# HG changeset patch
# User haftmann
# Date 1383216260 -3600
# Node ID 63fe59f64578b098dbec416fe82f9931ee4d7697
# Parent  07c0c121a8dce2cb99f86590e3e88089b4eec583
consolidated clone theory

diff -r 07c0c121a8dc -r 63fe59f64578 src/HOL/Decision_Procs/Polynomial_List.thy
--- a/src/HOL/Decision_Procs/Polynomial_List.thy	Thu Oct 31 11:48:45 2013 +0100
+++ b/src/HOL/Decision_Procs/Polynomial_List.thy	Thu Oct 31 11:44:20 2013 +0100
@@ -2,371 +2,379 @@
     Author:     Amine Chaieb
 *)
 
-header {* Univariate Polynomials as Lists *}
+header {* Univariate Polynomials as lists *}
 
 theory Polynomial_List
-imports Main
+imports Complex_Main
 begin
 
-text{* Application of polynomial as a real function. *}
+text{* Application of polynomial as a function. *}
 
-primrec poly :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a::comm_ring"
+primrec (in semiring_0) poly :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a"
 where
   poly_Nil:  "poly [] x = 0"
-| poly_Cons: "poly (h # t) x = h + x * poly t x"
+| poly_Cons: "poly (h#t) x = h + x * poly t x"
 
 
 subsection{*Arithmetic Operations on Polynomials*}
 
 text{*addition*}
-primrec padd :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a::comm_ring_1 list"  (infixl "+++" 65)
+
+primrec (in semiring_0) padd :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "+++" 65)
 where
   padd_Nil:  "[] +++ l2 = l2"
-| padd_Cons: "(h # t) +++ l2 = (if l2 = [] then h # t else (h + hd l2) # (t +++ tl l2))"
+| padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t else (h + hd l2)#(t +++ tl l2))"
 
 text{*Multiplication by a constant*}
-primrec cmult :: "'a::comm_ring_1 \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "%*" 70)
-where
+primrec (in semiring_0) cmult :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "%*" 70) where
   cmult_Nil:  "c %* [] = []"
-| cmult_Cons: "c %* (h # t) = (c * h) # (c %* t)"
+| cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
 
 text{*Multiplication by a polynomial*}
-primrec pmult :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a::comm_ring_1 list"  (infixl "***" 70)
+primrec (in semiring_0) pmult :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "***" 70)
 where
   pmult_Nil:  "[] *** l2 = []"
-| pmult_Cons: "(h # t) *** l2 =
-    (if t = [] then h %* l2 else (h %* l2) +++ (0 # (t *** l2)))"
+| pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2
+                              else (h %* l2) +++ ((0) # (t *** l2)))"
 
 text{*Repeated multiplication by a polynomial*}
-primrec mulexp :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a::comm_ring_1 list"
-where
+primrec (in semiring_0) mulexp :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a  list \<Rightarrow> 'a list" where
   mulexp_zero:  "mulexp 0 p q = q"
 | mulexp_Suc:   "mulexp (Suc n) p q = p *** mulexp n p q"
 
 text{*Exponential*}
-primrec pexp :: "'a list \<Rightarrow> nat \<Rightarrow> 'a::comm_ring_1 list"  (infixl "%^" 80)
-where
+primrec (in semiring_1) pexp :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list"  (infixl "%^" 80) where
   pexp_0:   "p %^ 0 = [1]"
 | pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
 
 text{*Quotient related value of dividing a polynomial by x + a*}
 (* Useful for divisor properties in inductive proofs *)
-primrec pquot :: "'a list \<Rightarrow> 'a::field \<Rightarrow> 'a list"
+primrec (in field) "pquot" :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list"
 where
-  pquot_Nil: "pquot [] a = []"
-| pquot_Cons: "pquot (h # t) a =
-    (if t = [] then [h] else (inverse a * (h - hd (pquot t a))) # pquot t a)"
-
+  pquot_Nil:  "pquot [] a= []"
+| pquot_Cons: "pquot (h#t) a =
+    (if t = [] then [h] else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))"
 
 text{*normalization of polynomials (remove extra 0 coeff)*}
-primrec pnormalize :: "'a::comm_ring_1 list \<Rightarrow> 'a list"
-where
+primrec (in semiring_0) pnormalize :: "'a list \<Rightarrow> 'a list" where
   pnormalize_Nil:  "pnormalize [] = []"
-| pnormalize_Cons: "pnormalize (h # p) =
-    (if (pnormalize p = []) then (if h = 0 then [] else [h])
-     else (h # pnormalize p))"
+| pnormalize_Cons: "pnormalize (h#p) =
+    (if pnormalize p = [] then (if h = 0 then [] else [h]) else h # pnormalize p)"
 
-definition "pnormal p = ((pnormalize p = p) \<and> p \<noteq> [])"
-definition "nonconstant p = (pnormal p \<and> (\<forall>x. p \<noteq> [x]))"
+definition (in semiring_0) "pnormal p = ((pnormalize p = p) \<and> p \<noteq> [])"
+definition (in semiring_0) "nonconstant p = (pnormal p \<and> (\<forall>x. p \<noteq> [x]))"
 text{*Other definitions*}
 
-definition poly_minus :: "'a list => ('a :: comm_ring_1) list"  ("-- _" [80] 80)
+definition (in ring_1) poly_minus :: "'a list \<Rightarrow> 'a list" ("-- _" [80] 80)
   where "-- p = (- 1) %* p"
 
-definition divides :: "'a::comm_ring_1 list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "divides" 70)
-  where "p1 divides p2 = (\<exists>q. poly p2 = poly (p1 *** q))"
+definition (in semiring_0) divides :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "divides" 70)
+  where "p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))"
+
+lemma (in semiring_0) dividesI:
+  "poly p2 = poly (p1 *** q) \<Longrightarrow> p1 divides p2"
+  by (auto simp add: divides_def)
 
-definition order :: "'a::comm_ring_1 \<Rightarrow> 'a list \<Rightarrow> nat" --{*order of a polynomial*}
-  where "order a p = (SOME n. ([-a, 1] %^ n) divides p & ~ (([-a, 1] %^ Suc n) divides p))"
+lemma (in semiring_0) dividesE:
+  assumes "p1 divides p2"
+  obtains q where "poly p2 = poly (p1 *** q)"
+  using assms by (auto simp add: divides_def)
 
-definition degree :: "'a::comm_ring_1 list \<Rightarrow> nat" --{*degree of a polynomial*}
+    --{*order of a polynomial*}
+definition (in ring_1) order :: "'a \<Rightarrow> 'a list \<Rightarrow> nat" where
+  "order a p = (SOME n. ([-a, 1] %^ n) divides p \<and> ~ (([-a, 1] %^ (Suc n)) divides p))"
+
+     --{*degree of a polynomial*}
+definition (in semiring_0) degree :: "'a list \<Rightarrow> nat"
   where "degree p = length (pnormalize p) - 1"
 
-definition rsquarefree :: "'a::comm_ring_1 list \<Rightarrow> bool"
-  where --{*squarefree polynomials --- NB with respect to real roots only.*}
-  "rsquarefree p = (poly p \<noteq> poly [] \<and> (\<forall>a. order a p = 0 \<or> order a p = 1))"
+     --{*squarefree polynomials --- NB with respect to real roots only.*}
+definition (in ring_1) rsquarefree :: "'a list \<Rightarrow> bool"
+  where "rsquarefree p \<longleftrightarrow> poly p \<noteq> poly [] \<and> (\<forall>a. order a p = 0 \<or> order a p = 1)"
 
-lemma padd_Nil2 [simp]: "p +++ [] = p"
+context semiring_0
+begin
+
+lemma padd_Nil2[simp]: "p +++ [] = p"
   by (induct p) auto
 
 lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
   by auto
 
-lemma pminus_Nil [simp]: "-- [] = []"
+lemma pminus_Nil: "-- [] = []"
   by (simp add: poly_minus_def)
 
-lemma pmult_singleton: "[h1] *** p1 = h1 %* p1"
-  by simp
+lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp
+
+end
 
-lemma poly_ident_mult [simp]: "1 %* t = t"
-  by (induct t) auto
+lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t" by (induct t) auto
 
-lemma poly_simple_add_Cons [simp]: "[a] +++ ((0)#t) = (a#t)"
+lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ ((0)#t) = (a#t)"
   by simp
 
 text{*Handy general properties*}
 
-lemma padd_commut: "b +++ a = a +++ b"
-  apply (induct b arbitrary: a)
-  apply auto
-  apply (rule padd_Cons [THEN ssubst])
-  apply (case_tac aa)
-  apply auto
+lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b"
+proof (induct b arbitrary: a)
+  case Nil
+  thus ?case by auto
+next
+  case (Cons b bs a)
+  thus ?case by (cases a) (simp_all add: add_commute)
+qed
+
+lemma (in comm_semiring_0) padd_assoc: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
+  apply (induct a)
+  apply (simp, clarify)
+  apply (case_tac b, simp_all add: add_ac)
   done
 
-lemma padd_assoc: "(a +++ b) +++ c = a +++ (b +++ c)"
-  apply (induct a arbitrary: b c)
+lemma (in semiring_0) poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)"
+  apply (induct p arbitrary: q)
   apply simp
-  apply (case_tac b)
-  apply simp_all
+  apply (case_tac q, simp_all add: distrib_left)
   done
 
-lemma poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)"
-  apply (induct p arbitrary: q)
+lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"
+  apply (induct t)
   apply simp
-  apply (case_tac q)
-  apply (simp_all add: distrib_left)
+  apply (auto simp add: padd_commut)
+  apply (case_tac t, auto)
   done
 
-lemma pmult_by_x [simp]: "[0, 1] *** t = ((0)#t)"
-  by (induct t) (auto simp add: padd_commut)
-
-
 text{*properties of evaluation of polynomials.*}
 
-lemma poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
-  apply (induct p1 arbitrary: p2)
-  apply auto
-  apply (case_tac "p2")
-  apply (auto simp add: distrib_left)
-  done
+lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
+proof(induct p1 arbitrary: p2)
+  case Nil
+  thus ?case by simp
+next
+  case (Cons a as p2)
+  thus ?case
+    by (cases p2) (simp_all  add: add_ac distrib_left)
+qed
 
-lemma poly_cmult: "poly (c %* p) x = c * poly p x"
+lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x"
   apply (induct p)
-  apply simp
-  apply (cases "x = 0")
+  apply (case_tac [2] "x = zero")
   apply (auto simp add: distrib_left mult_ac)
   done
 
-lemma poly_minus: "poly (-- p) x = - (poly p x)"
-  by (simp add: poly_minus_def poly_cmult)
+lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c*poly p x"
+  by (induct p) (auto simp add: distrib_left mult_ac)
 
-lemma poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
-  apply (induct p1 arbitrary: p2)
-  apply (case_tac p1)
-  apply (auto simp add: poly_cmult poly_add distrib_right distrib_left mult_ac)
+lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
+  apply (simp add: poly_minus_def)
+  apply (auto simp add: poly_cmult)
   done
 
-lemma poly_exp: "poly (p %^ n) (x::'a::comm_ring_1) = (poly p x) ^ n"
+lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
+proof (induct p1 arbitrary: p2)
+  case Nil
+  thus ?case by simp
+next
+  case (Cons a as p2)
+  thus ?case by (cases as)
+    (simp_all add: poly_cmult poly_add distrib_right distrib_left mult_ac)
+qed
+
+class idom_char_0 = idom + ring_char_0
+
+subclass (in field_char_0) idom_char_0 ..
+
+lemma (in comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n"
   by (induct n) (auto simp add: poly_cmult poly_mult)
 
 text{*More Polynomial Evaluation Lemmas*}
 
-lemma poly_add_rzero [simp]: "poly (a +++ []) x = poly a x"
+lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
   by simp
 
-lemma poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
+lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
   by (simp add: poly_mult mult_assoc)
 
-lemma poly_mult_Nil2 [simp]: "poly (p *** []) x = 0"
+lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0"
   by (induct p) auto
 
-lemma poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
+lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
   by (induct n) (auto simp add: poly_mult mult_assoc)
 
 subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
  @{term "p(x)"} *}
 
-lemma poly_linear_rem: "\<exists>q r. h # t = [r] +++ [-a, 1] *** q"
-  apply (induct t arbitrary: h)
-  apply (rule_tac x = "[]" in exI)
-  apply (rule_tac x = h in exI)
-  apply simp
-  apply (drule_tac x = aa in meta_spec)
-  apply safe
-  apply (rule_tac x = "r#q" in exI)
-  apply (rule_tac x = "a*r + h" in exI)
-  apply (case_tac q)
-  apply auto
-  done
+lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
+proof(induct t)
+  case Nil
+  { fix h have "[h] = [h] +++ [- a, 1] *** []" by simp }
+  thus ?case by blast
+next
+  case (Cons  x xs)
+  { fix h
+    from Cons.hyps[rule_format, of x]
+    obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
+    have "h#x#xs = [a*r + h] +++ [-a, 1] *** (r#q)"
+      using qr by (cases q) (simp_all add: algebra_simps)
+    hence "\<exists>q r. h#x#xs = [r] +++ [-a, 1] *** q" by blast}
+  thus ?case by blast
+qed
+
+lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
+  using lemma_poly_linear_rem [where t = t and a = a] by auto
+
 
