src/HOL/Power.thy
author immler@in.tum.de
Thu, 26 Feb 2009 10:13:43 +0100
changeset 30151 629f3a92863e
parent 30079 293b896b9c25
child 30242 aea5d7fa7ef5
permissions -rw-r--r--
removed global ref dfg_format
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
    ID:         $Id$
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
     4
    Copyright   1997  University of Cambridge
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
     5
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
     6
*)
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
     7
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 15251
diff changeset
     8
header{*Exponentiation*}
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
     9
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15066
diff changeset
    10
theory Power
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
    11
imports Nat
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15066
diff changeset
    12
begin
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    13
29608
564ea783ace8 no base sort in class import
haftmann
parents: 28131
diff changeset
    14
class power =
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24996
diff changeset
    15
  fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"            (infixr "^" 80)
24996
ebd5f4cc7118 moved class power to theory Power
haftmann
parents: 24376
diff changeset
    16
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 17149
diff changeset
    17
subsection{*Powers for Arbitrary Monoids*}
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    18
22390
378f34b1e380 now using "class"
haftmann
parents: 21456
diff changeset
    19
class recpower = monoid_mult + power +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24996
diff changeset
    20
  assumes power_0 [simp]: "a ^ 0       = 1"
af5ef0d4d655 global class syntax
haftmann
parents: 24996
diff changeset
    21
  assumes 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
    22
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 17149
diff changeset
    23
lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"
23183
haftmann
parents: 22991
diff changeset
    24
  by (simp add: power_Suc)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    25
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    26
text{*It looks plausible as a simprule, but its effect can be strange.*}
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 17149
diff changeset
    27
lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::{recpower,semiring_0}))"
23183
haftmann
parents: 22991
diff changeset
    28
  by (induct n) simp_all
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    29
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
    30
lemma power_one [simp]: "1^n = (1::'a::recpower)"
23183
haftmann
parents: 22991
diff changeset
    31
  by (induct n) (simp_all add: power_Suc)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    32
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
    33
lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
    34
  unfolding One_nat_def by (simp add: power_Suc)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    35
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 17149
diff changeset
    36
lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
23183
haftmann
parents: 22991
diff changeset
    37
  by (induct n) (simp_all add: power_Suc mult_assoc)
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 17149
diff changeset
    38
28131
3130d7b3149d add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents: 25874
diff changeset
    39
lemma power_Suc2: "(a::'a::recpower) ^ Suc n = a ^ n * a"
3130d7b3149d add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents: 25874
diff changeset
    40
  by (simp add: power_Suc power_commutes)
3130d7b3149d add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents: 25874
diff changeset
    41
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
    42
lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
23183
haftmann
parents: 22991
diff changeset
    43
  by (induct m) (simp_all add: power_Suc mult_ac)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    44
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
    45
lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
23183
haftmann
parents: 22991
diff changeset
    46
  by (induct n) (simp_all add: power_Suc power_add)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    47
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 17149
diff changeset
    48
lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"
23183
haftmann
parents: 22991
diff changeset
    49
  by (induct n) (simp_all add: power_Suc mult_ac)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    50
25874
14819a95cf75 added simp attributes
nipkow
parents: 25836
diff changeset
    51
lemma zero_less_power[simp]:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
    52
     "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
    53
apply (induct "n")
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16733
diff changeset
    54
apply (simp_all add: power_Suc zero_less_one mult_pos_pos)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    55
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    56
25874
14819a95cf75 added simp attributes
nipkow
parents: 25836
diff changeset
    57
lemma zero_le_power[simp]:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
    58
     "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    59
apply (simp add: order_le_less)
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
    60
apply (erule disjE)
25874
14819a95cf75 added simp attributes
nipkow
parents: 25836
diff changeset
    61
apply (simp_all add: zero_less_one power_0_left)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    62
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    63
25874
14819a95cf75 added simp attributes
nipkow
parents: 25836
diff changeset
    64
lemma one_le_power[simp]:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
    65
     "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
    66
apply (induct "n")
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    67
apply (simp_all add: power_Suc)
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
    68
apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
    69
apply (simp_all add: zero_le_one order_trans [OF zero_le_one])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    70
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    71
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14577
diff changeset
    72
lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semidom)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    73
  by (simp add: order_trans [OF zero_le_one order_less_imp_le])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    74
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    75
lemma power_gt1_lemma:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
    76
  assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
    77
  shows "1 < a * a^n"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    78
proof -
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
    79
  have "1*1 < a*1" using gt1 by simp
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
    80
  also have "\<dots> \<le> a * a^n" using gt1
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
    81
    by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
    82
        zero_le_one order_refl)
