paulson@3390
|
1 |
(* Title: HOL/Power.thy
|
paulson@3390
|
2 |
ID: $Id$
|
paulson@3390
|
3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
|
paulson@3390
|
4 |
Copyright 1997 University of Cambridge
|
paulson@3390
|
5 |
|
paulson@3390
|
6 |
*)
|
paulson@3390
|
7 |
|
nipkow@16733
|
8 |
header{*Exponentiation*}
|
paulson@14348
|
9 |
|
nipkow@15131
|
10 |
theory Power
|
haftmann@21413
|
11 |
imports Nat
|
nipkow@15131
|
12 |
begin
|
paulson@14348
|
13 |
|
haftmann@24996
|
14 |
class power = type +
|
haftmann@25062
|
15 |
fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80)
|
haftmann@24996
|
16 |
|
krauss@21199
|
17 |
subsection{*Powers for Arbitrary Monoids*}
|
paulson@14348
|
18 |
|
haftmann@22390
|
19 |
class recpower = monoid_mult + power +
|
haftmann@25062
|
20 |
assumes power_0 [simp]: "a ^ 0 = 1"
|
haftmann@25062
|
21 |
assumes power_Suc: "a ^ Suc n = a * (a ^ n)"
|
paulson@14348
|
22 |
|
krauss@21199
|
23 |
lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"
|
haftmann@23183
|
24 |
by (simp add: power_Suc)
|
paulson@14348
|
25 |
|
paulson@14348
|
26 |
text{*It looks plausible as a simprule, but its effect can be strange.*}
|
krauss@21199
|
27 |
lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::{recpower,semiring_0}))"
|
haftmann@23183
|
28 |
by (induct n) simp_all
|
paulson@14348
|
29 |
|
paulson@15004
|
30 |
lemma power_one [simp]: "1^n = (1::'a::recpower)"
|
haftmann@23183
|
31 |
by (induct n) (simp_all add: power_Suc)
|
paulson@14348
|
32 |
|
paulson@15004
|
33 |
lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
|
haftmann@23183
|
34 |
by (simp add: power_Suc)
|
paulson@14348
|
35 |
|
krauss@21199
|
36 |
lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
|
haftmann@23183
|
37 |
by (induct n) (simp_all add: power_Suc mult_assoc)
|
krauss@21199
|
38 |
|
paulson@15004
|
39 |
lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
|
haftmann@23183
|
40 |
by (induct m) (simp_all add: power_Suc mult_ac)
|
paulson@14348
|
41 |
|
paulson@15004
|
42 |
lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
|
haftmann@23183
|
43 |
by (induct n) (simp_all add: power_Suc power_add)
|
paulson@14348
|
44 |
|
krauss@21199
|
45 |
lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"
|
haftmann@23183
|
46 |
by (induct n) (simp_all add: power_Suc mult_ac)
|
paulson@14348
|
47 |
|
nipkow@25874
|
48 |
lemma zero_less_power[simp]:
|
paulson@15004
|
49 |
"0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"
|
paulson@15251
|
50 |
apply (induct "n")
|
avigad@16775
|
51 |
apply (simp_all add: power_Suc zero_less_one mult_pos_pos)
|
paulson@14348
|
52 |
done
|
paulson@14348
|
53 |
|
nipkow@25874
|
54 |
lemma zero_le_power[simp]:
|
paulson@15004
|
55 |
"0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"
|
paulson@14348
|
56 |
apply (simp add: order_le_less)
|
wenzelm@14577
|
57 |
apply (erule disjE)
|
nipkow@25874
|
58 |
apply (simp_all add: zero_less_one power_0_left)
|
paulson@14348
|
59 |
done
|
paulson@14348
|
60 |
|
nipkow@25874
|
61 |
lemma one_le_power[simp]:
|
paulson@15004
|
62 |
"1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"
|
paulson@15251
|
63 |
apply (induct "n")
|
paulson@14348
|
64 |
