author | hoelzl |
Thu, 05 Feb 2009 11:34:42 +0100 | |
changeset 29803 | c56a5571f60a |
parent 29654 | 24e73987bfe2 |
child 30056 | 0a35bee25c20 |
permissions | -rw-r--r-- |
21263 | 1 |
(* Title: HOL/Library/Parity.thy |
25600 | 2 |
Author: Jeremy Avigad, Jacques D. Fleuriot |
21256 | 3 |
*) |
4 |
||
5 |
header {* Even and Odd for int and nat *} |
|
6 |
||
7 |
theory Parity |
|
29654
24e73987bfe2
Plain, Main form meeting points in import hierarchy
haftmann
parents:
29608
diff
changeset
|
8 |
imports Plain Presburger Main |
21256 | 9 |
begin |
10 |
||
29608 | 11 |
class even_odd = |
22390 | 12 |
fixes even :: "'a \<Rightarrow> bool" |
21256 | 13 |
|
14 |
abbreviation |
|
22390 | 15 |
odd :: "'a\<Colon>even_odd \<Rightarrow> bool" where |
16 |
"odd x \<equiv> \<not> even x" |
|
17 |
||
26259 | 18 |
instantiation nat and int :: even_odd |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
19 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
20 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
21 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
22 |
even_def [presburger]: "even x \<longleftrightarrow> (x\<Colon>int) mod 2 = 0" |
22390 | 23 |
|
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
24 |
definition |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
25 |
even_nat_def [presburger]: "even x \<longleftrightarrow> even (int x)" |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
26 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
27 |
instance .. |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
28 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25502
diff
changeset
|
29 |
end |
21256 | 30 |
|
31 |
||
32 |
subsection {* Even and odd are mutually exclusive *} |
|
33 |
||
21263 | 34 |
lemma int_pos_lt_two_imp_zero_or_one: |
21256 | 35 |
"0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1" |
23522 | 36 |
by presburger |
21256 | 37 |
|
23522 | 38 |
lemma neq_one_mod_two [simp, presburger]: |
39 |
"((x::int) mod 2 ~= 0) = (x mod 2 = 1)" by presburger |
|
21256 | 40 |
|
25600 | 41 |
|
21256 | 42 |
subsection {* Behavior under integer arithmetic operations *} |
27668 | 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 |
|
21256 | 48 |
|
49 |
lemma even_times_anything: "even (x::int) ==> even (x * y)" |
|
27668 | 50 |
by algebra |
21256 | 51 |
|
27668 | 52 |
lemma anything_times_even: "even (y::int) ==> even (x * y)" by algebra |
21256 | 53 |
|
27668 | 54 |
lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)" |
21256 | 55 |
by (simp add: even_def zmod_zmult1_eq) |
56 |
||
23522 | 57 |
lemma even_product[presburger]: "even((x::int) * y) = (even x | even y)" |
21263 | 58 |
apply (auto simp add: even_times_anything anything_times_even) |
21256 | 59 |
apply (rule ccontr) |
60 |
apply (auto simp add: odd_times_odd) |
|
61 |
done |
|
62 |
||
63 |
lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)" |
|
23522 | 64 |
by presburger |
21256 | 65 |
|
66 |
lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)" |
|
23522 | 67 |
by presburger |
21256 | 68 |
|
69 |
lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)" |
|
23522 | 70 |
by presburger |
21256 | 71 |
|
23522 | 72 |
lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger |
21256 | 73 |
|
23522 | 74 |
lemma even_sum[presburger]: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))" |
75 |
by presburger |
|
21256 | 76 |
|
27668 | 77 |
lemma even_neg[presburger, algebra]: "even (-(x::int)) = even x" by presburger |
21256 | 78 |
|
21263 | 79 |
lemma even_difference: |
23522 | 80 |
"even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger |
21256 | 81 |
|
21263 | 82 |
lemma even_pow_gt_zero: |
83 |
"even (x::int) ==> 0 < n ==> even (x^n)" |
|
84 |
by (induct n) (auto simp add: even_product) |
|
21256 | 85 |
|
27668 | 86 |
lemma odd_pow_iff[presburger, algebra]: |
87 |
"odd ((x::int) ^ n) \<longleftrightarrow> (n = 0 \<or> odd x)" |
|
23522 | 88 |
apply (induct n, simp_all) |
