src/HOL/Power.thy
 author paulson Fri Nov 13 12:27:13 2015 +0000 (2015-11-13) changeset 61649 268d88ec9087 parent 61531 ab2e862263e7 child 61694 6571c78c9667 permissions -rw-r--r--
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
```     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 Equiv_Relations
```
```    10 begin
```
```    11
```
```    12 context linordered_ring (* TODO: move *)
```
```    13 begin
```
```    14
```
```    15 lemma sum_squares_ge_zero:
```
```    16   "0 \<le> x * x + y * y"
```
```    17   by (intro add_nonneg_nonneg zero_le_square)
```
```    18
```
```    19 lemma not_sum_squares_lt_zero:
```
```    20   "\<not> x * x + y * y < 0"
```
```    21   by (simp add: not_less sum_squares_ge_zero)
```
```    22
```
```    23 end
```
```    24
```
```    25 subsection \<open>Powers for Arbitrary Monoids\<close>
```
```    26
```
```    27 class power = one + times
```
```    28 begin
```
```    29
```
```    30 primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where
```
```    31     power_0: "a ^ 0 = 1"
```
```    32   | power_Suc: "a ^ Suc n = a * a ^ n"
```
```    33
```
```    34 notation (latex output)
```
```    35   power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
```
```    36
```
```    37 text \<open>Special syntax for squares.\<close>
```
```    38
```
```    39 abbreviation (xsymbols)
```
```    40   power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999) where
```
```    41   "x\<^sup>2 \<equiv> x ^ 2"
```
```    42
```
```    43 notation (latex output)
```
```    44   power2  ("(_\<^sup>2)" [1000] 999)
```
```    45
```
```    46 end
```
```    47
```
```    48 context monoid_mult
```
```    49 begin
```
```    50
```
```    51 subclass power .
```
```    52
```
```    53 lemma power_one [simp]:
```
```    54   "1 ^ n = 1"
```
```    55   by (induct n) simp_all
```
```    56
```
```    57 lemma power_one_right [simp]:
```
```    58   "a ^ 1 = a"
```
```    59   by simp
```
```    60
```
```    61 lemma power_Suc0_right [simp]:
```
```    62   "a ^ Suc 0 = a"
```
```    63   by simp
```
```    64
```
```    65 lemma power_commutes:
```
```    66   "a ^ n * a = a * a ^ n"
```
```    67   by (induct n) (simp_all add: mult.assoc)
```
```    68
```
```    69 lemma power_Suc2:
```
```    70   "a ^ Suc n = a ^ n * a"
```
```    71   by (simp add: power_commutes)
```
```    72
```
```    73 lemma power_add:
```
```    74   "a ^ (m + n) = a ^ m * a ^ n"
```
```    75   by (induct m) (simp_all add: algebra_simps)
```
```    76
```
```    77 lemma power_mult:
```
```    78   "a ^ (m * n) = (a ^ m) ^ n"
```
```    79   by (induct n) (simp_all add: power_add)
```
```    80
```
```    81 lemma power2_eq_square: "a\<^sup>2 = a * a"
```
```    82   by (simp add: numeral_2_eq_2)
```
```    83
```
```    84 lemma power3_eq_cube: "a ^ 3 = a * a * a"
```
```    85   by (simp add: numeral_3_eq_3 mult.assoc)
```
```    86
```
```    87 lemma power_even_eq:
```
```    88   "a ^ (2 * n) = (a ^ n)\<^sup>2"
```
```    89   by (subst mult.commute) (simp add: power_mult)
```
```    90
```
```    91 lemma power_odd_eq:
```
```    92   "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
```
```    93   by (simp add: power_even_eq)
```
```    94
```
```    95 lemma power_numeral_even:
```
```    96   "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
```
```    97   unfolding numeral_Bit0 power_add Let_def ..
```
```    98
```
```    99 lemma power_numeral_odd:
```
```   100   "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
```
```   101   unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
```
```   102   unfolding power_Suc power_add Let_def mult.assoc ..
```
```   103
```
```   104 lemma funpow_times_power:
```
```   105   "(times x ^^ f x) = times (x ^ f x)"
```
```   106 proof (induct "f x" arbitrary: f)
```
```   107   case 0 then show ?case by (simp add: fun_eq_iff)
```
```   108 next
```
```   109   case (Suc n)
```
```   110   def g \<equiv> "\<lambda>x. f x - 1"
```
```   111   with Suc have "n = g x" by simp
```
```   112   with Suc have "times x ^^ g x = times (x ^ g x)" by simp
```
```   113   moreover from Suc g_def have "f x = g x + 1" by simp
```
```   114   ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
```
```   115 qed
```
```   116
```
```   117 lemma power_commuting_commutes:
```
```   118   assumes "x * y = y * x"
```
```   119   shows "x ^ n * y = y * x ^n"
```
```   120 proof (induct n)
```
```   121   case (Suc n)
```
```   122   have "x ^ Suc n * y = x ^ n * y * x"
```
```   123     by (subst power_Suc2) (simp add: assms ac_simps)
```
```   124   also have "\<dots> = y * x ^ Suc n"
```
```   125     unfolding Suc power_Suc2
```
```   126     by (simp add: ac_simps)
```
```   127   finally show ?case .
