src/HOL/Power.thy
 author blanchet Wed Sep 24 15:45:55 2014 +0200 (2014-09-24) changeset 58425 246985c6b20b parent 58410 6d46ad54a2ab child 58437 8d124c73c37a permissions -rw-r--r--
simpler proof
```     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 Equiv_Relations
```
```    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"  ("(_\<^sup>2)" [1000] 999) where
```
```    31   "x\<^sup>2 \<equiv> x ^ 2"
```
```    32
```
```    33 notation (latex output)
```
```    34   power2  ("(_\<^sup>2)" [1000] 999)
```
```    35
```
```    36 notation (HTML output)
```
```    37   power2  ("(_\<^sup>2)" [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\<^sup>2 = 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)\<^sup>2"
```
```    78   by (subst mult.commute) (simp add: power_mult)
```
```    79
```
```    80 lemma power_odd_eq:
```
```    81   "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
```
```    82   by (simp add: power_even_eq)
```
```    83
```
```    84 lemma power_numeral_even:
```
```    85   "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
```
```    86   unfolding numeral_Bit0 power_add Let_def ..
```
```    87
```
```    88 lemma power_numeral_odd:
```
```    89   "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
```
```    90   unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
```
```    91   unfolding power_Suc power_add Let_def mult.assoc ..
```
```    92
```
```    93 lemma funpow_times_power:
```
```    94   "(times x ^^ f x) = times (x ^ f x)"
```
```    95 proof (induct "f x" arbitrary: f)
```
```    96   case 0 then show ?case by (simp add: fun_eq_iff)
```
```    97 next
```
```    98   case (Suc n)
```
```    99   def g \<equiv> "\<lambda>x. f x - 1"
```
```   100   with Suc have "n = g x" by simp
```
```   101   with Suc have "times x ^^ g x = times (x ^ g x)" by simp
```
```   102   moreover from Suc g_def have "f x = g x + 1" by simp
```
```   103   ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
```
```   104 qed
```
```   105
```
```   106 end
```
```   107
```
```   108 context comm_monoid_mult
```
```   109 begin
```
```   110
```
```   111 lemma power_mult_distrib [field_simps]:
```
```   112   "(a * b) ^ n = (a ^ n) * (b ^ n)"
```
```   113   by (induct n) (simp_all add: ac_simps)
```
```   114
```
```   115 end
```
```   116
```
```   117 context semiring_numeral
```
```   118 begin
```
```   119
```
```   120 lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
```
```   121   by (simp only: sqr_conv_mult numeral_mult)
```
```   122
```
```   123 lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
```
```   124   by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
```
```   125     numeral_sqr numeral_mult power_add power_one_right)
```
```   126
```
```   127 lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
```
```   128   by (rule numeral_pow [symmetric])
```
```   129
```
```   130 end
```
```   131
```
```   132 context semiring_1
```
```   133 begin
```
```   134
```
```   135 lemma of_nat_power:
```
```   136   "of_nat (m ^ n) = of_nat m ^ n"
```
```   137   by (induct n) (simp_all add: of_nat_mult)
```
```   138
```
```   139 lemma power_zero_numeral [simp]: "(0::'a) ^ numeral k = 0"
```
```   140   by (simp add: numeral_eq_Suc)
```
```   141
```
```   142 lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
```
```   143   by (rule power_zero_numeral)
```
```   144
```
```   145 lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
```
```   146   by (rule power_one)
```
```   147
```
```   148 end
```
```   149
```
```   150 context comm_semiring_1
```
```   151 begin
```
```   152
```
```   153 text {* The divides relation *}
```
```   154
```
```   155 lemma le_imp_power_dvd:
```
```   156   assumes "m \<le> n" shows "a ^ m dvd a ^ n"
```
```   157 proof
```
```   158   have "a ^ n = a ^ (m + (n - m))"
```
```   159     using `m \<le> n` by simp
```
```   160   also have "\<dots> = a ^ m * a ^ (n - m)"
```
```   161     by (rule power_add)
```
```   162   finally show "a ^ n = a ^ m * a ^ (n - m)" .
