src/HOL/Power.thy
 author haftmann Sat Jul 09 13:26:16 2016 +0200 (2016-07-09) changeset 63417 c184ec919c70 parent 63040 eb4ddd18d635 child 63648 f9f3006a5579 permissions -rw-r--r--
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
* * *
more rules for setsum, setprod on intervals
```     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]:
```
```    37   "1 ^ n = 1"
```
```    38   by (induct n) simp_all
```
```    39
```
```    40 lemma power_one_right [simp]:
```
```    41   "a ^ 1 = a"
```
```    42   by simp
```
```    43
```
```    44 lemma power_Suc0_right [simp]:
```
```    45   "a ^ Suc 0 = a"
```
```    46   by simp
```
```    47
```
```    48 lemma power_commutes:
```
```    49   "a ^ n * a = a * a ^ n"
```
```    50   by (induct n) (simp_all add: mult.assoc)
```
```    51
```
```    52 lemma power_Suc2:
```
```    53   "a ^ Suc n = a ^ n * a"
```
```    54   by (simp add: power_commutes)
```
```    55
```
```    56 lemma power_add:
```
```    57   "a ^ (m + n) = a ^ m * a ^ n"
```
```    58   by (induct m) (simp_all add: algebra_simps)
```
```    59
```
```    60 lemma power_mult:
```
```    61   "a ^ (m * n) = (a ^ m) ^ n"
```
```    62   by (induct n) (simp_all add: power_add)
```
```    63
```
```    64 lemma power2_eq_square: "a\<^sup>2 = a * a"
```
```    65   by (simp add: numeral_2_eq_2)
```
```    66
```
```    67 lemma power3_eq_cube: "a ^ 3 = a * a * a"
```
```    68   by (simp add: numeral_3_eq_3 mult.assoc)
```
```    69
```
```    70 lemma power_even_eq:
```
```    71   "a ^ (2 * n) = (a ^ n)\<^sup>2"
```
```    72   by (subst mult.commute) (simp add: power_mult)
```
```    73
```
```    74 lemma power_odd_eq:
```
```    75   "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
```
```    76   by (simp add: power_even_eq)
```
```    77
```
```    78 lemma power_numeral_even:
```
```    79   "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
```
```    80   unfolding numeral_Bit0 power_add Let_def ..
```
```    81
```
```    82 lemma power_numeral_odd:
```
```    83   "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
```
```    84   unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
```
```    85   unfolding power_Suc power_add Let_def mult.assoc ..
```
```    86
```
```    87 lemma funpow_times_power:
```
```    88   "(times x ^^ f x) = times (x ^ f x)"
```
```    89 proof (induct "f x" arbitrary: f)
```
```    90   case 0 then show ?case by (simp add: fun_eq_iff)
```
```    91 next
```
```    92   case (Suc n)
```
```    93   define g where "g x = f x - 1" for x
```
```    94   with Suc have "n = g x" by simp
```
```    95   with Suc have "times x ^^ g x = times (x ^ g x)" by simp
```
```    96   moreover from Suc g_def have "f x = g x + 1" by simp
```
```    97   ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
```
```    98 qed
```
```    99
```
```   100 lemma power_commuting_commutes:
```
```   101   assumes "x * y = y * x"
```
```   102   shows "x ^ n * y = y * x ^n"
```
```   103 proof (induct n)
```
```   104   case (Suc n)
```
```   105   have "x ^ Suc n * y = x ^ n * y * x"
```
```   106     by (subst power_Suc2) (simp add: assms ac_simps)
```
```   107   also have "\<dots> = y * x ^ Suc n"
```
```   108     unfolding Suc power_Suc2
```
```   109     by (simp add: ac_simps)
```
```   110   finally show ?case .
