src/HOL/Power.thy
author wenzelm
Wed Aug 10 22:05:36 2016 +0200 (2016-08-10)
changeset 63654 f90e3926e627
parent 63648 f9f3006a5579
child 63924 f91766530e13
permissions -rw-r--r--
misc tuning and modernization;
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
wenzelm@60758
     6
section \<open>Exponentiation\<close>
paulson@14348
     7
nipkow@15131
     8
theory Power
wenzelm@63654
     9
  imports Num
nipkow@15131
    10
begin
paulson@14348
    11
wenzelm@60758
    12
subsection \<open>Powers for Arbitrary Monoids\<close>
haftmann@30960
    13
haftmann@30996
    14
class power = one + times
haftmann@30960
    15
begin
haftmann@24996
    16
wenzelm@61955
    17
primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"  (infixr "^" 80)
wenzelm@63654
    18
  where
wenzelm@63654
    19
    power_0: "a ^ 0 = 1"
wenzelm@63654
    20
  | power_Suc: "a ^ Suc n = a * a ^ n"
paulson@14348
    21
haftmann@30996
    22
notation (latex output)
haftmann@30996
    23
  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30996
    24
wenzelm@60758
    25
text \<open>Special syntax for squares.\<close>
wenzelm@61955
    26
abbreviation power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999)
wenzelm@61955
    27
  where "x\<^sup>2 \<equiv> x ^ 2"
huffman@47192
    28
haftmann@30960
    29
end
paulson@14348
    30
haftmann@30996
    31
context monoid_mult
haftmann@30996
    32
begin
paulson@14348
    33
wenzelm@39438
    34
subclass power .
paulson@14348
    35
wenzelm@63654
    36
lemma power_one [simp]: "1 ^ n = 1"
huffman@30273
    37
  by (induct n) simp_all
paulson@14348
    38
wenzelm@63654
    39
lemma power_one_right [simp]: "a ^ 1 = a"
haftmann@30996
    40
  by simp
paulson@14348
    41
wenzelm@63654
    42
lemma power_Suc0_right [simp]: "a ^ Suc 0 = a"
lp15@59741
    43
  by simp
lp15@59741
    44
wenzelm@63654
    45
lemma power_commutes: "a ^ n * a = a * a ^ n"
haftmann@57512
    46
  by (induct n) (simp_all add: mult.assoc)
krauss@21199
    47
wenzelm@63654
    48
lemma power_Suc2: "a ^ Suc n = a ^ n * a"
huffman@30273
    49
  by (simp add: power_commutes)
huffman@28131
    50
wenzelm@63654
    51
lemma power_add: "a ^ (m + n) = a ^ m * a ^ n"
haftmann@30996
    52
  by (induct m) (simp_all add: algebra_simps)
paulson@14348
    53
wenzelm@63654
    54
lemma power_mult: "a ^ (m * n) = (a ^ m) ^ n"
huffman@30273
    55
  by (induct n) (simp_all add: power_add)
paulson@14348
    56
wenzelm@53015
    57
lemma power2_eq_square: "a\<^sup>2 = a * a"
huffman@47192
    58
  by (simp add: numeral_2_eq_2)
huffman@47192
    59
huffman@47192
    60
lemma power3_eq_cube: "a ^ 3 = a * a * a"
haftmann@57512
    61
  by (simp add: numeral_3_eq_3 mult.assoc)
huffman@47192
    62
wenzelm@63654
    63
lemma power_even_eq: "a ^ (2 * n) = (a ^ n)\<^sup>2"
haftmann@57512
    64
  by (subst mult.commute) (simp add: power_mult)
huffman@47192
    65
wenzelm@63654
    66
lemma power_odd_eq: "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
huffman@47192
    67
  by (simp add: power_even_eq)
huffman@47192
    68
wenzelm@63654
    69
lemma power_numeral_even: "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
wenzelm@63654
    70
  by (simp only: numeral_Bit0 power_add Let_def)
huffman@47255
    71
wenzelm@63654
    72
lemma power_numeral_odd: "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
wenzelm@63654
    73
  by (simp only: numeral_Bit1 One_nat_def add_Suc_right add_0_right
wenzelm@63654
    74
      power_Suc power_add Let_def mult.assoc)
huffman@47255
    75
wenzelm@63654
    76
lemma funpow_times_power: "(times x ^^ f x) = times (x ^ f x)"
haftmann@49824
    77
proof (induct "f x" arbitrary: f)
wenzelm@63654
    78
  case 0
wenzelm@63654
    79
  then show ?case by (simp add: fun_eq_iff)
haftmann@49824
    80
next
haftmann@49824
    81
  case (Suc n)
wenzelm@63040
    82
  define g where "g x = f x - 1" for x
haftmann@49824
    83
  with Suc have "n = g x" by simp
haftmann@49824
    84
  with Suc have "times x ^^ g x = times (x ^ g x)" by simp
haftmann@49824
    85
  moreover from Suc g_def have "f x = g x + 1" by simp
wenzelm@63654
    86
  ultimately show ?case
wenzelm@63654
    87
    by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
haftmann@49824
    88
qed
haftmann@49824
    89
immler@58656
    90
lemma power_commuting_commutes:
immler@58656
    91
  assumes "x * y = y * x"
immler@58656
    92
  shows "x ^ n * y = y * x ^n"
immler@58656
    93
proof (induct n)
wenzelm@63654
    94
  case 0
wenzelm@63654
    95
  then show ?case by simp
wenzelm@63654
    96
next
immler@58656
    97
  case (Suc n)
immler@58656
    98
  have "x ^ Suc n * y = x ^ n * y * x"
immler@58656
    99
    by (subst power_Suc2) (simp add: assms ac_simps)
immler@58656
   100
  also have "\<dots> = y * x ^ Suc n"
wenzelm@63654
   101
    by (simp only: Suc power_Suc2) (simp add: ac_simps)
immler@58656
   102
  finally show ?case .
