src/HOL/Power.thy
 author paulson Tue Apr 25 16:39:54 2017 +0100 (2017-04-25) changeset 65578 e4997c181cce parent 65057 799bbbb3a395 child 66912 a99a7cbf0fb5 permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
```     1 (*  Title:      HOL/Power.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1997  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 section \<open>Exponentiation\<close>
```
```     7
```
```     8 theory Power
```
```     9   imports Num
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Powers for Arbitrary Monoids\<close>
```
```    13
```
```    14 class power = one + times
```
```    15 begin
```
```    16
```
```    17 primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"  (infixr "^" 80)
```
```    18   where
```
```    19     power_0: "a ^ 0 = 1"
```
```    20   | power_Suc: "a ^ Suc n = a * a ^ n"
```
```    21
```
```    22 notation (latex output)
```
```    23   power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
```
```    24
```
```    25 text \<open>Special syntax for squares.\<close>
```
```    26 abbreviation power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999)
```
```    27   where "x\<^sup>2 \<equiv> x ^ 2"
```
```    28
```
```    29 end
```
```    30
```
```    31 context monoid_mult
```
```    32 begin
```
```    33
```
```    34 subclass power .
```
```    35
```
```    36 lemma power_one [simp]: "1 ^ n = 1"
```
```    37   by (induct n) simp_all
```
```    38
```
```    39 lemma power_one_right [simp]: "a ^ 1 = a"
```
```    40   by simp
```
```    41
```
```    42 lemma power_Suc0_right [simp]: "a ^ Suc 0 = a"
```
```    43   by simp
```
```    44
```
```    45 lemma power_commutes: "a ^ n * a = a * a ^ n"
```
```    46   by (induct n) (simp_all add: mult.assoc)
```
```    47
```
```    48 lemma power_Suc2: "a ^ Suc n = a ^ n * a"
```
```    49   by (simp add: power_commutes)
```
```    50
```
```    51 lemma power_add: "a ^ (m + n) = a ^ m * a ^ n"
```
```    52   by (induct m) (simp_all add: algebra_simps)
```
```    53
```
```    54 lemma power_mult: "a ^ (m * n) = (a ^ m) ^ n"
```
```    55   by (induct n) (simp_all add: power_add)
```
```    56
```
```    57 lemma power2_eq_square: "a\<^sup>2 = a * a"
```
```    58   by (simp add: numeral_2_eq_2)
```
```    59
```
```    60 lemma power3_eq_cube: "a ^ 3 = a * a * a"
```
```    61   by (simp add: numeral_3_eq_3 mult.assoc)
```
```    62
```
```    63 lemma power_even_eq: "a ^ (2 * n) = (a ^ n)\<^sup>2"
```
```    64   by (subst mult.commute) (simp add: power_mult)
```
```    65
```
```    66 lemma power_odd_eq: "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
```
```    67   by (simp add: power_even_eq)
```
```    68
```
```    69 lemma power_numeral_even: "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
```
```    70   by (simp only: numeral_Bit0 power_add Let_def)
```
```    71
```
```    72 lemma power_numeral_odd: "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
```
```    73   by (simp only: numeral_Bit1 One_nat_def add_Suc_right add_0_right
```
```    74       power_Suc power_add Let_def mult.assoc)
```
```    75
```
```    76 lemma funpow_times_power: "(times x ^^ f x) = times (x ^ f x)"
```
```    77 proof (induct "f x" arbitrary: f)
```
```    78   case 0
```
```    79   then show ?case by (simp add: fun_eq_iff)
```
```    80 next
```
```    81   case (Suc n)
```
```    82   define g where "g x = f x - 1" for x
```
```    83   with Suc have "n = g x" by simp
```
```    84   with Suc have "times x ^^ g x = times (x ^ g x)" by simp
```
```    85   moreover from Suc g_def have "f x = g x + 1" by simp
```
```    86   ultimately show ?case
```
```    87     by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
```
```    88 qed
```
```    89
```
```    90 lemma power_commuting_commutes:
```
```    91   assumes "x * y = y * x"
```
```    92   shows "x ^ n * y = y * x ^n"
```
```    93 proof (induct n)
```
```    94   case 0
```
```    95   then show ?case by simp
```
```    96 next
```
```    97   case (Suc n)
```
```    98   have "x ^ Suc n * y = x ^ n * y * x"
```
```    99     by (subst power_Suc2) (simp add: assms ac_simps)
```
```   100   also have "\<dots> = y * x ^ Suc n"
```
```   101     by (simp only: Suc power_Suc2) (simp add: ac_simps)
```
```   102   finally show ?case .