apply (simp_all add: power_Suc)
|
wenzelm@14577
|
65 |
apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
|
wenzelm@14577
|
66 |
apply (simp_all add: zero_le_one order_trans [OF zero_le_one])
|
paulson@14348
|
67 |
done
|
paulson@14348
|
68 |
|
obua@14738
|
69 |
lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semidom)"
|
paulson@14348
|
70 |
by (simp add: order_trans [OF zero_le_one order_less_imp_le])
|
paulson@14348
|
71 |
|
paulson@14348
|
72 |
lemma power_gt1_lemma:
|
paulson@15004
|
73 |
assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})"
|
wenzelm@14577
|
74 |
shows "1 < a * a^n"
|
paulson@14348
|
75 |
proof -
|
wenzelm@14577
|
76 |
have "1*1 < a*1" using gt1 by simp
|
wenzelm@14577
|
77 |
also have "\<dots> \<le> a * a^n" using gt1
|
wenzelm@14577
|
78 |
by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le
|
wenzelm@14577
|
79 |
zero_le_one order_refl)
|
wenzelm@14577
|
80 |
finally show ?thesis by simp
|
paulson@14348
|
81 |
qed
|
paulson@14348
|
82 |
|
nipkow@25874
|
83 |
lemma one_less_power[simp]:
|
huffman@24376
|
84 |
"\<lbrakk>1 < (a::'a::{ordered_semidom,recpower}); 0 < n\<rbrakk> \<Longrightarrow> 1 < a ^ n"
|
huffman@24376
|
85 |
by (cases n, simp_all add: power_gt1_lemma power_Suc)
|
huffman@24376
|
86 |
|
paulson@14348
|
87 |
lemma power_gt1:
|
paulson@15004
|
88 |
"1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
|
paulson@14348
|
89 |
by (simp add: power_gt1_lemma power_Suc)
|
paulson@14348
|
90 |
|
paulson@14348
|
91 |
lemma power_le_imp_le_exp:
|
paulson@15004
|
92 |
assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"
|
wenzelm@14577
|
93 |
shows "!!n. a^m \<le> a^n ==> m \<le> n"
|
wenzelm@14577
|
94 |
proof (induct m)
|
paulson@14348
|
95 |
case 0
|
wenzelm@14577
|
96 |
show ?case by simp
|
paulson@14348
|
97 |
next
|
paulson@14348
|
98 |
case (Suc m)
|
wenzelm@14577
|
99 |
show ?case
|
wenzelm@14577
|
100 |
proof (cases n)
|
wenzelm@14577
|
101 |
case 0
|
wenzelm@14577
|
102 |
from prems have "a * a^m \<le> 1" by (simp add: power_Suc)
|
wenzelm@14577
|
103 |
with gt1 show ?thesis
|
wenzelm@14577
|
104 |
by (force simp only: power_gt1_lemma
|
wenzelm@14577
|
105 |
linorder_not_less [symmetric])
|
wenzelm@14577
|
106 |
next
|
wenzelm@14577
|
107 |
case (Suc n)
|
wenzelm@14577
|
108 |
from prems show ?thesis
|
wenzelm@14577
|
109 |
by (force dest: mult_left_le_imp_le
|
wenzelm@14577
|
110 |
simp add: power_Suc order_less_trans [OF zero_less_one gt1])
|
wenzelm@14577
|
111 |
qed
|
paulson@14348
|
112 |
qed
|
paulson@14348
|
113 |
|
wenzelm@14577
|
114 |
text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
|
paulson@14348
|
115 |
lemma power_inject_exp [simp]:
|
paulson@15004
|
116 |
"1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)"
|
wenzelm@14577
|
117 |
by (force simp add: order_antisym power_le_imp_le_exp)
|
paulson@14348
|
118 |
|
paulson@14348
|
119 |
text{*Can relax the first premise to @{term "0<a"} in the case of the
|
paulson@14348
|
120 |
natural numbers.*}
|
paulson@14348
|
121 |
lemma power_less_imp_less_exp:
|
paulson@15004
|
122 |
"[| (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"
|
wenzelm@14577
|
123 |
by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
|
