src/HOL/Polynomial.thy
author huffman
Mon Jan 12 12:09:54 2009 -0800 (2009-01-12)
changeset 29460 ad87e5d1488b
parent 29457 2eadbc24de8c
child 29462 dc97c6188a7a
permissions -rw-r--r--
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
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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]:
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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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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)
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lemma coeff_diff [simp]:
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  "coeff (p - q) n = coeff p n - coeff q n"
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  unfolding minus_poly_def
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  by (simp add: Abs_poly_inverse coeff Poly_diff)
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instance proof
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  fix p q :: "'a poly"
huffman@29451
   328
  show "- p + p = 0"
huffman@29451
   329
    by (simp add: expand_poly_eq)
huffman@29451
   330
  show "p - q = p + - q"
huffman@29451
   331
    by (simp add: expand_poly_eq diff_minus)
huffman@29451
   332
qed
huffman@29451
   333
huffman@29451
   334
end
huffman@29451
   335
huffman@29451
   336
lemma add_pCons [simp]:
huffman@29451
   337
  "pCons a p + pCons b q = pCons (a + b) (p + q)"
huffman@29451
   338
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
huffman@29451
   339
huffman@29451
   340
lemma minus_pCons [simp]:
huffman@29451
   341
  "- pCons a p = pCons (- a) (- p)"
huffman@29451
   342
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
huffman@29451
   343
huffman@29451
   344
lemma diff_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 degree_add_le: "degree (p + q) \<le> max (degree p) (degree q)"
huffman@29451
   349
  by (rule degree_le, auto simp add: coeff_eq_0)
huffman@29451
   350
huffman@29453
   351
lemma degree_add_less:
huffman@29453
   352
  "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
huffman@29453
   353
  by (auto intro: le_less_trans degree_add_le)
huffman@29453
   354
huffman@29451
   355
lemma degree_add_eq_right:
huffman@29451
   356
  "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
huffman@29451
   357
  apply (cases "q = 0", simp)
huffman@29451
   358
  apply (rule order_antisym)
huffman@29451
   359
  apply (rule ord_le_eq_trans [OF degree_add_le])
huffman@29451
   360
  apply simp
huffman@29451
   361
  apply (rule le_degree)
huffman@29451
   362
  apply (simp add: coeff_eq_0)
huffman@29451
   363
  done
huffman@29451
   364
huffman@29451
   365
lemma degree_add_eq_left:
huffman@29451
   366
  "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
huffman@29451
   367
  using degree_add_eq_right [of q p]
huffman@29451
   368
  by (simp add: add_commute)
huffman@29451
   369
huffman@29451
   370
lemma degree_minus [simp]: "degree (- p) = degree p"
huffman@29451
   371
  unfolding degree_def by simp
huffman@29451
   372
huffman@29451
   373
lemma degree_diff_le: "degree (p - q) \<le> max (degree p) (degree q)"
huffman@29451
   374
  using degree_add_le [where p=p and q="-q"]
huffman@29451
   375
  by (simp add: diff_minus)
huffman@29451
   376
huffman@29453
   377
lemma degree_diff_less:
huffman@29453
   378
  "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
huffman@29453
   379
  by (auto intro: le_less_trans degree_diff_le)
huffman@29453
   380
huffman@29451
   381
lemma add_monom: "monom a n + monom b n = monom (a + b) n"
huffman@29451
   382
  by (rule poly_ext) simp
huffman@29451
   383
huffman@29451
   384
lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
huffman@29451
   385
  by (rule poly_ext) simp
huffman@29451
   386
huffman@29451
   387
lemma minus_monom: "- monom a n = monom (-a) n"
huffman@29451
   388
  by (rule poly_ext) simp
huffman@29451
   389
huffman@29451
   390
lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
huffman@29451
   391
  by (cases "finite A", induct set: finite, simp_all)
huffman@29451
   392
huffman@29451
   393
lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
huffman@29451
   394
  by (rule poly_ext) (simp add: coeff_setsum)
huffman@29451
   395
huffman@29451
   396
huffman@29451
   397
subsection {* Multiplication by a constant *}
huffman@29451
   398
huffman@29451
   399
definition
huffman@29451
   400
  smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
huffman@29451
   401
  "smult a p = Abs_poly (\<lambda>n. a * coeff p n)"
huffman@29451
   402
