(* Title: HOL/Library/Poly_Deriv.thy
Author: Amine Chaieb
Author: Brian Huffman
*)
section\<open>Polynomials and Differentiation\<close>
theory Poly_Deriv
imports Deriv Polynomial
begin
subsection \<open>Derivatives of univariate polynomials\<close>
function pderiv :: "'a::real_normed_field poly \<Rightarrow> 'a poly"
where
[simp del]: "pderiv (pCons a p) = (if p = 0 then 0 else p + pCons 0 (pderiv p))"
by (auto intro: pCons_cases)
termination pderiv
by (relation "measure degree") simp_all
lemma pderiv_0 [simp]:
"pderiv 0 = 0"
using pderiv.simps [of 0 0] by simp
lemma pderiv_pCons:
"pderiv (pCons a p) = p + pCons 0 (pderiv p)"
by (simp add: pderiv.simps)
lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
by (induct p arbitrary: n)
(auto simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
primrec pderiv_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list"
where
"pderiv_coeffs [] = []"
| "pderiv_coeffs (x # xs) = plus_coeffs xs (cCons 0 (pderiv_coeffs xs))"
lemma coeffs_pderiv [code abstract]:
"coeffs (pderiv p) = pderiv_coeffs (coeffs p)"
by (rule sym, induct p) (simp_all add: pderiv_pCons coeffs_plus_eq_plus_coeffs cCons_def)
lemma pderiv_eq_0_iff: "pderiv p = 0 \<longleftrightarrow> degree p = 0"
apply (rule iffI)
apply (cases p, simp)
apply (simp add: poly_eq_iff coeff_pderiv del: of_nat_Suc)
apply (simp add: poly_eq_iff coeff_pderiv coeff_eq_0)
done
lemma degree_pderiv: "degree (pderiv p) = degree p - 1"
apply (rule order_antisym [OF degree_le])
apply (simp add: coeff_pderiv coeff_eq_0)
apply (cases "degree p", simp)
apply (rule le_degree)
apply (simp add: coeff_pderiv del: of_nat_Suc)
apply (metis degree_0 leading_coeff_0_iff nat.distinct(1))
done
lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"
by (simp add: pderiv_pCons)
lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
lemma pderiv_minus: "pderiv (- p) = - pderiv p"
by (rule poly_eqI, simp add: coeff_pderiv)
lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
by (rule poly_eqI, simp add: coeff_pderiv algebra_simps)
lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
by (induct p) (auto simp: pderiv_add pderiv_smult pderiv_pCons algebra_simps)
lemma pderiv_power_Suc:
"pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
apply (induct n)
apply simp
apply (subst power_Suc)
apply (subst pderiv_mult)
apply (erule ssubst)
apply (simp only: of_nat_Suc smult_add_left smult_1_left)
apply (simp add: algebra_simps)
done
lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
by (rule DERIV_cong, rule DERIV_pow, simp)
declare DERIV_pow2 [simp] DERIV_pow [simp]
lemma DERIV_add_const: "DERIV f x :> D ==> DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
by (rule DERIV_cong, rule DERIV_add, auto)
lemma poly_DERIV[simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
by (induct p, auto intro!: derivative_eq_intros simp add: pderiv_pCons)
lemma continuous_on_poly [continuous_intros]:
fixes p :: "'a :: {real_normed_field} poly"
assumes "continuous_on A f"
shows "continuous_on A (\<lambda>x. poly p (f x))"
proof -
have "continuous_on A (\<lambda>x. (\<Sum>i\<le>degree p. (f x) ^ i * coeff p i))"
by (intro continuous_intros assms)
also have "\<dots> = (\<lambda>x. poly p (f x))" by (intro ext) (simp add: poly_altdef mult_ac)
finally show ?thesis .