-lemma poly_linear_divides: "poly p a = 0 \<longleftrightarrow> p = [] \<or> (\<exists>q. p = [-a, 1] *** q)"
-  apply (auto simp add: poly_add poly_cmult distrib_left)
-  apply (case_tac p)
-  apply simp
-  apply (cut_tac h = aa and t = list and a = a in poly_linear_rem)
-  apply safe
-  apply (case_tac q)
-  apply auto
-  apply (drule_tac x = "[]" in spec)
-  apply simp
-  apply (auto simp add: poly_add poly_cmult add_assoc)
-  apply (drule_tac x = "aa#lista" in spec)
-  apply auto
-  done
+lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))"
+proof -
+  { assume p: "p = []" hence ?thesis by simp }
+  moreover
+  {
+    fix x xs assume p: "p = x#xs"
+    {
+      fix q assume "p = [-a, 1] *** q"
+      hence "poly p a = 0" by (simp add: poly_add poly_cmult)
+    }
+    moreover
+    { assume p0: "poly p a = 0"
+      from poly_linear_rem[of x xs a] obtain q r
+      where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
+      have "r = 0" using p0 by (simp only: p qr poly_mult poly_add) simp
+      hence "\<exists>q. p = [- a, 1] *** q"
+        using p qr
+        apply -
+        apply (rule exI[where x=q])
+        apply auto
+        apply (cases q)
+        apply auto
+        done
+    }
+    ultimately have ?thesis using p by blast
+  }
+  ultimately show ?thesis by (cases p) auto
+qed
 
-lemma lemma_poly_length_mult [simp]: "length (k %* p +++  (h # (a %* p))) = Suc (length p)"
-  by (induct p arbitrary: h k a) auto
+lemma (in semiring_0) lemma_poly_length_mult[simp]: "\<forall>h k a. length (k %* p +++  (h # (a %* p))) = Suc (length p)"
+  by (induct p) auto
 
-lemma lemma_poly_length_mult2 [simp]: "length (k %* p +++  (h # p)) = Suc (length p)"
-  by (induct p arbitrary: h k) auto
+lemma (in semiring_0) lemma_poly_length_mult2[simp]: "\<forall>h k. length (k %* p +++  (h # p)) = Suc (length p)"
+  by (induct p) auto
 
-lemma poly_length_mult [simp]: "length([-a, 1] *** q) = Suc (length q)"
+lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"
   by auto
 
-
 subsection{*Polynomial length*}
 
-lemma poly_cmult_length [simp]: "length (a %* p) = length p"
+lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p"
   by (induct p) auto
 
-lemma poly_add_length:
-  "length (p1 +++ p2) = (if (length p1 < length p2) then length p2 else length p1)"
-  by (induct p1 arbitrary: p2) auto
+lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)"
+  by (induct p1 arbitrary: p2) (simp_all, arith)
 
-lemma poly_root_mult_length [simp]: "length ([a, b] *** p) = Suc (length p)"
-  by simp
+lemma (in semiring_0) poly_root_mult_length[simp]: "length([a,b] *** p) = Suc (length p)"
+  by (simp add: poly_add_length)
 
-lemma poly_mult_not_eq_poly_Nil [simp]:
-  "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] (x::'a::idom)"
+lemma (in idom) poly_mult_not_eq_poly_Nil[simp]:
+  "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x"
   by (auto simp add: poly_mult)
 
-lemma poly_mult_eq_zero_disj: "poly (p *** q) (x::'a::idom) = 0 \<longleftrightarrow> poly p x = 0 \<or> poly q x = 0"
+lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 \<longleftrightarrow> poly p x = 0 \<or> poly q x = 0"
   by (auto simp add: poly_mult)
 
 text{*Normalisation Properties*}
 
-lemma poly_normalized_nil: "pnormalize p = [] \<longrightarrow> poly p x = 0"
+lemma (in semiring_0) poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)"
   by (induct p) auto
 
 text{*A nontrivial polynomial of degree n has no more than n roots*}
+lemma (in idom) poly_roots_index_lemma:
+   assumes p: "poly p x \<noteq> poly [] x" and n: "length p = n"
+  shows "\<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
+  using p n
+proof (induct n arbitrary: p x)
+  case 0
+  thus ?case by simp
+next
+  case (Suc n p x)
+  {
+    assume C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)"
+    from Suc.prems have p0: "poly p x \<noteq> 0" "p\<noteq> []" by auto
+    from p0(1)[unfolded poly_linear_divides[of p x]]
+    have "\<forall>q. p \<noteq> [- x, 1] *** q" by blast
+    from C obtain a where a: "poly p a = 0" by blast
+    from a[unfolded poly_linear_divides[of p a]] p0(2)
+    obtain q where q: "p = [-a, 1] *** q" by blast
+    have lg: "length q = n" using q Suc.prems(2) by simp
+    from q p0 have qx: "poly q x \<noteq> poly [] x"
+      by (auto simp add: poly_mult poly_add poly_cmult)
+    from Suc.hyps[OF qx lg] obtain i where
+      i: "\<forall>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)" by blast
+    let ?i = "\<lambda>m. if m = Suc n then a else i m"
+    from C[of ?i] obtain y where y: "poly p y = 0" "\<forall>m\<le> Suc n. y \<noteq> ?i m"
+      by blast
+    from y have "y = a \<or> poly q y = 0"
+      by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: algebra_simps)
+    with i[rule_format, of y] y(1) y(2) have False
+      apply auto
+      apply (erule_tac x = "m" in allE)
+      apply auto
+      done
+  }
+  thus ?case by blast
+qed
 
-lemma poly_roots_index_lemma0 [rule_format]:
-   "\<forall>p x. poly p x \<noteq> poly [] x & length p = n
-    --> (\<exists>i. \<forall>x. (poly p x = (0::'a::idom)) --> (\<exists>m. (m \<le> n & x = i m)))"
-  apply (induct n)
-  apply safe
-  apply (rule ccontr)
-  apply (subgoal_tac "\<exists>a. poly p a = 0")
-  apply safe
-  apply (drule poly_linear_divides [THEN iffD1])
-  apply safe
-  apply (drule_tac x = q in spec)
-  apply (drule_tac x = x in spec)
-  apply (simp del: poly_Nil pmult_Cons)
-  apply (erule exE)
-  apply (drule_tac x = "%m. if m = Suc n then a else i m" in spec)
-  apply safe
-  apply (drule poly_mult_eq_zero_disj [THEN iffD1])
-  apply safe
-  apply (drule_tac x = "Suc (length q)" in spec)
-  apply (auto simp add: field_simps)
-  apply (drule_tac x = xa in spec)
-  apply (clarsimp simp add: field_simps)
-  apply (drule_tac x = m in spec)
-  apply (auto simp add:field_simps)
-  done
-lemmas poly_roots_index_lemma1 = conjI [THEN poly_roots_index_lemma0]
 
-lemma poly_roots_index_length0:
-  "poly p (x::'a::idom) \<noteq> poly [] x \<Longrightarrow>
-    \<exists>i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. n \<le> length p & x = i n)"
-  by (blast intro: poly_roots_index_lemma1)
+lemma (in idom) poly_roots_index_length:
+  "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. n \<le> length p \<and> x = i n)"
+  by (blast intro: poly_roots_index_lemma)
 
-lemma poly_roots_finite_lemma:
-  "poly p (x::'a::idom) \<noteq> poly [] x \<Longrightarrow>
-    \<exists>N i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. (n::nat) < N & x = i n)"
-  apply (drule poly_roots_index_length0)
-  apply safe
+lemma (in idom) poly_roots_finite_lemma1:
+  "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>N i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. (n::nat) < N \<and> x = i n)"
+  apply (drule poly_roots_index_length, safe)
   apply (rule_tac x = "Suc (length p)" in exI)
   apply (rule_tac x = i in exI)
   apply (simp add: less_Suc_eq_le)
   done
 
-
-lemma real_finite_lemma:
-  assumes "\<forall>x. P x \<longrightarrow> (\<exists>n. n < length j & x = j!n)"
-  shows "finite {(x::'a::idom). P x}"
+lemma (in idom) idom_finite_lemma:
+  assumes P: "\<forall>x. P x --> (\<exists>n. n < length j \<and> x = j!n)"
+  shows "finite {x. P x}"
 proof -
   let ?M = "{x. P x}"
   let ?N = "set j"
-  have "?M \<subseteq> ?N" using assms by auto
-  then show ?thesis using finite_subset by auto
+  have "?M \<subseteq> ?N" using P by auto
+  thus ?thesis using finite_subset by auto
 qed
 
-lemma poly_roots_index_lemma [rule_format]:
-  "\<forall>p x. poly p x \<noteq> poly [] x & length p = n
-    \<longrightarrow> (\<exists>i. \<forall>x. (poly p x = (0::'a::{idom})) \<longrightarrow> x \<in> set i)"
-  apply (induct n)
-  apply safe
-  apply (rule ccontr)
-  apply (subgoal_tac "\<exists>a. poly p a = 0")
-  apply safe
-  apply (drule poly_linear_divides [THEN iffD1])
-  apply safe
-  apply (drule_tac x = q in spec)
-  apply (drule_tac x = x in spec)
-  apply (auto simp del: poly_Nil pmult_Cons)
-  apply (drule_tac x = "a#i" in spec)
-  apply (auto simp only: poly_mult List.list.size)
-  apply (drule_tac x = xa in spec)
-  apply (clarsimp simp add: field_simps)
-  done
-
-lemmas poly_roots_index_lemma2 = conjI [THEN poly_roots_index_lemma]
-
-lemma poly_roots_index_length:
-  "poly p (x::'a::idom) \<noteq> poly [] x \<Longrightarrow>
-    \<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i"
-  by (blast intro: poly_roots_index_lemma2)
-
-lemma poly_roots_finite_lemma':
-  "poly p (x::'a::idom) \<noteq> poly [] x \<Longrightarrow>
-    \<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i"
-  apply (drule poly_roots_index_length)
-  apply auto
+lemma (in idom) poly_roots_finite_lemma2:
+  "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> x \<in> set i"
+  apply (drule poly_roots_index_length, safe)
+  apply (rule_tac x="map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI)
+  apply (auto simp add: image_iff)
+  apply (erule_tac x="x" in allE, clarsimp)
+  apply (case_tac "n = length p")
+  apply (auto simp add: order_le_less)
   done
 
-lemma UNIV_nat_infinite: "\<not> finite (UNIV :: nat set)"
-  unfolding finite_conv_nat_seg_image
-proof (auto simp add: set_eq_iff image_iff)
-  fix n::nat and f:: "nat \<Rightarrow> nat"
-  let ?N = "{i. i < n}"
-  let ?fN = "f ` ?N"
-  let ?y = "Max ?fN + 1"
-  from nat_seg_image_imp_finite[of "?fN" "f" n]
-  have thfN: "finite ?fN" by simp
-  { assume "n =0" hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto }
-  moreover
-  { assume nz: "n \<noteq> 0"
-    hence thne: "?fN \<noteq> {}" by auto
-    have "\<forall>x\<in> ?fN. Max ?fN \<ge> x" using nz Max_ge_iff[OF thfN thne] by auto
-    hence "\<forall>x\<in> ?fN. ?y > x" by (auto simp add: less_Suc_eq_le)
-    hence "?y \<notin> ?fN" by auto
-    hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto }
-  ultimately show "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by blast
+lemma (in ring_char_0) UNIV_ring_char_0_infinte: "\<not> (finite (UNIV:: 'a set))"
+proof
+  assume F: "finite (UNIV :: 'a set)"
+  have "finite (UNIV :: nat set)"
+  proof (rule finite_imageD)
+    have "of_nat ` UNIV \<subseteq> UNIV" by simp
+    then show "finite (of_nat ` UNIV :: 'a set)" using F by (rule finite_subset)
+    show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: inj_on_def)
+  qed
+  with infinite_UNIV_nat show False ..
 qed
 
-lemma UNIV_ring_char_0_infinte: "\<not> finite (UNIV:: ('a::ring_char_0) set)"
+lemma (in idom_char_0) poly_roots_finite: "poly p \<noteq> poly [] \<longleftrightarrow> finite {x. poly p x = 0}"
 proof
-  assume F: "finite (UNIV :: 'a set)"
-  have th0: "of_nat ` UNIV \<subseteq> (UNIV :: 'a set)" by simp
-  from finite_subset[OF th0 F] have th: "finite (of_nat ` UNIV :: 'a set)" .
-  have th': "inj_on (of_nat::nat \<Rightarrow> 'a) UNIV"
-    unfolding inj_on_def by auto
-  from finite_imageD[OF th th'] UNIV_nat_infinite
-  show False by blast
-qed
-
-lemma poly_roots_finite: "poly p \<noteq> poly [] \<longleftrightarrow> finite {x. poly p x = (0::'a::{idom,ring_char_0})}"
-proof
-  assume "poly p \<noteq> poly []"
-  then show "finite {x. poly p x = (0::'a)}"
+  assume H: "poly p \<noteq> poly []"
+  show "finite {x. poly p x = (0::'a)}"
+    using H
     apply -
-    apply (erule contrapos_np)
-    apply (rule ext)
+    apply (erule contrapos_np, rule ext)
     apply (rule ccontr)
-    apply (clarify dest!: poly_roots_finite_lemma')
+    apply (clarify dest!: poly_roots_finite_lemma2)
     using finite_subset
   proof -
     fix x i
@@ -377,119 +385,145 @@
     with finite_subset F show False by auto
   qed
 next
-  assume "finite {x. poly p x = (0\<Colon>'a)}"
-  then show "poly p \<noteq> poly []"
-    using UNIV_ring_char_0_infinte by auto
+  assume F: "finite {x. poly p x = (0\<Colon>'a)}"
+  show "poly p \<noteq> poly []" using F UNIV_ring_char_0_infinte by auto
 qed
 
 text{*Entirety and Cancellation for polynomials*}
 
-lemma poly_entire_lemma:
-  "poly (p:: ('a::{idom,ring_char_0}) list) \<noteq> poly [] \<Longrightarrow> poly q \<noteq> poly [] \<Longrightarrow>
-    poly (p *** q) \<noteq> poly []"
-  by (auto simp add: poly_roots_finite poly_mult Collect_disj_eq)
+lemma (in idom_char_0) poly_entire_lemma2:
+  assumes p0: "poly p \<noteq> poly []"
+    and q0: "poly q \<noteq> poly []"
+  shows "poly (p***q) \<noteq> poly []"
+proof -
+  let ?S = "\<lambda>p. {x. poly p x = 0}"
+  have "?S (p *** q) = ?S p \<union> ?S q" by (auto simp add: poly_mult)
+  with p0 q0 show ?thesis  unfolding poly_roots_finite by auto
+qed
 
