by (subst sum_shift_bounds_cl_nat_ivl [symmetric]) (simp add: mult_2)
940   also have "\<dots> = (\<Sum>k = 0..m. (2 * m + 1 choose (m - k)))"
941     by (intro sum.cong[OF refl], subst binomial_symmetric) simp_all
942   also have "\<dots> = (\<Sum>k = 0..m. (2 * m + 1 choose k))"
943     using sum.atLeast_atMost_rev [of "\<lambda>k. 2 * m + 1 choose (m - k)" 0 m]
944     by simp
945   also have "\<dots> + \<dots> = 2 * \<dots>"
946     by simp
947   finally show ?thesis
948     by (subst (asm) mult_cancel1) (simp add: atLeast0AtMost)
949 qed
951 lemma gbinomial_r_part_sum: "(\<Sum>k\<le>m. (2 * (of_nat m) + 1 gchoose k)) = 2 ^ (2 * m)"
952   (is "?lhs = ?rhs")
953 proof -
954   have "?lhs = of_nat (\<Sum>k\<le>m. (2 * m + 1) choose k)"
956   also have "\<dots> = of_nat (2 ^ (2 * m))"
957     by (subst binomial_r_part_sum) (rule refl)
958   finally show ?thesis by simp
959 qed
961 lemma gbinomial_sum_nat_pow2:
962   "(\<Sum>k\<le>m. (of_nat (m + k) gchoose k :: 'a::field_char_0) / 2 ^ k) = 2 ^ m"
963   (is "?lhs = ?rhs")
964 proof -
965   have "2 ^ m * 2 ^ m = (2 ^ (2*m) :: 'a)"
966     by (induct m) simp_all
967   also have "\<dots> = (\<Sum>k\<le>m. (2 * (of_nat m) + 1 gchoose k))"
968     using gbinomial_r_part_sum ..
969   also have "\<dots> = (\<Sum>k\<le>m. (of_nat (m + k) gchoose k) * 2 ^ (m - k))"
970     using gbinomial_partial_sum_poly_xpos[where x="1" and y="1" and r="of_nat m + 1" and m="m"]
972   also have "\<dots> = 2 ^ m * (\<Sum>k\<le>m. (of_nat (m + k) gchoose k) / 2 ^ k)"
973     by (subst sum_distrib_left) (simp add: algebra_simps power_diff)
974   finally show ?thesis
975     by (subst (asm) mult_left_cancel) simp_all
976 qed
978 lemma gbinomial_trinomial_revision:
979   assumes "k \<le> m"
980   shows "(r gchoose m) * (of_nat m gchoose k) = (r gchoose k) * (r - of_nat k gchoose (m - k))"
981 proof -
982   have "(r gchoose m) * (of_nat m gchoose k) = (r gchoose m) * fact m / (fact k * fact (m - k))"
983     using assms by (simp add: binomial_gbinomial [symmetric] binomial_fact)
984   also have "\<dots> = (r gchoose k) * (r - of_nat k gchoose (m - k))"
985     using assms by (simp add: gbinomial_pochhammer power_diff pochhammer_product)
986   finally show ?thesis .
