author haftmann Sat Jul 02 20:22:25 2016 +0200 (2016-07-02) changeset 63367 6c731c8b7f03 parent 63366 209c4cbbc4cd child 63368 e9e677b73011 child 63385 370cce7ad9b9
simplified definitions of combinatorial functions
```     1.1 --- a/src/HOL/Binomial.thy	Sat Jul 02 15:02:24 2016 +0200
1.2 +++ b/src/HOL/Binomial.thy	Sat Jul 02 20:22:25 2016 +0200
1.3 @@ -14,29 +14,38 @@
1.4
1.5  subsection \<open>Factorial\<close>
1.6
1.7 -fun (in semiring_char_0) fact :: "nat \<Rightarrow> 'a"
1.8 -  where "fact 0 = 1" | "fact (Suc n) = of_nat (Suc n) * fact n"
1.9 +definition (in semiring_char_0) fact :: "nat \<Rightarrow> 'a"
1.10 +where
1.11 +  "fact n = of_nat (\<Prod>{1..n})"
1.12 +
1.13 +lemma fact_altdef': "fact n = of_nat (\<Prod>{1..n})"
1.14 +  by (fact fact_def)
1.15
1.16 -lemmas fact_Suc = fact.simps(2)
1.17 +lemma fact_altdef_nat: "fact n = \<Prod>{1..n}"
1.18 +  by (simp add: fact_def)
1.19 +
1.20 +lemma fact_altdef: "fact n = (\<Prod>i=1..n. of_nat i)"
1.21 +  by (simp add: fact_def)
1.22 +
1.23 +lemma fact_0 [simp]: "fact 0 = 1"
1.24 +  by (simp add: fact_def)
1.25
1.26  lemma fact_1 [simp]: "fact 1 = 1"
1.27 -  by simp
1.28 +  by (simp add: fact_def)
1.29
1.30  lemma fact_Suc_0 [simp]: "fact (Suc 0) = Suc 0"
1.31 -  by simp
1.32 +  by (simp add: fact_def)
1.33 +
1.34 +lemma fact_Suc [simp]: "fact (Suc n) = of_nat (Suc n) * fact n"
1.35 +  by (simp add: fact_def atLeastAtMostSuc_conv algebra_simps)
1.36
1.37  lemma of_nat_fact [simp]:
1.38    "of_nat (fact n) = fact n"
1.39 -  by (induct n) (auto simp add: algebra_simps)
1.40 +  by (simp add: fact_def)
1.41
1.42  lemma of_int_fact [simp]:
1.43    "of_int (fact n) = fact n"
1.44 -proof -
1.45 -  have "of_int (of_nat (fact n)) = fact n"
1.46 -    unfolding of_int_of_nat_eq by simp
1.47 -  then show ?thesis
1.48 -    by simp
1.49 -qed
1.50 +  by (simp only: fact_def of_int_of_nat_eq)
1.51
1.52  lemma fact_reduce: "n > 0 \<Longrightarrow> fact n = of_nat n * fact (n - 1)"
1.53    by (cases n) auto
1.54 @@ -61,7 +70,7 @@
1.55      by (metis of_nat_fact of_nat_le_iff fact_mono_nat)
1.56
1.57    lemma fact_ge_1 [simp]: "fact n \<ge> (1 :: 'a)"
1.58 -    by (metis le0 fact.simps(1) fact_mono)
1.59 +    by (metis le0 fact_0 fact_mono)
1.60
1.61    lemma fact_gt_zero [simp]: "fact n > (0 :: 'a)"
1.62      using fact_ge_1 less_le_trans zero_less_one by blast
1.63 @@ -107,15 +116,6 @@
1.64  lemma fact_ge_self: "fact n \<ge> n"
1.65    by (cases "n = 0") (simp_all add: dvd_imp_le dvd_fact)
1.66
1.67 -lemma fact_altdef_nat: "fact n = \<Prod>{1..n}"
1.68 -  by (induct n) (auto simp: atLeastAtMostSuc_conv)
1.69 -
1.70 -lemma fact_altdef: "fact n = (\<Prod>i=1..n. of_nat i)"
1.71 -  by (induct n) (auto simp: atLeastAtMostSuc_conv)
1.72 -
1.73 -lemma fact_altdef': "fact n = of_nat (\<Prod>{1..n})"
1.74 -  by (subst fact_altdef_nat [symmetric]) simp
1.75 -
1.76  lemma fact_dvd: "n \<le> m \<Longrightarrow> fact n dvd (fact m :: 'a :: {semiring_div,linordered_semidom})"
1.77    by (induct m) (auto simp: le_Suc_eq)
1.78
1.79 @@ -164,7 +164,7 @@
1.80
1.81  lemma fact_numeral:  \<comment>\<open>Evaluation for specific numerals\<close>
1.82    "fact (numeral k) = (numeral k) * (fact (pred_numeral k))"
1.83 -  by (metis fact.simps(2) numeral_eq_Suc of_nat_numeral)
1.84 +  by (metis fact_Suc numeral_eq_Suc of_nat_numeral)
1.85
1.86
1.87  text \<open>This development is based on the work of Andy Gordon and
1.88 @@ -469,49 +469,44 @@
1.89
1.90  text \<open>See @{url "http://en.wikipedia.org/wiki/Pochhammer_symbol"}\<close>
1.91
1.92 -definition (in comm_semiring_1) "pochhammer (a :: 'a) n =
1.93 -  (if n = 0 then 1 else setprod (\<lambda>n. a + of_nat n) {0 .. n - 1})"
1.94 +definition (in comm_semiring_1) pochhammer :: "'a \<Rightarrow> nat \<Rightarrow> 'a"
1.95 +where
1.96 +  "pochhammer (a :: 'a) n = setprod (\<lambda>n. a + of_nat n) {..<n}"
1.97
1.98 +lemma pochhammer_Suc_setprod:
1.99 +  "pochhammer a (Suc n) = setprod (\<lambda>n. a + of_nat n) {..n}"
1.100 +  by (simp add: pochhammer_def lessThan_Suc_atMost)
1.101 +
1.102  lemma pochhammer_0 [simp]: "pochhammer a 0 = 1"
1.103    by (simp add: pochhammer_def)
1.104 -
1.105 +
1.106  lemma pochhammer_1 [simp]: "pochhammer a 1 = a"
1.107 -  by (simp add: pochhammer_def)
1.108 -
1.109 +  by (simp add: pochhammer_def lessThan_Suc)
1.110 +
