src/HOL/Library/Poly_Deriv.thy
author wenzelm
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(*  Title:      HOL/Library/Poly_Deriv.thy
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    Author:     Amine Chaieb
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    Author:     Brian Huffman
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*)
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section\<open>Polynomials and Differentiation\<close>
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theory Poly_Deriv
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imports Deriv Polynomial
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begin
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subsection \<open>Derivatives of univariate polynomials\<close>
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function pderiv :: "'a::real_normed_field poly \<Rightarrow> 'a poly"
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where
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  [simp del]: "pderiv (pCons a p) = (if p = 0 then 0 else p + pCons 0 (pderiv p))"
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  by (auto intro: pCons_cases)
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termination pderiv
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  by (relation "measure degree") simp_all
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lemma pderiv_0 [simp]:
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  "pderiv 0 = 0"
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  using pderiv.simps [of 0 0] by simp
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lemma pderiv_pCons:
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  "pderiv (pCons a p) = p + pCons 0 (pderiv p)"
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  by (simp add: pderiv.simps)
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lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
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  by (induct p arbitrary: n) 
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     (auto simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
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primrec pderiv_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list"
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where
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  "pderiv_coeffs [] = []"
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| "pderiv_coeffs (x # xs) = plus_coeffs xs (cCons 0 (pderiv_coeffs xs))"
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lemma coeffs_pderiv [code abstract]:
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  "coeffs (pderiv p) = pderiv_coeffs (coeffs p)"
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  by (rule sym, induct p) (simp_all add: pderiv_pCons coeffs_plus_eq_plus_coeffs cCons_def)
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lemma pderiv_eq_0_iff: "pderiv p = 0 \<longleftrightarrow> degree p = 0"
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  apply (rule iffI)
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  apply (cases p, simp)
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  apply (simp add: poly_eq_iff coeff_pderiv del: of_nat_Suc)
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  apply (simp add: poly_eq_iff coeff_pderiv coeff_eq_0)
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  done
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lemma degree_pderiv: "degree (pderiv p) = degree p - 1"
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  apply (rule order_antisym [OF degree_le])
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  apply (simp add: coeff_pderiv coeff_eq_0)
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  apply (cases "degree p", simp)
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  apply (rule le_degree)
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  apply (simp add: coeff_pderiv del: of_nat_Suc)
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  apply (metis degree_0 leading_coeff_0_iff nat.distinct(1))
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  done
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lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"
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by (simp add: pderiv_pCons)
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lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
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by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
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lemma pderiv_minus: "pderiv (- p) = - pderiv p"
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by (rule poly_eqI, simp add: coeff_pderiv)
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lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
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by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
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lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
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by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
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lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
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by (induct p) (auto simp: pderiv_add pderiv_smult pderiv_pCons algebra_simps)
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lemma pderiv_power_Suc:
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  "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
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apply (induct n)
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apply simp
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apply (subst power_Suc)
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apply (subst pderiv_mult)
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apply (erule ssubst)
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apply (simp only: of_nat_Suc smult_add_left smult_1_left)
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apply (simp add: algebra_simps)
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done
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lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
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by (rule DERIV_cong, rule DERIV_pow, simp)
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declare DERIV_pow2 [simp] DERIV_pow [simp]
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lemma DERIV_add_const: "DERIV f x :> D ==>  DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
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by (rule DERIV_cong, rule DERIV_add, auto)
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lemma poly_DERIV[simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
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  by (induct p, auto intro!: derivative_eq_intros simp add: pderiv_pCons)
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text\<open>Consequences of the derivative theorem above\<close>
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lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (at x::real filter)"
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apply (simp add: real_differentiable_def)
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apply (blast intro: poly_DERIV)
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done
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lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
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by (rule poly_DERIV [THEN DERIV_isCont])
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lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
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      ==> \<exists>x. a < x & x < b & (poly p x = 0)"
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using IVT_objl [of "poly p" a 0 b]
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by (auto simp add: order_le_less)
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lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
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      ==> \<exists>x. a < x & x < b & (poly p x = 0)"
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by (insert poly_IVT_pos [where p = "- p" ]) simp
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lemma poly_MVT: "(a::real) < b ==>
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     \<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
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using MVT [of a b "poly p"]
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apply auto
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apply (rule_tac x = z in exI)
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apply (auto simp add: mult_left_cancel poly_DERIV [THEN DERIV_unique])
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done
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text\<open>Lemmas for Derivatives\<close>
