src/HOL/Parity.thy
 author hoelzl Thu Feb 05 11:34:42 2009 +0100 (2009-02-05) changeset 29803 c56a5571f60a parent 29654 24e73987bfe2 child 30056 0a35bee25c20 permissions -rw-r--r--
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
1 (*  Title:      HOL/Library/Parity.thy
2     Author:     Jeremy Avigad, Jacques D. Fleuriot
3 *)
5 header {* Even and Odd for int and nat *}
7 theory Parity
8 imports Plain Presburger Main
9 begin
11 class even_odd =
12   fixes even :: "'a \<Rightarrow> bool"
14 abbreviation
15   odd :: "'a\<Colon>even_odd \<Rightarrow> bool" where
16   "odd x \<equiv> \<not> even x"
18 instantiation nat and int  :: even_odd
19 begin
21 definition
22   even_def [presburger]: "even x \<longleftrightarrow> (x\<Colon>int) mod 2 = 0"
24 definition
25   even_nat_def [presburger]: "even x \<longleftrightarrow> even (int x)"
27 instance ..
29 end
32 subsection {* Even and odd are mutually exclusive *}
34 lemma int_pos_lt_two_imp_zero_or_one:
35     "0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1"
36   by presburger
38 lemma neq_one_mod_two [simp, presburger]:
39   "((x::int) mod 2 ~= 0) = (x mod 2 = 1)" by presburger
42 subsection {* Behavior under integer arithmetic operations *}
43 declare dvd_def[algebra]
44 lemma nat_even_iff_2_dvd[algebra]: "even (x::nat) \<longleftrightarrow> 2 dvd x"
45   by (presburger add: even_nat_def even_def)
46 lemma int_even_iff_2_dvd[algebra]: "even (x::int) \<longleftrightarrow> 2 dvd x"
47   by presburger
49 lemma even_times_anything: "even (x::int) ==> even (x * y)"
50   by algebra
52 lemma anything_times_even: "even (y::int) ==> even (x * y)" by algebra
54 lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)"
55   by (simp add: even_def zmod_zmult1_eq)
57 lemma even_product[presburger]: "even((x::int) * y) = (even x | even y)"
58   apply (auto simp add: even_times_anything anything_times_even)
59   apply (rule ccontr)
60   apply (auto simp add: odd_times_odd)
61   done
63 lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"
64   by presburger
66 lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"
67   by presburger
69 lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"
70   by presburger
72 lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger
74 lemma even_sum[presburger]: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
75   by presburger
77 lemma even_neg[presburger, algebra]: "even (-(x::int)) = even x" by presburger
79 lemma even_difference:
80     "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger
82 lemma even_pow_gt_zero:
83     "even (x::int) ==> 0 < n ==> even (x^n)"
84   by (induct n) (auto simp add: even_product)
86 lemma odd_pow_iff[presburger, algebra]:
87   "odd ((x::int) ^ n) \<longleftrightarrow> (n = 0 \<or> odd x)"
88   apply (induct n, simp_all)
89   apply presburger
90   apply (case_tac n, auto)
92   done
94 lemma odd_pow: "odd x ==> odd((x::int)^n)" by (simp add: odd_pow_iff)
96 lemma even_power[presburger]: "even ((x::int)^n) = (even x & 0 < n)"
97   apply (auto simp add: even_pow_gt_zero)
98   apply (erule contrapos_pp, erule odd_pow)
99   apply (erule contrapos_pp, simp add: even_def)
100   done
102 lemma even_zero[presburger]: "even (0::int)" by presburger
104 lemma odd_one[presburger]: "odd (1::int)" by presburger
106 lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero
107   odd_one even_product even_sum even_neg even_difference even_power
110 subsection {* Equivalent definitions *}
112 lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x"
113   by presburger
115 lemma two_times_odd_div_two_plus_one: "odd (x::int) ==>
116     2 * (x div 2) + 1 = x" by presburger
118 lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger
120 lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" by presburger
122 subsection {* even and odd for nats *}
124 lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
127 lemma even_nat_product[presburger, algebra]: "even((x::nat) * y) = (even x | even y)"
128   by (simp add: even_nat_def int_mult)
130 lemma even_nat_sum[presburger, algebra]: "even ((x::nat) + y) =
131     ((even x & even y) | (odd x & odd y))" by presburger
133 lemma even_nat_difference[presburger, algebra]:
134     "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
135 by presburger
137 lemma even_nat_Suc[presburger, algebra]: "even (Suc x) = odd x" by presburger
139 lemma even_nat_power[presburger, algebra]: "even ((x::nat)^y) = (even x & 0 < y)"
140   by (simp add: even_nat_def int_power)
142 lemma even_nat_zero[presburger]: "even (0::nat)" by presburger
144 lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard]
145   even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power
148 subsection {* Equivalent definitions *}
