author  huffman 
Wed, 18 Feb 2009 12:24:06 0800  
changeset 29979  666f5f72dbb5 
parent 29977  d76b830366bc 
child 29980  17ddfd0c3506 
permissions  rwrr 
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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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Plain, Main form meeting points in import hierarchy
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imports Plain SetInterval 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 {* 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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instance poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add 
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proof 
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fix p q r :: "'a poly" 

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assume "p + q = p + r" thus "q = r" 

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by (simp add: expand_poly_eq) 

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qed 

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instantiation poly :: (ab_group_add) ab_group_add 
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begin 

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definition 

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uminus_poly_def [code del]: 

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

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definition 

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minus_poly_def [code del]: 

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

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lemma Poly_minus: 

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

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

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unfolding Poly_def by simp 

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lemma Poly_diff: 

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

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

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unfolding diff_minus by (simp add: Poly_add Poly_minus) 

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

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unfolding uminus_poly_def 

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by (simp add: Abs_poly_inverse coeff Poly_minus) 

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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_max: "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_le: 
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"\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n" 

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by (auto intro: order_trans degree_add_le_max) 

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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_max) 
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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 (simp add: degree_add_le) 
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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_max: "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_le: 
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"\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p  q) \<le> n" 

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by (simp add: diff_minus degree_add_le) 

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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 (simp add: diff_minus degree_add_less) 
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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" 

29667  444 
by (rule poly_ext, simp add: algebra_simps) 
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lemma smult_add_left: 

447 
"smult (a + b) p = smult a p + smult b p" 

29667  448 
by (rule poly_ext, simp add: algebra_simps) 
29451  449 

29457  450 
lemma smult_minus_right [simp]: 
29451  451 
"smult (a::'a::comm_ring) ( p) =  smult a p" 
452 
by (rule poly_ext, simp) 

453 

29457  454 
lemma smult_minus_left [simp]: 
29451  455 
"smult ( a::'a::comm_ring) p =  smult a p" 
456 
by (rule poly_ext, simp) 

457 

458 
lemma smult_diff_right: 

459 
"smult (a::'a::comm_ring) (p  q) = smult a p  smult a q" 

29667  460 
by (rule poly_ext, simp add: algebra_simps) 
29451  461 

462 
lemma smult_diff_left: 

463 
"smult (a  b::'a::comm_ring) p = smult a p  smult b p" 

29667  464 
by (rule poly_ext, simp add: algebra_simps) 
29451  465 

29472  466 
lemmas smult_distribs = 
467 
smult_add_left smult_add_right 

468 
smult_diff_left smult_diff_right 

469 

29451  470 
lemma smult_pCons [simp]: 
471 
"smult a (pCons b p) = pCons (a * b) (smult a p)" 

472 
by (rule poly_ext, simp add: coeff_pCons split: nat.split) 

473 

474 
lemma smult_monom: "smult a (monom b n) = monom (a * b) n" 

475 
by (induct n, simp add: monom_0, simp add: monom_Suc) 

476 

29659  477 
lemma degree_smult_eq [simp]: 
478 
fixes a :: "'a::idom" 

479 
shows "degree (smult a p) = (if a = 0 then 0 else degree p)" 

480 
by (cases "a = 0", simp, simp add: degree_def) 

481 

482 
lemma smult_eq_0_iff [simp]: 

483 
fixes a :: "'a::idom" 

484 
shows "smult a p = 0 \<longleftrightarrow> a = 0 \<or> p = 0" 

485 
by (simp add: expand_poly_eq) 

486 

29451  487 

488 
subsection {* Multiplication of polynomials *} 

489 

29474  490 
text {* TODO: move to SetInterval.thy *} 
29451  491 
lemma setsum_atMost_Suc_shift: 
492 
fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add" 

493 
shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))" 

494 
proof (induct n) 

495 
case 0 show ?case by simp 

496 
next 

497 
case (Suc n) note IH = this 

498 
have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))" 

499 
by (rule setsum_atMost_Suc) 

500 
also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))" 

501 
by (rule IH) 

502 
also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = 

503 
f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))" 

504 
by (rule add_assoc) 

505 
also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))" 

506 
by (rule setsum_atMost_Suc [symmetric]) 

507 
finally show ?case . 

508 
qed 

509 

510 
instantiation poly :: (comm_semiring_0) comm_semiring_0 

511 
begin 

512 

513 
definition 

29475  514 
times_poly_def [code del]: 
29474  515 
"p * q = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p" 
516 

517 
lemma mult_poly_0_left: "(0::'a poly) * q = 0" 

518 
unfolding times_poly_def by (simp add: poly_rec_0) 

519 

520 
lemma mult_pCons_left [simp]: 

521 
"pCons a p * q = smult a q + pCons 0 (p * q)" 

522 
unfolding times_poly_def by (simp add: poly_rec_pCons) 

523 

524 
lemma mult_poly_0_right: "p * (0::'a poly) = 0" 

525 
by (induct p, simp add: mult_poly_0_left, simp) 

29451  526 

29474  527 
lemma mult_pCons_right [simp]: 
528 
"p * pCons a q = smult a p + pCons 0 (p * q)" 

29667  529 
by (induct p, simp add: mult_poly_0_left, simp add: algebra_simps) 
29474  530 

531 
lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right 

532 

533 
lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)" 

534 
by (induct p, simp add: mult_poly_0, simp add: smult_add_right) 

535 

536 
lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)" 

537 
by (induct q, simp add: mult_poly_0, simp add: smult_add_right) 

