(*  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 => 'a  => '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 => '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))"

     --{*order of a polynomial*}

 definition (in ring_1) order :: "'a => 'a list => nat" where

   "order a p = (SOME n. ([-a, 1] %^ n) divides p &

                       ~ (([-a, 1] %^ (Suc n)) divides p))"

      --{*degree of a polynomial*}

 definition (in semiring_0) degree :: "'a list => nat" where

   "degree p = length (pnormalize p) - 1"

      --{*squarefree polynomials --- NB with respect to real roots only.*}

 definition (in ring_1)

   rsquarefree :: "'a list => bool" where

   "rsquarefree p = (poly p \<noteq> poly [] &

                      (\<forall>a. (order a p = 0) | (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[simp]: "-- [] = []"

 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, simp)

 apply (case_tac q, simp_all add: right_distrib)

 done

 lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"

 apply (induct "t", simp)

 apply (auto simp add: mult_zero_left poly_ident_mult 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 right_distrib)

 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: right_distrib 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: right_distrib 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 minus_mult_left[symmetric])

 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 left_distrib right_distrib 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"

 apply (induct "n")

 apply (auto simp add: poly_cmult poly_mult power_Suc)

 done

 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"

 apply (induct "n")

 apply (auto simp add: poly_mult mult_assoc)

 done

 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 diff_minus[symmetric]

         minus_mult_left[symmetric] right_minus)

     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"

 by (cut_tac t = t and a = a in lemma_poly_linear_rem, 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 minus_mult_left[symmetric])}

     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)"

 apply (induct p1 arbitrary: p2, simp_all)

 apply arith

 done

 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 ==>

       \<exists>i. \<forall>x. (poly p x = 0) --> (\<exists>n. n \<le> length p & x = i n)"

 by (blast intro: poly_roots_index_lemma)

 lemma (in idom) poly_roots_finite_lemma1: "poly p x \<noteq> poly [] x ==>

       \<exists>N i. \<forall>x. (poly p x = 0) --> (\<exists>n. (n::nat) < N & 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 & 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 ==>

       \<exists>i. \<forall>x. (poly p x = 0) --> 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)

 by (case_tac "n=length p", auto simp add: order_le_less)

 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 []) =

   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 []) = ((poly p \<noteq> poly []) & (poly q \<noteq> poly []))"

 by (simp add: poly_entire)

 lemma fun_eq: " (f = g) = (\<forall>x. f x = g x)"

 by (auto intro!: ext)

 lemma (in comm_ring_1) poly_add_minus_zero_iff: "(poly (p +++ -- q) = poly []) = (poly p = poly q)"

 by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult minus_mult_left[symmetric])

 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 right_distrib minus_mult_left[symmetric] minus_mult_right[symmetric])

 subclass (in idom_char_0) comm_ring_1 ..

 lemma (in idom_char_0) poly_mult_left_cancel: "(poly (p *** q) = poly (p *** r)) = (poly p = poly [] | 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 [] | poly q = poly r"

     by (auto intro: ext 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 []) = (poly p = poly [] & 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 semiring_1) one_neq_zero[simp]: "1 \<noteq> 0" using zero_neq_one by blast

 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 [] ==> h = 0 & 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", 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)) = ([a, 1] divides p | [a, 1] divides q)"

 apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult left_distrib [symmetric])

 apply (drule_tac x = "uminus a" in spec)

 apply (simp add: poly_linear_divides poly_add poly_cmult left_distrib [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: left_distrib [symmetric])

 apply clarsimp

 apply (auto simp add: right_minus poly_linear_divides poly_add poly_cmult left_distrib [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; q divides r |] ==> 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 ==> (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;  m\<le>n |] ==> (p %^ m) divides q"

 by (blast intro: poly_divides_exp poly_divides_trans)

 lemma (in comm_semiring_0) poly_divides_add:

    "[| p divides q; p divides r |] ==> 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 right_distrib)

 done

 lemma (in comm_ring_1) poly_divides_diff:

    "[| p divides q; p divides (q +++ r) |] ==> 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; p divides (q +++ r) |] ==> 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 [] ==> 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)

   {assume p0: "poly p a = 0"

     from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []" by auto

     hence pN: "p \<noteq> []" by auto

     from p0[unfolded poly_linear_divides] pN  obtain q where

       q: "p = [-a, 1] *** q" by blast

     from q h p0 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 minus_mult_left[symmetric])

       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

     hence ?case by blast}

   moreover

   {assume p0: "poly p a \<noteq> 0"

     hence ?case using Suc.prems apply simp by (rule exI[where x="0::nat"], simp)}

   ultimately show ?case by blast

 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 power_Suc 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 add: power_Suc)

 lemma (in idom_char_0) poly_order_exists:

   assumes lp: "length p = d" and p0: "poly p \<noteq> poly []"

   shows "\<exists>n. ([-a, 1] %^ n) divides p & ~(([-a, 1] %^ (Suc n)) divides p)"

 proof-

 let ?poly = poly

 let ?mulexp = mulexp

 let ?pexp = pexp

 from lp p0

 show ?thesis

 apply -

 apply (drule poly_order_exists_lemma [where a=a], assumption, clarify)

 apply (rule_tac x = n in exI, safe)

 apply (unfold divides_def)

 apply (rule_tac x = q in exI)

 apply (induct_tac "n", simp)

 apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult right_distrib mult_ac)

 apply safe

 apply (subgoal_tac "?poly (?mulexp n [uminus a, one] q) \<noteq> ?poly (pmult (?pexp [uminus a, one] (Suc n)) qa)")

 apply simp

 apply (induct_tac "n")

 apply (simp del: pmult_Cons pexp_Suc)

 apply (erule_tac Q = "?poly q a = zero" in contrapos_np)

 apply (simp add: poly_add poly_cmult minus_mult_left[symmetric])

