src/HOL/Power.thy
 author huffman Thu Mar 29 11:47:30 2012 +0200 (2012-03-29) changeset 47191 ebd8c46d156b parent 45231 d85a2fdc586c child 47192 0c0501cb6da6 permissions -rw-r--r--
bootstrap Num.thy before Power.thy;
move lemmas about powers into Power.thy
```     1 (*  Title:      HOL/Power.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1997  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 header {* Exponentiation *}
```
```     7
```
```     8 theory Power
```
```     9 imports Num
```
```    10 begin
```
```    11
```
```    12 subsection {* Powers for Arbitrary Monoids *}
```
```    13
```
```    14 class power = one + times
```
```    15 begin
```
```    16
```
```    17 primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where
```
```    18     power_0: "a ^ 0 = 1"
```
```    19   | power_Suc: "a ^ Suc n = a * a ^ n"
```
```    20
```
```    21 notation (latex output)
```
```    22   power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
```
```    23
```
```    24 notation (HTML output)
```
```    25   power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
```
```    26
```
```    27 end
```
```    28
```
```    29 context monoid_mult
```
```    30 begin
```
```    31
```
```    32 subclass power .
```
```    33
```
```    34 lemma power_one [simp]:
```
```    35   "1 ^ n = 1"
```
```    36   by (induct n) simp_all
```
```    37
```
```    38 lemma power_one_right [simp]:
```
```    39   "a ^ 1 = a"
```
```    40   by simp
```
```    41
```
```    42 lemma power_commutes:
```
```    43   "a ^ n * a = a * a ^ n"
```
```    44   by (induct n) (simp_all add: mult_assoc)
```
```    45
```
```    46 lemma power_Suc2:
```
```    47   "a ^ Suc n = a ^ n * a"
```
```    48   by (simp add: power_commutes)
```
```    49
```
```    50 lemma power_add:
```
```    51   "a ^ (m + n) = a ^ m * a ^ n"
```
```    52   by (induct m) (simp_all add: algebra_simps)
```
```    53
```
```    54 lemma power_mult:
```
```    55   "a ^ (m * n) = (a ^ m) ^ n"
```
```    56   by (induct n) (simp_all add: power_add)
```
```    57
```
```    58 end
```
```    59
```
```    60 context comm_monoid_mult
```
```    61 begin
```
```    62
```
```    63 lemma power_mult_distrib:
```
```    64   "(a * b) ^ n = (a ^ n) * (b ^ n)"
```
```    65   by (induct n) (simp_all add: mult_ac)
```
```    66
```
```    67 end
```
```    68
```
```    69 context semiring_numeral
```
```    70 begin
```
```    71
```
```    72 lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
```
```    73   by (simp only: sqr_conv_mult numeral_mult)
```
```    74
```
```    75 lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
```
```    76   by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
```
```    77     numeral_sqr numeral_mult power_add power_one_right)
```
```    78
```
```    79 lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
```
```    80   by (rule numeral_pow [symmetric])
```
```    81
```
```    82 end
```
```    83
```
```    84 context semiring_1
```
```    85 begin
```
```    86
```
```    87 lemma of_nat_power:
```
```    88   "of_nat (m ^ n) = of_nat m ^ n"
```
```    89   by (induct n) (simp_all add: of_nat_mult)
```
```    90
```
```    91 lemma power_zero_numeral [simp]: "(0::'a) ^ numeral k = 0"
```
```    92   by (cases "numeral k :: nat", simp_all)
```
```    93
```
```    94 end
```
```    95
```
```    96 context comm_semiring_1
```
```    97 begin
```
```    98
```
```    99 text {* The divides relation *}
```
```   100
```
```   101 lemma le_imp_power_dvd:
```
```   102   assumes "m \<le> n" shows "a ^ m dvd a ^ n"
```
```   103 proof
```
```   104   have "a ^ n = a ^ (m + (n - m))"
```
```   105     using `m \<le> n` by simp
```
```   106   also have "\<dots> = a ^ m * a ^ (n - m)"
```
```   107     by (rule power_add)
```
```   108   finally show "a ^ n = a ^ m * a ^ (n - m)" .
