src/HOL/Library/Polynomial.thy
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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 Main
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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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  "[:x:]" <= "CONST pCons x (_constrain 0 t)"
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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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   179
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   180
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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   184
proof (induct p rule: measure_induct_rule [where f=degree])
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  case (less p)
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   186
  obtain a q where "p = pCons a q" by (rule pCons_cases)
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   187
  have "P q"
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   188
  proof (cases "q = 0")
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    case True
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   190
    then show "P q" by (simp add: zero)
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  next
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    case False
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   193
    then have "degree (pCons a q) = Suc (degree q)"
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   194
      by (rule degree_pCons_eq)
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   195
    then have "degree q < degree p"
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   196
      using `p = pCons a q` by simp
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   197
    then show "P q"
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      by (rule less.hyps)
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  qed
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   200
  then have "P (pCons a q)"
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    by (rule pCons)
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  then show ?case
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    using `p = pCons a q` by simp
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qed
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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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c06d1b0a970f declare more definitions [code del]
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  poly_rec_pCons_eq_if [simp del, code del]:
29454
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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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   223
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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   226
  by (simp add: poly_rec_pCons_eq_if poly_rec_0)
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   227
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
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29451
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subsection {* Monomials *}
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definition
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  monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly" where
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  "monom a m = Abs_poly (\<lambda>n. if m = n then a else 0)"
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lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)"
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  unfolding monom_def
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  by (subst Abs_poly_inverse, auto simp add: Poly_def)
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   238
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   239
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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   241
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   242
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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   244
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   245
lemma monom_eq_0 [simp]: "monom 0 n = 0"
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   246
  by (rule poly_ext) simp
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   247
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lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0"
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  by (simp add: expand_poly_eq)
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   250
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lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b"
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  by (simp add: expand_poly_eq)
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   253
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   254
lemma degree_monom_le: "degree (monom a n) \<le> n"
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  by (rule degree_le, simp)
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   256
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   257
lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n"
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   258
  apply (rule order_antisym [OF degree_monom_le])
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   259
  apply (rule le_degree, simp)
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  done
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   261
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   262
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   263
subsection {* Addition and subtraction *}
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   264
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   265
instantiation poly :: (comm_monoid_add) comm_monoid_add
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   266
begin
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   267
5f0cb3fa530d new theory of polynomials
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   268
definition
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   269
  plus_poly_def [code del]:
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   270
    "p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)"
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   271
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   272
lemma Poly_add:
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   273
  fixes f g :: "nat \<Rightarrow> 'a"
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   274
  shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n + g n) \<in> Poly"
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   275
  unfolding Poly_def
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   276
  apply (clarify, rename_tac m n)
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   277
  apply (rule_tac x="max m n" in exI, simp)
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   278
  done
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   279
5f0cb3fa530d new theory of polynomials
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   280
lemma coeff_add [simp]:
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   281
  "coeff (p + q) n = coeff p n + coeff q n"
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   282
  unfolding plus_poly_def
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   283
  by (simp add: Abs_poly_inverse coeff Poly_add)
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   284
5f0cb3fa530d new theory of polynomials
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   285
instance proof
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   286
  fix p q r :: "'a poly"
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   287
  show "(p + q) + r = p + (q + r)"
5f0cb3fa530d new theory of polynomials
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   288
    by (simp add: expand_poly_eq add_assoc)
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   289
  show "p + q = q + p"
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   290
    by (simp add: expand_poly_eq add_commute)
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   291
  show "0 + p = p"
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   292
    by (simp add: expand_poly_eq)
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   293
qed
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   294
5f0cb3fa530d new theory of polynomials
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   295
end
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   296
29904
856f16a3b436 add class cancel_comm_monoid_add
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   297
instance poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add
29540
8858d197a9b6 more instance declarations for poly
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   298
proof
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   299
  fix p q r :: "'a poly"
8858d197a9b6 more instance declarations for poly
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   300
  assume "p + q = p + r" thus "q = r"
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   301
    by (simp add: expand_poly_eq)
8858d197a9b6 more instance declarations for poly
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   302
qed
8858d197a9b6 more instance declarations for poly
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diff changeset
   303
29451
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   304
instantiation poly :: (ab_group_add) ab_group_add
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   305
begin
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   306
5f0cb3fa530d new theory of polynomials
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   307
definition
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   308
  uminus_poly_def [code del]:
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   309
    "- p = Abs_poly (\<lambda>n. - coeff p n)"
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   310
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   311
definition
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   312
  minus_poly_def [code del]:
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   313
    "p - q = Abs_poly (\<lambda>n. coeff p n - coeff q n)"
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   314
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   315
lemma Poly_minus:
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   316
  fixes f :: "nat \<Rightarrow> 'a"
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   317
  shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. - f n) \<in> Poly"
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diff changeset
   318
  unfolding Poly_def by simp
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diff changeset
   319
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diff changeset
   320
lemma Poly_diff:
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   321
  fixes f g :: "nat \<Rightarrow> 'a"
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diff changeset
   322
  shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n - g n) \<in> Poly"
5f0cb3fa530d new theory of polynomials
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parents:
diff changeset
   323
  unfolding diff_minus by (simp add: Poly_add Poly_minus)
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diff changeset
   324
5f0cb3fa530d new theory of polynomials
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   325
lemma coeff_minus [simp]: "coeff (- p) n = - coeff p n"
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parents:
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   326
  unfolding uminus_poly_def
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   327
  by (simp add: Abs_poly_inverse coeff Poly_minus)
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parents:
diff changeset
   328
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   329
lemma coeff_diff [simp]:
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   330
  "coeff (p - q) n = coeff p n - coeff q n"
5f0cb3fa530d new theory of polynomials
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parents:
diff changeset
   331
  unfolding minus_poly_def
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   332
  by (simp add: Abs_poly_inverse coeff Poly_diff)
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parents:
diff changeset
   333
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   334
instance proof
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diff changeset
   335
  fix p q :: "'a poly"
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parents:
diff changeset
   336
  show "- p + p = 0"
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parents:
diff changeset
   337
    by (simp add: expand_poly_eq)
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parents:
diff changeset
   338
  show "p - q = p + - q"
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parents:
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   339
    by (simp add: expand_poly_eq diff_minus)
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   340
qed
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parents:
diff changeset
   341
5f0cb3fa530d new theory of polynomials
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diff changeset
   342
end
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parents:
diff changeset
   343
5f0cb3fa530d new theory of polynomials
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diff changeset
   344
lemma add_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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parents:
diff changeset
   348
