1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Library/Poly_Deriv.thy Wed Feb 18 19:32:26 2009 -0800
1.3 @@ -0,0 +1,316 @@
1.4 +(* Title: Poly_Deriv.thy
1.5 + Author: Amine Chaieb
1.6 + Ported to new Polynomial library by Brian Huffman
1.7 +*)
1.8 +
1.9 +header{* Polynomials and Differentiation *}
1.10 +
1.11 +theory Poly_Deriv
1.12 +imports Deriv Polynomial
1.13 +begin
1.14 +
1.15 +subsection {* Derivatives of univariate polynomials *}
1.16 +
1.17 +definition
1.18 + pderiv :: "'a::real_normed_field poly \<Rightarrow> 'a poly" where
1.19 + "pderiv = poly_rec 0 (\<lambda>a p p'. p + pCons 0 p')"
1.20 +
1.21 +lemma pderiv_0 [simp]: "pderiv 0 = 0"
1.22 + unfolding pderiv_def by (simp add: poly_rec_0)
1.23 +
1.24 +lemma pderiv_pCons: "pderiv (pCons a p) = p + pCons 0 (pderiv p)"
1.25 + unfolding pderiv_def by (simp add: poly_rec_pCons)
1.26 +
1.27 +lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
1.28 + apply (induct p arbitrary: n, simp)
1.29 + apply (simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
1.30 + done
1.31 +
1.32 +lemma pderiv_eq_0_iff: "pderiv p = 0 \<longleftrightarrow> degree p = 0"
1.33 + apply (rule iffI)
1.34 + apply (cases p, simp)
1.35 + apply (simp add: expand_poly_eq coeff_pderiv del: of_nat_Suc)
1.36 + apply (simp add: expand_poly_eq coeff_pderiv coeff_eq_0)
1.37 + done
1.38 +
1.39 +lemma degree_pderiv: "degree (pderiv p) = degree p - 1"
1.40 + apply (rule order_antisym [OF degree_le])
1.41 + apply (simp add: coeff_pderiv coeff_eq_0)
1.42 + apply (cases "degree p", simp)
1.43 + apply (rule le_degree)
1.44 + apply (simp add: coeff_pderiv del: of_nat_Suc)
1.45 + apply (rule subst, assumption)
1.46 + apply (rule leading_coeff_neq_0, clarsimp)
1.47 + done
1.48 +
1.49 +lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"
1.50 +by (simp add: pderiv_pCons)
1.51 +
1.52 +lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
1.53 +by (rule poly_ext, simp add: coeff_pderiv algebra_simps)
1.54 +
1.55 +lemma pderiv_minus: "pderiv (- p) = - pderiv p"
1.56 +by (rule poly_ext, simp add: coeff_pderiv)
1.57 +
1.58 +lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
1.59 +by (rule poly_ext, simp add: coeff_pderiv algebra_simps)
1.60 +
1.61 +lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
1.62 +by (rule poly_ext, simp add: coeff_pderiv algebra_simps)
1.63 +
1.64 +lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
1.65 +apply (induct p)
1.66 +apply simp
1.67 +apply (simp add: pderiv_add pderiv_smult pderiv_pCons algebra_simps)
1.68 +done
1.69 +
1.70 +lemma pderiv_power_Suc:
1.71 + "pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
1.72 +apply (induct n)
1.73 +apply simp
1.74 +apply (subst power_Suc)
1.75 +apply (subst pderiv_mult)
1.76 +apply (erule ssubst)
1.77 +apply (simp add: smult_add_left algebra_simps)
1.78 +done
1.79 +
1.80 +lemma DERIV_cmult2: "DERIV f x :> D ==> DERIV (%x. (f x) * c :: real) x :> D * c"
1.81 +by (simp add: DERIV_cmult mult_commute [of _ c])
1.82 +
1.83 +lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
1.84 +by (rule lemma_DERIV_subst, rule DERIV_pow, simp)
1.85 +declare DERIV_pow2 [simp] DERIV_pow [simp]
1.86 +
1.87 +lemma DERIV_add_const: "DERIV f x :> D ==> DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
1.88 +by (rule lemma_DERIV_subst, rule DERIV_add, auto)
1.89 +
1.90 +lemma poly_DERIV[simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
1.91 +apply (induct p)
1.92 +apply simp
1.93 +apply (simp add: pderiv_pCons)
1.94 +apply (rule lemma_DERIV_subst)
1.95 +apply (rule DERIV_add DERIV_mult DERIV_const DERIV_ident | assumption)+
1.96 +apply simp
1.97 +done
1.98 +
1.99 +text{* Consequences of the derivative theorem above*}
1.100 +
1.101 +lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (x::real)"
1.102 +apply (simp add: differentiable_def)
1.103 +apply (blast intro: poly_DERIV)
1.104 +done
1.105 +
1.106 +lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
1.107 +by (rule poly_DERIV [THEN DERIV_isCont])
1.108 +
1.109 +lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
