src/HOL/Power.thy
 author paulson Wed Apr 29 14:04:22 2015 +0100 (2015-04-29) changeset 60155 91477b3a2d6b parent 59867 58043346ca64 child 60685 cb21b7022b00 permissions -rw-r--r--
Tidying. Improved simplification for numerals, esp in exponents.
```     1 (*  Title:      HOL/Power.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1997  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 section {* 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_Suc0_right [simp]:
```
```    55   "a ^ Suc 0 = a"
```
```    56   by simp
```
```    57
```
```    58 lemma power_commutes:
```
```    59   "a ^ n * a = a * a ^ n"
```
```    60   by (induct n) (simp_all add: mult.assoc)
```
```    61
```
```    62 lemma power_Suc2:
```
```    63   "a ^ Suc n = a ^ n * a"
```
```    64   by (simp add: power_commutes)
```
```    65
```
```    66 lemma power_add:
```
```    67   "a ^ (m + n) = a ^ m * a ^ n"
```
```    68   by (induct m) (simp_all add: algebra_simps)
```
```    69
```
```    70 lemma power_mult:
```
```    71   "a ^ (m * n) = (a ^ m) ^ n"
```
```    72   by (induct n) (simp_all add: power_add)
```
```    73
```
```    74 lemma power2_eq_square: "a\<^sup>2 = a * a"
```
```    75   by (simp add: numeral_2_eq_2)
```
```    76
```
```    77 lemma power3_eq_cube: "a ^ 3 = a * a * a"
```
```    78   by (simp add: numeral_3_eq_3 mult.assoc)
```
```    79
```
```    80 lemma power_even_eq:
```
```    81   "a ^ (2 * n) = (a ^ n)\<^sup>2"
```
```    82   by (subst mult.commute) (simp add: power_mult)
```
```    83
```
```    84 lemma power_odd_eq:
```
```    85   "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
```
```    86   by (simp add: power_even_eq)
```
```    87
```
```    88 lemma power_numeral_even:
```
```    89   "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
```
```    90   unfolding numeral_Bit0 power_add Let_def ..
```
```    91
```
```    92 lemma power_numeral_odd:
```
```    93   "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
```
```    94   unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
```
```    95   unfolding power_Suc power_add Let_def mult.assoc ..
```
```    96
```
```    97 lemma funpow_times_power:
```
```    98   "(times x ^^ f x) = times (x ^ f x)"
```
```    99 proof (induct "f x" arbitrary: f)
```
```   100   case 0 then show ?case by (simp add: fun_eq_iff)
```
```   101 next
```
```   102   case (Suc n)
```
```   103   def g \<equiv> "\<lambda>x. f x - 1"
```
```   104   with Suc have "n = g x" by simp
```
```   105   with Suc have "times x ^^ g x = times (x ^ g x)" by simp
```
```   106   moreover from Suc g_def have "f x = g x + 1" by simp
```
```   107   ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
```
```   108 qed
```
```   109
```
```   110 lemma power_commuting_commutes:
```
```   111   assumes "x * y = y * x"
```
```   112   shows "x ^ n * y = y * x ^n"
```
```   113 proof (induct n)
```
```   114   case (Suc n)
```
```   115   have "x ^ Suc n * y = x ^ n * y * x"
```
```   116     by (subst power_Suc2) (simp add: assms ac_simps)
```
```   117   also have "\<dots> = y * x ^ Suc n"
```
```   118     unfolding Suc power_Suc2
```
```   119     by (simp add: ac_simps)
```
```   120   finally show ?case .