wenzelm@63654
   103
qed
immler@58656
   104
wenzelm@63654
   105
lemma power_minus_mult: "0 < n \<Longrightarrow> a ^ (n - 1) * a = a ^ n"
nipkow@63648
   106
  by (simp add: power_commutes split: nat_diff_split)
haftmann@62347
   107
haftmann@30996
   108
end
haftmann@30996
   109
haftmann@30996
   110
context comm_monoid_mult
haftmann@30996
   111
begin
haftmann@30996
   112
wenzelm@63654
   113
lemma power_mult_distrib [field_simps]: "(a * b) ^ n = (a ^ n) * (b ^ n)"
haftmann@57514
   114
  by (induct n) (simp_all add: ac_simps)
paulson@14348
   115
haftmann@30996
   116
end
haftmann@30996
   117
wenzelm@63654
   118
text \<open>Extract constant factors from powers.\<close>
lp15@59741
   119
declare power_mult_distrib [where a = "numeral w" for w, simp]
lp15@59741
   120
declare power_mult_distrib [where b = "numeral w" for w, simp]
lp15@59741
   121
wenzelm@63654
   122
lemma power_add_numeral [simp]: "a^numeral m * a^numeral n = a^numeral (m + n)"
wenzelm@63654
   123
  for a :: "'a::monoid_mult"
lp15@60155
   124
  by (simp add: power_add [symmetric])
lp15@60155
   125
wenzelm@63654
   126
lemma power_add_numeral2 [simp]: "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
wenzelm@63654
   127
  for a :: "'a::monoid_mult"
lp15@60155
   128
  by (simp add: mult.assoc [symmetric])
lp15@60155
   129
wenzelm@63654
   130
lemma power_mult_numeral [simp]: "(a^numeral m)^numeral n = a^numeral (m * n)"
wenzelm@63654
   131
  for a :: "'a::monoid_mult"
lp15@60155
   132
  by (simp only: numeral_mult power_mult)
lp15@60155
   133
huffman@47191
   134
context semiring_numeral
huffman@47191
   135
begin
huffman@47191
   136
huffman@47191
   137
lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
huffman@47191
   138
  by (simp only: sqr_conv_mult numeral_mult)
huffman@47191
   139
huffman@47191
   140
lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
wenzelm@63654
   141
  by (induct l)
wenzelm@63654
   142
    (simp_all only: numeral_class.numeral.simps pow.simps
wenzelm@63654
   143
      numeral_sqr numeral_mult power_add power_one_right)
huffman@47191
   144
huffman@47191
   145
lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
huffman@47191
   146
  by (rule numeral_pow [symmetric])
huffman@47191
   147
huffman@47191
   148
end
huffman@47191
   149
haftmann@30996
   150
context semiring_1
haftmann@30996
   151
begin
haftmann@30996
   152
wenzelm@63654
   153
lemma of_nat_power [simp]: "of_nat (m ^ n) = of_nat m ^ n"
haftmann@63417
   154
  by (induct n) simp_all
haftmann@30996
   155
wenzelm@63654
   156
lemma zero_power: "0 < n \<Longrightarrow> 0 ^ n = 0"
haftmann@59009
   157
  by (cases n) simp_all
haftmann@59009
   158
wenzelm@63654
   159
lemma power_zero_numeral [simp]: "0 ^ numeral k = 0"
huffman@47209
   160
  by (simp add: numeral_eq_Suc)
huffman@47191
   161
wenzelm@53015
   162
lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
huffman@47192
   163
  by (rule power_zero_numeral)
huffman@47192
   164
wenzelm@53015
   165
lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
huffman@47192
   166
  by (rule power_one)
huffman@47192
   167
wenzelm@63654
   168
lemma power_0_Suc [simp]: "0 ^ Suc n = 0"
haftmann@60867
   169
  by simp
haftmann@60867
   170
wenzelm@63654
   171
text \<open>It looks plausible as a simprule, but its effect can be strange.\<close>
wenzelm@63654
   172
lemma power_0_left: "0 ^ n = (if n = 0 then 1 else 0)"
haftmann@60867
   173
  by (cases n) simp_all
haftmann@60867
   174
haftmann@30996
   175
end
haftmann@30996
   176
haftmann@30996
   177
context comm_semiring_1
haftmann@30996
   178
begin
haftmann@30996
   179
wenzelm@63654
   180
text \<open>The divides relation.\<close>
haftmann@30996
   181
haftmann@30996
   182
lemma le_imp_power_dvd:
wenzelm@63654
   183
  assumes "m \<le> n"
wenzelm@63654
   184
  shows "a ^ m dvd a ^ n"
haftmann@30996
   185
proof
wenzelm@63654
   186
  from assms have "a ^ n = a ^ (m + (n - m))" by simp
wenzelm@63654
   187
  also have "\<dots> = a ^ m * a ^ (n - m)" by (rule power_add)
haftmann@30996
   188
  finally show "a ^ n = a ^ m * a ^ (n - m)" .