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
    83
  finally show ?thesis by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    84
qed
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    85
25874
14819a95cf75 added simp attributes
nipkow
parents: 25836
diff changeset
    86
lemma one_less_power[simp]:
24376
e403ab5c9415 add lemma one_less_power
huffman
parents: 24286
diff changeset
    87
  "\<lbrakk>1 < (a::'a::{ordered_semidom,recpower}); 0 < n\<rbrakk> \<Longrightarrow> 1 < a ^ n"
e403ab5c9415 add lemma one_less_power
huffman
parents: 24286
diff changeset
    88
by (cases n, simp_all add: power_gt1_lemma power_Suc)
e403ab5c9415 add lemma one_less_power
huffman
parents: 24286
diff changeset
    89
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    90
lemma power_gt1:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
    91
     "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    92
by (simp add: power_gt1_lemma power_Suc)
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    93
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    94
lemma power_le_imp_le_exp:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
    95
  assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
    96
  shows "!!n. a^m \<le> a^n ==> m \<le> n"
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
    97
proof (induct m)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
    98
  case 0
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
    99
  show ?case by simp
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   100
next
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   101
  case (Suc m)
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   102
  show ?case
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   103
  proof (cases n)
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   104
    case 0
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   105
    from prems have "a * a^m \<le> 1" by (simp add: power_Suc)
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   106
    with gt1 show ?thesis
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   107
      by (force simp only: power_gt1_lemma
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   108
          linorder_not_less [symmetric])
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   109
  next
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   110
    case (Suc n)
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   111
    from prems show ?thesis
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   112
      by (force dest: mult_left_le_imp_le
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   113
          simp add: power_Suc order_less_trans [OF zero_less_one gt1])
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   114
  qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   115
qed
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   116
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   117
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
   118
lemma power_inject_exp [simp]:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   119
     "1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   120
  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
   121
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   122
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
   123
natural numbers.*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   124
lemma power_less_imp_less_exp:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   125
     "[| (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   126
by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   127
              power_le_imp_le_exp)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   128
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   129
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   130
lemma power_mono:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   131
     "[|a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   132
apply (induct "n")
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   133
apply (simp_all add: power_Suc)
25874
14819a95cf75 added simp attributes
nipkow
parents: 25836
diff changeset
   134
apply (auto intro: mult_mono order_trans [of 0 a b])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   135
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   136
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   137
lemma power_strict_mono [rule_format]:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   138
     "[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   139
      ==> 0 < n --> a^n < b^n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   140
apply (induct "n")
25874
14819a95cf75 added simp attributes
nipkow
parents: 25836
diff changeset
   141
apply (auto simp add: mult_strict_mono power_Suc
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   142
                      order_le_less_trans [of 0 a b])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   143
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   144
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   145
lemma power_eq_0_iff [simp]:
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   146
  "(a^n = 0) \<longleftrightarrow>
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   147
   (a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,recpower}) & n\<noteq>0)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   148
apply (induct "n")
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   149
apply (auto simp add: power_Suc zero_neq_one [THEN not_sym] no_zero_divisors)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   150
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   151
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   152
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   153
lemma field_power_not_zero:
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   154
  "a \<noteq> (0::'a::{ring_1_no_zero_divisors,recpower}) ==> a^n \<noteq> 0"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   155
by force
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   156
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   157
lemma nonzero_power_inverse:
22991
b9e2a133e84e generalize class restrictions on some lemmas
huffman
parents: 22988
diff changeset
   158
  fixes a :: "'a::{division_ring,recpower}"
b9e2a133e84e generalize class restrictions on some lemmas
huffman
parents: 22988
diff changeset
   159
  shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   160
apply (induct "n")
22988
f6b8184f5b4a generalize some lemmas from field to division_ring
huffman
parents: 22957
diff changeset
   161
apply (auto simp add: power_Suc nonzero_inverse_mult_distrib power_commutes)
22991
b9e2a133e84e generalize class restrictions on some lemmas
huffman
parents: 22988
diff changeset
   162
done (* TODO: reorient or rename to nonzero_inverse_power *)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   163
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   164
text{*Perhaps these should be simprules.*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   165
lemma power_inverse:
22991
b9e2a133e84e generalize class restrictions on some lemmas
huffman
parents: 22988
diff changeset
   166
  fixes a :: "'a::{division_ring,division_by_zero,recpower}"
b9e2a133e84e generalize class restrictions on some lemmas
huffman
parents: 22988
diff changeset
   167
  shows "inverse (a ^ n) = (inverse a) ^ n"