```
```   121 qed simp
```
```   122
```
```   123 end
```
```   124
```
```   125 context comm_monoid_mult
```
```   126 begin
```
```   127
```
```   128 lemma power_mult_distrib [field_simps]:
```
```   129   "(a * b) ^ n = (a ^ n) * (b ^ n)"
```
```   130   by (induct n) (simp_all add: ac_simps)
```
```   131
```
```   132 end
```
```   133
```
```   134 text{*Extract constant factors from powers*}
```
```   135 declare power_mult_distrib [where a = "numeral w" for w, simp]
```
```   136 declare power_mult_distrib [where b = "numeral w" for w, simp]
```
```   137
```
```   138 lemma power_add_numeral [simp]:
```
```   139   fixes a :: "'a :: monoid_mult"
```
```   140   shows "a^numeral m * a^numeral n = a^numeral (m + n)"
```
```   141   by (simp add: power_add [symmetric])
```
```   142
```
```   143 lemma power_add_numeral2 [simp]:
```
```   144   fixes a :: "'a :: monoid_mult"
```
```   145   shows "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
```
```   146   by (simp add: mult.assoc [symmetric])
```
```   147
```
```   148 lemma power_mult_numeral [simp]:
```
```   149   fixes a :: "'a :: monoid_mult"
```
```   150   shows"(a^numeral m)^numeral n = a^numeral (m * n)"
```
```   151   by (simp only: numeral_mult power_mult)
```
```   152
```
```   153 context semiring_numeral
```
```   154 begin
```
```   155
```
```   156 lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
```
```   157   by (simp only: sqr_conv_mult numeral_mult)
```
```   158
```
```   159 lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
```
```   160   by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
```
```   161     numeral_sqr numeral_mult power_add power_one_right)
```
```   162
```
```   163 lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
```
```   164   by (rule numeral_pow [symmetric])
```
```   165
```
```   166 end
```
```   167
```
```   168 context semiring_1
```
```   169 begin
```
```   170
```
```   171 lemma of_nat_power:
```
```   172   "of_nat (m ^ n) = of_nat m ^ n"
```
```   173   by (induct n) (simp_all add: of_nat_mult)
```
```   174
```
```   175 lemma zero_power:
```
```   176   "0 < n \<Longrightarrow> 0 ^ n = 0"
```
```   177   by (cases n) simp_all
```
```   178
```
```   179 lemma power_zero_numeral [simp]:
```
```   180   "0 ^ numeral k = 0"
```
```   181   by (simp add: numeral_eq_Suc)
```
```   182
```
```   183 lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
```
```   184   by (rule power_zero_numeral)
```
```   185
```
```   186 lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
```
```   187   by (rule power_one)
```
```   188
```
```   189 end
```
```   190
```
```   191 context comm_semiring_1
```
```   192 begin
```
```   193
```
```   194 text {* The divides relation *}
```
```   195
```
```   196 lemma le_imp_power_dvd:
```
```   197   assumes "m \<le> n" shows "a ^ m dvd a ^ n"
```
```   198 proof
```
```   199   have "a ^ n = a ^ (m + (n - m))"
```
```   200     using `m \<le> n` by simp
```
```   201   also have "\<dots> = a ^ m * a ^ (n - m)"
```
```   202     by (rule power_add)
```
```   203   finally show "a ^ n = a ^ m * a ^ (n - m)" .
```
```   204 qed
```
```   205
```
```   206 lemma power_le_dvd:
```
```   207   "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
```
```   208   by (rule dvd_trans [OF le_imp_power_dvd])
```
```   209
```
```   210 lemma dvd_power_same:
```
```   211   "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
```
```   212   by (induct n) (auto simp add: mult_dvd_mono)
```
```   213
```
```   214 lemma dvd_power_le:
```
```   215   "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
```
```   216   by (rule power_le_dvd [OF dvd_power_same])
```
```   217
```
```   218 lemma dvd_power [simp]:
```
```   219   assumes "n > (0::nat) \<or> x = 1"
```
```   220   shows "x dvd (x ^ n)"
```
```   221 using assms proof
```
```   222   assume "0 < n"
```
```   223   then have "x ^ n = x ^ Suc (n - 1)" by simp
```
```   224   then show "x dvd (x ^ n)" by simp
```
```   225 next
```
```   226   assume "x = 1"
```
```   227   then show "x dvd (x ^ n)" by simp
```
```   228 qed
```
```   229
```
```   230 end
```
```   231
```
```   232 context ring_1
```
```   233 begin
```
```   234
```
```   235 lemma power_minus:
```
```   236   "(- a) ^ n = (- 1) ^ n * a ^ n"
```
```   237 proof (induct n)
```
```   238   case 0 show ?case by simp
```
```   239 next
```
```   240   case (Suc n) then show ?case
```
```   241     by (simp del: power_Suc add: power_Suc2 mult.assoc)
```
```   242 qed
```
```   243
```
