src/HOL/Library/Univ_Poly.thy
author nipkow
Tue Sep 07 10:05:19 2010 +0200 (2010-09-07)
changeset 39198 f967a16dfcdd
parent 37887 2ae085b07f2f
child 39302 d7728f65b353
permissions -rw-r--r--
expand_fun_eq -> ext_iff
expand_set_eq -> set_ext_iff
Naming in line now with multisets
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(*  Title:      HOL/Library/Univ_Poly.thy
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    Author:     Amine Chaieb
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*)
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header {* Univariate Polynomials *}
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theory Univ_Poly
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imports Main
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begin
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text{* Application of polynomial as a function. *}
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primrec (in semiring_0) poly :: "'a list => 'a  => 'a" where
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  poly_Nil:  "poly [] x = 0"
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| poly_Cons: "poly (h#t) x = h + x * poly t x"
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subsection{*Arithmetic Operations on Polynomials*}
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text{*addition*}
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primrec (in semiring_0) padd :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "+++" 65)
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where
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  padd_Nil:  "[] +++ l2 = l2"
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| padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t
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                            else (h + hd l2)#(t +++ tl l2))"
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text{*Multiplication by a constant*}
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primrec (in semiring_0) cmult :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "%*" 70) where
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   cmult_Nil:  "c %* [] = []"
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|  cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
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text{*Multiplication by a polynomial*}
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primrec (in semiring_0) pmult :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixl "***" 70)
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where
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   pmult_Nil:  "[] *** l2 = []"
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|  pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2
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                              else (h %* l2) +++ ((0) # (t *** l2)))"
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text{*Repeated multiplication by a polynomial*}
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primrec (in semiring_0) mulexp :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a  list \<Rightarrow> 'a list" where
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   mulexp_zero:  "mulexp 0 p q = q"
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|  mulexp_Suc:   "mulexp (Suc n) p q = p *** mulexp n p q"
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text{*Exponential*}
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primrec (in semiring_1) pexp :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list"  (infixl "%^" 80) where
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   pexp_0:   "p %^ 0 = [1]"
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|  pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
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text{*Quotient related value of dividing a polynomial by x + a*}
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(* Useful for divisor properties in inductive proofs *)
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primrec (in field) "pquot" :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list" where
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   pquot_Nil:  "pquot [] a= []"
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|  pquot_Cons: "pquot (h#t) a = (if t = [] then [h]
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                   else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))"
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text{*normalization of polynomials (remove extra 0 coeff)*}
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primrec (in semiring_0) pnormalize :: "'a list \<Rightarrow> 'a list" where
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  pnormalize_Nil:  "pnormalize [] = []"
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| pnormalize_Cons: "pnormalize (h#p) = (if ( (pnormalize p) = [])
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                                     then (if (h = 0) then [] else [h])
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                                     else (h#(pnormalize p)))"
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definition (in semiring_0) "pnormal p = ((pnormalize p = p) \<and> p \<noteq> [])"
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definition (in semiring_0) "nonconstant p = (pnormal p \<and> (\<forall>x. p \<noteq> [x]))"
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text{*Other definitions*}
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definition (in ring_1)
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  poly_minus :: "'a list => 'a list" ("-- _" [80] 80) where
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  "-- p = (- 1) %* p"
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definition (in semiring_0)
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  divides :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  (infixl "divides" 70) where
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  "p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))"
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    --{*order of a polynomial*}
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definition (in ring_1) order :: "'a => 'a list => nat" where
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  "order a p = (SOME n. ([-a, 1] %^ n) divides p &
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                      ~ (([-a, 1] %^ (Suc n)) divides p))"
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     --{*degree of a polynomial*}
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definition (in semiring_0) degree :: "'a list => nat" where
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  "degree p = length (pnormalize p) - 1"
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     --{*squarefree polynomials --- NB with respect to real roots only.*}
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definition (in ring_1)
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  rsquarefree :: "'a list => bool" where
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  "rsquarefree p = (poly p \<noteq> poly [] &
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                     (\<forall>a. (order a p = 0) | (order a p = 1)))"
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context semiring_0
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begin
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lemma padd_Nil2[simp]: "p +++ [] = p"
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by (induct p) auto
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lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
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by auto
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lemma pminus_Nil[simp]: "-- [] = []"
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by (simp add: poly_minus_def)
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lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp
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end
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lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t" by (induct "t", auto)
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lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ ((0)#t) = (a#t)"
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by simp
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text{*Handy general properties*}
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lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b"
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proof(induct b arbitrary: a)
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  case Nil thus ?case by auto
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next
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  case (Cons b bs a) thus ?case by (cases a, simp_all add: add_commute)
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qed
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lemma (in comm_semiring_0) padd_assoc: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
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apply (induct a arbitrary: b c)
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apply (simp, clarify)
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apply (case_tac b, simp_all add: add_ac)
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done
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lemma (in semiring_0) poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)"
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apply (induct p arbitrary: q,simp)
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apply (case_tac q, simp_all add: right_distrib)
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done
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lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"
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apply (induct "t", simp)
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apply (auto simp add: mult_zero_left poly_ident_mult padd_commut)
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apply (case_tac t, auto)
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done
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text{*properties of evaluation of polynomials.*}
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lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
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proof(induct p1 arbitrary: p2)
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  case Nil thus ?case by simp
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next
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  case (Cons a as p2) thus ?case
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    by (cases p2, simp_all  add: add_ac right_distrib)
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qed
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lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x"
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apply (induct "p")
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apply (case_tac [2] "x=zero")
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apply (auto simp add: right_distrib mult_ac)
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done
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lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c*poly p x"
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  by (induct p, auto simp add: right_distrib mult_ac)
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lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
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apply (simp add: poly_minus_def)
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apply (auto simp add: poly_cmult minus_mult_left[symmetric])
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done
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lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
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proof(induct p1 arbitrary: p2)
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  case Nil thus ?case by simp
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next
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  case (Cons a as p2)
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  thus ?case by (cases as,
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    simp_all add: poly_cmult poly_add left_distrib right_distrib mult_ac)
