src/HOL/Polynomial.thy
author huffman
Tue, 13 Jan 2009 13:48:21 -0800
changeset 29478 4a2482e16934
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child 29480 4e08ee896e81
permissions -rw-r--r--
code generation for polynomials
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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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   183
proof (induct p rule: measure_induct_rule [where f=degree])
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  case (less p)
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   185
  obtain a q where "p = pCons a q" by (rule pCons_cases)
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   186
  have "P q"
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  proof (cases "q = 0")
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    case True
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   189
    then show "P q" by (simp add: zero)
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  next
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    case False
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   192
    then have "degree (pCons a q) = Suc (degree q)"
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      by (rule degree_pCons_eq)
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   194
    then have "degree q < degree p"
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   195
      using `p = pCons a q` by simp
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   196
    then show "P q"
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      by (rule less.hyps)
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  qed
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   199
  then have "P (pCons a q)"
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    by (rule pCons)
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   201
  then show ?case
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    using `p = pCons a q` by simp
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qed
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   204
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29454
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subsection {* Recursion combinator for polynomials *}
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b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
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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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b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
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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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b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
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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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   222
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
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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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   225
  by (simp add: poly_rec_pCons_eq_if poly_rec_0)
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   226
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
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   227
29451
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   228
subsection {* Monomials *}
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   229
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   230
definition
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   231
  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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   233
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   234
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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   237
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   238
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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   240
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   241
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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   243
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   244
lemma monom_eq_0 [simp]: "monom 0 n = 0"
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   245
  by (rule poly_ext) simp
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   246
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   247
lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
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   248
  by (simp add: expand_poly_eq)
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   249
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   250
lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
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   251
  by (simp add: expand_poly_eq)
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   252
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   253
lemma degree_monom_le: "degree (monom a n) \<le> n"
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   254
  by (rule degree_le, simp)
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   255
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   256
lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
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   257
  apply (rule order_antisym [OF degree_monom_le])
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   258
  apply (rule le_degree, simp)
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   259
  done
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   260
5f0cb3fa530d new theory of polynomials
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   261
5f0cb3fa530d new theory of polynomials
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   262
subsection {* Addition and subtraction *}
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   263
5f0cb3fa530d new theory of polynomials
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   264
instantiation poly :: (comm_monoid_add) comm_monoid_add
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   265
begin
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   266
5f0cb3fa530d new theory of polynomials
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   267
definition
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   268
  plus_poly_def [code del]:
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   269
    "p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)"
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   270
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   271
lemma Poly_add:
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   272
  fixes f g :: "nat \<Rightarrow> 'a"
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   273
  shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n + g n) \<in> Poly"
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   274
  unfolding Poly_def
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   275
  apply (clarify, rename_tac m n)
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   276
  apply (rule_tac x="max m n" in exI, simp)
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   277
  done
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   278
5f0cb3fa530d new theory of polynomials
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   279
lemma coeff_add [simp]:
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   280
  "coeff (p + q) n = coeff p n + coeff q n"
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   281
  unfolding plus_poly_def
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   282
  by (simp add: Abs_poly_inverse coeff Poly_add)
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   283
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   284
instance proof
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   285
  fix p q r :: "'a poly"
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   286
  show "(p + q) + r = p + (q + r)"
5f0cb3fa530d new theory of polynomials
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   287
    by (simp add: expand_poly_eq add_assoc)
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   288
  show "p + q = q + p"
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   289
    by (simp add: expand_poly_eq add_commute)
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   290
  show "0 + p = p"
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   291
    by (simp add: expand_poly_eq)
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   292
qed
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   293
5f0cb3fa530d new theory of polynomials
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   294
end
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   295
5f0cb3fa530d new theory of polynomials
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   296
instantiation poly :: (ab_group_add) ab_group_add
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   297
begin
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   298
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   299
definition
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   300
  uminus_poly_def [code del]:
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   301
    "- p = Abs_poly (\<lambda>n. - coeff p n)"
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   302
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   303
definition
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   304
  minus_poly_def [code del]:
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   305
    "p - q = Abs_poly (\<lambda>n. coeff p n - coeff q n)"
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   306
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   307
lemma Poly_minus:
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   308
  fixes f :: "nat \<Rightarrow> 'a"
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   309
  shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. - f n) \<in> Poly"
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   310
  unfolding Poly_def by simp
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diff changeset
   311
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   312
lemma Poly_diff:
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   313
  fixes f g :: "nat \<Rightarrow> 'a"
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   314
  shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n - g n) \<in> Poly"
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parents:
diff changeset
   315
  unfolding diff_minus by (simp add: Poly_add Poly_minus)
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diff changeset
   316
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   317
lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
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   318
  unfolding uminus_poly_def
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   319
  by (simp add: Abs_poly_inverse coeff Poly_minus)
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diff changeset
   320
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   321
lemma coeff_diff [simp]:
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   322
  "coeff (p - q) n = coeff p n - coeff q n"
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   323
  unfolding minus_poly_def
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   324
  by (simp add: Abs_poly_inverse coeff Poly_diff)
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   325
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   326
instance proof
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   327
  fix p q :: "'a poly"
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   328
  show "- p + p = 0"
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parents:
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   329
    by (simp add: expand_poly_eq)
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parents:
diff changeset
   330
  show "p - q = p + - q"
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parents:
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   331
    by (simp add: expand_poly_eq diff_minus)
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   332
qed
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parents:
diff changeset
   333
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diff changeset
   334
end
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parents:
diff changeset
   335
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   336
lemma add_pCons [simp]:
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   337
  "pCons a p + pCons b q = pCons (a + b) (p + q)"
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   338
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
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parents:
diff changeset
   339
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   340
lemma minus_pCons [simp]:
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   341
  "- pCons a p = pCons (- a) (- p)"
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   342
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
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diff changeset
   343
5f0cb3fa530d new theory of polynomials
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   344
lemma diff_pCons [simp]:
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   345
  "pCons a p - pCons b q = pCons (a - b) (p - q)"
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   346
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
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parents:
diff changeset
   347
5f0cb3fa530d new theory of polynomials
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   348
lemma degree_add_le: "degree (p + q) \<le> max (degree p) (degree q)"
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   349
  by (rule degree_le, auto simp add: coeff_eq_0)
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parents:
diff changeset
   350
29453
de4f26f59135 add lemmas degree_{add,diff}_less
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diff changeset
   351
lemma degree_add_less:
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diff changeset
   352
  "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
de4f26f59135 add lemmas degree_{add,diff}_less
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parents: 29451
diff changeset
   353
  by (auto intro: le_less_trans degree_add_le)
de4f26f59135 add lemmas degree_{add,diff}_less
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parents: 29451
diff changeset
   354
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   355
lemma degree_add_eq_right:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   356
  "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   357
  apply (cases "q = 0", simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   358
  apply (rule order_antisym)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   359
  apply (rule ord_le_eq_trans [OF degree_add_le])
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   360
  apply simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   361
  apply (rule le_degree)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   362
  apply (simp add: coeff_eq_0)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   363
  done
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   364
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   365
lemma degree_add_eq_left:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   366
  "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   367
  using degree_add_eq_right [of q p]
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   368
  by (simp add: add_commute)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   369
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   370
lemma degree_minus [simp]: "degree (- p) = degree p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   371
  unfolding degree_def by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   372
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   373
lemma degree_diff_le: "degree (p - q) \<le> max (degree p) (degree q)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   374
  using degree_add_le [where p=p and q="-q"]
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   375
  by (simp add: diff_minus)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   376
29453
de4f26f59135 add lemmas degree_{add,diff}_less
huffman
parents: 29451
diff changeset
   377
lemma degree_diff_less:
de4f26f59135 add lemmas degree_{add,diff}_less
huffman
parents: 29451
diff changeset
   378
  "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
de4f26f59135 add lemmas degree_{add,diff}_less
huffman
parents: 29451
diff changeset
   379
  by (auto intro: le_less_trans degree_diff_le)
de4f26f59135 add lemmas degree_{add,diff}_less
huffman
parents: 29451
diff changeset
   380
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   381
lemma add_monom: "monom a n + monom b n = monom (a + b) n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   382
  by (rule poly_ext) simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   383
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   384
lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   385
  by (rule poly_ext) simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   386
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   387
lemma minus_monom: "- monom a n = monom (-a) n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   388
  by (rule poly_ext) simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   389
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   390
lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   391
  by (cases "finite A", induct set: finite, simp_all)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   392
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   393
lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   394
  by (rule poly_ext) (simp add: coeff_setsum)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   395
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   396
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   397
subsection {* Multiplication by a constant *}
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   398
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   399
definition
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   400
  smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   401
  "smult a p = Abs_poly (\<lambda>n. a * coeff p n)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   402
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   403
lemma Poly_smult:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   404
  fixes f :: "nat \<Rightarrow> 'a::comm_semiring_0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   405
  shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. a * f n) \<in> Poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   406
  unfolding Poly_def
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   407
  by (clarify, rule_tac x=n in exI, simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   408
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   409
lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   410
  unfolding smult_def
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   411
  by (simp add: Abs_poly_inverse Poly_smult coeff)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   412
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   413
lemma degree_smult_le: "degree (smult a p) \<le> degree p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   414
  by (rule degree_le, simp add: coeff_eq_0)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   415
29472
a63a2e46cec9 declare smult rules [simp]
huffman
parents: 29471
diff changeset
   416
lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   417
  by (rule poly_ext, simp add: mult_assoc)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   418
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   419
