src/HOL/Integ/Parity.thy
author wenzelm
Sun Apr 09 18:51:13 2006 +0200 (2006-04-09)
changeset 19380 b808efaa5828
parent 18648 22f96cd085d5
permissions -rw-r--r--
tuned syntax/abbreviations;
     1 (*  Title:      Parity.thy
     2     ID:         $Id$
     3     Author:     Jeremy Avigad
     4 *)
     5 
     6 header {* Even and Odd for ints and nats*}
     7 
     8 theory Parity
     9 imports Divides IntDiv NatSimprocs
    10 begin
    11 
    12 axclass even_odd < type
    13 
    14 consts
    15   even :: "'a::even_odd => bool"
    16 
    17 instance int :: even_odd ..
    18 instance nat :: even_odd ..
    19 
    20 defs (overloaded)
    21   even_def: "even (x::int) == x mod 2 = 0"
    22   even_nat_def: "even (x::nat) == even (int x)"
    23 
    24 abbreviation
    25   odd :: "'a::even_odd => bool"
    26   "odd x == \<not> even x"
    27 
    28 
    29 subsection {* Even and odd are mutually exclusive *}
    30 
    31 lemma int_pos_lt_two_imp_zero_or_one: 
    32     "0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1"
    33   by auto
    34 
    35 lemma neq_one_mod_two [simp]: "((x::int) mod 2 ~= 0) = (x mod 2 = 1)"
    36   apply (subgoal_tac "x mod 2 = 0 | x mod 2 = 1", force)
    37   apply (rule int_pos_lt_two_imp_zero_or_one, auto)
    38   done
    39 
    40 subsection {* Behavior under integer arithmetic operations *}
    41 
    42 lemma even_times_anything: "even (x::int) ==> even (x * y)"
    43   by (simp add: even_def zmod_zmult1_eq')
    44 
    45 lemma anything_times_even: "even (y::int) ==> even (x * y)"
    46   by (simp add: even_def zmod_zmult1_eq)
    47 
    48 lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)"
    49   by (simp add: even_def zmod_zmult1_eq)
    50 
    51 lemma even_product: "even((x::int) * y) = (even x | even y)"
    52   apply (auto simp add: even_times_anything anything_times_even) 
    53   apply (rule ccontr)
    54   apply (auto simp add: odd_times_odd)
    55   done
    56 
    57 lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"
    58   by (simp add: even_def zmod_zadd1_eq)
    59 
    60 lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"
    61   by (simp add: even_def zmod_zadd1_eq)
    62 
    63 lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"
    64   by (simp add: even_def zmod_zadd1_eq)
    65 
    66 lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)"
    67   by (simp add: even_def zmod_zadd1_eq)
    68 
    69 lemma even_sum: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
    70   apply (auto intro: even_plus_even odd_plus_odd)
    71   apply (rule ccontr, simp add: even_plus_odd)
    72   apply (rule ccontr, simp add: odd_plus_even)
    73   done
    74 
    75 lemma even_neg: "even (-(x::int)) = even x"
    76   by (auto simp add: even_def zmod_zminus1_eq_if)
    77 
    78 lemma even_difference: 
    79   "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))"
    80   by (simp only: diff_minus even_sum even_neg)
    81 
    82 lemma even_pow_gt_zero [rule_format]: 
    83     "even (x::int) ==> 0 < n --> even (x^n)"
    84   apply (induct n)
    85   apply (auto simp add: even_product)
    86   done
    87 
    88 lemma odd_pow: "odd x ==> odd((x::int)^n)"
    89   apply (induct n)
    90   apply (simp add: even_def)
    91   apply (simp add: even_product)
    92   done
    93 
    94 lemma even_power: "even ((x::int)^n) = (even x & 0 < n)"
    95   apply (auto simp add: even_pow_gt_zero) 
    96   apply (erule contrapos_pp, erule odd_pow)
    97   apply (erule contrapos_pp, simp add: even_def)
    98   done
    99 
   100 lemma even_zero: "even (0::int)"
   101   by (simp add: even_def)
   102 
   103 lemma odd_one: "odd (1::int)"
   104   by (simp add: even_def)
   105 
   106 lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero 
   107   odd_one even_product even_sum even_neg even_difference even_power
   108 
   109 
   110 subsection {* Equivalent definitions *}
   111 
   112 lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x" 
   113   by (auto simp add: even_def)
   114 
   115 lemma two_times_odd_div_two_plus_one: "odd (x::int) ==> 
   116     2 * (x div 2) + 1 = x"
