src/HOL/Library/Binomial.thy
 author Andreas Lochbihler Fri Sep 20 10:09:16 2013 +0200 (2013-09-20) changeset 53745 788730ab7da4 parent 52903 6c89225ddeba child 54489 03ff4d1e6784 permissions -rw-r--r--
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```     1 (*  Title:      HOL/Library/Binomial.thy
```
```     2     Author:     Lawrence C Paulson, Amine Chaieb
```
```     3     Copyright   1997  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 header {* Binomial Coefficients *}
```
```     7
```
```     8 theory Binomial
```
```     9 imports Complex_Main
```
```    10 begin
```
```    11
```
```    12 text {* This development is based on the work of Andy Gordon and
```
```    13   Florian Kammueller. *}
```
```    14
```
```    15 primrec binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65)
```
```    16 where
```
```    17   "0 choose k = (if k = 0 then 1 else 0)"
```
```    18 | "Suc n choose k = (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
```
```    19
```
```    20 lemma binomial_n_0 [simp]: "(n choose 0) = 1"
```
```    21   by (cases n) simp_all
```
```    22
```
```    23 lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
```
```    24   by simp
```
```    25
```
```    26 lemma binomial_Suc_Suc [simp]: "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
```
```    27   by simp
```
```    28
```
```    29 lemma binomial_eq_0: "n < k \<Longrightarrow> n choose k = 0"
```
```    30   by (induct n arbitrary: k) auto
```
```    31
```
```    32 declare binomial.simps [simp del]
```
```    33
```
```    34 lemma binomial_n_n [simp]: "n choose n = 1"
```
```    35   by (induct n) (simp_all add: binomial_eq_0)
```
```    36
```
```    37 lemma binomial_Suc_n [simp]: "Suc n choose n = Suc n"
```
```    38   by (induct n) simp_all
```
```    39
```
```    40 lemma binomial_1 [simp]: "n choose Suc 0 = n"
```
```    41   by (induct n) simp_all
```
```    42
```
```    43 lemma zero_less_binomial: "k \<le> n \<Longrightarrow> n choose k > 0"
```
```    44   by (induct n k rule: diff_induct) simp_all
```
```    45
```
```    46 lemma binomial_eq_0_iff: "n choose k = 0 \<longleftrightarrow> n < k"
```
```    47   apply (safe intro!: binomial_eq_0)
```
```    48   apply (erule contrapos_pp)
```
```    49   apply (simp add: zero_less_binomial)
```
```    50   done
```
```    51
```
```    52 lemma zero_less_binomial_iff: "n choose k > 0 \<longleftrightarrow> k \<le> n"
```
```    53   by (simp add: linorder_not_less binomial_eq_0_iff neq0_conv[symmetric] del: neq0_conv)
```
```    54
```
```    55 (*Might be more useful if re-oriented*)
```
```    56 lemma Suc_times_binomial_eq:
```
```    57   "k \<le> n \<Longrightarrow> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
```
```    58   apply (induct n arbitrary: k)
```
```    59    apply (simp add: binomial.simps)
```
```    60    apply (case_tac k)
```
```    61   apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0)
```
```    62   done
```
```    63
```
```    64 text{*This is the well-known version, but it's harder to use because of the
```
```    65   need to reason about division.*}
```
```    66 lemma binomial_Suc_Suc_eq_times:
```
```    67     "k \<le> n \<Longrightarrow> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
```
```    68   by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right)
```
```    69
```
```    70 text{*Another version, with -1 instead of Suc.*}
```
```    71 lemma times_binomial_minus1_eq:
```
```    72   "k \<le> n \<Longrightarrow> 0 < k \<Longrightarrow> (n choose k) * k = n * ((n - 1) choose (k - 1))"
```
```    73   using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 1"]
```
```    74   by (auto split add: nat_diff_split)
```
```    75
```
```    76
```
```    77 subsection {* Theorems about @{text "choose"} *}
```
```    78
```
```    79 text {*
```
```    80   \medskip Basic theorem about @{text "choose"}.  By Florian
```
```    81   Kamm\"uller, tidied by LCP.