```
```   111 qed simp
```
```   112
```
```   113 lemma power_minus_mult:
```
```   114   "0 < n \<Longrightarrow> a ^ (n - 1) * a = a ^ n"
```
```   115   by (simp add: power_commutes split add: nat_diff_split)
```
```   116
```
```   117 end
```
```   118
```
```   119 context comm_monoid_mult
```
```   120 begin
```
```   121
```
```   122 lemma power_mult_distrib [field_simps]:
```
```   123   "(a * b) ^ n = (a ^ n) * (b ^ n)"
```
```   124   by (induct n) (simp_all add: ac_simps)
```
```   125
```
```   126 end
```
```   127
```
```   128 text\<open>Extract constant factors from powers\<close>
```
```   129 declare power_mult_distrib [where a = "numeral w" for w, simp]
```
```   130 declare power_mult_distrib [where b = "numeral w" for w, simp]
```
```   131
```
```   132 lemma power_add_numeral [simp]:
```
```   133   fixes a :: "'a :: monoid_mult"
```
```   134   shows "a^numeral m * a^numeral n = a^numeral (m + n)"
```
```   135   by (simp add: power_add [symmetric])
```
```   136
```
```   137 lemma power_add_numeral2 [simp]:
```
```   138   fixes a :: "'a :: monoid_mult"
```
```   139   shows "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
```
```   140   by (simp add: mult.assoc [symmetric])
```
```   141
```
```   142 lemma power_mult_numeral [simp]:
```
```   143   fixes a :: "'a :: monoid_mult"
```
```   144   shows"(a^numeral m)^numeral n = a^numeral (m * n)"
```
```   145   by (simp only: numeral_mult power_mult)
```
```   146
```
```   147 context semiring_numeral
```
```   148 begin
```
```   149
```
```   150 lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
```
```   151   by (simp only: sqr_conv_mult numeral_mult)
```
```   152
```
```   153 lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
```
```   154   by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
```
```   155     numeral_sqr numeral_mult power_add power_one_right)
```
```   156
```
```   157 lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
```
```   158   by (rule numeral_pow [symmetric])
```
```   159
```
```   160 end
```
```   161
```
```   162 context semiring_1
```
```   163 begin
```
```   164
```
```   165 lemma of_nat_power [simp]:
```
```   166   "of_nat (m ^ n) = of_nat m ^ n"
```
```   167   by (induct n) simp_all
```
```   168
```
```   169 lemma zero_power:
```
```   170   "0 < n \<Longrightarrow> 0 ^ n = 0"
```
```   171   by (cases n) simp_all
```
```   172
```
```   173 lemma power_zero_numeral [simp]:
```
```   174   "0 ^ numeral k = 0"
```
```   175   by (simp add: numeral_eq_Suc)
```
```   176
```
```   177 lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
```
```   178   by (rule power_zero_numeral)
```
```   179
```
```   180 lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
```
```   181   by (rule power_one)
```
```   182
```
```   183 lemma power_0_Suc [simp]:
```
```   184   "0 ^ Suc n = 0"
```
```   185   by simp
```
```   186
```
```   187 text\<open>It looks plausible as a simprule, but its effect can be strange.\<close>
```
```   188 lemma power_0_left:
```
```   189   "0 ^ n = (if n = 0 then 1 else 0)"
```
```   190   by (cases n) simp_all
```
```   191
```
```   192 end
```
```   193
```
```   194 context comm_semiring_1
```
```   195 begin
```
```   196
```
```   197 text \<open>The divides relation\<close>
```
```   198
```
```   199 lemma le_imp_power_dvd:
```
```   200   assumes "m \<le> n" shows "a ^ m dvd a ^ n"
```
```   201 proof
```
```   202   have "a ^ n = a ^ (m + (n - m))"
```
```   203     using \<open>m \<le> n\<close> by simp
```
```   204   also have "\<dots> = a ^ m * a ^ (n - m)"
```
```   205     by (rule power_add)
```
```   206   finally show "a ^ n = a ^ m * a ^ (n - m)" .
```
```   207 qed
```
```   208
```
```   209 lemma power_le_dvd:
```
```   210   "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
```
```   211   by (rule dvd_trans [OF le_imp_power_dvd])
```
```   212
```
```   213 lemma dvd_power_same:
```
```   214   "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
```
```   215   by (induct n) (auto simp add: mult_dvd_mono)
```
```   216
```
```   217 lemma dvd_power_le:
```
```   218   "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
```
```   219   by (rule power_le_dvd [OF dvd_power_same])
```
```   220
```
```   221 lemma dvd_power [simp]:
```
```   222   assumes "n > (0::nat) \<or> x = 1"
```
```   223   shows "x dvd (x ^ n)"
```
```   224 using assms proof
```
```   225   assume "0 < n"
```
```   226   then have "x ^ n = x ^ Suc (n - 1)" by simp
```
```   227   then show "x dvd (x ^ n)" by simp
```
```   228 next
```
```   229   assume "x = 1"
```
```   230   then show "x dvd (x ^ n)" by simp
```
```   231 qed
```
```   232
```
```   233 end
```
```   234
```
```   235 context semiring_1_no_zero_divisors
```
```   236 begin
```
```   237
```
```   238 subclass power .
