src/HOL/Power.thy
author immler
Mon Oct 13 18:55:05 2014 +0200 (2014-10-13)
changeset 58656 7f14d5d9b933
parent 58437 8d124c73c37a
child 58787 af9eb5e566dd
permissions -rw-r--r--
relaxed class constraints for exp
paulson@3390
     1
(*  Title:      HOL/Power.thy
paulson@3390
     2
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@3390
     3
    Copyright   1997  University of Cambridge
paulson@3390
     4
*)
paulson@3390
     5
haftmann@30960
     6
header {* Exponentiation *}
paulson@14348
     7
nipkow@15131
     8
theory Power
traytel@55096
     9
imports Num Equiv_Relations
nipkow@15131
    10
begin
paulson@14348
    11
haftmann@30960
    12
subsection {* Powers for Arbitrary Monoids *}
haftmann@30960
    13
haftmann@30996
    14
class power = one + times
haftmann@30960
    15
begin
haftmann@24996
    16
haftmann@30960
    17
primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where
haftmann@30960
    18
    power_0: "a ^ 0 = 1"
haftmann@30960
    19
  | power_Suc: "a ^ Suc n = a * a ^ n"
paulson@14348
    20
haftmann@30996
    21
notation (latex output)
haftmann@30996
    22
  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30996
    23
haftmann@30996
    24
notation (HTML output)
haftmann@30996
    25
  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30996
    26
huffman@47192
    27
text {* Special syntax for squares. *}
huffman@47192
    28
huffman@47192
    29
abbreviation (xsymbols)
wenzelm@53015
    30
  power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999) where
wenzelm@53015
    31
  "x\<^sup>2 \<equiv> x ^ 2"
huffman@47192
    32
huffman@47192
    33
notation (latex output)
wenzelm@53015
    34
  power2  ("(_\<^sup>2)" [1000] 999)
huffman@47192
    35
huffman@47192
    36
notation (HTML output)
wenzelm@53015
    37
  power2  ("(_\<^sup>2)" [1000] 999)
huffman@47192
    38
haftmann@30960
    39
end
paulson@14348
    40
haftmann@30996
    41
context monoid_mult
haftmann@30996
    42
begin
paulson@14348
    43
wenzelm@39438
    44
subclass power .
paulson@14348
    45
haftmann@30996
    46
lemma power_one [simp]:
haftmann@30996
    47
  "1 ^ n = 1"
huffman@30273
    48
  by (induct n) simp_all
paulson@14348
    49
haftmann@30996
    50
lemma power_one_right [simp]:
haftmann@31001
    51
  "a ^ 1 = a"
haftmann@30996
    52
  by simp
paulson@14348
    53
haftmann@30996
    54
lemma power_commutes:
haftmann@30996
    55
  "a ^ n * a = a * a ^ n"
haftmann@57512
    56
  by (induct n) (simp_all add: mult.assoc)
krauss@21199
    57
haftmann@30996
    58
lemma power_Suc2:
haftmann@30996
    59
  "a ^ Suc n = a ^ n * a"
huffman@30273
    60
  by (simp add: power_commutes)
huffman@28131
    61
haftmann@30996
    62
lemma power_add:
haftmann@30996
    63
  "a ^ (m + n) = a ^ m * a ^ n"
haftmann@30996
    64
  by (induct m) (simp_all add: algebra_simps)
paulson@14348
    65
haftmann@30996
    66
lemma power_mult:
haftmann@30996
    67
  "a ^ (m * n) = (a ^ m) ^ n"
huffman@30273
    68
  by (induct n) (simp_all add: power_add)
paulson@14348
    69
wenzelm@53015
    70
lemma power2_eq_square: "a\<^sup>2 = a * a"
huffman@47192
    71
  by (simp add: numeral_2_eq_2)
huffman@47192
    72
huffman@47192
    73
lemma power3_eq_cube: "a ^ 3 = a * a * a"
haftmann@57512
    74
  by (simp add: numeral_3_eq_3 mult.assoc)
huffman@47192
    75
huffman@47192
    76
lemma power_even_eq:
wenzelm@53076
    77
  "a ^ (2 * n) = (a ^ n)\<^sup>2"
haftmann@57512
    78
  by (subst mult.commute) (simp add: power_mult)
huffman@47192
    79
huffman@47192
    80
lemma power_odd_eq:
wenzelm@53076
    81
  "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
huffman@47192
    82
  by (simp add: power_even_eq)
huffman@47192
    83
huffman@47255
    84
lemma power_numeral_even:
huffman@47255
    85
  "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
huffman@47255
    86
  unfolding numeral_Bit0 power_add Let_def ..
huffman@47255
    87
huffman@47255
    88
lemma power_numeral_odd:
huffman@47255
    89
  "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
huffman@47255
    90
  unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
haftmann@57512
    91
  unfolding power_Suc power_add Let_def mult.assoc ..
