author  huffman 
Wed, 04 Mar 2009 17:12:23 0800  
changeset 30273  ecd6f0ca62ea 
parent 30242  aea5d7fa7ef5 
child 30313  b2441b0c8d38 
permissions  rwrr 
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Defining the type class "ringpower" and deleting superseded theorems for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  10 
theory Power 
21413  11 
imports Nat 
15131  12 
begin 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

13 

29608  14 
class power = 
25062  15 
fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) 
24996  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Defining the type class "ringpower" and deleting superseded theorems for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  19 
class recpower = monoid_mult + power + 
25062  20 
assumes power_0 [simp]: "a ^ 0 = 1" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

21 
assumes power_Suc [simp]: "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Defining the type class "ringpower" and deleting superseded theorems for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" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

24 
by simp 
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  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  30 
lemma power_one [simp]: "1^n = (1::'a::recpower)" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

31 
by (induct n) simp_all 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

32 

15004  33 
lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

34 
unfolding One_nat_def by simp 
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" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

37 
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

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" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

40 
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

41 

15004  42 
lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

43 
by (induct m) (simp_all add: mult_ac) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

44 

15004  45 
lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

46 
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

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)" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

49 
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

50 

25874  51 
lemma zero_less_power[simp]: 
15004  52 
"0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

53 
by (induct n) (simp_all add: mult_pos_pos) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

54 

25874  55 
lemma zero_le_power[simp]: 
15004  56 
"0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

57 
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

58 

25874  59 
lemma one_le_power[simp]: 
15004  60 
"1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n" 
15251  61 
apply (induct "n") 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

62 
apply simp_all 
14577  63 
apply (rule order_trans [OF _ mult_mono [of 1 _ 1]]) 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

64 
apply (simp_all add: order_trans [OF zero_le_one]) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

65 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

66 

14738  67 
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

68 
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

69 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

70 
lemma power_gt1_lemma: 
15004  71 
assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})" 
14577  72 
shows "1 < a * a^n" 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

73 
proof  
14577  74 
have "1*1 < a*1" using gt1 by simp 
75 
also have "\<dots> \<le> a * a^n" using gt1 

76 
by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le 

77 
zero_le_one order_refl) 

78 
finally show ?thesis by simp 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

79 
qed 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

80 

25874  81 
lemma one_less_power[simp]: 
24376  82 
"\<lbrakk>1 < (a::'a::{ordered_semidom,recpower}); 0 < n\<rbrakk> \<Longrightarrow> 1 < a ^ n" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

83 
by (cases n, simp_all add: power_gt1_lemma) 
24376  84 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

85 
lemma power_gt1: 
15004  86 
"1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

87 
by (simp add: power_gt1_lemma) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

88 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

89 
lemma power_le_imp_le_exp: 
15004  90 
assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a" 
14577  91 
shows "!!n. a^m \<le> a^n ==> m \<le> n" 
92 
proof (induct m) 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

93 
case 0 
14577  94 
show ?case by simp 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

95 
next 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

96 
case (Suc m) 
14577  97 
show ?case 
98 
proof (cases n) 

99 
case 0 

30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

100 
from prems have "a * a^m \<le> 1" by simp 
14577  101 
with gt1 show ?thesis 
102 
by (force simp only: power_gt1_lemma 

103 
linorder_not_less [symmetric]) 

104 
next 

105 
case (Suc n) 

106 
from prems show ?thesis 

107 
by (force dest: mult_left_le_imp_le 

30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

108 
simp add: order_less_trans [OF zero_less_one gt1]) 
14577  109 
qed 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

110 
qed 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

111 

14577  112 
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

113 
lemma power_inject_exp [simp]: 
15004  114 
"1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)" 
14577  115 
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

116 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

117 
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

118 
natural numbers.*} 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

119 
lemma power_less_imp_less_exp: 
15004  120 
"[ (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n ] ==> m < n" 
14577  121 
by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"] 
122 
power_le_imp_le_exp) 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

123 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

124 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

125 
lemma power_mono: 
15004  126 
"[a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a] ==> a^n \<le> b^n" 
15251  127 
apply (induct "n") 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

128 
apply simp_all 
25874  129 
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