```
```   239
```
```   240 lemma power_eq_0_iff [simp]:
```
```   241   "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
```
```   242   by (induct n) auto
```
```   243
```
```   244 lemma power_not_zero:
```
```   245   "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
```
```   246   by (induct n) auto
```
```   247
```
```   248 lemma zero_eq_power2 [simp]:
```
```   249   "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
```
```   250   unfolding power2_eq_square by simp
```
```   251
```
```   252 end
```
```   253
```
```   254 context ring_1
```
```   255 begin
```
```   256
```
```   257 lemma power_minus:
```
```   258   "(- a) ^ n = (- 1) ^ n * a ^ n"
```
```   259 proof (induct n)
```
```   260   case 0 show ?case by simp
```
```   261 next
```
```   262   case (Suc n) then show ?case
```
```   263     by (simp del: power_Suc add: power_Suc2 mult.assoc)
```
```   264 qed
```
```   265
```
```   266 lemma power_minus': "NO_MATCH 1 x \<Longrightarrow> (-x) ^ n = (-1)^n * x ^ n"
```
```   267   by (rule power_minus)
```
```   268
```
```   269 lemma power_minus_Bit0:
```
```   270   "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
```
```   271   by (induct k, simp_all only: numeral_class.numeral.simps power_add
```
```   272     power_one_right mult_minus_left mult_minus_right minus_minus)
```
```   273
```
```   274 lemma power_minus_Bit1:
```
```   275   "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
```
```   276   by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
```
```   277
```
```   278 lemma power2_minus [simp]:
```
```   279   "(- a)\<^sup>2 = a\<^sup>2"
```
```   280   by (fact power_minus_Bit0)
```
```   281
```
```   282 lemma power_minus1_even [simp]:
```
```   283   "(- 1) ^ (2*n) = 1"
```
```   284 proof (induct n)
```
```   285   case 0 show ?case by simp
```
```   286 next
```
```   287   case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
```
```   288 qed
```
```   289
```
```   290 lemma power_minus1_odd:
```
```   291   "(- 1) ^ Suc (2*n) = -1"
```
```   292   by simp
```
```   293
```
```   294 lemma power_minus_even [simp]:
```
```   295   "(-a) ^ (2*n) = a ^ (2*n)"
```
```   296   by (simp add: power_minus [of a])
```
```   297
```
```   298 end
```
```   299
```
```   300 context ring_1_no_zero_divisors
```
```   301 begin
```
```   302
```
```   303 lemma power2_eq_1_iff:
```
```   304   "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
```
```   305   using square_eq_1_iff [of a] by (simp add: power2_eq_square)
```
```   306
```
```   307 end
```
```   308
```
```   309 context idom
```
```   310 begin
```
```   311
```
```   312 lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
```
```   313   unfolding power2_eq_square by (rule square_eq_iff)
```
```   314
```
```   315 end
```
```   316
```
```   317 context algebraic_semidom
```
```   318 begin
```
```   319
```
```   320 lemma div_power:
```
```   321   assumes "b dvd a"
```
```   322   shows "(a div b) ^ n = a ^ n div b ^ n"
```
```   323   using assms by (induct n) (simp_all add: div_mult_div_if_dvd dvd_power_same)
```
```   324
```
```   325 lemma is_unit_power_iff:
```
```   326   "is_unit (a ^ n) \<longleftrightarrow> is_unit a \<or> n = 0"
```
```   327   by (induct n) (auto simp add: is_unit_mult_iff)
```
```   328
```
```   329 end
```
```   330
```
```   331 context normalization_semidom
```
```   332 begin
```
```   333
```
```   334 lemma normalize_power:
```
```   335   "normalize (a ^ n) = normalize a ^ n"
```
```   336   by (induct n) (simp_all add: normalize_mult)
```
```   337
```
```   338 lemma unit_factor_power:
```
```   339   "unit_factor (a ^ n) = unit_factor a ^ n"
```
```   340   by (induct n) (simp_all add: unit_factor_mult)
```
```   341
```
```   342 end
```
```   343
```
```   344 context division_ring
```
```   345 begin
```
```   346
```
```   347 text\<open>Perhaps these should be simprules.\<close>
```
```   348 lemma power_inverse [field_simps, divide_simps]:
```
```   349   "inverse a ^ n = inverse (a ^ n)"
```
```   350 proof (cases "a = 0")
```
```   351   case True then show ?thesis by (simp add: power_0_left)
```
```   352 next
```
```   353   case False then have "inverse (a ^ n) = inverse a ^ n"
```
```   354     by (induct n) (simp_all add: nonzero_inverse_mult_distrib power_commutes)
```
```   355   then show ?thesis by simp
```
```   356 qed
```
```   357
```
