src/HOL/Parity.thy
 author haftmann Tue Oct 14 08:23:23 2014 +0200 (2014-10-14) changeset 58680 6b2fa479945f parent 58679 33c90658448a child 58681 a478a0742a8e permissions -rw-r--r--
more algebraic deductions for facts on even/odd
```     1 (*  Title:      HOL/Parity.thy
```
```     2     Author:     Jeremy Avigad
```
```     3     Author:     Jacques D. Fleuriot
```
```     4 *)
```
```     5
```
```     6 header {* Even and Odd for int and nat *}
```
```     7
```
```     8 theory Parity
```
```     9 imports Main
```
```    10 begin
```
```    11
```
```    12 subsection {* Preliminaries about divisibility on @{typ nat} and @{typ int} *}
```
```    13
```
```    14 lemma two_dvd_Suc_Suc_iff [simp]:
```
```    15   "2 dvd Suc (Suc n) \<longleftrightarrow> 2 dvd n"
```
```    16   using dvd_add_triv_right_iff [of 2 n] by simp
```
```    17
```
```    18 lemma two_dvd_Suc_iff:
```
```    19   "2 dvd Suc n \<longleftrightarrow> \<not> 2 dvd n"
```
```    20   by (induct n) auto
```
```    21
```
```    22 lemma two_dvd_diff_iff:
```
```    23   fixes k l :: int
```
```    24   shows "2 dvd k - l \<longleftrightarrow> 2 dvd k + l"
```
```    25   using dvd_add_times_triv_right_iff [of 2 "k - l" l] by (simp add: ac_simps)
```
```    26
```
```    27 lemma two_dvd_abs_add_iff:
```
```    28   fixes k l :: int
```
```    29   shows "2 dvd \<bar>k\<bar> + l \<longleftrightarrow> 2 dvd k + l"
```
```    30   by (cases "k \<ge> 0") (simp_all add: two_dvd_diff_iff ac_simps)
```
```    31
```
```    32 lemma two_dvd_add_abs_iff:
```
```    33   fixes k l :: int
```
```    34   shows "2 dvd k + \<bar>l\<bar> \<longleftrightarrow> 2 dvd k + l"
```
```    35   using two_dvd_abs_add_iff [of l k] by (simp add: ac_simps)
```
```    36
```
```    37
```
```    38 subsection {* Ring structures with parity *}
```
```    39
```
```    40 class semiring_parity = semiring_dvd + semiring_numeral +
```
```    41   assumes two_not_dvd_one [simp]: "\<not> 2 dvd 1"
```
```    42   assumes not_dvd_not_dvd_dvd_add: "\<not> 2 dvd a \<Longrightarrow> \<not> 2 dvd b \<Longrightarrow> 2 dvd a + b"
```
```    43   assumes two_is_prime: "2 dvd a * b \<Longrightarrow> 2 dvd a \<or> 2 dvd b"
```
```    44   assumes not_dvd_ex_decrement: "\<not> 2 dvd a \<Longrightarrow> \<exists>b. a = b + 1"
```
```    45 begin
```
```    46
```
```    47 lemma two_dvd_plus_one_iff [simp]:
```
```    48   "2 dvd a + 1 \<longleftrightarrow> \<not> 2 dvd a"
```
```    49   by (auto simp add: dvd_add_right_iff intro: not_dvd_not_dvd_dvd_add)
```
```    50
```
```    51 lemma not_two_dvdE [elim?]:
```
```    52   assumes "\<not> 2 dvd a"
```
```    53   obtains b where "a = 2 * b + 1"
```
```    54 proof -
```
```    55   from assms obtain b where *: "a = b + 1"
```
```    56     by (blast dest: not_dvd_ex_decrement)
```
```    57   with assms have "2 dvd b + 2" by simp
```
```    58   then have "2 dvd b" by simp
```
```    59   then obtain c where "b = 2 * c" ..
```
```    60   with * have "a = 2 * c + 1" by simp
```
```    61   with that show thesis .