haftmann@30996
   189
qed
haftmann@30996
   190
wenzelm@63654
   191
lemma power_le_dvd: "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
haftmann@30996
   192
  by (rule dvd_trans [OF le_imp_power_dvd])
haftmann@30996
   193
wenzelm@63654
   194
lemma dvd_power_same: "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
haftmann@30996
   195
  by (induct n) (auto simp add: mult_dvd_mono)
haftmann@30996
   196
wenzelm@63654
   197
lemma dvd_power_le: "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
haftmann@30996
   198
  by (rule power_le_dvd [OF dvd_power_same])
paulson@14348
   199
haftmann@30996
   200
lemma dvd_power [simp]:
wenzelm@63654
   201
  fixes n :: nat
wenzelm@63654
   202
  assumes "n > 0 \<or> x = 1"
haftmann@30996
   203
  shows "x dvd (x ^ n)"
wenzelm@63654
   204
  using assms
wenzelm@63654
   205
proof
haftmann@30996
   206
  assume "0 < n"
haftmann@30996
   207
  then have "x ^ n = x ^ Suc (n - 1)" by simp
haftmann@30996
   208
  then show "x dvd (x ^ n)" by simp
haftmann@30996
   209
next
haftmann@30996
   210
  assume "x = 1"
haftmann@30996
   211
  then show "x dvd (x ^ n)" by simp
haftmann@30996
   212
qed
haftmann@30996
   213
haftmann@30996
   214
end
haftmann@30996
   215
haftmann@62481
   216
context semiring_1_no_zero_divisors
haftmann@60867
   217
begin
haftmann@60867
   218
haftmann@60867
   219
subclass power .
haftmann@60867
   220
wenzelm@63654
   221
lemma power_eq_0_iff [simp]: "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
haftmann@60867
   222
  by (induct n) auto
haftmann@60867
   223
wenzelm@63654
   224
lemma power_not_zero: "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
haftmann@60867
   225
  by (induct n) auto
haftmann@60867
   226
wenzelm@63654
   227
lemma zero_eq_power2 [simp]: "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
haftmann@60867
   228
  unfolding power2_eq_square by simp
haftmann@60867
   229
haftmann@60867
   230
end
haftmann@60867
   231
haftmann@30996
   232
context ring_1
haftmann@30996
   233
begin
haftmann@30996
   234
wenzelm@63654
   235
lemma power_minus: "(- a) ^ n = (- 1) ^ n * a ^ n"
haftmann@30996
   236
proof (induct n)
wenzelm@63654
   237
  case 0
wenzelm@63654
   238
  show ?case by simp
haftmann@30996
   239
next
wenzelm@63654
   240
  case (Suc n)
wenzelm@63654
   241
  then show ?case
haftmann@57512
   242
    by (simp del: power_Suc add: power_Suc2 mult.assoc)
haftmann@30996
   243
qed
haftmann@30996
   244
eberlm@61531
   245
lemma power_minus': "NO_MATCH 1 x \<Longrightarrow> (-x) ^ n = (-1)^n * x ^ n"
eberlm@61531
   246
  by (rule power_minus)
eberlm@61531
   247
wenzelm@63654
   248
lemma power_minus_Bit0: "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
huffman@47191
   249
  by (induct k, simp_all only: numeral_class.numeral.simps power_add
huffman@47191
   250
    power_one_right mult_minus_left mult_minus_right minus_minus)
huffman@47191
   251
wenzelm@63654
   252
lemma power_minus_Bit1: "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
huffman@47220
   253
  by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
huffman@47191
   254
wenzelm@63654
   255
lemma power2_minus [simp]: "(- a)\<^sup>2 = a\<^sup>2"
haftmann@60867
   256
  by (fact power_minus_Bit0)
huffman@47192
   257
wenzelm@63654
   258
lemma power_minus1_even [simp]: "(- 1) ^ (2*n) = 1"
huffman@47192
   259
proof (induct n)
wenzelm@63654
   260
  case 0
wenzelm@63654
   261
  show ?case by simp
huffman@47192
   262
next
wenzelm@63654
   263
  case (Suc n)
wenzelm@63654
   264
  then show ?case by (simp add: power_add power2_eq_square)
huffman@47192
   265
qed
huffman@47192
   266
wenzelm@63654
   267
lemma power_minus1_odd: "(- 1) ^ Suc (2*n) = -1"
huffman@47192
   268
  by simp
lp15@61649
   269
wenzelm@63654
   270
lemma power_minus_even [simp]: "(-a) ^ (2*n) = a ^ (2*n)"
huffman@47192
   271
  by (simp add: power_minus [of a])
huffman@47192
   272
huffman@47192
   273
end
huffman@47192
   274
huffman@47192
   275
context ring_1_no_zero_divisors
huffman@47192
   276
begin
huffman@47192
   277
wenzelm@63654
   278
lemma power2_eq_1_iff: "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
haftmann@60867
   279
  using square_eq_1_iff [of a] by (simp add: power2_eq_square)
huffman@47192
   280
huffman@47192
   281
end
huffman@47192
   282
huffman@47192
   283
context idom
huffman@47192
   284
begin
huffman@47192
   285
wenzelm@53015
   286
lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
huffman@47192
   287
  unfolding power2_eq_square by (rule square_eq_iff)
huffman@47192
   288
huffman@47192
   289
end
huffman@47192
   290
haftmann@60867
   291
context algebraic_semidom
haftmann@60867
   292
begin
haftmann@60867
   293
wenzelm@63654
   294
lemma div_power: "b dvd a \<Longrightarrow> (a div b) ^ n = a ^ n div b ^ n"
wenzelm@63654
   295
  by (induct n) (simp_all add: div_mult_div_if_dvd dvd_power_same)
haftmann@60867
   296
wenzelm@63654
   297
lemma is_unit_power_iff: "is_unit (a ^ n) \<longleftrightarrow> is_unit a \<or> n = 0"
haftmann@62366
   298
  by (induct n) (auto simp add: is_unit_mult_iff)
haftmann@62366
   299
haftmann@60867
   300
end
haftmann@60867
   301
haftmann@60685
   302
context normalization_semidom
haftmann@60685
   303
begin
haftmann@60685
   304
wenzelm@63654
   305
lemma normalize_power: "normalize (a ^ n) = normalize a ^ n"
haftmann@60685
   306
  by (induct n) (simp_all add: normalize_mult)
haftmann@60685
   307
wenzelm@63654
