author | eberlm |
Tue, 05 Jan 2016 17:54:10 +0100 | |
changeset 62065 | 1546a042e87b |
parent 60867 | 86e7560e07d0 |
child 62072 | bf3d9f113474 |
permissions | -rw-r--r-- |
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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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|
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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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lemma continuous_on_poly [continuous_intros]: |
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fixes p :: "'a :: {real_normed_field} poly" |
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assumes "continuous_on A f" |
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shows "continuous_on A (\<lambda>x. poly p (f x))" |
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proof - |
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have "continuous_on A (\<lambda>x. (\<Sum>i\<le>degree p. (f x) ^ i * coeff p i))" |
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by (intro continuous_intros assms) |
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also have "\<dots> = (\<lambda>x. poly p (f x))" by (intro ext) (simp add: poly_altdef mult_ac) |
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finally show ?thesis . |
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qed |
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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_IVT: |
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fixes p::"real poly" |
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assumes "a<b" and "poly p a * poly p b < 0" |
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shows "\<exists>x>a. x < b \<and> poly p x = 0" |
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by (metis assms(1) assms(2) less_not_sym mult_less_0_iff poly_IVT_neg poly_IVT_pos) |
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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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lemma poly_MVT': |
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assumes "{min a b..max a b} \<subseteq> A" |
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shows "\<exists>x\<in>A. poly p b - poly p a = (b - a) * poly (pderiv p) (x::real)" |
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proof (cases a b rule: linorder_cases) |
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case less |
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from poly_MVT[OF less, of p] guess x by (elim exE conjE) |
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thus ?thesis by (intro bexI[of _ x]) (auto intro!: subsetD[OF assms]) |
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next |
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case greater |
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from poly_MVT[OF greater, of p] guess x by (elim exE conjE) |
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thus ?thesis by (intro bexI[of _ x]) (auto simp: algebra_simps intro!: subsetD[OF assms]) |
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qed (insert assms, auto) |
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lemma poly_pinfty_gt_lc: |
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fixes p:: "real poly" |
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assumes "lead_coeff p > 0" |
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shows "\<exists> n. \<forall> x \<ge> n. poly p x \<ge> lead_coeff p" using assms |
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proof (induct p) |
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case 0 |
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thus ?case by auto |
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next |
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case (pCons a p) |
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have "\<lbrakk>a\<noteq>0;p=0\<rbrakk> \<Longrightarrow> ?case" by auto |
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moreover have "p\<noteq>0 \<Longrightarrow> ?case" |
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proof - |
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assume "p\<noteq>0" |
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then obtain n1 where gte_lcoeff:"\<forall>x\<ge>n1. lead_coeff p \<le> poly p x" using that pCons by auto |
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have gt_0:"lead_coeff p >0" using pCons(3) `p\<noteq>0` by auto |
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def n\<equiv>"max n1 (1+ \<bar>a\<bar>/(lead_coeff p))" |
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show ?thesis |
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proof (rule_tac x=n in exI,rule,rule) |
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fix x assume "n \<le> x" |
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hence "lead_coeff p \<le> poly p x" |
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using gte_lcoeff unfolding n_def by auto |
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hence " \<bar>a\<bar>/(lead_coeff p) \<ge> \<bar>a\<bar>/(poly p x)" and "poly p x>0" using gt_0 |
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by (intro frac_le,auto) |
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hence "x\<ge>1+ \<bar>a\<bar>/(poly p x)" using `n\<le>x`[unfolded n_def] by auto |
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thus "lead_coeff (pCons a p) \<le> poly (pCons a p) x" |
|
181 |
using `lead_coeff p \<le> poly p x` `poly p x>0` `p\<noteq>0` |
|
182 |
by (auto simp add:field_simps) |
|
183 |
qed |
|
184 |
qed |
|
185 |
ultimately show ?case by fastforce |
|
186 |
qed |
|
187 |
||
188 |
||
60500 | 189 |
text\<open>Lemmas for Derivatives\<close> |
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changeset
|
190 |
|
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changeset
|
191 |
lemma order_unique_lemma: |
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changeset
|
192 |
