wneuper/isa: src/HOL/Integ/Parity.thy@83f1a514dcb4

     1 (*  Title:      Parity.thy

     2     ID:         $Id$

     3     Author:     Jeremy Avigad

     4     License:    GPL (GNU GENERAL PUBLIC LICENSE)

     5 *)

     7 header {* Parity: Even and Odd for ints and nats*}

     9 theory Parity = Divides + IntDiv + NatSimprocs:

    11 axclass even_odd < type

    13 instance int :: even_odd ..

    14 instance nat :: even_odd ..

    16 consts

    17   even :: "'a::even_odd => bool"

    19 syntax

    20   odd :: "'a::even_odd => bool"

    22 translations

    23   "odd x" == "~even x"

    25 defs (overloaded)

    26   even_def: "even (x::int) == x mod 2 = 0"

    27   even_nat_def: "even (x::nat) == even (int x)"

    30 subsection {* Casting a nat power to an integer *}

    32 lemma zpow_int: "int (x^y) = (int x)^y"

    33   apply (induct_tac y)

    34   apply (simp, simp add: zmult_int [THEN sym])

    35   done

    37 subsection {* Even and odd are mutually exclusive *}

    39 lemma int_pos_lt_two_imp_zero_or_one:

    40     "0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1"

    41   by auto

    43 lemma neq_one_mod_two [simp]: "((x::int) mod 2 ~= 0) = (x mod 2 = 1)"

    44   apply (subgoal_tac "x mod 2 = 0 | x mod 2 = 1", force)

    45   apply (rule int_pos_lt_two_imp_zero_or_one, auto)

    46   done

    48 subsection {* Behavior under integer arithmetic operations *}

    50 lemma even_times_anything: "even (x::int) ==> even (x * y)"

    51   by (simp add: even_def zmod_zmult1_eq')

    53 lemma anything_times_even: "even (y::int) ==> even (x * y)"

    54   by (simp add: even_def zmod_zmult1_eq)

    56 lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)"

    57   by (simp add: even_def zmod_zmult1_eq)

    59 lemma even_product: "even((x::int) * y) = (even x | even y)"

    60   apply (auto simp add: even_times_anything anything_times_even)

    61   apply (rule ccontr)

    62   apply (auto simp add: odd_times_odd)

    63   done

    65 lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"

    66   by (simp add: even_def zmod_zadd1_eq)

    68 lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"

    69   by (simp add: even_def zmod_zadd1_eq)

    71 lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"

    72   by (simp add: even_def zmod_zadd1_eq)

    74 lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)"

    75   by (simp add: even_def zmod_zadd1_eq)

    77 lemma even_sum: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"

    78   apply (auto intro: even_plus_even odd_plus_odd)

    79   apply (rule ccontr, simp add: even_plus_odd)

    80   apply (rule ccontr, simp add: odd_plus_even)

    81   done

    83 lemma even_neg: "even (-(x::int)) = even x"

    84   by (auto simp add: even_def zmod_zminus1_eq_if)

    86 lemma even_difference:

    87   "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))"

    88   by (simp only: diff_minus even_sum even_neg)

    90 lemma even_pow_gt_zero [rule_format]:

    91     "even (x::int) ==> 0 < n --> even (x^n)"

    92   apply (induct_tac n)

    93   apply (auto simp add: even_product)

    94   done

    96 lemma odd_pow: "odd x ==> odd((x::int)^n)"

    97   apply (induct_tac n)

    98   apply (simp add: even_def)

    99   apply (simp add: even_product)

   100   done

   102 lemma even_power: "even ((x::int)^n) = (even x & 0 < n)"

   103   apply (auto simp add: even_pow_gt_zero)

   104   apply (erule contrapos_pp, erule odd_pow)

   105   apply (erule contrapos_pp, simp add: even_def)

   106   done

   108 lemma even_zero: "even (0::int)"

   109   by (simp add: even_def)

   111 lemma odd_one: "odd (1::int)"

   112   by (simp add: even_def)

   114 lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero

   115   odd_one even_product even_sum even_neg even_difference even_power

   118 subsection {* Equivalent definitions *}

   120 lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x"

   121   by (auto simp add: even_def)

   123 lemma two_times_odd_div_two_plus_one: "odd (x::int) ==>

   124     2 * (x div 2) + 1 = x"

   125   apply (insert zmod_zdiv_equality [of x 2, THEN sym])

   126   by (simp add: even_def)

   128 lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)"

   129   apply auto

   130   apply (rule exI)

   131   by (erule two_times_even_div_two [THEN sym])

   133 lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)"

   134   apply auto

   135   apply (rule exI)

   136   by (erule two_times_odd_div_two_plus_one [THEN sym])

   139 subsection {* even and odd for nats *}

   141 lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"

   142   by (simp add: even_nat_def)

   144 lemma even_nat_product: "even((x::nat) * y) = (even x | even y)"

   145   by (simp add: even_nat_def zmult_int [THEN sym])

   147 lemma even_nat_sum: "even ((x::nat) + y) =

   148     ((even x & even y) | (odd x & odd y))"

   149   by (unfold even_nat_def, simp)

   151 lemma even_nat_difference:

   152     "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"

   153   apply (auto simp add: even_nat_def zdiff_int [THEN sym])

   154   apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym])

   155   apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym])

   156   done

   158 lemma even_nat_Suc: "even (Suc x) = odd x"

   159   by (simp add: even_nat_def)