538 

539 
lemma mult_poly_add_left: 

540 
fixes p q r :: "'a poly" 

541 
shows "(p + q) * r = p * r + q * r" 

542 
by (induct r, simp add: mult_poly_0, 

29667  543 
simp add: smult_distribs algebra_simps) 
29451  544 

545 
instance proof 

546 
fix p q r :: "'a poly" 

547 
show 0: "0 * p = 0" 

29474  548 
by (rule mult_poly_0_left) 
29451  549 
show "p * 0 = 0" 
29474  550 
by (rule mult_poly_0_right) 
29451  551 
show "(p + q) * r = p * r + q * r" 
29474  552 
by (rule mult_poly_add_left) 
29451  553 
show "(p * q) * r = p * (q * r)" 
29474  554 
by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left) 
29451  555 
show "p * q = q * p" 
29474  556 
by (induct p, simp add: mult_poly_0, simp) 
29451  557 
qed 
558 

559 
end 

560 

29540  561 
instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel .. 
562 

29474  563 
lemma coeff_mult: 
564 
"coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (ni))" 

565 
proof (induct p arbitrary: n) 

566 
case 0 show ?case by simp 

567 
next 

568 
case (pCons a p n) thus ?case 

569 
by (cases n, simp, simp add: setsum_atMost_Suc_shift 

570 
del: setsum_atMost_Suc) 

571 
qed 

29451  572 

29474  573 
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q" 
574 
apply (rule degree_le) 

575 
apply (induct p) 

576 
apply simp 

577 
apply (simp add: coeff_eq_0 coeff_pCons split: nat.split) 

29451  578 
done 
579 

580 
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)" 

581 
by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc) 

582 

583 

584 
subsection {* The unit polynomial and exponentiation *} 

585 

586 
instantiation poly :: (comm_semiring_1) comm_semiring_1 

587 
begin 

588 

589 
definition 

590 
one_poly_def: 

591 
"1 = pCons 1 0" 

592 

593 
instance proof 

594 
fix p :: "'a poly" show "1 * p = p" 

595 
unfolding one_poly_def 

596 
by simp 

597 
next 

598 
show "0 \<noteq> (1::'a poly)" 

599 
unfolding one_poly_def by simp 

600 
qed 

601 

602 
end 

603 

29540  604 
instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel .. 
605 

29451  606 
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)" 
607 
unfolding one_poly_def 

608 
by (simp add: coeff_pCons split: nat.split) 

609 

610 
lemma degree_1 [simp]: "degree 1 = 0" 

611 
unfolding one_poly_def 

612 
by (rule degree_pCons_0) 

613 

29979  614 
text {* Lemmas about divisibility *} 
615 

616 
lemma dvd_smult: "p dvd q \<Longrightarrow> p dvd smult a q" 

617 
proof  

618 
assume "p dvd q" 

619 
then obtain k where "q = p * k" .. 

620 
then have "smult a q = p * smult a k" by simp 

621 
then show "p dvd smult a q" .. 

622 
qed 

623 

624 
lemma dvd_smult_cancel: 

625 
fixes a :: "'a::field" 

626 
shows "p dvd smult a q \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> p dvd q" 

627 
by (drule dvd_smult [where a="inverse a"]) simp 

628 

629 
lemma dvd_smult_iff: 

630 
fixes a :: "'a::field" 

631 
shows "a \<noteq> 0 \<Longrightarrow> p dvd smult a q \<longleftrightarrow> p dvd q" 

632 
by (safe elim!: dvd_smult dvd_smult_cancel) 

633 

29451  634 
instantiation poly :: (comm_semiring_1) recpower 
635 
begin 

636 

637 
primrec power_poly where 

638 
power_poly_0: "(p::'a poly) ^ 0 = 1" 

639 
 power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n" 

640 

641 
instance 

642 
by default simp_all 

643 

644 
end 

645 

29979  646 
lemma degree_power_le: "degree (p ^ n) \<le> degree p * n" 
647 
by (induct n, simp, auto intro: order_trans degree_mult_le) 

648 

29451  649 
instance poly :: (comm_ring) comm_ring .. 
650 

651 
instance poly :: (comm_ring_1) comm_ring_1 .. 

652 

653 
instantiation poly :: (comm_ring_1) number_ring 

654 
begin 

655 

656 
definition 

657 
"number_of k = (of_int k :: 'a poly)" 

658 

659 
instance 

660 
by default (rule number_of_poly_def) 

661 

662 
end 

663 

664 

665 
subsection {* Polynomials form an integral domain *} 

666 

667 
lemma coeff_mult_degree_sum: 

668 
"coeff (p * q) (degree p + degree q) = 

669 
coeff p (degree p) * coeff q (degree q)" 

29471  670 
by (induct p, simp, simp add: coeff_eq_0) 
29451  671 

672 
instance poly :: (idom) idom 

673 
proof 

674 
fix p q :: "'a poly" 

675 
assume "p \<noteq> 0" and "q \<noteq> 0" 

676 
have "coeff (p * q) (degree p + degree q) = 

677 
coeff p (degree p) * coeff q (degree q)" 

678 
by (rule coeff_mult_degree_sum) 

679 
also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0" 

680 
using `p \<noteq> 0` and `q \<noteq> 0` by simp 

681 
finally have "\<exists>n. coeff (p * q) n \<noteq> 0" .. 