lemma minus_pCons [simp]:
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   349
  "- pCons a p = pCons (- a) (- p)"
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diff changeset
   350
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
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parents:
diff changeset
   351
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parents:
diff changeset
   352
lemma diff_pCons [simp]:
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   353
  "pCons a p - pCons b q = pCons (a - b) (p - q)"
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diff changeset
   354
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   355
29539
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   356
lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   357
  by (rule degree_le, auto simp add: coeff_eq_0)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   358
29539
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   359
lemma degree_add_le:
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   360
  "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n"
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   361
  by (auto intro: order_trans degree_add_le_max)
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   362
29453
de4f26f59135 add lemmas degree_{add,diff}_less
huffman
parents: 29451
diff changeset
   363
lemma degree_add_less:
de4f26f59135 add lemmas degree_{add,diff}_less
huffman
parents: 29451
diff changeset
   364
  "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
29539
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   365
  by (auto intro: le_less_trans degree_add_le_max)
29453
de4f26f59135 add lemmas degree_{add,diff}_less
huffman
parents: 29451
diff changeset
   366
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   367
lemma degree_add_eq_right:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   368
  "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   369
  apply (cases "q = 0", simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   370
  apply (rule order_antisym)
29539
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   371
  apply (simp add: degree_add_le)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   372
  apply (rule le_degree)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   373
  apply (simp add: coeff_eq_0)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   374
  done
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   375
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   376
lemma degree_add_eq_left:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   377
  "degree q < degree p \<Longrightarrow> degree (p + q) = degree p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   378
  using degree_add_eq_right [of q p]
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   379
  by (simp add: add_commute)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   380
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   381
lemma degree_minus [simp]: "degree (- p) = degree p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   382
  unfolding degree_def by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   383
29539
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   384
lemma degree_diff_le_max: "degree (p - q) \<le> max (degree p) (degree q)"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   385
  using degree_add_le [where p=p and q="-q"]
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   386
  by (simp add: diff_minus)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   387
29539
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   388
lemma degree_diff_le:
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   389
  "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p - q) \<le> n"
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   390
  by (simp add: diff_minus degree_add_le)
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   391
29453
de4f26f59135 add lemmas degree_{add,diff}_less
huffman
parents: 29451
diff changeset
   392
lemma degree_diff_less:
de4f26f59135 add lemmas degree_{add,diff}_less
huffman
parents: 29451
diff changeset
   393
  "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
29539
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   394
  by (simp add: diff_minus degree_add_less)
29453
de4f26f59135 add lemmas degree_{add,diff}_less
huffman
parents: 29451
diff changeset
   395
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   396
lemma add_monom: "monom a n + monom b n = monom (a + b) n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   397
  by (rule poly_ext) simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   398
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   399
lemma diff_monom: "monom a n - monom b n = monom (a - b) n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   400
  by (rule poly_ext) simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   401
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   402
lemma minus_monom: "- monom a n = monom (-a) n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   403
  by (rule poly_ext) simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   404
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   405
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
   406
  by (cases "finite A", induct set: finite, simp_all)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   407
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   408
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
   409
  by (rule poly_ext) (simp add: coeff_setsum)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   410
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   411
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   412
subsection {* Multiplication by a constant *}
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   413
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   414
definition
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   415
  smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   416
  "smult a p = Abs_poly (\<lambda>n. a * coeff p n)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   417
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   418
lemma Poly_smult:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   419
  fixes f :: "nat \<Rightarrow> 'a::comm_semiring_0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   420
  shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. a * f n) \<in> Poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   421
  unfolding Poly_def
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   422
  by (clarify, rule_tac x=n in exI, simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   423
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   424
lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   425
  unfolding smult_def
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   426
  by (simp add: Abs_poly_inverse Poly_smult coeff)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   427
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   428
lemma degree_smult_le: "degree (smult a p) \<le> degree p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   429
  by (rule degree_le, simp add: coeff_eq_0)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   430
29472
a63a2e46cec9 declare smult rules [simp]
huffman
parents: 29471
diff changeset
   431
lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   432
  by (rule poly_ext, simp add: mult_assoc)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   433
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   434
lemma smult_0_right [simp]: "smult a 0 = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   435
  by (rule poly_ext, simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   436
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   437
lemma smult_0_left [simp]: "smult 0 p = 0"
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
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   440
lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   441
  by (rule poly_ext, simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   442
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   443
lemma smult_add_right:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   444
  "smult a (p + q) = smult a p + smult a q"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29540
diff changeset
   445
  by (rule poly_ext, simp add: algebra_simps)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   446
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   447
lemma smult_add_left:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   448
  "smult (a + b) p = smult a p + smult b p"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29540
diff changeset
   449
  by (rule poly_ext, simp add: algebra_simps)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   450
29457
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   451
lemma smult_minus_right [simp]:
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   452
  "smult (a::'a::comm_ring) (- p) = - smult a p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   453
  by (rule poly_ext, simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   454
29457
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
   455
lemma smult_minus_left [simp]:
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   456
  "smult (- a::'a::comm_ring) p = - smult a p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   457
  by (rule poly_ext, simp)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   458
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   459
lemma smult_diff_right:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   460
  "smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29540
diff changeset
   461
  by (rule poly_ext, simp add: algebra_simps)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   462
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   463
lemma smult_diff_left:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   464
  "smult (a - b::'a::comm_ring) p = smult a p - smult b p"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29540
diff changeset
   465
  by (rule poly_ext, simp add: algebra_simps)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   466
29472
a63a2e46cec9 declare smult rules [simp]
huffman
parents: 29471
diff changeset
   467
lemmas smult_distribs =
a63a2e46cec9 declare smult rules [simp]
huffman
parents: 29471
diff changeset
   468
  smult_add_left smult_add_right
a63a2e46cec9 declare smult rules [simp]
huffman
parents: 29471
diff changeset
   469
  smult_diff_left smult_diff_right
a63a2e46cec9 declare smult rules [simp]
huffman
parents: 29471
diff changeset
   470
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   471
lemma smult_pCons [simp]:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   472
  "smult a (pCons b p) = pCons (a * b) (smult a p)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   473
  by (rule poly_ext, simp add: coeff_pCons split: nat.split)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   474
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   475
lemma smult_monom: "smult a (monom b n) = monom (a * b) n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   476
  by (induct n, simp add: monom_0, simp add: monom_Suc)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   477
29659
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
   478
lemma degree_smult_eq [simp]:
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
   479
  fixes a :: "'a::idom"
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
   480
  shows "degree (smult a p) = (if a = 0 then 0 else degree p)"
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
   481
  by (cases "a = 0", simp, simp add: degree_def)
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
   482
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
   483
lemma smult_eq_0_iff [simp]:
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
   484
  fixes a :: "'a::idom"
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
   485
  shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0"
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
   486
  by (simp add: expand_poly_eq)
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
   487
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   488
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   489
subsection {* Multiplication of polynomials *}
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   490
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   491
text {* TODO: move to SetInterval.thy *}
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   492
lemma setsum_atMost_Suc_shift:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   493
  fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   494
  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
   495
proof (induct n)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   496
  case 0 show ?case by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   497
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   498
  case (Suc n) note IH = this
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   499
  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
   500
    by (rule setsum_atMost_Suc)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   501
  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
   502
    by (rule IH)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   503
  also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   504
             f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   505
    by (rule add_assoc)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   506
  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
   507
    by (rule setsum_atMost_Suc [symmetric])
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   508
  finally show ?case .