b9e2a133e84e generalize class restrictions on some lemmas
huffman
parents: 22988
diff changeset
   168
apply (cases "a = 0")
b9e2a133e84e generalize class restrictions on some lemmas
huffman
parents: 22988
diff changeset
   169
apply (simp add: power_0_left)
b9e2a133e84e generalize class restrictions on some lemmas
huffman
parents: 22988
diff changeset
   170
apply (simp add: nonzero_power_inverse)
b9e2a133e84e generalize class restrictions on some lemmas
huffman
parents: 22988
diff changeset
   171
done (* TODO: reorient or rename to inverse_power *)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   172
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16733
diff changeset
   173
lemma power_one_over: "1 / (a::'a::{field,division_by_zero,recpower})^n = 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16733
diff changeset
   174
    (1 / a)^n"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16733
diff changeset
   175
apply (simp add: divide_inverse)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16733
diff changeset
   176
apply (rule power_inverse)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16733
diff changeset
   177
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16733
diff changeset
   178
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   179
lemma nonzero_power_divide:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   180
    "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   181
by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   182
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   183
lemma power_divide:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   184
    "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   185
apply (case_tac "b=0", simp add: power_0_left)
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   186
apply (rule nonzero_power_divide)
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   187
apply assumption
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   188
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   189
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   190
lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   191
apply (induct "n")
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   192
apply (auto simp add: power_Suc abs_mult)
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   193
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   194
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23544
diff changeset
   195
lemma zero_less_power_abs_iff [simp,noatp]:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   196
     "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   197
proof (induct "n")
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   198
  case 0
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   199
    show ?case by (simp add: zero_less_one)
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   200
next
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   201
  case (Suc n)
25231
1aa9c8f022d0 simplified proof
haftmann
parents: 25162
diff changeset
   202
    show ?case by (auto simp add: prems power_Suc zero_less_mult_iff
1aa9c8f022d0 simplified proof
haftmann
parents: 25162
diff changeset
   203
      abs_zero)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   204
qed
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   205
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   206
lemma zero_le_power_abs [simp]:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   207
     "(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"
22957
huffman
parents: 22955
diff changeset
   208
by (rule zero_le_power [OF abs_ge_zero])
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   209
28131
3130d7b3149d add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents: 25874
diff changeset
   210
lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{ring_1,recpower}) ^ n"
3130d7b3149d add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents: 25874
diff changeset
   211
proof (induct n)
3130d7b3149d add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents: 25874
diff changeset
   212
  case 0 show ?case by simp
3130d7b3149d add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents: 25874
diff changeset
   213
next
3130d7b3149d add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents: 25874
diff changeset
   214
  case (Suc n) then show ?case
3130d7b3149d add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents: 25874
diff changeset
   215
    by (simp add: power_Suc2 mult_assoc)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   216
qed
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   217
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   218
text{*Lemma for @{text power_strict_decreasing}*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   219
lemma power_Suc_less:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   220
     "[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|]
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   221
      ==> a * a^n < a^n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   222
apply (induct n)
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   223
apply (auto simp add: power_Suc mult_strict_left_mono)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   224
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   225
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   226
lemma power_strict_decreasing:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   227
     "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   228
      ==> a^N < a^n"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   229
apply (erule rev_mp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   230
apply (induct "N")
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   231
apply (auto simp add: power_Suc power_Suc_less less_Suc_eq)
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   232
apply (rename_tac m)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   233
apply (subgoal_tac "a * a^m < 1 * a^n", simp)
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   234
apply (rule mult_strict_mono)
25874
14819a95cf75 added simp attributes
nipkow
parents: 25836
diff changeset
   235
apply (auto simp add: zero_less_one order_less_imp_le)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   236
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   237
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   238
text{*Proof resembles that of @{text power_strict_decreasing}*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   239
lemma power_decreasing:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   240
     "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   241
      ==> a^N \<le> a^n"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   242
apply (erule rev_mp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   243
apply (induct "N")
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   244
apply (auto simp add: power_Suc  le_Suc_eq)
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   245
apply (rename_tac m)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   246
apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   247
apply (rule mult_mono)
25874
14819a95cf75 added simp attributes
nipkow
parents: 25836
diff changeset
   248
apply (auto simp add: zero_le_one)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   249
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   250
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   251
