author  huffman 
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changeset 29539  abfe2af6883e 
parent 29537  50345a0f9df8 
child 29540  8858d197a9b6 
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_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" 

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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" 
449 
by (rule poly_ext, simp) 

450 

451 
lemma smult_diff_right: 

452 
"smult (a::'a::comm_ring) (p  q) = smult a p  smult a q" 

453 
by (rule poly_ext, simp add: ring_simps) 

454 

455 
lemma smult_diff_left: 

456 
"smult (a  b::'a::comm_ring) p = smult a p  smult b p" 

457 
by (rule poly_ext, simp add: ring_simps) 

458 

29472  459 
lemmas smult_distribs = 
460 
smult_add_left smult_add_right 

461 
smult_diff_left smult_diff_right 

462 

29451  463 
lemma smult_pCons [simp]: 
464 
"smult a (pCons b p) = pCons (a * b) (smult a p)" 

465 
by (rule poly_ext, simp add: coeff_pCons split: nat.split) 

466 

467 
lemma smult_monom: "smult a (monom b n) = monom (a * b) n" 

468 
by (induct n, simp add: monom_0, simp add: monom_Suc) 

469 

470 

471 
subsection {* Multiplication of polynomials *} 

472 

29474  473 
text {* TODO: move to SetInterval.thy *} 
29451  474 
lemma setsum_atMost_Suc_shift: 
475 
fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add" 

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

477 
proof (induct n) 

478 
case 0 show ?case by simp 

479 
next 

480 
case (Suc n) note IH = this 

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

482 
by (rule setsum_atMost_Suc) 

483 
also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))" 

484 
by (rule IH) 

485 
also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = 

486 
f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))" 

487 
by (rule add_assoc) 

488 
also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))" 

489 
by (rule setsum_atMost_Suc [symmetric]) 

490 
finally show ?case . 

491 
qed 

492 

493 
instantiation poly :: (comm_semiring_0) comm_semiring_0 

494 
begin 

495 

496 
definition 

29475  497 
times_poly_def [code del]: 
29474  498 
"p * q = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p" 
499 

500 
lemma mult_poly_0_left: "(0::'a poly) * q = 0" 

501 
unfolding times_poly_def by (simp add: poly_rec_0) 

502 

503 
lemma mult_pCons_left [simp]: 

504 
"pCons a p * q = smult a q + pCons 0 (p * q)" 

505 
unfolding times_poly_def by (simp add: poly_rec_pCons) 

506 

507 
lemma mult_poly_0_right: "p * (0::'a poly) = 0" 

508 
by (induct p, simp add: mult_poly_0_left, simp) 

29451  509 

29474  510 
lemma mult_pCons_right [simp]: 
511 
"p * pCons a q = smult a p + pCons 0 (p * q)" 

512 
by (induct p, simp add: mult_poly_0_left, simp add: ring_simps) 

513 

514 
lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right 

515 

516 
lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)" 

517 
by (induct p, simp add: mult_poly_0, simp add: smult_add_right) 

518 

519 
lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)" 

520 
by (induct q, simp add: mult_poly_0, simp add: smult_add_right) 

521 

522 
lemma mult_poly_add_left: 

523 
fixes p q r :: "'a poly" 

524 
shows "(p + q) * r = p * r + q * r" 

525 
by (induct r, simp add: mult_poly_0, 

526 
simp add: smult_distribs group_simps) 

29451  527 

528 
instance proof 

529 
fix p q r :: "'a poly" 

530 
show 0: "0 * p = 0" 

29474  531 
by (rule mult_poly_0_left) 
29451  532 
show "p * 0 = 0" 
29474  533 
by (rule mult_poly_0_right) 
29451  534 
show "(p + q) * r = p * r + q * r" 
29474  535 
by (rule mult_poly_add_left) 
29451  536 
show "(p * q) * r = p * (q * r)" 
29474  537 
by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left) 
29451  538 
show "p * q = q * p" 
29474  539 
by (induct p, simp add: mult_poly_0, simp) 
29451  540 
qed 
541 