987 qed
989 text \<open>Versions of the theorems above for the natural-number version of "choose"\<close>
990 lemma binomial_altdef_of_nat:
991   "k \<le> n \<Longrightarrow> of_nat (n choose k) = (\<Prod>i = 0..<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
992   for n k :: nat and x :: "'a::field_char_0"
993   by (simp add: gbinomial_altdef_of_nat binomial_gbinomial of_nat_diff)
995 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)"
996   for k n :: nat and x :: "'a::linordered_field"
997   by (simp add: gbinomial_ge_n_over_k_pow_k binomial_gbinomial of_nat_diff)
999 lemma binomial_le_pow:
1000   assumes "r \<le> n"
1001   shows "n choose r \<le> n ^ r"
1002 proof -
1003   have "n choose r \<le> fact n div fact (n - r)"
1004     using assms by (subst binomial_fact_lemma[symmetric]) auto
1005   with fact_div_fact_le_pow [OF assms] show ?thesis
1006     by auto
1007 qed
1009 lemma binomial_altdef_nat: "k \<le> n \<Longrightarrow> n choose k = fact n div (fact k * fact (n - k))"
1010   for k n :: nat
1011   by (subst binomial_fact_lemma [symmetric]) auto
1013 lemma choose_dvd:
1014   "k \<le> n \<Longrightarrow> fact k * fact (n - k) dvd (fact n :: 'a::linordered_semidom)"
1015   unfolding dvd_def
1016   apply (rule exI [where x="of_nat (n choose k)"])
1017   using binomial_fact_lemma [of k n, THEN arg_cong [where f = of_nat and 'b='a]]
1018   apply auto
1019   done
1021 lemma fact_fact_dvd_fact:
1022   "fact k * fact n dvd (fact (k + n) :: 'a::linordered_semidom)"
1025 lemma choose_mult_lemma:
1026   "((m + r + k) choose (m + k)) * ((m + k) choose k) = ((m + r + k) choose k) * ((m + r) choose m)"
1027   (is "?lhs = _")
1028 proof -
1029   have "?lhs =
1030       fact (m + r + k) div (fact (m + k) * fact (m + r - m)) * (fact (m + k) div (fact k * fact m))"
1032   also have "\<dots> = fact (m + r + k) div (fact r * (fact k * fact m))"
1033     apply (subst div_mult_div_if_dvd)
1034     apply (auto simp: algebra_simps fact_fact_dvd_fact)
1036     done
1037   also have "\<dots> = (fact (m + r + k) * fact (m + r)) div (fact r * (fact k * fact m) * fact (m + r))"
1038     apply (subst div_mult_div_if_dvd [symmetric])
1039     apply (auto simp add: algebra_simps)
1040     apply (metis fact_fact_dvd_fact dvd_trans nat_mult_dvd_cancel_disj)
1041     done
1042   also have "\<dots> =
1043       (fact (m + r + k) div (fact k * fact (m + r)) * (fact (m + r) div (fact r * fact m)))"
1044     apply (subst div_mult_div_if_dvd)
1045     apply (auto simp: fact_fact_dvd_fact algebra_simps)
1046     done
1047   finally show ?thesis
1048     by (simp add: binomial_altdef_nat mult.commute)
1049 qed
1051 text \<open>The "Subset of a Subset" identity.\<close>
1052 lemma choose_mult:
1053   "k \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> (n choose m) * (m choose k) = (n choose k) * ((n - k) choose (m - k))"
1054   using choose_mult_lemma [of "m-k" "n-m" k] by simp
1057 subsection \<open>More on Binomial Coefficients\<close>
1059 lemma choose_one: "n choose 1 = n" for n :: nat
1060   by simp
1062 lemma card_UNION:
1063   assumes "finite A"
1064     and "\<forall>k \<in> A. finite k"
1065   shows "card (\<Union>A) = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * int (card (\<Inter>I)))"
1066   (is "?lhs = ?rhs")
1067 proof -
1068   have "?rhs = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * (\<Sum>_\<in>\<Inter>I. 1))"
1069     by simp
1070   also have "\<dots> = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (\<Sum>_\<in>\<Inter>I. (- 1) ^ (card I + 1)))"
1071     (is "_ = nat ?rhs")
1072     by (subst sum_distrib_left) simp
1073   also have "?rhs = (\<Sum>(I, _)\<in>Sigma {I. I \<subseteq> A \<and> I \<noteq> {}} Inter. (- 1) ^ (card I + 1))"
1074     using assms by (subst sum.Sigma) auto
1075   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))"
1076     by (rule sum.reindex_cong [where l = "\<lambda>(x, y). (y, x)"]) (auto intro: inj_onI simp add: split_beta)
