1 (* Title: HOL/Power.thy
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
4 Copyright 1997 University of Cambridge
8 header{*Exponentiation*}
15 fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80)
17 subsection{*Powers for Arbitrary Monoids*}
19 class recpower = monoid_mult + power +
20 assumes power_0 [simp]: "a ^ 0 = 1"
21 assumes power_Suc: "a ^ Suc n = a * (a ^ n)"
23 lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"
24 by (simp add: power_Suc)
26 text{*It looks plausible as a simprule, but its effect can be strange.*}
27 lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::{recpower,semiring_0}))"
28 by (induct n) simp_all
30 lemma power_one [simp]: "1^n = (1::'a::recpower)"
31 by (induct n) (simp_all add: power_Suc)
33 lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
34 by (simp add: power_Suc)
36 lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
37 by (induct n) (simp_all add: power_Suc mult_assoc)
39 lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
40 by (induct m) (simp_all add: power_Suc mult_ac)
42 lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
43 by (induct n) (simp_all add: power_Suc power_add)
45 lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"
46 by (induct n) (simp_all add: power_Suc mult_ac)
48 lemma zero_less_power:
49 "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"
51 apply (simp_all add: power_Suc zero_less_one mult_pos_pos)
55 "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"
56 apply (simp add: order_le_less)
58 apply (simp_all add: zero_less_power zero_less_one power_0_left)
62 "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"
64 apply (simp_all add: power_Suc)
65 apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
66 apply (simp_all add: zero_le_one order_trans [OF zero_le_one])
69 lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semidom)"
70 by (simp add: order_trans [OF zero_le_one order_less_imp_le])
72 lemma power_gt1_lemma:
73 assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})"
76 have "1*1 < a*1" using gt1 by simp
77 also have "\<dots> \<le> a * a^n" using gt1
78 by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le
79 zero_le_one order_refl)
80 finally show ?thesis by simp
84 "\<lbrakk>1 < (a::'a::{ordered_semidom,recpower}); 0 < n\<rbrakk> \<Longrightarrow> 1 < a ^ n"
85 by (cases n, simp_all add: power_gt1_lemma power_Suc)
88 "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
89 by (simp add: power_gt1_lemma power_Suc)
91 lemma power_le_imp_le_exp:
92 assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"
93 shows "!!n. a^m \<le> a^n ==> m \<le> n"
102 from prems have "a * a^m \<le> 1" by (simp add: power_Suc)
103 with gt1 show ?thesis
104 by (force simp only: power_gt1_lemma
105 linorder_not_less [symmetric])
108 from prems show ?thesis
109 by (force dest: mult_left_le_imp_le
110 simp add: power_Suc order_less_trans [OF zero_less_one gt1])
114 text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
115 lemma power_inject_exp [simp]:
116 "1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)"
117 by (force simp add: order_antisym power_le_imp_le_exp)
119 text{*Can relax the first premise to @{term "0<a"} in the case of the
121 lemma power_less_imp_less_exp:
122 "[| (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"
123 by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
128 "[|a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"
130 apply (simp_all add: power_Suc)
131 apply (auto intro: mult_mono zero_le_power order_trans [of 0 a b])
134 lemma power_strict_mono [rule_format]:
135 "[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]
136 ==> 0 < n --> a^n < b^n"
138 apply (auto simp add: mult_strict_mono zero_le_power power_Suc
139 order_le_less_trans [of 0 a b])
142 lemma power_eq_0_iff [simp]:
143 "(a^n = 0) = (a = (0::'a::{ring_1_no_zero_divisors,recpower}) & 0<n)"
145 apply (auto simp add: power_Suc zero_neq_one [THEN not_sym])
148 lemma field_power_eq_0_iff:
149 "(a^n = 0) = (a = (0::'a::{division_ring,recpower}) & 0<n)"
150 by simp (* TODO: delete *)
152 lemma field_power_not_zero: "a \<noteq> (0::'a::{ring_1_no_zero_divisors,recpower}) ==> a^n \<noteq> 0"
155 lemma nonzero_power_inverse:
156 fixes a :: "'a::{division_ring,recpower}"
157 shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n"
159 apply (auto simp add: power_Suc nonzero_inverse_mult_distrib power_commutes)
160 done (* TODO: reorient or rename to nonzero_inverse_power *)
162 text{*Perhaps these should be simprules.*}
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 *)