682 
thus "p * q \<noteq> 0" by (simp add: expand_poly_eq) 

683 
qed 

684 

685 
lemma degree_mult_eq: 

686 
fixes p q :: "'a::idom poly" 

687 
shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q" 

688 
apply (rule order_antisym [OF degree_mult_le le_degree]) 

689 
apply (simp add: coeff_mult_degree_sum) 

690 
done 

691 

692 
lemma dvd_imp_degree_le: 

693 
fixes p q :: "'a::idom poly" 

694 
shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q" 

695 
by (erule dvdE, simp add: degree_mult_eq) 

696 

697 

29878  698 
subsection {* Polynomials form an ordered integral domain *} 
699 

700 
definition 

701 
pos_poly :: "'a::ordered_idom poly \<Rightarrow> bool" 

702 
where 

703 
"pos_poly p \<longleftrightarrow> 0 < coeff p (degree p)" 

704 

705 
lemma pos_poly_pCons: 

706 
"pos_poly (pCons a p) \<longleftrightarrow> pos_poly p \<or> (p = 0 \<and> 0 < a)" 

707 
unfolding pos_poly_def by simp 

708 

709 
lemma not_pos_poly_0 [simp]: "\<not> pos_poly 0" 

710 
unfolding pos_poly_def by simp 

711 

712 
lemma pos_poly_add: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p + q)" 

713 
apply (induct p arbitrary: q, simp) 

714 
apply (case_tac q, force simp add: pos_poly_pCons add_pos_pos) 

715 
done 

716 

717 
lemma pos_poly_mult: "\<lbrakk>pos_poly p; pos_poly q\<rbrakk> \<Longrightarrow> pos_poly (p * q)" 

718 
unfolding pos_poly_def 

719 
apply (subgoal_tac "p \<noteq> 0 \<and> q \<noteq> 0") 

720 
apply (simp add: degree_mult_eq coeff_mult_degree_sum mult_pos_pos) 

721 
apply auto 

722 
done 

723 

724 
lemma pos_poly_total: "p = 0 \<or> pos_poly p \<or> pos_poly ( p)" 

725 
by (induct p) (auto simp add: pos_poly_pCons) 

726 

727 
instantiation poly :: (ordered_idom) ordered_idom 

728 
begin 

729 

730 
definition 

731 
[code del]: 

732 
"x < y \<longleftrightarrow> pos_poly (y  x)" 

733 

734 
definition 

735 
[code del]: 

736 
"x \<le> y \<longleftrightarrow> x = y \<or> pos_poly (y  x)" 

737 

738 
definition 

739 
[code del]: 

740 
"abs (x::'a poly) = (if x < 0 then  x else x)" 

741 

742 
definition 

743 
[code del]: 

744 
"sgn (x::'a poly) = (if x = 0 then 0 else if 0 < x then 1 else  1)" 

745 

746 
instance proof 

747 
fix x y :: "'a poly" 

748 
show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" 

749 
unfolding less_eq_poly_def less_poly_def 

750 
apply safe 

751 
apply simp 

752 
apply (drule (1) pos_poly_add) 

753 
apply simp 

754 
done 

755 
next 

756 
fix x :: "'a poly" show "x \<le> x" 

757 
unfolding less_eq_poly_def by simp 

758 
next 

759 
fix x y z :: "'a poly" 

760 
assume "x \<le> y" and "y \<le> z" thus "x \<le> z" 

761 
unfolding less_eq_poly_def 

762 
apply safe 

763 
apply (drule (1) pos_poly_add) 

764 
apply (simp add: algebra_simps) 

765 
done 

766 
next 

767 
fix x y :: "'a poly" 

768 
assume "x \<le> y" and "y \<le> x" thus "x = y" 

769 
unfolding less_eq_poly_def 

770 
apply safe 

771 
apply (drule (1) pos_poly_add) 

772 
apply simp 

773 
done 

774 
next 

775 
fix x y z :: "'a poly" 

776 
assume "x \<le> y" thus "z + x \<le> z + y" 

777 
unfolding less_eq_poly_def 

778 
apply safe 

779 
apply (simp add: algebra_simps) 

780 
done 

781 
next 

782 
fix x y :: "'a poly" 

783 
show "x \<le> y \<or> y \<le> x" 

784 
unfolding less_eq_poly_def 

785 
using pos_poly_total [of "x  y"] 

786 
by auto 

787 
next 

788 
fix x y z :: "'a poly" 

789 
assume "x < y" and "0 < z" 

790 
thus "z * x < z * y" 

791 
unfolding less_poly_def 

792 
by (simp add: right_diff_distrib [symmetric] pos_poly_mult) 

793 
next 

794 
fix x :: "'a poly" 

795 
show "\<bar>x\<bar> = (if x < 0 then  x else x)" 

796 
by (rule abs_poly_def) 

797 
next 

798 
fix x :: "'a poly" 

799 
show "sgn x = (if x = 0 then 0 else if 0 < x then 1 else  1)" 

800 
by (rule sgn_poly_def) 

801 
qed 

802 

803 
end 

804 

805 
text {* TODO: Simplification rules for comparisons *} 

806 

807 

29451  808 
subsection {* Long division of polynomials *} 
809 

810 
definition 

29537  811 
pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool" 
29451  812 
where 
29475  813 
[code del]: 
29537  814 
"pdivmod_rel x y q r \<longleftrightarrow> 
29451  815 
x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)" 
816 

29537  817 
lemma pdivmod_rel_0: 
818 
"pdivmod_rel 0 y 0 0" 

819 
unfolding pdivmod_rel_def by simp 

29451  820 

29537  821 
lemma pdivmod_rel_by_0: 
822 
"pdivmod_rel x 0 0 x" 

823 
unfolding pdivmod_rel_def by simp 

29451  824 

825 
lemma eq_zero_or_degree_less: 

826 
assumes "degree p \<le> n" and "coeff p n = 0" 

827 
shows "p = 0 \<or> degree p < n" 

828 
proof (cases n) 

829 
case 0 

830 
with `degree p \<le> n` and `coeff p n = 0` 

831 
have "coeff p (degree p) = 0" by simp 

832 
then have "p = 0" by simp 

833 
then show ?thesis .. 

