src/HOL/Binomial.thy
 author wenzelm Tue Oct 10 19:23:03 2017 +0200 (23 months ago) changeset 66831 29ea2b900a05 parent 66806 a4e82b58d833 child 67299 ba52a058942f permissions -rw-r--r--
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     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 *)

     8

     9 section \<open>Binomial Coefficients and Binomial Theorem\<close>

    10

    11 theory Binomial

    12   imports Presburger Factorial

    13 begin

    14

    15 subsection \<open>Binomial coefficients\<close>

    16

    17 text \<open>This development is based on the work of Andy Gordon and Florian Kammueller.\<close>

    18

    19 text \<open>Combinatorial definition\<close>

    20

    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}"

    23

    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")

    42     by (simp add: card_image)

    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

    48     by (simp add: binomial_def)

    49 qed

    50

    51 text \<open>Recursive characterization\<close>

    52

    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

    58     by (simp add: binomial_def)

    59 qed

    60

    61 lemma binomial_0_Suc [simp, code]: "0 choose Suc k = 0"

    62   by (simp add: binomial_def)

    63

    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

    96

    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)

    99

   100 lemma zero_less_binomial: "k \<le> n \<Longrightarrow> n choose k > 0"

   101   by (induct n k rule: diff_induct) simp_all

   102

   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)

   105

   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)

   108

   109 lemma binomial_n_n [simp]: "n choose n = 1"

   110   by (induct n) (simp_all add: binomial_eq_0)

   111

   112 lemma binomial_Suc_n [simp]: "Suc n choose n = Suc n"

   113   by (induct n) simp_all

   114

   115 lemma binomial_1 [simp]: "n choose Suc 0 = n"

   116   by (induct n) simp_all

   117

   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

   122

   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)

   128    apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0)

   129   done

   130

   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

   137   apply (simp add: add_le_mono mult_2)

   138   done

   139

   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

   143

   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)

   148

   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)

   153

   154

   155 subsection \<open>The binomial theorem (courtesy of Tobias Nipkow):\<close>

   156

   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))"

   183     by (simp add: decomp2)

   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

   192

   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)

   198

   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))"

   229       by (simp add: field_simps)

   230     also have "\<dots> = (k + (m - h)) * fact m"

   231       using H[rule_format, OF mn hm'] H[rule_format, OF mn km]

   232       by (simp add: field_simps)

   233     finally show ?thesis

   234       using k n km by simp

   235   qed

   236 qed

   237

   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)

   243

   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]

   248   apply (simp add: field_simps)

   249   apply (metis mult.commute of_nat_fact of_nat_mult)

   250   done

   251

   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)

   256

   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

   271     by (simp add: binomial_fact')

   272 qed

   273

   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)

   276

   277 lemma sum_choose_lower: "(\<Sum>k=0..n. (r+k) choose k) = Suc (r+n) choose n"

   278   by (induct n) auto

   279

   280 lemma sum_choose_upper: "(\<Sum>k=0..n. k choose m) = Suc n choose Suc m"

   281   by (induct n) auto

   282

   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)

   287

   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))"

   296     by (simp add: sum.distrib)

   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

   301

   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))"

   310     by (simp add: sum_subtractf)

   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

   315

   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)

   318

   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

   333

   334

   335 subsection \<open>Generalized binomial coefficients\<close>

   336

   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"

   339

   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)

   344

   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)

   347

   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)

   351

   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)

   355

   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)

   359     (simp_all add: pochhammer_minus,

   360      simp_all add: gbinomial_prod_rev pochhammer_prod_rev

   361        power_mult_distrib [symmetric] atLeastLessThanSuc_atLeastAtMost

   362        prod.atLeast_Suc_atMost_Suc_shift of_nat_diff)

   363

   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

   374

   375 lemma gbinomial_binomial: "n gchoose k = n choose k"

   376 proof (cases "k \<le> n")

   377   case False

   378   then have "n < k"

   379     by (simp add: not_le)

   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

   395

   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)"

   407     by (simp add: fact_prod_rev)

   408   also have "\<dots> = ((\<Prod>i\<in>{0..<k} \<union> {k..<m + k}. of_nat (m + k - i)) :: 'a)"

   409     by (simp add: ivl_disj_un)

   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)"

   414     by (simp add: n)

   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

   420

   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)

   423

   424 setup

   425   \<open>Sign.add_const_constraint (@{const_name gbinomial}, SOME @{typ "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a"})\<close>

   426

   427 lemma gbinomial_1[simp]: "a gchoose 1 = a"

   428   by (simp add: gbinomial_prod_rev lessThan_Suc)

   429

   430 lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a"