171 lemma power_one_over: "1 / (a::'a::{field,division_by_zero,recpower})^n =
173 apply (simp add: divide_inverse)
174 apply (rule power_inverse)
177 lemma nonzero_power_divide:
178 "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"
179 by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
182 "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"
183 apply (case_tac "b=0", simp add: power_0_left)
184 apply (rule nonzero_power_divide)
188 lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"
190 apply (auto simp add: power_Suc abs_mult)
193 lemma zero_less_power_abs_iff [simp,noatp]:
194 "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
197 show ?case by (simp add: zero_less_one)
200 show ?case by (force simp add: prems power_Suc zero_less_mult_iff)
203 lemma zero_le_power_abs [simp]:
204 "(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"
205 by (rule zero_le_power [OF abs_ge_zero])
207 lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{comm_ring_1,recpower}) ^ n"
209 have "-a = (- 1) * a" by (simp add: minus_mult_left [symmetric])
210 thus ?thesis by (simp only: power_mult_distrib)
213 text{*Lemma for @{text power_strict_decreasing}*}
214 lemma power_Suc_less:
215 "[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|]
218 apply (auto simp add: power_Suc mult_strict_left_mono)
221 lemma power_strict_decreasing:
222 "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]
226 apply (auto simp add: power_Suc power_Suc_less less_Suc_eq)
228 apply (subgoal_tac "a * a^m < 1 * a^n", simp)
229 apply (rule mult_strict_mono)
230 apply (auto simp add: zero_le_power zero_less_one order_less_imp_le)
233 text{*Proof resembles that of @{text power_strict_decreasing}*}
234 lemma power_decreasing:
235 "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]
239 apply (auto simp add: power_Suc le_Suc_eq)
241 apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
242 apply (rule mult_mono)
243 apply (auto simp add: zero_le_power zero_le_one)
246 lemma power_Suc_less_one:
247 "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1"
248 apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
251 text{*Proof again resembles that of @{text power_strict_decreasing}*}
252 lemma power_increasing:
253 "[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"
256 apply (auto simp add: power_Suc le_Suc_eq)
258 apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
259 apply (rule mult_mono)
260 apply (auto simp add: order_trans [OF zero_le_one] zero_le_power)
263 text{*Lemma for @{text power_strict_increasing}*}
264 lemma power_less_power_Suc:
265 "(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n"
267 apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one])
270 lemma power_strict_increasing:
271 "[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N"
274 apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq)
276 apply (subgoal_tac "1 * a^n < a * a^m", simp)
277 apply (rule mult_strict_mono)
278 apply (auto simp add: order_less_trans [OF zero_less_one] zero_le_power
282 lemma power_increasing_iff [simp]:
283 "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
284 by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le)
286 lemma power_strict_increasing_iff [simp]:
287 "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"
288 by (blast intro: power_less_imp_less_exp power_strict_increasing)
290 lemma power_le_imp_le_base:
291 assumes le: "a ^ Suc n \<le> b ^ Suc n"
292 and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b"
296 then have "b < a" by (simp only: linorder_not_le)
297 then have "b ^ Suc n < a ^ Suc n"
298 by (simp only: prems power_strict_mono)
299 from le and this show "False"
300 by (simp add: linorder_not_less [symmetric])
303 lemma power_less_imp_less_base:
304 fixes a b :: "'a::{ordered_semidom,recpower}"
305 assumes less: "a ^ n < b ^ n"
306 assumes nonneg: "0 \<le> b"
308 proof (rule contrapos_pp [OF less])
310 hence "b \<le> a" by (simp only: linorder_not_less)
311 hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
312 thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less)
315 lemma power_inject_base:
316 "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
317 ==> a = (b::'a::{ordered_semidom,recpower})"
318 by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
320 lemma power_eq_imp_eq_base:
321 fixes a b :: "'a::{ordered_semidom,recpower}"
322 shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b"
323 by (cases n, simp_all, rule power_inject_base)
326 subsection{*Exponentiation for the Natural Numbers*}
328 instance nat :: power ..
