# HG changeset patch
# User haftmann
# Date 1455742318 -3600
# Node ID 35a9e1cbb5b345cca9921973b58abd26151dfb6a
# Parent  fd049b54ad68d9e5e6d39a86051e68510a83d2b2
separated potentially conflicting type class instance into separate theory

diff -r fd049b54ad68 -r 35a9e1cbb5b3 NEWS
--- a/NEWS	Wed Feb 17 21:51:58 2016 +0100
+++ b/NEWS	Wed Feb 17 21:51:58 2016 +0100
@@ -37,6 +37,12 @@
 
 * Class semiring_Lcd merged into semiring_Gcd.  INCOMPATIBILITY.
 
+* Library/Polynomial.thy contains also derivation of polynomials
+but not gcd/lcm on polynomials over fields.  This has been moved
+to a separate theory Library/Polynomial_GCD_euclidean.thy, to
+pave way for a possible future different type class instantiation
+for polynomials over factorial rings.  INCOMPATIBILITY.
+
 * Dropped various legacy fact bindings, whose replacements are often
 of a more general type also:
   lcm_left_commute_nat ~> lcm.left_commute
diff -r fd049b54ad68 -r 35a9e1cbb5b3 src/HOL/Codegenerator_Test/Candidates.thy
--- a/src/HOL/Codegenerator_Test/Candidates.thy	Wed Feb 17 21:51:58 2016 +0100
+++ b/src/HOL/Codegenerator_Test/Candidates.thy	Wed Feb 17 21:51:58 2016 +0100
@@ -8,6 +8,7 @@
   Complex_Main
   "~~/src/HOL/Library/Library"
   "~~/src/HOL/Library/Sublist_Order"
+  "~~/src/HOL/Library/Polynomial_GCD_euclidean"
   "~~/src/HOL/Data_Structures/Tree_Map"
   "~~/src/HOL/Data_Structures/Tree_Set"
   "~~/src/HOL/Number_Theory/Eratosthenes"
diff -r fd049b54ad68 -r 35a9e1cbb5b3 src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Wed Feb 17 21:51:58 2016 +0100
+++ b/src/HOL/Library/Library.thy	Wed Feb 17 21:51:58 2016 +0100
@@ -53,7 +53,6 @@
   Parallel
   Permutation
   Permutations
-  Poly_Deriv
   Polynomial
   Preorder
   Product_Vector
diff -r fd049b54ad68 -r 35a9e1cbb5b3 src/HOL/Library/Poly_Deriv.thy
--- a/src/HOL/Library/Poly_Deriv.thy	Wed Feb 17 21:51:58 2016 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,634 +0,0 @@
-(*  Title:      HOL/Library/Poly_Deriv.thy
-    Author:     Amine Chaieb
-    Author:     Brian Huffman
-*)
-
-section\<open>Polynomials and Differentiation\<close>
-
-theory Poly_Deriv
-imports Deriv Polynomial
-begin
-
-subsection \<open>Derivatives of univariate polynomials\<close>
-
-function pderiv :: "('a :: semidom) poly \<Rightarrow> 'a poly"
-where
-  [simp del]: "pderiv (pCons a p) = (if p = 0 then 0 else p + pCons 0 (pderiv p))"
-  by (auto intro: pCons_cases)
-
-termination pderiv
-  by (relation "measure degree") simp_all
-
-lemma pderiv_0 [simp]:
-  "pderiv 0 = 0"
-  using pderiv.simps [of 0 0] by simp
-
-lemma pderiv_pCons:
-  "pderiv (pCons a p) = p + pCons 0 (pderiv p)"
-  by (simp add: pderiv.simps)
-
-lemma pderiv_1 [simp]: "pderiv 1 = 0" 
-  unfolding one_poly_def by (simp add: pderiv_pCons)
-
-lemma pderiv_of_nat  [simp]: "pderiv (of_nat n) = 0"
-  and pderiv_numeral [simp]: "pderiv (numeral m) = 0"
-  by (simp_all add: of_nat_poly numeral_poly pderiv_pCons)
-
-lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
-  by (induct p arbitrary: n) 
-     (auto simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
-
-fun pderiv_coeffs_code :: "('a :: semidom) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
-  "pderiv_coeffs_code f (x # xs) = cCons (f * x) (pderiv_coeffs_code (f+1) xs)"
-| "pderiv_coeffs_code f [] = []"
-
-definition pderiv_coeffs :: "('a :: semidom) list \<Rightarrow> 'a list" where
-  "pderiv_coeffs xs = pderiv_coeffs_code 1 (tl xs)"
-
-(* Efficient code for pderiv contributed by René Thiemann and Akihisa Yamada *)
-lemma pderiv_coeffs_code: 
-  "nth_default 0 (pderiv_coeffs_code f xs) n = (f + of_nat n) * (nth_default 0 xs n)"
-proof (induct xs arbitrary: f n)
-  case (Cons x xs f n)
-  show ?case 
-  proof (cases n)
-    case 0
-    thus ?thesis by (cases "pderiv_coeffs_code (f + 1) xs = [] \<and> f * x = 0", auto simp: cCons_def)
-  next
-    case (Suc m) note n = this
-    show ?thesis 
-    proof (cases "pderiv_coeffs_code (f + 1) xs = [] \<and> f * x = 0")
-      case False
-      hence "nth_default 0 (pderiv_coeffs_code f (x # xs)) n = 
-               nth_default 0 (pderiv_coeffs_code (f + 1) xs) m" 
-        by (auto simp: cCons_def n)
-      also have "\<dots> = (f + of_nat n) * (nth_default 0 xs m)" 
-        unfolding Cons by (simp add: n add_ac)
-      finally show ?thesis by (simp add: n)
-    next
-      case True
-      {
-        fix g 
-        have "pderiv_coeffs_code g xs = [] \<Longrightarrow> g + of_nat m = 0 \<or> nth_default 0 xs m = 0"
-        proof (induct xs arbitrary: g m)
-          case (Cons x xs g)
-          from Cons(2) have empty: "pderiv_coeffs_code (g + 1) xs = []"
-                            and g: "(g = 0 \<or> x = 0)"
-            by (auto simp: cCons_def split: if_splits)
-          note IH = Cons(1)[OF empty]
-          from IH[of m] IH[of "m - 1"] g
-          show ?case by (cases m, auto simp: field_simps)
-        qed simp
-      } note empty = this
-      from True have "nth_default 0 (pderiv_coeffs_code f (x # xs)) n = 0"
-        by (auto simp: cCons_def n)
-      moreover have "(f + of_nat n) * nth_default 0 (x # xs) n = 0" using True
-        by (simp add: n, insert empty[of "f+1"], auto simp: field_simps)
-      ultimately show ?thesis by simp
-    qed
-  qed
-qed simp
-
-lemma map_upt_Suc: "map f [0 ..< Suc n] = f 0 # map (\<lambda> i. f (Suc i)) [0 ..< n]"
-  by (induct n arbitrary: f, auto)
-
-lemma coeffs_pderiv_code [code abstract]:
-  "coeffs (pderiv p) = pderiv_coeffs (coeffs p)" unfolding pderiv_coeffs_def
-proof (rule coeffs_eqI, unfold pderiv_coeffs_code coeff_pderiv, goal_cases)
-  case (1 n)
-  have id: "coeff p (Suc n) = nth_default 0 (map (\<lambda>i. coeff p (Suc i)) [0..<degree p]) n"
-    by (cases "n < degree p", auto simp: nth_default_def coeff_eq_0)
-  show ?case unfolding coeffs_def map_upt_Suc by (auto simp: id)
-next
-  case 2
-  obtain n xs where id: "tl (coeffs p) = xs" "(1 :: 'a) = n" by auto
-  from 2 show ?case
-    unfolding id by (induct xs arbitrary: n, auto simp: cCons_def)
-qed
-
-context
-  assumes "SORT_CONSTRAINT('a::{semidom, semiring_char_0})"
-begin
-
-lemma pderiv_eq_0_iff: 
-  "pderiv (p :: 'a poly) = 0 \<longleftrightarrow> degree p = 0"
-  apply (rule iffI)
-  apply (cases p, simp)
-  apply (simp add: poly_eq_iff coeff_pderiv del: of_nat_Suc)
-  apply (simp add: poly_eq_iff coeff_pderiv coeff_eq_0)
-  done
-
-lemma degree_pderiv: "degree (pderiv (p :: 'a poly)) = degree p - 1"
-  apply (rule order_antisym [OF degree_le])
-  apply (simp add: coeff_pderiv coeff_eq_0)
-  apply (cases "degree p", simp)
-  apply (rule le_degree)
-  apply (simp add: coeff_pderiv del: of_nat_Suc)
-  apply (metis degree_0 leading_coeff_0_iff nat.distinct(1))
-  done
-
-lemma not_dvd_pderiv: 
-  assumes "degree (p :: 'a poly) \<noteq> 0"
-  shows "\<not> p dvd pderiv p"
-proof
-  assume dvd: "p dvd pderiv p"
-  then obtain q where p: "pderiv p = p * q" unfolding dvd_def by auto
-  from dvd have le: "degree p \<le> degree (pderiv p)"
-    by (simp add: assms dvd_imp_degree_le pderiv_eq_0_iff)
-  from this[unfolded degree_pderiv] assms show False by auto
-qed
-
-lemma dvd_pderiv_iff [simp]: "(p :: 'a poly) dvd pderiv p \<longleftrightarrow> degree p = 0"
-  using not_dvd_pderiv[of p] by (auto simp: pderiv_eq_0_iff [symmetric])
-
-end
-
-lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"
-by (simp add: pderiv_pCons)
-
-lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
-by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
-
-lemma pderiv_minus: "pderiv (- p :: 'a :: idom poly) = - pderiv p"
-by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
-
-lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
-by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
-
-lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
-by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
-
-lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
-by (induct p) (auto simp: pderiv_add pderiv_smult pderiv_pCons algebra_simps)
-
-lemma pderiv_power_Suc:
-  "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
-apply (induct n)
-apply simp
-apply (subst power_Suc)
-apply (subst pderiv_mult)
-apply (erule ssubst)
-apply (simp only: of_nat_Suc smult_add_left smult_1_left)
-apply (simp add: algebra_simps)
-done
-
-lemma pderiv_setprod: "pderiv (setprod f (as)) = 
-  (\<Sum>a \<in> as. setprod f (as - {a}) * pderiv (f a))"
-proof (induct as rule: infinite_finite_induct)
-  case (insert a as)
-  hence id: "setprod f (insert a as) = f a * setprod f as" 
-    "\<And> g. setsum g (insert a as) = g a + setsum g as"
-    "insert a as - {a} = as"
-    by auto
-  {
-    fix b
-    assume "b \<in> as"
-    hence id2: "insert a as - {b} = insert a (as - {b})" using \<open>a \<notin> as\<close> by auto
-    have "setprod f (insert a as - {b}) = f a * setprod f (as - {b})"
-      unfolding id2
-      by (subst setprod.insert, insert insert, auto)
-  } note id2 = this
-  show ?case
-    unfolding id pderiv_mult insert(3) setsum_right_distrib
-    by (auto simp add: ac_simps id2 intro!: setsum.cong)
-qed auto
-
-lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
-by (rule DERIV_cong, rule DERIV_pow, simp)
-declare DERIV_pow2 [simp] DERIV_pow [simp]
-
-lemma DERIV_add_const: "DERIV f x :> D ==>  DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
-by (rule DERIV_cong, rule DERIV_add, auto)
-
-lemma poly_DERIV [simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
-  by (induct p, auto intro!: derivative_eq_intros simp add: pderiv_pCons)
-
-lemma continuous_on_poly [continuous_intros]: 
-  fixes p :: "'a :: {real_normed_field} poly"
-  assumes "continuous_on A f"
-  shows   "continuous_on A (\<lambda>x. poly p (f x))"
-proof -
-  have "continuous_on A (\<lambda>x. (\<Sum>i\<le>degree p. (f x) ^ i * coeff p i))" 
-    by (intro continuous_intros assms)
-  also have "\<dots> = (\<lambda>x. poly p (f x))" by (intro ext) (simp add: poly_altdef mult_ac)
-  finally show ?thesis .
