src/HOL/Power.thy
 author huffman Fri Mar 30 12:32:35 2012 +0200 (2012-03-30) changeset 47220 52426c62b5d0 parent 47209 4893907fe872 child 47241 243b33052e34 permissions -rw-r--r--
replace lemmas eval_nat_numeral with a simpler reformulation
```     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 text {* Special syntax for squares. *}
```
```    28
```
```    29 abbreviation (xsymbols)
```
```    30   power2 :: "'a \<Rightarrow> 'a"  ("(_\<twosuperior>)" [1000] 999) where
```
```    31   "x\<twosuperior> \<equiv> x ^ 2"
```
```    32
```
```    33 notation (latex output)
```
```    34   power2  ("(_\<twosuperior>)" [1000] 999)
```
```    35
```
```    36 notation (HTML output)
```
```    37   power2  ("(_\<twosuperior>)" [1000] 999)
```
```    38
```
```    39 end
```
```    40
```
```    41 context monoid_mult
```
```    42 begin
```
```    43
```
```    44 subclass power .
```
```    45
```
```    46 lemma power_one [simp]:
```
```    47   "1 ^ n = 1"
```
```    48   by (induct n) simp_all
```
```    49
```
```    50 lemma power_one_right [simp]:
```
```    51   "a ^ 1 = a"
```
```    52   by simp
```
```    53
```
```    54 lemma power_commutes:
```
```    55   "a ^ n * a = a * a ^ n"
```
```    56   by (induct n) (simp_all add: mult_assoc)
```
```    57
```
```    58 lemma power_Suc2:
```
```    59   "a ^ Suc n = a ^ n * a"
```
```    60   by (simp add: power_commutes)
```
```    61
```
```    62 lemma power_add:
```
```    63   "a ^ (m + n) = a ^ m * a ^ n"
```
```    64   by (induct m) (simp_all add: algebra_simps)
```
```    65
```
```    66 lemma power_mult:
```
```    67   "a ^ (m * n) = (a ^ m) ^ n"
```
```    68   by (induct n) (simp_all add: power_add)
```
```    69
```
```    70 lemma power2_eq_square: "a\<twosuperior> = a * a"
```
```    71   by (simp add: numeral_2_eq_2)
```
```    72
```
```    73 lemma power3_eq_cube: "a ^ 3 = a * a * a"
```
```    74   by (simp add: numeral_3_eq_3 mult_assoc)
```
```    75
```
```    76 lemma power_even_eq:
```
```    77   "a ^ (2*n) = (a ^ n) ^ 2"
```
```    78   by (subst mult_commute) (simp add: power_mult)
```
```    79
```
```    80 lemma power_odd_eq:
```
```    81   "a ^ Suc (2*n) = a * (a ^ n) ^ 2"
```
```    82   by (simp add: power_even_eq)
```
```    83
```
```    84 end
```
```    85
```
```    86 context comm_monoid_mult
```
```    87 begin
```
```    88
```
```    89 lemma power_mult_distrib:
```
```    90   "(a * b) ^ n = (a ^ n) * (b ^ n)"
```
```    91   by (induct n) (simp_all add: mult_ac)
```
```    92
```
```    93 end
```
```    94
```
```    95 context semiring_numeral
```
```    96 begin
```
```    97
```
```    98 lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
```
```    99   by (simp only: sqr_conv_mult numeral_mult)
```
```   100
```
```   101 lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
```
```   102   by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
```
```   103     numeral_sqr numeral_mult power_add power_one_right)
```
```   104
```
```   105 lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
```
```   106   by (rule numeral_pow [symmetric])
```
```   107
```
```   108 end
```
```   109
```
```   110 context semiring_1
```
```   111 begin
```
```   112
```
```   113 lemma of_nat_power:
```
```   114   "of_nat (m ^ n) = of_nat m ^ n"
```
```   115   by (induct n) (simp_all add: of_nat_mult)
```
```   116
```
```   117 lemma power_zero_numeral [simp]: "(0::'a) ^ numeral k = 0"
```
```   118   by (simp add: numeral_eq_Suc)
```
```   119
```
```   120 lemma zero_power2: "0\<twosuperior> = 0" (* delete? *)
```
```   121   by (rule power_zero_numeral)
```
```   122
```
```   123 lemma one_power2: "1\<twosuperior> = 1" (* delete? *)
```
```   124   by (rule power_one)
```
```   125
```
```   126 end
```
```   127
```
```   128 context comm_semiring_1
```
```   129 begin
```
```   130
```
```   131 text {* The divides relation *}
```
```   132
```
```   133 lemma le_imp_power_dvd:
```
```   134   assumes "m \<le> n" shows "a ^ m dvd a ^ n"
```
```   135 proof
```
```   136   have "a ^ n = a ^ (m + (n - m))"
```
```   137     using `m \<le> n` by simp
```
```   138   also have "\<dots> = a ^ m * a ^ (n - m)"
```
```   139     by (rule power_add)
```
```   140   finally show "a ^ n = a ^ m * a ^ (n - m)" .
