src/HOL/Binomial.thy
 author wenzelm Mon Aug 31 21:28:08 2015 +0200 (2015-08-31) changeset 61070 b72a990adfe2 parent 60758 d8d85a8172b5 child 61076 bdc1e2f0a86a permissions -rw-r--r--
prefer symbols;
```     1 (*  Title       : Binomial.thy
```
```     2     Author      : Jacques D. Fleuriot
```
```     3     Copyright   : 1998  University of Cambridge
```
```     4     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
```
```     5     The integer version of factorial and other additions by Jeremy Avigad.
```
```     6 *)
```
```     7
```
```     8 section\<open>Factorial Function, Binomial Coefficients and Binomial Theorem\<close>
```
```     9
```
```    10 theory Binomial
```
```    11 imports Main
```
```    12 begin
```
```    13
```
```    14 subsection \<open>Factorial\<close>
```
```    15
```
```    16 fun fact :: "nat \<Rightarrow> ('a::semiring_char_0)"
```
```    17   where "fact 0 = 1" | "fact (Suc n) = of_nat (Suc n) * fact n"
```
```    18
```
```    19 lemmas fact_Suc = fact.simps(2)
```
```    20
```
```    21 lemma fact_1 [simp]: "fact 1 = 1"
```
```    22   by simp
```
```    23
```
```    24 lemma fact_Suc_0 [simp]: "fact (Suc 0) = Suc 0"
```
```    25   by simp
```
```    26
```
```    27 lemma of_nat_fact [simp]: "of_nat (fact n) = fact n"
```
```    28   by (induct n) (auto simp add: algebra_simps of_nat_mult)
```
```    29
```
```    30 lemma fact_reduce: "n > 0 \<Longrightarrow> fact n = of_nat n * fact (n - 1)"
```
```    31   by (cases n) auto
```
```    32
```
```    33 lemma fact_nonzero [simp]: "fact n \<noteq> (0::'a::{semiring_char_0,semiring_no_zero_divisors})"
```
```    34   apply (induct n)
```
```    35   apply auto
```
```    36   using of_nat_eq_0_iff by fastforce
```
```    37
```
```    38 lemma fact_mono_nat: "m \<le> n \<Longrightarrow> fact m \<le> (fact n :: nat)"
```
```    39   by (induct n) (auto simp: le_Suc_eq)
```
```    40
```
```    41 context
```
```    42   assumes "SORT_CONSTRAINT('a::linordered_semidom)"
```
```    43 begin
```
```    44
```
```    45   lemma fact_mono: "m \<le> n \<Longrightarrow> fact m \<le> (fact n :: 'a)"
```
```    46     by (metis of_nat_fact of_nat_le_iff fact_mono_nat)
```
```    47
```
```    48   lemma fact_ge_1 [simp]: "fact n \<ge> (1 :: 'a)"
```
```    49     by (metis le0 fact.simps(1) fact_mono)
```
```    50
```
```    51   lemma fact_gt_zero [simp]: "fact n > (0 :: 'a)"
```
```    52     using fact_ge_1 less_le_trans zero_less_one by blast
```
```    53
```
```    54   lemma fact_ge_zero [simp]: "fact n \<ge> (0 :: 'a)"
```
```    55     by (simp add: less_imp_le)
```
```    56
```
```    57   lemma fact_not_neg [simp]: "~ (fact n < (0 :: 'a))"
```
```    58     by (simp add: not_less_iff_gr_or_eq)
```
```    59
```
```    60   lemma fact_le_power:
```
```    61       "fact n \<le> (of_nat (n^n) ::'a)"
```
```    62   proof (induct n)
```
```    63     case (Suc n)
```
```    64     then have *: "fact n \<le> (of_nat (Suc n ^ n) ::'a)"
```
```    65       by (rule order_trans) (simp add: power_mono)
```
```    66     have "fact (Suc n) = (of_nat (Suc n) * fact n ::'a)"
```
```    67       by (simp add: algebra_simps)
```
```    68     also have "... \<le> (of_nat (Suc n) * of_nat (Suc n ^ n) ::'a)"
```
```    69       by (simp add: "*" ordered_comm_semiring_class.comm_mult_left_mono)
```
```    70     also have "... \<le> (of_nat (Suc n ^ Suc n) ::'a)"
```
```    71       by (metis of_nat_mult order_refl power_Suc)
```
```    72     finally show ?case .
```
```    73   qed simp
```
```    74
```
```    75 end
```
```    76
```
```    77 text\<open>Note that @{term "fact 0 = fact 1"}\<close>
```
```    78 lemma fact_less_mono_nat: "\<lbrakk>0 < m; m < n\<rbrakk> \<Longrightarrow> fact m < (fact n :: nat)"
```
```    79   by (induct n) (auto simp: less_Suc_eq)
```
```    80
```
```    81 lemma fact_less_mono:
```
```    82   "\<lbrakk>0 < m; m < n\<rbrakk> \<Longrightarrow> fact m < (fact n :: 'a::linordered_semidom)"
```
```    83   by (metis of_nat_fact of_nat_less_iff fact_less_mono_nat)
```
```    84
```
```    85 lemma fact_ge_Suc_0_nat [simp]: "fact n \<ge> Suc 0"
```
```    86   by (metis One_nat_def fact_ge_1)
```
```    87
```
```    88 lemma dvd_fact:
```
```    89   shows "1 \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> m dvd fact n"
```
```    90   by (induct n) (auto simp: dvdI le_Suc_eq)
```
```    91
```
```    92 lemma fact_altdef_nat: "fact n = \<Prod>{1..n}"
```
```    93   by (induct n) (auto simp: atLeastAtMostSuc_conv)
```
```    94
```
```    95 lemma fact_altdef: "fact n = setprod of_nat {1..n}"
```
```    96   by (induct n) (auto simp: atLeastAtMostSuc_conv)
```
```    97
```
```    98 lemma fact_dvd: "n \<le> m \<Longrightarrow> fact n dvd (fact m :: 'a :: {semiring_div,linordered_semidom})"
```
```    99   by (induct m) (auto simp: le_Suc_eq)
```
```   100
```
```   101 lemma fact_mod: "m \<le> n \<Longrightarrow> fact n mod (fact m :: 'a :: {semiring_div,linordered_semidom}) = 0"
```
```   102   by (auto simp add: fact_dvd)
```
```   103
```
```   104 lemma fact_div_fact:
```
```   105   assumes "m \<ge> n"
```
```   106   shows "(fact m) div (fact n) = \<Prod>{n + 1..m}"
```
```   107 proof -
```
```   108   obtain d where "d = m - n" by auto
```
```   109   from assms this have "m = n + d" by auto
```
```   110   have "fact (n + d) div (fact n) = \<Prod>{n + 1..n + d}"
```
```   111   proof (induct d)
```
```   112     case 0
```
```   113     show ?case by simp
```
```   114   next
```
```   115     case (Suc d')
```
```   116     have "fact (n + Suc d') div fact n = Suc (n + d') * fact (n + d') div fact n"
```
```   117       by simp
```
```   118     also from Suc.hyps have "... = Suc (n + d') * \<Prod>{n + 1..n + d'}"
```
```   119       unfolding div_mult1_eq[of _ "fact (n + d')"] by (simp add: fact_mod)
```
```   120     also have "... = \<Prod>{n + 1..n + Suc d'}"
```
```   121       by (simp add: atLeastAtMostSuc_conv)
```
```   122     finally show ?case .