```
```   103 qed
```
```   104
```
```   105 lemma power_minus_mult: "0 < n \<Longrightarrow> a ^ (n - 1) * a = a ^ n"
```
```   106   by (simp add: power_commutes split: nat_diff_split)
```
```   107
```
```   108 end
```
```   109
```
```   110 context comm_monoid_mult
```
```   111 begin
```
```   112
```
```   113 lemma power_mult_distrib [field_simps]: "(a * b) ^ n = (a ^ n) * (b ^ n)"
```
```   114   by (induct n) (simp_all add: ac_simps)
```
```   115
```
```   116 end
```
```   117
```
```   118 text \<open>Extract constant factors from powers.\<close>
```
```   119 declare power_mult_distrib [where a = "numeral w" for w, simp]
```
```   120 declare power_mult_distrib [where b = "numeral w" for w, simp]
```
```   121
```
```   122 lemma power_add_numeral [simp]: "a^numeral m * a^numeral n = a^numeral (m + n)"
```
```   123   for a :: "'a::monoid_mult"
```
```   124   by (simp add: power_add [symmetric])
```
```   125
```
```   126 lemma power_add_numeral2 [simp]: "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
```
```   127   for a :: "'a::monoid_mult"
```
```   128   by (simp add: mult.assoc [symmetric])
```
```   129
```
```   130 lemma power_mult_numeral [simp]: "(a^numeral m)^numeral n = a^numeral (m * n)"
```
```   131   for a :: "'a::monoid_mult"
```
```   132   by (simp only: numeral_mult power_mult)
```
```   133
```
```   134 context semiring_numeral
```
```   135 begin
```
```   136
```
```   137 lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
```
```   138   by (simp only: sqr_conv_mult numeral_mult)
```
```   139
```
```   140 lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
```
```   141   by (induct l)
```
```   142     (simp_all only: numeral_class.numeral.simps pow.simps
```
```   143       numeral_sqr numeral_mult power_add power_one_right)
```
```   144
```
```   145 lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
```
```   146   by (rule numeral_pow [symmetric])
```
```   147
```
```   148 end
```
```   149
```
```   150 context semiring_1
```
```   151 begin
```
```   152
```
```   153 lemma of_nat_power [simp]: "of_nat (m ^ n) = of_nat m ^ n"
```
```   154   by (induct n) simp_all
```
```   155
```
```   156 lemma zero_power: "0 < n \<Longrightarrow> 0 ^ n = 0"
```
```   157   by (cases n) simp_all
```
```   158
```
```   159 lemma power_zero_numeral [simp]: "0 ^ numeral k = 0"
```
```   160   by (simp add: numeral_eq_Suc)
```
```   161
```
```   162 lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
```
```   163   by (rule power_zero_numeral)
```
```   164
```
```   165 lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
```
```   166   by (rule power_one)
```
```   167
```
```   168 lemma power_0_Suc [simp]: "0 ^ Suc n = 0"
```
```   169   by simp
```
```   170
```
```   171 text \<open>It looks plausible as a simprule, but its effect can be strange.\<close>
```
```   172 lemma power_0_left: "0 ^ n = (if n = 0 then 1 else 0)"
```
```   173   by (cases n) simp_all
```
```   174
```
```   175 end
```
```   176
```
```   177 context comm_semiring_1
```
```   178 begin
```
```   179
```
```   180 text \<open>The divides relation.\<close>
```
```   181
```
```   182 lemma le_imp_power_dvd:
```
```   183   assumes "m \<le> n"
```
```   184   shows "a ^ m dvd a ^ n"
```
```   185 proof
```
```   186   from assms have "a ^ n = a ^ (m + (n - m))" by simp
```
```   187   also have "\<dots> = a ^ m * a ^ (n - m)" by (rule power_add)
```
```   188   finally show "a ^ n = a ^ m * a ^ (n - m)" .
```
```   189 qed
```
```   190
```
```   191 lemma power_le_dvd: "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
```
```   192   by (rule dvd_trans [OF le_imp_power_dvd])
```
```   193
```
```   194 lemma dvd_power_same: "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
```
```   195   by (induct n) (auto simp add: mult_dvd_mono)
```
```   196
```
```   197 lemma dvd_power_le: "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
```
```   198   by (rule power_le_dvd [OF dvd_power_same])
```
```   199
```
```   200 lemma dvd_power [simp]:
```
```   201   fixes n :: nat
```
```   202   assumes "n > 0 \<or> x = 1"
```
```   203   shows "x dvd (x ^ n)"
```
```   204   using assms
```
```   205 proof
```
```   206   assume "0 < n"
```
```   207   then have "x ^ n = x ^ Suc (n - 1)" by simp
```
```   208   then show "x dvd (x ^ n)" by simp
```
```   209 next
```
```   210   assume "x = 1"
```
```   211   then show "x dvd (x ^ n)" by simp
```
```   212 qed
```
```   213
```
```   214 end
```
```   215
```
```   216 context semiring_1_no_zero_divisors
```
```   217 begin
```
```   218
```
```   219 subclass power .