lemma power_Suc_less_one:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   252
     "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   253
apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   254
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   255
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   256
text{*Proof again resembles that of @{text power_strict_decreasing}*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   257
lemma power_increasing:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   258
     "[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   259
apply (erule rev_mp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   260
apply (induct "N")
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   261
apply (auto simp add: power_Suc le_Suc_eq)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   262
apply (rename_tac m)
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   263
apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   264
apply (rule mult_mono)
25874
14819a95cf75 added simp attributes
nipkow
parents: 25836
diff changeset
   265
apply (auto simp add: order_trans [OF zero_le_one])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   266
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   267
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   268
text{*Lemma for @{text power_strict_increasing}*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   269
lemma power_less_power_Suc:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   270
     "(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   271
apply (induct n)
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   272
apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   273
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   274
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   275
lemma power_strict_increasing:
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   276
     "[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   277
apply (erule rev_mp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   278
apply (induct "N")
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   279
apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   280
apply (rename_tac m)
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   281
apply (subgoal_tac "1 * a^n < a * a^m", simp)
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   282
apply (rule mult_strict_mono)
25874
14819a95cf75 added simp attributes
nipkow
parents: 25836
diff changeset
   283
apply (auto simp add: order_less_trans [OF zero_less_one] order_less_imp_le)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   284
done
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   285
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   286
lemma power_increasing_iff [simp]:
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   287
  "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   288
by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) 
15066
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   289
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   290
lemma power_strict_increasing_iff [simp]:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   291
  "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   292
by (blast intro: power_less_imp_less_exp power_strict_increasing) 
15066
d2f2b908e0a4 two new results
paulson
parents: 15004
diff changeset
   293
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   294
lemma power_le_imp_le_base:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   295
assumes le: "a ^ Suc n \<le> b ^ Suc n"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   296
    and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   297
shows "a \<le> b"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   298
proof (rule ccontr)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   299
  assume "~ a \<le> b"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   300
  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
   301
  then have "b ^ Suc n < a ^ Suc n"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   302
    by (simp only: prems power_strict_mono)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   303
  from le and this show "False"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   304
    by (simp add: linorder_not_less [symmetric])
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   305
qed
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   306
22853
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   307
lemma power_less_imp_less_base:
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   308
  fixes a b :: "'a::{ordered_semidom,recpower}"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   309
  assumes less: "a ^ n < b ^ n"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   310
  assumes nonneg: "0 \<le> b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   311
  shows "a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   312
proof (rule contrapos_pp [OF less])
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   313
  assume "~ a < b"
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   314
  hence "b \<le> a" by (simp only: linorder_not_less)
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   315
  hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   316
  thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less)
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   317
qed
7f000a385606 add lemma power_less_imp_less_base
huffman
parents: 22624
diff changeset
   318
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   319
lemma power_inject_base:
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   320
     "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
15004
44ac09ba7213 ringpower to recpower
paulson
parents: 14738
diff changeset
   321
      ==> a = (b::'a::{ordered_semidom,recpower})"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   322
by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   323
22955
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   324
lemma power_eq_imp_eq_base:
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   325
  fixes a b :: "'a::{ordered_semidom,recpower}"
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   326
  shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b"
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   327
by (cases n, simp_all, rule power_inject_base)
48dc37776d1e add lemma power_eq_imp_eq_base
huffman
parents: 22853
diff changeset
   328
29978
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   329
text {* The divides relation *}
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   330
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   331
lemma le_imp_power_dvd:
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   332
  fixes a :: "'a::{comm_semiring_1,recpower}"
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   333
  assumes "m \<le> n" shows "a^m dvd a^n"
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   334
proof
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   335
  have "a^n = a^(m + (n - m))"
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   336
    using `m \<le> n` by simp
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   337
  also have "\<dots> = a^m * a^(n - m)"
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   338
    by (rule power_add)
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   339
  finally show "a^n = a^m * a^(n - m)" .