wenzelm@14577
|
124 |
power_le_imp_le_exp)
|
paulson@14348
|
125 |
|
paulson@14348
|
126 |
|
paulson@14348
|
127 |
lemma power_mono:
|
paulson@15004
|
128 |
"[|a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"
|
paulson@15251
|
129 |
apply (induct "n")
|
paulson@14348
|
130 |
apply (simp_all add: power_Suc)
|
nipkow@25874
|
131 |
apply (auto intro: mult_mono order_trans [of 0 a b])
|
paulson@14348
|
132 |
done
|
paulson@14348
|
133 |
|
paulson@14348
|
134 |
lemma power_strict_mono [rule_format]:
|
paulson@15004
|
135 |
"[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]
|
wenzelm@14577
|
136 |
==> 0 < n --> a^n < b^n"
|
paulson@15251
|
137 |
apply (induct "n")
|
nipkow@25874
|
138 |
apply (auto simp add: mult_strict_mono power_Suc
|
paulson@14348
|
139 |
order_le_less_trans [of 0 a b])
|
paulson@14348
|
140 |
done
|
paulson@14348
|
141 |
|
paulson@14348
|
142 |
lemma power_eq_0_iff [simp]:
|
nipkow@25162
|
143 |
"(a^n = 0) = (a = (0::'a::{ring_1_no_zero_divisors,recpower}) & n>0)"
|
paulson@15251
|
144 |
apply (induct "n")
|
paulson@14348
|
145 |
apply (auto simp add: power_Suc zero_neq_one [THEN not_sym])
|
paulson@14348
|
146 |
done
|
paulson@14348
|
147 |
|
nipkow@25134
|
148 |
lemma field_power_not_zero:
|
nipkow@25134
|
149 |
"a \<noteq> (0::'a::{ring_1_no_zero_divisors,recpower}) ==> a^n \<noteq> 0"
|
paulson@14348
|
150 |
by force
|
paulson@14348
|
151 |
|
paulson@14353
|
152 |
lemma nonzero_power_inverse:
|
huffman@22991
|
153 |
fixes a :: "'a::{division_ring,recpower}"
|
huffman@22991
|
154 |
shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n"
|
paulson@15251
|
155 |
apply (induct "n")
|
huffman@22988
|
156 |
apply (auto simp add: power_Suc nonzero_inverse_mult_distrib power_commutes)
|
huffman@22991
|
157 |
done (* TODO: reorient or rename to nonzero_inverse_power *)
|
paulson@14353
|
158 |
|
paulson@14348
|
159 |
text{*Perhaps these should be simprules.*}
|
paulson@14348
|
160 |
lemma power_inverse:
|
huffman@22991
|
161 |
fixes a :: "'a::{division_ring,division_by_zero,recpower}"
|
huffman@22991
|
162 |
shows "inverse (a ^ n) = (inverse a) ^ n"
|
huffman@22991
|
163 |
apply (cases "a = 0")
|
huffman@22991
|
164 |
apply (simp add: power_0_left)
|
huffman@22991
|
165 |
apply (simp add: nonzero_power_inverse)
|
huffman@22991
|
166 |
done (* TODO: reorient or rename to inverse_power *)
|
paulson@14348
|
167 |
|
avigad@16775
|
168 |
lemma power_one_over: "1 / (a::'a::{field,division_by_zero,recpower})^n =
|
avigad@16775
|
169 |
(1 / a)^n"
|
avigad@16775
|
170 |
apply (simp add: divide_inverse)
|
avigad@16775
|
171 |
apply (rule power_inverse)
|
avigad@16775
|
172 |
done
|
avigad@16775
|
173 |
|
wenzelm@14577
|
174 |
lemma nonzero_power_divide:
|
paulson@15004
|
175 |
"b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"
|
paulson@14353
|
176 |
by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
|
paulson@14353
|
177 |
|
wenzelm@14577
|
178 |
lemma power_divide:
|
paulson@15004
|
179 |
"(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"
|
paulson@14353
|
180 |
apply (case_tac "b=0", simp add: power_0_left)
|
wenzelm@14577
|
181 |
apply (rule nonzero_power_divide)
|
wenzelm@14577
|
182 |
apply assumption
|
