src/HOL/Polynomial.thy
author huffman
Wed Jan 21 20:20:56 2009 -0800 (2009-01-21)
changeset 29659 f8d2c03ecfd8
parent 29540 8858d197a9b6
child 29660 d59918e668b7
permissions -rw-r--r--
add lemmas about smult
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(*  Title:      HOL/Polynomial.thy
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    Author:     Brian Huffman
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                Based on an earlier development by Clemens Ballarin
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*)
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header {* Univariate Polynomials *}
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theory Polynomial
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imports Plain SetInterval
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begin
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subsection {* Definition of type @{text poly} *}
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typedef (Poly) 'a poly = "{f::nat \<Rightarrow> 'a::zero. \<exists>n. \<forall>i>n. f i = 0}"
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  morphisms coeff Abs_poly
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  by auto
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lemma expand_poly_eq: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)"
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by (simp add: coeff_inject [symmetric] expand_fun_eq)
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lemma poly_ext: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q"
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by (simp add: expand_poly_eq)
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subsection {* Degree of a polynomial *}
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definition
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  degree :: "'a::zero poly \<Rightarrow> nat" where
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  "degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"
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lemma coeff_eq_0: "degree p < n \<Longrightarrow> coeff p n = 0"
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proof -
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  have "coeff p \<in> Poly"
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    by (rule coeff)
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  hence "\<exists>n. \<forall>i>n. coeff p i = 0"
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    unfolding Poly_def by simp
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  hence "\<forall>i>degree p. coeff p i = 0"
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    unfolding degree_def by (rule LeastI_ex)
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  moreover assume "degree p < n"
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  ultimately 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 {* The zero polynomial *}
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instantiation poly :: (zero) zero
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begin
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definition
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  zero_poly_def: "0 = Abs_poly (\<lambda>n. 0)"
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instance ..
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end
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lemma coeff_0 [simp]: "coeff 0 n = 0"
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  unfolding zero_poly_def
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  by (simp add: Abs_poly_inverse Poly_def)
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lemma degree_0 [simp]: "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" 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 `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0"
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    by (simp add: expand_poly_eq)
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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 `coeff p n \<noteq> 0` and `degree p = 0`
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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 `degree p = Suc n` 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 `degree p = Suc n` and `n < i` have "degree p \<le> i" by simp
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  also from `coeff p i \<noteq> 0` have "i \<le> degree p" by (rule le_degree)
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  finally have "degree p = i" .
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  with `coeff p i \<noteq> 0` show "coeff p (degree p) \<noteq> 0" by simp
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qed
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lemma leading_coeff_0_iff [simp]: "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 {* List-style constructor for polynomials *}
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definition
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  pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
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where
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  [code del]: "pCons a p = Abs_poly (nat_case a (coeff p))"
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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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lemma Poly_nat_case: "f \<in> Poly \<Longrightarrow> nat_case a f \<in> Poly"
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  unfolding Poly_def by (auto split: nat.split)
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lemma coeff_pCons:
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  "coeff (pCons a p) = nat_case a (coeff p)"
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  unfolding pCons_def
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  by (simp add: Abs_poly_inverse Poly_nat_case coeff)
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lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a"
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  by (simp add: coeff_pCons)
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lemma coeff_pCons_Suc [simp]: "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: "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: "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]: "pCons 0 0 = 0"
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by (rule poly_ext, 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: expand_poly_eq)
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qed
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lemma pCons_eq_0_iff [simp]: "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 Poly_Suc: "f \<in> Poly \<Longrightarrow> (\<lambda>n. f (Suc n)) \<in> Poly"
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  unfolding Poly_def
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  by (clarify, rule_tac x=n in exI, 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 (rule poly_ext)
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       (simp add: Abs_poly_inverse Poly_Suc coeff coeff_pCons
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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. 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 `p = pCons a q` 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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  then have "P (pCons a q)"
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    by (rule pCons)
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  then show ?case
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    using `p = pCons a q` by simp
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qed
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subsection {* Recursion combinator for polynomials *}
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function
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  poly_rec :: "'b \<Rightarrow> ('a::zero \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b"
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where
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  poly_rec_pCons_eq_if [simp del, code del]:
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    "poly_rec z f (pCons a p) = f a p (if p = 0 then z else poly_rec z f p)"
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by (case_tac x, rename_tac q, case_tac q, auto)