```
```   123   qed
```
```   124   from this \<open>m = n + d\<close> show ?thesis by simp
```
```   125 qed
```
```   126
```
```   127 lemma fact_num_eq_if:
```
```   128     "fact m = (if m=0 then 1 else of_nat m * fact (m - 1))"
```
```   129 by (cases m) auto
```
```   130
```
```   131 lemma fact_eq_rev_setprod_nat: "fact k = (\<Prod>i<k. k - i)"
```
```   132   unfolding fact_altdef_nat
```
```   133   by (rule setprod.reindex_bij_witness[where i="\<lambda>i. k - i" and j="\<lambda>i. k - i"]) auto
```
```   134
```
```   135 lemma fact_div_fact_le_pow:
```
```   136   assumes "r \<le> n" shows "fact n div fact (n - r) \<le> n ^ r"
```
```   137 proof -
```
```   138   have "\<And>r. r \<le> n \<Longrightarrow> \<Prod>{n - r..n} = (n - r) * \<Prod>{Suc (n - r)..n}"
```
```   139     by (subst setprod.insert[symmetric]) (auto simp: atLeastAtMost_insertL)
```
```   140   with assms show ?thesis
```
```   141     by (induct r rule: nat.induct) (auto simp add: fact_div_fact Suc_diff_Suc mult_le_mono)
```
```   142 qed
```
```   143
```
```   144 lemma fact_numeral:  --\<open>Evaluation for specific numerals\<close>
```
```   145   "fact (numeral k) = (numeral k) * (fact (pred_numeral k))"
```
```   146   by (metis fact.simps(2) numeral_eq_Suc of_nat_numeral)
```
```   147
```
```   148
```
```   149 text \<open>This development is based on the work of Andy Gordon and
```
```   150   Florian Kammueller.\<close>
```
```   151
```
```   152 subsection \<open>Basic definitions and lemmas\<close>
```
```   153
```
```   154 primrec binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65)
```
```   155 where
```
```   156   "0 choose k = (if k = 0 then 1 else 0)"
```
```   157 | "Suc n choose k = (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
```
```   158
```
```   159 lemma binomial_n_0 [simp]: "(n choose 0) = 1"
```
```   160   by (cases n) simp_all
```
```   161
```
```   162 lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
```
```   163   by simp
```
```   164
```
```   165 lemma binomial_Suc_Suc [simp]: "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
```
```   166   by simp
```
```   167
```
```   168 lemma choose_reduce_nat:
```
```   169   "0 < (n::nat) \<Longrightarrow> 0 < k \<Longrightarrow>
```
```   170     (n choose k) = ((n - 1) choose (k - 1)) + ((n - 1) choose k)"
```
```   171   by (metis Suc_diff_1 binomial.simps(2) neq0_conv)
```
```   172
```
```   173 lemma binomial_eq_0: "n < k \<Longrightarrow> n choose k = 0"
```
```   174   by (induct n arbitrary: k) auto
```
```   175
```
```   176 declare binomial.simps [simp del]
```
```   177
```
```   178 lemma binomial_n_n [simp]: "n choose n = 1"
```
```   179   by (induct n) (simp_all add: binomial_eq_0)
```
```   180
```
```   181 lemma binomial_Suc_n [simp]: "Suc n choose n = Suc n"
```
```   182   by (induct n) simp_all
```
```   183
```
```   184 lemma binomial_1 [simp]: "n choose Suc 0 = n"
```
```   185   by (induct n) simp_all
```
```   186
```
```   187 lemma zero_less_binomial: "k \<le> n \<Longrightarrow> n choose k > 0"
```
```   188   by (induct n k rule: diff_induct) simp_all
```
```   189
```
```   190 lemma binomial_eq_0_iff [simp]: "n choose k = 0 \<longleftrightarrow> n < k"
```
```   191   by (metis binomial_eq_0 less_numeral_extra(3) not_less zero_less_binomial)
```
```   192
```
```   193 lemma zero_less_binomial_iff [simp]: "n choose k > 0 \<longleftrightarrow> k \<le> n"
```
```   194   by (metis binomial_eq_0_iff not_less0 not_less zero_less_binomial)
```
```   195
```
```   196 lemma Suc_times_binomial_eq:
```
```   197   "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
```
```   198   apply (induct n arbitrary: k, simp add: binomial.simps)
```
```   199   apply (case_tac k)
```
```   200    apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0)
```
```   201   done
```
```   202
```
```   203 lemma binomial_le_pow2: "n choose k \<le> 2^n"
```
```   204   apply (induction n arbitrary: k)
```
```   205   apply (simp add: binomial.simps)
```
```   206   apply (case_tac k)
```
```   207   apply (auto simp: power_Suc)
```
```   208   by (simp add: add_le_mono mult_2)
```
```   209
```
```   210 text\<open>The absorption property\<close>
```
```   211 lemma Suc_times_binomial:
```
```   212   "Suc k * (Suc n choose Suc k) = Suc n * (n choose k)"
```
```   213   using Suc_times_binomial_eq by auto
```
```   214
```
```   215 text\<open>This is the well-known version of absorption, but it's harder to use because of the
```
```   216   need to reason about division.\<close>
```
```   217 lemma binomial_Suc_Suc_eq_times:
```
```   218     "(Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
```
```   219   by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right)
```
```   220
```
```   221 text\<open>Another version of absorption, with -1 instead of Suc.\<close>
```
```   222 lemma times_binomial_minus1_eq:
```
```   223   "0 < k \<Longrightarrow> k * (n choose k) = n * ((n - 1) choose (k - 1))"
```
```   224   using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 1"]
```
```   225   by (auto split add: nat_diff_split)
```
```   226
```
```   227
```
```   228 subsection \<open>Combinatorial theorems involving @{text "choose"}\<close>
```
```   229
```
```   230 text \<open>By Florian Kamm\"uller, tidied by LCP.\<close>
```
```   231
```
```   232 lemma card_s_0_eq_empty: "finite A \<Longrightarrow> card {B. B \<subseteq> A & card B = 0} = 1"
```
```   233   by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
```
```   234
```
```   235 lemma choose_deconstruct: "finite M \<Longrightarrow> x \<notin> M \<Longrightarrow>
```
```   236     {s. s \<subseteq> insert x M \<and> card s = Suc k} =
```
```   237     {s. s \<subseteq> M \<and> card s = Suc k} \<union> {s. \<exists>t. t \<subseteq> M \<and> card t = k \<and> s = insert x t}"
```
```   238   apply safe
```
```   239      apply (auto intro: finite_subset [THEN card_insert_disjoint])
```
```   240   by (metis (full_types) Diff_insert_absorb Set.set_insert Zero_neq_Suc card_Diff_singleton_if
```
```   241      card_eq_0_iff diff_Suc_1 in_mono subset_insert_iff)
```
```   242
```
```   243 lemma finite_bex_subset [simp]:
```
```   244   assumes "finite B"
```
```   245     and "\<And>A. A \<subseteq> B \<Longrightarrow> finite {x. P x A}"
```
```   246   shows "finite {x. \<exists>A \<subseteq> B. P x A}"
```
```   247   by (metis (no_types) assms finite_Collect_bounded_ex finite_Collect_subsets)
```
```   248
```
```   249 text\<open>There are as many subsets of @{term A} having cardinality @{term k}
```
```   250  as there are sets obtained from the former by inserting a fixed element
```
```   251  @{term x} into each.\<close>
```
```   252 lemma constr_bij:
```
```   253    "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow>
```
```   254     card {B. \<exists>C. C \<subseteq> A \<and> card C = k \<and> B = insert x C} =
```
```   255     card {B. B \<subseteq> A & card(B) = k}"
```
```   256   apply (rule card_bij_eq [where f = "\<lambda>s. s - {x}" and g = "insert x"])
```
```   257   apply (auto elim!: equalityE simp add: inj_on_def)
```
```   258   apply (metis card_Diff_singleton_if finite_subset in_mono)
```
```   259   done
```
```   260
```
```   261 text \<open>
```
```   262   Main theorem: combinatorial statement about number of subsets of a set.