1.111  lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a"
1.112 -  by (simp add: pochhammer_def)
1.113 -
1.114 -lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\<lambda>n. a + of_nat n) {0 .. n}"
1.115 -  by (simp add: pochhammer_def)
1.116 -
1.117 +  by (simp add: pochhammer_def lessThan_Suc)
1.118 +
1.119 +lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
1.120 +  by (simp add: pochhammer_def lessThan_Suc ac_simps)
1.121 +
1.122  lemma pochhammer_of_nat: "pochhammer (of_nat x) n = of_nat (pochhammer x n)"
1.123    by (simp add: pochhammer_def)
1.124
1.125  lemma pochhammer_of_int: "pochhammer (of_int x) n = of_int (pochhammer x n)"
1.126    by (simp add: pochhammer_def)
1.127
1.128 -lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc n)"
1.129 +lemma setprod_nat_ivl_Suc: "setprod f {.. Suc n} = setprod f {..n} * f (Suc n)"
1.130  proof -
1.131 -  have "{0..Suc n} = {0..n} \<union> {Suc n}" by auto
1.132 +  have "{..Suc n} = {..n} \<union> {Suc n}" by auto
1.133    then show ?thesis by (simp add: field_simps)
1.134  qed
1.135
1.136 -lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}"
1.137 +lemma setprod_nat_ivl_1_Suc: "setprod f {.. Suc n} = f 0 * setprod f {1.. Suc n}"
1.138  proof -
1.139 -  have "{0..Suc n} = {0} \<union> {1 .. Suc n}" by auto
1.140 +  have "{..Suc n} = {0} \<union> {1 .. Suc n}" by auto
1.141    then show ?thesis by simp
1.142  qed
1.143
1.144 -
1.145 -lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
1.146 -proof (cases n)
1.147 -  case 0
1.148 -  then show ?thesis by simp
1.149 -next
1.150 -  case (Suc n)
1.151 -  show ?thesis unfolding Suc pochhammer_Suc_setprod setprod_nat_ivl_Suc ..
1.152 -qed
1.153 -
1.154  lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n"
1.155  proof (cases "n = 0")
1.156    case True
1.157 @@ -519,14 +514,14 @@
1.158  next
1.159    case False
1.160    have *: "finite {1 .. n}" "0 \<notin> {1 .. n}" by auto
1.161 -  have eq: "insert 0 {1 .. n} = {0..n}" by auto
1.162 -  have **: "(\<Prod>n\<in>{1::nat..n}. a + of_nat n) = (\<Prod>n\<in>{0::nat..n - 1}. a + 1 + of_nat n)"
1.163 +  have eq: "insert 0 {1 .. n} = {..n}" by auto
1.164 +  have **: "(\<Prod>n\<in>{1..n}. a + of_nat n) = (\<Prod>n\<in>{..<n}. a + 1 + of_nat n)"
1.165      apply (rule setprod.reindex_cong [where l = Suc])
1.166      using False
1.167 -    apply (auto simp add: fun_eq_iff field_simps)
1.168 +    apply (auto simp add: fun_eq_iff field_simps image_Suc_lessThan)
1.169      done
1.170    show ?thesis
1.171 -    apply (simp add: pochhammer_def)
1.172 +    apply (simp add: pochhammer_def lessThan_Suc_atMost)
1.173      unfolding setprod.insert [OF *, unfolded eq]
1.174      using ** apply (simp add: field_simps)
1.175      done
1.176 @@ -545,27 +540,15 @@
1.177  qed simp_all
1.178
1.179  lemma pochhammer_fact: "fact n = pochhammer 1 n"
1.180 -  unfolding fact_altdef
1.181 -  apply (cases n)
1.182 -   apply (simp_all add: pochhammer_Suc_setprod)
1.183 +  apply (auto simp add: pochhammer_def fact_altdef)
1.184    apply (rule setprod.reindex_cong [where l = Suc])
1.185 -    apply (auto simp add: fun_eq_iff)
1.186 +  apply (auto simp add: image_Suc_lessThan)
1.187    done
1.188
1.189  lemma pochhammer_of_nat_eq_0_lemma:
1.190    assumes "k > n"
1.191    shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
1.192 -proof (cases "n = 0")
1.193 -  case True
1.194 -  then show ?thesis
1.195 -    using assms by (cases k) (simp_all add: pochhammer_rec)
1.196 -next
1.197 -  case False
1.198 -  from assms obtain h where "k = Suc h" by (cases k) auto
1.199 -  then show ?thesis
1.200 -    by (simp add: pochhammer_Suc_setprod)
1.201 -       (metis Suc_leI Suc_le_mono assms atLeastAtMost_iff less_eq_nat.simps(1))
1.202 -qed
1.203 +  using assms by (auto simp add: pochhammer_def)
1.204
1.205  lemma pochhammer_of_nat_eq_0_lemma':
1.206    assumes kn: "k \<le> n"
1.207 @@ -589,11 +572,7 @@
1.208    by (auto simp add: not_le[symmetric])
1.209
1.210  lemma pochhammer_eq_0_iff: "pochhammer a n = (0::'a::field_char_0) \<longleftrightarrow> (\<exists>k < n. a = - of_nat k)"
1.211 -  apply (auto simp add: pochhammer_of_nat_eq_0_iff)
1.212 -  apply (cases n)
1.214 -  apply (metis leD not_less_eq)
1.215 -  done
1.216 +  by (auto simp add: pochhammer_def eq_neg_iff_add_eq_0)
1.217
1.218  lemma pochhammer_eq_0_mono:
1.219    "pochhammer a n = (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a m = 0"
1.220 @@ -610,8 +589,8 @@
1.221    then show ?thesis by simp
1.222  next
1.223    case (Suc h)
1.224 -  have eq: "((- 1) ^ Suc h :: 'a) = (\<Prod>i=0..h. - 1)"
1.225 -    using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"]
1.226 +  have eq: "((- 1) ^ Suc h :: 'a) = (\<Prod>i\<le>h. - 1)"
1.227 +    using setprod_constant[where A="{.. h}" and y="- 1 :: 'a"]
1.228      by auto
1.229    show ?thesis
1.230      unfolding Suc pochhammer_Suc_setprod eq setprod.distrib[symmetric]
1.231 @@ -650,7 +629,7 @@
1.232
1.233  lemma pochhammer_times_pochhammer_half:
1.234    fixes z :: "'a :: field_char_0"
1.235 -  shows "pochhammer z (Suc n) * pochhammer (z + 1/2) (Suc n) = (\<Prod>k=0..2*n+1. z + of_nat k / 2)"
1.236 +  shows "pochhammer z (Suc n) * pochhammer (z + 1/2) (Suc n) = (\<Prod>k\<le>2*n+1. z + of_nat k / 2)"
1.237  proof (induction n)
1.238    case (Suc n)
1.239    define n' where "n' = Suc n"
1.240 @@ -661,7 +640,7 @@
1.241    also have "?A = (z + of_nat (Suc (2 * n + 1)) / 2) * (z + of_nat (Suc (Suc (2 * n + 1))) / 2)"
1.242      (is "_ = ?A") by (simp add: field_simps n'_def)
1.243    also note Suc[folded n'_def]
1.244 -  also have "(\<Prod>k = 0..2 * n + 1. z + of_nat k / 2) * ?A = (\<Prod>k = 0..2 * Suc n + 1. z + of_nat k / 2)"
1.245 +  also have "(\<Prod>k\<le>2 * n + 1. z + of_nat k / 2) * ?A = (\<Prod>k\<le>2 * Suc n + 1. z + of_nat k / 2)"
1.246      by (simp add: setprod_nat_ivl_Suc)
1.247    finally show ?case by (simp add: n'_def)
1.248  qed (simp add: setprod_nat_ivl_Suc)
1.249 @@ -699,8 +678,12 @@
1.250  subsection\<open>Generalized binomial coefficients\<close>
1.251
1.252  definition (in field_char_0) gbinomial :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
1.253 -  where "a gchoose n =
1.254 -    (if n = 0 then 1 else (setprod (\<lambda>i. a - of_nat i) {0 .. n - 1}) / (fact n))"
1.255 +where
1.256 +  "a gchoose n = setprod (\<lambda>i. a - of_nat i) {..<n} / fact n"
1.257 +
1.258 +lemma gbinomial_Suc:
1.259 +  "a gchoose (Suc k) = setprod (\<lambda>i. a - of_nat i) {..k} / fact (Suc k)"
1.260 +  by (simp add: gbinomial_def lessThan_Suc_atMost)
1.261
1.262  lemma gbinomial_0 [simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0"
1.263    by (simp_all add: gbinomial_def)
1.264 @@ -711,7 +694,7 @@
1.265    then show ?thesis by simp
1.266  next
1.267    case False
1.268 -  then have eq: "(- 1) ^ n = (\<Prod>i = 0..n - 1. - 1)"
1.269 +  then have eq: "(- 1) ^ n = (\<Prod>i<n. - 1)"
1.270      by (auto simp add: setprod_constant)
1.271    from False show ?thesis
1.272      by (simp add: pochhammer_def gbinomial_def field_simps
1.273 @@ -740,9 +723,9 @@
1.274    { assume kn: "k \<le> n" and k0: "k\<noteq> 0"
1.275      from k0 obtain h where h: "k = Suc h" by (cases k) auto
1.276      from h
1.277 -    have eq:"(- 1 :: 'a) ^ k = setprod (\<lambda>i. - 1) {0..h}"
1.278 +    have eq:"(- 1 :: 'a) ^ k = setprod (\<lambda>i. - 1) {..h}"
1.279        by (subst setprod_constant) auto
1.280 -    have eq': "(\<Prod>i\<in>{0..h}. of_nat n + - (of_nat i :: 'a)) = (\<Prod>i\<in>{n - h..n}. of_nat i)"
1.281 +    have eq': "(\<Prod>i\<le>h. of_nat n + - (of_nat i :: 'a)) = (\<Prod>i\<in>{n - h..n}. of_nat i)"
1.282          using h kn
1.283        by (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"])
1.284           (auto simp: of_nat_diff)
1.285 @@ -770,10 +753,10 @@
1.286  qed
1.287
1.288  lemma gbinomial_1[simp]: "a gchoose 1 = a"
1.289 -  by (simp add: gbinomial_def)
1.290 +  by (simp add: gbinomial_def lessThan_Suc)
1.291
1.292  lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a"
1.293 -  by (simp add: gbinomial_def)
1.294 +  by (simp add: gbinomial_def lessThan_Suc)
1.295
1.296  lemma gbinomial_mult_1:
1.297    fixes a :: "'a :: field_char_0"
1.298 @@ -783,7 +766,7 @@
1.299    have "?r = ((- 1) ^n * pochhammer (- a) n / (fact n)) * (of_nat n - (- a + of_nat n))"
1.300      unfolding gbinomial_pochhammer
1.301        pochhammer_Suc right_diff_distrib power_Suc
1.302 -    apply (simp del: of_nat_Suc fact.simps)
1.303 +    apply (simp del: of_nat_Suc fact_Suc)
1.304      apply (auto simp add: field_simps simp del: of_nat_Suc)
1.305      done
1.306    also have "\<dots> = ?l" unfolding gbinomial_pochhammer
1.307 @@ -796,20 +779,16 @@