```   244 lemma power_minus_Bit0:
```
```   245   "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
```
```   246   by (induct k, simp_all only: numeral_class.numeral.simps power_add
```
```   247     power_one_right mult_minus_left mult_minus_right minus_minus)
```
```   248
```
```   249 lemma power_minus_Bit1:
```
```   250   "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
```
```   251   by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
```
```   252
```
```   253 lemma power2_minus [simp]:
```
```   254   "(- a)\<^sup>2 = a\<^sup>2"
```
```   255   by (rule power_minus_Bit0)
```
```   256
```
```   257 lemma power_minus1_even [simp]:
```
```   258   "(- 1) ^ (2*n) = 1"
```
```   259 proof (induct n)
```
```   260   case 0 show ?case by simp
```
```   261 next
```
```   262   case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
```
```   263 qed
```
```   264
```
```   265 lemma power_minus1_odd:
```
```   266   "(- 1) ^ Suc (2*n) = -1"
```
```   267   by simp
```
```   268
```
```   269 lemma power_minus_even [simp]:
```
```   270   "(-a) ^ (2*n) = a ^ (2*n)"
```
```   271   by (simp add: power_minus [of a])
```
```   272
```
```   273 end
```
```   274
```
```   275 lemma power_eq_0_nat_iff [simp]:
```
```   276   fixes m n :: nat
```
```   277   shows "m ^ n = 0 \<longleftrightarrow> m = 0 \<and> n > 0"
```
```   278   by (induct n) auto
```
```   279
```
```   280 context ring_1_no_zero_divisors
```
```   281 begin
```
```   282
```
```   283 lemma power_eq_0_iff [simp]:
```
```   284   "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
```
```   285   by (induct n) auto
```
```   286
```
```   287 lemma field_power_not_zero:
```
```   288   "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
```
```   289   by (induct n) auto
```
```   290
```
```   291 lemma zero_eq_power2 [simp]:
```
```   292   "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
```
```   293   unfolding power2_eq_square by simp
```
```   294
```
```   295 lemma power2_eq_1_iff:
```
```   296   "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
```
```   297   unfolding power2_eq_square by (rule square_eq_1_iff)
```
```   298
```
```   299 end
```
```   300
```
```   301 context idom
```
```   302 begin
```
```   303
```
```   304 lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
```
```   305   unfolding power2_eq_square by (rule square_eq_iff)
```
```   306
```
```   307 end
```
```   308
```
```   309 context division_ring
```
```   310 begin
```
```   311
```
```   312 text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
```
```   313 lemma nonzero_power_inverse:
```
```   314   "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
```
```   315   by (induct n)
```
```   316     (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
```
```   317
```
```   318 end
```
```   319
```
```   320 context field
```
```   321 begin
```
```   322
```
```   323 lemma nonzero_power_divide:
```
```   324   "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
```
```   325   by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
```
```   326
```
```   327 declare nonzero_power_divide [where b = "numeral w" for w, simp]
```
```   328
```
```   329 end
```
```   330
```
```   331
```
```   332 subsection {* Exponentiation on ordered types *}
```
```   333
```
```   334 context linordered_ring (* TODO: move *)
```
```   335 begin
```
```   336
```
```   337 lemma sum_squares_ge_zero:
```
```   338   "0 \<le> x * x + y * y"
```
```   339   by (intro add_nonneg_nonneg zero_le_square)
```
```   340
```
```   341 lemma not_sum_squares_lt_zero:
```
```   342   "\<not> x * x + y * y < 0"
```
```   343   by (simp add: not_less sum_squares_ge_zero)
```
```   344
```
```   345 end
```
```   346
```
```   347 context linordered_semidom
```
```   348 begin
```
```   349
```
```   350 lemma zero_less_power [simp]:
```
```   351   "0 < a \<Longrightarrow> 0 < a ^ n"
```
```   352   by (induct n) simp_all
```
```   353
```
```   354 lemma zero_le_power [simp]:
```
```   355   "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
```
```   356   by (induct n) simp_all
```
```   357
```
```   358 lemma power_mono:
```
```   359   "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
```
```   360   by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
```
```   361
```
```   362 lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
```
```   363   using power_mono [of 1 a n] by simp
```
```   364
```
```   365 lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
```
```   366   using power_mono [of a 1 n] by simp
