src/HOL/Integ/Parity.thy
 author paulson Tue Jun 28 15:27:45 2005 +0200 (2005-06-28) changeset 16587 b34c8aa657a5 parent 16413 47ffc49c7d7b child 16775 c1b87ef4a1c3 permissions -rw-r--r--
Constant "If" is now local
```     1 (*  Title:      Parity.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Jeremy Avigad
```
```     4 *)
```
```     5
```
```     6 header {* Parity: 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 instance int :: even_odd ..
```
```    15 instance nat :: even_odd ..
```
```    16
```
```    17 consts
```
```    18   even :: "'a::even_odd => bool"
```
```    19
```
```    20 syntax
```
```    21   odd :: "'a::even_odd => bool"
```
```    22
```
```    23 translations
```
```    24   "odd x" == "~even x"
```
```    25
```
```    26 defs (overloaded)
```
```    27   even_def: "even (x::int) == x mod 2 = 0"
```
```    28   even_nat_def: "even (x::nat) == even (int x)"
```
```    29
```
```    30
```
```    31 subsection {* Even and odd are mutually exclusive *}
```
```    32
```
```    33 lemma int_pos_lt_two_imp_zero_or_one:
```
```    34     "0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1"
```
```    35   by auto
```
```    36
```
```    37 lemma neq_one_mod_two [simp]: "((x::int) mod 2 ~= 0) = (x mod 2 = 1)"
```
```    38   apply (subgoal_tac "x mod 2 = 0 | x mod 2 = 1", force)
```
```    39   apply (rule int_pos_lt_two_imp_zero_or_one, auto)
```
```    40   done
```
```    41
```
```    42 subsection {* Behavior under integer arithmetic operations *}
```
```    43
```
```    44 lemma even_times_anything: "even (x::int) ==> even (x * y)"
```
```    45   by (simp add: even_def zmod_zmult1_eq')
```
```    46
```
```    47 lemma anything_times_even: "even (y::int) ==> even (x * y)"
```
```    48   by (simp add: even_def zmod_zmult1_eq)
```
```    49
```
```    50 lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)"
```
```    51   by (simp add: even_def zmod_zmult1_eq)
```
```    52
```
```    53 lemma even_product: "even((x::int) * y) = (even x | even y)"
```
```    54   apply (auto simp add: even_times_anything anything_times_even)
```
```    55   apply (rule ccontr)
```
```    56   apply (auto simp add: odd_times_odd)
```
```    57   done
```
```    58
```
```    59 lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"
```
```    60   by (simp add: even_def zmod_zadd1_eq)
```
```    61
```
```    62 lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"
```
```    63   by (simp add: even_def zmod_zadd1_eq)
```
```    64
```
```    65 lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"
```
```    66   by (simp add: even_def zmod_zadd1_eq)
```
```    67
```
```    68 lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)"
```
```    69   by (simp add: even_def zmod_zadd1_eq)
```
```    70
```
```    71 lemma even_sum: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
```
```    72   apply (auto intro: even_plus_even odd_plus_odd)
```
```    73   apply (rule ccontr, simp add: even_plus_odd)
```
```    74   apply (rule ccontr, simp add: odd_plus_even)
```
```    75   done
```
```    76
```
```    77 lemma even_neg: "even (-(x::int)) = even x"
```
```    78   by (auto simp add: even_def zmod_zminus1_eq_if)
```
```    79
```
```    80 lemma even_difference:
```
```    81   "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))"
```
```    82   by (simp only: diff_minus even_sum even_neg)
```
```    83
```
```    84 lemma even_pow_gt_zero [rule_format]:
```
```    85     "even (x::int) ==> 0 < n --> even (x^n)"
```
```    86   apply (induct n)
```
```    87   apply (auto simp add: even_product)
```
```    88   done
```
```    89
```
```    90 lemma odd_pow: "odd x ==> odd((x::int)^n)"
```
```    91   apply (induct n)
```
```    92   apply (simp add: even_def)
```
```    93   apply (simp add: even_product)
```
```    94   done
```
```    95
```
```    96 lemma even_power: "even ((x::int)^n) = (even x & 0 < n)"
```
```    97   apply (auto simp add: even_pow_gt_zero)
```
```    98   apply (erule contrapos_pp, erule odd_pow)
```
```    99   apply (erule contrapos_pp, simp add: even_def)
```
```   100   done
```
```   101
```
```   102 lemma even_zero: "even (0::int)"
```
```   103   by (simp add: even_def)
```
```   104
```
```   105 lemma odd_one: "odd (1::int)"
```
```   106   by (simp add: even_def)
```
```   107
```
```   108 lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero
```