paulson@14353
|
183 |
done
|
paulson@14353
|
184 |
|
paulson@15004
|
185 |
lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"
|
paulson@15251
|
186 |
apply (induct "n")
|
paulson@14348
|
187 |
apply (auto simp add: power_Suc abs_mult)
|
paulson@14348
|
188 |
done
|
paulson@14348
|
189 |
|
paulson@24286
|
190 |
lemma zero_less_power_abs_iff [simp,noatp]:
|
paulson@15004
|
191 |
"(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
|
paulson@14353
|
192 |
proof (induct "n")
|
paulson@14353
|
193 |
case 0
|
paulson@14353
|
194 |
show ?case by (simp add: zero_less_one)
|
paulson@14353
|
195 |
next
|
paulson@14353
|
196 |
case (Suc n)
|
haftmann@25231
|
197 |
show ?case by (auto simp add: prems power_Suc zero_less_mult_iff
|
haftmann@25231
|
198 |
abs_zero)
|
paulson@14353
|
199 |
qed
|
paulson@14353
|
200 |
|
paulson@14353
|
201 |
lemma zero_le_power_abs [simp]:
|
paulson@15004
|
202 |
"(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"
|
huffman@22957
|
203 |
by (rule zero_le_power [OF abs_ge_zero])
|
paulson@14353
|
204 |
|
paulson@15004
|
205 |
lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{comm_ring_1,recpower}) ^ n"
|
paulson@14348
|
206 |
proof -
|
paulson@14348
|
207 |
have "-a = (- 1) * a" by (simp add: minus_mult_left [symmetric])
|
paulson@14348
|
208 |
thus ?thesis by (simp only: power_mult_distrib)
|
paulson@14348
|
209 |
qed
|
paulson@14348
|
210 |
|
paulson@14348
|
211 |
text{*Lemma for @{text power_strict_decreasing}*}
|
paulson@14348
|
212 |
lemma power_Suc_less:
|
paulson@15004
|
213 |
"[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|]
|
paulson@14348
|
214 |
==> a * a^n < a^n"
|
paulson@15251
|
215 |
apply (induct n)
|
wenzelm@14577
|
216 |
apply (auto simp add: power_Suc mult_strict_left_mono)
|
paulson@14348
|
217 |
done
|
paulson@14348
|
218 |
|
paulson@14348
|
219 |
lemma power_strict_decreasing:
|
paulson@15004
|
220 |
"[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]
|
paulson@14348
|
221 |
==> a^N < a^n"
|
wenzelm@14577
|
222 |
apply (erule rev_mp)
|
paulson@15251
|
223 |
apply (induct "N")
|
wenzelm@14577
|
224 |
apply (auto simp add: power_Suc power_Suc_less less_Suc_eq)
|
wenzelm@14577
|
225 |
apply (rename_tac m)
|
paulson@14348
|
226 |
apply (subgoal_tac "a * a^m < 1 * a^n", simp)
|
wenzelm@14577
|
227 |
apply (rule mult_strict_mono)
|
nipkow@25874
|
228 |
apply (auto simp add: zero_less_one order_less_imp_le)
|
paulson@14348
|
229 |
done
|
paulson@14348
|
230 |
|
paulson@14348
|
231 |
text{*Proof resembles that of @{text power_strict_decreasing}*}
|
paulson@14348
|
232 |
lemma power_decreasing:
|
paulson@15004
|
233 |
"[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]
|
paulson@14348
|
234 |
==> a^N \<le> a^n"
|
wenzelm@14577
|
235 |
apply (erule rev_mp)
|
paulson@15251
|
236 |
apply (induct "N")
|
wenzelm@14577
|
237 |
apply (auto simp add: power_Suc le_Suc_eq)
|
wenzelm@14577
|
238 |
apply (rename_tac m)
|
paulson@14348
|
239 |
apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
|
wenzelm@14577
|
240 |
apply (rule mult_mono)
|
nipkow@25874
|
241 |
apply (auto simp add: zero_le_one)
|
paulson@14348
|
242 |
done
|
paulson@14348
|
243 |
|
paulson@14348
|
244 |
lemma power_Suc_less_one:
|
paulson@15004
|
245 |
"[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1"
|
wenzelm@14577
|
246 |
apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
|
paulson@14348
|
247 |
done
|
paulson@14348
|
248 |
|
paulson@14348
|
249 |
text{*Proof again resembles that of @{text power_strict_decreasing}*}
|
paulson@14348
|
250 |
lemma power_increasing:
|
paulson@15004
|
251 |
"[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"
|
wenzelm@14577
|
252 |
apply (erule rev_mp)
|
paulson@15251
|
253 |
apply (induct "N")
|
wenzelm@14577
|
254 |
apply (auto simp add: power_Suc le_Suc_eq)
|
paulson@14348
|
255 |
apply (rename_tac m)
|
paulson@14348
|
256 |
apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
|
wenzelm@14577
|
257 |
apply (rule mult_mono)
|
nipkow@25874
|
258 |
apply (auto simp add: order_trans [OF zero_le_one])
|
paulson@14348
|
259 |
done
|
paulson@14348
|
260 |
|
paulson@14348
|
261 |
text{*Lemma for @{text power_strict_increasing}*}
|
paulson@14348
|
262 |
lemma power_less_power_Suc:
|
paulson@15004
|
263 |
"(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n"
|
paulson@15251
|
264 |
apply (induct n)
|
wenzelm@14577
|
265 |
apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one])
|
paulson@14348
|
266 |
done
|
paulson@14348
|
267 |
|
paulson@14348
|
268 |
lemma power_strict_increasing:
|
paulson@15004
|
269 |
"[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N"
|
wenzelm@14577
|
270 |
apply (erule rev_mp)
|
paulson@15251
|
271 |
apply (induct "N")
|
wenzelm@14577
|
272 |
apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq)
|
paulson@14348
|
273 |
apply (rename_tac m)
|
paulson@14348
|
274 |
apply (subgoal_tac "1 * a^n < a * a^m", simp)
|
wenzelm@14577
|
275 |
apply (rule mult_strict_mono)
|
nipkow@25874
|
276 |
apply (auto simp add: order_less_trans [OF zero_less_one] order_less_imp_le)
|
paulson@14348
|
277 |
done
|
paulson@14348
|
278 |
|
nipkow@25134
|
279 |
lemma power_increasing_iff [simp]:
|
nipkow@25134
|
280 |
"1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
|
nipkow@25134
|
281 |
by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le)
|
paulson@15066
|
282 |
|
paulson@15066
|
283 |
lemma power_strict_increasing_iff [simp]:
|
nipkow@25134
|
284 |
"1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"
|
nipkow@25134
|
285 |
by (blast intro: power_less_imp_less_exp power_strict_increasing)
|
paulson@15066
|
286 |
|
paulson@14348
|
287 |
lemma power_le_imp_le_base:
|
nipkow@25134
|
288 |
assumes le: "a ^ Suc n \<le> b ^ Suc n"
|
nipkow@25134
|
289 |
and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b"
|
nipkow@25134
|
290 |
shows "a \<le> b"
|
nipkow@25134
|
291 |
proof (rule ccontr)
|
nipkow@25134
|
292 |
assume "~ a \<le> b"
|
nipkow@25134
|
293 |
then have "b < a" by (simp only: linorder_not_le)
|
nipkow@25134
|
294 |
then have "b ^ Suc n < a ^ Suc n"
|
nipkow@25134
|
295 |
by (simp only: prems power_strict_mono)
|
nipkow@25134
|
296 |
from le and this show "False"
|
nipkow@25134
|
297 |
by (simp add: linorder_not_less [symmetric])
|
nipkow@25134
|
298 |
qed
|
wenzelm@14577
|
299 |
|
huffman@22853
|
300 |
lemma power_less_imp_less_base:
|
huffman@22853
|
301 |
