author  huffman 
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changeset 29475  c06d1b0a970f 
parent 29474  674a21226c5a 
child 29478  4a2482e16934 
permissions  rwrr 
29451  1 
(* Title: HOL/Polynomial.thy 
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Author: Brian Huffman 

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Based on an earlier development by Clemens Ballarin 

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*) 

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header {* Univariate Polynomials *} 

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theory Polynomial 

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imports Plain SetInterval 

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begin 

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subsection {* Definition of type @{text poly} *} 

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typedef (Poly) 'a poly = "{f::nat \<Rightarrow> 'a::zero. \<exists>n. \<forall>i>n. f i = 0}" 

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morphisms coeff Abs_poly 

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by auto 

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lemma expand_poly_eq: "p = q \<longleftrightarrow> (\<forall>n. coeff p n = coeff q n)" 

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by (simp add: coeff_inject [symmetric] expand_fun_eq) 

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lemma poly_ext: "(\<And>n. coeff p n = coeff q n) \<Longrightarrow> p = q" 

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by (simp add: expand_poly_eq) 

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subsection {* Degree of a polynomial *} 

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definition 

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degree :: "'a::zero poly \<Rightarrow> nat" where 

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"degree p = (LEAST n. \<forall>i>n. coeff p i = 0)" 

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lemma coeff_eq_0: "degree p < n \<Longrightarrow> coeff p n = 0" 

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proof  

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have "coeff p \<in> Poly" 

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by (rule coeff) 

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hence "\<exists>n. \<forall>i>n. coeff p i = 0" 

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unfolding Poly_def by simp 

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hence "\<forall>i>degree p. coeff p i = 0" 

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unfolding degree_def by (rule LeastI_ex) 

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moreover assume "degree p < n" 

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ultimately show ?thesis by simp 

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qed 

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lemma le_degree: "coeff p n \<noteq> 0 \<Longrightarrow> n \<le> degree p" 

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by (erule contrapos_np, rule coeff_eq_0, simp) 

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lemma degree_le: "\<forall>i>n. coeff p i = 0 \<Longrightarrow> degree p \<le> n" 

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unfolding degree_def by (erule Least_le) 

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lemma less_degree_imp: "n < degree p \<Longrightarrow> \<exists>i>n. coeff p i \<noteq> 0" 

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unfolding degree_def by (drule not_less_Least, simp) 

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subsection {* The zero polynomial *} 

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instantiation poly :: (zero) zero 

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begin 

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definition 

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zero_poly_def: "0 = Abs_poly (\<lambda>n. 0)" 

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instance .. 

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end 

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lemma coeff_0 [simp]: "coeff 0 n = 0" 

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unfolding zero_poly_def 

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by (simp add: Abs_poly_inverse Poly_def) 

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lemma degree_0 [simp]: "degree 0 = 0" 

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by (rule order_antisym [OF degree_le le0]) simp 

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lemma leading_coeff_neq_0: 

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assumes "p \<noteq> 0" shows "coeff p (degree p) \<noteq> 0" 

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proof (cases "degree p") 

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case 0 

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from `p \<noteq> 0` have "\<exists>n. coeff p n \<noteq> 0" 

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by (simp add: expand_poly_eq) 

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then obtain n where "coeff p n \<noteq> 0" .. 

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hence "n \<le> degree p" by (rule le_degree) 

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with `coeff p n \<noteq> 0` and `degree p = 0` 

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show "coeff p (degree p) \<noteq> 0" by simp 

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next 

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case (Suc n) 

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from `degree p = Suc n` have "n < degree p" by simp 

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hence "\<exists>i>n. coeff p i \<noteq> 0" by (rule less_degree_imp) 

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then obtain i where "n < i" and "coeff p i \<noteq> 0" by fast 

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from `degree p = Suc n` and `n < i` have "degree p \<le> i" by simp 

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also from `coeff p i \<noteq> 0` have "i \<le> degree p" by (rule le_degree) 

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finally have "degree p = i" . 

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with `coeff p i \<noteq> 0` show "coeff p (degree p) \<noteq> 0" by simp 

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qed 

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lemma leading_coeff_0_iff [simp]: "coeff p (degree p) = 0 \<longleftrightarrow> p = 0" 

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by (cases "p = 0", simp, simp add: leading_coeff_neq_0) 

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subsection {* Liststyle constructor for polynomials *} 

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definition 

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pCons :: "'a::zero \<Rightarrow> 'a poly \<Rightarrow> 'a poly" 

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where 

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[code del]: "pCons a p = Abs_poly (nat_case a (coeff p))" 

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syntax 
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"_poly" :: "args \<Rightarrow> 'a poly" ("[:(_):]") 

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translations 

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"[:x, xs:]" == "CONST pCons x [:xs:]" 

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"[:x:]" == "CONST pCons x 0" 

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lemma Poly_nat_case: "f \<in> Poly \<Longrightarrow> nat_case a f \<in> Poly" 
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unfolding Poly_def by (auto split: nat.split) 

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lemma coeff_pCons: 

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"coeff (pCons a p) = nat_case a (coeff p)" 

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unfolding pCons_def 

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by (simp add: Abs_poly_inverse Poly_nat_case coeff) 

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lemma coeff_pCons_0 [simp]: "coeff (pCons a p) 0 = a" 

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by (simp add: coeff_pCons) 

