Cleaned up Parity a little
authornipkow
Thu May 14 15:39:15 2009 +0200 (2009-05-14)
changeset 311487ba7c1f8bc22
parent 31144 bdc1504ad456
child 31149 7be8054639cd
Cleaned up Parity a little
src/HOL/Decision_Procs/Approximation.thy
src/HOL/Library/Formal_Power_Series.thy
src/HOL/MacLaurin.thy
src/HOL/Parity.thy
src/HOL/Transcendental.thy
     1.1 --- a/src/HOL/Decision_Procs/Approximation.thy	Thu May 14 08:22:07 2009 +0200
     1.2 +++ b/src/HOL/Decision_Procs/Approximation.thy	Thu May 14 15:39:15 2009 +0200
     1.3 @@ -460,7 +460,7 @@
     1.4  proof (cases "even n")
     1.5    case True
     1.6    obtain n' where "Suc n' = get_odd n" and "odd (Suc n')" using get_odd_ex by auto
     1.7 -  hence "even n'" unfolding even_nat_Suc by auto
     1.8 +  hence "even n'" unfolding even_Suc by auto
     1.9    have "arctan (real x) \<le> real (x * ub_arctan_horner prec (get_odd n) 1 (x * x))"
    1.10      unfolding `Suc n' = get_odd n`[symmetric] using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even n'`] by auto
    1.11    moreover
    1.12 @@ -470,7 +470,7 @@
    1.13  next
    1.14    case False hence "0 < n" by (rule odd_pos)
    1.15    from gr0_implies_Suc[OF this] obtain n' where "n = Suc n'" ..
    1.16 -  from False[unfolded this even_nat_Suc]
    1.17 +  from False[unfolded this even_Suc]
    1.18    have "even n'" and "even (Suc (Suc n'))" by auto
    1.19    have "get_odd n = Suc n'" unfolding get_odd_def if_P[OF False] using `n = Suc n'` .
    1.20  
     2.1 --- a/src/HOL/Library/Formal_Power_Series.thy	Thu May 14 08:22:07 2009 +0200
     2.2 +++ b/src/HOL/Library/Formal_Power_Series.thy	Thu May 14 15:39:15 2009 +0200
     2.3 @@ -917,8 +917,7 @@
     2.4  proof-
     2.5    have eq: "(1 + X) * ?r = 1"
     2.6      unfolding minus_one_power_iff
     2.7 -    apply (auto simp add: ring_simps fps_eq_iff)
     2.8 -    by presburger+
     2.9 +    by (auto simp add: ring_simps fps_eq_iff)
    2.10    show ?thesis by (auto simp add: eq intro: fps_inverse_unique)
    2.11  qed
    2.12  
    2.13 @@ -2286,9 +2285,7 @@
    2.14    (is "inverse ?l = ?r")
    2.15  proof-
    2.16    have th: "?l * ?r = 1"
    2.17 -    apply (auto simp add: ring_simps fps_eq_iff X_mult_nth  minus_one_power_iff)
    2.18 -    apply presburger+
    2.19 -    done
    2.20 +    by (auto simp add: ring_simps fps_eq_iff minus_one_power_iff)
    2.21    have th': "?l $ 0 \<noteq> 0" by (simp add: )
    2.22    from fps_inverse_unique[OF th' th] show ?thesis .
