src/HOL/Power.thy
 author paulson Fri Jan 09 10:46:18 2004 +0100 (2004-01-09) changeset 14348 744c868ee0b7 parent 8844 db71c334e854 child 14353 79f9fbef9106 permissions -rw-r--r--
Defining the type class "ringpower" and deleting superseded theorems for
types nat, int, real, hypreal
```     1 (*  Title:      HOL/Power.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1997  University of Cambridge
```
```     5
```
```     6 *)
```
```     7
```
```     8 header{*Exponentiation and Binomial Coefficients*}
```
```     9
```
```    10 theory Power = Divides:
```
```    11
```
```    12 subsection{*Powers for Arbitrary (Semi)Rings*}
```
```    13
```
```    14 axclass ringpower \<subseteq> semiring, power
```
```    15   power_0 [simp]:   "a ^ 0       = 1"
```
```    16   power_Suc: "a ^ (Suc n) = a * (a ^ n)"
```
```    17
```
```    18 lemma power_0_Suc [simp]: "(0::'a::ringpower) ^ (Suc n) = 0"
```
```    19 by (simp add: power_Suc)
```
```    20
```
```    21 text{*It looks plausible as a simprule, but its effect can be strange.*}
```
```    22 lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::ringpower))"
```
```    23 by (induct_tac "n", auto)
```
```    24
```
```    25 lemma power_one [simp]: "1^n = (1::'a::ringpower)"
```
```    26 apply (induct_tac "n")
```
```    27 apply (auto simp add: power_Suc)
```
```    28 done
```
```    29
```
```    30 lemma power_one_right [simp]: "(a::'a::ringpower) ^ 1 = a"
```
```    31 by (simp add: power_Suc)
```
```    32
```
```    33 lemma power_add: "(a::'a::ringpower) ^ (m+n) = (a^m) * (a^n)"
```
```    34 apply (induct_tac "n")
```
```    35 apply (simp_all add: power_Suc mult_ac)
```
```    36 done
```
```    37
```
```    38 lemma power_mult: "(a::'a::ringpower) ^ (m*n) = (a^m) ^ n"
```
```    39 apply (induct_tac "n")
```
```    40 apply (simp_all add: power_Suc power_add)
```
```    41 done
```
```    42
```
```    43 lemma power_mult_distrib: "((a::'a::ringpower) * b) ^ n = (a^n) * (b^n)"
```
```    44 apply (induct_tac "n")
```
```    45 apply (auto simp add: power_Suc mult_ac)
```
```    46 done
```
```    47
```
```    48 lemma zero_less_power:
```
```    49      "0 < (a::'a::{ordered_semiring,ringpower}) ==> 0 < a^n"
```
```    50 apply (induct_tac "n")
```
```    51 apply (simp_all add: power_Suc zero_less_one mult_pos)
```
```    52 done
```
```    53
```
```    54 lemma zero_le_power:
```
```    55      "0 \<le> (a::'a::{ordered_semiring,ringpower}) ==> 0 \<le> a^n"
```
```    56 apply (simp add: order_le_less)
```
```    57 apply (erule disjE)
```
```    58 apply (simp_all add: zero_less_power zero_less_one power_0_left)
```
```    59 done
```
```    60
```
```    61 lemma one_le_power:
```
```    62      "1 \<le> (a::'a::{ordered_semiring,ringpower}) ==> 1 \<le> a^n"
```
```    63 apply (induct_tac "n")
```
```    64 apply (simp_all add: power_Suc)
```
```    65 apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
```
```    66 apply (simp_all add: zero_le_one order_trans [OF zero_le_one])
```
```    67 done
```
```    68
```
```    69 lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semiring)"
```
```    70   by (simp add: order_trans [OF zero_le_one order_less_imp_le])
```
```    71
```
```    72 lemma power_gt1_lemma:
```