huffman@47255
    92
haftmann@49824
    93
lemma funpow_times_power:
haftmann@49824
    94
  "(times x ^^ f x) = times (x ^ f x)"
haftmann@49824
    95
proof (induct "f x" arbitrary: f)
haftmann@49824
    96
  case 0 then show ?case by (simp add: fun_eq_iff)
haftmann@49824
    97
next
haftmann@49824
    98
  case (Suc n)
haftmann@49824
    99
  def g \<equiv> "\<lambda>x. f x - 1"
haftmann@49824
   100
  with Suc have "n = g x" by simp
haftmann@49824
   101
  with Suc have "times x ^^ g x = times (x ^ g x)" by simp
haftmann@49824
   102
  moreover from Suc g_def have "f x = g x + 1" by simp
haftmann@57512
   103
  ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
haftmann@49824
   104
qed
haftmann@49824
   105
immler@58656
   106
lemma power_commuting_commutes:
immler@58656
   107
  assumes "x * y = y * x"
immler@58656
   108
  shows "x ^ n * y = y * x ^n"
immler@58656
   109
proof (induct n)
immler@58656
   110
  case (Suc n)
immler@58656
   111
  have "x ^ Suc n * y = x ^ n * y * x"
immler@58656
   112
    by (subst power_Suc2) (simp add: assms ac_simps)
immler@58656
   113
  also have "\<dots> = y * x ^ Suc n"
immler@58656
   114
    unfolding Suc power_Suc2
immler@58656
   115
    by (simp add: ac_simps)
immler@58656
   116
  finally show ?case .
immler@58656
   117
qed simp
immler@58656
   118
haftmann@30996
   119
end
haftmann@30996
   120
haftmann@30996
   121
context comm_monoid_mult
haftmann@30996
   122
begin
haftmann@30996
   123
hoelzl@56480
   124
lemma power_mult_distrib [field_simps]:
haftmann@30996
   125
  "(a * b) ^ n = (a ^ n) * (b ^ n)"
haftmann@57514
   126
  by (induct n) (simp_all add: ac_simps)
paulson@14348
   127
haftmann@30996
   128
end
haftmann@30996
   129
huffman@47191
   130
context semiring_numeral
huffman@47191
   131
begin
huffman@47191
   132
huffman@47191
   133
lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
huffman@47191
   134
  by (simp only: sqr_conv_mult numeral_mult)
huffman@47191
   135
huffman@47191
   136
lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
huffman@47191
   137
  by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
huffman@47191
   138
    numeral_sqr numeral_mult power_add power_one_right)
huffman@47191
   139
huffman@47191
   140
lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
huffman@47191
   141
  by (rule numeral_pow [symmetric])
huffman@47191
   142
huffman@47191
   143
end
huffman@47191
   144
haftmann@30996
   145
context semiring_1
haftmann@30996
   146
begin
haftmann@30996
   147
haftmann@30996
   148
lemma of_nat_power:
haftmann@30996
   149
  "of_nat (m ^ n) = of_nat m ^ n"
haftmann@30996
   150
  by (induct n) (simp_all add: of_nat_mult)
haftmann@30996
   151
huffman@47191
   152
lemma power_zero_numeral [simp]: "(0::'a) ^ numeral k = 0"
huffman@47209
   153
  by (simp add: numeral_eq_Suc)
huffman@47191
   154
wenzelm@53015
   155
lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
huffman@47192
   156
  by (rule power_zero_numeral)
huffman@47192
   157
wenzelm@53015
   158
lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
huffman@47192
   159
  by (rule power_one)
huffman@47192
   160
haftmann@30996
   161
end
haftmann@30996
   162
haftmann@30996
   163
context comm_semiring_1
haftmann@30996
   164
begin
haftmann@30996
   165
haftmann@30996
   166
text {* The divides relation *}
haftmann@30996
   167
haftmann@30996
   168
lemma le_imp_power_dvd:
haftmann@30996
   169
  assumes "m \<le> n" shows "a ^ m dvd a ^ n"
haftmann@30996
   170
proof
haftmann@30996
   171
  have "a ^ n = a ^ (m + (n - m))"
haftmann@30996
   172
    using `m \<le> n` by simp
haftmann@30996
   173
  also have "\<dots> = a ^ m * a ^ (n - m)"
haftmann@30996
   174
    by (rule power_add)
haftmann@30996
   175
  finally show "a ^ n = a ^ m * a ^ (n - m)" .
