author  huffman 
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changeset 29455  0139c9a01ca4 
parent 29454  b0f586f38dd7 
child 29456  3f8b85444512 
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: 

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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]: 
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"poly_rec z f (pCons a p) = f a p (if p = 0 then z else poly_rec z f p)" 
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by (case_tac x, rename_tac q, case_tac q, auto) 
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termination poly_rec 
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by (relation "measure (degree \<circ> snd \<circ> snd)", simp) 
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(simp add: degree_pCons_eq) 
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lemma poly_rec_0: 
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"f 0 0 z = z \<Longrightarrow> poly_rec z f 0 = z" 
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using poly_rec_pCons_eq_if [of z f 0 0] by simp 
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lemma poly_rec_pCons: 
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"f 0 0 z = z \<Longrightarrow> poly_rec z f (pCons a p) = f a p (poly_rec z f p)" 
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by (simp add: poly_rec_pCons_eq_if poly_rec_0) 
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29451  228 
subsection {* Monomials *} 
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definition 

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

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

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

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unfolding monom_def 

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by (subst Abs_poly_inverse, auto simp add: Poly_def) 

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

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

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

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

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

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by (rule poly_ext) simp 

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

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by (simp add: expand_poly_eq) 

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

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by (simp add: expand_poly_eq) 

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

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by (rule degree_le, simp) 

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

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apply (rule order_antisym [OF degree_monom_le]) 

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apply (rule le_degree, simp) 

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done 

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subsection {* Addition and subtraction *} 

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instantiation poly :: (comm_monoid_add) comm_monoid_add 

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begin 

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definition 

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plus_poly_def [code del]: 

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

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lemma Poly_add: 

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

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

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unfolding Poly_def 

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apply (clarify, rename_tac m n) 

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

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done 

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lemma coeff_add [simp]: 

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

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unfolding plus_poly_def 

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by (simp add: Abs_poly_inverse coeff Poly_add) 

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instance proof 

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fix p q r :: "'a poly" 

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

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by (simp add: expand_poly_eq add_assoc) 

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show "p + q = q + p" 

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by (simp add: expand_poly_eq add_commute) 

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show "0 + p = p" 

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by (simp add: expand_poly_eq) 

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qed 

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end 

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instantiation poly :: (ab_group_add) ab_group_add 

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begin 

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definition 

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uminus_poly_def [code del]: 

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

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definition 

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minus_poly_def [code del]: 

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

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lemma Poly_minus: 

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

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

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unfolding Poly_def by simp 

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lemma Poly_diff: 

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

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

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unfolding diff_minus by (simp add: Poly_add Poly_minus) 

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

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unfolding uminus_poly_def 

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by (simp add: Abs_poly_inverse coeff Poly_minus) 

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lemma coeff_diff [simp]: 

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

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unfolding minus_poly_def 

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by (simp add: Abs_poly_inverse coeff Poly_diff) 

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instance proof 

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fix p q :: "'a poly" 

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show " p + p = 0" 

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by (simp add: expand_poly_eq) 

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show "p  q = p +  q" 

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by (simp add: expand_poly_eq diff_minus) 

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qed 

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end 

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lemma add_pCons [simp]: 

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

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

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lemma minus_pCons [simp]: 

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

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

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lemma diff_pCons [simp]: 

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

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

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

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by (rule degree_le, auto simp add: coeff_eq_0) 

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

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by (auto intro: le_less_trans degree_add_le) 

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

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apply (cases "q = 0", simp) 

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apply (rule order_antisym) 

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apply (rule ord_le_eq_trans [OF degree_add_le]) 

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apply simp 

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apply (rule le_degree) 

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apply (simp add: coeff_eq_0) 

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done 

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lemma degree_add_eq_left: 

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

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using degree_add_eq_right [of q p] 

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by (simp add: add_commute) 

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

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unfolding degree_def by simp 

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

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

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by (simp add: diff_minus) 

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

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by (auto intro: le_less_trans degree_diff_le) 