qed
text\<open>Consequences of the derivative theorem above\<close>
lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (at x::real filter)"
apply (simp add: real_differentiable_def)
apply (blast intro: poly_DERIV)
done
lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
by (rule poly_DERIV [THEN DERIV_isCont])
lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
==> \<exists>x. a < x & x < b & (poly p x = 0)"
using IVT_objl [of "poly p" a 0 b]
by (auto simp add: order_le_less)
lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
==> \<exists>x. a < x & x < b & (poly p x = 0)"
by (insert poly_IVT_pos [where p = "- p" ]) simp
lemma poly_IVT:
fixes p::"real poly"
assumes "a<b" and "poly p a * poly p b < 0"
shows "\<exists>x>a. x < b \<and> poly p x = 0"
by (metis assms(1) assms(2) less_not_sym mult_less_0_iff poly_IVT_neg poly_IVT_pos)
lemma poly_MVT: "(a::real) < b ==>
\<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
using MVT [of a b "poly p"]
apply auto
apply (rule_tac x = z in exI)
apply (auto simp add: mult_left_cancel poly_DERIV [THEN DERIV_unique])
done
lemma poly_MVT':
assumes "{min a b..max a b} \<subseteq> A"
shows "\<exists>x\<in>A. poly p b - poly p a = (b - a) * poly (pderiv p) (x::real)"
proof (cases a b rule: linorder_cases)
case less
from poly_MVT[OF less, of p] guess x by (elim exE conjE)
thus ?thesis by (intro bexI[of _ x]) (auto intro!: subsetD[OF assms])
next
case greater
from poly_MVT[OF greater, of p] guess x by (elim exE conjE)
thus ?thesis by (intro bexI[of _ x]) (auto simp: algebra_simps intro!: subsetD[OF assms])
qed (insert assms, auto)
lemma poly_pinfty_gt_lc:
fixes p:: "real poly"
assumes "lead_coeff p > 0"
shows "\<exists> n. \<forall> x \<ge> n. poly p x \<ge> lead_coeff p" using assms
proof (induct p)
case 0
thus ?case by auto
next
case (pCons a p)
have "\<lbrakk>a\<noteq>0;p=0\<rbrakk> \<Longrightarrow> ?case" by auto
moreover have "p\<noteq>0 \<Longrightarrow> ?case"
proof -
assume "p\<noteq>0"
then obtain n1 where gte_lcoeff:"\<forall>x\<ge>n1. lead_coeff p \<le> poly p x" using that pCons by auto
have gt_0:"lead_coeff p >0" using pCons(3) `p\<noteq>0` by auto
def n\<equiv>"max n1 (1+ \<bar>a\<bar>/(lead_coeff p))"
show ?thesis
proof (rule_tac x=n in exI,rule,rule)
fix x assume "n \<le> x"
hence "lead_coeff p \<le> poly p x"
using gte_lcoeff unfolding n_def by auto
hence " \<bar>a\<bar>/(lead_coeff p) \<ge> \<bar>a\<bar>/(poly p x)" and "poly p x>0" using gt_0
by (intro frac_le,auto)
hence "x\<ge>1+ \<bar>a\<bar>/(poly p x)" using `n\<le>x`[unfolded n_def] by auto
thus "lead_coeff (pCons a p) \<le> poly (pCons a p) x"
using `lead_coeff p \<le> poly p x` `poly p x>0` `p\<noteq>0`
by (auto simp add:field_simps)
qed
qed
ultimately show ?case by fastforce
qed
text\<open>Lemmas for Derivatives\<close>
lemma order_unique_lemma:
fixes p :: "'a::idom poly"
assumes "[:-a, 1:] ^ n dvd p" "\<not> [:-a, 1:] ^ Suc n dvd p"
shows "n = order a p"
unfolding Polynomial.order_def
apply (rule Least_equality [symmetric])
apply (fact assms)
apply (rule classical)
apply (erule notE)
unfolding not_less_eq_eq
using assms(1) apply (rule power_le_dvd)
apply assumption
done
lemma lemma_order_pderiv1:
"pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
apply (simp only: pderiv_mult pderiv_power_Suc)
apply (simp del: power_Suc of_nat_Suc add: pderiv_pCons)
done
lemma dvd_add_cancel1:
fixes a b c :: "'a::comm_ring_1"
shows "a dvd b + c \<Longrightarrow> a dvd b \<Longrightarrow> a dvd c"
by (drule (1) Rings.dvd_diff, simp)
lemma lemma_order_pderiv:
assumes n: "0 < n"
and pd: "pderiv p \<noteq> 0"
and pe: "p = [:- a, 1:] ^ n * q"
and nd: "~ [:- a, 1:] dvd q"
shows "n = Suc (order a (pderiv p))"
using n
