# HG changeset patch
# User huffman
# Date 1232633381 28800
# Node ID 8096dc59d7794c128d2782a2caf5cf9d94d0f258
# Parent  d59918e668b7a3f632a4d73de61c7242391d0991# Parent  b36bcbc1be3a93b6b0c10ab2be5c68e4d0504ed7
merged

diff -r b36bcbc1be3a -r 8096dc59d779 src/HOL/Polynomial.thy
--- a/src/HOL/Polynomial.thy	Thu Jan 22 11:23:15 2009 +0100
+++ b/src/HOL/Polynomial.thy	Thu Jan 22 06:09:41 2009 -0800
@@ -475,6 +475,16 @@
 lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
   by (induct n, simp add: monom_0, simp add: monom_Suc)
 
+lemma degree_smult_eq [simp]:
+  fixes a :: "'a::idom"
+  shows "degree (smult a p) = (if a = 0 then 0 else degree p)"
+  by (cases "a = 0", simp, simp add: degree_def)
+
+lemma smult_eq_0_iff [simp]:
+  fixes a :: "'a::idom"
+  shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0"
+  by (simp add: expand_poly_eq)
+
 
 subsection {* Multiplication of polynomials *}
 
@@ -777,6 +787,12 @@
   qed
 qed
 
+lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0"
+by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0)
+
+lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x"
+by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0)
+
 lemmas pdivmod_rel_unique_div =
   pdivmod_rel_unique [THEN conjunct1, standard]
 
@@ -861,6 +877,54 @@
   thus "x mod y = x" by (rule mod_poly_eq)
 qed
 
+lemma pdivmod_rel_smult_left:
+  "pdivmod_rel x y q r
+    \<Longrightarrow> pdivmod_rel (smult a x) y (smult a q) (smult a r)"
+  unfolding pdivmod_rel_def by (simp add: smult_add_right)
+
+lemma div_smult_left: "(smult a x) div y = smult a (x div y)"
+  by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
+
+lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)"
+  by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
+
+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"
+  unfolding pdivmod_rel_def by simp
+
+lemma div_smult_right:
+  "a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)"
+  by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
+
+lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y"
+  by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
+
+lemma pdivmod_rel_mult:
+  "\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk>
+    \<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)"
+apply (cases "z = 0", simp add: pdivmod_rel_def)
+apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff)
+apply (cases "r = 0")
+apply (cases "r' = 0")
+apply (simp add: pdivmod_rel_def)
+apply (simp add: pdivmod_rel_def ring_simps degree_mult_eq)
+apply (cases "r' = 0")
+apply (simp add: pdivmod_rel_def degree_mult_eq)
+apply (simp add: pdivmod_rel_def ring_simps)
+apply (simp add: degree_mult_eq degree_add_less)
+done
+
+lemma poly_div_mult_right:
+  fixes x y z :: "'a::field poly"
+  shows "x div (y * z) = (x div y) div z"
+  by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
+
+lemma poly_mod_mult_right:
+  fixes x y z :: "'a::field poly"
+  shows "x mod (y * z) = y * (x div y mod z) + x mod y"
+  by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
+
 lemma mod_pCons:
   fixes a and x
   assumes y: "y \<noteq> 0"