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

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

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by (rule poly_ext) simp 

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

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by (rule poly_ext) simp 

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

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

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

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by (rule poly_ext) (simp add: coeff_setsum) 

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subsection {* Multiplication by a constant *} 

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definition 

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

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

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lemma Poly_smult: 

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

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

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unfolding Poly_def 

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

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lemma coeff_smult [simp]: "coeff (smult a p) n = a * coeff p n" 

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unfolding smult_def 

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by (simp add: Abs_poly_inverse Poly_smult coeff) 

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

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by (rule degree_le, simp add: coeff_eq_0) 

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lemma smult_smult: "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: 

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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: 

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

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by (rule poly_ext, simp) 

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lemma smult_diff_right: 

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

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by (rule poly_ext, simp add: ring_simps) 

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lemma smult_diff_left: 

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

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by (rule poly_ext, simp add: ring_simps) 

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lemma smult_pCons [simp]: 

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"smult a (pCons b p) = pCons (a * b) (smult a p)" 

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

455 

456 
lemma smult_monom: "smult a (monom b n) = monom (a * b) n" 

457 
by (induct n, simp add: monom_0, simp add: monom_Suc) 

458 

459 

460 
subsection {* Multiplication of polynomials *} 

461 

462 
lemma Poly_mult_lemma: 

463 
fixes f g :: "nat \<Rightarrow> 'a::comm_semiring_0" and m n :: nat 

464 
assumes "\<forall>i>m. f i = 0" 

465 
assumes "\<forall>j>n. g j = 0" 

466 
shows "\<forall>k>m+n. (\<Sum>i\<le>k. f i * g (ki)) = 0" 

467 
proof (clarify) 

468 
fix k :: nat 

469 
assume "m + n < k" 

470 
show "(\<Sum>i\<le>k. f i * g (k  i)) = 0" 

471 
proof (rule setsum_0' [rule_format]) 

472 
fix i :: nat 

473 
assume "i \<in> {..k}" hence "i \<le> k" by simp 

474 
with `m + n < k` have "m < i \<or> n < k  i" by arith 

475 
thus "f i * g (k  i) = 0" 

476 
using prems by auto 

477 
qed 

478 
qed 

479 

480 
lemma Poly_mult: 

481 
fixes f g :: "nat \<Rightarrow> 'a::comm_semiring_0" 

482 
shows "\<lbrakk>f \<in> Poly; g \<in> Poly\<rbrakk> \<Longrightarrow> (\<lambda>n. \<Sum>i\<le>n. f i * g (ni)) \<in> Poly" 

483 
unfolding Poly_def 

484 
by (safe, rule exI, rule Poly_mult_lemma) 

485 

486 
lemma poly_mult_assoc_lemma: 

487 
fixes k :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add" 

488 
shows "(\<Sum>j\<le>k. \<Sum>i\<le>j. f i (j  i) (n  j)) = 

489 
(\<Sum>j\<le>k. \<Sum>i\<le>k  j. f j i (n  j  i))" 

490 
proof (induct k) 

491 
case 0 show ?case by simp 

492 
next 

493 
case (Suc k) thus ?case 

494 
by (simp add: Suc_diff_le setsum_addf add_assoc 

495 
cong: strong_setsum_cong) 

496 
qed 

497 

498 
lemma poly_mult_commute_lemma: 

499 
fixes n :: nat and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_monoid_add" 

500 
shows "(\<Sum>i\<le>n. f i (n  i)) = (\<Sum>i\<le>n. f (n  i) i)" 

501 
proof (rule setsum_reindex_cong) 

502 
show "inj_on (\<lambda>i. n  i) {..n}" 

503 
by (rule inj_onI) simp 

504 
show "{..n} = (\<lambda>i. n  i) ` {..n}" 

505 
by (auto, rule_tac x="n  x" in image_eqI, simp_all) 

506 
next 

507 
fix i assume "i \<in> {..n}" 

