# Theory Binomial

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theory Binomial
imports Complex_Main
`(*  Title:      HOL/Library/Binomial.thy    Author:     Lawrence C Paulson, Amine Chaieb    Copyright   1997  University of Cambridge*)header {* Binomial Coefficients *}theory Binomialimports Complex_Mainbegintext {* This development is based on the work of Andy Gordon and  Florian Kammueller. *}primrec binomial :: "nat => nat => 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_alllemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"  by simplemma binomial_Suc_Suc [simp]:  "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"  by simplemma binomial_eq_0: "!!k. n < k ==> (n choose k) = 0"  by (induct n) autodeclare 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_alllemma binomial_1 [simp]: "(n choose Suc 0) = n"  by (induct n) simp_alllemma zero_less_binomial: "k ≤ n ==> (n choose k) > 0"  by (induct n k rule: diff_induct) simp_alllemma 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)  donelemma zero_less_binomial_iff: "(n choose k > 0) = (k≤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 ≤ 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)  donetext{*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 ≤ 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 ≤ 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)  donesubsection {* 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 ⊆ A & card B = 0} = 1"  by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])lemma choose_deconstruct: "finite M ==> x ∉ 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 ∉ 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 simplemma Collect_ex_eqlemma "{x. EX y. P x y} = (UN y. {x. P x y})"apply blast*)lemma finite_bex_subset[simp]:  "finite B ==> (!!A. A<=B ==> finite{x. P x A}) ==> 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  donetext{*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 ∉ 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)  donetext {*  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])  donetheorem 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 = (∑k=0..n. (n choose k) * a^k * b^(n-k))"proof (induct n)  case 0 thus ?case by simpnext  case (Suc n)  have decomp: "{0..n+1} = {0} ∪ {n+1} ∪ {1..n}"    by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)  have decomp2: "{0..n} = {0} ∪ {1..n}"    by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)  have "(a+b::nat)^(n+1) = (a+b) * (∑k=0..n. (n choose k) * a^k * b^(n-k))"    using Suc by simp  also have "… =  a*(∑k=0..n. (n choose k) * a^k * b^(n-k)) +                   b*(∑k=0..n. (n choose k) * a^k * b^(n-k))"    by (rule nat_distrib)  also have "… = (∑k=0..n. (n choose k) * a^(k+1) * b^(n-k)) +                  (∑k=0..n. (n choose k) * a^k * b^(n-k+1))"    by (simp add: setsum_right_distrib mult_ac)  also have "… = (∑k=0..n. (n choose k) * a^k * b^(n+1-k)) +                  (∑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 "… = a^(n+1) + b^(n+1) +                  (∑k=1..n. (n choose (k - 1)) * a^k * b^(n+1-k)) +                  (∑k=1..n. (n choose k) * a^k * b^(n+1-k))"    by (simp add: decomp2)  also have      "… = a^(n+1) + b^(n+1) + (∑k=1..n. (n+1 choose k) * a^k * b^(n+1-k))"    by (simp add: nat_distrib setsum_addf binomial.simps)  also have "… = (∑k=0..n+1. (n+1 choose k) * a^k * b^(n+1-k))"    using decomp by simp  finally show ?case by simpqedsubsection{* Pochhammer's symbol : generalized raising factorial*}definition "pochhammer (a::'a::comm_semiring_1) n = (if n = 0 then 1 else setprod (λn. 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 (λn. 