lemma smult_0_right [simp]: "smult a 0 = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   420
  by (rule poly_ext, simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   421
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   422
lemma smult_0_left [simp]: "smult 0 p = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   423
  by (rule poly_ext, simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   424
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   425
lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   426
  by (rule poly_ext, simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   427
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   428
lemma smult_add_right:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   429
  "smult a (p + q) = smult a p + smult a q"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   430
  by (rule poly_ext, simp add: ring_simps)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   431
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   432
lemma smult_add_left:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   433
  "smult (a + b) p = smult a p + smult b p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   434
  by (rule poly_ext, simp add: ring_simps)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   435
29457
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   436
lemma smult_minus_right [simp]:
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   437
  "smult (a::'a::comm_ring) (- p) = - smult a p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   438
  by (rule poly_ext, simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   439
29457
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   440
lemma smult_minus_left [simp]:
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   441
  "smult (- a::'a::comm_ring) p = - smult a p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   442
  by (rule poly_ext, simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   443
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   444
lemma smult_diff_right:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   445
  "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   446
  by (rule poly_ext, simp add: ring_simps)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   447
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   448
lemma smult_diff_left:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   449
  "smult (a - b::'a::comm_ring) p = smult a p - smult b p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   450
  by (rule poly_ext, simp add: ring_simps)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   451
29472
a63a2e46cec9 declare smult rules [simp]
huffman
parents: 29471
diff changeset
   452
lemmas smult_distribs =
a63a2e46cec9 declare smult rules [simp]
huffman
parents: 29471
diff changeset
   453
  smult_add_left smult_add_right
a63a2e46cec9 declare smult rules [simp]
huffman
parents: 29471
diff changeset
   454
  smult_diff_left smult_diff_right
a63a2e46cec9 declare smult rules [simp]
huffman
parents: 29471
diff changeset
   455
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   456
lemma smult_pCons [simp]:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   457
  "smult a (pCons b p) = pCons (a * b) (smult a p)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   458
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   459
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   460
lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   461
  by (induct n, simp add: monom_0, simp add: monom_Suc)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   462
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   463
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   464
subsection {* Multiplication of polynomials *}
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   465
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   466
text {* TODO: move to SetInterval.thy *}
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   467
lemma setsum_atMost_Suc_shift:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   468
  fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   469
  shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   470
proof (induct n)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   471
  case 0 show ?case by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   472
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   473
  case (Suc n) note IH = this
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   474
  have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   475
    by (rule setsum_atMost_Suc)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   476
  also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   477
    by (rule IH)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   478
  also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   479
             f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   480
    by (rule add_assoc)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   481
  also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   482
    by (rule setsum_atMost_Suc [symmetric])
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   483
  finally show ?case .
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   484
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   485
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   486
instantiation poly :: (comm_semiring_0) comm_semiring_0
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   487
begin
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   488
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   489
definition
29475
c06d1b0a970f declare more definitions [code del]
huffman
parents: 29474
diff changeset
   490
  times_poly_def [code del]:
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   491
    "p * q = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   492
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   493
lemma mult_poly_0_left: "(0::'a poly) * q = 0"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   494
  unfolding times_poly_def by (simp add: poly_rec_0)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   495
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   496
lemma mult_pCons_left [simp]:
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   497
  "pCons a p * q = smult a q + pCons 0 (p * q)"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   498
  unfolding times_poly_def by (simp add: poly_rec_pCons)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   499
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   500
lemma mult_poly_0_right: "p * (0::'a poly) = 0"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   501
  by (induct p, simp add: mult_poly_0_left, simp)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   502
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   503
lemma mult_pCons_right [simp]:
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   504
  "p * pCons a q = smult a p + pCons 0 (p * q)"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   505
  by (induct p, simp add: mult_poly_0_left, simp add: ring_simps)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   506
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   507
lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   508
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   509
lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   510
  by (induct p, simp add: mult_poly_0, simp add: smult_add_right)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   511
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   512
lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   513
  by (induct q, simp add: mult_poly_0, simp add: smult_add_right)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   514
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   515
lemma mult_poly_add_left:
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   516
  fixes p q r :: "'a poly"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   517
  shows "(p + q) * r = p * r + q * r"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   518
  by (induct r, simp add: mult_poly_0,
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   519
                simp add: smult_distribs group_simps)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   520
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   521
instance proof
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   522
  fix p q r :: "'a poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   523
  show 0: "0 * p = 0"
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   524
    by (rule mult_poly_0_left)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   525
  show "p * 0 = 0"
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   526
    by (rule mult_poly_0_right)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   527
  show "(p + q) * r = p * r + q * r"
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   528
    by (rule mult_poly_add_left)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   529
  show "(p * q) * r = p * (q * r)"
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   530
    by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   531
  show "p * q = q * p"
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   532
    by (induct p, simp add: mult_poly_0, simp)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   533
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   534
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   535
end
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   536
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   537
lemma coeff_mult:
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   538
  "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   539
proof (induct p arbitrary: n)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   540
  case 0 show ?case by simp
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   541
next
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   542
  case (pCons a p n) thus ?case