542 
end 

543 

29474  544 
lemma coeff_mult: 
545 
"coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (ni))" 

546 
proof (induct p arbitrary: n) 

547 
case 0 show ?case by simp 

548 
next 

549 
case (pCons a p n) thus ?case 

550 
by (cases n, simp, simp add: setsum_atMost_Suc_shift 

551 
del: setsum_atMost_Suc) 

552 
qed 

29451  553 

29474  554 
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q" 
555 
apply (rule degree_le) 

556 
apply (induct p) 

557 
apply simp 

558 
apply (simp add: coeff_eq_0 coeff_pCons split: nat.split) 

29451  559 
done 
560 

561 
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)" 

562 
by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc) 

563 

564 

565 
subsection {* The unit polynomial and exponentiation *} 

566 

567 
instantiation poly :: (comm_semiring_1) comm_semiring_1 

568 
begin 

569 

570 
definition 

571 
one_poly_def: 

572 
"1 = pCons 1 0" 

573 

574 
instance proof 

575 
fix p :: "'a poly" show "1 * p = p" 

576 
unfolding one_poly_def 

577 
by simp 

578 
next 

579 
show "0 \<noteq> (1::'a poly)" 

580 
unfolding one_poly_def by simp 

581 
qed 

582 

583 
end 

584 

585 
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)" 

586 
unfolding one_poly_def 

587 
by (simp add: coeff_pCons split: nat.split) 

588 

589 
lemma degree_1 [simp]: "degree 1 = 0" 

590 
unfolding one_poly_def 

591 
by (rule degree_pCons_0) 

592 

593 
instantiation poly :: (comm_semiring_1) recpower 

594 
begin 

595 

596 
primrec power_poly where 

597 
power_poly_0: "(p::'a poly) ^ 0 = 1" 

598 
 power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n" 

599 

600 
instance 

601 
by default simp_all 

602 

603 
end 

604 

605 
instance poly :: (comm_ring) comm_ring .. 

606 

607 
instance poly :: (comm_ring_1) comm_ring_1 .. 

608 

609 
instantiation poly :: (comm_ring_1) number_ring 

610 
begin 

611 

612 
definition 

613 
"number_of k = (of_int k :: 'a poly)" 

614 

615 
instance 

616 
by default (rule number_of_poly_def) 

617 

618 
end 

619 

620 

621 
subsection {* Polynomials form an integral domain *} 

622 

623 
lemma coeff_mult_degree_sum: 

624 
"coeff (p * q) (degree p + degree q) = 

625 
coeff p (degree p) * coeff q (degree q)" 

29471  626 
by (induct p, simp, simp add: coeff_eq_0) 
29451  627 

628 
instance poly :: (idom) idom 

629 
proof 

630 
fix p q :: "'a poly" 

631 
assume "p \<noteq> 0" and "q \<noteq> 0" 

632 
have "coeff (p * q) (degree p + degree q) = 

633 
coeff p (degree p) * coeff q (degree q)" 

634 
by (rule coeff_mult_degree_sum) 

635 
also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0" 

636 
using `p \<noteq> 0` and `q \<noteq> 0` by simp 

637 
finally have "\<exists>n. coeff (p * q) n \<noteq> 0" .. 