-lemma poly_entire:
-  "poly (p *** q) = poly ([]::('a::{idom,ring_char_0}) list) \<longleftrightarrow>
-    (poly p = poly [] \<or> poly q = poly [])"
-  apply (auto dest: fun_cong simp add: poly_entire_lemma poly_mult)
-  apply (blast intro: ccontr dest: poly_entire_lemma poly_mult [THEN subst])
-  done
+lemma (in idom_char_0) poly_entire:
+  "poly (p *** q) = poly [] \<longleftrightarrow> poly p = poly [] \<or> poly q = poly []"
+  using poly_entire_lemma2[of p q]
+  by (auto simp add: fun_eq_iff poly_mult)
 
-lemma poly_entire_neg:
-  "poly (p *** q) \<noteq> poly ([]::('a::{idom,ring_char_0}) list) \<longleftrightarrow>
-    poly p \<noteq> poly [] \<and> poly q \<noteq> poly []"
+lemma (in idom_char_0) poly_entire_neg:
+  "poly (p *** q) \<noteq> poly [] \<longleftrightarrow> poly p \<noteq> poly [] \<and> poly q \<noteq> poly []"
   by (simp add: poly_entire)
 
 lemma fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
   by auto
 
-lemma poly_add_minus_zero_iff: "poly (p +++ -- q) = poly [] \<longleftrightarrow> poly p = poly q"
-  by (auto simp add: field_simps poly_add poly_minus_def fun_eq poly_cmult)
+lemma (in comm_ring_1) poly_add_minus_zero_iff:
+  "poly (p +++ -- q) = poly [] \<longleftrightarrow> poly p = poly q"
+  by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult)
 
-lemma poly_add_minus_mult_eq: "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
+lemma (in comm_ring_1) poly_add_minus_mult_eq:
+  "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
   by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult distrib_left)
 
-lemma poly_mult_left_cancel:
-  "(poly (p *** q) = poly (p *** r)) =
-    (poly p = poly ([]::('a::{idom,ring_char_0}) list) | poly q = poly r)"
-  apply (rule_tac p1 = "p *** q" in poly_add_minus_zero_iff [THEN subst])
-  apply (auto simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
-  done
+subclass (in idom_char_0) comm_ring_1 ..
 
-lemma poly_exp_eq_zero [simp]:
-  "poly (p %^ n) = poly ([]::('a::idom) list) \<longleftrightarrow> poly p = poly [] \<and> n \<noteq> 0"
+lemma (in idom_char_0) poly_mult_left_cancel:
+  "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r"
+proof -
+  have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []"
+    by (simp only: poly_add_minus_zero_iff)
+  also have "\<dots> \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r"
+    by (auto intro: simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
+  finally show ?thesis .
+qed
+
+lemma (in idom) poly_exp_eq_zero[simp]:
+  "poly (p %^ n) = poly [] \<longleftrightarrow> poly p = poly [] \<and> n \<noteq> 0"
   apply (simp only: fun_eq add: HOL.all_simps [symmetric])
   apply (rule arg_cong [where f = All])
   apply (rule ext)
-  apply (induct_tac n)
-  apply (auto simp add: poly_mult)
+  apply (induct n)
+  apply (auto simp add: poly_exp poly_mult)
   done
 
-lemma poly_prime_eq_zero [simp]: "poly [a, 1::'a::comm_ring_1] \<noteq> poly []"
+lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a,1] \<noteq> poly []"
   apply (simp add: fun_eq)
-  apply (rule_tac x = "1 - a" in exI)
+  apply (rule_tac x = "minus one a" in exI)
+  apply (unfold diff_minus)
+  apply (subst add_commute)
+  apply (subst add_assoc)
   apply simp
   done
 
-lemma poly_exp_prime_eq_zero [simp]: "poly ([a, (1::'a::idom)] %^ n) \<noteq> poly []"
+lemma (in idom) poly_exp_prime_eq_zero: "poly ([a, 1] %^ n) \<noteq> poly []"
   by auto
 
 text{*A more constructive notion of polynomials being trivial*}
 
-lemma poly_zero_lemma':
-  "poly (h # t) = poly [] \<Longrightarrow> h = (0::'a::{idom,ring_char_0}) & poly t = poly []"
+lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] \<Longrightarrow> h = 0 \<and> poly t = poly []"
   apply (simp add: fun_eq)
-  apply (case_tac "h = 0")
-  apply (drule_tac [2] x = 0 in spec)
-  apply auto
-  apply (case_tac "poly t = poly []")
-  apply simp
+  apply (case_tac "h = zero")
+  apply (drule_tac [2] x = zero in spec, auto)
+  apply (cases "poly t = poly []", simp)
 proof -
   fix x
-  assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)"  and pnz: "poly t \<noteq> poly []"
+  assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)"
+    and pnz: "poly t \<noteq> poly []"
   let ?S = "{x. poly t x = 0}"
   from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast
   hence th: "?S \<supseteq> UNIV - {0}" by auto
   from poly_roots_finite pnz have th': "finite ?S" by blast
-  from finite_subset[OF th th'] UNIV_ring_char_0_infinte[where ?'a = 'a]
-  show "poly t x = (0\<Colon>'a)" by simp
+  from finite_subset[OF th th'] UNIV_ring_char_0_infinte show "poly t x = (0\<Colon>'a)"
+    by simp
 qed
 
-lemma poly_zero: "poly p = poly [] \<longleftrightarrow> list_all (\<lambda>c. c = (0::'a::{idom,ring_char_0})) p"
+lemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p"
   apply (induct p)
   apply simp
   apply (rule iffI)
-  apply (drule poly_zero_lemma')
-  apply auto
+  apply (drule poly_zero_lemma', auto)
   done
 
+lemma (in idom_char_0) poly_0: "list_all (\<lambda>c. c = 0) p \<Longrightarrow> poly p x = 0"
+  unfolding poly_zero[symmetric] by simp
+
+
 
 text{*Basics of divisibility.*}
 
-lemma poly_primes: "[a, (1::'a::idom)] divides (p *** q) \<longleftrightarrow> [a, 1] divides p \<or> [a, 1] divides q"
+lemma (in idom) poly_primes:
+  "[a, 1] divides (p *** q) \<longleftrightarrow> [a, 1] divides p \<or> [a, 1] divides q"
   apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult distrib_right [symmetric])
-  apply (drule_tac x = "-a" in spec)
+  apply (drule_tac x = "uminus a" in spec)
+  apply (simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
+  apply (cases "p = []")
+  apply (rule exI[where x="[]"])
+  apply simp
+  apply (cases "q = []")
+  apply (erule allE[where x="[]"], simp)
+
+  apply clarsimp
+  apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)")
+  apply (clarsimp simp add: poly_add poly_cmult)
+  apply (rule_tac x="qa" in exI)
+  apply (simp add: distrib_right [symmetric])
+  apply clarsimp
+
   apply (auto simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
-  apply (rule_tac x = "qa *** q" in exI)
-  apply (rule_tac [2] x = "p *** qa" in exI)
+  apply (rule_tac x = "pmult qa q" in exI)
+  apply (rule_tac [2] x = "pmult p qa" in exI)
   apply (auto simp add: poly_add poly_mult poly_cmult mult_ac)
   done
 
-lemma poly_divides_refl [simp]: "p divides p"
+lemma (in comm_semiring_1) poly_divides_refl[simp]: "p divides p"
   apply (simp add: divides_def)
-  apply (rule_tac x = "[1]" in exI)
+  apply (rule_tac x = "[one]" in exI)
   apply (auto simp add: poly_mult fun_eq)
   done
 
-lemma poly_divides_trans: "p divides q \<Longrightarrow> q divides r \<Longrightarrow> p divides r"
-  apply (simp add: divides_def)
-  apply safe
-  apply (rule_tac x = "qa *** qaa" in exI)
+lemma (in comm_semiring_1) poly_divides_trans: "p divides q \<Longrightarrow> q divides r \<Longrightarrow> p divides r"
+  apply (simp add: divides_def, safe)
+  apply (rule_tac x = "pmult qa qaa" in exI)
   apply (auto simp add: poly_mult fun_eq mult_assoc)
   done
 
-lemma poly_divides_exp: "m \<le> n \<Longrightarrow> (p %^ m) divides (p %^ n)"
+lemma (in comm_semiring_1) poly_divides_exp: "m \<le> n \<Longrightarrow> (p %^ m) divides (p %^ n)"
   apply (auto simp add: le_iff_add)
   apply (induct_tac k)
   apply (rule_tac [2] poly_divides_trans)
@@ -498,34 +532,37 @@
   apply (auto simp add: poly_mult fun_eq mult_ac)
   done
 
-lemma poly_exp_divides: "(p %^ n) divides q \<Longrightarrow> m \<le> n \<Longrightarrow> (p %^ m) divides q"
+lemma (in comm_semiring_1) poly_exp_divides:
+  "(p %^ n) divides q \<Longrightarrow> m \<le> n \<Longrightarrow> (p %^ m) divides q"
   by (blast intro: poly_divides_exp poly_divides_trans)
 
-lemma poly_divides_add: "p divides q \<Longrightarrow> p divides r \<Longrightarrow> p divides (q +++ r)"
-  apply (simp add: divides_def)
-  apply auto
-  apply (rule_tac x = "qa +++ qaa" in exI)
+lemma (in comm_semiring_0) poly_divides_add:
+  "p divides q \<Longrightarrow> p divides r \<Longrightarrow> p divides (q +++ r)"
+  apply (simp add: divides_def, auto)
+  apply (rule_tac x = "padd qa qaa" in exI)
   apply (auto simp add: poly_add fun_eq poly_mult distrib_left)
   done
 
-lemma poly_divides_diff: "p divides q \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides r"
-  apply (auto simp add: divides_def)
-  apply (rule_tac x = "qaa +++ -- qa" in exI)
+lemma (in comm_ring_1) poly_divides_diff:
+  "p divides q \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides r"
+  apply (simp add: divides_def, auto)
+  apply (rule_tac x = "padd qaa (poly_minus qa)" in exI)
   apply (auto simp add: poly_add fun_eq poly_mult poly_minus algebra_simps)
   done
 
-lemma poly_divides_diff2: "p divides r \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides q"
+lemma (in comm_ring_1) poly_divides_diff2:
+  "p divides r \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides q"
   apply (erule poly_divides_diff)
   apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac)
   done
 
-lemma poly_divides_zero: "poly p = poly [] \<Longrightarrow> q divides p"
+lemma (in semiring_0) poly_divides_zero: "poly p = poly [] \<Longrightarrow> q divides p"
   apply (simp add: divides_def)
-  apply (rule exI [where x = "[]"])
+  apply (rule exI[where x="[]"])
   apply (auto simp add: fun_eq poly_mult)
   done
 
-lemma poly_divides_zero2 [simp]: "q divides []"
+lemma (in semiring_0) poly_divides_zero2 [simp]: "q divides []"
   apply (simp add: divides_def)
   apply (rule_tac x = "[]" in exI)
   apply (auto simp add: fun_eq)
@@ -533,195 +570,256 @@
 
 text{*At last, we can consider the order of a root.*}
 
-lemma poly_order_exists_lemma [rule_format]:
-  "\<forall>p. length p = d \<longrightarrow> poly p \<noteq> poly [] \<longrightarrow>
-    (\<exists>n q. p = mulexp n [-a, (1::'a::{idom,ring_char_0})] q & poly q a \<noteq> 0)"
-  apply (induct "d")
-  apply (simp add: fun_eq)
-  apply safe
-  apply (case_tac "poly p a = 0")
-  apply (drule_tac poly_linear_divides [THEN iffD1])
-  apply safe
-  apply (drule_tac x = q in spec)
-  apply (drule_tac poly_entire_neg [THEN iffD1])
-  apply safe
-  apply force
-  apply (rule_tac x = "Suc n" in exI)
-  apply (rule_tac x = qa in exI)
-  apply (simp del: pmult_Cons)
-  apply (rule_tac x = 0 in exI)
-  apply force
-  done
+lemma (in idom_char_0) poly_order_exists_lemma:
+  assumes lp: "length p = d"
+    and p: "poly p \<noteq> poly []"
+  shows "\<exists>n q. p = mulexp n [-a, 1] q \<and> poly q a \<noteq> 0"
+  using lp p
+proof (induct d arbitrary: p)
+  case 0
+  thus ?case by simp
+next
+  case (Suc n p)
+  show ?case
+  proof (cases "poly p a = 0")
+    case True
+    from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []" by auto
+    hence pN: "p \<noteq> []" by auto
+    from True[unfolded poly_linear_divides] pN obtain q where q: "p = [-a, 1] *** q"
+      by blast
+    from q h True have qh: "length q = n" "poly q \<noteq> poly []"
+      apply -
+      apply simp
+      apply (simp only: fun_eq)
+      apply (rule ccontr)
+      apply (simp add: fun_eq poly_add poly_cmult)
+      done
+    from Suc.hyps[OF qh] obtain m r where mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0"
+      by blast
+    from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0" by simp
+    then show ?thesis by blast
+  next
+    case False
+    then show ?thesis
+      using Suc.prems
+      apply simp
+      apply (rule exI[where x="0::nat"])
+      apply simp
+      done
+  qed
+qed
+
+
+lemma (in comm_semiring_1) poly_mulexp: "poly (mulexp n p q) x = (poly p x) ^ n * poly q x"
+  by (induct n) (auto simp add: poly_mult mult_ac)
+
+lemma (in comm_semiring_1) divides_left_mult:
+  assumes d:"(p***q) divides r" shows "p divides r \<and> q divides r"
+proof-
+  from d obtain t where r:"poly r = poly (p***q *** t)"
+    unfolding divides_def by blast
+  hence "poly r = poly (p *** (q *** t))"
+    "poly r = poly (q *** (p***t))" by(auto simp add: fun_eq poly_mult mult_ac)
+  thus ?thesis unfolding divides_def by blast
+qed
+
 