```    73      assumes gt1: "1 < (a::'a::{ordered_semiring,ringpower})"
```
```    74         shows     "1 < a * a^n"
```
```    75 proof -
```
```    76   have "1*1 < a * a^n"
```
```    77     proof (rule order_less_le_trans)
```
```    78       show "1*1 < a*1" by (simp add: gt1)
```
```    79       show  "a*1 \<le> a * a^n"
```
```    80    by (simp only: mult_mono gt1 gt1_imp_ge0 one_le_power order_less_imp_le
```
```    81                   zero_le_one order_refl)
```
```    82     qed
```
```    83   thus ?thesis by simp
```
```    84 qed
```
```    85
```
```    86 lemma power_gt1:
```
```    87      "1 < (a::'a::{ordered_semiring,ringpower}) ==> 1 < a ^ (Suc n)"
```
```    88 by (simp add: power_gt1_lemma power_Suc)
```
```    89
```
```    90 lemma power_le_imp_le_exp:
```
```    91      assumes gt1: "(1::'a::{ringpower,ordered_semiring}) < a"
```
```    92        shows      "!!n. a^m \<le> a^n ==> m \<le> n"
```
```    93 proof (induct "m")
```
```    94   case 0
```
```    95     show ?case by simp
```
```    96 next
```
```    97   case (Suc m)
```
```    98     show ?case
```
```    99       proof (cases n)
```
```   100         case 0
```
```   101           from prems have "a * a^m \<le> 1" by (simp add: power_Suc)
```
```   102           with gt1 show ?thesis
```
```   103             by (force simp only: power_gt1_lemma
```
```   104                                  linorder_not_less [symmetric])
```
```   105       next
```
```   106         case (Suc n)
```
```   107           from prems show ?thesis
```
```   108 	    by (force dest: mult_left_le_imp_le
```
```   109 	          simp add: power_Suc order_less_trans [OF zero_less_one gt1])
```
```   110       qed
```
```   111 qed
```
```   112
```
```   113 text{*Surely we can strengthen this? It holds for 0<a<1 too.*}
```
```   114 lemma power_inject_exp [simp]:
```
```   115      "1 < (a::'a::{ordered_semiring,ringpower}) ==> (a^m = a^n) = (m=n)"
```
```   116   by (force simp add: order_antisym power_le_imp_le_exp)
```
```   117
```
```   118 text{*Can relax the first premise to @{term "0<a"} in the case of the
```
```   119 natural numbers.*}
```
```   120 lemma power_less_imp_less_exp:
```
```   121      "[| (1::'a::{ringpower,ordered_semiring}) < a; a^m < a^n |] ==> m < n"
```
```   122 by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
```
```   123               power_le_imp_le_exp)
```
```   124
```
```   125
```
```   126 lemma power_mono:
```
```   127      "[|a \<le> b; (0::'a::{ringpower,ordered_semiring}) \<le> a|] ==> a^n \<le> b^n"
```
```   128 apply (induct_tac "n")
```
```   129 apply (simp_all add: power_Suc)
```
```   130 apply (auto intro: mult_mono zero_le_power order_trans [of 0 a b])
```
```   131 done
```
```   132
```
```   133 lemma power_strict_mono [rule_format]:
```
```   134      "[|a < b; (0::'a::{ringpower,ordered_semiring}) \<le> a|]
```
```   135       ==> 0 < n --> a^n < b^n"
```
```   136 apply (induct_tac "n")
```
```   137 apply (auto simp add: mult_strict_mono zero_le_power power_Suc
```
```   138                       order_le_less_trans [of 0 a b])
```
```   139 done
```
```   140
```
```   141 lemma power_eq_0_iff [simp]:
```
```   142      "(a^n = 0) = (a = (0::'a::{ordered_ring,ringpower}) & 0<n)"
```