150 lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==>
151     x = 0 | x = Suc 0" by presburger
153 lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
154   by presburger
156 lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
157 by presburger
159 lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
160   by presburger
162 lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
163   by presburger
165 lemma even_nat_div_two_times_two: "even (x::nat) ==>
166     Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger
168 lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>
169     Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger
171 lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
172   by presburger
174 lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
175   by presburger
178 subsection {* Parity and powers *}
180 lemma  minus_one_even_odd_power:
181      "(even x --> (- 1::'a::{comm_ring_1,recpower})^x = 1) &
182       (odd x --> (- 1::'a)^x = - 1)"
183   apply (induct x)
184   apply (rule conjI)
185   apply simp
186   apply (insert even_nat_zero, blast)
188   done
190 lemma minus_one_even_power [simp]:
191     "even x ==> (- 1::'a::{comm_ring_1,recpower})^x = 1"
192   using minus_one_even_odd_power by blast
194 lemma minus_one_odd_power [simp]:
195     "odd x ==> (- 1::'a::{comm_ring_1,recpower})^x = - 1"
196   using minus_one_even_odd_power by blast
198 lemma neg_one_even_odd_power:
199      "(even x --> (-1::'a::{number_ring,recpower})^x = 1) &
200       (odd x --> (-1::'a)^x = -1)"
201   apply (induct x)
202   apply (simp, simp add: power_Suc)
203   done
205 lemma neg_one_even_power [simp]:
206     "even x ==> (-1::'a::{number_ring,recpower})^x = 1"
207   using neg_one_even_odd_power by blast
209 lemma neg_one_odd_power [simp]:
210     "odd x ==> (-1::'a::{number_ring,recpower})^x = -1"
211   using neg_one_even_odd_power by blast
213 lemma neg_power_if:
214      "(-x::'a::{comm_ring_1,recpower}) ^ n =
215       (if even n then (x ^ n) else -(x ^ n))"
216   apply (induct n)
217   apply (simp_all split: split_if_asm add: power_Suc)
218   done
220 lemma zero_le_even_power: "even n ==>
221     0 <= (x::'a::{recpower,ordered_ring_strict}) ^ n"
223   apply (erule exE)
224   apply (erule ssubst)
226   apply (rule zero_le_square)
227   done
229 lemma zero_le_odd_power: "odd n ==>
230     (0 <= (x::'a::{recpower,ordered_idom}) ^ n) = (0 <= x)"
232   apply (erule exE)
233   apply (erule ssubst)
234   apply (subst power_Suc)
236   apply (subst zero_le_mult_iff)
237   apply auto
238   apply (subgoal_tac "x = 0 & y > 0")
239   apply (erule conjE, assumption)
240   apply (subst power_eq_0_iff [symmetric])
241   apply (subgoal_tac "0 <= x^y * x^y")
242   apply simp
243   apply (rule zero_le_square)+
244   done
246 lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) =
247     (even n | (odd n & 0 <= x))"
248   apply auto
249   apply (subst zero_le_odd_power [symmetric])
250   apply assumption+
251   apply (erule zero_le_even_power)
252   done
254 lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) =
255     (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
257   unfolding order_less_le zero_le_power_eq by auto
259 lemma power_less_zero_eq[presburger]: "((x::'a::{recpower,ordered_idom}) ^ n < 0) =
260     (odd n & x < 0)"
261   apply (subst linorder_not_le [symmetric])+
262   apply (subst zero_le_power_eq)
263   apply auto
264   done
266 lemma power_le_zero_eq[presburger]: "((x::'a::{recpower,ordered_idom}) ^ n <= 0) =
267     (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"
268   apply (subst linorder_not_less [symmetric])+
269   apply (subst zero_less_power_eq)
270   apply auto
271   done
273 lemma power_even_abs: "even n ==>
274     (abs (x::'a::{recpower,ordered_idom}))^n = x^n"
275   apply (subst power_abs [symmetric])
277   done
279 lemma zero_less_power_nat_eq[presburger]: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)"
280   by (induct n) auto
282 lemma power_minus_even [simp]: "even n ==>
283     (- x)^n = (x^n::'a::{recpower,comm_ring_1})"
284   apply (subst power_minus)
285   apply simp
286   done
288 lemma power_minus_odd [simp]: "odd n ==>
289     (- x)^n = - (x^n::'a::{recpower,comm_ring_1})"
290   apply (subst power_minus)
291   apply simp
292   done
294 lemma power_mono_even: fixes x y :: "'a :: {recpower, ordered_idom}"
295   assumes "even n" and "\<bar>x\<bar> \<le> \<bar>y\<bar>"
296   shows "x^n \<le> y^n"
297 proof -
298   have "0 \<le> \<bar>x\<bar>" by auto
299   with `\<bar>x\<bar> \<le> \<bar>y\<bar>`
300   have "\<bar>x\<bar>^n \<le> \<bar>y\<bar>^n" by (rule power_mono)
301   thus ?thesis unfolding power_even_abs[OF `even n`] .
