diff -r 39bcdf19dd14 -r 608c157d743d src/HOL/Library/Binomial.thy
--- a/src/HOL/Library/Binomial.thy	Mon Jan 27 17:13:33 2014 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,606 +0,0 @@
-(*  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. *}
-
-subsection {* Basic definitions and lemmas *}
-
-primrec binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65)
-where
-  "0 choose k = (if k = 0 then 1 else 0)"
-| "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 choose_reduce_nat: 
-  "0 < (n::nat) \<Longrightarrow> 0 < k \<Longrightarrow>
-    (n choose k) = ((n - 1) choose k) + ((n - 1) choose (k - 1))"
-  by (metis Suc_diff_1 binomial.simps(2) nat_add_commute neq0_conv)
-
-lemma binomial_eq_0: "n < k \<Longrightarrow> n choose k = 0"
-  by (induct n arbitrary: k) auto
-
-declare binomial.simps [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 \<Longrightarrow> n choose k > 0"
-  by (induct n k rule: diff_induct) simp_all
-
-lemma binomial_eq_0_iff [simp]: "n choose k = 0 \<longleftrightarrow> n < k"
-  by (metis binomial_eq_0 less_numeral_extra(3) not_less zero_less_binomial)
-
-lemma zero_less_binomial_iff [simp]: "n choose k > 0 \<longleftrightarrow> k \<le> n"
-  by (metis binomial_eq_0_iff not_less0 not_less zero_less_binomial)
-
-(*Might be more useful if re-oriented*)
-lemma Suc_times_binomial_eq:
-  "k \<le> n \<Longrightarrow> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
-  apply (induct n arbitrary: k)
-   apply (simp add: binomial.simps)
-   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 \<Longrightarrow> (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 \<Longrightarrow> 0 < k \<Longrightarrow> (n choose k) * k = n * ((n - 1) choose (k - 1))"
-  using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 1"]
-  by (auto split add: nat_diff_split)
-
-
-subsection {* Combinatorial theorems involving @{text "choose"} *}
-
-text {*By Florian Kamm\"uller, tidied by LCP.*}
-
-lemma card_s_0_eq_empty: "finite A \<Longrightarrow> 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 \<Longrightarrow> x \<notin> M \<Longrightarrow>
-    {s. s \<subseteq> insert x M \<and> card s = Suc k} =
-    {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}"
-  apply safe
-     apply (auto intro: finite_subset [THEN card_insert_disjoint])
-  by (metis (full_types) Diff_insert_absorb Set.set_insert Zero_neq_Suc card_Diff_singleton_if 
-     card_eq_0_iff diff_Suc_1 in_mono subset_insert_iff)
-
-lemma finite_bex_subset [simp]:
-  assumes "finite B"
-    and "\<And>A. A \<subseteq> B \<Longrightarrow> finite {x. P x A}"
-  shows "finite {x. \<exists>A \<subseteq> B. P x A}"
-  by (metis (no_types) assms finite_Collect_bounded_ex finite_Collect_subsets)
-
-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 \<Longrightarrow> x \<notin> A \<Longrightarrow>
-    card {B. \<exists>C. C \<subseteq> A \<and> card C = k \<and> B = insert x C} =
-    card {B. B \<subseteq> A & card(B) = k}"
-  apply (rule card_bij_eq [where f = "\<lambda>s. s - {x}" and g = "insert x"])
-  apply (auto elim!: equalityE simp add: inj_on_def)
-  apply (metis card_Diff_singleton_if finite_subset in_mono)
-  done
-
-text {*
-  Main theorem: combinatorial statement about number of subsets of a set.