```
```   367
```
```   368 lemma power_gt1_lemma:
```
```   369   assumes gt1: "1 < a"
```
```   370   shows "1 < a * a ^ n"
```
```   371 proof -
```
```   372   from gt1 have "0 \<le> a"
```
```   373     by (fact order_trans [OF zero_le_one less_imp_le])
```
```   374   have "1 * 1 < a * 1" using gt1 by simp
```
```   375   also have "\<dots> \<le> a * a ^ n" using gt1
```
```   376     by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
```
```   377         zero_le_one order_refl)
```
```   378   finally show ?thesis by simp
```
```   379 qed
```
```   380
```
```   381 lemma power_gt1:
```
```   382   "1 < a \<Longrightarrow> 1 < a ^ Suc n"
```
```   383   by (simp add: power_gt1_lemma)
```
```   384
```
```   385 lemma one_less_power [simp]:
```
```   386   "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
```
```   387   by (cases n) (simp_all add: power_gt1_lemma)
```
```   388
```
```   389 lemma power_le_imp_le_exp:
```
```   390   assumes gt1: "1 < a"
```
```   391   shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
```
```   392 proof (induct m arbitrary: n)
```
```   393   case 0
```
```   394   show ?case by simp
```
```   395 next
```
```   396   case (Suc m)
```
```   397   show ?case
```
```   398   proof (cases n)
```
```   399     case 0
```
```   400     with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
```
```   401     with gt1 show ?thesis
```
```   402       by (force simp only: power_gt1_lemma
```
```   403           not_less [symmetric])
```
```   404   next
```
```   405     case (Suc n)
```
```   406     with Suc.prems Suc.hyps show ?thesis
```
```   407       by (force dest: mult_left_le_imp_le
```
```   408           simp add: less_trans [OF zero_less_one gt1])
```
```   409   qed
```
```   410 qed
```
```   411
```
```   412 text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
```
```   413 lemma power_inject_exp [simp]:
```
```   414   "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
```
```   415   by (force simp add: order_antisym power_le_imp_le_exp)
```
```   416
```
```   417 text{*Can relax the first premise to @{term "0<a"} in the case of the
```
```   418 natural numbers.*}
```
```   419 lemma power_less_imp_less_exp:
```
```   420   "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
```
```   421   by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
```
```   422     power_le_imp_le_exp)
```
```   423
```
```   424 lemma power_strict_mono [rule_format]:
```
```   425   "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
```
```   426   by (induct n)
```
```   427    (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
```
```   428
```
```   429 text{*Lemma for @{text power_strict_decreasing}*}
```
```   430 lemma power_Suc_less:
```
```   431   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
```
```   432   by (induct n)
```
```   433     (auto simp add: mult_strict_left_mono)
```
```   434
```
```   435 lemma power_strict_decreasing [rule_format]:
```
```   436   "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
```
```   437 proof (induct N)
```
```   438   case 0 then show ?case by simp
```
```   439 next
```
```   440   case (Suc N) then show ?case
```
```   441   apply (auto simp add: power_Suc_less less_Suc_eq)
```
```   442   apply (subgoal_tac "a * a^N < 1 * a^n")
```
```   443   apply simp
```
```   444   apply (rule mult_strict_mono) apply auto
```
```   445   done
```
```   446 qed
```
```   447
```
```   448 text{*Proof resembles that of @{text power_strict_decreasing}*}
```
```   449 lemma power_decreasing [rule_format]:
```
```   450   "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
```
```   451 proof (induct N)
```
```   452   case 0 then show ?case by simp
```
```   453 next
```
```   454   case (Suc N) then show ?case
```
```   455   apply (auto simp add: le_Suc_eq)
```
```   456   apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
```
```   457   apply (rule mult_mono) apply auto
```
```   458   done
```
```   459 qed
```
```   460
```
```   461 lemma power_Suc_less_one:
```
```   462   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
```
```   463   using power_strict_decreasing [of 0 "Suc n" a] by simp
```
```   464
```
```   465 text{*Proof again resembles that of @{text power_strict_decreasing}*}
```
```   466 lemma power_increasing [rule_format]:
```
```   467   "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
```
```   468 proof (induct N)
```
```   469   case 0 then show ?case by simp
```
```   470 next
```
```   471   case (Suc N) then show ?case