huffman@29451
   403
lemma Poly_smult:
huffman@29451
   404
  fixes f :: "nat \<Rightarrow> 'a::comm_semiring_0"
huffman@29451
   405
  shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. a * f n) \<in> Poly"
huffman@29451
   406
  unfolding Poly_def
huffman@29451
   407
  by (clarify, rule_tac x=n in exI, simp)
huffman@29451
   408
huffman@29451
   409
lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n"
huffman@29451
   410
  unfolding smult_def
huffman@29451
   411
  by (simp add: Abs_poly_inverse Poly_smult coeff)
huffman@29451
   412
huffman@29451
   413
lemma degree_smult_le: "degree (smult a p) \<le> degree p"
huffman@29451
   414
  by (rule degree_le, simp add: coeff_eq_0)
huffman@29451
   415
huffman@29451
   416
lemma smult_smult: "smult a (smult b p) = smult (a * b) p"
huffman@29451
   417
  by (rule poly_ext, simp add: mult_assoc)
huffman@29451
   418
huffman@29451
   419
lemma smult_0_right [simp]: "smult a 0 = 0"
huffman@29451
   420
  by (rule poly_ext, simp)
huffman@29451
   421
huffman@29451
   422
lemma smult_0_left [simp]: "smult 0 p = 0"
huffman@29451
   423
  by (rule poly_ext, simp)
huffman@29451
   424
huffman@29451
   425
lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
huffman@29451
   426
  by (rule poly_ext, simp)
huffman@29451
   427
huffman@29451
   428
lemma smult_add_right:
huffman@29451
   429
  "smult a (p + q) = smult a p + smult a q"
huffman@29451
   430
  by (rule poly_ext, simp add: ring_simps)
huffman@29451
   431
huffman@29451
   432
lemma smult_add_left:
huffman@29451
   433
  "smult (a + b) p = smult a p + smult b p"
huffman@29451
   434
  by (rule poly_ext, simp add: ring_simps)
huffman@29451
   435
huffman@29457
   436
lemma smult_minus_right [simp]:
huffman@29451
   437
  "smult (a::'a::comm_ring) (- p) = - smult a p"
huffman@29451
   438
  by (rule poly_ext, simp)
huffman@29451
   439
huffman@29457
   440
lemma smult_minus_left [simp]:
huffman@29451
   441
  "smult (- a::'a::comm_ring) p = - smult a p"
huffman@29451
   442
  by (rule poly_ext, simp)
huffman@29451
   443
huffman@29451
   444
lemma smult_diff_right:
huffman@29451
   445
  "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
huffman@29451
   446
  by (rule poly_ext, simp add: ring_simps)
huffman@29451
   447
huffman@29451
   448
lemma smult_diff_left:
huffman@29451
   449
  "smult (a - b::'a::comm_ring) p = smult a p - smult b p"
huffman@29451
   450
  by (rule poly_ext, simp add: ring_simps)
huffman@29451
   451
huffman@29451
   452
lemma smult_pCons [simp]:
huffman@29451
   453
  "smult a (pCons b p) = pCons (a * b) (smult a p)"
huffman@29451
   454
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
huffman@29451
   455
huffman@29451
   456
lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
huffman@29451
   457
  by (induct n, simp add: monom_0, simp add: monom_Suc)
huffman@29451
   458
huffman@29451
   459
huffman@29451
   460
subsection {* Multiplication of polynomials *}
huffman@29451
   461
huffman@29451
   462
lemma Poly_mult_lemma:
huffman@29451
   463
  fixes f g :: "nat \<Rightarrow> 'a::comm_semiring_0" and m n :: nat
huffman@29451
   464
  assumes "\<forall>i>m. f i = 0"
huffman@29451
   465
  assumes "\<forall>j>n. g j = 0"
huffman@29451
   466
  shows "\<forall>k>m+n. (\<Sum>i\<le>k. f i * g (k-i)) = 0"
huffman@29451
   467
proof (clarify)
huffman@29451
   468
  fix k :: nat
huffman@29451
   469
  assume "m + n < k"
huffman@29451
   470
  show "(\<Sum>i\<le>k. f i * g (k - i)) = 0"
huffman@29451
   471
  proof (rule setsum_0' [rule_format])
huffman@29451
   472
    fix i :: nat
huffman@29451
   473
    assume "i \<in> {..k}" hence "i \<le> k" by simp
huffman@29451
   474
    with `m + n < k` have "m < i \<or> n < k - i" by arith
huffman@29451
   475
    thus "f i * g (k - i) = 0"
huffman@29451
   476
      using prems by auto
huffman@29451
   477
  qed
huffman@29451
   478
qed
huffman@29451
   479
huffman@29451
   480
lemma Poly_mult:
huffman@29451
   481
  fixes f g :: "nat \<Rightarrow> 'a::comm_semiring_0"
huffman@29451
   482
  shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. \<Sum>i\<le>n. f i * g (n-i)) \<in> Poly"
huffman@29451
   483
  unfolding Poly_def
huffman@29451
   484
  by (safe, rule exI, rule Poly_mult_lemma)
huffman@29451
   485
huffman@29451
   486
lemma poly_mult_assoc_lemma:
huffman@29451
   487
  fixes k :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
huffman@29451
   488
  shows "(\<Sum>j\<le>k. \<Sum>i\<le>j. f i (j - i) (n - j)) =
huffman@29451
   489
         (\<Sum>j\<le>k. \<Sum>i\<le>k - j. f j i (n - j - i))"
huffman@29451
   490
proof (induct k)