1.308    shows "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
1.309    by (simp add: mult.commute gbinomial_mult_1)
1.310
1.311 -lemma gbinomial_Suc:
1.312 -    "a gchoose (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k}) / (fact (Suc k))"
1.313 -  by (simp add: gbinomial_def)
1.314 -
1.315  lemma gbinomial_mult_fact:
1.316    fixes a :: "'a::field_char_0"
1.317    shows
1.318     "fact (Suc k) * (a gchoose (Suc k)) =
1.319 -    (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
1.320 -  by (simp_all add: gbinomial_Suc field_simps del: fact.simps)
1.321 +    (setprod (\<lambda>i. a - of_nat i) {.. k})"
1.322 +  by (simp_all add: gbinomial_Suc field_simps del: fact_Suc)
1.323
1.324  lemma gbinomial_mult_fact':
1.325    fixes a :: "'a::field_char_0"
1.326 -  shows "(a gchoose (Suc k)) * fact (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
1.327 +  shows "(a gchoose (Suc k)) * fact (Suc k) = (setprod (\<lambda>i. a - of_nat i) {.. k})"
1.328    using gbinomial_mult_fact[of k a]
1.329    by (subst mult.commute)
1.330
1.331 @@ -821,36 +800,37 @@
1.332    then show ?thesis by simp
1.333  next
1.334    case (Suc h)
1.335 -  have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)"
1.336 +  have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{..h}. a - of_nat i)"
1.337      apply (rule setprod.reindex_cong [where l = Suc])
1.338        using Suc
1.339 -      apply auto
1.340 +      apply (auto simp add: image_Suc_atMost)
1.341      done
1.342    have "fact (Suc k) * (a gchoose k + (a gchoose (Suc k))) =
1.343          (a gchoose Suc h) * (fact (Suc (Suc h))) +
1.344          (a gchoose Suc (Suc h)) * (fact (Suc (Suc h)))"
1.345 -    by (simp add: Suc field_simps del: fact.simps)
1.346 +    by (simp add: Suc field_simps del: fact_Suc)
1.347    also have "... = (a gchoose Suc h) * of_nat (Suc (Suc h) * fact (Suc h)) +
1.348 -                   (\<Prod>i = 0..Suc h. a - of_nat i)"
1.349 -    by (metis fact.simps(2) gbinomial_mult_fact' of_nat_fact of_nat_id)
1.350 +                   (\<Prod>i\<le>Suc h. a - of_nat i)"
1.351 +    by (metis fact_Suc gbinomial_mult_fact' of_nat_fact of_nat_id)
1.352    also have "... = (fact (Suc h) * (a gchoose Suc h)) * of_nat (Suc (Suc h)) +
1.353 -                   (\<Prod>i = 0..Suc h. a - of_nat i)"
1.354 -    by (simp only: fact.simps(2) mult.commute mult.left_commute of_nat_fact of_nat_id of_nat_mult)
1.355 -  also have "... =  of_nat (Suc (Suc h)) * (\<Prod>i = 0..h. a - of_nat i) +
1.356 -                    (\<Prod>i = 0..Suc h. a - of_nat i)"
1.357 +                   (\<Prod>i\<le>Suc h. a - of_nat i)"
1.358 +    by (simp only: fact_Suc mult.commute mult.left_commute of_nat_fact of_nat_id of_nat_mult)
1.359 +  also have "... =  of_nat (Suc (Suc h)) * (\<Prod>i\<le>h. a - of_nat i) +
1.360 +                    (\<Prod>i\<le>Suc h. a - of_nat i)"
1.361      by (metis gbinomial_mult_fact mult.commute)
1.362 -  also have "... = (\<Prod>i = 0..Suc h. a - of_nat i) +
1.363 -                   (of_nat h * (\<Prod>i = 0..h. a - of_nat i) + 2 * (\<Prod>i = 0..h. a - of_nat i))"
1.364 +  also have "... = (\<Prod>i\<le>Suc h. a - of_nat i) +
1.365 +                   (of_nat h * (\<Prod>i\<le>h. a - of_nat i) + 2 * (\<Prod>i\<le>h. a - of_nat i))"
1.366      by (simp add: field_simps)
1.367    also have "... =
1.368 -    ((a gchoose Suc h) * (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{0::nat..Suc h}. a - of_nat i)"
1.369 +    ((a gchoose Suc h) * (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{..Suc h}. a - of_nat i)"
1.370      unfolding gbinomial_mult_fact'
1.371      by (simp add: comm_semiring_class.distrib field_simps Suc)
1.372 -  also have "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 1)"
1.373 +  also have "\<dots> = (\<Prod>i\<in>{..h}. a - of_nat i) * (a + 1)"
1.374      unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc
1.375 +      atMost_Suc
1.376      by (simp add: field_simps Suc)
1.377 -  also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)"
1.378 -    using eq0
1.379 +  also have "\<dots> = (\<Prod>i\<in>{..k}. (a + 1) - of_nat i)"
1.380 +    using eq0 setprod_nat_ivl_1_Suc
1.381      by (simp add: Suc setprod_nat_ivl_1_Suc)
1.382    also have "\<dots> = (fact (Suc k)) * ((a + 1) gchoose (Suc k))"
1.383      unfolding gbinomial_mult_fact ..
