src/HOL/Power.thy
 author nipkow Fri Mar 06 17:38:47 2009 +0100 (2009-03-06) changeset 30313 b2441b0c8d38 parent 30273 ecd6f0ca62ea child 30516 68b4a06cbd5c permissions -rw-r--r--
```     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*}
```
```     9
```
```    10 theory Power
```
```    11 imports Nat
```
```    12 begin
```
```    13
```
```    14 class power =
```
```    15   fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"            (infixr "^" 80)
```
```    16
```
```    17 subsection{*Powers for Arbitrary Monoids*}
```
```    18
```
```    19 class recpower = monoid_mult + power +
```
```    20   assumes power_0 [simp]: "a ^ 0       = 1"
```
```    21   assumes power_Suc [simp]: "a ^ Suc n = a * (a ^ n)"
```
```    22
```
```    23 lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"
```
```    24   by simp
```
```    25
```
```    26 text{*It looks plausible as a simprule, but its effect can be strange.*}
```
```    27 lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::{recpower,semiring_0}))"
```
```    28   by (induct n) simp_all
```
```    29
```
```    30 lemma power_one [simp]: "1^n = (1::'a::recpower)"
```
```    31   by (induct n) simp_all
```
```    32
```
```    33 lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
```
```    34   unfolding One_nat_def by simp
```
```    35
```
```    36 lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
```
```    37   by (induct n) (simp_all add: mult_assoc)
```
```    38
```
```    39 lemma power_Suc2: "(a::'a::recpower) ^ Suc n = a ^ n * a"
```
```    40   by (simp add: power_commutes)
```
```    41
```
```    42 lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
```
```    43   by (induct m) (simp_all add: mult_ac)
```
```    44
```
```    45 lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
```
```    46   by (induct n) (simp_all add: power_add)
```
```    47
```
```    48 lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"
```
```    49   by (induct n) (simp_all add: mult_ac)
```
```    50
```
```    51 lemma zero_less_power[simp]:
```
```    52      "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"
```
```    53 by (induct n) (simp_all add: mult_pos_pos)
```
```    54
```
```    55 lemma zero_le_power[simp]:
```
```    56      "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"
```
```    57 by (induct n) (simp_all add: mult_nonneg_nonneg)
```
```    58
```
```    59 lemma one_le_power[simp]:
```
```    60      "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"
```
```    61 apply (induct "n")
```
```    62 apply simp_all
```
```    63 apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
```
```    64 apply (simp_all add: order_trans [OF zero_le_one])
```
```    65 done
```
```    66
```
```    67 lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semidom)"
```
```    68   by (simp add: order_trans [OF zero_le_one order_less_imp_le])
```
```    69
```
```    70 lemma power_gt1_lemma:
```
```    71   assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})"
```
```    72   shows "1 < a * a^n"
```
```    73 proof -
```
```    74   have "1*1 < a*1" using gt1 by simp
```
```    75   also have "\<dots> \<le> a * a^n" using gt1
```
```    76     by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le
```
```    77         zero_le_one order_refl)
```
```    78   finally show ?thesis by simp
```
```    79 qed
```
```    80
```
```    81 lemma one_less_power[simp]:
```
```    82   "\<lbrakk>1 < (a::'a::{ordered_semidom,recpower}); 0 < n\<rbrakk> \<Longrightarrow> 1 < a ^ n"
```
```    83 by (cases n, simp_all add: power_gt1_lemma)
```
```    84
```
```    85 lemma power_gt1:
```
```    86      "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
```
```    87 by (simp add: power_gt1_lemma)
```
```    88
```
```    89 lemma power_le_imp_le_exp:
```
```    90   assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"
```
```    91   shows "!!n. a^m \<le> a^n ==> m \<le> n"
```
```    92 proof (induct m)
```
```    93   case 0
```
```    94   show ?case by simp
```
```    95 next
```
```    96   case (Suc m)
```
```    97   show ?case
```
```    98   proof (cases n)
```
```    99     case 0
```
```   100     from prems have "a * a^m \<le> 1" by simp
```
```   101     with gt1 show ?thesis
