src/HOL/Parity.thy
 author haftmann Thu Oct 09 22:43:48 2014 +0200 (2014-10-09) changeset 58645 94bef115c08f parent 54489 03ff4d1e6784 child 58678 398e05aa84d4 permissions -rw-r--r--
more foundational definition for predicate even
```     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 = semiring_div_parity
```
```    13 begin
```
```    14
```
```    15 definition even :: "'a \<Rightarrow> bool"
```
```    16 where
```
```    17   [algebra]: "even a \<longleftrightarrow> 2 dvd a"
```
```    18
```
```    19 lemmas even_iff_2_dvd = even_def
```
```    20
```
```    21 lemma even_iff_mod_2_eq_zero [presburger]:
```
```    22   "even a \<longleftrightarrow> a mod 2 = 0"
```
```    23   by (simp add: even_def dvd_eq_mod_eq_0)
```
```    24
```
```    25 lemma even_zero [simp]:
```
```    26   "even 0"
```
```    27   by (simp add: even_iff_mod_2_eq_zero)
```
```    28
```
```    29 lemma even_times_anything:
```
```    30   "even a \<Longrightarrow> even (a * b)"
```
```    31   by (simp add: even_iff_2_dvd)
```
```    32
```
```    33 lemma anything_times_even:
```
```    34   "even a \<Longrightarrow> even (b * a)"
```
```    35   by (simp add: even_iff_2_dvd)
```
```    36
```
```    37 abbreviation odd :: "'a \<Rightarrow> bool"
```
```    38 where
```
```    39   "odd a \<equiv> \<not> even a"
```
```    40
```
```    41 lemma odd_times_odd:
```
```    42   "odd a \<Longrightarrow> odd b \<Longrightarrow> odd (a * b)"
```
```    43   by (auto simp add: even_iff_mod_2_eq_zero mod_mult_left_eq)
```
```    44
```
```    45 lemma even_product [simp, presburger]:
```
```    46   "even (a * b) \<longleftrightarrow> even a \<or> even b"
```
```    47   apply (auto simp add: even_times_anything anything_times_even)
```
```    48   apply (rule ccontr)
```
```    49   apply (auto simp add: odd_times_odd)
```
```    50   done
```
```    51
```
```    52 end
```
```    53
```
```    54 instance nat and int  :: even_odd ..
```
```    55
```
```    56 lemma even_nat_def [presburger]:
```
```    57   "even x \<longleftrightarrow> even (int x)"
```
```    58   by (auto simp add: even_iff_mod_2_eq_zero int_eq_iff int_mult nat_mult_distrib)
```
```    59
```
```    60 lemma transfer_int_nat_relations:
```
```    61   "even (int x) \<longleftrightarrow> even x"
```
```    62   by (simp add: even_nat_def)
```
```    63
```
```    64 declare transfer_morphism_int_nat[transfer add return:
```
```    65   transfer_int_nat_relations
```
```    66 ]
```
```    67
```
```    68 lemma odd_one_int [simp]:
```
```    69   "odd (1::int)"
```
```    70   by presburger
```
```    71
```
```    72 lemma odd_1_nat [simp]:
```
```    73   "odd (1::nat)"
```
```    74   by presburger
```
```    75
```
```    76 lemma even_numeral_int [simp]: "even (numeral (Num.Bit0 k) :: int)"
```
```    77   unfolding even_iff_mod_2_eq_zero by simp
```
```    78
```
```    79 lemma odd_numeral_int [simp]: "odd (numeral (Num.Bit1 k) :: int)"
```
```    80   unfolding even_iff_mod_2_eq_zero by simp
```
```    81
```
```    82 (* TODO: proper simp rules for Num.Bit0, Num.Bit1 *)
```
```    83 declare even_iff_mod_2_eq_zero [of "- numeral v", simp] for v
```
```    84
```
```    85 lemma even_numeral_nat [simp]: "even (numeral (Num.Bit0 k) :: nat)"
```
```    86   unfolding even_nat_def by simp
```
```    87
```
```    88 lemma odd_numeral_nat [simp]: "odd (numeral (Num.Bit1 k) :: nat)"
```
```    89   unfolding even_nat_def by simp
```
```    90
```
```    91 subsection {* Even and odd are mutually exclusive *}
```
```    92
```
```    93
```
```    94 subsection {* Behavior under integer arithmetic operations *}
```
```    95
```
```    96 lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"
```
```    97 by presburger
```
```    98
```
```    99 lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"
```
```   100 by presburger
```
```   101
```
```   102 lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"
```
```   103 by presburger
```
```   104
```
```   105 lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger
```
```   106