   308
lemma unit_factor_power: "unit_factor (a ^ n) = unit_factor a ^ n"
haftmann@60685
   309
  by (induct n) (simp_all add: unit_factor_mult)
haftmann@60685
   310
haftmann@60685
   311
end
haftmann@60685
   312
huffman@47192
   313
context division_ring
huffman@47192
   314
begin
huffman@47192
   315
wenzelm@63654
   316
text \<open>Perhaps these should be simprules.\<close>
wenzelm@63654
   317
lemma power_inverse [field_simps, divide_simps]: "inverse a ^ n = inverse (a ^ n)"
haftmann@60867
   318
proof (cases "a = 0")
wenzelm@63654
   319
  case True
wenzelm@63654
   320
  then show ?thesis by (simp add: power_0_left)
haftmann@60867
   321
next
wenzelm@63654
   322
  case False
wenzelm@63654
   323
  then have "inverse (a ^ n) = inverse a ^ n"
haftmann@60867
   324
    by (induct n) (simp_all add: nonzero_inverse_mult_distrib power_commutes)
haftmann@60867
   325
  then show ?thesis by simp
haftmann@60867
   326
qed
huffman@47192
   327
wenzelm@63654
   328
lemma power_one_over [field_simps, divide_simps]: "(1 / a) ^ n = 1 / a ^ n"
haftmann@60867
   329
  using power_inverse [of a] by (simp add: divide_inverse)
haftmann@60867
   330
lp15@61649
   331
end
huffman@47192
   332
huffman@47192
   333
context field
huffman@47192
   334
begin
huffman@47192
   335
haftmann@60867
   336
lemma power_diff:
wenzelm@63654
   337
  assumes "a \<noteq> 0"
haftmann@60867
   338
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
wenzelm@63654
   339
  by (induct m n rule: diff_induct) (simp_all add: assms power_not_zero)
huffman@47192
   340
wenzelm@63654
   341
lemma power_divide [field_simps, divide_simps]: "(a / b) ^ n = a ^ n / b ^ n"
haftmann@60867
   342
  by (induct n) simp_all
haftmann@60867
   343
huffman@47192
   344
end
huffman@47192
   345
huffman@47192
   346
wenzelm@60758
   347
subsection \<open>Exponentiation on ordered types\<close>
huffman@47192
   348
haftmann@35028
   349
context linordered_semidom
haftmann@30996
   350
begin
haftmann@30996
   351
wenzelm@63654
   352
lemma zero_less_power [simp]: "0 < a \<Longrightarrow> 0 < a ^ n"
nipkow@56544
   353
  by (induct n) simp_all
haftmann@30996
   354
wenzelm@63654
   355
lemma zero_le_power [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
nipkow@56536
   356
  by (induct n) simp_all
paulson@14348
   357
wenzelm@63654
   358
lemma power_mono: "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
huffman@47241
   359
  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
huffman@47241
   360
huffman@47241
   361
lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
huffman@47241
   362
  using power_mono [of 1 a n] by simp
huffman@47241
   363
wenzelm@63654
   364
lemma power_le_one: "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ n \<le> 1"
huffman@47241
   365
  using power_mono [of a 1 n] by simp
paulson@14348
   366
paulson@14348
   367
lemma power_gt1_lemma:
haftmann@30996
   368
  assumes gt1: "1 < a"
haftmann@30996
   369
  shows "1 < a * a ^ n"
paulson@14348
   370
proof -
haftmann@30996
   371
  from gt1 have "0 \<le> a"
haftmann@30996
   372
    by (fact order_trans [OF zero_le_one less_imp_le])
wenzelm@63654
   373
  from gt1 have "1 * 1 < a * 1" by simp
wenzelm@63654
   374
  also from gt1 have "\<dots> \<le> a * a ^ n"
wenzelm@63654
   375
    by (simp only: mult_mono \<open>0 \<le> a\<close> one_le_power order_less_imp_le zero_le_one order_refl)
wenzelm@14577
   376
  finally show ?thesis by simp
paulson@14348
   377
qed
paulson@14348
   378
wenzelm@63654
   379
lemma power_gt1: "1 < a \<Longrightarrow> 1 < a ^ Suc n"
haftmann@30996
   380
  by (simp add: power_gt1_lemma)
huffman@24376
   381
wenzelm@63654
   382
lemma one_less_power [simp]: "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
haftmann@30996
   383
  by (cases n) (simp_all add: power_gt1_lemma)
paulson@14348
   384
paulson@14348
   385
lemma power_le_imp_le_exp:
haftmann@30996
   386
  assumes gt1: "1 < a"
haftmann@30996
   387
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
haftmann@30996
   388
proof (induct m arbitrary: n)
paulson@14348
   389
  case 0
wenzelm@14577
   390
  show ?case by simp
paulson@14348
   391
next
paulson@14348
   392
  case (Suc m)
wenzelm@14577
   393
  show ?case
wenzelm@14577
   394
  proof (cases n)
wenzelm@14577
   395
    case 0
wenzelm@63654
   396
    with Suc have "a * a ^ m \<le> 1" by simp
wenzelm@14577
   397
    with gt1 show ?thesis
wenzelm@63654
   398
      by (force simp only: power_gt1_lemma not_less [symmetric])
wenzelm@14577
   399
  next
wenzelm@14577
   400
    case (Suc n)
haftmann@30996
   401
    with Suc.prems Suc.hyps show ?thesis
wenzelm@63654
   402
      by (force dest: mult_left_le_imp_le simp add: less_trans [OF zero_less_one gt1])
wenzelm@14577
   403
  qed
paulson@14348
   404
qed
paulson@14348
   405
wenzelm@63654
   406
lemma of_nat_zero_less_power_iff [simp]: "of_nat x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
lp15@61649
   407
  by (induct n) auto
lp15@61649
   408
wenzelm@63654
   409
text \<open>Surely we can strengthen this? It holds for \<open>0<a<1\<close> too.\<close>
wenzelm@63654
   410
lemma power_inject_exp [simp]: "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
wenzelm@14577
   411
  by (force simp add: order_antisym power_le_imp_le_exp)
paulson@14348
   412
wenzelm@63654
   413
text \<open>
wenzelm@63654
   414
  Can relax the first premise to @{term "0<a"} in the case of the
wenzelm@63654
   415
  natural numbers.
