src/HOL/Power.thy
author hoelzl
Fri Feb 19 13:40:50 2016 +0100 (2016-02-19)
changeset 62378 85ed00c1fe7c
parent 62366 95c6cf433c91
child 62481 b5d8e57826df
permissions -rw-r--r--
generalize more theorems to support enat and ennreal
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
traytel@55096
     9
imports Num Equiv_Relations
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@61955
    18
where
wenzelm@61955
    19
  power_0: "a ^ 0 = 1"
wenzelm@61955
    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
haftmann@30996
    36
lemma power_one [simp]:
haftmann@30996
    37
  "1 ^ n = 1"
huffman@30273
    38
  by (induct n) simp_all
paulson@14348
    39
haftmann@30996
    40
lemma power_one_right [simp]:
haftmann@31001
    41
  "a ^ 1 = a"
haftmann@30996
    42
  by simp
paulson@14348
    43
lp15@59741
    44
lemma power_Suc0_right [simp]:
lp15@59741
    45
  "a ^ Suc 0 = a"
lp15@59741
    46
  by simp
lp15@59741
    47
haftmann@30996
    48
lemma power_commutes:
haftmann@30996
    49
  "a ^ n * a = a * a ^ n"
haftmann@57512
    50
  by (induct n) (simp_all add: mult.assoc)
krauss@21199
    51
haftmann@30996
    52
lemma power_Suc2:
haftmann@30996
    53
  "a ^ Suc n = a ^ n * a"
huffman@30273
    54
  by (simp add: power_commutes)
huffman@28131
    55
haftmann@30996
    56
lemma power_add:
haftmann@30996
    57
  "a ^ (m + n) = a ^ m * a ^ n"
haftmann@30996
    58
  by (induct m) (simp_all add: algebra_simps)
paulson@14348
    59
haftmann@30996
    60
lemma power_mult:
haftmann@30996
    61
  "a ^ (m * n) = (a ^ m) ^ n"
huffman@30273
    62
  by (induct n) (simp_all add: power_add)
paulson@14348
    63
wenzelm@53015
    64
lemma power2_eq_square: "a\<^sup>2 = a * a"
huffman@47192
    65
  by (simp add: numeral_2_eq_2)
huffman@47192
    66
huffman@47192
    67
lemma power3_eq_cube: "a ^ 3 = a * a * a"
haftmann@57512
    68
  by (simp add: numeral_3_eq_3 mult.assoc)
huffman@47192
    69
huffman@47192
    70
lemma power_even_eq:
wenzelm@53076
    71
  "a ^ (2 * n) = (a ^ n)\<^sup>2"
haftmann@57512
    72
  by (subst mult.commute) (simp add: power_mult)
huffman@47192
    73
huffman@47192
    74
lemma power_odd_eq:
wenzelm@53076
    75
  "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
huffman@47192
    76
  by (simp add: power_even_eq)
huffman@47192
    77
huffman@47255
    78
lemma power_numeral_even:
huffman@47255
    79
  "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
huffman@47255
    80
  unfolding numeral_Bit0 power_add Let_def ..
huffman@47255
    81
huffman@47255
    82
lemma power_numeral_odd:
huffman@47255
    83
  "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
huffman@47255
    84
  unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
haftmann@57512
    85
  unfolding power_Suc power_add Let_def mult.assoc ..
huffman@47255
    86
haftmann@49824
    87
lemma funpow_times_power:
haftmann@49824
    88
  "(times x ^^ f x) = times (x ^ f x)"
haftmann@49824
    89
proof (induct "f x" arbitrary: f)
haftmann@49824
    90
  case 0 then show ?case by (simp add: fun_eq_iff)
haftmann@49824
    91
next
haftmann@49824
    92
  case (Suc n)
haftmann@49824
    93
  def g \<equiv> "\<lambda>x. f x - 1"
haftmann@49824
    94
  with Suc have "n = g x" by simp
haftmann@49824
    95
  with Suc have "times x ^^ g x = times (x ^ g x)" by simp
haftmann@49824
    96
  moreover from Suc g_def have "f x = g x + 1" by simp
haftmann@57512
    97
  ultimately show ?case by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
haftmann@49824
    98
qed
haftmann@49824
    99
immler@58656
   100
lemma power_commuting_commutes:
immler@58656
   101
  assumes "x * y = y * x"
immler@58656
   102
  shows "x ^ n * y = y * x ^n"
immler@58656
   103
proof (induct n)
immler@58656
   104
  case (Suc n)
immler@58656
   105
  have "x ^ Suc n * y = x ^ n * y * x"
immler@58656
   106
    by (subst power_Suc2) (simp add: assms ac_simps)
immler@58656
   107
  also have "\<dots> = y * x ^ Suc n"
immler@58656
   108
    unfolding Suc power_Suc2
immler@58656
   109
    by (simp add: ac_simps)
immler@58656
   110
  finally show ?case .