508 
hence "n  (n  i) = i" by simp 

509 
thus "f (n  i) i = f (n  i) (n  (n  i))" by simp 

510 
qed 

511 

512 
text {* TODO: move to appropriate theory *} 

513 
lemma setsum_atMost_Suc_shift: 

514 
fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add" 

515 
shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))" 

516 
proof (induct n) 

517 
case 0 show ?case by simp 

518 
next 

519 
case (Suc n) note IH = this 

520 
have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))" 

521 
by (rule setsum_atMost_Suc) 

522 
also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))" 

523 
by (rule IH) 

524 
also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = 

525 
f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))" 

526 
by (rule add_assoc) 

527 
also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))" 

528 
by (rule setsum_atMost_Suc [symmetric]) 

529 
finally show ?case . 

530 
qed 

531 

532 
instantiation poly :: (comm_semiring_0) comm_semiring_0 

533 
begin 

534 

535 
definition 

536 
times_poly_def: 

537 
"p * q = Abs_poly (\<lambda>n. \<Sum>i\<le>n. coeff p i * coeff q (ni))" 

538 

539 
lemma coeff_mult: 

540 
"coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (ni))" 

541 
unfolding times_poly_def 

542 
by (simp add: Abs_poly_inverse coeff Poly_mult) 

543 

544 
instance proof 

545 
fix p q r :: "'a poly" 

546 
show 0: "0 * p = 0" 

547 
by (simp add: expand_poly_eq coeff_mult) 

548 
show "p * 0 = 0" 

549 
by (simp add: expand_poly_eq coeff_mult) 

550 
show "(p + q) * r = p * r + q * r" 

551 
by (simp add: expand_poly_eq coeff_mult left_distrib setsum_addf) 

552 
show "(p * q) * r = p * (q * r)" 

553 
proof (rule poly_ext) 

554 
fix n :: nat 

555 
have "(\<Sum>j\<le>n. \<Sum>i\<le>j. coeff p i * coeff q (j  i) * coeff r (n  j)) = 

556 
(\<Sum>j\<le>n. \<Sum>i\<le>n  j. coeff p j * coeff q i * coeff r (n  j  i))" 

557 
by (rule poly_mult_assoc_lemma) 

558 
thus "coeff ((p * q) * r) n = coeff (p * (q * r)) n" 

559 
by (simp add: coeff_mult setsum_right_distrib 

560 
setsum_left_distrib mult_assoc) 

561 
qed 

562 
show "p * q = q * p" 

563 
proof (rule poly_ext) 

564 
fix n :: nat 

565 
have "(\<Sum>i\<le>n. coeff p i * coeff q (n  i)) = 

566 
(\<Sum>i\<le>n. coeff p (n  i) * coeff q i)" 

567 
by (rule poly_mult_commute_lemma) 

568 
thus "coeff (p * q) n = coeff (q * p) n" 

569 
by (simp add: coeff_mult mult_commute) 

570 
qed 

571 
qed 

572 

573 
end 

574 

575 
lemma degree_mult_le: "degree (p * q) \<le> degree p + degree q" 

576 
apply (rule degree_le, simp add: coeff_mult) 

577 
apply (rule Poly_mult_lemma) 

578 
apply (simp_all add: coeff_eq_0) 

579 
done 

580 

581 
lemma mult_pCons_left [simp]: 

582 
"pCons a p * q = smult a q + pCons 0 (p * q)" 

583 
apply (rule poly_ext) 

584 
apply (case_tac n) 

585 
apply (simp add: coeff_mult) 

586 
apply (simp add: coeff_mult setsum_atMost_Suc_shift 

587 
del: setsum_atMost_Suc) 

588 
done 

589 

590 
lemma mult_pCons_right [simp]: 

591 
"p * pCons a q = smult a p + pCons 0 (p * q)" 

592 
using mult_pCons_left [of a q p] by (simp add: mult_commute) 

593 

594 
lemma mult_smult_left: "smult a p * q = smult a (p * q)" 