```   143 apply (induct_tac "n")
```
```   144 apply (auto simp add: power_Suc zero_neq_one [THEN not_sym])
```
```   145 done
```
```   146
```
```   147 lemma field_power_eq_0_iff [simp]:
```
```   148      "(a^n = 0) = (a = (0::'a::{field,ringpower}) & 0<n)"
```
```   149 apply (induct_tac "n")
```
```   150 apply (auto simp add: power_Suc field_mult_eq_0_iff zero_neq_one[THEN not_sym])
```
```   151 done
```
```   152
```
```   153 lemma field_power_not_zero: "a \<noteq> (0::'a::{field,ringpower}) ==> a^n \<noteq> 0"
```
```   154 by force
```
```   155
```
```   156 text{*Perhaps these should be simprules.*}
```
```   157 lemma power_inverse:
```
```   158   "inverse ((a::'a::{field,division_by_zero,ringpower}) ^ n) = (inverse a) ^ n"
```
```   159 apply (induct_tac "n")
```
```   160 apply (auto simp add: power_Suc inverse_mult_distrib)
```
```   161 done
```
```   162
```
```   163 lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_field,ringpower}) ^ n"
```
```   164 apply (induct_tac "n")
```
```   165 apply (auto simp add: power_Suc abs_mult)
```
```   166 done
```
```   167
```
```   168 lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{ring,ringpower}) ^ n"
```
```   169 proof -
```
```   170   have "-a = (- 1) * a"  by (simp add: minus_mult_left [symmetric])
```
```   171   thus ?thesis by (simp only: power_mult_distrib)
```
```   172 qed
```
```   173
```
```   174 text{*Lemma for @{text power_strict_decreasing}*}
```
```   175 lemma power_Suc_less:
```
```   176      "[|(0::'a::{ordered_semiring,ringpower}) < a; a < 1|]
```
```   177       ==> a * a^n < a^n"
```
```   178 apply (induct_tac n)
```
```   179 apply (auto simp add: power_Suc mult_strict_left_mono)
```
```   180 done
```
```   181
```
```   182 lemma power_strict_decreasing:
```
```   183      "[|n < N; 0 < a; a < (1::'a::{ordered_semiring,ringpower})|]
```
```   184       ==> a^N < a^n"
```
```   185 apply (erule rev_mp)
```
```   186 apply (induct_tac "N")
```
```   187 apply (auto simp add: power_Suc power_Suc_less less_Suc_eq)
```
```   188 apply (rename_tac m)
```
```   189 apply (subgoal_tac "a * a^m < 1 * a^n", simp)
```
```   190 apply (rule mult_strict_mono)
```
```   191 apply (auto simp add: zero_le_power zero_less_one order_less_imp_le)
```
```   192 done
```
```   193
```
```   194 text{*Proof resembles that of @{text power_strict_decreasing}*}
```
```   195 lemma power_decreasing:
```
```   196      "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semiring,ringpower})|]
```
```   197       ==> a^N \<le> a^n"
```
```   198 apply (erule rev_mp)
```
```   199 apply (induct_tac "N")
```
```   200 apply (auto simp add: power_Suc  le_Suc_eq)
```
```   201 apply (rename_tac m)
```
```   202 apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
```
```   203 apply (rule mult_mono)
```
```   204 apply (auto simp add: zero_le_power zero_le_one)
```
```   205 done
```
```   206
```
```   207 lemma power_Suc_less_one:
```
```   208      "[| 0 < a; a < (1::'a::{ordered_semiring,ringpower}) |] ==> a ^ Suc n < 1"
```
```   209 apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
```
```   210 done
```
```   211
```
```   212 text{*Proof again resembles that of @{text power_strict_decreasing}*}
```
```   213 lemma power_increasing:
```
```   214      "[|n \<le> N; (1::'a::{ordered_semiring,ringpower}) \<le> a|] ==> a^n \<le> a^N"
```
```   215 apply (erule rev_mp)
```
```   216 apply (induct_tac "N")
```
```   217 apply (auto simp add: power_Suc le_Suc_eq)
```
```   218 apply (rename_tac m)
```
```   219 apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
```
```   220 apply (rule mult_mono)
```
```   221 apply (auto simp add: order_trans [OF zero_le_one] zero_le_power)
```
```   222 done
```
```   223
```
```   224 text{*Lemma for @{text power_strict_increasing}*}
```
```   225 lemma power_less_power_Suc:
```
```   226      "(1::'a::{ordered_semiring,ringpower}) < a ==> a^n < a * a^n"
```
```   227 apply (induct_tac n)
```
```   228 apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one])
```
```   229 done
```
```   230
```
```   231 lemma power_strict_increasing:
```
```   232      "[|n < N; (1::'a::{ordered_semiring,ringpower}) < a|] ==> a^n < a^N"
```
```   233 apply (erule rev_mp)
```
```   234 apply (induct_tac "N")
```
```   235 apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq)
```
```   236 apply (rename_tac m)
```
```   237 apply (subgoal_tac "1 * a^n < a * a^m", simp)
```
```   238 apply (rule mult_strict_mono)
```
```   239 apply (auto simp add: order_less_trans [OF zero_less_one] zero_le_power
```
```   240                  order_less_imp_le)
```
```   241 done
```
```   242
```
```   243 lemma power_le_imp_le_base:
```
```   244   assumes le: "a ^ Suc n \<le> b ^ Suc n"
```
```   245       and xnonneg: "(0::'a::{ordered_semiring,ringpower}) \<le> a"
```
```   246       and ynonneg: "0 \<le> b"
```
```   247   shows "a \<le> b"
```
```   248  proof (rule ccontr)
```
```   249    assume "~ a \<le> b"
```
```   250    then have "b < a" by (simp only: linorder_not_le)
```
```   251    then have "b ^ Suc n < a ^ Suc n"
```
```   252      by (simp only: prems power_strict_mono)
```
```   253    from le and this show "False"
```
```   254       by (simp add: linorder_not_less [symmetric])
```
```   255  qed
```
```   256
```
```   257 lemma power_inject_base:
```
```   258      "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
```
```   259       ==> a = (b::'a::{ordered_semiring,ringpower})"
```
```   260 by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
```
```   261
```
```   262
```
```   263 subsection{*Exponentiation for the Natural Numbers*}
```
```   264
```
```   265 primrec (power)
```
```   266   "p ^ 0 = 1"
```
```   267   "p ^ (Suc n) = (p::nat) * (p ^ n)"
```
```   268
```
```   269 instance nat :: ringpower
```
```   270 proof
```
```   271   fix z :: nat
```
```   272   fix n :: nat
```
```   273   show "z^0 = 1" by simp
```
```   274   show "z^(Suc n) = z * (z^n)" by simp
```
```   275 qed
```
```   276
```
```   277 lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n"
```
```   278 by (insert one_le_power [of i n], simp)
```
```   279
```
```   280 lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
```
```   281 apply (unfold dvd_def)
```
```   282 apply (erule not_less_iff_le [THEN iffD2, THEN add_diff_inverse, THEN subst])
```
```   283 apply (simp add: power_add)
```
```   284 done
```
```   285
```
```   286 text{*Valid for the naturals, but what if @{text"0<i<1"}?