immler@58656
   111
qed simp
immler@58656
   112
haftmann@62347
   113
lemma power_minus_mult:
haftmann@62347
   114
  "0 < n \<Longrightarrow> a ^ (n - 1) * a = a ^ n"
haftmann@62347
   115
  by (simp add: power_commutes split add: nat_diff_split)
haftmann@62347
   116
haftmann@30996
   117
end
haftmann@30996
   118
haftmann@30996
   119
context comm_monoid_mult
haftmann@30996
   120
begin
haftmann@30996
   121
hoelzl@56480
   122
lemma power_mult_distrib [field_simps]:
haftmann@30996
   123
  "(a * b) ^ n = (a ^ n) * (b ^ n)"
haftmann@57514
   124
  by (induct n) (simp_all add: ac_simps)
paulson@14348
   125
haftmann@30996
   126
end
haftmann@30996
   127
wenzelm@60758
   128
text\<open>Extract constant factors from powers\<close>
lp15@59741
   129
declare power_mult_distrib [where a = "numeral w" for w, simp]
lp15@59741
   130
declare power_mult_distrib [where b = "numeral w" for w, simp]
lp15@59741
   131
lp15@60155
   132
lemma power_add_numeral [simp]:
lp15@60155
   133
  fixes a :: "'a :: monoid_mult"
lp15@60155
   134
  shows "a^numeral m * a^numeral n = a^numeral (m + n)"
lp15@60155
   135
  by (simp add: power_add [symmetric])
lp15@60155
   136
lp15@60155
   137
lemma power_add_numeral2 [simp]:
lp15@60155
   138
  fixes a :: "'a :: monoid_mult"
lp15@60155
   139
  shows "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
lp15@60155
   140
  by (simp add: mult.assoc [symmetric])
lp15@60155
   141
lp15@60155
   142
lemma power_mult_numeral [simp]:
lp15@60155
   143
  fixes a :: "'a :: monoid_mult"
lp15@60155
   144
  shows"(a^numeral m)^numeral n = a^numeral (m * n)"
lp15@60155
   145
  by (simp only: numeral_mult power_mult)
lp15@60155
   146
huffman@47191
   147
context semiring_numeral
huffman@47191
   148
begin
huffman@47191
   149
huffman@47191
   150
lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
huffman@47191
   151
  by (simp only: sqr_conv_mult numeral_mult)
huffman@47191
   152
huffman@47191
   153
lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
huffman@47191
   154
  by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
huffman@47191
   155
    numeral_sqr numeral_mult power_add power_one_right)
huffman@47191
   156
huffman@47191
   157
lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
huffman@47191
   158
  by (rule numeral_pow [symmetric])
huffman@47191
   159
huffman@47191
   160
end
huffman@47191
   161
haftmann@30996
   162
context semiring_1
haftmann@30996
   163
begin
haftmann@30996
   164
lp15@61649
   165
lemma of_nat_power [simp]:
haftmann@30996
   166
  "of_nat (m ^ n) = of_nat m ^ n"
haftmann@30996
   167
  by (induct n) (simp_all add: of_nat_mult)
haftmann@30996
   168
haftmann@59009
   169
lemma zero_power:
haftmann@59009
   170
  "0 < n \<Longrightarrow> 0 ^ n = 0"
haftmann@59009
   171
  by (cases n) simp_all
haftmann@59009
   172
haftmann@59009
   173
lemma power_zero_numeral [simp]:
haftmann@59009
   174
  "0 ^ numeral k = 0"
huffman@47209
   175
  by (simp add: numeral_eq_Suc)
huffman@47191
   176
wenzelm@53015
   177
lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
huffman@47192
   178
  by (rule power_zero_numeral)
huffman@47192
   179
wenzelm@53015
   180
lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
huffman@47192
   181
  by (rule power_one)
huffman@47192
   182
haftmann@60867
   183
lemma power_0_Suc [simp]:
haftmann@60867
   184
  "0 ^ Suc n = 0"
haftmann@60867
   185
  by simp
haftmann@60867
   186
haftmann@60867
   187
text\<open>It looks plausible as a simprule, but its effect can be strange.\<close>
haftmann@60867
   188
lemma power_0_left:
haftmann@60867
   189
  "0 ^ n = (if n = 0 then 1 else 0)"
haftmann@60867
   190
  by (cases n) simp_all
haftmann@60867
   191
haftmann@30996
   192
end
haftmann@30996
   193
haftmann@30996
   194
context comm_semiring_1
haftmann@30996
   195
begin
haftmann@30996
   196
wenzelm@60758
   197
text \<open>The divides relation\<close>
haftmann@30996
   198
haftmann@30996
   199
lemma le_imp_power_dvd:
haftmann@30996
   200
  assumes "m \<le> n" shows "a ^ m dvd a ^ n"
haftmann@30996
   201
proof
haftmann@30996
   202
  have "a ^ n = a ^ (m + (n - m))"
wenzelm@60758
   203
    using \<open>m \<le> n\<close> by simp
haftmann@30996
   204
  also have "\<dots> = a ^ m * a ^ (n - m)"
haftmann@30996
   205
    by (rule power_add)
haftmann@30996
   206
  finally show "a ^ n = a ^ m * a ^ (n - m)" .
haftmann@30996
   207
qed
haftmann@30996
   208
haftmann@30996
   209
lemma power_le_dvd:
haftmann@30996
   210
  "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
haftmann@30996
   211
  by (rule dvd_trans [OF le_imp_power_dvd])
haftmann@30996
   212
haftmann@30996
   213
lemma dvd_power_same:
haftmann@30996
   214
  "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
haftmann@30996
   215
  by (induct n) (auto simp add: mult_dvd_mono)
haftmann@30996
   216
haftmann@30996
   217
lemma dvd_power_le:
haftmann@30996
   218
  "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
haftmann@30996
   219
  by (rule power_le_dvd [OF dvd_power_same])
paulson@14348
   220
haftmann@30996
   221
lemma dvd_power [simp]:
haftmann@30996
   222
  assumes "n > (0::nat) \<or> x = 1"
haftmann@30996
   223
  shows "x dvd (x ^ n)"
haftmann@30996
   224
using assms proof
haftmann@30996
   225
  assume "0 < n"
haftmann@30996
   226
  then have "x ^ n = x ^ Suc (n - 1)" by simp
haftmann@30996
   227
  then show "x dvd (x ^ n)" by simp
haftmann@30996
   228
next
haftmann@30996
   229
  assume "x = 1"
haftmann@30996
   230
  then show "x dvd (x ^ n)" by simp
haftmann@30996
   231
qed
haftmann@30996
   232
haftmann@30996
   233
end
haftmann@30996
   234
haftmann@60867
   235
class semiring_1_no_zero_divisors = semiring_1 + semiring_no_zero_divisors
haftmann@60867
   236
begin
haftmann@60867
   237
haftmann@60867
   238
subclass power .
