src/HOL/Library/Polynomial.thy
author wenzelm
Mon Dec 28 01:28:28 2015 +0100 (2015-12-28)
changeset 61945 1135b8de26c3
parent 61605 1bf7b186542e
child 62065 1546a042e87b
permissions -rw-r--r--
more symbols;
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(*  Title:      HOL/Library/Polynomial.thy
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    Author:     Brian Huffman
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    Author:     Clemens Ballarin
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    Author:     Florian Haftmann
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*)
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section \<open>Polynomials as type over a ring structure\<close>
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theory Polynomial
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imports Main "~~/src/HOL/GCD" "~~/src/HOL/Library/More_List" "~~/src/HOL/Library/Infinite_Set"
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begin
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subsection \<open>Auxiliary: operations for lists (later) representing coefficients\<close>
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definition cCons :: "'a::zero \<Rightarrow> 'a list \<Rightarrow> 'a list"  (infixr "##" 65)
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where
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  "x ## xs = (if xs = [] \<and> x = 0 then [] else x # xs)"
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lemma cCons_0_Nil_eq [simp]:
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  "0 ## [] = []"
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  by (simp add: cCons_def)
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lemma cCons_Cons_eq [simp]:
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  "x ## y # ys = x # y # ys"
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  by (simp add: cCons_def)
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lemma cCons_append_Cons_eq [simp]:
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  "x ## xs @ y # ys = x # xs @ y # ys"
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  by (simp add: cCons_def)
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lemma cCons_not_0_eq [simp]:
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  "x \<noteq> 0 \<Longrightarrow> x ## xs = x # xs"
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  by (simp add: cCons_def)
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lemma strip_while_not_0_Cons_eq [simp]:
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  "strip_while (\<lambda>x. x = 0) (x # xs) = x ## strip_while (\<lambda>x. x = 0) xs"
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proof (cases "x = 0")
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  case False then show ?thesis by simp
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next
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  case True show ?thesis
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  proof (induct xs rule: rev_induct)
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    case Nil with True show ?case by simp
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  next
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    case (snoc y ys) then show ?case
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      by (cases "y = 0") (simp_all add: append_Cons [symmetric] del: append_Cons)
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  qed
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qed
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lemma tl_cCons [simp]:
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  "tl (x ## xs) = xs"
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  by (simp add: cCons_def)
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subsection \<open>Definition of type \<open>poly\<close>\<close>
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typedef (overloaded) 'a poly = "{f :: nat \<Rightarrow> 'a::zero. \<forall>\<^sub>\<infinity> n. f n = 0}"
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  morphisms coeff Abs_poly by (auto intro!: ALL_MOST)
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setup_lifting type_definition_poly
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lemma poly_eq_iff: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
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  by (simp add: coeff_inject [symmetric] fun_eq_iff)
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lemma poly_eqI: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
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  by (simp add: poly_eq_iff)
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lemma MOST_coeff_eq_0: "\<forall>\<^sub>\<infinity> n. coeff p n = 0"
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  using coeff [of p] by simp
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subsection \<open>Degree of a polynomial\<close>
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definition degree :: "'a::zero poly \<Rightarrow> nat"
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where
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  "degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"
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lemma coeff_eq_0:
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  assumes "degree p < n"
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  shows "coeff p n = 0"
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proof -
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  have "\<exists>n. \<forall>i>n. coeff p i = 0"
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    using MOST_coeff_eq_0 by (simp add: MOST_nat)
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  then have "\<forall>i>degree p. coeff p i = 0"
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    unfolding degree_def by (rule LeastI_ex)
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  with assms show ?thesis by simp
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qed
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lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p"
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  by (erule contrapos_np, rule coeff_eq_0, simp)
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lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n"
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  unfolding degree_def by (erule Least_le)
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lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0"
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  unfolding degree_def by (drule not_less_Least, simp)
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subsection \<open>The zero polynomial\<close>
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instantiation poly :: (zero) zero
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begin
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lift_definition zero_poly :: "'a poly"
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  is "\<lambda>_. 0" by (rule MOST_I) simp
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instance ..
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end
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lemma coeff_0 [simp]:
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  "coeff 0 n = 0"
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  by transfer rule
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lemma degree_0 [simp]:
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  "degree 0 = 0"
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  by (rule order_antisym [OF degree_le le0]) simp
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lemma leading_coeff_neq_0:
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  assumes "p \<noteq> 0"
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  shows "coeff p (degree p) \<noteq> 0"
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proof (cases "degree p")
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  case 0
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  from \<open>p \<noteq> 0\<close> have "\<exists>n. coeff p n \<noteq> 0"
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    by (simp add: poly_eq_iff)
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  then obtain n where "coeff p n \<noteq> 0" ..
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  hence "n \<le> degree p" by (rule le_degree)
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  with \<open>coeff p n \<noteq> 0\<close> and \<open>degree p = 0\<close>
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  show "coeff p (degree p) \<noteq> 0" by simp
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next
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  case (Suc n)
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  from \<open>degree p = Suc n\<close> have "n < degree p" by simp
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  hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp)
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  then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast
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  from \<open>degree p = Suc n\<close> and \<open>n < i\<close> have "degree p \<le> i" by simp
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  also from \<open>coeff p i \<noteq> 0\<close> have "i \<le> degree p" by (rule le_degree)
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  finally have "degree p = i" .
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  with \<open>coeff p i \<noteq> 0\<close> show "coeff p (degree p) \<noteq> 0" by simp
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qed
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lemma leading_coeff_0_iff [simp]:
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  "coeff p (degree p) = 0 \<longleftrightarrow> p = 0"
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  by (cases "p = 0", simp, simp add: leading_coeff_neq_0)
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subsection \<open>List-style constructor for polynomials\<close>
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lift_definition pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
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  is "\<lambda>a p. case_nat a (coeff p)"
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  by (rule MOST_SucD) (simp add: MOST_coeff_eq_0)
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lemmas coeff_pCons = pCons.rep_eq
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lemma coeff_pCons_0 [simp]:
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  "coeff (pCons a p) 0 = a"
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  by transfer simp
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lemma coeff_pCons_Suc [simp]:
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  "coeff (pCons a p) (Suc n) = coeff p n"
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  by (simp add: coeff_pCons)
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lemma degree_pCons_le:
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  "degree (pCons a p) \<le> Suc (degree p)"
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  by (rule degree_le) (simp add: coeff_eq_0 coeff_pCons split: nat.split)
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lemma degree_pCons_eq:
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  "p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)"
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  apply (rule order_antisym [OF degree_pCons_le])
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  apply (rule le_degree, simp)
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  done
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lemma degree_pCons_0:
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  "degree (pCons a 0) = 0"
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  apply (rule order_antisym [OF _ le0])
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  apply (rule degree_le, simp add: coeff_pCons split: nat.split)
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  done
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lemma degree_pCons_eq_if [simp]:
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  "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
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  apply (cases "p = 0", simp_all)
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  apply (rule order_antisym [OF _ le0])
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  apply (rule degree_le, simp add: coeff_pCons split: nat.split)
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  apply (rule order_antisym [OF degree_pCons_le])
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  apply (rule le_degree, simp)
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  done
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lemma pCons_0_0 [simp]:
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  "pCons 0 0 = 0"
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  by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
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lemma pCons_eq_iff [simp]:
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  "pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q"
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proof safe
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  assume "pCons a p = pCons b q"
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  then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp
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  then show "a = b" by simp
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next
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  assume "pCons a p = pCons b q"
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  then have "\<forall>n. coeff (pCons a p) (Suc n) =
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                 coeff (pCons b q) (Suc n)" by simp
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  then show "p = q" by (simp add: poly_eq_iff)
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qed
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lemma pCons_eq_0_iff [simp]:
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  "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0"
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  using pCons_eq_iff [of a p 0 0] by simp
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lemma pCons_cases [cases type: poly]:
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  obtains (pCons) a q where "p = pCons a q"
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proof
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  show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))"
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    by transfer
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       (simp_all add: MOST_inj[where f=Suc and P="\<lambda>n. p n = 0" for p] fun_eq_iff Abs_poly_inverse
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                 split: nat.split)
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qed
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lemma pCons_induct [case_names 0 pCons, induct type: poly]:
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  assumes zero: "P 0"
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  assumes pCons: "\<And>a p. a \<noteq> 0 \<or> p \<noteq> 0 \<Longrightarrow> P p \<Longrightarrow> P (pCons a p)"
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  shows "P p"
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proof (induct p rule: measure_induct_rule [where f=degree])
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  case (less p)
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  obtain a q where "p = pCons a q" by (rule pCons_cases)
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  have "P q"
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  proof (cases "q = 0")
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    case True
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    then show "P q" by (simp add: zero)
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  next
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    case False
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    then have "degree (pCons a q) = Suc (degree q)"
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      by (rule degree_pCons_eq)
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    then have "degree q < degree p"
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      using \<open>p = pCons a q\<close> by simp
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    then show "P q"
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      by (rule less.hyps)
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  qed
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  have "P (pCons a q)"
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  proof (cases "a \<noteq> 0 \<or> q \<noteq> 0")
