src/HOL/Parity.thy
 author haftmann Thu Oct 31 11:44:20 2013 +0100 (2013-10-31) changeset 54227 63b441f49645 parent 47225 650318981557 child 54228 229282d53781 permissions -rw-r--r--
moving generic lemmas out of theory parity, disregarding some unused auxiliary lemmas;
tuned presburger
```     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 begin
```
```    15
```
```    16 abbreviation odd :: "'a \<Rightarrow> bool"
```
```    17 where
```
```    18   "odd x \<equiv> \<not> even x"
```
```    19
```
```    20 end
```
```    21
```
```    22 instantiation nat and int  :: even_odd
```
```    23 begin
```
```    24
```
```    25 definition
```
```    26   even_def [presburger]: "even x \<longleftrightarrow> (x\<Colon>int) mod 2 = 0"
```
```    27
```
```    28 definition
```
```    29   even_nat_def [presburger]: "even x \<longleftrightarrow> even (int x)"
```
```    30
```
```    31 instance ..
```
```    32
```
```    33 end
```
```    34
```
```    35 lemma transfer_int_nat_relations:
```
```    36   "even (int x) \<longleftrightarrow> even x"
```
```    37   by (simp add: even_nat_def)
```
```    38
```
```    39 declare transfer_morphism_int_nat[transfer add return:
```
```    40   transfer_int_nat_relations
```
```    41 ]
```
```    42
```
```    43 lemma even_zero_int[simp]: "even (0::int)" by presburger
```
```    44
```
```    45 lemma odd_one_int[simp]: "odd (1::int)" by presburger
```
```    46
```
```    47 lemma even_zero_nat[simp]: "even (0::nat)" by presburger
```
```    48
```
```    49 lemma odd_1_nat [simp]: "odd (1::nat)" by presburger
```
```    50
```
```    51 lemma even_numeral_int [simp]: "even (numeral (Num.Bit0 k) :: int)"
```
```    52   unfolding even_def by simp
```
```    53
```
```    54 lemma odd_numeral_int [simp]: "odd (numeral (Num.Bit1 k) :: int)"
```
```    55   unfolding even_def by simp
```
```    56
```
```    57 (* TODO: proper simp rules for Num.Bit0, Num.Bit1 *)
```
```    58 declare even_def [of "neg_numeral v", simp] for v
```
```    59
```
```    60 lemma even_numeral_nat [simp]: "even (numeral (Num.Bit0 k) :: nat)"
```
```    61   unfolding even_nat_def by simp
```
```    62
```
```    63 lemma odd_numeral_nat [simp]: "odd (numeral (Num.Bit1 k) :: nat)"
```
```    64   unfolding even_nat_def by simp
```
```    65
```
```    66 subsection {* Even and odd are mutually exclusive *}
```
```    67
```
```    68
```
```    69 subsection {* Behavior under integer arithmetic operations *}
```
```    70 declare dvd_def[algebra]
```
```    71 lemma nat_even_iff_2_dvd[algebra]: "even (x::nat) \<longleftrightarrow> 2 dvd x"
```
```    72   by presburger
```
```    73 lemma int_even_iff_2_dvd[algebra]: "even (x::int) \<longleftrightarrow> 2 dvd x"
```
```    74   by presburger
```
```    75
```
```    76 lemma even_times_anything: "even (x::int) ==> even (x * y)"
```
```    77   by algebra
```
```    78
```
```    79 lemma anything_times_even: "even (y::int) ==> even (x * y)" by algebra
```
```    80
```
```    81 lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)"
```
```    82   by (simp add: even_def mod_mult_right_eq)
```
```    83
```
```    84 lemma even_product[simp,presburger]: "even((x::int) * y) = (even x | even y)"
```
```    85   apply (auto simp add: even_times_anything anything_times_even)
```
```    86   apply (rule ccontr)
```
```    87   apply (auto simp add: odd_times_odd)
```
```    88   done
```
```    89
```
```    90 lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"
```
```    91 by presburger
```
```    92
```
```    93 lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"
```
```    94 by presburger
```
```    95
```
```    96 lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"
```
```    97 by presburger
```
```    98
```
```    99 lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger
```
```   100
```
```   101 lemma even_sum[simp,presburger]:
```
```   102   "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