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qed
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class idom_char_0 = idom + ring_char_0
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lemma (in comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n"
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apply (induct "n")
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apply (auto simp add: poly_cmult poly_mult power_Suc)
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done
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text{*More Polynomial Evaluation Lemmas*}
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lemma  (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
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by simp
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lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
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  by (simp add: poly_mult mult_assoc)
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lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0"
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by (induct "p", auto)
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lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
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apply (induct "n")
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apply (auto simp add: poly_mult mult_assoc)
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done
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subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
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 @{term "p(x)"} *}
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lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
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proof(induct t)
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  case Nil
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  {fix h have "[h] = [h] +++ [- a, 1] *** []" by simp}
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  thus ?case by blast
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next
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  case (Cons  x xs)
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  {fix h
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    from Cons.hyps[rule_format, of x]
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    obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
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    have "h#x#xs = [a*r + h] +++ [-a, 1] *** (r#q)"
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      using qr by(cases q, simp_all add: algebra_simps diff_minus[symmetric]
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        minus_mult_left[symmetric] right_minus)
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    hence "\<exists>q r. h#x#xs = [r] +++ [-a, 1] *** q" by blast}
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  thus ?case by blast
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qed
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lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
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by (cut_tac t = t and a = a in lemma_poly_linear_rem, auto)
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lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))"
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proof-
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  {assume p: "p = []" hence ?thesis by simp}
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  moreover
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  {fix x xs assume p: "p = x#xs"
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    {fix q assume "p = [-a, 1] *** q" hence "poly p a = 0"
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        by (simp add: poly_add poly_cmult minus_mult_left[symmetric])}
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    moreover
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    {assume p0: "poly p a = 0"
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      from poly_linear_rem[of x xs a] obtain q r
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      where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
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      have "r = 0" using p0 by (simp only: p qr poly_mult poly_add) simp
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      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}
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    ultimately have ?thesis using p by blast}
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  ultimately show ?thesis by (cases p, auto)
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qed
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lemma (in semiring_0) lemma_poly_length_mult[simp]: "\<forall>h k a. length (k %* p +++  (h # (a %* p))) = Suc (length p)"
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by (induct "p", auto)
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lemma (in semiring_0) lemma_poly_length_mult2[simp]: "\<forall>h k. length (k %* p +++  (h # p)) = Suc (length p)"
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by (induct "p", auto)
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lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"
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by auto
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subsection{*Polynomial length*}
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lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p"
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by (induct "p", auto)
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lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)"
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apply (induct p1 arbitrary: p2, simp_all)
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apply arith
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done
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lemma (in semiring_0) poly_root_mult_length[simp]: "length([a,b] *** p) = Suc (length p)"
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by (simp add: poly_add_length)
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lemma (in idom) poly_mult_not_eq_poly_Nil[simp]:
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 "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x"
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by (auto simp add: poly_mult)
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lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 \<longleftrightarrow> poly p x = 0 \<or> poly q x = 0"
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by (auto simp add: poly_mult)
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text{*Normalisation Properties*}
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lemma (in semiring_0) poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)"
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by (induct "p", auto)
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text{*A nontrivial polynomial of degree n has no more than n roots*}
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lemma (in idom) poly_roots_index_lemma:
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   assumes p: "poly p x \<noteq> poly [] x" and n: "length p = n"
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  shows "\<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
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  using p n
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proof(induct n arbitrary: p x)
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  case 0 thus ?case by simp
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next
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  case (Suc n p x)
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  {assume C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)"
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    from Suc.prems have p0: "poly p x \<noteq> 0" "p\<noteq> []" by auto
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    from p0(1)[unfolded poly_linear_divides[of p x]]
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    have "\<forall>q. p \<noteq> [- x, 1] *** q" by blast
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    from C obtain a where a: "poly p a = 0" by blast
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    from a[unfolded poly_linear_divides[of p a]] p0(2)
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    obtain q where q: "p = [-a, 1] *** q" by blast
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    have lg: "length q = n" using q Suc.prems(2) by simp
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    from q p0 have qx: "poly q x \<noteq> poly [] x"
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      by (auto simp add: poly_mult poly_add poly_cmult)
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    from Suc.hyps[OF qx lg] obtain i where
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      i: "\<forall>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)" by blast
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    let ?i = "\<lambda>m. if m = Suc n then a else i m"
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    from C[of ?i] obtain y where y: "poly p y = 0" "\<forall>m\<le> Suc n. y \<noteq> ?i m"
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      by blast
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    from y have "y = a \<or> poly q y = 0"
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      by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: algebra_simps)
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   294
    with i[rule_format, of y] y(1) y(2) have False apply auto
chaieb@26124
   295
      apply (erule_tac x="m" in allE)
chaieb@26124
   296
      apply auto
chaieb@26124
   297
      done}
chaieb@26124
   298
  thus ?case by blast
chaieb@26124
   299
qed
chaieb@26124
   300
chaieb@26124
   301
chaieb@26124
   302
lemma (in idom) poly_roots_index_length: "poly p x \<noteq> poly [] x ==>
chaieb@26124
   303
      \<exists>i. \<forall>x. (poly p x = 0) --> (\<exists>n. n \<le> length p & x = i n)"
chaieb@26124
   304
by (blast intro: poly_roots_index_lemma)
chaieb@26124
   305
wenzelm@26313
   306
lemma (in idom) poly_roots_finite_lemma1: "poly p x \<noteq> poly [] x ==>
chaieb@26124
   307
      \<exists>N i. \<forall>x. (poly p x = 0) --> (\<exists>n. (n::nat) < N & x = i n)"
chaieb@26124
   308
apply (drule poly_roots_index_length, safe)
chaieb@26124
   309
apply (rule_tac x = "Suc (length p)" in exI)
huffman@30488
   310
apply (rule_tac x = i in exI)
chaieb@26124
   311
apply (simp add: less_Suc_eq_le)
chaieb@26124
   312
done
chaieb@26124
   313
chaieb@26124
   314
chaieb@26124
   315
lemma (in idom) idom_finite_lemma:
chaieb@26124
   316
  assumes P: "\<forall>x. P x --> (\<exists>n. n < length j & x = j!n)"
chaieb@26124
   317
  shows "finite {x. P x}"
chaieb@26124
   318
proof-
chaieb@26124
   319
  let ?M = "{x. P x}"
chaieb@26124
   320
  let ?N = "set j"
chaieb@26124
   321
  have "?M \<subseteq> ?N" using P by auto
chaieb@26124
   322
  thus ?thesis using finite_subset by auto
chaieb@26124
   323
qed
chaieb@26124
   324
chaieb@26124
   325
wenzelm@26313
   326
lemma (in idom) poly_roots_finite_lemma2: "poly p x \<noteq> poly [] x ==>
chaieb@26124
   327
      \<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i"
chaieb@26124
   328
apply (drule poly_roots_index_length, safe)
chaieb@26124
   329
apply (rule_tac x="map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI)
chaieb@26124
   330
apply (auto simp add: image_iff)
chaieb@26124
   331
apply (erule_tac x="x" in allE, clarsimp)
chaieb@26124
   332
by (case_tac "n=length p", auto simp add: order_le_less)
chaieb@26124
   333
huffman@30488
   334
lemma (in ring_char_0) UNIV_ring_char_0_infinte:
huffman@30488
   335
  "\<not> (finite (UNIV:: 'a set))"
chaieb@26124
   336
proof
chaieb@26124
   337
  assume F: "finite (UNIV :: 'a set)"
wenzelm@29292
   338
  have "finite (UNIV :: nat set)"
wenzelm@29292
   339
  proof (rule finite_imageD)
wenzelm@29292
   340
    have "of_nat ` UNIV \<subseteq> UNIV" by simp
wenzelm@29292
   341
    then show "finite (of_nat ` UNIV :: 'a set)" using F by (rule finite_subset)
wenzelm@29292
   342
    show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: inj_on_def)
wenzelm@29292
   343
  qed
nipkow@29879
   344
  with infinite_UNIV_nat show False ..