638 
thus "p * q \<noteq> 0" by (simp add: expand_poly_eq) 

639 
qed 

640 

641 
lemma degree_mult_eq: 

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

643 
shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q" 

644 
apply (rule order_antisym [OF degree_mult_le le_degree]) 

645 
apply (simp add: coeff_mult_degree_sum) 

646 
done 

647 

648 
lemma dvd_imp_degree_le: 

649 
fixes p q :: "'a::idom poly" 

650 
shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q" 

651 
by (erule dvdE, simp add: degree_mult_eq) 

652 

653 

654 
subsection {* Long division of polynomials *} 

655 

656 
definition 

29537  657 
pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool" 
29451  658 
where 
29475  659 
[code del]: 
29537  660 
"pdivmod_rel x y q r \<longleftrightarrow> 
29451  661 
x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)" 
662 

29537  663 
lemma pdivmod_rel_0: 
664 
"pdivmod_rel 0 y 0 0" 

665 
unfolding pdivmod_rel_def by simp 

29451  666 

29537  667 
lemma pdivmod_rel_by_0: 
668 
"pdivmod_rel x 0 0 x" 

669 
unfolding pdivmod_rel_def by simp 

29451  670 

671 
lemma eq_zero_or_degree_less: 

672 
assumes "degree p \<le> n" and "coeff p n = 0" 

673 
shows "p = 0 \<or> degree p < n" 

674 
proof (cases n) 

675 
case 0 

676 
with `degree p \<le> n` and `coeff p n = 0` 

677 
have "coeff p (degree p) = 0" by simp 

678 
then have "p = 0" by simp 

679 
then show ?thesis .. 

680 
next 

681 
case (Suc m) 

682 
have "\<forall>i>n. coeff p i = 0" 

683 
using `degree p \<le> n` by (simp add: coeff_eq_0) 

684 
then have "\<forall>i\<ge>n. coeff p i = 0" 

685 
using `coeff p n = 0` by (simp add: le_less) 

686 
then have "\<forall>i>m. coeff p i = 0" 

687 
using `n = Suc m` by (simp add: less_eq_Suc_le) 

688 
then have "degree p \<le> m" 

689 
by (rule degree_le) 

690 
then have "degree p < n" 

691 
using `n = Suc m` by (simp add: less_Suc_eq_le) 

692 
then show ?thesis .. 

693 
qed 

694 

29537  695 
lemma pdivmod_rel_pCons: 
696 
assumes rel: "pdivmod_rel x y q r" 

29451  697 
assumes y: "y \<noteq> 0" 
698 
assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)" 

29537  699 
shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r  smult b y)" 
700 
(is "pdivmod_rel ?x y ?q ?r") 

29451  701 
proof  
702 
have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y" 

29537  703 
using assms unfolding pdivmod_rel_def by simp_all 
29451  704 

705 
have 1: "?x = ?q * y + ?r" 

706 
using b x by simp 

707 

708 
have 2: "?r = 0 \<or> degree ?r < degree y" 

709 
proof (rule eq_zero_or_degree_less) 

29539  710 
show "degree ?r \<le> degree y" 
711 
proof (rule degree_diff_le) 

29451  712 
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

713 
using r by auto 
29451  714 
show "degree (smult b y) \<le> degree y" 
715 
by (rule degree_smult_le) 

716 
qed 

717 
next 

718 
show "coeff ?r (degree y) = 0" 

719 
using `y \<noteq> 0` unfolding b by simp 

720 
qed 

721 

722 
from 1 2 show ?thesis 

29537  723 
unfolding pdivmod_rel_def 
29451  724 
using `y \<noteq> 0` by simp 
725 
qed 

726 

29537  727 
lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r" 
29451  728 
apply (cases "y = 0") 
29537  729 
apply (fast intro!: pdivmod_rel_by_0) 
29451  730 
apply (induct x) 
29537  731 
apply (fast intro!: pdivmod_rel_0) 
732 
apply (fast intro!: pdivmod_rel_pCons) 

29451  733 
done 
734 

29537  735 
lemma pdivmod_rel_unique: 
736 
assumes 1: "pdivmod_rel x y q1 r1" 

737 
assumes 2: "pdivmod_rel x y q2 r2" 

29451  738 
shows "q1 = q2 \<and> r1 = r2" 
739 
proof (cases "y = 0") 

740 
assume "y = 0" with assms show ?thesis 

29537  741 
by (simp add: pdivmod_rel_def) 
29451  742 
next 
743 
assume [simp]: "y \<noteq> 0" 