```
```   220
```
```   221 lemma power_eq_0_iff [simp]: "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
```
```   222   by (induct n) auto
```
```   223
```
```   224 lemma power_not_zero: "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
```
```   225   by (induct n) auto
```
```   226
```
```   227 lemma zero_eq_power2 [simp]: "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
```
```   228   unfolding power2_eq_square by simp
```
```   229
```
```   230 end
```
```   231
```
```   232 context ring_1
```
```   233 begin
```
```   234
```
```   235 lemma power_minus: "(- a) ^ n = (- 1) ^ n * a ^ n"
```
```   236 proof (induct n)
```
```   237   case 0
```
```   238   show ?case by simp
```
```   239 next
```
```   240   case (Suc n)
```
```   241   then show ?case
```
```   242     by (simp del: power_Suc add: power_Suc2 mult.assoc)
```
```   243 qed
```
```   244
```
```   245 lemma power_minus': "NO_MATCH 1 x \<Longrightarrow> (-x) ^ n = (-1)^n * x ^ n"
```
```   246   by (rule power_minus)
```
```   247
```
```   248 lemma power_minus_Bit0: "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
```
```   249   by (induct k, simp_all only: numeral_class.numeral.simps power_add
```
```   250     power_one_right mult_minus_left mult_minus_right minus_minus)
```
```   251
```
```   252 lemma power_minus_Bit1: "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
```
```   253   by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
```
```   254
```
```   255 lemma power2_minus [simp]: "(- a)\<^sup>2 = a\<^sup>2"
```
```   256   by (fact power_minus_Bit0)
```
```   257
```
```   258 lemma power_minus1_even [simp]: "(- 1) ^ (2*n) = 1"
```
```   259 proof (induct n)
```
```   260   case 0
```
```   261   show ?case by simp
```
```   262 next
```
```   263   case (Suc n)
```
```   264   then show ?case by (simp add: power_add power2_eq_square)
```
```   265 qed
```
```   266
```
```   267 lemma power_minus1_odd: "(- 1) ^ Suc (2*n) = -1"
```
```   268   by simp
```
```   269
```
```   270 lemma power_minus_even [simp]: "(-a) ^ (2*n) = a ^ (2*n)"
```
```   271   by (simp add: power_minus [of a])
```
```   272
```
```   273 end
```
```   274
```
```   275 context ring_1_no_zero_divisors
```
```   276 begin
```
```   277
```
```   278 lemma power2_eq_1_iff: "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
```
```   279   using square_eq_1_iff [of a] by (simp add: power2_eq_square)
```
```   280
```
```   281 end
```
```   282
```
```   283 context idom
```
```   284 begin
```
```   285
```
```   286 lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
```
```   287   unfolding power2_eq_square by (rule square_eq_iff)
```
```   288
```
```   289 end
```
```   290
```
```   291 context algebraic_semidom
```
```   292 begin
```
```   293
```
```   294 lemma div_power: "b dvd a \<Longrightarrow> (a div b) ^ n = a ^ n div b ^ n"
```
```   295   by (induct n) (simp_all add: div_mult_div_if_dvd dvd_power_same)
```
```   296
```
```   297 lemma is_unit_power_iff: "is_unit (a ^ n) \<longleftrightarrow> is_unit a \<or> n = 0"
```
```   298   by (induct n) (auto simp add: is_unit_mult_iff)
```
```   299
```
```   300 lemma dvd_power_iff:
```
```   301   assumes "x \<noteq> 0"
```
```   302   shows   "x ^ m dvd x ^ n \<longleftrightarrow> is_unit x \<or> m \<le> n"
```
```   303 proof
```
```   304   assume *: "x ^ m dvd x ^ n"
```
```   305   {
```
```   306     assume "m > n"
```
```   307     note *
```
```   308     also have "x ^ n = x ^ n * 1" by simp
```
```   309     also from \<open>m > n\<close> have "m = n + (m - n)" by simp
```
```   310     also have "x ^ \<dots> = x ^ n * x ^ (m - n)" by (rule power_add)
```
```   311     finally have "x ^ (m - n) dvd 1"
```
```   312       by (subst (asm) dvd_times_left_cancel_iff) (insert assms, simp_all)
```
```   313     with \<open>m > n\<close> have "is_unit x" by (simp add: is_unit_power_iff)
```
```   314   }
```
```   315   thus "is_unit x \<or> m \<le> n" by force
```
```   316 qed (auto intro: unit_imp_dvd simp: is_unit_power_iff le_imp_power_dvd)
```
```   317
```
```   318
```
```   319 end
```
```   320
```
```   321 context normalization_semidom
```
```   322 begin
```
```   323
```
```   324 lemma normalize_power: "normalize (a ^ n) = normalize a ^ n"
```
```   325   by (induct n) (simp_all add: normalize_mult)
```
```   326
```
```   327 lemma unit_factor_power: "unit_factor (a ^ n) = unit_factor a ^ n"
```
```   328   by (induct n) (simp_all add: unit_factor_mult)
```
```   329
```
```   330 end