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    case True
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    with \<open>P q\<close> show ?thesis by (auto intro: pCons)
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  next
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    case False
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    with zero show ?thesis by simp
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  qed
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  then show ?case
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    using \<open>p = pCons a q\<close> by simp
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qed
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lemma degree_eq_zeroE:
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  fixes p :: "'a::zero poly"
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  assumes "degree p = 0"
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  obtains a where "p = pCons a 0"
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proof -
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  obtain a q where p: "p = pCons a q" by (cases p)
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  with assms have "q = 0" by (cases "q = 0") simp_all
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  with p have "p = pCons a 0" by simp
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  with that show thesis .
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qed
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subsection \<open>List-style syntax for polynomials\<close>
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syntax
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  "_poly" :: "args \<Rightarrow> 'a poly"  ("[:(_):]")
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translations
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  "[:x, xs:]" == "CONST pCons x [:xs:]"
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  "[:x:]" == "CONST pCons x 0"
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  "[:x:]" <= "CONST pCons x (_constrain 0 t)"
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subsection \<open>Representation of polynomials by lists of coefficients\<close>
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primrec Poly :: "'a::zero list \<Rightarrow> 'a poly"
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where
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  [code_post]: "Poly [] = 0"
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| [code_post]: "Poly (a # as) = pCons a (Poly as)"
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lemma Poly_replicate_0 [simp]:
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  "Poly (replicate n 0) = 0"
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  by (induct n) simp_all
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lemma Poly_eq_0:
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  "Poly as = 0 \<longleftrightarrow> (\<exists>n. as = replicate n 0)"
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  by (induct as) (auto simp add: Cons_replicate_eq)
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definition coeffs :: "'a poly \<Rightarrow> 'a::zero list"
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where
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  "coeffs p = (if p = 0 then [] else map (\<lambda>i. coeff p i) [0 ..< Suc (degree p)])"
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lemma coeffs_eq_Nil [simp]:
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  "coeffs p = [] \<longleftrightarrow> p = 0"
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  by (simp add: coeffs_def)
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lemma not_0_coeffs_not_Nil:
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  "p \<noteq> 0 \<Longrightarrow> coeffs p \<noteq> []"
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  by simp
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lemma coeffs_0_eq_Nil [simp]:
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  "coeffs 0 = []"
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  by simp
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lemma coeffs_pCons_eq_cCons [simp]:
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  "coeffs (pCons a p) = a ## coeffs p"
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proof -
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  { fix ms :: "nat list" and f :: "nat \<Rightarrow> 'a" and x :: "'a"
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    assume "\<forall>m\<in>set ms. m > 0"
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    then have "map (case_nat x f) ms = map f (map (\<lambda>n. n - 1) ms)"
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      by (induct ms) (auto split: nat.split)
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  }
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  note * = this
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  show ?thesis
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    by (simp add: coeffs_def * upt_conv_Cons coeff_pCons map_decr_upt del: upt_Suc)
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qed
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lemma not_0_cCons_eq [simp]:
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  "p \<noteq> 0 \<Longrightarrow> a ## coeffs p = a # coeffs p"
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  by (simp add: cCons_def)
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lemma Poly_coeffs [simp, code abstype]:
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  "Poly (coeffs p) = p"
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  by (induct p) auto
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lemma coeffs_Poly [simp]:
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  "coeffs (Poly as) = strip_while (HOL.eq 0) as"
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proof (induct as)
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  case Nil then show ?case by simp
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next
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  case (Cons a as)
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  have "(\<forall>n. as \<noteq> replicate n 0) \<longleftrightarrow> (\<exists>a\<in>set as. a \<noteq> 0)"
haftmann@52380
   329
    using replicate_length_same [of as 0] by (auto dest: sym [of _ as])
haftmann@52380
   330
  with Cons show ?case by auto
haftmann@52380
   331
qed
haftmann@52380
   332
haftmann@52380
   333
lemma last_coeffs_not_0:
haftmann@52380
   334
  "p \<noteq> 0 \<Longrightarrow> last (coeffs p) \<noteq> 0"
haftmann@52380
   335
  by (induct p) (auto simp add: cCons_def)
haftmann@52380
   336
haftmann@52380
   337
lemma strip_while_coeffs [simp]:
haftmann@52380
   338
  "strip_while (HOL.eq 0) (coeffs p) = coeffs p"
haftmann@52380
   339
  by (cases "p = 0") (auto dest: last_coeffs_not_0 intro: strip_while_not_last)
haftmann@52380
   340
haftmann@52380
   341
lemma coeffs_eq_iff:
haftmann@52380
   342
  "p = q \<longleftrightarrow> coeffs p = coeffs q" (is "?P \<longleftrightarrow> ?Q")
haftmann@52380
   343
proof
haftmann@52380
   344
  assume ?P then show ?Q by simp
haftmann@52380
   345
next
haftmann@52380
   346
  assume ?Q
haftmann@52380
   347
  then have "Poly (coeffs p) = Poly (coeffs q)" by simp
haftmann@52380
   348
  then show ?P by simp
haftmann@52380
   349
qed
haftmann@52380
   350
haftmann@52380
   351
lemma coeff_Poly_eq:
haftmann@52380
   352
  "coeff (Poly xs) n = nth_default 0 xs n"
haftmann@52380
   353
  apply (induct xs arbitrary: n) apply simp_all
blanchet@55642
   354
  by (metis nat.case not0_implies_Suc nth_default_Cons_0 nth_default_Cons_Suc pCons.rep_eq)
huffman@29454
   355
haftmann@52380
   356
lemma nth_default_coeffs_eq:
haftmann@52380
   357
  "nth_default 0 (coeffs p) = coeff p"
haftmann@52380
   358
  by (simp add: fun_eq_iff coeff_Poly_eq [symmetric])
haftmann@52380
   359
haftmann@52380
   360
lemma [code]:
haftmann@52380
   361
  "coeff p = nth_default 0 (coeffs p)"
haftmann@52380
   362
  by (simp add: nth_default_coeffs_eq)
haftmann@52380
   363
haftmann@52380
   364
lemma coeffs_eqI:
haftmann@52380
   365
  assumes coeff: "\<And>n. coeff p n = nth_default 0 xs n"
haftmann@52380
   366
  assumes zero: "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0"
haftmann@52380
   367
  shows "coeffs p = xs"
haftmann@52380
   368
proof -
haftmann@52380
   369
  from coeff have "p = Poly xs" by (simp add: poly_eq_iff coeff_Poly_eq)
haftmann@52380
   370
  with zero show ?thesis by simp (cases xs, simp_all)
haftmann@52380
   371
qed
haftmann@52380
   372
haftmann@52380
   373
lemma degree_eq_length_coeffs [code]:
haftmann@52380
   374
  "degree p = length (coeffs p) - 1"
haftmann@52380
   375
  by (simp add: coeffs_def)
haftmann@52380
   376
haftmann@52380
   377
lemma length_coeffs_degree:
haftmann@52380
   378
  "p \<noteq> 0 \<Longrightarrow> length (coeffs p) = Suc (degree p)"
haftmann@52380
   379
  by (induct p) (auto simp add: cCons_def)
haftmann@52380
   380
haftmann@52380
   381
lemma [code abstract]:
haftmann@52380
   382
  "coeffs 0 = []"
haftmann@52380
   383
  by (fact coeffs_0_eq_Nil)
haftmann@52380
   384
haftmann@52380
   385
lemma [code abstract]:
haftmann@52380
   386
  "coeffs (pCons a p) = a ## coeffs p"
haftmann@52380
   387
  by (fact coeffs_pCons_eq_cCons)
haftmann@52380
   388
haftmann@52380
   389
instantiation poly :: ("{zero, equal}") equal
haftmann@52380
   390
begin
haftmann@52380
   391
haftmann@52380
   392
definition
haftmann@52380
   393
  [code]: "HOL.equal (p::'a poly) q \<longleftrightarrow> HOL.equal (coeffs p) (coeffs q)"
haftmann@52380
   394
wenzelm@60679
   395
instance
wenzelm@60679
   396
  by standard (simp add: equal equal_poly_def coeffs_eq_iff)
haftmann@52380
   397
haftmann@52380
   398
end
haftmann@52380
   399
wenzelm@60679
   400
lemma [code nbe]: "HOL.equal (p :: _ poly) p \<longleftrightarrow> True"
haftmann@52380
   401
  by (fact equal_refl)
huffman@29454
   402
haftmann@52380
   403
definition is_zero :: "'a::zero poly \<Rightarrow> bool"
haftmann@52380
   404
where
haftmann@52380
   405
  [code]: "is_zero p \<longleftrightarrow> List.null (coeffs p)"
haftmann@52380
   406
haftmann@52380
   407
lemma is_zero_null [code_abbrev]:
haftmann@52380
   408
  "is_zero p \<longleftrightarrow> p = 0"
haftmann@52380
   409
  by (simp add: is_zero_def null_def)
haftmann@52380
   410
haftmann@52380
   411
wenzelm@60500
   412
subsection \<open>Fold combinator for polynomials\<close>
haftmann@52380
   413
haftmann@52380
   414
definition fold_coeffs :: "('a::zero \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b"
haftmann@52380
   415
where
haftmann@52380
   416
  "fold_coeffs f p = foldr f (coeffs p)"
haftmann@52380
   417
haftmann@52380
   418
lemma fold_coeffs_0_eq [simp]:
haftmann@52380
   419
  "fold_coeffs f 0 = id"
haftmann@52380
   420
  by (simp add: fold_coeffs_def)
haftmann@52380
   421
haftmann@52380
   422
lemma fold_coeffs_pCons_eq [simp]:
haftmann@52380
   423
  "f 0 = id \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
haftmann@52380
   424
  by (simp add: fold_coeffs_def cCons_def fun_eq_iff)
huffman@29454
   425
haftmann@52380
   426
lemma fold_coeffs_pCons_0_0_eq [simp]:
haftmann@52380
   427
  "fold_coeffs f (pCons 0 0) = id"
haftmann@52380
   428
  by (simp add: fold_coeffs_def)
haftmann@52380
   429
haftmann@52380
   430
lemma fold_coeffs_pCons_coeff_not_0_eq [simp]:
haftmann@52380
   431
  "a \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
haftmann@52380
   432
  by (simp add: fold_coeffs_def)
haftmann@52380
   433
haftmann@52380
   434
lemma fold_coeffs_pCons_not_0_0_eq [simp]:
haftmann@52380
   435
  "p \<noteq> 0 \<Longrightarrow> fold_coeffs f (pCons a p) = f a \<circ> fold_coeffs f p"
haftmann@52380
   436
  by (simp add: fold_coeffs_def)
haftmann@52380
   437
haftmann@52380
   438
wenzelm@60500
   439
subsection \<open>Canonical morphism on polynomials -- evaluation\<close>
haftmann@52380
   440
haftmann@52380
   441
definition poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@52380
   442
where
wenzelm@61585
   443
  "poly p = fold_coeffs (\<lambda>a f x. a + x * f x) p (\<lambda>x. 0)" \<comment> \<open>The Horner Schema\<close>
haftmann@52380
   444
haftmann@52380
   445
lemma poly_0 [simp]:
haftmann@52380
   446
  "poly 0 x = 0"
haftmann@52380
   447
  by (simp add: poly_def)
haftmann@52380
   448
haftmann@52380
   449
lemma poly_pCons [simp]:
haftmann@52380
   450
  "poly (pCons a p) x = a + x * poly p x"
haftmann@52380
   451
  by (cases "p = 0 \<and> a = 0") (auto simp add: poly_def)
huffman@29454
   452
huffman@29454
   453
wenzelm@60500
   454
subsection \<open>Monomials\<close>
huffman@29451
   455
haftmann@52380
   456
lift_definition monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly"
haftmann@52380
   457
  is "\<lambda>a m n. if m = n then a else 0"
hoelzl@59983
   458
  by (simp add: MOST_iff_cofinite)
haftmann@52380
   459
haftmann@52380
   460
lemma coeff_monom [simp]:
haftmann@52380
   461
  "coeff (monom a m) n = (if m = n then a else 0)"
haftmann@52380
   462
  by transfer rule
huffman@29451
   463
haftmann@52380
   464
lemma monom_0:
haftmann@52380
   465
  "monom a 0 = pCons a 0"
haftmann@52380
   466
  by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
huffman@29451
   467
haftmann@52380
   468
lemma monom_Suc:
haftmann@52380
   469
  "monom a (Suc n) = pCons 0 (monom a n)"
haftmann@52380
   470
  by (rule poly_eqI) (simp add: coeff_pCons split: nat.split)
huffman@29451
   471
huffman@29451
   472
lemma monom_eq_0 [simp]: "monom 0 n = 0"
haftmann@52380
   473
  by (rule poly_eqI) simp
huffman@29451
   474
huffman@29451
   475
lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
haftmann@52380
   476
  by (simp add: poly_eq_iff)
huffman@29451
   477
huffman@29451
   478
lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
haftmann@52380
   479
  by (simp add: poly_eq_iff)
huffman@29451
   480
huffman@29451
   481
lemma degree_monom_le: "degree (monom a n) \<le> n"
huffman@29451
   482
  by (rule degree_le, simp)
huffman@29451
   483
huffman@29451
   484
lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
huffman@29451
   485
  apply (rule order_antisym [OF degree_monom_le])
huffman@29451
   486
  apply (rule le_degree, simp)
huffman@29451
   487
  done
huffman@29451
   488
haftmann@52380
   489
lemma coeffs_monom [code abstract]:
haftmann@52380
   490
  "coeffs (monom a n) = (if a = 0 then [] else replicate n 0 @ [a])"
haftmann@52380
   491
  by (induct n) (simp_all add: monom_0 monom_Suc)
haftmann@52380
   492
haftmann@52380
   493
lemma fold_coeffs_monom [simp]:
haftmann@52380
   494
  "a \<noteq> 0 \<Longrightarrow> fold_coeffs f (monom a n) = f 0 ^^ n \<circ> f a"
haftmann@52380
   495
  by (simp add: fold_coeffs_def coeffs_monom fun_eq_iff)
haftmann@52380
   496
haftmann@52380
   497
lemma poly_monom:
haftmann@52380
   498
  fixes a x :: "'a::{comm_semiring_1}"
haftmann@52380
   499
  shows "poly (monom a n) x = a * x ^ n"
haftmann@52380
   500
  by (cases "a = 0", simp_all)
haftmann@52380
   501
    (induct n, simp_all add: mult.left_commute poly_def)
haftmann@52380
   502
huffman@29451
   503
wenzelm@60500
   504
subsection \<open>Addition and subtraction\<close>
huffman@29451
   505
huffman@29451
   506
instantiation poly :: (comm_monoid_add) comm_monoid_add
huffman@29451
   507
begin
huffman@29451
   508
haftmann@52380
   509
lift_definition plus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
haftmann@52380
   510
  is "\<lambda>p q n. coeff p n + coeff q n"
hoelzl@60040
   511
proof -
wenzelm@60679
   512
  fix q p :: "'a poly"
wenzelm@60679
   513
  show "\<forall>\<^sub>\<infinity>n. coeff p n + coeff q n = 0"
hoelzl@60040
   514
    using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp
haftmann@52380
   515
qed
huffman@29451
   516
wenzelm@60679
   517
lemma coeff_add [simp]: "coeff (p + q) n = coeff p n + coeff q n"
haftmann@52380
   518
  by (simp add: plus_poly.rep_eq)
huffman@29451
   519
wenzelm@60679
   520
instance
wenzelm@60679
   521
proof
huffman@29451
   522
  fix p q r :: "'a poly"
huffman@29451
   523
  show "(p + q) + r = p + (q + r)"
haftmann@57512
   524
    by (simp add: poly_eq_iff add.assoc)
huffman@29451
   525
  show "p + q = q + p"
haftmann@57512
   526
    by (simp add: poly_eq_iff add.commute)
huffman@29451
   527
  show "0 + p = p"
haftmann@52380
   528
    by (simp add: poly_eq_iff)
huffman@29451
   529
qed
huffman@29451
   530
huffman@29451
   531
end
huffman@29451
   532
haftmann@59815
   533
instantiation poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add
haftmann@59815
   534
begin
haftmann@59815
   535
haftmann@59815
   536
lift_definition minus_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
haftmann@59815
   537
  is "\<lambda>p q n. coeff p n - coeff q n"
hoelzl@60040
   538
proof -
wenzelm@60679
   539
  fix q p :: "'a poly"
wenzelm@60679
   540
  show "\<forall>\<^sub>\<infinity>n. coeff p n - coeff q n = 0"
hoelzl@60040
   541
    using MOST_coeff_eq_0[of p] MOST_coeff_eq_0[of q] by eventually_elim simp
haftmann@59815
   542
qed
haftmann@59815
   543