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   509
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   510
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   511
instantiation poly :: (comm_semiring_0) comm_semiring_0
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   512
begin
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   513
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   514
definition
29475
c06d1b0a970f declare more definitions [code del]
huffman
parents: 29474
diff changeset
   515
  times_poly_def [code del]:
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   516
    "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
   517
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   518
lemma mult_poly_0_left: "(0::'a poly) * q = 0"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   519
  unfolding times_poly_def by (simp add: poly_rec_0)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   520
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   521
lemma mult_pCons_left [simp]:
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   522
  "pCons a p * q = smult a q + pCons 0 (p * q)"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   523
  unfolding times_poly_def by (simp add: poly_rec_pCons)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   524
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   525
lemma mult_poly_0_right: "p * (0::'a poly) = 0"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   526
  by (induct p, simp add: mult_poly_0_left, simp)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   527
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   528
lemma mult_pCons_right [simp]:
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   529
  "p * pCons a q = smult a p + pCons 0 (p * q)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29540
diff changeset
   530
  by (induct p, simp add: mult_poly_0_left, simp add: algebra_simps)
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   531
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   532
lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   533
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   534
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
   535
  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
   536
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   537
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
   538
  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
   539
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   540
lemma mult_poly_add_left:
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   541
  fixes p q r :: "'a poly"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   542
  shows "(p + q) * r = p * r + q * r"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   543
  by (induct r, simp add: mult_poly_0,
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29540
diff changeset
   544
                simp add: smult_distribs algebra_simps)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   545
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   546
instance proof
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   547
  fix p q r :: "'a poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   548
  show 0: "0 * p = 0"
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   549
    by (rule mult_poly_0_left)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   550
  show "p * 0 = 0"
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   551
    by (rule mult_poly_0_right)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   552
  show "(p + q) * r = p * r + q * r"
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   553
    by (rule mult_poly_add_left)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   554
  show "(p * q) * r = p * (q * r)"
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   555
    by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   556
  show "p * q = q * p"
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   557
    by (induct p, simp add: mult_poly_0, simp)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   558
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   559
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   560
end
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   561
29540
8858d197a9b6 more instance declarations for poly
huffman
parents: 29539
diff changeset
   562
instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
8858d197a9b6 more instance declarations for poly
huffman
parents: 29539
diff changeset
   563
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   564
lemma coeff_mult:
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   565
  "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
   566
proof (induct p arbitrary: n)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   567
  case 0 show ?case by simp
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   568
next
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   569
  case (pCons a p n) thus ?case
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   570
    by (cases n, simp, simp add: setsum_atMost_Suc_shift
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   571
                            del: setsum_atMost_Suc)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   572
qed
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   573
29474
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   574
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q"
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   575
apply (rule degree_le)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   576
apply (induct p)
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   577
apply simp
674a21226c5a define polynomial multiplication using poly_rec
huffman
parents: 29472
diff changeset
   578
apply (simp add: coeff_eq_0 coeff_pCons split: nat.split)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   579
done
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   580
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   581
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   582
  by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   583
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   584
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   585
subsection {* The unit polynomial and exponentiation *}
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   586
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   587
instantiation poly :: (comm_semiring_1) comm_semiring_1
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   588
begin
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   589
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   590
definition
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   591
  one_poly_def:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   592
    "1 = pCons 1 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   593
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   594
instance proof
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   595
  fix p :: "'a poly" show "1 * p = p"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   596
    unfolding one_poly_def
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   597
    by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   598
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   599
  show "0 \<noteq> (1::'a poly)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   600
    unfolding one_poly_def by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   601
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   602
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   603
end
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   604
29540
8858d197a9b6 more instance declarations for poly
huffman
parents: 29539
diff changeset
   605
instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..
8858d197a9b6 more instance declarations for poly
huffman
parents: 29539
diff changeset
   606
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   607
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   608
  unfolding one_poly_def
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   609
  by (simp add: coeff_pCons split: nat.split)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   610
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   611
lemma degree_1 [simp]: "degree 1 = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   612
  unfolding one_poly_def
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   613
  by (rule degree_pCons_0)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   614
29979
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   615
text {* Lemmas about divisibility *}
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   616
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   617
lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q"
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   618
proof -
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   619
  assume "p dvd q"
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   620
  then obtain k where "q = p * k" ..
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   621
  then have "smult a q = p * smult a k" by simp
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   622
  then show "p dvd smult a q" ..
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   623
qed
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   624
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   625
lemma dvd_smult_cancel:
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   626
  fixes a :: "'a::field"
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   627
  shows "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q"
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   628
  by (drule dvd_smult [where a="inverse a"]) simp
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   629
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   630
lemma dvd_smult_iff:
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   631
  fixes a :: "'a::field"
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   632
  shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q"
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   633
  by (safe elim!: dvd_smult dvd_smult_cancel)
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   634
31663
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   635
lemma smult_dvd_cancel:
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   636
  "smult a p dvd q \<Longrightarrow> p dvd q"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   637
proof -
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   638
  assume "smult a p dvd q"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   639
  then obtain k where "q = smult a p * k" ..
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   640
  then have "q = p * smult a k" by simp
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   641
  then show "p dvd q" ..
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   642
qed
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   643
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   644
lemma smult_dvd:
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   645
  fixes a :: "'a::field"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   646
  shows "p dvd q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> smult a p dvd q"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   647
  by (rule smult_dvd_cancel [where a="inverse a"]) simp
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   648
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   649
lemma smult_dvd_iff:
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   650
  fixes a :: "'a::field"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   651
  shows "smult a p dvd q \<longleftrightarrow> (if a = 0 then q = 0 else p dvd q)"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   652
  by (auto elim: smult_dvd smult_dvd_cancel)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
   653
29979
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   654
lemma degree_power_le: "degree (p ^ n) \<le> degree p * n"
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   655
by (induct n, simp, auto intro: order_trans degree_mult_le)
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
   656
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   657
instance poly :: (comm_ring) comm_ring ..
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   658
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   659
instance poly :: (comm_ring_1) comm_ring_1 ..