```
```   287 Premises cannot be weakened: consider the case where @{term "i=0"},
```
```   288 @{term "m=1"} and @{term "n=0"}.*}
```
```   289 lemma nat_power_less_imp_less: "!!i::nat. [| 0 < i; i^m < i^n |] ==> m < n"
```
```   290 apply (rule ccontr)
```
```   291 apply (drule leI [THEN le_imp_power_dvd, THEN dvd_imp_le, THEN leD])
```
```   292 apply (erule zero_less_power, auto)
```
```   293 done
```
```   294
```
```   295 lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
```
```   296 by (induct_tac "n", auto)
```
```   297
```
```   298 lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
```
```   299 apply (induct_tac "j")
```
```   300 apply (simp_all add: le_Suc_eq)
```
```   301 apply (blast dest!: dvd_mult_right)
```
```   302 done
```
```   303
```
```   304 lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
```
```   305 apply (rule power_le_imp_le_exp, assumption)
```
```   306 apply (erule dvd_imp_le, simp)
```
```   307 done
```
```   308
```
```   309
```
```   310 subsection{*Binomial Coefficients*}
```
```   311
```
```   312 text{*This development is based on the work of Andy Gordon and
```
```   313 Florian Kammueller*}
```
```   314
```
```   315 consts
```
```   316   binomial :: "[nat,nat] => nat"      (infixl "choose" 65)
```
```   317
```
```   318 primrec
```
```   319   binomial_0:   "(0     choose k) = (if k = 0 then 1 else 0)"
```
```   320
```
```   321   binomial_Suc: "(Suc n choose k) =
```
```   322                  (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
```
```   323
```
```   324 lemma binomial_n_0 [simp]: "(n choose 0) = 1"
```
```   325 by (case_tac "n", simp_all)
```
```   326
```
```   327 lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
```
```   328 by simp
```
```   329
```
```   330 lemma binomial_Suc_Suc [simp]:
```
```   331      "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
```
```   332 by simp
```
```   333
```
```   334 lemma binomial_eq_0 [rule_format]: "\<forall>k. n < k --> (n choose k) = 0"
```
```   335 apply (induct_tac "n", auto)
```
```   336 apply (erule allE)
```
```   337 apply (erule mp, arith)
```
```   338 done
```
```   339
```
```   340 declare binomial_0 [simp del] binomial_Suc [simp del]
```
```   341
```
```   342 lemma binomial_n_n [simp]: "(n choose n) = 1"
```
```   343 apply (induct_tac "n")
```
```   344 apply (simp_all add: binomial_eq_0)
```
```   345 done
```
```   346
```
```   347 lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n"
```
```   348 by (induct_tac "n", simp_all)
```
```   349
```
```   350 lemma binomial_1 [simp]: "(n choose Suc 0) = n"
```
```   351 by (induct_tac "n", simp_all)
```
```   352
```
```   353 lemma zero_less_binomial [rule_format]: "k \<le> n --> 0 < (n choose k)"
```
```   354 by (rule_tac m = n and n = k in diff_induct, simp_all)
```
```   355
```
```   356 lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)"
```
```   357 apply (safe intro!: binomial_eq_0)
```
```   358 apply (erule contrapos_pp)
```
```   359 apply (simp add: zero_less_binomial)
```
```   360 done
```
```   361
```
```   362 lemma zero_less_binomial_iff: "(0 < n choose k) = (k\<le>n)"
```
```   363 by (simp add: linorder_not_less [symmetric] binomial_eq_0_iff [symmetric])
```
```   364
```
```   365 (*Might be more useful if re-oriented*)
```
```   366 lemma Suc_times_binomial_eq [rule_format]:
```
```   367      "\<forall>k. k \<le> n --> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
```
```   368 apply (induct_tac "n")
```
```   369 apply (simp add: binomial_0, clarify)
```
```   370 apply (case_tac "k")
```
```   371 apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq
```
```   372                       binomial_eq_0)
```
```   373 done
```
```   374
```
```   375 text{*This is the well-known version, but it's harder to use because of the
```
```   376   need to reason about division.*}
```
```   377 lemma binomial_Suc_Suc_eq_times:
```