595 
by (induct p, simp, simp add: smult_add_right smult_smult) 

596 

597 
lemma mult_smult_right: "p * smult a q = smult a (p * q)" 

598 
using mult_smult_left [of a q p] by (simp add: mult_commute) 

599 

600 
lemma mult_monom: "monom a m * monom b n = monom (a * b) (m + n)" 

601 
by (induct m, simp add: monom_0 smult_monom, simp add: monom_Suc) 

602 

603 

604 
subsection {* The unit polynomial and exponentiation *} 

605 

606 
instantiation poly :: (comm_semiring_1) comm_semiring_1 

607 
begin 

608 

609 
definition 

610 
one_poly_def: 

611 
"1 = pCons 1 0" 

612 

613 
instance proof 

614 
fix p :: "'a poly" show "1 * p = p" 

615 
unfolding one_poly_def 

616 
by simp 

617 
next 

618 
show "0 \<noteq> (1::'a poly)" 

619 
unfolding one_poly_def by simp 

620 
qed 

621 

622 
end 

623 

624 
lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)" 

625 
unfolding one_poly_def 

626 
by (simp add: coeff_pCons split: nat.split) 

627 

628 
lemma degree_1 [simp]: "degree 1 = 0" 

629 
unfolding one_poly_def 

630 
by (rule degree_pCons_0) 

631 

632 
instantiation poly :: (comm_semiring_1) recpower 

633 
begin 

634 

635 
primrec power_poly where 

636 
power_poly_0: "(p::'a poly) ^ 0 = 1" 

637 
 power_poly_Suc: "(p::'a poly) ^ (Suc n) = p * p ^ n" 

638 

639 
instance 

640 
by default simp_all 

641 

642 
end 

643 

644 
instance poly :: (comm_ring) comm_ring .. 

645 

646 
instance poly :: (comm_ring_1) comm_ring_1 .. 

647 

648 
instantiation poly :: (comm_ring_1) number_ring 

649 
begin 

650 

651 
definition 

652 
"number_of k = (of_int k :: 'a poly)" 

653 

654 
instance 

655 
by default (rule number_of_poly_def) 

656 

657 
end 

658 

659 

660 
subsection {* Polynomials form an integral domain *} 

661 

662 
lemma coeff_mult_degree_sum: 

663 
"coeff (p * q) (degree p + degree q) = 

664 
coeff p (degree p) * coeff q (degree q)" 

665 
apply (simp add: coeff_mult) 

666 
apply (subst setsum_diff1' [where a="degree p"]) 

667 
apply simp 

668 
apply simp 

669 
apply (subst setsum_0' [rule_format]) 

670 
apply clarsimp 

671 
apply (subgoal_tac "degree p < a \<or> degree q < degree p + degree q  a") 

672 
apply (force simp add: coeff_eq_0) 

673 
apply arith 

674 
apply simp 

675 
done 

676 

677 
instance poly :: (idom) idom 

678 
proof 

679 
fix p q :: "'a poly" 

680 
assume "p \<noteq> 0" and "q \<noteq> 0" 

681 
have "coeff (p * q) (degree p + degree q) = 

682 
coeff p (degree p) * coeff q (degree q)" 

683 
by (rule coeff_mult_degree_sum) 

684 
also have "coeff p (degree p) * coeff q (degree q) \<noteq> 0" 

685 
using `p \<noteq> 0` and `q \<noteq> 0` by simp 

686 
finally have "\<exists>n. coeff (p * q) n \<noteq> 0" .. 

