1 (* Title: HOL/Power.thy
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
4 Copyright 1997 University of Cambridge
8 header{*Exponentiation*}
14 subsection{*Powers for Arbitrary Monoids*}
16 class recpower = monoid_mult + power +
17 assumes power_0 [simp]: "a \<^loc>^ 0 = \<^loc>1"
18 assumes power_Suc: "a \<^loc>^ Suc n = a \<^loc>* (a \<^loc>^ n)"
20 lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"
21 by (simp add: power_Suc)
23 text{*It looks plausible as a simprule, but its effect can be strange.*}
24 lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::{recpower,semiring_0}))"
25 by (induct n) simp_all
27 lemma power_one [simp]: "1^n = (1::'a::recpower)"
28 by (induct n) (simp_all add: power_Suc)
30 lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
31 by (simp add: power_Suc)
33 lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
34 by (induct n) (simp_all add: power_Suc mult_assoc)
36 lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
37 by (induct m) (simp_all add: power_Suc mult_ac)
39 lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
40 by (induct n) (simp_all add: power_Suc power_add)
42 lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"
43 by (induct n) (simp_all add: power_Suc mult_ac)
45 lemma zero_less_power:
46 "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"
48 apply (simp_all add: power_Suc zero_less_one mult_pos_pos)
52 "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"
53 apply (simp add: order_le_less)
55 apply (simp_all add: zero_less_power zero_less_one power_0_left)
59 "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"
61 apply (simp_all add: power_Suc)
62 apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
63 apply (simp_all add: zero_le_one order_trans [OF zero_le_one])
66 lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semidom)"
67 by (simp add: order_trans [OF zero_le_one order_less_imp_le])
69 lemma power_gt1_lemma:
70 assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})"
73 have "1*1 < a*1" using gt1 by simp
74 also have "\<dots> \<le> a * a^n" using gt1
75 by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le
76 zero_le_one order_refl)
77 finally show ?thesis by simp
81 "\<lbrakk>1 < (a::'a::{ordered_semidom,recpower}); 0 < n\<rbrakk> \<Longrightarrow> 1 < a ^ n"
82 by (cases n, simp_all add: power_gt1_lemma power_Suc)
85 "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
86 by (simp add: power_gt1_lemma power_Suc)
88 lemma power_le_imp_le_exp:
89 assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"
90 shows "!!n. a^m \<le> a^n ==> m \<le> n"
99 from prems have "a * a^m \<le> 1" by (simp add: power_Suc)
100 with gt1 show ?thesis
101 by (force simp only: power_gt1_lemma
102 linorder_not_less [symmetric])
105 from prems show ?thesis
106 by (force dest: mult_left_le_imp_le
107 simp add: power_Suc order_less_trans [OF zero_less_one gt1])
111 text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
112 lemma power_inject_exp [simp]:
113 "1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)"
114 by (force simp add: order_antisym power_le_imp_le_exp)
116 text{*Can relax the first premise to @{term "0<a"} in the case of the
118 lemma power_less_imp_less_exp:
119 "[| (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"
120 by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
125 "[|a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"
127 apply (simp_all add: power_Suc)
128 apply (auto intro: mult_mono zero_le_power order_trans [of 0 a b])
131 lemma power_strict_mono [rule_format]:
132 "[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]
133 ==> 0 < n --> a^n < b^n"
135 apply (auto simp add: mult_strict_mono zero_le_power power_Suc
136 order_le_less_trans [of 0 a b])
139 lemma power_eq_0_iff [simp]:
140 "(a^n = 0) = (a = (0::'a::{ring_1_no_zero_divisors,recpower}) & 0<n)"
142 apply (auto simp add: power_Suc zero_neq_one [THEN not_sym])
145 lemma field_power_eq_0_iff:
146 "(a^n = 0) = (a = (0::'a::{division_ring,recpower}) & 0<n)"
147 by simp (* TODO: delete *)
149 lemma field_power_not_zero: "a \<noteq> (0::'a::{ring_1_no_zero_divisors,recpower}) ==> a^n \<noteq> 0"
152 lemma nonzero_power_inverse:
153 fixes a :: "'a::{division_ring,recpower}"
154 shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n"
156 apply (auto simp add: power_Suc nonzero_inverse_mult_distrib power_commutes)
157 done (* TODO: reorient or rename to nonzero_inverse_power *)
159 text{*Perhaps these should be simprules.*}