-*}
-
-theorem n_subsets: "finite A \<Longrightarrow> card {B. B \<subseteq> A \<and> card B = k} = (card A choose k)"
-proof (induct k arbitrary: A)
-  case 0 then show ?case by (simp add: card_s_0_eq_empty)
-next
-  case (Suc k)
-  show ?case using `finite A`
-  proof (induct A)
-    case empty show ?case by (simp add: card_s_0_eq_empty)
-  next
-    case (insert x A)
-    then show ?case using Suc.hyps
-      apply (simp 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)" for F])
-      apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
-      done
-  qed
-qed
-
-
-subsection {* The binomial theorem (courtesy of Tobias Nipkow): *}
-
-text{* Avigad's version, generalized to any commutative ring *}
-theorem binomial_ring: "(a+b::'a::{comm_ring_1,power})^n = 
-  (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" (is "?P n")
-proof (induct n)
-  case 0 then show "?P 0" by simp
-next
-  case (Suc n)
-  have decomp: "{0..n+1} = {0} Un {n+1} Un {1..n}"
-    by auto
-  have decomp2: "{0..n} = {0} Un {1..n}"
-    by auto
-  have "(a+b)^(n+1) = 
-      (a+b) * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
-    using Suc.hyps by simp
-  also have "\<dots> = a*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k)) +
-                   b*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
-    by (rule distrib)
-  also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) +
-                  (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k+1))"
-    by (auto simp add: setsum_right_distrib mult_ac)
-  also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n+1-k)) +
-                  (\<Sum>k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n+1-k))"
-    by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps  
-        del:setsum_cl_ivl_Suc)
-  also have "\<dots> = a^(n+1) + b^(n+1) +
-                  (\<Sum>k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-k)) +
-                  (\<Sum>k=1..n. of_nat (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. of_nat(n+1 choose k) * a^k * b^(n+1-k))"
-    by (auto simp add: field_simps setsum_addf [symmetric] choose_reduce_nat)
-  also have "\<dots> = (\<Sum>k=0..n+1. of_nat (n+1 choose k) * a^k * b^(n+1-k))"
-    using decomp by (simp add: field_simps)
-  finally show "?P (Suc n)" by simp
-qed
-
-text{* Original version for the naturals *}
-corollary binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))"
-    using binomial_ring [of "int a" "int b" n]
-  by (simp only: of_nat_add [symmetric] of_nat_mult [symmetric] of_nat_power [symmetric]
-           of_nat_setsum [symmetric]
-           of_nat_eq_iff of_nat_id)
-
-subsection{* Pochhammer's symbol : generalized rising factorial *}
-
-text {* See @{url "http://en.wikipedia.org/wiki/Pochhammer_symbol"} *}
-
-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 "{0..Suc n} = {0..n} \<union> {Suc n}" by auto
-  then show ?thesis 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 "{0..Suc n} = {0} \<union> {1 .. Suc n}" by auto
-  then show ?thesis by simp
-qed
-
-
-lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
-proof (cases n)
-  case 0
-  then show ?thesis by simp
-next
-  case (Suc n)
-  show ?thesis unfolding Suc pochhammer_Suc_setprod setprod_nat_ivl_Suc ..
-qed
-
-lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n"
-proof (cases "n = 0")
-  case True
-  then show ?thesis by (simp add: pochhammer_Suc_setprod)
-next
-  case False
-  have *: "finite {1 .. n}" "0 \<notin> {1 .. n}" by auto
-  have eq: "insert 0 {1 .. n} = {0..n}" by auto
-  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)"
-    apply (rule setprod_reindex_cong [where f = Suc])
-    using False
-    apply (auto simp add: fun_eq_iff field_simps)
-    done
-  show ?thesis
-    apply (simp add: pochhammer_def)
-    unfolding setprod_insert [OF *, unfolded eq]
-    using ** apply (simp add: field_simps)
-    done
-qed
-
-lemma 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)
-  done
-
-lemma pochhammer_of_nat_eq_0_lemma:
-  assumes "k > n"
-  shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
-proof (cases "n = 0")
-  case True
-  then show ?thesis
-    using assms by (cases k) (simp_all add: pochhammer_rec)
-next
-  case False
-  from assms obtain h where "k = Suc h" by (cases k) auto
-  then show ?thesis
-    by (simp add: pochhammer_Suc_setprod)
-       (metis Suc_leI Suc_le_mono assms atLeastAtMost_iff less_eq_nat.simps(1))
-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 (cases k)
-  case 0
-  then show ?thesis by simp
-next
-  case (Suc h)
-  then show ?thesis
-    apply (simp add: pochhammer_Suc_setprod)
-    using Suc kn apply (auto simp add: algebra_simps)
-    done