```
```    82 *}
```
```    83
```
```    84 lemma card_s_0_eq_empty: "finite A \<Longrightarrow> card {B. B \<subseteq> A & card B = 0} = 1"
```
```    85   by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
```
```    86
```
```    87 lemma choose_deconstruct: "finite M \<Longrightarrow> x \<notin> M \<Longrightarrow>
```
```    88     {s. s \<subseteq> insert x M \<and> card s = Suc k} =
```
```    89     {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}"
```
```    90   apply safe
```
```    91      apply (auto intro: finite_subset [THEN card_insert_disjoint])
```
```    92   apply (drule_tac x = "xa - {x}" in spec)
```
```    93   apply (subgoal_tac "x \<notin> xa")
```
```    94    apply auto
```
```    95   apply (erule rev_mp, subst card_Diff_singleton)
```
```    96     apply (auto intro: finite_subset)
```
```    97   done
```
```    98 (*
```
```    99 lemma "finite(UN y. {x. P x y})"
```
```   100 apply simp
```
```   101 lemma Collect_ex_eq
```
```   102
```
```   103 lemma "{x. \<exists>y. P x y} = (UN y. {x. P x y})"
```
```   104 apply blast
```
```   105 *)
```
```   106
```
```   107 lemma finite_bex_subset [simp]:
```
```   108   assumes "finite B"
```
```   109     and "\<And>A. A \<subseteq> B \<Longrightarrow> finite {x. P x A}"
```
```   110   shows "finite {x. \<exists>A \<subseteq> B. P x A}"
```
```   111 proof -
```
```   112   have "{x. \<exists>A\<subseteq>B. P x A} = (\<Union>A \<in> Pow B. {x. P x A})" by blast
```
```   113   with assms show ?thesis by simp
```
```   114 qed
```
```   115
```
```   116 text{*There are as many subsets of @{term A} having cardinality @{term k}
```
```   117  as there are sets obtained from the former by inserting a fixed element
```
```   118  @{term x} into each.*}
```
```   119 lemma constr_bij:
```
```   120    "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow>
```
```   121     card {B. \<exists>C. C \<subseteq> A \<and> card C = k \<and> B = insert x C} =
```
```   122     card {B. B \<subseteq> A & card(B) = k}"
```
```   123   apply (rule_tac f = "\<lambda>s. s - {x}" and g = "insert x" in card_bij_eq)
```
```   124        apply (auto elim!: equalityE simp add: inj_on_def)
```
```   125   apply (subst Diff_insert0)
```
```   126    apply auto
```
```   127   done
```
```   128
```
```   129 text {*
```
```   130   Main theorem: combinatorial statement about number of subsets of a set.
```
```   131 *}
```
```   132
```
```   133 lemma n_sub_lemma:
```
```   134     "finite A \<Longrightarrow> card {B. B \<subseteq> A \<and> card B = k} = (card A choose k)"
```
```   135   apply (induct k arbitrary: A)
```
```   136    apply (simp add: card_s_0_eq_empty)
```
```   137    apply atomize
```
```   138   apply (rotate_tac -1)
```
```   139   apply (erule finite_induct)
```
```   140    apply (simp_all (no_asm_simp) cong add: conj_cong
```
```   141      add: card_s_0_eq_empty choose_deconstruct)
```
```   142   apply (subst card_Un_disjoint)
```
```   143      prefer 4 apply (force simp add: constr_bij)
```
```   144     prefer 3 apply force
```
```   145    prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
```
```   146      finite_subset [of _ "Pow (insert x F)", standard])
```
```   147   apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
```
```   148   done
```
```   149
```
```   150 theorem n_subsets: "finite A \<Longrightarrow> card {B. B \<subseteq> A \<and> card B = k} = (card A choose k)"
```
```   151   by (simp add: n_sub_lemma)
```
```   152
```
```   153
```
```   154 text{* The binomial theorem (courtesy of Tobias Nipkow): *}
```
```   155
```
```   156 theorem binomial: "(a + b::nat)^n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n - k))"
