src/HOL/Parity.thy
 author haftmann Thu Oct 31 11:44:20 2013 +0100 (2013-10-31) changeset 54228 229282d53781 parent 54227 63b441f49645 child 54489 03ff4d1e6784 permissions -rw-r--r--
purely algebraic foundation for even/odd
1 (*  Title:      HOL/Parity.thy
3     Author:     Jacques D. Fleuriot
4 *)
6 header {* Even and Odd for int and nat *}
8 theory Parity
9 imports Main
10 begin
12 class even_odd = semiring_div_parity
13 begin
15 definition even :: "'a \<Rightarrow> bool"
16 where
17   even_def [presburger]: "even a \<longleftrightarrow> a mod 2 = 0"
19 lemma even_iff_2_dvd [algebra]:
20   "even a \<longleftrightarrow> 2 dvd a"
21   by (simp add: even_def dvd_eq_mod_eq_0)
23 lemma even_zero [simp]:
24   "even 0"
27 lemma even_times_anything:
28   "even a \<Longrightarrow> even (a * b)"
31 lemma anything_times_even:
32   "even a \<Longrightarrow> even (b * a)"
35 abbreviation odd :: "'a \<Rightarrow> bool"
36 where
37   "odd a \<equiv> \<not> even a"
39 lemma odd_times_odd:
40   "odd a \<Longrightarrow> odd b \<Longrightarrow> odd (a * b)"
41   by (auto simp add: even_def mod_mult_left_eq)
43 lemma even_product [simp, presburger]:
44   "even (a * b) \<longleftrightarrow> even a \<or> even b"
45   apply (auto simp add: even_times_anything anything_times_even)
46   apply (rule ccontr)
47   apply (auto simp add: odd_times_odd)
48   done
50 end
52 instance nat and int  :: even_odd ..
54 lemma even_nat_def [presburger]:
55   "even x \<longleftrightarrow> even (int x)"
56   by (auto simp add: even_def int_eq_iff int_mult nat_mult_distrib)
58 lemma transfer_int_nat_relations:
59   "even (int x) \<longleftrightarrow> even x"
63   transfer_int_nat_relations
64 ]
66 lemma odd_one_int [simp]:
67   "odd (1::int)"
68   by presburger
70 lemma odd_1_nat [simp]:
71   "odd (1::nat)"
72   by presburger
74 lemma even_numeral_int [simp]: "even (numeral (Num.Bit0 k) :: int)"
75   unfolding even_def by simp
77 lemma odd_numeral_int [simp]: "odd (numeral (Num.Bit1 k) :: int)"
78   unfolding even_def by simp
80 (* TODO: proper simp rules for Num.Bit0, Num.Bit1 *)
81 declare even_def [of "neg_numeral v", simp] for v
83 lemma even_numeral_nat [simp]: "even (numeral (Num.Bit0 k) :: nat)"
84   unfolding even_nat_def by simp
86 lemma odd_numeral_nat [simp]: "odd (numeral (Num.Bit1 k) :: nat)"
87   unfolding even_nat_def by simp
89 subsection {* Even and odd are mutually exclusive *}
92 subsection {* Behavior under integer arithmetic operations *}
94 lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"
95 by presburger
97 lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"
98 by presburger
100 lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"
101 by presburger
103 lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger
105 lemma even_sum[simp,presburger]:
106   "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
107 by presburger
109 lemma even_neg[simp,presburger,algebra]: "even (-(x::int)) = even x"
110 by presburger
112 lemma even_difference[simp]:
113     "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger
115 lemma even_power[simp,presburger]: "even ((x::int)^n) = (even x & n \<noteq> 0)"
116 by (induct n) auto
118 lemma odd_pow: "odd x ==> odd((x::int)^n)" by simp
121 subsection {* Equivalent definitions *}
123 lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x"
124 by presburger
126 lemma two_times_odd_div_two_plus_one:
127   "odd (x::int) ==> 2 * (x div 2) + 1 = x"
128 by presburger
130 lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger
132 lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" by presburger
134 subsection {* even and odd for nats *}
136 lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
139 lemma even_product_nat[simp,presburger,algebra]:
140   "even((x::nat) * y) = (even x | even y)"
141 by (simp add: even_nat_def int_mult)
143 lemma even_sum_nat[simp,presburger,algebra]:
144   "even ((x::nat) + y) = ((even x & even y) | (odd x & odd y))"
145 by presburger