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   543
    by (cases n, simp, simp add: setsum_atMost_Suc_shift
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   544
                            del: setsum_atMost_Suc)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   545
qed
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   546
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   547
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   548
apply (rule degree_le)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   549
apply (induct p)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   550
apply simp
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   551
apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   552
done
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   553
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   554
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   555
  by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   556
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   557
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   558
subsection {* The unit polynomial and exponentiation *}
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   559
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   560
instantiation poly :: (comm_semiring_1) comm_semiring_1
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   561
begin
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   562
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   563
definition
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   564
  one_poly_def:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   565
    "1 = pCons 1 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   566
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   567
instance proof
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   568
  fix p :: "'a poly" show "1 * p = p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   569
    unfolding one_poly_def
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   570
    by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   571
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   572
  show "0 \<noteq> (1::'a poly)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   573
    unfolding one_poly_def by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   574
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   575
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   576
end
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   577
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   578
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   579
  unfolding one_poly_def
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   580
  by (simp add: coeff_pCons split: nat.split)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   581
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   582
lemma degree_1 [simp]: "degree 1 = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   583
  unfolding one_poly_def
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   584
  by (rule degree_pCons_0)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   585
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   586
instantiation poly :: (comm_semiring_1) recpower
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   587
begin
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   588
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   589
primrec power_poly where
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   590
  power_poly_0: "(p::'a poly) ^ 0 = 1"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   591
| power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   592
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   593
instance
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   594
  by default simp_all
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   595
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   596
end
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   597
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   598
instance poly :: (comm_ring) comm_ring ..
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   599
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   600
instance poly :: (comm_ring_1) comm_ring_1 ..
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   601
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   602
instantiation poly :: (comm_ring_1) number_ring
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   603
begin
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   604
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   605
definition
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   606
  "number_of k = (of_int k :: 'a poly)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   607
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   608
instance
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   609
  by default (rule number_of_poly_def)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   610
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   611
end
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   612
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   613
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   614
subsection {* Polynomials form an integral domain *}
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   615
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   616
lemma coeff_mult_degree_sum:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   617
  "coeff (p * q) (degree p + degree q) =
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   618
   coeff p (degree p) * coeff q (degree q)"
29471
6a46a13ce1f9 simplify proof of coeff_mult_degree_sum
huffman
parents: 29462
diff changeset
   619
  by (induct p, simp, simp add: coeff_eq_0)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   620
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   621
instance poly :: (idom) idom
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   622
proof
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   623
  fix p q :: "'a poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   624
  assume "p \<noteq> 0" and "q \<noteq> 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   625
  have "coeff (p * q) (degree p + degree q) =
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   626
        coeff p (degree p) * coeff q (degree q)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   627
    by (rule coeff_mult_degree_sum)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   628
  also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   629
    using `p \<noteq> 0` and `q \<noteq> 0` by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   630
  finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   631
  thus "p * q \<noteq> 0" by (simp add: expand_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   632
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   633
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   634
lemma degree_mult_eq:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   635
  fixes p q :: "'a::idom poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   636
  shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   637
apply (rule order_antisym [OF degree_mult_le le_degree])
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   638
apply (simp add: coeff_mult_degree_sum)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   639
done
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   640
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   641
lemma dvd_imp_degree_le:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   642
  fixes p q :: "'a::idom poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   643
  shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   644
  by (erule dvdE, simp add: degree_mult_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   645
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   646
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   647
subsection {* Long division of polynomials *}
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   648
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   649
definition
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   650
  divmod_poly_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   651
where
29475
c06d1b0a970f declare more definitions [code del]
huffman
parents: 29474
diff changeset
   652
  [code del]:
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   653
  "divmod_poly_rel x y q r \<longleftrightarrow>
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   654
    x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   655
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   656
lemma divmod_poly_rel_0:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   657
  "divmod_poly_rel 0 y 0 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   658
  unfolding divmod_poly_rel_def by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   659
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   660
lemma divmod_poly_rel_by_0:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   661
  "divmod_poly_rel x 0 0 x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   662
  unfolding divmod_poly_rel_def by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   663
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   664
lemma eq_zero_or_degree_less:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   665
  assumes "degree p \<le> n" and "coeff p n = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   666
  shows "p = 0 \<or> degree p < n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   667
proof (cases n)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   668
  case 0
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   669
  with `degree p \<le> n` and `coeff p n = 0`
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   670
  have "coeff p (degree p) = 0" by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   671
  then have "p = 0" by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   672
  then show ?thesis ..
