author  huffman 
Wed, 18 Feb 2009 10:24:48 0800  
changeset 29978  33df3c4eb629 
parent 29608  564ea783ace8 
child 30056  0a35bee25c20 
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Defining the type class "ringpower" and deleting superseded theorems for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Defining the type class "ringpower" and deleting superseded theorems for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Defining the type class "ringpower" and deleting superseded theorems for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" 
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Defining the type class "ringpower" and deleting superseded theorems for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  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  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)" 
23183  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  33 
lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a" 
23183  34 
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  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  42 
lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)" 
23183  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  45 
lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n" 
23183  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  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  51 
lemma zero_less_power[simp]: 
15004  52 
"0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n" 
15251  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  57 
lemma zero_le_power[simp]: 
15004  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  60 
apply (erule disjE) 
25874  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  64 
lemma one_le_power[simp]: 
15004  65 
"1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n" 
15251  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  68 
apply (rule order_trans [OF _ mult_mono [of 1 _ 1]]) 
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  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  76 
assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})" 
14577  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  79 
have "1*1 < a*1" using gt1 by simp 
80 
also have "\<dots> \<le> a * a^n" using gt1 

81 
by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le 

82 
zero_le_one order_refl) 

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  86 
lemma one_less_power[simp]: 
24376  87 
"\<lbrakk>1 < (a::'a::{ordered_semidom,recpower}); 0 < n\<rbrakk> \<Longrightarrow> 1 < a ^ n" 
88 
by (cases n, simp_all add: power_gt1_lemma power_Suc) 

89 

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

90 
lemma power_gt1: 
15004  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  95 
assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a" 
14577  96 
shows "!!n. a^m \<le> a^n ==> m \<le> n" 
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  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  102 
show ?case 
103 
proof (cases n) 

104 
case 0 

105 
from prems have "a * a^m \<le> 1" by (simp add: power_Suc) 

106 
with gt1 show ?thesis 

107 
by (force simp only: power_gt1_lemma 

108 
linorder_not_less [symmetric]) 

109 
next 

110 
case (Suc n) 

111 
from prems show ?thesis 

112 
by (force dest: mult_left_le_imp_le 

113 
simp add: power_Suc order_less_trans [OF zero_less_one gt1]) 

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  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  119 
"1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)" 
14577  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  125 
"[ (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n ] ==> m < n" 
14577  126 
by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"] 
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  131 
"[a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a] ==> a^n \<le> b^n" 
15251  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  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  138 
"[a < b; (0::'a::{recpower,ordered_semidom}) \<le> a] 
14577  139 
==> 0 < n > a^n < b^n" 
15251  140 
apply (induct "n") 
25874  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]: 
25162  146 
"(a^n = 0) = (a = (0::'a::{ring_1_no_zero_divisors,recpower}) & n>0)" 
15251  147 
apply (induct "n") 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

148 
apply (auto simp add: power_Suc zero_neq_one [THEN not_sym]) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

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

150 

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

151 
lemma field_power_not_zero: 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

152 
"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

153 
by force 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

154 

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

155 
lemma nonzero_power_inverse: 
22991  156 
fixes a :: "'a::{division_ring,recpower}" 
157 
shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n" 

15251  158 
apply (induct "n") 
22988
f6b8184f5b4a
generalize some lemmas from field to division_ring
huffman
parents:
22957
diff
changeset

159 
apply (auto simp add: power_Suc nonzero_inverse_mult_distrib power_commutes) 
22991  160 
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

161 

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

162 
text{*Perhaps these should be simprules.*} 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

163 
lemma power_inverse: 
22991  164 
fixes a :: "'a::{division_ring,division_by_zero,recpower}" 
165 
shows "inverse (a ^ n) = (inverse a) ^ n" 

166 
apply (cases "a = 0") 

167 
apply (simp add: power_0_left) 

168 
apply (simp add: nonzero_power_inverse) 

169 
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

170 

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

171 
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

172 
(1 / a)^n" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16733
diff
changeset

173 
apply (simp add: divide_inverse) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16733
diff
changeset

174 
apply (rule power_inverse) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16733
diff
changeset

175 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16733
diff
changeset

176 

14577  177 
lemma nonzero_power_divide: 
15004  178 
"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

179 
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

180 

14577  181 
lemma power_divide: 
15004  182 
"(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

183 
apply (case_tac "b=0", simp add: power_0_left) 
14577  184 
apply (rule nonzero_power_divide) 
185 
apply assumption 

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

186 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

187 

15004  188 
lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n" 
15251  189 
apply (induct "n") 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

