author  huffman 
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changeset 29540  8858d197a9b6 
parent 29539  abfe2af6883e 
child 29654  24e73987bfe2 
child 29659  f8d2c03ecfd8 
child 29667  53103fc8ffa3 
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 

266 

267 
definition 

268 
plus_poly_def [code del]: 

269 
"p + q = Abs_poly (\<lambda>n. coeff p n + coeff q n)" 

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271 
lemma Poly_add: 

272 
fixes f g :: "nat \<Rightarrow> 'a" 

273 
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 

275 
apply (clarify, rename_tac m n) 

276 
apply (rule_tac x="max m n" in exI, simp) 

277 
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 

285 
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) 

292 
qed 

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end 

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29540  296 
instance poly :: 
297 
("{cancel_ab_semigroup_add,comm_monoid_add}") cancel_ab_semigroup_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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29451  304 
instantiation poly :: (ab_group_add) ab_group_add 
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begin 

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307 
definition 

308 
uminus_poly_def [code del]: 

309 
" p = Abs_poly (\<lambda>n.  coeff p n)" 

310 

311 
definition 

312 
minus_poly_def [code del]: 

313 
"p  q = Abs_poly (\<lambda>n. coeff p n  coeff q n)" 

314 

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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 

415 
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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29472  431 
lemma smult_smult [simp]: "smult a (smult b p) = smult (a * b) p" 
29451  432 
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: 

448 
"smult (a + b) p = smult a p + smult b p" 

449 
by (rule poly_ext, simp add: ring_simps) 

450 

29457  451 
lemma smult_minus_right [simp]: 
29451  452 
"smult (a::'a::comm_ring) ( p) =  smult a p" 
453 
by (rule poly_ext, simp) 

454 

29457  455 
lemma smult_minus_left [simp]: 
29451  456 
"smult ( a::'a::comm_ring) p =  smult a p" 
457 
by (rule poly_ext, simp) 

458 

459 
lemma smult_diff_right: 

460 
"smult (a::'a::comm_ring) (p  q) = smult a p  smult a q" 

461 
by (rule poly_ext, simp add: ring_simps) 

462 

463 
lemma smult_diff_left: 

464 
"smult (a  b::'a::comm_ring) p = smult a p  smult b p" 

465 
by (rule poly_ext, simp add: ring_simps) 

466 

29472  467 
lemmas smult_distribs = 
468 
smult_add_left smult_add_right 

469 
smult_diff_left smult_diff_right 

470 

29451  471 
lemma smult_pCons [simp]: 
472 
"smult a (pCons b p) = pCons (a * b) (smult a p)" 

473 
by (rule poly_ext, simp add: coeff_pCons split: nat.split) 

474 

475 
lemma smult_monom: "smult a (monom b n) = monom (a * b) n" 

476 
by (induct n, simp add: monom_0, simp add: monom_Suc) 

477 

478 

479 
subsection {* Multiplication of polynomials *} 

480 

29474  481 
text {* TODO: move to SetInterval.thy *} 
29451  482 
lemma setsum_atMost_Suc_shift: 
483 
fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add" 

484 
shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))" 

485 
proof (induct n) 

486 
case 0 show ?case by simp 

487 
next 

488 
case (Suc n) note IH = this 

489 
have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))" 

490 
by (rule setsum_atMost_Suc) 

491 
also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))" 

492 
by (rule IH) 

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

494 
f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))" 

495 
by (rule add_assoc) 

496 
also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))" 

497 
by (rule setsum_atMost_Suc [symmetric]) 

498 
finally show ?case . 

499 
qed 

500 

501 
instantiation poly :: (comm_semiring_0) comm_semiring_0 

502 
begin 

503 

504 
definition 

29475  505 
times_poly_def [code del]: 
29474  506 
"p * q = poly_rec 0 (\<lambda>a p pq. smult a q + pCons 0 pq) p" 
507 

508 
lemma mult_poly_0_left: "(0::'a poly) * q = 0" 

509 
unfolding times_poly_def by (simp add: poly_rec_0) 

510 

511 
lemma mult_pCons_left [simp]: 

512 
"pCons a p * q = smult a q + pCons 0 (p * q)" 

513 
unfolding times_poly_def by (simp add: poly_rec_pCons) 

514 

515 
lemma mult_poly_0_right: "p * (0::'a poly) = 0" 

