src/HOL/Presburger.thy

   proof (cases)
     assume "x mod d = 0"
     hence "P 0" using P Pmod by simp
     moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
     ultimately have "P d" by simp
-    moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
+    moreover have "d : {1..d}" using dpos by simp
     ultimately show ?RHS ..
   next
     assume not0: "x mod d \<noteq> 0"
-    have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
+    have "P(x mod d)" using dpos P Pmod by simp
     moreover have "x mod d : {1..d}"
     proof -
       from dpos have "0 \<le> x mod d" by(rule pos_mod_sign)
       moreover from dpos have "x mod d < d" by(rule pos_mod_bound)
-      ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
+      ultimately show ?thesis using not0 by simp
     qed
     ultimately show ?RHS ..
   qed
 qed auto