huffman@29451
   491
  case 0 show ?case by simp
huffman@29451
   492
next
huffman@29451
   493
  case (Suc k) thus ?case
huffman@29451
   494
    by (simp add: Suc_diff_le setsum_addf add_assoc
huffman@29451
   495
             cong: strong_setsum_cong)
huffman@29451
   496
qed
huffman@29451
   497
huffman@29451
   498
lemma poly_mult_commute_lemma:
huffman@29451
   499
  fixes n :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add"
huffman@29451
   500
  shows "(\<Sum>i\<le>n. f i (n - i)) = (\<Sum>i\<le>n. f (n - i) i)"
huffman@29451
   501
proof (rule setsum_reindex_cong)
huffman@29451
   502
  show "inj_on (\<lambda>i. n - i) {..n}"
huffman@29451
   503
    by (rule inj_onI) simp
huffman@29451
   504
  show "{..n} = (\<lambda>i. n - i) ` {..n}"
huffman@29451
   505
    by (auto, rule_tac x="n - x" in image_eqI, simp_all)
huffman@29451
   506
next
huffman@29451
   507
  fix i assume "i \<in> {..n}"
huffman@29451
   508
  hence "n - (n - i) = i" by simp
huffman@29451
   509
  thus "f (n - i) i = f (n - i) (n - (n - i))" by simp
huffman@29451
   510
qed
huffman@29451
   511
huffman@29451
   512
text {* TODO: move to appropriate theory *}
huffman@29451
   513
lemma setsum_atMost_Suc_shift:
huffman@29451
   514
  fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
huffman@29451
   515
  shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
huffman@29451
   516
proof (induct n)
huffman@29451
   517
  case 0 show ?case by simp
huffman@29451
   518
next
huffman@29451
   519
  case (Suc n) note IH = this
huffman@29451
   520
  have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
huffman@29451
   521
    by (rule setsum_atMost_Suc)
huffman@29451
   522
  also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
huffman@29451
   523
    by (rule IH)
huffman@29451
   524
  also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
huffman@29451
   525
             f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
huffman@29451
   526
    by (rule add_assoc)
huffman@29451
   527
  also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
huffman@29451
   528
    by (rule setsum_atMost_Suc [symmetric])
huffman@29451
   529
  finally show ?case .
huffman@29451
   530
qed
huffman@29451
   531
huffman@29451
   532
instantiation poly :: (comm_semiring_0) comm_semiring_0
huffman@29451
   533
begin
huffman@29451
   534
huffman@29451
   535
definition
huffman@29451
   536
  times_poly_def:
huffman@29451
   537
    "p * q = Abs_poly (\<lambda>n. \<Sum>i\<le>n. coeff p i * coeff q (n-i))"
huffman@29451
   538
huffman@29451
   539
lemma coeff_mult:
huffman@29451
   540
  "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
huffman@29451
   541
  unfolding times_poly_def
huffman@29451
   542
  by (simp add: Abs_poly_inverse coeff Poly_mult)
huffman@29451
   543
huffman@29451
   544
instance proof
huffman@29451
   545
  fix p q r :: "'a poly"
huffman@29451
   546
  show 0: "0 * p = 0"
huffman@29451
   547
    by (simp add: expand_poly_eq coeff_mult)
huffman@29451
   548
  show "p * 0 = 0"
huffman@29451
   549
    by (simp add: expand_poly_eq coeff_mult)
huffman@29451
   550
  show "(p + q) * r = p * r + q * r"
huffman@29451
   551
    by (simp add: expand_poly_eq coeff_mult left_distrib setsum_addf)
huffman@29451
   552
  show "(p * q) * r = p * (q * r)"
huffman@29451
   553
  proof (rule poly_ext)
huffman@29451
   554
    fix n :: nat
huffman@29451
   555
    have "(\<Sum>j\<le>n. \<Sum>i\<le>j. coeff p i * coeff q (j - i) * coeff r (n - j)) =
huffman@29451
   556
          (\<Sum>j\<le>n. \<Sum>i\<le>n - j. coeff p j * coeff q i * coeff r (n - j - i))"
huffman@29451
   557
      by (rule poly_mult_assoc_lemma)
huffman@29451
   558
    thus "coeff ((p * q) * r) n = coeff (p * (q * r)) n"
huffman@29451
   559
      by (simp add: coeff_mult setsum_right_distrib
huffman@29451
   560
                    setsum_left_distrib mult_assoc)
huffman@29451
   561
  qed
huffman@29451
   562
  show "p * q = q * p"
huffman@29451
   563
  proof (rule poly_ext)
huffman@29451
   564
    fix n :: nat
huffman@29451
   565
    have "(\<Sum>i\<le>n. coeff p i * coeff q (n - i)) =
huffman@29451
   566
          (\<Sum>i\<le>n. coeff p (n - i) * coeff q i)"
huffman@29451
   567
      by (rule poly_mult_commute_lemma)
huffman@29451
   568
    thus "coeff (p * q) n = coeff (q * p) n"
huffman@29451
   569
      by (simp add: coeff_mult mult_commute)
huffman@29451
   570
  qed
huffman@29451
   571
qed
huffman@29451
   572
huffman@29451
   573
end
huffman@29451
   574
huffman@29451
   575
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
huffman@29451
   576