chaieb@26124
   345
qed
chaieb@26124
   346
huffman@30488
   347
lemma (in idom_char_0) poly_roots_finite: "(poly p \<noteq> poly []) =
chaieb@26124
   348
  finite {x. poly p x = 0}"
chaieb@26124
   349
proof
chaieb@26124
   350
  assume H: "poly p \<noteq> poly []"
chaieb@26124
   351
  show "finite {x. poly p x = (0::'a)}"
chaieb@26124
   352
    using H
chaieb@26124
   353
    apply -
chaieb@26124
   354
    apply (erule contrapos_np, rule ext)
chaieb@26124
   355
    apply (rule ccontr)
wenzelm@26313
   356
    apply (clarify dest!: poly_roots_finite_lemma2)
chaieb@26124
   357
    using finite_subset
chaieb@26124
   358
  proof-
chaieb@26124
   359
    fix x i
huffman@30488
   360
    assume F: "\<not> finite {x. poly p x = (0\<Colon>'a)}"
chaieb@26124
   361
      and P: "\<forall>x. poly p x = (0\<Colon>'a) \<longrightarrow> x \<in> set i"
chaieb@26124
   362
    let ?M= "{x. poly p x = (0\<Colon>'a)}"
chaieb@26124
   363
    from P have "?M \<subseteq> set i" by auto
chaieb@26124
   364
    with finite_subset F show False by auto
chaieb@26124
   365
  qed
chaieb@26124
   366
next
chaieb@26124
   367
  assume F: "finite {x. poly p x = (0\<Colon>'a)}"
huffman@30488
   368
  show "poly p \<noteq> poly []" using F UNIV_ring_char_0_infinte by auto
chaieb@26124
   369
qed
chaieb@26124
   370
chaieb@26124
   371
text{*Entirety and Cancellation for polynomials*}
chaieb@26124
   372
huffman@30488
   373
lemma (in idom_char_0) poly_entire_lemma2:
wenzelm@26313
   374
  assumes p0: "poly p \<noteq> poly []" and q0: "poly q \<noteq> poly []"
wenzelm@26313
   375
  shows "poly (p***q) \<noteq> poly []"
wenzelm@26313
   376
proof-
wenzelm@26313
   377
  let ?S = "\<lambda>p. {x. poly p x = 0}"
wenzelm@26313
   378
  have "?S (p *** q) = ?S p \<union> ?S q" by (auto simp add: poly_mult)
wenzelm@26313
   379
  with p0 q0 show ?thesis  unfolding poly_roots_finite by auto
wenzelm@26313
   380
qed
chaieb@26124
   381
huffman@30488
   382
lemma (in idom_char_0) poly_entire:
wenzelm@26313
   383
  "poly (p *** q) = poly [] \<longleftrightarrow> poly p = poly [] \<or> poly q = poly []"
nipkow@29667
   384
using poly_entire_lemma2[of p q]
nipkow@39198
   385
by (auto simp add: ext_iff poly_mult)
chaieb@26124
   386
chaieb@26124
   387
lemma (in idom_char_0) poly_entire_neg: "(poly (p *** q) \<noteq> poly []) = ((poly p \<noteq> poly []) & (poly q \<noteq> poly []))"
chaieb@26124
   388
by (simp add: poly_entire)
chaieb@26124
   389
chaieb@26124
   390
lemma fun_eq: " (f = g) = (\<forall>x. f x = g x)"
chaieb@26124
   391
by (auto intro!: ext)
chaieb@26124
   392
chaieb@26124
   393
lemma (in comm_ring_1) poly_add_minus_zero_iff: "(poly (p +++ -- q) = poly []) = (poly p = poly q)"
nipkow@29667
   394
by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult minus_mult_left[symmetric])
chaieb@26124
   395
chaieb@26124
   396
lemma (in comm_ring_1) poly_add_minus_mult_eq: "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
chaieb@26124
   397
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])
chaieb@26124
   398
haftmann@28823
   399
subclass (in idom_char_0) comm_ring_1 ..
chaieb@26124
   400
lemma (in idom_char_0) poly_mult_left_cancel: "(poly (p *** q) = poly (p *** r)) = (poly p = poly [] | poly q = poly r)"
chaieb@26124
   401
proof-
chaieb@26124
   402
  have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []" by (simp only: poly_add_minus_zero_iff)
chaieb@26124
   403
  also have "\<dots> \<longleftrightarrow> poly p = poly [] | poly q = poly r"
chaieb@26124
   404
    by (auto intro: ext simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
chaieb@26124
   405
  finally show ?thesis .
chaieb@26124
   406
qed
chaieb@26124
   407
haftmann@31021
   408
lemma (in idom) poly_exp_eq_zero[simp]:
chaieb@26124
   409
     "(poly (p %^ n) = poly []) = (poly p = poly [] & n \<noteq> 0)"
haftmann@37598
   410
apply (simp only: fun_eq add: HOL.all_simps [symmetric])
huffman@30488
   411
apply (rule arg_cong [where f = All])
chaieb@26124
   412
apply (rule ext)
haftmann@26194
   413
apply (induct n)
chaieb@26124
   414
apply (auto simp add: poly_exp poly_mult)
chaieb@26124
   415
done
chaieb@26124
   416
chaieb@26124
   417
lemma (in semiring_1) one_neq_zero[simp]: "1 \<noteq> 0" using zero_neq_one by blast
chaieb@26124
   418
lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a,1] \<noteq> poly []"
chaieb@26124
   419
apply (simp add: fun_eq)
chaieb@26124
   420
apply (rule_tac x = "minus one a" in exI)
chaieb@26124
   421
apply (unfold diff_minus)
chaieb@26124
   422
apply (subst add_commute)
chaieb@26124
   423
apply (subst add_assoc)
chaieb@26124
   424
apply simp
huffman@30488
   425
done
chaieb@26124
   426
haftmann@31021
   427
lemma (in idom) poly_exp_prime_eq_zero: "(poly ([a, 1] %^ n) \<noteq> poly [])"
chaieb@26124
   428
by auto
chaieb@26124
   429
chaieb@26124
   430
text{*A more constructive notion of polynomials being trivial*}
chaieb@26124
   431
chaieb@26124
   432
lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] ==> h = 0 & poly t = poly []"
chaieb@26124
   433
apply(simp add: fun_eq)
chaieb@26124
   434
apply (case_tac "h = zero")
huffman@30488
   435
apply (drule_tac [2] x = zero in spec, auto)
huffman@30488
   436
apply (cases "poly t = poly []", simp)
chaieb@26124
   437
proof-
chaieb@26124
   438
  fix x
chaieb@26124
   439
  assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)"  and pnz: "poly t \<noteq> poly []"
chaieb@26124
   440
  let ?S = "{x. poly t x = 0}"
chaieb@26124
   441
  from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast
chaieb@26124
   442
  hence th: "?S \<supseteq> UNIV - {0}" by auto
chaieb@26124
   443
  from poly_roots_finite pnz have th': "finite ?S" by blast
chaieb@26124
   444
  from finite_subset[OF th th'] UNIV_ring_char_0_infinte
chaieb@26124
   445
  show "poly t x = (0\<Colon>'a)" by simp
chaieb@26124
   446
  qed
chaieb@26124
   447
chaieb@26124
   448
lemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p"
chaieb@26124
   449
apply (induct "p", simp)
chaieb@26124
   450
apply (rule iffI)
chaieb@26124
   451
apply (drule poly_zero_lemma', auto)
chaieb@26124
   452
done
chaieb@26124
   453
chaieb@26124
   454
lemma (in idom_char_0) poly_0: "list_all (\<lambda>c. c = 0) p \<Longrightarrow> poly p x = 0"
chaieb@26124
   455
  unfolding poly_zero[symmetric] by simp
chaieb@26124
   456
chaieb@26124
   457
chaieb@26124
   458
chaieb@26124
   459
text{*Basics of divisibility.*}
chaieb@26124
   460
chaieb@26124