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   673
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   674
  case (Suc m)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   675
  have "\<forall>i>n. coeff p i = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   676
    using `degree p \<le> n` by (simp add: coeff_eq_0)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   677
  then have "\<forall>i\<ge>n. coeff p i = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   678
    using `coeff p n = 0` by (simp add: le_less)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   679
  then have "\<forall>i>m. coeff p i = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   680
    using `n = Suc m` by (simp add: less_eq_Suc_le)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   681
  then have "degree p \<le> m"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   682
    by (rule degree_le)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   683
  then have "degree p < n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   684
    using `n = Suc m` by (simp add: less_Suc_eq_le)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   685
  then show ?thesis ..
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   686
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   687
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   688
lemma divmod_poly_rel_pCons:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   689
  assumes rel: "divmod_poly_rel x y q r"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   690
  assumes y: "y \<noteq> 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   691
  assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   692
  shows "divmod_poly_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   693
    (is "divmod_poly_rel ?x y ?q ?r")
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   694
proof -
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   695
  have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   696
    using assms unfolding divmod_poly_rel_def by simp_all
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   697
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   698
  have 1: "?x = ?q * y + ?r"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   699
    using b x by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   700
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   701
  have 2: "?r = 0 \<or> degree ?r < degree y"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   702
  proof (rule eq_zero_or_degree_less)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   703
    have "degree ?r \<le> max (degree (pCons a r)) (degree (smult b y))"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   704
      by (rule degree_diff_le)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   705
    also have "\<dots> \<le> degree y"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   706
    proof (rule min_max.le_supI)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   707
      show "degree (pCons a r) \<le> degree y"
29460
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   708
        using r by auto
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   709
      show "degree (smult b y) \<le> degree y"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   710
        by (rule degree_smult_le)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   711
    qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   712
    finally show "degree ?r \<le> degree y" .
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   713
  next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   714
    show "coeff ?r (degree y) = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   715
      using `y \<noteq> 0` unfolding b by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   716
  qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   717
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   718
  from 1 2 show ?thesis
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   719
    unfolding divmod_poly_rel_def
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   720
    using `y \<noteq> 0` by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   721
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   722
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   723
lemma divmod_poly_rel_exists: "\<exists>q r. divmod_poly_rel x y q r"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   724
apply (cases "y = 0")
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   725
apply (fast intro!: divmod_poly_rel_by_0)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   726
apply (induct x)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   727
apply (fast intro!: divmod_poly_rel_0)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   728
apply (fast intro!: divmod_poly_rel_pCons)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   729
done
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   730
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   731
lemma divmod_poly_rel_unique:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   732
  assumes 1: "divmod_poly_rel x y q1 r1"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   733
  assumes 2: "divmod_poly_rel x y q2 r2"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   734
  shows "q1 = q2 \<and> r1 = r2"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   735
proof (cases "y = 0")
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   736
  assume "y = 0" with assms show ?thesis
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   737
    by (simp add: divmod_poly_rel_def)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   738
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   739
  assume [simp]: "y \<noteq> 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   740
  from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   741
    unfolding divmod_poly_rel_def by simp_all
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   742
  from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   743
    unfolding divmod_poly_rel_def by simp_all
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   744
  from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   745
    by (simp add: ring_simps)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   746
  from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
29453
de4f26f59135 add lemmas degree_{add,diff}_less
huffman
parents: 29451
diff changeset
   747
    by (auto intro: degree_diff_less)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   748
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   749
  show "q1 = q2 \<and> r1 = r2"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   750
  proof (rule ccontr)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   751
    assume "\<not> (q1 = q2 \<and> r1 = r2)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   752
    with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   753
    with r3 have "degree (r2 - r1) < degree y" by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   754
    also have "degree y \<le> degree (q1 - q2) + degree y" by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   755
    also have "\<dots> = degree ((q1 - q2) * y)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   756
      using `q1 \<noteq> q2` by (simp add: degree_mult_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   757
    also have "\<dots> = degree (r2 - r1)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   758
      using q3 by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   759
    finally have "degree (r2 - r1) < degree (r2 - r1)" .
