src/HOL/Parity.thy
 author haftmann Tue Oct 14 08:23:23 2014 +0200 (2014-10-14) changeset 58679 33c90658448a parent 58678 398e05aa84d4 child 58680 6b2fa479945f 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 begin
```
```    45
```
```    46 lemma two_dvd_plus_one_iff [simp]:
```
```    47   "2 dvd a + 1 \<longleftrightarrow> \<not> 2 dvd a"
```
```    48   by (auto simp add: dvd_add_right_iff intro: not_dvd_not_dvd_dvd_add)
```
```    49
```
```    50 end
```
```    51
```
```    52 instance nat :: semiring_parity
```
```    53 proof
```
```    54   show "\<not> (2 :: nat) dvd 1"
```
```    55     by (rule notI, erule dvdE) simp
```
```    56 next
```
```    57   fix m n :: nat
```
```    58   assume "\<not> 2 dvd m"
```
```    59   moreover assume "\<not> 2 dvd n"
```
```    60   ultimately have *: "2 dvd Suc m \<and> 2 dvd Suc n"
```
```    61     by (simp add: two_dvd_Suc_iff)
```
```    62   then have "2 dvd Suc m + Suc n"
```
```    63     by (blast intro: dvd_add)
```
```    64   also have "Suc m + Suc n = m + n + 2"
```
```    65     by simp
```
```    66   finally show "2 dvd m + n"
```
```    67     using dvd_add_triv_right_iff [of 2 "m + n"] by simp
```
```    68 next
```
```    69   fix m n :: nat
```
```    70   assume *: "2 dvd m * n"
```
```    71   show "2 dvd m \<or> 2 dvd n"
```
```    72   proof (rule disjCI)
```
```    73     assume "\<not> 2 dvd n"
```
```    74     then have "2 dvd Suc n" by (simp add: two_dvd_Suc_iff)
```
```    75     then obtain r where "Suc n = 2 * r" ..
```
```    76     moreover from * obtain s where "m * n = 2 * s" ..
```
```    77     then have "2 * s + m = m * Suc n" by simp
```
```    78     ultimately have " 2 * s + m = 2 * (m * r)" by simp
```
```    79     then have "m = 2 * (m * r - s)" by simp
```
```    80     then show "2 dvd m" ..
```
```    81   qed
```
```    82 qed
```
```    83
```
```    84 class ring_parity = comm_ring_1 + semiring_parity
```
```    85
```
```    86 instance int :: ring_parity
```
```    87 proof
```
```    88   show "\<not> (2 :: int) dvd 1" by (simp add: dvd_int_unfold_dvd_nat)
```
```    89   fix k l :: int
```
```    90   assume "\<not> 2 dvd k"
```
```    91   moreover assume "\<not> 2 dvd l"
```
```    92   ultimately have "2 dvd nat \<bar>k\<bar> + nat \<bar>l\<bar>"
```
```    93     by (auto simp add: dvd_int_unfold_dvd_nat intro: not_dvd_not_dvd_dvd_add)
```
```    94   then have "2 dvd \<bar>k\<bar> + \<bar>l\<bar>"
```
```    95     by (simp add: dvd_int_unfold_dvd_nat nat_add_distrib)
```
```    96   then show "2 dvd k + l"
```
```    97     by (simp add: two_dvd_abs_add_iff two_dvd_add_abs_iff)
```
```    98 qed (simp add: dvd_int_unfold_dvd_nat two_is_prime nat_abs_mult_distrib)
```
```    99
```
```   100 context semiring_div_parity
```
```   101 begin
```
```   102
```
```   103 subclass semiring_parity
```
```   104 proof (unfold_locales, unfold dvd_eq_mod_eq_0 not_mod_2_eq_0_eq_1)