```   109   odd_one even_product even_sum even_neg even_difference even_power
```
```   110
```
```   111
```
```   112 subsection {* Equivalent definitions *}
```
```   113
```
```   114 lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x"
```
```   115   by (auto simp add: even_def)
```
```   116
```
```   117 lemma two_times_odd_div_two_plus_one: "odd (x::int) ==>
```
```   118     2 * (x div 2) + 1 = x"
```
```   119   apply (insert zmod_zdiv_equality [of x 2, THEN sym])
```
```   120   by (simp add: even_def)
```
```   121
```
```   122 lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)"
```
```   123   apply auto
```
```   124   apply (rule exI)
```
```   125   by (erule two_times_even_div_two [THEN sym])
```
```   126
```
```   127 lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)"
```
```   128   apply auto
```
```   129   apply (rule exI)
```
```   130   by (erule two_times_odd_div_two_plus_one [THEN sym])
```
```   131
```
```   132
```
```   133 subsection {* even and odd for nats *}
```
```   134
```
```   135 lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
```
```   136   by (simp add: even_nat_def)
```
```   137
```
```   138 lemma even_nat_product: "even((x::nat) * y) = (even x | even y)"
```
```   139   by (simp add: even_nat_def int_mult)
```
```   140
```
```   141 lemma even_nat_sum: "even ((x::nat) + y) =
```
```   142     ((even x & even y) | (odd x & odd y))"
```
```   143   by (unfold even_nat_def, simp)
```
```   144
```
```   145 lemma even_nat_difference:
```
```   146     "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
```
```   147   apply (auto simp add: even_nat_def zdiff_int [THEN sym])
```
```   148   apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym])
```
```   149   apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym])
```
```   150   done
```
```   151
```
```   152 lemma even_nat_Suc: "even (Suc x) = odd x"
```
```   153   by (simp add: even_nat_def)
```
```   154
```
```   155 text{*Compatibility, in case Avigad uses this*}
```
```   156 lemmas even_nat_suc = even_nat_Suc
```
```   157
```
```   158 lemma even_nat_power: "even ((x::nat)^y) = (even x & 0 < y)"
```
```   159   by (simp add: even_nat_def int_power)
```
```   160
```
```   161 lemma even_nat_zero: "even (0::nat)"
```
```   162   by (simp add: even_nat_def)
```
```   163
```
```   164 lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard]
```
```   165   even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power
```
```   166
```
```   167
```
```   168 subsection {* Equivalent definitions *}
```
```   169
```
```   170 lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==>
```
```   171     x = 0 | x = Suc 0"
```
```   172   by auto
```
```   173
```
```   174 lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
```
```   175   apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
```
```   176   apply (drule subst, assumption)
```
```   177   apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0")
```
```   178   apply force
```
```   179   apply (subgoal_tac "0 < Suc (Suc 0)")
```
```   180   apply (frule mod_less_divisor [of "Suc (Suc 0)" x])
```
```   181   apply (erule nat_lt_two_imp_zero_or_one, auto)
```
```   182   done
```
```   183
```
```   184 lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
```
```   185   apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
```
```   186   apply (drule subst, assumption)
```
```   187   apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0")
```
```   188   apply force
```
```   189   apply (subgoal_tac "0 < Suc (Suc 0)")
```
```   190   apply (frule mod_less_divisor [of "Suc (Suc 0)" x])
```
```   191   apply (erule nat_lt_two_imp_zero_or_one, auto)
```
```   192   done
```
```   193
```
```   194 lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
```
```   195   apply (rule iffI)
```
```   196   apply (erule even_nat_mod_two_eq_zero)
```
```   197   apply (insert odd_nat_mod_two_eq_one [of x], auto)
```
```   198   done
```
```   199
```
```   200 lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
```
```   201   apply (auto simp add: even_nat_equiv_def)
```
```   202   apply (subgoal_tac "x mod (Suc (Suc 0)) < Suc (Suc 0)")
```
```   203   apply (frule nat_lt_two_imp_zero_or_one, auto)
```
```   204   done
```
```   205
```
```   206 lemma even_nat_div_two_times_two: "even (x::nat) ==>
```
```   207     Suc (Suc 0) * (x div Suc (Suc 0)) = x"