wenzelm@60679
   544
lemma coeff_diff [simp]: "coeff (p - q) n = coeff p n - coeff q n"
haftmann@59815
   545
  by (simp add: minus_poly.rep_eq)
haftmann@59815
   546
wenzelm@60679
   547
instance
wenzelm@60679
   548
proof
huffman@29540
   549
  fix p q r :: "'a poly"
haftmann@59815
   550
  show "p + q - p = q"
haftmann@52380
   551
    by (simp add: poly_eq_iff)
haftmann@59815
   552
  show "p - q - r = p - (q + r)"
haftmann@59815
   553
    by (simp add: poly_eq_iff diff_diff_eq)
huffman@29540
   554
qed
huffman@29540
   555
haftmann@59815
   556
end
haftmann@59815
   557
huffman@29451
   558
instantiation poly :: (ab_group_add) ab_group_add
huffman@29451
   559
begin
huffman@29451
   560
haftmann@52380
   561
lift_definition uminus_poly :: "'a poly \<Rightarrow> 'a poly"
haftmann@52380
   562
  is "\<lambda>p n. - coeff p n"
hoelzl@60040
   563
proof -
wenzelm@60679
   564
  fix p :: "'a poly"
wenzelm@60679
   565
  show "\<forall>\<^sub>\<infinity>n. - coeff p n = 0"
hoelzl@60040
   566
    using MOST_coeff_eq_0 by simp
haftmann@52380
   567
qed
huffman@29451
   568
huffman@29451
   569
lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
haftmann@52380
   570
  by (simp add: uminus_poly.rep_eq)
huffman@29451
   571
wenzelm@60679
   572
instance
wenzelm@60679
   573
proof
huffman@29451
   574
  fix p q :: "'a poly"
huffman@29451
   575
  show "- p + p = 0"
haftmann@52380
   576
    by (simp add: poly_eq_iff)
huffman@29451
   577
  show "p - q = p + - q"
haftmann@54230
   578
    by (simp add: poly_eq_iff)
huffman@29451
   579
qed
huffman@29451
   580
huffman@29451
   581
end
huffman@29451
   582
huffman@29451
   583
lemma add_pCons [simp]:
huffman@29451
   584
  "pCons a p + pCons b q = pCons (a + b) (p + q)"
haftmann@52380
   585
  by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
huffman@29451
   586
huffman@29451
   587
lemma minus_pCons [simp]:
huffman@29451
   588
  "- pCons a p = pCons (- a) (- p)"
haftmann@52380
   589
  by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
huffman@29451
   590
huffman@29451
   591
lemma diff_pCons [simp]:
huffman@29451
   592
  "pCons a p - pCons b q = pCons (a - b) (p - q)"
haftmann@52380
   593
  by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
huffman@29451
   594
huffman@29539
   595
lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)"
huffman@29451
   596
  by (rule degree_le, auto simp add: coeff_eq_0)
huffman@29451
   597
huffman@29539
   598
lemma degree_add_le:
huffman@29539
   599
  "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n"
huffman@29539
   600
  by (auto intro: order_trans degree_add_le_max)
huffman@29539
   601
huffman@29453
   602
lemma degree_add_less:
huffman@29453
   603
  "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
huffman@29539
   604
  by (auto intro: le_less_trans degree_add_le_max)
huffman@29453
   605
huffman@29451
   606
lemma degree_add_eq_right:
huffman@29451
   607
  "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
huffman@29451
   608
  apply (cases "q = 0", simp)
huffman@29451
   609
  apply (rule order_antisym)
huffman@29539
   610
  apply (simp add: degree_add_le)
huffman@29451
   611
  apply (rule le_degree)
huffman@29451
   612
  apply (simp add: coeff_eq_0)
huffman@29451
   613
  done
huffman@29451
   614
huffman@29451
   615
lemma degree_add_eq_left:
huffman@29451
   616
  "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
huffman@29451
   617
  using degree_add_eq_right [of q p]
haftmann@57512
   618
  by (simp add: add.commute)
huffman@29451
   619
haftmann@59815
   620
lemma degree_minus [simp]:
haftmann@59815
   621
  "degree (- p) = degree p"
huffman@29451
   622
  unfolding degree_def by simp
huffman@29451
   623
haftmann@59815
   624
lemma degree_diff_le_max:
haftmann@59815
   625
  fixes p q :: "'a :: ab_group_add poly"
haftmann@59815
   626
  shows "degree (p - q) \<le> max (degree p) (degree q)"
huffman@29451
   627
  using degree_add_le [where p=p and q="-q"]
haftmann@54230
   628
  by simp
huffman@29451
   629
huffman@29539
   630
lemma degree_diff_le:
haftmann@59815
   631
  fixes p q :: "'a :: ab_group_add poly"
haftmann@59815
   632
  assumes "degree p \<le> n" and "degree q \<le> n"
haftmann@59815
   633
  shows "degree (p - q) \<le> n"
haftmann@59815
   634
  using assms degree_add_le [of p n "- q"] by simp
huffman@29539
   635
huffman@29453
   636
lemma degree_diff_less:
haftmann@59815
   637
  fixes p q :: "'a :: ab_group_add poly"
haftmann@59815
   638
  assumes "degree p < n" and "degree q < n"
haftmann@59815
   639
  shows "degree (p - q) < n"
haftmann@59815
   640
  using assms degree_add_less [of p n "- q"] by simp
huffman@29453
   641
huffman@29451
   642
lemma add_monom: "monom a n + monom b n = monom (a + b) n"
haftmann@52380
   643
  by (rule poly_eqI) simp
huffman@29451
   644
huffman@29451
   645
lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
haftmann@52380
   646
  by (rule poly_eqI) simp
huffman@29451
   647
huffman@29451
   648
lemma minus_monom: "- monom a n = monom (-a) n"
haftmann@52380
   649
  by (rule poly_eqI) simp
huffman@29451
   650
huffman@29451
   651
lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
huffman@29451
   652
  by (cases "finite A", induct set: finite, simp_all)
huffman@29451
   653
huffman@29451
   654
lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
haftmann@52380
   655
  by (rule poly_eqI) (simp add: coeff_setsum)
haftmann@52380
   656
haftmann@52380
   657
fun plus_coeffs :: "'a::comm_monoid_add list \<Rightarrow> 'a list \<Rightarrow> 'a list"
haftmann@52380
   658
where
haftmann@52380
   659
  "plus_coeffs xs [] = xs"
haftmann@52380
   660
| "plus_coeffs [] ys = ys"
haftmann@52380
   661
| "plus_coeffs (x # xs) (y # ys) = (x + y) ## plus_coeffs xs ys"
haftmann@52380
   662
haftmann@52380
   663
lemma coeffs_plus_eq_plus_coeffs [code abstract]:
haftmann@52380
   664
  "coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)"
haftmann@52380
   665
proof -
haftmann@52380
   666
  { fix xs ys :: "'a list" and n
haftmann@52380
   667
    have "nth_default 0 (plus_coeffs xs ys) n = nth_default 0 xs n + nth_default 0 ys n"
haftmann@52380
   668
    proof (induct xs ys arbitrary: n rule: plus_coeffs.induct)
wenzelm@60679
   669
      case (3 x xs y ys n)
wenzelm@60679
   670
      then show ?case by (cases n) (auto simp add: cCons_def)
haftmann@52380
   671
    qed simp_all }
haftmann@52380
   672
  note * = this
haftmann@52380
   673
  { fix xs ys :: "'a list"
haftmann@52380
   674
    assume "xs \<noteq> [] \<Longrightarrow> last xs \<noteq> 0" and "ys \<noteq> [] \<Longrightarrow> last ys \<noteq> 0"
haftmann@52380
   675
    moreover assume "plus_coeffs xs ys \<noteq> []"
haftmann@52380
   676
    ultimately have "last (plus_coeffs xs ys) \<noteq> 0"
haftmann@52380
   677
    proof (induct xs ys rule: plus_coeffs.induct)
haftmann@52380
   678
      case (3 x xs y ys) then show ?case by (auto simp add: cCons_def) metis
haftmann@52380
   679
    qed simp_all }
haftmann@52380
   680
  note ** = this
haftmann@52380
   681
  show ?thesis
haftmann@52380
   682
    apply (rule coeffs_eqI)
haftmann@52380
   683
    apply (simp add: * nth_default_coeffs_eq)
haftmann@52380
   684
    apply (rule **)
haftmann@52380
   685
    apply (auto dest: last_coeffs_not_0)
haftmann@52380
   686
    done
haftmann@52380
   687
qed
haftmann@52380
   688
haftmann@52380
   689
lemma coeffs_uminus [code abstract]:
haftmann@52380
   690
  "coeffs (- p) = map (\<lambda>a. - a) (coeffs p)"
haftmann@52380
   691
  by (rule coeffs_eqI)
haftmann@52380
   692
    (simp_all add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq)
haftmann@52380
   693
haftmann@52380
   694
lemma [code]:
haftmann@52380
   695
  fixes p q :: "'a::ab_group_add poly"
haftmann@52380
   696
  shows "p - q = p + - q"
haftmann@59557
   697
  by (fact diff_conv_add_uminus)
haftmann@52380
   698
haftmann@52380
   699
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
haftmann@52380
   700
  apply (induct p arbitrary: q, simp)
haftmann@52380
   701
  apply (case_tac q, simp, simp add: algebra_simps)
haftmann@52380
   702
  done
haftmann@52380
   703
haftmann@52380
   704
lemma poly_minus [simp]:
haftmann@52380
   705
  fixes x :: "'a::comm_ring"
haftmann@52380
   706
  shows "poly (- p) x = - poly p x"
haftmann@52380
   707
  by (induct p) simp_all
haftmann@52380
   708
haftmann@52380
   709
lemma poly_diff [simp]:
haftmann@52380
   710
  fixes x :: "'a::comm_ring"
haftmann@52380
   711
  shows "poly (p - q) x = poly p x - poly q x"
haftmann@54230
   712
  using poly_add [of p "- q" x] by simp
haftmann@52380
   713
haftmann@52380
   714
lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
haftmann@52380
   715
  by (induct A rule: infinite_finite_induct) simp_all
huffman@29451
   716
huffman@29451
   717
wenzelm@60500
   718
subsection \<open>Multiplication by a constant, polynomial multiplication and the unit polynomial\<close>
huffman@29451
   719
haftmann@52380
   720
lift_definition smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
haftmann@52380
   721
  is "\<lambda>a p n. a * coeff p n"
hoelzl@60040
   722
proof -
hoelzl@60040
   723
  fix a :: 'a and p :: "'a poly" show "\<forall>\<^sub>\<infinity> i. a * coeff p i = 0"
hoelzl@60040
   724
    using MOST_coeff_eq_0[of p] by eventually_elim simp
haftmann@52380
   725
qed
huffman@29451
   726
haftmann@52380
   727
lemma coeff_smult [simp]:
haftmann@52380
   728
  "coeff (smult a p) n = a * coeff p n"
haftmann@52380
   729
  by (simp add: smult.rep_eq)
huffman@29451
   730
huffman@29451
   731
lemma degree_smult_le: "degree (smult a p) \<le> degree p"
huffman@29451
   732
  by (rule degree_le, simp add: coeff_eq_0)
huffman@29451
   733
huffman@29472
   734
lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
haftmann@57512
   735
  by (rule poly_eqI, simp add: mult.assoc)
huffman@29451
   736
huffman@29451
   737
lemma smult_0_right [simp]: "smult a 0 = 0"
haftmann@52380
   738
  by (rule poly_eqI, simp)
huffman@29451
   739
huffman@29451
   740
lemma smult_0_left [simp]: "smult 0 p = 0"
haftmann@52380
   741
  by (rule poly_eqI, simp)
huffman@29451
   742
huffman@29451
   743
lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
haftmann@52380
   744
  by (rule poly_eqI, simp)
huffman@29451
   745
huffman@29451
   746
lemma smult_add_right:
huffman@29451
   747
  "smult a (p + q) = smult a p + smult a q"
haftmann@52380
   748
  by (rule poly_eqI, simp add: algebra_simps)
huffman@29451
   749
huffman@29451
   750
lemma smult_add_left:
huffman@29451
   751
  "smult (a + b) p = smult a p + smult b p"
haftmann@52380
   752
  by (rule poly_eqI, simp add: algebra_simps)
huffman@29451
   753
huffman@29457
   754
lemma smult_minus_right [simp]:
huffman@29451
   755
  "smult (a::'a::comm_ring) (- p) = - smult a p"
haftmann@52380
   756
  by (rule poly_eqI, simp)
huffman@29451
   757
huffman@29457
   758
lemma smult_minus_left [simp]:
huffman@29451
   759
  "smult (- a::'a::comm_ring) p = - smult a p"
haftmann@52380
   760
  by (rule poly_eqI, simp)
huffman@29451
   761
huffman@29451
   762
lemma smult_diff_right:
huffman@29451
   763
  "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
haftmann@52380
   764
  by (rule poly_eqI, simp add: algebra_simps)
huffman@29451
   765
huffman@29451
   766
lemma smult_diff_left:
huffman@29451
   767
  "smult (a - b::'a::comm_ring) p = smult a p - smult b p"
haftmann@52380
   768
  by (rule poly_eqI, simp add: algebra_simps)
huffman@29451
   769
huffman@29472
   770
lemmas smult_distribs =
huffman@29472
   771
  smult_add_left smult_add_right
huffman@29472
   772
  smult_diff_left smult_diff_right
huffman@29472
   773
huffman@29451
   774
lemma smult_pCons [simp]:
huffman@29451
   775
  "smult a (pCons b p) = pCons (a * b) (smult a p)"
haftmann@52380
   776
  by (rule poly_eqI, simp add: coeff_pCons split: nat.split)
huffman@29451
   777
huffman@29451
   778
lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
huffman@29451
   779
  by (induct n, simp add: monom_0, simp add: monom_Suc)
huffman@29451
   780
huffman@29659
   781
lemma degree_smult_eq [simp]:
huffman@29659
   782
  fixes a :: "'a::idom"
huffman@29659
   783
  shows "degree (smult a p) = (if a = 0 then 0 else degree p)"
huffman@29659
   784
  by (cases "a = 0", simp, simp add: degree_def)
huffman@29659
   785
huffman@29659
   786
lemma smult_eq_0_iff [simp]:
huffman@29659
   787
  fixes a :: "'a::idom"
huffman@29659
   788
  shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0"
haftmann@52380
   789
  by (simp add: poly_eq_iff)
huffman@29451
   790
haftmann@52380
   791
lemma coeffs_smult [code abstract]:
haftmann@52380
   792
  fixes p :: "'a::idom poly"
haftmann@52380
   793
  shows "coeffs (smult a p) = (if a = 0 then [] else map (Groups.times a) (coeffs p))"
haftmann@52380
   794
  by (rule coeffs_eqI)
haftmann@52380
   795
    (auto simp add: not_0_coeffs_not_Nil last_map last_coeffs_not_0 nth_default_map_eq nth_default_coeffs_eq)
huffman@29451
   796
huffman@29451
   797
instantiation poly :: (comm_semiring_0) comm_semiring_0
huffman@29451
   798
begin
huffman@29451
   799
huffman@29451
   800
definition
haftmann@52380
   801
  "p * q = fold_coeffs (\<lambda>a p. smult a q + pCons 0 p) p 0"
huffman@29474
   802
huffman@29474
   803
lemma mult_poly_0_left: "(0::'a poly) * q = 0"
haftmann@52380
   804
  by (simp add: times_poly_def)
huffman@29474
   805
huffman@29474
   806
lemma mult_pCons_left [simp]:
huffman@29474
   807
  "pCons a p * q = smult a q + pCons 0 (p * q)"
haftmann@52380
   808
  by (cases "p = 0 \<and> a = 0") (auto simp add: times_poly_def)
huffman@29474
   809
huffman@29474
   810
lemma mult_poly_0_right: "p * (0::'a poly) = 0"
haftmann@52380
   811
  by (induct p) (simp add: mult_poly_0_left, simp)
huffman@29451
   812
huffman@29474
   813
lemma mult_pCons_right [simp]:
huffman@29474
   814
  "p * pCons a q = smult a p + pCons 0 (p * q)"
haftmann@52380
   815
  by (induct p) (simp add: mult_poly_0_left, simp add: algebra_simps)
huffman@29474
   816
huffman@29474
   817
lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right
huffman@29474
   818
haftmann@52380
   819
lemma mult_smult_left [simp]:
haftmann@52380
   820
  "smult a p * q = smult a (p * q)"
haftmann@52380
   821
  by (induct p) (simp add: mult_poly_0, simp add: smult_add_right)
huffman@29474
   822
haftmann@52380
   823
lemma mult_smult_right [simp]:
haftmann@52380
   824
  "p * smult a q = smult a (p * q)"
haftmann@52380
   825
  by (induct q) (simp add: mult_poly_0, simp add: smult_add_right)
huffman@29474
   826
huffman@29474
   827
lemma mult_poly_add_left:
huffman@29474
   828
  fixes p q r :: "'a poly"
huffman@29474
   829
  shows "(p + q) * r = p * r + q * r"
haftmann@52380
   830
  by (induct r) (simp add: mult_poly_0, simp add: smult_distribs algebra_simps)
huffman@29451
   831
wenzelm@60679
   832
instance
wenzelm@60679
   833
proof
huffman@29451
   834
  fix p q r :: "'a poly"
huffman@29451
   835
  show 0: "0 * p = 0"
huffman@29474
   836
    by (rule mult_poly_0_left)
huffman@29451
   837
  show "p * 0 = 0"
huffman@29474
   838
    by (rule mult_poly_0_right)
huffman@29451
   839
  show "(p + q) * r = p * r + q * r"
huffman@29474
   840
    by (rule mult_poly_add_left)
huffman@29451
   841
  show "(p * q) * r = p * (q * r)"
huffman@29474
   842
    by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left)
huffman@29451
   843
  show "p * q = q * p"
huffman@29474
   844
    by (induct p, simp add: mult_poly_0, simp)
huffman@29451
   845
qed
huffman@29451
   846
huffman@29451
   847
end
huffman@29451
   848
huffman@29540
   849
instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
huffman@29540
   850
huffman@29474
   851
lemma coeff_mult:
huffman@29474
   852
  "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
huffman@29474
   853
proof (induct p arbitrary: n)
huffman@29474
   854
  case 0 show ?case by simp
huffman@29474
   855
next
huffman@29474
   856
  case (pCons a p n) thus ?case
huffman@29474
   857
    by (cases n, simp, simp add: setsum_atMost_Suc_shift
huffman@29474
   858
                            del: setsum_atMost_Suc)
huffman@29474
   859
qed
huffman@29451
   860
huffman@29474
   861
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
huffman@29474
   862
apply (rule degree_le)
huffman@29474
   863
apply (induct p)
huffman@29474
   864
apply simp
huffman@29474
   865
apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)
huffman@29451
   866
done
huffman@29451
   867
huffman@29451
   868
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
wenzelm@60679
   869
  by (induct m) (simp add: monom_0 smult_monom, simp add: monom_Suc)
huffman@29451
   870
huffman@29451
   871
instantiation poly :: (comm_semiring_1) comm_semiring_1
huffman@29451
   872
begin
huffman@29451
   873
wenzelm@60679
   874
definition one_poly_def: "1 = pCons 1 0"
huffman@29451
   875
wenzelm@60679
   876
instance
wenzelm@60679
   877
proof
wenzelm@60679
   878
  show "1 * p = p" for p :: "'a poly"
haftmann@52380
   879
    unfolding one_poly_def by simp
huffman@29451
   880
  show "0 \<noteq> (1::'a poly)"
huffman@29451
   881
    unfolding one_poly_def by simp
huffman@29451
   882
qed
huffman@29451
   883
huffman@29451
   884
end
huffman@29451
   885
haftmann@52380
   886
instance poly :: (comm_ring) comm_ring ..
