(*  Title:      HOL/Library/Polynomial.thy

    Author:     Brian Huffman

    Author:     Clemens Ballarin

*)

header {* Univariate Polynomials *}

theory Polynomial

imports Main

begin

subsection {* Definition of type @{text poly} *}

definition "Poly = {f::nat \<Rightarrow> 'a::zero. \<exists>n. \<forall>i>n. f i = 0}"

typedef (open) 'a poly = "Poly :: (nat => 'a::zero) set"

  morphisms coeff Abs_poly

  unfolding Poly_def by auto

(* FIXME should be named poly_eq_iff *)

lemma expand_poly_eq: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"

  by (simp add: coeff_inject [symmetric] fun_eq_iff)

(* FIXME should be named poly_eqI *)

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

  "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"

  "[:x:]" <= "CONST pCons x (_constrain 0 t)"

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, code_post]: "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:

    "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

instance poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add

proof

  fix p q r :: "'a poly"

  assume "p + q = p + r" thus "q = r"

    by (simp add: expand_poly_eq)

qed

instantiation poly :: (ab_group_add) ab_group_add

begin

definition

  uminus_poly_def:

    "- p = Abs_poly (\<lambda>n. - coeff p n)"

definition

  minus_poly_def:

    "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_max: "degree (p + q) \<le> max (degree p) (degree q)"

  by (rule degree_le, auto simp add: coeff_eq_0)

lemma degree_add_le:

  "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n"

  by (auto intro: order_trans degree_add_le_max)

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_max)

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 (simp add: degree_add_le)

  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_max: "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_le:

  "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p - q) \<le> n"

  by (simp add: diff_minus degree_add_le)

lemma degree_diff_less:

  "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"

  by (simp add: diff_minus degree_add_less)

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: algebra_simps)

lemma smult_add_left:

  "smult (a + b) p = smult a p + smult b p"

  by (rule poly_ext, simp add: algebra_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: algebra_simps)

lemma smult_diff_left:

  "smult (a - b::'a::comm_ring) p = smult a p - smult b p"

  by (rule poly_ext, simp add: algebra_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)

lemma degree_smult_eq [simp]:

  fixes a :: "'a::idom"

  shows "degree (smult a p) = (if a = 0 then 0 else degree p)"

  by (cases "a = 0", simp, simp add: degree_def)

lemma smult_eq_0_iff [simp]:

  fixes a :: "'a::idom"

  shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0"

  by (simp add: expand_poly_eq)

subsection {* Multiplication of polynomials *}

text {* TODO: move to SetInterval.thy *}

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 = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p"

lemma mult_poly_0_left: "(0::'a poly) * q = 0"

  unfolding times_poly_def by (simp add: poly_rec_0)

lemma mult_pCons_left [simp]:

  "pCons a p * q = smult a q + pCons 0 (p * q)"

  unfolding times_poly_def by (simp add: poly_rec_pCons)

lemma mult_poly_0_right: "p * (0::'a poly) = 0"

  by (induct p, simp add: mult_poly_0_left, simp)

lemma mult_pCons_right [simp]:

  "p * pCons a q = smult a p + pCons 0 (p * q)"

  by (induct p, simp add: mult_poly_0_left, simp add: algebra_simps)

lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right

lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)"

  by (induct p, simp add: mult_poly_0, simp add: smult_add_right)

lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)"

  by (induct q, simp add: mult_poly_0, simp add: smult_add_right)

lemma mult_poly_add_left:

  fixes p q r :: "'a poly"

  shows "(p + q) * r = p * r + q * r"

  by (induct r, simp add: mult_poly_0,

                simp add: smult_distribs algebra_simps)

instance proof

  fix p q r :: "'a poly"

  show 0: "0 * p = 0"

    by (rule mult_poly_0_left)

  show "p * 0 = 0"

    by (rule mult_poly_0_right)

  show "(p + q) * r = p * r + q * r"

    by (rule mult_poly_add_left)

  show "(p * q) * r = p * (q * r)"

    by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left)

  show "p * q = q * p"

    by (induct p, simp add: mult_poly_0, simp)

qed

end

instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..

lemma coeff_mult:

  "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"

proof (induct p arbitrary: n)

  case 0 show ?case by simp

next

  case (pCons a p n) thus ?case

    by (cases n, simp, simp add: setsum_atMost_Suc_shift

                            del: setsum_atMost_Suc)

qed

lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"

apply (rule degree_le)

apply (induct p)

apply simp

apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)

done

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

instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..

