author | paulson <lp15@cam.ac.uk> |
Mon, 01 Apr 2019 17:02:43 +0100 | |
changeset 70019 | 095dce9892e8 |
parent 69712 | dc85b5b3a532 |
child 70160 | 8e9100dcde52 |
permissions | -rw-r--r-- |
68582 | 1 |
(* Title: HOL/Algebra/Polynomials.thy |
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Author: Paulo EmÃlio de Vilhena |
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*) |
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theory Polynomials |
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imports Ring Ring_Divisibility Subrings |
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begin |
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section \<open>Polynomials\<close> |
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subsection \<open>Definitions\<close> |
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abbreviation lead_coeff :: "'a list \<Rightarrow> 'a" |
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where "lead_coeff \<equiv> hd" |
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abbreviation degree :: "'a list \<Rightarrow> nat" |
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where "degree p \<equiv> length p - 1" |
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definition polynomial :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a list \<Rightarrow> bool" ("polynomial\<index>") |
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where "polynomial\<^bsub>R\<^esub> K p \<longleftrightarrow> p = [] \<or> (set p \<subseteq> K \<and> lead_coeff p \<noteq> \<zero>\<^bsub>R\<^esub>)" |
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definition (in ring) monom :: "'a \<Rightarrow> nat \<Rightarrow> 'a list" |
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where "monom a n = a # (replicate n \<zero>\<^bsub>R\<^esub>)" |
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fun (in ring) eval :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a" |
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where |
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"eval [] = (\<lambda>_. \<zero>)" |
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| "eval p = (\<lambda>x. ((lead_coeff p) \<otimes> (x [^] (degree p))) \<oplus> (eval (tl p) x))" |
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fun (in ring) coeff :: "'a list \<Rightarrow> nat \<Rightarrow> 'a" |
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where |
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"coeff [] = (\<lambda>_. \<zero>)" |
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| "coeff p = (\<lambda>i. if i = degree p then lead_coeff p else (coeff (tl p)) i)" |
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fun (in ring) normalize :: "'a list \<Rightarrow> 'a list" |
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where |
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"normalize [] = []" |
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| "normalize p = (if lead_coeff p \<noteq> \<zero> then p else normalize (tl p))" |
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fun (in ring) poly_add :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" |
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where "poly_add p1 p2 = |
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(if length p1 \<ge> length p2 |
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then normalize (map2 (\<oplus>) p1 ((replicate (length p1 - length p2) \<zero>) @ p2)) |
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else poly_add p2 p1)" |
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fun (in ring) poly_mult :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" |
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where |
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"poly_mult [] p2 = []" |
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| "poly_mult p1 p2 = |
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poly_add ((map (\<lambda>a. lead_coeff p1 \<otimes> a) p2) @ (replicate (degree p1) \<zero>)) (poly_mult (tl p1) p2)" |
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fun (in ring) dense_repr :: "'a list \<Rightarrow> ('a \<times> nat) list" |
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where |
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"dense_repr [] = []" |
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| "dense_repr p = (if lead_coeff p \<noteq> \<zero> |
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then (lead_coeff p, degree p) # (dense_repr (tl p)) |
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else (dense_repr (tl p)))" |
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fun (in ring) of_dense :: "('a \<times> nat) list \<Rightarrow> 'a list" |
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where "of_dense dl = foldr (\<lambda>(a, n) l. poly_add (monom a n) l) dl []" |
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subsection \<open>Basic Properties\<close> |
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context ring |
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begin |
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lemma polynomialI [intro]: "\<lbrakk> set p \<subseteq> K; lead_coeff p \<noteq> \<zero> \<rbrakk> \<Longrightarrow> polynomial K p" |
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unfolding polynomial_def by auto |
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lemma polynomial_incl: "polynomial K p \<Longrightarrow> set p \<subseteq> K" |
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unfolding polynomial_def by auto |
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lemma monom_in_carrier [intro]: "a \<in> carrier R \<Longrightarrow> set (monom a n) \<subseteq> carrier R" |
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unfolding monom_def by auto |
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lemma lead_coeff_not_zero: "polynomial K (a # p) \<Longrightarrow> a \<in> K - { \<zero> }" |
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unfolding polynomial_def by simp |
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lemma zero_is_polynomial [intro]: "polynomial K []" |
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unfolding polynomial_def by simp |
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lemma const_is_polynomial [intro]: "a \<in> K - { \<zero> } \<Longrightarrow> polynomial K [ a ]" |
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unfolding polynomial_def by auto |
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lemma normalize_gives_polynomial: "set p \<subseteq> K \<Longrightarrow> polynomial K (normalize p)" |
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by (induction p) (auto simp add: polynomial_def) |
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lemma normalize_in_carrier: "set p \<subseteq> carrier R \<Longrightarrow> set (normalize p) \<subseteq> carrier R" |
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by (induction p) (auto) |
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lemma normalize_polynomial: "polynomial K p \<Longrightarrow> normalize p = p" |
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unfolding polynomial_def by (cases p) (auto) |
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lemma normalize_idem: "normalize ((normalize p) @ q) = normalize (p @ q)" |
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by (induct p) (auto) |
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lemma normalize_length_le: "length (normalize p) \<le> length p" |
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by (induction p) (auto) |
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lemma eval_in_carrier: "\<lbrakk> set p \<subseteq> carrier R; x \<in> carrier R \<rbrakk> \<Longrightarrow> (eval p) x \<in> carrier R" |
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by (induction p) (auto) |
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lemma coeff_in_carrier [simp]: "set p \<subseteq> carrier R \<Longrightarrow> (coeff p) i \<in> carrier R" |
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by (induction p) (auto) |
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lemma lead_coeff_simp [simp]: "p \<noteq> [] \<Longrightarrow> (coeff p) (degree p) = lead_coeff p" |
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by (metis coeff.simps(2) list.exhaust_sel) |
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lemma coeff_list: "map (coeff p) (rev [0..< length p]) = p" |
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proof (induction p) |
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case Nil thus ?case by simp |
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next |
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case (Cons a p) |
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have "map (coeff (a # p)) (rev [0..<length (a # p)]) = |
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a # (map (coeff p) (rev [0..<length p]))" |
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by auto |
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also have " ... = a # p" |
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using Cons by simp |
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finally show ?case . |
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qed |
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lemma coeff_nth: "i < length p \<Longrightarrow> (coeff p) i = p ! (length p - 1 - i)" |
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proof - |
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assume i_lt: "i < length p" |
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hence "(coeff p) i = (map (coeff p) [0..< length p]) ! i" |
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by simp |
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also have " ... = (rev (map (coeff p) (rev [0..< length p]))) ! i" |
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by (simp add: rev_map) |
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also have " ... = (map (coeff p) (rev [0..< length p])) ! (length p - 1 - i)" |
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using coeff_list i_lt rev_nth by auto |
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also have " ... = p ! (length p - 1 - i)" |
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using coeff_list[of p] by simp |
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finally show "(coeff p) i = p ! (length p - 1 - i)" . |
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qed |
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lemma coeff_iff_length_cond: |
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assumes "length p1 = length p2" |
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shows "p1 = p2 \<longleftrightarrow> coeff p1 = coeff p2" |
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proof |
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show "p1 = p2 \<Longrightarrow> coeff p1 = coeff p2" |
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by simp |
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next |
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assume A: "coeff p1 = coeff p2" |
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have "p1 = map (coeff p1) (rev [0..< length p1])" |
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using coeff_list[of p1] by simp |
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also have " ... = map (coeff p2) (rev [0..< length p2])" |
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using A assms by simp |
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also have " ... = p2" |
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using coeff_list[of p2] by simp |
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finally show "p1 = p2" . |
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qed |
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lemma coeff_img_restrict: "(coeff p) ` {..< length p} = set p" |
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using coeff_list[of p] by (metis atLeast_upt image_set set_rev) |
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lemma coeff_length: "\<And>i. i \<ge> length p \<Longrightarrow> (coeff p) i = \<zero>" |
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by (induction p) (auto) |
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lemma coeff_degree: "\<And>i. i > degree p \<Longrightarrow> (coeff p) i = \<zero>" |
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using coeff_length by (simp) |
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lemma replicate_zero_coeff [simp]: "coeff (replicate n \<zero>) = (\<lambda>_. \<zero>)" |
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by (induction n) (auto) |
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lemma scalar_coeff: "a \<in> carrier R \<Longrightarrow> coeff (map (\<lambda>b. a \<otimes> b) p) = (\<lambda>i. a \<otimes> (coeff p) i)" |
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by (induction p) (auto) |
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lemma monom_coeff: "coeff (monom a n) = (\<lambda>i. if i = n then a else \<zero>)" |
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unfolding monom_def by (induction n) (auto) |
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lemma coeff_img: |
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"(coeff p) ` {..< length p} = set p" |
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"(coeff p) ` { length p ..} = { \<zero> }" |
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"(coeff p) ` UNIV = (set p) \<union> { \<zero> }" |
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using coeff_img_restrict |
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proof (simp) |
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show coeff_img_up: "(coeff p) ` { length p ..} = { \<zero> }" |
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using coeff_length[of p] by force |
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from coeff_img_up and coeff_img_restrict[of p] |
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show "(coeff p) ` UNIV = (set p) \<union> { \<zero> }" |
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by force |
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qed |
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lemma degree_def': |
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assumes "polynomial K p" |
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shows "degree p = (LEAST n. \<forall>i. i > n \<longrightarrow> (coeff p) i = \<zero>)" |
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proof (cases p) |
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case Nil thus ?thesis by auto |
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next |
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define P where "P = (\<lambda>n. \<forall>i. i > n \<longrightarrow> (coeff p) i = \<zero>)" |
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case (Cons a ps) |
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hence "(coeff p) (degree p) \<noteq> \<zero>" |
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using assms unfolding polynomial_def by auto |
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hence "\<And>n. n < degree p \<Longrightarrow> \<not> P n" |
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unfolding P_def by auto |
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moreover have "P (degree p)" |
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unfolding P_def using coeff_degree[of p] by simp |
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ultimately have "degree p = (LEAST n. P n)" |
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by (meson LeastI nat_neq_iff not_less_Least) |
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thus ?thesis unfolding P_def . |
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qed |
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lemma coeff_iff_polynomial_cond: |
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assumes "polynomial K p1" and "polynomial K p2" |
68578 | 209 |
shows "p1 = p2 \<longleftrightarrow> coeff p1 = coeff p2" |
210 |
proof |
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show "p1 = p2 \<Longrightarrow> coeff p1 = coeff p2" |
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by simp |
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next |
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assume coeff_eq: "coeff p1 = coeff p2" |
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hence deg_eq: "degree p1 = degree p2" |
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using degree_def'[OF assms(1)] degree_def'[OF assms(2)] by auto |
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thus "p1 = p2" |
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proof (cases) |
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assume "p1 \<noteq> [] \<and> p2 \<noteq> []" |
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hence "length p1 = length p2" |
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using deg_eq by (simp add: Nitpick.size_list_simp(2)) |
68578 | 222 |
thus ?thesis |
223 |
using coeff_iff_length_cond[of p1 p2] coeff_eq by simp |
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224 |
next |
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{ fix p1 p2 assume A: "p1 = []" "coeff p1 = coeff p2" "polynomial K p2" |
68578 | 226 |
have "p2 = []" |
227 |
proof (rule ccontr) |
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228 |
assume "p2 \<noteq> []" |
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229 |
hence "(coeff p2) (degree p2) \<noteq> \<zero>" |
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230 |
using A(3) unfolding polynomial_def |
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231 |
by (metis coeff.simps(2) list.collapse) |
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232 |
moreover have "(coeff p1) ` UNIV = { \<zero> }" |
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233 |
using A(1) by auto |
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234 |
hence "(coeff p2) ` UNIV = { \<zero> }" |
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235 |
using A(2) by simp |
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236 |
ultimately show False |
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237 |
by blast |
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238 |
qed } note aux_lemma = this |
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239 |
assume "\<not> (p1 \<noteq> [] \<and> p2 \<noteq> [])" |
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hence "p1 = [] \<or> p2 = []" by simp |
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241 |
thus ?thesis |
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242 |
using assms coeff_eq aux_lemma[of p1 p2] aux_lemma[of p2 p1] by auto |
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243 |
qed |
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244 |
qed |
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245 |
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lemma normalize_lead_coeff: |
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247 |
assumes "length (normalize p) < length p" |
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248 |
shows "lead_coeff p = \<zero>" |
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249 |
proof (cases p) |
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250 |
case Nil thus ?thesis |
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251 |
using assms by simp |
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252 |
next |
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253 |
case (Cons a ps) thus ?thesis |
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using assms by (cases "a = \<zero>") (auto) |
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255 |
qed |
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256 |
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lemma normalize_length_lt: |
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258 |
assumes "lead_coeff p = \<zero>" and "length p > 0" |
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259 |
shows "length (normalize p) < length p" |
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260 |
proof (cases p) |
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261 |
case Nil thus ?thesis |
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262 |
using assms by simp |
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263 |
next |
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264 |
case (Cons a ps) thus ?thesis |
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265 |
using normalize_length_le[of ps] assms by simp |
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266 |
qed |
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lemma normalize_length_eq: |
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269 |
assumes "lead_coeff p \<noteq> \<zero>" |
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shows "length (normalize p) = length p" |
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using normalize_length_le[of p] assms nat_less_le normalize_lead_coeff by auto |
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lemma normalize_replicate_zero: "normalize ((replicate n \<zero>) @ p) = normalize p" |
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by (induction n) (auto) |
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lemma normalize_def': |
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shows "p = (replicate (length p - length (normalize p)) \<zero>) @ |
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(drop (length p - length (normalize p)) p)" (is ?statement1) |
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and "normalize p = drop (length p - length (normalize p)) p" (is ?statement2) |
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proof - |
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show ?statement1 |
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282 |
proof (induction p) |
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case Nil thus ?case by simp |
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284 |
next |
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case (Cons a p) thus ?case |
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proof (cases "a = \<zero>") |
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287 |
assume "a \<noteq> \<zero>" thus ?case |