-qed
-
-text\<open>Consequences of the derivative theorem above\<close>
-
-lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (at x::real filter)"
-apply (simp add: real_differentiable_def)
-apply (blast intro: poly_DERIV)
-done
-
-lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
-by (rule poly_DERIV [THEN DERIV_isCont])
-
-lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
-      ==> \<exists>x. a < x & x < b & (poly p x = 0)"
-using IVT_objl [of "poly p" a 0 b]
-by (auto simp add: order_le_less)
-
-lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
-      ==> \<exists>x. a < x & x < b & (poly p x = 0)"
-by (insert poly_IVT_pos [where p = "- p" ]) simp
-
-lemma poly_IVT:
-  fixes p::"real poly"
-  assumes "a<b" and "poly p a * poly p b < 0"
-  shows "\<exists>x>a. x < b \<and> poly p x = 0"
-by (metis assms(1) assms(2) less_not_sym mult_less_0_iff poly_IVT_neg poly_IVT_pos)
-
-lemma poly_MVT: "(a::real) < b ==>
-     \<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
-using MVT [of a b "poly p"]
-apply auto
-apply (rule_tac x = z in exI)
-apply (auto simp add: mult_left_cancel poly_DERIV [THEN DERIV_unique])
-done
-
-lemma poly_MVT':
-  assumes "{min a b..max a b} \<subseteq> A"
-  shows   "\<exists>x\<in>A. poly p b - poly p a = (b - a) * poly (pderiv p) (x::real)"
-proof (cases a b rule: linorder_cases)
-  case less
-  from poly_MVT[OF less, of p] guess x by (elim exE conjE)
-  thus ?thesis by (intro bexI[of _ x]) (auto intro!: subsetD[OF assms])
-
-next
-  case greater
-  from poly_MVT[OF greater, of p] guess x by (elim exE conjE)
-  thus ?thesis by (intro bexI[of _ x]) (auto simp: algebra_simps intro!: subsetD[OF assms])
-qed (insert assms, auto)
-
-lemma poly_pinfty_gt_lc:
-  fixes p:: "real poly"
-  assumes  "lead_coeff p > 0" 
-  shows "\<exists> n. \<forall> x \<ge> n. poly p x \<ge> lead_coeff p" using assms
-proof (induct p)
-  case 0
-  thus ?case by auto
-next
-  case (pCons a p)
-  have "\<lbrakk>a\<noteq>0;p=0\<rbrakk> \<Longrightarrow> ?case" by auto
-  moreover have "p\<noteq>0 \<Longrightarrow> ?case"
-    proof -
-      assume "p\<noteq>0"
-      then obtain n1 where gte_lcoeff:"\<forall>x\<ge>n1. lead_coeff p \<le> poly p x" using that pCons by auto
-      have gt_0:"lead_coeff p >0" using pCons(3) \<open>p\<noteq>0\<close> by auto
-      def n\<equiv>"max n1 (1+ \<bar>a\<bar>/(lead_coeff p))"
-      show ?thesis 
-        proof (rule_tac x=n in exI,rule,rule) 
-          fix x assume "n \<le> x"
-          hence "lead_coeff p \<le> poly p x" 
-            using gte_lcoeff unfolding n_def by auto
-          hence " \<bar>a\<bar>/(lead_coeff p) \<ge> \<bar>a\<bar>/(poly p x)" and "poly p x>0" using gt_0
-            by (intro frac_le,auto)
-          hence "x\<ge>1+ \<bar>a\<bar>/(poly p x)" using \<open>n\<le>x\<close>[unfolded n_def] by auto
-          thus "lead_coeff (pCons a p) \<le> poly (pCons a p) x"
-            using \<open>lead_coeff p \<le> poly p x\<close> \<open>poly p x>0\<close> \<open>p\<noteq>0\<close>
-            by (auto simp add:field_simps)
-        qed
-    qed
-  ultimately show ?case by fastforce
-qed
-
-
-subsection \<open>Algebraic numbers\<close>
-
-text \<open>
-  Algebraic numbers can be defined in two equivalent ways: all real numbers that are 
-  roots of rational polynomials or of integer polynomials. The Algebraic-Numbers AFP entry 
-  uses the rational definition, but we need the integer definition.
-
-  The equivalence is obvious since any rational polynomial can be multiplied with the 
-  LCM of its coefficients, yielding an integer polynomial with the same roots.
-\<close>
-subsection \<open>Algebraic numbers\<close>
-
-definition algebraic :: "'a :: field_char_0 \<Rightarrow> bool" where
-  "algebraic x \<longleftrightarrow> (\<exists>p. (\<forall>i. coeff p i \<in> \<int>) \<and> p \<noteq> 0 \<and> poly p x = 0)"
-
-lemma algebraicI:
-  assumes "\<And>i. coeff p i \<in> \<int>" "p \<noteq> 0" "poly p x = 0"
-  shows   "algebraic x"
-  using assms unfolding algebraic_def by blast
-  
-lemma algebraicE:
-  assumes "algebraic x"
-  obtains p where "\<And>i. coeff p i \<in> \<int>" "p \<noteq> 0" "poly p x = 0"
-  using assms unfolding algebraic_def by blast
-
-lemma quotient_of_denom_pos': "snd (quotient_of x) > 0"
-  using quotient_of_denom_pos[OF surjective_pairing] .