```
```   331
```
```   332 context division_ring
```
```   333 begin
```
```   334
```
```   335 text \<open>Perhaps these should be simprules.\<close>
```
```   336 lemma power_inverse [field_simps, divide_simps]: "inverse a ^ n = inverse (a ^ n)"
```
```   337 proof (cases "a = 0")
```
```   338   case True
```
```   339   then show ?thesis by (simp add: power_0_left)
```
```   340 next
```
```   341   case False
```
```   342   then have "inverse (a ^ n) = inverse a ^ n"
```
```   343     by (induct n) (simp_all add: nonzero_inverse_mult_distrib power_commutes)
```
```   344   then show ?thesis by simp
```
```   345 qed
```
```   346
```
```   347 lemma power_one_over [field_simps, divide_simps]: "(1 / a) ^ n = 1 / a ^ n"
```
```   348   using power_inverse [of a] by (simp add: divide_inverse)
```
```   349
```
```   350 end
```
```   351
```
```   352 context field
```
```   353 begin
```
```   354
```
```   355 lemma power_diff:
```
```   356   assumes "a \<noteq> 0"
```
```   357   shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
```
```   358   by (induct m n rule: diff_induct) (simp_all add: assms power_not_zero)
```
```   359
```
```   360 lemma power_divide [field_simps, divide_simps]: "(a / b) ^ n = a ^ n / b ^ n"
```
```   361   by (induct n) simp_all
```
```   362
```
```   363 end
```
```   364
```
```   365
```
```   366 subsection \<open>Exponentiation on ordered types\<close>
```
```   367
```
```   368 context linordered_semidom
```
```   369 begin
```
```   370
```
```   371 lemma zero_less_power [simp]: "0 < a \<Longrightarrow> 0 < a ^ n"
```
```   372   by (induct n) simp_all
```
```   373
```
```   374 lemma zero_le_power [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
```
```   375   by (induct n) simp_all
```
```   376
```
```   377 lemma power_mono: "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
```
```   378   by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
```
```   379
```
```   380 lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
```
```   381   using power_mono [of 1 a n] by simp
```
```   382
```
```   383 lemma power_le_one: "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ n \<le> 1"
```
```   384   using power_mono [of a 1 n] by simp
```
```   385
```
```   386 lemma power_gt1_lemma:
```
```   387   assumes gt1: "1 < a"
```
```   388   shows "1 < a * a ^ n"
```
```   389 proof -
```
```   390   from gt1 have "0 \<le> a"
```
```   391     by (fact order_trans [OF zero_le_one less_imp_le])
```
```   392   from gt1 have "1 * 1 < a * 1" by simp
```
```   393   also from gt1 have "\<dots> \<le> a * a ^ n"
```
```   394     by (simp only: mult_mono \<open>0 \<le> a\<close> one_le_power order_less_imp_le zero_le_one order_refl)
```
```   395   finally show ?thesis by simp
```
```   396 qed
```
```   397
```
```   398 lemma power_gt1: "1 < a \<Longrightarrow> 1 < a ^ Suc n"
```
```   399   by (simp add: power_gt1_lemma)
```
```   400
```
```   401 lemma one_less_power [simp]: "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
```
```   402   by (cases n) (simp_all add: power_gt1_lemma)
```
```   403
```
```   404 lemma power_le_imp_le_exp:
```
```   405   assumes gt1: "1 < a"
```
```   406   shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
```
```   407 proof (induct m arbitrary: n)
```
```   408   case 0
```
```   409   show ?case by simp
```
```   410 next
```
```   411   case (Suc m)
```
```   412   show ?case
```
```   413   proof (cases n)
```
```   414     case 0
```
```   415     with Suc have "a * a ^ m \<le> 1" by simp
```
```   416     with gt1 show ?thesis
```
```   417       by (force simp only: power_gt1_lemma not_less [symmetric])
```
```   418   next
```
```   419     case (Suc n)
```
```   420     with Suc.prems Suc.hyps show ?thesis
```
```   421       by (force dest: mult_left_le_imp_le simp add: less_trans [OF zero_less_one gt1])
```
```   422   qed
```
```   423 qed
```
```   424
```
```   425 lemma of_nat_zero_less_power_iff [simp]: "of_nat x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
```
```   426   by (induct n) auto
```
```   427
```
```   428 text \<open>Surely we can strengthen this? It holds for \<open>0<a<1\<close> too.\<close>
```
```   429 lemma power_inject_exp [simp]: "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
```
```   430   by (force simp add: order_antisym power_le_imp_le_exp)
```
```   431
```
```   432 text \<open>
```
```   433   Can relax the first premise to @{term "0<a"} in the case of the
```
```   434   natural numbers.