744 
from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y" 

29537  745 
unfolding pdivmod_rel_def by simp_all 
29451  746 
from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y" 
29537  747 
unfolding pdivmod_rel_def by simp_all 
29451  748 
from q1 q2 have q3: "(q1  q2) * y = r2  r1" 
749 
by (simp add: ring_simps) 

750 
from r1 r2 have r3: "(r2  r1) = 0 \<or> degree (r2  r1) < degree y" 

29453  751 
by (auto intro: degree_diff_less) 
29451  752 

753 
show "q1 = q2 \<and> r1 = r2" 

754 
proof (rule ccontr) 

755 
assume "\<not> (q1 = q2 \<and> r1 = r2)" 

756 
with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto 

757 
with r3 have "degree (r2  r1) < degree y" by simp 

758 
also have "degree y \<le> degree (q1  q2) + degree y" by simp 

759 
also have "\<dots> = degree ((q1  q2) * y)" 

760 
using `q1 \<noteq> q2` by (simp add: degree_mult_eq) 

761 
also have "\<dots> = degree (r2  r1)" 

762 
using q3 by simp 

763 
finally have "degree (r2  r1) < degree (r2  r1)" . 

764 
then show "False" by simp 

765 
qed 

766 
qed 

767 

29537  768 
lemmas pdivmod_rel_unique_div = 
769 
pdivmod_rel_unique [THEN conjunct1, standard] 

29451  770 

29537  771 
lemmas pdivmod_rel_unique_mod = 
772 
pdivmod_rel_unique [THEN conjunct2, standard] 

29451  773 

774 
instantiation poly :: (field) ring_div 

775 
begin 

776 

777 
definition div_poly where 

29537  778 
[code del]: "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)" 
29451  779 

780 
definition mod_poly where 

29537  781 
[code del]: "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)" 
29451  782 

783 
lemma div_poly_eq: 

29537  784 
"pdivmod_rel x y q r \<Longrightarrow> x div y = q" 
29451  785 
unfolding div_poly_def 
29537  786 
by (fast elim: pdivmod_rel_unique_div) 
29451  787 

788 
lemma mod_poly_eq: 

29537  789 
"pdivmod_rel x y q r \<Longrightarrow> x mod y = r" 
29451  790 
unfolding mod_poly_def 
29537  791 
by (fast elim: pdivmod_rel_unique_mod) 
29451  792 

29537  793 
lemma pdivmod_rel: 
794 
"pdivmod_rel x y (x div y) (x mod y)" 

29451  795 
proof  
29537  796 
from pdivmod_rel_exists 
797 
obtain q r where "pdivmod_rel x y q r" by fast 

29451  798 
thus ?thesis 
799 
by (simp add: div_poly_eq mod_poly_eq) 

800 
qed 

801 

802 
instance proof 

803 
fix x y :: "'a poly" 

804 
show "x div y * y + x mod y = x" 

29537  805 
using pdivmod_rel [of x y] 
806 
by (simp add: pdivmod_rel_def) 

29451  807 
next 
808 
fix x :: "'a poly" 

29537  809 
have "pdivmod_rel x 0 0 x" 
810 
by (rule pdivmod_rel_by_0) 

29451  811 
thus "x div 0 = 0" 
812 
by (rule div_poly_eq) 

813 
next 

814 
fix y :: "'a poly" 

29537  815 
have "pdivmod_rel 0 y 0 0" 
816 
by (rule pdivmod_rel_0) 

29451  817 
thus "0 div y = 0" 
818 
by (rule div_poly_eq) 

819 
next 

820 
fix x y z :: "'a poly" 

821 
assume "y \<noteq> 0" 

29537  822 
hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)" 
823 
using pdivmod_rel [of x y] 

824 
by (simp add: pdivmod_rel_def left_distrib) 

29451  825 
thus "(x + z * y) div y = z + x div y" 
826 
by (rule div_poly_eq) 