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)

text {* Lemmas about divisibility *}

lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q"

proof -

  assume "p dvd q"

  then obtain k where "q = p * k" ..

  then have "smult a q = p * smult a k" by simp

  then show "p dvd smult a q" ..

qed

lemma dvd_smult_cancel:

  fixes a :: "'a::field"

  shows "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q"

  by (drule dvd_smult [where a="inverse a"]) simp

lemma dvd_smult_iff:

  fixes a :: "'a::field"

  shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q"

  by (safe elim!: dvd_smult dvd_smult_cancel)

lemma smult_dvd_cancel:

  "smult a p dvd q \<Longrightarrow> p dvd q"

proof -

  assume "smult a p dvd q"

  then obtain k where "q = smult a p * k" ..

  then have "q = p * smult a k" by simp

  then show "p dvd q" ..

qed

lemma smult_dvd:

  fixes a :: "'a::field"

  shows "p dvd q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> smult a p dvd q"

  by (rule smult_dvd_cancel [where a="inverse a"]) simp

lemma smult_dvd_iff:

  fixes a :: "'a::field"

  shows "smult a p dvd q \<longleftrightarrow> (if a = 0 then q = 0 else p dvd q)"

  by (auto elim: smult_dvd smult_dvd_cancel)

lemma degree_power_le: "degree (p ^ n) \<le> degree p * n"

by (induct n, simp, auto intro: order_trans degree_mult_le)

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 {* Polynomials form an ordered integral domain *}

definition

  pos_poly :: "'a::linordered_idom poly \<Rightarrow> bool"

where

  "pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)"

lemma pos_poly_pCons:

  "pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)"

  unfolding pos_poly_def by simp

lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0"

  unfolding pos_poly_def by simp

lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)"

  apply (induct p arbitrary: q, simp)

  apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos)

  done

lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)"

  unfolding pos_poly_def

  apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0")

  apply (simp add: degree_mult_eq coeff_mult_degree_sum mult_pos_pos)

  apply auto

  done

lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"

by (induct p) (auto simp add: pos_poly_pCons)

instantiation poly :: (linordered_idom) linordered_idom

begin

definition

  "x < y \<longleftrightarrow> pos_poly (y - x)"

definition

  "x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)"

definition

  "abs (x::'a poly) = (if x < 0 then - x else x)"

definition

  "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)"

instance proof

  fix x y :: "'a poly"

  show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"

    unfolding less_eq_poly_def less_poly_def

    apply safe

    apply simp

    apply (drule (1) pos_poly_add)

    apply simp

    done

next

  fix x :: "'a poly" show "x \<le> x"

    unfolding less_eq_poly_def by simp

next

  fix x y z :: "'a poly"

  assume "x \<le> y" and "y \<le> z" thus "x \<le> z"

    unfolding less_eq_poly_def

    apply safe

    apply (drule (1) pos_poly_add)

    apply (simp add: algebra_simps)

    done

next

  fix x y :: "'a poly"

  assume "x \<le> y" and "y \<le> x" thus "x = y"

    unfolding less_eq_poly_def

    apply safe

    apply (drule (1) pos_poly_add)

    apply simp

    done

next

  fix x y z :: "'a poly"

  assume "x \<le> y" thus "z + x \<le> z + y"

    unfolding less_eq_poly_def

    apply safe

    apply (simp add: algebra_simps)

    done

next

  fix x y :: "'a poly"