haftmann@60867
   239
haftmann@60867
   240
lemma power_eq_0_iff [simp]:
haftmann@60867
   241
  "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
haftmann@60867
   242
  by (induct n) auto
haftmann@60867
   243
haftmann@60867
   244
lemma power_not_zero:
haftmann@60867
   245
  "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
haftmann@60867
   246
  by (induct n) auto
haftmann@60867
   247
haftmann@60867
   248
lemma zero_eq_power2 [simp]:
haftmann@60867
   249
  "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
haftmann@60867
   250
  unfolding power2_eq_square by simp
haftmann@60867
   251
haftmann@60867
   252
end
haftmann@60867
   253
haftmann@60867
   254
context semidom
haftmann@60867
   255
begin
haftmann@60867
   256
haftmann@60867
   257
subclass semiring_1_no_zero_divisors ..
haftmann@60867
   258
haftmann@60867
   259
end
haftmann@60867
   260
haftmann@30996
   261
context ring_1
haftmann@30996
   262
begin
haftmann@30996
   263
haftmann@30996
   264
lemma power_minus:
haftmann@30996
   265
  "(- a) ^ n = (- 1) ^ n * a ^ n"
haftmann@30996
   266
proof (induct n)
haftmann@30996
   267
  case 0 show ?case by simp
haftmann@30996
   268
next
haftmann@30996
   269
  case (Suc n) then show ?case
haftmann@57512
   270
    by (simp del: power_Suc add: power_Suc2 mult.assoc)
haftmann@30996
   271
qed
haftmann@30996
   272
eberlm@61531
   273
lemma power_minus': "NO_MATCH 1 x \<Longrightarrow> (-x) ^ n = (-1)^n * x ^ n"
eberlm@61531
   274
  by (rule power_minus)
eberlm@61531
   275
huffman@47191
   276
lemma power_minus_Bit0:
huffman@47191
   277
  "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
huffman@47191
   278
  by (induct k, simp_all only: numeral_class.numeral.simps power_add
huffman@47191
   279
    power_one_right mult_minus_left mult_minus_right minus_minus)
huffman@47191
   280
huffman@47191
   281
lemma power_minus_Bit1:
huffman@47191
   282
  "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
huffman@47220
   283
  by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
huffman@47191
   284
huffman@47192
   285
lemma power2_minus [simp]:
wenzelm@53015
   286
  "(- a)\<^sup>2 = a\<^sup>2"
haftmann@60867
   287
  by (fact power_minus_Bit0)
huffman@47192
   288
huffman@47192
   289
lemma power_minus1_even [simp]:
haftmann@58410
   290
  "(- 1) ^ (2*n) = 1"
huffman@47192
   291
proof (induct n)
huffman@47192
   292
  case 0 show ?case by simp
huffman@47192
   293
next
huffman@47192
   294
  case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
huffman@47192
   295
qed
huffman@47192
   296
huffman@47192
   297
lemma power_minus1_odd:
haftmann@58410
   298
  "(- 1) ^ Suc (2*n) = -1"
huffman@47192
   299
  by simp
lp15@61649
   300
huffman@47192
   301
lemma power_minus_even [simp]:
huffman@47192
   302
  "(-a) ^ (2*n) = a ^ (2*n)"
huffman@47192
   303
  by (simp add: power_minus [of a])
huffman@47192
   304
huffman@47192
   305
end
huffman@47192
   306
huffman@47192
   307
context ring_1_no_zero_divisors
huffman@47192
   308
begin
huffman@47192
   309
lp15@61649
   310
subclass semiring_1_no_zero_divisors ..
huffman@47192
   311
huffman@47192
   312
lemma power2_eq_1_iff:
wenzelm@53015
   313
  "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
haftmann@60867
   314
  using square_eq_1_iff [of a] by (simp add: power2_eq_square)
huffman@47192
   315
huffman@47192
   316
end
huffman@47192
   317
huffman@47192
   318
context idom
huffman@47192
   319
begin
huffman@47192
   320
wenzelm@53015
   321
lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
huffman@47192
   322
  unfolding power2_eq_square by (rule square_eq_iff)
huffman@47192
   323
huffman@47192
   324
end
huffman@47192
   325
haftmann@60867
   326
context algebraic_semidom
haftmann@60867
   327
begin
haftmann@60867
   328
haftmann@60867
   329
lemma div_power:
haftmann@60867
   330
  assumes "b dvd a"
haftmann@60867
   331
  shows "(a div b) ^ n = a ^ n div b ^ n"
haftmann@60867
   332
  using assms by (induct n) (simp_all add: div_mult_div_if_dvd dvd_power_same)
haftmann@60867
   333
haftmann@62366
   334
lemma is_unit_power_iff:
haftmann@62366
   335
  "is_unit (a ^ n) \<longleftrightarrow> is_unit a \<or> n = 0"
haftmann@62366
   336
  by (induct n) (auto simp add: is_unit_mult_iff)
haftmann@62366
   337
haftmann@60867
   338
end
haftmann@60867
   339
haftmann@60685
   340
context normalization_semidom
haftmann@60685
   341
begin
haftmann@60685
   342
haftmann@60685
   343
lemma normalize_power:
haftmann@60685
   344
  "normalize (a ^ n) = normalize a ^ n"
haftmann@60685
   345
  by (induct n) (simp_all add: normalize_mult)
haftmann@60685
   346
haftmann@60685
   347
lemma unit_factor_power:
haftmann@60685
   348
  "unit_factor (a ^ n) = unit_factor a ^ n"
haftmann@60685
   349
  by (induct n) (simp_all add: unit_factor_mult)
haftmann@60685
   350
haftmann@60685
   351
end
haftmann@60685
   352
huffman@47192
   353
context division_ring
huffman@47192
   354
begin
huffman@47192
   355
haftmann@60867
   356
text\<open>Perhaps these should be simprules.\<close>
haftmann@60867
   357
lemma power_inverse [field_simps, divide_simps]:
haftmann@60867
   358
  "inverse a ^ n = inverse (a ^ n)"
haftmann@60867
   359
proof (cases "a = 0")
haftmann@60867
   360
  case True then show ?thesis by (simp add: power_0_left)
haftmann@60867
   361
next
haftmann@60867
   362
  case False then have "inverse (a ^ n) = inverse a ^ n"
haftmann@60867
   363
    by (induct n) (simp_all add: nonzero_inverse_mult_distrib power_commutes)
haftmann@60867
   364
  then show ?thesis by simp
haftmann@60867
