(*  Title:      HOL/Library/Binomial.thy

    Author:     Lawrence C Paulson, Amine Chaieb

    Copyright   1997  University of Cambridge

*)

header {* Binomial Coefficients *}

theory Binomial

imports Complex_Main

begin

text {* This development is based on the work of Andy Gordon and

  Florian Kammueller. *}

primrec binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65) where

  binomial_0: "(0 choose k) = (if k = 0 then 1 else 0)"

  | binomial_Suc: "(Suc n choose k) =

                 (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"

lemma binomial_n_0 [simp]: "(n choose 0) = 1"

by (cases n) simp_all

lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"

by simp

lemma binomial_Suc_Suc [simp]:

  "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"

by simp

lemma binomial_eq_0: "!!k. n < k ==> (n choose k) = 0"

by (induct n) auto

declare binomial_0 [simp del] binomial_Suc [simp del]

lemma binomial_n_n [simp]: "(n choose n) = 1"

by (induct n) (simp_all add: binomial_eq_0)

lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n"

by (induct n) simp_all

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

by (induct n) simp_all

lemma zero_less_binomial: "k \<le> n ==> (n choose k) > 0"

by (induct n k rule: diff_induct) simp_all

lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)"

apply (safe intro!: binomial_eq_0)

apply (erule contrapos_pp)

apply (simp add: zero_less_binomial)

done

lemma zero_less_binomial_iff: "(n choose k > 0) = (k\<le>n)"

by(simp add: linorder_not_less binomial_eq_0_iff neq0_conv[symmetric]

        del:neq0_conv)

(*Might be more useful if re-oriented*)

lemma Suc_times_binomial_eq:

  "!!k. k \<le> n ==> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"

apply (induct n)

apply (simp add: binomial_0)

apply (case_tac k)

apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq

    binomial_eq_0)

done

text{*This is the well-known version, but it's harder to use because of the

  need to reason about division.*}

lemma binomial_Suc_Suc_eq_times:

    "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"

  by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right)

text{*Another version, with -1 instead of Suc.*}

lemma times_binomial_minus1_eq:

    "[|k \<le> n;  0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"

  apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)

  apply (simp split add: nat_diff_split, auto)

  done

subsection {* Theorems about @{text "choose"} *}

text {*

  \medskip Basic theorem about @{text "choose"}.  By Florian

  Kamm\"uller, tidied by LCP.

*}

lemma card_s_0_eq_empty:

    "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"

by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])

lemma choose_deconstruct: "finite M ==> x \<notin> M

  ==> {s. s <= insert x M & card(s) = Suc k}

       = {s. s <= M & card(s) = Suc k} Un

         {s. EX t. t <= M & card(t) = k & s = insert x t}"

  apply safe

   apply (auto intro: finite_subset [THEN card_insert_disjoint])

  apply (drule_tac x = "xa - {x}" in spec)

  apply (subgoal_tac "x \<notin> xa", auto)

  apply (erule rev_mp, subst card_Diff_singleton)

  apply (auto intro: finite_subset)

  done

(*

lemma "finite(UN y. {x. P x y})"

apply simp

lemma Collect_ex_eq

lemma "{x. EX y. P x y} = (UN y. {x. P x y})"

apply blast

*)

lemma finite_bex_subset[simp]:

  "finite B \<Longrightarrow> (!!A. A<=B \<Longrightarrow> finite{x. P x A}) \<Longrightarrow> finite{x. EX A<=B. P x A}"

apply(subgoal_tac "{x. EX A<=B. P x A} = (UN A:Pow B. {x. P x A})")

 apply simp

apply blast

done

text{*There are as many subsets of @{term A} having cardinality @{term k}

 as there are sets obtained from the former by inserting a fixed element

 @{term x} into each.*}

lemma constr_bij:

   "[|finite A; x \<notin> A|] ==>

    card {B. EX C. C <= A & card(C) = k & B = insert x C} =

    card {B. B <= A & card(B) = k}"

apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)

     apply (auto elim!: equalityE simp add: inj_on_def)

apply (subst Diff_insert0, auto)

done

text {*

  Main theorem: combinatorial statement about number of subsets of a set.