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lemma order_unique_lemma:
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  fixes p :: "'a::idom poly"
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  assumes "[:-a, 1:] ^ n dvd p" "\<not> [:-a, 1:] ^ Suc n dvd p"
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  shows "n = order a p"
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unfolding Polynomial.order_def
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apply (rule Least_equality [symmetric])
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apply (fact assms)
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apply (rule classical)
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apply (erule notE)
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unfolding not_less_eq_eq
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   137
using assms(1) apply (rule power_le_dvd)
5fbe474b5da8 explicit theory with additional, less commonly used list operations
haftmann
parents: 56383
diff changeset
   138
apply assumption
5fbe474b5da8 explicit theory with additional, less commonly used list operations
haftmann
parents: 56383
diff changeset
   139
done
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   140
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   141
lemma lemma_order_pderiv1:
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   142
  "pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   143
    smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   144
apply (simp only: pderiv_mult pderiv_power_Suc)
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 29985
diff changeset
   145
apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons)
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   146
done
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   147
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   148
lemma dvd_add_cancel1:
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   149
  fixes a b c :: "'a::comm_ring_1"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   150
  shows "a dvd b + c \<Longrightarrow> a dvd b \<Longrightarrow> a dvd c"
35050
9f841f20dca6 renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents: 31881
diff changeset
   151
  by (drule (1) Rings.dvd_diff, simp)
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   152
56383
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   153
lemma lemma_order_pderiv:
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   154
  assumes n: "0 < n" 
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   155
      and pd: "pderiv p \<noteq> 0" 
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   156
      and pe: "p = [:- a, 1:] ^ n * q" 
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   157
      and nd: "~ [:- a, 1:] dvd q"
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   158
    shows "n = Suc (order a (pderiv p))"
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   159
using n 
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   160
proof -
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   161
  have "pderiv ([:- a, 1:] ^ n * q) \<noteq> 0"
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   162
    using assms by auto
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   163
  obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) \<noteq> 0"
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   164
    using assms by (cases n) auto
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   165
  then have *: "!!k l. k dvd k * pderiv q + smult (of_nat (Suc n')) l \<Longrightarrow> k dvd l"
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   166
    by (metis dvd_add_cancel1 dvd_smult_iff dvd_triv_left of_nat_eq_0_iff old.nat.distinct(2))
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   167
  have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))" 
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   168
  proof (rule order_unique_lemma)
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   169
    show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   170
      apply (subst lemma_order_pderiv1)
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   171
      apply (rule dvd_add)
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   172
      apply (metis dvdI dvd_mult2 power_Suc2)
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   173
      apply (metis dvd_smult dvd_triv_right)
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   174
      done
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   175
  next
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   176
    show "\<not> [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   177
     apply (subst lemma_order_pderiv1)
60867
86e7560e07d0 slight cleanup of lemmas
haftmann
parents: 60688
diff changeset
   178
     by (metis * nd dvd_mult_cancel_right power_not_zero pCons_eq_0_iff power_Suc zero_neq_one)
56383
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   179
  qed
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   180
  then show ?thesis
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   181
    by (metis \<open>n = Suc n'\<close> pe)
56383
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   182
qed
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   183
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   184
lemma order_decomp:
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   185
  assumes "p \<noteq> 0"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   186
  shows "\<exists>q. p = [:- a, 1:] ^ order a p * q \<and> \<not> [:- a, 1:] dvd q"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   187
proof -
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   188
  from assms have A: "[:- a, 1:] ^ order a p dvd p"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   189
    and B: "\<not> [:- a, 1:] ^ Suc (order a p) dvd p" by (auto dest: order)
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   190
  from A obtain q where C: "p = [:- a, 1:] ^ order a p * q" ..
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   191
  with B have "\<not> [:- a, 1:] ^ Suc (order a p) dvd [:- a, 1:] ^ order a p * q"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   192
    by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   193
  then have "\<not> [:- a, 1:] ^ order a p * [:- a, 1:] dvd [:- a, 1:] ^ order a p * q"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   194
    by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   195
  then have D: "\<not> [:- a, 1:] dvd q"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   196
    using idom_class.dvd_mult_cancel_left [of "[:- a, 1:] ^ order a p" "[:- a, 1:]" q]
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   197
    by auto
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   198
  from C D show ?thesis by blast
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60500
diff changeset
   199
qed
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   200
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   201
lemma order_pderiv: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   202
      ==> (order a p = Suc (order a (pderiv p)))"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   203
apply (case_tac "p = 0", simp)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   204
apply (drule_tac a = a and p = p in order_decomp)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   205
using neq0_conv
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   206
apply (blast intro: lemma_order_pderiv)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   207
done
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   208
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   209
lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   210
proof -
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   211
  def i \<equiv> "order a p"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   212
  def j \<equiv> "order a q"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   213
  def t \<equiv> "[:-a, 1:]"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   214
  have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   215
    unfolding t_def by (simp add: dvd_iff_poly_eq_0)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   216
  assume "p * q \<noteq> 0"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   217
  then show "order a (p * q) = i + j"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   218
    apply clarsimp
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   219
    apply (drule order [where a=a and p=p, folded i_def t_def])
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   220
    apply (drule order [where a=a and p=q, folded j_def t_def])
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   221
    apply clarify
56383
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   222
    apply (erule dvdE)+
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   223
    apply (rule order_unique_lemma [symmetric], fold t_def)
