src/HOL/Old_Number_Theory/EvenOdd.thy
 author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 44766 d4d33a4d7548 child 49962 a8cc904a6820 permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
1 (*  Title:      HOL/Old_Number_Theory/EvenOdd.thy
3 *)
5 header {*Parity: Even and Odd Integers*}
7 theory EvenOdd
8 imports Int2
9 begin
11 definition zOdd :: "int set"
12   where "zOdd = {x. \<exists>k. x = 2 * k + 1}"
14 definition zEven :: "int set"
15   where "zEven = {x. \<exists>k. x = 2 * k}"
17 subsection {* Some useful properties about even and odd *}
19 lemma zOddI [intro?]: "x = 2 * k + 1 \<Longrightarrow> x \<in> zOdd"
20   and zOddE [elim?]: "x \<in> zOdd \<Longrightarrow> (!!k. x = 2 * k + 1 \<Longrightarrow> C) \<Longrightarrow> C"
21   by (auto simp add: zOdd_def)
23 lemma zEvenI [intro?]: "x = 2 * k \<Longrightarrow> x \<in> zEven"
24   and zEvenE [elim?]: "x \<in> zEven \<Longrightarrow> (!!k. x = 2 * k \<Longrightarrow> C) \<Longrightarrow> C"
25   by (auto simp add: zEven_def)
27 lemma one_not_even: "~(1 \<in> zEven)"
28 proof
29   assume "1 \<in> zEven"
30   then obtain k :: int where "1 = 2 * k" ..
31   then show False by arith
32 qed
34 lemma even_odd_conj: "~(x \<in> zOdd & x \<in> zEven)"
35 proof -
36   {
37     fix a b
38     assume "2 * (a::int) = 2 * (b::int) + 1"
39     then have "2 * (a::int) - 2 * (b :: int) = 1"
40       by arith
41     then have "2 * (a - b) = 1"
42       by (auto simp add: left_diff_distrib)
43     moreover have "(2 * (a - b)):zEven"
44       by (auto simp only: zEven_def)
45     ultimately have False
46       by (auto simp add: one_not_even)
47   }
48   then show ?thesis
49     by (auto simp add: zOdd_def zEven_def)
50 qed
52 lemma even_odd_disj: "(x \<in> zOdd | x \<in> zEven)"
53   by (simp add: zOdd_def zEven_def) arith
55 lemma not_odd_impl_even: "~(x \<in> zOdd) ==> x \<in> zEven"
56   using even_odd_disj by auto
58 lemma odd_mult_odd_prop: "(x*y):zOdd ==> x \<in> zOdd"
59 proof (rule classical)
60   assume "\<not> ?thesis"
61   then have "x \<in> zEven" by (rule not_odd_impl_even)
62   then obtain a where a: "x = 2 * a" ..
63   assume "x * y : zOdd"
64   then obtain b where "x * y = 2 * b + 1" ..
65   with a have "2 * a * y = 2 * b + 1" by simp
66   then have "2 * a * y - 2 * b = 1"
67     by arith
68   then have "2 * (a * y - b) = 1"
69     by (auto simp add: left_diff_distrib)
70   moreover have "(2 * (a * y - b)):zEven"
71     by (auto simp only: zEven_def)
72   ultimately have False
73     by (auto simp add: one_not_even)
74   then show ?thesis ..