```
```   163 qed
```
```   164
```
```   165 lemma power_le_dvd:
```
```   166   "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
```
```   167   by (rule dvd_trans [OF le_imp_power_dvd])
```
```   168
```
```   169 lemma dvd_power_same:
```
```   170   "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
```
```   171   by (induct n) (auto simp add: mult_dvd_mono)
```
```   172
```
```   173 lemma dvd_power_le:
```
```   174   "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
```
```   175   by (rule power_le_dvd [OF dvd_power_same])
```
```   176
```
```   177 lemma dvd_power [simp]:
```
```   178   assumes "n > (0::nat) \<or> x = 1"
```
```   179   shows "x dvd (x ^ n)"
```
```   180 using assms proof
```
```   181   assume "0 < n"
```
```   182   then have "x ^ n = x ^ Suc (n - 1)" by simp
```
```   183   then show "x dvd (x ^ n)" by simp
```
```   184 next
```
```   185   assume "x = 1"
```
```   186   then show "x dvd (x ^ n)" by simp
```
```   187 qed
```
```   188
```
```   189 end
```
```   190
```
```   191 context ring_1
```
```   192 begin
```
```   193
```
```   194 lemma power_minus:
```
```   195   "(- a) ^ n = (- 1) ^ n * a ^ n"
```
```   196 proof (induct n)
```
```   197   case 0 show ?case by simp
```
```   198 next
```
```   199   case (Suc n) then show ?case
```
```   200     by (simp del: power_Suc add: power_Suc2 mult.assoc)
```
```   201 qed
```
```   202
```
```   203 lemma power_minus_Bit0:
```
```   204   "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
```
```   205   by (induct k, simp_all only: numeral_class.numeral.simps power_add
```
```   206     power_one_right mult_minus_left mult_minus_right minus_minus)
```
```   207
```
```   208 lemma power_minus_Bit1:
```
```   209   "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
```
```   210   by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
```
```   211
```
```   212 lemma power2_minus [simp]:
```
```   213   "(- a)\<^sup>2 = a\<^sup>2"
```
```   214   by (rule power_minus_Bit0)
```
```   215
```
```   216 lemma power_minus1_even [simp]:
```
```   217   "(- 1) ^ (2*n) = 1"
```
```   218 proof (induct n)
```
```   219   case 0 show ?case by simp
```
```   220 next
```
```   221   case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
```
```   222 qed
```
```   223
```
```   224 lemma power_minus1_odd:
```
```   225   "(- 1) ^ Suc (2*n) = -1"
```
```   226   by simp
```
```   227
```
```   228 lemma power_minus_even [simp]:
```
```   229   "(-a) ^ (2*n) = a ^ (2*n)"
```
```   230   by (simp add: power_minus [of a])
```
```   231
```
```   232 end
```
```   233
```
```   234 context ring_1_no_zero_divisors
```
```   235 begin
```
```   236
```
```   237 lemma field_power_not_zero:
```
```   238   "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
```
```   239   by (induct n) auto
```
```   240
```
```   241 lemma zero_eq_power2 [simp]:
```
```   242   "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
```
```   243   unfolding power2_eq_square by simp
```
```   244
```
```   245 lemma power2_eq_1_iff:
```
```   246   "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
```
```   247   unfolding power2_eq_square by (rule square_eq_1_iff)
```
```   248
```
```   249 end
```
```   250
```
```   251 context idom
```
```   252 begin
```
```   253
```
```   254 lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
```
```   255   unfolding power2_eq_square by (rule square_eq_iff)
```
```   256
```
```   257 end
```
```   258
```
```   259 context division_ring
```
```   260 begin
```
```   261
```
```   262 text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
```
```   263 lemma nonzero_power_inverse:
```
```   264   "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
```
```   265   by (induct n)
```
```   266     (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
```
```   267
```
```   268 end
```
```   269
```
```   270 context field
```
```   271 begin
```
```   272
```
```   273 lemma nonzero_power_divide:
```
```   274   "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
```
```   275   by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
```
```   276
```
```   277 end
```
```   278
```
```   279
```
```   280 subsection {* Exponentiation on ordered types *}
```
```   281
```
```   282 context linordered_ring (* TODO: move *)
```
```   283 begin
```
```   284
```
```   285 lemma sum_squares_ge_zero:
```
```   286   "0 \<le> x * x + y * y"
```
```   287   by (intro add_nonneg_nonneg zero_le_square)
```
```   288
```
```   289 lemma not_sum_squares_lt_zero:
```