   117   apply (insert zmod_zdiv_equality [of x 2, THEN sym])
   118   by (simp add: even_def)
   119 
   120 lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)"
   121   apply auto
   122   apply (rule exI)
   123   by (erule two_times_even_div_two [THEN sym])
   124 
   125 lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)"
   126   apply auto
   127   apply (rule exI)
   128   by (erule two_times_odd_div_two_plus_one [THEN sym])
   129 
   130 
   131 subsection {* even and odd for nats *}
   132 
   133 lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
   134   by (simp add: even_nat_def)
   135 
   136 lemma even_nat_product: "even((x::nat) * y) = (even x | even y)"
   137   by (simp add: even_nat_def int_mult)
   138 
   139 lemma even_nat_sum: "even ((x::nat) + y) = 
   140     ((even x & even y) | (odd x & odd y))"
   141   by (unfold even_nat_def, simp)
   142 
   143 lemma even_nat_difference: 
   144     "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
   145   apply (auto simp add: even_nat_def zdiff_int [THEN sym])
   146   apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym])
   147   apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym])
   148   done
   149 
   150 lemma even_nat_Suc: "even (Suc x) = odd x"
   151   by (simp add: even_nat_def)
   152 
   153 lemma even_nat_power: "even ((x::nat)^y) = (even x & 0 < y)"
   154   by (simp add: even_nat_def int_power)
   155 
   156 lemma even_nat_zero: "even (0::nat)"
   157   by (simp add: even_nat_def)
   158 
   159 lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard] 
   160   even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power
   161 
   162 
   163 subsection {* Equivalent definitions *}
   164 
   165 lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==> 
   166     x = 0 | x = Suc 0"
   167   by auto
   168 
   169 lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
   170   apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
   171   apply (drule subst, assumption)
   172   apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0")
   173   apply force
   174   apply (subgoal_tac "0 < Suc (Suc 0)")
   175   apply (frule mod_less_divisor [of "Suc (Suc 0)" x])
   176   apply (erule nat_lt_two_imp_zero_or_one, auto)
   177   done
   178 
   179 lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
   180   apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
   181   apply (drule subst, assumption)
   182   apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0")
   183   apply force 
   184   apply (subgoal_tac "0 < Suc (Suc 0)")
   185   apply (frule mod_less_divisor [of "Suc (Suc 0)" x])
   186   apply (erule nat_lt_two_imp_zero_or_one, auto)
   187   done
   188 
   189 lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)" 
   190   apply (rule iffI)
   191   apply (erule even_nat_mod_two_eq_zero)
   192   apply (insert odd_nat_mod_two_eq_one [of x], auto)
   193   done
   194 
   195 lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
   196   apply (auto simp add: even_nat_equiv_def)
   197   apply (subgoal_tac "x mod (Suc (Suc 0)) < Suc (Suc 0)")
   198   apply (frule nat_lt_two_imp_zero_or_one, auto)
   199   done
   200 
   201 lemma even_nat_div_two_times_two: "even (x::nat) ==> 
   202     Suc (Suc 0) * (x div Suc (Suc 0)) = x"
   203   apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
   204   apply (drule even_nat_mod_two_eq_zero, simp)
   205   done
   206 
   207 lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==> 
   208     Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x"  
   209   apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
   210   apply (drule odd_nat_mod_two_eq_one, simp)
   211   done
   212 
   213 lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
   214   apply (rule iffI, rule exI)
   215   apply (erule even_nat_div_two_times_two [THEN sym], auto)
   216   done
   217 
   218 lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
   219   apply (rule iffI, rule exI)
   220   apply (erule odd_nat_div_two_times_two_plus_one [THEN sym], auto)
   221   done
   222 
   223 subsection {* Parity and powers *}