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lemma coeff_pCons_Suc [simp]: "coeff (pCons a p) (Suc n) = coeff p n" 

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by (simp add: coeff_pCons) 

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lemma degree_pCons_le: "degree (pCons a p) \<le> Suc (degree p)" 

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by (rule degree_le, simp add: coeff_eq_0 coeff_pCons split: nat.split) 

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lemma degree_pCons_eq: 

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"p \<noteq> 0 \<Longrightarrow> degree (pCons a p) = Suc (degree p)" 

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apply (rule order_antisym [OF degree_pCons_le]) 

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apply (rule le_degree, simp) 

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done 

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lemma degree_pCons_0: "degree (pCons a 0) = 0" 

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apply (rule order_antisym [OF _ le0]) 

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apply (rule degree_le, simp add: coeff_pCons split: nat.split) 

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done 

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lemma degree_pCons_eq_if [simp]: 
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"degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))" 
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apply (cases "p = 0", simp_all) 

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apply (rule order_antisym [OF _ le0]) 

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apply (rule degree_le, simp add: coeff_pCons split: nat.split) 

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apply (rule order_antisym [OF degree_pCons_le]) 

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apply (rule le_degree, simp) 

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done 

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lemma pCons_0_0 [simp]: "pCons 0 0 = 0" 

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by (rule poly_ext, simp add: coeff_pCons split: nat.split) 

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lemma pCons_eq_iff [simp]: 

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"pCons a p = pCons b q \<longleftrightarrow> a = b \<and> p = q" 

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proof (safe) 

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assume "pCons a p = pCons b q" 

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then have "coeff (pCons a p) 0 = coeff (pCons b q) 0" by simp 

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then show "a = b" by simp 

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next 

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assume "pCons a p = pCons b q" 

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then have "\<forall>n. coeff (pCons a p) (Suc n) = 

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coeff (pCons b q) (Suc n)" by simp 

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then show "p = q" by (simp add: expand_poly_eq) 

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qed 

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lemma pCons_eq_0_iff [simp]: "pCons a p = 0 \<longleftrightarrow> a = 0 \<and> p = 0" 

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using pCons_eq_iff [of a p 0 0] by simp 

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lemma Poly_Suc: "f \<in> Poly \<Longrightarrow> (\<lambda>n. f (Suc n)) \<in> Poly" 

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unfolding Poly_def 

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by (clarify, rule_tac x=n in exI, simp) 

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lemma pCons_cases [cases type: poly]: 

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obtains (pCons) a q where "p = pCons a q" 

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proof 

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show "p = pCons (coeff p 0) (Abs_poly (\<lambda>n. coeff p (Suc n)))" 

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by (rule poly_ext) 

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(simp add: Abs_poly_inverse Poly_Suc coeff coeff_pCons 

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split: nat.split) 

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qed 

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lemma pCons_induct [case_names 0 pCons, induct type: poly]: 

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assumes zero: "P 0" 

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assumes pCons: "\<And>a p. P p \<Longrightarrow> P (pCons a p)" 

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shows "P p" 

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proof (induct p rule: measure_induct_rule [where f=degree]) 

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case (less p) 

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obtain a q where "p = pCons a q" by (rule pCons_cases) 

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have "P q" 

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proof (cases "q = 0") 

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case True 

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then show "P q" by (simp add: zero) 

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next 

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case False 

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then have "degree (pCons a q) = Suc (degree q)" 

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by (rule degree_pCons_eq) 

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then have "degree q < degree p" 

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using `p = pCons a q` by simp 

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then show "P q" 

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by (rule less.hyps) 

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qed 

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then have "P (pCons a q)" 

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by (rule pCons) 

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then show ?case 

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using `p = pCons a q` by simp 

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qed 

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subsection {* Recursion combinator for polynomials *} 
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function 
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poly_rec :: "'b \<Rightarrow> ('a::zero \<Rightarrow> 'a poly \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a poly \<Rightarrow> 'b" 
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where 
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poly_rec_pCons_eq_if [simp del, code del]: 
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"poly_rec z f (pCons a p) = f a p (if p = 0 then z else poly_rec z f p)" 
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by (case_tac x, rename_tac q, case_tac q, auto) 
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termination poly_rec 
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by (relation "measure (degree \<circ> snd \<circ> snd)", simp) 
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(simp add: degree_pCons_eq) 
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lemma poly_rec_0: 
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"f 0 0 z = z \<Longrightarrow> poly_rec z f 0 = z" 
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using poly_rec_pCons_eq_if [of z f 0 0] by simp 
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lemma poly_rec_pCons: 
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"f 0 0 z = z \<Longrightarrow> poly_rec z f (pCons a p) = f a p (poly_rec z f p)" 
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by (simp add: poly_rec_pCons_eq_if poly_rec_0) 
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29451  228 
subsection {* Monomials *} 
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definition 

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monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a::zero poly" where 

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"monom a m = Abs_poly (\<lambda>n. if m = n then a else 0)" 

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lemma coeff_monom [simp]: "coeff (monom a m) n = (if m=n then a else 0)" 

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unfolding monom_def 

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by (subst Abs_poly_inverse, auto simp add: Poly_def) 

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lemma monom_0: "monom a 0 = pCons a 0" 

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by (rule poly_ext, simp add: coeff_pCons split: nat.split) 