haftmann@52380
   887
haftmann@52380
   888
instance poly :: (comm_ring_1) comm_ring_1 ..
haftmann@52380
   889
huffman@29451
   890
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
huffman@29451
   891
  unfolding one_poly_def
huffman@29451
   892
  by (simp add: coeff_pCons split: nat.split)
huffman@29451
   893
haftmann@60570
   894
lemma monom_eq_1 [simp]:
haftmann@60570
   895
  "monom 1 0 = 1"
haftmann@60570
   896
  by (simp add: monom_0 one_poly_def)
haftmann@60570
   897
  
huffman@29451
   898
lemma degree_1 [simp]: "degree 1 = 0"
huffman@29451
   899
  unfolding one_poly_def
huffman@29451
   900
  by (rule degree_pCons_0)
huffman@29451
   901
haftmann@52380
   902
lemma coeffs_1_eq [simp, code abstract]:
haftmann@52380
   903
  "coeffs 1 = [1]"
haftmann@52380
   904
  by (simp add: one_poly_def)
haftmann@52380
   905
haftmann@52380
   906
lemma degree_power_le:
haftmann@52380
   907
  "degree (p ^ n) \<le> degree p * n"
haftmann@52380
   908
  by (induct n) (auto intro: order_trans degree_mult_le)
haftmann@52380
   909
haftmann@52380
   910
lemma poly_smult [simp]:
haftmann@52380
   911
  "poly (smult a p) x = a * poly p x"
haftmann@52380
   912
  by (induct p, simp, simp add: algebra_simps)
haftmann@52380
   913
haftmann@52380
   914
lemma poly_mult [simp]:
haftmann@52380
   915
  "poly (p * q) x = poly p x * poly q x"
haftmann@52380
   916
  by (induct p, simp_all, simp add: algebra_simps)
haftmann@52380
   917
haftmann@52380
   918
lemma poly_1 [simp]:
haftmann@52380
   919
  "poly 1 x = 1"
haftmann@52380
   920
  by (simp add: one_poly_def)
haftmann@52380
   921
haftmann@52380
   922
lemma poly_power [simp]:
haftmann@52380
   923
  fixes p :: "'a::{comm_semiring_1} poly"
haftmann@52380
   924
  shows "poly (p ^ n) x = poly p x ^ n"
haftmann@52380
   925
  by (induct n) simp_all
haftmann@52380
   926
haftmann@52380
   927
wenzelm@60500
   928
subsection \<open>Lemmas about divisibility\<close>
huffman@29979
   929
huffman@29979
   930
lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q"
huffman@29979
   931
proof -
huffman@29979
   932
  assume "p dvd q"
huffman@29979
   933
  then obtain k where "q = p * k" ..
huffman@29979
   934
  then have "smult a q = p * smult a k" by simp
huffman@29979
   935
  then show "p dvd smult a q" ..
huffman@29979
   936
qed
huffman@29979
   937
huffman@29979
   938
lemma dvd_smult_cancel:
huffman@29979
   939
  fixes a :: "'a::field"
huffman@29979
   940
  shows "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q"
huffman@29979
   941
  by (drule dvd_smult [where a="inverse a"]) simp
huffman@29979
   942
huffman@29979
   943
lemma dvd_smult_iff:
huffman@29979
   944
  fixes a :: "'a::field"
huffman@29979
   945
  shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q"
huffman@29979
   946
  by (safe elim!: dvd_smult dvd_smult_cancel)
huffman@29979
   947
huffman@31663
   948
lemma smult_dvd_cancel:
huffman@31663
   949
  "smult a p dvd q \<Longrightarrow> p dvd q"
huffman@31663
   950
proof -
huffman@31663
   951
  assume "smult a p dvd q"
huffman@31663
   952
  then obtain k where "q = smult a p * k" ..
huffman@31663
   953
  then have "q = p * smult a k" by simp
huffman@31663
   954
  then show "p dvd q" ..
huffman@31663
   955
qed
huffman@31663
   956
huffman@31663
   957
lemma smult_dvd:
huffman@31663
   958
  fixes a :: "'a::field"
huffman@31663
   959
  shows "p dvd q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> smult a p dvd q"
huffman@31663
   960
  by (rule smult_dvd_cancel [where a="inverse a"]) simp
huffman@31663
   961
huffman@31663
   962
lemma smult_dvd_iff:
huffman@31663
   963
  fixes a :: "'a::field"
huffman@31663
   964
  shows "smult a p dvd q \<longleftrightarrow> (if a = 0 then q = 0 else p dvd q)"
huffman@31663
   965
  by (auto elim: smult_dvd smult_dvd_cancel)
huffman@31663
   966
huffman@29451
   967
wenzelm@60500
   968
subsection \<open>Polynomials form an integral domain\<close>
huffman@29451
   969
huffman@29451
   970
lemma coeff_mult_degree_sum:
huffman@29451
   971
  "coeff (p * q) (degree p + degree q) =
huffman@29451
   972
   coeff p (degree p) * coeff q (degree q)"
huffman@29471
   973
  by (induct p, simp, simp add: coeff_eq_0)
huffman@29451
   974
huffman@29451
   975
instance poly :: (idom) idom
huffman@29451
   976
proof
huffman@29451
   977
  fix p q :: "'a poly"
huffman@29451
   978
  assume "p \<noteq> 0" and "q \<noteq> 0"
huffman@29451
   979
  have "coeff (p * q) (degree p + degree q) =
huffman@29451
   980
        coeff p (degree p) * coeff q (degree q)"
huffman@29451
   981
    by (rule coeff_mult_degree_sum)
huffman@29451
   982
  also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
wenzelm@60500
   983
    using \<open>p \<noteq> 0\<close> and \<open>q \<noteq> 0\<close> by simp
huffman@29451
   984
  finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
haftmann@52380
   985
  thus "p * q \<noteq> 0" by (simp add: poly_eq_iff)
huffman@29451
   986
qed
huffman@29451
   987
huffman@29451
   988
lemma degree_mult_eq:
huffman@29451
   989
  fixes p q :: "'a::idom poly"
huffman@29451
   990
  shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
huffman@29451
   991
apply (rule order_antisym [OF degree_mult_le le_degree])
huffman@29451
   992
apply (simp add: coeff_mult_degree_sum)
huffman@29451
   993
done
huffman@29451
   994
haftmann@60570
   995
lemma degree_mult_right_le:
haftmann@60570
   996
  fixes p q :: "'a::idom poly"
haftmann@60570
   997
  assumes "q \<noteq> 0"
haftmann@60570
   998
  shows "degree p \<le> degree (p * q)"
haftmann@60570
   999
  using assms by (cases "p = 0") (simp_all add: degree_mult_eq)
haftmann@60570
  1000
haftmann@60570
  1001
lemma coeff_degree_mult:
haftmann@60570
  1002
  fixes p q :: "'a::idom poly"
haftmann@60570
  1003
  shows "coeff (p * q) (degree (p * q)) =
haftmann@60570
  1004
    coeff q (degree q) * coeff p (degree p)"
haftmann@60570
  1005
  by (cases "p = 0 \<or> q = 0") (auto simp add: degree_mult_eq coeff_mult_degree_sum)
haftmann@60570
  1006
huffman@29451
  1007
lemma dvd_imp_degree_le:
huffman@29451
  1008
  fixes p q :: "'a::idom poly"
huffman@29451
  1009
  shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
huffman@29451
  1010
  by (erule dvdE, simp add: degree_mult_eq)
huffman@29451
  1011
huffman@29451
  1012
wenzelm@60500
  1013
subsection \<open>Polynomials form an ordered integral domain\<close>
huffman@29878
  1014
haftmann@52380
  1015
definition pos_poly :: "'a::linordered_idom poly \<Rightarrow> bool"
huffman@29878
  1016
where
huffman@29878
  1017
  "pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)"
huffman@29878
  1018
huffman@29878
  1019
lemma pos_poly_pCons:
huffman@29878
  1020
  "pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)"
huffman@29878
  1021
  unfolding pos_poly_def by simp
huffman@29878
  1022
huffman@29878
  1023
lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0"
huffman@29878
  1024
  unfolding pos_poly_def by simp
huffman@29878
  1025
huffman@29878
  1026
lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)"
huffman@29878
  1027
  apply (induct p arbitrary: q, simp)
huffman@29878
  1028
  apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos)
huffman@29878
  1029
  done
huffman@29878
  1030
huffman@29878
  1031
lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)"
huffman@29878
  1032
  unfolding pos_poly_def
huffman@29878
  1033
  apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0")
nipkow@56544
  1034
  apply (simp add: degree_mult_eq coeff_mult_degree_sum)
huffman@29878
  1035
  apply auto
huffman@29878
  1036
  done
huffman@29878
  1037
huffman@29878
  1038
lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
huffman@29878
  1039
by (induct p) (auto simp add: pos_poly_pCons)
huffman@29878
  1040
haftmann@52380
  1041
lemma last_coeffs_eq_coeff_degree:
haftmann@52380
  1042
  "p \<noteq> 0 \<Longrightarrow> last (coeffs p) = coeff p (degree p)"
haftmann@52380
  1043
  by (simp add: coeffs_def)
haftmann@52380
  1044
haftmann@52380
  1045
lemma pos_poly_coeffs [code]:
haftmann@52380
  1046
  "pos_poly p \<longleftrightarrow> (let as = coeffs p in as \<noteq> [] \<and> last as > 0)" (is "?P \<longleftrightarrow> ?Q")
haftmann@52380
  1047
proof
haftmann@52380
  1048
  assume ?Q then show ?P by (auto simp add: pos_poly_def last_coeffs_eq_coeff_degree)
haftmann@52380
  1049
next
haftmann@52380
  1050
  assume ?P then have *: "0 < coeff p (degree p)" by (simp add: pos_poly_def)
haftmann@52380
  1051
  then have "p \<noteq> 0" by auto
haftmann@52380
  1052
  with * show ?Q by (simp add: last_coeffs_eq_coeff_degree)
haftmann@52380
  1053
qed
haftmann@52380
  1054
haftmann@35028
  1055
instantiation poly :: (linordered_idom) linordered_idom
huffman@29878
  1056
begin
huffman@29878
  1057
huffman@29878
  1058
definition
haftmann@37765
  1059
  "x < y \<longleftrightarrow> pos_poly (y - x)"
huffman@29878
  1060
huffman@29878
  1061
definition
haftmann@37765
  1062
  "x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)"
huffman@29878
  1063
huffman@29878
  1064
definition
wenzelm@61945
  1065
  "\<bar>x::'a poly\<bar> = (if x < 0 then - x else x)"
huffman@29878
  1066
huffman@29878
  1067
definition
haftmann@37765
  1068
  "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
huffman@29878
  1069
wenzelm@60679
  1070
instance
wenzelm@60679
  1071
proof
wenzelm@60679
  1072
  fix x y z :: "'a poly"
huffman@29878
  1073
  show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
huffman@29878
  1074
    unfolding less_eq_poly_def less_poly_def
huffman@29878
  1075
    apply safe
huffman@29878
  1076
    apply simp
huffman@29878
  1077
    apply (drule (1) pos_poly_add)
huffman@29878
  1078
    apply simp
huffman@29878
  1079
    done
wenzelm@60679
  1080
  show "x \<le> x"
huffman@29878
  1081
    unfolding less_eq_poly_def by simp
wenzelm@60679
  1082
  show "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
huffman@29878
  1083
    unfolding less_eq_poly_def
huffman@29878
  1084
    apply safe
huffman@29878
  1085
    apply (drule (1) pos_poly_add)
huffman@29878
  1086
    apply (simp add: algebra_simps)
huffman@29878
  1087
    done
wenzelm@60679
  1088
  show "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
huffman@29878
  1089
    unfolding less_eq_poly_def
huffman@29878
  1090
    apply safe
huffman@29878
  1091
    apply (drule (1) pos_poly_add)
huffman@29878
  1092
    apply simp
huffman@29878
  1093
    done
wenzelm@60679
  1094
  show "x \<le> y \<Longrightarrow> z + x \<le> z + y"
huffman@29878
  1095
    unfolding less_eq_poly_def
huffman@29878
  1096
    apply safe
huffman@29878
  1097
    apply (simp add: algebra_simps)
huffman@29878
  1098
    done
huffman@29878
  1099
  show "x \<le> y \<or> y \<le> x"
huffman@29878
  1100
    unfolding less_eq_poly_def
huffman@29878
  1101
    using pos_poly_total [of "x - y"]
huffman@29878
  1102
    by auto
wenzelm@60679
  1103
  show "x < y \<Longrightarrow> 0 < z \<Longrightarrow> z * x < z * y"
huffman@29878
  1104
    unfolding less_poly_def
huffman@29878
  1105
    by (simp add: right_diff_distrib [symmetric] pos_poly_mult)
huffman@29878
  1106
  show "\<bar>x\<bar> = (if x < 0 then - x else x)"
huffman@29878
  1107
    by (rule abs_poly_def)
huffman@29878
  1108
  show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
huffman@29878
  1109
    by (rule sgn_poly_def)
huffman@29878
  1110
qed
huffman@29878
  1111
huffman@29878
  1112
end
huffman@29878
  1113
wenzelm@60500
  1114
text \<open>TODO: Simplification rules for comparisons\<close>
huffman@29878
  1115
huffman@29878
  1116
wenzelm@60500
  1117
subsection \<open>Synthetic division and polynomial roots\<close>
haftmann@52380
  1118
wenzelm@60500
  1119
text \<open>
haftmann@52380
  1120
  Synthetic division is simply division by the linear polynomial @{term "x - c"}.