89 |
apply presburger |
|
90 |
apply (case_tac n, auto) |
|
91 |
apply (simp_all add: even_product) |
|
21256 | 92 |
done |
93 |
||
23522 | 94 |
lemma odd_pow: "odd x ==> odd((x::int)^n)" by (simp add: odd_pow_iff) |
95 |
||
96 |
lemma even_power[presburger]: "even ((x::int)^n) = (even x & 0 < n)" |
|
21263 | 97 |
apply (auto simp add: even_pow_gt_zero) |
21256 | 98 |
apply (erule contrapos_pp, erule odd_pow) |
99 |
apply (erule contrapos_pp, simp add: even_def) |
|
100 |
done |
|
101 |
||
23522 | 102 |
lemma even_zero[presburger]: "even (0::int)" by presburger |
21256 | 103 |
|
23522 | 104 |
lemma odd_one[presburger]: "odd (1::int)" by presburger |
21256 | 105 |
|
21263 | 106 |
lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero |
21256 | 107 |
odd_one even_product even_sum even_neg even_difference even_power |
108 |
||
109 |
||
110 |
subsection {* Equivalent definitions *} |
|
111 |
||
23522 | 112 |
lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x" |
113 |
by presburger |
|
21256 | 114 |
|
21263 | 115 |
lemma two_times_odd_div_two_plus_one: "odd (x::int) ==> |
23522 | 116 |
2 * (x div 2) + 1 = x" by presburger |
21256 | 117 |
|
23522 | 118 |
lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger |
21256 | 119 |
|
23522 | 120 |
lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" by presburger |
21256 | 121 |
|
122 |
subsection {* even and odd for nats *} |
|
123 |
||
124 |
lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)" |
|
125 |
by (simp add: even_nat_def) |
|
126 |
||
27668 | 127 |
lemma even_nat_product[presburger, algebra]: "even((x::nat) * y) = (even x | even y)" |
23431
25ca91279a9b
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents:
23309
diff
changeset
|
128 |
by (simp add: even_nat_def int_mult) |
21256 | 129 |
|
27668 | 130 |
lemma even_nat_sum[presburger, algebra]: "even ((x::nat) + y) = |
23522 | 131 |
((even x & even y) | (odd x & odd y))" by presburger |
21256 | 132 |
|
27668 | 133 |
lemma even_nat_difference[presburger, algebra]: |
21256 | 134 |
"even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))" |
23522 | 135 |
by presburger |
21256 | 136 |
|
27668 | 137 |
lemma even_nat_Suc[presburger, algebra]: "even (Suc x) = odd x" by presburger |
21256 | 138 |
|
27668 | 139 |
lemma even_nat_power[presburger, algebra]: "even ((x::nat)^y) = (even x & 0 < y)" |
23431
25ca91279a9b
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents:
23309
diff
changeset
|
140 |
by (simp add: even_nat_def int_power) |
21256 | 141 |
|
23522 | 142 |
lemma even_nat_zero[presburger]: "even (0::nat)" by presburger |
21256 | 143 |
|
21263 | 144 |
lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard] |
21256 | 145 |
even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power |
146 |
||
147 |
||
148 |
subsection {* Equivalent definitions *} |
|
149 |
||
21263 | 150 |
lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==> |
23522 | 151 |
x = 0 | x = Suc 0" by presburger |
21256 | 152 |
|
153 |
lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0" |
|
23522 | 154 |
by presburger |
21256 | 155 |
|
156 |
lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0" |
|
23522 | 157 |
by presburger |
21256 | 158 |
|
21263 | 159 |
lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)" |
23522 | 160 |
by presburger |
21256 | 161 |
|
162 |
lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)" |
|
23522 | 163 |
by presburger |
21256 | 164 |
|
21263 | 165 |
lemma even_nat_div_two_times_two: "even (x::nat) ==> |
23522 | 166 |
Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger |
21256 | 167 |
|
21263 | 168 |
lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==> |
23522 | 169 |
Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger |
21256 | 170 |
|
171 |
lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)" |
|
23522 | 172 |
by presburger |
21256 | 173 |
|
174 |
lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))" |