516 
by (induct p, simp add: mult_poly_0_left, simp) 

29451  517 

29474  518 
lemma mult_pCons_right [simp]: 
519 
"p * pCons a q = smult a p + pCons 0 (p * q)" 

520 
by (induct p, simp add: mult_poly_0_left, simp add: ring_simps) 

521 

522 
lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right 

523 

524 
lemma mult_smult_left [simp]: "smult a p * q = smult a (p * q)" 

525 
by (induct p, simp add: mult_poly_0, simp add: smult_add_right) 

526 

527 
lemma mult_smult_right [simp]: "p * smult a q = smult a (p * q)" 

528 
by (induct q, simp add: mult_poly_0, simp add: smult_add_right) 

529 

530 
lemma mult_poly_add_left: 

531 
fixes p q r :: "'a poly" 

532 
shows "(p + q) * r = p * r + q * r" 

533 
by (induct r, simp add: mult_poly_0, 

534 
simp add: smult_distribs group_simps) 

29451  535 

536 
instance proof 

537 
fix p q r :: "'a poly" 

538 
show 0: "0 * p = 0" 

29474  539 
by (rule mult_poly_0_left) 
29451  540 
show "p * 0 = 0" 
29474  541 
by (rule mult_poly_0_right) 
29451  542 
show "(p + q) * r = p * r + q * r" 
29474  543 
by (rule mult_poly_add_left) 
29451  544 
show "(p * q) * r = p * (q * r)" 
29474  545 
by (induct p, simp add: mult_poly_0, simp add: mult_poly_add_left) 
29451  546 
show "p * q = q * p" 
29474  547 
by (induct p, simp add: mult_poly_0, simp) 
29451  548 
qed 
549 

550 
end 

551 

29540  552 
instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel .. 
553 

29474  554 
lemma coeff_mult: 
555 
"coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (ni))" 

556 
proof (induct p arbitrary: n) 

557 
case 0 show ?case by simp 

558 
next 

559 
case (pCons a p n) thus ?case 

560 
by (cases n, simp, simp add: setsum_atMost_Suc_shift 

561 
del: setsum_atMost_Suc) 

562 
qed 

29451  563 

29474  564 
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q" 
565 
apply (rule degree_le) 

566 
apply (induct p) 

567 
apply simp 

568 
apply (simp add: coeff_eq_0 coeff_pCons split: nat.split) 

29451  569 
done 
570 

571 
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)" 

572 
by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc) 

573 

574 

575 
subsection {* The unit polynomial and exponentiation *} 

576 

577 
instantiation poly :: (comm_semiring_1) comm_semiring_1 

578 
begin 

579 

580 
definition 

581 
one_poly_def: 

582 
"1 = pCons 1 0" 

583 

584 
instance proof 

585 
fix p :: "'a poly" show "1 * p = p" 

586 
unfolding one_poly_def 

587 
by simp 

588 
next 

589 
show "0 \<noteq> (1::'a poly)" 

590 
unfolding one_poly_def by simp 

591 
qed 

592 

593 
end 

594 

29540  595 
instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel .. 
596 

29451  597 
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)" 
598 
unfolding one_poly_def 

599 
by (simp add: coeff_pCons split: nat.split) 

600 

601 
lemma degree_1 [simp]: "degree 1 = 0" 

602 
unfolding one_poly_def 

603 
by (rule degree_pCons_0) 

604 

605 
instantiation poly :: (comm_semiring_1) recpower 

606 
begin 

607 

608 
primrec power_poly where 

609 
power_poly_0: "(p::'a poly) ^ 0 = 1" 

610 
 power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n" 

611 

612 
instance 

613 
by default simp_all 

614 

615 
end 

616 

617 
instance poly :: (comm_ring) comm_ring .. 

618 

619 
instance poly :: (comm_ring_1) comm_ring_1 .. 