1.384 @@ -1024,12 +1004,12 @@
1.385  proof (cases b)
1.386    case (Suc b)
1.387    hence "((a + 1) gchoose (Suc (Suc b))) =
1.388 -             (\<Prod>i = 0..Suc b. a + (1 - of_nat i)) / fact (b + 2)"
1.389 -    by (simp add: field_simps gbinomial_def)
1.390 -  also have "(\<Prod>i = 0..Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i = 0..b. a - of_nat i)"
1.391 -    by (simp add: setprod_nat_ivl_1_Suc setprod_shift_bounds_cl_Suc_ivl)
1.392 +             (\<Prod>i\<le>Suc b. a + (1 - of_nat i)) / fact (b + 2)"
1.393 +    by (simp add: field_simps gbinomial_def lessThan_Suc_atMost)
1.394 +  also have "(\<Prod>i\<le>Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i\<le>b. a - of_nat i)"
1.395 +    by (simp add: setprod_nat_ivl_1_Suc setprod_shift_bounds_cl_Suc_ivl atLeast0AtMost)
1.396    also have "... / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)"
1.397 -    by (simp_all add: gbinomial_def setprod_nat_ivl_1_Suc setprod_shift_bounds_cl_Suc_ivl)
1.398 +    by (simp_all add: gbinomial_def setprod_nat_ivl_1_Suc setprod_shift_bounds_cl_Suc_ivl lessThan_Suc_atMost)
1.399    finally show ?thesis by (simp add: Suc field_simps del: of_nat_Suc)
1.400  qed simp
1.401
1.402 @@ -1038,12 +1018,12 @@
1.403  proof (cases b)
1.404    case (Suc b)
1.405    hence "((a + 1) gchoose (Suc (Suc b))) =
1.406 -             (\<Prod>i = 0..Suc b. a + (1 - of_nat i)) / fact (b + 2)"
1.407 -    by (simp add: field_simps gbinomial_def)
1.408 -  also have "(\<Prod>i = 0..Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i = 0..b. a - of_nat i)"
1.409 +             (\<Prod>i\<le>Suc b. a + (1 - of_nat i)) / fact (b + 2)"
1.410 +    by (simp add: field_simps gbinomial_def lessThan_Suc_atMost)
1.411 +  also have "(\<Prod>i\<le>Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i = 0..b. a - of_nat i)"
1.412      by (simp add: setprod_nat_ivl_1_Suc setprod_shift_bounds_cl_Suc_ivl)
1.413    also have "... / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)"
1.414 -    by (simp_all add: gbinomial_def setprod_nat_ivl_1_Suc setprod_shift_bounds_cl_Suc_ivl)
1.415 +    by (simp_all add: gbinomial_def setprod_nat_ivl_1_Suc setprod_shift_bounds_cl_Suc_ivl lessThan_Suc_atMost atLeast0AtMost)
1.416    finally show ?thesis by (simp add: Suc)
1.417  qed simp
1.418
1.419 @@ -1379,8 +1359,7 @@
1.420    apply (case_tac "k = 0")
1.421    apply auto
1.422    apply (case_tac "k = Suc n")
1.423 -  apply auto
1.424 -  apply (metis Suc_le_eq fact.cases le_Suc_eq le_eq_less_or_eq)
1.425 +  apply (auto simp add: le_Suc_eq elim: lessE)
1.426    done
1.427
1.428  lemma card_UNION:
1.429 @@ -1579,15 +1558,20 @@
1.430    finally show ?thesis .
1.431  qed
1.432
1.433 +lemma setprod_lessThan_fold_atLeastAtMost_nat:
1.434 +  "setprod f {..<Suc n} = fold_atLeastAtMost_nat (times \<circ> f) 0 n 1"
1.435 +  by (simp add: lessThan_Suc_atMost atLeast0AtMost [symmetric] setprod_atLeastAtMost_code comp_def)
1.436 +
1.437 +
1.438  lemma pochhammer_code [code]:
1.439    "pochhammer a n = (if n = 0 then 1 else
1.440         fold_atLeastAtMost_nat (\<lambda>n acc. (a + of_nat n) * acc) 0 (n - 1) 1)"
1.441 -  by (simp add: setprod_atLeastAtMost_code pochhammer_def)
1.442 +  by (cases n) (simp_all add: pochhammer_def setprod_lessThan_fold_atLeastAtMost_nat comp_def)
1.443
1.444  lemma gbinomial_code [code]:
1.445    "a gchoose n = (if n = 0 then 1 else
1.446       fold_atLeastAtMost_nat (\<lambda>n acc. (a - of_nat n) * acc) 0 (n - 1) 1 / fact n)"
1.447 -  by (simp add: setprod_atLeastAtMost_code gbinomial_def)
1.448 +  by (cases n) (simp_all add: gbinomial_def setprod_lessThan_fold_atLeastAtMost_nat comp_def)
1.449
1.450  (*TODO: This code equation breaks Scala code generation in HOL-Codegenerator_Test. We have to figure out why and how to prevent that. *)
1.451
```
```     2.1 --- a/src/HOL/Library/Formal_Power_Series.thy	Sat Jul 02 15:02:24 2016 +0200
2.2 +++ b/src/HOL/Library/Formal_Power_Series.thy	Sat Jul 02 20:22:25 2016 +0200
2.3 @@ -3700,7 +3700,7 @@
2.4  proof -
2.5    have "?l\$n = ?r \$ n" for n
2.6      apply (auto simp add: E_def field_simps power_Suc[symmetric]
2.7 -      simp del: fact.simps of_nat_Suc power_Suc)
2.8 +      simp del: fact_Suc of_nat_Suc power_Suc)
2.9      apply (simp add: of_nat_mult field_simps)
2.10      done
2.11    then show ?thesis
2.12 @@ -4154,7 +4154,7 @@
2.13          case False
2.14          with kn have kn': "k < n"
2.15            by simp
2.16 -        have m1nk: "?m1 n = setprod (\<lambda>i. - 1) {0..m}" "?m1 k = setprod (\<lambda>i. - 1) {0..h}"
2.17 +        have m1nk: "?m1 n = setprod (\<lambda>i. - 1) {..m}" "?m1 k = setprod (\<lambda>i. - 1) {0..h}"
2.18            by (simp_all add: setprod_constant m h)
2.19          have bnz0: "pochhammer (b - of_nat n + 1) k \<noteq> 0"
2.20            using bn0 kn
2.21 @@ -4163,27 +4163,19 @@
2.22            apply (erule_tac x= "n - ka - 1" in allE)