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lemma monom_Suc: "monom a (Suc n) = pCons 0 (monom a n)" 

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by (rule poly_ext, simp add: coeff_pCons split: nat.split) 

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lemma monom_eq_0 [simp]: "monom 0 n = 0" 

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by (rule poly_ext) simp 

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lemma monom_eq_0_iff [simp]: "monom a n = 0 \<longleftrightarrow> a = 0" 

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by (simp add: expand_poly_eq) 

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lemma monom_eq_iff [simp]: "monom a n = monom b n \<longleftrightarrow> a = b" 

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by (simp add: expand_poly_eq) 

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lemma degree_monom_le: "degree (monom a n) \<le> n" 

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by (rule degree_le, simp) 

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lemma degree_monom_eq: "a \<noteq> 0 \<Longrightarrow> degree (monom a n) = n" 

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apply (rule order_antisym [OF degree_monom_le]) 

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apply (rule le_degree, simp) 

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done 

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subsection {* Addition and subtraction *} 

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instantiation poly :: (comm_monoid_add) comm_monoid_add 

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begin 

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definition 

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plus_poly_def [code del]: 

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"p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)" 

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lemma Poly_add: 

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fixes f g :: "nat \<Rightarrow> 'a" 

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shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n + g n) \<in> Poly" 

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unfolding Poly_def 

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apply (clarify, rename_tac m n) 

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apply (rule_tac x="max m n" in exI, simp) 

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done 

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lemma coeff_add [simp]: 

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"coeff (p + q) n = coeff p n + coeff q n" 

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unfolding plus_poly_def 

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by (simp add: Abs_poly_inverse coeff Poly_add) 

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instance proof 

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fix p q r :: "'a poly" 

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show "(p + q) + r = p + (q + r)" 

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by (simp add: expand_poly_eq add_assoc) 

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show "p + q = q + p" 

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by (simp add: expand_poly_eq add_commute) 

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show "0 + p = p" 

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by (simp add: expand_poly_eq) 

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qed 

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end 

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instantiation poly :: (ab_group_add) ab_group_add 

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begin 

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definition 

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uminus_poly_def [code del]: 

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" p = Abs_poly (\<lambda>n.  coeff p n)" 

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definition 

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minus_poly_def [code del]: 

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"p  q = Abs_poly (\<lambda>n. coeff p n  coeff q n)" 

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lemma Poly_minus: 

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fixes f :: "nat \<Rightarrow> 'a" 

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shows "f \<in> Poly \<Longrightarrow> (\<lambda>n.  f n) \<in> Poly" 

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unfolding Poly_def by simp 

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lemma Poly_diff: 

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fixes f g :: "nat \<Rightarrow> 'a" 

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shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. f n  g n) \<in> Poly" 

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unfolding diff_minus by (simp add: Poly_add Poly_minus) 

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lemma coeff_minus [simp]: "coeff ( p) n =  coeff p n" 

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unfolding uminus_poly_def 

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by (simp add: Abs_poly_inverse coeff Poly_minus) 

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lemma coeff_diff [simp]: 

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"coeff (p  q) n = coeff p n  coeff q n" 

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unfolding minus_poly_def 

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by (simp add: Abs_poly_inverse coeff Poly_diff) 

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instance proof 

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fix p q :: "'a poly" 

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show " p + p = 0" 

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by (simp add: expand_poly_eq) 

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show "p  q = p +  q" 

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by (simp add: expand_poly_eq diff_minus) 

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qed 

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end 

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lemma add_pCons [simp]: 

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"pCons a p + pCons b q = pCons (a + b) (p + q)" 

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by (rule poly_ext, simp add: coeff_pCons split: nat.split) 

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lemma minus_pCons [simp]: 

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" pCons a p = pCons ( a) ( p)" 

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by (rule poly_ext, simp add: coeff_pCons split: nat.split) 

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lemma diff_pCons [simp]: 

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"pCons a p  pCons b q = pCons (a  b) (p  q)" 

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by (rule poly_ext, simp add: coeff_pCons split: nat.split) 

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lemma degree_add_le: "degree (p + q) \<le> max (degree p) (degree q)" 

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by (rule degree_le, auto simp add: coeff_eq_0) 

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lemma degree_add_less: 
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"\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n" 

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by (auto intro: le_less_trans degree_add_le) 

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lemma degree_add_eq_right: 
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"degree p < degree q \<Longrightarrow> degree (p + q) = degree q" 

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apply (cases "q = 0", simp) 

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apply (rule order_antisym) 

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apply (rule ord_le_eq_trans [OF degree_add_le]) 

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apply simp 

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apply (rule le_degree) 

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apply (simp add: coeff_eq_0) 

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done 

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lemma degree_add_eq_left: 

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"degree q < degree p \<Longrightarrow> degree (p + q) = degree p" 

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using degree_add_eq_right [of q p] 

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by (simp add: add_commute) 

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lemma degree_minus [simp]: "degree ( p) = degree p" 

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unfolding degree_def by simp 

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lemma degree_diff_le: "degree (p  q) \<le> max (degree p) (degree q)" 

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using degree_add_le [where p=p and q="q"] 

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by (simp add: diff_minus) 

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lemma degree_diff_less: 
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"\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p  q) < n" 

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by (auto intro: le_less_trans degree_diff_le) 