-  
-lemma of_int_div_in_Ints: 
-  "b dvd a \<Longrightarrow> of_int a div of_int b \<in> (\<int> :: 'a :: ring_div set)"
-proof (cases "of_int b = (0 :: 'a)")
-  assume "b dvd a" "of_int b \<noteq> (0::'a)"
-  then obtain c where "a = b * c" by (elim dvdE)
-  with \<open>of_int b \<noteq> (0::'a)\<close> show ?thesis by simp
-qed auto
-
-lemma of_int_divide_in_Ints: 
-  "b dvd a \<Longrightarrow> of_int a / of_int b \<in> (\<int> :: 'a :: field set)"
-proof (cases "of_int b = (0 :: 'a)")
-  assume "b dvd a" "of_int b \<noteq> (0::'a)"
-  then obtain c where "a = b * c" by (elim dvdE)
-  with \<open>of_int b \<noteq> (0::'a)\<close> show ?thesis by simp
-qed auto
-
-lemma algebraic_altdef:
-  fixes p :: "'a :: field_char_0 poly"
-  shows "algebraic x \<longleftrightarrow> (\<exists>p. (\<forall>i. coeff p i \<in> \<rat>) \<and> p \<noteq> 0 \<and> poly p x = 0)"
-proof safe
-  fix p assume rat: "\<forall>i. coeff p i \<in> \<rat>" and root: "poly p x = 0" and nz: "p \<noteq> 0"
-  def cs \<equiv> "coeffs p"
-  from rat have "\<forall>c\<in>range (coeff p). \<exists>c'. c = of_rat c'" unfolding Rats_def by blast
-  then obtain f where f: "\<And>i. coeff p i = of_rat (f (coeff p i))" 
-    by (subst (asm) bchoice_iff) blast
-  def cs' \<equiv> "map (quotient_of \<circ> f) (coeffs p)"
-  def d \<equiv> "Lcm (set (map snd cs'))"
-  def p' \<equiv> "smult (of_int d) p"
-  
-  have "\<forall>n. coeff p' n \<in> \<int>"
-  proof
-    fix n :: nat
-    show "coeff p' n \<in> \<int>"
-    proof (cases "n \<le> degree p")
-      case True
-      def c \<equiv> "coeff p n"
-      def a \<equiv> "fst (quotient_of (f (coeff p n)))" and b \<equiv> "snd (quotient_of (f (coeff p n)))"
-      have b_pos: "b > 0" unfolding b_def using quotient_of_denom_pos' by simp
-      have "coeff p' n = of_int d * coeff p n" by (simp add: p'_def)
-      also have "coeff p n = of_rat (of_int a / of_int b)" unfolding a_def b_def
-        by (subst quotient_of_div [of "f (coeff p n)", symmetric])
-           (simp_all add: f [symmetric])
-      also have "of_int d * \<dots> = of_rat (of_int (a*d) / of_int b)"
-        by (simp add: of_rat_mult of_rat_divide)
-      also from nz True have "b \<in> snd ` set cs'" unfolding cs'_def
-        by (force simp: o_def b_def coeffs_def simp del: upt_Suc)
-      hence "b dvd (a * d)" unfolding d_def by simp
-      hence "of_int (a * d) / of_int b \<in> (\<int> :: rat set)"
-        by (rule of_int_divide_in_Ints)
-      hence "of_rat (of_int (a * d) / of_int b) \<in> \<int>" by (elim Ints_cases) auto
-      finally show ?thesis .
-    qed (auto simp: p'_def not_le coeff_eq_0)
-  qed
-  
-  moreover have "set (map snd cs') \<subseteq> {0<..}"
-    unfolding cs'_def using quotient_of_denom_pos' by (auto simp: coeffs_def simp del: upt_Suc) 
-  hence "d \<noteq> 0" unfolding d_def by (induction cs') simp_all
-  with nz have "p' \<noteq> 0" by (simp add: p'_def)
-  moreover from root have "poly p' x = 0" by (simp add: p'_def)
-  ultimately show "algebraic x" unfolding algebraic_def by blast
-next
-
-  assume "algebraic x"
-  then obtain p where p: "\<And>i. coeff p i \<in> \<int>" "poly p x = 0" "p \<noteq> 0" 
-    by (force simp: algebraic_def)
-  moreover have "coeff p i \<in> \<int> \<Longrightarrow> coeff p i \<in> \<rat>" for i by (elim Ints_cases) simp
-  ultimately show  "(\<exists>p. (\<forall>i. coeff p i \<in> \<rat>) \<and> p \<noteq> 0 \<and> poly p x = 0)" by auto
-qed
-
-
-text\<open>Lemmas for Derivatives\<close>
-
-lemma order_unique_lemma:
-  fixes p :: "'a::idom poly"
-  assumes "[:-a, 1:] ^ n dvd p" "\<not> [:-a, 1:] ^ Suc n dvd p"
-  shows "n = order a p"
-unfolding Polynomial.order_def
-apply (rule Least_equality [symmetric])
-apply (fact assms)
-apply (rule classical)
-apply (erule notE)
-unfolding not_less_eq_eq
-using assms(1) apply (rule power_le_dvd)
-apply assumption
-done
-
-lemma lemma_order_pderiv1:
-  "pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
-    smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
-apply (simp only: pderiv_mult pderiv_power_Suc)
-apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons)
-done
-
-lemma lemma_order_pderiv:
-  fixes p :: "'a :: field_char_0 poly"
-  assumes n: "0 < n" 
-      and pd: "pderiv p \<noteq> 0" 
-      and pe: "p = [:- a, 1:] ^ n * q" 
-      and nd: "~ [:- a, 1:] dvd q"
-    shows "n = Suc (order a (pderiv p))"
-using n 
-proof -
-  have "pderiv ([:- a, 1:] ^ n * q) \<noteq> 0"
-    using assms by auto
-  obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) \<noteq> 0"
-    using assms by (cases n) auto
-  have *: "!!k l. k dvd k * pderiv q + smult (of_nat (Suc n')) l \<Longrightarrow> k dvd l"
-    by (auto simp del: of_nat_Suc simp: dvd_add_right_iff dvd_smult_iff)
-  have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))" 
-  proof (rule order_unique_lemma)
-    show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
-      apply (subst lemma_order_pderiv1)
-      apply (rule dvd_add)
-      apply (metis dvdI dvd_mult2 power_Suc2)
-      apply (metis dvd_smult dvd_triv_right)
-      done
-  next
-    show "\<not> [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
-     apply (subst lemma_order_pderiv1)
-     by (metis * nd dvd_mult_cancel_right power_not_zero pCons_eq_0_iff power_Suc zero_neq_one)
-  qed
-  then show ?thesis
-    by (metis \<open>n = Suc n'\<close> pe)
-qed
-
-lemma order_decomp:
-  assumes "p \<noteq> 0"
-  shows "\<exists>q. p = [:- a, 1:] ^ order a p * q \<and> \<not> [:- a, 1:] dvd q"
-proof -
-  from assms have A: "[:- a, 1:] ^ order a p dvd p"
-    and B: "\<not> [:- a, 1:] ^ Suc (order a p) dvd p" by (auto dest: order)
-  from A obtain q where C: "p = [:- a, 1:] ^ order a p * q" ..
-  with B have "\<not> [:- a, 1:] ^ Suc (order a p) dvd [:- a, 1:] ^ order a p * q"
-    by simp
-  then have "\<not> [:- a, 1:] ^ order a p * [:- a, 1:] dvd [:- a, 1:] ^ order a p * q"
-    by simp
-  then have D: "\<not> [:- a, 1:] dvd q"
-    using idom_class.dvd_mult_cancel_left [of "[:- a, 1:] ^ order a p" "[:- a, 1:]" q]
-    by auto
-  from C D show ?thesis by blast
-qed
-
-lemma order_pderiv:
-  "\<lbrakk>pderiv p \<noteq> 0; order a (p :: 'a :: field_char_0 poly) \<noteq> 0\<rbrakk> \<Longrightarrow>
-     (order a p = Suc (order a (pderiv p)))"
-apply (case_tac "p = 0", simp)
-apply (drule_tac a = a and p = p in order_decomp)
-using neq0_conv
-apply (blast intro: lemma_order_pderiv)
-done
-
-lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q"
-proof -
-  def i \<equiv> "order a p"
-  def j \<equiv> "order a q"
-  def t \<equiv> "[:-a, 1:]"
-  have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0"
-    unfolding t_def by (simp add: dvd_iff_poly_eq_0)
-  assume "p * q \<noteq> 0"
-  then show "order a (p * q) = i + j"
-    apply clarsimp
-    apply (drule order [where a=a and p=p, folded i_def t_def])
-    apply (drule order [where a=a and p=q, folded j_def t_def])
-    apply clarify
-    apply (erule dvdE)+
-    apply (rule order_unique_lemma [symmetric], fold t_def)
-    apply (simp_all add: power_add t_dvd_iff)
-    done
-qed
-
-lemma order_smult:
-  assumes "c \<noteq> 0" 
-  shows "order x (smult c p) = order x p"
-proof (cases "p = 0")
-  case False
-  have "smult c p = [:c:] * p" by simp
-  also from assms False have "order x \<dots> = order x [:c:] + order x p" 
-    by (subst order_mult) simp_all
-  also from assms have "order x [:c:] = 0" by (intro order_0I) auto
-  finally show ?thesis by simp
-qed simp
-
-(* Next two lemmas contributed by Wenda Li *)
-lemma order_1_eq_0 [simp]:"order x 1 = 0" 
-  by (metis order_root poly_1 zero_neq_one)
-
-lemma order_power_n_n: "order a ([:-a,1:]^n)=n" 
-proof (induct n) (*might be proved more concisely using nat_less_induct*)
-  case 0
-  thus ?case by (metis order_root poly_1 power_0 zero_neq_one)
-next 
-  case (Suc n)
-  have "order a ([:- a, 1:] ^ Suc n)=order a ([:- a, 1:] ^ n) + order a [:-a,1:]" 
-    by (metis (no_types, hide_lams) One_nat_def add_Suc_right monoid_add_class.add.right_neutral 