wenzelm@63654
   416
\<close>
wenzelm@63654
   417
lemma power_less_imp_less_exp: "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
wenzelm@63654
   418
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"] power_le_imp_le_exp)
paulson@14348
   419
wenzelm@63654
   420
lemma power_strict_mono [rule_format]: "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
wenzelm@63654
   421
  by (induct n) (auto simp: mult_strict_mono le_less_trans [of 0 a b])
paulson@14348
   422
wenzelm@61799
   423
text\<open>Lemma for \<open>power_strict_decreasing\<close>\<close>
wenzelm@63654
   424
lemma power_Suc_less: "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
wenzelm@63654
   425
  by (induct n) (auto simp: mult_strict_left_mono)
paulson@14348
   426
wenzelm@63654
   427
lemma power_strict_decreasing [rule_format]: "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
haftmann@30996
   428
proof (induct N)
wenzelm@63654
   429
  case 0
wenzelm@63654
   430
  then show ?case by simp
haftmann@30996
   431
next
wenzelm@63654
   432
  case (Suc N)
wenzelm@63654
   433
  then show ?case
wenzelm@63654
   434
    apply (auto simp add: power_Suc_less less_Suc_eq)
wenzelm@63654
   435
    apply (subgoal_tac "a * a^N < 1 * a^n")
wenzelm@63654
   436
     apply simp
wenzelm@63654
   437
    apply (rule mult_strict_mono)
wenzelm@63654
   438
       apply auto
wenzelm@63654
   439
    done
haftmann@30996
   440
qed
paulson@14348
   441
wenzelm@63654
   442
text \<open>Proof resembles that of \<open>power_strict_decreasing\<close>.\<close>
wenzelm@63654
   443
lemma power_decreasing: "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ N \<le> a ^ n"
haftmann@30996
   444
proof (induct N)
wenzelm@63654
   445
  case 0
wenzelm@63654
   446
  then show ?case by simp
haftmann@30996
   447
next
wenzelm@63654
   448
  case (Suc N)
wenzelm@63654
   449
  then show ?case
wenzelm@63654
   450
    apply (auto simp add: le_Suc_eq)
wenzelm@63654
   451
    apply (subgoal_tac "a * a^N \<le> 1 * a^n")
wenzelm@63654
   452
     apply simp
wenzelm@63654
   453
    apply (rule mult_mono)
wenzelm@63654
   454
       apply auto
wenzelm@63654
   455
    done
haftmann@30996
   456
qed
paulson@14348
   457
wenzelm@63654
   458
lemma power_Suc_less_one: "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
haftmann@30996
   459
  using power_strict_decreasing [of 0 "Suc n" a] by simp
paulson@14348
   460
wenzelm@63654
   461
text \<open>Proof again resembles that of \<open>power_strict_decreasing\<close>.\<close>
wenzelm@63654
   462
lemma power_increasing: "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
haftmann@30996
   463
proof (induct N)
wenzelm@63654
   464
  case 0
wenzelm@63654
   465
  then show ?case by simp
haftmann@30996
   466
next
wenzelm@63654
   467
  case (Suc N)
wenzelm@63654
   468
  then show ?case
wenzelm@63654
   469
    apply (auto simp add: le_Suc_eq)
wenzelm@63654
   470
    apply (subgoal_tac "1 * a^n \<le> a * a^N")
wenzelm@63654
   471
     apply simp
wenzelm@63654
   472
    apply (rule mult_mono)
wenzelm@63654
   473
       apply (auto simp add: order_trans [OF zero_le_one])
wenzelm@63654
   474
    done
haftmann@30996
   475
qed
paulson@14348
   476
wenzelm@63654
   477
text \<open>Lemma for \<open>power_strict_increasing\<close>.\<close>
wenzelm@63654
   478
lemma power_less_power_Suc: "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
wenzelm@63654
   479
  by (induct n) (auto simp: mult_strict_left_mono less_trans [OF zero_less_one])
paulson@14348
   480
wenzelm@63654
   481
lemma power_strict_increasing: "n < N \<Longrightarrow> 1 < a \<Longrightarrow> a ^ n < a ^ N"
haftmann@30996
   482
proof (induct N)
wenzelm@63654
   483
  case 0
wenzelm@63654
   484
  then show ?case by simp
haftmann@30996
   485
next
wenzelm@63654
   486
  case (Suc N)
wenzelm@63654
   487
  then show ?case
wenzelm@63654
   488
    apply (auto simp add: power_less_power_Suc less_Suc_eq)
wenzelm@63654
   489
    apply (subgoal_tac "1 * a^n < a * a^N")
wenzelm@63654
   490
     apply simp
wenzelm@63654
   491
    apply (rule mult_strict_mono)
wenzelm@63654
   492
    apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
wenzelm@63654
   493
    done
haftmann@30996
   494
qed
paulson@14348
   495
wenzelm@63654
   496
lemma power_increasing_iff [simp]: "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
haftmann@30996
   497
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
paulson@15066
   498
wenzelm@63654
   499
lemma power_strict_increasing_iff [simp]: "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
wenzelm@63654
   500
  by (blast intro: power_less_imp_less_exp power_strict_increasing)
paulson@15066
   501
paulson@14348
   502
lemma power_le_imp_le_base:
haftmann@30996
   503
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
wenzelm@63654
   504
    and "0 \<le> b"
haftmann@30996
   505
  shows "a \<le> b"
nipkow@25134
   506
proof (rule ccontr)
wenzelm@63654
   507
  assume "\<not> ?thesis"
nipkow@25134
   508
  then have "b < a" by (simp only: linorder_not_le)
nipkow@25134
   509
  then have "b ^ Suc n < a ^ Suc n"