147 lemma even_difference_nat[simp,presburger,algebra]:
148   "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
149 by presburger
151 lemma even_Suc[simp,presburger,algebra]: "even (Suc x) = odd x"
152 by presburger
154 lemma even_power_nat[simp,presburger,algebra]:
155   "even ((x::nat)^y) = (even x & 0 < y)"
156 by (simp add: even_nat_def int_power)
159 subsection {* Equivalent definitions *}
161 lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
162 by presburger
164 lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
165 by presburger
167 lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
168 by presburger
170 lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
171 by presburger
173 lemma even_nat_div_two_times_two: "even (x::nat) ==>
174     Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger
176 lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>
177     Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger
179 lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
180 by presburger
182 lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
183 by presburger
186 subsection {* Parity and powers *}
188 lemma (in comm_ring_1) neg_power_if:
189   "(- a) ^ n = (if even n then (a ^ n) else - (a ^ n))"
190   by (induct n) simp_all
192 lemma (in comm_ring_1)
193   shows minus_one_even_power [simp]: "even n \<Longrightarrow> (- 1) ^ n = 1"
194   and minus_one_odd_power [simp]: "odd n \<Longrightarrow> (- 1) ^ n = - 1"
195   by (simp_all add: neg_power_if del: minus_one)
197 lemma (in comm_ring_1)
198   shows neg_one_even_power [simp]: "even n \<Longrightarrow> (-1) ^ n = 1"
199   and neg_one_odd_power [simp]: "odd n \<Longrightarrow> (-1) ^ n = - 1"
200   by (simp_all add: minus_one [symmetric] del: minus_one)
202 lemma zero_le_even_power: "even n ==>
203     0 <= (x::'a::{linordered_ring,monoid_mult}) ^ n"
205   apply (erule exE)
206   apply (erule ssubst)
208   apply (rule zero_le_square)
209   done
211 lemma zero_le_odd_power: "odd n ==>
212     (0 <= (x::'a::{linordered_idom}) ^ n) = (0 <= x)"
213 apply (auto simp: odd_nat_equiv_def2 power_add zero_le_mult_iff)
214 apply (metis field_power_not_zero divisors_zero order_antisym_conv zero_le_square)
215 done
217 lemma zero_le_power_eq [presburger]: "(0 <= (x::'a::{linordered_idom}) ^ n) =
218     (even n | (odd n & 0 <= x))"
219   apply auto
220   apply (subst zero_le_odd_power [symmetric])
221   apply assumption+
222   apply (erule zero_le_even_power)
223   done
225 lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{linordered_idom}) ^ n) =
226     (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
228   unfolding order_less_le zero_le_power_eq by auto
230 lemma power_less_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n < 0) =
231     (odd n & x < 0)"
232   apply (subst linorder_not_le [symmetric])+
233   apply (subst zero_le_power_eq)
234   apply auto
235   done
237 lemma power_le_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n <= 0) =
238     (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"
239   apply (subst linorder_not_less [symmetric])+
240   apply (subst zero_less_power_eq)
241   apply auto
242   done
244 lemma power_even_abs: "even n ==>
245     (abs (x::'a::{linordered_idom}))^n = x^n"
246   apply (subst power_abs [symmetric])
248   done
250 lemma power_minus_even [simp]: "even n ==>
251     (- x)^n = (x^n::'a::{comm_ring_1})"
252   apply (subst power_minus)
253   apply simp
254   done
256 lemma power_minus_odd [simp]: "odd n ==>
257     (- x)^n = - (x^n::'a::{comm_ring_1})"
258   apply (subst power_minus)
259   apply simp
260   done
262 lemma power_mono_even: fixes x y :: "'a :: {linordered_idom}"
263   assumes "even n" and "\<bar>x\<bar> \<le> \<bar>y\<bar>"
264   shows "x^n \<le> y^n"
265 proof -
266   have "0 \<le> \<bar>x\<bar>" by auto
267   with `\<bar>x\<bar> \<le> \<bar>y\<bar>`
268   have "\<bar>x\<bar>^n \<le> \<bar>y\<bar>^n" by (rule power_mono)
269   thus ?thesis unfolding power_even_abs[OF `even n`] .