-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> (\<exists>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 (metis leD not_less_eq)
-  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 (cases k)
-  case 0
-  then show ?thesis by simp
-next
-  case (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
-  show ?thesis
-    unfolding Suc pochhammer_Suc_setprod eq setprod_timesf[symmetric]
-    apply (rule strong_setprod_reindex_cong[where f = "%i. h - i"])
-    using Suc
-    apply (auto simp add: inj_on_def image_def of_nat_diff)
-    apply (metis atLeast0AtMost atMost_iff diff_diff_cancel diff_le_self)
-    done
-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]
-  by simp
-
-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 (cases "n = 0")
-  case True
-  then show ?thesis by simp
-next
-  case False
-  from this setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"]
-  have eq: "(- (1\<Colon>'a)) ^ n = setprod (\<lambda>i. - 1) {0 .. n - 1}"
-    by auto
-  from False show ?thesis
-    by (simp add: pochhammer_def gbinomial_def field_simps
-      eq setprod_timesf[symmetric])
-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> (\<exists>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"
-    then have ?thesis
-      by (subst binomial_eq_0[OF kn]) 
-         (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
-      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} \<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)
-      apply (simp add: pochhammer_Suc_setprod fact_altdef_nat h
-        of_nat_setprod setprod_timesf[symmetric] eq' del: One_nat_def power_Suc)
-      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
-      apply (rule of_nat_diff)
-      apply simp
-      done
-  }
-  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})"
-  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 (\<lambda>i. a - of_nat i) {0 .. k})"
-  using gbinomial_mult_fact[of k a]
-  by (subst mult_commute)
-
-
-lemma gbinomial_Suc_Suc:
-  "((a::'a::field_char_0) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
-proof (cases k)
-  case 0
-  then show ?thesis by simp
-next
-  case (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 Suc
-      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)) + (\<Prod>i\<in>{0\<Colon>nat..Suc h}. a - of_nat i)"
-    apply (simp add: Suc 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 "\<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 Suc)
-  also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)"
-    using eq0
-    by (simp add: Suc setprod_nat_ivl_1_Suc)
-  also have "\<dots> = of_nat (fact (Suc k)) * ((a + 1) gchoose (Suc k))"
-    unfolding gbinomial_mult_fact ..
-  finally show ?thesis by (simp del: fact_Suc)
-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
-
-(* Contributed by Manuel Eberl *)
-(* Alternative definition of the binomial coefficient as \<Prod>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 \<le> n"
-  shows "of_nat (n choose k) = (\<Prod>i<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
-proof (cases "0 < k")
-  case True
-  then have "(of_nat (n choose k) :: 'a) = (\<Prod>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 "\<dots> = (\<Prod>i<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
-    using `k \<le> 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 .
-next
-  case False
-  then show ?thesis by simp
-qed
-
-lemma binomial_ge_n_over_k_pow_k:
-  fixes k n :: nat
-    and x :: "'a :: linordered_field_inverse_zero"
-  assumes "0 < k"
-    and "k \<le> n"
-  shows "(of_nat n / of_nat k :: 'a) ^ k \<le> of_nat (n choose k)"
-proof -
-  have "(of_nat n / of_nat k :: 'a) ^ k = (\<Prod>i<k. of_nat n / of_nat k :: 'a)"
-    by (simp add: setprod_constant)
-  also have "\<dots> \<le> of_nat (n choose k)"
-    unfolding binomial_altdef_of_nat[OF `k\<le>n`]
-  proof (safe intro!: setprod_mono)
-    fix i :: nat
-    assume  "i < k"
-    from assms have "n * i \<ge> i * k" by simp
-    then have "n * k - n * i \<le> n * k - i * k" by arith
-    then have "n * (k - i) \<le> (n - i) * k"
-      by (simp add: diff_mult_distrib2 nat_mult_commute)
-    then have "of_nat n * of_nat (k - i) \<le> 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 \<le> 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 .
-qed
-
-lemma binomial_le_pow:
-  assumes "r \<le> n"
-  shows "n choose r \<le> n ^ r"
-proof -
-  have "n choose r \<le> fact n div fact (n - r)"
-    using `r \<le> n` by (subst binomial_fact_lemma[symmetric]) auto
-  with fact_div_fact_le_pow [OF assms] show ?thesis by auto
-qed
-
-lemma binomial_altdef_nat: "(k::nat) \<le> n \<Longrightarrow>
-    n choose k = fact n div (fact k * fact (n - k))"
- by (subst binomial_fact_lemma [symmetric]) auto
-
-end