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   660
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   661
instantiation poly :: (comm_ring_1) number_ring
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   662
begin
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   663
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   664
definition
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   665
  "number_of k = (of_int k :: 'a poly)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   666
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   667
instance
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   668
  by default (rule number_of_poly_def)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   669
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   670
end
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   671
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   672
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   673
subsection {* Polynomials form an integral domain *}
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   674
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   675
lemma coeff_mult_degree_sum:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   676
  "coeff (p * q) (degree p + degree q) =
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   677
   coeff p (degree p) * coeff q (degree q)"
29471
6a46a13ce1f9 simplify proof of coeff_mult_degree_sum
huffman
parents: 29462
diff changeset
   678
  by (induct p, simp, simp add: coeff_eq_0)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   679
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   680
instance poly :: (idom) idom
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   681
proof
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   682
  fix p q :: "'a poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   683
  assume "p \<noteq> 0" and "q \<noteq> 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   684
  have "coeff (p * q) (degree p + degree q) =
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   685
        coeff p (degree p) * coeff q (degree q)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   686
    by (rule coeff_mult_degree_sum)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   687
  also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   688
    using `p \<noteq> 0` and `q \<noteq> 0` by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   689
  finally have "\<exists>n. coeff (p * q) n \<noteq> 0" ..
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   690
  thus "p * q \<noteq> 0" by (simp add: expand_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   691
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   692
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   693
lemma degree_mult_eq:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   694
  fixes p q :: "'a::idom poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   695
  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
   696
apply (rule order_antisym [OF degree_mult_le le_degree])
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   697
apply (simp add: coeff_mult_degree_sum)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   698
done
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   699
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   700
lemma dvd_imp_degree_le:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   701
  fixes p q :: "'a::idom poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   702
  shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   703
  by (erule dvdE, simp add: degree_mult_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   704
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   705
29878
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   706
subsection {* Polynomials form an ordered integral domain *}
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   707
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   708
definition
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   709
  pos_poly :: "'a::ordered_idom poly \<Rightarrow> bool"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   710
where
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   711
  "pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   712
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   713
lemma pos_poly_pCons:
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   714
  "pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   715
  unfolding pos_poly_def by simp
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   716
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   717
lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   718
  unfolding pos_poly_def by simp
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   719
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   720
lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   721
  apply (induct p arbitrary: q, simp)
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   722
  apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos)
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   723
  done
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   724
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   725
lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   726
  unfolding pos_poly_def
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   727
  apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0")
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   728
  apply (simp add: degree_mult_eq coeff_mult_degree_sum mult_pos_pos)
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   729
  apply auto
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   730
  done
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   731
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   732
lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   733
by (induct p) (auto simp add: pos_poly_pCons)
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   734
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   735
instantiation poly :: (ordered_idom) ordered_idom
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   736
begin
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   737
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   738
definition
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   739
  [code del]:
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   740
    "x < y \<longleftrightarrow> pos_poly (y - x)"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   741
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   742
definition
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   743
  [code del]:
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   744
    "x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y - x)"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   745
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   746
definition
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   747
  [code del]:
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   748
    "abs (x::'a poly) = (if x < 0 then - x else x)"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   749
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   750
definition
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   751
  [code del]:
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   752
    "sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   753
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   754
instance proof
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   755
  fix x y :: "'a poly"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   756
  show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   757
    unfolding less_eq_poly_def less_poly_def
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   758
    apply safe
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   759
    apply simp
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   760
    apply (drule (1) pos_poly_add)
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   761
    apply simp
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   762
    done
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   763
next
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   764
  fix x :: "'a poly" show "x \<le> x"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   765
    unfolding less_eq_poly_def by simp
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   766
next
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   767
  fix x y z :: "'a poly"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   768
  assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   769
    unfolding less_eq_poly_def
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   770
    apply safe
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   771
    apply (drule (1) pos_poly_add)
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   772
    apply (simp add: algebra_simps)
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   773
    done
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   774
next
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   775
  fix x y :: "'a poly"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   776
  assume "x \<le> y" and "y \<le> x" thus "x = y"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   777
    unfolding less_eq_poly_def
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   778
    apply safe
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   779
    apply (drule (1) pos_poly_add)
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   780
    apply simp
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   781
    done
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   782
next
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   783
  fix x y z :: "'a poly"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   784
  assume "x \<le> y" thus "z + x \<le> z + y"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   785
    unfolding less_eq_poly_def
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   786
    apply safe
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   787
    apply (simp add: algebra_simps)
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   788
    done
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   789
next
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   790
  fix x y :: "'a poly"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   791
  show "x \<le> y \<or> y \<le> x"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   792
    unfolding less_eq_poly_def
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   793
    using pos_poly_total [of "x - y"]
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   794
    by auto
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   795
next
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   796
  fix x y z :: "'a poly"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   797
  assume "x < y" and "0 < z"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   798
  thus "z * x < z * y"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   799
    unfolding less_poly_def
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   800
    by (simp add: right_diff_distrib [symmetric] pos_poly_mult)
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   801
next
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   802
  fix x :: "'a poly"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   803
  show "\<bar>x\<bar> = (if x < 0 then - x else x)"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   804
    by (rule abs_poly_def)
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   805
next
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   806
  fix x :: "'a poly"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   807
  show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   808
    by (rule sgn_poly_def)
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   809
qed
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   810
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   811
end
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   812
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   813
text {* TODO: Simplification rules for comparisons *}
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   814
06efd6e731c6 ordered_idom instance for polynomials
huffman
parents: 29668
diff changeset
   815
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   816
subsection {* Long division of polynomials *}
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   817
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   818
definition
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   819
  pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   820
where
29475
c06d1b0a970f declare more definitions [code del]
huffman
parents: 29474
diff changeset
   821
  [code del]:
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   822
  "pdivmod_rel x y q r \<longleftrightarrow>
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   823
    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
   824
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   825
lemma pdivmod_rel_0:
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   826
  "pdivmod_rel 0 y 0 0"
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   827
  unfolding pdivmod_rel_def by simp
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   828
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   829
lemma pdivmod_rel_by_0:
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   830
  "pdivmod_rel x 0 0 x"
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   831
  unfolding pdivmod_rel_def by simp
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   832
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   833
lemma eq_zero_or_degree_less:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   834
  assumes "degree p \<le> n" and "coeff p n = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   835
  shows "p = 0 \<or> degree p < n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   836
proof (cases n)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   837
  case 0
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   838
  with `degree p \<le> n` and `coeff p n = 0`
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   839
  have "coeff p (degree p) = 0" by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   840
  then have "p = 0" by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   841
  then show ?thesis ..
