src/HOL/Parity.thy
author huffman
Sun Apr 01 16:09:58 2012 +0200 (2012-04-01)
changeset 47255 30a1692557b0
parent 47225 650318981557
child 54227 63b441f49645
permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
     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 class even_odd = 
    13   fixes even :: "'a \<Rightarrow> bool"
    14 
    15 abbreviation
    16   odd :: "'a\<Colon>even_odd \<Rightarrow> bool" where
    17   "odd x \<equiv> \<not> even x"
    18 
    19 instantiation nat and int  :: even_odd
    20 begin
    21 
    22 definition
    23   even_def [presburger]: "even x \<longleftrightarrow> (x\<Colon>int) mod 2 = 0"
    24 
    25 definition
    26   even_nat_def [presburger]: "even x \<longleftrightarrow> even (int x)"
    27 
    28 instance ..
    29 
    30 end
    31 
    32 lemma transfer_int_nat_relations:
    33   "even (int x) \<longleftrightarrow> even x"
    34   by (simp add: even_nat_def)
    35 
    36 declare transfer_morphism_int_nat[transfer add return:
    37   transfer_int_nat_relations
    38 ]
    39 
    40 lemma even_zero_int[simp]: "even (0::int)" by presburger
    41 
    42 lemma odd_one_int[simp]: "odd (1::int)" by presburger
    43 
    44 lemma even_zero_nat[simp]: "even (0::nat)" by presburger
    45 
    46 lemma odd_1_nat [simp]: "odd (1::nat)" by presburger
    47 
    48 lemma even_numeral_int [simp]: "even (numeral (Num.Bit0 k) :: int)"
    49   unfolding even_def by simp
    50 
    51 lemma odd_numeral_int [simp]: "odd (numeral (Num.Bit1 k) :: int)"
    52   unfolding even_def by simp
    53 
    54 (* TODO: proper simp rules for Num.Bit0, Num.Bit1 *)
    55 declare even_def[of "neg_numeral v", simp] for v
    56 
    57 lemma even_numeral_nat [simp]: "even (numeral (Num.Bit0 k) :: nat)"
    58   unfolding even_nat_def by simp
    59 
    60 lemma odd_numeral_nat [simp]: "odd (numeral (Num.Bit1 k) :: nat)"
    61   unfolding even_nat_def by simp
    62 
    63 subsection {* Even and odd are mutually exclusive *}
    64 
    65 lemma int_pos_lt_two_imp_zero_or_one:
    66     "0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1"
    67   by presburger
    68 
    69 lemma neq_one_mod_two [simp, presburger]: 
    70   "((x::int) mod 2 ~= 0) = (x mod 2 = 1)" by presburger
    71 
    72 
    73 subsection {* Behavior under integer arithmetic operations *}
    74 declare dvd_def[algebra]
    75 lemma nat_even_iff_2_dvd[algebra]: "even (x::nat) \<longleftrightarrow> 2 dvd x"
    76   by presburger
    77 lemma int_even_iff_2_dvd[algebra]: "even (x::int) \<longleftrightarrow> 2 dvd x"
    78   by presburger
    79 
    80 lemma even_times_anything: "even (x::int) ==> even (x * y)"
    81   by algebra
    82 
    83 lemma anything_times_even: "even (y::int) ==> even (x * y)" by algebra
    84 
    85 lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)" 
    86   by (simp add: even_def mod_mult_right_eq)
    87 
    88 lemma even_product[simp,presburger]: "even((x::int) * y) = (even x | even y)"
    89   apply (auto simp add: even_times_anything anything_times_even)
    90   apply (rule ccontr)
    91   apply (auto simp add: odd_times_odd)
    92   done
    93 
    94 lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"
    95 by presburger
    96 
    97 lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"
    98 by presburger
    99 
   100 lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"
   101 by presburger
   102 
   103 lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger
   104 
   105 lemma even_sum[simp,presburger]:
   106   "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
   107 by presburger
   108 
   109 lemma even_neg[simp,presburger,algebra]: "even (-(x::int)) = even x"
   110 by presburger
   111 
   112 lemma even_difference[simp]:
   113     "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger
   114 
   115 lemma even_power[simp,presburger]: "even ((x::int)^n) = (even x & n \<noteq> 0)"
   116 by (induct n) auto
   117 
   118 lemma odd_pow: "odd x ==> odd((x::int)^n)" by simp
   119 
   120 
   121 subsection {* Equivalent definitions *}
   122 
   123 lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x" 
   124 by presburger
   125 
   126 lemma two_times_odd_div_two_plus_one:
   127   "odd (x::int) ==> 2 * (x div 2) + 1 = x"
   128 by presburger
   129 
   130 lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger
   131 
   132 lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" by presburger
   133 
   134 subsection {* even and odd for nats *}
   135 
   136 lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
   137 by (simp add: even_nat_def)
   138 
   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)
   142 
   143 lemma even_sum_nat[simp,presburger,algebra]:
   144   "even ((x::nat) + y) = ((even x & even y) | (odd x & odd y))"
   145 by presburger
   146 
   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
   150 
   151 lemma even_Suc[simp,presburger,algebra]: "even (Suc x) = odd x"
   152 by presburger
   153 
   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)
   157 
   158 