-      one_neq_zero order_mult pCons_eq_0_iff power_add power_eq_0_iff power_one_right)
-  moreover have "order a [:-a,1:]=1" unfolding order_def
-    proof (rule Least_equality,rule ccontr)
-      assume  "\<not> \<not> [:- a, 1:] ^ Suc 1 dvd [:- a, 1:]"
-      hence "[:- a, 1:] ^ Suc 1 dvd [:- a, 1:]" by simp
-      hence "degree ([:- a, 1:] ^ Suc 1) \<le> degree ([:- a, 1:] )" 
-        by (rule dvd_imp_degree_le,auto) 
-      thus False by auto
-    next
-      fix y assume asm:"\<not> [:- a, 1:] ^ Suc y dvd [:- a, 1:]"
-      show "1 \<le> y" 
-        proof (rule ccontr)
-          assume "\<not> 1 \<le> y"
-          hence "y=0" by auto
-          hence "[:- a, 1:] ^ Suc y dvd [:- a, 1:]" by auto
-          thus False using asm by auto
-        qed
-    qed
-  ultimately show ?case using Suc by auto
-qed
-
-text\<open>Now justify the standard squarefree decomposition, i.e. f / gcd(f,f').\<close>
-
-lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p"
-apply (cases "p = 0", auto)
-apply (drule order_2 [where a=a and p=p])
-apply (metis not_less_eq_eq power_le_dvd)
-apply (erule power_le_dvd [OF order_1])
-done
-
-lemma poly_squarefree_decomp_order:
-  assumes "pderiv (p :: 'a :: field_char_0 poly) \<noteq> 0"
-  and p: "p = q * d"
-  and p': "pderiv p = e * d"
-  and d: "d = r * p + s * pderiv p"
-  shows "order a q = (if order a p = 0 then 0 else 1)"
-proof (rule classical)
-  assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)"
-  from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
-  with p have "order a p = order a q + order a d"
-    by (simp add: order_mult)
-  with 1 have "order a p \<noteq> 0" by (auto split: if_splits)
-  have "order a (pderiv p) = order a e + order a d"
-    using \<open>pderiv p \<noteq> 0\<close> \<open>pderiv p = e * d\<close> by (simp add: order_mult)
-  have "order a p = Suc (order a (pderiv p))"
-    using \<open>pderiv p \<noteq> 0\<close> \<open>order a p \<noteq> 0\<close> by (rule order_pderiv)
-  have "d \<noteq> 0" using \<open>p \<noteq> 0\<close> \<open>p = q * d\<close> by simp
-  have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
-    apply (simp add: d)
-    apply (rule dvd_add)
-    apply (rule dvd_mult)
-    apply (simp add: order_divides \<open>p \<noteq> 0\<close>
-           \<open>order a p = Suc (order a (pderiv p))\<close>)
-    apply (rule dvd_mult)
-    apply (simp add: order_divides)
-    done
-  then have "order a (pderiv p) \<le> order a d"
-    using \<open>d \<noteq> 0\<close> by (simp add: order_divides)
-  show ?thesis
-    using \<open>order a p = order a q + order a d\<close>
-    using \<open>order a (pderiv p) = order a e + order a d\<close>
-    using \<open>order a p = Suc (order a (pderiv p))\<close>
-    using \<open>order a (pderiv p) \<le> order a d\<close>
-    by auto
-qed
-
-lemma poly_squarefree_decomp_order2: 
-     "\<lbrakk>pderiv p \<noteq> (0 :: 'a :: field_char_0 poly);
-       p = q * d;
-       pderiv p = e * d;
-       d = r * p + s * pderiv p
-      \<rbrakk> \<Longrightarrow> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
-by (blast intro: poly_squarefree_decomp_order)
-
-lemma order_pderiv2: 
-  "\<lbrakk>pderiv p \<noteq> 0; order a (p :: 'a :: field_char_0 poly) \<noteq> 0\<rbrakk>
-      \<Longrightarrow> (order a (pderiv p) = n) = (order a p = Suc n)"
-by (auto dest: order_pderiv)
-
-definition
-  rsquarefree :: "'a::idom poly => bool" where
-  "rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))"
-
-lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h :: 'a :: {semidom,semiring_char_0}:]"
-apply (simp add: pderiv_eq_0_iff)
-apply (case_tac p, auto split: if_splits)
-done
-
-lemma rsquarefree_roots:
-  fixes p :: "'a :: field_char_0 poly"
-  shows "rsquarefree p = (\<forall>a. \<not>(poly p a = 0 \<and> poly (pderiv p) a = 0))"
-apply (simp add: rsquarefree_def)
-apply (case_tac "p = 0", simp, simp)
-apply (case_tac "pderiv p = 0")
-apply simp
-apply (drule pderiv_iszero, clarsimp)
-apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree)
-apply (force simp add: order_root order_pderiv2)
-done
-
-lemma poly_squarefree_decomp:
-  assumes "pderiv (p :: 'a :: field_char_0 poly) \<noteq> 0"
-    and "p = q * d"
-    and "pderiv p = e * d"
-    and "d = r * p + s * pderiv p"
-  shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
-proof -
-  from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
-  with \<open>p = q * d\<close> have "q \<noteq> 0" by simp
-  have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
-    using assms by (rule poly_squarefree_decomp_order2)
-  with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> show ?thesis
-    by (simp add: rsquarefree_def order_root)
-qed
-
-end
diff -r fd049b54ad68 -r 35a9e1cbb5b3 src/HOL/Library/Polynomial.thy
--- a/src/HOL/Library/Polynomial.thy	Wed Feb 17 21:51:58 2016 +0100
+++ b/src/HOL/Library/Polynomial.thy	Wed Feb 17 21:51:58 2016 +0100
@@ -1,13 +1,15 @@
 (*  Title:      HOL/Library/Polynomial.thy
     Author:     Brian Huffman
     Author:     Clemens Ballarin
+    Author:     Amine Chaieb
     Author:     Florian Haftmann
 *)
 
 section \<open>Polynomials as type over a ring structure\<close>
 
 theory Polynomial
-imports Main "~~/src/HOL/GCD" "~~/src/HOL/Library/More_List" "~~/src/HOL/Library/Infinite_Set"
+imports Main "~~/src/HOL/Deriv" "~~/src/HOL/Library/More_List"
+  "~~/src/HOL/Library/Infinite_Set"
 begin
 
 subsection \<open>Auxiliary: operations for lists (later) representing coefficients\<close>
@@ -1900,133 +1902,6 @@
   by (subst (asm) order_root) auto
 
 
-subsection \<open>GCD of polynomials\<close>
-
-instantiation poly :: (field) gcd
-begin
-
-function gcd_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
-where
-  "gcd (x::'a poly) 0 = smult (inverse (coeff x (degree x))) x"
-| "y \<noteq> 0 \<Longrightarrow> gcd (x::'a poly) y = gcd y (x mod y)"
-by auto
-
-termination "gcd :: _ poly \<Rightarrow> _"
-by (relation "measure (\<lambda>(x, y). if y = 0 then 0 else Suc (degree y))")
-   (auto dest: degree_mod_less)
-
-declare gcd_poly.simps [simp del]
-
-definition lcm_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
-where
-  "lcm_poly a b = a * b div smult (coeff a (degree a) * coeff b (degree b)) (gcd a b)"
-
-instance ..
-
-end
-
-lemma
-  fixes x y :: "_ poly"
-  shows poly_gcd_dvd1 [iff]: "gcd x y dvd x"
-    and poly_gcd_dvd2 [iff]: "gcd x y dvd y"
-  apply (induct x y rule: gcd_poly.induct)
-  apply (simp_all add: gcd_poly.simps)
-  apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
-  apply (blast dest: dvd_mod_imp_dvd)
-  done
-
-lemma poly_gcd_greatest:
-  fixes k x y :: "_ poly"
-  shows "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd gcd x y"
-  by (induct x y rule: gcd_poly.induct)
-     (simp_all add: gcd_poly.simps dvd_mod dvd_smult)
-
-lemma dvd_poly_gcd_iff [iff]:
-  fixes k x y :: "_ poly"
-  shows "k dvd gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
-  by (auto intro!: poly_gcd_greatest intro: dvd_trans [of _ "gcd x y"])
-
-lemma poly_gcd_monic:
-  fixes x y :: "_ poly"
-  shows "coeff (gcd x y) (degree (gcd x y)) =
-    (if x = 0 \<and> y = 0 then 0 else 1)"
-  by (induct x y rule: gcd_poly.induct)
-     (simp_all add: gcd_poly.simps nonzero_imp_inverse_nonzero)
-
-lemma poly_gcd_zero_iff [simp]:
-  fixes x y :: "_ poly"
-  shows "gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
-  by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)
-
-lemma poly_gcd_0_0 [simp]:
-  "gcd (0::_ poly) 0 = 0"
-  by simp
-
-lemma poly_dvd_antisym:
-  fixes p q :: "'a::idom poly"
-  assumes coeff: "coeff p (degree p) = coeff q (degree q)"
-  assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
-proof (cases "p = 0")
-  case True with coeff show "p = q" by simp
-next
-  case False with coeff have "q \<noteq> 0" by auto
-  have degree: "degree p = degree q"
-    using \<open>p dvd q\<close> \<open>q dvd p\<close> \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close>
-    by (intro order_antisym dvd_imp_degree_le)
-
-  from \<open>p dvd q\<close> obtain a where a: "q = p * a" ..
