author | blanchet |
Wed, 24 Sep 2014 15:45:55 +0200 | |
changeset 58425 | 246985c6b20b |
parent 58199 | 5fbe474b5da8 |
child 58881 | b9556a055632 |
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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header{* Polynomials and Differentiation *} |
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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 {* Derivatives of univariate polynomials *} |
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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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||
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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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||
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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{* Consequences of the derivative theorem above*} |
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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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|
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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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|
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text{*Lemmas for Derivatives*} |
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|
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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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using assms(1) apply (rule power_le_dvd) |
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apply assumption |
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done |
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|
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lemma lemma_order_pderiv1: |
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"pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q + |
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smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)" |
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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 | 153 |
lemma lemma_order_pderiv: |
154 |
assumes n: "0 < n" |
|
155 |
and pd: "pderiv p \<noteq> 0" |
|
156 |
and pe: "p = [:- a, 1:] ^ n * q" |
|
157 |
and nd: "~ [:- a, 1:] dvd q" |
|
158 |
shows "n = Suc (order a (pderiv p))" |
|
159 |
using n |
|
160 |
proof - |
|
161 |
have "pderiv ([:- a, 1:] ^ n * q) \<noteq> 0" |
|
162 |
using assms by auto |
|
163 |
obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) \<noteq> 0" |
|
164 |
using assms by (cases n) auto |
|
165 |
then have *: "!!k l. k dvd k * pderiv q + smult (of_nat (Suc n')) l \<Longrightarrow> k dvd l" |
|
166 |
by (metis dvd_add_cancel1 dvd_smult_iff dvd_triv_left of_nat_eq_0_iff old.nat.distinct(2)) |
|
167 |
have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))" |
|
168 |
proof (rule order_unique_lemma) |
|
169 |
show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)" |
|
170 |
apply (subst lemma_order_pderiv1) |
|
171 |
apply (rule dvd_add) |
|
172 |
apply (metis dvdI dvd_mult2 power_Suc2) |
|
173 |
apply (metis dvd_smult dvd_triv_right) |
|
174 |
done |
|
175 |
next |
|
176 |
show "\<not> [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)" |
|
177 |
apply (subst lemma_order_pderiv1) |
|
178 |
by (metis * nd dvd_mult_cancel_right field_power_not_zero pCons_eq_0_iff power_Suc zero_neq_one) |
|
179 |
qed |
|
180 |
then show ?thesis |
|
181 |
by (metis `n = Suc n'` pe) |
|
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: |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
185 |
"p \<noteq> 0 |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
186 |
==> \<exists>q. p = [:-a, 1:] ^ (order a p) * q & |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
187 |
~([:-a, 1:] dvd q)" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
188 |
apply (drule order [where a=a]) |
56383 | 189 |
by (metis dvdE dvd_mult_cancel_left power_Suc2) |
29985
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
190 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
191 |
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
|
192 |
==> (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
|
193 |
apply (case_tac "p = 0", simp) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
194 |
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
|
195 |
using neq0_conv |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
196 |
apply (blast intro: lemma_order_pderiv) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
197 |
done |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
198 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
199 |
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
|
200 |
proof - |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
201 |
def i \<equiv> "order a p" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
202 |
def j \<equiv> "order a q" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
203 |
def t \<equiv> "[:-a, 1:]" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
204 |
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
|
205 |
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
|
206 |
assume "p * q \<noteq> 0" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
207 |
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
|
208 |
apply clarsimp |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
209 |
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
|
210 |
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
|
211 |
apply clarify |
56383 | 212 |
apply (erule dvdE)+ |
29985
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
213 |
apply (rule order_unique_lemma [symmetric], fold t_def) |
56383 | 214 |
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
|
215 |
done |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
216 |
qed |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
217 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
218 |
text{*Now justify the standard squarefree decomposition, i.e. f / gcd(f,f'). *} |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
219 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
220 |
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
|
221 |
apply (cases "p = 0", auto) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
222 |
apply (drule order_2 [where a=a and p=p]) |
56383 | 223 |
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
|
224 |
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
|
225 |
done |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
226 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
227 |
lemma poly_squarefree_decomp_order: |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
228 |
assumes "pderiv p \<noteq> 0" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
229 |
and p: "p = q * d" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
230 |
and p': "pderiv p = e * d" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
231 |
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
|
232 |
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
|