```
```   472   apply (auto simp add: le_Suc_eq)
```
```   473   apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
```
```   474   apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
```
```   475   done
```
```   476 qed
```
```   477
```
```   478 text{*Lemma for @{text power_strict_increasing}*}
```
```   479 lemma power_less_power_Suc:
```
```   480   "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
```
```   481   by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
```
```   482
```
```   483 lemma power_strict_increasing [rule_format]:
```
```   484   "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
```
```   485 proof (induct N)
```
```   486   case 0 then show ?case by simp
```
```   487 next
```
```   488   case (Suc N) then show ?case
```
```   489   apply (auto simp add: power_less_power_Suc less_Suc_eq)
```
```   490   apply (subgoal_tac "1 * a^n < a * a^N", simp)
```
```   491   apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
```
```   492   done
```
```   493 qed
```
```   494
```
```   495 lemma power_increasing_iff [simp]:
```
```   496   "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
```
```   497   by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
```
```   498
```
```   499 lemma power_strict_increasing_iff [simp]:
```
```   500   "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
```
```   501 by (blast intro: power_less_imp_less_exp power_strict_increasing)
```
```   502
```
```   503 lemma power_le_imp_le_base:
```
```   504   assumes le: "a ^ Suc n \<le> b ^ Suc n"
```
```   505     and ynonneg: "0 \<le> b"
```
```   506   shows "a \<le> b"
```
```   507 proof (rule ccontr)
```
```   508   assume "~ a \<le> b"
```
```   509   then have "b < a" by (simp only: linorder_not_le)
```
```   510   then have "b ^ Suc n < a ^ Suc n"
```
```   511     by (simp only: assms power_strict_mono)
```
```   512   from le and this show False
```
```   513     by (simp add: linorder_not_less [symmetric])
```
```   514 qed
```
```   515
```
```   516 lemma power_less_imp_less_base:
```
```   517   assumes less: "a ^ n < b ^ n"
```
```   518   assumes nonneg: "0 \<le> b"
```
```   519   shows "a < b"
```
```   520 proof (rule contrapos_pp [OF less])
```
```   521   assume "~ a < b"
```
```   522   hence "b \<le> a" by (simp only: linorder_not_less)
```
```   523   hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
```
```   524   thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
```
```   525 qed
```
```   526
```
```   527 lemma power_inject_base:
```
```   528   "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
```
```   529 by (blast intro: power_le_imp_le_base antisym eq_refl sym)
```
```   530
```
```   531 lemma power_eq_imp_eq_base:
```
```   532   "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
```
```   533   by (cases n) (simp_all del: power_Suc, rule power_inject_base)
```
```   534
```
```   535 lemma power2_le_imp_le:
```
```   536   "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
```
```   537   unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
```
```   538
```
```   539 lemma power2_less_imp_less:
```
```   540   "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
```
```   541   by (rule power_less_imp_less_base)
```
```   542
```
```   543 lemma power2_eq_imp_eq:
```
```   544   "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
```
```   545   unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
```
```   546
```
```   547 end
```
```   548
```
```   549 context linordered_ring_strict
```
```   550 begin
```
```   551
```
```   552 lemma sum_squares_eq_zero_iff:
```
```   553   "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   554   by (simp add: add_nonneg_eq_0_iff)
```
```   555
```
```   556 lemma sum_squares_le_zero_iff:
```
```   557   "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   558   by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
```
```   559
```
```   560 lemma sum_squares_gt_zero_iff:
```
```   561   "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   562   by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
```
```   563
```
```   564 end
```
```   565
```
```   566 context linordered_idom
```
```   567 begin
```
```   568
```
```   569 lemma power_abs:
```
```   570   "abs (a ^ n) = abs a ^ n"
```
```   571   by (induct n) (auto simp add: abs_mult)
```
```   572
```
```   573 lemma abs_power_minus [simp]:
```
```   574   "abs ((-a) ^ n) = abs (a ^ n)"
```
```   575   by (simp add: power_abs)
```
```   576
```