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termination poly_rec
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by (relation "measure (degree \<circ> snd \<circ> snd)", simp)
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   (simp add: degree_pCons_eq)
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lemma poly_rec_0:
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  "f 0 0 z = z \<Longrightarrow> poly_rec z f 0 = z"
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  using poly_rec_pCons_eq_if [of z f 0 0] by simp
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lemma poly_rec_pCons:
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  "f 0 0 z = z \<Longrightarrow> poly_rec z f (pCons a p) = f a p (poly_rec z f p)"
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  by (simp add: poly_rec_pCons_eq_if poly_rec_0)
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subsection {* Monomials *}
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definition
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  monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly" where
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  "monom a m = Abs_poly (\<lambda>n. if m = n then a else 0)"
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lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)"
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  unfolding monom_def
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  by (subst Abs_poly_inverse, auto simp add: Poly_def)
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lemma monom_0: "monom a 0 = pCons a 0"
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  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
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lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)"
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  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
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lemma monom_eq_0 [simp]: "monom 0 n = 0"
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  by (rule poly_ext) simp
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lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
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  by (simp add: expand_poly_eq)
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lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
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  by (simp add: expand_poly_eq)
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lemma degree_monom_le: "degree (monom a n) \<le> n"
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  by (rule degree_le, simp)
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lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
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  apply (rule order_antisym [OF degree_monom_le])
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  apply (rule le_degree, simp)
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  done
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subsection {* Addition and subtraction *}
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instantiation poly :: (comm_monoid_add) comm_monoid_add
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begin
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definition
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  plus_poly_def [code del]:
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    "p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)"
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lemma Poly_add:
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  fixes f g :: "nat \<Rightarrow> 'a"
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  shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n + g n) \<in> Poly"
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  unfolding Poly_def
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  apply (clarify, rename_tac m n)
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  apply (rule_tac x="max m n" in exI, simp)
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  done
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lemma coeff_add [simp]:
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  "coeff (p + q) n = coeff p n + coeff q n"
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  unfolding plus_poly_def
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  by (simp add: Abs_poly_inverse coeff Poly_add)
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instance proof
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  fix p q r :: "'a poly"
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  show "(p + q) + r = p + (q + r)"
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    by (simp add: expand_poly_eq add_assoc)
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  show "p + q = q + p"
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    by (simp add: expand_poly_eq add_commute)
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  show "0 + p = p"
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    by (simp add: expand_poly_eq)
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qed
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end
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instance poly ::
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  ("{cancel_ab_semigroup_add,comm_monoid_add}") cancel_ab_semigroup_add
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proof
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  fix p q r :: "'a poly"
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  assume "p + q = p + r" thus "q = r"
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    by (simp add: expand_poly_eq)
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qed
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instantiation poly :: (ab_group_add) ab_group_add
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begin
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definition
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  uminus_poly_def [code del]:
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    "- p = Abs_poly (\<lambda>n. - coeff p n)"
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definition
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  minus_poly_def [code del]:
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    "p - q = Abs_poly (\<lambda>n. coeff p n - coeff q n)"
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lemma Poly_minus:
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  fixes f :: "nat \<Rightarrow> 'a"
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  shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. - f n) \<in> Poly"
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  unfolding Poly_def by simp
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lemma Poly_diff:
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  fixes f g :: "nat \<Rightarrow> 'a"
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  shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n - g n) \<in> Poly"
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  unfolding diff_minus by (simp add: Poly_add Poly_minus)
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lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
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  unfolding uminus_poly_def
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  by (simp add: Abs_poly_inverse coeff Poly_minus)
huffman@29451
   328
huffman@29451
   329
lemma coeff_diff [simp]:
huffman@29451
   330
  "coeff (p - q) n = coeff p n - coeff q n"
huffman@29451
   331
  unfolding minus_poly_def
huffman@29451
   332
  by (simp add: Abs_poly_inverse coeff Poly_diff)
huffman@29451
   333
huffman@29451
   334
instance proof
huffman@29451
   335
  fix p q :: "'a poly"
huffman@29451
   336
  show "- p + p = 0"
huffman@29451
   337
    by (simp add: expand_poly_eq)
huffman@29451
   338
  show "p - q = p + - q"
huffman@29451
   339
    by (simp add: expand_poly_eq diff_minus)
huffman@29451
   340
qed
huffman@29451
   341
huffman@29451
   342
end
huffman@29451
   343
huffman@29451
   344
lemma add_pCons [simp]:
huffman@29451
   345
  "pCons a p + pCons b q = pCons (a + b) (p + q)"
huffman@29451
   346
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
huffman@29451
   347
huffman@29451
   348
lemma minus_pCons [simp]:
huffman@29451
   349
  "- pCons a p = pCons (- a) (- p)"
huffman@29451
   350