```
```   435 \<close>
```
```   436 lemma power_less_imp_less_exp: "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
```
```   437   by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"] power_le_imp_le_exp)
```
```   438
```
```   439 lemma power_strict_mono [rule_format]: "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
```
```   440   by (induct n) (auto simp: mult_strict_mono le_less_trans [of 0 a b])
```
```   441
```
```   442 text\<open>Lemma for \<open>power_strict_decreasing\<close>\<close>
```
```   443 lemma power_Suc_less: "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
```
```   444   by (induct n) (auto simp: mult_strict_left_mono)
```
```   445
```
```   446 lemma power_strict_decreasing [rule_format]: "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
```
```   447 proof (induct N)
```
```   448   case 0
```
```   449   then show ?case by simp
```
```   450 next
```
```   451   case (Suc N)
```
```   452   then show ?case
```
```   453     apply (auto simp add: power_Suc_less less_Suc_eq)
```
```   454     apply (subgoal_tac "a * a^N < 1 * a^n")
```
```   455      apply simp
```
```   456     apply (rule mult_strict_mono)
```
```   457        apply auto
```
```   458     done
```
```   459 qed
```
```   460
```
```   461 text \<open>Proof resembles that of \<open>power_strict_decreasing\<close>.\<close>
```
```   462 lemma power_decreasing: "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ N \<le> a ^ n"
```
```   463 proof (induct N)
```
```   464   case 0
```
```   465   then show ?case by simp
```
```   466 next
```
```   467   case (Suc N)
```
```   468   then show ?case
```
```   469     apply (auto simp add: le_Suc_eq)
```
```   470     apply (subgoal_tac "a * a^N \<le> 1 * a^n")
```
```   471      apply simp
```
```   472     apply (rule mult_mono)
```
```   473        apply auto
```
```   474     done
```
```   475 qed
```
```   476
```
```   477 lemma power_Suc_less_one: "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
```
```   478   using power_strict_decreasing [of 0 "Suc n" a] by simp
```
```   479
```
```   480 text \<open>Proof again resembles that of \<open>power_strict_decreasing\<close>.\<close>
```
```   481 lemma power_increasing: "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
```
```   482 proof (induct N)
```
```   483   case 0
```
```   484   then show ?case by simp
```
```   485 next
```
```   486   case (Suc N)
```
```   487   then show ?case
```
```   488     apply (auto simp add: le_Suc_eq)
```
```   489     apply (subgoal_tac "1 * a^n \<le> a * a^N")
```
```   490      apply simp
```
```   491     apply (rule mult_mono)
```
```   492        apply (auto simp add: order_trans [OF zero_le_one])
```
```   493     done
```
```   494 qed
```
```   495
```
```   496 text \<open>Lemma for \<open>power_strict_increasing\<close>.\<close>
```
```   497 lemma power_less_power_Suc: "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
```
```   498   by (induct n) (auto simp: mult_strict_left_mono less_trans [OF zero_less_one])
```
```   499
```
```   500 lemma power_strict_increasing: "n < N \<Longrightarrow> 1 < a \<Longrightarrow> a ^ n < a ^ N"
```
```   501 proof (induct N)
```
```   502   case 0
```
```   503   then show ?case by simp
```
```   504 next
```
```   505   case (Suc N)
```
```   506   then show ?case
```
```   507     apply (auto simp add: power_less_power_Suc less_Suc_eq)
```
```   508     apply (subgoal_tac "1 * a^n < a * a^N")
```
```   509      apply simp
```
```   510     apply (rule mult_strict_mono)
```
```   511     apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
```
```   512     done
```
```   513 qed
```
```   514
```
```   515 lemma power_increasing_iff [simp]: "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
```
```   516   by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
```
```   517
```
```   518 lemma power_strict_increasing_iff [simp]: "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
```
```   519   by (blast intro: power_less_imp_less_exp power_strict_increasing)
```
```   520
```
```   521 lemma power_le_imp_le_base:
```
```   522   assumes le: "a ^ Suc n \<le> b ^ Suc n"
```
```   523     and "0 \<le> b"
```
```   524   shows "a \<le> b"
```
```   525 proof (rule ccontr)
```
```   526   assume "\<not> ?thesis"
```
```   527   then have "b < a" by (simp only: linorder_not_le)
```
```   528   then have "b ^ Suc n < a ^ Suc n"
```
```   529     by (simp only: assms(2) power_strict_mono)
```
```   530   with le show False
```
```   531     by (simp add: linorder_not_less [symmetric])
```
```   532 qed
```
```   533
```
```   534 lemma power_less_imp_less_base:
```
```   535   assumes less: "a ^ n < b ^ n"
```
```   536   assumes nonneg: "0 \<le> b"
```
```   537   shows "a < b"
```
```   538 proof (rule contrapos_pp [OF less])
```
```   539   assume "\<not> ?thesis"
```
```   540   then have "b \<le> a" by (simp only: linorder_not_less)
```
```   541   from this nonneg have "b ^ n \<le> a ^ n" by (rule power_mono)
```
```   542   then show "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
```
```   543 qed
```
```   544
```
```   545 lemma power_inject_base: "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
```
```   546   by (blast intro: power_le_imp_le_base antisym eq_refl sym)