620 

621 
instantiation poly :: (comm_ring_1) number_ring 

622 
begin 

623 

624 
definition 

625 
"number_of k = (of_int k :: 'a poly)" 

626 

627 
instance 

628 
by default (rule number_of_poly_def) 

629 

630 
end 

631 

632 

633 
subsection {* Polynomials form an integral domain *} 

634 

635 
lemma coeff_mult_degree_sum: 

636 
"coeff (p * q) (degree p + degree q) = 

637 
coeff p (degree p) * coeff q (degree q)" 

29471  638 
by (induct p, simp, simp add: coeff_eq_0) 
29451  639 

640 
instance poly :: (idom) idom 

641 
proof 

642 
fix p q :: "'a poly" 

643 
assume "p \<noteq> 0" and "q \<noteq> 0" 

644 
have "coeff (p * q) (degree p + degree q) = 

645 
coeff p (degree p) * coeff q (degree q)" 

646 
by (rule coeff_mult_degree_sum) 

647 
also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0" 

648 
using `p \<noteq> 0` and `q \<noteq> 0` by simp 

649 
finally have "\<exists>n. coeff (p * q) n \<noteq> 0" .. 

650 
thus "p * q \<noteq> 0" by (simp add: expand_poly_eq) 

651 
qed 

652 

653 
lemma degree_mult_eq: 

654 
fixes p q :: "'a::idom poly" 

655 
shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q" 

656 
apply (rule order_antisym [OF degree_mult_le le_degree]) 

657 
apply (simp add: coeff_mult_degree_sum) 

658 
done 

659 

660 
lemma dvd_imp_degree_le: 

661 
fixes p q :: "'a::idom poly" 

662 
shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q" 

663 
by (erule dvdE, simp add: degree_mult_eq) 

664 

665 

666 
subsection {* Long division of polynomials *} 

667 

668 
definition 

29537  669 
pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool" 
29451  670 
where 
29475  671 
[code del]: 
29537  672 
"pdivmod_rel x y q r \<longleftrightarrow> 
29451  673 
x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)" 
674 

29537  675 
lemma pdivmod_rel_0: 
676 
"pdivmod_rel 0 y 0 0" 

677 
unfolding pdivmod_rel_def by simp 

29451  678 

29537  679 
lemma pdivmod_rel_by_0: 
680 
"pdivmod_rel x 0 0 x" 

681 
unfolding pdivmod_rel_def by simp 

29451  682 

683 
lemma eq_zero_or_degree_less: 

684 
assumes "degree p \<le> n" and "coeff p n = 0" 

685 
shows "p = 0 \<or> degree p < n" 

686 
proof (cases n) 

687 
case 0 

688 
with `degree p \<le> n` and `coeff p n = 0` 

689 
have "coeff p (degree p) = 0" by simp 

690 
then have "p = 0" by simp 

691 
then show ?thesis .. 

692 
next 

693 
case (Suc m) 

694 
have "\<forall>i>n. coeff p i = 0" 

695 
using `degree p \<le> n` by (simp add: coeff_eq_0) 

696 
then have "\<forall>i\<ge>n. coeff p i = 0" 

697 
using `coeff p n = 0` by (simp add: le_less) 

698 
then have "\<forall>i>m. coeff p i = 0" 

699 
using `n = Suc m` by (simp add: less_eq_Suc_le) 

700 
then have "degree p \<le> m" 

701 
by (rule degree_le) 

702 
then have "degree p < n" 

703 
using `n = Suc m` by (simp add: less_Suc_eq_le) 

704 
then show ?thesis .. 

705 
qed 

706 

29537  707 
lemma pdivmod_rel_pCons: 
708 
assumes rel: "pdivmod_rel x y q r" 

29451  709 
assumes y: "y \<noteq> 0" 
710 
assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)" 

29537  711 
shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r  smult b y)" 
712 
(is "pdivmod_rel ?x y ?q ?r") 

29451  713 
proof  
714 
have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y" 

29537  715 
using assms unfolding pdivmod_rel_def by simp_all 
29451  716 

717 
have 1: "?x = ?q * y + ?r" 

718 
using b x by simp 

719 

720 
have 2: "?r = 0 \<or> degree ?r < degree y" 

721 
proof (rule eq_zero_or_degree_less) 

29539  722 
show "degree ?r \<le> degree y" 
723 
proof (rule degree_diff_le) 

29451  724 
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

725 
using r by auto 
29451  726 
show "degree (smult b y) \<le> degree y" 
727 
by (rule degree_smult_le) 

728 
qed 

729 
next 

730 
show "coeff ?r (degree y) = 0" 

731 
using `y \<noteq> 0` unfolding b by simp 

732 
qed 

733 

734 
from 1 2 show ?thesis 

29537  735 
unfolding pdivmod_rel_def 
29451  736 
using `y \<noteq> 0` by simp 
737 
qed 