 (* FIXME: Tidy up *)
-lemma poly_order_exists:
-  "length p = d \<Longrightarrow> poly p \<noteq> poly [] \<Longrightarrow>
-    \<exists>n. ([-a, 1] %^ n) divides p \<and> \<not> (([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p)"
-  apply (drule poly_order_exists_lemma [where a=a])
-  apply assumption
-  apply clarify
-  apply (rule_tac x = n in exI)
-  apply safe
-  apply (unfold divides_def)
-  apply (rule_tac x = q in exI)
-  apply (induct_tac n)
-  apply simp
-  apply (simp add: poly_add poly_cmult poly_mult distrib_left mult_ac)
-  apply safe
-  apply (subgoal_tac "poly (mulexp n [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc n *** qa)")
-  apply simp
-  apply (induct_tac n)
-  apply (simp del: pmult_Cons pexp_Suc)
-  apply (erule_tac Q = "poly q a = 0" in contrapos_np)
-  apply (simp add: poly_add poly_cmult)
-  apply (rule pexp_Suc [THEN ssubst])
-  apply (rule ccontr)
-  apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc)
-  done
+
+lemma (in semiring_1) zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"
+  by (induct n) simp_all
 
-lemma poly_one_divides [simp]: "[1] divides p"
-  by (auto simp: divides_def)
+lemma (in idom_char_0) poly_order_exists:
+  assumes "length p = d" and "poly p \<noteq> poly []"
+  shows "\<exists>n. [- a, 1] %^ n divides p \<and> \<not> [- a, 1] %^ Suc n divides p"
+proof -
+  from assms have "\<exists>n q. p = mulexp n [- a, 1] q \<and> poly q a \<noteq> 0"
+    by (rule poly_order_exists_lemma)
+  then obtain n q where p: "p = mulexp n [- a, 1] q" and "poly q a \<noteq> 0" by blast
+  have "[- a, 1] %^ n divides mulexp n [- a, 1] q"
+  proof (rule dividesI)
+    show "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ n *** q)"
+      by (induct n) (simp_all add: poly_add poly_cmult poly_mult distrib_left mult_ac)
+  qed
+  moreover have "\<not> [- a, 1] %^ Suc n divides mulexp n [- a, 1] q"
+  proof
+    assume "[- a, 1] %^ Suc n divides mulexp n [- a, 1] q"
+    then obtain m where "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ Suc n *** m)"
+      by (rule dividesE)
+    moreover have "poly (mulexp n [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc n *** m)"
+    proof (induct n)
+      case 0 show ?case
+      proof (rule ccontr)
+        assume "\<not> poly (mulexp 0 [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc 0 *** m)"
+        then have "poly q a = 0"
+          by (simp add: poly_add poly_cmult)
+        with `poly q a \<noteq> 0` show False by simp
+      qed
+    next
+      case (Suc n) show ?case
+        by (rule pexp_Suc [THEN ssubst], rule ccontr)
+          (simp add: poly_mult_left_cancel poly_mult_assoc Suc del: pmult_Cons pexp_Suc)
+    qed
+    ultimately show False by simp
+  qed
+  ultimately show ?thesis by (auto simp add: p)
+qed
 
-lemma poly_order: "poly p \<noteq> poly [] \<Longrightarrow>
-    \<exists>! n. ([-a, (1::'a::{idom,ring_char_0})] %^ n) divides p \<and> \<not> (([-a, 1] %^ Suc n) divides p)"
+lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"
+  by (auto simp add: divides_def)
+
+lemma (in idom_char_0) poly_order:
+  "poly p \<noteq> poly [] \<Longrightarrow> \<exists>!n. ([-a, 1] %^ n) divides p \<and> \<not> (([-a, 1] %^ Suc n) divides p)"
   apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
   apply (cut_tac x = y and y = n in less_linear)
   apply (drule_tac m = n in poly_exp_divides)
   apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
-    simp del: pmult_Cons pexp_Suc)
+              simp del: pmult_Cons pexp_Suc)
   done
 
 text{*Order*}
 
-lemma some1_equalityD: "n = (SOME n. P n) \<Longrightarrow> EX! n. P n \<Longrightarrow> P n"
+lemma some1_equalityD: "n = (SOME n. P n) \<Longrightarrow> \<exists>!n. P n \<Longrightarrow> P n"
   by (blast intro: someI2)
 
-lemma order:
-  "(([-a, (1::'a::{idom,ring_char_0})] %^ n) divides p &
-    ~(([-a, 1] %^ (Suc n)) divides p)) =
-    ((n = order a p) & ~(poly p = poly []))"
+lemma (in idom_char_0) order:
+      "(([-a, 1] %^ n) divides p \<and>
+        ~(([-a, 1] %^ (Suc n)) divides p)) =
+        ((n = order a p) \<and> ~(poly p = poly []))"
   apply (unfold order_def)
   apply (rule iffI)
   apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
   apply (blast intro!: poly_order [THEN [2] some1_equalityD])
   done
 
-lemma order2: "poly p \<noteq> poly [] \<Longrightarrow>
-  ([-a, (1::'a::{idom,ring_char_0})] %^ (order a p)) divides p &
-    ~(([-a, 1] %^ (Suc(order a p))) divides p)"
+lemma (in idom_char_0) order2:
+  "poly p \<noteq> poly [] \<Longrightarrow>
+    ([-a, 1] %^ (order a p)) divides p \<and> \<not> (([-a, 1] %^ (Suc (order a p))) divides p)"
   by (simp add: order del: pexp_Suc)
 
-lemma order_unique: "poly p \<noteq> poly [] \<Longrightarrow> ([-a, 1] %^ n) divides p \<Longrightarrow>
-  \<not> (([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p) \<Longrightarrow> n = order a p"
+lemma (in idom_char_0) order_unique:
+  "poly p \<noteq> poly [] \<Longrightarrow> ([-a, 1] %^ n) divides p \<Longrightarrow> ~(([-a, 1] %^ (Suc n)) divides p) \<Longrightarrow>
+    n = order a p"
   using order [of a n p] by auto
 
-lemma order_unique_lemma:
-  "(poly p \<noteq> poly [] \<and> ([-a, 1] %^ n) divides p \<and>
-    \<not> (([-a, (1::'a::{idom,ring_char_0})] %^ (Suc n)) divides p)) \<Longrightarrow>
+lemma (in idom_char_0) order_unique_lemma:
+  "poly p \<noteq> poly [] \<and> ([-a, 1] %^ n) divides p \<and> ~(([-a, 1] %^ (Suc n)) divides p) \<Longrightarrow>
     n = order a p"
   by (blast intro: order_unique)
 
-lemma order_poly: "poly p = poly q ==> order a p = order a q"
+lemma (in ring_1) order_poly: "poly p = poly q \<Longrightarrow> order a p = order a q"
   by (auto simp add: fun_eq divides_def poly_mult order_def)
 
-lemma pexp_one [simp]: "p %^ (Suc 0) = p"
-  by (induct p) simp_all
+lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p"
+  by (induct "p") auto
+
+lemma (in comm_ring_1) lemma_order_root:
+  "0 < n \<and> [- a, 1] %^ n divides p \<and> ~ [- a, 1] %^ (Suc n) divides p \<Longrightarrow> poly p a = 0"
+  by (induct n arbitrary: a p) (auto simp add: divides_def poly_mult simp del: pmult_Cons)
 
-lemma lemma_order_root:
-  "0 < n & [- a, 1] %^ n divides p & ~ [- a, 1] %^ (Suc n) divides p \<Longrightarrow> poly p a = 0"
-  apply (induct n arbitrary: p a)
-  apply blast
-  apply (auto simp add: divides_def poly_mult simp del: pmult_Cons)
+lemma (in idom_char_0) order_root:
+  "poly p a = 0 \<longleftrightarrow> poly p = poly [] \<or> order a p \<noteq> 0"
+  apply (cases "poly p = poly []")
+  apply auto
+  apply (simp add: poly_linear_divides del: pmult_Cons, safe)
+  apply (drule_tac [!] a = a in order2)
+  apply (rule ccontr)
+  apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast)
+  using neq0_conv
+  apply (blast intro: lemma_order_root)
   done
 
-lemma order_root: "poly p a = (0::'a::{idom,ring_char_0}) \<longleftrightarrow> poly p = poly [] \<or> order a p \<noteq> 0"
-  apply (cases "poly p = poly []")
-  apply auto
-  apply (simp add: poly_linear_divides del: pmult_Cons)
-  apply safe
-  apply (drule_tac [!] a = a in order2)
-  apply (rule ccontr)
-  apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons)
-  apply blast
-  using neq0_conv apply (blast intro: lemma_order_root)
-  done
-
-lemma order_divides: "([-a, 1::'a::{idom,ring_char_0}] %^ n) divides p \<longleftrightarrow>
-    poly p = poly [] \<or> n \<le> order a p"
+lemma (in idom_char_0) order_divides:
+  "([-a, 1] %^ n) divides p \<longleftrightarrow> poly p = poly [] \<or> n \<le> order a p"
   apply (cases "poly p = poly []")
   apply auto
   apply (simp add: divides_def fun_eq poly_mult)
   apply (rule_tac x = "[]" in exI)
-  apply (auto dest!: order2 [where a = a] intro: poly_exp_divides simp del: pexp_Suc)
+  apply (auto dest!: order2 [where a=a] intro: poly_exp_divides simp del: pexp_Suc)
   done
 
-lemma order_decomp:
-  "poly p \<noteq> poly [] \<Longrightarrow>
-    \<exists>q. poly p = poly (([-a, 1] %^ (order a p)) *** q) \<and>
-      \<not> ([-a, 1::'a::{idom,ring_char_0}] divides q)"
+lemma (in idom_char_0) order_decomp:
+  "poly p \<noteq> poly [] \<Longrightarrow> \<exists>q. poly p = poly (([-a, 1] %^ (order a p)) *** q) \<and> ~([-a, 1] divides q)"
   apply (unfold divides_def)
   apply (drule order2 [where a = a])
-  apply (simp add: divides_def del: pexp_Suc pmult_Cons)
-  apply safe
-  apply (rule_tac x = q in exI)
-  apply safe
+  apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
+  apply (rule_tac x = q in exI, safe)
   apply (drule_tac x = qa in spec)
   apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons)
   done
 
 text{*Important composition properties of orders.*}
-
-lemma order_mult: "poly (p *** q) \<noteq> poly [] \<Longrightarrow>
-  order a (p *** q) = order a p + order (a::'a::{idom,ring_char_0}) q"
-  apply (cut_tac a = a and p = "p***q" and n = "order a p + order a q" in order)
+lemma order_mult:
+  "poly (p *** q) \<noteq> poly [] \<Longrightarrow>
+    order a (p *** q) = order a p + order (a::'a::{idom_char_0}) q"
+  apply (cut_tac a = a and p = "p *** q" and n = "order a p + order a q" in order)
   apply (auto simp add: poly_entire simp del: pmult_Cons)
   apply (drule_tac a = a in order2)+
   apply safe
-  apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons)
-  apply safe
+  apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
   apply (rule_tac x = "qa *** qaa" in exI)
   apply (simp add: poly_mult mult_ac del: pmult_Cons)
   apply (drule_tac a = a in order_decomp)+
   apply safe
-  apply (subgoal_tac "[-a, 1] divides (qa *** qaa) ")
+  apply (subgoal_tac "[-a,1] divides (qa *** qaa) ")
   apply (simp add: poly_primes del: pmult_Cons)
   apply (auto simp add: divides_def simp del: pmult_Cons)
   apply (rule_tac x = qb in exI)
-  apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) =
-    poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
-  apply (drule poly_mult_left_cancel [THEN iffD1])
-  apply force
-  apply (subgoal_tac "poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** (qa *** qaa))) =
-    poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))) ")
-  apply (drule poly_mult_left_cancel [THEN iffD1])
-  apply force
+  apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
+  apply (drule poly_mult_left_cancel [THEN iffD1], force)
+  apply (subgoal_tac "poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** (qa *** qaa))) = poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))) ")
+  apply (drule poly_mult_left_cancel [THEN iffD1], force)
   apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
   done
 
-lemma order_root2: "poly p \<noteq> poly [] \<Longrightarrow> poly p a = 0 \<longleftrightarrow> order (a::'a::{idom,ring_char_0}) p \<noteq> 0"
+lemma (in idom_char_0) order_mult:
+  assumes "poly (p *** q) \<noteq> poly []"
+  shows "order a (p *** q) = order a p + order a q"
+  using assms
+  apply (cut_tac a = a and p = "pmult p q" and n = "order a p + order a q" in order)
+  apply (auto simp add: poly_entire simp del: pmult_Cons)
+  apply (drule_tac a = a in order2)+
+  apply safe
+  apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
+  apply (rule_tac x = "pmult qa qaa" in exI)
+  apply (simp add: poly_mult mult_ac del: pmult_Cons)
+  apply (drule_tac a = a in order_decomp)+
+  apply safe
+  apply (subgoal_tac "[uminus a, one] divides pmult qa qaa")
+  apply (simp add: poly_primes del: pmult_Cons)
+  apply (auto simp add: divides_def simp del: pmult_Cons)
+  apply (rule_tac x = qb in exI)
+  apply (subgoal_tac "poly (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa)) =
+    poly (pmult (pexp [uminus a, one] (?order a p)) (pmult [uminus a, one] qb))")
+  apply (drule poly_mult_left_cancel [THEN iffD1], force)
+  apply (subgoal_tac "poly (pmult (pexp [uminus a, one] (order a q))
+      (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa))) =
+    poly (pmult (pexp [uminus a, one] (order a q))
+      (pmult (pexp [uminus a, one] (order a p)) (pmult [uminus a, one] qb)))")
+  apply (drule poly_mult_left_cancel [THEN iffD1], force)
+  apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
+  done
+
+lemma (in idom_char_0) order_root2: "poly p \<noteq> poly [] \<Longrightarrow> poly p a = 0 \<longleftrightarrow> order a p \<noteq> 0"
   by (rule order_root [THEN ssubst]) auto
 