827 
qed 

828 

829 
end 

830 

831 
lemma degree_mod_less: 

832 
"y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y" 

29537  833 
using pdivmod_rel [of x y] 
834 
unfolding pdivmod_rel_def by simp 

29451  835 

836 
lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0" 

837 
proof  

838 
assume "degree x < degree y" 

29537  839 
hence "pdivmod_rel x y 0 x" 
840 
by (simp add: pdivmod_rel_def) 

29451  841 
thus "x div y = 0" by (rule div_poly_eq) 
842 
qed 

843 

844 
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x" 

845 
proof  

846 
assume "degree x < degree y" 

29537  847 
hence "pdivmod_rel x y 0 x" 
848 
by (simp add: pdivmod_rel_def) 

29451  849 
thus "x mod y = x" by (rule mod_poly_eq) 
850 
qed 

851 

852 
lemma mod_pCons: 

853 
fixes a and x 

854 
assumes y: "y \<noteq> 0" 

855 
defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)" 

856 
shows "(pCons a x) mod y = (pCons a (x mod y)  smult b y)" 

857 
unfolding b 

858 
apply (rule mod_poly_eq) 

29537  859 
apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl]) 
29451  860 
done 
861 

862 

863 
subsection {* Evaluation of polynomials *} 

864 

865 
definition 

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

866 
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

867 
"poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)" 
29451  868 

869 
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

870 
unfolding poly_def by (simp add: poly_rec_0) 
29451  871 

872 
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

873 
unfolding poly_def by (simp add: poly_rec_pCons) 
29451  874 

875 
lemma poly_1 [simp]: "poly 1 x = 1" 

876 
unfolding one_poly_def by simp 

877 

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

878 
lemma poly_monom: 
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset

879 
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

880 
shows "poly (monom a n) x = a * x ^ n" 
29451  881 
by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac) 
882 

883 
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x" 

884 
apply (induct p arbitrary: q, simp) 

885 
apply (case_tac q, simp, simp add: ring_simps) 

886 
done 

887 

888 
lemma poly_minus [simp]: 

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

889 
fixes x :: "'a::comm_ring" 
29451  890 
shows "poly ( p) x =  poly p x" 
891 
by (induct p, simp_all) 

892 

893 
lemma poly_diff [simp]: 

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

894 
fixes x :: "'a::comm_ring" 
29451  895 
shows "poly (p  q) x = poly p x  poly q x" 
896 
by (simp add: diff_minus) 

897 

898 
lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)" 

899 
by (cases "finite A", induct set: finite, simp_all) 

900 

901 
lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x" 

902 
by (induct p, simp, simp add: ring_simps) 

903 

904 
lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x" 

905 
by (induct p, simp_all, simp add: ring_simps) 

906 

29462  907 
lemma poly_power [simp]: 
908 
fixes p :: "'a::{comm_semiring_1,recpower} poly" 

909 
shows "poly (p ^ n) x = poly p x ^ n" 

910 
by (induct n, simp, simp add: power_Suc) 

911 

29456  912 

913 
subsection {* Synthetic division *} 

914 

915 
definition 

916 
synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a" 

29478  917 
where [code del]: 
29456  918 
"synthetic_divmod p c = 
919 
poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p" 

920 

921 
definition 

922 
synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly" 

923 
where 

924 
"synthetic_div p c = fst (synthetic_divmod p c)" 

925 

926 
lemma synthetic_divmod_0 [simp]: 

927 
"synthetic_divmod 0 c = (0, 0)" 

928 
unfolding synthetic_divmod_def 

929 
by (simp add: poly_rec_0) 

930 

931 
lemma synthetic_divmod_pCons [simp]: 

932 
"synthetic_divmod (pCons a p) c = 

933 
(\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)" 

934 
unfolding synthetic_divmod_def 

935 
by (simp add: poly_rec_pCons) 