haftmann@30996
   176
qed
haftmann@30996
   177
haftmann@30996
   178
lemma power_le_dvd:
haftmann@30996
   179
  "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
haftmann@30996
   180
  by (rule dvd_trans [OF le_imp_power_dvd])
haftmann@30996
   181
haftmann@30996
   182
lemma dvd_power_same:
haftmann@30996
   183
  "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
haftmann@30996
   184
  by (induct n) (auto simp add: mult_dvd_mono)
haftmann@30996
   185
haftmann@30996
   186
lemma dvd_power_le:
haftmann@30996
   187
  "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
haftmann@30996
   188
  by (rule power_le_dvd [OF dvd_power_same])
paulson@14348
   189
haftmann@30996
   190
lemma dvd_power [simp]:
haftmann@30996
   191
  assumes "n > (0::nat) \<or> x = 1"
haftmann@30996
   192
  shows "x dvd (x ^ n)"
haftmann@30996
   193
using assms proof
haftmann@30996
   194
  assume "0 < n"
haftmann@30996
   195
  then have "x ^ n = x ^ Suc (n - 1)" by simp
haftmann@30996
   196
  then show "x dvd (x ^ n)" by simp
haftmann@30996
   197
next
haftmann@30996
   198
  assume "x = 1"
haftmann@30996
   199
  then show "x dvd (x ^ n)" by simp
haftmann@30996
   200
qed
haftmann@30996
   201
haftmann@30996
   202
end
haftmann@30996
   203
haftmann@30996
   204
context ring_1
haftmann@30996
   205
begin
haftmann@30996
   206
haftmann@30996
   207
lemma power_minus:
haftmann@30996
   208
  "(- a) ^ n = (- 1) ^ n * a ^ n"
haftmann@30996
   209
proof (induct n)
haftmann@30996
   210
  case 0 show ?case by simp
haftmann@30996
   211
next
haftmann@30996
   212
  case (Suc n) then show ?case
haftmann@57512
   213
    by (simp del: power_Suc add: power_Suc2 mult.assoc)
haftmann@30996
   214
qed
haftmann@30996
   215
huffman@47191
   216
lemma power_minus_Bit0:
huffman@47191
   217
  "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
huffman@47191
   218
  by (induct k, simp_all only: numeral_class.numeral.simps power_add
huffman@47191
   219
    power_one_right mult_minus_left mult_minus_right minus_minus)
huffman@47191
   220
huffman@47191
   221
lemma power_minus_Bit1:
huffman@47191
   222
  "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
huffman@47220
   223
  by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
huffman@47191
   224
huffman@47192
   225
lemma power2_minus [simp]:
wenzelm@53015
   226
  "(- a)\<^sup>2 = a\<^sup>2"
huffman@47192
   227
  by (rule power_minus_Bit0)
huffman@47192
   228
huffman@47192
   229
lemma power_minus1_even [simp]:
haftmann@58410
   230
  "(- 1) ^ (2*n) = 1"
huffman@47192
   231
proof (induct n)
huffman@47192
   232
  case 0 show ?case by simp
huffman@47192
   233
next
huffman@47192
   234
  case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
huffman@47192
   235
qed
huffman@47192
   236
huffman@47192
   237
lemma power_minus1_odd:
haftmann@58410
   238
  "(- 1) ^ Suc (2*n) = -1"
huffman@47192
   239
  by simp
huffman@47192
   240
huffman@47192
   241
lemma power_minus_even [simp]:
huffman@47192
   242
  "(-a) ^ (2*n) = a ^ (2*n)"
huffman@47192
   243
  by (simp add: power_minus [of a])
huffman@47192
   244
huffman@47192
   245
end
huffman@47192
   246
huffman@47192
   247
context ring_1_no_zero_divisors
huffman@47192
   248
begin
huffman@47192
   249
huffman@47192
   250
lemma field_power_not_zero:
huffman@47192
   251
  "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
huffman@47192
   252
  by (induct n) auto
huffman@47192
   253
huffman@47192
   254
lemma zero_eq_power2 [simp]:
wenzelm@53015
   255
  "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
huffman@47192
   256
  unfolding power2_eq_square by simp
huffman@47192
   257
huffman@47192
   258
lemma power2_eq_1_iff:
wenzelm@53015
   259
  "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
huffman@47192
   260
  unfolding power2_eq_square by (rule square_eq_1_iff)
huffman@47192
   261
huffman@47192
   262
end
huffman@47192
   263
huffman@47192
   264
context idom
huffman@47192
   265
begin
huffman@47192
   266
wenzelm@53015
   267
lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
huffman@47192
   268
  unfolding power2_eq_square by (rule square_eq_iff)
huffman@47192
   269
huffman@47192
   270
end
huffman@47192
   271
huffman@47192
   272
context division_ring
huffman@47192
   273
begin
huffman@47192
   274
huffman@47192
   275
text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
huffman@47192
   276
lemma nonzero_power_inverse:
huffman@47192
   277
  "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
huffman@47192
   278
  by (induct n)
huffman@47192
   279
    (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
huffman@47192
   280
huffman@47192
   281
end
huffman@47192
   282
huffman@47192
   283
context field
huffman@47192
   284
begin
huffman@47192
   285
huffman@47192
   286
lemma nonzero_power_divide:
huffman@47192
   287
  "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
huffman@47192
   288
  by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
huffman@47192
   289
huffman@47192
   290
end
huffman@47192
   291
huffman@47192
   292
huffman@47192
   293
subsection {* Exponentiation on ordered types *}
huffman@47192
   294
huffman@47192
   295
context linordered_ring (* TODO: move *)
huffman@47192
   296
begin
huffman@47192
   297
huffman@47192
   298
lemma sum_squares_ge_zero:
huffman@47192
   299
  "0 \<le> x * x + y * y"