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using Cons by simp |
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289 |
next |
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assume eq_zero: "a = \<zero>" |
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hence len_eq: |
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292 |
"Suc (length p - length (normalize p)) = length (a # p) - length (normalize (a # p))" |
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293 |
by (simp add: Suc_diff_le normalize_length_le) |
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294 |
have "a # p = \<zero> # (replicate (length p - length (normalize p)) \<zero> @ |
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295 |
drop (length p - length (normalize p)) p)" |
|
296 |
using eq_zero Cons by simp |
|
297 |
also have " ... = (replicate (Suc (length p - length (normalize p))) \<zero> @ |
|
298 |
drop (Suc (length p - length (normalize p))) (a # p))" |
|
299 |
by simp |
|
300 |
also have " ... = (replicate (length (a # p) - length (normalize (a # p))) \<zero> @ |
|
301 |
drop (length (a # p) - length (normalize (a # p))) (a # p))" |
|
302 |
using len_eq by simp |
|
303 |
finally show ?case . |
|
304 |
qed |
|
305 |
qed |
|
306 |
next |
|
307 |
show ?statement2 |
|
308 |
proof - |
|
309 |
have "\<exists>m. normalize p = drop m p" |
|
310 |
proof (induction p) |
|
311 |
case Nil thus ?case by simp |
|
312 |
next |
|
313 |
case (Cons a p) thus ?case |
|
314 |
apply (cases "a = \<zero>") |
|
315 |
apply (auto) |
|
316 |
apply (metis drop_Suc_Cons) |
|
317 |
apply (metis drop0) |
|
318 |
done |
|
319 |
qed |
|
320 |
then obtain m where m: "normalize p = drop m p" by auto |
|
321 |
hence "length (normalize p) = length p - m" by simp |
|
322 |
thus ?thesis |
|
323 |
using m by (metis rev_drop rev_rev_ident take_rev) |
|
324 |
qed |
|
325 |
qed |
|
326 |
||
327 |
lemma normalize_coeff: "coeff p = coeff (normalize p)" |
|
328 |
proof (induction p) |
|
329 |
case Nil thus ?case by simp |
|
330 |
next |
|
331 |
case (Cons a p) |
|
332 |
have "coeff (normalize p) (length p) = \<zero>" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
333 |
using normalize_length_le[of p] coeff_degree[of "normalize p"] |
68578 | 334 |
by (metis One_nat_def coeff.simps(1) diff_less length_0_conv |
335 |
less_imp_diff_less nat_neq_iff neq0_conv not_le zero_less_Suc) |
|
336 |
then show ?case |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
337 |
using Cons by (cases "a = \<zero>") (auto) |
68578 | 338 |
qed |
339 |
||
340 |
lemma append_coeff: |
|
341 |
"coeff (p @ q) = (\<lambda>i. if i < length q then (coeff q) i else (coeff p) (i - length q))" |
|
342 |
proof (induction p) |
|
343 |
case Nil thus ?case |
|
344 |
using coeff_length[of q] by auto |
|
345 |
next |
|
346 |
case (Cons a p) |
|
347 |
have "coeff ((a # p) @ q) = (\<lambda>i. if i = length p + length q then a else (coeff (p @ q)) i)" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
348 |
by auto |
68578 | 349 |
also have " ... = (\<lambda>i. if i = length p + length q then a |
350 |
else if i < length q then (coeff q) i |
|
351 |
else (coeff p) (i - length q))" |
|
352 |
using Cons by auto |
|
353 |
also have " ... = (\<lambda>i. if i < length q then (coeff q) i |
|
354 |
else if i = length p + length q then a else (coeff p) (i - length q))" |
|
355 |
by auto |
|
356 |
also have " ... = (\<lambda>i. if i < length q then (coeff q) i |
|
357 |
else if i - length q = length p then a else (coeff p) (i - length q))" |
|
358 |
by fastforce |
|
359 |
also have " ... = (\<lambda>i. if i < length q then (coeff q) i else (coeff (a # p)) (i - length q))" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
360 |
by auto |
68578 | 361 |
finally show ?case . |
362 |
qed |
|
363 |
||
364 |
lemma prefix_replicate_zero_coeff: "coeff p = coeff ((replicate n \<zero>) @ p)" |
|
365 |
using append_coeff[of "replicate n \<zero>" p] replicate_zero_coeff[of n] coeff_length[of p] by auto |
|
366 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
367 |
(* ========================================================================== *) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
368 |
context |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
369 |
fixes K :: "'a set" assumes K: "subring K R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
370 |
begin |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
371 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
372 |
lemma polynomial_in_carrier [intro]: "polynomial K p \<Longrightarrow> set p \<subseteq> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
373 |
unfolding polynomial_def using subringE(1)[OF K] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
374 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
375 |
lemma carrier_polynomial [intro]: "polynomial K p \<Longrightarrow> polynomial (carrier R) p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
376 |
unfolding polynomial_def using subringE(1)[OF K] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
377 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
378 |
lemma append_is_polynomial: "\<lbrakk> polynomial K p; p \<noteq> [] \<rbrakk> \<Longrightarrow> polynomial K (p @ (replicate n \<zero>))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
379 |
unfolding polynomial_def using subringE(2)[OF K] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
380 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
381 |
lemma lead_coeff_in_carrier: "polynomial K (a # p) \<Longrightarrow> a \<in> carrier R - { \<zero> }" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
382 |
unfolding polynomial_def using subringE(1)[OF K] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
383 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
384 |
lemma monom_is_polynomial [intro]: "a \<in> K - { \<zero> } \<Longrightarrow> polynomial K (monom a n)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
385 |
unfolding polynomial_def monom_def using subringE(2)[OF K] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
386 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
387 |
lemma eval_poly_in_carrier: "\<lbrakk> polynomial K p; x \<in> carrier R \<rbrakk> \<Longrightarrow> (eval p) x \<in> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
388 |
using eval_in_carrier[OF polynomial_in_carrier] . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
389 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
390 |
lemma poly_coeff_in_carrier [simp]: "polynomial K p \<Longrightarrow> coeff p i \<in> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
391 |
using coeff_in_carrier[OF polynomial_in_carrier] . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
392 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
393 |
end (* of fixed K context. *) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
394 |
(* ========================================================================== *) |
68578 | 395 |
|
396 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
397 |
subsection \<open>Polynomial Addition\<close> |
68578 | 398 |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
399 |
(* ========================================================================== *) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
400 |
context |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
401 |
fixes K :: "'a set" assumes K: "subring K R" |
68578 | 402 |
begin |
403 |
||
404 |
lemma poly_add_is_polynomial: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
405 |
assumes "set p1 \<subseteq> K" and "set p2 \<subseteq> K" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
406 |
shows "polynomial K (poly_add p1 p2)" |
68578 | 407 |
proof - |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
408 |
{ fix p1 p2 assume A: "set p1 \<subseteq> K" "set p2 \<subseteq> K" "length p1 \<ge> length p2" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
409 |
hence "polynomial K (poly_add p1 p2)" |
68578 | 410 |
proof - |
411 |
define p2' where "p2' = (replicate (length p1 - length p2) \<zero>) @ p2" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
412 |
hence "set p2' \<subseteq> K" and "length p1 = length p2'" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
413 |
using A(2-3) subringE(2)[OF K] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
414 |
hence "set (map2 (\<oplus>) p1 p2') \<subseteq> K" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
415 |
using A(1) subringE(7)[OF K] |
69712 | 416 |
by (induct p1) (auto, metis set_ConsD subsetD set_zip_leftD set_zip_rightD) |
68578 | 417 |
thus ?thesis |
418 |
unfolding p2'_def using normalize_gives_polynomial A(3) by simp |
|
419 |
qed } |
|
420 |
thus ?thesis |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
421 |
using assms by auto |
68578 | 422 |
qed |
423 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
424 |
lemma poly_add_closed: "\<lbrakk> polynomial K p1; polynomial K p2 \<rbrakk> \<Longrightarrow> polynomial K (poly_add p1 p2)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
425 |
using poly_add_is_polynomial polynomial_incl by simp |
68578 | 426 |
|
427 |
lemma poly_add_length_eq: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
428 |
assumes "polynomial K p1" "polynomial K p2" and "length p1 \<noteq> length p2" |
68578 | 429 |
shows "length (poly_add p1 p2) = max (length p1) (length p2)" |
430 |
proof - |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
431 |
{ fix p1 p2 assume A: "polynomial K p1" "polynomial K p2" "length p1 > length p2" |
68578 | 432 |
hence "length (poly_add p1 p2) = max (length p1) (length p2)" |
433 |
proof - |
|
434 |
let ?p2 = "(replicate (length p1 - length p2) \<zero>) @ p2" |
|
435 |
have p1: "p1 \<noteq> []" and p2: "?p2 \<noteq> []" |
|
436 |
using A(3) by auto |
|
68605 | 437 |
then have "zip p1 (replicate (length p1 - length p2) \<zero> @ p2) = zip (lead_coeff p1 # tl p1) (lead_coeff (replicate (length p1 - length p2) \<zero> @ p2) # tl (replicate (length p1 - length p2) \<zero> @ p2))" |
438 |
by auto |
|
68578 | 439 |
hence "lead_coeff (map2 (\<oplus>) p1 ?p2) = lead_coeff p1 \<oplus> lead_coeff ?p2" |
68605 | 440 |
by simp |
68578 | 441 |
moreover have "lead_coeff p1 \<in> carrier R" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
442 |
using p1 A(1) lead_coeff_in_carrier[OF K, of "hd p1" "tl p1"] by auto |
68578 | 443 |
ultimately have "lead_coeff (map2 (\<oplus>) p1 ?p2) = lead_coeff p1" |
444 |
using A(3) by auto |
|
445 |
moreover have "lead_coeff p1 \<noteq> \<zero>" |
|
446 |
using p1 A(1) unfolding polynomial_def by simp |
|
447 |
ultimately have "length (normalize (map2 (\<oplus>) p1 ?p2)) = length p1" |
|
448 |
using normalize_length_eq by auto |
|
449 |
thus ?thesis |
|
450 |
using A(3) by auto |
|
451 |
qed } |
|
452 |
thus ?thesis |
|
453 |
using assms by auto |
|
454 |
qed |
|
455 |
||
456 |
lemma poly_add_degree_eq: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
457 |
assumes "polynomial K p1" "polynomial K p2" and "degree p1 \<noteq> degree p2" |
68578 | 458 |
shows "degree (poly_add p1 p2) = max (degree p1) (degree p2)" |
459 |
using poly_add_length_eq[of p1 p2] assms |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
460 |
by (metis (no_types, lifting) diff_le_mono max.absorb_iff1 max_def) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
461 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
462 |
end (* of fixed K context. *) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
463 |
(* ========================================================================== *) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
464 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
465 |
lemma poly_add_in_carrier: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
466 |
"\<lbrakk> set p1 \<subseteq> carrier R; set p2 \<subseteq> carrier R \<rbrakk> \<Longrightarrow> set (poly_add p1 p2) \<subseteq> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
467 |
using polynomial_incl[OF poly_add_is_polynomial[OF carrier_is_subring]] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
468 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
469 |
lemma poly_add_length_le: "length (poly_add p1 p2) \<le> max (length p1) (length p2)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
470 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
471 |
{ fix p1 p2 :: "'a list" assume A: "length p1 \<ge> length p2" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
472 |
let ?p2 = "(replicate (length p1 - length p2) \<zero>) @ p2" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
473 |
have "length (poly_add p1 p2) \<le> max (length p1) (length p2)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
474 |
using normalize_length_le[of "map2 (\<oplus>) p1 ?p2"] A by auto } |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
475 |
thus ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
476 |
by (metis le_cases max.commute poly_add.simps) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
477 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
478 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
479 |
lemma poly_add_degree: "degree (poly_add p1 p2) \<le> max (degree p1) (degree p2)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
480 |
using poly_add_length_le by (meson diff_le_mono le_max_iff_disj) |
68578 | 481 |
|
482 |
lemma poly_add_coeff_aux: |
|
483 |
assumes "length p1 \<ge> length p2" |
|
484 |
shows "coeff (poly_add p1 p2) = (\<lambda>i. ((coeff p1) i) \<oplus> ((coeff p2) i))" |
|
485 |
proof |
|
486 |
fix i |
|
487 |
have "i < length p1 \<Longrightarrow> (coeff (poly_add p1 p2)) i = ((coeff p1) i) \<oplus> ((coeff p2) i)" |
|
488 |
proof - |
|
489 |
let ?p2 = "(replicate (length p1 - length p2) \<zero>) @ p2" |
|
490 |
have len_eqs: "length p1 = length ?p2" "length (map2 (\<oplus>) p1 ?p2) = length p1" |
|
491 |
using assms by auto |
|
492 |
assume i_lt: "i < length p1" |
|
493 |
have "(coeff (poly_add p1 p2)) i = (coeff (map2 (\<oplus>) p1 ?p2)) i" |
|
494 |
using normalize_coeff[of "map2 (\<oplus>) p1 ?p2"] assms by auto |
|
495 |
also have " ... = (map2 (\<oplus>) p1 ?p2) ! (length p1 - 1 - i)" |
|
496 |
using coeff_nth[of i "map2 (\<oplus>) p1 ?p2"] len_eqs(2) i_lt by auto |
|
497 |
also have " ... = (p1 ! (length p1 - 1 - i)) \<oplus> (?p2 ! (length ?p2 - 1 - i))" |
|
498 |
using len_eqs i_lt by auto |
|
499 |
also have " ... = ((coeff p1) i) \<oplus> ((coeff ?p2) i)" |
|
500 |
using coeff_nth[of i p1] coeff_nth[of i ?p2] i_lt len_eqs(1) by auto |
|
501 |
also have " ... = ((coeff p1) i) \<oplus> ((coeff p2) i)" |
|
502 |
using prefix_replicate_zero_coeff by simp |
|
503 |
finally show "(coeff (poly_add p1 p2)) i = ((coeff p1) i) \<oplus> ((coeff p2) i)" . |
|
504 |
qed |
|
505 |
moreover |
|
506 |
have "i \<ge> length p1 \<Longrightarrow> (coeff (poly_add p1 p2)) i = ((coeff p1) i) \<oplus> ((coeff p2) i)" |
|
507 |
using coeff_length[of "poly_add p1 p2"] coeff_length[of p1] coeff_length[of p2] |
|
508 |
poly_add_length_le[of p1 p2] assms by auto |
|
509 |
ultimately show "(coeff (poly_add p1 p2)) i = ((coeff p1) i) \<oplus> ((coeff p2) i)" |
|
510 |
using not_le by blast |
|
511 |
qed |
|
512 |
||
513 |
lemma poly_add_coeff: |
|
514 |
assumes "set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" |
|
515 |
shows "coeff (poly_add p1 p2) = (\<lambda>i. ((coeff p1) i) \<oplus> ((coeff p2) i))" |
|
516 |
proof - |
|
517 |
have "length p1 \<ge> length p2 \<or> length p2 > length p1" |
|
518 |
by auto |
|
519 |
thus ?thesis |
|
520 |
proof |
|
521 |
assume "length p1 \<ge> length p2" thus ?thesis |
|
522 |
using poly_add_coeff_aux by simp |
|
523 |
next |
|
524 |
assume "length p2 > length p1" |
|
525 |
hence "coeff (poly_add p1 p2) = (\<lambda>i. ((coeff p2) i) \<oplus> ((coeff p1) i))" |
|
526 |
using poly_add_coeff_aux by simp |
|
527 |
thus ?thesis |
|
528 |
using assms by (simp add: add.m_comm) |
|
529 |
qed |
|
530 |
qed |
|
531 |
||
532 |
lemma poly_add_comm: |
|
533 |
assumes "set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" |
|
534 |
shows "poly_add p1 p2 = poly_add p2 p1" |
|
535 |
proof - |
|
536 |
have "coeff (poly_add p1 p2) = coeff (poly_add p2 p1)" |
|
537 |
using poly_add_coeff[OF assms] poly_add_coeff[OF assms(2) assms(1)] |
|
538 |
coeff_in_carrier[OF assms(1)] coeff_in_carrier[OF assms(2)] add.m_comm by auto |
|
539 |
thus ?thesis |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
540 |
using coeff_iff_polynomial_cond[OF |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
541 |
poly_add_is_polynomial[OF carrier_is_subring assms] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
542 |
poly_add_is_polynomial[OF carrier_is_subring assms(2,1)]] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
543 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
544 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
545 |
lemma poly_add_monom: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
546 |
assumes "set p \<subseteq> carrier R" and "a \<in> carrier R - { \<zero> }" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
547 |
shows "poly_add (monom a (length p)) p = a # p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
548 |
unfolding monom_def using assms by (induction p) (auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
549 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
550 |
lemma poly_add_append_replicate: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
551 |
assumes "set p \<subseteq> carrier R" "set q \<subseteq> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
552 |
shows "poly_add (p @ (replicate (length q) \<zero>)) q = normalize (p @ q)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
553 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
554 |
have "map2 (\<oplus>) (p @ (replicate (length q) \<zero>)) ((replicate (length p) \<zero>) @ q) = p @ q" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
555 |
using assms by (induct p) (induct q, auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
556 |
thus ?thesis by simp |
68578 | 557 |
qed |
558 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
559 |
lemma poly_add_append_zero: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
560 |
assumes "set p \<subseteq> carrier R" "set q \<subseteq> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
561 |
shows "poly_add (p @ [ \<zero> ]) (q @ [ \<zero> ]) = normalize ((poly_add p q) @ [ \<zero> ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
562 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
563 |
have in_carrier: "set (p @ [ \<zero> ]) \<subseteq> carrier R" "set (q @ [ \<zero> ]) \<subseteq> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
564 |
using assms by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
565 |
have "coeff (poly_add (p @ [ \<zero> ]) (q @ [ \<zero> ])) = coeff ((poly_add p q) @ [ \<zero> ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
566 |
using append_coeff[of p "[ \<zero> ]"] poly_add_coeff[OF in_carrier] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
567 |
append_coeff[of q "[ \<zero> ]"] append_coeff[of "poly_add p q" "[ \<zero> ]"] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
568 |
poly_add_coeff[OF assms] assms[THEN coeff_in_carrier] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
569 |
hence "coeff (poly_add (p @ [ \<zero> ]) (q @ [ \<zero> ])) = coeff (normalize ((poly_add p q) @ [ \<zero> ]))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
570 |
using normalize_coeff by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
571 |
moreover have "set ((poly_add p q) @ [ \<zero> ]) \<subseteq> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
572 |
using poly_add_in_carrier[OF assms] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
573 |
ultimately show ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
574 |
using coeff_iff_polynomial_cond[OF poly_add_is_polynomial[OF carrier_is_subring in_carrier] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
575 |
normalize_gives_polynomial] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
576 |
qed |
68578 | 577 |
|
578 |
lemma poly_add_normalize_aux: |
|
579 |
assumes "set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" |
|
580 |
shows "poly_add p1 p2 = poly_add (normalize p1) p2" |
|
581 |
proof - |
|
582 |
{ fix n p1 p2 assume "set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" |
|
583 |
hence "poly_add p1 p2 = poly_add ((replicate n \<zero>) @ p1) p2" |
|
584 |
proof (induction n) |
|
585 |
case 0 thus ?case by simp |
|
586 |
next |
|
587 |
{ fix p1 p2 :: "'a list" |
|
588 |
assume in_carrier: "set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" |
|
589 |
have "poly_add p1 p2 = poly_add (\<zero> # p1) p2" |
|
590 |
proof - |
|
591 |
have "length p1 \<ge> length p2 \<Longrightarrow> ?thesis" |
|
592 |
proof - |
|
593 |
assume A: "length p1 \<ge> length p2" |
|
594 |
let ?p2 = "\<lambda>n. (replicate n \<zero>) @ p2" |
|
595 |
have "poly_add p1 p2 = normalize (map2 (\<oplus>) (\<zero> # p1) (\<zero> # ?p2 (length p1 - length p2)))" |
|
596 |
using A by simp |
|
597 |
also have " ... = normalize (map2 (\<oplus>) (\<zero> # p1) (?p2 (length (\<zero> # p1) - length p2)))" |
|
598 |
by (simp add: A Suc_diff_le) |
|
599 |
also have " ... = poly_add (\<zero> # p1) p2" |
|
600 |
using A by simp |
|
601 |
finally show ?thesis . |
|
602 |
qed |
|
603 |
||
604 |
moreover have "length p2 > length p1 \<Longrightarrow> ?thesis" |
|
605 |
proof - |
|
606 |
assume A: "length p2 > length p1" |
|
607 |
let ?f = "\<lambda>n p. (replicate n \<zero>) @ p" |
|
608 |
have "poly_add p1 p2 = poly_add p2 p1" |
|
609 |
using A by simp |
|
610 |
also have " ... = normalize (map2 (\<oplus>) p2 (?f (length p2 - length p1) p1))" |
|
611 |
using A by simp |
|
612 |
also have " ... = normalize (map2 (\<oplus>) p2 (?f (length p2 - Suc (length p1)) (\<zero> # p1)))" |
|
613 |
by (metis A Suc_diff_Suc append_Cons replicate_Suc replicate_app_Cons_same) |
|
614 |
also have " ... = poly_add p2 (\<zero> # p1)" |
|
615 |
using A by simp |
|
616 |
also have " ... = poly_add (\<zero> # p1) p2" |
|
617 |
using poly_add_comm[of p2 "\<zero> # p1"] in_carrier by auto |
|
618 |
finally show ?thesis . |
|
619 |
qed |
|
620 |
||
621 |
ultimately show ?thesis by auto |
|
622 |
qed } note aux_lemma = this |
|
623 |
||
624 |
case (Suc n) |
|
625 |
hence in_carrier: "set (replicate n \<zero> @ p1) \<subseteq> carrier R" |
|
626 |
by auto |
|
627 |
have "poly_add p1 p2 = poly_add (replicate n \<zero> @ p1) p2" |
|
628 |
using Suc by simp |
|
629 |
also have " ... = poly_add (replicate (Suc n) \<zero> @ p1) p2" |
|
630 |
using aux_lemma[OF in_carrier Suc(3)] by simp |
|
631 |
finally show ?case . |
|
632 |
qed } note aux_lemma = this |
|
633 |
||
634 |
have "poly_add p1 p2 = |
|