```
```   263 \<close>
```
```   264
```
```   265 theorem n_subsets: "finite A \<Longrightarrow> card {B. B \<subseteq> A \<and> card B = k} = (card A choose k)"
```
```   266 proof (induct k arbitrary: A)
```
```   267   case 0 then show ?case by (simp add: card_s_0_eq_empty)
```
```   268 next
```
```   269   case (Suc k)
```
```   270   show ?case using \<open>finite A\<close>
```
```   271   proof (induct A)
```
```   272     case empty show ?case by (simp add: card_s_0_eq_empty)
```
```   273   next
```
```   274     case (insert x A)
```
```   275     then show ?case using Suc.hyps
```
```   276       apply (simp add: card_s_0_eq_empty choose_deconstruct)
```
```   277       apply (subst card_Un_disjoint)
```
```   278          prefer 4 apply (force simp add: constr_bij)
```
```   279         prefer 3 apply force
```
```   280        prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
```
```   281          finite_subset [of _ "Pow (insert x F)" for F])
```
```   282       apply (blast intro: finite_Pow_iff [THEN iffD2, THEN  finite_subset])
```
```   283       done
```
```   284   qed
```
```   285 qed
```
```   286
```
```   287
```
```   288 subsection \<open>The binomial theorem (courtesy of Tobias Nipkow):\<close>
```
```   289
```
```   290 text\<open>Avigad's version, generalized to any commutative ring\<close>
```
```   291 theorem binomial_ring: "(a+b::'a::{comm_ring_1,power})^n =
```
```   292   (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" (is "?P n")
```
```   293 proof (induct n)
```
```   294   case 0 then show "?P 0" by simp
```
```   295 next
```
```   296   case (Suc n)
```
```   297   have decomp: "{0..n+1} = {0} Un {n+1} Un {1..n}"
```
```   298     by auto
```
```   299   have decomp2: "{0..n} = {0} Un {1..n}"
```
```   300     by auto
```
```   301   have "(a+b)^(n+1) =
```
```   302       (a+b) * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
```
```   303     using Suc.hyps by simp
```
```   304   also have "\<dots> = a*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k)) +
```
```   305                    b*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
```
```   306     by (rule distrib_right)
```
```   307   also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) +
```
```   308                   (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k+1))"
```
```   309     by (auto simp add: setsum_right_distrib ac_simps)
```
```   310   also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n+1-k)) +
```
```   311                   (\<Sum>k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n+1-k))"
```
```   312     by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps
```
```   313         del:setsum_cl_ivl_Suc)
```
```   314   also have "\<dots> = a^(n+1) + b^(n+1) +
```
```   315                   (\<Sum>k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-k)) +
```
```   316                   (\<Sum>k=1..n. of_nat (n choose k) * a^k * b^(n+1-k))"
```
```   317     by (simp add: decomp2)
```
```   318   also have
```
```   319       "\<dots> = a^(n+1) + b^(n+1) +
```
```   320             (\<Sum>k=1..n. of_nat(n+1 choose k) * a^k * b^(n+1-k))"
```
```   321     by (auto simp add: field_simps setsum.distrib [symmetric] choose_reduce_nat)
```
```   322   also have "\<dots> = (\<Sum>k=0..n+1. of_nat (n+1 choose k) * a^k * b^(n+1-k))"
```
```   323     using decomp by (simp add: field_simps)
```
```   324   finally show "?P (Suc n)" by simp
```
```   325 qed
```
```   326
```
```   327 text\<open>Original version for the naturals\<close>
```
```   328 corollary binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))"
```
```   329     using binomial_ring [of "int a" "int b" n]
```
```   330   by (simp only: of_nat_add [symmetric] of_nat_mult [symmetric] of_nat_power [symmetric]
```
```   331            of_nat_setsum [symmetric]
```
```   332            of_nat_eq_iff of_nat_id)
```
```   333
```
```   334 lemma binomial_fact_lemma: "k \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n"
```
```   335 proof (induct n arbitrary: k rule: nat_less_induct)
```
```   336   fix n k assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) =
```
```   337                       fact m" and kn: "k \<le> n"
```
```   338   let ?ths = "fact k * fact (n - k) * (n choose k) = fact n"
```
```   339   { assume "n=0" then have ?ths using kn by simp }
```
```   340   moreover
```
```   341   { assume "k=0" then have ?ths using kn by simp }
```
```   342   moreover
```
```   343   { assume nk: "n=k" then have ?ths by simp }
```
```   344   moreover
```
```   345   { fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < m"
```
```   346     from n have mn: "m < n" by arith
```
```   347     from hm have hm': "h \<le> m" by arith
```
```   348     from hm h n kn have km: "k \<le> m" by arith
```
```   349     have "m - h = Suc (m - Suc h)" using  h km hm by arith
```
```   350     with km h have th0: "fact (m - h) = (m - h) * fact (m - k)"
```
```   351       by simp
```
```   352     from n h th0
```
```   353     have "fact k * fact (n - k) * (n choose k) =
```
```   354         k * (fact h * fact (m - h) * (m choose h)) +
```
```   355         (m - h) * (fact k * fact (m - k) * (m choose k))"
```
```   356       by (simp add: field_simps)
```
```   357     also have "\<dots> = (k + (m - h)) * fact m"
```
```   358       using H[rule_format, OF mn hm'] H[rule_format, OF mn km]
```
```   359       by (simp add: field_simps)
```
```   360     finally have ?ths using h n km by simp }
```
```   361   moreover have "n=0 \<or> k = 0 \<or> k = n \<or> (\<exists>m h. n = Suc m \<and> k = Suc h \<and> h < m)"
```
```   362     using kn by presburger
```
```   363   ultimately show ?ths by blast
```
```   364 qed
```
```   365
```
```   366 lemma binomial_fact:
```
```   367   assumes kn: "k \<le> n"
```
```   368   shows "(of_nat (n choose k) :: 'a::field_char_0) =
```
```   369          (fact n) / (fact k * fact(n - k))"
```
```   370   using binomial_fact_lemma[OF kn]
```
```   371   apply (simp add: field_simps)
```
```   372   by (metis mult.commute of_nat_fact of_nat_mult)
```
```   373
```
```   374 lemma choose_row_sum: "(\<Sum>k=0..n. n choose k) = 2^n"
```
```   375   using binomial [of 1 "1" n]
```
```   376   by (simp add: numeral_2_eq_2)
```
```   377
```
```   378 lemma sum_choose_lower: "(\<Sum>k=0..n. (r+k) choose k) = Suc (r+n) choose n"
```
```   379   by (induct n) auto
```
```   380
```
```   381 lemma sum_choose_upper: "(\<Sum>k=0..n. k choose m) = Suc n choose Suc m"
```
```   382   by (induct n) auto
```
```   383
```
```   384 lemma natsum_reverse_index:
```
```   385   fixes m::nat
```
```   386   shows "(\<And>k. m \<le> k \<Longrightarrow> k \<le> n \<Longrightarrow> g k = f (m + n - k)) \<Longrightarrow> (\<Sum>k=m..n. f k) = (\<Sum>k=m..n. g k)"
```
```   387   by (rule setsum.reindex_bij_witness[where i="\<lambda>k. m+n-k" and j="\<lambda>k. m+n-k"]) auto
```
```   388
```
```   389 text\<open>NW diagonal sum property\<close>
```
```   390 lemma sum_choose_diagonal:
```
```   391   assumes "m\<le>n" shows "(\<Sum>k=0..m. (n-k) choose (m-k)) = Suc n choose m"
```
```   392 proof -
```
```   393   have "(\<Sum>k=0..m. (n-k) choose (m-k)) = (\<Sum>k=0..m. (n-m+k) choose k)"
```
```   394     by (rule natsum_reverse_index) (simp add: assms)
```
```   395   also have "... = Suc (n-m+m) choose m"
```
```   396     by (rule sum_choose_lower)
```
```   397   also have "... = Suc n choose m" using assms
```
```   398     by simp
```
```   399   finally show ?thesis .