2.23            apply (auto simp add: algebra_simps of_nat_diff)
2.24            done
2.25 -        have eq1: "setprod (\<lambda>k. (1::'a) + of_nat m - of_nat k) {0 .. h} =
2.26 +        have eq1: "setprod (\<lambda>k. (1::'a) + of_nat m - of_nat k) {..h} =
2.27            setprod of_nat {Suc (m - h) .. Suc m}"
2.28            using kn' h m
2.29            by (intro setprod.reindex_bij_witness[where i="\<lambda>k. Suc m - k" and j="\<lambda>k. Suc m - k"])
2.30               (auto simp: of_nat_diff)
2.31 -
2.32          have th1: "(?m1 k * ?p (of_nat n) k) / ?f n = 1 / of_nat(fact (n - k))"
2.33            unfolding m1nk
2.34 -          unfolding m h pochhammer_Suc_setprod
2.35 -          apply (simp add: field_simps del: fact_Suc)
2.36 -          unfolding fact_altdef id_def
2.37 -          unfolding of_nat_setprod
2.38 -          unfolding setprod.distrib[symmetric]
2.39 -          apply auto
2.40 -          unfolding eq1
2.41 -          apply (subst setprod.union_disjoint[symmetric])
2.42 -          apply (auto)
2.43 -          apply (rule setprod.cong)
2.44 -          apply auto
2.45 +          apply (simp add: field_simps m h pochhammer_Suc_setprod del: fact_Suc)
2.46 +          apply (simp add: fact_altdef id_def atLeast0AtMost setprod.distrib [symmetric] eq1)
2.47 +          apply (subst setprod.union_disjoint [symmetric])
2.48 +          apply (auto intro!: setprod.cong)
2.49            done
2.50 -        have th20: "?m1 n * ?p b n = setprod (\<lambda>i. b - of_nat i) {0..m}"
2.51 +        have th20: "?m1 n * ?p b n = setprod (\<lambda>i. b - of_nat i) {..m}"
2.52            unfolding m1nk
2.53            unfolding m h pochhammer_Suc_setprod
2.54            unfolding setprod.distrib[symmetric]
2.55 @@ -4216,7 +4208,10 @@
2.56            by (simp add: field_simps)
2.57          also have "\<dots> = b gchoose (n - k)"
2.58            unfolding th1 th2
2.59 -          using kn' by (simp add: gbinomial_def)
2.60 +          using kn' apply (simp add: gbinomial_def atLeast0AtMost)
2.61 +            apply (rule setprod.cong)
2.62 +            apply auto
2.63 +            done
2.64          finally show ?thesis by simp
2.65        qed
2.66      qed
```
```     3.1 --- a/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Sat Jul 02 15:02:24 2016 +0200
3.2 +++ b/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Sat Jul 02 20:22:25 2016 +0200
3.3 @@ -5906,7 +5906,7 @@
3.4
3.5
3.6  lemma bb: "Suc n choose k = (n choose k) + (if k = 0 then 0 else (n choose (k - 1)))"
3.7 -  by (simp add: Binomial.binomial.simps)
3.8 +  by (cases k) simp_all
3.9
3.10  proposition higher_deriv_mult:
3.11    fixes z::complex
3.12 @@ -5924,7 +5924,7 @@
3.13    have sumeq: "(\<Sum>i = 0..n.
3.14                 of_nat (n choose i) * (deriv ((deriv ^^ i) f) z * (deriv ^^ (n - i)) g z + deriv ((deriv ^^ (n - i)) g) z * (deriv ^^ i) f z)) =
3.15              g z * deriv ((deriv ^^ n) f) z + (\<Sum>i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))"
3.16 -    apply (simp add: bb distrib_right algebra_simps setsum.distrib)
3.17 +    apply (simp add: bb algebra_simps setsum.distrib)
3.18      apply (subst (4) setsum_Suc_reindex)
3.19      apply (auto simp: algebra_simps Suc_diff_le intro: setsum.cong)
3.20      done
```
```     4.1 --- a/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Sat Jul 02 15:02:24 2016 +0200
4.2 +++ b/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Sat Jul 02 20:22:25 2016 +0200
4.3 @@ -1106,19 +1106,19 @@
4.4              ((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) +
4.5              ((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) -
4.6              ((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))"
4.7 -          by (simp add: algebra_simps del: fact.simps)
4.8 +          by (simp add: algebra_simps del: fact_Suc)
4.9          also have "... = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (fact n) +
4.10                           (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
4.11                           (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
4.12 -          by (simp del: fact.simps)
4.13 +          by (simp del: fact_Suc)
4.14          also have "... = (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
4.15                           (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
4.16                           (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
4.17 -          by (simp only: fact.simps of_nat_mult ac_simps) simp
4.18 +          by (simp only: fact_Suc of_nat_mult ac_simps) simp
4.19          also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
4.20            by (simp add: algebra_simps)
4.21          finally show ?thesis
4.22 -        by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact.simps)
4.23 +        by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact_Suc)
4.24        qed
4.25        finally show ?case .