```
```   105   fix a b c
```
```   106   show "(c * a + b) mod a = 0 \<longleftrightarrow> b mod a = 0"
```
```   107     by simp
```
```   108 next
```
```   109   fix a b c
```
```   110   assume "(b + c) mod a = 0"
```
```   111   with mod_add_eq [of b c a]
```
```   112   have "(b mod a + c mod a) mod a = 0"
```
```   113     by simp
```
```   114   moreover assume "b mod a = 0"
```
```   115   ultimately show "c mod a = 0"
```
```   116     by simp
```
```   117 next
```
```   118   show "1 mod 2 = 1"
```
```   119     by (fact one_mod_two_eq_one)
```
```   120 next
```
```   121   fix a b
```
```   122   assume "a mod 2 = 1"
```
```   123   moreover assume "b mod 2 = 1"
```
```   124   ultimately show "(a + b) mod 2 = 0"
```
```   125     using mod_add_eq [of a b 2] by simp
```
```   126 next
```
```   127   fix a b
```
```   128   assume "(a * b) mod 2 = 0"
```
```   129   then have "(a mod 2) * (b mod 2) = 0"
```
```   130     by (cases "a mod 2 = 0") (simp_all add: mod_mult_eq [of a b 2])
```
```   131   then show "a mod 2 = 0 \<or> b mod 2 = 0"
```
```   132     by (rule divisors_zero)
```
```   133 qed
```
```   134
```
```   135 end
```
```   136
```
```   137
```
```   138 subsection {* Dedicated @{text even}/@{text odd} predicate *}
```
```   139
```
```   140 context semiring_parity
```
```   141 begin
```
```   142
```
```   143 definition even :: "'a \<Rightarrow> bool"
```
```   144 where
```
```   145   [algebra]: "even a \<longleftrightarrow> 2 dvd a"
```
```   146
```
```   147 abbreviation odd :: "'a \<Rightarrow> bool"
```
```   148 where
```
```   149   "odd a \<equiv> \<not> even a"
```
```   150
```
```   151 lemma even_times_iff [simp, presburger, algebra]:
```
```   152   "even (a * b) \<longleftrightarrow> even a \<or> even b"
```
```   153   by (auto simp add: even_def dest: two_is_prime)
```
```   154
```
```   155 lemma even_zero [simp]:
```
```   156   "even 0"
```
```   157   by (simp add: even_def)
```
```   158
```
```   159 lemma odd_one [simp]:
```
```   160   "odd 1"
```
```   161   by (simp add: even_def)
```
```   162
```
```   163 lemma even_numeral [simp]:
```
```   164   "even (numeral (Num.Bit0 n))"
```
```   165 proof -
```
```   166   have "even (2 * numeral n)"
```
```   167     unfolding even_times_iff by (simp add: even_def)
```
```   168   then have "even (numeral n + numeral n)"
```
```   169     unfolding mult_2 .
```
```   170   then show ?thesis
```
```   171     unfolding numeral.simps .
```
```   172 qed
```
```   173
```
```   174 lemma odd_numeral [simp]:
```
```   175   "odd (numeral (Num.Bit1 n))"
```
```   176 proof
```
```   177   assume "even (numeral (num.Bit1 n))"
```
```   178   then have "even (numeral n + numeral n + 1)"
```
```   179     unfolding numeral.simps .
```
```   180   then have "even (2 * numeral n + 1)"
```
```   181     unfolding mult_2 .