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   340
qed
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   341
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   342
lemma power_le_dvd:
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   343
  fixes a b :: "'a::{comm_semiring_1,recpower}"
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   344
  shows "a^n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a^m dvd b"
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   345
  by (rule dvd_trans [OF le_imp_power_dvd])
33df3c4eb629 generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents: 29608
diff changeset
   346
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   347
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   348
subsection{*Exponentiation for the Natural Numbers*}
3390
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
   349
25836
f7771e4f7064 more instantiation
haftmann
parents: 25231
diff changeset
   350
instantiation nat :: recpower
f7771e4f7064 more instantiation
haftmann
parents: 25231
diff changeset
   351
begin
21456
1c2b9df41e98 cleanup
haftmann
parents: 21413
diff changeset
   352
25836
f7771e4f7064 more instantiation
haftmann
parents: 25231
diff changeset
   353
primrec power_nat where
f7771e4f7064 more instantiation
haftmann
parents: 25231
diff changeset
   354
  "p ^ 0 = (1\<Colon>nat)"
f7771e4f7064 more instantiation
haftmann
parents: 25231
diff changeset
   355
  | "p ^ (Suc n) = (p\<Colon>nat) * (p ^ n)"
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   356
25836
f7771e4f7064 more instantiation
haftmann
parents: 25231
diff changeset
   357
instance proof
14438
paulson
parents: 14353
diff changeset
   358
  fix z n :: nat
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   359
  show "z^0 = 1" by simp
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   360
  show "z^(Suc n) = z * (z^n)" by simp
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   361
qed
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   362
25836
f7771e4f7064 more instantiation
haftmann
parents: 25231
diff changeset
   363
end
f7771e4f7064 more instantiation
haftmann
parents: 25231
diff changeset
   364
23305
8ae6f7b0903b add lemma of_nat_power
huffman
parents: 23183
diff changeset
   365
lemma of_nat_power:
8ae6f7b0903b add lemma of_nat_power
huffman
parents: 23183
diff changeset
   366
  "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"
23431
25ca91279a9b change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents: 23305
diff changeset
   367
by (induct n, simp_all add: power_Suc of_nat_mult)
23305
8ae6f7b0903b add lemma of_nat_power
huffman
parents: 23183
diff changeset
   368
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   369
lemma nat_one_le_power [simp]: "Suc 0 \<le> i ==> Suc 0 \<le> i^n"
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   370
by (rule one_le_power [of i n, unfolded One_nat_def])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   371
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   372
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   373
by (induct "n", auto)
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   374
30056
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   375
lemma nat_power_eq_Suc_0_iff [simp]: 
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   376
  "((x::nat)^m = Suc 0) = (m = 0 | x = Suc 0)"
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   377
by (induct_tac m, auto)
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   378
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   379
lemma power_Suc_0[simp]: "(Suc 0)^n = Suc 0"
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   380
by simp
0a35bee25c20 added lemmas
nipkow
parents: 29978
diff changeset
   381
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   382
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
   383
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
   384
@{term "m=1"} and @{term "n=0"}.*}
21413
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   385
lemma nat_power_less_imp_less:
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   386
  assumes nonneg: "0 < (i\<Colon>nat)"
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   387
  assumes less: "i^m < i^n"
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   388
  shows "m < n"
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   389
proof (cases "i = 1")
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   390
  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
   391
next
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   392
  case False with nonneg have "1 < i" by auto
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   393
  from power_strict_increasing_iff [OF this] less show ?thesis ..
