1 (* Title: HOL/Power.thy
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
4 Copyright 1997 University of Cambridge
8 header{*Exponentiation and Binomial Coefficients*}
10 theory Power = Divides:
12 subsection{*Powers for Arbitrary Semirings*}
14 axclass recpower \<subseteq> comm_semiring_1_cancel, power
15 power_0 [simp]: "a ^ 0 = 1"
16 power_Suc: "a ^ (Suc n) = a * (a ^ n)"
18 lemma power_0_Suc [simp]: "(0::'a::recpower) ^ (Suc n) = 0"
19 by (simp add: power_Suc)
21 text{*It looks plausible as a simprule, but its effect can be strange.*}
22 lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::recpower))"
23 by (induct_tac "n", auto)
25 lemma power_one [simp]: "1^n = (1::'a::recpower)"
26 apply (induct_tac "n")
27 apply (auto simp add: power_Suc)
30 lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
31 by (simp add: power_Suc)
33 lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
34 apply (induct_tac "n")
35 apply (simp_all add: power_Suc mult_ac)
38 lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
39 apply (induct_tac "n")
40 apply (simp_all add: power_Suc power_add)
43 lemma power_mult_distrib: "((a::'a::recpower) * b) ^ n = (a^n) * (b^n)"
44 apply (induct_tac "n")
45 apply (auto simp add: power_Suc mult_ac)
48 lemma zero_less_power:
49 "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"
50 apply (induct_tac "n")
51 apply (simp_all add: power_Suc zero_less_one mult_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"
63 apply (induct_tac "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 "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
85 by (simp add: power_gt1_lemma power_Suc)
87 lemma power_le_imp_le_exp:
88 assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"
89 shows "!!n. a^m \<le> a^n ==> m \<le> n"
98 from prems have "a * a^m \<le> 1" by (simp add: power_Suc)
100 by (force simp only: power_gt1_lemma
101 linorder_not_less [symmetric])
104 from prems show ?thesis
105 by (force dest: mult_left_le_imp_le
106 simp add: power_Suc order_less_trans [OF zero_less_one gt1])
110 text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
111 lemma power_inject_exp [simp]:
112 "1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)"
113 by (force simp add: order_antisym power_le_imp_le_exp)
115 text{*Can relax the first premise to @{term "0<a"} in the case of the
117 lemma power_less_imp_less_exp:
118 "[| (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"
119 by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
124 "[|a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"
125 apply (induct_tac "n")
126 apply (simp_all add: power_Suc)
127 apply (auto intro: mult_mono zero_le_power order_trans [of 0 a b])
130 lemma power_strict_mono [rule_format]:
131 "[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]
132 ==> 0 < n --> a^n < b^n"
133 apply (induct_tac "n")
134 apply (auto simp add: mult_strict_mono zero_le_power power_Suc
135 order_le_less_trans [of 0 a b])
138 lemma power_eq_0_iff [simp]:
139 "(a^n = 0) = (a = (0::'a::{ordered_idom,recpower}) & 0<n)"
140 apply (induct_tac "n")
141 apply (auto simp add: power_Suc zero_neq_one [THEN not_sym])
144 lemma field_power_eq_0_iff [simp]:
145 "(a^n = 0) = (a = (0::'a::{field,recpower}) & 0<n)"
146 apply (induct_tac "n")
147 apply (auto simp add: power_Suc field_mult_eq_0_iff zero_neq_one[THEN not_sym])
150 lemma field_power_not_zero: "a \<noteq> (0::'a::{field,recpower}) ==> a^n \<noteq> 0"
153 lemma nonzero_power_inverse:
154 "a \<noteq> 0 ==> inverse ((a::'a::{field,recpower}) ^ n) = (inverse a) ^ n"
155 apply (induct_tac "n")
156 apply (auto simp add: power_Suc nonzero_inverse_mult_distrib mult_commute)
159 text{*Perhaps these should be simprules.*}
161 "inverse ((a::'a::{field,division_by_zero,recpower}) ^ n) = (inverse a) ^ n"
162 apply (induct_tac "n")
163 apply (auto simp add: power_Suc inverse_mult_distrib)
166 lemma nonzero_power_divide:
167 "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"
168 by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
171 "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"
172 apply (case_tac "b=0", simp add: power_0_left)
173 apply (rule nonzero_power_divide)
177 lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"
178 apply (induct_tac "n")
179 apply (auto simp add: power_Suc abs_mult)
182 lemma zero_less_power_abs_iff [simp]:
183 "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
186 show ?case by (simp add: zero_less_one)
189 show ?case by (force simp add: prems power_Suc zero_less_mult_iff)
192 lemma zero_le_power_abs [simp]:
193 "(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"
194 apply (induct_tac "n")
195 apply (auto simp add: zero_le_one zero_le_power)
198 lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{comm_ring_1,recpower}) ^ n"