130 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

131 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

132 
lemma power_strict_mono [rule_format]: 
15004  133 
"[a < b; (0::'a::{recpower,ordered_semidom}) \<le> a] 
14577  134 
==> 0 < n > a^n < b^n" 
15251  135 
apply (induct "n") 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

136 
apply (auto simp add: mult_strict_mono order_le_less_trans [of 0 a b]) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

137 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

138 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

139 
lemma power_eq_0_iff [simp]: 
30056  140 
"(a^n = 0) \<longleftrightarrow> 
141 
(a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,recpower}) & n\<noteq>0)" 

15251  142 
apply (induct "n") 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

143 
apply (auto simp add: no_zero_divisors) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

144 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

145 

30056  146 

25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

147 
lemma field_power_not_zero: 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

148 
"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

149 
by force 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

150 

14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

151 
lemma nonzero_power_inverse: 
22991  152 
fixes a :: "'a::{division_ring,recpower}" 
153 
shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n" 

15251  154 
apply (induct "n") 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

155 
apply (auto simp add: nonzero_inverse_mult_distrib power_commutes) 
22991  156 
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

157 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

158 
text{*Perhaps these should be simprules.*} 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

159 
lemma power_inverse: 
22991  160 
fixes a :: "'a::{division_ring,division_by_zero,recpower}" 
161 
shows "inverse (a ^ n) = (inverse a) ^ n" 

162 
apply (cases "a = 0") 

163 
apply (simp add: power_0_left) 

164 
apply (simp add: nonzero_power_inverse) 

165 
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

166 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16733
diff
changeset

167 
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

168 
(1 / a)^n" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16733
diff
changeset

169 
apply (simp add: divide_inverse) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16733
diff
changeset

170 
apply (rule power_inverse) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16733
diff
changeset

171 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16733
diff
changeset

172 

14577  173 
lemma nonzero_power_divide: 
15004  174 
"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

175 
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

176 

14577  177 
lemma power_divide: 
15004  178 
"(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

179 
apply (case_tac "b=0", simp add: power_0_left) 
14577  180 
apply (rule nonzero_power_divide) 
181 
apply assumption 

14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

182 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

183 

15004  184 
lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n" 
15251  185 
apply (induct "n") 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

186 
apply (auto simp add: abs_mult) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

187 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

188 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23544
diff
changeset

189 
lemma zero_less_power_abs_iff [simp,noatp]: 
15004  190 
"(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

191 
proof (induct "n") 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

192 
case 0 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

193 
show ?case by simp 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

194 
next 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

195 
case (Suc n) 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

196 
show ?case by (auto simp add: prems zero_less_mult_iff) 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

197 
qed 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

198 

79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

199 
lemma zero_le_power_abs [simp]: 
15004  200 
"(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n" 
22957  201 
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

202 

28131
3130d7b3149d
add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents:
25874
diff
changeset

203 
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

204 
proof (induct n) 
3130d7b3149d
add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents:
25874
diff
changeset

205 
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

206 
next 
3130d7b3149d
add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents:
25874
diff
changeset

207 
case (Suc n) then show ?case 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

208 
by (simp del: power_Suc add: power_Suc2 mult_assoc) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

209 
qed 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

210 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

211 
text{*Lemma for @{text power_strict_decreasing}*} 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

212 
lemma power_Suc_less: 
15004  213 
"[(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

214 
==> a * a^n < a^n" 
15251  215 
apply (induct n) 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

216 
apply (auto simp add: mult_strict_left_mono) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

217 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

218 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

219 
lemma power_strict_decreasing: 
15004  220 
"[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

221 
==> a^N < a^n" 
14577  222 
apply (erule rev_mp) 
15251  223 
apply (induct "N") 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

224 
apply (auto simp add: power_Suc_less less_Suc_eq) 
14577  225 
apply (rename_tac m) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

226 
apply (subgoal_tac "a * a^m < 1 * a^n", simp) 
14577  227 
apply (rule mult_strict_mono) 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

228 
apply (auto simp add: order_less_imp_le) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

229 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

230 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

231 
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

232 
lemma power_decreasing: 
15004  233 
"[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

234 
==> a^N \<le> a^n" 
14577  235 
apply (erule rev_mp) 
15251  236 
apply (induct "N") 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