apply (rule degree_le, simp add: coeff_mult)
huffman@29451
   577
apply (rule Poly_mult_lemma)
huffman@29451
   578
apply (simp_all add: coeff_eq_0)
huffman@29451
   579
done
huffman@29451
   580
huffman@29451
   581
lemma mult_pCons_left [simp]:
huffman@29451
   582
  "pCons a p * q = smult a q + pCons 0 (p * q)"
huffman@29451
   583
apply (rule poly_ext)
huffman@29451
   584
apply (case_tac n)
huffman@29451
   585
apply (simp add: coeff_mult)
huffman@29451
   586
apply (simp add: coeff_mult setsum_atMost_Suc_shift
huffman@29451
   587
            del: setsum_atMost_Suc)
huffman@29451
   588
done
huffman@29451
   589
huffman@29451
   590
lemma mult_pCons_right [simp]:
huffman@29451
   591
  "p * pCons a q = smult a p + pCons 0 (p * q)"
huffman@29451
   592
  using mult_pCons_left [of a q p] by (simp add: mult_commute)
huffman@29451
   593
huffman@29451
   594
lemma mult_smult_left: "smult a p * q = smult a (p * q)"
huffman@29451
   595
  by (induct p, simp, simp add: smult_add_right smult_smult)
huffman@29451
   596
huffman@29451
   597
lemma mult_smult_right: "p * smult a q = smult a (p * q)"
huffman@29451
   598
  using mult_smult_left [of a q p] by (simp add: mult_commute)
huffman@29451
   599
huffman@29451
   600
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
huffman@29451
   601
  by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)
huffman@29451
   602
huffman@29451
   603
huffman@29451
   604
subsection {* The unit polynomial and exponentiation *}
huffman@29451
   605
huffman@29451
   606
instantiation poly :: (comm_semiring_1) comm_semiring_1
huffman@29451
   607
begin
huffman@29451
   608
huffman@29451
   609
definition
huffman@29451
   610
  one_poly_def:
huffman@29451
   611
    "1 = pCons 1 0"
huffman@29451
   612
huffman@29451
   613
instance proof
huffman@29451
   614
  fix p :: "'a poly" show "1 * p = p"
huffman@29451
   615
    unfolding one_poly_def
huffman@29451
   616
    by simp
huffman@29451
   617
next
huffman@29451
   618
  show "0 \<noteq> (1::'a poly)"
huffman@29451
   619
    unfolding one_poly_def by simp
huffman@29451
   620
qed
huffman@29451
   621
huffman@29451
   622
end
huffman@29451
   623
huffman@29451
   624
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
huffman@29451
   625
  unfolding one_poly_def
huffman@29451
   626
  by (simp add: coeff_pCons split: nat.split)
huffman@29451
   627
huffman@29451
   628
lemma degree_1 [simp]: "degree 1 = 0"
huffman@29451
   629
  unfolding one_poly_def
huffman@29451
   630
  by (rule degree_pCons_0)
huffman@29451
   631
huffman@29451
   632
instantiation poly :: (comm_semiring_1) recpower
huffman@29451
   633
begin
huffman@29451
   634
huffman@29451
   635
primrec power_poly where
huffman@29451
   636
  power_poly_0: "(p::'a poly) ^ 0 = 1"
huffman@29451
   637
| power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n"
huffman@29451
   638
huffman@29451
   639
instance
huffman@29451
   640
  by default simp_all
huffman@29451
   641
huffman@29451
   642
end
huffman@29451
   643
huffman@29451
   644
instance poly :: (comm_ring) comm_ring ..
huffman@29451
   645
huffman@29451
   646
instance poly :: (comm_ring_1) comm_ring_1 ..
huffman@29451
   647
huffman@29451
   648
instantiation poly :: (comm_ring_1) number_ring
huffman@29451
   649
begin
huffman@29451
   650
huffman@29451
   651
definition
huffman@29451
   652
  "number_of k = (of_int k :: 'a poly)"
huffman@29451
   653
huffman@29451
   654
instance
huffman@29451
   655
  by default (rule number_of_poly_def)
huffman@29451
   656
huffman@29451
   657
end
huffman@29451
   658
huffman@29451
   659
huffman@29451
   660
subsection {* Polynomials form an integral domain *}
huffman@29451
   661
huffman@29451
   662
lemma coeff_mult_degree_sum:
huffman@29451
   663
  "coeff (p * q) (degree p + degree q) =
huffman@29451
   664
   coeff p (degree p) * coeff q (degree q)"
huffman@29451
   665
 apply (simp add: coeff_mult)
huffman@29451
   666
 apply (subst setsum_diff1' [where a="degree p"])
huffman@29451
   667
   apply simp
huffman@29451
   668
  apply simp
huffman@29451
   669
 apply (subst setsum_0' [rule_format])
huffman@29451
   670
  apply clarsimp
huffman@29451
   671
  apply (subgoal_tac "degree p < a \<or> degree q < degree p + degree q - a")
huffman@29451
   672
   apply (force simp add: coeff_eq_0)
huffman@29451
   673
  apply arith
huffman@29451
   674
 apply simp
huffman@29451
   675
done
huffman@29451
   676
huffman@29451
   677
instance poly :: (idom) idom
huffman@29451
   678
proof
huffman@29451
   679
  fix p q :: "'a poly"
huffman@29451
   680
  assume "p \<noteq> 0" and "q \<noteq> 0"
huffman@29451