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} ∪ {Suc n}" by auto  show ?thesis unfolding eq by (simp add: field_simps)qedlemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}"proof-  have eq: "{0..Suc n} = {0} ∪ {1 .. Suc n}" by auto  show ?thesis unfolding eq by simpqedlemma 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) autoqedlemma 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 ≠ 0"    have th0: "finite {1 .. n}" "0 ∉ {1 .. n}" by auto    have eq: "insert 0 {1 .. n} = {0..n}" by auto    have th1: "(∏n∈{1::nat..n}. a + of_nat n) =      (∏n∈{0::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 blastqedlemma pochhammer_fact: "of_nat (fact n) = pochhammer 1 n"  unfolding fact_altdef_nat  apply (cases n)   apply (simp_all add: of_nat_setprod pochhammer_Suc_setprod)  apply (rule setprod_reindex_cong[where f=Suc])    apply (auto simp add: fun_eq_iff)  donelemma 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 ≠ 0"    then have ?thesis      apply (simp add: h pochhammer_Suc_setprod)      apply (rule_tac x="n" in bexI)      using h kn      apply auto      done }  ultimately show ?thesis by blastqedlemma pochhammer_of_nat_eq_0_lemma': assumes kn: "k ≤ n"  shows "pochhammer (- (of_nat n :: 'a:: {idom, ring_char_0})) k ≠ 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) autoqedlemma pochhammer_of_nat_eq_0_iff:  shows "pochhammer (- (of_nat n :: 'a:: {idom, ring_char_0})) k = 0 <-> 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) <-> (EX k < n . a = - of_nat k) "  apply (auto simp add: pochhammer_of_nat_eq_0_iff)  apply (cases n)   apply (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  donelemma pochhammer_eq_0_mono:  "pochhammer a n = (0::'a::field_char_0) ==> m ≥ n ==> pochhammer a m = 0"  unfolding pochhammer_eq_0_iff by autolemma pochhammer_neq_0_mono:  "pochhammer a m ≠ (0::'a::field_char_0) ==> m ≥ n ==> pochhammer a n ≠ 0"  unfolding pochhammer_eq_0_iff by autolemma pochhammer_minus:  assumes kn: "k ≤ 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)      apply (auto simp add: fun_eq_iff h of_nat_diff)      done }  ultimately show ?thesis by (cases k) autoqedlemma pochhammer_minus':  assumes kn: "k ≤ 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  donelemma 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 => nat => 'a" (infixl "gchoose" 65)  where "a gchoose n =    (if n = 0 then 1 else (setprod (λi. 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 "(∏i::nat∈{0::nat..n}. - of_nat i) = (0::'b)")   apply (simp del:setprod_zero_iff)  apply simp  donelemma 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≠0"    from n0 setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"]    have eq: "(- (1::'a)) ^ n = setprod (λi. - 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 blastqedlemma binomial_fact_lemma: "k ≤ n ==> fact k * fact (n - k) * (n choose k) = fact n"proof (induct n arbitrary: k rule: nat_less_induct)  fix n k assume H: "∀m<n. ∀x≤m. fact x * fact (m - x) * (m choose x) =                      fact m" and kn: "k ≤ 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 ≤ m" by arith    from hm h n kn have km: "k ≤ 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 "… = (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 ∨ k = 0 ∨ k = n ∨ (EX m h. n=Suc m ∧ k = Suc h ∧ h < m)"    using kn by presburger  ultimately show ?ths by blastqedlemma binomial_fact:  assumes kn: "k ≤ 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 ≤ n" and k0: "k≠ 0"    from k0 obtain h where h: "k = Suc h" by (cases k) auto    from h    have eq:"(- 1 :: 'a) ^ k = setprod (λi. - 1) {0..h}"      by (subst setprod_constant, auto)    have eq': "(∏i∈{0..h}. of_nat n + - (of_nat i :: 'a)) = (∏i∈{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      apply (simp add: fun_eq_iff field_simps of_nat_add[symmetric] del: of_nat_add)      done    have th0: "finite {1..n - Suc h}" "finite {n - h .. n}"        "{1..n - Suc h} ∩ {n - h .. n} = {}" and        eq3: "{1..n - Suc h} ∪ {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 => '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      apply (rule of_nat_diff)      apply simp      done  }  moreover  have "k > n ∨ k = 0 ∨ (k ≤ n ∧ k ≠ 0)" by arith  ultimately show ?thesis by blastqedlemma 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 "… = ?l" unfolding gbinomial_pochhammer    by (simp add: field_simps)  finally show ?thesis ..qedlemma 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 (λi. 