190 
apply (auto simp add: power_Suc abs_mult) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

191 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

192 

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

193 
lemma zero_less_power_abs_iff [simp,noatp]: 
15004  194 
"(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

195 
proof (induct "n") 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

196 
case 0 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

197 
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

198 
next 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

199 
case (Suc n) 
25231  200 
show ?case by (auto simp add: prems power_Suc zero_less_mult_iff 
201 
abs_zero) 

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

202 
qed 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

203 

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

204 
lemma zero_le_power_abs [simp]: 
15004  205 
"(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n" 
22957  206 
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

207 

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

208 
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

209 
proof (induct n) 
3130d7b3149d
add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents:
25874
diff
changeset

210 
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

211 
next 
3130d7b3149d
add lemma power_Suc2; generalize power_minus from class comm_ring_1 to ring_1
huffman
parents:
25874
diff
changeset

212 
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

213 
by (simp add: power_Suc2 mult_assoc) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

214 
qed 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

215 

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

216 
text{*Lemma for @{text power_strict_decreasing}*} 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

217 
lemma power_Suc_less: 
15004  218 
"[(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

219 
==> a * a^n < a^n" 
15251  220 
apply (induct n) 
14577  221 
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

222 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

223 

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

224 
lemma power_strict_decreasing: 
15004  225 
"[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

226 
==> a^N < a^n" 
14577  227 
apply (erule rev_mp) 
15251  228 
apply (induct "N") 
14577  229 
apply (auto simp add: power_Suc power_Suc_less less_Suc_eq) 
230 
apply (rename_tac m) 

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

231 
apply (subgoal_tac "a * a^m < 1 * a^n", simp) 
14577  232 
apply (rule mult_strict_mono) 
25874  233 
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

234 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

235 

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

236 
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

237 
lemma power_decreasing: 
15004  238 
"[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

239 
==> a^N \<le> a^n" 
14577  240 
apply (erule rev_mp) 
15251  241 
apply (induct "N") 
14577  242 
apply (auto simp add: power_Suc le_Suc_eq) 
243 
apply (rename_tac m) 

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

244 
apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp) 
14577  245 
apply (rule mult_mono) 
25874  246 
apply (auto simp add: zero_le_one) 
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 
lemma power_Suc_less_one: 
15004  250 
"[ 0 < a; a < (1::'a::{ordered_semidom,recpower}) ] ==> a ^ Suc n < 1" 
14577  251 
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

252 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

253 

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

254 
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

255 
lemma power_increasing: 
15004  256 
"[n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a] ==> a^n \<le> a^N" 
14577  257 
apply (erule rev_mp) 
15251  258 
apply (induct "N") 
14577  259 
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

260 
apply (rename_tac m) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

261 
apply (subgoal_tac "1 * a^n \<le> a * a^m", simp) 
14577  262 
apply (rule mult_mono) 
25874  263 
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

264 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

265 

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

266 
text{*Lemma for @{text power_strict_increasing}*} 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

267 
lemma power_less_power_Suc: 
15004  268 
"(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n" 
15251  269 
apply (induct n) 
14577  270 
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

271 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

272 

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

273 
lemma power_strict_increasing: 
15004  274 
"[n < N; (1::'a::{ordered_semidom,recpower}) < a] ==> a^n < a^N" 
14577  275 
apply (erule rev_mp) 
15251  276 
apply (induct "N") 
14577  277 
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

278 
apply (rename_tac m) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

279 
apply (subgoal_tac "1 * a^n < a * a^m", simp) 
14577  280 
apply (rule mult_strict_mono) 
25874  281 
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

282 
done 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

283 

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

284 
lemma power_increasing_iff [simp]: 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

285 
"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

286 
by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) 
15066  287 

288 
lemma power_strict_increasing_iff [simp]: 

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

289 
"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

290 
by (blast intro: power_less_imp_less_exp power_strict_increasing) 
15066  291 

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

292 
lemma power_le_imp_le_base: 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

293 
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

294 
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

295 
shows "a \<le> b" 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

296 
proof (rule ccontr) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

297 
assume "~ a \<le> b" 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

298 
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

299 
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

300 
by (simp only: prems power_strict_mono) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

301 
from le and this show "False" 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

302 
by (simp add: linorder_not_less [symmetric]) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25062
diff
changeset

303 
qed 
14577  304 

22853  305 
lemma power_less_imp_less_base: 
306 
fixes a b :: "'a::{ordered_semidom,recpower}" 

307 
assumes less: "a ^ n < b ^ n" 

308 
assumes nonneg: "0 \<le> b" 

309 
shows "a < b" 