huffman@47192
   300
  by (intro add_nonneg_nonneg zero_le_square)
huffman@47192
   301
huffman@47192
   302
lemma not_sum_squares_lt_zero:
huffman@47192
   303
  "\<not> x * x + y * y < 0"
huffman@47192
   304
  by (simp add: not_less sum_squares_ge_zero)
huffman@47192
   305
haftmann@30996
   306
end
haftmann@30996
   307
haftmann@35028
   308
context linordered_semidom
haftmann@30996
   309
begin
haftmann@30996
   310
haftmann@30996
   311
lemma zero_less_power [simp]:
haftmann@30996
   312
  "0 < a \<Longrightarrow> 0 < a ^ n"
nipkow@56544
   313
  by (induct n) simp_all
haftmann@30996
   314
haftmann@30996
   315
lemma zero_le_power [simp]:
haftmann@30996
   316
  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
nipkow@56536
   317
  by (induct n) simp_all
paulson@14348
   318
huffman@47241
   319
lemma power_mono:
huffman@47241
   320
  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
huffman@47241
   321
  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
huffman@47241
   322
huffman@47241
   323
lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
huffman@47241
   324
  using power_mono [of 1 a n] by simp
huffman@47241
   325
huffman@47241
   326
lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
huffman@47241
   327
  using power_mono [of a 1 n] by simp
paulson@14348
   328
paulson@14348
   329
lemma power_gt1_lemma:
haftmann@30996
   330
  assumes gt1: "1 < a"
haftmann@30996
   331
  shows "1 < a * a ^ n"
paulson@14348
   332
proof -
haftmann@30996
   333
  from gt1 have "0 \<le> a"
haftmann@30996
   334
    by (fact order_trans [OF zero_le_one less_imp_le])
haftmann@30996
   335
  have "1 * 1 < a * 1" using gt1 by simp
haftmann@30996
   336
  also have "\<dots> \<le> a * a ^ n" using gt1
haftmann@30996
   337
    by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
wenzelm@14577
   338
        zero_le_one order_refl)
wenzelm@14577
   339
  finally show ?thesis by simp
paulson@14348
   340
qed
paulson@14348
   341
haftmann@30996
   342
lemma power_gt1:
haftmann@30996
   343
  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
haftmann@30996
   344
  by (simp add: power_gt1_lemma)
huffman@24376
   345
haftmann@30996
   346
lemma one_less_power [simp]:
haftmann@30996
   347
  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
haftmann@30996
   348
  by (cases n) (simp_all add: power_gt1_lemma)
paulson@14348
   349
paulson@14348
   350
lemma power_le_imp_le_exp:
haftmann@30996
   351
  assumes gt1: "1 < a"
haftmann@30996
   352
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
haftmann@30996
   353
proof (induct m arbitrary: n)
paulson@14348
   354
  case 0
wenzelm@14577
   355
  show ?case by simp
paulson@14348
   356
next
paulson@14348
   357
  case (Suc m)
wenzelm@14577
   358
  show ?case
wenzelm@14577
   359
  proof (cases n)
wenzelm@14577
   360
    case 0
haftmann@30996
   361
    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
wenzelm@14577
   362
    with gt1 show ?thesis
wenzelm@14577
   363
      by (force simp only: power_gt1_lemma
haftmann@30996
   364
          not_less [symmetric])
wenzelm@14577
   365
  next
wenzelm@14577
   366
    case (Suc n)
haftmann@30996
   367
    with Suc.prems Suc.hyps show ?thesis
wenzelm@14577
   368
      by (force dest: mult_left_le_imp_le
haftmann@30996
   369
          simp add: less_trans [OF zero_less_one gt1])
wenzelm@14577
   370
  qed
paulson@14348
   371
qed
paulson@14348
   372
wenzelm@14577
   373
text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
paulson@14348
   374
lemma power_inject_exp [simp]:
haftmann@30996
   375
  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
wenzelm@14577
   376
  by (force simp add: order_antisym power_le_imp_le_exp)
paulson@14348
   377
paulson@14348
   378
text{*Can relax the first premise to @{term "0<a"} in the case of the
paulson@14348
   379
natural numbers.*}
paulson@14348
   380
lemma power_less_imp_less_exp:
haftmann@30996
   381
  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
haftmann@30996
   382
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
haftmann@30996
   383
    power_le_imp_le_exp)
paulson@14348
   384
paulson@14348
   385
lemma power_strict_mono [rule_format]:
haftmann@30996
   386
  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
haftmann@30996
   387
  by (induct n)
haftmann@30996
   388
   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
paulson@14348
   389
paulson@14348
   390
text{*Lemma for @{text power_strict_decreasing}*}
paulson@14348
   391
lemma power_Suc_less:
haftmann@30996
   392
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
haftmann@30996
   393
  by (induct n)
haftmann@30996
   394
    (auto simp add: mult_strict_left_mono)
paulson@14348
   395
haftmann@30996
   396
lemma power_strict_decreasing [rule_format]:
haftmann@30996
   397
  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
haftmann@30996
   398
proof (induct N)
haftmann@30996
   399
  case 0 then show ?case by simp
haftmann@30996
   400
next
haftmann@30996
   401
  case (Suc N) then show ?case 
haftmann@30996
   402
  apply (auto simp add: power_Suc_less less_Suc_eq)
haftmann@30996
   403
  apply (subgoal_tac "a * a^N < 1 * a^n")
haftmann@30996
   404
  apply simp
haftmann@30996
   405
  apply (rule mult_strict_mono) apply auto
haftmann@30996
   406
  done
haftmann@30996
   407
qed
paulson@14348