```
```    62 qed
```
```    63
```
```    64 end
```
```    65
```
```    66 instance nat :: semiring_parity
```
```    67 proof
```
```    68   show "\<not> (2 :: nat) dvd 1"
```
```    69     by (rule notI, erule dvdE) simp
```
```    70 next
```
```    71   fix m n :: nat
```
```    72   assume "\<not> 2 dvd m"
```
```    73   moreover assume "\<not> 2 dvd n"
```
```    74   ultimately have *: "2 dvd Suc m \<and> 2 dvd Suc n"
```
```    75     by (simp add: two_dvd_Suc_iff)
```
```    76   then have "2 dvd Suc m + Suc n"
```
```    77     by (blast intro: dvd_add)
```
```    78   also have "Suc m + Suc n = m + n + 2"
```
```    79     by simp
```
```    80   finally show "2 dvd m + n"
```
```    81     using dvd_add_triv_right_iff [of 2 "m + n"] by simp
```
```    82 next
```
```    83   fix m n :: nat
```
```    84   assume *: "2 dvd m * n"
```
```    85   show "2 dvd m \<or> 2 dvd n"
```
```    86   proof (rule disjCI)
```
```    87     assume "\<not> 2 dvd n"
```
```    88     then have "2 dvd Suc n" by (simp add: two_dvd_Suc_iff)
```
```    89     then obtain r where "Suc n = 2 * r" ..
```
```    90     moreover from * obtain s where "m * n = 2 * s" ..
```
```    91     then have "2 * s + m = m * Suc n" by simp
```
```    92     ultimately have " 2 * s + m = 2 * (m * r)" by simp
```
```    93     then have "m = 2 * (m * r - s)" by simp
```
```    94     then show "2 dvd m" ..
```
```    95   qed
```
```    96 next
```
```    97   fix n :: nat
```
```    98   assume "\<not> 2 dvd n"
```
```    99   then show "\<exists>m. n = m + 1"
```
```   100     by (cases n) simp_all
```
```   101 qed
```
```   102
```
```   103 class ring_parity = comm_ring_1 + semiring_parity
```
```   104
```
```   105 instance int :: ring_parity
```
```   106 proof
```
```   107   show "\<not> (2 :: int) dvd 1" by (simp add: dvd_int_unfold_dvd_nat)
```
```   108   fix k l :: int
```
```   109   assume "\<not> 2 dvd k"
```
```   110   moreover assume "\<not> 2 dvd l"
```
```   111   ultimately have "2 dvd nat \<bar>k\<bar> + nat \<bar>l\<bar>"
```
```   112     by (auto simp add: dvd_int_unfold_dvd_nat intro: not_dvd_not_dvd_dvd_add)
```
```   113   then have "2 dvd \<bar>k\<bar> + \<bar>l\<bar>"
```
```   114     by (simp add: dvd_int_unfold_dvd_nat nat_add_distrib)
```
```   115   then show "2 dvd k + l"
```
```   116     by (simp add: two_dvd_abs_add_iff two_dvd_add_abs_iff)
```
```   117 next
```
```   118   fix k l :: int
```
```   119   assume "2 dvd k * l"
```
```   120   then show "2 dvd k \<or> 2 dvd l"
```
```   121     by (simp add: dvd_int_unfold_dvd_nat two_is_prime nat_abs_mult_distrib)
```
```   122 next
```
```   123   fix k :: int
```
```   124   have "k = (k - 1) + 1" by simp
```
```   125   then show "\<exists>l. k = l + 1" ..
```
```   126 qed
```
```   127
```
```   128 context semiring_div_parity
```
```   129 begin
```
```   130
```
```   131 subclass semiring_parity
```
```   132 proof (unfold_locales, unfold dvd_eq_mod_eq_0 not_mod_2_eq_0_eq_1)
```
```   133   fix a b c
```
```   134   show "(c * a + b) mod a = 0 \<longleftrightarrow> b mod a = 0"
```
```   135     by simp
```
```   136 next
```
```   137   fix a b c
```
```   138   assume "(b + c) mod a = 0"
```
```   139   with mod_add_eq [of b c a]
```
```   140   have "(b mod a + c mod a) mod a = 0"
```
```   141     by simp
```
```   142   moreover assume "b mod a = 0"
```
```   143   ultimately show "c mod a = 0"
```
```   144     by simp
```
```   145 next
```
```   146   show "1 mod 2 = 1"
```
```   147     by (fact one_mod_two_eq_one)
```
```   148 next
```
```   149   fix a b
```
```   150   assume "a mod 2 = 1"
```
```   151   moreover assume "b mod 2 = 1"
```
```   152   ultimately show "(a + b) mod 2 = 0"
```
```   153     using mod_add_eq [of a b 2] by simp
```
```   154 next
```
```   155   fix a b
```
```   156   assume "(a * b) mod 2 = 0"
```
```   157   then have "(a mod 2) * (b mod 2) = 0"
```
```   158     by (cases "a mod 2 = 0") (simp_all add: mod_mult_eq [of a b 2])
```
```   159   then show "a mod 2 = 0 \<or> b mod 2 = 0"
```
```   160     by (rule divisors_zero)
```
```   161 next
```
```   162   fix a
```
```   163   assume "a mod 2 = 1"
```
```   164   then have "a = a div 2 * 2 + 1" using mod_div_equality [of a 2] by simp
```
```   165   then show "\<exists>b. a = b + 1" ..