```
```   109 qed
```
```   110
```
```   111 lemma power_le_dvd:
```
```   112   "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
```
```   113   by (rule dvd_trans [OF le_imp_power_dvd])
```
```   114
```
```   115 lemma dvd_power_same:
```
```   116   "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
```
```   117   by (induct n) (auto simp add: mult_dvd_mono)
```
```   118
```
```   119 lemma dvd_power_le:
```
```   120   "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
```
```   121   by (rule power_le_dvd [OF dvd_power_same])
```
```   122
```
```   123 lemma dvd_power [simp]:
```
```   124   assumes "n > (0::nat) \<or> x = 1"
```
```   125   shows "x dvd (x ^ n)"
```
```   126 using assms proof
```
```   127   assume "0 < n"
```
```   128   then have "x ^ n = x ^ Suc (n - 1)" by simp
```
```   129   then show "x dvd (x ^ n)" by simp
```
```   130 next
```
```   131   assume "x = 1"
```
```   132   then show "x dvd (x ^ n)" by simp
```
```   133 qed
```
```   134
```
```   135 end
```
```   136
```
```   137 context ring_1
```
```   138 begin
```
```   139
```
```   140 lemma power_minus:
```
```   141   "(- a) ^ n = (- 1) ^ n * a ^ n"
```
```   142 proof (induct n)
```
```   143   case 0 show ?case by simp
```
```   144 next
```
```   145   case (Suc n) then show ?case
```
```   146     by (simp del: power_Suc add: power_Suc2 mult_assoc)
```
```   147 qed
```
```   148
```
```   149 lemma power_minus_Bit0:
```
```   150   "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
```
```   151   by (induct k, simp_all only: numeral_class.numeral.simps power_add
```
```   152     power_one_right mult_minus_left mult_minus_right minus_minus)
```
```   153
```
```   154 lemma power_minus_Bit1:
```
```   155   "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
```
```   156   by (simp only: nat_number(4) power_Suc power_minus_Bit0 mult_minus_left)
```
```   157
```
```   158 lemma power_neg_numeral_Bit0 [simp]:
```
```   159   "neg_numeral k ^ numeral (Num.Bit0 l) = numeral (Num.pow k (Num.Bit0 l))"
```
```   160   by (simp only: neg_numeral_def power_minus_Bit0 power_numeral)
```
```   161
```
```   162 lemma power_neg_numeral_Bit1 [simp]:
```
```   163   "neg_numeral k ^ numeral (Num.Bit1 l) = neg_numeral (Num.pow k (Num.Bit1 l))"
```
```   164   by (simp only: neg_numeral_def power_minus_Bit1 power_numeral pow.simps)
```
```   165
```
```   166 end
```
```   167
```
```   168 context linordered_semidom
```
```   169 begin
```
```   170
```
```   171 lemma zero_less_power [simp]:
```
```   172   "0 < a \<Longrightarrow> 0 < a ^ n"
```
```   173   by (induct n) (simp_all add: mult_pos_pos)
```
```   174
```
```   175 lemma zero_le_power [simp]:
```
```   176   "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
```
```   177   by (induct n) (simp_all add: mult_nonneg_nonneg)
```
```   178
```
```   179 lemma one_le_power[simp]:
```
```   180   "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
```
```   181   apply (induct n)
```
```   182   apply simp_all
```
```   183   apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
```
```   184   apply (simp_all add: order_trans [OF zero_le_one])
```
```   185   done
```
```   186
```
```   187 lemma power_gt1_lemma:
```
```   188   assumes gt1: "1 < a"
```
```   189   shows "1 < a * a ^ n"
```
```   190 proof -
```
```   191   from gt1 have "0 \<le> a"
```
```   192     by (fact order_trans [OF zero_le_one less_imp_le])
```
```   193   have "1 * 1 < a * 1" using gt1 by simp
```
```   194   also have "\<dots> \<le> a * a ^ n" using gt1
```
```   195     by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
```
```   196         zero_le_one order_refl)
```
```   197   finally show ?thesis by simp
```
```   198 qed
```
```   199
```
```   200 lemma power_gt1:
```
```   201   "1 < a \<Longrightarrow> 1 < a ^ Suc n"
```
```   202   by (simp add: power_gt1_lemma)
```
```   203
```
```   204 lemma one_less_power [simp]:
```
```   205   "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
```
```   206   by (cases n) (simp_all add: power_gt1_lemma)
```
```   207
```
```   208 lemma power_le_imp_le_exp:
```
```   209   assumes gt1: "1 < a"
```