```
```   128 qed simp
```
```   129
```
```   130 end
```
```   131
```
```   132 context comm_monoid_mult
```
```   133 begin
```
```   134
```
```   135 lemma power_mult_distrib [field_simps]:
```
```   136   "(a * b) ^ n = (a ^ n) * (b ^ n)"
```
```   137   by (induct n) (simp_all add: ac_simps)
```
```   138
```
```   139 end
```
```   140
```
```   141 text\<open>Extract constant factors from powers\<close>
```
```   142 declare power_mult_distrib [where a = "numeral w" for w, simp]
```
```   143 declare power_mult_distrib [where b = "numeral w" for w, simp]
```
```   144
```
```   145 lemma power_add_numeral [simp]:
```
```   146   fixes a :: "'a :: monoid_mult"
```
```   147   shows "a^numeral m * a^numeral n = a^numeral (m + n)"
```
```   148   by (simp add: power_add [symmetric])
```
```   149
```
```   150 lemma power_add_numeral2 [simp]:
```
```   151   fixes a :: "'a :: monoid_mult"
```
```   152   shows "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
```
```   153   by (simp add: mult.assoc [symmetric])
```
```   154
```
```   155 lemma power_mult_numeral [simp]:
```
```   156   fixes a :: "'a :: monoid_mult"
```
```   157   shows"(a^numeral m)^numeral n = a^numeral (m * n)"
```
```   158   by (simp only: numeral_mult power_mult)
```
```   159
```
```   160 context semiring_numeral
```
```   161 begin
```
```   162
```
```   163 lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
```
```   164   by (simp only: sqr_conv_mult numeral_mult)
```
```   165
```
```   166 lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
```
```   167   by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
```
```   168     numeral_sqr numeral_mult power_add power_one_right)
```
```   169
```
```   170 lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
```
```   171   by (rule numeral_pow [symmetric])
```
```   172
```
```   173 end
```
```   174
```
```   175 context semiring_1
```
```   176 begin
```
```   177
```
```   178 lemma of_nat_power [simp]:
```
```   179   "of_nat (m ^ n) = of_nat m ^ n"
```
```   180   by (induct n) (simp_all add: of_nat_mult)
```
```   181
```
```   182 lemma zero_power:
```
```   183   "0 < n \<Longrightarrow> 0 ^ n = 0"
```
```   184   by (cases n) simp_all
```
```   185
```
```   186 lemma power_zero_numeral [simp]:
```
```   187   "0 ^ numeral k = 0"
```
```   188   by (simp add: numeral_eq_Suc)
```
```   189
```
```   190 lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
```
```   191   by (rule power_zero_numeral)
```
```   192
```
```   193 lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
```
```   194   by (rule power_one)
```
```   195
```
```   196 lemma power_0_Suc [simp]:
```
```   197   "0 ^ Suc n = 0"
```
```   198   by simp
```
```   199
```
```   200 text\<open>It looks plausible as a simprule, but its effect can be strange.\<close>
```
```   201 lemma power_0_left:
```
```   202   "0 ^ n = (if n = 0 then 1 else 0)"
```
```   203   by (cases n) simp_all
```
```   204
```
```   205 end
```
```   206
```
```   207 context comm_semiring_1
```
```   208 begin
```
```   209
```
```   210 text \<open>The divides relation\<close>
```
```   211
```
```   212 lemma le_imp_power_dvd:
```
```   213   assumes "m \<le> n" shows "a ^ m dvd a ^ n"
```
```   214 proof
```
```   215   have "a ^ n = a ^ (m + (n - m))"
```
```   216     using \<open>m \<le> n\<close> by simp
```
```   217   also have "\<dots> = a ^ m * a ^ (n - m)"
```
```   218     by (rule power_add)
```
```   219   finally show "a ^ n = a ^ m * a ^ (n - m)" .