|
23522 | 175 |
by presburger |
21256 | 176 |
|
25600 | 177 |
|
21256 | 178 |
subsection {* Parity and powers *} |
179 |
||
21263 | 180 |
lemma minus_one_even_odd_power: |
181 |
"(even x --> (- 1::'a::{comm_ring_1,recpower})^x = 1) & |
|
21256 | 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) |
|
187 |
apply (simp add: power_Suc) |
|
21263 | 188 |
done |
21256 | 189 |
|
190 |
lemma minus_one_even_power [simp]: |
|
21263 | 191 |
"even x ==> (- 1::'a::{comm_ring_1,recpower})^x = 1" |
192 |
using minus_one_even_odd_power by blast |
|
21256 | 193 |
|
194 |
lemma minus_one_odd_power [simp]: |
|
21263 | 195 |
"odd x ==> (- 1::'a::{comm_ring_1,recpower})^x = - 1" |
196 |
using minus_one_even_odd_power by blast |
|
21256 | 197 |
|
198 |
lemma neg_one_even_odd_power: |
|
21263 | 199 |
"(even x --> (-1::'a::{number_ring,recpower})^x = 1) & |
21256 | 200 |
(odd x --> (-1::'a)^x = -1)" |
201 |
apply (induct x) |
|
202 |
apply (simp, simp add: power_Suc) |
|
203 |
done |
|
204 |
||
205 |
lemma neg_one_even_power [simp]: |
|
21263 | 206 |
"even x ==> (-1::'a::{number_ring,recpower})^x = 1" |
207 |
using neg_one_even_odd_power by blast |
|
21256 | 208 |
|
209 |
lemma neg_one_odd_power [simp]: |
|
21263 | 210 |
"odd x ==> (-1::'a::{number_ring,recpower})^x = -1" |
211 |
using neg_one_even_odd_power by blast |
|
21256 | 212 |
|
213 |
lemma neg_power_if: |
|
21263 | 214 |
"(-x::'a::{comm_ring_1,recpower}) ^ n = |
21256 | 215 |
(if even n then (x ^ n) else -(x ^ n))" |
21263 | 216 |
apply (induct n) |
217 |
apply (simp_all split: split_if_asm add: power_Suc) |
|
218 |
done |
|
21256 | 219 |
|
21263 | 220 |
lemma zero_le_even_power: "even n ==> |
21256 | 221 |
0 <= (x::'a::{recpower,ordered_ring_strict}) ^ n" |
222 |
apply (simp add: even_nat_equiv_def2) |
|
223 |
apply (erule exE) |
|
224 |
apply (erule ssubst) |
|
225 |
apply (subst power_add) |
|
226 |
apply (rule zero_le_square) |
|
227 |
done |
|
228 |
||
21263 | 229 |
lemma zero_le_odd_power: "odd n ==> |
21256 | 230 |
(0 <= (x::'a::{recpower,ordered_idom}) ^ n) = (0 <= x)" |
231 |
apply (simp add: odd_nat_equiv_def2) |
|
232 |
apply (erule exE) |
|
233 |
apply (erule ssubst) |
|
234 |
apply (subst power_Suc) |
|
235 |
apply (subst power_add) |
|
236 |
apply (subst zero_le_mult_iff) |
|
237 |
apply auto |
|
25162 | 238 |
apply (subgoal_tac "x = 0 & y > 0") |
21256 | 239 |
apply (erule conjE, assumption) |
21263 | 240 |
apply (subst power_eq_0_iff [symmetric]) |
21256 | 241 |
apply (subgoal_tac "0 <= x^y * x^y") |
242 |
apply simp |
|
243 |
apply (rule zero_le_square)+ |
|
21263 | 244 |
done |
21256 | 245 |
|
23522 | 246 |
lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) = |
21256 | 247 |
(even n | (odd n & 0 <= x))" |
248 |
apply auto |
|
21263 | 249 |
apply (subst zero_le_odd_power [symmetric]) |
21256 | 250 |
apply assumption+ |
251 |
apply (erule zero_le_even_power) |
|
21263 | 252 |
done |
21256 | 253 |
|
23522 | 254 |
lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) = |
21256 | 255 |
(n = 0 | (even n & x ~= 0) | (odd n & 0 < x))" |
27668 | 256 |
|
257 |
unfolding order_less_le zero_le_power_eq by auto |
|
21256 | 258 |
|
23522 | 259 |
lemma power_less_zero_eq[presburger]: "((x::'a::{recpower,ordered_idom}) ^ n < 0) = |
27668 | 260 |
(odd n & x < 0)" |
21263 | 261 |
apply (subst linorder_not_le [symmetric])+ |
21256 | 262 |
apply (subst zero_le_power_eq) |
263 |
apply auto |
|
21263 | 264 |
done |
21256 | 265 |
|
23522 | 266 |
lemma power_le_zero_eq[presburger]: "((x::'a::{recpower,ordered_idom}) ^ n <= 0) = |
21256 | 267 |
(n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))" |
21263 | 268 |
apply (subst linorder_not_less [symmetric])+ |
21256 | 269 |
apply (subst zero_less_power_eq) |
270 |
apply auto |
|
21263 | 271 |
done |
21256 | 272 |
|
21263 | 273 |
lemma power_even_abs: "even n ==> |
21256 | 274 |
(abs (x::'a::{recpower,ordered_idom}))^n = x^n" |
21263 | 275 |
apply (subst power_abs [symmetric]) |