```   358 lemma power_one_over [field_simps, divide_simps]:
```
```   359   "(1 / a) ^ n = 1 / a ^ n"
```
```   360   using power_inverse [of a] by (simp add: divide_inverse)
```
```   361
```
```   362 end
```
```   363
```
```   364 context field
```
```   365 begin
```
```   366
```
```   367 lemma power_diff:
```
```   368   assumes nz: "a \<noteq> 0"
```
```   369   shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
```
```   370   by (induct m n rule: diff_induct) (simp_all add: nz power_not_zero)
```
```   371
```
```   372 lemma power_divide [field_simps, divide_simps]:
```
```   373   "(a / b) ^ n = a ^ n / b ^ n"
```
```   374   by (induct n) simp_all
```
```   375
```
```   376 end
```
```   377
```
```   378
```
```   379 subsection \<open>Exponentiation on ordered types\<close>
```
```   380
```
```   381 context linordered_semidom
```
```   382 begin
```
```   383
```
```   384 lemma zero_less_power [simp]:
```
```   385   "0 < a \<Longrightarrow> 0 < a ^ n"
```
```   386   by (induct n) simp_all
```
```   387
```
```   388 lemma zero_le_power [simp]:
```
```   389   "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
```
```   390   by (induct n) simp_all
```
```   391
```
```   392 lemma power_mono:
```
```   393   "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
```
```   394   by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
```
```   395
```
```   396 lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
```
```   397   using power_mono [of 1 a n] by simp
```
```   398
```
```   399 lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
```
```   400   using power_mono [of a 1 n] by simp
```
```   401
```
```   402 lemma power_gt1_lemma:
```
```   403   assumes gt1: "1 < a"
```
```   404   shows "1 < a * a ^ n"
```
```   405 proof -
```
```   406   from gt1 have "0 \<le> a"
```
```   407     by (fact order_trans [OF zero_le_one less_imp_le])
```
```   408   have "1 * 1 < a * 1" using gt1 by simp
```
```   409   also have "\<dots> \<le> a * a ^ n" using gt1
```
```   410     by (simp only: mult_mono \<open>0 \<le> a\<close> one_le_power order_less_imp_le
```
```   411         zero_le_one order_refl)
```
```   412   finally show ?thesis by simp
```
```   413 qed
```
```   414
```
```   415 lemma power_gt1:
```
```   416   "1 < a \<Longrightarrow> 1 < a ^ Suc n"
```
```   417   by (simp add: power_gt1_lemma)
```
```   418
```
```   419 lemma one_less_power [simp]:
```
```   420   "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
```
```   421   by (cases n) (simp_all add: power_gt1_lemma)
```
```   422
```
```   423 lemma power_le_imp_le_exp:
```
```   424   assumes gt1: "1 < a"
```
```   425   shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
```
```   426 proof (induct m arbitrary: n)
```
```   427   case 0
```
```   428   show ?case by simp
```
```   429 next
```
```   430   case (Suc m)
```
```   431   show ?case
```
```   432   proof (cases n)
```
```   433     case 0
```
```   434     with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
```
```   435     with gt1 show ?thesis
```
```   436       by (force simp only: power_gt1_lemma
```
```   437           not_less [symmetric])
```
```   438   next
```
```   439     case (Suc n)
```
```   440     with Suc.prems Suc.hyps show ?thesis
```
```   441       by (force dest: mult_left_le_imp_le
```
```   442           simp add: less_trans [OF zero_less_one gt1])
```
```   443   qed
```
```   444 qed
```
```   445
```
```   446 lemma of_nat_zero_less_power_iff [simp]:
```
```   447   "of_nat x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
```
```   448   by (induct n) auto
```
```   449
```
```   450 text\<open>Surely we can strengthen this? It holds for \<open>0<a<1\<close> too.\<close>
```
```   451 lemma power_inject_exp [simp]:
```
```   452   "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
```
```   453   by (force simp add: order_antisym power_le_imp_le_exp)
```
```   454
```
```   455 text\<open>Can relax the first premise to @{term "0<a"} in the case of the
```
```   456 natural numbers.\<close>
```
```   457 lemma power_less_imp_less_exp:
```
```   458   "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
```
```   459   by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
```
```   460     power_le_imp_le_exp)
```
```   461
```
```   462 lemma power_strict_mono [rule_format]:
```
```   463   "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