   461
lemma (in idom) poly_primes: "([a, 1] divides (p *** q)) = ([a, 1] divides p | [a, 1] divides q)"
chaieb@26124
   462
apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult left_distrib [symmetric])
chaieb@26124
   463
apply (drule_tac x = "uminus a" in spec)
chaieb@26124
   464
apply (simp add: poly_linear_divides poly_add poly_cmult left_distrib [symmetric])
chaieb@26124
   465
apply (cases "p = []")
chaieb@26124
   466
apply (rule exI[where x="[]"])
chaieb@26124
   467
apply simp
chaieb@26124
   468
apply (cases "q = []")
chaieb@26124
   469
apply (erule allE[where x="[]"], simp)
chaieb@26124
   470
chaieb@26124
   471
apply clarsimp
chaieb@26124
   472
apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)")
chaieb@26124
   473
apply (clarsimp simp add: poly_add poly_cmult)
chaieb@26124
   474
apply (rule_tac x="qa" in exI)
chaieb@26124
   475
apply (simp add: left_distrib [symmetric])
chaieb@26124
   476
apply clarsimp
chaieb@26124
   477
chaieb@26124
   478
apply (auto simp add: right_minus poly_linear_divides poly_add poly_cmult left_distrib [symmetric])
chaieb@26124
   479
apply (rule_tac x = "pmult qa q" in exI)
chaieb@26124
   480
apply (rule_tac [2] x = "pmult p qa" in exI)
chaieb@26124
   481
apply (auto simp add: poly_add poly_mult poly_cmult mult_ac)
chaieb@26124
   482
done
chaieb@26124
   483
chaieb@26124
   484
lemma (in comm_semiring_1) poly_divides_refl[simp]: "p divides p"
chaieb@26124
   485
apply (simp add: divides_def)
chaieb@26124
   486
apply (rule_tac x = "[one]" in exI)
chaieb@26124
   487
apply (auto simp add: poly_mult fun_eq)
chaieb@26124
   488
done
chaieb@26124
   489
chaieb@26124
   490
lemma (in comm_semiring_1) poly_divides_trans: "[| p divides q; q divides r |] ==> p divides r"
chaieb@26124
   491
apply (simp add: divides_def, safe)
chaieb@26124
   492
apply (rule_tac x = "pmult qa qaa" in exI)
chaieb@26124
   493
apply (auto simp add: poly_mult fun_eq mult_assoc)
chaieb@26124
   494
done
chaieb@26124
   495
chaieb@26124
   496
haftmann@31021
   497
lemma (in comm_semiring_1) poly_divides_exp: "m \<le> n ==> (p %^ m) divides (p %^ n)"
chaieb@26124
   498
apply (auto simp add: le_iff_add)
chaieb@26124
   499
apply (induct_tac k)
chaieb@26124
   500
apply (rule_tac [2] poly_divides_trans)
chaieb@26124
   501
apply (auto simp add: divides_def)
chaieb@26124
   502
apply (rule_tac x = p in exI)
chaieb@26124
   503
apply (auto simp add: poly_mult fun_eq mult_ac)
chaieb@26124
   504
done
chaieb@26124
   505
haftmann@31021
   506
lemma (in comm_semiring_1) poly_exp_divides: "[| (p %^ n) divides q;  m\<le>n |] ==> (p %^ m) divides q"
chaieb@26124
   507
by (blast intro: poly_divides_exp poly_divides_trans)
chaieb@26124
   508
chaieb@26124
   509
lemma (in comm_semiring_0) poly_divides_add:
chaieb@26124
   510
   "[| p divides q; p divides r |] ==> p divides (q +++ r)"
chaieb@26124
   511
apply (simp add: divides_def, auto)
chaieb@26124
   512
apply (rule_tac x = "padd qa qaa" in exI)
chaieb@26124
   513
apply (auto simp add: poly_add fun_eq poly_mult right_distrib)
chaieb@26124
   514
done
chaieb@26124
   515
chaieb@26124
   516
lemma (in comm_ring_1) poly_divides_diff:
chaieb@26124
   517
   "[| p divides q; p divides (q +++ r) |] ==> p divides r"
chaieb@26124
   518
apply (simp add: divides_def, auto)
chaieb@26124
   519
apply (rule_tac x = "padd qaa (poly_minus qa)" in exI)
nipkow@29667
   520
apply (auto simp add: poly_add fun_eq poly_mult poly_minus algebra_simps)
chaieb@26124
   521
done
chaieb@26124
   522
chaieb@26124
   523
lemma (in comm_ring_1) poly_divides_diff2: "[| p divides r; p divides (q +++ r) |] ==> p divides q"
chaieb@26124
   524
apply (erule poly_divides_diff)
chaieb@26124
   525
apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac)
chaieb@26124
   526
done
chaieb@26124
   527
chaieb@26124
   528
lemma (in semiring_0) poly_divides_zero: "poly p = poly [] ==> q divides p"
chaieb@26124
   529
apply (simp add: divides_def)
chaieb@26124
   530
apply (rule exI[where x="[]"])
chaieb@26124
   531
apply (auto simp add: fun_eq poly_mult)
chaieb@26124
   532
done
chaieb@26124
   533
chaieb@26124
   534
lemma (in semiring_0) poly_divides_zero2[simp]: "q divides []"
chaieb@26124
   535
apply (simp add: divides_def)
chaieb@26124
   536
apply (rule_tac x = "[]" in exI)
chaieb@26124
   537
apply (auto simp add: fun_eq)
chaieb@26124
   538
done
chaieb@26124
   539
chaieb@26124
   540
text{*At last, we can consider the order of a root.*}
chaieb@26124
   541
chaieb@26124
   542
lemma (in idom_char_0)  poly_order_exists_lemma:
chaieb@26124
   543
  assumes lp: "length p = d" and p: "poly p \<noteq> poly []"
chaieb@26124
   544
  shows "\<exists>n q. p = mulexp n [-a, 1] q \<and> poly q a \<noteq> 0"
chaieb@26124
   545
using lp p
chaieb@26124
   546
proof(induct d arbitrary: p)
chaieb@26124
   547
  case 0 thus ?case by simp
chaieb@26124
   548
next
chaieb@26124
   549
  case (Suc n p)
chaieb@26124
   550
  {assume p0: "poly p a = 0"
wenzelm@29292
   551
    from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []" by auto
wenzelm@29292
   552
    hence pN: "p \<noteq> []" by auto
huffman@30488
   553
    from p0[unfolded poly_linear_divides] pN  obtain q where
chaieb@26124
   554
      q: "p = [-a, 1] *** q" by blast
huffman@30488
   555
    from q h p0 have qh: "length q = n" "poly q \<noteq> poly []"
chaieb@26124
   556
      apply -
chaieb@26124
   557
      apply simp
chaieb@26124
   558
      apply (simp only: fun_eq)
chaieb@26124
   559
      apply (rule ccontr)
chaieb@26124
   560
      apply (simp add: fun_eq poly_add poly_cmult minus_mult_left[symmetric])
chaieb@26124
   561
      done
huffman@30488
   562
    from Suc.hyps[OF qh] obtain m r where
huffman@30488
   563
      mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0" by blast
chaieb@26124
   564
    from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0" by simp
chaieb@26124
   565
    hence ?case by blast}
chaieb@26124
   566
  moreover
chaieb@26124
   567
  {assume p0: "poly p a \<noteq> 0"
chaieb@26124
   568
    hence ?case using Suc.prems apply simp by (rule exI[where x="0::nat"], simp)}
chaieb@26124
   569
  ultimately show ?case by blast
chaieb@26124
   570
qed
chaieb@26124
   571
chaieb@26124
   572
haftmann@31021
   573
lemma (in comm_semiring_1) poly_mulexp: "poly (mulexp n p q) x = (poly p x) ^ n * poly q x"
chaieb@26124
   574
by(induct n, auto simp add: poly_mult power_Suc mult_ac)
chaieb@26124
   575
chaieb@26124
   576