635 |
poly_add ((replicate (length p1 - length (normalize p1)) \<zero>) @ normalize p1) p2" |
|
636 |
using normalize_def'[of p1] by simp |
|
637 |
also have " ... = poly_add (normalize p1) p2" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
638 |
using aux_lemma[OF normalize_in_carrier[OF assms(1)] assms(2)] by simp |
68578 | 639 |
finally show ?thesis . |
640 |
qed |
|
641 |
||
642 |
lemma poly_add_normalize: |
|
643 |
assumes "set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" |
|
644 |
shows "poly_add p1 p2 = poly_add (normalize p1) p2" |
|
645 |
and "poly_add p1 p2 = poly_add p1 (normalize p2)" |
|
646 |
and "poly_add p1 p2 = poly_add (normalize p1) (normalize p2)" |
|
647 |
proof - |
|
648 |
show "poly_add p1 p2 = poly_add p1 (normalize p2)" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
649 |
unfolding poly_add_comm[OF assms] poly_add_normalize_aux[OF assms(2) assms(1)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
650 |
poly_add_comm[OF normalize_in_carrier[OF assms(2)] assms(1)] by simp |
68578 | 651 |
next |
652 |
show "poly_add p1 p2 = poly_add (normalize p1) p2" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
653 |
using poly_add_normalize_aux[OF assms] . |
68578 | 654 |
also have " ... = poly_add (normalize p2) (normalize p1)" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
655 |
unfolding poly_add_comm[OF normalize_in_carrier[OF assms(1)] assms(2)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
656 |
poly_add_normalize_aux[OF assms(2) normalize_in_carrier[OF assms(1)]] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
657 |
finally show "poly_add p1 p2 = poly_add (normalize p1) (normalize p2)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
658 |
unfolding poly_add_comm[OF assms[THEN normalize_in_carrier]] . |
68578 | 659 |
qed |
660 |
||
661 |
lemma poly_add_zero': |
|
662 |
assumes "set p \<subseteq> carrier R" |
|
663 |
shows "poly_add p [] = normalize p" and "poly_add [] p = normalize p" |
|
664 |
proof - |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
665 |
have "map2 (\<oplus>) p (replicate (length p) \<zero>) = p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
666 |
using assms by (induct p) (auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
667 |
thus "poly_add p [] = normalize p" and "poly_add [] p = normalize p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
668 |
using poly_add_comm[OF assms, of "[]"] by simp+ |
68578 | 669 |
qed |
670 |
||
671 |
lemma poly_add_zero: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
672 |
assumes "subring K R" "polynomial K p" |
68578 | 673 |
shows "poly_add p [] = p" and "poly_add [] p = p" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
674 |
using poly_add_zero' normalize_polynomial polynomial_in_carrier assms by auto |
68578 | 675 |
|
676 |
lemma poly_add_replicate_zero': |
|
677 |
assumes "set p \<subseteq> carrier R" |
|
678 |
shows "poly_add p (replicate n \<zero>) = normalize p" and "poly_add (replicate n \<zero>) p = normalize p" |
|
679 |
proof - |
|
680 |
have "poly_add p (replicate n \<zero>) = poly_add p []" |
|
681 |
using poly_add_normalize(2)[OF assms, of "replicate n \<zero>"] |
|
682 |
normalize_replicate_zero[of n "[]"] by force |
|
683 |
also have " ... = normalize p" |
|
684 |
using poly_add_zero'[OF assms] by simp |
|
685 |
finally show "poly_add p (replicate n \<zero>) = normalize p" . |
|
686 |
thus "poly_add (replicate n \<zero>) p = normalize p" |
|
687 |
using poly_add_comm[OF assms, of "replicate n \<zero>"] by force |
|
688 |
qed |
|
689 |
||
690 |
lemma poly_add_replicate_zero: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
691 |
assumes "subring K R" "polynomial K p" |
68578 | 692 |
shows "poly_add p (replicate n \<zero>) = p" and "poly_add (replicate n \<zero>) p = p" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
693 |
using poly_add_replicate_zero' normalize_polynomial polynomial_in_carrier assms by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
694 |
|
68578 | 695 |
|
696 |
||
697 |
subsection \<open>Dense Representation\<close> |
|
698 |
||
699 |
lemma dense_repr_replicate_zero: "dense_repr ((replicate n \<zero>) @ p) = dense_repr p" |
|
700 |
by (induction n) (auto) |
|
701 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
702 |
lemma dense_repr_normalize: "dense_repr (normalize p) = dense_repr p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
703 |
by (induct p) (auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
704 |
|
68578 | 705 |
lemma polynomial_dense_repr: |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
706 |
assumes "polynomial K p" and "p \<noteq> []" |
68578 | 707 |
shows "dense_repr p = (lead_coeff p, degree p) # dense_repr (normalize (tl p))" |
708 |
proof - |
|
709 |
let ?len = length and ?norm = normalize |
|
710 |
obtain a p' where p: "p = a # p'" |
|
711 |
using assms(2) list.exhaust_sel by blast |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
712 |
hence a: "a \<in> K - { \<zero> }" and p': "set p' \<subseteq> K" |
68578 | 713 |
using assms(1) unfolding p by (auto simp add: polynomial_def) |
714 |
hence "dense_repr p = (lead_coeff p, degree p) # dense_repr p'" |
|
715 |
unfolding p by simp |
|
716 |
also have " ... = |
|
717 |
(lead_coeff p, degree p) # dense_repr ((replicate (?len p' - ?len (?norm p')) \<zero>) @ ?norm p')" |
|
718 |
using normalize_def' dense_repr_replicate_zero by simp |
|
719 |
also have " ... = (lead_coeff p, degree p) # dense_repr (?norm p')" |
|
720 |
using dense_repr_replicate_zero by simp |
|
721 |
finally show ?thesis |
|
722 |
unfolding p by simp |
|
723 |
qed |
|
724 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
725 |
lemma monom_decomp: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
726 |
assumes "subring K R" "polynomial K p" |
68578 | 727 |
shows "p = of_dense (dense_repr p)" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
728 |
using assms(2) |
68578 | 729 |
proof (induct "length p" arbitrary: p rule: less_induct) |
730 |
case less thus ?case |
|
731 |
proof (cases p) |
|
732 |
case Nil thus ?thesis by simp |
|
733 |
next |
|
734 |
case (Cons a l) |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
735 |
hence a: "a \<in> carrier R - { \<zero> }" and l: "set l \<subseteq> carrier R" "set l \<subseteq> K" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
736 |
using less(2) subringE(1)[OF assms(1)] by (auto simp add: polynomial_def) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
737 |
hence "a # l = poly_add (monom a (degree (a # l))) l" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
738 |
using poly_add_monom[of l a] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
739 |
also have " ... = poly_add (monom a (degree (a # l))) (normalize l)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
740 |
using poly_add_normalize(2)[of "monom a (degree (a # l))", OF _ l(1)] a |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
741 |
unfolding monom_def by force |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
742 |
also have " ... = poly_add (monom a (degree (a # l))) (of_dense (dense_repr (normalize l)))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
743 |
using less(1)[OF _ normalize_gives_polynomial[OF l(2)]] normalize_length_le[of l] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
744 |
unfolding Cons by simp |
68578 | 745 |
also have " ... = of_dense ((a, degree (a # l)) # dense_repr (normalize l))" |
746 |
by simp |
|
747 |
also have " ... = of_dense (dense_repr (a # l))" |
|
748 |
using polynomial_dense_repr[OF less(2)] unfolding Cons by simp |
|
749 |
finally show ?thesis |
|
750 |
unfolding Cons by simp |
|
751 |
qed |
|
752 |
qed |
|
753 |
||
754 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
755 |
subsection \<open>Polynomial Multiplication\<close> |
68578 | 756 |
|
757 |
lemma poly_mult_is_polynomial: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
758 |
assumes "subring K R" "set p1 \<subseteq> K" and "set p2 \<subseteq> K" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
759 |
shows "polynomial K (poly_mult p1 p2)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
760 |
using assms(2-3) |
68578 | 761 |
proof (induction p1) |
762 |
case Nil thus ?case |
|
763 |
by (simp add: polynomial_def) |
|
764 |
next |
|
765 |
case (Cons a p1) |
|
766 |
let ?a_p2 = "(map (\<lambda>b. a \<otimes> b) p2) @ (replicate (degree (a # p1)) \<zero>)" |
|
767 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
768 |
have "set (poly_mult p1 p2) \<subseteq> K" |
68578 | 769 |
using Cons unfolding polynomial_def by auto |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
770 |
moreover have "set ?a_p2 \<subseteq> K" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
771 |
using assms(3) Cons(2) subringE(1-2,6)[OF assms(1)] by(induct p2) (auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
772 |
ultimately have "polynomial K (poly_add ?a_p2 (poly_mult p1 p2))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
773 |
using poly_add_is_polynomial[OF assms(1)] by blast |
68578 | 774 |
thus ?case by simp |
775 |
qed |
|
776 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
777 |
lemma poly_mult_closed: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
778 |
assumes "subring K R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
779 |
shows "\<lbrakk> polynomial K p1; polynomial K p2 \<rbrakk> \<Longrightarrow> polynomial K (poly_mult p1 p2)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
780 |
using poly_mult_is_polynomial polynomial_incl assms by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
781 |
|
68578 | 782 |
lemma poly_mult_in_carrier: |
783 |
"\<lbrakk> set p1 \<subseteq> carrier R; set p2 \<subseteq> carrier R \<rbrakk> \<Longrightarrow> set (poly_mult p1 p2) \<subseteq> carrier R" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
784 |
using poly_mult_is_polynomial polynomial_in_carrier carrier_is_subring by simp |
68578 | 785 |
|
786 |
lemma poly_mult_coeff: |
|
787 |
assumes "set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" |
|
788 |
shows "coeff (poly_mult p1 p2) = (\<lambda>i. \<Oplus> k \<in> {..i}. (coeff p1) k \<otimes> (coeff p2) (i - k))" |
|
789 |
using assms(1) |
|
790 |
proof (induction p1) |
|
791 |
case Nil thus ?case using assms(2) by auto |
|
792 |
next |
|
793 |
case (Cons a p1) |
|
794 |
hence in_carrier: |
|
795 |
"a \<in> carrier R" "\<And>i. (coeff p1) i \<in> carrier R" "\<And>i. (coeff p2) i \<in> carrier R" |
|
796 |
using coeff_in_carrier assms(2) by auto |
|
797 |
||
798 |
let ?a_p2 = "(map (\<lambda>b. a \<otimes> b) p2) @ (replicate (degree (a # p1)) \<zero>)" |
|
799 |
have "coeff (replicate (degree (a # p1)) \<zero>) = (\<lambda>_. \<zero>)" |
|
800 |
and "length (replicate (degree (a # p1)) \<zero>) = length p1" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
801 |
using prefix_replicate_zero_coeff[of "[]" "length p1"] by auto |
68578 | 802 |
hence "coeff ?a_p2 = (\<lambda>i. if i < length p1 then \<zero> else (coeff (map (\<lambda>b. a \<otimes> b) p2)) (i - length p1))" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
803 |
using append_coeff[of "map (\<lambda>b. a \<otimes> b) p2" "replicate (length p1) \<zero>"] by auto |
68578 | 804 |
also have " ... = (\<lambda>i. if i < length p1 then \<zero> else a \<otimes> ((coeff p2) (i - length p1)))" |
805 |
proof - |
|
806 |
have "\<And>i. i < length p2 \<Longrightarrow> (coeff (map (\<lambda>b. a \<otimes> b) p2)) i = a \<otimes> ((coeff p2) i)" |
|
807 |
proof - |
|
808 |
fix i assume i_lt: "i < length p2" |
|
809 |
hence "(coeff (map (\<lambda>b. a \<otimes> b) p2)) i = (map (\<lambda>b. a \<otimes> b) p2) ! (length p2 - 1 - i)" |
|
810 |
using coeff_nth[of i "map (\<lambda>b. a \<otimes> b) p2"] by auto |
|
811 |
also have " ... = a \<otimes> (p2 ! (length p2 - 1 - i))" |
|
812 |
using i_lt by auto |
|
813 |
also have " ... = a \<otimes> ((coeff p2) i)" |
|
814 |
using coeff_nth[OF i_lt] by simp |
|
815 |
finally show "(coeff (map (\<lambda>b. a \<otimes> b) p2)) i = a \<otimes> ((coeff p2) i)" . |
|
816 |
qed |
|
817 |
moreover have "\<And>i. i \<ge> length p2 \<Longrightarrow> (coeff (map (\<lambda>b. a \<otimes> b) p2)) i = a \<otimes> ((coeff p2) i)" |
|
818 |
using coeff_length[of p2] coeff_length[of "map (\<lambda>b. a \<otimes> b) p2"] in_carrier by auto |
|
819 |
ultimately show ?thesis by (meson not_le) |
|
820 |
qed |
|
821 |
also have " ... = (\<lambda>i. \<Oplus> k \<in> {..i}. (if k = length p1 then a else \<zero>) \<otimes> (coeff p2) (i - k))" |
|
822 |
(is "?f1 = (\<lambda>i. (\<Oplus> k \<in> {..i}. ?f2 k \<otimes> ?f3 (i - k)))") |
|
823 |
proof |
|
824 |
fix i |
|
825 |
have "\<And>k. k \<in> {..i} \<Longrightarrow> ?f2 k \<otimes> ?f3 (i - k) = \<zero>" if "i < length p1" |
|
826 |
using in_carrier that by auto |
|
827 |
hence "(\<Oplus> k \<in> {..i}. ?f2 k \<otimes> ?f3 (i - k)) = \<zero>" if "i < length p1" |
|
828 |
using that in_carrier |
|
829 |
add.finprod_cong'[of "{..i}" "{..i}" "\<lambda>k. ?f2 k \<otimes> ?f3 (i - k)" "\<lambda>i. \<zero>"] |
|
830 |
by auto |
|
831 |
hence eq_lt: "?f1 i = (\<lambda>i. (\<Oplus> k \<in> {..i}. ?f2 k \<otimes> ?f3 (i - k))) i" if "i < length p1" |
|
832 |
using that by auto |
|
833 |
||
834 |
have "\<And>k. k \<in> {..i} \<Longrightarrow> |
|
835 |
?f2 k \<otimes>\<^bsub>R\<^esub> ?f3 (i - k) = (if length p1 = k then a \<otimes> coeff p2 (i - k) else \<zero>)" |
|
836 |
using in_carrier by auto |
|
837 |
hence "(\<Oplus> k \<in> {..i}. ?f2 k \<otimes> ?f3 (i - k)) = |
|
838 |
(\<Oplus> k \<in> {..i}. (if length p1 = k then a \<otimes> coeff p2 (i - k) else \<zero>))" |
|
839 |
using in_carrier |
|
840 |
add.finprod_cong'[of "{..i}" "{..i}" "\<lambda>k. ?f2 k \<otimes> ?f3 (i - k)" |
|
841 |
"\<lambda>k. (if length p1 = k then a \<otimes> coeff p2 (i - k) else \<zero>)"] |
|
842 |
by fastforce |
|
843 |
also have " ... = a \<otimes> (coeff p2) (i - length p1)" if "i \<ge> length p1" |
|
844 |
using add.finprod_singleton[of "length p1" "{..i}" "\<lambda>j. a \<otimes> (coeff p2) (i - j)"] |
|
845 |
in_carrier that by auto |
|
846 |
finally |
|
847 |
have "(\<Oplus> k \<in> {..i}. ?f2 k \<otimes> ?f3 (i - k)) = a \<otimes> (coeff p2) (i - length p1)" if "i \<ge> length p1" |
|
848 |
using that by simp |
|
849 |
hence eq_ge: "?f1 i = (\<lambda>i. (\<Oplus> k \<in> {..i}. ?f2 k \<otimes> ?f3 (i - k))) i" if "i \<ge> length p1" |
|
850 |
using that by auto |
|
851 |
||
852 |
from eq_lt eq_ge show "?f1 i = (\<lambda>i. (\<Oplus> k \<in> {..i}. ?f2 k \<otimes> ?f3 (i - k))) i" by auto |
|
853 |
qed |
|
854 |
||
855 |
finally have coeff_a_p2: |
|
856 |
"coeff ?a_p2 = (\<lambda>i. \<Oplus> k \<in> {..i}. (if k = length p1 then a else \<zero>) \<otimes> (coeff p2) (i - k))" . |
|
857 |
||
858 |
have "set ?a_p2 \<subseteq> carrier R" |
|
859 |
using in_carrier(1) assms(2) by auto |
|
860 |
||
861 |
moreover have "set (poly_mult p1 p2) \<subseteq> carrier R" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
862 |
using poly_mult_in_carrier[OF _ assms(2)] Cons(2) by simp |
68578 | 863 |
|
864 |
ultimately |
|
865 |
have "coeff (poly_mult (a # p1) p2) = (\<lambda>i. ((coeff ?a_p2) i) \<oplus> ((coeff (poly_mult p1 p2)) i))" |
|
866 |
using poly_add_coeff[of ?a_p2 "poly_mult p1 p2"] by simp |
|
867 |
also have " ... = (\<lambda>i. (\<Oplus> k \<in> {..i}. (if k = length p1 then a else \<zero>) \<otimes> (coeff p2) (i - k)) \<oplus> |
|
868 |
(\<Oplus> k \<in> {..i}. (coeff p1) k \<otimes> (coeff p2) (i - k)))" |
|
869 |
using Cons coeff_a_p2 by simp |
|
870 |
also have " ... = (\<lambda>i. (\<Oplus> k \<in> {..i}. ((if k = length p1 then a else \<zero>) \<otimes> (coeff p2) (i - k)) \<oplus> |
|
871 |
((coeff p1) k \<otimes> (coeff p2) (i - k))))" |
|
872 |
using add.finprod_multf in_carrier by auto |
|
873 |
also have " ... = (\<lambda>i. (\<Oplus> k \<in> {..i}. (coeff (a # p1) k) \<otimes> (coeff p2) (i - k)))" |
|
874 |
(is "(\<lambda>i. (\<Oplus> k \<in> {..i}. ?f i k)) = (\<lambda>i. (\<Oplus> k \<in> {..i}. ?g i k))") |
|
875 |
proof |
|
876 |
fix i |
|
877 |
have "\<And>k. ?f i k = ?g i k" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
878 |
using in_carrier coeff_length[of p1] by auto |
68578 | 879 |
thus "(\<Oplus> k \<in> {..i}. ?f i k) = (\<Oplus> k \<in> {..i}. ?g i k)" by simp |
880 |
qed |
|
881 |
finally show ?case . |
|
882 |
qed |
|
883 |
||
884 |
lemma poly_mult_zero: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
885 |
assumes "set p \<subseteq> carrier R" |
68578 | 886 |
shows "poly_mult [] p = []" and "poly_mult p [] = []" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
887 |
proof (simp) |
68578 | 888 |
have "coeff (poly_mult p []) = (\<lambda>_. \<zero>)" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
889 |
using poly_mult_coeff[OF assms, of "[]"] coeff_in_carrier[OF assms] by auto |
68578 | 890 |
thus "poly_mult p [] = []" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
891 |
using coeff_iff_polynomial_cond[OF |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
892 |
poly_mult_is_polynomial[OF carrier_is_subring assms] zero_is_polynomial] by simp |
68578 | 893 |
qed |
894 |
||
895 |
lemma poly_mult_l_distr': |
|
896 |
assumes "set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" "set p3 \<subseteq> carrier R" |
|
897 |
shows "poly_mult (poly_add p1 p2) p3 = poly_add (poly_mult p1 p3) (poly_mult p2 p3)" |
|
898 |
proof - |
|
899 |
let ?c1 = "coeff p1" and ?c2 = "coeff p2" and ?c3 = "coeff p3" |
|
900 |
have in_carrier: |
|
901 |
"\<And>i. ?c1 i \<in> carrier R" "\<And>i. ?c2 i \<in> carrier R" "\<And>i. ?c3 i \<in> carrier R" |
|
902 |
using assms coeff_in_carrier by auto |
|
903 |
||
904 |
have "coeff (poly_mult (poly_add p1 p2) p3) = (\<lambda>n. \<Oplus>i \<in> {..n}. (?c1 i \<oplus> ?c2 i) \<otimes> ?c3 (n - i))" |
|
905 |
using poly_mult_coeff[of "poly_add p1 p2" p3] poly_add_coeff[OF assms(1-2)] |
|
906 |
poly_add_in_carrier[OF assms(1-2)] assms by auto |
|
907 |
also have " ... = (\<lambda>n. \<Oplus>i \<in> {..n}. (?c1 i \<otimes> ?c3 (n - i)) \<oplus> (?c2 i \<otimes> ?c3 (n - i)))" |
|
908 |
using in_carrier l_distr by auto |
|
909 |
also |
|
910 |
have " ... = (\<lambda>n. (\<Oplus>i \<in> {..n}. (?c1 i \<otimes> ?c3 (n - i))) \<oplus> (\<Oplus>i \<in> {..n}. (?c2 i \<otimes> ?c3 (n - i))))" |
|
911 |
using add.finprod_multf in_carrier by auto |
|
912 |
also have " ... = coeff (poly_add (poly_mult p1 p3) (poly_mult p2 p3))" |
|
913 |
using poly_mult_coeff[OF assms(1) assms(3)] poly_mult_coeff[OF assms(2-3)] |
|
914 |
poly_add_coeff[OF poly_mult_in_carrier[OF assms(1) assms(3)]] |
|
915 |
poly_mult_in_carrier[OF assms(2-3)] by simp |
|
916 |
finally have "coeff (poly_mult (poly_add p1 p2) p3) = |
|
917 |
coeff (poly_add (poly_mult p1 p3) (poly_mult p2 p3))" . |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
918 |
moreover have "polynomial (carrier R) (poly_mult (poly_add p1 p2) p3)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
919 |
and "polynomial (carrier R) (poly_add (poly_mult p1 p3) (poly_mult p2 p3))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
920 |
using assms poly_add_is_polynomial poly_mult_is_polynomial polynomial_in_carrier |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
921 |
carrier_is_subring by auto |
68578 | 922 |
ultimately show ?thesis |
923 |
using coeff_iff_polynomial_cond by auto |
|
924 |
qed |
|
925 |
||
926 |
lemma poly_mult_l_distr: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
927 |
assumes "subring K R" "polynomial K p1" "polynomial K p2" "polynomial K p3" |
68578 | 928 |
shows "poly_mult (poly_add p1 p2) p3 = poly_add (poly_mult p1 p3) (poly_mult p2 p3)" |
929 |
using poly_mult_l_distr' polynomial_in_carrier assms by auto |
|
930 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
931 |
lemma poly_mult_prepend_replicate_zero: |
68578 | 932 |
assumes "set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" |
933 |
shows "poly_mult p1 p2 = poly_mult ((replicate n \<zero>) @ p1) p2" |
|
934 |
proof - |
|
935 |
{ fix p1 p2 assume A: "set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" |
|
936 |
hence "poly_mult p1 p2 = poly_mult (\<zero> # p1) p2" |
|
937 |
proof - |
|
938 |
let ?a_p2 = "(map ((\<otimes>) \<zero>) p2) @ (replicate (length p1) \<zero>)" |
|
939 |
have "?a_p2 = replicate (length p2 + length p1) \<zero>" |
|
940 |
using A(2) by (induction p2) (auto) |
|
941 |
hence "poly_mult (\<zero> # p1) p2 = poly_add (replicate (length p2 + length p1) \<zero>) (poly_mult p1 p2)" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
942 |
by simp |
68578 | 943 |
also have " ... = poly_add (normalize (replicate (length p2 + length p1) \<zero>)) (poly_mult p1 p2)" |
944 |
using poly_add_normalize(1)[of "replicate (length p2 + length p1) \<zero>" "poly_mult p1 p2"] |
|
945 |
poly_mult_in_carrier[OF A] by force |
|
946 |
also have " ... = poly_mult p1 p2" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
947 |
using poly_add_zero(2)[OF _ poly_mult_is_polynomial[OF _ A]] carrier_is_subring |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
948 |
normalize_replicate_zero[of "length p2 + length p1" "[]"] by simp |
68578 | 949 |
finally show ?thesis by auto |
950 |
qed } note aux_lemma = this |
|
951 |
||
952 |
from assms show ?thesis |
|
953 |
proof (induction n) |
|
954 |
case 0 thus ?case by simp |
|
955 |
next |
|
956 |
case (Suc n) thus ?case |
|
957 |
using aux_lemma[of "replicate n \<zero> @ p1" p2] by force |
|
958 |
qed |
|
959 |
qed |
|
960 |
||
961 |
lemma poly_mult_normalize: |
|
962 |
assumes "set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" |
|
963 |
shows "poly_mult p1 p2 = poly_mult (normalize p1) p2" |
|
964 |
proof - |
|
965 |
let ?replicate = "replicate (length p1 - length (normalize p1)) \<zero>" |
|
966 |
have "poly_mult p1 p2 = poly_mult (?replicate @ (normalize p1)) p2" |
|
967 |
using normalize_def'[of p1] by simp |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
968 |
thus ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
969 |
using poly_mult_prepend_replicate_zero normalize_in_carrier assms by auto |
68578 | 970 |
qed |
971 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
972 |
lemma poly_mult_append_zero: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
973 |
assumes "set p \<subseteq> carrier R" "set q \<subseteq> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
974 |
shows "poly_mult (p @ [ \<zero> ]) q = normalize ((poly_mult p q) @ [ \<zero> ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
975 |
using assms(1) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
976 |
proof (induct p) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
977 |
case Nil thus ?case |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
978 |
using poly_mult_normalize[OF _ assms(2), of "[] @ [ \<zero> ]"] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