  show "x \<le> y \<or> y \<le> x"

    unfolding less_eq_poly_def

    using pos_poly_total [of "x - y"]

    by auto

next

  fix x y z :: "'a poly"

  assume "x < y" and "0 < z"

  thus "z * x < z * y"

    unfolding less_poly_def

    by (simp add: right_diff_distrib [symmetric] pos_poly_mult)

next

  fix x :: "'a poly"

  show "\<bar>x\<bar> = (if x < 0 then - x else x)"

    by (rule abs_poly_def)

next

  fix x :: "'a poly"

  show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"

    by (rule sgn_poly_def)

qed

end

text {* TODO: Simplification rules for comparisons *}

subsection {* Long division of polynomials *}

definition

  pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"

where

  "pdivmod_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 pdivmod_rel_0:

  "pdivmod_rel 0 y 0 0"

  unfolding pdivmod_rel_def by simp

lemma pdivmod_rel_by_0:

  "pdivmod_rel x 0 0 x"

  unfolding pdivmod_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 pdivmod_rel_pCons:

  assumes rel: "pdivmod_rel x y q r"

  assumes y: "y \<noteq> 0"

  assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"

  shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"

    (is "pdivmod_rel ?x y ?q ?r")

proof -

  have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"

    using assms unfolding pdivmod_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)

    show "degree ?r \<le> degree y"

    proof (rule degree_diff_le)

      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

  next

    show "coeff ?r (degree y) = 0"

      using `y \<noteq> 0` unfolding b by simp

  qed

  from 1 2 show ?thesis

    unfolding pdivmod_rel_def

    using `y \<noteq> 0` by simp

qed

lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r"

apply (cases "y = 0")

apply (fast intro!: pdivmod_rel_by_0)

apply (induct x)

apply (fast intro!: pdivmod_rel_0)

apply (fast intro!: pdivmod_rel_pCons)

done

lemma pdivmod_rel_unique:

  assumes 1: "pdivmod_rel x y q1 r1"

  assumes 2: "pdivmod_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: pdivmod_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 pdivmod_rel_def by simp_all

  from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"

    unfolding pdivmod_rel_def by simp_all

  from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"

    by (simp add: algebra_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

lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0"

by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0)

lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x"

by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0)

lemmas pdivmod_rel_unique_div = pdivmod_rel_unique [THEN conjunct1]

lemmas pdivmod_rel_unique_mod = pdivmod_rel_unique [THEN conjunct2]

instantiation poly :: (field) ring_div

begin

definition div_poly where

  "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)"

definition mod_poly where

  "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)"

lemma div_poly_eq:

  "pdivmod_rel x y q r \<Longrightarrow> x div y = q"

unfolding div_poly_def

by (fast elim: pdivmod_rel_unique_div)

lemma mod_poly_eq:

  "pdivmod_rel x y q r \<Longrightarrow> x mod y = r"

unfolding mod_poly_def

by (fast elim: pdivmod_rel_unique_mod)

lemma pdivmod_rel:

  "pdivmod_rel x y (x div y) (x mod y)"

proof -

  from pdivmod_rel_exists

    obtain q r where "pdivmod_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 pdivmod_rel [of x y]

    by (simp add: pdivmod_rel_def)

next

  fix x :: "'a poly"

  have "pdivmod_rel x 0 0 x"

    by (rule pdivmod_rel_by_0)

  thus "x div 0 = 0"

    by (rule div_poly_eq)

next

  fix y :: "'a poly"

  have "pdivmod_rel 0 y 0 0"

    by (rule pdivmod_rel_0)

  thus "0 div y = 0"

    by (rule div_poly_eq)

next

  fix x y z :: "'a poly"

  assume "y \<noteq> 0"

  hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"

    using pdivmod_rel [of x y]

    by (simp add: pdivmod_rel_def left_distrib)

  thus "(x + z * y) div y = z + x div y"

    by (rule div_poly_eq)

next

  fix x y z :: "'a poly"