src/HOL/Power.thy
 author haftmann Mon Nov 29 13:44:54 2010 +0100 (2010-11-29) changeset 40815 6e2d17cc0d1d parent 39438 c5ece2a7a86e child 41550 efa734d9b221 permissions -rw-r--r--
equivI has replaced equiv.intro
```     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 Nat
```
```    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_1
```
```    70 begin
```
```    71
```
```    72 lemma of_nat_power:
```
```    73   "of_nat (m ^ n) = of_nat m ^ n"
```
```    74   by (induct n) (simp_all add: of_nat_mult)
```
```    75
```
```    76 end
```
```    77
```
```    78 context comm_semiring_1
```
```    79 begin
```
```    80
```
```    81 text {* The divides relation *}
```
```    82
```
```    83 lemma le_imp_power_dvd:
```
```    84   assumes "m \<le> n" shows "a ^ m dvd a ^ n"
```
```    85 proof
```
```    86   have "a ^ n = a ^ (m + (n - m))"
```
```    87     using `m \<le> n` by simp
```
```    88   also have "\<dots> = a ^ m * a ^ (n - m)"
```
```    89     by (rule power_add)
```
```    90   finally show "a ^ n = a ^ m * a ^ (n - m)" .
```
```    91 qed
```
```    92
```
```    93 lemma power_le_dvd:
```
```    94   "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
```
```    95   by (rule dvd_trans [OF le_imp_power_dvd])
```
```    96
```
```    97 lemma dvd_power_same:
```
```    98   "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
```
```    99   by (induct n) (auto simp add: mult_dvd_mono)
```
```   100
```
```   101 lemma dvd_power_le:
```
```   102   "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
```
```   103   by (rule power_le_dvd [OF dvd_power_same])
```
```   104
```
```   105 lemma dvd_power [simp]:
```
```   106   assumes "n > (0::nat) \<or> x = 1"
```
```   107   shows "x dvd (x ^ n)"
```
```   108 using assms proof
```
```   109   assume "0 < n"
```
```   110   then have "x ^ n = x ^ Suc (n - 1)" by simp
```
```   111   then show "x dvd (x ^ n)" by simp
```
```   112 next
```
```   113   assume "x = 1"
```
```   114   then show "x dvd (x ^ n)" by simp
```
```   115 qed
```
```   116
```
```   117 end
```
```   118
```
```   119 context ring_1
```
```   120 begin
```
```   121
```
```   122 lemma power_minus:
```
```   123   "(- a) ^ n = (- 1) ^ n * a ^ n"
```
```   124 proof (induct n)
```
```   125   case 0 show ?case by simp
```
```   126 next
```
```   127   case (Suc n) then show ?case
```
```   128     by (simp del: power_Suc add: power_Suc2 mult_assoc)
```
```   129 qed
```
```   130
```
```   131 end
```
```   132
```
```   133 context linordered_semidom
```
```   134 begin
```
```   135
```
```   136 lemma zero_less_power [simp]:
```
```   137   "0 < a \<Longrightarrow> 0 < a ^ n"
```
```   138   by (induct n) (simp_all add: mult_pos_pos)
```
```   139
```
```   140 lemma zero_le_power [simp]:
```
```   141   "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
```
```   142   by (induct n) (simp_all add: mult_nonneg_nonneg)
```
```   143
```
```   144 lemma one_le_power[simp]:
```
```   145   "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
```
```   146   apply (induct n)
```
```   147   apply simp_all
```
```   148   apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
```
```   149   apply (simp_all add: order_trans [OF zero_le_one])
```
```   150   done
```
```   151
```
```   152 lemma power_gt1_lemma:
```
```   153   assumes gt1: "1 < a"
```
```   154   shows "1 < a * a ^ n"
```
```   155 proof -
```
```   156   from gt1 have "0 \<le> a"
```
```   157     by (fact order_trans [OF zero_le_one less_imp_le])
```
```   158   have "1 * 1 < a * 1" using gt1 by simp
```
```   159   also have "\<dots> \<le> a * a ^ n" using gt1
```
```   160     by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
```
```   161         zero_le_one order_refl)
```
```   162   finally show ?thesis by simp
```
```   163 qed
```
```   164
```
```   165 lemma power_gt1:
```
```   166   "1 < a \<Longrightarrow> 1 < a ^ Suc n"
```
```   167   by (simp add: power_gt1_lemma)
```
```   168
```
```   169 lemma one_less_power [simp]:
```
```   170   "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
```
```   171   by (cases n) (simp_all add: power_gt1_lemma)
```
```   172
```
```   173 lemma power_le_imp_le_exp:
```