738 

29537  739 
lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r" 
29451  740 
apply (cases "y = 0") 
29537  741 
apply (fast intro!: pdivmod_rel_by_0) 
29451  742 
apply (induct x) 
29537  743 
apply (fast intro!: pdivmod_rel_0) 
744 
apply (fast intro!: pdivmod_rel_pCons) 

29451  745 
done 
746 

29537  747 
lemma pdivmod_rel_unique: 
748 
assumes 1: "pdivmod_rel x y q1 r1" 

749 
assumes 2: "pdivmod_rel x y q2 r2" 

29451  750 
shows "q1 = q2 \<and> r1 = r2" 
751 
proof (cases "y = 0") 

752 
assume "y = 0" with assms show ?thesis 

29537  753 
by (simp add: pdivmod_rel_def) 
29451  754 
next 
755 
assume [simp]: "y \<noteq> 0" 

756 
from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y" 

29537  757 
unfolding pdivmod_rel_def by simp_all 
29451  758 
from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y" 
29537  759 
unfolding pdivmod_rel_def by simp_all 
29451  760 
from q1 q2 have q3: "(q1  q2) * y = r2  r1" 
761 
by (simp add: ring_simps) 

762 
from r1 r2 have r3: "(r2  r1) = 0 \<or> degree (r2  r1) < degree y" 

29453  763 
by (auto intro: degree_diff_less) 
29451  764 

765 
show "q1 = q2 \<and> r1 = r2" 

766 
proof (rule ccontr) 

767 
assume "\<not> (q1 = q2 \<and> r1 = r2)" 

768 
with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto 

769 
with r3 have "degree (r2  r1) < degree y" by simp 

770 
also have "degree y \<le> degree (q1  q2) + degree y" by simp 

771 
also have "\<dots> = degree ((q1  q2) * y)" 

772 
using `q1 \<noteq> q2` by (simp add: degree_mult_eq) 

773 
also have "\<dots> = degree (r2  r1)" 

774 
using q3 by simp 

775 
finally have "degree (r2  r1) < degree (r2  r1)" . 

776 
then show "False" by simp 

777 
qed 

778 
qed 

779 

29537  780 
lemmas pdivmod_rel_unique_div = 
781 
pdivmod_rel_unique [THEN conjunct1, standard] 

29451  782 

29537  783 
lemmas pdivmod_rel_unique_mod = 
784 
pdivmod_rel_unique [THEN conjunct2, standard] 

29451  785 

786 
instantiation poly :: (field) ring_div 

787 
begin 

788 

789 
definition div_poly where 

29537  790 
[code del]: "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)" 
29451  791 

792 
definition mod_poly where 

29537  793 
[code del]: "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)" 
29451  794 

795 
lemma div_poly_eq: 

29537  796 
"pdivmod_rel x y q r \<Longrightarrow> x div y = q" 
29451  797 
unfolding div_poly_def 
29537  798 
by (fast elim: pdivmod_rel_unique_div) 
29451  799 

800 
lemma mod_poly_eq: 

29537  801 
"pdivmod_rel x y q r \<Longrightarrow> x mod y = r" 
29451  802 
unfolding mod_poly_def 
29537  803 
by (fast elim: pdivmod_rel_unique_mod) 
29451  804 

29537  805 
lemma pdivmod_rel: 
806 
"pdivmod_rel x y (x div y) (x mod y)" 

29451  807 
proof  
29537  808 
from pdivmod_rel_exists 
809 
obtain q r where "pdivmod_rel x y q r" by fast 

29451  810 
thus ?thesis 
811 
by (simp add: div_poly_eq mod_poly_eq) 

812 
qed 

813 

814 
instance proof 

815 
fix x y :: "'a poly" 

816 
show "x div y * y + x mod y = x" 

29537  817 
using pdivmod_rel [of x y] 
818 
by (simp add: pdivmod_rel_def) 

29451  819 
next 
820 
fix x :: "'a poly" 

29537  821 
have "pdivmod_rel x 0 0 x" 
822 
by (rule pdivmod_rel_by_0) 

29451  823 
thus "x div 0 = 0" 
824 
by (rule div_poly_eq) 

825 
next 

826 
fix y :: "'a poly" 

29537  827 
have "pdivmod_rel 0 y 0 0" 
828 
by (rule pdivmod_rel_0) 