wenzelm@63654
   510
    by (simp only: assms(2) power_strict_mono)
wenzelm@63654
   511
  with le show False
nipkow@25134
   512
    by (simp add: linorder_not_less [symmetric])
nipkow@25134
   513
qed
wenzelm@14577
   514
huffman@22853
   515
lemma power_less_imp_less_base:
huffman@22853
   516
  assumes less: "a ^ n < b ^ n"
huffman@22853
   517
  assumes nonneg: "0 \<le> b"
huffman@22853
   518
  shows "a < b"
huffman@22853
   519
proof (rule contrapos_pp [OF less])
wenzelm@63654
   520
  assume "\<not> ?thesis"
wenzelm@63654
   521
  then have "b \<le> a" by (simp only: linorder_not_less)
wenzelm@63654
   522
  from this nonneg have "b ^ n \<le> a ^ n" by (rule power_mono)
wenzelm@63654
   523
  then show "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
huffman@22853
   524
qed
huffman@22853
   525
wenzelm@63654
   526
lemma power_inject_base: "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
wenzelm@63654
   527
  by (blast intro: power_le_imp_le_base antisym eq_refl sym)
paulson@14348
   528
wenzelm@63654
   529
lemma power_eq_imp_eq_base: "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
haftmann@30996
   530
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
huffman@22955
   531
wenzelm@63654
   532
lemma power_eq_iff_eq_base: "0 < n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a ^ n = b ^ n \<longleftrightarrow> a = b"
haftmann@62347
   533
  using power_eq_imp_eq_base [of a n b] by auto
haftmann@62347
   534
wenzelm@63654
   535
lemma power2_le_imp_le: "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
huffman@47192
   536
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
huffman@47192
   537
wenzelm@63654
   538
lemma power2_less_imp_less: "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
huffman@47192
   539
  by (rule power_less_imp_less_base)
huffman@47192
   540
wenzelm@63654
   541
lemma power2_eq_imp_eq: "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
huffman@47192
   542
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
huffman@47192
   543
wenzelm@63654
   544
lemma power_Suc_le_self: "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ Suc n \<le> a"
haftmann@62347
   545
  using power_decreasing [of 1 "Suc n" a] by simp
haftmann@62347
   546
huffman@47192
   547
end
huffman@47192
   548
huffman@47192
   549
context linordered_ring_strict
huffman@47192
   550
begin
huffman@47192
   551
wenzelm@63654
   552
lemma sum_squares_eq_zero_iff: "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   553
  by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   554
wenzelm@63654
   555
lemma sum_squares_le_zero_iff: "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   556
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
huffman@47192
   557
wenzelm@63654
   558
lemma sum_squares_gt_zero_iff: "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   559
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
huffman@47192
   560
haftmann@30996
   561
end
haftmann@30996
   562
haftmann@35028
   563
context linordered_idom
haftmann@30996
   564
begin
huffman@29978
   565
wenzelm@61944
   566
lemma power_abs: "\<bar>a ^ n\<bar> = \<bar>a\<bar> ^ n"
haftmann@30996
   567
  by (induct n) (auto simp add: abs_mult)
haftmann@30996
   568
wenzelm@61944
   569
lemma abs_power_minus [simp]: "\<bar>(-a) ^ n\<bar> = \<bar>a ^ n\<bar>"
huffman@35216
   570
  by (simp add: power_abs)
haftmann@30996
   571
wenzelm@61944
   572
lemma zero_less_power_abs_iff [simp]: "0 < \<bar>a\<bar> ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
haftmann@30996
   573
proof (induct n)
wenzelm@63654
   574
  case 0
wenzelm@63654
   575
  show ?case by simp
haftmann@30996
   576
next
wenzelm@63654
   577
  case Suc
wenzelm@63654
   578
  then show ?case by (auto simp: zero_less_mult_iff)
huffman@29978
   579
qed
huffman@29978
   580
wenzelm@61944
   581
lemma zero_le_power_abs [simp]: "0 \<le> \<bar>a\<bar> ^ n"
haftmann@30996
   582
  by (rule zero_le_power [OF abs_ge_zero])
haftmann@30996
   583
wenzelm@63654
   584
lemma zero_le_power2 [simp]: "0 \<le> a\<^sup>2"
huffman@47192
   585
  by (simp add: power2_eq_square)
huffman@47192
   586
wenzelm@63654
   587
lemma zero_less_power2 [simp]: "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
huffman@47192
   588
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
huffman@47192
   589
wenzelm@63654
   590
lemma power2_less_0 [simp]: "\<not> a\<^sup>2 < 0"
huffman@47192
   591
  by (force simp add: power2_eq_square mult_less_0_iff)
huffman@47192
   592
wenzelm@63654
   593
lemma power2_less_eq_zero_iff [simp]: "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
haftmann@58787
   594
  by (simp add: le_less)
haftmann@58787
   595
wenzelm@61944
   596
lemma abs_power2 [simp]: "\<bar>a\<^sup>2\<bar> = a\<^sup>2"
haftmann@63417
   597
  by (simp add: power2_eq_square)
huffman@47192
   598
wenzelm@61944
   599
lemma power2_abs [simp]: "\<bar>a\<bar>\<^sup>2 = a\<^sup>2"
haftmann@63417
   600
  by (simp add: power2_eq_square)
huffman@47192
   601
wenzelm@63654
   602
lemma odd_power_less_zero: "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