834 
next 

835 
case (Suc m) 

836 
have "\<forall>i>n. coeff p i = 0" 

837 
using `degree p \<le> n` by (simp add: coeff_eq_0) 

838 
then have "\<forall>i\<ge>n. coeff p i = 0" 

839 
using `coeff p n = 0` by (simp add: le_less) 

840 
then have "\<forall>i>m. coeff p i = 0" 

841 
using `n = Suc m` by (simp add: less_eq_Suc_le) 

842 
then have "degree p \<le> m" 

843 
by (rule degree_le) 

844 
then have "degree p < n" 

845 
using `n = Suc m` by (simp add: less_Suc_eq_le) 

846 
then show ?thesis .. 

847 
qed 

848 

29537  849 
lemma pdivmod_rel_pCons: 
850 
assumes rel: "pdivmod_rel x y q r" 

29451  851 
assumes y: "y \<noteq> 0" 
852 
assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)" 

29537  853 
shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r  smult b y)" 
854 
(is "pdivmod_rel ?x y ?q ?r") 

29451  855 
proof  
856 
have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y" 

29537  857 
using assms unfolding pdivmod_rel_def by simp_all 
29451  858 

859 
have 1: "?x = ?q * y + ?r" 

860 
using b x by simp 

861 

862 
have 2: "?r = 0 \<or> degree ?r < degree y" 

863 
proof (rule eq_zero_or_degree_less) 

29539  864 
show "degree ?r \<le> degree y" 
865 
proof (rule degree_diff_le) 

29451  866 
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

867 
using r by auto 
29451  868 
show "degree (smult b y) \<le> degree y" 
869 
by (rule degree_smult_le) 

870 
qed 

871 
next 

872 
show "coeff ?r (degree y) = 0" 

873 
using `y \<noteq> 0` unfolding b by simp 

874 
qed 

875 

876 
from 1 2 show ?thesis 

29537  877 
unfolding pdivmod_rel_def 
29451  878 
using `y \<noteq> 0` by simp 
879 
qed 

880 

29537  881 
lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r" 
29451  882 
apply (cases "y = 0") 
29537  883 
apply (fast intro!: pdivmod_rel_by_0) 
29451  884 
apply (induct x) 
29537  885 
apply (fast intro!: pdivmod_rel_0) 
886 
apply (fast intro!: pdivmod_rel_pCons) 

29451  887 
done 
888 

29537  889 
lemma pdivmod_rel_unique: 
890 
assumes 1: "pdivmod_rel x y q1 r1" 

891 
assumes 2: "pdivmod_rel x y q2 r2" 

29451  892 
shows "q1 = q2 \<and> r1 = r2" 
893 
proof (cases "y = 0") 

894 
assume "y = 0" with assms show ?thesis 

29537  895 
by (simp add: pdivmod_rel_def) 
29451  896 
next 
897 
assume [simp]: "y \<noteq> 0" 

898 
from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y" 

29537  899 
unfolding pdivmod_rel_def by simp_all 
29451  900 
from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y" 
29537  901 
unfolding pdivmod_rel_def by simp_all 
29451  902 
from q1 q2 have q3: "(q1  q2) * y = r2  r1" 
29667  903 
by (simp add: algebra_simps) 
29451  904 
from r1 r2 have r3: "(r2  r1) = 0 \<or> degree (r2  r1) < degree y" 
29453  905 
by (auto intro: degree_diff_less) 
29451  906 

907 
show "q1 = q2 \<and> r1 = r2" 

908 
proof (rule ccontr) 

909 
assume "\<not> (q1 = q2 \<and> r1 = r2)" 

910 
with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto 

911 
with r3 have "degree (r2  r1) < degree y" by simp 

912 
also have "degree y \<le> degree (q1  q2) + degree y" by simp 

913 
also have "\<dots> = degree ((q1  q2) * y)" 

914 
using `q1 \<noteq> q2` by (simp add: degree_mult_eq) 

915 
also have "\<dots> = degree (r2  r1)" 

916 
using q3 by simp 

917 
finally have "degree (r2  r1) < degree (r2  r1)" . 

