--- a/src/HOL/Library/Polynomial.thy Tue Jul 01 23:02:25 2014 +0200
+++ b/src/HOL/Library/Polynomial.thy Tue Jul 01 21:57:08 2014 -0700
@@ -1588,6 +1588,35 @@
shows "(- x) mod y = - (x mod y)"
using mod_smult_left [of "- 1::'a"] by simp
+lemma pdivmod_rel_add_left:
+ assumes "pdivmod_rel x y q r"
+ assumes "pdivmod_rel x' y q' r'"
+ shows "pdivmod_rel (x + x') y (q + q') (r + r')"
+ using assms unfolding pdivmod_rel_def
+ by (auto simp add: distrib degree_add_less)
+
+lemma poly_div_add_left:
+ fixes x y z :: "'a::field poly"
+ shows "(x + y) div z = x div z + y div z"
+ using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel]
+ by (rule div_poly_eq)
+
+lemma poly_mod_add_left:
+ fixes x y z :: "'a::field poly"
+ shows "(x + y) mod z = x mod z + y mod z"
+ using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel]
+ by (rule mod_poly_eq)
+
+lemma poly_div_diff_left:
+ fixes x y z :: "'a::field poly"
+ shows "(x - y) div z = x div z - y div z"
+ by (simp only: diff_conv_add_uminus poly_div_add_left poly_div_minus_left)
+
+lemma poly_mod_diff_left:
+ fixes x y z :: "'a::field poly"
+ shows "(x - y) mod z = x mod z - y mod z"
+ by (simp only: diff_conv_add_uminus poly_mod_add_left poly_mod_minus_left)
+
lemma pdivmod_rel_smult_right:
"\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>
\<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r"