237 
apply (auto simp add: le_Suc_eq) 
14577  238 
apply (rename_tac m) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

239 
apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp) 
14577  240 
apply (rule mult_mono) 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

241 
apply auto 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

242 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

243 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

244 
lemma power_Suc_less_one: 
15004  245 
"[ 0 < a; a < (1::'a::{ordered_semidom,recpower}) ] ==> a ^ Suc n < 1" 
14577  246 
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

247 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

248 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

249 
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

250 
lemma power_increasing: 
15004  251 
"[n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a] ==> a^n \<le> a^N" 
14577  252 
apply (erule rev_mp) 
15251  253 
apply (induct "N") 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

254 
apply (auto simp add: le_Suc_eq) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

255 
apply (rename_tac m) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

256 
apply (subgoal_tac "1 * a^n \<le> a * a^m", simp) 
14577  257 
apply (rule mult_mono) 
25874  258 
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

259 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

260 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

261 
text{*Lemma for @{text power_strict_increasing}*} 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

262 
lemma power_less_power_Suc: 
15004  263 
"(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n" 
15251  264 
apply (induct n) 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

265 
apply (auto simp add: 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

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 
lemma power_strict_increasing: 
15004  269 
"[n < N; (1::'a::{ordered_semidom,recpower}) < a] ==> a^n < a^N" 
14577  270 
apply (erule rev_mp) 
15251  271 
apply (induct "N") 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

272 
apply (auto simp add: power_less_power_Suc less_Suc_eq) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

273 
apply (rename_tac m) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

274 
apply (subgoal_tac "1 * a^n < a * a^m", simp) 
14577  275 
apply (rule mult_strict_mono) 
25874  276 
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

277 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

278 

25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

279 
lemma power_increasing_iff [simp]: 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

280 
"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

281 
by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) 
15066  282 

283 
lemma power_strict_increasing_iff [simp]: 

25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

284 
"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

285 
by (blast intro: power_less_imp_less_exp power_strict_increasing) 
15066  286 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

287 
lemma power_le_imp_le_base: 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

288 
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

289 
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

290 
shows "a \<le> b" 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

291 
proof (rule ccontr) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

292 
assume "~ a \<le> b" 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

293 
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

294 
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

295 
by (simp only: prems power_strict_mono) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

296 
from le and this show "False" 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

297 
by (simp add: linorder_not_less [symmetric]) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

298 
qed 
14577  299 

22853  300 
lemma power_less_imp_less_base: 
301 
fixes a b :: "'a::{ordered_semidom,recpower}" 

302 
assumes less: "a ^ n < b ^ n" 

303 
assumes nonneg: "0 \<le> b" 

304 
shows "a < b" 

305 
proof (rule contrapos_pp [OF less]) 

306 
assume "~ a < b" 

307 
hence "b \<le> a" by (simp only: linorder_not_less) 

308 
hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono) 

309 
thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less) 

310 
qed 

311 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

312 
lemma power_inject_base: 
14577  313 
"[ a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b ] 
15004  314 
==> a = (b::'a::{ordered_semidom,recpower})" 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

315 
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

316 

22955  317 
lemma power_eq_imp_eq_base: 
318 
fixes a b :: "'a::{ordered_semidom,recpower}" 

319 
shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b" 

30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

320 
by (cases n, simp_all del: power_Suc, rule power_inject_base) 
22955  321 

29978
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset

322 
text {* The divides relation *} 
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset

323 

33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset

324 
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

325 
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

326 
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

327 
proof 
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset

328 
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

329 
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

330 
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

331 
by (rule power_add) 
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset

332 
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

333 
qed 
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset

334 

33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset

335 
lemma power_le_dvd: 
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset

336 
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

337 
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

338 
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

339 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

340 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

341 
subsection{*Exponentiation for the Natural Numbers*} 
3390
0c7625196d95
New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff
changeset

342 

25836  343 
instantiation nat :: recpower 
344 
begin 

21456  345 

25836  346 
primrec power_nat where 
347 
"p ^ 0 = (1\<Colon>nat)" 

348 
 "p ^ (Suc n) = (p\<Colon>nat) * (p ^ n)" 