   161 text{*Compatibility, in case Avigad uses this*}

   162 lemmas even_nat_suc = even_nat_Suc

   164 lemma even_nat_power: "even ((x::nat)^y) = (even x & 0 < y)"

   165   by (simp add: even_nat_def zpow_int)

   167 lemma even_nat_zero: "even (0::nat)"

   168   by (simp add: even_nat_def)

   170 lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard]

   171   even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power

   174 subsection {* Equivalent definitions *}

   176 lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==>

   177     x = 0 | x = Suc 0"

   178   by auto

   180 lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"

   181   apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])

   182   apply (drule subst, assumption)

   183   apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0")

   184   apply force

   185   apply (subgoal_tac "0 < Suc (Suc 0)")

   186   apply (frule mod_less_divisor [of "Suc (Suc 0)" x])

   187   apply (erule nat_lt_two_imp_zero_or_one, auto)

   188   done

   190 lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"

   191   apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])

   192   apply (drule subst, assumption)

   193   apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0")

   194   apply force

   195   apply (subgoal_tac "0 < Suc (Suc 0)")

   196   apply (frule mod_less_divisor [of "Suc (Suc 0)" x])

   197   apply (erule nat_lt_two_imp_zero_or_one, auto)

   198   done

   200 lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"

   201   apply (rule iffI)

   202   apply (erule even_nat_mod_two_eq_zero)

   203   apply (insert odd_nat_mod_two_eq_one [of x], auto)

   204   done

   206 lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"

   207   apply (auto simp add: even_nat_equiv_def)

   208   apply (subgoal_tac "x mod (Suc (Suc 0)) < Suc (Suc 0)")

   209   apply (frule nat_lt_two_imp_zero_or_one, auto)

   210   done

   212 lemma even_nat_div_two_times_two: "even (x::nat) ==>

   213     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 even_nat_mod_two_eq_zero, simp)

   216   done

   218 lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==>

   219     Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x"

   220   apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym])

   221   apply (drule odd_nat_mod_two_eq_one, simp)

   222   done

   224 lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"

   225   apply (rule iffI, rule exI)

   226   apply (erule even_nat_div_two_times_two [THEN sym], auto)

   227   done

   229 lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"

   230   apply (rule iffI, rule exI)

   231   apply (erule odd_nat_div_two_times_two_plus_one [THEN sym], auto)

   232   done

   234 subsection {* Powers of negative one *}

   236 lemma neg_one_even_odd_power:

   237      "(even x --> (-1::'a::{number_ring,ringpower})^x = 1) &

   238       (odd x --> (-1::'a)^x = -1)"

   239   apply (induct_tac x)

   240   apply (simp, simp add: power_Suc)

   241   done

   243 lemma neg_one_even_power [simp]:

   244      "even x ==> (-1::'a::{number_ring,ringpower})^x = 1"

   245   by (rule neg_one_even_odd_power [THEN conjunct1, THEN mp], assumption)

   247 lemma neg_one_odd_power [simp]:

   248      "odd x ==> (-1::'a::{number_ring,ringpower})^x = -1"

   249   by (rule neg_one_even_odd_power [THEN conjunct2, THEN mp], assumption)

   251 lemma neg_power_if:

   252      "(-x::'a::{comm_ring_1,ringpower}) ^ n =

   253       (if even n then (x ^ n) else -(x ^ n))"

   254   by (induct n, simp_all split: split_if_asm add: power_Suc)

   257 subsection {* An Equivalence for @{term "0 \<le> a^n"} *}

   259 lemma even_power_le_0_imp_0:

   260      "a ^ (2*k) \<le> (0::'a::{ordered_idom,ringpower}) ==> a=0"

   261 apply (induct k)

   262 apply (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc)

   263 done

   265 lemma zero_le_power_iff:

   266      "(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,ringpower}) | even n)"

   267       (is "?P n")

   268 proof cases

   269   assume even: "even n"

   270   then obtain k where "n = 2*k"

   271     by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2)

   272   thus ?thesis by (simp add: zero_le_even_power even)

   273 next

   274   assume odd: "odd n"

   275   then obtain k where "n = Suc(2*k)"

   276     by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2)

   277   thus ?thesis

   278     by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power

   279              dest!: even_power_le_0_imp_0)

   280 qed

   282 subsection {* Miscellaneous *}

   284 lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2"

   285   apply (subst zdiv_zadd1_eq)

   286   apply (simp add: even_def)

   287   done

   289 lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1"

   290   apply (subst zdiv_zadd1_eq)

   291   apply (simp add: even_def)

   292   done

   294 lemma div_Suc: "Suc a div c = a div c + Suc 0 div c +

   295     (a mod c + Suc 0 mod c) div c"

   296   apply (subgoal_tac "Suc a = a + Suc 0")

   297   apply (erule ssubst)

   298   apply (rule div_add1_eq, simp)

   299   done

   301 lemma even_nat_plus_one_div_two: "even (x::nat) ==>

   302    (Suc x) div Suc (Suc 0) = x div Suc (Suc 0)"

   303   apply (subst div_Suc)

   304   apply (simp add: even_nat_equiv_def)

   305   done

   307 lemma odd_nat_plus_one_div_two: "odd (x::nat) ==>

   308     (Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))"

   309   apply (subst div_Suc)

   310   apply (simp add: odd_nat_equiv_def)

   311   done

   313 end

author	obua
	Tue, 11 May 2004 20:11:08 +0200
changeset 14738	83f1a514dcb4
parent 14473	846c237bd9b3
child 14981	e73f8140af78
permissions	-rw-r--r--