```   378      "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
```
```   379 by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc
```
```   380         del: mult_Suc mult_Suc_right)
```
```   381
```
```   382 text{*Another version, with -1 instead of Suc.*}
```
```   383 lemma times_binomial_minus1_eq:
```
```   384      "[|k \<le> n;  0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"
```
```   385 apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)
```
```   386 apply (simp split add: nat_diff_split, auto)
```
```   387 done
```
```   388
```
```   389 text{*ML bindings for the general exponentiation theorems*}
```
```   390 ML
```
```   391 {*
```
```   392 val power_0 = thm"power_0";
```
```   393 val power_Suc = thm"power_Suc";
```
```   394 val power_0_Suc = thm"power_0_Suc";
```
```   395 val power_0_left = thm"power_0_left";
```
```   396 val power_one = thm"power_one";
```
```   397 val power_one_right = thm"power_one_right";
```
```   398 val power_add = thm"power_add";
```
```   399 val power_mult = thm"power_mult";
```
```   400 val power_mult_distrib = thm"power_mult_distrib";
```
```   401 val zero_less_power = thm"zero_less_power";
```
```   402 val zero_le_power = thm"zero_le_power";
```
```   403 val one_le_power = thm"one_le_power";
```
```   404 val gt1_imp_ge0 = thm"gt1_imp_ge0";
```
```   405 val power_gt1_lemma = thm"power_gt1_lemma";
```
```   406 val power_gt1 = thm"power_gt1";
```
```   407 val power_le_imp_le_exp = thm"power_le_imp_le_exp";
```
```   408 val power_inject_exp = thm"power_inject_exp";
```
```   409 val power_less_imp_less_exp = thm"power_less_imp_less_exp";
```
```   410 val power_mono = thm"power_mono";
```
```   411 val power_strict_mono = thm"power_strict_mono";
```
```   412 val power_eq_0_iff = thm"power_eq_0_iff";
```
```   413 val field_power_eq_0_iff = thm"field_power_eq_0_iff";
```
```   414 val field_power_not_zero = thm"field_power_not_zero";
```
```   415 val power_inverse = thm"power_inverse";
```
```   416 val power_abs = thm"power_abs";
```
```   417 val power_minus = thm"power_minus";
```
```   418 val power_Suc_less = thm"power_Suc_less";
```
```   419 val power_strict_decreasing = thm"power_strict_decreasing";
```
```   420 val power_decreasing = thm"power_decreasing";
```
```   421 val power_Suc_less_one = thm"power_Suc_less_one";
```
```   422 val power_increasing = thm"power_increasing";
```
```   423 val power_strict_increasing = thm"power_strict_increasing";
```
```   424 val power_le_imp_le_base = thm"power_le_imp_le_base";
```
```   425 val power_inject_base = thm"power_inject_base";
```
```   426 *}
```
```   427
```
```   428 text{*ML bindings for the remaining theorems*}
```
```   429 ML
```
```   430 {*
```
```   431 val nat_one_le_power = thm"nat_one_le_power";
```
```   432 val le_imp_power_dvd = thm"le_imp_power_dvd";
```
```   433 val nat_power_less_imp_less = thm"nat_power_less_imp_less";
```
```   434 val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";
```
```   435 val power_le_dvd = thm"power_le_dvd";
```
```   436 val power_dvd_imp_le = thm"power_dvd_imp_le";
```
```   437 val binomial_n_0 = thm"binomial_n_0";
```
```   438 val binomial_0_Suc = thm"binomial_0_Suc";
```
```   439 val binomial_Suc_Suc = thm"binomial_Suc_Suc";
```
```   440 val binomial_eq_0 = thm"binomial_eq_0";
```
```   441 val binomial_n_n = thm"binomial_n_n";
```
```   442 val binomial_Suc_n = thm"binomial_Suc_n";
```
```   443 val binomial_1 = thm"binomial_1";
```
```   444 val zero_less_binomial = thm"zero_less_binomial";
```
```   445 val binomial_eq_0_iff = thm"binomial_eq_0_iff";
```
```   446 val zero_less_binomial_iff = thm"zero_less_binomial_iff";
```
```   447 val Suc_times_binomial_eq = thm"Suc_times_binomial_eq";
```
```   448 val binomial_Suc_Suc_eq_times = thm"binomial_Suc_Suc_eq_times";
```
```   449 val times_binomial_minus1_eq = thm"times_binomial_minus1_eq";
```
```   450 *}
```
```   451
```
```   452 end
```
```   453
```