979 |
poly_mult_zero(1) poly_mult_zero(1)[of "q @ [ \<zero> ]"] assms(2) by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
980 |
next |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
981 |
case (Cons a p) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
982 |
let ?q_a = "\<lambda>n. (map ((\<otimes>) a) q) @ (replicate n \<zero>)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
983 |
have set_q_a: "\<And>n. set (?q_a n) \<subseteq> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
984 |
using Cons(2) assms(2) by (induct q) (auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
985 |
have set_poly_mult: "set ((poly_mult p q) @ [ \<zero> ]) \<subseteq> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
986 |
using poly_mult_in_carrier[OF _ assms(2)] Cons(2) by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
987 |
have "poly_mult ((a # p) @ [\<zero>]) q = poly_add (?q_a (Suc (length p))) (poly_mult (p @ [\<zero>]) q)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
988 |
by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
989 |
also have " ... = poly_add (?q_a (Suc (length p))) (normalize ((poly_mult p q) @ [ \<zero> ]))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
990 |
using Cons by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
991 |
also have " ... = poly_add ((?q_a (length p)) @ [ \<zero> ]) ((poly_mult p q) @ [ \<zero> ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
992 |
using poly_add_normalize(2)[OF set_q_a[of "Suc (length p)"] set_poly_mult] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
993 |
by (simp add: replicate_append_same) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
994 |
also have " ... = normalize ((poly_add (?q_a (length p)) (poly_mult p q)) @ [ \<zero> ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
995 |
using poly_add_append_zero[OF set_q_a[of "length p"] poly_mult_in_carrier[OF _ assms(2)]] Cons(2) by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
996 |
also have " ... = normalize ((poly_mult (a # p) q) @ [ \<zero> ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
997 |
by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
998 |
finally show ?case . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
999 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1000 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1001 |
end (* of ring context. *) |
68578 | 1002 |
|
1003 |
||
1004 |
subsection \<open>Properties Within a Domain\<close> |
|
1005 |
||
1006 |
context domain |
|
1007 |
begin |
|
1008 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1009 |
lemma one_is_polynomial [intro]: "subring K R \<Longrightarrow> polynomial K [ \<one> ]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1010 |
unfolding polynomial_def using subringE(3) by auto |
68578 | 1011 |
|
1012 |
lemma poly_mult_comm: |
|
1013 |
assumes "set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" |
|
1014 |
shows "poly_mult p1 p2 = poly_mult p2 p1" |
|
1015 |
proof - |
|
1016 |
let ?c1 = "coeff p1" and ?c2 = "coeff p2" |
|
1017 |
have "\<And>i. (\<Oplus>k \<in> {..i}. ?c1 k \<otimes> ?c2 (i - k)) = (\<Oplus>k \<in> {..i}. ?c2 k \<otimes> ?c1 (i - k))" |
|
1018 |
proof - |
|
1019 |
fix i :: nat |
|
1020 |
let ?f = "\<lambda>k. ?c1 k \<otimes> ?c2 (i - k)" |
|
1021 |
have in_carrier: "\<And>i. ?c1 i \<in> carrier R" "\<And>i. ?c2 i \<in> carrier R" |
|
1022 |
using coeff_in_carrier[OF assms(1)] coeff_in_carrier[OF assms(2)] by auto |
|
1023 |
||
1024 |
have reindex_inj: "inj_on (\<lambda>k. i - k) {..i}" |
|
1025 |
using inj_on_def by force |
|
1026 |
moreover have "(\<lambda>k. i - k) ` {..i} \<subseteq> {..i}" by auto |
|
1027 |
hence "(\<lambda>k. i - k) ` {..i} = {..i}" |
|
1028 |
using reindex_inj endo_inj_surj[of "{..i}" "\<lambda>k. i - k"] by simp |
|
1029 |
ultimately have "(\<Oplus>k \<in> {..i}. ?f k) = (\<Oplus>k \<in> {..i}. ?f (i - k))" |
|
1030 |
using add.finprod_reindex[of ?f "\<lambda>k. i - k" "{..i}"] in_carrier by auto |
|
1031 |
||
1032 |
moreover have "\<And>k. k \<in> {..i} \<Longrightarrow> ?f (i - k) = ?c2 k \<otimes> ?c1 (i - k)" |
|
1033 |
using in_carrier m_comm by auto |
|
1034 |
hence "(\<Oplus>k \<in> {..i}. ?f (i - k)) = (\<Oplus>k \<in> {..i}. ?c2 k \<otimes> ?c1 (i - k))" |
|
1035 |
using add.finprod_cong'[of "{..i}" "{..i}"] in_carrier by auto |
|
1036 |
ultimately show "(\<Oplus>k \<in> {..i}. ?f k) = (\<Oplus>k \<in> {..i}. ?c2 k \<otimes> ?c1 (i - k))" |
|
1037 |
by simp |
|
1038 |
qed |
|
1039 |
hence "coeff (poly_mult p1 p2) = coeff (poly_mult p2 p1)" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1040 |
using poly_mult_coeff[OF assms] poly_mult_coeff[OF assms(2,1)] by simp |
68578 | 1041 |
thus ?thesis |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1042 |
using coeff_iff_polynomial_cond[OF poly_mult_is_polynomial[OF _ assms] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1043 |
poly_mult_is_polynomial[OF _ assms(2,1)]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1044 |
carrier_is_subring by simp |
68578 | 1045 |
qed |
1046 |
||
1047 |
lemma poly_mult_r_distr': |
|
1048 |
assumes "set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" "set p3 \<subseteq> carrier R" |
|
1049 |
shows "poly_mult p1 (poly_add p2 p3) = poly_add (poly_mult p1 p2) (poly_mult p1 p3)" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1050 |
unfolding poly_mult_comm[OF assms(1) poly_add_in_carrier[OF assms(2-3)]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1051 |
poly_mult_l_distr'[OF assms(2-3,1)] assms(2-3)[THEN poly_mult_comm[OF _ assms(1)]] .. |
68578 | 1052 |
|
1053 |
lemma poly_mult_r_distr: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1054 |
assumes "subring K R" "polynomial K p1" "polynomial K p2" "polynomial K p3" |
68578 | 1055 |
shows "poly_mult p1 (poly_add p2 p3) = poly_add (poly_mult p1 p2) (poly_mult p1 p3)" |
1056 |
using poly_mult_r_distr' polynomial_in_carrier assms by auto |
|
1057 |
||
1058 |
lemma poly_mult_replicate_zero: |
|
1059 |
assumes "set p \<subseteq> carrier R" |
|
1060 |
shows "poly_mult (replicate n \<zero>) p = []" |
|
1061 |
and "poly_mult p (replicate n \<zero>) = []" |
|
1062 |
proof - |
|
1063 |
have in_carrier: "\<And>n. set (replicate n \<zero>) \<subseteq> carrier R" by auto |
|
1064 |
show "poly_mult (replicate n \<zero>) p = []" using assms |
|
1065 |
proof (induction n) |
|
1066 |
case 0 thus ?case by simp |
|
1067 |
next |
|
1068 |
case (Suc n) |
|
1069 |
hence "poly_mult (replicate (Suc n) \<zero>) p = poly_mult (\<zero> # (replicate n \<zero>)) p" |
|
1070 |
by simp |
|
1071 |
also have " ... = poly_add ((map (\<lambda>a. \<zero> \<otimes> a) p) @ (replicate n \<zero>)) []" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1072 |
using Suc by simp |
68578 | 1073 |
also have " ... = poly_add ((map (\<lambda>a. \<zero>) p) @ (replicate n \<zero>)) []" |
68605 | 1074 |
proof - |
1075 |
have "map ((\<otimes>) \<zero>) p = map (\<lambda>a. \<zero>) p" |
|
1076 |
using Suc.prems by auto |
|
1077 |
then show ?thesis |
|
1078 |
by presburger |
|
1079 |
qed |
|
68578 | 1080 |
also have " ... = poly_add (replicate (length p + n) \<zero>) []" |
1081 |
by (simp add: map_replicate_const replicate_add) |
|
1082 |
also have " ... = poly_add [] []" |
|
1083 |
using poly_add_normalize(1)[of "replicate (length p + n) \<zero>" "[]"] |
|
1084 |
normalize_replicate_zero[of "length p + n" "[]"] by auto |
|
1085 |
also have " ... = []" by simp |
|
1086 |
finally show ?case . |
|
1087 |
qed |
|
1088 |
thus "poly_mult p (replicate n \<zero>) = []" |
|
1089 |
using poly_mult_comm[OF assms in_carrier] by simp |
|
1090 |
qed |
|
1091 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1092 |
lemma poly_mult_const': |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1093 |
assumes "set p \<subseteq> carrier R" "a \<in> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1094 |
shows "poly_mult [ a ] p = normalize (map (\<lambda>b. a \<otimes> b) p)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1095 |
and "poly_mult p [ a ] = normalize (map (\<lambda>b. a \<otimes> b) p)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1096 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1097 |
have "map2 (\<oplus>) (map ((\<otimes>) a) p) (replicate (length p) \<zero>) = map ((\<otimes>) a) p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1098 |
using assms by (induction p) (auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1099 |
thus "poly_mult [ a ] p = normalize (map (\<lambda>b. a \<otimes> b) p)" by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1100 |
thus "poly_mult p [ a ] = normalize (map (\<lambda>b. a \<otimes> b) p)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1101 |
using poly_mult_comm[OF assms(1), of "[ a ]"] assms(2) by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1102 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1103 |
|
68578 | 1104 |
lemma poly_mult_const: |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1105 |
assumes "subring K R" "polynomial K p" "a \<in> K - { \<zero> }" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1106 |
shows "poly_mult [ a ] p = map (\<lambda>b. a \<otimes> b) p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1107 |
and "poly_mult p [ a ] = map (\<lambda>b. a \<otimes> b) p" |
68578 | 1108 |
proof - |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1109 |
have in_carrier: "set p \<subseteq> carrier R" "a \<in> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1110 |
using polynomial_in_carrier[OF assms(1-2)] assms(3) subringE(1)[OF assms(1)] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1111 |
|
68578 | 1112 |
show "poly_mult [ a ] p = map (\<lambda>b. a \<otimes> b) p" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1113 |
proof (cases p) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1114 |
case Nil thus ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1115 |
using poly_mult_const'(1) in_carrier by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1116 |
next |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1117 |
case (Cons b q) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1118 |
have "lead_coeff (map (\<lambda>b. a \<otimes> b) p) \<noteq> \<zero>" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1119 |
using assms subringE(1)[OF assms(1)] integral[of a b] Cons lead_coeff_in_carrier by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1120 |
hence "normalize (map (\<lambda>b. a \<otimes> b) p) = (map (\<lambda>b. a \<otimes> b) p)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1121 |
unfolding Cons by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1122 |
thus ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1123 |
using poly_mult_const'(1) in_carrier by auto |
68578 | 1124 |
qed |
1125 |
thus "poly_mult p [ a ] = map (\<lambda>b. a \<otimes> b) p" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1126 |
using poly_mult_comm[OF in_carrier(1)] in_carrier(2) by auto |
68578 | 1127 |
qed |
1128 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1129 |
lemma poly_mult_semiassoc: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1130 |
assumes "set p \<subseteq> carrier R" "set q \<subseteq> carrier R" and "a \<in> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1131 |
shows "poly_mult (poly_mult [ a ] p) q = poly_mult [ a ] (poly_mult p q)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1132 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1133 |
let ?cp = "coeff p" and ?cq = "coeff q" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1134 |
have "coeff (poly_mult [ a ] p) = (\<lambda>i. (a \<otimes> ?cp i))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1135 |
using poly_mult_const'(1)[OF assms(1,3)] normalize_coeff scalar_coeff[OF assms(3)] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1136 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1137 |
hence "coeff (poly_mult (poly_mult [ a ] p) q) = (\<lambda>i. (\<Oplus>j \<in> {..i}. (a \<otimes> ?cp j) \<otimes> ?cq (i - j)))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1138 |
using poly_mult_coeff[OF poly_mult_in_carrier[OF _ assms(1)] assms(2), of "[ a ]"] assms(3) by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1139 |
also have " ... = (\<lambda>i. a \<otimes> (\<Oplus>j \<in> {..i}. ?cp j \<otimes> ?cq (i - j)))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1140 |
proof |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1141 |
fix i show "(\<Oplus>j \<in> {..i}. (a \<otimes> ?cp j) \<otimes> ?cq (i - j)) = a \<otimes> (\<Oplus>j \<in> {..i}. ?cp j \<otimes> ?cq (i - j))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1142 |
using finsum_rdistr[OF _ assms(3), of _ "\<lambda>j. ?cp j \<otimes> ?cq (i - j)"] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1143 |
assms(1-2)[THEN coeff_in_carrier] by (simp add: assms(3) m_assoc) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1144 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1145 |
also have " ... = coeff (poly_mult [ a ] (poly_mult p q))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1146 |
unfolding poly_mult_const'(1)[OF poly_mult_in_carrier[OF assms(1-2)] assms(3)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1147 |
using scalar_coeff[OF assms(3), of "poly_mult p q"] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1148 |
poly_mult_coeff[OF assms(1-2)] normalize_coeff by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1149 |
finally have "coeff (poly_mult (poly_mult [ a ] p) q) = coeff (poly_mult [ a ] (poly_mult p q))" . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1150 |
moreover have "polynomial (carrier R) (poly_mult (poly_mult [ a ] p) q)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1151 |
and "polynomial (carrier R) (poly_mult [ a ] (poly_mult p q))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1152 |
using poly_mult_is_polynomial[OF _ poly_mult_in_carrier[OF _ assms(1)] assms(2)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1153 |
poly_mult_is_polynomial[OF _ _ poly_mult_in_carrier[OF assms(1-2)]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1154 |
carrier_is_subring assms(3) by (auto simp del: poly_mult.simps) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1155 |
ultimately show ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1156 |
using coeff_iff_polynomial_cond by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1157 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1158 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1159 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1160 |
text \<open>Note that "polynomial (carrier R) p" and "subring K p; polynomial K p" are "equivalent" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1161 |
assumptions for any lemma in ring which the result doesn't depend on K, because carrier |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1162 |
is a subring and a polynomial for a subset of the carrier is a carrier polynomial. The |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1163 |
decision between one of them should be based on how the lemma is going to be used and |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1164 |
proved. These are some tips: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1165 |
(a) Lemmas about the algebraic structure of polynomials should use the latter option. |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1166 |
(b) Also, if the lemma deals with lots of polynomials, then the latter option is preferred. |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1167 |
(c) If the proof is going to be much easier with the first option, do not hesitate. \<close> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1168 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1169 |
lemma poly_mult_monom': |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1170 |
assumes "set p \<subseteq> carrier R" "a \<in> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1171 |
shows "poly_mult (monom a n) p = normalize ((map ((\<otimes>) a) p) @ (replicate n \<zero>))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1172 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1173 |
have set_map: "set ((map ((\<otimes>) a) p) @ (replicate n \<zero>)) \<subseteq> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1174 |
using assms by (induct p) (auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1175 |
show ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1176 |
using poly_mult_replicate_zero(1)[OF assms(1), of n] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1177 |
poly_add_zero'(1)[OF set_map] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1178 |
unfolding monom_def by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1179 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1180 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1181 |
lemma poly_mult_monom: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1182 |
assumes "polynomial (carrier R) p" "a \<in> carrier R - { \<zero> }" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1183 |
shows "poly_mult (monom a n) p = |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1184 |
(if p = [] then [] else (poly_mult [ a ] p) @ (replicate n \<zero>))" |
68578 | 1185 |
proof (cases p) |
1186 |
case Nil thus ?thesis |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1187 |
using poly_mult_zero(2)[of "monom a n"] assms(2) monom_def by fastforce |
68578 | 1188 |
next |
1189 |
case (Cons b ps) |
|
1190 |
hence "lead_coeff ((map (\<lambda>b. a \<otimes> b) p) @ (replicate n \<zero>)) \<noteq> \<zero>" |
|
1191 |
using Cons assms integral[of a b] unfolding polynomial_def by auto |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1192 |
thus ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1193 |
using poly_mult_monom'[OF polynomial_incl[OF assms(1)], of a n] assms(2) Cons |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1194 |
unfolding poly_mult_const(1)[OF carrier_is_subring assms] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1195 |
qed |
68578 | 1196 |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1197 |
lemma poly_mult_one': |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1198 |
assumes "set p \<subseteq> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1199 |
shows "poly_mult [ \<one> ] p = normalize p" and "poly_mult p [ \<one> ] = normalize p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1200 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1201 |
have "map2 (\<oplus>) (map ((\<otimes>) \<one>) p) (replicate (length p) \<zero>) = p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1202 |
using assms by (induct p) (auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1203 |
thus "poly_mult [ \<one> ] p = normalize p" and "poly_mult p [ \<one> ] = normalize p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1204 |
using poly_mult_comm[OF assms, of "[ \<one> ]"] by auto |
68578 | 1205 |
qed |
1206 |
||
1207 |
lemma poly_mult_one: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1208 |
assumes "subring K R" "polynomial K p" |
68578 | 1209 |
shows "poly_mult [ \<one> ] p = p" and "poly_mult p [ \<one> ] = p" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1210 |
using poly_mult_one'[OF polynomial_in_carrier[OF assms]] normalize_polynomial[OF assms(2)] by auto |
68578 | 1211 |
|
1212 |
lemma poly_mult_lead_coeff_aux: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1213 |
assumes "subring K R" "polynomial K p1" "polynomial K p2" and "p1 \<noteq> []" and "p2 \<noteq> []" |
68578 | 1214 |
shows "(coeff (poly_mult p1 p2)) (degree p1 + degree p2) = (lead_coeff p1) \<otimes> (lead_coeff p2)" |
1215 |
proof - |
|
1216 |
have p1: "lead_coeff p1 \<in> carrier R - { \<zero> }" and p2: "lead_coeff p2 \<in> carrier R - { \<zero> }" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1217 |
using assms(2-5) lead_coeff_in_carrier[OF assms(1)] by (metis list.collapse)+ |
68578 | 1218 |
|
1219 |
have "(coeff (poly_mult p1 p2)) (degree p1 + degree p2) = |
|
1220 |
(\<Oplus> k \<in> {..((degree p1) + (degree p2))}. |
|
1221 |
(coeff p1) k \<otimes> (coeff p2) ((degree p1) + (degree p2) - k))" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1222 |
using poly_mult_coeff[OF assms(2-3)[THEN polynomial_in_carrier[OF assms(1)]]] by simp |
68578 | 1223 |
also have " ... = (lead_coeff p1) \<otimes> (lead_coeff p2)" |
1224 |
proof - |
|
1225 |
let ?f = "\<lambda>i. (coeff p1) i \<otimes> (coeff p2) ((degree p1) + (degree p2) - i)" |
|
1226 |
have in_carrier: "\<And>i. (coeff p1) i \<in> carrier R" "\<And>i. (coeff p2) i \<in> carrier R" |
|
1227 |
using coeff_in_carrier assms by auto |
|
1228 |
have "\<And>i. i < degree p1 \<Longrightarrow> ?f i = \<zero>" |
|
1229 |
using coeff_degree[of p2] in_carrier by auto |
|
1230 |
moreover have "\<And>i. i > degree p1 \<Longrightarrow> ?f i = \<zero>" |
|
1231 |
using coeff_degree[of p1] in_carrier by auto |
|
1232 |
moreover have "?f (degree p1) = (lead_coeff p1) \<otimes> (lead_coeff p2)" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1233 |
using assms(4-5) lead_coeff_simp by simp |
68578 | 1234 |
ultimately have "?f = (\<lambda>i. if degree p1 = i then (lead_coeff p1) \<otimes> (lead_coeff p2) else \<zero>)" |
1235 |
using nat_neq_iff by auto |
|
1236 |
thus ?thesis |
|
1237 |
using add.finprod_singleton[of "degree p1" "{..((degree p1) + (degree p2))}" |
|
1238 |
"\<lambda>i. (lead_coeff p1) \<otimes> (lead_coeff p2)"] p1 p2 by auto |
|
1239 |
qed |
|
1240 |
finally show ?thesis . |
|
1241 |
qed |
|
1242 |
||
1243 |
lemma poly_mult_degree_eq: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1244 |
assumes "subring K R" "polynomial K p1" "polynomial K p2" |
68578 | 1245 |
shows "degree (poly_mult p1 p2) = (if p1 = [] \<or> p2 = [] then 0 else (degree p1) + (degree p2))" |
1246 |
proof (cases p1) |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1247 |
case Nil thus ?thesis by simp |
68578 | 1248 |
next |
1249 |
case (Cons a p1') note p1 = Cons |
|
1250 |
show ?thesis |
|
1251 |
proof (cases p2) |
|
1252 |
case Nil thus ?thesis |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1253 |
using poly_mult_zero(2)[OF polynomial_in_carrier[OF assms(1-2)]] by simp |
68578 | 1254 |
next |
1255 |
case (Cons b p2') note p2 = Cons |
|
1256 |
have a: "a \<in> carrier R" and b: "b \<in> carrier R" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1257 |
using p1 p2 polynomial_in_carrier[OF assms(1-2)] polynomial_in_carrier[OF assms(1,3)] by auto |
68578 | 1258 |
have "(coeff (poly_mult p1 p2)) ((degree p1) + (degree p2)) = a \<otimes> b" |
1259 |
using poly_mult_lead_coeff_aux[OF assms] p1 p2 by simp |
|
68605 | 1260 |
