src/HOL/Library/Polynomial.thy
changeset 62352 35a9e1cbb5b3
parent 62351 fd049b54ad68
child 62422 4aa35fd6c152
     1.1 --- a/src/HOL/Library/Polynomial.thy	Wed Feb 17 21:51:58 2016 +0100
     1.2 +++ b/src/HOL/Library/Polynomial.thy	Wed Feb 17 21:51:58 2016 +0100
     1.3 @@ -1,13 +1,15 @@
     1.4  (*  Title:      HOL/Library/Polynomial.thy
     1.5      Author:     Brian Huffman
     1.6      Author:     Clemens Ballarin
     1.7 +    Author:     Amine Chaieb
     1.8      Author:     Florian Haftmann
     1.9  *)
    1.10  
    1.11  section \<open>Polynomials as type over a ring structure\<close>
    1.12  
    1.13  theory Polynomial
    1.14 -imports Main "~~/src/HOL/GCD" "~~/src/HOL/Library/More_List" "~~/src/HOL/Library/Infinite_Set"
    1.15 +imports Main "~~/src/HOL/Deriv" "~~/src/HOL/Library/More_List"
    1.16 +  "~~/src/HOL/Library/Infinite_Set"
    1.17  begin
    1.18  
    1.19  subsection \<open>Auxiliary: operations for lists (later) representing coefficients\<close>
    1.20 @@ -1900,133 +1902,6 @@
    1.21    by (subst (asm) order_root) auto
    1.22  
    1.23  
    1.24 -subsection \<open>GCD of polynomials\<close>
    1.25 -
    1.26 -instantiation poly :: (field) gcd
    1.27 -begin
    1.28 -
    1.29 -function gcd_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
    1.30 -where
    1.31 -  "gcd (x::'a poly) 0 = smult (inverse (coeff x (degree x))) x"
    1.32 -| "y \<noteq> 0 \<Longrightarrow> gcd (x::'a poly) y = gcd y (x mod y)"
    1.33 -by auto
    1.34 -
    1.35 -termination "gcd :: _ poly \<Rightarrow> _"
    1.36 -by (relation "measure (\<lambda>(x, y). if y = 0 then 0 else Suc (degree y))")
    1.37 -   (auto dest: degree_mod_less)
    1.38 -
    1.39 -declare gcd_poly.simps [simp del]
    1.40 -
    1.41 -definition lcm_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
    1.42 -where
    1.43 -  "lcm_poly a b = a * b div smult (coeff a (degree a) * coeff b (degree b)) (gcd a b)"
    1.44 -
    1.45 -instance ..
    1.46 -
    1.47 -end
    1.48 -
    1.49 -lemma
    1.50 -  fixes x y :: "_ poly"
    1.51 -  shows poly_gcd_dvd1 [iff]: "gcd x y dvd x"
    1.52 -    and poly_gcd_dvd2 [iff]: "gcd x y dvd y"
    1.53 -  apply (induct x y rule: gcd_poly.induct)
    1.54 -  apply (simp_all add: gcd_poly.simps)
    1.55 -  apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
    1.56 -  apply (blast dest: dvd_mod_imp_dvd)
    1.57 -  done
    1.58 -
    1.59 -lemma poly_gcd_greatest:
    1.60 -  fixes k x y :: "_ poly"
    1.61 -  shows "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd gcd x y"
    1.62 -  by (induct x y rule: gcd_poly.induct)
    1.63 -     (simp_all add: gcd_poly.simps dvd_mod dvd_smult)
    1.64 -
    1.65 -lemma dvd_poly_gcd_iff [iff]:
    1.66 -  fixes k x y :: "_ poly"
    1.67 -  shows "k dvd gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
    1.68 -  by (auto intro!: poly_gcd_greatest intro: dvd_trans [of _ "gcd x y"])
    1.69 -
    1.70 -lemma poly_gcd_monic:
    1.71 -  fixes x y :: "_ poly"
    1.72 -  shows "coeff (gcd x y) (degree (gcd x y)) =
    1.73 -    (if x = 0 \<and> y = 0 then 0 else 1)"
    1.74 -  by (induct x y rule: gcd_poly.induct)
    1.75 -     (simp_all add: gcd_poly.simps nonzero_imp_inverse_nonzero)
    1.76 -
    1.77 -lemma poly_gcd_zero_iff [simp]:
    1.78 -  fixes x y :: "_ poly"
    1.79 -  shows "gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
    1.80 -  by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)
    1.81 -
    1.82 -lemma poly_gcd_0_0 [simp]:
    1.83 -  "gcd (0::_ poly) 0 = 0"
    1.84 -  by simp
    1.85 -
    1.86 -lemma poly_dvd_antisym:
    1.87 -  fixes p q :: "'a::idom poly"
    1.88 -  assumes coeff: "coeff p (degree p) = coeff q (degree q)"
    1.89 -  assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
    1.90 -proof (cases "p = 0")
    1.91 -  case True with coeff show "p = q" by simp
    1.92 -next
    1.93 -  case False with coeff have "q \<noteq> 0" by auto
    1.94 -  have degree: "degree p = degree q"
    1.95 -    using \<open>p dvd q\<close> \<open>q dvd p\<close> \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close>
    1.96 -    by (intro order_antisym dvd_imp_degree_le)
    1.97 -
    1.98 -  from \<open>p dvd q\<close> obtain a where a: "q = p * a" ..
