src/HOL/Divides.thy

 next
   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
   thus  ?lhs by simp
 qed
+
+lemma take_bit_greater_eq:
+  \<open>k + 2 ^ n \<le> take_bit n k\<close> if \<open>k < 0\<close> for k :: int
+proof -
+  have \<open>k + 2 ^ n \<le> take_bit n (k + 2 ^ n)\<close>
+  proof (cases \<open>k > - (2 ^ n)\<close>)
+    case False
+    then have \<open>k + 2 ^ n \<le> 0\<close>
+      by simp
+    also note take_bit_nonnegative
+    finally show ?thesis .
+  next
+    case True
+    with that have \<open>0 \<le> k + 2 ^ n\<close> and \<open>k + 2 ^ n < 2 ^ n\<close>
+      by simp_all
+    then show ?thesis
+      by (simp only: take_bit_eq_mod mod_pos_pos_trivial)
+  qed
+  then show ?thesis
+    by (simp add: take_bit_eq_mod)
+qed
+
+lemma take_bit_less_eq:
+  \<open>take_bit n k \<le> k - 2 ^ n\<close> if \<open>2 ^ n \<le> k\<close> and \<open>n > 0\<close> for k :: int
+  using that zmod_le_nonneg_dividend [of \<open>k - 2 ^ n\<close> \<open>2 ^ n\<close>]
+  by (simp add: take_bit_eq_mod)
 
 
 subsection \<open>Numeral division with a pragmatic type class\<close>
 
 text \<open>