-  with \<open>q \<noteq> 0\<close> have "a \<noteq> 0" by auto
-  with degree a \<open>p \<noteq> 0\<close> have "degree a = 0"
-    by (simp add: degree_mult_eq)
-  with coeff a show "p = q"
-    by (cases a, auto split: if_splits)
-qed
-
-lemma poly_gcd_unique:
-  fixes d x y :: "_ poly"
-  assumes dvd1: "d dvd x" and dvd2: "d dvd y"
-    and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
-    and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
-  shows "gcd x y = d"
-proof -
-  have "coeff (gcd x y) (degree (gcd x y)) = coeff d (degree d)"
-    by (simp_all add: poly_gcd_monic monic)
-  moreover have "gcd x y dvd d"
-    using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
-  moreover have "d dvd gcd x y"
-    using dvd1 dvd2 by (rule poly_gcd_greatest)
-  ultimately show ?thesis
-    by (rule poly_dvd_antisym)
-qed
-
-instance poly :: (field) semiring_gcd
-proof
-  fix p q :: "'a::field poly"
-  show "normalize (gcd p q) = gcd p q"
-    by (induct p q rule: gcd_poly.induct)
-      (simp_all add: gcd_poly.simps normalize_poly_def)
-  show "lcm p q = normalize (p * q) div gcd p q"
-    by (simp add: coeff_degree_mult div_smult_left div_smult_right lcm_poly_def normalize_poly_def)
-      (metis (no_types, lifting) div_smult_right inverse_mult_distrib inverse_zero mult.commute pdivmod_rel pdivmod_rel_def smult_eq_0_iff)
-qed simp_all
-
-lemma poly_gcd_1_left [simp]: "gcd 1 y = (1 :: _ poly)"
-by (rule poly_gcd_unique) simp_all
-
-lemma poly_gcd_1_right [simp]: "gcd x 1 = (1 :: _ poly)"
-by (rule poly_gcd_unique) simp_all
-
-lemma poly_gcd_minus_left [simp]: "gcd (- x) y = gcd x (y :: _ poly)"
-by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
-
-lemma poly_gcd_minus_right [simp]: "gcd x (- y) = gcd x (y :: _ poly)"
-by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
-
-lemma poly_gcd_code [code]:
-  "gcd x y = (if y = 0 then normalize x else gcd y (x mod (y :: _ poly)))"
-  by (simp add: gcd_poly.simps)
-
-
 subsection \<open>Additional induction rules on polynomials\<close>
 
 text \<open>
@@ -2338,8 +2213,630 @@
 
 lemma lead_coeff_nonzero: "p \<noteq> 0 \<Longrightarrow> lead_coeff p \<noteq> 0"
   by (simp add: lead_coeff_def)
+
+
+subsection \<open>Derivatives of univariate polynomials\<close>
+
+function pderiv :: "('a :: semidom) poly \<Rightarrow> 'a poly"
+where
+  [simp del]: "pderiv (pCons a p) = (if p = 0 then 0 else p + pCons 0 (pderiv p))"
+  by (auto intro: pCons_cases)
+
+termination pderiv
+  by (relation "measure degree") simp_all
+
+lemma pderiv_0 [simp]:
+  "pderiv 0 = 0"
+  using pderiv.simps [of 0 0] by simp
+
+lemma pderiv_pCons:
+  "pderiv (pCons a p) = p + pCons 0 (pderiv p)"
+  by (simp add: pderiv.simps)
+
+lemma pderiv_1 [simp]: "pderiv 1 = 0" 
+  unfolding one_poly_def by (simp add: pderiv_pCons)
+
+lemma pderiv_of_nat  [simp]: "pderiv (of_nat n) = 0"
+  and pderiv_numeral [simp]: "pderiv (numeral m) = 0"
+  by (simp_all add: of_nat_poly numeral_poly pderiv_pCons)
+
+lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
+  by (induct p arbitrary: n) 
+     (auto simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
+
+fun pderiv_coeffs_code :: "('a :: semidom) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
+  "pderiv_coeffs_code f (x # xs) = cCons (f * x) (pderiv_coeffs_code (f+1) xs)"
+| "pderiv_coeffs_code f [] = []"
+
+definition pderiv_coeffs :: "('a :: semidom) list \<Rightarrow> 'a list" where
+  "pderiv_coeffs xs = pderiv_coeffs_code 1 (tl xs)"
+
+(* Efficient code for pderiv contributed by René Thiemann and Akihisa Yamada *)
+lemma pderiv_coeffs_code: 
+  "nth_default 0 (pderiv_coeffs_code f xs) n = (f + of_nat n) * (nth_default 0 xs n)"
+proof (induct xs arbitrary: f n)
+  case (Cons x xs f n)
+  show ?case 
+  proof (cases n)
+    case 0
+    thus ?thesis by (cases "pderiv_coeffs_code (f + 1) xs = [] \<and> f * x = 0", auto simp: cCons_def)
+  next
+    case (Suc m) note n = this
+    show ?thesis 
+    proof (cases "pderiv_coeffs_code (f + 1) xs = [] \<and> f * x = 0")
+      case False
+      hence "nth_default 0 (pderiv_coeffs_code f (x # xs)) n = 
+               nth_default 0 (pderiv_coeffs_code (f + 1) xs) m" 
+        by (auto simp: cCons_def n)
+      also have "\<dots> = (f + of_nat n) * (nth_default 0 xs m)" 
+        unfolding Cons by (simp add: n add_ac)
+      finally show ?thesis by (simp add: n)
+    next
+      case True
+      {
+        fix g 
+        have "pderiv_coeffs_code g xs = [] \<Longrightarrow> g + of_nat m = 0 \<or> nth_default 0 xs m = 0"
+        proof (induct xs arbitrary: g m)
+          case (Cons x xs g)
+          from Cons(2) have empty: "pderiv_coeffs_code (g + 1) xs = []"
+                            and g: "(g = 0 \<or> x = 0)"
+            by (auto simp: cCons_def split: if_splits)
+          note IH = Cons(1)[OF empty]
+          from IH[of m] IH[of "m - 1"] g
+          show ?case by (cases m, auto simp: field_simps)
+        qed simp
+      } note empty = this
+      from True have "nth_default 0 (pderiv_coeffs_code f (x # xs)) n = 0"
+        by (auto simp: cCons_def n)
+      moreover have "(f + of_nat n) * nth_default 0 (x # xs) n = 0" using True
+        by (simp add: n, insert empty[of "f+1"], auto simp: field_simps)
+      ultimately show ?thesis by simp
+    qed
+  qed
+qed simp
+
+lemma map_upt_Suc: "map f [0 ..< Suc n] = f 0 # map (\<lambda> i. f (Suc i)) [0 ..< n]"
+  by (induct n arbitrary: f, auto)
+
+lemma coeffs_pderiv_code [code abstract]:
+  "coeffs (pderiv p) = pderiv_coeffs (coeffs p)" unfolding pderiv_coeffs_def
+proof (rule coeffs_eqI, unfold pderiv_coeffs_code coeff_pderiv, goal_cases)
+  case (1 n)
+  have id: "coeff p (Suc n) = nth_default 0 (map (\<lambda>i. coeff p (Suc i)) [0..<degree p]) n"
+    by (cases "n < degree p", auto simp: nth_default_def coeff_eq_0)
+  show ?case unfolding coeffs_def map_upt_Suc by (auto simp: id)
+next
+  case 2
+  obtain n xs where id: "tl (coeffs p) = xs" "(1 :: 'a) = n" by auto
+  from 2 show ?case
+    unfolding id by (induct xs arbitrary: n, auto simp: cCons_def)
+qed
+
+context
+  assumes "SORT_CONSTRAINT('a::{semidom, semiring_char_0})"
+begin
+
+lemma pderiv_eq_0_iff: 
+  "pderiv (p :: 'a poly) = 0 \<longleftrightarrow> degree p = 0"
+  apply (rule iffI)
+  apply (cases p, simp)
+  apply (simp add: poly_eq_iff coeff_pderiv del: of_nat_Suc)
+  apply (simp add: poly_eq_iff coeff_pderiv coeff_eq_0)
+  done
+
+lemma degree_pderiv: "degree (pderiv (p :: 'a poly)) = degree p - 1"
+  apply (rule order_antisym [OF degree_le])
+  apply (simp add: coeff_pderiv coeff_eq_0)
+  apply (cases "degree p", simp)
+  apply (rule le_degree)
+  apply (simp add: coeff_pderiv del: of_nat_Suc)
+  apply (metis degree_0 leading_coeff_0_iff nat.distinct(1))
+  done
+
+lemma not_dvd_pderiv: 
+  assumes "degree (p :: 'a poly) \<noteq> 0"
+  shows "\<not> p dvd pderiv p"
+proof
+  assume dvd: "p dvd pderiv p"
+  then obtain q where p: "pderiv p = p * q" unfolding dvd_def by auto
+  from dvd have le: "degree p \<le> degree (pderiv p)"
+    by (simp add: assms dvd_imp_degree_le pderiv_eq_0_iff)
+  from this[unfolded degree_pderiv] assms show False by auto
+qed
+
+lemma dvd_pderiv_iff [simp]: "(p :: 'a poly) dvd pderiv p \<longleftrightarrow> degree p = 0"
+  using not_dvd_pderiv[of p] by (auto simp: pderiv_eq_0_iff [symmetric])
+
+end
+
+lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"
+by (simp add: pderiv_pCons)
+
+lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
+by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
+
+lemma pderiv_minus: "pderiv (- p :: 'a :: idom poly) = - pderiv p"
+by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
+
+lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
+by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
+
+lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
+by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
+
+lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
+by (induct p) (auto simp: pderiv_add pderiv_smult pderiv_pCons algebra_simps)
+
+lemma pderiv_power_Suc:
+  "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
+apply (induct n)
+apply simp
+apply (subst power_Suc)
+apply (subst pderiv_mult)
+apply (erule ssubst)
+apply (simp only: of_nat_Suc smult_add_left smult_1_left)
+apply (simp add: algebra_simps)
+done
+
+lemma pderiv_setprod: "pderiv (setprod f (as)) = 
+  (\<Sum>a \<in> as. setprod f (as - {a}) * pderiv (f a))"
+proof (induct as rule: infinite_finite_induct)
+  case (insert a as)
+  hence id: "setprod f (insert a as) = f a * setprod f as" 
+    "\<And> g. setsum g (insert a as) = g a + setsum g as"
+    "insert a as - {a} = as"
+    by auto
+  {
+    fix b
+    assume "b \<in> as"
+    hence id2: "insert a as - {b} = insert a (as - {b})" using \<open>a \<notin> as\<close> by auto
+    have "setprod f (insert a as - {b}) = f a * setprod f (as - {b})"
+      unfolding id2
+      by (subst setprod.insert, insert insert, auto)
+  } note id2 = this
+  show ?case
+    unfolding id pderiv_mult insert(3) setsum_right_distrib
+    by (auto simp add: ac_simps id2 intro!: setsum.cong)
+qed auto
+
+lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
+by (rule DERIV_cong, rule DERIV_pow, simp)
+declare DERIV_pow2 [simp] DERIV_pow [simp]
+
+lemma DERIV_add_const: "DERIV f x :> D ==>  DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
+by (rule DERIV_cong, rule DERIV_add, auto)
+
+lemma poly_DERIV [simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
+  by (induct p, auto intro!: derivative_eq_intros simp add: pderiv_pCons)
+
+lemma continuous_on_poly [continuous_intros]: 
+  fixes p :: "'a :: {real_normed_field} poly"
+  assumes "continuous_on A f"
+  shows   "continuous_on A (\<lambda>x. poly p (f x))"
+proof -
+  have "continuous_on A (\<lambda>x. (\<Sum>i\<le>degree p. (f x) ^ i * coeff p i))" 
+    by (intro continuous_intros assms)
+  also have "\<dots> = (\<lambda>x. poly p (f x))" by (intro ext) (simp add: poly_altdef mult_ac)
+  finally show ?thesis .