```
```   157 proof (induct n)
```
```   158   case 0
```
```   159   then show ?case by simp
```
```   160 next
```
```   161   case (Suc n)
```
```   162   have decomp: "{0..n+1} = {0} \<union> {n+1} \<union> {1..n}"
```
```   163     by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)
```
```   164   have decomp2: "{0..n} = {0} \<union> {1..n}"
```
```   165     by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)
```
```   166   have "(a + b)^(n + 1) = (a + b) * (\<Sum>k=0..n. (n choose k) * a^k * b^(n - k))"
```
```   167     using Suc by simp
```
```   168   also have "\<dots> =  a*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k)) +
```
```   169                    b*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
```
```   170     by (rule nat_distrib)
```
```   171   also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^(k+1) * b^(n-k)) +
```
```   172                   (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k+1))"
```
```   173     by (simp add: setsum_right_distrib mult_ac)
```
```   174   also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^k * b^(n+1-k)) +
```
```   175                   (\<Sum>k=1..n+1. (n choose (k - 1)) * a^k * b^(n+1-k))"
```
```   176     by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le
```
```   177              del:setsum_cl_ivl_Suc)
```
```   178   also have "\<dots> = a^(n+1) + b^(n+1) +
```
```   179                   (\<Sum>k=1..n. (n choose (k - 1)) * a^k * b^(n+1-k)) +
```
```   180                   (\<Sum>k=1..n. (n choose k) * a^k * b^(n+1-k))"
```
```   181     by (simp add: decomp2)
```
```   182   also have
```
```   183       "\<dots> = a^(n+1) + b^(n+1) + (\<Sum>k=1..n. (n+1 choose k) * a^k * b^(n+1-k))"
```
```   184     by (simp add: nat_distrib setsum_addf binomial.simps)
```
```   185   also have "\<dots> = (\<Sum>k=0..n+1. (n+1 choose k) * a^k * b^(n+1-k))"
```
```   186     using decomp by simp
```
```   187   finally show ?case by simp
```
```   188 qed
```
```   189
```
```   190 subsection{* Pochhammer's symbol : generalized raising factorial*}
```
```   191
```
```   192 definition "pochhammer (a::'a::comm_semiring_1) n =
```
```   193   (if n = 0 then 1 else setprod (\<lambda>n. a + of_nat n) {0 .. n - 1})"
```
```   194
```
```   195 lemma pochhammer_0 [simp]: "pochhammer a 0 = 1"
```
```   196   by (simp add: pochhammer_def)
```
```   197
```
```   198 lemma pochhammer_1 [simp]: "pochhammer a 1 = a"
```
```   199   by (simp add: pochhammer_def)
```
```   200
```
```   201 lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a"
```
```   202   by (simp add: pochhammer_def)
```
```   203
```
```   204 lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\<lambda>n. a + of_nat n) {0 .. n}"
```
```   205   by (simp add: pochhammer_def)
```
```   206
```
```   207 lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc n)"
```
```   208 proof -
```
```   209   have "{0..Suc n} = {0..n} \<union> {Suc n}" by auto
```
```   210   then show ?thesis by (simp add: field_simps)
```
```   211 qed
```
```   212
```
```   213 lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}"
```
```   214 proof -
```
```   215   have "{0..Suc n} = {0} \<union> {1 .. Suc n}" by auto
```
```   216   then show ?thesis by simp
```
```   217 qed
```
```   218
```
```   219
```
```   220 lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
```
```   221 proof (cases n)
```
```   222   case 0
```
```   223   then show ?thesis by simp
```
```   224 next
```
```   225   case (Suc n)
```
```   226   show ?thesis unfolding Suc pochhammer_Suc_setprod setprod_nat_ivl_Suc ..