```
```   107 lemma even_sum[simp,presburger]:
```
```   108   "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
```
```   109 by presburger
```
```   110
```
```   111 lemma even_neg[simp,presburger,algebra]: "even (-(x::int)) = even x"
```
```   112 by presburger
```
```   113
```
```   114 lemma even_difference[simp]:
```
```   115     "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger
```
```   116
```
```   117 lemma even_power[simp,presburger]: "even ((x::int)^n) = (even x & n \<noteq> 0)"
```
```   118 by (induct n) auto
```
```   119
```
```   120 lemma odd_pow: "odd x ==> odd((x::int)^n)" by simp
```
```   121
```
```   122
```
```   123 subsection {* Equivalent definitions *}
```
```   124
```
```   125 lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x"
```
```   126 by presburger
```
```   127
```
```   128 lemma two_times_odd_div_two_plus_one:
```
```   129   "odd (x::int) ==> 2 * (x div 2) + 1 = x"
```
```   130 by presburger
```
```   131
```
```   132 lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger
```
```   133
```
```   134 lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" by presburger
```
```   135
```
```   136 subsection {* even and odd for nats *}
```
```   137
```
```   138 lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
```
```   139 by (simp add: even_nat_def)
```
```   140
```
```   141 lemma even_product_nat[simp,presburger,algebra]:
```
```   142   "even((x::nat) * y) = (even x | even y)"
```
```   143 by (simp add: even_nat_def int_mult)
```
```   144
```
```   145 lemma even_sum_nat[simp,presburger,algebra]:
```
```   146   "even ((x::nat) + y) = ((even x & even y) | (odd x & odd y))"
```
```   147 by presburger
```
```   148
```
```   149 lemma even_difference_nat[simp,presburger,algebra]:
```
```   150   "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
```
```   151 by presburger
```
```   152
```
```   153 lemma even_Suc[simp,presburger,algebra]: "even (Suc x) = odd x"
```
```   154 by presburger
```
```   155
```
```   156 lemma even_power_nat[simp,presburger,algebra]:
```
```   157   "even ((x::nat)^y) = (even x & 0 < y)"
```
```   158 by (simp add: even_nat_def int_power)
```
```   159
```
```   160
```
```   161 subsection {* Equivalent definitions *}
```
```   162
```
```   163 lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
```
```   164 by presburger
```
```   165
```
```   166 lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
```
```   167 by presburger
```
```   168
```
```   169 lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
```
```   170 by presburger
```
```   171
```
```   172 lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
```
```   173 by presburger
```
```   174
```
```   175 lemma even_nat_div_two_times_two: "even (x::nat) ==>
```
```   176     Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger
```
```   177
```
```   178 lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>
```
```   179     Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger
```
```   180
```
```   181 lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
```
```   182 by presburger
```
```   183
```
```   184 lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
```
```   185 by presburger
```
```   186
```
```   187
```
```   188 subsection {* Parity and powers *}
```
```   189
```
```   190 lemma (in comm_ring_1) neg_power_if:
```
```   191   "(- a) ^ n = (if even n then (a ^ n) else - (a ^ n))"
```
```   192   by (induct n) simp_all
```
```   193
```
```   194 lemma (in comm_ring_1)
```
```   195   shows neg_one_even_power [simp]: "even n \<Longrightarrow> (- 1) ^ n = 1"
```
```   196   and neg_one_odd_power [simp]: "odd n \<Longrightarrow> (- 1) ^ n = - 1"
```
```   197   by (simp_all add: neg_power_if)
```
```   198
```
```   199 lemma zero_le_even_power: "even n ==>
```
```   200     0 <= (x::'a::{linordered_ring,monoid_mult}) ^ n"
```
```   201   apply (simp add: even_nat_equiv_def2)
```
```   202   apply (erule exE)
```
```   203   apply (erule ssubst)
```
```   204   apply (subst power_add)