```
```   102       by (force simp only: power_gt1_lemma
```
```   103           linorder_not_less [symmetric])
```
```   104   next
```
```   105     case (Suc n)
```
```   106     from prems show ?thesis
```
```   107       by (force dest: mult_left_le_imp_le
```
```   108           simp add: order_less_trans [OF zero_less_one gt1])
```
```   109   qed
```
```   110 qed
```
```   111
```
```   112 text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
```
```   113 lemma power_inject_exp [simp]:
```
```   114      "1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)"
```
```   115   by (force simp add: order_antisym power_le_imp_le_exp)
```
```   116
```
```   117 text{*Can relax the first premise to @{term "0<a"} in the case of the
```
```   118 natural numbers.*}
```
```   119 lemma power_less_imp_less_exp:
```
```   120      "[| (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"
```
```   121 by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
```
```   122               power_le_imp_le_exp)
```
```   123
```
```   124
```
```   125 lemma power_mono:
```
```   126      "[|a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"
```
```   127 apply (induct "n")
```
```   128 apply simp_all
```
```   129 apply (auto intro: mult_mono order_trans [of 0 a b])
```
```   130 done
```
```   131
```
```   132 lemma power_strict_mono [rule_format]:
```
```   133      "[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]
```
```   134       ==> 0 < n --> a^n < b^n"
```
```   135 apply (induct "n")
```
```   136 apply (auto simp add: mult_strict_mono order_le_less_trans [of 0 a b])
```
```   137 done
```
```   138
```
```   139 lemma power_eq_0_iff [simp]:
```
```   140   "(a^n = 0) \<longleftrightarrow>
```
```   141    (a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,recpower}) & n\<noteq>0)"
```
```   142 apply (induct "n")
```
```   143 apply (auto simp add: no_zero_divisors)
```
```   144 done
```
```   145
```
```   146
```
```   147 lemma field_power_not_zero:
```
```   148   "a \<noteq> (0::'a::{ring_1_no_zero_divisors,recpower}) ==> a^n \<noteq> 0"
```
```   149 by force
```
```   150
```
```   151 lemma nonzero_power_inverse:
```
```   152   fixes a :: "'a::{division_ring,recpower}"
```
```   153   shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n"
```
```   154 apply (induct "n")
```
```   155 apply (auto simp add: nonzero_inverse_mult_distrib power_commutes)
```
```   156 done (* TODO: reorient or rename to nonzero_inverse_power *)
```
```   157
```
```   158 text{*Perhaps these should be simprules.*}
```
```   159 lemma power_inverse:
```
```   160   fixes a :: "'a::{division_ring,division_by_zero,recpower}"
```
```   161   shows "inverse (a ^ n) = (inverse a) ^ n"
```
```   162 apply (cases "a = 0")
```
```   163 apply (simp add: power_0_left)
```
```   164 apply (simp add: nonzero_power_inverse)
```
```   165 done (* TODO: reorient or rename to inverse_power *)
```
```   166
```
```   167 lemma power_one_over: "1 / (a::'a::{field,division_by_zero,recpower})^n =
```
```   168     (1 / a)^n"
```
```   169 apply (simp add: divide_inverse)
```
```   170 apply (rule power_inverse)
```
```   171 done
```
```   172
```
```   173 lemma nonzero_power_divide:
```
```   174     "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"
```
```   175 by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
```
```   176
```
```   177 lemma power_divide:
```
```   178     "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"
```
```   179 apply (case_tac "b=0", simp add: power_0_left)
```
```   180 apply (rule nonzero_power_divide)
```
```   181 apply assumption
```
```   182 done
```
```   183
```
```   184 lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"
```
```   185 apply (induct "n")
```
```   186 apply (auto simp add: abs_mult)
```
```   187 done
```
```   188
```
```   189 lemma zero_less_power_abs_iff [simp,noatp]:
```
```   190      "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
```
```   191 proof (induct "n")
```
```   192   case 0
```
```   193     show ?case by simp
```
```   194 next
```
```   195   case (Suc n)
```
```   196     show ?case by (auto simp add: prems zero_less_mult_iff)
```
```   197 qed
```
```   198
```
```   199 lemma zero_le_power_abs [simp]:
```
```   200      "(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"
```
```   201 by (rule zero_le_power [OF abs_ge_zero])
```
```   202
```
```   203 lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{ring_1,recpower}) ^ n"
```
```   204 proof (induct n)
```
```   205   case 0 show ?case by simp
```
```   206 next
```
```   207   case (Suc n) then show ?case
```
```   208     by (simp del: power_Suc add: power_Suc2 mult_assoc)
```
```   209 qed
```
```   210
```
```   211 text{*Lemma for @{text power_strict_decreasing}*}
```
```   212 lemma power_Suc_less:
```
```   213      "[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|]
```
```   214       ==> a * a^n < a^n"
```
```   215 apply (induct n)
```
```   216 apply (auto simp add: mult_strict_left_mono)
```
```   217 done
```
```   218
```
```   219 lemma power_strict_decreasing:
```
```   220      "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]
```
```   221       ==> a^N < a^n"
```
```   222 apply (erule rev_mp)
```
```   223 apply (induct "N")
```
```   224 apply (auto simp add: power_Suc_less less_Suc_eq)
```
```   225 apply (rename_tac m)
```
```   226 apply (subgoal_tac "a * a^m < 1 * a^n", simp)
```
```   227 apply (rule mult_strict_mono)
```
```   228 apply (auto simp add: order_less_imp_le)
```
```   229 done
```
```   230
```
```   231 text{*Proof resembles that of @{text power_strict_decreasing}*}
```
```   232 lemma power_decreasing:
```
```   233      "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]
```
```   234       ==> a^N \<le> a^n"
```
```   235 apply (erule rev_mp)
```
```   236 apply (induct "N")
```
```   237 apply (auto simp add: le_Suc_eq)
```
```   238 apply (rename_tac m)
```
```   239 apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
```
```   240 apply (rule mult_mono)
```
```   241 apply auto
```
```   242 done
```
```   243
```
```   244 lemma power_Suc_less_one:
```
```   245      "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1"
```
```   246 apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
```
```   247 done
```
```   248
```
```   249 text{*Proof again resembles that of @{text power_strict_decreasing}*}
```
```   250 lemma power_increasing:
```
```   251      "[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"
```
```   252 apply (erule rev_mp)
```
```   253 apply (induct "N")
```
```   254 apply (auto simp add: le_Suc_eq)
```
```   255 apply (rename_tac m)
```
```   256 apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
```
```   257 apply (rule mult_mono)
```
```   258 apply (auto simp add: order_trans [OF zero_le_one])
```
```   259 done
```
```   260
```
```   261 text{*Lemma for @{text power_strict_increasing}*}
```
```   262 lemma power_less_power_Suc:
```
```   263      "(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n"
```
```   264 apply (induct n)
```
```   265 apply (auto simp add: mult_strict_left_mono order_less_trans [OF zero_less_one])
```
```   266 done
```
```   267
```
```   268 lemma power_strict_increasing:
```
```   269      "[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N"
```
```   270 apply (erule rev_mp)
```
```   271 apply (induct "N")
```
```   272 apply (auto simp add: power_less_power_Suc less_Suc_eq)
```
```   273 apply (rename_tac m)
```
```   274 apply (subgoal_tac "1 * a^n < a * a^m", simp)
```
```   275 apply (rule mult_strict_mono)
```
```   276 apply (auto simp add: order_less_trans [OF zero_less_one] order_less_imp_le)
```
```   277 done
```
```   278
```
```   279 lemma power_increasing_iff [simp]:
```
```   280   "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
```
```   281 by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le)
```
```   282
```
```   283 lemma power_strict_increasing_iff [simp]:
```
```   284   "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"
```
```   285 by (blast intro: power_less_imp_less_exp power_strict_increasing)
```
```   286
```
```   287 lemma power_le_imp_le_base:
```
```   288 assumes le: "a ^ Suc n \<le> b ^ Suc n"
```
```   289     and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b"
```
```   290 shows "a \<le> b"
```
```   291 proof (rule ccontr)
```
```   292   assume "~ a \<le> b"
```
```   293   then have "b < a" by (simp only: linorder_not_le)
