author huffman Sun, 11 Jan 2009 12:05:50 -0800 changeset 29451 5f0cb3fa530d parent 29448 34b9652b2f45 child 29452 b81ae415873d
new theory of polynomials
 src/HOL/Polynomial.thy file | annotate | diff | comparison | revisions
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+++ b/src/HOL/Polynomial.thy	Sun Jan 11 12:05:50 2009 -0800
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+(*  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
+
+  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))"
+
+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:
+  "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 {* 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 *}
+
+begin
+
+definition
+  plus_poly_def [code del]:
+    "p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)"
+
+  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
+
+  "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)"
+  show "p + q = q + p"
+  show "0 + p = p"
+    by (simp add: expand_poly_eq)
+qed
+
+end
+
+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
+
+  "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)
+
+  "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
+
+  "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
+  using degree_add_eq_right [of q p]
+
+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 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: "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)
+
+  "smult a (p + q) = smult a p + smult a q"
+  by (rule poly_ext, simp add: ring_simps)
+
+  "smult (a + b) p = smult a p + smult b p"
+  by (rule poly_ext, simp add: ring_simps)
+
+lemma smult_minus_right:
+  "smult (a::'a::comm_ring) (- p) = - smult a p"
+  by (rule poly_ext, simp)
+
+lemma smult_minus_left:
+  "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)
+
+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
+             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: "smult a p * q = smult a (p * q)"
+  by (induct p, simp, simp add: smult_add_right smult_smult)
+
+lemma mult_smult_right: "p * smult a q = smult a (p * q)"
+  using mult_smult_left [of a q p] by (simp add: mult_commute)
+
+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)"
+ apply (simp add: coeff_mult)
+ apply (subst setsum_diff1' [where a="degree p"])
+   apply simp
+  apply simp
+ apply (subst setsum_0' [rule_format])
+  apply clarsimp
+  apply (subgoal_tac "degree p < a \<or> degree q < degree p + degree q - a")
+   apply (force simp add: coeff_eq_0)
+  apply arith
+ apply simp
+done
+
+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 simp add: degree_pCons_eq_if)
+      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: le_less_trans [OF degree_diff_le])
+
+  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_1,recpower} poly \<Rightarrow> 'a \<Rightarrow> 'a" where
+  "poly p = (\<lambda>x. \<Sum>n\<le>degree p. coeff p n * x ^ n)"
+
+lemma poly_0 [simp]: "poly 0 x = 0"
+  unfolding poly_def by simp
+
+lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x"
+  unfolding poly_def
+  by (simp add: degree_pCons_eq_if setsum_atMost_Suc_shift power_Suc
+                setsum_left_distrib setsum_right_distrib mult_ac
+           del: setsum_atMost_Suc)
+
+lemma poly_1 [simp]: "poly 1 x = 1"
+  unfolding one_poly_def by simp
+
+lemma poly_monom: "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_1,recpower}"
+  shows "poly (- p) x = - poly p x"
+  by (induct p, simp_all)
+
+lemma poly_diff [simp]:
+  fixes x :: "'a::{comm_ring_1,recpower}"
+  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)
+
+end```