```   577 lemma zero_less_power_abs_iff [simp]:
```
```   578   "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
```
```   579 proof (induct n)
```
```   580   case 0 show ?case by simp
```
```   581 next
```
```   582   case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
```
```   583 qed
```
```   584
```
```   585 lemma zero_le_power_abs [simp]:
```
```   586   "0 \<le> abs a ^ n"
```
```   587   by (rule zero_le_power [OF abs_ge_zero])
```
```   588
```
```   589 lemma zero_le_power2 [simp]:
```
```   590   "0 \<le> a\<^sup>2"
```
```   591   by (simp add: power2_eq_square)
```
```   592
```
```   593 lemma zero_less_power2 [simp]:
```
```   594   "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
```
```   595   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
```
```   596
```
```   597 lemma power2_less_0 [simp]:
```
```   598   "\<not> a\<^sup>2 < 0"
```
```   599   by (force simp add: power2_eq_square mult_less_0_iff)
```
```   600
```
```   601 lemma power2_less_eq_zero_iff [simp]:
```
```   602   "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
```
```   603   by (simp add: le_less)
```
```   604
```
```   605 lemma abs_power2 [simp]:
```
```   606   "abs (a\<^sup>2) = a\<^sup>2"
```
```   607   by (simp add: power2_eq_square abs_mult abs_mult_self)
```
```   608
```
```   609 lemma power2_abs [simp]:
```
```   610   "(abs a)\<^sup>2 = a\<^sup>2"
```
```   611   by (simp add: power2_eq_square abs_mult_self)
```
```   612
```
```   613 lemma odd_power_less_zero:
```
```   614   "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
```
```   615 proof (induct n)
```
```   616   case 0
```
```   617   then show ?case by simp
```
```   618 next
```
```   619   case (Suc n)
```
```   620   have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
```
```   621     by (simp add: ac_simps power_add power2_eq_square)
```
```   622   thus ?case
```
```   623     by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
```
```   624 qed
```
```   625
```
```   626 lemma odd_0_le_power_imp_0_le:
```
```   627   "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
```
```   628   using odd_power_less_zero [of a n]
```
```   629     by (force simp add: linorder_not_less [symmetric])
```
```   630
```
```   631 lemma zero_le_even_power'[simp]:
```
```   632   "0 \<le> a ^ (2*n)"
```
```   633 proof (induct n)
```
```   634   case 0
```
```   635     show ?case by simp
```
```   636 next
```
```   637   case (Suc n)
```
```   638     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
```
```   639       by (simp add: ac_simps power_add power2_eq_square)
```
```   640     thus ?case
```
```   641       by (simp add: Suc zero_le_mult_iff)
```
```   642 qed
```
```   643
```
```   644 lemma sum_power2_ge_zero:
```
```   645   "0 \<le> x\<^sup>2 + y\<^sup>2"
```
```   646   by (intro add_nonneg_nonneg zero_le_power2)
```
```   647
```
```   648 lemma not_sum_power2_lt_zero:
```
```   649   "\<not> x\<^sup>2 + y\<^sup>2 < 0"
```
```   650   unfolding not_less by (rule sum_power2_ge_zero)
```
```   651
```
```   652 lemma sum_power2_eq_zero_iff:
```
```   653   "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   654   unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
```
```   655
```
```   656 lemma sum_power2_le_zero_iff:
```
```   657   "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   658   by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
```
```   659
```
```   660 lemma sum_power2_gt_zero_iff:
```
```   661   "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   662   unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
```
```   663
```
```   664 lemma abs_le_square_iff:
```
```   665    "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
```
```   666 proof
```
```   667   assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
```
```   668   then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
```
```   669   then show "x\<^sup>2 \<le> y\<^sup>2" by simp
```
```   670 next
```
```   671   assume "x\<^sup>2 \<le> y\<^sup>2"
```
```   672   then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
```
```   673     by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
```
```   674 qed
```
```   675
```
```   676 lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> abs(x) \<le> 1"
```
```   677   using abs_le_square_iff [of x 1]
```
```   678   by simp
```
```   679
```
```   680 lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> abs(x) = 1"
```
```   681   by (auto simp add: abs_if power2_eq_1_iff)
```
```   682
```