lemma (in comm_semiring_1) divides_left_mult:
chaieb@26124
   577
  assumes d:"(p***q) divides r" shows "p divides r \<and> q divides r"
chaieb@26124
   578
proof-
chaieb@26124
   579
  from d obtain t where r:"poly r = poly (p***q *** t)"
chaieb@26124
   580
    unfolding divides_def by blast
chaieb@26124
   581
  hence "poly r = poly (p *** (q *** t))"
chaieb@26124
   582
    "poly r = poly (q *** (p***t))" by(auto simp add: fun_eq poly_mult mult_ac)
chaieb@26124
   583
  thus ?thesis unfolding divides_def by blast
chaieb@26124
   584
qed
chaieb@26124
   585
chaieb@26124
   586
chaieb@26124
   587
chaieb@26124
   588
(* FIXME: Tidy up *)
chaieb@26124
   589
haftmann@31021
   590
lemma (in semiring_1)
chaieb@26124
   591
  zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"
chaieb@26124
   592
  by (induct n, simp_all add: power_Suc)
chaieb@26124
   593
haftmann@31021
   594
lemma (in idom_char_0) poly_order_exists:
chaieb@26124
   595
  assumes lp: "length p = d" and p0: "poly p \<noteq> poly []"
chaieb@26124
   596
  shows "\<exists>n. ([-a, 1] %^ n) divides p & ~(([-a, 1] %^ (Suc n)) divides p)"
chaieb@26124
   597
proof-
chaieb@26124
   598
let ?poly = poly
chaieb@26124
   599
let ?mulexp = mulexp
chaieb@26124
   600
let ?pexp = pexp
chaieb@26124
   601
from lp p0
chaieb@26124
   602
show ?thesis
chaieb@26124
   603
apply -
huffman@30488
   604
apply (drule poly_order_exists_lemma [where a=a], assumption, clarify)
chaieb@26124
   605
apply (rule_tac x = n in exI, safe)
chaieb@26124
   606
apply (unfold divides_def)
chaieb@26124
   607
apply (rule_tac x = q in exI)
chaieb@26124
   608
apply (induct_tac "n", simp)
chaieb@26124
   609
apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult right_distrib mult_ac)
chaieb@26124
   610
apply safe
huffman@30488
   611
apply (subgoal_tac "?poly (?mulexp n [uminus a, one] q) \<noteq> ?poly (pmult (?pexp [uminus a, one] (Suc n)) qa)")
huffman@30488
   612
apply simp
chaieb@26124
   613
apply (induct_tac "n")
chaieb@26124
   614
apply (simp del: pmult_Cons pexp_Suc)
chaieb@26124
   615
apply (erule_tac Q = "?poly q a = zero" in contrapos_np)
chaieb@26124
   616
apply (simp add: poly_add poly_cmult minus_mult_left[symmetric])
chaieb@26124
   617
apply (rule pexp_Suc [THEN ssubst])
chaieb@26124
   618
apply (rule ccontr)
chaieb@26124
   619
apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc)
chaieb@26124
   620
done
chaieb@26124
   621
qed
chaieb@26124
   622
chaieb@26124
   623
chaieb@26124
   624
lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"
chaieb@26124
   625
by (simp add: divides_def, auto)
chaieb@26124
   626
haftmann@31021
   627
lemma (in idom_char_0) poly_order: "poly p \<noteq> poly []
chaieb@26124
   628
      ==> EX! n. ([-a, 1] %^ n) divides p &
chaieb@26124
   629
                 ~(([-a, 1] %^ (Suc n)) divides p)"
chaieb@26124
   630
apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
chaieb@26124
   631
apply (cut_tac x = y and y = n in less_linear)
chaieb@26124
   632
apply (drule_tac m = n in poly_exp_divides)
chaieb@26124
   633
apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
chaieb@26124
   634
            simp del: pmult_Cons pexp_Suc)
chaieb@26124
   635
done
chaieb@26124
   636
chaieb@26124
   637
text{*Order*}
chaieb@26124
   638
chaieb@26124
   639
lemma some1_equalityD: "[| n = (@n. P n); EX! n. P n |] ==> P n"
chaieb@26124
   640
by (blast intro: someI2)
chaieb@26124
   641
haftmann@31021
   642
lemma (in idom_char_0) order:
chaieb@26124
   643
      "(([-a, 1] %^ n) divides p &
chaieb@26124
   644
        ~(([-a, 1] %^ (Suc n)) divides p)) =
chaieb@26124
   645
        ((n = order a p) & ~(poly p = poly []))"
chaieb@26124
   646
apply (unfold order_def)
chaieb@26124
   647
apply (rule iffI)
chaieb@26124
   648
apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
chaieb@26124
   649
apply (blast intro!: poly_order [THEN [2] some1_equalityD])
chaieb@26124
   650
done
chaieb@26124
   651
haftmann@31021
   652
lemma (in idom_char_0) order2: "[| poly p \<noteq> poly [] |]
chaieb@26124
   653
      ==> ([-a, 1] %^ (order a p)) divides p &
chaieb@26124
   654
              ~(([-a, 1] %^ (Suc(order a p))) divides p)"
chaieb@26124
   655
by (simp add: order del: pexp_Suc)
chaieb@26124
   656
haftmann@31021
   657
lemma (in idom_char_0) order_unique: "[| poly p \<noteq> poly []; ([-a, 1] %^ n) divides p;
chaieb@26124
   658
         ~(([-a, 1] %^ (Suc n)) divides p)
chaieb@26124
   659
      |] ==> (n = order a p)"
huffman@30488
   660
by (insert order [of a n p], auto)
chaieb@26124
   661
haftmann@31021
   662
lemma (in idom_char_0) order_unique_lemma: "(poly p \<noteq> poly [] & ([-a, 1] %^ n) divides p &
chaieb@26124
   663
         ~(([-a, 1] %^ (Suc n)) divides p))
chaieb@26124
   664
      ==> (n = order a p)"
chaieb@26124
   665
by (blast intro: order_unique)
chaieb@26124
   666
chaieb@26124
   667
lemma (in ring_1) order_poly: "poly p = poly q ==> order a p = order a q"
chaieb@26124
   668
by (auto simp add: fun_eq divides_def poly_mult order_def)
chaieb@26124
   669
chaieb@26124
   670
lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p"
chaieb@26124
   671
apply (induct "p")
chaieb@26124
   672
apply (auto simp add: numeral_1_eq_1)
chaieb@26124
   673
done
chaieb@26124
   674
chaieb@26124
   675
lemma (in comm_ring_1) lemma_order_root:
chaieb@26124
   676
     " 0 < n & [- a, 1] %^ n divides p & ~ [- a, 1] %^ (Suc n) divides p
chaieb@26124
   677
             \<Longrightarrow> poly p a = 0"
chaieb@26124
   678
apply (induct n arbitrary: a p, blast)
chaieb@26124
   679
apply (auto simp add: divides_def poly_mult simp del: pmult_Cons)
chaieb@26124
   680
done
chaieb@26124
   681
haftmann@31021
   682
lemma (in idom_char_0) order_root: "(poly p a = 0) = ((poly p = poly []) | order a p \<noteq> 0)"
chaieb@26124
   683
proof-
chaieb@26124
   684
  let ?poly = poly
huffman@30488
   685
  show ?thesis
chaieb@26124
   686
apply (case_tac "?poly p = ?poly []", auto)
chaieb@26124
   687
apply (simp add: poly_linear_divides del: pmult_Cons, safe)
chaieb@26124
   688
apply (drule_tac [!] a = a in order2)
chaieb@26124
   689
apply (rule ccontr)
chaieb@26124
   690
apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast)
chaieb@26124
   691
using neq0_conv
chaieb@26124
   692
apply (blast intro: lemma_order_root)
chaieb@26124
   693
done
chaieb@26124
   694
qed
chaieb@26124
   695
haftmann@31021
   696
lemma (in idom_char_0) order_divides: "(([-a, 1] %^ n) divides p) = ((poly p = poly []) | n \<le> order a p)"