302 qed
304 lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger
306 lemma power_mono_odd: fixes x y :: "'a :: {recpower, ordered_idom}"
307   assumes "odd n" and "x \<le> y"
308   shows "x^n \<le> y^n"
309 proof (cases "y < 0")
310   case True with `x \<le> y` have "-y \<le> -x" and "0 \<le> -y" by auto
311   hence "(-y)^n \<le> (-x)^n" by (rule power_mono)
312   thus ?thesis unfolding power_minus_odd[OF `odd n`] by auto
313 next
314   case False
315   show ?thesis
316   proof (cases "x < 0")
317     case True hence "n \<noteq> 0" and "x \<le> 0" using `odd n`[THEN odd_pos] by auto
318     hence "x^n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto
319     moreover
320     from `\<not> y < 0` have "0 \<le> y" by auto
321     hence "0 \<le> y^n" by auto
322     ultimately show ?thesis by auto
323   next
324     case False hence "0 \<le> x" by auto
325     with `x \<le> y` show ?thesis using power_mono by auto
326   qed
327 qed
329 subsection {* General Lemmas About Division *}
331 lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"
332 apply (induct "m")
334 done
336 declare Suc_times_mod_eq [of "number_of w", standard, simp]
338 lemma [simp]: "n div k \<le> (Suc n) div k"
341 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
342 by arith
344 lemma div_2_gt_zero [simp]: "(1::nat) < n ==> 0 < n div 2"
345 by arith
347   (* Potential use of algebra : Equality modulo n*)
348 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
351 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
352 proof -
353   have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
354   also have "... = Suc m mod n" by (rule mod_mult_self3)
355   finally show ?thesis .
356 qed
358 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
359 apply (subst mod_Suc [of m])
360 apply (subst mod_Suc [of "m mod n"], simp)
361 done
364 subsection {* More Even/Odd Results *}
366 lemma even_mult_two_ex: "even(n) = (\<exists>m::nat. n = 2*m)" by presburger
367 lemma odd_Suc_mult_two_ex: "odd(n) = (\<exists>m. n = Suc (2*m))" by presburger
368 lemma even_add [simp]: "even(m + n::nat) = (even m = even n)"  by presburger
370 lemma odd_add [simp]: "odd(m + n::nat) = (odd m \<noteq> odd n)" by presburger
372 lemma div_Suc: "Suc a div c = a div c + Suc 0 div c +
373     (a mod c + Suc 0 mod c) div c"
374   apply (subgoal_tac "Suc a = a + Suc 0")
375   apply (erule ssubst)
377   done
379 lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" by presburger
381 lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)"
382 by presburger
384 lemma even_num_iff: "0 < n ==> even n = (~ even(n - 1 :: nat))"  by presburger
385 lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)" by presburger
387 lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n - 1) div 2)" by presburger
389 lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n - 1) div 2)"
390   by presburger
392 text {* Simplify, when the exponent is a numeral *}
394 lemmas power_0_left_number_of = power_0_left [of "number_of w", standard]
395 declare power_0_left_number_of [simp]
397 lemmas zero_le_power_eq_number_of [simp] =
398     zero_le_power_eq [of _ "number_of w", standard]
400 lemmas zero_less_power_eq_number_of [simp] =
401     zero_less_power_eq [of _ "number_of w", standard]
403 lemmas power_le_zero_eq_number_of [simp] =
404     power_le_zero_eq [of _ "number_of w", standard]
406 lemmas power_less_zero_eq_number_of [simp] =
407     power_less_zero_eq [of _ "number_of w", standard]
409 lemmas zero_less_power_nat_eq_number_of [simp] =
410     zero_less_power_nat_eq [of _ "number_of w", standard]
412 lemmas power_eq_0_iff_number_of [simp] = power_eq_0_iff [of _ "number_of w", standard]
414 lemmas power_even_abs_number_of [simp] = power_even_abs [of "number_of w" _, standard]
417 subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *}
419 lemma even_power_le_0_imp_0:
420     "a ^ (2*k) \<le> (0::'a::{ordered_idom,recpower}) ==> a=0"
421   by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc)
423 lemma zero_le_power_iff[presburger]:
424   "(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even n)"
425 proof cases
426   assume even: "even n"
427   then obtain k where "n = 2*k"
428     by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2)
429   thus ?thesis by (simp add: zero_le_even_power even)
430 next
431   assume odd: "odd n"
432   then obtain k where "n = Suc(2*k)"
433     by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)
434   thus ?thesis
435     by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power
436              dest!: even_power_le_0_imp_0)
437 qed
440 subsection {* Miscellaneous *}
442 lemma [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger
443 lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger
444 lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2"  by presburger
445 lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger
447 lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger
448 lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger
449 lemma even_nat_plus_one_div_two: "even (x::nat) ==>
450     (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger
452 lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>
453     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger
455 end