161 fixes a :: "'a::{division_ring,division_by_zero,recpower}"
162 shows "inverse (a ^ n) = (inverse a) ^ n"
163 apply (cases "a = 0")
164 apply (simp add: power_0_left)
165 apply (simp add: nonzero_power_inverse)
166 done (* TODO: reorient or rename to inverse_power *)
168 lemma power_one_over: "1 / (a::'a::{field,division_by_zero,recpower})^n =
170 apply (simp add: divide_inverse)
171 apply (rule power_inverse)
174 lemma nonzero_power_divide:
175 "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"
176 by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
179 "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"
180 apply (case_tac "b=0", simp add: power_0_left)
181 apply (rule nonzero_power_divide)
185 lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"
187 apply (auto simp add: power_Suc abs_mult)
190 lemma zero_less_power_abs_iff [simp,noatp]:
191 "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
194 show ?case by (simp add: zero_less_one)
197 show ?case by (force simp add: prems power_Suc zero_less_mult_iff)
200 lemma zero_le_power_abs [simp]:
201 "(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"
202 by (rule zero_le_power [OF abs_ge_zero])
204 lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{comm_ring_1,recpower}) ^ n"
206 have "-a = (- 1) * a" by (simp add: minus_mult_left [symmetric])
207 thus ?thesis by (simp only: power_mult_distrib)
210 text{*Lemma for @{text power_strict_decreasing}*}
211 lemma power_Suc_less:
212 "[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|]
215 apply (auto simp add: power_Suc mult_strict_left_mono)
218 lemma power_strict_decreasing:
219 "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]
223 apply (auto simp add: power_Suc power_Suc_less less_Suc_eq)
225 apply (subgoal_tac "a * a^m < 1 * a^n", simp)
226 apply (rule mult_strict_mono)
227 apply (auto simp add: zero_le_power zero_less_one order_less_imp_le)
230 text{*Proof resembles that of @{text power_strict_decreasing}*}
231 lemma power_decreasing:
232 "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]
236 apply (auto simp add: power_Suc le_Suc_eq)
238 apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
239 apply (rule mult_mono)
240 apply (auto simp add: zero_le_power zero_le_one)
243 lemma power_Suc_less_one:
244 "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1"
245 apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
248 text{*Proof again resembles that of @{text power_strict_decreasing}*}
249 lemma power_increasing:
250 "[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"
253 apply (auto simp add: power_Suc le_Suc_eq)
255 apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
256 apply (rule mult_mono)
257 apply (auto simp add: order_trans [OF zero_le_one] zero_le_power)
260 text{*Lemma for @{text power_strict_increasing}*}
261 lemma power_less_power_Suc:
262 "(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n"
264 apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one])
267 lemma power_strict_increasing:
268 "[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N"
271 apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq)
273 apply (subgoal_tac "1 * a^n < a * a^m", simp)
274 apply (rule mult_strict_mono)
275 apply (auto simp add: order_less_trans [OF zero_less_one] zero_le_power
279 lemma power_increasing_iff [simp]:
280 "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
281 by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le)
283 lemma power_strict_increasing_iff [simp]:
284 "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"
285 by (blast intro: power_less_imp_less_exp power_strict_increasing)
287 lemma power_le_imp_le_base:
288 assumes le: "a ^ Suc n \<le> b ^ Suc n"
289 and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b"
293 then have "b < a" by (simp only: linorder_not_le)
294 then have "b ^ Suc n < a ^ Suc n"
295 by (simp only: prems power_strict_mono)
296 from le and this show "False"
297 by (simp add: linorder_not_less [symmetric])
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"
305 proof (rule contrapos_pp [OF less])
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)
312 lemma power_inject_base:
313 "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
314 ==> a = (b::'a::{ordered_semidom,recpower})"
315 by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
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"
320 by (cases n, simp_all, rule power_inject_base)
323 subsection{*Exponentiation for the Natural Numbers*}
325 instance nat :: power ..