    1.99 -  with \<open>q \<noteq> 0\<close> have "a \<noteq> 0" by auto
   1.100 -  with degree a \<open>p \<noteq> 0\<close> have "degree a = 0"
   1.101 -    by (simp add: degree_mult_eq)
   1.102 -  with coeff a show "p = q"
   1.103 -    by (cases a, auto split: if_splits)
   1.104 -qed
   1.105 -
   1.106 -lemma poly_gcd_unique:
   1.107 -  fixes d x y :: "_ poly"
   1.108 -  assumes dvd1: "d dvd x" and dvd2: "d dvd y"
   1.109 -    and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
   1.110 -    and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
   1.111 -  shows "gcd x y = d"
   1.112 -proof -
   1.113 -  have "coeff (gcd x y) (degree (gcd x y)) = coeff d (degree d)"
   1.114 -    by (simp_all add: poly_gcd_monic monic)
   1.115 -  moreover have "gcd x y dvd d"
   1.116 -    using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
   1.117 -  moreover have "d dvd gcd x y"
   1.118 -    using dvd1 dvd2 by (rule poly_gcd_greatest)
   1.119 -  ultimately show ?thesis
   1.120 -    by (rule poly_dvd_antisym)
   1.121 -qed
   1.122 -
   1.123 -instance poly :: (field) semiring_gcd
   1.124 -proof
   1.125 -  fix p q :: "'a::field poly"
   1.126 -  show "normalize (gcd p q) = gcd p q"
   1.127 -    by (induct p q rule: gcd_poly.induct)
   1.128 -      (simp_all add: gcd_poly.simps normalize_poly_def)
   1.129 -  show "lcm p q = normalize (p * q) div gcd p q"
   1.130 -    by (simp add: coeff_degree_mult div_smult_left div_smult_right lcm_poly_def normalize_poly_def)
   1.131 -      (metis (no_types, lifting) div_smult_right inverse_mult_distrib inverse_zero mult.commute pdivmod_rel pdivmod_rel_def smult_eq_0_iff)
   1.132 -qed simp_all
   1.133 -
   1.134 -lemma poly_gcd_1_left [simp]: "gcd 1 y = (1 :: _ poly)"
   1.135 -by (rule poly_gcd_unique) simp_all
   1.136 -
   1.137 -lemma poly_gcd_1_right [simp]: "gcd x 1 = (1 :: _ poly)"
   1.138 -by (rule poly_gcd_unique) simp_all
   1.139 -
   1.140 -lemma poly_gcd_minus_left [simp]: "gcd (- x) y = gcd x (y :: _ poly)"
   1.141 -by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
   1.142 -
   1.143 -lemma poly_gcd_minus_right [simp]: "gcd x (- y) = gcd x (y :: _ poly)"
   1.144 -by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
   1.145 -
   1.146 -lemma poly_gcd_code [code]:
   1.147 -  "gcd x y = (if y = 0 then normalize x else gcd y (x mod (y :: _ poly)))"
   1.148 -  by (simp add: gcd_poly.simps)
   1.149 -
   1.150 -
   1.151  subsection \<open>Additional induction rules on polynomials\<close>
   1.152  
   1.153  text \<open>
   1.154 @@ -2338,8 +2213,630 @@
   1.155  
   1.156  lemma lead_coeff_nonzero: "p \<noteq> 0 \<Longrightarrow> lead_coeff p \<noteq> 0"
   1.157    by (simp add: lead_coeff_def)
   1.158 +
   1.159 +
   1.160 +subsection \<open>Derivatives of univariate polynomials\<close>
   1.161 +
   1.162 +function pderiv :: "('a :: semidom) poly \<Rightarrow> 'a poly"
   1.163 +where
   1.164 +  [simp del]: "pderiv (pCons a p) = (if p = 0 then 0 else p + pCons 0 (pderiv p))"
   1.165 +  by (auto intro: pCons_cases)
   1.166 +
   1.167 +termination pderiv
   1.168 +  by (relation "measure degree") simp_all
   1.169 +
   1.170 +lemma pderiv_0 [simp]:
   1.171 +  "pderiv 0 = 0"
   1.172 +  using pderiv.simps [of 0 0] by simp
   1.173 +
   1.174 +lemma pderiv_pCons:
   1.175 +  "pderiv (pCons a p) = p + pCons 0 (pderiv p)"
   1.176 +  by (simp add: pderiv.simps)
   1.177 +
   1.178 +lemma pderiv_1 [simp]: "pderiv 1 = 0" 
   1.179 +  unfolding one_poly_def by (simp add: pderiv_pCons)
   1.180 +
   1.181 +lemma pderiv_of_nat  [simp]: "pderiv (of_nat n) = 0"
   1.182 +  and pderiv_numeral [simp]: "pderiv (numeral m) = 0"
   1.183 +  by (simp_all add: of_nat_poly numeral_poly pderiv_pCons)
   1.184 +
   1.185 +lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
   1.186 +  by (induct p arbitrary: n) 
   1.187 +     (auto simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
   1.188 +
   1.189 +fun pderiv_coeffs_code :: "('a :: semidom) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   1.190 +  "pderiv_coeffs_code f (x # xs) = cCons (f * x) (pderiv_coeffs_code (f+1) xs)"
   1.191 +| "pderiv_coeffs_code f [] = []"
   1.192 +
   1.193 +definition pderiv_coeffs :: "('a :: semidom) list \<Rightarrow> 'a list" where
   1.194 +  "pderiv_coeffs xs = pderiv_coeffs_code 1 (tl xs)"
   1.195 +
   1.196 +(* Efficient code for pderiv contributed by René Thiemann and Akihisa Yamada *)
   1.197 +lemma pderiv_coeffs_code: 
   1.198 +  "nth_default 0 (pderiv_coeffs_code f xs) n = (f + of_nat n) * (nth_default 0 xs n)"
   1.199 +proof (induct xs arbitrary: f n)
   1.200 +  case (Cons x xs f n)
   1.201 +  show ?case 
   1.202 +  proof (cases n)
   1.203 +    case 0
   1.204 +    thus ?thesis by (cases "pderiv_coeffs_code (f + 1) xs = [] \<and> f * x = 0", auto simp: cCons_def)
   1.205 +  next
   1.206 +    case (Suc m) note n = this
   1.207 +    show ?thesis 