0951647209f2 moved dvd stuff to theory Divides
haftmann
parents: 21199
diff changeset
   394
qed
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   395
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16796
diff changeset
   396
lemma power_diff:
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16796
diff changeset
   397
  assumes nz: "a ~= 0"
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16796
diff changeset
   398
  shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)"
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16796
diff changeset
   399
  by (induct m n rule: diff_induct)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16796
diff changeset
   400
    (simp_all add: power_Suc nonzero_mult_divide_cancel_left nz)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16796
diff changeset
   401
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16796
diff changeset
   402
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   403
text{*ML bindings for the general exponentiation theorems*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   404
ML
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   405
{*
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   406
val power_0 = thm"power_0";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   407
val power_Suc = thm"power_Suc";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   408
val power_0_Suc = thm"power_0_Suc";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   409
val power_0_left = thm"power_0_left";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   410
val power_one = thm"power_one";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   411
val power_one_right = thm"power_one_right";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   412
val power_add = thm"power_add";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   413
val power_mult = thm"power_mult";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   414
val power_mult_distrib = thm"power_mult_distrib";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   415
val zero_less_power = thm"zero_less_power";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   416
val zero_le_power = thm"zero_le_power";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   417
val one_le_power = thm"one_le_power";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   418
val gt1_imp_ge0 = thm"gt1_imp_ge0";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   419
val power_gt1_lemma = thm"power_gt1_lemma";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   420
val power_gt1 = thm"power_gt1";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   421
val power_le_imp_le_exp = thm"power_le_imp_le_exp";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   422
val power_inject_exp = thm"power_inject_exp";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   423
val power_less_imp_less_exp = thm"power_less_imp_less_exp";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   424
val power_mono = thm"power_mono";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   425
val power_strict_mono = thm"power_strict_mono";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   426
val power_eq_0_iff = thm"power_eq_0_iff";
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25062
diff changeset
   427
val field_power_eq_0_iff = thm"power_eq_0_iff";
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   428
val field_power_not_zero = thm"field_power_not_zero";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   429
val power_inverse = thm"power_inverse";
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   430
val nonzero_power_divide = thm"nonzero_power_divide";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   431
val power_divide = thm"power_divide";
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   432
val power_abs = thm"power_abs";
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   433
val zero_less_power_abs_iff = thm"zero_less_power_abs_iff";
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   434
val zero_le_power_abs = thm "zero_le_power_abs";
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   435
val power_minus = thm"power_minus";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   436
val power_Suc_less = thm"power_Suc_less";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   437
val power_strict_decreasing = thm"power_strict_decreasing";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   438
val power_decreasing = thm"power_decreasing";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   439
val power_Suc_less_one = thm"power_Suc_less_one";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   440
val power_increasing = thm"power_increasing";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   441
val power_strict_increasing = thm"power_strict_increasing";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   442
val power_le_imp_le_base = thm"power_le_imp_le_base";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   443
val power_inject_base = thm"power_inject_base";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   444
*}
14577
dbb95b825244 tuned document;
wenzelm
parents: 14438
diff changeset
   445
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   446
text{*ML bindings for the remaining theorems*}
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   447
ML
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   448
{*
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   449
val nat_one_le_power = thm"nat_one_le_power";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   450
val nat_power_less_imp_less = thm"nat_power_less_imp_less";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   451
val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 8844
diff changeset
   452
*}
3390
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
   453
0c7625196d95 New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff changeset
   454
end