```
```   464   by (induct n)
```
```   465    (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
```
```   466
```
```   467 text\<open>Lemma for \<open>power_strict_decreasing\<close>\<close>
```
```   468 lemma power_Suc_less:
```
```   469   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
```
```   470   by (induct n)
```
```   471     (auto simp add: mult_strict_left_mono)
```
```   472
```
```   473 lemma power_strict_decreasing [rule_format]:
```
```   474   "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
```
```   475 proof (induct N)
```
```   476   case 0 then show ?case by simp
```
```   477 next
```
```   478   case (Suc N) then show ?case
```
```   479   apply (auto simp add: power_Suc_less less_Suc_eq)
```
```   480   apply (subgoal_tac "a * a^N < 1 * a^n")
```
```   481   apply simp
```
```   482   apply (rule mult_strict_mono) apply auto
```
```   483   done
```
```   484 qed
```
```   485
```
```   486 text\<open>Proof resembles that of \<open>power_strict_decreasing\<close>\<close>
```
```   487 lemma power_decreasing [rule_format]:
```
```   488   "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
```
```   489 proof (induct N)
```
```   490   case 0 then show ?case by simp
```
```   491 next
```
```   492   case (Suc N) then show ?case
```
```   493   apply (auto simp add: le_Suc_eq)
```
```   494   apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
```
```   495   apply (rule mult_mono) apply auto
```
```   496   done
```
```   497 qed
```
```   498
```
```   499 lemma power_Suc_less_one:
```
```   500   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
```
```   501   using power_strict_decreasing [of 0 "Suc n" a] by simp
```
```   502
```
```   503 text\<open>Proof again resembles that of \<open>power_strict_decreasing\<close>\<close>
```
```   504 lemma power_increasing [rule_format]:
```
```   505   "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
```
```   506 proof (induct N)
```
```   507   case 0 then show ?case by simp
```
```   508 next
```
```   509   case (Suc N) then show ?case
```
```   510   apply (auto simp add: le_Suc_eq)
```
```   511   apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
```
```   512   apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
```
```   513   done
```
```   514 qed
```
```   515
```
```   516 text\<open>Lemma for \<open>power_strict_increasing\<close>\<close>
```
```   517 lemma power_less_power_Suc:
```
```   518   "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
```
```   519   by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
```
```   520
```
```   521 lemma power_strict_increasing [rule_format]:
```
```   522   "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
```
```   523 proof (induct N)
```
```   524   case 0 then show ?case by simp
```
```   525 next
```
```   526   case (Suc N) then show ?case
```
```   527   apply (auto simp add: power_less_power_Suc less_Suc_eq)
```
```   528   apply (subgoal_tac "1 * a^n < a * a^N", simp)
```
```   529   apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
```
```   530   done
```
```   531 qed
```
```   532
```
```   533 lemma power_increasing_iff [simp]:
```
```   534   "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
```
```   535   by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
```
```   536
```
```   537 lemma power_strict_increasing_iff [simp]:
```
```   538   "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
```
```   539 by (blast intro: power_less_imp_less_exp power_strict_increasing)
```
```   540
```
```   541 lemma power_le_imp_le_base:
```
```   542   assumes le: "a ^ Suc n \<le> b ^ Suc n"
```
```   543     and ynonneg: "0 \<le> b"
```
```   544   shows "a \<le> b"
```
```   545 proof (rule ccontr)
```
```   546   assume "~ a \<le> b"
```
```   547   then have "b < a" by (simp only: linorder_not_le)
```
```   548   then have "b ^ Suc n < a ^ Suc n"
```
```   549     by (simp only: assms power_strict_mono)
```
```   550   from le and this show False
```
```   551     by (simp add: linorder_not_less [symmetric])
```
```   552 qed
```
```   553
```
```   554 lemma power_less_imp_less_base:
```
```   555   assumes less: "a ^ n < b ^ n"
```
```   556   assumes nonneg: "0 \<le> b"
```
```   557   shows "a < b"
```
```   558 proof (rule contrapos_pp [OF less])
```
```   559   assume "~ a < b"
```
```   560   hence "b \<le> a" by (simp only: linorder_not_less)