```
```   166 qed
```
```   167
```
```   168 end
```
```   169
```
```   170
```
```   171 subsection {* Dedicated @{text even}/@{text odd} predicate *}
```
```   172
```
```   173 subsubsection {* Properties *}
```
```   174
```
```   175 context semiring_parity
```
```   176 begin
```
```   177
```
```   178 definition even :: "'a \<Rightarrow> bool"
```
```   179 where
```
```   180   [algebra]: "even a \<longleftrightarrow> 2 dvd a"
```
```   181
```
```   182 abbreviation odd :: "'a \<Rightarrow> bool"
```
```   183 where
```
```   184   "odd a \<equiv> \<not> even a"
```
```   185
```
```   186 lemma odd_dvdE [elim?]:
```
```   187   assumes "odd a"
```
```   188   obtains b where "a = 2 * b + 1"
```
```   189 proof -
```
```   190   from assms have "\<not> 2 dvd a" by (simp add: even_def)
```
```   191   then show thesis using that by (rule not_two_dvdE)
```
```   192 qed
```
```   193
```
```   194 lemma even_times_iff [simp, presburger, algebra]:
```
```   195   "even (a * b) \<longleftrightarrow> even a \<or> even b"
```
```   196   by (auto simp add: even_def dest: two_is_prime)
```
```   197
```
```   198 lemma even_zero [simp]:
```
```   199   "even 0"
```
```   200   by (simp add: even_def)
```
```   201
```
```   202 lemma odd_one [simp]:
```
```   203   "odd 1"
```
```   204   by (simp add: even_def)
```
```   205
```
```   206 lemma even_numeral [simp]:
```
```   207   "even (numeral (Num.Bit0 n))"
```
```   208 proof -
```
```   209   have "even (2 * numeral n)"
```
```   210     unfolding even_times_iff by (simp add: even_def)
```
```   211   then have "even (numeral n + numeral n)"
```
```   212     unfolding mult_2 .
```
```   213   then show ?thesis
```
```   214     unfolding numeral.simps .
```
```   215 qed
```
```   216
```
```   217 lemma odd_numeral [simp]:
```
```   218   "odd (numeral (Num.Bit1 n))"
```
```   219 proof
```
```   220   assume "even (numeral (num.Bit1 n))"
```
```   221   then have "even (numeral n + numeral n + 1)"
```
```   222     unfolding numeral.simps .
```
```   223   then have "even (2 * numeral n + 1)"
```
```   224     unfolding mult_2 .
```
```   225   then have "2 dvd numeral n * 2 + 1"
```
```   226     unfolding even_def by (simp add: ac_simps)
```
```   227   with dvd_add_times_triv_left_iff [of 2 "numeral n" 1]
```
```   228     have "2 dvd 1"
```
```   229     by simp
```
```   230   then show False by simp
```
```   231 qed
```
```   232
```
```   233 lemma even_add [simp]:
```
```   234   "even (a + b) \<longleftrightarrow> (even a \<longleftrightarrow> even b)"
```
```   235   by (auto simp add: even_def dvd_add_right_iff dvd_add_left_iff not_dvd_not_dvd_dvd_add)
```
```   236
```
```   237 lemma odd_add [simp]:
```
```   238   "odd (a + b) \<longleftrightarrow> (\<not> (odd a \<longleftrightarrow> odd b))"
```
```   239   by simp
```
```   240
```
```   241 lemma even_power [simp, presburger]:
```
```   242   "even (a ^ n) \<longleftrightarrow> even a \<and> n \<noteq> 0"
```
```   243   by (induct n) auto
```
```   244
```
```   245 end
```
```   246
```
```   247 context ring_parity
```
```   248 begin
```
```   249
```
```   250 lemma even_minus [simp, presburger, algebra]:
```
```   251   "even (- a) \<longleftrightarrow> even a"
```
```   252   by (simp add: even_def)
```
```   253
```
```   254 lemma even_diff [simp]:
```
```   255   "even (a - b) \<longleftrightarrow> even (a + b)"
```
```   256   using even_add [of a "- b"] by simp
```
```   257
```
```   258 end
```
```   259
```
```   260 context semiring_div_parity
```
```   261 begin
```
```   262
```
```   263 lemma even_iff_mod_2_eq_zero [presburger]:
```