```
```   227 qed
```
```   228
```
```   229 lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n"
```
```   230 proof (cases "n = 0")
```
```   231   case True
```
```   232   then show ?thesis by (simp add: pochhammer_Suc_setprod)
```
```   233 next
```
```   234   case False
```
```   235   have *: "finite {1 .. n}" "0 \<notin> {1 .. n}" by auto
```
```   236   have eq: "insert 0 {1 .. n} = {0..n}" by auto
```
```   237   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)"
```
```   238     apply (rule setprod_reindex_cong [where f = Suc])
```
```   239     using False
```
```   240     apply (auto simp add: fun_eq_iff field_simps)
```
```   241     done
```
```   242   show ?thesis
```
```   243     apply (simp add: pochhammer_def)
```
```   244     unfolding setprod_insert [OF *, unfolded eq]
```
```   245     using ** apply (simp add: field_simps)
```
```   246     done
```
```   247 qed
```
```   248
```
```   249 lemma pochhammer_fact: "of_nat (fact n) = pochhammer 1 n"
```
```   250   unfolding fact_altdef_nat
```
```   251   apply (cases n)
```
```   252    apply (simp_all add: of_nat_setprod pochhammer_Suc_setprod)
```
```   253   apply (rule setprod_reindex_cong[where f=Suc])
```
```   254     apply (auto simp add: fun_eq_iff)
```
```   255   done
```
```   256
```
```   257 lemma pochhammer_of_nat_eq_0_lemma:
```
```   258   assumes "k > n"
```
```   259   shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
```
```   260 proof (cases "n = 0")
```
```   261   case True
```
```   262   then show ?thesis
```
```   263     using assms by (cases k) (simp_all add: pochhammer_rec)
```
```   264 next
```
```   265   case False
```
```   266   from assms obtain h where "k = Suc h" by (cases k) auto
```
```   267   then show ?thesis
```
```   268     apply (simp add: pochhammer_Suc_setprod)
```
```   269     apply (rule_tac x="n" in bexI)
```
```   270     using assms
```
```   271     apply auto
```
```   272     done
```
```   273 qed
```
```   274
```
```   275 lemma pochhammer_of_nat_eq_0_lemma':
```
```   276   assumes kn: "k \<le> n"
```
```   277   shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k \<noteq> 0"
```
```   278 proof (cases k)
```
```   279   case 0
```
```   280   then show ?thesis by simp
```
```   281 next
```
```   282   case (Suc h)
```
```   283   then show ?thesis
```
```   284     apply (simp add: pochhammer_Suc_setprod)
```
```   285     using Suc kn apply (auto simp add: algebra_simps)
```
```   286     done
```
```   287 qed
```
```   288
```
```   289 lemma pochhammer_of_nat_eq_0_iff:
```
```   290   shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k = 0 \<longleftrightarrow> k > n"
```
```   291   (is "?l = ?r")
```
```   292   using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a]
```
```   293     pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a]
```
```   294   by (auto simp add: not_le[symmetric])
```
```   295
```
```   296
```
```   297 lemma pochhammer_eq_0_iff: "pochhammer a n = (0::'a::field_char_0) \<longleftrightarrow> (\<exists>k < n. a = - of_nat k)"
```
```   298   apply (auto simp add: pochhammer_of_nat_eq_0_iff)
```
```   299   apply (cases n)
```
```   300    apply (auto simp add: pochhammer_def algebra_simps group_add_class.eq_neg_iff_add_eq_0)
```
```   301   apply (rule_tac x=x in exI)
```
```   302   apply auto
```
```   303   done
```
```   304
```
```   305
```
```   306 lemma pochhammer_eq_0_mono:
```
```   307   "pochhammer a n = (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a m = 0"
```
```   308   unfolding pochhammer_eq_0_iff by auto
```
```   309
```
```   310 lemma pochhammer_neq_0_mono:
```
```   311   "pochhammer a m \<noteq> (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a n \<noteq> 0"
```
```   312   unfolding pochhammer_eq_0_iff by auto
```
```   313
```
```   314 lemma pochhammer_minus:
```
```   315   assumes kn: "k \<le> n"
```
```   316   shows "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k"
```
```   317 proof (cases k)
```
```   318   case 0
```
```   319   then show ?thesis by simp
```
```   320 next
```
```   321   case (Suc h)
```
```   322   have eq: "((- 1) ^ Suc h :: 'a) = setprod (%i. - 1) {0 .. h}"
```