-lemma pmult_one [simp]: "[1] *** p = p"
-  by auto
+lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p" by auto
 
-lemma poly_Nil_zero: "poly [] = poly [0]"
+lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]"
   by (simp add: fun_eq)
 
-lemma rsquarefree_decomp:
-  "rsquarefree p \<Longrightarrow> poly p a = (0::'a::{idom,ring_char_0}) \<Longrightarrow>
+lemma (in idom_char_0) rsquarefree_decomp:
+  "rsquarefree p \<Longrightarrow> poly p a = 0 \<Longrightarrow>
     \<exists>q. poly p = poly ([-a, 1] *** q) \<and> poly q a \<noteq> 0"
-  apply (simp add: rsquarefree_def)
-  apply safe
+  apply (simp add: rsquarefree_def, safe)
   apply (frule_tac a = a in order_decomp)
   apply (drule_tac x = a in spec)
   apply (drule_tac a = a in order_root2 [symmetric])
   apply (auto simp del: pmult_Cons)
-  apply (rule_tac x = q in exI)
-  apply safe
+  apply (rule_tac x = q in exI, safe)
   apply (simp add: poly_mult fun_eq)
   apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
-  apply (simp add: divides_def del: pmult_Cons)
-  apply safe
+  apply (simp add: divides_def del: pmult_Cons, safe)
   apply (drule_tac x = "[]" in spec)
   apply (auto simp add: fun_eq)
   done
@@ -729,72 +827,222 @@
 
 text{*Normalization of a polynomial.*}
 
-lemma poly_normalize [simp]: "poly (pnormalize p) = poly p"
+lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p"
   by (induct p) (auto simp add: fun_eq)
 
-
 text{*The degree of a polynomial.*}
 
-lemma lemma_degree_zero: "list_all (\<lambda>c. c = 0) p \<longleftrightarrow> pnormalize p = []"
+lemma (in semiring_0) lemma_degree_zero: "list_all (%c. c = 0) p \<longleftrightarrow> pnormalize p = []"
   by (induct p) auto
 
-lemma degree_zero: "poly p = poly ([] :: 'a::{idom,ring_char_0} list) \<Longrightarrow> degree p = 0"
-  apply (cases "pnormalize p = []")
-  apply (simp add: degree_def)
-  apply (auto simp add: poly_zero lemma_degree_zero)
-  done
+lemma (in idom_char_0) degree_zero:
+  assumes "poly p = poly []"
+  shows "degree p = 0"
+  using assms
+  by (cases "pnormalize p = []") (auto simp add: degree_def poly_zero lemma_degree_zero)
 
-lemma pnormalize_sing: "pnormalize [x] = [x] \<longleftrightarrow> x \<noteq> 0"
+lemma (in semiring_0) pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0"
+  by simp
+
+lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])"
   by simp
 
-lemma pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])"
-  by simp
-
-lemma pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c # p)"
+lemma (in semiring_0) pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)"
   unfolding pnormal_def by simp
 
-lemma pnormal_tail: "p \<noteq> [] \<Longrightarrow> pnormal (c # p) \<Longrightarrow> pnormal p"
-  unfolding pnormal_def
-  apply (cases "pnormalize p = []")
+lemma (in semiring_0) pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p"
+  unfolding pnormal_def by(auto split: split_if_asm)
+
+
+lemma (in semiring_0) pnormal_last_nonzero: "pnormal p \<Longrightarrow> last p \<noteq> 0"
+  by (induct p) (simp_all add: pnormal_def split: split_if_asm)
+
+lemma (in semiring_0) pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
+  unfolding pnormal_def length_greater_0_conv by blast
+
+lemma (in semiring_0) pnormal_last_length: "0 < length p \<Longrightarrow> last p \<noteq> 0 \<Longrightarrow> pnormal p"
+  by (induct p) (auto simp: pnormal_def  split: split_if_asm)
+
+
+lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> 0 < length p \<and> last p \<noteq> 0"
+  using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
+
+lemma (in idom_char_0) poly_Cons_eq:
+  "poly (c # cs) = poly (d # ds) \<longleftrightarrow> c = d \<and> poly cs = poly ds"
+  (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+  assume eq: ?lhs
+  hence "\<And>x. poly ((c#cs) +++ -- (d#ds)) x = 0"
+    by (simp only: poly_minus poly_add algebra_simps) simp
+  hence "poly ((c#cs) +++ -- (d#ds)) = poly []" by(simp add: fun_eq_iff)
+  hence "c = d \<and> list_all (\<lambda>x. x=0) ((cs +++ -- ds))"
+    unfolding poly_zero by (simp add: poly_minus_def algebra_simps)
+  hence "c = d \<and> (\<forall>x. poly (cs +++ -- ds) x = 0)"
+    unfolding poly_zero[symmetric] by simp
+  then show ?rhs by (simp add: poly_minus poly_add algebra_simps fun_eq_iff)
+next
+  assume ?rhs
+  then show ?lhs by(simp add:fun_eq_iff)
+qed
+
+lemma (in idom_char_0) pnormalize_unique: "poly p = poly q \<Longrightarrow> pnormalize p = pnormalize q"
+proof (induct q arbitrary: p)
+  case Nil
+  thus ?case by (simp only: poly_zero lemma_degree_zero) simp
+next
+  case (Cons c cs p)
+  thus ?case
+  proof (induct p)
+    case Nil
+    hence "poly [] = poly (c#cs)" by blast
+    then have "poly (c#cs) = poly [] " by simp
+    thus ?case by (simp only: poly_zero lemma_degree_zero) simp
+  next
+    case (Cons d ds)
+    hence eq: "poly (d # ds) = poly (c # cs)" by blast
+    hence eq': "\<And>x. poly (d # ds) x = poly (c # cs) x" by simp
+    hence "poly (d # ds) 0 = poly (c # cs) 0" by blast
+    hence dc: "d = c" by auto
+    with eq have "poly ds = poly cs"
+      unfolding  poly_Cons_eq by simp
+    with Cons.prems have "pnormalize ds = pnormalize cs" by blast
+    with dc show ?case by simp
+  qed
+qed
+
+lemma (in idom_char_0) degree_unique:
+  assumes pq: "poly p = poly q"
+  shows "degree p = degree q"
+  using pnormalize_unique[OF pq] unfolding degree_def by simp
+
+lemma (in semiring_0) pnormalize_length:
+  "length (pnormalize p) \<le> length p" by (induct p) auto
+
+lemma (in semiring_0) last_linear_mul_lemma:
+  "last ((a %* p) +++ (x#(b %* p))) = (if p = [] then x else b * last p)"
+  apply (induct p arbitrary: a x b)
   apply auto
-  apply (cases "c = 0")
+  apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []")
+  apply simp
+  apply (induct_tac p)
   apply auto
   done
 
-lemma pnormal_last_nonzero: "pnormal p \<Longrightarrow> last p \<noteq> 0"
-  apply (induct p)
-  apply (auto simp add: pnormal_def)
-  apply (case_tac "pnormalize p = []")
-  apply auto
-  apply (case_tac "a = 0")
-  apply auto
-  done
+lemma (in semiring_1) last_linear_mul:
+  assumes p: "p \<noteq> []"
+  shows "last ([a,1] *** p) = last p"
+proof -
+  from p obtain c cs where cs: "p = c#cs" by (cases p) auto
+  from cs have eq: "[a,1] *** p = (a %* (c#cs)) +++ (0#(1 %* (c#cs)))"
+    by (simp add: poly_cmult_distr)
+  show ?thesis using cs
+    unfolding eq last_linear_mul_lemma by simp
+qed
+
+lemma (in semiring_0) pnormalize_eq: "last p \<noteq> 0 \<Longrightarrow> pnormalize p = p"
+  by (induct p) (auto split: split_if_asm)
+
+lemma (in semiring_0) last_pnormalize: "pnormalize p \<noteq> [] \<Longrightarrow> last (pnormalize p) \<noteq> 0"
+  by (induct p) auto
+
+lemma (in semiring_0) pnormal_degree: "last p \<noteq> 0 \<Longrightarrow> degree p = length p - 1"
+  using pnormalize_eq[of p] unfolding degree_def by simp
 
-lemma  pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
-  unfolding pnormal_def length_greater_0_conv by blast
+lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)"
+  by (rule ext) simp
+
+lemma (in idom_char_0) linear_mul_degree:
+  assumes p: "poly p \<noteq> poly []"
+  shows "degree ([a,1] *** p) = degree p + 1"
+proof -
+  from p have pnz: "pnormalize p \<noteq> []"
+    unfolding poly_zero lemma_degree_zero .
+
+  from last_linear_mul[OF pnz, of a] last_pnormalize[OF pnz]
+  have l0: "last ([a, 1] *** pnormalize p) \<noteq> 0" by simp
+  from last_pnormalize[OF pnz] last_linear_mul[OF pnz, of a]
+    pnormal_degree[OF l0] pnormal_degree[OF last_pnormalize[OF pnz]] pnz
+
+  have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1"
+    by simp
+
+  have eqs: "poly ([a,1] *** pnormalize p) = poly ([a,1] *** p)"
+    by (rule ext) (simp add: poly_mult poly_add poly_cmult)
+  from degree_unique[OF eqs] th
+  show ?thesis by (simp add: degree_unique[OF poly_normalize])
+qed
 
-lemma pnormal_last_length: "0 < length p \<Longrightarrow> last p \<noteq> 0 \<Longrightarrow> pnormal p"
-  apply (induct p)
-  apply auto
-  apply (case_tac "p = []")
-  apply auto
-  apply (simp add: pnormal_def)
-  apply (rule pnormal_cons)
-  apply auto
-  done
+lemma (in idom_char_0) linear_pow_mul_degree:
+  "degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)"
+proof (induct n arbitrary: a p)
+  case (0 a p)
+  show ?case
+  proof (cases "poly p = poly []")
+    case True
+    then show ?thesis
+      using degree_unique[OF True] by (simp add: degree_def)
+  next
+    case False
+    then show ?thesis by (auto simp add: poly_Nil_ext)
+  qed
+next
+  case (Suc n a p)
+  have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1] %^ n *** ([a,1] *** p))"
+    apply (rule ext)
+    apply (simp add: poly_mult poly_add poly_cmult)
+    apply (simp add: mult_ac add_ac distrib_left)
+    done
+  note deq = degree_unique[OF eq]
+  show ?case
+  proof (cases "poly p = poly []")
+    case True
+    with eq have eq': "poly ([a,1] %^(Suc n) *** p) = poly []"
+      apply -
+      apply (rule ext)
+      apply (simp add: poly_mult poly_cmult poly_add)
+      done
+    from degree_unique[OF eq'] True show ?thesis
+      by (simp add: degree_def)
+  next
+    case False
+    then have ap: "poly ([a,1] *** p) \<noteq> poly []"
+      using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto
+    have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1]%^n *** ([a,1] *** p))"
+      by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps)
+    from ap have ap': "(poly ([a,1] *** p) = poly []) = False"
+      by blast
+    have th0: "degree ([a,1]%^n *** ([a,1] *** p)) = degree ([a,1] *** p) + n"
+      apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap')
+      apply simp
+      done
+    from degree_unique[OF eq] ap False th0 linear_mul_degree[OF False, of a]
+    show ?thesis by (auto simp del: poly.simps)
+  qed
+qed
 
-lemma pnormal_id: "pnormal p \<longleftrightarrow> 0 < length p \<and> last p \<noteq> 0"
-  using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
+lemma (in idom_char_0) order_degree:
+  assumes p0: "poly p \<noteq> poly []"
+  shows "order a p \<le> degree p"
+proof -
+  from order2[OF p0, unfolded divides_def]
+  obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)" by blast
+  {
+    assume "poly q = poly []"
+    with q p0 have False by (simp add: poly_mult poly_entire)
+  }
+  with degree_unique[OF q, unfolded linear_pow_mul_degree] show ?thesis
+    by auto
+qed
 
 text{*Tidier versions of finiteness of roots.*}
 
-lemma poly_roots_finite_set:
-  "poly p \<noteq> poly [] \<Longrightarrow> finite {x::'a::{idom,ring_char_0}. poly p x = 0}"
+lemma (in idom_char_0) poly_roots_finite_set:
+  "poly p \<noteq> poly [] \<Longrightarrow> finite {x. poly p x = 0}"
   unfolding poly_roots_finite .
 
 text{*bound for polynomial.*}
 
-lemma poly_mono: "abs x \<le> k \<Longrightarrow> abs (poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"
+lemma poly_mono: "abs(x) \<le> k \<Longrightarrow> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"
   apply (induct p)
   apply auto
   apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
@@ -802,7 +1050,6 @@
   apply (auto intro!: mult_mono simp add: abs_mult)
   done
 