200 have "-a = (- 1) * a" by (simp add: minus_mult_left [symmetric])
201 thus ?thesis by (simp only: power_mult_distrib)
204 text{*Lemma for @{text power_strict_decreasing}*}
205 lemma power_Suc_less:
206 "[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|]
209 apply (auto simp add: power_Suc mult_strict_left_mono)
212 lemma power_strict_decreasing:
213 "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]
216 apply (induct_tac "N")
217 apply (auto simp add: power_Suc power_Suc_less less_Suc_eq)
219 apply (subgoal_tac "a * a^m < 1 * a^n", simp)
220 apply (rule mult_strict_mono)
221 apply (auto simp add: zero_le_power zero_less_one order_less_imp_le)
224 text{*Proof resembles that of @{text power_strict_decreasing}*}
225 lemma power_decreasing:
226 "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]
229 apply (induct_tac "N")
230 apply (auto simp add: power_Suc le_Suc_eq)
232 apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
233 apply (rule mult_mono)
234 apply (auto simp add: zero_le_power zero_le_one)
237 lemma power_Suc_less_one:
238 "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1"
239 apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
242 text{*Proof again resembles that of @{text power_strict_decreasing}*}
243 lemma power_increasing:
244 "[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"
246 apply (induct_tac "N")
247 apply (auto simp add: power_Suc le_Suc_eq)
249 apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
250 apply (rule mult_mono)
251 apply (auto simp add: order_trans [OF zero_le_one] zero_le_power)
254 text{*Lemma for @{text power_strict_increasing}*}
255 lemma power_less_power_Suc:
256 "(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n"
258 apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one])
261 lemma power_strict_increasing:
262 "[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N"
264 apply (induct_tac "N")
265 apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq)
267 apply (subgoal_tac "1 * a^n < a * a^m", simp)
268 apply (rule mult_strict_mono)
269 apply (auto simp add: order_less_trans [OF zero_less_one] zero_le_power
273 lemma power_increasing_iff [simp]:
274 "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
275 by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le)
277 lemma power_strict_increasing_iff [simp]:
278 "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"
279 by (blast intro: power_less_imp_less_exp power_strict_increasing)
281 lemma power_le_imp_le_base:
282 assumes le: "a ^ Suc n \<le> b ^ Suc n"
283 and xnonneg: "(0::'a::{ordered_semidom,recpower}) \<le> a"
284 and ynonneg: "0 \<le> b"
288 then have "b < a" by (simp only: linorder_not_le)
289 then have "b ^ Suc n < a ^ Suc n"
290 by (simp only: prems power_strict_mono)
291 from le and this show "False"
292 by (simp add: linorder_not_less [symmetric])
295 lemma power_inject_base:
296 "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
297 ==> a = (b::'a::{ordered_semidom,recpower})"
298 by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
301 subsection{*Exponentiation for the Natural Numbers*}
305 "p ^ (Suc n) = (p::nat) * (p ^ n)"
307 instance nat :: recpower
310 show "z^0 = 1" by simp
311 show "z^(Suc n) = z * (z^n)" by simp
314 lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n"
315 by (insert one_le_power [of i n], simp)
317 lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
318 apply (unfold dvd_def)
319 apply (erule not_less_iff_le [THEN iffD2, THEN add_diff_inverse, THEN subst])
320 apply (simp add: power_add)
323 text{*Valid for the naturals, but what if @{text"0<i<1"}?
324 Premises cannot be weakened: consider the case where @{term "i=0"},
325 @{term "m=1"} and @{term "n=0"}.*}
326 lemma nat_power_less_imp_less: "!!i::nat. [| 0 < i; i^m < i^n |] ==> m < n"
328 apply (drule leI [THEN le_imp_power_dvd, THEN dvd_imp_le, THEN leD])
329 apply (erule zero_less_power, auto)
332 lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
333 by (induct_tac "n", auto)
335 lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
336 apply (induct_tac "j")
337 apply (simp_all add: le_Suc_eq)
338 apply (blast dest!: dvd_mult_right)
341 lemma power_dvd_imp_le: "[|i^m dvd i^n; (1::nat) < i|] ==> m \<le> n"
342 apply (rule power_le_imp_le_exp, assumption)
343 apply (erule dvd_imp_le, simp)
347 subsection{*Binomial Coefficients*}
349 text{*This development is based on the work of Andy Gordon and
353 binomial :: "[nat,nat] => nat" (infixl "choose" 65)
356 binomial_0: "(0 choose k) = (if k = 0 then 1 else 0)"
358 binomial_Suc: "(Suc n choose k) =
359 (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
361 lemma binomial_n_0 [simp]: "(n choose 0) = 1"
362 by (case_tac "n", simp_all)
364 lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
367 lemma binomial_Suc_Suc [simp]:
368 "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
371 lemma binomial_eq_0 [rule_format]: "\<forall>k. n < k --> (n choose k) = 0"
372 apply (induct_tac "n", auto)
374 apply (erule mp, arith)
377 declare binomial_0 [simp del] binomial_Suc [simp del]
379 lemma binomial_n_n [simp]: "(n choose n) = 1"
380 apply (induct_tac "n")
381 apply (simp_all add: binomial_eq_0)
384 lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n"
385 by (induct_tac "n", simp_all)
387 lemma binomial_1 [simp]: "(n choose Suc 0) = n"
388 by (induct_tac "n", simp_all)
390 lemma zero_less_binomial [rule_format]: "k \<le> n --> 0 < (n choose k)"
391 by (rule_tac m = n and n = k in diff_induct, simp_all)
393 lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)"
394 apply (safe intro!: binomial_eq_0)
395 apply (erule contrapos_pp)
396 apply (simp add: zero_less_binomial)
399 lemma zero_less_binomial_iff: "(0 < n choose k) = (k\<le>n)"
400 by (simp add: linorder_not_less [symmetric] binomial_eq_0_iff [symmetric])
402 (*Might be more useful if re-oriented*)
403 lemma Suc_times_binomial_eq [rule_format]:
404 "\<forall>k. k \<le> n --> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
405 apply (induct_tac "n")
406 apply (simp add: binomial_0, clarify)
408 apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq
412 text{*This is the well-known version, but it's harder to use because of the
413 need to reason about division.*}
414 lemma binomial_Suc_Suc_eq_times:
415 "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
416 by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc
417 del: mult_Suc mult_Suc_right)
419 text{*Another version, with -1 instead of Suc.*}
420 lemma times_binomial_minus1_eq:
421 "[|k \<le> n; 0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"
422 apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)
423 apply (simp split add: nat_diff_split, auto)
426 text{*ML bindings for the general exponentiation theorems*}
429 val power_0 = thm"power_0";
430 val power_Suc = thm"power_Suc";
431 val power_0_Suc = thm"power_0_Suc";
432 val power_0_left = thm"power_0_left";
433 val power_one = thm"power_one";
434 val power_one_right = thm"power_one_right";
435 val power_add = thm"power_add";
436 val power_mult = thm"power_mult";
437 val power_mult_distrib = thm"power_mult_distrib";
438 val zero_less_power = thm"zero_less_power";
439 val zero_le_power = thm"zero_le_power";
440 val one_le_power = thm"one_le_power";
441 val gt1_imp_ge0 = thm"gt1_imp_ge0";
442 val power_gt1_lemma = thm"power_gt1_lemma";
443 val power_gt1 = thm"power_gt1";
444 val power_le_imp_le_exp = thm"power_le_imp_le_exp";
445 val power_inject_exp = thm"power_inject_exp";
446 val power_less_imp_less_exp = thm"power_less_imp_less_exp";
447 val power_mono = thm"power_mono";
448 val power_strict_mono = thm"power_strict_mono";
449 val power_eq_0_iff = thm"power_eq_0_iff";
450 val field_power_eq_0_iff = thm"field_power_eq_0_iff";
451 val field_power_not_zero = thm"field_power_not_zero";
452 val power_inverse = thm"power_inverse";
453 val nonzero_power_divide = thm"nonzero_power_divide";
454 val power_divide = thm"power_divide";
455 val power_abs = thm"power_abs";
456 val zero_less_power_abs_iff = thm"zero_less_power_abs_iff";
457 val zero_le_power_abs = thm "zero_le_power_abs";
458 val power_minus = thm"power_minus";
459 val power_Suc_less = thm"power_Suc_less";
460 val power_strict_decreasing = thm"power_strict_decreasing";
461 val power_decreasing = thm"power_decreasing";
462 val power_Suc_less_one = thm"power_Suc_less_one";
463 val power_increasing = thm"power_increasing";
464 val power_strict_increasing = thm"power_strict_increasing";
465 val power_le_imp_le_base = thm"power_le_imp_le_base";
466 val power_inject_base = thm"power_inject_base";
469 text{*ML bindings for the remaining theorems*}
472 val nat_one_le_power = thm"nat_one_le_power";
473 val le_imp_power_dvd = thm"le_imp_power_dvd";
474 val nat_power_less_imp_less = thm"nat_power_less_imp_less";
475 val nat_zero_less_power_iff = thm"nat_zero_less_power_iff";
476 val power_le_dvd = thm"power_le_dvd";
477 val power_dvd_imp_le = thm"power_dvd_imp_le";
478 val binomial_n_0 = thm"binomial_n_0";
479 val binomial_0_Suc = thm"binomial_0_Suc";
480 val binomial_Suc_Suc = thm"binomial_Suc_Suc";
481 val binomial_eq_0 = thm"binomial_eq_0";
482 val binomial_n_n = thm"binomial_n_n";
483 val binomial_Suc_n = thm"binomial_Suc_n";
484 val binomial_1 = thm"binomial_1";
485 val zero_less_binomial = thm"zero_less_binomial";
486 val binomial_eq_0_iff = thm"binomial_eq_0_iff";
487 val zero_less_binomial_iff = thm"zero_less_binomial_iff";
488 val Suc_times_binomial_eq = thm"Suc_times_binomial_eq";
489 val binomial_Suc_Suc_eq_times = thm"binomial_Suc_Suc_eq_times";
490 val times_binomial_minus1_eq = thm"times_binomial_minus1_eq";