```
```   561   hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
```
```   562   thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
```
```   563 qed
```
```   564
```
```   565 lemma power_inject_base:
```
```   566   "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
```
```   567 by (blast intro: power_le_imp_le_base antisym eq_refl sym)
```
```   568
```
```   569 lemma power_eq_imp_eq_base:
```
```   570   "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
```
```   571   by (cases n) (simp_all del: power_Suc, rule power_inject_base)
```
```   572
```
```   573 lemma power_eq_iff_eq_base:
```
```   574   "0 < n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a ^ n = b ^ n \<longleftrightarrow> a = b"
```
```   575   using power_eq_imp_eq_base [of a n b] by auto
```
```   576
```
```   577 lemma power2_le_imp_le:
```
```   578   "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
```
```   579   unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
```
```   580
```
```   581 lemma power2_less_imp_less:
```
```   582   "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
```
```   583   by (rule power_less_imp_less_base)
```
```   584
```
```   585 lemma power2_eq_imp_eq:
```
```   586   "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
```
```   587   unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
```
```   588
```
```   589 lemma power_Suc_le_self:
```
```   590   shows "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ Suc n \<le> a"
```
```   591   using power_decreasing [of 1 "Suc n" a] by simp
```
```   592
```
```   593 end
```
```   594
```
```   595 context linordered_ring_strict
```
```   596 begin
```
```   597
```
```   598 lemma sum_squares_eq_zero_iff:
```
```   599   "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   600   by (simp add: add_nonneg_eq_0_iff)
```
```   601
```
```   602 lemma sum_squares_le_zero_iff:
```
```   603   "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   604   by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
```
```   605
```
```   606 lemma sum_squares_gt_zero_iff:
```
```   607   "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   608   by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
```
```   609
```
```   610 end
```
```   611
```
```   612 context linordered_idom
```
```   613 begin
```
```   614
```
```   615 lemma power_abs: "\<bar>a ^ n\<bar> = \<bar>a\<bar> ^ n"
```
```   616   by (induct n) (auto simp add: abs_mult)
```
```   617
```
```   618 lemma abs_power_minus [simp]: "\<bar>(-a) ^ n\<bar> = \<bar>a ^ n\<bar>"
```
```   619   by (simp add: power_abs)
```
```   620
```
```   621 lemma zero_less_power_abs_iff [simp]: "0 < \<bar>a\<bar> ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
```
```   622 proof (induct n)
```
```   623   case 0 show ?case by simp
```
```   624 next
```
```   625   case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
```
```   626 qed
```
```   627
```
```   628 lemma zero_le_power_abs [simp]: "0 \<le> \<bar>a\<bar> ^ n"
```
```   629   by (rule zero_le_power [OF abs_ge_zero])
```
```   630
```
```   631 lemma zero_le_power2 [simp]:
```
```   632   "0 \<le> a\<^sup>2"
```
```   633   by (simp add: power2_eq_square)
```
```   634
```
```   635 lemma zero_less_power2 [simp]:
```
```   636   "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
```
```   637   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
```
```   638
```
```   639 lemma power2_less_0 [simp]:
```
```   640   "\<not> a\<^sup>2 < 0"
```
```   641   by (force simp add: power2_eq_square mult_less_0_iff)
```
```   642
```
```   643 lemma power2_less_eq_zero_iff [simp]:
```
```   644   "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
```
```   645   by (simp add: le_less)
```
```   646
```
```   647 lemma abs_power2 [simp]: "\<bar>a\<^sup>2\<bar> = a\<^sup>2"
```
```   648   by (simp add: power2_eq_square)
```
```   649
```
```   650 lemma power2_abs [simp]: "\<bar>a\<bar>\<^sup>2 = a\<^sup>2"
```
```   651   by (simp add: power2_eq_square)
```
```   652
```
```   653 lemma odd_power_less_zero:
```
```   654   "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
```
```   655 proof (induct n)
```
```   656   case 0
```
```   657   then show ?case by simp
```
```   658 next
```
```   659   case (Suc n)
```
```   660   have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
```