```   290   "\<not> x * x + y * y < 0"
```
```   291   by (simp add: not_less sum_squares_ge_zero)
```
```   292
```
```   293 end
```
```   294
```
```   295 context linordered_semidom
```
```   296 begin
```
```   297
```
```   298 lemma zero_less_power [simp]:
```
```   299   "0 < a \<Longrightarrow> 0 < a ^ n"
```
```   300   by (induct n) simp_all
```
```   301
```
```   302 lemma zero_le_power [simp]:
```
```   303   "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
```
```   304   by (induct n) simp_all
```
```   305
```
```   306 lemma power_mono:
```
```   307   "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
```
```   308   by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
```
```   309
```
```   310 lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
```
```   311   using power_mono [of 1 a n] by simp
```
```   312
```
```   313 lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
```
```   314   using power_mono [of a 1 n] by simp
```
```   315
```
```   316 lemma power_gt1_lemma:
```
```   317   assumes gt1: "1 < a"
```
```   318   shows "1 < a * a ^ n"
```
```   319 proof -
```
```   320   from gt1 have "0 \<le> a"
```
```   321     by (fact order_trans [OF zero_le_one less_imp_le])
```
```   322   have "1 * 1 < a * 1" using gt1 by simp
```
```   323   also have "\<dots> \<le> a * a ^ n" using gt1
```
```   324     by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
```
```   325         zero_le_one order_refl)
```
```   326   finally show ?thesis by simp
```
```   327 qed
```
```   328
```
```   329 lemma power_gt1:
```
```   330   "1 < a \<Longrightarrow> 1 < a ^ Suc n"
```
```   331   by (simp add: power_gt1_lemma)
```
```   332
```
```   333 lemma one_less_power [simp]:
```
```   334   "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
```
```   335   by (cases n) (simp_all add: power_gt1_lemma)
```
```   336
```
```   337 lemma power_le_imp_le_exp:
```
```   338   assumes gt1: "1 < a"
```
```   339   shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
```
```   340 proof (induct m arbitrary: n)
```
```   341   case 0
```
```   342   show ?case by simp
```
```   343 next
```
```   344   case (Suc m)
```
```   345   show ?case
```
```   346   proof (cases n)
```
```   347     case 0
```
```   348     with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
```
```   349     with gt1 show ?thesis
```
```   350       by (force simp only: power_gt1_lemma
```
```   351           not_less [symmetric])
```
```   352   next
```
```   353     case (Suc n)
```
```   354     with Suc.prems Suc.hyps show ?thesis
```
```   355       by (force dest: mult_left_le_imp_le
```
```   356           simp add: less_trans [OF zero_less_one gt1])
```
```   357   qed
```
```   358 qed
```
```   359
```
```   360 text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
```
```   361 lemma power_inject_exp [simp]:
```
```   362   "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
```
```   363   by (force simp add: order_antisym power_le_imp_le_exp)
```
```   364
```
```   365 text{*Can relax the first premise to @{term "0<a"} in the case of the
```
```   366 natural numbers.*}
```
```   367 lemma power_less_imp_less_exp:
```
```   368   "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
```
```   369   by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
```
```   370     power_le_imp_le_exp)
```
```   371
```
```   372 lemma power_strict_mono [rule_format]:
```
```   373   "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
```
```   374   by (induct n)
```
```   375    (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
```
```   376
```
```   377 text{*Lemma for @{text power_strict_decreasing}*}
```
```   378 lemma power_Suc_less:
```
```   379   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
```
```   380   by (induct n)
```
```   381     (auto simp add: mult_strict_left_mono)
```
```   382
```
```   383 lemma power_strict_decreasing [rule_format]:
```
```   384   "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
```
```   385 proof (induct N)
```
```   386   case 0 then show ?case by simp
```
```   387 next
```
```   388   case (Suc N) then show ?case
```
```   389   apply (auto simp add: power_Suc_less less_Suc_eq)
```
```   390   apply (subgoal_tac "a * a^N < 1 * a^n")
```
```   391   apply simp
```
```   392   apply (rule mult_strict_mono) apply auto
```
```   393   done
```
```   394 qed
```
```   395
```
```   396 text{*Proof resembles that of @{text power_strict_decreasing}*}
```
```   397 lemma power_decreasing [rule_format]:
```