-lemma poly_Sing: "poly [c] x = c"
-  by simp
+lemma (in semiring_0) poly_Sing: "poly [c] x = c" by simp
 
 end
diff -r 07c0c121a8dc -r 63fe59f64578 src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Thu Oct 31 11:48:45 2013 +0100
+++ b/src/HOL/Library/Library.thy	Thu Oct 31 11:44:20 2013 +0100
@@ -65,7 +65,6 @@
   Sublist
   Sum_of_Squares
   Transitive_Closure_Table
-  Univ_Poly
   Wfrec
   While_Combinator
   Zorn
diff -r 07c0c121a8dc -r 63fe59f64578 src/HOL/Library/Univ_Poly.thy
--- a/src/HOL/Library/Univ_Poly.thy	Thu Oct 31 11:48:45 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1053 +0,0 @@
-(*  Title:      HOL/Library/Univ_Poly.thy
-    Author:     Amine Chaieb
-*)
-
-header {* Univariate Polynomials *}
-
-theory Univ_Poly
-imports Main
-begin
-
-text{* Application of polynomial as a function. *}
-
-primrec (in semiring_0) poly :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a"
-where
-  poly_Nil:  "poly [] x = 0"
-| poly_Cons: "poly (h#t) x = h + x * poly t x"
-
-
-subsection{*Arithmetic Operations on Polynomials*}
-
-text{*addition*}
-
-primrec (in semiring_0) padd :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "+++" 65)
-where
-  padd_Nil:  "[] +++ l2 = l2"
-| padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t else (h + hd l2)#(t +++ tl l2))"
-
-text{*Multiplication by a constant*}
-primrec (in semiring_0) cmult :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "%*" 70) where
-  cmult_Nil:  "c %* [] = []"
-| cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
-
-text{*Multiplication by a polynomial*}
-primrec (in semiring_0) pmult :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "***" 70)
-where
-  pmult_Nil:  "[] *** l2 = []"
-| pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2
-                              else (h %* l2) +++ ((0) # (t *** l2)))"
-
-text{*Repeated multiplication by a polynomial*}
-primrec (in semiring_0) mulexp :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a  list \<Rightarrow> 'a list" where
-  mulexp_zero:  "mulexp 0 p q = q"
-| mulexp_Suc:   "mulexp (Suc n) p q = p *** mulexp n p q"
-
-text{*Exponential*}
-primrec (in semiring_1) pexp :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list"  (infixl "%^" 80) where
-  pexp_0:   "p %^ 0 = [1]"
-| pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
-
-text{*Quotient related value of dividing a polynomial by x + a*}
-(* Useful for divisor properties in inductive proofs *)
-primrec (in field) "pquot" :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list"
-where
-  pquot_Nil:  "pquot [] a= []"
-| pquot_Cons: "pquot (h#t) a =
-    (if t = [] then [h] else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))"
-
-text{*normalization of polynomials (remove extra 0 coeff)*}
-primrec (in semiring_0) pnormalize :: "'a list \<Rightarrow> 'a list" where
-  pnormalize_Nil:  "pnormalize [] = []"
-| pnormalize_Cons: "pnormalize (h#p) =
-    (if pnormalize p = [] then (if h = 0 then [] else [h]) else h # pnormalize p)"
-
-definition (in semiring_0) "pnormal p = ((pnormalize p = p) \<and> p \<noteq> [])"
-definition (in semiring_0) "nonconstant p = (pnormal p \<and> (\<forall>x. p \<noteq> [x]))"
-text{*Other definitions*}
-
-definition (in ring_1) poly_minus :: "'a list \<Rightarrow> 'a list" ("-- _" [80] 80)
-  where "-- p = (- 1) %* p"
-
-definition (in semiring_0) divides :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "divides" 70)
-  where "p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))"
-
-lemma (in semiring_0) dividesI:
-  "poly p2 = poly (p1 *** q) \<Longrightarrow> p1 divides p2"
-  by (auto simp add: divides_def)
-
-lemma (in semiring_0) dividesE:
-  assumes "p1 divides p2"
-  obtains q where "poly p2 = poly (p1 *** q)"
-  using assms by (auto simp add: divides_def)
-
-    --{*order of a polynomial*}
-definition (in ring_1) order :: "'a \<Rightarrow> 'a list \<Rightarrow> nat" where
-  "order a p = (SOME n. ([-a, 1] %^ n) divides p \<and> ~ (([-a, 1] %^ (Suc n)) divides p))"
-
-     --{*degree of a polynomial*}
-definition (in semiring_0) degree :: "'a list \<Rightarrow> nat"
-  where "degree p = length (pnormalize p) - 1"
-
-     --{*squarefree polynomials --- NB with respect to real roots only.*}
-definition (in ring_1) rsquarefree :: "'a list \<Rightarrow> bool"
-  where "rsquarefree p \<longleftrightarrow> poly p \<noteq> poly [] \<and> (\<forall>a. order a p = 0 \<or> order a p = 1)"
-
-context semiring_0
-begin
-
-lemma padd_Nil2[simp]: "p +++ [] = p"
-  by (induct p) auto
-
-lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
-  by auto
-
-lemma pminus_Nil: "-- [] = []"
-  by (simp add: poly_minus_def)
-
-lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp
-
-end
-
-lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t" by (induct t) auto
-
-lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ ((0)#t) = (a#t)"
-  by simp
-
-text{*Handy general properties*}
-
-lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b"
-proof (induct b arbitrary: a)
-  case Nil
-  thus ?case by auto
-next
-  case (Cons b bs a)
-  thus ?case by (cases a) (simp_all add: add_commute)
-qed
-
-lemma (in comm_semiring_0) padd_assoc: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
-  apply (induct a)
-  apply (simp, clarify)
-  apply (case_tac b, simp_all add: add_ac)
-  done
-
-lemma (in semiring_0) poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)"
-  apply (induct p arbitrary: q)
-  apply simp
-  apply (case_tac q, simp_all add: distrib_left)
-  done
-
-lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"
-  apply (induct t)
-  apply simp
-  apply (auto simp add: padd_commut)
-  apply (case_tac t, auto)
-  done
-
-text{*properties of evaluation of polynomials.*}
-
-lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
-proof(induct p1 arbitrary: p2)
-  case Nil
-  thus ?case by simp
-next
-  case (Cons a as p2)
-  thus ?case
-    by (cases p2) (simp_all  add: add_ac distrib_left)
-qed
-
-lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x"
-  apply (induct p)
-  apply (case_tac [2] "x = zero")
-  apply (auto simp add: distrib_left mult_ac)
-  done
-
-lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c*poly p x"
-  by (induct p) (auto simp add: distrib_left mult_ac)
-
-lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
-  apply (simp add: poly_minus_def)
-  apply (auto simp add: poly_cmult)
-  done
-
-lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
-proof (induct p1 arbitrary: p2)
-  case Nil
-  thus ?case by simp
-next
-  case (Cons a as p2)
-  thus ?case by (cases as)
-    (simp_all add: poly_cmult poly_add distrib_right distrib_left mult_ac)
-qed
-
-class idom_char_0 = idom + ring_char_0
-
-lemma (in comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n"
-  by (induct n) (auto simp add: poly_cmult poly_mult)
-
-text{*More Polynomial Evaluation Lemmas*}
-
-lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
-  by simp
-
-lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
-  by (simp add: poly_mult mult_assoc)
-
-lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0"
-  by (induct p) auto
-
-lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
-  by (induct n) (auto simp add: poly_mult mult_assoc)
-
-subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
- @{term "p(x)"} *}
-
-lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
-proof(induct t)
-  case Nil
-  { fix h have "[h] = [h] +++ [- a, 1] *** []" by simp }
-  thus ?case by blast
-next
-  case (Cons  x xs)
-  { fix h
-    from Cons.hyps[rule_format, of x]
-    obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
-    have "h#x#xs = [a*r + h] +++ [-a, 1] *** (r#q)"
-      using qr by (cases q) (simp_all add: algebra_simps)
-    hence "\<exists>q r. h#x#xs = [r] +++ [-a, 1] *** q" by blast}
-  thus ?case by blast
-qed
-
-lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
-  using lemma_poly_linear_rem [where t = t and a = a] by auto
-
-
-lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))"
-proof -
-  { assume p: "p = []" hence ?thesis by simp }
-  moreover
-  {
-    fix x xs assume p: "p = x#xs"
-    {
-      fix q assume "p = [-a, 1] *** q"
-      hence "poly p a = 0" by (simp add: poly_add poly_cmult)
-    }
-    moreover
-    { assume p0: "poly p a = 0"
-      from poly_linear_rem[of x xs a] obtain q r
-      where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
-      have "r = 0" using p0 by (simp only: p qr poly_mult poly_add) simp
-      hence "\<exists>q. p = [- a, 1] *** q"
-        using p qr
-        apply -
-        apply (rule exI[where x=q])
-        apply auto
-        apply (cases q)
-        apply auto
-        done
-    }
-    ultimately have ?thesis using p by blast
-  }
-  ultimately show ?thesis by (cases p) auto
-qed
-
-lemma (in semiring_0) lemma_poly_length_mult[simp]: "\<forall>h k a. length (k %* p +++  (h # (a %* p))) = Suc (length p)"
-  by (induct p) auto
-
-lemma (in semiring_0) lemma_poly_length_mult2[simp]: "\<forall>h k. length (k %* p +++  (h # p)) = Suc (length p)"
-  by (induct p) auto
-
-lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"
-  by auto
-
-subsection{*Polynomial length*}
-
-lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p"
-  by (induct p) auto
-
-lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)"
-  by (induct p1 arbitrary: p2) (simp_all, arith)
-
-lemma (in semiring_0) poly_root_mult_length[simp]: "length([a,b] *** p) = Suc (length p)"
-  by (simp add: poly_add_length)
-
-lemma (in idom) poly_mult_not_eq_poly_Nil[simp]:
-  "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x"
-  by (auto simp add: poly_mult)
-
-lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 \<longleftrightarrow> poly p x = 0 \<or> poly q x = 0"
-  by (auto simp add: poly_mult)
-
-text{*Normalisation Properties*}
-
-lemma (in semiring_0) poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)"
-  by (induct p) auto
-
-text{*A nontrivial polynomial of degree n has no more than n roots*}
-lemma (in idom) poly_roots_index_lemma:
-   assumes p: "poly p x \<noteq> poly [] x" and n: "length p = n"
-  shows "\<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
-  using p n
-proof (induct n arbitrary: p x)
-  case 0
-  thus ?case by simp
-next
-  case (Suc n p x)
-  {
-    assume C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)"
-    from Suc.prems have p0: "poly p x \<noteq> 0" "p\<noteq> []" by auto
-    from p0(1)[unfolded poly_linear_divides[of p x]]
-    have "\<forall>q. p \<noteq> [- x, 1] *** q" by blast
-    from C obtain a where a: "poly p a = 0" by blast
-    from a[unfolded poly_linear_divides[of p a]] p0(2)
-    obtain q where q: "p = [-a, 1] *** q" by blast
-    have lg: "length q = n" using q Suc.prems(2) by simp
-    from q p0 have qx: "poly q x \<noteq> poly [] x"
-      by (auto simp add: poly_mult poly_add poly_cmult)
-    from Suc.hyps[OF qx lg] obtain i where
-      i: "\<forall>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)" by blast
-    let ?i = "\<lambda>m. if m = Suc n then a else i m"
-    from C[of ?i] obtain y where y: "poly p y = 0" "\<forall>m\<le> Suc n. y \<noteq> ?i m"
-      by blast
-    from y have "y = a \<or> poly q y = 0"
-      by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: algebra_simps)
-    with i[rule_format, of y] y(1) y(2) have False
-      apply auto
-      apply (erule_tac x = "m" in allE)
-      apply auto
-      done
-  }
-  thus ?case by blast
-qed
-
-
-lemma (in idom) poly_roots_index_length:
-  "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. n \<le> length p \<and> x = i n)"
-  by (blast intro: poly_roots_index_lemma)
-
-lemma (in idom) poly_roots_finite_lemma1:
-  "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>N i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. (n::nat) < N \<and> x = i n)"
-  apply (drule poly_roots_index_length, safe)
-  apply (rule_tac x = "Suc (length p)" in exI)
-  apply (rule_tac x = i in exI)
-  apply (simp add: less_Suc_eq_le)
-  done
-
-lemma (in idom) idom_finite_lemma:
-  assumes P: "\<forall>x. P x --> (\<exists>n. n < length j \<and> x = j!n)"
-  shows "finite {x. P x}"
-proof -
-  let ?M = "{x. P x}"
-  let ?N = "set j"
-  have "?M \<subseteq> ?N" using P by auto
-  thus ?thesis using finite_subset by auto
-qed
-
-lemma (in idom) poly_roots_finite_lemma2:
-  "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> x \<in> set i"
-  apply (drule poly_roots_index_length, safe)
-  apply (rule_tac x="map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI)
-  apply (auto simp add: image_iff)
-  apply (erule_tac x="x" in allE, clarsimp)
-  apply (case_tac "n = length p")
-  apply (auto simp add: order_le_less)
-  done
-
-lemma (in ring_char_0) UNIV_ring_char_0_infinte: "\<not> (finite (UNIV:: 'a set))"
-proof
-  assume F: "finite (UNIV :: 'a set)"
-  have "finite (UNIV :: nat set)"
-  proof (rule finite_imageD)
-    have "of_nat ` UNIV \<subseteq> UNIV" by simp
-    then show "finite (of_nat ` UNIV :: 'a set)" using F by (rule finite_subset)
-    show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: inj_on_def)
-  qed
-  with infinite_UNIV_nat show False ..