   681
  have "coeff (p * q) (degree p + degree q) =
huffman@29451
   682
        coeff p (degree p) * coeff q (degree q)"
huffman@29451
   683
    by (rule coeff_mult_degree_sum)
huffman@29451
   684
  also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
huffman@29451
   685
    using `p \<noteq> 0` and `q \<noteq> 0` by simp
huffman@29451
   686
  finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
huffman@29451
   687
  thus "p * q \<noteq> 0" by (simp add: expand_poly_eq)
huffman@29451
   688
qed
huffman@29451
   689
huffman@29451
   690
lemma degree_mult_eq:
huffman@29451
   691
  fixes p q :: "'a::idom poly"
huffman@29451
   692
  shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
huffman@29451
   693
apply (rule order_antisym [OF degree_mult_le le_degree])
huffman@29451
   694
apply (simp add: coeff_mult_degree_sum)
huffman@29451
   695
done
huffman@29451
   696
huffman@29451
   697
lemma dvd_imp_degree_le:
huffman@29451
   698
  fixes p q :: "'a::idom poly"
huffman@29451
   699
  shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
huffman@29451
   700
  by (erule dvdE, simp add: degree_mult_eq)
huffman@29451
   701
huffman@29451
   702
huffman@29451
   703
subsection {* Long division of polynomials *}
huffman@29451
   704
huffman@29451
   705
definition
huffman@29451
   706
  divmod_poly_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
huffman@29451
   707
where
huffman@29451
   708
  "divmod_poly_rel x y q r \<longleftrightarrow>
huffman@29451
   709
    x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
huffman@29451
   710
huffman@29451
   711
lemma divmod_poly_rel_0:
huffman@29451
   712
  "divmod_poly_rel 0 y 0 0"
huffman@29451
   713
  unfolding divmod_poly_rel_def by simp
huffman@29451
   714
huffman@29451
   715
lemma divmod_poly_rel_by_0:
huffman@29451
   716
  "divmod_poly_rel x 0 0 x"
huffman@29451
   717
  unfolding divmod_poly_rel_def by simp
huffman@29451
   718
huffman@29451
   719
lemma eq_zero_or_degree_less:
huffman@29451
   720
  assumes "degree p \<le> n" and "coeff p n = 0"
huffman@29451
   721
  shows "p = 0 \<or> degree p < n"
huffman@29451
   722
proof (cases n)
huffman@29451
   723
  case 0
huffman@29451
   724
  with `degree p \<le> n` and `coeff p n = 0`
huffman@29451
   725
  have "coeff p (degree p) = 0" by simp
huffman@29451
   726
  then have "p = 0" by simp
huffman@29451
   727
  then show ?thesis ..
huffman@29451
   728
next
huffman@29451
   729
  case (Suc m)
huffman@29451
   730
  have "\<forall>i>n. coeff p i = 0"
huffman@29451
   731
    using `degree p \<le> n` by (simp add: coeff_eq_0)
huffman@29451
   732
  then have "\<forall>i\<ge>n. coeff p i = 0"
huffman@29451
   733
    using `coeff p n = 0` by (simp add: le_less)
huffman@29451
   734
  then have "\<forall>i>m. coeff p i = 0"
huffman@29451
   735
    using `n = Suc m` by (simp add: less_eq_Suc_le)
huffman@29451
   736
  then have "degree p \<le> m"
huffman@29451
   737
    by (rule degree_le)
huffman@29451
   738
  then have "degree p < n"
huffman@29451
   739
    using `n = Suc m` by (simp add: less_Suc_eq_le)
huffman@29451
   740
  then show ?thesis ..
huffman@29451
   741
qed
huffman@29451
   742
huffman@29451
   743
lemma divmod_poly_rel_pCons:
huffman@29451
   744
  assumes rel: "divmod_poly_rel x y q r"
huffman@29451
   745
  assumes y: "y \<noteq> 0"
huffman@29451
   746
  assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
huffman@29451
   747
  shows "divmod_poly_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
huffman@29451
   748
    (is "divmod_poly_rel ?x y ?q ?r")
huffman@29451
   749
proof -
huffman@29451
   750
  have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
huffman@29451
   751
    using assms unfolding divmod_poly_rel_def by simp_all
huffman@29451
   752
huffman@29451
   753
  have 1: "?x = ?q * y + ?r"
huffman@29451
   754
    using b x by simp
huffman@29451
   755
huffman@29451
   756
  have 2: "?r = 0 \<or> degree ?r < degree y"
huffman@29451
   757
  proof (rule eq_zero_or_degree_less)
huffman@29451
   758
    have "degree ?r \<le> max (degree (pCons a r)) (degree (smult b y))"
huffman@29451
   759
      by (rule degree_diff_le)
huffman@29451
   760
    also have "\<dots> \<le> degree y"
huffman@29451
   761
    proof (rule min_max.le_supI)
huffman@29451
   762
      show "degree (pCons a r) \<le> degree y"
huffman@29460
   763
        using r by auto
huffman@29451
   764
      show "degree (smult b y) \<le> degree y"
huffman@29451
   765
        by (rule degree_smult_le)
huffman@29451
   766
    qed
huffman@29451
   767
    finally show "degree ?r \<le> degree y" .