```   174   assumes gt1: "1 < a"
```
```   175   shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
```
```   176 proof (induct m arbitrary: n)
```
```   177   case 0
```
```   178   show ?case by simp
```
```   179 next
```
```   180   case (Suc m)
```
```   181   show ?case
```
```   182   proof (cases n)
```
```   183     case 0
```
```   184     with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
```
```   185     with gt1 show ?thesis
```
```   186       by (force simp only: power_gt1_lemma
```
```   187           not_less [symmetric])
```
```   188   next
```
```   189     case (Suc n)
```
```   190     with Suc.prems Suc.hyps show ?thesis
```
```   191       by (force dest: mult_left_le_imp_le
```
```   192           simp add: less_trans [OF zero_less_one gt1])
```
```   193   qed
```
```   194 qed
```
```   195
```
```   196 text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
```
```   197 lemma power_inject_exp [simp]:
```
```   198   "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
```
```   199   by (force simp add: order_antisym power_le_imp_le_exp)
```
```   200
```
```   201 text{*Can relax the first premise to @{term "0<a"} in the case of the
```
```   202 natural numbers.*}
```
```   203 lemma power_less_imp_less_exp:
```
```   204   "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
```
```   205   by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
```
```   206     power_le_imp_le_exp)
```
```   207
```
```   208 lemma power_mono:
```
```   209   "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
```
```   210   by (induct n)
```
```   211     (auto intro: mult_mono order_trans [of 0 a b])
```
```   212
```
```   213 lemma power_strict_mono [rule_format]:
```
```   214   "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
```
```   215   by (induct n)
```
```   216    (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
```
```   217
```
```   218 text{*Lemma for @{text power_strict_decreasing}*}
```
```   219 lemma power_Suc_less:
```
```   220   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
```
```   221   by (induct n)
```
```   222     (auto simp add: mult_strict_left_mono)
```
```   223
```
```   224 lemma power_strict_decreasing [rule_format]:
```
```   225   "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
```
```   226 proof (induct N)
```
```   227   case 0 then show ?case by simp
```
```   228 next
```
```   229   case (Suc N) then show ?case
```
```   230   apply (auto simp add: power_Suc_less less_Suc_eq)
```
```   231   apply (subgoal_tac "a * a^N < 1 * a^n")
```
```   232   apply simp
```
```   233   apply (rule mult_strict_mono) apply auto
```
```   234   done
```
```   235 qed
```
```   236
```
```   237 text{*Proof resembles that of @{text power_strict_decreasing}*}
```
```   238 lemma power_decreasing [rule_format]:
```
```   239   "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
```
```   240 proof (induct N)
```
```   241   case 0 then show ?case by simp
```
```   242 next
```
```   243   case (Suc N) then show ?case
```
```   244   apply (auto simp add: le_Suc_eq)
```
```   245   apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
```
```   246   apply (rule mult_mono) apply auto
```
```   247   done
```
```   248 qed
```
```   249
```
```   250 lemma power_Suc_less_one:
```
```   251   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
```
```   252   using power_strict_decreasing [of 0 "Suc n" a] by simp
```
```   253
```
```   254 text{*Proof again resembles that of @{text power_strict_decreasing}*}
```
```   255 lemma power_increasing [rule_format]:
```
```   256   "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
```
```   257 proof (induct N)
```
```   258   case 0 then show ?case by simp
```
```   259 next
```
```   260   case (Suc N) then show ?case
```
```   261   apply (auto simp add: le_Suc_eq)
```
```   262   apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
```
```   263   apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
```
```   264   done
```
```   265 qed
```
```   266
```
```   267 text{*Lemma for @{text power_strict_increasing}*}
```
```   268 lemma power_less_power_Suc:
```
```   269   "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