```
```   220 qed
```
```   221
```
```   222 lemma power_le_dvd:
```
```   223   "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
```
```   224   by (rule dvd_trans [OF le_imp_power_dvd])
```
```   225
```
```   226 lemma dvd_power_same:
```
```   227   "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
```
```   228   by (induct n) (auto simp add: mult_dvd_mono)
```
```   229
```
```   230 lemma dvd_power_le:
```
```   231   "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
```
```   232   by (rule power_le_dvd [OF dvd_power_same])
```
```   233
```
```   234 lemma dvd_power [simp]:
```
```   235   assumes "n > (0::nat) \<or> x = 1"
```
```   236   shows "x dvd (x ^ n)"
```
```   237 using assms proof
```
```   238   assume "0 < n"
```
```   239   then have "x ^ n = x ^ Suc (n - 1)" by simp
```
```   240   then show "x dvd (x ^ n)" by simp
```
```   241 next
```
```   242   assume "x = 1"
```
```   243   then show "x dvd (x ^ n)" by simp
```
```   244 qed
```
```   245
```
```   246 end
```
```   247
```
```   248 class semiring_1_no_zero_divisors = semiring_1 + semiring_no_zero_divisors
```
```   249 begin
```
```   250
```
```   251 subclass power .
```
```   252
```
```   253 lemma power_eq_0_iff [simp]:
```
```   254   "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
```
```   255   by (induct n) auto
```
```   256
```
```   257 lemma power_not_zero:
```
```   258   "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
```
```   259   by (induct n) auto
```
```   260
```
```   261 lemma zero_eq_power2 [simp]:
```
```   262   "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
```
```   263   unfolding power2_eq_square by simp
```
```   264
```
```   265 end
```
```   266
```
```   267 context semidom
```
```   268 begin
```
```   269
```
```   270 subclass semiring_1_no_zero_divisors ..
```
```   271
```
```   272 end
```
```   273
```
```   274 context ring_1
```
```   275 begin
```
```   276
```
```   277 lemma power_minus:
```
```   278   "(- a) ^ n = (- 1) ^ n * a ^ n"
```
```   279 proof (induct n)
```
```   280   case 0 show ?case by simp
```
```   281 next
```
```   282   case (Suc n) then show ?case
```
```   283     by (simp del: power_Suc add: power_Suc2 mult.assoc)
```
```   284 qed
```
```   285
```
```   286 lemma power_minus': "NO_MATCH 1 x \<Longrightarrow> (-x) ^ n = (-1)^n * x ^ n"
```
```   287   by (rule power_minus)
```
```   288
```
```   289 lemma power_minus_Bit0:
```
```   290   "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
```
```   291   by (induct k, simp_all only: numeral_class.numeral.simps power_add
```
```   292     power_one_right mult_minus_left mult_minus_right minus_minus)
```
```   293
```
```   294 lemma power_minus_Bit1:
```
```   295   "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
```
```   296   by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
```
```   297
```
```   298 lemma power2_minus [simp]:
```
```   299   "(- a)\<^sup>2 = a\<^sup>2"
```
```   300   by (fact power_minus_Bit0)
```
```   301
```
```   302 lemma power_minus1_even [simp]:
```
```   303   "(- 1) ^ (2*n) = 1"
```
```   304 proof (induct n)
```
```   305   case 0 show ?case by simp
```
```   306 next
```
```   307   case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
```
```   308 qed
```
```   309
```
```   310 lemma power_minus1_odd:
```
```   311   "(- 1) ^ Suc (2*n) = -1"
```
```   312   by simp
```
```   313
```
```   314 lemma power_minus_even [simp]:
```
```   315   "(-a) ^ (2*n) = a ^ (2*n)"
```
```   316   by (simp add: power_minus [of a])
```
```   317
```
```   318 end
```
```   319
```
```   320 context ring_1_no_zero_divisors
```
```   321 begin
```
```   322
```
```   323 subclass semiring_1_no_zero_divisors ..