hence neq0: "(coeff (poly_mult p1 p2)) ((degree p1) + (degree p2)) \<noteq> \<zero>" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1261 |
using assms(2-3) integral[of a b] lead_coeff_in_carrier[OF assms(1)] p1 p2 by auto |
68605 | 1262 |
moreover have eq0: "\<And>i. i > (degree p1) + (degree p2) \<Longrightarrow> (coeff (poly_mult p1 p2)) i = \<zero>" |
68578 | 1263 |
proof - |
1264 |
have aux_lemma: "degree (poly_mult p1 p2) \<le> (degree p1) + (degree p2)" |
|
1265 |
proof (induct p1) |
|
1266 |
case Nil |
|
1267 |
then show ?case by simp |
|
1268 |
next |
|
1269 |
case (Cons a p1) |
|
1270 |
let ?a_p2 = "(map (\<lambda>b. a \<otimes> b) p2) @ (replicate (degree (a # p1)) \<zero>)" |
|
1271 |
have "poly_mult (a # p1) p2 = poly_add ?a_p2 (poly_mult p1 p2)" by simp |
|
1272 |
hence "degree (poly_mult (a # p1) p2) \<le> max (degree ?a_p2) (degree (poly_mult p1 p2))" |
|
1273 |
using poly_add_degree[of ?a_p2 "poly_mult p1 p2"] by simp |
|
1274 |
also have " ... \<le> max ((degree (a # p1)) + (degree p2)) (degree (poly_mult p1 p2))" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1275 |
by auto |
68578 | 1276 |
also have " ... \<le> max ((degree (a # p1)) + (degree p2)) ((degree p1) + (degree p2))" |
1277 |
using Cons by simp |
|
1278 |
also have " ... \<le> (degree (a # p1)) + (degree p2)" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1279 |
by auto |
68578 | 1280 |
finally show ?case . |
1281 |
qed |
|
1282 |
fix i show "i > (degree p1) + (degree p2) \<Longrightarrow> (coeff (poly_mult p1 p2)) i = \<zero>" |
|
1283 |
using coeff_degree aux_lemma by simp |
|
1284 |
qed |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1285 |
moreover have "polynomial K (poly_mult p1 p2)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1286 |
by (simp add: assms poly_mult_closed) |
68578 | 1287 |
ultimately have "degree (poly_mult p1 p2) = degree p1 + degree p2" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1288 |
by (metis (no_types) assms(1) coeff.simps(1) coeff_degree domain.poly_mult_one(1) domain_axioms eq0 lead_coeff_simp length_greater_0_conv neq0 normalize_length_lt not_less_iff_gr_or_eq poly_mult_one'(1) polynomial_in_carrier) |
68578 | 1289 |
thus ?thesis |
1290 |
using p1 p2 by auto |
|
1291 |
qed |
|
1292 |
qed |
|
1293 |
||
1294 |
lemma poly_mult_integral: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1295 |
assumes "subring K R" "polynomial K p1" "polynomial K p2" |
68578 | 1296 |
shows "poly_mult p1 p2 = [] \<Longrightarrow> p1 = [] \<or> p2 = []" |
1297 |
proof (rule ccontr) |
|
1298 |
assume A: "poly_mult p1 p2 = []" "\<not> (p1 = [] \<or> p2 = [])" |
|
1299 |
hence "degree (poly_mult p1 p2) = degree p1 + degree p2" |
|
1300 |
using poly_mult_degree_eq[OF assms] by simp |
|
1301 |
hence "length p1 = 1 \<and> length p2 = 1" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1302 |
using A Suc_diff_Suc by fastforce |
68578 | 1303 |
then obtain a b where p1: "p1 = [ a ]" and p2: "p2 = [ b ]" |
1304 |
by (metis One_nat_def length_0_conv length_Suc_conv) |
|
1305 |
hence "a \<in> carrier R - { \<zero> }" and "b \<in> carrier R - { \<zero> }" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1306 |
using assms lead_coeff_in_carrier by auto |
68578 | 1307 |
hence "poly_mult [ a ] [ b ] = [ a \<otimes> b ]" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1308 |
using integral by auto |
68578 | 1309 |
thus False using A(1) p1 p2 by simp |
1310 |
qed |
|
1311 |
||
1312 |
lemma poly_mult_lead_coeff: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1313 |
assumes "subring K R" "polynomial K p1" "polynomial K p2" and "p1 \<noteq> []" and "p2 \<noteq> []" |
68578 | 1314 |
shows "lead_coeff (poly_mult p1 p2) = (lead_coeff p1) \<otimes> (lead_coeff p2)" |
1315 |
proof - |
|
1316 |
have "poly_mult p1 p2 \<noteq> []" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1317 |
using poly_mult_integral[OF assms(1-3)] assms(4-5) by auto |
68578 | 1318 |
hence "lead_coeff (poly_mult p1 p2) = (coeff (poly_mult p1 p2)) (degree p1 + degree p2)" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1319 |
using poly_mult_degree_eq[OF assms(1-3)] assms(4-5) by (metis coeff.simps(2) list.collapse) |
68578 | 1320 |
thus ?thesis |
1321 |
using poly_mult_lead_coeff_aux[OF assms] by simp |
|
1322 |
qed |
|
1323 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1324 |
lemma poly_mult_append_zero_lcancel: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1325 |
assumes "subring K R" and "polynomial K p" "polynomial K q" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1326 |
shows "poly_mult (p @ [ \<zero> ]) q = r @ [ \<zero> ] \<Longrightarrow> poly_mult p q = r" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1327 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1328 |
note in_carrier = assms(2-3)[THEN polynomial_in_carrier[OF assms(1)]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1329 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1330 |
assume pmult: "poly_mult (p @ [ \<zero> ]) q = r @ [ \<zero> ]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1331 |
have "poly_mult (p @ [ \<zero> ]) q = []" if "q = []" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1332 |
using poly_mult_zero(2)[of "p @ [ \<zero> ]"] that in_carrier(1) by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1333 |
moreover have "poly_mult (p @ [ \<zero> ]) q = []" if "p = []" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1334 |
using poly_mult_normalize[OF _ in_carrier(2), of "p @ [ \<zero> ]"] poly_mult_zero[OF in_carrier(2)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1335 |
unfolding that by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1336 |
ultimately have "p \<noteq> []" and "q \<noteq> []" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1337 |
using pmult by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1338 |
hence "poly_mult p q \<noteq> []" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1339 |
using poly_mult_integral[OF assms] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1340 |
hence "normalize ((poly_mult p q) @ [ \<zero> ]) = (poly_mult p q) @ [ \<zero> ]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1341 |
using normalize_polynomial[OF append_is_polynomial[OF assms(1) poly_mult_closed[OF assms], of "Suc 0"]] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1342 |
thus "poly_mult p q = r" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1343 |
using poly_mult_append_zero[OF assms(2-3)[THEN polynomial_in_carrier[OF assms(1)]]] pmult by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1344 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1345 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1346 |
lemma poly_mult_append_zero_rcancel: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1347 |
assumes "subring K R" and "polynomial K p" "polynomial K q" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1348 |
shows "poly_mult p (q @ [ \<zero> ]) = r @ [ \<zero> ] \<Longrightarrow> poly_mult p q = r" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1349 |
using poly_mult_append_zero_lcancel[OF assms(1,3,2)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1350 |
poly_mult_comm[of p "q @ [ \<zero> ]"] poly_mult_comm[of p q] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1351 |
assms(2-3)[THEN polynomial_in_carrier[OF assms(1)]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1352 |
by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1353 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1354 |
end (* of domain context. *) |
68578 | 1355 |
|
1356 |
||
1357 |
subsection \<open>Algebraic Structure of Polynomials\<close> |
|
1358 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1359 |
definition univ_poly :: "('a, 'b) ring_scheme \<Rightarrow>'a set \<Rightarrow> ('a list) ring" ("_ [X]\<index>" 80) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1360 |
where "univ_poly R K = |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1361 |
\<lparr> carrier = { p. polynomial\<^bsub>R\<^esub> K p }, |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1362 |
mult = ring.poly_mult R, |
68578 | 1363 |
one = [ \<one>\<^bsub>R\<^esub> ], |
1364 |
zero = [], |
|
1365 |
add = ring.poly_add R \<rparr>" |
|
1366 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1367 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1368 |
text \<open>These lemmas allow you to unfold one field of the record at a time. \<close> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1369 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1370 |
lemma univ_poly_carrier: "polynomial\<^bsub>R\<^esub> K p \<longleftrightarrow> p \<in> carrier (K[X]\<^bsub>R\<^esub>)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1371 |
unfolding univ_poly_def by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1372 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1373 |
lemma univ_poly_mult: "mult (K[X]\<^bsub>R\<^esub>) = ring.poly_mult R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1374 |
unfolding univ_poly_def by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1375 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1376 |
lemma univ_poly_one: "one (K[X]\<^bsub>R\<^esub>) = [ \<one>\<^bsub>R\<^esub> ]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1377 |
unfolding univ_poly_def by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1378 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1379 |
lemma univ_poly_zero: "zero (K[X]\<^bsub>R\<^esub>) = []" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1380 |
unfolding univ_poly_def by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1381 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1382 |
lemma univ_poly_add: "add (K[X]\<^bsub>R\<^esub>) = ring.poly_add R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1383 |
unfolding univ_poly_def by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1384 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1385 |
|
68578 | 1386 |
context domain |
1387 |
begin |
|
1388 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1389 |
lemma poly_mult_monom_assoc: |
68578 | 1390 |
assumes "set p \<subseteq> carrier R" "set q \<subseteq> carrier R" and "a \<in> carrier R" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1391 |
shows "poly_mult (poly_mult (monom a n) p) q = |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1392 |
poly_mult (monom a n) (poly_mult p q)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1393 |
proof (induct n) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1394 |
case 0 thus ?case |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1395 |
unfolding monom_def using poly_mult_semiassoc[OF assms] by (auto simp del: poly_mult.simps) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1396 |
next |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1397 |
case (Suc n) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1398 |
have "poly_mult (poly_mult (monom a (Suc n)) p) q = |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1399 |
poly_mult (normalize ((poly_mult (monom a n) p) @ [ \<zero> ])) q" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1400 |
using poly_mult_append_zero[OF monom_in_carrier[OF assms(3), of n] assms(1)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1401 |
unfolding monom_def by (auto simp del: poly_mult.simps simp add: replicate_append_same) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1402 |
also have " ... = normalize ((poly_mult (poly_mult (monom a n) p) q) @ [ \<zero> ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1403 |
using poly_mult_normalize[OF _ assms(2)] poly_mult_append_zero[OF _ assms(2)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1404 |
poly_mult_in_carrier[OF monom_in_carrier[OF assms(3), of n] assms(1)] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1405 |
also have " ... = normalize ((poly_mult (monom a n) (poly_mult p q)) @ [ \<zero> ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1406 |
using Suc by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1407 |
also have " ... = poly_mult (monom a (Suc n)) (poly_mult p q)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1408 |
using poly_mult_append_zero[OF monom_in_carrier[OF assms(3), of n] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1409 |
poly_mult_in_carrier[OF assms(1-2)]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1410 |
unfolding monom_def by (simp add: replicate_append_same) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1411 |
finally show ?case . |
68578 | 1412 |
qed |
1413 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1414 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1415 |
context |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1416 |
fixes K :: "'a set" assumes K: "subring K R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1417 |
begin |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1418 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1419 |
lemma univ_poly_is_monoid: "monoid (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1420 |
unfolding univ_poly_def using poly_mult_one[OF K] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1421 |
proof (auto simp add: K poly_add_closed poly_mult_closed one_is_polynomial monoid_def) |
68578 | 1422 |
fix p1 p2 p3 |
1423 |
let ?P = "poly_mult (poly_mult p1 p2) p3 = poly_mult p1 (poly_mult p2 p3)" |
|
1424 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1425 |
assume A: "polynomial K p1" "polynomial K p2" "polynomial K p3" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1426 |
show ?P using polynomial_in_carrier[OF K A(1)] |
68578 | 1427 |
proof (induction p1) |
1428 |
case Nil thus ?case by simp |
|
1429 |
next |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1430 |
next |
68578 | 1431 |
case (Cons a p1) thus ?case |
1432 |
proof (cases "a = \<zero>") |
|
1433 |
assume eq_zero: "a = \<zero>" |
|
1434 |
have p1: "set p1 \<subseteq> carrier R" |
|
1435 |
using Cons(2) by simp |
|
1436 |
have "poly_mult (poly_mult (a # p1) p2) p3 = poly_mult (poly_mult p1 p2) p3" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1437 |
using poly_mult_prepend_replicate_zero[OF p1 polynomial_in_carrier[OF K A(2)], of "Suc 0"] |
68578 | 1438 |
eq_zero by simp |
1439 |
also have " ... = poly_mult p1 (poly_mult p2 p3)" |
|
1440 |
using p1[THEN Cons(1)] by simp |
|
1441 |
also have " ... = poly_mult (a # p1) (poly_mult p2 p3)" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1442 |
using poly_mult_prepend_replicate_zero[OF p1 |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1443 |
poly_mult_in_carrier[OF A(2-3)[THEN polynomial_in_carrier[OF K]]], of "Suc 0"] eq_zero |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1444 |
by simp |
68578 | 1445 |
finally show ?thesis . |
1446 |
next |
|
1447 |
assume "a \<noteq> \<zero>" hence in_carrier: |
|
1448 |
"set p1 \<subseteq> carrier R" "set p2 \<subseteq> carrier R" "set p3 \<subseteq> carrier R" "a \<in> carrier R - { \<zero> }" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1449 |
using A(2-3) polynomial_in_carrier[OF K] Cons by auto |
68578 | 1450 |
|
1451 |
let ?a_p2 = "(map (\<lambda>b. a \<otimes> b) p2) @ (replicate (length p1) \<zero>)" |
|
1452 |
have a_p2_in_carrier: "set ?a_p2 \<subseteq> carrier R" |
|
1453 |
using in_carrier by auto |
|
1454 |
||
1455 |
have "poly_mult (poly_mult (a # p1) p2) p3 = poly_mult (poly_add ?a_p2 (poly_mult p1 p2)) p3" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1456 |
by simp |
68578 | 1457 |
also have " ... = poly_add (poly_mult ?a_p2 p3) (poly_mult (poly_mult p1 p2) p3)" |
1458 |
using poly_mult_l_distr'[OF a_p2_in_carrier poly_mult_in_carrier[OF in_carrier(1-2)] in_carrier(3)] . |
|
1459 |
also have " ... = poly_add (poly_mult ?a_p2 p3) (poly_mult p1 (poly_mult p2 p3))" |
|
1460 |
using Cons(1)[OF in_carrier(1)] by simp |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1461 |
also have " ... = poly_add (poly_mult (normalize ?a_p2) p3) (poly_mult p1 (poly_mult p2 p3))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1462 |
using poly_mult_normalize[OF a_p2_in_carrier in_carrier(3)] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1463 |
also have " ... = poly_add (poly_mult (poly_mult (monom a (length p1)) p2) p3) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1464 |
(poly_mult p1 (poly_mult p2 p3))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1465 |
using poly_mult_monom'[OF in_carrier(2), of a "length p1"] in_carrier(4) by simp |
68578 | 1466 |
also have " ... = poly_add (poly_mult (a # (replicate (length p1) \<zero>)) (poly_mult p2 p3)) |
1467 |
(poly_mult p1 (poly_mult p2 p3))" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1468 |
using poly_mult_monom_assoc[of p2 p3 a "length p1"] in_carrier unfolding monom_def by simp |
68578 | 1469 |
also have " ... = poly_mult (poly_add (a # (replicate (length p1) \<zero>)) p1) (poly_mult p2 p3)" |
1470 |
using poly_mult_l_distr'[of "a # (replicate (length p1) \<zero>)" p1 "poly_mult p2 p3"] |
|
1471 |
poly_mult_in_carrier[OF in_carrier(2-3)] in_carrier by force |
|
1472 |
also have " ... = poly_mult (a # p1) (poly_mult p2 p3)" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1473 |
using poly_add_monom[OF in_carrier(1) in_carrier(4)] unfolding monom_def by simp |
68578 | 1474 |
finally show ?thesis . |
1475 |
qed |
|
1476 |
qed |
|
1477 |
qed |
|
1478 |
||
1479 |
declare poly_add.simps[simp del] |
|
1480 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1481 |
lemma univ_poly_is_abelian_monoid: "abelian_monoid (K[X])" |
68578 | 1482 |
unfolding univ_poly_def |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1483 |
using poly_add_closed poly_add_zero zero_is_polynomial K |
68578 | 1484 |
proof (auto simp add: abelian_monoid_def comm_monoid_def monoid_def comm_monoid_axioms_def) |
1485 |
fix p1 p2 p3 |
|
1486 |
let ?c = "\<lambda>p. coeff p" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1487 |
assume A: "polynomial K p1" "polynomial K p2" "polynomial K p3" |
68578 | 1488 |
hence |
1489 |
p1: "\<And>i. (?c p1) i \<in> carrier R" "set p1 \<subseteq> carrier R" and |
|
1490 |
p2: "\<And>i. (?c p2) i \<in> carrier R" "set p2 \<subseteq> carrier R" and |
|
1491 |
p3: "\<And>i. (?c p3) i \<in> carrier R" "set p3 \<subseteq> carrier R" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1492 |
using A[THEN polynomial_in_carrier[OF K]] coeff_in_carrier by auto |
68578 | 1493 |
have "?c (poly_add (poly_add p1 p2) p3) = (\<lambda>i. (?c p1 i \<oplus> ?c p2 i) \<oplus> (?c p3 i))" |
1494 |
using poly_add_coeff[OF poly_add_in_carrier[OF p1(2) p2(2)] p3(2)] |
|
1495 |
poly_add_coeff[OF p1(2) p2(2)] by simp |
|
1496 |
also have " ... = (\<lambda>i. (?c p1 i) \<oplus> ((?c p2 i) \<oplus> (?c p3 i)))" |
|
1497 |
using p1 p2 p3 add.m_assoc by simp |
|
1498 |
also have " ... = ?c (poly_add p1 (poly_add p2 p3))" |
|
1499 |
using poly_add_coeff[OF p1(2) poly_add_in_carrier[OF p2(2) p3(2)]] |
|
1500 |
poly_add_coeff[OF p2(2) p3(2)] by simp |
|
1501 |
finally have "?c (poly_add (poly_add p1 p2) p3) = ?c (poly_add p1 (poly_add p2 p3))" . |
|
1502 |
thus "poly_add (poly_add p1 p2) p3 = poly_add p1 (poly_add p2 p3)" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1503 |
using coeff_iff_polynomial_cond poly_add_closed[OF K] A by meson |
68578 | 1504 |
show "poly_add p1 p2 = poly_add p2 p1" |
1505 |
using poly_add_comm[OF p1(2) p2(2)] . |
|
1506 |
qed |
|
1507 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1508 |
lemma univ_poly_is_abelian_group: "abelian_group (K[X])" |
68578 | 1509 |
proof - |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1510 |
interpret abelian_monoid "K[X]" |
68578 | 1511 |
using univ_poly_is_abelian_monoid . |
1512 |
show ?thesis |
|
1513 |
proof (unfold_locales) |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1514 |
show "carrier (add_monoid (K[X])) \<subseteq> Units (add_monoid (K[X]))" |
68578 | 1515 |
unfolding univ_poly_def Units_def |
1516 |
proof (auto) |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1517 |
fix p assume p: "polynomial K p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1518 |
have "polynomial K [ \<ominus> \<one> ]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1519 |
unfolding polynomial_def using r_neg subringE(3,5)[OF K] by force |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1520 |
hence cond0: "polynomial K (poly_mult [ \<ominus> \<one> ] p)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1521 |
using poly_mult_closed[OF K, of "[ \<ominus> \<one> ]" p] p by simp |
68578 | 1522 |
|
1523 |
have "poly_add p (poly_mult [ \<ominus> \<one> ] p) = poly_add (poly_mult [ \<one> ] p) (poly_mult [ \<ominus> \<one> ] p)" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1524 |
using poly_mult_one[OF K p] by simp |
68578 | 1525 |
also have " ... = poly_mult (poly_add [ \<one> ] [ \<ominus> \<one> ]) p" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1526 |
using poly_mult_l_distr' polynomial_in_carrier[OF K p] by auto |
68578 | 1527 |
also have " ... = poly_mult [] p" |
1528 |
using poly_add.simps[of "[ \<one> ]" "[ \<ominus> \<one> ]"] |
|
1529 |
by (simp add: case_prod_unfold r_neg) |
|
1530 |
also have " ... = []" by simp |
|
1531 |
finally have cond1: "poly_add p (poly_mult [ \<ominus> \<one> ] p) = []" . |
|
1532 |
||
1533 |
have "poly_add (poly_mult [ \<ominus> \<one> ] p) p = poly_add (poly_mult [ \<ominus> \<one> ] p) (poly_mult [ \<one> ] p)" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1534 |
using poly_mult_one[OF K p] by simp |
68578 | 1535 |
also have " ... = poly_mult (poly_add [ \<ominus> \<one> ] [ \<one> ]) p" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1536 |
using poly_mult_l_distr' polynomial_in_carrier[OF K p] by auto |
68578 | 1537 |
also have " ... = poly_mult [] p" |
1538 |
using \<open>poly_mult (poly_add [\<one>] [\<ominus> \<one>]) p = poly_mult [] p\<close> poly_add_comm by auto |
|
1539 |
also have " ... = []" by simp |
|
1540 |
finally have cond2: "poly_add (poly_mult [ \<ominus> \<one> ] p) p = []" . |
|
1541 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1542 |
from cond0 cond1 cond2 show "\<exists>q. polynomial K q \<and> poly_add q p = [] \<and> poly_add p q = []" |
68578 | 1543 |
by auto |
1544 |
qed |
|
1545 |
qed |
|
1546 |
qed |
|
1547 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1548 |
lemma univ_poly_is_ring: "ring (K[X])" |
68578 | 1549 |
proof - |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1550 |
interpret UP: abelian_group "K[X]" + monoid "K[X]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1551 |
using univ_poly_is_abelian_group univ_poly_is_monoid . |
68578 | 1552 |
show ?thesis |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1553 |
by (unfold_locales) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1554 |