```
```   547
```
```   548 lemma power_eq_imp_eq_base: "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
```
```   549   by (cases n) (simp_all del: power_Suc, rule power_inject_base)
```
```   550
```
```   551 lemma power_eq_iff_eq_base: "0 < n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a ^ n = b ^ n \<longleftrightarrow> a = b"
```
```   552   using power_eq_imp_eq_base [of a n b] by auto
```
```   553
```
```   554 lemma power2_le_imp_le: "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
```
```   555   unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
```
```   556
```
```   557 lemma power2_less_imp_less: "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
```
```   558   by (rule power_less_imp_less_base)
```
```   559
```
```   560 lemma power2_eq_imp_eq: "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
```
```   561   unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
```
```   562
```
```   563 lemma power_Suc_le_self: "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ Suc n \<le> a"
```
```   564   using power_decreasing [of 1 "Suc n" a] by simp
```
```   565
```
```   566 lemma power2_eq_iff_nonneg [simp]:
```
```   567   assumes "0 \<le> x" "0 \<le> y"
```
```   568   shows "(x ^ 2 = y ^ 2) \<longleftrightarrow> x = y"
```
```   569 using assms power2_eq_imp_eq by blast
```
```   570
```
```   571 end
```
```   572
```
```   573 context linordered_ring_strict
```
```   574 begin
```
```   575
```
```   576 lemma sum_squares_eq_zero_iff: "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   577   by (simp add: add_nonneg_eq_0_iff)
```
```   578
```
```   579 lemma sum_squares_le_zero_iff: "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   580   by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
```
```   581
```
```   582 lemma sum_squares_gt_zero_iff: "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   583   by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
```
```   584
```
```   585 end
```
```   586
```
```   587 context linordered_idom
```
```   588 begin
```
```   589
```
```   590 lemma zero_le_power2 [simp]: "0 \<le> a\<^sup>2"
```
```   591   by (simp add: power2_eq_square)
```
```   592
```
```   593 lemma zero_less_power2 [simp]: "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
```
```   594   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
```
```   595
```
```   596 lemma power2_less_0 [simp]: "\<not> a\<^sup>2 < 0"
```
```   597   by (force simp add: power2_eq_square mult_less_0_iff)
```
```   598
```
```   599 lemma power_abs: "\<bar>a ^ n\<bar> = \<bar>a\<bar> ^ n" -- \<open>FIXME simp?\<close>
```
```   600   by (induct n) (simp_all add: abs_mult)
```
```   601
```
```   602 lemma power_sgn [simp]: "sgn (a ^ n) = sgn a ^ n"
```
```   603   by (induct n) (simp_all add: sgn_mult)
```
```   604
```
```   605 lemma abs_power_minus [simp]: "\<bar>(- a) ^ n\<bar> = \<bar>a ^ n\<bar>"
```
```   606   by (simp add: power_abs)
```
```   607
```
```   608 lemma zero_less_power_abs_iff [simp]: "0 < \<bar>a\<bar> ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
```
```   609 proof (induct n)
```
```   610   case 0
```
```   611   show ?case by simp
```
```   612 next
```
```   613   case Suc
```
```   614   then show ?case by (auto simp: zero_less_mult_iff)
```
```   615 qed
```
```   616
```
```   617 lemma zero_le_power_abs [simp]: "0 \<le> \<bar>a\<bar> ^ n"
```
```   618   by (rule zero_le_power [OF abs_ge_zero])
```
```   619
```
```   620 lemma power2_less_eq_zero_iff [simp]: "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
```
```   621   by (simp add: le_less)
```
```   622
```
```   623 lemma abs_power2 [simp]: "\<bar>a\<^sup>2\<bar> = a\<^sup>2"
```
```   624   by (simp add: power2_eq_square)
```
```   625
```
```   626 lemma power2_abs [simp]: "\<bar>a\<bar>\<^sup>2 = a\<^sup>2"
```
```   627   by (simp add: power2_eq_square)
```
```   628
```
```   629 lemma odd_power_less_zero: "a < 0 \<Longrightarrow> a ^ Suc (2 * n) < 0"
```
```   630 proof (induct n)
```
```   631   case 0
```
```   632   then show ?case by simp
```
```   633 next
```
```   634   case (Suc n)
```
```   635   have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
```
```   636     by (simp add: ac_simps power_add power2_eq_square)
```
```   637   then show ?case
```
```   638     by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
```
```   639 qed
```
```   640
```
```   641 lemma odd_0_le_power_imp_0_le: "0 \<le> a ^ Suc (2 * n) \<Longrightarrow> 0 \<le> a"
```
```   642   using odd_power_less_zero [of a n]
```
```   643   by (force simp add: linorder_not_less [symmetric])
```
```   644
```
```   645 lemma zero_le_even_power'[simp]: "0 \<le> a ^ (2 * n)"
```
```   646 proof (induct n)
```
```   647   case 0
```
```   648   show ?case by simp
```
```   649 next
```
```   650   case (Suc n)
```
```   651   have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
```
```   652     by (simp add: ac_simps power_add power2_eq_square)
```
```   653   then show ?case
```
```   654     by (simp add: Suc zero_le_mult_iff)
```
```   655 qed
```
```   656
```
```   657 lemma sum_power2_ge_zero: "0 \<le> x\<^sup>2 + y\<^sup>2"
```
```   658   by (intro add_nonneg_nonneg zero_le_power2)
```
```   659
```