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lemma add_monom: "monom a n + monom b n = monom (a + b) n" 
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by (rule poly_ext) simp 

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lemma diff_monom: "monom a n  monom b n = monom (a  b) n" 

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by (rule poly_ext) simp 

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lemma minus_monom: " monom a n = monom (a) n" 

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by (rule poly_ext) simp 

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lemma coeff_setsum: "coeff (\<Sum>x\<in>A. p x) i = (\<Sum>x\<in>A. coeff (p x) i)" 

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by (cases "finite A", induct set: finite, simp_all) 

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lemma monom_setsum: "monom (\<Sum>x\<in>A. a x) n = (\<Sum>x\<in>A. monom (a x) n)" 

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by (rule poly_ext) (simp add: coeff_setsum) 

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subsection {* Multiplication by a constant *} 

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definition 

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smult :: "'a::comm_semiring_0 \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where 

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"smult a p = Abs_poly (\<lambda>n. a * coeff p n)" 

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lemma Poly_smult: 

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fixes f :: "nat \<Rightarrow> 'a::comm_semiring_0" 

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shows "f \<in> Poly \<Longrightarrow> (\<lambda>n. a * f 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 coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n" 

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unfolding smult_def 

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by (simp add: Abs_poly_inverse Poly_smult coeff) 

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lemma degree_smult_le: "degree (smult a p) \<le> degree p" 

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by (rule degree_le, simp add: coeff_eq_0) 

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lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p" 
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by (rule poly_ext, simp add: mult_assoc) 
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lemma smult_0_right [simp]: "smult a 0 = 0" 

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by (rule poly_ext, simp) 

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lemma smult_0_left [simp]: "smult 0 p = 0" 

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by (rule poly_ext, simp) 

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lemma smult_1_left [simp]: "smult (1::'a::comm_semiring_1) p = p" 

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by (rule poly_ext, simp) 

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lemma smult_add_right: 

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"smult a (p + q) = smult a p + smult a q" 

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by (rule poly_ext, simp add: ring_simps) 

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lemma smult_add_left: 

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"smult (a + b) p = smult a p + smult b p" 

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by (rule poly_ext, simp add: ring_simps) 

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lemma smult_minus_right [simp]: 
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"smult (a::'a::comm_ring) ( p) =  smult a p" 
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by (rule poly_ext, simp) 

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lemma smult_minus_left [simp]: 
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"smult ( a::'a::comm_ring) p =  smult a p" 
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by (rule poly_ext, simp) 

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lemma smult_diff_right: 

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"smult (a::'a::comm_ring) (p  q) = smult a p  smult a q" 

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by (rule poly_ext, simp add: ring_simps) 

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lemma smult_diff_left: 

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"smult (a  b::'a::comm_ring) p = smult a p  smult b p" 

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by (rule poly_ext, simp add: ring_simps) 

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lemmas smult_distribs = 
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smult_add_left smult_add_right 

454 
smult_diff_left smult_diff_right 

455 

29451  456 
lemma smult_pCons [simp]: 
457 
"smult a (pCons b p) = pCons (a * b) (smult a p)" 

458 
by (rule poly_ext, simp add: coeff_pCons split: nat.split) 

459 

460 
lemma smult_monom: "smult a (monom b n) = monom (a * b) n" 

461 
by (induct n, simp add: monom_0, simp add: monom_Suc) 

462 

463 

464 
subsection {* Multiplication of polynomials *} 

465 

29474  466 
text {* TODO: move to SetInterval.thy *} 
29451  467 
lemma setsum_atMost_Suc_shift: 
468 
fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add" 

469 
shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))" 

470 
proof (induct n) 

471 
case 0 show ?case by simp 

472 
next 

473 
case (Suc n) note IH = this 

474 
have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))" 

475 
by (rule setsum_atMost_Suc) 

476 
also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))" 

477 
by (rule IH) 

478 
also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = 

479 
f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))" 

480 
by (rule add_assoc) 

481 
also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))" 

482 
by (rule setsum_atMost_Suc [symmetric]) 

483 
finally show ?case . 

484 
qed 

485 

486 
instantiation poly :: (comm_semiring_0) comm_semiring_0 

487 
begin 

488 

489 
definition 

29475  490 
times_poly_def [code del]: 
29474  491 
"p * q = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p" 
492 

493 
lemma mult_poly_0_left: "(0::'a poly) * q = 0" 

494 
unfolding times_poly_def by (simp add: poly_rec_0) 

495 

496 
lemma mult_pCons_left [simp]: 

497 
"pCons a p * q = smult a q + pCons 0 (p * q)" 

498 
unfolding times_poly_def by (simp add: poly_rec_pCons) 

499 

500 
lemma mult_poly_0_right: "p * (0::'a poly) = 0" 

501 
by (induct p, simp add: mult_poly_0_left, simp) 

29451  502 

29474  503 
lemma mult_pCons_right [simp]: 
504 
"p * pCons a q = smult a p + pCons 0 (p * q)" 

505 
by (induct p, simp add: mult_poly_0_left, simp add: ring_simps) 

506 

507 
lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right 

508 

509 
lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)" 

510 
by (induct p, simp add: mult_poly_0, simp add: smult_add_right) 

511 

512 
lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)" 

513 
by (induct q, simp add: mult_poly_0, simp add: smult_add_right) 