 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

 qed

 lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"

 by (simp add: divides_def, auto)

 lemma (in idom_char_0) poly_order: "poly p \<noteq> poly []

       ==> EX! n. ([-a, 1] %^ n) divides p &

                  ~(([-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 = (@n. P n); EX! n. P n |] ==> P n"

 by (blast intro: someI2)

 lemma (in idom_char_0) order:

       "(([-a, 1] %^ n) divides p &

         ~(([-a, 1] %^ (Suc n)) divides p)) =

         ((n = order a p) & ~(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 [] |]

       ==> ([-a, 1] %^ (order a p)) divides p &

               ~(([-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 []; ([-a, 1] %^ n) divides p;

          ~(([-a, 1] %^ (Suc n)) divides p)

       |] ==> (n = order a p)"

 by (insert order [of a n p], auto)

 lemma (in idom_char_0) order_unique_lemma: "(poly p \<noteq> poly [] & ([-a, 1] %^ n) divides p &

          ~(([-a, 1] %^ (Suc n)) divides p))

       ==> (n = order a p)"

 by (blast intro: order_unique)

 lemma (in ring_1) order_poly: "poly p = poly q ==> 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"

 apply (induct "p")

 apply (auto simp add: numeral_1_eq_1)

 done

 lemma (in comm_ring_1) 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: a p, blast)

 apply (auto simp add: divides_def poly_mult simp del: pmult_Cons)

 done

 lemma (in idom_char_0) order_root: "(poly p a = 0) = ((poly p = poly []) | order a p \<noteq> 0)"

 proof-

   let ?poly = poly

   show ?thesis

 apply (case_tac "?poly p = ?poly []", 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

 qed

 lemma (in idom_char_0) order_divides: "(([-a, 1] %^ n) divides p) = ((poly p = poly []) | n \<le> order a p)"

 proof-

   let ?poly = poly

   show ?thesis

 apply (case_tac "?poly p = ?poly []", 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

 qed

 lemma (in idom_char_0) order_decomp:

      "poly p \<noteq> poly []

       ==> \<exists>q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) &

                 ~([-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 []

       ==> 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 pq0: "poly (p *** q) \<noteq> poly []"

   shows "order a (p *** q) = order a p + order a q"

 proof-

   let ?order = order

   let ?divides = "op divides"

   let ?poly = poly

 from pq0

 show ?thesis

 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 "?divides [uminus a,one ] (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

 qed

 lemma (in idom_char_0) order_root2: "poly p \<noteq> poly [] ==> (poly p a = 0) = (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; poly p a = 0 |]

       ==> \<exists>q. (poly p = poly ([-a, 1] *** q)) & 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"

 apply (induct "p")

 apply (auto simp add: fun_eq)

 done

 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 pN: "poly p = poly []" shows"degree p = 0"

 proof-

   let ?pn = pnormalize

   from pN

   show ?thesis

     apply (simp add: degree_def)

     apply (case_tac "?pn p = []")

     apply (auto simp add: poly_zero lemma_degree_zero )

     done

 qed

 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 ==> 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: "\<lbrakk>0 < length p ; last p \<noteq> 0\<rbrakk> \<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

   thus ?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, auto)

 apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []", simp)

 apply (induct_tac p, 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 (auto simp add: poly_length_mult)

   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)

   {assume p: "poly p = poly []"

     hence ?case using degree_unique[OF p] by (simp add: degree_def)}

   moreover

   {assume p: "poly p \<noteq> poly []" hence ?case by (auto simp add: poly_Nil_ext) }

   ultimately show ?case by blast

 next

   case (Suc n a p)

   have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1] %^ n *** ([a,1] *** p))"

     apply (rule ext, simp add: poly_mult poly_add poly_cmult)

     by (simp add: mult_ac add_ac right_distrib)

   note deq = degree_unique[OF eq]

   {assume p: "poly p = poly []"

     with eq have eq': "poly ([a,1] %^(Suc n) *** p) = poly []"

       by - (rule ext,simp add: poly_mult poly_cmult poly_add)

     from degree_unique[OF eq'] p have ?case by (simp add: degree_def)}

   moreover

   {assume p: "poly p \<noteq> poly []"

     from p 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')

      by simp

    from degree_unique[OF eq] ap p th0 linear_mul_degree[OF p, of a]

    have ?case by (auto simp del: poly.simps)}

   ultimately show ?case by blast

 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 [] ==> finite {x. poly p x = 0}"

 unfolding poly_roots_finite .

 text{*bound for polynomial.*}

 lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"

 apply (induct "p", 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