   1.208 +    proof (cases "pderiv_coeffs_code (f + 1) xs = [] \<and> f * x = 0")
   1.209 +      case False
   1.210 +      hence "nth_default 0 (pderiv_coeffs_code f (x # xs)) n = 
   1.211 +               nth_default 0 (pderiv_coeffs_code (f + 1) xs) m" 
   1.212 +        by (auto simp: cCons_def n)
   1.213 +      also have "\<dots> = (f + of_nat n) * (nth_default 0 xs m)" 
   1.214 +        unfolding Cons by (simp add: n add_ac)
   1.215 +      finally show ?thesis by (simp add: n)
   1.216 +    next
   1.217 +      case True
   1.218 +      {
   1.219 +        fix g 
   1.220 +        have "pderiv_coeffs_code g xs = [] \<Longrightarrow> g + of_nat m = 0 \<or> nth_default 0 xs m = 0"
   1.221 +        proof (induct xs arbitrary: g m)
   1.222 +          case (Cons x xs g)
   1.223 +          from Cons(2) have empty: "pderiv_coeffs_code (g + 1) xs = []"
   1.224 +                            and g: "(g = 0 \<or> x = 0)"
   1.225 +            by (auto simp: cCons_def split: if_splits)
   1.226 +          note IH = Cons(1)[OF empty]
   1.227 +          from IH[of m] IH[of "m - 1"] g
   1.228 +          show ?case by (cases m, auto simp: field_simps)
   1.229 +        qed simp
   1.230 +      } note empty = this
   1.231 +      from True have "nth_default 0 (pderiv_coeffs_code f (x # xs)) n = 0"
   1.232 +        by (auto simp: cCons_def n)
   1.233 +      moreover have "(f + of_nat n) * nth_default 0 (x # xs) n = 0" using True
   1.234 +        by (simp add: n, insert empty[of "f+1"], auto simp: field_simps)
   1.235 +      ultimately show ?thesis by simp
   1.236 +    qed
   1.237 +  qed
   1.238 +qed simp
   1.239 +
   1.240 +lemma map_upt_Suc: "map f [0 ..< Suc n] = f 0 # map (\<lambda> i. f (Suc i)) [0 ..< n]"
   1.241 +  by (induct n arbitrary: f, auto)
   1.242 +
   1.243 +lemma coeffs_pderiv_code [code abstract]:
   1.244 +  "coeffs (pderiv p) = pderiv_coeffs (coeffs p)" unfolding pderiv_coeffs_def
   1.245 +proof (rule coeffs_eqI, unfold pderiv_coeffs_code coeff_pderiv, goal_cases)
   1.246 +  case (1 n)
   1.247 +  have id: "coeff p (Suc n) = nth_default 0 (map (\<lambda>i. coeff p (Suc i)) [0..<degree p]) n"
   1.248 +    by (cases "n < degree p", auto simp: nth_default_def coeff_eq_0)
   1.249 +  show ?case unfolding coeffs_def map_upt_Suc by (auto simp: id)
   1.250 +next
   1.251 +  case 2
   1.252 +  obtain n xs where id: "tl (coeffs p) = xs" "(1 :: 'a) = n" by auto
   1.253 +  from 2 show ?case
   1.254 +    unfolding id by (induct xs arbitrary: n, auto simp: cCons_def)
   1.255 +qed
   1.256 +
   1.257 +context
   1.258 +  assumes "SORT_CONSTRAINT('a::{semidom, semiring_char_0})"
   1.259 +begin
   1.260 +
   1.261 +lemma pderiv_eq_0_iff: 
   1.262 +  "pderiv (p :: 'a poly) = 0 \<longleftrightarrow> degree p = 0"
   1.263 +  apply (rule iffI)
   1.264 +  apply (cases p, simp)
   1.265 +  apply (simp add: poly_eq_iff coeff_pderiv del: of_nat_Suc)
   1.266 +  apply (simp add: poly_eq_iff coeff_pderiv coeff_eq_0)
   1.267 +  done
   1.268 +
   1.269 +lemma degree_pderiv: "degree (pderiv (p :: 'a poly)) = degree p - 1"
   1.270 +  apply (rule order_antisym [OF degree_le])
   1.271 +  apply (simp add: coeff_pderiv coeff_eq_0)
   1.272 +  apply (cases "degree p", simp)
   1.273 +  apply (rule le_degree)
   1.274 +  apply (simp add: coeff_pderiv del: of_nat_Suc)
   1.275 +  apply (metis degree_0 leading_coeff_0_iff nat.distinct(1))
   1.276 +  done
   1.277 +
   1.278 +lemma not_dvd_pderiv: 
   1.279 +  assumes "degree (p :: 'a poly) \<noteq> 0"
   1.280 +  shows "\<not> p dvd pderiv p"
   1.281 +proof
   1.282 +  assume dvd: "p dvd pderiv p"
   1.283 +  then obtain q where p: "pderiv p = p * q" unfolding dvd_def by auto
   1.284 +  from dvd have le: "degree p \<le> degree (pderiv p)"
   1.285 +    by (simp add: assms dvd_imp_degree_le pderiv_eq_0_iff)
   1.286 +  from this[unfolded degree_pderiv] assms show False by auto
   1.287 +qed
   1.288 +
   1.289 +lemma dvd_pderiv_iff [simp]: "(p :: 'a poly) dvd pderiv p \<longleftrightarrow> degree p = 0"
   1.290 +  using not_dvd_pderiv[of p] by (auto simp: pderiv_eq_0_iff [symmetric])
   1.291 +
   1.292 +end
   1.293 +
   1.294 +lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"
   1.295 +by (simp add: pderiv_pCons)
   1.296 +
   1.297 +lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
   1.298 +by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
   1.299 +
   1.300 +lemma pderiv_minus: "pderiv (- p :: 'a :: idom poly) = - pderiv p"
   1.301 +by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
   1.302 +
   1.303 +lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
   1.304 +by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
   1.305 +
   1.306 +lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
   1.307 +by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
   1.308 +
   1.309 +lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