56383
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   224
    apply (simp_all add: power_add t_dvd_iff)
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   225
    done
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   226
qed
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   227
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   228
text\<open>Now justify the standard squarefree decomposition, i.e. f / gcd(f,f').\<close>
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   229
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   230
lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   231
apply (cases "p = 0", auto)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   232
apply (drule order_2 [where a=a and p=p])
56383
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   233
apply (metis not_less_eq_eq power_le_dvd)
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   234
apply (erule power_le_dvd [OF order_1])
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   235
done
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   236
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   237
lemma poly_squarefree_decomp_order:
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   238
  assumes "pderiv p \<noteq> 0"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   239
  and p: "p = q * d"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   240
  and p': "pderiv p = e * d"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   241
  and d: "d = r * p + s * pderiv p"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   242
  shows "order a q = (if order a p = 0 then 0 else 1)"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   243
proof (rule classical)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   244
  assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)"
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   245
  from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   246
  with p have "order a p = order a q + order a d"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   247
    by (simp add: order_mult)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   248
  with 1 have "order a p \<noteq> 0" by (auto split: if_splits)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   249
  have "order a (pderiv p) = order a e + order a d"
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   250
    using \<open>pderiv p \<noteq> 0\<close> \<open>pderiv p = e * d\<close> by (simp add: order_mult)
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   251
  have "order a p = Suc (order a (pderiv p))"
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   252
    using \<open>pderiv p \<noteq> 0\<close> \<open>order a p \<noteq> 0\<close> by (rule order_pderiv)
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   253
  have "d \<noteq> 0" using \<open>p \<noteq> 0\<close> \<open>p = q * d\<close> by simp
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   254
  have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   255
    apply (simp add: d)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   256
    apply (rule dvd_add)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   257
    apply (rule dvd_mult)
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   258
    apply (simp add: order_divides \<open>p \<noteq> 0\<close>
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   259
           \<open>order a p = Suc (order a (pderiv p))\<close>)
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   260
    apply (rule dvd_mult)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   261
    apply (simp add: order_divides)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   262
    done
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   263
  then have "order a (pderiv p) \<le> order a d"
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   264
    using \<open>d \<noteq> 0\<close> by (simp add: order_divides)
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   265
  show ?thesis
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   266
    using \<open>order a p = order a q + order a d\<close>
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   267
    using \<open>order a (pderiv p) = order a e + order a d\<close>
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   268
    using \<open>order a p = Suc (order a (pderiv p))\<close>
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   269
    using \<open>order a (pderiv p) \<le> order a d\<close>
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   270
    by auto
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   271
qed
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   272
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
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   273
lemma poly_squarefree_decomp_order2: "[| pderiv p \<noteq> 0;
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
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   274
         p = q * d;
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   275
         pderiv p = e * d;
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   276
         d = r * p + s * pderiv p
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   277
      |] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
56383
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   278
by (blast intro: poly_squarefree_decomp_order)
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   279
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   280
lemma order_pderiv2: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   281
      ==> (order a (pderiv p) = n) = (order a p = Suc n)"
56383
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   282
by (auto dest: order_pderiv)
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   283
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   284
definition
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   285
  rsquarefree :: "'a::idom poly => bool" where
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   286
  "rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   287
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   288
lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h:]"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   289
apply (simp add: pderiv_eq_0_iff)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   290
apply (case_tac p, auto split: if_splits)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   291
done
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   292
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   293
lemma rsquarefree_roots:
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   294
  "rsquarefree p = (\<forall>a. ~(poly p a = 0 & poly (pderiv p) a = 0))"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   295
apply (simp add: rsquarefree_def)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   296
apply (case_tac "p = 0", simp, simp)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   297
apply (case_tac "pderiv p = 0")
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   298
apply simp
56383
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   299
apply (drule pderiv_iszero, clarsimp)
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   300
apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree)
8e7052e9fda4 Cleaned up some messy proofs
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   301
apply (force simp add: order_root order_pderiv2)
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   302
done
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   303
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   304
lemma poly_squarefree_decomp:
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   305
  assumes "pderiv p \<noteq> 0"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   306
    and "p = q * d"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   307
    and "pderiv p = e * d"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   308
    and "d = r * p + s * pderiv p"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   309
  shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   310
proof -
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   311
  from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   312
  with \<open>p = q * d\<close> have "q \<noteq> 0" by simp
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   313
  have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   314
    using assms by (rule poly_squarefree_decomp_order2)
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 58881
diff changeset
   315
  with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> show ?thesis
29985
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   316
    by (simp add: rsquarefree_def order_root)
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   317
qed
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   318
57975b45ab70 split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff changeset
   319
end