src/HOL/Parity.thy
changeset 31148 7ba7c1f8bc22
parent 31017 2c227493ea56
child 31718 7715d4d3586f
     1.1 --- a/src/HOL/Parity.thy	Thu May 14 08:22:07 2009 +0200
     1.2 +++ b/src/HOL/Parity.thy	Thu May 14 15:39:15 2009 +0200
     1.3 @@ -29,6 +29,18 @@
     1.4  end
     1.5  
     1.6  
     1.7 +lemma even_zero_int[simp]: "even (0::int)" by presburger
     1.8 +
     1.9 +lemma odd_one_int[simp]: "odd (1::int)" by presburger
    1.10 +
    1.11 +lemma even_zero_nat[simp]: "even (0::nat)" by presburger
    1.12 +
    1.13 +lemma odd_zero_nat [simp]: "odd (1::nat)" by presburger
    1.14 +
    1.15 +declare even_def[of "number_of v", standard, simp]
    1.16 +
    1.17 +declare even_nat_def[of "number_of v", standard, simp]
    1.18 +
    1.19  subsection {* Even and odd are mutually exclusive *}
    1.20  
    1.21  lemma int_pos_lt_two_imp_zero_or_one:
    1.22 @@ -54,66 +66,47 @@
    1.23  lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)" 
    1.24    by (simp add: even_def zmod_zmult1_eq)
    1.25  
    1.26 -lemma even_product[presburger]: "even((x::int) * y) = (even x | even y)"
    1.27 +lemma even_product[simp,presburger]: "even((x::int) * y) = (even x | even y)"
    1.28    apply (auto simp add: even_times_anything anything_times_even)
    1.29    apply (rule ccontr)
    1.30    apply (auto simp add: odd_times_odd)
    1.31    done
    1.32  
    1.33  lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"
    1.34 -  by presburger
    1.35 +by presburger
    1.36  
    1.37  lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"
    1.38 -  by presburger
    1.39 +by presburger
    1.40  
    1.41  lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"
    1.42 -  by presburger
    1.43 +by presburger
    1.44  
    1.45  lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger
    1.46  
    1.47 -lemma even_sum[presburger]: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
    1.48 -  by presburger
    1.49 +lemma even_sum[simp,presburger]:
    1.50 +  "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
    1.51 +by presburger
    1.52  
    1.53 -lemma even_neg[presburger, algebra]: "even (-(x::int)) = even x" by presburger
    1.54 +lemma even_neg[simp,presburger,algebra]: "even (-(x::int)) = even x"
    1.55 +by presburger
    1.56  
    1.57 -lemma even_difference:
    1.58 +lemma even_difference[simp]:
    1.59      "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger
    1.60  
    1.61 -lemma even_pow_gt_zero:
    1.62 -    "even (x::int) ==> 0 < n ==> even (x^n)"
    1.63 -  by (induct n) (auto simp add: even_product)
    1.64 -
    1.65 -lemma odd_pow_iff[presburger, algebra]: 
    1.66 -  "odd ((x::int) ^ n) \<longleftrightarrow> (n = 0 \<or> odd x)"
    1.67 -  apply (induct n, simp_all)
    1.68 -  apply presburger
    1.69 -  apply (case_tac n, auto)
    1.70 -  apply (simp_all add: even_product)
    1.71 -  done
    1.72 +lemma even_power[simp,presburger]: "even ((x::int)^n) = (even x & n \<noteq> 0)"
    1.73 +by (induct n) auto
    1.74  
    1.75 -lemma odd_pow: "odd x ==> odd((x::int)^n)" by (simp add: odd_pow_iff)
    1.76 -
    1.77 -lemma even_power[presburger]: "even ((x::int)^n) = (even x & 0 < n)"
    1.78 -  apply (auto simp add: even_pow_gt_zero)
    1.79 -  apply (erule contrapos_pp, erule odd_pow)
    1.80 -  apply (erule contrapos_pp, simp add: even_def)
    1.81 -  done
    1.82 -
    1.83 -lemma even_zero[presburger]: "even (0::int)" by presburger
    1.84 -
    1.85 -lemma odd_one[presburger]: "odd (1::int)" by presburger
    1.86 -
    1.87 -lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero
    1.88 -  odd_one even_product even_sum even_neg even_difference even_power
    1.89 +lemma odd_pow: "odd x ==> odd((x::int)^n)" by simp
    1.90  
    1.91  
    1.92  subsection {* Equivalent definitions *}
    1.93  
    1.94  lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x" 
