--- a/src/HOL/Presburger.thy Thu Jul 19 21:47:42 2007 +0200
+++ b/src/HOL/Presburger.thy Thu Jul 19 21:47:43 2007 +0200
@@ -703,4 +703,37 @@
less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1
less_number_of
+
+lemma of_int_num [code func]:
+ "of_int k = (if k = 0 then 0 else if k < 0 then
+ - of_int (- k) else let
+ (l, m) = divAlg (k, 2);
+ l' = of_int l
+ in if m = 0 then l' + l' else l' + l' + 1)"
+proof -
+ have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow>
+ of_int k = of_int (k div 2 * 2 + 1)"
+ proof -
+ assume "k mod 2 \<noteq> 0"
+ then have "k mod 2 = 1" by arith
+ moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
+ ultimately show ?thesis by auto
+ qed
+ have aux2: "\<And>x. of_int 2 * x = x + x"
+ proof -
+ fix x
+ have int2: "(2::int) = 1 + 1" by arith
+ show "of_int 2 * x = x + x"
+ unfolding int2 of_int_add left_distrib by simp
+ qed
+ have aux3: "\<And>x. x * of_int 2 = x + x"
+ proof -
+ fix x
+ have int2: "(2::int) = 1 + 1" by arith
+ show "x * of_int 2 = x + x"
+ unfolding int2 of_int_add right_distrib by simp
+ qed
+ from aux1 show ?thesis by (auto simp add: divAlg_mod_div Let_def aux2 aux3)
+qed
+
end