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   842
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   843
  case (Suc m)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   844
  have "\<forall>i>n. coeff p i = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   845
    using `degree p \<le> n` by (simp add: coeff_eq_0)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   846
  then have "\<forall>i\<ge>n. coeff p i = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   847
    using `coeff p n = 0` by (simp add: le_less)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   848
  then have "\<forall>i>m. coeff p i = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   849
    using `n = Suc m` by (simp add: less_eq_Suc_le)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   850
  then have "degree p \<le> m"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   851
    by (rule degree_le)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   852
  then have "degree p < n"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   853
    using `n = Suc m` by (simp add: less_Suc_eq_le)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   854
  then show ?thesis ..
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   855
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   856
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   857
lemma pdivmod_rel_pCons:
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   858
  assumes rel: "pdivmod_rel x y q r"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   859
  assumes y: "y \<noteq> 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   860
  assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   861
  shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   862
    (is "pdivmod_rel ?x y ?q ?r")
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   863
proof -
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   864
  have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   865
    using assms unfolding pdivmod_rel_def by simp_all
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   866
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   867
  have 1: "?x = ?q * y + ?r"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   868
    using b x by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   869
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   870
  have 2: "?r = 0 \<or> degree ?r < degree y"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   871
  proof (rule eq_zero_or_degree_less)
29539
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   872
    show "degree ?r \<le> degree y"
abfe2af6883e add lemmas about degree
huffman
parents: 29537
diff changeset
   873
    proof (rule degree_diff_le)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   874
      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
   875
        using r by auto
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   876
      show "degree (smult b y) \<le> degree y"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   877
        by (rule degree_smult_le)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   878
    qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   879
  next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   880
    show "coeff ?r (degree y) = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   881
      using `y \<noteq> 0` unfolding b by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   882
  qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   883
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   884
  from 1 2 show ?thesis
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   885
    unfolding pdivmod_rel_def
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   886
    using `y \<noteq> 0` by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   887
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   888
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   889
lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   890
apply (cases "y = 0")
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   891
apply (fast intro!: pdivmod_rel_by_0)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   892
apply (induct x)
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   893
apply (fast intro!: pdivmod_rel_0)
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   894
apply (fast intro!: pdivmod_rel_pCons)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   895
done
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   896
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   897
lemma pdivmod_rel_unique:
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   898
  assumes 1: "pdivmod_rel x y q1 r1"
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   899
  assumes 2: "pdivmod_rel x y q2 r2"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   900
  shows "q1 = q2 \<and> r1 = r2"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   901
proof (cases "y = 0")
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   902
  assume "y = 0" with assms show ?thesis
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   903
    by (simp add: pdivmod_rel_def)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   904
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   905
  assume [simp]: "y \<noteq> 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   906
  from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   907
    unfolding pdivmod_rel_def by simp_all
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   908
  from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   909
    unfolding pdivmod_rel_def by simp_all
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   910
  from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29540
diff changeset
   911
    by (simp add: algebra_simps)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   912
  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
   913
    by (auto intro: degree_diff_less)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   914
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   915
  show "q1 = q2 \<and> r1 = r2"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   916
  proof (rule ccontr)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   917
    assume "\<not> (q1 = q2 \<and> r1 = r2)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   918
    with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   919
    with r3 have "degree (r2 - r1) < degree y" by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   920
    also have "degree y \<le> degree (q1 - q2) + degree y" by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   921
    also have "\<dots> = degree ((q1 - q2) * y)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   922
      using `q1 \<noteq> q2` by (simp add: degree_mult_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   923
    also have "\<dots> = degree (r2 - r1)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   924
      using q3 by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   925
    finally have "degree (r2 - r1) < degree (r2 - r1)" .
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   926
    then show "False" by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   927
  qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   928
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   929
29660
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
   930
lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0"
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
   931
by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0)
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
   932
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
   933
lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x"
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
   934
by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0)
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
   935
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   936
lemmas pdivmod_rel_unique_div =
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   937
  pdivmod_rel_unique [THEN conjunct1, standard]
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   938
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   939
lemmas pdivmod_rel_unique_mod =
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   940
  pdivmod_rel_unique [THEN conjunct2, standard]
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   941
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   942
instantiation poly :: (field) ring_div
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   943
begin
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   944
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   945
definition div_poly where
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   946
  [code del]: "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   947
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   948
definition mod_poly where
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   949
  [code del]: "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   950
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   951
lemma div_poly_eq:
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   952
  "pdivmod_rel x y q r \<Longrightarrow> x div y = q"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   953
unfolding div_poly_def
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   954
by (fast elim: pdivmod_rel_unique_div)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   955
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   956
lemma mod_poly_eq:
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   957
  "pdivmod_rel x y q r \<Longrightarrow> x mod y = r"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   958
unfolding mod_poly_def
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   959
by (fast elim: pdivmod_rel_unique_mod)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   960
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   961
lemma pdivmod_rel:
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   962
  "pdivmod_rel x y (x div y) (x mod y)"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   963
proof -
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   964
  from pdivmod_rel_exists
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   965
    obtain q r where "pdivmod_rel x y q r" by fast
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   966
  thus ?thesis
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   967
    by (simp add: div_poly_eq mod_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   968
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   969
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   970
instance proof
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   971
  fix x y :: "'a poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   972
  show "x div y * y + x mod y = x"
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   973
    using pdivmod_rel [of x y]
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   974
    by (simp add: pdivmod_rel_def)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   975
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   976
  fix x :: "'a poly"
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   977
  have "pdivmod_rel x 0 0 x"
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   978
    by (rule pdivmod_rel_by_0)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   979
  thus "x div 0 = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   980
    by (rule div_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   981
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   982
  fix y :: "'a poly"
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   983
  have "pdivmod_rel 0 y 0 0"
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   984
    by (rule pdivmod_rel_0)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   985
  thus "0 div y = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   986
    by (rule div_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   987
next
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   988
  fix x y z :: "'a poly"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   989
  assume "y \<noteq> 0"
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   990
  hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   991
    using pdivmod_rel [of x y]
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
   992
    by (simp add: pdivmod_rel_def left_distrib)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   993
  thus "(x + z * y) div y = z + x div y"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
   994
    by (rule div_poly_eq)
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
   995
next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
   996
  fix x y z :: "'a poly"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
   997
  assume "x \<noteq> 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
   998
  show "(x * y) div (x * z) = y div z"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
   999
  proof (cases "y \<noteq> 0 \<and> z \<noteq> 0")
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1000
    have "\<And>x::'a poly. pdivmod_rel x 0 0 x"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1001
      by (rule pdivmod_rel_by_0)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1002
    then have [simp]: "\<And>x::'a poly. x div 0 = 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1003
      by (rule div_poly_eq)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1004
    have "\<And>x::'a poly. pdivmod_rel 0 x 0 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1005
      by (rule pdivmod_rel_0)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1006
    then have [simp]: "\<And>x::'a poly. 0 div x = 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1007
      by (rule div_poly_eq)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1008
    case False then show ?thesis by auto
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1009
  next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1010
    case True then have "y \<noteq> 0" and "z \<noteq> 0" by auto
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1011
    with `x \<noteq> 0`
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1012
    have "\<And>q r. pdivmod_rel y z q r \<Longrightarrow> pdivmod_rel (x * y) (x * z) q (x * r)"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1013
      by (auto simp add: pdivmod_rel_def algebra_simps)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1014
        (rule classical, simp add: degree_mult_eq)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1015
    moreover from pdivmod_rel have "pdivmod_rel y z (y div z) (y mod z)" .