```
```   270   by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
```
```   271
```
```   272 lemma power_strict_increasing [rule_format]:
```
```   273   "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
```
```   274 proof (induct N)
```
```   275   case 0 then show ?case by simp
```
```   276 next
```
```   277   case (Suc N) then show ?case
```
```   278   apply (auto simp add: power_less_power_Suc less_Suc_eq)
```
```   279   apply (subgoal_tac "1 * a^n < a * a^N", simp)
```
```   280   apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
```
```   281   done
```
```   282 qed
```
```   283
```
```   284 lemma power_increasing_iff [simp]:
```
```   285   "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
```
```   286   by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
```
```   287
```
```   288 lemma power_strict_increasing_iff [simp]:
```
```   289   "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
```
```   290 by (blast intro: power_less_imp_less_exp power_strict_increasing)
```
```   291
```
```   292 lemma power_le_imp_le_base:
```
```   293   assumes le: "a ^ Suc n \<le> b ^ Suc n"
```
```   294     and ynonneg: "0 \<le> b"
```
```   295   shows "a \<le> b"
```
```   296 proof (rule ccontr)
```
```   297   assume "~ a \<le> b"
```
```   298   then have "b < a" by (simp only: linorder_not_le)
```
```   299   then have "b ^ Suc n < a ^ Suc n"
```
```   300     by (simp only: prems power_strict_mono)
```
```   301   from le and this show False
```
```   302     by (simp add: linorder_not_less [symmetric])
```
```   303 qed
```
```   304
```
```   305 lemma power_less_imp_less_base:
```
```   306   assumes less: "a ^ n < b ^ n"
```
```   307   assumes nonneg: "0 \<le> b"
```
```   308   shows "a < b"
```
```   309 proof (rule contrapos_pp [OF less])
```
```   310   assume "~ a < b"
```
```   311   hence "b \<le> a" by (simp only: linorder_not_less)
```
```   312   hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
```
```   313   thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
```
```   314 qed
```
```   315
```
```   316 lemma power_inject_base:
```
```   317   "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
```
```   318 by (blast intro: power_le_imp_le_base antisym eq_refl sym)
```
```   319
```
```   320 lemma power_eq_imp_eq_base:
```
```   321   "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
```
```   322   by (cases n) (simp_all del: power_Suc, rule power_inject_base)
```
```   323
```
```   324 end
```
```   325
```
```   326 context linordered_idom
```
```   327 begin
```
```   328
```
```   329 lemma power_abs:
```
```   330   "abs (a ^ n) = abs a ^ n"
```
```   331   by (induct n) (auto simp add: abs_mult)
```
```   332
```
```   333 lemma abs_power_minus [simp]:
```
```   334   "abs ((-a) ^ n) = abs (a ^ n)"
```
```   335   by (simp add: power_abs)
```
```   336
```
```   337 lemma zero_less_power_abs_iff [simp, no_atp]:
```
```   338   "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
```
```   339 proof (induct n)
```
```   340   case 0 show ?case by simp
```
```   341 next
```
```   342   case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
```
```   343 qed
```
```   344
```
```   345 lemma zero_le_power_abs [simp]:
```
```   346   "0 \<le> abs a ^ n"
```
```   347   by (rule zero_le_power [OF abs_ge_zero])
```
```   348
```
```   349 end
```
```   350
```
```   351 context ring_1_no_zero_divisors
```
```   352 begin
```
```   353
```
```   354 lemma field_power_not_zero:
```
```   355   "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
```
```   356   by (induct n) auto
```
```   357
```
```   358 end
```
```   359
```
```   360 context division_ring
```
```   361 begin
```
```   362
```
```   363 text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
```
```   364 lemma nonzero_power_inverse:
```
```   365   "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
```
```   366   by (induct n)
```
```   367     (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
```
```   368
```
```   369 end
```
```   370
```
```   371 context field
```
```   372 begin
```
```   373
```
```   374 lemma nonzero_power_divide:
```
```   375   "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