514 

515 
lemma mult_poly_add_left: 

516 
fixes p q r :: "'a poly" 

517 
shows "(p + q) * r = p * r + q * r" 

518 
by (induct r, simp add: mult_poly_0, 

519 
simp add: smult_distribs group_simps) 

29451  520 

521 
instance proof 

522 
fix p q r :: "'a poly" 

523 
show 0: "0 * p = 0" 

29474  524 
by (rule mult_poly_0_left) 
29451  525 
show "p * 0 = 0" 
29474  526 
by (rule mult_poly_0_right) 
29451  527 
show "(p + q) * r = p * r + q * r" 
29474  528 
by (rule mult_poly_add_left) 
29451  529 
show "(p * q) * r = p * (q * r)" 
29474  530 
by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left) 
29451  531 
show "p * q = q * p" 
29474  532 
by (induct p, simp add: mult_poly_0, simp) 
29451  533 
qed 
534 

535 
end 

536 

29474  537 
lemma coeff_mult: 
538 
"coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (ni))" 

539 
proof (induct p arbitrary: n) 

540 
case 0 show ?case by simp 

541 
next 

542 
case (pCons a p n) thus ?case 

543 
by (cases n, simp, simp add: setsum_atMost_Suc_shift 

544 
del: setsum_atMost_Suc) 

545 
qed 

29451  546 

29474  547 
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q" 
548 
apply (rule degree_le) 

549 
apply (induct p) 

550 
apply simp 

551 
apply (simp add: coeff_eq_0 coeff_pCons split: nat.split) 

29451  552 
done 
553 

554 
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)" 

555 
by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc) 

556 

557 

558 
subsection {* The unit polynomial and exponentiation *} 

559 

560 
instantiation poly :: (comm_semiring_1) comm_semiring_1 

561 
begin 

562 

563 
definition 

564 
one_poly_def: 

565 
"1 = pCons 1 0" 

566 

567 
instance proof 

568 
fix p :: "'a poly" show "1 * p = p" 

569 
unfolding one_poly_def 

570 
by simp 

571 
next 

572 
show "0 \<noteq> (1::'a poly)" 

573 
unfolding one_poly_def by simp 

574 
qed 

575 

576 
end 

577 

578 
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)" 

579 
unfolding one_poly_def 

580 
by (simp add: coeff_pCons split: nat.split) 

581 

582 
lemma degree_1 [simp]: "degree 1 = 0" 

583 
unfolding one_poly_def 

584 
by (rule degree_pCons_0) 

585 

586 
instantiation poly :: (comm_semiring_1) recpower 

587 
begin 

588 

589 
primrec power_poly where 

590 
power_poly_0: "(p::'a poly) ^ 0 = 1" 

591 
 power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n" 

592 

593 
instance 

594 
by default simp_all 

595 

596 
end 

597 

598 
instance poly :: (comm_ring) comm_ring .. 

599 

600 
instance poly :: (comm_ring_1) comm_ring_1 .. 

601 

602 
instantiation poly :: (comm_ring_1) number_ring 

603 
begin 

604 

605 
definition 

606 
"number_of k = (of_int k :: 'a poly)" 

607 

608 
instance 

609 
by default (rule number_of_poly_def) 

610 

611 
end 

612 

613 

614 
subsection {* Polynomials form an integral domain *} 

615 

616 
lemma coeff_mult_degree_sum: 

617 
"coeff (p * q) (degree p + degree q) = 

618 
coeff p (degree p) * coeff q (degree q)" 

29471  619 
by (induct p, simp, simp add: coeff_eq_0) 
29451  620 

621 
instance poly :: (idom) idom 

622 
proof 

623 
fix p q :: "'a poly" 

624 
assume "p \<noteq> 0" and "q \<noteq> 0" 

625 
have "coeff (p * q) (degree p + degree q) = 

626 
coeff p (degree p) * coeff q (degree q)" 

627 
by (rule coeff_mult_degree_sum) 

628 
also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0" 

629 
using `p \<noteq> 0` and `q \<noteq> 0` by simp 

630 
finally have "\<exists>n. coeff (p * q) n \<noteq> 0" .. 

631 
thus "p * q \<noteq> 0" by (simp add: expand_poly_eq) 

632 
qed 

633 

634 
lemma degree_mult_eq: 

635 
fixes p q :: "'a::idom poly" 

636 
shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q" 

637 
apply (rule order_antisym [OF degree_mult_le le_degree]) 

638 
apply (simp add: coeff_mult_degree_sum) 

639 
done 

640 

641 
lemma dvd_imp_degree_le: 

642 
fixes p q :: "'a::idom poly" 

643 
shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q" 

644 
by (erule dvdE, simp add: degree_mult_eq) 

645 

646 

647 
subsection {* Long division of polynomials *} 

648 

649 
definition 

650 
divmod_poly_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool" 

651 
where 

29475  652 
[code del]: 
29451  653 
"divmod_poly_rel x y q r \<longleftrightarrow> 
654 
x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)" 

655 

656 
lemma divmod_poly_rel_0: 

657 
"divmod_poly_rel 0 y 0 0" 

658 
unfolding divmod_poly_rel_def by simp 

659 

660 
lemma divmod_poly_rel_by_0: 

661 
"divmod_poly_rel x 0 0 x" 

662 
unfolding divmod_poly_rel_def by simp 

663 

664 
lemma eq_zero_or_degree_less: 

665 
assumes "degree p \<le> n" and "coeff p n = 0" 

666 
shows "p = 0 \<or> degree p < n" 

667 
proof (cases n) 

668 
case 0 

669 
with `degree p \<le> n` and `coeff p n = 0` 

670 
have "coeff p (degree p) = 0" by simp 

671 
then have "p = 0" by simp 

672 
then show ?thesis .. 

