src/HOL/Power.thy
author huffman
Sat, 31 Mar 2012 19:10:58 +0200
changeset 47241 243b33052e34
parent 47220 52426c62b5d0
child 47255 30a1692557b0
permissions -rw-r--r--
add lemma power_le_one
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
3390
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
     1
(*  Title:      HOL/Power.thy
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
     2
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
     3
    Copyright   1997  University of Cambridge
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
     4
*)
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
     5
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
     6
header {* Exponentiation *}
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
     7
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15066
diff changeset
     8
theory Power
47191
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
     9
imports Num
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15066
diff changeset
    10
begin
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    11
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
    12
subsection {* Powers for Arbitrary Monoids *}
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
    13
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    14
class power = one + times
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
    15
begin
24996
ebd5f4cc7118 moved class power to theory Power
haftmann
parents: 24376
diff changeset
    16
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
    17
primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
    18
    power_0: "a ^ 0 = 1"
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
    19
  | power_Suc: "a ^ Suc n = a * a ^ n"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    20
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    21
notation (latex output)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    22
  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    23
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    24
notation (HTML output)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    25
  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    26
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    27
text {* Special syntax for squares. *}
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    28
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    29
abbreviation (xsymbols)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    30
  power2 :: "'a \<Rightarrow> 'a"  ("(_\<twosuperior>)" [1000] 999) where
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    31
  "x\<twosuperior> \<equiv> x ^ 2"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    32
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    33
notation (latex output)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    34
  power2  ("(_\<twosuperior>)" [1000] 999)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    35
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    36
notation (HTML output)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    37
  power2  ("(_\<twosuperior>)" [1000] 999)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    38
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
    39
end
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    40
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    41
context monoid_mult
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    42
begin
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    43
39438
c5ece2a7a86e Isar "default" step needs to fail for solved problems, for clear distinction of '.' and '..' for example -- amending lapse introduced in 9de4d64eee3b (April 2004);
wenzelm
parents: 36409
diff changeset
    44
subclass power .
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    45
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    46
lemma power_one [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    47
  "1 ^ n = 1"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
    48
  by (induct n) simp_all
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    49
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    50
lemma power_one_right [simp]:
31001
7e6ffd8f51a9 cleaned up theory power further
haftmann
parents: 30997
diff changeset
    51
  "a ^ 1 = a"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    52
  by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    53
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    54
lemma power_commutes:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    55
  "a ^ n * a = a * a ^ n"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
    56
  by (induct n) (simp_all add: mult_assoc)
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 17149
diff changeset
    57
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    58
lemma power_Suc2:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    59
  "a ^ Suc n = a ^ n * a"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
    60
  by (simp add: power_commutes)
28131
3130d7b3149d add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents: 25874
diff changeset
    61
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    62
lemma power_add:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    63
  "a ^ (m + n) = a ^ m * a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    64
  by (induct m) (simp_all add: algebra_simps)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    65
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    66
lemma power_mult:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    67
  "a ^ (m * n) = (a ^ m) ^ n"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
    68
  by (induct n) (simp_all add: power_add)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    69
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    70
lemma power2_eq_square: "a\<twosuperior> = a * a"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    71
  by (simp add: numeral_2_eq_2)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    72
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    73
lemma power3_eq_cube: "a ^ 3 = a * a * a"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    74
  by (simp add: numeral_3_eq_3 mult_assoc)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    75
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    76
lemma power_even_eq:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    77
  "a ^ (2*n) = (a ^ n) ^ 2"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    78
  by (subst mult_commute) (simp add: power_mult)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    79
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    80
lemma power_odd_eq:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    81
  "a ^ Suc (2*n) = a * (a ^ n) ^ 2"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    82
  by (simp add: power_even_eq)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
    83
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    84
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    85
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    86
context comm_monoid_mult
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    87
begin
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    88
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    89
lemma power_mult_distrib:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    90
  "(a * b) ^ n = (a ^ n) * (b ^ n)"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
    91
  by (induct n) (simp_all add: mult_ac)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    92
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    93
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
    94
47191
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
    95
context semiring_numeral
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
    96
begin
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
    97
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
    98
lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
    99
  by (simp only: sqr_conv_mult numeral_mult)
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   100
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   101
lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   102
  by (induct l, simp_all only: numeral_class.numeral.simps pow.simps
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   103
    numeral_sqr numeral_mult power_add power_one_right)
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   104
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   105
lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   106
  by (rule numeral_pow [symmetric])
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   107
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   108
end
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   109
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   110
context semiring_1
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   111
begin
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   112
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   113
lemma of_nat_power:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   114
  "of_nat (m ^ n) = of_nat m ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   115
  by (induct n) (simp_all add: of_nat_mult)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   116