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1016
    ultimately have "pdivmod_rel (x * y) (x * z) (y div z) (x * (y mod z))" .
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1017
    then show ?thesis by (simp add: div_poly_eq)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30738
diff changeset
  1018
  qed
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1019
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1020
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1021
end
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1022
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1023
lemma degree_mod_less:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1024
  "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
  1025
  using pdivmod_rel [of x y]
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
  1026
  unfolding pdivmod_rel_def by simp
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1027
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1028
lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1029
proof -
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1030
  assume "degree x < degree y"
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
  1031
  hence "pdivmod_rel x y 0 x"
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
  1032
    by (simp add: pdivmod_rel_def)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1033
  thus "x div y = 0" by (rule div_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1034
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1035
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1036
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1037
proof -
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1038
  assume "degree x < degree y"
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
  1039
  hence "pdivmod_rel x y 0 x"
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
  1040
    by (simp add: pdivmod_rel_def)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1041
  thus "x mod y = x" by (rule mod_poly_eq)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1042
qed
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1043
29659
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1044
lemma pdivmod_rel_smult_left:
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1045
  "pdivmod_rel x y q r
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1046
    \<Longrightarrow> pdivmod_rel (smult a x) y (smult a q) (smult a r)"
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1047
  unfolding pdivmod_rel_def by (simp add: smult_add_right)
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1048
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1049
lemma div_smult_left: "(smult a x) div y = smult a (x div y)"
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1050
  by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1051
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1052
lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)"
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1053
  by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel)
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1054
30072
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1055
lemma poly_div_minus_left [simp]:
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1056
  fixes x y :: "'a::field poly"
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1057
  shows "(- x) div y = - (x div y)"
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1058
  using div_smult_left [of "- 1::'a"] by simp
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1059
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1060
lemma poly_mod_minus_left [simp]:
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1061
  fixes x y :: "'a::field poly"
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1062
  shows "(- x) mod y = - (x mod y)"
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1063
  using mod_smult_left [of "- 1::'a"] by simp
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1064
29659
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1065
lemma pdivmod_rel_smult_right:
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1066
  "\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk>
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1067
    \<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r"
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1068
  unfolding pdivmod_rel_def by simp
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1069
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1070
lemma div_smult_right:
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1071
  "a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)"
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1072
  by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1073
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1074
lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y"
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1075
  by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel)
f8d2c03ecfd8 add lemmas about smult
huffman
parents: 29540
diff changeset
  1076
30072
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1077
lemma poly_div_minus_right [simp]:
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1078
  fixes x y :: "'a::field poly"
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1079
  shows "x div (- y) = - (x div y)"
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1080
  using div_smult_right [of "- 1::'a"]
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1081
  by (simp add: nonzero_inverse_minus_eq)
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1082
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1083
lemma poly_mod_minus_right [simp]:
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1084
  fixes x y :: "'a::field poly"
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1085
  shows "x mod (- y) = x mod y"
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1086
  using mod_smult_right [of "- 1::'a"] by simp
4eecd8b9b6cf add lemmas poly_{div,mod}_minus_{left,right}
huffman
parents: 29987
diff changeset
  1087
29660
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1088
lemma pdivmod_rel_mult:
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1089
  "\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk>
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1090
    \<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)"
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1091
apply (cases "z = 0", simp add: pdivmod_rel_def)
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1092
apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff)
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1093
apply (cases "r = 0")
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1094
apply (cases "r' = 0")
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1095
apply (simp add: pdivmod_rel_def)
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1096
apply (simp add: pdivmod_rel_def ring_simps degree_mult_eq)
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1097
apply (cases "r' = 0")
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1098
apply (simp add: pdivmod_rel_def degree_mult_eq)
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1099
apply (simp add: pdivmod_rel_def ring_simps)
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1100
apply (simp add: degree_mult_eq degree_add_less)
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1101
done
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1102
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1103
lemma poly_div_mult_right:
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1104
  fixes x y z :: "'a::field poly"
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1105
  shows "x div (y * z) = (x div y) div z"
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1106
  by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1107
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1108
lemma poly_mod_mult_right:
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1109
  fixes x y z :: "'a::field poly"
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1110
  shows "x mod (y * z) = y * (x div y mod z) + x mod y"
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1111
  by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+)
d59918e668b7 add lemmas about div/mod with multiplication
huffman
parents: 29659
diff changeset
  1112
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1113
lemma mod_pCons:
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1114
  fixes a and x
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1115
  assumes y: "y \<noteq> 0"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1116
  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
  1117
  shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1118
unfolding b
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1119
apply (rule mod_poly_eq)
29537
50345a0f9df8 rename divmod_poly to pdivmod
huffman
parents: 29480
diff changeset
  1120
apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1121
done
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1122
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1123
31663
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1124
subsection {* GCD of polynomials *}
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1125
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1126
function
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1127
  poly_gcd :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1128
  "poly_gcd x 0 = smult (inverse (coeff x (degree x))) x"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1129
| "y \<noteq> 0 \<Longrightarrow> poly_gcd x y = poly_gcd y (x mod y)"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1130
by auto
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1131
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1132
termination poly_gcd
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1133
by (relation "measure (\<lambda>(x, y). if y = 0 then 0 else Suc (degree y))")
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1134
   (auto dest: degree_mod_less)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1135
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1136
declare poly_gcd.simps [simp del, code del]
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1137
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1138
lemma poly_gcd_dvd1 [iff]: "poly_gcd x y dvd x"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1139
  and poly_gcd_dvd2 [iff]: "poly_gcd x y dvd y"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1140
  apply (induct x y rule: poly_gcd.induct)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1141
  apply (simp_all add: poly_gcd.simps)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1142
  apply (fastsimp simp add: smult_dvd_iff dest: inverse_zero_imp_zero)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1143
  apply (blast dest: dvd_mod_imp_dvd)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1144
  done
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1145
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1146
lemma poly_gcd_greatest: "k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd poly_gcd x y"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1147
  by (induct x y rule: poly_gcd.induct)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1148
     (simp_all add: poly_gcd.simps dvd_mod dvd_smult)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1149
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1150
lemma dvd_poly_gcd_iff [iff]:
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1151
  "k dvd poly_gcd x y \<longleftrightarrow> k dvd x \<and> k dvd y"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1152
  by (blast intro!: poly_gcd_greatest intro: dvd_trans)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1153
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1154
lemma poly_gcd_monic:
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1155
  "coeff (poly_gcd x y) (degree (poly_gcd x y)) =
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1156
    (if x = 0 \<and> y = 0 then 0 else 1)"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1157
  by (induct x y rule: poly_gcd.induct)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1158
     (simp_all add: poly_gcd.simps nonzero_imp_inverse_nonzero)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1159
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1160
lemma poly_gcd_zero_iff [simp]:
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1161
  "poly_gcd x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1162
  by (simp only: dvd_0_left_iff [symmetric] dvd_poly_gcd_iff)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1163
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1164
lemma poly_gcd_0_0 [simp]: "poly_gcd 0 0 = 0"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1165
  by simp
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1166
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1167
lemma poly_dvd_antisym:
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1168
  fixes p q :: "'a::idom poly"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1169
  assumes coeff: "coeff p (degree p) = coeff q (degree q)"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1170
  assumes dvd1: "p dvd q" and dvd2: "q dvd p" shows "p = q"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1171
proof (cases "p = 0")
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1172
  case True with coeff show "p = q" by simp
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1173
next
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1174
  case False with coeff have "q \<noteq> 0" by auto
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1175
  have degree: "degree p = degree q"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1176
    using `p dvd q` `q dvd p` `p \<noteq> 0` `q \<noteq> 0`
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1177
    by (intro order_antisym dvd_imp_degree_le)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1178
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1179
  from `p dvd q` obtain a where a: "q = p * a" ..
