(* Title: HOL/Polynomial.thy
Author: Brian Huffman
Based on an earlier development by Clemens Ballarin
*)
header {* Univariate Polynomials *}
theory Polynomial
imports Plain SetInterval
begin
subsection {* Definition of type @{text poly} *}
typedef (Poly) 'a poly = "{f::nat \<Rightarrow> 'a::zero. \<exists>n. \<forall>i>n. f i = 0}"
morphisms coeff Abs_poly
by auto
lemma expand_poly_eq: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
by (simp add: coeff_inject [symmetric] expand_fun_eq)
lemma poly_ext: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
by (simp add: expand_poly_eq)
subsection {* Degree of a polynomial *}
definition
degree :: "'a::zero poly \<Rightarrow> nat" where
"degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"
lemma coeff_eq_0: "degree p < n \<Longrightarrow> coeff p n = 0"
proof -
have "coeff p \<in> Poly"
by (rule coeff)
hence "\<exists>n. \<forall>i>n. coeff p i = 0"
unfolding Poly_def by simp
hence "\<forall>i>degree p. coeff p i = 0"
unfolding degree_def by (rule LeastI_ex)
moreover assume "degree p < n"
ultimately show ?thesis by simp
qed
lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p"
by (erule contrapos_np, rule coeff_eq_0, simp)
lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n"
unfolding degree_def by (erule Least_le)
lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0"
unfolding degree_def by (drule not_less_Least, simp)
subsection {* The zero polynomial *}
instantiation poly :: (zero) zero
begin
definition
zero_poly_def: "0 = Abs_poly (\<lambda>n. 0)"
instance ..
end
lemma coeff_0 [simp]: "coeff 0 n = 0"
unfolding zero_poly_def
by (simp add: Abs_poly_inverse Poly_def)
lemma degree_0 [simp]: "degree 0 = 0"
by (rule order_antisym [OF degree_le le0]) simp
lemma leading_coeff_neq_0:
assumes "p \<noteq> 0" shows "coeff p (degree p) \<noteq> 0"
proof (cases "degree p")
case 0
from `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0"
by (simp add: expand_poly_eq)
then obtain n where "coeff p n \<noteq> 0" ..
hence "n \<le> degree p" by (rule le_degree)
with `coeff p n \<noteq> 0` and `degree p = 0`
show "coeff p (degree p) \<noteq> 0" by simp
next
case (Suc n)
from `degree p = Suc n` have "n < degree p" by simp
hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp)
then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast
from `degree p = Suc n` and `n < i` have "degree p \<le> i" by simp
also from `coeff p i \<noteq> 0` have "i \<le> degree p" by (rule le_degree)
finally have "degree p = i" .
with `coeff p i \<noteq> 0` show "coeff p (degree p) \<noteq> 0" by simp
qed
lemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 \<longleftrightarrow> p = 0"
by (cases "p = 0", simp, simp add: leading_coeff_neq_0)
subsection {* List-style constructor for polynomials *}
definition
pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
where
[code del]: "pCons a p = Abs_poly (nat_case a (coeff p))"
syntax
"_poly" :: "args \<Rightarrow> 'a poly" ("[:(_):]")
translations
"[:x, xs:]" == "CONST pCons x [:xs:]"
"[:x:]" == "CONST pCons x 0"
lemma Poly_nat_case: "f \<in> Poly \<Longrightarrow> nat_case a f \<in> Poly"
unfolding Poly_def by (auto split: nat.split)
lemma coeff_pCons:
"coeff (pCons a p) = nat_case a (coeff p)"
unfolding pCons_def
by (simp add: Abs_poly_inverse Poly_nat_case coeff)
lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a"
by (simp add: coeff_pCons)
lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n"
by (simp add: coeff_pCons)
lemma degree_pCons_le: "degree (pCons a p) \<le> Suc (degree p)"
by (rule degree_le, simp add: coeff_eq_0 coeff_pCons split: nat.split)
lemma degree_pCons_eq:
"p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)"
apply (rule order_antisym [OF degree_pCons_le])
apply (rule le_degree, simp)
done