47191
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   117
lemma power_zero_numeral [simp]: "(0::'a) ^ numeral k = 0"
47209
4893907fe872 add constant pred_numeral k = numeral k - (1::nat);
huffman
parents: 47192
diff changeset
   118
  by (simp add: numeral_eq_Suc)
47191
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   119
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   120
lemma zero_power2: "0\<twosuperior> = 0" (* delete? *)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   121
  by (rule power_zero_numeral)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   122
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   123
lemma one_power2: "1\<twosuperior> = 1" (* delete? *)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   124
  by (rule power_one)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   125
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   126
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   127
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   128
context comm_semiring_1
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   129
begin
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   130
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   131
text {* The divides relation *}
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   132
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   133
lemma le_imp_power_dvd:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   134
  assumes "m \<le> n" shows "a ^ m dvd a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   135
proof
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   136
  have "a ^ n = a ^ (m + (n - m))"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   137
    using `m \<le> n` by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   138
  also have "\<dots> = a ^ m * a ^ (n - m)"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   139
    by (rule power_add)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   140
  finally show "a ^ n = a ^ m * a ^ (n - m)" .
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   141
qed
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   142
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   143
lemma power_le_dvd:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   144
  "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   145
  by (rule dvd_trans [OF le_imp_power_dvd])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   146
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   147
lemma dvd_power_same:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   148
  "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   149
  by (induct n) (auto simp add: mult_dvd_mono)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   150
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   151
lemma dvd_power_le:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   152
  "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   153
  by (rule power_le_dvd [OF dvd_power_same])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   154
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   155
lemma dvd_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   156
  assumes "n > (0::nat) \<or> x = 1"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   157
  shows "x dvd (x ^ n)"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   158
using assms proof
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   159
  assume "0 < n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   160
  then have "x ^ n = x ^ Suc (n - 1)" by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   161
  then show "x dvd (x ^ n)" by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   162
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   163
  assume "x = 1"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   164
  then show "x dvd (x ^ n)" by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   165
qed
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   166
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   167
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   168
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   169
context ring_1
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   170
begin
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   171
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   172
lemma power_minus:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   173
  "(- a) ^ n = (- 1) ^ n * a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   174
proof (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   175
  case 0 show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   176
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   177
  case (Suc n) then show ?case
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   178
    by (simp del: power_Suc add: power_Suc2 mult_assoc)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   179
qed
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   180
47191
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   181
lemma power_minus_Bit0:
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   182
  "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   183
  by (induct k, simp_all only: numeral_class.numeral.simps power_add
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   184
    power_one_right mult_minus_left mult_minus_right minus_minus)
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   185
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   186
lemma power_minus_Bit1:
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   187
  "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
47220
52426c62b5d0 replace lemmas eval_nat_numeral with a simpler reformulation
huffman
parents: 47209
diff changeset
   188
  by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
47191
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   189
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   190
lemma power_neg_numeral_Bit0 [simp]:
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   191
  "neg_numeral k ^ numeral (Num.Bit0 l) = numeral (Num.pow k (Num.Bit0 l))"
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   192
  by (simp only: neg_numeral_def power_minus_Bit0 power_numeral)
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   193
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   194
lemma power_neg_numeral_Bit1 [simp]:
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   195
  "neg_numeral k ^ numeral (Num.Bit1 l) = neg_numeral (Num.pow k (Num.Bit1 l))"
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   196
  by (simp only: neg_numeral_def power_minus_Bit1 power_numeral pow.simps)
ebd8c46d156b bootstrap Num.thy before Power.thy;
huffman
parents: 45231
diff changeset
   197
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   198
lemma power2_minus [simp]:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   199
  "(- a)\<twosuperior> = a\<twosuperior>"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   200
  by (rule power_minus_Bit0)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   201
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   202
lemma power_minus1_even [simp]:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   203
  "-1 ^ (2*n) = 1"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   204
proof (induct n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   205
  case 0 show ?case by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   206
next
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   207
  case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   208
qed
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   209
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   210
lemma power_minus1_odd:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   211
  "-1 ^ Suc (2*n) = -1"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   212
  by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   213
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   214
lemma power_minus_even [simp]:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   215
  "(-a) ^ (2*n) = a ^ (2*n)"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   216
  by (simp add: power_minus [of a])
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   217
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   218
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   219
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   220
context ring_1_no_zero_divisors
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   221
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   222
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   223
lemma field_power_not_zero:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   224
  "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   225
  by (induct n) auto
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   226
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   227
lemma zero_eq_power2 [simp]:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   228
  "a\<twosuperior> = 0 \<longleftrightarrow> a = 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   229
  unfolding power2_eq_square by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   230
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   231
lemma power2_eq_1_iff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   232