```
```   182   then have "2 dvd numeral n * 2 + 1"
```
```   183     unfolding even_def by (simp add: ac_simps)
```
```   184   with dvd_add_times_triv_left_iff [of 2 "numeral n" 1]
```
```   185     have "2 dvd 1"
```
```   186     by simp
```
```   187   then show False by simp
```
```   188 qed
```
```   189
```
```   190 end
```
```   191
```
```   192 context ring_parity
```
```   193 begin
```
```   194
```
```   195 lemma even_minus [simp, presburger, algebra]:
```
```   196   "even (- a) \<longleftrightarrow> even a"
```
```   197   by (simp add: even_def)
```
```   198
```
```   199 end
```
```   200
```
```   201 context semiring_div_parity
```
```   202 begin
```
```   203
```
```   204 lemma even_iff_mod_2_eq_zero [presburger]:
```
```   205   "even a \<longleftrightarrow> a mod 2 = 0"
```
```   206   by (simp add: even_def dvd_eq_mod_eq_0)
```
```   207
```
```   208 end
```
```   209
```
```   210 lemma even_int_iff:
```
```   211   "even (int n) \<longleftrightarrow> even n"
```
```   212   by (simp add: even_def dvd_int_iff)
```
```   213
```
```   214 declare transfer_morphism_int_nat [transfer add return:
```
```   215   even_int_iff
```
```   216 ]
```
```   217
```
```   218 lemma [presburger]:
```
```   219   "even n \<longleftrightarrow> even (int n)"
```
```   220   using even_int_iff [of n] by simp
```
```   221
```
```   222
```
```   223 subsection {* Behavior under integer arithmetic operations *}
```
```   224
```
```   225 lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"
```
```   226 by presburger
```
```   227
```
```   228 lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"
```
```   229 by presburger
```
```   230
```
```   231 lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"
```
```   232 by presburger
```
```   233
```
```   234 lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger
```
```   235
```
```   236 lemma even_sum[simp,presburger]:
```
```   237   "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
```
```   238 by presburger
```
```   239
```
```   240 lemma even_difference[simp]:
```
```   241     "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger
```
```   242
```
```   243 lemma even_power[simp,presburger]: "even ((x::int)^n) = (even x & n \<noteq> 0)"
```
```   244 by (induct n) auto
```
```   245
```
```   246 lemma odd_pow: "odd x ==> odd((x::int)^n)" by simp
```
```   247
```
```   248
```
```   249 subsection {* Equivalent definitions *}
```
```   250
```
```   251 lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x"
```
```   252 by presburger
```
```   253
```
```   254 lemma two_times_odd_div_two_plus_one:
```
```   255   "odd (x::int) ==> 2 * (x div 2) + 1 = x"
```
```   256 by presburger
```
```   257
```
```   258 lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger
```
```   259
```
```   260 lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" by presburger
```
```   261
```
```   262 subsection {* even and odd for nats *}
```
```   263
```
```   264 lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
```
```   265 by (simp add: even_int_iff [symmetric])
```
```   266
```
```   267 lemma even_sum_nat[simp,presburger,algebra]:
```
```   268   "even ((x::nat) + y) = ((even x & even y) | (odd x & odd y))"
```
```   269 by presburger
```
```   270
```
```   271 lemma even_difference_nat[simp,presburger,algebra]:
```
```   272   "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
```
```   273 by presburger
```
```   274
```
```   275 lemma even_Suc[simp,presburger,algebra]: "even (Suc x) = odd x"
```
```   276 by presburger
```
```   277
```
```   278 lemma even_power_nat[simp,presburger,algebra]:
```
```   279   "even ((x::nat)^y) = (even x & 0 < y)"
```
```   280 by (simp add: even_int_iff [symmetric] int_power)
```
```   281
```
```   282
```
```   283 subsection {* Equivalent definitions *}
```
```   284
```
```   285 lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
```
```   286 by presburger
```
```   287
```
```   288 lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
```
```   289 by presburger
```
```   290
```
```   291 lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
```
```   292 by presburger
```
```   293
```