```   661     by (simp add: ac_simps power_add power2_eq_square)
```
```   662   thus ?case
```
```   663     by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
```
```   664 qed
```
```   665
```
```   666 lemma odd_0_le_power_imp_0_le:
```
```   667   "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
```
```   668   using odd_power_less_zero [of a n]
```
```   669     by (force simp add: linorder_not_less [symmetric])
```
```   670
```
```   671 lemma zero_le_even_power'[simp]:
```
```   672   "0 \<le> a ^ (2*n)"
```
```   673 proof (induct n)
```
```   674   case 0
```
```   675     show ?case by simp
```
```   676 next
```
```   677   case (Suc n)
```
```   678     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
```
```   679       by (simp add: ac_simps power_add power2_eq_square)
```
```   680     thus ?case
```
```   681       by (simp add: Suc zero_le_mult_iff)
```
```   682 qed
```
```   683
```
```   684 lemma sum_power2_ge_zero:
```
```   685   "0 \<le> x\<^sup>2 + y\<^sup>2"
```
```   686   by (intro add_nonneg_nonneg zero_le_power2)
```
```   687
```
```   688 lemma not_sum_power2_lt_zero:
```
```   689   "\<not> x\<^sup>2 + y\<^sup>2 < 0"
```
```   690   unfolding not_less by (rule sum_power2_ge_zero)
```
```   691
```
```   692 lemma sum_power2_eq_zero_iff:
```
```   693   "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   694   unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
```
```   695
```
```   696 lemma sum_power2_le_zero_iff:
```
```   697   "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   698   by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
```
```   699
```
```   700 lemma sum_power2_gt_zero_iff:
```
```   701   "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   702   unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
```
```   703
```
```   704 lemma abs_le_square_iff:
```
```   705    "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
```
```   706 proof
```
```   707   assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
```
```   708   then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
```
```   709   then show "x\<^sup>2 \<le> y\<^sup>2" by simp
```
```   710 next
```
```   711   assume "x\<^sup>2 \<le> y\<^sup>2"
```
```   712   then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
```
```   713     by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
```
```   714 qed
```
```   715
```
```   716 lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> \<bar>x\<bar> \<le> 1"
```
```   717   using abs_le_square_iff [of x 1]
```
```   718   by simp
```
```   719
```
```   720 lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
```
```   721   by (auto simp add: abs_if power2_eq_1_iff)
```
```   722
```
```   723 lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> \<bar>x\<bar> < 1"
```
```   724   using  abs_square_eq_1 [of x] abs_square_le_1 [of x]
```
```   725   by (auto simp add: le_less)
```
```   726
```
```   727 end
```
```   728
```
```   729
```
```   730 subsection \<open>Miscellaneous rules\<close>
```
```   731
```
```   732 lemma (in linordered_semidom) self_le_power:
```
```   733   "1 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a ^ n"
```
```   734   using power_increasing [of 1 n a] power_one_right [of a] by auto
```
```   735
```
```   736 lemma (in power) power_eq_if:
```
```   737   "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
```
```   738   unfolding One_nat_def by (cases m) simp_all
```
```   739
```
```   740 lemma (in comm_semiring_1) power2_sum:
```
```   741   "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
```
```   742   by (simp add: algebra_simps power2_eq_square mult_2_right)
```
```   743
```
```   744 lemma (in comm_ring_1) power2_diff:
```
```   745   "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
```
```   746   by (simp add: algebra_simps power2_eq_square mult_2_right)
```
```   747
```
```   748 lemma (in comm_ring_1) power2_commute:
```
```   749   "(x - y)\<^sup>2 = (y - x)\<^sup>2"
```
```   750   by (simp add: algebra_simps power2_eq_square)
```
```   751
```
```   752 lemma (in comm_ring_1) minus_power_mult_self:
```
```   753   "(- a) ^ n * (- a) ^ n = a ^ (2 * n)"
```
```   754   by (simp add: power_mult_distrib [symmetric]) (simp add: power2_eq_square [symmetric] power_mult [symmetric])
```
```   755
```
```   756 lemma (in comm_ring_1) minus_one_mult_self [simp]:
```