   365
qed
huffman@47192
   366
haftmann@60867
   367
lemma power_one_over [field_simps, divide_simps]:
haftmann@60867
   368
  "(1 / a) ^ n = 1 / a ^ n"
haftmann@60867
   369
  using power_inverse [of a] by (simp add: divide_inverse)
haftmann@60867
   370
lp15@61649
   371
end
huffman@47192
   372
huffman@47192
   373
context field
huffman@47192
   374
begin
huffman@47192
   375
haftmann@60867
   376
lemma power_diff:
haftmann@60867
   377
  assumes nz: "a \<noteq> 0"
haftmann@60867
   378
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
haftmann@60867
   379
  by (induct m n rule: diff_induct) (simp_all add: nz power_not_zero)
huffman@47192
   380
haftmann@60867
   381
lemma power_divide [field_simps, divide_simps]:
haftmann@60867
   382
  "(a / b) ^ n = a ^ n / b ^ n"
haftmann@60867
   383
  by (induct n) simp_all
haftmann@60867
   384
huffman@47192
   385
end
huffman@47192
   386
huffman@47192
   387
wenzelm@60758
   388
subsection \<open>Exponentiation on ordered types\<close>
huffman@47192
   389
haftmann@35028
   390
context linordered_semidom
haftmann@30996
   391
begin
haftmann@30996
   392
haftmann@30996
   393
lemma zero_less_power [simp]:
haftmann@30996
   394
  "0 < a \<Longrightarrow> 0 < a ^ n"
nipkow@56544
   395
  by (induct n) simp_all
haftmann@30996
   396
haftmann@30996
   397
lemma zero_le_power [simp]:
haftmann@30996
   398
  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
nipkow@56536
   399
  by (induct n) simp_all
paulson@14348
   400
huffman@47241
   401
lemma power_mono:
huffman@47241
   402
  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
huffman@47241
   403
  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
huffman@47241
   404
huffman@47241
   405
lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
huffman@47241
   406
  using power_mono [of 1 a n] by simp
huffman@47241
   407
huffman@47241
   408
lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
huffman@47241
   409
  using power_mono [of a 1 n] by simp
paulson@14348
   410
paulson@14348
   411
lemma power_gt1_lemma:
haftmann@30996
   412
  assumes gt1: "1 < a"
haftmann@30996
   413
  shows "1 < a * a ^ n"
paulson@14348
   414
proof -
haftmann@30996
   415
  from gt1 have "0 \<le> a"
haftmann@30996
   416
    by (fact order_trans [OF zero_le_one less_imp_le])
haftmann@30996
   417
  have "1 * 1 < a * 1" using gt1 by simp
haftmann@30996
   418
  also have "\<dots> \<le> a * a ^ n" using gt1
wenzelm@60758
   419
    by (simp only: mult_mono \<open>0 \<le> a\<close> one_le_power order_less_imp_le
wenzelm@14577
   420
        zero_le_one order_refl)
wenzelm@14577
   421
  finally show ?thesis by simp
paulson@14348
   422
qed
paulson@14348
   423
haftmann@30996
   424
lemma power_gt1:
haftmann@30996
   425
  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
haftmann@30996
   426
  by (simp add: power_gt1_lemma)
huffman@24376
   427
haftmann@30996
   428
lemma one_less_power [simp]:
haftmann@30996
   429
  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
haftmann@30996
   430
  by (cases n) (simp_all add: power_gt1_lemma)
paulson@14348
   431
paulson@14348
   432
lemma power_le_imp_le_exp:
haftmann@30996
   433
  assumes gt1: "1 < a"
haftmann@30996
   434
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
haftmann@30996
   435
proof (induct m arbitrary: n)
paulson@14348
   436
  case 0
wenzelm@14577
   437
  show ?case by simp
paulson@14348
   438
next
paulson@14348
   439
  case (Suc m)
wenzelm@14577
   440
  show ?case
wenzelm@14577
   441
  proof (cases n)
wenzelm@14577
   442
    case 0
haftmann@30996
   443
    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
wenzelm@14577
   444
    with gt1 show ?thesis
wenzelm@14577
   445
      by (force simp only: power_gt1_lemma
haftmann@30996
   446
          not_less [symmetric])
wenzelm@14577
   447
  next
wenzelm@14577
   448
    case (Suc n)
haftmann@30996
   449
    with Suc.prems Suc.hyps show ?thesis
wenzelm@14577
   450
      by (force dest: mult_left_le_imp_le
haftmann@30996
   451
          simp add: less_trans [OF zero_less_one gt1])
wenzelm@14577
   452
  qed
paulson@14348
   453
qed
paulson@14348
   454
lp15@61649
   455
lemma of_nat_zero_less_power_iff [simp]:
lp15@61649
   456
  "of_nat x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
lp15@61649
   457
  by (induct n) auto
lp15@61649
   458
wenzelm@61799
   459
text\<open>Surely we can strengthen this? It holds for \<open>0<a<1\<close> too.\<close>
paulson@14348
   460
lemma power_inject_exp [simp]:
haftmann@30996
   461
  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
wenzelm@14577
   462
  by (force simp add: order_antisym power_le_imp_le_exp)
paulson@14348
   463
wenzelm@60758
   464
text\<open>Can relax the first premise to @{term "0<a"} in the case of the
wenzelm@60758
   465
natural numbers.\<close>
paulson@14348
   466
lemma power_less_imp_less_exp:
haftmann@30996
   467
  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
haftmann@30996
   468
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
haftmann@30996
   469
    power_le_imp_le_exp)
paulson@14348
   470
paulson@14348
   471
lemma power_strict_mono [rule_format]:
haftmann@30996
   472
  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
haftmann@30996
   473
  by (induct n)
haftmann@30996
   474
   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
paulson@14348
   475
wenzelm@61799
   476
text\<open>Lemma for \<open>power_strict_decreasing\<close>\<close>
paulson@14348
   477
lemma power_Suc_less:
haftmann@30996
   478
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