```
```   400 qed
```
```   401
```
```   402 subsection\<open>Pochhammer's symbol : generalized rising factorial\<close>
```
```   403
```
```   404 text \<open>See @{url "http://en.wikipedia.org/wiki/Pochhammer_symbol"}\<close>
```
```   405
```
```   406 definition "pochhammer (a::'a::comm_semiring_1) n =
```
```   407   (if n = 0 then 1 else setprod (\<lambda>n. a + of_nat n) {0 .. n - 1})"
```
```   408
```
```   409 lemma pochhammer_0 [simp]: "pochhammer a 0 = 1"
```
```   410   by (simp add: pochhammer_def)
```
```   411
```
```   412 lemma pochhammer_1 [simp]: "pochhammer a 1 = a"
```
```   413   by (simp add: pochhammer_def)
```
```   414
```
```   415 lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a"
```
```   416   by (simp add: pochhammer_def)
```
```   417
```
```   418 lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\<lambda>n. a + of_nat n) {0 .. n}"
```
```   419   by (simp add: pochhammer_def)
```
```   420
```
```   421 lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc n)"
```
```   422 proof -
```
```   423   have "{0..Suc n} = {0..n} \<union> {Suc n}" by auto
```
```   424   then show ?thesis by (simp add: field_simps)
```
```   425 qed
```
```   426
```
```   427 lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}"
```
```   428 proof -
```
```   429   have "{0..Suc n} = {0} \<union> {1 .. Suc n}" by auto
```
```   430   then show ?thesis by simp
```
```   431 qed
```
```   432
```
```   433
```
```   434 lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
```
```   435 proof (cases n)
```
```   436   case 0
```
```   437   then show ?thesis by simp
```
```   438 next
```
```   439   case (Suc n)
```
```   440   show ?thesis unfolding Suc pochhammer_Suc_setprod setprod_nat_ivl_Suc ..
```
```   441 qed
```
```   442
```
```   443 lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n"
```
```   444 proof (cases "n = 0")
```
```   445   case True
```
```   446   then show ?thesis by (simp add: pochhammer_Suc_setprod)
```
```   447 next
```
```   448   case False
```
```   449   have *: "finite {1 .. n}" "0 \<notin> {1 .. n}" by auto
```
```   450   have eq: "insert 0 {1 .. n} = {0..n}" by auto
```
```   451   have **: "(\<Prod>n\<in>{1\<Colon>nat..n}. a + of_nat n) = (\<Prod>n\<in>{0\<Colon>nat..n - 1}. a + 1 + of_nat n)"
```
```   452     apply (rule setprod.reindex_cong [where l = Suc])
```
```   453     using False
```
```   454     apply (auto simp add: fun_eq_iff field_simps)
```
```   455     done
```
```   456   show ?thesis
```
```   457     apply (simp add: pochhammer_def)
```
```   458     unfolding setprod.insert [OF *, unfolded eq]
```
```   459     using ** apply (simp add: field_simps)
```
```   460     done
```
```   461 qed
```
```   462
```
```   463 lemma pochhammer_fact: "fact n = pochhammer 1 n"
```
```   464   unfolding fact_altdef
```
```   465   apply (cases n)
```
```   466    apply (simp_all add: of_nat_setprod pochhammer_Suc_setprod)
```
```   467   apply (rule setprod.reindex_cong [where l = Suc])
```
```   468     apply (auto simp add: fun_eq_iff)
```
```   469   done
```
```   470
```
```   471 lemma pochhammer_of_nat_eq_0_lemma:
```
```   472   assumes "k > n"
```
```   473   shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
```
```   474 proof (cases "n = 0")
```
```   475   case True
```
```   476   then show ?thesis
```
```   477     using assms by (cases k) (simp_all add: pochhammer_rec)
```
```   478 next
```
```   479   case False
```
```   480   from assms obtain h where "k = Suc h" by (cases k) auto
```
```   481   then show ?thesis
```
```   482     by (simp add: pochhammer_Suc_setprod)
```
```   483        (metis Suc_leI Suc_le_mono assms atLeastAtMost_iff less_eq_nat.simps(1))
```
```   484 qed
```
```   485
```
```   486 lemma pochhammer_of_nat_eq_0_lemma':
```
```   487   assumes kn: "k \<le> n"
```
```   488   shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k \<noteq> 0"
```
```   489 proof (cases k)
```
```   490   case 0
```
```   491   then show ?thesis by simp
```
```   492 next
```
```   493   case (Suc h)
```
```   494   then show ?thesis
```
```   495     apply (simp add: pochhammer_Suc_setprod)
```
```   496     using Suc kn apply (auto simp add: algebra_simps)
```
```   497     done
```
```   498 qed
```
```   499
```
```   500 lemma pochhammer_of_nat_eq_0_iff:
```
```   501   shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k = 0 \<longleftrightarrow> k > n"
```
```   502   (is "?l = ?r")
```
```   503   using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a]
```
```   504     pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a]
```
```   505   by (auto simp add: not_le[symmetric])
```
```   506
```
```   507 lemma pochhammer_eq_0_iff: "pochhammer a n = (0::'a::field_char_0) \<longleftrightarrow> (\<exists>k < n. a = - of_nat k)"
```
```   508   apply (auto simp add: pochhammer_of_nat_eq_0_iff)
```
```   509   apply (cases n)
```
```   510    apply (auto simp add: pochhammer_def algebra_simps group_add_class.eq_neg_iff_add_eq_0)
```
```   511   apply (metis leD not_less_eq)
```
```   512   done
```
```   513
```
```   514 lemma pochhammer_eq_0_mono:
```
```   515   "pochhammer a n = (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a m = 0"
```
```   516   unfolding pochhammer_eq_0_iff by auto
```
```   517
```
```   518 lemma pochhammer_neq_0_mono:
```
```   519   "pochhammer a m \<noteq> (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a n \<noteq> 0"
```
```   520   unfolding pochhammer_eq_0_iff by auto
```
```   521
```
```   522 lemma pochhammer_minus:
```
```   523     "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k"
```
```   524 proof (cases k)
```
```   525   case 0
```
```   526   then show ?thesis by simp
```
```   527 next
```
```   528   case (Suc h)
```
```   529   have eq: "((- 1) ^ Suc h :: 'a) = (\<Prod>i=0..h. - 1)"
```
```   530     using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"]
```
```   531     by auto
```
```   532   show ?thesis
```
```   533     unfolding Suc pochhammer_Suc_setprod eq setprod.distrib[symmetric]
```
```   534     by (rule setprod.reindex_bij_witness[where i="op - h" and j="op - h"])
```
```   535        (auto simp: of_nat_diff)
```
```   536 qed
```
```   537
```
```   538 lemma pochhammer_minus':
```
```   539     "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
```
```   540   unfolding pochhammer_minus[where b=b]
```
```   541   unfolding mult.assoc[symmetric]
```
```   542   unfolding power_add[symmetric]
```
```   543   by simp
```
```   544
```
```   545 lemma pochhammer_same: "pochhammer (- of_nat n) n =
```
```   546     ((- 1) ^ n :: 'a::{semiring_char_0,comm_ring_1,semiring_no_zero_divisors}) * (fact n)"
```
```   547   unfolding pochhammer_minus
```
```   548   by (simp add: of_nat_diff pochhammer_fact)
```
```   549
```
```   550
```
```   551 subsection\<open>Generalized binomial coefficients\<close>
```
```   552
```
```   553 definition gbinomial :: "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
```
```   554   where "a gchoose n =
```
```   555     (if n = 0 then 1 else (setprod (\<lambda>i. a - of_nat i) {0 .. n - 1}) / (fact n))"
```
```   556
```
```   557 lemma gbinomial_0 [simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0"
```
```   558   by (simp_all add: gbinomial_def)
```
```   559
```
```   560 lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / (fact n)"
```
```   561 proof (cases "n = 0")
```
```   562   case True
```