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   760
    then show "False" by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   761
  qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   762
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   763
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   764
lemmas divmod_poly_rel_unique_div =
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   765
  divmod_poly_rel_unique [THEN conjunct1, standard]
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   766
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   767
lemmas divmod_poly_rel_unique_mod =
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   768
  divmod_poly_rel_unique [THEN conjunct2, standard]
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   769
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   770
instantiation poly :: (field) ring_div
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   771
begin
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   772
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   773
definition div_poly where
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   774
  [code del]: "x div y = (THE q. \<exists>r. divmod_poly_rel x y q r)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   775
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   776
definition mod_poly where
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   777
  [code del]: "x mod y = (THE r. \<exists>q. divmod_poly_rel x y q r)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   778
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   779
lemma div_poly_eq:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   780
  "divmod_poly_rel x y q r \<Longrightarrow> x div y = q"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   781
unfolding div_poly_def
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   782
by (fast elim: divmod_poly_rel_unique_div)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   783
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   784
lemma mod_poly_eq:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   785
  "divmod_poly_rel x y q r \<Longrightarrow> x mod y = r"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   786
unfolding mod_poly_def
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   787
by (fast elim: divmod_poly_rel_unique_mod)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   788
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   789
lemma divmod_poly_rel:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   790
  "divmod_poly_rel x y (x div y) (x mod y)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   791
proof -
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   792
  from divmod_poly_rel_exists
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   793
    obtain q r where "divmod_poly_rel x y q r" by fast
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   794
  thus ?thesis
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   795
    by (simp add: div_poly_eq mod_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   796
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   797
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   798
instance proof
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   799
  fix x y :: "'a poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   800
  show "x div y * y + x mod y = x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   801
    using divmod_poly_rel [of x y]
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   802
    by (simp add: divmod_poly_rel_def)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   803
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   804
  fix x :: "'a poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   805
  have "divmod_poly_rel x 0 0 x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   806
    by (rule divmod_poly_rel_by_0)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   807
  thus "x div 0 = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   808
    by (rule div_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   809
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   810
  fix y :: "'a poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   811
  have "divmod_poly_rel 0 y 0 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   812
    by (rule divmod_poly_rel_0)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   813
  thus "0 div y = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   814
    by (rule div_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   815
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   816
  fix x y z :: "'a poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   817
  assume "y \<noteq> 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   818
  hence "divmod_poly_rel (x + z * y) y (z + x div y) (x mod y)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   819
    using divmod_poly_rel [of x y]
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   820
    by (simp add: divmod_poly_rel_def left_distrib)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   821
  thus "(x + z * y) div y = z + x div y"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   822
    by (rule div_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   823
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   824
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   825
end
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   826
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   827
lemma degree_mod_less:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   828
  "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   829
  using divmod_poly_rel [of x y]
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   830
  unfolding divmod_poly_rel_def by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   831
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   832
lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   833
proof -
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   834
  assume "degree x < degree y"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   835
  hence "divmod_poly_rel x y 0 x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   836
    by (simp add: divmod_poly_rel_def)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   837
  thus "x div y = 0" by (rule div_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   838
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   839
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   840
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   841
proof -
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   842
  assume "degree x < degree y"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   843
  hence "divmod_poly_rel x y 0 x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   844
    by (simp add: divmod_poly_rel_def)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   845
  thus "x mod y = x" by (rule mod_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   846
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   847
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   848
lemma mod_pCons:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   849
  fixes a and x
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   850
  assumes y: "y \<noteq> 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   851
  defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   852
  shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   853
unfolding b
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   854
apply (rule mod_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   855
apply (rule divmod_poly_rel_pCons [OF divmod_poly_rel y refl])
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   856
done
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   857
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   858
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   859
subsection {* Evaluation of polynomials *}
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   860
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   861
definition
29454
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
   862
  poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a" where
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
   863
  "poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   864
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   865
lemma poly_0 [simp]: "poly 0 x = 0"
29454
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
   866
  unfolding poly_def by (simp add: poly_rec_0)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   867
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   868
lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x"
29454
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
   869
  unfolding poly_def by (simp add: poly_rec_pCons)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   870
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   871
lemma poly_1 [simp]: "poly 1 x = 1"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   872
  unfolding one_poly_def by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   873
29454
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
   874
lemma poly_monom:
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
   875
  fixes a x :: "'a::{comm_semiring_1,recpower}"
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
   876
  shows "poly (monom a n) x = a * x ^ n"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   877
  by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   878
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   879
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   880
  apply (induct p arbitrary: q, simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   881
  apply (case_tac q, simp, simp add: ring_simps)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   882
  done
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   883
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   884
lemma poly_minus [simp]:
29454
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
   885
  fixes x :: "'a::comm_ring"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   886