chaieb@26124
   697
proof-
chaieb@26124
   698
  let ?poly = poly
huffman@30488
   699
  show ?thesis
chaieb@26124
   700
apply (case_tac "?poly p = ?poly []", auto)
chaieb@26124
   701
apply (simp add: divides_def fun_eq poly_mult)
chaieb@26124
   702
apply (rule_tac x = "[]" in exI)
chaieb@26124
   703
apply (auto dest!: order2 [where a=a]
wenzelm@32960
   704
            intro: poly_exp_divides simp del: pexp_Suc)
chaieb@26124
   705
done
chaieb@26124
   706
qed
chaieb@26124
   707
haftmann@31021
   708
lemma (in idom_char_0) order_decomp:
chaieb@26124
   709
     "poly p \<noteq> poly []
chaieb@26124
   710
      ==> \<exists>q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) &
chaieb@26124
   711
                ~([-a, 1] divides q)"
chaieb@26124
   712
apply (unfold divides_def)
chaieb@26124
   713
apply (drule order2 [where a = a])
chaieb@26124
   714
apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
chaieb@26124
   715
apply (rule_tac x = q in exI, safe)
chaieb@26124
   716
apply (drule_tac x = qa in spec)
chaieb@26124
   717
apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons)
chaieb@26124
   718
done
chaieb@26124
   719
chaieb@26124
   720
text{*Important composition properties of orders.*}
chaieb@26124
   721
lemma order_mult: "poly (p *** q) \<noteq> poly []
haftmann@31021
   722
      ==> order a (p *** q) = order a p + order (a::'a::{idom_char_0}) q"
chaieb@26124
   723
apply (cut_tac a = a and p = "p *** q" and n = "order a p + order a q" in order)
chaieb@26124
   724
apply (auto simp add: poly_entire simp del: pmult_Cons)
chaieb@26124
   725
apply (drule_tac a = a in order2)+
chaieb@26124
   726
apply safe
chaieb@26124
   727
apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
chaieb@26124
   728
apply (rule_tac x = "qa *** qaa" in exI)
chaieb@26124
   729
apply (simp add: poly_mult mult_ac del: pmult_Cons)
chaieb@26124
   730
apply (drule_tac a = a in order_decomp)+
chaieb@26124
   731
apply safe
chaieb@26124
   732
apply (subgoal_tac "[-a,1] divides (qa *** qaa) ")
chaieb@26124
   733
apply (simp add: poly_primes del: pmult_Cons)
chaieb@26124
   734
apply (auto simp add: divides_def simp del: pmult_Cons)
chaieb@26124
   735
apply (rule_tac x = qb in exI)
chaieb@26124
   736
apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
chaieb@26124
   737
apply (drule poly_mult_left_cancel [THEN iffD1], force)
chaieb@26124
   738
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))) ")
chaieb@26124
   739
apply (drule poly_mult_left_cancel [THEN iffD1], force)
chaieb@26124
   740
apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
chaieb@26124
   741
done
chaieb@26124
   742
haftmann@31021
   743
lemma (in idom_char_0) order_mult:
chaieb@26124
   744
  assumes pq0: "poly (p *** q) \<noteq> poly []"
chaieb@26124
   745
  shows "order a (p *** q) = order a p + order a q"
chaieb@26124
   746
proof-
chaieb@26124
   747
  let ?order = order
chaieb@26124
   748
  let ?divides = "op divides"
chaieb@26124
   749
  let ?poly = poly
huffman@30488
   750
from pq0
chaieb@26124
   751
show ?thesis
chaieb@26124
   752
apply (cut_tac a = a and p = "pmult p q" and n = "?order a p + ?order a q" in order)
chaieb@26124
   753
apply (auto simp add: poly_entire simp del: pmult_Cons)
chaieb@26124
   754
apply (drule_tac a = a in order2)+
chaieb@26124
   755
apply safe
chaieb@26124
   756
apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
chaieb@26124
   757
apply (rule_tac x = "pmult qa qaa" in exI)
chaieb@26124
   758
apply (simp add: poly_mult mult_ac del: pmult_Cons)
chaieb@26124
   759
apply (drule_tac a = a in order_decomp)+
chaieb@26124
   760
apply safe
chaieb@26124
   761
apply (subgoal_tac "?divides [uminus a,one ] (pmult qa qaa) ")
chaieb@26124
   762
apply (simp add: poly_primes del: pmult_Cons)
chaieb@26124
   763
apply (auto simp add: divides_def simp del: pmult_Cons)
chaieb@26124
   764
apply (rule_tac x = qb in exI)
chaieb@26124
   765
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))")
chaieb@26124
   766
apply (drule poly_mult_left_cancel [THEN iffD1], force)
chaieb@26124
   767
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))) ")
chaieb@26124
   768
apply (drule poly_mult_left_cancel [THEN iffD1], force)
chaieb@26124
   769
apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
chaieb@26124
   770
done
chaieb@26124
   771
qed
chaieb@26124
   772
haftmann@31021
   773
lemma (in idom_char_0) order_root2: "poly p \<noteq> poly [] ==> (poly p a = 0) = (order a p \<noteq> 0)"
chaieb@26124
   774
by (rule order_root [THEN ssubst], auto)
chaieb@26124
   775
chaieb@26124
   776
lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p" by auto
chaieb@26124
   777
chaieb@26124
   778
lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]"
chaieb@26124
   779
by (simp add: fun_eq)
chaieb@26124
   780
haftmann@31021
   781
lemma (in idom_char_0) rsquarefree_decomp:
chaieb@26124
   782
     "[| rsquarefree p; poly p a = 0 |]
chaieb@26124
   783
      ==> \<exists>q. (poly p = poly ([-a, 1] *** q)) & poly q a \<noteq> 0"
chaieb@26124
   784
apply (simp add: rsquarefree_def, safe)
chaieb@26124
   785
apply (frule_tac a = a in order_decomp)
chaieb@26124
   786
apply (drule_tac x = a in spec)
chaieb@26124
   787
apply (drule_tac a = a in order_root2 [symmetric])
chaieb@26124
   788
apply (auto simp del: pmult_Cons)
chaieb@26124
   789
apply (rule_tac x = q in exI, safe)
chaieb@26124
   790
apply (simp add: poly_mult fun_eq)
chaieb@26124
   791
apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
chaieb@26124
   792
apply (simp add: divides_def del: pmult_Cons, safe)
chaieb@26124
   793
apply (drule_tac x = "[]" in spec)
chaieb@26124
   794
apply (auto simp add: fun_eq)
chaieb@26124
   795
done
chaieb@26124
   796
chaieb@26124
   797
chaieb@26124
   798
text{*Normalization of a polynomial.*}
chaieb@26124
   799
chaieb@26124
   800
lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p"
chaieb@26124
   801
apply (induct "p")
chaieb@26124
   802
apply (auto simp add: fun_eq)
chaieb@26124
   803
done
chaieb@26124
   804
chaieb@26124
   805
text{*The degree of a polynomial.*}
chaieb@26124
   806
chaieb@26124
   807
lemma (in semiring_0) lemma_degree_zero:
chaieb@26124
   808
     "list_all (%c. c = 0) p \<longleftrightarrow>  pnormalize p = []"
chaieb@26124
   809
by (induct "p", auto)
chaieb@26124
   810
huffman@30488
   811
lemma (in idom_char_0) degree_zero:
chaieb@26124
   812
  assumes pN: "poly p = poly []" shows"degree p = 0"