```   563   then show ?thesis by simp
```
```   564 next
```
```   565   case False
```
```   566   from this setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"]
```
```   567   have eq: "(- (1\<Colon>'a)) ^ n = setprod (\<lambda>i. - 1) {0 .. n - 1}"
```
```   568     by auto
```
```   569   from False show ?thesis
```
```   570     by (simp add: pochhammer_def gbinomial_def field_simps
```
```   571       eq setprod.distrib[symmetric])
```
```   572 qed
```
```   573
```
```   574 lemma binomial_gbinomial:
```
```   575     "of_nat (n choose k) = (of_nat n gchoose k :: 'a::field_char_0)"
```
```   576 proof -
```
```   577   { assume kn: "k > n"
```
```   578     then have ?thesis
```
```   579       by (subst binomial_eq_0[OF kn])
```
```   580          (simp add: gbinomial_pochhammer field_simps  pochhammer_of_nat_eq_0_iff) }
```
```   581   moreover
```
```   582   { assume "k=0" then have ?thesis by simp }
```
```   583   moreover
```
```   584   { assume kn: "k \<le> n" and k0: "k\<noteq> 0"
```
```   585     from k0 obtain h where h: "k = Suc h" by (cases k) auto
```
```   586     from h
```
```   587     have eq:"(- 1 :: 'a) ^ k = setprod (\<lambda>i. - 1) {0..h}"
```
```   588       by (subst setprod_constant) auto
```
```   589     have eq': "(\<Prod>i\<in>{0..h}. of_nat n + - (of_nat i :: 'a)) = (\<Prod>i\<in>{n - h..n}. of_nat i)"
```
```   590         using h kn
```
```   591       by (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"])
```
```   592          (auto simp: of_nat_diff)
```
```   593     have th0: "finite {1..n - Suc h}" "finite {n - h .. n}"
```
```   594         "{1..n - Suc h} \<inter> {n - h .. n} = {}" and
```
```   595         eq3: "{1..n - Suc h} \<union> {n - h .. n} = {1..n}"
```
```   596       using h kn by auto
```
```   597     from eq[symmetric]
```
```   598     have ?thesis using kn
```
```   599       apply (simp add: binomial_fact[OF kn, where ?'a = 'a]
```
```   600         gbinomial_pochhammer field_simps pochhammer_Suc_setprod)
```
```   601       apply (simp add: pochhammer_Suc_setprod fact_altdef h
```
```   602         of_nat_setprod setprod.distrib[symmetric] eq' del: One_nat_def power_Suc)
```
```   603       unfolding setprod.union_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \<Rightarrow> 'a"] eq[unfolded h]
```
```   604       unfolding mult.assoc
```
```   605       unfolding setprod.distrib[symmetric]
```
```   606       apply simp
```
```   607       apply (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"])
```
```   608       apply (auto simp: of_nat_diff)
```
```   609       done
```
```   610   }
```
```   611   moreover
```
```   612   have "k > n \<or> k = 0 \<or> (k \<le> n \<and> k \<noteq> 0)" by arith
```
```   613   ultimately show ?thesis by blast
```
```   614 qed
```
```   615
```
```   616 lemma gbinomial_1[simp]: "a gchoose 1 = a"
```
```   617   by (simp add: gbinomial_def)
```
```   618
```
```   619 lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a"
```
```   620   by (simp add: gbinomial_def)
```
```   621
```
```   622 lemma gbinomial_mult_1:
```
```   623   fixes a :: "'a :: field_char_0"
```
```   624   shows "a * (a gchoose n) =
```
```   625     of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"  (is "?l = ?r")
```
```   626 proof -
```
```   627   have "?r = ((- 1) ^n * pochhammer (- a) n / (fact n)) * (of_nat n - (- a + of_nat n))"
```
```   628     unfolding gbinomial_pochhammer
```
```   629       pochhammer_Suc of_nat_mult right_diff_distrib power_Suc
```
```   630     apply (simp del: of_nat_Suc fact.simps)
```
```   631     apply (auto simp add: field_simps simp del: of_nat_Suc)
```
```   632     done
```
```   633   also have "\<dots> = ?l" unfolding gbinomial_pochhammer
```
```   634     by (simp add: field_simps)
```
```   635   finally show ?thesis ..
```
```   636 qed
```
```   637
```
```   638 lemma gbinomial_mult_1':
```
```   639   fixes a :: "'a :: field_char_0"
```
```   640   shows "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
```
```   641   by (simp add: mult.commute gbinomial_mult_1)
```
```   642
```
```   643 lemma gbinomial_Suc:
```
```   644     "a gchoose (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k}) / (fact (Suc k))"
```
```   645   by (simp add: gbinomial_def)
```
```   646
```
```   647 lemma gbinomial_mult_fact:
```
```   648   fixes a :: "'a::field_char_0"
```
```   649   shows
```
```   650    "fact (Suc k) * (a gchoose (Suc k)) =
```
```   651     (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
```
```   652   by (simp_all add: gbinomial_Suc field_simps del: fact.simps)
```
```   653
```
```   654 lemma gbinomial_mult_fact':
```
```   655   fixes a :: "'a::field_char_0"
```
```   656   shows "(a gchoose (Suc k)) * fact (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
```
```   657   using gbinomial_mult_fact[of k a]
```
```   658   by (subst mult.commute)
```
```   659
```
```   660 lemma gbinomial_Suc_Suc:
```
```   661   fixes a :: "'a :: field_char_0"
```
```   662   shows "(a + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
```
```   663 proof (cases k)
```
```   664   case 0
```
```   665   then show ?thesis by simp
```
```   666 next
```
```   667   case (Suc h)
```
```   668   have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)"
```
```   669     apply (rule setprod.reindex_cong [where l = Suc])
```
```   670       using Suc
```
```   671       apply auto
```
```   672     done
```
```   673   have "fact (Suc k) * (a gchoose k + (a gchoose (Suc k))) =
```
```   674         (a gchoose Suc h) * (fact (Suc (Suc h))) +
```
```   675         (a gchoose Suc (Suc h)) * (fact (Suc (Suc h)))"
```
```   676     by (simp add: Suc field_simps del: fact.simps)
```
```   677   also have "... = (a gchoose Suc h) * of_nat (Suc (Suc h) * fact (Suc h)) +
```
```   678                    (\<Prod>i = 0..Suc h. a - of_nat i)"
```
```   679     by (metis fact.simps(2) gbinomial_mult_fact' of_nat_fact of_nat_id)
```
```   680   also have "... = (fact (Suc h) * (a gchoose Suc h)) * of_nat (Suc (Suc h)) +
```
```   681                    (\<Prod>i = 0..Suc h. a - of_nat i)"
```
```   682     by (simp only: fact.simps(2) mult.commute mult.left_commute of_nat_fact of_nat_id of_nat_mult)
```
```   683   also have "... =  of_nat (Suc (Suc h)) * (\<Prod>i = 0..h. a - of_nat i) +
```
```   684                     (\<Prod>i = 0..Suc h. a - of_nat i)"
```
```   685     by (metis gbinomial_mult_fact mult.commute)
```
```   686   also have "... = (\<Prod>i = 0..Suc h. a - of_nat i) +
```
```   687                    (of_nat h * (\<Prod>i = 0..h. a - of_nat i) + 2 * (\<Prod>i = 0..h. a - of_nat i))"
```
```   688     by (simp add: field_simps)
```
```   689   also have "... =
```
```   690     ((a gchoose Suc h) * (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{0\<Colon>nat..Suc h}. a - of_nat i)"
```
```   691     unfolding gbinomial_mult_fact'
```
```   692     by (simp add: comm_semiring_class.distrib field_simps Suc)
```
```   693   also have "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 1)"
```
```   694     unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc
```
```   695     by (simp add: field_simps Suc)
```
```   696   also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)"
```
```   697     using eq0
```
```   698     by (simp add: Suc setprod_nat_ivl_1_Suc)
```
```   699   also have "\<dots> = (fact (Suc k)) * ((a + 1) gchoose (Suc k))"
```
```   700     unfolding gbinomial_mult_fact ..