    2.23  qed
     3.1 --- a/src/HOL/MacLaurin.thy	Thu May 14 08:22:07 2009 +0200
     3.2 +++ b/src/HOL/MacLaurin.thy	Thu May 14 15:39:15 2009 +0200
     3.3 @@ -552,10 +552,6 @@
     3.4      "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
     3.5  by auto
     3.6  
     3.7 -text {* TODO: move to Parity.thy *}
     3.8 -lemma nat_odd_1 [simp]: "odd (1::nat)"
     3.9 -  unfolding even_nat_def by simp
    3.10 -
    3.11  lemma Maclaurin_sin_bound:
    3.12    "abs(sin x - (\<Sum>m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
    3.13    x ^ m))  \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
     4.1 --- a/src/HOL/Parity.thy	Thu May 14 08:22:07 2009 +0200
     4.2 +++ b/src/HOL/Parity.thy	Thu May 14 15:39:15 2009 +0200
     4.3 @@ -29,6 +29,18 @@
     4.4  end
     4.5  
     4.6  
     4.7 +lemma even_zero_int[simp]: "even (0::int)" by presburger
     4.8 +
     4.9 +lemma odd_one_int[simp]: "odd (1::int)" by presburger
    4.10 +
    4.11 +lemma even_zero_nat[simp]: "even (0::nat)" by presburger
    4.12 +
    4.13 +lemma odd_zero_nat [simp]: "odd (1::nat)" by presburger
    4.14 +
    4.15 +declare even_def[of "number_of v", standard, simp]
    4.16 +
    4.17 +declare even_nat_def[of "number_of v", standard, simp]
    4.18 +
    4.19  subsection {* Even and odd are mutually exclusive *}
    4.20  
    4.21  lemma int_pos_lt_two_imp_zero_or_one:
    4.22 @@ -54,66 +66,47 @@
    4.23  lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)" 
    4.24    by (simp add: even_def zmod_zmult1_eq)
    4.25  
    4.26 -lemma even_product[presburger]: "even((x::int) * y) = (even x | even y)"
    4.27 +lemma even_product[simp,presburger]: "even((x::int) * y) = (even x | even y)"
    4.28    apply (auto simp add: even_times_anything anything_times_even)
    4.29    apply (rule ccontr)
    4.30    apply (auto simp add: odd_times_odd)
    4.31    done
    4.32  
    4.33  lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)"
    4.34 -  by presburger
    4.35 +by presburger
    4.36  
    4.37  lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)"
    4.38 -  by presburger
    4.39 +by presburger
    4.40  
    4.41  lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)"
    4.42 -  by presburger
    4.43 +by presburger
    4.44  
    4.45  lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" by presburger
    4.46  
    4.47 -lemma even_sum[presburger]: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
    4.48 -  by presburger
    4.49 +lemma even_sum[simp,presburger]:
    4.50 +  "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))"
    4.51 +by presburger
    4.52  
    4.53 -lemma even_neg[presburger, algebra]: "even (-(x::int)) = even x" by presburger
    4.54 +lemma even_neg[simp,presburger,algebra]: "even (-(x::int)) = even x"
    4.55 +by presburger
    4.56  
    4.57 -lemma even_difference:
    4.58 +lemma even_difference[simp]:
    4.59      "even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" by presburger
    4.60  
    4.61 -lemma even_pow_gt_zero:
    4.62 -    "even (x::int) ==> 0 < n ==> even (x^n)"
    4.63 -  by (induct n) (auto simp add: even_product)
    4.64 -
    4.65 -lemma odd_pow_iff[presburger, algebra]: 
    4.66 -  "odd ((x::int) ^ n) \<longleftrightarrow> (n = 0 \<or> odd x)"
    4.67 -  apply (induct n, simp_all)
    4.68 -  apply presburger
    4.69 -  apply (case_tac n, auto)
    4.70 -  apply (simp_all add: even_product)
    4.71 -  done
    4.72 +lemma even_power[simp,presburger]: "even ((x::int)^n) = (even x & n \<noteq> 0)"
    4.73 +by (induct n) auto
    4.74  
    4.75 -lemma odd_pow: "odd x ==> odd((x::int)^n)" by (simp add: odd_pow_iff)
    4.76 -
    4.77 -lemma even_power[presburger]: "even ((x::int)^n) = (even x & 0 < n)"
    4.78 -  apply (auto simp add: even_pow_gt_zero)
    4.79 -  apply (erule contrapos_pp, erule odd_pow)
    4.80 -  apply (erule contrapos_pp, simp add: even_def)
    4.81 -  done
    4.82 -
    4.83 -lemma even_zero[presburger]: "even (0::int)" by presburger
    4.84 -
    4.85 -lemma odd_one[presburger]: "odd (1::int)" by presburger
    4.86 -
    4.87 -lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero
    4.88 -  odd_one even_product even_sum even_neg even_difference even_power
    4.89 +lemma odd_pow: "odd x ==> odd((x::int)^n)" by simp
    4.90  
    4.91  
    4.92  subsection {* Equivalent definitions *}
    4.93  
    4.94  lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x" 
    4.95 -  by presburger
    4.96 +by presburger