   1.310 +by (induct p) (auto simp: pderiv_add pderiv_smult pderiv_pCons algebra_simps)
   1.311 +
   1.312 +lemma pderiv_power_Suc:
   1.313 +  "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
   1.314 +apply (induct n)
   1.315 +apply simp
   1.316 +apply (subst power_Suc)
   1.317 +apply (subst pderiv_mult)
   1.318 +apply (erule ssubst)
   1.319 +apply (simp only: of_nat_Suc smult_add_left smult_1_left)
   1.320 +apply (simp add: algebra_simps)
   1.321 +done
   1.322 +
   1.323 +lemma pderiv_setprod: "pderiv (setprod f (as)) = 
   1.324 +  (\<Sum>a \<in> as. setprod f (as - {a}) * pderiv (f a))"
   1.325 +proof (induct as rule: infinite_finite_induct)
   1.326 +  case (insert a as)
   1.327 +  hence id: "setprod f (insert a as) = f a * setprod f as" 
   1.328 +    "\<And> g. setsum g (insert a as) = g a + setsum g as"
   1.329 +    "insert a as - {a} = as"
   1.330 +    by auto
   1.331 +  {
   1.332 +    fix b
   1.333 +    assume "b \<in> as"
   1.334 +    hence id2: "insert a as - {b} = insert a (as - {b})" using \<open>a \<notin> as\<close> by auto
   1.335 +    have "setprod f (insert a as - {b}) = f a * setprod f (as - {b})"
   1.336 +      unfolding id2
   1.337 +      by (subst setprod.insert, insert insert, auto)
   1.338 +  } note id2 = this
   1.339 +  show ?case
   1.340 +    unfolding id pderiv_mult insert(3) setsum_right_distrib
   1.341 +    by (auto simp add: ac_simps id2 intro!: setsum.cong)
   1.342 +qed auto
   1.343 +
   1.344 +lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
   1.345 +by (rule DERIV_cong, rule DERIV_pow, simp)
   1.346 +declare DERIV_pow2 [simp] DERIV_pow [simp]
   1.347 +
   1.348 +lemma DERIV_add_const: "DERIV f x :> D ==>  DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
   1.349 +by (rule DERIV_cong, rule DERIV_add, auto)
   1.350 +
   1.351 +lemma poly_DERIV [simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
   1.352 +  by (induct p, auto intro!: derivative_eq_intros simp add: pderiv_pCons)
   1.353 +
   1.354 +lemma continuous_on_poly [continuous_intros]: 
   1.355 +  fixes p :: "'a :: {real_normed_field} poly"
   1.356 +  assumes "continuous_on A f"
   1.357 +  shows   "continuous_on A (\<lambda>x. poly p (f x))"
   1.358 +proof -
   1.359 +  have "continuous_on A (\<lambda>x. (\<Sum>i\<le>degree p. (f x) ^ i * coeff p i))" 
   1.360 +    by (intro continuous_intros assms)
   1.361 +  also have "\<dots> = (\<lambda>x. poly p (f x))" by (intro ext) (simp add: poly_altdef mult_ac)
   1.362 +  finally show ?thesis .
   1.363 +qed
   1.364 +
   1.365 +text\<open>Consequences of the derivative theorem above\<close>
   1.366 +
   1.367 +lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (at x::real filter)"
   1.368 +apply (simp add: real_differentiable_def)
   1.369 +apply (blast intro: poly_DERIV)
   1.370 +done
   1.371 +
   1.372 +lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
   1.373 +by (rule poly_DERIV [THEN DERIV_isCont])
   1.374 +
   1.375 +lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
   1.376 +      ==> \<exists>x. a < x & x < b & (poly p x = 0)"
   1.377 +using IVT_objl [of "poly p" a 0 b]
   1.378 +by (auto simp add: order_le_less)
   1.379 +
   1.380 +lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
   1.381 +      ==> \<exists>x. a < x & x < b & (poly p x = 0)"
   1.382 +by (insert poly_IVT_pos [where p = "- p" ]) simp
   1.383 +
   1.384 +lemma poly_IVT:
   1.385 +  fixes p::"real poly"
   1.386 +  assumes "a<b" and "poly p a * poly p b < 0"
   1.387 +  shows "\<exists>x>a. x < b \<and> poly p x = 0"
   1.388 +by (metis assms(1) assms(2) less_not_sym mult_less_0_iff poly_IVT_neg poly_IVT_pos)
   1.389 +
   1.390 +lemma poly_MVT: "(a::real) < b ==>
   1.391 +     \<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
   1.392 +using MVT [of a b "poly p"]
   1.393 +apply auto
   1.394 +apply (rule_tac x = z in exI)
   1.395 +apply (auto simp add: mult_left_cancel poly_DERIV [THEN DERIV_unique])
   1.396 +done
   1.397 +
   1.398 +lemma poly_MVT':
   1.399 +  assumes "{min a b..max a b} \<subseteq> A"
   1.400 +  shows   "\<exists>x\<in>A. poly p b - poly p a = (b - a) * poly (pderiv p) (x::real)"
   1.401 +proof (cases a b rule: linorder_cases)
   1.402 +  case less
   1.403 +  from poly_MVT[OF less, of p] guess x by (elim exE conjE)
   1.404 +  thus ?thesis by (intro bexI[of _ x]) (auto intro!: subsetD[OF assms])
   1.405 +
   1.406 +next
   1.407 +  case greater
   1.408 +  from poly_MVT[OF greater, of p] guess x by (elim exE conjE)
   1.409 +  thus ?thesis by (intro bexI[of _ x]) (auto simp: algebra_simps intro!: subsetD[OF assms])
   1.410 +qed (insert assms, auto)
   1.411 +
   1.412 +lemma poly_pinfty_gt_lc:
   1.413 +  fixes p:: "real poly"
   1.414 +  assumes  "lead_coeff p > 0" 
   1.415 +  shows "\<exists> n. \<forall> x \<ge> n. poly p x \<ge> lead_coeff p" using assms
   1.416 +proof (induct p)
   1.417 +  case 0