1.110 + ==> \<exists>x. a < x & x < b & (poly p x = 0)"
1.111 +apply (cut_tac f = "%x. poly p x" and a = a and b = b and y = 0 in IVT_objl)
1.112 +apply (auto simp add: order_le_less)
1.113 +done
1.114 +
1.115 +lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
1.116 + ==> \<exists>x. a < x & x < b & (poly p x = 0)"
1.117 +by (insert poly_IVT_pos [where p = "- p" ]) simp
1.118 +
1.119 +lemma poly_MVT: "(a::real) < b ==>
1.120 + \<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
1.121 +apply (drule_tac f = "poly p" in MVT, auto)
1.122 +apply (rule_tac x = z in exI)
1.123 +apply (auto simp add: real_mult_left_cancel poly_DERIV [THEN DERIV_unique])
1.124 +done
1.125 +
1.126 +text{*Lemmas for Derivatives*}
1.127 +
1.128 +lemma order_unique_lemma:
1.129 + fixes p :: "'a::idom poly"
1.130 + assumes "[:-a, 1:] ^ n dvd p \<and> \<not> [:-a, 1:] ^ Suc n dvd p"
1.131 + shows "n = order a p"
1.132 +unfolding Polynomial.order_def
1.133 +apply (rule Least_equality [symmetric])
1.134 +apply (rule assms [THEN conjunct2])
1.135 +apply (erule contrapos_np)
1.136 +apply (rule power_le_dvd)
1.137 +apply (rule assms [THEN conjunct1])
1.138 +apply simp
1.139 +done
1.140 +
1.141 +lemma lemma_order_pderiv1:
1.142 + "pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
1.143 + smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
1.144 +apply (simp only: pderiv_mult pderiv_power_Suc)
1.145 +apply (simp del: power_poly_Suc of_nat_Suc add: pderiv_pCons)
1.146 +done
1.147 +
1.148 +lemma dvd_add_cancel1:
1.149 + fixes a b c :: "'a::comm_ring_1"
1.150 + shows "a dvd b + c \<Longrightarrow> a dvd b \<Longrightarrow> a dvd c"
1.151 + by (drule (1) Ring_and_Field.dvd_diff, simp)
1.152 +
1.153 +lemma lemma_order_pderiv [rule_format]:
1.154 + "\<forall>p q a. 0 < n &
1.155 + pderiv p \<noteq> 0 &
1.156 + p = [:- a, 1:] ^ n * q & ~ [:- a, 1:] dvd q
1.157 + --> n = Suc (order a (pderiv p))"
1.158 + apply (cases "n", safe, rename_tac n p q a)
1.159 + apply (rule order_unique_lemma)
1.160 + apply (rule conjI)
1.161 + apply (subst lemma_order_pderiv1)
1.162 + apply (rule dvd_add)
1.163 + apply (rule dvd_mult2)
1.164 + apply (rule le_imp_power_dvd, simp)
1.165 + apply (rule dvd_smult)
1.166 + apply (rule dvd_mult)
1.167 + apply (rule dvd_refl)
1.168 + apply (subst lemma_order_pderiv1)
1.169 + apply (erule contrapos_nn) back
1.170 + apply (subgoal_tac "[:- a, 1:] ^ Suc n dvd q * [:- a, 1:] ^ n")
1.171 + apply (simp del: mult_pCons_left)
1.172 + apply (drule dvd_add_cancel1)
1.173 + apply (simp del: mult_pCons_left)
1.174 + apply (drule dvd_smult_cancel, simp del: of_nat_Suc)
1.175 + apply assumption
1.176 +done
1.177 +
1.178 +lemma order_decomp:
1.179 + "p \<noteq> 0
1.180 + ==> \<exists>q. p = [:-a, 1:] ^ (order a p) * q &
1.181 + ~([:-a, 1:] dvd q)"
1.182 +apply (drule order [where a=a])
1.183 +apply (erule conjE)
1.184 +apply (erule dvdE)
1.185 +apply (rule exI)
1.186 +apply (rule conjI, assumption)
1.187 +apply (erule contrapos_nn)
1.188 +apply (erule ssubst) back
1.189 +apply (subst power_Suc2)
1.190 +apply (erule mult_dvd_mono [OF dvd_refl])
1.191 +done
1.192 +
1.193 +lemma order_pderiv: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
1.194 + ==> (order a p = Suc (order a (pderiv p)))"
1.195 +apply (case_tac "p = 0", simp)
1.196 +apply (drule_tac a = a and p = p in order_decomp)
1.197 +using neq0_conv
1.198 +apply (blast intro: lemma_order_pderiv)
1.199 +done
1.200 +
1.201 +lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q"
1.202 +proof -
1.203 + def i \<equiv> "order a p"
1.204 + def j \<equiv> "order a q"
1.205 + def t \<equiv> "[:-a, 1:]"
1.206 + have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0"
1.207 + unfolding t_def by (simp add: dvd_iff_poly_eq_0)
1.208 + assume "p * q \<noteq> 0"
1.209 + then show "order a (p * q) = i + j"
1.210 + apply clarsimp
1.211 + apply (drule order [where a=a and p=p, folded i_def t_def])
1.212 + apply (drule order [where a=a and p=q, folded j_def t_def])
1.213 + apply clarify
1.214 + apply (rule order_unique_lemma [symmetric], fold t_def)
1.215 + apply (erule dvdE)+