```
```   701   finally show ?thesis
```
```   702     by (metis fact_nonzero mult_cancel_left)
```
```   703 qed
```
```   704
```
```   705 lemma gbinomial_reduce_nat:
```
```   706   fixes a :: "'a :: field_char_0"
```
```   707   shows "0 < k \<Longrightarrow> a gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)"
```
```   708   by (metis Suc_pred' diff_add_cancel gbinomial_Suc_Suc)
```
```   709
```
```   710 lemma gchoose_row_sum_weighted:
```
```   711   fixes r :: "'a::field_char_0"
```
```   712   shows "(\<Sum>k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))"
```
```   713 proof (induct m)
```
```   714   case 0 show ?case by simp
```
```   715 next
```
```   716   case (Suc m)
```
```   717   from Suc show ?case
```
```   718     by (simp add: algebra_simps distrib gbinomial_mult_1)
```
```   719 qed
```
```   720
```
```   721 lemma binomial_symmetric:
```
```   722   assumes kn: "k \<le> n"
```
```   723   shows "n choose k = n choose (n - k)"
```
```   724 proof-
```
```   725   from kn have kn': "n - k \<le> n" by arith
```
```   726   from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']
```
```   727   have "fact k * fact (n - k) * (n choose k) =
```
```   728     fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" by simp
```
```   729   then show ?thesis using kn by simp
```
```   730 qed
```
```   731
```
```   732 text\<open>Contributed by Manuel Eberl, generalised by LCP.
```
```   733   Alternative definition of the binomial coefficient as @{term "\<Prod>i<k. (n - i) / (k - i)"}\<close>
```
```   734 lemma gbinomial_altdef_of_nat:
```
```   735   fixes k :: nat
```
```   736     and x :: "'a :: {field_char_0,field}"
```
```   737   shows "x gchoose k = (\<Prod>i<k. (x - of_nat i) / of_nat (k - i) :: 'a)"
```
```   738 proof -
```
```   739   have "(x gchoose k) = (\<Prod>i<k. x - of_nat i) / of_nat (fact k)"
```
```   740     unfolding gbinomial_def
```
```   741     by (auto simp: gr0_conv_Suc lessThan_Suc_atMost atLeast0AtMost)
```
```   742   also have "\<dots> = (\<Prod>i<k. (x - of_nat i) / of_nat (k - i) :: 'a)"
```
```   743     unfolding fact_eq_rev_setprod_nat of_nat_setprod
```
```   744     by (auto simp add: setprod_dividef intro!: setprod.cong of_nat_diff[symmetric])
```
```   745   finally show ?thesis .
```
```   746 qed
```
```   747
```
```   748 lemma gbinomial_ge_n_over_k_pow_k:
```
```   749   fixes k :: nat
```
```   750     and x :: "'a :: linordered_field"
```
```   751   assumes "of_nat k \<le> x"
```
```   752   shows "(x / of_nat k :: 'a) ^ k \<le> x gchoose k"
```
```   753 proof -
```
```   754   have x: "0 \<le> x"
```
```   755     using assms of_nat_0_le_iff order_trans by blast
```
```   756   have "(x / of_nat k :: 'a) ^ k = (\<Prod>i<k. x / of_nat k :: 'a)"
```
```   757     by (simp add: setprod_constant)
```
```   758   also have "\<dots> \<le> x gchoose k"
```
```   759     unfolding gbinomial_altdef_of_nat
```
```   760   proof (safe intro!: setprod_mono)
```
```   761     fix i :: nat
```
```   762     assume ik: "i < k"
```
```   763     from assms have "x * of_nat i \<ge> of_nat (i * k)"
```
```   764       by (metis mult.commute mult_le_cancel_right of_nat_less_0_iff of_nat_mult)
```
```   765     then have "x * of_nat k - x * of_nat i \<le> x * of_nat k - of_nat (i * k)" by arith
```
```   766     then have "x * of_nat (k - i) \<le> (x - of_nat i) * of_nat k"
```
```   767       using ik
```
```   768       by (simp add: algebra_simps zero_less_mult_iff of_nat_diff of_nat_mult)
```
```   769     then have "x * of_nat (k - i) \<le> (x - of_nat i) * (of_nat k :: 'a)"
```
```   770       unfolding of_nat_mult[symmetric] of_nat_le_iff .
```
```   771     with assms show "x / of_nat k \<le> (x - of_nat i) / (of_nat (k - i) :: 'a)"
```
```   772       using \<open>i < k\<close> by (simp add: field_simps)
```
```   773   qed (simp add: x zero_le_divide_iff)
```
```   774   finally show ?thesis .
```
```   775 qed
```
```   776
```
```   777 text\<open>Versions of the theorems above for the natural-number version of "choose"\<close>
```
```   778 lemma binomial_altdef_of_nat:
```
```   779   fixes n k :: nat
```
```   780     and x :: "'a :: {field_char_0,field}"  --\<open>the point is to constrain @{typ 'a}\<close>
```
```   781   assumes "k \<le> n"
```
```   782   shows "of_nat (n choose k) = (\<Prod>i<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
```
```   783 using assms
```
```   784 by (simp add: gbinomial_altdef_of_nat binomial_gbinomial of_nat_diff)
```
```   785
```
```   786 lemma binomial_ge_n_over_k_pow_k:
```
```   787   fixes k n :: nat
```
```   788     and x :: "'a :: linordered_field"
```
```   789   assumes "k \<le> n"
```
```   790   shows "(of_nat n / of_nat k :: 'a) ^ k \<le> of_nat (n choose k)"
```
```   791 by (simp add: assms gbinomial_ge_n_over_k_pow_k binomial_gbinomial of_nat_diff)
```
```   792
```
```   793 lemma binomial_le_pow:
```
```   794   assumes "r \<le> n"
```
```   795   shows "n choose r \<le> n ^ r"
```
```   796 proof -
```
```   797   have "n choose r \<le> fact n div fact (n - r)"
```
```   798     using \<open>r \<le> n\<close> by (subst binomial_fact_lemma[symmetric]) auto
```
```   799   with fact_div_fact_le_pow [OF assms] show ?thesis by auto
```
```   800 qed
```
```   801
```
```   802 lemma binomial_altdef_nat: "(k::nat) \<le> n \<Longrightarrow>
```
```   803     n choose k = fact n div (fact k * fact (n - k))"
```
```   804  by (subst binomial_fact_lemma [symmetric]) auto
```
```   805
```
```   806 lemma choose_dvd: "k \<le> n \<Longrightarrow> fact k * fact (n - k) dvd (fact n :: 'a :: {semiring_div,linordered_semidom})"
```
```   807   unfolding dvd_def
```
```   808   apply (rule exI [where x="of_nat (n choose k)"])
```
```   809   using binomial_fact_lemma [of k n, THEN arg_cong [where f = of_nat and 'b='a]]
```
```   810   apply (auto simp: of_nat_mult)
```
```   811   done
```
```   812
```
```   813 lemma fact_fact_dvd_fact:
```
```   814     "fact k * fact n dvd (fact (k+n) :: 'a :: {semiring_div,linordered_semidom})"
```
```   815 by (metis add.commute add_diff_cancel_left' choose_dvd le_add2)
```
```   816
```
```   817 lemma choose_mult_lemma:
```
```   818      "((m+r+k) choose (m+k)) * ((m+k) choose k) = ((m+r+k) choose k) * ((m+r) choose m)"
```
```   819 proof -
```
```   820   have "((m+r+k) choose (m+k)) * ((m+k) choose k) =
```
```   821         fact (m+r + k) div (fact (m + k) * fact (m+r - m)) * (fact (m + k) div (fact k * fact m))"
```
```   822     by (simp add: assms binomial_altdef_nat)
```
```   823   also have "... = fact (m+r+k) div (fact r * (fact k * fact m))"
```
```   824     apply (subst div_mult_div_if_dvd)
```
```   825     apply (auto simp: algebra_simps fact_fact_dvd_fact)
```
```   826     apply (metis add.assoc add.commute fact_fact_dvd_fact)
```
```   827     done
```
```   828   also have "... = (fact (m+r+k) * fact (m+r)) div (fact r * (fact k * fact m) * fact (m+r))"
```