```
```   205   apply (rule zero_le_square)
```
```   206   done
```
```   207
```
```   208 lemma zero_le_odd_power: "odd n ==>
```
```   209     (0 <= (x::'a::{linordered_idom}) ^ n) = (0 <= x)"
```
```   210 apply (auto simp: odd_nat_equiv_def2 power_add zero_le_mult_iff)
```
```   211 apply (metis field_power_not_zero divisors_zero order_antisym_conv zero_le_square)
```
```   212 done
```
```   213
```
```   214 lemma zero_le_power_eq [presburger]: "(0 <= (x::'a::{linordered_idom}) ^ n) =
```
```   215     (even n | (odd n & 0 <= x))"
```
```   216   apply auto
```
```   217   apply (subst zero_le_odd_power [symmetric])
```
```   218   apply assumption+
```
```   219   apply (erule zero_le_even_power)
```
```   220   done
```
```   221
```
```   222 lemma zero_less_power_eq[presburger]: "(0 < (x::'a::{linordered_idom}) ^ n) =
```
```   223     (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
```
```   224
```
```   225   unfolding order_less_le zero_le_power_eq by auto
```
```   226
```
```   227 lemma power_less_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n < 0) =
```
```   228     (odd n & x < 0)"
```
```   229   apply (subst linorder_not_le [symmetric])+
```
```   230   apply (subst zero_le_power_eq)
```
```   231   apply auto
```
```   232   done
```
```   233
```
```   234 lemma power_le_zero_eq[presburger]: "((x::'a::{linordered_idom}) ^ n <= 0) =
```
```   235     (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"
```
```   236   apply (subst linorder_not_less [symmetric])+
```
```   237   apply (subst zero_less_power_eq)
```
```   238   apply auto
```
```   239   done
```
```   240
```
```   241 lemma power_even_abs: "even n ==>
```
```   242     (abs (x::'a::{linordered_idom}))^n = x^n"
```
```   243   apply (subst power_abs [symmetric])
```
```   244   apply (simp add: zero_le_even_power)
```
```   245   done
```
```   246
```
```   247 lemma power_minus_even [simp]: "even n ==>
```
```   248     (- x)^n = (x^n::'a::{comm_ring_1})"
```
```   249   apply (subst power_minus)
```
```   250   apply simp
```
```   251   done
```
```   252
```
```   253 lemma power_minus_odd [simp]: "odd n ==>
```
```   254     (- x)^n = - (x^n::'a::{comm_ring_1})"
```
```   255   apply (subst power_minus)
```
```   256   apply simp
```
```   257   done
```
```   258
```
```   259 lemma power_mono_even: fixes x y :: "'a :: {linordered_idom}"
```
```   260   assumes "even n" and "\<bar>x\<bar> \<le> \<bar>y\<bar>"
```
```   261   shows "x^n \<le> y^n"
```
```   262 proof -
```
```   263   have "0 \<le> \<bar>x\<bar>" by auto
```
```   264   with `\<bar>x\<bar> \<le> \<bar>y\<bar>`
```
```   265   have "\<bar>x\<bar>^n \<le> \<bar>y\<bar>^n" by (rule power_mono)
```
```   266   thus ?thesis unfolding power_even_abs[OF `even n`] .
```
```   267 qed
```
```   268
```
```   269 lemma odd_pos: "odd (n::nat) \<Longrightarrow> 0 < n" by presburger
```
```   270
```
```   271 lemma power_mono_odd: fixes x y :: "'a :: {linordered_idom}"
```
```   272   assumes "odd n" and "x \<le> y"
```
```   273   shows "x^n \<le> y^n"
```
```   274 proof (cases "y < 0")
```
```   275   case True with `x \<le> y` have "-y \<le> -x" and "0 \<le> -y" by auto
```
```   276   hence "(-y)^n \<le> (-x)^n" by (rule power_mono)
```
```   277   thus ?thesis unfolding power_minus_odd[OF `odd n`] by auto
```
```   278 next
```
```   279   case False
```
```   280   show ?thesis
```
```   281   proof (cases "x < 0")
```
```   282     case True hence "n \<noteq> 0" and "x \<le> 0" using `odd n`[THEN odd_pos] by auto
```
```   283     hence "x^n \<le> 0" unfolding power_le_zero_eq using `odd n` by auto
```
```   284     moreover
```
```   285     from `\<not> y < 0` have "0 \<le> y" by auto
```
```   286     hence "0 \<le> y^n" by auto
```
```   287     ultimately show ?thesis by auto
```
```   288   next
```
```   289     case False hence "0 \<le> x" by auto
```
```   290     with `x \<le> y` show ?thesis using power_mono by auto
```
```   291   qed
```
```   292 qed
```
```   293
```
```   294
```