918 
then show "False" by simp 

919 
qed 

920 
qed 

921 

29660  922 
lemma pdivmod_rel_0_iff: "pdivmod_rel 0 y q r \<longleftrightarrow> q = 0 \<and> r = 0" 
923 
by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_0) 

924 

925 
lemma pdivmod_rel_by_0_iff: "pdivmod_rel x 0 q r \<longleftrightarrow> q = 0 \<and> r = x" 

926 
by (auto dest: pdivmod_rel_unique intro: pdivmod_rel_by_0) 

927 

29537  928 
lemmas pdivmod_rel_unique_div = 
929 
pdivmod_rel_unique [THEN conjunct1, standard] 

29451  930 

29537  931 
lemmas pdivmod_rel_unique_mod = 
932 
pdivmod_rel_unique [THEN conjunct2, standard] 

29451  933 

934 
instantiation poly :: (field) ring_div 

935 
begin 

936 

937 
definition div_poly where 

29537  938 
[code del]: "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)" 
29451  939 

940 
definition mod_poly where 

29537  941 
[code del]: "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)" 
29451  942 

943 
lemma div_poly_eq: 

29537  944 
"pdivmod_rel x y q r \<Longrightarrow> x div y = q" 
29451  945 
unfolding div_poly_def 
29537  946 
by (fast elim: pdivmod_rel_unique_div) 
29451  947 

948 
lemma mod_poly_eq: 

29537  949 
"pdivmod_rel x y q r \<Longrightarrow> x mod y = r" 
29451  950 
unfolding mod_poly_def 
29537  951 
by (fast elim: pdivmod_rel_unique_mod) 
29451  952 

29537  953 
lemma pdivmod_rel: 
954 
"pdivmod_rel x y (x div y) (x mod y)" 

29451  955 
proof  
29537  956 
from pdivmod_rel_exists 
957 
obtain q r where "pdivmod_rel x y q r" by fast 

29451  958 
thus ?thesis 
959 
by (simp add: div_poly_eq mod_poly_eq) 

960 
qed 

961 

962 
instance proof 

963 
fix x y :: "'a poly" 

964 
show "x div y * y + x mod y = x" 

29537  965 
using pdivmod_rel [of x y] 
966 
by (simp add: pdivmod_rel_def) 

29451  967 
next 
968 
fix x :: "'a poly" 

29537  969 
have "pdivmod_rel x 0 0 x" 
970 
by (rule pdivmod_rel_by_0) 

29451  971 
thus "x div 0 = 0" 
972 
by (rule div_poly_eq) 

973 
next 

974 
fix y :: "'a poly" 

29537  975 
have "pdivmod_rel 0 y 0 0" 
976 
by (rule pdivmod_rel_0) 

29451  977 
thus "0 div y = 0" 
978 
by (rule div_poly_eq) 

979 
next 

980 
fix x y z :: "'a poly" 

981 
assume "y \<noteq> 0" 

29537  982 
hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)" 
983 
using pdivmod_rel [of x y] 

984 
by (simp add: pdivmod_rel_def left_distrib) 

29451  985 
thus "(x + z * y) div y = z + x div y" 
986 
by (rule div_poly_eq) 

987 
qed 

988 

989 
end 

990 

991 
lemma degree_mod_less: 

992 
"y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y" 

29537  993 
using pdivmod_rel [of x y] 
994 
unfolding pdivmod_rel_def by simp 

29451  995 

996 
lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0" 

997 
proof  

998 
assume "degree x < degree y" 

29537  999 
hence "pdivmod_rel x y 0 x" 
1000 
by (simp add: pdivmod_rel_def) 

29451  1001 
thus "x div y = 0" by (rule div_poly_eq) 
1002 
qed 

1003 

1004 
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x" 

1005 
proof  

1006 
assume "degree x < degree y" 

29537  1007 
hence "pdivmod_rel x y 0 x" 
1008 
by (simp add: pdivmod_rel_def) 

29451  1009 
thus "x mod y = x" by (rule mod_poly_eq) 
1010 
qed 

1011 

29659  1012 
lemma pdivmod_rel_smult_left: 
1013 
"pdivmod_rel x y q r 

1014 
\<Longrightarrow> pdivmod_rel (smult a x) y (smult a q) (smult a r)" 

1015 
unfolding pdivmod_rel_def by (simp add: smult_add_right) 

1016 

1017 
lemma div_smult_left: "(smult a x) div y = smult a (x div y)" 

1018 
by (rule div_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel) 

1019 

1020 
lemma mod_smult_left: "(smult a x) mod y = smult a (x mod y)" 

1021 
by (rule mod_poly_eq, rule pdivmod_rel_smult_left, rule pdivmod_rel) 

1022 

1023 
lemma pdivmod_rel_smult_right: 

1024 
"\<lbrakk>a \<noteq> 0; pdivmod_rel x y q r\<rbrakk> 

1025 
\<Longrightarrow> pdivmod_rel x (smult a y) (smult (inverse a) q) r" 

1026 
unfolding pdivmod_rel_def by simp 

1027 

1028 
lemma div_smult_right: 

1029 
"a \<noteq> 0 \<Longrightarrow> x div (smult a y) = smult (inverse a) (x div y)" 

1030 
by (rule div_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel) 

1031 

1032 
lemma mod_smult_right: "a \<noteq> 0 \<Longrightarrow> x mod (smult a y) = x mod y" 

1033 
by (rule mod_poly_eq, erule pdivmod_rel_smult_right, rule pdivmod_rel) 

1034 

29660  1035 
lemma pdivmod_rel_mult: 
1036 
"\<lbrakk>pdivmod_rel x y q r; pdivmod_rel q z q' r'\<rbrakk> 

1037 
\<Longrightarrow> pdivmod_rel x (y * z) q' (y * r' + r)" 

1038 
apply (cases "z = 0", simp add: pdivmod_rel_def) 

1039 
apply (cases "y = 0", simp add: pdivmod_rel_by_0_iff pdivmod_rel_0_iff) 

1040 
apply (cases "r = 0") 

1041 
apply (cases "r' = 0") 

1042 
apply (simp add: pdivmod_rel_def) 

1043 
apply (simp add: pdivmod_rel_def ring_simps degree_mult_eq) 

