src/HOL/Library/Poly_Deriv.thy
 author blanchet Wed Sep 24 15:45:55 2014 +0200 (2014-09-24) changeset 58425 246985c6b20b parent 58199 5fbe474b5da8 child 58881 b9556a055632 permissions -rw-r--r--
simpler proof
1 (*  Title:      HOL/Library/Poly_Deriv.thy
2     Author:     Amine Chaieb
3     Author:     Brian Huffman
4 *)
6 header{* Polynomials and Differentiation *}
8 theory Poly_Deriv
9 imports Deriv Polynomial
10 begin
12 subsection {* Derivatives of univariate polynomials *}
14 function pderiv :: "'a::real_normed_field poly \<Rightarrow> 'a poly"
15 where
16   [simp del]: "pderiv (pCons a p) = (if p = 0 then 0 else p + pCons 0 (pderiv p))"
17   by (auto intro: pCons_cases)
19 termination pderiv
20   by (relation "measure degree") simp_all
22 lemma pderiv_0 [simp]:
23   "pderiv 0 = 0"
24   using pderiv.simps [of 0 0] by simp
26 lemma pderiv_pCons:
27   "pderiv (pCons a p) = p + pCons 0 (pderiv p)"
30 lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
31   by (induct p arbitrary: n)
32      (auto simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
34 primrec pderiv_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list"
35 where
36   "pderiv_coeffs [] = []"
37 | "pderiv_coeffs (x # xs) = plus_coeffs xs (cCons 0 (pderiv_coeffs xs))"
39 lemma coeffs_pderiv [code abstract]:
40   "coeffs (pderiv p) = pderiv_coeffs (coeffs p)"
41   by (rule sym, induct p) (simp_all add: pderiv_pCons coeffs_plus_eq_plus_coeffs cCons_def)
43 lemma pderiv_eq_0_iff: "pderiv p = 0 \<longleftrightarrow> degree p = 0"
44   apply (rule iffI)
45   apply (cases p, simp)
46   apply (simp add: poly_eq_iff coeff_pderiv del: of_nat_Suc)
47   apply (simp add: poly_eq_iff coeff_pderiv coeff_eq_0)
48   done
50 lemma degree_pderiv: "degree (pderiv p) = degree p - 1"
51   apply (rule order_antisym [OF degree_le])
52   apply (simp add: coeff_pderiv coeff_eq_0)
53   apply (cases "degree p", simp)
54   apply (rule le_degree)
55   apply (simp add: coeff_pderiv del: of_nat_Suc)
56   apply (metis degree_0 leading_coeff_0_iff nat.distinct(1))
57   done
59 lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"
62 lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
63 by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
65 lemma pderiv_minus: "pderiv (- p) = - pderiv p"
66 by (rule poly_eqI, simp add: coeff_pderiv)
68 lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
69 by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
71 lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
72 by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
74 lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
75 by (induct p) (auto simp: pderiv_add pderiv_smult pderiv_pCons algebra_simps)
77 lemma pderiv_power_Suc:
78   "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
79 apply (induct n)
80 apply simp
81 apply (subst power_Suc)
82 apply (subst pderiv_mult)
83 apply (erule ssubst)
84 apply (simp only: of_nat_Suc smult_add_left smult_1_left)
86 done
88 lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
89 by (rule DERIV_cong, rule DERIV_pow, simp)
90 declare DERIV_pow2 [simp] DERIV_pow [simp]
92 lemma DERIV_add_const: "DERIV f x :> D ==>  DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
93 by (rule DERIV_cong, rule DERIV_add, auto)
95 lemma poly_DERIV[simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
96   by (induct p, auto intro!: derivative_eq_intros simp add: pderiv_pCons)
98 text{* Consequences of the derivative theorem above*}
100 lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (at x::real filter)"
102 apply (blast intro: poly_DERIV)
103 done
105 lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
106 by (rule poly_DERIV [THEN DERIV_isCont])
108 lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
109       ==> \<exists>x. a < x & x < b & (poly p x = 0)"
110 using IVT_objl [of "poly p" a 0 b]
111 by (auto simp add: order_le_less)
113 lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
114       ==> \<exists>x. a < x & x < b & (poly p x = 0)"
115 by (insert poly_IVT_pos [where p = "- p" ]) simp
117 lemma poly_MVT: "(a::real) < b ==>
118      \<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
119 using MVT [of a b "poly p"]
120 apply auto
121 apply (rule_tac x = z in exI)
122 apply (auto simp add: mult_left_cancel poly_DERIV [THEN DERIV_unique])
123 done
125 text{*Lemmas for Derivatives*}
127 lemma order_unique_lemma:
128   fixes p :: "'a::idom poly"
129   assumes "[:-a, 1:] ^ n dvd p" "\<not> [:-a, 1:] ^ Suc n dvd p"
130   shows "n = order a p"
131 unfolding Polynomial.order_def
132 apply (rule Least_equality [symmetric])
133 apply (fact assms)
134 apply (rule classical)
135 apply (erule notE)
136 unfolding not_less_eq_eq
137 using assms(1) apply (rule power_le_dvd)
138 apply assumption
139 done
141 lemma lemma_order_pderiv1:
142   "pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
143     smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