```   398   "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
```
```   399 proof (induct N)
```
```   400   case 0 then show ?case by simp
```
```   401 next
```
```   402   case (Suc N) then show ?case
```
```   403   apply (auto simp add: le_Suc_eq)
```
```   404   apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
```
```   405   apply (rule mult_mono) apply auto
```
```   406   done
```
```   407 qed
```
```   408
```
```   409 lemma power_Suc_less_one:
```
```   410   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
```
```   411   using power_strict_decreasing [of 0 "Suc n" a] by simp
```
```   412
```
```   413 text{*Proof again resembles that of @{text power_strict_decreasing}*}
```
```   414 lemma power_increasing [rule_format]:
```
```   415   "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
```
```   416 proof (induct N)
```
```   417   case 0 then show ?case by simp
```
```   418 next
```
```   419   case (Suc N) then show ?case
```
```   420   apply (auto simp add: le_Suc_eq)
```
```   421   apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
```
```   422   apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
```
```   423   done
```
```   424 qed
```
```   425
```
```   426 text{*Lemma for @{text power_strict_increasing}*}
```
```   427 lemma power_less_power_Suc:
```
```   428   "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
```
```   429   by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
```
```   430
```
```   431 lemma power_strict_increasing [rule_format]:
```
```   432   "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
```
```   433 proof (induct N)
```
```   434   case 0 then show ?case by simp
```
```   435 next
```
```   436   case (Suc N) then show ?case
```
```   437   apply (auto simp add: power_less_power_Suc less_Suc_eq)
```
```   438   apply (subgoal_tac "1 * a^n < a * a^N", simp)
```
```   439   apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
```
```   440   done
```
```   441 qed
```
```   442
```
```   443 lemma power_increasing_iff [simp]:
```
```   444   "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
```
```   445   by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
```
```   446
```
```   447 lemma power_strict_increasing_iff [simp]:
```
```   448   "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
```
```   449 by (blast intro: power_less_imp_less_exp power_strict_increasing)
```
```   450
```
```   451 lemma power_le_imp_le_base:
```
```   452   assumes le: "a ^ Suc n \<le> b ^ Suc n"
```
```   453     and ynonneg: "0 \<le> b"
```
```   454   shows "a \<le> b"
```
```   455 proof (rule ccontr)
```
```   456   assume "~ a \<le> b"
```
```   457   then have "b < a" by (simp only: linorder_not_le)
```
```   458   then have "b ^ Suc n < a ^ Suc n"
```
```   459     by (simp only: assms power_strict_mono)
```
```   460   from le and this show False
```
```   461     by (simp add: linorder_not_less [symmetric])
```
```   462 qed
```
```   463
```
```   464 lemma power_less_imp_less_base:
```
```   465   assumes less: "a ^ n < b ^ n"
```
```   466   assumes nonneg: "0 \<le> b"
```
```   467   shows "a < b"
```
```   468 proof (rule contrapos_pp [OF less])
```
```   469   assume "~ a < b"
```
```   470   hence "b \<le> a" by (simp only: linorder_not_less)
```
```   471   hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
```
```   472   thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
```
```   473 qed
```
```   474
```
```   475 lemma power_inject_base:
```
```   476   "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
```
```   477 by (blast intro: power_le_imp_le_base antisym eq_refl sym)
```
```   478
```
```   479 lemma power_eq_imp_eq_base:
```
```   480   "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
```
```   481   by (cases n) (simp_all del: power_Suc, rule power_inject_base)
```
```   482
```
```   483 lemma power2_le_imp_le:
```
```   484   "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
```
```   485   unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
```
```   486
```
```   487 lemma power2_less_imp_less:
```
```   488   "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
```
```   489   by (rule power_less_imp_less_base)
```
```   490
```
```   491 lemma power2_eq_imp_eq:
```
```   492   "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
```
```   493   unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
```
```   494
```
```   495 end
```
```   496
```
```   497 context linordered_ring_strict
```
```   498 begin
```
```   499
```
```   500 lemma sum_squares_eq_zero_iff:
```
```   501   "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   502   by (simp add: add_nonneg_eq_0_iff)
```
```   503
```
```   504 lemma sum_squares_le_zero_iff:
```
```   505   "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   506   by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
```
```   507
```
```   508 lemma sum_squares_gt_zero_iff:
```
```   509   "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   510   by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
```
```   511
```
```   512 end
```
```   513
```
```   514 context linordered_idom
```
```   515 begin
```
```   516
```
```   517 lemma power_abs:
```
```   518   "abs (a ^ n) = abs a ^ n"
```
```   519   by (induct n) (auto simp add: abs_mult)
```
```   520
```
```   521 lemma abs_power_minus [simp]:
```
```   522   "abs ((-a) ^ n) = abs (a ^ n)"
```
```   523   by (simp add: power_abs)
```
```   524
```
```   525 lemma zero_less_power_abs_iff [simp]:
```
```   526   "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
```
```   527 proof (induct n)
```
```   528   case 0 show ?case by simp
```
```   529 next
```
```   530   case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
```
```   531 qed
```
```   532
```
```   533 lemma zero_le_power_abs [simp]:
```
```   534   "0 \<le> abs a ^ n"
```
```   535   by (rule zero_le_power [OF abs_ge_zero])
```
```   536
```
```   537 lemma zero_le_power2 [simp]:
```
```   538   "0 \<le> a\<^sup>2"
```
```   539   by (simp add: power2_eq_square)
```
```   540
```
```   541 lemma zero_less_power2 [simp]:
```
```   542   "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
```
```   543   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
```
```   544
```
```   545 lemma power2_less_0 [simp]:
```
```   546   "\<not> a\<^sup>2 < 0"
```
```   547   by (force simp add: power2_eq_square mult_less_0_iff)
```
```   548
```
```   549 lemma abs_power2 [simp]:
```
```   550   "abs (a\<^sup>2) = a\<^sup>2"
```
```   551   by (simp add: power2_eq_square abs_mult abs_mult_self)
```
```   552
```
```   553 lemma power2_abs [simp]:
```
```   554   "(abs a)\<^sup>2 = a\<^sup>2"
```
```   555   by (simp add: power2_eq_square abs_mult_self)
```
```   556
```
```   557 lemma odd_power_less_zero:
```
```   558   "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
```
```   559 proof (induct n)
```
```   560   case 0
```
```   561   then show ?case by simp
```
```   562 next
```
```   563   case (Suc n)
```
```   564   have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
```
```   565     by (simp add: ac_simps power_add power2_eq_square)
```
```   566   thus ?case
```
```   567     by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
```
```   568 qed
```
```   569
```
```   570 lemma odd_0_le_power_imp_0_le:
```
```   571   "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
```
```   572   using odd_power_less_zero [of a n]
```
```   573     by (force simp add: linorder_not_less [symmetric])
```
```   574
```
```   575 lemma zero_le_even_power'[simp]:
```
```   576   "0 \<le> a ^ (2*n)"
```
```   577 proof (induct n)
```
```   578   case 0
```
```   579     show ?case by simp
```
```   580 next
```
```   581   case (Suc n)
```
```   582     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
```
```   583       by (simp add: ac_simps power_add power2_eq_square)
```
```   584     thus ?case
```
```   585       by (simp add: Suc zero_le_mult_iff)
```
```   586 qed
```
```   587
```
```   588 lemma sum_power2_ge_zero:
```
```   589   "0 \<le> x\<^sup>2 + y\<^sup>2"
```
```   590   by (intro add_nonneg_nonneg zero_le_power2)
```
```   591
```
```   592 lemma not_sum_power2_lt_zero:
```
```   593   "\<not> x\<^sup>2 + y\<^sup>2 < 0"
```
```   594   unfolding not_less by (rule sum_power2_ge_zero)
```
```   595
```
```   596 lemma sum_power2_eq_zero_iff:
```
```   597   "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   598   unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
```
```   599
```
```   600 lemma sum_power2_le_zero_iff:
```
```   601   "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   602   by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
```
```   603
```
```   604 lemma sum_power2_gt_zero_iff:
```
```   605   "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   606   unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
```
```   607
```
```   608 end
```
```   609
```
```   610
```
```   611 subsection {* Miscellaneous rules *}
```
```   612
```
```   613 lemma self_le_power:
```
```   614   fixes x::"'a::linordered_semidom"
```