   1.418 +  thus ?case by auto
   1.419 +next
   1.420 +  case (pCons a p)
   1.421 +  have "\<lbrakk>a\<noteq>0;p=0\<rbrakk> \<Longrightarrow> ?case" by auto
   1.422 +  moreover have "p\<noteq>0 \<Longrightarrow> ?case"
   1.423 +    proof -
   1.424 +      assume "p\<noteq>0"
   1.425 +      then obtain n1 where gte_lcoeff:"\<forall>x\<ge>n1. lead_coeff p \<le> poly p x" using that pCons by auto
   1.426 +      have gt_0:"lead_coeff p >0" using pCons(3) \<open>p\<noteq>0\<close> by auto
   1.427 +      def n\<equiv>"max n1 (1+ \<bar>a\<bar>/(lead_coeff p))"
   1.428 +      show ?thesis 
   1.429 +        proof (rule_tac x=n in exI,rule,rule) 
   1.430 +          fix x assume "n \<le> x"
   1.431 +          hence "lead_coeff p \<le> poly p x" 
   1.432 +            using gte_lcoeff unfolding n_def by auto
   1.433 +          hence " \<bar>a\<bar>/(lead_coeff p) \<ge> \<bar>a\<bar>/(poly p x)" and "poly p x>0" using gt_0
   1.434 +            by (intro frac_le,auto)
   1.435 +          hence "x\<ge>1+ \<bar>a\<bar>/(poly p x)" using \<open>n\<le>x\<close>[unfolded n_def] by auto
   1.436 +          thus "lead_coeff (pCons a p) \<le> poly (pCons a p) x"
   1.437 +            using \<open>lead_coeff p \<le> poly p x\<close> \<open>poly p x>0\<close> \<open>p\<noteq>0\<close>
   1.438 +            by (auto simp add:field_simps)
   1.439 +        qed
   1.440 +    qed
   1.441 +  ultimately show ?case by fastforce
   1.442 +qed
   1.443 +
   1.444 +
   1.445 +subsection \<open>Algebraic numbers\<close>
   1.446 +
   1.447 +text \<open>
   1.448 +  Algebraic numbers can be defined in two equivalent ways: all real numbers that are 
   1.449 +  roots of rational polynomials or of integer polynomials. The Algebraic-Numbers AFP entry 
   1.450 +  uses the rational definition, but we need the integer definition.
   1.451 +
   1.452 +  The equivalence is obvious since any rational polynomial can be multiplied with the 
   1.453 +  LCM of its coefficients, yielding an integer polynomial with the same roots.
   1.454 +\<close>
   1.455 +subsection \<open>Algebraic numbers\<close>
   1.456 +
   1.457 +definition algebraic :: "'a :: field_char_0 \<Rightarrow> bool" where
   1.458 +  "algebraic x \<longleftrightarrow> (\<exists>p. (\<forall>i. coeff p i \<in> \<int>) \<and> p \<noteq> 0 \<and> poly p x = 0)"
   1.459 +
   1.460 +lemma algebraicI:
   1.461 +  assumes "\<And>i. coeff p i \<in> \<int>" "p \<noteq> 0" "poly p x = 0"
   1.462 +  shows   "algebraic x"
   1.463 +  using assms unfolding algebraic_def by blast
   1.464    
   1.465 +lemma algebraicE:
   1.466 +  assumes "algebraic x"
   1.467 +  obtains p where "\<And>i. coeff p i \<in> \<int>" "p \<noteq> 0" "poly p x = 0"
   1.468 +  using assms unfolding algebraic_def by blast
   1.469 +
   1.470 +lemma quotient_of_denom_pos': "snd (quotient_of x) > 0"
   1.471 +  using quotient_of_denom_pos[OF surjective_pairing] .
   1.472    
   1.473 +lemma of_int_div_in_Ints: 
   1.474 +  "b dvd a \<Longrightarrow> of_int a div of_int b \<in> (\<int> :: 'a :: ring_div set)"
   1.475 +proof (cases "of_int b = (0 :: 'a)")
   1.476 +  assume "b dvd a" "of_int b \<noteq> (0::'a)"
   1.477 +  then obtain c where "a = b * c" by (elim dvdE)
   1.478 +  with \<open>of_int b \<noteq> (0::'a)\<close> show ?thesis by simp
   1.479 +qed auto
   1.480 +
   1.481 +lemma of_int_divide_in_Ints: 
   1.482 +  "b dvd a \<Longrightarrow> of_int a / of_int b \<in> (\<int> :: 'a :: field set)"
   1.483 +proof (cases "of_int b = (0 :: 'a)")
   1.484 +  assume "b dvd a" "of_int b \<noteq> (0::'a)"
   1.485 +  then obtain c where "a = b * c" by (elim dvdE)
   1.486 +  with \<open>of_int b \<noteq> (0::'a)\<close> show ?thesis by simp
   1.487 +qed auto
   1.488 +
   1.489 +lemma algebraic_altdef:
   1.490 +  fixes p :: "'a :: field_char_0 poly"
   1.491 +  shows "algebraic x \<longleftrightarrow> (\<exists>p. (\<forall>i. coeff p i \<in> \<rat>) \<and> p \<noteq> 0 \<and> poly p x = 0)"
   1.492 +proof safe
   1.493 +  fix p assume rat: "\<forall>i. coeff p i \<in> \<rat>" and root: "poly p x = 0" and nz: "p \<noteq> 0"
   1.494 +  def cs \<equiv> "coeffs p"
   1.495 +  from rat have "\<forall>c\<in>range (coeff p). \<exists>c'. c = of_rat c'" unfolding Rats_def by blast
   1.496 +  then obtain f where f: "\<And>i. coeff p i = of_rat (f (coeff p i))" 
   1.497 +    by (subst (asm) bchoice_iff) blast
   1.498 +  def cs' \<equiv> "map (quotient_of \<circ> f) (coeffs p)"
   1.499 +  def d \<equiv> "Lcm (set (map snd cs'))"
   1.500 +  def p' \<equiv> "smult (of_int d) p"
   1.501 +  
   1.502 +  have "\<forall>n. coeff p' n \<in> \<int>"
   1.503 +  proof
   1.504 +    fix n :: nat
   1.505 +    show "coeff p' n \<in> \<int>"
   1.506 +    proof (cases "n \<le> degree p")
   1.507 +      case True
   1.508 +      def c \<equiv> "coeff p n"
   1.509 +      def a \<equiv> "fst (quotient_of (f (coeff p n)))" and b \<equiv> "snd (quotient_of (f (coeff p n)))"