```
```   103 by presburger
```
```   104
```
```   105 lemma even_neg[simp,presburger,algebra]: "even (-(x::int)) = even x"
```
```   106 by presburger
```
```   107
```
```   108 lemma even_difference[simp]:
```
```   109     "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger
```
```   110
```
```   111 lemma even_power[simp,presburger]: "even ((x::int)^n) = (even x & n \<noteq> 0)"
```
```   112 by (induct n) auto
```
```   113
```
```   114 lemma odd_pow: "odd x ==> odd((x::int)^n)" by simp
```
```   115
```
```   116
```
```   117 subsection {* Equivalent definitions *}
```
```   118
```
```   119 lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x"
```
```   120 by presburger
```
```   121
```
```   122 lemma two_times_odd_div_two_plus_one:
```
```   123   "odd (x::int) ==> 2 * (x div 2) + 1 = x"
```
```   124 by presburger
```
```   125
```
```   126 lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger
```
```   127
```
```   128 lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" by presburger
```
```   129
```
```   130 subsection {* even and odd for nats *}
```
```   131
```
```   132 lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
```
```   133 by (simp add: even_nat_def)
```
```   134
```
```   135 lemma even_product_nat[simp,presburger,algebra]:
```
```   136   "even((x::nat) * y) = (even x | even y)"
```
```   137 by (simp add: even_nat_def int_mult)
```
```   138
```
```   139 lemma even_sum_nat[simp,presburger,algebra]:
```
```   140   "even ((x::nat) + y) = ((even x & even y) | (odd x & odd y))"
```
```   141 by presburger
```
```   142
```
```   143 lemma even_difference_nat[simp,presburger,algebra]:
```
```   144   "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
```
```   145 by presburger
```
```   146
```
```   147 lemma even_Suc[simp,presburger,algebra]: "even (Suc x) = odd x"
```
```   148 by presburger
```
```   149
```
```   150 lemma even_power_nat[simp,presburger,algebra]:
```
```   151   "even ((x::nat)^y) = (even x & 0 < y)"
```
```   152 by (simp add: even_nat_def int_power)
```
```   153
```
```   154
```
```   155 subsection {* Equivalent definitions *}
```
```   156
```
```   157 lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
```
```   158 by presburger
```
```   159
```
```   160 lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
```
```   161 by presburger
```
```   162
```
```   163 lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
```
```   164 by presburger
```
```   165
```
```   166 lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
```
```   167 by presburger
```
```   168
```
```   169 lemma even_nat_div_two_times_two: "even (x::nat) ==>
```
```   170     Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger
```
```   171
```
```   172 lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>
```
```   173     Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger
```
```   174
```
```   175 lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
```
```   176 by presburger
```
```   177
```
```   178 lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
```
```   179 by presburger
```
```   180
```
```   181
```
```   182 subsection {* Parity and powers *}
```
```   183
```
```   184 lemma  minus_one_even_odd_power:
```
```   185      "(even x --> (- 1::'a::{comm_ring_1})^x = 1) &
```
```   186       (odd x --> (- 1::'a)^x = - 1)"
```
```   187   apply (induct x)
```
```   188   apply (rule conjI)
```
```   189   apply simp
```
```   190   apply (insert even_zero_nat, blast)
```
```   191   apply simp
```
```   192   done
```
```   193
```
```   194 lemma minus_one_even_power [simp]:
```
```   195     "even x ==> (- 1::'a::{comm_ring_1})^x = 1"
```
```   196   using minus_one_even_odd_power by blast
```
```   197
```
```   198 lemma minus_one_odd_power [simp]:
```
```   199     "odd x ==> (- 1::'a::{comm_ring_1})^x = - 1"
```
```   200   using minus_one_even_odd_power by blast