```   615   shows "1 \<le> x \<Longrightarrow> 0 < n \<Longrightarrow> x \<le> x ^ n"
```
```   616   using power_increasing[of 1 n x] power_one_right[of x] by auto
```
```   617
```
```   618 lemma power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
```
```   619   unfolding One_nat_def by (cases m) simp_all
```
```   620
```
```   621 lemma power2_sum:
```
```   622   fixes x y :: "'a::comm_semiring_1"
```
```   623   shows "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
```
```   624   by (simp add: algebra_simps power2_eq_square mult_2_right)
```
```   625
```
```   626 lemma power2_diff:
```
```   627   fixes x y :: "'a::comm_ring_1"
```
```   628   shows "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
```
```   629   by (simp add: ring_distribs power2_eq_square mult_2) (rule mult.commute)
```
```   630
```
```   631 lemma power_0_Suc [simp]:
```
```   632   "(0::'a::{power, semiring_0}) ^ Suc n = 0"
```
```   633   by simp
```
```   634
```
```   635 text{*It looks plausible as a simprule, but its effect can be strange.*}
```
```   636 lemma power_0_left:
```
```   637   "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
```
```   638   by (induct n) simp_all
```
```   639
```
```   640 lemma power_eq_0_iff [simp]:
```
```   641   "a ^ n = 0 \<longleftrightarrow>
```
```   642      a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
```
```   643   by (induct n)
```
```   644     (auto simp add: no_zero_divisors elim: contrapos_pp)
```
```   645
```
```   646 lemma (in field) power_diff:
```
```   647   assumes nz: "a \<noteq> 0"
```
```   648   shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
```
```   649   by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
```
```   650
```
```   651 text{*Perhaps these should be simprules.*}
```
```   652 lemma power_inverse:
```
```   653   fixes a :: "'a::division_ring_inverse_zero"
```
```   654   shows "inverse (a ^ n) = inverse a ^ n"
```
```   655 apply (cases "a = 0")
```
```   656 apply (simp add: power_0_left)
```
```   657 apply (simp add: nonzero_power_inverse)
```
```   658 done (* TODO: reorient or rename to inverse_power *)
```
```   659
```
```   660 lemma power_one_over:
```
```   661   "1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ n"
```
```   662   by (simp add: divide_inverse) (rule power_inverse)
```
```   663
```
```   664 lemma power_divide [field_simps, divide_simps]:
```
```   665   "(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n"
```
```   666 apply (cases "b = 0")
```
```   667 apply (simp add: power_0_left)
```
```   668 apply (rule nonzero_power_divide)
```
```   669 apply assumption
```
```   670 done
```
```   671
```
```   672 text {* Simprules for comparisons where common factors can be cancelled. *}
```
```   673
```
```   674 lemmas zero_compare_simps =
```
```   675     add_strict_increasing add_strict_increasing2 add_increasing
```
```   676     zero_le_mult_iff zero_le_divide_iff
```
```   677     zero_less_mult_iff zero_less_divide_iff
```
```   678     mult_le_0_iff divide_le_0_iff
```
```   679     mult_less_0_iff divide_less_0_iff
```
```   680     zero_le_power2 power2_less_0
```
```   681
```
```   682
```
```   683 subsection {* Exponentiation for the Natural Numbers *}
```
```   684
```
```   685 lemma nat_one_le_power [simp]:
```
```   686   "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
```
```   687   by (rule one_le_power [of i n, unfolded One_nat_def])
```
```   688
```
```   689 lemma nat_zero_less_power_iff [simp]:
```
```   690   "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
```
```   691   by (induct n) auto
```
```   692
```
```   693 lemma nat_power_eq_Suc_0_iff [simp]:
```
```   694   "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
```
```   695   by (induct m) auto
```
```   696
```
```   697 lemma power_Suc_0 [simp]:
```
```   698   "Suc 0 ^ n = Suc 0"
```
```   699   by simp
```
```   700
```
```   701 text{*Valid for the naturals, but what if @{text"0<i<1"}?
```
```   702 Premises cannot be weakened: consider the case where @{term "i=0"},
```
```   703 @{term "m=1"} and @{term "n=0"}.*}
```
```   704 lemma nat_power_less_imp_less:
```
```   705   assumes nonneg: "0 < (i\<Colon>nat)"
```
```   706   assumes less: "i ^ m < i ^ n"
```
```   707   shows "m < n"
```
```   708 proof (cases "i = 1")
```
```   709   case True with less power_one [where 'a = nat] show ?thesis by simp
```
```   710 next
```
```   711   case False with nonneg have "1 < i" by auto
```
```   712   from power_strict_increasing_iff [OF this] less show ?thesis ..