```   210   shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
```
```   211 proof (induct m arbitrary: n)
```
```   212   case 0
```
```   213   show ?case by simp
```
```   214 next
```
```   215   case (Suc m)
```
```   216   show ?case
```
```   217   proof (cases n)
```
```   218     case 0
```
```   219     with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
```
```   220     with gt1 show ?thesis
```
```   221       by (force simp only: power_gt1_lemma
```
```   222           not_less [symmetric])
```
```   223   next
```
```   224     case (Suc n)
```
```   225     with Suc.prems Suc.hyps show ?thesis
```
```   226       by (force dest: mult_left_le_imp_le
```
```   227           simp add: less_trans [OF zero_less_one gt1])
```
```   228   qed
```
```   229 qed
```
```   230
```
```   231 text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
```
```   232 lemma power_inject_exp [simp]:
```
```   233   "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
```
```   234   by (force simp add: order_antisym power_le_imp_le_exp)
```
```   235
```
```   236 text{*Can relax the first premise to @{term "0<a"} in the case of the
```
```   237 natural numbers.*}
```
```   238 lemma power_less_imp_less_exp:
```
```   239   "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
```
```   240   by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
```
```   241     power_le_imp_le_exp)
```
```   242
```
```   243 lemma power_mono:
```
```   244   "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
```
```   245   by (induct n)
```
```   246     (auto intro: mult_mono order_trans [of 0 a b])
```
```   247
```
```   248 lemma power_strict_mono [rule_format]:
```
```   249   "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
```
```   250   by (induct n)
```
```   251    (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
```
```   252
```
```   253 text{*Lemma for @{text power_strict_decreasing}*}
```
```   254 lemma power_Suc_less:
```
```   255   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
```
```   256   by (induct n)
```
```   257     (auto simp add: mult_strict_left_mono)
```
```   258
```
```   259 lemma power_strict_decreasing [rule_format]:
```
```   260   "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
```
```   261 proof (induct N)
```
```   262   case 0 then show ?case by simp
```
```   263 next
```
```   264   case (Suc N) then show ?case
```
```   265   apply (auto simp add: power_Suc_less less_Suc_eq)
```
```   266   apply (subgoal_tac "a * a^N < 1 * a^n")
```
```   267   apply simp
```
```   268   apply (rule mult_strict_mono) apply auto
```
```   269   done
```
```   270 qed
```
```   271
```
```   272 text{*Proof resembles that of @{text power_strict_decreasing}*}
```
```   273 lemma power_decreasing [rule_format]:
```
```   274   "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
```
```   275 proof (induct N)
```
```   276   case 0 then show ?case by simp
```
```   277 next
```
```   278   case (Suc N) then show ?case
```
```   279   apply (auto simp add: le_Suc_eq)
```
```   280   apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
```
```   281   apply (rule mult_mono) apply auto
```
```   282   done
```
```   283 qed
```
```   284
```
```   285 lemma power_Suc_less_one:
```
```   286   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
```
```   287   using power_strict_decreasing [of 0 "Suc n" a] by simp
```
```   288
```
```   289 text{*Proof again resembles that of @{text power_strict_decreasing}*}
```
```   290 lemma power_increasing [rule_format]:
```
```   291   "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
```
```   292 proof (induct N)
```
```   293   case 0 then show ?case by simp
```
```   294 next
```
```   295   case (Suc N) then show ?case
```
```   296   apply (auto simp add: le_Suc_eq)
```
```   297   apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
```
```   298   apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
```
```   299   done
```
```   300 qed
```
```   301
```
```   302 text{*Lemma for @{text power_strict_increasing}*}
```
```   303 lemma power_less_power_Suc:
```
```   304   "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
```