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 (λi. a - of_nat i) {0 .. k})"  by (simp_all add: gbinomial_Suc field_simps del: fact_Suc)lemma gbinomial_mult_fact':  "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) =    (setprod (λi. a - of_nat i) {0 .. k})"  using gbinomial_mult_fact[of k a]  apply (subst mult_commute)  apply assumption  donelemma 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: "(∏i∈{1..k}. (a + 1) - of_nat i) = (∏i∈{0..h}. a - of_nat i)"      apply (rule strong_setprod_reindex_cong[where f = Suc])        using h        apply auto      done    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)) + (∏i∈{0::nat..Suc h}. a - of_nat i)"      apply (simp add: h 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      apply (simp add: field_simps)      done    also have "… = (∏i∈{0..h}. a - of_nat i) * (a + 1)"      unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc      by (simp add: field_simps h)    also have "… = (∏i∈{0..k}. (a + 1) - of_nat i)"      using eq0      by (simp add: h setprod_nat_ivl_1_Suc)    also have "… = 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) autoqedlemma binomial_symmetric:  assumes kn: "k ≤ n"  shows "n choose k = n choose (n - k)"proof-  from kn have kn': "n - k ≤ 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 simpqed(* Contributed by Manuel Eberl *)(* Alternative definition of the binomial coefficient as ∏i<k. (n - i) / (k - i) *)lemma binomial_altdef_of_nat:  fixes n k :: nat and x :: "'a :: {field_char_0, field_inverse_zero}"  assumes "k ≤ n" shows "of_nat (n choose k) = (∏i<k. of_nat (n - i) / of_nat (k - i) :: 'a)"proof cases  assume "0 < k"  then have "(of_nat (n choose k) :: 'a) = (∏i<k. of_nat n - of_nat i) / of_nat (fact k)"    unfolding binomial_gbinomial gbinomial_def    by (auto simp: gr0_conv_Suc lessThan_Suc_atMost atLeast0AtMost)  also have "… = (∏i<k. of_nat (n - i) / of_nat (k - i) :: 'a)"    using `k ≤ n` unfolding fact_eq_rev_setprod_nat of_nat_setprod    by (auto simp add: setprod_dividef intro!: setprod_cong of_nat_diff[symmetric])  finally show ?thesis .qed simplemma binomial_ge_n_over_k_pow_k:  fixes k n :: nat and x :: "'a :: linordered_field_inverse_zero"  assumes "0 < k" and "k ≤ n" shows "(of_nat n / of_nat k :: 'a) ^ k ≤ of_nat (n choose k)"proof -  have "(of_nat n / of_nat k :: 'a) ^ k = (∏i<k. of_nat n / of_nat k :: 'a)"    by (simp add: setprod_constant)  also have "… ≤ of_nat (n choose k)"    unfolding binomial_altdef_of_nat[OF `k≤n`]  proof (safe intro!: setprod_mono)    fix i::nat  assume  "i < k"    from assms have "n * i ≥ i * k" by simp    hence "n * k - n * i ≤ n * k - i * k" by arith    hence "n * (k - i) ≤ (n - i) * k"      by (simp add: diff_mult_distrib2 nat_mult_commute)    hence "of_nat n * of_nat (k - i) ≤ of_nat (n - i) * (of_nat k :: 'a)"      unfolding of_nat_mult[symmetric] of_nat_le_iff .    with assms show "of_nat n / of_nat k ≤ of_nat (n - i) / (of_nat (k - i) :: 'a)"      using `i < k` by (simp add: field_simps)  qed (simp add: zero_le_divide_iff)  finally show ?thesis .qedlemma binomial_le_pow:  assumes "r ≤ n" shows "n choose r ≤ n ^ r"proof -  have "n choose r ≤ fact n div fact (n - r)"    using `r ≤ n` by (subst binomial_fact_lemma[symmetric]) auto  with fact_div_fact_le_pow[OF assms] show ?thesis by autoqedlemma binomial_altdef_nat: "(k::nat) ≤ n ==>    n choose k = fact n div (fact k * fact (n - k))" by (subst binomial_fact_lemma[symmetric]) autoend`