   408
paulson@14348
   409
text{*Proof resembles that of @{text power_strict_decreasing}*}
haftmann@30996
   410
lemma power_decreasing [rule_format]:
haftmann@30996
   411
  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
haftmann@30996
   412
proof (induct N)
haftmann@30996
   413
  case 0 then show ?case by simp
haftmann@30996
   414
next
haftmann@30996
   415
  case (Suc N) then show ?case 
haftmann@30996
   416
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   417
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
haftmann@30996
   418
  apply (rule mult_mono) apply auto
haftmann@30996
   419
  done
haftmann@30996
   420
qed
paulson@14348
   421
paulson@14348
   422
lemma power_Suc_less_one:
haftmann@30996
   423
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
haftmann@30996
   424
  using power_strict_decreasing [of 0 "Suc n" a] by simp
paulson@14348
   425
paulson@14348
   426
text{*Proof again resembles that of @{text power_strict_decreasing}*}
haftmann@30996
   427
lemma power_increasing [rule_format]:
haftmann@30996
   428
  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
haftmann@30996
   429
proof (induct N)
haftmann@30996
   430
  case 0 then show ?case by simp
haftmann@30996
   431
next
haftmann@30996
   432
  case (Suc N) then show ?case 
haftmann@30996
   433
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   434
  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
haftmann@30996
   435
  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
haftmann@30996
   436
  done
haftmann@30996
   437
qed
paulson@14348
   438
paulson@14348
   439
text{*Lemma for @{text power_strict_increasing}*}
paulson@14348
   440
lemma power_less_power_Suc:
haftmann@30996
   441
  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
haftmann@30996
   442
  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
paulson@14348
   443
haftmann@30996
   444
lemma power_strict_increasing [rule_format]:
haftmann@30996
   445
  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
haftmann@30996
   446
proof (induct N)
haftmann@30996
   447
  case 0 then show ?case by simp
haftmann@30996
   448
next
haftmann@30996
   449
  case (Suc N) then show ?case 
haftmann@30996
   450
  apply (auto simp add: power_less_power_Suc less_Suc_eq)
haftmann@30996
   451
  apply (subgoal_tac "1 * a^n < a * a^N", simp)
haftmann@30996
   452
  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
haftmann@30996
   453
  done
haftmann@30996
   454
qed
paulson@14348
   455
nipkow@25134
   456
lemma power_increasing_iff [simp]:
haftmann@30996
   457
  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
haftmann@30996
   458
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
paulson@15066
   459
paulson@15066
   460
lemma power_strict_increasing_iff [simp]:
haftmann@30996
   461
  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
nipkow@25134
   462
by (blast intro: power_less_imp_less_exp power_strict_increasing) 
paulson@15066
   463
paulson@14348
   464
lemma power_le_imp_le_base:
haftmann@30996
   465
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
haftmann@30996
   466
    and ynonneg: "0 \<le> b"
haftmann@30996
   467
  shows "a \<le> b"
nipkow@25134
   468
proof (rule ccontr)
nipkow@25134
   469
  assume "~ a \<le> b"
nipkow@25134
   470
  then have "b < a" by (simp only: linorder_not_le)
nipkow@25134
   471
  then have "b ^ Suc n < a ^ Suc n"
wenzelm@41550
   472
    by (simp only: assms power_strict_mono)
haftmann@30996
   473
  from le and this show False
nipkow@25134
   474
    by (simp add: linorder_not_less [symmetric])
nipkow@25134
   475
qed
wenzelm@14577
   476
huffman@22853
   477
lemma power_less_imp_less_base:
huffman@22853
   478
  assumes less: "a ^ n < b ^ n"
huffman@22853
   479
  assumes nonneg: "0 \<le> b"
huffman@22853
   480
  shows "a < b"
huffman@22853
   481
proof (rule contrapos_pp [OF less])
huffman@22853
   482
  assume "~ a < b"
huffman@22853
   483
  hence "b \<le> a" by (simp only: linorder_not_less)
huffman@22853
   484
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
haftmann@30996
   485
  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
huffman@22853
   486
qed
huffman@22853
   487
paulson@14348
   488
lemma power_inject_base:
haftmann@30996
   489
  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
haftmann@30996
   490
by (blast intro: power_le_imp_le_base antisym eq_refl sym)
paulson@14348
   491
huffman@22955
   492
lemma power_eq_imp_eq_base:
haftmann@30996
   493
  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
haftmann@30996
   494
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
huffman@22955
   495
huffman@47192
   496
lemma power2_le_imp_le:
wenzelm@53015
   497
  "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
huffman@47192
   498
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
huffman@47192
   499
huffman@47192
   500
lemma power2_less_imp_less:
wenzelm@53015
   501
  "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
huffman@47192
   502
  by (rule power_less_imp_less_base)
huffman@47192
   503
huffman@47192
   504
lemma power2_eq_imp_eq:
wenzelm@53015
   505
  "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
huffman@47192
   506
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
huffman@47192
   507
huffman@47192
   508
end
huffman@47192
   509
