diff -r dfda74619509 -r ddd97d9dfbfb src/HOL/Presburger.thy
--- a/src/HOL/Presburger.thy	Thu Oct 29 08:14:39 2009 +0100
+++ b/src/HOL/Presburger.thy	Thu Oct 29 11:41:36 2009 +0100
@@ -385,20 +385,6 @@
 
 text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
 
-lemma all_nat: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x\<ge>0. P (nat x))"
-  by (simp split add: split_nat)
-
-lemma ex_nat: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> P (nat x))"
-proof
-  assume "\<exists>x. P x"
-  then obtain x where "P x" ..
-  then have "int x \<ge> 0 \<and> P (nat (int x))" by simp
-  then show "\<exists>x\<ge>0. P (nat x)" ..
-next
-  assume "\<exists>x\<ge>0. P (nat x)"
-  then show "\<exists>x. P x" by auto
-qed
-
 lemma zdiff_int_split: "P (int (x - y)) =
   ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
   by (case_tac "y \<le> x", simp_all add: zdiff_int)