wenzelm@60500
  1121
\<close>
haftmann@52380
  1122
haftmann@52380
  1123
definition synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
haftmann@52380
  1124
where
haftmann@52380
  1125
  "synthetic_divmod p c = fold_coeffs (\<lambda>a (q, r). (pCons r q, a + c * r)) p (0, 0)"
haftmann@52380
  1126
haftmann@52380
  1127
definition synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
haftmann@52380
  1128
where
haftmann@52380
  1129
  "synthetic_div p c = fst (synthetic_divmod p c)"
haftmann@52380
  1130
haftmann@52380
  1131
lemma synthetic_divmod_0 [simp]:
haftmann@52380
  1132
  "synthetic_divmod 0 c = (0, 0)"
haftmann@52380
  1133
  by (simp add: synthetic_divmod_def)
haftmann@52380
  1134
haftmann@52380
  1135
lemma synthetic_divmod_pCons [simp]:
haftmann@52380
  1136
  "synthetic_divmod (pCons a p) c = (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
haftmann@52380
  1137
  by (cases "p = 0 \<and> a = 0") (auto simp add: synthetic_divmod_def)
haftmann@52380
  1138
haftmann@52380
  1139
lemma synthetic_div_0 [simp]:
haftmann@52380
  1140
  "synthetic_div 0 c = 0"
haftmann@52380
  1141
  unfolding synthetic_div_def by simp
haftmann@52380
  1142
haftmann@52380
  1143
lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
haftmann@52380
  1144
by (induct p arbitrary: a) simp_all
haftmann@52380
  1145
haftmann@52380
  1146
lemma snd_synthetic_divmod:
haftmann@52380
  1147
  "snd (synthetic_divmod p c) = poly p c"
haftmann@52380
  1148
  by (induct p, simp, simp add: split_def)
haftmann@52380
  1149
haftmann@52380
  1150
lemma synthetic_div_pCons [simp]:
haftmann@52380
  1151
  "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
haftmann@52380
  1152
  unfolding synthetic_div_def
haftmann@52380
  1153
  by (simp add: split_def snd_synthetic_divmod)
haftmann@52380
  1154
haftmann@52380
  1155
lemma synthetic_div_eq_0_iff:
haftmann@52380
  1156
  "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
haftmann@52380
  1157
  by (induct p, simp, case_tac p, simp)
haftmann@52380
  1158
haftmann@52380
  1159
lemma degree_synthetic_div:
haftmann@52380
  1160
  "degree (synthetic_div p c) = degree p - 1"
haftmann@52380
  1161
  by (induct p, simp, simp add: synthetic_div_eq_0_iff)
haftmann@52380
  1162
haftmann@52380
  1163
lemma synthetic_div_correct:
haftmann@52380
  1164
  "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
haftmann@52380
  1165
  by (induct p) simp_all
haftmann@52380
  1166
haftmann@52380
  1167
lemma synthetic_div_unique:
haftmann@52380
  1168
  "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
haftmann@52380
  1169
apply (induct p arbitrary: q r)
haftmann@52380
  1170
apply (simp, frule synthetic_div_unique_lemma, simp)
haftmann@52380
  1171
apply (case_tac q, force)
haftmann@52380
  1172
done
haftmann@52380
  1173
haftmann@52380
  1174
lemma synthetic_div_correct':
haftmann@52380
  1175
  fixes c :: "'a::comm_ring_1"
haftmann@52380
  1176
  shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
haftmann@52380
  1177
  using synthetic_div_correct [of p c]
haftmann@52380
  1178
  by (simp add: algebra_simps)
haftmann@52380
  1179
haftmann@52380
  1180
lemma poly_eq_0_iff_dvd:
haftmann@52380
  1181
  fixes c :: "'a::idom"
haftmann@52380
  1182
  shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
haftmann@52380
  1183
proof
haftmann@52380
  1184
  assume "poly p c = 0"
haftmann@52380
  1185
  with synthetic_div_correct' [of c p]
haftmann@52380
  1186
  have "p = [:-c, 1:] * synthetic_div p c" by simp
haftmann@52380
  1187
  then show "[:-c, 1:] dvd p" ..
haftmann@52380
  1188
next
haftmann@52380
  1189
  assume "[:-c, 1:] dvd p"
haftmann@52380
  1190
  then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
haftmann@52380
  1191
  then show "poly p c = 0" by simp
haftmann@52380
  1192
qed
haftmann@52380
  1193
haftmann@52380
  1194
lemma dvd_iff_poly_eq_0:
haftmann@52380
  1195
  fixes c :: "'a::idom"
haftmann@52380
  1196
  shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
haftmann@52380
  1197
  by (simp add: poly_eq_0_iff_dvd)
haftmann@52380
  1198
haftmann@52380
  1199
lemma poly_roots_finite:
haftmann@52380
  1200
  fixes p :: "'a::idom poly"
haftmann@52380
  1201
  shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
haftmann@52380
  1202
proof (induct n \<equiv> "degree p" arbitrary: p)
haftmann@52380
  1203
  case (0 p)
haftmann@52380
  1204
  then obtain a where "a \<noteq> 0" and "p = [:a:]"
haftmann@52380
  1205
    by (cases p, simp split: if_splits)
haftmann@52380
  1206
  then show "finite {x. poly p x = 0}" by simp
haftmann@52380
  1207
next
haftmann@52380
  1208
  case (Suc n p)
haftmann@52380
  1209
  show "finite {x. poly p x = 0}"
haftmann@52380
  1210
  proof (cases "\<exists>x. poly p x = 0")
haftmann@52380
  1211
    case False
haftmann@52380
  1212
    then show "finite {x. poly p x = 0}" by simp
haftmann@52380
  1213
  next
haftmann@52380
  1214
    case True
haftmann@52380
  1215
    then obtain a where "poly p a = 0" ..
haftmann@52380
  1216
    then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
haftmann@52380
  1217
    then obtain k where k: "p = [:-a, 1:] * k" ..
wenzelm@60500
  1218
    with \<open>p \<noteq> 0\<close> have "k \<noteq> 0" by auto
haftmann@52380
  1219
    with k have "degree p = Suc (degree k)"
haftmann@52380
  1220
      by (simp add: degree_mult_eq del: mult_pCons_left)
wenzelm@60500
  1221
    with \<open>Suc n = degree p\<close> have "n = degree k" by simp
wenzelm@60500
  1222
    then have "finite {x. poly k x = 0}" using \<open>k \<noteq> 0\<close> by (rule Suc.hyps)
haftmann@52380
  1223
    then have "finite (insert a {x. poly k x = 0})" by simp
haftmann@52380
  1224
    then show "finite {x. poly p x = 0}"
wenzelm@57862
  1225
      by (simp add: k Collect_disj_eq del: mult_pCons_left)
haftmann@52380
  1226
  qed
haftmann@52380
  1227
qed
haftmann@52380
  1228
haftmann@52380
  1229
lemma poly_eq_poly_eq_iff:
haftmann@52380
  1230
  fixes p q :: "'a::{idom,ring_char_0} poly"
haftmann@52380
  1231
  shows "poly p = poly q \<longleftrightarrow> p = q" (is "?P \<longleftrightarrow> ?Q")
haftmann@52380
  1232
proof
haftmann@52380
  1233
  assume ?Q then show ?P by simp
haftmann@52380
  1234
next
haftmann@52380
  1235
  { fix p :: "'a::{idom,ring_char_0} poly"
haftmann@52380
  1236
    have "poly p = poly 0 \<longleftrightarrow> p = 0"
haftmann@52380
  1237
      apply (cases "p = 0", simp_all)
haftmann@52380
  1238
      apply (drule poly_roots_finite)
haftmann@52380
  1239
      apply (auto simp add: infinite_UNIV_char_0)
haftmann@52380
  1240
      done
haftmann@52380
  1241
  } note this [of "p - q"]
haftmann@52380
  1242
  moreover assume ?P
haftmann@52380
  1243
  ultimately show ?Q by auto
haftmann@52380
  1244
qed
haftmann@52380
  1245
haftmann@52380
  1246
lemma poly_all_0_iff_0:
haftmann@52380
  1247
  fixes p :: "'a::{ring_char_0, idom} poly"
haftmann@52380
  1248
  shows "(\<forall>x. poly p x = 0) \<longleftrightarrow> p = 0"
haftmann@52380
  1249
  by (auto simp add: poly_eq_poly_eq_iff [symmetric])
haftmann@52380
  1250
haftmann@52380
  1251
wenzelm@60500
  1252
subsection \<open>Long division of polynomials\<close>
huffman@29451
  1253
haftmann@52380
  1254
definition pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
huffman@29451
  1255
where
huffman@29537
  1256
  "pdivmod_rel x y q r \<longleftrightarrow>
huffman@29451
  1257
    x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
huffman@29451
  1258
huffman@29537
  1259
lemma pdivmod_rel_0:
huffman@29537
  1260
  "pdivmod_rel 0 y 0 0"
huffman@29537
  1261
  unfolding pdivmod_rel_def by simp
huffman@29451
  1262
huffman@29537
  1263
lemma pdivmod_rel_by_0:
huffman@29537
  1264
  "pdivmod_rel x 0 0 x"
huffman@29537
  1265
  unfolding pdivmod_rel_def by simp
huffman@29451
  1266
huffman@29451
  1267
lemma eq_zero_or_degree_less:
huffman@29451
  1268
  assumes "degree p \<le> n" and "coeff p n = 0"
huffman@29451
  1269
  shows "p = 0 \<or> degree p < n"
huffman@29451
  1270
proof (cases n)
huffman@29451
  1271
  case 0
wenzelm@60500
  1272
  with \<open>degree p \<le> n\<close> and \<open>coeff p n = 0\<close>
huffman@29451
  1273
  have "coeff p (degree p) = 0" by simp
huffman@29451
  1274
  then have "p = 0" by simp
huffman@29451
  1275
  then show ?thesis ..
huffman@29451
  1276
next
huffman@29451
  1277
  case (Suc m)
huffman@29451
  1278
  have "\<forall>i>n. coeff p i = 0"
wenzelm@60500
  1279
    using \<open>degree p \<le> n\<close> by (simp add: coeff_eq_0)
huffman@29451
  1280
  then have "\<forall>i\<ge>n. coeff p i = 0"
wenzelm@60500
  1281
    using \<open>coeff p n = 0\<close> by (simp add: le_less)
huffman@29451
  1282
  then have "\<forall>i>m. coeff p i = 0"
wenzelm@60500
  1283
    using \<open>n = Suc m\<close> by (simp add: less_eq_Suc_le)
huffman@29451
  1284
  then have "degree p \<le> m"
huffman@29451
  1285
    by (rule degree_le)
huffman@29451
  1286
  then have "degree p < n"
wenzelm@60500
  1287
    using \<open>n = Suc m\<close> by (simp add: less_Suc_eq_le)
huffman@29451
  1288
  then show ?thesis ..