```
```   141 qed
```
```   142
```
```   143 lemma power_le_dvd:
```
```   144   "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
```
```   145   by (rule dvd_trans [OF le_imp_power_dvd])
```
```   146
```
```   147 lemma dvd_power_same:
```
```   148   "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
```
```   149   by (induct n) (auto simp add: mult_dvd_mono)
```
```   150
```
```   151 lemma dvd_power_le:
```
```   152   "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
```
```   153   by (rule power_le_dvd [OF dvd_power_same])
```
```   154
```
```   155 lemma dvd_power [simp]:
```
```   156   assumes "n > (0::nat) \<or> x = 1"
```
```   157   shows "x dvd (x ^ n)"
```
```   158 using assms proof
```
```   159   assume "0 < n"
```
```   160   then have "x ^ n = x ^ Suc (n - 1)" by simp
```
```   161   then show "x dvd (x ^ n)" by simp
```
```   162 next
```
```   163   assume "x = 1"
```
```   164   then show "x dvd (x ^ n)" by simp
```
```   165 qed
```
```   166
```
```   167 end
```
```   168
```
```   169 context ring_1
```
```   170 begin
```
```   171
```
```   172 lemma power_minus:
```
```   173   "(- a) ^ n = (- 1) ^ n * a ^ n"
```
```   174 proof (induct n)
```
```   175   case 0 show ?case by simp
```
```   176 next
```
```   177   case (Suc n) then show ?case
```
```   178     by (simp del: power_Suc add: power_Suc2 mult_assoc)
```
```   179 qed
```
```   180
```
```   181 lemma power_minus_Bit0:
```
```   182   "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
```
```   183   by (induct k, simp_all only: numeral_class.numeral.simps power_add
```
```   184     power_one_right mult_minus_left mult_minus_right minus_minus)
```
```   185
```
```   186 lemma power_minus_Bit1:
```
```   187   "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
```
```   188   by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
```
```   189
```
```   190 lemma power_neg_numeral_Bit0 [simp]:
```
```   191   "neg_numeral k ^ numeral (Num.Bit0 l) = numeral (Num.pow k (Num.Bit0 l))"
```
```   192   by (simp only: neg_numeral_def power_minus_Bit0 power_numeral)
```
```   193
```
```   194 lemma power_neg_numeral_Bit1 [simp]:
```
```   195   "neg_numeral k ^ numeral (Num.Bit1 l) = neg_numeral (Num.pow k (Num.Bit1 l))"
```
```   196   by (simp only: neg_numeral_def power_minus_Bit1 power_numeral pow.simps)
```
```   197
```
```   198 lemma power2_minus [simp]:
```
```   199   "(- a)\<twosuperior> = a\<twosuperior>"
```
```   200   by (rule power_minus_Bit0)
```
```   201
```
```   202 lemma power_minus1_even [simp]:
```
```   203   "-1 ^ (2*n) = 1"
```
```   204 proof (induct n)
```
```   205   case 0 show ?case by simp
```
```   206 next
```
```   207   case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
```
```   208 qed
```
```   209
```
```   210 lemma power_minus1_odd:
```
```   211   "-1 ^ Suc (2*n) = -1"
```
```   212   by simp
```
```   213
```
```   214 lemma power_minus_even [simp]:
```
```   215   "(-a) ^ (2*n) = a ^ (2*n)"
```
```   216   by (simp add: power_minus [of a])
```
```   217
```
```   218 end
```
```   219
```
```   220 context ring_1_no_zero_divisors
```
```   221 begin
```
```   222
```
```   223 lemma field_power_not_zero:
```
```   224   "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
```
```   225   by (induct n) auto
```
```   226
```
```   227 lemma zero_eq_power2 [simp]:
```
```   228   "a\<twosuperior> = 0 \<longleftrightarrow> a = 0"
```
```   229   unfolding power2_eq_square by simp
```
```   230
```
```   231 lemma power2_eq_1_iff:
```
```   232   "a\<twosuperior> = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
```
```   233   unfolding power2_eq_square by (rule square_eq_1_iff)
```
```   234
```
```   235 end
```
```   236
```
```   237 context idom
```
```   238 begin
```
```   239
```
```   240 lemma power2_eq_iff: "x\<twosuperior> = y\<twosuperior> \<longleftrightarrow> x = y \<or> x = - y"
```
```   241   unfolding power2_eq_square by (rule square_eq_iff)