proof -
have "pderiv ([:- a, 1:] ^ n * q) \<noteq> 0"
using assms by auto
obtain n' where "n = Suc n'" "0 < Suc n'" "pderiv ([:- a, 1:] ^ Suc n' * q) \<noteq> 0"
using assms by (cases n) auto
then have *: "!!k l. k dvd k * pderiv q + smult (of_nat (Suc n')) l \<Longrightarrow> k dvd l"
by (metis dvd_add_cancel1 dvd_smult_iff dvd_triv_left of_nat_eq_0_iff old.nat.distinct(2))
have "n' = order a (pderiv ([:- a, 1:] ^ Suc n' * q))"
proof (rule order_unique_lemma)
show "[:- a, 1:] ^ n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
apply (subst lemma_order_pderiv1)
apply (rule dvd_add)
apply (metis dvdI dvd_mult2 power_Suc2)
apply (metis dvd_smult dvd_triv_right)
done
next
show "\<not> [:- a, 1:] ^ Suc n' dvd pderiv ([:- a, 1:] ^ Suc n' * q)"
apply (subst lemma_order_pderiv1)
by (metis * nd dvd_mult_cancel_right power_not_zero pCons_eq_0_iff power_Suc zero_neq_one)
qed
then show ?thesis
by (metis \<open>n = Suc n'\<close> pe)
qed
lemma order_decomp:
assumes "p \<noteq> 0"
shows "\<exists>q. p = [:- a, 1:] ^ order a p * q \<and> \<not> [:- a, 1:] dvd q"
proof -
from assms have A: "[:- a, 1:] ^ order a p dvd p"
and B: "\<not> [:- a, 1:] ^ Suc (order a p) dvd p" by (auto dest: order)
from A obtain q where C: "p = [:- a, 1:] ^ order a p * q" ..
with B have "\<not> [:- a, 1:] ^ Suc (order a p) dvd [:- a, 1:] ^ order a p * q"
by simp
then have "\<not> [:- a, 1:] ^ order a p * [:- a, 1:] dvd [:- a, 1:] ^ order a p * q"
by simp
then have D: "\<not> [:- a, 1:] dvd q"
using idom_class.dvd_mult_cancel_left [of "[:- a, 1:] ^ order a p" "[:- a, 1:]" q]
by auto
from C D show ?thesis by blast
qed
lemma order_pderiv: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
==> (order a p = Suc (order a (pderiv p)))"
apply (case_tac "p = 0", simp)
apply (drule_tac a = a and p = p in order_decomp)
using neq0_conv
apply (blast intro: lemma_order_pderiv)
done
lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q"
proof -
def i \<equiv> "order a p"
def j \<equiv> "order a q"
def t \<equiv> "[:-a, 1:]"
have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0"
unfolding t_def by (simp add: dvd_iff_poly_eq_0)
assume "p * q \<noteq> 0"
then show "order a (p * q) = i + j"
apply clarsimp
apply (drule order [where a=a and p=p, folded i_def t_def])
apply (drule order [where a=a and p=q, folded j_def t_def])
apply clarify
apply (erule dvdE)+
apply (rule order_unique_lemma [symmetric], fold t_def)
apply (simp_all add: power_add t_dvd_iff)
done
qed
lemma order_smult:
assumes "c \<noteq> 0"
shows "order x (smult c p) = order x p"
proof (cases "p = 0")
case False
have "smult c p = [:c:] * p" by simp
also from assms False have "order x \<dots> = order x [:c:] + order x p"
by (subst order_mult) simp_all
also from assms have "order x [:c:] = 0" by (intro order_0I) auto
finally show ?thesis by simp
qed simp
(* Next two lemmas contributed by Wenda Li *)
lemma order_1_eq_0 [simp]:"order x 1 = 0"
by (metis order_root poly_1 zero_neq_one)
lemma order_power_n_n: "order a ([:-a,1:]^n)=n"
proof (induct n) (*might be proved more concisely using nat_less_induct*)
case 0
thus ?case by (metis order_root poly_1 power_0 zero_neq_one)
next
case (Suc n)
have "order a ([:- a, 1:] ^ Suc n)=order a ([:- a, 1:] ^ n) + order a [:-a,1:]"
by (metis (no_types, hide_lams) One_nat_def add_Suc_right monoid_add_class.add.right_neutral
one_neq_zero order_mult pCons_eq_0_iff power_add power_eq_0_iff power_one_right)
moreover have "order a [:-a,1:]=1" unfolding order_def
proof (rule Least_equality,rule ccontr)
assume "\<not> \<not> [:- a, 1:] ^ Suc 1 dvd [:- a, 1:]"
hence "[:- a, 1:] ^ Suc 1 dvd [:- a, 1:]" by simp
hence "degree ([:- a, 1:] ^ Suc 1) \<le> degree ([:- a, 1:] )"
by (rule dvd_imp_degree_le,auto)