673 
next 

674 
case (Suc m) 

675 
have "\<forall>i>n. coeff p i = 0" 

676 
using `degree p \<le> n` by (simp add: coeff_eq_0) 

677 
then have "\<forall>i\<ge>n. coeff p i = 0" 

678 
using `coeff p n = 0` by (simp add: le_less) 

679 
then have "\<forall>i>m. coeff p i = 0" 

680 
using `n = Suc m` by (simp add: less_eq_Suc_le) 

681 
then have "degree p \<le> m" 

682 
by (rule degree_le) 

683 
then have "degree p < n" 

684 
using `n = Suc m` by (simp add: less_Suc_eq_le) 

685 
then show ?thesis .. 

686 
qed 

687 

688 
lemma divmod_poly_rel_pCons: 

689 
assumes rel: "divmod_poly_rel x y q r" 

690 
assumes y: "y \<noteq> 0" 

691 
assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)" 

692 
shows "divmod_poly_rel (pCons a x) y (pCons b q) (pCons a r  smult b y)" 

693 
(is "divmod_poly_rel ?x y ?q ?r") 

694 
proof  

695 
have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y" 

696 
using assms unfolding divmod_poly_rel_def by simp_all 

697 

698 
have 1: "?x = ?q * y + ?r" 

699 
using b x by simp 

700 

701 
have 2: "?r = 0 \<or> degree ?r < degree y" 

702 
proof (rule eq_zero_or_degree_less) 

703 
have "degree ?r \<le> max (degree (pCons a r)) (degree (smult b y))" 

704 
by (rule degree_diff_le) 

705 
also have "\<dots> \<le> degree y" 

706 
proof (rule min_max.le_supI) 

707 
show "degree (pCons a r) \<le> degree y" 

29460
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

708 
using r by auto 
29451  709 
show "degree (smult b y) \<le> degree y" 
710 
by (rule degree_smult_le) 

711 
qed 

712 
finally show "degree ?r \<le> degree y" . 

713 
next 

714 
show "coeff ?r (degree y) = 0" 

715 
using `y \<noteq> 0` unfolding b by simp 

716 
qed 

717 

718 
from 1 2 show ?thesis 

719 
unfolding divmod_poly_rel_def 

720 
using `y \<noteq> 0` by simp 

721 
qed 

722 

723 
lemma divmod_poly_rel_exists: "\<exists>q r. divmod_poly_rel x y q r" 

724 
apply (cases "y = 0") 

725 
apply (fast intro!: divmod_poly_rel_by_0) 

726 
apply (induct x) 

727 
apply (fast intro!: divmod_poly_rel_0) 

728 
apply (fast intro!: divmod_poly_rel_pCons) 

729 
done 

730 

731 
lemma divmod_poly_rel_unique: 

732 
assumes 1: "divmod_poly_rel x y q1 r1" 

733 
assumes 2: "divmod_poly_rel x y q2 r2" 

734 
shows "q1 = q2 \<and> r1 = r2" 

735 
proof (cases "y = 0") 

736 
assume "y = 0" with assms show ?thesis 

737 
by (simp add: divmod_poly_rel_def) 

738 
next 

739 
assume [simp]: "y \<noteq> 0" 

740 
from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y" 

741 
unfolding divmod_poly_rel_def by simp_all 

742 
from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y" 

743 
unfolding divmod_poly_rel_def by simp_all 

744 
from q1 q2 have q3: "(q1  q2) * y = r2  r1" 

745 
by (simp add: ring_simps) 

746 
from r1 r2 have r3: "(r2  r1) = 0 \<or> degree (r2  r1) < degree y" 

29453  747 
by (auto intro: degree_diff_less) 
29451  748 

749 
show "q1 = q2 \<and> r1 = r2" 

750 
proof (rule ccontr) 

751 
assume "\<not> (q1 = q2 \<and> r1 = r2)" 

752 
with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto 

753 
with r3 have "degree (r2  r1) < degree y" by simp 

754 
also have "degree y \<le> degree (q1  q2) + degree y" by simp 

755 
also have "\<dots> = degree ((q1  q2) * y)" 

756 
using `q1 \<noteq> q2` by (simp add: degree_mult_eq) 

757 
also have "\<dots> = degree (r2  r1)" 

758 
using q3 by simp 

759 
finally have "degree (r2  r1) < degree (r2  r1)" . 