(auto simp add: univ_poly_def poly_mult_r_distr[OF K] poly_mult_l_distr[OF K]) |
68578 | 1555 |
qed |
1556 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1557 |
lemma univ_poly_is_cring: "cring (K[X])" |
68578 | 1558 |
proof - |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1559 |
interpret UP: ring "K[X]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1560 |
using univ_poly_is_ring . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1561 |
have "\<And>p q. \<lbrakk> p \<in> carrier (K[X]); q \<in> carrier (K[X]) \<rbrakk> \<Longrightarrow> p \<otimes>\<^bsub>K[X]\<^esub> q = q \<otimes>\<^bsub>K[X]\<^esub> p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1562 |
unfolding univ_poly_def using poly_mult_comm polynomial_in_carrier[OF K] by auto |
68578 | 1563 |
thus ?thesis |
1564 |
by unfold_locales auto |
|
1565 |
qed |
|
1566 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1567 |
lemma univ_poly_is_domain: "domain (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1568 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1569 |
interpret UP: cring "K[X]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1570 |
using univ_poly_is_cring . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1571 |
show ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1572 |
by (unfold_locales, auto simp add: univ_poly_def poly_mult_integral[OF K]) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1573 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1574 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1575 |
declare poly_add.simps[simp] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1576 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1577 |
lemma univ_poly_a_inv_def': |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1578 |
assumes "p \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1579 |
shows "\<ominus>\<^bsub>K[X]\<^esub> p = map (\<lambda>a. \<ominus> a) p" |
68578 | 1580 |
proof - |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1581 |
have aux_lemma: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1582 |
"\<And>p. p \<in> carrier (K[X]) \<Longrightarrow> p \<oplus>\<^bsub>K[X]\<^esub> (map (\<lambda>a. \<ominus> a) p) = []" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1583 |
"\<And>p. p \<in> carrier (K[X]) \<Longrightarrow> (map (\<lambda>a. \<ominus> a) p) \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1584 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1585 |
fix p assume p: "p \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1586 |
hence set_p: "set p \<subseteq> K" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1587 |
unfolding univ_poly_def using polynomial_incl by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1588 |
show "(map (\<lambda>a. \<ominus> a) p) \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1589 |
proof (cases "p = []") |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1590 |
assume "p = []" thus ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1591 |
unfolding univ_poly_def polynomial_def by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1592 |
next |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1593 |
assume not_nil: "p \<noteq> []" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1594 |
hence "lead_coeff p \<noteq> \<zero>" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1595 |
using p unfolding univ_poly_def polynomial_def by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1596 |
moreover have "lead_coeff (map (\<lambda>a. \<ominus> a) p) = \<ominus> (lead_coeff p)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1597 |
using not_nil by (simp add: hd_map) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1598 |
ultimately have "lead_coeff (map (\<lambda>a. \<ominus> a) p) \<noteq> \<zero>" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1599 |
using hd_in_set local.minus_zero not_nil set_p subringE(1)[OF K] by force |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1600 |
moreover have "set (map (\<lambda>a. \<ominus> a) p) \<subseteq> K" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1601 |
using set_p subringE(5)[OF K] by (induct p) (auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1602 |
ultimately show ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1603 |
unfolding univ_poly_def polynomial_def by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1604 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1605 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1606 |
have "map2 (\<oplus>) p (map (\<lambda>a. \<ominus> a) p) = replicate (length p) \<zero>" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1607 |
using set_p subringE(1)[OF K] by (induct p) (auto simp add: r_neg) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1608 |
thus "p \<oplus>\<^bsub>K[X]\<^esub> (map (\<lambda>a. \<ominus> a) p) = []" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1609 |
unfolding univ_poly_def using normalize_replicate_zero[of "length p" "[]"] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1610 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1611 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1612 |
interpret UP: ring "K[X]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1613 |
using univ_poly_is_ring . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1614 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1615 |
from aux_lemma |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1616 |
have "\<And>p. p \<in> carrier (K[X]) \<Longrightarrow> \<ominus>\<^bsub>K[X]\<^esub> p = map (\<lambda>a. \<ominus> a) p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1617 |
by (metis Nil_is_map_conv UP.add.inv_closed UP.l_zero UP.r_neg1 UP.r_zero UP.zero_closed) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1618 |
thus ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1619 |
using assms by simp |
68578 | 1620 |
qed |
1621 |
||
1622 |
||
1623 |
subsection \<open>Long Division Theorem\<close> |
|
1624 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1625 |
lemma long_division_theorem: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1626 |
assumes "polynomial K p" and "polynomial K b" "b \<noteq> []" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1627 |
and "lead_coeff b \<in> Units (R \<lparr> carrier := K \<rparr>)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1628 |
shows "\<exists>q r. polynomial K q \<and> polynomial K r \<and> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1629 |
p = (b \<otimes>\<^bsub>K[X]\<^esub> q) \<oplus>\<^bsub>K[X]\<^esub> r \<and> (r = [] \<or> degree r < degree b)" |
68578 | 1630 |
(is "\<exists>q r. ?long_division p q r") |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1631 |
using assms(1) |
68578 | 1632 |
proof (induct "length p" arbitrary: p rule: less_induct) |
1633 |
case less thus ?case |
|
1634 |
proof (cases p) |
|
1635 |
case Nil |
|
1636 |
hence "?long_division p [] []" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1637 |
using zero_is_polynomial poly_mult_zero[OF polynomial_in_carrier[OF K assms(2)]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1638 |
by (simp add: univ_poly_def) |
68578 | 1639 |
thus ?thesis by blast |
1640 |
next |
|
1641 |
case (Cons a p') thus ?thesis |
|
1642 |
proof (cases "length b > length p") |
|
1643 |
assume "length b > length p" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1644 |
hence "p = [] \<or> degree p < degree b" |
68578 | 1645 |
by (meson diff_less_mono length_0_conv less_one not_le) |
1646 |
hence "?long_division p [] p" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1647 |
using poly_mult_zero(2)[OF polynomial_in_carrier[OF K assms(2)]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1648 |
poly_add_zero(2)[OF K less(2)] zero_is_polynomial less(2) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1649 |
by (simp add: univ_poly_def) |
68578 | 1650 |
thus ?thesis by blast |
1651 |
next |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1652 |
interpret UP: cring "K[X]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1653 |
using univ_poly_is_cring . |
68578 | 1654 |
|
1655 |
assume "\<not> length b > length p" |
|
1656 |
hence len_ge: "length p \<ge> length b" by simp |
|
1657 |
obtain c b' where b: "b = c # b'" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1658 |
using assms(3) list.exhaust_sel by blast |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1659 |
then obtain c' where c': "c' \<in> carrier R" "c' \<in> K" "c' \<otimes> c = \<one>" "c \<otimes> c' = \<one>" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1660 |
using assms(4) subringE(1)[OF K] unfolding Units_def by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1661 |
have c: "c \<in> carrier R" "c \<in> K" "c \<noteq> \<zero>" and a: "a \<in> carrier R" "a \<in> K" "a \<noteq> \<zero>" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1662 |
using less(2) assms(2) lead_coeff_not_zero subringE(1)[OF K] b Cons by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1663 |
hence lc: "c' \<otimes> (\<ominus> a) \<in> K - { \<zero> }" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1664 |
using subringE(5-6)[OF K] c' add.inv_solve_right integral_iff by fastforce |
68578 | 1665 |
|
1666 |
let ?len = "length" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1667 |
define s where "s = monom (c' \<otimes> (\<ominus> a)) (?len p - ?len b)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1668 |
hence s: "polynomial K s" "s \<noteq> []" "degree s = ?len p - ?len b" "length s \<ge> 1" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1669 |
using monom_is_polynomial[OF K lc] unfolding monom_def by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1670 |
hence is_polynomial: "polynomial K (p \<oplus>\<^bsub>K[X]\<^esub> (b \<otimes>\<^bsub>K[X]\<^esub> s))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1671 |
using poly_add_closed[OF K less(2) poly_mult_closed[OF K assms(2), of s]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1672 |
by (simp add: univ_poly_def) |
68578 | 1673 |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1674 |
have "lead_coeff (b \<otimes>\<^bsub>K[X]\<^esub> s) = \<ominus> a" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1675 |
using poly_mult_lead_coeff[OF K assms(2) s(1) assms(3) s(2)] c c' a |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1676 |
unfolding b s_def monom_def univ_poly_def by (auto simp del: poly_mult.simps, algebra) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1677 |
then obtain s' where s': "b \<otimes>\<^bsub>K[X]\<^esub> s = (\<ominus> a) # s'" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1678 |
using poly_mult_integral[OF K assms(2) s(1)] assms(2-3) s(2) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1679 |
by (simp add: univ_poly_def, metis hd_Cons_tl) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1680 |
moreover have "degree p = degree (b \<otimes>\<^bsub>K[X]\<^esub> s)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1681 |
using poly_mult_degree_eq[OF K assms(2) s(1)] assms(3) s(2-4) len_ge b Cons |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1682 |
by (auto simp add: univ_poly_def) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1683 |
hence "?len p = ?len (b \<otimes>\<^bsub>K[X]\<^esub> s)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1684 |
unfolding Cons s' by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1685 |
hence "?len (p \<oplus>\<^bsub>K[X]\<^esub> (b \<otimes>\<^bsub>K[X]\<^esub> s)) < ?len p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1686 |
unfolding Cons s' using a normalize_length_le[of "map2 (\<oplus>) p' s'"] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1687 |
by (auto simp add: univ_poly_def r_neg) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1688 |
then obtain q' r' where l_div: "?long_division (p \<oplus>\<^bsub>K[X]\<^esub> (b \<otimes>\<^bsub>K[X]\<^esub> s)) q' r'" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1689 |
using less(1)[OF _ is_polynomial] by blast |
68578 | 1690 |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1691 |
have in_carrier: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1692 |
"p \<in> carrier (K[X])" "b \<in> carrier (K[X])" "s \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1693 |
"q' \<in> carrier (K[X])" "r' \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1694 |
using l_div assms less(2) s unfolding univ_poly_def by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1695 |
have "(p \<oplus>\<^bsub>K[X]\<^esub> (b \<otimes>\<^bsub>K[X]\<^esub> s)) \<ominus>\<^bsub>K[X]\<^esub> (b \<otimes>\<^bsub>K[X]\<^esub> s) = |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1696 |
((b \<otimes>\<^bsub>K[X]\<^esub> q') \<oplus>\<^bsub>K[X]\<^esub> r') \<ominus>\<^bsub>K[X]\<^esub> (b \<otimes>\<^bsub>K[X]\<^esub> s)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1697 |
using l_div by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1698 |
hence "p = (b \<otimes>\<^bsub>K[X]\<^esub> (q' \<ominus>\<^bsub>K[X]\<^esub> s)) \<oplus>\<^bsub>K[X]\<^esub> r'" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1699 |
using in_carrier by algebra |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1700 |
moreover have "q' \<ominus>\<^bsub>K[X]\<^esub> s \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1701 |
using in_carrier by algebra |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1702 |
hence "polynomial K (q' \<ominus>\<^bsub>K[X]\<^esub> s)" |
68578 | 1703 |
unfolding univ_poly_def by simp |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1704 |
ultimately have "?long_division p (q' \<ominus>\<^bsub>K[X]\<^esub> s) r'" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1705 |
using l_div by auto |
68578 | 1706 |
thus ?thesis by blast |
1707 |
qed |
|
1708 |
qed |
|
1709 |
qed |
|
1710 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1711 |
end (* of fixed K context. *) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1712 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1713 |
end (* of domain context. *) |
68578 | 1714 |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1715 |
lemma (in field) field_long_division_theorem: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1716 |
assumes "subfield K R" "polynomial K p" and "polynomial K b" "b \<noteq> []" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1717 |
shows "\<exists>q r. polynomial K q \<and> polynomial K r \<and> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1718 |
p = (b \<otimes>\<^bsub>K[X]\<^esub> q) \<oplus>\<^bsub>K[X]\<^esub> r \<and> (r = [] \<or> degree r < degree b)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1719 |
using long_division_theorem[OF subfieldE(1)[OF assms(1)] assms(2-4)] assms(3-4) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1720 |
subfield.subfield_Units[OF assms(1)] lead_coeff_not_zero[of K "hd b" "tl b"] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1721 |
by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1722 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1723 |
text \<open>The same theorem as above, but now, everything is in a shell. \<close> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1724 |
lemma (in field) field_long_division_theorem_shell: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1725 |
assumes "subfield K R" "p \<in> carrier (K[X])" and "b \<in> carrier (K[X])" "b \<noteq> \<zero>\<^bsub>K[X]\<^esub>" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1726 |
shows "\<exists>q r. q \<in> carrier (K[X]) \<and> r \<in> carrier (K[X]) \<and> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1727 |
p = (b \<otimes>\<^bsub>K[X]\<^esub> q) \<oplus>\<^bsub>K[X]\<^esub> r \<and> (r = \<zero>\<^bsub>K[X]\<^esub> \<or> degree r < degree b)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1728 |
using field_long_division_theorem assms by (auto simp add: univ_poly_def) |
68578 | 1729 |
|
1730 |
||
1731 |
subsection \<open>Consistency Rules\<close> |
|
1732 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1733 |
lemma polynomial_consistent [simp]: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1734 |
shows "polynomial\<^bsub>(R \<lparr> carrier := K \<rparr>)\<^esub> K p \<Longrightarrow> polynomial\<^bsub>R\<^esub> K p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1735 |
unfolding polynomial_def by auto |
68578 | 1736 |
|
1737 |
lemma (in ring) eval_consistent [simp]: |
|
1738 |
assumes "subring K R" shows "ring.eval (R \<lparr> carrier := K \<rparr>) = eval" |
|
1739 |
proof |
|
1740 |
fix p show "ring.eval (R \<lparr> carrier := K \<rparr>) p = eval p" |
|
1741 |
using nat_pow_consistent ring.eval.simps[OF subring_is_ring[OF assms]] by (induct p) (auto) |
|
1742 |
qed |
|
1743 |
||
1744 |
lemma (in ring) coeff_consistent [simp]: |
|
1745 |
assumes "subring K R" shows "ring.coeff (R \<lparr> carrier := K \<rparr>) = coeff" |
|
1746 |
proof |
|
1747 |
fix p show "ring.coeff (R \<lparr> carrier := K \<rparr>) p = coeff p" |
|
1748 |
using ring.coeff.simps[OF subring_is_ring[OF assms]] by (induct p) (auto) |
|
1749 |
qed |
|
1750 |
||
1751 |
lemma (in ring) normalize_consistent [simp]: |
|
1752 |
assumes "subring K R" shows "ring.normalize (R \<lparr> carrier := K \<rparr>) = normalize" |
|
1753 |
proof |
|
1754 |
fix p show "ring.normalize (R \<lparr> carrier := K \<rparr>) p = normalize p" |
|
1755 |
using ring.normalize.simps[OF subring_is_ring[OF assms]] by (induct p) (auto) |
|
1756 |
qed |
|
1757 |
||
1758 |
lemma (in ring) poly_add_consistent [simp]: |
|
1759 |
assumes "subring K R" shows "ring.poly_add (R \<lparr> carrier := K \<rparr>) = poly_add" |
|
1760 |
proof - |
|
1761 |
have "\<And>p q. ring.poly_add (R \<lparr> carrier := K \<rparr>) p q = poly_add p q" |
|
1762 |
proof - |
|
1763 |
fix p q show "ring.poly_add (R \<lparr> carrier := K \<rparr>) p q = poly_add p q" |
|
1764 |
using ring.poly_add.simps[OF subring_is_ring[OF assms]] normalize_consistent[OF assms] by auto |
|
1765 |
qed |
|
1766 |
thus ?thesis by (auto simp del: poly_add.simps) |
|
1767 |
qed |
|
1768 |
||
1769 |
lemma (in ring) poly_mult_consistent [simp]: |
|
1770 |
assumes "subring K R" shows "ring.poly_mult (R \<lparr> carrier := K \<rparr>) = poly_mult" |
|
1771 |
proof - |
|
1772 |
have "\<And>p q. ring.poly_mult (R \<lparr> carrier := K \<rparr>) p q = poly_mult p q" |
|
1773 |
proof - |
|
1774 |
fix p q show "ring.poly_mult (R \<lparr> carrier := K \<rparr>) p q = poly_mult p q" |
|
1775 |
using ring.poly_mult.simps[OF subring_is_ring[OF assms]] poly_add_consistent[OF assms] |
|
1776 |
by (induct p) (auto) |
|
1777 |
qed |
|
1778 |
thus ?thesis by auto |
|
1779 |
qed |
|
1780 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1781 |
lemma (in domain) univ_poly_a_inv_consistent: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1782 |
assumes "subring K R" "p \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1783 |
shows "\<ominus>\<^bsub>K[X]\<^esub> p = \<ominus>\<^bsub>(carrier R)[X]\<^esub> p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1784 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1785 |
have in_carrier: "p \<in> carrier ((carrier R)[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1786 |
using assms carrier_polynomial by (auto simp add: univ_poly_def) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1787 |
show ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1788 |
using univ_poly_a_inv_def'[OF assms] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1789 |
univ_poly_a_inv_def'[OF carrier_is_subring in_carrier] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1790 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1791 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1792 |
lemma (in domain) univ_poly_a_minus_consistent: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1793 |
assumes "subring K R" "q \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1794 |
shows "p \<ominus>\<^bsub>K[X]\<^esub> q = p \<ominus>\<^bsub>(carrier R)[X]\<^esub> q" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1795 |
using univ_poly_a_inv_consistent[OF assms] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1796 |
unfolding a_minus_def univ_poly_def by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1797 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1798 |
lemma (in ring) univ_poly_consistent: |
68578 | 1799 |
assumes "subring K R" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1800 |
shows "univ_poly (R \<lparr> carrier := K \<rparr>) = univ_poly R" |
68578 | 1801 |
unfolding univ_poly_def polynomial_def |
1802 |
using poly_add_consistent[OF assms] |
|
1803 |
poly_mult_consistent[OF assms] |
|
1804 |
subringE(1)[OF assms] |
|
1805 |
by auto |
|
1806 |
||
1807 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1808 |
subsubsection \<open>Corollaries\<close> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1809 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1810 |
corollary (in ring) subfield_long_division_theorem_shell: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1811 |
assumes "subfield K R" "p \<in> carrier (K[X])" and "b \<in> carrier (K[X])" "b \<noteq> \<zero>\<^bsub>K[X]\<^esub>" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1812 |
shows "\<exists>q r. q \<in> carrier (K[X]) \<and> r \<in> carrier (K[X]) \<and> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1813 |
p = (b \<otimes>\<^bsub>K[X]\<^esub> q) \<oplus>\<^bsub>K[X]\<^esub> r \<and> (r = \<zero>\<^bsub>K[X]\<^esub> \<or> degree r < degree b)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1814 |
using field.field_long_division_theorem_shell[OF subfield_iff(2)[OF assms(1)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1815 |
field.carrier_is_subfield[OF subfield_iff(2)[OF assms(1)]]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1816 |
univ_poly_consistent[OF subfieldE(1)[OF assms(1)]] assms(2-4) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1817 |
by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1818 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1819 |
corollary (in domain) univ_poly_is_euclidean: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1820 |
assumes "subfield K R" shows "euclidean_domain (K[X]) degree" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1821 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1822 |
interpret UP: domain "K[X]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1823 |
using univ_poly_is_domain[OF subfieldE(1)[OF assms]] field_def by blast |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1824 |
show ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1825 |
using subfield_long_division_theorem_shell[OF assms] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1826 |