310 
proof (rule contrapos_pp [OF less]) 

311 
assume "~ a < b" 

312 
hence "b \<le> a" by (simp only: linorder_not_less) 

313 
hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono) 

314 
thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less) 

315 
qed 

316 

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

317 
lemma power_inject_base: 
14577  318 
"[ a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b ] 
15004  319 
==> a = (b::'a::{ordered_semidom,recpower})" 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

320 
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

321 

22955  322 
lemma power_eq_imp_eq_base: 
323 
fixes a b :: "'a::{ordered_semidom,recpower}" 

324 
shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b" 

325 
by (cases n, simp_all, rule power_inject_base) 

326 

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

327 
text {* The divides relation *} 
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset

328 

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

329 
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

330 
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

331 
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

332 
proof 
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset

333 
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

334 
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

335 
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

336 
by (rule power_add) 
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset

337 
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

338 
qed 
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset

339 

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

340 
lemma power_le_dvd: 
33df3c4eb629
generalize le_imp_power_dvd and power_le_dvd; move from Divides to Power
huffman
parents:
29608
diff
changeset

341 
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

342 
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

343 
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

344 

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

345 

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

346 
subsection{*Exponentiation for the Natural Numbers*} 
3390
0c7625196d95
New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff
changeset

347 

25836  348 
instantiation nat :: recpower 
349 
begin 

21456  350 

25836  351 
primrec power_nat where 
352 
"p ^ 0 = (1\<Colon>nat)" 

353 
 "p ^ (Suc n) = (p\<Colon>nat) * (p ^ n)" 

14577  354 

25836  355 
instance proof 
14438  356 
fix z n :: nat 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

357 
show "z^0 = 1" by simp 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

358 
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

359 
qed 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

360 

25836  361 
end 
362 

23305  363 
lemma of_nat_power: 
364 
"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

365 
by (induct n, simp_all add: power_Suc of_nat_mult) 
23305  366 

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

367 
lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n" 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

368 
by (insert one_le_power [of i n], simp) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

369 

25162  370 
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat)  n=0)" 
21413  371 
by (induct "n", auto) 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

372 

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

373 
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

374 
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

375 
@{term "m=1"} and @{term "n=0"}.*} 
21413  376 
lemma nat_power_less_imp_less: 
377 
assumes nonneg: "0 < (i\<Colon>nat)" 

378 
assumes less: "i^m < i^n" 

379 
shows "m < n" 

380 
proof (cases "i = 1") 

381 
case True with less power_one [where 'a = nat] show ?thesis by simp 

382 
next 

383 
case False with nonneg have "1 < i" by auto 

384 
from power_strict_increasing_iff [OF this] less show ?thesis .. 

385 
qed 

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

386 

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

387 
lemma power_diff: 
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
16796
diff
changeset

388 
assumes nz: "a ~= 0" 
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
16796
diff
changeset

389 
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

390 
by (induct m n rule: diff_induct) 
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
16796
diff
changeset

391 
(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

392 

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

393 

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

394 
text{*ML bindings for the general exponentiation theorems*} 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

395 
ML 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

396 
{* 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

397 
val power_0 = thm"power_0"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

398 
val power_Suc = thm"power_Suc"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

399 
val power_0_Suc = thm"power_0_Suc"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

400 
val power_0_left = thm"power_0_left"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

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

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

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

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

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

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

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

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

409 
val gt1_imp_ge0 = thm"gt1_imp_ge0"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

410 
val power_gt1_lemma = thm"power_gt1_lemma"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

411 
val power_gt1 = thm"power_gt1"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

412 
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

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

414 
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

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

416 
val power_strict_mono = thm"power_strict_mono"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

417 
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

418 
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

419 
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

420 
val power_inverse = thm"power_inverse"; 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

421 
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

422 
val power_divide = thm"power_divide"; 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

423 
val power_abs = thm"power_abs"; 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

424 
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

425 
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

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

427 
val power_Suc_less = thm"power_Suc_less"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

428 
val power_strict_decreasing = thm"power_strict_decreasing"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

429 
val power_decreasing = thm"power_decreasing"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

430 
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

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

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

433 
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

434 
val power_inject_base = thm"power_inject_base"; 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

435 
*} 
14577  436 

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

437 
text{*ML bindings for the remaining theorems*} 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

438 
ML 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

439 
{* 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
8844
diff
changeset

440 
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

441 
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

442 
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

443 
*} 
3390
0c7625196d95
New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff
changeset

444 

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

445 
end 
0c7625196d95
New theory "Power" of exponentiation (and binomial coefficients)
paulson
parents:
diff
changeset

446 