```   264   "even a \<longleftrightarrow> a mod 2 = 0"
```
```   265   by (simp add: even_def dvd_eq_mod_eq_0)
```
```   266
```
```   267 end
```
```   268
```
```   269
```
```   270 subsubsection {* Particularities for @{typ nat}/@{typ int} *}
```
```   271
```
```   272 lemma even_int_iff:
```
```   273   "even (int n) \<longleftrightarrow> even n"
```
```   274   by (simp add: even_def dvd_int_iff)
```
```   275
```
```   276 declare transfer_morphism_int_nat [transfer add return:
```
```   277   even_int_iff
```
```   278 ]
```
```   279
```
```   280
```
```   281 subsubsection {* Tools setup *}
```
```   282
```
```   283 lemma [presburger]:
```
```   284   "even n \<longleftrightarrow> even (int n)"
```
```   285   using even_int_iff [of n] by simp
```
```   286
```
```   287 lemma [presburger]:
```
```   288   "even (a + b) \<longleftrightarrow> even a \<and> even b \<or> odd a \<and> odd b"
```
```   289   by auto
```
```   290
```
```   291
```
```   292 subsubsection {* Legacy cruft *}
```
```   293
```
```   294 lemma even_plus_even:
```
```   295   "even (x::int) ==> even y ==> even (x + y)"
```
```   296   by simp
```
```   297
```
```   298 lemma odd_plus_odd:
```
```   299   "odd (x::int) ==> odd y ==> even (x + y)"
```
```   300   by simp
```
```   301
```
```   302 lemma even_sum_nat:
```
```   303   "even ((x::nat) + y) = ((even x & even y) | (odd x & odd y))"
```
```   304   by auto
```
```   305
```
```   306 lemma odd_pow:
```
```   307   "odd x ==> odd((x::int)^n)"
```
```   308   by simp
```
```   309
```
```   310 lemma even_equiv_def:
```
```   311   "even (x::int) = (EX y. x = 2 * y)"
```
```   312   by presburger
```
```   313
```
```   314
```
```   315 subsubsection {* Equivalent definitions *}
```
```   316
```
```   317 lemma two_times_even_div_two:
```
```   318   "even (x::int) ==> 2 * (x div 2) = x"
```
```   319   by presburger
```
```   320
```
```   321 lemma two_times_odd_div_two_plus_one:
```
```   322   "odd (x::int) ==> 2 * (x div 2) + 1 = x"
```
```   323   by presburger
```
```   324
```
```   325
```
```   326 subsubsection {* even and odd for nats *}
```
```   327
```
```   328 lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
```
```   329 by (simp add: even_int_iff [symmetric])
```
```   330
```
```   331 lemma even_difference_nat [simp,presburger,algebra]:
```
```   332   "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
```
```   333   by presburger
```
```   334
```
```   335 lemma even_Suc [simp ,presburger, algebra]:
```
```   336   "even (Suc x) = odd x"
```
```   337   by presburger
```
```   338
```
```   339 lemma even_power_nat[simp,presburger,algebra]:
```
```   340   "even ((x::nat)^y) = (even x & 0 < y)"
```
```   341   by simp
```
```   342
```
```   343
```
```   344 subsubsection {* Equivalent definitions *}
```
```   345
```
```   346 lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
```
```   347 by presburger
```
```   348
```
```   349 lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
```
```   350 by presburger
```
```   351
```
```   352 lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
```
```   353 by presburger
```
```   354
```
```   355 lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
```
```   356 by presburger
```
```   357
```
```   358 lemma even_nat_div_two_times_two: "even (x::nat) ==>
```
```   359     Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger
```
```   360
```
```   361 lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>
```
```   362     Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger
```
```   363
```
```   364 lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
```