```
```   324
```
```   325 lemma power2_eq_1_iff:
```
```   326   "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
```
```   327   using square_eq_1_iff [of a] by (simp add: power2_eq_square)
```
```   328
```
```   329 end
```
```   330
```
```   331 context idom
```
```   332 begin
```
```   333
```
```   334 lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
```
```   335   unfolding power2_eq_square by (rule square_eq_iff)
```
```   336
```
```   337 end
```
```   338
```
```   339 context algebraic_semidom
```
```   340 begin
```
```   341
```
```   342 lemma div_power:
```
```   343   assumes "b dvd a"
```
```   344   shows "(a div b) ^ n = a ^ n div b ^ n"
```
```   345   using assms by (induct n) (simp_all add: div_mult_div_if_dvd dvd_power_same)
```
```   346
```
```   347 end
```
```   348
```
```   349 context normalization_semidom
```
```   350 begin
```
```   351
```
```   352 lemma normalize_power:
```
```   353   "normalize (a ^ n) = normalize a ^ n"
```
```   354   by (induct n) (simp_all add: normalize_mult)
```
```   355
```
```   356 lemma unit_factor_power:
```
```   357   "unit_factor (a ^ n) = unit_factor a ^ n"
```
```   358   by (induct n) (simp_all add: unit_factor_mult)
```
```   359
```
```   360 end
```
```   361
```
```   362 context division_ring
```
```   363 begin
```
```   364
```
```   365 text\<open>Perhaps these should be simprules.\<close>
```
```   366 lemma power_inverse [field_simps, divide_simps]:
```
```   367   "inverse a ^ n = inverse (a ^ n)"
```
```   368 proof (cases "a = 0")
```
```   369   case True then show ?thesis by (simp add: power_0_left)
```
```   370 next
```
```   371   case False then have "inverse (a ^ n) = inverse a ^ n"
```
```   372     by (induct n) (simp_all add: nonzero_inverse_mult_distrib power_commutes)
```
```   373   then show ?thesis by simp
```
```   374 qed
```
```   375
```
```   376 lemma power_one_over [field_simps, divide_simps]:
```
```   377   "(1 / a) ^ n = 1 / a ^ n"
```
```   378   using power_inverse [of a] by (simp add: divide_inverse)
```
```   379
```
```   380 end
```
```   381
```
```   382 context field
```
```   383 begin
```
```   384
```
```   385 lemma power_diff:
```
```   386   assumes nz: "a \<noteq> 0"
```
```   387   shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
```
```   388   by (induct m n rule: diff_induct) (simp_all add: nz power_not_zero)
```
```   389
```
```   390 lemma power_divide [field_simps, divide_simps]:
```
```   391   "(a / b) ^ n = a ^ n / b ^ n"
```
```   392   by (induct n) simp_all
```
```   393
```
```   394 declare power_divide [where b = "numeral w" for w, simp]
```
```   395
```
```   396 end
```
```   397
```
```   398
```
```   399 subsection \<open>Exponentiation on ordered types\<close>
```
```   400
```
```   401 context linordered_semidom
```
```   402 begin
```
```   403
```
```   404 lemma zero_less_power [simp]:
```
```   405   "0 < a \<Longrightarrow> 0 < a ^ n"
```
```   406   by (induct n) simp_all
```
```   407
```
```   408 lemma zero_le_power [simp]:
```
```   409   "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
```
```   410   by (induct n) simp_all
```
```   411
```
```   412 lemma power_mono:
```
```   413   "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
```
```   414   by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
```
```   415
```
```   416 lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
```
```   417   using power_mono [of 1 a n] by simp
```
```   418
```
```   419 lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
```
```   420   using power_mono [of a 1 n] by simp
```
```   421
```
```   422 lemma power_gt1_lemma:
```
```   423   assumes gt1: "1 < a"
```
```   424   shows "1 < a * a ^ n"
```
```   425 proof -
```
```   426   from gt1 have "0 \<le> a"
```
```   427     by (fact order_trans [OF zero_le_one less_imp_le])
```
```   428   have "1 * 1 < a * 1" using gt1 by simp
```
```   429   also have "\<dots> \<le> a * a ^ n" using gt1
```
```   430     by (simp only: mult_mono \<open>0 \<le> a\<close> one_le_power order_less_imp_le
```
```   431         zero_le_one order_refl)
```
```   432   finally show ?thesis by simp
```
```   433 qed
```
```   434
```
```   435 lemma power_gt1:
```
```   436   "1 < a \<Longrightarrow> 1 < a ^ Suc n"
```
```   437   by (simp add: power_gt1_lemma)
```
```   438
```
```   439 lemma one_less_power [simp]:
```
```   440   "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
```
```   441   by (cases n) (simp_all add: power_gt1_lemma)
```
```   442
```
```   443 lemma power_le_imp_le_exp:
```
```   444   assumes gt1: "1 < a"
```
```   445   shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
```
```   446 proof (induct m arbitrary: n)
```
```   447   case 0
```
```   448   show ?case by simp
```
```   449 next
```
```   450   case (Suc m)
```
```   451   show ?case
```
```   452   proof (cases n)
```
```   453     case 0
```
```   454     with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
```
```   455     with gt1 show ?thesis
```
```   456       by (force simp only: power_gt1_lemma
```
```   457           not_less [symmetric])
```
```   458   next
```
```   459     case (Suc n)
```