   224 
   225 lemma minus_one_even_odd_power:
   226      "(even x --> (- 1::'a::{comm_ring_1,recpower})^x = 1) & 
   227       (odd x --> (- 1::'a)^x = - 1)"
   228   apply (induct x)
   229   apply (rule conjI)
   230   apply simp
   231   apply (insert even_nat_zero, blast)
   232   apply (simp add: power_Suc)
   233 done
   234 
   235 lemma minus_one_even_power [simp]:
   236      "even x ==> (- 1::'a::{comm_ring_1,recpower})^x = 1"
   237   by (rule minus_one_even_odd_power [THEN conjunct1, THEN mp], assumption)
   238 
   239 lemma minus_one_odd_power [simp]:
   240      "odd x ==> (- 1::'a::{comm_ring_1,recpower})^x = - 1"
   241   by (rule minus_one_even_odd_power [THEN conjunct2, THEN mp], assumption)
   242 
   243 lemma neg_one_even_odd_power:
   244      "(even x --> (-1::'a::{number_ring,recpower})^x = 1) & 
   245       (odd x --> (-1::'a)^x = -1)"
   246   apply (induct x)
   247   apply (simp, simp add: power_Suc)
   248   done
   249 
   250 lemma neg_one_even_power [simp]:
   251      "even x ==> (-1::'a::{number_ring,recpower})^x = 1"
   252   by (rule neg_one_even_odd_power [THEN conjunct1, THEN mp], assumption)
   253 
   254 lemma neg_one_odd_power [simp]:
   255      "odd x ==> (-1::'a::{number_ring,recpower})^x = -1"
   256   by (rule neg_one_even_odd_power [THEN conjunct2, THEN mp], assumption)
   257 
   258 lemma neg_power_if:
   259      "(-x::'a::{comm_ring_1,recpower}) ^ n = 
   260       (if even n then (x ^ n) else -(x ^ n))"
   261   by (induct n, simp_all split: split_if_asm add: power_Suc) 
   262 
   263 lemma zero_le_even_power: "even n ==> 
   264     0 <= (x::'a::{recpower,ordered_ring_strict}) ^ n"
   265   apply (simp add: even_nat_equiv_def2)
   266   apply (erule exE)
   267   apply (erule ssubst)
   268   apply (subst power_add)
   269   apply (rule zero_le_square)
   270   done
   271 
   272 lemma zero_le_odd_power: "odd n ==> 
   273     (0 <= (x::'a::{recpower,ordered_idom}) ^ n) = (0 <= x)"
   274   apply (simp add: odd_nat_equiv_def2)
   275   apply (erule exE)
   276   apply (erule ssubst)
   277   apply (subst power_Suc)
   278   apply (subst power_add)
   279   apply (subst zero_le_mult_iff)
   280   apply auto
   281   apply (subgoal_tac "x = 0 & 0 < y")
   282   apply (erule conjE, assumption)
   283   apply (subst power_eq_0_iff [THEN sym])
   284   apply (subgoal_tac "0 <= x^y * x^y")
   285   apply simp
   286   apply (rule zero_le_square)+
   287 done
   288 
   289 lemma zero_le_power_eq: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) = 
   290     (even n | (odd n & 0 <= x))"
   291   apply auto
   292   apply (subst zero_le_odd_power [THEN sym])
   293   apply assumption+
   294   apply (erule zero_le_even_power)
   295   apply (subst zero_le_odd_power) 
   296   apply assumption+
   297 done
   298 
   299 lemma zero_less_power_eq: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) = 
   300     (n = 0 | (even n & x ~= 0) | (odd n & 0 < x))"
   301   apply (rule iffI)
   302   apply clarsimp
   303   apply (rule conjI)
   304   apply clarsimp
   305   apply (rule ccontr)
   306   apply (subgoal_tac "~ (0 <= x^n)")
   307   apply simp
   308   apply (subst zero_le_odd_power)
   309   apply assumption 
   310   apply simp
   311   apply (rule notI)
   312   apply (simp add: power_0_left)
   313   apply (rule notI)
   314   apply (simp add: power_0_left)
   315   apply auto
   316   apply (subgoal_tac "0 <= x^n")
   317   apply (frule order_le_imp_less_or_eq)
   318   apply simp
   319   apply (erule zero_le_even_power)
   320   apply (subgoal_tac "0 <= x^n")
   321   apply (frule order_le_imp_less_or_eq)
   322   apply auto
   323   apply (subst zero_le_odd_power)
   324   apply assumption
   325   apply (erule order_less_imp_le)
   326 done
   327 
   328 lemma power_less_zero_eq: "((x::'a::{recpower,ordered_idom}) ^ n < 0) =
   329     (odd n & x < 0)" 
   330   apply (subst linorder_not_le [THEN sym])+
   331   apply (subst zero_le_power_eq)
   332   apply auto
   333 done
   334 
   335 lemma power_le_zero_eq: "((x::'a::{recpower,ordered_idom}) ^ n <= 0) =
   336     (n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))"