   1.510 +      have b_pos: "b > 0" unfolding b_def using quotient_of_denom_pos' by simp
   1.511 +      have "coeff p' n = of_int d * coeff p n" by (simp add: p'_def)
   1.512 +      also have "coeff p n = of_rat (of_int a / of_int b)" unfolding a_def b_def
   1.513 +        by (subst quotient_of_div [of "f (coeff p n)", symmetric])
   1.514 +           (simp_all add: f [symmetric])
   1.515 +      also have "of_int d * \<dots> = of_rat (of_int (a*d) / of_int b)"
   1.516 +        by (simp add: of_rat_mult of_rat_divide)
   1.517 +      also from nz True have "b \<in> snd ` set cs'" unfolding cs'_def
   1.518 +        by (force simp: o_def b_def coeffs_def simp del: upt_Suc)
   1.519 +      hence "b dvd (a * d)" unfolding d_def by simp
   1.520 +      hence "of_int (a * d) / of_int b \<in> (\<int> :: rat set)"
   1.521 +        by (rule of_int_divide_in_Ints)
   1.522 +      hence "of_rat (of_int (a * d) / of_int b) \<in> \<int>" by (elim Ints_cases) auto
   1.523 +      finally show ?thesis .
   1.524 +    qed (auto simp: p'_def not_le coeff_eq_0)
   1.525 +  qed
   1.526 +  
   1.527 +  moreover have "set (map snd cs') \<subseteq> {0<..}"
   1.528 +    unfolding cs'_def using quotient_of_denom_pos' by (auto simp: coeffs_def simp del: upt_Suc) 
   1.529 +  hence "d \<noteq> 0" unfolding d_def by (induction cs') simp_all
   1.530 +  with nz have "p' \<noteq> 0" by (simp add: p'_def)
   1.531 +  moreover from root have "poly p' x = 0" by (simp add: p'_def)
   1.532 +  ultimately show "algebraic x" unfolding algebraic_def by blast
   1.533 +next
   1.534 +
   1.535 +  assume "algebraic x"
   1.536 +  then obtain p where p: "\<And>i. coeff p i \<in> \<int>" "poly p x = 0" "p \<noteq> 0" 
   1.537 +    by (force simp: algebraic_def)
   1.538 +  moreover have "coeff p i \<in> \<int> \<Longrightarrow> coeff p i \<in> \<rat>" for i by (elim Ints_cases) simp
   1.539 +  ultimately show  "(\<exists>p. (\<forall>i. coeff p i \<in> \<rat>) \<and> p \<noteq> 0 \<and> poly p x = 0)" by auto
   1.540 +qed
   1.541 +
   1.542 +
   1.543 +text\<open>Lemmas for Derivatives\<close>
   1.544 +
   1.545 +lemma order_unique_lemma:
   1.546 +  fixes p :: "'a::idom poly"
   1.547 +  assumes "[:-a, 1:] ^ n dvd p" "\<not> [:-a, 1:] ^ Suc n dvd p"
   1.548 +  shows "n = order a p"
   1.549 +unfolding Polynomial.order_def
   1.550 +apply (rule Least_equality [symmetric])
   1.551 +apply (fact assms)
   1.552 +apply (rule classical)
   1.553 +apply (erule notE)
   1.554 +unfolding not_less_eq_eq
   1.555 +using assms(1) apply (rule power_le_dvd)
   1.556 +apply assumption
   1.557 +done
   1.558 +
   1.559 +lemma lemma_order_pderiv1:
   1.560 +  "pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
   1.561 +    smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
   1.562 +apply (simp only: pderiv_mult pderiv_power_Suc)
   1.563 +apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons)
   1.564 +done
   1.565 +
   1.566 +lemma lemma_order_pderiv:
   1.567 +  fixes p :: "'a :: field_char_0 poly"
   1.568 +  assumes n: "0 < n" 
   1.569 +      and pd: "pderiv p \<noteq> 0" 
   1.570 +      and pe: "p = [:- a, 1:] ^ n * q" 
   1.571 +      and nd: "~ [:- a, 1:] dvd q"
   1.572 +    shows "n = Suc (order a (pderiv p))"
   1.573 +using n 
   1.574 +proof -
   1.575 +  have "pderiv ([:- a, 1:] ^ n * q) \<noteq> 0"
   1.576 +    using assms by auto
   1.577 +  obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) \<noteq> 0"
   1.578 +    using assms by (cases n) auto
   1.579 +  have *: "!!k l. k dvd k * pderiv q + smult (of_nat (Suc n')) l \<Longrightarrow> k dvd l"
   1.580 +    by (auto simp del: of_nat_Suc simp: dvd_add_right_iff dvd_smult_iff)
   1.581 +  have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))" 
   1.582 +  proof (rule order_unique_lemma)
   1.583 +    show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
   1.584 +      apply (subst lemma_order_pderiv1)
   1.585 +      apply (rule dvd_add)
   1.586 +      apply (metis dvdI dvd_mult2 power_Suc2)
   1.587 +      apply (metis dvd_smult dvd_triv_right)
   1.588 +      done
   1.589 +  next
   1.590 +    show "\<not> [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
   1.591 +     apply (subst lemma_order_pderiv1)
   1.592 +     by (metis * nd dvd_mult_cancel_right power_not_zero pCons_eq_0_iff power_Suc zero_neq_one)
   1.593 +  qed
   1.594 +  then show ?thesis
   1.595 +    by (metis \<open>n = Suc n'\<close> pe)
   1.596 +qed
   1.597 +
   1.598 +lemma order_decomp:
   1.599 +  assumes "p \<noteq> 0"
   1.600 +  shows "\<exists>q. p = [:- a, 1:] ^ order a p * q \<and> \<not> [:- a, 1:] dvd q"
   1.601 +proof -
   1.602 +  from assms have A: "[:- a, 1:] ^ order a p dvd p"
   1.603 +    and B: "\<not> [:- a, 1:] ^ Suc (order a p) dvd p" by (auto dest: order)
   1.604 +  from A obtain q where C: "p = [:- a, 1:] ^ order a p * q" ..