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
huffman@29451
   351
huffman@29451
   352
lemma diff_pCons [simp]:
huffman@29451
   353
  "pCons a p - pCons b q = pCons (a - b) (p - q)"
huffman@29451
   354
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
huffman@29451
   355
huffman@29539
   356
lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)"
huffman@29451
   357
  by (rule degree_le, auto simp add: coeff_eq_0)
huffman@29451
   358
huffman@29539
   359
lemma degree_add_le:
huffman@29539
   360
  "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n"
huffman@29539
   361
  by (auto intro: order_trans degree_add_le_max)
huffman@29539
   362
huffman@29453
   363
lemma degree_add_less:
huffman@29453
   364
  "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
huffman@29539
   365
  by (auto intro: le_less_trans degree_add_le_max)
huffman@29453
   366
huffman@29451
   367
lemma degree_add_eq_right:
huffman@29451
   368
  "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
huffman@29451
   369
  apply (cases "q = 0", simp)
huffman@29451
   370
  apply (rule order_antisym)
huffman@29539
   371
  apply (simp add: degree_add_le)
huffman@29451
   372
  apply (rule le_degree)
huffman@29451
   373
  apply (simp add: coeff_eq_0)
huffman@29451
   374
  done
huffman@29451
   375
huffman@29451
   376
lemma degree_add_eq_left:
huffman@29451
   377
  "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
huffman@29451
   378
  using degree_add_eq_right [of q p]
huffman@29451
   379
  by (simp add: add_commute)
huffman@29451
   380
huffman@29451
   381
lemma degree_minus [simp]: "degree (- p) = degree p"
huffman@29451
   382
  unfolding degree_def by simp
huffman@29451
   383
huffman@29539
   384
lemma degree_diff_le_max: "degree (p - q) \<le> max (degree p) (degree q)"
huffman@29451
   385
  using degree_add_le [where p=p and q="-q"]
huffman@29451
   386
  by (simp add: diff_minus)
huffman@29451
   387
huffman@29539
   388
lemma degree_diff_le:
huffman@29539
   389
  "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p - q) \<le> n"
huffman@29539
   390
  by (simp add: diff_minus degree_add_le)
huffman@29539
   391
huffman@29453
   392
lemma degree_diff_less:
huffman@29453
   393
  "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
huffman@29539
   394
  by (simp add: diff_minus degree_add_less)
huffman@29453
   395
huffman@29451
   396
lemma add_monom: "monom a n + monom b n = monom (a + b) n"
huffman@29451
   397
  by (rule poly_ext) simp
huffman@29451
   398
huffman@29451
   399
lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
huffman@29451
   400
  by (rule poly_ext) simp
huffman@29451
   401
huffman@29451
   402
lemma minus_monom: "- monom a n = monom (-a) n"
huffman@29451
   403
  by (rule poly_ext) simp
huffman@29451
   404
huffman@29451
   405
lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
huffman@29451
   406
  by (cases "finite A", induct set: finite, simp_all)
huffman@29451
   407
huffman@29451
   408
lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
huffman@29451
   409
  by (rule poly_ext) (simp add: coeff_setsum)
huffman@29451
   410
huffman@29451
   411
huffman@29451
   412
subsection {* Multiplication by a constant *}
huffman@29451
   413
huffman@29451
   414
definition
huffman@29451
   415
  smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
huffman@29451
   416
  "smult a p = Abs_poly (\<lambda>n. a * coeff p n)"
huffman@29451
   417
huffman@29451
   418
lemma Poly_smult:
huffman@29451
   419
  fixes f :: "nat \<Rightarrow> 'a::comm_semiring_0"
huffman@29451
   420
  shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. a * f n) \<in> Poly"
huffman@29451
   421
  unfolding Poly_def
huffman@29451
   422
  by (clarify, rule_tac x=n in exI, simp)
huffman@29451
   423
huffman@29451
   424
lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n"
huffman@29451
   425
  unfolding smult_def
huffman@29451
   426
  by (simp add: Abs_poly_inverse Poly_smult coeff)
huffman@29451
   427
huffman@29451
   428
lemma degree_smult_le: "degree (smult a p) \<le> degree p"
huffman@29451
   429
  by (rule degree_le, simp add: coeff_eq_0)
huffman@29451
   430
huffman@29472
   431
lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
huffman@29451
   432
  by (rule poly_ext, simp add: mult_assoc)
huffman@29451
   433
huffman@29451
   434
lemma smult_0_right [simp]: "smult a 0 = 0"
huffman@29451
   435
  by (rule poly_ext, simp)
huffman@29451
   436
huffman@29451
   437
lemma smult_0_left [simp]: "smult 0 p = 0"
huffman@29451
   438
  by (rule poly_ext, simp)
huffman@29451
   439
huffman@29451
   440
lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
huffman@29451
   441
  by (rule poly_ext, simp)
huffman@29451
   442
huffman@29451
   443
lemma smult_add_right:
huffman@29451
   444
  "smult a (p + q) = smult a p + smult a q"
huffman@29451
   445
  by (rule poly_ext, simp add: ring_simps)
huffman@29451
   446
huffman@29451
   447
lemma smult_add_left:
huffman@29451
   448
  "smult (a + b) p = smult a p + smult b p"
huffman@29451
   449
  by (rule poly_ext, simp add: ring_simps)
huffman@29451
   450
huffman@29457
   451
lemma smult_minus_right [simp]:
huffman@29451
   452
  "smult (a::'a::comm_ring) (- p) = - smult a p"
huffman@29451
   453
  by (rule poly_ext, simp)
huffman@29451
   454
huffman@29457
   455
lemma smult_minus_left [simp]:
huffman@29451
   456
  "smult (- a::'a::comm_ring) p = - smult a p"
huffman@29451
   457
  by (rule poly_ext, simp)
huffman@29451
   458
huffman@29451
   459
lemma smult_diff_right:
huffman@29451
   460
  "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
huffman@29451
   461
  by (rule poly_ext, simp add: ring_simps)
huffman@29451
   462
huffman@29451
   463
lemma smult_diff_left:
huffman@29451
   464
  "smult (a - b::'a::comm_ring) p = smult a p - smult b p"
huffman@29451
   465
  by (rule poly_ext, simp add: ring_simps)
huffman@29451
   466
huffman@29472
   467
lemmas smult_distribs =
huffman@29472
   468
  smult_add_left smult_add_right
huffman@29472
   469
  smult_diff_left smult_diff_right
huffman@29472
   470
huffman@29451
   471
lemma smult_pCons [simp]:
huffman@29451
   472
  "smult a (pCons b p) = pCons (a * b) (smult a p)"
huffman@29451
   473
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
huffman@29451
   474
huffman@29451
   475
lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
huffman@29451
   476
  by (induct n, simp add: monom_0, simp add: monom_Suc)
huffman@29451
   477
huffman@29659
   478
lemma degree_smult_eq [simp]:
huffman@29659
   479
  fixes a :: "'a::idom"
huffman@29659
   480
  shows "degree (smult a p) = (if a = 0 then 0 else degree p)"
huffman@29659
   481
  by (cases "a = 0", simp, simp add: degree_def)
huffman@29659
   482
huffman@29659
   483
lemma smult_eq_0_iff [simp]:
huffman@29659
   484
  fixes a :: "'a::idom"
huffman@29659
   485
  shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0"
huffman@29659
   486
  by (simp add: expand_poly_eq)
huffman@29659
   487
huffman@29451
   488
huffman@29451
   489
subsection {* Multiplication of polynomials *}
huffman@29451
   490
huffman@29474
   491
text {* TODO: move to SetInterval.thy *}
huffman@29451
   492
lemma setsum_atMost_Suc_shift:
huffman@29451
   493
  fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
huffman@29451
   494
  shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
huffman@29451
   495
proof (induct n)
huffman@29451
   496
  case 0 show ?case by simp
huffman@29451
   497
next
huffman@29451
   498
  case (Suc n) note IH = this
huffman@29451
   499
  have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
huffman@29451
   500
    by (rule setsum_atMost_Suc)
huffman@29451
   501
  also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
huffman@29451
   502
    by (rule IH)
huffman@29451
   503
  also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
huffman@29451
   504
             f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
huffman@29451
   505
    by (rule add_assoc)
huffman@29451
   506
  also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
huffman@29451
   507
    by (rule setsum_atMost_Suc [symmetric])
huffman@29451
   508
  finally show ?case .
huffman@29451
   509
qed
huffman@29451
   510
huffman@29451
   511
instantiation poly :: (comm_semiring_0) comm_semiring_0
huffman@29451
   512
begin
huffman@29451
   513
huffman@29451
   514
definition
huffman@29475
   515
  times_poly_def [code del]:
huffman@29474
   516
    "p * q = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p"
huffman@29474
   517
huffman@29474
   518