huffman@29451
   768
  next
huffman@29451
   769
    show "coeff ?r (degree y) = 0"
huffman@29451
   770
      using `y \<noteq> 0` unfolding b by simp
huffman@29451
   771
  qed
huffman@29451
   772
huffman@29451
   773
  from 1 2 show ?thesis
huffman@29451
   774
    unfolding divmod_poly_rel_def
huffman@29451
   775
    using `y \<noteq> 0` by simp
huffman@29451
   776
qed
huffman@29451
   777
huffman@29451
   778
lemma divmod_poly_rel_exists: "\<exists>q r. divmod_poly_rel x y q r"
huffman@29451
   779
apply (cases "y = 0")
huffman@29451
   780
apply (fast intro!: divmod_poly_rel_by_0)
huffman@29451
   781
apply (induct x)
huffman@29451
   782
apply (fast intro!: divmod_poly_rel_0)
huffman@29451
   783
apply (fast intro!: divmod_poly_rel_pCons)
huffman@29451
   784
done
huffman@29451
   785
huffman@29451
   786
lemma divmod_poly_rel_unique:
huffman@29451
   787
  assumes 1: "divmod_poly_rel x y q1 r1"
huffman@29451
   788
  assumes 2: "divmod_poly_rel x y q2 r2"
huffman@29451
   789
  shows "q1 = q2 \<and> r1 = r2"
huffman@29451
   790
proof (cases "y = 0")
huffman@29451
   791
  assume "y = 0" with assms show ?thesis
huffman@29451
   792
    by (simp add: divmod_poly_rel_def)
huffman@29451
   793
next
huffman@29451
   794
  assume [simp]: "y \<noteq> 0"
huffman@29451
   795
  from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
huffman@29451
   796
    unfolding divmod_poly_rel_def by simp_all
huffman@29451
   797
  from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
huffman@29451
   798
    unfolding divmod_poly_rel_def by simp_all
huffman@29451
   799
  from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
huffman@29451
   800
    by (simp add: ring_simps)
huffman@29451
   801
  from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
huffman@29453
   802
    by (auto intro: degree_diff_less)
huffman@29451
   803
huffman@29451
   804
  show "q1 = q2 \<and> r1 = r2"
huffman@29451
   805
  proof (rule ccontr)
huffman@29451
   806
    assume "\<not> (q1 = q2 \<and> r1 = r2)"
huffman@29451
   807
    with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
huffman@29451
   808
    with r3 have "degree (r2 - r1) < degree y" by simp
huffman@29451
   809
    also have "degree y \<le> degree (q1 - q2) + degree y" by simp
huffman@29451
   810
    also have "\<dots> = degree ((q1 - q2) * y)"
huffman@29451
   811
      using `q1 \<noteq> q2` by (simp add: degree_mult_eq)
huffman@29451
   812
    also have "\<dots> = degree (r2 - r1)"
huffman@29451
   813
      using q3 by simp
huffman@29451
   814
    finally have "degree (r2 - r1) < degree (r2 - r1)" .
huffman@29451
   815
    then show "False" by simp
huffman@29451
   816
  qed
huffman@29451
   817
qed
huffman@29451
   818
huffman@29451
   819
lemmas divmod_poly_rel_unique_div =
huffman@29451
   820
  divmod_poly_rel_unique [THEN conjunct1, standard]
huffman@29451
   821
huffman@29451
   822
lemmas divmod_poly_rel_unique_mod =
huffman@29451
   823
  divmod_poly_rel_unique [THEN conjunct2, standard]
huffman@29451
   824
huffman@29451
   825
instantiation poly :: (field) ring_div
huffman@29451
   826
begin
huffman@29451
   827
huffman@29451
   828
definition div_poly where
huffman@29451
   829
  [code del]: "x div y = (THE q. \<exists>r. divmod_poly_rel x y q r)"
huffman@29451
   830
huffman@29451
   831
definition mod_poly where
huffman@29451
   832
  [code del]: "x mod y = (THE r. \<exists>q. divmod_poly_rel x y q r)"
huffman@29451
   833
huffman@29451
   834
lemma div_poly_eq:
huffman@29451
   835
  "divmod_poly_rel x y q r \<Longrightarrow> x div y = q"
huffman@29451
   836
unfolding div_poly_def
huffman@29451
   837
by (fast elim: divmod_poly_rel_unique_div)
huffman@29451
   838
huffman@29451
   839
lemma mod_poly_eq:
huffman@29451
   840
  "divmod_poly_rel x y q r \<Longrightarrow> x mod y = r"
huffman@29451
   841
unfolding mod_poly_def
huffman@29451
   842
by (fast elim: divmod_poly_rel_unique_mod)
huffman@29451
   843
huffman@29451
   844
lemma divmod_poly_rel:
huffman@29451
   845
  "divmod_poly_rel x y (x div y) (x mod y)"
huffman@29451
   846
proof -
huffman@29451
   847
  from divmod_poly_rel_exists
huffman@29451
   848
    obtain q r where "divmod_poly_rel x y q r" by fast
huffman@29451
   849
  thus ?thesis
huffman@29451
   850
    by (simp add: div_poly_eq mod_poly_eq)
huffman@29451
   851
qed
huffman@29451
   852
huffman@29451
   853
instance proof
huffman@29451
   854
  fix x y :: "'a poly"
huffman@29451
   855
  show "x div y * y + x mod y = x"
huffman@29451
   856