-qed
-
-lemma (in idom_char_0) poly_roots_finite: "poly p \<noteq> poly [] \<longleftrightarrow> finite {x. poly p x = 0}"
-proof
-  assume H: "poly p \<noteq> poly []"
-  show "finite {x. poly p x = (0::'a)}"
-    using H
-    apply -
-    apply (erule contrapos_np, rule ext)
-    apply (rule ccontr)
-    apply (clarify dest!: poly_roots_finite_lemma2)
-    using finite_subset
-  proof -
-    fix x i
-    assume F: "\<not> finite {x. poly p x = (0\<Colon>'a)}"
-      and P: "\<forall>x. poly p x = (0\<Colon>'a) \<longrightarrow> x \<in> set i"
-    let ?M= "{x. poly p x = (0\<Colon>'a)}"
-    from P have "?M \<subseteq> set i" by auto
-    with finite_subset F show False by auto
-  qed
-next
-  assume F: "finite {x. poly p x = (0\<Colon>'a)}"
-  show "poly p \<noteq> poly []" using F UNIV_ring_char_0_infinte by auto
-qed
-
-text{*Entirety and Cancellation for polynomials*}
-
-lemma (in idom_char_0) poly_entire_lemma2:
-  assumes p0: "poly p \<noteq> poly []"
-    and q0: "poly q \<noteq> poly []"
-  shows "poly (p***q) \<noteq> poly []"
-proof -
-  let ?S = "\<lambda>p. {x. poly p x = 0}"
-  have "?S (p *** q) = ?S p \<union> ?S q" by (auto simp add: poly_mult)
-  with p0 q0 show ?thesis  unfolding poly_roots_finite by auto
-qed
-
-lemma (in idom_char_0) poly_entire:
-  "poly (p *** q) = poly [] \<longleftrightarrow> poly p = poly [] \<or> poly q = poly []"
-  using poly_entire_lemma2[of p q]
-  by (auto simp add: fun_eq_iff poly_mult)
-
-lemma (in idom_char_0) poly_entire_neg:
-  "poly (p *** q) \<noteq> poly [] \<longleftrightarrow> poly p \<noteq> poly [] \<and> poly q \<noteq> poly []"
-  by (simp add: poly_entire)
-
-lemma fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
-  by auto
-
-lemma (in comm_ring_1) poly_add_minus_zero_iff:
-  "poly (p +++ -- q) = poly [] \<longleftrightarrow> poly p = poly q"
-  by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult)
-
-lemma (in comm_ring_1) poly_add_minus_mult_eq:
-  "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
-  by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult distrib_left)
-
-subclass (in idom_char_0) comm_ring_1 ..
-
-lemma (in idom_char_0) poly_mult_left_cancel:
-  "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r"
-proof -
-  have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []"
-    by (simp only: poly_add_minus_zero_iff)
-  also have "\<dots> \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r"
-    by (auto intro: simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
-  finally show ?thesis .
-qed
-
-lemma (in idom) poly_exp_eq_zero[simp]:
-  "poly (p %^ n) = poly [] \<longleftrightarrow> poly p = poly [] \<and> n \<noteq> 0"
-  apply (simp only: fun_eq add: HOL.all_simps [symmetric])
-  apply (rule arg_cong [where f = All])
-  apply (rule ext)
-  apply (induct n)
-  apply (auto simp add: poly_exp poly_mult)
-  done
-
-lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a,1] \<noteq> poly []"
-  apply (simp add: fun_eq)
-  apply (rule_tac x = "minus one a" in exI)
-  apply (unfold diff_minus)
-  apply (subst add_commute)
-  apply (subst add_assoc)
-  apply simp
-  done
-
-lemma (in idom) poly_exp_prime_eq_zero: "poly ([a, 1] %^ n) \<noteq> poly []"
-  by auto
-
-text{*A more constructive notion of polynomials being trivial*}
-
-lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] \<Longrightarrow> h = 0 \<and> poly t = poly []"
-  apply (simp add: fun_eq)
-  apply (case_tac "h = zero")
-  apply (drule_tac [2] x = zero in spec, auto)
-  apply (cases "poly t = poly []", simp)
-proof -
-  fix x
-  assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)"
-    and pnz: "poly t \<noteq> poly []"
-  let ?S = "{x. poly t x = 0}"
-  from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast
-  hence th: "?S \<supseteq> UNIV - {0}" by auto
-  from poly_roots_finite pnz have th': "finite ?S" by blast
-  from finite_subset[OF th th'] UNIV_ring_char_0_infinte show "poly t x = (0\<Colon>'a)"
-    by simp
-qed
-
-lemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p"
-  apply (induct p)
-  apply simp
-  apply (rule iffI)
-  apply (drule poly_zero_lemma', auto)
-  done
-
-lemma (in idom_char_0) poly_0: "list_all (\<lambda>c. c = 0) p \<Longrightarrow> poly p x = 0"
-  unfolding poly_zero[symmetric] by simp
-
-
-
-text{*Basics of divisibility.*}
-
-lemma (in idom) poly_primes:
-  "[a, 1] divides (p *** q) \<longleftrightarrow> [a, 1] divides p \<or> [a, 1] divides q"
-  apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult distrib_right [symmetric])
-  apply (drule_tac x = "uminus a" in spec)
-  apply (simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
-  apply (cases "p = []")
-  apply (rule exI[where x="[]"])
-  apply simp
-  apply (cases "q = []")
-  apply (erule allE[where x="[]"], simp)
-
-  apply clarsimp
-  apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)")
-  apply (clarsimp simp add: poly_add poly_cmult)
-  apply (rule_tac x="qa" in exI)
-  apply (simp add: distrib_right [symmetric])
-  apply clarsimp
-
-  apply (auto simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric])
-  apply (rule_tac x = "pmult qa q" in exI)
-  apply (rule_tac [2] x = "pmult p qa" in exI)
-  apply (auto simp add: poly_add poly_mult poly_cmult mult_ac)
-  done
-
-lemma (in comm_semiring_1) poly_divides_refl[simp]: "p divides p"
-  apply (simp add: divides_def)
-  apply (rule_tac x = "[one]" in exI)
-  apply (auto simp add: poly_mult fun_eq)
-  done
-
-lemma (in comm_semiring_1) poly_divides_trans: "p divides q \<Longrightarrow> q divides r \<Longrightarrow> p divides r"
-  apply (simp add: divides_def, safe)
-  apply (rule_tac x = "pmult qa qaa" in exI)
-  apply (auto simp add: poly_mult fun_eq mult_assoc)
-  done
-
-lemma (in comm_semiring_1) poly_divides_exp: "m \<le> n \<Longrightarrow> (p %^ m) divides (p %^ n)"
-  apply (auto simp add: le_iff_add)
-  apply (induct_tac k)
-  apply (rule_tac [2] poly_divides_trans)
-  apply (auto simp add: divides_def)
-  apply (rule_tac x = p in exI)
-  apply (auto simp add: poly_mult fun_eq mult_ac)
-  done
-
-lemma (in comm_semiring_1) poly_exp_divides:
-  "(p %^ n) divides q \<Longrightarrow> m \<le> n \<Longrightarrow> (p %^ m) divides q"
-  by (blast intro: poly_divides_exp poly_divides_trans)
-
-lemma (in comm_semiring_0) poly_divides_add:
-  "p divides q \<Longrightarrow> p divides r \<Longrightarrow> p divides (q +++ r)"
-  apply (simp add: divides_def, auto)
-  apply (rule_tac x = "padd qa qaa" in exI)
-  apply (auto simp add: poly_add fun_eq poly_mult distrib_left)
-  done
-
-lemma (in comm_ring_1) poly_divides_diff:
-  "p divides q \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides r"
-  apply (simp add: divides_def, auto)
-  apply (rule_tac x = "padd qaa (poly_minus qa)" in exI)
-  apply (auto simp add: poly_add fun_eq poly_mult poly_minus algebra_simps)
-  done
-
-lemma (in comm_ring_1) poly_divides_diff2:
-  "p divides r \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides q"
-  apply (erule poly_divides_diff)
-  apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac)
-  done
-
-lemma (in semiring_0) poly_divides_zero: "poly p = poly [] \<Longrightarrow> q divides p"
-  apply (simp add: divides_def)
-  apply (rule exI[where x="[]"])
-  apply (auto simp add: fun_eq poly_mult)
-  done
-
-lemma (in semiring_0) poly_divides_zero2 [simp]: "q divides []"
-  apply (simp add: divides_def)
-  apply (rule_tac x = "[]" in exI)
-  apply (auto simp add: fun_eq)
-  done
-
-text{*At last, we can consider the order of a root.*}
-
-lemma (in idom_char_0) poly_order_exists_lemma:
-  assumes lp: "length p = d"
-    and p: "poly p \<noteq> poly []"
-  shows "\<exists>n q. p = mulexp n [-a, 1] q \<and> poly q a \<noteq> 0"
-  using lp p
-proof (induct d arbitrary: p)
-  case 0
-  thus ?case by simp
-next
-  case (Suc n p)
-  show ?case
-  proof (cases "poly p a = 0")
-    case True
-    from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []" by auto
-    hence pN: "p \<noteq> []" by auto
-    from True[unfolded poly_linear_divides] pN obtain q where q: "p = [-a, 1] *** q"
-      by blast
-    from q h True have qh: "length q = n" "poly q \<noteq> poly []"
-      apply -
-      apply simp
-      apply (simp only: fun_eq)
-      apply (rule ccontr)
-      apply (simp add: fun_eq poly_add poly_cmult)
-      done
-    from Suc.hyps[OF qh] obtain m r where mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0"
-      by blast
-    from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0" by simp
-    then show ?thesis by blast
-  next
-    case False
-    then show ?thesis
-      using Suc.prems
-      apply simp
-      apply (rule exI[where x="0::nat"])
-      apply simp
-      done
-  qed
-qed
-
-
-lemma (in comm_semiring_1) poly_mulexp: "poly (mulexp n p q) x = (poly p x) ^ n * poly q x"
-  by (induct n) (auto simp add: poly_mult mult_ac)
-
-lemma (in comm_semiring_1) divides_left_mult:
-  assumes d:"(p***q) divides r" shows "p divides r \<and> q divides r"
-proof-
-  from d obtain t where r:"poly r = poly (p***q *** t)"
-    unfolding divides_def by blast
-  hence "poly r = poly (p *** (q *** t))"
-    "poly r = poly (q *** (p***t))" by(auto simp add: fun_eq poly_mult mult_ac)
-  thus ?thesis unfolding divides_def by blast
-qed
-
-
-(* FIXME: Tidy up *)
-
-lemma (in semiring_1) zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"
-  by (induct n) simp_all
-
-lemma (in idom_char_0) poly_order_exists:
-  assumes "length p = d" and "poly p \<noteq> poly []"
-  shows "\<exists>n. [- a, 1] %^ n divides p \<and> \<not> [- a, 1] %^ Suc n divides p"
-proof -
-  from assms have "\<exists>n q. p = mulexp n [- a, 1] q \<and> poly q a \<noteq> 0"
-    by (rule poly_order_exists_lemma)
-  then obtain n q where p: "p = mulexp n [- a, 1] q" and "poly q a \<noteq> 0" by blast
-  have "[- a, 1] %^ n divides mulexp n [- a, 1] q"
-  proof (rule dividesI)
-    show "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ n *** q)"
-      by (induct n) (simp_all add: poly_add poly_cmult poly_mult distrib_left mult_ac)
-  qed
-  moreover have "\<not> [- a, 1] %^ Suc n divides mulexp n [- a, 1] q"
-  proof
-    assume "[- a, 1] %^ Suc n divides mulexp n [- a, 1] q"
-    then obtain m where "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ Suc n *** m)"
-      by (rule dividesE)
-    moreover have "poly (mulexp n [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc n *** m)"
-    proof (induct n)
-      case 0 show ?case
-      proof (rule ccontr)
-        assume "\<not> poly (mulexp 0 [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc 0 *** m)"
-        then have "poly q a = 0"
-          by (simp add: poly_add poly_cmult)
-        with `poly q a \<noteq> 0` show False by simp
-      qed
-    next
-      case (Suc n) show ?case
-        by (rule pexp_Suc [THEN ssubst], rule ccontr)
-          (simp add: poly_mult_left_cancel poly_mult_assoc Suc del: pmult_Cons pexp_Suc)
-    qed
-    ultimately show False by simp
-  qed
-  ultimately show ?thesis by (auto simp add: p)
-qed
-
-lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"
-  by (auto simp add: divides_def)
-
-lemma (in idom_char_0) poly_order:
-  "poly p \<noteq> poly [] \<Longrightarrow> \<exists>!n. ([-a, 1] %^ n) divides p \<and> \<not> (([-a, 1] %^ Suc n) divides p)"
-  apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
-  apply (cut_tac x = y and y = n in less_linear)
-  apply (drule_tac m = n in poly_exp_divides)
-  apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
-              simp del: pmult_Cons pexp_Suc)
-  done
-
-text{*Order*}
-
-lemma some1_equalityD: "n = (SOME n. P n) \<Longrightarrow> \<exists>!n. P n \<Longrightarrow> P n"
-  by (blast intro: someI2)
-
-lemma (in idom_char_0) order:
-      "(([-a, 1] %^ n) divides p \<and>
-        ~(([-a, 1] %^ (Suc n)) divides p)) =
-        ((n = order a p) \<and> ~(poly p = poly []))"
-  apply (unfold order_def)
-  apply (rule iffI)
-  apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
-  apply (blast intro!: poly_order [THEN [2] some1_equalityD])
-  done
-
-lemma (in idom_char_0) order2:
-  "poly p \<noteq> poly [] \<Longrightarrow>
-    ([-a, 1] %^ (order a p)) divides p \<and> \<not> (([-a, 1] %^ (Suc (order a p))) divides p)"
-  by (simp add: order del: pexp_Suc)
-
-lemma (in idom_char_0) order_unique:
-  "poly p \<noteq> poly [] \<Longrightarrow> ([-a, 1] %^ n) divides p \<Longrightarrow> ~(([-a, 1] %^ (Suc n)) divides p) \<Longrightarrow>
-    n = order a p"
-  using order [of a n p] by auto
-
-lemma (in idom_char_0) order_unique_lemma:
-  "poly p \<noteq> poly [] \<and> ([-a, 1] %^ n) divides p \<and> ~(([-a, 1] %^ (Suc n)) divides p) \<Longrightarrow>
-    n = order a p"
-  by (blast intro: order_unique)
-
-lemma (in ring_1) order_poly: "poly p = poly q \<Longrightarrow> order a p = order a q"
-  by (auto simp add: fun_eq divides_def poly_mult order_def)
-
-lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p"
-  by (induct "p") auto
-
-lemma (in comm_ring_1) lemma_order_root:
-  "0 < n \<and> [- a, 1] %^ n divides p \<and> ~ [- a, 1] %^ (Suc n) divides p \<Longrightarrow> poly p a = 0"