```   323     using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"]
```
```   324     by auto
```
```   325   show ?thesis
```
```   326     unfolding Suc pochhammer_Suc_setprod eq setprod_timesf[symmetric]
```
```   327     apply (rule strong_setprod_reindex_cong[where f = "%i. h - i"])
```
```   328     using Suc
```
```   329     apply (auto simp add: inj_on_def image_def)
```
```   330     apply (rule_tac x="h - x" in bexI)
```
```   331     apply (auto simp add: fun_eq_iff of_nat_diff)
```
```   332     done
```
```   333 qed
```
```   334
```
```   335 lemma pochhammer_minus':
```
```   336   assumes kn: "k \<le> n"
```
```   337   shows "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
```
```   338   unfolding pochhammer_minus[OF kn, where b=b]
```
```   339   unfolding mult_assoc[symmetric]
```
```   340   unfolding power_add[symmetric]
```
```   341   by simp
```
```   342
```
```   343 lemma pochhammer_same: "pochhammer (- of_nat n) n =
```
```   344     ((- 1) ^ n :: 'a::comm_ring_1) * of_nat (fact n)"
```
```   345   unfolding pochhammer_minus[OF le_refl[of n]]
```
```   346   by (simp add: of_nat_diff pochhammer_fact)
```
```   347
```
```   348
```
```   349 subsection{* Generalized binomial coefficients *}
```
```   350
```
```   351 definition gbinomial :: "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
```
```   352   where "a gchoose n =
```
```   353     (if n = 0 then 1 else (setprod (\<lambda>i. a - of_nat i) {0 .. n - 1}) / of_nat (fact n))"
```
```   354
```
```   355 lemma gbinomial_0 [simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0"
```
```   356   apply (simp_all add: gbinomial_def)
```
```   357   apply (subgoal_tac "(\<Prod>i\<Colon>nat\<in>{0\<Colon>nat..n}. - of_nat i) = (0::'b)")
```
```   358    apply (simp del:setprod_zero_iff)
```
```   359   apply simp
```
```   360   done
```
```   361
```
```   362 lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / of_nat (fact n)"
```
```   363 proof (cases "n = 0")
```
```   364   case True
```
```   365   then show ?thesis by simp
```
```   366 next
```
```   367   case False
```
```   368   from this setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"]
```
```   369   have eq: "(- (1\<Colon>'a)) ^ n = setprod (\<lambda>i. - 1) {0 .. n - 1}"
```
```   370     by auto
```
```   371   from False show ?thesis
```
```   372     by (simp add: pochhammer_def gbinomial_def field_simps
```
```   373       eq setprod_timesf[symmetric] del: minus_one)
```
```   374 qed
```
```   375
```
```   376 lemma binomial_fact_lemma: "k \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n"
```
```   377 proof (induct n arbitrary: k rule: nat_less_induct)
```
```   378   fix n k assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) =
```
```   379                       fact m" and kn: "k \<le> n"
```
```   380   let ?ths = "fact k * fact (n - k) * (n choose k) = fact n"
```
```   381   { assume "n=0" then have ?ths using kn by simp }
```
```   382   moreover
```
```   383   { assume "k=0" then have ?ths using kn by simp }
```
```   384   moreover
```
```   385   { assume nk: "n=k" then have ?ths by simp }
```
```   386   moreover
```
```   387   { fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < m"
```
```   388     from n have mn: "m < n" by arith
```
```   389     from hm have hm': "h \<le> m" by arith
```
```   390     from hm h n kn have km: "k \<le> m" by arith
```
```   391     have "m - h = Suc (m - Suc h)" using  h km hm by arith
```
```   392     with km h have th0: "fact (m - h) = (m - h) * fact (m - k)"
```
```   393       by simp
```
```   394     from n h th0
```
```   395     have "fact k * fact (n - k) * (n choose k) =
```
```   396         k * (fact h * fact (m - h) * (m choose h)) +
```
```   397         (m - h) * (fact k * fact (m - k) * (m choose k))"
```
```   398       by (simp add: field_simps)
```
```   399     also have "\<dots> = (k + (m - h)) * fact m"
```
```   400       using H[rule_format, OF mn hm'] H[rule_format, OF mn km]
```
```   401       by (simp add: field_simps)
```
```   402     finally have ?ths using h n km by simp }
```
```   403   moreover have "n=0 \<or> k = 0 \<or> k = n \<or> (\<exists>m h. n = Suc m \<and> k = Suc h \<and> h < m)"