14577  349 

25836  350 
instance proof 
14438  351 
fix z n :: nat 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

352 
show "z^0 = 1" by simp 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

353 
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

354 
qed 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

355 

30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

356 
declare power_nat.simps [simp del] 
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

357 

25836  358 
end 
359 

23305  360 
lemma of_nat_power: 
361 
"of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n" 

30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

362 
by (induct n, simp_all add: of_nat_mult) 
23305  363 

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

364 
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

365 
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

366 

25162  367 
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat)  n=0)" 
21413  368 
by (induct "n", auto) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

369 

30056  370 
lemma nat_power_eq_Suc_0_iff [simp]: 
371 
"((x::nat)^m = Suc 0) = (m = 0  x = Suc 0)" 

372 
by (induct_tac m, auto) 

373 

374 
lemma power_Suc_0[simp]: "(Suc 0)^n = Suc 0" 

375 
by simp 

376 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

377 
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

378 
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

379 
@{term "m=1"} and @{term "n=0"}.*} 
21413  380 
lemma nat_power_less_imp_less: 
381 
assumes nonneg: "0 < (i\<Colon>nat)" 

382 
assumes less: "i^m < i^n" 

383 
shows "m < n" 

384 
proof (cases "i = 1") 

385 
case True with less power_one [where 'a = nat] show ?thesis by simp 

386 
next 

387 
case False with nonneg have "1 < i" by auto 

388 
from power_strict_increasing_iff [OF this] less show ?thesis .. 

389 
qed 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

390 

17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
16796
diff
changeset

391 
lemma power_diff: 
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
16796
diff
changeset

392 
assumes nz: "a ~= 0" 
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
16796
diff
changeset

393 
shows "n <= m ==> (a::'a::{recpower, field}) ^ (mn) = (a^m) / (a^n)" 
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
16796
diff
changeset

394 
by (induct m n rule: diff_induct) 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30242
diff
changeset

395 
(simp_all add: nonzero_mult_divide_cancel_left nz) 
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
16796
diff
changeset

396 

e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
16796
diff
changeset

397 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

398 
text{*ML bindings for the general exponentiation theorems*} 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

399 
ML 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

400 
{* 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

401 
val power_0 = thm"power_0"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

402 
val power_Suc = thm"power_Suc"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

403 
val power_0_Suc = thm"power_0_Suc"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

404 
val power_0_left = thm"power_0_left"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

405 
val power_one = thm"power_one"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

406 
val power_one_right = thm"power_one_right"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

407 
val power_add = thm"power_add"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

408 
val power_mult = thm"power_mult"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

409 
val power_mult_distrib = thm"power_mult_distrib"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

410 
val zero_less_power = thm"zero_less_power"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

411 
val zero_le_power = thm"zero_le_power"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

412 
val one_le_power = thm"one_le_power"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

413 
val gt1_imp_ge0 = thm"gt1_imp_ge0"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

414 
val power_gt1_lemma = thm"power_gt1_lemma"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

415 
val power_gt1 = thm"power_gt1"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

416 
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

417 
val power_inject_exp = thm"power_inject_exp"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

418 
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

419 
val power_mono = thm"power_mono"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

420 
val power_strict_mono = thm"power_strict_mono"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

421 
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

422 
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

423 
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

424 
val power_inverse = thm"power_inverse"; 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

425 
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

426 
val power_divide = thm"power_divide"; 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

427 
val power_abs = thm"power_abs"; 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

428 
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

429 
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

430 
val power_minus = thm"power_minus"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

431 
val power_Suc_less = thm"power_Suc_less"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

432 
val power_strict_decreasing = thm"power_strict_decreasing"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

433 
val power_decreasing = thm"power_decreasing"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

434 
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

435 
val power_increasing = thm"power_increasing"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

436 
val power_strict_increasing = thm"power_strict_increasing"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

437 
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

438 
val power_inject_base = thm"power_inject_base"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

439 
*} 
14577  440 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

441 
text{*ML bindings for the remaining theorems*} 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

442 
ML 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

443 
{* 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

444 
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

445 
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

446 
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

447 
*} 
3390
0c7625196d95
New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff
changeset

448 

0c7625196d95
New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff
changeset

449 
end 