```   460     with Suc.prems Suc.hyps show ?thesis
```
```   461       by (force dest: mult_left_le_imp_le
```
```   462           simp add: less_trans [OF zero_less_one gt1])
```
```   463   qed
```
```   464 qed
```
```   465
```
```   466 lemma of_nat_zero_less_power_iff [simp]:
```
```   467   "of_nat x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
```
```   468   by (induct n) auto
```
```   469
```
```   470 text\<open>Surely we can strengthen this? It holds for @{text "0<a<1"} too.\<close>
```
```   471 lemma power_inject_exp [simp]:
```
```   472   "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
```
```   473   by (force simp add: order_antisym power_le_imp_le_exp)
```
```   474
```
```   475 text\<open>Can relax the first premise to @{term "0<a"} in the case of the
```
```   476 natural numbers.\<close>
```
```   477 lemma power_less_imp_less_exp:
```
```   478   "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
```
```   479   by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
```
```   480     power_le_imp_le_exp)
```
```   481
```
```   482 lemma power_strict_mono [rule_format]:
```
```   483   "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
```
```   484   by (induct n)
```
```   485    (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
```
```   486
```
```   487 text\<open>Lemma for @{text power_strict_decreasing}\<close>
```
```   488 lemma power_Suc_less:
```
```   489   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
```
```   490   by (induct n)
```
```   491     (auto simp add: mult_strict_left_mono)
```
```   492
```
```   493 lemma power_strict_decreasing [rule_format]:
```
```   494   "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
```
```   495 proof (induct N)
```
```   496   case 0 then show ?case by simp
```
```   497 next
```
```   498   case (Suc N) then show ?case
```
```   499   apply (auto simp add: power_Suc_less less_Suc_eq)
```
```   500   apply (subgoal_tac "a * a^N < 1 * a^n")
```
```   501   apply simp
```
```   502   apply (rule mult_strict_mono) apply auto
```
```   503   done
```
```   504 qed
```
```   505
```
```   506 text\<open>Proof resembles that of @{text power_strict_decreasing}\<close>
```
```   507 lemma power_decreasing [rule_format]:
```
```   508   "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
```
```   509 proof (induct N)
```
```   510   case 0 then show ?case by simp
```
```   511 next
```
```   512   case (Suc N) then show ?case
```
```   513   apply (auto simp add: le_Suc_eq)
```
```   514   apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
```
```   515   apply (rule mult_mono) apply auto
```
```   516   done
```
```   517 qed
```
```   518
```
```   519 lemma power_Suc_less_one:
```
```   520   "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
```
```   521   using power_strict_decreasing [of 0 "Suc n" a] by simp
```
```   522
```
```   523 text\<open>Proof again resembles that of @{text power_strict_decreasing}\<close>
```
```   524 lemma power_increasing [rule_format]:
```
```   525   "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
```
```   526 proof (induct N)
```
```   527   case 0 then show ?case by simp
```
```   528 next
```
```   529   case (Suc N) then show ?case
```
```   530   apply (auto simp add: le_Suc_eq)
```
```   531   apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
```
```   532   apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
```
```   533   done
```
```   534 qed
```
```   535
```
```   536 text\<open>Lemma for @{text power_strict_increasing}\<close>
```
```   537 lemma power_less_power_Suc:
```
```   538   "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
```
```   539   by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
```
```   540
```
```   541 lemma power_strict_increasing [rule_format]:
```
```   542   "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
```
```   543 proof (induct N)
```
```   544   case 0 then show ?case by simp
```
```   545 next
```
```   546   case (Suc N) then show ?case
```
```   547   apply (auto simp add: power_less_power_Suc less_Suc_eq)
```
```   548   apply (subgoal_tac "1 * a^n < a * a^N", simp)
```
```   549   apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
```
```   550   done
```
```   551 qed
```
```   552
```
```   553 lemma power_increasing_iff [simp]:
```
```   554   "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
```
```   555   by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
```
```   556
```
```   557 lemma power_strict_increasing_iff [simp]:
```
```   558   "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
```
```   559 by (blast intro: power_less_imp_less_exp power_strict_increasing)
```
```   560
```
```   561 lemma power_le_imp_le_base:
```
```   562   assumes le: "a ^ Suc n \<le> b ^ Suc n"
```
```   563     and ynonneg: "0 \<le> b"
```
```   564   shows "a \<le> b"
```
```   565 proof (rule ccontr)
```
```   566   assume "~ a \<le> b"
```
```   567   then have "b < a" by (simp only: linorder_not_le)
```
```   568   then have "b ^ Suc n < a ^ Suc n"
```
```   569     by (simp only: assms power_strict_mono)
```
```   570   from le and this show False
```