687 
thus "p * q \<noteq> 0" by (simp add: expand_poly_eq) 

688 
qed 

689 

690 
lemma degree_mult_eq: 

691 
fixes p q :: "'a::idom poly" 

692 
shows "\<lbrakk>p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree (p * q) = degree p + degree q" 

693 
apply (rule order_antisym [OF degree_mult_le le_degree]) 

694 
apply (simp add: coeff_mult_degree_sum) 

695 
done 

696 

697 
lemma dvd_imp_degree_le: 

698 
fixes p q :: "'a::idom poly" 

699 
shows "\<lbrakk>p dvd q; q \<noteq> 0\<rbrakk> \<Longrightarrow> degree p \<le> degree q" 

700 
by (erule dvdE, simp add: degree_mult_eq) 

701 

702 

703 
subsection {* Long division of polynomials *} 

704 

705 
definition 

706 
divmod_poly_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool" 

707 
where 

708 
"divmod_poly_rel x y q r \<longleftrightarrow> 

709 
x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)" 

710 

711 
lemma divmod_poly_rel_0: 

712 
"divmod_poly_rel 0 y 0 0" 

713 
unfolding divmod_poly_rel_def by simp 

714 

715 
lemma divmod_poly_rel_by_0: 

716 
"divmod_poly_rel x 0 0 x" 

717 
unfolding divmod_poly_rel_def by simp 

718 

719 
lemma eq_zero_or_degree_less: 

720 
assumes "degree p \<le> n" and "coeff p n = 0" 

721 
shows "p = 0 \<or> degree p < n" 

722 
proof (cases n) 

723 
case 0 

724 
with `degree p \<le> n` and `coeff p n = 0` 

725 
have "coeff p (degree p) = 0" by simp 

726 
then have "p = 0" by simp 

727 
then show ?thesis .. 

728 
next 

729 
case (Suc m) 

730 
have "\<forall>i>n. coeff p i = 0" 

731 
using `degree p \<le> n` by (simp add: coeff_eq_0) 

732 
then have "\<forall>i\<ge>n. coeff p i = 0" 

733 
using `coeff p n = 0` by (simp add: le_less) 

734 
then have "\<forall>i>m. coeff p i = 0" 

735 
using `n = Suc m` by (simp add: less_eq_Suc_le) 

736 
then have "degree p \<le> m" 

737 
by (rule degree_le) 

738 
then have "degree p < n" 

739 
using `n = Suc m` by (simp add: less_Suc_eq_le) 

740 
then show ?thesis .. 

741 
qed 

742 

743 
lemma divmod_poly_rel_pCons: 

744 
assumes rel: "divmod_poly_rel x y q r" 

745 
assumes y: "y \<noteq> 0" 

746 
assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)" 

747 
shows "divmod_poly_rel (pCons a x) y (pCons b q) (pCons a r  smult b y)" 

748 
(is "divmod_poly_rel ?x y ?q ?r") 

749 
proof  

750 
have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y" 

751 
using assms unfolding divmod_poly_rel_def by simp_all 

752 

753 
have 1: "?x = ?q * y + ?r" 

754 
using b x by simp 

755 

756 
have 2: "?r = 0 \<or> degree ?r < degree y" 

757 
proof (rule eq_zero_or_degree_less) 

758 
have "degree ?r \<le> max (degree (pCons a r)) (degree (smult b y))" 

759 
by (rule degree_diff_le) 

760 
also have "\<dots> \<le> degree y" 

761 
proof (rule min_max.le_supI) 

762 
show "degree (pCons a r) \<le> degree y" 

763 
using r by (auto simp add: degree_pCons_eq_if) 

764 
show "degree (smult b y) \<le> degree y" 

765 
by (rule degree_smult_le) 

766 
qed 

767 
finally show "degree ?r \<le> degree y" . 

768 
next 

769 
show "coeff ?r (degree y) = 0" 

770 
using `y \<noteq> 0` unfolding b by simp 

771 
qed 

772 

773 
from 1 2 show ?thesis 

774 
unfolding divmod_poly_rel_def 

775 
using `y \<noteq> 0` by simp 

776 
qed 

777 

778 
lemma divmod_poly_rel_exists: "\<exists>q r. divmod_poly_rel x y q r" 

779 
apply (cases "y = 0") 

780 
apply (fast intro!: divmod_poly_rel_by_0) 

781 
apply (induct x) 

782 
apply (fast intro!: divmod_poly_rel_0) 