    1.95 -  by presburger
    1.96 +by presburger
    1.97  
    1.98 -lemma two_times_odd_div_two_plus_one: "odd (x::int) ==>
    1.99 -    2 * (x div 2) + 1 = x" by presburger
   1.100 +lemma two_times_odd_div_two_plus_one:
   1.101 +  "odd (x::int) ==> 2 * (x div 2) + 1 = x"
   1.102 +by presburger
   1.103  
   1.104  lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger
   1.105  
   1.106 @@ -122,45 +115,45 @@
   1.107  subsection {* even and odd for nats *}
   1.108  
   1.109  lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
   1.110 -  by (simp add: even_nat_def)
   1.111 -
   1.112 -lemma even_nat_product[presburger, algebra]: "even((x::nat) * y) = (even x | even y)"
   1.113 -  by (simp add: even_nat_def int_mult)
   1.114 +by (simp add: even_nat_def)
   1.115  
   1.116 -lemma even_nat_sum[presburger, algebra]: "even ((x::nat) + y) =
   1.117 -    ((even x & even y) | (odd x & odd y))" by presburger
   1.118 +lemma even_product_nat[simp,presburger,algebra]:
   1.119 +  "even((x::nat) * y) = (even x | even y)"
   1.120 +by (simp add: even_nat_def int_mult)
   1.121  
   1.122 -lemma even_nat_difference[presburger, algebra]:
   1.123 -    "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
   1.124 +lemma even_sum_nat[simp,presburger,algebra]:
   1.125 +  "even ((x::nat) + y) = ((even x & even y) | (odd x & odd y))"
   1.126  by presburger
   1.127  
   1.128 -lemma even_nat_Suc[presburger, algebra]: "even (Suc x) = odd x" by presburger
   1.129 -
   1.130 -lemma even_nat_power[presburger, algebra]: "even ((x::nat)^y) = (even x & 0 < y)"
   1.131 -  by (simp add: even_nat_def int_power)
   1.132 +lemma even_difference_nat[simp,presburger,algebra]:
   1.133 +  "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
   1.134 +by presburger
   1.135  
   1.136 -lemma even_nat_zero[presburger]: "even (0::nat)" by presburger
   1.137 +lemma even_Suc[simp,presburger,algebra]: "even (Suc x) = odd x"
   1.138 +by presburger
   1.139  
   1.140 -lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard]
   1.141 -  even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power
   1.142 +lemma even_power_nat[simp,presburger,algebra]:
   1.143 +  "even ((x::nat)^y) = (even x & 0 < y)"
   1.144 +by (simp add: even_nat_def int_power)
   1.145  
   1.146  
   1.147  subsection {* Equivalent definitions *}
   1.148  
   1.149 -lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==>
   1.150 -    x = 0 | x = Suc 0" by presburger
   1.151 +lemma nat_lt_two_imp_zero_or_one:
   1.152 +  "(x::nat) < Suc (Suc 0) ==> x = 0 | x = Suc 0"
   1.153 +by presburger
   1.154  
   1.155  lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
   1.156 -  by presburger
   1.157 +by presburger
   1.158  
   1.159  lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
   1.160  by presburger
   1.161  
   1.162  lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
   1.163 -  by presburger
   1.164 +by presburger
   1.165  
   1.166  lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
   1.167 -  by presburger
   1.168 +by presburger
   1.169  
   1.170  lemma even_nat_div_two_times_two: "even (x::nat) ==>
   1.171      Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger
   1.172 @@ -169,10 +162,10 @@
   1.173      Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger
   1.174  
   1.175  lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
   1.176 -  by presburger
   1.177 +by presburger
   1.178  
   1.179  lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
   1.180 -  by presburger
   1.181 +by presburger
   1.182  
   1.183  
   1.184  subsection {* Parity and powers *}
   1.185 @@ -183,7 +176,7 @@
   1.186    apply (induct x)
   1.187    apply (rule conjI)
   1.188    apply simp
   1.189 -  apply (insert even_nat_zero, blast)
   1.190 +  apply (insert even_zero_nat, blast)
   1.191    apply (simp add: power_Suc)
   1.192    done
   1.193