huffman@47192
   510
context linordered_ring_strict
huffman@47192
   511
begin
huffman@47192
   512
huffman@47192
   513
lemma sum_squares_eq_zero_iff:
huffman@47192
   514
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   515
  by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   516
huffman@47192
   517
lemma sum_squares_le_zero_iff:
huffman@47192
   518
  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   519
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
huffman@47192
   520
huffman@47192
   521
lemma sum_squares_gt_zero_iff:
huffman@47192
   522
  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   523
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
huffman@47192
   524
haftmann@30996
   525
end
haftmann@30996
   526
haftmann@35028
   527
context linordered_idom
haftmann@30996
   528
begin
huffman@29978
   529
haftmann@30996
   530
lemma power_abs:
haftmann@30996
   531
  "abs (a ^ n) = abs a ^ n"
haftmann@30996
   532
  by (induct n) (auto simp add: abs_mult)
haftmann@30996
   533
haftmann@30996
   534
lemma abs_power_minus [simp]:
haftmann@30996
   535
  "abs ((-a) ^ n) = abs (a ^ n)"
huffman@35216
   536
  by (simp add: power_abs)
haftmann@30996
   537
blanchet@54147
   538
lemma zero_less_power_abs_iff [simp]:
haftmann@30996
   539
  "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
haftmann@30996
   540
proof (induct n)
haftmann@30996
   541
  case 0 show ?case by simp
haftmann@30996
   542
next
haftmann@30996
   543
  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
huffman@29978
   544
qed
huffman@29978
   545
haftmann@30996
   546
lemma zero_le_power_abs [simp]:
haftmann@30996
   547
  "0 \<le> abs a ^ n"
haftmann@30996
   548
  by (rule zero_le_power [OF abs_ge_zero])
haftmann@30996
   549
huffman@47192
   550
lemma zero_le_power2 [simp]:
wenzelm@53015
   551
  "0 \<le> a\<^sup>2"
huffman@47192
   552
  by (simp add: power2_eq_square)
huffman@47192
   553
huffman@47192
   554
lemma zero_less_power2 [simp]:
wenzelm@53015
   555
  "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
huffman@47192
   556
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
huffman@47192
   557
huffman@47192
   558
lemma power2_less_0 [simp]:
wenzelm@53015
   559
  "\<not> a\<^sup>2 < 0"
huffman@47192
   560
  by (force simp add: power2_eq_square mult_less_0_iff)
huffman@47192
   561
huffman@47192
   562
lemma abs_power2 [simp]:
wenzelm@53015
   563
  "abs (a\<^sup>2) = a\<^sup>2"
huffman@47192
   564
  by (simp add: power2_eq_square abs_mult abs_mult_self)
huffman@47192
   565
huffman@47192
   566
lemma power2_abs [simp]:
wenzelm@53015
   567
  "(abs a)\<^sup>2 = a\<^sup>2"
huffman@47192
   568
  by (simp add: power2_eq_square abs_mult_self)
huffman@47192
   569
huffman@47192
   570
lemma odd_power_less_zero:
huffman@47192
   571
  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
huffman@47192
   572
proof (induct n)
huffman@47192
   573
  case 0
huffman@47192
   574
  then show ?case by simp
huffman@47192
   575
next
huffman@47192
   576
  case (Suc n)
huffman@47192
   577
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
haftmann@57514
   578
    by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   579
  thus ?case
huffman@47192
   580
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
huffman@47192
   581
qed
haftmann@30996
   582
huffman@47192
   583
lemma odd_0_le_power_imp_0_le:
huffman@47192
   584
  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
huffman@47192
   585
  using odd_power_less_zero [of a n]
huffman@47192
   586
    by (force simp add: linorder_not_less [symmetric]) 
huffman@47192
   587
huffman@47192
   588
lemma zero_le_even_power'[simp]:
huffman@47192
   589
  "0 \<le> a ^ (2*n)"
huffman@47192
   590
proof (induct n)
huffman@47192
   591
  case 0
huffman@47192
   592
    show ?case by simp
huffman@47192
   593
next
huffman@47192
   594
  case (Suc n)
huffman@47192
   595
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
haftmann@57514
   596
      by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   597
    thus ?case
huffman@47192
   598
      by (simp add: Suc zero_le_mult_iff)
huffman@47192
   599
qed
haftmann@30996
   600
huffman@47192
   601
lemma sum_power2_ge_zero:
wenzelm@53015
   602
  "0 \<le> x\<^sup>2 + y\<^sup>2"
huffman@47192
   603
  by (intro add_nonneg_nonneg zero_le_power2)
huffman@47192
   604
huffman@47192
   605
lemma not_sum_power2_lt_zero:
wenzelm@53015
   606
  "\<not> x\<^sup>2 + y\<^sup>2 < 0"
huffman@47192
   607
  unfolding not_less by (rule sum_power2_ge_zero)
huffman@47192
   608
huffman@47192
   609
lemma sum_power2_eq_zero_iff:
wenzelm@53015
   610
  "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   611
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   612
huffman@47192
   613
lemma sum_power2_le_zero_iff:
wenzelm@53015
   614
  "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   615
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
huffman@47192
   616
huffman@47192
   617
lemma sum_power2_gt_zero_iff:
wenzelm@53015
   618
  "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   619
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
haftmann@30996
   620
haftmann@30996
   621
end
haftmann@30996
   622
huffman@29978
   623
huffman@47192
   624
subsection {* Miscellaneous rules *}
paulson@14348
   625
lp15@55718
   626