```
```   242
```
```   243 end
```
```   244
```
```   245 context division_ring
```
```   246 begin
```
```   247
```
```   248 text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
```
```   249 lemma nonzero_power_inverse:
```
```   250   "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
```
```   251   by (induct n)
```
```   252     (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
```
```   253
```
```   254 end
```
```   255
```
```   256 context field
```
```   257 begin
```
```   258
```
```   259 lemma nonzero_power_divide:
```
```   260   "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
```
```   261   by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
```
```   262
```
```   263 end
```
```   264
```
```   265
```
```   266 subsection {* Exponentiation on ordered types *}
```
```   267
```
```   268 context linordered_ring (* TODO: move *)
```
```   269 begin
```
```   270
```
```   271 lemma sum_squares_ge_zero:
```
```   272   "0 \<le> x * x + y * y"
```
```   273   by (intro add_nonneg_nonneg zero_le_square)
```
```   274
```
```   275 lemma not_sum_squares_lt_zero:
```
```   276   "\<not> x * x + y * y < 0"
```
```   277   by (simp add: not_less sum_squares_ge_zero)
```
```   278
```
```   279 end
```
```   280
```
```   281 context linordered_semidom
```
```   282 begin
```
```   283
```
```   284 lemma zero_less_power [simp]:
```
```   285   "0 < a \<Longrightarrow> 0 < a ^ n"
```
```   286   by (induct n) (simp_all add: mult_pos_pos)
```
```   287
```
```   288 lemma zero_le_power [simp]:
```
```   289   "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
```
```   290   by (induct n) (simp_all add: mult_nonneg_nonneg)
```
```   291
```
```   292 lemma one_le_power[simp]:
```
```   293   "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
```
```   294   apply (induct n)
```
```   295   apply simp_all
```
```   296   apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
```
```   297   apply (simp_all add: order_trans [OF zero_le_one])
```
```   298   done
```
```   299
```
```   300 lemma power_gt1_lemma:
```
```   301   assumes gt1: "1 < a"
```
```   302   shows "1 < a * a ^ n"
```
```   303 proof -
```
```   304   from gt1 have "0 \<le> a"
```
```   305     by (fact order_trans [OF zero_le_one less_imp_le])
```
```   306   have "1 * 1 < a * 1" using gt1 by simp
```
```   307   also have "\<dots> \<le> a * a ^ n" using gt1
```
```   308     by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
```
```   309         zero_le_one order_refl)
```
```   310   finally show ?thesis by simp
```
```   311 qed
```
```   312
```
```   313 lemma power_gt1:
```
```   314   "1 < a \<Longrightarrow> 1 < a ^ Suc n"
```
```   315   by (simp add: power_gt1_lemma)
```
```   316
```
```   317 lemma one_less_power [simp]:
```
```   318   "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
```
```   319   by (cases n) (simp_all add: power_gt1_lemma)
```
```   320
```
```   321 lemma power_le_imp_le_exp:
```
```   322   assumes gt1: "1 < a"
```
```   323   shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
```
```   324 proof (induct m arbitrary: n)
```
```   325   case 0
```
```   326   show ?case by simp
```
```   327 next
```
```   328   case (Suc m)
```
```   329   show ?case
```
```   330   proof (cases n)
```
```   331     case 0
```
```   332     with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
```
```   333     with gt1 show ?thesis
```
```   334       by (force simp only: power_gt1_lemma
```
```   335           not_less [symmetric])
```
```   336   next
```
```   337     case (Suc n)
```
```   338     with Suc.prems Suc.hyps show ?thesis
```
```   339       by (force dest: mult_left_le_imp_le
```
```   340           simp add: less_trans [OF zero_less_one gt1])
```
```   341   qed
```
```   342 qed
```
```   343
```
```   344 text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
```
```   345 lemma power_inject_exp [simp]:
```
```   346   "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
```
```   347   by (force simp add: order_antisym power_le_imp_le_exp)
```
```   348
```
```   349 text{*Can relax the first premise to @{term "0<a"} in the case of the
```
```   350 natural numbers.*}
```
```   351 lemma power_less_imp_less_exp:
```
```   352   "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
```
```   353   by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
```
```   354     power_le_imp_le_exp)
```
```   355
```
```   356 lemma power_mono:
```
```   357   "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
```
```   358   by (induct n)
```
```   359     (auto intro: mult_mono order_trans [of 0 a b])
```
```   360
```
```   361 lemma power_strict_mono [rule_format]:
```
```   362   "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
```
```   363   by (induct n)
```
```   364    (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
```
```   365
```
```   366 text{*Lemma for @{text power_strict_decreasing}*}
```
```   367 lemma power_Suc_less:
```
```   368   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
```
```   369   by (induct n)
```
```   370     (auto simp add: mult_strict_left_mono)
```
```   371
```
```   372 lemma power_strict_decreasing [rule_format]:
```
```   373   "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
```
```   374 proof (induct N)
```
```   375   case 0 then show ?case by simp
```
```   376 next
```
```   377   case (Suc N) then show ?case
```
```   378   apply (auto simp add: power_Suc_less less_Suc_eq)
```
```   379   apply (subgoal_tac "a * a^N < 1 * a^n")
```
```   380   apply simp
```
```   381   apply (rule mult_strict_mono) apply auto
```
```   382   done
```
```   383 qed
```
```   384
```
```   385 text{*Proof resembles that of @{text power_strict_decreasing}*}
```
```   386 lemma power_decreasing [rule_format]:
```
```   387   "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
```
```   388 proof (induct N)
```
```   389   case 0 then show ?case by simp
```
```   390 next
```
```   391   case (Suc N) then show ?case
```
```   392   apply (auto simp add: le_Suc_eq)
```
```   393   apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
```
```   394   apply (rule mult_mono) apply auto
```
```   395   done
```
```   396 qed
```
```   397
```
```   398 lemma power_Suc_less_one:
```
```   399   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
```
```   400   using power_strict_decreasing [of 0 "Suc n" a] by simp
```
```   401
```
```   402 text{*Proof again resembles that of @{text power_strict_decreasing}*}
```
```   403 lemma power_increasing [rule_format]:
```
```   404   "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
```
```   405 proof (induct N)
```
```   406   case 0 then show ?case by simp
```
```   407 next
```
```   408   case (Suc N) then show ?case
```
```   409   apply (auto simp add: le_Suc_eq)
```
```   410   apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
```
```   411   apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
```
```   412   done
```
```   413 qed
```
```   414
```
```   415 text{*Lemma for @{text power_strict_increasing}*}
```
```   416 lemma power_less_power_Suc:
```
```   417   "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
```
```   418   by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
```
```   419
```
```   420 lemma power_strict_increasing [rule_format]:
```
```   421   "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
```
```   422 proof (induct N)
```
```   423   case 0 then show ?case by simp
```
```   424 next
```
```   425   case (Suc N) then show ?case
```
```   426   apply (auto simp add: power_less_power_Suc less_Suc_eq)
```
```   427   apply (subgoal_tac "1 * a^n < a * a^N", simp)
```
```   428   apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
```
```   429   done
```
```   430 qed
```
```   431
```
```   432 lemma power_increasing_iff [simp]:
```
```   433   "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
```
```   434   by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
```
```   435
```
```   436 lemma power_strict_increasing_iff [simp]:
```
```   437   "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
```
```   438 by (blast intro: power_less_imp_less_exp power_strict_increasing)
```
```   439
```
```   440 lemma power_le_imp_le_base:
```
```   441   assumes le: "a ^ Suc n \<le> b ^ Suc n"
```
```   442     and ynonneg: "0 \<le> b"
```
```   443   shows "a \<le> b"
```
```   444 proof (rule ccontr)
```
```   445   assume "~ a \<le> b"
```
```   446   then have "b < a" by (simp only: linorder_not_le)
```
```   447   then have "b ^ Suc n < a ^ Suc n"
```
```   448     by (simp only: assms power_strict_mono)
```
```   449   from le and this show False
```
```   450     by (simp add: linorder_not_less [symmetric])
```
```   451 qed
```
```   452
```
```   453 lemma power_less_imp_less_base:
```
```   454   assumes less: "a ^ n < b ^ n"
```
```   455   assumes nonneg: "0 \<le> b"
```
```   456   shows "a < b"
```
```   457 proof (rule contrapos_pp [OF less])
```
```   458   assume "~ a < b"
```
```   459   hence "b \<le> a" by (simp only: linorder_not_less)
```
```   460   hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
```
```   461   thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
```
```   462 qed
```
```   463
```
```   464 lemma power_inject_base:
```
```   465   "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
```
```   466 by (blast intro: power_le_imp_le_base antisym eq_refl sym)
```
```   467
```
```   468 lemma power_eq_imp_eq_base:
```
```   469   "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
```
```   470   by (cases n) (simp_all del: power_Suc, rule power_inject_base)
```
```   471
```
```   472 lemma power2_le_imp_le:
```
```   473   "x\<twosuperior> \<le> y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
```
```   474   unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
```
```   475
```
```   476 lemma power2_less_imp_less:
```
```   477   "x\<twosuperior> < y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
```
```   478   by (rule power_less_imp_less_base)
```
```   479
```
```   480 lemma power2_eq_imp_eq:
```
```   481   "x\<twosuperior> = y\<twosuperior> \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
```
```   482   unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
```
```   483
```
```   484 end
```
```   485
```
```   486 context linordered_ring_strict
```
```   487 begin
```
```   488
```
```   489 lemma sum_squares_eq_zero_iff:
```
```   490   "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   491   by (simp add: add_nonneg_eq_0_iff)
```
```   492
```
```   493 lemma sum_squares_le_zero_iff:
```
```   494   "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   495   by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
```
```   496
```
```   497 lemma sum_squares_gt_zero_iff:
```
```   498   "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   499   by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
```
```   500
```
```   501 end
```
```   502
```
```   503 context linordered_idom
```
```   504 begin
```
```   505
```
```   506 lemma power_abs:
```
```   507   "abs (a ^ n) = abs a ^ n"
```
```   508   by (induct n) (auto simp add: abs_mult)
```
```   509
```
```   510 lemma abs_power_minus [simp]:
```
```   511   "abs ((-a) ^ n) = abs (a ^ n)"
```
```   512   by (simp add: power_abs)
```
```   513
```
```   514 lemma zero_less_power_abs_iff [simp, no_atp]:
```
```   515   "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
```
```   516 proof (induct n)
```
```   517   case 0 show ?case by simp
```
```   518 next
```
```   519   case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
```
```   520 qed
```
```   521
```
```   522 lemma zero_le_power_abs [simp]:
```
```   523   "0 \<le> abs a ^ n"
```