fixes a b :: "'a::{ordered_semidom,recpower}"
|
huffman@22853
|
302 |
assumes less: "a ^ n < b ^ n"
|
huffman@22853
|
303 |
assumes nonneg: "0 \<le> b"
|
huffman@22853
|
304 |
shows "a < b"
|
huffman@22853
|
305 |
proof (rule contrapos_pp [OF less])
|
huffman@22853
|
306 |
assume "~ a < b"
|
huffman@22853
|
307 |
hence "b \<le> a" by (simp only: linorder_not_less)
|
huffman@22853
|
308 |
hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
|
huffman@22853
|
309 |
thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less)
|
huffman@22853
|
310 |
qed
|
huffman@22853
|
311 |
|
paulson@14348
|
312 |
lemma power_inject_base:
|
wenzelm@14577
|
313 |
"[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
|
paulson@15004
|
314 |
==> a = (b::'a::{ordered_semidom,recpower})"
|
paulson@14348
|
315 |
by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
|
paulson@14348
|
316 |
|
huffman@22955
|
317 |
lemma power_eq_imp_eq_base:
|
huffman@22955
|
318 |
fixes a b :: "'a::{ordered_semidom,recpower}"
|
huffman@22955
|
319 |
shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b"
|
huffman@22955
|
320 |
by (cases n, simp_all, rule power_inject_base)
|
huffman@22955
|
321 |
|
paulson@14348
|
322 |
|
paulson@14348
|
323 |
subsection{*Exponentiation for the Natural Numbers*}
|
paulson@3390
|
324 |
|
haftmann@25836
|
325 |
instantiation nat :: recpower
|
haftmann@25836
|
326 |
begin
|
haftmann@21456
|
327 |
|
haftmann@25836
|
328 |
primrec power_nat where
|
haftmann@25836
|
329 |
"p ^ 0 = (1\<Colon>nat)"
|
haftmann@25836
|
330 |
| "p ^ (Suc n) = (p\<Colon>nat) * (p ^ n)"
|
wenzelm@14577
|
331 |
|
haftmann@25836
|
332 |
instance proof
|
paulson@14438
|
333 |
fix z n :: nat
|
paulson@14348
|
334 |
show "z^0 = 1" by simp
|
paulson@14348
|
335 |
show "z^(Suc n) = z * (z^n)" by simp
|
paulson@14348
|
336 |
qed
|
paulson@14348
|
337 |
|
haftmann@25836
|
338 |
end
|
haftmann@25836
|
339 |
|
huffman@23305
|
340 |
lemma of_nat_power:
|
huffman@23305
|
341 |
"of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"
|
huffman@23431
|
342 |
by (induct n, simp_all add: power_Suc of_nat_mult)
|
huffman@23305
|
343 |
|
paulson@14348
|
344 |
lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n"
|
paulson@14348
|
345 |
by (insert one_le_power [of i n], simp)
|
paulson@14348
|
346 |
|
nipkow@25162
|
347 |
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
|
haftmann@21413
|
348 |
by (induct "n", auto)
|
paulson@14348
|
349 |
|
paulson@14348
|
350 |
text{*Valid for the naturals, but what if @{text"0<i<1"}?
|
paulson@14348
|
351 |
Premises cannot be weakened: consider the case where @{term "i=0"},
|
paulson@14348
|
352 |
@{term "m=1"} and @{term "n=0"}.*}
|
haftmann@21413
|
353 |
lemma nat_power_less_imp_less:
|
haftmann@21413
|
354 |
assumes nonneg: "0 < (i\<Colon>nat)"
|
haftmann@21413
|
355 |
assumes less: "i^m < i^n"
|
haftmann@21413
|
356 |
shows "m < n"
|
haftmann@21413
|
357 |
proof (cases "i = 1")
|
haftmann@21413
|
358 |
case True with less power_one [where 'a = nat] show ?thesis by simp
|
haftmann@21413
|
359 |
next
|
haftmann@21413
|
360 |
case False with nonneg have "1 < i" by auto
|
haftmann@21413
|
361 |
from power_strict_increasing_iff [OF this] less show ?thesis ..