*}

lemma n_sub_lemma:

    "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"

  apply (induct k)

   apply (simp add: card_s_0_eq_empty, atomize)

  apply (rotate_tac -1, erule finite_induct)

   apply (simp_all (no_asm_simp) cong add: conj_cong

     add: card_s_0_eq_empty choose_deconstruct)

  apply (subst card_Un_disjoint)

     prefer 4 apply (force simp add: constr_bij)

    prefer 3 apply force

   prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]

     finite_subset [of _ "Pow (insert x F)", standard])

  apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])

  done

theorem n_subsets:

    "finite A ==> card {B. B <= A & card B = k} = (card A choose k)"

  by (simp add: n_sub_lemma)

text{* The binomial theorem (courtesy of Tobias Nipkow): *}

theorem binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"

proof (induct n)

  case 0 thus ?case by simp

next

  case (Suc n)

  have decomp: "{0..n+1} = {0} \<union> {n+1} \<union> {1..n}"

    by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)

  have decomp2: "{0..n} = {0} \<union> {1..n}"

    by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)

  have "(a+b::nat)^(n+1) = (a+b) * (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"

    using Suc by simp

  also have "\<dots> =  a*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k)) +

                   b*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"

    by (rule nat_distrib)

  also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^(k+1) * b^(n-k)) +

                  (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k+1))"

    by (simp add: setsum_right_distrib mult_ac)

  also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^k * b^(n+1-k)) +

                  (\<Sum>k=1..n+1. (n choose (k - 1)) * a^k * b^(n+1-k))"

    by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le

             del:setsum_cl_ivl_Suc)

  also have "\<dots> = a^(n+1) + b^(n+1) +

                  (\<Sum>k=1..n. (n choose (k - 1)) * a^k * b^(n+1-k)) +

                  (\<Sum>k=1..n. (n choose k) * a^k * b^(n+1-k))"

    by (simp add: decomp2)

  also have

      "\<dots> = a^(n+1) + b^(n+1) + (\<Sum>k=1..n. (n+1 choose k) * a^k * b^(n+1-k))"

    by (simp add: nat_distrib setsum_addf binomial.simps)

  also have "\<dots> = (\<Sum>k=0..n+1. (n+1 choose k) * a^k * b^(n+1-k))"

    using decomp by simp

  finally show ?case by simp

qed

subsection{* Pochhammer's symbol : generalized raising factorial*}

definition "pochhammer (a::'a::comm_semiring_1) n = (if n = 0 then 1 else setprod (\<lambda>n. a + of_nat n) {0 .. n - 1})"

lemma pochhammer_0[simp]: "pochhammer a 0 = 1" 

  by (simp add: pochhammer_def)

lemma pochhammer_1[simp]: "pochhammer a 1 = a" by (simp add: pochhammer_def)

lemma pochhammer_Suc0[simp]: "pochhammer a (Suc 0) = a" 

  by (simp add: pochhammer_def)

lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\<lambda>n. a + of_nat n) {0 .. n}"

  by (simp add: pochhammer_def)

lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc n)"

proof-

  have eq: "{0..Suc n} = {0..n} \<union> {Suc n}" by auto

  show ?thesis unfolding eq by (simp add: field_simps)

qed

lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}"

proof-

  have eq: "{0..Suc n} = {0} \<union> {1 .. Suc n}" by auto

  show ?thesis unfolding eq by simp

qed

lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"

proof-

  {assume "n=0" then have ?thesis by simp}

  moreover

  {fix m assume m: "n = Suc m"

    have ?thesis unfolding m pochhammer_Suc_setprod setprod_nat_ivl_Suc ..}

  ultimately show ?thesis by (cases n, auto)

qed 

lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n"

proof-

  {assume "n=0" then have ?thesis by (simp add: pochhammer_Suc_setprod)}

  moreover

  {assume n0: "n \<noteq> 0"

    have th0: "finite {1 .. n}" "0 \<notin> {1 .. n}" by auto

    have eq: "insert 0 {1 .. n} = {0..n}" by auto

    have th1: "(\<Prod>n\<in>{1\<Colon>nat..n}. a + of_nat n) =

      (\<Prod>n\<in>{0\<Colon>nat..n - 1}. a + 1 + of_nat n)"