huffman@47192
   603
proof (induct n)
huffman@47192
   604
  case 0
huffman@47192
   605
  then show ?case by simp
huffman@47192
   606
next
huffman@47192
   607
  case (Suc n)
huffman@47192
   608
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
haftmann@57514
   609
    by (simp add: ac_simps power_add power2_eq_square)
wenzelm@63654
   610
  then show ?case
huffman@47192
   611
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
huffman@47192
   612
qed
haftmann@30996
   613
wenzelm@63654
   614
lemma odd_0_le_power_imp_0_le: "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
huffman@47192
   615
  using odd_power_less_zero [of a n]
wenzelm@63654
   616
  by (force simp add: linorder_not_less [symmetric])
huffman@47192
   617
wenzelm@63654
   618
lemma zero_le_even_power'[simp]: "0 \<le> a ^ (2*n)"
huffman@47192
   619
proof (induct n)
huffman@47192
   620
  case 0
wenzelm@63654
   621
  show ?case by simp
huffman@47192
   622
next
huffman@47192
   623
  case (Suc n)
wenzelm@63654
   624
  have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
wenzelm@63654
   625
    by (simp add: ac_simps power_add power2_eq_square)
wenzelm@63654
   626
  then show ?case
wenzelm@63654
   627
    by (simp add: Suc zero_le_mult_iff)
huffman@47192
   628
qed
haftmann@30996
   629
wenzelm@63654
   630
lemma sum_power2_ge_zero: "0 \<le> x\<^sup>2 + y\<^sup>2"
huffman@47192
   631
  by (intro add_nonneg_nonneg zero_le_power2)
huffman@47192
   632
wenzelm@63654
   633
lemma not_sum_power2_lt_zero: "\<not> x\<^sup>2 + y\<^sup>2 < 0"
huffman@47192
   634
  unfolding not_less by (rule sum_power2_ge_zero)
huffman@47192
   635
wenzelm@63654
   636
lemma sum_power2_eq_zero_iff: "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   637
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   638
wenzelm@63654
   639
lemma sum_power2_le_zero_iff: "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   640
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
huffman@47192
   641
wenzelm@63654
   642
lemma sum_power2_gt_zero_iff: "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   643
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
haftmann@30996
   644
wenzelm@63654
   645
lemma abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
wenzelm@63654
   646
  (is "?lhs \<longleftrightarrow> ?rhs")
lp15@59865
   647
proof
wenzelm@63654
   648
  assume ?lhs
wenzelm@63654
   649
  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono) simp
wenzelm@63654
   650
  then show ?rhs by simp
lp15@59865
   651
next
wenzelm@63654
   652
  assume ?rhs
wenzelm@63654
   653
  then show ?lhs
lp15@59865
   654
    by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
lp15@59865
   655
qed
lp15@59865
   656
wenzelm@61944
   657
lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> \<bar>x\<bar> \<le> 1"
wenzelm@63654
   658
  using abs_le_square_iff [of x 1] by simp
lp15@59865
   659
wenzelm@61944
   660
lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
lp15@59865
   661
  by (auto simp add: abs_if power2_eq_1_iff)
lp15@61649
   662
wenzelm@61944
   663
lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> \<bar>x\<bar> < 1"
wenzelm@63654
   664
  using  abs_square_eq_1 [of x] abs_square_le_1 [of x] by (auto simp add: le_less)
lp15@59865
   665
haftmann@30996
   666
end
haftmann@30996
   667
huffman@29978
   668
wenzelm@60758
   669
subsection \<open>Miscellaneous rules\<close>
paulson@14348
   670
wenzelm@63654
   671
lemma (in linordered_semidom) self_le_power: "1 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a ^ n"
haftmann@60867
   672
  using power_increasing [of 1 n a] power_one_right [of a] by auto
lp15@55718
   673
wenzelm@63654
   674
lemma (in power) power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
huffman@47255
   675
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
   676
wenzelm@63654
   677
lemma (in comm_semiring_1) power2_sum: "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
huffman@47192
   678
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   679
wenzelm@63654
   680
context comm_ring_1
wenzelm@63654
   681
begin
wenzelm@63654
   682
wenzelm@63654
   683
lemma power2_diff: "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
haftmann@58787
   684
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   685
wenzelm@63654
   686
lemma power2_commute: "(x - y)\<^sup>2 = (y - x)\<^sup>2"
lp15@60974
   687
  by (simp add: algebra_simps power2_eq_square)
lp15@60974
   688
wenzelm@63654
   689
lemma minus_power_mult_self: "(- a) ^ n * (- a) ^ n = a ^ (2 * n)"
wenzelm@63654
   690
  by (simp add: power_mult_distrib [symmetric])
wenzelm@63654
   691
    (simp add: power2_eq_square [symmetric] power_mult [symmetric])
wenzelm@63654
   692
wenzelm@63654
   693
lemma minus_one_mult_self [simp]: "(- 1) ^ n * (- 1) ^ n = 1"
haftmann@63417
   694
  using minus_power_mult_self [of 1 n] by simp
haftmann@63417
   695
wenzelm@63654
   696
lemma left_minus_one_mult_self [simp]: "(- 1) ^ n * ((- 1) ^ n * a) = a"
haftmann@63417
   697
  by (simp add: mult.assoc [symmetric])