```
```   404     using kn by presburger
```
```   405   ultimately show ?ths by blast
```
```   406 qed
```
```   407
```
```   408 lemma binomial_fact:
```
```   409   assumes kn: "k \<le> n"
```
```   410   shows "(of_nat (n choose k) :: 'a::field_char_0) =
```
```   411     of_nat (fact n) / (of_nat (fact k) * of_nat (fact (n - k)))"
```
```   412   using binomial_fact_lemma[OF kn]
```
```   413   by (simp add: field_simps of_nat_mult [symmetric])
```
```   414
```
```   415 lemma binomial_gbinomial: "of_nat (n choose k) = of_nat n gchoose k"
```
```   416 proof -
```
```   417   { assume kn: "k > n"
```
```   418     from kn binomial_eq_0[OF kn] have ?thesis
```
```   419       by (simp add: gbinomial_pochhammer field_simps  pochhammer_of_nat_eq_0_iff) }
```
```   420   moreover
```
```   421   { assume "k=0" then have ?thesis by simp }
```
```   422   moreover
```
```   423   { assume kn: "k \<le> n" and k0: "k\<noteq> 0"
```
```   424     from k0 obtain h where h: "k = Suc h" by (cases k) auto
```
```   425     from h
```
```   426     have eq:"(- 1 :: 'a) ^ k = setprod (\<lambda>i. - 1) {0..h}"
```
```   427       by (subst setprod_constant) auto
```
```   428     have eq': "(\<Prod>i\<in>{0..h}. of_nat n + - (of_nat i :: 'a)) = (\<Prod>i\<in>{n - h..n}. of_nat i)"
```
```   429       apply (rule strong_setprod_reindex_cong[where f="op - n"])
```
```   430         using h kn
```
```   431         apply (simp_all add: inj_on_def image_iff Bex_def set_eq_iff)
```
```   432         apply clarsimp
```
```   433         apply presburger
```
```   434        apply presburger
```
```   435       apply (simp add: fun_eq_iff field_simps of_nat_add[symmetric] del: of_nat_add)
```
```   436       done
```
```   437     have th0: "finite {1..n - Suc h}" "finite {n - h .. n}"
```
```   438         "{1..n - Suc h} \<inter> {n - h .. n} = {}" and
```
```   439         eq3: "{1..n - Suc h} \<union> {n - h .. n} = {1..n}"
```
```   440       using h kn by auto
```
```   441     from eq[symmetric]
```
```   442     have ?thesis using kn
```
```   443       apply (simp add: binomial_fact[OF kn, where ?'a = 'a]
```
```   444         gbinomial_pochhammer field_simps pochhammer_Suc_setprod del: minus_one)
```
```   445       apply (simp add: pochhammer_Suc_setprod fact_altdef_nat h
```
```   446         of_nat_setprod setprod_timesf[symmetric] eq' del: One_nat_def power_Suc del: minus_one)
```
```   447       unfolding setprod_Un_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \<Rightarrow> 'a"] eq[unfolded h]
```
```   448       unfolding mult_assoc[symmetric]
```
```   449       unfolding setprod_timesf[symmetric]
```
```   450       apply simp
```
```   451       apply (rule strong_setprod_reindex_cong[where f= "op - n"])
```
```   452         apply (auto simp add: inj_on_def image_iff Bex_def)
```
```   453        apply presburger
```
```   454       apply (subgoal_tac "(of_nat (n - x) :: 'a) = of_nat n - of_nat x")
```
```   455        apply simp
```
```   456       apply (rule of_nat_diff)
```
```   457       apply simp
```
```   458       done
```
```   459   }
```
```   460   moreover
```
```   461   have "k > n \<or> k = 0 \<or> (k \<le> n \<and> k \<noteq> 0)" by arith
```
```   462   ultimately show ?thesis by blast
```
```   463 qed
```
```   464
```
```   465 lemma gbinomial_1[simp]: "a gchoose 1 = a"
```
```   466   by (simp add: gbinomial_def)
```
```   467
```
```   468 lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a"
```
```   469   by (simp add: gbinomial_def)
```
```   470
```
```   471 lemma gbinomial_mult_1:
```
```   472   "a * (a gchoose n) =
```
```   473     of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"  (is "?l = ?r")
```
```   474 proof -
```
```   475   have "?r = ((- 1) ^n * pochhammer (- a) n / of_nat (fact n)) * (of_nat n - (- a + of_nat n))"
```
```   476     unfolding gbinomial_pochhammer
```
```   477       pochhammer_Suc fact_Suc of_nat_mult right_diff_distrib power_Suc
```
```   478     by (simp add:  field_simps del: of_nat_Suc)
```
```   479   also have "\<dots> = ?l" unfolding gbinomial_pochhammer
```
```   480     by (simp add: field_simps)
```
```   481   finally show ?thesis ..