      apply (rule setprod_reindex_cong [where f = Suc])

      using n0 by (auto simp add: fun_eq_iff field_simps)

    have ?thesis apply (simp add: pochhammer_def)

    unfolding setprod_insert[OF th0, unfolded eq]

    using th1 by (simp add: field_simps)}

ultimately show ?thesis by blast

qed

lemma pochhammer_fact: "of_nat (fact n) = pochhammer 1 n"

  unfolding fact_altdef_nat

  apply (cases n, simp_all add: of_nat_setprod pochhammer_Suc_setprod)

  apply (rule setprod_reindex_cong[where f=Suc])

  by (auto simp add: fun_eq_iff)

lemma pochhammer_of_nat_eq_0_lemma: assumes kn: "k > n"

  shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0"

proof-

  from kn obtain h where h: "k = Suc h" by (cases k, auto)

  {assume n0: "n=0" then have ?thesis using kn 

      by (cases k) (simp_all add: pochhammer_rec)}

  moreover

  {assume n0: "n \<noteq> 0"

    then have ?thesis apply (simp add: h pochhammer_Suc_setprod)

  apply (rule_tac x="n" in bexI)

  using h kn by auto}

ultimately show ?thesis by blast

qed

lemma pochhammer_of_nat_eq_0_lemma': assumes kn: "k \<le> n"

  shows "pochhammer (- (of_nat n :: 'a:: {idom, ring_char_0})) k \<noteq> 0"

proof-

  {assume "k=0" then have ?thesis by simp}

  moreover

  {fix h assume h: "k = Suc h"

    then have ?thesis apply (simp add: pochhammer_Suc_setprod)

      using h kn by (auto simp add: algebra_simps)}

  ultimately show ?thesis by (cases k, auto)

qed

lemma pochhammer_of_nat_eq_0_iff: 

  shows "pochhammer (- (of_nat n :: 'a:: {idom, ring_char_0})) k = 0 \<longleftrightarrow> k > n"

  (is "?l = ?r")

  using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a] 

    pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a]

  by (auto simp add: not_le[symmetric])

lemma pochhammer_eq_0_iff: 

  "pochhammer a n = (0::'a::field_char_0) \<longleftrightarrow> (EX k < n . a = - of_nat k) "

  apply (auto simp add: pochhammer_of_nat_eq_0_iff)

  apply (cases n, auto simp add: pochhammer_def algebra_simps group_add_class.eq_neg_iff_add_eq_0)

  apply (rule_tac x=x in exI)

  apply auto

  done

lemma pochhammer_eq_0_mono: 

  "pochhammer a n = (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a m = 0"

  unfolding pochhammer_eq_0_iff by auto 

lemma pochhammer_neq_0_mono: 

  "pochhammer a m \<noteq> (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a n \<noteq> 0"

  unfolding pochhammer_eq_0_iff by auto 

lemma pochhammer_minus:

  assumes kn: "k \<le> n" 

  shows "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k"

proof-

  {assume k0: "k = 0" then have ?thesis by simp}

  moreover 

  {fix h assume h: "k = Suc h"

    have eq: "((- 1) ^ Suc h :: 'a) = setprod (%i. - 1) {0 .. h}"

      using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"]

      by auto

    have ?thesis

      unfolding h pochhammer_Suc_setprod eq setprod_timesf[symmetric]

      apply (rule strong_setprod_reindex_cong[where f = "%i. h - i"])

      apply (auto simp add: inj_on_def image_def h )

      apply (rule_tac x="h - x" in bexI)

      by (auto simp add: fun_eq_iff h of_nat_diff)}

  ultimately show ?thesis by (cases k, auto)

qed

lemma pochhammer_minus':

  assumes kn: "k \<le> n" 

  shows "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"

  unfolding pochhammer_minus[OF kn, where b=b]

  unfolding mult_assoc[symmetric]

  unfolding power_add[symmetric]

  apply simp

  done

lemma pochhammer_same: "pochhammer (- of_nat n) n = ((- 1) ^ n :: 'a::comm_ring_1) * of_nat (fact n)"

  unfolding pochhammer_minus[OF le_refl[of n]]

  by (simp add: of_nat_diff pochhammer_fact)