-  by (induct n arbitrary: a p) (auto simp add: divides_def poly_mult simp del: pmult_Cons)
-
-lemma (in idom_char_0) order_root:
-  "poly p a = 0 \<longleftrightarrow> poly p = poly [] \<or> order a p \<noteq> 0"
-  apply (cases "poly p = poly []")
-  apply auto
-  apply (simp add: poly_linear_divides del: pmult_Cons, safe)
-  apply (drule_tac [!] a = a in order2)
-  apply (rule ccontr)
-  apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast)
-  using neq0_conv
-  apply (blast intro: lemma_order_root)
-  done
-
-lemma (in idom_char_0) order_divides:
-  "([-a, 1] %^ n) divides p \<longleftrightarrow> poly p = poly [] \<or> n \<le> order a p"
-  apply (cases "poly p = poly []")
-  apply auto
-  apply (simp add: divides_def fun_eq poly_mult)
-  apply (rule_tac x = "[]" in exI)
-  apply (auto dest!: order2 [where a=a] intro: poly_exp_divides simp del: pexp_Suc)
-  done
-
-lemma (in idom_char_0) order_decomp:
-  "poly p \<noteq> poly [] \<Longrightarrow> \<exists>q. poly p = poly (([-a, 1] %^ (order a p)) *** q) \<and> ~([-a, 1] divides q)"
-  apply (unfold divides_def)
-  apply (drule order2 [where a = a])
-  apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
-  apply (rule_tac x = q in exI, safe)
-  apply (drule_tac x = qa in spec)
-  apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons)
-  done
-
-text{*Important composition properties of orders.*}
-lemma order_mult:
-  "poly (p *** q) \<noteq> poly [] \<Longrightarrow>
-    order a (p *** q) = order a p + order (a::'a::{idom_char_0}) q"
-  apply (cut_tac a = a and p = "p *** q" and n = "order a p + order a q" in order)
-  apply (auto simp add: poly_entire simp del: pmult_Cons)
-  apply (drule_tac a = a in order2)+
-  apply safe
-  apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
-  apply (rule_tac x = "qa *** qaa" in exI)
-  apply (simp add: poly_mult mult_ac del: pmult_Cons)
-  apply (drule_tac a = a in order_decomp)+
-  apply safe
-  apply (subgoal_tac "[-a,1] divides (qa *** qaa) ")
-  apply (simp add: poly_primes del: pmult_Cons)
-  apply (auto simp add: divides_def simp del: pmult_Cons)
-  apply (rule_tac x = qb in exI)
-  apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
-  apply (drule poly_mult_left_cancel [THEN iffD1], force)
-  apply (subgoal_tac "poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** (qa *** qaa))) = poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))) ")
-  apply (drule poly_mult_left_cancel [THEN iffD1], force)
-  apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
-  done
-
-lemma (in idom_char_0) order_mult:
-  assumes "poly (p *** q) \<noteq> poly []"
-  shows "order a (p *** q) = order a p + order a q"
-  using assms
-  apply (cut_tac a = a and p = "pmult p q" and n = "order a p + order a q" in order)
-  apply (auto simp add: poly_entire simp del: pmult_Cons)
-  apply (drule_tac a = a in order2)+
-  apply safe
-  apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
-  apply (rule_tac x = "pmult qa qaa" in exI)
-  apply (simp add: poly_mult mult_ac del: pmult_Cons)
-  apply (drule_tac a = a in order_decomp)+
-  apply safe
-  apply (subgoal_tac "[uminus a, one] divides pmult qa qaa")
-  apply (simp add: poly_primes del: pmult_Cons)
-  apply (auto simp add: divides_def simp del: pmult_Cons)
-  apply (rule_tac x = qb in exI)
-  apply (subgoal_tac "poly (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa)) =
-    poly (pmult (pexp [uminus a, one] (?order a p)) (pmult [uminus a, one] qb))")
-  apply (drule poly_mult_left_cancel [THEN iffD1], force)
-  apply (subgoal_tac "poly (pmult (pexp [uminus a, one] (order a q))
-      (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa))) =
-    poly (pmult (pexp [uminus a, one] (order a q))
-      (pmult (pexp [uminus a, one] (order a p)) (pmult [uminus a, one] qb)))")
-  apply (drule poly_mult_left_cancel [THEN iffD1], force)
-  apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
-  done
-
-lemma (in idom_char_0) order_root2: "poly p \<noteq> poly [] \<Longrightarrow> poly p a = 0 \<longleftrightarrow> order a p \<noteq> 0"
-  by (rule order_root [THEN ssubst]) auto
-
-lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p" by auto
-
-lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]"
-  by (simp add: fun_eq)
-
-lemma (in idom_char_0) rsquarefree_decomp:
-  "rsquarefree p \<Longrightarrow> poly p a = 0 \<Longrightarrow>
-    \<exists>q. poly p = poly ([-a, 1] *** q) \<and> poly q a \<noteq> 0"
-  apply (simp add: rsquarefree_def, safe)
-  apply (frule_tac a = a in order_decomp)
-  apply (drule_tac x = a in spec)
-  apply (drule_tac a = a in order_root2 [symmetric])
-  apply (auto simp del: pmult_Cons)
-  apply (rule_tac x = q in exI, safe)
-  apply (simp add: poly_mult fun_eq)
-  apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
-  apply (simp add: divides_def del: pmult_Cons, safe)
-  apply (drule_tac x = "[]" in spec)
-  apply (auto simp add: fun_eq)
-  done
-
-
-text{*Normalization of a polynomial.*}
-
-lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p"
-  by (induct p) (auto simp add: fun_eq)
-
-text{*The degree of a polynomial.*}
-
-lemma (in semiring_0) lemma_degree_zero: "list_all (%c. c = 0) p \<longleftrightarrow> pnormalize p = []"
-  by (induct p) auto
-
-lemma (in idom_char_0) degree_zero:
-  assumes "poly p = poly []"
-  shows "degree p = 0"
-  using assms
-  by (cases "pnormalize p = []") (auto simp add: degree_def poly_zero lemma_degree_zero)
-
-lemma (in semiring_0) pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0"
-  by simp
-
-lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])"
-  by simp
-
-lemma (in semiring_0) pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)"
-  unfolding pnormal_def by simp
-
-lemma (in semiring_0) pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p"
-  unfolding pnormal_def by(auto split: split_if_asm)
-
-
-lemma (in semiring_0) pnormal_last_nonzero: "pnormal p \<Longrightarrow> last p \<noteq> 0"
-  by (induct p) (simp_all add: pnormal_def split: split_if_asm)
-
-lemma (in semiring_0) pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
-  unfolding pnormal_def length_greater_0_conv by blast
-
-lemma (in semiring_0) pnormal_last_length: "0 < length p \<Longrightarrow> last p \<noteq> 0 \<Longrightarrow> pnormal p"
-  by (induct p) (auto simp: pnormal_def  split: split_if_asm)
-
-
-lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> 0 < length p \<and> last p \<noteq> 0"
-  using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
-
-lemma (in idom_char_0) poly_Cons_eq:
-  "poly (c # cs) = poly (d # ds) \<longleftrightarrow> c = d \<and> poly cs = poly ds"
-  (is "?lhs \<longleftrightarrow> ?rhs")
-proof
-  assume eq: ?lhs
-  hence "\<And>x. poly ((c#cs) +++ -- (d#ds)) x = 0"
-    by (simp only: poly_minus poly_add algebra_simps) simp
-  hence "poly ((c#cs) +++ -- (d#ds)) = poly []" by(simp add: fun_eq_iff)
-  hence "c = d \<and> list_all (\<lambda>x. x=0) ((cs +++ -- ds))"
-    unfolding poly_zero by (simp add: poly_minus_def algebra_simps)
-  hence "c = d \<and> (\<forall>x. poly (cs +++ -- ds) x = 0)"
-    unfolding poly_zero[symmetric] by simp
-  then show ?rhs by (simp add: poly_minus poly_add algebra_simps fun_eq_iff)
-next
-  assume ?rhs
-  then show ?lhs by(simp add:fun_eq_iff)
-qed
-
-lemma (in idom_char_0) pnormalize_unique: "poly p = poly q \<Longrightarrow> pnormalize p = pnormalize q"
-proof (induct q arbitrary: p)
-  case Nil
-  thus ?case by (simp only: poly_zero lemma_degree_zero) simp
-next
-  case (Cons c cs p)
-  thus ?case
-  proof (induct p)
-    case Nil
-    hence "poly [] = poly (c#cs)" by blast
-    then have "poly (c#cs) = poly [] " by simp
-    thus ?case by (simp only: poly_zero lemma_degree_zero) simp
-  next
-    case (Cons d ds)
-    hence eq: "poly (d # ds) = poly (c # cs)" by blast
-    hence eq': "\<And>x. poly (d # ds) x = poly (c # cs) x" by simp
-    hence "poly (d # ds) 0 = poly (c # cs) 0" by blast
-    hence dc: "d = c" by auto
-    with eq have "poly ds = poly cs"
-      unfolding  poly_Cons_eq by simp
-    with Cons.prems have "pnormalize ds = pnormalize cs" by blast
-    with dc show ?case by simp
-  qed
-qed
-
-lemma (in idom_char_0) degree_unique:
-  assumes pq: "poly p = poly q"
-  shows "degree p = degree q"
-  using pnormalize_unique[OF pq] unfolding degree_def by simp
-
-lemma (in semiring_0) pnormalize_length:
-  "length (pnormalize p) \<le> length p" by (induct p) auto
-
-lemma (in semiring_0) last_linear_mul_lemma:
-  "last ((a %* p) +++ (x#(b %* p))) = (if p = [] then x else b * last p)"
-  apply (induct p arbitrary: a x b)
-  apply auto
-  apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []")
-  apply simp
-  apply (induct_tac p)
-  apply auto
-  done
-
-lemma (in semiring_1) last_linear_mul:
-  assumes p: "p \<noteq> []"
-  shows "last ([a,1] *** p) = last p"
-proof -
-  from p obtain c cs where cs: "p = c#cs" by (cases p) auto
-  from cs have eq: "[a,1] *** p = (a %* (c#cs)) +++ (0#(1 %* (c#cs)))"
-    by (simp add: poly_cmult_distr)
-  show ?thesis using cs
-    unfolding eq last_linear_mul_lemma by simp
-qed
-
-lemma (in semiring_0) pnormalize_eq: "last p \<noteq> 0 \<Longrightarrow> pnormalize p = p"
-  by (induct p) (auto split: split_if_asm)
-
-lemma (in semiring_0) last_pnormalize: "pnormalize p \<noteq> [] \<Longrightarrow> last (pnormalize p) \<noteq> 0"
-  by (induct p) auto
-
-lemma (in semiring_0) pnormal_degree: "last p \<noteq> 0 \<Longrightarrow> degree p = length p - 1"
-  using pnormalize_eq[of p] unfolding degree_def by simp
-
-lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)"
-  by (rule ext) simp
-
-lemma (in idom_char_0) linear_mul_degree:
-  assumes p: "poly p \<noteq> poly []"
-  shows "degree ([a,1] *** p) = degree p + 1"
-proof -
-  from p have pnz: "pnormalize p \<noteq> []"
-    unfolding poly_zero lemma_degree_zero .
-
-  from last_linear_mul[OF pnz, of a] last_pnormalize[OF pnz]
-  have l0: "last ([a, 1] *** pnormalize p) \<noteq> 0" by simp
-  from last_pnormalize[OF pnz] last_linear_mul[OF pnz, of a]
-    pnormal_degree[OF l0] pnormal_degree[OF last_pnormalize[OF pnz]] pnz
-
-  have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1"
-    by simp
-
-  have eqs: "poly ([a,1] *** pnormalize p) = poly ([a,1] *** p)"
-    by (rule ext) (simp add: poly_mult poly_add poly_cmult)
-  from degree_unique[OF eqs] th
-  show ?thesis by (simp add: degree_unique[OF poly_normalize])
-qed
-
-lemma (in idom_char_0) linear_pow_mul_degree:
-  "degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)"
-proof (induct n arbitrary: a p)
-  case (0 a p)
-  show ?case
-  proof (cases "poly p = poly []")
-    case True
-    then show ?thesis
-      using degree_unique[OF True] by (simp add: degree_def)
-  next
-    case False
-    then show ?thesis by (auto simp add: poly_Nil_ext)
-  qed
-next
-  case (Suc n a p)
-  have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1] %^ n *** ([a,1] *** p))"
-    apply (rule ext)
-    apply (simp add: poly_mult poly_add poly_cmult)
-    apply (simp add: mult_ac add_ac distrib_left)
-    done
-  note deq = degree_unique[OF eq]
-  show ?case
-  proof (cases "poly p = poly []")
-    case True
-    with eq have eq': "poly ([a,1] %^(Suc n) *** p) = poly []"
-      apply -
-      apply (rule ext)
-      apply (simp add: poly_mult poly_cmult poly_add)
-      done
-    from degree_unique[OF eq'] True show ?thesis
-      by (simp add: degree_def)
-  next
-    case False
-    then have ap: "poly ([a,1] *** p) \<noteq> poly []"
-      using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto
-    have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1]%^n *** ([a,1] *** p))"
-      by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps)
-    from ap have ap': "(poly ([a,1] *** p) = poly []) = False"
-      by blast
-    have th0: "degree ([a,1]%^n *** ([a,1] *** p)) = degree ([a,1] *** p) + n"
-      apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap')
-      apply simp
-      done
-    from degree_unique[OF eq] ap False th0 linear_mul_degree[OF False, of a]
-    show ?thesis by (auto simp del: poly.simps)
-  qed
-qed
-
-lemma (in idom_char_0) order_degree:
-  assumes p0: "poly p \<noteq> poly []"
-  shows "order a p \<le> degree p"
-proof -
-  from order2[OF p0, unfolded divides_def]
-  obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)" by blast
-  {
-    assume "poly q = poly []"
-    with q p0 have False by (simp add: poly_mult poly_entire)
-  }
-  with degree_unique[OF q, unfolded linear_pow_mul_degree] show ?thesis
-    by auto
-qed
-
-text{*Tidier versions of finiteness of roots.*}
-
-lemma (in idom_char_0) poly_roots_finite_set:
-  "poly p \<noteq> poly [] \<Longrightarrow> finite {x. poly p x = 0}"
-  unfolding poly_roots_finite .
-
-text{*bound for polynomial.*}
-
-lemma poly_mono: "abs(x) \<le> k \<Longrightarrow> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"
-  apply (induct p)
-  apply auto
-  apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
-  apply (rule abs_triangle_ineq)
-  apply (auto intro!: mult_mono simp add: abs_mult)
-  done
-
-lemma (in semiring_0) poly_Sing: "poly [c] x = c" by simp
-
-end