332 "p ^ (Suc n) = (p::nat) * (p ^ n)"
334 instance nat :: recpower
337 show "z^0 = 1" by simp
338 show "z^(Suc n) = z * (z^n)" by simp
342 "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"
343 by (induct n, simp_all add: power_Suc of_nat_mult)
345 lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n"
346 by (insert one_le_power [of i n], simp)
348 lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
349 by (induct "n", auto)
351 text{*Valid for the naturals, but what if @{text"0<i<1"}?
352 Premises cannot be weakened: consider the case where @{term "i=0"},
353 @{term "m=1"} and @{term "n=0"}.*}
354 lemma nat_power_less_imp_less:
355 assumes nonneg: "0 < (i\<Colon>nat)"
356 assumes less: "i^m < i^n"
358 proof (cases "i = 1")
359 case True with less power_one [where 'a = nat] show ?thesis by simp
361 case False with nonneg have "1 < i" by auto
362 from power_strict_increasing_iff [OF this] less show ?thesis ..
367 shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)"
368 by (induct m n rule: diff_induct)
369 (simp_all add: power_Suc nonzero_mult_divide_cancel_left nz)
372 text{*ML bindings for the general exponentiation theorems*}
375 val power_0 = thm"power_0";
376 val power_Suc = thm"power_Suc";
377 val power_0_Suc = thm"power_0_Suc";
378 val power_0_left = thm"power_0_left";
379 val power_one = thm"power_one";
380 val power_one_right = thm"power_one_right";
381 val power_add = thm"power_add";
382 val power_mult = thm"power_mult";
383 val power_mult_distrib = thm"power_mult_distrib";
384 val zero_less_power = thm"zero_less_power";
385 val zero_le_power = thm"zero_le_power";
386 val one_le_power = thm"one_le_power";
387 val gt1_imp_ge0 = thm"gt1_imp_ge0";
388 val power_gt1_lemma = thm"power_gt1_lemma";
389 val power_gt1 = thm"power_gt1";
390 val power_le_imp_le_exp = thm"power_le_imp_le_exp";
391 val power_inject_exp = thm"power_inject_exp";
392 val power_less_imp_less_exp = thm"power_less_imp_less_exp";
393 val power_mono = thm"power_mono";
394 val power_strict_mono = thm"power_strict_mono";
395 val power_eq_0_iff = thm"power_eq_0_iff";
396 val field_power_eq_0_iff = thm"field_power_eq_0_iff";
397 val field_power_not_zero = thm"field_power_not_zero";
398 val power_inverse = thm"power_inverse";
399 val nonzero_power_divide = thm"nonzero_power_divide";
400 val power_divide = thm"power_divide";
401 val power_abs = thm"power_abs";
402 val zero_less_power_abs_iff = thm"zero_less_power_abs_iff";
403 val zero_le_power_abs = thm "zero_le_power_abs";
404 val power_minus = thm"power_minus";
405 val power_Suc_less = thm"power_Suc_less";
406 val power_strict_decreasing = thm"power_strict_decreasing";
407 val power_decreasing = thm"power_decreasing";
408 val power_Suc_less_one = thm"power_Suc_less_one";
409 val power_increasing = thm"power_increasing";
410 val power_strict_increasing = thm"power_strict_increasing";
411 val power_le_imp_le_base = thm"power_le_imp_le_base";
412 val power_inject_base = thm"power_inject_base";
415 text{*ML bindings for the remaining theorems*}
418 val nat_one_le_power = thm"nat_one_le_power";
419 val nat_power_less_imp_less = thm"nat_power_less_imp_less";
420 val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";