huffman@29451
  1289
qed
huffman@29451
  1290
huffman@29537
  1291
lemma pdivmod_rel_pCons:
huffman@29537
  1292
  assumes rel: "pdivmod_rel x y q r"
huffman@29451
  1293
  assumes y: "y \<noteq> 0"
huffman@29451
  1294
  assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
huffman@29537
  1295
  shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
huffman@29537
  1296
    (is "pdivmod_rel ?x y ?q ?r")
huffman@29451
  1297
proof -
huffman@29451
  1298
  have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
huffman@29537
  1299
    using assms unfolding pdivmod_rel_def by simp_all
huffman@29451
  1300
huffman@29451
  1301
  have 1: "?x = ?q * y + ?r"
huffman@29451
  1302
    using b x by simp
huffman@29451
  1303
huffman@29451
  1304
  have 2: "?r = 0 \<or> degree ?r < degree y"
huffman@29451
  1305
  proof (rule eq_zero_or_degree_less)
huffman@29539
  1306
    show "degree ?r \<le> degree y"
huffman@29539
  1307
    proof (rule degree_diff_le)
huffman@29451
  1308
      show "degree (pCons a r) \<le> degree y"
huffman@29460
  1309
        using r by auto
huffman@29451
  1310
      show "degree (smult b y) \<le> degree y"
huffman@29451
  1311
        by (rule degree_smult_le)
huffman@29451
  1312
    qed
huffman@29451
  1313
  next
huffman@29451
  1314
    show "coeff ?r (degree y) = 0"
wenzelm@60500
  1315
      using \<open>y \<noteq> 0\<close> unfolding b by simp
huffman@29451
  1316
  qed
huffman@29451
  1317
huffman@29451
  1318
  from 1 2 show ?thesis
huffman@29537
  1319
    unfolding pdivmod_rel_def
wenzelm@60500
  1320
    using \<open>y \<noteq> 0\<close> by simp
huffman@29451
  1321
qed
huffman@29451
  1322
huffman@29537
  1323
lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r"
huffman@29451
  1324
apply (cases "y = 0")
huffman@29537
  1325
apply (fast intro!: pdivmod_rel_by_0)
huffman@29451
  1326
apply (induct x)
huffman@29537
  1327
apply (fast intro!: pdivmod_rel_0)
huffman@29537
  1328
apply (fast intro!: pdivmod_rel_pCons)
huffman@29451
  1329
done
huffman@29451
  1330
huffman@29537
  1331
lemma pdivmod_rel_unique:
huffman@29537
  1332
  assumes 1: "pdivmod_rel x y q1 r1"
huffman@29537
  1333
  assumes 2: "pdivmod_rel x y q2 r2"
huffman@29451
  1334
  shows "q1 = q2 \<and> r1 = r2"
huffman@29451
  1335
proof (cases "y = 0")
huffman@29451
  1336
  assume "y = 0" with assms show ?thesis
huffman@29537
  1337
    by (simp add: pdivmod_rel_def)
huffman@29451
  1338
next
huffman@29451
  1339
  assume [simp]: "y \<noteq> 0"
huffman@29451
  1340
  from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
huffman@29537
  1341
    unfolding pdivmod_rel_def by simp_all
huffman@29451
  1342
  from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
huffman@29537
  1343
    unfolding pdivmod_rel_def by simp_all
huffman@29451
  1344
  from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
nipkow@29667
  1345
    by (simp add: algebra_simps)
huffman@29451
  1346
  from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
huffman@29453
  1347
    by (auto intro: degree_diff_less)
huffman@29451
  1348
huffman@29451
  1349
  show "q1 = q2 \<and> r1 = r2"
huffman@29451
  1350
  proof (rule ccontr)
huffman@29451
  1351
    assume "\<not> (q1 = q2 \<and> r1 = r2)"
huffman@29451
  1352
    with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
huffman@29451
  1353
    with r3 have "degree (r2 - r1) < degree y" by simp
huffman@29451
  1354
    also have "degree y \<le> degree (q1 - q2) + degree y" by simp
huffman@29451
  1355
    also have "\<dots> = degree ((q1 - q2) * y)"
wenzelm@60500
  1356
      using \<open>q1 \<noteq> q2\<close> by (simp add: degree_mult_eq)
huffman@29451
  1357
    also have "\<dots> = degree (r2 - r1)"
huffman@29451
  1358
      using q3 by simp
huffman@29451
  1359
    finally have "degree (r2 - r1) < degree (r2 - r1)" .
huffman@29451
  1360
    then show "False" by simp
huffman@29451
  1361
  qed
huffman@29451
  1362
qed
huffman@29451
  1363
huffman@29660
  1364
lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0"
huffman@29660
  1365
by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0)
huffman@29660
  1366
huffman@29660
  1367
lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x"
huffman@29660
  1368
by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0)
huffman@29660
  1369
wenzelm@45605
  1370
lemmas pdivmod_rel_unique_div = pdivmod_rel_unique [THEN conjunct1]
huffman@29451
  1371
wenzelm@45605
  1372
lemmas pdivmod_rel_unique_mod = pdivmod_rel_unique [THEN conjunct2]
huffman@29451
  1373
huffman@29451
  1374
instantiation poly :: (field) ring_div
huffman@29451
  1375
begin
huffman@29451
  1376
haftmann@60352
  1377
definition divide_poly where
haftmann@60429
  1378
  div_poly_def: "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)"
huffman@29451
  1379
huffman@29451
  1380
definition mod_poly where
haftmann@37765
  1381
  "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)"
huffman@29451
  1382
huffman@29451
  1383
lemma div_poly_eq:
haftmann@60429
  1384
  "pdivmod_rel x y q r \<Longrightarrow> x div y = q"
huffman@29451
  1385
unfolding div_poly_def
huffman@29537
  1386
by (fast elim: pdivmod_rel_unique_div)
huffman@29451
  1387
huffman@29451
  1388
lemma mod_poly_eq:
huffman@29537
  1389
  "pdivmod_rel x y q r \<Longrightarrow> x mod y = r"
huffman@29451
  1390
unfolding mod_poly_def
huffman@29537
  1391
by (fast elim: pdivmod_rel_unique_mod)
huffman@29451
  1392
huffman@29537
  1393
lemma pdivmod_rel:
haftmann@60429
  1394
  "pdivmod_rel x y (x div y) (x mod y)"
huffman@29451
  1395
proof -
huffman@29537
  1396
  from pdivmod_rel_exists
huffman@29537
  1397
    obtain q r where "pdivmod_rel x y q r" by fast
huffman@29451
  1398
  thus ?thesis
huffman@29451
  1399
    by (simp add: div_poly_eq mod_poly_eq)
huffman@29451
  1400
qed
huffman@29451
  1401
wenzelm@60679
  1402
instance
wenzelm@60679
  1403
proof
huffman@29451
  1404
  fix x y :: "'a poly"
haftmann@60429
  1405
  show "x div y * y + x mod y = x"
huffman@29537
  1406
    using pdivmod_rel [of x y]
huffman@29537
  1407
    by (simp add: pdivmod_rel_def)
huffman@29451
  1408
next
huffman@29451
  1409
  fix x :: "'a poly"
huffman@29537
  1410
  have "pdivmod_rel x 0 0 x"
huffman@29537
  1411
    by (rule pdivmod_rel_by_0)
haftmann@60429
  1412
  thus "x div 0 = 0"
huffman@29451
  1413
    by (rule div_poly_eq)
huffman@29451
  1414
next
huffman@29451
  1415
  fix y :: "'a poly"
huffman@29537
  1416
  have "pdivmod_rel 0 y 0 0"
huffman@29537
  1417
    by (rule pdivmod_rel_0)
haftmann@60429
  1418
  thus "0 div y = 0"
huffman@29451
  1419
    by (rule div_poly_eq)
huffman@29451
  1420
next
huffman@29451
  1421
  fix x y z :: "'a poly"
huffman@29451
  1422
  assume "y \<noteq> 0"
haftmann@60429
  1423
  hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"
huffman@29537
  1424
    using pdivmod_rel [of x y]
webertj@49962
  1425
    by (simp add: pdivmod_rel_def distrib_right)
haftmann@60429
  1426
  thus "(x + z * y) div y = z + x div y"
huffman@29451
  1427
    by (rule div_poly_eq)
haftmann@30930
  1428
next
haftmann@30930
  1429
  fix x y z :: "'a poly"
haftmann@30930
  1430
  assume "x \<noteq> 0"
haftmann@60429
  1431
  show "(x * y) div (x * z) = y div z"
haftmann@30930
  1432
  proof (cases "y \<noteq> 0 \<and> z \<noteq> 0")
haftmann@30930
  1433
    have "\<And>x::'a poly. pdivmod_rel x 0 0 x"
haftmann@30930
  1434
      by (rule pdivmod_rel_by_0)
haftmann@60429
  1435
    then have [simp]: "\<And>x::'a poly. x div 0 = 0"
haftmann@30930
  1436
      by (rule div_poly_eq)
haftmann@30930
  1437
    have "\<And>x::'a poly. pdivmod_rel 0 x 0 0"
haftmann@30930
  1438
      by (rule pdivmod_rel_0)
haftmann@60429
  1439
    then have [simp]: "\<And>x::'a poly. 0 div x = 0"
haftmann@30930
  1440
      by (rule div_poly_eq)
haftmann@30930
  1441
    case False then show ?thesis by auto
haftmann@30930
  1442
  next
haftmann@30930
  1443
    case True then have "y \<noteq> 0" and "z \<noteq> 0" by auto
wenzelm@60500
  1444
    with \<open>x \<noteq> 0\<close>
haftmann@30930
  1445
    have "\<And>q r. pdivmod_rel y z q r \<Longrightarrow> pdivmod_rel (x * y) (x * z) q (x * r)"
haftmann@30930
  1446
      by (auto simp add: pdivmod_rel_def algebra_simps)
haftmann@30930
  1447
        (rule classical, simp add: degree_mult_eq)
haftmann@60429
  1448
    moreover from pdivmod_rel have "pdivmod_rel y z (y div z) (y mod z)" .
haftmann@60429
  1449
    ultimately have "pdivmod_rel (x * y) (x * z) (y div z) (x * (y mod z))" .
haftmann@30930
  1450
    then show ?thesis by (simp add: div_poly_eq)
haftmann@30930
  1451
  qed
huffman@29451
  1452
qed
huffman@29451
  1453
huffman@29451
  1454
end
huffman@29451
  1455
haftmann@60570
  1456
lemma is_unit_monom_0:
haftmann@60570
  1457
  fixes a :: "'a::field"
haftmann@60570
  1458
  assumes "a \<noteq> 0"
haftmann@60570
  1459
  shows "is_unit (monom a 0)"
haftmann@60570
  1460
proof
haftmann@60570
  1461
  from assms show "1 = monom a 0 * monom (1 / a) 0"
haftmann@60570
  1462
    by (simp add: mult_monom)
haftmann@60570
  1463
qed
haftmann@60570
  1464
haftmann@60570
  1465
lemma is_unit_triv:
haftmann@60570
  1466
  fixes a :: "'a::field"
haftmann@60570
  1467
  assumes "a \<noteq> 0"
haftmann@60570
  1468
  shows "is_unit [:a:]"
haftmann@60570
  1469
  using assms by (simp add: is_unit_monom_0 monom_0 [symmetric])
haftmann@60570
  1470
haftmann@60570
  1471
lemma is_unit_iff_degree:
haftmann@60570
  1472
  assumes "p \<noteq> 0"
haftmann@60570
  1473
  shows "is_unit p \<longleftrightarrow> degree p = 0" (is "?P \<longleftrightarrow> ?Q")
haftmann@60570
  1474
proof
haftmann@60570
  1475
  assume ?Q
haftmann@60570
  1476
  then obtain a where "p = [:a:]" by (rule degree_eq_zeroE)
haftmann@60570
  1477
  with assms show ?P by (simp add: is_unit_triv)
haftmann@60570
  1478
next
haftmann@60570
  1479
  assume ?P
haftmann@60570
  1480
  then obtain q where "q \<noteq> 0" "p * q = 1" ..