lemma mult_poly_0_left: "(0::'a poly) * q = 0"
huffman@29474
   519
  unfolding times_poly_def by (simp add: poly_rec_0)
huffman@29474
   520
huffman@29474
   521
lemma mult_pCons_left [simp]:
huffman@29474
   522
  "pCons a p * q = smult a q + pCons 0 (p * q)"
huffman@29474
   523
  unfolding times_poly_def by (simp add: poly_rec_pCons)
huffman@29474
   524
huffman@29474
   525
lemma mult_poly_0_right: "p * (0::'a poly) = 0"
huffman@29474
   526
  by (induct p, simp add: mult_poly_0_left, simp)
huffman@29451
   527
huffman@29474
   528
lemma mult_pCons_right [simp]:
huffman@29474
   529
  "p * pCons a q = smult a p + pCons 0 (p * q)"
huffman@29474
   530
  by (induct p, simp add: mult_poly_0_left, simp add: ring_simps)
huffman@29474
   531
huffman@29474
   532
lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right
huffman@29474
   533
huffman@29474
   534
lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)"
huffman@29474
   535
  by (induct p, simp add: mult_poly_0, simp add: smult_add_right)
huffman@29474
   536
huffman@29474
   537
lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)"
huffman@29474
   538
  by (induct q, simp add: mult_poly_0, simp add: smult_add_right)
huffman@29474
   539
huffman@29474
   540
lemma mult_poly_add_left:
huffman@29474
   541
  fixes p q r :: "'a poly"
huffman@29474
   542
  shows "(p + q) * r = p * r + q * r"
huffman@29474
   543
  by (induct r, simp add: mult_poly_0,
huffman@29474
   544
                simp add: smult_distribs group_simps)
huffman@29451
   545
huffman@29451
   546
instance proof
huffman@29451
   547
  fix p q r :: "'a poly"
huffman@29451
   548
  show 0: "0 * p = 0"
huffman@29474
   549
    by (rule mult_poly_0_left)
huffman@29451
   550
  show "p * 0 = 0"
huffman@29474
   551
    by (rule mult_poly_0_right)
huffman@29451
   552
  show "(p + q) * r = p * r + q * r"
huffman@29474
   553
    by (rule mult_poly_add_left)
huffman@29451
   554
  show "(p * q) * r = p * (q * r)"
huffman@29474
   555
    by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left)
huffman@29451
   556
  show "p * q = q * p"
huffman@29474
   557
    by (induct p, simp add: mult_poly_0, simp)
huffman@29451
   558
qed
huffman@29451
   559
huffman@29451
   560
end
huffman@29451
   561
huffman@29540
   562
instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
huffman@29540
   563
huffman@29474
   564
lemma coeff_mult:
huffman@29474
   565
  "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
huffman@29474
   566
proof (induct p arbitrary: n)
huffman@29474
   567
  case 0 show ?case by simp
huffman@29474
   568
next
huffman@29474
   569
  case (pCons a p n) thus ?case
huffman@29474
   570
    by (cases n, simp, simp add: setsum_atMost_Suc_shift
huffman@29474
   571
                            del: setsum_atMost_Suc)
huffman@29474
   572
qed
huffman@29451
   573
huffman@29474
   574
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
huffman@29474
   575
apply (rule degree_le)
huffman@29474
   576
apply (induct p)
huffman@29474
   577
apply simp
huffman@29474
   578
apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)
huffman@29451
   579
done
huffman@29451
   580
huffman@29451
   581
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
huffman@29451
   582
  by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)
huffman@29451
   583
huffman@29451
   584
huffman@29451
   585
subsection {* The unit polynomial and exponentiation *}
huffman@29451
   586
huffman@29451
   587
instantiation poly :: (comm_semiring_1) comm_semiring_1
huffman@29451
   588
begin
huffman@29451
   589
huffman@29451
   590
definition
huffman@29451
   591
  one_poly_def:
huffman@29451
   592
    "1 = pCons 1 0"
huffman@29451
   593
huffman@29451
   594
instance proof
huffman@29451
   595
  fix p :: "'a poly" show "1 * p = p"
huffman@29451
   596
    unfolding one_poly_def
huffman@29451
   597
    by simp
huffman@29451
   598
next
huffman@29451
   599
  show "0 \<noteq> (1::'a poly)"
huffman@29451
   600
    unfolding one_poly_def by simp
huffman@29451
   601
qed
huffman@29451
   602
huffman@29451
   603
end
huffman@29451
   604
huffman@29540
   605
instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..
huffman@29540
   606
huffman@29451
   607
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
huffman@29451
   608
  unfolding one_poly_def
huffman@29451
   609
  by (simp add: coeff_pCons split: nat.split)
huffman@29451
   610
huffman@29451
   611
lemma degree_1 [simp]: "degree 1 = 0"
huffman@29451
   612
  unfolding one_poly_def
huffman@29451
   613
  by (rule degree_pCons_0)
huffman@29451
   614
huffman@29451
   615
instantiation poly :: (comm_semiring_1) recpower
huffman@29451
   616
begin
huffman@29451
   617
huffman@29451
   618
primrec power_poly where
huffman@29451
   619
  power_poly_0: "(p::'a poly) ^ 0 = 1"
huffman@29451
   620
| power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n"
huffman@29451
   621
huffman@29451
   622
instance
huffman@29451
   623
  by default simp_all
huffman@29451
   624
huffman@29451
   625
end
huffman@29451
   626
huffman@29451
   627
instance poly :: (comm_ring) comm_ring ..
huffman@29451
   628
huffman@29451
   629
instance poly :: (comm_ring_1) comm_ring_1 ..
huffman@29451
   630
huffman@29451
   631
instantiation poly :: (comm_ring_1) number_ring
huffman@29451
   632
begin
huffman@29451
   633
huffman@29451
   634
definition
huffman@29451
   635
  "number_of k = (of_int k :: 'a poly)"
huffman@29451
   636
huffman@29451
   637
instance
huffman@29451
   638
  by default (rule number_of_poly_def)
huffman@29451
   639
huffman@29451
   640
end
huffman@29451
   641
huffman@29451
   642
huffman@29451
   643
subsection {* Polynomials form an integral domain *}
huffman@29451
   644
huffman@29451
   645
lemma coeff_mult_degree_sum:
huffman@29451
   646
  "coeff (p * q) (degree p + degree q) =
huffman@29451
   647
   coeff p (degree p) * coeff q (degree q)"
huffman@29471
   648
  by (induct p, simp, simp add: coeff_eq_0)
huffman@29451
   649
huffman@29451
   650
instance poly :: (idom) idom
huffman@29451
   651
proof
huffman@29451
   652
  fix p q :: "'a poly"
huffman@29451
   653
  assume "p \<noteq> 0" and "q \<noteq> 0"
huffman@29451
   654
  have "coeff (p * q) (degree p + degree q) =
huffman@29451
   655
        coeff p (degree p) * coeff q (degree q)"
huffman@29451
   656
    by (rule coeff_mult_degree_sum)
huffman@29451
   657
  also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
huffman@29451
   658
    using `p \<noteq> 0` and `q \<noteq> 0` by simp
huffman@29451
   659
  finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
huffman@29451
   660
  thus "p * q \<noteq> 0" by (simp add: expand_poly_eq)
huffman@29451
   661
qed
huffman@29451
   662
huffman@29451
   663
lemma degree_mult_eq:
huffman@29451
   664
  fixes p q :: "'a::idom poly"
huffman@29451
   665
  shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
huffman@29451
   666
apply (rule order_antisym [OF degree_mult_le le_degree])
huffman@29451
   667
apply (simp add: coeff_mult_degree_sum)
huffman@29451
   668
done
huffman@29451
   669
huffman@29451
   670
lemma dvd_imp_degree_le:
huffman@29451
   671
  fixes p q :: "'a::idom poly"
huffman@29451
   672
  shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
huffman@29451
   673
  by (erule dvdE, simp add: degree_mult_eq)
huffman@29451
   674
huffman@29451
   675
huffman@29451
   676
subsection {* Long division of polynomials *}
huffman@29451
   677
huffman@29451
   678
definition
huffman@29537
   679
  pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
huffman@29451
   680
where
huffman@29475
   681
  [code del]:
huffman@29537
   682
  "pdivmod_rel x y q r \<longleftrightarrow>
huffman@29451
   683
    x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
huffman@29451
   684
huffman@29537
   685
lemma pdivmod_rel_0:
huffman@29537
   686
  "pdivmod_rel 0 y 0 0"
huffman@29537
   687
  unfolding pdivmod_rel_def by simp
huffman@29451
   688
huffman@29537
   689
lemma pdivmod_rel_by_0:
huffman@29537
   690
  "pdivmod_rel x 0 0 x"
huffman@29537
   691
  unfolding pdivmod_rel_def by simp
huffman@29451
   692
huffman@29451
   693
lemma eq_zero_or_degree_less:
huffman@29451
   694
  assumes "degree p \<le> n" and "coeff p n = 0"
huffman@29451
   695
  shows "p = 0 \<or> degree p < n"
huffman@29451
   696
proof (cases n)
huffman@29451
   697
  case 0
huffman@29451
   698
  with `degree p \<le> n` and `coeff p n = 0`
huffman@29451
   699
  have "coeff p (degree p) = 0" by simp
huffman@29451
   700
  then have "p = 0" by simp
huffman@29451
   701
  then show ?thesis ..