lemma degree_pCons_0: "degree (pCons a 0) = 0"
apply (rule order_antisym [OF _ le0])
apply (rule degree_le, simp add: coeff_pCons split: nat.split)
done
lemma degree_pCons_eq_if [simp]:
"degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
apply (cases "p = 0", simp_all)
apply (rule order_antisym [OF _ le0])
apply (rule degree_le, simp add: coeff_pCons split: nat.split)
apply (rule order_antisym [OF degree_pCons_le])
apply (rule le_degree, simp)
done
lemma pCons_0_0 [simp]: "pCons 0 0 = 0"
by (rule poly_ext, simp add: coeff_pCons split: nat.split)
lemma pCons_eq_iff [simp]:
"pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q"
proof (safe)
assume "pCons a p = pCons b q"
then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp
then show "a = b" by simp
next
assume "pCons a p = pCons b q"
then have "\<forall>n. coeff (pCons a p) (Suc n) =
coeff (pCons b q) (Suc n)" by simp
then show "p = q" by (simp add: expand_poly_eq)
qed
lemma pCons_eq_0_iff [simp]: "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
using pCons_eq_iff [of a p 0 0] by simp
lemma Poly_Suc: "f \<in> Poly \<Longrightarrow> (\<lambda>n. f (Suc n)) \<in> Poly"
unfolding Poly_def
by (clarify, rule_tac x=n in exI, simp)
lemma pCons_cases [cases type: poly]:
obtains (pCons) a q where "p = pCons a q"
proof
show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))"
by (rule poly_ext)
(simp add: Abs_poly_inverse Poly_Suc coeff coeff_pCons
split: nat.split)
qed
lemma pCons_induct [case_names 0 pCons, induct type: poly]:
assumes zero: "P 0"
assumes pCons: "\<And>a p. P p \<Longrightarrow> P (pCons a p)"
shows "P p"
proof (induct p rule: measure_induct_rule [where f=degree])
case (less p)
obtain a q where "p = pCons a q" by (rule pCons_cases)
have "P q"
proof (cases "q = 0")
case True
then show "P q" by (simp add: zero)
next
case False
then have "degree (pCons a q) = Suc (degree q)"
by (rule degree_pCons_eq)
then have "degree q < degree p"
using `p = pCons a q` by simp
then show "P q"
by (rule less.hyps)
qed
then have "P (pCons a q)"
by (rule pCons)
then show ?case
using `p = pCons a q` by simp
qed
subsection {* Recursion combinator for polynomials *}
function
poly_rec :: "'b \<Rightarrow> ('a::zero \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b"
where
poly_rec_pCons_eq_if [simp del]:
"poly_rec z f (pCons a p) = f a p (if p = 0 then z else poly_rec z f p)"
by (case_tac x, rename_tac q, case_tac q, auto)
termination poly_rec
by (relation "measure (degree \<circ> snd \<circ> snd)", simp)
(simp add: degree_pCons_eq)
lemma poly_rec_0:
"f 0 0 z = z \<Longrightarrow> poly_rec z f 0 = z"
using poly_rec_pCons_eq_if [of z f 0 0] by simp
lemma poly_rec_pCons:
"f 0 0 z = z \<Longrightarrow> poly_rec z f (pCons a p) = f a p (poly_rec z f p)"
by (simp add: poly_rec_pCons_eq_if poly_rec_0)
subsection {* Monomials *}
definition
monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly" where
"monom a m = Abs_poly (\<lambda>n. if m = n then a else 0)"
lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)"
unfolding monom_def
by (subst Abs_poly_inverse, auto simp add: Poly_def)
lemma monom_0: "monom a 0 = pCons a 0"
by (rule poly_ext, simp add: coeff_pCons split: nat.split)
lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)"
by (rule poly_ext, simp add: coeff_pCons split: nat.split)
lemma monom_eq_0 [simp]: "monom 0 n = 0"
by (rule poly_ext) simp
lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
by (simp add: expand_poly_eq)
lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
by (simp add: expand_poly_eq)
lemma degree_monom_le: "degree (monom a n) \<le> n"
by (rule degree_le, simp)
lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
apply (rule order_antisym [OF degree_monom_le])
apply (rule le_degree, simp)
done
subsection {* Addition and subtraction *}