  shows "poly (- p) x = - poly p x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   887
  by (induct p, simp_all)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   888
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   889
lemma poly_diff [simp]:
29454
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
   890
  fixes x :: "'a::comm_ring"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   891
  shows "poly (p - q) x = poly p x - poly q x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   892
  by (simp add: diff_minus)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   893
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   894
lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   895
  by (cases "finite A", induct set: finite, simp_all)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   896
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   897
lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   898
  by (induct p, simp, simp add: ring_simps)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   899
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   900
lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   901
  by (induct p, simp_all, simp add: ring_simps)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   902
29462
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   903
lemma poly_power [simp]:
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   904
  fixes p :: "'a::{comm_semiring_1,recpower} poly"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   905
  shows "poly (p ^ n) x = poly p x ^ n"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   906
  by (induct n, simp, simp add: power_Suc)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   907
29456
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   908
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   909
subsection {* Synthetic division *}
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   910
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   911
definition
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   912
  synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a"
29478
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
   913
where [code del]:
29456
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   914
  "synthetic_divmod p c =
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   915
    poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   916
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   917
definition
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   918
  synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   919
where
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   920
  "synthetic_div p c = fst (synthetic_divmod p c)"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   921
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   922
lemma synthetic_divmod_0 [simp]:
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   923
  "synthetic_divmod 0 c = (0, 0)"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   924
  unfolding synthetic_divmod_def
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   925
  by (simp add: poly_rec_0)
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   926
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   927
lemma synthetic_divmod_pCons [simp]:
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   928
  "synthetic_divmod (pCons a p) c =
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   929
    (\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   930
  unfolding synthetic_divmod_def
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   931
  by (simp add: poly_rec_pCons)
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   932
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   933
lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   934
  by (induct p, simp, simp add: split_def)
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   935
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   936
lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   937
  unfolding synthetic_div_def by simp
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   938
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   939
lemma synthetic_div_pCons [simp]:
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   940
  "synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   941
  unfolding synthetic_div_def
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   942
  by (simp add: split_def snd_synthetic_divmod)
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   943
29460
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   944
lemma synthetic_div_eq_0_iff:
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   945
  "synthetic_div p c = 0 \<longleftrightarrow> degree p = 0"
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   946
  by (induct p, simp, case_tac p, simp)
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   947
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   948
lemma degree_synthetic_div:
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   949
  "degree (synthetic_div p c) = degree p - 1"
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   950
  by (induct p, simp, simp add: synthetic_div_eq_0_iff)
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   951
29457
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   952
lemma synthetic_div_correct:
29456
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   953
  "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   954
  by (induct p) simp_all
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
   955
29457
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   956
lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0"
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   957
by (induct p arbitrary: a) simp_all
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   958
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   959
lemma synthetic_div_unique:
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   960
  "p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c"
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   961
apply (induct p arbitrary: q r)
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   962
apply (simp, frule synthetic_div_unique_lemma, simp)
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   963
apply (case_tac q, force)
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   964
done
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   965
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   966
lemma synthetic_div_correct':
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   967
  fixes c :: "'a::comm_ring_1"
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   968
  shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   969
  using synthetic_div_correct [of p c]
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   970
  by (simp add: group_simps)
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   971
29460
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   972
lemma poly_eq_0_iff_dvd:
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   973
  fixes c :: "'a::idom"
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   974
  shows "poly p c = 0 \<longleftrightarrow> [:-c, 1:] dvd p"
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   975
proof
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   976
  assume "poly p c = 0"
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   977
  with synthetic_div_correct' [of c p]
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   978
  have "p = [:-c, 1:] * synthetic_div p c" by simp
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   979
  then show "[:-c, 1:] dvd p" ..
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   980
next
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   981
  assume "[:-c, 1:] dvd p"
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   982
  then obtain k where "p = [:-c, 1:] * k" by (rule dvdE)
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   983
  then show "poly p c = 0" by simp
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   984
qed
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   985
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   986
lemma dvd_iff_poly_eq_0:
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   987
  fixes c :: "'a::idom"
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   988
  shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (-c) = 0"
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   989
  by (simp add: poly_eq_0_iff_dvd)
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
   990
29462
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   991
lemma poly_roots_finite:
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   992
  fixes p :: "'a::idom poly"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   993
  shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   994
proof (induct n \<equiv> "degree p" arbitrary: p)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   995
  case (0 p)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   996
  then obtain a where "a \<noteq> 0" and "p = [:a:]"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   997
    by (cases p, simp split: if_splits)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   998
  then show "finite {x. poly p x = 0}" by simp
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
   999
next
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1000
  case (Suc n p)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1001
  show "finite {x. poly p x = 0}"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1002
  proof (cases "\<exists>x. poly p x = 0")
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1003
    case False
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1004
    then show "finite {x. poly p x = 0}" by simp
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1005
  next
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1006
    case True
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1007
    then obtain a where "poly p a = 0" ..
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1008
    then have "[:-a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1009
    then obtain k where k: "p = [:-a, 1:] * k" ..