```
```   713 qed
```
```   714
```
```   715 lemma power_dvd_imp_le:
```
```   716   "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
```
```   717   apply (rule power_le_imp_le_exp, assumption)
```
```   718   apply (erule dvd_imp_le, simp)
```
```   719   done
```
```   720
```
```   721 lemma power2_nat_le_eq_le:
```
```   722   fixes m n :: nat
```
```   723   shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
```
```   724   by (auto intro: power2_le_imp_le power_mono)
```
```   725
```
```   726 lemma power2_nat_le_imp_le:
```
```   727   fixes m n :: nat
```
```   728   assumes "m\<^sup>2 \<le> n"
```
```   729   shows "m \<le> n"
```
```   730 proof (cases m)
```
```   731   case 0 then show ?thesis by simp
```
```   732 next
```
```   733   case (Suc k)
```
```   734   show ?thesis
```
```   735   proof (rule ccontr)
```
```   736     assume "\<not> m \<le> n"
```
```   737     then have "n < m" by simp
```
```   738     with assms Suc show False
```
```   739       by (auto simp add: algebra_simps) (simp add: power2_eq_square)
```
```   740   qed
```
```   741 qed
```
```   742
```
```   743 subsubsection {* Cardinality of the Powerset *}
```
```   744
```
```   745 lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
```
```   746   unfolding UNIV_bool by simp
```
```   747
```
```   748 lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
```
```   749 proof (induct rule: finite_induct)
```
```   750   case empty
```
```   751     show ?case by auto
```
```   752 next
```
```   753   case (insert x A)
```
```   754   then have "inj_on (insert x) (Pow A)"
```
```   755     unfolding inj_on_def by (blast elim!: equalityE)
```
```   756   then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A"
```
```   757     by (simp add: mult_2 card_image Pow_insert insert.hyps)
```
```   758   then show ?case using insert
```
```   759     apply (simp add: Pow_insert)
```
```   760     apply (subst card_Un_disjoint, auto)
```
```   761     done
```
```   762 qed
```
```   763
```
```   764
```
```   765 subsubsection {* Generalized sum over a set *}
```
```   766
```
```   767 lemma setsum_zero_power [simp]:
```
```   768   fixes c :: "nat \<Rightarrow> 'a::division_ring"
```
```   769   shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
```
```   770 apply (cases "finite A")
```
```   771   by (induction A rule: finite_induct) auto
```
```   772
```
```   773 lemma setsum_zero_power' [simp]:
```
```   774   fixes c :: "nat \<Rightarrow> 'a::field"
```
```   775   shows "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
```
```   776   using setsum_zero_power [of "\<lambda>i. c i / d i" A]
```
```   777   by auto
```
```   778
```
```   779
```
```   780 subsubsection {* Generalized product over a set *}
```
```   781
```
```   782 lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
```
```   783 apply (erule finite_induct)
```
```   784 apply auto
```
```   785 done
```
```   786
```
```   787 lemma setprod_power_distrib:
```
```   788   fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
```
```   789   shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
```
```   790 proof (cases "finite A")
```
```   791   case True then show ?thesis
```
```   792     by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
```
```   793 next
```
```   794   case False then show ?thesis
```
```   795     by simp
```
```   796 qed
```
```   797
```
```   798 lemma setprod_gen_delta:
```
```   799   assumes fS: "finite S"
```
```   800   shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::comm_monoid_mult) * c^ (card S - 1) else c^ card S)"
```
```   801 proof-
```
```   802   let ?f = "(\<lambda>k. if k=a then b k else c)"
```
```   803   {assume a: "a \<notin> S"
```
```   804     hence "\<forall> k\<in> S. ?f k = c" by simp
```
```   805     hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
```
```   806   moreover
```
```   807   {assume a: "a \<in> S"
```
```   808     let ?A = "S - {a}"
```
```   809     let ?B = "{a}"
```
```   810     have eq: "S = ?A \<union> ?B" using a by blast
```
```   811     have dj: "?A \<inter> ?B = {}" by simp
```
```   812     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   813     have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
```
```   814       apply (rule setprod.cong) by auto
```
```   815     have cA: "card ?A = card S - 1" using fS a by auto
```
```   816     have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
```
```   817     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
```
```   818       using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```   819       by simp
```
```   820     then have ?thesis using a cA
```
```   821       by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
```
```   822   ultimately show ?thesis by blast
```
```   823 qed
```
```   824
```
```   825 subsection {* Code generator tweak *}
```
```   826
```
```   827 lemma power_power_power [code]:
```
```   828   "power = power.power (1::'a::{power}) (op *)"
```
```   829   unfolding power_def power.power_def ..
```
```   830
```
```   831 declare power.power.simps [code]
```
```   832
```
```   833 code_identifier
```
```   834   code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```   835
```
```   836 end
```
```   837
```