```
```   294   then have "b ^ Suc n < a ^ Suc n"
```
```   295     by (simp only: prems power_strict_mono)
```
```   296   from le and this show "False"
```
```   297     by (simp add: linorder_not_less [symmetric])
```
```   298 qed
```
```   299
```
```   300 lemma power_less_imp_less_base:
```
```   301   fixes a b :: "'a::{ordered_semidom,recpower}"
```
```   302   assumes less: "a ^ n < b ^ n"
```
```   303   assumes nonneg: "0 \<le> b"
```
```   304   shows "a < b"
```
```   305 proof (rule contrapos_pp [OF less])
```
```   306   assume "~ a < b"
```
```   307   hence "b \<le> a" by (simp only: linorder_not_less)
```
```   308   hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
```
```   309   thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less)
```
```   310 qed
```
```   311
```
```   312 lemma power_inject_base:
```
```   313      "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
```
```   314       ==> a = (b::'a::{ordered_semidom,recpower})"
```
```   315 by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
```
```   316
```
```   317 lemma power_eq_imp_eq_base:
```
```   318   fixes a b :: "'a::{ordered_semidom,recpower}"
```
```   319   shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b"
```
```   320 by (cases n, simp_all del: power_Suc, rule power_inject_base)
```
```   321
```
```   322 text {* The divides relation *}
```
```   323
```
```   324 lemma le_imp_power_dvd:
```
```   325   fixes a :: "'a::{comm_semiring_1,recpower}"
```
```   326   assumes "m \<le> n" shows "a^m dvd a^n"
```
```   327 proof
```
```   328   have "a^n = a^(m + (n - m))"
```
```   329     using `m \<le> n` by simp
```
```   330   also have "\<dots> = a^m * a^(n - m)"
```
```   331     by (rule power_add)
```
```   332   finally show "a^n = a^m * a^(n - m)" .
```
```   333 qed
```
```   334
```
```   335 lemma power_le_dvd:
```
```   336   fixes a b :: "'a::{comm_semiring_1,recpower}"
```
```   337   shows "a^n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a^m dvd b"
```
```   338   by (rule dvd_trans [OF le_imp_power_dvd])
```
```   339
```
```   340
```
```   341 lemma dvd_power_same:
```
```   342   "(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> x^n dvd y^n"
```
```   343 by (induct n) (auto simp add: mult_dvd_mono)
```
```   344
```
```   345 lemma dvd_power_le:
```
```   346   "(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> m >= n \<Longrightarrow> x^n dvd y^m"
```
```   347 by(rule power_le_dvd[OF dvd_power_same])
```
```   348
```
```   349 lemma dvd_power [simp]:
```
```   350   "n > 0 | (x::'a::{comm_semiring_1,recpower}) = 1 \<Longrightarrow> x dvd x^n"
```
```   351 apply (erule disjE)
```
```   352  apply (subgoal_tac "x ^ n = x^(Suc (n - 1))")
```
```   353   apply (erule ssubst)
```
```   354   apply (subst power_Suc)
```
```   355   apply auto
```
```   356 done
```
```   357
```
```   358
```
```   359 subsection{*Exponentiation for the Natural Numbers*}
```
```   360
```
```   361 instantiation nat :: recpower
```
```   362 begin
```
```   363
```
```   364 primrec power_nat where
```
```   365   "p ^ 0 = (1\<Colon>nat)"
```
```   366   | "p ^ (Suc n) = (p\<Colon>nat) * (p ^ n)"
```
```   367
```
```   368 instance proof
```
```   369   fix z n :: nat
```
```   370   show "z^0 = 1" by simp
```
```   371   show "z^(Suc n) = z * (z^n)" by simp
```
```   372 qed
```
```   373
```
```   374 declare power_nat.simps [simp del]
```
```   375
```
```   376 end
```
```   377
```
```   378 lemma of_nat_power:
```
```   379   "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"
```
```   380 by (induct n, simp_all add: of_nat_mult)
```
```   381
```
```   382 lemma nat_one_le_power [simp]: "Suc 0 \<le> i ==> Suc 0 \<le> i^n"
```
```   383 by (rule one_le_power [of i n, unfolded One_nat_def])
```
```   384
```
```   385 lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
```
```   386 by (induct "n", auto)
```
```   387
```
```   388 lemma nat_power_eq_Suc_0_iff [simp]:
```
```   389   "((x::nat)^m = Suc 0) = (m = 0 | x = Suc 0)"
```
```   390 by (induct_tac m, auto)
```
```   391
```
```   392 lemma power_Suc_0[simp]: "(Suc 0)^n = Suc 0"
```
```   393 by simp
```
```   394
```
```   395 text{*Valid for the naturals, but what if @{text"0<i<1"}?