    4.97  
    4.98 -lemma two_times_odd_div_two_plus_one: "odd (x::int) ==>
    4.99 -    2 * (x div 2) + 1 = x" by presburger
   4.100 +lemma two_times_odd_div_two_plus_one:
   4.101 +  "odd (x::int) ==> 2 * (x div 2) + 1 = x"
   4.102 +by presburger
   4.103  
   4.104  lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" by presburger
   4.105  
   4.106 @@ -122,45 +115,45 @@
   4.107  subsection {* even and odd for nats *}
   4.108  
   4.109  lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)"
   4.110 -  by (simp add: even_nat_def)
   4.111 -
   4.112 -lemma even_nat_product[presburger, algebra]: "even((x::nat) * y) = (even x | even y)"
   4.113 -  by (simp add: even_nat_def int_mult)
   4.114 +by (simp add: even_nat_def)
   4.115  
   4.116 -lemma even_nat_sum[presburger, algebra]: "even ((x::nat) + y) =
   4.117 -    ((even x & even y) | (odd x & odd y))" by presburger
   4.118 +lemma even_product_nat[simp,presburger,algebra]:
   4.119 +  "even((x::nat) * y) = (even x | even y)"
   4.120 +by (simp add: even_nat_def int_mult)
   4.121  
   4.122 -lemma even_nat_difference[presburger, algebra]:
   4.123 -    "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
   4.124 +lemma even_sum_nat[simp,presburger,algebra]:
   4.125 +  "even ((x::nat) + y) = ((even x & even y) | (odd x & odd y))"
   4.126  by presburger
   4.127  
   4.128 -lemma even_nat_Suc[presburger, algebra]: "even (Suc x) = odd x" by presburger
   4.129 -
   4.130 -lemma even_nat_power[presburger, algebra]: "even ((x::nat)^y) = (even x & 0 < y)"
   4.131 -  by (simp add: even_nat_def int_power)
   4.132 +lemma even_difference_nat[simp,presburger,algebra]:
   4.133 +  "even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))"
   4.134 +by presburger
   4.135  
   4.136 -lemma even_nat_zero[presburger]: "even (0::nat)" by presburger
   4.137 +lemma even_Suc[simp,presburger,algebra]: "even (Suc x) = odd x"
   4.138 +by presburger
   4.139  
   4.140 -lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard]
   4.141 -  even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power
   4.142 +lemma even_power_nat[simp,presburger,algebra]:
   4.143 +  "even ((x::nat)^y) = (even x & 0 < y)"
   4.144 +by (simp add: even_nat_def int_power)
   4.145  
   4.146  
   4.147  subsection {* Equivalent definitions *}
   4.148  
   4.149 -lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==>
   4.150 -    x = 0 | x = Suc 0" by presburger
   4.151 +lemma nat_lt_two_imp_zero_or_one:
   4.152 +  "(x::nat) < Suc (Suc 0) ==> x = 0 | x = Suc 0"
   4.153 +by presburger
   4.154  
   4.155  lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0"
   4.156 -  by presburger
   4.157 +by presburger
   4.158  
   4.159  lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0"
   4.160  by presburger
   4.161  
   4.162  lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)"
   4.163 -  by presburger
   4.164 +by presburger
   4.165  
   4.166  lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)"
   4.167 -  by presburger
   4.168 +by presburger
   4.169  
   4.170  lemma even_nat_div_two_times_two: "even (x::nat) ==>
   4.171      Suc (Suc 0) * (x div Suc (Suc 0)) = x" by presburger
   4.172 @@ -169,10 +162,10 @@
   4.173      Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" by presburger
   4.174  
   4.175  lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)"
   4.176 -  by presburger
   4.177 +by presburger
   4.178  
   4.179  lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))"
   4.180 -  by presburger
   4.181 +by presburger
   4.182  
   4.183  
   4.184  subsection {* Parity and powers *}
   4.185 @@ -183,7 +176,7 @@
   4.186    apply (induct x)
   4.187    apply (rule conjI)
   4.188    apply simp
   4.189 -  apply (insert even_nat_zero, blast)
   4.190 +  apply (insert even_zero_nat, blast)
   4.191    apply (simp add: power_Suc)
   4.192    done
   4.193  
     5.1 --- a/src/HOL/Transcendental.thy	Thu May 14 08:22:07 2009 +0200
     5.2 +++ b/src/HOL/Transcendental.thy	Thu May 14 15:39:15 2009 +0200
     5.3 @@ -173,7 +173,7 @@
     5.4      have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
     5.5        unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
     5.6                  image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
     5.7 -                even_nat_Suc Suc_m1 if_eq .
     5.8 +                even_Suc Suc_m1 if_eq .
     5.9    } from sums_add[OF g_sums this]
    5.10    show ?thesis unfolding if_sum .
    5.11  qed