```
```   482 qed
```
```   483
```
```   484 lemma gbinomial_mult_1':
```
```   485     "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
```
```   486   by (simp add: mult_commute gbinomial_mult_1)
```
```   487
```
```   488 lemma gbinomial_Suc:
```
```   489     "a gchoose (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k}) / of_nat (fact (Suc k))"
```
```   490   by (simp add: gbinomial_def)
```
```   491
```
```   492 lemma gbinomial_mult_fact:
```
```   493   "(of_nat (fact (Suc k)) :: 'a) * ((a::'a::field_char_0) gchoose (Suc k)) =
```
```   494     (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
```
```   495   by (simp_all add: gbinomial_Suc field_simps del: fact_Suc)
```
```   496
```
```   497 lemma gbinomial_mult_fact':
```
```   498   "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) =
```
```   499     (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
```
```   500   using gbinomial_mult_fact[of k a]
```
```   501   by (subst mult_commute)
```
```   502
```
```   503
```
```   504 lemma gbinomial_Suc_Suc:
```
```   505   "((a::'a::field_char_0) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
```
```   506 proof (cases k)
```
```   507   case 0
```
```   508   then show ?thesis by simp
```
```   509 next
```
```   510   case (Suc h)
```
```   511   have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)"
```
```   512     apply (rule strong_setprod_reindex_cong[where f = Suc])
```
```   513       using Suc
```
```   514       apply auto
```
```   515     done
```
```   516
```
```   517   have "of_nat (fact (Suc k)) * (a gchoose k + (a gchoose (Suc k))) =
```
```   518     ((a gchoose Suc h) * of_nat (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{0\<Colon>nat..Suc h}. a - of_nat i)"
```
```   519     apply (simp add: Suc field_simps del: fact_Suc)
```
```   520     unfolding gbinomial_mult_fact'
```
```   521     apply (subst fact_Suc)
```
```   522     unfolding of_nat_mult
```
```   523     apply (subst mult_commute)
```
```   524     unfolding mult_assoc
```
```   525     unfolding gbinomial_mult_fact
```
```   526     apply (simp add: field_simps)
```
```   527     done
```
```   528   also have "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 1)"
```
```   529     unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc
```
```   530     by (simp add: field_simps Suc)
```
```   531   also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)"
```
```   532     using eq0
```
```   533     by (simp add: Suc setprod_nat_ivl_1_Suc)
```
```   534   also have "\<dots> = of_nat (fact (Suc k)) * ((a + 1) gchoose (Suc k))"
```
```   535     unfolding gbinomial_mult_fact ..