lemma self_le_power:
lp15@55718
   627
  fixes x::"'a::linordered_semidom" 
lp15@55718
   628
  shows "1 \<le> x \<Longrightarrow> 0 < n \<Longrightarrow> x \<le> x ^ n"
traytel@55811
   629
  using power_increasing[of 1 n x] power_one_right[of x] by auto
lp15@55718
   630
huffman@47255
   631
lemma power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
huffman@47255
   632
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
   633
huffman@47192
   634
lemma power2_sum:
huffman@47192
   635
  fixes x y :: "'a::comm_semiring_1"
wenzelm@53015
   636
  shows "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
huffman@47192
   637
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   638
huffman@47192
   639
lemma power2_diff:
huffman@47192
   640
  fixes x y :: "'a::comm_ring_1"
wenzelm@53015
   641
  shows "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
haftmann@57512
   642
  by (simp add: ring_distribs power2_eq_square mult_2) (rule mult.commute)
haftmann@30996
   643
haftmann@30996
   644
lemma power_0_Suc [simp]:
haftmann@30996
   645
  "(0::'a::{power, semiring_0}) ^ Suc n = 0"
haftmann@30996
   646
  by simp
nipkow@30313
   647
haftmann@30996
   648
text{*It looks plausible as a simprule, but its effect can be strange.*}
haftmann@30996
   649
lemma power_0_left:
haftmann@30996
   650
  "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
haftmann@30996
   651
  by (induct n) simp_all
haftmann@30996
   652
haftmann@30996
   653
lemma power_eq_0_iff [simp]:
haftmann@30996
   654
  "a ^ n = 0 \<longleftrightarrow>
haftmann@30996
   655
     a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
haftmann@30996
   656
  by (induct n)
haftmann@30996
   657
    (auto simp add: no_zero_divisors elim: contrapos_pp)
haftmann@30996
   658
haftmann@36409
   659
lemma (in field) power_diff:
haftmann@30996
   660
  assumes nz: "a \<noteq> 0"
haftmann@30996
   661
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
haftmann@36409
   662
  by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
nipkow@30313
   663
haftmann@30996
   664
text{*Perhaps these should be simprules.*}
haftmann@30996
   665
lemma power_inverse:
haftmann@36409
   666
  fixes a :: "'a::division_ring_inverse_zero"
haftmann@36409
   667
  shows "inverse (a ^ n) = inverse a ^ n"
haftmann@30996
   668
apply (cases "a = 0")
haftmann@30996
   669
apply (simp add: power_0_left)
haftmann@30996
   670
apply (simp add: nonzero_power_inverse)
haftmann@30996
   671
done (* TODO: reorient or rename to inverse_power *)
haftmann@30996
   672
haftmann@30996
   673
lemma power_one_over:
haftmann@36409
   674
  "1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ n"
haftmann@30996
   675
  by (simp add: divide_inverse) (rule power_inverse)
haftmann@30996
   676
hoelzl@56481
   677
lemma power_divide [field_simps, divide_simps]:
haftmann@36409
   678
  "(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n"
haftmann@30996
   679
apply (cases "b = 0")
haftmann@30996
   680
apply (simp add: power_0_left)
haftmann@30996
   681
apply (rule nonzero_power_divide)
haftmann@30996
   682
apply assumption
nipkow@30313
   683
done
nipkow@30313
   684
huffman@47255
   685
text {* Simprules for comparisons where common factors can be cancelled. *}
huffman@47255
   686
huffman@47255
   687
lemmas zero_compare_simps =
huffman@47255
   688
    add_strict_increasing add_strict_increasing2 add_increasing
huffman@47255
   689
    zero_le_mult_iff zero_le_divide_iff 
huffman@47255
   690
    zero_less_mult_iff zero_less_divide_iff 
huffman@47255
   691
    mult_le_0_iff divide_le_0_iff 
huffman@47255
   692
    mult_less_0_iff divide_less_0_iff 
huffman@47255
   693
    zero_le_power2 power2_less_0
huffman@47255
   694
nipkow@30313
   695
haftmann@30960
   696
subsection {* Exponentiation for the Natural Numbers *}
wenzelm@14577
   697
haftmann@30996
   698
lemma nat_one_le_power [simp]:
haftmann@30996
   699
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
haftmann@30996
   700
  by (rule one_le_power [of i n, unfolded One_nat_def])
huffman@23305
   701
haftmann@30996
   702
lemma nat_zero_less_power_iff [simp]:
haftmann@30996
   703
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
haftmann@30996
   704
  by (induct n) auto
paulson@14348
   705
nipkow@30056
   706
lemma nat_power_eq_Suc_0_iff [simp]: 
haftmann@30996
   707
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
haftmann@30996
   708
  by (induct m) auto
nipkow@30056
   709
haftmann@30996
   710
lemma power_Suc_0 [simp]:
haftmann@30996
   711
  "Suc 0 ^ n = Suc 0"
haftmann@30996
   712
  by simp
nipkow@30056
   713
paulson@14348
   714
text{*Valid for the naturals, but what if @{text"0<i<1"}?
paulson@14348
   715
Premises cannot be weakened: consider the case where @{term "i=0"},
paulson@14348
   716
@{term "m=1"} and @{term "n=0"}.*}
haftmann@21413
   717
lemma nat_power_less_imp_less:
haftmann@21413
   718
  assumes nonneg: "0 < (i\<Colon>nat)"
haftmann@30996
   719
  assumes less: "i ^ m < i ^ n"
haftmann@21413
   720
  shows "m < n"
haftmann@21413
   721
proof (cases "i = 1")
haftmann@21413
   722
  case True with less power_one [where 'a = nat] show ?thesis by simp
haftmann@21413
   723
next
haftmann@21413
   724
  case False with nonneg have "1 < i" by auto
haftmann@21413
   725
  from power_strict_increasing_iff [OF this] less show ?thesis ..