```   305   by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
```
```   306
```
```   307 lemma power_strict_increasing [rule_format]:
```
```   308   "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
```
```   309 proof (induct N)
```
```   310   case 0 then show ?case by simp
```
```   311 next
```
```   312   case (Suc N) then show ?case
```
```   313   apply (auto simp add: power_less_power_Suc less_Suc_eq)
```
```   314   apply (subgoal_tac "1 * a^n < a * a^N", simp)
```
```   315   apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
```
```   316   done
```
```   317 qed
```
```   318
```
```   319 lemma power_increasing_iff [simp]:
```
```   320   "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
```
```   321   by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
```
```   322
```
```   323 lemma power_strict_increasing_iff [simp]:
```
```   324   "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
```
```   325 by (blast intro: power_less_imp_less_exp power_strict_increasing)
```
```   326
```
```   327 lemma power_le_imp_le_base:
```
```   328   assumes le: "a ^ Suc n \<le> b ^ Suc n"
```
```   329     and ynonneg: "0 \<le> b"
```
```   330   shows "a \<le> b"
```
```   331 proof (rule ccontr)
```
```   332   assume "~ a \<le> b"
```
```   333   then have "b < a" by (simp only: linorder_not_le)
```
```   334   then have "b ^ Suc n < a ^ Suc n"
```
```   335     by (simp only: assms power_strict_mono)
```
```   336   from le and this show False
```
```   337     by (simp add: linorder_not_less [symmetric])
```
```   338 qed
```
```   339
```
```   340 lemma power_less_imp_less_base:
```
```   341   assumes less: "a ^ n < b ^ n"
```
```   342   assumes nonneg: "0 \<le> b"
```
```   343   shows "a < b"
```
```   344 proof (rule contrapos_pp [OF less])
```
```   345   assume "~ a < b"
```
```   346   hence "b \<le> a" by (simp only: linorder_not_less)
```
```   347   hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
```
```   348   thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
```
```   349 qed
```
```   350
```
```   351 lemma power_inject_base:
```
```   352   "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
```
```   353 by (blast intro: power_le_imp_le_base antisym eq_refl sym)
```
```   354
```
```   355 lemma power_eq_imp_eq_base:
```
```   356   "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
```
```   357   by (cases n) (simp_all del: power_Suc, rule power_inject_base)
```
```   358
```
```   359 end
```
```   360
```
```   361 context linordered_idom
```
```   362 begin
```
```   363
```
```   364 lemma power_abs:
```
```   365   "abs (a ^ n) = abs a ^ n"
```
```   366   by (induct n) (auto simp add: abs_mult)
```
```   367
```
```   368 lemma abs_power_minus [simp]:
```
```   369   "abs ((-a) ^ n) = abs (a ^ n)"
```
```   370   by (simp add: power_abs)
```
```   371
```
```   372 lemma zero_less_power_abs_iff [simp, no_atp]:
```
```   373   "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
```
```   374 proof (induct n)
```
```   375   case 0 show ?case by simp
```
```   376 next
```
```   377   case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
```
```   378 qed
```
```   379
```
```   380 lemma zero_le_power_abs [simp]:
```
```   381   "0 \<le> abs a ^ n"
```
```   382   by (rule zero_le_power [OF abs_ge_zero])
```
```   383
```
```   384 end
```
```   385
```
```   386 context ring_1_no_zero_divisors
```
```   387 begin
```
```   388
```
```   389 lemma field_power_not_zero:
```
```   390   "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
```
```   391   by (induct n) auto
```
```   392
```
```   393 end
```
```   394
```
```   395 context division_ring
```
```   396 begin
```
```   397
```
```   398 text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
```
```   399 lemma nonzero_power_inverse:
```
```   400   "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
```
```   401   by (induct n)
```
```   402     (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
```
```   403
```
```   404 end
```
```   405
```
```   406 context field
```
```   407 begin
```
```   408
```
```   409 lemma nonzero_power_divide:
```
```   410   "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