21256 | 276 |
apply (simp add: zero_le_even_power) |
21263 | 277 |
done |
21256 | 278 |
|
23522 | 279 |
lemma zero_less_power_nat_eq[presburger]: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)" |
21263 | 280 |
by (induct n) auto |
21256 | 281 |
|
21263 | 282 |
lemma power_minus_even [simp]: "even n ==> |
21256 | 283 |
(- x)^n = (x^n::'a::{recpower,comm_ring_1})" |
284 |
apply (subst power_minus) |
|
285 |
apply simp |
|
21263 | 286 |
done |
21256 | 287 |
|
21263 | 288 |
lemma power_minus_odd [simp]: "odd n ==> |
21256 | 289 |
(- x)^n = - (x^n::'a::{recpower,comm_ring_1})" |
290 |
apply (subst power_minus) |
|
291 |
apply simp |
|
21263 | 292 |
done |
21256 | 293 |
|
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
294 |
lemma power_mono_even: fixes x y :: "'a :: {recpower, ordered_idom}" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
295 |
assumes "even n" and "\<bar>x\<bar> \<le> \<bar>y\<bar>" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
296 |
shows "x^n \<le> y^n" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
297 |
proof - |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
298 |
have "0 \<le> \<bar>x\<bar>" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
299 |
with `\<bar>x\<bar> \<le> \<bar>y\<bar>` |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
300 |
have "\<bar>x\<bar>^n \<le> \<bar>y\<bar>^n" by (rule power_mono) |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
301 |
thus ?thesis unfolding power_even_abs[OF `even n`] . |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
302 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
303 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
304 |
lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
305 |
|
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
306 |
lemma power_mono_odd: fixes x y :: "'a :: {recpower, ordered_idom}" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
307 |
assumes "odd n" and "x \<le> y" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
308 |
shows "x^n \<le> y^n" |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
309 |
proof (cases "y < 0") |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
310 |
case True with `x \<le> y` have "-y \<le> -x" and "0 \<le> -y" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
311 |
hence "(-y)^n \<le> (-x)^n" by (rule power_mono) |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
312 |
thus ?thesis unfolding power_minus_odd[OF `odd n`] by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
313 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
314 |
case False |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
315 |
show ?thesis |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
316 |
proof (cases "x < 0") |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
317 |
case True hence "n \<noteq> 0" and "x \<le> 0" using `odd n`[THEN odd_pos] by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
318 |
hence "x^n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
319 |
moreover |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
320 |
from `\<not> y < 0` have "0 \<le> y" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
321 |
hence "0 \<le> y^n" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
322 |
ultimately show ?thesis by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
323 |
next |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
324 |
case False hence "0 \<le> x" by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
325 |
with `x \<le> y` show ?thesis using power_mono by auto |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
326 |
qed |
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29654
diff
changeset
|
327 |
qed |
21263 | 328 |
|
25600 | 329 |
subsection {* General Lemmas About Division *} |
330 |
||
331 |
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" |
|
332 |
apply (induct "m") |
|
333 |
apply (simp_all add: mod_Suc) |