4.26      qed
```
```     5.1 --- a/src/HOL/Multivariate_Analysis/Gamma.thy	Sat Jul 02 15:02:24 2016 +0200
5.2 +++ b/src/HOL/Multivariate_Analysis/Gamma.thy	Sat Jul 02 20:22:25 2016 +0200
5.3 @@ -512,9 +512,10 @@
5.4      by (intro setprod.cong[OF refl], subst exp_Ln) (auto simp: field_simps plus_of_nat_eq_0_imp)
5.5    also have "... = (\<Prod>k=1..n. z + k) / fact n" unfolding fact_altdef
5.6      by (subst setprod_dividef [symmetric]) (simp_all add: field_simps)
5.7 -  also from assms have "z * ... = (\<Prod>k=0..n. z + k) / fact n"
5.8 +  also from assms have "z * ... = (\<Prod>k\<le>n. z + k) / fact n"
5.9      by (cases n) (simp_all add: setprod_nat_ivl_1_Suc)
5.10 -  also have "(\<Prod>k=0..n. z + k) = pochhammer z (Suc n)" unfolding pochhammer_def by simp
5.11 +  also have "(\<Prod>k\<le>n. z + k) = pochhammer z (Suc n)" unfolding pochhammer_def
5.12 +    by (simp add: lessThan_Suc_atMost)
5.13    also have "of_nat n powr z / (pochhammer z (Suc n) / fact n) = Gamma_series z n"
5.14      unfolding Gamma_series_def using assms by (simp add: divide_simps powr_def Ln_of_nat)
5.15    finally show ?thesis .
5.16 @@ -999,7 +1000,7 @@
5.17    hence "z \<noteq> - of_nat n" for n by auto
5.18    from rGamma_series_aux[OF this] show ?thesis
5.19      by (simp add: rGamma_series_def[abs_def] fact_altdef pochhammer_Suc_setprod
5.20 -                  exp_def of_real_def[symmetric] suminf_def sums_def[abs_def])
5.21 +                  exp_def of_real_def[symmetric] suminf_def sums_def[abs_def] atLeast0AtMost)
5.22  qed (insert rGamma_eq_zero_iff[of z], simp_all add: rGamma_series_nonpos_Ints_LIMSEQ)
5.23
5.24  lemma Gamma_series_LIMSEQ [tendsto_intros]:
5.25 @@ -1364,7 +1365,7 @@
5.26              pochhammer' = \<lambda>a n. \<Prod>n = 0..n. a + of_nat n
5.27          in  (\<lambda>n. pochhammer' z n / (fact' n * exp (z * ln (real_of_nat n) *\<^sub>R 1))) \<longlonglongrightarrow> rGamma z"
5.28      by (simp add: fact_altdef pochhammer_Suc_setprod rGamma_series_def [abs_def] exp_def
5.29 -                  of_real_def [symmetric] suminf_def sums_def [abs_def])
5.30 +                  of_real_def [symmetric] suminf_def sums_def [abs_def] atLeast0AtMost)
5.31  qed
5.32
5.33  end
5.34 @@ -1497,7 +1498,7 @@
5.35              pochhammer' = \<lambda>a n. \<Prod>n = 0..n. a + of_nat n
5.36          in  (\<lambda>n. pochhammer' x n / (fact' n * exp (x * ln (real_of_nat n) *\<^sub>R 1))) \<longlonglongrightarrow> rGamma x"
5.37      by (simp add: fact_altdef pochhammer_Suc_setprod rGamma_series_def [abs_def] exp_def
5.38 -                  of_real_def [symmetric] suminf_def sums_def [abs_def])
5.39 +                  of_real_def [symmetric] suminf_def sums_def [abs_def] atLeast0AtMost)
5.40  qed
5.41
5.42  end
5.43 @@ -2424,7 +2425,7 @@
5.44                          setprod_inversef[symmetric] divide_inverse)
5.45      also have "(\<Prod>k=1..n. (1 + z / of_nat k)) = pochhammer (z + 1) n / fact n"
5.46        by (cases n) (simp_all add: pochhammer_def fact_altdef setprod_shift_bounds_cl_Suc_ivl
5.47 -                                  setprod_dividef[symmetric] divide_simps add_ac)
5.48 +                                  setprod_dividef[symmetric] divide_simps add_ac atLeast0AtMost lessThan_Suc_atMost)
5.49      also have "z * \<dots> = pochhammer z (Suc n) / fact n" by (simp add: pochhammer_rec)
5.50      finally show "?r n = Gamma_series_euler' z n / Gamma_series z n" by simp
5.51    qed
```
```     6.1 --- a/src/HOL/Multivariate_Analysis/Generalised_Binomial_Theorem.thy	Sat Jul 02 15:02:24 2016 +0200
6.2 +++ b/src/HOL/Multivariate_Analysis/Generalised_Binomial_Theorem.thy	Sat Jul 02 20:22:25 2016 +0200
6.3 @@ -27,11 +27,12 @@
6.4      show "eventually (\<lambda>n. ?f n = (a gchoose n) /(a gchoose Suc n)) sequentially"
6.5    proof eventually_elim
6.6      fix n :: nat assume n: "n > 0"
6.7 -    let ?P = "\<Prod>i = 0..n - 1. a - of_nat i"
6.8 +    let ?P = "\<Prod>i<n. a - of_nat i"
6.9      from n have "(a gchoose n) / (a gchoose Suc n) = (of_nat (Suc n) :: 'a) *
6.10 -                   (?P / (\<Prod>i = 0..n. a - of_nat i))" by (simp add: gbinomial_def)
6.11 -    also from n have "(\<Prod>i = 0..n. a - of_nat i) = ?P * (a - of_nat n)"
6.12 -      by (cases n) (simp_all add: setprod_nat_ivl_Suc)
6.13 +                   (?P / (\<Prod>i\<le>n. a - of_nat i))"
6.14 +      by (simp add: gbinomial_def lessThan_Suc_atMost)