haftmann@21413
   726
qed
paulson@14348
   727
haftmann@33274
   728
lemma power_dvd_imp_le:
haftmann@33274
   729
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
haftmann@33274
   730
  apply (rule power_le_imp_le_exp, assumption)
haftmann@33274
   731
  apply (erule dvd_imp_le, simp)
haftmann@33274
   732
  done
haftmann@33274
   733
haftmann@51263
   734
lemma power2_nat_le_eq_le:
haftmann@51263
   735
  fixes m n :: nat
wenzelm@53015
   736
  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
haftmann@51263
   737
  by (auto intro: power2_le_imp_le power_mono)
haftmann@51263
   738
haftmann@51263
   739
lemma power2_nat_le_imp_le:
haftmann@51263
   740
  fixes m n :: nat
wenzelm@53015
   741
  assumes "m\<^sup>2 \<le> n"
haftmann@51263
   742
  shows "m \<le> n"
haftmann@54249
   743
proof (cases m)
haftmann@54249
   744
  case 0 then show ?thesis by simp
haftmann@54249
   745
next
haftmann@54249
   746
  case (Suc k)
haftmann@54249
   747
  show ?thesis
haftmann@54249
   748
  proof (rule ccontr)
haftmann@54249
   749
    assume "\<not> m \<le> n"
haftmann@54249
   750
    then have "n < m" by simp
haftmann@54249
   751
    with assms Suc show False
haftmann@54249
   752
      by (auto simp add: algebra_simps) (simp add: power2_eq_square)
haftmann@54249
   753
  qed
haftmann@54249
   754
qed
haftmann@51263
   755
traytel@55096
   756
subsubsection {* Cardinality of the Powerset *}
traytel@55096
   757
traytel@55096
   758
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
traytel@55096
   759
  unfolding UNIV_bool by simp
traytel@55096
   760
traytel@55096
   761
lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
traytel@55096
   762
proof (induct rule: finite_induct)
traytel@55096
   763
  case empty 
traytel@55096
   764
    show ?case by auto
traytel@55096
   765
next
traytel@55096
   766
  case (insert x A)
traytel@55096
   767
  then have "inj_on (insert x) (Pow A)" 
traytel@55096
   768
    unfolding inj_on_def by (blast elim!: equalityE)
traytel@55096
   769
  then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A" 
traytel@55096
   770
    by (simp add: mult_2 card_image Pow_insert insert.hyps)
traytel@55096
   771
  then show ?case using insert
traytel@55096
   772
    apply (simp add: Pow_insert)
traytel@55096
   773
    apply (subst card_Un_disjoint, auto)
traytel@55096
   774
    done
traytel@55096
   775
qed
traytel@55096
   776
haftmann@57418
   777
haftmann@57418
   778
subsubsection {* Generalized sum over a set *}
haftmann@57418
   779
haftmann@57418
   780
lemma setsum_zero_power [simp]:
haftmann@57418
   781
  fixes c :: "nat \<Rightarrow> 'a::division_ring"
haftmann@57418
   782
  shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
haftmann@57418
   783
apply (cases "finite A")
haftmann@57418
   784
  by (induction A rule: finite_induct) auto
haftmann@57418
   785
haftmann@57418
   786
lemma setsum_zero_power' [simp]:
haftmann@57418
   787
  fixes c :: "nat \<Rightarrow> 'a::field"
haftmann@57418
   788
  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)"
haftmann@57418
   789
  using setsum_zero_power [of "\<lambda>i. c i / d i" A]
haftmann@57418
   790
  by auto
haftmann@57418
   791
haftmann@57418
   792
traytel@55096
   793
subsubsection {* Generalized product over a set *}
traytel@55096
   794
traytel@55096
   795
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
traytel@55096
   796
apply (erule finite_induct)
traytel@55096
   797
apply auto
traytel@55096
   798
done
traytel@55096
   799
haftmann@57418
   800
lemma setprod_power_distrib:
haftmann@57418
   801
  fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
haftmann@57418
   802
  shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
haftmann@57418
   803
proof (cases "finite A") 
haftmann@57418
   804
  case True then show ?thesis 
haftmann@57418
   805
    by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
haftmann@57418
   806
next
haftmann@57418
   807
  case False then show ?thesis 
haftmann@57418
   808
    by simp
haftmann@57418
   809
qed
haftmann@57418
   810
haftmann@58437
   811
lemma power_setsum:
haftmann@58437
   812
  "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
haftmann@58437
   813
  by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
haftmann@58437
   814
traytel@55096
   815
lemma setprod_gen_delta:
traytel@55096
   816
  assumes fS: "finite S"
traytel@55096
   817
  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)"
traytel@55096
   818
proof-
traytel@55096
   819
  let ?f = "(\<lambda>k. if k=a then b k else c)"
traytel@55096
   820
  {assume a: "a \<notin> S"
traytel@55096
   821
    hence "\<forall> k\<in> S. ?f k = c" by simp
traytel@55096
   822
    hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
traytel@55096
   823
  moreover 
traytel@55096
   824
  {assume a: "a \<in> S"
traytel@55096
   825
    let ?A = "S - {a}"
traytel@55096
   826
    let ?B = "{a}"
traytel@55096
   827
    have eq: "S = ?A \<union> ?B" using a by blast 
traytel@55096
   828
    have dj: "?A \<inter> ?B = {}" by simp
traytel@55096
   829
    from fS have fAB: "finite ?A" "finite ?B" by auto  
traytel@55096
   830
    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
haftmann@57418
   831
      apply (rule setprod.cong) by auto
traytel@55096
   832
    have cA: "card ?A = card S - 1" using fS a by auto
traytel@55096
   833
    have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
traytel@55096
   834
    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
haftmann@57418
   835
      using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
traytel@55096
   836
      by simp
traytel@55096
   837
    then have ?thesis using a cA
haftmann@57418
   838
      by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
traytel@55096
   839
  ultimately show ?thesis by blast
traytel@55096
   840
qed
traytel@55096
   841
haftmann@31155
   842
subsection {* Code generator tweak *}
haftmann@31155
   843
bulwahn@45231
   844
lemma power_power_power [code]:
haftmann@31155
   845
  "power = power.power (1::'a::{power}) (op *)"
haftmann@31155
   846
  unfolding power_def power.power_def ..
haftmann@31155
   847
haftmann@31155
   848
declare power.power.simps [code]
haftmann@31155
   849
haftmann@52435
   850
code_identifier
haftmann@52435
   851
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
   852
paulson@3390
   853
end
haftmann@49824
   854