|
334 |
done |
|
335 |
||
336 |
declare Suc_times_mod_eq [of "number_of w", standard, simp] |
|
337 |
||
338 |
lemma [simp]: "n div k \<le> (Suc n) div k" |
|
339 |
by (simp add: div_le_mono) |
|
340 |
||
341 |
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2" |
|
342 |
by arith |
|
343 |
||
344 |
lemma div_2_gt_zero [simp]: "(1::nat) < n ==> 0 < n div 2" |
|
345 |
by arith |
|
346 |
||
27668 | 347 |
(* Potential use of algebra : Equality modulo n*) |
25600 | 348 |
lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)" |
349 |
by (simp add: mult_ac add_ac) |
|
350 |
||
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 |
|
357 |
||
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 |
|
362 |
||
363 |
||
364 |
subsection {* More Even/Odd Results *} |
|
365 |
||
27668 | 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 |
|
25600 | 369 |
|
27668 | 370 |
lemma odd_add [simp]: "odd(m + n::nat) = (odd m \<noteq> odd n)" by presburger |
25600 | 371 |
|
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) |
|
376 |
apply (rule div_add1_eq, simp) |
|
377 |
done |
|
378 |
||
27668 | 379 |
lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" by presburger |
25600 | 380 |
|
381 |
lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)" |
|
27668 | 382 |
by presburger |
25600 | 383 |
|
27668 | 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 |
|
25600 | 386 |
|
27668 | 387 |
lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n - 1) div 2)" by presburger |
25600 | 388 |
|
389 |
lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n - 1) div 2)" |
|
27668 | 390 |
by presburger |
25600 | 391 |
|
21263 | 392 |
text {* Simplify, when the exponent is a numeral *} |
21256 | 393 |
|
394 |
lemmas power_0_left_number_of = power_0_left [of "number_of w", standard] |
|
395 |
declare power_0_left_number_of [simp] |
|
396 |
||
21263 | 397 |
lemmas zero_le_power_eq_number_of [simp] = |
21256 | 398 |
zero_le_power_eq [of _ "number_of w", standard] |
399 |
||
21263 | 400 |
lemmas zero_less_power_eq_number_of [simp] = |
21256 | 401 |
zero_less_power_eq [of _ "number_of w", standard] |
402 |
||
21263 | 403 |
lemmas power_le_zero_eq_number_of [simp] = |
21256 | 404 |
power_le_zero_eq [of _ "number_of w", standard] |
405 |
||
21263 | 406 |
lemmas power_less_zero_eq_number_of [simp] = |
21256 | 407 |
power_less_zero_eq [of _ "number_of w", standard] |
408 |
||
21263 | 409 |
lemmas zero_less_power_nat_eq_number_of [simp] = |
21256 | 410 |
zero_less_power_nat_eq [of _ "number_of w", standard] |
411 |
||
21263 | 412 |
lemmas power_eq_0_iff_number_of [simp] = power_eq_0_iff [of _ "number_of w", standard] |
21256 | 413 |
|
21263 | 414 |
lemmas power_even_abs_number_of [simp] = power_even_abs [of "number_of w" _, standard] |
21256 | 415 |
|
416 |
||
417 |
subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *} |
|
418 |
||
419 |
lemma even_power_le_0_imp_0: |
|
21263 | 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) |
|
21256 | 422 |
|
23522 | 423 |
lemma zero_le_power_iff[presburger]: |
21263 | 424 |
"(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even n)" |
21256 | 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) |
|
21263 | 429 |
thus ?thesis by (simp add: zero_le_even_power even) |
21256 | 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 |
|
21263 | 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 |
|
438 |
||
21256 | 439 |
|
440 |
subsection {* Miscellaneous *} |
|
441 |
||
23522 | 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 |
|
21256 | 446 |
|
23522 | 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 |
|
21263 | 449 |
lemma even_nat_plus_one_div_two: "even (x::nat) ==> |
23522 | 450 |
(Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger |
21256 | 451 |
|
21263 | 452 |
lemma odd_nat_plus_one_div_two: "odd (x::nat) ==> |
23522 | 453 |
(Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger |
21256 | 454 |
|
455 |
end |