```   683 lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> abs(x) < 1"
```
```   684   using  abs_square_eq_1 [of x] abs_square_le_1 [of x]
```
```   685   by (auto simp add: le_less)
```
```   686
```
```   687 end
```
```   688
```
```   689
```
```   690 subsection {* Miscellaneous rules *}
```
```   691
```
```   692 lemma self_le_power:
```
```   693   fixes x::"'a::linordered_semidom"
```
```   694   shows "1 \<le> x \<Longrightarrow> 0 < n \<Longrightarrow> x \<le> x ^ n"
```
```   695   using power_increasing[of 1 n x] power_one_right[of x] by auto
```
```   696
```
```   697 lemma power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
```
```   698   unfolding One_nat_def by (cases m) simp_all
```
```   699
```
```   700 lemma (in comm_semiring_1) power2_sum:
```
```   701   "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
```
```   702   by (simp add: algebra_simps power2_eq_square mult_2_right)
```
```   703
```
```   704 lemma (in comm_ring_1) power2_diff:
```
```   705   "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
```
```   706   by (simp add: algebra_simps power2_eq_square mult_2_right)
```
```   707
```
```   708 lemma power_0_Suc [simp]:
```
```   709   "(0::'a::{power, semiring_0}) ^ Suc n = 0"
```
```   710   by simp
```
```   711
```
```   712 text{*It looks plausible as a simprule, but its effect can be strange.*}
```
```   713 lemma power_0_left:
```
```   714   "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
```
```   715   by (induct n) simp_all
```
```   716
```
```   717 lemma (in field) power_diff:
```
```   718   assumes nz: "a \<noteq> 0"
```
```   719   shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
```
```   720   by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
```
```   721
```
```   722 text{*Perhaps these should be simprules.*}
```
```   723 lemma power_inverse:
```
```   724   fixes a :: "'a::division_ring"
```
```   725   shows "inverse (a ^ n) = inverse a ^ n"
```
```   726 apply (cases "a = 0")
```
```   727 apply (simp add: power_0_left)
```
```   728 apply (simp add: nonzero_power_inverse)
```
```   729 done (* TODO: reorient or rename to inverse_power *)
```
```   730
```
```   731 lemma power_one_over:
```
```   732   "1 / (a::'a::{field, power}) ^ n =  (1 / a) ^ n"
```
```   733   by (simp add: divide_inverse) (rule power_inverse)
```
```   734
```
```   735 lemma power_divide [field_simps, divide_simps]:
```
```   736   "(a / b) ^ n = (a::'a::field) ^ n / b ^ n"
```
```   737 apply (cases "b = 0")
```
```   738 apply (simp add: power_0_left)
```
```   739 apply (rule nonzero_power_divide)
```
```   740 apply assumption
```
```   741 done
```
```   742
```
```   743 text {* Simprules for comparisons where common factors can be cancelled. *}
```
```   744
```
```   745 lemmas zero_compare_simps =
```
```   746     add_strict_increasing add_strict_increasing2 add_increasing
```
```   747     zero_le_mult_iff zero_le_divide_iff
```
```   748     zero_less_mult_iff zero_less_divide_iff
```
```   749     mult_le_0_iff divide_le_0_iff
```
```   750     mult_less_0_iff divide_less_0_iff
```
```   751     zero_le_power2 power2_less_0
```
```   752
```
```   753
```
```   754 subsection {* Exponentiation for the Natural Numbers *}
```
```   755
```
```   756 lemma nat_one_le_power [simp]:
```
```   757   "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
```
```   758   by (rule one_le_power [of i n, unfolded One_nat_def])
```
```   759
```
```   760 lemma nat_zero_less_power_iff [simp]:
```
```   761   "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
```
```   762   by (induct n) auto
```
```   763
```
```   764 lemma nat_power_eq_Suc_0_iff [simp]:
```
```   765   "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
```
```   766   by (induct m) auto
```
```   767
```
```   768 lemma power_Suc_0 [simp]:
```
```   769   "Suc 0 ^ n = Suc 0"
```
```   770   by simp
```
```   771
```
```   772 text{*Valid for the naturals, but what if @{text"0<i<1"}?
```
```   773 Premises cannot be weakened: consider the case where @{term "i=0"},
```
```   774 @{term "m=1"} and @{term "n=0"}.*}
```
```   775 lemma nat_power_less_imp_less:
```
```   776   assumes nonneg: "0 < (i\<Colon>nat)"
```
```   777   assumes less: "i ^ m < i ^ n"
```
```   778   shows "m < n"
```
```   779 proof (cases "i = 1")
```
```   780   case True with less power_one [where 'a = nat] show ?thesis by simp
```
```   781 next
```
```   782   case False with nonneg have "1 < i" by auto
```
```   783   from power_strict_increasing_iff [OF this] less show ?thesis ..