783 
apply (fast intro!: divmod_poly_rel_pCons) 

784 
done 

785 

786 
lemma divmod_poly_rel_unique: 

787 
assumes 1: "divmod_poly_rel x y q1 r1" 

788 
assumes 2: "divmod_poly_rel x y q2 r2" 

789 
shows "q1 = q2 \<and> r1 = r2" 

790 
proof (cases "y = 0") 

791 
assume "y = 0" with assms show ?thesis 

792 
by (simp add: divmod_poly_rel_def) 

793 
next 

794 
assume [simp]: "y \<noteq> 0" 

795 
from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y" 

796 
unfolding divmod_poly_rel_def by simp_all 

797 
from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y" 

798 
unfolding divmod_poly_rel_def by simp_all 

799 
from q1 q2 have q3: "(q1  q2) * y = r2  r1" 

800 
by (simp add: ring_simps) 

801 
from r1 r2 have r3: "(r2  r1) = 0 \<or> degree (r2  r1) < degree y" 

29453  802 
by (auto intro: degree_diff_less) 
29451  803 

804 
show "q1 = q2 \<and> r1 = r2" 

805 
proof (rule ccontr) 

806 
assume "\<not> (q1 = q2 \<and> r1 = r2)" 

807 
with q3 have "q1 \<noteq> q2" and "r1 \<noteq> r2" by auto 

808 
with r3 have "degree (r2  r1) < degree y" by simp 

809 
also have "degree y \<le> degree (q1  q2) + degree y" by simp 

810 
also have "\<dots> = degree ((q1  q2) * y)" 

811 
using `q1 \<noteq> q2` by (simp add: degree_mult_eq) 

812 
also have "\<dots> = degree (r2  r1)" 

813 
using q3 by simp 

814 
finally have "degree (r2  r1) < degree (r2  r1)" . 

815 
then show "False" by simp 

816 
qed 

817 
qed 

818 

819 
lemmas divmod_poly_rel_unique_div = 

820 
divmod_poly_rel_unique [THEN conjunct1, standard] 

821 

822 
lemmas divmod_poly_rel_unique_mod = 

823 
divmod_poly_rel_unique [THEN conjunct2, standard] 

824 

825 
instantiation poly :: (field) ring_div 

826 
begin 

827 

828 
definition div_poly where 

829 
[code del]: "x div y = (THE q. \<exists>r. divmod_poly_rel x y q r)" 

830 

831 
definition mod_poly where 

832 
[code del]: "x mod y = (THE r. \<exists>q. divmod_poly_rel x y q r)" 

833 

834 
lemma div_poly_eq: 

835 
"divmod_poly_rel x y q r \<Longrightarrow> x div y = q" 

836 
unfolding div_poly_def 

837 
by (fast elim: divmod_poly_rel_unique_div) 

838 

839 
lemma mod_poly_eq: 

840 
"divmod_poly_rel x y q r \<Longrightarrow> x mod y = r" 

841 
unfolding mod_poly_def 

842 
by (fast elim: divmod_poly_rel_unique_mod) 

843 

844 
lemma divmod_poly_rel: 

845 
"divmod_poly_rel x y (x div y) (x mod y)" 

846 
proof  

847 
from divmod_poly_rel_exists 

848 
obtain q r where "divmod_poly_rel x y q r" by fast 

849 
thus ?thesis 

850 
by (simp add: div_poly_eq mod_poly_eq) 

851 
qed 

852 

853 
instance proof 

854 
fix x y :: "'a poly" 

855 
show "x div y * y + x mod y = x" 

856 
using divmod_poly_rel [of x y] 

857 
by (simp add: divmod_poly_rel_def) 

858 
next 

859 
fix x :: "'a poly" 

860 
have "divmod_poly_rel x 0 0 x" 

861 
by (rule divmod_poly_rel_by_0) 

862 
thus "x div 0 = 0" 

863 
by (rule div_poly_eq) 