by (auto intro!: UP.euclidean_domainI) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1827 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1828 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1829 |
corollary (in domain) univ_poly_is_principal: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1830 |
assumes "subfield K R" shows "principal_domain (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1831 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1832 |
interpret UP: euclidean_domain "K[X]" degree |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1833 |
using univ_poly_is_euclidean[OF assms] . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1834 |
show ?thesis .. |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1835 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1836 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1837 |
|
68578 | 1838 |
subsection \<open>The Evaluation Homomorphism\<close> |
1839 |
||
1840 |
lemma (in ring) eval_replicate: |
|
1841 |
assumes "set p \<subseteq> carrier R" "a \<in> carrier R" |
|
1842 |
shows "eval ((replicate n \<zero>) @ p) a = eval p a" |
|
1843 |
using assms eval_in_carrier by (induct n) (auto) |
|
1844 |
||
1845 |
lemma (in ring) eval_normalize: |
|
1846 |
assumes "set p \<subseteq> carrier R" "a \<in> carrier R" |
|
1847 |
shows "eval (normalize p) a = eval p a" |
|
1848 |
using eval_replicate[OF normalize_in_carrier] normalize_def'[of p] assms by metis |
|
1849 |
||
1850 |
lemma (in ring) eval_poly_add_aux: |
|
1851 |
assumes "set p \<subseteq> carrier R" "set q \<subseteq> carrier R" and "length p = length q" and "a \<in> carrier R" |
|
1852 |
shows "eval (poly_add p q) a = (eval p a) \<oplus> (eval q a)" |
|
1853 |
proof - |
|
1854 |
have "eval (map2 (\<oplus>) p q) a = (eval p a) \<oplus> (eval q a)" |
|
1855 |
using assms |
|
1856 |
proof (induct p arbitrary: q) |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1857 |
case Nil thus ?case by simp |
68578 | 1858 |
next |
1859 |
case (Cons b1 p') |
|
1860 |
then obtain b2 q' where q: "q = b2 # q'" |
|
1861 |
by (metis length_Cons list.exhaust list.size(3) nat.simps(3)) |
|
1862 |
show ?case |
|
1863 |
using eval_in_carrier[OF _ Cons(5), of q'] |
|
1864 |
eval_in_carrier[OF _ Cons(5), of p'] Cons unfolding q |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1865 |
by (auto simp add: ring_simprules(7,13,22)) |
68578 | 1866 |
qed |
1867 |
moreover have "set (map2 (\<oplus>) p q) \<subseteq> carrier R" |
|
1868 |
using assms(1-2) |
|
1869 |
by (induct p arbitrary: q) (auto, metis add.m_closed in_set_zipE set_ConsD subsetCE) |
|
1870 |
ultimately show ?thesis |
|
1871 |
using assms(3) eval_normalize[OF _ assms(4), of "map2 (\<oplus>) p q"] by auto |
|
1872 |
qed |
|
1873 |
||
1874 |
lemma (in ring) eval_poly_add: |
|
1875 |
assumes "set p \<subseteq> carrier R" "set q \<subseteq> carrier R" and "a \<in> carrier R" |
|
1876 |
shows "eval (poly_add p q) a = (eval p a) \<oplus> (eval q a)" |
|
1877 |
proof - |
|
1878 |
{ fix p q assume A: "set p \<subseteq> carrier R" "set q \<subseteq> carrier R" "length p \<ge> length q" |
|
1879 |
hence "eval (poly_add p ((replicate (length p - length q) \<zero>) @ q)) a = |
|
1880 |
(eval p a) \<oplus> (eval ((replicate (length p - length q) \<zero>) @ q) a)" |
|
1881 |
using eval_poly_add_aux[OF A(1) _ _ assms(3), of "(replicate (length p - length q) \<zero>) @ q"] by force |
|
1882 |
hence "eval (poly_add p q) a = (eval p a) \<oplus> (eval q a)" |
|
1883 |
using eval_replicate[OF A(2) assms(3)] A(3) by auto } |
|
1884 |
note aux_lemma = this |
|
1885 |
||
1886 |
have ?thesis if "length q \<ge> length p" |
|
1887 |
using assms(1-2)[THEN eval_in_carrier[OF _ assms(3)]] poly_add_comm[OF assms(1-2)] |
|
1888 |
aux_lemma[OF assms(2,1) that] |
|
1889 |
by (auto simp del: poly_add.simps simp add: add.m_comm) |
|
1890 |
moreover have ?thesis if "length p \<ge> length q" |
|
1891 |
using aux_lemma[OF assms(1-2) that] . |
|
1892 |
ultimately show ?thesis by auto |
|
1893 |
qed |
|
1894 |
||
1895 |
lemma (in ring) eval_append_aux: |
|
1896 |
assumes "set p \<subseteq> carrier R" and "b \<in> carrier R" and "a \<in> carrier R" |
|
1897 |
shows "eval (p @ [ b ]) a = ((eval p a) \<otimes> a) \<oplus> b" |
|
1898 |
using assms(1) |
|
1899 |
proof (induct p) |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1900 |
case Nil thus ?case by (auto simp add: assms(2-3)) |
68578 | 1901 |
next |
1902 |
case (Cons l q) |
|
1903 |
have "a [^] length q \<in> carrier R" "eval q a \<in> carrier R" |
|
1904 |
using eval_in_carrier Cons(2) assms(2-3) by auto |
|
1905 |
thus ?case |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1906 |
using Cons assms(2-3) by (auto, algebra) |
68578 | 1907 |
qed |
1908 |
||
1909 |
lemma (in ring) eval_append: |
|
1910 |
assumes "set p \<subseteq> carrier R" "set q \<subseteq> carrier R" and "a \<in> carrier R" |
|
1911 |
shows "eval (p @ q) a = ((eval p a) \<otimes> (a [^] (length q))) \<oplus> (eval q a)" |
|
1912 |
using assms(2) |
|
1913 |
proof (induct "length q" arbitrary: q) |
|
1914 |
case 0 thus ?case |
|
1915 |
using eval_in_carrier[OF assms(1,3)] by auto |
|
1916 |
next |
|
1917 |
case (Suc n) |
|
1918 |
then obtain b q' where q: "q = q' @ [ b ]" |
|
1919 |
by (metis length_Suc_conv list.simps(3) rev_exhaust) |
|
1920 |
hence in_carrier: "eval p a \<in> carrier R" "eval q' a \<in> carrier R" |
|
1921 |
"a [^] (length q') \<in> carrier R" "b \<in> carrier R" |
|
1922 |
using assms(1,3) Suc(3) eval_in_carrier[OF _ assms(3)] by auto |
|
1923 |
||
1924 |
have "eval (p @ q) a = ((eval (p @ q') a) \<otimes> a) \<oplus> b" |
|
1925 |
using eval_append_aux[OF _ _ assms(3), of "p @ q'" b] assms(1) Suc(3) unfolding q by auto |
|
1926 |
also have " ... = ((((eval p a) \<otimes> (a [^] (length q'))) \<oplus> (eval q' a)) \<otimes> a) \<oplus> b" |
|
1927 |
using Suc unfolding q by auto |
|
1928 |
also have " ... = (((eval p a) \<otimes> ((a [^] (length q')) \<otimes> a))) \<oplus> (((eval q' a) \<otimes> a) \<oplus> b)" |
|
1929 |
using assms(3) in_carrier by algebra |
|
1930 |
also have " ... = (eval p a) \<otimes> (a [^] (length q)) \<oplus> (eval q a)" |
|
1931 |
using eval_append_aux[OF _ in_carrier(4) assms(3), of q'] Suc(3) unfolding q by auto |
|
1932 |
finally show ?case . |
|
1933 |
qed |
|
1934 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1935 |
lemma (in ring) eval_monom: |
68578 | 1936 |
assumes "b \<in> carrier R" and "a \<in> carrier R" |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1937 |
shows "eval (monom b n) a = b \<otimes> (a [^] n)" |
68578 | 1938 |
proof (induct n) |
1939 |
case 0 thus ?case |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1940 |
using assms unfolding monom_def by auto |
68578 | 1941 |
next |
1942 |
case (Suc n) |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1943 |
have "monom b (Suc n) = (monom b n) @ [ \<zero> ]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1944 |
unfolding monom_def by (simp add: replicate_append_same) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1945 |
hence "eval (monom b (Suc n)) a = ((eval (monom b n) a) \<otimes> a) \<oplus> \<zero>" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1946 |
using eval_append_aux[OF monom_in_carrier[OF assms(1)] zero_closed assms(2), of n] by simp |
68578 | 1947 |
also have " ... = b \<otimes> (a [^] (Suc n))" |
1948 |
using Suc assms m_assoc by auto |
|
1949 |
finally show ?case . |
|
1950 |
qed |
|
1951 |
||
1952 |
lemma (in cring) eval_poly_mult: |
|
1953 |
assumes "set p \<subseteq> carrier R" "set q \<subseteq> carrier R" and "a \<in> carrier R" |
|
1954 |
shows "eval (poly_mult p q) a = (eval p a) \<otimes> (eval q a)" |
|
1955 |
using assms(1) |
|
1956 |
proof (induct p) |
|
1957 |
case Nil thus ?case |
|
1958 |
using eval_in_carrier[OF assms(2-3)] by simp |
|
1959 |
next |
|
1960 |
{ fix n b assume b: "b \<in> carrier R" |
|
1961 |
hence "set (map ((\<otimes>) b) q) \<subseteq> carrier R" and "set (replicate n \<zero>) \<subseteq> carrier R" |
|
1962 |
using assms(2) by (induct q) (auto) |
|
1963 |
hence "eval ((map ((\<otimes>) b) q) @ (replicate n \<zero>)) a = (eval ((map ((\<otimes>) b) q)) a) \<otimes> (a [^] n) \<oplus> \<zero>" |
|
1964 |
using eval_append[OF _ _ assms(3), of "map ((\<otimes>) b) q" "replicate n \<zero>"] |
|
1965 |
eval_replicate[OF _ assms(3), of "[]"] by auto |
|
1966 |
moreover have "eval (map ((\<otimes>) b) q) a = b \<otimes> eval q a" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1967 |
using assms(2-3) eval_in_carrier b by(induct q) (auto simp add: m_assoc r_distr) |
68578 | 1968 |
ultimately have "eval ((map ((\<otimes>) b) q) @ (replicate n \<zero>)) a = (b \<otimes> eval q a) \<otimes> (a [^] n) \<oplus> \<zero>" |
1969 |
by simp |
|
1970 |
also have " ... = (b \<otimes> (a [^] n)) \<otimes> (eval q a)" |
|
1971 |
using eval_in_carrier[OF assms(2-3)] b assms(3) m_assoc m_comm by auto |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1972 |
finally have "eval ((map ((\<otimes>) b) q) @ (replicate n \<zero>)) a = (eval (monom b n) a) \<otimes> (eval q a)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1973 |
using eval_monom[OF b assms(3)] by simp } |
68578 | 1974 |
note aux_lemma = this |
1975 |
||
1976 |
case (Cons b p) |
|
1977 |
hence in_carrier: |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1978 |
"eval (monom b (length p)) a \<in> carrier R" "eval p a \<in> carrier R" "eval q a \<in> carrier R" "b \<in> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1979 |
using eval_in_carrier monom_in_carrier assms by auto |
68578 | 1980 |
have set_map: "set ((map ((\<otimes>) b) q) @ (replicate (length p) \<zero>)) \<subseteq> carrier R" |
1981 |
using in_carrier(4) assms(2) by (induct q) (auto) |
|
1982 |
have set_poly: "set (poly_mult p q) \<subseteq> carrier R" |
|
1983 |
using poly_mult_in_carrier[OF _ assms(2), of p] Cons(2) by auto |
|
1984 |
have "eval (poly_mult (b # p) q) a = |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1985 |
((eval (monom b (length p)) a) \<otimes> (eval q a)) \<oplus> ((eval p a) \<otimes> (eval q a))" |
68578 | 1986 |
using eval_poly_add[OF set_map set_poly assms(3)] aux_lemma[OF in_carrier(4), of "length p"] Cons |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1987 |
by (auto simp del: poly_add.simps) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1988 |
also have " ... = ((eval (monom b (length p)) a) \<oplus> (eval p a)) \<otimes> (eval q a)" |
68578 | 1989 |
using l_distr[OF in_carrier(1-3)] by simp |
1990 |
also have " ... = (eval (b # p) a) \<otimes> (eval q a)" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1991 |
unfolding eval_monom[OF in_carrier(4) assms(3), of "length p"] by auto |
68578 | 1992 |
finally show ?case . |
1993 |
qed |
|
1994 |
||
1995 |
proposition (in cring) eval_is_hom: |
|
1996 |
assumes "subring K R" and "a \<in> carrier R" |
|
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1997 |
shows "(\<lambda>p. (eval p) a) \<in> ring_hom (K[X]) R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1998 |
unfolding univ_poly_def |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
1999 |
using polynomial_in_carrier[OF assms(1)] eval_in_carrier |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2000 |
eval_poly_add eval_poly_mult assms(2) |
68578 | 2001 |
by (auto intro!: ring_hom_memI |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2002 |
simp add: univ_poly_carrier |
68578 | 2003 |
simp del: poly_add.simps poly_mult.simps) |
2004 |
||
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2005 |
theorem (in domain) eval_cring_hom: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2006 |
assumes "subring K R" and "a \<in> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2007 |
shows "ring_hom_cring (K[X]) R (\<lambda>p. (eval p) a)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2008 |
unfolding ring_hom_cring_def ring_hom_cring_axioms_def |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2009 |
using domain.axioms(1)[OF univ_poly_is_domain[OF assms(1)]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2010 |
eval_is_hom[OF assms] cring_axioms by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2011 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2012 |
corollary (in domain) eval_ring_hom: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2013 |
assumes "subring K R" and "a \<in> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2014 |
shows "ring_hom_ring (K[X]) R (\<lambda>p. (eval p) a)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2015 |
using eval_cring_hom[OF assms] ring_hom_ringI2 |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2016 |
unfolding ring_hom_cring_def ring_hom_cring_axioms_def cring_def by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2017 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2018 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2019 |
subsection \<open>The X Variable\<close> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2020 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2021 |
definition var :: "_ \<Rightarrow> 'a list" ("X\<index>") |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2022 |
where "X\<^bsub>R\<^esub> = [ \<one>\<^bsub>R\<^esub>, \<zero>\<^bsub>R\<^esub> ]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2023 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2024 |
lemma (in ring) eval_var: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2025 |
assumes "x \<in> carrier R" shows "eval X x = x" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2026 |
using assms unfolding var_def by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2027 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2028 |
lemma (in domain) var_closed: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2029 |
assumes "subring K R" shows "X \<in> carrier (K[X])" and "polynomial K X" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2030 |
using subringE(2-3)[OF assms] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2031 |
by (auto simp add: var_def univ_poly_def polynomial_def) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2032 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2033 |
lemma (in domain) poly_mult_var': |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2034 |
assumes "set p \<subseteq> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2035 |
shows "poly_mult X p = normalize (p @ [ \<zero> ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2036 |
and "poly_mult p X = normalize (p @ [ \<zero> ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2037 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2038 |
from \<open>set p \<subseteq> carrier R\<close> have "poly_mult [ \<one> ] p = normalize p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2039 |
using poly_mult_one' by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2040 |
thus "poly_mult X p = normalize (p @ [ \<zero> ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2041 |
using poly_mult_append_zero[OF _ assms, of "[ \<one> ]"] normalize_idem |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2042 |
unfolding var_def by (auto simp del: poly_mult.simps) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2043 |
thus "poly_mult p X = normalize (p @ [ \<zero> ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2044 |
using poly_mult_comm[OF assms] unfolding var_def by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2045 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2046 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2047 |
lemma (in domain) poly_mult_var: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2048 |
assumes "subring K R" "p \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2049 |
shows "p \<otimes>\<^bsub>K[X]\<^esub> X = (if p = [] then [] else p @ [ \<zero> ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2050 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2051 |
have is_poly: "polynomial K p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2052 |
using assms(2) unfolding univ_poly_def by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2053 |
hence "polynomial K (p @ [ \<zero> ])" if "p \<noteq> []" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2054 |
using that subringE(2)[OF assms(1)] unfolding polynomial_def by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2055 |
thus ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2056 |
using poly_mult_var'(2)[OF polynomial_in_carrier[OF assms(1) is_poly]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2057 |
normalize_polynomial[of K "p @ [ \<zero> ]"] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2058 |
by (auto simp add: univ_poly_mult[of R K]) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2059 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2060 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2061 |
lemma (in domain) var_pow_closed: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2062 |
assumes "subring K R" shows "X [^]\<^bsub>K[X]\<^esub> (n :: nat) \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2063 |
using monoid.nat_pow_closed[OF univ_poly_is_monoid[OF assms] var_closed(1)[OF assms]] . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2064 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2065 |
lemma (in domain) unitary_monom_eq_var_pow: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2066 |
assumes "subring K R" shows "monom \<one> n = X [^]\<^bsub>K[X]\<^esub> n" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2067 |
using poly_mult_var[OF assms var_pow_closed[OF assms]] unfolding nat_pow_def monom_def |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2068 |
by (induct n) (auto simp add: univ_poly_one, metis append_Cons replicate_append_same) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2069 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2070 |
lemma (in domain) monom_eq_var_pow: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2071 |
assumes "subring K R" "a \<in> carrier R - { \<zero> }" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2072 |
shows "monom a n = [ a ] \<otimes>\<^bsub>K[X]\<^esub> (X [^]\<^bsub>K[X]\<^esub> n)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2073 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2074 |
have "monom a n = map ((\<otimes>) a) (monom \<one> n)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2075 |
unfolding monom_def using assms(2) by (induct n) (auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2076 |
also have " ... = poly_mult [ a ] (monom \<one> n)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2077 |
using poly_mult_const(1)[OF _ monom_is_polynomial assms(2)] carrier_is_subring by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2078 |
also have " ... = [ a ] \<otimes>\<^bsub>K[X]\<^esub> (X [^]\<^bsub>K[X]\<^esub> n)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2079 |
unfolding unitary_monom_eq_var_pow[OF assms(1)] univ_poly_mult[of R K] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2080 |
finally show ?thesis . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2081 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2082 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2083 |
lemma (in ring) dense_repr_set_fst: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2084 |
assumes "set p \<subseteq> K" shows "fst ` (set (dense_repr p)) \<subseteq> K - { \<zero> }" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2085 |
using assms by (induct p) (auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2086 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2087 |
lemma (in ring) dense_repr_set_snd: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2088 |
shows "snd ` (set (dense_repr p)) \<subseteq> {..< length p}" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2089 |
by (induct p) (auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2090 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2091 |
lemma (in domain) dense_repr_monom_closed: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2092 |
assumes "subring K R" "set p \<subseteq> K" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2093 |
shows "t \<in> set (dense_repr p) \<Longrightarrow> monom (fst t) (snd t) \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2094 |
using dense_repr_set_fst[OF assms(2)] monom_is_polynomial[OF assms(1)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2095 |
by (auto simp add: univ_poly_carrier) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2096 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2097 |
lemma (in domain) monom_finsum_decomp: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2098 |
assumes "subring K R" "p \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2099 |
shows "p = (\<Oplus>\<^bsub>K[X]\<^esub> t \<in> set (dense_repr p). monom (fst t) (snd t))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2100 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2101 |
interpret UP: domain "K[X]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2102 |
using univ_poly_is_domain[OF assms(1)] . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2103 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2104 |
from \<open>p \<in> carrier (K[X])\<close> show ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2105 |
proof (induct "length p" arbitrary: p rule: less_induct) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2106 |
case less thus ?case |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2107 |
proof (cases p) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2108 |
case Nil thus ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2109 |
using UP.finsum_empty univ_poly_zero[of R K] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2110 |
next |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2111 |
case (Cons a l) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2112 |
hence in_carrier: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2113 |
"normalize l \<in> carrier (K[X])" "polynomial K (normalize l)" "polynomial K (a # l)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2114 |
using normalize_gives_polynomial polynomial_incl[of K p] less(2) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2115 |
unfolding univ_poly_carrier by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2116 |
have len_lt: "length (local.normalize l) < length p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2117 |
using normalize_length_le by (simp add: Cons le_imp_less_Suc) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2118 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2119 |
have a: "a \<in> K - { \<zero> }" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2120 |
