diff --git a/src/HOL/Library/Rounded_Division.thy b/src/HOL/Library/Rounded_Division.thy
--- a/src/HOL/Library/Rounded_Division.thy
+++ b/src/HOL/Library/Rounded_Division.thy
@@ -1,84 +1,84 @@
 (*  Author:  Florian Haftmann, TU Muenchen
 *)
 
-subsection \<open>Rounded division: modulus centered towars zero.\<close>
+subsection \<open>Rounded division: modulus centered towards zero.\<close>
 
 theory Rounded_Division
   imports Main
 begin
 
 definition rounded_divide :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close>  (infixl \<open>rdiv\<close> 70)
   where \<open>k rdiv l = (k + l div 2 + of_bool (l < 0)) div l\<close>
 
 definition rounded_modulo :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close>  (infixl \<open>rmod\<close> 70)
   where \<open>k rmod l = k - k rdiv l * l\<close>
 
 lemma rdiv_mult_rmod_eq:
   \<open>k rdiv l * l + k rmod l = k\<close>
   by (simp add: rounded_divide_def rounded_modulo_def)
 
 lemma mult_rdiv_rmod_eq:
   \<open>l * (k rdiv l) + k rmod l = k\<close>
   using rdiv_mult_rmod_eq [of k l] by (simp add: ac_simps)
 
 lemma rmod_rdiv_mult_eq:
   \<open>k rmod l + k rdiv l * l = k\<close>
   using rdiv_mult_rmod_eq [of k l] by (simp add: ac_simps)
 
 lemma rmod_mult_rdiv_eq:
   \<open>k rmod l + l * (k rdiv l) = k\<close>
   using rdiv_mult_rmod_eq [of k l] by (simp add: ac_simps)
 
 lemma minus_rdiv_mult_eq_rmod:
   \<open>k - k rdiv l * l = k rmod l\<close>
   by (rule add_implies_diff [symmetric]) (fact rmod_rdiv_mult_eq)
 
 lemma minus_mult_rdiv_eq_rmod:
   \<open>k - l * (k rdiv l) = k rmod l\<close>
   by (rule add_implies_diff [symmetric]) (fact rmod_mult_rdiv_eq)
 
 lemma minus_rmod_eq_rdiv_mult:
   \<open>k - k rmod l = k rdiv l * l\<close>
   by (rule add_implies_diff [symmetric]) (fact rdiv_mult_rmod_eq)
 
 lemma minus_rmod_eq_mult_rdiv:
   \<open>k - k rmod l = l * (k rdiv l)\<close>
   by (rule add_implies_diff [symmetric]) (fact mult_rdiv_rmod_eq)
 
 lemma rdiv_0_eq [simp]:
   \<open>k rdiv 0 = 0\<close>
   by (simp add: rounded_divide_def)
 
 lemma rmod_0_eq [simp]:
   \<open>k rmod 0 = k\<close>
   by (simp add: rounded_modulo_def)
 
 lemma rdiv_1_eq [simp]:
   \<open>k rdiv 1 = k\<close>
   by (simp add: rounded_divide_def)
 
 lemma rmod_1_eq [simp]:
   \<open>k rmod 1 = 0\<close>
   by (simp add: rounded_modulo_def)
 
 lemma zero_rdiv_eq [simp]:
   \<open>0 rdiv k = 0\<close>
   by (auto simp add: rounded_divide_def not_less zdiv_eq_0_iff)
 
 lemma zero_rmod_eq [simp]:
   \<open>0 rmod k = 0\<close>
   by (simp add: rounded_modulo_def)
 
 lemma nonzero_mult_rdiv_cancel_right:
   \<open>k * l rdiv l = k\<close> if \<open>l \<noteq> 0\<close>
   using that by (auto simp add: rounded_divide_def ac_simps)
 
 lemma rdiv_self_eq [simp]:
   \<open>k rdiv k = 1\<close> if \<open>k \<noteq> 0\<close>
   using that nonzero_mult_rdiv_cancel_right [of k 1] by simp
 
 lemma rmod_self_eq [simp]:
   \<open>k rmod k = 0\<close>
   by (cases \<open>k = 0\<close>) (simp_all add: rounded_modulo_def)
 
 end