   159 subsection {* Equivalent definitions *}
   160 
   161 lemma nat_lt_two_imp_zero_or_one:
   162   "(x::nat) < Suc (Suc 0) ==> x = 0 | x = Suc 0"
   163 by presburger
   164 
   165 lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
   166 by presburger
   167 
   168 lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
   169 by presburger
   170 
   171 lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
   172 by presburger
   173 
   174 lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
   175 by presburger
   176 
   177 lemma even_nat_div_two_times_two: "even (x::nat) ==>
   178     Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger
   179 
   180 lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>
   181     Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger
   182 
   183 lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
   184 by presburger
   185 
   186 lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
   187 by presburger
   188 
   189 
   190 subsection {* Parity and powers *}
   191 
   192 lemma  minus_one_even_odd_power:
   193      "(even x --> (- 1::'a::{comm_ring_1})^x = 1) &
   194       (odd x --> (- 1::'a)^x = - 1)"
   195   apply (induct x)
   196   apply (rule conjI)
   197   apply simp
   198   apply (insert even_zero_nat, blast)
   199   apply simp
   200   done
   201 
   202 lemma minus_one_even_power [simp]:
   203     "even x ==> (- 1::'a::{comm_ring_1})^x = 1"
   204   using minus_one_even_odd_power by blast
   205 
   206 lemma minus_one_odd_power [simp]:
   207     "odd x ==> (- 1::'a::{comm_ring_1})^x = - 1"
   208   using minus_one_even_odd_power by blast
   209 
   210 lemma neg_one_even_odd_power:
   211      "(even x --> (-1::'a::{comm_ring_1})^x = 1) &
   212       (odd x --> (-1::'a)^x = -1)"
   213   apply (induct x)
   214   apply (simp, simp)
   215   done
   216 
   217 lemma neg_one_even_power [simp]:
   218     "even x ==> (-1::'a::{comm_ring_1})^x = 1"
   219   using neg_one_even_odd_power by blast
   220 
   221 lemma neg_one_odd_power [simp]:
   222     "odd x ==> (-1::'a::{comm_ring_1})^x = -1"
   223   using neg_one_even_odd_power by blast
   224 
   225 lemma neg_power_if:
   226      "(-x::'a::{comm_ring_1}) ^ n =
   227       (if even n then (x ^ n) else -(x ^ n))"
   228   apply (induct n)
   229   apply simp_all
   230   done
   231 
   232 lemma zero_le_even_power: "even n ==>
   233     0 <= (x::'a::{linordered_ring,monoid_mult}) ^ n"
   234   apply (simp add: even_nat_equiv_def2)
   235   apply (erule exE)
   236   apply (erule ssubst)
   237   apply (subst power_add)
   238   apply (rule zero_le_square)
   239   done
   240 
   241 lemma zero_le_odd_power: "odd n ==>
   242     (0 <= (x::'a::{linordered_idom}) ^ n) = (0 <= x)"
   243 apply (auto simp: odd_nat_equiv_def2 power_add zero_le_mult_iff)
   244 apply (metis field_power_not_zero divisors_zero order_antisym_conv zero_le_square)
   245 done
   246 
   247 lemma zero_le_power_eq[presburger]: "(0 <= (x::'a::{linordered_idom}) ^ n) =
   248     (even n | (odd n & 0 <= x))"
   249   apply auto
   250   apply (subst zero_le_odd_power [symmetric])
   251   apply assumption+
   252   apply (erule zero_le_even_power)
   253   done
   254 
   255 lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{linordered_idom}) ^ n) =
   256     (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
   257 
   258   unfolding order_less_le zero_le_power_eq by auto
   259 
   260 lemma power_less_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n < 0) =
   261     (odd n & x < 0)"
   262   apply (subst linorder_not_le [symmetric])+
   263   apply (subst zero_le_power_eq)
   264   apply auto
   265   done
   266 
   267 lemma power_le_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n <= 0) =
   268     (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"
   269   apply (subst linorder_not_less [symmetric])+
   270   apply (subst zero_less_power_eq)
   271   apply auto
   272   done
   273 
   274 lemma power_even_abs: "even n ==>
   275     (abs (x::'a::{linordered_idom}))^n = x^n"
   276   apply (subst power_abs [symmetric])
   277   apply (simp add: zero_le_even_power)
   278   done
   279 
   280 lemma zero_less_power_nat_eq[presburger]: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)"
   281   by (induct n) auto
   282 
   283 lemma power_minus_even [simp]: "even n ==>
   284     (- x)^n = (x^n::'a::{comm_ring_1})"
   285   apply (subst power_minus)
   286   apply simp
   287   done
   288 
   289 lemma power_minus_odd [simp]: "odd n ==>
   290     (- x)^n = - (x^n::'a::{comm_ring_1})"
   291   apply (subst power_minus)
   292   apply simp
   293   done
   294 
   295 lemma power_mono_even: fixes x y :: "'a :: {linordered_idom}"
   296   assumes "even n" and "\<bar>x\<bar> \<le> \<bar>y\<bar>"
   297   shows "x^n \<le> y^n"
   298 proof -
   299   have "0 \<le> \<bar>x\<bar>" by auto
   300   with `\<bar>x\<bar> \<le> \<bar>y\<bar>`
   301   have "\<bar>x\<bar>^n \<le> \<bar>y\<bar>^n" by (rule power_mono)
   302   thus ?thesis unfolding power_even_abs[OF `even n`] .