```   829     apply (subst div_mult_div_if_dvd [symmetric])
```
```   830     apply (auto simp add: algebra_simps)
```
```   831     apply (metis fact_fact_dvd_fact dvd.order.trans nat_mult_dvd_cancel_disj)
```
```   832     done
```
```   833   also have "... = (fact (m+r+k) div (fact k * fact (m+r)) * (fact (m+r) div (fact r * fact m)))"
```
```   834     apply (subst div_mult_div_if_dvd)
```
```   835     apply (auto simp: fact_fact_dvd_fact algebra_simps)
```
```   836     done
```
```   837   finally show ?thesis
```
```   838     by (simp add: binomial_altdef_nat mult.commute)
```
```   839 qed
```
```   840
```
```   841 text\<open>The "Subset of a Subset" identity\<close>
```
```   842 lemma choose_mult:
```
```   843   assumes "k\<le>m" "m\<le>n"
```
```   844     shows "(n choose m) * (m choose k) = (n choose k) * ((n-k) choose (m-k))"
```
```   845 using assms choose_mult_lemma [of "m-k" "n-m" k]
```
```   846 by simp
```
```   847
```
```   848
```
```   849 subsection \<open>Binomial coefficients\<close>
```
```   850
```
```   851 lemma choose_one: "(n::nat) choose 1 = n"
```
```   852   by simp
```
```   853
```
```   854 (*FIXME: messy and apparently unused*)
```
```   855 lemma binomial_induct [rule_format]: "(ALL (n::nat). P n n) \<longrightarrow>
```
```   856     (ALL n. P (Suc n) 0) \<longrightarrow> (ALL n. (ALL k < n. P n k \<longrightarrow> P n (Suc k) \<longrightarrow>
```
```   857     P (Suc n) (Suc k))) \<longrightarrow> (ALL k <= n. P n k)"
```
```   858   apply (induct n)
```
```   859   apply auto
```
```   860   apply (case_tac "k = 0")
```
```   861   apply auto
```
```   862   apply (case_tac "k = Suc n")
```
```   863   apply auto
```
```   864   apply (metis Suc_le_eq fact.cases le_Suc_eq le_eq_less_or_eq)
```
```   865   done
```
```   866
```
```   867 lemma card_UNION:
```
```   868   assumes "finite A" and "\<forall>k \<in> A. finite k"
```
```   869   shows "card (\<Union>A) = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * int (card (\<Inter>I)))"
```
```   870   (is "?lhs = ?rhs")
```
```   871 proof -
```
```   872   have "?rhs = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * (\<Sum>_\<in>\<Inter>I. 1))" by simp
```
```   873   also have "\<dots> = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (\<Sum>_\<in>\<Inter>I. (- 1) ^ (card I + 1)))" (is "_ = nat ?rhs")
```
```   874     by(subst setsum_right_distrib) simp
```
```   875   also have "?rhs = (\<Sum>(I, _)\<in>Sigma {I. I \<subseteq> A \<and> I \<noteq> {}} Inter. (- 1) ^ (card I + 1))"
```
```   876     using assms by(subst setsum.Sigma)(auto)
```
```   877   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))"
```
```   878     by (rule setsum.reindex_cong [where l = "\<lambda>(x, y). (y, x)"]) (auto intro: inj_onI simp add: split_beta)
```
```   879   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))"
```
```   880     using assms by(auto intro!: setsum.mono_neutral_cong_right finite_SigmaI2 intro: finite_subset[where B="\<Union>A"])
```
```   881   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)))"
```
```   882     using assms by(subst setsum.Sigma) auto
```
```   883   also have "\<dots> = (\<Sum>_\<in>\<Union>A. 1)" (is "setsum ?lhs _ = _")
```
```   884   proof(rule setsum.cong[OF refl])
```
```   885     fix x
```
```   886     assume x: "x \<in> \<Union>A"
```
```   887     def K \<equiv> "{X \<in> A. x \<in> X}"
```
```   888     with \<open>finite A\<close> have K: "finite K" by auto
```
```   889     let ?I = "\<lambda>i. {I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I}"
```
```   890     have "inj_on snd (SIGMA i:{1..card A}. ?I i)"
```
```   891       using assms by(auto intro!: inj_onI)
```
```   892     moreover have [symmetric]: "snd ` (SIGMA i:{1..card A}. ?I i) = {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}"
```
```   893       using assms by(auto intro!: rev_image_eqI[where x="(card a, a)" for a]
```
```   894         simp add: card_gt_0_iff[folded Suc_le_eq]
```
```   895         dest: finite_subset intro: card_mono)
```
```   896     ultimately have "?lhs x = (\<Sum>(i, I)\<in>(SIGMA i:{1..card A}. ?I i). (- 1) ^ (i + 1))"
```
```   897       by (rule setsum.reindex_cong [where l = snd]) fastforce
```
```   898     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)))"
```
```   899       using assms by(subst setsum.Sigma) auto
```
```   900     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))"
```
```   901       by(subst setsum_right_distrib) simp
```
```   902     also have "\<dots> = (\<Sum>i=1..card K. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> K \<and> card I = i. 1))" (is "_ = ?rhs")
```
```   903     proof(rule setsum.mono_neutral_cong_right[rule_format])
```
```   904       show "{1..card K} \<subseteq> {1..card A}" using \<open>finite A\<close>
```
```   905         by(auto simp add: K_def intro: card_mono)
```
```   906     next
```
```   907       fix i
```
```   908       assume "i \<in> {1..card A} - {1..card K}"
```
```   909       hence i: "i \<le> card A" "card K < i" by auto
```
```   910       have "{I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I} = {I. I \<subseteq> K \<and> card I = i}"
```
```   911         by(auto simp add: K_def)
```
```   912       also have "\<dots> = {}" using \<open>finite A\<close> i
```
```   913         by(auto simp add: K_def dest: card_mono[rotated 1])
```
```   914       finally show "(- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1 :: int) = 0"
```
```   915         by(simp only:) simp
```
```   916     next
```
```   917       fix i
```
```   918       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)"
```
```   919         (is "?lhs = ?rhs")
```
```   920         by(rule setsum.cong)(auto simp add: K_def)
```
```   921       thus "(- 1) ^ (i + 1) * ?lhs = (- 1) ^ (i + 1) * ?rhs" by simp
```
```   922     qed simp
```
```   923     also have "{I. I \<subseteq> K \<and> card I = 0} = {{}}" using assms
```
```   924       by(auto simp add: card_eq_0_iff K_def dest: finite_subset)
```
```   925     hence "?rhs = (\<Sum>i = 0..card K. (- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> K \<and> card I = i. 1 :: int)) + 1"
```
```   926       by(subst (2) setsum_head_Suc)(simp_all )
```
```   927     also have "\<dots> = (\<Sum>i = 0..card K. (- 1) * ((- 1) ^ i * int (card K choose i))) + 1"
```
```   928       using K by(subst n_subsets[symmetric]) simp_all
```
```   929     also have "\<dots> = - (\<Sum>i = 0..card K. (- 1) ^ i * int (card K choose i)) + 1"
```
```   930       by(subst setsum_right_distrib[symmetric]) simp
```
```   931     also have "\<dots> =  - ((-1 + 1) ^ card K) + 1"
```
```   932       by(subst binomial_ring)(simp add: ac_simps)
```
```   933     also have "\<dots> = 1" using x K by(auto simp add: K_def card_gt_0_iff)
```
```   934     finally show "?lhs x = 1" .
```
```   935   qed
```
```   936   also have "nat \<dots> = card (\<Union>A)" by simp
```
```   937   finally show ?thesis ..