```
```   208   apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
```
```   209   apply (drule even_nat_mod_two_eq_zero, simp)
```
```   210   done
```
```   211
```
```   212 lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>
```
```   213     Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x"
```
```   214   apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])
```
```   215   apply (drule odd_nat_mod_two_eq_one, simp)
```
```   216   done
```
```   217
```
```   218 lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
```
```   219   apply (rule iffI, rule exI)
```
```   220   apply (erule even_nat_div_two_times_two [THEN sym], auto)
```
```   221   done
```
```   222
```
```   223 lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
```
```   224   apply (rule iffI, rule exI)
```
```   225   apply (erule odd_nat_div_two_times_two_plus_one [THEN sym], auto)
```
```   226   done
```
```   227
```
```   228 subsection {* Powers of negative one *}
```
```   229
```
```   230 lemma neg_one_even_odd_power:
```
```   231      "(even x --> (-1::'a::{number_ring,recpower})^x = 1) &
```
```   232       (odd x --> (-1::'a)^x = -1)"
```
```   233   apply (induct x)
```
```   234   apply (simp, simp add: power_Suc)
```
```   235   done
```
```   236
```
```   237 lemma neg_one_even_power [simp]:
```
```   238      "even x ==> (-1::'a::{number_ring,recpower})^x = 1"
```
```   239   by (rule neg_one_even_odd_power [THEN conjunct1, THEN mp], assumption)
```
```   240
```
```   241 lemma neg_one_odd_power [simp]:
```
```   242      "odd x ==> (-1::'a::{number_ring,recpower})^x = -1"
```
```   243   by (rule neg_one_even_odd_power [THEN conjunct2, THEN mp], assumption)
```
```   244
```
```   245 lemma neg_power_if:
```
```   246      "(-x::'a::{comm_ring_1,recpower}) ^ n =
```
```   247       (if even n then (x ^ n) else -(x ^ n))"
```
```   248   by (induct n, simp_all split: split_if_asm add: power_Suc)
```
```   249
```
```   250
```
```   251 subsection {* An Equivalence for @{term "0 \<le> a^n"} *}
```
```   252
```
```   253 lemma even_power_le_0_imp_0:
```
```   254      "a ^ (2*k) \<le> (0::'a::{ordered_idom,recpower}) ==> a=0"
```
```   255 apply (induct k)
```
```   256 apply (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc)
```
```   257 done
```
```   258
```
```   259 lemma zero_le_power_iff:
```
```   260      "(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even n)"
```
```   261       (is "?P n")
```
```   262 proof cases
```
```   263   assume even: "even n"
```
```   264   then obtain k where "n = 2*k"
```
```   265     by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2)
```
```   266   thus ?thesis by (simp add: zero_le_even_power even)
```
```   267 next
```
```   268   assume odd: "odd n"
```
```   269   then obtain k where "n = Suc(2*k)"
```
```   270     by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)
```
```   271   thus ?thesis
```
```   272     by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power
```
```   273              dest!: even_power_le_0_imp_0)
```
```   274 qed
```
```   275
```
```   276 subsection {* Miscellaneous *}
```
```   277
```
```   278 lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2"
```
```   279   apply (subst zdiv_zadd1_eq)
```
```   280   apply (simp add: even_def)
```
```   281   done
```
```   282
```
```   283 lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1"
```
```   284   apply (subst zdiv_zadd1_eq)
```
```   285   apply (simp add: even_def)
```
```   286   done
```
```   287
```
```   288 lemma div_Suc: "Suc a div c = a div c + Suc 0 div c +
```
```   289     (a mod c + Suc 0 mod c) div c"
```
```   290   apply (subgoal_tac "Suc a = a + Suc 0")
```
```   291   apply (erule ssubst)
```
```   292   apply (rule div_add1_eq, simp)
```
```   293   done
```
```   294
```
```   295 lemma even_nat_plus_one_div_two: "even (x::nat) ==>
```
```   296    (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)"
```
```   297   apply (subst div_Suc)
```
```   298   apply (simp add: even_nat_equiv_def)
```
```   299   done
```
```   300
```
```   301 lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>
```
```   302     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))"
```
```   303   apply (subst div_Suc)
```
```   304   apply (simp add: odd_nat_equiv_def)
```
```   305   done
```
```   306
```
```   307 end
```