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1010
    with `p \<noteq> 0` have "k \<noteq> 0" by auto
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1011
    with k have "degree p = Suc (degree k)"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1012
      by (simp add: degree_mult_eq del: mult_pCons_left)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1013
    with `Suc n = degree p` have "n = degree k" by simp
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1014
    with `k \<noteq> 0` have "finite {x. poly k x = 0}" by (rule Suc.hyps)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1015
    then have "finite (insert a {x. poly k x = 0})" by simp
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1016
    then show "finite {x. poly p x = 0}"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1017
      by (simp add: k uminus_add_conv_diff Collect_disj_eq
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1018
               del: mult_pCons_left)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1019
  qed
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1020
qed
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1021
29478
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1022
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1023
subsection {* Configuration of the code generator *}
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1024
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1025
code_datatype "0::'a::zero poly" pCons
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1026
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1027
instantiation poly :: ("{zero,eq}") eq
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1028
begin
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1029
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1030
definition [code del]:
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1031
  "eq_class.eq (p::'a poly) q \<longleftrightarrow> p = q"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1032
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1033
instance
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1034
  by default (rule eq_poly_def)
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1035
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1036
end
29478
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1037
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1038
lemma eq_poly_code [code]:
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1039
  "eq_class.eq (0::_ poly) (0::_ poly) \<longleftrightarrow> True"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1040
  "eq_class.eq (0::_ poly) (pCons b q) \<longleftrightarrow> eq_class.eq 0 b \<and> eq_class.eq 0 q"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1041
  "eq_class.eq (pCons a p) (0::_ poly) \<longleftrightarrow> eq_class.eq a 0 \<and> eq_class.eq p 0"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1042
  "eq_class.eq (pCons a p) (pCons b q) \<longleftrightarrow> eq_class.eq a b \<and> eq_class.eq p q"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1043
unfolding eq by simp_all
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1044
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1045
lemmas coeff_code [code] =
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1046
  coeff_0 coeff_pCons_0 coeff_pCons_Suc
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1047
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1048
lemmas degree_code [code] =
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1049
  degree_0 degree_pCons_eq_if
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1050
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1051
lemmas monom_poly_code [code] =
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1052
  monom_0 monom_Suc
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1053
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1054
lemma add_poly_code [code]:
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1055
  "0 + q = (q :: _ poly)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1056
  "p + 0 = (p :: _ poly)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1057
  "pCons a p + pCons b q = pCons (a + b) (p + q)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1058
by simp_all
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1059
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1060
lemma minus_poly_code [code]:
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1061
  "- 0 = (0 :: _ poly)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1062
  "- pCons a p = pCons (- a) (- p)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1063
by simp_all
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1064
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1065
lemma diff_poly_code [code]:
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1066
  "0 - q = (- q :: _ poly)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1067
  "p - 0 = (p :: _ poly)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1068
  "pCons a p - pCons b q = pCons (a - b) (p - q)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1069
by simp_all
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1070
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1071
lemmas smult_poly_code [code] =
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1072
  smult_0_right smult_pCons
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1073
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1074
lemma mult_poly_code [code]:
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1075
  "0 * q = (0 :: _ poly)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1076
  "pCons a p * q = smult a q + pCons 0 (p * q)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1077
by simp_all
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1078
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1079
lemmas poly_code [code] =
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1080
  poly_0 poly_pCons
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1081
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1082
lemmas synthetic_divmod_code [code] =
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1083
  synthetic_divmod_0 synthetic_divmod_pCons
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1084
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1085
text {* code generator setup for div and mod *}
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1086
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1087
definition
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1088
  divmod_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1089
where
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1090
  [code del]: "divmod_poly x y = (x div y, x mod y)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1091
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1092
lemma div_poly_code [code]: "x div y = fst (divmod_poly x y)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1093
  unfolding divmod_poly_def by simp
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1094
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1095
lemma mod_poly_code [code]: "x mod y = snd (divmod_poly x y)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1096
  unfolding divmod_poly_def by simp
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1097
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1098
lemma divmod_poly_0 [code]: "divmod_poly 0 y = (0, 0)"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1099
  unfolding divmod_poly_def by simp
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1100
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1101
lemma divmod_poly_pCons [code]:
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1102
  "divmod_poly (pCons a x) y =
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1103
    (if y = 0 then (0, pCons a x) else
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1104
      (let (q, r) = divmod_poly x y;
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1105
           b = coeff (pCons a r) (degree y) / coeff y (degree y)
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1106
        in (pCons b q, pCons a r - smult b y)))"
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1107
apply (simp add: divmod_poly_def Let_def, safe)
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1108
apply (rule div_poly_eq)
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1109
apply (erule divmod_poly_rel_pCons [OF divmod_poly_rel _ refl])
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1110
apply (rule mod_poly_eq)
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1111
apply (erule divmod_poly_rel_pCons [OF divmod_poly_rel _ refl])
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1112
done
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1113
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1114
end