329 "p ^ (Suc n) = (p::nat) * (p ^ n)"
331 instance nat :: recpower
334 show "z^0 = 1" by simp
335 show "z^(Suc n) = z * (z^n)" by simp
339 "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"
340 by (induct n, simp_all add: power_Suc of_nat_mult)
342 lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n"
343 by (insert one_le_power [of i n], simp)
345 lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
346 by (induct "n", auto)
348 text{*Valid for the naturals, but what if @{text"0<i<1"}?
349 Premises cannot be weakened: consider the case where @{term "i=0"},
350 @{term "m=1"} and @{term "n=0"}.*}
351 lemma nat_power_less_imp_less:
352 assumes nonneg: "0 < (i\<Colon>nat)"
353 assumes less: "i^m < i^n"
355 proof (cases "i = 1")
356 case True with less power_one [where 'a = nat] show ?thesis by simp
358 case False with nonneg have "1 < i" by auto
359 from power_strict_increasing_iff [OF this] less show ?thesis ..
364 shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)"
365 by (induct m n rule: diff_induct)
366 (simp_all add: power_Suc nonzero_mult_divide_cancel_left nz)
369 text{*ML bindings for the general exponentiation theorems*}
372 val power_0 = thm"power_0";
373 val power_Suc = thm"power_Suc";
374 val power_0_Suc = thm"power_0_Suc";
375 val power_0_left = thm"power_0_left";
376 val power_one = thm"power_one";
377 val power_one_right = thm"power_one_right";
378 val power_add = thm"power_add";
379 val power_mult = thm"power_mult";
380 val power_mult_distrib = thm"power_mult_distrib";
381 val zero_less_power = thm"zero_less_power";
382 val zero_le_power = thm"zero_le_power";
383 val one_le_power = thm"one_le_power";
384 val gt1_imp_ge0 = thm"gt1_imp_ge0";
385 val power_gt1_lemma = thm"power_gt1_lemma";
386 val power_gt1 = thm"power_gt1";
387 val power_le_imp_le_exp = thm"power_le_imp_le_exp";
388 val power_inject_exp = thm"power_inject_exp";
389 val power_less_imp_less_exp = thm"power_less_imp_less_exp";
390 val power_mono = thm"power_mono";
391 val power_strict_mono = thm"power_strict_mono";
392 val power_eq_0_iff = thm"power_eq_0_iff";
393 val field_power_eq_0_iff = thm"field_power_eq_0_iff";
394 val field_power_not_zero = thm"field_power_not_zero";
395 val power_inverse = thm"power_inverse";
396 val nonzero_power_divide = thm"nonzero_power_divide";
397 val power_divide = thm"power_divide";
398 val power_abs = thm"power_abs";
399 val zero_less_power_abs_iff = thm"zero_less_power_abs_iff";
400 val zero_le_power_abs = thm "zero_le_power_abs";
401 val power_minus = thm"power_minus";
402 val power_Suc_less = thm"power_Suc_less";
403 val power_strict_decreasing = thm"power_strict_decreasing";
404 val power_decreasing = thm"power_decreasing";
405 val power_Suc_less_one = thm"power_Suc_less_one";
406 val power_increasing = thm"power_increasing";
407 val power_strict_increasing = thm"power_strict_increasing";
408 val power_le_imp_le_base = thm"power_le_imp_le_base";
409 val power_inject_base = thm"power_inject_base";
412 text{*ML bindings for the remaining theorems*}
415 val nat_one_le_power = thm"nat_one_le_power";
416 val nat_power_less_imp_less = thm"nat_power_less_imp_less";
417 val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";