29451  829 
thus "0 div y = 0" 
830 
by (rule div_poly_eq) 

831 
next 

832 
fix x y z :: "'a poly" 

833 
assume "y \<noteq> 0" 

29537  834 
hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)" 
835 
using pdivmod_rel [of x y] 

836 
by (simp add: pdivmod_rel_def left_distrib) 

29451  837 
thus "(x + z * y) div y = z + x div y" 
838 
by (rule div_poly_eq) 

839 
qed 

840 

841 
end 

842 

843 
lemma degree_mod_less: 

844 
"y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y" 

29537  845 
using pdivmod_rel [of x y] 
846 
unfolding pdivmod_rel_def by simp 

29451  847 

848 
lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0" 

849 
proof  

850 
assume "degree x < degree y" 

29537  851 
hence "pdivmod_rel x y 0 x" 
852 
by (simp add: pdivmod_rel_def) 

29451  853 
thus "x div y = 0" by (rule div_poly_eq) 
854 
qed 

855 

856 
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x" 

857 
proof  

858 
assume "degree x < degree y" 

29537  859 
hence "pdivmod_rel x y 0 x" 
860 
by (simp add: pdivmod_rel_def) 

29451  861 
thus "x mod y = x" by (rule mod_poly_eq) 
862 
qed 

863 

864 
lemma mod_pCons: 

865 
fixes a and x 

866 
assumes y: "y \<noteq> 0" 

867 
defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)" 

868 
shows "(pCons a x) mod y = (pCons a (x mod y)  smult b y)" 

869 
unfolding b 

870 
apply (rule mod_poly_eq) 

29537  871 
apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl]) 
29451  872 
done 
873 

874 

875 
subsection {* Evaluation of polynomials *} 

876 

877 
definition 

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

878 
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

879 
"poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)" 
29451  880 

881 
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

882 
unfolding poly_def by (simp add: poly_rec_0) 
29451  883 

884 
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

885 
unfolding poly_def by (simp add: poly_rec_pCons) 
29451  886 

887 
lemma poly_1 [simp]: "poly 1 x = 1" 

888 
unfolding one_poly_def by simp 

889 

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

890 
lemma poly_monom: 
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset

891 
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

892 
shows "poly (monom a n) x = a * x ^ n" 
29451  893 
by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac) 
894 

895 
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x" 

896 
apply (induct p arbitrary: q, simp) 

897 
apply (case_tac q, simp, simp add: ring_simps) 

898 
done 

899 

900 
lemma poly_minus [simp]: 

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

901 
fixes x :: "'a::comm_ring" 
29451  902 
shows "poly ( p) x =  poly p x" 
903 
by (induct p, simp_all) 

904 

905 
lemma poly_diff [simp]: 

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

906 
fixes x :: "'a::comm_ring" 
29451  907 
shows "poly (p  q) x = poly p x  poly q x" 
908 
by (simp add: diff_minus) 

909 

910 
lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)" 

911 
by (cases "finite A", induct set: finite, simp_all) 

912 

913 
lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x" 

914 
by (induct p, simp, simp add: ring_simps) 

915 

916 
lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x" 

917 
by (induct p, simp_all, simp add: ring_simps) 

918 

29462  919 
lemma poly_power [simp]: 
920 
fixes p :: "'a::{comm_semiring_1,recpower} poly" 

921 
shows "poly (p ^ n) x = poly p x ^ n" 

922 
by (induct n, simp, simp add: power_Suc) 

923 

29456  924 

925 
subsection {* Synthetic division *} 

926 

927 
definition 

928 
synthetic_divmod :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly \<times> 'a" 

29478  929 
where [code del]: 
29456  930 
"synthetic_divmod p c = 
931 
poly_rec (0, 0) (\<lambda>a p (q, r). (pCons r q, a + c * r)) p" 

932 

933 
definition 

934 
synthetic_div :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly" 

935 
where 

936 
"synthetic_div p c = fst (synthetic_divmod p c)" 

937 

938 
lemma synthetic_divmod_0 [simp]: 

939 
"synthetic_divmod 0 c = (0, 0)" 

940 
unfolding synthetic_divmod_def 

941 
by (simp add: poly_rec_0) 

942 

943 
lemma synthetic_divmod_pCons [simp]: 