haftmann@30996
   479
  by (induct n)
haftmann@30996
   480
    (auto simp add: mult_strict_left_mono)
paulson@14348
   481
haftmann@30996
   482
lemma power_strict_decreasing [rule_format]:
haftmann@30996
   483
  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
haftmann@30996
   484
proof (induct N)
haftmann@30996
   485
  case 0 then show ?case by simp
haftmann@30996
   486
next
lp15@61649
   487
  case (Suc N) then show ?case
haftmann@30996
   488
  apply (auto simp add: power_Suc_less less_Suc_eq)
haftmann@30996
   489
  apply (subgoal_tac "a * a^N < 1 * a^n")
haftmann@30996
   490
  apply simp
haftmann@30996
   491
  apply (rule mult_strict_mono) apply auto
haftmann@30996
   492
  done
haftmann@30996
   493
qed
paulson@14348
   494
wenzelm@61799
   495
text\<open>Proof resembles that of \<open>power_strict_decreasing\<close>\<close>
haftmann@30996
   496
lemma power_decreasing [rule_format]:
haftmann@30996
   497
  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
haftmann@30996
   498
proof (induct N)
haftmann@30996
   499
  case 0 then show ?case by simp
haftmann@30996
   500
next
lp15@61649
   501
  case (Suc N) then show ?case
haftmann@30996
   502
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   503
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
haftmann@30996
   504
  apply (rule mult_mono) apply auto
haftmann@30996
   505
  done
haftmann@30996
   506
qed
paulson@14348
   507
paulson@14348
   508
lemma power_Suc_less_one:
haftmann@30996
   509
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
haftmann@30996
   510
  using power_strict_decreasing [of 0 "Suc n" a] by simp
paulson@14348
   511
wenzelm@61799
   512
text\<open>Proof again resembles that of \<open>power_strict_decreasing\<close>\<close>
haftmann@30996
   513
lemma power_increasing [rule_format]:
haftmann@30996
   514
  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
haftmann@30996
   515
proof (induct N)
haftmann@30996
   516
  case 0 then show ?case by simp
haftmann@30996
   517
next
lp15@61649
   518
  case (Suc N) then show ?case
haftmann@30996
   519
  apply (auto simp add: le_Suc_eq)
haftmann@30996
   520
  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
haftmann@30996
   521
  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
haftmann@30996
   522
  done
haftmann@30996
   523
qed
paulson@14348
   524
wenzelm@61799
   525
text\<open>Lemma for \<open>power_strict_increasing\<close>\<close>
paulson@14348
   526
lemma power_less_power_Suc:
haftmann@30996
   527
  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
haftmann@30996
   528
  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
paulson@14348
   529
haftmann@30996
   530
lemma power_strict_increasing [rule_format]:
haftmann@30996
   531
  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
haftmann@30996
   532
proof (induct N)
haftmann@30996
   533
  case 0 then show ?case by simp
haftmann@30996
   534
next
lp15@61649
   535
  case (Suc N) then show ?case
haftmann@30996
   536
  apply (auto simp add: power_less_power_Suc less_Suc_eq)
haftmann@30996
   537
  apply (subgoal_tac "1 * a^n < a * a^N", simp)
haftmann@30996
   538
  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
haftmann@30996
   539
  done
haftmann@30996
   540
qed
paulson@14348
   541
nipkow@25134
   542
lemma power_increasing_iff [simp]:
haftmann@30996
   543
  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
haftmann@30996
   544
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
paulson@15066
   545
paulson@15066
   546
lemma power_strict_increasing_iff [simp]:
haftmann@30996
   547
  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
lp15@61649
   548
by (blast intro: power_less_imp_less_exp power_strict_increasing)
paulson@15066
   549
paulson@14348
   550
lemma power_le_imp_le_base:
haftmann@30996
   551
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
haftmann@30996
   552
    and ynonneg: "0 \<le> b"
haftmann@30996
   553
  shows "a \<le> b"
nipkow@25134
   554
proof (rule ccontr)
nipkow@25134
   555
  assume "~ a \<le> b"
nipkow@25134
   556
  then have "b < a" by (simp only: linorder_not_le)
nipkow@25134
   557
  then have "b ^ Suc n < a ^ Suc n"
wenzelm@41550
   558
    by (simp only: assms power_strict_mono)
haftmann@30996
   559
  from le and this show False
nipkow@25134
   560
    by (simp add: linorder_not_less [symmetric])
nipkow@25134
   561
qed
wenzelm@14577
   562
huffman@22853
   563
lemma power_less_imp_less_base:
huffman@22853
   564
  assumes less: "a ^ n < b ^ n"
huffman@22853
   565
  assumes nonneg: "0 \<le> b"
huffman@22853
   566
  shows "a < b"
huffman@22853
   567
proof (rule contrapos_pp [OF less])
huffman@22853
   568
  assume "~ a < b"
huffman@22853
   569
  hence "b \<le> a" by (simp only: linorder_not_less)
huffman@22853
   570
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
haftmann@30996
   571
  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
huffman@22853
   572
qed
huffman@22853
   573
paulson@14348
   574
lemma power_inject_base:
haftmann@30996
   575
  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
haftmann@30996
   576
by (blast intro: power_le_imp_le_base antisym eq_refl sym)
paulson@14348
   577
huffman@22955
   578
lemma power_eq_imp_eq_base:
haftmann@30996
   579
  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
haftmann@30996
   580
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
huffman@22955
   581
haftmann@62347
   582
lemma power_eq_iff_eq_base:
haftmann@62347
   583