1077   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))"
1078     using assms
1079     by (auto intro!: sum.mono_neutral_cong_right finite_SigmaI2 intro: finite_subset[where B="\<Union>A"])
1080   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)))"
1081     using assms by (subst sum.Sigma) auto
1082   also have "\<dots> = (\<Sum>_\<in>\<Union>A. 1)" (is "sum ?lhs _ = _")
1083   proof (rule sum.cong[OF refl])
1084     fix x
1085     assume x: "x \<in> \<Union>A"
1086     define K where "K = {X \<in> A. x \<in> X}"
1087     with \<open>finite A\<close> have K: "finite K"
1088       by auto
1089     let ?I = "\<lambda>i. {I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I}"
1090     have "inj_on snd (SIGMA i:{1..card A}. ?I i)"
1091       using assms by (auto intro!: inj_onI)
1092     moreover have [symmetric]: "snd  (SIGMA i:{1..card A}. ?I i) = {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}"
1093       using assms
1094       by (auto intro!: rev_image_eqI[where x="(card a, a)" for a]
1096         dest: finite_subset intro: card_mono)
1097     ultimately have "?lhs x = (\<Sum>(i, I)\<in>(SIGMA i:{1..card A}. ?I i). (- 1) ^ (i + 1))"
1098       by (rule sum.reindex_cong [where l = snd]) fastforce
1099     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)))"
1100       using assms by (subst sum.Sigma) auto
1101     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))"
1102       by (subst sum_distrib_left) simp
1103     also have "\<dots> = (\<Sum>i=1..card K. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> K \<and> card I = i. 1))"
1104       (is "_ = ?rhs")
1105     proof (rule sum.mono_neutral_cong_right[rule_format])
1106       show "finite {1..card A}"
1107         by simp
1108       show "{1..card K} \<subseteq> {1..card A}"
1109         using \<open>finite A\<close> by (auto simp add: K_def intro: card_mono)
1110     next
1111       fix i
1112       assume "i \<in> {1..card A} - {1..card K}"
1113       then have i: "i \<le> card A" "card K < i"
1114         by auto
1115       have "{I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I} = {I. I \<subseteq> K \<and> card I = i}"
1116         by (auto simp add: K_def)
1117       also have "\<dots> = {}"
1118         using \<open>finite A\<close> i by (auto simp add: K_def dest: card_mono[rotated 1])
1119       finally show "(- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1 :: int) = 0"
1120         by (simp only:) simp
1121     next
1122       fix i
1123       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)"
1124         (is "?lhs = ?rhs")
1125         by (rule sum.cong) (auto simp add: K_def)
1126       then show "(- 1) ^ (i + 1) * ?lhs = (- 1) ^ (i + 1) * ?rhs"
1127         by simp
1128     qed
1129     also have "{I. I \<subseteq> K \<and> card I = 0} = {{}}"
1130       using assms by (auto simp add: card_eq_0_iff K_def dest: finite_subset)
1131     then have "?rhs = (\<Sum>i = 0..card K. (- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> K \<and> card I = i. 1 :: int)) + 1"
1132       by (subst (2) sum_head_Suc) simp_all
1133     also have "\<dots> = (\<Sum>i = 0..card K. (- 1) * ((- 1) ^ i * int (card K choose i))) + 1"
1134       using K by (subst n_subsets[symmetric]) simp_all
1135     also have "\<dots> = - (\<Sum>i = 0..card K. (- 1) ^ i * int (card K choose i)) + 1"
1136       by (subst sum_distrib_left[symmetric]) simp
1137     also have "\<dots> =  - ((-1 + 1) ^ card K) + 1"
1138       by (subst binomial_ring) (simp add: ac_simps)
1139     also have "\<dots> = 1"
1140       using x K by (auto simp add: K_def card_gt_0_iff)
1141     finally show "?lhs x = 1" .
1142   qed
1143   also have "nat \<dots> = card (\<Union>A)"
1144     by simp
1145   finally show ?thesis ..
1146 qed
1148 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>
1149 lemma card_length_sum_list_rec:
1150   assumes "m \<ge> 1"