instantiation poly :: (comm_monoid_add) comm_monoid_add
begin
definition
plus_poly_def [code del]:
"p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)"
lemma Poly_add:
fixes f g :: "nat \<Rightarrow> 'a"
shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n + g n) \<in> Poly"
unfolding Poly_def
apply (clarify, rename_tac m n)
apply (rule_tac x="max m n" in exI, simp)
done
lemma coeff_add [simp]:
"coeff (p + q) n = coeff p n + coeff q n"
unfolding plus_poly_def
by (simp add: Abs_poly_inverse coeff Poly_add)
instance proof
fix p q r :: "'a poly"
show "(p + q) + r = p + (q + r)"
by (simp add: expand_poly_eq add_assoc)
show "p + q = q + p"
by (simp add: expand_poly_eq add_commute)
show "0 + p = p"
by (simp add: expand_poly_eq)
qed
end
instantiation poly :: (ab_group_add) ab_group_add
begin
definition
uminus_poly_def [code del]:
"- p = Abs_poly (\<lambda>n. - coeff p n)"
definition
minus_poly_def [code del]:
"p - q = Abs_poly (\<lambda>n. coeff p n - coeff q n)"
lemma Poly_minus:
fixes f :: "nat \<Rightarrow> 'a"
shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. - f n) \<in> Poly"
unfolding Poly_def by simp
lemma Poly_diff:
fixes f g :: "nat \<Rightarrow> 'a"
shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n - g n) \<in> Poly"
unfolding diff_minus by (simp add: Poly_add Poly_minus)
lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
unfolding uminus_poly_def
by (simp add: Abs_poly_inverse coeff Poly_minus)
lemma coeff_diff [simp]:
"coeff (p - q) n = coeff p n - coeff q n"
unfolding minus_poly_def
by (simp add: Abs_poly_inverse coeff Poly_diff)
instance proof
fix p q :: "'a poly"
show "- p + p = 0"
by (simp add: expand_poly_eq)
show "p - q = p + - q"
by (simp add: expand_poly_eq diff_minus)
qed
end
lemma add_pCons [simp]:
"pCons a p + pCons b q = pCons (a + b) (p + q)"
by (rule poly_ext, simp add: coeff_pCons split: nat.split)
lemma minus_pCons [simp]:
"- pCons a p = pCons (- a) (- p)"
by (rule poly_ext, simp add: coeff_pCons split: nat.split)
lemma diff_pCons [simp]:
"pCons a p - pCons b q = pCons (a - b) (p - q)"
by (rule poly_ext, simp add: coeff_pCons split: nat.split)
lemma degree_add_le: "degree (p + q) \<le> max (degree p) (degree q)"
by (rule degree_le, auto simp add: coeff_eq_0)
lemma degree_add_less:
"\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
by (auto intro: le_less_trans degree_add_le)
lemma degree_add_eq_right:
"degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
apply (cases "q = 0", simp)
apply (rule order_antisym)
apply (rule ord_le_eq_trans [OF degree_add_le])
apply simp
apply (rule le_degree)
apply (simp add: coeff_eq_0)
done
lemma degree_add_eq_left:
"degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
using degree_add_eq_right [of q p]
by (simp add: add_commute)
lemma degree_minus [simp]: "degree (- p) = degree p"
unfolding degree_def by simp
lemma degree_diff_le: "degree (p - q) \<le> max (degree p) (degree q)"
using degree_add_le [where p=p and q="-q"]
by (simp add: diff_minus)
lemma degree_diff_less:
"\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
by (auto intro: le_less_trans degree_diff_le)
lemma add_monom: "monom a n + monom b n = monom (a + b) n"
by (rule poly_ext) simp
lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
by (rule poly_ext) simp
lemma minus_monom: "- monom a n = monom (-a) n"
by (rule poly_ext) simp
lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
by (cases "finite A", induct set: finite, simp_all)
lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
by (rule poly_ext) (simp add: coeff_setsum)
subsection {* Multiplication by a constant *}
definition
smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
"smult a p = Abs_poly (\<lambda>n. a * coeff p n)"
lemma Poly_smult:
fixes f :: "nat \<Rightarrow> 'a::comm_semiring_0"
shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. a * f n) \<in> Poly"
unfolding Poly_def
by (clarify, rule_tac x=n in exI, simp)
lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n"
unfolding smult_def
by (simp add: Abs_poly_inverse Poly_smult coeff)
lemma degree_smult_le: "degree (smult a p) \<le> degree p"
by (rule degree_le, simp add: coeff_eq_0)
lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
by (rule poly_ext, simp add: mult_assoc)
lemma smult_0_right [simp]: "smult a 0 = 0"
by (rule poly_ext, simp)
lemma smult_0_left [simp]: "smult 0 p = 0"
by (rule poly_ext, simp)
lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
by (rule poly_ext, simp)
lemma smult_add_right:
"smult a (p + q) = smult a p + smult a q"
by (rule poly_ext, simp add: ring_simps)
lemma smult_add_left:
"smult (a + b) p = smult a p + smult b p"
by (rule poly_ext, simp add: ring_simps)
lemma smult_minus_right [simp]:
"smult (a::'a::comm_ring) (- p) = - smult a p"
by (rule poly_ext, simp)
lemma smult_minus_left [simp]:
"smult (- a::'a::comm_ring) p = - smult a p"
by (rule poly_ext, simp)
lemma smult_diff_right:
"smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
by (rule poly_ext, simp add: ring_simps)
lemma smult_diff_left:
"smult (a - b::'a::comm_ring) p = smult a p - smult b p"
by (rule poly_ext, simp add: ring_simps)
lemmas smult_distribs =
smult_add_left smult_add_right
smult_diff_left smult_diff_right
lemma smult_pCons [simp]:
"smult a (pCons b p) = pCons (a * b) (smult a p)"
by (rule poly_ext, simp add: coeff_pCons split: nat.split)
lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
by (induct n, simp add: monom_0, simp add: monom_Suc)
subsection {* Multiplication of polynomials *}
lemma Poly_mult_lemma:
fixes f g :: "nat \<Rightarrow> 'a::comm_semiring_0" and m n :: nat
assumes "\<forall>i>m. f i = 0"
assumes "\<forall>j>n. g j = 0"
shows "\<forall>k>m+n. (\<Sum>i\<le>k. f i * g (k-i)) = 0"
proof (clarify)
fix k :: nat
assume "m + n < k"
show "(\<Sum>i\<le>k. f i * g (k - i)) = 0"
proof (rule setsum_0' [rule_format])
fix i :: nat
assume "i \<in> {..k}" hence "i \<le> k" by simp
with `m + n < k` have "m < i \<or> n < k - i" by arith
thus "f i * g (k - i) = 0"
using prems by auto
qed
qed
lemma Poly_mult:
fixes f g :: "nat \<Rightarrow> 'a::comm_semiring_0"
shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. \<Sum>i\<le>n. f i * g (n-i)) \<in> Poly"
unfolding Poly_def
by (safe, rule exI, rule Poly_mult_lemma)
lemma poly_mult_assoc_lemma:
fixes k :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
shows "(\<Sum>j\<le>k. \<Sum>i\<le>j. f i (j - i) (n - j)) =
(\<Sum>j\<le>k. \<Sum>i\<le>k - j. f j i (n - j - i))"
proof (induct k)
case 0 show ?case by simp
next
case (Suc k) thus ?case
by (simp add: Suc_diff_le setsum_addf add_assoc
cong: strong_setsum_cong)
qed
lemma poly_mult_commute_lemma:
fixes n :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
shows "(\<Sum>i\<le>n. f i (n - i)) = (\<Sum>i\<le>n. f (n - i) i)"
proof (rule setsum_reindex_cong)
show "inj_on (\<lambda>i. n - i) {..n}"
by (rule inj_onI) simp
show "{..n} = (\<lambda>i. n - i) ` {..n}"
by (auto, rule_tac x="n - x" in image_eqI, simp_all)
next
fix i assume "i \<in> {..n}"
hence "n - (n - i) = i" by simp
thus "f (n - i) i = f (n - i) (n - (n - i))" by simp
qed
text {* TODO: move to appropriate theory *}
lemma setsum_atMost_Suc_shift:
fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n) note IH = this
have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
by (rule setsum_atMost_Suc)
also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
by (rule IH)
also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
by (rule add_assoc)
also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
by (rule setsum_atMost_Suc [symmetric])
finally show ?case .