  "0 < n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a ^ n = b ^ n \<longleftrightarrow> a = b"
haftmann@62347
   584
  using power_eq_imp_eq_base [of a n b] by auto
haftmann@62347
   585
huffman@47192
   586
lemma power2_le_imp_le:
wenzelm@53015
   587
  "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
huffman@47192
   588
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
huffman@47192
   589
huffman@47192
   590
lemma power2_less_imp_less:
wenzelm@53015
   591
  "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
huffman@47192
   592
  by (rule power_less_imp_less_base)
huffman@47192
   593
huffman@47192
   594
lemma power2_eq_imp_eq:
wenzelm@53015
   595
  "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
huffman@47192
   596
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
huffman@47192
   597
haftmann@62347
   598
lemma power_Suc_le_self:
haftmann@62347
   599
  shows "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ Suc n \<le> a"
haftmann@62347
   600
  using power_decreasing [of 1 "Suc n" a] by simp
haftmann@62347
   601
huffman@47192
   602
end
huffman@47192
   603
huffman@47192
   604
context linordered_ring_strict
huffman@47192
   605
begin
huffman@47192
   606
huffman@47192
   607
lemma sum_squares_eq_zero_iff:
huffman@47192
   608
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   609
  by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   610
huffman@47192
   611
lemma sum_squares_le_zero_iff:
huffman@47192
   612
  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   613
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
huffman@47192
   614
huffman@47192
   615
lemma sum_squares_gt_zero_iff:
huffman@47192
   616
  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   617
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
huffman@47192
   618
haftmann@30996
   619
end
haftmann@30996
   620
haftmann@35028
   621
context linordered_idom
haftmann@30996
   622
begin
huffman@29978
   623
wenzelm@61944
   624
lemma power_abs: "\<bar>a ^ n\<bar> = \<bar>a\<bar> ^ n"
haftmann@30996
   625
  by (induct n) (auto simp add: abs_mult)
haftmann@30996
   626
wenzelm@61944
   627
lemma abs_power_minus [simp]: "\<bar>(-a) ^ n\<bar> = \<bar>a ^ n\<bar>"
huffman@35216
   628
  by (simp add: power_abs)
haftmann@30996
   629
wenzelm@61944
   630
lemma zero_less_power_abs_iff [simp]: "0 < \<bar>a\<bar> ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
haftmann@30996
   631
proof (induct n)
haftmann@30996
   632
  case 0 show ?case by simp
haftmann@30996
   633
next
haftmann@30996
   634
  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
huffman@29978
   635
qed
huffman@29978
   636
wenzelm@61944
   637
lemma zero_le_power_abs [simp]: "0 \<le> \<bar>a\<bar> ^ n"
haftmann@30996
   638
  by (rule zero_le_power [OF abs_ge_zero])
haftmann@30996
   639
huffman@47192
   640
lemma zero_le_power2 [simp]:
wenzelm@53015
   641
  "0 \<le> a\<^sup>2"
huffman@47192
   642
  by (simp add: power2_eq_square)
huffman@47192
   643
huffman@47192
   644
lemma zero_less_power2 [simp]:
wenzelm@53015
   645
  "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
huffman@47192
   646
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
huffman@47192
   647
huffman@47192
   648
lemma power2_less_0 [simp]:
wenzelm@53015
   649
  "\<not> a\<^sup>2 < 0"
huffman@47192
   650
  by (force simp add: power2_eq_square mult_less_0_iff)
huffman@47192
   651
haftmann@58787
   652
lemma power2_less_eq_zero_iff [simp]:
haftmann@58787
   653
  "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
haftmann@58787
   654
  by (simp add: le_less)
haftmann@58787
   655
wenzelm@61944
   656
lemma abs_power2 [simp]: "\<bar>a\<^sup>2\<bar> = a\<^sup>2"
huffman@47192
   657
  by (simp add: power2_eq_square abs_mult abs_mult_self)
huffman@47192
   658
wenzelm@61944
   659
lemma power2_abs [simp]: "\<bar>a\<bar>\<^sup>2 = a\<^sup>2"
huffman@47192
   660
  by (simp add: power2_eq_square abs_mult_self)
huffman@47192
   661
huffman@47192
   662
lemma odd_power_less_zero:
huffman@47192
   663
  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
huffman@47192
   664
proof (induct n)
huffman@47192
   665
  case 0
huffman@47192
   666
  then show ?case by simp
huffman@47192
   667
next
huffman@47192
   668
  case (Suc n)
huffman@47192
   669
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
haftmann@57514
   670
    by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   671
  thus ?case
huffman@47192
   672
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
huffman@47192
   673
qed
haftmann@30996
   674
huffman@47192
   675
lemma odd_0_le_power_imp_0_le:
huffman@47192
   676
  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
huffman@47192
   677
  using odd_power_less_zero [of a n]
lp15@61649
   678
    by (force simp add: linorder_not_less [symmetric])
huffman@47192
   679
huffman@47192
   680
lemma zero_le_even_power'[simp]:
huffman@47192
   681
  "0 \<le> a ^ (2*n)"
huffman@47192
   682
proof (induct n)
huffman@47192
   683
  case 0
huffman@47192
   684
    show ?case by simp
huffman@47192
   685
next
huffman@47192
   686
  case (Suc n)
lp15@61649
   687
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
haftmann@57514
   688
      by (simp add: ac_simps power_add power2_eq_square)
huffman@47192
   689
    thus ?case
huffman@47192
   690
      by (simp add: Suc zero_le_mult_iff)
huffman@47192
   691
qed
haftmann@30996
   692
huffman@47192
   693
lemma sum_power2_ge_zero:
wenzelm@53015
   694
  "0 \<le> x\<^sup>2 + y\<^sup>2"
huffman@47192
   695
  by (intro add_nonneg_nonneg zero_le_power2)