944 
"synthetic_divmod (pCons a p) c = 

945 
(\<lambda>(q, r). (pCons r q, a + c * r)) (synthetic_divmod p c)" 

946 
unfolding synthetic_divmod_def 

947 
by (simp add: poly_rec_pCons) 

948 

949 
lemma snd_synthetic_divmod: "snd (synthetic_divmod p c) = poly p c" 

950 
by (induct p, simp, simp add: split_def) 

951 

952 
lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0" 

953 
unfolding synthetic_div_def by simp 

954 

955 
lemma synthetic_div_pCons [simp]: 

956 
"synthetic_div (pCons a p) c = pCons (poly p c) (synthetic_div p c)" 

957 
unfolding synthetic_div_def 

958 
by (simp add: split_def snd_synthetic_divmod) 

959 

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

960 
lemma synthetic_div_eq_0_iff: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

961 
"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

962 
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

963 

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

964 
lemma degree_synthetic_div: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

965 
"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

966 
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

967 

29457  968 
lemma synthetic_div_correct: 
29456  969 
"p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)" 
970 
by (induct p) simp_all 

971 

29457  972 
lemma synthetic_div_unique_lemma: "smult c p = pCons a p \<Longrightarrow> p = 0" 
973 
by (induct p arbitrary: a) simp_all 

974 

975 
lemma synthetic_div_unique: 

976 
"p + smult c q = pCons r q \<Longrightarrow> r = poly p c \<and> q = synthetic_div p c" 

977 
apply (induct p arbitrary: q r) 

978 
apply (simp, frule synthetic_div_unique_lemma, simp) 

979 
apply (case_tac q, force) 

980 
done 

981 

982 
lemma synthetic_div_correct': 

983 
fixes c :: "'a::comm_ring_1" 

984 
shows "[:c, 1:] * synthetic_div p c + [:poly p c:] = p" 

985 
using synthetic_div_correct [of p c] 

986 
by (simp add: group_simps) 

987 

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

988 
lemma poly_eq_0_iff_dvd: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

989 
fixes c :: "'a::idom" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

990 
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

991 
proof 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

992 
assume "poly p c = 0" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

993 
with synthetic_div_correct' [of c p] 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

994 
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

995 
then show "[:c, 1:] dvd p" .. 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

996 
next 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

997 
assume "[:c, 1:] dvd p" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

998 
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

999 
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

1000 
qed 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1001 

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

1002 
lemma dvd_iff_poly_eq_0: 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1003 
fixes c :: "'a::idom" 
ad87e5d1488b
new lemmas about synthetic_div; declare degree_pCons_eq_if [simp]
huffman
parents:
29457
diff
changeset

1004 
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

1005 
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

1006 

29462  1007 
lemma poly_roots_finite: 
1008 
fixes p :: "'a::idom poly" 

1009 
shows "p \<noteq> 0 \<Longrightarrow> finite {x. poly p x = 0}" 

1010 
proof (induct n \<equiv> "degree p" arbitrary: p) 

1011 
case (0 p) 

1012 
then obtain a where "a \<noteq> 0" and "p = [:a:]" 

1013 
by (cases p, simp split: if_splits) 

1014 
then show "finite {x. poly p x = 0}" by simp 

1015 
next 

1016 
case (Suc n p) 

1017 
show "finite {x. poly p x = 0}" 

1018 
proof (cases "\<exists>x. poly p x = 0") 

1019 
case False 

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

1021 
next 

1022 
case True 

1023 
then obtain a where "poly p a = 0" .. 

1024 
then have "[:a, 1:] dvd p" by (simp only: poly_eq_0_iff_dvd) 

1025 
then obtain k where k: "p = [:a, 1:] * k" .. 

1026 
with `p \<noteq> 0` have "k \<noteq> 0" by auto 

1027 
with k have "degree p = Suc (degree k)" 

1028 
by (simp add: degree_mult_eq del: mult_pCons_left) 

1029 
with `Suc n = degree p` have "n = degree k" by simp 

1030 
with `k \<noteq> 0` have "finite {x. poly k x = 0}" by (rule Suc.hyps) 

1031 
then have "finite (insert a {x. poly k x = 0})" by simp 

1032 
then show "finite {x. poly p x = 0}" 

1033 
by (simp add: k uminus_add_conv_diff Collect_disj_eq 

1034 
del: mult_pCons_left) 