huffman@29451
   702
next
huffman@29451
   703
  case (Suc m)
huffman@29451
   704
  have "\<forall>i>n. coeff p i = 0"
huffman@29451
   705
    using `degree p \<le> n` by (simp add: coeff_eq_0)
huffman@29451
   706
  then have "\<forall>i\<ge>n. coeff p i = 0"
huffman@29451
   707
    using `coeff p n = 0` by (simp add: le_less)
huffman@29451
   708
  then have "\<forall>i>m. coeff p i = 0"
huffman@29451
   709
    using `n = Suc m` by (simp add: less_eq_Suc_le)
huffman@29451
   710
  then have "degree p \<le> m"
huffman@29451
   711
    by (rule degree_le)
huffman@29451
   712
  then have "degree p < n"
huffman@29451
   713
    using `n = Suc m` by (simp add: less_Suc_eq_le)
huffman@29451
   714
  then show ?thesis ..
huffman@29451
   715
qed
huffman@29451
   716
huffman@29537
   717
lemma pdivmod_rel_pCons:
huffman@29537
   718
  assumes rel: "pdivmod_rel x y q r"
huffman@29451
   719
  assumes y: "y \<noteq> 0"
huffman@29451
   720
  assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
huffman@29537
   721
  shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
huffman@29537
   722
    (is "pdivmod_rel ?x y ?q ?r")
huffman@29451
   723
proof -
huffman@29451
   724
  have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
huffman@29537
   725
    using assms unfolding pdivmod_rel_def by simp_all
huffman@29451
   726
huffman@29451
   727
  have 1: "?x = ?q * y + ?r"
huffman@29451
   728
    using b x by simp
huffman@29451
   729
huffman@29451
   730
  have 2: "?r = 0 \<or> degree ?r < degree y"
huffman@29451
   731
  proof (rule eq_zero_or_degree_less)
huffman@29539
   732
    show "degree ?r \<le> degree y"
huffman@29539
   733
    proof (rule degree_diff_le)
huffman@29451
   734
      show "degree (pCons a r) \<le> degree y"
huffman@29460
   735
        using r by auto
huffman@29451
   736
      show "degree (smult b y) \<le> degree y"
huffman@29451
   737
        by (rule degree_smult_le)
huffman@29451
   738
    qed
huffman@29451
   739
  next
huffman@29451
   740
    show "coeff ?r (degree y) = 0"
huffman@29451
   741
      using `y \<noteq> 0` unfolding b by simp
huffman@29451
   742
  qed
huffman@29451
   743
huffman@29451
   744
  from 1 2 show ?thesis
huffman@29537
   745
    unfolding pdivmod_rel_def
huffman@29451
   746
    using `y \<noteq> 0` by simp
huffman@29451
   747
qed
huffman@29451
   748
huffman@29537
   749
lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r"
huffman@29451
   750
apply (cases "y = 0")
huffman@29537
   751
apply (fast intro!: pdivmod_rel_by_0)
huffman@29451
   752
apply (induct x)
huffman@29537
   753
apply (fast intro!: pdivmod_rel_0)
huffman@29537
   754
apply (fast intro!: pdivmod_rel_pCons)
huffman@29451
   755
done
huffman@29451
   756
huffman@29537
   757
lemma pdivmod_rel_unique:
huffman@29537
   758
  assumes 1: "pdivmod_rel x y q1 r1"
huffman@29537
   759
  assumes 2: "pdivmod_rel x y q2 r2"
huffman@29451
   760
  shows "q1 = q2 \<and> r1 = r2"
huffman@29451
   761
proof (cases "y = 0")
huffman@29451
   762
  assume "y = 0" with assms show ?thesis
huffman@29537
   763
    by (simp add: pdivmod_rel_def)
huffman@29451
   764
next
huffman@29451
   765
  assume [simp]: "y \<noteq> 0"
huffman@29451
   766
  from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
huffman@29537
   767
    unfolding pdivmod_rel_def by simp_all
huffman@29451
   768
  from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
huffman@29537
   769
    unfolding pdivmod_rel_def by simp_all
huffman@29451
   770
  from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
huffman@29451
   771
    by (simp add: ring_simps)
huffman@29451
   772
  from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
huffman@29453
   773
    by (auto intro: degree_diff_less)
huffman@29451
   774
huffman@29451
   775
  show "q1 = q2 \<and> r1 = r2"
huffman@29451
   776
  proof (rule ccontr)
huffman@29451
   777
    assume "\<not> (q1 = q2 \<and> r1 = r2)"
huffman@29451
   778
    with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
huffman@29451
   779
    with r3 have "degree (r2 - r1) < degree y" by simp
huffman@29451
   780
    also have "degree y \<le> degree (q1 - q2) + degree y" by simp
huffman@29451
   781
    also have "\<dots> = degree ((q1 - q2) * y)"
huffman@29451
   782
      using `q1 \<noteq> q2` by (simp add: degree_mult_eq)
huffman@29451
   783
    also have "\<dots> = degree (r2 - r1)"
huffman@29451
   784
      using q3 by simp
huffman@29451
   785
    finally have "degree (r2 - r1) < degree (r2 - r1)" .