qed
instantiation poly :: (comm_semiring_0) comm_semiring_0
begin
definition
times_poly_def:
"p * q = Abs_poly (\<lambda>n. \<Sum>i\<le>n. coeff p i * coeff q (n-i))"
lemma coeff_mult:
"coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
unfolding times_poly_def
by (simp add: Abs_poly_inverse coeff Poly_mult)
instance proof
fix p q r :: "'a poly"
show 0: "0 * p = 0"
by (simp add: expand_poly_eq coeff_mult)
show "p * 0 = 0"
by (simp add: expand_poly_eq coeff_mult)
show "(p + q) * r = p * r + q * r"
by (simp add: expand_poly_eq coeff_mult left_distrib setsum_addf)
show "(p * q) * r = p * (q * r)"
proof (rule poly_ext)
fix n :: nat
have "(\<Sum>j\<le>n. \<Sum>i\<le>j. coeff p i * coeff q (j - i) * coeff r (n - j)) =
(\<Sum>j\<le>n. \<Sum>i\<le>n - j. coeff p j * coeff q i * coeff r (n - j - i))"
by (rule poly_mult_assoc_lemma)
thus "coeff ((p * q) * r) n = coeff (p * (q * r)) n"
by (simp add: coeff_mult setsum_right_distrib
setsum_left_distrib mult_assoc)
qed
show "p * q = q * p"
proof (rule poly_ext)
fix n :: nat
have "(\<Sum>i\<le>n. coeff p i * coeff q (n - i)) =
(\<Sum>i\<le>n. coeff p (n - i) * coeff q i)"
by (rule poly_mult_commute_lemma)
thus "coeff (p * q) n = coeff (q * p) n"
by (simp add: coeff_mult mult_commute)
qed
qed
end
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
apply (rule degree_le, simp add: coeff_mult)
apply (rule Poly_mult_lemma)
apply (simp_all add: coeff_eq_0)
done
lemma mult_pCons_left [simp]:
"pCons a p * q = smult a q + pCons 0 (p * q)"
apply (rule poly_ext)
apply (case_tac n)
apply (simp add: coeff_mult)
apply (simp add: coeff_mult setsum_atMost_Suc_shift
del: setsum_atMost_Suc)
done
lemma mult_pCons_right [simp]:
"p * pCons a q = smult a p + pCons 0 (p * q)"
using mult_pCons_left [of a q p] by (simp add: mult_commute)
lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)"
by (induct p, simp, simp add: smult_add_right)
lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)"
by (induct q, simp, simp add: smult_add_right)
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)
subsection {* The unit polynomial and exponentiation *}
instantiation poly :: (comm_semiring_1) comm_semiring_1
begin
definition
one_poly_def:
"1 = pCons 1 0"
instance proof
fix p :: "'a poly" show "1 * p = p"
unfolding one_poly_def
by simp
next
show "0 \<noteq> (1::'a poly)"
unfolding one_poly_def by simp
qed
end
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
unfolding one_poly_def
by (simp add: coeff_pCons split: nat.split)
lemma degree_1 [simp]: "degree 1 = 0"
unfolding one_poly_def
by (rule degree_pCons_0)
instantiation poly :: (comm_semiring_1) recpower
begin
primrec power_poly where
power_poly_0: "(p::'a poly) ^ 0 = 1"
| power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n"
instance
by default simp_all
end
instance poly :: (comm_ring) comm_ring ..
instance poly :: (comm_ring_1) comm_ring_1 ..