233 |
proof (rule classical) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
234 |
assume 1: "order a q \<noteq> (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
|
235 |
from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
236 |
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
|
237 |
by (simp add: order_mult) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
238 |
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
|
239 |
have "order a (pderiv p) = order a e + order a d" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
240 |
using `pderiv p \<noteq> 0` `pderiv p = e * d` by (simp add: order_mult) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
241 |
have "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
|
242 |
using `pderiv p \<noteq> 0` `order a p \<noteq> 0` by (rule order_pderiv) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
243 |
have "d \<noteq> 0" using `p \<noteq> 0` `p = q * d` by simp |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
244 |
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
|
245 |
apply (simp add: d) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
246 |
apply (rule dvd_add) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
247 |
apply (rule dvd_mult) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
248 |
apply (simp add: order_divides `p \<noteq> 0` |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
249 |
`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
|
250 |
apply (rule dvd_mult) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
251 |
apply (simp add: order_divides) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
252 |
done |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
253 |
then have "order a (pderiv p) \<le> order a d" |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
254 |
using `d \<noteq> 0` by (simp add: order_divides) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
255 |
show ?thesis |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
256 |
using `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
|
257 |
using `order a (pderiv p) = order a e + order a d` |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
258 |
using `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
|
259 |
using `order a (pderiv p) \<le> order a d` |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
260 |
by auto |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
261 |
qed |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
262 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
263 |
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
|
264 |
p = q * d; |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
265 |
pderiv p = e * d; |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
266 |
d = r * p + s * pderiv p |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
267 |
|] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)" |
56383 | 268 |
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
|
269 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
270 |
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
|
271 |
==> (order a (pderiv p) = n) = (order a p = Suc n)" |
56383 | 272 |
by (auto dest: order_pderiv) |
29985
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
273 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
274 |
definition |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
275 |
rsquarefree :: "'a::idom poly => bool" where |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
276 |
"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
|
277 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
278 |
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
|
279 |
apply (simp add: pderiv_eq_0_iff) |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
280 |
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
|
281 |
done |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
282 |
|
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
283 |
lemma rsquarefree_roots: |
57975b45ab70
split polynomial-related stuff from Deriv.thy into Library/Poly_Deriv.thy
huffman
parents:
diff
changeset
|
284 |
"rsquarefree p = (\<forall>a. ~(poly p a = 0 & poly (pderiv p) a = 0))" |
57975b45ab70
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285 |
apply (simp add: rsquarefree_def) |
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parents:
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286 |
apply (case_tac "p = 0", simp, simp) |
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287 |
apply (case_tac "pderiv p = 0") |
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288 |
apply simp |
56383 | 289 |
apply (drule pderiv_iszero, clarsimp) |
290 |
apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree) |
|
291 |
apply (force simp add: order_root order_pderiv2) |
|
29985
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parents:
diff
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292 |
done |
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parents:
diff
changeset
|
293 |
|
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|
294 |
lemma poly_squarefree_decomp: |
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295 |
assumes "pderiv p \<noteq> 0" |
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296 |
and "p = q * d" |
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297 |
and "pderiv p = e * d" |
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298 |
and "d = r * p + s * pderiv p" |
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parents:
diff
changeset
|
299 |
shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))" |
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parents:
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|
300 |
proof - |
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301 |
from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto |
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parents:
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302 |
with `p = q * d` have "q \<noteq> 0" by simp |
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parents:
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changeset
|
303 |
have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)" |
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parents:
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304 |
using assms by (rule poly_squarefree_decomp_order2) |
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parents:
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changeset
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305 |
with `p \<noteq> 0` `q \<noteq> 0` show ?thesis |
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parents:
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306 |
by (simp add: rsquarefree_def order_root) |
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parents:
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|
307 |
qed |
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parents:
diff
changeset
|
308 |
|
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parents:
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changeset
|
309 |
end |