  "a\<twosuperior> = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   233
  unfolding power2_eq_square by (rule square_eq_1_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   234
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   235
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   236
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   237
context idom
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   238
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   239
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   240
lemma power2_eq_iff: "x\<twosuperior> = y\<twosuperior> \<longleftrightarrow> x = y \<or> x = - y"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   241
  unfolding power2_eq_square by (rule square_eq_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   242
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   243
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   244
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   245
context division_ring
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   246
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   247
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   248
text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   249
lemma nonzero_power_inverse:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   250
  "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   251
  by (induct n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   252
    (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   253
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   254
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   255
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   256
context field
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   257
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   258
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   259
lemma nonzero_power_divide:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   260
  "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   261
  by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   262
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   263
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   264
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   265
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   266
subsection {* Exponentiation on ordered types *}
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   267
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   268
context linordered_ring (* TODO: move *)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   269
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   270
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   271
lemma sum_squares_ge_zero:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   272
  "0 \<le> x * x + y * y"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   273
  by (intro add_nonneg_nonneg zero_le_square)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   274
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   275
lemma not_sum_squares_lt_zero:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   276
  "\<not> x * x + y * y < 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   277
  by (simp add: not_less sum_squares_ge_zero)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   278
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   279
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   280
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33364
diff changeset
   281
context linordered_semidom
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   282
begin
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   283
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   284
lemma zero_less_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   285
  "0 < a \<Longrightarrow> 0 < a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   286
  by (induct n) (simp_all add: mult_pos_pos)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   287
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   288
lemma zero_le_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   289
  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   290
  by (induct n) (simp_all add: mult_nonneg_nonneg)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   291
47241
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   292
lemma power_mono:
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   293
  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   294
  by (induct n) (auto intro: mult_mono order_trans [of 0 a b])
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   295
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   296
lemma one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   297
  using power_mono [of 1 a n] by simp
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   298
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   299
lemma power_le_one: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
243b33052e34 add lemma power_le_one
huffman
parents: 47220
diff changeset
   300
  using power_mono [of a 1 n] by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   301
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   302
lemma power_gt1_lemma:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   303
  assumes gt1: "1 < a"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   304
  shows "1 < a * a ^ n"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   305
proof -
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   306
  from gt1 have "0 \<le> a"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   307
    by (fact order_trans [OF zero_le_one less_imp_le])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   308
  have "1 * 1 < a * 1" using gt1 by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   309
  also have "\<dots> \<le> a * a ^ n" using gt1
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   310
    by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   311
        zero_le_one order_refl)
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   312
  finally show ?thesis by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   313
qed
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   314
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   315
lemma power_gt1:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   316
  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   317
  by (simp add: power_gt1_lemma)
24376
e403ab5c9415 add lemma one_less_power
huffman
parents: 24286
diff changeset
   318
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   319
lemma one_less_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   320
  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   321
  by (cases n) (simp_all add: power_gt1_lemma)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   322
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   323
lemma power_le_imp_le_exp:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   324
  assumes gt1: "1 < a"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   325
  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   326
proof (induct m arbitrary: n)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   327
  case 0
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   328
  show ?case by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   329
next
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   330
  case (Suc m)
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   331
  show ?case
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   332
  proof (cases n)
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   333
    case 0
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   334
    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   335
    with gt1 show ?thesis
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   336
      by (force simp only: power_gt1_lemma
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   337
          not_less [symmetric])
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   338
  next
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   339
    case (Suc n)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   340
    with Suc.prems Suc.hyps show ?thesis
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   341
      by (force dest: mult_left_le_imp_le
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   342
          simp add: less_trans [OF zero_less_one gt1])
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   343
  qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   344
qed
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   345
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   346
text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   347
lemma power_inject_exp [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   348
  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   349
  by (force simp add: order_antisym power_le_imp_le_exp)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   350
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   351
text{*Can relax the first premise to @{term "0<a"} in the case of the
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   352
natural numbers.*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   353
lemma power_less_imp_less_exp:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   354
  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   355