270 qed
272 lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger
274 lemma power_mono_odd: fixes x y :: "'a :: {linordered_idom}"
275   assumes "odd n" and "x \<le> y"
276   shows "x^n \<le> y^n"
277 proof (cases "y < 0")
278   case True with `x \<le> y` have "-y \<le> -x" and "0 \<le> -y" by auto
279   hence "(-y)^n \<le> (-x)^n" by (rule power_mono)
280   thus ?thesis unfolding power_minus_odd[OF `odd n`] by auto
281 next
282   case False
283   show ?thesis
284   proof (cases "x < 0")
285     case True hence "n \<noteq> 0" and "x \<le> 0" using `odd n`[THEN odd_pos] by auto
286     hence "x^n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto
287     moreover
288     from `\<not> y < 0` have "0 \<le> y" by auto
289     hence "0 \<le> y^n" by auto
290     ultimately show ?thesis by auto
291   next
292     case False hence "0 \<le> x" by auto
293     with `x \<le> y` show ?thesis using power_mono by auto
294   qed
295 qed
298 subsection {* More Even/Odd Results *}
300 lemma even_mult_two_ex: "even(n) = (\<exists>m::nat. n = 2*m)" by presburger
301 lemma odd_Suc_mult_two_ex: "odd(n) = (\<exists>m. n = Suc (2*m))" by presburger
302 lemma even_add [simp]: "even(m + n::nat) = (even m = even n)"  by presburger
304 lemma odd_add [simp]: "odd(m + n::nat) = (odd m \<noteq> odd n)" by presburger
306 lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" by presburger
308 lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)"
309 by presburger
311 lemma even_num_iff: "0 < n ==> even n = (~ even(n - 1 :: nat))"  by presburger
312 lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)" by presburger
314 lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n - 1) div 2)" by presburger
316 lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n - 1) div 2)"
317   by presburger
319 text {* Simplify, when the exponent is a numeral *}
321 lemmas zero_le_power_eq_numeral [simp] =
322   zero_le_power_eq [of _ "numeral w"] for w
324 lemmas zero_less_power_eq_numeral [simp] =
325   zero_less_power_eq [of _ "numeral w"] for w
327 lemmas power_le_zero_eq_numeral [simp] =
328   power_le_zero_eq [of _ "numeral w"] for w
330 lemmas power_less_zero_eq_numeral [simp] =
331   power_less_zero_eq [of _ "numeral w"] for w
333 lemmas zero_less_power_nat_eq_numeral [simp] =
334   nat_zero_less_power_iff [of _ "numeral w"] for w
336 lemmas power_eq_0_iff_numeral [simp] =
337   power_eq_0_iff [of _ "numeral w"] for w
339 lemmas power_even_abs_numeral [simp] =
340   power_even_abs [of "numeral w" _] for w
343 subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *}
345 lemma zero_le_power_iff[presburger]:
346   "(0 \<le> a^n) = (0 \<le> (a::'a::{linordered_idom}) | even n)"
347 proof cases
348   assume even: "even n"
349   then obtain k where "n = 2*k"
350     by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2)
351   thus ?thesis by (simp add: zero_le_even_power even)
352 next
353   assume odd: "odd n"
354   then obtain k where "n = Suc(2*k)"
355     by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)
356   moreover have "a ^ (2 * k) \<le> 0 \<Longrightarrow> a = 0"
357     by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff)
358   ultimately show ?thesis
359     by (auto simp add: zero_le_mult_iff zero_le_even_power)
360 qed
363 subsection {* Miscellaneous *}
365 lemma [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger
366 lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger
367 lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2"  by presburger
368 lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger
370 lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger
371 lemma even_nat_plus_one_div_two: "even (x::nat) ==>
372     (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger
374 lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>
375     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger
377 end