instantiation poly :: (comm_ring_1) number_ring
begin
definition
"number_of k = (of_int k :: 'a poly)"
instance
by default (rule number_of_poly_def)
end
subsection {* Polynomials form an integral domain *}
lemma coeff_mult_degree_sum:
"coeff (p * q) (degree p + degree q) =
coeff p (degree p) * coeff q (degree q)"
by (induct p, simp, simp add: coeff_eq_0)
instance poly :: (idom) idom
proof
fix p q :: "'a poly"
assume "p \<noteq> 0" and "q \<noteq> 0"
have "coeff (p * q) (degree p + degree q) =
coeff p (degree p) * coeff q (degree q)"
by (rule coeff_mult_degree_sum)
also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
using `p \<noteq> 0` and `q \<noteq> 0` by simp
finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
thus "p * q \<noteq> 0" by (simp add: expand_poly_eq)
qed
lemma degree_mult_eq:
fixes p q :: "'a::idom poly"
shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
apply (rule order_antisym [OF degree_mult_le le_degree])
apply (simp add: coeff_mult_degree_sum)
done
lemma dvd_imp_degree_le:
fixes p q :: "'a::idom poly"
shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
by (erule dvdE, simp add: degree_mult_eq)
subsection {* Long division of polynomials *}
definition
divmod_poly_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
where
"divmod_poly_rel x y q r \<longleftrightarrow>
x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
lemma divmod_poly_rel_0:
"divmod_poly_rel 0 y 0 0"
unfolding divmod_poly_rel_def by simp
lemma divmod_poly_rel_by_0:
"divmod_poly_rel x 0 0 x"
unfolding divmod_poly_rel_def by simp
lemma eq_zero_or_degree_less:
assumes "degree p \<le> n" and "coeff p n = 0"
shows "p = 0 \<or> degree p < n"
proof (cases n)
case 0
with `degree p \<le> n` and `coeff p n = 0`
have "coeff p (degree p) = 0" by simp
then have "p = 0" by simp
then show ?thesis ..
next
case (Suc m)
have "\<forall>i>n. coeff p i = 0"
using `degree p \<le> n` by (simp add: coeff_eq_0)
then have "\<forall>i\<ge>n. coeff p i = 0"
using `coeff p n = 0` by (simp add: le_less)
then have "\<forall>i>m. coeff p i = 0"
using `n = Suc m` by (simp add: less_eq_Suc_le)
then have "degree p \<le> m"
by (rule degree_le)
then have "degree p < n"
using `n = Suc m` by (simp add: less_Suc_eq_le)
then show ?thesis ..
qed
lemma divmod_poly_rel_pCons:
assumes rel: "divmod_poly_rel x y q r"
assumes y: "y \<noteq> 0"
assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
shows "divmod_poly_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
(is "divmod_poly_rel ?x y ?q ?r")
proof -
have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
using assms unfolding divmod_poly_rel_def by simp_all
have 1: "?x = ?q * y + ?r"
using b x by simp
have 2: "?r = 0 \<or> degree ?r < degree y"
proof (rule eq_zero_or_degree_less)
have "degree ?r \<le> max (degree (pCons a r)) (degree (smult b y))"
by (rule degree_diff_le)
also have "\<dots> \<le> degree y"
proof (rule min_max.le_supI)
show "degree (pCons a r) \<le> degree y"
using r by auto
show "degree (smult b y) \<le> degree y"
by (rule degree_smult_le)
qed
finally show "degree ?r \<le> degree y" .
next
show "coeff ?r (degree y) = 0"
using `y \<noteq> 0` unfolding b by simp
qed
from 1 2 show ?thesis
unfolding divmod_poly_rel_def
using `y \<noteq> 0` by simp
qed
lemma divmod_poly_rel_exists: "\<exists>q r. divmod_poly_rel x y q r"
apply (cases "y = 0")
apply (fast intro!: divmod_poly_rel_by_0)
apply (induct x)
apply (fast intro!: divmod_poly_rel_0)
apply (fast intro!: divmod_poly_rel_pCons)
done
lemma divmod_poly_rel_unique:
assumes 1: "divmod_poly_rel x y q1 r1"
assumes 2: "divmod_poly_rel x y q2 r2"
shows "q1 = q2 \<and> r1 = r2"
proof (cases "y = 0")
assume "y = 0" with assms show ?thesis
by (simp add: divmod_poly_rel_def)
next
assume [simp]: "y \<noteq> 0"
from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
unfolding divmod_poly_rel_def by simp_all
from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
unfolding divmod_poly_rel_def by simp_all
from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
by (simp add: ring_simps)
from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
by (auto intro: degree_diff_less)
show "q1 = q2 \<and> r1 = r2"
proof (rule ccontr)
assume "\<not> (q1 = q2 \<and> r1 = r2)"
with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
with r3 have "degree (r2 - r1) < degree y" by simp
also have "degree y \<le> degree (q1 - q2) + degree y" by simp
also have "\<dots> = degree ((q1 - q2) * y)"
using `q1 \<noteq> q2` by (simp add: degree_mult_eq)
also have "\<dots> = degree (r2 - r1)"
using q3 by simp
finally have "degree (r2 - r1) < degree (r2 - r1)" .