|
haftmann@21413
|
362 |
qed
|
paulson@14348
|
363 |
|
ballarin@17149
|
364 |
lemma power_diff:
|
ballarin@17149
|
365 |
assumes nz: "a ~= 0"
|
ballarin@17149
|
366 |
shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)"
|
ballarin@17149
|
367 |
by (induct m n rule: diff_induct)
|
ballarin@17149
|
368 |
(simp_all add: power_Suc nonzero_mult_divide_cancel_left nz)
|
ballarin@17149
|
369 |
|
ballarin@17149
|
370 |
|
paulson@14348
|
371 |
text{*ML bindings for the general exponentiation theorems*}
|
paulson@14348
|
372 |
ML
|
paulson@14348
|
373 |
{*
|
paulson@14348
|
374 |
val power_0 = thm"power_0";
|
paulson@14348
|
375 |
val power_Suc = thm"power_Suc";
|
paulson@14348
|
376 |
val power_0_Suc = thm"power_0_Suc";
|
paulson@14348
|
377 |
val power_0_left = thm"power_0_left";
|
paulson@14348
|
378 |
val power_one = thm"power_one";
|
paulson@14348
|
379 |
val power_one_right = thm"power_one_right";
|
paulson@14348
|
380 |
val power_add = thm"power_add";
|
paulson@14348
|
381 |
val power_mult = thm"power_mult";
|
paulson@14348
|
382 |
val power_mult_distrib = thm"power_mult_distrib";
|
paulson@14348
|
383 |
val zero_less_power = thm"zero_less_power";
|
paulson@14348
|
384 |
val zero_le_power = thm"zero_le_power";
|
paulson@14348
|
385 |
val one_le_power = thm"one_le_power";
|
paulson@14348
|
386 |
val gt1_imp_ge0 = thm"gt1_imp_ge0";
|
paulson@14348
|
387 |
val power_gt1_lemma = thm"power_gt1_lemma";
|
paulson@14348
|
388 |
val power_gt1 = thm"power_gt1";
|
paulson@14348
|
389 |
val power_le_imp_le_exp = thm"power_le_imp_le_exp";
|
paulson@14348
|
390 |
val power_inject_exp = thm"power_inject_exp";
|
paulson@14348
|
391 |
val power_less_imp_less_exp = thm"power_less_imp_less_exp";
|
paulson@14348
|
392 |
val power_mono = thm"power_mono";
|
paulson@14348
|
393 |
val power_strict_mono = thm"power_strict_mono";
|
paulson@14348
|
394 |
val power_eq_0_iff = thm"power_eq_0_iff";
|
nipkow@25134
|
395 |
val field_power_eq_0_iff = thm"power_eq_0_iff";
|
paulson@14348
|
396 |
val field_power_not_zero = thm"field_power_not_zero";
|
paulson@14348
|
397 |
val power_inverse = thm"power_inverse";
|
paulson@14353
|
398 |
val nonzero_power_divide = thm"nonzero_power_divide";
|
paulson@14353
|
399 |
val power_divide = thm"power_divide";
|
paulson@14348
|
400 |
val power_abs = thm"power_abs";
|
paulson@14353
|
401 |
val zero_less_power_abs_iff = thm"zero_less_power_abs_iff";
|
paulson@14353
|
402 |
val zero_le_power_abs = thm "zero_le_power_abs";
|
paulson@14348
|
403 |
val power_minus = thm"power_minus";
|
paulson@14348
|
404 |
val power_Suc_less = thm"power_Suc_less";
|
paulson@14348
|
405 |
val power_strict_decreasing = thm"power_strict_decreasing";
|
paulson@14348
|
406 |
val power_decreasing = thm"power_decreasing";
|
paulson@14348
|
407 |
val power_Suc_less_one = thm"power_Suc_less_one";
|
paulson@14348
|
408 |
val power_increasing = thm"power_increasing";
|
paulson@14348
|
409 |
val power_strict_increasing = thm"power_strict_increasing";
|
paulson@14348
|
410 |
val power_le_imp_le_base = thm"power_le_imp_le_base";
|
paulson@14348
|
411 |
val power_inject_base = thm"power_inject_base";
|
paulson@14348
|
412 |
*}
|
wenzelm@14577
|
413 |
|
paulson@14348
|
414 |
text{*ML bindings for the remaining theorems*}
|
paulson@14348
|
415 |
ML
|
paulson@14348
|
416 |
{*
|
paulson@14348
|
417 |
val nat_one_le_power = thm"nat_one_le_power";
|
paulson@14348
|
418 |
val nat_power_less_imp_less = thm"nat_power_less_imp_less";
|
paulson@14348
|
419 |
val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";
|
paulson@14348
|
420 |
*}
|
paulson@3390
|
421 |
|
paulson@3390
|
422 |
end
|
paulson@3390
|
423 |
|