haftmann@60570
  1481
  then have "degree (p * q) = degree 1"
haftmann@60570
  1482
    by simp
haftmann@60570
  1483
  with \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close> have "degree p + degree q = 0"
haftmann@60570
  1484
    by (simp add: degree_mult_eq)
haftmann@60570
  1485
  then show ?Q by simp
haftmann@60570
  1486
qed
haftmann@60570
  1487
haftmann@60570
  1488
lemma is_unit_pCons_iff:
haftmann@60570
  1489
  "is_unit (pCons a p) \<longleftrightarrow> p = 0 \<and> a \<noteq> 0" (is "?P \<longleftrightarrow> ?Q")
haftmann@60570
  1490
  by (cases "p = 0") (auto simp add: is_unit_triv is_unit_iff_degree)
haftmann@60570
  1491
haftmann@60570
  1492
lemma is_unit_monom_trival:
haftmann@60570
  1493
  fixes p :: "'a::field poly"
haftmann@60570
  1494
  assumes "is_unit p"
haftmann@60570
  1495
  shows "monom (coeff p (degree p)) 0 = p"
haftmann@60570
  1496
  using assms by (cases p) (simp_all add: monom_0 is_unit_pCons_iff)
haftmann@60570
  1497
haftmann@60685
  1498
lemma is_unit_polyE:
haftmann@60685
  1499
  assumes "is_unit p"
haftmann@60685
  1500
  obtains a where "p = monom a 0" and "a \<noteq> 0"
haftmann@60685
  1501
proof -
haftmann@60685
  1502
  obtain a q where "p = pCons a q" by (cases p)
haftmann@60685
  1503
  with assms have "p = [:a:]" and "a \<noteq> 0"
haftmann@60685
  1504
    by (simp_all add: is_unit_pCons_iff)
haftmann@60685
  1505
  with that show thesis by (simp add: monom_0)
haftmann@60685
  1506
qed
haftmann@60685
  1507
haftmann@60685
  1508
instantiation poly :: (field) normalization_semidom
haftmann@60685
  1509
begin
haftmann@60685
  1510
haftmann@60685
  1511
definition normalize_poly :: "'a poly \<Rightarrow> 'a poly"
haftmann@60685
  1512
  where "normalize_poly p = smult (1 / coeff p (degree p)) p"
haftmann@60685
  1513
haftmann@60685
  1514
definition unit_factor_poly :: "'a poly \<Rightarrow> 'a poly"
haftmann@60685
  1515
  where "unit_factor_poly p = monom (coeff p (degree p)) 0"
haftmann@60685
  1516
haftmann@60685
  1517
instance
haftmann@60685
  1518
proof
haftmann@60685
  1519
  fix p :: "'a poly"
haftmann@60685
  1520
  show "unit_factor p * normalize p = p"
haftmann@60685
  1521
    by (simp add: normalize_poly_def unit_factor_poly_def)
haftmann@60685
  1522
      (simp only: mult_smult_left [symmetric] smult_monom, simp)
haftmann@60685
  1523
next
haftmann@60685
  1524
  show "normalize 0 = (0::'a poly)"
haftmann@60685
  1525
    by (simp add: normalize_poly_def)
haftmann@60685
  1526
next
haftmann@60685
  1527
  show "unit_factor 0 = (0::'a poly)"
haftmann@60685
  1528
    by (simp add: unit_factor_poly_def)
haftmann@60685
  1529
next
haftmann@60685
  1530
  fix p :: "'a poly"
haftmann@60685
  1531
  assume "is_unit p"
haftmann@60685
  1532
  then obtain a where "p = monom a 0" and "a \<noteq> 0"
haftmann@60685
  1533
    by (rule is_unit_polyE)
haftmann@60685
  1534
  then show "normalize p = 1"
haftmann@60685
  1535
    by (auto simp add: normalize_poly_def smult_monom degree_monom_eq)
haftmann@60685
  1536
next
haftmann@60685
  1537
  fix p q :: "'a poly"
haftmann@60685
  1538
  assume "q \<noteq> 0"
haftmann@60685
  1539
  from \<open>q \<noteq> 0\<close> have "is_unit (monom (coeff q (degree q)) 0)"
haftmann@60685
  1540
    by (auto intro: is_unit_monom_0)
haftmann@60685
  1541
  then show "is_unit (unit_factor q)"
haftmann@60685
  1542
    by (simp add: unit_factor_poly_def)
haftmann@60685
  1543
next
haftmann@60685
  1544
  fix p q :: "'a poly"
haftmann@60685
  1545
  have "monom (coeff (p * q) (degree (p * q))) 0 =
haftmann@60685
  1546
    monom (coeff p (degree p)) 0 * monom (coeff q (degree q)) 0"
haftmann@60685
  1547
    by (simp add: monom_0 coeff_degree_mult)
haftmann@60685
  1548
  then show "unit_factor (p * q) =
haftmann@60685
  1549
    unit_factor p * unit_factor q"
haftmann@60685
  1550
    by (simp add: unit_factor_poly_def)
haftmann@60685
  1551
qed
haftmann@60685
  1552
haftmann@60685
  1553
end
haftmann@60685
  1554
huffman@29451
  1555
lemma degree_mod_less:
huffman@29451
  1556
  "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
huffman@29537
  1557
  using pdivmod_rel [of x y]
huffman@29537
  1558
  unfolding pdivmod_rel_def by simp
huffman@29451
  1559
huffman@29451
  1560
lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
huffman@29451
  1561
proof -
huffman@29451
  1562
  assume "degree x < degree y"
huffman@29537
  1563
  hence "pdivmod_rel x y 0 x"
huffman@29537
  1564
    by (simp add: pdivmod_rel_def)
huffman@29451
  1565
  thus "x div y = 0" by (rule div_poly_eq)
huffman@29451
  1566
qed
huffman@29451
  1567
huffman@29451
  1568
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
huffman@29451
  1569
proof -
huffman@29451
  1570
  assume "degree x < degree y"
huffman@29537
  1571
  hence "pdivmod_rel x y 0 x"
huffman@29537
  1572
    by (simp add: pdivmod_rel_def)
huffman@29451
  1573
  thus "x mod y = x" by (rule mod_poly_eq)
huffman@29451
  1574
qed
huffman@29451
  1575
huffman@29659
  1576
lemma pdivmod_rel_smult_left:
huffman@29659
  1577
  "pdivmod_rel x y q r
huffman@29659
  1578
    \<Longrightarrow> pdivmod_rel (smult a x) y (smult a q) (smult a r)"
huffman@29659
  1579
  unfolding pdivmod_rel_def by (simp add: smult_add_right)
huffman@29659
  1580
huffman@29659
  1581
lemma div_smult_left: "(smult a x) div y = smult a (x div y)"
huffman@29659
  1582
  by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
huffman@29659
  1583
huffman@29659
  1584
lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)"
huffman@29659
  1585
  by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
huffman@29659
  1586
huffman@30072
  1587
lemma poly_div_minus_left [simp]:
huffman@30072
  1588
  fixes x y :: "'a::field poly"
huffman@30072
  1589
  shows "(- x) div y = - (x div y)"
haftmann@54489
  1590
  using div_smult_left [of "- 1::'a"] by simp
huffman@30072
  1591
huffman@30072
  1592
lemma poly_mod_minus_left [simp]:
huffman@30072
  1593
  fixes x y :: "'a::field poly"
huffman@30072
  1594
  shows "(- x) mod y = - (x mod y)"
haftmann@54489
  1595
  using mod_smult_left [of "- 1::'a"] by simp
huffman@30072
  1596
huffman@57482
  1597
lemma pdivmod_rel_add_left:
huffman@57482
  1598
  assumes "pdivmod_rel x y q r"
huffman@57482
  1599
  assumes "pdivmod_rel x' y q' r'"
huffman@57482
  1600
  shows "pdivmod_rel (x + x') y (q + q') (r + r')"
huffman@57482
  1601
  using assms unfolding pdivmod_rel_def
haftmann@59557
  1602
  by (auto simp add: algebra_simps degree_add_less)
huffman@57482
  1603
huffman@57482
  1604
lemma poly_div_add_left:
huffman@57482
  1605
  fixes x y z :: "'a::field poly"
huffman@57482
  1606
  shows "(x + y) div z = x div z + y div z"
huffman@57482
  1607
  using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel]
huffman@57482
  1608
  by (rule div_poly_eq)
huffman@57482
  1609
huffman@57482
  1610
lemma poly_mod_add_left:
huffman@57482
  1611
  fixes x y z :: "'a::field poly"
huffman@57482
  1612
  shows "(x + y) mod z = x mod z + y mod z"
huffman@57482
  1613
  using pdivmod_rel_add_left [OF pdivmod_rel pdivmod_rel]
huffman@57482
  1614
  by (rule mod_poly_eq)
huffman@57482
  1615
huffman@57482
  1616
lemma poly_div_diff_left:
huffman@57482
  1617
  fixes x y z :: "'a::field poly"
huffman@57482
  1618
  shows "(x - y) div z = x div z - y div z"
huffman@57482
  1619
  by (simp only: diff_conv_add_uminus poly_div_add_left poly_div_minus_left)
huffman@57482
  1620
huffman@57482
  1621
lemma poly_mod_diff_left:
huffman@57482
  1622
  fixes x y z :: "'a::field poly"
huffman@57482
  1623
  shows "(x - y) mod z = x mod z - y mod z"
huffman@57482
  1624
  by (simp only: diff_conv_add_uminus poly_mod_add_left poly_mod_minus_left)
huffman@57482
  1625
huffman@29659
  1626
lemma pdivmod_rel_smult_right:
huffman@29659
  1627
  "\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>
huffman@29659
  1628
    \<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r"
huffman@29659
  1629
  unfolding pdivmod_rel_def by simp
huffman@29659
  1630
huffman@29659
  1631
lemma div_smult_right:
huffman@29659
  1632
  "a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)"
huffman@29659
  1633
  by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
huffman@29659
  1634
huffman@29659
  1635
lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y"
huffman@29659
  1636
  by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
huffman@29659
  1637
huffman@30072
  1638
lemma poly_div_minus_right [simp]:
huffman@30072
  1639
  fixes x y :: "'a::field poly"
huffman@30072
  1640
  shows "x div (- y) = - (x div y)"
haftmann@54489
  1641
  using div_smult_right [of "- 1::'a"] by (simp add: nonzero_inverse_minus_eq)
huffman@30072
  1642
huffman@30072
  1643
lemma poly_mod_minus_right [simp]:
huffman@30072
  1644
  fixes x y :: "'a::field poly"
huffman@30072
  1645
  shows "x mod (- y) = x mod y"
haftmann@54489
  1646
  using mod_smult_right [of "- 1::'a"] by simp
huffman@30072
  1647
huffman@29660
  1648
lemma pdivmod_rel_mult:
huffman@29660
  1649
  "\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk>
huffman@29660
  1650
    \<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)"
huffman@29660
  1651
apply (cases "z = 0", simp add: pdivmod_rel_def)
huffman@29660
  1652
apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff)
huffman@29660
  1653
apply (cases "r = 0")
huffman@29660
  1654
apply (cases "r' = 0")
huffman@29660
  1655
apply (simp add: pdivmod_rel_def)
haftmann@36350
  1656
apply (simp add: pdivmod_rel_def field_simps degree_mult_eq)
huffman@29660
  1657
apply (cases "r' = 0")
huffman@29660
  1658
apply (simp add: pdivmod_rel_def degree_mult_eq)
haftmann@36350
  1659
apply (simp add: pdivmod_rel_def field_simps)
huffman@29660
  1660
apply (simp add: degree_mult_eq degree_add_less)
huffman@29660
  1661
done
huffman@29660
  1662
huffman@29660
  1663
lemma poly_div_mult_right:
huffman@29660
  1664
  fixes x y z :: "'a::field poly"
huffman@29660
  1665
  shows "x div (y * z) = (x div y) div z"
huffman@29660
  1666
  by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
huffman@29660
  1667
huffman@29660
  1668
lemma poly_mod_mult_right:
huffman@29660
  1669
  fixes x y z :: "'a::field poly"
huffman@29660
  1670
  shows "x mod (y * z) = y * (x div y mod z) + x mod y"
huffman@29660
  1671
  by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
huffman@29660
  1672
huffman@29451
  1673
lemma mod_pCons:
huffman@29451
  1674
  fixes a and x
huffman@29451
  1675
  assumes y: "y \<noteq> 0"
huffman@29451
  1676
  defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
huffman@29451
  1677
  shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
huffman@29451
  1678
unfolding b
huffman@29451
  1679
apply (rule mod_poly_eq)
huffman@29537
  1680
apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
huffman@29451
  1681
done
huffman@29451
  1682
haftmann@52380
  1683
definition pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
haftmann@52380
  1684
where
haftmann@52380
  1685
  "pdivmod p q = (p div q, p mod q)"
huffman@31663
  1686
haftmann@52380
  1687
lemma div_poly_code [code]: 
haftmann@52380
  1688
  "p div q = fst (pdivmod p q)"
haftmann@52380
  1689
  by (simp add: pdivmod_def)
huffman@31663
  1690
haftmann@52380
  1691
lemma mod_poly_code [code]:
haftmann@52380
  1692
  "p mod q = snd (pdivmod p q)"
haftmann@52380
  1693
  by (simp add: pdivmod_def)
huffman@31663
  1694
haftmann@52380
  1695
lemma pdivmod_0:
haftmann@52380
  1696
  "pdivmod 0 q = (0, 0)"
haftmann@52380
  1697
  by (simp add: pdivmod_def)
huffman@31663
  1698
haftmann@52380
  1699
lemma pdivmod_pCons:
haftmann@52380
  1700
  "pdivmod (pCons a p) q =
haftmann@52380
  1701
    (if q = 0 then (0, pCons a p) else
haftmann@52380
  1702
      (let (s, r) = pdivmod p q;
haftmann@52380
  1703
           b = coeff (pCons a r) (degree q) / coeff q (degree q)
haftmann@52380
  1704
        in (pCons b s, pCons a r - smult b q)))"
haftmann@52380
  1705
  apply (simp add: pdivmod_def Let_def, safe)
haftmann@52380
  1706
  apply (rule div_poly_eq)
haftmann@52380
  1707
  apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
haftmann@52380
  1708
  apply (rule mod_poly_eq)
haftmann@52380
  1709
  apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
huffman@29451
  1710
  done
huffman@29451
  1711
haftmann@52380
  1712
lemma pdivmod_fold_coeffs [code]:
haftmann@52380
  1713
  "pdivmod p q = (if q = 0 then (0, p)
haftmann@52380
  1714
    else fold_coeffs (\<lambda>a (s, r).