fixes p :: "'a::idom poly" |
56383 | 193 |
assumes "[:-a, 1:] ^ n dvd p" "\<not> [:-a, 1:] ^ Suc n dvd p" |
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changeset
|
194 |
shows "n = order a p" |
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diff
changeset
|
195 |
unfolding Polynomial.order_def |
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changeset
|
196 |
apply (rule Least_equality [symmetric]) |
58199
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|
197 |
apply (fact assms) |
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diff
changeset
|
198 |
apply (rule classical) |
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changeset
|
199 |
apply (erule notE) |
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changeset
|
200 |
unfolding not_less_eq_eq |
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|
201 |
using assms(1) apply (rule power_le_dvd) |
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changeset
|
202 |
apply assumption |
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changeset
|
203 |
done |
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changeset
|
204 |
|
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changeset
|
205 |
lemma lemma_order_pderiv1: |
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|
206 |
"pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q + |
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parents:
diff
changeset
|
207 |
smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)" |
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parents:
diff
changeset
|
208 |
apply (simp only: pderiv_mult pderiv_power_Suc) |
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changeset
|
209 |
apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons) |
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changeset
|
210 |
done |
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parents:
diff
changeset
|
211 |
|
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parents:
diff
changeset
|
212 |
lemma dvd_add_cancel1: |
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parents:
diff
changeset
|
213 |
fixes a b c :: "'a::comm_ring_1" |
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parents:
diff
changeset
|
214 |
shows "a dvd b + c \<Longrightarrow> a dvd b \<Longrightarrow> a dvd c" |
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31881
diff
changeset
|
215 |
by (drule (1) Rings.dvd_diff, simp) |
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changeset
|
216 |
|
56383 | 217 |
lemma lemma_order_pderiv: |
218 |
assumes n: "0 < n" |
|
219 |
and pd: "pderiv p \<noteq> 0" |
|
220 |
and pe: "p = [:- a, 1:] ^ n * q" |
|
221 |
and nd: "~ [:- a, 1:] dvd q" |
|
222 |
shows "n = Suc (order a (pderiv p))" |
|
223 |
using n |
|
224 |
proof - |
|
225 |
have "pderiv ([:- a, 1:] ^ n * q) \<noteq> 0" |
|
226 |
using assms by auto |
|
227 |
obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) \<noteq> 0" |
|
228 |
using assms by (cases n) auto |
|
229 |
then have *: "!!k l. k dvd k * pderiv q + smult (of_nat (Suc n')) l \<Longrightarrow> k dvd l" |
|
230 |
by (metis dvd_add_cancel1 dvd_smult_iff dvd_triv_left of_nat_eq_0_iff old.nat.distinct(2)) |
|
231 |
have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))" |
|
232 |
proof (rule order_unique_lemma) |
|
233 |
show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)" |
|
234 |
apply (subst lemma_order_pderiv1) |
|
235 |
apply (rule dvd_add) |
|
236 |
apply (metis dvdI dvd_mult2 power_Suc2) |
|
237 |
apply (metis dvd_smult dvd_triv_right) |
|
238 |
done |
|
239 |
next |
|
240 |
show "\<not> [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)" |
|
241 |
apply (subst lemma_order_pderiv1) |
|
60867 | 242 |
by (metis * nd dvd_mult_cancel_right power_not_zero pCons_eq_0_iff power_Suc zero_neq_one) |
56383 | 243 |
qed |
244 |
then show ?thesis |
|
60500 | 245 |
by (metis \<open>n = Suc n'\<close> pe) |
56383 | 246 |
qed |
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parents:
diff
changeset
|
247 |
|
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changeset
|
248 |
lemma order_decomp: |
60688
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|
249 |
assumes "p \<noteq> 0" |
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changeset
|
250 |
shows "\<exists>q. p = [:- a, 1:] ^ order a p * q \<and> \<not> [:- a, 1:] dvd q" |
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|
251 |
proof - |
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|
252 |
from assms have A: "[:- a, 1:] ^ order a p dvd p" |
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|
253 |
and B: "\<not> [:- a, 1:] ^ Suc (order a p) dvd p" by (auto dest: order) |
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changeset
|
254 |
from A obtain q where C: "p = [:- a, 1:] ^ order a p * q" .. |
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changeset
|
255 |
with B have "\<not> [:- a, 1:] ^ Suc (order a p) dvd [:- a, 1:] ^ order a p * q" |
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parents:
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changeset
|
256 |
by simp |
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parents:
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changeset
|
257 |
then have "\<not> [:- a, 1:] ^ order a p * [:- a, 1:] dvd [:- a, 1:] ^ order a p * q" |
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parents:
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changeset
|
258 |
by simp |
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parents:
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changeset
|
259 |
then have D: "\<not> [:- a, 1:] dvd q" |
01488b559910
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parents:
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changeset
|
260 |
using idom_class.dvd_mult_cancel_left [of "[:- a, 1:] ^ order a p" "[:- a, 1:]" q] |
01488b559910
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parents:
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changeset
|
261 |
by auto |
01488b559910
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parents:
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changeset
|
262 |
from C D show ?thesis by blast |
01488b559910
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parents:
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changeset
|
263 |
qed |
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parents:
diff
changeset
|
264 |
|
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parents:
diff
changeset
|
265 |
lemma order_pderiv: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |] |
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parents:
diff
changeset
|
266 |
==> (order a p = Suc (order a (pderiv p)))" |
57975b45ab70
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parents:
diff
changeset
|
267 |
apply (case_tac "p = 0", simp) |
57975b45ab70
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huffman
parents:
diff
changeset
|
268 |
apply (drule_tac a = a and p = p in order_decomp) |
57975b45ab70
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huffman
parents:
diff
changeset
|
269 |
using neq0_conv |
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huffman
parents:
diff
changeset
|
270 |
apply (blast intro: lemma_order_pderiv) |
57975b45ab70
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huffman
parents:
diff
changeset
|
271 |
done |
57975b45ab70
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huffman
parents:
diff
changeset
|
272 |
|
57975b45ab70
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huffman
parents:
diff
changeset
|
273 |
lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q" |
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huffman
parents:
diff
changeset
|
274 |
proof - |
57975b45ab70
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huffman
parents:
diff
changeset
|
275 |
def i \<equiv> "order a p" |
57975b45ab70
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huffman
parents:
diff
changeset
|
276 |
def j \<equiv> "order a q" |
57975b45ab70
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huffman
parents:
diff
changeset
|
277 |
def t \<equiv> "[:-a, 1:]" |
57975b45ab70
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huffman
parents:
diff
changeset
|
278 |
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
|
279 |
unfolding t_def by (simp add: dvd_iff_poly_eq_0) |
57975b45ab70
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huffman
parents:
diff
changeset
|
280 |
assume "p * q \<noteq> 0" |
57975b45ab70
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huffman
parents:
diff
changeset
|
281 |
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
|
282 |
apply clarsimp |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
283 |
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
|
284 |
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
|
285 |
apply clarify |
56383 | 286 |
apply (erule dvdE)+ |
29985
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huffman
parents:
diff
changeset
|
287 |
apply (rule order_unique_lemma [symmetric], fold t_def) |
56383 | 288 |
apply (simp_all add: power_add t_dvd_iff) |
29985
57975b45ab70
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huffman
parents:
diff
changeset
|
289 |
done |
57975b45ab70
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huffman
parents:
diff
changeset
|
290 |
qed |
57975b45ab70
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huffman
parents:
diff
changeset
|
291 |
|
62065 | 292 |
lemma order_smult: |
293 |
assumes "c \<noteq> 0" |
|
294 |
shows "order x (smult c p) = order x p" |
|
295 |
proof (cases "p = 0") |
|
296 |
case False |
|
297 |
have "smult c p = [:c:] * p" by simp |
|
298 |
also from assms False have "order x \<dots> = order x [:c:] + order x p" |
|
299 |
by (subst order_mult) simp_all |
|
300 |
also from assms have "order x [:c:] = 0" by (intro order_0I) auto |
|
301 |
finally show ?thesis by simp |
|
302 |
qed simp |
|
303 |
||
304 |
(* Next two lemmas contributed by Wenda Li *) |
|
305 |
lemma order_1_eq_0 [simp]:"order x 1 = 0" |
|
306 |
by (metis order_root poly_1 zero_neq_one) |
|
307 |
||
308 |
lemma order_power_n_n: "order a ([:-a,1:]^n)=n" |
|
309 |
proof (induct n) (*might be proved more concisely using nat_less_induct*) |
|
310 |
case 0 |
|
311 |
thus ?case by (metis order_root poly_1 power_0 zero_neq_one) |
|
312 |
next |
|
313 |
case (Suc n) |
|
314 |
have "order a ([:- a, 1:] ^ Suc n)=order a ([:- a, 1:] ^ n) + order a [:-a,1:]" |
|
315 |
by (metis (no_types, hide_lams) One_nat_def add_Suc_right monoid_add_class.add.right_neutral |
|
316 |
one_neq_zero order_mult pCons_eq_0_iff power_add power_eq_0_iff power_one_right) |
|
317 |
moreover have "order a [:-a,1:]=1" unfolding order_def |
|
318 |
proof (rule Least_equality,rule ccontr) |
|
319 |
assume "\<not> \<not> [:- a, 1:] ^ Suc 1 dvd [:- a, 1:]" |
|
320 |
hence "[:- a, 1:] ^ Suc 1 dvd [:- a, 1:]" by simp |
|
321 |
hence "degree ([:- a, 1:] ^ Suc 1) \<le> degree ([:- a, 1:] )" |
|
322 |
by (rule dvd_imp_degree_le,auto) |
|
323 |
thus False by auto |
|
324 |
next |
|
325 |
fix y assume asm:"\<not> [:- a, 1:] ^ Suc y dvd [:- a, 1:]" |
|
326 |
show "1 \<le> y" |
|
327 |
proof (rule ccontr) |
|
328 |
assume "\<not> 1 \<le> y" |
|
329 |
hence "y=0" by auto |
|
330 |
hence "[:- a, 1:] ^ Suc y dvd [:- a, 1:]" by auto |
|
331 |
thus False using asm by auto |
|
332 |
qed |
|
333 |
qed |
|
334 |
ultimately show ?case using Suc by auto |
|
335 |
qed |
|
336 |
||
60500 | 337 |
text\<open>Now justify the standard squarefree decomposition, i.e. f / gcd(f,f').\<close> |
29985
57975b45ab70
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huffman
parents:
diff
changeset
|
338 |
|
57975b45ab70
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huffman
parents:
diff
changeset
|
339 |
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
|
340 |
apply (cases "p = 0", auto) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
341 |
apply (drule order_2 [where a=a and p=p]) |
56383 | 342 |
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
|
343 |
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
|
344 |
done |
57975b45ab70