```   365   by presburger
```
```   366
```
```   367
```
```   368 subsubsection {* Parity and powers *}
```
```   369
```
```   370 lemma (in comm_ring_1) neg_power_if:
```
```   371   "(- a) ^ n = (if even n then (a ^ n) else - (a ^ n))"
```
```   372   by (induct n) simp_all
```
```   373
```
```   374 lemma (in comm_ring_1)
```
```   375   shows neg_one_even_power [simp]: "even n \<Longrightarrow> (- 1) ^ n = 1"
```
```   376   and neg_one_odd_power [simp]: "odd n \<Longrightarrow> (- 1) ^ n = - 1"
```
```   377   by (simp_all add: neg_power_if)
```
```   378
```
```   379 lemma zero_le_even_power: "even n ==>
```
```   380     0 <= (x::'a::{linordered_ring,monoid_mult}) ^ n"
```
```   381   apply (simp add: even_def)
```
```   382   apply (erule dvdE)
```
```   383   apply (erule ssubst)
```
```   384   unfolding mult.commute [of 2]
```
```   385   unfolding power_mult power2_eq_square
```
```   386   apply (rule zero_le_square)
```
```   387   done
```
```   388
```
```   389 lemma zero_le_odd_power: "odd n ==>
```
```   390     (0 <= (x::'a::{linordered_idom}) ^ n) = (0 <= x)"
```
```   391 apply (auto simp: odd_nat_equiv_def2 power_add zero_le_mult_iff)
```
```   392 apply (metis field_power_not_zero divisors_zero order_antisym_conv zero_le_square)
```
```   393 done
```
```   394
```
```   395 lemma zero_le_power_eq [presburger]: "(0 <= (x::'a::{linordered_idom}) ^ n) =
```
```   396     (even n | (odd n & 0 <= x))"
```
```   397   apply auto
```
```   398   apply (subst zero_le_odd_power [symmetric])
```
```   399   apply assumption+
```
```   400   apply (erule zero_le_even_power)
```
```   401   done
```
```   402
```
```   403 lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{linordered_idom}) ^ n) =
```
```   404     (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
```
```   405   unfolding order_less_le zero_le_power_eq by auto
```
```   406
```
```   407 lemma power_less_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n < 0) =
```
```   408     (odd n & x < 0)"
```
```   409   apply (subst linorder_not_le [symmetric])+
```
```   410   apply (subst zero_le_power_eq)
```
```   411   apply auto
```
```   412   done
```
```   413
```
```   414 lemma power_le_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n <= 0) =
```
```   415     (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"
```
```   416   apply (subst linorder_not_less [symmetric])+
```
```   417   apply (subst zero_less_power_eq)
```
```   418   apply auto
```
```   419   done
```
```   420
```
```   421 lemma power_even_abs: "even n ==>
```
```   422     (abs (x::'a::{linordered_idom}))^n = x^n"
```
```   423   apply (subst power_abs [symmetric])
```
```   424   apply (simp add: zero_le_even_power)
```
```   425   done
```
```   426
```
```   427 lemma power_minus_even [simp]: "even n ==>
```
```   428     (- x)^n = (x^n::'a::{comm_ring_1})"
```
```   429   apply (subst power_minus)
```
```   430   apply simp
```
```   431   done
```
```   432
```
```   433 lemma power_minus_odd [simp]: "odd n ==>
```
```   434     (- x)^n = - (x^n::'a::{comm_ring_1})"
```
```   435   apply (subst power_minus)
```
```   436   apply simp
```
```   437   done
```
```   438
```
```   439 lemma power_mono_even: fixes x y :: "'a :: {linordered_idom}"
```
```   440   assumes "even n" and "\<bar>x\<bar> \<le> \<bar>y\<bar>"
```
```   441   shows "x^n \<le> y^n"
```
```   442 proof -
```
```   443   have "0 \<le> \<bar>x\<bar>" by auto
```
```   444   with `\<bar>x\<bar> \<le> \<bar>y\<bar>`
```
```   445   have "\<bar>x\<bar>^n \<le> \<bar>y\<bar>^n" by (rule power_mono)
```
```   446   thus ?thesis unfolding power_even_abs[OF `even n`] .