subsection{* Generalized binomial coefficients *}

definition gbinomial :: "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)

  where "a gchoose n = (if n = 0 then 1 else (setprod (\<lambda>i. a - of_nat i) {0 .. n - 1}) / of_nat (fact n))"

lemma gbinomial_0[simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0"

apply (simp_all add: gbinomial_def)

apply (subgoal_tac "(\<Prod>i\<Colon>nat\<in>{0\<Colon>nat..n}. - of_nat i) = (0::'b)")

 apply (simp del:setprod_zero_iff)

apply simp

done

lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / of_nat (fact n)"

proof-

  {assume "n=0" then have ?thesis by simp}

  moreover

  {assume n0: "n\<noteq>0"

    from n0 setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"]

    have eq: "(- (1\<Colon>'a)) ^ n = setprod (\<lambda>i. - 1) {0 .. n - 1}"

      by auto

    from n0 have ?thesis 

      by (simp add: pochhammer_def gbinomial_def field_simps eq setprod_timesf[symmetric] del: minus_one) (* FIXME: del: minus_one *)}

  ultimately show ?thesis by blast

qed

lemma binomial_fact_lemma:

  "k \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n"

proof(induct n arbitrary: k rule: nat_less_induct)

  fix n k assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) =

                      fact m" and kn: "k \<le> n"

    let ?ths = "fact k * fact (n - k) * (n choose k) = fact n"

  {assume "n=0" then have ?ths using kn by simp}

  moreover

  {assume "k=0" then have ?ths using kn by simp}

  moreover

  {assume nk: "n=k" then have ?ths by simp}

  moreover

  {fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < m"

    from n have mn: "m < n" by arith

    from hm have hm': "h \<le> m" by arith

    from hm h n kn have km: "k \<le> m" by arith

    have "m - h = Suc (m - Suc h)" using  h km hm by arith 

    with km h have th0: "fact (m - h) = (m - h) * fact (m - k)"

      by simp

    from n h th0 

    have "fact k * fact (n - k) * (n choose k) = k * (fact h * fact (m - h) * (m choose h)) +  (m - h) * (fact k * fact (m - k) * (m choose k))"

      by (simp add: field_simps)

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

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

      by (simp add: field_simps)

    finally have ?ths using h n km by simp}

  moreover have "n=0 \<or> k = 0 \<or> k = n \<or> (EX m h. n=Suc m \<and> k = Suc h \<and> h < m)" using kn by presburger

  ultimately show ?ths by blast

qed

lemma binomial_fact: 

  assumes kn: "k \<le> n" 

  shows "(of_nat (n choose k) :: 'a::field_char_0) = of_nat (fact n) / (of_nat (fact k) * of_nat (fact (n - k)))"

  using binomial_fact_lemma[OF kn]

  by (simp add: field_simps of_nat_mult [symmetric])

lemma binomial_gbinomial: "of_nat (n choose k) = of_nat n gchoose k"

proof-

  {assume kn: "k > n" 

    from kn binomial_eq_0[OF kn] have ?thesis 

      by (simp add: gbinomial_pochhammer field_simps

        pochhammer_of_nat_eq_0_iff)}

  moreover

  {assume "k=0" then have ?thesis by simp}

  moreover

  {assume kn: "k \<le> n" and k0: "k\<noteq> 0"

    from k0 obtain h where h: "k = Suc h" by (cases k, auto)

    from h

    have eq:"(- 1 :: 'a) ^ k = setprod (\<lambda>i. - 1) {0..h}"

      by (subst setprod_constant, auto)

    have eq': "(\<Prod>i\<in>{0..h}. of_nat n + - (of_nat i :: 'a)) = (\<Prod>i\<in>{n - h..n}. of_nat i)"

      apply (rule strong_setprod_reindex_cong[where f="op - n"])

      using h kn 

      apply (simp_all add: inj_on_def image_iff Bex_def set_eq_iff)

      apply clarsimp

      apply (presburger)

      apply presburger

      by (simp add: fun_eq_iff field_simps of_nat_add[symmetric] del: of_nat_add)

    have th0: "finite {1..n - Suc h}" "finite {n - h .. n}" 