864 
next 

865 
fix y :: "'a poly" 

866 
have "divmod_poly_rel 0 y 0 0" 

867 
by (rule divmod_poly_rel_0) 

868 
thus "0 div y = 0" 

869 
by (rule div_poly_eq) 

870 
next 

871 
fix x y z :: "'a poly" 

872 
assume "y \<noteq> 0" 

873 
hence "divmod_poly_rel (x + z * y) y (z + x div y) (x mod y)" 

874 
using divmod_poly_rel [of x y] 

875 
by (simp add: divmod_poly_rel_def left_distrib) 

876 
thus "(x + z * y) div y = z + x div y" 

877 
by (rule div_poly_eq) 

878 
qed 

879 

880 
end 

881 

882 
lemma degree_mod_less: 

883 
"y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y" 

884 
using divmod_poly_rel [of x y] 

885 
unfolding divmod_poly_rel_def by simp 

886 

887 
lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0" 

888 
proof  

889 
assume "degree x < degree y" 

890 
hence "divmod_poly_rel x y 0 x" 

891 
by (simp add: divmod_poly_rel_def) 

892 
thus "x div y = 0" by (rule div_poly_eq) 

893 
qed 

894 

895 
lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x" 

896 
proof  

897 
assume "degree x < degree y" 

898 
hence "divmod_poly_rel x y 0 x" 

899 
by (simp add: divmod_poly_rel_def) 

900 
thus "x mod y = x" by (rule mod_poly_eq) 

901 
qed 

902 

903 
lemma mod_pCons: 

904 
fixes a and x 

905 
assumes y: "y \<noteq> 0" 

906 
defines b: "b \<equiv> coeff (pCons a (x mod y)) (degree y) / coeff y (degree y)" 

907 
shows "(pCons a x) mod y = (pCons a (x mod y)  smult b y)" 

908 
unfolding b 

909 
apply (rule mod_poly_eq) 

910 
apply (rule divmod_poly_rel_pCons [OF divmod_poly_rel y refl]) 

911 
done 

912 

913 

914 
subsection {* Evaluation of polynomials *} 

915 

916 
definition 

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

917 
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

918 
"poly = poly_rec (\<lambda>x. 0) (\<lambda>a p f x. a + x * f x)" 
29451  919 

920 
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

921 
unfolding poly_def by (simp add: poly_rec_0) 
29451  922 

923 
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

924 
unfolding poly_def by (simp add: poly_rec_pCons) 
29451  925 

926 
lemma poly_1 [simp]: "poly 1 x = 1" 

927 
unfolding one_poly_def by simp 

928 

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

929 
lemma poly_monom: 
b0f586f38dd7
add recursion combinator poly_rec; define poly function using poly_rec
huffman
parents:
29453
diff
changeset

930 
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

931 
shows "poly (monom a n) x = a * x ^ n" 
29451  932 
by (induct n, simp add: monom_0, simp add: monom_Suc power_Suc mult_ac) 
933 

934 
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x" 

935 
apply (induct p arbitrary: q, simp) 

936 
apply (case_tac q, simp, simp add: ring_simps) 

937 
done 

938 

939 
lemma poly_minus [simp]: 

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

940 
fixes x :: "'a::comm_ring" 
29451  941 
shows "poly ( p) x =  poly p x" 
942 
by (induct p, simp_all) 

943 

944 
lemma poly_diff [simp]: 

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

945 
fixes x :: "'a::comm_ring" 
29451  946 
shows "poly (p  q) x = poly p x  poly q x" 
947 
by (simp add: diff_minus) 

948 

949 
lemma poly_setsum: "poly (\<Sum>k\<in>A. p k) x = (\<Sum>k\<in>A. poly (p k) x)" 

950 
by (cases "finite A", induct set: finite, simp_all) 

951 

952 
lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x" 

953 
by (induct p, simp, simp add: ring_simps) 

954 

955 
lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x" 

956 
by (induct p, simp_all, simp add: ring_simps) 

957 

958 
end 