1044 
apply (cases "r' = 0") 

1045 
apply (simp add: pdivmod_rel_def degree_mult_eq) 

1046 
apply (simp add: pdivmod_rel_def ring_simps) 

1047 
apply (simp add: degree_mult_eq degree_add_less) 

1048 
done 

1049 

1050 
lemma poly_div_mult_right: 

1051 
fixes x y z :: "'a::field poly" 

1052 
shows "x div (y * z) = (x div y) div z" 

1053 
by (rule div_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+) 

1054 

1055 
lemma poly_mod_mult_right: 

1056 
fixes x y z :: "'a::field poly" 

1057 
shows "x mod (y * z) = y * (x div y mod z) + x mod y" 

1058 
by (rule mod_poly_eq, rule pdivmod_rel_mult, (rule pdivmod_rel)+) 

1059 

29451  1060 
lemma mod_pCons: 
1061 
fixes a and x 

1062 
assumes y: "y \<noteq> 0" 

1063 
defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)" 

1064 
shows "(pCons a x) mod y = (pCons a (x mod y)  smult b y)" 

1065 
unfolding b 

1066 
apply (rule mod_poly_eq) 

29537  1067 
apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl]) 
29451  1068 
done 
1069 

1070 

1071 
subsection {* Evaluation of polynomials *} 

1072 

1073 
definition 

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

1074 
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

1075 
"poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)" 
29451  1076 

1077 
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

1078 
unfolding poly_def by (simp add: poly_rec_0) 
29451  1079 

1080 
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

1081 
unfolding poly_def by (simp add: poly_rec_pCons) 
29451  1082 

1083 
lemma poly_1 [simp]: "poly 1 x = 1" 

1084 
unfolding one_poly_def by simp 

1085 

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

1086 
lemma poly_monom: 
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset

1087 
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

1088 
shows "poly (monom a n) x = a * x ^ n" 
29451  1089 
by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac) 
1090 

1091 
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x" 

1092 
apply (induct p arbitrary: q, simp) 

29667  1093 
apply (case_tac q, simp, simp add: algebra_simps) 
29451  1094 
done 
1095 

1096 
lemma poly_minus [simp]: 

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

1097 
fixes x :: "'a::comm_ring" 
29451  1098 
shows "poly ( p) x =  poly p x" 
1099 
by (induct p, simp_all) 

1100 

1101 
lemma poly_diff [simp]: 

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

1102 
fixes x :: "'a::comm_ring" 
29451  1103 
shows "poly (p  q) x = poly p x  poly q x" 
1104 
by (simp add: diff_minus) 

1105 

1106 
lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)" 

1107 
by (cases "finite A", induct set: finite, simp_all) 

1108 

1109 
lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x" 

29667  1110 
by (induct p, simp, simp add: algebra_simps) 
29451  1111 

1112 
lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x" 

29667  1113 
by (induct p, simp_all, simp add: algebra_simps) 
29451  1114 

29462  1115 
lemma poly_power [simp]: 
1116 
fixes p :: "'a::{comm_semiring_1,recpower} poly" 

1117 
shows "poly (p ^ n) x = poly p x ^ n" 

1118 
by (induct n, simp, simp add: power_Suc) 

1119 

29456  1120 

1121 
subsection {* Synthetic division *} 

1122 

1123 
definition 

1124 
synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a" 

29478  1125 
where [code del]: 
29456  1126 
"synthetic_divmod p c = 
1127 
poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p" 

1128 

1129 
definition 

1130 
synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly" 

1131 
where 

1132 
"synthetic_div p c = fst (synthetic_divmod p c)" 

1133 

1134 
lemma synthetic_divmod_0 [simp]: 

1135 
"synthetic_divmod 0 c = (0, 0)" 

1136 
unfolding synthetic_divmod_def 

1137 
by (simp add: poly_rec_0) 

1138 

1139 
lemma synthetic_divmod_pCons [simp]: 

1140 
"synthetic_divmod (pCons a p) c = 

1141 
(\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)" 

1142 
unfolding synthetic_divmod_def 

1143 
by (simp add: poly_rec_pCons) 

1144 

1145 
lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c" 

1146 
by (induct p, simp, simp add: split_def) 

1147 

1148 
lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0" 

1149 
unfolding synthetic_div_def by simp 

1150 

1151 
lemma synthetic_div_pCons [simp]: 

1152 
"synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)" 

1153 
unfolding synthetic_div_def 

1154 
by (simp add: split_def snd_synthetic_divmod) 

1155 

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

1156 
lemma synthetic_div_eq_0_iff: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1157 
"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

1158 
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

1159 

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

1160 
lemma degree_synthetic_div: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1161 
"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

1162 
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

1163 

29457  1164 
lemma synthetic_div_correct: 
29456  1165 
"p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)" 
1166 
by (induct p) simp_all 

1167 

29457  1168 
lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0" 
1169 
by (induct p arbitrary: a) simp_all 

1170 

1171 
lemma synthetic_div_unique: 

1172 
"p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c" 

1173 
apply (induct p arbitrary: q r) 

1174 
apply (simp, frule synthetic_div_unique_lemma, simp) 

1175 
apply (case_tac q, force) 

1176 
done 

1177 

1178 
lemma synthetic_div_correct': 

1179 
fixes c :: "'a::comm_ring_1" 

1180 
shows "[:c, 1:] * synthetic_div p c + [:poly p c:] = p" 