   337   apply (subst linorder_not_less [THEN sym])+
   338   apply (subst zero_less_power_eq)
   339   apply auto
   340 done
   341 
   342 lemma power_even_abs: "even n ==> 
   343     (abs (x::'a::{recpower,ordered_idom}))^n = x^n"
   344   apply (subst power_abs [THEN sym])
   345   apply (simp add: zero_le_even_power)
   346 done
   347 
   348 lemma zero_less_power_nat_eq: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)"
   349   by (induct n, auto)
   350 
   351 lemma power_minus_even [simp]: "even n ==> 
   352     (- x)^n = (x^n::'a::{recpower,comm_ring_1})"
   353   apply (subst power_minus)
   354   apply simp
   355 done
   356 
   357 lemma power_minus_odd [simp]: "odd n ==> 
   358     (- x)^n = - (x^n::'a::{recpower,comm_ring_1})"
   359   apply (subst power_minus)
   360   apply simp
   361 done
   362 
   363 (* Simplify, when the exponent is a numeral *)
   364 
   365 lemmas power_0_left_number_of = power_0_left [of "number_of w", standard]
   366 declare power_0_left_number_of [simp]
   367 
   368 lemmas zero_le_power_eq_number_of =
   369     zero_le_power_eq [of _ "number_of w", standard]
   370 declare zero_le_power_eq_number_of [simp]
   371 
   372 lemmas zero_less_power_eq_number_of =
   373     zero_less_power_eq [of _ "number_of w", standard]
   374 declare zero_less_power_eq_number_of [simp]
   375 
   376 lemmas power_le_zero_eq_number_of =
   377     power_le_zero_eq [of _ "number_of w", standard]
   378 declare power_le_zero_eq_number_of [simp]
   379 
   380 lemmas power_less_zero_eq_number_of =
   381     power_less_zero_eq [of _ "number_of w", standard]
   382 declare power_less_zero_eq_number_of [simp]
   383 
   384 lemmas zero_less_power_nat_eq_number_of =
   385     zero_less_power_nat_eq [of _ "number_of w", standard]
   386 declare zero_less_power_nat_eq_number_of [simp]
   387 
   388 lemmas power_eq_0_iff_number_of = power_eq_0_iff [of _ "number_of w", standard]
   389 declare power_eq_0_iff_number_of [simp]
   390 
   391 lemmas power_even_abs_number_of = power_even_abs [of "number_of w" _, standard]
   392 declare power_even_abs_number_of [simp]
   393 
   394 
   395 subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *}
   396 
   397 lemma even_power_le_0_imp_0:
   398      "a ^ (2*k) \<le> (0::'a::{ordered_idom,recpower}) ==> a=0"
   399 apply (induct k) 
   400 apply (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc)  
   401 done
   402 
   403 lemma zero_le_power_iff:
   404      "(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even n)"
   405       (is "?P n")
   406 proof cases
   407   assume even: "even n"
   408   then obtain k where "n = 2*k"
   409     by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2)
   410   thus ?thesis by (simp add: zero_le_even_power even) 
   411 next
   412   assume odd: "odd n"
   413   then obtain k where "n = Suc(2*k)"
   414     by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)
   415   thus ?thesis
   416     by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power 
   417              dest!: even_power_le_0_imp_0) 
   418 qed 
   419 
   420 subsection {* Miscellaneous *}
   421 
   422 lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2"
   423   apply (subst zdiv_zadd1_eq)
   424   apply (simp add: even_def)
   425   done
   426 
   427 lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1"
   428   apply (subst zdiv_zadd1_eq)
   429   apply (simp add: even_def)
   430   done
   431 
   432 lemma div_Suc: "Suc a div c = a div c + Suc 0 div c + 
   433     (a mod c + Suc 0 mod c) div c"
   434   apply (subgoal_tac "Suc a = a + Suc 0")
   435   apply (erule ssubst)
   436   apply (rule div_add1_eq, simp)
   437   done
   438 
   439 lemma even_nat_plus_one_div_two: "even (x::nat) ==> 
   440    (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)"
   441   apply (subst div_Suc)
   442   apply (simp add: even_nat_equiv_def)
   443   done
   444 
   445 lemma odd_nat_plus_one_div_two: "odd (x::nat) ==> 
   446     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))"
   447   apply (subst div_Suc)
   448   apply (simp add: odd_nat_equiv_def)
   449   done
   450 
   451 end