```   294 lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
```
```   295 by presburger
```
```   296
```
```   297 lemma even_nat_div_two_times_two: "even (x::nat) ==>
```
```   298     Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger
```
```   299
```
```   300 lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>
```
```   301     Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger
```
```   302
```
```   303 lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
```
```   304 by presburger
```
```   305
```
```   306 lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
```
```   307 by presburger
```
```   308
```
```   309
```
```   310 subsection {* Parity and powers *}
```
```   311
```
```   312 lemma (in comm_ring_1) neg_power_if:
```
```   313   "(- a) ^ n = (if even n then (a ^ n) else - (a ^ n))"
```
```   314   by (induct n) simp_all
```
```   315
```
```   316 lemma (in comm_ring_1)
```
```   317   shows neg_one_even_power [simp]: "even n \<Longrightarrow> (- 1) ^ n = 1"
```
```   318   and neg_one_odd_power [simp]: "odd n \<Longrightarrow> (- 1) ^ n = - 1"
```
```   319   by (simp_all add: neg_power_if)
```
```   320
```
```   321 lemma zero_le_even_power: "even n ==>
```
```   322     0 <= (x::'a::{linordered_ring,monoid_mult}) ^ n"
```
```   323   apply (simp add: even_nat_equiv_def2)
```
```   324   apply (erule exE)
```
```   325   apply (erule ssubst)
```
```   326   apply (subst power_add)
```
```   327   apply (rule zero_le_square)
```
```   328   done
```
```   329
```
```   330 lemma zero_le_odd_power: "odd n ==>
```
```   331     (0 <= (x::'a::{linordered_idom}) ^ n) = (0 <= x)"
```
```   332 apply (auto simp: odd_nat_equiv_def2 power_add zero_le_mult_iff)
```
```   333 apply (metis field_power_not_zero divisors_zero order_antisym_conv zero_le_square)
```
```   334 done
```
```   335
```
```   336 lemma zero_le_power_eq [presburger]: "(0 <= (x::'a::{linordered_idom}) ^ n) =
```
```   337     (even n | (odd n & 0 <= x))"
```
```   338   apply auto
```
```   339   apply (subst zero_le_odd_power [symmetric])
```
```   340   apply assumption+
```
```   341   apply (erule zero_le_even_power)
```
```   342   done
```
```   343
```
```   344 lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{linordered_idom}) ^ n) =
```
```   345     (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
```
```   346
```
```   347   unfolding order_less_le zero_le_power_eq by auto
```
```   348
```
```   349 lemma power_less_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n < 0) =
```
```   350     (odd n & x < 0)"
```
```   351   apply (subst linorder_not_le [symmetric])+
```
```   352   apply (subst zero_le_power_eq)
```
```   353   apply auto
```
```   354   done
```
```   355
```
```   356 lemma power_le_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n <= 0) =
```
```   357     (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"
```
```   358   apply (subst linorder_not_less [symmetric])+
```
```   359   apply (subst zero_less_power_eq)
```
```   360   apply auto
```
```   361   done
```
```   362
```
```   363 lemma power_even_abs: "even n ==>
```
```   364     (abs (x::'a::{linordered_idom}))^n = x^n"
```
```   365   apply (subst power_abs [symmetric])
```
```   366   apply (simp add: zero_le_even_power)
```
```   367   done
```
```   368
```
```   369 lemma power_minus_even [simp]: "even n ==>
```
```   370     (- x)^n = (x^n::'a::{comm_ring_1})"
```
```   371   apply (subst power_minus)
```
```   372   apply simp
```
```   373   done
```
```   374
```
```   375 lemma power_minus_odd [simp]: "odd n ==>
```
```   376     (- x)^n = - (x^n::'a::{comm_ring_1})"
```
```   377   apply (subst power_minus)
```
```   378   apply simp
```
```   379   done
```
```   380
```
```   381 lemma power_mono_even: fixes x y :: "'a :: {linordered_idom}"
```
```   382   assumes "even n" and "\<bar>x\<bar> \<le> \<bar>y\<bar>"
```
```   383   shows "x^n \<le> y^n"
```
```   384 proof -
```
```   385   have "0 \<le> \<bar>x\<bar>" by auto
```
```   386   with `\<bar>x\<bar> \<le> \<bar>y\<bar>`
```
```   387   have "\<bar>x\<bar>^n \<le> \<bar>y\<bar>^n" by (rule power_mono)
```
```   388   thus ?thesis unfolding power_even_abs[OF `even n`] .