huffman@47192
   696
huffman@47192
   697
lemma not_sum_power2_lt_zero:
wenzelm@53015
   698
  "\<not> x\<^sup>2 + y\<^sup>2 < 0"
huffman@47192
   699
  unfolding not_less by (rule sum_power2_ge_zero)
huffman@47192
   700
huffman@47192
   701
lemma sum_power2_eq_zero_iff:
wenzelm@53015
   702
  "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   703
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
huffman@47192
   704
huffman@47192
   705
lemma sum_power2_le_zero_iff:
wenzelm@53015
   706
  "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@47192
   707
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
huffman@47192
   708
huffman@47192
   709
lemma sum_power2_gt_zero_iff:
wenzelm@53015
   710
  "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
huffman@47192
   711
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
haftmann@30996
   712
lp15@59865
   713
lemma abs_le_square_iff:
lp15@59865
   714
   "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
lp15@59865
   715
proof
lp15@59865
   716
  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
lp15@59865
   717
  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
lp15@59865
   718
  then show "x\<^sup>2 \<le> y\<^sup>2" by simp
lp15@59865
   719
next
lp15@59865
   720
  assume "x\<^sup>2 \<le> y\<^sup>2"
lp15@59865
   721
  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
lp15@59865
   722
    by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
lp15@59865
   723
qed
lp15@59865
   724
wenzelm@61944
   725
lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> \<bar>x\<bar> \<le> 1"
lp15@59865
   726
  using abs_le_square_iff [of x 1]
lp15@59865
   727
  by simp
lp15@59865
   728
wenzelm@61944
   729
lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
lp15@59865
   730
  by (auto simp add: abs_if power2_eq_1_iff)
lp15@61649
   731
wenzelm@61944
   732
lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> \<bar>x\<bar> < 1"
lp15@59865
   733
  using  abs_square_eq_1 [of x] abs_square_le_1 [of x]
lp15@59865
   734
  by (auto simp add: le_less)
lp15@59865
   735
haftmann@30996
   736
end
haftmann@30996
   737
huffman@29978
   738
wenzelm@60758
   739
subsection \<open>Miscellaneous rules\<close>
paulson@14348
   740
haftmann@60867
   741
lemma (in linordered_semidom) self_le_power:
haftmann@60867
   742
  "1 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a ^ n"
haftmann@60867
   743
  using power_increasing [of 1 n a] power_one_right [of a] by auto
lp15@55718
   744
haftmann@60867
   745
lemma (in power) power_eq_if:
haftmann@60867
   746
  "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
huffman@47255
   747
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
   748
haftmann@58787
   749
lemma (in comm_semiring_1) power2_sum:
haftmann@58787
   750
  "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
huffman@47192
   751
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   752
haftmann@58787
   753
lemma (in comm_ring_1) power2_diff:
haftmann@58787
   754
  "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
haftmann@58787
   755
  by (simp add: algebra_simps power2_eq_square mult_2_right)
haftmann@30996
   756
lp15@60974
   757
lemma (in comm_ring_1) power2_commute:
lp15@60974
   758
  "(x - y)\<^sup>2 = (y - x)\<^sup>2"
lp15@60974
   759
  by (simp add: algebra_simps power2_eq_square)
lp15@60974
   760
wenzelm@60758
   761
text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
huffman@47255
   762
huffman@47255
   763
lemmas zero_compare_simps =
huffman@47255
   764
    add_strict_increasing add_strict_increasing2 add_increasing
lp15@61649
   765
    zero_le_mult_iff zero_le_divide_iff
lp15@61649
   766
    zero_less_mult_iff zero_less_divide_iff
lp15@61649
   767
    mult_le_0_iff divide_le_0_iff
lp15@61649
   768
    mult_less_0_iff divide_less_0_iff
huffman@47255
   769
    zero_le_power2 power2_less_0
huffman@47255
   770
nipkow@30313
   771
wenzelm@60758
   772
subsection \<open>Exponentiation for the Natural Numbers\<close>
wenzelm@14577
   773
haftmann@30996
   774
lemma nat_one_le_power [simp]:
haftmann@30996
   775
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
haftmann@30996
   776
  by (rule one_le_power [of i n, unfolded One_nat_def])
huffman@23305
   777
haftmann@30996
   778
lemma nat_zero_less_power_iff [simp]:
haftmann@30996
   779
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
haftmann@30996
   780
  by (induct n) auto
paulson@14348
   781
lp15@61649
   782
lemma nat_power_eq_Suc_0_iff [simp]:
haftmann@30996
   783
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
haftmann@30996
   784
  by (induct m) auto
nipkow@30056
   785
haftmann@30996
   786
lemma power_Suc_0 [simp]:
haftmann@30996
   787
  "Suc 0 ^ n = Suc 0"
haftmann@30996
   788
  by simp
nipkow@30056
   789
wenzelm@61799
   790
text\<open>Valid for the naturals, but what if \<open>0<i<1\<close>?
paulson@14348
   791
Premises cannot be weakened: consider the case where @{term "i=0"},
wenzelm@60758
   792
@{term "m=1"} and @{term "n=0"}.\<close>
haftmann@21413
   793
lemma nat_power_less_imp_less:
wenzelm@61076
   794
  assumes nonneg: "0 < (i::nat)"
haftmann@30996
   795
  assumes less: "i ^ m < i ^ n"
haftmann@21413
   796
  shows "m < n"
haftmann@21413
   797
proof (cases "i = 1")
haftmann@21413
   798
  case True with less power_one [where 'a = nat] show ?thesis by simp
haftmann@21413
   799
next
haftmann@21413
   800
  case False with nonneg have "1 < i" by auto
haftmann@21413
   801
  from power_strict_increasing_iff [OF this] less show ?thesis ..