1181 
using synthetic_div_correct [of p c] 

29667  1182 
by (simp add: algebra_simps) 
29457  1183 

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

1184 
lemma poly_eq_0_iff_dvd: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1185 
fixes c :: "'a::idom" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1186 
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

1187 
proof 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1188 
assume "poly p c = 0" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1189 
with synthetic_div_correct' [of c p] 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1190 
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

1191 
then show "[:c, 1:] dvd p" .. 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1192 
next 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1193 
assume "[:c, 1:] dvd p" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1194 
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

1195 
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

1196 
qed 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1197 

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

1198 
lemma dvd_iff_poly_eq_0: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1199 
fixes c :: "'a::idom" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1200 
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

1201 
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

1202 

29462  1203 
lemma poly_roots_finite: 
1204 
fixes p :: "'a::idom poly" 

1205 
shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}" 

1206 
proof (induct n \<equiv> "degree p" arbitrary: p) 

1207 
case (0 p) 

1208 
then obtain a where "a \<noteq> 0" and "p = [:a:]" 

1209 
by (cases p, simp split: if_splits) 

1210 
then show "finite {x. poly p x = 0}" by simp 

1211 
next 

1212 
case (Suc n p) 

1213 
show "finite {x. poly p x = 0}" 

1214 
proof (cases "\<exists>x. poly p x = 0") 

1215 
case False 

1216 
then show "finite {x. poly p x = 0}" by simp 

1217 
next 

1218 
case True 

1219 
then obtain a where "poly p a = 0" .. 

1220 
then have "[:a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd) 

1221 
then obtain k where k: "p = [:a, 1:] * k" .. 

1222 
with `p \<noteq> 0` have "k \<noteq> 0" by auto 

1223 
with k have "degree p = Suc (degree k)" 

1224 
by (simp add: degree_mult_eq del: mult_pCons_left) 

1225 
with `Suc n = degree p` have "n = degree k" by simp 

1226 
with `k \<noteq> 0` have "finite {x. poly k x = 0}" by (rule Suc.hyps) 

1227 
then have "finite (insert a {x. poly k x = 0})" by simp 

1228 
then show "finite {x. poly p x = 0}" 

1229 
by (simp add: k uminus_add_conv_diff Collect_disj_eq 

1230 
del: mult_pCons_left) 

1231 
qed 

1232 
qed 

1233 

29979  1234 
lemma poly_zero: 
1235 
fixes p :: "'a::{idom,ring_char_0} poly" 

1236 
shows "poly p = poly 0 \<longleftrightarrow> p = 0" 

1237 
apply (cases "p = 0", simp_all) 

1238 
apply (drule poly_roots_finite) 

1239 
apply (auto simp add: infinite_UNIV_char_0) 

1240 
done 

1241 

1242 
lemma poly_eq_iff: 

1243 
fixes p q :: "'a::{idom,ring_char_0} poly" 

1244 
shows "poly p = poly q \<longleftrightarrow> p = q" 

1245 
using poly_zero [of "p  q"] 

1246 
by (simp add: expand_fun_eq) 

1247 

29478  1248 

29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1249 
subsection {* Order of polynomial roots *} 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1250 

d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1251 
definition 
29979  1252 
order :: "'a::idom \<Rightarrow> 'a poly \<Rightarrow> nat" 
29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1253 
where 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1254 
[code del]: 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1255 
"order a p = (LEAST n. \<not> [:a, 1:] ^ Suc n dvd p)" 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1256 

d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1257 
lemma coeff_linear_power: 
29979  1258 
fixes a :: "'a::comm_semiring_1" 
29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1259 
shows "coeff ([:a, 1:] ^ n) n = 1" 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1260 
apply (induct n, simp_all) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1261 
apply (subst coeff_eq_0) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1262 
apply (auto intro: le_less_trans degree_power_le) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1263 
done 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1264 

d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1265 
lemma degree_linear_power: 
29979  1266 
fixes a :: "'a::comm_semiring_1" 
29977
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1267 
shows "degree ([:a, 1:] ^ n) = n" 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1268 
apply (rule order_antisym) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1269 
apply (rule ord_le_eq_trans [OF degree_power_le], simp) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1270 
apply (rule le_degree, simp add: coeff_linear_power) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1271 
done 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1272 

d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1273 
lemma order_1: "[:a, 1:] ^ order a p dvd p" 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1274 
apply (cases "p = 0", simp) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1275 
apply (cases "order a p", simp) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1276 
apply (subgoal_tac "nat < (LEAST n. \<not> [:a, 1:] ^ Suc n dvd p)") 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1277 
apply (drule not_less_Least, simp) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1278 
apply (fold order_def, simp) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1279 
done 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1280 

d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1281 
lemma order_2: "p \<noteq> 0 \<Longrightarrow> \<not> [:a, 1:] ^ Suc (order a p) dvd p" 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1282 
unfolding order_def 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1283 
apply (rule LeastI_ex) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1284 
apply (rule_tac x="degree p" in exI) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1285 
apply (rule notI) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1286 
apply (drule (1) dvd_imp_degree_le) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1287 
apply (simp only: degree_linear_power) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1288 
done 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1289 

d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1290 
lemma order: 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1291 
"p \<noteq> 0 \<Longrightarrow> [:a, 1:] ^ order a p dvd p \<and> \<not> [:a, 1:] ^ Suc (order a p) dvd p" 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1292 
by (rule conjI [OF order_1 order_2]) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1293 