1.216 + apply (simp add: power_add t_dvd_iff)
1.217 + done
1.218 +qed
1.219 +
1.220 +text{*Now justify the standard squarefree decomposition, i.e. f / gcd(f,f'). *}
1.221 +
1.222 +lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p"
1.223 +apply (cases "p = 0", auto)
1.224 +apply (drule order_2 [where a=a and p=p])
1.225 +apply (erule contrapos_np)
1.226 +apply (erule power_le_dvd)
1.227 +apply simp
1.228 +apply (erule power_le_dvd [OF order_1])
1.229 +done
1.230 +
1.231 +lemma poly_squarefree_decomp_order:
1.232 + assumes "pderiv p \<noteq> 0"
1.233 + and p: "p = q * d"
1.234 + and p': "pderiv p = e * d"
1.235 + and d: "d = r * p + s * pderiv p"
1.236 + shows "order a q = (if order a p = 0 then 0 else 1)"
1.237 +proof (rule classical)
1.238 + assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)"
1.239 + from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto
1.240 + with p have "order a p = order a q + order a d"
1.241 + by (simp add: order_mult)
1.242 + with 1 have "order a p \<noteq> 0" by (auto split: if_splits)
1.243 + have "order a (pderiv p) = order a e + order a d"
1.244 + using `pderiv p \<noteq> 0` `pderiv p = e * d` by (simp add: order_mult)
1.245 + have "order a p = Suc (order a (pderiv p))"
1.246 + using `pderiv p \<noteq> 0` `order a p \<noteq> 0` by (rule order_pderiv)
1.247 + have "d \<noteq> 0" using `p \<noteq> 0` `p = q * d` by simp
1.248 + have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
1.249 + apply (simp add: d)
1.250 + apply (rule dvd_add)
1.251 + apply (rule dvd_mult)
1.252 + apply (simp add: order_divides `p \<noteq> 0`
1.253 + `order a p = Suc (order a (pderiv p))`)
1.254 + apply (rule dvd_mult)
1.255 + apply (simp add: order_divides)
1.256 + done
1.257 + then have "order a (pderiv p) \<le> order a d"
1.258 + using `d \<noteq> 0` by (simp add: order_divides)
1.259 + show ?thesis
1.260 + using `order a p = order a q + order a d`
1.261 + using `order a (pderiv p) = order a e + order a d`
1.262 + using `order a p = Suc (order a (pderiv p))`
1.263 + using `order a (pderiv p) \<le> order a d`
1.264 + by auto
1.265 +qed
1.266 +
1.267 +lemma poly_squarefree_decomp_order2: "[| pderiv p \<noteq> 0;
1.268 + p = q * d;
1.269 + pderiv p = e * d;
1.270 + d = r * p + s * pderiv p
1.271 + |] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
1.272 +apply (blast intro: poly_squarefree_decomp_order)
1.273 +done
1.274 +
1.275 +lemma order_pderiv2: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
1.276 + ==> (order a (pderiv p) = n) = (order a p = Suc n)"
1.277 +apply (auto dest: order_pderiv)
1.278 +done
1.279 +
1.280 +definition
1.281 + rsquarefree :: "'a::idom poly => bool" where
1.282 + "rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))"
1.283 +
1.284 +lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h:]"
1.285 +apply (simp add: pderiv_eq_0_iff)
1.286 +apply (case_tac p, auto split: if_splits)
1.287 +done
1.288 +
1.289 +lemma rsquarefree_roots:
1.290 + "rsquarefree p = (\<forall>a. ~(poly p a = 0 & poly (pderiv p) a = 0))"
1.291 +apply (simp add: rsquarefree_def)
1.292 +apply (case_tac "p = 0", simp, simp)
1.293 +apply (case_tac "pderiv p = 0")
1.294 +apply simp
1.295 +apply (drule pderiv_iszero, clarify)
1.296 +apply simp
1.297 +apply (rule allI)
1.298 +apply (cut_tac p = "[:h:]" and a = a in order_root)
1.299 +apply simp
1.300 +apply (auto simp add: order_root order_pderiv2)
1.301 +apply (erule_tac x="a" in allE, simp)
1.302 +done
1.303 +
1.304 +lemma poly_squarefree_decomp:
1.305 + assumes "pderiv p \<noteq> 0"
1.306 + and "p = q * d"
1.307 + and "pderiv p = e * d"
1.308 + and "d = r * p + s * pderiv p"
1.309 + shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
1.310 +proof -
1.311 + from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto
1.312 + with `p = q * d` have "q \<noteq> 0" by simp
1.313 + have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
1.314 + using assms by (rule poly_squarefree_decomp_order2)
1.315 + with `p \<noteq> 0` `q \<noteq> 0` show ?thesis
1.316 + by (simp add: rsquarefree_def order_root)
1.317 +qed
1.318 +
1.319 +end