    using divmod_poly_rel [of x y]
huffman@29451
   857
    by (simp add: divmod_poly_rel_def)
huffman@29451
   858
next
huffman@29451
   859
  fix x :: "'a poly"
huffman@29451
   860
  have "divmod_poly_rel x 0 0 x"
huffman@29451
   861
    by (rule divmod_poly_rel_by_0)
huffman@29451
   862
  thus "x div 0 = 0"
huffman@29451
   863
    by (rule div_poly_eq)
huffman@29451
   864
next
huffman@29451
   865
  fix y :: "'a poly"
huffman@29451
   866
  have "divmod_poly_rel 0 y 0 0"
huffman@29451
   867
    by (rule divmod_poly_rel_0)
huffman@29451
   868
  thus "0 div y = 0"
huffman@29451
   869
    by (rule div_poly_eq)
huffman@29451
   870
next
huffman@29451
   871
  fix x y z :: "'a poly"
huffman@29451
   872
  assume "y \<noteq> 0"
huffman@29451
   873
  hence "divmod_poly_rel (x + z * y) y (z + x div y) (x mod y)"
huffman@29451
   874
    using divmod_poly_rel [of x y]
huffman@29451
   875
    by (simp add: divmod_poly_rel_def left_distrib)
huffman@29451
   876
  thus "(x + z * y) div y = z + x div y"
huffman@29451
   877
    by (rule div_poly_eq)
huffman@29451
   878
qed
huffman@29451
   879
huffman@29451
   880
end
huffman@29451
   881
huffman@29451
   882
lemma degree_mod_less:
huffman@29451
   883
  "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
huffman@29451
   884
  using divmod_poly_rel [of x y]
huffman@29451
   885
  unfolding divmod_poly_rel_def by simp
huffman@29451
   886
huffman@29451
   887
lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
huffman@29451
   888
proof -
huffman@29451
   889
  assume "degree x < degree y"
huffman@29451
   890
  hence "divmod_poly_rel x y 0 x"
huffman@29451
   891
    by (simp add: divmod_poly_rel_def)
huffman@29451
   892
  thus "x div y = 0" by (rule div_poly_eq)
huffman@29451
   893
qed
huffman@29451
   894
huffman@29451
   895
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
huffman@29451
   896
proof -
huffman@29451
   897
  assume "degree x < degree y"
huffman@29451
   898
  hence "divmod_poly_rel x y 0 x"
huffman@29451
   899
    by (simp add: divmod_poly_rel_def)
huffman@29451
   900
  thus "x mod y = x" by (rule mod_poly_eq)
huffman@29451
   901
qed
huffman@29451
   902
huffman@29451
   903
lemma mod_pCons:
huffman@29451
   904
  fixes a and x
huffman@29451
   905
  assumes y: "y \<noteq> 0"
huffman@29451
   906
  defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
huffman@29451
   907
  shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
huffman@29451
   908
unfolding b
huffman@29451
   909
apply (rule mod_poly_eq)
huffman@29451
   910
apply (rule divmod_poly_rel_pCons [OF divmod_poly_rel y refl])
huffman@29451
   911
done
huffman@29451
   912
huffman@29451
   913
huffman@29451
   914
subsection {* Evaluation of polynomials *}
huffman@29451
   915
huffman@29451
   916
definition
huffman@29454
   917
  poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a" where
huffman@29454
   918
  "poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)"
huffman@29451
   919
huffman@29451
   920
lemma poly_0 [simp]: "poly 0 x = 0"
huffman@29454
   921
  unfolding poly_def by (simp add: poly_rec_0)
huffman@29451
   922
huffman@29451
   923
lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x"
huffman@29454
   924
  unfolding poly_def by (simp add: poly_rec_pCons)
huffman@29451
   925
huffman@29451
   926
lemma poly_1 [simp]: "poly 1 x = 1"
huffman@29451
   927
  unfolding one_poly_def by simp
huffman@29451
   928
huffman@29454
   929
lemma poly_monom:
huffman@29454
   930
  fixes a x :: "'a::{comm_semiring_1,recpower}"
huffman@29454
   931
  shows "poly (monom a n) x = a * x ^ n"
huffman@29451
   932
  by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac)
huffman@29451
   933
huffman@29451
   934
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
huffman@29451
   935
  apply (induct p arbitrary: q, simp)
huffman@29451
   936
  apply (case_tac q, simp, simp add: ring_simps)
huffman@29451
   937
  done
huffman@29451
   938
huffman@29451
   939
lemma poly_minus [simp]:
huffman@29454
   940
  fixes x :: "'a::comm_ring"
huffman@29451
   941
  shows "poly (- p) x = - poly p x"
huffman@29451
   942
  by (induct p, simp_all)
huffman@29451
   943
huffman@29451
   944
lemma poly_diff [simp]:
huffman@29454
   945
  fixes x :: "'a::comm_ring"
huffman@29451
   946
  shows "poly (p - q) x = poly p x - poly q x"
huffman@29451
   947
  by (simp add: diff_minus)
huffman@29451
   948
huffman@29451
   949
lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
huffman@29451
   950