760 
then show "False" by simp 

761 
qed 

762 
qed 

763 

764 
lemmas divmod_poly_rel_unique_div = 

765 
divmod_poly_rel_unique [THEN conjunct1, standard] 

766 

767 
lemmas divmod_poly_rel_unique_mod = 

768 
divmod_poly_rel_unique [THEN conjunct2, standard] 

769 

770 
instantiation poly :: (field) ring_div 

771 
begin 

772 

773 
definition div_poly where 

774 
[code del]: "x div y = (THE q. \<exists>r. divmod_poly_rel x y q r)" 

775 

776 
definition mod_poly where 

777 
[code del]: "x mod y = (THE r. \<exists>q. divmod_poly_rel x y q r)" 

778 

779 
lemma div_poly_eq: 

780 
"divmod_poly_rel x y q r \<Longrightarrow> x div y = q" 

781 
unfolding div_poly_def 

782 
by (fast elim: divmod_poly_rel_unique_div) 

783 

784 
lemma mod_poly_eq: 

785 
"divmod_poly_rel x y q r \<Longrightarrow> x mod y = r" 

786 
unfolding mod_poly_def 

787 
by (fast elim: divmod_poly_rel_unique_mod) 

788 

789 
lemma divmod_poly_rel: 

790 
"divmod_poly_rel x y (x div y) (x mod y)" 

791 
proof  

792 
from divmod_poly_rel_exists 

793 
obtain q r where "divmod_poly_rel x y q r" by fast 

794 
thus ?thesis 

795 
by (simp add: div_poly_eq mod_poly_eq) 

796 
qed 

797 

798 
instance proof 

799 
fix x y :: "'a poly" 

800 
show "x div y * y + x mod y = x" 

801 
using divmod_poly_rel [of x y] 

802 
by (simp add: divmod_poly_rel_def) 

803 
next 

804 
fix x :: "'a poly" 

805 
have "divmod_poly_rel x 0 0 x" 

806 
by (rule divmod_poly_rel_by_0) 

807 
thus "x div 0 = 0" 

808 
by (rule div_poly_eq) 

809 
next 

810 
fix y :: "'a poly" 

811 
have "divmod_poly_rel 0 y 0 0" 

812 
by (rule divmod_poly_rel_0) 

813 
thus "0 div y = 0" 

814 
by (rule div_poly_eq) 

815 
next 

816 
fix x y z :: "'a poly" 

817 
assume "y \<noteq> 0" 

818 
hence "divmod_poly_rel (x + z * y) y (z + x div y) (x mod y)" 

819 
using divmod_poly_rel [of x y] 

820 
by (simp add: divmod_poly_rel_def left_distrib) 

821 
thus "(x + z * y) div y = z + x div y" 

822 
by (rule div_poly_eq) 

823 
qed 

824 

825 
end 

826 

827 
lemma degree_mod_less: 

828 
"y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y" 

829 
using divmod_poly_rel [of x y] 

830 
unfolding divmod_poly_rel_def by simp 

831 

832 
lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0" 

833 
proof  

834 
assume "degree x < degree y" 

835 
hence "divmod_poly_rel x y 0 x" 

836 
by (simp add: divmod_poly_rel_def) 

837 
thus "x div y = 0" by (rule div_poly_eq) 

838 
qed 

839 

840 
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x" 

841 
proof  

842 
assume "degree x < degree y" 

843 
hence "divmod_poly_rel x y 0 x" 

844 
by (simp add: divmod_poly_rel_def) 

845 
thus "x mod y = x" by (rule mod_poly_eq) 

846 
qed 

847 

848 
lemma mod_pCons: 

849 
fixes a and x 

850 
assumes y: "y \<noteq> 0" 

851 
defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)" 

852 
shows "(pCons a x) mod y = (pCons a (x mod y)  smult b y)" 

853 
unfolding b 

854 
apply (rule mod_poly_eq) 

855 
apply (rule divmod_poly_rel_pCons [OF divmod_poly_rel y refl]) 

856 
done 

857 

858 

859 
subsection {* Evaluation of polynomials *} 

860 

861 
definition 

29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset

862 
poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a" where 
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset

863 
"poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)" 
29451  864 

865 
lemma poly_0 [simp]: "poly 0 x = 0" 

29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset

866 
unfolding poly_def by (simp add: poly_rec_0) 
29451  867 

868 
lemma poly_pCons [simp]: "poly (pCons a p) x = a + x * poly p x" 

29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset

869 
unfolding poly_def by (simp add: poly_rec_pCons) 
29451  870 

871 
lemma poly_1 [simp]: "poly 1 x = 1" 

872 
unfolding one_poly_def by simp 

873 

29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset

874 
lemma poly_monom: 
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset

875 
fixes a x :: "'a::{comm_semiring_1,recpower}" 
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset

876 
shows "poly (monom a n) x = a * x ^ n" 
29451  877 
by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac) 
878 

879 
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x" 

880 
apply (induct p arbitrary: q, simp) 

881 
apply (case_tac q, simp, simp add: ring_simps) 

882 
done 

883 

884 
lemma poly_minus [simp]: 

29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset

885 
fixes x :: "'a::comm_ring" 
29451  886 
shows "poly ( p) x =  poly p x" 
887 
by (induct p, simp_all) 

888 

889 
lemma poly_diff [simp]: 

29454
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset

890 
fixes x :: "'a::comm_ring" 
29451  891 
shows "poly (p  q) x = poly p x  poly q x" 
892 
by (simp add: diff_minus) 

893 

894 
lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)" 

895 
by (cases "finite A", induct set: finite, simp_all) 

896 

897 
lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x" 

898 
by (induct p, simp, simp add: ring_simps) 

899 

900 
lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x" 

901 
by (induct p, simp_all, simp add: ring_simps) 