chaieb@26124
   813
proof-
chaieb@26124
   814
  let ?pn = pnormalize
chaieb@26124
   815
  from pN
huffman@30488
   816
  show ?thesis
chaieb@26124
   817
    apply (simp add: degree_def)
chaieb@26124
   818
    apply (case_tac "?pn p = []")
chaieb@26124
   819
    apply (auto simp add: poly_zero lemma_degree_zero )
chaieb@26124
   820
    done
chaieb@26124
   821
qed
chaieb@26124
   822
nipkow@32456
   823
lemma (in semiring_0) pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0"
nipkow@32456
   824
by simp
chaieb@26124
   825
lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])" by simp
huffman@30488
   826
lemma (in semiring_0) pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)"
chaieb@26124
   827
  unfolding pnormal_def by simp
chaieb@26124
   828
lemma (in semiring_0) pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p"
nipkow@32456
   829
  unfolding pnormal_def by(auto split: split_if_asm)
chaieb@26124
   830
chaieb@26124
   831
chaieb@26124
   832
lemma (in semiring_0) pnormal_last_nonzero: "pnormal p ==> last p \<noteq> 0"
nipkow@32456
   833
by(induct p) (simp_all add: pnormal_def split: split_if_asm)
chaieb@26124
   834
chaieb@26124
   835
lemma (in semiring_0) pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
chaieb@26124
   836
  unfolding pnormal_def length_greater_0_conv by blast
chaieb@26124
   837
chaieb@26124
   838
lemma (in semiring_0) pnormal_last_length: "\<lbrakk>0 < length p ; last p \<noteq> 0\<rbrakk> \<Longrightarrow> pnormal p"
nipkow@32456
   839
by (induct p) (auto simp: pnormal_def  split: split_if_asm)
nipkow@32456
   840
chaieb@26124
   841
chaieb@26124
   842
lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> (0 < length p \<and> last p \<noteq> 0)"
chaieb@26124
   843
  using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
chaieb@26124
   844
chaieb@26124
   845
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")
chaieb@26124
   846
proof
chaieb@26124
   847
  assume eq: ?lhs
chaieb@26124
   848
  hence "\<And>x. poly ((c#cs) +++ -- (d#ds)) x = 0"
nipkow@29667
   849
    by (simp only: poly_minus poly_add algebra_simps) simp
nipkow@39198
   850
  hence "poly ((c#cs) +++ -- (d#ds)) = poly []" by(simp add: ext_iff)
chaieb@26124
   851
  hence "c = d \<and> list_all (\<lambda>x. x=0) ((cs +++ -- ds))"
nipkow@29667
   852
    unfolding poly_zero by (simp add: poly_minus_def algebra_simps)
chaieb@26124
   853
  hence "c = d \<and> (\<forall>x. poly (cs +++ -- ds) x = 0)"
huffman@30488
   854
    unfolding poly_zero[symmetric] by simp
nipkow@39198
   855
  thus ?rhs  by (simp add: poly_minus poly_add algebra_simps ext_iff)
chaieb@26124
   856
next
nipkow@39198
   857
  assume ?rhs then show ?lhs by(simp add:ext_iff)
chaieb@26124
   858
qed
huffman@30488
   859
chaieb@26124
   860
lemma (in idom_char_0) pnormalize_unique: "poly p = poly q \<Longrightarrow> pnormalize p = pnormalize q"
chaieb@26124
   861
proof(induct q arbitrary: p)
chaieb@26124
   862
  case Nil thus ?case by (simp only: poly_zero lemma_degree_zero) simp
chaieb@26124
   863
next
chaieb@26124
   864
  case (Cons c cs p)
chaieb@26124
   865
  thus ?case
chaieb@26124
   866
  proof(induct p)
chaieb@26124
   867
    case Nil
chaieb@26124
   868
    hence "poly [] = poly (c#cs)" by blast
huffman@30488
   869
    then have "poly (c#cs) = poly [] " by simp
chaieb@26124
   870
    thus ?case by (simp only: poly_zero lemma_degree_zero) simp
chaieb@26124
   871
  next
chaieb@26124
   872
    case (Cons d ds)
chaieb@26124
   873
    hence eq: "poly (d # ds) = poly (c # cs)" by blast
chaieb@26124
   874
    hence eq': "\<And>x. poly (d # ds) x = poly (c # cs) x" by simp
chaieb@26124
   875
    hence "poly (d # ds) 0 = poly (c # cs) 0" by blast
chaieb@26124
   876
    hence dc: "d = c" by auto
chaieb@26124
   877
    with eq have "poly ds = poly cs"
chaieb@26124
   878
      unfolding  poly_Cons_eq by simp
chaieb@26124
   879
    with Cons.prems have "pnormalize ds = pnormalize cs" by blast
chaieb@26124
   880
    with dc show ?case by simp
chaieb@26124
   881
  qed
chaieb@26124
   882
qed
chaieb@26124
   883
chaieb@26124
   884
lemma (in idom_char_0) degree_unique: assumes pq: "poly p = poly q"
chaieb@26124
   885
  shows "degree p = degree q"
chaieb@26124
   886
using pnormalize_unique[OF pq] unfolding degree_def by simp
chaieb@26124
   887
chaieb@26124
   888
lemma (in semiring_0) pnormalize_length: "length (pnormalize p) \<le> length p" by (induct p, auto)
chaieb@26124
   889
huffman@30488
   890
lemma (in semiring_0) last_linear_mul_lemma:
chaieb@26124
   891
  "last ((a %* p) +++ (x#(b %* p))) = (if p=[] then x else b*last p)"
chaieb@26124
   892
chaieb@26124
   893
apply (induct p arbitrary: a x b, auto)
chaieb@26124
   894
apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []", simp)
chaieb@26124
   895
apply (induct_tac p, auto)
chaieb@26124
   896
done
chaieb@26124
   897
chaieb@26124
   898
lemma (in semiring_1) last_linear_mul: assumes p:"p\<noteq>[]" shows "last ([a,1] *** p) = last p"
chaieb@26124
   899
proof-
chaieb@26124
   900
  from p obtain c cs where cs: "p = c#cs" by (cases p, auto)
chaieb@26124
   901
  from cs have eq:"[a,1] *** p = (a %* (c#cs)) +++ (0#(1 %* (c#cs)))"
chaieb@26124
   902
    by (simp add: poly_cmult_distr)
chaieb@26124
   903
  show ?thesis using cs
chaieb@26124
   904
    unfolding eq last_linear_mul_lemma by simp
chaieb@26124
   905
qed
chaieb@26124
   906
chaieb@26124
   907
lemma (in semiring_0) pnormalize_eq: "last p \<noteq> 0 \<Longrightarrow> pnormalize p = p"
nipkow@32456
   908
by (induct p) (auto split: split_if_asm)
chaieb@26124
   909
chaieb@26124
   910
lemma (in semiring_0) last_pnormalize: "pnormalize p \<noteq> [] \<Longrightarrow> last (pnormalize p) \<noteq> 0"
chaieb@26124
   911
  by (induct p, auto)
chaieb@26124
   912
chaieb@26124
   913
lemma (in semiring_0) pnormal_degree: "last p \<noteq> 0 \<Longrightarrow> degree p = length p - 1"
chaieb@26124
   914
  using pnormalize_eq[of p] unfolding degree_def by simp
chaieb@26124
   915
wenzelm@26313
   916
lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)" by (rule ext) simp
chaieb@26124
   917
chaieb@26124
   918
lemma (in idom_char_0) linear_mul_degree: assumes p: "poly p \<noteq> poly []"
chaieb@26124
   919
  shows "degree ([a,1] *** p) = degree p + 1"
chaieb@26124
   920
proof-
chaieb@26124
   921
  from p have pnz: "pnormalize p \<noteq> []"
chaieb@26124
   922
    unfolding poly_zero lemma_degree_zero .
