src/HOL/Library/Poly_Deriv.thy
 author haftmann Fri Aug 27 19:34:23 2010 +0200 (2010-08-27 ago) changeset 38857 97775f3e8722 parent 35050 9f841f20dca6 child 41959 b460124855b8 permissions -rw-r--r--
renamed class/constant eq to equal; tuned some instantiations
1 (*  Title:      Poly_Deriv.thy
2     Author:     Amine Chaieb
3                 Ported to new Polynomial library by 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 definition
15   pderiv :: "'a::real_normed_field poly \<Rightarrow> 'a poly" where
16   "pderiv = poly_rec 0 (\<lambda>a p p'. p + pCons 0 p')"
18 lemma pderiv_0 [simp]: "pderiv 0 = 0"
19   unfolding pderiv_def by (simp add: poly_rec_0)
21 lemma pderiv_pCons: "pderiv (pCons a p) = p + pCons 0 (pderiv p)"
22   unfolding pderiv_def by (simp add: poly_rec_pCons)
24 lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
25   apply (induct p arbitrary: n, simp)
26   apply (simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
27   done
29 lemma pderiv_eq_0_iff: "pderiv p = 0 \<longleftrightarrow> degree p = 0"
30   apply (rule iffI)
31   apply (cases p, simp)
32   apply (simp add: expand_poly_eq coeff_pderiv del: of_nat_Suc)
33   apply (simp add: expand_poly_eq coeff_pderiv coeff_eq_0)
34   done
36 lemma degree_pderiv: "degree (pderiv p) = degree p - 1"
37   apply (rule order_antisym [OF degree_le])
38   apply (simp add: coeff_pderiv coeff_eq_0)
39   apply (cases "degree p", simp)
40   apply (rule le_degree)
41   apply (simp add: coeff_pderiv del: of_nat_Suc)
42   apply (rule subst, assumption)
44   done
46 lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"
49 lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
50 by (rule poly_ext, simp add: coeff_pderiv algebra_simps)
52 lemma pderiv_minus: "pderiv (- p) = - pderiv p"
53 by (rule poly_ext, simp add: coeff_pderiv)
55 lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
56 by (rule poly_ext, simp add: coeff_pderiv algebra_simps)
58 lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
59 by (rule poly_ext, simp add: coeff_pderiv algebra_simps)
61 lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
62 apply (induct p)
63 apply simp
65 done
67 lemma pderiv_power_Suc:
68   "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
69 apply (induct n)
70 apply simp
71 apply (subst power_Suc)
72 apply (subst pderiv_mult)
73 apply (erule ssubst)
75 done
77 lemma DERIV_cmult2: "DERIV f x :> D ==> DERIV (%x. (f x) * c :: real) x :> D * c"
78 by (simp add: DERIV_cmult mult_commute [of _ c])
80 lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
81 by (rule lemma_DERIV_subst, rule DERIV_pow, simp)
82 declare DERIV_pow2 [simp] DERIV_pow [simp]
84 lemma DERIV_add_const: "DERIV f x :> D ==>  DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
85 by (rule lemma_DERIV_subst, rule DERIV_add, auto)
87 lemma poly_DERIV[simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
88   by (induct p, auto intro!: DERIV_intros simp add: pderiv_pCons)
90 text{* Consequences of the derivative theorem above*}
92 lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (x::real)"
94 apply (blast intro: poly_DERIV)
95 done
97 lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
98 by (rule poly_DERIV [THEN DERIV_isCont])
100 lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
101       ==> \<exists>x. a < x & x < b & (poly p x = 0)"
102 apply (cut_tac f = "%x. poly p x" and a = a and b = b and y = 0 in IVT_objl)
103 apply (auto simp add: order_le_less)
104 done
106 lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
107       ==> \<exists>x. a < x & x < b & (poly p x = 0)"
108 by (insert poly_IVT_pos [where p = "- p" ]) simp
110 lemma poly_MVT: "(a::real) < b ==>
111      \<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
112 apply (drule_tac f = "poly p" in MVT, auto)
113 apply (rule_tac x = z in exI)
114 apply (auto simp add: real_mult_left_cancel poly_DERIV [THEN DERIV_unique])
115 done
117 text{*Lemmas for Derivatives*}
119 lemma order_unique_lemma:
120   fixes p :: "'a::idom poly"
121   assumes "[:-a, 1:] ^ n dvd p \<and> \<not> [:-a, 1:] ^ Suc n dvd p"
122   shows "n = order a p"
123 unfolding Polynomial.order_def
124 apply (rule Least_equality [symmetric])
125 apply (rule assms [THEN conjunct2])
126 apply (erule contrapos_np)
127 apply (rule power_le_dvd)
128 apply (rule assms [THEN conjunct1])
129 apply simp
130 done
132 lemma lemma_order_pderiv1:
133   "pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
134     smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
135 apply (simp only: pderiv_mult pderiv_power_Suc)
136 apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons)
137 done
140   fixes a b c :: "'a::comm_ring_1"
141   shows "a dvd b + c \<Longrightarrow> a dvd b \<Longrightarrow> a dvd c"
142   by (drule (1) Rings.dvd_diff, simp)
144 lemma lemma_order_pderiv [rule_format]:
145      "\<forall>p q a. 0 < n &
146        pderiv p \<noteq> 0 &
147        p = [:- a, 1:] ^ n * q & ~ [:- a, 1:] dvd q
148        --> n = Suc (order a (pderiv p))"
149  apply (cases "n", safe, rename_tac n p q a)
150  apply (rule order_unique_lemma)
151  apply (rule conjI)
152   apply (subst lemma_order_pderiv1)
154    apply (rule dvd_mult2)
155    apply (rule le_imp_power_dvd, simp)
156   apply (rule dvd_smult)
157   apply (rule dvd_mult)