```
```   201
```
```   202 lemma neg_one_even_odd_power:
```
```   203      "(even x --> (-1::'a::{comm_ring_1})^x = 1) &
```
```   204       (odd x --> (-1::'a)^x = -1)"
```
```   205   apply (induct x)
```
```   206   apply (simp, simp)
```
```   207   done
```
```   208
```
```   209 lemma neg_one_even_power [simp]:
```
```   210     "even x ==> (-1::'a::{comm_ring_1})^x = 1"
```
```   211   using neg_one_even_odd_power by blast
```
```   212
```
```   213 lemma neg_one_odd_power [simp]:
```
```   214     "odd x ==> (-1::'a::{comm_ring_1})^x = -1"
```
```   215   using neg_one_even_odd_power by blast
```
```   216
```
```   217 lemma neg_power_if:
```
```   218      "(-x::'a::{comm_ring_1}) ^ n =
```
```   219       (if even n then (x ^ n) else -(x ^ n))"
```
```   220   apply (induct n)
```
```   221   apply simp_all
```
```   222   done
```
```   223
```
```   224 lemma zero_le_even_power: "even n ==>
```
```   225     0 <= (x::'a::{linordered_ring,monoid_mult}) ^ n"
```
```   226   apply (simp add: even_nat_equiv_def2)
```
```   227   apply (erule exE)
```
```   228   apply (erule ssubst)
```
```   229   apply (subst power_add)
```
```   230   apply (rule zero_le_square)
```
```   231   done
```
```   232
```
```   233 lemma zero_le_odd_power: "odd n ==>
```
```   234     (0 <= (x::'a::{linordered_idom}) ^ n) = (0 <= x)"
```
```   235 apply (auto simp: odd_nat_equiv_def2 power_add zero_le_mult_iff)
```
```   236 apply (metis field_power_not_zero divisors_zero order_antisym_conv zero_le_square)
```
```   237 done
```
```   238
```
```   239 lemma zero_le_power_eq [presburger]: "(0 <= (x::'a::{linordered_idom}) ^ n) =
```
```   240     (even n | (odd n & 0 <= x))"
```
```   241   apply auto
```
```   242   apply (subst zero_le_odd_power [symmetric])
```
```   243   apply assumption+
```
```   244   apply (erule zero_le_even_power)
```
```   245   done
```
```   246
```
```   247 lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{linordered_idom}) ^ n) =
```
```   248     (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
```
```   249
```
```   250   unfolding order_less_le zero_le_power_eq by auto
```
```   251
```
```   252 lemma power_less_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n < 0) =
```
```   253     (odd n & x < 0)"
```
```   254   apply (subst linorder_not_le [symmetric])+
```
```   255   apply (subst zero_le_power_eq)
```
```   256   apply auto
```
```   257   done
```
```   258
```
```   259 lemma power_le_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n <= 0) =
```
```   260     (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"
```
```   261   apply (subst linorder_not_less [symmetric])+
```
```   262   apply (subst zero_less_power_eq)
```
```   263   apply auto
```
```   264   done
```
```   265
```
```   266 lemma power_even_abs: "even n ==>
```
```   267     (abs (x::'a::{linordered_idom}))^n = x^n"
```
```   268   apply (subst power_abs [symmetric])
```
```   269   apply (simp add: zero_le_even_power)
```
```   270   done
```
```   271
```
```   272 lemma power_minus_even [simp]: "even n ==>
```
```   273     (- x)^n = (x^n::'a::{comm_ring_1})"
```
```   274   apply (subst power_minus)
```
```   275   apply simp
```
```   276   done
```
```   277
```
```   278 lemma power_minus_odd [simp]: "odd n ==>
```
```   279     (- x)^n = - (x^n::'a::{comm_ring_1})"
```
```   280   apply (subst power_minus)
```
```   281   apply simp
```
```   282   done
```
```   283
```
```   284 lemma power_mono_even: fixes x y :: "'a :: {linordered_idom}"
```
```   285   assumes "even n" and "\<bar>x\<bar> \<le> \<bar>y\<bar>"
```
```   286   shows "x^n \<le> y^n"
```
```   287 proof -
```
```   288   have "0 \<le> \<bar>x\<bar>" by auto
```
```   289   with `\<bar>x\<bar> \<le> \<bar>y\<bar>`
```
```   290   have "\<bar>x\<bar>^n \<le> \<bar>y\<bar>^n" by (rule power_mono)
```
```   291   thus ?thesis unfolding power_even_abs[OF `even n`] .