1151   shows "card {l::nat list. length l = m \<and> sum_list l = N} =
1152       card {l. length l = (m - 1) \<and> sum_list l = N} +
1153       card {l. length l = m \<and> sum_list l + 1 = N}"
1154     (is "card ?C = card ?A + card ?B")
1155 proof -
1156   let ?A' = "{l. length l = m \<and> sum_list l = N \<and> hd l = 0}"
1157   let ?B' = "{l. length l = m \<and> sum_list l = N \<and> hd l \<noteq> 0}"
1158   let ?f = "\<lambda>l. 0 # l"
1159   let ?g = "\<lambda>l. (hd l + 1) # tl l"
1160   have 1: "xs \<noteq> [] \<Longrightarrow> x = hd xs \<Longrightarrow> x # tl xs = xs" for x :: nat and xs
1161     by simp
1162   have 2: "xs \<noteq> [] \<Longrightarrow> sum_list(tl xs) = sum_list xs - hd xs" for xs :: "nat list"
1163     by (auto simp add: neq_Nil_conv)
1164   have f: "bij_betw ?f ?A ?A'"
1165     apply (rule bij_betw_byWitness[where f' = tl])
1166     using assms
1167     apply (auto simp: 2 length_0_conv[symmetric] 1 simp del: length_0_conv)
1168     done
1169   have 3: "xs \<noteq> [] \<Longrightarrow> hd xs + (sum_list xs - hd xs) = sum_list xs" for xs :: "nat list"
1170     by (metis 1 sum_list_simps(2) 2)
1171   have g: "bij_betw ?g ?B ?B'"
1172     apply (rule bij_betw_byWitness[where f' = "\<lambda>l. (hd l - 1) # tl l"])
1173     using assms
1174     by (auto simp: 2 length_0_conv[symmetric] intro!: 3
1175         simp del: length_greater_0_conv length_0_conv)
1176   have fin: "finite {xs. size xs = M \<and> set xs \<subseteq> {0..<N}}" for M N :: nat
1177     using finite_lists_length_eq[OF finite_atLeastLessThan] conj_commute by auto
1178   have fin_A: "finite ?A" using fin[of _ "N+1"]
1179     by (intro finite_subset[where ?A = "?A" and ?B = "{xs. size xs = m - 1 \<and> set xs \<subseteq> {0..<N+1}}"])
1180       (auto simp: member_le_sum_list less_Suc_eq_le)
1181   have fin_B: "finite ?B"
1182     by (intro finite_subset[where ?A = "?B" and ?B = "{xs. size xs = m \<and> set xs \<subseteq> {0..<N}}"])
1183       (auto simp: member_le_sum_list less_Suc_eq_le fin)
1184   have uni: "?C = ?A' \<union> ?B'"
1185     by auto
1186   have disj: "?A' \<inter> ?B' = {}" by blast
1187   have "card ?C = card(?A' \<union> ?B')"
1188     using uni by simp
1189   also have "\<dots> = card ?A + card ?B"
1190     using card_Un_disjoint[OF _ _ disj] bij_betw_finite[OF f] bij_betw_finite[OF g]
1191       bij_betw_same_card[OF f] bij_betw_same_card[OF g] fin_A fin_B
1192     by presburger
1193   finally show ?thesis .
1194 qed
1196 lemma card_length_sum_list: "card {l::nat list. size l = m \<and> sum_list l = N} = (N + m - 1) choose N"
1197   \<comment> "by Holden Lee, tidied by Tobias Nipkow"
1198 proof (cases m)
1199   case 0
1200   then show ?thesis
1201     by (cases N) (auto cong: conj_cong)
1202 next
1203   case (Suc m')
1204   have m: "m \<ge> 1"
1206   then show ?thesis
1207   proof (induct "N + m - 1" arbitrary: N m)
1208     case 0  \<comment> "In the base case, the only solution is [0]."
1209     have [simp]: "{l::nat list. length l = Suc 0 \<and> (\<forall>n\<in>set l. n = 0)} = {[0]}"
1210       by (auto simp: length_Suc_conv)
1211     have "m = 1 \<and> N = 0"
1212       using 0 by linarith
1213     then show ?case
1214       by simp
1215   next
1216     case (Suc k)
1217     have c1: "card {l::nat list. size l = (m - 1) \<and> sum_list l =  N} = (N + (m - 1) - 1) choose N"
1218     proof (cases "m = 1")
1219       case True
1220       with Suc.hyps have "N \<ge> 1"
1221         by auto
1222       with True show ?thesis
1224     next
1225       case False
1226       then show ?thesis
1227         using Suc by fastforce
1228     qed
1229     from Suc have c2: "card {l::nat list. size l = m \<and> sum_list l + 1 = N} =
1230       (if N > 0 then ((N - 1) + m - 1) choose (N - 1) else 0)"
1231     proof -
1232       have *: "n > 0 \<Longrightarrow> Suc m = n \<longleftrightarrow> m = n - 1" for m n
1233         by arith