902 

29462  903 
lemma poly_power [simp]: 
904 
fixes p :: "'a::{comm_semiring_1,recpower} poly" 

905 
shows "poly (p ^ n) x = poly p x ^ n" 

906 
by (induct n, simp, simp add: power_Suc) 

907 

29456  908 

909 
subsection {* Synthetic division *} 

910 

911 
definition 

912 
synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a" 

913 
where 

914 
"synthetic_divmod p c = 

915 
poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p" 

916 

917 
definition 

918 
synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly" 

919 
where 

920 
"synthetic_div p c = fst (synthetic_divmod p c)" 

921 

922 
lemma synthetic_divmod_0 [simp]: 

923 
"synthetic_divmod 0 c = (0, 0)" 

924 
unfolding synthetic_divmod_def 

925 
by (simp add: poly_rec_0) 

926 

927 
lemma synthetic_divmod_pCons [simp]: 

928 
"synthetic_divmod (pCons a p) c = 

929 
(\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)" 

930 
unfolding synthetic_divmod_def 

931 
by (simp add: poly_rec_pCons) 

932 

933 
lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c" 

934 
by (induct p, simp, simp add: split_def) 

935 

936 
lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0" 

937 
unfolding synthetic_div_def by simp 

938 

939 
lemma synthetic_div_pCons [simp]: 

940 
"synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)" 

941 
unfolding synthetic_div_def 

942 
by (simp add: split_def snd_synthetic_divmod) 

943 

29460
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

944 
lemma synthetic_div_eq_0_iff: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

945 
"synthetic_div p c = 0 \<longleftrightarrow> degree p = 0" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

946 
by (induct p, simp, case_tac p, simp) 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

947 

ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

948 
lemma degree_synthetic_div: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

949 
"degree (synthetic_div p c) = degree p  1" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

950 
by (induct p, simp, simp add: synthetic_div_eq_0_iff) 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

951 

29457  952 
lemma synthetic_div_correct: 
29456  953 
"p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)" 
954 
by (induct p) simp_all 

955 

29457  956 
lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0" 
957 
by (induct p arbitrary: a) simp_all 

958 

959 
lemma synthetic_div_unique: 

960 
"p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c" 

961 
apply (induct p arbitrary: q r) 

962 
apply (simp, frule synthetic_div_unique_lemma, simp) 

963 
apply (case_tac q, force) 

964 
done 

965 

966 
lemma synthetic_div_correct': 

967 
fixes c :: "'a::comm_ring_1" 

968 
shows "[:c, 1:] * synthetic_div p c + [:poly p c:] = p" 

969 
using synthetic_div_correct [of p c] 

970 
by (simp add: group_simps) 

971 

29460
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

972 
lemma poly_eq_0_iff_dvd: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

973 
fixes c :: "'a::idom" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

974 
shows "poly p c = 0 \<longleftrightarrow> [:c, 1:] dvd p" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

975 
proof 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

976 
assume "poly p c = 0" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

977 
with synthetic_div_correct' [of c p] 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

978 
have "p = [:c, 1:] * synthetic_div p c" by simp 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

979 
then show "[:c, 1:] dvd p" .. 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

980 
next 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

981 
assume "[:c, 1:] dvd p" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

982 
then obtain k where "p = [:c, 1:] * k" by (rule dvdE) 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

983 
then show "poly p c = 0" by simp 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

984 
qed 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

985 

ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

986 
lemma dvd_iff_poly_eq_0: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

987 
fixes c :: "'a::idom" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

988 
shows "[:c, 1:] dvd p \<longleftrightarrow> poly p (c) = 0" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

989 
by (simp add: poly_eq_0_iff_dvd) 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

990 

29462  991 
lemma poly_roots_finite: 
992 
fixes p :: "'a::idom poly" 

993 
shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}" 

994 
proof (induct n \<equiv> "degree p" arbitrary: p) 

995 
case (0 p) 

996 
then obtain a where "a \<noteq> 0" and "p = [:a:]" 

997 
by (cases p, simp split: if_splits) 

998 
then show "finite {x. poly p x = 0}" by simp 

999 
next 

1000 
case (Suc n p) 

1001 
show "finite {x. poly p x = 0}" 

1002 
proof (cases "\<exists>x. poly p x = 0") 

1003 
case False 

1004 
then show "finite {x. poly p x = 0}" by simp 

1005 
next 

1006 
case True 

1007 
then obtain a where "poly p a = 0" .. 

1008 
then have "[:a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd) 

1009 
then obtain k where k: "p = [:a, 1:] * k" .. 

1010 
with `p \<noteq> 0` have "k \<noteq> 0" by auto 

1011 
with k have "degree p = Suc (degree k)" 

1012 
by (simp add: degree_mult_eq del: mult_pCons_left) 

1013 
with `Suc n = degree p` have "n = degree k" by simp 

1014 
with `k \<noteq> 0` have "finite {x. poly k x = 0}" by (rule Suc.hyps) 

1015 
then have "finite (insert a {x. poly k x = 0})" by simp 

1016 
then show "finite {x. poly p x = 0}" 

1017 
by (simp add: k uminus_add_conv_diff Collect_disj_eq 

1018 
del: mult_pCons_left) 

1019 
qed 

1020 
qed 

1021 

29451  1022 
end 