huffman@29451
   786
    then show "False" by simp
huffman@29451
   787
  qed
huffman@29451
   788
qed
huffman@29451
   789
huffman@29537
   790
lemmas pdivmod_rel_unique_div =
huffman@29537
   791
  pdivmod_rel_unique [THEN conjunct1, standard]
huffman@29451
   792
huffman@29537
   793
lemmas pdivmod_rel_unique_mod =
huffman@29537
   794
  pdivmod_rel_unique [THEN conjunct2, standard]
huffman@29451
   795
huffman@29451
   796
instantiation poly :: (field) ring_div
huffman@29451
   797
begin
huffman@29451
   798
huffman@29451
   799
definition div_poly where
huffman@29537
   800
  [code del]: "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)"
huffman@29451
   801
huffman@29451
   802
definition mod_poly where
huffman@29537
   803
  [code del]: "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)"
huffman@29451
   804
huffman@29451
   805
lemma div_poly_eq:
huffman@29537
   806
  "pdivmod_rel x y q r \<Longrightarrow> x div y = q"
huffman@29451
   807
unfolding div_poly_def
huffman@29537
   808
by (fast elim: pdivmod_rel_unique_div)
huffman@29451
   809
huffman@29451
   810
lemma mod_poly_eq:
huffman@29537
   811
  "pdivmod_rel x y q r \<Longrightarrow> x mod y = r"
huffman@29451
   812
unfolding mod_poly_def
huffman@29537
   813
by (fast elim: pdivmod_rel_unique_mod)
huffman@29451
   814
huffman@29537
   815
lemma pdivmod_rel:
huffman@29537
   816
  "pdivmod_rel x y (x div y) (x mod y)"
huffman@29451
   817
proof -
huffman@29537
   818
  from pdivmod_rel_exists
huffman@29537
   819
    obtain q r where "pdivmod_rel x y q r" by fast
huffman@29451
   820
  thus ?thesis
huffman@29451
   821
    by (simp add: div_poly_eq mod_poly_eq)
huffman@29451
   822
qed
huffman@29451
   823
huffman@29451
   824
instance proof
huffman@29451
   825
  fix x y :: "'a poly"
huffman@29451
   826
  show "x div y * y + x mod y = x"
huffman@29537
   827
    using pdivmod_rel [of x y]
huffman@29537
   828
    by (simp add: pdivmod_rel_def)
huffman@29451
   829
next
huffman@29451
   830
  fix x :: "'a poly"
huffman@29537
   831
  have "pdivmod_rel x 0 0 x"
huffman@29537
   832
    by (rule pdivmod_rel_by_0)
huffman@29451
   833
  thus "x div 0 = 0"
huffman@29451
   834
    by (rule div_poly_eq)
huffman@29451
   835
next
huffman@29451
   836
  fix y :: "'a poly"
huffman@29537
   837
  have "pdivmod_rel 0 y 0 0"
huffman@29537
   838
    by (rule pdivmod_rel_0)
huffman@29451
   839
  thus "0 div y = 0"
huffman@29451
   840
    by (rule div_poly_eq)
huffman@29451
   841
next
huffman@29451
   842
  fix x y z :: "'a poly"
huffman@29451
   843
  assume "y \<noteq> 0"
huffman@29537
   844
  hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"
huffman@29537
   845
    using pdivmod_rel [of x y]
huffman@29537
   846
    by (simp add: pdivmod_rel_def left_distrib)
huffman@29451
   847
  thus "(x + z * y) div y = z + x div y"
huffman@29451
   848
    by (rule div_poly_eq)
huffman@29451
   849
qed
huffman@29451
   850
huffman@29451
   851
end
huffman@29451
   852
huffman@29451
   853
lemma degree_mod_less:
huffman@29451
   854
  "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
huffman@29537
   855
  using pdivmod_rel [of x y]
huffman@29537
   856
  unfolding pdivmod_rel_def by simp
huffman@29451
   857
huffman@29451
   858
lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
huffman@29451
   859
proof -
huffman@29451
   860
  assume "degree x < degree y"
huffman@29537
   861
  hence "pdivmod_rel x y 0 x"
huffman@29537
   862
    by (simp add: pdivmod_rel_def)
huffman@29451
   863
  thus "x div y = 0" by (rule div_poly_eq)
huffman@29451
   864
qed
huffman@29451
   865
huffman@29451
   866
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
huffman@29451
   867
proof -
huffman@29451
   868
  assume "degree x < degree y"
huffman@29537
   869
  hence "pdivmod_rel x y 0 x"
huffman@29537
   870
    by (simp add: pdivmod_rel_def)
huffman@29451
   871
  thus "x mod y = x" by (rule mod_poly_eq)
huffman@29451
   872
qed
huffman@29451
   873
huffman@29659
   874
lemma pdivmod_rel_smult_left:
huffman@29659
   875
  "pdivmod_rel x y q r
huffman@29659
   876
    \<Longrightarrow> pdivmod_rel (smult a x) y (smult a q) (smult a r)"
huffman@29659
   877
  unfolding pdivmod_rel_def by (simp add: smult_add_right)
huffman@29659
   878
huffman@29659
   879
lemma div_smult_left: "(smult a x) div y = smult a (x div y)"
huffman@29659
   880
  by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
huffman@29659
   881
huffman@29659
   882
lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)"
huffman@29659
   883
  by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
huffman@29659
   884
huffman@29659
   885
lemma pdivmod_rel_smult_right:
huffman@29659
   886
  "\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>
huffman@29659
   887
    \<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r"
huffman@29659
   888
  unfolding pdivmod_rel_def by simp
huffman@29659
   889
huffman@29659
   890
lemma div_smult_right:
huffman@29659
   891
  "a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)"
huffman@29659
   892
  by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
huffman@29659
   893
huffman@29659
   894
lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y"
huffman@29659
   895
  by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
huffman@29659
   896
huffman@29451
   897
lemma mod_pCons:
huffman@29451
   898
  fixes a and x
huffman@29451
   899
  assumes y: "y \<noteq> 0"
huffman@29451
   900
  defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
huffman@29451
   901
  shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
huffman@29451
   902
unfolding b
huffman@29451
   903
apply (rule mod_poly_eq)
huffman@29537
   904
apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
huffman@29451
   905
done
huffman@29451
   906
huffman@29451
   907
huffman@29451
   908
subsection {* Evaluation of polynomials *}
huffman@29451
   909
huffman@29451
   910
definition
huffman@29454
   911
  poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a" where
huffman@29454
   912
  "poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)"
huffman@29451
   913
huffman@29451
   914
lemma poly_0 [simp]: "poly 0 x = 0"
huffman@29454
   915
  unfolding poly_def by (simp add: poly_rec_0)
huffman@29451
   916
huffman@29451
   917
lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x"
huffman@29454
   918
  unfolding poly_def by (simp add: poly_rec_pCons)
huffman@29451
   919
huffman@29451
   920
lemma poly_1 [simp]: "poly 1 x = 1"
huffman@29451
   921
  unfolding one_poly_def by simp
huffman@29451
   922
huffman@29454
   923
lemma poly_monom:
huffman@29454
   924
  fixes a x :: "'a::{comm_semiring_1,recpower}"