```
```   536   finally show ?thesis by (simp del: fact_Suc)
```
```   537 qed
```
```   538
```
```   539
```
```   540 lemma binomial_symmetric:
```
```   541   assumes kn: "k \<le> n"
```
```   542   shows "n choose k = n choose (n - k)"
```
```   543 proof-
```
```   544   from kn have kn': "n - k \<le> n" by arith
```
```   545   from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']
```
```   546   have "fact k * fact (n - k) * (n choose k) =
```
```   547     fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" by simp
```
```   548   then show ?thesis using kn by simp
```
```   549 qed
```
```   550
```
```   551 (* Contributed by Manuel Eberl *)
```
```   552 (* Alternative definition of the binomial coefficient as \<Prod>i<k. (n - i) / (k - i) *)
```
```   553 lemma binomial_altdef_of_nat:
```
```   554   fixes n k :: nat
```
```   555     and x :: "'a :: {field_char_0,field_inverse_zero}"
```
```   556   assumes "k \<le> n"
```
```   557   shows "of_nat (n choose k) = (\<Prod>i<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
```
```   558 proof (cases "0 < k")
```
```   559   case True
```
```   560   then have "(of_nat (n choose k) :: 'a) = (\<Prod>i<k. of_nat n - of_nat i) / of_nat (fact k)"
```
```   561     unfolding binomial_gbinomial gbinomial_def
```
```   562     by (auto simp: gr0_conv_Suc lessThan_Suc_atMost atLeast0AtMost)
```
```   563   also have "\<dots> = (\<Prod>i<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
```
```   564     using `k \<le> n` unfolding fact_eq_rev_setprod_nat of_nat_setprod
```
```   565     by (auto simp add: setprod_dividef intro!: setprod_cong of_nat_diff[symmetric])
```
```   566   finally show ?thesis .
```
```   567 next
```
```   568   case False
```
```   569   then show ?thesis by simp
```
```   570 qed
```
```   571
```
```   572 lemma binomial_ge_n_over_k_pow_k:
```
```   573   fixes k n :: nat
```
```   574     and x :: "'a :: linordered_field_inverse_zero"
```
```   575   assumes "0 < k"
```
```   576     and "k \<le> n"
```
```   577   shows "(of_nat n / of_nat k :: 'a) ^ k \<le> of_nat (n choose k)"
```
```   578 proof -
```
```   579   have "(of_nat n / of_nat k :: 'a) ^ k = (\<Prod>i<k. of_nat n / of_nat k :: 'a)"
```
```   580     by (simp add: setprod_constant)
```
```   581   also have "\<dots> \<le> of_nat (n choose k)"
```
```   582     unfolding binomial_altdef_of_nat[OF `k\<le>n`]
```
```   583   proof (safe intro!: setprod_mono)
```
```   584     fix i :: nat
```
```   585     assume  "i < k"
```
```   586     from assms have "n * i \<ge> i * k" by simp
```
```   587     then have "n * k - n * i \<le> n * k - i * k" by arith
```
```   588     then have "n * (k - i) \<le> (n - i) * k"
```
```   589       by (simp add: diff_mult_distrib2 nat_mult_commute)
```
```   590     then have "of_nat n * of_nat (k - i) \<le> of_nat (n - i) * (of_nat k :: 'a)"
```
```   591       unfolding of_nat_mult[symmetric] of_nat_le_iff .
```
```   592     with assms show "of_nat n / of_nat k \<le> of_nat (n - i) / (of_nat (k - i) :: 'a)"
```
```   593       using `i < k` by (simp add: field_simps)
```
```   594   qed (simp add: zero_le_divide_iff)
```
```   595   finally show ?thesis .
```
```   596 qed
```
```   597
```
```   598 lemma binomial_le_pow:
```
```   599   assumes "r \<le> n"
```
```   600   shows "n choose r \<le> n ^ r"
```
```   601 proof -
```
```   602   have "n choose r \<le> fact n div fact (n - r)"
```
```   603     using `r \<le> n` by (subst binomial_fact_lemma[symmetric]) auto
```
```   604   with fact_div_fact_le_pow [OF assms] show ?thesis by auto
```
```   605 qed
```
```   606
```
```   607 lemma binomial_altdef_nat: "(k::nat) \<le> n \<Longrightarrow>
```
```   608     n choose k = fact n div (fact k * fact (n - k))"
```
```   609  by (subst binomial_fact_lemma [symmetric]) auto
```
```   610
```
```   611 end
```