haftmann@63417
   698
wenzelm@63654
   699
end
wenzelm@63654
   700
wenzelm@60758
   701
text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
huffman@47255
   702
huffman@47255
   703
lemmas zero_compare_simps =
wenzelm@63654
   704
  add_strict_increasing add_strict_increasing2 add_increasing
wenzelm@63654
   705
  zero_le_mult_iff zero_le_divide_iff
wenzelm@63654
   706
  zero_less_mult_iff zero_less_divide_iff
wenzelm@63654
   707
  mult_le_0_iff divide_le_0_iff
wenzelm@63654
   708
  mult_less_0_iff divide_less_0_iff
wenzelm@63654
   709
  zero_le_power2 power2_less_0
huffman@47255
   710
nipkow@30313
   711
wenzelm@60758
   712
subsection \<open>Exponentiation for the Natural Numbers\<close>
wenzelm@14577
   713
wenzelm@63654
   714
lemma nat_one_le_power [simp]: "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
haftmann@30996
   715
  by (rule one_le_power [of i n, unfolded One_nat_def])
huffman@23305
   716
wenzelm@63654
   717
lemma nat_zero_less_power_iff [simp]: "x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
wenzelm@63654
   718
  for x :: nat
haftmann@30996
   719
  by (induct n) auto
paulson@14348
   720
wenzelm@63654
   721
lemma nat_power_eq_Suc_0_iff [simp]: "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
haftmann@30996
   722
  by (induct m) auto
nipkow@30056
   723
wenzelm@63654
   724
lemma power_Suc_0 [simp]: "Suc 0 ^ n = Suc 0"
haftmann@30996
   725
  by simp
nipkow@30056
   726
wenzelm@63654
   727
text \<open>
wenzelm@63654
   728
  Valid for the naturals, but what if \<open>0 < i < 1\<close>? Premises cannot be
wenzelm@63654
   729
  weakened: consider the case where \<open>i = 0\<close>, \<open>m = 1\<close> and \<open>n = 0\<close>.
wenzelm@63654
   730
\<close>
wenzelm@63654
   731
haftmann@21413
   732
lemma nat_power_less_imp_less:
wenzelm@63654
   733
  fixes i :: nat
wenzelm@63654
   734
  assumes nonneg: "0 < i"
haftmann@30996
   735
  assumes less: "i ^ m < i ^ n"
haftmann@21413
   736
  shows "m < n"
haftmann@21413
   737
proof (cases "i = 1")
wenzelm@63654
   738
  case True
wenzelm@63654
   739
  with less power_one [where 'a = nat] show ?thesis by simp
haftmann@21413
   740
next
wenzelm@63654
   741
  case False
wenzelm@63654
   742
  with nonneg have "1 < i" by auto
haftmann@21413
   743
  from power_strict_increasing_iff [OF this] less show ?thesis ..
haftmann@21413
   744
qed
paulson@14348
   745
wenzelm@63654
   746
lemma power_dvd_imp_le: "i ^ m dvd i ^ n \<Longrightarrow> 1 < i \<Longrightarrow> m \<le> n"
wenzelm@63654
   747
  for i m n :: nat
wenzelm@63654
   748
  apply (rule power_le_imp_le_exp)
wenzelm@63654
   749
   apply assumption
wenzelm@63654
   750
  apply (erule dvd_imp_le)
wenzelm@63654
   751
  apply simp
haftmann@33274
   752
  done
haftmann@33274
   753
wenzelm@63654
   754
lemma power2_nat_le_eq_le: "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
wenzelm@63654
   755
  for m n :: nat
haftmann@51263
   756
  by (auto intro: power2_le_imp_le power_mono)
haftmann@51263
   757
haftmann@51263
   758
lemma power2_nat_le_imp_le:
haftmann@51263
   759
  fixes m n :: nat
wenzelm@53015
   760
  assumes "m\<^sup>2 \<le> n"
haftmann@51263
   761
  shows "m \<le> n"
haftmann@54249
   762
proof (cases m)
wenzelm@63654
   763
  case 0
wenzelm@63654
   764
  then show ?thesis by simp
haftmann@54249
   765
next
haftmann@54249
   766
  case (Suc k)
haftmann@54249
   767
  show ?thesis
haftmann@54249
   768
  proof (rule ccontr)
wenzelm@63654
   769
    assume "\<not> ?thesis"
haftmann@54249
   770
    then have "n < m" by simp
haftmann@54249
   771
    with assms Suc show False
haftmann@60867
   772
      by (simp add: power2_eq_square)
haftmann@54249
   773
  qed
haftmann@54249
   774
qed
haftmann@51263
   775
wenzelm@63654
   776
wenzelm@60758
   777
subsubsection \<open>Cardinality of the Powerset\<close>
traytel@55096
   778
traytel@55096
   779
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
traytel@55096
   780
  unfolding UNIV_bool by simp
traytel@55096
   781
traytel@55096
   782
lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
traytel@55096
   783
proof (induct rule: finite_induct)
lp15@61649
   784
  case empty
wenzelm@63654
   785
  show ?case by auto
traytel@55096
   786
next
traytel@55096
   787
  case (insert x A)
lp15@61649
   788
  then have "inj_on (insert x) (Pow A)"
traytel@55096
   789
    unfolding inj_on_def by (blast elim!: equalityE)
lp15@61649
   790
  then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A"
traytel@55096
   791
    by (simp add: mult_2 card_image Pow_insert insert.hyps)
wenzelm@63654
   792
  with insert show ?case
traytel@55096
   793
    apply (simp add: Pow_insert)
wenzelm@63654
   794
    apply (subst card_Un_disjoint)
wenzelm@63654
   795
       apply auto
traytel@55096
   796
    done
traytel@55096
   797
qed
traytel@55096
   798
haftmann@57418
   799
wenzelm@60758
   800
subsection \<open>Code generator tweak\<close>
haftmann@31155
   801
haftmann@52435
   802
code_identifier
haftmann@52435
   803
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
   804
paulson@3390
   805
end