using less(2) subringE(1)[OF assms(1)] unfolding Cons univ_poly_def polynomial_def by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2121 |
hence "p = (monom a (length l)) \<oplus>\<^bsub>K[X]\<^esub> (of_dense (dense_repr (normalize l)))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2122 |
using monom_decomp[OF assms(1), of p] less(2) dense_repr_normalize |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2123 |
unfolding univ_poly_add univ_poly_carrier Cons by (auto simp del: poly_add.simps) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2124 |
also have " ... = (monom a (length l)) \<oplus>\<^bsub>K[X]\<^esub> (normalize l)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2125 |
using monom_decomp[OF assms(1) in_carrier(2)] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2126 |
finally have "p = monom a (length l) \<oplus>\<^bsub>K[X]\<^esub> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2127 |
(\<Oplus>\<^bsub>K[X]\<^esub> t \<in> set (dense_repr l). monom (fst t) (snd t))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2128 |
using less(1)[OF len_lt in_carrier(1)] dense_repr_normalize by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2129 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2130 |
moreover have "(a, (length l)) \<notin> set (dense_repr l)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2131 |
using dense_repr_set_snd[of l] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2132 |
moreover have "monom a (length l) \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2133 |
using monom_is_polynomial[OF assms(1) a] unfolding univ_poly_carrier by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2134 |
moreover have "\<And>t. t \<in> set (dense_repr l) \<Longrightarrow> monom (fst t) (snd t) \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2135 |
using dense_repr_monom_closed[OF assms(1)] polynomial_incl[OF in_carrier(3)] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2136 |
ultimately have "p = (\<Oplus>\<^bsub>K[X]\<^esub> t \<in> set (dense_repr (a # l)). monom (fst t) (snd t))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2137 |
using UP.add.finprod_insert a by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2138 |
thus ?thesis unfolding Cons . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2139 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2140 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2141 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2142 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2143 |
lemma (in domain) var_pow_finsum_decomp: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2144 |
assumes "subring K R" "p \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2145 |
shows "p = (\<Oplus>\<^bsub>K[X]\<^esub> t \<in> set (dense_repr p). [ fst t ] \<otimes>\<^bsub>K[X]\<^esub> (X [^]\<^bsub>K[X]\<^esub> (snd t)))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2146 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2147 |
let ?f = "\<lambda>t. monom (fst t) (snd t)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2148 |
let ?g = "\<lambda>t. [ fst t ] \<otimes>\<^bsub>K[X]\<^esub> (X [^]\<^bsub>K[X]\<^esub> (snd t))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2149 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2150 |
interpret UP: domain "K[X]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2151 |
using univ_poly_is_domain[OF assms(1)] . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2152 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2153 |
have set_p: "set p \<subseteq> K" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2154 |
using polynomial_incl assms(2) by (simp add: univ_poly_carrier) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2155 |
hence f: "?f \<in> set (dense_repr p) \<rightarrow> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2156 |
using dense_repr_monom_closed[OF assms(1)] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2157 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2158 |
moreover |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2159 |
have "\<And>t. t \<in> set (dense_repr p) \<Longrightarrow> fst t \<in> carrier R - { \<zero> }" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2160 |
using dense_repr_set_fst[OF set_p] subringE(1)[OF assms(1)] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2161 |
hence "\<And>t. t \<in> set (dense_repr p) \<Longrightarrow> monom (fst t) (snd t) = [ fst t ] \<otimes>\<^bsub>K[X]\<^esub> (X [^]\<^bsub>K[X]\<^esub> (snd t))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2162 |
using monom_eq_var_pow[OF assms(1)] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2163 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2164 |
ultimately show ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2165 |
using UP.add.finprod_cong[of _ _ ?f ?g] monom_finsum_decomp[OF assms] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2166 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2167 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2168 |
corollary (in domain) hom_var_pow_finsum: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2169 |
assumes "subring K R" and "p \<in> carrier (K[X])" "ring_hom_ring (K[X]) A h" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2170 |
shows "h p = (\<Oplus>\<^bsub>A\<^esub> t \<in> set (dense_repr p). h [ fst t ] \<otimes>\<^bsub>A\<^esub> (h X [^]\<^bsub>A\<^esub> (snd t)))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2171 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2172 |
let ?f = "\<lambda>t. [ fst t ] \<otimes>\<^bsub>K[X]\<^esub> (X [^]\<^bsub>K[X]\<^esub> (snd t))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2173 |
let ?g = "\<lambda>t. h [ fst t ] \<otimes>\<^bsub>A\<^esub> (h X [^]\<^bsub>A\<^esub> (snd t))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2174 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2175 |
interpret UP: domain "K[X]" + A: ring A |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2176 |
using univ_poly_is_domain[OF assms(1)] ring_hom_ring.axioms(2)[OF assms(3)] by simp+ |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2177 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2178 |
have const_in_carrier: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2179 |
"\<And>t. t \<in> set (dense_repr p) \<Longrightarrow> [ fst t ] \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2180 |
using dense_repr_set_fst[OF polynomial_incl, of K p] assms(2) const_is_polynomial[of _ K] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2181 |
by (auto simp add: univ_poly_carrier) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2182 |
hence f: "?f: set (dense_repr p) \<rightarrow> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2183 |
using UP.m_closed[OF _ var_pow_closed[OF assms(1)]] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2184 |
hence h: "h \<circ> ?f: set (dense_repr p) \<rightarrow> carrier A" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2185 |
using ring_hom_memE(1)[OF ring_hom_ring.homh[OF assms(3)]] by (auto simp add: Pi_def) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2186 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2187 |
have hp: "h p = (\<Oplus>\<^bsub>A\<^esub> t \<in> set (dense_repr p). (h \<circ> ?f) t)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2188 |
using ring_hom_ring.hom_finsum[OF assms(3) f] var_pow_finsum_decomp[OF assms(1-2)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2189 |
by (auto, meson o_apply) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2190 |
have eq: "\<And>t. t \<in> set (dense_repr p) \<Longrightarrow> h [ fst t ] \<otimes>\<^bsub>A\<^esub> (h X [^]\<^bsub>A\<^esub> (snd t)) = (h \<circ> ?f) t" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2191 |
using ring_hom_memE(2)[OF ring_hom_ring.homh[OF assms(3)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2192 |
const_in_carrier var_pow_closed[OF assms(1)]] |
70019
095dce9892e8
A few results in Algebra, and bits for Analysis
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2193 |
ring_hom_ring.hom_nat_pow[OF assms(3) var_closed(1)[OF assms(1)]] by auto |
68664
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2194 |
show ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2195 |
using A.add.finprod_cong'[OF _ h eq] hp by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2196 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2197 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2198 |
corollary (in domain) determination_of_hom: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2199 |
assumes "subring K R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2200 |
and "ring_hom_ring (K[X]) A h" "ring_hom_ring (K[X]) A g" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2201 |
and "\<And>k. k \<in> K \<Longrightarrow> h [ k ] = g [ k ]" and "h X = g X" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2202 |
shows "\<And>p. p \<in> carrier (K[X]) \<Longrightarrow> h p = g p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2203 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2204 |
interpret A: ring A |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2205 |
using ring_hom_ring.axioms(2)[OF assms(2)] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2206 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2207 |
fix p assume p: "p \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2208 |
hence |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2209 |
"\<And>t. t \<in> set (dense_repr p) \<Longrightarrow> [ fst t ] \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2210 |
using dense_repr_set_fst[OF polynomial_incl, of K p] const_is_polynomial[of _ K] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2211 |
by (auto simp add: univ_poly_carrier) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2212 |
hence f: "(\<lambda>t. h [ fst t ] \<otimes>\<^bsub>A\<^esub> (h X [^]\<^bsub>A\<^esub> (snd t))): set (dense_repr p) \<rightarrow> carrier A" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2213 |
using ring_hom_memE(1)[OF ring_hom_ring.homh[OF assms(2)]] var_closed(1)[OF assms(1)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2214 |
A.m_closed[OF _ A.nat_pow_closed] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2215 |
by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2216 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2217 |
have eq: "\<And>t. t \<in> set (dense_repr p) \<Longrightarrow> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2218 |
g [ fst t ] \<otimes>\<^bsub>A\<^esub> (g X [^]\<^bsub>A\<^esub> (snd t)) = h [ fst t ] \<otimes>\<^bsub>A\<^esub> (h X [^]\<^bsub>A\<^esub> (snd t))" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2219 |
using dense_repr_set_fst[OF polynomial_incl, of K p] p assms(4-5) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2220 |
by (auto simp add: univ_poly_carrier) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2221 |
show "h p = g p" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2222 |
unfolding assms(2-3)[THEN hom_var_pow_finsum[OF assms(1) p]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2223 |
using A.add.finprod_cong'[OF _ f eq] by simp |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2224 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2225 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2226 |
corollary (in domain) eval_as_unique_hom: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2227 |
assumes "subring K R" "x \<in> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2228 |
and "ring_hom_ring (K[X]) R h" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2229 |
and "\<And>k. k \<in> K \<Longrightarrow> h [ k ] = k" and "h X = x" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2230 |
shows "\<And>p. p \<in> carrier (K[X]) \<Longrightarrow> h p = eval p x" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2231 |
using determination_of_hom[OF assms(1,3) eval_ring_hom[OF assms(1-2)]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2232 |
eval_var[OF assms(2)] assms(4-5) subringE(1)[OF assms(1)] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2233 |
by fastforce |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2234 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2235 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2236 |
subsection \<open>The Constant Term\<close> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2237 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2238 |
definition (in ring) const_term :: "'a list \<Rightarrow> 'a" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2239 |
where "const_term p = eval p \<zero>" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2240 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2241 |
lemma (in ring) const_term_eq_last: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2242 |
assumes "set p \<subseteq> carrier R" and "a \<in> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2243 |
shows "const_term (p @ [ a ]) = a" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2244 |
using assms by (induct p) (auto simp add: const_term_def) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2245 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2246 |
lemma (in ring) const_term_not_zero: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2247 |
assumes "const_term p \<noteq> \<zero>" shows "p \<noteq> []" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2248 |
using assms by (auto simp add: const_term_def) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2249 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2250 |
lemma (in ring) const_term_explicit: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2251 |
assumes "set p \<subseteq> carrier R" "p \<noteq> []" and "const_term p = a" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2252 |
obtains p' where "set p' \<subseteq> carrier R" and "p = p' @ [ a ]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2253 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2254 |
obtain a' p' where p: "p = p' @ [ a' ]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2255 |
using assms(2) rev_exhaust by blast |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2256 |
have p': "set p' \<subseteq> carrier R" and a: "a = a'" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2257 |
using assms const_term_eq_last[of p' a'] unfolding p by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2258 |
show thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2259 |
using p p' that unfolding a by blast |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2260 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2261 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2262 |
lemma (in ring) const_term_zero: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2263 |
assumes "subring K R" "polynomial K p" "p \<noteq> []" and "const_term p = \<zero>" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2264 |
obtains p' where "polynomial K p'" "p' \<noteq> []" and "p = p' @ [ \<zero> ]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2265 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2266 |
obtain p' where p': "p = p' @ [ \<zero> ]" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2267 |
using const_term_explicit[OF polynomial_in_carrier[OF assms(1-2)] assms(3-4)] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2268 |
have "polynomial K p'" "p' \<noteq> []" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2269 |
using assms(2) unfolding p' polynomial_def by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2270 |
thus thesis using p' .. |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2271 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2272 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2273 |
lemma (in cring) const_term_simprules: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2274 |
shows "\<And>p. set p \<subseteq> carrier R \<Longrightarrow> const_term p \<in> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2275 |
and "\<And>p q. \<lbrakk> set p \<subseteq> carrier R; set q \<subseteq> carrier R \<rbrakk> \<Longrightarrow> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2276 |
const_term (poly_mult p q) = const_term p \<otimes> const_term q" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2277 |
and "\<And>p q. \<lbrakk> set p \<subseteq> carrier R; set q \<subseteq> carrier R \<rbrakk> \<Longrightarrow> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2278 |
const_term (poly_add p q) = const_term p \<oplus> const_term q" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2279 |
using eval_poly_mult eval_poly_add eval_in_carrier zero_closed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2280 |
unfolding const_term_def by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2281 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2282 |
lemma (in domain) const_term_simprules_shell: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2283 |
assumes "subring K R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2284 |
shows "\<And>p. p \<in> carrier (K[X]) \<Longrightarrow> const_term p \<in> K" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2285 |
and "\<And>p q. \<lbrakk> p \<in> carrier (K[X]); q \<in> carrier (K[X]) \<rbrakk> \<Longrightarrow> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2286 |
const_term (p \<otimes>\<^bsub>K[X]\<^esub> q) = const_term p \<otimes> const_term q" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2287 |
and "\<And>p q. \<lbrakk> p \<in> carrier (K[X]); q \<in> carrier (K[X]) \<rbrakk> \<Longrightarrow> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2288 |
const_term (p \<oplus>\<^bsub>K[X]\<^esub> q) = const_term p \<oplus> const_term q" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2289 |
and "\<And>p. p \<in> carrier (K[X]) \<Longrightarrow> const_term (\<ominus>\<^bsub>K[X]\<^esub> p) = \<ominus> (const_term p)" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2290 |
using eval_is_hom[OF assms(1) zero_closed] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2291 |
unfolding ring_hom_def const_term_def |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2292 |
proof (auto) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2293 |
fix p assume p: "p \<in> carrier (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2294 |
hence "set p \<subseteq> carrier R" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2295 |
using polynomial_in_carrier[OF assms(1)] by (auto simp add: univ_poly_def) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2296 |
thus "eval (\<ominus>\<^bsub>K [X]\<^esub> p) \<zero> = \<ominus> local.eval p \<zero>" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2297 |
unfolding univ_poly_a_inv_def'[OF assms(1) p] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2298 |
by (induct p) (auto simp add: eval_in_carrier l_minus local.minus_add) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2299 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2300 |
have "set p \<subseteq> K" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2301 |
using p by (auto simp add: univ_poly_def polynomial_def) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2302 |
thus "eval p \<zero> \<in> K" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2303 |
using subringE(1-2,6-7)[OF assms] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2304 |
by (induct p) (auto, metis assms nat_pow_0 nat_pow_zero subringE(3)) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2305 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2306 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2307 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2308 |
subsection \<open>The Canonical Embedding of K in K[X]\<close> |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2309 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2310 |
lemma (in field) univ_poly_carrier_subfield_of_consts: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2311 |
"subfield { p \<in> carrier ((carrier R)[X]). degree p = 0 } ((carrier R)[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2312 |
proof - |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2313 |
have ring_hom: "ring_hom_ring R ((carrier R)[X]) (\<lambda>k. normalize [ k ])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2314 |
by (rule ring_hom_ringI[OF ring_axioms univ_poly_is_ring[OF carrier_is_subring]]) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2315 |
(auto simp add: univ_poly_def) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2316 |
have subfield: "subfield ((\<lambda>k. normalize [ k ]) ` (carrier R)) ((carrier R)[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2317 |
using ring_hom_ring.img_is_subfield(2)[OF ring_hom carrier_is_subfield] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2318 |
unfolding univ_poly_def by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2319 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2320 |
have "(\<lambda>k. normalize [ k ]) ` (carrier R) = insert [] { [ k ] | k. k \<in> carrier R - { \<zero> } }" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2321 |
by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2322 |
also have " ... = { p \<in> carrier ((carrier R)[X]). degree p = 0 }" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2323 |
unfolding univ_poly_def polynomial_def |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2324 |
by (auto, metis le_Suc_eq le_zero_eq length_0_conv length_Suc_conv list.sel(1) list.set_sel(1) subsetCE) |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2325 |
finally have "(\<lambda>k. normalize [ k ]) ` (carrier R) = { p \<in> carrier ((carrier R)[X]). degree p = 0 }" . |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2326 |
thus ?thesis |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2327 |
using subfield by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2328 |
qed |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2329 |
|
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2330 |
proposition (in ring) univ_poly_subfield_of_consts: |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2331 |
assumes "subfield K R" shows "subfield { p \<in> carrier (K[X]). degree p = 0 } (K[X])" |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2332 |
using field.univ_poly_carrier_subfield_of_consts[OF subfield_iff(2)[OF assms]] |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2333 |
univ_poly_consistent[OF subfieldE(1)[OF assms]] by auto |
bd0df72c16d5
updated material concerning Algebra
paulson <lp15@cam.ac.uk>
parents:
68605
diff
changeset
|
2334 |
|
68583 | 2335 |
end |