  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   356
    power_le_imp_le_exp)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   357
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   358
lemma power_strict_mono [rule_format]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   359
  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   360
  by (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   361
   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   362
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   363
text{*Lemma for @{text power_strict_decreasing}*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   364
lemma power_Suc_less:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   365
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   366
  by (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   367
    (auto simp add: mult_strict_left_mono)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   368
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   369
lemma power_strict_decreasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   370
  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   371
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   372
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   373
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   374
  case (Suc N) then show ?case 
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   375
  apply (auto simp add: power_Suc_less less_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   376
  apply (subgoal_tac "a * a^N < 1 * a^n")
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   377
  apply simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   378
  apply (rule mult_strict_mono) apply auto
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   379
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   380
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   381
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   382
text{*Proof resembles that of @{text power_strict_decreasing}*}
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   383
lemma power_decreasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   384
  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   385
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   386
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   387
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   388
  case (Suc N) then show ?case 
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   389
  apply (auto simp add: le_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   390
  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   391
  apply (rule mult_mono) apply auto
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   392
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   393
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   394
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   395
lemma power_Suc_less_one:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   396
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   397
  using power_strict_decreasing [of 0 "Suc n" a] by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   398
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   399
text{*Proof again resembles that of @{text power_strict_decreasing}*}
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   400
lemma power_increasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   401
  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   402
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   403
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   404
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   405
  case (Suc N) then show ?case 
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   406
  apply (auto simp add: le_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   407
  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   408
  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   409
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   410
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   411
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   412
text{*Lemma for @{text power_strict_increasing}*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   413
lemma power_less_power_Suc:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   414
  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   415
  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   416
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   417
lemma power_strict_increasing [rule_format]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   418
  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   419
proof (induct N)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   420
  case 0 then show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   421
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   422
  case (Suc N) then show ?case 
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   423
  apply (auto simp add: power_less_power_Suc less_Suc_eq)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   424
  apply (subgoal_tac "1 * a^n < a * a^N", simp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   425
  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   426
  done
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   427
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   428
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   429
lemma power_increasing_iff [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   430
  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   431
  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
15066
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   432
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   433
lemma power_strict_increasing_iff [simp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   434
  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   435
by (blast intro: power_less_imp_less_exp power_strict_increasing) 
15066
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   436
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   437
lemma power_le_imp_le_base:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   438
  assumes le: "a ^ Suc n \<le> b ^ Suc n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   439
    and ynonneg: "0 \<le> b"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   440
  shows "a \<le> b"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   441
proof (rule ccontr)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   442
  assume "~ a \<le> b"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   443
  then have "b < a" by (simp only: linorder_not_le)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   444
  then have "b ^ Suc n < a ^ Suc n"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 39438
diff changeset
   445
    by (simp only: assms power_strict_mono)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   446
  from le and this show False
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   447
    by (simp add: linorder_not_less [symmetric])
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   448
qed
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   449
22853
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   450
lemma power_less_imp_less_base:
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   451
  assumes less: "a ^ n < b ^ n"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   452
  assumes nonneg: "0 \<le> b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   453
  shows "a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   454
proof (rule contrapos_pp [OF less])
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   455
  assume "~ a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   456
  hence "b \<le> a" by (simp only: linorder_not_less)
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   457
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   458
  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
22853
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   459
qed
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   460
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   461
lemma power_inject_base:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   462
  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   463
by (blast intro: power_le_imp_le_base antisym eq_refl sym)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   464
22955
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   465
lemma power_eq_imp_eq_base:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   466
  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   467
  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
22955
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   468
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   469
lemma power2_le_imp_le:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   470
  "x\<twosuperior> \<le> y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   471
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   472
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   473
lemma power2_less_imp_less:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   474
  "x\<twosuperior> < y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   475
  by (rule power_less_imp_less_base)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   476