thus False by auto
next
fix y assume asm:"\<not> [:- a, 1:] ^ Suc y dvd [:- a, 1:]"
show "1 \<le> y"
proof (rule ccontr)
assume "\<not> 1 \<le> y"
hence "y=0" by auto
hence "[:- a, 1:] ^ Suc y dvd [:- a, 1:]" by auto
thus False using asm by auto
qed
qed
ultimately show ?case using Suc by auto
qed
text\<open>Now justify the standard squarefree decomposition, i.e. f / gcd(f,f').\<close>
lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p"
apply (cases "p = 0", auto)
apply (drule order_2 [where a=a and p=p])
apply (metis not_less_eq_eq power_le_dvd)
apply (erule power_le_dvd [OF order_1])
done
lemma poly_squarefree_decomp_order:
assumes "pderiv p \<noteq> 0"
and p: "p = q * d"
and p': "pderiv p = e * d"
and d: "d = r * p + s * pderiv p"
shows "order a q = (if order a p = 0 then 0 else 1)"
proof (rule classical)
assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)"
from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
with p have "order a p = order a q + order a d"
by (simp add: order_mult)
with 1 have "order a p \<noteq> 0" by (auto split: if_splits)
have "order a (pderiv p) = order a e + order a d"
using \<open>pderiv p \<noteq> 0\<close> \<open>pderiv p = e * d\<close> by (simp add: order_mult)
have "order a p = Suc (order a (pderiv p))"
using \<open>pderiv p \<noteq> 0\<close> \<open>order a p \<noteq> 0\<close> by (rule order_pderiv)
have "d \<noteq> 0" using \<open>p \<noteq> 0\<close> \<open>p = q * d\<close> by simp
have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
apply (simp add: d)
apply (rule dvd_add)
apply (rule dvd_mult)
apply (simp add: order_divides \<open>p \<noteq> 0\<close>
\<open>order a p = Suc (order a (pderiv p))\<close>)
apply (rule dvd_mult)
apply (simp add: order_divides)
done
then have "order a (pderiv p) \<le> order a d"
using \<open>d \<noteq> 0\<close> by (simp add: order_divides)
show ?thesis
using \<open>order a p = order a q + order a d\<close>
using \<open>order a (pderiv p) = order a e + order a d\<close>
using \<open>order a p = Suc (order a (pderiv p))\<close>
using \<open>order a (pderiv p) \<le> order a d\<close>
by auto
qed
lemma poly_squarefree_decomp_order2: "[| pderiv p \<noteq> 0;
p = q * d;
pderiv p = e * d;
d = r * p + s * pderiv p
|] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
by (blast intro: poly_squarefree_decomp_order)
lemma order_pderiv2: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
==> (order a (pderiv p) = n) = (order a p = Suc n)"
by (auto dest: order_pderiv)
definition
rsquarefree :: "'a::idom poly => bool" where
"rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))"
lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h:]"
apply (simp add: pderiv_eq_0_iff)
apply (case_tac p, auto split: if_splits)
done
lemma rsquarefree_roots:
"rsquarefree p = (\<forall>a. ~(poly p a = 0 & poly (pderiv p) a = 0))"
apply (simp add: rsquarefree_def)
apply (case_tac "p = 0", simp, simp)
apply (case_tac "pderiv p = 0")
apply simp
apply (drule pderiv_iszero, clarsimp)
apply (metis coeff_0 coeff_pCons_0 degree_pCons_0 le0 le_antisym order_degree)
apply (force simp add: order_root order_pderiv2)
done
lemma poly_squarefree_decomp:
assumes "pderiv p \<noteq> 0"
and "p = q * d"
and "pderiv p = e * d"
and "d = r * p + s * pderiv p"
shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
proof -
from \<open>pderiv p \<noteq> 0\<close> have "p \<noteq> 0" by auto
with \<open>p = q * d\<close> have "q \<noteq> 0" by simp
have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
using assms by (rule poly_squarefree_decomp_order2)
with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> show ?thesis
by (simp add: rsquarefree_def order_root)
qed
end