```   295 subsection {* More Even/Odd Results *}
```
```   296
```
```   297 lemma even_mult_two_ex: "even(n) = (\<exists>m::nat. n = 2*m)" by presburger
```
```   298 lemma odd_Suc_mult_two_ex: "odd(n) = (\<exists>m. n = Suc (2*m))" by presburger
```
```   299 lemma even_add [simp]: "even(m + n::nat) = (even m = even n)"  by presburger
```
```   300
```
```   301 lemma odd_add [simp]: "odd(m + n::nat) = (odd m \<noteq> odd n)" by presburger
```
```   302
```
```   303 lemma lemma_even_div2 [simp]: "even (n::nat) ==> (n + 1) div 2 = n div 2" by presburger
```
```   304
```
```   305 lemma lemma_not_even_div2 [simp]: "~even n ==> (n + 1) div 2 = Suc (n div 2)"
```
```   306 by presburger
```
```   307
```
```   308 lemma even_num_iff: "0 < n ==> even n = (~ even(n - 1 :: nat))"  by presburger
```
```   309 lemma even_even_mod_4_iff: "even (n::nat) = even (n mod 4)" by presburger
```
```   310
```
```   311 lemma lemma_odd_mod_4_div_2: "n mod 4 = (3::nat) ==> odd((n - 1) div 2)" by presburger
```
```   312
```
```   313 lemma lemma_even_mod_4_div_2: "n mod 4 = (1::nat) ==> even ((n - 1) div 2)"
```
```   314   by presburger
```
```   315
```
```   316 text {* Simplify, when the exponent is a numeral *}
```
```   317
```
```   318 lemmas zero_le_power_eq_numeral [simp] =
```
```   319   zero_le_power_eq [of _ "numeral w"] for w
```
```   320
```
```   321 lemmas zero_less_power_eq_numeral [simp] =
```
```   322   zero_less_power_eq [of _ "numeral w"] for w
```
```   323
```
```   324 lemmas power_le_zero_eq_numeral [simp] =
```
```   325   power_le_zero_eq [of _ "numeral w"] for w
```
```   326
```
```   327 lemmas power_less_zero_eq_numeral [simp] =
```
```   328   power_less_zero_eq [of _ "numeral w"] for w
```
```   329
```
```   330 lemmas zero_less_power_nat_eq_numeral [simp] =
```
```   331   nat_zero_less_power_iff [of _ "numeral w"] for w
```
```   332
```
```   333 lemmas power_eq_0_iff_numeral [simp] =
```
```   334   power_eq_0_iff [of _ "numeral w"] for w
```
```   335
```
```   336 lemmas power_even_abs_numeral [simp] =
```
```   337   power_even_abs [of "numeral w" _] for w
```
```   338
```
```   339
```
```   340 subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *}
```
```   341
```
```   342 lemma zero_le_power_iff[presburger]:
```
```   343   "(0 \<le> a^n) = (0 \<le> (a::'a::{linordered_idom}) | even n)"
```
```   344 proof cases
```
```   345   assume even: "even n"
```
```   346   then obtain k where "n = 2*k"
```
```   347     by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2)
```
```   348   thus ?thesis by (simp add: zero_le_even_power even)
```
```   349 next
```
```   350   assume odd: "odd n"
```
```   351   then obtain k where "n = Suc(2*k)"
```
```   352     by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)
```
```   353   moreover have "a ^ (2 * k) \<le> 0 \<Longrightarrow> a = 0"
```
```   354     by (induct k) (auto simp add: zero_le_mult_iff mult_le_0_iff)
```
```   355   ultimately show ?thesis
```
```   356     by (auto simp add: zero_le_mult_iff zero_le_even_power)
```
```   357 qed
```
```   358
```
```   359
```
```   360 subsection {* Miscellaneous *}
```
```   361
```
```   362 lemma [presburger]:"(x + 1) div 2 = x div 2 \<longleftrightarrow> even (x::int)" by presburger
```
```   363 lemma [presburger]: "(x + 1) div 2 = x div 2 + 1 \<longleftrightarrow> odd (x::int)" by presburger
```
```   364 lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2"  by presburger
```
```   365 lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" by presburger
```
```   366
```
```   367 lemma [presburger]: "(Suc x) div Suc (Suc 0) = x div Suc (Suc 0) \<longleftrightarrow> even x" by presburger
```
```   368 lemma even_nat_plus_one_div_two: "even (x::nat) ==>
```
```   369     (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" by presburger
```
```   370
```
```   371 lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>
```
```   372     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" by presburger
```
```   373
```
```   374 end
```
```   375
```