6.15 +    also from n have "(\<Prod>i\<le>n. a - of_nat i) = ?P * (a - of_nat n)"
6.16 +      by (cases n) (simp_all add: setprod_nat_ivl_Suc lessThan_Suc_atMost)
6.17      also have "?P / \<dots> = (?P / ?P) / (a - of_nat n)" by (rule divide_divide_eq_left[symmetric])
6.18      also from assms have "?P / ?P = 1" by auto
6.19      also have "of_nat (Suc n) * (1 / (a - of_nat n)) =
```
```     7.1 --- a/src/HOL/NthRoot.thy	Sat Jul 02 15:02:24 2016 +0200
7.2 +++ b/src/HOL/NthRoot.thy	Sat Jul 02 20:22:25 2016 +0200
7.3 @@ -655,7 +655,7 @@
7.4      { fix n :: nat assume "2 < n"
7.5        have "1 + (real (n - 1) * n) / 2 * x n^2 = 1 + of_nat (n choose 2) * x n^2"
7.6          using \<open>2 < n\<close> unfolding gbinomial_def binomial_gbinomial
7.7 -        by (simp add: atLeast0AtMost atMost_Suc field_simps of_nat_diff numeral_2_eq_2)
7.8 +        by (simp add: atLeast0AtMost lessThan_Suc field_simps of_nat_diff numeral_2_eq_2)
7.9        also have "\<dots> \<le> (\<Sum>k\<in>{0, 2}. of_nat (n choose k) * x n^k)"
7.10          by (simp add: x_def)
7.11        also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)"
7.12 @@ -692,7 +692,8 @@
7.13             (simp_all add: at_infinity_eq_at_top_bot)
7.14        { fix n :: nat assume "1 < n"
7.15          have "1 + x n * n = 1 + of_nat (n choose 1) * x n^1"
7.16 -          using \<open>1 < n\<close> unfolding gbinomial_def binomial_gbinomial by simp
7.17 +          using \<open>1 < n\<close> unfolding gbinomial_def binomial_gbinomial
7.18 +            by (simp add: lessThan_Suc)
7.19          also have "\<dots> \<le> (\<Sum>k\<in>{0, 1}. of_nat (n choose k) * x n^k)"
7.20            by (simp add: x_def)
7.21          also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)"
```
```     8.1 --- a/src/HOL/Transcendental.thy	Sat Jul 02 15:02:24 2016 +0200
8.2 +++ b/src/HOL/Transcendental.thy	Sat Jul 02 20:22:25 2016 +0200
8.3 @@ -1758,7 +1758,7 @@
8.4        by (rule mult_mono)
8.5          (rule mult_mono, simp_all add: power_le_one a b)
8.6      hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)"
8.7 -      unfolding power_add by (simp add: ac_simps del: fact.simps) }
8.8 +      unfolding power_add by (simp add: ac_simps del: fact_Suc) }
8.9    note aux1 = this
8.10    have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))"
8.11      by (intro sums_mult geometric_sums, simp)
8.12 @@ -3319,7 +3319,7 @@
8.13  lemma cos_two_less_zero [simp]:
8.14    "cos 2 < (0::real)"
8.15  proof -
8.16 -  note fact.simps(2) [simp del]
8.17 +  note fact_Suc [simp del]
8.18    from sums_minus [OF cos_paired]
8.19    have *: "(\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / fact (2 * n))) sums - cos (2::real)"
8.20      by simp
8.21 @@ -3335,7 +3335,7 @@
8.22        have "(4::real) * (fact (?six4d)) < (Suc (Suc (?six4d)) * fact (Suc (?six4d)))"
8.23          unfolding of_nat_mult   by (rule mult_strict_mono) (simp_all add: fact_less_mono)
8.24        then have "(4::real) * (fact (?six4d)) < (fact (Suc (Suc (?six4d))))"
8.25 -        by (simp only: fact.simps(2) [of "Suc (?six4d)"] of_nat_mult of_nat_fact)
8.26 +        by (simp only: fact_Suc [of "Suc (?six4d)"] of_nat_mult of_nat_fact)
8.27        then have "(4::real) * inverse (fact (Suc (Suc (?six4d)))) < inverse (fact (?six4d))"
8.28          by (simp add: inverse_eq_divide less_divide_eq)
8.29      }
```
```     9.1 --- a/src/HOL/ex/Sum_of_Powers.thy	Sat Jul 02 15:02:24 2016 +0200
9.2 +++ b/src/HOL/ex/Sum_of_Powers.thy	Sat Jul 02 20:22:25 2016 +0200
9.3 @@ -147,7 +147,7 @@
9.4
9.5  lemma binomial_unroll:
9.6    "n > 0 \<Longrightarrow> (n choose k) = (if k = 0 then 1 else (n - 1) choose (k - 1) + ((n - 1) choose k))"
9.7 -by (cases n) (auto simp add: binomial.simps(2))
9.8 +  by (auto simp add: gr0_conv_Suc)
9.9
9.10  lemma setsum_unroll:
9.11    "(\<Sum>k\<le>n::nat. f k) = (if n = 0 then f 0 else f n + (\<Sum>k\<le>n - 1. f k))"
9.12 @@ -157,7 +157,7 @@
9.13    "n > 0 \<Longrightarrow> bernoulli n = - 1 / (real n + 1) * (\<Sum>k\<le>n - 1. real (n + 1 choose k) * bernoulli k)"
9.14  by (cases n) (simp add: bernoulli.simps One_nat_def)+
9.15
9.16 -lemmas unroll = binomial.simps(1) binomial_unroll
9.17 +lemmas unroll = binomial_unroll
9.18    bernoulli.simps(1) bernoulli_unroll setsum_unroll bernpoly_def
9.19
9.20  lemma sum_of_squares: "(\<Sum>k\<le>n::nat. k ^ 2) = (2 * n ^ 3 + 3 * n ^ 2 + n) / 6"
```