1.1 --- a/src/HOL/Library/Polynomial.thy	Tue Jul 01 23:02:25 2014 +0200
     1.2 +++ b/src/HOL/Library/Polynomial.thy	Tue Jul 01 21:57:08 2014 -0700
     1.3 @@ -1588,6 +1588,35 @@
     1.4    shows "(- x) mod y = - (x mod y)"
     1.5    using mod_smult_left [of "- 1::'a"] by simp
     1.6  
     1.7 +lemma pdivmod_rel_add_left:
     1.8 +  assumes "pdivmod_rel x y q r"
     1.9 +  assumes "pdivmod_rel x' y q' r'"
    1.10 +  shows "pdivmod_rel (x + x') y (q + q') (r + r')"
    1.11 +  using assms unfolding pdivmod_rel_def
    1.12 +  by (auto simp add: distrib degree_add_less)
    1.13 +
    1.14 +lemma poly_div_add_left:
    1.15 +  fixes x y z :: "'a::field poly"
    1.16 +  shows "(x + y) div z = x div z + y div z"
    1.17 +  using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel]
    1.18 +  by (rule div_poly_eq)
    1.19 +
    1.20 +lemma poly_mod_add_left:
    1.21 +  fixes x y z :: "'a::field poly"
    1.22 +  shows "(x + y) mod z = x mod z + y mod z"
    1.23 +  using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel]
    1.24 +  by (rule mod_poly_eq)
    1.25 +
    1.26 +lemma poly_div_diff_left:
    1.27 +  fixes x y z :: "'a::field poly"
    1.28 +  shows "(x - y) div z = x div z - y div z"
    1.29 +  by (simp only: diff_conv_add_uminus poly_div_add_left poly_div_minus_left)
    1.30 +
    1.31 +lemma poly_mod_diff_left:
    1.32 +  fixes x y z :: "'a::field poly"
    1.33 +  shows "(x - y) mod z = x mod z - y mod z"
    1.34 +  by (simp only: diff_conv_add_uminus poly_mod_add_left poly_mod_minus_left)
    1.35 +
    1.36  lemma pdivmod_rel_smult_right:
    1.37    "\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>
    1.38      \<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r"