haftmann@52380
  1715
      let b = coeff (pCons a r) (degree q) / coeff q (degree q)
haftmann@52380
  1716
      in (pCons b s, pCons a r - smult b q)
haftmann@52380
  1717
   ) p (0, 0))"
haftmann@52380
  1718
  apply (cases "q = 0")
haftmann@52380
  1719
  apply (simp add: pdivmod_def)
haftmann@52380
  1720
  apply (rule sym)
haftmann@52380
  1721
  apply (induct p)
haftmann@52380
  1722
  apply (simp_all add: pdivmod_0 pdivmod_pCons)
haftmann@52380
  1723
  apply (case_tac "a = 0 \<and> p = 0")
haftmann@52380
  1724
  apply (auto simp add: pdivmod_def)
haftmann@52380
  1725
  done
huffman@29980
  1726
huffman@29980
  1727
wenzelm@60500
  1728
subsection \<open>Order of polynomial roots\<close>
huffman@29977
  1729
haftmann@52380
  1730
definition order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat"
huffman@29977
  1731
where
huffman@29977
  1732
  "order a p = (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)"
huffman@29977
  1733
huffman@29977
  1734
lemma coeff_linear_power:
huffman@29979
  1735
  fixes a :: "'a::comm_semiring_1"
huffman@29977
  1736
  shows "coeff ([:a, 1:] ^ n) n = 1"
huffman@29977
  1737
apply (induct n, simp_all)
huffman@29977
  1738
apply (subst coeff_eq_0)
huffman@29977
  1739
apply (auto intro: le_less_trans degree_power_le)
huffman@29977
  1740
done
huffman@29977
  1741
huffman@29977
  1742
lemma degree_linear_power:
huffman@29979
  1743
  fixes a :: "'a::comm_semiring_1"
huffman@29977
  1744
  shows "degree ([:a, 1:] ^ n) = n"
huffman@29977
  1745
apply (rule order_antisym)
huffman@29977
  1746
apply (rule ord_le_eq_trans [OF degree_power_le], simp)
huffman@29977
  1747
apply (rule le_degree, simp add: coeff_linear_power)
huffman@29977
  1748
done
huffman@29977
  1749
huffman@29977
  1750
lemma order_1: "[:-a, 1:] ^ order a p dvd p"
huffman@29977
  1751
apply (cases "p = 0", simp)
huffman@29977
  1752
apply (cases "order a p", simp)
huffman@29977
  1753
apply (subgoal_tac "nat < (LEAST n. \<not> [:-a, 1:] ^ Suc n dvd p)")
huffman@29977
  1754
apply (drule not_less_Least, simp)
huffman@29977
  1755
apply (fold order_def, simp)
huffman@29977
  1756
done
huffman@29977
  1757
huffman@29977
  1758
lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
huffman@29977
  1759
unfolding order_def
huffman@29977
  1760
apply (rule LeastI_ex)
huffman@29977
  1761
apply (rule_tac x="degree p" in exI)
huffman@29977
  1762
apply (rule notI)
huffman@29977
  1763
apply (drule (1) dvd_imp_degree_le)
huffman@29977
  1764
apply (simp only: degree_linear_power)
huffman@29977
  1765
done
huffman@29977
  1766
huffman@29977
  1767
lemma order:
huffman@29977
  1768
  "p \<noteq> 0 \<Longrightarrow> [:-a, 1:] ^ order a p dvd p \<and> \<not> [:-a, 1:] ^ Suc (order a p) dvd p"
huffman@29977
  1769
by (rule conjI [OF order_1 order_2])
huffman@29977
  1770
huffman@29977
  1771
lemma order_degree:
huffman@29977
  1772
  assumes p: "p \<noteq> 0"
huffman@29977
  1773
  shows "order a p \<le> degree p"
huffman@29977
  1774
proof -
huffman@29977
  1775
  have "order a p = degree ([:-a, 1:] ^ order a p)"
huffman@29977
  1776
    by (simp only: degree_linear_power)
huffman@29977
  1777
  also have "\<dots> \<le> degree p"
huffman@29977
  1778
    using order_1 p by (rule dvd_imp_degree_le)
huffman@29977
  1779
  finally show ?thesis .
huffman@29977
  1780
qed
huffman@29977
  1781
huffman@29977
  1782
lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0"
huffman@29977
  1783
apply (cases "p = 0", simp_all)
huffman@29977
  1784
apply (rule iffI)
lp15@56383
  1785
apply (metis order_2 not_gr0 poly_eq_0_iff_dvd power_0 power_Suc_0 power_one_right)
lp15@56383
  1786
unfolding poly_eq_0_iff_dvd
lp15@56383
  1787
apply (metis dvd_power dvd_trans order_1)
huffman@29977
  1788
done
huffman@29977
  1789
huffman@29977
  1790
wenzelm@60500
  1791
subsection \<open>GCD of polynomials\<close>
huffman@29478
  1792
haftmann@52380
  1793
instantiation poly :: (field) gcd
huffman@29478
  1794
begin
huffman@29478
  1795
haftmann@52380
  1796
function gcd_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
haftmann@52380
  1797
where
haftmann@52380
  1798
  "gcd (x::'a poly) 0 = smult (inverse (coeff x (degree x))) x"
haftmann@52380
  1799
| "y \<noteq> 0 \<Longrightarrow> gcd (x::'a poly) y = gcd y (x mod y)"
haftmann@52380
  1800
by auto
huffman@29478
  1801
haftmann@52380
  1802
termination "gcd :: _ poly \<Rightarrow> _"
haftmann@52380
  1803
by (relation "measure (\<lambda>(x, y). if y = 0 then 0 else Suc (degree y))")
haftmann@52380
  1804
   (auto dest: degree_mod_less)
haftmann@52380
  1805
haftmann@52380
  1806
declare gcd_poly.simps [simp del]
haftmann@52380
  1807
haftmann@58513
  1808
definition lcm_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
haftmann@58513
  1809
where
haftmann@58513
  1810
  "lcm_poly a b = a * b div smult (coeff a (degree a) * coeff b (degree b)) (gcd a b)"
haftmann@58513
  1811
haftmann@52380
  1812
instance ..
huffman@29478
  1813
huffman@29451
  1814
end
huffman@29478
  1815
haftmann@52380
  1816
lemma
haftmann@52380
  1817
  fixes x y :: "_ poly"
haftmann@52380
  1818
  shows poly_gcd_dvd1 [iff]: "gcd x y dvd x"
haftmann@52380
  1819
    and poly_gcd_dvd2 [iff]: "gcd x y dvd y"
haftmann@52380
  1820
  apply (induct x y rule: gcd_poly.induct)
haftmann@52380
  1821
  apply (simp_all add: gcd_poly.simps)
haftmann@52380
  1822
  apply (fastforce simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
haftmann@52380
  1823
  apply (blast dest: dvd_mod_imp_dvd)
haftmann@52380
  1824
  done
haftmann@38857
  1825
haftmann@52380
  1826
lemma poly_gcd_greatest:
haftmann@52380
  1827
  fixes k x y :: "_ poly"
haftmann@52380
  1828
  shows "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd gcd x y"
haftmann@52380
  1829
  by (induct x y rule: gcd_poly.induct)
haftmann@52380
  1830
     (simp_all add: gcd_poly.simps dvd_mod dvd_smult)
huffman@29478
  1831
haftmann@52380
  1832
lemma dvd_poly_gcd_iff [iff]:
haftmann@52380
  1833
  fixes k x y :: "_ poly"
haftmann@52380
  1834
  shows "k dvd gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
haftmann@60686
  1835
  by (auto intro!: poly_gcd_greatest intro: dvd_trans [of _ "gcd x y"])
huffman@29478
  1836
haftmann@52380
  1837
lemma poly_gcd_monic:
haftmann@52380
  1838
  fixes x y :: "_ poly"
haftmann@52380
  1839
  shows "coeff (gcd x y) (degree (gcd x y)) =
haftmann@52380
  1840
    (if x = 0 \<and> y = 0 then 0 else 1)"
haftmann@52380
  1841
  by (induct x y rule: gcd_poly.induct)
haftmann@52380
  1842
     (simp_all add: gcd_poly.simps nonzero_imp_inverse_nonzero)
huffman@29478
  1843
haftmann@52380
  1844
lemma poly_gcd_zero_iff [simp]:
haftmann@52380
  1845
  fixes x y :: "_ poly"
haftmann@52380
  1846
  shows "gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
haftmann@52380
  1847
  by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)
huffman@29478
  1848
haftmann@52380
  1849
lemma poly_gcd_0_0 [simp]:
haftmann@52380
  1850
  "gcd (0::_ poly) 0 = 0"
haftmann@52380
  1851
  by simp
huffman@29478
  1852
haftmann@52380
  1853
lemma poly_dvd_antisym:
haftmann@52380
  1854
  fixes p q :: "'a::idom poly"
haftmann@52380
  1855
  assumes coeff: "coeff p (degree p) = coeff q (degree q)"
haftmann@52380
  1856
  assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
haftmann@52380
  1857
proof (cases "p = 0")
haftmann@52380
  1858
  case True with coeff show "p = q" by simp
haftmann@52380
  1859
next
haftmann@52380
  1860
  case False with coeff have "q \<noteq> 0" by auto
haftmann@52380
  1861
  have degree: "degree p = degree q"
wenzelm@60500
  1862
    using \<open>p dvd q\<close> \<open>q dvd p\<close> \<open>p \<noteq> 0\<close> \<open>q \<noteq> 0\<close>
haftmann@52380
  1863
    by (intro order_antisym dvd_imp_degree_le)
huffman@29478
  1864
wenzelm@60500
  1865
  from \<open>p dvd q\<close> obtain a where a: "q = p * a" ..
wenzelm@60500
  1866
  with \<open>q \<noteq> 0\<close> have "a \<noteq> 0" by auto
wenzelm@60500
  1867
  with degree a \<open>p \<noteq> 0\<close> have "degree a = 0"
haftmann@52380
  1868
    by (simp add: degree_mult_eq)
haftmann@52380
  1869
  with coeff a show "p = q"
haftmann@52380
  1870
    by (cases a, auto split: if_splits)
haftmann@52380
  1871
qed
huffman@29478
  1872
haftmann@52380
  1873
lemma poly_gcd_unique:
haftmann@52380
  1874
  fixes d x y :: "_ poly"
haftmann@52380
  1875
  assumes dvd1: "d dvd x" and dvd2: "d dvd y"
haftmann@52380
  1876
    and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
haftmann@52380
  1877
    and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
haftmann@52380
  1878
  shows "gcd x y = d"
haftmann@52380
  1879
proof -
haftmann@52380
  1880
  have "coeff (gcd x y) (degree (gcd x y)) = coeff d (degree d)"
haftmann@52380
  1881
    by (simp_all add: poly_gcd_monic monic)
haftmann@52380
  1882
  moreover have "gcd x y dvd d"
haftmann@52380
  1883
    using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
haftmann@52380
  1884
  moreover have "d dvd gcd x y"
haftmann@52380
  1885
    using dvd1 dvd2 by (rule poly_gcd_greatest)
haftmann@52380
  1886
  ultimately show ?thesis
haftmann@52380
  1887
    by (rule poly_dvd_antisym)
haftmann@52380
  1888
qed
huffman@29478
  1889
wenzelm@61605
  1890
interpretation gcd_poly: abel_semigroup "gcd :: _ poly \<Rightarrow> _"
haftmann@52380
  1891
proof
haftmann@52380
  1892
  fix x y z :: "'a poly"
haftmann@52380
  1893
  show "gcd (gcd x y) z = gcd x (gcd y z)"
haftmann@52380
  1894
    by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic)
haftmann@52380
  1895
  show "gcd x y = gcd y x"
haftmann@52380
  1896
    by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
haftmann@52380
  1897
qed
huffman@29478
  1898
haftmann@52380
  1899
lemmas poly_gcd_assoc = gcd_poly.assoc
haftmann@52380
  1900
lemmas poly_gcd_commute = gcd_poly.commute
haftmann@52380
  1901
lemmas poly_gcd_left_commute = gcd_poly.left_commute
huffman@29478
  1902
haftmann@52380
  1903
lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute
haftmann@52380
  1904
haftmann@52380
  1905
lemma poly_gcd_1_left [simp]: "gcd 1 y = (1 :: _ poly)"
haftmann@52380
  1906
by (rule poly_gcd_unique) simp_all
huffman@29478
  1907
haftmann@52380
  1908
lemma poly_gcd_1_right [simp]: "gcd x 1 = (1 :: _ poly)"
haftmann@52380
  1909
by (rule poly_gcd_unique) simp_all
haftmann@52380
  1910
haftmann@52380
  1911
lemma poly_gcd_minus_left [simp]: "gcd (- x) y = gcd x (y :: _ poly)"
haftmann@52380
  1912
by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
huffman@29478
  1913
haftmann@52380
  1914
lemma poly_gcd_minus_right [simp]: "gcd x (- y) = gcd x (y :: _ poly)"
haftmann@52380
  1915
by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
huffman@29478
  1916
haftmann@52380
  1917
lemma poly_gcd_code [code]:
haftmann@52380
  1918
  "gcd x y = (if y = 0 then smult (inverse (coeff x (degree x))) x else gcd y (x mod (y :: _ poly)))"
haftmann@52380
  1919
  by (simp add: gcd_poly.simps)
haftmann@52380
  1920
haftmann@52380
  1921
wenzelm@60500
  1922
subsection \<open>Composition of polynomials\<close>
huffman@29478
  1923
haftmann@52380
  1924
definition pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
haftmann@52380
  1925
where
haftmann@52380
  1926
  "pcompose p q = fold_coeffs (\<lambda>a c. [:a:] + q * c) p 0"
haftmann@52380
  1927
haftmann@52380
  1928
lemma pcompose_0 [simp]:
haftmann@52380
  1929
  "pcompose 0 q = 0"
haftmann@52380
  1930
  by (simp add: pcompose_def)
haftmann@52380
  1931
haftmann@52380
  1932
lemma pcompose_pCons:
haftmann@52380
  1933
  "pcompose (pCons a p) q = [:a:] + q * pcompose p q"
haftmann@52380
  1934
  by (cases "p = 0 \<and> a = 0") (auto simp add: pcompose_def)
haftmann@52380
  1935
haftmann@52380
  1936
lemma poly_pcompose:
haftmann@52380
  1937
  "poly (pcompose p q) x = poly p (poly q x)"
haftmann@52380
  1938
  by (induct p) (simp_all add: pcompose_pCons)
haftmann@52380
  1939
haftmann@52380
  1940
lemma degree_pcompose_le:
haftmann@52380
  1941
  "degree (pcompose p q) \<le> degree p * degree q"
haftmann@52380
  1942
apply (induct p, simp)
haftmann@52380
  1943
apply (simp add: pcompose_pCons, clarify)
haftmann@52380
  1944
apply (rule degree_add_le, simp)
haftmann@52380
  1945
apply (rule order_trans [OF degree_mult_le], simp)
huffman@29478
  1946
done
huffman@29478
  1947
haftmann@52380
  1948
haftmann@52380
  1949
no_notation cCons (infixr "##" 65)
huffman@31663
  1950
huffman@29478
  1951
end