1234       from Suc have "N > 0 \<Longrightarrow>
1235         card {l::nat list. size l = m \<and> sum_list l + 1 = N} =
1236           ((N - 1) + m - 1) choose (N - 1)"
1238       then show ?thesis
1239         by auto
1240     qed
1241     from Suc.prems have "(card {l::nat list. size l = (m - 1) \<and> sum_list l = N} +
1242           card {l::nat list. size l = m \<and> sum_list l + 1 = N}) = (N + m - 1) choose N"
1243       by (auto simp: c1 c2 choose_reduce_nat[of "N + m - 1" N] simp del: One_nat_def)
1244     then show ?case
1245       using card_length_sum_list_rec[OF Suc.prems] by auto
1246   qed
1247 qed
1249 lemma card_disjoint_shuffle:
1250   assumes "set xs \<inter> set ys = {}"
1251   shows   "card (shuffle xs ys) = (length xs + length ys) choose length xs"
1252 using assms
1253 proof (induction xs ys rule: shuffle.induct)
1254   case (3 x xs y ys)
1255   have "shuffle (x # xs) (y # ys) = op # x  shuffle xs (y # ys) \<union> op # y  shuffle (x # xs) ys"
1256     by (rule shuffle.simps)
1257   also have "card \<dots> = card (op # x  shuffle xs (y # ys)) + card (op # y  shuffle (x # xs) ys)"
1258     by (rule card_Un_disjoint) (insert "3.prems", auto)
1259   also have "card (op # x  shuffle xs (y # ys)) = card (shuffle xs (y # ys))"
1260     by (rule card_image) auto
1261   also have "\<dots> = (length xs + length (y # ys)) choose length xs"
1262     using "3.prems" by (intro "3.IH") auto
1263   also have "card (op # y  shuffle (x # xs) ys) = card (shuffle (x # xs) ys)"
1264     by (rule card_image) auto
1265   also have "\<dots> = (length (x # xs) + length ys) choose length (x # xs)"
1266     using "3.prems" by (intro "3.IH") auto
1267   also have "length xs + length (y # ys) choose length xs + \<dots> =
1268                (length (x # xs) + length (y # ys)) choose length (x # xs)" by simp
1269   finally show ?case .
1270 qed auto
1272 lemma Suc_times_binomial_add: "Suc a * (Suc (a + b) choose Suc a) = Suc b * (Suc (a + b) choose a)"
1273   \<comment> \<open>by Lukas Bulwahn\<close>
1274 proof -
1275   have dvd: "Suc a * (fact a * fact b) dvd fact (Suc (a + b))" for a b
1276     using fact_fact_dvd_fact[of "Suc a" "b", where 'a=nat]
1277     by (simp only: fact_Suc add_Suc[symmetric] of_nat_id mult.assoc)
1278   have "Suc a * (fact (Suc (a + b)) div (Suc a * fact a * fact b)) =
1279       Suc a * fact (Suc (a + b)) div (Suc a * (fact a * fact b))"
1280     by (subst div_mult_swap[symmetric]; simp only: mult.assoc dvd)
1281   also have "\<dots> = Suc b * fact (Suc (a + b)) div (Suc b * (fact a * fact b))"
1282     by (simp only: div_mult_mult1)
1283   also have "\<dots> = Suc b * (fact (Suc (a + b)) div (Suc b * (fact a * fact b)))"
1284     using dvd[of b a] by (subst div_mult_swap[symmetric]; simp only: ac_simps dvd)
1285   finally show ?thesis
1286     by (subst (1 2) binomial_altdef_nat)
1287       (simp_all only: ac_simps diff_Suc_Suc Suc_diff_le diff_add_inverse fact_Suc of_nat_id)
1288 qed
1291 subsection \<open>Misc\<close>
1293 lemma gbinomial_code [code]:
1294   "a gchoose n =
1295     (if n = 0 then 1
1296      else fold_atLeastAtMost_nat (\<lambda>n acc. (a - of_nat n) * acc) 0 (n - 1) 1 / fact n)"
1297   by (cases n)
1298     (simp_all add: gbinomial_prod_rev prod_atLeastAtMost_code [symmetric]
1299       atLeastLessThanSuc_atLeastAtMost)
1301 declare [[code drop: binomial]]
1303 lemma binomial_code [code]:
1304   "(n choose k) =
1305       (if k > n then 0
1306        else if 2 * k > n then (n choose (n - k))
1307        else (fold_atLeastAtMost_nat (op * ) (n-k+1) n 1 div fact k))"
1308 proof -
1309   {
1310     assume "k \<le> n"
1311     then have "{1..n} = {1..n-k} \<union> {n-k+1..n}" by auto
1312     then have "(fact n :: nat) = fact (n-k) * \<Prod>{n-k+1..n}"
1313       by (simp add: prod.union_disjoint fact_prod)
1314   }
1315   then show ?thesis by (auto simp: binomial_altdef_nat mult_ac prod_atLeastAtMost_code)
1316 qed
1318 end