```   571     by (simp add: linorder_not_less [symmetric])
```
```   572 qed
```
```   573
```
```   574 lemma power_less_imp_less_base:
```
```   575   assumes less: "a ^ n < b ^ n"
```
```   576   assumes nonneg: "0 \<le> b"
```
```   577   shows "a < b"
```
```   578 proof (rule contrapos_pp [OF less])
```
```   579   assume "~ a < b"
```
```   580   hence "b \<le> a" by (simp only: linorder_not_less)
```
```   581   hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
```
```   582   thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
```
```   583 qed
```
```   584
```
```   585 lemma power_inject_base:
```
```   586   "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
```
```   587 by (blast intro: power_le_imp_le_base antisym eq_refl sym)
```
```   588
```
```   589 lemma power_eq_imp_eq_base:
```
```   590   "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
```
```   591   by (cases n) (simp_all del: power_Suc, rule power_inject_base)
```
```   592
```
```   593 lemma power2_le_imp_le:
```
```   594   "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
```
```   595   unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
```
```   596
```
```   597 lemma power2_less_imp_less:
```
```   598   "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
```
```   599   by (rule power_less_imp_less_base)
```
```   600
```
```   601 lemma power2_eq_imp_eq:
```
```   602   "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
```
```   603   unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
```
```   604
```
```   605 end
```
```   606
```
```   607 context linordered_ring_strict
```
```   608 begin
```
```   609
```
```   610 lemma sum_squares_eq_zero_iff:
```
```   611   "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   612   by (simp add: add_nonneg_eq_0_iff)
```
```   613
```
```   614 lemma sum_squares_le_zero_iff:
```
```   615   "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   616   by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
```
```   617
```
```   618 lemma sum_squares_gt_zero_iff:
```
```   619   "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   620   by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
```
```   621
```
```   622 end
```
```   623
```
```   624 context linordered_idom
```
```   625 begin
```
```   626
```
```   627 lemma power_abs:
```
```   628   "abs (a ^ n) = abs a ^ n"
```
```   629   by (induct n) (auto simp add: abs_mult)
```
```   630
```
```   631 lemma abs_power_minus [simp]:
```
```   632   "abs ((-a) ^ n) = abs (a ^ n)"
```
```   633   by (simp add: power_abs)
```
```   634
```
```   635 lemma zero_less_power_abs_iff [simp]:
```
```   636   "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
```
```   637 proof (induct n)
```
```   638   case 0 show ?case by simp
```
```   639 next
```
```   640   case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
```
```   641 qed
```
```   642
```
```   643 lemma zero_le_power_abs [simp]:
```
```   644   "0 \<le> abs a ^ n"
```
```   645   by (rule zero_le_power [OF abs_ge_zero])
```
```   646
```
```   647 lemma zero_le_power2 [simp]:
```
```   648   "0 \<le> a\<^sup>2"
```
```   649   by (simp add: power2_eq_square)
```
```   650
```
```   651 lemma zero_less_power2 [simp]:
```
```   652   "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
```
```   653   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
```
```   654
```
```   655 lemma power2_less_0 [simp]:
```
```   656   "\<not> a\<^sup>2 < 0"
```
```   657   by (force simp add: power2_eq_square mult_less_0_iff)
```
```   658
```
```   659 lemma power2_less_eq_zero_iff [simp]:
```
```   660   "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
```
```   661   by (simp add: le_less)
```
```   662
```
```   663 lemma abs_power2 [simp]:
```
```   664   "abs (a\<^sup>2) = a\<^sup>2"
```
```   665   by (simp add: power2_eq_square abs_mult abs_mult_self)
```
```   666
```
```   667 lemma power2_abs [simp]:
```
```   668   "(abs a)\<^sup>2 = a\<^sup>2"
```
```   669   by (simp add: power2_eq_square abs_mult_self)
```
```   670
```
```   671 lemma odd_power_less_zero:
```
```   672   "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
```
```   673 proof (induct n)
```
```   674   case 0
```
```   675   then show ?case by simp
```
```   676 next
```
```   677   case (Suc n)
```
```   678   have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
```
```   679     by (simp add: ac_simps power_add power2_eq_square)
```
```   680   thus ?case
```
```   681     by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
```
```   682 qed
```
```   683
```
```   684 lemma odd_0_le_power_imp_0_le:
```
```   685   "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
```
```   686   using odd_power_less_zero [of a n]
```
```   687     by (force simp add: linorder_not_less [symmetric])
```
```   688
```
```   689 lemma zero_le_even_power'[simp]:
```
```   690   "0 \<le> a ^ (2*n)"
```
```   691 proof (induct n)
```
```   692   case 0
```
```   693     show ?case by simp
```
```   694 next
```
```   695   case (Suc n)
```
```   696     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
```
```   697       by (simp add: ac_simps power_add power2_eq_square)
```