75 qed
77 lemma odd_minus_one_even: "x \<in> zOdd ==> (x - 1):zEven"
78   by (auto simp add: zOdd_def zEven_def)
80 lemma even_div_2_prop1: "x \<in> zEven ==> (x mod 2) = 0"
81   by (auto simp add: zEven_def)
83 lemma even_div_2_prop2: "x \<in> zEven ==> (2 * (x div 2)) = x"
84   by (auto simp add: zEven_def)
86 lemma even_plus_even: "[| x \<in> zEven; y \<in> zEven |] ==> x + y \<in> zEven"
87   apply (auto simp add: zEven_def)
88   apply (auto simp only: right_distrib [symmetric])
89   done
91 lemma even_times_either: "x \<in> zEven ==> x * y \<in> zEven"
92   by (auto simp add: zEven_def)
94 lemma even_minus_even: "[| x \<in> zEven; y \<in> zEven |] ==> x - y \<in> zEven"
95   apply (auto simp add: zEven_def)
96   apply (auto simp only: right_diff_distrib [symmetric])
97   done
99 lemma odd_minus_odd: "[| x \<in> zOdd; y \<in> zOdd |] ==> x - y \<in> zEven"
100   apply (auto simp add: zOdd_def zEven_def)
101   apply (auto simp only: right_diff_distrib [symmetric])
102   done
104 lemma even_minus_odd: "[| x \<in> zEven; y \<in> zOdd |] ==> x - y \<in> zOdd"
105   apply (auto simp add: zOdd_def zEven_def)
106   apply (rule_tac x = "k - ka - 1" in exI)
107   apply auto
108   done
110 lemma odd_minus_even: "[| x \<in> zOdd; y \<in> zEven |] ==> x - y \<in> zOdd"
111   apply (auto simp add: zOdd_def zEven_def)
112   apply (auto simp only: right_diff_distrib [symmetric])
113   done
115 lemma odd_times_odd: "[| x \<in> zOdd;  y \<in> zOdd |] ==> x * y \<in> zOdd"
116   apply (auto simp add: zOdd_def left_distrib right_distrib)
117   apply (rule_tac x = "2 * ka * k + ka + k" in exI)
118   apply (auto simp add: left_distrib)
119   done
121 lemma odd_iff_not_even: "(x \<in> zOdd) = (~ (x \<in> zEven))"
122   using even_odd_conj even_odd_disj by auto
124 lemma even_product: "x * y \<in> zEven ==> x \<in> zEven | y \<in> zEven"
125   using odd_iff_not_even odd_times_odd by auto
127 lemma even_diff: "x - y \<in> zEven = ((x \<in> zEven) = (y \<in> zEven))"
128 proof
129   assume xy: "x - y \<in> zEven"
130   {
131     assume x: "x \<in> zEven"
132     have "y \<in> zEven"
133     proof (rule classical)
134       assume "\<not> ?thesis"
135       then have "y \<in> zOdd"
137       with x have "x - y \<in> zOdd"
139       with xy have False
140         by (auto simp add: odd_iff_not_even)
141       then show ?thesis ..
142     qed
143   } moreover {
144     assume y: "y \<in> zEven"
145     have "x \<in> zEven"
146     proof (rule classical)
147       assume "\<not> ?thesis"
148       then have "x \<in> zOdd"
149         by (auto simp add: odd_iff_not_even)
150       with y have "x - y \<in> zOdd"
152       with xy have False
153         by (auto simp add: odd_iff_not_even)
154       then show ?thesis ..
155     qed
156   }
157   ultimately show "(x \<in> zEven) = (y \<in> zEven)"
158     by (auto simp add: odd_iff_not_even even_minus_even odd_minus_odd
159       even_minus_odd odd_minus_even)
160 next
161   assume "(x \<in> zEven) = (y \<in> zEven)"
162   then show "x - y \<in> zEven"
163     by (auto simp add: odd_iff_not_even even_minus_even odd_minus_odd
164       even_minus_odd odd_minus_even)
165 qed
167 lemma neg_one_even_power: "[| x \<in> zEven; 0 \<le> x |] ==> (-1::int)^(nat x) = 1"
168 proof -
169   assume "x \<in> zEven" and "0 \<le> x"
170   from `x \<in> zEven` obtain a where "x = 2 * a" ..