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1180
  with `q \<noteq> 0` have "a \<noteq> 0" by auto
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1181
  with degree a `p \<noteq> 0` have "degree a = 0"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1182
    by (simp add: degree_mult_eq)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1183
  with coeff a show "p = q"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1184
    by (cases a, auto split: if_splits)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1185
qed
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1186
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1187
lemma poly_gcd_unique:
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1188
  assumes dvd1: "d dvd x" and dvd2: "d dvd y"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1189
    and greatest: "\<And>k. k dvd x \<Longrightarrow> k dvd y \<Longrightarrow> k dvd d"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1190
    and monic: "coeff d (degree d) = (if x = 0 \<and> y = 0 then 0 else 1)"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1191
  shows "poly_gcd x y = d"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1192
proof -
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1193
  have "coeff (poly_gcd x y) (degree (poly_gcd x y)) = coeff d (degree d)"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1194
    by (simp_all add: poly_gcd_monic monic)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1195
  moreover have "poly_gcd x y dvd d"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1196
    using poly_gcd_dvd1 poly_gcd_dvd2 by (rule greatest)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1197
  moreover have "d dvd poly_gcd x y"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1198
    using dvd1 dvd2 by (rule poly_gcd_greatest)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1199
  ultimately show ?thesis
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1200
    by (rule poly_dvd_antisym)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1201
qed
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1202
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1203
lemma poly_gcd_commute: "poly_gcd x y = poly_gcd y x"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1204
by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1205
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1206
lemma poly_gcd_assoc: "poly_gcd (poly_gcd x y) z = poly_gcd x (poly_gcd y z)"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1207
by (rule poly_gcd_unique) (auto intro: dvd_trans simp add: poly_gcd_monic)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1208
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1209
lemmas poly_gcd_left_commute =
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1210
  mk_left_commute [where f=poly_gcd, OF poly_gcd_assoc poly_gcd_commute]
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1211
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1212
lemmas poly_gcd_ac = poly_gcd_assoc poly_gcd_commute poly_gcd_left_commute
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1213
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1214
lemma poly_gcd_1_left [simp]: "poly_gcd 1 y = 1"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1215
by (rule poly_gcd_unique) simp_all
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1216
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1217
lemma poly_gcd_1_right [simp]: "poly_gcd x 1 = 1"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1218
by (rule poly_gcd_unique) simp_all
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1219
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1220
lemma poly_gcd_minus_left [simp]: "poly_gcd (- x) y = poly_gcd x y"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1221
by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1222
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1223
lemma poly_gcd_minus_right [simp]: "poly_gcd x (- y) = poly_gcd x y"
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1224
by (rule poly_gcd_unique) (simp_all add: poly_gcd_monic)
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1225
5eb82f064630 smult_dvd lemmas; polynomial gcd
huffman
parents: 31021
diff changeset
  1226
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1227
subsection {* Evaluation of polynomials *}
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1228
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1229
definition
29454
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
  1230
  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
  1231
  "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
  1232
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1233
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
  1234
  unfolding poly_def by (simp add: poly_rec_0)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1235
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1236
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
  1237
  unfolding poly_def by (simp add: poly_rec_pCons)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1238
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1239
lemma poly_1 [simp]: "poly 1 x = 1"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1240
  unfolding one_poly_def by simp
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1241
29454
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
  1242
lemma poly_monom:
31021
53642251a04f farewell to class recpower
haftmann
parents: 30960
diff changeset
  1243
  fixes a x :: "'a::{comm_semiring_1}"
29454
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
  1244
  shows "poly (monom a n) x = a * x ^ n"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1245
  by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1246
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1247
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1248
  apply (induct p arbitrary: q, simp)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29540
diff changeset
  1249
  apply (case_tac q, simp, simp add: algebra_simps)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1250
  done
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1251
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1252
lemma poly_minus [simp]:
29454
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
  1253
  fixes x :: "'a::comm_ring"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1254
  shows "poly (- p) x = - poly p x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1255
  by (induct p, simp_all)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1256
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1257
lemma poly_diff [simp]:
29454
b0f586f38dd7 add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents: 29453
diff changeset
  1258
  fixes x :: "'a::comm_ring"
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1259
  shows "poly (p - q) x = poly p x - poly q x"
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1260
  by (simp add: diff_minus)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1261
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1262
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
  1263
  by (cases "finite A", induct set: finite, simp_all)
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1264
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1265
lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29540
diff changeset
  1266
  by (induct p, simp, simp add: algebra_simps)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1267
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1268
lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29540
diff changeset
  1269
  by (induct p, simp_all, simp add: algebra_simps)
29451
5f0cb3fa530d new theory of polynomials
huffman
parents:
diff changeset
  1270
29462
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1271
lemma poly_power [simp]:
31021
53642251a04f farewell to class recpower
haftmann
parents: 30960
diff changeset
  1272
  fixes p :: "'a::{comm_semiring_1} poly"
29462
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1273
  shows "poly (p ^ n) x = poly p x ^ n"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1274
  by (induct n, simp, simp add: power_Suc)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1275
29456
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1276
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1277
subsection {* Synthetic division *}
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1278
29980
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1279
text {*
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1280
  Synthetic division is simply division by the
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1281
  linear polynomial @{term "x - c"}.