158   apply (rule dvd_refl)
159  apply (subst lemma_order_pderiv1)
160  apply (erule contrapos_nn) back
161  apply (subgoal_tac "[:- a, 1:] ^ Suc n dvd q * [:- a, 1:] ^ n")
162   apply (simp del: mult_pCons_left)
164   apply (simp del: mult_pCons_left)
165  apply (drule dvd_smult_cancel, simp del: of_nat_Suc)
166  apply assumption
167 done
169 lemma order_decomp:
170      "p \<noteq> 0
171       ==> \<exists>q. p = [:-a, 1:] ^ (order a p) * q &
172                 ~([:-a, 1:] dvd q)"
173 apply (drule order [where a=a])
174 apply (erule conjE)
175 apply (erule dvdE)
176 apply (rule exI)
177 apply (rule conjI, assumption)
178 apply (erule contrapos_nn)
179 apply (erule ssubst) back
180 apply (subst power_Suc2)
181 apply (erule mult_dvd_mono [OF dvd_refl])
182 done
184 lemma order_pderiv: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
185       ==> (order a p = Suc (order a (pderiv p)))"
186 apply (case_tac "p = 0", simp)
187 apply (drule_tac a = a and p = p in order_decomp)
188 using neq0_conv
189 apply (blast intro: lemma_order_pderiv)
190 done
192 lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q"
193 proof -
194   def i \<equiv> "order a p"
195   def j \<equiv> "order a q"
196   def t \<equiv> "[:-a, 1:]"
197   have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0"
198     unfolding t_def by (simp add: dvd_iff_poly_eq_0)
199   assume "p * q \<noteq> 0"
200   then show "order a (p * q) = i + j"
201     apply clarsimp
202     apply (drule order [where a=a and p=p, folded i_def t_def])
203     apply (drule order [where a=a and p=q, folded j_def t_def])
204     apply clarify
205     apply (rule order_unique_lemma [symmetric], fold t_def)
206     apply (erule dvdE)+
208     done
209 qed
211 text{*Now justify the standard squarefree decomposition, i.e. f / gcd(f,f'). *}
213 lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p"
214 apply (cases "p = 0", auto)
215 apply (drule order_2 [where a=a and p=p])
216 apply (erule contrapos_np)
217 apply (erule power_le_dvd)
218 apply simp
219 apply (erule power_le_dvd [OF order_1])
220 done
222 lemma poly_squarefree_decomp_order:
223   assumes "pderiv p \<noteq> 0"
224   and p: "p = q * d"
225   and p': "pderiv p = e * d"
226   and d: "d = r * p + s * pderiv p"
227   shows "order a q = (if order a p = 0 then 0 else 1)"
228 proof (rule classical)
229   assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)"
230   from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto
231   with p have "order a p = order a q + order a d"
233   with 1 have "order a p \<noteq> 0" by (auto split: if_splits)
234   have "order a (pderiv p) = order a e + order a d"
235     using `pderiv p \<noteq> 0` `pderiv p = e * d` by (simp add: order_mult)
236   have "order a p = Suc (order a (pderiv p))"
237     using `pderiv p \<noteq> 0` `order a p \<noteq> 0` by (rule order_pderiv)
238   have "d \<noteq> 0" using `p \<noteq> 0` `p = q * d` by simp
239   have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
242     apply (rule dvd_mult)
243     apply (simp add: order_divides `p \<noteq> 0`
244            `order a p = Suc (order a (pderiv p))`)
245     apply (rule dvd_mult)
247     done
248   then have "order a (pderiv p) \<le> order a d"
249     using `d \<noteq> 0` by (simp add: order_divides)
250   show ?thesis
251     using `order a p = order a q + order a d`
252     using `order a (pderiv p) = order a e + order a d`
253     using `order a p = Suc (order a (pderiv p))`
254     using `order a (pderiv p) \<le> order a d`
255     by auto
256 qed
258 lemma poly_squarefree_decomp_order2: "[| pderiv p \<noteq> 0;
259          p = q * d;
260          pderiv p = e * d;
261          d = r * p + s * pderiv p
262       |] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
263 apply (blast intro: poly_squarefree_decomp_order)
264 done
266 lemma order_pderiv2: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
267       ==> (order a (pderiv p) = n) = (order a p = Suc n)"
268 apply (auto dest: order_pderiv)
269 done
271 definition
272   rsquarefree :: "'a::idom poly => bool" where
273   "rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))"
275 lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h:]"
277 apply (case_tac p, auto split: if_splits)
278 done
280 lemma rsquarefree_roots:
281   "rsquarefree p = (\<forall>a. ~(poly p a = 0 & poly (pderiv p) a = 0))"
283 apply (case_tac "p = 0", simp, simp)
284 apply (case_tac "pderiv p = 0")
285 apply simp
286 apply (drule pderiv_iszero, clarify)
287 apply simp
288 apply (rule allI)
289 apply (cut_tac p = "[:h:]" and a = a in order_root)
290 apply simp
291 apply (auto simp add: order_root order_pderiv2)
292 apply (erule_tac x="a" in allE, simp)
293 done
295 lemma poly_squarefree_decomp:
296   assumes "pderiv p \<noteq> 0"
297     and "p = q * d"
298     and "pderiv p = e * d"
299     and "d = r * p + s * pderiv p"
300   shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
301 proof -
302   from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto
303   with `p = q * d` have "q \<noteq> 0" by simp
304   have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
305     using assms by (rule poly_squarefree_decomp_order2)
306   with `p \<noteq> 0` `q \<noteq> 0` show ?thesis
307     by (simp add: rsquarefree_def order_root)
308 qed
310 end