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   477
lemma power2_eq_imp_eq:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   478
  "x\<twosuperior> = y\<twosuperior> \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   479
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   480
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   481
end
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   482
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   483
context linordered_ring_strict
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   484
begin
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   485
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   486
lemma sum_squares_eq_zero_iff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   487
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   488
  by (simp add: add_nonneg_eq_0_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   489
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   490
lemma sum_squares_le_zero_iff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   491
  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   492
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   493
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   494
lemma sum_squares_gt_zero_iff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   495
  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   496
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   497
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   498
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   499
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33364
diff changeset
   500
context linordered_idom
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   501
begin
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   502
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   503
lemma power_abs:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   504
  "abs (a ^ n) = abs a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   505
  by (induct n) (auto simp add: abs_mult)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   506
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   507
lemma abs_power_minus [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   508
  "abs ((-a) ^ n) = abs (a ^ n)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
   509
  by (simp add: power_abs)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   510
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35216
diff changeset
   511
lemma zero_less_power_abs_iff [simp, no_atp]:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   512
  "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   513
proof (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   514
  case 0 show ?case by simp
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   515
next
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   516
  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   517
qed
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   518
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   519
lemma zero_le_power_abs [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   520
  "0 \<le> abs a ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   521
  by (rule zero_le_power [OF abs_ge_zero])
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   522
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   523
lemma zero_le_power2 [simp]:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   524
  "0 \<le> a\<twosuperior>"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   525
  by (simp add: power2_eq_square)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   526
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   527
lemma zero_less_power2 [simp]:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   528
  "0 < a\<twosuperior> \<longleftrightarrow> a \<noteq> 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   529
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   530
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   531
lemma power2_less_0 [simp]:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   532
  "\<not> a\<twosuperior> < 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   533
  by (force simp add: power2_eq_square mult_less_0_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   534
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   535
lemma abs_power2 [simp]:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   536
  "abs (a\<twosuperior>) = a\<twosuperior>"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   537
  by (simp add: power2_eq_square abs_mult abs_mult_self)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   538
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   539
lemma power2_abs [simp]:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   540
  "(abs a)\<twosuperior> = a\<twosuperior>"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   541
  by (simp add: power2_eq_square abs_mult_self)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   542
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   543
lemma odd_power_less_zero:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   544
  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   545
proof (induct n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   546
  case 0
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   547
  then show ?case by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   548
next
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   549
  case (Suc n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   550
  have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   551
    by (simp add: mult_ac power_add power2_eq_square)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   552
  thus ?case
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   553
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   554
qed
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   555
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   556
lemma odd_0_le_power_imp_0_le:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   557
  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   558
  using odd_power_less_zero [of a n]
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   559
    by (force simp add: linorder_not_less [symmetric]) 
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   560
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   561
lemma zero_le_even_power'[simp]:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   562
  "0 \<le> a ^ (2*n)"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   563
proof (induct n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   564
  case 0
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   565
    show ?case by simp
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   566
next
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   567
  case (Suc n)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   568
    have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   569
      by (simp add: mult_ac power_add power2_eq_square)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   570
    thus ?case
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   571
      by (simp add: Suc zero_le_mult_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   572
qed
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   573
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   574
lemma sum_power2_ge_zero:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   575
  "0 \<le> x\<twosuperior> + y\<twosuperior>"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   576
  by (intro add_nonneg_nonneg zero_le_power2)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   577
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   578
lemma not_sum_power2_lt_zero:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   579
  "\<not> x\<twosuperior> + y\<twosuperior> < 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   580
  unfolding not_less by (rule sum_power2_ge_zero)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   581
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   582
lemma sum_power2_eq_zero_iff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   583
  "x\<twosuperior> + y\<twosuperior> = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   584
  unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   585
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   586
lemma sum_power2_le_zero_iff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   587
  "x\<twosuperior> + y\<twosuperior> \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   588
  by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   589
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   590
lemma sum_power2_gt_zero_iff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   591
  "0 < x\<twosuperior> + y\<twosuperior> \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   592
  unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   593
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   594
end
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   595
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   596
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   597
subsection {* Miscellaneous rules *}
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   598
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   599