144 apply (simp only: pderiv_mult pderiv_power_Suc)
145 apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons)
146 done
149   fixes a b c :: "'a::comm_ring_1"
150   shows "a dvd b + c \<Longrightarrow> a dvd b \<Longrightarrow> a dvd c"
151   by (drule (1) Rings.dvd_diff, simp)
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)
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
184 lemma order_decomp:
185      "p \<noteq> 0
186       ==> \<exists>q. p = [:-a, 1:] ^ (order a p) * q &
187                 ~([:-a, 1:] dvd q)"
188 apply (drule order [where a=a])
189 by (metis dvdE dvd_mult_cancel_left power_Suc2)
191 lemma order_pderiv: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
192       ==> (order a p = Suc (order a (pderiv p)))"
193 apply (case_tac "p = 0", simp)
194 apply (drule_tac a = a and p = p in order_decomp)
195 using neq0_conv
196 apply (blast intro: lemma_order_pderiv)
197 done
199 lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q"
200 proof -
201   def i \<equiv> "order a p"
202   def j \<equiv> "order a q"
203   def t \<equiv> "[:-a, 1:]"
204   have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0"
205     unfolding t_def by (simp add: dvd_iff_poly_eq_0)
206   assume "p * q \<noteq> 0"
207   then show "order a (p * q) = i + j"
208     apply clarsimp
209     apply (drule order [where a=a and p=p, folded i_def t_def])
210     apply (drule order [where a=a and p=q, folded j_def t_def])
211     apply clarify
212     apply (erule dvdE)+
213     apply (rule order_unique_lemma [symmetric], fold t_def)
215     done
216 qed
218 text{*Now justify the standard squarefree decomposition, i.e. f / gcd(f,f'). *}
220 lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p"
221 apply (cases "p = 0", auto)
222 apply (drule order_2 [where a=a and p=p])
223 apply (metis not_less_eq_eq power_le_dvd)
224 apply (erule power_le_dvd [OF order_1])
225 done
227 lemma poly_squarefree_decomp_order:
228   assumes "pderiv p \<noteq> 0"
229   and p: "p = q * d"
230   and p': "pderiv p = e * d"
231   and d: "d = r * p + s * pderiv p"
232   shows "order a q = (if order a p = 0 then 0 else 1)"
233 proof (rule classical)
234   assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)"
235   from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto
236   with p have "order a p = order a q + order a d"
238   with 1 have "order a p \<noteq> 0" by (auto split: if_splits)
239   have "order a (pderiv p) = order a e + order a d"
240     using `pderiv p \<noteq> 0` `pderiv p = e * d` by (simp add: order_mult)
241   have "order a p = Suc (order a (pderiv p))"
242     using `pderiv p \<noteq> 0` `order a p \<noteq> 0` by (rule order_pderiv)
243   have "d \<noteq> 0" using `p \<noteq> 0` `p = q * d` by simp
244   have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
247     apply (rule dvd_mult)
248     apply (simp add: order_divides `p \<noteq> 0`
249            `order a p = Suc (order a (pderiv p))`)
250     apply (rule dvd_mult)
252     done
253   then have "order a (pderiv p) \<le> order a d"
254     using `d \<noteq> 0` by (simp add: order_divides)
255   show ?thesis
256     using `order a p = order a q + order a d`
257     using `order a (pderiv p) = order a e + order a d`
258     using `order a p = Suc (order a (pderiv p))`
259     using `order a (pderiv p) \<le> order a d`
260     by auto
261 qed
263 lemma poly_squarefree_decomp_order2: "[| pderiv p \<noteq> 0;
264          p = q * d;
265          pderiv p = e * d;
266          d = r * p + s * pderiv p
267       |] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
268 by (blast intro: poly_squarefree_decomp_order)
270 lemma order_pderiv2: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
271       ==> (order a (pderiv p) = n) = (order a p = Suc n)"
272 by (auto dest: order_pderiv)
274 definition
275   rsquarefree :: "'a::idom poly => bool" where
276   "rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))"
278 lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h:]"
280 apply (case_tac p, auto split: if_splits)
281 done
283 lemma rsquarefree_roots:
284   "rsquarefree p = (\<forall>a. ~(poly p a = 0 & poly (pderiv p) a = 0))"
286 apply (case_tac "p = 0", simp, simp)
287 apply (case_tac "pderiv p = 0")
288 apply simp
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)
292 done
294 lemma poly_squarefree_decomp:
295   assumes "pderiv p \<noteq> 0"
296     and "p = q * d"
297     and "pderiv p = e * d"
298     and "d = r * p + s * pderiv p"
299   shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
300 proof -
301   from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto
302   with `p = q * d` have "q \<noteq> 0" by simp
303   have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
304     using assms by (rule poly_squarefree_decomp_order2)
305   with `p \<noteq> 0` `q \<noteq> 0` show ?thesis
306     by (simp add: rsquarefree_def order_root)
307 qed
309 end