  by (cases "finite A", induct set: finite, simp_all)
huffman@29451
   951
huffman@29451
   952
lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
huffman@29451
   953
  by (induct p, simp, simp add: ring_simps)
huffman@29451
   954
huffman@29451
   955
lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
huffman@29451
   956
  by (induct p, simp_all, simp add: ring_simps)
huffman@29451
   957
huffman@29456
   958
huffman@29456
   959
subsection {* Synthetic division *}
huffman@29456
   960
huffman@29456
   961
definition
huffman@29456
   962
  synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
huffman@29456
   963
where
huffman@29456
   964
  "synthetic_divmod p c =
huffman@29456
   965
    poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p"
huffman@29456
   966
huffman@29456
   967
definition
huffman@29456
   968
  synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
huffman@29456
   969
where
huffman@29456
   970
  "synthetic_div p c = fst (synthetic_divmod p c)"
huffman@29456
   971
huffman@29456
   972
lemma synthetic_divmod_0 [simp]:
huffman@29456
   973
  "synthetic_divmod 0 c = (0, 0)"
huffman@29456
   974
  unfolding synthetic_divmod_def
huffman@29456
   975
  by (simp add: poly_rec_0)
huffman@29456
   976
huffman@29456
   977
lemma synthetic_divmod_pCons [simp]:
huffman@29456
   978
  "synthetic_divmod (pCons a p) c =
huffman@29456
   979
    (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
huffman@29456
   980
  unfolding synthetic_divmod_def
huffman@29456
   981
  by (simp add: poly_rec_pCons)
huffman@29456
   982
huffman@29456
   983
lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c"
huffman@29456
   984
  by (induct p, simp, simp add: split_def)
huffman@29456
   985
huffman@29456
   986
lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0"
huffman@29456
   987
  unfolding synthetic_div_def by simp
huffman@29456
   988
huffman@29456
   989
lemma synthetic_div_pCons [simp]:
huffman@29456
   990
  "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
huffman@29456
   991
  unfolding synthetic_div_def
huffman@29456
   992
  by (simp add: split_def snd_synthetic_divmod)
huffman@29456
   993
huffman@29460
   994
lemma synthetic_div_eq_0_iff:
huffman@29460
   995
  "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
huffman@29460
   996
  by (induct p, simp, case_tac p, simp)
huffman@29460
   997
huffman@29460
   998
lemma degree_synthetic_div:
huffman@29460
   999
  "degree (synthetic_div p c) = degree p - 1"
huffman@29460
  1000
  by (induct p, simp, simp add: synthetic_div_eq_0_iff)
huffman@29460
  1001
huffman@29457
  1002
lemma synthetic_div_correct:
huffman@29456
  1003
  "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
huffman@29456
  1004
  by (induct p) simp_all
huffman@29456
  1005
huffman@29457
  1006
lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
huffman@29457
  1007
by (induct p arbitrary: a) simp_all
huffman@29457
  1008
huffman@29457
  1009
lemma synthetic_div_unique:
huffman@29457
  1010
  "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
huffman@29457
  1011
apply (induct p arbitrary: q r)
huffman@29457
  1012
apply (simp, frule synthetic_div_unique_lemma, simp)
huffman@29457
  1013
apply (case_tac q, force)
huffman@29457
  1014
done
huffman@29457
  1015
huffman@29457
  1016
lemma synthetic_div_correct':
huffman@29457
  1017
  fixes c :: "'a::comm_ring_1"
huffman@29457
  1018
  shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
huffman@29457
  1019
  using synthetic_div_correct [of p c]
huffman@29457
  1020
  by (simp add: group_simps)
huffman@29457
  1021
huffman@29460
  1022
lemma poly_eq_0_iff_dvd:
huffman@29460
  1023
  fixes c :: "'a::idom"
huffman@29460
  1024
  shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
huffman@29460
  1025
proof
huffman@29460
  1026
  assume "poly p c = 0"
huffman@29460
  1027
  with synthetic_div_correct' [of c p]
huffman@29460
  1028
  have "p = [:-c, 1:] * synthetic_div p c" by simp
huffman@29460
  1029
  then show "[:-c, 1:] dvd p" ..
huffman@29460
  1030
next
huffman@29460
  1031
  assume "[:-c, 1:] dvd p"
huffman@29460
  1032
  then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
huffman@29460
  1033
  then show "poly p c = 0" by simp
huffman@29460
  1034
qed
huffman@29460
  1035
huffman@29460
  1036
lemma dvd_iff_poly_eq_0:
huffman@29460
  1037
  fixes c :: "'a::idom"
huffman@29460
  1038
  shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
huffman@29460
  1039
  by (simp add: poly_eq_0_iff_dvd)
huffman@29460
  1040
huffman@29451
  1041
end