+qed
+
+text\<open>Consequences of the derivative theorem above\<close>
+
+lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (at x::real filter)"
+apply (simp add: real_differentiable_def)
+apply (blast intro: poly_DERIV)
+done
+
+lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
+by (rule poly_DERIV [THEN DERIV_isCont])
+
+lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
+      ==> \<exists>x. a < x & x < b & (poly p x = 0)"
+using IVT_objl [of "poly p" a 0 b]
+by (auto simp add: order_le_less)
+
+lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
+      ==> \<exists>x. a < x & x < b & (poly p x = 0)"
+by (insert poly_IVT_pos [where p = "- p" ]) simp
+
+lemma poly_IVT:
+  fixes p::"real poly"
+  assumes "a<b" and "poly p a * poly p b < 0"
+  shows "\<exists>x>a. x < b \<and> poly p x = 0"
+by (metis assms(1) assms(2) less_not_sym mult_less_0_iff poly_IVT_neg poly_IVT_pos)
+
+lemma poly_MVT: "(a::real) < b ==>
+     \<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
+using MVT [of a b "poly p"]
+apply auto
+apply (rule_tac x = z in exI)
+apply (auto simp add: mult_left_cancel poly_DERIV [THEN DERIV_unique])
+done
+
+lemma poly_MVT':
+  assumes "{min a b..max a b} \<subseteq> A"
+  shows   "\<exists>x\<in>A. poly p b - poly p a = (b - a) * poly (pderiv p) (x::real)"
+proof (cases a b rule: linorder_cases)
+  case less
+  from poly_MVT[OF less, of p] guess x by (elim exE conjE)
+  thus ?thesis by (intro bexI[of _ x]) (auto intro!: subsetD[OF assms])
+
+next
+  case greater
+  from poly_MVT[OF greater, of p] guess x by (elim exE conjE)
+  thus ?thesis by (intro bexI[of _ x]) (auto simp: algebra_simps intro!: subsetD[OF assms])
+qed (insert assms, auto)
+
+lemma poly_pinfty_gt_lc:
+  fixes p:: "real poly"
+  assumes  "lead_coeff p > 0" 
+  shows "\<exists> n. \<forall> x \<ge> n. poly p x \<ge> lead_coeff p" using assms
+proof (induct p)
+  case 0
+  thus ?case by auto
+next
+  case (pCons a p)
+  have "\<lbrakk>a\<noteq>0;p=0\<rbrakk> \<Longrightarrow> ?case" by auto
+  moreover have "p\<noteq>0 \<Longrightarrow> ?case"
+    proof -
+      assume "p\<noteq>0"
+      then obtain n1 where gte_lcoeff:"\<forall>x\<ge>n1. lead_coeff p \<le> poly p x" using that pCons by auto
+      have gt_0:"lead_coeff p >0" using pCons(3) \<open>p\<noteq>0\<close> by auto
+      def n\<equiv>"max n1 (1+ \<bar>a\<bar>/(lead_coeff p))"
+      show ?thesis 
+        proof (rule_tac x=n in exI,rule,rule) 
+          fix x assume "n \<le> x"
+          hence "lead_coeff p \<le> poly p x" 
+            using gte_lcoeff unfolding n_def by auto
+          hence " \<bar>a\<bar>/(lead_coeff p) \<ge> \<bar>a\<bar>/(poly p x)" and "poly p x>0" using gt_0
+            by (intro frac_le,auto)
+          hence "x\<ge>1+ \<bar>a\<bar>/(poly p x)" using \<open>n\<le>x\<close>[unfolded n_def] by auto
+          thus "lead_coeff (pCons a p) \<le> poly (pCons a p) x"
+            using \<open>lead_coeff p \<le> poly p x\<close> \<open>poly p x>0\<close> \<open>p\<noteq>0\<close>
+            by (auto simp add:field_simps)
+        qed
+    qed
+  ultimately show ?case by fastforce
+qed
+
+
+subsection \<open>Algebraic numbers\<close>
+
+text \<open>
+  Algebraic numbers can be defined in two equivalent ways: all real numbers that are 
+  roots of rational polynomials or of integer polynomials. The Algebraic-Numbers AFP entry 
+  uses the rational definition, but we need the integer definition.
+
+  The equivalence is obvious since any rational polynomial can be multiplied with the 
+  LCM of its coefficients, yielding an integer polynomial with the same roots.
+\<close>
+subsection \<open>Algebraic numbers\<close>
+
+definition algebraic :: "'a :: field_char_0 \<Rightarrow> bool" where
+  "algebraic x \<longleftrightarrow> (\<exists>p. (\<forall>i. coeff p i \<in> \<int>) \<and> p \<noteq> 0 \<and> poly p x = 0)"
+
+lemma algebraicI:
+  assumes "\<And>i. coeff p i \<in> \<int>" "p \<noteq> 0" "poly p x = 0"
+  shows   "algebraic x"
+  using assms unfolding algebraic_def by blast
   
+lemma algebraicE:
+  assumes "algebraic x"
+  obtains p where "\<And>i. coeff p i \<in> \<int>" "p \<noteq> 0" "poly p x = 0"
+  using assms unfolding algebraic_def by blast
+
+lemma quotient_of_denom_pos': "snd (quotient_of x) > 0"
+  using quotient_of_denom_pos[OF surjective_pairing] .
   
+lemma of_int_div_in_Ints: 
+  "b dvd a \<Longrightarrow> of_int a div of_int b \<in> (\<int> :: 'a :: ring_div set)"
+proof (cases "of_int b = (0 :: 'a)")
+  assume "b dvd a" "of_int b \<noteq> (0::'a)"
+  then obtain c where "a = b * c" by (elim dvdE)
+  with \<open>of_int b \<noteq> (0::'a)\<close> show ?thesis by simp
+qed auto
+
+lemma of_int_divide_in_Ints: 
+  "b dvd a \<Longrightarrow> of_int a / of_int b \<in> (\<int> :: 'a :: field set)"
+proof (cases "of_int b = (0 :: 'a)")
+  assume "b dvd a" "of_int b \<noteq> (0::'a)"
+  then obtain c where "a = b * c" by (elim dvdE)
+  with \<open>of_int b \<noteq> (0::'a)\<close> show ?thesis by simp
+qed auto
+
+lemma algebraic_altdef:
+  fixes p :: "'a :: field_char_0 poly"
+  shows "algebraic x \<longleftrightarrow> (\<exists>p. (\<forall>i. coeff p i \<in> \<rat>) \<and> p \<noteq> 0 \<and> poly p x = 0)"
+proof safe
+  fix p assume rat: "\<forall>i. coeff p i \<in> \<rat>" and root: "poly p x = 0" and nz: "p \<noteq> 0"
+  def cs \<equiv> "coeffs p"
+  from rat have "\<forall>c\<in>range (coeff p). \<exists>c'. c = of_rat c'" unfolding Rats_def by blast
+  then obtain f where f: "\<And>i. coeff p i = of_rat (f (coeff p i))" 
+    by (subst (asm) bchoice_iff) blast
+  def cs' \<equiv> "map (quotient_of \<circ> f) (coeffs p)"
+  def d \<equiv> "Lcm (set (map snd cs'))"
+  def p' \<equiv> "smult (of_int d) p"
+  
+  have "\<forall>n. coeff p' n \<in> \<int>"
+  proof
+    fix n :: nat
+    show "coeff p' n \<in> \<int>"
+    proof (cases "n \<le> degree p")
+      case True
+      def c \<equiv> "coeff p n"
+      def a \<equiv> "fst (quotient_of (f (coeff p n)))" and b \<equiv> "snd (quotient_of (f (coeff p n)))"
+      have b_pos: "b > 0" unfolding b_def using quotient_of_denom_pos' by simp
+      have "coeff p' n = of_int d * coeff p n" by (simp add: p'_def)
+      also have "coeff p n = of_rat (of_int a / of_int b)" unfolding a_def b_def
+        by (subst quotient_of_div [of "f (coeff p n)", symmetric])
+           (simp_all add: f [symmetric])
+      also have "of_int d * \<dots> = of_rat (of_int (a*d) / of_int b)"
+        by (simp add: of_rat_mult of_rat_divide)
+      also from nz True have "b \<in> snd ` set cs'" unfolding cs'_def
+        by (force simp: o_def b_def coeffs_def simp del: upt_Suc)
+      hence "b dvd (a * d)" unfolding d_def by simp
+      hence "of_int (a * d) / of_int b \<in> (\<int> :: rat set)"
+        by (rule of_int_divide_in_Ints)
+      hence "of_rat (of_int (a * d) / of_int b) \<in> \<int>" by (elim Ints_cases) auto
+      finally show ?thesis .