   303 qed
   304 
   305 lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger
   306 
   307 lemma power_mono_odd: fixes x y :: "'a :: {linordered_idom}"
   308   assumes "odd n" and "x \<le> y"
   309   shows "x^n \<le> y^n"
   310 proof (cases "y < 0")
   311   case True with `x \<le> y` have "-y \<le> -x" and "0 \<le> -y" by auto
   312   hence "(-y)^n \<le> (-x)^n" by (rule power_mono)
   313   thus ?thesis unfolding power_minus_odd[OF `odd n`] by auto
   314 next
   315   case False
   316   show ?thesis
   317   proof (cases "x < 0")
   318     case True hence "n \<noteq> 0" and "x \<le> 0" using `odd n`[THEN odd_pos] by auto
   319     hence "x^n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto
   320     moreover
   321     from `\<not> y < 0` have "0 \<le> y" by auto
   322     hence "0 \<le> y^n" by auto
   323     ultimately show ?thesis by auto
   324   next
   325     case False hence "0 \<le> x" by auto
   326     with `x \<le> y` show ?thesis using power_mono by auto
   327   qed
   328 qed
   329 
   330 
   331 subsection {* More Even/Odd Results *}
   332  
   333 lemma even_mult_two_ex: "even(n) = (\<exists>m::nat. n = 2*m)" by presburger
   334 lemma odd_Suc_mult_two_ex: "odd(n) = (\<exists>m. n = Suc (2*m))" by presburger
   335 lemma even_add [simp]: "even(m + n::nat) = (even m = even n)"  by presburger
   336 
   337 lemma odd_add [simp]: "odd(m + n::nat) = (odd m \<noteq> odd n)" by presburger
   338 
   339 lemma div_Suc: "Suc a div c = a div c + Suc 0 div c +
   340     (a mod c + Suc 0 mod c) div c" 
   341   apply (subgoal_tac "Suc a = a + Suc 0")
   342   apply (erule ssubst)
   343   apply (rule div_add1_eq, simp)
   344   done
   345 
   346 lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" by presburger
   347 
   348 lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)"
   349 by presburger
   350 
   351 lemma even_num_iff: "0 < n ==> even n = (~ even(n - 1 :: nat))"  by presburger
   352 lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)" by presburger
   353 
   354 lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n - 1) div 2)" by presburger
   355 
   356 lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n - 1) div 2)"
   357   by presburger
   358 
   359 text {* Simplify, when the exponent is a numeral *}
   360 
   361 lemmas zero_le_power_eq_numeral [simp] =
   362     zero_le_power_eq [of _ "numeral w"] for w
   363 
   364 lemmas zero_less_power_eq_numeral [simp] =
   365     zero_less_power_eq [of _ "numeral w"] for w
   366 
   367 lemmas power_le_zero_eq_numeral [simp] =
   368     power_le_zero_eq [of _ "numeral w"] for w
   369 
   370 lemmas power_less_zero_eq_numeral [simp] =
   371     power_less_zero_eq [of _ "numeral w"] for w
   372 
   373 lemmas zero_less_power_nat_eq_numeral [simp] =
   374     zero_less_power_nat_eq [of _ "numeral w"] for w
   375 
   376 lemmas power_eq_0_iff_numeral [simp] = power_eq_0_iff [of _ "numeral w"] for w
   377 
   378 lemmas power_even_abs_numeral [simp] = power_even_abs [of "numeral w" _] for w
   379 
   380 
   381 subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *}
   382 
   383 lemma even_power_le_0_imp_0:
   384     "a ^ (2*k) \<le> (0::'a::{linordered_idom}) ==> a=0"
   385   by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff)
   386 
   387 lemma zero_le_power_iff[presburger]:
   388   "(0 \<le> a^n) = (0 \<le> (a::'a::{linordered_idom}) | even n)"
   389 proof cases
   390   assume even: "even n"
   391   then obtain k where "n = 2*k"
   392     by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2)
   393   thus ?thesis by (simp add: zero_le_even_power even)
   394 next
   395   assume odd: "odd n"
   396   then obtain k where "n = Suc(2*k)"
   397     by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)
   398   thus ?thesis
   399     by (auto simp add: zero_le_mult_iff zero_le_even_power
   400              dest!: even_power_le_0_imp_0)
   401 qed
   402 
   403 
   404 subsection {* Miscellaneous *}
   405 
   406 lemma [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger
   407 lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger
   408 lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2"  by presburger
   409 lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger
   410 
   411 lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger
   412 lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger
   413 lemma even_nat_plus_one_div_two: "even (x::nat) ==>
   414     (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger
   415 
   416 lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>
   417     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger
   418 
   419 end