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1282
*}
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1283
29456
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1284
definition
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1285
  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
  1286
where [code del]:
29456
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1287
  "synthetic_divmod p c =
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1288
    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
  1289
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1290
definition
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1291
  synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1292
where
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1293
  "synthetic_div p c = fst (synthetic_divmod p c)"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1294
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1295
lemma synthetic_divmod_0 [simp]:
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1296
  "synthetic_divmod 0 c = (0, 0)"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1297
  unfolding synthetic_divmod_def
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1298
  by (simp add: poly_rec_0)
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1299
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1300
lemma synthetic_divmod_pCons [simp]:
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1301
  "synthetic_divmod (pCons a p) c =
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1302
    (\<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
  1303
  unfolding synthetic_divmod_def
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1304
  by (simp add: poly_rec_pCons)
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1305
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1306
lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1307
  by (induct p, simp, simp add: split_def)
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1308
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1309
lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0"
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1310
  unfolding synthetic_div_def by simp
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1311
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1312
lemma synthetic_div_pCons [simp]:
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1313
  "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
  1314
  unfolding synthetic_div_def
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1315
  by (simp add: split_def snd_synthetic_divmod)
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1316
29460
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1317
lemma synthetic_div_eq_0_iff:
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1318
  "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
  1319
  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
  1320
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1321
lemma degree_synthetic_div:
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1322
  "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
  1323
  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
  1324
29457
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1325
lemma synthetic_div_correct:
29456
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1326
  "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
  1327
  by (induct p) simp_all
3f8b85444512 add synthetic division algorithm for polynomials
huffman
parents: 29455
diff changeset
  1328
29457
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1329
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
  1330
by (induct p arbitrary: a) simp_all
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1331
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1332
lemma synthetic_div_unique:
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1333
  "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
  1334
apply (induct p arbitrary: q r)
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1335
apply (simp, frule synthetic_div_unique_lemma, simp)
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1336
apply (case_tac q, force)
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1337
done
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1338
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1339
lemma synthetic_div_correct':
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1340
  fixes c :: "'a::comm_ring_1"
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1341
  shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1342
  using synthetic_div_correct [of p c]
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29540
diff changeset
  1343
  by (simp add: algebra_simps)
29457
2eadbc24de8c correctness and uniqueness of synthetic division
huffman
parents: 29456
diff changeset
  1344
29460
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1345
lemma poly_eq_0_iff_dvd:
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1346
  fixes c :: "'a::idom"
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1347
  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
  1348
proof
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1349
  assume "poly p c = 0"
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1350
  with synthetic_div_correct' [of c p]
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1351
  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
  1352
  then show "[:-c, 1:] dvd p" ..
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1353
next
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1354
  assume "[:-c, 1:] dvd p"
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1355
  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
  1356
  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
  1357
qed
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1358
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1359
lemma dvd_iff_poly_eq_0:
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1360
  fixes c :: "'a::idom"
ad87e5d1488b new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents: 29457
diff changeset
  1361
  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
  1362
  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
  1363
29462
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1364
lemma poly_roots_finite:
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1365
  fixes p :: "'a::idom poly"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1366
  shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1367
proof (induct n \<equiv> "degree p" arbitrary: p)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1368
  case (0 p)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1369
  then obtain a where "a \<noteq> 0" and "p = [:a:]"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1370
    by (cases p, simp split: if_splits)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1371
  then show "finite {x. poly p x = 0}" by simp
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1372
next
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1373
  case (Suc n p)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1374
  show "finite {x. poly p x = 0}"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1375
  proof (cases "\<exists>x. poly p x = 0")
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1376
    case False
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1377
    then show "finite {x. poly p x = 0}" by simp
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1378
  next
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1379
    case True
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1380
    then obtain a where "poly p a = 0" ..
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1381
    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
  1382
    then obtain k where k: "p = [:-a, 1:] * k" ..
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1383
    with `p \<noteq> 0` have "k \<noteq> 0" by auto
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1384
    with k have "degree p = Suc (degree k)"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1385
      by (simp add: degree_mult_eq del: mult_pCons_left)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1386
    with `Suc n = degree p` have "n = degree k" by simp
34915
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 31998
diff changeset
  1387
    then have "finite {x. poly k x = 0}" using `k \<noteq> 0` by (rule Suc.hyps)
29462
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1388
    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
  1389
    then show "finite {x. poly p x = 0}"
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1390
      by (simp add: k uminus_add_conv_diff Collect_disj_eq
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1391
               del: mult_pCons_left)
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1392
  qed
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1393
qed
dc97c6188a7a add lemmas poly_power, poly_roots_finite
huffman
parents: 29460
diff changeset
  1394
29979
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1395
lemma poly_zero:
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1396
  fixes p :: "'a::{idom,ring_char_0} poly"
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1397
  shows "poly p = poly 0 \<longleftrightarrow> p = 0"
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1398
apply (cases "p = 0", simp_all)
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1399
apply (drule poly_roots_finite)
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1400
apply (auto simp add: infinite_UNIV_char_0)
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1401
done
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1402
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1403
lemma poly_eq_iff:
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1404
  fixes p q :: "'a::{idom,ring_char_0} poly"
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1405
  shows "poly p = poly q \<longleftrightarrow> p = q"
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1406
  using poly_zero [of "p - q"]
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1407
  by (simp add: expand_fun_eq)
666f5f72dbb5 add some lemmas, cleaned up
huffman
parents: 29977
diff changeset
  1408
29478
4a2482e16934 code generation for polynomials
huffman
parents: 29475
diff changeset
  1409
29980
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1410
subsection {* Composition of polynomials *}
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1411
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1412
definition
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1413
  pcompose :: "'a::comm_semiring_0 poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly"
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1414
where
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1415
  "pcompose p q = poly_rec 0 (\<lambda>a _ c. [:a:] + q * c) p"
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1416
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1417
lemma pcompose_0 [simp]: "pcompose 0 q = 0"
17ddfd0c3506 composition of polynomials
huffman
parents: 29979
diff changeset
  1418
  unfolding pcompose_def by (simp add: poly_rec_0)
17ddfd0c3506 composition of polynomials
huffman
parents: 29979