```
```   784 qed
```
```   785
```
```   786 lemma power_dvd_imp_le:
```
```   787   "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
```
```   788   apply (rule power_le_imp_le_exp, assumption)
```
```   789   apply (erule dvd_imp_le, simp)
```
```   790   done
```
```   791
```
```   792 lemma power2_nat_le_eq_le:
```
```   793   fixes m n :: nat
```
```   794   shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
```
```   795   by (auto intro: power2_le_imp_le power_mono)
```
```   796
```
```   797 lemma power2_nat_le_imp_le:
```
```   798   fixes m n :: nat
```
```   799   assumes "m\<^sup>2 \<le> n"
```
```   800   shows "m \<le> n"
```
```   801 proof (cases m)
```
```   802   case 0 then show ?thesis by simp
```
```   803 next
```
```   804   case (Suc k)
```
```   805   show ?thesis
```
```   806   proof (rule ccontr)
```
```   807     assume "\<not> m \<le> n"
```
```   808     then have "n < m" by simp
```
```   809     with assms Suc show False
```
```   810       by (auto simp add: algebra_simps) (simp add: power2_eq_square)
```
```   811   qed
```
```   812 qed
```
```   813
```
```   814 subsubsection {* Cardinality of the Powerset *}
```
```   815
```
```   816 lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
```
```   817   unfolding UNIV_bool by simp
```
```   818
```
```   819 lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
```
```   820 proof (induct rule: finite_induct)
```
```   821   case empty
```
```   822     show ?case by auto
```
```   823 next
```
```   824   case (insert x A)
```
```   825   then have "inj_on (insert x) (Pow A)"
```
```   826     unfolding inj_on_def by (blast elim!: equalityE)
```
```   827   then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A"
```
```   828     by (simp add: mult_2 card_image Pow_insert insert.hyps)
```
```   829   then show ?case using insert
```
```   830     apply (simp add: Pow_insert)
```
```   831     apply (subst card_Un_disjoint, auto)
```
```   832     done
```
```   833 qed
```
```   834
```
```   835
```
```   836 subsubsection {* Generalized sum over a set *}
```
```   837
```
```   838 lemma setsum_zero_power [simp]:
```
```   839   fixes c :: "nat \<Rightarrow> 'a::division_ring"
```
```   840   shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
```
```   841 apply (cases "finite A")
```
```   842   by (induction A rule: finite_induct) auto
```
```   843
```
```   844 lemma setsum_zero_power' [simp]:
```
```   845   fixes c :: "nat \<Rightarrow> 'a::field"
```
```   846   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)"
```
```   847   using setsum_zero_power [of "\<lambda>i. c i / d i" A]
```
```   848   by auto
```
```   849
```
```   850
```
```   851 subsubsection {* Generalized product over a set *}
```
```   852
```
```   853 lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
```
```   854 apply (erule finite_induct)
```
```   855 apply auto
```
```   856 done
```
```   857
```
```   858 lemma setprod_power_distrib:
```
```   859   fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
```
```   860   shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
```
```   861 proof (cases "finite A")
```
```   862   case True then show ?thesis
```
```   863     by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
```
```   864 next
```
```   865   case False then show ?thesis
```
```   866     by simp
```
```   867 qed
```
```   868
```
```   869 lemma power_setsum:
```
```   870   "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
```
```   871   by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
```
```   872
```
```   873 lemma setprod_gen_delta:
```
```   874   assumes fS: "finite S"
```
```   875   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)"
```
```   876 proof-
```
```   877   let ?f = "(\<lambda>k. if k=a then b k else c)"
```
```   878   {assume a: "a \<notin> S"
```
```   879     hence "\<forall> k\<in> S. ?f k = c" by simp
```
```   880     hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
```
```   881   moreover
```
```   882   {assume a: "a \<in> S"
```
```   883     let ?A = "S - {a}"
```
```   884     let ?B = "{a}"
```
```   885     have eq: "S = ?A \<union> ?B" using a by blast
```
```   886     have dj: "?A \<inter> ?B = {}" by simp
```
```   887     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   888     have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
```
```   889       apply (rule setprod.cong) by auto
```
```   890     have cA: "card ?A = card S - 1" using fS a by auto
```
```   891     have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
```
```   892     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
```
```   893       using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```   894       by simp
```
```   895     then have ?thesis using a cA
```
```   896       by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
```
```   897   ultimately show ?thesis by blast
```
```   898 qed
```
```   899
```
```   900 subsection {* Code generator tweak *}
```
```   901
```
```   902 lemma power_power_power [code]:
```
```   903   "power = power.power (1::'a::{power}) (op *)"
```
```   904   unfolding power_def power.power_def ..
```
```   905
```
```   906 declare power.power.simps [code]
```
```   907
```
```   908 code_identifier
```
```   909   code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```   910
```
```   911 end
```
```   912
```