```
```   396 Premises cannot be weakened: consider the case where @{term "i=0"},
```
```   397 @{term "m=1"} and @{term "n=0"}.*}
```
```   398 lemma nat_power_less_imp_less:
```
```   399   assumes nonneg: "0 < (i\<Colon>nat)"
```
```   400   assumes less: "i^m < i^n"
```
```   401   shows "m < n"
```
```   402 proof (cases "i = 1")
```
```   403   case True with less power_one [where 'a = nat] show ?thesis by simp
```
```   404 next
```
```   405   case False with nonneg have "1 < i" by auto
```
```   406   from power_strict_increasing_iff [OF this] less show ?thesis ..
```
```   407 qed
```
```   408
```
```   409 lemma power_diff:
```
```   410   assumes nz: "a ~= 0"
```
```   411   shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)"
```
```   412   by (induct m n rule: diff_induct)
```
```   413     (simp_all add: nonzero_mult_divide_cancel_left nz)
```
```   414
```
```   415
```
```   416 text{*ML bindings for the general exponentiation theorems*}
```
```   417 ML
```
```   418 {*
```
```   419 val power_0 = thm"power_0";
```
```   420 val power_Suc = thm"power_Suc";
```
```   421 val power_0_Suc = thm"power_0_Suc";
```
```   422 val power_0_left = thm"power_0_left";
```
```   423 val power_one = thm"power_one";
```
```   424 val power_one_right = thm"power_one_right";
```
```   425 val power_add = thm"power_add";
```
```   426 val power_mult = thm"power_mult";
```
```   427 val power_mult_distrib = thm"power_mult_distrib";
```
```   428 val zero_less_power = thm"zero_less_power";
```
```   429 val zero_le_power = thm"zero_le_power";
```
```   430 val one_le_power = thm"one_le_power";
```
```   431 val gt1_imp_ge0 = thm"gt1_imp_ge0";
```
```   432 val power_gt1_lemma = thm"power_gt1_lemma";
```
```   433 val power_gt1 = thm"power_gt1";
```
```   434 val power_le_imp_le_exp = thm"power_le_imp_le_exp";
```
```   435 val power_inject_exp = thm"power_inject_exp";
```
```   436 val power_less_imp_less_exp = thm"power_less_imp_less_exp";
```
```   437 val power_mono = thm"power_mono";
```
```   438 val power_strict_mono = thm"power_strict_mono";
```
```   439 val power_eq_0_iff = thm"power_eq_0_iff";
```
```   440 val field_power_eq_0_iff = thm"power_eq_0_iff";
```
```   441 val field_power_not_zero = thm"field_power_not_zero";
```
```   442 val power_inverse = thm"power_inverse";
```
```   443 val nonzero_power_divide = thm"nonzero_power_divide";
```
```   444 val power_divide = thm"power_divide";
```
```   445 val power_abs = thm"power_abs";
```
```   446 val zero_less_power_abs_iff = thm"zero_less_power_abs_iff";
```
```   447 val zero_le_power_abs = thm "zero_le_power_abs";
```
```   448 val power_minus = thm"power_minus";
```
```   449 val power_Suc_less = thm"power_Suc_less";
```
```   450 val power_strict_decreasing = thm"power_strict_decreasing";
```
```   451 val power_decreasing = thm"power_decreasing";
```
```   452 val power_Suc_less_one = thm"power_Suc_less_one";
```
```   453 val power_increasing = thm"power_increasing";
```
```   454 val power_strict_increasing = thm"power_strict_increasing";
```
```   455 val power_le_imp_le_base = thm"power_le_imp_le_base";
```
```   456 val power_inject_base = thm"power_inject_base";
```
```   457 *}
```
```   458
```
```   459 text{*ML bindings for the remaining theorems*}
```
```   460 ML
```
```   461 {*
```
```   462 val nat_one_le_power = thm"nat_one_le_power";
```
```   463 val nat_power_less_imp_less = thm"nat_power_less_imp_less";
```
```   464 val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";
```
```   465 *}
```
```   466
```
```   467 end
```