171   with `0 \<le> x` have "0 \<le> a" by simp
172   from `0 \<le> x` and `x = 2 * a` have "nat x = nat (2 * a)"
173     by simp
174   also from `x = 2 * a` have "nat (2 * a) = 2 * nat a"
176   finally have "(-1::int)^nat x = (-1)^(2 * nat a)"
177     by simp
178   also have "... = ((-1::int)^2)^ (nat a)"
179     by (simp add: zpower_zpower [symmetric])
180   also have "(-1::int)^2 = 1"
181     by simp
182   finally show ?thesis
183     by simp
184 qed
186 lemma neg_one_odd_power: "[| x \<in> zOdd; 0 \<le> x |] ==> (-1::int)^(nat x) = -1"
187 proof -
188   assume "x \<in> zOdd" and "0 \<le> x"
189   from `x \<in> zOdd` obtain a where "x = 2 * a + 1" ..
190   with `0 \<le> x` have a: "0 \<le> a" by simp
191   with `0 \<le> x` and `x = 2 * a + 1` have "nat x = nat (2 * a + 1)"
192     by simp
193   also from a have "nat (2 * a + 1) = 2 * nat a + 1"
195   finally have "(-1::int)^nat x = (-1)^(2 * nat a + 1)"
196     by simp
197   also have "... = ((-1::int)^2)^ (nat a) * (-1)^1"
199   also have "(-1::int)^2 = 1"
200     by simp
201   finally show ?thesis
202     by simp
203 qed
205 lemma neg_one_power_parity: "[| 0 \<le> x; 0 \<le> y; (x \<in> zEven) = (y \<in> zEven) |] ==>
206     (-1::int)^(nat x) = (-1::int)^(nat y)"
207   using even_odd_disj [of x] even_odd_disj [of y]
208   by (auto simp add: neg_one_even_power neg_one_odd_power)
211 lemma one_not_neg_one_mod_m: "2 < m ==> ~([1 = -1] (mod m))"
212   by (auto simp add: zcong_def zdvd_not_zless)
214 lemma even_div_2_l: "[| y \<in> zEven; x < y |] ==> x div 2 < y div 2"
215 proof -
216   assume "y \<in> zEven" and "x < y"
217   from `y \<in> zEven` obtain k where k: "y = 2 * k" ..
218   with `x < y` have "x < 2 * k" by simp
219   then have "x div 2 < k" by (auto simp add: div_prop1)
220   also have "k = (2 * k) div 2" by simp
221   finally have "x div 2 < 2 * k div 2" by simp
222   with k show ?thesis by simp
223 qed
225 lemma even_sum_div_2: "[| x \<in> zEven; y \<in> zEven |] ==> (x + y) div 2 = x div 2 + y div 2"
226   by (auto simp add: zEven_def)
228 lemma even_prod_div_2: "[| x \<in> zEven |] ==> (x * y) div 2 = (x div 2) * y"
229   by (auto simp add: zEven_def)
231 (* An odd prime is greater than 2 *)
233 lemma zprime_zOdd_eq_grt_2: "zprime p ==> (p \<in> zOdd) = (2 < p)"
234   apply (auto simp add: zOdd_def zprime_def)
235   apply (drule_tac x = 2 in allE)
236   using odd_iff_not_even [of p]
237   apply (auto simp add: zOdd_def zEven_def)
238   done
240 (* Powers of -1 and parity *)
242 lemma neg_one_special: "finite A ==>
243     ((-1 :: int) ^ card A) * (-1 ^ card A) = 1"
244   by (induct set: finite) auto
246 lemma neg_one_power: "(-1::int)^n = 1 | (-1::int)^n = -1"
247   by (induct n) auto
249 lemma neg_one_power_eq_mod_m: "[| 2 < m; [(-1::int)^j = (-1::int)^k] (mod m) |]
250     ==> ((-1::int)^j = (-1::int)^k)"
251   using neg_one_power [of j] and ListMem.insert neg_one_power [of k]
252   by (auto simp add: one_not_neg_one_mod_m zcong_sym)
254 end