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huffman
parents:
diff
changeset
|
345 |
|
57975b45ab70
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huffman
parents:
diff
changeset
|
346 |
lemma poly_squarefree_decomp_order: |
57975b45ab70
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huffman
parents:
diff
changeset
|
347 |
assumes "pderiv p \<noteq> 0" |
57975b45ab70
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huffman
parents:
diff
changeset
|
348 |
and p: "p = q * d" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
349 |
and p': "pderiv p = e * d" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
350 |
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
|
351 |
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
|
352 |
proof (rule classical) |
57975b45ab70
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huffman
parents:
diff
changeset
|
353 |
assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)" |
60500 | 354 |
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
|
355 |
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
|
356 |
by (simp add: order_mult) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
357 |
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
|
358 |
have "order a (pderiv p) = order a e + order a d" |
60500 | 359 |
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
|
360 |
have "order a p = Suc (order a (pderiv p))" |
60500 | 361 |
using \<open>pderiv p \<noteq> 0\<close> \<open>order a p \<noteq> 0\<close> by (rule order_pderiv) |
362 |
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
|
363 |
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
|
364 |
apply (simp add: d) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
365 |
apply (rule dvd_add) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
366 |
apply (rule dvd_mult) |
60500 | 367 |
apply (simp add: order_divides \<open>p \<noteq> 0\<close> |
368 |
\<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
|
369 |
apply (rule dvd_mult) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
370 |
apply (simp add: order_divides) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
371 |
done |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
372 |
then have "order a (pderiv p) \<le> order a d" |
60500 | 373 |
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
|
374 |
show ?thesis |
60500 | 375 |
using \<open>order a p = order a q + order a d\<close> |
376 |
using \<open>order a (pderiv p) = order a e + order a d\<close> |
|
377 |
using \<open>order a p = Suc (order a (pderiv p))\<close> |
|
378 |
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
|
379 |
by auto |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
380 |
qed |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
381 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
382 |
lemma poly_squarefree_decomp_order2: "[| pderiv p \<noteq> 0; |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
383 |
p = q * d; |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
384 |
pderiv p = e * d; |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
385 |
d = r * p + s * pderiv p |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
386 |
|] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)" |
56383 | 387 |
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
|
388 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
389 |
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
|
390 |
==> (order a (pderiv p) = n) = (order a p = Suc n)" |
56383 | 391 |
by (auto dest: order_pderiv) |
29985
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
392 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
393 |
definition |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
394 |
rsquarefree :: "'a::idom poly => bool" where |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
395 |
"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
|
396 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
397 |
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
|
398 |
apply (simp add: pderiv_eq_0_iff) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
399 |
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
|
400 |
done |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
401 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
402 |
lemma rsquarefree_roots: |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
403 |
"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
|
404 |
apply (simp add: rsquarefree_def) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
405 |
apply (case_tac "p = 0", simp, simp) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
406 |
apply (case_tac "pderiv p = 0") |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
407 |
apply simp |
56383 | 408 |
apply (drule pderiv_iszero, clarsimp) |
409 |
apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree) |
|
410 |
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
|
411 |
done |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
412 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
413 |
lemma poly_squarefree_decomp: |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
414 |
assumes "pderiv p \<noteq> 0" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
415 |
and "p = q * d" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
416 |
and "pderiv p = e * d" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
417 |
and "d = r * p + s * pderiv p" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
418 |
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
|
419 |
proof - |
60500 | 420 |
from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto |
421 |
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
|
422 |
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
|
423 |
using assms by (rule poly_squarefree_decomp_order2) |
60500 | 424 |
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
|
425 |
by (simp add: rsquarefree_def order_root) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
426 |
qed |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
427 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
428 |
end |