d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1294 
lemma order_degree: 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1295 
assumes p: "p \<noteq> 0" 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1296 
shows "order a p \<le> degree p" 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1297 
proof  
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1298 
have "order a p = degree ([:a, 1:] ^ order a p)" 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1299 
by (simp only: degree_linear_power) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1300 
also have "\<dots> \<le> degree p" 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1301 
using order_1 p by (rule dvd_imp_degree_le) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1302 
finally show ?thesis . 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1303 
qed 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1304 

d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1305 
lemma order_root: "poly p a = 0 \<longleftrightarrow> p = 0 \<or> order a p \<noteq> 0" 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1306 
apply (cases "p = 0", simp_all) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1307 
apply (rule iffI) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1308 
apply (rule ccontr, simp) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1309 
apply (frule order_2 [where a=a], simp) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1310 
apply (simp add: poly_eq_0_iff_dvd) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1311 
apply (simp add: poly_eq_0_iff_dvd) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1312 
apply (simp only: order_def) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1313 
apply (drule not_less_Least, simp) 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1314 
done 
d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1315 

d76b830366bc
move polynomial order stuff from Fundamental_Theorem_Algebra to Polynomial
huffman
parents:
29904
diff
changeset

1316 

29478  1317 
subsection {* Configuration of the code generator *} 
1318 

1319 
code_datatype "0::'a::zero poly" pCons 

1320 

29480  1321 
declare pCons_0_0 [code post] 
1322 

29478  1323 
instantiation poly :: ("{zero,eq}") eq 
1324 
begin 

1325 

1326 
definition [code del]: 

1327 
"eq_class.eq (p::'a poly) q \<longleftrightarrow> p = q" 

1328 

1329 
instance 

1330 
by default (rule eq_poly_def) 

1331 

29451  1332 
end 
29478  1333 

1334 
lemma eq_poly_code [code]: 

1335 
"eq_class.eq (0::_ poly) (0::_ poly) \<longleftrightarrow> True" 

1336 
"eq_class.eq (0::_ poly) (pCons b q) \<longleftrightarrow> eq_class.eq 0 b \<and> eq_class.eq 0 q" 

1337 
"eq_class.eq (pCons a p) (0::_ poly) \<longleftrightarrow> eq_class.eq a 0 \<and> eq_class.eq p 0" 

1338 
"eq_class.eq (pCons a p) (pCons b q) \<longleftrightarrow> eq_class.eq a b \<and> eq_class.eq p q" 

1339 
unfolding eq by simp_all 

1340 

1341 
lemmas coeff_code [code] = 

1342 
coeff_0 coeff_pCons_0 coeff_pCons_Suc 

1343 

1344 
lemmas degree_code [code] = 

1345 
degree_0 degree_pCons_eq_if 

1346 

1347 
lemmas monom_poly_code [code] = 

1348 
monom_0 monom_Suc 

1349 

1350 
lemma add_poly_code [code]: 

1351 
"0 + q = (q :: _ poly)" 

1352 
"p + 0 = (p :: _ poly)" 

1353 
"pCons a p + pCons b q = pCons (a + b) (p + q)" 

1354 
by simp_all 

1355 

1356 
lemma minus_poly_code [code]: 

1357 
" 0 = (0 :: _ poly)" 

1358 
" pCons a p = pCons ( a) ( p)" 

1359 
by simp_all 

1360 

1361 
lemma diff_poly_code [code]: 

1362 
"0  q = ( q :: _ poly)" 

1363 
"p  0 = (p :: _ poly)" 

1364 
"pCons a p  pCons b q = pCons (a  b) (p  q)" 

1365 
by simp_all 

1366 

1367 
lemmas smult_poly_code [code] = 

1368 
smult_0_right smult_pCons 

1369 

1370 
lemma mult_poly_code [code]: 

1371 
"0 * q = (0 :: _ poly)" 

1372 
"pCons a p * q = smult a q + pCons 0 (p * q)" 

1373 
by simp_all 

1374 

1375 
lemmas poly_code [code] = 

1376 
poly_0 poly_pCons 

1377 

1378 
lemmas synthetic_divmod_code [code] = 

1379 
synthetic_divmod_0 synthetic_divmod_pCons 

1380 

1381 
text {* code generator setup for div and mod *} 

1382 

1383 
definition 

29537  1384 
pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly" 
29478  1385 
where 
29537  1386 
[code del]: "pdivmod x y = (x div y, x mod y)" 
29478  1387 

29537  1388 
lemma div_poly_code [code]: "x div y = fst (pdivmod x y)" 
1389 
unfolding pdivmod_def by simp 

29478  1390 

29537  1391 
lemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)" 
1392 
unfolding pdivmod_def by simp 

29478  1393 

29537  1394 
lemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)" 
1395 
unfolding pdivmod_def by simp 

29478  1396 

29537  1397 
lemma pdivmod_pCons [code]: 
1398 
"pdivmod (pCons a x) y = 

29478  1399 
(if y = 0 then (0, pCons a x) else 
29537  1400 
(let (q, r) = pdivmod x y; 
29478  1401 
b = coeff (pCons a r) (degree y) / coeff y (degree y) 
1402 
in (pCons b q, pCons a r  smult b y)))" 

29537  1403 
apply (simp add: pdivmod_def Let_def, safe) 
29478  1404 
apply (rule div_poly_eq) 
29537  1405 
apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) 
29478  1406 
apply (rule mod_poly_eq) 
29537  1407 
apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) 
29478  1408 
done 
1409 

1410 
end 