haftmann@21413
   802
qed
paulson@14348
   803
haftmann@33274
   804
lemma power_dvd_imp_le:
haftmann@33274
   805
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
haftmann@33274
   806
  apply (rule power_le_imp_le_exp, assumption)
haftmann@33274
   807
  apply (erule dvd_imp_le, simp)
haftmann@33274
   808
  done
haftmann@33274
   809
haftmann@51263
   810
lemma power2_nat_le_eq_le:
haftmann@51263
   811
  fixes m n :: nat
wenzelm@53015
   812
  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
haftmann@51263
   813
  by (auto intro: power2_le_imp_le power_mono)
haftmann@51263
   814
haftmann@51263
   815
lemma power2_nat_le_imp_le:
haftmann@51263
   816
  fixes m n :: nat
wenzelm@53015
   817
  assumes "m\<^sup>2 \<le> n"
haftmann@51263
   818
  shows "m \<le> n"
haftmann@54249
   819
proof (cases m)
haftmann@54249
   820
  case 0 then show ?thesis by simp
haftmann@54249
   821
next
haftmann@54249
   822
  case (Suc k)
haftmann@54249
   823
  show ?thesis
haftmann@54249
   824
  proof (rule ccontr)
haftmann@54249
   825
    assume "\<not> m \<le> n"
haftmann@54249
   826
    then have "n < m" by simp
haftmann@54249
   827
    with assms Suc show False
haftmann@60867
   828
      by (simp add: power2_eq_square)
haftmann@54249
   829
  qed
haftmann@54249
   830
qed
haftmann@51263
   831
wenzelm@60758
   832
subsubsection \<open>Cardinality of the Powerset\<close>
traytel@55096
   833
traytel@55096
   834
lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
traytel@55096
   835
  unfolding UNIV_bool by simp
traytel@55096
   836
traytel@55096
   837
lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
traytel@55096
   838
proof (induct rule: finite_induct)
lp15@61649
   839
  case empty
traytel@55096
   840
    show ?case by auto
traytel@55096
   841
next
traytel@55096
   842
  case (insert x A)
lp15@61649
   843
  then have "inj_on (insert x) (Pow A)"
traytel@55096
   844
    unfolding inj_on_def by (blast elim!: equalityE)
lp15@61649
   845
  then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A"
traytel@55096
   846
    by (simp add: mult_2 card_image Pow_insert insert.hyps)
traytel@55096
   847
  then show ?case using insert
traytel@55096
   848
    apply (simp add: Pow_insert)
traytel@55096
   849
    apply (subst card_Un_disjoint, auto)
traytel@55096
   850
    done
traytel@55096
   851
qed
traytel@55096
   852
haftmann@57418
   853
wenzelm@60758
   854
subsubsection \<open>Generalized sum over a set\<close>
haftmann@57418
   855
haftmann@57418
   856
lemma setsum_zero_power [simp]:
haftmann@57418
   857
  fixes c :: "nat \<Rightarrow> 'a::division_ring"
haftmann@57418
   858
  shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
haftmann@57418
   859
apply (cases "finite A")
haftmann@57418
   860
  by (induction A rule: finite_induct) auto
haftmann@57418
   861
haftmann@57418
   862
lemma setsum_zero_power' [simp]:
haftmann@57418
   863
  fixes c :: "nat \<Rightarrow> 'a::field"
haftmann@57418
   864
  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
   865
  using setsum_zero_power [of "\<lambda>i. c i / d i" A]
haftmann@57418
   866
  by auto
haftmann@57418
   867
haftmann@57418
   868
wenzelm@60758
   869
subsubsection \<open>Generalized product over a set\<close>
traytel@55096
   870
traytel@55096
   871
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
traytel@55096
   872
apply (erule finite_induct)
traytel@55096
   873
apply auto
traytel@55096
   874
done
traytel@55096
   875
haftmann@57418
   876
lemma setprod_power_distrib:
haftmann@57418
   877
  fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
haftmann@57418
   878
  shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
lp15@61649
   879
proof (cases "finite A")
lp15@61649
   880
  case True then show ?thesis
haftmann@57418
   881
    by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
haftmann@57418
   882
next
lp15@61649
   883
  case False then show ?thesis
haftmann@57418
   884
    by simp
haftmann@57418
   885
qed
haftmann@57418
   886
haftmann@58437
   887
lemma power_setsum:
haftmann@58437
   888
  "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
haftmann@58437
   889
  by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
haftmann@58437
   890
traytel@55096
   891
lemma setprod_gen_delta:
traytel@55096
   892
  assumes fS: "finite S"
traytel@55096
   893
  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
   894
proof-
traytel@55096
   895
  let ?f = "(\<lambda>k. if k=a then b k else c)"
traytel@55096
   896
  {assume a: "a \<notin> S"
traytel@55096
   897
    hence "\<forall> k\<in> S. ?f k = c" by simp
traytel@55096
   898
    hence ?thesis  using a setprod_constant[OF fS, of c] by simp }
lp15@61649
   899
  moreover
traytel@55096
   900
  {assume a: "a \<in> S"
traytel@55096
   901
    let ?A = "S - {a}"
traytel@55096
   902
    let ?B = "{a}"
lp15@61649
   903
    have eq: "S = ?A \<union> ?B" using a by blast
traytel@55096
   904
    have dj: "?A \<inter> ?B = {}" by simp
lp15@61649
   905
    from fS have fAB: "finite ?A" "finite ?B" by auto
traytel@55096
   906
    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
haftmann@57418
   907
      apply (rule setprod.cong) by auto
traytel@55096
   908
    have cA: "card ?A = card S - 1" using fS a by auto
traytel@55096
   909
    have fA1: "setprod ?f ?A = c ^ card ?A"  unfolding fA0 apply (rule setprod_constant) using fS by auto
traytel@55096
   910
    have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
haftmann@57418
   911
      using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
traytel@55096
   912
      by simp
traytel@55096
   913
    then have ?thesis using a cA
haftmann@57418
   914
      by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
traytel@55096
   915
  ultimately show ?thesis by blast
traytel@55096
   916
qed
traytel@55096
   917
wenzelm@60758
   918
subsection \<open>Code generator tweak\<close>
haftmann@31155
   919
haftmann@52435
   920
code_identifier
haftmann@52435
   921
  code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
   922
paulson@3390
   923
end