```   524   by (rule zero_le_power [OF abs_ge_zero])
```
```   525
```
```   526 lemma zero_le_power2 [simp]:
```
```   527   "0 \<le> a\<twosuperior>"
```
```   528   by (simp add: power2_eq_square)
```
```   529
```
```   530 lemma zero_less_power2 [simp]:
```
```   531   "0 < a\<twosuperior> \<longleftrightarrow> a \<noteq> 0"
```
```   532   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
```
```   533
```
```   534 lemma power2_less_0 [simp]:
```
```   535   "\<not> a\<twosuperior> < 0"
```
```   536   by (force simp add: power2_eq_square mult_less_0_iff)
```
```   537
```
```   538 lemma abs_power2 [simp]:
```
```   539   "abs (a\<twosuperior>) = a\<twosuperior>"
```
```   540   by (simp add: power2_eq_square abs_mult abs_mult_self)
```
```   541
```
```   542 lemma power2_abs [simp]:
```
```   543   "(abs a)\<twosuperior> = a\<twosuperior>"
```
```   544   by (simp add: power2_eq_square abs_mult_self)
```
```   545
```
```   546 lemma odd_power_less_zero:
```
```   547   "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
```
```   548 proof (induct n)
```
```   549   case 0
```
```   550   then show ?case by simp
```
```   551 next
```
```   552   case (Suc n)
```
```   553   have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
```
```   554     by (simp add: mult_ac power_add power2_eq_square)
```
```   555   thus ?case
```
```   556     by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
```
```   557 qed
```
```   558
```
```   559 lemma odd_0_le_power_imp_0_le:
```
```   560   "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
```
```   561   using odd_power_less_zero [of a n]
```
```   562     by (force simp add: linorder_not_less [symmetric])
```
```   563
```
```   564 lemma zero_le_even_power'[simp]:
```
```   565   "0 \<le> a ^ (2*n)"
```
```   566 proof (induct n)
```
```   567   case 0
```
```   568     show ?case by simp
```
```   569 next
```
```   570   case (Suc n)
```
```   571     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
```
```   572       by (simp add: mult_ac power_add power2_eq_square)
```
```   573     thus ?case
```
```   574       by (simp add: Suc zero_le_mult_iff)
```
```   575 qed
```
```   576
```
```   577 lemma sum_power2_ge_zero:
```
```   578   "0 \<le> x\<twosuperior> + y\<twosuperior>"
```
```   579   by (intro add_nonneg_nonneg zero_le_power2)
```
```   580
```
```   581 lemma not_sum_power2_lt_zero:
```
```   582   "\<not> x\<twosuperior> + y\<twosuperior> < 0"
```
```   583   unfolding not_less by (rule sum_power2_ge_zero)
```
```   584
```
```   585 lemma sum_power2_eq_zero_iff:
```
```   586   "x\<twosuperior> + y\<twosuperior> = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   587   unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
```
```   588
```
```   589 lemma sum_power2_le_zero_iff:
```
```   590   "x\<twosuperior> + y\<twosuperior> \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   591   by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
```
```   592
```
```   593 lemma sum_power2_gt_zero_iff:
```
```   594   "0 < x\<twosuperior> + y\<twosuperior> \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   595   unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
```
```   596
```
```   597 end
```
```   598
```
```   599
```
```   600 subsection {* Miscellaneous rules *}
```
```   601
```
```   602 lemma power2_sum:
```
```   603   fixes x y :: "'a::comm_semiring_1"
```
```   604   shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y"
```
```   605   by (simp add: algebra_simps power2_eq_square mult_2_right)
```
```   606
```
```   607 lemma power2_diff:
```
```   608   fixes x y :: "'a::comm_ring_1"
```
```   609   shows "(x - y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> - 2 * x * y"
```
```   610   by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute)
```
```   611
```
```   612 lemma power_0_Suc [simp]:
```
```   613   "(0::'a::{power, semiring_0}) ^ Suc n = 0"
```
```   614   by simp