```
```   389 qed
```
```   390
```
```   391 lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger
```
```   392
```
```   393 lemma power_mono_odd: fixes x y :: "'a :: {linordered_idom}"
```
```   394   assumes "odd n" and "x \<le> y"
```
```   395   shows "x^n \<le> y^n"
```
```   396 proof (cases "y < 0")
```
```   397   case True with `x \<le> y` have "-y \<le> -x" and "0 \<le> -y" by auto
```
```   398   hence "(-y)^n \<le> (-x)^n" by (rule power_mono)
```
```   399   thus ?thesis unfolding power_minus_odd[OF `odd n`] by auto
```
```   400 next
```
```   401   case False
```
```   402   show ?thesis
```
```   403   proof (cases "x < 0")
```
```   404     case True hence "n \<noteq> 0" and "x \<le> 0" using `odd n`[THEN odd_pos] by auto
```
```   405     hence "x^n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto
```
```   406     moreover
```
```   407     from `\<not> y < 0` have "0 \<le> y" by auto
```
```   408     hence "0 \<le> y^n" by auto
```
```   409     ultimately show ?thesis by auto
```
```   410   next
```
```   411     case False hence "0 \<le> x" by auto
```
```   412     with `x \<le> y` show ?thesis using power_mono by auto
```
```   413   qed
```
```   414 qed
```
```   415
```
```   416
```
```   417 subsection {* More Even/Odd Results *}
```
```   418
```
```   419 lemma even_mult_two_ex: "even(n) = (\<exists>m::nat. n = 2*m)" by presburger
```
```   420 lemma odd_Suc_mult_two_ex: "odd(n) = (\<exists>m. n = Suc (2*m))" by presburger
```
```   421 lemma even_add [simp]: "even(m + n::nat) = (even m = even n)"  by presburger
```
```   422
```
```   423 lemma odd_add [simp]: "odd(m + n::nat) = (odd m \<noteq> odd n)" by presburger
```
```   424
```
```   425 lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" by presburger
```
```   426
```
```   427 lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)"
```
```   428 by presburger
```
```   429
```
```   430 lemma even_num_iff: "0 < n ==> even n = (~ even(n - 1 :: nat))"  by presburger
```
```   431 lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)" by presburger
```
```   432
```
```   433 lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n - 1) div 2)" by presburger
```
```   434
```
```   435 lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n - 1) div 2)"
```
```   436   by presburger
```
```   437
```
```   438 text {* Simplify, when the exponent is a numeral *}
```
```   439
```
```   440 lemmas zero_le_power_eq_numeral [simp] =
```
```   441   zero_le_power_eq [of _ "numeral w"] for w
```
```   442
```
```   443 lemmas zero_less_power_eq_numeral [simp] =
```
```   444   zero_less_power_eq [of _ "numeral w"] for w
```
```   445
```
```   446 lemmas power_le_zero_eq_numeral [simp] =
```
```   447   power_le_zero_eq [of _ "numeral w"] for w
```
```   448
```
```   449 lemmas power_less_zero_eq_numeral [simp] =
```
```   450   power_less_zero_eq [of _ "numeral w"] for w
```
```   451
```
```   452 lemmas zero_less_power_nat_eq_numeral [simp] =
```
```   453   nat_zero_less_power_iff [of _ "numeral w"] for w
```
```   454
```
```   455 lemmas power_eq_0_iff_numeral [simp] =
```
```   456   power_eq_0_iff [of _ "numeral w"] for w
```
```   457
```
```   458 lemmas power_even_abs_numeral [simp] =
```
```   459   power_even_abs [of "numeral w" _] for w
```
```   460
```
```   461
```
```   462 subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *}
```
```   463
```
```   464 lemma zero_le_power_iff[presburger]:
```
```   465   "(0 \<le> a^n) = (0 \<le> (a::'a::{linordered_idom}) | even n)"
```
```   466 proof cases
```
```   467   assume even: "even n"
```
```   468   then obtain k where "n = 2*k"
```
```   469     by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2)
```
```   470   thus ?thesis by (simp add: zero_le_even_power even)
```
```   471 next
```
```   472   assume odd: "odd n"
```
```   473   then obtain k where "n = Suc(2*k)"
```
```   474     by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)
```
```   475   moreover have "a ^ (2 * k) \<le> 0 \<Longrightarrow> a = 0"
```
```   476     by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff)
```
```   477   ultimately show ?thesis
```
```   478     by (auto simp add: zero_le_mult_iff zero_le_even_power)
```
```   479 qed
```
```   480
```
```   481
```
```   482 subsection {* Miscellaneous *}
```
```   483
```
```   484 lemma [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger
```
```   485 lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger
```
```   486 lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2"  by presburger
```
```   487 lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger
```
```   488
```
```   489 lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger
```
```   490 lemma even_nat_plus_one_div_two: "even (x::nat) ==>
```
```   491     (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger
```
```   492
```
```   493 lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>
```
```   494     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger
```
```   495
```
```   496 end
```
```   497
```