+    qed (auto simp: p'_def not_le coeff_eq_0)
+  qed
+  
+  moreover have "set (map snd cs') \<subseteq> {0<..}"
+    unfolding cs'_def using quotient_of_denom_pos' by (auto simp: coeffs_def simp del: upt_Suc) 
+  hence "d \<noteq> 0" unfolding d_def by (induction cs') simp_all
+  with nz have "p' \<noteq> 0" by (simp add: p'_def)
+  moreover from root have "poly p' x = 0" by (simp add: p'_def)
+  ultimately show "algebraic x" unfolding algebraic_def by blast
+next
+
+  assume "algebraic x"
+  then obtain p where p: "\<And>i. coeff p i \<in> \<int>" "poly p x = 0" "p \<noteq> 0" 
+    by (force simp: algebraic_def)
+  moreover have "coeff p i \<in> \<int> \<Longrightarrow> coeff p i \<in> \<rat>" for i by (elim Ints_cases) simp
+  ultimately show  "(\<exists>p. (\<forall>i. coeff p i \<in> \<rat>) \<and> p \<noteq> 0 \<and> poly p x = 0)" by auto
+qed
+
+
+text\<open>Lemmas for Derivatives\<close>
+
+lemma order_unique_lemma:
+  fixes p :: "'a::idom poly"
+  assumes "[:-a, 1:] ^ n dvd p" "\<not> [:-a, 1:] ^ Suc n dvd p"
+  shows "n = order a p"
+unfolding Polynomial.order_def
+apply (rule Least_equality [symmetric])
+apply (fact assms)
+apply (rule classical)
+apply (erule notE)
+unfolding not_less_eq_eq
+using assms(1) apply (rule power_le_dvd)
+apply assumption
+done
+
+lemma lemma_order_pderiv1:
+  "pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
+    smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
+apply (simp only: pderiv_mult pderiv_power_Suc)
+apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons)
+done
+
+lemma lemma_order_pderiv:
+  fixes p :: "'a :: field_char_0 poly"
+  assumes n: "0 < n" 
+      and pd: "pderiv p \<noteq> 0" 
+      and pe: "p = [:- a, 1:] ^ n * q" 
+      and nd: "~ [:- a, 1:] dvd q"
+    shows "n = Suc (order a (pderiv p))"
+using n 
+proof -
+  have "pderiv ([:- a, 1:] ^ n * q) \<noteq> 0"
+    using assms by auto
+  obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) \<noteq> 0"
+    using assms by (cases n) auto
+  have *: "!!k l. k dvd k * pderiv q + smult (of_nat (Suc n')) l \<Longrightarrow> k dvd l"
+    by (auto simp del: of_nat_Suc simp: dvd_add_right_iff dvd_smult_iff)
+  have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))" 
+  proof (rule order_unique_lemma)
+    show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
+      apply (subst lemma_order_pderiv1)
+      apply (rule dvd_add)
+      apply (metis dvdI dvd_mult2 power_Suc2)
+      apply (metis dvd_smult dvd_triv_right)
+      done
+  next
+    show "\<not> [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
+     apply (subst lemma_order_pderiv1)
+     by (metis * nd dvd_mult_cancel_right power_not_zero pCons_eq_0_iff power_Suc zero_neq_one)
+  qed
+  then show ?thesis
+    by (metis \<open>n = Suc n'\<close> pe)
+qed
+
+lemma order_decomp:
+  assumes "p \<noteq> 0"
+  shows "\<exists>q. p = [:- a, 1:] ^ order a p * q \<and> \<not> [:- a, 1:] dvd q"
+proof -
+  from assms have A: "[:- a, 1:] ^ order a p dvd p"
+    and B: "\<not> [:- a, 1:] ^ Suc (order a p) dvd p" by (auto dest: order)
+  from A obtain q where C: "p = [:- a, 1:] ^ order a p * q" ..
+  with B have "\<not> [:- a, 1:] ^ Suc (order a p) dvd [:- a, 1:] ^ order a p * q"
+    by simp
+  then have "\<not> [:- a, 1:] ^ order a p * [:- a, 1:] dvd [:- a, 1:] ^ order a p * q"
+    by simp
+  then have D: "\<not> [:- a, 1:] dvd q"
+    using idom_class.dvd_mult_cancel_left [of "[:- a, 1:] ^ order a p" "[:- a, 1:]" q]
+    by auto
+  from C D show ?thesis by blast
+qed
+
+lemma order_pderiv:
+  "\<lbrakk>pderiv p \<noteq> 0; order a (p :: 'a :: field_char_0 poly) \<noteq> 0\<rbrakk> \<Longrightarrow>
+     (order a p = Suc (order a (pderiv p)))"
+apply (case_tac "p = 0", simp)
+apply (drule_tac a = a and p = p in order_decomp)
+using neq0_conv
+apply (blast intro: lemma_order_pderiv)
+done
+
+lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q"
+proof -
+  def i \<equiv> "order a p"
+  def j \<equiv> "order a q"
+  def t \<equiv> "[:-a, 1:]"
+  have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0"
+    unfolding t_def by (simp add: dvd_iff_poly_eq_0)
+  assume "p * q \<noteq> 0"
+  then show "order a (p * q) = i + j"
+    apply clarsimp
+    apply (drule order [where a=a and p=p, folded i_def t_def])
+    apply (drule order [where a=a and p=q, folded j_def t_def])
+    apply clarify
+    apply (erule dvdE)+
+    apply (rule order_unique_lemma [symmetric], fold t_def)
+    apply (simp_all add: power_add t_dvd_iff)
+    done
+qed
+
+lemma order_smult:
+  assumes "c \<noteq> 0" 
+  shows "order x (smult c p) = order x p"
+proof (cases "p = 0")
+  case False
+  have "smult c p = [:c:] * p" by simp
+  also from assms False have "order x \<dots> = order x [:c:] + order x p" 
+    by (subst order_mult) simp_all
+  also from assms have "order x [:c:] = 0" by (intro order_0I) auto
+  finally show ?thesis by simp
+qed simp
+
+(* Next two lemmas contributed by Wenda Li *)
+lemma order_1_eq_0 [simp]:"order x 1 = 0" 
+  by (metis order_root poly_1 zero_neq_one)
+
+lemma order_power_n_n: "order a ([:-a,1:]^n)=n" 
+proof (induct n) (*might be proved more concisely using nat_less_induct*)
+  case 0
+  thus ?case by (metis order_root poly_1 power_0 zero_neq_one)
+next 
+  case (Suc n)
+  have "order a ([:- a, 1:] ^ Suc n)=order a ([:- a, 1:] ^ n) + order a [:-a,1:]" 
+    by (metis (no_types, hide_lams) One_nat_def add_Suc_right monoid_add_class.add.right_neutral 
+      one_neq_zero order_mult pCons_eq_0_iff power_add power_eq_0_iff power_one_right)
+  moreover have "order a [:-a,1:]=1" unfolding order_def
+    proof (rule Least_equality,rule ccontr)
+      assume  "\<not> \<not> [:- a, 1:] ^ Suc 1 dvd [:- a, 1:]"
+      hence "[:- a, 1:] ^ Suc 1 dvd [:- a, 1:]" by simp
+      hence "degree ([:- a, 1:] ^ Suc 1) \<le> degree ([:- a, 1:] )" 
+        by (rule dvd_imp_degree_le,auto) 
+      thus False by auto
+    next
+      fix y assume asm:"\<not> [:- a, 1:] ^ Suc y dvd [:- a, 1:]"
+      show "1 \<le> y" 
+        proof (rule ccontr)
+          assume "\<not> 1 \<le> y"
+          hence "y=0" by auto
+          hence "[:- a, 1:] ^ Suc y dvd [:- a, 1:]" by auto
+          thus False using asm by auto
+        qed
+    qed
+  ultimately show ?case using Suc by auto
+qed
+
+text\<open>Now justify the standard squarefree decomposition, i.e. f / gcd(f,f').\<close>
+
+lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p"
+apply (cases "p = 0", auto)
+apply (drule order_2 [where a=a and p=p])
+apply (metis not_less_eq_eq power_le_dvd)
+apply (erule power_le_dvd [OF order_1])
+done
+
+lemma poly_squarefree_decomp_order:
+  assumes "pderiv (p :: 'a :: field_char_0 poly) \<noteq> 0"
+  and p: "p = q * d"
+  and p': "pderiv p = e * d"
+  and d: "d = r * p + s * pderiv p"
+  shows "order a q = (if order a p = 0 then 0 else 1)"
+proof (rule classical)
+  assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)"
+  from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
+  with p have "order a p = order a q + order a d"
+    by (simp add: order_mult)
+  with 1 have "order a p \<noteq> 0" by (auto split: if_splits)
+  have "order a (pderiv p) = order a e + order a d"
+    using \<open>pderiv p \<noteq> 0\<close> \<open>pderiv p = e * d\<close> by (simp add: order_mult)
+  have "order a p = Suc (order a (pderiv p))"
+    using \<open>pderiv p \<noteq> 0\<close> \<open>order a p \<noteq> 0\<close> by (rule order_pderiv)
+  have "d \<noteq> 0" using \<open>p \<noteq> 0\<close> \<open>p = q * d\<close> by simp
+  have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
+    apply (simp add: d)
+    apply (rule dvd_add)
+    apply (rule dvd_mult)
+    apply (simp add: order_divides \<open>p \<noteq> 0\<close>
+           \<open>order a p = Suc (order a (pderiv p))\<close>)
+    apply (rule dvd_mult)