```   660 lemma not_sum_power2_lt_zero: "\<not> x\<^sup>2 + y\<^sup>2 < 0"
```
```   661   unfolding not_less by (rule sum_power2_ge_zero)
```
```   662
```
```   663 lemma sum_power2_eq_zero_iff: "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   664   unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
```
```   665
```
```   666 lemma sum_power2_le_zero_iff: "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   667   by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
```
```   668
```
```   669 lemma sum_power2_gt_zero_iff: "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   670   unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
```
```   671
```
```   672 lemma abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
```
```   673   (is "?lhs \<longleftrightarrow> ?rhs")
```
```   674 proof
```
```   675   assume ?lhs
```
```   676   then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono) simp
```
```   677   then show ?rhs by simp
```
```   678 next
```
```   679   assume ?rhs
```
```   680   then show ?lhs
```
```   681     by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
```
```   682 qed
```
```   683
```
```   684 lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> \<bar>x\<bar> \<le> 1"
```
```   685   using abs_le_square_iff [of x 1] by simp
```
```   686
```
```   687 lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
```
```   688   by (auto simp add: abs_if power2_eq_1_iff)
```
```   689
```
```   690 lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> \<bar>x\<bar> < 1"
```
```   691   using  abs_square_eq_1 [of x] abs_square_le_1 [of x] by (auto simp add: le_less)
```
```   692
```
```   693 end
```
```   694
```
```   695
```
```   696 subsection \<open>Miscellaneous rules\<close>
```
```   697
```
```   698 lemma (in linordered_semidom) self_le_power: "1 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a ^ n"
```
```   699   using power_increasing [of 1 n a] power_one_right [of a] by auto
```
```   700
```
```   701 lemma (in power) power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
```
```   702   unfolding One_nat_def by (cases m) simp_all
```
```   703
```
```   704 lemma (in comm_semiring_1) power2_sum: "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
```
```   705   by (simp add: algebra_simps power2_eq_square mult_2_right)
```
```   706
```
```   707 context comm_ring_1
```
```   708 begin
```
```   709
```
```   710 lemma power2_diff: "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
```
```   711   by (simp add: algebra_simps power2_eq_square mult_2_right)
```
```   712
```
```   713 lemma power2_commute: "(x - y)\<^sup>2 = (y - x)\<^sup>2"
```
```   714   by (simp add: algebra_simps power2_eq_square)
```
```   715
```
```   716 lemma minus_power_mult_self: "(- a) ^ n * (- a) ^ n = a ^ (2 * n)"
```
```   717   by (simp add: power_mult_distrib [symmetric])
```
```   718     (simp add: power2_eq_square [symmetric] power_mult [symmetric])
```
```   719
```
```   720 lemma minus_one_mult_self [simp]: "(- 1) ^ n * (- 1) ^ n = 1"
```
```   721   using minus_power_mult_self [of 1 n] by simp
```
```   722
```
```   723 lemma left_minus_one_mult_self [simp]: "(- 1) ^ n * ((- 1) ^ n * a) = a"
```
```   724   by (simp add: mult.assoc [symmetric])
```
```   725
```
```   726 end
```
```   727
```
```   728 text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
```
```   729
```
```   730 lemmas zero_compare_simps =
```
```   731   add_strict_increasing add_strict_increasing2 add_increasing
```
```   732   zero_le_mult_iff zero_le_divide_iff
```
```   733   zero_less_mult_iff zero_less_divide_iff
```
```   734   mult_le_0_iff divide_le_0_iff
```
```   735   mult_less_0_iff divide_less_0_iff
```
```   736   zero_le_power2 power2_less_0
```
```   737
```
```   738
```
```   739 subsection \<open>Exponentiation for the Natural Numbers\<close>
```
```   740
```
```   741 lemma nat_one_le_power [simp]: "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
```
```   742   by (rule one_le_power [of i n, unfolded One_nat_def])
```
```   743
```
```   744 lemma nat_zero_less_power_iff [simp]: "x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
```
```   745   for x :: nat
```
```   746   by (induct n) auto
```
```   747
```
```   748 lemma nat_power_eq_Suc_0_iff [simp]: "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
```
```   749   by (induct m) auto
```
```   750
```
```   751 lemma power_Suc_0 [simp]: "Suc 0 ^ n = Suc 0"
```
```   752   by simp
```
```   753
```
```   754 text \<open>
```
```   755   Valid for the naturals, but what if \<open>0 < i < 1\<close>? Premises cannot be
```
```   756   weakened: consider the case where \<open>i = 0\<close>, \<open>m = 1\<close> and \<open>n = 0\<close>.
```
```   757 \<close>
```
```   758
```
```   759 lemma nat_power_less_imp_less:
```
```   760   fixes i :: nat
```
```   761   assumes nonneg: "0 < i"
```
```   762   assumes less: "i ^ m < i ^ n"
```
```   763   shows "m < n"
```
```   764 proof (cases "i = 1")
```
```   765   case True
```
```   766   with less power_one [where 'a = nat] show ?thesis by simp
```
```   767 next
```
```   768   case False
```
```   769   with nonneg have "1 < i" by auto
```
```   770   from power_strict_increasing_iff [OF this] less show ?thesis ..