```
```   411   by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
```
```   412
```
```   413 end
```
```   414
```
```   415 lemma power_0_Suc [simp]:
```
```   416   "(0::'a::{power, semiring_0}) ^ Suc n = 0"
```
```   417   by simp
```
```   418
```
```   419 text{*It looks plausible as a simprule, but its effect can be strange.*}
```
```   420 lemma power_0_left:
```
```   421   "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
```
```   422   by (induct n) simp_all
```
```   423
```
```   424 lemma power_eq_0_iff [simp]:
```
```   425   "a ^ n = 0 \<longleftrightarrow>
```
```   426      a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
```
```   427   by (induct n)
```
```   428     (auto simp add: no_zero_divisors elim: contrapos_pp)
```
```   429
```
```   430 lemma (in field) power_diff:
```
```   431   assumes nz: "a \<noteq> 0"
```
```   432   shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
```
```   433   by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
```
```   434
```
```   435 text{*Perhaps these should be simprules.*}
```
```   436 lemma power_inverse:
```
```   437   fixes a :: "'a::division_ring_inverse_zero"
```
```   438   shows "inverse (a ^ n) = inverse a ^ n"
```
```   439 apply (cases "a = 0")
```
```   440 apply (simp add: power_0_left)
```
```   441 apply (simp add: nonzero_power_inverse)
```
```   442 done (* TODO: reorient or rename to inverse_power *)
```
```   443
```
```   444 lemma power_one_over:
```
```   445   "1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ n"
```
```   446   by (simp add: divide_inverse) (rule power_inverse)
```
```   447
```
```   448 lemma power_divide:
```
```   449   "(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n"
```
```   450 apply (cases "b = 0")
```
```   451 apply (simp add: power_0_left)
```
```   452 apply (rule nonzero_power_divide)
```
```   453 apply assumption
```
```   454 done
```
```   455
```
```   456
```
```   457 subsection {* Exponentiation for the Natural Numbers *}
```
```   458
```
```   459 lemma nat_one_le_power [simp]:
```
```   460   "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
```
```   461   by (rule one_le_power [of i n, unfolded One_nat_def])
```
```   462
```
```   463 lemma nat_zero_less_power_iff [simp]:
```
```   464   "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
```
```   465   by (induct n) auto
```
```   466
```
```   467 lemma nat_power_eq_Suc_0_iff [simp]:
```
```   468   "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
```
```   469   by (induct m) auto
```
```   470
```
```   471 lemma power_Suc_0 [simp]:
```
```   472   "Suc 0 ^ n = Suc 0"
```
```   473   by simp
```
```   474
```
```   475 text{*Valid for the naturals, but what if @{text"0<i<1"}?
```
```   476 Premises cannot be weakened: consider the case where @{term "i=0"},
```
```   477 @{term "m=1"} and @{term "n=0"}.*}
```
```   478 lemma nat_power_less_imp_less:
```
```   479   assumes nonneg: "0 < (i\<Colon>nat)"
```
```   480   assumes less: "i ^ m < i ^ n"
```
```   481   shows "m < n"
```
```   482 proof (cases "i = 1")
```
```   483   case True with less power_one [where 'a = nat] show ?thesis by simp
```
```   484 next
```
```   485   case False with nonneg have "1 < i" by auto
```
```   486   from power_strict_increasing_iff [OF this] less show ?thesis ..
```
```   487 qed
```
```   488
```
```   489 lemma power_dvd_imp_le:
```
```   490   "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
```
```   491   apply (rule power_le_imp_le_exp, assumption)
```
```   492   apply (erule dvd_imp_le, simp)
```
```   493   done
```
```   494
```
```   495
```
```   496 subsection {* Code generator tweak *}
```
```   497
```
```   498 lemma power_power_power [code]:
```
```   499   "power = power.power (1::'a::{power}) (op *)"
```
```   500   unfolding power_def power.power_def ..
```
```   501
```
```   502 declare power.power.simps [code]
```
```   503
```
```   504 code_modulename SML
```
```   505   Power Arith
```
```   506
```
```   507 code_modulename OCaml
```
```   508   Power Arith
```
```   509
```
```   510 code_modulename Haskell
```
```   511   Power Arith
```
```   512
```
```   513 end
```