"{1..n - Suc h} \<inter> {n - h .. n} = {}" and eq3: "{1..n - Suc h} \<union> {n - h .. n} = {1..n}" using h kn by auto

    from eq[symmetric]

    have ?thesis using kn

      apply (simp add: binomial_fact[OF kn, where ?'a = 'a] 

        gbinomial_pochhammer field_simps pochhammer_Suc_setprod del: minus_one)

      apply (simp add: pochhammer_Suc_setprod fact_altdef_nat h of_nat_setprod setprod_timesf[symmetric] eq' del: One_nat_def power_Suc del: minus_one)

      unfolding setprod_Un_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \<Rightarrow> 'a"] eq[unfolded h]

      unfolding mult_assoc[symmetric] 

      unfolding setprod_timesf[symmetric]

      apply simp

      apply (rule strong_setprod_reindex_cong[where f= "op - n"])

      apply (auto simp add: inj_on_def image_iff Bex_def)

      apply presburger

      apply (subgoal_tac "(of_nat (n - x) :: 'a) = of_nat n - of_nat x")

      apply simp

      by (rule of_nat_diff, simp)

  }

  moreover

  have "k > n \<or> k = 0 \<or> (k \<le> n \<and> k \<noteq> 0)" by arith

  ultimately show ?thesis by blast

qed

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

  by (simp add: gbinomial_def)

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

  by (simp add: gbinomial_def)

lemma gbinomial_mult_1: "a * (a gchoose n) = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))" (is "?l = ?r")

proof-

  have "?r = ((- 1) ^n * pochhammer (- a) n / of_nat (fact n)) * (of_nat n - (- a + of_nat n))"

    unfolding gbinomial_pochhammer

    pochhammer_Suc fact_Suc of_nat_mult right_diff_distrib power_Suc

    by (simp add:  field_simps del: of_nat_Suc)

  also have "\<dots> = ?l" unfolding gbinomial_pochhammer

    by (simp add: field_simps)

  finally show ?thesis ..

qed

lemma gbinomial_mult_1': "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"

  by (simp add: mult_commute gbinomial_mult_1)

lemma gbinomial_Suc: "a gchoose (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k}) / of_nat (fact (Suc k))"

  by (simp add: gbinomial_def)

lemma gbinomial_mult_fact:

  "(of_nat (fact (Suc k)) :: 'a) * ((a::'a::field_char_0) gchoose (Suc k)) = (setprod (\<lambda>i. a - of_nat i) {0 .. k})"

  unfolding gbinomial_Suc

  by (simp_all add: field_simps del: fact_Suc)

lemma gbinomial_mult_fact':

  "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) = (setprod (\<lambda>i. a - of_nat i) {0 .. k})"

  using gbinomial_mult_fact[of k a]

  apply (subst mult_commute) .

lemma gbinomial_Suc_Suc: "((a::'a::field_char_0) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"

proof-

  {assume "k = 0" then have ?thesis by simp}

  moreover

  {fix h assume h: "k = Suc h"

   have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)"

     apply (rule strong_setprod_reindex_cong[where f = Suc])

     using h by auto

    have "of_nat (fact (Suc k)) * (a gchoose k + (a gchoose (Suc k))) = ((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)" 

      unfolding h

      apply (simp add: field_simps del: fact_Suc)

      unfolding gbinomial_mult_fact'

      apply (subst fact_Suc)

      unfolding of_nat_mult 

      apply (subst mult_commute)

      unfolding mult_assoc

      unfolding gbinomial_mult_fact

      by (simp add: field_simps)

    also have "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 1)"

      unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc

      by (simp add: field_simps h)

    also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)"

      using eq0

      unfolding h  setprod_nat_ivl_1_Suc

      by simp

    also have "\<dots> = of_nat (fact (Suc k)) * ((a + 1) gchoose (Suc k))"

      unfolding gbinomial_mult_fact ..

    finally have ?thesis by (simp del: fact_Suc) }

  ultimately show ?thesis by (cases k, auto)

qed

lemma binomial_symmetric: assumes kn: "k \<le> n" 

  shows "n choose k = n choose (n - k)"

proof-

  from kn have kn': "n - k \<le> n" by arith

  from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']

  have "fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" by simp

  then show ?thesis using kn by simp

qed

end