```
```   771 qed
```
```   772
```
```   773 lemma power_dvd_imp_le: "i ^ m dvd i ^ n \<Longrightarrow> 1 < i \<Longrightarrow> m \<le> n"
```
```   774   for i m n :: nat
```
```   775   apply (rule power_le_imp_le_exp)
```
```   776    apply assumption
```
```   777   apply (erule dvd_imp_le)
```
```   778   apply simp
```
```   779   done
```
```   780
```
```   781 lemma power2_nat_le_eq_le: "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
```
```   782   for m n :: nat
```
```   783   by (auto intro: power2_le_imp_le power_mono)
```
```   784
```
```   785 lemma power2_nat_le_imp_le:
```
```   786   fixes m n :: nat
```
```   787   assumes "m\<^sup>2 \<le> n"
```
```   788   shows "m \<le> n"
```
```   789 proof (cases m)
```
```   790   case 0
```
```   791   then show ?thesis by simp
```
```   792 next
```
```   793   case (Suc k)
```
```   794   show ?thesis
```
```   795   proof (rule ccontr)
```
```   796     assume "\<not> ?thesis"
```
```   797     then have "n < m" by simp
```
```   798     with assms Suc show False
```
```   799       by (simp add: power2_eq_square)
```
```   800   qed
```
```   801 qed
```
```   802
```
```   803 lemma ex_power_ivl1: fixes b k :: nat assumes "b \<ge> 2"
```
```   804 shows "k \<ge> 1 \<Longrightarrow> \<exists>n. b^n \<le> k \<and> k < b^(n+1)" (is "_ \<Longrightarrow> \<exists>n. ?P k n")
```
```   805 proof(induction k)
```
```   806   case 0 thus ?case by simp
```
```   807 next
```
```   808   case (Suc k)
```
```   809   show ?case
```
```   810   proof cases
```
```   811     assume "k=0"
```
```   812     hence "?P (Suc k) 0" using assms by simp
```
```   813     thus ?case ..
```
```   814   next
```
```   815     assume "k\<noteq>0"
```
```   816     with Suc obtain n where IH: "?P k n" by auto
```
```   817     show ?case
```
```   818     proof (cases "k = b^(n+1) - 1")
```
```   819       case True
```
```   820       hence "?P (Suc k) (n+1)" using assms
```
```   821         by (simp add: power_less_power_Suc)
```
```   822       thus ?thesis ..
```
```   823     next
```
```   824       case False
```
```   825       hence "?P (Suc k) n" using IH by auto
```
```   826       thus ?thesis ..
```
```   827     qed
```
```   828   qed
```
```   829 qed
```
```   830
```
```   831 lemma ex_power_ivl2: fixes b k :: nat assumes "b \<ge> 2" "k \<ge> 2"
```
```   832 shows "\<exists>n. b^n < k \<and> k \<le> b^(n+1)"
```
```   833 proof -
```
```   834   have "1 \<le> k - 1" using assms(2) by arith
```
```   835   from ex_power_ivl1[OF assms(1) this]
```
```   836   obtain n where "b ^ n \<le> k - 1 \<and> k - 1 < b ^ (n + 1)" ..
```
```   837   hence "b^n < k \<and> k \<le> b^(n+1)" using assms by auto
```
```   838   thus ?thesis ..
```
```   839 qed
```
```   840
```
```   841
```
```   842 subsubsection \<open>Cardinality of the Powerset\<close>
```
```   843
```
```   844 lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
```
```   845   unfolding UNIV_bool by simp
```
```   846
```
```   847 lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
```
```   848 proof (induct rule: finite_induct)
```
```   849   case empty
```
```   850   show ?case by simp
```
```   851 next
```
```   852   case (insert x A)
```
```   853   from \<open>x \<notin> A\<close> have disjoint: "Pow A \<inter> insert x ` Pow A = {}" by blast
```
```   854   from \<open>x \<notin> A\<close> have inj_on: "inj_on (insert x) (Pow A)"
```
```   855     unfolding inj_on_def by auto
```
```   856
```
```   857   have "card (Pow (insert x A)) = card (Pow A \<union> insert x ` Pow A)"
```
```   858     by (simp only: Pow_insert)
```
```   859   also have "\<dots> = card (Pow A) + card (insert x ` Pow A)"
```
```   860     by (rule card_Un_disjoint) (use \<open>finite A\<close> disjoint in simp_all)
```
```   861   also from inj_on have "card (insert x ` Pow A) = card (Pow A)"
```
```   862     by (rule card_image)
```
```   863   also have "\<dots> + \<dots> = 2 * \<dots>" by (simp add: mult_2)
```
```   864   also from insert(3) have "\<dots> = 2 ^ Suc (card A)" by simp
```
```   865   also from insert(1,2) have "Suc (card A) = card (insert x A)"
```
```   866     by (rule card_insert_disjoint [symmetric])
```
```   867   finally show ?case .
```
```   868 qed
```
```   869
```
```   870
```
```   871 subsection \<open>Code generator tweak\<close>
```
```   872
```
```   873 code_identifier
```
```   874   code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```   875
```
```   876 end
```