then show "False" by simp
qed
qed
lemmas divmod_poly_rel_unique_div =
divmod_poly_rel_unique [THEN conjunct1, standard]
lemmas divmod_poly_rel_unique_mod =
divmod_poly_rel_unique [THEN conjunct2, standard]
instantiation poly :: (field) ring_div
begin
definition div_poly where
[code del]: "x div y = (THE q. \<exists>r. divmod_poly_rel x y q r)"
definition mod_poly where
[code del]: "x mod y = (THE r. \<exists>q. divmod_poly_rel x y q r)"
lemma div_poly_eq:
"divmod_poly_rel x y q r \<Longrightarrow> x div y = q"
unfolding div_poly_def
by (fast elim: divmod_poly_rel_unique_div)
lemma mod_poly_eq:
"divmod_poly_rel x y q r \<Longrightarrow> x mod y = r"
unfolding mod_poly_def
by (fast elim: divmod_poly_rel_unique_mod)
lemma divmod_poly_rel:
"divmod_poly_rel x y (x div y) (x mod y)"
proof -
from divmod_poly_rel_exists
obtain q r where "divmod_poly_rel x y q r" by fast
thus ?thesis
by (simp add: div_poly_eq mod_poly_eq)
qed
instance proof
fix x y :: "'a poly"
show "x div y * y + x mod y = x"
using divmod_poly_rel [of x y]
by (simp add: divmod_poly_rel_def)
next
fix x :: "'a poly"
have "divmod_poly_rel x 0 0 x"
by (rule divmod_poly_rel_by_0)
thus "x div 0 = 0"
by (rule div_poly_eq)
next
fix y :: "'a poly"
have "divmod_poly_rel 0 y 0 0"
by (rule divmod_poly_rel_0)
thus "0 div y = 0"
by (rule div_poly_eq)
next
fix x y z :: "'a poly"
assume "y \<noteq> 0"
hence "divmod_poly_rel (x + z * y) y (z + x div y) (x mod y)"
using divmod_poly_rel [of x y]
by (simp add: divmod_poly_rel_def left_distrib)
thus "(x + z * y) div y = z + x div y"
by (rule div_poly_eq)
qed
end
lemma degree_mod_less:
"y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
using divmod_poly_rel [of x y]
unfolding divmod_poly_rel_def by simp
lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
proof -
assume "degree x < degree y"
hence "divmod_poly_rel x y 0 x"
by (simp add: divmod_poly_rel_def)
thus "x div y = 0" by (rule div_poly_eq)
qed
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
proof -
assume "degree x < degree y"
hence "divmod_poly_rel x y 0 x"
by (simp add: divmod_poly_rel_def)
thus "x mod y = x" by (rule mod_poly_eq)
qed
lemma mod_pCons:
fixes a and x
assumes y: "y \<noteq> 0"
defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
unfolding b
apply (rule mod_poly_eq)
apply (rule divmod_poly_rel_pCons [OF divmod_poly_rel y refl])
done
subsection {* Evaluation of polynomials *}
definition
poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a" where
"poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)"
lemma poly_0 [simp]: "poly 0 x = 0"
unfolding poly_def by (simp add: poly_rec_0)
lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x"
unfolding poly_def by (simp add: poly_rec_pCons)
lemma poly_1 [simp]: "poly 1 x = 1"
unfolding one_poly_def by simp
lemma poly_monom:
fixes a x :: "'a::{comm_semiring_1,recpower}"
shows "poly (monom a n) x = a * x ^ n"
by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac)
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
apply (induct p arbitrary: q, simp)
apply (case_tac q, simp, simp add: ring_simps)
done
lemma poly_minus [simp]:
fixes x :: "'a::comm_ring"
shows "poly (- p) x = - poly p x"
by (induct p, simp_all)
lemma poly_diff [simp]:
fixes x :: "'a::comm_ring"
shows "poly (p - q) x = poly p x - poly q x"
by (simp add: diff_minus)
lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
by (cases "finite A", induct set: finite, simp_all)
lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
by (induct p, simp, simp add: ring_simps)
lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