```
```   615
```
```   616 text{*It looks plausible as a simprule, but its effect can be strange.*}
```
```   617 lemma power_0_left:
```
```   618   "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
```
```   619   by (induct n) simp_all
```
```   620
```
```   621 lemma power_eq_0_iff [simp]:
```
```   622   "a ^ n = 0 \<longleftrightarrow>
```
```   623      a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
```
```   624   by (induct n)
```
```   625     (auto simp add: no_zero_divisors elim: contrapos_pp)
```
```   626
```
```   627 lemma (in field) power_diff:
```
```   628   assumes nz: "a \<noteq> 0"
```
```   629   shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
```
```   630   by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
```
```   631
```
```   632 text{*Perhaps these should be simprules.*}
```
```   633 lemma power_inverse:
```
```   634   fixes a :: "'a::division_ring_inverse_zero"
```
```   635   shows "inverse (a ^ n) = inverse a ^ n"
```
```   636 apply (cases "a = 0")
```
```   637 apply (simp add: power_0_left)
```
```   638 apply (simp add: nonzero_power_inverse)
```
```   639 done (* TODO: reorient or rename to inverse_power *)
```
```   640
```
```   641 lemma power_one_over:
```
```   642   "1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ n"
```
```   643   by (simp add: divide_inverse) (rule power_inverse)
```
```   644
```
```   645 lemma power_divide:
```
```   646   "(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n"
```
```   647 apply (cases "b = 0")
```
```   648 apply (simp add: power_0_left)
```
```   649 apply (rule nonzero_power_divide)
```
```   650 apply assumption
```
```   651 done
```
```   652
```
```   653
```
```   654 subsection {* Exponentiation for the Natural Numbers *}
```
```   655
```
```   656 lemma nat_one_le_power [simp]:
```
```   657   "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
```
```   658   by (rule one_le_power [of i n, unfolded One_nat_def])
```
```   659
```
```   660 lemma nat_zero_less_power_iff [simp]:
```
```   661   "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
```
```   662   by (induct n) auto
```
```   663
```
```   664 lemma nat_power_eq_Suc_0_iff [simp]:
```
```   665   "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
```
```   666   by (induct m) auto
```
```   667
```
```   668 lemma power_Suc_0 [simp]:
```
```   669   "Suc 0 ^ n = Suc 0"
```
```   670   by simp
```
```   671
```
```   672 text{*Valid for the naturals, but what if @{text"0<i<1"}?
```
```   673 Premises cannot be weakened: consider the case where @{term "i=0"},
```
```   674 @{term "m=1"} and @{term "n=0"}.*}
```
```   675 lemma nat_power_less_imp_less:
```
```   676   assumes nonneg: "0 < (i\<Colon>nat)"
```
```   677   assumes less: "i ^ m < i ^ n"
```
```   678   shows "m < n"
```
```   679 proof (cases "i = 1")
```
```   680   case True with less power_one [where 'a = nat] show ?thesis by simp
```
```   681 next
```
```   682   case False with nonneg have "1 < i" by auto
```
```   683   from power_strict_increasing_iff [OF this] less show ?thesis ..
```
```   684 qed
```
```   685
```
```   686 lemma power_dvd_imp_le:
```
```   687   "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
```
```   688   apply (rule power_le_imp_le_exp, assumption)
```
```   689   apply (erule dvd_imp_le, simp)
```
```   690   done
```
```   691
```
```   692
```
```   693 subsection {* Code generator tweak *}
```
```   694
```
```   695 lemma power_power_power [code]:
```
```   696   "power = power.power (1::'a::{power}) (op *)"
```
```   697   unfolding power_def power.power_def ..
```
```   698
```
```   699 declare power.power.simps [code]
```
```   700
```
```   701 code_modulename SML
```
```   702   Power Arith
```
```   703
```
```   704 code_modulename OCaml
```
```   705   Power Arith
```
```   706
```
```   707 code_modulename Haskell
```
```   708   Power Arith
```
```   709
```
```   710 end
```