```   757   "(- 1) ^ n * (- 1) ^ n = 1"
```
```   758   using minus_power_mult_self [of 1 n] by simp
```
```   759
```
```   760 lemma (in comm_ring_1) left_minus_one_mult_self [simp]:
```
```   761   "(- 1) ^ n * ((- 1) ^ n * a) = a"
```
```   762   by (simp add: mult.assoc [symmetric])
```
```   763
```
```   764 text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
```
```   765
```
```   766 lemmas zero_compare_simps =
```
```   767     add_strict_increasing add_strict_increasing2 add_increasing
```
```   768     zero_le_mult_iff zero_le_divide_iff
```
```   769     zero_less_mult_iff zero_less_divide_iff
```
```   770     mult_le_0_iff divide_le_0_iff
```
```   771     mult_less_0_iff divide_less_0_iff
```
```   772     zero_le_power2 power2_less_0
```
```   773
```
```   774
```
```   775 subsection \<open>Exponentiation for the Natural Numbers\<close>
```
```   776
```
```   777 lemma nat_one_le_power [simp]:
```
```   778   "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
```
```   779   by (rule one_le_power [of i n, unfolded One_nat_def])
```
```   780
```
```   781 lemma nat_zero_less_power_iff [simp]:
```
```   782   "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
```
```   783   by (induct n) auto
```
```   784
```
```   785 lemma nat_power_eq_Suc_0_iff [simp]:
```
```   786   "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
```
```   787   by (induct m) auto
```
```   788
```
```   789 lemma power_Suc_0 [simp]:
```
```   790   "Suc 0 ^ n = Suc 0"
```
```   791   by simp
```
```   792
```
```   793 text\<open>Valid for the naturals, but what if \<open>0<i<1\<close>?
```
```   794 Premises cannot be weakened: consider the case where @{term "i=0"},
```
```   795 @{term "m=1"} and @{term "n=0"}.\<close>
```
```   796 lemma nat_power_less_imp_less:
```
```   797   assumes nonneg: "0 < (i::nat)"
```
```   798   assumes less: "i ^ m < i ^ n"
```
```   799   shows "m < n"
```
```   800 proof (cases "i = 1")
```
```   801   case True with less power_one [where 'a = nat] show ?thesis by simp
```
```   802 next
```
```   803   case False with nonneg have "1 < i" by auto
```
```   804   from power_strict_increasing_iff [OF this] less show ?thesis ..
```
```   805 qed
```
```   806
```
```   807 lemma power_dvd_imp_le:
```
```   808   "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
```
```   809   apply (rule power_le_imp_le_exp, assumption)
```
```   810   apply (erule dvd_imp_le, simp)
```
```   811   done
```
```   812
```
```   813 lemma power2_nat_le_eq_le:
```
```   814   fixes m n :: nat
```
```   815   shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
```
```   816   by (auto intro: power2_le_imp_le power_mono)
```
```   817
```
```   818 lemma power2_nat_le_imp_le:
```
```   819   fixes m n :: nat
```
```   820   assumes "m\<^sup>2 \<le> n"
```
```   821   shows "m \<le> n"
```
```   822 proof (cases m)
```
```   823   case 0 then show ?thesis by simp
```
```   824 next
```
```   825   case (Suc k)
```
```   826   show ?thesis
```
```   827   proof (rule ccontr)
```
```   828     assume "\<not> m \<le> n"
```
```   829     then have "n < m" by simp
```
```   830     with assms Suc show False
```
```   831       by (simp add: power2_eq_square)
```
```   832   qed
```
```   833 qed
```
```   834
```
```   835 subsubsection \<open>Cardinality of the Powerset\<close>
```
```   836
```
```   837 lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
```
```   838   unfolding UNIV_bool by simp
```
```   839
```
```   840 lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
```
```   841 proof (induct rule: finite_induct)
```
```   842   case empty
```
```   843     show ?case by auto
```
```   844 next
```
```   845   case (insert x A)
```
```   846   then have "inj_on (insert x) (Pow A)"
```
```   847     unfolding inj_on_def by (blast elim!: equalityE)
```
```   848   then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A"
```
```   849     by (simp add: mult_2 card_image Pow_insert insert.hyps)
```
```   850   then show ?case using insert
```
```   851     apply (simp add: Pow_insert)
```
```   852     apply (subst card_Un_disjoint, auto)
```
```   853     done
```
```   854 qed
```
```   855
```
```   856
```
```   857 subsection \<open>Code generator tweak\<close>
```
```   858
```
```   859 code_identifier
```
```   860   code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```   861
```
```   862 end
```