1035 
qed 

1036 
qed 

1037 

29478  1038 

1039 
subsection {* Configuration of the code generator *} 

1040 

1041 
code_datatype "0::'a::zero poly" pCons 

1042 

29480  1043 
declare pCons_0_0 [code post] 
1044 

29478  1045 
instantiation poly :: ("{zero,eq}") eq 
1046 
begin 

1047 

1048 
definition [code del]: 

1049 
"eq_class.eq (p::'a poly) q \<longleftrightarrow> p = q" 

1050 

1051 
instance 

1052 
by default (rule eq_poly_def) 

1053 

29451  1054 
end 
29478  1055 

1056 
lemma eq_poly_code [code]: 

1057 
"eq_class.eq (0::_ poly) (0::_ poly) \<longleftrightarrow> True" 

1058 
"eq_class.eq (0::_ poly) (pCons b q) \<longleftrightarrow> eq_class.eq 0 b \<and> eq_class.eq 0 q" 

1059 
"eq_class.eq (pCons a p) (0::_ poly) \<longleftrightarrow> eq_class.eq a 0 \<and> eq_class.eq p 0" 

1060 
"eq_class.eq (pCons a p) (pCons b q) \<longleftrightarrow> eq_class.eq a b \<and> eq_class.eq p q" 

1061 
unfolding eq by simp_all 

1062 

1063 
lemmas coeff_code [code] = 

1064 
coeff_0 coeff_pCons_0 coeff_pCons_Suc 

1065 

1066 
lemmas degree_code [code] = 

1067 
degree_0 degree_pCons_eq_if 

1068 

1069 
lemmas monom_poly_code [code] = 

1070 
monom_0 monom_Suc 

1071 

1072 
lemma add_poly_code [code]: 

1073 
"0 + q = (q :: _ poly)" 

1074 
"p + 0 = (p :: _ poly)" 

1075 
"pCons a p + pCons b q = pCons (a + b) (p + q)" 

1076 
by simp_all 

1077 

1078 
lemma minus_poly_code [code]: 

1079 
" 0 = (0 :: _ poly)" 

1080 
" pCons a p = pCons ( a) ( p)" 

1081 
by simp_all 

1082 

1083 
lemma diff_poly_code [code]: 

1084 
"0  q = ( q :: _ poly)" 

1085 
"p  0 = (p :: _ poly)" 

1086 
"pCons a p  pCons b q = pCons (a  b) (p  q)" 

1087 
by simp_all 

1088 

1089 
lemmas smult_poly_code [code] = 

1090 
smult_0_right smult_pCons 

1091 

1092 
lemma mult_poly_code [code]: 

1093 
"0 * q = (0 :: _ poly)" 

1094 
"pCons a p * q = smult a q + pCons 0 (p * q)" 

1095 
by simp_all 

1096 

1097 
lemmas poly_code [code] = 

1098 
poly_0 poly_pCons 

1099 

1100 
lemmas synthetic_divmod_code [code] = 

1101 
synthetic_divmod_0 synthetic_divmod_pCons 

1102 

1103 
text {* code generator setup for div and mod *} 

1104 

1105 
definition 

29537  1106 
pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly" 
29478  1107 
where 
29537  1108 
[code del]: "pdivmod x y = (x div y, x mod y)" 
29478  1109 

29537  1110 
lemma div_poly_code [code]: "x div y = fst (pdivmod x y)" 
1111 
unfolding pdivmod_def by simp 

29478  1112 

29537  1113 
lemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)" 
1114 
unfolding pdivmod_def by simp 

29478  1115 

29537  1116 
lemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)" 
1117 
unfolding pdivmod_def by simp 

29478  1118 

29537  1119 
lemma pdivmod_pCons [code]: 
1120 
"pdivmod (pCons a x) y = 

29478  1121 
(if y = 0 then (0, pCons a x) else 
29537  1122 
(let (q, r) = pdivmod x y; 
29478  1123 
b = coeff (pCons a r) (degree y) / coeff y (degree y) 
1124 
in (pCons b q, pCons a r  smult b y)))" 

29537  1125 
apply (simp add: pdivmod_def Let_def, safe) 
29478  1126 
apply (rule div_poly_eq) 
29537  1127 
apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) 
29478  1128 
apply (rule mod_poly_eq) 
29537  1129 
apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl]) 
29478  1130 
done 
1131 

1132 
end 