936 

937 
lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c" 

938 
by (induct p, simp, simp add: split_def) 

939 

940 
lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0" 

941 
unfolding synthetic_div_def by simp 

942 

943 
lemma synthetic_div_pCons [simp]: 

944 
"synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)" 

945 
unfolding synthetic_div_def 

946 
by (simp add: split_def snd_synthetic_divmod) 

947 

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

948 
lemma synthetic_div_eq_0_iff: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

949 
"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

950 
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

951 

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

952 
lemma degree_synthetic_div: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

953 
"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

954 
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

955 

29457  956 
lemma synthetic_div_correct: 
29456  957 
"p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)" 
958 
by (induct p) simp_all 

959 

29457  960 
lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0" 
961 
by (induct p arbitrary: a) simp_all 

962 

963 
lemma synthetic_div_unique: 

964 
"p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c" 

965 
apply (induct p arbitrary: q r) 

966 
apply (simp, frule synthetic_div_unique_lemma, simp) 

967 
apply (case_tac q, force) 

968 
done 

969 

970 
lemma synthetic_div_correct': 

971 
fixes c :: "'a::comm_ring_1" 

972 
shows "[:c, 1:] * synthetic_div p c + [:poly p c:] = p" 

973 
using synthetic_div_correct [of p c] 

974 
by (simp add: group_simps) 

975 

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

976 
lemma poly_eq_0_iff_dvd: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

977 
fixes c :: "'a::idom" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

978 
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

979 
proof 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

980 
assume "poly p c = 0" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

981 
with synthetic_div_correct' [of c p] 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

982 
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

983 
then show "[:c, 1:] dvd p" .. 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

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

985 
assume "[:c, 1:] dvd p" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

986 
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

987 
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

988 
qed 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

989 

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

990 
lemma dvd_iff_poly_eq_0: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

991 
fixes c :: "'a::idom" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

992 
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

993 
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

994 

29462  995 
lemma poly_roots_finite: 
996 
fixes p :: "'a::idom poly" 

997 
shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}" 

998 
proof (induct n \<equiv> "degree p" arbitrary: p) 

999 
case (0 p) 

1000 
then obtain a where "a \<noteq> 0" and "p = [:a:]" 

1001 
by (cases p, simp split: if_splits) 

1002 
then show "finite {x. poly p x = 0}" by simp 

1003 
next 

1004 
case (Suc n p) 

1005 
show "finite {x. poly p x = 0}" 

1006 
proof (cases "\<exists>x. poly p x = 0") 

1007 
case False 

1008 
then show "finite {x. poly p x = 0}" by simp 

1009 
next 

1010 
case True 

1011 
then obtain a where "poly p a = 0" .. 

1012 
then have "[:a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd) 

1013 
then obtain k where k: "p = [:a, 1:] * k" .. 

1014 
with `p \<noteq> 0` have "k \<noteq> 0" by auto 

1015 
with k have "degree p = Suc (degree k)" 

1016 
by (simp add: degree_mult_eq del: mult_pCons_left) 

1017 
with `Suc n = degree p` have "n = degree k" by simp 

1018 
with `k \<noteq> 0` have "finite {x. poly k x = 0}" by (rule Suc.hyps) 

1019 
then have "finite (insert a {x. poly k x = 0})" by simp 

1020 
then show "finite {x. poly p x = 0}" 

1021 
by (simp add: k uminus_add_conv_diff Collect_disj_eq 

1022 
del: mult_pCons_left) 

1023 
qed 

1024 
qed 

1025 

29478  1026 

1027 
subsection {* Configuration of the code generator *} 

1028 

1029 
code_datatype "0::'a::zero poly" pCons 

1030 

29480  1031 
declare pCons_0_0 [code post] 
1032 

29478  1033 
instantiation poly :: ("{zero,eq}") eq 
1034 
begin 

1035 

1036 
definition [code del]: 

1037 
"eq_class.eq (p::'a poly) q \<longleftrightarrow> p = q" 