lemma power2_sum:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   600
  fixes x y :: "'a::comm_semiring_1"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   601
  shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   602
  by (simp add: algebra_simps power2_eq_square mult_2_right)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   603
47192
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   604
lemma power2_diff:
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   605
  fixes x y :: "'a::comm_ring_1"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   606
  shows "(x - y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> - 2 * x * y"
0c0501cb6da6 move many lemmas from Nat_Numeral.thy to Power.thy or Num.thy
huffman
parents: 47191
diff changeset
   607
  by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute)
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   608
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   609
lemma power_0_Suc [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   610
  "(0::'a::{power, semiring_0}) ^ Suc n = 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   611
  by simp
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   612
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   613
text{*It looks plausible as a simprule, but its effect can be strange.*}
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   614
lemma power_0_left:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   615
  "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   616
  by (induct n) simp_all
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   617
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   618
lemma power_eq_0_iff [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   619
  "a ^ n = 0 \<longleftrightarrow>
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   620
     a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   621
  by (induct n)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   622
    (auto simp add: no_zero_divisors elim: contrapos_pp)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   623
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   624
lemma (in field) power_diff:
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   625
  assumes nz: "a \<noteq> 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   626
  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   627
  by (induct m n rule: diff_induct) (simp_all add: nz field_power_not_zero)
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   628
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   629
text{*Perhaps these should be simprules.*}
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   630
lemma power_inverse:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   631
  fixes a :: "'a::division_ring_inverse_zero"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   632
  shows "inverse (a ^ n) = inverse a ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   633
apply (cases "a = 0")
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   634
apply (simp add: power_0_left)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   635
apply (simp add: nonzero_power_inverse)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   636
done (* TODO: reorient or rename to inverse_power *)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   637
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   638
lemma power_one_over:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   639
  "1 / (a::'a::{field_inverse_zero, power}) ^ n =  (1 / a) ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   640
  by (simp add: divide_inverse) (rule power_inverse)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   641
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   642
lemma power_divide:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   643
  "(a / b) ^ n = (a::'a::field_inverse_zero) ^ n / b ^ n"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   644
apply (cases "b = 0")
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   645
apply (simp add: power_0_left)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   646
apply (rule nonzero_power_divide)
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   647
apply assumption
30313
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   648
done
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   649
b2441b0c8d38 added lemmas
nipkow
parents: 30273
diff changeset
   650
30960
fec1a04b7220 power operation defined generic
haftmann
parents: 30730
diff changeset
   651
subsection {* Exponentiation for the Natural Numbers *}
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   652
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   653
lemma nat_one_le_power [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   654
  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   655
  by (rule one_le_power [of i n, unfolded One_nat_def])
23305
8ae6f7b0903b add lemma of_nat_power
huffman
parents: 23183
diff changeset
   656
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   657
lemma nat_zero_less_power_iff [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   658
  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   659
  by (induct n) auto
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   660
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   661
lemma nat_power_eq_Suc_0_iff [simp]: 
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   662
  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   663
  by (induct m) auto
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   664
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   665
lemma power_Suc_0 [simp]:
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   666
  "Suc 0 ^ n = Suc 0"
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   667
  by simp
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   668
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   669
text{*Valid for the naturals, but what if @{text"0<i<1"}?
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   670
Premises cannot be weakened: consider the case where @{term "i=0"},
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   671
@{term "m=1"} and @{term "n=0"}.*}
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   672
lemma nat_power_less_imp_less:
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   673
  assumes nonneg: "0 < (i\<Colon>nat)"
30996
648d02b124d8 cleaned up Power theory
haftmann
parents: 30960
diff changeset
   674
  assumes less: "i ^ m < i ^ n"
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   675
  shows "m < n"
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   676
proof (cases "i = 1")
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   677
  case True with less power_one [where 'a = nat] show ?thesis by simp
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   678
next
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   679
  case False with nonneg have "1 < i" by auto
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   680
  from power_strict_increasing_iff [OF this] less show ?thesis ..
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   681
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   682
33274
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   683
lemma power_dvd_imp_le:
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   684
  "i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n"
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   685
  apply (rule power_le_imp_le_exp, assumption)
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   686
  apply (erule dvd_imp_le, simp)
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   687
  done
b6ff7db522b5 moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents: 31998
diff changeset
   688
31155
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   689
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   690
subsection {* Code generator tweak *}
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   691
45231
d85a2fdc586c replacing code_inline by code_unfold, removing obsolete code_unfold, code_inline del now that the ancient code generator is removed
bulwahn
parents: 41550
diff changeset
   692
lemma power_power_power [code]:
31155
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   693
  "power = power.power (1::'a::{power}) (op *)"
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   694
  unfolding power_def power.power_def ..
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   695
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   696
declare power.power.simps [code]
92d8ff6af82c monomorphic code generation for power operations
haftmann
parents: 31021
diff changeset
   697
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   698
code_modulename SML
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   699
  Power Arith
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   700
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   701
code_modulename OCaml
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   702
  Power Arith
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   703
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   704
code_modulename Haskell
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   705
  Power Arith
2bd12592c5e8 tuned code setup
haftmann
parents: 33274
diff changeset
   706
3390
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
   707
end