by (induct p, simp_all, simp add: ring_simps)
lemma poly_power [simp]:
fixes p :: "'a::{comm_semiring_1,recpower} poly"
shows "poly (p ^ n) x = poly p x ^ n"
by (induct n, simp, simp add: power_Suc)
subsection {* Synthetic division *}
definition
synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
where
"synthetic_divmod p c =
poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p"
definition
synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
where
"synthetic_div p c = fst (synthetic_divmod p c)"
lemma synthetic_divmod_0 [simp]:
"synthetic_divmod 0 c = (0, 0)"
unfolding synthetic_divmod_def
by (simp add: poly_rec_0)
lemma synthetic_divmod_pCons [simp]:
"synthetic_divmod (pCons a p) c =
(\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
unfolding synthetic_divmod_def
by (simp add: poly_rec_pCons)
lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c"
by (induct p, simp, simp add: split_def)
lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0"
unfolding synthetic_div_def by simp
lemma synthetic_div_pCons [simp]:
"synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
unfolding synthetic_div_def
by (simp add: split_def snd_synthetic_divmod)
lemma synthetic_div_eq_0_iff:
"synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
by (induct p, simp, case_tac p, simp)
lemma degree_synthetic_div:
"degree (synthetic_div p c) = degree p - 1"
by (induct p, simp, simp add: synthetic_div_eq_0_iff)
lemma synthetic_div_correct:
"p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
by (induct p) simp_all
lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
by (induct p arbitrary: a) simp_all
lemma synthetic_div_unique:
"p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
apply (induct p arbitrary: q r)
apply (simp, frule synthetic_div_unique_lemma, simp)
apply (case_tac q, force)
done
lemma synthetic_div_correct':
fixes c :: "'a::comm_ring_1"
shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
using synthetic_div_correct [of p c]
by (simp add: group_simps)
lemma poly_eq_0_iff_dvd:
fixes c :: "'a::idom"
shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
proof
assume "poly p c = 0"
with synthetic_div_correct' [of c p]
have "p = [:-c, 1:] * synthetic_div p c" by simp
then show "[:-c, 1:] dvd p" ..
next
assume "[:-c, 1:] dvd p"
then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
then show "poly p c = 0" by simp
qed
lemma dvd_iff_poly_eq_0:
fixes c :: "'a::idom"
shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
by (simp add: poly_eq_0_iff_dvd)
lemma poly_roots_finite:
fixes p :: "'a::idom poly"
shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
proof (induct n \<equiv> "degree p" arbitrary: p)
case (0 p)
then obtain a where "a \<noteq> 0" and "p = [:a:]"
by (cases p, simp split: if_splits)
then show "finite {x. poly p x = 0}" by simp
next
case (Suc n p)
show "finite {x. poly p x = 0}"
proof (cases "\<exists>x. poly p x = 0")
case False
then show "finite {x. poly p x = 0}" by simp
next
case True
then obtain a where "poly p a = 0" ..
then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
then obtain k where k: "p = [:-a, 1:] * k" ..
with `p \<noteq> 0` have "k \<noteq> 0" by auto
with k have "degree p = Suc (degree k)"
by (simp add: degree_mult_eq del: mult_pCons_left)
with `Suc n = degree p` have "n = degree k" by simp
with `k \<noteq> 0` have "finite {x. poly k x = 0}" by (rule Suc.hyps)
then have "finite (insert a {x. poly k x = 0})" by simp
then show "finite {x. poly p x = 0}"
by (simp add: k uminus_add_conv_diff Collect_disj_eq
del: mult_pCons_left)
qed
qed
end