```   698     thus ?case
```
```   699       by (simp add: Suc zero_le_mult_iff)
```
```   700 qed
```
```   701
```
```   702 lemma sum_power2_ge_zero:
```
```   703   "0 \<le> x\<^sup>2 + y\<^sup>2"
```
```   704   by (intro add_nonneg_nonneg zero_le_power2)
```
```   705
```
```   706 lemma not_sum_power2_lt_zero:
```
```   707   "\<not> x\<^sup>2 + y\<^sup>2 < 0"
```
```   708   unfolding not_less by (rule sum_power2_ge_zero)
```
```   709
```
```   710 lemma sum_power2_eq_zero_iff:
```
```   711   "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   712   unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
```
```   713
```
```   714 lemma sum_power2_le_zero_iff:
```
```   715   "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
```
```   716   by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
```
```   717
```
```   718 lemma sum_power2_gt_zero_iff:
```
```   719   "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
```
```   720   unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
```
```   721
```
```   722 lemma abs_le_square_iff:
```
```   723    "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
```
```   724 proof
```
```   725   assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
```
```   726   then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
```
```   727   then show "x\<^sup>2 \<le> y\<^sup>2" by simp
```
```   728 next
```
```   729   assume "x\<^sup>2 \<le> y\<^sup>2"
```
```   730   then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
```
```   731     by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
```
```   732 qed
```
```   733
```
```   734 lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> abs(x) \<le> 1"
```
```   735   using abs_le_square_iff [of x 1]
```
```   736   by simp
```
```   737
```
```   738 lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> abs(x) = 1"
```
```   739   by (auto simp add: abs_if power2_eq_1_iff)
```
```   740
```
```   741 lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> abs(x) < 1"
```
```   742   using  abs_square_eq_1 [of x] abs_square_le_1 [of x]
```
```   743   by (auto simp add: le_less)
```
```   744
```
```   745 end
```
```   746
```
```   747
```
```   748 subsection \<open>Miscellaneous rules\<close>
```
```   749
```
```   750 lemma (in linordered_semidom) self_le_power:
```
```   751   "1 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a ^ n"
```
```   752   using power_increasing [of 1 n a] power_one_right [of a] by auto
```
```   753
```
```   754 lemma (in power) power_eq_if:
```
```   755   "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
```
```   756   unfolding One_nat_def by (cases m) simp_all
```
```   757
```
```   758 lemma (in comm_semiring_1) power2_sum:
```
```   759   "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
```
```   760   by (simp add: algebra_simps power2_eq_square mult_2_right)
```
```   761
```
```   762 lemma (in comm_ring_1) power2_diff:
```
```   763   "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
```
```   764   by (simp add: algebra_simps power2_eq_square mult_2_right)
```
```   765
```
```   766 lemma (in comm_ring_1) power2_commute:
```
```   767   "(x - y)\<^sup>2 = (y - x)\<^sup>2"
```
```   768   by (simp add: algebra_simps power2_eq_square)
```
```   769
```
```   770
```
```   771 text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
```
```   772
```
```   773 lemmas zero_compare_simps =
```
```   774     add_strict_increasing add_strict_increasing2 add_increasing
```
```   775     zero_le_mult_iff zero_le_divide_iff
```
```   776     zero_less_mult_iff zero_less_divide_iff
```
```   777     mult_le_0_iff divide_le_0_iff
```
```   778     mult_less_0_iff divide_less_0_iff
```
```   779     zero_le_power2 power2_less_0
```
```   780
```
```   781
```
```   782 subsection \<open>Exponentiation for the Natural Numbers\<close>
```
```   783
```
```   784 lemma nat_one_le_power [simp]:
```
```   785   "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
```
```   786   by (rule one_le_power [of i n, unfolded One_nat_def])
```
```   787
```
```   788 lemma nat_zero_less_power_iff [simp]:
```
```   789   "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
```
```   790   by (induct n) auto
```
```   791
```
```   792 lemma nat_power_eq_Suc_0_iff [simp]:
```
```   793   "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
```
```   794   by (induct m) auto
```
```   795
```
```   796 lemma power_Suc_0 [simp]:
```
```   797   "Suc 0 ^ n = Suc 0"
```
```   798   by simp
```
```   799
```
```   800 text\<open>Valid for the naturals, but what if @{text"0<i<1"}?
```
```   801 Premises cannot be weakened: consider the case where @{term "i=0"},
```
```   802 @{term "m=1"} and @{term "n=0"}.\<close>
```
```   803 lemma nat_power_less_imp_less:
```
```   804   assumes nonneg: "0 < (i::nat)"
```
```   805   assumes less: "i ^ m < i ^ n"
```
```   806   shows "m < n"
```
```   807 proof (cases "i = 1")
```
```   808   case True with less power_one [where 'a = nat] show ?thesis by simp
```
```   809 next
```
```   810   case False with nonneg have "1 < i" by auto
```
```   811   from power_strict_increasing_iff [OF this] less show ?thesis ..