   1.605 +  with B have "\<not> [:- a, 1:] ^ Suc (order a p) dvd [:- a, 1:] ^ order a p * q"
   1.606 +    by simp
   1.607 +  then have "\<not> [:- a, 1:] ^ order a p * [:- a, 1:] dvd [:- a, 1:] ^ order a p * q"
   1.608 +    by simp
   1.609 +  then have D: "\<not> [:- a, 1:] dvd q"
   1.610 +    using idom_class.dvd_mult_cancel_left [of "[:- a, 1:] ^ order a p" "[:- a, 1:]" q]
   1.611 +    by auto
   1.612 +  from C D show ?thesis by blast
   1.613 +qed
   1.614 +
   1.615 +lemma order_pderiv:
   1.616 +  "\<lbrakk>pderiv p \<noteq> 0; order a (p :: 'a :: field_char_0 poly) \<noteq> 0\<rbrakk> \<Longrightarrow>
   1.617 +     (order a p = Suc (order a (pderiv p)))"
   1.618 +apply (case_tac "p = 0", simp)
   1.619 +apply (drule_tac a = a and p = p in order_decomp)
   1.620 +using neq0_conv
   1.621 +apply (blast intro: lemma_order_pderiv)
   1.622 +done
   1.623 +
   1.624 +lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q"
   1.625 +proof -
   1.626 +  def i \<equiv> "order a p"
   1.627 +  def j \<equiv> "order a q"
   1.628 +  def t \<equiv> "[:-a, 1:]"
   1.629 +  have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0"
   1.630 +    unfolding t_def by (simp add: dvd_iff_poly_eq_0)
   1.631 +  assume "p * q \<noteq> 0"
   1.632 +  then show "order a (p * q) = i + j"
   1.633 +    apply clarsimp
   1.634 +    apply (drule order [where a=a and p=p, folded i_def t_def])
   1.635 +    apply (drule order [where a=a and p=q, folded j_def t_def])
   1.636 +    apply clarify
   1.637 +    apply (erule dvdE)+
   1.638 +    apply (rule order_unique_lemma [symmetric], fold t_def)
   1.639 +    apply (simp_all add: power_add t_dvd_iff)
   1.640 +    done
   1.641 +qed
   1.642 +
   1.643 +lemma order_smult:
   1.644 +  assumes "c \<noteq> 0" 
   1.645 +  shows "order x (smult c p) = order x p"
   1.646 +proof (cases "p = 0")
   1.647 +  case False
   1.648 +  have "smult c p = [:c:] * p" by simp
   1.649 +  also from assms False have "order x \<dots> = order x [:c:] + order x p" 
   1.650 +    by (subst order_mult) simp_all
   1.651 +  also from assms have "order x [:c:] = 0" by (intro order_0I) auto
   1.652 +  finally show ?thesis by simp
   1.653 +qed simp
   1.654 +
   1.655 +(* Next two lemmas contributed by Wenda Li *)
   1.656 +lemma order_1_eq_0 [simp]:"order x 1 = 0" 
   1.657 +  by (metis order_root poly_1 zero_neq_one)
   1.658 +
   1.659 +lemma order_power_n_n: "order a ([:-a,1:]^n)=n" 
   1.660 +proof (induct n) (*might be proved more concisely using nat_less_induct*)
   1.661 +  case 0
   1.662 +  thus ?case by (metis order_root poly_1 power_0 zero_neq_one)
   1.663 +next 
   1.664 +  case (Suc n)
   1.665 +  have "order a ([:- a, 1:] ^ Suc n)=order a ([:- a, 1:] ^ n) + order a [:-a,1:]" 
   1.666 +    by (metis (no_types, hide_lams) One_nat_def add_Suc_right monoid_add_class.add.right_neutral 
   1.667 +      one_neq_zero order_mult pCons_eq_0_iff power_add power_eq_0_iff power_one_right)
   1.668 +  moreover have "order a [:-a,1:]=1" unfolding order_def
   1.669 +    proof (rule Least_equality,rule ccontr)
   1.670 +      assume  "\<not> \<not> [:- a, 1:] ^ Suc 1 dvd [:- a, 1:]"
   1.671 +      hence "[:- a, 1:] ^ Suc 1 dvd [:- a, 1:]" by simp
   1.672 +      hence "degree ([:- a, 1:] ^ Suc 1) \<le> degree ([:- a, 1:] )" 
   1.673 +        by (rule dvd_imp_degree_le,auto) 
   1.674 +      thus False by auto
   1.675 +    next
   1.676 +      fix y assume asm:"\<not> [:- a, 1:] ^ Suc y dvd [:- a, 1:]"
   1.677 +      show "1 \<le> y" 
   1.678 +        proof (rule ccontr)
   1.679 +          assume "\<not> 1 \<le> y"
   1.680 +          hence "y=0" by auto
   1.681 +          hence "[:- a, 1:] ^ Suc y dvd [:- a, 1:]" by auto
   1.682 +          thus False using asm by auto
   1.683 +        qed
   1.684 +    qed
   1.685 +  ultimately show ?case using Suc by auto
   1.686 +qed
   1.687 +
   1.688 +text\<open>Now justify the standard squarefree decomposition, i.e. f / gcd(f,f').\<close>
   1.689 +
   1.690 +lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p"
   1.691 +apply (cases "p = 0", auto)
   1.692 +apply (drule order_2 [where a=a and p=p])
   1.693 +apply (metis not_less_eq_eq power_le_dvd)
   1.694 +apply (erule power_le_dvd [OF order_1])
   1.695 +done
   1.696 +
   1.697 +lemma poly_squarefree_decomp_order:
   1.698 +  assumes "pderiv (p :: 'a :: field_char_0 poly) \<noteq> 0"
   1.699 +  and p: "p = q * d"
   1.700 +  and p': "pderiv p = e * d"
   1.701 +  and d: "d = r * p + s * pderiv p"
   1.702 +  shows "order a q = (if order a p = 0 then 0 else 1)"
   1.703 +proof (rule classical)
   1.704 +  assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)"
   1.705 +  from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
   1.706 +  with p have "order a p = order a q + order a d"
   1.707 +    by (simp add: order_mult)
   1.708 +  with 1 have "order a p \<noteq> 0" by (auto split: if_splits)
   1.709 +  have "order a (pderiv p) = order a e + order a d"
   1.710 +    using \<open>pderiv p \<noteq> 0\<close> \<open>pderiv p = e * d\<close> by (simp add: order_mult)
   1.711 +  have "order a p = Suc (order a (pderiv p))"
   1.712 +    using \<open>pderiv p \<noteq> 0\<close> \<open>order a p \<noteq> 0\<close> by (rule order_pderiv)
   1.713 +  have "d \<noteq> 0" using \<open>p \<noteq> 0\<close> \<open>p = q * d\<close> by simp
   1.714 +  have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
   1.715 +    apply (simp add: d)
   1.716 +    apply (rule dvd_add)
   1.717 +    apply (rule dvd_mult)
   1.718 +    apply (simp add: order_divides \<open>p \<noteq> 0\<close>
   1.719 +           \<open>order a p = Suc (order a (pderiv p))\<close>)
   1.720 +    apply (rule dvd_mult)
   1.721 +    apply (simp add: order_divides)
   1.722 +    done
   1.723 +  then have "order a (pderiv p) \<le> order a d"
   1.724 +    using \<open>d \<noteq> 0\<close> by (simp add: order_divides)
   1.725 +  show ?thesis
   1.726 +    using \<open>order a p = order a q + order a d\<close>
   1.727 +    using \<open>order a (pderiv p) = order a e + order a d\<close>
   1.728 +    using \<open>order a p = Suc (order a (pderiv p))\<close>
   1.729 +    using \<open>order a (pderiv p) \<le> order a d\<close>
   1.730 +    by auto
   1.731 +qed
   1.732 +
   1.733 +lemma poly_squarefree_decomp_order2: 
   1.734 +     "\<lbrakk>pderiv p \<noteq> (0 :: 'a :: field_char_0 poly);
   1.735 +       p = q * d;
   1.736 +       pderiv p = e * d;
   1.737 +       d = r * p + s * pderiv p
   1.738 +      \<rbrakk> \<Longrightarrow> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
   1.739 +by (blast intro: poly_squarefree_decomp_order)
   1.740 +
   1.741 +lemma order_pderiv2: 
   1.742 +  "\<lbrakk>pderiv p \<noteq> 0; order a (p :: 'a :: field_char_0 poly) \<noteq> 0\<rbrakk>
   1.743 +      \<Longrightarrow> (order a (pderiv p) = n) = (order a p = Suc n)"
   1.744 +by (auto dest: order_pderiv)
   1.745 +
   1.746 +definition
   1.747 +  rsquarefree :: "'a::idom poly => bool" where
   1.748 +  "rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))"
   1.749 +
   1.750 +lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h :: 'a :: {semidom,semiring_char_0}:]"
   1.751 +apply (simp add: pderiv_eq_0_iff)
   1.752 +apply (case_tac p, auto split: if_splits)
   1.753 +done
   1.754 +
   1.755 +lemma rsquarefree_roots:
   1.756 +  fixes p :: "'a :: field_char_0 poly"
   1.757 +  shows "rsquarefree p = (\<forall>a. \<not>(poly p a = 0 \<and> poly (pderiv p) a = 0))"
   1.758 +apply (simp add: rsquarefree_def)
   1.759 +apply (case_tac "p = 0", simp, simp)
   1.760 +apply (case_tac "pderiv p = 0")
   1.761 +apply simp
   1.762 +apply (drule pderiv_iszero, clarsimp)
   1.763 +apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree)
   1.764 +apply (force simp add: order_root order_pderiv2)
   1.765 +done
   1.766 +
   1.767 +lemma poly_squarefree_decomp:
   1.768 +  assumes "pderiv (p :: 'a :: field_char_0 poly) \<noteq> 0"
   1.769 +    and "p = q * d"
   1.770 +    and "pderiv p = e * d"
   1.771 +    and "d = r * p + s * pderiv p"
   1.772 +  shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
   1.773 +proof -
   1.774 +  from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
   1.775 +  with \<open>p = q * d\<close> have "q \<noteq> 0" by simp
   1.776 +  have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
   1.777 +    using assms by (rule poly_squarefree_decomp_order2)
   1.778 +  with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> show ?thesis
   1.779 +    by (simp add: rsquarefree_def order_root)
   1.780 +qed
   1.781 +
   1.782  
   1.783  no_notation cCons (infixr "##" 65)
   1.784