+    apply (simp add: order_divides)
+    done
+  then have "order a (pderiv p) \<le> order a d"
+    using \<open>d \<noteq> 0\<close> by (simp add: order_divides)
+  show ?thesis
+    using \<open>order a p = order a q + order a d\<close>
+    using \<open>order a (pderiv p) = order a e + order a d\<close>
+    using \<open>order a p = Suc (order a (pderiv p))\<close>
+    using \<open>order a (pderiv p) \<le> order a d\<close>
+    by auto
+qed
+
+lemma poly_squarefree_decomp_order2: 
+     "\<lbrakk>pderiv p \<noteq> (0 :: 'a :: field_char_0 poly);
+       p = q * d;
+       pderiv p = e * d;
+       d = r * p + s * pderiv p
+      \<rbrakk> \<Longrightarrow> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
+by (blast intro: poly_squarefree_decomp_order)
+
+lemma order_pderiv2: 
+  "\<lbrakk>pderiv p \<noteq> 0; order a (p :: 'a :: field_char_0 poly) \<noteq> 0\<rbrakk>
+      \<Longrightarrow> (order a (pderiv p) = n) = (order a p = Suc n)"
+by (auto dest: order_pderiv)
+
+definition
+  rsquarefree :: "'a::idom poly => bool" where
+  "rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))"
+
+lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h :: 'a :: {semidom,semiring_char_0}:]"
+apply (simp add: pderiv_eq_0_iff)
+apply (case_tac p, auto split: if_splits)
+done
+
+lemma rsquarefree_roots:
+  fixes p :: "'a :: field_char_0 poly"
+  shows "rsquarefree p = (\<forall>a. \<not>(poly p a = 0 \<and> poly (pderiv p) a = 0))"
+apply (simp add: rsquarefree_def)
+apply (case_tac "p = 0", simp, simp)
+apply (case_tac "pderiv p = 0")
+apply simp
+apply (drule pderiv_iszero, clarsimp)
+apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree)
+apply (force simp add: order_root order_pderiv2)
+done
+
+lemma poly_squarefree_decomp:
+  assumes "pderiv (p :: 'a :: field_char_0 poly) \<noteq> 0"
+    and "p = q * d"
+    and "pderiv p = e * d"
+    and "d = r * p + s * pderiv p"
+  shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
+proof -
+  from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
+  with \<open>p = q * d\<close> have "q \<noteq> 0" by simp
+  have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
+    using assms by (rule poly_squarefree_decomp_order2)
+  with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> show ?thesis
+    by (simp add: rsquarefree_def order_root)
+qed
+
 
 no_notation cCons (infixr "##" 65)
 
diff -r fd049b54ad68 -r 35a9e1cbb5b3 src/HOL/Library/Polynomial_GCD_euclidean.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Polynomial_GCD_euclidean.thy	Wed Feb 17 21:51:58 2016 +0100
@@ -0,0 +1,140 @@
+(*  Title:      HOL/Library/Polynomial_GCD_euclidean.thy
+    Author:     Brian Huffman
+    Author:     Clemens Ballarin
+    Author:     Amine Chaieb
+    Author:     Florian Haftmann
+*)
+
+section \<open>GCD and LCM on polynomials over a field\<close>
+
+theory Polynomial_GCD_euclidean
+imports Main "~~/src/HOL/GCD" "~~/src/HOL/Library/Polynomial"
+begin
+
+subsection \<open>GCD of polynomials\<close>
+
+instantiation poly :: (field) gcd
+begin
+
+function gcd_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
+where
+  "gcd (x::'a poly) 0 = smult (inverse (coeff x (degree x))) x"
+| "y \<noteq> 0 \<Longrightarrow> gcd (x::'a poly) y = gcd y (x mod y)"
+by auto
+
+termination "gcd :: _ poly \<Rightarrow> _"
+by (relation "measure (\<lambda>(x, y). if y = 0 then 0 else Suc (degree y))")
+   (auto dest: degree_mod_less)
+
+declare gcd_poly.simps [simp del]
+
+definition lcm_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
+where
+  "lcm_poly a b = a * b div smult (coeff a (degree a) * coeff b (degree b)) (gcd a b)"
+
+instance ..
+
+end
+
+lemma
+  fixes x y :: "_ poly"
+  shows poly_gcd_dvd1 [iff]: "gcd x y dvd x"
+    and poly_gcd_dvd2 [iff]: "gcd x y dvd y"
+  apply (induct x y rule: gcd_poly.induct)
+  apply (simp_all add: gcd_poly.simps)
+  apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
+  apply (blast dest: dvd_mod_imp_dvd)
+  done
+
+lemma poly_gcd_greatest:
+  fixes k x y :: "_ poly"
+  shows "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd gcd x y"
+  by (induct x y rule: gcd_poly.induct)
+     (simp_all add: gcd_poly.simps dvd_mod dvd_smult)
+
+lemma dvd_poly_gcd_iff [iff]:
+  fixes k x y :: "_ poly"
+  shows "k dvd gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
+  by (auto intro!: poly_gcd_greatest intro: dvd_trans [of _ "gcd x y"])
+
+lemma poly_gcd_monic:
+  fixes x y :: "_ poly"
+  shows "coeff (gcd x y) (degree (gcd x y)) =
+    (if x = 0 \<and> y = 0 then 0 else 1)"
+  by (induct x y rule: gcd_poly.induct)
+     (simp_all add: gcd_poly.simps nonzero_imp_inverse_nonzero)
+
+lemma poly_gcd_zero_iff [simp]:
+  fixes x y :: "_ poly"
+  shows "gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
+  by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)
+
+lemma poly_gcd_0_0 [simp]:
+  "gcd (0::_ poly) 0 = 0"
+  by simp
+
+lemma poly_dvd_antisym:
+  fixes p q :: "'a::idom poly"
+  assumes coeff: "coeff p (degree p) = coeff q (degree q)"
+  assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
+proof (cases "p = 0")
+  case True with coeff show "p = q" by simp
+next
+  case False with coeff have "q \<noteq> 0" by auto
+  have degree: "degree p = degree q"
+    using \<open>p dvd q\<close> \<open>q dvd p\<close> \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close>
+    by (intro order_antisym dvd_imp_degree_le)
+
+  from \<open>p dvd q\<close> obtain a where a: "q = p * a" ..
+  with \<open>q \<noteq> 0\<close> have "a \<noteq> 0" by auto
+  with degree a \<open>p \<noteq> 0\<close> have "degree a = 0"
+    by (simp add: degree_mult_eq)
+  with coeff a show "p = q"
+    by (cases a, auto split: if_splits)
+qed
+
+lemma poly_gcd_unique:
+  fixes d x y :: "_ poly"
+  assumes dvd1: "d dvd x" and dvd2: "d dvd y"
+    and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
+    and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
+  shows "gcd x y = d"
+proof -
+  have "coeff (gcd x y) (degree (gcd x y)) = coeff d (degree d)"
+    by (simp_all add: poly_gcd_monic monic)
+  moreover have "gcd x y dvd d"
+    using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
+  moreover have "d dvd gcd x y"
+    using dvd1 dvd2 by (rule poly_gcd_greatest)
+  ultimately show ?thesis
+    by (rule poly_dvd_antisym)
+qed
+
+instance poly :: (field) semiring_gcd
+proof
+  fix p q :: "'a::field poly"
+  show "normalize (gcd p q) = gcd p q"
+    by (induct p q rule: gcd_poly.induct)
+      (simp_all add: gcd_poly.simps normalize_poly_def)
+  show "lcm p q = normalize (p * q) div gcd p q"
+    by (simp add: coeff_degree_mult div_smult_left div_smult_right lcm_poly_def normalize_poly_def)
+      (metis (no_types, lifting) div_smult_right inverse_mult_distrib inverse_zero mult.commute pdivmod_rel pdivmod_rel_def smult_eq_0_iff)
+qed simp_all
+
+lemma poly_gcd_1_left [simp]: "gcd 1 y = (1 :: _ poly)"
+by (rule poly_gcd_unique) simp_all
+
+lemma poly_gcd_1_right [simp]: "gcd x 1 = (1 :: _ poly)"
+by (rule poly_gcd_unique) simp_all
+
+lemma poly_gcd_minus_left [simp]: "gcd (- x) y = gcd x (y :: _ poly)"
+by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
+
+lemma poly_gcd_minus_right [simp]: "gcd x (- y) = gcd x (y :: _ poly)"
+by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
+
+lemma poly_gcd_code [code]:
+  "gcd x y = (if y = 0 then normalize x else gcd y (x mod (y :: _ poly)))"
+  by (simp add: gcd_poly.simps)
+
+end
diff -r fd049b54ad68 -r 35a9e1cbb5b3 src/HOL/ROOT
--- a/src/HOL/ROOT	Wed Feb 17 21:51:58 2016 +0100
+++ b/src/HOL/ROOT	Wed Feb 17 21:51:58 2016 +0100
@@ -38,6 +38,7 @@
     Product_Lexorder
     Product_Order
     Finite_Lattice
+    Polynomial_GCD_euclidean
     (*data refinements and dependent applications*)
     AList_Mapping
     Code_Binary_Nat