```
```   376   by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
```
```   377
```
```   378 end
```
```   379
```
```   380 lemma power_0_Suc [simp]:
```
```   381   "(0::'a::{power, semiring_0}) ^ Suc n = 0"
```
```   382   by simp
```
```   383
```
```   384 text{*It looks plausible as a simprule, but its effect can be strange.*}
```
```   385 lemma power_0_left:
```
```   386   "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
```
```   387   by (induct n) simp_all
```
```   388
```
```   389 lemma power_eq_0_iff [simp]:
```
```   390   "a ^ n = 0 \<longleftrightarrow>
```
```   391      a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
```
```   392   by (induct n)
```
```   393     (auto simp add: no_zero_divisors elim: contrapos_pp)
```
```   394
```
```   395 lemma (in field) power_diff:
```
```   396   assumes nz: "a \<noteq> 0"
```
```   397   shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
```
```   398   by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
```
```   399
```
```   400 text{*Perhaps these should be simprules.*}
```
```   401 lemma power_inverse:
```
```   402   fixes a :: "'a::division_ring_inverse_zero"
```
```   403   shows "inverse (a ^ n) = inverse a ^ n"
```
```   404 apply (cases "a = 0")
```
```   405 apply (simp add: power_0_left)
```
```   406 apply (simp add: nonzero_power_inverse)
```
```   407 done (* TODO: reorient or rename to inverse_power *)
```
```   408
```
```   409 lemma power_one_over:
```
```   410   "1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ n"
```
```   411   by (simp add: divide_inverse) (rule power_inverse)
```
```   412
```
```   413 lemma power_divide:
```
```   414   "(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n"
```
```   415 apply (cases "b = 0")
```
```   416 apply (simp add: power_0_left)
```
```   417 apply (rule nonzero_power_divide)
```
```   418 apply assumption
```
```   419 done
```
```   420
```
```   421
```
```   422 subsection {* Exponentiation for the Natural Numbers *}
```
```   423
```
```   424 lemma nat_one_le_power [simp]:
```
```   425   "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
```
```   426   by (rule one_le_power [of i n, unfolded One_nat_def])
```
```   427
```
```   428 lemma nat_zero_less_power_iff [simp]:
```
```   429   "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
```
```   430   by (induct n) auto
```
```   431
```
```   432 lemma nat_power_eq_Suc_0_iff [simp]:
```
```   433   "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
```
```   434   by (induct m) auto
```
```   435
```
```   436 lemma power_Suc_0 [simp]:
```
```   437   "Suc 0 ^ n = Suc 0"
```
```   438   by simp
```
```   439
```
```   440 text{*Valid for the naturals, but what if @{text"0<i<1"}?
```
```   441 Premises cannot be weakened: consider the case where @{term "i=0"},
```
```   442 @{term "m=1"} and @{term "n=0"}.*}
```
```   443 lemma nat_power_less_imp_less:
```
```   444   assumes nonneg: "0 < (i\<Colon>nat)"
```
```   445   assumes less: "i ^ m < i ^ n"
```
```   446   shows "m < n"
```
```   447 proof (cases "i = 1")
```
```   448   case True with less power_one [where 'a = nat] show ?thesis by simp
```
```   449 next
```
```   450   case False with nonneg have "1 < i" by auto
```
```   451   from power_strict_increasing_iff [OF this] less show ?thesis ..
```
```   452 qed
```
```   453
```
```   454 lemma power_dvd_imp_le:
```
```   455   "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
```
```   456   apply (rule power_le_imp_le_exp, assumption)
```
```   457   apply (erule dvd_imp_le, simp)
```
```   458   done
```
```   459
```
```   460
```
```   461 subsection {* Code generator tweak *}
```
```   462
```
```   463 lemma power_power_power [code, code_unfold, code_inline del]:
```
```   464   "power = power.power (1::'a::{power}) (op *)"
```
```   465   unfolding power_def power.power_def ..
```
```   466
```
```   467 declare power.power.simps [code]
```
```   468
```
```   469 code_modulename SML
```
```   470   Power Arith
```
```   471
```
```   472 code_modulename OCaml
```
```   473   Power Arith
```
```   474
```
```   475 code_modulename Haskell
```
```   476   Power Arith
```
```   477
```
```   478 end
```