huffman@29454
   925
  shows "poly (monom a n) x = a * x ^ n"
huffman@29451
   926
  by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac)
huffman@29451
   927
huffman@29451
   928
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
huffman@29451
   929
  apply (induct p arbitrary: q, simp)
huffman@29451
   930
  apply (case_tac q, simp, simp add: ring_simps)
huffman@29451
   931
  done
huffman@29451
   932
huffman@29451
   933
lemma poly_minus [simp]:
huffman@29454
   934
  fixes x :: "'a::comm_ring"
huffman@29451
   935
  shows "poly (- p) x = - poly p x"
huffman@29451
   936
  by (induct p, simp_all)
huffman@29451
   937
huffman@29451
   938
lemma poly_diff [simp]:
huffman@29454
   939
  fixes x :: "'a::comm_ring"
huffman@29451
   940
  shows "poly (p - q) x = poly p x - poly q x"
huffman@29451
   941
  by (simp add: diff_minus)
huffman@29451
   942
huffman@29451
   943
lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
huffman@29451
   944
  by (cases "finite A", induct set: finite, simp_all)
huffman@29451
   945
huffman@29451
   946
lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
huffman@29451
   947
  by (induct p, simp, simp add: ring_simps)
huffman@29451
   948
huffman@29451
   949
lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
huffman@29451
   950
  by (induct p, simp_all, simp add: ring_simps)
huffman@29451
   951
huffman@29462
   952
lemma poly_power [simp]:
huffman@29462
   953
  fixes p :: "'a::{comm_semiring_1,recpower} poly"
huffman@29462
   954
  shows "poly (p ^ n) x = poly p x ^ n"
huffman@29462
   955
  by (induct n, simp, simp add: power_Suc)
huffman@29462
   956
huffman@29456
   957
huffman@29456
   958
subsection {* Synthetic division *}
huffman@29456
   959
huffman@29456
   960
definition
huffman@29456
   961
  synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
huffman@29478
   962
where [code del]:
huffman@29456
   963
  "synthetic_divmod p c =
huffman@29456
   964
    poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p"
huffman@29456
   965
huffman@29456
   966
definition
huffman@29456
   967
  synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
huffman@29456
   968
where
huffman@29456
   969
  "synthetic_div p c = fst (synthetic_divmod p c)"
huffman@29456
   970
huffman@29456
   971
lemma synthetic_divmod_0 [simp]:
huffman@29456
   972
  "synthetic_divmod 0 c = (0, 0)"
huffman@29456
   973
  unfolding synthetic_divmod_def
huffman@29456
   974
  by (simp add: poly_rec_0)
huffman@29456
   975
huffman@29456
   976
lemma synthetic_divmod_pCons [simp]:
huffman@29456
   977
  "synthetic_divmod (pCons a p) c =
huffman@29456
   978
    (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
huffman@29456
   979
  unfolding synthetic_divmod_def
huffman@29456
   980
  by (simp add: poly_rec_pCons)
huffman@29456
   981
huffman@29456
   982
lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c"
huffman@29456
   983
  by (induct p, simp, simp add: split_def)
huffman@29456
   984
huffman@29456
   985
lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0"
huffman@29456
   986
  unfolding synthetic_div_def by simp
huffman@29456
   987
huffman@29456
   988
lemma synthetic_div_pCons [simp]:
huffman@29456
   989
  "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
huffman@29456
   990
  unfolding synthetic_div_def
huffman@29456
   991
  by (simp add: split_def snd_synthetic_divmod)
huffman@29456
   992
huffman@29460
   993
lemma synthetic_div_eq_0_iff:
huffman@29460
   994
  "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
huffman@29460
   995
  by (induct p, simp, case_tac p, simp)
huffman@29460
   996
huffman@29460
   997
lemma degree_synthetic_div:
huffman@29460
   998
  "degree (synthetic_div p c) = degree p - 1"
huffman@29460
   999
  by (induct p, simp, simp add: synthetic_div_eq_0_iff)
huffman@29460
  1000
huffman@29457
  1001
lemma synthetic_div_correct:
huffman@29456
  1002
  "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
huffman@29456
  1003
  by (induct p) simp_all
huffman@29456
  1004
huffman@29457
  1005
lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
huffman@29457
  1006
by (induct p arbitrary: a) simp_all
huffman@29457
  1007
huffman@29457
  1008
lemma synthetic_div_unique:
huffman@29457
  1009
  "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
huffman@29457
  1010
apply (induct p arbitrary: q r)
huffman@29457
  1011
apply (simp, frule synthetic_div_unique_lemma, simp)
huffman@29457
  1012
apply (case_tac q, force)
huffman@29457
  1013
done
huffman@29457
  1014
huffman@29457
  1015
lemma synthetic_div_correct':
huffman@29457
  1016
  fixes c :: "'a::comm_ring_1"
huffman@29457
  1017
  shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
huffman@29457
  1018
  using synthetic_div_correct [of p c]
huffman@29457
  1019
  by (simp add: group_simps)
huffman@29457
  1020
huffman@29460
  1021
lemma poly_eq_0_iff_dvd:
huffman@29460
  1022
  fixes c :: "'a::idom"
huffman@29460
  1023
  shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
huffman@29460
  1024
proof
huffman@29460
  1025
  assume "poly p c = 0"
huffman@29460
  1026
  with synthetic_div_correct' [of c p]
huffman@29460
  1027
  have "p = [:-c, 1:] * synthetic_div p c" by simp
huffman@29460
  1028
  then show "[:-c, 1:] dvd p" ..
huffman@29460
  1029
next
huffman@29460
  1030
  assume "[:-c, 1:] dvd p"
huffman@29460
  1031
  then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
huffman@29460
  1032
  then show "poly p c = 0" by simp
huffman@29460
  1033
qed
huffman@29460
  1034
huffman@29460
  1035
lemma dvd_iff_poly_eq_0:
huffman@29460
  1036
  fixes c :: "'a::idom"
huffman@29460
  1037
  shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
huffman@29460
  1038
  by (simp add: poly_eq_0_iff_dvd)
huffman@29460
  1039
huffman@29462
  1040
lemma poly_roots_finite:
huffman@29462
  1041
  fixes p :: "'a::idom poly"
huffman@29462
  1042
  shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
huffman@29462
  1043
proof (induct n \<equiv> "degree p" arbitrary: p)
huffman@29462
  1044
  case (0 p)
huffman@29462
  1045
  then obtain a where "a \<noteq> 0" and "p = [:a:]"
huffman@29462
  1046
    by (cases p, simp split: if_splits)
huffman@29462
  1047
  then show "finite {x. poly p x = 0}" by simp
huffman@29462
  1048
next
huffman@29462
  1049
  case (Suc n p)
huffman@29462
  1050
  show "finite {x. poly p x = 0}"
huffman@29462
  1051
  proof (cases "\<exists>x. poly p x = 0")
huffman@29462
  1052
    case False
huffman@29462
  1053
    then show "finite {x. poly p x = 0}" by simp
huffman@29462
  1054
  next
huffman@29462
  1055
    case True
huffman@29462
  1056
    then obtain a where "poly p a = 0" ..