```
```   292 qed
```
```   293
```
```   294 lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger
```
```   295
```
```   296 lemma power_mono_odd: fixes x y :: "'a :: {linordered_idom}"
```
```   297   assumes "odd n" and "x \<le> y"
```
```   298   shows "x^n \<le> y^n"
```
```   299 proof (cases "y < 0")
```
```   300   case True with `x \<le> y` have "-y \<le> -x" and "0 \<le> -y" by auto
```
```   301   hence "(-y)^n \<le> (-x)^n" by (rule power_mono)
```
```   302   thus ?thesis unfolding power_minus_odd[OF `odd n`] by auto
```
```   303 next
```
```   304   case False
```
```   305   show ?thesis
```
```   306   proof (cases "x < 0")
```
```   307     case True hence "n \<noteq> 0" and "x \<le> 0" using `odd n`[THEN odd_pos] by auto
```
```   308     hence "x^n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto
```
```   309     moreover
```
```   310     from `\<not> y < 0` have "0 \<le> y" by auto
```
```   311     hence "0 \<le> y^n" by auto
```
```   312     ultimately show ?thesis by auto
```
```   313   next
```
```   314     case False hence "0 \<le> x" by auto
```
```   315     with `x \<le> y` show ?thesis using power_mono by auto
```
```   316   qed
```
```   317 qed
```
```   318
```
```   319
```
```   320 subsection {* More Even/Odd Results *}
```
```   321
```
```   322 lemma even_mult_two_ex: "even(n) = (\<exists>m::nat. n = 2*m)" by presburger
```
```   323 lemma odd_Suc_mult_two_ex: "odd(n) = (\<exists>m. n = Suc (2*m))" by presburger
```
```   324 lemma even_add [simp]: "even(m + n::nat) = (even m = even n)"  by presburger
```
```   325
```
```   326 lemma odd_add [simp]: "odd(m + n::nat) = (odd m \<noteq> odd n)" by presburger
```
```   327
```
```   328 lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" by presburger
```
```   329
```
```   330 lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)"
```
```   331 by presburger
```
```   332
```
```   333 lemma even_num_iff: "0 < n ==> even n = (~ even(n - 1 :: nat))"  by presburger
```
```   334 lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)" by presburger
```
```   335
```
```   336 lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n - 1) div 2)" by presburger
```
```   337
```
```   338 lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n - 1) div 2)"
```
```   339   by presburger
```
```   340
```
```   341 text {* Simplify, when the exponent is a numeral *}
```
```   342
```
```   343 lemmas zero_le_power_eq_numeral [simp] =
```
```   344   zero_le_power_eq [of _ "numeral w"] for w
```
```   345
```
```   346 lemmas zero_less_power_eq_numeral [simp] =
```
```   347   zero_less_power_eq [of _ "numeral w"] for w
```
```   348
```
```   349 lemmas power_le_zero_eq_numeral [simp] =
```
```   350   power_le_zero_eq [of _ "numeral w"] for w
```
```   351
```
```   352 lemmas power_less_zero_eq_numeral [simp] =
```
```   353   power_less_zero_eq [of _ "numeral w"] for w
```
```   354
```
```   355 lemmas zero_less_power_nat_eq_numeral [simp] =
```
```   356   nat_zero_less_power_iff [of _ "numeral w"] for w
```
```   357
```
```   358 lemmas power_eq_0_iff_numeral [simp] =
```
```   359   power_eq_0_iff [of _ "numeral w"] for w
```
```   360
```
```   361 lemmas power_even_abs_numeral [simp] =
```
```   362   power_even_abs [of "numeral w" _] for w
```
```   363
```
```   364
```
```   365 subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *}
```
```   366
```
```   367 lemma zero_le_power_iff[presburger]:
```
```   368   "(0 \<le> a^n) = (0 \<le> (a::'a::{linordered_idom}) | even n)"
```
```   369 proof cases
```
```   370   assume even: "even n"
```
```   371   then obtain k where "n = 2*k"
```
```   372     by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2)
```
```   373   thus ?thesis by (simp add: zero_le_even_power even)
```
```   374 next
```
```   375   assume odd: "odd n"
```
```   376   then obtain k where "n = Suc(2*k)"
```
```   377     by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)
```
```   378   moreover have "a ^ (2 * k) \<le> 0 \<Longrightarrow> a = 0"
```
```   379     by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff)
```
```   380   ultimately show ?thesis
```
```   381     by (auto simp add: zero_le_mult_iff zero_le_even_power)
```
```   382 qed
```
```   383
```
```   384
```
```   385 subsection {* Miscellaneous *}
```
```   386
```
```   387 lemma [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger
```
```   388 lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger
```
```   389 lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2"  by presburger
```
```   390 lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger
```
```   391
```
```   392 lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger
```
```   393 lemma even_nat_plus_one_div_two: "even (x::nat) ==>
```
```   394     (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger
```
```   395
```
```   396 lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>
```
```   397     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger
```
```   398
```
```   399 end
```
```   400
```