```
```   938 qed
```
```   939
```
```   940 text\<open>The number of nat lists of length @{text m} summing to @{text N} is
```
```   941 @{term "(N + m - 1) choose N"}:\<close>
```
```   942
```
```   943 lemma card_length_listsum_rec:
```
```   944   assumes "m\<ge>1"
```
```   945   shows "card {l::nat list. length l = m \<and> listsum l = N} =
```
```   946     (card {l. length l = (m - 1) \<and> listsum l = N} +
```
```   947     card {l. length l = m \<and> listsum l + 1 =  N})"
```
```   948     (is "card ?C = (card ?A + card ?B)")
```
```   949 proof -
```
```   950   let ?A'="{l. length l = m \<and> listsum l = N \<and> hd l = 0}"
```
```   951   let ?B'="{l. length l = m \<and> listsum l = N \<and> hd l \<noteq> 0}"
```
```   952   let ?f ="\<lambda> l. 0#l"
```
```   953   let ?g ="\<lambda> l. (hd l + 1) # tl l"
```
```   954   have 1: "\<And>xs x. xs \<noteq> [] \<Longrightarrow> x = hd xs \<Longrightarrow> x # tl xs = xs" by simp
```
```   955   have 2: "\<And>xs. (xs::nat list) \<noteq> [] \<Longrightarrow> listsum(tl xs) = listsum xs - hd xs"
```
```   956     by(auto simp add: neq_Nil_conv)
```
```   957   have f: "bij_betw ?f ?A ?A'"
```
```   958     apply(rule bij_betw_byWitness[where f' = tl])
```
```   959     using assms
```
```   960     by (auto simp: 2 length_0_conv[symmetric] 1 simp del: length_0_conv)
```
```   961   have 3: "\<And>xs:: nat list. xs \<noteq> [] \<Longrightarrow> hd xs + (listsum xs - hd xs) = listsum xs"
```
```   962     by (metis 1 listsum_simps(2) 2)
```
```   963   have g: "bij_betw ?g ?B ?B'"
```
```   964     apply(rule bij_betw_byWitness[where f' = "\<lambda> l. (hd l - 1) # tl l"])
```
```   965     using assms
```
```   966     by (auto simp: 2 length_0_conv[symmetric] intro!: 3
```
```   967       simp del: length_greater_0_conv length_0_conv)
```
```   968   { fix M N :: nat have "finite {xs. size xs = M \<and> set xs \<subseteq> {0..<N}}"
```
```   969     using finite_lists_length_eq[OF finite_atLeastLessThan] conj_commute by auto }
```
```   970     note fin = this
```
```   971   have fin_A: "finite ?A" using fin[of _ "N+1"]
```
```   972     by (intro finite_subset[where ?A = "?A" and ?B = "{xs. size xs = m - 1 \<and> set xs \<subseteq> {0..<N+1}}"],
```
```   973       auto simp: member_le_listsum_nat less_Suc_eq_le)
```
```   974   have fin_B: "finite ?B"
```
```   975     by (intro finite_subset[where ?A = "?B" and ?B = "{xs. size xs = m \<and> set xs \<subseteq> {0..<N}}"],
```
```   976       auto simp: member_le_listsum_nat less_Suc_eq_le fin)
```
```   977   have uni: "?C = ?A' \<union> ?B'" by auto
```
```   978   have disj: "?A' \<inter> ?B' = {}" by auto
```
```   979   have "card ?C = card(?A' \<union> ?B')" using uni by simp
```
```   980   also have "\<dots> = card ?A + card ?B"
```
```   981     using card_Un_disjoint[OF _ _ disj] bij_betw_finite[OF f] bij_betw_finite[OF g]
```
```   982       bij_betw_same_card[OF f] bij_betw_same_card[OF g] fin_A fin_B
```
```   983     by presburger
```
```   984   finally show ?thesis .
```
```   985 qed
```
```   986
```
```   987 lemma card_length_listsum: --"By Holden Lee, tidied by Tobias Nipkow"
```
```   988   "card {l::nat list. size l = m \<and> listsum l = N} = (N + m - 1) choose N"
```
```   989 proof (cases m)
```
```   990   case 0 then show ?thesis
```
```   991     by (cases N) (auto simp: cong: conj_cong)
```
```   992 next
```
```   993   case (Suc m')
```
```   994     have m: "m\<ge>1" by (simp add: Suc)
```
```   995     then show ?thesis
```
```   996     proof (induct "N + m - 1" arbitrary: N m)
```
```   997       case 0   -- "In the base case, the only solution is ."
```
```   998       have [simp]: "{l::nat list. length l = Suc 0 \<and> (\<forall>n\<in>set l. n = 0)} = {}"
```
```   999         by (auto simp: length_Suc_conv)
```
```  1000       have "m=1 \<and> N=0" using 0 by linarith
```
```  1001       then show ?case by simp
```
```  1002     next
```
```  1003       case (Suc k)
```
```  1004
```
```  1005       have c1: "card {l::nat list. size l = (m - 1) \<and> listsum l =  N} =
```
```  1006         (N + (m - 1) - 1) choose N"
```
```  1007       proof cases
```
```  1008         assume "m = 1"
```
```  1009         with Suc.hyps have "N\<ge>1" by auto
```
```  1010         with \<open>m = 1\<close> show ?thesis by (simp add: binomial_eq_0)
```
```  1011       next
```
```  1012         assume "m \<noteq> 1" thus ?thesis using Suc by fastforce
```
```  1013       qed
```
```  1014
```
```  1015       from Suc have c2: "card {l::nat list. size l = m \<and> listsum l + 1 = N} =
```
```  1016         (if N>0 then ((N - 1) + m - 1) choose (N - 1) else 0)"
```
```  1017       proof -
```
```  1018         have aux: "\<And>m n. n > 0 \<Longrightarrow> Suc m = n \<longleftrightarrow> m = n - 1" by arith
```
```  1019         from Suc have "N>0 \<Longrightarrow>
```
```  1020           card {l::nat list. size l = m \<and> listsum l + 1 = N} =
```
```  1021           ((N - 1) + m - 1) choose (N - 1)" by (simp add: aux)
```
```  1022         thus ?thesis by auto
```
```  1023       qed
```
```  1024
```
```  1025       from Suc.prems have "(card {l::nat list. size l = (m - 1) \<and> listsum l = N} +
```
```  1026           card {l::nat list. size l = m \<and> listsum l + 1 = N}) = (N + m - 1) choose N"
```
```  1027         by (auto simp: c1 c2 choose_reduce_nat[of "N + m - 1" N] simp del: One_nat_def)
```
```  1028       thus ?case using card_length_listsum_rec[OF Suc.prems] by auto
```
```  1029     qed
```
```  1030 qed
```
```  1031
```
```  1032
```
```  1033 lemma Suc_times_binomial_add: -- \<open>by Lukas Bulwahn\<close>
```
```  1034   "Suc a * (Suc (a + b) choose Suc a) = Suc b * (Suc (a + b) choose a)"
```
```  1035 proof -
```
```  1036   have dvd: "Suc a * (fact a * fact b) dvd fact (Suc (a + b))" for a b
```
```  1037     using fact_fact_dvd_fact[of "Suc a" "b", where 'a=nat]
```
```  1038     by (simp only: fact_Suc add_Suc[symmetric] of_nat_id mult.assoc)
```
```  1039
```
```  1040   have "Suc a * (fact (Suc (a + b)) div (Suc a * fact a * fact b)) =
```
```  1041       Suc a * fact (Suc (a + b)) div (Suc a * (fact a * fact b))"
```
```  1042     by (subst div_mult_swap[symmetric]; simp only: mult.assoc dvd)
```
```  1043   also have "\<dots> = Suc b * fact (Suc (a + b)) div (Suc b * (fact a * fact b))"
```
```  1044     by (simp only: div_mult_mult1)
```
```  1045   also have "\<dots> = Suc b * (fact (Suc (a + b)) div (Suc b * (fact a * fact b)))"
```
```  1046     using dvd[of b a] by (subst div_mult_swap[symmetric]; simp only: ac_simps dvd)
```
```  1047   finally show ?thesis
```
```  1048     by (subst (1 2) binomial_altdef_nat)
```
```  1049        (simp_all only: ac_simps diff_Suc_Suc Suc_diff_le diff_add_inverse fact_Suc of_nat_id)
```
```  1050 qed
```
```  1051
```
```  1052 end
```