   431   by (simp add: gbinomial_prod_rev lessThan_Suc)

   432

   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

   447

   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)

   452

   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))"

   483     by (simp add: field_simps)

   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

   499

   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)

   503

   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)

   508

   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

   521

   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

   532

   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))"

   540     by (simp add: Suc)

   541   also have "\<dots> = Suc m * 2 ^ m"

   542     by (simp only: sum_atMost_Suc_shift Suc_times_binomial sum_distrib_left[symmetric])

   543        (simp add: choose_row_sum')

   544   finally show ?thesis

   545     using Suc by simp

   546 qed

   547

   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))"

   560     by (simp add: Suc)

   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)

   565        (simp add: algebra_simps)

   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

   571

   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"

   578     by (simp add: sum.delta)

   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

   586

   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])

   590

   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))"

   610     apply (subst sum_head_Suc)

   611     apply simp

   612     apply (subst sum_shift_bounds_cl_Suc_ivl)

   613     apply (simp add: atLeast0AtMost)

   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

   626

   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)

   632

   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)"

   642     by (simp add: prod_constant)

   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

   665

   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)

   668

   669 lemma gbinomial_minus: "((-a) gchoose b) = (-1) ^ b * ((a + of_nat b - 1) gchoose b)"

   670   by (subst gbinomial_negated_upper) (simp add: add_ac)

   671

   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)"

   681     by (simp add: prod.atLeast0_atMost_Suc_shift)

   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

   686

   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)"

   696     by (simp add: prod.atLeast0_atMost_Suc_shift)

   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

   700     by (simp add: Suc)

   701 qed

   702

   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)

   706

   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])

   709

   710

   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)

   718

   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)

   727

   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)

   731

   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

   750

   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)

   754

   755 lemma gbinomial_addition_formula:

   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)

   758

   759 lemma binomial_addition_formula:

   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

   762

   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]

   779     by (simp add: add_ac)

   780 qed

   781

   782

   783 subsubsection \<open>Summation on the upper index\<close>

   784

   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]

   801     by (simp add: add_ac)

   802 qed

   803

   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

   816

   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

   828

   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"

   838     by (simp add: field_simps)

   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

   845

   846 lemma sum_bounds_lt_plus1: "(\<Sum>k<mm. f (Suc k)) = (\<Sum>k=1..mm. f k)"

   847   by (induct mm) simp_all

   848

   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 ..

   862

   863   have "S (Suc mm) = G (Suc mm) 0 + (\<Sum>k=Suc 0..Suc mm. G (Suc mm) k)"

   864     using SG_def by (simp add: sum_head_Suc atLeast0AtMost [symmetric])

   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)"

   895     by (simp add: G_def)

   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)"

   898     by (simp add: ring_distribs)

   899   also have "G mm 0 + (\<Sum>k=1..mm. (G mm k)) = S mm"

   900     by (simp add: sum_head_Suc[symmetric] SG_def atLeast0AtMost)

   901   finally have "S (Suc mm) = (x + y) * (S mm) + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm)"

   902     by (simp add: algebra_simps)

   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

   918

   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

   927

   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"]

   936     by (simp add: mult_2)

   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

   950

   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)"

   955     by (simp add: binomial_gbinomial add_ac)

   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

   960

   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"]

   971     by (simp add: add_ac)

   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

   977

   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

   988

   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)

   994

   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)

   998

   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

  1008

  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

  1012

  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

  1020

  1021 lemma fact_fact_dvd_fact:

  1022   "fact k * fact n dvd (fact (k + n) :: 'a::linordered_semidom)"

  1023   by (metis add.commute add_diff_cancel_left' choose_dvd le_add2)

  1024

  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))"

  1031     by (simp add: binomial_altdef_nat)

  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)

  1035     apply (metis add.assoc add.commute 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

  1050

  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

  1055

  1056

  1057 subsection \<open>More on Binomial Coefficients\<close>

  1058

  1059 lemma choose_one: "n choose 1 = n" for n :: nat

  1060   by simp

  1061

  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]

  1095         simp add: card_gt_0_iff[folded Suc_le_eq]

  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

  1147

  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

  1195

  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"

  1205     by (simp add: Suc)

  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 ."

  1209     have [simp]: "{l::nat list. length l = Suc 0 \<and> (\<forall>n\<in>set l. n = 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

  1223         by (simp add: binomial_eq_0)

  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)"

  1237         by (simp add: *)

  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

  1248

  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

  1271

  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

  1289

  1290

  1291 subsection \<open>Misc\<close>

  1292

  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)

  1300

  1301 declare [[code drop: binomial]]

  1302

  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

  1317

  1318 end