```
```   812 qed
```
```   813
```
```   814 lemma power_dvd_imp_le:
```
```   815   "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
```
```   816   apply (rule power_le_imp_le_exp, assumption)
```
```   817   apply (erule dvd_imp_le, simp)
```
```   818   done
```
```   819
```
```   820 lemma power2_nat_le_eq_le:
```
```   821   fixes m n :: nat
```
```   822   shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
```
```   823   by (auto intro: power2_le_imp_le power_mono)
```
```   824
```
```   825 lemma power2_nat_le_imp_le:
```
```   826   fixes m n :: nat
```
```   827   assumes "m\<^sup>2 \<le> n"
```
```   828   shows "m \<le> n"
```
```   829 proof (cases m)
```
```   830   case 0 then show ?thesis by simp
```
```   831 next
```
```   832   case (Suc k)
```
```   833   show ?thesis
```
```   834   proof (rule ccontr)
```
```   835     assume "\<not> m \<le> n"
```
```   836     then have "n < m" by simp
```
```   837     with assms Suc show False
```
```   838       by (simp add: power2_eq_square)
```
```   839   qed
```
```   840 qed
```
```   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 auto
```
```   851 next
```
```   852   case (insert x A)
```
```   853   then have "inj_on (insert x) (Pow A)"
```
```   854     unfolding inj_on_def by (blast elim!: equalityE)
```
```   855   then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A"
```
```   856     by (simp add: mult_2 card_image Pow_insert insert.hyps)
```
```   857   then show ?case using insert
```
```   858     apply (simp add: Pow_insert)
```
```   859     apply (subst card_Un_disjoint, auto)
```
```   860     done
```
```   861 qed
```
```   862
```
```   863
```
```   864 subsubsection \<open>Generalized sum over a set\<close>
```
```   865
```
```   866 lemma setsum_zero_power [simp]:
```
```   867   fixes c :: "nat \<Rightarrow> 'a::division_ring"
```
```   868   shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
```
```   869 apply (cases "finite A")
```
```   870   by (induction A rule: finite_induct) auto
```
```   871
```
```   872 lemma setsum_zero_power' [simp]:
```
```   873   fixes c :: "nat \<Rightarrow> 'a::field"
```
```   874   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)"
```
```   875   using setsum_zero_power [of "\<lambda>i. c i / d i" A]
```
```   876   by auto
```
```   877
```
```   878
```
```   879 subsubsection \<open>Generalized product over a set\<close>
```
```   880
```
```   881 lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
```
```   882 apply (erule finite_induct)
```
```   883 apply auto
```
```   884 done
```
```   885
```
```   886 lemma setprod_power_distrib:
```
```   887   fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
```
```   888   shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
```
```   889 proof (cases "finite A")
```
```   890   case True then show ?thesis
```
```   891     by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
```
```   892 next
```
```   893   case False then show ?thesis
```
```   894     by simp
```
```   895 qed
```
```   896
```
```   897 lemma power_setsum:
```
```   898   "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
```
```   899   by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
```
```   900
```
```   901 lemma setprod_gen_delta:
```
```   902   assumes fS: "finite S"
```
```   903   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)"
```
```   904 proof-
```
```   905   let ?f = "(\<lambda>k. if k=a then b k else c)"
```
```   906   {assume a: "a \<notin> S"
```
```   907     hence "\<forall> k\<in> S. ?f k = c" by simp
```
```   908     hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
```
```   909   moreover
```
```   910   {assume a: "a \<in> S"
```
```   911     let ?A = "S - {a}"
```
```   912     let ?B = "{a}"
```
```   913     have eq: "S = ?A \<union> ?B" using a by blast
```
```   914     have dj: "?A \<inter> ?B = {}" by simp
```
```   915     from fS have fAB: "finite ?A" "finite ?B" by auto
```
```   916     have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
```
```   917       apply (rule setprod.cong) by auto
```
```   918     have cA: "card ?A = card S - 1" using fS a by auto
```
```   919     have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
```
```   920     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
```
```   921       using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
```
```   922       by simp
```
```   923     then have ?thesis using a cA
```
```   924       by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
```
```   925   ultimately show ?thesis by blast
```
```   926 qed
```
```   927
```
```   928 subsection \<open>Code generator tweak\<close>
```
```   929
```
```   930 code_identifier
```
```   931   code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```   932
```
```   933 end
```