1038 

1039 
instance 

1040 
by default (rule eq_poly_def) 

1041 

29451  1042 
end 
29478  1043 

1044 
lemma eq_poly_code [code]: 

1045 
"eq_class.eq (0::_ poly) (0::_ poly) \<longleftrightarrow> True" 

1046 
"eq_class.eq (0::_ poly) (pCons b q) \<longleftrightarrow> eq_class.eq 0 b \<and> eq_class.eq 0 q" 

1047 
"eq_class.eq (pCons a p) (0::_ poly) \<longleftrightarrow> eq_class.eq a 0 \<and> eq_class.eq p 0" 

1048 
"eq_class.eq (pCons a p) (pCons b q) \<longleftrightarrow> eq_class.eq a b \<and> eq_class.eq p q" 

1049 
unfolding eq by simp_all 

1050 

1051 
lemmas coeff_code [code] = 

1052 
coeff_0 coeff_pCons_0 coeff_pCons_Suc 

1053 

1054 
lemmas degree_code [code] = 

1055 
degree_0 degree_pCons_eq_if 

1056 

1057 
lemmas monom_poly_code [code] = 

1058 
monom_0 monom_Suc 

1059 

1060 
lemma add_poly_code [code]: 

1061 
"0 + q = (q :: _ poly)" 

1062 
"p + 0 = (p :: _ poly)" 

1063 
"pCons a p + pCons b q = pCons (a + b) (p + q)" 

1064 
by simp_all 

1065 

1066 
lemma minus_poly_code [code]: 

1067 
" 0 = (0 :: _ poly)" 

1068 
" pCons a p = pCons ( a) ( p)" 

1069 
by simp_all 

1070 

1071 
lemma diff_poly_code [code]: 

1072 
"0  q = ( q :: _ poly)" 

1073 
"p  0 = (p :: _ poly)" 

1074 
"pCons a p  pCons b q = pCons (a  b) (p  q)" 

1075 
by simp_all 

1076 

1077 
lemmas smult_poly_code [code] = 

1078 
smult_0_right smult_pCons 

1079 

1080 
lemma mult_poly_code [code]: 

1081 
"0 * q = (0 :: _ poly)" 

1082 
"pCons a p * q = smult a q + pCons 0 (p * q)" 

1083 
by simp_all 

1084 

1085 
lemmas poly_code [code] = 

1086 
poly_0 poly_pCons 

1087 

1088 
lemmas synthetic_divmod_code [code] = 

1089 
synthetic_divmod_0 synthetic_divmod_pCons 

1090 

1091 
text {* code generator setup for div and mod *} 

1092 

1093 
definition 

29537  1094 
pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly" 
29478  1095 
where 
29537  1096 
[code del]: "pdivmod x y = (x div y, x mod y)" 
29478  1097 

29537  1098 
lemma div_poly_code [code]: "x div y = fst (pdivmod x y)" 
1099 
unfolding pdivmod_def by simp 

29478  1100 

29537  1101 
lemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)" 
1102 
unfolding pdivmod_def by simp 

29478  1103 

29537  1104 
lemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)" 
1105 
unfolding pdivmod_def by simp 

29478  1106 

29537  1107 
lemma pdivmod_pCons [code]: 
1108 
"pdivmod (pCons a x) y = 

29478  1109 
(if y = 0 then (0, pCons a x) else 
29537  1110 
(let (q, r) = pdivmod x y; 
29478  1111 
b = coeff (pCons a r) (degree y) / coeff y (degree y) 
1112 
in (pCons b q, pCons a r  smult b y)))" 

29537  1113 
apply (simp add: pdivmod_def Let_def, safe) 
29478  1114 
apply (rule div_poly_eq) 
29537  1115 
apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) 
29478  1116 
apply (rule mod_poly_eq) 
29537  1117 
apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) 
29478  1118 
done 
1119 

1120 
end 