huffman@30488
   923
chaieb@26124
   924
  from last_linear_mul[OF pnz, of a] last_pnormalize[OF pnz]
chaieb@26124
   925
  have l0: "last ([a, 1] *** pnormalize p) \<noteq> 0" by simp
chaieb@26124
   926
  from last_pnormalize[OF pnz] last_linear_mul[OF pnz, of a]
chaieb@26124
   927
    pnormal_degree[OF l0] pnormal_degree[OF last_pnormalize[OF pnz]] pnz
huffman@30488
   928
chaieb@26124
   929
huffman@30488
   930
  have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1"
chaieb@26124
   931
    by (auto simp add: poly_length_mult)
chaieb@26124
   932
chaieb@26124
   933
  have eqs: "poly ([a,1] *** pnormalize p) = poly ([a,1] *** p)"
chaieb@26124
   934
    by (rule ext) (simp add: poly_mult poly_add poly_cmult)
chaieb@26124
   935
  from degree_unique[OF eqs] th
chaieb@26124
   936
  show ?thesis by (simp add: degree_unique[OF poly_normalize])
chaieb@26124
   937
qed
chaieb@26124
   938
chaieb@26124
   939
lemma (in idom_char_0) linear_pow_mul_degree:
chaieb@26124
   940
  "degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)"
chaieb@26124
   941
proof(induct n arbitrary: a p)
chaieb@26124
   942
  case (0 a p)
chaieb@26124
   943
  {assume p: "poly p = poly []"
chaieb@26124
   944
    hence ?case using degree_unique[OF p] by (simp add: degree_def)}
chaieb@26124
   945
  moreover
wenzelm@26313
   946
  {assume p: "poly p \<noteq> poly []" hence ?case by (auto simp add: poly_Nil_ext) }
chaieb@26124
   947
  ultimately show ?case by blast
chaieb@26124
   948
next
chaieb@26124
   949
  case (Suc n a p)
chaieb@26124
   950
  have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1] %^ n *** ([a,1] *** p))"
chaieb@26124
   951
    apply (rule ext, simp add: poly_mult poly_add poly_cmult)
chaieb@26124
   952
    by (simp add: mult_ac add_ac right_distrib)
chaieb@26124
   953
  note deq = degree_unique[OF eq]
chaieb@26124
   954
  {assume p: "poly p = poly []"
huffman@30488
   955
    with eq have eq': "poly ([a,1] %^(Suc n) *** p) = poly []"
chaieb@26124
   956
      by - (rule ext,simp add: poly_mult poly_cmult poly_add)
chaieb@26124
   957
    from degree_unique[OF eq'] p have ?case by (simp add: degree_def)}
chaieb@26124
   958
  moreover
chaieb@26124
   959
  {assume p: "poly p \<noteq> poly []"
chaieb@26124
   960
    from p have ap: "poly ([a,1] *** p) \<noteq> poly []"
huffman@30488
   961
      using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto
chaieb@26124
   962
    have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1]%^n *** ([a,1] *** p))"
nipkow@29667
   963
     by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps)
chaieb@26124
   964
   from ap have ap': "(poly ([a,1] *** p) = poly []) = False" by blast
chaieb@26124
   965
   have  th0: "degree ([a,1]%^n *** ([a,1] *** p)) = degree ([a,1] *** p) + n"
chaieb@26124
   966
     apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap')
chaieb@26124
   967
     by simp
huffman@30488
   968
chaieb@26124
   969
   from degree_unique[OF eq] ap p th0 linear_mul_degree[OF p, of a]
chaieb@26124
   970
   have ?case by (auto simp del: poly.simps)}
chaieb@26124
   971
  ultimately show ?case by blast
chaieb@26124
   972
qed
chaieb@26124
   973
haftmann@31021
   974
lemma (in idom_char_0) order_degree:
chaieb@26124
   975
  assumes p0: "poly p \<noteq> poly []"
chaieb@26124
   976
  shows "order a p \<le> degree p"
chaieb@26124
   977
proof-
chaieb@26124
   978
  from order2[OF p0, unfolded divides_def]
chaieb@26124
   979
  obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)" by blast
chaieb@26124
   980
  {assume "poly q = poly []"
chaieb@26124
   981
    with q p0 have False by (simp add: poly_mult poly_entire)}
huffman@30488
   982
  with degree_unique[OF q, unfolded linear_pow_mul_degree]
chaieb@26124
   983
  show ?thesis by auto
chaieb@26124
   984
qed
chaieb@26124
   985
chaieb@26124
   986
text{*Tidier versions of finiteness of roots.*}
chaieb@26124
   987
chaieb@26124
   988
lemma (in idom_char_0) poly_roots_finite_set: "poly p \<noteq> poly [] ==> finite {x. poly p x = 0}"
chaieb@26124
   989
unfolding poly_roots_finite .
chaieb@26124
   990
chaieb@26124
   991
text{*bound for polynomial.*}
chaieb@26124
   992
haftmann@35028
   993
lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k"
chaieb@26124
   994
apply (induct "p", auto)
chaieb@26124
   995
apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
chaieb@26124
   996
apply (rule abs_triangle_ineq)
chaieb@26124
   997
apply (auto intro!: mult_mono simp add: abs_mult)
chaieb@26124
   998
done
chaieb@26124
   999
chaieb@26124
  1000
lemma (in semiring_0) poly_Sing: "poly [c] x = c" by simp
chaieb@26124
  1001
chaieb@26124
  1002
end