huffman@29462
  1057
    then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
huffman@29462
  1058
    then obtain k where k: "p = [:-a, 1:] * k" ..
huffman@29462
  1059
    with `p \<noteq> 0` have "k \<noteq> 0" by auto
huffman@29462
  1060
    with k have "degree p = Suc (degree k)"
huffman@29462
  1061
      by (simp add: degree_mult_eq del: mult_pCons_left)
huffman@29462
  1062
    with `Suc n = degree p` have "n = degree k" by simp
huffman@29462
  1063
    with `k \<noteq> 0` have "finite {x. poly k x = 0}" by (rule Suc.hyps)
huffman@29462
  1064
    then have "finite (insert a {x. poly k x = 0})" by simp
huffman@29462
  1065
    then show "finite {x. poly p x = 0}"
huffman@29462
  1066
      by (simp add: k uminus_add_conv_diff Collect_disj_eq
huffman@29462
  1067
               del: mult_pCons_left)
huffman@29462
  1068
  qed
huffman@29462
  1069
qed
huffman@29462
  1070
huffman@29478
  1071
huffman@29478
  1072
subsection {* Configuration of the code generator *}
huffman@29478
  1073
huffman@29478
  1074
code_datatype "0::'a::zero poly" pCons
huffman@29478
  1075
huffman@29480
  1076
declare pCons_0_0 [code post]
huffman@29480
  1077
huffman@29478
  1078
instantiation poly :: ("{zero,eq}") eq
huffman@29478
  1079
begin
huffman@29478
  1080
huffman@29478
  1081
definition [code del]:
huffman@29478
  1082
  "eq_class.eq (p::'a poly) q \<longleftrightarrow> p = q"
huffman@29478
  1083
huffman@29478
  1084
instance
huffman@29478
  1085
  by default (rule eq_poly_def)
huffman@29478
  1086
huffman@29451
  1087
end
huffman@29478
  1088
huffman@29478
  1089
lemma eq_poly_code [code]:
huffman@29478
  1090
  "eq_class.eq (0::_ poly) (0::_ poly) \<longleftrightarrow> True"
huffman@29478
  1091
  "eq_class.eq (0::_ poly) (pCons b q) \<longleftrightarrow> eq_class.eq 0 b \<and> eq_class.eq 0 q"
huffman@29478
  1092
  "eq_class.eq (pCons a p) (0::_ poly) \<longleftrightarrow> eq_class.eq a 0 \<and> eq_class.eq p 0"
huffman@29478
  1093
  "eq_class.eq (pCons a p) (pCons b q) \<longleftrightarrow> eq_class.eq a b \<and> eq_class.eq p q"
huffman@29478
  1094
unfolding eq by simp_all
huffman@29478
  1095
huffman@29478
  1096
lemmas coeff_code [code] =
huffman@29478
  1097
  coeff_0 coeff_pCons_0 coeff_pCons_Suc
huffman@29478
  1098
huffman@29478
  1099
lemmas degree_code [code] =
huffman@29478
  1100
  degree_0 degree_pCons_eq_if
huffman@29478
  1101
huffman@29478
  1102
lemmas monom_poly_code [code] =
huffman@29478
  1103
  monom_0 monom_Suc
huffman@29478
  1104
huffman@29478
  1105
lemma add_poly_code [code]:
huffman@29478
  1106
  "0 + q = (q :: _ poly)"
huffman@29478
  1107
  "p + 0 = (p :: _ poly)"
huffman@29478
  1108
  "pCons a p + pCons b q = pCons (a + b) (p + q)"
huffman@29478
  1109
by simp_all
huffman@29478
  1110
huffman@29478
  1111
lemma minus_poly_code [code]:
huffman@29478
  1112
  "- 0 = (0 :: _ poly)"
huffman@29478
  1113
  "- pCons a p = pCons (- a) (- p)"
huffman@29478
  1114
by simp_all
huffman@29478
  1115
huffman@29478
  1116
lemma diff_poly_code [code]:
huffman@29478
  1117
  "0 - q = (- q :: _ poly)"
huffman@29478
  1118
  "p - 0 = (p :: _ poly)"
huffman@29478
  1119
  "pCons a p - pCons b q = pCons (a - b) (p - q)"
huffman@29478
  1120
by simp_all
huffman@29478
  1121
huffman@29478
  1122
lemmas smult_poly_code [code] =
huffman@29478
  1123
  smult_0_right smult_pCons
huffman@29478
  1124
huffman@29478
  1125
lemma mult_poly_code [code]:
huffman@29478
  1126
  "0 * q = (0 :: _ poly)"
huffman@29478
  1127
  "pCons a p * q = smult a q + pCons 0 (p * q)"
huffman@29478
  1128
by simp_all
huffman@29478
  1129
huffman@29478
  1130
lemmas poly_code [code] =
huffman@29478
  1131
  poly_0 poly_pCons
huffman@29478
  1132
huffman@29478
  1133
lemmas synthetic_divmod_code [code] =
huffman@29478
  1134
  synthetic_divmod_0 synthetic_divmod_pCons
huffman@29478
  1135
huffman@29478
  1136
text {* code generator setup for div and mod *}
huffman@29478
  1137
huffman@29478
  1138
definition
huffman@29537
  1139
  pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
huffman@29478
  1140
where
huffman@29537
  1141
  [code del]: "pdivmod x y = (x div y, x mod y)"
huffman@29478
  1142
huffman@29537
  1143
lemma div_poly_code [code]: "x div y = fst (pdivmod x y)"
huffman@29537
  1144
  unfolding pdivmod_def by simp
huffman@29478
  1145
huffman@29537
  1146
lemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)"
huffman@29537
  1147
  unfolding pdivmod_def by simp
huffman@29478
  1148
huffman@29537
  1149
lemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)"
huffman@29537
  1150
  unfolding pdivmod_def by simp
huffman@29478
  1151
huffman@29537
  1152
lemma pdivmod_pCons [code]:
huffman@29537
  1153
  "pdivmod (pCons a x) y =
huffman@29478
  1154
    (if y = 0 then (0, pCons a x) else
huffman@29537
  1155
      (let (q, r) = pdivmod x y;
huffman@29478
  1156
           b = coeff (pCons a r) (degree y) / coeff y (degree y)
huffman@29478
  1157
        in (pCons b q, pCons a r - smult b y)))"
huffman@29537
  1158
apply (simp add: pdivmod_def Let_def, safe)
huffman@29478
  1159
apply (rule div_poly_eq)
huffman@29537
  1160
apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
huffman@29478
  1161
apply (rule mod_poly_eq)
huffman@29537
  1162
apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
huffman@29478
  1163
done
huffman@29478
  1164
huffman@29478
  1165
end