```
```   447 qed
```
```   448
```
```   449 lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger
```
```   450
```
```   451 lemma power_mono_odd: fixes x y :: "'a :: {linordered_idom}"
```
```   452   assumes "odd n" and "x \<le> y"
```
```   453   shows "x^n \<le> y^n"
```
```   454 proof (cases "y < 0")
```
```   455   case True with `x \<le> y` have "-y \<le> -x" and "0 \<le> -y" by auto
```
```   456   hence "(-y)^n \<le> (-x)^n" by (rule power_mono)
```
```   457   thus ?thesis unfolding power_minus_odd[OF `odd n`] by auto
```
```   458 next
```
```   459   case False
```
```   460   show ?thesis
```
```   461   proof (cases "x < 0")
```
```   462     case True hence "n \<noteq> 0" and "x \<le> 0" using `odd n`[THEN odd_pos] by auto
```
```   463     hence "x^n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto
```
```   464     moreover
```
```   465     from `\<not> y < 0` have "0 \<le> y" by auto
```
```   466     hence "0 \<le> y^n" by auto
```
```   467     ultimately show ?thesis by auto
```
```   468   next
```
```   469     case False hence "0 \<le> x" by auto
```
```   470     with `x \<le> y` show ?thesis using power_mono by auto
```
```   471   qed
```
```   472 qed
```
```   473
```
```   474
```
```   475 subsubsection {* More Even/Odd Results *}
```
```   476
```
```   477 lemma even_mult_two_ex: "even(n) = (\<exists>m::nat. n = 2*m)" by presburger
```
```   478 lemma odd_Suc_mult_two_ex: "odd(n) = (\<exists>m. n = Suc (2*m))" by presburger
```
```   479
```
```   480 lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" by presburger
```
```   481
```
```   482 lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)"
```
```   483 by presburger
```
```   484
```
```   485 lemma even_num_iff: "0 < n ==> even n = (~ even(n - 1 :: nat))"  by presburger
```
```   486 lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)" by presburger
```
```   487
```
```   488 lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n - 1) div 2)" by presburger
```
```   489
```
```   490 lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n - 1) div 2)"
```
```   491   by presburger
```
```   492
```
```   493 text {* Simplify, when the exponent is a numeral *}
```
```   494
```
```   495 lemmas zero_le_power_eq_numeral [simp] =
```
```   496   zero_le_power_eq [of _ "numeral w"] for w
```
```   497
```
```   498 lemmas zero_less_power_eq_numeral [simp] =
```
```   499   zero_less_power_eq [of _ "numeral w"] for w
```
```   500
```
```   501 lemmas power_le_zero_eq_numeral [simp] =
```
```   502   power_le_zero_eq [of _ "numeral w"] for w
```
```   503
```
```   504 lemmas power_less_zero_eq_numeral [simp] =
```
```   505   power_less_zero_eq [of _ "numeral w"] for w
```
```   506
```
```   507 lemmas zero_less_power_nat_eq_numeral [simp] =
```
```   508   nat_zero_less_power_iff [of _ "numeral w"] for w
```
```   509
```
```   510 lemmas power_eq_0_iff_numeral [simp] =
```
```   511   power_eq_0_iff [of _ "numeral w"] for w
```
```   512
```
```   513 lemmas power_even_abs_numeral [simp] =
```
```   514   power_even_abs [of "numeral w" _] for w
```
```   515
```
```   516
```
```   517 subsubsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *}
```
```   518
```
```   519 lemma zero_le_power_iff[presburger]:
```
```   520   "(0 \<le> a^n) = (0 \<le> (a::'a::{linordered_idom}) | even n)"
```
```   521 proof cases
```
```   522   assume even: "even n"
```
```   523   then have "2 dvd n" by (simp add: even_def)
```
```   524   then obtain k where "n = 2 * k" ..
```
```   525   thus ?thesis by (simp add: zero_le_even_power even)
```
```   526 next
```
```   527   assume odd: "odd n"
```
```   528   then obtain k where "n = Suc(2*k)"
```
```   529     by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)
```
```   530   moreover have "a ^ (2 * k) \<le> 0 \<Longrightarrow> a = 0"
```
```   531     by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff)
```
```   532   ultimately show ?thesis
```
```   533     by (auto simp add: zero_le_mult_iff zero_le_even_power)
```
```   534 qed
```
```   535
```
```   536
```
```   537 subsubsection {* Miscellaneous *}
```
```   538
```
```   539 lemma [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger
```
```   540 lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger
```
```   541 lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2"  by presburger
```
```   542 lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger
```
```   543
```
```   544 lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger
```
```   545 lemma even_nat_plus_one_div_two: "even (x::nat) ==>
```
```   546     (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger
```
```   547
```
```   548 lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>
```
```   549     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger
```
```   550
```
```   551 end
```
```   552
```