src/HOL/Real/RComplete.thy

   then obtain n where "inverse (real (Suc n)) < inverse x"
     using reals_Archimedean by blast
   hence "inverse (real (Suc n)) * x < inverse x * x"
     using x_greater_zero by (rule mult_strict_right_mono)
   hence "inverse (real (Suc n)) * x < 1"
-    using x_greater_zero by (simp add: real_mult_inverse_left mult_commute)
+    using x_greater_zero by simp
   hence "real (Suc n) * (inverse (real (Suc n)) * x) < real (Suc n) * 1"
     by (rule mult_strict_left_mono) simp
   hence "x < real (Suc n)"
-    by (simp add: mult_commute ring_eq_simps real_mult_inverse_left)
+    by (simp add: ring_eq_simps)
   thus "\<exists>(n::nat). x < real n" ..
 qed
 
 lemma reals_Archimedean3:
   assumes x_greater_zero: "0 < x"

   obtain n where "y * inverse x < real (n::nat)"
     using reals_Archimedean2 ..
   hence "y * inverse x * x < real n * x"
     using x_greater_zero by (simp add: mult_strict_right_mono)
   hence "x * inverse x * y < x * real n"
-    by (simp add: mult_commute ring_eq_simps)
+    by (simp add: ring_eq_simps)
   hence "y < real (n::nat) * x"
-    using x_not_zero by (simp add: real_mult_inverse_left ring_eq_simps)
+    using x_not_zero by (simp add: ring_eq_simps)
   thus "\<exists>(n::nat). y < real n * x" ..
 qed
 
 lemma reals_Archimedean6:
      "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"